Stars with masses 130 M
⊙ ≲ M
ZAMS ≲ 250 M
⊙ are subject to the pair instability, which disrupts them completely, and hence no BH forms. Stars with M
ZAMS ≳ 250 M
⊙ can collapse directly to IMBHs with a mass ≳135 M
⊙. Thus, in the standard picture, there should be an upper stellar-mass BH gap in the range [65, 135] M
⊙, and any BHs observed in this range (e.g., the primary BH of GW190521) have to form via other formation channels—for example, through hierarchical coalescence of smaller BHs or direct collapse of a stellar merger between an evolved star and a main-sequence companion (e.g., Quinlan & Shapiro 1989; Portegies Zwart & McMillan 2000; Ebisuzaki et al. 2001; Miller & Hamilton 2002; O’Leary et al. 2006; Gerosa & Berti 2017; Antonini et al. 2019; Di Carlo et al. 2019, 2020; Rodriguez et al. 2019; Gayathri et al. 2020a; Kimball et al. 2021; Mapelli et al. 2021). However, the exact mass boundaries of the gap depend on parameters that are uncertain. For example, the 12C(α, γ)16O nuclear reaction rate, which converts carbon to oxygen in the core, can affect the boundary significantly (Takahashi 2018; Farmer et al. 2020; Costa et al. 2021; Woosley & Heger 2021). Here, to better determine whether future GW observations will be able to observe BHs in the mass gap, we recompute the mass-gap boundaries with updated 12C(α, γ)16O reaction rates and increased mass and temporal resolution. The complexity of this system has made a reliable analysis of the reaction a decades-old challenge.
7

7
The reaction rate for 12C(α, γ)16O is determined by the quantum structure of the compound nucleus 16O as an α cluster system. It is characterized by the interfering ℓ = 1 waves of the J

π
 = 1− resonances and sub-threshold levels defining an E1 component for the reaction cross section as well as by the ℓ = 2 components and interference from broad J

π
 = 2+ resonances and the nonresonant E2 external capture to the ground state of 16O. In addition to these two main E1 and E2 ground-state components, transitions to higher lying excited states occur that also add to the total cross section (see, e.g., Buchmann & Barnes 2006; deBoer et al. 2017). The rapidly declining cross section at low energies has prohibited a direct measurement of the reaction at stellar temperatures, and the reaction rate is entirely based on the theoretical analysis and extrapolation of the experimental data toward lower energies. The rate newly derived by deBoer et al. (2017), using a multichannel analysis approach, derives for the first time a reliable prediction for the interference patterns within the reaction components by taking into account all available experimental data sets that cover the near-threshold energy range of the 12C(α, γ)16O process.


We gathered four independent measurements of

 from the literature: 2.312 ± 0.095 (Carter et al. 1988), 2.237 ± 0.013 (Beuing et al. 2002), 2.292 ± 0.0429

9
Wegner et al. (2003) listed a value within a 0.595

 kpc radius aperture, and this value is converted to the value for an aperture with 1/8 of

 using Equation (1) of Cappellari et al. (2006).
 (Wegner et al. 2003), and 2.212 ± 0.045 (Ogando et al. 2008). We adopt a weighted mean value,

, of the four independent measurements. Using the

 values from Table 1 after applying the Galactic extinction correction and the FP coefficients from various sources, we derive distances to NGC 4993 in seven different bands. Table 3 summarizes the result. In total, 17 different estimates are derived. Two of the FP relations are based on

, and z (Bernardi et al. 2003; La Barbera et al. 2010), and for these cases, the

, and I values are converted to

, and i values while keeping the structural parameters in the corresponding bands. Uncertainties in the distances are dominated by the intrinsic dispersion in the FP relation, which has errors of about 7 to 12 Mpc (or 23% of the derived value; e.g., Bernardi et al. 2003). The errors from the observed quantities amount to only ≲10% of the FP distance, and we ignore them. The uncertainty in the Hubble constant is only a few percent or less according to recent estimates (e.g., Riess et al. 2016) and also can be neglected. We note that the rms dispersion of the 17 estimates is 5.3 Mpc, smaller than the FP distance error of each estimate. This suggests that the 17 estimates are not independent quantities (e.g., they share a

 value and are based on an identical object), and the rms dispersion of 5.3 Mpc is an uncertainty related to the wavelength and the adopted FP relation parameters for each measurement. Therefore, we consider a typical intrinsic scatter in the FP relation along

 of 0.09 dex (23% of the value) as our distance error. Considering these factors, we adopt 37.7 ± 8.7

 Mpc as the FP-based distance. Note that this is an angular diameter distance, and at z = 0.009783 of NGC 4993 (Levan et al. 2017), the luminosity distance is 2% larger at 38.4 ± 8.9

 Mpc. We also note that the mean angular diameter distance of 37.7 Mpc is accurate to about

 Mpc in regard to uncertainties due to wavelengths and FP parameter sets.


For the transition range around ion scales, the turbulent energy is believed to further cascade and the kinetic physics begins (Alexandrova et al. 2013; Kiyani et al. 2015). Various forms of coherent structures (i.e., current sheet, vortex filaments) are usually found to reside near this range (i.e., between the end of the MHD range and proton scales in Lion et al. 2016; Perrone et al. 2016, 2017). Hence, in the following analysis we focus on fluctuations of similar scales, where the frequency ranges is chosen as [0.4, 3] Hz, and the timescale is [0.3, 3] s. It has been shown recently that, in case of a collisionless turbulent system as the solar wind, the intermittency, non-Gaussian fluctuations, and phase coherence of magnetic field components are interrelated (Perrone et al. 2017). We expect this relation to be present here and thus take similar procedures as in Perrone et al. (2017) to search for the intermittent events. The first step is to reconstruct the fluctuations using the bandpass filter based on wavelet transforms (Torrence & Compo 1998; He et al. 2012; Wang et al. 2014; Perrone et al. 2016). The magnetic field fluctuations are thus defined as
1

where i represent the magnetic field components, j represent the scale index,

 is the constant scales step,

 is the real part of the wavelet coefficient

, Cδ = 0.776.

 is the Morlet mother function and

 at time t = 0 (Torrence & Compo 1998),

 and

 are taken to be 0.3 and 3 s, respectively. The second step is to determine the threshold energy as defined by

, where

 is the standard deviation of the Gaussian fit for each magnetic field fluctuation component.

 are fitted to be 0.36, 0.32, and 0.49 nT, leading to a

 ∼  4.3 nT2. From a statistical point of view, 99.7% of all the values in Gaussian distribution are within

 from the mean. In other words, the events whose total energy

 are larger than

 could contribute to the non-Gaussian part of the distributions. Figure 1(e) presents the PDFs of the normalized magnetic field fluctuations

 together with their Gaussian fits. The presence of clear non-Gaussian tails suggest the abundance of intermittent events during the whole interval. The last step is to locate these events. As seen in Figure 1(b), there are approximately 14 events (with at least 10 s in duration) with

 larger than 5 nT2. We have picked one interval during 10:59:41.5–10:59:48.5 UT for further study. This event is characterized by large magnetic energy ∼150  nT2 and strong flow vorticity up to ∼1.4 /s as compared to the mean value of ∼0.4 /s. Furthermore, the local intermittency measure (LIM) exhibits an extension of temporal scale from several tens of seconds to sub-seconds, as seen in the LIMtotal of Figure 1(c). The LIM spectrogram, as a function of time and scales, is computed as the instantaneous energy of fluctuations normalized to its mean value over the studied time interval (Farge 1992):
2

where

 is the total fluctuation energy. For the coherent structures (space or time localized energetic events), one of its intrinsic properties is the energy coupling over many scales (Farge 1992; Frisch 1995), which is usually manifested in the spanning of LIM at a wide range of spatial scales (or temporal scales if the Taylor frozen-in hypothesis is assumed; Lion et al. 2016; Perrone et al. 2017). While for the wave phenomenon, the energy distribution is typically focused around a certain frequency, i.e., Alfvén ion cyclotron (Alexandrova et al. 2004) or electron cyclotron waves (Lacombe et al. 2014). Therefore, the extension of LIM provides evidence of coupling from MHD to sub-ion scales and hence implies the presence of a coherent structure (Farge 1992; Frisch 1995; Lion et al. 2016; Perrone et al. 2017). Note that because

,

, and LIMtotal exhibit nearly the same features in this event, only LIMtotal is presented here.


A triple system consists of an inner binary with masses m1 and m2 (defined as m1 ≥ m2), and a third object with mass m3 (in our case, the third object is always the central SMBH). It is convenient to describe the orbit using Jacobi coordinates (see e.g. Murray & Dermott 2000, p. 441). In this treatment, the dominant motion of the triple system can be divided into two separate Keplerian orbits: one is the motion of objects 1 and 2 relative to each other, and the other one is the motion of object 3 relative to the centre of mass of the inner binary. We denote the relative position vectors, semimajor axes, and eccentricities of these systems by $\boldsymbol {r}_1$ and $\boldsymbol {r}_2$, a1 and a2, and e1 and e2, respectively. If the third object is sufficiently distant from the inner binary, a perturbative approach can be used to describe the evolution of the system. In the usual secular approximation (e.g. Marchal 1990), the two orbits exchange angular momentum, but not energy. Hence, the eccentricities and orientations of the orbits can change, but the semimajor axes cannot. The coupling term in the Hamiltonian can be written in a power series expansion of parameter α = a1/a2 (see e.g. Harrington 1968), where the Hamiltonian is
(16)
\begin{eqnarray*}
\mathcal {H} &=&\, \frac{m_1 m_2}{2a_1} + \frac{(m_1+m_2)m_3}{2a_2} \nonumber \\
&&+\, \frac{1}{a_2} \sum \limits _{j=2}^{\infty } \alpha ^j M_\mathrm{ j} \left(\frac{r_1}{a_1} \right)^j \left(\frac{a_2}{r_2} \right)^{j+1} P_\mathrm{ j}(\mathrm{cos}\Phi) ,
\end{eqnarray*}
Pj is the jth-degree Legendre polynomial, Φ is the angle between $\boldsymbol {r}_1$ and $\boldsymbol {r}_2$, and
(17)
\begin{eqnarray*}
M_\mathrm{ j} = m_1 m_2 m_3 \frac{m_1^{j-1}+(-1)^j(m_2)^{j-1}}{(m_1+m_2)^j} .
\end{eqnarray*}
When analysing these hierarchical triple systems, it is particularly convenient to adopt the canonical coordinates known as Delaunay’s elements (e.g. Valtonen & Karttunen 2006). The coordinates are the mean anomalies, l1 and l2, the longitudes of ascending nodes, h1 and h2, and the arguments of periastron, g1 and g2. One can then eliminate the short-period terms by a canonical transformation to get equations that describe the long-term evolution of the triple system. This technique is known as the Von Zeipel transformation (see e.g. Brouwer 1959). Naoz et al. (2013a) showed that going beyond the quadrupole-order approximation is necessary in order to acquire highly eccentric Kozai excitations. They also pointed out a common mistake in the Hamiltonian treatment of these hierarchical systems in previous studies, which could lead to erroneous conclusions in the test particle limit. The full octupole-order equations of motion can be found in Naoz et al. (2013a).


A detailed description of the dynamo model can be found in our previous paper (Pipin & Kosovichev 2019, hereafter PK19). The model describes the dynamo generation of large-scale magnetic fields (LSMFs) in the bulk of the solar convective zone (CZ). The model is based on the mean-field induction equation (Krause & Rädler 1980):
1

where the induction vector of the LSMF,

, is represented as the sum of the toroidal and poloidal components:

where r is the radial distance, θ is the polar angle, and

 is the unit vector in the azimuthal direction. The mean electromotive force

 describes the turbulent generation effects, pumping, and diffusion:
2

where the symmetric tensor αij stands for the turbulent generation of the LSMF by kinetic and magnetic helicities, the antisymmetric tensor γij describes the turbulent pumping effect, and the anisotropic (in the general case) tensor ηijk is the eddy diffusivity of the LSMF (Pipin 2018). The large-scale (LS) flow field,

, produces the LS toroidal magnetic field from the LS poloidal field by means of the differential rotation,

. The meridional circulation,

, advects the LSMF in the convection zone. The angular momentum conservation and the equation for the azimuthal component of large-scale vorticity,

, determine distributions of the differential rotation and meridional circulation:
3

4

where

 is the turbulent stress tensor:
5

(see detailed description in PK19). Also,

 is the mean density,

 is the mean entropy, and

 is the gradient along the axis of rotation. The mean heat transport equation determines the mean entropy variations from the reference state due to the generation and dissipation of LSMF and large-scale flows (Pipin & Kitchatinov 2000):
6

where

 is the mean temperature, Fr is the radiative heat flux, and Fc is the anisotropic convective flux. An analytical mean-field expression for Fc takes into account the effect of the Coriolis force, and the influence of the LSMF on the turbulent convection (see PK19). The last two terms in Equation (6) take into account the convective energy gain and sink caused by the generation and dissipation of LSMF and large-scale flows. The reference profiles of mean thermodynamic parameters, such as entropy, density, and temperature are determined from the stellar interior model MESA (Paxton et al. 2011, 2013). The radial profile of the typical convective turnover time, τc, is determined from the MESA code, as well. We assume that τc does not depend on the magnetic field and global flows. The convective rms velocity is determined from the mixing-length approximation
7

where

 is the mixing length, αMLT = 1.9 is the mixing-length parameter, and Hp is the pressure height scale. Equation (7) determines the reference profiles for the eddy heat conductivity,

, eddy viscosity, νT, and eddy diffusivity, ηT, as follows:
8

9

10

The model gives the best agreement of the angular velocity profile with helioseismology results for

 (PK19). Also, the dynamo model reproduces the solar magnetic cycle period, ∼22 years, if

.


Let the data be denoted $\mathscr {D}=\lbrace \boldsymbol {\hat{z}},\boldsymbol {\hat{\theta }},\boldsymbol {\hat{s}},\boldsymbol {\hat{\imath }}\rbrace$ (and possibly the experimental covariance matrix), and $\mathcal {S}$ be the proposition that we have observed some data thus passing the selection criteria. The probability that a given data set is observed depends on the selection criteria. The likelihood of observing $\mathscr {D}$ given ϑ and that we have observed some data is (Loredo 2004; Mandel, Farr & Gair 2019)
(54)$$\begin{eqnarray*}
p(\mathscr {D}\mid \mathcal {S},\vartheta) =\frac{p(\mathcal {S}\mid \mathscr {D},\vartheta)\, p(\mathscr {D}\mid \vartheta)}{p(\mathcal {S}\mid \vartheta)} =\frac{p(\mathcal {S}\mid \mathscr {D},\vartheta)\, p(\vartheta \mid \mathscr {D})}{\int \mathrm{d}\mathscr {D}^{\prime }\, p(\mathcal {S}\mid \mathscr {D}^{\prime },\vartheta)\, p(\vartheta \mid \mathscr {D}^{\prime })},
\end{eqnarray*}$$where $p(\mathcal {S}|\mathscr {D},\vartheta)$ is the selection function, and we used Bayes’ theorem in the second equality so $p(\vartheta |\mathscr {D}) \propto p(\mathscr {D}|\vartheta)p(\vartheta)$. If we assume that all objects that exceed some threshold are successfully observed then $p(\mathcal {S}|\mathscr {D},\vartheta)=1$. In the ideal scenario of no selection effects (see Section 3) clearly all possible data sets are observable. In the case of a cut-off, equivalent to replacing with a truncated distribution, as in (5), we have $p(\mathcal {S}|\mathscr {D},\vartheta)=1$ if it exceeds some threshold, and $p(\mathcal {S}|\mathscr {D},\vartheta)=0$ otherwise. Thus, $p(\mathcal {S}|\vartheta)$ is the fraction of all possible data sets that are observable given parameters ϑ. From (54) we can see that the form of the posterior distribution subject to selection effects, $p(\vartheta |\mathscr {D},\mathcal {S})$, is simply that of $p(\vartheta |\mathscr {D})$ given by (44) but with a different normalization.


The gas inside the filaments may be turbulent and the dissipation of this turbulence could, in principle, also contribute to the filament powering. In order to obtain a very rough estimate of the turbulent dissipation rate, we assume that the velocity dispersion σ in the filaments is at best comparable to the sound speed in the Hα-emitting filament gas. Otherwise, there should be evidence for shock heating but that is not observed. In general, shock heating should lead to a correlation between [N ii]/Hα emission line ratios and the velocity dispersion of the gas, but such correlations have not been detected in the Perseus and Centaurus clusters where this issue was studied (Hatch et al. 2006; Canning et al. 2011; albeit the caveat that these models did not incorporate non-thermal pressure which could affect the nature of the shocks). The absence of shocks implies that
19

where fg is fraction of thermal pressure support in the filament and cs,0 is the sound speed in the absence of any non-thermal pressure in the filament for 104 K. For fg = fcr = fB = 1/3, we get σ ∼ 24 km s−1. The turbulent power

. Typical masses of filaments are in the range from Mfil ∼ 104 to 106 M⊙ (Conselice et al. 2001). Using a filament mass of 106 M⊙ and assuming a turbulence injection scale comparable to the filament width lturb = 60 pc (c.f. Canning et al. 2016 who use smaller value), we get Lturb ∼ 2×1038 erg s−1, which is a few percent of the Hα luminosity of the resolved Horseshoe or Northern filaments in Perseus. This is not a strict upper limit on the contribution of turbulent dissipation as the filament may consist of a number of subfilaments. However, inside the filaments the Hα phase, while possibly not completely volume filling, may be more volume filling than the phase corresponding to the dense molecular gas. Furthermore, the power contributed by turbulent dissipation depends on the uncertain mass in the Hα phase in the filament, and that mass is smaller than the total Mfil mass of the filament adopted above. Most importantly however, the measurements of internal turbulence in the filaments are very difficult because of limited spatial resolution. Current measurements of the velocity dispersion inside the filaments are very likely to be significantly overestimated due to filament or subfilament confusion (Canning et al. 2016) and consequently velocity dispersions could be consistent with values lower than those adopted above. We thus conclude that it is at least plausible that turbulent heating is not the dominant powering mechanism.


On 2017 August 17, a gravitational wave (GW) event (GW170817; Abbott et al. 2017a, 2017b) was linked with a faint gamma-ray prompt emission of GRB 170817A (Goldstein et al. 2017; Savchenko et al. 2017). Immediately, GRB 170817A was followed by an extensive observational campaign covering radio, optical, and X-ray bands (see, e.g., Troja et al. 2017; Abbott et al. 2017b; Kisaka et al. 2018; D’Avanzo et al. 2018a, and references therein). The observations of the nonthermal spectrum of GRB 170817A gathered during the first ≈ 900 days after the initial merger were analyzed by several authors. It was shown that they were consistent with synchrotron forward-shock emission generated by the deceleration of an off-axis structured jet with an opening angle θ

j
 ≈ 5° that was observed from a viewing angle in the range of 15° ≤ θ
obs ≤ 25° (Troja et al. 2017; Kasliwal et al. 2017b; Lamb & Kobayashi 2017, 2018; Resmi et al. 2018; Margutti et al. 2017a; Lazzati et al. 2017; Gottlieb et al. 2018b; Fraija et al. 2019b; Gottlieb et al. 2018b; Hotokezaka et al. 2018; Fraija et al. 2019c). In some proposed models, the off-axis structured jet is formed with an off-axis jet with a cocoon (Lazzati et al. 2017; Gottlieb et al. 2018b; Fraija et al. 2019b) and a shock breakout (Gottlieb et al. 2018b; Hotokezaka et al. 2018; Fraija et al. 2019c; Urrutia et al. 2021). Recently, Hajela et al. (2022) analyzed the latest X-ray and radio observations of GRB 170817A collected with the Chandra X-ray Observatory, the Very Large Array (VLA), and the MeerKAT radio interferometer about 3.3 yr after the initial merger. These new observations did not agree with the best-fit synchrotron curves from the off-axis jet model, thus reporting evidence of a new X-ray emission component. Given these contrasting observations, the authors offered the solution to explain this phenomenon in the context of either radiation from accretion processes on the compact-object remnant or a KN afterglow.


No studies have yet provided an extensive and conclusive kinematics analysis of populations of simulated star-forming galaxies that can be broadly contrasted to the results from slit-like and IFS-like current observations. So far, large uniform-volume simulations have been analysed in order to shed light on the dichotomy and formation of fast and slow rotators (e.g. Penoyre et al. 2017; Li et al. 2018; Schulze et al. 2018, with Illustris and Magneticum) and to quantify the connection between galaxy morphology and spin on the one side and AGN feedback, environment and/or mergers on the other (Dubois et al. 2016; Rodriguez-Gomez et al. 2017; Choi et al. 2018; Lagos et al. 2018a,b, with Illustris, Horizon-AGN and EAGLE). Recently, van de Sande et al. (2019) have compared structural and kinematic properties of simulated galaxies from EAGLE, Hydrangea (Bahé et al. 2017), Horizon-AGN (Dubois et al. 2016), and Magneticum (Dolag et al., in preparation) with observed galaxies in the SAMI, ATLAS3D, CALIFA, and MASSIVE surveys. However, in most cases these works have principally focused on massive early-type galaxies at $z$ ∼ 0, which are expected to be mostly quenched. This choice has been dictated by the limited numerical resolution of such models, which is generally thought to be insufficient to properly capture even the kiloparsec-scale kinematics of intermediate and low-mass galaxies that are observationally accessible, especially in the low-redshift Universe. Other analyses have investigated the relation between stellar shapes and stellar kinematics with EAGLE and across galaxy types (Thob et al. 2019). Furthermore, there is a rich body of works on the problem of the evolution and conservation of galaxies angular momentum, also informed by the results of large uniform-volume simulations and across galaxy types (e.g. more recently Genel et al. 2015; Lagos et al. 2017, with Illustris and EAGLE). However, in most cases so far, the focus has been placed on the final $z$ = 0 outcome, or on the stellar components, or the analyses have been carried out via theoretical characterizations of galaxies properties (e.g. resolution element-based angular momenta) that cannot be easily contrasted to observations. In fact, so far, no quantitative analysis of the spatially averaged or map-based internal kinematics of star-forming galaxies within large uniform-volume simulations exists.


The primary focus of this Letter is the spatial distribution of galaxies and their orientations projected onto the sky. While the former is characterized by the fluctuations of number density, denoted by

, the latter is quantified by the two-component ellipticity, (γ+ , γ× ), defined with the minor-to-major-axis ratio q on the celestial sphere:
1

with ϕx being the misalignment angle relative to the reference axis. We will below set q to zero for simplicity, which corresponds to the galaxy being assumed to be a line along its major axis (Okumura et al. 2009). In the weak-lensing measurements, a more convenient characterization of the ellipticity distribution is the rotation-invariant decomposition called E-/B-modes,

 (Kamionkowski et al. 1998; Crittenden et al. 2002), and these are defined, in Fourier space, by

, where

 are the Fourier counterpart of the ellipticity fields, and ϕk is the azimuthal angle of the wavevector projected on the celestial sphere, measured from the x-axis. Then, we consider the two-point statistics among

 and

. To quantify the cosmological information encoded in these statistics, we adopt the LA model as mentioned above. In Fourier space, it is given by
2

with

 being the redshift-dependent coefficient (Okumura & Taruya 2020). Here we used the Poisson equation to relate the gravitational potential to the mass density field,

. Note that the observable ellipticities are density-weighted, i.e.,

, but at large scales, the term

 is higher order and can be ignored. Then Equation (2) leads to γB = 0, and the nonvanishing two-point statistics in Fourier space become the auto-power spectra of the galaxy density and E-mode ellipticity, and their cross power spectrum, which we respectively denote by Pgg, PEE, and PgE. In redshift space, where the line-of-sight position of galaxies is determined by the redshift, the observed galaxy density field is affected by the effect of RSD. Furthermore, the ellipticity of galaxies is measured on the celestial sphere normal to the line of sight. Thus, all the power spectra considered here exhibit anisotropies along the line-of-sight direction, and denoting the directional cosine between the wavevector and line-of-sight direction by μ, they are expressed as a function of k and μ. In the linear theory limit, we have (see Okumura & Taruya 2020, for their configuration-space counterparts)
3

4

5

Here, we assume the linear bias relation between the galaxy and matter density fields, and b1 is the coefficient. The quantity f is the linear growth rate defined by

 with a and D being, respectively, the scale factor of the universe and linear growth factor, and

 is the linear-order matter power spectrum at the redshift z.


Figures 4 and 5 display two enlarged regions from Figure 3, Region 1 being quiet Sun and Region 2 being plage. We have plotted a variety of statistics in these plots that measure the components of the profiles. The emission at h1v generally originates in the upper photosphere. Based on inspection of this and additional higher spatial resolution IRIS data sets, we determine that the structures are a mix of reverse granulation and grains. At h2v, much of the small granular structure disappears and a diffuse halo of bright structures overlay and surround magnetic concentrations. The brightest plage is almost twice as bright as the brightest network, and the maximum h2v emission occurs near the center of the magnetic flux concentration. Single-peaked profiles are relativity common (∼10%) in plage, as depicted by the contiguous red regions in Figure 4. Figure 6(a) shows an example of a single-peaked plage profile. While our algorithm separates between single-peaked and double-peaked profiles to aid in statistical analysis, there is in fact a smooth continuum of weakly separated, weakly reversed profiles that bridge those two profile categories (Figure 6(c) shows an example). In the semi-empirical models of the solar atmosphere, the Mg ii h source function above the temperature minimum has a local maximum at an altitude of 1.2 Mm (see Figure 7 in Leenaarts et al. 2013a), while the line core forms near a height of 2 Mm. To produce a single-peaked profile the atmosphere must be structured such that source function does not vary significantly between

 at h2 and h3. At high densities, the source function closely adheres to the Planck function, which rises through the chromosphere. These profiles are consistent with the model that hot high-density loops are rooted in plage. Most sunspot profiles are single peaked, and an example is displayed in Figure 6(e). Single-peaked umbral profiles were previously reported in Lites & Skumanich (1982) and Morrill et al. (2001). Sunspot profiles are half as bright at h2 and 20% narrower at h1 than plage. The far wing intensity (beyond h1) is lower in sunspots than plage or internetwork. Single-peaked profiles also occur in internetwork regions, albeit at much lower frequency (∼1%). Figure 6(b) shows a typical internetwork single-peaked profile. Although this profile contains a single maximum, it still requires a substantial contribution from the negative Gaussian amplitude (h in Equation (1)) to achieve a low

 Figures 6(d) and (f) show profiles that occur at low frequencies in the internetwork (

). These profiles represent some of the extreme variations from the mean profile shape that we observe in this large data set. The profile of Figure 6(d) could be produced by extreme upflows in the upper chromosphere that completely mask the photons in the nominal position of the h2v peak and wing. Figure 6(f) illustrates Mg ii h in absorption, which requires that the source function monotonically decrease from the photosphere precluding the mid-chromospheric temperature rise.


For case 1, we find that most of the stellar values of the MW halo ([Fe/H]  −1) reside between the evolutionary tracks explored here and also in the coloured region. Note that the carbon stars are excluded in Fig. 5, since they might have been contaminated by the binary mass transfer from former asymptotic-giant-branch stars. The smaller SFE for a less-massive galaxy leads to an increase of [Eu/Fe] at a lower [Fe/H]. It is notable that our model predicts that almost all highly r-process-enhanced stars ([Eu/Fe] > 1 at [Fe/H] ∼ −3) originate from UFD-sized (M*  105) building blocks [but see a caution in Section 4 for the applicability of the mass–metallicity relation of Kirby et al. (2013) to UFD-sized systems], as also suggested in Ishimaru et al. (2015) and Ojima et al. (2018). For instance, our model suggests that J1521−3538 (star, the highest measured Eu/Fe, Cain et al. 2020) and CS 31082−001 (diamond, Hill et al. 2002) were born in building-block galaxies of M* ∼ 103 and 5 × 103 (in M⊙), respectively. This is a consequence of the fact that a less-massive galaxy contains a smaller amount of gas to be mixed with Eu from an NSM. In fact, the highly eccentric orbit of the former indicates that J1521−3538 originates from a building-block galaxy subsequently accreted by the MW (Cain et al. 2020). In contrast, the r-process-deficient stars may have been born in the most-massive building-block galaxies of M* ∼ 108 M⊙, like HD 4306 (circle, the lowest measured Eu/Fe, Ishimaru et al. 2004) and HD 122563 (triangle, Honda et al. 2006). Note that our model predicts the presence of even further r-process-deficient stars with [Eu/Fe]  −1 at [Fe/H] ≲ −3. The absence of such stars with measured Eu is likely due to the current detection limit for Eu (Ishimaru et al. 2004). Overall, the result for case 1 (a mean delay of 0.1 Gyr in the lognormal term for NSMs) appears in qualitative agreement with those in Ishimaru et al. (2015) and Ojima et al. (2018) with a fixed delay of 0.1 Gyr (see Appendix A for additional tests).


As to the chemical enrichment history of individual galaxies, we have exploited the standard code che-evo incorporated into GRASIL (Silva et al. 1998, 2011; Bressan et al. 2002; Panuzzo et al. 2003; Vega et al. 2005). For spheroidal galaxies, it reproduces the observed local relationship between stellar metallicity and stellar mass and its weak evolution with redshift (see Arrigoni et al. 2010; Spolaor et al. 2010; Gallazzi et al. 2014). For disk galaxies at z ≲ 2, it reproduces the observed relationship between gas metallicity and stellar mass, including its appreciable redshift dependence (see Andrews & Martini 2013; Zahid et al. 2014; de la Rosa et al. 2016; Onodera et al. 2016). In both cases, within an individual galaxy the metallicity behavior is closely approximated by
4

i.e., it increases from Z = 0 almost linearly with the galactic age, and then after a time τ = Δτψ it saturates to the value Zsat. The dependence of Zsat and Δ on SFR/stellar mass can be described with an expression inspired by analytic chemical evolution models (see Cai et al. 2013; Feldmann 2015); this yields
5

here s ≈ 3 is the ratio between the dynamical timescale of the infalling gas and the star formation timescale,

 is the recycling stellar mass fraction,

 is the metal yield (assuming the Romano et al. 2010 stellar yields), and out is the mass-loading factor of galactic outflows from stellar winds and SN explosions. In the above equation we have provided a fit for out as a function of the final stellar mass M⋆ from the results of the che-evo code; a similar expression concurrently describes the time-averaged outcome from hydrodynamical simulations of stellar feedback (e.g., Hopkins et al. 2012). As a result, typical values

 are obtained for galaxies with final stellar masses in the range M⋆ ∼ 109–1011 M⊙, respectively (see, e.g., Chruslinska et al. 2019); the related quantity Δ ∼ 0.1–0.3 specifies how quickly the metallicity saturates to such values as a consequence of the interplay between cooling, dilution, and feedback processes. Note that several chemical evolution codes present in the literature, reproducing comparably well observations on the chemical abundances in galaxies of different stellar masses, also share a similar age-dependent metallicity behavior.


The unsmoothed normalized higher order cumulants or $S_N = \langle \delta ^N \rangle _c/\langle \delta ^2 \rangle ^{N-1}_c$ can be expressed in terms of the tree-level vertices denoted as νN using the following expressions (Bernardeau et al. 2002a):
(20)$$\begin{eqnarray*}
&& S_3 = 3\nu _2; \quad S_4 = 4\nu _3+ 12\nu _2^2; \quad S_5 = 5\nu _4+ 60\nu _3\nu _2 + 60\nu _2^3. \quad
\end{eqnarray*}$$The vertices νN are the angular averages of the mode-coupling kernels FN defined in equation (1c) i.e. νN = N!〈 FN〉 introduced in Section   2.2 in the Fourier domain.
(21)$$\begin{eqnarray*}
&& \nu _N = N!\langle F_N \rangle = N! \int {\mathrm{ d}{{\bf \widehat{\Omega }}}_{k_1} \over 4\pi } \cdots \int {\mathrm{ d}{{\bf \widehat{\Omega }}}_{k_N} \over 4\pi } F_N({\bf k}_1.\cdots , {\bf k}_N);\quad \mathrm{ d}{{\bf \widehat{\Omega }}}_k = \sin \theta _k \mathrm{ d}\theta _k \mathrm{ d}\varphi _k.
\end{eqnarray*}$$The following generating function approach was introduced in Bernardeau (1992, 1994b). The generating functions ${\cal G}_{\delta }(\tau)$ are solved using the equations of gravitational dynamics encapsulated in Euler–Continuity–Poisson equations. Here, τ plays the role of a dummy variable. In the perturbative regime, the νN parameter can be computed for an arbitrary N(22)$$\begin{eqnarray*}
&& {\cal G}_{\delta }(\tau) = \sum _{n}{\nu _N \over N!}\tau ^N = -\tau + {12 \over 14} \tau ^2 - {29 \over 42} \tau ^3 + {79 \over 147} \tau ^4 - {2085 \over 5096} \tau ^5 + \cdots
\end{eqnarray*}$$Next, using equation (20), the one-point cumulants in 2D (Munshi et al. 1999b), denoted as ΣN as opposed to SN parameters which represent the cumulants in 3D, can be used to compute the cumulants to arbitrary order in 2D (Munshi et al. 1999b)
(23)$$\begin{eqnarray*}
&& \Sigma _3 = {36\over 7}; \quad \Sigma _4={2540 \over 49}; \quad \Sigma _5={793}; \quad \Sigma _6=16370;
\end{eqnarray*}$$The generalization of the one-point cumulants i.e the SN parameters to the two-point cumulant correlators $C_{pq} = \langle \delta ^p_1\delta _2^q\rangle _c/\langle \delta ^2\rangle _c^{p+q-1}\langle \delta _1\delta _2\rangle _c$ or Cpq parameters was introduced in Bernardeau (1996a). The lower order normalized cumulant correlators can also be expressed in terms of the tree-level vertices νN just as the one-point cumulants introduced in equation (20)
(24)$$\begin{eqnarray*}
&& C_{21} = 2\nu _2; \quad C_{31} = 3\nu _3+ 6\nu _2^2; \quad C_{41} = 4\nu _4 + 36 \nu _3\nu _2 + 24 \nu _2^3.
\end{eqnarray*}$$The corresponding quantities for convergence maps can be defined in an analogous manner and are denoted with a superscript κ i.e. $C^{\kappa }_{pq}$. To compare with observed or simulated data smoothing of the field is necessary. The smoothed generating function ${\cal G}^s_{\delta }$ can be computed from the unsmoothed generating function ${\cal G}_{\delta }$. The generating functions ${\cal G}^s_{\delta }$ and ${\cal G}_{\delta }$ are related by the following implicit relation (Bernardeau 1995):
(25)$$\begin{eqnarray*}
&& {\cal G}_{\delta }^s(\tau)= {\cal G}_{\delta }(\tau [1+ {\cal G}_{\delta }^s]^{-(2+n)/4}).
\end{eqnarray*}$$A top hat smoothing window is assumed and the power spectrum is approximated locally as a power law P(k) ∝ kn (Bernardeau 1995; Munshi et al. 1999b). For other window functions, e.g. Gaussian window generic results are not possible for arbitrary N. However, an order-by-order approach can be adopted to obtain the lower order cumulants (Matsubara 2003). Notice that the smoothed power law depends on the spectral index while unsmoothed vertices depend solely on the gravitational collapse in 3D spherical or cylindrical in 2D. The smoothed vertices can be recovered by Taylor-expanding the smoothed generating function ${\cal G}^s$. Using these vertices, it is possible to now compute the 2D skewness Σ3 and kurtosis Σ4 can be computed (Munshi et al. 1999b)
(26a)$$\begin{eqnarray*}
&& \Sigma _3 = {36 \over 7} -{3\over 2} (n+2);
\end{eqnarray*}$$(26b)$$\begin{eqnarray*}
&& \Sigma _4 = {2540 \over 49} - 33(n+2) + {21 \over 4}(n+2)^2.
\end{eqnarray*}$$


Interestingly, the presence of such an underdense librating region at the leading side of the planet has also been observed in simulations of giant planets which undergo inward runaway migration (Artymowicz 2004; D'angelo & Lubow 2008). Here, it is worth noting that the low-density region does not result from the gap opening process but is rather due to radiative effects. Never the less, using the terminology employed in the framework of Type III migration, Fig. 3 shows that radiative effects can give rise to a coorbital mass deficit given by (Masset & Papaloizou 2003)

(8)

\begin{equation}
\delta \mathrm{m}=4\pi a x_\mathrm{s}\left(\Sigma _\mathrm{s}-\Sigma _h\right),
\end{equation}

where Σs is the surface density at the upstream separatrix and Σh the mean surface density in the librating region. In order to provide an estimation for the coorbital mass deficit in the librating region, we make in the following the assumption that the only contribution to the density perturbation in the librating region results from entropy advection. In reality, however, the density perturbation features an additional contribution linked to the production of vortensity at the outgoing separatrix (Paardekooper et al. 2011). Ignoring this contribution gives the following expression for the surface density Σ− in the librating region (Baruteau & Masset 2008; Paardekooper et al. 2011):

(9)

\begin{equation}
\Sigma _-=\Sigma _\mathrm{s}\left(1-2\frac{\mid \xi \mid x}{\gamma a}\right) \quad \mathrm{for} \quad 0<x<x_\mathrm{s}.
\end{equation}

Here, we have also ignored the radial density gradient and assumed that the corotation torque was initially unsaturated. This gives a mean surface density in the librating region:

(10)

\begin{equation}
\Sigma _h=\Sigma _\mathrm{s}(1-\frac{\mid \xi \mid x_\mathrm{s}}{\gamma a}).
\end{equation}

The coorbital mass deficit then becomes:

(11)

\begin{equation}
\delta \mathrm{m} = 4\pi \mid \xi \mid x_\mathrm{s}^2 \Sigma _\mathrm{s}/\gamma ,
\end{equation}

which shows that the coorbital mass deficit scales with the entropy gradient and the width of the horseshoe region. We note in passing that at the boundary corresponding to $\dot{a} \,{\sim }\, \dot{a_f}$, we expect the corotation torque exerted on the planet to be given by (e.g. Papaloizou et al. 2007):

(12)

\begin{equation}
\Gamma _{\mathrm{CR}}=\frac{1}{2}\dot{a_f}\Omega _\mathrm{p}a \delta \mathrm{m}.
\end{equation}

Using the expression for the coorbital mass deficit given by equation (11), the previous equation then becomes:

(13)

\begin{equation}
\Gamma _{\mathrm{CR}}=\frac{3}{4}\frac{\xi }{\gamma }\Sigma _\mathrm{s} x_\mathrm{s}^4\Omega _\mathrm{p}^2,
\end{equation}

so that we recover the standard expression for the entropy-related horseshoe drag (Baruteau & Masset 2008). For $\dot{a} > \dot{a_f}$, however, the expression for the unsaturated corotation torque given by equation (12) is no longer valid and should be modified (Papaloizou et al. 2007). In that case, we expect the corotation torque to be prevented from saturation since the horseshoe region does not longer extend to the full 2π in azimuth, so that phase-mixing cannot occur. In isothermal discs with a large-scale vortensity gradient, Ogilvie & Lubow (2006) found that drift rates with $\dot{a} > \dot{a_f}$ can even lead to torques that are much larger than the unsaturated value in absence of migration. They showed that this arises due to corotational torques caused by a vortensity asymmetry in the coorbital region between gas on the leading and trailing sides of the planet. In the limit of low viscosity, the trapped material tends to conserve its initial vortensity as the planet migrates, leading to a possibly significant vortensity contrast between the librating region and the local disc. In that case, the impact on migration depends in fact on the sign of the vortensity deficit (Paardekooper 2014) which is defined as

(14)

\begin{equation}
\delta (\omega /\Sigma )=1-\frac{\omega _a/\Sigma _a}{\omega _{0}/\Sigma _{0}},
\end{equation}

where ωa/Σa is the local disc vortensity at the position of the planet and ω0/Σ0 the vortensity at the initial location of the planet. In the case where the planet migrates in the direction set by the vortensity-related horseshoe drag, the vortensity deficit is positive and this yields a positive feedback on migration, whereas if the planet migrates in the opposite direction to that set by the corotation torque, the vortensity deficit is negative and this gives rise to a negative feedback on migration (Paardekooper 2014).


Standard data reduction was applied to the data taken with HIPO (red and blue) and FPI+. This includes bias and dark subtraction, and in the case of FPI+ also flat field correction. For FLITECAM we did not acquire bias frames, as bias contributions are generally very low for this type of NIR detector array. Furthermore, it is complicated and time consuming to obtain a reliable flat field on such a narrow band filter as the 1.9 μm Paschen-α continuum filter. Since it was not possible to take long enough exposures during this campaign, we used K-band flat fields taken on the same flight before our observation run, which, however, did not improve the photometric precision significantly. Similarly, dark subtraction did not show any improvement in photometric precision. Additionally, we corrected for the sky background using dithered images taken during the observation run. Stars were detected with the Source Extractor by Bertin & Arnouts (1996). Aperture photometry was applied using Image Reduction and Analysis Facility’s (IRAF Tody 1993) DAOPHOT (Stetson 1987) using circular apertures. The optimal aperture radius with the lowest noise level was found to be six pixels for FLITECAM and 12 pixel for HIPO and FPI+. As part of the DAOPHOT routine, an annulus around the target was used to estimate the sky background in each exposure. Next to GJ 1214, we extracted the light curves of two additional bright stars within our field of view (see Table 1 and Fig. 1). To identify stars in the images we calculated a rough astrometric solution for each image using data provided by Astrometry.net (Barron et al. 2008). In the FLIPO setup SOFIA does not provide an image rotator to compensate field rotation during long integrations. This introduces a rotation of the images over time. Due to SOFIA’s unique setup, the telescope must periodically undergo so-called “line-of-sight (LOS) rewinds”. The required frequency of LOS rewinds depends on the rate of field rotation experienced by the target, which is a complex function of the position of the target in the sky relative to that of the aircraft heading. These need to be carefully timed with regard to the transit observation, to not interfere with, for example, ingress or egress. While we kept the target star, GJ 1214, in boresight, the comparison stars moved over the CCD due to this field rotation. This is one of the main factors introducing systematic noise and limiting the photometric precision of the instrument and is another reason why reliable flat fields are crucial for this kind of time series observation. After comparing the light curves, the brighter star was selected as comparison star to correct for first-order systematic effects present in all light curves.


Dust grains are modelled using Lagrangian tracer particles, where each simulated particle represents a collection of dust grains with similar properties and motions. Trajectories of the particles are integrated using the fully implicit method of Bai & Stone (2010), which we have incorporated into the VL integrator in Athena. In a Cartesian coordinate system, Athena solves an equation of motion for each particle given by

(12)

\begin{equation}
\frac{{\rm d} {{\boldsymbol v}_i}}{{\rm d}t} = -\frac{{{\boldsymbol v}_i} - {\boldsymbol u}}{t_{\rm stop}},
\end{equation}

with ${{\boldsymbol v}_i}$ the velocity vector of particle i, ${\boldsymbol u}$ the local gas velocity vector, and tstop the particle stopping time due to gas drag. Neglecting grain charges and assuming only pure hydrogen gas, the (collisional) drag law is given by (Draine & Salpeter 1979)

(13)

\begin{equation}
\frac{{\rm d} {\boldsymbol v}_i}{{\rm d}t} \approx -\frac{2 \pi a^2 n k_{\rm B} T G_0(s)}{(4/3)\pi \rho _{\rm d} a^3},
\end{equation}

with

(14)

\begin{equation}
G_0(s) \approx \frac{8 s}{3 \sqrt{\pi }} \left(1 + \frac{9\pi }{64} s^2\right)^{1/2}
\end{equation}

and

(15)

\begin{equation}
s \equiv \left(\frac{m_{\rm H} {\boldsymbol v}_{\rm rel}^2}{2 k_{\rm B} T}\right)^{1/2},
\end{equation}

where a is the dust grain radius, kB is the Boltzmann constant, T is the temperature of the gas, n is the gas number density, ρd is the internal density of the dust (which we treat as constant at ρd = 3.0 g cm−3), mH is the mass of hydrogen, and ${\boldsymbol v}_{\rm rel} \equiv {{\boldsymbol v}_i} - {\boldsymbol u}$ is the relative velocity difference between the dust and gas. The stopping distance is evaluated as

(16)

\begin{equation}
t_{\rm stop} = \frac{\sqrt{\pi }}{2\sqrt{2}} \frac{a \rho _{\rm d}}{n \sqrt{m_{\rm H} k_{\rm B} T}} \left(1 + \frac{9\pi m_{\rm H}}{128 k_{\rm B} T} {\boldsymbol v}^2_{\rm rel}\right)^{-1/2}.
\end{equation}

The gas properties (n, T, ${\boldsymbol u}$) at each particle's location are calculated from nearby grid points using a triangular-shaped cloud (TSC) interpolation scheme (Hockney & Eastwood 1988). There is no momentum feedback from the particles on the gas.


The bottom panel of Fig. 9 shows the mass accretion histories for Z = 10−2 (-2Sol), 10−1, (-1Sol_from-2Sol), and Z⊙ (Sol_from-2Sol), starting from the envelope structure of Z = 10−2 Z⊙. In the early phase of M* ≲ 100 M⊙, while radiative feedback is still weak, the accretion histories are mostly determined by the initial condition and very similar to each other. At later time, radiative feedback starts operating and the accretion history depends on the metallicity in a way that the final mass is lower, or the feedback is stronger, at lower metallicities. This metallicity dependence comes from the dust attenuation of ionizing photons, which can be seen in the following analytic argument. The photoionization feedback quenches the accretion by the photoevaporation of an accretion disc, which proceeds most efficiently near the gravitational radius
(19)$$\begin{eqnarray*}
r_{\rm g} &=& \frac{GM_{*}}{c_{\rm i}^2} \nonumber \\
&\simeq & 1.1 \times 10^3 ~{\rm au} \left(\frac{M_*}{500~\mathrm{ M}_{\odot }} \right)
\end{eqnarray*}$$(e.g. Hollenbach et al. 1994; McKee & Tan 2008), where ci is the sound speed of the ionized gas for which we assume T = 3 × 104 K. Equations (15) and (19) indicate that the gravitational radius is located outside the dust destruction front and thus the ionizing photons around the gravitational radius is attenuated by dust grains. Using the disc surface density at r ∼ rg set by the ionization balance,
(20)$$\begin{eqnarray*}
n_0 &=& \left(\frac{3 S_*}{4 \pi r_{\rm g}^3 \alpha _{\rm B}} \right)^{1/2} \nonumber \\
&\simeq & 2 \times 10^{7} ~{\rm cm^{-3}} \left(\frac{S_*}{10^{51} ~{\rm s^{-1}}} \right)^{1/2} \left(\frac{M_*}{500~\mathrm{ M}_{\odot }} \right)^{-3/2},
\end{eqnarray*}$$where S* is the stellar emissivity of ionizing photons, we find the optical depth from the star to the gravitational radius,
(21)$$\begin{eqnarray*}
\tau _{\rm d} &=& \rho \kappa _{\rm d} r_{\rm g} \nonumber \\
&=& 3.0 \left(\frac{S_*}{10^{51} ~{\rm s^{-1}}} \right)^{1/2} \left(\frac{M_*}{200~M_{\odot }} \right)^{-1/2} \left(\frac{\kappa _{\rm d}}{350~{\rm cm^2 g^{-1}}} \right) \nonumber\\
&&\times \,\left(\frac{Z}{10^{-2}~Z_{\odot }} \right),
\end{eqnarray*}$$larger than unity for Z ≳ 10−2 Z⊙, where κd is the dust opacity for Lyman-limit frequency at the solar metallicity, meaning that the photoionizing radiation is substantially attenuated by the dust, especially at higher metallicity. This explains the trend of weaker feedback at higher metallicity seen in Fig. 9. According to recent studies (Tanaka, Tan & Zhang 2017; Kuiper & Hosokawa 2018; Tanaka et al. 2018), the impact of photoionization is significantly diminished by the dust attenuation at the solar metallicity. In fact, in our cases with Z = 0.1 (-1Sol_from-2Sol) and 1 Z⊙ (Sol_from-2Sol), the photoionization effect being disabled by this process, the accretion continues until the star becomes very massive with >300 M⊙ and the radiation-pressure feedback finally shuts off the inflow. In this process, the radiation force by the infrared dust re-emission also contributes in terminating the accretion (see equation 17). In particular, it can also repel the flow coming from the shade of the disc, for which the radiation force by the stellar direct light is ineffective (e.g. Yorke & Bodenheimer 1999; Kuiper et al. 2010b; Kuiper & Yorke 2013). Our results suggest that the metallicity dependence of the radiation-pressure effect is weaker than that of the photoionization owing to the dust attenuation in metal-enriched environments. As a result, the final stellar mass increases with increasing metallicity for Z > 10−2 Z⊙.


We scale the maps of NGC 4710 to a system that can be directly compared with the Milky Way bulge and compare the resulting maps with those constructed by Zoccali et al. (2014) based on interpolation of the GIBS survey rotational curves. In this comparison we found that the rotation map of NGC 4710 and the rotation map of the Milky Way bulge show a remarkable similarity. Although the velocity dispersion map of the bulge of NGC 4710 appears noisier than that of the Milky Way bulge, owing to the way the maps are constructed, we see that the velocity dispersion map of the Milky Way bulge is more vertically elongated in the centre than in the velocity dispersion map of NGC 4710. We see a similar change in the velocity dispersion maps constructed using different bar orientation angles in the simulation from Cole et al. (2014). This is also in good agreement with the features identified in the LOS kinematic maps of BP bulges modelled by Iannuzzi & Athanassoula (2015). We thus suggest that central increase in the velocity dispersion profile of the Milky Way bulge can be partially the consequence of the Milky Way bar and BP bulge viewing angle of 27° with respect to the Sun-Galactic centre LOS. However, a comparison of the rotation and velocity dispersion curves at different heights from the plane, used by Zoccali et al. (2014) to obtain the rotation maps, are very similar to those of the bulge of NGC 4710. These 1D rotation profiles are also in good agreement with the simulation of a galaxy hosting a pure BP bulge. The differences identified between the inner regions of the σ maps of the Milky Way and NGC 4710 are not seen in the 1D curves. This suggests that the vertical elongation observed in the central region of the σ map of the Milky Way bulge could be an artefact from the plane interpolation method to account for the σ peak observed at b = −2. Instead, the σ peak in the Milky Way bulge would be limited to Galactic latitudes | b |   2 (~0.28 kpc).


The third and higher harmonic plasma emissions, at the frequency of multiples of the local plasma frequency, were also observed in SRBs (Takakura & Yousef 1974; Cairns 1986; Reiner & MacDowall 2019). Various schemes have been proposed to explain the formation of higher harmonic plasma emission. Zlotnik (1978) put forward that the third harmonic emission H
3 is generated by the coalescence process of the Langmuir waves and the harmonic emission H, i.e., L + H → H
3. Cairns (1988) generalized this idea to explain higher harmonics of transverse wave emissions, claiming that nth harmonic waves H

n
 could be generated by a coalescence of beam-generated Langmuir waves L and adjacent transverse waves H

n−1, i.e., L + H

n−1 → H

n
. An alternative way for the generation of harmonic transverse waves would be an interaction of Langmuir waves L and harmonic electrostatic waves L

n−1, i.e., L + L

n−1 → H

n
 (Gaelzer et al. 2003; Yi et al. 2007). Ziebell et al. (2015) numerically solved the weak-turbulence equations and theoretically demonstrate the process of multiple-harmonic plasma emission, including the nonlinear conversion from Langmuir turbulence to electromagnetic radiation. Rhee et al. (2009), Ziebell et al. (2015), Yao et al. (2021) investigated both schemes of multiple-harmonic plasma emissions by means of Particle-In-Cell (PIC) code simulations. They found that Cairns (1988)'s scheme can qualitatively explain the third and fourth harmonic emissions while Yi et al. (2007)'s theory is appropriate to explain the simultaneously generated harmonics of electrostatic waves. PIC code simulations of electron beams demonstrated that fundamental transverse waves at the plasma frequency can be generated because of the wave–wave interaction processes involving ion-acoustic waves (Thurgood & Tsiklauri 2015; Henri et al. 2019). The PIC simulations of ring-beam EVDFs by Zhou et al. (2020) also provided evidence for the generation of the second and third harmonic emission via wave–wave interactions. Annenkov et al. (2019) used PIC code simulations to describe localized beams and found that the emissions at harmonics of the plasma frequency can be generated due to an antenna mechanism.


Measurements of the scale length (h

R
) and scale height (h

Z
) are important to trace the structure, size, mass distribution, and radial luminosity profile of the Galactic disk components (e.g., Dehnen & Binney 1998). In order to calculate h

R
 and h

Z
 of our Atari disk sample, we solve the fundamental collisionless Boltzmann equation of axisymmetric systems, which is expressed as follows (see Equation (4.12) in Binney & Tremaine 2008):
10

∂f∂t+vR∂f∂R+vϕR2∂f∂ϕ+vz∂f∂z−∂Φ∂R−vϕ2R3∂f∂vR−∂Φ∂ϕ∂f∂vϕ−∂Φ∂z∂f∂vz=0,

where f is the number of objects in a small volume and Φ is the gravitational potential. It is then convenient to derive the Jeans equation from the Boltzmann equation in the radial and Z-component directions as follows (see Equation (9) in Gilmore et al. 1989):
11

ρKR=1R∂(RρσVR2)∂R+∂(ρσVR,Z2)∂Z−ρσVϕ2R−ρRVϕ¯2

12

ρKZ=∂(ρσVZ2)∂Z+1R∂(RρσVR,Z2)∂R,

where ρ(R, Z) is the space density of the stars in the thick disk and K

R
 =

∂ϕ∂R

 and K

Z
 =

∂ϕ∂Z

 are the derivatives of the potential. Assuming an exponential density profile, the radial Jeans equation can be rewritten as follows (Li et al. 2018):
13

σVϕ2σVR2−2+2RhR−Vc2−Vϕ¯2σVR2+σVZ2σVR2=0,

where h

R
 is the scale length. By substituting our calculated velocity dispersions from the Atari disk sample, within ≈3 kpc of the Sun in the cylindrical R coordinate and ≈2 kpc above or below the Galactic plane (6347 stars), into Equation (13), we obtain a radial scale length of h

R
 = 2.48 kpc. Calculating the scale length using different metallicity bins shows a small increase from 2.38 kpc among the higher-metallicity stars up to 2.91 kpc for the low-metallicity stars. The results are detailed in Table 3. In general, these results point to the Atari disk being comparable in size in the radial direction to the thick and thin disk. For reference, the scale length of the canonical thick disk has been measured as 2.0 kpc (Bensby et al. 2011), 2.2 kpc (Carollo et al. 2010), and 2.31 kpc (Sanders & Binney 2015), although larger values have also been reported previously (Chiba & Beers 2000; de Jong et al. 2010). Thin disk values refer to an overall similar spatial distribution, although it is likely somewhat more extended (h

R
 > 3.0 kpc;  e.g., Bensby et al. 2011; Sanders & Binney 2015). See Table 2 for further details.


Our fitting is performed based on the Markov Chain Monte Carlo (MCMC) method to produce posterior predictions for the model parameters. The MCMC method is widely used in finding the best set of parameters for a specified model, e.g., GRB 080413B (Geng et al. 2016); GRBs 100418A, 100901A, 120326A, and 120404A (Laskar et al. 2015). In our work, fitting the afterglows of the four bursts with the MCMC method is to test whether or not the late simultaneous bumps followed by a steep decay can be explained with an external-forward shock in a free-to-shocked wind environment. The posterior probability density functions for the physical parameters, i.e., Ek,iso,on, Γ0, θc, θjet/θc, θv/θc, p, ϵe, ϵB, A*, and Rtr, are presented in Figure 6, where only the fitting result of GRB 120326A is shown as an example. The optimal result from MCMC fitting is shown in Figure 5 with the blue line (XRT) and red line (optical), and the obtained parameters at the 1σ confidence level are reported in Table 1, where the values of the transition radius

R

tr

 (i.e., 1.05 × 1017, 1.91 × 1017, 3.98 ×1017, and 6.31 × 1016 cm for GRBs 120326A, 100901A, 100814A, and 120404A, respectively) are consistent with those found in other bursts (e.g., Ramirez-Ruiz et al. 2001; Kong et al. 2010; Feng & Dai 2011; Li et al. 2020). It can be found that both the X-ray afterglow and the optical afterglow of these bumps can be well modeled with an external-forward shock in a free-to-shocked wind environment.

1.
The theoretical light curves do not well fit the transition behavior from the free-wind phase to the shocked-wind phase in the afterglows of GRB 120326A. This may imply that the density jump factor ξ from the free-wind medium to the shocked-wind medium may be less than 4, i.e., ξ  4. Figure 1 reveals that the external-forward shock propagating in a shocked-wind environment can yield a plateau before the late simultaneous bumps in the case of θv = 4θc. Then, we try to fit the afterglows of GRB 120326A with the emission of the external-forward shock propagating in a homogeneous medium, where the priors of

log
10

(

E

k
,
iso
,
on

erg

-
1

)

,

log

10

Γ

0

, θc, θjet/θc, θv/θc, p,

log

10

ϵ

e

,

log

10

ϵ

B

, and

log

10

n

0

 are set as a uniform distribution in the range of (52, 55), (1.5, 3.0), (0.3, 8.0), (2.5, 4.5), (0.0, 8.0), (2.1, 2.9), (−3.2, −0.5), (−6.5, −2.0), and (0.0, 1.7) in our MCMC fittings, respectively. However, no well-fitting result is found. The fitting result with minimum reduced χ2 is shown with dashed lines in the top left panel of Figure 5. It can be found that the early plateau of GRB 120326A could not be well fitted based on the emission of the external-forward shock in a homogeneous medium.

2.
There seems to be a plateau at tobs ∼ 600 s in the optical afterglow of GRB 100814A, which may indicate an energy injection into the external shock. If so, our model could not well describe the afterglows of GRB 100814A in its early phase. This may be the reason for the deviation in our theoretical results relative to the observations in the early phase. In addition, the X-ray and optical afterglows of GRB 100814A show achromatic behavior, which could not be explained only by our scenario (i.e., an off-core observed structured jet propagating into a free-to-shocked wind environment) since the appearance of the late simultaneous bump/plateau in our scenario is achromatic. Other emission rather than that of the external-forward shock may contribute to the long X-ray shallow decay. In addition, our model could not well describe the X-ray emission at tobs ≳ 3 × 105 s in GRB 100901A, which seems to be another component in this phase.

3.
The fitting result of GRB 120404A shows that this burst is observed in the core. Based on the fitting result of this burst, the deceleration radius of the (0, 0) jet can be estimated and is 2.43 × 1016 cm, which is at around the transition radius

R

tr

=
6.31
×

10

16

 cm of this burst.


For the cosmic variance term, any two samples are correlated so the off-diagonal
            element of Gij
              (i ≠
              j) is non-zero. But for the measurement noise term,
            small scale, and intrinsic dispersion, the two samples are not correlated, so they only
            contribute to the diagonal term in the Gij matrix. The first
            term is the real space velocity correlation function, which is related to the matter
            power spectrum in Fourier space (Watkins et al.
              2009; Ma & Scott 2013),
              (10)\begin{eqnarray} \bigl\langle \bigl(\hat{{\vec r}}_{i} \cdot {\vec v}({\vec r}_{i})\bigr)  \bigl(\hat{{\vec r}}_{j} \cdot {\vec v}({\vec r}_{j})\bigr) \bigr\rangle  =\frac{H^{2}_{0}f^{2}(z=0)}{2 \pi^{2}}\int \mathrm{\ } \der k  \mathrm{\ } P(k) \mathrm{\ } F_{ij}(k), \label{vel_rnrm1} \end{eqnarray}

r

ˆ

i

·

v

(

r

i

)

r

ˆ

j

·

v

(

r

j

)

=

H

0

2

f

2

(
z
=
0
)

2

π

2

∫

d
k

P
(
k
)

F

ij

(
k
)
,

where P(k) is
            the matter power spectrum that we output from public code camb (Lewis et al. 2000). The f(z) =
                dlnD/ dlna is
            the growth rate function that characterizes how fast the structures grow at different
            epochs of the Universe. Since the Cosmicflows-2  samples peak at the redshift
              z ≃
              0.0167 (lower panel of Fig. 2),
            we use the zero-redshift growth function \hbox{$f(z=0)=\Omega_{\rm m}^{0.55}$}
f
(
z
=
0
)
=

Ω
m
0.55

 (Watkins
              et al. 2009; Ma & Scott 2013) in Eq.
              (10). In the future, if the survey
            probes deeper region of the space, one should use the corresponding growth function
              f(z) in Eq. (10), so that the joint constraints on
              f(z)σ8
            can be obtained, which constitutes a sensitive test of modify gravity models (Hudson & Turnbull 2012; Planck Collaboration XIII 2015).


Solar flares are one of the most spectacular activities in the solar system. Up to 1029–1032 erg magnetic free energy is impulsively released, accompanied by rapid increases of emissions in various wavelengths (Shibata & Magara 2011). It is generally believed that magnetic reconnection plays a key role in the reconfiguration of magnetic field lines and the conversion of magnetic energy into kinetic and thermal energies of plasma (Forbes & Acton 1996; Priest & Forbes 2000). In the context of the standard solar flare model (Carmichael 1964; Sturrock 1966; Hirayama 1974; Kopp & Pneuman 1976), accelerated nonthermal electrons propagate downward and heat the chromosphere, forming bright flare ribbons in the Hα, Ca ii H, ultraviolet (UV), and extreme-ultraviolet (EUV) wavelengths. The flare ribbons have diverse shapes, with two parallel flare ribbons (PFRs) being the most commonplace (Li & Zhang 2015). Sometimes, circular flare ribbons (CFRs) appear accompanied by remote brightenings, which are generally believed to be associated with a magnetic null point (

) and a dome-like spine–fan topology (Masson et al. 2009; Zhang et al. 2012; Pontin et al. 2016). More complex flare ribbons have also been observed (Joshi et al. 2015). Apart from the localized heating, hard X-ray (HXR) emissions are generated via Coulomb collisions (Brown 1971). If open magnetic field lines are involved, it is likely that the nonthermal electrons at speeds of 0.06–0.25c (c is the speed of light) escape from the Sun into interplanetary space and generate type III radio bursts with frequencies ranging from 0.2 to hundreds of MHz (Dulk et al. 1987; Aschwanden et al. 1995; Zhang et al. 2015). Sometimes, the HXR and radio emissions of a flare show quasi-periodic pulsations (QPPs), with their periods ranging from milliseconds (Tan et al. 2010) through a few seconds (Kliem et al. 2000; Asai et al. 2001; Ning et al. 2005; Nakariakov et al. 2010; Hayes et al. 2016) to several minutes (Ofman & Sui 2006; Nakariakov & Melnikov 2009; Sych et al. 2009; Ning 2014). So far, QPPs have been extensively explored using both imaging (Su et al. 2012a, 2012b) and spectral observations (Mariska 2005). Li et al. (2015b) studied the X1.6 flare on 2014 September 10. Four-minute QPPs are evident not only in the HXR, EUV, UV, and radio light curves, but also in the temporal evolutions of the Doppler velocities and line widths of the C i, O iv, Si iv, and Fe xxi lines. Brosius & Daw (2015) studied the M7.3 flare on 2014 April 18. The chromospheric and transition region line emissions show quasi-periodic intensity and velocity fluctuations with periods of ∼3 minutes during the first four peaks and ∼1.5 minutes during the last four peaks.


Figures 2 and 3 demonstrate that the GW alert system is capable of providing GW alerts before merger, but they do not consider the prospects for detection from an astrophysical source population. We generate a population of simulated BNS signals, henceforth referred to as injections, using the TaylorF2 (Sathyaprakash & Dhurandhar 1991; Blanchet et al. 1995, 2005; Buonanno et al. 2009) waveform model. Both source-frame component masses are drawn from a Gaussian distribution between 1.0 M⊙ m1, m2  2.0 M⊙ with mean mass of 1.33 M⊙ and standard deviation of 0.09 M⊙, modeled after observations of galactic BNSs (Özel & Freire 2016).40

40
Note that if GW190425 is a BNS, then galactic measurements are not representative of neutron star masses.
 The neutron stars in the population are nonspinning, motivated by the low spins of BNSs expected to merge within a Hubble time (Burgay et al. 2003; Zhu et al. 2018). The signals are distributed uniformly in comoving volume up to a redshift of z = 0.2. We consider a network of four GW detectors: LIGO-Hanford, LIGO-Livingston, Virgo, and KAGRA at their projected O4 sensitivities.41

41

https://dcc.ligo.org/LIGO-T2000012/public

 We simulate the results of an early warning matched-filtering pipeline by considering six different discrete frequency cutoffs: 29, 32, 38, 49, 56, and 1024 Hz to analyze signal recovery at (approximately) 58, 44, 28, 14, 10, and 0 s before merger, motivated by Sachdev et al. (2020). We calculate the network S/N of each injection at each frequency cutoff and consider the events that pass an S/N cutoff of 12.0 as “detected.” We then calculate the sky posteriors for each of the detected signals by using BAYESTAR (Singer & Price 2016). We use the most recent BNS local merger rate from Abbott et al. (2020b) of

 to estimate the number of events detected per year in the detector network. In Figure 4(a) we see that our optimistic scenario predicts

 GCN will be received 1 s before merger per year, while our pessimistic scenario predicts

 GCN will be received 1 s before merger per year considering the higher end of the BNS rate. Figure 4(b) predicts that ∼9 events will be detected per year, out of which ∼20% (∼1.3%) will be detected 10 s (60 s) before merger. Further, ∼3% of the detectable events (∼1 BNS every 3–4 yr) will be detected 10 s prior to merger and have a localization less than 100 deg2 at O4 sensitivities. This highlights the need for continued latency improvements in advance of O4 to maximize the potential of capturing prompt emission.


Solar system planetary bodies are generally volatile depleted compared to chondritic meteorites (e.g., O’Neill & Palme 2008; Braukmüller et al. 2019). For example, Rb–Sr and K–U are elemental pairs with similar geochemical behaviors, but Rb and K are much more volatile than Sr and U, and Earth has Rb/Sr and K/U ratios that are ∼10 times lower than that of Ivuna-like carbonaceous (CI) chondrites (Allègre et al. 2001; Halliday & Porcelli 2001). However, the mechanisms and conditions that led to this relative volatile-element depletion remain ambiguous. The stable isotopes of elements can be fractionated by volatilization processes and the shifts in their isotopic compositions have been applied to understand volatile-element depletion processes (Humayun & Clayton 1995; Luck et al. 2005; Hin et al. 2017; Kato & Moynier 2017b; Pringle & Moynier 2017; Young et al. 2019; Bloom et al. 2020; Hellmann et al. 2021). Cr is a moderately volatile element (Lodders 2003; Wood et al. 2019) that may show a special behavior during volatilization at higher temperatures. Under moderately reducing to oxidizing conditions (fO2 > IW+1), relevant during evaporation processes from magma oceans on planets (Visscher & Fegley 2013), Cr may be present in the gas as oxide species CrO(g), CrO2(g) and CrO3(g) (Chase 1998). Equilibrium isotopic fractionation between these gas phase species and melt will cause enrichment of the lighter Cr isotopes in the condensed phases, in which Cr is either di- or trivalent, relative to the vapor (Sossi et al. 2018). This contrasts with its speciation under more reduced nebular conditions in which Cr may be present as Cr2+ (major; in silicates) or Cr0 (minor; in Fe–Ni metal) in condensed phases and Cr0 in the gas (Grossman et al. 2008), in which case, through either equilibrium or kinetic fractionation, the condensed phase will be enriched in the heavy Cr isotopes. Solids or magmas that evaporated significant quantities of moderately volatile elements such as Zn, K, Rb, or Ga normally show enrichments of heavy isotopes of these elements, both for kinetic and equilibrium isotope fractionation processes (Paniello et al. 2012; Wang & Jacobsen 2016; Kato & Moynier 2017a; Pringle & Moynier 2017). The behavior of Cr during evaporation has been a motivation to apply this element to study the origin of volatile depletion in the Moon and of the eucrite parent body (likely asteroid 4-Vesta) (Sossi et al. 2018; Zhu et al. 2019b). Both lunar samples and HED meteorites from Vesta are enriched in the lighter isotopes of Cr compared to Earth and chondrites. These isotopic enrichments have been interpreted to reflect equilibrium Cr isotope fractionation between melt (reduced, and enriched in CrO) and gas (oxidized, and enriched in CrO2) during evaporation from a magma ocean. Given that at equilibrium, isotopic fractionation scales with 1/T
2, the Cr evaporation from the Moon and Vesta was estimated to have occurred under relatively low temperatures (∼2000 K), which in the case of the Moon is comparable to the surface temperature of a lunar magma ocean (Sossi et al. 2018).


The semiforbidden [C iii] λ1907 + C iii] λ1909 doublet (hereafter C iii] λ1909 or C iii]) is one of the strongest nebular emission line features observed in the rest-frame ultraviolet (rest-UV) spectrum of low-metallicity, 12+log(O/H) ≤ 8.4 (Z  0.5 Z⊙) star-forming galaxies (SFGs) both locally (Garnett et al. 1995; Rigby et al. 2015; Berg et al. 2016, 2019; Senchyna et al. 2017, 2019) and at high redshifts (Fosbury et al. 2003; Erb et al. 2010; Bayliss et al. 2014; Stark et al. 2014, 2015a, 2017; de Barros et al. 2016; Vanzella et al. 2016; Amorín et al. 2017; Laporte et al. 2017; Maseda et al. 2017; Berg et al. 2018; Hutchison et al. 2019; Le Fèvre et al. 2019). The C iii] emission line is frequently observed in gravitationally lensed SFGs at z > 2–6 as the strongest line after Lyα λ1216 emission. Since the Lyα emission at z > 6 is expected to be severely quenched by the increasingly neutral intergalactic medium (IGM), C iii] is emerging as an alternative redshift indicator for galaxies in the reionization era (Stark et al. 2015a, 2017; Ding et al. 2017). At z > 6, the rest-UV spectrum of galaxies is redshifted to the near-infrared (NIR) wavelengths and is accessible to the spectrographs on upcoming facilities (such as the James Webb Space Telescope (JWST) and the >20 m class Extremely Large Telescopes), which will address the key goal of identifying the sources of reionization. In recent years, there has been considerable effort to develop spectral diagnostics involving the C iii] emission line and other UV spectral features (e.g., C iv λλ1548, 1550, He ii λ1640, and O iii] λλ1661, 1666 (hereafter O iii] λ1663 or O iii])) that can be used to understand the nature of the reionizers, their hard ionizing continua, and the physical conditions in their ISM (Feltre et al. 2016; Gutkin et al. 2016; Jaskot & Ravindranath 2016, hereafter JR16; Byler et al. 2018; Nakajima et al. 2018b). The C iii] emission observed at high redshifts is mostly from strongly lensed low-metallicity (0.5 Z⊙), low-mass (1010 M⊙) galaxies with high specific star formation rates (SSFRs ≳ 2 Gyr−1), and their C iii] equivalent widths (EWs) show a broad range: EW(C iii]) ∼ 3–25 Å (Fosbury et al. 2003; Erb et al. 2010; Bayliss et al. 2014; Stark et al. 2014; de Barros et al. 2016; Vanzella et al. 2016; Berg et al. 2018). The composite rest-UV spectra of Lyman-break galaxies at z ∼ 3 (Shapley et al. 2003) and SFGs at z = 1–2.5 (Steidel et al. 2016; Amorín et al. 2017) also reveal the C iii] λ1909 and O iii] λ1663 nebular lines at subsolar metallicities.


While the properties of the SLSN hosts themselves are of interest, they are most diagnostic when compared to other classes of extragalactic transient whose progenitors are better understood. To this end, we employ a comparison sample of LGRB and CCSN3 host galaxies. In principle, CCSNe should trace all core-collapse events, although the mass function means they will be dominated by stars at the lower mass end (∼8 M⊙ to ∼25 M⊙). There also remains a possibility that some very massive stars can undergo core collapse without yielding a LSNe (e.g. Smartt 2009; Ugliano et al. 2012; Kochanek 2014) such that CCSNe samples might only provide a census of lower mass core collapsing stars (e.g. 8  M*  20 M⊙). Indeed, constraints from explosion parameters have shown the majority of CCSNe to be consistent with lower mass progenitors, as opposed to more massive Wolf–Rayet stars (Cano 2013; Lyman et al. 2016) GRBs likely represent a population with rather larger initial masses (Larsson et al. 2007; Raskin et al. 2008). LGRBs are now known to be associated with the core collapse of massive stars, and broad line SN Ic are near ubiquitously associated with low-z events (where such signatures can be seen; Hjorth et al. 2012). When compared to the hosts of CCSNe they are generally smaller and of lower luminosity, consistent with an origin in galaxies of lower metallicity (Fruchter et al. 2006; Svensson et al. 2010). In relatively local examples, where spatially resolved gas phase metallicities can be obtained, these indeed appear to be lower for GRBs than for CCSNe, even in cases where the luminosity of the galaxy is relatively high (i.e. the GRB host galaxies lie off the mass–metallicity relation; Modjaz et al. 2008; Graham & Fruchter 2013). Hence, comparing the hosts of SLSNe to these events allows us to test the large-scale environments of SLSNe against those of the bulk core-collapse population and a subset which appears to derive largely from massive stars at lower metallicity, although we note that agreement on this matter is not complete (e.g. Podsiadlowski, Joss & Hsu 1992; Eldridge, Izzard & Tout 2008; Smartt 2009; Drout et al. 2011). By exploiting both LGRB and CCSN host samples we can ascertain if there is a strong metallicity dependence in SLSN production, and if this is more or less extreme than that observed in GRB hosts.


The distribution of optical depth values at 10 au and 1.3 mm (Figure 7, top) has its maximum at

log

τ

1.3

mm
,
10

au

=
−
0.25

 (corresponding to

τ

1.3

mm
,
10

au

∼
0.5

), and a secondary peak at

log

τ

1.3

mm
,
10

au

=
−
1

. We note that the shape of this distribution is determined mostly by Taurus, given the lack of sufficient long-wavelength data for most objects in Chamaeleon I and Ophiuchus, and the distributions in these regions appear to be broader than the one in Taurus (again, this interpretation is limited by the small number statistics in these regions). For comparison, reasonable assumptions about the dust opacity and surface density based on observations of the solar system bodies yield

τ

1

mm

=
1

 at ∼10 au for the Minimum Mass Solar Nebula (Davis 2005), suggesting that several of the modeled protoplanetary disks may have optical depth profiles (and hence possibly surface densities) similar to that of the parental disk of the solar system. In the case of β, values smaller than the one measured for the ISM (∼1.6–2; see Draine 2006 and references therein) imply some degree of grain growth. Almost the entirety of the Taurus and Ophiuchus distributions (and part of Chamaeleon I) are constrained within that value, in correspondence with the observational result discussed in Section 3.3. As with

α

mm

, Chamaeleon I shows a different behavior (an excess of high β values) that will be discussed further on. We note that the distributions of β should be considered with caution, not only due to the aforementioned caveats, but also because degenerate cases have been removed from the analysis. Because these occur when

α
=
2

, this procedure inevitably discards objects with

β
∼
0

. There is a tentative bimodality in both distributions, especially in the case of β, with a tentative secondary peak occurring at ∼1.4–1.5. Given the limited size of the modeled sample and simplicity of the models used, we do not investigate this issue in detail here. However, we speculate that, if real, it may hint at a quick transition from micron-sized grains (large β values) to mm/cm-sized dust (smaller β).


Our SBs and MSs are located in a dense environment. To better understand the environmental effect on gas turbulence, we compare the four cluster members with field galaxies from the literature. The data included in Figs. 5 and 6 contain molecular and ionized gas observations. The ionized gas observations of the star-forming galaxies, sorted by redshifts from the lowest to the highest, are from surveys GHASP (log M*[M⊙]=9.4–11.0; log

M

∗

avg

$ M_{\mathrm{*}}^{\mathrm{avg}} $

[M⊙]=10.6; Epinat et al. 2010), DYNAMO (log M*[M⊙]=9.0–11.8; log

M

∗

avg

$ M_{\mathrm{*}}^{\mathrm{avg}} $

[M⊙]=10.3; Green et al. 2014), MUSE and KMOS (log M*[M⊙]=8.0-11.1; log

M

∗

avg

$ M_{\mathrm{*}}^{\mathrm{avg}} $

[M⊙]=9.4, 9.8 at z ∼ 0.7, 1.3; Swinbank et al. 2017), KROSS (log M*[M⊙] = 8.7–11.0; log

M

∗

avg

$ M_{\mathrm{*}}^{\mathrm{avg}} $

[M⊙]=9.9; Johnson et al. 2018), KMOS3D (log M*[M⊙]=9.0–11.7; log

M

∗

avg

$ M_{\mathrm{*}}^{\mathrm{avg}} $

[M⊙]=10.5, 10.6, 10.7 at z ∼ 0.9, 1.5, 2.3; Wisnioski et al. 2015; Übler et al. 2019), MASSIV (log M*[M⊙]=9.4–11.0; log

M

∗

avg

$ M_{\mathrm{*}}^{\mathrm{avg}} $

[M⊙]=10.2; Epinat et al. 2012), SIGMA (log M*[M⊙]=9.2–11.8; log

M

∗

avg

$ M_{\mathrm{*}}^{\mathrm{avg}} $

[M⊙]=10.0; Simons et al. 2016), SINS (log M*[M⊙]=9.8–11.5; log

M

∗

avg

$ M_{\mathrm{*}}^{\mathrm{avg}} $

[M⊙]=10.8; Förster Schreiber et al. 2009; Cresci et al. 2009), LAW09 (log M*[M⊙]=9.0–10.9; log

M

∗

avg

$ M_{\mathrm{*}}^{\mathrm{avg}} $

[M⊙]=10.3; Law et al. 2009), AMAZE (log M*[M⊙]=9.2–10.6; log

M

∗

avg

$ M_{\mathrm{*}}^{\mathrm{avg}} $

[M⊙]=10.0; Gnerucci et al. 2011), and KDS (log M*[M⊙]=9.0–10.5; log

M

∗

avg

$ M_{\mathrm{*}}^{\mathrm{avg}} $

[M⊙]=9.8; Turner et al. 2017). The molecular gas observations of the star-forming galaxies, sorted by redshifts, are from the HERACLES survey (log M*[M⊙]=7.1–10.9; log

M

∗

avg

$ M_{\mathrm{*}}^{\mathrm{avg}} $

[M⊙]=10.5; Leroy et al. 2008, 2009) and the PHIBSS survey (log M*[M⊙]=10.6-11.2; log

M

∗

avg

$ M_{\mathrm{*}}^{\mathrm{avg}} $

[M⊙]=11.0; Tacconi et al. 2013). We note that although the velocity dispersion measured from the molecular gas is ∼10−15 km s−1 lower than from the ionized gas (with extra contributions from thermal broadening and expansion of the HII regions) in the local Universe, this difference becomes smaller with increasing redshift (Übler et al. 2019). Some field SBs with individual CO observations (Calistro Rivera et al. 2018; Barro et al. 2017; Swinbank et al. 2011; Tadaki et al. 2017, 2019) are indicated with blue triangles in Figs. 5 and 6. When possible, we also identified the field starbursts (orange symbols in Figs. 5 and 6) within the above star-forming samples, requiring their SFR to be at least 0.5 dex higher than the MS. In general, the gas in these field starbursts is slightly more turbulent than in main-sequence galaxies, showing a higher σ0, especially at 2   z   4. A similar trend of increasing σ0 as galaxies move above the MS at a fixed stellar mass is shown for star-forming galaxies at z = 0−3 (e.g., Perna et al. 2022; Wisnioski et al. 2015), suggesting that mergers or interactions would increase gas σ0 as they enhance star formation.


Analytical expressions for the IP and OP spectra obtained by IR–MAIRS are theorized by Itoh et al. (2009) and Shioya et al. (2019). In short, Shioya et al. showed both experimentally and theoretically that the IP spectrum (A
IP) quantitatively corresponds to the normal-incidence transmission spectrum,

Athinθ=0

 (see the Appendix for details; Shioya et al. 2019).
5

AIP=Athinθ=0=8πdaλln102n2k2xy.

A
OP is expressed similarly to A
IP, while the TO energy-loss function is replaced with a longitudinal optic (LO) energy-loss function: i.e., (2n
2
k
2z
)/(

n22

 + k
2z

2)2 (Shioya et al. 2019; Hama et al. 2020).
6

AOP=8πdaHλln10(2n2k2z)n22+k2z22,

where H is a substrate-specific correction factor (0.33 for Si) that accounts for the intensity ratio of the electric fields at the optical interface along the surface-parallel and surface-perpendicular directions (Shioya et al. 2019). Because amorphous H2O vapor-deposited at 80–100 K has values of k
2 that are approximately 2 orders of magnitude smaller than n
2 (k
2 ≪n
2) at 3750–3650 cm−1 in the bulk (Mastrapa et al. 2009), Equation (6) simplifies to
7

AOP≈8πdaHλln10(2n2k2z)n24.

From Equations (5) and (7), k
2xy
 and k
2z
 are derived as
8

k2xy=AIPλln1016πdan2

9

k2z=n24HAOPλln1016πdan2.

Therefore, optically isotropic k
2 and σ can be obtained from A
IP and A
OP even for a thin sample (Shioya et al. 2020):
10

k2=2k2xy+k2z3=2AIP+n24HAOP3λln1016πdan2

11

σ=4πdk2λN=4πdλN2AIP+n24HAOP3λln1016πdan2=2AIP+n24HAOP3ln104an2N.

The band strength β (cm molecule−1) can be also obtained by integrating

σ(ν˜)

 at

ν˜=

 3710–3680 cm−1.
12

β=∫σν˜dν˜=23∫AIPν˜dν˜+n24H3∫AOPν˜dν˜ln104an2N,

where

AIPν˜

 and

AOPν˜

 is the IP and OP absorbance at a given wavenumber (

ν˜

), respectively. This study uses n
1 = 1 for vacuum, n
2 = 1.26 for amorphous water at 90 K (Kofman et al. 2019), and n
3=3.41 for Si (Tasumi 2014; Shioya et al. 2019). Hence, a = 0.290 and

n24

H = 0.832. Seven independent measurements for amorphous water formed after 32 minutes of exposure at 90 K gave A
IP  = 0 [

∫AIPν˜dν˜=0

 (cm−1)] and

n24HAOP

 = 6.2 ± 0.4 × 10−5 at 3696 cm−1 [

n24H∫AOPν˜dν˜

 = 8.4 ± 0.4 × 10−4 (cm−1)], considering amorphous water formed on both sides of the Si substrate. Therefore, σ is derived as 1.0 ± 0.2 × 10−18 cm2 at 3696 cm−1 from Equation (11) by adopting N = 3.3 ± 0.6 × 1013 molecules cm−2. β is also derived as 1.4 ± 0.3 × 10−17 cm molecule−1 at

ν̃

 = 3710–3680 cm−1 from Equation (12).


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).


To investigate this further, we performed an additional test where we froze the slopes for both the fossils and groups to 3.0 and only fit the normalisation for both samples. The value of 3.0 for the slope was chosen as generally most groups and clusters can be described well by this particular slope, after factoring in selection effects. The fossils now have a normalisation of 0.38 ± 0.11 and the groups sample have a normalisation of 0.22 ± 0.13. We now proceeded to remove the influence of selection effects on the normalisation of the scaling relation for both samples by generating mock samples of objects as in Bharadwaj et al. (2015) by varying the input normalisations. For each mock sample, the slope was always fixed to 3 and the intrinsic scatter was fixed to the observed values. Flux and luminosity cuts were applied to both the 400d sample and the groups sample to match the true sample of objects. For the 400d fossil sample, the selection criteria were taken from Voevodkin et al. (2010), and can be described as follows: a lower flux cut of 1.4 ×  10-13 erg / s / cm2, an upper redshift cut of 0.2, and a lower luminosity cut of 1043 erg/s. Here, fluxes and luminosities are in the ROSAT (0.5–2.0 keV) band. After performing the bias corrections, the fossils scaling relation has a normalisation of 0.30 ± 0.10 vs. 0.0078 ± 0.13 for the groups, indicating that the large normalisations seem to persist even after accounting for selection effects. Despite a nearly 2.3σ higher normalisation for fossils with respect to non-fossils, we still only treat this finding as an indication because on top of the statistical uncertainties in both quantities (which is large currently for the 400d fossils), there could also be an additional systematic uncertainty introduced due to differences in the luminosity determination in the different parent catalogues. Secondly, the archival nature of this subsample of 400d objects is biased towards systems lacking a SCC (Table 2) and, assuming that the remaining systems are SCC, then adding these objects could potentially increase the normalisation for the scaling relation, and the difference between the fossils and the groups sample would be higher than what we demonstrate here. We plan to explore this in greater detail in a future study of fossils scaling relations.


In this work, we focus on constraining the variability of the fine-structure constant, α ≡ e2/ℏc, which represents the coupling strength of the electromagnetic force. In the last 15 years, there have been many attempts to measure this constant in absorption systems along the lines of sight to distant quasars. The approach called the ‘Many Multiplet’ (MM) method, pioneered by Dzuba, Flambaum & Webb (1999) and Webb et al. (1999), compares the relative velocity spacing between different metal ion transitions and relates it to possible variation in α. For example, considering just a single transition, variation in α is related to the velocity shift Δvi of a transition

(1)

\begin{equation}
\Delta \alpha /\alpha \equiv \frac{\alpha _{\rm obs}-\alpha _{\rm lab}}{\alpha _{\rm lab}}\approx \frac{-\Delta v_i}{2c}\frac{\omega _i}{q_i},
\end{equation}

where c is the speed of light, qi is sensitivity of the transition to α-variation, calculated from many body-relativistic corrections to the energy levels of ions and ωi is its wavenumber measured in the laboratory. There have been two MM method studies of large absorber samples: the Keck High Resolution Echelle Spectrometer (HIRES/Keck) sample of 143 absorption systems (Webb et al. 1999, 2001; Murphy et al. 2001a, 2004; Murphy, Webb & Flambaum 2003b) and the Ultraviolet and Visual Echelle Spectrograph on the Very Large Telescope (UVES/VLT) sample of 154 absorption systems (Webb et al. 2011; King et al. 2012). These studies reported weighted mean values of Δα/α = −5.7 ± 1.1 parts per million (ppm) and 2.29 ± 0.95 ppm, respectively. Webb et al. (2011) and King et al. (2012) also combined both samples and found a 4.1σ statistical preference for a dipole-like variation in α across the sky. Contrary to these studies, several other attempts to measure α from individual absorbers (Quast, Reimers & Levshakov 2004; Levshakov et al. 2005, 2006, 2007; Chand et al. 2006) or smaller samples of absorption systems (Chand et al. 2004; Srianand et al. 2004) reported Δα/α consistent with zero. However, these analyses contained shortcomings that produced biased values and/or considerably underestimated error bars (Murphy, Webb & Flambaum 2007a, 2008b; cf. Srianand et al., 2007). For example, a reanalysis, including remodelling of the Chand et al. (2004) spectra by Wilczynska et al. (2015), yielded a weighted mean of Δα/α = 2.2 ± 2.3 ppm which, given the distribution of the 18 quasars across the sky, was not inconsistent with the evidence for dipole-like α-variation.


Even when imaged into 40 km s−1 channels, CO(2−1) absorption is apparent in NGC 4374 and IC 4296 against their strong nuclear continuum sources. While extragalactic CO absorption is not frequently observed due to low equivalent widths, ALMA’s high angular resolution and sensitivity allow its detection against some nearby active nuclei (e.g., Davis et al. 2014; Rangwala et al. 2015). To better characterize the central absorption features, we also imaged their visibility data (without continuum subtraction) into cubes with 1.28 km s−1 channels. The primary absorption in IC 4296 is consistent with the galaxy’s assumed systemic velocity (Table 1), although there may be a small secondary absorption feature ∼15 km s−1 redward of

. The CO(1−2) transition seen in the nucleus of NGC 4374, however, is redshifted by ∼100 km s−1 from the

 value derived using stellar kinematics (Emsellem et al. 2004) and by ∼50 km s−1 with respect to the best-fitting

 from ionized gas dynamical modeling (e.g., Walsh et al. 2010). The absorption lines in these two galaxies have narrow widths (≲7 km s−1 FWHM) that are only slightly smaller than the minimum line widths in our CO-bright subsample. Maximum absorption depths are (−3.26 ± 0.74) and (−5.25 ± 0.35) K with integrated absorption intensities

 of (19.4 ± 2.5) and (44.3 ± 1.4) K km s−1 for NGC 4374 and IC 4296, respectively. From these

 measurements, we estimate corresponding H2 column densities

 of

 and

 cm−2 after assuming an absorption depth ratio CO(1−2)/CO(0−1) ≈ 0.6 (based on 13CO absorption line ratios; e.g., see Eckart et al. 1990) and a CO-to-H2 conversion factor

 corresponding to

 M pc−2 (K km s−1)−1 (see Section 3.1.2; Sandstrom et al. 2013). These

 estimates are much more uncertain than indicated by the formal associated uncertainties, since the

 and CO(1−2)/CO(1−0) values carry large systematic uncertainties, and faint nuclear CO emission may somewhat dilute these equivalent widths.


In order to examine how our qcr(M1) would affect the brightness distribution of the violent merger scenario, we adopt the same assumptions as in Ruiter et al. (2013) except for qcr(M1). For the primary at the time of merging, we use the same WD mass distribution as Ruiter et al. (2013; see also Ruiter et al. 2011), although it is highly uncertain whether the WD can increase its mass by avoiding the formation of a common envelope during the very rapid accretion from the He star companion. We also assume the same

m

WD

–

M

bol

 relation of SNe Ia as Ruiter et al. (2013; see Figure 4 in Ruiter et al. 2013; Sim et al. 2010) and a flat mass ratio distribution of DD systems for simplicity. Using either our critical mass ratio or theirs, we calculated the brightness distribution and compared them with each other (Figure 11). We find that there is no significant difference between them qualitatively. This result implies that the qcr of the violent merger-induced explosion is not so crucial for the brightness distribution of SNe Ia, as mentioned in Ruiter et al. (2013). Our results are closest to the η = 1.5 case among Ruiter et al.'s three cases. In the cases of our qcr and Ruiter et al.'s η = 1.5, the brightness distribution concentrates around −19.0 mag. In Figure 11, we add the observational volume-limited brightness distributions of SNe Ia. One in panel (a) is derived by the Lick Observatory Supernova Search (LOSS; Li et al. 2011), and the other in panel (b) is obtained by ROTSE-IIIb (see Table 1 in Quimby et al. 2012). The fraction of faint events in our models is lower than the observation of LOSS. This discrepancy might decrease if we consider the viewing angle effects (see also the discussion in Ruiter et al. 2013), which might be increasingly important especially for a large value of qcr. On the other hand, the results of ROTSE-IIIb are consistent with ours, except most luminous (≲−19.5 mag) events. It should be noted, however, that there are large uncertainties for the observational brightness distribution (e.g., Quimby et al. 2012) and our models are too simple to be compared with the observational results. More detailed studies are required to reach a definitive conclusion.


Since its discovery in 1983, Phaethon has remained a well-studied yet poorly understood object, known to exhibit both cometary and asteroidal characteristics, and widely presumed to be the parent to the Earth-crossing Geminid meteor shower (Green & Kowal 1983; Gustafson 1989; Williams & Wu 1993). Significant efforts have been made to study the physical properties of the asteroid such as size (6 km in diameter, Taylor et al. 2019), spectral shape (B type, e.g., Licandro et al. 2007), and albedo (0.11, Green et al. 1985), as well as the Geminids (e.g., Hughes & McBride 1989; Jenniskens 1994; Arlt & Rendtel 2006; Blaauw 2017), their relationship to one another (e.g., Gustafson 1989; de León et al. 2010), and the proposed relationship to asteroids 2005 UD and 1999 YC (e.g., Ohtsuka et al. 2006, 2008; Kareta et al. 2021). Several Earth-based studies of Phaethon have sought to investigate the perihelion activity of the asteroid, attempting—but failing—to detect small fragments (Jewitt et al. 2018; Ye et al. 2018) or nearby dust grains (Jewitt et al. 2019) released during its close (0.14 au) passage by the Sun. Despite this, spacecraft observations of Phaethon’s activity at perihelion obtained by the Sun-Earth Connection Coronal and Heliospheric Investigation (Howard et al. 2008) on NASA’s Solar Terrestrial Relations Observatory (Kaiser et al. 2008) during the asteroid’s perihelion passages in 2009, 2012, 2016, and 2022, have shown that the asteroid is clearly active at perihelion, displaying a short tail and reported to be releasing mass on the order ∼2.5 × 108
a
1 kg (where a
1 is the grain radius in millimeters, and noting that the size of dust released at perihelion remains unknown; Jewitt & Li 2010; Jewitt et al. 2013; Hui & Li 2017). This is insufficient by orders of magnitude to support the best estimates for the mass of the Geminids. Shortly before submission of this manuscript, Phaethon was also observed by the Solar and Heliospheric Observatory Large Angle Spectrometric Coronagraph (Brueckner et al. 1995) C2 coronagraph during its 2022 May 15 perihelion passage, constituting the first known detection of Phaethon by this instrument.