The X-ray sources in the BCG, NGC 1399, have been extensively studied with a focus on their connection with GCs


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


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


The supernova explosion has driven a forward shock into the circumstellar material, thought to be from the stellar wind of the progenitor


while the ejecta from the supernova itself crosses the reverse shock as it expands


The ejecta is mostly comprised of heavy elements, principally oxygen


As well as the X-ray emitting gas, the shocked ejecta also consists of denser clumps or knots, emitting in the optical and IR


Electron densities in the shocked clumps are $n_{\rm e}\sim 10^3{\text{-}}10^5\, {\rm cm}^{-3}$


inferred a pre-shock temperature of ${\sim } 100 \, {\rm K}$ from [Si i] IR emission lines.


while the gas temperatures are of order $10^4 \, {\rm K}$


estimated a maximum electron density of $n_{\rm e}\lesssim 100 \, {\rm cm}^{-3}$ for the unshocked ejecta based on forbidden line ratios


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

(1)

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

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

(2)

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

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

(3)

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

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

(4)

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

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


With these assumptions, the equation can be solved analytically and the solution for the size of the radius as a function of time takes the following parametric form

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

where A, B, and T are constants that can be fixed once the initial conditions are fixed and θ is an indicator of time.


For voids with the same initial settings, the analytical solutions can also be found by taking an inverse top-hat model for the density profile

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


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

λem=ρκemaradTe4

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

b

T

e
/m

e

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

b

T

e
/m

e

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

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

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

e
 ≳ 108 K within r ≲ 5r

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


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

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

where x denotes the nondimensional frequency and

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

. In Equation (1),

 denotes the CMF term, which has the form
2

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

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

) velocities. In Equation (1),

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

where the line source vector is of the form
5

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

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

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

where

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

The Rayleigh phase matrix

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

where
9


The CMF polarized PRD transfer equation for a spherically symmetric medium in (p, z) representation under the nonrelativistic limit is given by (see also Equation (17) of

1

where x denotes the nondimensional frequency and

 denotes the irreducible Stokes vector with “+” and “−” referring to the outgoing and incoming rays, respectively.


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


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

(44)

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

where the scale factors are P, Q and R are

(45)

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

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

(46)

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

with

(47)

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

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

(48)

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

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

(49)

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

Integrating up, this gives us

(50)

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

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


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

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

 and

. This implies

, with

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

, one can next recover the following quantities:
25

26

27

28

29

We want to solve for four quantities:

,

,

 (or, equivalently,

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

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

 and

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

, given the effect of the GC cluster potential on the

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

 and

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

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

 (which also depends on the average moment of inertia

,

, and

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

 or

 for a given value of

 and keeping other parameters fixed. If we require

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

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

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

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

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


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

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

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


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

(24)

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

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

(25)

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

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


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


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

 km s−1 (red), −21 km s−1

 km s−1 (central), and −25 km s−1

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


associated with known star-forming subregions, namely East 1, East 2, and the Central region (e.g.,


found three molecular gas components (i.e., −18 km s−1

 km s−1 (red), −21 km s−1

 km s−1 (central), and −25 km s−1

 km s−1 (blue)) in the direction of the S235 complex.


Previous studies indicated that the S235 H ii region is interacting with its surrounding molecular cloud and the S235 complex has been cited as a possible site of triggered star formation


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

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

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

2

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

 transfer):
3

4

followed by reaction (2) or exothermic proton transfer of

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


The initially assumed gas-phase model involving an ion–molecule reaction of protonated methanol (

) with formaldehyde (H2CO) fails to reproduce the large MF column density, because this pathway demands a significant activation barrier (∼128 kJ mol−1) for one of the intermediate steps


However, this reaction scheme fails to reproduce the huge MF concentration observed in hot cores


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

 and assuming an intensity ratio of

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

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

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

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


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


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


21 cm Intensity Mapping (IM) is a promising technique to map the cosmological large scale distribution of matter through the observation of 21 cm radio emission/absorption of neutral hydrogen gas (H i), while not requiring the detection of individual sources


Such surveys require instruments with large instantaneous bandwidth and field of view and several groups have built dense interferometric arrays to explore IM, such as CHIME


have also been built to explore specific technical aspects of these arrays, as well as transit mode operation and calibration.


Intensity Mapping is also being used to search for signals from the EoR and the cosmic dawn, at redshifts above z ≳ 10, by several large radio-interferometers, such as


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

 (B/50 μG)

 (E/1 keV)

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


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

R

(
θ
)

=

R

0

csc

θ

3
(
1
−
θ

cot

θ
)

,

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

m

w

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

R

0

=

m

w

v

w

4
π

ρ

A

v

p

2

1

/

2

.

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

m

w

v

w

=

E

/

c

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

θ

0

=
56.3

mas

sin

2

i

n

A

1

/

2

E

33

1

/

2

μ

100

D

kpc

2

,

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

E

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

E

=
4

π

2

I

P

/

P

3

, where I ≈ 1045 g cm2. The period derivative

P

 can be corrected for the Schklovskii effect using
10

P

=

P

obs

−
2.43
×

10

−
21

P

μ

masy

2

D

kpc

,

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


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

θ

0

=
56.3

mas

sin

2

i

n

A

1

/

2

E

33

1

/

2

μ

100

D

kpc

2

,

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

E

 is the spin-down luminosity in erg per second, D is the distance to the pulsar in kiloparsecs, and μ100 is the pulsar proper motion in 100 mas yr−1


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


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

, the lower limit of number density of each ionized cloud

 can be given by
8

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

 since the optical depth is given by

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

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

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

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


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


The astronomical observed trends in the elemental abundance distribution across the Galaxy serve as a major constraint on the GCE models in order to understand the accretional growth rate and the dynamical evolution of the Galaxy with its three main components, i.e. the Galactic halo, the thick disc, and the thin disc


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


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


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

A
(
t
)

$ \sqrt{A(t)} $

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


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

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

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

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


One of the most intriguing structures is the phase-space spiral discovered by Antoja et al. (2018) and studied in more detail in subsequent studies (e.g.,


More careful analyses in later studies (e.g.,

have nailed down the interaction time to ∼500 Myr ago.


The one-armed spiral makes two to three complete wraps, resembling a snail shell, and is interpreted as a signature of phase mixing in the vertical direction following a perturbation, which

estimate to have occurred between 300 and 900 Myr ago.


At large galactocentric radii (>10 kpc) this includes, among others,

and streams of stars kicked up from the disk that undergo phase mixing, sometimes referred to as “feathers” (e.g.,


Similar oscillations and vertical asymmetries have also been reported in the solar vicinity (e.g.,


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

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

 and a 2D component with

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


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

β
N
,  f

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

β
B
,  f

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

γ
N
,  f

γ
N
]) and G
Hγ, B
([λ
0, γ
B
,  σ

γ
B
,  f

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

c1([λ
0, c1,  σ

c1,  f

c1])and G

c2([λ
0, c2,  σ

c2,  f

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

e1([λ
0, e1,  σ

e1,  f

e1]) and G

e2([λ
0, e2,  σ

e2,  f

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

c3([λ
0, c3,  σ

c3,  f

c3]) and G

e3([λ
0, e3,  σ

e3,  f

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

A×λ4100ÅB

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

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

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

c1 to G

c2 to G

c3 are fixed to be
2

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

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

e1 to G

e2 to G

e3 are fixed to be
3

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

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


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


In general, the more energy is injected during a SNF cycle, the shorter the time is over which that energy is injected, and the more concentrated the baryonic mass is to the centre of the DM halo

the more efficient the SNF-induced cusp-core transformation will be.


Recent studies suggest that, in cosmological simulations, the efficiency of SNF at flattening the cusps of dwarf-size DM haloes is mainly determined by one model parameter, the gas density threshold for star formation


In a given dwarf galaxy, larger star formation thresholds lead to more bursty star formation, more concentrated and impulsive feedback, and a stronger contribution of baryons to the central potential, and hence to enhanced core formation


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


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


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


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

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

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

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

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

 and the LOS,

 is the cosine of the angle between

 and the LOS, and

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

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

 with respect to the vertical (with

 between 0° and 90°) and azimuth

 (with

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

4

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

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

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

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

 for

 and

 for

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

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

 in Equations (3) and (4).


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

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

 (or Mbary ≤ 1010.3

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

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


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