Tidal heating of Io has been shown to be responsible for its widespread volcanism. The tidal heating rate of Jupiter’s tidally locked moon,

, driven by forced eccentricities, e, locked by Europa and Ganymede’s Laplace resonance with Io, is the dominant interior heating source. Similarly, the tidal heating of an exomoon will likely dominate the interior energy budget due to the additional stellar tide. Consequently, the tidal heating rate is orders of magnitude higher than at Io, which for an exo-Io of similar rheological properties (

, Rs = RIo, ρs = ρIo) can be written as (Cassidy et al. 2009; Equations (19) and (20))
3

where υ = 3 × 10−7 cm3 erg−1, and τs = τp/5 based on the tidal stability criterion discussed in Section 2. For utility, we describe the exo-Io’s tidal efficiency as

, which can readily be computed for any three-body system as tabulated in Table 4. The enhanced tidal heating described in Equation (3) will also contribute to the surface temperature T0 = Teq + ΔT0, which is very roughly approximated as
4

where σsb is the Stefan–Boltzmann constant and Teq. At Io, the total neutral volcanic content (SO2, SO, NaCl, KCl, Cl, and dissociation products) ejected to space (Section 4.2.1) by the incident plasma is estimated to be, on average, ∼1000 kg s−1 (e.g., Thomas et al. 2004), varying within an order of magnitude over decades of observations (Burger et al. 2001; Wilson et al. 2002; Thomas et al. 2004). While the source of the dominant gas SO2 is ultimately tidally driven volcanism, the near-surface atmosphere is mostly dominated by the sublimation of SO2 frost (Tsang et al. 2016). By observing the atmospheric evolution of the SO2 column density with heliocentric distance, Tsang et al. (2013) estimated the direct volcanic component to be Nvolc ∼ 6.5 × 1016 cm−2, typically

 of the total observed SO2 column density. Ingersoll (1989) demonstrated the relative contributions due to both sublimation and volcanic sources in maintaining Io’s atmosphere and established a relationship relating the volcanic source rate to the volcanically supplied atmospheric pressure:
5

This expression also gives the volcanic column density

, where g is the acceleration due to gravity. Adopting an observed atmospheric temperature of Tatm = 170 K by Lellouch et al. (2015) corresponding to an atmospheric scale height of H = 12 km, a thermal velocity

 equal to 150 m s−1, and a sticking coefficient α = 0.5 for the SO2 mass of 64 amu yields a volcanic source rate of

 ∼ 6.9 × 106 kg s−1 of SO2 integrated over Io’s mass MIo. The average volumetric mixing ratio for NaCl to SO2 at Io is observed to be XNaCl ∼ 3 × 10−3 (Lellouch et al. 2003). This leads to a source rate of

 ∼ 7.4 × 103 kg s−1 of NaCl, somewhat larger than but reasonably consistent with the direct measurement of the NaCl volcanic source rate of (0.8–3.1) × 103 kg s−1 (Lellouch et al. 2003). From these estimates, we will adopt ∼3 × 103 kg s−1 of Na i as the volcanic source rate for Io.


Observations have shown that the shape of the Lyα line is diverse. It includes broad damped absorption profiles, P-Cygni profiles, double-peak profiles, pure symmetric emission profiles, and combinations thereof (Kunth et al. 1998; Mas-Hesse et al. 2003; Shapley et al. 2003; Møller et al. 2004; Noll et al. 2004; Tapken et al. 2004; Venemans et al. 2005; Wilman et al. 2005). This variety can be understood through a detailed radiative transfer calculation, which is analytically solvable only for simple cases (e.g., a static, plane-parallel slab by Harrington 1973 and Neufeld 1990, and a static uniform sphere by Dijkstra et al. 2006). Later, numerical algorithms based on Monte Carlo techniques were developed to solve radiative transfer for more general cases. Now theoretical studies mostly rely on them (e.g., Spaans 1996; Loeb & Rybicki 1999; Ahn et al. 2000, 2002; Zheng & Miralda-Escudé 2002; Richling 2003; Cantalupo et al. 2005; Dijkstra et al. 2006; Hansen & Oh 2006; Tasitsiomi 2006; Verhamme et al. 2006, 2015; Laursen et al. 2013; Behrens et al. 2014; Duval et al. 2014; Gronke et al. 2015; Smith et al. 2019; Lao & Smith 2020; Michel-Dansac et al. 2020). Meanwhile, a galaxy model needs to be constructed to perform such a radiative transfer calculation. One can adopt a realistic galaxy model from hydrodynamical simulations. Galaxies from such simulations can be useful for performing a statistical study of Lyα properties, but they cannot be directly used to model individual galaxies in observations. Therefore it would be better to adopt a simple but manageable toy model for the purpose of reproducing observations. One example for such models is a shell model, in which a central Lyα source is surrounded by a constantly expanding, homogeneous, spherical shell of H i medium with dust. Although this shell model has surprisingly well reproduced many observed Lyα line profiles (e.g., Ahn 2004; Schaerer & Verhamme 2008; Verhamme et al. 2008; Schaerer et al. 2011; Gronke et al. 2015; Yang et al. 2016; Gronke 2017; Karman et al. 2017), because of its extreme simplicity and contrivance, there is still room for improvement (e.g., see Section 7.2 in Yang et al. 2016; Orlitová et al. 2018).


Along with the time-domain detections, we identified J173438.35-504550.4 as a potential host galaxy for FRB 20201123A using robust statistical treatment given the relatively small localization error region. At face value, the low redshift of J173438.35-504550.4 appears at odds with the large dispersion measure for FRB 20201123A (${\rm DM}_{\rm FRB}\approx 434 \, {\rm pc \, cm^{-3}}$). Our Galaxy, however, contributes ${\rm DM}_{\rm ISM}\approx 200 \, {\rm pc \, cm^{-3}}$ (NE2001 gives 229 ${\rm pc \, cm^{-3}}$ and YMW16 gives 162 ${\rm pc \, cm^{-3}}$) from its interstellar medium and a presumed ${\rm DM}_{\rm Halo}\sim 50 \, {\rm pc \, cm^{-3}}$ from its halo (Prochaska & Zheng 2019). This leaves ${\approx}180 \, {\rm pc \, cm^{-3}}$ for the cosmos (DMcosmic) and the host (DMhost). At z = 0.05, the average cosmic contribution is $\langle {\rm DM}_{\rm cosmic}\rangle \sim 42\, {\rm pc \, cm^{-3}}$ (Macquart et al. 2020) but the intrinsic scatter in this quantity is predicted to be large. Adopting the best-fitting model to the Macquart relation by Macquart et al. (2020), the 95 per cent confidence interval is ${\rm DM}_{\rm cosmic}= [15,125] \, {\rm pc \, cm^{-3}}$. Allowing for the maximum value of this interval (which would imply a significant foreground galaxy halo), we recover a minimum host contribution of ${\rm DM}_{\rm host, min} \approx 60~\rm pc~cm^{-3}$. This is consistent with estimates for host galaxy contributions from theoretical and empirical treatments (Prochaska & Zheng 2019; James et al. 2022). For a true DMcosmic value of this sightline closer to (or below) the mean, the host contribution would exceed $100 \, {\rm pc \, cm^{-3}}$. Such values are inferred for other FRB hosts (e.g. FRB 20121102A; Tendulkar et al. 2017). In conclusion, we find no strong evidence to rule out the association with J173438.35-504550.4 based on its redshift and DMFRB. The significant host contribution to the DM, combined with the scattering in FRB 20201123A possibly originating in the host, shows that it shares similarities with other highly active, repeating FRBs like FRB 20121102A and FRB 20190520A and potentially resides in a turbulent and dense environment within the host.


It is generally thought that the stellar halo of the Milky Way (MW) contains the fossil record of the MW’s formation history imprinted in its kinematical and chemical properties. Since Eggen et al. (1962) showed a model of the MW’s formation process from the kinematic analysis of halo stars in the solar neighborhood, many studies of this Galactic fossil component have been carried out using a large number of newly available halo samples, with their assembly requiring considerable observational efforts over the past decades. These include, for example, the Hipparcos Catalog, obtained from the first astrometry satellite (Perryman et al. 1997), and spectroscopic catalogs such as RAVE (Steinmetz et al. 2006), SEGUE (Yanny et al. 2009), LAMOST (Cui et al. 2012; Zhao et al. 2012), GALAH (De Silva et al. 2015), APOGEE (Majewski et al. 2017), H3 (Conroy et al. 2019), and more. Perhaps the most significant impacts on this field of research have been brought about by the second astrometry satellite, Gaia. The Gaia catalog (Gaia Collaboration et al. 2016, 2018, 2021) provides trigonometric parallaxes and proper motions for billions of Galactic stars with unprecedented high accuracy. Based on the astrometry data of the stars in the Gaia catalog combined with that of the spectroscopic catalogs, the MW’s new dynamical maps have been drawn (e.g., Belokurov et al. 2018; Helmi et al. 2018; Myeong et al. 2018a, 2018b; Beane et al. 2019; Hagen et al. 2019; Iorio & Belokurov 2019; Anguiano et al. 2020; Cordoni et al. 2021; Koppelman et al. 2021), and new aspects of the MW’s formation and evolution history have been revealed (e.g., Antoja et al. 2018; Sestito et al. 2019; Wyse 2019). In particular, the Gaia catalog has enabled the discovery of new substructures in the MW’s stellar halo (e.g., Koppelman et al. 2019; Myeong et al. 2019; Li et al. 2020) that are remnants of past merging/accretion events associated with the MW’s formation history (e.g., Fernández-Trincado et al. 2020; Naidu et al. 2020, 2021).


In the upper panel of Fig. 17, we demonstrate that the mass distribution of GMCs in each galaxy reproduces the upper limit of ∼3 to 8 × 106 M⊙ observed by Rosolowsky et al. (2003) in M33, by Freeman et al. (2017) in M83 and by Miville-Deschênes et al. (2017) and Colombo et al. (2019) in the Milky Way. This upper limit has been predicted to arise due to a combination of centrifugal forces and stellar feedback (Reina-Campos & Kruijssen 2017). We also find a turnover in the mass spectrum between 104.8 and 105 M⊙, consistent with the behaviour of the GMC mass distribution in the Milky Way (Miville-Deschênes et al. 2017), although we cannot rule out the possibility that the turnover we see in the simulations is influenced by their limited mass resolution. Above the turnover, the GMC mass function has a power-law form with β ∼ 1.9, close to the observed range of β ∈ [1.6, 1.8] for clouds in the Milky Way (Solomon et al. 1987; Williams & McKee 1997; Heyer et al. 2009; Roman-Duval et al. 2010; Miville-Deschênes et al. 2017; Colombo et al. 2019) over the same mass range (log M ∈ [4.8, 6.5]). In the lower panel of Fig. 17, we display the spectrum of GMC sizes for each simulated galactic disc, given by the effective cloud radius ℓeff, such that
(42)$$\begin{eqnarray*}
\ell _{\rm eff} = 1.91 \sqrt{\Delta \ell _{\rm maj}^2 + \Delta \ell _{\rm min}^2},
\end{eqnarray*}$$where Δℓmaj and Δℓmin are the second moments of an ellipse fitted to the footprint of each cloud in the galactic mid-plane, using astrodendro. We adopt this definition of the cloud size in order to make a direct comparison to works in the existing observational literature (e.g. Solomon et al. 1987; Bertoldi & McKee 1992; Rosolowsky & Leroy 2006; Colombo et al. 2019). The factor of 1.91 is the correction first defined by Solomon et al. (1987) for converting the RMS cloud extent to an estimate of the spherical cloud size. The smallest resolved cloud has a diameter of 18 pc, so we do not capture the observed turnover of the distribution at ∼30 pc (Miville-Deschênes et al. 2017). Likewise, our largest clouds slightly exceed the truncation size of 70 pc observed by Colombo et al. (2019), with a maximum diameter of ∼200 pc. Importantly, we do approximately reproduce the observed power-law slope of $\mathrm{d}N/\mathrm{d}R \sim R^{-\beta _\ell }$ with βℓ ∼ 2.8 (Colombo et al. 2019). This is given by the black line in Fig. 17, while our best fit to the simulation data over the observed range of cloud sizes ℓeff ∈ [18, 70] pc is given by the purple line, with a slightly shallower slope of βℓ = 2.43 ± 0.06.


In the following, we consider an NS with mass MNS = 1.5 M⊙ and radius RNS = 12 km, which is compatible with expectations from modern equations of state such as APR or BSk21 models (Akmal, Pandharipande & Ravenhall 1998; Goriely, Chamel & Pearson 2010). The value of the radius is also in agreement with the estimates derived by Sartore et al. (2012) and Ho et al. (2007), assuming a source distance of 120 pc (Walter et al. 2010). This choice translates into a gravitational redshift factor at the star surface 1 + z = 1.26. The rotational period of RX J1856 is P = 7 s and the X-ray pulsed fraction is the lowest among the XDINSs, ∼1.3 per cent (Tiengo & Mereghetti 2007). The polar strength of the dipole field is taken to be Bp = 1013 G, a value which is somehow intermediate between the spin-down measure and the estimates from spectral fitting (Ho et al. 2007; van Kerkwijk & Kaplan 2008). We assume that the magnetic field is dipolar (see Section 2) and that the surface temperature distribution is that induced by the core-centred dipole. Since for fields higher than ∼1011 G, electron conduction across the field lines is strongly suppressed, the meridional temperature variation is Tdip ≃ Tp|cos θB|1/2, where Tp is the polar value of the temperature (e.g. Greenstein & Hartke 1983). We checked that this simple expression for Tdip differs only slightly ( ≲ 6  per  cent) from the more accurate formula by Potekhin, Pons & Page (2015) for θ ≲ 80°. However, taken face value, the previous expression for Tdip yields vanishingly small values near the magnetic equator. The analysis of Sartore et al. (2012) has shown that the X-ray spectrum of RX J1856 is best modelled in terms of two blackbody components with $kT^\infty _\mathrm{h}\sim 60$ eV and $kT^\infty _\mathrm{c}\sim 40$ eV. To account for this in a simple way, we actually adopt a temperature profile given by Ts = max (Tdip, Tc) with Tp = Th, where $T_\mathrm{h,c}=T^\infty _\mathrm{h,c}/(1+z)$.


We show the mass-loading factors required to simultaneously reproduce the stellar metallicity and SFR of passive galaxies in the bottom panel of Fig. 9. We find that the mass-loading factor strongly decreases with increasing stellar mass. This anticorrelation between stellar mass and λeff is qualitatively consistent with the mass dependence in theoretical models, which predict that $\lambda \propto M_*^{-1/3}$ (Murray, Quataert & Thompson 2005) for momentum-driven winds and $\lambda \propto M_*^{-2/3}$ for energy-driven winds (e.g. Dekel & Silk 1986), as well as other observational evidence (e.g. Heckman et al. 2015; Chisholm et al. 2017; Fluetsch et al. 2019). Our results indicate that ‘effective’ outflows (which are capable of permanently removing gas from the galaxy) are, together with starvation, of increasing importance in low-mass galaxies. In particular, since the rate at which gas is locked up into long-lived stars is given by (1 − R)Ψ = 0.575Ψ, and the rate at which gas is ejected from the galaxy is given by λeffΨ, then, for galaxies with log (M*/M⊙)  10.2, the rate at which gas is lost through galactic winds is roughly 1–3 times larger than the rate at which gas is locked up into long-lived stars. Clearly outflows play an essential role in depleting the gas reservoir of low-mass galaxies during the quenching phase. Outflows are relatively weaker (λeff ≤ 0.6) in more massive galaxies (log (M*/M⊙) > 10.2), with starvation becoming the dominant quenching mechanism, as illustrated in Fig. 9. However, outflows still play an important role in quenching star formation, as comparable amounts of gas are removed through galactic winds and through the formation of long-lived stars. Although these massive galaxies may be ejecting large amounts of gas in the form of outflows (i.e. a large λ), our results suggest that these ejection events are short lived and/or most of the outflowing gas does not escape the galaxy and is instead recycled (i.e. a relatively small λeff).


The fifth force, propagated by the scalar degree of freedom, affects the Poisson equations associated to the Newtonian potential Φ, as well as the relativistic one, Ψ, according to (Kobayashi, Watanabe & Yamauchi 2015; Crisostomi & Koyama 2018; Dima & Vernizzi 2018),
(1)$$\begin{eqnarray*}
\frac{\text{d} \Phi (r)}{\text{d}r} = \frac{G M(r)}{r^2} \left[1+\frac{3}{4}Y_1\left(\frac{\rho (r)}{\bar{\rho }(r)}\right)\left(2+\frac{\text{d}\ln \rho }{\text{d}\ln r}\right)\right],
\end{eqnarray*}
$$(2)$$\begin{eqnarray*}
\frac{\text{d} \Psi (r)}{\text{d}r} =\frac{G M(r)}{r^2}\left[1-\frac{15}{4}Y_2\left(\frac{\rho (r)}{\bar{\rho }(r)}\right)\right].
\end{eqnarray*}
$$In the above equations, we have assumed spherical symmetry. $\bar{\rho }(r)$ is the (spatially) average density at radius r from the centre of the galaxy cluster, and Y1, Y2 correspond to the dimensionless fifth-force couplings. Finally, G is the Newton’s constant. Although the dynamics of member galaxies in the cluster is governed by the potential Φ, lensing is sourced by the combination
(3)$$\begin{eqnarray*}
\frac{\mathrm{ d}}{\mathrm{ d}r} \Phi _{\rm {lens}} = \frac{1}{2}\frac{\mathrm{ d}}{\mathrm{ d}r}(\Phi + \Psi).
\end{eqnarray*}
$$Therefore, kinematical observations allow for contraints on Y1, while lensing constrains both Y1 and Y2. The right-hand side of above equation can be expressed in terms of the density profile ρ(r) according to the relevant equations for Φ and Ψ above. The dominant source of pressureless matter density in the cluster comes from dark matter, which density we choose to model with a Navarro-Frenk-White (NFW) of Navarro, Frenk & White (1997) profile as
(4)$$\begin{eqnarray*}
\rho (r)=\frac{\rho _\text{s}}{r/r_\text{s}(1+r/r_\text{s})^2},
\end{eqnarray*}
$$with ρs is a characteristic density and rs the radius at which the logarithmic derivative of the density profile takes the value −2. The NFW profile has been shown to provide an overall good agreement with observations and simulations over a broad range of scales in GR (e.g. Biviano et al. 2013; Umetsu et al. 2016; Peirani et al. 2017) and in MG (e.g. Lombriser et al. 2012a; Wilcox et al. 2016). Moreover, the GR analyses with lensing and internal kinematics of both clusters indicate that the total mass profile is well fitted by the NFW model (Biviano et al. 2013; Umetsu et al. 2016; Caminha et al. 2017; Sartoris et al. 2020). Under the assumption of a NFW profile, we can re-write the equation for the potential Φ in an effective way as
(5)$$\begin{eqnarray*}
\frac{\text{d}\Phi }{\text{d}r} \equiv \frac{G M_{\text{dyn}}}{r^2}=\frac{G}{r^2}\left[ M_{\rm {NFW}}(r)+M_1(r)\right],
\end{eqnarray*}
$$which serves as a definition of the dynamical mass Mdyn. Notice that, G here is still Newton’s constant as measure in the Solar system. The fifth-force contribution M1 is defined in terms of the NFW parameters as
(6)$$\begin{eqnarray*}
M_1(r)= M_{200}\frac{Y_1}{4}\frac{r^2(r_\text{s}-r)}{(r_\text{s}+r)^3}\times [\ln (1+c)- c/(1+c)]^{-1}.
\end{eqnarray*}
$$where c = r/rs is the concentration and M200 is the mass of a sphere of radius r200 enclosing an average density 200 times the critical density of the universe at that redshift. In a similar fashion, the relevant expression for the lensing mass can be found by computing
(7)$$\begin{eqnarray*}
M_{\text{lens}}(r) =\frac{r^2}{2G}\left[\frac{\text{d}\Psi }{\text{d}r}+\frac{\text{d}\Phi }{\text{d}r}\right].
\end{eqnarray*}
$$$$\begin{eqnarray*}
M_{\text{lens}}=M_{\text{NFW}}+\frac{r^2M_{200}\left[Y_1(r_\text{s}-r)-5Y_2(r_\text{s}+r)\right]}{4[\log (1+c_{200})-c_{200}/(1+c_{200})]}\frac{1}{(r_\text{s}+r)^{3}},
\end{eqnarray*}
$$which can be effectively re-expressed in terms of the dynamical mass as
(8)$$\begin{eqnarray*}
M_{\text{lens}} \equiv M_{\text{dyn}}+M_2,
\end{eqnarray*}
$$with M2 the contribution from the fifth force defined through
(9)$$\begin{eqnarray*}
M_2=\frac{r^2M_{200}}{8(r_\text{s}+r)^{3}}\frac{Y_1(r-r_\text{s})-5Y_2(r_\text{s}+r)}{[\ln (1+c)-c/(1+c)]}.
\end{eqnarray*}
$$In view of the above equations, it is important to emphasize again that, although the fifth force effect enters the dynamical mass only through the coupling Y1, the lensing mass is affected by both Y1 and Y2. This is expected, since lensing is sourced by the combination of the two potentials Φ and Ψ, equation (3). Note also that, with gravitational lensing observations, one reconstructs the projected surface mass density profile Σ(R), where R is the projected radius from the cluster centre. We refer to e.g. Umetsu (2020) for an explicit discussion of the physics and mathematical framework.


As we describe in Section 2 the case of TNDWs propagating in a plasma with equal mass fraction of 12C and 16O (CO) and density of ρ0,7 ≈ 1,1 which is typical for Type Ia supernovae, is particularly challenging for full-star simulations. In addition to the problem that the burning length-scale is much smaller than the typical cell size, near detailed balance is obtained for many isotopes while NSE is not reached. We test in Section 2 two available one-dimensional (1D) codes: a modified version of the 1D, Lagrangian version of the vulcan code (hereafter V1D; for details, see Livne 1993) and a modified version of the Eulerian, 1D hydrodynamic flash4.0 code with thermonuclear burning (Fryxell et al. 2000; Dubey et al. 2009), against the ρ0,7 = 1 case. We show that with resolutions that are typical for multidimensional full-star simulations, the V1D and the flash results are not satisfactory (up to $50{{\ \rm per\ cent}}$ error in V1D and up to $20{{\ \rm per\ cent}}$ error in flash). We demonstrate in Section 3 the operation of a new numerical scheme for thermonuclear burning that can be implemented in multidimensional full-star simulations. The new scheme allows an accurate calculation of TNDWs in a consistent way (i.e. without pre-describing the position and/or the conditions behind the TNDW) with all thermonuclear burning taking place in situ (without post-processing) for an arbitrary reaction network with hundreds of isotopes. The new scheme contains two important ingredients: (1) a burning limiter (a variant of Kushnir et al. 2013), which guarantees that the thermodynamic variables and the composition are accurate for the resolved scales, while keeping the numerical thermodynamic trajectory for unresolved scales within some controlled error from the true thermodynamic trajectory, and (2) adaptive statistical equilibrium (ASE) burning, which groups isotopes that are in detailed balance into one effective isotope, where the ratio between the isotope abundances inside the group is given from equilibrium conditions (this is an extension of the earlier attempts of Hix et al. 2007; Parete-Koon & Hix 2008; Parete-Koon, Hix & Thielemann 2008, 2010).


Over the last 15 yr, our understanding of how galaxies form and evolve has improved substantially. The combination of technological and theoretical progress has brought this field into a new era: advances in observational techniques (e.g. multi-object spectroscopy, efficient near-infrared CCDs) have enabled multiwavelength observations of large samples of galaxies out to the highest redshifts, while the steady rise of computational power and refinement of numerical techniques have fostered new approaches (e.g. semi-analytic models, hydro-dynamic simulations) to model the formation and evolution of galaxies. This progress has led to a general consensus about the main physical ingredients required to describe the evolution of the galaxy population (e.g. Gonzalez-Perez et al. 2014; Lu et al. 2014; Vogelsberger et al. 2014; Henriques et al. 2015; Schaye et al. 2015): collapse and hierarchical growth of dark matter haloes; accretion of baryons on to these haloes; conversion of baryons into stars; feedback of massive stars and active galactic nuclei (AGN) on star formation; supernova- and AGN-driven outflows of metal-enriched gas; infall of both pristine and metal-enriched gas on to galaxies. The large-scale environment can also affect galaxy properties, in particular, by providing quenching mechanisms (e.g. tidal or ram-pressure stripping, strangulation; e.g. Lagos et al. 2014; Rafieferantsoa et al. 2015), and through its influence on the merger rate (e.g. Lackner et al. 2012; Rafieferantsoa et al. 2015) and galactic spins (e.g. Hahn, Teyssier & Carollo 2010; Codis et al. 2012). Although these different ingredients are present in many galaxy formation models, we still lack a detailed quantification of their respective roles in shaping the properties of galaxies. This is because of the complexity inherent in galaxy physics, which combines gravity, radiation hydro-dynamics, magnetic fields and high-energy physics, acting on scales from less than a pc (e.g. for the formation of proto-stellar cores) to over a Mpc (e.g. for environmental effects). For this reason, ‘first-principles’ simulations of galaxy formation remain far beyond the reach of current computational capabilities. Instead, small-scale baryonic physics is generally subsumed into sub-grid prescriptions, which vary from model to model (e.g. Scannapieco et al. 2012; Haas et al. 2013a,b; Vogelsberger et al. 2013; Torrey et al. 2014; Crain et al. 2015). The appropriateness of such prescriptions, and hence, our ability to understand galaxy formation, must be assessed by comparing simulated and observed galaxy properties.


Thanks to their short spectral lags, cosmological distances, and very high-energy photons, GRBs have been viewed as the most promising sources for studying  LIV effects (Amelino-Camelia et al. 1998; Ellis et al. 2006; Jacob & Piran 2008). In the past, various limits on LIV have been obtained from the spectral time lags of individual GRBs or a large sample of GRBs (e.g., Amelino-Camelia et al. 1998; Coleman & Glashow 1999; Schaefer 1999; Ellis et al. 2003, 2006; Boggs et al. 2004; Kahniashvili et al. 2006; Jacob & Piran 2008; Abdo et al. 2009a, 2009b; Biesiada & Piórkowska 2009; Xiao & Ma 2009; Shao et al. 2010; Chang et al. 2012, 2016; Nemiroff et al. 2012; Ellis & Mavromatos 2013; Kostelecký & Mewes 2013; Vasileiou et al. 2013, 2015; Pan et al. 2015; Zhang & Ma 2015; Wei et al. 2016). In particular, Abdo et al. (2009a) used the time lag of the highest energy (13.2 GeV) photon from GRB 080916C to constrain the linear LIV energy scale (

E

QG
,
1

) and presented a stringent limit of

1.3
×

10

18

 GeV, improving the previous limits by at least one order of magnitude. Abdo et al. (2009b) set the current strictest limits on both the linear and quadratic LIV energy scales by analyzing the arrival time delay between a 31 GeV photon and the low-energy (trigger) photons from GRB 090510. The limits set are

E

QG
,
1

>
(
1
−
10
)

E

Pl

 and

E

QG
,
2

>
1.3
×

10

11

 GeV. However, these limits were based on the rough time lag of a single GeV-scale photon. It is necessary to consider using the true spectral time lags of bunches of high-energy photons (i.e., the lags of high-quality, high-energy light curves) to constrain the LIV. Furthermore, since the emission mechanism of GRBs is still poorly understood, it is difficult to distinguish an intrinsic time delay at the source from a delay induced by propagation in a vacuum to the observer. That is, the method of the flight-time difference used for testing LIV is hindered by our ignorance concerning the intrinsic time delay in different energy bands (see, e.g., Ellis et al. 2006; Biesiada & Piórkowska 2009).


The differences in composition are reflected in the final abundances after r-process nucleosynthesis, shown in Fig. 15. The abundances are obtained using a grid of pre-computed trajectories with SkyNet (Lippuner & Roberts 2017), as discussed in detail in Radice et al. (2018b). We normalize the relative abundances by fixing the height of the third r-process peak (A ≃ 190). We also report Solar r-process abundances from Arlandini et al. (1999) in the same figure. However, we emphasize that even if NS mergers were the sole contributor of r-process elements, there is no reason to expect that every merger should produce ejecta with relative abundances close to Solar. Indeed, variability between the yields of different mergers is required to explain observed abundances in metal-poor stars (Holmbeck et al. 2019). Overall, the simulations span a factor ∼2 in the ratio of A ≃ 100 to third r-process peak. However, the difference between the M0 and M1 at the SR resolution, which is the resolution we use for production simulation, are modest compared to the systematic uncertainties from the unknown NS EOS and to the variability due to the binary mass ratio (Radice et al. 2018b; Nedora et al. 2021b). Clearly, strong conclusions cannot be drawn from this limited study alone, but our simulations suggest that the uncertainties in the yields from mergers arising from neutrino radiation treatment are modest. This is also supported by the results of Foucart et al. (2020). They compared M1 and Monte Carlo neutrino transport in the context of NS mergers and reported only a modest ${\sim }10{{\ \rm per\ cent}}$ difference in the Ye of the ejecta between the two schemes. Interestingly, they reported that M1 systematically overestimates the Ye of the ejecta, so we cannot exclude that the M0+Leakage results are actually more accurate than the results obtained with THC_M1. That said, it is important to emphasize that this comparisons has only been made for the dynamical ejecta and not for the secular ejecta, which we discuss in Section 6.5.


Theoretical single-ionization cross-sections for the Ar2+ ion are compared to the experimental data in Fig. 2. The cross-sections for other two levels, 3P1 and 3P0, are in close agreement with the values from 3P2 and are therefore not presented. The cross-sections of the direct process are evaluated in the potential of the ionized ion. Experimental data obtained by Diserens et al. (1988) and Man et al. (1993) are only presented in Fig. 2. These experimental data showed the lowest contribution from the metastable fraction in the ion beam. Measurements from Muller et al. (1980), Danjo et al. (1984), Mueller et al. (1985), and Matsumoto et al. (1990) demonstrated onsets below the ionization threshold. In addition, the CADW calculations (Loch et al. 2007) are presented for comparison in Fig. 2. The DW cross-sections for the ground level are in good agreement with the experimental results at low energies (Diserens et al. 1988; Man et al. 1993). However, the calculations are below the measurements at the high energies. The theoretical cross-sections are above the experimental values for the two highest levels of the ground configuration from the ionization threshold up to 200–300 eV, which is well beyond the peak region. On the other hand, good agreement with the measurements for these two levels is obtained at the higher energies. It should be noted that the CADW cross-sections (Loch et al. 2007) underestimate the measurements at the low electron energies. Excitations from the 2p subshell are investigated in the CADW calculations of these latter authors, but it is not clear whether the EA channels corresponding to the excitations from the 3s and 3p subshells are included in their study. On the other hand, decays of the excited Ar2+ 2p 53s 23p4 nl (n ≥ 4, l n) configurations lead to states of the Ar4+ ion. The total excitation cross-sections from the 2p subshell amount to ~1 Mb at the peakenergy of 280 eV in our study. Therefore, the excitations from the 2p subshell are not investigated in this work.


We now analyse whether release of this entrapped CO is possible, and quantify it using simple assumptions. In the Fomalhaut ring, the rate at which large planetesimals in the belt are being ground down, which is the same as the rate at which small dust grains are being replenished, has been estimated as 6.0 × 1017 kg yr−1 (Wyatt & Dent 2002). Assuming the planetesimal composition to be similar to that observed in Solar system comets (ice/rock ratio of about 1, CO/H2O ice ratios between 0.4 and 30 per cent), we can use it to estimate the rate of CO release, which corresponds to between 1.9 × 1015 and 8.0 × 1016 kg yr−1. The mechanisms that can be responsible for this release are collisions and photodesorption. Collisions can contribute through dust vaporization (Czechowski & Mann 2007), planetesimal breakup (Zuckerman & Song 2012) and giant impacts (Jackson et al. 2014). Photodesorption, on the other hand, will affect the H2O ice on the surface of solids, in turn exposing CO and allowing it to escape on very short time-scales. The rate at which water vapour will be released is da/dt ∼ 1.5 × 10−3 μm yr−1 (scaled to the distance of the ring, from the result of Grigorieva et al. 2007), where a is the vertical thickness of the layer. The total water mass released in this manner will then be

(19)

\begin{equation}
\frac{\mathrm{d}M_{\rm H_2O}}{\mathrm{d}t} = 4\sigma _{\rm tot}\rho _{\rm gr} \frac{\mathrm{d}a}{\mathrm{d}t},
\end{equation}

where ρgr is the grain density, which we take as that of water ice (∼1 × 10−15 kg μm−3) and σtot is the total cross-section of icy grains in the Fomalhaut ring (in μm2). Under the assumption that all the grains are fully icy, we can use the total cross-sectional area of the Fomalhaut ring (33.7 au2; Wyatt & Dent 2002) to obtain an H2O production rate of 4.6 × 1018 kg yr−1. This is higher than the rate at which mass is being passed down the collisional cascade (again, 6.0 × 1017 kg yr−1), pointing towards a more realistic scenario where grains are not fully icy, but made up of a mixture of ice and rock. We therefore conclude that photodesorption of H2O might contribute significantly to the release of trapped CO gas from planetesimals in the ring, though the extent of this contribution depends on how much of the cross-sectional area of the Fomalhaut disc is icy. In any case, the exact mechanism for CO gas production is unimportant as long as there is one that can feasibly explain its release.


Although the method developed here is not identical to the means by which SONG determines radial velocity from the observed spectra, it simulates the SONG observations sufficiently well. First, the set of fictitious Fe i lines carefully chosen in this work is able to represent the properties of most Fe i lines seen between 4400 and 6900 Å, which constitute a large part of all lines in this wavelength interval. Secondly, the procedure to extract radial velocity from theoretical spectral lines is similar to how radial velocities are typically obtained from observed spectra. Thirdly, the evaluation of our final radial velocity amplitude includes the information of many spectral lines that span the whole range in observation. The major uncertainty in our method is associated with the linear relationship between radial velocity amplitude and equivalent width. Due to the complicated physical processes involved in spectral line formation in a 3D atmosphere (e.g. Asplund et al. 2000), it is difficult to quantify higher order effects beyond the linear relation between $\mathfrak {v}$ and Wλ; that is, the systematic uncertainty of the linear fitting. Nevertheless, it is still illuminating to provide the statistical uncertainty. The statistical uncertainty is quantified using the bootstrap method. The data set considered here is the radial velocity amplitude and equivalent width of 49 fictitious Fe i lines. We conduct 10 000 bootstrap samplings, that is, generating 10 000 data sets each containing 49 randomly sampled $\mathfrak {v}$ and Wλ pairs. A linear regression between equivalent width and radial velocity is then performed for each re-sampled data set. For each fitting, we compute the equivalent width weighted mean radial velocity amplitude for all selected Fe i lines. The bootstrap method therefore results in 10 000 weighted mean radial velocity amplitudes, their mean and variance is the desired final radial velocity amplitude and its statistical uncertainty, which is 72.2 ± 0.5 m s−1 for the 3D solar model and 93.2 ± 0.3 m s−1 for the ϵ Tau model.


Filamentary structures have been found at almost all size scales in the Galaxy. Massive, long filamentary dark clouds are commonly found inside giant molecular clouds (GMCs; e.g. Bergin & Tafalla 2007; André et al. 2014, and references therein), such as the dark clouds L1495 in the Taurus cloud complex (e.g. Chapman et al. 2011) and the Serpens South cloud in the Serpens region (e.g. Dhabal et al. 2018). Filamentary clouds of 4–6 pc length are common, and possibly longer than 10 pc. Some of these clouds are dark at infrared wavelengths. The line–width size relation observed for molecular gas indicates that the thermal Mach number would exceed 10 at such size scales. The long-term survival of these filamentary structures requires a reinforcing mechanism. As shown in the ideal magnetohydrodynamical (MHD) simulations of Li & Klein (2019), a moderately strong, large-scale magnetic field (Alfv$\acute{\rm e}$n Mach number, ${{\cal M}_{\rm A}}\sim 1$) can provide such a mechanism. In the weak-field model with ${{\cal M}_{\rm A}}=10$, the appearance of molecular clouds is clumpy, rather than the long and slender filamentary clouds seen in moderately strong field models. High-resolution images of massive molecular clouds from the Herschel space telescope reveal complex filamentary substructures (e.g. André et al. 2014). The characteristic inner width of molecular filaments found with Herschel is about ∼0.1 pc (Arzoumanian 2011; Arzoumanian et al. 2019). Dense cores, where stars form, are located along or at the intersections of some of these fine substructures (e.g. Könyves et al. 2015; Tafalla & Hacar 2015). From these observations of molecular cloud structures at different size scales, one can visualize an evolutionary sequence of star formation starting from highly supersonic, magnetized GMCs, continuing on to filamentary dark clouds that form within them, and then on to finer filamentary substructures. Fragmentation of these filamentary structures and substructures leads to the clumps and dense cores that form protostellar clusters and protostars. Knowing the physical conditions inside filamentary clouds would provide crucial information on the formation of filamentary substructures and dense cores, and on the origin of the initial mass function (IMF) and the star formation rate. Particularly important is the characterization of the physical properties of transcritical filamentary structures whose mass per unit length is within a factor of ∼2 of the critical line mass ${M_{\rm crit,\, th,\, \ell }}=2\, c_{\rm s}^2/G$ of nearly isothermal cylindrical filaments (e.g. Ostriker 1964; Inutsuka & Miyama 1997), where cs is the isothermal sound speed. Indeed, Herschel observations suggest that transcritical filamentary structures dominate the mass function of star-forming filaments and that their fragmentation may set the peak of the prestellar core mass function and perhaps ultimately the peak of the IMF (André et al. 2019). In this paper, we report the results of polarimetric observations of the pristine section B211 of one such transcritical filament, the Taurus B211/B213 filament, using the High-resolution Airborne Wideband Campera plus (HAWC+) onboard Stratospheric Observatory For Infrared Astronomy (SOFIA). We determine the magnetic field structure inside a filamentary cloud with filamentary substructures.


Star formation is an inefficient process, as evidenced by observed gas depletion times,1 which are two orders of magnitude above the dynamical time, both in galaxies (e.g. Leroy et al. 2017; Utomo et al. 2018), and in individual giant molecular clouds (GMCs) (e.g. Krumholz & Tan 2007; Evans, Heiderman & Vutisalchavakul 2014; Heyer et al. 2016; Pokhrel et al. 2020; Hu et al. 2022). Theoretical models explain this inefficiency through a combination of mechanisms that provide support against gravitational collapse, including turbulence, magnetic fields, stellar feedback, and dynamical stabilization (Krumholz & McKee 2005; Ostriker, McKee & Leroy 2010; Federrath & Klessen 2012; Krumholz, Klein & McKee 2012b; Federrath 2013b; Padoan et al. 2014; Federrath 2015; Burkhart 2018; Meidt et al. 2018; Krumholz & Federrath 2019; Evans, Kim & Ostriker 2022). Recent progress in both theory and observations have highlighted the pivotal role that feedback, especially due to massive (main-sequence) stars, plays in star/star-cluster formation (Krumholz et al. 2014; Krumholz, McKee & Bland-Hawthorn 2019), and the lifecycle of GMCs (see Chevance et al. 2020, 2022a for reviews). This massive-star feedback has been suggested to be largely responsible for limiting the integrated star formation efficiency (ϵ*) to low values in typical environments, where ϵ* is given by
(1)$$\begin{eqnarray}
\epsilon _* = \frac{M_{*}}{M_{\mathrm{gas}}},
\end{eqnarray}$$which quantifies the net efficiency of star formation over the lifetime of a GMC, i.e. the ratio of the final stellar mass M* and the available gas mass in the parent molecular cloud Mgas. Feedback achieves this by (i) disrupting GMCs in order ∼ unity dynamical time-scales, through the momentum and energy carried by feedback processes (e.g. Grudić et al. 2018), and (ii) driving turbulent motions that could further provide support against collapse (e.g. Mac Low & Klessen 2004; Krumholz, Matzner & McKee 2006; Elmegreen 2009; Gritschneder et al. 2009; Federrath et al. 2010; Wibking, Thompson & Krumholz 2018; Gallegos-Garcia et al. 2020; Menon, Federrath & Kuiper 2020; Menon et al. 2021).


The stellar metallicity in the Universe evolves with redshift (Mannucci et al. 2010; Sommariva et al. 2012; Krumholz & Dekel 2012; Dayal, Ferrara & Dunlop 2013; Madau & Dickinson 2014). The metallicity at a high redshift (z > 2) is much smaller in comparison to the low redshift Universe z  2. The first-generation stars contaminate the interstellar medium and cause a chemical evolution of the Universe. We can treat the metallicity evolution with redshift by a relation
(2)$$\begin{eqnarray*}
\log _{10}(Z(z))= \gamma z +\zeta ,
\end{eqnarray*}$$where γ captures the redshift dependence and ζ captures the metallicity value at z = 0 (Mannucci et al. 2010; Madau & Dickinson 2014). This relation captures the metallicity of the parent star or the gas cloud from which a star has formed. It is written to express only a mean evolution of the metallicity. Along with the mean metallicity evolution of the Universe, there is going to be a scatter in the metallicity depending on the galaxy properties. Such a source of uncertainty brings additional stochasticity to the metallicity relation. Currently, a limited number of observations (Gallazzi et al. 2008; Mannucci et al. 2010; Krumholz & Dekel 2012) are available to explore the environment dependence of the metallicity, and most of our current understandings are based on simulations(Genel 2016; Torrey et al. 2019). These studies show that the overall median metallicity dependence of the galaxies at different redshifts can be explained by power form (Pei, Fall & Hauser 1999; Young & Fryer 2007; Torrey et al. 2019). Several studies of GW merger rates and mass distribution are performed (Belczynski et al. 2002; Dominik et al. 2012, 2015; Mapelli et al. 2017; Giacobbo, Mapelli & Spera 2018; Toffano et al. 2019; van Son et al. 2022) which are motivated by these studies and show that the black hole mass distribution can exhibit a redshift dependence. The existence of any stochasticity in the galaxy metallicity distribution will also influence the mass distribution but is currently not well known. However, as the relation given in equation (1) is in terms of the logarithm of metallicity, so the impact of fluctuation around the median value depending on the individual galaxy properties is going to be a small (logarithmic) change. As we are unable to measure the host of the BBH due to a large sky localization error of the BBH, we cannot directly associate the properties of galaxies with BBH source properties. So, we can only infer an ensemble average mass distribution from the GW data and the additional stochasticity (which will depend on the host properties) will appear as an additional uncertainty in the measurement of MPISN. As a result, we consider a median distribution of galaxy metallicity and the dependence of MPISN on it.


Although most galaxies have an implied SFR that scatters within a factor two around the MS, some do show a significantly higher SFR. Those objects also exhibit a higher gas content, shorter gas depletion times (e.g., Genzel et al. 2015; Schinnerer et al. 2016; Tacconi et al. 2013, 2018), and higher dust temperatures (e.g., Magnelli et al. 2014). Likewise, the stellar-light radial distribution is different in these two galaxy populations; while MS galaxies are closely approximated by exponential disks (e.g., Bremer et al. 2018), those above (and below) it exhibit a higher central mass concentration (e.g., Wuyts et al. 2011). Based on this dichotomy and the parametrization of the MS over cosmic time, a scenario has been proposed to explain the evolutionary path of galaxies along the MS. Since the normalization of the MS, the gas fraction of galaxies, and cosmic molecular gas density decrease from z ∼ 2.5 to 0 at a similar pace (e.g., Speagle et al. 2014; Decarli et al. 2016; Tacconi et al. 2018), it is thought that MS galaxies evolved through a steady mode of star formation, possibly regulated by the accretion of cool gas from the intergalactic medium (e.g., Dekel et al. 2009; Kereš et al. 2009; Davé et al. 2010; Hodge et al. 2012; Romano-Díaz et al. 2014, 2017; Feng et al. 2015; Anglés-Alcázar et al. 2017). From theoretical predictions, the scatter of the MS could thus be explained as the result of a fluctuating gas inflow rate that is different in each galaxy (e.g., Tacchella et al. 2016; Mitra et al. 2017). In this context, a galaxy enhances its SFR and moves towards the upper envelope of the MS due to gas compaction. As the gas is depleted, the SFR decreases and the galaxy falls below the MS. As long as a SFG is replenished with fresh gas within a timescale shorter than its depletion time, it will be confined within the MS (Tacchella et al. 2016). On the other hand, the enhanced star formation efficiency of galaxies above the MS has been linked to mergers (e.g., Walter et al. 2009; Narayanan et al. 2010; Hayward et al. 2011; Alaghband-Zadeh et al. 2012; Riechers et al. 2013, 2014) and instability episodes in gas-rich disks (particularly at high redshift; e.g., Davé et al. 2010; Hodge et al. 2012; Wang et al. 2019).


Using the CASSIS2 software, we detected 8 lines of glycolaldehyde, 31 lines of the aGg′ conformer of ethylene glycol, and 26 lines of methyl formate (see Table 1). The glycolaldehyde and methyl formate transitions are taken from the JPL spectroscopic database (Pickett et al. 1998), while the ethylene glycol transitions are from the CDMS catalog (Müller et al. 2001, 2005). The predictions are based on experimental data from Butler et al. (2001), Widicus Weaver et al. (2005) and Carroll et al. (2010) for glycolaldehyde, Christen et al. (1995) and Christen & Müller (2003) for ethylene glycol, and Ilyushin et al. (2009) for methyl formate. The frequencies of five of the detected glycolaldehyde lines were directly measured in the laboratory (Butler et al. 2001). Some of the lines result from a blending of several transitions of the same species. The lines that are strongly blended with other species are not listed in Table 1. All three species are emitted very compactly at the position of the continuum peak (\hbox{$\alpha_{2000}=03^{\rm h}28^{\rm m}55\fs57$}α2000=03h28m55.s57, \hbox{$\delta_{2000}=31\degr14\arcmin37\farcs1$}δ2000 = 31°14′37 .̋ 1). The angular sizes obtained with a circular Gaussian fit in the (u, ν) plane vary from a point source to a maximum of 1″ depending on the transition. The line fluxes listed in Table 1 were measured at the continuum peak position with the CASSIS software using a Gaussian fitting method (Levenberg-Marquardt algorithm). The lines that are contaminated in the wings by other transitions are consequently fitted with a sum of Gaussians. We carefully checked that the derived full widths at half maximum (FWHM) are consistent with the other line measurements. The average FWHM is about 4.5 km s-1 at 317 GHz, and 5.0 km s-1 at 225 and 242 GHz. The widths of the methyl formate lines at 87 GHz are quite broad (~12 km s-1). It is consequently difficult to completely exclude an additional flux contribution from other species. The variation of FWHM with the frequency can be explained by the spectral resolution of the observations that decreases toward the lower frequencies.


It has been generally believed (see Kormendy & Ho 2013; Kormendy 2016, for recent reviews) that, unlike massive galaxies at high redshifts (e.g.,

) whose evolution is driven by a major merger, low-redshift galaxies largely evolve through secular evolution (i.e., in a slow and gentle manner) driven by internal processes within galactic disks and/or by environmental effects such as harassment or nurturing (see Kormendy & Kennicutt 2004 for details; see also Cowie et al. 1996; Conselice et al. 2014 for the cosmic evolution). It has also been generally believed accordingly that most activity (namely active galactic nuclei—AGNs) of supermassive black holes (BHs) at the centers of galaxies at low redshifts (e.g.,

, Cisternas et al. 2011; and even up to

, Kocevski et al. 2012) appear to be fueled by the random accretion of gas via internal, secular processes working close to the BH (say, within a few hundred parsec, Cisternas et al. 2013; also see Hopkins & Hernquist 2006). In these cases, there is no connection between AGN activity and major mergers of their host galaxies. This non-connection appears to be particularly true for low-z AGNs hosting intermediate-mass black holes (IMBHs):12

12
 Following Greene & Ho (2007) and Dong et al. (2012), hereinafter we refer to BHs with

 at the centers of galaxies as “low-mass” or “intermediate-mass” BHs; accordingly, for the ease of narration wherever it is not ambiguous, hereinafter we refer to AGNs hosting low-mass BHs as low-mass AGNs or IMBH AGNs. Normally, we prefer “intermediate-mass BHs” to “low-mass BHs” because of the possible confusion of the latter with the stellar-mass BHs in low-mass X-ray binaries (LMXBs).
 according to the analysis of their images by the Hubble Space Telescope (HST), the majority of low-mass AGNs live in late-type disk galaxies without a classical bulge (Greene et al. 2008; Jiang et al. 2011). Indeed, the accretion rate for an IMBH is so tiny (0.05

 yr−1 even if at the maximum Eddington accretion) that a steady supply of fuel, in the form of stellar mass loss from evolved stars or Bondi accretion of hot gas in the innermost regions, is available readily much more than required (Ho 2008). “In fact, the paradox for local BHs is not whether there is enough fuel to light them up. Rather, the puzzle is how to keep them so dim despite the ready abundance of in situ gas.” (Kormendy & Ho 2013; see also Ho 2008). That is, the real problem seems to be this (see Ho 2009): there must be some mechanism (yet to know) to hinder the innermost fueling process.13

13
 According to W.-M. Gu (2016, private communication), AGN outflows, which can be launched even at very low accretion rates (e.g., Wang et al. 2013; Gu 2015), provide such a mechanism.


After the gravitational collapse and if the total mass of the individual cloud is approximately the mass of the Sun (2 × 1030 Kg; (van Dishoeck 2014), a new astrophysical system forms that is dominated gravitationally by a low-mass protostar known as a young stellar object (YSO). The protostar is in the center of the system and is surrounded by a Keplerian-rotating envelope of dust and gas, that is gravitationally connected and in which the angular momentum is conserved (Cassen & Moosman 1981). The distance to the star will define the surrounding energy, which is dominantly thermal and capable of heating the more distant dustgrain mantles. The processes progressively inject chemical constituents into the gas-phase (Ceccarelli et al. 2001)and determine what types of physical-chemical processes govern in every region of the disk. Circumstellar envelopes oflow-mass protostars (CELMP) are environments that are extraordinarily rich in organic molecules as H2CO or HNCO and in iCOMs such as CH3CN, CH3CHO, and C2 H5OH (Schöier et al. 2002), in addition to other O- and N-bearing complexes (Jørgensen et al. 2012). IRAS 16293-2422 (hereafter IRAS 16293) was chosen for this work as the quintessential example of such an object. IRAS 16293 is adequate for this role because formaldehyde has previously been detected with notable variations in abundances in three differentiated regions of its protodisk (Ceccarelli et al. 2001; Jørgensen et al. 2012; Jaber et al. 2014; van der Wiel et al. 2019). Additionally, glycolaldehyde (C2 H4O2), the simplest form of sugar and one of the first intermediates in the formose reaction (Larralde et al. 1995), has also been detected in this object for the first time in space in 2012 (Jørgensen et al. 2012). The more distant regions of the disk (>4000 AU) contain species such as CO, H2CO, CH3OH, or H2O (van Dishoeck et al. 1995) at low temperatures (10–20 K) where the gas is slightly warmer than the dust grains as they are tightly coupled to them. Energy from collisions among dust and gas is therefore considered to be the main heating mechanism in this region. Nevertheless, the thermal energy generated by those collisions is not enough to activate the formation of molecules with barriers at or below 20 K (0.0397 kcal mol−1). The gas column density of molecular hydrogen has been estimated to be N(H2) = 1.3 × 1023 cm−2 (Ward-Thompson et al. 1999), with a fractional abundance of formaldehyde – N(H2CO)/N(H2) – ~ 4 × 10−10 cm−3 in the gas-phase (Ceccarelli et al. 2001). When the temperature rises above ~ 20 K, CO starts to desorb from the ice grains and enters the gas-phase with an increase of ~ 103 cm−3 in detected densities with respectto H2 (Cassen & Moosman 1981; Aikawa et al. 2015). Formaldehyde starts to deplete from frozen grains at around ~ 40 K and it is fully desorbed at ~ 60 K (Ceccarelli et al. 2001). The additional H2CO mixes with the existing circumstellar mass of gas, which may justify why at ~ 700 AU from the core and at gas temperatures between 80 and 100 K ~ 50 kcal mol−1), the detected fractional abundances of formaldehyde reach N(H2CO)/N(H2) = ~ 4.0 × 10−9 cm−3 (Ceccarelli et al. 2001). This implies an H2 column density that is estimated to be N(H2) = ~ 5.0 × 1021 cm−2 (Bottinelli et al. 2014). The inner part of the envelope at ~ 150 AU in a region with temperatures of 100–150 K has a higher density of formaldehyde with an N(H2CO)/N(H2) of ~ 10 × 10−7 cm−3, as well as an increase in fractional abundances for H2O (which desorbs from iced mantles at ~80 K) (Ceccarelli et al. 2001). This region also produces new molecules principally due to the thermal energy emitted from the YSO (99.99 kcal mol−1). One example is trans-HONO. This chemical compound has recently been detected for first time in space and in this part of the disk (Coutens et al. 2019). Its proposed formation has inspired some reactions proposed in this work that may lead to H2CO. In this region, a column density for molecular hydrogen is considered like that in region II (N(H2) = ~ 5.0 × 1021 cm−2). The three regions of IRAS 16293 dictate the physical parameters for the presentation of our computations which are defined as follows:

Region I/d ~ 4000 AU, Tgas = 20 K, Pgas = 2.29 × 107 K cm−3;
Region II/d~ 700 AU, Tgas = 80 K, Pgas = 7.06 × 107 K cm−3;
Region III/d~150 AU, Tgas = 150 K, Pgas = 6.08 × 108 K cm−3.


Various groups around the world have proposed different models to explain the formation of these two peculiar classes of SNe Ia. Sub-Chandrasekhar limiting-mass WDs were believed to be formed by merging two sub-Chandrasekhar mass WDs (double-degenerate scenario), leading to another sub-Chandrasekhar mass WD, exploding due to accretion of a helium layer (Hillebrandt & Niemeyer 2000; Pakmor et al. 2010). On the other hand, the super-Chandrasekhar WDs were often explained by incorporating different physics, such as a double-degenerate scenario (Hicken et al. 2007), presence of magnetic fields (Das & Mukhopadhyay 2013, 2014), presence of a differential rotation (Hachisu et al. 2012), presence of charge in the WDs (Liu et al. 2014), ungravity effect (Bertolami & Mariji 2016), lepton number violation in magnetized WD (Belyaev et al. 2015), generalized Heisenberg uncertainty principle (Ong 2018), and many more. However, none of these theories can self-consistently explain both of the peculiar classes of WDs. Moreover, each of these has some caveats or incompleteness, mostly based on the stability (Komatsu et al. 1989; Braithwaite 2009). Furthermore, numerical simulations showed that a merger of two massive WDs could never lead to a mass as high as 2.8M⊙ owing to the off-center ignition and formation of a neutron star rather than an (overluminous) SN Ia (Saio & Nomoto 2004; Martin et al. 2006). Hence, all the conventional pictures failed to explain the inferred masses of both the sub- and super-Chandrasekhar progenitor WDs and also both classes of progenitor WDs simultaneously by invoking the same physics. Moreover, each of the theories can explain only one regime of SN Ia, but it seems more likely that the nature would prefer only one scenario/physics to exhibit the same class of SNe. Whether it be an under- or overluminous SN Ia, other physics such as the presence of Si, etc., remains the same. Therefore, we seem to require just one theory to explain all the SNe Ia.


Currently, there is an extensive data base of observations of magnetic activity on the main-sequence stars (see e.g. reviews by Donati & Landstreet 2009; Reiners 2012). Cool stars with outer convective envelope are of particular interests because they are Sun-like. It is believed that the magnetic activity on the solar-like stars results from large-scale dynamo processes driven by turbulent convection and rotation (Brandenburg & Subramanian 2005). Observations (e.g. Böhm-Vitense 2007; Donati & Landstreet 2009; Katsova et al. 2010; Saar 2011; Katsova, Livshits & Mishenina 2013; Marsden et al. 2014; Vidotto et al. 2014), as well as the 2D mean-field models of the angular momentum balance (Ruediger 1989; Kitchatinov & Rüdiger 1999; Kitchatinov 2013) and the 3D numerical simulations (Miesch & Toomre 2009; Hotta & Yokoyama 2011; Guerrero et al. 2013b; Brun et al. 2014; Käpylä, Käpylä & Brandenburg 2014) show that parameters of the differential rotation and convection, e.g. the typical size and turnover time of convective flows, depend on the general stellar parameters, such as mass, age, the spectral class and the rotation rate. The mass of a star and its Rossby number, which is the ratio of the period of rotation and a typical turnover time of convection, are likely the most important parameters governing the stellar dynamo (Donati & Landstreet 2009; Morin et al. 2013). The diagram 3 in the paper by Donati & Landstreet (2009) shows an increase of the magnetic activity with decrease of the Rossby number and the mass of a star. These parameters determine the topology of the large-scale magnetic field, as well. It is found that the axisymmetric solar-type dynamo can operate in stars with mass about 1 M⊙, and with periods of rotation longer than 10 d. Observations, also show pieces of evidence that solar analogues with period of rotation smaller than 10 d, may have the substantial non-axisymmetric components of the large-scale magnetic field (Donati & Landstreet 2009; Folsom et al. 2014).


Even though the understanding of high-mass star formation has made tremendous progress over the past decade (Beuther et al. 2007; Zinnecker & Yorke 2007; Klessen 2011; Tan et al. 2014), the initial conditions are still poorly constrained. It is known, that stars predominantly form in clusters (Lada & Lada 2003), which is especially true for high-mass stars (e.g., de Wit et al. 2005; Gvaramadze et al. 2012). On observational grounds, the first evolutionary stage of such clusters spawning future high-mass stars might in general be termed pre-protocluster cores (Evans et al. 2002). Their appearance and morphology can be different depending on the environment. Objects of this sort have been found in several (sub-)millimeter surveys (e.g., Klein et al. 2005; Beuther & Sridharan 2007). Also the ISO satellite mission has resulted in a list of such objects revealed by far-infrared observations at 170 μm within the ISO Serendipity Survey (ISOSS, Bogun et al. 1996). For these clumps, cold dust and gas temperatures have been established in follow-up investigations (Krause et al. 2003, 2004). The most prominent variety of young massive clumps are the infrared dark clouds (IRDCs). They were discovered as dark silhouettes against the galactic background at 8 and 15 μm with the Midcourse Space Experiment (MSX, Egan et al. 1998) and ISO (Perault et al. 1996). ISO, MSX, the Spitzer Space telescope, and further (sub-)millimeter observations have helped for studying these objects in detail (e.g., Simon et al. 2006; Hennemann et al. 2008; Vasyunina et al. 2009; Ragan et al. 2009; Peretto & Fuller 2009). Because IRDCs can only be seen in absorption against a strong infrared background, their location is mainly within the disk toward the inner quadrants of our Milky Way, whereas ISOSS sources are more widely distributed. In fact, the Milky Way midplane had to be avoided for ISOSS because of saturation. IRDCs can have masses up to several thousand solar masses, and the more massive ones are explicitly thought to be progenitors of star clusters (e.g., Wyrowski 2008; Rathborne et al. 2010; Henning et al. 2010; Ragan et al. 2012a). On average the ISOSS sources have masses less than 500 M⊙ (Hennemann et al. 2008). While their masses often are somewhat lower than for large high-contrast IRDCs, the general characteristics such as low temperature and low turbulent linewidth are similar to IRDCs. Therefore, ISOSS sources are also considered to be early evolutionary stages of intermediate- to high-mass star formation in more isolated regions. Consequently, the better characterization of such clumps (ISOSS and IRDCs) is an important step toward understanding the initial conditions of high-mass star formation.


For the first time, we detected [C ii] 158-μm emission from a GRB host galaxy at z > 2. This is the second detection of [C ii] 158-μm emission among known GRB host galaxies, following GRB 980425 (Michałowski et al. 2016). The [C ii] 158-μm fine structure line is the dominant cooling line of the cool interstellar medium, arising from photodissociation regions (PDR) on molecular cloud surfaces. It is one of the brightest emission lines from star-forming galaxies from FIR to metre wavelengths, almost unaffected by dust extinction. [C ii] 158-μm luminosity, L[C II], has been discussed as an indicator of SFR (e.g. Stacey et al. 2010). If L[C II] scales linearly with SFR, the ratio to FIR luminosity, L[C II]/LFIR, is expected to be constant, since LFIR is a linear function of SFR (e.g. Kennicutt 1998a). However, LC II/LFIR is not constant, but declines with increasing LFIR, known as the ‘[C ii] deficit’ (e.g. Luhman et al. 1998, 2003; Malhotra et al. 2001; Sargsyan et al. 2012; Díaz-Santos et al. 2013, 2017; Spilker et al. 2016). The [C ii] deficit persists when including high-z galaxies (e.g. Stacey et al. 2010; Wang et al. 2013; Rawle et al. 2014). In Fig. 5, we compare the [C ii] deficit in the GRB 080207 host and other star-forming galaxies. Two GRB hosts are shown by stars: GRB 080207 (orange star) and 980425 (blue star). The comparison sample is compiled from the literature up to z ∼ 3 (Malhotra et al. 2001; Cormier et al. 2010, 2014; Ivison et al. 2010; Stacey et al. 2010; Sargsyan et al. 2012; Farrah et al. 2013; Magdis et al. 2014; Brisbin et al. 2015; Gullberg et al. 2015; Schaerer et al. 2015). Active galactic nuclei are separated from star-forming galaxies based on either (i) the explicit description in the literature or (ii) EWPAH 6.2μm  0.1 (Sargsyan et al. 2012). As reported by previous studies (e.g. Maiolino et al. 2009; Stacey et al. 2010), high-z galaxies are located at a different place from local galaxies in the L[C II]/LFIR–LFIR plane.


X-rays emerging from active galactic nuclei (AGNs) are the result of an inverse-Compton process occurring in the proximity of the central black hole (BH), where optical-UV photons arising from the accretion disc are inverse-Compton scattered by hot electrons in an optically thin, compact corona (e.g. Haardt & Maraschi 1991, 1993; Haardt et al. 1994, for details on the two-phase model). Such a Comptonisation mechanism accounts for the power-law-like shape of the X-ray primary continuum emission and the high-energy roll-over observed in various nearby AGNs (e.g. Nicastro et al. 2000; Perola et al. 2002; De Rosa et al. 2002; Molina et al. 2009, 2013; Malizia et al. 2014; Ricci et al. 2018). Broadband X-ray spectral investigations are of primary importance in studying AGN Comptonisation properties. Indeed, as extensively discussed in the literature (e.g. Ghisellini 2013) both the photon index (Γ) and high-energy cut-off (Ec) of the X-ray primary emission depend on the intrinsic properties of the Comptonising medium, namely its temperature, optical depth, and geometry. Therefore, the interplay between coronal parameters and the AGN X-ray spectral shape has been the object of several investigations, especially with observatories capable of detecting hard X-rays. Dadina (2007), using BeppoSAX data, collected and studied the photon index and Ec of a sample of AGNs (see also Perola et al. 2002, for previous results), while similar works were performed on INTEGRAL data (e.g. Bassani et al. 2006; Molina et al. 2009; Malizia et al. 2014), and, in the context of the BAT AGN Spectroscopic Survey, by Ricci et al. (2018). Subsequently, NuSTAR (Harrison et al. 2013), thanks to its unprecedented effective area above 10 keV, greatly helped in studying the exponential cut-offs of the nuclear continuum in several AGNs (see e.g. Fabian et al. 2015, 2017; Tortosa et al. 2018a). These space missions gave rise to a substantial corpus of high-energy cut-off and photon index measurements.


Even though Ω is only a function of Ψ, the physical connection between the two at the foot-point is not very obvious.9

9
They may be constrained by proper boundary conditions (e.g., Contopoulos et al. 2013).
 Likely most researchers, here we assume that they generally follow the ansatz
13

In the region approaching the polar axis, one expects that the magnetic flux vanishes and thus

. Therefore, in the region near the polar axis, mathematically λ ≥ 0 is required to guarantee a finite value of Ω (notice that λ > 0 is likely unphysical). Magnetic field lines just around the polar axis would connect to the central CO, which is expected to rotate with a roughly constant angular velocity10

10
In the case of threading a BH, magnetic field lines at different polar angles may not rotate with exactly the same frequency; see discussion in Section 8.
 (see details below), implying λ = 0 for the case of magnetic field lines threading the CO. In the region where magnetic field lines are threading the AD, one usually expects λ  0. The choice of Φ, which determines the toroidal magnetic field, is important (e.g., Camenzind 1987; Sulkanen & Lovelace 1990). Physically, it is the rotation that develops a toroidal magnetic field and a poloidal electric field in the lab frame (or the inertial frame), which seems to imply that Φ would not be chosen arbitrarily and would be self-consistently determined by the MHD equations, given the Ψ and Ω specified (e.g., Beskin & Tchekhovskoy 2005; Contopoulos et al. 2013; Huang et al. 2019, 2020). To solve the rotation term, Equation (11), one can see that in the case of
14

the function Ψ can be variable-separated,
15

where the subscript “r” denotes “rotation.” The negative sign in Equation (14) exists to guarantee that there is always a swept-back magnetic field line with respect to the direction of rotation. This relation implies that the strength of the toroidal magnetic field would be proportional to the angular velocity and the enclosed magnetic flux, which can be reasonably understood because it is the rotation of the poloidal magnetic field (related to the magnetic flux) that produces the toroidal magnetic field. In terms of Equation (15), Equation (11) can be expressed as
16

where

,

,

, and

. The left-hand side of the above equation is only a function of r (the r component), and the right-hand side is only a function of θ (the θ component). Both equal the same constant. Let us set this constant as

, a choice making the solution of the r component equation concise:
17

We introduce a new variable,
18

which makes the θ-component equation become
19

where

 and

. Let us first consider its asymptotic properties. In the case of θ ≪ 1 (keeping in mind

 when

), the leading-order terms give
20

It is clear that this equation has a solution of the form
21

Therefore, a magnetic field line forms a “general parabolic” configuration11

11
In the case of ν = 0, Ψ is only a function of θ, which presents a monopole solution. The case of ν = β leads to Ψ ∝ R, which gives a cylindrical solution.
 at θ ≪ 1, that is, Ψ ∝ rνθβ. Now, let us consider what value β might take. In order to do this, we have to consider the higher-order terms of Equation (19), which may be comparable to the nonrotation term. Therefore, we have to consider the original Equation (10). Let us substitute a general form of

 (with coefficients a1, a2, a3... to be determined) into the original Equation (10). One gets the first two leading-order terms (note that a1 = 0):
22

It can be seen that, for any values of β, λ, and ν, one always has an a2 to make the second term vanish, whereas the first term yields β ≈ 2 (β = 0 corresponds to the nonrotation case). We note that this choice cannot guarantee that the higher-order terms vanish, so this solution is only an approximation. In the case of β  2ν (i.e., ν ≳ 1), the first term dominates over the second term in Equation (22), whereas in the case of β > 2ν (i.e., ν ≲ 1), the second term dominates over the first one. We therefore expect that the approximation β ≈ 2 would be more efficient in the former case (i.e., ν ≳ 1). In another aspect, from Equation (2), one has

, which implies that the choice of β = 2 can avoid singularity or vanishing magnetic fields on the magnetic polar axis (either Bθ or Bϕ vanishes). This choice is also supported by some numerical simulations, which showed that this choice corresponds to a minimum torque (i.e., the least amount of toroidal magnetic fields) and is the one picked by a “real” system (see, e.g., Michel 1969; Contopoulos 1995; Narayan et al. 2007; Tchekhovskoy et al. 2008). Although the coefficient β ≈ 2 is derived asymptotically (θ ≪ 1), which must hold throughout the jet region because Φ, Ω, and Ψ are each conserved along magnetic field lines. In Appendix A, we come back to this question and present another proof of the relation Φ ≈ −2ΩΨ based on a more physical consideration, showing that this relation is only valid in the limit of a highly magnetized jet flow (e.g., Lyubarsky 2009).


In this context, MWC 758 (HD 36112) offers a unique environment to probe the existence of planetary companions and to explore the connection between disk structures and planet formation. MWC 758 is a young stellar object (3.5 ± 2 Myr, Meeus et al. 2012) at a distance of 151
$^{+9}_{-8}$

151

−9

+8

4

 pc (Gaia Collaboration 2016) close to the edge of the Taurus star forming region (stellar properties are given in Table 1). Measurements of resolved CO emission around the star determined the stellar mass to be 2.0 ± 0.2 M⊙ and the disk to have an inclination of 21° ± 2° and a position angle of the semi-major axis of 65° ± 7° (Isella et al. 2010). The mass and age estimates were based on the previously adopted hipparcos distances of 200 pc (van den Ancker et al. 1998) and 279 pc (van Leeuwen 2007). Given the revised Gaia distance, the star could be older and lighter than previously thought. In this paper, we assume a stellar mass of 1.5 ±0.2 M⊙, reflecting the scaling of the dynamical mass estimate to the new Gaia distance. Based on its SED, MWC 758 has been classified as a pre-transition disk (Grady et al. 2013). Although a cavity of 55 astronomical units (au) in radius has been inferred from dust millimeter emission (Andrews et al. 2011), infrared polarized intensity observations have found no clear evidence for a cavity in scattered light (Grady et al. 2013; Benisty et al. 2015). Using Ks-band (2.15 μm) direct imaging andH-band (1.65 μm) polarimetric imaging with the High-Contrast Instrument with Adaptive Optics (HiCIAO) at the Subaru Telescope, Grady et al. (2013) detected two spiral arms and polarized light down to 0.′′ 1 (15 au) from the star. Recent VLT Spectro-Polarimetric High-contrast Exoplanet REsearch (SPHERE) observations in the Y band (1.04 μm) have confirmed the presence of scattered light at least down to 14 au (Benisty et al. 2015). The asymmetries observed by Isella et al. (2010) in the mm-dust distribution and in CO emission suggest that the disk may be gravitationally perturbed by a low-mass companion orbiting within a radius of 23 au (assuming a distance of 151 pc). The asymmetric cm-dust distribution was shown to follow the location of the mm-dust (Marino et al. 2015a), hinting towards the hypothesis of a dust trap, which could also be created by a companion in the gap through the Rossby wave instability (e.g., Pinilla et al. 2012b). Hydrodynamical simulations of the disk indicate that the observed spirals could instead be launched by a massive planet or brown dwarf at larger separations (~ 100 au based on the revised Gaia distance, Dong et al. 2015b). The presence of stellar companions down to a mass limit of 12 MJup at 0.′′ 25 and of planets outside 0.′′5 (5 MJup at 0.′′ 5, and 3 MJup at 1′′ , according to the BT-SETTL models; Allard et al. 2012) has been ruled out based on a combination of sparse aperture masking observations at L′ band and angular differential imaging at K′ and Ks bands (Grady et al. 2013).


It is also possible that these clusters have a smaller extent and can just be missed by our extent selection as our detection algorithm sets the extent to zero if it is smaller than 6 (Brunner et al. 2022). Following the method presented in Ghirardini et al. (2021a), we estimated several dynamical properties of the clusters in the point source sample and compared them with the extent-selected sample presented in Ghirardini et al. (2021a). In Fig. 5 we compare the distributions of the core radii (Rcore) constrained by the V06 model and the concentration parameter (cSB) between these two samples. The concentration parameter is defined as the ratio of the surface brightness within 0.1 R500 to the surface brightness within R500 (Ghirardini et al. 2021a; Santos et al. 2008; Maughan et al. 2012). Intuitively, the expectation is that the smaller the core radius, the more compact the cluster. The left panel of Fig. 5 clearly shows that the clusters in the point source sample have relatively smaller core radii, hence the emission is more concentrated in a smaller area. Consistently, the concentration of the point source sample shows a clear excess in higher values than the extent-selected sample, indicating that a significantly larger fraction of cool-core clusters and clusters host a central AGN. We performed the same experiment by applying cuts in flux 1.5 × 10−14 ergs s−1 cm−2 and in detection likelihood to test whether the clusters are missed by the extent selection because they are fainter and/or more compact than the extent-selected clusters. The distribution of number density, core radius, and concentration parameter remains the same, indicating that the population of clusters in the point source catalog is more compact than the extent-selected sample. The extent-selected sample does not show a clear bias toward cool-core clusters or clusters with a central AGN, but contains the fraction of cool-cores is similar to that of SZ surveys (Ghirardini et al. 2021a). In this sample, we observe the opposite trend.


Lastly, we model the evolution of the ejecta discussed in Section 2.2 into circumstellar profiles discussed in Section 2.3. We use our cosmic-ray hydrodynamics code, hereafter called ChN to model the evolution of the ejecta to an age of

t

SNR

=
400

 yr. ChN is a Lagrangian hydrodynamics code that includes a prescription for diffusive shock acceleration (DSA; Ellison et al. 2007; Lee et al. 2012). We have modified the code to include the effects of DSA on non-equilibrium ionization (Patnaude et al. 2009, 2010) and have coupled the code to SN ejecta models (Lee et al. 2014; Patnaude et al. 2015). We have also included radiative losses via forbidden line cooling (Lee et al. 2015). This effect will be important in the evolution of the SN shock with a nearby CSM shell, or if we choose to model the radiative shock that could form in the ejecta during early SN evolution (Nymark et al. 2006). However, we begin our simulations at an age of 5 yr, and over the lifetime of the simulation the shocks remain adiabatic, so we do not consider the radiative shock model presented in our previous work here. Since ChN couples nonlinear particle acceleration to the SNR shock dynamics, we are able to reproduce the broadband thermal and nonthermal emission (Ellison et al. 2010, 2012; Castro et al. 2012; Slane et al. 2014; Lee et al. 2013). The diffusive shock acceleration process is an integral part of ChN, and some injection of thermal particles into the acceleration process is always assumed. Here we set the injection parameter to the test particle limit, though we note that the interaction of a strong shock with a massive CSM shell or cloud will lead to enhanced particle acceleration (e.g., Ellison et al. 2012; Lee et al. 2014), and the differing CSM configurations, combined with the differing ejecta profiles and compositions, may result in differences in the broadband nonthermal emission. The study of nonthermal emission in evolving SNe is sufficiently broad that we defer its study to future papers.


In addition to being useful laboratories for studying the evolution of galaxies in dense environments, galaxy clusters can be effective gravitational lenses (for a review, see Kneib & Natarajan 2011). Gravitational lensing is the deflection of light by intervening mass, that can produce highly magnified and distorted images of background galaxies. The large mass and solid angle covered by highly concentrated galaxy clusters make them ideal gravitational lenses, which can be used as ‘cosmic telescopes’ to observe very distant galaxies (e.g. Richard et al. 2009). The amplification of sources through gravitational lensing has been extremely effective in observing faint, distant sources across the electromagnetic spectrum, including continuum and spectral line emission in the radio domain (e.g. Carilli & Walter 2013, and references therein). While molecular gas has been studied across the Universe up to z ∼ 1 and beyond, H i emission remains undetected through lensing, with two searches for galaxy–galaxy lensed H i sources at z ∼ 0.4 (Hunt, Pisano & Edel 2016; Blecher et al. 2019). Gravitational lensing conserves surface brightness while increasing the solid angle of the source, boosting the observed flux. This amplification μ can facilitate the detection of unresolved lensed sources, which maximises their detection probability, and reduces the integration time needed for a given source by μ2. Next-generation cm-wavelength interferometers are now sensitive enough to observe the higher redshift H i Universe, and the detection of gravitationally lensed H i is probable in new surveys (Deane, Obreschkow & Heywood 2015). The detection of lensed H i behind intermediate-redshift galaxy clusters will provide a deep cosmic view of H i emission in galaxies, pre-SKA era, within a fraction of the observation time of unlensed detections. Successful lensed H i detections, along with readily detected CO emission lines, will constrain the H i/H2 ratio at these redshifts, an important parameter in understanding galaxy evolution over cosmic time (Obreschkow & Rawlings 2009).


The present Universe is full of galaxies that come in various shapes and sizes with different mass, luminosity, colour, star formation rate (SFR), metallicity, and H i content. Understanding the galaxy properties and their evolution is an important goal of cosmology. The modern galaxy surveys, 2dF Galaxy Redshift Survey (2dFGRS; Colless et al. 2001) and Sloan Digital Sky Survey (SDSS; Strauss et al. 2002) reveal that the galaxies are distributed in the cosmic web (Bond, Kofman & Pogosyan 1996), which is an interconnected web-like network comprising of different types of environments such as filaments, sheets, knots, and voids. The galaxy properties vary across the different environments in the cosmic web. For example, the well-known density–morphology relation reveals that the ellipticals are preferably found inside the dense groups and clusters, whereas the spirals are intermittently distributed in the fields (Hubble 1936; Zwicky 1968; Oemler 1974; Dressler 1980; Goto et al. 2003). These findings are further supported by other studies with two-point correlation function (Willmer, da Costa & Pellegrini 1998; Brown, Webstar & Boyle 2000; Zehavi et al. 2005), genus statistics (Hoyle et al. 2002; Park et al. 2005), and filamentarity (Pandey & Bharadwaj 2005, 2006) of the galaxy distribution. It is now well known that many other galaxy properties are strongly sensitive to their environment (Davis & Geller 1976; Guzzo et al. 1997; Zehavi et al. 2002; Blanton et al. 2003; Einasto et al. 2003a; Hogg et al. 2003; Kauffmann et al. 2004; Mouhcine, Baldry & Bamford 2007; Koyama et al. 2013). The formation and evolution of galaxies are known to be driven by accretion, tidal interaction, merger, and various other secular processes. These physical processes are largely determined by the environment of the galaxies. The environment thus play a central role in the formation and evolution of galaxies and the study of the environmental dependence of the galaxy properties provides crucial inputs to the theories of galaxy formation and evolution.


We recall that the velocity anisotropy profile is given by a combination of the radial and tangential velocity dispersion:

(15)

\begin{equation}
\beta _{\rm ani}(r)\equiv 1-\frac{\bar{v_{\theta }^2}(r)}{\bar{v_r^2}(r)}\,.
\end{equation}

Due to the lack of observational constraints on this quantity, the first anisotropy profiles discussed in the literature were based on analytical studies aiming at building dynamical models (in spherical symmetry) with self-consistent stellar phase-space distribution functions. Many such models have simple anisotropy profiles that are either constant or change from isotropic near the centre to radial at large radius (e.g. Osipkov 1979; Merritt 1985, see below). More recently, indications of radial anisotropy in the outer regions of DM haloes have been obtained from numerical simulations (e.g. Diemand, Moore & Stadel 2004). In the inner region, a strong anisotropy can be generated by dynamical formation and evolution processes. To better describe these profiles, Baes & van Hese (2007) introduced a technique to construct dynamical models with arbitrary mass density and anisotropy profiles. These three different families of anisotropy profiles are described below and will be explored in Section 5.

The constant anisotropy modelling (e.g. used by Charbonnier et al. 2011) simply reads

(16)

\begin{equation}
\beta _{\rm ani}^{\rm Cst}(r)=\beta _0.
\end{equation}

The Osipkov–Merritt profile (Osipkov 1979; Merritt 1985) is parametrized as

(17)

\begin{equation}
\beta _{\rm ani}^{\rm Osipkov}(r)=\frac{r^2}{r^2+r_a^2},
\end{equation}

with a single free scale parameter ra which locates the transition from βani = 0 in the inner parts (isotropic) to 1 at large radii (full radial anisotropy).
The Baes and van Hese profile (Baes & van Hese 2007) is more general and is written as

(18)

\begin{equation}
\beta _{\rm ani}^{\rm Baes}(r) =\frac{\beta _0 + \beta _\infty (r/r_a)^\eta }{1+(r/r_a)^\eta }\,,
\end{equation}

where the four parameters are the central anisotropy β0, the anisotropy at large radii β∞, and the sharpness of the transition η at the scale radius ra. The Osipkov–Merritt profile is recovered when using β0 = 0, β∞ = 1 and η = 2.


We have performed the first detailed dissection of the stellar populations of the Milky Way disc in age, [Fe/H] and $\mathrm{[ \alpha \mathrm{/Fe]}}$ space, bridging the gap between the detailed observational understanding of MAPs (e.g. Bovy et al. 2012b, 2016b) and the plethora of studies of co-eval stellar populations in simulated galaxies (e.g Bird et al. 2013; Stinson et al. 2013; Martig et al. 2014a). We have placed novel constraints on models for the formation of the Milky Way disc by combining detailed density models fit to the mono-age, mono-[Fe/H] populations of the low and high $\mathrm{[ \alpha \mathrm{/Fe]}}$ disc, with surface mass density contributions calculated on the basis of these density fits and stellar evolution models. We summarize our key results as follows:
Radial and vertical profiles: The mono-age, mono-[Fe/H] populations of the $\mathrm{[ \alpha \mathrm{/Fe]}}$ poor disc are well fitted by a radially broken exponential, with a peak radius, Rpeak, that varies as a function of age and [Fe/H]. We find that the distance between Rpeak's of the low and high [Fe/H] populations increases with age, which we interpret as evidence for a decreasing [Fe/H] gradient with time (e.g. Anders et al. 2017). The radial variation of the stellar surface density of the high $\mathrm{[ \alpha \mathrm{/Fe]}}$ mono-age populations is found to have insignificant breaks, and they are better fit by a single exponential in this disc region. As these populations are the oldest, this may be a sign of the disc evolution washing out the density peak over time, or may point to a different formation scenario for high $\mathrm{[ \alpha \mathrm{/Fe]}}$ stars, where no density peak ever existed. These findings are in good agreement with earlier studies of MAPs (Bovy et al. 2016b). We measure an average high $\mathrm{[ \alpha \mathrm{/Fe]}}$ population scalelength of hR, in = 1.9 ± 0.1 kpc, and find scaleheights between 600 and 1000 pc, in good agreement with current measures of the $\mathrm{[ \alpha \mathrm{/Fe]}}$ rich disc scalelength and scaleheight (e.g. those outlined in Bland-Hawthorn & Gerhard 2016).Profile broadening: We show that the radial surface density profile of the low $\mathrm{[ \alpha \mathrm{/Fe]}}$ populations broadens with age in a given [Fe/H] bin, which we interpret as evidence of the gradual dispersal of mono-[Fe/H] populations, presumably due to radial migration and radial heating. The variation in shape of the broken exponential profile changes differently depending on the population [Fe/H], with low [Fe/H] populations inner profiles flattening faster, whereas the high [Fe/H] outer profiles flatten faster. We interpret this effect as tentative evidence for [Fe/H] dependent radial migration arising from pre-existing [Fe/H] gradients in the star-forming disc. We showed that our results qualitatively reproduce those of Hayden et al. (2015), finding a skewed MDF that varies as a function of R.Flaring: We find that flaring seems to be present in almost all mono-age populations, at differing levels. We have shown that the inverse flaring scalelength Rflare− 1 increases with age, meaning that the youngest populations flare most strongly. This finding appears inconsistent with that above, under the assumption that flaring is the result of radial migration. However, these results may be reconciled by invoking a more active accretion history in the early life of the disc, which could have suppressed flaring (e.g. Minchev et al. 2014b).The surface-mass density at R0: We have measured the surface mass density at the solar radius for each mono-age, mono-[Fe/H] population, finding a total surface mass density of $\Sigma _{R_0, {\rm tot}} = 20.0_{-2.9}^{+2.4}\mathrm{(stat.)}_{-2.4}^{+5.0}\mathrm{(syst.)}\ \mathrm{M_{{\odot }} \ pc^{-2}}$. Before allowing for systematics, this value is less than current estimates (e.g. Flynn et al. 2006; Bovy et al. 2012a; McKee et al. 2015), however, the systematic uncertainties are large, mainly due to a mismatch between the log g scales in APOGEE and the PARSEC models, and as such, we find our value to be consistent within the uncertainties. The relative contribution of high to low $\mathrm{[ \alpha \mathrm{/Fe]}}$ populations, $f_\Sigma$, is 18 per cent ± 5 per cent, which is consistent with existing measurements (e.g. Bland-Hawthorn & Gerhard 2016).The hZ distribution at R0: The shape of the mass-weighted hZ distribution found by this study is in good agreement with that of Bovy et al. (2012a), calling into question the existence of a vertical structural discontinuity in the Milky Way disc. The reconciliation of this finding with the discontinuity in chemical space (e.g. the bimodality in $\mathrm{[ \alpha \mathrm{/Fe]}}$ at fixed [Fe/H]: Nidever et al. 2014; Hayden et al. 2015) may shed new light on our understanding of the formation of the Galactic disc.The surface-mass density profile of the Milky Way: We have found the combined (from mono-age, mono-[Fe/H] populations at low and high $\mathrm{[ \alpha \mathrm{/Fe]}}$) surface-mass density-weighted profiles of the Milky Way disc as a function of $\mathrm{[ \alpha \mathrm{/Fe]}}$, age and [Fe/H], and found that the total surface density is also described by a broken exponential. We find that our results fail to determine the sign of the inner exponential to high significance out to ∼10 kpc, but detect a turnover to a declining exponential, at high significance, thereafter. We find evidence of a radial mean age and [Fe/H] gradient driven by the changing dominant population as a function of radius. A detailed comparison of these findings with numerical simulations is necessary for a proper interpretation. Our finding of a decline in stellar density may be consistent with that found in other studies (e.g. Reylé et al. 2009; Sale et al. 2010), albeit at shorter radii.


Ring systems are a ubiquitous feature in planetary systems – all the gas giants in the Solar System have ring systems around them of varying optical depths (see, e.g., Tiscareno 2013; Charnoz et al. 2018), and ring systems have been detected around minor planets (e.g., Chariklo; Braga-Ribas et al. 2014), so it is reasonable that exoplanets and substellar objects host ring systems as well. Long-period eclipsing binary star systems, where one star is surrounded by an extended dark disk-like structure that periodically eclipses the other component, have already been observed, such as EE Cep (Mikolajewski & Graczyk 1999), ϵ Aurigae (Guinan & Dewarf 2002), and TYC 2505-672-1, with a companion period of 69 yr (Lipunov et al. 2016; Rodriguez et al. 2016). A large ring-like structure around a substellar companion was proposed to explain observations from 2007 from the J1407 system (Mamajek et al. 2012). 1SWASP J140747.93-394 542.6 (V1400 Cen; hereafter called “J1407”) is a young, pre-main-sequence star in the Sco-Cen OB association (Mamajek et al. 2012) with spectral type K5 IV(e) Li and is similar in size and mass to the Sun. In 2007, it displayed a complex symmetric dimming pattern of up to ~ 3 magnitudes during a 56 day eclipse. This has been attributed to the transit of a substellar companion (called “J1407 b”) with a mass of 60–100 MJup (Rieder & Kenworthy 2016) surrounded by an exoring system consisting of at least 37 rings and extending out to 0.6 au in radius (Kenworthy & Mamajek 2015). For these rings to show detectable transit signatures, they must be significantly misaligned with respect to the orbital plane of J1407 b (Zanazzi & Lai 2017). This potential ring system would be considerably larger than the ring system of Saturn, which is located within the planet’s tidal disruption radius. The proposed rings around J1407 b would even cover a significant fraction of the companion’s Hill sphere and would not be expected to be stable over gigayear timescales. If the candidate ringed companion is in a bound orbit around the star, this orbit must be moderately eccentric in order for no othereclipses to have been detected to date (Kenworthy et al. 2015), raising the possibility that there might be a second as yet undetected companion in the system that causes the implied orbital eccentricity for J1407 b. Radial velocity measurements are overwhelmed by the chromospheric noise of the star and do not place strong constraints on other substellar companions (Kenworthy et al. 2015). The transit of J1407 suggests that its orbital plane has a high inclination to our line of sight – if there are other planets inside the orbit of J1407 b, their orbits may well be coplanar with J1407 b and there is a high chance that these companions may transit J1407.


In order to try to get insights into such systematics, we produced several variants of the ICRF3 S/X band frame by changing the reference epoch of the catalog or alternately by not considering Galactic acceleration in the modeling. Interestingly, the D2 and D3 glide terms for these variants were found to vary by several tens of microarcseconds in the comparison to ICRF2, in line with the level of the systematics observed for those terms. Such findings are not unexpected since Galactic acceleration manifests itself as a dipolar deformation in the source coordinates (e.g., Titov & Lambert 2013). Moving further, and noting that Mignard et al. (2016) mentioned this phenomenon as a possibility for explaining the observed glide between ICRF2 and the Gaia Data Release 1 (Gaia DR1) auxiliary quasar solution, we decided to reproduce an equivalent of ICRF2 by considering only the stretch of data used for ICRF2 (i.e., including only the VLBI sessions up to March 2009 in the solution) and to make a variant that adds Galactic acceleration in the modeling, as implemented for ICRF3. To guarantee the maximum consistency, those two analyses were conducted by employing the same software package as that used for ICRF2, namely CALC-SOLVE (see Fey et al. 2015). Looking at the results, we first observed that our “reproduced” ICRF2 shows similar deformations as the original ICRF2 when compared to ICRF3, hence ruling out the possibility that ICRF2 was in error. Most importantly, the ICRF2 variant that incorporates Galactic acceleration modeling was found to have much reduced glide terms compared to the original or reproduced ICRF2. This is illustrated by the bar chart in Fig. 16 which shows that the D2 term has now vanished while the D3 term has been cut by more than half (down to a value of −39 ± 4 μas) in this variant, hence indicating that the deformations between the two frames, in large part, stem from Galactic acceleration not accounted for in ICRF2. This is somehow not surprising since the data set for ICRF2 already covered 30 years, enough for Galactic acceleration effects to emerge, even though the accuracy of the frame was lower than that of ICRF3.


NGC 3557 is a bona fide southern-sky elliptical galaxy (E3) at a distance of 40 Mpc (1″ is 198 pc at this distance7

7
For distance derivations, we have assumed a cosmology model with H0 =73 km s−1 Mpc−1, Ωmatter = 0.27, and Ωvacuum = 0.73.
), and a member of a small group of galaxies (Brough et al. 2006). It has been classified as a LINER (e.g., Annibali et al. 2010) and as a flat-spectrum radio galaxy (Healey et al. 2007), with a jet that bends at distances of a few arcmin from the center (Schmitt et al. 2002). Detections of several of the ISM components of this object have been reported in the literature. Dust has been observed both as FIR emission (Pasquale et al. 2009) and as absorption against the central stellar continuum (Lauer et al. 2005). Nuclear optical line emission has been reported from spectroscopy (Veron-Cetty & Veron 1986; Rampazzo et al. 2005) as well as narrowband photometry (Goudfrooij et al. 1994). The MIR spectrum of NGC 3557 is of the most common Class-2 type, which is currently associated with a post-star-formation phase (Vega et al. 2009). Regarding the cold gas component, atomic gas (H i) has been reported as nondetection in several works (Serra & Osterloo 2010), while molecular gas emission has been detected in single-dish observations (Prandoni et al. 2010). NGC 3557 was included in our sample because of its relative proximity and CO(2–1) brightness (as reported by Prandoni et al. 2010), which suggested the possibility to study the molecular structures in detail. Since the object is considered to be in a stage where little current star formation may be happening, it was considered important as a representative of the molecular gas structures to be expected in post-star-formation scenarios. Furthermore, the disk-like nuclear dust distribution seen in the photometry by Lauer et al. (2005) indicated the possible presence of a molecular gas disk, which is a very interesting dynamical structure linked to the presence of organized angular momentum at those spatial scales.