In SID/Milgromian dynamics there appears an interesting effect that leads to an observable prediction which does not exist in the Newtonian plus DM model. It is relevant for a satellite object falling within the gravitational field of a host such that the host field does not vary significantly across the satellite. This external gravitational field effectively truncates the isothermal PDMH, as deduced by an observer who interprets the observations within Newtonian dynamics. The Milgromian dynamics/MOND equation for a spherical, axisymmetric or cylindrical system embedded in an external field is (Milgrom 1983a; Sanders & McGaugh 2002)

(24)

\begin{equation}
{\boldsymbol g}_{{\rm N,i}}+{\boldsymbol g}_{{\rm N,e}}=\mu (| ({\boldsymbol g}_{\rm i} + {\boldsymbol g}_{\rm e})|/a_0) ({\boldsymbol g}_{\rm i}+{\boldsymbol g}_{\rm e}),
\end{equation}

where ${\boldsymbol g}_{{\rm N,i}}$ is the Newtonian acceleration from the baryonic matter of the internal system, ${\boldsymbol g}_{{\rm N,e}}$ is the Newtonian acceleration from the baryonic matter of the external gravitational source generating the background uniform field and ${\boldsymbol g}_{\rm i}$ and ${\boldsymbol g}_{\rm e}$ are the internal and external gravitational accelerations. For an external field dominated system, $g_{\rm e}=|{\boldsymbol g}_{\rm e}| \gg g_{\rm i}=|{\boldsymbol g}_{\rm i}|$, equation (24) is expanded around ge to lowest order as

(25)

\begin{eqnarray}
g_{{\rm N,e}}&=&\mu (|{\boldsymbol g}_{\rm e}|/a_0) {\boldsymbol g}_{\rm e},\nonumber \\
g_{{\rm N,i}}&=&\mu (|{\boldsymbol g}_{\rm e}|/a_0) {\boldsymbol g}_{\rm i},
\end{eqnarray}

with a dilation factor of $\Delta _1=(1+{\rm d}\ln \mu /{\rm d} \ln x)_{x=g_{\rm e}/a_0}$. Δ1 approaches 1 in the Newtonian limit and approaches 2 in the deep-MOND limit. The value of Δ1 also depends on the direction relative to the external field (for more details see Milgrom 1983a; Bekenstein & Milgrom 1984; Zhao & Tian 2006). μ(x) = x in the low-acceleration regime where SID is valid (i.e. in deep-MOND limit).6


Our motivation for this work is manifold. First, we have incorporated the gas-accretion models of Papaloizou  Nelson (2005) into the combined N-body and protoplanetary disc evolution code presented in Coleman  Nelson (2014, 2016a), in order to increase the realism of the planetary system formation simulations that can be computed with this numerical tool. The purpose of this paper is to provide a simplified examination of gas accretion on to isolated, non-interacting planetary cores at a variety of orbital locations using the evolving disc models from Coleman  Nelson (2016a), as a precursor to presenting simulations that consider the contemporaneous accretion of solids and gas on to protoplanets under the influence of migration within the evolving disc models. A second motivation is to examine how accreting planets with different initial core masses at different locations in the disc evolve within disc lifetimes. Observations indicate the presence of exoplanets (super-Earths, Neptunes and Jupiters) that are likely to be gas bearing with a broad range of orbital radii, covering the interval 0.04au  ap  100au (e.g. Marois etal. 2008; Borucki etal. 2011). Analysis of the occurrence rates as a function of planet radius in the Kepler data suggests that planets with radii in the range 0.85  Rp  4R are present around 50percent of Sun-like stars (e.g. Fressin etal. 2013), which is consonant with the results from earlier radial velocity surveys that focused on low-mass planets (Mayor etal. 2011). Transit timing variations (Wu  Lithwick 2013; Hadden  Lithwick 2016) and radial velocity measurements (Marcy etal. 2014; Weiss etal. 2016) indicate that the super-Earths and Neptunes in the Kepler data with radii Rp4R have masses in the range 2  Mp  15M, and show a tendency for the bulk density to increase for Rp1.5R and decrease for 1.5Rp4R, indicating that the smaller planets are composed almost entirely of solids, whereas the larger planets have larger radii due to the presence of significant gaseous envelopes that have been accreted during formation. Comparing theoretical models with the data suggests that the gaseous envelopes contribute little to the total masses of the super-Earths and Neptunes (Lopez  Fortney 2014), with recent estimates indicating that planets with Rp  1.2R contain 1percent gas by mass, rising to 5percent for planets with Rp3R (Wolfgang  Lopez 2015). Considering the gas giant exoplanets, numerous attempts have been made to constrain the core masses and total heavy element abundances of those transiting planets that have mass estimates from radial velocity measurements using evolutionary models of the internal structure (Guillot etal. 2006; Burrows etal. 2007; Miller  Fortney 2011; Thorngren etal. 2016). These studies provide a strong indication that the majority of gas giant exoplanets are substantially enriched with heavy elements relative to their host stars, with heavy element masses typically being 20M or larger. These results are consistent with the idea that giant planets form when a sufficiently massive (i.e. in the range 1015M) core forms that is able to accrete gas and undergo runaway gas accretion sufficiently early in the disc lifetime, in basic agreement with the theoretical core accretion models. In the absence of a sufficiently massive core that forms early enough, a more common outcome is the formation of super-Earths or Neptunes that fail to become gas giants because of the relatively slow rate of gas accretion until the envelope mass becomes comparable to the core mass. It is clearly of interest to address the above observational results through theoretical modelling of in situ planet formation, and to compare the outcomes of our calculations with recent work that has considered gas accretion on to planets with short (Lee, Chiang  Ormel 2014; Batygin, Bodenheimer  Laughlin 2016) and long orbital periods (Piso  Youdin 2014; Piso, Youdin  Murray-Clay 2015). Our third motivation is to consider the longer term cooling and contraction of the gaseous envelopes for those planets that remain in a state of quasi-static equilibrium throughout the period of gas accretion when the disc is present. Since the observations are typically relevant to exoplanets that have existed around their stars for Gyrs, they will have evolved for time periods of this duration after the epoch of formation. This is the first time that our models have been used to examine this longer term evolution, and our approach is similar to that used in recent population synthesis studies (see the recent review by Mordasini etal. 2015).


The extended star-forming region S235 is known to be a part of the giant molecular cloud G174+2.5 in the Perseus Spiral Arm (e.g., Heyer et al. 1996) and contains two known sites: the S235 complex and the S235AB region (see Figure 1 in Dewangan & Anandarao 2011 and also Figure 1 in Kirsanova et al. 2014). The present work is focused on the S235 complex and does not include the S235AB region. Different values of the distance (1.36, 1.59, 1.8, 2.1, and 2.5 kpc) to the extended star-forming region S235 have been reported in the literature (e.g., Georgelin et al. 1973; Israel & Felli 1978; Evans & Blair 1981; Brand & Blitz 1993; Burns et al. 2015; Foster & Brunt 2015). In the present work, we have chosen a distance of 1.8 kpc following Evans & Blair (1981), which is an intermediate value of the published distance range. The H ii region associated with the S235 complex is predominantly ionized by a single massive star, BD+35o1201, of O9.5V type (Georgelin et al. 1973). The S235 complex has been studied using multiple data sets spanning near-infrared (NIR) to radio wavelengths. The S235 complex is known as an active site of star formation, harboring young stellar clusters (e.g., Kirsanova et al. 2008; Camargo et al. 2011; Dewangan & Anandarao 2011; Chavarría et al. 2014) associated with known star-forming subregions, namely East 1, East 2, and the Central region (e.g., Kirsanova et al. 2008). In our previous work on the S235 complex using Spitzer-IRAC data (Dewangan & Anandarao 2011, hereafter Paper I) we detected several young stellar objects (YSOs) including a high mass protostellar object (HMPO) candidate as well as signatures of outflow activities. Using 13CO (1−0) line data, Kirsanova et al. (2008) found three molecular gas components (i.e., −18 km s−1

 km s−1 (red), −21 km s−1

 km s−1 (central), and −25 km s−1

 km s−1 (blue)) in the direction of the S235 complex. However, the complex is well traced in mainly two molecular gas components (central and blue). More recently, Kirsanova et al. (2014) derived physical parameters of dense gas (i.e., gas density and temperature) in subregions of the complex using ammonia (NH3) line observations. However, the properties of dense gas have not been explored with respect to the ionizing star location. Previous studies indicated that the S235 H ii region is interacting with its surrounding molecular cloud and the S235 complex has been cited as a possible site of triggered star formation (Kirsanova et al. 2008, 2014; Camargo et al. 2011).


The first interstellar discovery of MF took place in 1975 from the microwave emission spectrum of Sgr B2, which agrees with the rotational constants of the syn isomer (Curl 1959; Brown et al. 1975; Nummelin et al. 2000). Subsequently, it was detected in several other sources such as comet Hale–Bopp, and protostars and hot corinos (e.g., in Orion A and the protoplanetary nebula CRL 618) (Ellder et al. 1980; Ikeda et al. 2001; Cazaux et al. 2003; Bottinelli et al. 2004a, 2004b; Remijan et al. 2005, 2006; Remijan & Hollis 2006). Later, the less stable trans rotamer was detected in both the laboratory and the interstellar medium (ISM; Ruschin & Bauer 1980; Blom & Günthard 1981; Müller et al. 1983; Neill et al. 2011, 2012). The detailed mechanistic route for the interstellar synthesis of MF is a subject of considerable debate (Horn et al. 2004; Herbst 2005; Garrod & Herbst 2006; Snyder 2006; Bennett & Kaiser 2007; Occhiogrosso et al. 2011; Lawson et al. 2012). The initially assumed gas-phase model involving an ion–molecule reaction of protonated methanol (

) with formaldehyde (H2CO) fails to reproduce the large MF column density, because this pathway demands a significant activation barrier (∼128 kJ mol−1) for one of the intermediate steps (Horn et al. 2004). Instead, gas-phase radiative association between the methyl cation (

) and formic acid (HCOOH) seems energetically promising to produce H+MF and thereafter neutral MF via dissociative recombination with electrons:
1

2

However, this reaction scheme fails to reproduce the huge MF concentration observed in hot cores (Horn et al. 2004). Alternative routes suggested more recently involve ion–molecule reactions with a low barrier or no barrier between protonated methanol and formic acid (acid-catalyzed Fischer esterification) or between protonated formic acid and methanol (

 transfer):
3

4

followed by reaction (2) or exothermic proton transfer of

 (H+MF) to a base with a higher proton affinity than MF (Ehrenfreund & Charnley 2000; Neill et al. 2011, 2012).


The third scenario to consider is emission from the quasar host galaxy. The approach of using the residual flux in DLA troughs to observe the Lyα emission associated with star formation in the host galaxies has been studied extensively (e.g., Kulkarni et al. 2006; Fynbo et al. 2010; Finley et al. 2013; Noterdaeme et al. 2014). Using the Kennicutt et al. (2008) calibration

 and assuming an intensity ratio of

 for the case B recombination, we estimate a star formation ratio of

 based on the luminosity of the residual Lyα emission. This is much higher than estimates of the SFR in quasar host galaxies (

 e.g., Ho 2005; Cai et al. 2014) and in the high-redshift Lyman break galaxies and Lyα emitters (

 e.g., Shapley et al. 2003; Erb et al. 2006; Gronwall et al. 2007). To further check whether the residual Lyα emission originated from the star formation, we compare the colors (the flux density ratios: FUV/NUV and NUV/Lyα) of the residues with those of the galaxies with active star formation. The comparison galaxies are selected from the starburst original 1999 data set (Leitherer et al. 1999). The spectral energy distributions (SEDs) with “Topic—stellar continua” and “Quantity—stellar emission only” (Figures 7–12 on the Starburst99 website http://www.stsci.edu/science/starburst99/docs/table-index.html) are used to compute the galaxy’s colors. The starlight models present in a homogeneous way for five metallicities between Z = 0.040 and 0.001 and three choices of the initial mass function. The age coverage is from 1 Myr to 1 Gyr, and both star formation laws (instantaneous and continuous) are contained. The SEDs placed at the quasar’s redshift are extracted from the flux densities at the effective wavelengths of the FUV/NUV filters and 1215.6 Å. Figure 3(a) shows that SDSS J1259+6212 seriously deviates from the star-forming galaxy group. At wavelengths of a few hundred Angstroms, SDSS J1259+6212 has much bluer slopes than the bluest galaxy continuum. In Fathivavsari et al. (2018), strong and narrow Lyα emission is reported in 155 eclipsing damped Lyα systems, which is revealed with certainty as the narrow emission line from the host galaxy. However, the residual Lyα emission of SDSS J1259+6212 fills in the sub-DLA trough rather, forming a narrow peaked profile, and its luminosity is also higher than that from Fathivavsari et al., eclipsing the DLA sample. Thus, the host galaxy scenario is questionable.


In both models, as we will see, the thick disc scale length is about a factor of two shorter than that of the thin disc, in agreement with the results by Bovy et al. (2012a). The choice of presenting two mass models for the mass distribution of our Galaxy is mainly dictated by two reasons. First, the need to add a central bulge to the global gravitational potential to reproduce the rotation curve in the inner kpcs of the Milky Way strongly depends on the observational data with which one compares the theoretical curve: to reproduce the rise observed in the inner kpcs (see the observational data adopted by Caldwell & Ostriker 1981), Allen & Santillan (1991) introduced a central mass concentration, whose mass is about 15% of the disc mass. However, the central rise observed in the rotation of the molecular gas in the inner Galaxy (for more recent estimates see, for example,  Sofue 2012) may be an effect of non circular motions generated by large scale asymmetries like the bar, as has been shown recently by Chemin et al. (2015). Moreover, this feature is not reported in all the observational studies (see, for example, Reid et al. 2014). Secondly, there is growing evidence that the mass of any classical bulge, if present in the Milky Way, must be small (Shen et al. 2010; Kunder et al. 2012, 2016; Di Matteo et al. 2014, 2015). For these reasons, we prefer to present a second model, our Model II, which does not include any spherical central component, and which is still compatible with the rotation curve of the Galaxy, as given by Reid et al. (2014). Because it has been widely used in the last decades, and due to the facility of its implementation, we explicitly aim at generating Galactic models similar to the Allen & Santillan (1991) model, so to make any implementation of these new models, and any comparison with Allen & Santillan (1991), straightforward. As for the model proposed by Allen & Santillan (1991), Models I and II are axisymmetric and time-independent, and do not include stellar asymmetries such as a bar or spiral arms. No truncation is assumed for the discs, while the halo is truncated at 100 kpc, in agreement with the choice of Allen & Santillan (1991). As we describe in the following section, the analytic forms for the discs, halo, and bulge potentials are the same as those adopted by Allen & Santillan (1991). To allow an easy comparison with the Allen & Santillan (1991) model, in the following we will make use of the same system of units adopted by these authors: the potential is given in units of 100 km2/ s2, lengths are in kpc, masses in units of 2.32 × 107M⊙, time in units of 0.1 Gyr, velocities in units of 10 km s-1 and the vertical force in units of 10-9 cm s2. In these units, the gravitational constant G is equal to 1 and the mass volume density is in units of 2.32 × 107M⊙/ kpc3.


21 cm Intensity Mapping (IM) is a promising technique to map the cosmological large scale distribution of matter through the observation of 21 cm radio emission/absorption of neutral hydrogen gas (H i), while not requiring the detection of individual sources (Bharadwaj, Nath & Sethi 2001; Battye, Davies & Weller 2004) and has been largely explored in the context of the search for the EoR (Epoch of Reionization) signal (Pritchard & Loeb 2008; Morales & Wyithe 2010). Subsequently, it was suggested that post-EoR 21 cm Intensity Mapping surveys could be used to constrain dark energy through the measurement of the Baryon Acoustic Oscillations (BAO) scale (Chang et al. 2008; Seo et al. 2010; Ansari et al. 2012) in the large scale structure (LSS) distribution, over a broad redshift range (z ≲ 6). Such surveys require instruments with large instantaneous bandwidth and field of view and several groups have built dense interferometric arrays to explore IM, such as CHIME (Bandura et al. 2014) or Tianlai (Chen 2012) in the last decade. Smaller instruments such as PAON4 (Ansari et al. 2020) and BMX (O’Connor et al. 2021) have also been built to explore specific technical aspects of these arrays, as well as transit mode operation and calibration. CHIME has proved to be a powerful fast radio burst and pulsar observation machine (The CHIME/FRB Collaboration 2021) and has motivated the design and construction of larger, dish-based, dense interferometric arrays such as HIRAX (Newburgh et al. 2016) and CHORD (Vanderlinde et al. 2019). Non-interferometric surveys have also been considered, such as BINGO (Battye et al. 2016; Wuensche et al. 2022), FAST (Hu et al. 2020) or using the SKA precursor MeerKAT (Wang et al. 2021). Intensity Mapping is also being used to search for signals from the EoR and the cosmic dawn, at redshifts above z ≳ 10, by several large radio-interferometers, such as LOFAR (van Haarlem et al. 2013), MWA (Tingay et al. 2013), HERA (DeBoer et al. 2017), and LWA (Eastwood et al. 2018). It is also planned to be used with SKA-low (Mondal et al. 2020).


We found two nebulae associated with J2055, with angular dimensions of 12′ × 20″ and 250″ × 30″, respectively, for the brightest and faintest. The observed extensions of the nebulae, at a distance of 600 pc, would correspond to physical dimensions of ∼2.1 × 0.05 pc and ∼0.7 × 0.09 pc, respectively (assuming no inclination with respect to the plane of the sky). The luminosities of the nebulae in the 0.3–10 keV energy range (assuming d = 600 pc) are 1.2 × 1031 erg s−1 and 2.4 × 1030 erg s−1, corresponding to fractions of 2.4 × 10−3 and 3.0 × 10−4 of the pulsar spin-down luminosity. A few elongated tails of X-ray emission associated with rotation-powered pulsars have been discovered in the past (Gaensler et al. 2004; Becker et al. 2006; McGowan et al. 2006; Kargaltsev & Pavlov 2008b). Although for our pulsar we have no information about the proper motion, the bow-shock PWN scenario would seem the most natural explanation for one of the nebulae. Luminosity values are fully compatible with that measured for other synchrotron nebulae, for which the pulsars channel into their tails 10−2 to 10−4 of their rotational energy loss. In the case of synchrotron emission tails, if we assume the optimistic maximum Lorentz factor of ∼108 for electron acceleration in such a low-energetic pulsar magnetosphere and the high value of the ambient magnetic field of ∼50 μG (following considerations in De Luca et al. 2011), it is possible to estimate the synchrotron cooling time of the emitting electrons as

 (B/50 μG)

 (E/1 keV)

 yr. Coupling this value with the estimated physical length of the feature yields an estimate of the bulk flow speed of the emitting particles of ∼20,000 and 3000 km s−1, assuming no inclination with respect to the plane of the sky. The first value is only marginally consistent with results in the literature, while the second one is fully consistent (Kargaltsev & Pavlov 2008b). Taking into account the energetics, both nebulae are at least marginally consistent with a classical synchrotron nebula explanation. For classical synchrotron nebulae, we expect a relatively bright diffuse emission surrounding the pulsar, where the emission from the wind termination shock is brightest. The model of Gaensler & Slane (2006) predicts a low-scale termination shock of ∼06 fo J2055 (at 600 pc), thus not resolved by XMM-Newton, even in the case of typical ambient densities (0.01 atoms cm−3) and pulsar velocities (some hundreds of kilometers per second). In that case, part of the flux we assign to the pulsar comes instead from the termination shock (Kargaltsev & Pavlov 2008a). Along their main axis, the brightness profiles of both  nebulae are consistent with an almost flat behavior, with a sudden decrease at the end. This is acceptable for synchrotron emission nebulae: in fact, the pulsar wind is more energetic in the surroundings of the pulsar, where the loss of energy through synchrotron emission is higher, decreasing in flux and energy along the pulsar trail. Taking also into account an inhomogeneous medium and/or magnetic field, this behavior is expected, and it is observed for many objects in the literature (see, e.g., Kargaltsev & Pavlov 2008b). Such a synchrotron cooling of the particles injected at the termination shock induces a significant softening of the emission spectrum as a function of the distance from the pulsar in bow-shock PWNe. Taking into account the upper limit variation of the photon index we found, for the brightest tail of J2055 we do not have the predicted spectral variation. Classical theories of ram pressure particle confinement in pulsar tails are hardly put to the test by parsec-long nebulae, which would require much higher efficiencies than in all the other cases. The tightness of the main feature of J2055, if confirmed to be a synchrotron nebula from its shape and the pulsar motion, cannot be accounted for by any model in the literature. We conclude that the low energetics of the pulsar, the lack of any spatial-spectral variability, and the tightness of the brighter nebula disfavor the classical synchrotron emission nebula. This model can explain the characteristics of the secondary feature.


The shape of the bow shock nose can be directly inferred from the Hα images and used to constrain the bow shock standoff radius. In the thin-shell limit, the radial shape of the bow shock can be expressed as (Wilkin 1996)
7

R

(
θ
)

=

R

0

csc

θ

3
(
1
−
θ

cot

θ
)

,

where R0 is the standoff radius and θ represents the angle between the pulsar’s velocity and a point R(θ) along the bow shock. The standoff radius is dictated by the pressure balance between the ambient ISM and the neutron star wind and directly related to the interstellar density ρA, the pulsar wind velocity vw and mass-loss rate

m

w

, and the pulsar velocity vp (Wilkin 1996):
8

R

0

=

m

w

v

w

4
π

ρ

A

v

p

2

1

/

2

.

Equation (8) assumes an isotropic stellar wind, which is not generally true of a relativistic neutron star wind. When the pulsar rotation vector is skewed relative to the pulsar proper motion and magnetic field, the pulsar wind will be quasi-isotropic, and its exact behavior will depend on both the orientations of these vectors and the opening angle of the wind. For B2224+65, the light-cylinder radius is significantly smaller than the bow shock standoff radius, suggesting that an isotropic wind is an adequate assumption without additional evidence for anisotropy (for a detailed discussion, see Chatterjee & Cordes 2002). Assuming that the spin-down energy loss is entirely carried by the relativistic wind,

m

w

v

w

=

E

/

c

. The standoff radius can be conveniently reformulated as an angle given by
9

θ

0

=
56.3

mas

sin

2

i

n

A

1

/

2

E

33

1

/

2

μ

100

D

kpc

2

,

where i is the inclination angle, nA = ρA/mH is the total number density of the interstellar hydrogen and helium mixture in atomic mass units per cubic centimeter,

E

 is the spin-down luminosity in erg per second, D is the distance to the pulsar in kiloparsecs, and μ100 is the pulsar proper motion in 100 mas yr−1 (Chatterjee & Cordes 2002). The spin-down luminosity is

E

=
4

π

2

I

P

/

P

3

, where I ≈ 1045 g cm2. The period derivative

P

 can be corrected for the Schklovskii effect using
10

P

=

P

obs

−
2.43
×

10

−
21

P

μ

masy

2

D

kpc

,

where μmasy is the proper motion in milliarcseconds per year (Shklovskii 1970; Brownsberger & Romani 2014). This correction is negligible for B2224+65. The atomic number density nA is converted to an electron density ne assuming a cosmic abundance γH = 1.37, where nA = nHγH and nH ≈ ne. The standoff angle that is inferred from the outer edge of the Hα emission corresponds to a forward shock that lies slightly upstream of the contact discontinuity at an angular distance θa ≈ 1.3θ0 (Aldcroft et al. 1992; Bucciantini 2002). The period, period derivative, and spin-down luminosity for B2224+65 are shown in Table 1.


In a pure scattering medium, the radiative transfer equations written in terms of the photon number intensity ni (i = O, X) take the form (see e.g. Alexander, Mészáros & Bussard 1989; Mészáros, Pavlov & Shibanov 1989)
(15)
\begin{eqnarray}
\mu _z\frac{\mathrm{d}n_i}{\mathrm{d}\tau }&=& \sum _{k=\mathrm{O,X}}\int \bigg \lbrace -\sigma _{ik}(\alpha \rightarrow \alpha ^{\prime })n_i(\alpha )\left [1+n_k(\alpha ^{\prime })\right] \nonumber \\
&&+ \,\sigma _{ki}(\alpha ^{\prime }\rightarrow \alpha )\bigg (\frac{\varepsilon ^{\prime }}{\varepsilon }\bigg )^2n_k(\alpha ^{\prime })\left[1+n_i(\alpha )\right ]\bigg \rbrace \mathrm{d}\varepsilon ^{\prime }\mathrm{d}\Omega ^{\prime },
\end{eqnarray}
where stimulated scattering is accounted for. Here, dτ = neσTds is the infinitesimal Thomson depth, with s being a parameter along the photon propagation direction. However, the previous expression can be considerably simplified. In fact, in LTE, the scattering cross-sections (1) must obey the detailed balance condition (see e.g. Alexander et al. 1989; Mészáros et al. 1989; Alexander & Mészáros 1991):
(16)
\begin{eqnarray}
\sigma _{ki}(\alpha ^{\prime }\rightarrow \alpha ) =\bigg (\frac{\varepsilon }{\varepsilon ^{\prime }}\bigg )^2\exp \left[-(\varepsilon -\varepsilon ^{\prime })/kT\right]\sigma _{ik}(\alpha \rightarrow \alpha ^{\prime })\,;
\end{eqnarray}
in this way, equation (15) becomes
(17)
\begin{eqnarray}
\mu _z\frac{\mathrm{d}n_i}{\mathrm{d}\tau } &=& \sum _{k=\mathrm{O,X}}\int \bigg \lbrace -\sigma _{ik}(\alpha \rightarrow \alpha ^{\prime })n_i(\alpha ) \nonumber \\
 &&+ \, \sigma _{ik}(\alpha \rightarrow \alpha ^{\prime })F_i(\alpha ,\varepsilon ^{\prime })n_k(\alpha ^{\prime })\bigg \rbrace \mathrm{d}\varepsilon ^{\prime }\mathrm{d}\Omega ^{\prime },
\end{eqnarray}
where
(18)
\begin{eqnarray}
F_i(\alpha ,\varepsilon ^{\prime })\equiv \exp \left [-(\varepsilon -\varepsilon ^{\prime })/kT\right]\left(1+n_i(\alpha )\right)-n_i(\alpha ).
\end{eqnarray}
Finally, under the assumption of conservative scattering (ε = ε΄) and substituting the expressions (1), one obtains
(19)
\begin{eqnarray}
\mu _z\frac{\mathrm{d}n_\mathrm{O}}{\mathrm{d}\tau } &=& -\bigg [1-\mu ^2_\mathrm{Bk}+\frac{3\mu ^2_\mathrm{Bk}}{4}\bigg (\frac{\varepsilon }{\varepsilon _\mathrm{B}}\bigg )^2\bigg ]n_\mathrm{O}(\alpha ) \nonumber \\
&&+\,\frac{3}{8\pi }\int _{4\pi }\bigg [\left(1-\mu _\mathrm{Bk}^2\right)\left(1-\mu _\mathrm{Bk}^{\prime 2}\right)n_\mathrm{O}(\alpha ^{\prime }) \nonumber \\
&&+\,\bigg (\frac{\varepsilon }{\varepsilon _\mathrm{B}}\bigg )^2\mu ^2_\mathrm{Bk}\cos ^2\left(\phi _\mathrm{Bk}-\phi ^{\prime }_\mathrm{Bk}\right)n_\mathrm{X}(\alpha ^{\prime })\bigg ]\mathrm{d}\Omega ^{\prime }, \nonumber \\
\ \nonumber \\
\mu _z\frac{\mathrm{d}n_\mathrm{X}}{\mathrm{d}\tau } &=& -\bigg (\frac{\varepsilon }{\varepsilon _\mathrm{B}}\bigg )^2n_\mathrm{X}(\alpha ) \nonumber \\
&&+ \,\frac{3}{8\pi }\bigg (\frac{\varepsilon }{\varepsilon _\mathrm{B}}\bigg )^2\int _{4\pi }\left[\sin ^2\left(\phi _\mathrm{Bk}-\phi ^{\prime }_\mathrm{Bk}\right)n_\mathrm{X}(\alpha ^{\prime })\right. \nonumber \\
&&\left.+\,\mu ^{\prime 2}_\mathrm{Bk}\cos ^2\left(\phi _\mathrm{Bk}-\phi ^{\prime }_\mathrm{Bk}\right)n_\mathrm{O}(\alpha ^{\prime })\right]\mathrm{d}\Omega ^{\prime }.
\end{eqnarray}


To constrain the number density of the clump, getting the size of the clump (rc) should be required. The size of the self-gravitating clump in the context of AGNs has been estimated in the literature (e.g., Krolik & Begelman 1988; Hönig & Beckert 2007; Kawaguchi & Mori 2011). According to the authors of these works, we have rc ≤ 0.05 pc at 1 pc from the central SMBH with MBH = 8 × 108M⊙ (Scharwächter et al. 2013). On the other hand, the size of clumps can be constrained by the observations of transient X-ray absorption events in nearby AGNs, i.e., the typical size is 0.002 pc (e.g., Markowitz et al. 2014; Tanimoto et al. 2019). By adopting Nc and rc, taking into account their uncertainties of 3 ≤ Nc ≤ 15 and

, the lower limit of number density of each ionized cloud

 can be given by
8

In the case of ionized gas clumps, the typical Thomson scattering opacity of each cloud can be estimated as

 since the optical depth is given by

, where σT is the Thomson cross section. If we adopt the opacity ratio of the dusty gas and the ionized gas is ≃103 at the UV band for the AGN radiation (e.g., Umemura et al. 1998; Ohsuga & Umemura 2001; Wada 2012), the optical depth of each cloud is

, which is consistent with the lower value for nearby Seyfert galaxies (e.g., Table 10 in Ramos Almeida et al. 2011). This might indicate that the ionized gas is also clumpy within the 1 pc region of 3C 84. Wada et al. (2018) examine properties of the ionized gas irradiated by less luminous AGN such as Seyfert galaxies based on their “radiation-driven fountain” model (Wada 2012). They found that the ionized region show nonuniform internal structures, corresponding to the clumpy fountain flows caused by the radiation pressure on dusty gas, although the typical density (

) is smaller than our estimate. In addition, by the optical/NIR observations, the existence of dense clumps with

 cm−3 has been reported from the detection of coronal lines within NLR (e.g., Murayama & Taniguchi 1998). These high density clouds in NLRs might contribute the absorption feature of N1.


The origin of the Milky Way Galaxy presumably commenced within the initial one billion years (Gyr) subsequent to the Big Bang origin of the Universe around 13.7 Gyr ago. The formation of the Galaxy initiated with the merging of the initial diffuse neutral hydrogen gas clouds. This was followed by the gravitational merging of the colliding initial protogalaxies, thereby leading to the gradual accretional evolution of the galaxies in a hierarchical manner. The formation and evolution of the successive generations of stars inside the Galaxy led to the gradual abundance evolution of the elemental (and isotopic) inventories of the Galaxy. Galactic chemical evolution (GCE) models deal with understanding the origin of the distribution and gradual evolution of the elemental (and isotopic) abundances across the Galaxy over the Galactic time-scale of ∼13.7 Gyr (Matteucci and Franҫois 1989; Rana 1991; Chiappini, Matteucci & Gratton 1997; Pagel 1997; Chang et al. 1999; Goswami & Prantzos 2000; Alibés, Labay & Canal 2001; Chiappini, Matteucci & Romano 2001; Matteucci 2003, 2014; Franҫois et al. 2004; Kobayashi et al. 2006; Kobayashi and Nomoto 2009; Matteucci et al. 2009; Kobayashi and Nakasato 2011; Kobayashi, Karakas & Umeda 2011; Micali, Matteucci & Romano 2013; Minchev, Chiappini & Martig, 2013, 2014; Mott, Spitoni & Matteucci 2013; Sahijpal 2013, 2014; Sahijpal & Gupta 2013; Sahijpal 2014; Minchev et al. 2015; Spitoni et al. 2015). The astronomical observed trends in the elemental abundance distribution across the Galaxy serve as a major constraint on the GCE models in order to understand the accretional growth rate and the dynamical evolution of the Galaxy with its three main components, i.e. the Galactic halo, the thick disc, and the thin disc (Marochnik & Suchkov 1996; Matteucci 2003; Sparke & Gallagher 2007; Chiappini et al. 2015). The GCE model provides a detailed evolutionary account of the star formation rate (SFR), the average prevalent initial mass function (IMF) for star formation, the supernova (SN Ia, Ib/c, II) rates, and the stellar nucleosynthetic contributions of stars with distinct mass and metallicity.


Most of these OH absorbers show an OH column density about the order of $10^{15} \rm\, cm^{-2}$. Gupta et al. (2018) show that galaxies with z  0.4 could have much lower OH column densities, i.e. below $10^{14} \rm\, cm^{-2}$, using a detected OH absorber and upper limits of eight non-detections and assuming the OH excitation temperature $T_x^{\rm OH} = 3.5\,$ K, however, a more commonly used $T_x^{\rm OH}$ for OH absorbers at cosmological distances is 10 K (e.g. Kanekar & Chengalur 2002; Kanekar et al. 2003; Curran et al. 2011). The Gupta et al. (2018) sample contains only intervening absorbers with relatively low redshifts. Grasha et al. (2020) searched 11 intervening and five associated H i absorbers for OH absorptions and report no new detections except for a re-detection of the 1667 MHz OH absorption towards PKS 1830−211. Grasha et al. (2019) searched a large sample of compact radio sources for associated H i and OH absorptions but only six of them, which have relatively larger redshifts (z ∼ 0.6), have H i absorption detections and none of them have OH absorption detection. Intervening absorbers are mostly from foreground galaxies or H i dark clouds. Associated absorbers can be found in AGN outflows, circumnuclear discs, and cold gas clouds in the host galaxy or merger relics (Maccagni et al. 2017; Allison et al. 2019; Oosterloo et al. 2019). The cold molecular gas in these systems is not well studied due to a low detection number (Allison et al. 2019) and they could potentially have very different OH column densities and different excitation temperatures (Curran et al. 2016). Therefore, we observe a sample of low-redshift (z  0.3) galaxies with associated H i 21-cm absorption detections, with an extra target with a deep intervening H i 21-cm absorption for comparison, using the Five-hundred-meter Aperture Spherical radio Telescope (FAST; Nan et al. 2011; Li & Pan 2016; Li et al. 2018a; Jiang et al. 2020) aiming to study their cold gas parameters through OH absorptions.


In general, the time evolution of the PSR is governed by,
(3)$$\begin{eqnarray*}
{T^{\mu \nu }}_{;\nu } = 0,
\end{eqnarray*}$$where Tμν is the energy–momentum tensor. In the MPD formulation, a multipole expansion of the energy–momentum tensor is undertaken. This constructs the ‘gravitational skeleton’. In the extreme mass ratio limit, the pole and dipole terms are dominant and terms greater than the quadrupole can be neglected. The PSR dynamics are then entirely determined by the background gravitational field and the dynamical spin interaction with this field. The mass monopole is described by the 4-momentum pμ whilst the spin dipole is sμν. The corresponding equations of motion are (Mathisson 1937; Papapetrou 1951; Dixon 1974),
(4)$$\begin{eqnarray*}
\frac{D p^{\mu }}{\mathrm{ d} \tau } = - \frac{1}{2} {R^{\mu }}_{\nu \alpha \beta } u^{\nu } s^{\alpha \beta },
\end{eqnarray*}$$(5)$$\begin{eqnarray*}
\frac{D s^{\mu \nu }}{\mathrm{ d} \tau } = p^{\mu } u^{\nu } -p^{\nu } u^{\mu },
\end{eqnarray*}$$for proper time parametrization τ along the PSR worldline zα(τ), D/dτ denotes a covariant derivative whilst uν is the PSR 4-velocity and Rμναβ the Riemann curvature tensor. In order to first construct the gravitational skeleton requires first specifying a reference worldline with which to define the expansion. Typically, such a choice would be the worldline described by the centroid (centre of mass) of the body. However, in GR the centroid choice is observer-dependent. This uncertainty is evidenced in the system of equations (4) and (5), which is not a determinate set since there exist more unknowns than equations. This is related to the uncertainty in choosing a reference world line for the multipole expansion. Explicitly choosing an observer with respect to which the centre of mass is defined renders the system of equations determinate. Such a choice is known as the spin supplementary condition (SSC). For this work we adopt the Tulczyjew-Dixon (TD) condition which specifies the centroid to be that measured in zero 3-momentum frame:
(6)$$\begin{eqnarray*}
s^{\mu \nu } p_{\nu } = 0,
\end{eqnarray*}$$(Tulczyjew 1959; Dixon 1964). A key advantage of this SSC is that it specifies a unique worldline. In contrast, other choices of SSC are infinitely degenerate (for discussion, including application to EMRIs, see Babak, Gair & Cole 2014; Filipe Costa & Natário 2014). With this choice of SSC both the mass of the MSP,
(7)$$\begin{eqnarray*}
m = \sqrt{- p^{\mu } p_{\mu }}
\end{eqnarray*}$$and the scalar contraction of the spin vector
(8)$$\begin{eqnarray*}
s = s^{\mu } s_{\mu }
\end{eqnarray*}$$are constants of the motion. In turn, the spin vector is a contraction of the spin tensor,
(9)$$\begin{eqnarray*}
s_{\mu } = -\frac{1}{2m} \epsilon _{\alpha \beta \mu \nu } p^{\nu } s^{\alpha \beta }
.
\end{eqnarray*}$$


To understand the physics of sunspots, one has to study their temporal evolution. Formation and decaying phases play an important role in sunspot evolution. We refer to Martínez Pillet (2002) for a review of decaying sunspot evolution. For a long time, it was only possible to study the decay of the morphological changes of the area of sunspots because of the lack of inversion codes to interpret full-Stokes measurements, although the magnetic flux is the more important parameter. Bumba (1963) found a linear decrease of the area of the sunspot with time. Martinez Pillet et al. (1993) confirmed this linear decay, but obtained a different coeffcient. In contrast, Petrovay & van Driel-Gesztelyi (1997) found a parabolic decay with a rate proportional to

A
(
t
)

$ \sqrt{A(t)} $

, where A is the area of the spot. Linear decay rates of the areas of 32 sunspots are found by Chapman et al. (2003). Baumann & Solanki (2005) investigate the Greenwich sunspot group record, but they were not able to distinguish between a linear and a quadratic decay law. Hathaway & Choudhary (2008) publish an almost constant decay rate of 3.65μHemispheres day−1. Gafeira et al. (2014) study four sunspots and find an approximately linear decay of the areas similar for umbra and penumbra. In both studies, the decay rates are different for individual spots. Case studies resulting in a linear decrease of the magnetic flux during the decay phase are presented by Deng et al. (2007) and Verma et al. (2016). In the latter publication, the development of the area is non-monotonic. Sheeley et al. (2017) investigated the development of the magnetic flux in 36 sunspots, but not all of them were in the decaying phase. Some spots showed a nearly linear decay, but they found also indications of a bursty decay. In a 100 h numerical simulation of a mature sunspot, Rempel (2015) find a linear decay of the magnetic flux in the umbra for the last 80 h. The penumbral magnetic flux remained almost constant during this period.


Disk galaxies typically reveal out-of-equilibrium features due to incomplete equilibration. These may appear in the form of bars and spiral arms, which are large-scale perturbations in the radial and azimuthal directions, responsible for a slow, secular evolution of the disk. In the vertical direction, disks often reveal warps (Binney 1992). In the case of the Milky Way (MW) disk, which can be studied in much greater detail than any other system, recent data from astrometric and radial velocity surveys such as SEGUE (Yanny et al. 2009), RAVE (Steinmetz et al. 2006), GALAH (Bland-Hawthorn et al. 2019), LAMOST (Cui et al. 2012), and above all Gaia (Gaia Collaboration et al. 2016, 2018a, 2018b) have revealed a variety of additional vertical distortions. At large galactocentric radii (>10 kpc) this includes, among others, oscillations and corrugations (Xu et al. 2015; Schönrich & Dehnen 2018), and streams of stars kicked up from the disk that undergo phase mixing, sometimes referred to as “feathers” (e.g., Price-Whelan et al. 2015; Thomas et al. 2019; Laporte et al. 2022). Similar oscillations and vertical asymmetries have also been reported in the solar vicinity (e.g., Widrow et al. 2012; Williams et al. 2013; Yanny & Gardner 2013; Gaia Collaboration et al. 2018b; Quillen et al. 2018; Bennett & Bovy 2019; Carrillo et al. 2019). One of the most intriguing structures is the phase-space spiral discovered by Antoja et al. (2018) and studied in more detail in subsequent studies (e.g., Bland-Hawthorn et al. 2019; Li 2021; Li & Widrow 2021; Gandhi et al. 2022). Using data from Gaia DR2 (Gaia Collaboration et al. 2018a), Antoja et al. (2018) selected ∼900,000 stars within a narrow range of galactocentric radius and azimuthal angle centered around the Sun. When plotting the density of stars in the (z, v

z
)-plane of vertical position, z, and vertical velocity, v

z
, they noticed a faint, unexpected spiral pattern, which became more enhanced when color-coding the (z, v

z
)-“pixels” by the median radial or azimuthal velocities. The one-armed spiral makes two to three complete wraps, resembling a snail shell, and is interpreted as a signature of phase mixing in the vertical direction following a perturbation, which Antoja et al. (2018) estimate to have occurred between 300 and 900 Myr ago. More careful analyses in later studies (e.g., Bland-Hawthorn et al. 2019; Li 2021) have nailed down the interaction time to ∼500 Myr ago.


The spectral ratio method has some shortcomings, e.g., one has to assume that the transverse fluctuations consist of a slab component and a 2D component only, and that these two components have the same spectral shape over the wavenumber range considered. This is why new techniques have recently been developed to investigate the slab and 2D turbulence. In particular, Horbury et al. (2008) proposed a different approach: a local mean magnetic field can be defined, at any given time and scale, as the convolution between the instantaneous magnetic field B0 and a Gaussian function (normalized to unity) whose width is the scale of interest. Fluctuations at a particular scale and at a range of angles to the local mean magnetic field can be gathered to study the solar wind turbulence in that direction. To obtain an estimate for the component of the turbulence that might provide the largest contribution to the scattering of the electrons, most likely fluctuations with wavevectors nearly parallel to the magnetic field (i.e., the “slab” component) with which the electrons can interact resonantly, we apply the method of wavelet transforms (e.g., Bruno & Telloni 2015). Figure 11 shows the power spectral density of the z-component of the magnetic field during 2002 October 20 as a function of the angle between the orientations of the local mean magnetic field and the sampling direction. The angular distribution of the normalized magnetic helicity spectrum of the magnetic field fluctuations is presented in Figure 12. At wavenumbers around

 km−1 left- and right-handed polarized magnetic fluctuations, commonly associated with ion cyclotron waves and kinetic Alfvén waves, respectively (Podesta & Gary 2011; Telloni et al. 2015), are clearly resolved in quasi-perpendicular and quasi-parallel directions with respect to the local mean magnetic field. Figure 13 shows the total power spectral density, Pzz(k), of the Bz component computed by means of the wavelet transforms (black curve), which is in good agreement with the spectral density determined with the standard FFT method as described above. The red curve, which makes up ≈50% of the transverse power, shows our estimate for the slab component. It is obtained by integrating the angular distribution of power spectral density of Bz shown in Figure 11, between 0° and 10°, i.e., over the angle range where the magnetic field vector can be considered quasi-parallel to the radial direction. In other words, the slab component is obtained by averaging, at each scale, the power spectral density found at those time instants when the magnetic field and the velocity vectors are, at that particular scale, aligned within 10°. It is worth noting that, since the data points involved in these averages can be very few (in an Alfvénic fast wind, such as the one considered in the present analysis, the magnetic field vector is mostly perpendicular or highly oblique to the velocity vector), the uncertainty related to the averages can be very large. As for the spectral ratio test, the underlying assumption is that Pzz(k) consists of two disjoint parts—a slab component with

 and a 2D component with

—and the more the real geometry of the fluctuations deviates from the above, the larger the uncertainty in the amount of the slab component will become. Nevertheless, the above method seems to provide the best estimate of the slab component of the solar wind fluctuations that can be provided by single-spacecraft measurements at the moment (e.g., Chen et al. 2010; Wicks et al. 2010). In the following, we adopt as a working hypothesis that a 50% slab component is responsible for the scattering of the electrons and we neglect effects of the magnetic helicity.


For Model A, which assumes the very broad component around 4450 Å as coming from the Balmer emission regions, the following model functions are included. A narrow plus a broad Gaussian function (second moment smaller or larger than 500 km s−1) G
Hβ, N
([λ
0, β
N
,  σ

β
N
,  f

β
N
]) (parameter λ
0 and σ as central wavelength and second moment in units of angstrom) and G
Hβ, B
([λ
0, β
B
,  σ

β
B
,  f

β
B
]) are applied to describe the Hβ. Narrow and broad Gaussian functions G
Hγ,N
([λ
0, γ
N
,  σ

γ
N
,  f

γ
N
]) and G
Hγ, B
([λ
0, γ
B
,  σ

γ
B
,  f

γ
B
]) are applied to describe the Hγ. Two narrow Gaussian functions G

c1([λ
0, c1,  σ

c1,  f

c1])and G

c2([λ
0, c2,  σ

c2,  f

c2]) are applied to describe the core components of the [O iii]λ4959, 5007 Å doublet. Two broad Gaussian functions G

e1([λ
0, e1,  σ

e1,  f

e1]) and G

e2([λ
0, e2,  σ

e2,  f

e2]) are applied to describe the extended components of the [O iii] doublet (Greene & Ho 2005a; Shen et al. 2011; Zhang 2021a), considering probably spatially extended emission regions for the outflow-related extended [O iii] components as discussed in Zakamska et al. (2016). A broad and a narrow Gaussian functions G

c3([λ
0, c3,  σ

c3,  f

c3]) and G

e3([λ
0, e3,  σ

e3,  f

e3]) are applied to describe the core and the extended components of the [O iii]λ4363 Å. A power-law component

A×λ4100ÅB

 is applied to describe the continuum emissions underneath the emission lines. When the model functions are applied, the following restrictions are accepted. First, the line flux and second moment of each Gaussian component are not smaller than zero. Second, ratios of the central wavelengths (in units of angstrom) and second moments (in units of angstrom) of G
Hβ, N
 to G
Hγ, N
 are fixed to be
1

λ0,γN4341.68Å=λ0,βN4862.68Åσ0,γN4341.68Å=σ0,βN4862.68Å,

leading the narrow Balmer lines to have the same redshift and line widths. Third, ratios of the central wavelengths, second moments, and fluxes of G

c1 to G

c2 to G

c3 are fixed to be
2

λ0,c15008.24Å=λ0,c24960.295Å=λ0,c34364.436Åσ0,c15008.24Å=σ0,c24960.295Å=σ0,c34364.436Åfc1=3×fc2(fc3free).

Fourth, the ratios of the central wavelengths, second moments, and fluxes of G

e1 to G

e2 to G

e3 are fixed to be
3

λ0,e15008.24Å=λ0,e24960.295Å=λ0,e34364.436Åσ0,e15008.24Å=σ0,e24960.295Å=σ0,e34364.436Åfe1=3×fe2(fe3free).

The model parameters of G
Hγ, B
 and G
Hβ, B
 are free to model parameters, which will provide further clues to the origin of the very broad components around 4450 Å from Balmer emissions or from extended [O iii] emissions or other physical origins. Then, through the Levenberg–Marquardt least-squares minimization technique, the best-fitting results to the emission lines obtained using Model A can be well determined, and are shown in Figure 2 with χ
2/dof ∼ 0.88 (where χ
2 and dof are the summed squared residuals for the best-fitting results and the degrees of freedom, respectively). Here, in order to show clearer results, the left panel of Figure 2 shows the best-fitting results to the emission lines around Hβ with a rest wavelength range from 4600–5400 Å, and the right panel of Figure 2 shows the best-fitting results for the emission lines around Hγ with the rest wavelength range from 4200–4600 Å.


To be a feasible mechanism of cusp-core transformation, SNF needs to fulfill a number of conditions. First and foremost, the total energy that is released by supernovae has to be sufficient to unbind the DM halo’s central cusp (Peñarrubia et al. 2012). A secondary condition is that SNF needs to be impulsive, i.e. SN-driven gas outflows need to give rise to sizeable changes of the gravitational potential on time-scales which are shorter than the typical dynamical times of DM particles in the inner halo (Pontzen & Governato 2012; Burger & Zavala 2021). From the observational side, there is evidence that starbursts in bright dwarfs, and thus, their associated supernova cycles, happen on time-scales that are comparable to the typical dynamical times of those galaxies (Kauffmann 2014). However, observations still lack the time resolution required to resolve starburst cycles on the smaller dynamical time-scales of the low-mass MW dwarf spheroidals (Weisz et al. 2014). In general, the more energy is injected during a SNF cycle, the shorter the time is over which that energy is injected, and the more concentrated the baryonic mass is to the centre of the DM halo (Burger & Zavala 2021), the more efficient the SNF-induced cusp-core transformation will be. In hydrodynamic simulations of galaxy formation, the implementations of SNF are calibrated to the resulting structural properties of larger galaxies. Recent studies suggest that, in cosmological simulations, the efficiency of SNF at flattening the cusps of dwarf-size DM haloes is mainly determined by one model parameter, the gas density threshold for star formation (Benítez-Llambay et al. 2019; Dutton et al. 2020). In a given dwarf galaxy, larger star formation thresholds lead to more bursty star formation, more concentrated and impulsive feedback, and a stronger contribution of baryons to the central potential, and hence to enhanced core formation (Benítez-Llambay et al. 2019; Bose et al. 2019).


Radio interferometric observations of the redshifted 21-cm signal directly measures the complex visibilities that are the Fourier components of the intensity distribution on the sky. The radio telescope typically has a finite beam that allows us to use the ‘flat-sky’ approximation. Ideally, the fields κ and δT are expanded in the basis of spherical harmonics. For convenience, we use a simplified expression for the angular power spectrum by considering the flat sky approximation whereby we can use the Fourier basis. Using this simplifying assumption, we may approximately write the cross-correlation angular power spectrum as (Dash & Guha Sarkar 2021)
$$\begin{eqnarray}
C^{ T \kappa }_\ell &=& \frac{1 }{\pi (\chi _2- \chi _1)} \sum _{\chi _1}^{\chi _2} \frac{\Delta \chi }{\chi ^2} ~ \mathcal {A}_T (\chi) \mathcal {A}_\kappa (\chi) D_{+}^2 (\chi) \int _0^{\infty } \mathrm{ d}k_{\parallel }\\
&&\times \,\left[ 1 + \beta _T(\chi) \frac{k_{\parallel }^2}{k^2} \right] P (k),
\end{eqnarray}$$where $k = \sqrt{k_{\parallel }^2 + \left(\frac{\ell }{\chi } \right)^2 }$, D+ is the growing mode of density fluctuations, and βT = f/bT is the redshift distortion factor – the ratio of the logarithmic growth rate f and the bias function and bT(k, z). The redshift-dependent function $\mathcal {A}_{T}$ is given by (Bharadwaj & Ali 2005; Datta, Choudhury & Bharadwaj 2007; Guha Sarkar et al. 2012)
(6)$$\begin{eqnarray}
\mathcal {A}_{T} = 4.0 \, {\rm {mK}} \, b_{T} \, {\bar{x}_{\rm H\,{\small I}}}(1 + z)^2\left(\frac{\Omega _{b0} h^2}{0.02} \right) \left(\frac{0.7}{h} \right) \left(\frac{H_0}{H(z)} \right) .
\end{eqnarray}$$The quantity bT(k, z) is the bias function defined as ratio of H i-21 cm power spectrum to dark matter power spectrum $b_T^2 = P_{\mathrm{ H}\,{\small I}}(z)/P(z)$. In the post-reionization epoch z  6, the neutral hydrogen fraction remains with a value ${\bar{x}_{\rm H\,{\small I}}} = 2.45 \times 10^{-2}$ (adopted from Noterdaeme et al. 2009; Zafar et al. 2013). The clustering of the post-reionization H i is quantified using bT. On sub-Jean’s length, the bias is scale dependent (Fang et al. 1993). However, on large-scale the bias is known to be scale independent. The scale above which the bias is linear, is however sensitive to the redshift. Post-reionization H i bias is studied extensively using N-body simulations (Bagla et al. 2010; Guha Sarkar et al. 2012; Sarkar et al. 2016; Carucci, Villaescusa-Navarro & Viel 2017). These simulations demonstrate that the large-scale linear bias increases with redshift for 1  z  4 (Marín et al. 2010). We have adopted the fitting formula for the bias bT(k, z) as a function of both redshift z and scale k (Guha Sarkar et al. 2012; Sarkar et al. 2016) of the post-reionization signal as
(7)$$\begin{eqnarray}
b_{T}(k,z) = \sum _{m=0}^{4} \sum _{n=0}^{2} c(m,n) k^{m}z^{n} .
\end{eqnarray}$$The coefficients c(m, n) in the fit function are adopted from Sarkar et al. (2016).


Relative time delays between multiple gravitationally lensed images provide a one-step distance anchor of the universe, and thus H
0. This probe is independent of the local distance ladder and the sound-horizon-physics anchors of the CMB and large-scale structure probes. The method, known as the time-delay cosmography, has been proposed more than half a century ago to utilize the transient nature of supernovae (SNe) for measuring the time delays (Refsdal 1964). Time-delay cosmography was first applied by measuring the time delays of multiply lensed quasars with multiseason monitoring campaigns (e.g., Kundić et al. 1997; Schechter et al. 1997; Fassnacht et al. 2002; Tewes et al. 2013; Courbin et al. 2018; Millon et al. 2020a). The discovery of numerous lensed quasar systems, follow-up monitoring, high-resolution imaging, and precise spectroscopic observations have led to a precise measurement of H
0 using seven multiply lensed quasars (Wong et al. 2020; Shajib et al. 2020; Millon et al. 2020b). These measurements assumed particular forms of the mass density profiles of the deflector galaxies. The mass-sheet degeneracy (MSD; see Falco et al. 1985; Schneider & Sluse 2013), an inherent transform leaving the lensing observables invariant while changing the time-delay prediction, poses limits in the precision of H
0 measurements in the absence of additional data. Birrer et al. (2020) introduced an additional degree of freedom to the mass density profiles to avoid constraining the lens model based on the specific form of the mass profiles previously chosen. Birrer et al. (2020) constrained the MSD solely by stellar kinematics observations of the deflector galaxy hierarchically on the deflector population-level mitigating covariances among the assumptions of individual lenses. For the achieved 5% precision measurement of H
0, Birrer et al. (2020) combined the seven TDCOSMO lenses with 33 galaxy–galaxy lenses from the Sloan Lens ACS (SLACS) survey (Bolton et al. 2008; Shajib et al. 2021). The interpretation of the kinematics measurements is impacted by the mass-anisotropy degeneracy (Binney & Mamon 1982; Dejonghe & Merritt 1992), and mitigating this degeneracy requires assumption on the stellar anisotropy distribution or spatially resolved kinematics measurements (e.g., Cappellari 2008; Barnabè et al. 2011; Yıldırım et al. 2020). A forecast for future constraints using kinematics observations in breaking the MSD within the assumptions of the Birrer et al. (2020) analysis is provided by Birrer & Treu (2021).


Some fraction of the Comptonized photons will be intercepted by the disc and reflected into the line of sight by further Compton scattering and florescence. These processes give rise to a complex spectral component that typically peaks at energies of 3050keV (the so-called Compton hump, see Sunyaev  Titarchuk 1980, figs 10a and c) and adds further fluorescent emission to the spectrum. The reflection component is present in both BH XBs (due to interaction of X-rays with the accretion disc and the surface of the normal star; Basko, Sunyaev  Titarchuk 1974) and active galactic nuclei (AGNs; Nandra  Pounds 1994), and complicates attempts to understand the dominant Comptonized emission owing to proximity of the Compton hump to the plausible location of the high-energy turn-off (corresponding to 3kTe). Typical attempts to take reflection into account involve adding sophisticated reflection components (such as the xspec models pexrav (Magdziarz  Zdziarski 1995) or reflionx (Ross  Fabian 2005)) to unsophisticated models of Comptonization, such as a cut-off power law. This inaccurate representation of the Comptonization means that interesting properties such as the electron temperature or Compton y-parameter are then inferred from the best fit of a cut-off power law to the data, rather than a direct treatment of these quantities. In addition to this, Ibragimov etal. (2005) showed that the strength of the reflected component R (defined in terms of the solid angle  subtended by the portion of the disc in line of sight to the corona) is systematically overestimated by such treatment. It has been known for some time that R correlates strongly with photon index  (Gilfanov, Churazov  Revnivtsev 1999; Revnivtsev, Gilfanov  Churazov 2001; Ibragimov etal. 2005), increasing as spectra become softer (implying a decreasing y-parameter). Similar behaviour has been observed for AGN spectra (Zdziarski, Lubiski  Smith 1999; Gilfanov, Churazov  Revnivtsev 2000; Zdziarski etal. 2003). This relationship clearly favours the accretion disc as the source of the seed photons for Comptonization, as the radiation being intercepted by the disc is increasing in tandem with the seed photon flux incident on the electron cloud. However, there is increasing evidence that at lower luminosities synchrotron photons from the magnetized corona may contribute a significant, perhaps dominant, population of seed photons, motivating the creation of so-called hybrid models. Such models typically comprise both a low-energy thermal population and an additional high-energy non-thermal population of electrons (see, e.g. Coppi 1999; Merloni  Fabian 2001; Del Santo etal. 2013).


The line-center photons of the hydrogen Lyα line originate just at the boundary of the model’s TR. In a 1D model atmosphere (e.g., Fontenla et al. 1993), the vector normal to the TR lies along the vertical, which coincides with the symmetry axis of the incident radiation field. In a 3D model atmosphere (e.g., Carlsson et al. 2016), the model’s TR delineates a corrugated surface, so that the vector normal to the model’s TR changes its direction from point to point (see Figure 7 of Štěpán et al. 2015). On the other hand, at each point on such corrugated surface the stratification of the physical quantities along the local normal vector

 is much more important than along the perpendicular direction. In order to estimate how the line-center fractional polarization of the hydrogen Lyα line is at each point of the field of view, it is reasonable to assume that the incident radiation field has axial symmetry around the direction of the normal vector

 corresponding to the spatial point under consideration (see also Štěpán et al. 2018). Taking a reference system with the Z-axis directed along the normal vector

, and recalling Equation (1), at each point of the corrugated TR surface we can estimate the line-center fractional polarization signals through the following formula
2

where the positive Stokes Q reference direction is now the perpendicular to the plane formed by

 and the LOS,

 is the cosine of the angle between

 and the LOS, and

 is the anisotropy factor calculated in the new reference system. Clearly, because the direction of

 changes as we move through the corrugated surface that delineates the TR, the Q/I signals estimated with Equation (2) do not share the same reference direction for the quantification of the linear polarization. In order to arrive at equations for Q/I and U/I having the parallel to the nearest limb as the positive Stokes Q reference direction, we have applied suitable rotations of the reference system. Specifying the direction of the local TR normal vector through its inclination

 with respect to the vertical (with

 between 0° and 90°) and azimuth

 (with

 between 0° and 360°), our analytical calculations show that the line-center Q/I and U/I signals of the hydrogen Lyα line can be estimated by the following formulas:
3

4

Figure 2 shows examples of CLV of the Q/I and U/I Lyα line-center signals for several topologies of the model’s TR, with random azimuth

. The first three rows of the figure correspond to the indicated fixed inclinations

, while the bottom row panels show the case in which also

 has random values (i.e., the probability distribution

 for

 and

 for

). Note that the CLV of the fractional linear polarization line-center signals is very sensitive to the geometry of the model’s TR, and that there is no CLV at all when the normal vector

 has random orientations. Moreover, it is also very important to point out that the Q/I and U/I amplitudes are sensitive to the magnetic field strength of the chromosphere-corona TR, through the Hanle factor

 in Equations (3) and (4).


Accretion of diffuse gas onto the disks of galaxies from the intergalactic medium (IGM) is a possible explanation for how the H i content of galaxies has remained relatively constant since z ∼ 2 while the star formation rate (SFR) was up to 10 times higher at high redshifts (Noterdaeme et al. 2012; Madau & Dickinson 2014). The constant H i content implies that galaxies have somehow replenished themselves with enough gas to fuel continuous star formation. And though not directly responsible for star formation, H i is an intermediate phase toward molecular hydrogen, which is the raw ingredient of the star formation fuel. If the star formation is to continue, external gas has to be accreted and pass through the H i phase at some stage in the accretion process. Observationally inferred accretion rates as traced by H i, however, fall between 0.1 and 0.2

 at low redshifts. This is a full order of magnitude lower than what is needed for galaxies to continually form stars at their current rates (Sancisi et al. 2008; Kauffmann et al. 2010). This discrepancy presents two intriguing scenarios: the cycle of star formation will eventually exhaust all of the available fuel within a few Gyr and star formation itself will gradually cease, or processes that refuel galaxies with the necessary gas have been missed by previous surveys. Numerical simulations have shown a likely mechanism for refueling star formation is through a quasi-spherical “hot” mode and filamentary “cold” mode (Birnboim & Dekel 2003; Kereš et al. 2005, 2009). “Cold” in the context of these numerical simulations refers to gas that has not been heated above the virial temperature of the galaxy’s potential well (∼105 K), and “hot” refers to gas that has virialized in a process akin to the classical theory of galaxy formation in which shock-heated, virialized gas with short cooling timescales accretes onto the central galaxy (e.g., Rees & Ostriker 1977). These simulations also suggest cold mode accretion was the dominant form of accretion at z ≥ 1 for all systems, and remains prevalent through z = 0 for galaxies in low-density environments (ngal ≲1 h3 Mpc−3) and Mhalo ≲ 1011.4

 (or Mbary ≤ 1010.3

). For perspective, our own Milky Way has a virial (and thus halo) mass on the order of 1012

. These cold flows should exist in the form of vast filaments of cold, diffuse gas that permeate through the hot halo (Kereš et al. 2005). Comparisons by Nelson et al. (2013) between the smoothed particle hydrodynamic (SPH) numerical scheme employed in Kereš et al. (2005, 2009) and more sophisticated adaptive mesh refinement (AMR) simulations revealed the relative contribution of the cold mode is likely overestimated in earlier SPH simulations due to inherent numerical deficiencies. Nevertheless, the AMR simulations do show some fraction of gas is accreted cold.


With the ALMA results, we now estimate such physical parameters as the temperature, density, and volume filling factor of warm ionized gas in the IRS 13E region. Assuming that that this gas is under the local thermodynamic equilibrium and its 232 GHz continuum and H30α line emissions are optically thin, the electron temperature Te can be expressed as
(4)$$\begin{eqnarray*}
T_{\rm e}({\rm K}) &=& \left[ \frac{6985}{\alpha (\nu ,T_{\rm e})} \left(\frac{\nu }{\rm GHz}\right)^{1.1} \frac{1}{1+N({\rm He})/N({\rm H})}\right. \nonumber\\
&&\left.\times \, \left(\frac{S_{\rm C}}{S\Delta V_{\rm H30\alpha }} \right) \right]^{0.87},
\end{eqnarray*}$$where the elemental number density ratio of $N({\rm He})/N(\rm H) \sim 0.08$ and α(ν, Te) ∼ 0.85 at ν = 232 GHz and Te ∼ 104 K (Mezger & Henderson 1967). For E3, the total flux densities of the continuum and H30α line emissions are SC = 28.5 ± 1.8 mJy and SΔVH30α = 2.37 ± 0.12 Jy km s−1. We thus have Te = 9260 ± 650 K, which is consistent with previous estimates, ranging from 6000 to 12 000 K (Zhao et al. 2010; Moser et al. 2017; Tsuboi et al. 2019). We can also obtain the volume emission measure (EM) as (Murchikova et al. 2019)
(5)$$\begin{eqnarray*}
{\rm EM} = S\Delta V_{\rm H30\alpha }\frac{4\pi D^2}{\epsilon _{\rm H30\alpha }(T_{\rm e},n_{\rm e})} \frac{\nu }{c} \sim 1 \times 10^{60}\rm ~cm^{-3},
\end{eqnarray*}$$where the H30α emissivity $\epsilon _{\rm H30\alpha }=1.25\times 10^{-31} \rm ~erg~s^{-1}$ for Te = 104 K and $n_{\rm e}=10^4\sim 10^7 \rm ~cm^{-3}$ (Storey & Hummer 1995) and the speed of light c are used. Furthermore, we find that both the continuum and H30α emission intensities of E3 can be characterized by a Gaussian with a deconvolved size of ∼0.01 pc, assuming a spherical symmetry. This together with the above EM value leads to the estimate of $n_{\rm e} \sim 3 \times 10^5\rm ~cm^{-3} f_V^{-1/2}$. By further assuming a thermal pressure balance between the warm and hot gases (e.g. P/k ∼ 2 × 1010 K cm−3; Section 3.2), we estimate the volume filling factor fV ∼ 0.1 and hence $n_{\rm e} \sim 10^6\rm ~cm^{-3}$. If similar temperature and pressure could be assumed for warm ionized gas on larger scales, its filling factor has to be very small. We estimate that the total H30α flux within a circular region of 1 arcsec radius centred on E3 (Fig. 7B) is ${\sim}12 \rm ~Jy~km~s^{-1}$, which corresponds to ${\rm EM} = 6\times 10^{60}\rm ~cm^{-3}$. The filling factor is then ∼8 × 10−4, while the mass of the warm ionized gas in this region is ∼mpEM/ne ∼ 1 × 10−2 M⊙. These estimates indicate that the volume is indeed dominated by hot gas and that warm gas traces dense clumps, probably their surfaces exposed to the ionizing radiation from massive stars in the region.


Still, to further rule out a stellar origin, we attempt to fit the transit spectrum with a stellar contamination model. Following Rackham et al. (2018) we consider the hypothesis that the observed transit depth variations are due to inhomogeneities on the stellar surface. The observed wavelength-dependent transit depths

D

λ
,
obs

 are modeled as:
4

D

λ
,
obs

=

D

λ

1
−

f

spot

1
−

F

λ
,
spot

F

λ
,
phot

−

f

fac

1
−

F

λ
,
fac

F

λ
,
phot

,

where Dλ is the geometric radius ratio

R

p

/

R

⋆

2

, assumed here to be constant with wavelength (=D), fspot and ffac are the spot and faculae covering fractions, and the Fλ refer to bandpass-integrated PHOENIX model spectra of the photosphere, spots, and faculae (Hauschildt et al. 1999). Using K2-18b stellar parameters (Table 2), we explore a wide range of possible stellar contamination spectra for M2–M3 stars with spot and faculae covering fractions consistent with 1% I-band variability. Based on Rackham et al. (2018), we then explore scenarios with giant spots and solar-like spots as well as with and without faculae, resulting in spot fractions of up to 20%. However, none of those models can explain the amplitude of the observed transit spectrum. The amplitude of resulting transit depth variations due to the stellar inhomogeneities are up to 3–8 parts-per-million (ppm), about an order of magnitude smaller than the observed 80–90 ppm transit depth variations in the observed WFC3 spectrum (Figure 3). Finally, we also explore the most extreme scenarios where the spot and faculae covering fractions can be as high as 100%, but even those stellar inhomogeneity models fail to explain the amplitude of the observed transit depth variation. They deliver an absolute maximum of 20 ppm at 1.4 μm, which still only corresponds to less than a quarter of the transit depth variation in the observations. We conclude that stellar inhomogeneities and activity cannot explain the measured transmission spectrum.


The results presented in this work answer a number of questions regarding the character of the GG Tau A system, while raising others and leaving others untouched. First among the questions raised by our results are the questions of whether the detailed morphology of features in the circumbinary torus as seen in our simulations are actually present in the GG Tau A system. The current best resolution observations of the torus provide tantalizing hints that such features do exist, but a definitive statement that such features are present must await higher resolution observations of quality similar to those described by the ALMA Partnership (2015), for the HL Tau circumstellar disk. Another important question concerns planet formation in multiple systems. Whether or not any theoretical mechanism presently exists to explain the formation of giant planets in binary systems, the fact remains that at least a few binary systems, such as γ Cephei (Neuhauser et al. 2007), GI86 (Queloz et al. 2000), and HD 41004 (Zucker et al. 2004), do harbor planets. Therefore, some formation mechanism does in fact exist. While we find that accretion into the circumstellar disks occurs rapidly enough so that they can survive for the comparatively long timescales needed to form planets, other conditions, such as temperatures, remain quite unfavorable. What are the mechanisms still missing from our models that permit such objects to form? Finally, our simulations model the evolution of the “A” component of the full GG Tau system over a time span extending over only a tiny fraction of its formation timescale and in only two spatial dimensions. We neglected the full dynamical effects expected to be present in the system, insofar as our results include neither the distant binary “B” component of the GG Tau system, nor the newly discovered tight binary nature of the GG Tau Ab component. Even so, we find large scale morphological changes even over this short time span, and restricted dimensionality and physical system. Given the vigor of the activity over such a short timescale, we would expect activity of similar scale to occur over longer time spans as well, with correspondingly large consequences on the system morphology. To what extent will 3D effects also play a role in the evolution? What will be the end state configuration of the GG Tau system as a whole? Will the components eventually break apart? Merge? Future investigations extending the work presented here will be required in order to answer these questions.


Our formula can be used to confront the curvature effect with observations and estimate the decay timescale of the steep decay phase. In observations, the steep decay phase has been observed in the decay phase of the prompt emission phase and flares in GRBs. Since the value of E0 (

∼
0.3
–
1

MeV

) in the prompt emission phase is significantly larger than

10

keV

, the β in the steep decay phase of prompt emission would be a linear function of observer time based on Equation (38). This behavior has been observed in a number of bursts, such as GRBs 050814 (Zhang et al. 2009, please see the spectral evolution in the

β
−

t

obs

 space), 051001 etc. The linear relation between β and

t

obs

 is also found in the steep decay of flares (e.g., Mu et al. 2016). For a linear function β of

t

obs

, one can obtain the slope of the linear function, which is equal to

7.9

κ

ˆ

/

t

c

=
2.37

keV

/

[

E

0

(

t

obs

=
0
)

t

c

]

 based on Equations (38) or (39). Then, the decay timescale

t

c

 of our studying phase can be estimated if the value of

E

0

(

t

obs

=
0
)

 is known. We fit the spectral evolution in the decay phase of a flare (

∼
172

s

) in GRB 060904B (Mu et al. 2016) with a linear function. The fitting result, i.e.,

β
=
0.84
+
0.020

t

obs

 with

t

obs

=

t

obs

−

t

p

, is shown in the left panel of Figure 4 with  a red solid line. Then, we have

E

0

(

t

obs

=
0
)

t

c

=
119

keV

s

, which is around that found in Mu et al. (2016), i.e.,

2.54

keV
×
65.39

s

. For a flare (

∼
116

s

) in GRB 131030A, the spectrum at the beginning of the steep decay phase can be fitted with a Band function. With the fitting result found in Mu et al. (2016), i.e.,

α

ˆ

=
−
0.91

,

β

ˆ

=
−
3.19

, and

E

0

(

t

obs

=
0
)
=
1.95

, we plot the evolution of β based on Equation (38) in the right panel of Figure 4 with a red solid line. Here, we adopt

t

c

=
54

s

 (Mu et al. 2016), which can also be roughly estimated based on the flux evolution. It can be found that Equation (38) describes the spectral evolution approximately, which may reveal that a more appropriate value of parameters (e.g.,

t

c

) may be required for this source. The agreement of our analytical formula and observational data shows that the assumption (2) given at the beginning of Section 2 (i.e., the jet shell has no θ-dependent spectral parameters or Lorentz factor) is applicable in reality.


In the single cloud simulations presented in Paper I and Paper II, the system was treated in isolation, i.e. with no external forces acting upon it. In reality, such a binary would be placed deep in the potential well of a galaxy, where stars and dark matter exert their gravitational influence. To mimic the presence of the galactic spheroid, we include an external potential with a Hernquist profile (Hernquist 1990), which can be written as
(1)
\begin{eqnarray*}
\rho (r) = \frac{M_*}{2\pi}\frac{a_*}{r}\frac{1}{(r+a_*)^3},
\end{eqnarray*}
and an enclosed mass given by
(2)
\begin{eqnarray*}
m(r) = M_*\frac{r^2}{(r+a_*)^2},
\end{eqnarray*}
where M* and a* are scaling constants that represent the total mass of the spheroid and the size of core, respectively. Based on a typical MBH–Mbulge relation (Magorrian et al. 1998) and a radius-to-stellar-mass relation (Dabringhausen, Hilker & Kroupa 2008), we choose MH = 4.78 × 102M0 and rH = 3.24 × 102a0, thus implying
(3)
\begin{eqnarray*}
m(a_0) \approx 5\times 10^{-3} M_0.
\end{eqnarray*}
As a consequence, this external force represents only a small perturbation to the dynamics of the interaction that is still dominated by the binary’s gravitational potential. Moreover, because this external potential is spherically symmetric, orbits experience precession only, where energy and angular momentum are conserved. In principle, this precession could enhance stream collisions, prompting more gas onto the binary, although the time-scales are too long with respect to the frequency of cloud infall, which is the dominant effect in our models. Based on this, we do not expect the oscillations induced by this potential to have a secular effect in the overall evolution of the system during the stages modelled. In practice, this potential maintains the binary close to the centre of the reference frame by applying a restoring force when it drifts away due to the interaction with the incoming material. It is important to mention that the gas also feels this external force.


For the new observations described in Sect. 3, the spectra are renormalised by applying a linear fit to two wavelength regions, 6455–6559 Å and 6567–6580 Å, on either side of the Hα line. The value for pEW (Hα) is measured over the wavelength range 6560–6565 Å. The selected wavelength range is sufficiently narrow to exclude nearby spectral lines and sufficiently broad to measure the Hα line of a fast rotating star. The value of pEW(Hα) is calculated from

(1)

${\rm{pEW}(H\alpha)} = \int^{\lambda_2}_{\lambda_1}
\left(1-\frac{F(\lambda)}{F_{\rm{pc}}}\right){\rm{d}}\lambda,$pEW(Hα)= ∫ λ1λ2 1-F(λ)Fpc dλ,

where Fpc is the average of the median flux in the pseudo-continuum in the ranges [6545:6559] and [6567:6580], λ1  = 6560Å, λ2  = 6565Å, and the error is determined with the method of Vollmann & Eversberg (2006), which is applied as follows:

(2)

$\sigma_{\rm{pEW}} = \frac{\mathrm\Delta \lambda -
\rm{pEW(H\alpha)}}{\rm{S/N}} \sqrt{1 + \frac{F_{\rm{pc}}}{F_{1,2}}},$σpEW=Δλ-pEW(Hα)S∕N1+  Fpc   F1,2,

where Δλ = λ2 − λ1 and F1,2  is the mean flux in the wavelength range between λ1 and λ2. For Hα active stars, i.e. where pEW(Hα) is negative, the normalised Hα luminosity can be expressed as

(3)

$\log \left(\frac{L_{{\rm{H}}\alpha}}{L_{\rm{bol}}}\right) = \log \chi
+ \log (-{\rm{pEW}}({\rm{H}}\alpha)),$logLHαLbol = logχ+ log(-pEW(Hα)),

where log χ is given as a function of the effective temperature (e.g. Reiners & Basri 2008). Lines with Hα in absorption have positive pEW(Hα) values and lines with Hα in emission have negative pEW(Hα) values. The minimum level of emission in Hα that we are sensitive to is pEW(Hα) ≤−0.5Å, which is the definition for an active star in this work and is indicated by the term Hα active. This detection threshold was determined by visually inspecting the spectra and is consistent with values obtained by Newton et al. (2017) and West et al. (2015). The values derived from the new observations are tabulated in Table A.2.


In this work, we consider four different galaxy models from Smirnov & Sotnikova (2018). Full details of the simulations can be found in that work and we refer an interested reader to it. Here, we briefly describe some important aspects of the simulations. Initially, each model consists of an axisymmetric exponential disc:
(11)$$\begin{eqnarray}
\rho (R,z) = \frac{M_\mathrm{d}}{4\pi R_\mathrm{d}^2 z_\mathrm{d}} \exp \left(-\frac{R}{R_\mathrm{d}}\right) \operatorname{sech}^2\left(\frac{z}{z_\mathrm{d}}\right) \, ,
\end{eqnarray}$$where Md is the total mass of the disc, Rd is the disc scale length, and zd is the disc scale height, plus spherically symmetric dark halo of NFW-type (Navarro, Frenk & White 1996). The halo is characterized by the ratio Mh(r  4Rd)/Md, where Mh(r  4Rd) is the dark halo mass inside the sphere with a radius of 4Rd. One model also includes a Hernquist type bulge (Hernquist 1990):
(12)$$\begin{eqnarray}
\rho _\mathrm{b}(r) = \frac{M_\mathrm{b}\, r_\mathrm{b}}{2\pi \, r\, (r_\mathrm{b} + r)^3} \, ,
\end{eqnarray}$$where rb is the scale parameter and Mb is the total bulge mass. Disc and halo components are represented by several millions of particles (see Table 1). Bulge particle number is chosen such that the mass of one particle from the bulge is equal to the mass of one particle from the disc. Some important physical parameters of the models are specified in Table 1. The initial equilibrium state of each multicomponent model was prepared via script mkgalaxy (McMillan & Dehnen 2007) from the toolbox for N-body simulation NEMO (Teuben 1995). Evolution of the models was followed for about 8 Gyr using the fastest N-body code for one CPU gyrfalcON (Dehnen 2002). Each model gives rise to a bar after 1−2 Gyr from the beginning of the simulation. Each bar gradually thickens in the vertical direction (clear buckling events can occur depending on the model) and takes a B/PS shape if seen side-on. Fig. 1 shows the end-state of the bulgeless model with μ = 1.5 and Q(2Rd) = 1.2. This model is our fiducial or ‘main’ model. On the example of this model, we consider below how various projection effects affect the extracted parameters of the B/PS bulge. For each of the models, we prepared a set of Flexible Image Transport System (FITS) images obtained from the corresponding density distribution of disc/bulge particles. Such images can then be decomposed using a usual decomposition procedure analogous to that used for images of real galaxies.


The poor theoretical knowledge of the stellar mass loss history represents one of the main uncertainties in the computation of AGB stellar models. Low- and intermediate-mass stars lose the majority of their mass during the red giant branch (RGB) and the AGB phases. In 1D stellar evolutionary codes, the mass-loss rate during the RGB phase is commonly parameterized according to the formulation proposed by Reimers (1975):
2

where

 is in units of M

 yr−1 and other quantities are in solar units. The uncertainty affecting this formula was originally quoted by Reimers to be at least a factor of two either way. Later, Fusi-Pecci  Renzini (1976) introduced a normalization constant in order to reproduce the horizontal branch morphology of globular clusters (

 = 0.4).10

10
FRUITY models adopt this value.
 Depending on the mass lost during the RGB phase (and thus on the value of

), stars attain the AGB phase with different envelope masses. Thus, in principle, the RGB mass loss could have an effect on the subsequent AGB nucleosynthesis. Those effects are expected to be important for low-mass stars (M

 1.5 M

) because they spend more time in the RGB phase than do larger masses. Moreover, their envelopes are thinner than in more massive stars, so even a small amount of material lost (e.g., 0.1 M

) can produce sizeable effects on the occurrence of TDU in the subsequent AGB phase (see, e.g., Straniero 2003). To properly determine the effects of RGB mass-loss rate on AGB nucleosynthesis (and thus on the solar s-only distribution), we calculate a set of M = 1.3 M

 models at different metallicities with

. In Figure 10 we report the variations of the surface abundances (

) with respect to the corresponding FRUITY cases. We find that

 are larger at low metallicities (in particular for the heaviest s-only isotopes). This is because at large metallicities this mass experiences a few TDU episodes, even using a milder RGB mass-loss rate. Thus, the final surface s-process enhancement is, in any case, low. By comparison, we also report data relative to a M = 2

 Z = 10−2 ([Fe/H] = −0.15) model. The low variations found in this case confirm that for massive-enough AGB stars a reduced mass-loss rate during the RGB phase has practically no effect. In Figure 11 we report a GCE model computed with

 in stars with M

M

 (hereinafter the Reimers case). We find minor variations in the s-only distribution (see also Table 1), with slightly larger enhancements for the heaviest s-only isotopes (

). Our results reinforce the evidence that the major contributors to the solar system s-process inventory are AGB stars in the mass range (1.5–3.0)

, as already inferred in Section 3. Their nucleosynthesis is strongly affected by the rate at which they lose mass during the AGB. A viable method to estimate AGB mass loss is based on the observed correlation with the pulsation period (Vassiliadis  Wood 1993). Because the evolution of the pulsation period depends on the variations of radius, luminosity, and mass, this relation provides a simple method to estimate the evolution of the mass-loss rate from basic stellar parameters. In our models, the AGB mass loss is determined according to a procedure similar to the one adopted by Vassiliadis  Wood (1993), but revising the mass loss–period and the period–luminosity relations, taking into account more recent infrared observations of solar-metallicity AGB stars (see Straniero et al. 2006 and references therein). It has been demonstrated that AGB mass-loss rates are mildly dependent on the metallicity (Groenewegen et al. 2007; Lagadec et al. 2008), and thus we applied the same period–mass loss relation for all AGB models present in the FRUITY database. Nevertheless, it is worth noting that, when a fixed period is defined, the observational data show quite a large scatter. In a period–mass loss plot, a theoretical curve constructed when reducing by a factor of two the mass-loss rate at a fixed period still lies within the observed spread (see Figure 8.10 of Cristallo 2006). This still holds for a mass-loss rate increased by a factor of two. In order to quantify the effects on the s-only distribution induced by a variation of the AGB mass-loss rate, we compute some AGB models with a milder and stronger period–mass loss relations. In Figure 12 we show the results on the final surface distributions of 2 M

 stellar models at various metallicities. In the plot,

 represents the difference between models computed with the standard

–period relation (Straniero et al. 2006) and the modified ones. Obviously, positive differences are obtained with a milder mass-loss rate, while negative differences with the stronger one. Heavy-element surface variations are below 0.1 dex (25%) for the whole s-process distribution, being slightly larger at low metallicity. Thus, we expect that a modified

–period relation in the AGB phase will produce an almost rigid shift (upward or downward, depending on the adopted mass-loss law) of the s-process isotopes. In order to verify this statement, we compute a GCE model with a milder

–period relation during the AGB phase (hereinafter the Mloss AGB case). The results are shown in Figure 13; the corresponding data are reported in Table 1. As expected, for s-only isotopes with A ≥ 96, there is an almost rigid upper shift of solar percentages (∼25%). In summary, a rigid shift (upward or downward) of the s-process isotopic inventory can be obtained by adopting a different prescription for the AGB mass-loss rate within the intrinsic observed scatter in the

–period relation.


In order to establish the completeness and purity of the RR Lyrae stars confirmed by the SOS Cep&RRL pipeline and to estimate the number of new discoveries by Gaia, we performed a deep and careful comparison with the literature. As a first step, the catalogue of 140 784 confirmed sources was cross-matched against all major catalogues of known RR Lyrae stars that are available. We primarily used the OGLE catalogues for RR Lyrae stars (version IV of the survey, Soszyński et al. 2014, 2016), but we also used RR Lyrae stars by CTRS (Drake et al. 2013a,b, 2014, 2017; Torrealba et al. 2015), ASAS (Pojmanski 1997; Richards et al. 2012), ASAS-SN (Jayasinghe et al. 2018), ATLAS (Tonry et al. 2018), IOMC (Alfonso-Garzón et al. 2012), LINEAR (Palaversa et al. 2013), NSVS (Kinemuchi et al. 2006), Pann-Stars (PS1 Sesar et al. 2017), and from the works based on Kepler/K2 (Debosscher et al. 2011; Nemec et al. 2011; Molnár et al. 2015a,b, 2016) and on the Simbad database (Wenger et al. 2000). These cross-matches returned a list of 88 578 known RR Lyrae stars in our sample of 140 784 stars. The SOS Cep&RRL confirmed RR Lyrae stars were also cross-matched against catalogues of candidate RR Lyrae stars discovered by the VVV survey (Gran et al. 2016; Minniti et al. 2017; D. Minniti, priv. comm.) in the MW disc and bulge. This returned 319 VVV cross-identified sources in the MW disc and 222 in the MW bulge. We thus confirm these VVV candidates. For known RR Lyrae stars in GCs, the main reference was the catalogue of Clement et al. (2001), which was updated to the latest literature as described in Garofalo et al. (in prep.). For variables in dSphs, we used the following references: Kaluzny et al. (1995), Clementini et al. (2005), Kinemuchi et al. (2008), Dall’Ora et al. (2012) and Garofalo et al. (2013). These latter cross-matches returned a list of 1986 further known RR Lyrae stars. At the end of this cross-match procedure, of the 140 784 RR Lyrae stars that are confirmed by the SOS Cep&RRL pipeline, 90 564 were shown to be known previously, and 50 220 are new discoveries by Gaia.


In Figure 7, we find a trend between

 and SFR over about 3 orders of magnitude in far-IR luminosity/SFR. The solid line shows the best-fit (a logarithmic linear least-squares bisector fit) correlation for the whole sample of QSOs, with a power law of

. The Spearman rank correlation coefficient for this correlation is 0.77 with a high significance (P  10−5) but substantial scatter. Our result is in general agreement with the correlation found between AGN activity and star formation, i.e.,

 (α ∼ 1.1–1.7; see Netzer 2009; Chen et al. 2013; Hickox et al. 2014; Duras et al. 2017; Dai et al. 2018; Izumi et al. 2019 and references therein). As noted in previous works (see e.g., Dai et al. 2018), the variation of slope might be attributed to different sample compositions and/or methods of analysis. However, all of these studies, including our own, reveal a significant trend between the SFR and BH accretion spanning a wide range of luminosity, implying that there may be a direct connection between the AGN and star formation activities over galaxy evolution timescales. In addition, there is evidence that SMBHs have grown in step with their host galaxies since z ∼ 2 (Daddi et al. 2007; Mullaney et al. 2012). The median value of

/SFR is 6.0 × 10−3 for IR QSOs, consistent with the value derived in Xia et al. (2012). In contrast, the

/SFR values for high-z QSOs are found to be about a factor of 10 higher (a median of 7.3 × 10−2) than the local IR QSOs. This is consistent with the “overmassive” BH revealed for z > 5 QSOs (Walter et al. 2004; Wang et al. 2010, 2013, 2016), for which a plausible interpretation is likely related to the different efficiency of gas accretion to the central SMBHs, i.e., faster gas consumption in the most massive SMBHs. Similar results have been reported by Hao et al. (2008), who found a trend that the SFR increases with the accretion rate for both low- and high-redshift IR-luminous QSOs, and the relative growth of BHs and their host spheroids may depend on the intensity of QSO activities.


There are several possible physical processes to reconcile these discrepancies between the models and the observations at z ∼ 10–13. As discussed in Harikane et al. (2021b), less efficient mass quenching and/or lower dust obscuration than assumed in the models can explain the existence of these UV-bright galaxies. Active galactic nucleus (AGN) activity may also boost the UV luminosity in these galaxies. Previous studies indicate that the AGN fraction starts to increase at M
UV ≃ −22 mag (Ono et al. 2018; Stevans et al. 2018; Adams et al. 2020; Harikane et al. 2022; see also Piana et al. 2022). If we assume that the UV luminosities of HD1 and HD2 are solely powered by black holes, the inferred black hole masses are ∼108
M
⊙, assuming accretion at the Eddington rate (Pacucci et al. 2022), in accordance with expectations for high-redshift quasars (see, e.g., Haiman & Menou 2000 and Willott et al. 2010). In addition, note that a ∼108
M
⊙ black hole at z ∼ 12 could be the progenitor of z ∼ 7 quasars, as the growth time to reach a mass of 109–1010
M
⊙, typical of z > 6 quasars detected thus far, is shorter than the cosmic time between z = 12–13 and z = 7, for an Eddington-limited accretion. It is also possible that the observed bright sources at z ∼ 10–13 are galaxies in a short-time starburst phase that is not captured in the models whose outputs are averaged over a time interval (see also Dayal et al. 2013). Finally, a top-heavier IMF would explain the discrepancies by producing more UV photons at the same stellar mass. It is possible that these bright galaxies (especially HD1 and HD2) are merging systems that are not resolved in the ground-based images. However, even if they are major mergers, the UV luminosity will decrease only by a factor of a few, which would not explain the discrepancy at z ∼ 13 (see also discussions in Harikane et al. 2022; Shibuya et al. 2022). In any case, if these bright z ∼ 10–13 galaxies are spectroscopically confirmed, the discrepancies will motivate the exploration of new physical processes that are responsible for driving the formation of these bright galaxies in the early universe.


For the distance to the source of HESS J1825−137 of 4 kpc (based on dispersion measure using the models for Galactic electrons by Taylor & Cordes 1993; Cordes & Lazio 2002), the measured angular size of the latter corresponds to the 70 pc linear size of the gamma-ray production region. Although the gamma-ray image deviates from the spherically symmetric shape; below for simplicity we will assume that the radius of the source is 35 pc. The detection of a Hα rim at a distance of

 from the pulsar PSR J1826-1334 has been interpreted as a signature of the progenitor SNR (see Stupar et al. 2008; Voisin et al. 2016). As illustrated in Figure 4, in the case of an ISM density gradient, the measured radius does not coincide with the radius, Rsnr . If the pulsar is still located inside the SNR then the SNR radius cannot be significantly smaller than the observed distance:

. Thus, with a factor of two uncertainty one can estimate the radius based on Sedov’s solution:
40

where

 are the density, the hydrogen number density, and the proton mass, respectively. Thus, the SNR shell can be located at the distance of

 pc, provided that the following condition is fulfilled:
41

where the numerical value corresponds to

. The ejecta energy at SN type II is constrained by

, while the age of the source is limited by

 (for

). Thus, Sedov’s solution is consistent with the observed size of the SNR when

. Note that de Jager & Djannati-Ataï (2009) derived a somewhat smaller upper limit for

 assuming the braking index n = 2. This upper limit seems to be below the typical density of the interstellar gas in the Galactic Plane. Moreover, Voisin et al. (2016) reported the presence of several molecular clouds in the small angular proximity to the nebula with an average number density

 and characteristic size

. It is likely that one of these clouds is responsible for the anisotropic expansion of the nebula, thus it should be located close to the nebula. The gas density in molecular clouds is expected to follow the King’s profile:
42

Thus, the density to the south of the pulsar can hardly be as low as it is required by Sedov’s solution. Instead, it should be comparable to the upper limit of

 obtained by Voisin et al. (2016).


In some cases, the precursor emission has a nonthermal spectrum, especially GRB111117510 and GRB160804180 with ΔBIC ≳ 6. These cases may be explained by the NS magnetospheric interaction model (assuming that the sources are nearby). For NS mergers with the surface magnetic field

 for the primary NS, the typical spectrum may be approximately described by a synchrotron radiation spectrum of a photon index around −2/3 peaking at ∼MeV, because of the effect of synchrotron-pair cascades (Wang et al. 2018a, 2018b). Such a model can well explain the photon indices and peak energies of the nonthermal precursor bursts, e.g., GRB111117510, GRB140209313, and GRB160804180 (Wang et al. 2018b). The precursor emission time for this magnetospheric interaction model roughly coincides with the gravitational-wave radiation chirp signal time. So the waiting time between the precursor and the main burst should correspond to the time delay between the GW signal and the SGRB signal. This timescale consists of three parts (Zhang 2019): the time (

) for the jet to be launched by the central engine, the time (Δtbo) for the jet to propagate through and break out from the circumburst medium, and the time (ΔtGRB) for the jet to reach the energy dissipation radius (e.g., the photospheric radius or the internal shock radius). The last term is

, while the first two terms depend on the jet-launch models. According to the Table 1 in Zhang (2019), for most models,

 = 0.01–1 s. Consequently, one would also expect

 ∼

. An exception is the SMNS/SNS magnetic model, in which a uniform-rotation-supported supramassive NS (SMNS) is formed after the NS merger, which subsequently becomes a stable NS (SNS). In this model, the waiting time is dominated by the term

, which is mainly contributed by the time needed to clean the environment to launch a relativistic jet (Metzger et al. 2011; Zhang 2019). In this case, one expects

 ≫

. In our sample, we find most events satisfy

 ∼

, except GRB191221802, which has

/

 ≈ 52 and

 s. We also notice that for GRB090510016, Troja et al. (2010) found two precursors in the Swift data, but only the second precursor can be found in Fermi data (consistent with our results). Its first precursor is found to be of

/

 ≈ 40 and

 ≈ 12 s, while its second precursor in our analysis is consistent with the photospheric radiation of the fireball. Therefore, its first precursor with a long waiting time (

/

 ≫ 1) could originate from NS magnetospheric interaction, and such long waiting times are caused by the jet-launch mechanism in the SMNS/SNS magnetic model. In conclusion, according to this model, an SNS engine might have been formed after the merger in events with

/

 ≫ 1, e.g., GRB090510016 and GRB191221802.