For the stars below

, our Ca abundances are determined from the only Cai line detected, the 4226.73 Å resonance line. This line is known to produce lower Ca abundances than other Ca i lines for EMP stars, at least in part because of NLTE effects (e.g., Spite et al. 2012), leading T10 to dismiss the significance of the even lower [Ca/Fe] ratio they derived for Scl 07-50. However, NLTE models do not agree well on the correction for the 4226.73 Å line for stars with similar atmospheric parameters to Scl 07-50 and Scl 11_1_4296, with recent predictions ranging from −0.02 dex (Starkenburg et al. 2010) to +0.21 dex (Mashonkina et al. 2007; L. Mashonkina 2014, private communication). We therefore attempted several additional tests to verify the low Ca abundances. First, we compared the Ca lines of both stars with those of the ultra metal-poor giant CD

 245 (Bessell  Norris 1984), which is comparable in temperature to Scl 11_1_4296 and ∼200 K warmer than Scl 07-50 (see Figure 2). Scl 11_1_4296 has weaker Ca i λ4226.73 Å and near-infrared Ca ii triplet lines than CD

 245, confirming its low Ca abundance. Scl 07-50 has similar Ca K and Cai λ4226.73 Å EWs to CD

 245, consistent with a lower Ca abundance given the temperature difference.5

5
The Ca triplet lines of Scl 07-50 are stronger than those of CD

 245, but NLTE and 3D corrections for those lines are much larger and even less well understood.
 Second, we compared to stars with similar parameters (after adjusting their spectroscopic temperatures according to the Frebel et al. 2013 formula) from Roederer et al. (2014). These stars were selected to have 4400 K

 4900 K,

,

, and a detection of the Ca i resonance line, resulting in a sample of 13 stars. For this sample, Ca abundances from Ca i 4226.73 Å are 0.09 dex lower than the mean abundance from all other Ca i lines.6

6
The only outlier where Ca i 4226.73 Å and the other lines have a significantly larger abundance difference is the coolest star, CS 22950–046, which is similar in temperature to Scl 07-50, raising the possibility of a sharp temperature dependence in the abundance derived from the resonance line. However, because the theoretical studies of Mashonkina et al. (2007) Merle et al. (2011), and Spite et al. (2012) do not indicate strong changes in NLTE corrections at

 K, we regard CS 22950–046 as a random outlier rather than a systematic one.
 This offset is in excellent agreement with the most recent NLTE corrections for Ca determined by Spite et al. (2012), which give 0.08 dex for

 K,

,

. Finally, we stacked the four strongest Ca i non-resonance lines and determined upper limits of

 dex and

 dex for Scl 07-50 and Scl 11_1_4296, respectively, by comparing to synthesized spectra. We therefore conclude that even after factoring in the uncertain NLTE effects, Scl 07-50 and Scl 11_1_4296 indeed have low Ca abundances.


Therefore, we now fit the two pn spectra between 3 and 10 keV using a power-law model with Galactic absorption, a narrow neutral Fe Kα core (En = 6.4 keV and σn = 0 eV), and a disc line model (RELLINE; Dauser et al. 2010, 2013) to account for the mildly broad line. For the third component, the energy was fixed at 6.4 keV and the spin at zero (since it is not constrained). All parameters of the Fe Kα narrow component and the emissivity index for the mild relativistic Fe K line component were tied between the two epochs. From this modelling (χ2/d.o.f. = 585.4/668), we infer an accretion disc emissivity index lower than 2.1 and a disc inclination lower than 24.6°, which is consistent with the inclination angle inferred from the variable O VII soft X-ray emission line (

θ
=
9
.

9

−
1.4

+
1.0

$ \theta=9.9^{+1.0}_{-1.4} $

) arising from the accretion disc too (Reeves et al. 2021a). For the narrow core of the Fe Kα line, we measure a EWn ≲ 20 eV, which is in the lower range of the values found for type-1 AGN (∼30−200 eV; Liu & Wang 2010; Shu et al. 2010; Fukazawa et al. 2011; Ricci et al. 2014). This is consistent with the very weak covering factor of 0.06 for the putative torus, which is inferred from the ratio of the infrared to the bolometric luminosity of the source (Ezhikode et al. 2017). Moreover, the torus covering factor measured for Mrk 110 is one of the lowest reported in the Ezhikode et al. (2017) sample; this is also in agreement with the lack of Compton hump. The equivalent widths of the moderately broad Fe Kα line are EWb = 61

−
27

+
23

$ ^{+23}_{-27} $

 eV and EWb = 43

−
27

+
22

$ ^{+22}_{-27} $

 eV for 2019 and 2020, respectively. Likewise, if we allow the inner disc radius to be free to vary and assume a standard value of three for the disc emissivity index, as well as fixing the inclination angle at 9.9° (χ2/d.o.f. = 585.1/668), we infer Rin = 120

−
67

+
263

$ ^{+263}_{-67} $

Rg. This strengthens that the moderately broad Fe line could originate from the accretion disc but not too close to the inner stable circular orbit (ISCO), as was found for the O VII soft X-ray emission line (Reeves et al. 2021a). Its origin from the outer accretion disc is also consistent with the timing analysis reported in Lobban et al. (in prep.) where a hint of an extra variability is found at the Fe Kα complex energy range in the fractional variability spectra.


When B
X is positive (see the center gray area in Figure 2(a), and corresponding area in Figures 2(e)–(f)), electrons at two energy ranges both present isotropic PAD. As early as 1972, Eastwood et al. (1972) have pointed out that a strong field line curvature would scatter particles. Such a nonadiabatic mechanism has been studied successfully. Büchner and Zelenyi 1989 were the first to use the curvature parameter κ
2 = R
min/ρ
max, the ratio between the minimum curvature radius R
min and the maximum particle Larmor radius ρ
max, to carry out the trapped particle motion in magnetic field reversals with arbitrary curvature radii in 1989. The parameter depends on the total kinetic energy of the particles and the field geometry, followed by the particle adiabatic invariants. κ ≫ 1 corresponds to the usual adiabatic case. As κ decreases toward unity, the particle motion becomes stochastic due to deterministic chaos. Such behavior, which for κ ≈ 1 becomes strongly chaotic, applies, e.g., to thermal electrons in Earth's magnetotail and makes its collisionless tearing mode instability possible. While for κ  1, it corresponds to the quasi-adiabatic regime, where new current sheet equilibriums can be built (e.g., Zelenyi et al. 2004). We use the curvature radius of magnetic field line R
C (Figure 2(a)), electron cyclotron radius R
e (Figure 2(a)) at energy range 6.52–8600 eV separated into two small channels, which is marked by color lines, and the improved adiabatic parameter 1/κ
2 = R
e/R
C, the ratio between R
e and R
C (Figure 2(c)), to investigate and approve the nonadiabatic motion of electrons (Büchner & Zelenyi 1989; Delcourt et al. 2017). Understandably, R
C of the magnetic field line in the current sheet is much smaller, showing a strongly curved shape, seen in Figure 2(b). R
e of electrons is much larger than outside, and the higher-energy range electrons have the larger R
e, seen in Figure 2(c). Typically, R
e of electrons and R
C of the magnetotail field line differ by 3 orders of magnitude (see the red lines in Figures 2(d), 4(d), (i), (m), and (r)). Electrons would be scattered inside the current sheet when the R
e and R
C can be compared. Lavraud et al. (2016) have observed a similar pitch-angle scattering process at the magnetopause during a reconnection event, and pointed out that the small curvature radius is the reason by using the κ parameter to analyze first electron MMS data. Thus, when the spacecraft start entering the current sheet, electrons would be scattered along the strongly curved magnetic lines (see the area marked by the first and fourth dotted lines in Figure 2(f)). Meanwhile, high-energy electrons are more easily scattered than low-energy electrons, as electrons at 6.52–35 eV are only scattered at the center of the current sheet (see the area marked by the second and third dotted lines in Figure 2(e)).


The electron-positron creation process may induce dynamical instability during the most advanced phases of massive star evolution. Pair creation absorbs part of the thermal energy of the plasma, consequently lowering the thermal pressure. The plasma is not a perfect gas anymore, and temperature variations do not lead to changes in pressure (Kippenhahn, Weigert & Weiss 2012). Regions in which this process happens become locally dynamically unstable. To check if a star is globally stable or not, a perturbation method should be adopted (Ledoux & Walraven 1958). However, Stothers (1999) showed that an evaluation of the first adiabatic exponent properly weighted and integrated over the whole star, 〈Γ1〉, is a very good approximation to determine the dynamical stability of a star. As done by other authors (e.g. Marchant et al. 2019; Farmer et al. 2019, 2020), we adopted the Stothers (1999) stability criterion, which states that a star is stable if
(4)$$\begin{eqnarray*}
\mbox{$\langle \Gamma _{1} \rangle _{\rm }$} = \frac{\int ^{M}_0 \frac{\Gamma _1 P}{\rho } dm}{\int ^{M}_0 \frac{P}{\rho } dm} \gt \frac{4}{3},
\end{eqnarray*}$$where Γ1 is the first adiabatic exponent, P is the pressure, ρ is the density and dm is the element of mass. We decided to compute the above integrals in two ways. In the first case, the integral is calculated from the centre of the star (M = 0) up to the mass of helium core (M = MHe), defining 〈Γ1〉Core; in the second case, we compute the integral from the centre to the surface of the star (M = MTOT), to include the contribution of the envelope, thus 〈Γ1〉TOT. These two values are computed at each time-step for all tracks. In the case of pure-He stars, 〈Γ1〉Core = 〈Γ1〉TOT. Since parsec is a hydro-static code, we cannot follow the evolution through the dynamical collapse; we stop the computation if the 〈Γ1〉 4/3 + 0.01, to be conservative, and label the star as a PI. Since we cannot follow the hydrodynamical evolution, we cannot distinguish between PPI and PISN: we classify both of them as PI and we assume that both of them leave no compact object. This makes our results for the BH mass even more conservative, because stars that undergo a PPI might still retain most of their mass after weak pulses and form a massive BH by core collapse ( Farmer et al. 2019; Marchant et al. 2019).


In regions of high C/O, planets form primarily from carbonates, and in regions of low C/O, the Mg/Si determines the types of silicates that dominate the compositions (e.g., Brewer et al. 2016, and references therein). This means that the C/O ratio controls the distribution of Si among carbide and oxide species. If C/O is greater than 0.8, Si exists in solid form primarily as SiC, and also graphite and TiC will be formed; for C/O ratios below 0.8, Si is present in rock-forming minerals as ${\rm{SiO}}_4^{4 - }$SiO44− or SiO2, serving as seeds for Mg silicates for which the exact composition will be controlled by the Mg/Si value (Bond et al. 2010; Thiabaud et al. 2015b). Moreover, Thiabaud et al. (2015a) have shown that the condensation of volatile species as a function of radial distance allows for C/O enrichment in specific parts of the protoplanetary disk of up to four times the solar values, leading to the formation of planets that can be enriched in C/O in their envelope up to three times the solar value. This is the case of HD209458b observed by Giacobbe et al. (2021) for which a scenario of planet formation beyond the water snowline and migration toward its host star through disk or disk-free migration was hypothesized (see also Brewer et al. 2017). At the same time, Mg/Si governs the distribution of silicates: for Mg/Si  1, Mg forms orthopyroxene (MgSiO3) and the excess Si is present as other silicate species such as feldspars (CaAl2Si2O8, NaAlSiO8) or olivine (Mg2SiO4); for Mg/Si values ranging from 1 to 2, Mg is distributed between olivine and pyroxene; for Mg/Si > 2, all available Si is consumed to form olivine with excess Mg available to bond with other minerals, mostly oxides such as MgO or MgS (Bond et al. 2010; Thiabaud et al. 2015b). The peak of the Mg/Si-C/O distribution for our targets is therefore consistent with Si which will take solid form as ${\rm{SiO}}_4^{4 - }$SiO44− and SiO2 and Mg equally distributed between pyroxene and olivine.


Adaptive optics (AO; Beckers 1993) has become a key technology for all the main existing telescopes and is considered an enabling technology for future Extremely Large Telescopes (ELTs; Ramsay et al. 2016; Boyer 2018; Bouchez et al. 2018). AO requires sufficiently bright reference sources for the measurement of the wavefront (WF) distortions introduced by the atmospheric turbulence: Sodium laser guide stars (LGS; Foy & Labeyrie 1985; Tallon & Foy 1990) are currently the only valid option to increase the low sky coverage allowed by the use of natural stars (Ellerbroek & Andersen 2008). A Sodium LGS is generated by a laser beam at 589 nm wavelength that excites the Sodium atoms concentrated in a ∼13 km thick layer at about 90 km above the ground (Pfrommer & Hickson 2014), returning a photon flux at the same wavelength (Happer et al. 1994; Thomas et al. 2008). The Sodium layer vertical extension and its variable density profile (Pfrommer, Hickson & She 2009; Holzlöhner et al. 2016) lead to some peculiar complications of the WF sensing process (Lombini 2011). The Shack–Hartmann WF Sensor (SHWS) is the current baseline for the LGS WF sensors (WSs) for the future ELT AO instruments  (Boyer & Ellerbroek 2016; Stuik et al. 2016; Males et al. 2018; Sauvage et al. 2018). The LGS spots appear elongated in the sub-apertures located off-axis with respect to the upward laser beam, due to the parallactic effect; the elongation points at the laser launch location and its amplitude increases with the distance from the launcher itself (Fig. 1). The LGS angular elongation ϵ can be approximated by
(1)$$\begin{eqnarray}
\epsilon \approx r \frac{\Delta H}{H^2} \text{cos}(\xi) ,
\end{eqnarray}$$where r is the sub-aperture distance from the launcher, H is the Sodium layer mean altitude, ΔH is the layer vertical extension, and ξ is the Zenith angle. The spot position angle ω is
(2)$$\begin{eqnarray}
\omega = \text{atan}\left(\frac{y_l-y_{s}}{x_l - x_{s}}\right) ,
\end{eqnarray}$$


The energy partition study of Emslie et al. (2012) was restricted to 38 large solar eruptive events. In a more comprehensive study of the global flare energetics we choose a data set that contains the 400 largest (GOES M- and X-class) flare events observed during the first 3.5 years of the SDO era. Previously, we determined the dissipated magnetic energies Emag in these flares based on fitting the vertical-current approximation of a nonlinear force-free field (NLFFF) solution to the loop geometries detected in EUV images from SDO/AIA, a new method that could be applied to 177 events with a heliographic longitude of

 (Paper I). We also determined the thermal energy Eth in the soft X-ray and EUV-emitting plasma during the flare peak times based on a multi-temperature DEM forward-fitting method to SDO/AIA image pixels with spatial synthesis, which was applicable to 391 events (Paper II). In the present study, we determined the nonthermal energy Ent contained in accelerated electrons based on spectral fits to RHESSI data using the OSPEX software, which was applicable to 191 events. The major conclusions of the new results emerging from this study are as follows.

1.
The (logarithmic) mean energy ratio of the nonthermal energy to the total magnetically dissipated flare energy is found to be

, with a logarithmic standard deviation corresponding to a factor of

, which yields an uncertainty of

 for the mean, i.e.,

. The majority (

) of the flare events fulfill the inequality

, which suggests that magnetic energy dissipation (most likely by a magnetic reconnection process) provides sufficient energy to accelerate the nonthermal electrons detected by bremsstrahlung in hard X-rays. Our results yield an order of magnitude higher electron acceleration efficiency than previous estimates, i.e.,

 (with N = 37, Emslie et al. 2012).

2.
The (logarithmic) mean of the thermal energy Eth to the nonthermal energy Ent is found to be

, with a logarithmic standard deviation corresponding to a factor of

. The fraction of flares with thermal energy smaller than the nonthermal energy, as expected in the thick-target model, is found to be the case for

 only. Therefore, the thick-target model is sufficient to explain the full amount of thermal energy in most flares, in the framework of the warm-target model. The cross-over method shows the opposite tendency, but we suspect that the cross-over method overestimates the low-energy cutoff and underestimates the nonthermal energies. Previous estimates yielded a similar ratio, i.e.,

 (Emslie et al. 2012).

3.
A corollary of the two previous conclusions is that the thermal to magnetic energy ratio is

. A total of 95% of flares fulfills the inequality

, indicating that all thermal energy in flares is supplied by magnetic energy. Previous estimates were a factor of 17 lower, i.e.,

 (Emslie et al. 2012), which would imply a very inefficient magnetic to thermal energy conversion process.

4.
The largest uncertainty in the calculation of nonthermal energies, the low-energy cutoff, is found to yield different values for two used methods, i.e.,

 keV for the warm thick-target model, versus

 for the thermal/nonthermal cross-over method. The calculation of the nonthermal energies is highly sensitive to the value of the low-energy cutoff, which strongly depends on the assumed (warm-target) temperature.

5.
The flare temperature can be characterized with three different definitions, for which we found the following (67%-standard deviation) ranges:

 MK for the AIA DEM peak temperature,

 MK for the emission measure-weighted temperatures, and

 MK for the RHESSI high-temperature DEM tails. The median ratios are found to be

 and

. The mean active region temperature evaluated from DEMs with AIA, Te = 8.6 MK, is used to estimate the low-energy cutoff ec of the nonthermal component according to the warm-target model, i.e.,

. The low-energy cutoff ec of the nonthermal spectrum has a strong functional dependence on the temperature TR.


Under the symmetry assumption of Friedmann–Lemaître cosmologies, all fluids are characterized by their density and their equation of state. In spatially flat cosmologies with the matter density parameter Ωm and the corresponding dark energy density 1 − Ωm one obtains for the Hubble function $H(a)=\dot{a}/a$ the expression,
(1)$$\begin{eqnarray*}
\frac{H^2(a)}{H_0^2} = \frac{\Omega _\mathrm{m}}{a^{3}} + \frac{1-\Omega _\mathrm{m}}{a^{3(1+w)}},
\end{eqnarray*}$$The comoving distance χ is related to the scale factor a through
(2)$$\begin{eqnarray*}
\chi = -c\int _1^a\:\frac{\mathrm{d}a}{a^2 H(a)},
\end{eqnarray*}$$where the Hubble distance χH = c/H0 sets the distance scale for cosmological distance measures. Small fluctuations δ in the distribution of dark matter grow, as long as they are in the linear regime |δ| ≪ 1, according to the growth function D+(a) (Linder & Jenkins 2003),
(3)$$\begin{eqnarray*}
\frac{\mathrm{d}^2}{\mathrm{d}a^2}D_+(a) + \frac{2-q}{a}\frac{\mathrm{d}}{\mathrm{d}a}D_+(a) - \frac{3}{2a^2}\Omega _\mathrm{m}(a) D_+(a) = 0,
\end{eqnarray*}$$and their statistics is characterized by the spectrum $\langle \delta (\boldsymbol {k})\delta ^*(\boldsymbol {k}^\prime)\rangle = (2\pi)^3\delta _D(\boldsymbol {k}-\boldsymbol {k}^\prime)P_\delta (k)$. Inflation generates a spectrum of the form $P_\delta (k)\propto k^{n_s}T^2(k)$ with the transfer function T(k) (Bardeen et al. 1986) which is normalized to the variance σ8 smoothed to the scale of 8 Mpc h−1,
(4)$$\begin{eqnarray*}
\sigma _8^2 = \int _0^\infty \frac{k^2\mathrm{d}k}{2\pi ^2}\: W^2(8 \mathrm{Mpc}/h\times k)\:P_\delta (k),
\end{eqnarray*}$$with a Fourier-transformed spherical top-hat W(x) = 3j1(x)/x as the filter function. From the CDM spectrum of the density perturbation, the spectrum of the dimensionless Newtonian gravitational potential Φ can be obtained,
(5)$$\begin{eqnarray*}
P_\Phi (k) \propto \left(\frac{3\Omega _\mathrm{m}}{2 (k\chi _H)^2}\right)^2\:P_\delta (k),
\end{eqnarray*}$$by applying the comoving Poisson equation $\Delta \Phi = 3\Omega _\mathrm{m}/(2\chi _H^2)\delta$ for deriving the gravitational potential Φ (in units of c2) from the density δ. Because our analysis relies on the assumption of Gaussianity, we need to avoid non-linearly evolving scales and will restrict our analysis to large angular and spatial scales, where the cosmic density field can be approximated to follow a linear evolution, conserving the near-Gaussianity of the initial conditions. We increase the variance of the weak lensing signal and of the intrinsic alignment signal of elliptical galaxies on small scales because of non-linear structure formation using the description of Smith et al. (2003) and Casarini et al. (2011, 2012). Consequently, the cross-correlation between weak lensing and elliptical galaxy shapes will likewise have increased variances on small scales. Shapes of spiral galaxies are set by the initial conditions of structure formation; therefore, we did not apply changes to the linear CDM-spectrum P(k).


(3) Extended cold-accretion model. The actual behavior of cold accretion will be more complicated than suggested by the analytical work of Dekel & Birnboim (2006), which investigated the behavior of mean halos under many simplifications. One possibility is that cold-accretion behavior depends upon the environment. Galaxies with the same halo mass may experience different accretion histories depending on whether they reside in highly crowded regions such as proto-galaxy clusters or in sparse regions that will eventually develop into fields or voids. Keres et al. (2005) used detailed numerical simulation to show that the intensity and time variation of cold-mode accretion indeed depends on the galactic environment. According to their results, there is a tendency for cold-mode accretion to persist into lower redshifts in the region of lower galaxy number densities. Keres et al. (2005) stated that this environmental dependence is driven largely by the galaxy mass dependence of the cold–hot accretion fraction, but they also recognized a truly environmental effect that massive galaxies in low-density environments tend to have higher cold-accretion fractions than those in denser regions. Because disk galaxies, especially those with larger stellar masses (

), tend to reside in low-density regions such as fields and poor groups (Blanton & Moustakas 2009), this effect may be important in modeling the evolution of such galaxies. We therefore implement the third type of cold accretion. Because the evaluation of cold accretion by Dekel & Birnboim (2006) is based on many simplified assumptions that make the problem analytically amenable, it is considered that there remains a considerable quantitative ambiguity. In view of this, we take a simple approach in specifying the cosmological history of cold accretion. Specifically, we parameterize the third type of cold accretion using four parameters: the characteristic time,

, when cold accretion gives way to hot accretion; the characteristic mass,

, which divides hot and cold accretion at

; and two parameters,

 and

, that describe the change in the characteristic mass before and after

, respectively. In this parameterization, the critical mass is specified as
18

19

If

, the halo gas is assumed to collapse to the disk plane with the freefall time (

). Otherwise, the gas is assumed to accrete over the larger of the cooling and freefall times (

). This specification, depending on the adopted parameters, accommodates a wide variety of accretion histories. We set

,

, and

 throughout this paper. This specification of parameters moves the end of cold accretion to more recent epochs than the prediction of Dekel & Birnboim (2006) for halos with

 and realizes the prolonged cold accretion suggested in Keres et al. (2005). Under this parameterization, the model gives the best match to the observed disk sizes (Figure 7) and galaxy colors (Figure 9) without changing other observational predictions to a significant degree.


One of the independent measures to the EM luminosity distance ${\rm d}_l^{\rm EM} (z)$ for a metric theory is through its relation with the geometric distance (the angular diameter distance) ${\rm d}_A(z)= {\rm d}_l^{\rm EM}(z)/(1+z)^2$, according to Etherington’s reciprocity theorem (Etherington 2007) or the distance duality relation. This relation is valid for EM probes if photon number is conserved, and photons propagate along null geodesics. The distance duality relation has been tested from several observations (Holanda, Lima & Ribeiro 2010; Holanda, Gonçalves & Alcaniz 2012; Liao et al. 2016) and will also be tested more stringently in the future (Liao et al. 2016; Arjona et al. 2020; Martinelli et al. 2020; Renzi et al. 2020). The angular diameter distance to any redshift z is related to the BAO scale θBAO in the matter correlation function by (Peebles & Yu 1970; Bond & Efstathiou 1984; Hu & Sugiyama 1996; Eisenstein & Hu 1998, 1997)
(4)$$\begin{eqnarray*}
\theta _{\rm BAO} (z)= \frac{r_{\rm s}}{(1+z){\rm d}_A(z)},
\end{eqnarray*}$$where, $r_{\rm s}= \int _{z_{\rm d}}^\infty {\rm d}z c_{\rm s}(z)/ H(z)$ is the sound horizon where zd denotes the drag redshift. As a result, we can relate the BAO scale with the EM luminosity distance ${\rm d}_l^{\rm EM}$ by the relation
(5)$$\begin{eqnarray*}
{\rm d}_l^{\rm EM} (z)= \frac{(1+z) r_{\rm s}}{\theta _{\rm BAO}(z)}.
\end{eqnarray*}$$Using equation (5) in equation (3), we can write
(6)$$\begin{eqnarray*}
{\rm d}^{\rm GW}_l(z)= \exp {\bigg (-\int {\rm d}z^{\prime } \frac{\gamma (z^{\prime })}{1+z^{\prime }}\bigg)}\frac{(1+z) r_{\rm s}}{\theta _{\rm BAO} (z)}.
\end{eqnarray*}$$This is the key equation of this paper. In this expression, the measurement of the term $\frac{r_{\rm s}}{\theta _{\rm BAO}(z)}$ comes from EM probes such as large-scale structure galaxy redshift surveys (Eisenstein et al. 2005; Dawson et al. 2013; Alam et al. 2017) and cosmic microwave background (CMB) (Spergel et al. 2003, 2007; Komatsu et al. 2011; Hinshaw et al. 2013; Planck Collaboration XIII 2016; Aghanim et al. 2020), and the measurement of $d_l^{GW}$ arises from the GW strain. We can write the above equation as
(7)$$\begin{eqnarray*}
{\rm d}^{\rm GW}_l(z)\theta _{\rm BAO}(z)= \exp {\bigg (-\int {\rm d}z^{\prime } \frac{\gamma (z^{\prime })}{1+z^{\prime }}\bigg)}(1+z) r_{\rm s}.
\end{eqnarray*}$$This relation shows that the product of the BAO angular scale θBAO(z) and the luminosity distance ${\rm d}^{\rm GW}_l(z)$ can measure the frictional term γ(z) as a function of redshift. So, the concordance between the EM geometric probes and the GW luminosity probes allows a way to test the theory of gravity. If the general theory of relativity is the correct theory of gravity, then the product between θBAO(z) and ${\rm d}^{\rm GW}_l(z)$ should vary with redshift as (1 + z). Any deviation from this scaling can be a signature of alternative theories of gravity. The quantities θBAO and ${\rm d}_l^{\rm GW}$ are measured from large-scale structure and GW data, and the value of rs depends on recombination physics and the sound speed in the baryon–photon fluid at the time of decoupling at redshift z ≈ 1100 (Silk 1968; Peebles & Yu 1970; Sunyaev & Zeldovich 1970; Hu, Sugiyama & Silk 1997). As a result, this relation is nearly model-independent, and can be written directly in terms of observables such as ${\rm d}_l^{\rm GW}(z)$, and θBAO(z).


Figure 3 shows the column densities of H2, c-C3H2 and CH3OH extracted along the dotted line present in Fig. 1, as well as the N(H2) calculated with the model of L1544 described in Keto & Caselli (2010) assuming a beam size of 40′′. The column densities of c-C3H2 and CH3OH were calculated assuming that the lines are optically thin, using the formula given in Eq. (1) of Spezzano et al. (2016). We assumed a Tex of 6 K for c-C3H2 and 6.5 K for CH3OH, as done in Spezzano et al. (2013) and Bizzocchi et al. (2014), respectively. The same works reported that both lines present moderate optical depths (τ  0.4). This plot shows that the decrease of N(H2) towards the south-west, where cyclopropenylidene is more abundant, is much steeper than towards the north-east, where methanol is more abundant. A full map of the N(c-C3H2)/N(CH3OH) column density ratio can be seen in Fig. A.1, showing a clear peak towards the south-east side of L1544. Figure 3 also shows that c-C3H2 and CH3OH both belong to the same dense core (identified by the brightest peak in N(H2)), but trace different parts of it. We obtain the same result by comparing the line-width and velocity maps of the two molecules, see Fig. B.1. Despite the different spatial distributions, the two molecules trace the same kinematic patterns (velocity gradient and amount of non-thermal motions). This indicates that the velocity fields are dominated by the bulk motions (gravitational contraction and rotation) of the prestellar core, which similarly affect the two sides of the core and do not depend on the chemical composition of the gas. In summary, c-C3H2 and CH3OH trace different parts of the same dense core and no velocity shift is present (unlike for the L1551 case; Swift et al. 2006). We note that the observed transitions of c-C3H2 and CH3OH have relatively high critical densities (between a few ×104 and 106 cm-3), so that these lines are not expected to trace the more diffuse filamentary material surrounding the prestellar core.


Figure 1 shows a schematic of the Planetary Haze Research (PHAZER) experimental setup at Johns Hopkins University (He et al. 2017; Hörst et al. 2018). The initial gas mixtures for our experiments are calculated from the chemical-equilibrium models of Moses et al. (2013), who examined the possible thermochemistry and photochemistry in the atmospheres of Neptune-sized and sub-Neptune-sized exoplanets. Most of the known super-Earth and mini-Neptune size planets known to date were discovered by the Kepler mission (Borucki et al. 2011; Fressin et al. 2013). However, follow-up of Kepler targets to determine planetary mass and atmospheric characterization has been limited due to the brightness of their host stars. The TESS mission (Ricker et al. 2014) will target stars that are 10–100 times brighter than those targeted with the Kepler mission and yield a large sample of super-Earth and mini-Neptune class planets with equilibrium temperatures (Teq) 1000 K (Sullivan et al. 2015). This planetary size (Rp  4 REarth) and temperature (Teq  1000 K) phase space has been recently explored by exoplanet atmospheric characterization efforts for a handful of planets such as GJ 1214b, HD 97658b, GJ 436b, and GJ 3470b, which all indicate that aerosols are shaping the planetary spectrum (e.g., Knutson et al. 2014a, 2014b; Kreidberg et al. 2014; Dragomir et al. 2015). Theoretical studies predict a diverse range of possibilities for the composition of atmospheric envelopes of mini-Neptunes and super-Earths (e.g., Fortney et al. 2013; Howe & Burrows 2015; Wolfgang & Lopez 2015). Broadly speaking, we expect that our hydrogen-rich gas mixtures are more relevant to larger mini-Neptune class planets and carbon dioxide-rich gas mixtures are more relevant to smaller super-Earth class planets, with water-rich gas mixtures being relevant to both classes of planets. However, further observational, theoretical, and laboratory work will be required to confirm or refute the potential diversity in atmospheric compositions for these classes of planets that do not exist in our solar system. The equilibrium calculations relevant to this investigation were performed at conditions of 300, 400, and 600 K at 1 mbar for 100×, 1000×, and 10,000× solar metallicity. Smaller planets are less efficient at accreting the H- and He-rich gas from the protoplanetary disk, and the lighter elements have a greater chance of escaping from smaller planets, which enhances the overall metallicity of planets that are Neptune-sized and smaller. As an example, the O/H metallicity on Uranus and Neptune is estimated to be 160 and 540 times solar, respectively (Cavalié et al. 2017). Various evolutionary processes could drive the relative abundances of the heavy elements away from solar, but these high-metallicity chemical-equilibrium models provide a reasonable starting point for our study. As listed in Table 1, we included gases with a calculated abundance of 1% or higher to maintain a manageable level of experimental complexity; this resulted in no nitrogen-bearing species in two cases and the exclusion of sulfur-bearing species. Sulfur-bearing species, like H2S, could be important for haze formation (Gao et al. 2017) and will be an avenue of future work. The pressure, temperature, and gas compositions used in the experiments are self-consistent based on the model calculations.


Recently, Sezestre et al. (2017) investigated this scenario in a quantitative way, using numerical simulations with test particles. The model is relatively straightforward and makes a minimal number of assumptions. It assumes grains released at a given distance of the star andat several epochs. The dynamical behavior of the grains is ruled by the ratio of the pressure to gravitational forces (β). In the case of AU Mic, the stellar wind exceeds the radiation pressure, so that β scales with the stellar mass loss. The authors considered two situations, one where the source of dust was fixed in the system, and one where it was in Kepleriancircular orbit. To match the positions of the features as well as their projected speeds, the grains must be characterized by high β values of atleast 6, which points to a combination of small grains (a few tenths of m) and large stellar mass loss (a few hundreds of

$\dot{M_{\odot}}$

M
⊙

.

), which is in rather good agreement with previous estimations (Fitzgerald et al. 2007; Schüppler et al. 2015). Although several geometries (position of the source, direction of emission, and periodicity) can still match the observations given that only three epochs were available, the fit of the model to the data indicates that the source of this dust production should be located at ~ 8–28 au from the star, that is, within the belt of planetesimals, the “birth ring”. To introduce the aim of this paper, we present in Fig. 1 a sketch of the hypothetical distribution of the features in the system as pictured from the first observations (Boccaletti et al. 2015), and we assume that they originate from an orbiting parent body. An alternative explanation is proposed by Chiang & Fung (2017), in which the fast-moving features would originate from a collisional avalanche site at the intersection of two belts: the main belt, and a belt resulting from the catastrophic disruption of a large asteroid-likebody. Small dust particles would be released from this place and expelled by the stellar wind. This theory involves several assumptions to qualitatively match the observations, but the dynamical behavior of the grains once released is quite similar to the case of a fixed source as proposed in Sezestre et al. (2017). From the photometry of the HST image, Chiang & Fung (2017) roughly estimated that one dust feature should contain ~ 4 ×10−7 M⊕.


The observed X-ray emission from AGNs may be modified by line-of-sight obscuration, resulting in lower observed X-ray fluxes than those expected from the α
OX–L
2500 Å relation. A common approach to parameterize the amount of X-ray weakness uses the Δα
OX parameter, defined as the difference between the observed α
OX value and the α
OX value expected from the α
OX–L
2500 Å relation; Δα
OX = − 0.3838 thus corresponds to a factor of X-ray weakness of 10 at rest-frame 2 keV. Type 2 AGNs are generally X-ray obscured, likely due to the dusty “torus” (e.g., Netzer 2015; Hickox & Alexander 2018). Type 1 AGNs may also have X-ray obscuration from largely dust-free gas.
13

13
Similar obscuration from dust-free gas might also be present in some of the type 2 AGNs, though usually not distinguishable from the torus obscuration. For example, broad absorption line (BAL) quasars, which are characterized by blueshifted broad UV absorption lines (e.g., C iv
λ1549), generally show weak X-ray emission (e.g., Gallagher et al. 2002, 2006; Fan et al. 2009; Gibson et al. 2009). One frequently adopted physical model for BAL quasars is the disk wind model, where the observed BALs originate from an outflowing equatorial wind launched from the accretion disk and radiatively driven by UV-line pressure (e.g., Murray et al. 1995; Proga et al. 2000; Matthews et al. 2016). This model usually invokes “shielding” gas between the wind and nucleus or a clumpy wind (e.g., Baskin et al. 2014; Matthews et al. 2016; Giustini & Proga 2019) to provide obscuration of the nuclear extreme-UV (EUV) and X-ray radiation, which might otherwise overionize the wind and hamper radiative acceleration. BAL quasars are considered generally to have larger inclination angles than non-BAL quasars, with the line of sight to the UV continuum region of the accretion disk intersecting the wind, leading to the observed BALs. The line of sight to the X-ray-emitting corona, though not necessarily the same as the UV line of sight, is likely also through the shielding gas or the clumpy wind, resulting in the often-observed X-ray weakness (e.g., Figure 1 of Luo et al. 2013). Besides BAL quasars, a small fraction (5.8% ± 0.7%) of non-BAL type 1 quasars have been found to be X-ray weak, likely due to absorption (e.g., Pu et al. 2020). They may share a similar nature to the BAL quasars; they do not show any UV BALs, probably due to geometric effects (e.g., small inclination angles) or a low velocity of the wind along the UV line of sight (e.g., Giustini & Proga 2019).


From these two observational campaigns, the inference on the nature of Sgr A* has been reached on the ground of novel theoretical understandings. A recent important result has been obtained in Matsumoto, Chan & Piran (2020) (see also Tursunov et al. 2020), by re-considering the flare emissions around Sgr A*, emphasizing both, that their motion is not purely geodesic, and establishing a limit on the spin of a putative Kerr BH mass of |a|  0.5. Soon after, in Becerra-Vergara et al. (2020), it was introduced an alternative model to the classical BH in Sgr A* by re-interpreting it as a high concentration of quantum self-gravitating DM made of fermions of about 56 keV c−2 rest mass. This alternative approach can explain the astrometric data of both the S2-star and the G2 object with similar accuracy than the Schwarzschild BH scenario, but without introducing a drag force on G2 which is needed in the BH case to reconcile it with the G2 post-pericentre passage velocity data. An underlying assumption about the nature of such a DM quantum core is its absence of rotation, which is well supported by recent upper bounds on the spin of the central BH of a  0.1, based on the spatial distribution of the S-stars (Ali et al. 2020; Fragione & Loeb 2020; Peißker et al. 2020). The first results obtained in Becerra-Vergara et al. (2020) within the DM scenario have been further extended in Becerra-Vergara et al. (2021) by considering the 17 best resolved stars orbiting Sgr A*, achieving an equally good fit than in the BH paradigm. Remarkably, such a dense DM core is the central region of a continuous distribution of DM whose diluted halo explains the Galactic rotation curves (Argüelles et al. 2018; Becerra-Vergara et al. 2020, 2021). Core-halo DM distributions of this kind are obtained from the solution of the Einstein equations for a self-gravitating, finite-temperature fluid of fermions in equilibrium following the Ruffini–Argüelles–Rueda (RAR) model (Ruffini, Argüelles & Rueda 2015; Argüelles et al. 2016, 2018, 2019, 2021; Gómez et al. 2016; Gómez & Rueda 2017; Becerra-Vergara et al. 2020, 2021; Penacchioni et al. 2020; Yunis et al. 2020). These novel core-halo DM profiles, as the ones applied in this Letter, have been shown to form and remain stable in cosmological time-scales, when accounting for the quantum nature of the particles within proper relaxation mechanisms of collisionless fermions (Argüelles et al. 2021). There are other alternative scenarios for Sgr A* involving a compact object of quantum nature, e.g. a boson star composed of ultralight scalars (see e.g. Torres et al. 2000). However, unlike the ∼50–345 keV c−2 fermionic DM RAR solutions, those boson stars do not explain the Galaxy rotation curves.


Argon (Ar) is also one of the most abundant rare gases in the Universe and Ar ion lines are observed in solar and astrophysical plasmas as well as in the laboratory. It is observed that the spectra of early B-type stars contain a large number of unblended Ar ii lines (Keenan et al. 1990; Lanz et al. 2008). Lanz et al. analysed the blue spectra of Ar ii in the 10 B-type main-sequence stars in the Orion association and inferred about the mean abundance of Ar in the star HD 35299. Direct observation of the fine structure splittings in the ground states of Ne ii and Ar ii were carried out by Yamada, Kanamori & Hirota (1985). These two transitions were measured with frequencies 780.4240 ± 0.0011 cm−1 and 1431.5831 ± 0.0007 cm−1 for Ne ii and Ar ii, respectively. They are advantageous for infrared spectroscopy over optical, radio frequency and microwave regions to estimate the abundance of Ne and Ar. Because these infrared lines are relatively unaffected by interstellar dust, they can be easily identified by high-resolution telescopes. In fact, the considered infrared line of Ne ii is the only line that can be observed using a ground-based observatory among all other most abundant elements in the Universe at their various stages of ionization. Krypton (Kr) ion spectra were also detected in the interstellar medium (Cardelli & Mayer 1997), in the galactic disc (Cartledge, Meyer & Lauroesch 2003) and in planetary nebulae (Dinerstein 2001). With the help of the Goddard high-resolution spectrograph in the Hubble Space Telescope, many Kr ii lines have also been observed in the interstellar medium (Cardelli, Sarage & Ebbets 1991). More importantly, the materials responsible for the formation of the young early-type stars are enriched with Kr (Leckrone et al. 1993). Extensive analysis of the Kr ii spectra has been carried out by deBruin, Humphreys & Meggers (1933), who quoted uncertainties of the sharp lines of about 0.01 Å. A second analysis was carried out by Minnhagen, Strihed & Petersson (1969) for two sets of wavelength ranges, 800–1350 and 2350–2450 Å. Humphreys & Paul (1970) also reported wavelengths of the 43 lines in Kr ii most precisely from the interferometric observations, using isotropically pure 86Kr. Following this, Bredice et al. (1988) obtained 36 additional lines between 1134 and 8652 Å using an energetically excited source with an accuracy of 0.01 Å. There have been numerous attempts made in the last two decades to discover accurate values of the transition probabilities in Kr ii (Koozekanani & Trusty 1969; Levchenko 1971; Miller, Roig & Bengtson 1972) for various inter-combination transitions. Similarly, the lines of singly charged xenon (Xe ii) have been identified in many HgMn young stars, such as k Cnc, 112 Her, 46 Aql (HD 186122), 33 Gem, HR 7143, HR 7245, HR 7361, HD 175640, etc. (Whitford 1962; Bidelman 1966), which were analysed using the available spectroscopic data of Xe ii (Castelli & Hubrig 2004; Dworetsky, Persaud & Patel 2008). Yüce, Castelli & Hubrig (2011) also investigated the Xe ii lines lying in the ranges 3900–4521, 4769–7542 and 7660–8000 Å in HgMn stars. Observations of the optical lines from heavy elements, especially from radioactive atoms, in the atmosphere of Procyon have been reported (Yushchenko & Gopka 1996). It happens that the wavelength of the emission line of the 7p 2P3/2 → 7p 2P1/2 transition in singly ionized radon (Rn ii) is about 3216 Å and could be useful for estimating the abundance of Rn in astronomical objects.


By combining observational constraints on the dynamics in galaxies with constraints from the Solar system, Milgrom (1983a) corrected the theory of gravitation at low acceleration, which could be a consequence of the quantum vacuum (Milgrom 1999; Pazy 2013; Smolin 2017; Verlinde 2017). In analogy with Newtonian dynamics, a non-relativistic Milgromian gravity theory (MOND) can be constructed by setting up a Lagrangian which, upon extremization of the action, yields a generalized Milgromian Poisson equation. Two such Lagrangians have been proposed, called AQUAL (Bekenstein & Milgrom 1984) and QUMOND (Milgrom 2010). The latter, which we adopt hereafter, yields the following generalized Poisson equation,
(1)$$\begin{eqnarray}
\Delta \Phi (\vec {x}) = 4\pi G\rho _{\rm b} (\vec {x}) + \vec {\nabla } \cdot [\tilde{v}(\left|\vec{\nabla } \phi \right|/a_0)\vec {\nabla } \phi (\vec {x})],
\end{eqnarray}$$or,
(2)$$\begin{eqnarray}
\Delta \Phi (\vec {x}) = 4\pi G [\rho _{\rm b} (\vec {x}) + \rho _{\rm ph}(\vec {x})],
\end{eqnarray}$$where $\rho _{\rm b}(\vec{x})$ is the baryonic density, $\phi (\vec{x})$ is the Newtonian potential that fulfils the standard Poisson equation, $\Delta \phi (\vec{x})$  = 4πG$\rho _{\rm b}(\vec{x})$ and Milgrom’s constant a0$\approx \rm 1.2 \times 10^{-10} \ ms^{-2} \approx 3.8 \ pc \ Myr^{-2}$. The phantom dark matter (PDM) density, $\rho _{ph}(\vec{x})$, is not a real density distribution but a mathematical function that arises out of the non-linearity of the Poisson equation. $\Phi (\vec{x})$ is the total gravitational potential from which the accelerations follow, $\vec{a} = -\vec{\nabla } \vec{\Phi }$, and $\tilde{v}(y)$ is a transition function characterizing the theory (see Milgrom 2008, 2010, 2014; Famaey & McGaugh 2012; and Banik & Zhao 2022 for detailed reviews on the theory). $\tilde{v}(y)$ has the limits,
(3)$$\begin{eqnarray}
\tilde{v}(y) \rightarrow 0 \,\,\rm {for}\,\, \it y\,\, \gg \rm 1 \,\,\rm {and}\,\, \it \tilde{v}(\it y)\,\, \rightarrow \it y^{\rm -1/2}\,\, \rm {for}\,\, \it y\,\, \ll \rm 1.
\end{eqnarray}$$The above formulation deals only with linear differential equations and is shown to emerge as a natural modification of a Palatini-type formulation of Newtonian gravity. It is a member of a larger class of bipotential theories (quasi-linear formulation of MOND; QUMOND Milgrom 2010). Bekenstein & Milgrom (1984) also noted a correspondence with some theories of quark confinement using a different form of the function $\tilde{v}$.


When considering only spaxels with S/N > 5 for each emission line contributing to a given BPT diagram, we are left only with spaxels on the galaxy discs of CGCG097–079 and CGCG097–073 and their immediate surroundings where the emission line flux is high enough to obtain robust line ratios in individual spaxels. We present these results in Fig. 8. The top panel shows the distribution of individual spaxels in the aforementioned BPT diagrams. Each spaxel is colour coded by its distance from the lines separating the region dominated by photo-ionization from the one dominated by active galactic nuclei (AGNs) or shocks, following the separation of Kauffmann et al. (2003). The distributions of line ratios from nuclear spectra of a sample of 50 000 SDSS galaxies at 0.03  z  0.08 whose emission lines are detected at S/N > 5 (Fossati et al. 2016) are shown with grey contours. The innermost to outermost contours include the 25, 50, 75, and 98 per cent of the full data set, respectively. However, this SDSS sample only represents the nuclear properties of galaxies which might be different from their outskirts which are likely providing most of the stripped gas. To overcome this possible issue, we add to the diagnostic diagrams the same density contours from a sample of individual spaxels from the DR17 of the Mapping Nearby Galaxies at APO (MaNGA) IFU survey of ∼10 000 galaxies (Abdurro’uf et al. 2021). For each BPT diagram, we selected spaxels outside the effective radius (R > 1 Re) of each individual galaxy whose emission lines are detected at S/N > 5. This sample includes roughly half a million spaxels in the BPT diagram using the fainter [O i] line and roughly 2 million spaxels in the other two BPT diagrams. Density contours from this sample (at the same levels mentioned above) are shown in pink in Fig. 8. We also verified that by using spaxels from the entire galaxy discs, the density contours would appear very similar to the SDSS sample, due larger weight given to the inner parts of the discs. Despite the differences seen in these two reference samples, the BPT diagnostic made using [O i]/H α shows enhanced [O i]/H α in the RPS tails that appear not to be typical neither of the galaxy nuclei nor of their disc outskirts. We will discuss this point in a more quantitative fashion in Section 5.1. In the middle and bottom panels, we show the spatial distribution of the spaxels from the two galaxies, with points colour coded as above.


The sensitivity of dwarfs to galaxy formation physics, like the gas density at which stars form (e.g. Kravtsov 2003; Saitoh et al. 2008) or the details of galactic outflows (e.g. Read et al. 2006b), makes them a natural ‘rosetta stone’ for constraining galaxy formation models. Early work simulating dwarf galaxies focused on high resolution small box simulations, stopping at high redshift (z ∼ 5–10) to avoid gravitational collapse on the scale of the box (e.g. Abel et al. 1998; Gnedin & Kravtsov 2006; Read et al. 2006b; Mashchenko, Wadsley & Couchman 2008). These simulations demonstrated that stellar winds, SN feedback, and ionizing radiation combine to prevent star formation in halo masses below ∼107–8 M⊙ (Read et al. 2006b; Bland-Hawthorn, Sutherland & Webster 2015). Furthermore, once cooling below 104 K is permitted and gas is allowed to reach high densities $n_{\rm max} {\gtrsim}10\,{\rm cm}^{-3}$ (requiring a spatial and mass resolution better than $\Delta x {\lesssim}100$ pc and mmin  103 M⊙; Pontzen & Governato 2012; Benitez-Llambay et al. 2019; Bose et al. 2019; Dutton et al. 2019), star formation becomes much more stochastic and violent (Mashchenko et al. 2008; Pontzen & Governato 2012; Dutton et al. 2019). The repeated action of gas cooling and blow-out due to feedback expels DM from the galaxy centre, transforming an initially dense DM cusp to a core (e.g. Read & Gilmore 2005; Pontzen & Governato 2012). Finally, independent of any internal sources of feedback energy, cosmic reionization can halt star formation in low-mass galaxies (e.g. Gnedin & Kravtsov 2006). Dwarfs that have not reached a mass of $M_{200} {\gtrsim}10^8$ M⊙ (see Section 2.4 for a definition of M200) by the redshift that reionization begins (z ∼ 8–10; Gnedin & Kaurov 2014; Ocvirk et al. 2018) are gradually starved of fresh cold gas, causing their star formation to shut down by a redshift of z ∼ 4 (Oñorbe et al. 2015). This is similar to the age of nearby ‘ultra-faint’ dwarf galaxies (UFDs) that have $M_* {\lesssim}10^5$ M⊙, suggesting that at least some of these are likely to be relics from reionization, inhabiting pre-infall halo masses in the range M200 ∼ 108–9 M⊙ (Gnedin & Kravtsov 2006; Bovill & Ricotti 2009, 2011; Brown et al. 2014; Weisz et al. 2014; Jethwa, Erkal & Belokurov 2018; Read & Erkal 2019).


The empirical relation between the mass of SMBHs and the properties of their host galaxies is considered to be one of the most important outcomes caused by their coevolution over the cosmic timescale (e.g., Magorrian et al. 1998; Ferrarese & Merritt 2000; Kormendy & Ho 2013). Theoretical models for explaining the tight correlations have been proposed, but the origin is still unclear. To understand the nature of these correlations, it is critically important to study them beyond the local universe, characterizing how and when the relations have been established and evolved until now. So far, a large number of observational studies have extensively investigated the redshift dependence of the BH-to-bulge mass ratio of M
•/M
⋆ and overall suggested its positive redshift dependence, i.e., the ratio increases with redshift (Bennert et al. 2011; Schramm & Silverman 2013; Ding et al. 2020). Beyond z ∼ 6, the Atacama Large Millimeter/submillimeter Array (ALMA) is a powerful tool to measure the dynamical mass of gas in quasar host galaxies and allows us to explore the early stage of the BH/galaxy correlation (e.g., Wang et al. 2010, 2013; Venemans et al. 2017). In addition, observations with the Subaru HSC provide low-luminosity and less-massive BH samples, which are unique populations to determine the M
•/M
⋆ ratio at z > 6 (Izumi et al. 2019, 2021). Figure 11 shows the distribution of z > 6 quasars compiled in Izumi et al. (2021), together with those in the local universe (Kormendy & Ho 2013). First, the brightest z > 6 quasars with M
1450  −25 mag tend to have M
•/M
⋆ ratios higher than those seen in the local universe. Namely, the mass ratio for those brightest objects is boosted by a factor of ∼10 (blue dashed line; Pensabene et al. 2020). On the other hand, the fainter quasars with M
1450 >−25 mag appear to follow the local relation, although those BHs are considered to grow at rates of ≳ 0.05 SFR and will be overmassive at lower redshifts.
15

15
In this paper, overmassive BHs are referred to as a BH population with a BH-to-galaxy mass ratio higher than that observed in the local universe;

M•/M⋆≳4.9−0.5+0.6×10−3

 (Kormendy & Ho 2013). We employ this terminology to clearly contrast the difference between the overmassive and undermassive BH population with respect to the local value (see Figure 11). Note that a previous study by Agarwal et al. (2013) used a term of “obese BH,” which refers to a BH population dominating over the stellar mass of its host galaxy at least in the initial growing stage (i.e., M
•/M
⋆ > 1). We note that for all of the z > 6 samples, the values of the x-axis are not the bulge mass of their host galaxies but the dynamical mass measured by [C ii] 158 μm lines. In general, the dynamical mass is considered to be higher than the true bulge mass. With a high-resolution ALMA observation, Izumi et al. (2021) found that the gas dynamics of the core component of a low-luminosity quasar at z = 7.07 (HSC J1243+0100) is governed by rotation associated with a compact bulge and estimated its mass as ∼50% of the [C ii]-based dynamical mass. Therefore, the correlation at z > 6 might be shifted to the left if the conversion factor from the dynamical mass to the bulge mass is taken into account.


Small fluctuations δ in the distribution of dark matter grow, as long as they are in the linear regime |δ| ≪ 1, according to the growth function D+(a) (Linder & Jenkins 2003; Wang & Steinhardt 1998),
(3)
\begin{equation}
\frac{\mathrm{d}^2}{\mathrm{d}a^2}D_+(a) + \frac{2-q}{a}\frac{\mathrm{d}}{\mathrm{d}a}D_+(a) - \frac{3}{2a^2}\Omega _\mathrm{m}(a) D_+(a) = 0,
\end{equation}
and their statistics is characterized by the spectrum $\langle \delta (\boldsymbol {k})\delta ^*(\boldsymbol {k}^\prime )\rangle = (2\pi )^3\delta _{\rm D}(\boldsymbol {k}-\boldsymbol {k}^\prime )P_\delta (k)$. Inflation generates a spectrum of the form $P_\delta (k)\propto k^{n_s}T^2(k)$ with the transfer function T(k) (Eisenstein & Hu 1999, 1998) which is normalized to the variance $\sigma _8^2$ smoothed to the scale of 8 Mpc h−1,
(4)
\begin{equation}
\sigma _8^2 = \int _0^\infty \frac{k^2\mathrm{d}k}{2\pi ^2}\, W^2(8\,{\rm }\mathrm{Mpc}\, h^{-1}\times k)\,P_\delta (k),
\end{equation}
with a Fourier-transformed spherical top-hat W(x) = 3j1(x)/x as the filter function. From the CDM spectrum of the density perturbation, the spectrum of the Newtonian gravitational potential Φ can be obtained,
(5)
\begin{equation}
P_\Phi (k; \chi ) = \left(\frac{3 \Omega _\mathrm{m} H_0^2}{2} \right)^2 k^{-4} P_\delta (k; \chi ) \propto \left(\frac{3\Omega _\mathrm{m}H_0^2}{2}\right)^2\,k^{n_s-4}\,T(k)^2,
\end{equation}
by applying the comoving Poisson equation $\Delta \Phi = \frac{3 \Omega _\mathrm{m}H_0^2}{2} \delta$ for deriving the gravitational potential Φ from the density δ. With equation (3) yielding a solution for the homogeneous growth of the density contrast. It should be noted that velocities at linear order are obtained from the continuity equation,
(6)
\begin{equation}
\boldsymbol {\nabla } \cdot \boldsymbol {\upsilon } = - a \dot{ \delta },
\end{equation}
such that in Fourier space,
(7)
\begin{equation}
\boldsymbol {\upsilon }_{\rm k} = a H(a) \frac{\mathrm{d}\ln ( D_+)}{\mathrm{d}\ln (a)} \frac{ \boldsymbol {k}}{k^2} \boldsymbol { \delta }_{\rm k}.
\end{equation}


Recent technological advances related to 3D Integral Field Spectroscopy (IFS), which produces data cubes where each pixel on the image has a spectrum, have opened a new window for examining the CGM gas. This approach combines the information gathered in absorption against background sources (whose lines of sight pass through a galaxy’s CGM) with traditional emission-based properties of galaxies. Following at least two decades of limited success in identifying the galaxies associated with quasar absorbers, IFS have open a new era in establishing the relation between absorption and emission with high success rates. Early efforts with near-infrared IFS VLT/SINFONI (Bouché et al. 2007; Péroux et al. 2011; Péroux et al. 2013, 2016) led to efficient discoveries of star-forming galaxies associated with Mg ii and H i absorbers at z ∼ 2 (see also Rudie, Newman & Murphy 2017; Joshi et al. 2021). The optical IFS VLT/MUSE (Bacon et al. 2010) has proved to be a true game-changer in the field. Early on, the MUSE Guaranteed Time Observations (GTO) team established surveys including MUSE-QuBES (Muzahid et al. 2020) and MEGAFLOW (Bouché et al. 2016; Schroetter et al. 2016; Schroetter et al. 2019; Zabl et al. 2019) to relate gas traced by absorbers to galaxies. In a parallel effort, the MAGG survey targets higher redshift galaxies (Fumagalli, O’Meara & Prochaska 2016; Dutta et al. 2020; Lofthouse et al. 2020). The Cosmic Ultraviolet Baryon Survey CUBS instead is absorption-blind and uncovers new quasar absorbers in a wide range of column densities (ranging from few times 16.0  $\rm{log}\,\,{\it N}(\rm{H\,\,{\small I}})$20.1) at z1 (Chen et al. 2020; Boettcher et al. 2021; Cooper et al. 2021; Zahedy et al. 2021). By extending to bluer wavelengths, the optical IFS Keck/KCWI (Martin et al. 2010) has enabled similar studies at higher spectral resolution (Martin et al. 2019; Nielsen et al. 2020). BlueMUSE, a blue-optimized, medium spectral resolution IFS based on the MUSE concept and proposed for the Very Large Telescope is also under planning (Richard et al. 2019). Contemporary to these works, ALMA - which can be viewed as an IFS at mm-wavelengths - has enabled the detections of both CO and [CII] emission in galaxies associated with strong quasar absorbers at intermediate and high redshifts, respectively (Neeleman et al. 2016; Kanekar et al. 2018; Klitsch et al. 2018; Neeleman et al. 2018; Neeleman et al. 2019; Péroux et al. 2019; Klitsch et al. 2021; Szakacs et al. 2021a). These lines enable us to trace the colder (∼100K) and denser phase of the neutral gas: the molecular hydrogen, H2. The molecular gas constitutes the ultimate phase of the gas reservoir from which stars form and hence is an essential link to the baryon cycle. Together, these IFS observations have provided unique information on the resolved galaxy kinematics which can then be combined with the gas dynamics to probe gas flows in the CGM regions (Bouché et al. 2013; Rahmani et al. 2018a; Schroetter et al. 2019; Zabl et al. 2019; Neeleman et al. 2020; Szakacs et al. 2021a).


The recently observed ultra-diffuse galaxy NGC 1052-DF2 with a dark matter mass 400 times smaller than theoretically expected based on an internal velocity dispersion of

σ
intr

=
3
.

2

+
5.5

−
3.2

$ \sigma_\mathrm{{intr}}=3.2_{+5.5}^{-3.2} $

 km s−1, seems to support the existence of dark matter-free galaxies in our Universe (van Dokkum et al. 2018a). van Dokkum et al. (2018b) derived a revised internal velocity dispersion of

σ
intr

=
7
.

8

+
5.2

−
2.2

$ \sigma_\mathrm{{intr}}=7.8_{+5.2}^{-2.2} $

 km s−1 using ten GCs surrounding this galaxy. Danieli et al. (2019) measured a stellar velocity dispersion of

σ
stars

=
8
.

5

+
2.3

−
3.1

$ \sigma_\mathrm{{stars}}=8.5_{+2.3}^{-3.1} $

 km s−1 with the Keck Cosmic Web Imager (KCWI). The high relative velocity to the nearby massive elliptical galaxy NGC 1052 underpins the theory that this observed dark matter-lacking galaxy is indeed a TDG. However, Martin et al. (2018) revised the internal velocity of NGC 1052-DF2 to a 90 percent upper limit of 17.3 km s−1 corresponding to a mass-to-light ratio of M/LV   8.1 Υ⊙, consistent with many Local Group dwarf galaxies. Emsellem et al. (2019) obtain M/LV in the range 3.5−3.9(±1.8) Υ⊙ using the Jeans model if located at D = 20 Mpc. This result would be close to the 2σ upper limit of the study from Martin et al. (2018). The lack of dark matter and the unusual high luminosity of ten globular cluster-like objects surrounding this galaxy only holds if NGC 1052-DF2 is located at a distance of around 20 Mpc (van Dokkum et al. 2018a). Danieli et al. (2019) confirmed that DF2 is dark matter deficient and concluded that it is an outlier to dwarf galaxies of the Local Group. In contrast to that, Trujillo et al. (2019) derived a revised distance to NGC 1052-DF2 of D = 13.0 ± 0.4 Mpc based on five redshift-independent measurements including the tip of the red giant branch and the surface brightness fluctuation method. Thus, NGC 1052-DF2 would be a dwarf galaxy with an ordinary dark matter content Mhalo/Mstellar >  20 and a normal globular cluster population. Meanwhile, van Dokkum et al. (2019) reported that the dwarf galaxy NGC 1052-DF4 also lacks dark matter and is found at a distance of D = 20 Mpc.

Danieli et al. (2019)
Danieli et al. (2019)


The angular dispersion function method (hereafter the ADF method; Falceta-Gonçalves et al. 2008; Hildebrand et al. 2009; Houde et al. 2009, 2016) analytically accounts for various of effects that may affect the measured angular dispersion. Based on different effects considered, the ADF methods can be divided into the structure function method (hereafter the Hil09 method; Hildebrand et al. 2009), the autocorrelation function method for single-dish observations (hereafter the Hou09 method; Houde et al. 2009), and the autocorrelation function method for interferometer observations (hereafter the Hou16 method; Houde et al. 2016). Similarly, we adopt the correction factors in Liu et al. (2021) for strong field models. For the Hil09 method, the corrected plane-of-sky uniform magnetic field strength is estimated as
6

Bposu,Hil09,est∼0.1μ0ρδvlos〈Bt2〉〈B02〉−12Hil09

and the corrected plane-of-sky total magnetic field strength is estimated as
7

Bpostot,Hil09,est∼0.21μ0ρδvlos〈Bt2〉〈B2〉−12Hil09,

where

((〈Bt2〉/〈B02〉)0.5)Hil09

 is the turbulent-to-ordered field strength ratio and

((〈Bt2〉/〈B2〉)0.5)Hil09

 is the turbulent-to-total field strength ratio derived from this method. We do not calculate

Bposu,Hil09,est

 if

((〈Bt2〉/〈B02〉)0.5)Hil09>0.1

 (Liu et al. 2021). For the Hou09 and Hou16 methods, the corrected plane-of-sky uniform magnetic field strength is estimated as
8

Bposu,Hou,est∼0.19μ0ρδvlos〈Bt2〉〈B02〉−12Hou

and the corrected plane-of-sky total magnetic field strength is estimated as
9

Bpostot,Hou,est∼0.39μ0ρδvlos〈Bt2〉〈B2〉−12Hou,

where

((〈Bt2〉/〈B02〉)0.5)Hou

 is the turbulent-to-ordered field strength ratio and

((〈Bt2〉/〈B2〉)0.5)Hou

 is the turbulent-to-total field strength ratio derived from the two methods. Due to the limitation that the angular dispersion cannot exceed the value expected for a random field, the maximum derivable turbulent-to-ordered field strength ratio from the ADF methods is 0.76 (Liu et al. 2021). Thus, if the derived

((〈Bt2〉/〈B02〉)0.5)Hou

 is greater than 0.76, the assumptions underlying the ADF methods on accounting for the line-of-sight signal integration may not be valid, which could lead to overestimation of the turbulent-to-ordered field strength ratio and underestimation of the field strength. In this situation, we adopt the turbulent-to-ordered field strength ratio

((〈Bt2〉/〈B02〉)0.5)nosiHou

 and the turbulent-to-total field strength ratio

((〈Bt2〉/〈B2〉)0.5)nosiHou

 without accounting for the line-of-sight signal integration and apply the same correction factors as those for the Hil09 method. Then the corrected plane-of-sky uniform magnetic field strength is estimated as
10

Bposu,Hou,est∼0.1μ0ρδvlos(〈Bt2〉〈B02〉−12)nosiHou

and the corrected plane-of-sky total magnetic field strength is estimated as
11

Bpostot,Hou,est∼0.21μ0ρδvlos(〈Bt2〉〈B2〉−12)nosiHou,

where

((〈Bt2〉/〈B02〉)0.5)nosiHou=((〈Bt2〉/〈B02〉)0.5)Hou/Nadf

. N
adf is the number of turbulent fluid elements along the line of sight. Similarly, we do not calculate

Bposu,Hou,est

 if

((〈Bt2〉/〈B02〉)0.5)nosiHou>0.1

 (Liu et al. 2021).


An estimate of a sputtering cross-section can be inferred from our measurements with σs ≈ V∕d, where V is the volume occupied by

${Y}_{\textrm{s}}^{\infty}$Ys∞
 molecules and d the depth of sputtering.

$\sigma_{\textrm{s}}\approx {Y}_{\textrm{s}}^{\infty}/l_{\textrm{d}}/\textrm{ml}$σs≈Ys∞/ld/ml
, where ml is the number of CO or CO2 molecules cm−2 in a monolayer (about 6.7 × 1014 cm−2 and 5.7 × 1014 cm−2, respectively,with the adopted ice densities). As is shown in Table 1, the sputtering radius rs would therefore be about 1.26 to 2.12 times larger than the radiolysis destruction radius rd in the case of the CO2 ice, and 2.03 to 2.36 for CO in the considered energy range (~0.5-1 MeV/u). The net radiolysis is the combined effect of the direct primary excitations and ionisations, the core of the energy deposition by the ion, and the so-called delta rays (energetic secondary electrons) travelling at much larger distances from the core; that is, several hundreds of nanometres at the considered energies in this work (e.g. Mozunder et al. 1968; Magee & Chatterjee 1980; Katz et al. 1990; Moribayashi 2014; Awad & Abu-Shady 2020). The effective radiolysis track radius that we calculate is lower than the sputtering one, which points towards a large fraction of the energy deposited in the core of the track. The scatter on the ratio of these radii is due to the lack of more precise data. It nevertheless allows to put a rough constraint on the estimate of Nd in the absence of additional depth measurements, with

${N}_{\textrm{d}} \lesssim {Y}_{\textrm{s}}^{\infty}/\sigma_{\textrm{d}}$Nd≲Ys∞/σd
. If the rs∕rd ratio is high, a large amount of species come from the thermal sublimation of an ice spot less affected by radiolysis, and the fraction of ejected intact molecules is higher. The aspect ratio corresponding to these experiments evolves between about ten and twenty for CO2 and CO, whereas for water ice at a stopping power of Se ≈ 3.6 × 103eV∕1015 H2 O molecules cm−2, we show that it is closer to one (Dartois et al. 2018). The depth of sputtering is much larger for CO and CO2 than for H2O at the same energy deposition, not only because their sublimation rate is higher, but also because they do not form OH bonds. For complex organic molecules embedded in ice mantles dominated by a CO or CO2 ice matrix, with the lack of OH bonding and the sputtering for trace species being driven by that of the matrix (in the astrophysical context), the co-desorption of complex organic molecules present in low proportions with respect to CO/CO2 cannot only be more efficient, but will thus arise from deeper layers.


For GRB 140629A, our analysis suggests that the optical and X-ray afterglows are from a narrow jet (

 rad) with a low B

 in a dense medium (n = 60 cm−3). In addition, the radiation efficiency of GRB 140629A is extremely low. We test whether or not it satisfies various empirical relations reported in the literature derived from observations of the prompt gamma-ray phase and the multi-wavelength afterglows. By estimating the jet opening angle with a jet-like break time tj in late multi-wavelength light curves, Ghirlanda et al. (2004a) derived a tight correlation between geometrically corrected jet energy

 and the peak energy

 of

 spectrum in the burst frame, i.e.,

. The

 value inferred from the Ghirlanda relation is 46 keV for GRB 140629A, which is definitely inconsistent with the data, i.e.,

. Liang & Zhang (2005) derived an empirical relation between

,

, and the jet break time (

) in the burst frame, i.e.,

/

/

/

Based on this relation, an isotropic energy

 erg is obtained, which is larger than that observed by more than one order of magnitude. These results suggest that GRB 140629A does not follow these two relations (Ghirlanda et al. 2004a; Liang & Zhang 2005), although both tight correlations have been used for measuring the cosmological parameters with GRBs (e.g., Dai et al. 2004; Ghirlanda et al. 2004b; Liang & Zhang 2005; Wang et al. 2015a). Note that the observed jet break time of GRB 140629A is much earlier, hence the inferred θj is much lower than those of the GRBs used to derive these relations (e.g., Frail et al. 2001; Bloom et al. 2003). It is unclear whether the violation of GRB 140629A is due to the selection effect or other physical reasons. For example, two-component jet models composed of a narrow and a wide component have been proposed to explain the data of some GRBs (e.g., Huang et al. 2004; Racusin et al. 2008). In these cases, the high-energy emission was proposed to be emitted by the narrow jet. However, one cannot exclude the possibility that the observed gamma-ray energy would be dominated by the wide jet component under certain conditions. Meanwhile, the early break time for GRB 140629A is likely due to the effect of the narrow jet component but not the wide one. If this is the case, the inconsistency between the jet energy and the opening angle would result in this violation of GRB 140629A. Liang et al. (2015) discovered a tight empirical correlation between Liso,

, and Γ0 to reveal the direct connection between the gamma-ray and afterglows,
7

Based on the equation above, we get

 for GRB 140629A, where the error is calculated from the uncertainties in

 and Γ0 only. The derived

 is well consistent with the observed one,

 erg s−1, as shown in Figure 6. Note that the initial Lorentz factor of the ejecta Γ0 is sensitive to the deceleration time (the peak time of the onset bump), but not strongly related to the jet break time. The onset of the afterglow bump is usually bright (Liang et al. 2010, 2013; Li et al. 2012; Wang et al. 2013), and it is easier to identify than the jet break time from an observed light curve.10

10
The jet break is usually detected in late optical afterglow light curves. It is dim and also contaminated by emission from the host galaxy and/or associated supernovae (e.g., Li et al. 2012). This is also an issue in identifying an observed jet break as the narrow or the wide component in the case of a two-component jet.
 The consistency of GRB 140629A with

 suggests that this relation is more robust than the Ghirlanda or Liang–Zhang relations, since it is not sensitive to the jet opening angle θj.


Bolometric magnitudes,

, are estimated as the sum of the

-band magnitudes,

, bolometric correction (

), and DMs, i.e.,

 =

−

+

-DM.

 are adopted from the work of Levesque et al. (2005), which is based on MARCS models. The new temperature scale by Davies et al. (2013) agrees within ±100 K with that of Levesque et al. (2005). Errors in the

 values are calculated by propagation.

 magnitude errors are provided in the 2MASS catalog (0.02 mag in average); errors in

 are described in Section 3.2 (0.02 mag on average); errors in

 are conservatively assumed to be 0.5 mag—the difference between the

 values provided by Levesque et al. (2005) for an M0Iab and M2Iab star amounts to ≈0.1 mag, while the typical difference between their

 values is 0.5 mag (e.g., Verhoelst et al. 2009); rare supergiant Ia can be more than 1 mag brighter. Adopted distance moduli, DMs, are those of Messineo et al. (2016), who estimated distances of 68 (out of 94) targets using extinction as indicator of distance, and calibrating the extinction–distance relation with clump stars surrounding the target (see also Drimmel et al. 2003; Messineo et al. 2014). Errors in the DM values range from 0.12 to 0.36 mag; they have been estimated as a sum of three terms: the uncertainty of the absolute magnitude of clump stars (≈0.11 mag), the error of the centroid of the Gaussian (0.04–0.25 mag,

 of the Gaussian divided by the square-root of the number of enclosed stars), and the uncertainty in the reddening of clump stars (

 ≈ 0.06 mag, taken as the half-width of the J − K bin times 0.537); only CMDs with a well visible trace were considered. To verify these bolometric magnitudes, which are based only on one magnitude (

) plus the bolometric corrections by Levesque et al. (2005), we calculated estimates of bolometric magnitudes (

) by directly integrating under their stellar energy distribution with the collected infrared measurements (see also Ortiz et al. 2002; Messineo et al. 2004, 2014). We extrapolated to zero the flux on the blue part (from 0 to 1.2 μm) of the spectrum with a blackbody, and to the red part (beyond 12–25 μm) with a linear fit passing through the last two available flux measurements. The differences between the two estimates of bolometric magnitudes yield an average of 0.09 mag with a σ = 0.03 mag. We obtained that the red extrapolation contributed only −0.004 to 0.002 mag to the total

; the blue extrapolation added −0.85 mag (for the direct integration only JHKs magnitudes were used and not the DENIS I magnitudes). When including the DENIS I magnitudes, i.e., blue from 0 to 0.8 μm and green from 0.8 to 12–25 μm, we obtained that the contribution of the blue extrapolation to the total

 would drop to −0.18 mag—extinction in the I-band

 is assumed to be 6.8 times

(Messineo et al. 2005). Table 7 lists the inferred values of

, bolometric magnitudes, and adopted DM values.


The Virgo, however, remains the only cluster in which intracluster X-ray sources have been probed and detected. The Chandra observations utilized by Hou et al. (2017) covered only a small portion (∼3 deg2) of Virgo, but its large angular size (∼100 deg2) renders a full mapping a challenging, if not infeasible, task with contemporary X-ray telescopes, thus limiting our ability to extend the search for intracluster X-ray sources in Virgo. Located at a distance of ∼20.0 Mpc (Blakeslee et al. 2009), the Fornax cluster is a dynamically more evolved, more compact and less massive system compared to Virgo. Like Virgo, Fornax has been an important laboratory to explore the physics of hierarchical structure growth. Recent optical surveys including the Hubble Space Telescope ACS Fornax Cluster Survey (ACSFCS; Jordán et al. 2007) and the Fornax Deep Survey with VST (FDS; Iodice et al. 2016) have significantly advanced our knowledge about the ETGs (Côté et al. 2007; Turner et al. 2012), dwarf galaxies (Venhola et al. 2017), GC populations (Jordán et al. 2015), as well as the ICL (Iodice et al. 2017) of this cluster. In the X-ray band, a survey of the inner ∼30′ region of Fornax has been conducted with Chandra observations (Scharf et al. 2005), which resulted in the detection of more than 700 point-like sources (including CXB sources) against an extended, asymmetric diffuse X-ray emission from the intracluster medium (Ikebe et al. 1996; Jones et al. 1997). The X-ray sources in the BCG, NGC 1399, have been extensively studied with a focus on their connection with GCs (Angelini et al. 2001; Paolillo et al. 2011; D’Ago et al. 2014). The X-ray sources located outside the main stellar content of NGC 1399, however, have received little attention so far, except for the work of Phillipps et al. (2013), which studied the incidence rate of X-ray sources in compact stellar systems, including GCs and the so-called ultra-compact dwarfs (UCDs; Phillipps et al. 2001).


In order to determine the dust emission from the remnant, several physical properties are required: the densities and temperatures of the electrons and nuclei, the dominant type of nucleus, and the radiation field strength and spectrum. Observations of Cas A reveal a complex structure, with material covering a wide range of densities and temperatures emitting at different wavelengths. The supernova explosion has driven a forward shock into the circumstellar material, thought to be from the stellar wind of the progenitor (Hwang & Laming 2009), while the ejecta from the supernova itself crosses the reverse shock as it expands (DeLaney et al. 2004). Both shocks are visible as X-ray emitting regions, with typical densities of $n \sim 1{\text{-}}10 \, {\rm cm}^{-3}$ and temperatures $T \gtrsim 10^7 \, {\rm K}$ (Willingale et al. 2003; Lazendic et al. 2006; Patnaude & Fesen 2014; Wang & Li 2016). The ejecta is mostly comprised of heavy elements, principally oxygen (Chevalier & Kirshner 1979; Willingale et al. 2003). As well as the X-ray emitting gas, the shocked ejecta also consists of denser clumps or knots, emitting in the optical and IR (Hurford & Fesen 1996; DeLaney et al. 2010; Patnaude & Fesen 2014) and associated with the dust emission (Arendt et al. 1999). Electron densities in the shocked clumps are $n_{\rm e}\sim 10^3{\text{-}}10^5\, {\rm cm}^{-3}$ (Smith et al. 2009; DeLaney et al. 2010; Lee et al. 2017), while the gas temperatures are of order $10^4 \, {\rm K}$ (Arendt et al. 1999; Docenko & Sunyaev 2010). The SNR also contains ejecta that has not yet encountered the reverse shock and is consequently much cooler. Smith et al. (2009) estimated a maximum electron density of $n_{\rm e}\lesssim 100 \, {\rm cm}^{-3}$ for the unshocked ejecta based on forbidden line ratios, while observations of radio absorption by DeLaney et al. (2014) and Arias et al. (2018) give $n_{\rm e}\sim 10 \, {\rm cm}^{-3}$ and $T \sim 100 \, {\rm K}$. Raymond et al. (2018) inferred a pre-shock temperature of ${\sim } 100 \, {\rm K}$ from [Si i] IR emission lines.


The spherical evolution model was originally introduced to model the evolution of overdensities (Gunn & Gott 1972). This model assumes a spherical underdensity ρi embedded in an expanding, homogeneous background with density $\bar{\rho }$. The evolution of each radius is determined by the total mass M contained within the proper radius R via the acceleration equation in the Newtonian regime.1 The model makes no assumption about the background cosmology with the evolution given as

(1)

\begin{equation}
\frac{\ddot{R}}{R} = \frac{-4\pi G}{3} \sum _{n}(\rho _n + 3p_n),
\end{equation}

where R is the proper radius, the double dot indicates the second derivative with respect to proper time t, G is the gravitational constant, and ρn and pn are the density and pressure components, respectively, of any contributing component i.e., radiation, matter, dark energy (Padmanabhan 1996). The same equation, known as the Friedmann equation, applies to an unperturbed region, which yields the expansion history of the universe. The spherical model has been applied to solve the evolution of overdensities and underdensities (e.g. Gunn & Gott 1972; Peebles 1980; Lilje & Lahav 1991; Sheth & van de Weygaert 2004). Using the spherical model, the evolution equation in a Λ = 0 universe becomes

(2)

\begin{equation}
\ddot{R} = -\frac{GM}{R^2}.
\end{equation}

To solve the above equation, the initial density and velocity profiles are needed. For the case of an overdensity, which eventually collapses and virializes as a halo, the initial density profile is usually taken to be a spherical top-hat and the initial velocity is assumed to be the Hubble flow at the initial time ti. We will use the subscript i to indicate quantities at the initial time throughout the paper. With these assumptions, the equation can be solved analytically and the solution for the size of the radius as a function of time takes the following parametric form (Gunn & Gott 1972; Lilje & Lahav 1991):

(3)

\begin{eqnarray}
R &=& A(1-\cos \theta ), \nonumber \\
t+T &=& B(\theta - \sin \theta ), \nonumber \\
A^3 &=& GMB^2,
\end{eqnarray}

where A, B, and T are constants that can be fixed once the initial conditions are fixed and θ is an indicator of time. For voids with the same initial settings, the analytical solutions can also be found by taking an inverse top-hat model for the density profile (Gunn & Gott 1972; Peebles 1980; Lilje & Lahav 1991; Sheth & van de Weygaert 2004):

(4)

\begin{eqnarray}
R &=& A(\cosh \theta -1), \nonumber \\
t+T &=& B(\sinh \theta -\theta ), \nonumber \\
A^3 &=& GMB^2.
\end{eqnarray}

Note that the parametric solutions above apply to any Λ = 0 universe. For this study we use the flat EdS cosmology.


Figure 4 shows the cooling rate of the plasma due to emission (

λem=ρκemaradTe4

) and inverse-Compton scattering (λ
sc = ρ
κ
es
E
rad 4k

b

T

e
/m

e

c
2), in addition to the heating rate through absorption (λ
abs = ρ
κ
abs
E
rad). As can be seen, the cooling is predominantly concentrated near the equatorial plane which has an optical depth of order unity (τ
es ∼ 1). Because the Compton scattering rate exceeds the emission rate, it is conceivable that most radiation actually gets Compton upscattered before leaving the corona. We estimate that the Compton-y parameter (y = τ
es 4k

b

T

e
/m

e

c
2), which measures the average fractional energy gain of photons due to Comptonization (e.g., Rybicki & Lightman 1986), can easily exceed y ≳ 1 in the T

e
 ≳ 5 × 108 K plasma near the midplane of the corona. Pinpointing the exact value of the Compton y parameter is nontrivial because absorption will play an important role in some optically thick patches near the midplane. A more detailed calculation that takes into account emission, absorption, and scattering in the plasma along photon geodesics will be required to further constrain the Compton-y parameter. Nevertheless, this work suggests that the corona will upscatter and significantly harden thermal radiation from the disk and cyclosynchrotron emission from the corona (see also Dexter et al. 2021; Scepi et al. 2021). In contrast, the bottom row in Figure 3 shows that in RADTOR, the ions and electrons are strongly coupled in the cold disk, which remains optically thick to both scattering and absorption. Because this disk extends all the way to the event horizon, we expect a strong thermal emission component in the emergent spectrum. Compton upscattering of disk photons by the winds sandwiching the inner disk to temperatures necessary to significantly harden the spectrum is not expected because these winds are predominantly cold (T

e
 ≲ 5 × 107 K). The hot pockets of gas with temperatures T

e
 ≳ 108 K within r ≲ 5r

g
 in Figure 3 are relatively rare and seem to be more prominent at later times when the disk becomes numerically underresolved due to thermal collapse (Appendix A.2).


In this paper, we consider resonance scattering on a two-level atom with an infinitely sharp and unpolarized lower level. As for the frequency redistribution, we consider both CRD and angle-averaged PRD. In an unmagnetized one-dimensional spherically symmetric atmosphere, the polarized radiation field is axially symmetric and is described by

. We also take into account the effects of radial velocity fields. We solve the concerned transfer equation in the CMF and in the nonrelativistic regime of velocity fields. To efficiently handle the sphericity effects, we solve the spherical transfer equation in the (p, z) coordinate system (Hummer & Rybicki 1971), which is also called the tangent-ray method. Here z is the distance along the tangent rays and p is impact parameter (see Figure 1 in Megha et al. 2019). Following Frisch (2007), we express the Stokes vector components in terms of their irreducible components. From here on we present all the basic equations in the irreducible basis (see Sampoorna & Trujillo Bueno 2010 for details). The CMF polarized PRD transfer equation for a spherically symmetric medium in (p, z) representation under the nonrelativistic limit is given by (see also Equation (17) of Megha et al. 2019)
1

where x denotes the nondimensional frequency and

 denotes the irreducible Stokes vector with “+” and “−” referring to the outgoing and incoming rays, respectively. The radial optical depth is defined as dτr = −χl(r)dr, where r is the radial distance and χl(r) is the line averaged absorption coefficient. In the CMF the monochromatic optical depth along the tangent ray is given by dτ = [φ(x) + βc]dτr/μ, where φ(x) is the line absorption profile function and βc = χc(r)/χl(r) with χc(r) denoting the continuum absorption coefficient. The direction cosine of the tangent ray about the radius vector of the intercepting radial shell is given by

. In Equation (1),

 denotes the CMF term, which has the form
2

where χ(r, x) = χl(r)φ(x) + χc(r), and
3

The symbol V denotes the ratio of radial (vr) to the thermal (

) velocities. In Equation (1),

 represents the irreducible CMF total source vector and is given by
4

where the line source vector is of the form
5

Here ϵ gives the probability of destruction of photons by inelastic collisions, Bν0 is the Planck function at the line center frequency ν0, and

. The continuum is assumed to be unpolarized. Therefore, the continuum source vector is given by

. The frequency-averaged PRD mean intensity vector is given by
6

where

 is the 2 × 2 nonmagnetic angle-averaged PRD matrix (Domke & Hubeny 1988; Bommier 1997), and
7

The Rayleigh phase matrix

 in the irreducible basis is given in Appendix A of Frisch (2007). It is useful to rewrite Equation (7) as
8

where
9


The number density of SMGs in either BOSS1244 or BOSS1542 is significantly higher than that of the blank fields. It is reasonable to attribute the excess of SMGs to the overdensity at z = 2.24. This has been successfully verified with H α emitters (Shi et al. 2021; Zheng et al. 2021). We link the excess of SMGs in our two MAMMOTH fields to the presence of 850 $\rm{\mu m}$-detected SMG overdensity at z = 2.24. There are 43 (54) SMGs detected over an effective area of 264 arcmin2 in the deep 850 $\rm{\mu m}$ map of BOSS1244 (BOSS1542). Considering the redshift distribution, we calculate the fraction of SMGs located in 2.2  z  2.3 to be 5.7 per cent comparing to the whole redshift range in the AS2UDS survey (Dudzevičiūtė et al. 2020), which has a survey area of ∼1 deg2 and make the cosmic variance negligible (Simpson et al. 2019). Then we obtain that the volume density of SMGs in this redshift slice is 2.6 × 10−5 cMpc−3 at S850 ≥ 4.0 mJy in the blank fields. If we simply assume the two overdensities span a narrow redshift range (δz  0.042 given by the band width of H2S(1) filter), the volume of the redshift slice of z = 2.246 ± 0.021 over 264 arcmin2 corresponds to a co-moving box of 54.3 × 665.6 (=36142) h−3 cMpc3. If the excess of SMGs (i.e. (1.8–1)/1.8 × 54 for BOSS1542) are associated with the z = 2.24 overdensities, we obtain that SMGs in the two overdensities may reach a volume density of ∼(40–70) × 10−5 cMpc−3, 15–30 times that of the blank fields. After enlarging the redshift slice of the two protoclusters to that of blank field (δz = 0.1), we still obtain the volume density of SMG to be 6–11 times that of the blank fields. The overdensity factor of SMGs, although with large uncertainties, is even higher than that of H α emission-line galaxies (HAEs) in these two protoclusters (Shi et al. 2021), indicating that there are more extreme starbursts with enhanced star formation in respect to normal SFGs (i.e. HAEs) in the two z = 2.24 protoclusters.


It is straightforward to generalize the proof to the instance of axisymmetric stellar systems with second velocity moments aligned in spheroidal coordinates (λ, μ, ϕ). This coordinate system has been introduced in equation (13) and is described in detail in, for example, Morse & Feshbach (1953) or Binney & Tremaine (2008). Here, we will show that if the second velocity moment ellipsoid is everywhere aligned in spheroidal coordinates (〈vλvϕ〉 = 〈vλvμ〉 = 〈vμvϕ〉 = 0), then the gravitational potential has Stäckel or separable form. Again, we construct the even part of the DF. Introducing canonical coordinates, the DF has the form ${F_{\rm e}}(p_\lambda ^2, p_\mu ^2, p_\phi ^2, \lambda ,\mu ,\phi )$. The Hamiltonian is

(44)

\begin{equation}
H = {1\over 2} \Bigl ( {p_\lambda ^2\over P^2} + {p_\mu ^2 \over Q^2} + {p_\phi ^2 \over R^2} \Bigr ) - \psi (\lambda ,\mu ,\phi ),
\end{equation}

where the scale factors are P, Q and R are

(45)

\begin{eqnarray}
P^2 &=& {\lambda -\mu \over 4(\lambda + a)(\lambda +b)},\qquad Q^2 = {\mu -\lambda \over 4(\mu + a)(\mu +b)},\nonumber \\
R^2 &=& {(\lambda +a)(\mu +a)\over a-b},
\end{eqnarray}

and a and b are constants (see for example the tables of Lynden-Bell 1962 or equation (6) of Evans & Lynden-Bell 1989 or Section 2 of de Zeeuw 1985). This implies that the DF can be re-written as ${F_{\rm e}}(H, p_\mu ^2, p_\phi ^2, \lambda ,\mu ,\phi ).$ Just as before, requiring the Poisson bracket {H, Fe} to vanish implies that ∂Fe/∂λ also vanishes and so Fe is independent of λ. This leaves us with the condition

(46)

\begin{equation}
A {\mathrm{\partial} {F_{\rm e}}\over \mathrm{\partial} p_\mu } - {\mathrm{\partial} \psi \over \mathrm{\partial} \phi } {\mathrm{\partial} {F_{\rm e}}\over \mathrm{\partial} p_\phi } = {p_\mu \over Q^2} {\mathrm{\partial} {F_{\rm e}}\over \mathrm{\partial} \mu } + {p_\phi \over R^2} {\mathrm{\partial} {F_{\rm e}}\over \mathrm{\partial} \phi }
\end{equation}

with

(47)

\begin{eqnarray}
&&{A ={1\over \lambda -\mu }}\nonumber\\
&&{\left[ {p_\mu ^2\over 2} {\mathrm{\partial} \over \mathrm{\partial} \mu }\left( {\lambda -\mu \over Q^2} \right) \!+\! {p_\phi ^2\over 2} {\mathrm{\partial} \over \mathrm{\partial} \mu } \left( {\lambda \!-\!\mu \over R^2} \right) - {\mathrm{\partial} \over \mathrm{\partial} \mu }\left( (\lambda -\mu ) \psi \right) +H\right].}\nonumber\\
\end{eqnarray}

Again, the equation must hold on transforming pμ → −pμ, so that in the general case (i.e. ignoring degenerate cases like isotropy), we must have

(48)

\begin{equation}
A {\mathrm{\partial} {F_{\rm e}}\over \mathrm{\partial} p_\mu } = {p_\mu \over Q^2} {\mathrm{\partial} {F_{\rm e}}\over \mathrm{\partial} \mu }, \qquad \qquad {\mathrm{\partial} \psi \over \mathrm{\partial} \phi } {\mathrm{\partial} {F_{\rm e}}\over \mathrm{\partial} p_\phi }= {p_\phi \over R^2} {\mathrm{\partial} {F_{\rm e}}\over \mathrm{\partial} \phi }.
\end{equation}

Multiplying the first equation by λ − μ, differentiating with respect to λ at constant H and then using the definitions of the scale factors gives us the simple result

(49)

\begin{equation}
{\mathrm{\partial} ^2 \over \mathrm{\partial} \lambda \mathrm{\partial} \mu } \left( (\lambda -\mu )\psi \right) =0.
\end{equation}

Integrating up, this gives us

(50)

\begin{equation}
\psi = {A(\lambda ,\phi ) - B (\mu ,\phi ) \over \lambda - \mu },
\end{equation}

where A(λ, ϕ) and B(μ, ϕ) are arbitrary.


In an attempt to perform a more robust analysis, potentially obtain stronger constraints on the MSP population, and break some parameter degeneracies, we consider a parameterized pulsar spin-down luminosity function

 (Johnston & Verbunt 1996). We will use this to balance the required X-ray energetics by assuming that

 and

. This implies

, with

 a normalization constant. Johnston & Verbunt (1996) inferred a typical GC value of γL ∼ 0.5, while Heinke et al. (2006) found γL ∼ 0.4–0.7 for Terzan 5, depending on the energy band. By defining

, one can next recover the following quantities:
25

26

27

28

29

We want to solve for four quantities:

,

,

 (or, equivalently,

), and γL; once these are fixed, we can infer the MSP population properties through the above equations. We note, however, that we are using this luminosity function to fit X-ray luminosities, which are integral quantities. We therefore expect to find degenerate solutions, as different combinations might yield the same integral luminosities. Thus, we need four constraints or measurements. We can use the same three constraints as before. Crucially, one needs to specify a fourth parameter ηX to convert from spin-down luminosities to X-ray luminosities. By fixing ηX, we implicitly fix the product

. As a first attempt, let us assume ηX = 0.05% (e.g., for

 and

 erg s−1 to make the calculation consistent with the previous estimate). It is difficult to obtain the actual value for

, given the effect of the GC cluster potential on the

 of each MSP (e.g., Bogdanov et al. 2008). If we could, this would further constrain the system via Equation (28). Heinke et al. (2006) noted that while they did not detect an X-ray MSP explicitly, one X-ray source could plausibly be an MSP based on the proximity to a radio MSP position; they also noted that more identifications of X-ray MSPs could be made as radio positions become available. We thus have additional constraints

 and

 (which may be used as checks on the consistency of the solutions we obtain). We do obtain a nonunique solution for each fixed value of ηX. However, ηX is not known, and the parameters that satisfy the other three constraints are quite degenerate, as expected. Table 2 indicates a number of parameter combinations that satisfy the observational constraints.25

25
A preliminary Markov chain Monte Carlo investigation (Foreman-Mackey et al. 2013) confirmed the degenerate nature of the free parameters (some are correlated), as well as their being quite unconstrained (reflected by asymmetrical and flat probability distributions, as well as elongated confidence contours). Best-fit values furthermore depend on the choice of priors/parameter bounds. The median values are, however, similar to those in Table 2.
 It is clear that a different choice of ηX will favor a different solution that will imply a different value of

 (which also depends on the average moment of inertia

,

, and

). For example, a higher value of ηX will yield a lower value of

 or

 for a given value of

 and keeping other parameters fixed. If we require

 to be the same as assumed in the model used to predict the pulsed emission, this may lead to unrealistic values for γL for a given ηX. Relaxing this requirement (which may easily be done, given other parameter uncertainties) implies more suitable values for the other parameters. It is therefore clear that the system of equations is very coupled and the parameters are degenerate, given the lack of suitable constraints. One may think to constrain the solution space by requiring

, the latter being the estimated total number of visible MSPs in Terzan 5 as inferred from the Fermi-measured GeV energy flux (Abdo et al. 2010). However, this estimate is quite uncertain and does not contain uncertainties in distance (the square of which determines the γ-ray luminosity Lγ) and conversion efficiency of

 to Lγ, so this does not seem to be a strong constraint. Likewise, we chose

 when fitting the CR component, but this value is also subject to other model assumptions, such as MSP geometry and gap width. Finally, it seems that the last few entries in Table 2 might be the more plausible combinations in view of the independent constraints on γL ∼ 0.4–0.7 and

 (i.e., probably relatively small numbers of visible X-ray pulsars) gleaned from the analysis of Heinke et al. (2006). Thus, the uncertainty in several parameters, particularly ηX, as well as parameter degeneracies preclude us from making definite statements about the MSP population properties. Yet we see that there are several plausible solutions that characterize and constrain the MSP population's energetics, implying that the scenario of MSPs being responsible for the broadband SED may be justified and thus plausible.


The newly discovered arcs and new spectroscopic redshifts have been incorporated into updated HFF+ versions of the Abell 2744 and MACSJ0416.1-2403 lensing models (Table 5); many of these have

 (see Figure 8 for a comparison of the arc redshift distributions adopted by the pre-HFF and new HFF+ lensing models; Cypriano et al. 2004; Okabe & Umetsu 2008; Zitrin et al. 2009; 2013; Okabe et al. 2010a, 2010b; Merten et al. 2011; Christensen et al. 2012; Mann & Ebeling 2012; Jauzac et al. 2014; Lam et al. 2014; Richard et al. 2014; Balestra et al. 2015; Diego et al. 2015; Grillo et al. 2015; Jauzac et al. 2015; Rodney et al. 2015; Wang et al. 2015; Kawamata et al. 2016). The incorporation of these new multiple image systems often results in a reduction in the statistical uncertainty in the galaxy magnifications for a given model. All of the public HFF lensing models provide a range of possible realizations from which the statistical uncertainty of a given model set may be calculated (typically 100 but no fewer than 30). We plot the cumulative distribution of the galaxy magnification uncertainties σ(model)/

, for the galaxies and photometric redshifts provided by the ASTRODEEP catalogs (Merlin et al. 2016; Castellano et al. 2016a) for Abell 2744 (Figure 9) and MACSJ0416.1-2403 (Figure 10). Generally, the statistical uncertainties are reduced for the models computed with the new HFF data sets, with more dramatic reductions for the methods that rely strongly upon the strong-lensing constraints. The parametric methods (CATS, Sharon, Zitrin, GLAFIC) report median statistical magnification errors of 0.2%–5%, while the non-parametric methods (Bradac Williams, Diego) report median statistical magnification errors of 2%–11% for the post-HFF calculations (green curves), versus 2%–22% and 2%–17% respectively for pre-HFF models (blue curves). (We note that the statistical errors for the MACSJ0416.1-2403 Bradac post-HFF models (Hoag et al. 2016) included additional uncertainties due to the photometric redshift uncertainties of the multiple images. These were not included in the pre-HFF Bradac model, and thus may explain why the post-HFF statistical errors are larger for this model.)