# Cloud-by-cloud Multiphase Investigation of the Circumgalactic Medium of Low-
redshift Galaxies

Sameer,1,2 Jane C. Charlton,1 Bart P. Wakker,3 Glenn G. Kacprzak,4,5 Nikole M.
Nielsen,4,5 Christopher W. Churchill,6 Philipp Richter,7 Sowgat Muzahid,8
Stephanie H. Ho,6 Hasti Nateghi,4,5 Benjamin Rosenwasser,9 Anand Narayanan,10
Rajib Ganguly11
1Department of Astronomy & Astrophysics, 525 Davey Lab, The Pennsylvania State
University, University Park, PA 16802, USA
2Department of Physics & Astronomy, Nieuwland Science Hall, The University of
Notre Dame, Notre Dame, IN 46556, USA
3Department of Astronomy, University of Wisconsin-Madison, 475 N. Charter
Street, Madison, WI 53706, USA
4Centre for Astrophysics and Supercomputing, Swinburne University of
Technology, Hawthorn, Victoria 3122, Australia
5 ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D)
6Department of Astronomy, New Mexico State University, Las Cruces, NM 88003,
USA
7 Institut für Physik und Astronomie, Universität Potsdam, Haus 28, Karl-
Liebknecht-Str. 24/25, D-14476, Potsdam, Germany
8 IUCAA, Post Bag-04, Ganeshkhind, Pune, India - 411007
9Department of Physics & Astronomy, The University of Toledo, Toledo, OH
43606, USA
10Department of Earth and Space Sciences, Indian Institute of Space Science &
Technology, Thiruvananthapuram 695547, Kerala, INDIA
11College of Innovation & Technology, University of Michigan-Flint, 196
Murchie Science Building, 303 Kearsley Street, Flint MI 48502, USA
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The pervasive presence of warm gas in galaxy halos suggests that the
circumgalactic medium (CGM) is multiphase in its ionization structure and
complex in its kinematics. Some recent state-of-the-art cosmological galaxy
simulations predict an azimuthal dependence of CGM metallicities. We
investigate the presence of such a trend by analyzing the distribution of gas
properties in the CGM around 47 $z<$ 0.7 galaxies from the Multiphase Galaxy
Halos Survey determined using a cloud-by-cloud, multiphase, ionization
modelling approach. We identify three distinct populations of absorbers: cool
clouds ($T\sim$ 104.1 K) in photoionization equilibrium, warm-hot
collisionally ionized clouds ($T\sim$ 104.5-5 K) affected by time-dependent
photoionization, and hotter clouds ($T\sim$ 105.4-6 K) with broad O vi and
Ly$\alpha$ absorption consistent with collisional ionization. We find that
fragmentation can play a role in the origin of cool clouds, that warm-hot
clouds are out of equilibrium due to rapid cooling, and that hotter clouds are
representative of virialized halo gas in all but the lowest mass galaxies. The
metallicities of clouds do not depend on the azimuthal angle or other galaxy
properties for any of these populations. At face value, this disagrees with
the simplistic model of the CGM with bipolar outflows and cold-mode planar
accretion. However, the number of clouds per sightline is significantly larger
close to the minor and major axes. This implies that the processes of outflows
and accretion are contributing to these CGM cloud populations, but that they
are well mixed in these low redshift galaxies.

###### keywords:

galaxies: groups: general – galaxies: interactions – quasars: absorption lines

††pubyear: 2024

##  1 Introduction

The circumgalactic medium (CGM) of a galaxy is the tenuous gaseous medium
permeating the space beyond the interstellar medium (ISM) and extending out to
its virial radius (Kacprzak et al., 2008, 2010; Chen et al., 2010; Steidel et
al., 2010; Tumlinson et al., 2011a, 2017; Rudie et al., 2012; Burchett et al.,
2013; Nielsen et al., 2013a, b; Werk et al., 2013; Johnson et al., 2015;
Lehner et al., 2020). The CGM plays a vital role in galaxy evolution by
moderating gas flows between the ISM and the intergalactic medium (IGM). A
diverse array of processes such as pristine inflows from the intergalactic
medium, galactic-scale outflows, recycled accretion via the galactic fountain
mechanism, tidal interactions via satellite galaxy mergers, and effects of
cosmic rays and magnetic fields, influence the properties of the CGM.
Understanding how these processes manifest in the observed properties of the
CGM is a crucial step in understanding galaxy formation and evolution.

The chief ingredients in galaxy formation and evolution models include: 1)
accretion of metal-poor inflowing gas from the IGM through filaments onto
galaxies (e.g., Oort 1970; Wakker & van Woerden 1997; Wakker et al. 1999;
Fumagalli et al. 2011; Stewart et al. 2011, 2013, 2017; Danovich et al. 2012,
2015; Oppenheimer et al. 2012; van de Voort & Schaye 2012; Shen et al. 2013;
Kacprzak et al. 2016; Borthakur 2022). 2) recycled accretion from previously
ejected metal-rich large-scale galactic outflows (Springel & Hernquist, 2003;
Kereš et al., 2005; Dekel et al., 2009; Oppenheimer et al., 2010; Davé et al.,
2011a, b, 2012; Stewart et al., 2011; Rubin et al., 2012; Ford et al., 2014;
Anglés-Alcázar et al., 2017; Zheng et al., 2017; Hafen et al., 2019; Lochhaas
et al., 2020; Pandya et al., 2020, 2021; Fielding & Bryan, 2022). Accretion,
in these forms, is necessary to sustain the formation of stars over billions
of years (e.g., van den Bergh 1962; Pagel & Patchett 1975; Pagel 1989;
Chiappini et al. 1997; Maller & Bullock 2004; Dekel & Birnboim 2006). 3)
Outflows from galaxies are needed to regulate the rate of star formation
within galaxies by removing gas from the ISM (e.g., Shapiro & Field 1976;
Bregman 1980; Kacprzak et al. 2008; Oppenheimer et al. 2010; Faucher-Giguère
et al. 2011; Oppenheimer et al. 2012). Outflows carry enriched material out of
the galaxy and into the CGM, and are characterized by relatively high
metallicities compared with material reaccreting onto the galaxy. On the other
hand, pristine accretion is expected to have even lower metallicities.

Recent cosmological magnetohydrodynamical simulations of galaxy formation
(e.g., Illustris TNG50, Nelson et al. 2019; Pillepich et al. 2019; Péroux et
al. 2020) find that gas with the highest metallicities preferentially aligns
with the minor axis, and the highest CGM gas metallicities are associated with
high-velocity outflowing gas from the concerted feedback action of supernovae
as well as from AGN. The lower metallicities, on the other hand, are
associated with accreting/inflowing gas along the major axis. Thus CGM
properties, and particularly the distribution of metal-rich gas in the CGM,
are anisotropic according to these high-resolution simulations, with
metallicity differences of $\sim$ 0.5 dex.

Contrary to these theoretical expectations, previous studies, e.g., Pointon et
al. (2019) (hereafter P19), have not found a dependence of metallicity on
galaxy azimuthal angle. This was unexpected given that a bimodality is found
in distribution of the azimuthal angle of Mg ii absorbers (Bordoloi et al.,
2011a, 2014; Bouché et al., 2012; Kacprzak et al., 2012b, 2015a; Rubin et al.,
2014; Lan & Mo, 2018; Martin et al., 2019; Dutta et al., 2021; Lundgren et
al., 2021). However, consistent with the results on outflow and accretion in
hydro-dynamical simulations, Wendt et al. (2021) identify metal enrichment of
$\lesssim$ 1 dex for sightlines along the minor axis compared with sightlines
along the major axis, using dust depletion as a proxy for metallicity. They
argue that metals deplete onto dust grains preferentially along the minor
axis, thus needing to account for the dust when estimating metallicities.
Their results are based on only 13 galaxy-Mg ii absorber pairs (9$-$81 kpc
distance) from the MusE GAs FLOw and Wind (MEGAFLOW) survey at 0.4 $<z<$ 1.4
(Wendt et al., 2021). However, this work is only applicable to damped
Ly$\alpha$ absorbers (DLAs) limiting the usefulness of this approach to other
lower column density H i-absorbers.

The findings of an association between galaxy azimuthal angle and metal
absorption properties are based primarily on observations using Mg ii
absorption. However, CGM gas is observed to be multiphase and must be
investigated accounting for the contribution of all the gas phases traced by
low, intermediate, and high ionization gas, to the H i Lyman series lines in
the metallicity determination. Information is lost by assuming an integrated
single-valued metallicity per sightline. For simulated absorption profiles,
Churchill et al. (2015) and Peeples et al. (2019) found that gas arises in a
wide range of structures and physical locations. In addition, there might be
little physical overlap between multi-phase gas structures that contribute to
the same absorption components (Marra et al., 2022).

Even a single line of sight through the CGM of a galaxy would be expected to
have complex multiphase structure with a range of properties. The latest
efforts to model observed systems find just that, with metallicities often
varying by one or two orders of magnitude along the line of sight through a
single absorption system (Lehner et al., 2019; Wotta et al., 2019; Zahedy et
al., 2019a; Sameer et al., 2021; Narayanan et al., 2021; Haislmaier et al.,
2021; Lehner et al., 2022). Observational investigations of individual
absorbers have identified a variety of metallicities within a single
absorption system, indicating that CGM characteristics vary along each sight-
line (e.g. Churchill et al., 2012; Crighton et al., 2015; Muzahid et al.,
2015; Rosenwasser et al., 2018; Zahedy et al., 2019a; Nielsen et al., 2022).
Therefore it is important to move beyond observational modelling efforts that
typically average together properties (e.g., Pointon et al. 2019) and conduct
a large statistical study to determine if the metallicities of any of the
multiple structures in the CGM depend on orientation. This will allow us to
disentangle the complex nature of the CGM and possibly identify a hidden
metallicity/azimuthal bimodality.

Ionization modelling in observational work is often limited to the
characterization of the low ionization absorption, which contributes the most
to the absorption seen in H i. The properties of the cooler gas phases which
typically dominate the H i optical depth are captured by single-phase
ionization models (Marra et al., 2021), while the warm-hot phases are often
left unaccounted for. Observationally, constraining the properties of high
ionization gas traced by O vi has been challenging because the O vi exhibits a
significantly broader velocity profile compared with the nearby H i absorbers,
and often with misaligned O vi kinematics (e.g., Fox et al. 2013; Savage et
al. 2014; Werk et al. 2016). Thus it is often excluded from single-phase
ionization modelling. The multiphase nature of the CGM can only be fully
understood by analyzing, in combination, the absorption line diagnostics from
low, intermediate, and high ions.

In this study, we reanalyze the P19 sample using a cloud-by-cloud, multiphase,
Bayesian ionization modelling method. This approach will allow us to delineate
the phase structure along the line of sight and ascertain the contribution of
multiphase gas to the H i absorption. The P19 sample is unique because all 47
galaxies at redshift $z<0.7$: 1) have galaxy imaging with HST, which is
essential for the determination of azimuthal angles; 2) have UV HST/COS
spectra in the MAST archive mostly from PIDs 13398 (P.I. Christopher
Churchill) and 11598 (P.I. Jason Tumlinson), with 36/47 absorption systems
supplemented with high-resolution optical spectra, covering a range of
ionization states including the H i series, O vi, C iv, and Mg ii using
optical spectra. This paper is organized as follows: in Section 2 we describe
the spectral observations that are analysed in this work; in Section 3.1 we
describe the methodology used to determine the physical conditions of an
absorber; in Section 4 we present the ionization modelling of all the
absorbers in our sample; in Section 5 we present the results from our
ionization modelling; in Section 6 we discuss the implications based on the
inferred absorption properties seen along these sightlines. We conclude in
Section 7. Throughout this work, we assume a $\Lambda$CDM cosmology with
$H_{\\_}{0}\,=\,70$ km s-1 Mpc-1, $\Omega_{\\_}{\rm M}\,=\,0.3$, and
$\Omega_{\\_}{\Lambda}\,=\,0.7.$ Metallicities are given in the notation $\log
Z/Z_{\sun}$ with solar relative abundances taken from Grevesse et al. (2010).
All the distances given are in physical units. All the logarithmic values are
presented in base-10.

##  2 OBSERVATIONS AND DATA

Table 1: Quasar Observations

(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12) | (13)
---|---|---|---|---|---|---|---|---|---|---|---|---
J-Name | Proper Name | $z_{\\_}{\rm qso}$ | RA | DEC | COS | COS | STIS | STIS | $FUSE$ | $FUSE$ | Optical | Optical
|  |  | (J2000) | (J2000) | Gratings | PID(s) | Gratings | PID(s) | Gratings | PID(s) | Spectrograph | PID(s)

  * a

For the J0456 sightline, we also use the GHRS G200M spectrum from the PID =
5961.

  * b

Spectra from Churchill & Vogt (2001).

We study the distribution of CGM properties using the “Multiphase Galaxy
Halos” Survey, which consists of the HST program with PID 13398 (Kacprzak et
al., 2015b, 2019; Muzahid et al., 2015, 2016; Nielsen et al., 2017; Ng et al.,
2019; Nateghi et al., 2021) and data from the literature (Yuan et al., 2002;
Danforth et al., 2010; Meiring et al., 2011; Churchill et al., 2012; Shull et
al., 2012; Tilton et al., 2012; Werk et al., 2013; Fox et al., 2013; Tilton &
Shull, 2013; Mathes et al., 2014). HST imaging and UV spectra are available
for all 29 quasar fields.

This sample consists of 47 galaxies with spectroscopic redshifts between 0.07
$<z<$ 0.66, which have an impact parameter range of 21 kpc $<D<$ 276 kpc from
a background quasar. Two absorption systems in P19, the $z=$ 0.2198 absorber
towards the quasar J1139 and the $z=$ 0.2013 absorber towards the quasar
J1342, are not considered in our analysis, as for these systems the strongest
observed potential H i line (Ly$\beta$ and Ly$\alpha$, respectively) cannot be
disentangled from a complex set of lines from higher-redshift systems, and
there are no associated metal lines. Two new systems not presented in P19 are
presented in this work. These are the $z=$ 0.1538 and $z=$ 0.3900 absorber
towards the quasar J2253. The properties of the galaxies associated with these
two systems were determined by and presented in Nateghi et al. (2023). The
properties of the remaining galaxies are adopted from P19. For the convenience
of the reader, we reproduce Table 2 from P19 and present it in Table
LABEL:tab:obsgal. Additionally, we have added the columns of virial radii of
the galaxies estimated using the halo masses, impact parameter scaled by the
virial radius, absorber velocity scaled by the circular velocity of the galaxy
(see § 5.2.4), and information on specific star formation. The propagation of
asymmetric uncertainties is carried out using the asymmetric_uncertainty
(Gobat, 2022) Python package. The galaxies in this sample are isolated such
that they have no neighbors within 100 kpc and have a line-of-sight velocity
separation $>500$ km s-1 from the nearest galaxy P19. This selection minimizes
the effect of mergers on the CGM. The halo mass range of the galaxies in the
sample is $10.8<\log M_{\\_}{h}/M_{\\_}{\odot}<12.5$, which is typical for
$L_{\\_}{\star}$ galaxies. Galaxy morphologies and other properties for each
of the galaxy-absorber pairs in the “Multiphase Galaxy Halos” survey are from
Kacprzak et al. 2015b, 2019; Muzahid et al. 2015, 2016; Nielsen et al. 2017;
Pointon et al. 2017, 2019; Ng et al. 2019, and from the literature (Chen et
al. 2001; Chen & Mulchaey 2009; Prochaska et al. 2011; Werk et al. 2012, 2013;
Johnson et al. 2013). We define $\Phi=$ 0∘ and $\Phi=$90∘ as the projected
major and minor axes. The survey reaches an average limiting magnitude of
$\approx$ 26 for a S/N of $\approx$ 10, in an average exposure time of
$\approx$ 800 s.

###  2.1 UV Quasar Spectra

The UV spectra in this study were acquired with the COS and STIS instruments
onboard the HST telescope, and the LiF and SiC2 channels onboard the $FUSE$
telescope. While P19 have used only the COS G130M and/or G160M when available,
and the STIS/E140M for the sightline towards J1704, we have complemented
HST/COS observations with $FUSE$ and HST/STIS observations whenever available.
The data from these additional gratings often provide better constraints on
the inferred absorber properties. The UV $HST$/COS spectra in this study are
downloaded from the Barbara A. Mikulski Archive for Space Telescopes (MAST).
We use COS data when both STIS and COS are available, owing to the
significantly higher S/N of COS data. We use STIS data when a COS observation
does not cover a transition of interest. In Table 1, we list the background
quasars and various instruments/gratings they have been observed with.

The COS G130M/G160M spectra have an average resolving power of $R\approx
20,000$ and cover a range of ions including the H i Lyman series, C ii, C iii,
C iv, Si ii, Si iii, Si iv, N ii, N iii, N v, and O vi. The HST/COS spectra
were reduced following the procedures in Wakker et al. (2015). In essence,
spectra are processed with the calcos v3.2.1 pipeline and are aligned using a
cross-correlation, and then shifted to ensure that (1) the velocities of the
interstellar lines match the 21cm H i profile, and (2) the velocities of the
lines in a single absorption system are aligned properly. The exposures are
then collectively combined by summing total counts per pixel prior to
converting to flux. The $FUSE$ observations were processed with the calfuse
pipeline (ver 3.2.3). A zero point offset correction was applied using the
methods in Wakker (2006). $FUSE$ spectra have an average resolving power of
$R\approx 15,000$. In some cases, where the absorption of interest is affected
by the O i $\lambda$1302 geocoronal emission, we use the timefilter module
from the costools package to filter unwanted data (SUN_ALT $>$ 0) and rerun
the calcos pipeline on the filtered data to generate the airglow emission
corrected spectrum for the affected portion of the spectrum. The STIS archival
observations are available for some of the sightlines with the E230M and E140M
echelle gratings. STIS data were processed using the calstis ver. 3.4.2
pipeline to reduce the raw data into 1-D spectrum. E230M and E140M spectra
have an average resolving power of $R\approx$ 30,000 and 45,000, respectively.

Continuum normalization was done by fitting a cubic spline to the spectrum.
The statistical uncertainties in the continuum fits were determined by using
“flux randomization” Monte Carlo simulations (e.g., Peterson et al. 1998),
varying the flux in each pixel of the spectrum by a random Gaussian deviate
based on the spectral uncertainty. The pixel-error weighted average and
standard deviation of 1000 iterations were adopted as the flux and
uncertainty, respectively.

###  2.2 Optical Quasar Spectra

The UV spectra are complemented with the optical spectra, which cover the low
ionization transitions of Mg i, Ca i, Na i, Fe i, Mg ii, Ca ii, Fe ii, Mn ii,
and Al ii. Some of these transitions are especially useful in resolving the
component structure. Optical spectra from Keck/HIRES or VLT/UVES are available
for 34 absorption systems with an average resolving power of $R\simeq 40,000$.
The observation IDs and instruments are listed in Table 1. The HIRES spectra
were reduced using the Mauna Kea Echelle Extraction (makee) package
111https://sites.astro.caltech.edu/~tb/makee/index.html. The UVES spectra were
reduced using the European Southern Observatory (ESO) pipeline (Dekker et al.,
2000) and the UVES Post-Pipeline Echelle Reduction (UVES POPLER) code (Murphy,
2016; Murphy et al., 2019).

##  3 Cloud-by-cloud multiphase Bayesian ionization modelling

To extract the properties of CGM absorbers, we employ a novel forward-
modelling Bayesian inference suite, Cloud-by-cloud, Multiphase, Bayesian,
ionization Modelling method (CMBM; Sameer et al., 2021, 2022), which couples
ionization modelling from cloudy with nested sampling Bayesian inference
techniques. This modelling approach has been tested using both observations
(Sameer et al., 2021, 2022) and large cosmological boxes to cosmological zoom-
in simulations of the CGM (Hafen, Sameer et al., 2023), and found to obtain
observational estimates accurate to within 0.1 dex of the true source values.
This method improves the efficiency of component by component modelling that
has been successful in recovering the physical conditions for various
individual absorbers (e.g. Churchill & Charlton, 1999; Charlton et al., 2000;
Ding et al., 2003a; Charlton et al., 2003; Ding et al., 2003b; Zonak et al.,
2004; Ding et al., 2005; Masiero et al., 2005; Lynch & Charlton, 2007; Misawa
et al., 2008; Lacki & Charlton, 2010; Jones et al., 2010; Muzahid et al.,
2015; Crighton et al., 2015; Richter et al., 2018; Rosenwasser et al., 2018;
Zahedy et al., 2019b; Norris et al., 2021). CMBM allows for self-consistently
modelling the multicomponent, multiphase absorption profiles seen in multiple
ionization states. It uses the observed profile shapes in all the available
line transitions, allowing us to better capture the multiphase nature of the
absorption systems. This procedure eliminates the need for assuming ionization
corrections in the estimation of the metallicities of absorbers as it
marginalizes over the unknown amounts of ionization corrections to be applied
during parameter inference.

Low-density diffuse gas is seldom found in galaxy halos where there are no
substantial stellar or extragalactic sources of ionizing radiation. This
photoionizing radiation modulates the ionization states, and thus affects the
absorption line signatures from CGM gas. A variety of extragalactic
photoionization background radiation (EBR) models exist in the literature. The
KS19 (Khaire et al., 2019) EBR model uses the latest QSO emissivities, galaxy
star formation rates, estimates of H i distribution in the IGM, dust
attenuation in galaxies, and the escape fraction of ionizing photons from
galaxies. An investigation of the systematic uncertainties in the inferred
parameters due to the choice of the radiation field models is presented in
Acharya & Khaire (2021) and Gibson et al. (2022). Acharya & Khaire (2021) find
a variation of 4–6.3 and 1.6–3.2 times in the inferred density and
metallicity, respectively, depending on whether the gas is high or low
density. For $z<1$, Gibson et al. (2022) find that the absorber metallicities
increase on average by $\sim 0.3$ dex as the EUV slope is increased from
$\alpha_{\\_}{\rm EUVB}=-2.0$ to $-1.4$. We adopt the KS19 default EUV slope
of $\alpha_{\\_}{\rm EUVB}=-1.8$.

###  3.1 Methodology

We first generate a grid of converged photoionization thermal equilibrium
(PIE) cloudy models on a 3–D grid, adopting the KS19 EBR model. The gas
conditions of CGM absorbers, which are subject to photoionization by the EBR,
are functions of the metallicity, density, and temperature. The axes of the
PIE grid are $\log Z/Z_{\sun}$, $\log n_{\rm H}$, and $\log N(\mbox{H\,{\sc
i}})$. The $\log N(\mbox{H\,{\sc i}})$ functions as a stopping criterion for a
given cloudy model. For a PIE model, cloudy determines an equilibrium
temperature, $T$, such that the absorber is in thermal equilibrium with the
background radiation field. The ranges for these parameters are $\log
Z/Z_{\\_}{\sun}$ $\in$ [$-3.0,1.5$], $\log n_{\\_}{\hbox{{\rm H}}}$ $\in$
[$-6.0,2.0$], and $\log N(\mbox{H\,{\sc i}})$ $\in$ [11.0, 21.0], and obtained
with a 0.1 dex step-size.

The galaxies in our sample have spectroscopic redshifts between 0.07 < $z$ <
0.66. Therefore, we construct the PIE grids at each $z\in$ [0.05, 0.65], with
a 0.05 step size. To create large grids of cloudy calculations, we use the
grid command in cloudy and generate grids on a distributed system–Stampede
222https://www.tacc.utexas.edu/systems/stampede2 cluster. We adopt the cloudy
default solar abundance pattern in model generation. We do not account for
depletion due to dust or deviations from solar abundance pattern in our model
generation. We report any deviations from the solar abundance pattern.

We start by inspecting the spectrum for low ionization transitions viz. Mg ii,
Si ii or C ii that can serve as constraining ions to determine the component
structure. We obtain a Voigt Profile (VP) fit to the constraining ion to
obtain the redshifts, z${}_{\\_}{i}$, and total Doppler broadening,
$b_{\\_}{i}$, parameters for each, $i$th, component of an absorption system.
The VPs are modelled using an analytic approximation (Tepper-García, 2006) and
implemented using the VoigtFit package (Krogager, 2018). With this assumed
component structure, we optimise over five parameters in each component, the
five parameters being $\log Z/Z_{\sun}$, $\log n_{\rm H}$, $\log
N(\mbox{H\,{\sc i}})$, b${}_{\\_}{nt}$, and $z$. Here b${}_{\\_}{nt}$ is the
broadening due to non-thermal processes, and $z$ is the redshift of the
absorption component. We use uniform priors for all five parameters, in the
range mentioned above for $\log Z/Z_{\sun}$, $\log n_{\rm H}$, and $\log
N(\mbox{H\,{\sc i}})$. For b${}_{\\_}{nt}$ we adopt the range $\in$ [0,
$b+3\sigma_{\\_}{b}$], where $b$ is the value determined from the VP fit, and
$\sigma_{\\_}{b}$ is the uncertainty on $b$ value, and for $z$ in the range
$\in$ [$z-3\sigma_{\\_}{z}$, $z+3\sigma_{\\_}{z}$], where z is the absorption
centroid for the component.

Typically, an absorption component in an absorption system is composed of one
or more “clouds”. Each cloud represents a parcel of gas in a specific spatial
location with its own unique properties. These different clouds could
potentially contribute to the absorption at a given velocity, albeit
potentially being spatially at disparate locations (e.g., Marra et al. 2022).
In some cases, asymmetry in the component profile can reveal the presence of
two clouds. While in other cases, the need for different ionization parameters
to explain the range of transitions observed at a given velocity can reveal
the need for two clouds to explain the component. It is generally the case
that a model that explains the absorption in low-ionization transitions cannot
fully account for the absorption seen in intermediate/high ionization states
(e.g., Si iii, C iii, Si iv, C iv, N v, O vi, Ne viii). When it is found that
there is absorption unaccounted for by the low-ionization gas phases, we
include higher ionization transitions as additional constraining ions. This
unaccounted absorption may arise in environments where particle-particle
interactions dominate over energy exchange with the radiation field. Earlier
works (e.g., Tripp et al. 2001, 2011; Narayanan et al. 2011, 2012) which have
undertaken the ionization modelling of high-ionization absorbers have shown
that this gas phase is dominated by collisional ionization, primarily based
upon the large Doppler broadening parameters of the absorption profiles.

Furthermore, in some instances collisional ionization equilibrium (CIE) cannot
sufficiently explain the ratios of observed column densities and/or profile
shapes (e.g., Savage et al. 2014). For highly ionized gas at temperatures
$\lesssim 5\times 10^{6}$ K, and in the presence of external photoionizing
radiation, recombination can slow down in comparison to cooling causing the
plasma to remain over-ionized compared with what would be predicted from CIE.
Departures from CIE become more prominent for high-metallicity gas. Gnat
(2017) (hereafter G17) examined the non-equilibrium evolution of photoionized
cooling gas and computed the fractional abundances of different ions as a
function of temperature, metallicity, and hydrogen number density. We thus
model higher ionization gas, by assuming cooling in the presence of external
photoionizing radiation, a more realistic scenario than CIE, using the time-
dependent photoionized (TDP) ion fractions from Table 11 of G17. In TDP
cooling gas, the abundances of the major coolants may be considered to be
either in PIE or affected by time-dependent collisional (TDC) processes,
depending on the density. However, the abundance of some specific species may
differ by a few factors from the TDC/PIE values, and thus we use the TDP ion
fractions. We optimize over five parameters in each cloud modelled as a TDP
cloud, the five parameters being $\log Z/Z_{\sun}$, $\log n_{\rm H}$, $\log
T$, b${}_{\\_}{nt}$, and $z$. We use uniform priors on all parameters in the
range - $\log Z/Z_{\\_}{\sun}$ $\in$ [$-3.0,0.3$], $\log n_{\hbox{{\rm
H}}}/\hbox{cm${}^{-3}$}$$\in$ [$-6.0,2.0$], and $\log T/K$ $\in$ [4.0, 7.0].
The bounds on these priors are set by the model limits obtained by G17. The
priors for b${}_{\\_}{nt}$ and $z$ are determined in a way similar to
discussed earlier. We do not adopt TDP models to model low ionization phases
(e.g., Mg ii, Si ii, C ii) because the G17 models have a lower bound on
temperature of 104 K, and, a posteriori in a number of cases we observe narrow
Mg ii absorption suggestive of even lower temperatures.

(a)

(b)

(c)

Figure 1: Ionization fractions of chosen low, intermediate, and higher
ionization ions from the models in G17 for the densities of $\log n_{\rm H}$ =
$-3$, $-2$, and $-1$, from top to down. CIE indicates collisional ionization
equilibrium. PIE indicates photoionization equilibrium, and TDP indicates
time-dependent photoionization. For CIE gas, cooling times are much longer
than the ionization and recombination timescales. On the other hand, for gas
out of ionization equilibrium, the cooling time is shorter than the ionization
and recombination timescales. Cooling is more efficient for higher metallicity
gas, where the departures from CIE are noticeably marked. For O vi, at
temperatures greater than T $\simeq$ 2.5 $\times$ 105 cooling times are longer
than the ionization timescales and independent of the gas metallicity such
that the CIE and non-equilibrium ion fractions are identical.

In Fig. 1, we compare the temperature dependence of ionization fractions for
gas traced by intermediate and high ionization ions under the various
conditions of photoionization equilibrium, collisional ionization equilibrium,
and time-dependent photoionization for metallicities $Z/Z_{\\_}{\odot}$ =
10-3, 10-2, 10-1, and 1, at the densities of 10-3 cm-2, 10-2 cm-2, 10-1 cm-2.
For the intermediate and higher ionization ions of C iii, C iv, N v, and O vi,
we observe that the departure between equilibrium models (PIE or CIE) and
time-dependent photoionization models is smallest for lower metallicity gas,
where the cooling times are longest. The PIE and TDP fractions diverge for
metal-enriched gas at temperature, $T\gtrsim$ 104 K. The ionizaton fractions
in Fig. 1 show that non-equilibrium ionization can produce significant
intermediate and higher ions in a temperature regime often associated with
photoionization. While it is possible for the intermediate and higher
ionization phases to arise at lower temperatures, by using profile shapes in
multiple ionization states in conjunction with the column densities we are
able to break the ionization fraction degeneracy. For O vi, at $T\gtrsim$
4$\times$105 K, the different ionization models are consistent with one
another and independent of the density and metallicity, suggesting that CIE is
a good approximation for the ionization of O vi gas.

For parameter estimation and model comparison, we use the nested sampling
algorithm, PyMultiNest (Buchner, 2014). PyMultiNest is well-suited for
tackling multi-dimensional and multimodal posteriors. During the log-
likelihood evaluation, the column densities and temperatures from the cloudy
grids at the sampled parameter values are calculated on the fly by
interpolating over the preconstructed cloudy grid. The interpolant is
constructed by triangulating the grid data using the Quick hull algorithm
(Barber et al., 1996), and on each triangle linear barycentric interpolation
is carried out. We create a “mask” over the regions contaminated by
interlopers and over noisy continuum-level regions of the spectrum. The masked
regions do not influence the parameter estimation. We perform deblending in
cases where there is substantial contribution from Si ii $\lambda$989 line to
the absorption in N iii $\lambda$989 line. For H i-only clouds with no metals
detected we adopt 3$\sigma$ upper limits on the metallicity measurement. To
confirm the detection (or lack thereof) of the absorption of interest at the
location of H i absorption, we calculate the limiting equivalent width and
column density using the pyND333https://github.com/jchowk/pyND software.

We compute the Bayes factor to compare models of increasing complexity (more
clouds), adopting the model with the highest log-evidence and the least model
complexity. In models of increasing complexity, the gain in likelihood does
not compensate for the increase in parameters of the model and is found to
yield clouds with unconstrained posteriors. We synthesize the expected
absorption profiles and compare them to the observed profiles in order to
infer the column densities, Doppler parameters, and physical conditions of the
absorbing gas. By using the shapes of the absorption profiles and centering of
individual components of constraining ions, the CMBM method allows for more
robust constraints on inferred model parameters than methods that average
together components and perform cloudy modelling of total column densities
derived from the data.

##  4 Absorption profiles and physical conditions

Here, we present the CMBM analysis of the “Multiphase Galaxy Halos” survey.
For each absorption system, we present a system plot (e.g., Fig. 21) showing
the absorption in various radiative transitions and the Maximum Likelihood
Estimate (MLE) ionization models are overlaid on the observed absorption
profiles. The observed profiles are presented as a function of velocity
relative to the zero-point defined by the redshift of the associated galaxy.
The detailed notes on the ionization modelling and system plots for all the
systems are presented in appendices B and C, respectively. A summary of the
cloud-by-cloud properties of all the individual absorption systems determined
by using our CMBM analysis is presented in Table 2.

Based on the models that explain the observed absorption in various phases, we
classify clouds into PIE clouds and TDP clouds. Furthermore, for the TDP
clouds, we subclassify them into two types–those that show absorption in the
high ionization ions of at least the O vi, and possibly N v and/or C iv, but
not in the low or intermediate ions, which we refer to as the TDP–High clouds,
and those that show absorption in intermediate and high ionization states, and
possibly also in low ionization states, which we refer to as the TDP–Low
clouds.

In the rightmost panel of every system plot, the inferred marginalised
posterior distributions representing the probability distributions of the
parameters of interest are presented. The parameters are $\log Z/Z_{\sun}$,
$\log n_{\rm H}$, $\log N(\mbox{H\,{\sc i}})$, $\log T$, $\log L$, and
b${}_{\\_}{nt}$. The distributions are presented in the form of violin plots
to visualise any skewness and multi-modality of the distributions. The region
enclosing 99.7% of the distribution is shown as a dotted boundary, and the
region enclosing the $16-84$ percentiles is shaded using a colour
corresponding to the modelled cloud. The MLE values are indicated as a $\star$
on top of the distributions. In many cases, the MLE provides a good point
estimate of the parameters of the distribution. However, it is possible for
the MLE to be outside the peak of the distribution. The MLE can be affected by
noise in the data which can pull the estimate away from the center of the
distribution towards the tail.

For each absorption system, we also discuss the various blends that affect the
absorption of interest, and these blended regions are excluded from the
modelling. Pixels that are contaminated by absorption from other intervening
systems are blanked out and shaded in grey. However, we do not discuss
contaminating blends that might appear to be present in a weaker transition
but absent in the stronger one, where the line strength is proportional to the
product of the oscillator strength and the rest wavelength of the line. We
exclude Ca ii from the modeling. Robertson et al. (1988), in the first census
of extragalactic Ca ii absorption, found that the Ca ii is severely
underabundant compared to both Fe ii and Mg ii, typically smaller by a factor
of 100. They argued that the underabundance suggests that Ca ii condenses more
readily onto dust grains than Mg ii and Fe ii. Richter et al. (2011) observed
a column-density-dependent dust depletion of calcium, finding enhanced dust
depletion of Ca ii in high H i column density systems.

Table 2: Cloud-by-cloud properties of the absorption systems

(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12) | (13)
---|---|---|---|---|---|---|---|---|---|---|---|---
J-Name | $z_{\\_}{\rm gal}$ | $V$ | $\log Z/Z_{\sun}$ | $\log n_{\hbox{{\rm H}}}/\hbox{cm${}^{-3}$}$ | $\log N(\rm H)/\hbox{cm${}^{-2}$}$ | $\log N(\mbox{H\,{\sc i}})/\hbox{cm${}^{-2}$}$ | $\log T/K$ | $\log L/kpc$ | bnT | b${}_{\rm{T}}(\rm{\mbox{H\,{\sc i}}})$ | b$(\rm{\mbox{H\,{\sc i}}})$ | Ionization Model
|  | km s-1 |  |  |  |  |  |  | km s-1 | km s-1 | km s-1 |

Properties of the different gas phases that contribute to the absorption
towards different sightlines. Notes: (1) J-Name of the quasar; (2) Redshift of
the galaxy; (3) Velocity of the cloud in the rest frame of the galaxy; (4)
cloud metallicity; (5) cloud total hydrogen number density; (6) cloud total
hydrogen column density; (7) cloud hydrogen column density; (8) cloud
temperature; (9) cloud line of sight thickness; (10) cloud non-thermal Doppler
broadening parameter (11) cloud thermal Doppler broadening parameter measured
for H i; (12) cloud total Doppler broadening parameter measured for H i; (13)
type of ionization model for the cloud; PIE - Photoionized Equilibrium, TDP -
Time Dependent Photoionized (see Section 3.1). The marginalised posterior
values of model parameters are given as the median along with the upper and
lower bounds corresponding to the 16–84 percentiles. The lower and upper
limits correspond to $\mu_{\\_}{1/2}-2\sigma$ and $\mu_{\\_}{1/2}+2\sigma$,
respectively. The rest of the table is presented in appendix B.1.

##  5 Results

We present the metallicity analysis of the “Multiphase Galaxy Halos” Survey
using CMBM analysis of the 47 absorption systems associated with 47 galaxies
with well characterized properties. We resolved the absorption in these
systems into 240 clouds. These absorbers were found to be explained by one of
three kinds of models PIE (157 absorbers in number), TDP–Low (35 absorbers in
number), or TDP–High (48 absorbers in number). We plot the various properties
to visualize the range and trends, summarize them using the interquartile
range (IQR) to capture the dispersion, and the median and the standard
deviation of the median from bootstrapped estimates to capture the central
tendency of the data. The IQR is calculated as the difference between the 75th
and 25th percentiles of the data. Table LABEL:tab:summarystat summarizes the
summary statistics of the properties of different cloud types.

In addition to the distribution of metallicities, we also consider the
relationship between metallicity and other intrinsic absorber properties
including H i column density, total hydrogen column density, temperature,
total hydrogen number density, line of sight thickness, and Doppler parameter.
We then compare cloud metallicities to galaxy properties including the impact
parameter scaled by the galaxy virial-radius, galaxy azimuthal angle, galaxy
colour, halo mass, and redshift. We then examine the connection between the
azimuthal angle and the CGM metallicities.

###  5.1 Metallicity and its relationship with absorber intrinsic properties

####  5.1.1 Metallicity distributions of clouds, $\log Z/Z_{\sun}$

We present the histograms of the metallicities of different cloud types from
our sample in Fig. 2. The summary statistics of metallicity distributions are
summarized in Table LABEL:tab:summarystat. The histogram for PIE clouds is
shown in panel (a). The median of the metallicity distribution of PIE clouds
is estimated to be $\log Z/Z_{\sun}$ = $-0.47\pm 0.10$.444We used the Kaplan-
Meier estimator, implemented using the survfit function in R language to
determine the median, accounting for the upper limits on some of the
metallicity measurements. To account for random variations in the
observations, we obtained 1000 bootstrapped estimates of the median to
ascertain the uncertainty on the median. The metallicity distribution for PIE
clouds is skewed left with a low fraction of very low metallicity clouds. The
hatched distribution is the PDF of the upper limits and corresponds to the
metallicity limits of H i-only clouds. These clouds have just H i detected
with no associated metal absorption, and we adopt 2$\sigma$ upper limits for
their metallicities. The upper limits on PIE clouds are modelled as uniform
distributions because of the inferred broad posterior distributions for these
clouds. The range of measured metallicities for PIE clouds is in close
agreement with the result of P19. However, they obtain a median metallicity of
[Si/H] $\approx-1.3$. P19 infer a metallicity value using only the low and
intermediate ionization phases, by measuring total column densities and
performing ionization corrections with the help of cloudy. With the CMBM
approach, we are able to measure how much of the H i is associated with the
various phases contributing to the absorption, and obtain separate
metallicities for various clouds and phases. This difference in modelling, and
potentially to some extent the choice of EBR, could explain the disagreement
between P19 and our result. P19 adopted HM12 (Haardt & Madau, 2012), while we
have used KS19 to model the PIE clouds.

Figure 2: The histograms of metallicity for various cloud types obtained from
the CMBM analysis. The cloud types are (a) Photoionized Equilibrium (PIE), (b)
Time-Dependent Photoionized - High (TDP–High), and (c) TDP–Low clouds. The
upper limits (2$\sigma$) in PIE clouds are shown as lightly shaded hatched
region. The metallicity distribution of TDP–High clouds only comprises upper
limits, and is shown as a uniform distribution. The number of clouds with
measurements and upper limits are also noted in each panel. The optimum bin
width for the displayed histograms is determined using Knuth’s rule (Knuth,
2006).

The metallicity distribution of TDP–Low clouds is shown in panel (b) of Fig.
2, and falls in the range [$-1.1$, $0.0$] with a median metallicity of $\log
Z/Z_{\sun}$ = $-0.56\pm 0.21$. For the TDP–High clouds we display a uniform
distribution for their metallicities spanning between [$-3$, 0.3] due to the
independency on metallicity for these clouds as indicated by their platykurtic
posterior distributions.

We compare the metallicity distributions in the context of previous studies
that investigated the properties of cool CGM gas. Prochaska et al. (2017)
modelled the cool CGM gas of absorbers associated with $z\approx$ 0.2
$L^{\star}$ galaxies and found a median metallicity of [Si/H] = $-0.51$, with
a range of $-1.9$ $\lesssim$ [Si/H] $\lesssim$ 1.0, adopting the HM12 ionizing
background. Zahedy et al. (2019a) found a median metallicity of [M/H] =
$-0.7\pm 0.2$, with a 1$\sigma$ confidence interval of [M/H] = ($-1.6$,
$-0.1$) for absorbers associated with the CGM of luminous red galaxies (LRGs)
at redshifts, $0.21\lesssim z\lesssim 0.55$. The median metallicity of PIE
clouds is consistent with the inferred metallicities from these previous
works.

####  5.1.2 Neutral and total hydrogen column density distributions of clouds,
$\log N(\mbox{H\,{\sc i}})$ and $\log N(\rm H)$

(a)  (b)

Figure 3: (a): The metallicity as a function of neutral hydrogen column
density for the three cloud types. The metallicity measurements are shown as
filled circles, while the upper limits on metallicity are shown as filled
squares with downward pointing arrows. The histogram shows the distribution of
neutral hydrogen column densities for different cloud types. The metallicity
of PIE clouds shows an anti-correlation with the neutral hydrogen column
density. (b): Same as on the left but for the total hydrogen column density.
The metallicity of PIE clouds also shows an anti-correlation with the total
hydrogen column density. Error bars show the 1$\sigma$ uncertainty.

The scatter plots in Figure 3a show the inferred metallicities plotted against
the inferred neutral hydrogen column densities for PIE, TDP–Low, and TDP–High
clouds. The H i column density of our sample spans several orders of
magnitude: $11.8\lesssim$ $\log N(\mbox{H\,{\sc i}})$ $\lesssim 19.7$. The
distributions of neutral hydrogen column densities are presented in the bottom
panel. The summary statistics of H i column density distributions for the
three cloud types are summarized in Table LABEL:tab:summarystat.

The metallicity of PIE clouds appears to indicate an anti-correlation with the
H i column density. We test for the null hypothesis of no correlation between
the H i column density and metallicity by computing the Akritas–Theil–Sen
(ATS) Kendall correlation coefficient (Akritas et al., 1995) and the
associated null-hypothesis probability555This test accounts for the upper
limits present on some of the metallicity measurements.. We obtain a $p$-value
of $\approx$ 0.18, suggesting that the null hypothesis cannot be rejected.
However, if we just consider those sightlines which are impacting within the
virial radius of the galaxy, we find evidence for significant anticorrelation
between metallicity and neutral hydrogen column density ($p$-value = 0.03).
P19 using this same sample of absorbers identified a significant anti-
correlation between metallicity and H i column density. However, they found
that the correlation vanishes when the HM05 (Haardt & Madau, 2001) EBR is used
in ionization modelling instead of HM12 (Haardt & Madau, 2012). Zahedy et al.
(2019a) investigated the [M/H] – N(H i) relationship for luminous red galaxies
and found no correlation when the HM05 background was used in ionization
modelling. Prochaska et al. (2017) also found significant evidence for an
anti-correlation between metallicity and H i column density in their analysis
of 32 absorption systems from the COS-Halos survey comprising of $L^{\star}$
galaxies using the HM12 EBR. An anticorrelation is also seen for a sample of
weak-Mg ii analogs at low-$z$ (Muzahid et al., 2018); they adopted the KS15
EBR (Khaire & Srianand, 2015). The existence of an anticorrelation could be
attributed to similarity in the EBR spectrum of KS19 and HM12, both of which
have harder spectrum compared with HM05.

Lehner et al. (2019) characterized the metallicities of cool, photoionized gas
in the H i-selected Cosmic Origins Spectrograph (COS) circumgalactic medium
compendium (CCC) survey (Lehner et al., 2018; Wotta et al., 2019; Lehner et
al., 2019), consisting of H i absorbers with column densities ranging between
15 $\leq$ $\log N(\mbox{H\,{\sc i}})$ $\leq$ 19 at $z\lesssim$ 1\. Within our
PIE sample, we separately tabulate the summary statistics for the
metallicities of SLFSs (15 $<$ $\log N(\mbox{H\,{\sc i}})$ $<$ 16.2), pLLSs
(16.2 $\leq$ $\log N(\mbox{H\,{\sc i}})$ $<$ 17.2), and LLSs (17.2 $\leq$
$\log N(\mbox{H\,{\sc i}})$ $\leq$ 19) in Table LABEL:tab:summarystat, as
these groupings of H i absorbers overlap with the H i-selected absorbers from
the CCC survey. We find that the observed range of our metallicity values are
consistent with the range of values estimated in the CCC survey. However, our
estimated mean values for SLFSs and pLLSs are higher compared to the
corresponding values for the CCC survey. The CCC survey obtains mean values of
$\log Z/Z_{\sun}$ = $-1.2$ and $-1.3$ for SLFSs and pLLSs, respectively. The
metallicities of our LLSs are lower compared with the LLSs in CCC; they obtain
a value of $\log Z/Z_{\sun}$ of $-0.6$. We attribute these differences to the
sample selection, our sample is galaxy-selected while the CCC survey is H
i-selected. Systematic differences can also stem from the choice of the
radiation field. The CCC survey adopted the HM05 radiation field.

The observed relationship between $\log Z/Z_{\sun}$ and neutral hydrogen
column density can be directly compared with simulations. Hafen et al. (2017)
find that the metallicity distribution of H i absorbers with $\log
N(\mbox{H\,{\sc i}})$ $\leq$ 15, the Ly$\alpha$ forest, extends to below $\log
Z/Z_{\sun}$ = $-2$, with many of these absorbers likely tracing IGM gas yet to
be enriched by galaxies. We find that a significant percentage ($57\pm 6$%) of
our absorbers with $\log N(\mbox{H\,{\sc i}})$ $\leq$ 15 are found beyond the
virial radius, with no associated metals detected, consistent with results of
Hafen et al. (2017). They also find that simulated absorbers with 16.2 $\leq$
$\log N(\mbox{H\,{\sc i}})$ $\leq$ 19 have a metallicity within 1 dex of the
mean metallicity of $\log Z/Z_{\sun}$ = $-0.9$, also consistent with our
estimates for this sample.

We also explore the relationship between the metallicity and H i column
density of the TDP–Low clouds and find a significant positive correlation
($p$-value = 0.007). While we have modelled these clouds assuming time-
dependent photoionization, many of these absorbers are found to be overlapping
in metallicity with the PIE clouds. We do not interpret this positive
correlation as it is quite possible that some of these clouds could be in PIE,
depending on their density, complicating the interpretation of the
correlation. We do not investigate the trend between the metallicity and
neutral hydrogen column density of TDP–High clouds, as their metallicities are
unconstrained.

In Fig. 3b, the metallicity is plotted as a function of the total hydrogen
column density for different absorber types. The total hydrogen column density
of our sample spans several orders of magnitude: $12.9\lesssim$ $\log N(\rm
H)$ $\lesssim 22.0$. The distribution of total hydrogen column densities for
PIE clouds seems to show a bimodality, with a smaller peak at $\log N(\rm H)$
$\approx$ 14.6, and a more prominent peak at $\approx$ 18.0. However,
Hartigan’s Dip test suggests that the null hypothesis of unimodality cannot be
rejected ($p-$value = 0.09). We determine that the $\log N(\rm H)$ of H i-only
PIE clouds ranges from 12.9 to 17.9, with an IQR of [14.2, 15.0].

The $\log N(\rm H)$ distribution of TDP–Low clouds is quite flat and spans a
narrow range of values. The predicted range and median for $\log N(\rm H)$ of
TDP–High clouds are consistent with the properties of low redshift, $z<0.5$,
warm O vi absorbers (5.4 $\lesssim$ $\log T$ $\lesssim$ 6.2) from Savage et
al. (2014) who obtained a range of 18.4 $<$ $\log N(\rm H)$ $<$ 20.4, and
median of $\log N(\rm H)$ = 19.35.

####  5.1.3 Temperature, $\log T$

(a)  (b)

Figure 4: (a): The metallicity as a function of temperature for the three
cloud types. The metallicity measurements are shown as filled circles, while
the upper limits on metallicity are shown as filled squares with downward
pointing arrows. The histogram shows the distribution of cloud temperatures,
which is reminiscent of the multiphase gas observed in high-resolution
cosmological simulations. Clouds in photoionization equilibrium show lower
temperatures compared to the clouds that are affected by time-dependent
photoionization. The top-axis shows the thermal Doppler broadening parameter
in log values. (b): The metallicity as a function of non-thermal Doppler
broadening parameter for the three cloud types. The histogram shows the
distribution of the non-thermal Doppler broadening parameter for these cloud
types in log values. Error bars reflect the 1$\sigma$ uncertainty.

In Figure 4a, we show metallicity as a function of temperature, $\log T$,
plotted on the bottom x-axis. The summary statistics of the temperature
distribution for the different cloud types is given in Table
LABEL:tab:summarystat. We find that the temperature ranges for these three
cloud types agree with the multiphase CGM manifested in hydrodynamic
simulations, which exhibit a mixture of cool, warm-hot, and hot phases. The
temperature distribution of PIE clouds appears to be right-skewed with a
majority of the absorbers concentrated around the median of $T\sim$ 104 K, and
a tail extending to $T\sim$ 105 K. The temperature distribution of TDP–Low and
TDP–High clouds appears to be flat between $T\sim$104.2-5.2 K) and
($T\sim$105.4-6.0 K), respectively. The temperatures of some of the TDP–High
clouds are consistent with Coronal Broad Lyman-$\alpha$ (CBLA) absorbers–the
collisionally ionized, million-degree gas hypothesized to be pervading the
extended, hot gaseous halos of low-redshift galaxies (e.g., Narayanan et al.
2010; Savage et al. 2010, 2011a, 2011b; Richter 2020).

We perform an ATS test between the $\log Z/Z_{\sun}$ and $\log T$ measurements
of the PIE clouds and TDP–Low clouds. For PIE clouds, we obtain a $p$-value of
0.01, indicating evidence for anti-correlation between these parameters. We
also see evidence for anti-correlation between the metallicity and temperature
of the TDP–Low clouds ($p$-value = 0.005). Higher metallicity allows the gas
to cool efficiently, leading to a lower temperature–causing the
anticorrelation.

####  5.1.4 Doppler parameters

The Doppler parameter, $b$, quantifies the line broadening mechanism, which is
influenced by thermal and/or turbulent processes. The thermal motions of the
particles result in Doppler broadening of the observed profiles. The thermal
Doppler broadening and temperature are related by $b_{\\_}{\rm
T}=\sqrt{2kT/m}$ km s-1. In Fig. 4a, we show the metallicity as a function of
the thermal Doppler broadening parameter, b${}_{\rm{T}}(\rm{\mbox{H\,{\sc
i}}})$, plotted on the top-axis in log values.

(a)  (b)

Figure 5: (a): The metallicity as a function of total Doppler broadening
parameter for the three cloud types. The metallicity measurements are shown as
filled circles, while the upper limits on metallicity are shown as filled
squares with downward pointing arrows. The histogram shows the distribution of
the total Doppler broadening parameter for these cloud types in log values.
(b): The metallicity as a function of total hydrogen number density for the
three cloud types. The histogram shows the distribution of total hydrogen
number densities. The lower limits in hydrogen number density for PIE clouds
are shown as right-slanted hatching. The hydrogen density for TDP–High clouds
is unconstrained and shown as a uniform distribution in red. Error bars
reflect the 1$\sigma$ uncertainty.

In Fig. 4b, we show the metallicity as a function of the non-thermal Doppler
broadening parameter, bnT, plotted in log values. The non-thermal Doppler
broadening parameter in absorbing structures is potentially mediated by post-
shock turbulence from outflows and/or accretion (e.g., Cen & Ostriker 2006;
Draganova 2015). Magnetohydrodynamic turbulence produced by collisionless
shocks could result in a significant non-thermal contribution to the observed
line profiles, as observed in TNG100 simulations (Nelson et al., 2018). The
non-thermal Doppler broadening distribution of PIE clouds is relatively flat
with a lack of pronounced mode, however, there are some clouds with smaller
values, suggesting that some of these clouds are predominantly thermally
broadened. The non-thermal Doppler broadening distribution of TDP clouds is
also flat, but the TDP clouds always show some amount of contribution from
non-thermal broadening to their line profiles.

In Figure 5a, we show the metallicity as a function of the total Doppler
broadening parameter, b$(\rm{\mbox{H\,{\sc i}}})$, and the
b$(\rm{\mbox{H\,{\sc i}}})$ distribution for different cloud types. The
distribution of b$(\rm{\mbox{H\,{\sc i}}})$ for PIE clouds appears broad and
ranges between 10–90 km s-1 with a median of 21.3 $\pm$ 1.2. The COS CGM
Compendium (CCC) survey (Lehner et al., 2018) comprising of H i-selected
absorbers found a median Doppler broadening parameter of b$(\rm{\mbox{H\,{\sc
i}}})$ $\approx$ 28 km s-1 for gas consistent with being primarily
photoionized. The distribution of b$(\rm{\mbox{H\,{\sc i}}})$ for TDP–Low
clouds overlaps with that of the right tail of PIE clouds. Richter (2020) who
systematically studied the extended, hot gaseous halos of low-redshift with
CBLAs, inferred Doppler parameters ranging from 70 – 200 km s-1. The total
Doppler broadening parameter, b$(\rm{\mbox{H\,{\sc i}}})$, for TDP–High clouds
in our sample is consistent with the range determined for CBLAs.

####  5.1.5 Total hydrogen density, $\log n_{\rm H}$

In our model clouds the total hydrogen density, $\log n_{\rm H}$, allows the
ionization parameter to be well-defined for the adopted EBR spectrum at the
redshift of the absorber. In Fig. 5b, we show the metallicity as a function of
total hydrogen number density and the histogram of $\log n_{\rm H}$. The
summary statistics of the hydrogen density distribution for the different
cloud types is given in Table LABEL:tab:summarystat. The hydrogen density
distribution of PIE clouds is right-skewed, with a pronounced peak at the
median value of $\log n_{\rm H}$ $\approx-2.5$. The TDP–Low clouds show a
range of hydrogen densities, but there is a higher frequency of these clouds
with low densities. For the TDP–High clouds, the posterior distribution of
$\log n_{\rm H}$ of the modelled clouds is unconstrained. For gas at $\log T$
$\gtrsim$ 5.4, the ionization of the gas is most likely dominated by
collisional ionization since heating by photoionizing radiation is unlikely to
produce such high temperatures. For such gas, the radiative cooling
efficiencies are functions of the temperature only and independent of density,
thus resulting in large model uncertainties, provided the density is high
enough ($\gtrsim$ 10-4 cm-3) for particle interactions to become more
important than photoionization (Gnat, 2017). Thus, for the TDP–High clouds we
adopt a lower limit on their $\log n_{\rm H}$ and their $\log n_{\rm H}$
distribution is presented as a uniform distribution in [$-4.0,2.0$].

####  5.1.6 Line of sight thickness, $\log L$

Figure 6: The metallicity as a function of the inferred line of sight
thickness. The metallicity measurements are shown as filled circles, while the
upper limits on metallicity are shown as filled squares with downward pointing
arrows. The histogram shows the distribution of the line of sight thickness.
Error bars reflect the 1$\sigma$ uncertainty.

Our models assume that the cloud geometry is plane-parallel, and the
ionization structure of the model cloud is determined using the boundary
condition set by the neutral hydrogen column density. Using the predicted
total hydrogen column density corresponding to the boundary condition, the
thickness can be determined using the relation $L=\frac{N_{\\_}{\rm
H}}{n_{\\_}{\rm H}}$, where N${}_{\\_}{\rm H}$ and n${}_{\\_}{\rm H}$ are the
total hydrogen column density and total hydrogen number density, respectively.
The summary statistics of the size distribution for the different cloud types
is given in Table LABEL:tab:summarystat. In Fig. 6, we show the metallicity as
a function of cloud size for the three cloud types. The $\log L$ distribution
of PIE clouds appears to be platykurtic, with a median value of $\log L/kpc$ =
$-1.4\pm 0.2$. The TDP–Low clouds show a range of sizes, but there is a higher
frequency of these clouds with large sizes. For the TDP–High clouds, because
of the dependence of $\log L$ on $\log n_{\rm H}$, we obtain upper limits on
the inferred $\log L$ values. We determine upper limits based on the lower
limit of $\log n_{\rm H}$ $\geq-4$ and the model predicted $\log N(\rm H)$.

###  5.2 Metallicity and its relationship with galaxy properties

We next examine the trends of metallicity with a set of galaxy properties.

####  5.2.1 M${}_{\\_}{\rm halo}$

(a)  (b)

Figure 7: (a): The metallicity as a function of galaxy halo mass for the three
cloud types. The metallicity measurements are shown as filled circles, while
the upper limits on metallicity are shown as filled squares with downward
pointing arrows. The histogram shows the distribution of galaxy halo masses.
(b): The metallicity as a function of galaxy redshift. The histogram shows the
distribution of galaxy redshifts. No trends are evident for metallicity as a
function of halo mass or galaxy redshift for any of the cloud types. Error
bars reflect the 1$\sigma$ uncertainty.

In Figure 7a, we plot the metallicity, $\log Z/Z_{\sun}$, as a function of the
galaxy halo mass, M${}_{\\_}{\rm halo}$. All the types of clouds are seen for
all halo masses. We examine the relationship between metallicity and halo mass
for any underlying trends. We do not find evidence to reject the null
hypothesis of no correlation between metallicity and halo mass for PIE,
TDP–Low, and TDP–High clouds ($p$-values $\approx$ 0.75, 0.92, and 0.82,
respectively). We also do not find evidence for significant differences in the
halo mass distributions of the three groupings.

####  5.2.2 Galaxy Redshift

Galaxies and quasars contribute most of the photoionizing radiation at all
redshifts and their contribution is known to evolve with redshift (e.g.,
Gibson et al. 2022). The galaxies in our sample span the redshift range of
0.07 $<z<$ 0.66. We investigate any trends due to the evolution of EBR on how
various absorber types would be manifest in our sample. In Figure 7b, we plot
the metallicity as a function of galaxy redshift, z${}_{\\_}{\rm gal}$. We do
not see an indication of trend between the metallicity and galaxy redshift for
PIE and TDP–Low clouds. These cloud types are found at all redshifts. We do
not expect and see the effect of EBR on the properties of TDP–High clouds as
these are dominated by collisional ionization.

####  5.2.3 Impact Parameter

(a)  (b)

Figure 8: (a): The metallicity as a function of the impact parameter scaled by
the virial radius of galaxy for the three cloud types. The metallicity
measurements are shown as filled circles, while the upper limits on
metallicity are shown as filled squares with downward pointing arrows. The
histogram shows the distribution of normalized impact parameters. (b): The
neutral hydrogen column density as a function of normalized impact parameter.
There is significant evidence for an anticorrelation between these parameters
for PIE and TDP–Low clouds, but not for the TDP–High clouds. The histogram
shows the distribution of impact parameters. Error bars reflect the 1$\sigma$
uncertainty.

In Figure 8a, we plot the metallicity, $\log Z/Z_{\sun}$, as a function of the
normalized impact parameter, D/$R_{\rm vir}$. The PIE clouds and TDP–clouds
are found up to 1.7 $R_{\rm vir}$. We determine that within $R_{\rm vir}$,
$84\pm 3$% of the clouds show detected metals at the 2$\sigma$ level, but
beyond $R_{\rm vir}$ only $36_{\\_}{-8}^{+9}$% of the clouds show detected
metals at the 2$\sigma$ level. This is indicative of the fact that most of the
metals are retained within the $R_{\rm vir}$ of galaxies. We investigate the
existence of any trends between these two parameters and find that none of the
groupings show evidence for a correlation.

In Figure 8b, we investigate the relationship between neutral hydrogen column
density and normalized impact parameter. Previous studies (e.g., Lanzetta et
al. 1995; Tripp et al. 1998; Chen et al. 2001; Wakker & Savage 2009; Rao et
al. 2011; Werk et al. 2014; Borthakur et al. 2015; Curran et al. 2016;
Prochaska et al. 2017; Kulkarni et al. 2022) found that the H i column density
declines with increasing impact parameter for CGM absorbers. We also observe a
similar trend, consistent with these previous works. The dispersion in neutral
hydrogen column densities for PIE and TDP–Low clouds decreases with the
normalized impact parameter, while the TDP–High clouds show a similar level of
dispersion. We test for correlation for each of the three groupings using the
ATS test. For the PIE clouds, mainly tracing the cool CGM gas, we see
significant anti-correlation ($p$-value $\approx$ 10-5) between $\log
N(\mbox{H\,{\sc i}})$ and D/$R_{\rm vir}$. TDP–Low clouds also show
significant anti-correlation ($p$-value $\approx$ 0.03), while the TDP–High
clouds do not show evidence for anti-correlation ($p$-value $\approx$ 0.73).

Figure 9: The hydrogen number density, $\log n_{\rm H}$, as a function of
normalized impact parameter for the three cloud types. The $\log n_{\rm H}$
measurements are shown as filled circles, while the lower limits on density
are shown as filled squares with upward pointing arrows.

Werk et al. (2014) found an indication of declining hydrogen density as a
function of impact parameter scaled to the galaxy virial radius for cool,
photoionized CGM gas using the COS-Halos (Tumlinson et al., 2011a) gas column
density measurements. In Fig. 9, we plot the hydrogen number density as a
function of the normalized impact parameter. For PIE clouds, we observe that
within $R_{\rm vir}$, a large range of hydrogen densities are seen spanning
between $\log n_{\hbox{{\rm H}}}/\hbox{cm${}^{-3}$}$ = $-3.7$ to 0.2. However,
for impact parameters $>$ $R_{\rm vir}$ we find a narrower range of densities
spanning between $\log n_{\hbox{{\rm H}}}/\hbox{cm${}^{-3}$}$ = $-4.0$ to
$-2.3$.

We perform an ATS test to investigate the null hypothesis of no correlation
between the hydrogen density and normalized impact parameter. We find
significant evidence of a negative correlation ($p$-value = 0.01) between
these parameters for PIE clouds, excluding the lower limits on hydrogen
density. We also find significant evidence for a negative correlation between
these parameters for TDP–Low clouds ($p$-value = 0.003).

####  5.2.4 Absorber velocities

Cosmological hydrodynamical simulations (e.g., TNG50; Péroux et al. 2020)
predict that the highest metallicities in the CGM are associated with high-
velocity outflowing gas expelled from the ISM. On the other hand, gas with
lower metallicities is associated with lower velocity accreting or inflowing
processes. Simulations (e.g., Ford et al. 2014) also predict that cool
photoionized gas traces recycled accretion, warm-hot gas traces both recycled
accretion and ancient outflows depending on impact parameter, while hotter gas
probes ancient outflows.

(a)  (b)

Figure 10: (a): The metallicity as a function of absorber velocities scaled by
the circular velocity of galaxy for the three cloud types. The metallicity
measurements are shown as filled circles, while the upper limits on
metallicity are shown as filled squares with downward pointing arrows. (b):
The neutral hydrogen column density as a function of absorber velocities
scaled by the circular velocity of galaxy for the three cloud type. The
histograms show the distribution of absorber velocities scaled by the circular
velocity of galaxy. Error bars reflect the 1$\sigma$ uncertainty.

In Figure 10a, we plot the metallicity, $\log Z/Z_{\sun}$, as a function of
absorber velocity scaled by the galaxy circular velocity measured at the
impact parameter, |$V_{\rm abs}$/$V_{\rm c}$(D)| for different cloud types.
Galaxy circular velocities are calculated at the impact parameter of
absorption using equation (5) of Navarro et al. (1996) and equation (B2) of
Churchill et al. (2013). The velocity zero point is the host galaxy systemic
velocity. Accretion is expected to roughly match the rotation velocity of the
host galaxy at a maximum, whereas outflows can sometimes exceed the escape
velocity of galaxies, especially for lower mass galaxies. But also more
massive galaxies can host absorption with higher velocities. Our sample of
galaxies spans a range of halo masses; thus, we scale the absorber velocities
shifted with respect to the galaxy by the circular velocity at the measured
impact parameter of the host galaxy, V${}_{\\_}{c}$(D) to account for the
galaxy halo mass (see e.g., Ng et al., 2019). We tabulate the
V${}_{\\_}{c}$(D) values in Table LABEL:tab:obsgal.

We observe that clouds with high |$V_{\rm abs}$/$V_{\rm c}$(D)| show both low
and high metallicities; similarly clouds with low |$V_{\rm abs}$/$V_{\rm
c}$(D)| also exhibit a range of metallicities. We determine that the metal
detection fraction for clouds with absolute absorber velocities $\leq$
V${}_{\\_}{c}$(D) is $85_{\\_}{-4}^{+3}$%, while for clouds with absorber
velocities $>$ V${}_{\\_}{c}$(D) it is only $54\pm 7$%. This observation
likely indicates that material beyond V${}_{\\_}{c}$(D) is not closely
associated with the host galaxy and is likely tracing IGM material. We further
investigate if there is an underlying trend by plotting the $\log
N(\mbox{H\,{\sc i}})$ as a function of |$V_{\rm abs}$/$V_{\rm c}$(D)| in
Figure 10b, we see that the data points with upper limits on metallicity are
the ones that have low $\log N(\mbox{H\,{\sc i}})$ values, $\lesssim 14.5$,
indicated by lightly shaded diamond markers. At these low $\log
N(\mbox{H\,{\sc i}})$ values the expected metal columns are quite low, which
are difficult to measure at the resolution of COS and S/N of the observed
data. These high-velocity clouds can be reasonably attributed to the fact that
the material likely at the periphery of galaxies or in the IGM is unlikely to
exhibit a direct connection with the kinematics of the galaxy.

####  5.2.5 Azimuthal Angle

(a)  (b)

Figure 11: (a): Barplot showing the total number of systems (black), clouds
(magenta), PIE clouds (blue), TDP–Low clouds (orange), and TDP–High clouds
(red), in different azimuthal angle bins for different cloud types. A greater
number of clouds are seen along the major axis compared with other azimuthal
angles. (b): The number of clouds per absorption system seen in all the phases
along major and minor axis (red), and along interjacent angles (blue).

In Fig. 11a, we show the azimuthal angle, $\phi$, distributions for the number
of clouds. We find that there are more PIE clouds along the major axis
compared with other angles, with additional enhancement along the minor axis.
We create four equal-width azimuthal angle bins: 0∘–22.5∘, 22.5∘–45∘,
45∘–67.5∘, and 67.5∘–90∘. Our choice of binning is to ensure that there are a
sufficient number of data points in each bin, and the number of systems is
nearly evenly distributed in these bins. The total number of absorption
systems is 16, 8, 12, and 11 in each of these azimuthal angle bins. The total
number of clouds is 101, 28, 52, and 59 clouds in these four bins. We compare
the number of clouds per absorption system along the major and minor axis
($0^{\circ}\leq\Phi\leq 22.5^{\circ}$ $\cup$ $67.5^{\circ}\leq\Phi\leq
90^{\circ}$) with the number of clouds per absorption system along the
interjacent angles ($22.5^{\circ}<\Phi<67.5^{\circ}$), and show their
distributions in Fig. 11b. We see that along the major and minor axes, there
is a relatively higher frequency of observing more than $\sim$5 clouds per
absorption system, compared with the interjacent angles. We compare their
distributions using a 2-sample Anderson-Darling test (Scholz & Stephens, 1987)
to test the null hypothesis that the distribution of the number of clouds per
system along the minor and major axis is different from the distribution of
the number of clouds per system along the interjacent angles. This test is
particularly suitable for sample sizes of less than 25 observations each. We
find statistically significant evidence to reject the null hypothesis
($p$-value = 0.03). Berg et al. (2023), in their bimodal absorption system
imaging campaign (BASIC) survey, aimed at characterizing the galaxy
environments of pLLSs and LLSs at $z\lesssim 1$, and found that metal-enriched
absorbers ($\rm[X/H]$ $>$ $-1.4$) are preferentially found along the major and
minor axes, consistent with our observations. We note that a significant
percentage, $89_{\\_}{-5}^{+4}$%, of our galaxy-selected pLLSs and LLSs
absorbers have metallicities, $\log Z/Z_{\sun}$ $>$ $-1.4$.

(a)  (b)

Figure 12: (a): The metallicity as a function of azimuthal angle for PIE and
TDP–Low cloud types. The metallicity measurements are shown as filled circles,
while the upper limits on metallicity are shown as filled squares with
downward pointing arrows. The bootstrapped mean of metallicity and the
standard error of the mean within the four azimuthal angle bins are shown in
black. The mean values across the various azimuthal angle bins are similar for
these clouds. (b): The difference between maximum and minimum metallicities in
a given absorption system as a function of azimuthal angle considering all
cloud types and color-coded by the inclination angle.

We next examine the spatial distribution of CGM metallicities in Fig. 12a with
the bootstrapped mean metallicities in the various azimuthal angle bins
overplotted in black. We observe a wide range of metallicities for all
azimuthal angles, and observe a similar dispersion in metallicity for all the
cloud types for all angles. We find that the mean metallicities are consistent
with one another within their uncertainties (standard error on the mean) for
each of the cloud types. The bootstrapped metallicity mean and its uncertainty
are tabulated in Table LABEL:tab:meanmetallicity_azi. We also perform a log-
rank test, accounting for the non-detections, to test the null hypothesis of
no difference in metallicities between any two chosen azimuthal angle bins for
the three different types of clouds. The inferred $p-$values from these tests
suggest that the null hypothesis of no difference in the metallicity
distribution of gas along the minor axis, major axis, and interjacent angles
cannot be rejected for all three cloud types.

Veilleux et al. (2005) suggested that outflows should have large cloud-cloud
variations in metallicity. Such signals are also observed in very strong Mg ii
absorbers (Bond et al., 2001) and at lower column densities (e.g., Rauch et
al., 2002; Zonak et al., 2004). In Fig. 12b, we examine this by plotting the
metallicity variation, the difference between maximum and minimum
metallicities in a given absorption system along a given sightline, with
points coloured by their inclination angle. We only plot points corresponding
to an absorption system with more than one cloud. We do not observe
significant differences in the metallicity variation along the minor or major
axis. In the case of TDP–Low clouds, the number of clouds is too low to
observe a trend. It is possible that the dependency on other galaxy or gas
properties is the cause of scatter in the metallicity distribution as a
function of the azimuthal angle. We investigate any such dependency while
taking into account the H i-column density, inclination angle, impact
parameter scaled by virial radius, $B-K$ galaxy colour, absorber velocity
scaled by the galaxy circular velocity, and the halo mass of the galaxy.

Figure 13: The metallicity as a function of azimuthal angle color-coded by
different galaxy properties for PIE clouds. The upper limits on metallicity
are indicated by downward pointing arrows. Color-coded by (a): neutral
hydrogen column density; (b): inclination angle; (c): normalized impact
parameter; (d): galaxy $B-K$ color with marker shapes of upward triangles
indicating star-forming galaxies, downward triangles indicating passive
galaxies, and circles for which the nature of galaxy is not determined; (e):
absorber velocity scaled by the galaxy circular velocity; (f): galaxy halo
mass. Error bars reflect the 1$\sigma$ uncertainty.  Figure 14: Same as in
Fig. 13 but for TDP–Low clouds.

In Figs. 13 and 14, we present the relationship between metallicity and
azimuthal angle for PIE clouds and TDP–Low clouds where the data points are
coloured by different galaxy properties. In Figure 13a, the data points are
coloured by neutral hydrogen column density. We observe that the lowest
neutral hydrogen column densities are often upper limits on metallicity, and
these low hydrogen column densities are seen at all azimuthal angles. There is
also no discernible trend in the association between metallicity and azimuthal
angle among the absorbers chosen by H i column density. A metallicity
bimodality is expected to be strongest for pLLS due to inflows and outflows
(Wotta et al., 2019). We compare the metallicities of pLLSs for the minor and
major axis groupings, however, we do not find evidence of a significant
difference in their metallicity distributions. This finding suggests that a
generalization that metal-poor accretion occurs along the major axis and
metal-enriched outflows along the minor axis is too simplistic. Instead, this
indicates that the CGM is evenly mixed across the entire range of azimuthal
angle and for all H i column densities. We also do not see any evidence of a
trend for TDP–Low as well (Fig. 14a).

Furthermore, we take into account how the inclination angle affects the
correlation between metallicity and azimuthal angle. Assuming the simple
picture of the CGM, if the galaxy were edge-on, outflows and inflows would be
easier to identify since the cross sections of the gas flows on the sky are
minimal and do not overlap. It becomes increasingly challenging to establish
whether the quasar sight-line probes the major or minor axis as the galaxy’s
inclination becomes more face-on, when the cross sections of the gas flows
grow, and there are potentially more structures along the line of sight
(Churchill et al., 2015; Kacprzak et al., 2019; Peeples et al., 2019). An
edge-on galaxy sample maximizes the prospect of finding a metallicity-
azimuthal angle bimodality. Therefore, it is crucial to account for the
galaxy’s inclination angle when determining if metallicity and azimuthal angle
show any correlation. In Figure 13b, the metallicity is plotted as a function
of azimuthal angle with points coloured by the corresponding galaxy
inclination angle. We observe that edge-on galaxies, $i\gtrsim$ 60∘, show a
range of metallicities, with no apparent trend between metallicity and
azimuthal angle. We also do not see any evidence of a trend for TDP–Low (Fig.
14b).

We also consider how the impact parameter might affect the metallicity and
azimuthal angle relationship. Previous works (Bordoloi et al., 2011b; Lan &
Mo, 2018) have shown that for impact parameters less than 50–100 kpc, the
equivalent width of Mg ii absorbers was strongest along the minor axis,
suggesting that the azimuthal distribution of the CGM metallicities is
potentially bimodal at smaller impact parameters. Consequently, absorption
clouds with lower impact parameters could have a metallicity dependence on the
azimuthal angle. In Figure 13c, the metallicity is plotted as a function of
azimuthal angle with points coloured by the corresponding normalized impact
parameter. We observe that clouds at low impact parameters
$\frac{D}{R_{\\_}{vir}}\lesssim 0.5$, show a range of metallicities, with no
apparent trend between metallicity and azimuthal angle. We also do not see any
evidence of a trend for TDP–Low (Figure 14c).

Previous works have also found a bimodality in the Mg ii absorber azimuthal-
angle distribution for blue star-forming galaxies (Bordoloi et al., 2011b;
Kacprzak et al., 2012b; Lan & Mo, 2018). This finding indicates that blue,
star-forming galaxies may be where the simple CGM model holds up best. In
Figure 13d, we plot the metallicity as a function of azimuthal angle coloured
by $B-K$ colour with markers indicating whether a galaxy is star-forming
(upward triangles), passive (downward triangles), undetermined (circles). The
blue star-forming galaxies, $B-K<1.5$, in our sample have quasar sight-lines
probing all azimuthal angles, but we observe similar metallicity spread across
all orientations. Even along the minor axis, where it is anticipated that high
metallicities due to outflows would be seen, we find clouds with the two
lowest metallicities in our sample. A similar observation is made for TDP–Low
(Figure 14d. Thus, a metallicity and azimuthal angle relationship does not
seem to exist for blue, star-forming galaxies. Though many of our galaxies are
star-forming, their specific star-formation rates (sSFRs) are more than an
order of magnitude lower than starburst galaxies. Starburst galaxies can
experience star formation rates of 100 M⊙/yr (Schneider, 2006) compared with
an average of only few times M⊙/yr in our sample. Starburst galaxies are
expected to have active galactic winds that would produce high metallicities
along their minor axes. We also do not see any evidence of a trend for TDP–Low
clouds (Figure 14d).

The TNG50 simulations (Péroux et al., 2020) predict that on average higher
metallicities are expected to be linked with rapidly moving outflows emanating
along the minor axis. Conversely, gas with lower metallicities is linked with
slower-moving processes involving either accretion into the galaxy or inflow.
In Figure 13 (e), we plot the metallicity as a function of the azimuthal angle
coloured by the absorber velocity scaled by the galaxy circular velocity
measured at the impact parameter. While the large majority of the absorbers
have velocities within the circular velocity of the galaxy, we observe that
these absorbers show a wide range of metallicities for all azimuthal angles. A
minority of the absorbers that have velocities greater than the circular
velocity of the galaxy neither show a preference for high metallicities nor
are predominantly found along the minor axis. A similar observation is made
for TDP–Low (Figure 14e).

The TNG50 simulations (Péroux et al., 2020) also find that the enhancement of
metallicities along the minor axis compared with the major axis is strongest
for halo masses $\log M_{\rm halo}/M_{\odot}$$\sim$ 11.5. In Figure 13f, we
plot the metallicity as a function of the azimuthal angle coloured by the halo
mass. We find a range of metallicities for all azimuthal angles in various
bins of galaxy halo masses. A similar observation is made for TDP–Low (Figure
14f).

Berg et al. (2023) also did not find any strong correlations between the
metallicities or H i column densities of the gas and galaxy properties, except
for stellar mass of the galaxies. For the cool, photoionized gas, they found
that low-metallicity ($\rm[X/H]$ $\leq$ $-1.4$) systems have a probability of
$0.39_{\\_}{-0.15}^{+0.16}$ for being associated with a host galaxy with $\log
M_{\star}/M_{\odot}$ $\geq$ 9.0 within 1.5$R_{\rm vir}$, while higher
metallicity absorbers ($\rm[X/H]$ $>$ $-1.4$) have a probability of
$0.78_{\\_}{-0.13}^{+0.10}$.

##  6 DISCUSSION

We applied the CMBM method to the absorbers from the “Multiphase Galaxy Halos”
Survey to investigate the properties of the different regions/clouds of gas
along the quasar sightlines through the CGM of known galaxies. The focus of
our study was to examine if there is a relationship between the metallicity of
the CGM clouds and the azimuthal angle for sightlines through $z<0.7$
$L^{\star}$ isolated galaxies. Our methodology provides a much more detailed
view of differing cloud properties along the sightlines, as compared with P19,
who only computed an average metallicity for each galaxy/absorption line
system. With our CMBM analysis, we determined that CGM sightlines pass through
multiple phases of gas, some consistent with photoionization equilibrium and
some requiring time-dependent photoionization to explain the line ratios and
profile shapes. The 240 absorbing clouds that we found in 47 absorption
systems are found to be broadly consistent with three different underlying
populations when characterized by their intrinsic properties.

Figure 15: A comparison of the inferred properties viz. $\log Z/Z_{\sun}$,
$\log n_{\rm H}$, and $\log N(\mbox{H\,{\sc i}})$, obtained using the CMBM for
the different clouds along each sightline with the values from P19. The PIE
clouds are shown in navy blue, TDP–Low in dark orange, and TDP–High clouds in
red. The values from P19 are shown as crosses in magenta. The upper limits on
metallicity are shown as downward-pointing arrows, and the lower limits on
density are shown as upward-pointing arrows. A single average value is not
particularly representative of any of the gas clouds.

###  6.1 Interpreting the Absence of Metallicity/Azimuthal Angle Relationship

In the introduction, we discussed the expected trends in galaxy metallicities,
anticipating higher metallicities along the minor axis and lower metallicities
along the major axis. This expectation is based on both intuition and
simulations, particularly those presented in the study of Péroux et al.
(2020). We also pointed out that no such trend was identified in the analysis
conducted by P19 on the same galaxy sample as the one examined in the present
paper. P19 delved into several potential reasons for this lack of agreement
with simple models of the CGM, emphasizing that simple models of the CGM do
not apply, those with bipolar outflows and cold-mode accretion in the disk
plane. For instance, regions with high metallicity gas can arise close to the
major axis due to the recycling of metal-enriched gas from outflows.
Additionally, large wind opening angles might dilute metallicities along the
minor axis. Furthermore, they pointed out that the signature of outflows would
likely be more prominent at higher redshifts than in our current sample, which
is limited to $z\lesssim 0.7$. P19 also suggested that low-metallicity gas
might accrete along filaments that are not aligned with the major axis of the
galaxy. Despite the various plausible explanations for the lack of a trend,
the results of P19 were limited by their characterization of each sightline by
just a single metallicity.

To compare the P19 results with the CMBM method, we show the derived
metallicities, densities, and $N({\mbox{H\,{\sc i}}})$ for the different
clouds along each sightline in Fig. 15. The single-valued properties or limits
derived for each system by Pointon et al. (2019) are shown as a magenta cross.
A typical sightline through a galaxy has several clouds in PIE and one
TDP–High cloud, and perhaps half also have one or more TDP–Low clouds. The
ranges of metallicities and densities can be up to two orders of magnitude
along a single galaxy sightline. This reflects the fact that most sightlines
pass through different parts of a galaxy, potentially having contributions in
absorption from outer disk, halo, outflows, recycled accretion, tidal streams,
satellite galaxies, etc. The single average metallicities are not
systematically larger or smaller than the range of multi-cloud modelling
results, showing that the situation is complicated, but that the single
average value is not particularly representative of any of the gas clouds. The
total system $N({\mbox{H\,{\sc i}}})$ values from P19 are often consistent
with the value for the highest $N({\mbox{H\,{\sc i}}})$ cloud in the system
using CMBM. The main cause for the differences in derived metallicities
between our approach and P19 is that it is not typical for all of the H i to
be associated with all of the metal line column.

Figure 16: The trend between metallicity and azimuthal angle for a chosen
subsample, $\log N(\mbox{H\,{\sc i}})$ $\geq$ 15.0, inclination angle $\geq$
60∘, and impact parameter $\leq R_{\\_}{\rm vir}$, most likely to show
difference in metallicity between major ($\phi$ = 0∘) and minor axis ($\phi$ =
90∘). The clouds associated with star-forming galaxies are shown as upward
triangles, passive galaxies are shown as downward triangles, and the clouds
without information on the galaxy type are shown as circles.

Even with the detailed cloud information about multiple phases, there is still
no indication of a relationship between azimuthal angle and metallicity as
seen in Figs 13 and 14. There is no apparent trend for the highest or for the
lowest metallicity cloud in a system, nor is there a trend for galaxies’ near-
edge-on inclinations, where the largest difference between the major and minor
axis is expected. Both high and low metallicity clouds are found along the
same sightlines. To investigate this further, we chose a subsample of
sightlines most likely to show differences between major and minor axes ($\log
N(\mbox{H\,{\sc i}})$ $\geq$ 15.0, inclination angle $\geq$ 60∘, and impact
parameter $\leq R_{\\_}{\rm vir}$). A plot of the metallicity vs. azimuthal
angle for the subsample, comprising of PIE and TDP–Low clouds, is shown in
Fig. 16. There are very few sightlines close to the major axis that satisfy
all the subsample criteria, but it is worth noting that the full range of
metallicities is apparent along the minor axis, including high values expected
for outflow clouds, as well as lower values that could be contributed by dwarf
satellites and their stripped material. The simplest interpretation is that
the gas in these $z<0.7$ galaxies is often well mixed, meaning that gas
originating from different processes is now located in various parts of the
galaxy.

A substantial insight into that interpretation is provided by the results of
the FIRE–2 simulations (Hafen et al., 2019). The trajectories of different
particles, characterized by their origins in IGM accretion, winds, or
satellite winds, were followed for the 1 Gyr prior to $z=0.25$ for galaxy
halos of mass $10^{12}$ M${}_{\\_}{\odot}$, $10^{11}$ M${}_{\\_}{\odot}$, and
$10^{10}$ M${}_{\\_}{\odot}$. The higher mass halos correspond to those in our
present sample. The gas of different origins in the $10^{12}$
M${}_{\\_}{\odot}$ halos appears to be relatively well mixed at $z=0.25$, with
all three particle origins seen over the full halo volume. When only particles
which originated recently, between 1 Gyr and 0.5 Gyr ago, are considered, we
do see specific regions that the three types of particles prefer. The wind
particles are close to the central galaxy, the satellite wind particles are
mostly associated with three halo substructures and the accretion avoids some
directions in the halo. When origins at all times are included, however, this
asymmetry is no longer apparent. In conclusion, based on the results of Hafen
et al. (2019), we should not have expected a relationship between metallicity
and azimuthal angle in our sample. Gas at $z=0.25$ is well mixed, with gas
from several origins present at all impact parameters and azimuthal angles.

At higher redshifts the contribution of outflows is larger (e.g., Muratov et
al. 2015; Mitchell et al. 2018), and it is more likely that their signatures
will be observable in absorption. Hafen et al. (2019) found that winds
provided more than 40% of the inner halo mass at $z=2$, and that the
contributions of winds, satellite winds, and IGM accretion are all quite
inhomogeneous and not well mixed. Thus if we were to perform this same
experiment at high redshift we would be more likely to see relationships
between azimuthal angle and metallicity for some of the clouds.

Though we did not see any trends of cloud properties with galaxy properties,
we should note that the galaxies in our sample are relatively isolated, with
no neighbors within 100 kpc and 500 km s-1. This would imply fewer active
interactions in these galaxies and less wind activity than for a more crowded
environment (P19).

###  6.2 Nature and Origin of Different Cloud Types

In this section, we discuss the nature and origin of different types of
clouds. One key principle to keep in mind is that every location along a
sightline has a specific density, temperature, and metallicity. It is also
important to note that together the thicknesses that we have derived for the
clouds must sum to something of order a halo size or less.

In Fig. 4a, the three types of clouds found in our modelling, with the blue
symbols representing cooler clouds in thermal equilibrium, the gold ones
representing clouds of intermediate temperature that are modelled with time-
dependent photoionization, and the red ones with detected O vi as hotter
clouds also modelled with time-dependent photoionization, but as we discuss
below, they are found to be in collisional ionization equilibrium.

####  6.2.1 PIE Clouds

First, we will discuss the PIE clouds. Fig. 6 showed that the majority of them
have densities between $-3.5\leq$ $\log n_{\hbox{{\rm H}}}/\hbox{cm${}^{-3}$}$
$\leq-2.5$, metallicities greater than solar, and small thicknesses less than
a kpc and even sometimes sub-parsec. Werk et al. (2013) argued that the large
scatter in the low-ion absorption strengths indicates that the cool CGM is
patchy and hence reveals a wide range of ionization conditions. Many of the
PIE clouds have weak, low ionization absorption detected, and their properties
thus coincide with the metal-rich weak Mg ii absorbers studied by Rigby et al.
(2002), Narayanan et al. (2005), and Muzahid et al. (2018). While it is
tempting to characterize this population of absorbers as having a single
origin, we must remember that gas of density $\log n_{\rm H}$ = $-3$ is surely
not rare in galaxies and will occur in disk material, in outflows, in tidal
debris, and in various other regions of the CGM. This population is common,
with a typical sightline passing through a few of them (see Fig. 15).

Figure 17: Relationship between the sizes and hydrogen densities for the PIE
clouds. The size scale of cool CGM clouds formed by the process of shattering
is expected to follow $L=0.1$pc/$n_{\\_}{\rm H}$ (McCourt et al., 2018), and
shown as a dashed line on the plot.

Theoretical arguments predict small, cold clouds in the CGM of galaxies in
several contexts. McCourt et al. (2018) describe a hydrodynamic process by
which cold gas fragments into a collection of smaller cloudlets by the process
of cloud shattering, achieving equilibrium in the process. They predict that
cold gas fragments down to a lengthscale given by $L=0.1$pc/$n_{\\_}{\rm H}$,
with individual cloudlets having a characteristic column density
$N_{\\_}{cloudlet}$ $\sim$ 1017 cm-2. This relationship is shown as a dashed
line on the plot of $\log L$ vs. $\log n_{\\_}H$ in Fig. 17. We find that the
line is consistent with being a lower bound in PIE cloud thickness at every
$\log n_{\rm H}$, and the column densities of several of the PIE clouds are in
remarkable agreement with $N_{\\_}{\rm H}$ $\sim$ 1017 cm-2, suggesting that
fragmentation plays a role in their origin.

Figure 18: Absorber velocity as a function of azimuthal angle for the
different cloud types. The marker size is indicative of the cloud size. The
PIE clouds are shown in blue circles, TDP–Low in gold circles, and TDP–High
clouds in red triangles. In some cases, clouds of different sizes are present
along the same sightline suggesting that smaller clouds form by fragmentation
from a parent cloud. However, in other instances, it is possible that the
parent cloud is not intercepted along the line of sight.

Several high-resolution numerical simulations also show small structures
pervasive in the CGM. For example, the Mg ii column density distribution in
the TNG50 simulation of Nelson et al. (2020) forms, through thermal
instability, tens of thousands of fragments that are each a few 100 parsecs in
size. Schneider et al. (2020) find many small fragments in the Cholla Outflow
Simulation, which has a cell size of 5 pc. The expected CGM can be
characterized as a galactic mist, with fragments from a larger cloud that are
moving through a hot halo (Gronke & Oh, 2020). The fragments/droplets are not
stable, but the larger cloud continues to produce smaller ones so they are
ubiquitous. We plot the cloud velocities seen along different sightlines in
our sample in Fig. 18 as a function of the azimuthal angle; this figure shows
that both larger clouds and several smaller clouds are present along the same
sightline. The larger clouds are, however, not always consistent in velocity
with the small ones, but this is to be expected that the parent cloud would
not always be intercepted along the same line of sight.

The idea of a galactic mist is often connected to an origin in outflows, which
at face value would predict higher metallicities along the minor axis. Since
we do not see such a trend between azimuthal angle and metallicity in our
sample, that might suggest galactic mist is not an important process. On the
other hand, if mixing is efficient and clouds can survive, this would not be a
concern. Furthermore, the idea of fragmentation of cooler clouds moving
through a hot halo is more universal than the specific idea of galactic mist.
The Milky Way Galaxy does not have active outflows on large scales (Fox et
al., 2019), but it does have a hot halo and multiphase high-velocity clouds
passing through the halo. This could lead to fragmentation and the type of PIE
clouds we are observing in these absorption systems.

####  6.2.2 TDP–Low Clouds

Figure 19: Column density ratio of O vi/C iii plotted against N v/C iii for
the three different cloud types. PIE clouds are shown as filled circles,
TDP–Low clouds are shown as filled diamonds, and TDP–High clouds are shown as
squares, and the data points are coloured by their (a) metallicity; (b)
temperature; and (c) hydrogen number density. The TDP–Low clouds, with
temperatures in the range $4.7<$ $\log T$ $<5.1$ and metallicities in the
range $-2.0<$ $\log Z/Z_{\sun}$ $<0$, have the lowest densities in our sample
with $\log n_{\rm H}$ $<-4$. For such low-density clouds, the time to achieve
collisional ionization equilibrium is longer, and this gives rise to the need
for TDP. We do not show the TDP–High clouds on the plots coloured by
metallicity and density because these properties are not constrained for these
clouds. The arrows are indicative of the upper limits on the column densities.

The TDP–Low clouds have considerable overlap with PIE clouds in the phase
space of metallicity, density, and temperature. It is interesting to consider
in what way they are out of thermal equilibrium in order to gain insight into
their origins. Roughly half of the TDP–Low clouds have temperatures
$4.7\lesssim$ $\log T/K$ $\lesssim 5.1$, overlapping with PIE clouds but on
the higher side, and metallicities $-2<$ $\log Z/Z_{\sun}$ $<0$, also
consistent with PIE clouds. This can be seen in Fig. 4. The biggest difference
between these TDP–Low clouds, with higher temperatures, and the PIE clouds is
that the densities of many of these TDP–Low clouds are the lowest of clouds in
our sample, only $-5<$ $\log n_{\rm H}$ $<-4$ (see Fig. 5b). These densities
correspond to large cloud thicknesses, in the range $10$–$100$ kpc, among the
largest in our sample (see Fig. 6). The Doppler parameters tend to be higher
than those of the PIE sample because of higher temperatures (see Fig. 4 and
5). These larger Doppler parameters were constrained in our modelling process
by the relative widths of lines from different ions. Perhaps the higher
temperatures are responsible for the non-equilibrium conditions.

In Fig. 19, we use the line of ratios of key intermediate and high ionization
transitions as another way to distinguish the TDP–Low clouds from the PIE
clouds. Most of the TDP–Low clouds have $\log$(O vi/C iii) ratio $\sim
0.5$–$1$ dex higher than the PIE clouds on a plot of $\log$(O vi/C iii) vs.
$\log$(N v/C iii). Fig. 19c shows that the TDP–Low clouds with the largest
column densities of N v and O vi have the lowest densities, $\log n_{\rm H}$
$<-4$, lower than any other clouds in our sample. For such low-density clouds,
the time to achieve collisional ionization equilibrium is longer, and this
gives rise to the need for TDP. This same low-density TDP–Low population
includes those clouds with the highest temperatures among the TDP–Low clouds,
in the interval $4.7\lesssim$ $\log T/K$ $\lesssim 5.1$. Especially at these
intermediate temperatures, cooling occurs rapidly, often without sufficient
time to attain equilibrium. The other TDP–Low clouds have temperatures similar
to PIE clouds, which implies some of them may be in or close to PIE. They were
modelled with TDP only because they had detected C iv, but we found that in
many of them the temperature needed to produce observed line ratios turned out
to be consistent with equilibrium conditions.

####  6.2.3 TDP–High Clouds

The TDP–High clouds are distinguished from the other classes by their broad O
vi profiles and higher derived temperatures in the range $5.5<$ $\log T$
$<6.0$. In a few clouds, a discernable contribution to the Ly$\alpha$ profile
is seen, but in most, the contribution to Ly$\alpha$ is negligible or
dominated by other clouds in a saturated velocity range. The metallicities,
densities, and sizes of these clouds are not constrained, as we discuss below.

Though our models allowed for non-equilibrium collisional ionization and for
contributions from time-dependent photoionization, most of these O vi clouds
are in collisional ionization equilibrium. In order to have a departure from
equilibrium, the collisional ionization time scale would have to be short
compared with the timescale for recombination. The resulting non-equilibrium,
“overionized” condition occurs only for low densities, $\log n_{\rm H}$ $<-4$
for collisional ionization timescales longer than 10 million years
(Oppenheimer & Schaye, 2013). In Fig. 1, we show that the ionization fraction
of O vi diverges from the CIE expectation only at $\log T$$\lesssim 5.2$, and
especially for high metallicities ($Z/Z_{\\_}{\odot}>1$). At those lower
temperatures, the cooling times are short enough to leave the gas
“overionized”. In Fig. 19b, we see that O vi/C iii for the TDP–High clouds is
elevated by roughly an order of magnitude over the values for PIE clouds with
the same large values of N v/C iii. This is also consistent with the origin of
O vi in collisionally ionized material at a higher temperature.

In the absence of constraints on cloud sizes, an important question is whether
the collisionally ionized O vi absorbing gas found through the TDP–High clouds
is characteristic of the entire galaxy halo or whether this O vi arises in
substructures within the halo. Such substructures could be small regions,
heated by local processes, or could occupy a larger fraction of the halo
(i.e., tens to 100 kpc). To understand the origin of these higher temperature
regions giving rise to the TDP–High clouds, we can compare to numerical
simulations and to known structures in the Milky Way galaxy and local
universe.

Fig. 2 in Hafen et al. (2023) shows the temperature-vs-density distribution of
CGM gas in FIRE-2 cosmological simulations of Milky Way-like galaxies. A
substantial phase space density is present at the temperatures found for the
collisionally ionized O vi clouds, and these clouds are at over an order of
magnitude higher temperatures than given by the balance of heating and
cooling. Appleby et al. (2023) have studied the physical conditions of CGM gas
at low redshift using the SIMBA simulation Davé et al. (2019). They find that
O vi absorbers arise in gas with a temperature distribution that peaks at
$\log(T/{\rm K})=5.5$, with at least 80% collisionally ionized, though there
is a tail to lower temperatures with an origin in photoionized gas. They find
that the majority of the O vi absorption arises from the inner CGM and
characterize it as arising in WHIM or hot halo gas. This would imply larger,
more diffuse structures.

Figure 20: Histograms showing (a): halo masses for absorption systems with at
least one TDP–High (red) and systems with no TDP–High clouds (blue), (b):
normalized impact parameter for absorption systems with at least one TDP–High
(red) and systems with no TDP–High clouds (blue).

A favoured temperature for the presence of O vi absorption predicts a
dependence of O vi on halo mass, as seen in a version of the eagle simulations
that includes non-equilibrium ionization (Oppenheimer et al., 2016). These
simulations show maximum O vi column densities for $L^{\star}$ galaxies with
halo mass around $10^{12}$ M⊙. More massive galaxies or groups have higher
virial temperatures which promotes much of the oxygen to O vii and O viii.
This halo mass preference for O vi absorption is seen observationally in the
analyses of Pointon et al. (2017) and Ng et al. (2019), which compare galaxy
properties to the measured column densities of O vi. (Nielsen et al., 2017;
Kacprzak et al., 2019) also confirm that there are no obvious kinematic
signatures in the O vi profiles that would signal contributions of outflow,
inflow, or co-rotation. This is consistent with O vi arising in gas
distributed over much of the galaxy halo.

We can more definitively confirm the presence of O vi halos around $L^{\star}$
galaxies by examining the galaxy properties for TDP–High clouds in our sample.
Fig. 20a compares the mass distributions for sightlines through the 30
galaxies with one or more TDP–High cloud to the 17 sightlines through galaxies
that have no detected TDP–High absorption. The interquartile range for the
mass of galaxies without any TDP–High cloud is $\log M_{\\_}{\rm
halo}/M_{\\_}{\odot}=11.4-11.9$, lower than the range of 11.7–12.1 for
galaxies with at least one TDP–High cloud. We perform a 2-sample Anderson-
Darling test to test the null hypothesis that the galaxies with detected
TDP–High absorption and the galaxies without detected TDP–High absorption are
drawn from the same population. We find that the null hypothesis can be
rejected at the 5% level ($p=0.022$). Fig. 20 (b) shows the distributions of
impact parameters comparing galaxies with at least one TDP–High cloud to
galaxies without TDP–High clouds. We find that these two samples are also
significantly different from one another ($p=0.038$). This shows that several
of the sightlines without TDP–High clouds are passing outside of the virial
radius of their galaxies. We conclude that it is more likely for isolated, low
redshift, more massive ($\sim L^{\star}$) galaxies to be surrounded by
collisionally ionized O vi halos extending to $0.6$–$1.2$ virial radii.

Using semi-analytic models, Richter (2020) explored expectations for coronal
broad Ly$\alpha$ lines from galaxy halos and predicted $\log(N({\mbox{H\,{\sc
i}}})/{\rm cm}^{-2})=12.4-13.4$ and $b({\mbox{H\,{\sc i}}})$ of 70–200 km s-1.
Although the Ly$\alpha$ is not often detected because of the presence of other
phases and the $S/N$ of the data, these values are consistent with the
expectations for our TDP–High clouds (see Figs. 3a and 5a).

Within the coronal O vi halos of $L^{\star}$ galaxies, and in the lower mass
galaxies in our sample, there can also be substructures that produce O vi
absorption. In the specialized simulations of Mandelker et al. (2020), the O
vi absorption arises in the turbulent mixing layer formed as a cold
filamentary stream falls through the hot CGM. In such cases, the O vi is thus
arising in kpc-scale structures, and the surrounding medium is too hot to give
rise to O vi absorption. These would be more analogous to the TDP–Low clouds
in our sample, those with $\log(T/{\rm K})\sim 5$ (Fig. 4a), which have O vi
along with lower ionization transitions due to over-ionization.

There is also evidence that the TDP–High clouds are large substructures within
halos. They must be large because, for massive enough galaxies within
$0.75R_{\\_}{\rm vir}$, the covering factor for TDP–High clouds is close to
unity in our sample. In addition, we find that 15 of the 30 sightlines with
one or more TDP–High clouds have two or more of these clouds along the
sightline. So these galaxies, with multiple components, have kinematically
distinct substructures within the hot halos that give rise to the broad O vi
lines. In other cases, there could be kinematically overlapping substructures
that produce the O vi absorption from different regions within the halo.
Fundamentally, there are likely to be many segments of gas along the sightline
through the whole halo, with relatively low densities and temperatures
consistent with the virial temperature, that shape the TDP–High O vi profiles
that we observe.

The most detailed view that we have of structures like those that may be
responsible for the O vi clouds (TDP–High and TDP–Low) is in the Local Group.
For example, Krishnarao et al. (2022) published a focused study of the corona
of the Large Magellanic Cloud (LMC). Their analysis provides evidence for a
$\log T$$\sim 5.5$ corona around the LMC within the larger $\log T$$\sim 6$
Milky Way corona. Cooler clouds ($\log T$ $\sim 4$) lie within the LMC corona,
and C iv and Si iv absorption is produced in interface layers between these
cool clouds and the LMC corona. Generalizing those measurements of the LMC
corona, there are likely to be many substructures in the CGM of the galaxies
in our sample in the form of dwarf galaxies, and other infalling structures.
There is also an extended O vi corona surrounding Milky Way (and Andromeda)
analogues in the High-resolution Environmental Simulations of The Immediate
Area (HESTIA) model, with predicted O vi similar to the galaxies in our sample
Damle et al. (2022).

We conclude that the TDP–High clouds are collisionally ionized. They are
prevalent in $\sim L^{\star}$ galaxies and absent in some of the lower-mass
galaxies in our sample. Their temperatures are consistent with the virial
temperature for these halos, and their kinematics are consistent with
expectations for one or two substructures of order the halo size. Our CMBM
analysis has made it possible to detect the diffuse halo of most isolated, low
redshift galaxies, separating it from other phases through the broad O vi
absorption that it produces.

##  7 CONCLUSION

We inferred CGM absorption properties of 47, $z<0.7$, isolated galaxy-absorber
pairs from the “Multiphase Galaxy Halos” Survey using a cloud-by-cloud
multiphase Bayesian ionization modelling approach. The metallicity of the CGM
was compared with intrinsic absorber properties, including neutral hydrogen
column density, temperature, Doppler parameter, hydrogen number density, and
inferred line of sight thickness. We then compared the metallicity of the CGM
to galaxy properties such as the galaxy inclination, azimuthal angle, impact
parameter, halo mass, colours, and redshift. The galaxy inclinations and
azimuthal angles were obtained by modelling the galaxies in HST images with
GIM2D by P19. Our main findings are summarized here:

  1. 1.

We identify three main populations of CGM absorbers based on the type of
models needed to best explain the observed multiphase absorption. The 47
absorption systems are found to comprise 240 clouds. The temperature
distribution of the 157 cool PIE clouds is right-skewed with a median of
$T\sim$ 104.1 K and a tail extending to $T\sim$ 105 K. The temperature
distribution of the 35 warm-hot TDP–Low and 48 hotter TDP–High clouds appears
to be flat between $T\sim$104.2-5.2 K) and ($T\sim$105.4-6.0 K).

  2. 2.

The metallicity for the PIE clouds ranges between $-2.6\lesssim$ $\log
Z/Z_{\sun}$ $\lesssim 0.8$, with a median metallicity of $\log Z/Z_{\sun}$ =
$-0.41\pm 0.09$. The TDP–Low clouds show a median metallicity of $-0.59\pm
0.19$, and their metallicity ranges between $-2.3\lesssim$ $\log Z/Z_{\sun}$
$\lesssim 0.2$. The TDP–High clouds are found to be in CIE and thus have
unconstrained metallicities.

  3. 3.

The H i column density range of cool clouds (PIE) spans several decades 11.8
$\lesssim$ $\log N(\mbox{H\,{\sc i}})$ $\lesssim 19.7$, and decreases as the
impact parameter increases. The H i column density of warm-hot (TDP–Low)
clouds ranges between $12.8\lesssim$ $\log N(\mbox{H\,{\sc i}})$ $\lesssim
17.2$, and the H i column density of hotter phases (TDP–High) ranges between
$12.0\lesssim$ $\log N(\mbox{H\,{\sc i}})$ $\lesssim 15.2$.

  4. 4.

The cool clouds are explained by models that assume photoionization
equilibrium (PIE). In agreement with theoretical predictions, we find that
some of the PIE clouds follow the $L=0.1$pc/$n_{\\_}{\rm H}$ relationship,
consistent with the expectation of characteristic size scale for clouds formed
by the process of fragmentation.

  5. 5.

About half of the warm-hot clouds, referred to as TDP–Low clouds, are found to
be affected by time-dependent photoionization. These clouds are different from
PIE clouds in their large ratios (0.5$-$1 dex higher) of $\log$(O vi/C iii).
The remaining TDP–Low clouds have properties similar to PIE clouds, and these
clouds may be in or close to PIE.

  6. 6.

The hotter clouds, referred to as TDP–High clouds, are distinguished from
other classes by their broad O vi profiles. Though our models allowed for non-
equilibrium ionization, most of the O vi clouds are found to be close to
collisional ionization equilibrium at temperatures consistent with the virial
temperatures of their galaxy halos. The lowest mass galaxies and sightlines at
high impact parameters do not produce TDP–High clouds, but the $T\sim$ 105.7 K
gas that produces them is abundant in $L^{\star}$ galaxy halos.

  7. 7.

A bimodality is observed in the azimuthal distribution of the total number of
clouds and mean number of clouds per absorption system. More clouds are seen
along minor and major axes ($\Phi\lesssim 22.5^{\circ}$ $\cup$ $\Phi\gtrsim
67.5^{\circ}$), compared with interjacent azimuthal angles
($22.5^{\circ}\lesssim\Phi\lesssim 67.5^{\circ}$).

  8. 8.

The metallicity is not related to the azimuthal angle. Additionally, the
inclination angle, impact parameter, $B-K$ colours, subpopulations of the H i
column density, or the absorber velocities with respect to the galaxy central
velocity, have no bearing on this connection. The lack of a link between CGM
metallicity and azimuthal angle in this low-redshift study challenges a simple
model of the CGM with outflows along the minor axis and inflows along the
major axis. A variety of other processes (e.g., disk material and recycled
accretion) are contributing and there has been time for mixing of the
different cloud populations in these galaxies.

## Acknowledgements

We thank Nicolas Lehner and Christopher Howk for helpful discussions during
the course of this work. S, J.C.C, B.W, acknowledges the support for this work
through the grant number HST-AR-16607. Computations for this research were
performed on the Pennsylvania State University’s Institute for Computational
and Data Sciences’ Roar supercomputer. The authors also acknowledge the Texas
Advanced Computing Center (TACC) at The University of Texas at Austin for
providing HPC resources that have contributed to the research results reported
within this paper. Parts of this research were supported by the Australian
Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions
(ASTRO 3D), through project number CE170100013. A.N acknowledges the support
for this work through grant number EMR/2017/002531 from the Department of
Science and Technology, Government of India. We are grateful to Gary Ferland
and collaborators for developing the cloudy photoionization code. We also
thank Orly Gnat for making the computational data on radiatively cooling
models public. This research made use of Astropy, a community-developed core
Python package for Astronomy (Astropy Collaboration et al., 2013), NumPy
(Harris et al., 2020), matplotlib (Hunter, 2007), and SciPy (Virtanen et al.,
2020). This research has made use of the HSLA database, developed and
maintained at STScI, Baltimore, USA. Some of the data presented herein were
obtained at the W. M. Keck Observatory, which is operated as a scientific
partnership among the California Institute of Technology, the University of
California and the National Aeronautics and Space Administration. Observations
were supported by Swinburne Keck programs 2014A_W178E, 2014B_W018E, 2015_W018E
and 2016A_W056E. The Observatory was made possible by the generous financial
support of the W. M. Keck Foundation. The authors wish to recognize and
acknowledge the very significant cultural role and reverence that the summit
of Maunakea has always had within the indigenous Hawaiian community. We are
most fortunate to have the opportunity to conduct observations from this
mountain. This research has made use of the NASA/IPAC Extragalactic Database
(NED), which is operated by the Jet Propulsion Laboratory, California
Institute of Technology, under contract with NASA. We acknowledge the work of
people involved in the design, construction and deployment of the COS on-board
the Hubble Space Telescope, and thank all those who obtained data for the
sight-lines studied in this paper. We thank KITP for hosting the Fundamentals
of Gaseous Halos workshop (supported by the NSF under Grant No. NSF
PHY-1748958) during which some of the ideas presented in this work were
formulated.

## Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

## References

  * Acharya & Khaire (2021) Acharya A., Khaire V., 2021, MNRAS,
  * Akritas et al. (1995) Akritas M. G., Murphy S. A., Lavalley M. P., 1995, Journal of the American Statistical Association, 90, 170
  * Anglés-Alcázar et al. (2017) Anglés-Alcázar D., Faucher-Giguère C.-A., Kereš D., Hopkins P. F., Quataert E., Murray N., 2017, MNRAS, 470, 4698
  * Appleby et al. (2023) Appleby S., Davé R., Sorini D., Cui W., Christiansen J., 2023, MNRAS, 519, 5514
  * Astropy Collaboration et al. (2013) Astropy Collaboration et al., 2013, A&A, 558, A33
  * Barber et al. (1996) Barber C. B., Dobkin D. P., Huhdanpaa H., 1996, ACM Trans. Math. Softw., 22, 469–483
  * Berg et al. (2023) Berg M. A., et al., 2023, ApJ, 944, 101
  * Bond et al. (2001) Bond N. A., Churchill C. W., Charlton J. C., Vogt S. S., 2001, ApJ, 562, 641
  * Bordoloi et al. (2011a) Bordoloi R., et al., 2011a, ApJ, 743, 10
  * Bordoloi et al. (2011b) Bordoloi R., et al., 2011b, ApJ, 743, 10
  * Bordoloi et al. (2014) Bordoloi R., et al., 2014, ApJ, 794, 130
  * Borthakur (2022) Borthakur S., 2022, ApJ, 924, 123
  * Borthakur et al. (2015) Borthakur S., et al., 2015, ApJ, 813, 46
  * Bouché et al. (2012) Bouché N., Hohensee W., Vargas R., Kacprzak G. G., Martin C. L., Cooke J., Churchill C. W., 2012, MNRAS, 426, 801
  * Bregman (1980) Bregman J. N., 1980, ApJ, 236, 577
  * Buchner (2014) Buchner J., 2014, arXiv e-prints, p. arXiv:1407.5459
  * Burchett et al. (2013) Burchett J. N., Tripp T. M., Werk J. K., Howk J. C., Prochaska J. X., Ford A. B., Davé R., 2013, ApJ, 779, L17
  * Cen & Ostriker (2006) Cen R., Ostriker J. P., 2006, ApJ, 650, 560
  * Charlton et al. (2000) Charlton J. C., Mellon R. R., Rigby J. R., Churchill C. W., 2000, ApJ, 545, 635
  * Charlton et al. (2003) Charlton J. C., Ding J., Zonak S. G., Churchill C. W., Bond N. A., Rigby J. R., 2003, ApJ, 589, 111
  * Chen & Mulchaey (2009) Chen H.-W., Mulchaey J. S., 2009, ApJ, 701, 1219
  * Chen et al. (2001) Chen H.-W., Lanzetta K. M., Webb J. K., 2001, ApJ, 556, 158
  * Chen et al. (2010) Chen H.-W., Wild V., Tinker J. L., Gauthier J.-R., Helsby J. E., Shectman S. A., Thompson I. B., 2010, ApJ, 724, L176
  * Chiappini et al. (1997) Chiappini C., Matteucci F., Gratton R., 1997, ApJ, 477, 765
  * Churchill & Charlton (1999) Churchill C. W., Charlton J. C., 1999, AJ, 118, 59
  * Churchill & Vogt (2001) Churchill C. W., Vogt S. S., 2001, AJ, 122, 679
  * Churchill et al. (2012) Churchill C. W., Kacprzak G. G., Steidel C. C., Spitler L. R., Holtzman J., Nielsen N. M., Trujillo-Gomez S., 2012, ApJ, 760, 68
  * Churchill et al. (2013) Churchill C. W., Trujillo-Gomez S., Nielsen N. M., Kacprzak G. G., 2013, ApJ, 779, 87
  * Churchill et al. (2015) Churchill C. W., Vander Vliet J. R., Trujillo-Gomez S., Kacprzak G. G., Klypin A., 2015, ApJ, 802, 10
  * Crighton et al. (2015) Crighton N. H. M., Hennawi J. F., Simcoe R. A., Cooksey K. L., Murphy M. T., Fumagalli M., Prochaska J. X., Shanks T., 2015, MNRAS, 446, 18
  * Curran et al. (2016) Curran S. J., Reeves S. N., Allison J. R., Sadler E. M., 2016, MNRAS, 459, 4136
  * Damle et al. (2022) Damle M., et al., 2022, Monthly Notices of the Royal Astronomical Society, 512, 3717
  * Danforth et al. (2010) Danforth C. W., Stocke J. T., Shull J. M., 2010, ApJ, 710, 613
  * Danovich et al. (2012) Danovich M., Dekel A., Hahn O., Teyssier R., 2012, MNRAS, 422, 1732
  * Danovich et al. (2015) Danovich M., Dekel A., Hahn O., Ceverino D., Primack J., 2015, MNRAS, 449, 2087
  * Davé et al. (2011a) Davé R., Oppenheimer B. D., Finlator K., 2011a, MNRAS, 415, 11
  * Davé et al. (2011b) Davé R., Finlator K., Oppenheimer B. D., 2011b, MNRAS, 416, 1354
  * Davé et al. (2012) Davé R., Finlator K., Oppenheimer B. D., 2012, MNRAS, 421, 98
  * Davé et al. (2019) Davé R., Anglés-Alcázar D., Narayanan D., Li Q., Rafieferantsoa M. H., Appleby S., 2019, MNRAS, 486, 2827
  * Dekel & Birnboim (2006) Dekel A., Birnboim Y., 2006, MNRAS, 368, 2
  * Dekel et al. (2009) Dekel A., et al., 2009, Nature, 457, 451
  * Dekker et al. (2000) Dekker H., D’Odorico S., Kaufer A., Delabre B., Kotzlowski H., 2000, in Iye M., Moorwood A. F., eds, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series Vol. 4008, Optical and IR Telescope Instrumentation and Detectors. pp 534–545, doi:10.1117/12.395512
  * Ding et al. (2003a) Ding J., Charlton J. C., Bond N. A., Zonak S. G., Churchill C. W., 2003a, ApJ, 587, 551
  * Ding et al. (2003b) Ding J., Charlton J. C., Churchill C. W., Palma C., 2003b, ApJ, 590, 746
  * Ding et al. (2005) Ding J., Charlton J. C., Churchill C. W., 2005, ApJ, 621, 615
  * Draganova (2015) Draganova N., 2015, arXiv e-prints, p. arXiv:1511.04604
  * Dutta et al. (2021) Dutta R., et al., 2021, MNRAS, 508, 4573
  * Faucher-Giguère et al. (2011) Faucher-Giguère C.-A., Kereš D., Ma C.-P., 2011, MNRAS, 417, 2982
  * Fielding & Bryan (2022) Fielding D. B., Bryan G. L., 2022, ApJ, 924, 82
  * Ford et al. (2014) Ford A. B., Davé R., Oppenheimer B. D., Katz N., Kollmeier J. A., Thompson R., Weinberg D. H., 2014, MNRAS, 444, 1260
  * Fox et al. (2013) Fox A. J., et al., 2013, ApJ, 778, 187
  * Fox et al. (2019) Fox A. J., Richter P., Ashley T., Heckman T. M., Lehner N., Werk J. K., Bordoloi R., Peeples M. S., 2019, ApJ, 884, 53
  * Fumagalli et al. (2011) Fumagalli M., O’Meara J. M., Prochaska J. X., 2011, Science, 334, 1245
  * Gibson et al. (2022) Gibson J. L., Lehner N., Oppenheimer B. D., Howk J. C., Cooksey K. L., Fox A. J., 2022, arXiv e-prints, p. arXiv:2204.07586
  * Gnat (2017) Gnat O., 2017, ApJS, 228, 11
  * Gobat (2022) Gobat C., 2022, Asymmetric Uncertainty: Handling nonstandard numerical uncertainties, Astrophysics Source Code Library, record ascl:2208.005 (ascl:2208.005)
  * Grevesse et al. (2010) Grevesse N., Asplund M., Sauval A. J., Scott P., 2010, Ap&SS, 328, 179
  * Gronke & Oh (2020) Gronke M., Oh S. P., 2020, MNRAS, 494, L27
  * Haardt & Madau (2001) Haardt F., Madau P., 2001, in Neumann D. M., Tran J. T. V., eds, Clusters of Galaxies and the High Redshift Universe Observed in X-rays. p. 64 (arXiv:astro-ph/0106018), doi:10.48550/arXiv.astro-ph/0106018
  * Haardt & Madau (2012) Haardt F., Madau P., 2012, ApJ, 746, 125
  * Hafen et al. (2017) Hafen Z., et al., 2017, MNRAS, 469, 2292
  * Hafen et al. (2019) Hafen Z., et al., 2019, MNRAS, 488, 1248
  * Hafen et al. (2023) Hafen Z., et al., 2023, arXiv e-prints, p. arXiv:2305.01842
  * Haislmaier et al. (2021) Haislmaier K. J., Tripp T. M., Katz N., Prochaska J. X., Burchett J. N., O’Meara J. M., Werk J. K., 2021, MNRAS, 502, 4993
  * Harris et al. (2020) Harris C. R., et al., 2020, Nature, 585, 357
  * Hunter (2007) Hunter J. D., 2007, Computing in Science & Engineering, 9, 90
  * Johnson et al. (2013) Johnson S. D., Chen H.-W., Mulchaey J. S., 2013, MNRAS, 434, 1765
  * Johnson et al. (2015) Johnson S. D., Chen H.-W., Mulchaey J. S., 2015, MNRAS, 449, 3263
  * Johnson et al. (2017) Johnson S. D., Chen H.-W., Mulchaey J. S., Schaye J., Straka L. A., 2017, ApJ, 850, L10
  * Jones et al. (2010) Jones T. M., Misawa T., Charlton J. C., Mshar A. C., Ferland G. J., 2010, ApJ, 715, 1497
  * Kacprzak et al. (2008) Kacprzak G. G., Churchill C. W., Steidel C. C., Murphy M. T., 2008, AJ, 135, 922
  * Kacprzak et al. (2010) Kacprzak G. G., Murphy M. T., Churchill C. W., 2010, MNRAS, 406, 445
  * Kacprzak et al. (2011) Kacprzak G. G., Churchill C. W., Barton E. J., Cooke J., 2011, ApJ, 733, 105
  * Kacprzak et al. (2012a) Kacprzak G. G., Churchill C. W., Steidel C. C., Spitler L. R., Holtzman J. A., 2012a, MNRAS, 427, 3029
  * Kacprzak et al. (2012b) Kacprzak G. G., Churchill C. W., Nielsen N. M., 2012b, ApJ, 760, L7
  * Kacprzak et al. (2015a) Kacprzak G. G., Muzahid S., Churchill C. W., Nielsen N. M., Charlton J. C., 2015a, ApJ, 815, 22
  * Kacprzak et al. (2015b) Kacprzak G. G., Muzahid S., Churchill C. W., Nielsen N. M., Charlton J. C., 2015b, ApJ, 815, 22
  * Kacprzak et al. (2016) Kacprzak G. G., et al., 2016, ApJ, 826, L11
  * Kacprzak et al. (2019) Kacprzak G. G., Pointon S. K., Nielsen N. M., Churchill C. W., Muzahid S., Charlton J. C., 2019, ApJ, 886, 91
  * Keeney et al. (2017) Keeney B. A., et al., 2017, ApJS, 230, 6
  * Keeney et al. (2018) Keeney B. A., et al., 2018, ApJS, 237, 11
  * Kereš et al. (2005) Kereš D., Katz N., Weinberg D. H., Davé R., 2005, MNRAS, 363, 2
  * Khaire & Srianand (2015) Khaire V., Srianand R., 2015, ApJ, 805, 33
  * Khaire et al. (2019) Khaire V., et al., 2019, MNRAS, 486, 769
  * Knuth (2006) Knuth K. H., 2006, arXiv e-prints, p. physics/0605197
  * Krishnarao et al. (2022) Krishnarao D., et al., 2022, Nature, 609, 915
  * Krogager (2018) Krogager J.-K., 2018, arXiv e-prints, p. arXiv:1803.01187
  * Kulkarni et al. (2022) Kulkarni V. P., Bowen D. V., Straka L. A., York D. G., Gupta N., Noterdaeme P., Srianand R., 2022, ApJ, 929, 150
  * Lacki & Charlton (2010) Lacki B. C., Charlton J. C., 2010, MNRAS, 403, 1556
  * Lan & Mo (2018) Lan T.-W., Mo H., 2018, ApJ, 866, 36
  * Lanzetta et al. (1995) Lanzetta K. M., Bowen D. V., Tytler D., Webb J. K., 1995, ApJ, 442, 538
  * Lehner et al. (2018) Lehner N., Wotta C. B., Howk J. C., O’Meara J. M., Oppenheimer B. D., Cooksey K. L., 2018, ApJ, 866, 33
  * Lehner et al. (2019) Lehner N., Wotta C. B., Howk J. C., O’Meara J. M., Oppenheimer B. D., Cooksey K. L., 2019, ApJ, 887, 5
  * Lehner et al. (2020) Lehner N., et al., 2020, ApJ, 900, 9
  * Lehner et al. (2022) Lehner N., et al., 2022, ApJ, 936, 156
  * Lochhaas et al. (2020) Lochhaas C., Bryan G. L., Li Y., Li M., Fielding D., 2020, MNRAS, 493, 1461
  * Lundgren et al. (2021) Lundgren B. F., et al., 2021, ApJ, 913, 50
  * Lynch & Charlton (2007) Lynch R. S., Charlton J. C., 2007, ApJ, 666, 64
  * Maller & Bullock (2004) Maller A. H., Bullock J. S., 2004, MNRAS, 355, 694
  * Mandelker et al. (2020) Mandelker N., van den Bosch F. C., Nagai D., Dekel A., Birnboim Y., Aung H., 2020, MNRAS, 498, 2415
  * Marra et al. (2021) Marra R., et al., 2021, MNRAS, 508, 4938
  * Marra et al. (2022) Marra R., Churchill C. W., Kacprzak G. G., Nielsen N. M., Trujillo-Gomez S., Lewis E., 2022, arXiv e-prints, p. arXiv:2202.12228
  * Martin et al. (2019) Martin C. L., Ho S. H., Kacprzak G. G., Churchill C. W., 2019, ApJ, 878, 84
  * Masiero et al. (2005) Masiero J. R., Charlton J. C., Ding J., Churchill C. W., Kacprzak G., 2005, ApJ, 623, 57