Equilibrium States of Galactic Atmospheres II: Interpretation and ImplicationsJournal: ApJ

Whether or not a galactic atmosphere ultimately expands or contracts depends on how $\varepsilon_{\rm CGM}$ compares with the mass weighted value of $\varepsilon_{\rm acc}$, averaged over a halo’s cosmic mass accretion history. If radiative cooling, galaxy formation, and feedback had never happened, then $\varepsilon_{\rm CGM}$ would be nearly equal to that average value, which is assigned to a halo’s baryons early in time through cosmological initial conditions. A galactic atmosphere’s current value of $\varepsilon_{\rm CGM}$ therefore reflects how radiative cooling, galaxy formation, and energy input from galactic feedback alter the initial specific energy of the halo’s baryons.

If cumulative feedback energy input exceeds cumulative radiative cooling, then $\varepsilon_{\rm CGM}$ exceeds $\varepsilon_{\rm acc}$, and the atmosphere’s equilibrium state is expanded relative to what its purely cosmological state would have been (i.e. without any radiative cooling or feedback). If instead cumulative radiative cooling exceeds cumulative feedback energy input, then the atmosphere’s equilibrium state is contracted relative to its purely cosmological state.

This paper’s discussion of atmospheric expansion and contraction assumes $\varepsilon_{\rm acc}\approx 4v_{\rm c}^{2}$, where $v_{\rm c}$ is the maximum circular velocity of the halo’s gravitational potential well.³³ 3 Paper I chooses the zero point of the halo’s gravitational potential to be at its central galaxy’s outer boundary, so that baryons have zero potential energy when going from the CGM into the ISM. Because of that choice, $\varepsilon_{\rm acc}$ is similar to the gravitational potential at the halo’s virial radius. The factor of 4 reflects the typical depth of a Navarro-Frenk-White potential well (Navarro et al. 1997). Getting out to the halo’s virial radius requires $\approx 3.5v_{\rm c}^{2}$, depending on how concentrated the halo’s mass is toward its center, and that is the specific potential energy that accreting matter has at the halo’s virial radius. The specific kinetic energy associated with infall to the virial radius is $\approx 0.5v_{\rm c}^{2}$, and the sum is $\varepsilon_{\rm acc}\approx 4v_{\rm c}^{2}$.

The minimalist regulator model of Paper I assumes that outflows of feedback energy are coupled with the CGM. For example, star formation at the rate $\dot{M}_{*}$ drives an outflow that transports ISM gas into the CGM at the rate

where $\eta_{M}$ is the model’s mass-loading parameter. The outflow then adds energy to the CGM at the rate

where $\varepsilon_{\rm SN}$ is the supernova energy output per unit mass of star formation and $\eta_{E}$ is the model’s energy-loading parameter, describing the fraction of supernova energy that couples with the CGM.

At the opposite extreme are uncoupled outflows that eject gas from a galaxy without transferring any of their energy or mass to the rest of the CGM. Whether or not the ejected gas returns to the galaxy depends on the outflow’s specific energy

Paper I briefly explored such uncoupled outflows, showing that they do not return to the galaxy for $\varepsilon_{\rm fb}\gg\varepsilon_{\rm acc}$ and lead to excessive recycling for $\varepsilon_{\rm fb}\ll\varepsilon_{\rm acc}$. This paper will mostly focus on coupled outflows except where specifically indicated.

Radiative cooling enables accreted gas to accumulate in a halo’s central galaxy.⁴⁴ 4 Accreting baryons need to shed a specific energy similar to $\varepsilon_{\rm acc}$ before they can enter the central galaxy and remain there. In Paper I, $\dot{M}_{\rm in}$ is the gas supply rate at which baryons pass from the CGM into the central galaxy’s ISM, and $\dot{E}_{\rm loss}$ is the rate of atmospheric energy loss. With those losses included, the net heating rate associated with star formation and supernova feedback is

If $\dot{M}_{\rm in}$ is steady and interstellar gas forms stars on a relatively short timescale, then $\dot{M}_{*}\approx\dot{M}_{\rm in}/(1+\eta_{M})$, and the net heating rate can be expressed as

where $\varepsilon_{\rm loss}\equiv\dot{E}_{\rm loss}/\dot{M}_{\rm in}$. Whether or not the supernova feedback loop results in a net gain or net loss of atmospheric energy therefore depends on how $\eta_{E}\varepsilon_{\rm SN}$ compares with $(1+\eta_{M})\varepsilon_{\rm loss}$.

The minimalist regulator model assumes that $\dot{M}_{\rm in}$ depends on $E_{\rm CGM}$ and $M_{\rm CGM}$. Here we will also assume that $\dot{M}_{\rm in}$ depends on a potentially large vector of parameters $\mathbf{q}=(q_{1},q_{2},...,q_{N})$ describing the structure of the CGM, the cosmological accretion rate, and the halo’s gravitational potential, so that the function representing the galaxy’s gas supply can be expressed as

When determining the feedback system’s equilibrium states, Paper I assumes that

so that increasing the atmosphere’s specific energy reduces the galaxy’s gas supply, as long as the model’s parameters do not change.

Now the stage is set for describing the minimalist regulator model’s equilibrium states. They are the states in which both $\varepsilon_{\rm CGM}$ and $f_{\rm CGM}$ remain constant, as long as the parameter set $\mathbf{q}$ remains unchanged. Under those conditions, the galaxy’s star formation rate converges toward

as long as the star formation timescale $t_{\rm SF}=M_{\rm ISM}/\dot{M}_{*}$ of the ISM is sufficiently short. In such an equilibrium state, $f_{\rm CGM}$ stays constant as long as $\dot{M}_{\rm in}$ is proportional to $\dot{M}_{\rm halo}$. Paper I shows that the halo’s stellar baryon fraction $f_{*}\equiv M_{*}/f_{\rm b}M_{\rm halo}$ then asymptotically approaches

in which $\varepsilon_{\rm eq}$ is the equilibrium value of $\varepsilon_{\rm CGM}$.

There are two distinct families of equilibrium states. One has $\varepsilon_{\rm eq}>\varepsilon_{\rm acc}$ and corresponds to atmospheres that have expanded, because cumulative feedback energy input exceeds cumulative radiative losses. The other has $\varepsilon_{\rm eq}<\varepsilon_{\rm acc}$ and corresponds to atmospheres that have contracted, because cumulative radiative losses exceed cumulative feedback energy input.

At any given moment, feedback is pushing a galaxy and its atmosphere toward one of those equilibrium states. For example, consider a particular model for $\dot{M}_{\rm in}$ that depends on a parameter set $\mathbf{q}$. As long as $\mathbf{q}$ remains constant, star formation asymptotically converges toward a value of $\dot{M}_{*}$ that depends on $\mathbf{q}$.

However, the parameter vector $\mathbf{q}$ is likely to fluctuate in response to variations in cosmological accretion, CGM structure, the halo’s gravitational potential, and perhaps other factors. The atmosphere’s equilibrium state therefore also fluctuates, meaning that the atmosphere’s trajectory in the $f_{\rm CGM}$–$\varepsilon_{\rm CGM}$ phase plane wanders around rather than steadily converging toward a single fixed point, as it does in Figure 5 of Paper I. Nevertheless, the trajectory remains close to the set of fixed points corresponding to the range of fluctuations in $\mathbf{q}$. Future papers in this series will explore those fluctuations and a galaxy’s responses to them in more depth.

To conclude our discussion of Paper I’s findings, we will consider their implications for how observations might be able to determine whether supernova feedback is primarily ejective or primarily preventative:

1. 1.

   Preventative. Supernova feedback in Paper I’s minimalist regulator model is primarily preventative, because the model assumes that supernova-driven feedback outflows couple with the CGM and add their energy to it. The left panel of Figure 1 shows how the model’s star-formation predictions respond to mass loading, via the specific energy $\varepsilon_{\rm fb}$, using Paper I’s “generic atmosphere” formula for $\dot{M}_{\rm in}$. Purple lines represent how a halo’s asymptotic stellar baryon fraction $f_{*,{\rm asy}}$ depends on the specific energy ratio $\varepsilon_{\rm fb}/v_{\rm c}^{2}$. Notice that they are nearly flat for $\varepsilon_{\rm fb}>\varepsilon_{\rm acc}$. In that regime, preventative supernova feedback reduces a galaxy’s gas supply by expanding its atmosphere, making star formation insensitive to $\eta_{M}$ for reasons explained in Paper I and summarized in §II.5.1. However, the purple lines rapidly rise once $\varepsilon_{\rm fb}$ drops below $\varepsilon_{\rm acc}$, because then supernova feedback cannot prevent atmospheric contraction, which sharply increases the galaxy’s gas supply and star formation rate.

2. 2.

   Ejective. The right-hand panel of Figure 1 presents a contrasting toy model from Paper I, the “ballistic circulation” model, in which supernova-driven outflows are uncoupled, meaning that they do not share their energy with the rest of the CGM. That feedback mode is purely ejective because it does nothing to prevent halo gas from entering the galaxy. Orange lines represent how the model’s star-formation predictions depend on $\varepsilon_{\rm fb}$. All of those lines have a minimum near where $\varepsilon_{\rm fb}=\varepsilon_{\rm acc}$. Star formation increases toward greater values of $\varepsilon_{\rm fb}$ because galactic outflows eject less gas when $\eta_{M}$ is smaller. Star formation increases toward smaller values of $\varepsilon_{\rm fb}$ because of recycling: Ejected gas ultimately falls back into the galaxy, and reducing $\varepsilon_{\rm fb}$ shortens the recycling timescale, thereby increasing the galaxy’s overall gas supply. Figure 1 therefore implies that ejective feedback needs to have $\varepsilon_{\rm fb}\approx\varepsilon_{\rm acc}$ in order to be optimally effective.

Comparing those two alternatives suggests that measurements showing how $\varepsilon_{\rm fb}$ depends on $v_{\rm c}^{2}$ might help distinguish between preventative and ejective feedback modes. Such measurements of both simulated and real galactic outflows need to include all forms of energy contributing to $\varepsilon_{\rm fb}$, including the thermal energy of a hot component that might be difficult to observe. Values of $\varepsilon_{\rm fb}$ consistently exceeding $\varepsilon_{\rm acc}\approx 4v_{\rm c}^{2}$ would signify modes of feedback that need to be preventative in order to be effective. Observations consistently implying $\varepsilon_{\rm fb}\approx 4v_{\rm c}^{2}$ would be more ambiguous, but a strong correlation between $\varepsilon_{\rm fb}$ and $v_{\rm c}^{2}$ could be interpreted as evidence favoring ejective feedback modes.

Traditional semi-analytic models for galaxy evolution track how a halo’s baryons flow through different reservoirs (White & Frenk 1991; Kauffmann et al. 1993; Cole & Lacey 1996; Somerville & Primack 1999, early examples include). They typically have the following features:

1. 1.

   Accretion of baryons onto the CGM tracks $f_{b}\dot{M}_{\rm halo}$, where $\dot{M}_{\rm halo}$ is the total mass growth rate of the gravitationally bound halo. The gas is assumed to reach the halo’s virial temperature when it accretes, and its initial composition is assumed to be primordial.

2. 2.

   Radiative cooling enables CGM gas to accrete onto the ISM. Its cooling rate depends on the temperature and density profile of the CGM. Usually, the CGM is assumed to be isothermal, with a temperature $\propto v_{\rm c}^{2}$ and a density profile $\propto r^{-2}$ (Lu et al. 2011; Monaco et al. 2014; Stevens et al. 2017; Hou et al. 2019, for discussions of other density profiles, see).

3. 3.

   Star formation converts ISM gas into stars, which enrich the ISM with heavier elements (i.e. “metals”). Some of those metals become locked into stars.

4. 4.

   Galactic winds eject some of the ISM gas, along with some metals, into either the CGM or an “ejected reservoir” beyond the CGM. Enrichment of the CGM affects its cooling rate.

5. 5.

   Some of the ejected gas and metals may “re-accrete” from the ejected reservoir onto the CGM.

Traditional semi-analytical models usually do not comprehensively track energy flows among these reservoirs, but energy budget arguments sometimes do play a role. For example, in the L-galaxies model and many of its relatives (Guo et al. 2011; Henriques et al. 2015; Zhong et al. 2023, e.g.,), supernova energy is able to eject gas from the CGM reservoir. First, supernova energy added to some ISM gas must unbind it from the galaxy. The excess energy is then added to the CGM, where it can unbind some of the CGM gas, removing it from the CGM reservoir (Lagos et al. 2013; Cousin et al. 2015, see also).

Also, later generations of semi-analytic models include energy input from black-hole accretion (Croton et al. 2006; Somerville et al. 2008, e.g.,), which is assumed to produce jets that deposit energy in the CGM and can offset radiative cooling. However, any energy input exceeding the radiative cooling rate is simply lost, and does not modify the temperature or profile of the CGM, meaning that it does not impact the next timestep beyond a one-time suppression of cooling and accretion.

This reliance on gas ejection to suppress star formation explains why most traditional semi-analytic models need to adopt extremely high mass loading factors to match observational constraints on stellar baryon fractions (Pandya et al. 2020; Mitchell et al. 2020; Carr et al. 2023, e.g.,). In order to reproduce observations of the $f_{*}$–$M_{\rm halo}$ relation, $\eta_{M}$ needs to depend strongly on $M_{\rm halo}$, becoming as large as $\eta_{M}\sim 10^{2}$ among low-mass halos. A few semi-analytic models have included preventative feedback algorithms that enable supernova energy to limit accretion onto the CGM and/or ISM (Lu et al. 2015; Hirschmann et al. 2016; Lu et al. 2017; Pandya et al. 2023, e.g.,). That approach naturally lowers the required mass-loading factors, but the ad hoc preventative feedback recipes are not self-consistently coupled to the atmosphere’s energy budget.

In contrast, the progenitors of Paper I’s minimalist regulator model explicitly track energy flows. Carr et al. 2023 track how energy input from both supernova feedback and cosmological accretion affects the CGM temperature and allow some of the CGM to escape if that temperature becomes too large. Pandya et al. 2023 include both of those energy sources and track evolution of both thermal and turbulent energy components in the CGM. However, neither Carr et al. 2023 nor Pandya et al. 2023 track gains or losses of gravitational potential energy as the CGM expands or contracts in response to changes in the balance between energy input and radiative cooling.

One semi-analytic model that does track potential energy gains and losses was proposed by Sharma & Theuns 2020. The details of their approach differ from the minimalist regulator model, but one of its outcomes is fundamentally similar. They find that star-formation rates at $M_{\rm halo}\lesssim 10^{12}\,M_{\odot}$ become linked to the halo’s cosmological accretion rate, as in equation (12), because stellar feedback is compensating for cosmological accretion, not radiative cooling.

Now we will take a closer look at the semi-analytic regulator model of Carr et al. 2023, who obtained $f_{*}$–$M_{\rm halo}$ predictions that were insensitive to the mass loading parameter $\eta_{M}$. They were also insensitive to heavy-element enrichment of the CGM. What mattered most was the energy loading parameter $\eta_{E}$.

Figure 2 illustrates how the $f_{*}$ predictions of Carr et al. 2023 depend on $M_{\rm halo}$ and $\eta_{M}$. All of the purple lines represent models with $\eta_{E}=0.3$ and values of $\eta_{M}$ ranging from 0.1 to 20. They result from integrating a regulator model similar to the minimalist regulator model of Paper I over cosmic time. Those lines are almost identical, showing that the predicted dependence of cumulative star formation on $\eta_{M}$ is generally weak, in alignment with the minimalist regulator model of Paper I, as expressed in equation (14) and as illustrated in the “expanding” region of Figure 1.

However, the Carr et al. model differs from the minimalist regulator model in three important ways:

1. 1.

   Baryons leave the CGM if atmospheric expansion pushes them beyond the halo’s virial radius ($R_{\rm halo}$).

2. 2.

   The model’s treatment of gravitational potential energy is less complete.

3. 3.

   A galaxy’s atmosphere cannot contract while radiative cooling exceeds feedback heating.

The first difference opens another channel for atmospheric energy loss, because baryons going beyond $R_{\rm halo}$ carry away whatever specific energy they had before leaving, thereby reducing both $E_{\rm CGM}$ and $M_{\rm CGM}$. The second difference means that the specific energy of baryons entering the halo via cosmic accretion does not reflect all of their potential energy. Baryons accreting onto the CGM in the Carr et al. model have a specific energy $\varepsilon_{\rm acc}=0.66v_{\rm c}^{2}$, and expansion of the CGM pushes baryons beyond $R_{\rm halo}$ and out of the CGM if $\varepsilon_{\rm CGM}$ exceeds $\varepsilon_{\rm acc}$. In contrast, accreting baryons in the minimalist regulator model are assumed to have $\varepsilon_{\rm acc}\approx 4v_{\rm c}^{2}$, because $\varepsilon_{\rm acc}$ includes gravitational potential energy, in addition to thermal and kinetic energy. The third difference means that excess radiative cooling cannot trigger a runaway star formation crisis.

A grey band in Figure 2 shows the $f_{*}$–$M_{\rm halo}$ relation that Behroozi et al. 2019 inferred from observations. In order to fit that relation, Carr et al. 2023 had to assume that $\eta_{E}$ was a function of halo mass, becoming smaller as halo mass increases. Using a power-law relation with

they were able to bring their predictions into good agreement with the observations.

Contemporary cosmological simulations of galaxy evolution often do something similar, by employing parametric formulae that indirectly make the efficiency of supernova feedback a function of halo mass (Schaye et al. 2015; Nelson et al. 2019; Pillepich et al. 2018a, e.g.,). Each formula’s parameters are tuned to reproduce the observable properties of galaxies. To obtain agreement, the simulations’ equivalent of $\eta_{E}$ needs to become smaller as $M_{\rm halo}$ rises toward $\sim 10^{12}\,M_{\odot}$ (Crain et al. 2015; Pillepich et al. 2018b, e.g.,).

A dashed red line in the figure shows the asymptotic $f_{*}$–$M_{\rm halo}$ relation predicted by the minimalist regulator model with $\xi=0.7$, $\eta_{M}=0.3$, and $\varepsilon_{\rm SN}=(700\,{\rm km\,s^{-1}})^{2}$ when the radiative-loss correction in equation (14) is ignored. It is similar to the Carr et al. predictions in the vicinity of $M_{\rm halo}\approx 10^{11}\,M_{\odot}$, but has a slightly steeper slope. At least two factors contribute to the difference in slope. First, early star formation in the low-mass halos of Carr et al. happens under conditions far from equilibrium. Second, the equilibrium specific energy of the CGM among low-mass halos in Carr et al. is significantly greater than $0.66v_{\rm c}^{2}$. Consequently, the atmospheric energy losses sustained when expansion pushes gas beyond the virial radius are larger than in the minimalist regulator model. More star formation (or larger $\eta_{E}$) is then required to drive a given amount of atmospheric expansion.

A simple power-law dependence of $\eta_{E}$ on $M_{\rm halo}$, shallower than the one used by Carr et al., brings the asymptotic $f_{*}$–$M_{\rm halo}$ relation from the minimalist regulator model into good agreement with the observations. The blue line in Figure 2 shows the result of using

instead of $\eta_{E}=0.3$. It matches the observations well, except near the peak of the observed $f_{*}$–$M_{\rm halo}$ relation, which we will discuss next. Note that any set of parameter choices with $\eta_{E}\propto M_{\rm halo}^{-1/3}$ and $\xi/\eta_{E}\approx 0.7/0.3$ near $M_{\rm halo}=10^{11}\,M_{\odot}$ would give similar results, because the parameters $\xi$ and $\eta_{M}$ are nearly degenerate.

Next we consider the well-known peak in the $f_{*}$–$M_{\rm halo}$ relation, observed to persist near $M_{\rm halo}\sim 10^{12}M_{\odot}$ across most of cosmic time (Behroozi et al. 2019, e.g.,). None of the models shown in Figure 2 reproduces it, which is not surprising because those models do not include feedback resulting from accretion onto a central supermassive black hole. Nevertheless, the mimimalist regulator model provides a clue about why black-hole feedback becomes important as $M_{\rm halo}$ approaches $10^{12}\,M_{\odot}$. Apparently, that is where supernova feedback starts to fail, permitting an atmospheric contraction crisis to fuel strong black-hole feedback (Sharma & Theuns 2020, see also).

So far, we have been discussing the peak in the $f_{*}$–$M_{\rm halo}$ relation as if it were a borderline in $v_{\rm c}$ instead of $M_{\rm halo}$. However, higher-redshift galaxies at fixed $M_{\rm halo}$ have greater $v_{\rm c}$. To remain near $M_{\rm halo}\sim 10^{12}M_{\odot}$, the borderline for atmospheric contraction therefore needs to be at larger $v_{\rm c}$ earlier in time. This section suggests that evolution of the surface density $\Sigma_{\rm SFR}$ of a galaxy’s star formation rate may be what keeps the borderline for atmospheric contraction close to $M_{\rm halo}\sim 10^{12}M_{\odot}$.

Small-box simulations designed to resolve the ISM physics at the root of supernova feedback (Kim et al. 2017; Li et al. 2017; Fielding et al. 2018; Armillotta et al. 2019, e.g.,) have explored how $\Sigma_{\rm SFR}$ is related to $\eta_{E}/\eta_{M}$. In the compilation of such simulations presented by Li & Bryan 2020, the best fit for hot outflows yields

Inserting that fit into equation (17) gives the borderline criterion

which implies that the tipping point for atmospheric contraction may depend weakly on the surface density of star formation within a halo’s central galaxy. For the Milky Way, observations within the solar circle show that $\Sigma_{\rm SFR}\approx 10^{-2}\,M_{\odot}\,{\rm kpc^{-2}\,s^{-1}}$ (Elia et al. 2022). Equation (18) then gives $\eta_{E}/\eta_{M}\approx 0.3$, and equation (19) puts the borderline between expansion and contraction near $200\,{\rm km\,s^{-1}}$, in accord with the Milky Way estimate in §III.4.

Higher-redshift galaxies with stellar masses and halo masses similar to the Milky Way’s have much greater star formation rates but somewhat smaller effective radii, and therefore have much greater $\Sigma_{\rm SFR}$ values (Salim et al. 2023, e.g.). More highly concentrated star formation could therefore be what keeps the peak of the $f_{*}$–$M_{\rm halo}$ relation near $10^{12}\,M_{\odot}$. For example, consider a high-redshift galaxy with $\Sigma_{\rm SFR}\sim 1\,M_{\odot}\,{\rm kpc^{-2}\,yr^{-1}}$. According to equation (19), its borderline circular velocity is near $300\,{\rm km\,s^{-1}}$, corresponding to $M_{\rm halo}\sim 3\times 10^{12}\,M_{\odot}$ at $z\sim 2$. In other words, the borderline above which supernova feedback cannot prevent atmospheric contraction around high-redshift galaxies is at greater $v_{\rm c}$, but the halo mass at a given value of $v_{\rm c}$ is smaller. Further investigation will be needed to determine whether those two countervailing effects do indeed combine to keep the peak of $f_{*}$ near $10^{12}\,M_{\odot}$.

Any given simulation contains a population of halos with measurable values of $M_{\rm CGM}$, $E_{\rm CGM}$, and $M_{\rm halo}$. Those measurements can be converted to $f_{\rm CGM}=M_{\rm CGM}/f_{\rm b}M_{\rm halo}$ and $\varepsilon_{\rm CGM}=E_{\rm CGM}/M_{\rm CGM}$ and plotted in the $f_{\rm CGM}$–$\varepsilon_{\rm CGM}$ phase plane. Mapping the flow of those points through the phase plane would then reveal how the atmospheres of galaxies respond to feedback. Also, the two-dimensional distribution of the phase-plane points at a particular moment can be compared with both observational constraints and the predictions of semi-analytic models (see Pandya et al. 2020; Pandya et al. 2023, for examples).

Applying the comprehensive accounting approach adopted in Paper I to particle-based simulations should start with identification of baryonic particles that were originally cospatial with a halo’s current dark matter particles. Summing the masses of baryonic particles that did not form stars and are not currently part of some galaxy’s interstellar medium then gives $M_{\rm CGM}$. However, assessing $E_{\rm CGM}$ requires another step beyond a simple sum of thermal and non-thermal particle energy, because the dominant contribution to $E_{\rm CGM}$ usually comes from gravitational potential energy. Comprehensive accounting algorithms to determine $E_{\rm CGM}$ therefore need to identify a halo’s center, calculate the gravitational potential with respect to that centroid, and designate an appropriate zero point.

Methods for measuring $M_{\rm CGM}$ and $E_{\rm CGM}$ in grid-based codes cannot be as direct, because baryons in those simulations do not retain a traceable identity. An alternative method for estimating $M_{\rm CGM}$ would be to measure the sum of $M_{*}$ and $M_{\rm ISM}$ within a halo and subtract it from $f_{\rm b}M_{\rm halo}$. The atmosphere’s total energy can then be estimated from the intergalactic baryons within a sphere containing gas mass $M_{\rm CGM}$. Summing the thermal, non-thermal, and gravitational potential energies of those atmospheric baryons yields an estimate of $E_{\rm CGM}$. However, that sum may underestimate the total amount of feedback energy added to the atmosphere if a significant fraction of the halo’s original baryons have been ejected far beyond that sphere and replaced by baryons falling toward the halo for the first time.

Obtaining observational constraints on $M_{\rm CGM}$ and $E_{\rm CGM}$ may be more challenging, because baryons near or beyond a halo’s virial radius are so difficult to detect. However, the distribution of atmospheric mass as a function of radius places a lower limit on $E_{\rm CGM}$ that can be useful for distinguishing among various feedback models. One possible proxy for $E_{\rm CGM}$ is the circumgalactic baryon fraction within the virial radius

(Wright et al. 2024, for comparisons of $f_{\rm CGM,vir}$ among simulations, see). Another is the closure radius (Ayromlou et al. 2023, e.g.,).

Perhaps more fundamentally, the minimalist regulator model clarifies how the specific energy of a feedback outflow determines the ultimate outcome of a cosmological galaxy-evolution simulation. According to Figure 1, supernova-driven outflows that couple with the CGM result in stellar baryon fractions that are insensitive to mass loading, as long as $\varepsilon_{\rm fb}$ exceeds $\varepsilon_{\rm acc}$. In that regime, a central galaxy’s long-term star formation rate becomes closely linked to the mass-accretion rate of its halo, because supernova feedback is mainly balancing baryon accretion, not radiative cooling. A halo’s eventual stellar baryon fraction therefore depends mostly on the fraction of supernova energy that couples with the CGM ($\eta_{E}$), because $\eta_{E}\varepsilon_{\rm SN}M_{*}$ is always comparable to $f_{\rm b}M_{\rm halo}v_{\rm c}^{2}$ (Sharma & Theuns 2020, see also).

When supernova feedback is balancing cosmological accretion, raising $\eta_{M}$ reduces atmospheric expansion, because more of the supernova energy going into the CGM is lost to radiative cooling (see Paper I for a more detailed explanation). The effect is minor until $\eta_{M}$ becomes large enough for $\varepsilon_{\rm fb}$ to fall below $\varepsilon_{\rm acc}$. When those two specific energies are comparable, then a galaxy’s atmosphere does not expand, because radiative cooling sheds all the energy input from supernovae and cosmological accretion.

To point the way toward more detailed comparisons with simulations, this concluding section takes a tiny step in that direction, briefly sketching out where the Illustris, EAGLE, and IllustrisTNG simulations reside within the parameter space of the minimalist regulator model. What it presents helps to explain how those simulations manage to produce galaxies with similar star-formation properties but strikingly different atmospheric characteristics.

Our overall point here is that the specific energy of supernova-driven outflows ($\varepsilon_{\rm fb}=\eta_{E}\varepsilon_{\rm SN}/\eta_{M}$) and the total energy of those outflows ($\eta_{E}\varepsilon_{\rm SN}M_{*}$) play different roles. According to the minimalist regulator model, outflows capable of expanding a galaxy’s atmosphere self-regulate so that $\eta_{E}\varepsilon_{\rm SN}M_{*}$ approaches $\approx f_{\rm b}M_{\rm halo}v_{\rm c}^{2}$. That may be why simulations producing outflows with quite different mass-loading factors, like EAGLE and IllustrisTNG, can produce galaxy populations that are quite similar.

However, outflows with similar power but different proportions of mass loading have different specific energies and therefore affect the CGM differently. EAGLE produces outflows with high specific energy, resulting in a more extended CGM and less recycling of gas through the central galaxy. IllustrisTNG produces outflows with less specific energy, resulting in a less extended CGM and more recycling, while greater mass loading compensates for the additional recycling.

We therefore suspect that specific energy may be the key to understanding why simulated galaxies having similar star-formation properties can have CGM characteristics that significantly differ. We also encourage other groups to perform similar specific-energy analyses on their own simulations, with more quantitative rigor than the sketch presented here.