# An Empirical Determination of the Dependence of the Circumgalactic Mass
Cooling Rate and Feedback Mass Loading Factor on Galactic Stellar Mass

Huanian Zhang (张华年)11affiliation: Steward Observatory, University of Arizona,
Tucson, AZ 85719, USA<EMAIL_ADDRESS>, Dennis
Zaritsky11affiliation: Steward Observatory, University of Arizona, Tucson, AZ
85719, USA<EMAIL_ADDRESS>, Karen Pardos Olsen 11affiliation:
Steward Observatory, University of Arizona, Tucson, AZ 85719, USA;
<EMAIL_ADDRESS>, Peter Behroozi 11affiliation: Steward
Observatory, University of Arizona, Tucson, AZ 85719, USA;
<EMAIL_ADDRESS>, Jessica Werk 22affiliation: Department of
Astronomy, University of Washington, Seattle, WA 98195, USA , Robert Kennicutt
11affiliation: Steward Observatory, University of Arizona, Tucson, AZ 85719,
USA<EMAIL_ADDRESS>33affiliation: George P. and Cynthia W.
Mitchell Institute for Fundamental Physics & Astronomy, Texas A&M University,
College Station, TX 77843-4242, USA , Lizhi Xie (谢利智) 44affiliation: Tianjin
Normal University, Binshuixidao 393, 300387, Tianjin, China , Xiaohu Yang
(杨小虎) 55affiliation: Department of Astronomy, School of Physics and Astronomy,
and Shanghai Key Laboratory for Particle Physics and Cosmology, Shanghai
JiaoTong University, Shanghai, 200240, China 66affiliation: Tsung-Dao Lee
Institute, and Key Laboratory for Particle Physics, Astrophysics and
Cosmology, Ministry of Education, Shanghai Jiao Tong University Shanghai,
200240, China , Taotao Fang (方陶陶) 77affiliation: Department of Astronomy,
Xiamen University, Xiamen, Fujian, China  , Gabriella De Lucia 88affiliation:
INAF - Astronomical Observatory of Trieste, via G.B. Tiepolo 11, I-34143
Trieste, Italy , Michaela Hirschmann 99affiliation: DARK, Niels Bohr
Institute, University of Copenhagen, Niels Bohr Building, Jagtvej 128, 2200
Copenhagen, Denmark , Fabio Fontanot 88affiliation:  INAF - Astronomical
Observatory of Trieste, via G.B. Tiepolo 11, I-34143 Trieste, Italy
1010affiliation: IFPU - Institute for Fundamental Physics of the Universe, via
Beirut 2, 34151, Trieste, Italy

###### Abstract

Using our measurements of the H$\alpha$ emission line flux originating in the
cool (T $\sim 10^{4}$ K) gas that populates the halos of galaxies, we build a
joint model to describe mass of the cool circumgalactic medium (CGM) as a
function of galactic stellar mass ($10^{9.5}<({\rm M_{*}/M}_{\odot})<10^{11}$)
and environment. Because the H$\alpha$ emission correlates with the main
cooling channel for this gas, we are able to estimate the rate at which the
CGM cools and becomes fuel for star formation in the central galaxy. We
describe this calculation, which uses our observations, previous measurements
of some critical CGM properties, and modeling of the cooling mechanism using
the Cloudy modeling suite. We find that the mass cooling rate is larger than
the star formation rates of the central galaxies by a factor of $\sim 4-90$,
empirically confirming that there is sufficient fuel to resolve the gas
consumption problem and that feedback is needed to avoid collecting too much
cold gas in galaxies. We find excellent agreement between our estimates of
both the mass cooling rates and mass loading factors and the predictions of
independent theoretical studies. The convergence in results that we find from
several completely different treatments of the problem, particularly at the
lower end of the galactic mass range, is a strong indication that we have a
relatively robust understanding of the quantitative effects of feedback across
this mass range.

###### Subject headings:

galaxies: kinematics and dynamics, structure, halos, intergalactic medium

##  1\. Introduction

The interstellar medium (ISM) alone can not sustain the star formation
activity of a galaxy for its entire lifespan, thus necessitating the inflow of
cool gas from the circumgalactic medium (CGM) to provide additional fuel
(Spitzer, 1956). Indeed, the COS-GASS survey found a correlation between
interstellar and circumgalactic H I, implying a physical connection between
the H I disk and the CGM such that the H I disk is nourished by accretion of
gas from the CGM (Borthakur et al., 2015). Through observations and
simulations, we have now arrived at the conclusion that the baryon cycling
between the CGM and central galaxy is the dominant factor driving the
observable properties of galaxies (see Tumlinson, Peeples & Werk, 2017, for a
review). As such, measuring the net gas mass inflow rate is key to our
understanding of galaxy evolution.

Hydrodynamic simulations have shed light on the delicate process of
precipitating cold gas out of the hot medium in the CGM. Small-box simulations
of a region of a galactic wind flow or the CGM can reach the resolution needed
to track cooling/precipitation and have been used to study the possible origin
of phenomena such as cold molecular clumps found in quasar outflows (e.g.
Ferrara & Scannapieco, 2016; Gronke & Oh, 2018; Liang & Remming, 2020). Larger
cosmological simulations provide the information required to understand more
global phenomena such as the observed O VI content of galactic haloes (e.g.
Suresh et al., 2017; Oppenheimer et al., 2018; Peeples et al., 2019;
vandeVoort et al., 2019; Hummels et al., 2019; Nelson et al., 2020). Among
others, one of the ultimate goals of these hydrodynamic simulations, combined
with more phenomenological treatments, is to determine the inflow rate of cold
gas into the central portion of galaxies and the fraction of that gas that
must be removed or reheated, so that together they fuel and throttle the
process of star formation on galactic scales (Muratov et al., 2015; Hayward &
Hopkins, 2017; Xie et al., 2017, 2020).

Among observational studies, most attempt to trace the CGM and establish its
basic characteristics (e.g., total mass, radial density profile, metallicity,
kinematics). There are relatively few empirical constraints on the cooling
rate itself which is more closely related with the star formation process in
galaxies (for one exception to this statement see Voit et al., 2019, which
models the edges of cold clouds to estimate the precipitation rate). The
reason for the absence of constraints on the cooling rate is that studies of
the CGM have come primarily from the study of absorption lines in the spectra
of bright background objects (e.g. Steidel et al., 2010; Chen et al., 2010;
Ménard et al., 2011; Bordoloi et al., 2011; Zhu & Ménard, 2013a, b; Werk et
al., 2013; Johnson, Chen & Mulchaey, 2013; Johnson et al., 2014; Werk et al.,
2014; Johnson, Chen & Mulchaey, 2015; Werk et al., 2016; Croft et al., 2016,
2018; Prochaska et al., 2017; Cai et al., 2017; Johnson et al., 2017; Chen et
al., 2010; Chen, 2017b, a; Lan & Mo, 2018; Joshi et al., 2018; Chen et al.,
2019; Zahedy et al., 2019). However, a growing set of studies is focusing on
emission lines. For example, there are now measurements of UV or soft X-ray
emission lines in either low-redshift starburst galaxies (Hayes et al., 2016)
or high-redshift galaxies (Vikhlinin, 2006; Cantalupo et al., 2014; Prescott,
Martin & Dey, 2015; Cai et al., 2019).

In emission lines, we are directly viewing the escaping energy from the CGM.
As such, the measurement of emission lines originating from the CGM offers the
potential opportunity to explore questions regarding the cooling rates. Of
course, the problem is more complex than any simple scenario. In the local
Universe, galactic coronae can be studied via their X-ray emission, but X-ray
observations of the Milky Way and nearby galaxies have shown that charge
exchange X-ray emission, generated at the interface between the cool and hot
gas in the halo, may also make a non-negligible contribution to the cooling
(e.g. Henley et al., 2015; Jiang et al., 2019). As such, the reader should
bear in mind that our discussion of the mass cooling process in the following
is likely to be oversimplified.

Optical emission lines provide an opportunity to explore the cooling rate of
the cool CGM, but are extremely challenging to measure. In the local universe,
H$\alpha$ is detected in individual galaxies only when the systems are extreme
and difficult to model (such as in the starburst/merger NGC 6240; Yoshida et
al., 2016). However, Zhang et al. (2016) presented the first detection of the
emission line flux of H$\alpha$ and [N II] $\lambda$6583, from low redshift,
normal galaxies extending out to $\sim$ 100 kpc projected radius by stacking a
sample of over 7 million lines of sight from the Sloan Digital Sky Survey
(SDSS DR12; Alam et al., 2015). Building on that first result, we have applied
the technique to characterize the nature of the emission of the CGM within 50
kpc in low redshift galaxies (Zhang et al., 2016; Zhang, Zaritsky & Behroozi,
2018; Zhang et al., 2018, 2019, 2020b, 2020a, hereafter, Papers I, II, III,
IV, V, VI). We restrict ourselves to the inner 50 kpc because the measured
emission beyond that is strongly affected by emission from nearby galaxies
(Paper II). In Paper IV and V, we studied and modeled the environmental
dependence and the stellar mass dependence, respectively.

Theoretical studies in the literature (e.g., Mathews & Bregman, 1978; Schure
et al., 2009; Lykins et al., 2013; Vasiliev, 2013) have examined the cooling
of the plasma within the temperature range of $10^{4}-10^{8}$ K, which can be
assumed to be in collisional ionization equilibrium, using the hydrogen and
metal line emission fluxes of the elements H, He, C, N, O, Ne, Mg, Si and Fe
for temperatures of $10^{4}-10^{8}$ K, among which the hydrogen emission flux
can be dominant for the temperatures of $10^{4}-10^{5}$ K. These existing,
extensive studies of cooling rates and the computational infrastructure
developed to study the ISM and CGM, such as Cloudy (Ferland et al., 1998,
2013, 2017), are key to our study.

We aim to convert the observed H$\alpha$ emission flux radiating from the CGM
of galaxies into an estimate of the mass cooling rate in units of M⊙ yr-1. We
will then use this information to infer the required rate at which infalling
gas must be reheated or ejected back into the CGM, under the hypothesis of
steady state equilibrium, by comparing the mass cooling rate to the star
formation rate. This will be compared to theoretical estimates of these
quantities derived from requiring a match between other observable properties
of the galaxies (stellar mass, star formation rates) in their models. Multiple
approaches in estimating these quantities are critical given the likely
presence of systematic errors in any individual approach. Before doing all of
this, we revisit previous empirical results and models of the CGM dependence
on stellar mass and environment using both an expanded data set and a slightly
more complicated model to ensure that we have the latest estimates of CGM
properties with which to estimate the mass cooling rate and feedback mass
loading factor.

The paper is organized as follows. In §2 we present the data analysis and
results, with a focus on how the emission flux depends on galactic stellar
mass and local environment. In §3.1 we discuss our joint modeling of the CGM
cool gas mass fraction on both galactic stellar mass and local environment. We
describe our calculation of the cooling rate in §3.2, including the
application of Cloudy in §3.2.1. We present our estimates of the mass cooling
rates and mass loading factors in §3.3. Lastly, we summarize in §4. Throughout
this paper, we adopt a $\Lambda$CDM cosmology with parameters $\Omega_{m}$ =
0.3, $\Omega_{\Lambda}=$ 0.7, $\Omega_{k}$ = 0 and the dimensionless Hubble
constant $h=$ 0.7 (cf. Riess et al., 2018; Planck Collaboration et al., 2018).

##  2\. Data Analysis

We follow the approach developed in Papers I through VI (Zhang et al., 2016;
Zhang, Zaritsky & Behroozi, 2018; Zhang et al., 2018, 2019, 2020b, 2020a,
respectively), by selecting galaxies that meet our criteria in redshift
($0.02<z<0.2$), half light radius ($1.5<R_{50}/{\rm kpc}<10$), and luminosity
($10^{9.5}<L/L_{\odot}<10^{11}$) as primary galaxies. For these galaxies, we
extract measurements of the Sérsic index ($n$) and $r$-band absolute magnitude
($M_{r}$) from Simard et al. (2011), stellar mass (M∗) from Kauffmann et al.
(2003a, b) and Gallazzi et al. (2005), and star formation rates (SFR) from the
MPA-JHU catalog (Brinchmann et al., 2004). The SFR estimates are aperture
corrected to account for the light outside the SDSS fiber aperture, which only
collects $\sim$ 1/3 of the total light for a typical galaxy at the median
redshift of the survey (for details see Brinchmann et al., 2004). Our
requirements for these ancillary measurements limit the primary sample to
galaxies from the 7th major data release of SDSS (DR7), but we utilize spectra
from DR16 to probe the CGM of these galaxies.

We collect the spectra from the SDSS DR16 (Ahumada et al., 2020) for all
lines-of-sight projected within a specified range of scaled projected radii
from any of our primary galaxies. As described in Paper V, we use scale
projected radii to account for the range in primary galaxy sizes. We define
the scaled projected radius, $r_{s}$, as the ratio between the physical
projected separation and the virial radius of the primary galaxy ($r_{s}\equiv
r_{p}/r_{\rm vir}$). To estimate the virial radius of the primary galaxy, we
use the scaling relation between the luminosity and the virial radius obtained
by fitting a high order polynomial to the results drawn from the
UniverseMachine (Behroozi et al., 2018). As discussed in more detail in Paper
III, we set a physical lower limit on the projected radius (10 kpc) we explore
to mitigate possible contamination of the line of sight spectra by the central
galaxy. We confirm, but discuss no further below, that increasing this lower
limit on the projected separation to 20 kpc produces statistically consistent
results except for those involving the innermost radial bin of the smallest
primary galaxies. Those measurements are affected because this change removes
almost all lines of sight from this bin.

Our procedure for the processing of the line-of-sight probes follows from our
previous papers. For each such spectrum, we fit and subtract a 10th order
polynomial to a 300 Å wide section surrounding the wavelength of H$\alpha$ at
the related primary galaxy redshift to remove the continuum. We only present
and discuss H$\alpha$ fluxes in this work because the theoretical modeling of
[N II] is more complicated and beyond the question we focus on here. We then
measure the residual H$\alpha$ flux within a velocity window corresponding to
the recessional velocity of the associated primary galaxy to capture the
majority of the emission flux from the gas surrounding that galaxy. We vary
the velocity window as a function of stellar mass as described in Paper V.

We do apply this procedure to all of the line of sight spectra. Specifically,
we have a set of spectral selection criteria. First, the continuum level must
be $<$ 3 $\times$ $10^{-17}$ erg cm-2 s-1 Å-1 to limit the noise introduced by
the actual SDSS spectral target. Second, we require that the measured emission
line flux be within 3$\sigma$ of the mean of the whole sample so as to remove
spectra of interloping strong emitters such as satellite galaxies or outer
disk HII regions. We have confirmed that using the mean or median of the
resulting set of flux measurements in our subsequent analysis produces
consistent results. We present the results of mean flux in the following
study.

We estimate the uncertainties in the mean flux values by randomly selecting
half of the individual spectra in the relevant subsample, calculating the mean
emission line flux, and repeating the process 1000 times to establish the
distribution of measurements from which we quote the values corresponding to
the 16.5, 50.0 and 83.5 percentiles. We compensate for using only half the
sample in each measurement by dividing the $1\sigma$ estimated uncertainties
by a factor of $\sqrt{2}$.

###  2.1. H$\alpha$ Emission Dependence on the Galaxy’s Stellar Mass

We begin our analysis by re-evaluating the dependence of the H$\alpha$ flux on
the primary galaxy stellar mass. We follow the procedure outlined in Paper V
and measure the H$\alpha$ flux for galaxies in different stellar mass bins.
The set of primary galaxies used here, taken from SDSS DR7, is somewhat
different from that used in Paper V, which was taken from the group catalog
from Yang et al. (2007). As in the previous study, we select lines-of-sight
with $0.05<r_{s}<0.25$ around each primary galaxy and split the data into two
bins with equal $\Delta\log r_{s}$. We present the H$\alpha$ flux results, as
well as the average SFR which will be used in the following discussion, in
Table 1 and Figure 1. N in Tables 1 and 2 refers to the number of galaxies in
the corresponding subsample. We use the average value of $\log$ M∗ and $r_{s}$
to reference each bin.

In broad terms, we find that the measured H$\alpha$ flux decreases as the
galaxy stellar mass increases in both $r_{s}$ bins, and that the flux is
consistently larger in the inner $r_{s}$ bin. These fluxes are quantitatively
consistent with those presented in Paper V except for the smallest stellar
mass bin. There are two substantive differences between this study and the
previous one. We have already mentioned the first, which is that the primary
galaxy samples come from slightly different sources. The second is that the
line of sight spectra came from DR12 for the previous study and come from DR16
for the current study. We confirm that using DR12 with the current primary
galaxy sample leads to insignificant differences, so we attribute the
discrepancy to differences in the primary galaxy samples. This difference may
be a statistical fluctuation or related to the nature of the group catalog.
For example, low mass galaxies in the group catalog are exclusively
satellites, while those in the more general SDSS population are likely to be a
more heterogeneous population.

Table 1The stacked H$\alpha$ emission flux vs stellar mass and radius $\log({\rm M}_{*}$/M⊙) |  $r_{s}$ aa$r_{s}$ is the ratio between the physical projected separation and the virial radius of the primary galaxy, $r_{s}\equiv r_{p}/r_{\rm vir}$. | N | $f$ | SFR
---|---|---|---|---
|  |  |  [$10^{-17}$ erg cm-2 s-1 Å-1] | [M⊙/yr]
9.44 | 0.09 | 53 | $0.038\pm 0.015$ | 0.70 $\pm$ 0.06
0.19 | 611 | $0.020\pm 0.004$
10.05 | 0.09 | 290 | $0.033\pm 0.006$ | 1.34 $\pm$ 0.14
0.19 | 2059 | $0.006\pm 0.002$
10.64 | 0.09 | 1265 | $0.013\pm 0.002$ | 1.86 $\pm$ 0.04
0.19 | 6486 | $-0.001\pm 0.001$
11.07 | 0.09 | 966 | $0.000\pm 0.002$ | 1.59 $\pm$ 0.06
0.19 | 4236 | $0.004\pm 0.001$
Figure 1.— Measured average H$\alpha$ flux, $f$, in two $r_{s}$ bins
($0.05<r_{s}\leq 0.11$ and $0.11<r_{s}\leq 0.25$) as a function of the stellar
mass of the primary galaxy. For easier visualization, we apply slight
horizontal offsets for the data points of the inner and outer $r_{s}$ bins.
The blue dashed horizontal line represents no net emission or absorption at
H$\alpha$. The solid black lines represent the modeled flux, and the
associated shaded regions correspond to 1$\sigma$ and 2$\sigma$ uncertainties
of the model described in Sec. 3.1.

###  2.2. The H$\alpha$ Emission Dependence on the Galaxy’s Local Environment

Quantifying environment is always a challenge, and part of that challenge is
whether to focus on a local or global measurement of environment. In Paper IV,
one of the environmental measurements we adopted was the projected proximity
to a massive (M∗ $>10^{11}$ M⊙) galaxy. We justified this choice in the
context of the scenario in which the CGM of a primary galaxy is removed once
it enters the sphere of influence of a massive neighbor. In that work, we also
explored other scenarios, such as one where the CGM of a primary galaxy is
removed once it enters a group or cluster environment. Although we also found
some evidence in favor of this latter scenario, the number of galaxies in rich
environments is small and exploring this scenario further with the current
sample proved difficult. Therefore, we proceed by further exploring only the
single massive neighbor scenario and caution that environmental effects beyond
that explored here may be at play.

We adjust our massive neighbor scenario slightly by lowering the required
minimum massive neighbor mass to $10^{10.5}$ M⊙ and $10^{10}$ M⊙, in two
separate trials, both to increase the sample size and to probe whether the
physical effect continues for lower mass “massive” neighbors. We counter the
absolute decrease in neighbor mass by requiring that every affected primary
galaxy be substantially less massive (a factor of 5) than their massive
neighbor. This additional criterion is set to ensure that we are still
considering systems where the primary galaxy is dominated by its neighbor. We
chose a factor of 5 because in our previous study (Paper VI ) we found that
galaxy pairs with a mass ratio less than 5 and low average stellar mass
($<10^{10.4}$ M⊙) exhibit enhanced rather than diminished H$\alpha$ emission,
perhaps due to contributions from tidal material.

As before, we quantify the effect using the projected separation, $R_{p}$,
between the affected primary and its associated massive neighbor, for
potential massive neighbors that satisfy the mass criteria described above and
are within recessional velocities offset of 1000 km s-1 of the primary. As we
did before for the projected separation of lines of sight to the primary, we
now scale the projected radius to the massive neighbor using the virial radius
of the massive neighbor. Also as before, we estimate the virial radius of the
massive galaxy using our derived scaling relation between the virial radius
and galactic stellar mass. We confirm that lowering the maximum velocity
difference to 500 km s-1 leads to consistent results, although it decreases
the sample size by approximately 30%.

A series of steps is required in order to assemble our environmentally
affected sample and control samples. When multiple, $i$, massive galaxies lie
within $R_{p,i}/R_{\rm vir,i}<4$ of the primary galaxy, we consider only the
one with the smallest $R_{p,i}/R_{\rm vir,i}$ as the massive neighbor. To
construct one control sample, we identify a sample of “isolated” primary
galaxies that have no neighbors that satisfy our massive neighbor criteria
projected within 5 Mpc and lie within 1000 km s-1 in velocity offset. For
reference, the average calculated virial radius of the massive neighbors in
our sample is $\sim$ 500 kpc, which implies that the typical galaxy in the
“isolated” galaxy sample is free of massive neighbors to $R_{p}/R_{\rm
vir}\sim 10$. Next, we require $R_{p}/R_{\rm vir}>0.05$ to remove pairs in
which emission from the massive neighbor is likely to contaminate our
measurement of the primary galaxy. We find that adopting twice as large a
limit makes an insignificant difference in the results. Finally, because the
primary galaxy mass distributions are different between the data sample and
the control sample, we force the mass distribution of the control sample to be
the same as that of the data sample at $R_{p}/R_{\rm vir}>1$. We do this by
randomly drawing from the larger control sample until we match the mass
distribution of the data. The H$\alpha$ fluxes of the mass-matched control
sample are $(0.028\pm 0.003)\times 10^{-17}$ erg cm-2 s-1 Å-1 and $(0.011\pm
0.001)\times 10^{-17}$ erg cm-2 s-1 Å-1, for the radial bins $0.05<r_{s}\leq
0.11$ and $0.11<r_{s}\leq 0.25$, respectively.

We find that the results are insensitive to our choice of the lower mass limit
for the neighbors, within the range we explore, so we adopt our lowest mass
limit, $10^{10}$ M⊙, to maximize the sample size. In Table 2 and Figure 2 we
present the stacked H$\alpha$ fluxes as a function of $R_{p}/R_{\rm vir}$ for
both the inner and outer $r_{s}$ bins around our primary galaxies and isolated
galaxy control sample. We refer to this control sample as having $R_{p}/R_{\rm
vir}\gtrsim 10$. We also present the mean SFRs for the primary galaxies in
each subsample. Finally, we construct a second control sample in which we
insert two artificial primaries at the same $R_{p}/R_{vir}$ as each primary,
but at different angles around the massive neighbor. We refer to this second
control sample as the “azimuthal” control sample with $\sim$ 6000 lines of
sight, in contrast to the “isolated” control sample with $\sim$ 8000 lines of
sight.

Table 2The H$\alpha$ emission flux vs. scaled distance from nearest massive neighboraaThe scaled distance refers to the projected separation between the primary and the massive neighbor divided by the estimated virial radius of the massive neighbor. A massive neighbor is defined to be at least 5 times as massive as the primary galaxy and have M${}_{*}\geq 10^{10}$ M⊙. $R_{p}/R_{\rm vir}$ | $r_{s}$ | N | $f$ | SFR
---|---|---|---|---
|  |  | [$10^{-17}$ erg cm-2 s-1 Å-1] | [M⊙/yr]
0.15 | 0.09 | 8 | $0.044\pm 0.065$ | 0.54 $\pm$ 0.07
0.19 | 86 | $-0.015\pm 0.009$
0.33 | 0.09 | 39 | $0.070\pm 0.014$ | 0.84 $\pm$ 0.06
0.19 | 219 | $-0.012\pm 0.005$
0.68 | 0.09 | 46 | $0.017\pm 0.011$ | 0.93 $\pm$ 0.05
0.19 | 350 | $0.005\pm 0.046$
1.41 | 0.09 | 67 | $0.031\pm 0.010$ | 1.14 $\pm$ 0.06
0.19 | 625 | $0.007\pm 0.003$
2.92 | 0.09 | 131 | $0.027\pm 0.008$ | 1.56 $\pm$ 0.20
0.19 | 1022 | $0.010\pm 0.003$
$\gtrsim 10.0$ | 0.09 | 1137 | $0.028\pm 0.003$ | 1.17 $\pm$ 0.02
0.19 | 6572 | $0.011\pm 0.001$

We draw several conclusions from these results (Table 2 and Figure 2). First,
and most significantly, for the outer $r_{s}$ bin around the primary galaxies,
the fluxes for primaries at the two smallest values of $R_{p}/R_{\rm vir}$ are
significantly lower than for those either at larger values of $R_{p}/R_{\rm
vir}$ or for those that are isolated. To confirm this impression, we calculate
the nonparametric rank-order Spearman correlation and find that the null
hypothesis of no correlation between flux and $R_{p}/R_{\rm vir}$ can be
rejected at $>$ 99% confidence in the outer $r_{s}$ bin. Second, less
significantly, there is the appearance that the effect exists even beyond
$R_{p}/R_{\rm vir}>1$. Third, the fluxes in the inner $r_{s}$ bin appear to be
unaffected by the presence of the massive neighbor. If anything, there may be
a slight flux increase rather than decrease at small $R_{p}/R_{\rm vir}$, but
the results for the inner $r_{s}$ bin have larger uncertainties due to the
smaller number of lines of sight. Fourth, for both $r_{s}$ bins, the flux
values for the outermost $R_{p}/R_{\rm vir}$ bin are consistent with those of
the isolated control sample, indicating that we have measured out to the
asymptotic value. Lastly, as expected if there is no contamination from the
massive neighbor, we find no mean H$\alpha$ flux and no systemic dependence of
the flux with $R_{p}/R_{\rm vir}$ for the azimuthal control sample for both
radius bins (Figure 2).

The one remaining puzzle is the slight systematic tendency to produce negative
fluxes at small $R_{p}/R_{\rm vir}$ in the larger $r_{s}$ bin. Because we do
not find the same trend in the azimuthal control sample, we exclude the
possibilities that the effect is due to a systematic overestimate of the
continuum in these environments or to the presence of an absorption line
spectrum contribution from the massive neighbor. At this point, given its
somewhat marginal statistical significance, we attribute the result to noise
but note that it could also reflect stellar absorption from tidal material
around the primary.

Figure 2.— Measured average H$\alpha$ flux for $0.05<r_{s}\leq 0.11$ (upper
panel) and $0.11<r_{s}\leq 0.25$ (lower panel) as a function of scaled
projected distance ($R_{p}/R_{\rm vir}$) from the nearest massive neighboring
galaxy which is at least 5 times as massive as the primary galaxy and has
stellar mass greater than $10^{10}$ M⊙. We also present the average H$\alpha$
fluxes for the azimuthal and isolated control samples (see text). For easier
visualization, we apply slight horizontal offsets between the data and control
in the same bins. The solid black lines represents the model flux and the
corresponding shaded region represent the 1 and 2$\sigma$ uncertainties of the
model described in Sec. 3.1. The flux for the isolated control sample is
presented as a point with error bar and an attached line indicating that the
value represents galaxies at large projected separations. Again, zero net flux
is presented as a blue dashed line to guide the eye.

##  3\. Models and Interpretation

We now present two different modeling efforts. First, we expand our previous
work of defining broad characteristics of the CGM by setting basic rules
within the construct of the UniverseMachine models and optimizing the
parameter values for those rules. Our effort here is distinguished from our
previous work in that we are jointly fitting the mass and environment
dependence to determine the interdependence of those, and in fitting
separately to the measurements in both $r_{s}$ bins. The latter is important
given that it is only the flux in the outer $r_{s}$ bin that appears affected
by the nearby massive neighbor. Second, we present a new analysis where we use
the observed H$\alpha$ fluxes and our derived CGM characterization to
constrain the gas mass cooling rate in the halos. In turn, when combined with
the independently measured star formation rates, we will use these results to
determine how much gas mass needs to be reheated or ejected in order that the
central galaxy not systematically gain or lose cool gas. We then compare our
measurement to independent theoretical predictions as a demonstration of where
future work might lead.

###  3.1. UniverseMachine and Global Trends

The UniverseMachine is a dark matter-only simulation with co-moving box length
of 250 Mpc $h^{-1}$, mass resolution of $1.8\times 10^{8}\ {\rm M}_{\odot}$
(20483 particles), and force resolution of $1$ kpc $h^{-1}$. Halos were found
with the Rockstar phase-space halo finder (Behroozi, Wechsler & Wu, 2013), and
merger trees were generated with the Consistent Trees code (Behroozi et al.,
2013). The code empirically determines the dependence of galaxy star formation
rates on the host dark matter halo mass, the halo accretion rate, and
redshift, and produced observables are matched to SDSS galaxies in stellar
mass and star formation rate. In particular, galaxy colors are matched based
on both stellar mass and specific SFR.

In previous studies, Papers IV and V, we studied the dependence of the CGM on
the environment and stellar mass separately. However, due to the common
multiparameter correlations present among galaxies, it is possible that
effects attributed to either environment or mass in those studies should
rightly be ascribed to the other. Here we perform a joint modeling exercise to
avoid “double-counting” any effect.

The central tenet of this model is the following description of the cool gas
fraction that includes a simple dependence on the stellar mass of the primary
galaxy, M∗, and environment taken in large part from our previous work:

$\mathcal{F}_{cool}(R,{\rm M_{*}})=\begin{cases}a({\rm
M_{*,10}})^{b}{\left(R\over{x\cdot R_{\rm vir}}\right)}&\ \text{if}~{}R<x\cdot
R_{\rm vir}\\\ a({\rm M_{*,10}})^{b}&\ \text{if}~{}R\geq x\cdot R_{\rm
vir}\end{cases}$ (1)

where $(x,a,b)$ are free parameters, $R$ is the three dimensional distance
between the primary galaxy and the massive neighboring galaxy, $R_{\rm vir}$
is the virial radius of the massive galaxy, which is defined to be the radius
of a spherical volume within which the mean density is $\Delta_{c}$ times the
critical density at that redshift (where
$\Delta_{\mathrm{c}}=18\pi^{2}+82x-39x^{2}$ and $x=\Omega_{m}(z)-1$),
$\mathcal{F}_{cool}$ describes the fraction of the cosmological baryon
fraction in the halo that is in the cool ($T\sim 10^{4-5}$K) gas phase, and
$M_{*,10}$ is the stellar mass in units of $10^{10}$ M⊙. Our assumption that
the baryon fraction of galaxy halos is the universal one has not yet been
empirically verified. While it is observed to hold roughly for higher mass
halos (Gonzalez et al., 2013; Lim et al., 2020), it does not appear to hold
universally for the lower mass halos of galaxy groups (e.g., Vikhlinin et al.,
2009; Gonzalez et al., 2013). On galaxy scales we have few constraints, but
the rough concurrence in the Milky Way’s dynamical mass and that inferred
assuming the cosmological baryon fraction indicates that the baryon fraction
for large galaxies, at least for our own galaxy, does not deviate grossly from
the universal value (Zaritsky & Courtois, 2017). To the degree that the baryon
fraction is lower than then universal value in our sample of galaxies, our
results underestimate $\mathcal{F}_{cool}$. The value of the parameter $x$
affects both the extent of the environmental and the amplitude of the effect
at a given physical radius. The environmental effect is only applied to the
gas in the outer $r_{s}$ bin, as there is no evidence for stripping of the CGM
in the inner $r_{s}$ bin. For gas in the inner $r_{s}$ bin we adopt
$\mathcal{F}_{cool}$ as given by the $R\geq x\cdot R_{\rm vir}$ condition in
Equation 1. Additionally, only $\sim$ 30% galaxies have neighbors that are at
least 5 times as massive as the primary galaxy within 4$R_{\rm vir}$ of the
massive neighbor. For those galaxies without such a massive neighbor,
$\mathcal{F}_{cool}$ in both $r_{s}$ bins is defined using the $R\geq x\cdot
R_{\rm vir}$ condition in Equation 1. Our cool gas fraction functional form
thus accounts for both a stellar mass and environmental dependence, which is
notionally consistent with what numerical simulations find when investigating
star formation/quenching (e.g., Wang et al., 2018).

We follow our earlier approach in Papers IV and V using UniverseMachine
(Behroozi et al., 2018) catalog, and adopt a gas mass density profile for the
primary galaxy consistent with that describing gas in an NFW potential
(Navarro, Frenk & White, 1996, 1997). As we discussed in Paper II, the
H$\alpha$ emission flux we detect comes from two contributions: the emission
from the central galaxy itself and that from the spatially associated halos
around the central galaxy. By constraining our measuring and modeling to
within projected radii of 50 kpc, or $r_{s}\sim$ 0.25, from the primary
galaxy, we naturally focus on the first of these components. In Paper IV, we
roughly approximated the effect of the second component of H$\alpha$ emission
in those calculations by assigning half of the measured emission to the second
component. Now, we account for both components in our models because we
include the flux contributions from nearby neighbors. Given the simplicity of
this approach, we neglect the higher order effect represented by the
interaction of massive galaxies with the CGM of the other neighbors. Moreover,
the agreement in measured fluxes between our outermost $R_{p}/R_{\rm vir}$
radial bin and the isolated sample, suggests that this effect is not
measurable at the precision available in the current data. Furthermore, by
modeling the radial profile of the emission, in addition to the total
amplitude, we are able to attempt to reproduce the measurements in our two
radial bins.

We perform a Bayesian analysis to derive confidence intervals on each of the
three parameters in Eq. 1. Based on the results from our previous studies we
adopt uniform priors over the given ranges as follows,

$0.25<x<4.0,\quad 0<a<0.5,\quad-1.0<b<0,$ (2)

and utilize the measured H$\alpha$ fluxes presented in Tables 1 and 2. There
is an implicit physical restriction on the model parameters that the cool gas
fraction can not exceed one. Therefore, we add that constraint by setting any
$\mathcal{F}_{cool}>1$ to 1. As standard, the posterior distribution,
$p(\Theta|{\rm data})$, is described as follows:

$p(\Theta|{\rm data})=\frac{p(\Theta)\cdot\Pi~{}p({\rm data}|\Theta)}{p({\rm
data})}$ (3)

where $\Theta$ is the parameter space $\Theta=(x,a,b)$, $p(\Theta)$ is the
prior distribution described in Eq. 2, $p({\rm data}|\Theta)$ is the
likelihood of the data and $p(\rm data)$ is the marginal probability for the
data. We define the likelihood of obtaining the data given a specific model
using the difference between the actual and model data, ${\rm
exp[-0.5\cdot(actual-model)^{2}/\sigma^{2}}]$, where $\sigma$ is the
observational uncertainty. We use “emcee” Foreman-Mackey et al. (2013) to
implement a Markov chain Monte Carlo sampling of the likelihoods across the
parameter space to calculate the posterior distribution. In Figure 3 we
display the posterior distributions of the three parameters, the median and
1$\sigma$ confidence levels (based on the 16th and 84th percentiles) for each,
and the correlations between the probability distributions.

Regarding the modeling of the environmental dependence, the preferred value
for $x$ is $1.92^{+1.05}_{-0.88}$, demonstrating that the effects of a massive
neighbor on the CGM of the primary galaxy are evident even when the primary
lies beyond the neighbor’s virial radius. Unfortunately, the precision on $x$
is low, presumably because the sample is still relatively small (we only have
1447 data points for $Rp/R_{\rm vir}<2$) and because we only have projected
distances between primaries and neighbors. Even so, the finding that $x>1$
suggests either that the physical effect begins at these large radii or that
our sample includes galaxies that have ‘splashed’-back, having already passed
within the virial radius and currently lying outside. Various modeling and
sample differences complicate the comparison to our previous result in Paper
IV, but there we estimated the effect to extend to a projected separation of
$\sim$ 724 kpc for a sample of neighbors with a mean virial radius of $\sim$
550 kpc, or alternatively an $x$ value of $\sim$ 1.32, which is consistent
with our current estimate.

Regarding the modeling of the mass dependence, we find $a=0.23\pm 0.01$, which
is only slightly smaller than, but consistent with, our previous estimate of
$0.28^{+0.07}_{-0.06}$ in Paper V and $b=-0.55\pm 0.03$, which is
significantly lower than our previous estimate of $-0.33\pm 0.06$ in Paper V.
To explore the cause of this marked change, we model the stellar mass
dependence and the environmental effect separately, as done in our previous
studies. Our new best fit for $b$ is $-0.55\pm 0.03$, demonstrating that the
difference with our previous result does not lie with the new joint analysis
of mass and environment.

We ascribe the significant difference in $b$ to two factors. First, there is a
statistically significant difference in the measured flux for the lowest mass
galaxies in the innermost $r_{s}$ bin between the two studies. The flux was
lower in the previous study and this might be related to the selection of
group galaxies in that previous work, most of which at these low masses would
be satellites and might differ from a more general sample of low mass
galaxies. Second, the basic power-law model we use is manifestly
overpredicting the measured flux for these low mass galaxies at the smallest
separations (Figure 1), suggesting that this model may not be an accurate
representation and that the dependence of $\mathcal{F}_{cool}$ with mass may
not be quite as steep as indicated by our new estimate of $b$.

Figure 3.— Posterior distribution for the cool gas fraction in the CGM of the
primary galaxy for the two radii bins of $0.05<r_{s}\leq 0.11$ and
$0.11<r_{s}\leq 0.25$, as a function of distance, $R$, from the massive
neighbor, and the stellar mass of the primary galaxies. The red vertical line
indicates the median value of the model parameters and the shaded region
highlights the 1$\sigma$ uncertainties.

We show in Figures 1 and 2 that the most likely model is also one that
reasonably reproduces the observational data. With regard to the mass
dependence of $\mathcal{F}_{cool}$, we find a strong decline in the cool gas
fraction with increasing stellar, and presumably total, mass is needed.

With regards to the environmental effects, we interpret the model as
indicative of the removal of the outer portions of the CGM of the primary
galaxy by the massive neighbor (Putman et al., 2021). The effect is
significant even when the primary galaxy falls in the potential well within
2$\times R_{\rm vir}$ of the massive neighbor, and most severe when the
primary galaxy lies within the virial radius of the massive neighbor. The
model suggests that satellites rapidly lose some of their gaseous haloes, or
that their hot gaseous halos quickly become unable to cool after reaching the
virial radius of a larger halo. The apparent onset of the effect at radii
larger than the virial radius could reflect the nature of the splashback
radius (for discussion see More, Diemer & Kravtsov, 2015), where galaxies that
have fallen inside of the virial radius return to larger radii as they execute
their orbits, or on the affected nature of mass accretion onto halos at these
radii (Behroozi et al., 2014). The significant removal of the gas but only
small decrease in the SFR might indicate the delayed quenching of those
satellites in rich environments (Wetzel et al., 2013).

####  3.1.1 An Aside on H$\alpha$ Emission and Galaxy’s Color

As is always the case for galaxy studies, a variety of properties are related
to each other and this complicates the analysis and interpretation. We just
measured that H$\alpha$ flux from the CGM is inversely related to the stellar
mass of the galaxy, but stellar mass is related to a number of other galaxy
properties, such as color and morphology. Is it possible that a relationship
between H$\alpha$ flux and galaxy color, which is a parameter that is
tabulated for our sample, provides an easy-to-access additional constraint on
our modeling?

At a given stellar mass, one might expect blue, star-forming galaxies to have
a richer gas reservoir. Indeed, we do find that the bluer galaxies have larger
H$\alpha$ emission, as shown in Figure 4 and that color is highly correlated
to the galaxy stellar mass (for $g-r$ bins centered on 0.34, 0.52, 0.73, 0.83,
the corresponding mean stellar masses are $10^{9.8}\rm M_{\odot}$,
$10^{10.34}\rm M_{\odot}$, $10^{10.76}\rm M_{\odot}$, $10^{10.91}\rm
M_{\odot}$, respectively).

We now use the UniverseMachine models to see if there is potential additional
information in the relationship between H$\alpha$ flux and galaxy color.
Galaxy color is included in the output of our previous modeling, so we now
compare the model predictions to the data in Figure 4. We find good agreement,
which is at a minimum a reassuring consistency check and suggests that galaxy
color is highly degenerate with galaxy stellar mass. The result does not
inform us as to whether stellar mass or color is the primary driver of the
observed trend in flux, only that ascribing it to one or the other is likely
to produce similar results. Given the expected rise in the virial temperature
of galaxy halos with halo mass, which is presumably tracked by stellar mass,
we prefer to base our simple modeling of $\mathcal{F}_{cool}$ on stellar mass
than on galaxy color.

Figure 4.— Measured average H$\alpha$ flux for $0.05<r_{s}<0.25$ as a function
of $g-r$. The solid black line represents the model flux and the corresponding
shaded regions represent to 1 and 2$\sigma$ uncertainties of the model
described in Sec. 3.1. Again, zero net flux is presented as a blue dashed line
to guide the eye.

###  3.2. Modeling the Cooling Mass Rate and Inferring the Mass Loading Factor

We do not have a quantitative, complete, empirically-validated understanding
of how star forming galaxies sustain their star formation. On one hand,
ongoing star formation in a large fraction of galaxies demonstrates that star
formation lasts at least $\sim$ 10 billion years in those systems, but on the
other hand, the depletion time of the molecular gas for low-redshift normal
galaxies is in the range of $1-2$ Gyr (Saintonge et al., 2011; Tacconi et al.,
2018). The natural solution is that the galaxy disk is being refueled and that
the source of that fuel is, by consensus, the CGM (Borthakur et al., 2015;
Tumlinson, Peeples & Werk, 2017).

Our measurement of the H$\alpha$ flux coming from the CGM provides an
opportunity to measure the rate at which the CGM is radiatively losing energy,
and hence cooling. Because the radiative energy loss must lead to the
equivalent loss of thermal energy in the gas, we will estimate the rate at
which CGM gas is cooling and, presumably, sinking into the central disk. A
complication that we address below is that H$\alpha$ is only one of the
emission lines through which the gas is cooling. Assuming that the disk
maintains a certain gas mass on average over some current modest time
interval, the excess of the mass inflow rate over the star formation rate is
the amount of gas that is reheated or ejected. This is the mass loading factor
of whatever feedback mechanism is at work regulating star formation.

Once we have calculated the total radiative losses across the entire
wavelength range (§3.2.1), we will set that equal to the thermal energy loss
in the gas. Assuming an ideal gas, the energy radiated per particle as it
changes in temperature by $\Delta T$ is

$\Delta E=\frac{3}{2}k\Delta T.$ (4)

In our calculation, we divide the entire temperature range into a few bins and
$\Delta T$ is the change in temperature of the gas parcel that is considered
to contribute to the measured H$\alpha$ flux. For a mean particle mass of $\mu
m_{p}$, where $m_{p}$ is the proton mass, the rate at which mass is cooling
(the mass cooling rate), $\dot{m}$ is:

$\dot{m}=\dot{E}/\Delta E\times\mu m_{p},$ (5)

where $\dot{E}$ is the radiated energy rate for the whole halo and is
calculated as follows:

$\dot{E}=R_{\rm fiber}\times f_{\rm H\alpha}\times C_{\rm Cl}$ (6)

where $R_{\rm fiber}$ is the ratio of the flux contained within the fiber
aperture to that contributed by the full halo, $f_{\rm H\alpha}$ is the
H$\alpha$ flux measurements described above in units of erg cm-2 s-1, and
$C_{\rm Cl}$ is the unitless correction factor derived from Cloudy modeling
(§3.2.1) that converts the H$\alpha$ radiative energy loss to that across all
emission lines. $R_{\rm fiber}$ is described as $\pi(D_{\rm A}r_{\rm
fiber})^{2}\times C_{\rm G}$, where $D_{\rm A}$ is the angular diameter
distance at the redshift of our primary galaxy sample, which on average is
approximately 0.1, $r_{\rm fiber}$ is the eBOSS angular fiber size, and
$C_{\rm G}$ is a unitless geometric correction factor that we describe in
§3.2.2.

####  3.2.1 Determining $C_{\rm Cl}$

We model the dependence of the emission spectrum from the CGM on the
ionization state ($\log{\rm U}$), the metallicity ([X/H]), the gas temperature
(T), the total hydrogen column density ($\log{\rm N_{H}}$) and hydrogen
density ($\rho_{\rm H}$) using Cloudy (Ferland et al., 1998, 2013, 2017)
version 17.02. For the ionization spectrum we adopt either the Haardt & Madau
(2001) background radiation field from quasars and galaxies (HM 2001) or the
Kurucz79 O star spectrum corresponding to an effective temperature of 30,000 K
of the star (Kurucz, 1979). We assume that the gas is in ionization
equilibrium.

We find that the results are relatively insensitive to the choice of
ionization source spectrum (resulting in $<10\%$ changes to the mass cooling
rate estimated below) and so we only present results from calculations using
the O star spectrum as the ionization source.

One complication that arises when converting the H$\alpha$ flux into an energy
loss rate is that the halo gas emitting H$\alpha$ is not at a single
temperature, but rather consists of gas at a large range of temperatures,
approximately $10^{4-5}$ K (e.g., Schure et al., 2009; Lykins et al., 2013),
all of which can contribute to the observed flux. We present results from a
calculation to illustrate how the flux and inferred cooling rate depend on the
gas temperature T. For this exercise we set the other parameters to those
inferred by Werk et al. (2014) and Prochaska et al. (2017): ionization
parameter $\log{\rm U}$ = $-$3.0, metallicity [X/H] = $-$0.75, total hydrogen
column density $\log({\rm N_{H}}$/cm-2) = 19.5, and hydrogen density
$\rho_{\rm H}$ = 10-3.0 cm-3. We do not include dust in the Cloudy modelling.
The interplay of total hydrogen column density and the hydrogen density
determines the geometry of the clouds, in other words, it sets the total
length of the gas cloud along the line of sight. We vary the gas temperature T
from 103.5 K to 106 K. The selected temperature range roughly covers all the
different phases for the halo gas in the CGM for a Milky Way like galaxy that
will emit H$\alpha$.

Figure 5.— The fraction of the total cooling rate attributed to the H$\alpha$
emission line (top), the H$\alpha$ line intensity (middle), and the total
cooling/heating rate (bottom) as functions of the gas temperature as
calculated by Cloudy for $\log\rm U=-3.0$, [X/H] = $-0.75,\log$ (NH/cm-2) =
19.5, and $\rho_{\rm H}$ = 10-3.0 cm-3.

The fraction of all the cooling radiation that comes from H$\alpha$ emission,
what we refer to as the H$\alpha$ fraction, the H$\alpha$ radiation intensity,
and the total cooling/heating rate are shown as functions of temperature in
Figure 5. At higher ionization ($\log U>-3.5$) and temperatures $<10^{4}$ K
the cooling rate drops by almost two orders of magnitude and the heating rate
exceeds the cooling rate, suggesting that this simple model is missing cooling
channels, such as molecular lines. Therefore, we set the lower boundary of the
temperature range we will consider at 104 K. At the high temperature end (T
$>10^{5}$ K), the H$\alpha$ intensity and the H$\alpha$ fraction drops
rapidly, so we also consider the contribution of this gas to H$\alpha$
negligible. At lower ionization ($\log U<-3.5$), the heating rate is orders of
magnitude lower than the cooling rate at all temperature. Although we have
limited the temperature range of the gas that is relevant to our calculation,
the range remains large, $10^{4-5}$ K, and H$\alpha$ fraction and intensity
vary significantly across this range, suggesting the estimate of the
distribution of gas across this range of temperatures is fundamental to
understand the origin of H$\alpha$ properties.

Unfortunately, this information is currently unavailable to the required
accuracy and, in order to make progress, we need to introduce additional
assumptions. Because the cooling rates vary with temperature, there are
naturally temperature regimes that the gas transitions through faster and
slower as it cools. To avoid gas piling up at certain temperatures, we enforce
a continuous cooling mass flow through the temperature regimes by scaling the
gas mass at the different temperatures inversely by its corresponding mass
cooling rate. As expected, this implies that most of the gas ends up at
temperatures near $10^{4}$ K, where the cooling rate falls steeply. This
approach also means that if one calculates the mass cooling rate at any single
temperature, that value is the mass cooling rate at all temperatures.

In addition, we also require the integrated H$\alpha$ flux in the temperature
range of $10^{4}-10^{5}$ K to match the observational values shown in Table 1
and 2. After scaling, the integrated H$\alpha$ intensity over the temperature
range of $10^{4}-10^{5}$ K is $0.81\times 10^{-9}$ erg cm-2 s-1. The aperture
size of eBOSS is 2 arcsec, which gives a conversion factor from intensity to
flux of $9.41\times 10^{-11}$, then the integrated H$\alpha$ flux over the
SDSS fiber size in the the temperature range of $10^{4}-10^{5}$ K is
$0.06\times 10^{-17}$ erg cm-2 s-1. The mass cooling rate corresponding to
this H$\alpha$ flux is estimated to be 0.15 M⊙/yr before geometric correction.

This approach allows us to estimate the mass cooling rate from the observed
H$\alpha$ fluxes, building up on the derived scaling relations between the
Cloudy H$\alpha$ flux and the Cloudy mass cooling rate. Nonetheless, this
result may be sensitive to our assumptions in the Cloudy modelling, as we
discuss below.

####  3.2.2 Determining $C_{\rm G}$

To determine the geometric correction factor, ${C_{G}}$, we take a two step
process. First, we correct for the fraction of the projected area covered by
the eBOSS fiber within the radial range probed by our study ($0.05<r_{s}\leq
0.25$). Because we average results from thousands of fibers scattered
throughout this area with no correlation to the primary galaxy, the mean flux
in a fiber aperture accurately represents a sampling of the flux within the
full area, modulo the ratio of the fiber aperture to the full aperture. As
such, we simply correct by the ratio of the two areas $(A_{r_{s}<0.25}/A_{\rm
fiber})$. The correction factor varies for different galaxies because the
physical radii are different even for the same scale radius range of
$0.05<r_{s}<0.25$. Then the mean correction factor is 14.3, 36.5, 128.9, 682.5
for the four stellar mass bins described above. Second, we consider the
correction for the ratio of the full halo to that which the $0.05<r_{s}\leq
0.25$ annulus represents, which we define as $\mathcal{F}_{corr}$. We find in
our model that although there is nearly as much baryonic mass in the gaseous
halo outside $r_{s}=0.25$ as inside, there is a negligible contribution to the
H$\alpha$ flux (see Figure 7 in (Zhang, Zaritsky & Behroozi, 2018)).
Therefore, $\mathcal{F}_{corr}$ is approximately equal to 1. Our calculation
for $C_{G}$ is

$C_{G}=\frac{A_{r_{s}<0.25}}{A_{\rm fiber}}\times\mathcal{F}_{corr},$ (7)

where we set $\mathcal{F}_{corr}=1$.

####  3.2.3 Uncertainties in the Derived Mass Cooling Rate

To develop intuition regarding the sensitivity of our results to various
parameter choices, we investigate the dependence of the total mass cooling
rate on the adopted ionization parameter ($U$), the metallicity ([X/H]), and
the hydrogen density ($\rho_{\rm H}$), while keeping the column density fixed.
Prochaska et al. (2017) presented a radial distribution of hydrogen column
density measurements in $L_{*}$ galaxies, roughly ranging from
$\log{\rm(N_{H}/{\rm cm}^{-2})}=20$ to $\log{\rm(N_{H}/{\rm cm}^{-2})}=19$ for
projected radius between 10 and 50 kpc, respectively, so we adopt
$\log{\rm(N_{H}/{\rm cm}^{-2})}=19.5$. For the other parameters we allow that:

  1. 1.

$\log{\rm U}$ vary from $-$6 to $-$1 (our preferred parameter range is from
$-$4 to $-$2),

  2. 2.

$[$X/H] vary from $-$2.5 to 0.5 (our preferred parameter range is from $-$1.5
to 0), and

  3. 3.

$\rho_{\rm H}$ vary from 10-2.0 cm-3 to 10-4.0 cm-3 (our preferred parameter
range is from $10^{-2.5}$ to $10^{-3.5}$ cm-3).

Our preferred parameter ranges are based on the constraints from absorption
line measurements (Werk et al., 2014; Prochaska et al., 2017) using QSO
spectra from the COS-Halos Survey (Werk et al., 2013). Given the fixed column
density, the range of hydrogen densities we probe corresponds to a column
length of the simulated halo between 10 kpc and 300 kpc.

##### Dependence on the Ionization Parameter

The ionization parameter, which measures the number of ionizing photons per
atom, affects the calculated heating and cooling rates, and therefore the
resulting H$\alpha$ flux. Here, we fix the other parameters to be [X/H] =
$-$0.75 and $\rho_{\rm H}$ = 10-3.0 cm-3, and vary only $\log\rm U$ and the
temperature. As described above, for each ionization parameter, we enforce
continuity in the mass gas flow rate at different temperatures (§3.2.1) and
then estimate the mass cooling rate.

The result is shown in Figure 6. We find that changes of about a factor of
three in the estimated mass cooling rate are possible, although the change is
rather sudden at about $\log{\rm U}\sim-3.5$. At our preferred value of
$\log{\rm U}=-3$ we expect to be calculating the comparably lowest likely
values of the mass cooling rate and plausible changes in $\log{\rm U}$ should
only produce higher estimates of the mass cooling rate.

Figure 6.— The mass cooling rate as a function of the ionization parameter,
$\log\rm U$, for fixed [X/H] = $-0.75,\log({\rm N}_{\rm H}$/cm-2) = 19.5, and
$\rho_{\rm H}$ = 10-3.0 cm-3. What we consider to be the plausible range of
$\log{\rm U}$, based on Werk et al. (2014), is identified by vertical dashed
lines and the corresponding range of plausible mass cooling rates as the
shaded region.

##### Dependence on Metallicity

Metals contribute significantly to the total cooling in the temperature range
of $10^{4-5}$ K (e.g., Schure et al., 2009; Lykins et al., 2013). Here we
investigate how the adopted gas metallicity affects the inferred mass cooling
rate. We adopt $\log{\rm U}=-3.0$, and $\rho_{\rm H}$ = 10-3.0 cm-3, and vary
the gas metallicity and the temperature. Again, we enforce gas cooling
continuity across temperature ranges and estimate the mass cooling rate, as
described in §3.2.1.

The result is shown in Figure 7. As expected, the mass cooling rate increases
as the metallicity increases for metallicity below solar metallicity,
resulting in the inferred mass cooling rate increasing significantly as we
adopt a higher metallicity of the gas. At our preferred value of [X/H] =
$-$0.75, we are near the lower limit of the inferred mass cooling rates. If
the CGM has solar metallicity we would be underestimating the mass cooling
rate by about a factor of 2 for our standard assumed [X/H], but it is highly
unlikely that the CGM reaches solar metallicity (Prochaska et al. (2017) find
that the typical CGM metallicity is $\sim-0.5$). For more plausible errors in
the metallicity, the corresponding errors are significantly smaller.

Figure 7.— The mass cooling rate as a function of the metallicity, [X/H], for
fixed $\log\rm U=-3.0$, $\log({\rm N}_{\rm H}$/cm-2) = 19.5 and $\rho_{\rm H}$
= 10-3.0 cm-3. What we consider to be the plausible range of [X/H], based on
Werk et al. (2014) and Prochaska et al. (2017), is identified by vertical
dashed lines and the corresponding range of plausible mass cooling rates as
the shaded region.

##### Dependence on the Density

Both the cooling rate and the H$\alpha$ flux are proportional to the gas
density squared. Because we have fixed the total hydrogen column density, our
inferred cooling rates and H$\alpha$ fluxes are proportional to the gas
density, not the density squared. In the Cloudy calculation, we have adopted a
uniform gas density as a simple approximation to the halo gas density profile,
which roughly follows a power law according to Werk et al. (2014) in other
simulations. Adopting $\log{\rm U}=-3.0$ and [X/H] = $-0.75$, we obtain the
results shown in Figure 8. As expected, the mass cooling rate is proportional
to the adopted gas density.

We close this section by noting that the model we present is a highly
oversimplified version of the CGM. Highlighting a few specific shortcomings,
we note that 1) the density we model with Cloudy is independent of the
galactocentric distance, whereas we know the true density of the galaxy halo
is decreasing significantly with radius, 2) we do not consider density
variations within a given column (i.e. clumpiness), and 3) we only consider
the variation of $\mathcal{F}_{cool}$ with galaxy mass and ignore other
potential mass dependencies, such as one between CGM metallicity and mass. The
model we present is best at establishing broad behavior and testing our
conceptual understanding rather than at providing a quantitative
understanding. Ultimately, the data presented here will provide the strongest
constraints when compared with forward-modeling of physically-motivated, high-
resolution CGM simulations.

Figure 8.— The mass cooling rate dependence on the gas density ($\rho_{\rm
H}$) for fixed $\log\rm U=-3.0$, [X/H] = $-0.75$ and $\log({\rm N}_{\rm
H}$/cm-2) = 19.5. What we consider to be the plausible range of $\rho_{\rm
H}$, based on Werk et al. (2014) and Prochaska et al. (2017), is identified by
vertical dashed lines and the corresponding range of plausible mass cooling
rates as the shaded region.

###  3.3. The Mass Cooling Rate

####  3.3.1 Empirical Results

We integrate the flux within the radial range of $0.05<r_{s}\leq 0.25$ over
the velocity window described in §2 and then apply the appropriate geometric
correction described in Eq. 7 to obtain the CGM H$\alpha$ emission within the
virial radii of galaxies in the different stellar mass bins. We also calculate
the CGM H$\alpha$ flux using our fitted model for $\mathcal{F}_{cool}$ in Eq.
1 and the best fit parameter values for the assumed $T=12,000$ K. We will
present both results.

To infer a mass cooling rate (MCR) from those emission fluxes, we adopt our
preferred parameter set ($\log\rm U=-3.0$, [X/H] = $-$0.75, $\log({\rm N}_{\rm
H}$/cm-2) = 19.5, and $\rho_{\rm H}=10^{-3.0}$ cm-3 (from Werk et al., 2014;
Prochaska et al., 2017). From Figures 6 to 8, we know that the mass cooling
rate depends significantly on the adopted Cloudy model parameters. To estimate
the effect of these uncertainties, we randomly uniformly distribute 10,000
points in the 3-D parameter space within the range of $(-4<\log{\rm
U}<-2,-1.5<{\rm[X/H]}<0,10^{-3.5}<\rm\rho_{H}/cm^{-3}<10^{-2.5})$. We follow
the steps discussed in Sec. 3.2.1 to calculate the mass cooling rate from the
H$\alpha$ flux derived from our model for $\mathcal{F}_{cool}$ using Cloudy.
We adopt the standard deviation among those 10,000 estimates of the mass
cooling rate as the 1$\sigma$ uncertainty of the modeled mass cooling rate.

We present our results for the mass cooling rates in Figure 9 and compare
those rates to the current SFR as a function of galaxy stellar mass. As
expected, the mass cooling rates are much higher across the board than the SFR
of the central galaxies. We draw several conclusions from this comparison: 1)
for our preferred parameters the mass cooling rate is well above what is
needed to maintain the current rate of star formation, thereby addressing the
gas consumption problem, 2) this estimated mass cooling rate significantly
exceeds what is needed and therefore supports the theoretical understanding
that the conversion of cooling gas into stars is inefficient and that a large
fraction of the gas must be reheated or ejected, and 3) the strength of this
“feedback” is the largest for the galaxies with the largest stellar masses,
but otherwise appears roughly constant as a function of stellar mass across
this limited mass range. The approximate linearity in log-log indicates that
the relation between $\log\rm MCR$ and $\log\rm M_{*,10}$, where $M_{*,10}$ is
the stellar mass in units of $10^{10}$ M⊙, can be approximated by a second
order polynomial function as follows:

$\log{\rm MCR}=0.81+0.48\log{\rm M_{*,10}+0.35\log^{2}M_{*,10}}$ (8)

When drawing quantitative conclusions from our results, such as inferring the
fraction of the gas that needs to be reheated or ejected to maintain the SFR
at a steady state, one must bear in mind the large systematic uncertainties
that exist. The error bars as plotted on the individual data points reflect
only the uncertainties due to the measurement errors on the H$\alpha$ fluxes.
The uncertainties plotted on the model predictions reflect systematic
uncertainties in the adopted parameters. In either case, our previous
qualitative conclusions drawn from Figure 9 hold, but these uncertainties,
particularly the systematic ones, obviously significantly affect a
quantitative estimate of the ratio of the cooling mass that must be reheated
or ejected relative to that resulting in new stars.

Figure 9.— The total mass cooling rates, MCRs, and star formation rate, SFRs,
of the primary galaxies as a function of stellar mass. The error bars on the
blue circles (the cooling rates calculated from our H$\alpha$ flux
measurements) reflect only the flux measurement uncertainties and do not
reflect any uncertainty in the adopted Cloudy parameters. In contrast, the
mass cooling rates calculated from our fitted model for $\mathcal{F}_{cool}$,
the solid green line, include the systematic modeling uncertainties (shaded
region). We also plot the mass cooling rates predicted by the GAEA simulation
(Xie et al., 2020) as triangles. The SFRs both empirical (red stars) and as
predicted in the GAEA realizations (yellow stars) provide benchmark
comparisons. Finally, as the red dashed line we also plot our polynomial fit
to our model (Equation 8).

####  3.3.2 Comparison to Previous Theoretical Results

Given the importance of gas cooling to galaxy evolution, there is a large set
of existing literature examining this topic. Of particular relevance to our
results, there is theoretical work estimating the gas cooling flow. For
example, Stern et al. (2019) derived the gas cooling flow rate for a Milky Way
like galaxy. Concisely put, their simulations are controlled numerical
experiments in which gas that is initially in hydrostatic equilibrium within
an external gravitational potential is allowed to cool radiatively. They
estimate the ratio between the cooling flow rate and the SFR ($\dot{M}$/SFR)
to be $\sim 10$ within 100 kpc for a Milky Way like galaxy of halo mass $\sim
10^{12}$ M⊙. This is broadly consistent with what we have shown are our
results in Figure 9. Again, broadly, this estimate is consistent with results
obtained utilizing the radiatively cooling gas traced by O VI (Mathews &
Prochaska, 2017; McQuinn & Werk, 2018; Stern et al., 2018). In particular,
McQuinn & Werk (2018) obtained the estimation of massive flow of cooling gas
with $\sim 10-30$ M⊙/yr from the O VI absorption for low redshift $L_{*}$
galaxies.

To pursue a more quantitative comparison and, in particular, to examine the
trend in cooling rate with galaxy mass, we explore a more direct comparison to
simulations. In particular, we consider the predictions of the GAlaxy
Evolution and Assembly semi-analytic model that has been developed over a
series of studies (GAEA; De Lucia & Blaizot, 2007; De Lucia et al., 2014;
Hirschmann, De Lucia & Fontanot, 2016; Xie et al., 2017; Zoldan et al., 2018;
Fontanot et al., 2018, 2020; Xie et al., 2020). It is characterised by an
accurate treatment of chemical enrichment and energy recycling; a realistic
stellar feedback model, a self-consistent partition of multi-phase cold gas;
and an explicit treatment of the environmental effects on satellite galaxies.
The model produces good agreement with the observed stellar mass function up
to $z\sim 7$, and the cosmic star formation history up to $z\sim 10$
(Fontanot, Hirschmann & De Lucia, 2017). It is also able to reproduce the HI
and H2 fractions, as well as the quenched fractions in the local universe for
central and satellite galaxies respectively (Xie et al., 2020). In particular,
the theoretical predictions in the following have been taken from the model
realization applied to the Millennium Simulation (Springel et al., 2005) and
discussed in Xie et al. (2020). In this semi-analytic model, baryons in
galaxies are distributed in discrete components: hot gas, cold gas, stellar
disk, stellar bulge, and central massive black hole. In detail, Xie et al.
(2020) assume that cold gas is partitioned into HI and H2, and that stars form
from the H2 component following the empirical law proposed by (Blitz &
Rosolowsky, 2006). This approach implies that the SFR is therefore a function
of mid-plane pressure of the gas disk. The gas disk consists only of cold gas
that cools either from a hydrostatic hot atmosphere when the cooling time is
longer than dynamical time, or comes directly from outside of the dark matter
halo when the cooling time is shorter. The former cooling regime dominates at
low redshift. When cooling from a hydrostatic hot gas halo, the cooling rate
is calculated using the cooling function (Sutherland & Dopita, 1993; De Lucia
et al., 2010). In the latter case, all of the matter accreted into the dark
matter halo cools onto the gas disk. Cooling is also allowed for satellite
galaxies in GAEA. The cold gas amassing in galactic discs is then supposed to
fuel star formation. The resulting feedback reheats the cold gas at a rate
that depends on the SFR, on the circular velocity and on redshift (Hirschmann,
De Lucia & Fontanot, 2016), using functional forms that have been inspired
from the results of high-resolution N-body simulations (Muratov et al., 2015).
Reheated gas is moved from the cold gas component to the hot gas component,
leading to a complex interplay between cooling and heating. Moreover, feedback
from radio-mode AGN feedback (due to inefficient gas accretion onto the
central massive black holes) is assumed to provide an additional heating
channel (Croton et al., 2006; Xie et al., 2017), that further prevents the hot
gas from cooling in the more massive systems.

The resulting mass cooling rates plotted in Figure 9 are the combination of
the simulated mass of the cooling gas retained by the disk and the mass of the
reheated gas. In addition, the models present calculated SFRs. The calculated
SFRs are naturally a good fit to the measured ones because the models are in
part tuned to the reproduce related constraints such as galaxy stellar masses.
More impressive, however, is the independent agreement between the model
predictions for the mass cooling rates and our measurements. The only
significant disagreement appears at the lowest galaxy masses, where we have
already noted our model may have a problem (§3.1) and the GAEA model may also
have a problem, partially due to limited resolution (Xie et al., 2017, 2020).

###  3.4. The Feedback Mass Loading Factor

####  3.4.1 Empirical Results

As is discussed above, the mass cooling rate is significantly higher than the
SFR, demonstrating that the conversion of cooling gas into stars is
inefficient and that a large fraction of the gas must be reheated or ejected.
We define the mass loading factor, $\eta$, to be the ratio between the net
mass cooling rate (the difference between the mass cooling rate and the SFR)
and the SFR. Note that this definition may differ from that in other studies
that focus on physical mass outflow. Our estimate includes the effects both of
outflow and of any gas reheating that prevents that gas from participating in
star formation.

The mass loading factors as calculated from our model of CGM H$\alpha$ fluxes
show a rise of about a factor of 10 across galaxy mass range we explore
(Figure 9) and in all cases have values that are significantly larger than 1.
The rise is steeper at the larger M∗ in our sample, in the log-log plot,
indicating that the dependence of $\eta$ on M∗ rises faster than a simple
power law. The relation in the limited mass range between $\log\eta$ and
$\log\rm M_{*,10}$, where M∗,10 is again the stellar mass in units of
$10^{10}$ M⊙, can be approximated by a second order polynomial function as
follows:

$\log\eta=0.66+0.27\log{\rm M_{*,10}+0.46\log^{2}M_{*,10}}$ (9)

Figure 10.— The feedback mass loading factor, $\eta$, calculated using our
model for $\mathcal{F}_{cool}$, with resulting systematic uncertainties
plotted as the shaded region, and the results from two different published
theoretical studies are presented versus primary galaxy stellar mass. The
dashed red line presents the second-order polynomial fit to our results
(Equation 9). For comparison, we plot the mass loading factors from the GAEA
(Xie et al., 2020) and FIRE simulations (Muratov et al., 2015; Hayward &
Hopkins, 2017) as triangles and squares, respectively. The apparent
discrepancy with the FIRE results is resolved in the text.

####  3.4.2 Comparison to Previous Theoretical Results

In Figure 10 we compare our estimates of the mass loading factors to two
theoretical predictions. First, we compare to the same GAEA models we
discussed previously. The agreement is excellent, which is as expected given
the agreement in both the predicted vs. observed SFRs and mass cooling rates
(Figure 9).

Next we compare to results presented from the FIRE (The Feedback In Realistic
Environments) collaboration (Hopkins et al., 2014, 2018). They executed a
series of high-resolution cosmological simulations of galaxy formation to z =
0, spanning halo masses $\sim 10^{8}-10^{13}$ M⊙, and stellar masses $\sim
10^{4}-10^{11}$ M⊙. These simulations explicitly treat the multi-phase ISM
with heating and cooling physics from gas at a range of temperatures $T\sim
10-10^{10}$ K, star formation restricted only to self-gravitating, self-
shielding, molecular, high density ($n_{\rm H}\gtrsim 5-50$ cm-3) gas, and
(most importantly) explicit treatment of stellar feedback including the
energy, momentum, mass, and metal fluxes from SNe Types Ia & II, stellar mass
loss (O-star and AGB), radiation pressure (UV and IR), and photo-ionization
and photo-electric heating. These sources of feedback reproduce the observed
relation between stellar and halo mass up to M${}_{halo}\sim 10^{12}$ M⊙,
roughly corresponding to $M_{*}\sim 10^{10.5}$ M⊙. Star formation rates agree
well with the observed Kennicutt relation at all redshifts.

In Muratov et al. (2015) and Hayward & Hopkins (2017), the investigators
identified whether a particle is considered as outflowing or inflowing by
their radial velocity. Once outflowing particles are identified, they
calculate outflow rates in M⊙ yr-1 by computing the instantaneous mass flux
through a thin spherical shell. Combining the SFR from the simulation, they
calculate the mass loading factors for low mass galaxies with
M${}_{*}<10^{10}$ M⊙ and upper limit for galaxies with M${}_{*}>10^{10}$ M⊙ at
redshift $\sim 0.25$, which are shown in squares in Figure 10. A more recent
analysis with FIRE-2 simulations (Pandya et al., 2021) is able to convert
these limits to measurements and those measurements lie slightly below the
previous limits. Note that these studies focus on outflowing gas and,
therefore, that the estimates of the mass loading factor, as defined here,
using their results is a lower bound on what we should be measuring.

At the low mass end with M${}_{*}<10^{10}$ M⊙, the FIRE simulation is
statistically consistent with our estimation. At M${}_{*}>10^{10}$ M⊙,
however, the FIRE simulation results significantly diverge from our estimates.
However, this disagreement does not necessarily reflect a fault in either our
measurements or the FIRE results. Again recall that their results only
represent true outflow and do not include reheating. Furthermore, their
simulations are intended to reproduce only galaxies with M${}_{halo}<10^{12}$
M⊙, which for our stellar mass- halo mass relation corresponds to a stellar
mass of $10^{10.5}$ M⊙. Lastly, the divergence in Figure 10 is likely also
reflecting the lack of feedback other than from stars in these models, e.g.,
AGN feedback, which is included in the GAEA modeling. The agreement among the
mass loading factors from our empirical results, the semi-analytic results
from GAEA, and the more detailed, physically motivated simulations from FIRE
at low masses shows an encouraging convergence on our understanding of mass
loading factors at those masses.

This comparison highlights a challenge in comparing observations and
theoretical results given the different definitions used for related
quantities. Similarly, there are other recent studies (e.g., Mitchell, Schaye
& Bower, 2020) that present quantities that are difficult to translate into
the quantities we measure. With the advent of measured H$\alpha$ fluxes, we
encourage theorists to also tabulate quantities that are directly observable.

##  4\. Summary

We present measurements of the mass cooling rates for CGM gas and infer a
feedback mass loading factors from the comparison of that rate to the measured
star formation rates as a function of galactic stellar mass.

To calculate these estimates, we first revisited our measurements of the
H$\alpha$ emission line flux as a function of stellar mass, M∗, and as a
function of the local environment using both newer data and an improved
treatment. We track the behavior of H$\alpha$ flux within two scaled annular
bins around our primary galaxies as a function of both stellar mass
($10^{9.5}<{\rm M}_{*}<10^{11}$) and scaled projected distance from a massive
neighbor ($0.05<R_{p}/R_{\rm vir}<4$). The flux in both annular bins drops as
M∗ increases, indicating a strong inverse relationship between the cool gas
fraction of the CGM, $f_{\rm cool}$ and M∗. Here $\mathcal{F}_{cool}$
describes the fraction of the cosmological baryon fraction in the halo that is
in the cool ($T\sim 10^{4-5}$K) gas phase. We also find that a strong
environmental effect for the outer annular bin at the inner two smallest
values of $R_{p}/R_{vir}$. We fit a model that includes both a mass and
environment dependence within the framework of the UniverseMachine modeling
(Behroozi et al., 2018) and find

$\mathcal{F}_{cool,{\rm isolated}}({\rm M_{*}})=(0.23\pm 0.01)\left({\rm
M_{*,10}}\right)^{(-0.55\pm 0.03)}$ (10)

for galaxies beyond the influence of a nearby, massive neighbor (as defined in
§3.1), and

$\mathcal{F}_{cool}(R,{\rm
M_{*}})={R\over{1.92^{+1.05}_{-0.88}R_{vir}}}\cdot\mathcal{F}_{cool,{\rm
isolated}}$ (11)

otherwise, where $R$ is the physical distance between the primary and the
massive neighbor and $R_{vir}$ is the virial radius of the massive neighbor.
Our prediction for the cool gas fraction is as high as 90% for the smallest
halos we probe (although we may be overestimating the fraction at the low mass
end, see §3.1), $23\pm 1$% for a Milky Way like galaxy, and only a few percent
for the most massive galaxies in our sample. These results, except for the
lowest mass galaxies, are consistent with our previous results in Paper IV and
V, despite changes to the data and treatment. The only category where there is
some tension with the previous work is for the inner annular bin for the
lowest mass galaxies.

Our measurement of the H$\alpha$ flux originating from the CGM, and our
models, provides us the opportunity to measure the rate at which the CGM is
radiatively losing energy, and hence cooling. We use the Cloudy modeling suite
(Ferland et al., 1998, 2013, 2017) to understand what fraction of the cooling
is contributed by the H$\alpha$ flux we measure. In doing this, we need to
account for gas at different temperatures and present arguments for why we
consider only gas between $10^{4}-10^{5}$ K. By enforcing a continuous cooling
mass flow through the different temperatures we are able to calculate the gas
mass at the different temperatures within this range and complete the
calculation.

We then estimate the rate at which the CGM gas is cooling and presumably
sinking into the central disk. We explore the dependence of the CGM emission
spectra on the ionization state, the metallicity, the hydrogen density, and
estimate the uncertainties in our recovered mass cooling rates. We present
both the gas mass cooling rate and effective mass loading factor required to
avoid growing the disk gas mass. The mass cooling rate is a factor of several
higher than the star formation rate of the central galaxies. The simple fitted
relations between the mass cooling rate, MCR, and the mass loading factor,
$\eta$ with galactic stellar mass are as follows:

$\displaystyle\log{\rm MCR}=0.81+0.48\log{\rm M_{*,10}+0.35\log^{2}M_{*,10}}$
(12) $\displaystyle\log\eta=0.66+0.27\log{\rm M_{*,10}+0.46\log^{2}M_{*,10}}$
(13)

where M∗,10 is the galactic stellar mass in units of $10^{10}$ M⊙.

Although we have used highly idealized models to transform our measured
H$\alpha$ flux values into estimates of the mass cooling rates and feedback
mass loading factors, it is heartening to find excellent agreement between our
estimates of those quantities and those derived from two theoretical
treatments that are themselves quite different in their approach (Xie et al.,
2018, 2020; Hopkins et al., 2014, 2018). The general agreement we find
suggests that we have a relatively robust quantitative understanding of the
cooling and feedback rates for galaxies in this mass range. The empirical
confirmation of these rates provided here supports further modeling of the
role of the CGM in galaxy evolution. It is perhaps somewhat surprising that
our highly idealized model produces results in such good agreement given the
likely complexity of CGM (Tumlinson, Peeples & Werk, 2017), but perhaps our
measurement, which involves massively averaging over thousands and thousands
of galaxy lines of sight provides a bulwark against the complexity of any
individual galaxy’s CGM. The field critically needs maps of the CGM in
individual galaxies to further refine our understanding of this key galactic
component.

##  5\. Acknowledgments

DZ and HZ acknowledge financial support from NSF grant AST-1311326. KPO is
funded by NASA under award No 80NSSC19K1651. XY is supported by the national
science foundation of China (grant Nos.11833005, 11890692, 11621303) and 111
project No.B20019. TF is supported by the National Key R&D Program of China
No. 2017YFA0402600. The authors gratefully acknowledge the SDSS III team for
providing a valuable resource to the community. Funding for SDSS-III has been
provided by the Alfred P. Sloan Foundation, the Participating I institutions,
the National Science Foundation, and the U.S. Department of Energy Office of
Science. The SDSS-III web site is http://www.sdss3.org/.

SDSS-III is managed by the Astrophysical Research Consortium for the
Participating Institutions of the SDSS-III Collaboration including the
University of Arizona, the Brazilian Participation Group, Brookhaven National
Laboratory, Carnegie Mellon University, University of Florida, the French
Participation Group, the German Participation Group, Harvard University, the
Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA
Participation Group, Johns Hopkins University, Lawrence Berkeley National
Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for
Extraterrestrial Physics, New Mexico State University, New York University,
Ohio State University, Pennsylvania State University, University of
Portsmouth, Princeton University, the Spanish Participation Group, University
of Tokyo, University of Utah, Vanderbilt University, University of Virginia,
University of Washington, and Yale University.

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