# Equilibrium States of Galactic Atmospheres I: The Flip Side of Mass Loading

G. Mark Voit Michigan State University, Department of Physics and Astronomy,
East Lansing, MI 48824, USA  Viraj Pandya Hubble Fellow  Columbia University,
Department of Astronomy, New York, NY 10027, USA  Drummond B. Fielding Center
for Computational Astrophysics, Flatiron Institute, New York, NY 10010, USA
Department of Astronomy, Cornell University, Ithaca, NY 14853, USA  Greg L.
Bryan Columbia University, Department of Astronomy, New York, NY 10027, USA
Christopher Carr Columbia University, Department of Astronomy, New York, NY
10027, USA  Megan Donahue Michigan State University, Department of Physics and
Astronomy, East Lansing, MI 48824, USA  Benjamin D. Oppenheimer University of
Colorado, Center for Astrophysics and Space Astronomy, 389 UCB, Boulder, CO
80309, USA  Rachel S. Somerville Center for Computational Astrophysics,
Flatiron Institute, New York, NY 10010, USA

(Received 11 Jun 2024; Revised —; Accepted —)

###### Abstract

This paper presents a new framework for understanding the relationship between
a galaxy and its circumgalactic medium (CGM). It focuses on how imbalances
between heating and cooling cause either expansion or contraction of the CGM.
It does this by tracking all of the mass and energy associated with a halo’s
baryons, including their gravitational potential energy, even if feedback has
pushed some of those baryons beyond the halo’s virial radius. We show how a
star-forming galaxy’s equilibrium state can be algebraically derived within
the context of this framework, and we analyze how the equilibrium star
formation rate depends on supernova feedback. We consider the consequences of
varying the mass loading parameter $\eta_{M}\equiv\dot{M}_{\rm
wind}/\dot{M}_{*}$ relating a galaxy’s gas mass outflow rate ($\dot{M}_{\rm
wind}$) to its star formation rate ($\dot{M}_{*}$) and obtain results that
challenge common assumptions. In particular, we find that equilibrium star
formation rates in low-mass galaxies are generally insensitive to mass
loading, and when mass loading does matter, increasing it actually results in
more star formation because more supernova energy is needed to resist
atmospheric contraction.

††journal: ApJ

##  1 Introduction

You are about to read a paper that radically oversimplifies the phenomenon of
galaxy evolution. Where you might expect to see a cutting edge numerical
simulation or a sophisticated semi-analytic model, you will instead find a
system of three ordinary differential equations, sometimes reduced to just
two. This radical reduction in complexity makes it possible to derive
algebraic solutions defining a star-forming galaxy’s equilibrium state and
clarifying how a galaxy’s time-averaged star formation rate depends on the
properties of supernova feedback. Our goal is therefore complementary to the
goals of cosmological numerical simulations and complex semi-analytic models:
We are trying to provide simple conceptual tools for interpreting and
comparing simulations, models, and observational data sets that are far more
complicated.

This paper is the third in a series starting with Carr et al. (2023) and
continuing through Pandya et al. (2023). All three present regulator models
for galaxy evolution that emphasize the role of the circumgalactic medium
(CGM) and its energy content in governing the supply of star-forming gas. You
can find more extensive discussion of the motivations for the series in the
first two papers. A companion paper (Voit et al. 2024, hereafter Paper II)
relates this paper’s models to previous semi-analytic models for the CGM and
suggests how those models may be used to understand differences between
cosmological numerical simulations of galaxy evolution.

The new regulator model differs from the previous two, and from many other
semi-analytic models for galaxy evolution, in two key respects:

  1. 1.

It applies a comprehensive accounting approach to the total energy of a halo’s
baryons, including the energy content of baryons that feedback may have pushed
far beyond the halo’s virial radius.

  2. 2.

It explicitly includes gravitational potential energy in the baryonic energy
budget, because imbalances between energy input and radiative losses
ultimately cause either expansion or contraction of the atmosphere surrounding
a galaxy.

Another difference from many other semi-analytic models, including Pandya et
al. (2023) but not Carr et al. (2023), is that the new model allows the CGM to
supply a galaxy with gas even if feedback energy input exceeds radiative
cooling of the galaxy’s atmosphere.

We will show that our approach helps to clarify some misconceptions about the
relationship between galactic winds and the fraction $f_{*}$ of a halo’s
baryons that become stars. Early discussions of that relationship (e.g.,
Larson, 1974; Chevalier & Clegg, 1985; Dekel & Silk, 1986; White & Frenk,
1991) suggested to many that galactic winds might limit star formation by
permanently ejecting a galaxy’s gas. Much of the published work on galactic
winds has therefore focused on the mass loading parameter
$\eta_{M}\equiv\dot{M}_{\rm wind}/\dot{M}_{*}$ relating a galaxy’s gas mass
outflow rate ($\dot{M}_{\rm wind}$) to its star formation rate
($\dot{M}_{*}$). If most of the baryons accreting onto a halo end up in its
central galaxy, then $\eta_{M}$ needs to depend strongly on halo mass ($M_{\rm
halo}$) in order to explain observations of the $f_{*}$–$M_{\rm halo}$
relation indicating that $f_{*}$ is nearly an order of magnitude smaller at
$M_{\rm halo}\sim 10^{11}\,M_{\odot}$ than at $M_{\rm halo}\sim
10^{12}\,M_{\odot}$ (e.g., McGaugh et al., 2010; Moster et al., 2010; Behroozi
et al., 2019).

Some simulations produce dwarf galaxies with outflows having $\eta_{M}\gtrsim
10$ (e.g., Vogelsberger et al., 2013; Muratov et al., 2015; Pandya et al.,
2021), but observational constraints on $\eta_{M}$ usually indicate smaller
proportions of mass loading (e.g., Martin, 1999; Heckman et al., 2015;
Chisholm et al., 2017; McQuinn et al., 2019). Similarly, simulations that
focus on the galactic disk and resolve individual supernova events tend to
find lower mass loading values (Girichidis et al., 2016; Fielding et al.,
2018; Li & Bryan, 2020; Kim et al., 2020) Also, semi-analytic models that rely
solely on gas ejection, as parameterized by $\eta_{M}$, tend to have trouble
reproducing the mass-metallicity relationships observed among low-mass
galaxies (e.g., Somerville & Davé, 2015). Those tensions have prompted
questions about how much of a halo’s gas actually cycles through its central
galaxy and have stimulated interest in preventative feedback modes that reduce
a galaxy’s gas supply (e.g., Oppenheimer et al., 2010; van de Voort et al.,
2011; Davé et al., 2012; Lu et al., 2015; Hirschmann et al., 2016; Lu et al.,
2017; Pandya et al., 2020; Mitchell & Schaye, 2022; Carr et al., 2023; Pandya
et al., 2023).

One interesting feature of the feedback models in Carr et al. (2023) is the
insensitivity of their predicted $f_{*}$–$M_{\rm halo}$ relations to
$\eta_{M}$. This paper explains why Carr et al. (2023) arrived at that result
and outlines the general conditions that can make $f_{*}$ insensitive to
$\eta_{M}$. It also discusses why large values of $\eta_{M}$ can have
counterintuitive effects, sometimes causing $f_{*}$ to increase. By presenting
and interpreting those findings, both here and in Paper II, we hope to reframe
how astronomers view the relationship between supernova feedback and mass
loading of the outflows it produces.

Here is how things proceed: The next section (§2) lays the paper’s conceptual
foundation and previews the findings that follow. Section 3 then makes the
conceptual reasoning more quantitative by introducing a “minimalist” regulator
model designed to represent as simply as possible how supernova-driven
galactic outflows shape the global properties of a circumgalactic atmosphere.
Section 4 analyzes the equilibrium states of that model, which illustrate how
convergence of a galactic atmosphere toward a quasi-steady state and the
resulting value of $f_{*}$ depend on the properties of supernova feedback.
Section 5 considers the contrasting case of supernova-driven outflows that do
not interact with the rest of a galaxy’s atmosphere. The concluding section
summarizes our findings. A glossary in the Appendix describes the many symbols
introduced along the way.

##  2 Conceptual Foundation

Figure 1: Schematic summary of the paper’s foundational concepts and key
results.

To prepare readers for the paper’s main arguments, we would like to make sure
that the following foundational concepts are crystal clear:

  * •

Specific Energy. The total specific energy $\varepsilon_{\rm CGM}$ of a
galaxy’s circumgalactic atmosphere consists of gravitational potential energy,
thermal energy, and non-thermal energy.111The kinetic energy of fluid flow,
including both bulk flows and turbulence, is part of the non-thermal category.

Specifying the gravitational potential’s zero point so that gas entering or
leaving a halo’s central galaxy has zero potential energy (instead of putting
the zero point at infinity) helps to clarify the gravitational potential’s
role.

With that zero point, gas accreting at time $t$ onto a cosmological halo with
circular velocity $v_{\rm c}$ has a specific energy $\varepsilon_{\rm
acc}(t)\approx 4v_{\rm c}^{2}$, consisting mostly of gravitational potential
energy.222The factor of four corresponds to a nearly isothermal halo potential
well containing a central galaxy having a radius $\sim 50$ times smaller than
the halo’s virial radius (because $\ln 50\approx 4$). A comparable factor
applies to an NFW potential (Navarro et al., 1997), which has a depth
approximately $4.6$ times its maximum value of $v_{\rm c}^{2}$. Halo gas must
therefore radiate an amount of specific energy similar to $\varepsilon_{\rm
acc}$ before it can descend into the halo’s central galaxy and form stars
there.

Feedback produces outflows of gas that have a specific energy
$\varepsilon_{\rm fb}$ when they leave the galaxy. Initially, the outflowing
gas has zero potential energy, but its thermal and kinetic energy convert into
gravitational potential energy as the outflow rises to greater altitudes.

Figure 1 schematically summarizes how feedback outcomes depend on the
relationship between $\varepsilon_{\rm fb}$ and $\varepsilon_{\rm acc}$. Cases
with $\varepsilon_{\rm fb}<\varepsilon_{\rm acc}$ are on the left, and cases
with $\varepsilon_{\rm fb}>\varepsilon_{\rm acc}$ are on the right.

  * •

Feedback Coupling. The fate of a feedback outflow depends on how closely it
couples with a galaxy’s circumgalactic atmosphere. Here we consider two
limiting cases:

    1. 1.

Uncoupled Outflows. At one extreme are feedback outflows that are uncoupled
because they never transfer any of their energy to the rest of the CGM and
therefore cannot reduce the supply of gas going from the CGM into a halo’s
central galaxy. The top panel of Figure 1 represents what happens to uncoupled
outflows.

If the specific energy $\varepsilon_{\rm fb}$ of an uncoupled outflow
significantly exceeds $\varepsilon_{\rm acc}(t)$, then the recycling timescale
$t_{\rm cyc}$ on which outflowing gas returns to the halo’s central galaxy is
longer than the cosmic time $t$. That is because $\varepsilon_{\rm acc}(t)$ is
the specific energy of a cosmological trajectory that reached its maximum
altitude at a cosmic time similar to $t/2$ and completes its descent at time
$t$. An outflow with greater specific energy will take even longer to rise to
its maximum altitude and then return. The fate of an uncoupled outflow with
$\varepsilon_{\rm fb}>\varepsilon_{\rm acc}$ is therefore expulsion from the
halo.

However, gas exiting a galaxy in an uncoupled outflow with $\varepsilon_{\rm
fb}<\varepsilon_{\rm acc}$ can fall back into the galaxy on a timescale
shorter than $t$. In that case, the result of feedback is recycling capable of
ejecting gas from a galaxy multiple times.

The number of times $N_{\rm cyc}\sim t/t_{\rm cyc}$ that galactic gas is
typically recycled determines how effectively uncoupled outflows suppress
galactic star formation. If $N_{\rm cyc}\ll 1$, then the amount of gas that
forms stars is $1/(1+\eta_{M})$ times the gas mass that has passed through the
galaxy. But if $N_{\rm cyc}$ is larger, then the proportion of cycling gas
that forms stars is $(N_{\rm cyc}+1)/(1+\eta_{M})$, as long as $N_{\rm
cyc}\ll\eta_{M}$.

Section 5 presents a more quantitative analysis of uncoupled outflows that
supports this line of reasoning.

    2. 2.

Coupled Outflows. An outflow that transfers energy to the rest of the CGM is
considered coupled because the energy it transfers can reduce the central
galaxy’s gas supply.

The rate of energy transfer from a coupled supernova-driven outflow to the CGM
can be expressed as $\eta_{E}\varepsilon_{\rm SN}\dot{M}_{*}$, in which
$\varepsilon_{\rm SN}$ is the specific supernova energy released by a stellar
population and $\eta_{E}$ is an energy-loading parameter specifying the
fraction of supernova energy transported into the CGM. The specific energy of
such a coupled outflow is $\varepsilon_{\rm fb}=\eta_{E}\varepsilon_{\rm
SN}/\eta_{M}$.

Sections 3 and 4 present quantitative models for coupled supernova outflows
showing that coupled outflows with $\varepsilon_{\rm fb}>\varepsilon_{\rm
acc}$ cause expansion of the CGM, while coupled outflows with
$\varepsilon_{\rm fb}<\varepsilon_{\rm acc}$ allow contraction of the CGM.

The middle panel of Figure 1 schematically illustrates the results of §3 and
§4 by depicting the relationship between a halo’s virial radius $R_{\rm halo}$
and the closure radius $R_{\rm closure}$ within which the proportion of mass
in baryons equals the cosmic baryon mass fraction $f_{\rm b}$ (see Ayromlou et
al., 2023). Expanded atmospheres have $R_{\rm closure}>R_{\rm halo}$, while
contracted atmospheres can have $R_{\rm closure}<R_{\rm halo}$.

  * •

Asymptotic Star Formation. Each of the four cases illustrated in the upper two
panels of Figure 1 converges toward an asymptotic star formation rate and
stellar baryon fraction that depend on either $\eta_{M}$ or $\eta_{E}$, or
both. Section 4 analyzes the star-forming properties of galaxies with coupled
outflows. Section 5 analyzes the star-forming properties of galaxies with
uncoupled outflows.

The figure’s bottom panel schematically summarizes three primary results of
those analyses:

    1. 1.

Uncoupled outflows with high specific energy ($\varepsilon_{\rm
fb}\gg\varepsilon_{\rm acc}$) lead to stellar baryon fractions that are
inversely proportional to $1+\eta_{M}$ and insensitive to $\eta_{E}$.

    2. 2.

Coupled outflows with high specific energy ($\varepsilon_{\rm
fb}\gg\varepsilon_{\rm acc}$) lead to stellar baryon fractions that are
inversely proportional to $\eta_{E}$ and insensitive to $\eta_{M}$.

    3. 3.

Both coupled and uncoupled outflows fail to limit star formation if $\eta_{M}$
is large enough to make $\varepsilon_{\rm fb}$ significantly smaller than
$\varepsilon_{\rm acc}$. Uncoupled outflows face a recycling crisis when
$t_{\rm cyc}$ becomes much smaller than $t$. Coupled outflows face a
contraction crisis when gas flowing out of the central galaxy has a specific
energy much less than $\varepsilon_{\rm acc}\approx 4v_{\rm c}^{2}$. In both
cases, the crisis leads to larger amounts of star formation than a galaxy with
$\varepsilon_{\rm fb}\approx\varepsilon_{\rm acc}$ would have.

While the supporting calculations in sections 3 through 5 are highly idealized
compared to sophisticated cosmological simulations of galaxy evolution, we
believe they provide important insights into the outcomes of those
simulations, for reasons explained in Paper II.

##  3 Coupled Outflows: Regulator Model

We will start by examining how coupled outflows self-regulate. The first step
is to develop a regulator model that mimics the interplay between a galaxy’s
star formation rate and its gas supply (e.g., Bouché et al., 2010; Davé et
al., 2012; Lilly et al., 2013).

###  3.1 Comprehensive Accounting

The regulator model presented here keeps track of all of the baryons that have
entered a cosmological halo. Accretion brings baryons into the halo at the
rate $\dot{M}_{\rm acc}$. Some of that mass ends up in stars ($M_{*}$). Some
stays in the interstellar medium ($M_{\rm ISM}$). The rest remains
circumgalactic. A comprehensive accounting approach to the baryon budget
therefore defines the mass of the central galaxy’s circumgalactic medium (CGM)
to be

$M_{\rm CGM}=M_{\rm acc}-M_{*}-M_{\rm ISM}\;\;.$ (1)

Unlike many previous approaches, there is no term representing ejection of gas
mass from the CGM. Accreted gas belongs either to the galaxy or its CGM and is
never lost from the system. However, feedback from coupled outflows can push
some of the CGM beyond $R_{\rm halo}$, causing $R_{\rm closure}$ to exceed
$R_{\rm halo}$.

Energy associated with the gas mass $M_{\rm CGM}$ can be either gravitational
potential energy ($E_{\varphi}$), thermal energy ($E_{\rm th}$), or non-
thermal energy ($E_{\rm nt}$). The total energy content of the circumgalactic
gas is therefore

$E_{\rm CGM}=E_{\varphi}+E_{\rm th}+E_{\rm nt}\;\;.$ (2)

Adding either thermal or non-thermal energy generally causes the
circumgalactic medium to expand. Expansion then converts the added energy into
gravitational potential energy. Likewise, losses of energy generally permit
atmospheric contraction that converts gravitational potential energy into
either thermal or non-thermal energy.

###  3.2 Energy Sources and Sinks

A circumgalactic atmosphere’s total energy evolves according to

$\dot{E}_{\rm CGM}=\dot{E}_{\rm acc}-\dot{E}_{\rm rad}-\dot{E}_{\rm
in}+\dot{E}_{\rm fb}+\dot{E}_{\varphi}\;\;.$ (3)

Cosmological accretion adds energy at the rate $\dot{E}_{\rm acc}$. Radiative
losses ($\dot{E}_{\rm rad}$) allow some of the accreted gas to sink into the
central galaxy. Gas entering the central galaxy’s interstellar medium removes
energy from the circumgalactic medium at the rate $\dot{E}_{\rm in}$. Star
formation fueled by interstellar gas then produces a feedback response that
adds energy to the circumgalactic atmosphere at the rate $\dot{E}_{\rm fb}$.
While all that is happening, the gravitational potential well confining the
circumgalactic gas evolves. The last source term

$\dot{E}_{\varphi}\equiv\int\dot{\varphi}\,dM_{\rm CGM}$ (4)

represents the change in atmospheric energy coming solely from time-dependent
changes in the gravitational potential $\varphi$.

###  3.3 Gravitational Potential

To simplify later calculations, we will approximate the halo’s gravitational
potential as a singular isothermal sphere:

$\varphi(r)=v_{\rm c}^{2}\ln\left(\frac{r}{r_{0}}\right)\;\;.$ (5)

This potential well has a constant circular velocity $v_{\rm c}$ and a zero
point at $r=r_{0}$. We will assume that $v_{\rm c}$ and $r_{0}$ both remain
constant with time, so that $\dot{E}_{\varphi}=0$.

Gas accreting through the virial radius $R_{\rm halo}$ of a galaxy’s
cosmological halo at the rate $\dot{M}_{\rm acc}$ has a specific energy
$\varepsilon_{\rm acc}\equiv\dot{E}_{\rm acc}/\dot{M}_{\rm acc}$ at least as
large as $\varphi(R_{\rm halo})$. Before accreting gas can enter the central
galaxy, it needs to lose some of that energy. Here we will choose $r_{0}$ to
represent the inner boundary of the circumgalactic medium, so that gas passing
from the circumgalactic medium into the interstellar medium has zero potential
energy. This choice makes $\varphi(R_{\rm halo})$ similar to the specific
energy that accreting gas needs to lose in order to enter the central galaxy
and remain within it. The magnitude of $\varphi(R_{\rm halo})$ ranges from
$3.9v_{\rm c}^{2}$ to $4.6v_{\rm c}^{2}$ as $R_{\rm halo}$ ranges from
$50r_{0}$ to $100r_{0}$ in an isothermal potential well.

###  3.4 Mass Exchange

Suppose that circumgalactic radiative cooling allows atmospheric gas to enter
the central galaxy at the rate $\dot{M}_{\rm in}$ and that feedback from star
formation expels gas from the central galaxy at a rate that is $\eta_{M}$
times the star formation rate $\dot{M}_{*}$. The interstellar gas mass then
evolves (neglecting mass return from stars for simplicity) according to

$\dot{M}_{\rm ISM}=\dot{M}_{\rm in}-\left(1+\eta_{M}\right)\dot{M}_{*}\;\;,$
(6)

and the circumgalactic gas mass evolves according to

$\dot{M}_{\rm CGM}=\dot{M}_{\rm acc}-\dot{M}_{\rm
in}+\eta_{M}\dot{M}_{*}\;\;.$ (7)

We will soon be paying closer attention to the mass loading parameter
$\eta_{M}$ that appears in both of these equations.

###  3.5 Supernova Feedback

Star formation in the central galaxy produces a feedback response that can
limit further star formation in two distinct ways. First, it can eject gas
from the galaxy, as represented by the mass loading parameter $\eta_{M}$.
Second, it can add energy to the circumgalactic medium, causing it to expand,
thereby reducing the galaxy’s gas supply $\dot{M}_{\rm in}$. Supernova
feedback energy enters the galaxy’s atmosphere at a rate

$\dot{E}_{\rm fb}=\eta_{E}\varepsilon_{\rm SN}\dot{M}_{*}$ (8)

in which the specific kinetic energy output from a typical stellar population
is

$\varepsilon_{\rm SN}\approx\frac{10^{51}\,{\rm
erg}}{100\,M_{\odot}}\approx\left(700\,{\rm km\,s^{-1}}\right)^{2}\;\;.$ (9)

The energy-loading parameter $\eta_{E}$ specifies the proportion of supernova
energy reaching the circumgalactic medium and coupling with it.

###  3.6 Radiative Losses

For expansion to be effective, the supernova energy coupling with the galaxy’s
atmosphere ($\dot{E}_{\rm fb}$) needs to be at least comparable to the
atmosphere’s radiative loss rate ($\dot{E}_{\rm rad}$). We can relate the
atmosphere’s radiative losses to the galaxy’s gas supply via the specific
energy

$\varepsilon_{\rm rad}\equiv\frac{\dot{E}_{\rm rad}}{\dot{M}_{\rm in}}\;\;.$
(10)

This quantity will turn out to be useful when we compare $\dot{E}_{\rm fb}$
with $\dot{E}_{\rm rad}$. Under most circumstances, we expect
$\varepsilon_{\rm rad}$ to be similar to $\varepsilon_{\rm
acc}\approx\varphi(R_{\rm halo})$, because the latter quantity is the specific
potential energy that accreted gas needs to lose in order to get into the
central galaxy. For completeness, we also define

$\varepsilon_{\rm in}\equiv\frac{\dot{E}_{\rm in}}{\dot{M}_{\rm in}}$ (11)

and

$\varepsilon_{\rm loss}\equiv\varepsilon_{\rm rad}+\varepsilon_{\rm in}$ (12)

so that the total rate of atmospheric energy loss becomes $\dot{E}_{\rm
loss}=\dot{M}_{\rm in}\varepsilon_{\rm loss}$.

###  3.7 Net Feedback Energy

Suppose we now integrate atmospheric energy losses and gains over a galaxy’s
entire history, to capture both the radiative losses needed to make a stellar
mass $M_{*}$ and the resulting feedback response. The cumulative amount of
supernova feedback energy added to the atmosphere by coupled outflows is
$E_{\rm fb}=\eta_{E}\varepsilon_{\rm SN}M_{*}$. The cumulative radiative
losses follow from integrating $\dot{E}_{\rm rad}=\dot{M}_{\rm
in}\varepsilon_{\rm rad}$ over time, giving

$E_{\rm rad}=\left[\langle 1+\eta_{M}\rangle M_{*}+M_{\rm
ISM}\right]\langle\varepsilon_{\rm rad}\rangle\;\;,$ (13)

in which

$\left\langle
1+\eta_{M}\right\rangle\equiv\frac{1}{M_{*}}\int(1+\eta_{M})\dot{M}_{*}\,dt\;\;.$
(14)

and

$\left\langle\varepsilon_{\rm rad}\right\rangle\equiv\frac{\int\dot{M}_{\rm
in}\varepsilon_{\rm rad}\,dt}{\int\dot{M}_{\rm in}\,dt}\;\;.$ (15)

The net effect of cooling and feedback on the total energy content of a
galaxy’s atmosphere is therefore

$\displaystyle E_{\rm fb}-E_{\rm rad}$
$\displaystyle=\left[\eta_{E}\varepsilon_{\rm SN}-\langle
1+\eta_{M}\rangle\left\langle\varepsilon_{\rm rad}\right\rangle\right]M_{*}$
$\displaystyle\quad\quad-\left\langle\varepsilon_{\rm rad}\right\rangle M_{\rm
ISM}\;\;.$ (16)

Meanwhile, the gas supply fueling galactic star formation transfers a total
energy

$E_{\rm in}=\left[\langle 1+\eta_{M}\rangle M_{*}+M_{\rm
ISM}\right]\left\langle\varepsilon_{\rm in}\right\rangle$ (17)

from the galaxy’s circumgalactic medium to its interstellar medium. Here,
$\langle\varepsilon_{\rm in}\rangle$ is a mass-averaged value of
$\varepsilon_{\rm in}$ similar to $\langle\varepsilon_{\rm rad}\rangle$.

###  3.8 The Flip Side of Mass Loading

Now we revisit the mass loading parameter $\eta_{M}$. Its most obvious effect
on star formation in the central galaxy is to suppress it by ejecting
potentially star forming gas from the galaxy’s ISM. However, the presence of
$\eta_{M}$ following a minus sign in Equation 16, the formula for net feedback
energy, suggests another side to the mass loading story.

According to our energy accounting exercise, increasing $\eta_{M}$ reduces the
supernova feedback cycle’s net energy input into a galaxy’s atmosphere. In
fact, a supernova feedback cycle with

$\langle 1+\eta_{M}\rangle\langle\varepsilon_{\rm
rad}\rangle>\eta_{E}\varepsilon_{\rm SN}$ (18)

results in a net atmospheric energy loss rather than a net energy gain.333This
condition is sufficient but not necessary, because it omits, for simplicity,
the additional energy loss terms proportional to $M_{\rm ISM}$ and
$\langle\varepsilon_{\rm in}\rangle$.

This underappreciated feature of mass-loaded outflows—that more mass loading
implies greater atmospheric radiative losses—is the key point of the entire
paper. It is consequential because a galaxy’s atmosphere responds to energy
loss by contracting and becoming denser, possibly increasing the galaxy’s gas
supply, its star formation rate, and its total stellar mass. The combined
effects of $\eta_{M}$ on $M_{*}$ are therefore interestingly coupled and
require closer examination.

###  3.9 Minimalist Regulator Model

Section 4 will explore the impact of mass loaded galactic outflows on galaxies
and their atmospheres in the context of a highly simplified evolutionary
model. Three coupled differential equations define the model:

$\displaystyle\dot{E}_{\rm CGM}\>$ $\displaystyle=\>\dot{M}_{\rm
acc}\varepsilon_{\rm acc}\>-\>\dot{M}_{\rm in}\varepsilon_{\rm
loss}\>+\>\eta_{E}\varepsilon_{\rm SN}\frac{M_{\rm ISM}}{t_{\rm SF}}$ (19)
$\displaystyle\dot{M}_{\rm CGM}\>$ $\displaystyle=\>\dot{M}_{\rm
acc}\>-\>\dot{M}_{\rm in}\>+\>\eta_{M}\frac{M_{\rm ISM}}{t_{\rm SF}}$ (20)
$\displaystyle\dot{M}_{\rm ISM}\>$ $\displaystyle=\>\dot{M}_{\rm
in}\>-\>\left(1+\eta_{M}\right)\frac{M_{\rm ISM}}{t_{\rm SF}}\;\;.$ (21)

We will call this system of equations the minimalist regulator model. For a
given halo, both $\dot{M}_{\rm acc}$ and $\varepsilon_{\rm acc}$ are
cosmological source terms that depend only on time. The galaxy’s star
formation timescale $t_{\rm SF}\equiv M_{\rm ISM}/\dot{M}_{*}$ can be either
constant, or a function of time, or a function of both time and $M_{\rm ISM}$.
Both the galaxy’s gas supply $\dot{M}_{\rm in}$ and the energy loss parameter
$\varepsilon_{\rm loss}$ are presumed to be functions of $E_{\rm CGM}$ and
$M_{\rm CGM}$. They may also depend explicitly on time.

Notice that the main effect of the $M_{\rm ISM}$ equation is to spread the
star formation fueled by incoming gas over a timescale $\sim t_{\rm
SF}/(1+\eta_{M})$. Integrating the $M_{\rm ISM}$ equation using the
integrating factor method gives

$M_{\rm ISM}(t)\>=\>\int_{0}^{t}\dot{M}_{\rm
in}(t^{\prime})\,\exp\left[\tau(t^{\prime})-\tau(t)\right]\,dt^{\prime}$ (22)

in which

$\tau(t)\equiv\int_{0}^{t}\frac{1+\eta_{M}}{t_{\rm
SF}(t^{\prime})}\,dt^{\prime}\\\ $ (23)

is a gas depletion parameter defined so that the exponential factor represents
the fraction of gas that entered the galaxy at time $t^{\prime}$ and still
remains in the interstellar medium at time $t$. The galaxy’s star formation
rate is therefore $\dot{M}_{*}=\langle\dot{M}_{\rm in}\rangle/(1+\eta_{M})$,
in which $\langle\dot{M}_{\rm in}\rangle\equiv(1+\eta_{M})M_{\rm ISM}/t_{\rm
SF}$ can be considered the average value of $\dot{M}_{\rm in}$ during a time
interval $\Delta t\sim t_{\rm SF}/(1+\eta_{M})$.

If $\dot{M}_{\rm in}$ and $t_{\rm SF}$ remain sufficiently steady, so that
$\dot{M}_{\rm in}=\langle\dot{M}_{\rm in}\rangle$, then the minimalist
regulator model reduces to

$\displaystyle\dot{E}_{\rm CGM}\>$ $\displaystyle=\>\dot{M}_{\rm
acc}\varepsilon_{\rm acc}\>+\>\left(\frac{\eta_{E}\varepsilon_{\rm
SN}}{1+\eta_{M}}-\varepsilon_{\rm loss}\right)\dot{M}_{\rm in}$ (24)
$\displaystyle\dot{M}_{\rm CGM}\>$ $\displaystyle=\>\dot{M}_{\rm
acc}\>-\>\frac{\dot{M}_{\rm in}}{1+\eta_{M}}\;\;.$ (25)

We will use the minimalist regulator model in this reduced form whenever
possible. Doing so enables an assessment of whether $\dot{M}_{\rm in}$ remains
sufficiently steady, at least in the context of the model. If the steady-
supply assumption is justified, then explicit integration of the $\dot{M}_{\rm
ISM}$ equation is not necessary. The exceptions are cases with either an
erratic gas supply or erratic variations in $t_{\rm SF}$. Explicit integration
of the $\dot{M}_{\rm ISM}$ equation then smooths the star-formation response
to the galaxy’s gas supply over a time interval $\sim t_{\rm
SF}/(1+\eta_{M})$.

Figure 2: Introduction to minimalism.

###  3.10 Generality of the Model

Before moving on, we would like to emphasize the generality of the minimalist
regulator model. It is primarily an accounting system that tracks how baryons
move from one reservoir to another and how cooling and feedback change the
atmosphere’s total energy. It becomes an integrable physical model once
$\dot{M}_{\rm in}$ and $\varepsilon_{\rm loss}$ are specified as functions of
$E_{\rm CGM}$, $M_{\rm CGM}$, and $t$. Those features make the model broadly
useful for testing how various assumptions about the structure and radiative
loss rate of a galaxy’s atmosphere, as expressed via $\dot{M}_{\rm in}$ and
$\varepsilon_{\rm loss}$, affect long-term predictions for galactic star
formation. We will discuss $\dot{M}_{\rm in}$ and $\varepsilon_{\rm loss}$
further in §4.4.

##  4 Coupled Outflows: Asymptotic States

Now we will use the minimalist regulator model to illuminate how mass loading
of coupled galactic outflows affects the stellar baryon fraction $f_{*}$ of a
galactic halo. At any given moment in time, there is a particular equilibrium
state toward which the model is converging. That equilibrium state depends on
the gas supply function $\dot{M}_{\rm in}$, the specific energy loss function
$\varepsilon_{\rm loss}$, the cosmological baryon accretion rate $\dot{M}_{\rm
acc}$, and the model’s parameters: $\eta_{M}$, $\eta_{E}$, $\varepsilon_{\rm
SN}$, and $\varepsilon_{\rm acc}$. In equilibrium, the star formation rate
$\dot{M}_{*}$ and atmospheric specific energy $\varepsilon_{\rm CGM}$ remain
constant. However, explicitly time-dependent changes in either of the
governing functions ($\dot{M}_{\rm in}$ or $\varepsilon_{\rm loss}$) or in the
model’s parameters can cause the equilibrium values of $\dot{M}_{*}$ and
$\varepsilon_{\rm CGM}$ to drift with time.444To be clear, both $\dot{M}_{\rm
in}$ and $\varepsilon_{\rm loss}$ may change with time because $E_{\rm CGM}$
and $M_{\rm CGM}$ are changing with time. Those temporal changes are implicit.
It is also possible for the dependences of those functions on $E_{\rm CGM}$
and $M_{\rm CGM}$ to change with time, and those kinds of temporal changes are
explicit. In other words, there are potentially two types of time dependence:
(1) asymptotic convergence toward an equilibrium state, and (2) temporal
changes in the asymptotic equilibrium state.

The asymptotic states fall into two distinct categories, depending on the sign
of the factor

$\frac{\eta_{E}\varepsilon_{\rm SN}}{1+\eta_{M}}-\varepsilon_{\rm loss}\;\;.$
(26)

If this factor is positive, then the net effect of a coupled supernova
feedback loop is to add energy to the surrounding atmosphere, causing the
atmosphere to expand and the central galaxy’s gas supply to decline. If this
factor is negative, then the atmosphere loses energy and contracts relative to
the state it would have without star formation and supernova feedback.

This section will demonstrate why asymptotic star formation rates within
expanded atmospheres are insensitive to $\eta_{M}$ and will also explain why
asymptotic star formation rates within contracted atmospheres are likely to be
larger than those within expanded atmospheres.

###  4.1 Asymptotic Expansion

Feedback that expands a galaxy’s atmosphere causes star formation to approach
a rate jointly determined by the cosmological accretion rate $\dot{M}_{\rm
acc}$ and the specific energy $\eta_{E}\varepsilon_{\rm SN}$. For example, if
$\varepsilon_{\rm loss}$ is negligible compared to $\eta_{E}\varepsilon_{\rm
SN}/(1+\eta_{M})$, then essentially all of the feedback energy accumulates in
the CGM, meaning that

$(\varepsilon_{\rm CGM}-\varepsilon_{\rm acc})(\dot{M}_{\rm
acc}-\dot{M}_{*})\>\approx\>\eta_{E}\varepsilon_{\rm SN}\dot{M}_{*}$ (27)

in which $\varepsilon_{\rm CGM}\equiv E_{\rm CGM}/M_{\rm CGM}$ is the mean
specific energy of the CGM. Significant expansion of the atmosphere generally
requires a value of $\varepsilon_{\rm CGM}-\varepsilon_{\rm acc}$ similar to
$v_{\rm c}^{2}$, and a value much greater than $v_{\rm c}^{2}$ would eject the
entire atmosphere. Consequently, the halo’s stellar baryon fraction
asymptotically approaches

$f_{*}\>\approx\>\frac{\xi v_{\rm c}^{2}}{\eta_{E}\varepsilon_{\rm SN}+\xi
v_{\rm c}^{2}}$ (28)

in which $\xi\equiv(\varepsilon_{\rm eq}-\varepsilon_{\rm acc})/v_{\rm c}^{2}$
is a dimensionless number of order unity and $\varepsilon_{\rm eq}$ is the
value of $\varepsilon_{\rm CGM}$ that makes $\dot{\varepsilon}_{\rm CGM}=0$.

The absence of the mass loading parameter $\eta_{M}$ from this estimate for
$f_{*}$ implies that the asymptotic stellar mass of the galaxy at the center
of an expanding atmosphere is insensitive to mass loading, as long as
radiative losses are negligible. Fundamentally, a supernova feedback system
with these properties tunes itself so that the galaxy’s energy output is
regulating its gas supply by keeping $\varepsilon_{\rm CGM}$ approximately
constant and greater than $\varepsilon_{\rm acc}$ as cosmological accretion
adds gas mass to the atmosphere. (See also Sharma & Theuns, 2020, who arrive
at a similar conclusion through a different approach.)

###  4.2 Asymptotic Contraction

Radiative losses come into play when $\varepsilon_{\rm loss}$ is comparable to
$\eta_{E}\varepsilon_{\rm SN}/(1+\eta_{M})$. If atmospheric energy losses
exceed feedback energy input, then they reduce $E_{\rm CGM}$ below the
cumulative energy input $E_{\rm acc}$ from cosmological accretion. Those
losses eventually lead to an atmosphere with a smaller mean radius and greater
mean density than an atmosphere without radiative cooling or galaxy formation
would have. Atmospheric contraction is therefore likely to increase the
central galaxy’s gas supply, because a denser atmosphere has both a shorter
cooling time and a shorter dynamical time.

Such an atmosphere is prone to runaway cooling and catastrophic collapse. The
risk of collapse remains high while $\varepsilon_{\rm loss}$ remains greater
than $\eta_{E}\varepsilon_{\rm SN}/(1+\eta_{M})$, because the net effect of
additional star formation and supernova feedback is even more atmospheric
energy loss. However, there are a few ways to prevent a catastrophe:

  1. 1.

Angular momentum can limit a galactic atmosphere’s collapse once it contracts
enough for orbital gas motions to prevent further contraction. The galaxy’s
gas supply then depends on torques that remove angular momentum from gas near
the galaxy and on the timescale for dissipation of orbital energy.

  2. 2.

Accretion onto a central black hole can release additional feedback energy
capable of exceeding radiative cooling and reversing the contraction.

  3. 3.

Atmospheric contraction itself can in principle reduce the specific energy
loss function $\varepsilon_{\rm loss}$ so that it becomes comparable to
$\eta_{E}\varepsilon_{\rm SN}/(1+\eta_{M})$. For example, consider gas in a
supernova-driven galactic fountain that manages to keep most of a halo’s
baryons aloft in the CGM rather than allowing them to collect in the central
galaxy. Gas in the fountain has a specific energy similar to $\varepsilon_{\rm
fb}=\eta_{E}\varepsilon_{\rm SN}/\eta_{M}$ and therefore radiates an amount of
energy comparable to $\varepsilon_{\rm fb}$ while returning to the galaxy.

The first two remedies are beyond the scope of the minimalist regulator model,
at least in the form presented in this paper. We will therefore focus here on
the third possibility.

The threshold condition $\varepsilon_{\rm loss}=\eta_{E}\varepsilon_{\rm
SN}/(1+\eta_{M})$ corresponds to a feedback energy input rate that balances
atmospheric energy losses. Star formation in such a balanced state proceeds at
a rate

$\dot{M}_{*}\>=\>\frac{\dot{M}_{\rm in}}{1+\eta_{M}}\>=\>\frac{\dot{E}_{\rm
loss}}{\eta_{E}\varepsilon_{\rm SN}}\;\;.$ (29)

The atmosphere’s energy loss rate $\dot{E}_{\rm loss}$ depends sensitively on
the atmosphere’s density and temperature structure. Consequently, the
asymptotic value of $\dot{M}_{*}$ at the center of a contracted atmosphere
also depends sensitively on atmospheric structure.

We will not attempt to model here the complex details that determine
$\dot{E}_{\rm loss}$ in a contracted atmosphere supported by supernova-driven
outflows. Even sophisticated numerical simulations have trouble with that
task. For the moment, we will simply point out two important global features
of contracted atmospheres: (1) larger values of $\eta_{M}$ make an outflow’s
specific energy $\varepsilon_{\rm fb}$ smaller, resulting in atmospheres with
smaller radii, and (2) denser atmospheres usually suffer greater radiative
losses. Coupled outflows with larger values of $\eta_{M}$ are therefore likely
to result in larger steady-state star formation rates. The next section tries
to make that inference more quantitative.

###  4.3 Analysis of Equilibria

Here we analyze how the minimalist regulator model for coupled outflows
evolves toward an equilibrium state and how that asymptotic state depends on
the model’s parameters. Our goal is to understand how star formation depends
on $\eta_{M}$ in such models.

####  4.3.1 Dynamical Equations

The following investigation of the model’s equilibrium states focuses on the
quantity

$f_{*,{\rm asy}}\>\equiv\>\frac{\dot{M}_{\rm in}}{(1+\eta_{M})\dot{M}_{\rm
acc}}\;\;,$ (30)

which is the asymptotic stellar baryon fraction of a halo with constant
$\dot{M}_{\rm acc}$ and $\dot{M}_{\rm in}$. When written in terms of
$f_{*,{\rm asy}}$, the mimimalist regulator model becomes

$\displaystyle\frac{dE_{\rm CGM}}{dM_{\rm acc}}$
$\displaystyle=\varepsilon_{\rm acc}\>+\>\left[\eta_{E}\varepsilon_{\rm
SN}-(1+\eta_{M})\varepsilon_{\rm loss}\right]f_{*,{\rm asy}}$
$\displaystyle\quad\quad\>+\>\eta_{E}\varepsilon_{\rm SN}\cdot\delta f_{*}$
(31) $\displaystyle\frac{dM_{\rm CGM}}{dM_{\rm acc}}$
$\displaystyle=1\>-\>f_{*,{\rm asy}}+\eta_{M}\cdot\delta f_{*}$ (32)
$\displaystyle\frac{dM_{\rm ISM}}{dM_{\rm acc}}$
$\displaystyle=-\,(1+\eta_{M})\cdot\delta f_{*}\;\;.$ (33)

In these equations, the total mass of accreted baryons ($M_{\rm acc}$) plays
the role of time and the quantity

$\delta f_{*}\>\equiv\>\frac{\dot{M}_{*}}{\dot{M}_{\rm acc}}\>-\>f_{*,{\rm
asy}}$ (34)

accounts for differences between $\dot{M}_{\rm in}/(1+\eta_{M})$ and
$\dot{M}_{*}$. Setting $\delta f_{*}$ equal to zero makes this system of
equations equivalent to the reduced version of the minimalist regulator model
defined by equations (24) and (25).

####  4.3.2 Equilibrium States

We would like to identify equilibrium states in which $\dot{M}_{*}$ remains
constant as long as all of the model’s parameters ($\varepsilon_{\rm acc}$,
$\varepsilon_{\rm SN}$, $\eta_{E}$, and $\eta_{M}$) remain constant. In such
an equilibrium state, $M_{\rm ISM}=\dot{M}_{*}t_{\rm SF}$ remains constant,
implying $\delta f_{*}=0$. The reduced version of the minimalist regulator
model then applies. Also, the mass-exchange rates ($\dot{M}_{*}$,
$\dot{M}_{\rm in}$, and $\dot{M}_{\rm acc}$) in such an equilibrium state must
all remain proportional to each other as $E_{\rm CGM}$ and $M_{\rm CGM}$
increase with time.

We will therefore restrict our attention to gas supply functions having the
separable form

$\dot{M}_{\rm in}=f_{\rm CGM}\dot{M}_{\rm acc}\cdot f_{\rm
in}(\varepsilon_{\rm CGM})$ (35)

so that $\dot{M}_{\rm in}/\dot{M}_{\rm acc}$ does not change as long as the
atmosphere’s specific energy ($\varepsilon_{\rm CGM}$) and circumgalactic
baryon fraction ($f_{\rm CGM}\equiv M_{\rm CGM}/M_{\rm acc}$) remain constant.
The quantity $f_{\rm CGM}\dot{M}_{\rm acc}=M_{\rm CGM}/(M_{\rm
acc}/\dot{M}_{\rm acc})$ corresponds to the gas supply rate provided by an
atmosphere of mass $M_{\rm CGM}$ that flows into a halo’s central galaxy on
the cosmological timescale $M_{\rm acc}/\dot{M}_{\rm acc}$. The dimensionless
function $f_{\rm in}(\varepsilon_{\rm CGM})$ accounts for how cooling and
feedback change the gas supply rate by causing the atmosphere either to expand
or to contract.

When expressed in terms of $\varepsilon_{\rm CGM}$ and $f_{\rm CGM}$, the
reduced version of the minimalist regulator model becomes

$\displaystyle\frac{d\varepsilon_{\rm CGM}}{d\ln M_{\rm acc}}$
$\displaystyle=\varepsilon_{\rm acc}\>+\>\left[\eta_{E}\varepsilon_{\rm
SN}-(1+\eta_{M})\varepsilon_{\rm loss}\right]f_{*,{\rm asy}}$
$\displaystyle\quad\quad\>-\>(1-f_{*,{\rm asy}})\varepsilon_{\rm CGM}$ (36)
$\displaystyle\frac{df_{\rm CGM}}{d\ln M_{\rm acc}}$
$\displaystyle=1\>-\>f_{*,{\rm asy}}-f_{\rm CGM}\;\;.$ (37)

A gas supply function with the form of equation (35) remains constant if
$\varepsilon_{\rm CGM}$, $f_{\rm CGM}$, and $\dot{M}_{\rm acc}$ all remain
constant. The equilibrium values of $\varepsilon_{\rm CGM}$ and $f_{\rm CGM}$
are then solutions to the system of equations

$\displaystyle\varepsilon_{\rm CGM}$ $\displaystyle\>=\>$
$\displaystyle\frac{\varepsilon_{\rm acc}\>+\>\left[\eta_{E}\varepsilon_{\rm
SN}-(1+\eta_{M})\varepsilon_{\rm loss}\right]f_{*,{\rm asy}}}{1-f_{*,{\rm
asy}}}$ (38) $\displaystyle f_{\rm CGM}$ $\displaystyle\>=\>$ $\displaystyle
1\>-\>f_{*,{\rm asy}}$ (39)

in which $f_{*,{\rm asy}}$ and $\varepsilon_{\rm loss}$ are functions of
$\varepsilon_{\rm CGM}$ and $f_{\rm CGM}$.

Combining equations (30), (35), and (39) gives the asymptotic stellar baryon
fraction

$f_{*,{\rm asy}}(\varepsilon_{\rm eq})\>=\>\frac{f_{\rm in}(\varepsilon_{\rm
eq})}{1+\eta_{M}+f_{\rm in}(\varepsilon_{\rm eq})}$ (40)

in which $\varepsilon_{\rm eq}$ is the equilibrium value of $\varepsilon_{\rm
CGM}$. It is also possible to obtain

$f_{*,{\rm asy}}(\varepsilon_{\rm eq})\>=\>\frac{\varepsilon_{\rm
eq}-\varepsilon_{\rm acc}}{\eta_{E}\varepsilon_{\rm SN}+\varepsilon_{\rm
eq}-(1+\eta_{M})\varepsilon_{\rm loss}(\varepsilon_{\rm eq})}$ (41)

by rearranging equation (38). The latter equation expresses $\varepsilon_{\rm
loss}$ as a function of just $\varepsilon_{\rm eq}$, because the equilibrium
value of $f_{\rm CGM}$ depends only on $\varepsilon_{\rm eq}$, via equations
(39) and (40).

####  4.3.3 Astrophysical Interpretation

Before proceeding with the analysis, it’s worth spending a moment considering
the astrophysical implications of equations (40) and (41). Notice that
equation (40) reflects the amount of recycling implicit in the minimalist
regulator model. For example, if $f_{\rm in}(\varepsilon_{\rm eq})$ is much
less than unity, then only a small fraction of a halo’s gas has cycled through
its central galaxy, and the proportion that has formed stars is $f_{\rm
in}(\varepsilon_{\rm eq})/(1+\eta_{M})$. Conversely, if $f_{\rm
in}(\varepsilon_{\rm eq})$ is much greater than unity, then halo gas typically
cycles through the central galaxy many times, and a fraction $1/(1+\eta_{M})$
forms stars during each cycle. Consequently, a halo’s stellar baryon fraction
approaches unity when $f_{\rm in}(\varepsilon_{\rm eq})$ exceeds $1+\eta_{M}$.

Equation (41) is a quantitative and succinct representation of the physical
reasoning in §4.1 and §4.2. Rewriting it terms of $\xi\equiv(\varepsilon_{\rm
eq}-\varepsilon_{\rm acc})/v_{\rm c}^{2}$ makes its relationship to equation
(28) more transparent:

$f_{*,{\rm asy}}=\frac{\xi v_{\rm c}^{2}}{\eta_{E}\varepsilon_{\rm SN}+\xi
v_{\rm c}^{2}+[\varepsilon_{\rm acc}-(1+\eta_{M})\varepsilon_{\rm
loss}]}\;\;.$ (42)

The term in square brackets accounts for radiative losses, meaning that
equation (41) reduces to equation (28) when that term is small. In that limit,
supernova feedback acts to maintain the expanded state of a galaxy’s
atmosphere as cosmological accretion proceeds. The galaxy’s equilibrium star
formation rate then depends more directly on $\eta_{E}$ than on $\eta_{M}$.
However, the nature of the atmosphere’s equilibrium state qualitatively
changes as radiative energy losses become comparable to supernova energy
input, for the reasons discussed in §4.2.

According to equation (41), the qualitative transition happens as the system
passes through the singular case with $\varepsilon_{\rm eq}=\varepsilon_{\rm
acc}$, which corresponds to a critical value of the mass-loading parameter:

$\eta_{M}=\frac{\eta_{E}\varepsilon_{\rm SN}+\varepsilon_{\rm
acc}-\varepsilon_{\rm loss}(\varepsilon_{\rm acc})}{\varepsilon_{\rm
loss}(\varepsilon_{\rm acc})}\;\;.$ (43)

When mass loading of coupled outflows exceeds this critical value, then a
galaxy’s atmosphere converges toward an equilibrium state with less specific
energy than the specific energy of accreting gas. Supernova heating must
somehow balance atmospheric radiative losses in that state. Furthermore, if we
assume that the specific energy losses necessary for gas to enter a galaxy are
similar to the specific energy of accreting gas, so that $\varepsilon_{\rm
loss}(\varepsilon_{\rm acc})\approx\varepsilon_{\rm acc}$, we then obtain the
approximation

$\eta_{M}\approx\frac{\eta_{E}\varepsilon_{\rm SN}}{\varepsilon_{\rm acc}}$
(44)

corresponding to the vertical line in Figure 1.

###  4.4 Atmosphere Models

So far, we have avoided proposing a physical model for the gas supply function
$\dot{M}_{\rm in}$ because the astrophysical details can be forbiddingly
complex. In preparation for implementing such a physical model, we have
constructed and analyzed an accounting framework—the minimalist regulator
model—that sorts a halo’s baryonic mass and energy into categories and tracks
how mass and energy move from one reservoir to another. While doing that, we
have made few astrophysical assumptions so that the framework broadly applies
to galactic baryon cycles dominated by supernova feedback.555Adding a feedback
term accounting for black-hole energy injection would make it even more
general.

Within this framework, all of the fearsome astrophysical details end up
bundled into the gas supply function $\dot{M}_{\rm in}$ and the energy-loss
function $\varepsilon_{\rm loss}$. Together, those two functions represent an
atmosphere model describing a particular astrophysical scenario, in which the
product $\dot{M}_{\rm in}\varepsilon_{\rm loss}$ is equal to the sum of the
atmosphere’s radiative loss rate $\dot{E}_{\rm rad}$ and the rate
$\dot{E}_{\rm in}$ at which gas flowing into the halo’s central galaxy
transports energy out of the CGM (see §3). Both $\dot{M}_{\rm in}$ and
$\varepsilon_{\rm loss}$ can be specified as functions of $E_{\rm CGM}$,
$M_{\rm CGM}$, and $t$, or equivalently, as functions of $\varepsilon_{\rm
CGM}$, $f_{\rm CGM}$, and $t$. Differing astrophysical scenarios666These could
include cooling flows, cold streams, precipitation models, or galactic
fountains, to name a few examples. can then be tested by inserting various
choices for the $\dot{M}_{\rm in}$ and $\varepsilon_{\rm loss}$ functions into
the minimalist regulator model and integrating it over a particular gas-mass
accretion history $\dot{M}_{\rm acc}(t)$.

Future papers in this series will present specific families of physically
motivated atmosphere models and will analyze how they evolve. Here we will
illustrate the general procedure by examining the properties of a generic
atmosphere model that provides some useful insights.

####  4.4.1 Generic Atmosphere Model

To build intuition about the dependence of a galactic atmosphere’s equilibrium
states on $\eta_{M}$ without having to construct a more complicated model, we
will examine the equilibria and phase plane trajectories of a toy atmosphere
model defined by $\varepsilon_{\rm loss}=\varepsilon_{\rm CGM}$ and

$f_{\rm in}(\varepsilon_{\rm CGM})=\exp\left(\frac{\varepsilon_{\rm
acc}-\varepsilon_{\rm CGM}}{v_{\rm c}^{2}}\right)$ (45)

Our motivation for making $f_{\rm in}$ an exponential function of
$\varepsilon_{\rm CGM}/v_{\rm c}^{2}$ is that the characteristic radius and
dynamical time of an atmosphere in an isothermal potential well are both
proportional to $\exp(\varepsilon_{\rm CGM}/v_{\rm c}^{2})$.

The normalization of the exponential function corresponds to a gas supply rate
that would transport gas into the halo’s central galaxy at a rate
$\dot{M}_{\rm in}\approx\dot{M}_{\rm acc}$ as long as $\varepsilon_{\rm
CGM}\approx\varepsilon_{\rm acc}$ and $f_{\rm CGM}\approx 1$. When feedback
heating exceeds radiative cooling, the gas supply exponentially declines as
$\varepsilon_{\rm CGM}$ rises. Conversely, when radiative cooling exceeds
feedback heating, the gas supply exponentially rises as $\varepsilon_{\rm
CGM}$ falls.

####  4.4.2 The Missing Details

Many readers who have reached this point in the paper may be wondering why
they have not yet seen a radiative cooling function, an assessment of heavy-
element enrichment, a description of atmospheric structure, or a breakdown of
multiphase atmospheric components and their interactions. We have not needed
to specify those details because the equilibrium states of the minimalist
regulator model can be identified and classified without them. Also, the
asymptotic stellar baryon fraction of a halo with an expanded atmosphere does
not depend much on the details, because equilibrium states with
$\varepsilon_{\rm fb}>\varepsilon_{\rm acc}$ are not sensitive to radiative
cooling.

The missing details become more important in contracted atmospheres, because
the gas supply rate $\dot{M}_{\rm in}(\varepsilon_{\rm eq})$ associated with
the atmosphere’s equilibrium state is then closely tied to its radiative
energy loss rate. However, galactic atmospheres in the most recent generation
of cosmological numerical simulations tend to be expanded, not contracted,
with closure radii that can be several times the halo’s virial radius (e.g.,
Ayromlou et al., 2023; Sorini et al., 2022; Wright et al., 2024). It therefore
appears likely that long-term galactic star formation rates depend mostly on
the self-regulating properties of atmospheric expansion, as described in §4.1.

To clarify, radiative cooling is still essential for star formation in a
galaxy at the center of an expanded atmosphere. Its gas supply and long-term
star formation rates are still proportional to the atmosphere’s radiative
cooling rate. But its equilibrium state depends on the interplay between
supernova feedback and cosmological accretion, not radiative cooling.

###  4.5 Sensitivity to $\eta_{M}$

Figure 3: Dependence of equilibrium specific energy $\varepsilon_{\rm eq}$
(solid lines) and asymptotic stellar baryon fraction $f_{*,{\rm asy}}$ (dashed
lines) on the mass loading parameter $\eta_{M}$ in the generic atmosphere
model of §4.4.1. Colors represent supernova feedback strength
$\eta_{E}\varepsilon_{\rm SN}/v_{\rm c}^{2}$ as labeled. A horizontal solid
line marks $\varepsilon_{\rm eq}=\varepsilon_{\rm acc}$. A horizontal dotted
line marks $\varepsilon_{\rm eq}=v_{\rm c}^{2}$. All examples have
$\varepsilon_{\rm acc}=4v_{\rm c}^{2}$. Atmospheres with $\varepsilon_{\rm
eq}$ in the red shaded area have expanded. Atmospheres with $\varepsilon_{\rm
eq}$ in the blue shaded area have contracted. Each line is truncated as
$\varepsilon_{\rm eq}$ drops into the grey region because the generic
atmosphere model breaks at small values of $\varepsilon_{\rm CGM}$.

Now we are ready to consider the role of mass loaded galactic winds. Figure 3
illustrates how the generic atmosphere’s equilibrium specific energy
($\varepsilon_{\rm eq}$) and asymptotic stellar baryon fraction ($f_{*,{\rm
asy}}$) depend on $\eta_{M}$. Four color-coded pairs of lines show their
equilibrium values for a particular choice of supernova feedback strength,
quantified by $\eta_{E}\varepsilon_{\rm SN}/v_{\rm c}^{2}$. Solid lines
represent $\varepsilon_{\rm eq}/\varepsilon_{\rm acc}$, dashed lines represent
$f_{*,{\rm asy}}$, and all examples have $\varepsilon_{\rm acc}=4v_{\rm
c}^{2}$.

As expected, halos with expanded atmospheres (i.e. $\varepsilon_{\rm
eq}>\varepsilon_{\rm acc}$) have stellar baryon fractions that are insensitive
to mass loading. When an atmosphere’s specific energy (solid line) is in the
red region of the figure, the corresponding dashed line representing
$f_{*,{\rm asy}}$ remains nearly flat as $\eta_{M}$ increases.

Also as expected, each of the solid lines falls below $\varepsilon_{\rm
eq}=\varepsilon_{\rm acc}$ and crosses into the blue (contracted) region of
the figure as $\eta_{M}$ surpasses $\eta_{E}\varepsilon_{\rm
SN}/\varepsilon_{\rm acc}$. When that happens, the corresponding value of
$f_{*,{\rm asy}}$ becomes more sensitive to the mass-loading parameter and
sharply rises as $\eta_{M}$ increases.

Each line is then truncated where $\varepsilon_{\rm eq}$ drops into the grey
region below $\varepsilon_{\rm eq}=v_{\rm c}^{2}$. That is where $\dot{M}_{\rm
in}$ becomes insensitive to $\varepsilon_{\rm CGM}$, according to the generic
atmosphere model. Qualitatively, the atmosphere enters a regime in which its
altitude is similar to the galaxy’s size. Ejected gas then falls back into the
galaxy on a timescale similar to the galaxy’s dynamical time. A different
approach for estimating $\dot{M}_{\rm in}$ is needed in this limit because the
CGM and ISM are becoming indistinguishable (see also §5.2).

Some readers may be surprised by how star formation rises as $\eta_{M}$
increases beyond $\eta_{E}\varepsilon_{\rm SN}/\varepsilon_{\rm acc}$, a trend
opposite to the usual intuitive assumptions about mass loading. It comes about
because greater mass loading factors result in outflows that fail to replace
the losses of atmospheric specific energy that lead to star formation (i.e.
$\varepsilon_{\rm fb}<\varepsilon_{\rm acc}$). When an outflow’s specific
energy is less than the atmosphere’s mean specific energy, it reduces the
atmosphere’s mean specific energy and promotes atmospheric contraction,
allowing recycling to be more rapid.

Recycling implied by the generic atmosphere model is exponentially sensitive
to $\varepsilon_{\rm CGM}$ because the recycling rate is proportional to
$f_{\rm in}(\varepsilon_{\rm CGM})$. Consequently, recycling is also
exponentially sensitive to the effects that $\eta_{M}$ has on
$\varepsilon_{\rm CGM}$. As a result, the galaxy’s gas supply $\dot{M}_{\rm
in}$ increases more quickly than $1+\eta_{M}$ as $\eta_{M}$ increases, and so
the steady-state star formation rate $\dot{M}_{*}=\dot{M}_{\rm
in}/(1+\eta_{M})$ also increases.

Many plausible atmosphere models share this qualitative property with the
generic atmosphere model. The key is exponential sensitivity of the gas supply
to $\eta_{M}$. It arises in the generic atmosphere model because the timescale
for recycling scales with the atmosphere’s mean radius, which is exponentially
sensitive to $\varepsilon_{\rm CGM}$. It would also arise in a cooling-flow
model that feeds gas into the central galaxy on a cooling time scale, because
cooling time is inversely proportional to density, and the mean density of a
galactic atmosphere is also exponentially sensitive to $\varepsilon_{\rm
CGM}$. Future papers investigating more sophisticated atmosphere models will
explore these relationships in greater detail.

###  4.6 Convergence toward Equilibrium

Figure 4: Trajectories of generic atmosphere models in the $M_{\rm
CGM}$–$E_{\rm CGM}$ plane. Both panels show models with $\varepsilon_{\rm
acc}/v_{\rm c}^{2}=4$ and $\eta_{E}\varepsilon_{\rm SN}/v_{\rm c}^{2}=10$.
Trajectories in the left panel (solid orange lines) have $\eta_{M}=0$, and
trajectories in the right panel have $\eta_{M}=5$. They all start at $M_{\rm
CGM}=M_{\rm CGM,init}$ with various values of $E_{\rm CGM}$ and converge
toward the dashed green lines along which $E_{\rm CGM}/M_{\rm
CGM}=\varepsilon_{\rm eq}$. Grey lines indicate $\varepsilon_{\rm
CGM}=\varepsilon_{\rm acc}$. Models with less mass loading end up in the red
(expanded) region with $\varepsilon_{\rm CGM}>\varepsilon_{\rm acc}$ Models
more mass loading end up in the blue (contracted) region with
$\varepsilon_{\rm CGM}<\varepsilon_{\rm acc}$.  Figure 5: Trajectories of
generic atmosphere models in the $f_{\rm CGM}$–$\varepsilon_{\rm CGM}$ phase
plane. All examples have $\varepsilon_{\rm acc}/v_{\rm c}^{2}=4$ and
$\eta_{E}\varepsilon_{\rm SN}/v_{\rm c}^{2}=10$. The panels show trajectories
for $\eta_{M}=0$, 1, 2, 3, 5, and 10, as labeled. Those trajectories (orange
lines) converge toward the green dots marking each atmosphere’s equilibrium
configuration. Excess heating leads to equilibria in the red (expanded) region
($\varepsilon_{\rm CGM}>\varepsilon_{\rm acc}$). Excess cooling leads to
equilibria in the blue (contracted) region ($\varepsilon_{\rm
CGM}<\varepsilon_{\rm acc}$).

We will conclude §4 with a look at the convergence properties of the
minimalist regulator model. Figure 4 shows model trajectories in the $M_{\rm
CGM}$–$E_{\rm CGM}$ plane for two different values of $\eta_{M}$. Each
trajectory comes from integrating a reduced version of the minimalist
regulator model that incorporates the generic atmosphere model of §4.4.1. The
integrations begin with various values of $E_{\rm CGM}$ and a circumgalactic
medium comprising all of the halo’s baryons, so that $f_{\rm CGM}=1$. All
models have $\varepsilon_{\rm acc}/v_{\rm c}^{2}=4$ and
$\eta_{E}\varepsilon_{\rm SN}/v_{\rm c}^{2}=10$. The dimensionless quantity
$\eta_{E}\varepsilon_{\rm SN}/v_{\rm c}^{2}$ determines how effectively
supernova feedback lifts gas within the halo’s potential well. The chosen
value of that quantity corresponds to a galaxy with $\eta_{E}\approx 1$ and a
potential well similar to the Milky Way’s.

Models with negligible mass loading ($\eta_{M}=0$) converge toward an
equilibrium value of atmospheric specific energy ($\varepsilon_{\rm eq}$)
exceeding the specific energy of accreting gas ($\varepsilon_{\rm acc}$)
because cumulative supernova heating of the atmosphere exceeds cumulative
radiative cooling. Those models consequently result in expanded atmospheres.
In contrast, the models with substantial mass loading ($\eta_{M}=5$) converge
toward an equilibrium specific energy smaller than $\varepsilon_{\rm acc}$,
because cumulative radiative cooling exceeds cumulative supernova heating,
resulting in a contracted atmosphere.

How an atmosphere model’s convergence toward equilibrium depends on $\eta_{M}$
is easier to appreciate in a phase plane with $f_{\rm CGM}$ on one axis and
$\varepsilon_{\rm CGM}$ on the other. An equilibrium state of the minimalist
regulator model is then a fixed point rather than a line. Figure 5 illustrates
six examples based on the generic atmosphere model of §4.4.1. All of them have
$\varepsilon_{\rm acc}/v_{\rm c}^{2}=4$ and $\eta_{E}\varepsilon_{\rm
SN}/v_{\rm c}^{2}=10$ but differing values of the mass loading parameter
$\eta_{M}$. As in Figure 4, each orange line represents an integration of a
reduced version of the minimalist regulator model incorporating the generic
atmosphere model.

A green dot in each panel marks the model’s equilibrium point, at which
$\varepsilon_{\rm CGM}$ and $f_{\rm CGM}$ remain constant, as long as the
halo’s accretion rate remains constant. Far from equilibrium, each example
evolves more rapidly in the vertical direction than in the horizontal
direction, meaning that either net heating or net cooling causes the
atmosphere’s specific energy to change more rapidly than the ratio of galactic
baryons to circumgalactic baryons. Once an atmosphere model is near its
equilibrium value of $\varepsilon_{\rm CGM}$, it converges horizontally toward
its equilibrium value of $f_{\rm CGM}$.

Mass loading is relatively modest in the upper row of examples, with
$\eta_{M}=0$, 1, and 2. Those examples have equilibrium points in the red
(expanded) region of each panel, with $\varepsilon_{\rm CGM}>\varepsilon_{\rm
acc}$, meaning that cumulative feedback heating exceeds cumulative radiative
cooling. In those examples the equilibrium value of the asymptotic stellar
baryon fraction ($f_{*,{\rm asy}}=1-f_{\rm CGM}\approx 0.2$) is insensitive to
the mass loading parameter.

However, further increases of $\eta_{M}$ reduce the proportion of accreted
baryons that remain circumgalactic and therefore increase the halo’s
asymptotic stellar baryon fraction. As $\eta_{M}$ increases from 2 to 10, the
atmospheric equilibrium point drops into the blue (contracted) region with
$\varepsilon_{\rm CGM}<\varepsilon_{\rm acc}$, meaning that cumulative
radiative cooling exceeds cumulative feedback heating. The asymptotic stellar
baryon fraction of the equilibrium state simultaneously rises, exceeding 50%
in the example with $\eta_{M}=10$.

##  5 Uncoupled Outflows and Circulation

All of the predictions discussed in §3 and §4 pertain to coupled supernova-
driven outflows that deposit energy into the circumgalactic medium. For
completeness, this section considers how mass loaded outflows affect star
formation if they are not coupled with the rest of a galaxy’s atmosphere and
therefore cannot reduce the galaxy’s gas supply. In particular, it focuses on
why uncoupled outflows with large mass-loading factors can result in excessive
recycling.

###  5.1 Recycling and Star Formation Rate

When recycling of a galaxy’s atmosphere happens, the relationship between
$\dot{M}_{*}$ and $\eta_{M}$ depends on the number of times baryons typically
cycle through the galaxy ($N_{\rm cyc}$). To make this statement more
quantitative, let us suppose that $\dot{M}_{\rm in,0}$ is the rate at which
halo gas enters a galaxy for the first time, fueling star formation that
ejects a proportion $\eta_{M}/(1+\eta_{M})$ of that gas. If the ejected gas
eventually falls back into the galaxy, its infall rate has been reduced by the
factor $\eta_{M}/(1+\eta_{M})$. After $n$ cycles, it has been reduced by the
factor $[\eta_{M}/(1+\eta_{M})]^{n}$. The total gas infall rate at time $t$ in
this idealized recycling model is therefore

$\dot{M}_{\rm in}(t)=\sum_{n=0}^{N_{\rm
cyc}}\left(\frac{\eta_{M}}{1+\eta_{M}}\right)^{n}\dot{M}_{\rm in,0}(t-nt_{\rm
cyc})\;\;,$ (46)

where $t_{\rm cyc}=t/N_{\rm cyc}$ is the time to complete one cycle.

Now suppose the initial gas supply rate $\dot{M}_{\rm in,0}$ is approximately
constant with time, so it can be taken out of the sum, leaving the polynomial
$1+x+...+x^{N_{\rm cyc}}$, where $x=\eta_{M}/(1+\eta_{M})$. In that case, the
sum reduces to $(1-x^{N_{\rm cyc}+1})/(1-x)$, leading to

$\dot{M}_{\rm
in}\approx(1+\eta_{M})\left[1-\left(\frac{\eta_{M}}{1+\eta_{M}}\right)^{N_{\rm
cyc}+1}\right]\dot{M}_{\rm in,0}\;\;.$ (47)

Dividing by $1+\eta_{M}$ then gives the steady-state star formation rate that
the galaxy asymptotically approaches:

$\dot{M}_{*}\>\approx\>\left[1-\left(\frac{\eta_{M}}{1+\eta_{M}}\right)^{N_{\rm
cyc}+1}\right]\dot{M}_{\rm in,0}\;\;.$ (48)

Notice that $\dot{M}_{*}$ converges toward $\dot{M}_{\rm in,0}$ when $N_{\rm
cyc}$ is sufficiently large, because excessive recycling ends up turning a
galaxy’s entire gas supply into stars, if given enough time.

This expression can be further simplified when $N_{\rm cyc}\ll\eta_{M}$. The
sum then reduces to $\dot{M}_{\rm in}\approx(N_{\rm cyc}+1)\dot{M}_{\rm
in,0}$, giving

$\frac{\dot{M}_{*}}{\dot{M}_{{\rm in},0}}\>\approx\>\frac{N_{\rm
cyc}+1}{1+\eta_{M}}\;\;,$ (49)

because each of the $N_{\rm cyc}+1$ terms in the sum is close to unity. This
result supports the more intuitive argument presented in §2 and may seem to
support the simple picture of decreasing star formation by increasing mass
loading. However, we still need to account for the hidden dependence of
$N_{\rm cyc}$ on $\eta_{M}$. The next section presents an estimate for that
relation.

###  5.2 Ballistic Circulation

Here we will consider uncoupled outflows that undergo what we will call
ballistic circulation. During ballistic circulation, a clump of ejected gas
transfers little to none of its ejection energy to the rest of the atmosphere.
Each clump then follows its own nearly ballistic trajectory, reaching a
maximum altitude determined by the specific energy

$\varepsilon_{\rm fb}=\frac{\eta_{E}\varepsilon_{\rm SN}}{\eta_{M}}$ (50)

it had upon leaving the galaxy. Given enough time, those clumps fall back
toward the galaxy and augment its gas supply, potentially entering and leaving
the galaxy several times if the outflows are significantly mass loaded.777Some
astronomers call this feedback mode “juggling.”

Ejected gas cannot formally escape the isothermal potential well we have
adopted, because $\varphi(r)$ logarithmically approaches infinity as $r$
increases. However, gas ejected with a specific energy much greater than the
gas currently accreting onto the halo does not fall back down on a timescale
short enough to be recycled, for reasons discussed in §2. Consequently,
ballistic gas in outflows with

$\eta_{M}>\frac{\eta_{E}\varepsilon_{\rm SN}}{\varepsilon_{\rm acc}}$ (51)

cycles back into a halo’s central galaxy, and ballistic outflows with less
mass loading decouple from the central galaxy’s gas supply.

In a nearly isothermal potential well, the timescale $t_{\rm cyc}$ for
ballistic recycling is approximately proportional to the maximum radius of a
ballistic trajectory with specific energy $\varepsilon_{\rm fb}$, which is
$\sim r_{0}\exp(\varepsilon_{\rm fb}/{v_{\rm c}^{2}})$. We therefore conclude
that

$N_{\rm cyc}=\frac{t}{t_{\rm cyc}}\sim\exp\left(\frac{\varepsilon_{\rm
acc}-\varepsilon_{\rm fb}}{v_{\rm c}^{2}}\right)$ (52)

for recycling within a nearly isothermal halo.

With this approximation for $N_{\rm cyc}$, we can take the derivative of
equation (49) with respect to $\eta_{M}$, finding

$\frac{d}{d\eta_{M}}\frac{\dot{M}_{*}}{\dot{M}_{\rm
in,0}}\>\sim\>\frac{1}{1+\eta_{M}}\left(\frac{N_{\rm
cyc}}{\eta_{M}}\frac{\varepsilon_{\rm fb}}{v_{\rm c}^{2}}-\frac{N_{\rm
cyc}+1}{1+\eta_{M}}\right)\;\;.$ (53)

The derivative is negative, implying that increased mass loading lowers the
galaxy’s star formation rate, as long as

$N_{\rm cyc}\lesssim\left(\frac{1+\eta_{M}}{\eta_{M}}\frac{\varepsilon_{\rm
fb}}{v_{\rm c}^{2}}-1\right)^{-1}\;\;.$ (54)

Using equation (52), we see that the derivative changes sign near where

$\displaystyle\varepsilon_{\rm fb}\approx\varepsilon_{\rm acc}+v_{\rm
c}^{2}\ln\left(\frac{1+\eta_{M}}{\eta_{M}}\frac{\varepsilon_{\rm fb}}{v_{\rm
c}^{2}}-1\right)\;\;.$ (55)

Below this critical value of specific feedback energy, further increases of
$\eta_{M}$ cause the galaxy’s steady-state star formation rate to increase.
That happens because the galaxy’s gas supply rate, enhanced by recycling,
increases more rapidly than $1+\eta_{M}$.

According to this approximation, recycling-driven enhancement of the galaxy’s
gas supply saturates as $\varepsilon_{\rm fb}$ approaches $v_{\rm c}^{2}$. But
heavily mass-loaded galactic outflows don’t propagate very far in that limit.
They reach a maximum altitude $\lesssim 3r_{0}$ and return to the galaxy on a
timescale similar to the galaxy’s rotation period. Such outflows are actually
low-altitude galactic fountains. If most of the gas accreting onto a halo
enters its central galaxy, perhaps through cold cosmological streams, then
almost all of it ends up circulating at radii similar to the galaxy’s radius.

Such a recycling crisis would make the predicted star formation rate several
times greater than the minimum value reached at
$\eta_{M}\approx\eta_{E}\varepsilon_{\rm SN}/\varepsilon_{\rm acc}$ (see
§5.3). In a halo suffering from a recycling crisis, the majority of the halo’s
baryons would be concentrated near its central galaxy. The rarity of low-
redshift galaxies observed to have that property therefore indicates that
ballistic circulation in the present-day universe, if it happens, results from
outflows with $\eta_{M}<\eta_{E}\varepsilon_{\rm SN}/v_{\rm c}^{2}$.

###  5.3 Comparison with Coupled Outflows

This schematic analysis of uncoupled outflows illustrates why excessive mass
loading results in a crisis regardless of whether supernova-driven outflows
from galaxies are coupled or uncoupled. In both cases, feedback with
$\eta_{M}>\eta_{E}\varepsilon_{\rm SN}/\varepsilon_{\rm acc}$ fails to push a
halo’s baryons beyond its virial radius. A crisis becomes inevitable because
galactic gas returning to the circumgalactic medium has less specific energy
than it had when it accreted onto the halo. Under those circumstances, the net
effect of radiative cooling, star formation, and supernova feedback is a
reduction in atmospheric energy that allows the atmosphere to contract. If no
other form of feedback intervenes, then the resulting increase in atmospheric
density and reduction in atmospheric dynamical time combine to boost the
central galaxy’s gas supply and star formation rate. Larger values of
$\eta_{M}$ just make the crisis worse. That is why both curves in the bottom
panel of Figure 1 rise toward larger $f_{*}$ as $\eta_{M}$ rises beyond
$\eta_{E}\varepsilon_{\rm SN}/\varepsilon_{\rm acc}$

However, those curves diverge as $\eta_{M}$ drops below
$\eta_{E}\varepsilon_{\rm SN}/\varepsilon_{\rm acc}$. Uncoupled outflows
become less effective at suppressing star formation because feedback energy
becomes concentrated in a smaller proportion of the halo’s baryons and fails
to suppress the galaxy’s gas supply. Coupled outflows, on the other hand,
become more effective because they are able to push more of a halo’s baryons
out to larger radii. Paper II builds upon those findings to explain how
different cosmological simulations can successfully produce similar galaxy
populations while making starkly different predictions for CGM properties.

##  6 Summary

This paper has presented a new regulator model for galaxy evolution that
prioritizes simplicity and generality over complexity and detail. Its core is
an accounting system—the minimalist regulator model—that tracks all of the
mass and energy associated with a halo’s baryons, regardless of where they are
located (§3). That approach enables the model to represent expansion and
contraction of a galaxy’s circumgalactic atmosphere driven by imbalances
between feedback heating and radiative cooling.

Here are the most important points that emerge from our analysis of that
simple model:

  1. 1.

Supernova feedback’s effects on galaxy evolution depend on whether the
outflows that supernovae generate are coupled to the CGM (§2). The minimalist
regulator model applies to coupled outflows that share their energy with the
rest of a galaxy’s atmosphere.

  2. 2.

Increasing the mass loading parameter $\eta_{M}$ in the minimalist regulator
model reduces the net atmospheric energy change coming from the supernova
feedback loop (§3.8). That happens because the specific energy of a galactic
wind ($\varepsilon_{\rm fb}=\eta_{E}\varepsilon_{\rm SN}/\eta_{M}$) is smaller
if a given amount of feedback energy is spread over a greater gas mass.

  3. 3.

Whether or not a galaxy’s atmosphere expands or contracts in response to
supernova feedback depends on how the atmosphere’s radiative cooling rate
compares with the rate at which supernova feedback adds energy to the
atmosphere (§4). The outcome hinges on whether the specific energy of gas
returning to the atmosphere ($\varepsilon_{\rm fb}$) exceeds the specific
energy it had when it first accreted onto the halo ($\varepsilon_{\rm acc}$).
Large values of the mass loading parameter that reduce $\varepsilon_{\rm fb}$
below $\varepsilon_{\rm acc}$ result in atmospheric contraction. Smaller
proportions of mass loading result in atmospheric expansion.

  4. 4.

Expanding galactic atmospheres asymptotically approach an equilibrium state in
which supernova energy input keeps the atmosphere’s specific energy constant
as cosmological accretion adds atmospheric mass (§4.1). A galaxy’s star
formation rate in that asymptotic equilibrium state is

$\dot{M_{*}}\approx\left(\frac{\xi v_{\rm c}^{2}}{\eta_{E}\varepsilon_{\rm
SN}+\xi v_{\rm c}^{2}}\right)\dot{M}_{\rm acc}$ (56)

in which $\xi$ is a model dependent dimensionless factor of order unity. The
stellar baryon fraction of the galaxy’s halo is insensitive to $\eta_{M}$
because the atmospheric energy input coming from supernova feedback is more
consequential than ejection of galactic gas.

  5. 5.

Contracting galactic atmospheres can come into equilibrium, at least in
principle, if supernova heating somehow manages to balance radiative cooling
of the atmosphere (§4.2). The halo’s equilibrium star formation rate then
depends on the atmosphere’s radiative cooling rate. However, such a state of
balance may be precarious because of the complex dependence of radiative
cooling on atmospheric structure and enrichment.

  6. 6.

Galactic outflows that do not share their energy with the rest of the CGM,
which we have called uncoupled outflows, cannot suppress the gas supply
flowing from the CGM into a halo’s central galaxy. They must regulate star
formation in that galaxy by ejecting gas from it. However, uncoupled outflows
with large proportions of mass loading cannot escape the halo and fall back
toward the central galaxy if $\varepsilon_{\rm fb}<\varepsilon_{\rm acc}$
(§5). Recycling of ejected gas that returns to the central galaxy boosts its
star formation rate. The long-term star formation rate of a galaxy with
uncoupled outflows then increases as $\eta_{M}$ rises.

Some of these findings may seem obvious in retrospect, but they were not all
obvious to the authors of this paper when we began to write it and are not
universally appreciated within the community studying supernova feedback.
Therein lie the benefits of a simple model.

Paper II discusses how the results of this paper help to explain differences
among the CGM predictions made by current cosmological simulations. Future
papers in this series will add more astrophysical content to the
framework.888The next paper in the series (Pandya et al., in preparation) will
call it the ExpCGM framework, short for Expandable CircumGalactic Medium. Our
objective will be to learn how various assumptions about the structure and
composition of galactic atmospheres and their halos affect model predictions
for both galaxy evolution and observable features of the CGM. The framework’s
most distinctive feature is its focus on expansion/contraction rather than
heating/cooling. This paper has tried to explain why expansion and contraction
of a galaxy’s atmosphere are more fundamental to the feedback loop than
balance between heating and cooling.

Meanwhile, we invite the community to use the minimalist regulator model to
interpret both observations and simulations of relationships between galaxy
evolution and the CGM. In particular, we are hoping that phase-plane analyses
like the idealized one shown in Figure 5 will help to reveal how the star-
formation histories of galaxies respond to the impact of feedback on their
atmospheres.999Future versions of these phase-plane plots may feature a
logarithmic axis showing $1-f_{\rm CGM}=(M_{*}+M_{\rm ISM})/f_{\rm b}M_{\rm
halo}$ instead of a linear axis showing $f_{\rm CGM}$, so that results for
low-mass halos with $f_{*}\ll 1$ are easier to visualize.

GMV acknowledges support from the NSF through grant AAG-2106575 and also
benefited from the Turbulence in Astrophysical Environments program, supported
in part by grant NSF PHY-2309135 to the Kavli Institute for Theoretical
Physics (KITP). NASA provides support for VP through the NASA Hubble
Fellowship grant HST-HF2-51489 awarded by the Space Telescope Science
Institute, which is operated by the Association of Universities for Research
in Astronomy, Inc., for NASA, under contract NAS5-26555. GLB acknowledges
support from the NSF (AST-2108470, ACCESS PHY2400043, AST-2307419), NASA TCAN
award 80NSSC21K1053, the Simons Foundation (grant 822237) and the Simons
Collaboration on Learning the Universe. The Flatiron Institute is a division
of the Simons Foundation. MD is grateful for partial support of this work from
NASA award NASA-80NSSC22K0476. BDO’s contribution was supported by Chandra
Grant TM2-23004X

##  Appendix A Descriptions of Symbols

To help readers keep track of the many symbols used in the paper, Table 1
provides a glossary, including the place where each symbol first appears.
Also, the following list groups some subsets of those symbols into categories.

  * •

Baryonic Energy Accounting: $E_{\rm CGM}$, $E_{\varphi}$, $E_{\rm th}$,
$E_{\rm nt}$

  * •

Baryonic Mass Accounting: $M_{\rm acc}$, $M_{\rm CGM}$, $M_{\rm ISM}$, $M_{*}$

  * •

Baryonic Mass Exchange Rates: $\dot{M}_{\rm acc}$, $\dot{M}_{\rm CGM}$,
$\dot{M}_{\rm in}$, $\dot{M}_{\rm ISM}$, $\dot{M}_{*}$

  * •

Cosmological Halo Description: $M_{\rm halo}$, $r_{0}$, $v_{\rm c}$,
$\varphi$, $\dot{\varphi}$

  * •

Equilibrium State Analysis: $f_{*,{\rm asy}}$, $\delta f_{*}$,
$\varepsilon_{\rm eq}$, $\xi$

  * •

Galactic Atmosphere Model Description: $\dot{E}_{\rm in}$, $\dot{E}_{\rm
rad}$, $f_{\rm CGM}$, $f_{\rm in}$, $\dot{M}_{\rm in}$, $\varepsilon_{\rm
CGM}$, $\varepsilon_{\rm in}$, $\varepsilon_{\rm loss}$, $\varepsilon_{\rm
rad}$

  * •

Minimalist Regulator Model Parameters: $t_{\rm SF}$, $\eta_{E}$,
$\varepsilon_{\rm SN}$, $\eta_{M}$

  * •

Recycling Model Description: $\dot{M}_{\rm in,0}$, $N_{\rm cyc}$, $t_{\rm
cyc}$

Table 1: Glossary of Symbols

Symbol | Description | First Used
---|---|---
$E_{\rm CGM}$ | Total CGM energy: $E_{\varphi}+E_{\rm th}+E_{\rm nt}$ | Eq. 2
$\dot{E}_{\rm fb}$ | Feedback energy output rate ($\eta_{E}\varepsilon_{\rm SN}\dot{M}_{*}$ for SN feedback) | Eq. 3
$\dot{E}_{\rm in}$ | Rate at which CGM loses energy as baryons move from CGM into ISM | Eq. 3
$E_{\rm nt}$ | Non-thermal CGM energy (including kinetic energy) | Eq. 2
$\dot{E}_{\rm rad}$ | Radiative energy loss rate from the CGM | Eq. 3
$E_{\rm th}$ | Thermal CGM energy | Eq. 2
$E_{\varphi}$ | Gravitational potential energy of the CGM | Eq. 2
$f_{\rm b}$ | Cosmic baryon mass fraction | §2
$f_{\rm CGM}$ | Fraction of accreted baryons in CGM: $M_{\rm CGM}/M_{\rm acc}$ | Eq. 35
$f_{\rm in}$ | Dimensionless function describing how $\dot{M}_{\rm in}$ depends on $\varepsilon_{\rm CGM}$ | Eq. 35
$f_{*}$ | Fraction of a halo’s baryons in stars | §1
$f_{*,{\rm asy}}$ | Asymptotic stellar baryon fraction: $\dot{M}_{\rm in}/(1+\eta_{M})\dot{M}_{\rm acc}$ | Eq. 30
$M_{\rm acc}$ | Total mass of accreted baryons | Eq. 1
$M_{\rm CGM}$ | Total CGM mass (including baryons pushed beyond $R_{\rm halo})$ | Eq. 1
$M_{\rm halo}$ | Mass of a galaxy’s cosmological halo | §1
$M_{\rm ISM}$ | Mass of a galaxy’s interstellar medium | Eq. 1
$\dot{M}_{\rm in}$ | Gas supply rate at which baryons flow from CGM into ISM | Eq. 6
$\dot{M}_{\rm in,0}$ | Rate at which baryons flow from CGM into ISM for the first time | §5
$\dot{M}_{\rm wind}$ | Mass outflow rate of galactic wind | §1
$M_{*}$ | Central galaxy’s stellar mass | Eq. 1
$\dot{M}_{*}$ | Central galaxy’s star formation rate | §1
$N_{\rm cyc}$ | Number of times a halo’s baryons cycle through the central galaxy | §2
$r_{0}$ | Inner radius of CGM, at which $\varphi=0$ | Eq.5
$R_{\rm closure}$ | Closure radius within which baryon mass fraction equals $f_{\rm b}$ | §2
$R_{\rm halo}$ | Halo (virial) radius | §2
$t$ | Cosmic time | §2
$t_{\rm cyc}$ | Recycling timescale for halo baryons | §2
$t_{\rm SF}$ | Star formation timescale for ISM: $M_{\rm ISM}/\dot{M}_{*}$ | §3.9
$v_{\rm c}$ | Circular velocity of cosmological halo | §2
$\delta f_{*}$ | Difference between $\dot{M}_{*}/\dot{M}_{\rm acc}$ and $f_{*,{\rm asy}}$ | §4.3
$\varepsilon_{\rm acc}$ | Specific energy of accreting baryons (including potential energy) | §2
$\varepsilon_{\rm CGM}$ | Specific energy of CGM: $E_{\rm CGM}/M_{\rm CGM}$ | §4.1
$\varepsilon_{\rm eq}$ | Specific energy of CGM in an equilibrium state | §4.3
$\varepsilon_{\rm fb}$ | Specific energy of galactic wind before coupling with CGM | §2
$\varepsilon_{\rm in}$ | Specific energy of baryons entering the central galaxy: $\dot{E}_{\rm in}/\dot{M}_{\rm in}$ | Eq. 11
$\varepsilon_{\rm loss}$ | Specific energy loss associated with the galaxy’s gas supply: $(\dot{E}_{\rm rad}+\dot{E}_{\rm in})/\dot{M}_{\rm in}$ | Eq. 12
$\varepsilon_{\rm rad}$ | Specific radiative energy loss associated with the galaxy’s gas supply: $\dot{E}_{\rm rad}/\dot{M}_{\rm in}$ | Eq. 10
$\varepsilon_{\rm SN}$ | Specific supernova energy output from a stellar population | §2
$\eta_{E}$ | Energy loading parameter: $\dot{E}_{\rm fb}/\dot{M}_{*}\varepsilon_{\rm SN}$ | §2
$\eta_{M}$ | Mass loading parameter: $\dot{M}_{\rm wind}/\dot{M}_{*}$ | §1
$\xi$ | Model-dependent dimensionless energy difference: $\xi=(\varepsilon_{\rm eq}-\varepsilon_{\rm acc})/v_{\rm c}^{2}$ | §4.3
$\tau$ | Dimensionless gas depletion parameter | §3.9
$\varphi$ | Gravitational potential of cosmological halo (with zero point at $r_{0}$) | Eq. 5
$\dot{\varphi}$ | Rate of change in the gravitational potential | Eq. 4

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