Figuring Out Gas & Galaxies In Enzo (FOGGIE) V: The Virial Temperature Does Not Describe Gas in a Virialized Galaxy Halo
Cassandra Lochhaas, Jason Tumlinson, Brian W. O'Shea, Molly S. Peeples, Britton D. Smith, Jessica K. Werk, Ramona Augustin, Raymond C. Simons
DDraft version February 18, 2021
Typeset using L A TEX twocolumn style in AASTeX63
Figuring Out Gas & Galaxies In Enzo (FOGGIE) V:The Virial Temperature Does Not Describe Gas in a Virialized Galaxy Halo
Cassandra Lochhaas, Jason Tumlinson,
1, 2
Brian W. O’Shea, Molly S. Peeples,
1, 2
Britton D. Smith, Jessica K. Werk, Ramona Augustin, and Raymond C. Simons Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218 Department of Physics & Astronomy, Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD 21218 Department of Computational Mathematics, Science, and Engineering, Department of Physics and Astronomy, NationalSuperconducting Cyclotron Laboratory, Michigan State University Royal Observatory, University of Edinburgh, United Kingdom Department of Astronomy, University of Washington, Seattle, WA 98195
Submitted to ApJABSTRACTThe classical definition of the virial temperature of a galaxy halo excludes a fundamental contributionto the energy partition of the halo: the kinetic energy of non-thermal gas motions. Using simulationsfrom the FOGGIE project (Figuring Out Gas & Galaxies In Enzo) that are optimized to resolve low-density gas, we show that the kinetic energy of non-thermal motions is roughly equal to the energy ofthermal motions. The simulated FOGGIE halos have ∼ × lower bulk temperatures than expectedfrom a classical virial equilibrium, owing to significant non-thermal kinetic energy that is formallyexcluded from the definition of T vir . We derive a modified virial temperature explicitly includingnon-thermal gas motions that provides a more accurate description of gas temperatures for simulatedhalos in virial equilibrium. Strong bursts of stellar feedback drive the simulated FOGGIE halos out ofvirial equilibrium, but the halo gas cannot be accurately described by the standard virial temperatureeven when in virial equilibrium. Compared to the standard virial temperature, the cooler modifiedvirial temperature implies other effects on halo gas: (i) the thermal gas pressure is lower, (ii) radiativecooling is more efficient, (iii) O VI absorbing gas that traces the virial temperature may be prevalentin halos of a higher mass than expected, (iv) gas mass estimates from X-ray surface brightness profilesmay be incorrect, and (v) turbulent motions make an important contribution to the energy balance ofa galaxy halo. Keywords: circumgalactic medium — galaxy evolution INTRODUCTIONThe current paradigm of hierarchical structure forma-tion has been in place for decades (Rees & Ostriker1977; Silk 1977; White & Rees 1978; White & Frenk1991). The general mechanism by which galaxies formstarts with random density fluctuations in dark mat-ter that collapse over cosmic time into massive darkmatter halos. Baryonic matter is similarly gravita-tionally attracted to dark matter halos, where it col-
Corresponding author: Cassandra [email protected] lects and eventually forms galaxies (for a review, seeBenson 2010). The standard paradigm of galaxy for-mation supposes that gas falling onto a massive halo, M halo (cid:38) few × M (cid:12) (Birnboim & Dekel 2003), shock-heats to the virial temperature before later cooling at thehalo center to form stars. More recent models (Kereˇset al. 2005; Dekel & Birnboim 2006; Dekel et al. 2009;Nelson et al. 2013) show that infalling gas need notshock-heat to high temperatures, but may instead beaccreted to the central galaxy along filaments while re-maining cold ( T (cid:46) − K). Modern simulations showthat both hot halos and cold filaments can exist sur-rounding a galaxy simultaneously, but the presence ofa hot, shock-heated gaseous halo is still expected sur- a r X i v : . [ a s t r o - ph . GA ] F e b Lochhaas et al. rounding massive galaxies, even if it is not the primarymode of gas accretion (Bennett & Sijacki 2020; Fieldinget al. 2017; Stern et al. 2020a).In order for gas to form stars at the center of ha-los, it must be cold (with T (cid:28) K). The standardparadigm assumes gas either cools radiatively before ac-creting onto the galaxy or flows onto the galaxy fromthe intergalactic medium (IGM) without being heated.However, the exact processes by which cold gas formsfrom, or interacts with, the expected hot halo are notfully understood. The conditions for the formation andsurvival of the cold gas are strongly dependent on theproperties of the hot, diffuse, volume-filling phase of thecircumgalactic medium (CGM).In models that attempt to describe hot gas observa-tions or cool gas formation and survival within the hothalo, the hot gas is usually assumed to exist in a simplehydrostatic equilibrium at the virial temperature, T vir ,of the halo (Maller & Bullock 2004; Anderson & Breg-man 2010; Miller & Bregman 2013; Faerman et al. 2017;Mathews & Prochaska 2017; McQuinn & Werk 2018; Qu& Bregman 2018; Stern et al. 2019; Voit 2019; Faermanet al. 2020). ‘Idealized’ simulations commonly adopt ahydrostatic hot halo at T vir as part of their initial con-ditions (Armillotta et al. 2017; Fielding et al. 2017; Li& Tonnesen 2020). Small, cold gas clouds may thencondense from the hot medium, seeded by thermal in-stabilities (McCourt et al. 2012; Voit et al. 2015). Coldgas may also be seeded by galactic winds, where thehot flow can entrain, precipitate, or carry cold clumpsinto the CGM (Thompson et al. 2016; Schneider et al.2018; Lochhaas et al. 2018) to re-accrete later. If coldCGM gas is instead in the form of extended filamen-tary structures, these structures may pierce through theexpected virial shock and hot halo (Kereˇs et al. 2005;Dekel & Birnboim 2006; Kereˇs et al. 2009; Dekel et al.2009; Bennett & Sijacki 2020), interacting with the hotdiffuse gas and creating hydrodynamical instabilities atthe hot-cold interface (Mandelker et al. 2016, 2020). Al-ternatively, cold accreting gas may take the form of acooling flow, where the hot halo undergoes bulk coolingas it is compressed on its journey to the central galaxy(Mathews & Bregman 1978; Fabian et al. 1984; Malagoliet al. 1987; Li & Bryan 2012; Stern et al. 2019, 2020a).Observations of the CGM typically find both hot andcold gas traced by high- and low-ionization state absorp-tion observed in the UV and optical (Wakker & Savage2009; Rudie et al. 2012; Werk et al. 2013; Stocke et al.2013; Bordoloi et al. 2014; Lehner et al. 2015; Borthakuret al. 2016; Heckman et al. 2017; Keeney et al. 2017;Chen et al. 2018; Berg et al. 2018, 2019; Rudie et al.2019; Chen et al. 2020; Lehner et al. 2020) or emis- sion in the X-ray (Anderson & Bregman 2010). Thedensities and temperatures are derived from ionizationmodeling, where generally it is assumed that high-ionabsorbers (O VI or O VII ) trace a warmer, collisionally-ionized gas phase than low-ionization state absorbers(e.g., Mg II , Si III ), which trace a cooler, photoionizedgas phase (e.g., Tumlinson et al. 2017). Studies thatfind a significant mass of cold gas in the CGM of L ∗ galaxies, ∼ M (cid:12) , have raised questions about howso much cold gas could be supported in the CGM (e.g.,Werk et al. 2014; Keeney et al. 2017). Fitting small,thermal-pressure-supported cold clouds into the stan-dard paradigm of a hydrostatic hot halo is difficult whilealso matching the cold and hot gas densities inferredfrom photoionized modeling (Werk et al. 2014; McQuinn& Werk 2018, but see Haislmaier et al. 2020). All suchmodeling is laden with assumptions about the thermalbalance of the CGM that could prove to be mistaken.At larger scales, galaxy cluster and intra-clustermedium (ICM) analytic, simulation, and observationalstudies have shown that the ICM is not in perfect hy-drostatic equilibrium because non-thermal kinetic gasmotions are crucial to the overall energy balance of thehalo. Bulk non-thermal gas motions, such as turbulence,contribute a significant fraction of pressure support tothe cluster gas (Shi et al. 2015, 2018; Simionescu et al.2019). This fraction is significant enough to produce a“hydrostatic mass bias”, i.e. the cluster mass derivedwithout including non-thermal pressure support differsfrom the “true” cluster mass by ∼
15% on average (Lauet al. 2013; Shi et al. 2016).At ∼ L ∗ galaxy halo scales, only recently have simu-lations begun to show that the standard picture of ahot gaseous halo in hydrostatic equilibrium may notbe accurate. Lochhaas et al. (2020) showed that evenin idealized L ∗ CGM simulations initiated with hydro-static hot halos, galactic feedback creates bulk flowsthat induce significant turbulence and rapidly drive thehalo out of hydrostatic equilibrium. Instead, the haloevolves toward a dynamical equilibrium in which non-thermal turbulent and ram pressure combine with theusual thermal pressure to hold the CGM up againstgravity. The simulations of Oppenheimer (2018) alsoshowed the importance non-thermal pressure supportof the CGM. Salem et al. (2016), Ji et al. (2020) andButsky et al. (2020) found that cosmic rays are also animportant non-thermal supporting pressure in the CGMof simulated galaxies. Clearly, the structure of the hotphase, which is so important to models of observed andsimulated cold CGM gas at the galaxy halo scale, war-rants further investigation beyond a simple assumptionof hydrostatic equilibrium at T vir . OGGIE V: Modified Virial Temperature ∼ L ∗ galaxy halos are in virialequilibrium. We find that dynamic gas motions drivethe temperature of the diffuse hot halo below the clas-sical T vir by a factor of ∼
2, even when the halo is in orclose to virial equilibrium. We derive a “modified” virialtemperature, which adds explicit treatment of bulk gasmotions to the classical definition of T vir . This modifiedvirial temperature more accurately describes the tem-perature of gas in the outskirts of the ∼ L (cid:63) FOGGIEgalaxy halos. A cooler than expected “hot” halo has sig-nificant implications on the thermal pressure and coolingrates of the gas as well as on inferences made from UVabsorption line and X-ray emission CGM observations.Section 2 provides the derivation of the modified virialtemperature and explains how it differs from the stan-dard virial temperature. Section 3 describes the FOG-GIE simulations and the basic analysis we use through-out the paper. Section 4 presents how we assess thevirial equilibrium of the FOGGIE halos ( § § § § § § § VI ion ( § § DERIVING VIRIAL TEMPERATUREIn the standard paradigm of galaxy formation, thegaseous halo bound to a galaxy is virialized within thepotential well of the dark matter halo such thatPE = − , (1)where PE is the potential energy of the galaxy and itsdark matter halo and KE is the kinetic energy of the halo gas. Gas falling into the halo is heated by passingthrough a virial shock at roughly the virial radius, so itis assumed that the kinetic energy of gas infall is com-pletely thermalized into a thermal energy, KE th . Then, virial equilibrium is defined by:KE th = 32 k B Tµm p , (2)where T is the temperature, µ = 0 . k B is the Boltzmann constant and m p is the mass of the proton. Note that Equation 2gives the specific thermal energy of the gas, which is theenergy per unit gas mass. Through the virial equation(Eq. 1), the (specific) potential energy of the gas is thusdirectly related to the temperature of the gas, and thistemperature is defined as virial temperature T vir (e.g.,Mo et al. 2010): T vir = 12 µm p k B GM R , (3)where M and R are the halo virial mass and radius,respectively, and G is the gravitational constant.Throughout this paper, we define R as the radiusenclosing an overdensity 200 times the critical densityof the universe, which evolves with redshift (althoughwe always use an overdensity factor of 200, regardlessof redshift). We show in Appendix A that our resultsare insensitive to the exact choice of overdensity in thedefinition of virial radius, and so robust against inconsis-tent practice for this quantity in the existing literature.Equation (3) assumes the specific potential energy of gasin a dark matter halo at the virial radius is described byPE = − GM R . (4)The definition of virial temperature (3) makes astrong, deeply embedded assumption about the energypartition in gas-filled dark-matter halos: that all the po-tential energy of gas flowing into the halo is fully ther-malized into internal thermal energy and that gas turbu-lence and bulk flows contribute nothing, by definition, tothe overall energy of the halo gas. To explore the con-sequences of explicitly including non-thermal motions In some systems there may also be significant gas pressure atthe outskirts that works in the same direction as gravity, so thatthe kinetic energy must also oppose this confining pressure forthe system to be in virial equilibrium. We checked if the sys-tems explored in this paper had sufficient external gas pressureto include the confining pressure term and found it only madea difference in the overall energy balance of the halo on the or-der of ∼ − Lochhaas et al. (such as turbulence and bulk flows) in the energy par-tition of the halo, we rewrite the virial equation (Eq. 1)to explicitly include kinetic energy from both thermaland non-thermal motions:PE = − th + KE nt ) (5)where PE is still given by Equation 4, the thermal ki-netic energy KE th is given by Equation 2, and the ki-netic energy due to non-thermal gas motions is KE nt .Plugging these in and rewriting, we find a modificationto the virial temperature, T mod , that explicitly includesnon-thermal gas motions, given by: T mod = 12 µm p k B GM R − µm p k B KE nt (6)or T mod = T vir − µm p k B KE nt . (7)Both T vir and T mod assume the virial equation (Eq. 1)holds. A halo in perfect virial equilibrium does not con-tain any sources or sinks of energy — the gas can onlytransfer energy between its potential and kinetic ener-gies. Star formation, feedback, and radiative coolingprovide sources and sinks of energy in the halo that candrive a departure from virial equilibrium. Therefore, weexpect the virial temperature (either the standard T vir or the modified T mod ) to be a good descriptor of halo gasonly far from the galaxy where these sources and sinksoperate and where the gas cooling time is longer than theHubble time, at galactocentric radii (cid:38) . R . In de-tail, there may be events in a galaxy’s history that leadto temporary departures from virial equilibrium evennear the virial radius: mergers may lead to especiallystrong bursts of star formation feedback that may un-bind a portion of the halo’s gas. Likewise, there may bespatially distinct parts of the halo that do not partici-pate in the overall balance of virial equilibrium, such ascosmological filaments that can pierce inward throughthe virial shock without being heated (e.g. Bennett &Sijacki 2020) or strong outflows faster than the escapevelocity of the halo (see § r (cid:38) . R ) tend to be set bycosmological structure formation whereas the propertiesof the inner CGM ( r (cid:46) . R ) tend to be set by feed-back processes in the central galaxy (a result also cor-roborated by Stern et al. 2020b), validating our choiceto focus on the outer CGM. We now have the definitionswe need to apply a virial analysis to simulated galaxies. l o g M a ss w i t h i n R [ M ] Tempest Total massDark matter massStellar massGas mass1.50 1.00 0.75 0.50 0.30 0.20 0.10 0.00Redshift4 6 8 10 12Time [Gyr]9.09.510.010.511.011.512.0 l o g M a ss w i t h i n R [ M ] Squall1.50 1.00 0.75 0.50 0.30 0.20 0.10 0.00Redshift4 6 8 10 12Time [Gyr]9.510.010.511.011.512.012.5 l o g M a ss w i t h i n R [ M ] Maelstrom1.50 1.00 0.75 0.50 0.30 0.20 0.10Redshift
Figure 1.
The total (black solid), dark matter (red dashed),stellar (blue dotted) and gaseous (green dash-dotted) masseswithin R for the three galaxies from the FOGGIE suiteconsidered in this work, arranged from least massive to mostmassive at z = 0 top to bottom. Note that Maelstrom (bot-tom panel) has been run only to z = 0 .
05 at the time ofwriting.
OGGIE V: Modified Virial Temperature Property Tempest Squall Maelstrom a R [kpc] 168.3 195.93 208.72 b M [10 M (cid:12) ] 5.04 8.02 10.11 c M DM , [10 M (cid:12) ] 4.26 6.56 8.45 d M (cid:63), [10 M (cid:12) ] 5.44 12.34 11.41 e M gas , [10 M (cid:12) ] 2.33 2.28 5.18 Table 1.
Properties of the three FOGGIE halos studied inthis paper at z = 0 for Tempest and Squall and z = 0 .
05 forMaelstrom. a Radius enclosing an average density of 200 × the criticaldensity of the universe at z = 0. b Total mass enclosed within R . c Dark matter mass enclosed within R . d Stellar mass enclosed within R . Includes satellites. e Gas mass enclosed within R . Includes ISM of centraland satellites.3. THE SIMULATIONS: FIGURING OUT GAS &GALAXIES IN ENZOTo explore energy partition in realistic halo simula-tions and assess the viabilty of the modified virial tem-perature for characterizing the bulk properties of theCGM, we use simulations from the Figuring Out Gas &Galaxies In Enzo (FOGGIE) project. These simulationsare described fully in the previous papers FOGGIE I –IV (Peeples et al. 2019; Corlies et al. 2020; Zheng et al.2020; Simons et al. 2020), but we briefly describe therelevant parts here for convenience.FOGGIE is run using the adaptive mesh refinement(AMR) code Enzo (Bryan et al. 2014; Brummel-Smithet al. 2019) . As introduced in Simons et al. (2020), sixhalos with roughly the Milky Way’s present day totalmass were selected from a cosmological volume 143.88comoving Mpc on a side to be re-simulated in “zoom-in” regions, where additional spatial refinement of atleast 1.10 comoving kpc is forced in a box 287.77 co-moving kpc on a side centered on the galaxy as it movesthrough the cosmological domain. Within this “forced-refinement” box, the resolution is refined further up to274.44 comoving pc using an adaptive “cooling refine-ment” criterion in which one cell is replaced with 8 cellswhen the product of gas cooling time and sound speedis smaller the original cell size. The cooling refinementscheme places high resolution elements where they areneeded most, in the high density and/or rapidly coolingcells, saving computational resources with less refine-ment in the hot and/or lowest density phases. How-ever, the forced refinement region keeps the warm, dif-fuse gas resolved to a high level even in the absence https://enzo-project.org of short cooling times, allowing detailed kinematics tobe resolved and reducing the degree of artificial mixingfrequently present in simulations with standard refine-ment schemes. In the outskirts of the halo that we focuson in this paper, the typical spatial resolution is set atthe fixed, minimum refinement of the forced-refinementtracking box, where cells are 1.10 comoving kpc on aside, because there is not much gas there with very shortcooling times.The galaxies chosen to be simulated at high resolutionhave their last significant merger ( < z = 2, to be similar to the Milky Way mergerhistory. This generally means they do not have strongbursts of star formation or feedback driving their halogas significantly away from equilibrium at low redshifts(see § z = 0, Tempestand Squall, and a third, Maelstrom, has been run to z ∼ .
05, so we focus on just these three in this paper(see Simons et al. 2020 for more information on thesehalos). However, we expect our results to be generallyapplicable and not specific to the properties of thesegalaxies and their halos.Figure 1 shows the build-up of gaseous, stellar, darkmatter, and total masses within R in these three halosover the redshift range considered here, z ∼ z = 0(or z = 0 .
05 for Maelstrom). Table 1 shows the finalproperties of each halo at z = 0 (or z = 0 .
05 for Mael-strom). By z ∼
1, each galaxy’s total mass is above thethreshold where a virial shock is expected to form andremain stable, M h ∼ few × M (cid:12) (Birnboim & Dekel2003). Maelstrom, being the most massive of the threegalaxies, surpasses this threshold by z ∼ .
5. In general,the build-up of all types of mass within R becomessmooth and slowly increasing at late times. Squall is anexception because it continues to undergo gas-rich mi-nor mergers at late times that drive the star formationrate up and lead to bursty changes in the stellar or gasmasses within R .We select ∼
190 snapshots in time between z = 2 and z = 0, separated by ∼
50 Myr, for each of the Tempestand Squall halos, and ∼
180 snapshots between z = 2and z = 0 .
05 for the Maelstrom halo. The FOGGIE runsare set up to output a snapshot every ∼ . R and 1 . R into 100 radial bins (ofwidth 0 . R ) to compute the properties of the CGMgas as functions of radius. In what follows, we take the Lochhaas et al. radial bin 0 . R < r < R as the bin representingthe gas near R . At low redshift, R for the FOGGIE halos exam-ined here falls partially inside and partially outside ofthe forced refinement region. We test the impact ofcombining high and low resolution cells within a singlespherical shell by re-calculating all results using only thehigh-resolution cells in the shell near R and find it hasa minor quantitative and no qualitative effect on our re-sults. The higher-resolution cells can better resolve thegas kinematics and thus contain somewhat more non-thermal kinetic energy overall than the low-resolutioncells, which serves to somewhat strengthen the differ-ence between T vir and T mod and strengthen our qualita-tive conclusions. We proceed with using all cells withinthe shell at R rather than only the high-resolutioncells, but note that perhaps a cosmological simulationwith higher forced resolution than FOGGIE would findan even stronger difference between T vir and T mod thanwe report here. This result shows the importance of highresolution within the diffuse CGM gas. VIRIAL ENERGY OF THE FOGGIE HALOSBecause both the standard virial temperature and themodified virial temperature are built on the assumptionof virial equilibrium, we start by assessing when andwhere the FOGGIE halos are in virial equilibrium.4.1.
Assessing Virial Equilibrium
First, we remove the parts of the CGM that we do notexpect to be virialized: satellites and filaments. Satel-lites are excised from the domain by removing all gascells with a density > × − g cm − and temper-ature < . × K. In some cases, this method doesnot perfectly remove all gas associated with a satellite,but it does eliminate confusion of satellite ISM gas withhost halo CGM gas. To remove filaments, we excise allgas with an inward radial velocity faster than half ofthe local free-fall velocity, v ff = r/ (cid:112) π/ (32 Gρ ), where r is the galactocentric radius of the gas parcel, G is thegravitational constant, ρ = π M enc ( 000 cells.We recalculated all results with a shell of width 0 . R andfound no qualitative difference in using shells of different widthsexcept for smoother radial profiles, so we continue with the thin-ner shell. most filament contamination. Again, this cut does notperfectly remove all filament gas, but it removes enoughthat the filament contamination to virialized CGM gasis small in most cases.Outflows, in the form of galactic winds launched fromthe central galaxy and any of its satellites, also shouldnot contribute to the overall virial energy balance of thehalo. However, outflows have a range of velocities, mak-ing them difficult to select and remove cleanly, and theymay also mix into the ambient CGM to become partof the virialized halo. Rather than attempt to removecoherent structures of outflows like we do with satellitesand filaments, we do not consider any gas with an out-ward radial velocity greater than the escape velocity atits location in the dark matter halo to be contributingto the virial balance of the halo. Near the virial radius,very little gas is moving fast enough to escape the haloat times when the halo is in virial equilibrium (see Fig-ure 5 and discussion in § z = 0, beforeand after the cuts to remove satellite ISM and filaments.In both cases, we remove the central 0 . R to eliminatethe galaxy and extended disk. This particular snapshotof this halo does not have any satellites in the planeof this slice, but the filament cut removes an extendedwedge-shaped filament in the bottom right of the panel,primarily gas with T (cid:46) K and v r < − 100 km s − .For Tempest at z = 0, the satellite cut removes 0 . . R and R while the fil-ament cut removes 37 . 4% of all gas mass in the sameregion. The total gas mass between 0 . R and R is5 . × M (cid:12) before any cuts and is 3 . × M (cid:12) after re-moving satellites and filaments. By volume, the satellitecut removes 0 . 01% of the volume between 0 . R and R and the filament cut removes 21 . 6% of the volume.Both the standard and the modified virial tempera-tures assume the halo (or at least the gas near R )is in virial equilibrium, i.e. that Equation (1) is sat-isfied. We generally expect this to be true unless thehalo is experiencing galaxy mergers or a strong burstof energy input in the form of feedback (Fielding et al.2020b; Stern et al. 2020b). Rather than assuming thevirial equation holds, we explicitly measure it within theFOGGIE halos.We measure directly whether the gas at R is invirial equilibrium by summing the potential, kinetic, andthermal energies of the gas in a thin spherical shell. Wefocus on the outer CGM, where we expect virial equi-librium to hold. We define the “virial energy”, VE, tobe this sum:VE = PE + (cid:88) nt + KE th ) (8) OGGIE V: Modified Virial Temperature 150 100 50 0 50 100 150 y (kpc) z ( k p c ) t = 13.8 Gyrz = 0.00 10 T e m p e r a t u r e ( K ) 150 100 50 0 50 100 150 y (kpc) z ( k p c ) t = 13.8 Gyrz = 0.00 20015010050050100150200 R a d i a l V e l o c i t y ( k m / s ) 150 100 50 0 50 100 150 y (kpc) z ( k p c ) t = 13.8 Gyrz = 0.00 10 T e m p e r a t u r e ( K ) 150 100 50 0 50 100 150 y (kpc) z ( k p c ) t = 13.8 Gyrz = 0.00 20015010050050100150200 R a d i a l V e l o c i t y ( k m / s ) Figure 2. Slices of gas temperature (left) and radial velocity (right) in the Tempest halo at z = 0 before and after cuttingsatellite ISM and filament gas (top and bottom panels, respectively). The black circle shows the location of R . The satellitecut removes a negligible amount of mass and volume from the domain for this halo at this redshift, but the filament cut removes ∼ 37% of the gas mass and ∼ 22% of the volume between 0 . R and R . where PE is given by Equation (4) multiplied by thegas mass in the shell and the thermal and non-thermalkinetic energies are obtained by direct sum over all cellsin the spherical shell. We use the total energies, not thespecific energies as we did in Eqs. (1) through Eq. (7),making Equation (8) a true measurement of the totalenergies of the gas contained within the shell. If the gasis in virial equilibrium, then VE = 0.4.2. Halos are in Virial Equilibrium Only WhenNon-Thermal Kinetic Energy is Included Figure 3 shows the virial energy VE (thick red line)given by Equation (8) in a radial shell 0 . R < r < . R over cosmic time as the halos evolve from z = 2to z = 0 ( z = 0 . 05 for Maelstrom). None of the ha- los are in perfect virial equilibrium for extended periodsof time; instead their VE oscillates as the halos evolve,approaching values near zero only at low redshift or dur-ing periods of low star formation rate (SFR). The SFRis plotted as the thin black line, with values marked onthe right axis, and is calculated as all new stars formedsince the previous time step within 20 kpc of the centerof the halo. It is clear that there is a correlation betweenstar formation bursts and when the gas near R is outof virial equilibrium, for example at z ∼ . z ∼ . R stabilizes and approaches virial equilibrium, butbursts of stellar feedback still drive the gas near R away from equilibrium temporarily (this direct cause- Lochhaas et al. E n e r g y a t R / P E ( R ) Tempest VirialThermal-only VirialKE th KE nt SFR (right axis)2.00 1.50 1.00 0.75 0.50 0.30 0.20 0.10 0.00Redshift 020406080100 S F R [ M / y r ] E n e r g y a t R / P E ( R ) Squall2.00 1.50 1.00 0.75 0.50 0.30 0.20 0.10 0.00Redshift 020406080100 S F R [ M / y r ] E n e r g y a t R / P E ( R ) Maelstrom2.00 1.50 1.00 0.75 0.50 0.30 0.20 0.10Redshift 020406080100 S F R [ M / y r ] Figure 3. Energies of the gas within 0 . R < r < R as a function of cosmic time (bottom axis) and redshift (topaxis). All energies are normalized by GM R . The virialenergy (Eq. 8) is plotted as the thick, red line. The thermalkinetic energy of the gas is plotted as the green dashed lineand the non-thermal kinetic energy is plotted as the bluedotted line. A thin solid red line indicates the virial energy ofthe gas near R if KE nt is neglected in the virial equation.The SFR of the central 20 kpc of the halo is shown as thethin black line, with values indicated on the right axis. and-effect relationship will be discussed further below,see § z (cid:46) . 2. Squall is more massive (see Figure 1), but hasvery strong bursts of star formation that drive it signif-icantly out of equilibrium. Nonetheless, it approachesvirial equilibrium somewhat earlier than Tempest, z (cid:46) . 4. Maelstrom, the most massive halo of the three, isroughly in virial equilibrium (thick red line close to zero)throughout much of its evolution z (cid:46) . 75, despite a sig-nificant number of star formation bursts. Maelstrom’sSFR peaks to higher values, and more frequently, thanTempest’s (at z (cid:46) R is dependent on the mass of the halo.We expect the gas near R to be at the virial tem-perature (either T vir or T mod ) only when the gas near R satisfies VE ∼ 0. However, these halos are not inperfect virial equilibrium throughout much of their evo-lution — instead, the virial energy oscillates near zeroand is perturbed by feedback events, especially at higherredshift.The thin red line in Figure 3 shows the virial energyof the gas near R if the non-thermal kinetic energy ofbulk flows is neglected, like in the standard definition ofvirial temperature. This curve falls below the VE = 0at all times other than following strong bursts of starformation for all halos. At late times, when the halosare massive enough to maintain virialized halos, neglect-ing the energy of bulk flows in the energy balance ofthe halo would lead to the conclusion that the halos areunder-virialized and under-supported and should be col-lapsing. It is only when the non-thermal kinetic energyof bulk flows is included that the halos can be said tobe close to virialized (even if perfect virial equilibriumis not achieved long-term).Figure 3 also shows the thermal and non-thermal bulkkinetic energies of the gas near R as the green dashedand blue dotted curves, respectively. This figure il-lustrates our basic finding that the gas near R hasroughly equal amounts of thermal and non-thermal ki-netic energy at nearly all times. Shortly after strongbursts of feedback, both the thermal energy and thenon-thermal kinetic energy increase, as feedback bothheats and accelerates gas.Figure 4 shows the same energy components as inFigure 3, but as a function of radius at a given snap-shot in time, over the radius range 0 . R to 1 . R OGGIE V: Modified Virial Temperature E n e r g y / P E z = 0.00Tempest R VirialThermal-only VirialKE th KE nt E n e r g y / P E z = 0.06Squall R E n e r g y / P E z = 0.07Maelstrom R Figure 4. Energies of the gas within the halos, as in Fig-ure 3, as a function of distance from the center of the haloand normalized by GM enc ( r ) r . The snapshots shown here arechosen as the latest times when the gas near R is close tovirial equilibrium (VE ≈ (note that the vertical scale differs from Figure 3). Eachhalo’s snapshot was chosen to reflect a time when thegas near R in each halo is roughly in virial equilib-rium, which is z = 0 for Tempest, z = 0 . 06 for Squall,and z = 0 . 07 for Maelstrom. The halo gas is closestto virial equilibrium for r (cid:38) . R , and again we seethat neglecting the non-thermal kinetic energy in thevirial equation leads to a configuration that is far out ofequilibrium. Feedback drives the gas away from equilib-rium in the inner regions of each halo, and some residualfeedback-driven outflows that traveled to the outer halocan push it out of virial equilibrium near R as well, asin the case of Squall (middle panel). Figure 4 shows thatat z = 0 we expect any temperature derived from thevirial equation to be a poor description of the gas within r (cid:46) . R , where the virial energy deviates stronglyfrom zero. However, the virial energy is not exactly zerofor most of the volume and time so we do not expect ei-ther T vir or T mod to be a perfect descriptor of the gastemperature.In summary, we find that while the FOGGIE halos arerarely in perfect virial equilibrium (VE = 0), their totalvirial energies (Equation 8) are close to zero for much ofthe later stages of their evolution, when they are massiveenough to be expected to host a hot halo (see discussionsurrounding Figure 1 in § R , ex-cept when strong bursts of star formation feedback tem-porarily drive the halo out of equilibrium, after whichit settles back into an equilibrium state. Neglecting thenon-thermal kinetic energy contribution to the overallenergy balance of the halo would suggest these halos arefurther out of virial equilibrium than they really are.We have shown that the non-thermal kinetic energy ofbulk flows and turbulence are important components ofenergy partition in halos and should not be neglected inanalyses that rely on accurate characterization of theirmain properties. MODIFIED VIRIAL TEMPERATURE INFOGGIE HALOSWith an understanding of when and where the CGMgas is in virial equilibrium and thus when and wherewe expect T mod to be a good descriptor of the gas tem-perature, we move on to calculating the modified virialtemperature for the FOGGIE halos. We measure thegas temperature in the simulations and compare to both T mod and T vir to determine if the modified virial tem-perature is a better descriptor of the gas temperature inthe CGM than the standard virial temperature.5.1. Calculating Modified Virial Temperature Lochhaas et al. 200 100 0 100 200 Velocity [km/s]0.0000.0050.0100.0150.020 P D F = 27.5= 39.3 z = 0.50 PDFBest fit 200 100 0 100 200 Velocity [km/s]0.0000.0050.0100.0150.020 P D F = 4.2= 51.9 200 0 200 400Radial Velocity [km/s]0.00000.00250.00500.00750.0100 P D F r = 87.3 r = 63.6 tan = 41.5200 100 0 100 200 Velocity [km/s]0.0000.0050.0100.015 P D F = 7.0= 35.0 z = 0.00 PDFBest fit 200 100 0 100 200 Velocity [km/s]0.0000.0020.0040.0060.008 P D F = 9.7= 55.7 200 0 200 400Radial Velocity [km/s]0.0000.0020.0040.0060.008 P D F r = 68.3 r = 49.0 tan = 35.9 Figure 5. Mass-weighted distributions of velocities of gas within 0 . R < r < R are shown as the black solid lines, splitinto the three spherical directions: θ (left), φ (middle), and radial (right). The top row shows gas near R at z = 0 . R at z = 0, both in the Tempest halo. The best-fit Gaussian model (to the two tangentialvelocity directions) and the best-fit double-Gaussian model (to the radial velocity direction) are shown as the blue dashed lines.For the double-Gaussian model, the two Gaussians that make it up are shown with thin blue dashed lines. Dotted vertical linesshow the location of the best-fit mean of the Gaussian and dotted horizontal lines show ± To compute T mod we need a measurement of the non-thermal kinetic energy KE nt , and thus the bulk flowvelocity of the gas. As gas velocities are tracked explic-itly, cell-by-cell, at runtime, we could use the simulatedvelocity fields to obtain a measure of the non-thermalkinetic energy as in § 4. However, rather than integrateall the cell-level data directly, we will use statistical de-scriptions of their distributions in velocity space withinradial shells, as gas velocities are typically more acces-sible in CGM observations than the unknown sum ofall kinetic energies. In doing so, we must be careful toconsider coherent bulk flows, such as filaments and fastoutflows, apart from localized turbulence or convectivemotions.First, we decompose the CGM velocity into sphericalcomponents: radial velocity v r and velocities v θ and v φ tangential to the radial direction (Figure 5). The tan-gential velocities are defined arbitrarily, not relative tothe disk of the galaxy. Near the virial radius, we findthat any rotation in the filament-removed CGM gas isnegligible at times when the halo is near virial equilib-rium (i.e. at z = 0 in the Tempest halo, Figure 5), sowe do not include bulk CGM rotation in our accounting of the halo’s non-thermal gas motions. The spread ofthe tangential velocity distributions will thus be a goodtracer for turbulent or convective non-thermal motions.We perform a least-squares fit of Gaussian distributionsof the form f tan ( v ) = A exp − ( v − µ ) σ (9)to the two tangential velocity distributions to obtainthe peak velocity, µ , and velocity dispersion, σ , of thetwo tangential velocity distributions (the amplitude A is a free parameter that has no physical meaning in thiscase, as all of the velocity distributions being fit arenormalized). This also allows us to confirm that thepeak of these distributions are close to zero, indicating asmall net rotation. The left and center panels of Figure 5show the two tangential velocity distributions with theirbest-fit Gaussian distributions for the gas near R inthe Tempest halo at z = 0 . z = 0, as an example. Observations and simulations alike have found the CGM to berotating within ∼ . R (Hodges-Kluck et al. 2016; Ho et al.2019; Martin et al. 2019; Ho et al. 2020), although Oppenheimer(2018) shows rotation is sub-dominant to other forms of non-thermal gas motions at the virial radius. OGGIE V: Modified Virial Temperature z = 0 . 5, the gas does show some bulk rotation in the θ direction as indicated by a non-zero µ = − 27 km s − ,but the rotation is not long-lived or coherent, generallyappearing and disappearing from snapshot to snapshotas the halo evolves. By z = 0, there is no significantrotation of the Tempest halo’s gas near R in eitherthe θ or φ directions, as indicated by µ ≈ f r ( v ) = A exp − v σ + A exp − ( v − µ r ) σ r (10)where the first term in the sum represents the contribu-tion due to turbulence and the second term representsthe contribution due to outflows. The turbulence Gaus-sian has its mean fixed to 0 km s − and its standard de-viation fixed to σ tan , where σ tan is the best-fit standarddeviation of a Gaussian fit to the tangential velocity dis-tribution (not shown), given by v tan = (cid:113) v θ + v φ . Theoutflow Gaussian’s mean and standard deviation are un-constrained, but it is defined to be zero for v < − so that it measures strictly the outflow componentof the radial velocity. Both Gaussians’ amplitudes areunconstrained. The right panels in Figure 5 show theradial velocity distribution along with the best-fit sumof Gaussians to the distribution. The cut to remove fil-aments removes all gas with v r (cid:46) 50 km s − (this valuechanges slightly with redshift), and galactic outflows canbe seen as the Gaussian component shifted toward largepositive velocities.With the velocity distributions characterized, we cancompute the non-thermal kinetic energy from bulk flowsand thus T mod . Because we do not expect significant ro-tation of the CGM near the virial radius and the meansof the best-fit Gaussians to the tangential velocity dis-tributions are close to zero, we use only the standarddeviation of the best-fit Gaussians to the tangential ve-locity distributions as a measure of turbulent velocity.Turbulence will always contribute to the virializationand non-thermal kinetic energy of the gas, but it is un-clear how much, if at all, outflows with velocities < v esc in the radial velocity distribution contribute to the viri-alization of the halo. It could be that outflows producea perturbation from virial equilibrium for the halo (like filaments and satellite galaxies) and thus should not beincluded in the derivation of the modified virial tem-perature, or it could be that outflows provide a neces-sary supporting force for the halo and thus should beincluded.Instead of attempting to characterize how much out-flows contribute to the virialized, non-thermal kineticenergy, we define two ways of calculating the modifiedvirial temperature: one with only turbulence, and onewith both turbulence and outflows. The total kineticenergy per mass (specific kinetic energy) of a velocitydistribution is (cid:82) v f ( v ) d v where f ( v ) is the probabil-ity of a parcel of gas having a velocity between v and v + d v , normalized such that (cid:82) f ( v ) d v = 1. For aGaussian velocity distribution, like we find for the tan-gential velocity distributions, the specific kinetic energyis simply σ where σ is the standard deviation of theGaussian, the velocity dispersion. For the radial velocitydistribution, we model f ( v ) as the sum of two Gaussians,one of which cuts off for v r < − , so we compute (cid:82) v r f ( v r ) d v r directly from the best-fit function, whichincludes both the radial-direction turbulent velocity dis-tribution and the outflow velocity distribution. The spe-cific kinetic energy of a velocity distribution that can bedescribed as a three-dimensional Gaussian is given byKE = ( σ θ + σ φ + σ r ), where each σ is the velocity dis-persion in one of the three directions. In our case, wehave measured velocity dispersions for the two tangen-tial dimensions and assume that the velocity dispersionin the third, radial dimension can be described as theaverage of the tangential dispersions, σ r = ( σ θ + σ φ ),which leads toKE turbnt = 12 (cid:18) σ θ + 32 σ φ (cid:19) (11)where σ θ and σ φ are the standard deviations of the best-fit Gaussians to the θ and φ velocity distributions, re-spectively. The non-thermal specific kinetic energy ofboth turbulence and outflows is given byKE turb+outnt = 12 (cid:0) σ θ + σ φ (cid:1) + 12 (cid:90) v esc − . v ff v r f ( v r ) d v r (12)where the lower bound of the integral reflects the cutmade to remove the inflow filaments (see § v esc assumes thatoutflows that are fast enough to escape the halo do notcontribute to the virialization of the halo. Note thatthe contribution of turbulence in the radial direction isincluded in f ( v r ) within the integral, so the factor of3 / T vir is a single tem-perature for all locations in the halo, by definition. How-ever, for T mod we can derive a radius-dependent form by2 Lochhaas et al. making some simple substitutions. In Equation (4), wereplace M with M enc ( < r ), the enclosed mass withina radius r , and replace R with r . We also measure σ θ , σ φ , and (cid:82) v r f ( v r ) d v r within radial bins. This gives T turbmod = 12 µm p k B (cid:20) GM enc ( < r ) r − (cid:0) σ θ ( r ) + σ φ ( r ) (cid:1)(cid:21) . (13)and T turb+outmod = 12 µm p k B (cid:20) GM enc ( < r ) r − (cid:32) σ θ ( r ) + σ φ ( r ) + (cid:90) v esc − . v ff ( r ) v r f ( v r , r ) d v r (cid:33)(cid:35) . (14)In equations (13) and (14), we find σ θ ( r ), σ φ ( r ), and f ( v r , r ) by fitting these functional forms to the velocitydistributions in bins of radius, from 0 . R to 1 . R ,of radial width 0 . R . We now have all the analytictools we need to compare T vir and T mod as descriptionsof simulated halo gas.5.2. T mod Better Describes the CGM Temperaturethan T vir Figure 6 shows two-dimensional mass-weighted his-tograms of the gas within 0 . R < r < R intemperature-time space. The standard virial temper-ature (Equation 3) is shown as the dashed orange lineand the modified virial temperature, calculated eitherwith or without the outflow kinetic energy (equations 14or 13, respectively) is shown as the dotted red and solidred lines, respectively. When there is a strong burst ofstar formation, the temperature histogram of gas near R shifts upward to higher temperatures as feedbackheats the CGM. At low redshift and during quiescentperiods, when the gas near R is closest to virial equi-librium (see Figure 3), T mod is a closer description of thepeak of the temperature distribution than the standard T vir , which over-estimates the temperature of the gasnear R by roughly a factor of two at nearly all timesunless feedback is coincidentally heating the CGM gas.Figure 7 shows a mass-weighted two-dimensional his-togram of temperature of CGM gas as a function of dis-tance from the halo center compared to the standardvirial temperature (orange dashed line) and the two cal-culations for the modified virial temperature (solid redfor turbulence-only, Equation 13, dotted red for turbu-lence and outflows, Equation 14), at the same time snap-shots as in Figure 4 for each halo. The distribution issmoother at smaller radii where more of the cells in agiven radial bin are refined to a higher resolution (see § l o g T e m p e r a t u r e a t R [ K ] Tempest T vir T turbmod T turb + outmod SFR (right axis)1.50 1.00 0.75 0.50 0.300.20 0.10 0.00Redshift log Mass020406080100 S F R [ M / y r ] l o g T e m p e r a t u r e a t R [ K ] Squall2.001.50 1.00 0.75 0.50 0.300.20 0.10 0.00Redshift log Mass020406080100 S F R [ M / y r ] l o g T e m p e r a t u r e a t R [ K ] Maelstrom2.00 1.50 1.00 0.75 0.50 0.30 0.20 0.10Redshift log Mass020406080100 S F R [ M / y r ] Figure 6. The temperature of gas with 0 . R < r < R is shown as a mass-weighted distribution as a function ofcosmic time (bottom axis) and redshift (top axis), with darkcolors indicating the peak of the mass-weighted temperaturedistribution. The orange dashed line shows the standardvirial temperature (Equation 3) and red solid and red dashedlines show the modified virial temperature, calculated usingthe kinetic energy due to turbulence only (Equation 13) orboth turbulence and outflows (Equation 14), respectively.The thin black line shows the star formation rate, with valueson the right axis. OGGIE V: Modified Virial Temperature T mod is a much bet-ter description of the mass-weighted temperature distri-bution than the standard T vir . The standard T vir over-estimates the temperature of the majority of the gas (bymass) by a factor of ∼ T vir to describe the gas temperaturebest. The modified virial temperature calculated includ-ing both turbulence and outflows, T turb+outmod , performssomewhat better in describing the peak of the tem-perature distribution near R than T turbmod calculatedfrom turbulence alone in Tempest and Squall, whereasit makes no difference in Maelstrom, perhaps indicatingthat the kinetic energy due to outflows is more impor-tant to include in the overall energy balance of lower-mass halos.When the kinetic energy due to outflows is included inthe energy balance of gas near R , the corresponding T mod is smaller because the KE nt term in Equation (7)is larger, driving a larger deviation below the standard T vir . Because it is unclear how much, if at all, outflowswith v < v esc contribute to the energy budget of theCGM gas, we report both T turbmod and T turb+outmod and donot pick one or the other as a better description of thetemperature of the bulk of the CGM gas. However, Fig-ure 6 indicates there are some times when it appears T turb+outmod is a better descriptor of the peak of the tem-perature distribution near R than T turbmod , which in-dicates there are some times in the halo’s history whenoutflows are an important contributor to the energy bud-get of the halo gas, and some times when they are not(although note that outflows faster than the escape ve-locity are never included in T turb+outmod ). In addition, atdifferent radii within the halo one or the other form of T mod may be a closer description of the peak of the gastemperature distribution, indicating that outflows mayonly be important to the energy budget at certain radii.We also see from Figure 7 that neither form of T mod is an appropriate descriptor of the gas temperature atsmall radii where the strong impact of feedback causesthe assumption of virial equilibrium to break down, asexpected. FEEDBACK DRIVES DEVIATIONS FROMEQUILIBRIUMWhile T mod is clearly a better descriptor of the gastemperature in the outskirts of the FOGGIE halos than T vir for most each halos’ evolution, there are times fol-lowing strong bursts of star formation when outflowspush the halo out of virial equilibrium and away from T mod or T vir (Figure 6). Here, we quantify this cross-correlation between the SFR and energy or temperatureof the gas near R explicitly. l o g T e m p e r a t u r e [ K ] R z = 0.00Tempest T vir T turbmod T turb + outmod log Mass0 50 100 150 200 250Radius [kpc]4.504.755.005.255.505.756.006.256.50 l o g T e m p e r a t u r e [ K ] R z = 0.06Squall log Mass0 50 100 150 200 250Radius [kpc]4.504.755.005.255.505.756.006.256.50 l o g T e m p e r a t u r e [ K ] R z = 0.07Maelstrom log Mass Figure 7. The mass-weighted temperature of the filament-and satellite-removed CGM gas at times when the regionnear R is in virial equilibrium (as indicated in each panel)is shown as a two-dimensional histogram in temperature-radius space, with dark colors indicating the peak of themass-weighted temperature distribution at a given radius.The standard virial temperature (Equation 3) is radius-independent and shown as the orange dashed line. The twoways of calculating the modified virial temperature, with andwithout outflows (Equations 14 and 13) are shown as thedotted and solid red lines, respectively. Lochhaas et al. C r o ss - c o rr e l a t i o n w i t h S F R Tempest VirialKE th KE nt C r o ss - c o rr e l a t i o n w i t h S F R Squall0 500 1000 1500 2000Time delay [Myr]1.000.750.500.250.000.250.500.751.00 C r o ss - c o rr e l a t i o n w i t h S F R Maelstrom Figure 8. The cross-correlation between SFR and variousenergies of the gas near R as a function of time delay(Equation 15). The correlation is positive for τ (cid:46) − . τ ∼ − 75 Myr, quantifying what is easyto see by eye in Figure 3, that the halo is driven out of virialequilibrium by strong bursts of feedback. Figure 8 shows a time-delay cross-correlation betweenthe central galaxy’s SFR and the energy of the gas at R . The cross-correlation is computed as ξ ( τ ) = 1 N N (cid:88) i [SFR( t i ) − SFR][E( t i − τ ) − E] σ SFR σ E (15)where SFR( t i ) is the star formation rate and E( t i ) is theenergy, which can be VE (Equation 8), KE th , or KE nt ,at time snapshot t i . SFR and E are the averages of SFRand energy over time, σ SFR and σ E are the standarddeviations of SFR and energy over time, τ is a time-delay shift of one function relative to the other, and thesum is taken over all N time snapshots. This function isnormalized such that ξ ( τ ) = 1 would indicate a perfectcorrelation between SFR and energy at a time delay of τ , ξ ( τ ) = − τ , and ξ ( τ ) = 0 indicates no correlation at τ . Weperform this calculation over every snapshot output bythe FOGGIE runs, separated by ∼ τ (cid:46) − . nt , and KE th (except for Mael-strom, see below) are all positively correlated, peakingat τ ∼ 25 Myr for Tempest and τ ∼ 75 Myr for Squalland Maelstrom. This means that a burst of star forma-tion drives up the energy of the gas near R ∼ − ∼ − . R to fully “relax” to what it was beforethe burst, as can also be seen in Figure 3. The strongcorrelation between SFR and gas energy at R con-firms what was suspected from Figure 3: strong burstsof feedback, driven by large SFRs, is what drives the gasnear R away from the halo’s “natural state” of virialequilibrium.Interestingly, Squall shows two prominent peaks in thecross-correlation, but the second peak at ∼ 800 Myris likely driven by the two extremely strong bursts ofstar formation at ∼ . ∼ . M (cid:12) yr − and re-calculating the cross-correlation, which greatly diminishes the strength of thesecond peak without affecting the primary peak at 75Myr (not shown). For large τ , the cross-correlation sam-ples fewer points and so it can be disproportionatelydriven by a handful of extreme events.Maelstrom shows the weakest correlation strength be-tween SFR and energy of the gas near R out of thethree halos examined here. It is also the most quies-cent of the three halos, with few very strong bursts of OGGIE V: Modified Virial Temperature nt and SFR, with a weak peakat ∼ 100 Myr, that declines over 1 Gyr. Interestingly,there is an anti-correlation between SFR and KE th inMaelstrom roughly constant with time-delay, which isnot seen in either Squall or Tempest, but a strong pos-itive correlation with KE nt . This indicates that despiteMaelstrom’s general quiescence compared to the othertwo halos where it appears bursts of star formation donot heat the gas much, the non-thermal kinetic motionstriggered by star formation feedback are still importantto the overall energy balance (or lack thereof) of the gasat R .Figure 9 shows a similar time-delay cross-correlation,this time correlating the SFR with the mass of gas at R within different temperature bins relative to thestandard T vir as marked on the figure (calculated withEquation 15, but replacing E( t i ) and E with the massin a temperature bin at t i and averaged over time, re-spectively). The two hottest temperature bins ( (cid:38) T vir )in Tempest and Squall are positively correlated withbursts of SFR, indicating that the presence of ∼ T vir gas at R is due to star formation feedback, not thatthe gas at R is naturally at T vir when the halo is fullyrelaxed. The two coolest temperature bins ( (cid:46) T vir ) areanticorrelated with the SFR in Tempest and Maelstrom,indicating that bursts of star formation remove cool gasmass from R . This trend can be seen by eye in Fig-ure 6, where the only time the gas at R is close toor greater than the standard T vir (orange dashed line inthat figure) is shortly following a burst of feedback, afterwhich the temperature drops again as the halo relaxes.Like the energy cross-correlation, the temperature cross-correlation takes ∼ − . R . Just like above with the energy-SFR cross-correlation, the secondary peak in the hot gas mass in Squall is driven by just two of the strongest starburstevents and is greatly reduced if we cap the SFR at 20 M (cid:12) yr − (not shown). Maelstrom shows the opposite trend,where the two hottest temperature bins are not particu-larly correlated with SFR while the two coolest temper-ature bins are strongly anti-correlated with SFR. Theanti-correlation of cool gas mass with SFR is expectedif feedback heats the gas near R , but lack of corre-lation with hot gas is unexpected and may just suggestthat there are not enough significant peaks in the SFR todrive the gas temperature near R significantly awayfrom its equilibrium value. This seems to be corrobo-rated by Figure 6 (bottom panel), where the tempera-ture of the gas near R does not exhibit as many shortdeviations to high temperatures as in the other halos. IMPLICATIONS OF A COOLER CGMIn Sections 4 through 6, we found that across cosmictime and throughout the outskirts of a galactic halo,non-thermal gas motions are critical to understandingthe energy partition and the temperature of the halogas. For the simulated halos studied here, virial equilib-rium holds near R only if non-thermal kinetic energiesare included in the energy balance and only when thehalo is not being perturbed by strong feedback. Theconsequence of this finding is that the standard virialtemperature T vir overestimates the peak of the gas tem-perature distribution by a factor of ∼ R .The end result is a somewhat cooler galactic halo thanexpected from standard galaxy formation theory.While we carefully calculate the contribution to theenergy budget of bulk flows here, this may not be possi-ble in many cases, such as in interpreting observations orin analytic models or idealized simulations of L ∗ galaxyhalos. In these cases, we suggest using a halo tempera-ture roughly a factor of 2 lower than the standard virialtemperature. The factor of two arises due to the roughlyequal contributions of thermal and non-thermal energiesto the halo’s energy budget throughout most of its evo-lution (see Figure 3), which we suggest as a general ruleof thumb in cases where emergent non-thermal kineticenergies cannot be explicitly calculated.Observations of the diffuse gas making up the CGM,especially in the outskirts of galaxy halos, are typicallydone in absorption line studies against the light froma bright background source. This generally restrictsthe derived information to at most a handful of linesof sight through any given galaxy’s halo (an importantexception is M31, for which multiple sightlines through6 Lochhaas et al. C r o ss - c o rr e l a t i o n w i t h S F R o f M a ss F r a c t i o n i n T b i n Tempest T < T vir T vir T vir T > T vir + 0.5 dex0 500 1000 1500 2000Time delay [Myr]1.000.750.500.250.000.250.500.751.00 C r o ss - c o rr e l a t i o n w i t h S F R o f M a ss F r a c t i o n i n T b i n Squall0 500 1000 1500 2000Time delay [Myr]1.000.750.500.250.000.250.500.751.00 C r o ss - c o rr e l a t i o n w i t h S F R o f M a ss F r a c t i o n i n T b i n Maelstrom Figure 9. The cross-correlation between SFR and mass indifferent temperature bins relative to the standard T vir asindicated in the figure, as a function of time delay τ (Equa-tion 15). The mass in the warmer temperature bins is posi-tively correlated with SFR while the mass in the cooler binsis anticorrelated, quantifying what is easy to see by eye inFigure 6, that temperatures (cid:38) T vir are only achievable fol-lowing a burst of SFR and not when the halo is in relaxedvirial equilibrium. the same galaxy’s CGM can be obtained, Lehner et al.2015, 2020), and restricts the dimensionality of that in-formation to only line-of-sight velocities. Galaxy forma-tion simulations or cosmological simulations can providemore information than these pencil-beam observations,but they may be under-resolving the small-scale struc-ture of the CGM, especially on scales far from the galaxy.In addition, the implementation of star formation andfeedback in the central galaxy varies across different sim-ulations and with resolution. It is necessary to developand analyze analytic models for the CGM to tie thediffering information from observations and simulationsback to the gas physics that govern the CGM, and it isin the context of these analytic models where a cooler-than-expected CGM has significant implications that weoutline below.7.1. Thermal Pressure of the CGM Absorption-line surveys of the CGM routinely discoverboth hot ( T ∼ K) and cool ( T ∼ K) gas, fre-quently in the same line of sight and at the same line-of-sight velocity (Tumlinson et al. 2013; Bordoloi et al.2014; Borthakur et al. 2015; Werk et al. 2016; Keeneyet al. 2017; Berg et al. 2018; Chen et al. 2018; Muzahidet al. 2018). Ionization modeling of low-ionization stateabsorption lines produces estimates of the gas densitythat tend to show that the cool gas density is similarto the hot gas density (Werk et al. 2014; Stern et al.2016) despite large temperature differences, and thusthe two gas phases are out of pressure equilibrium. How-ever, multiphase CGM models that pose the cold phaseis found in small clouds embedded within the volume-filling hot phase generally expect that the cool and hotgas are in pressure equilibrium at the pressure of thehot phase (Mo & Miralda-Escude 1996; Maller & Bul-lock 2004). An overall cooler CGM can help alleviatethis discrepancy somewhat, as a lower temperature forthe hot phase reduces its thermal pressure and thus re-duces the thermal pressure needed in the cold phase tomatch it, allowing the cold phase to be more diffuse. Afactor of two decrease in hot-phase temperature leadsto a factor of two decrease in the expected cold-phasedensity if the phases are in pressure equilibrium. Thedetailed multiphase ionization modeling of Haislmaieret al. (2020) finds the warm and cool gas phases mayactually be in pressure equilibrium in some cases, butout-of-equilibrium solutions are not entirely ruled out,and those authors also find that the thermal gas pressureof all phases (regardless of whether they are in pressureequilibrium with each other) is lower than typically ex-pected for ∼ L (cid:63) halos. A lower thermal pressure maybe explained by the cooler-than-expected volume-filling OGGIE V: Modified Virial Temperature Cooling Time of CGM Gas The efficiency of radiative cooling is strongly depen-dent on temperature and peaks around T ∼ K formetal-enriched gas. At these temperatures, CGM gasproduces many intermediate ions such as C IV , Si IV ,and potentially O VI , all species that are frequentlyfound in absorption in the CGM surrounding MilkyWay-like galaxies. This intermediate-temperature gasmay live in radiatively-cooling interface regions betweenhot and cold phases, which grow as hot and cold gas tur-bulently mix together (e.g., Begelman & Fabian 1990;Slavin et al. 1993; Wakker et al. 2012; Kwak et al. 2015;Ji et al. 2019; Fielding et al. 2020a; Tan et al. 2020). Acooler hot phase means less mixing is required for gasto reach intermediate temperatures and cool efficiently,perhaps allowing cooling to proceed more quickly thanwould be expected in a hotter medium. This could leadto more cool gas forming by entraining mass from thehot phase, explaining observations of the cool and inter-mediate phases.7.3. CGM Mass Estimates from X-Ray Observations X-ray studies observe the hot component of the CGMof both the Milky Way and external galaxies. X-rayemission is strongly dependent on the gas density, sotypically the CGM is only observed in X-ray emissionin the densest regions closest to massive galaxies. Apopular strategy for characterizing the hot CGM gasthat emits in X-ray is to fit its density profile with a β model, which is a power-law profile where the parame-ter β describes the power. To estimate the total massof hot gas in a galaxy’s CGM, the β model is extrapo-lated out to the virial radius and integrated (Anderson& Bregman 2010; Gupta et al. 2012; Das et al. 2019),finding ∼ − M (cid:12) of hot gas within the halo (Ander-son & Bregman 2010). This method assumes that thegas maintains its hot temperature out to the virial radiusand that it is only the decline in density that leads tothe decrease in X-ray surface brightness below detectionlimits in the outskirts of galaxy halos.Figure 7 shows that the gas temperature decreaseswith increasing radius approaching the virial radius, andis a factor of ∼ β models may not be accurate. If the decreaseof gas temperature with increasing radius is what drivesthe low X-ray surface brightness in the outskirts of galac-tic halos, rather than lack of gas mass, the gas mass inthe halo’s volume-filling phase may be higher than es-timated. However, the volume-filling gas is not “hot”( T > K) but “warm” ( T = 10 − K , see Fig-ure 7), so the hot gas contribution to the mass of thehalo may be lower than estimated even if the volume-filling gas phase contribution to the halo mass is higherthan estimated. A detailed analysis of the relative con-tribution of each gas phase to the mass budget of theCGM is beyond the scope of this paper.7.4. The Origin of O VI The UV doublet of O VI is the highest-ionizationtracer of warm gas that is readily accessible outside theX-ray. Tumlinson et al. (2011) presented correlations ofCGM O VI abundance with galaxy properties, findinga bimodality in presence of O VI that depends on SFRof the central galaxy: star-forming galaxies have moreO VI in their halos, implying that star formation activityleads to circumgalactic O VI . Oppenheimer et al. (2016)proposed a different source for gas traced by O VI : theionization fraction of O VI peaks near the virial tem-perature of roughly Milky Way-mass galaxies such thatthe O VI bimodality is actually a bimodality in halomass (and thus virial temperature), rather than relat-ing primarily to SFR. Galaxies living in massive haloshave virial temperatures too high for O VI to be preva-lent (oxygen ions are instead ionized further to O VII orO VIII ), and these galaxies are also typically quenched,thus explaining the O VI bimodality with SFR as well.However, McQuinn & Werk (2018) showed that thereis a slight offset between the halo mass where virial-temperature-tracing O VI is expected to be prevalentand the halo mass where O VI is most frequently ob-served, such that O VI is detected surrounding moremassive halos whose virial temperatures are too large,seemingly, for O VI to survive (see their Figure 3). Mc-Quinn & Werk (2018) proposed a spread in gas tem-perature around the virial temperature as one possiblesolution to this dilemma, such that some of the gas ex-ists at lower temperatures where O VI can be found.In this paper, we showed the temperature of CGM gasis lower than expected from the classical virial analysis.While we do not yet have enough halos simulated athigh resolution to examine this trend with halo mass,the prevalence of the hydrostatic mass bias in galaxyclusters (see § Lochhaas et al. perature. If the actual temperature of halo gas is lowerthan expected for halos of all masses, then the halo massat which the O VI ionization fraction peaks at the virialtemperature is larger than expected, potentially explain-ing why O VI is seen with such abundance surroundingmore massive galaxies than expected. The highest-masshalo explored in this paper, Maelstrom, does still showa temperature lower than the standard virial tempera-ture, so it seems likely that halos of all masses will havelower temperatures than expected. We note, however,that without a more rigorous study of what drives theturbulence and other non-thermal gas motions and howthose processes may change with halo mass, we cannotderive a halo mass scaling for the modified virial tem-perature to confirm this scenario.A substantial amount of the O VI in a galaxy halomay arise from a cooler, photoionized phase rather thanthe warm, volume-filling virialized phase (Stern et al.2018; Strawn et al. 2020). The scenario outlined in thissubsection assumes most of the observed O VI arisesfrom the warm phase in collisional ionization equilib-rium, rather than from a cool phase in photoionizationequilibrium. If O VI -hosting gas is primarily cool, then atrend of O VI column density with halo mass would notbe tracing the virial temperature of the halo but ratherthe amount of cool gas in a halo as a function of halomass. Reality is likely to be a mixture of both scenarios,and distinguishing between them is beyond the scope ofthis paper.7.5. The Importance of Turbulence The main result of this study is that non-thermal gasmotions, such as turbulence, are important to the energypartition of a virialized halo, and this has important con-sequences for the temperature of the CGM. Significantturbulent motions also have effects that go beyond justmodifying the virial temperature: turbulence can pro-vide pressure support to the CGM (Oppenheimer 2018;Lochhaas et al. 2020) and it can affect how cool gas con-denses out of the hot medium (Voit 2018) or mixes withthe hot medium to efficiently create more cool gas (e.g.,Fielding et al. 2020a).In particular, turbulent pressure drives halo gas awayfrom purely hydrostatic solutions where it is assumedthat thermal pressure exactly balances the gravitationalpotential. The importance of non-thermal pressure sup-port has been known for some time in galaxy clusters,where the idea of a “hydrostatic mass bias” is well known(Nagai et al. 2007; Piffaretti & Valdarnini 2008; Lauet al. 2009, 2013; Shi et al. 2015, 2016; Biffi et al. 2016;Shi et al. 2018; Simionescu et al. 2019; Gianfagna et al.2020). The hydrostatic mass bias is the difference be- tween inferred cluster mass from a hydrostatic assump-tion for cluster gas and the true cluster mass, and moststudies find differences on the order of ∼ − ∼ − 30% of the total pressure (Vazza et al. 2011; Nel-son et al. 2014; He et al. 2020). Clearly, non-thermal gasmotions are important in galaxy clusters, and there is noreason to suspect that galaxy-scale halos lack significantnon-thermal pressure or energy. Indeed, we have shownin this paper that non-thermal kinetic energy is a signif-icant contribution to the energy balance of galaxy-scalehalos, and that this has consequences for the temper-ature of the halo gas. The consequences of significantnon-thermal energy on the pressure of the halo gas willbe explored in a forthcoming paper.Rudie et al. (2019) observed the CGM of star-forminggalaxies at z ∼ VI , the highest ionization stateion probed in that study, was larger, roughly a few tensof km s − . If O VI traces the warmest phase of gas,this would indicate turbulence roughly on the scale ofthe simulated turbulence in the warmest gas phases inthe FOGGIE halos. Lehner et al. (2014) found a simi-lar result for O VI , but also found roughly half of C IV and Si IV absorption lines at high redshift were broaderthan would be expected from pure thermal broadening,indicating that there may be turbulence in the warmgas phase probed by these mid-ions as well. Rudie et al.(2019) found lower ionization state ions had narrowerabsorption lines, indicating less turbulent broadening.If lower ions are found in cool clouds embedded withina warmer halo, each individual cloud may not have sig-nificant internal turbulence, leading to narrow individ-ual absorption components, but the collection of coolclouds may trace the turbulence of the hot phase inwhich they are embedded. If that is the case, it is the ve-locity dispersion between individual cool-phase compo-nents that traces the hot-phase turbulence, which Rudieet al. (2019) find to be ∼ − 200 km s − . Some of thesecomponents may be tracing fast-moving coherent struc-tures like outflows or accretion filaments (and indeedthey find a subset of absorbers with velocities above theescape velocity of their host halo). Zahedy et al. (2019)carried out a similar analysis at lower redshift ( z ∼ . OGGIE V: Modified Virial Temperature ∼ 80% in turbulent energy nearthe virial radius with increasing resolution (see their Fig-ure 13), which was balanced by a decrease in thermalenergy and thus likely temperature, although they donot discuss temperature explicitly. Assuming a turbu-lent cascade from large to small scales, the majority ofthe turbulent energy is located in the large scales onwhich turbulence is first driven, so it may be that thisdriving scale is all that needs to be resolved in order tocapture the majority of the turbulent energy. Li et al.(2020) found the driving scale for turbulence in galaxyclusters to be on the order of the scale of feedback, soan analysis of the impact of feedback at different scalesin CGM simulations may specify the driving scale andthus enlighten the resolution needed to resolve the bulkof the turbulent energy in the CGM. SUMMARY AND CONCLUSIONSIn this paper, we derived a modified virial tempera-ture by explicitly including the kinetic energy of non-thermal bulk gas motions in the virial equation for agalaxy halo (Equation 7). We made two estimates forthe non-thermal kinetic energy: one that includes onlyturbulence (Equation 11) and one that includes bothturbulence and bulk outflows (Equation 12). We usedthe Figuring Out Gas & Galaxies In Enzo simulationsto show how non-thermal kinetic energy contributes togalaxy halos roughly equally to thermal kinetic energy,motivating the need for non-thermal kinetic energy inconsiderations of virial equilibrium. Even when all formsof energy are accounted for, galaxy halos are generallynot in virial equilibrium throughout much of their evo-lution (Figure 3) and only approach equilibrium at lowredshifts when the halo mass surpasses few × M (cid:12) and only when strong bursts of stellar feedback are notperturbing the halo gas (Figure 8). Finally, we showedthat the modified virial temperature is a better descrip-tion of the gas temperature for most of the gas mass inthe outskirts of a galaxy halo than the standard virialtemperature, which is ∼ × too large, even when the halo is virialized (Figures 6 and 7). The only timeswhen the standard T vir is a good descriptor of the gasnear R is for a short time following a strong burst offeedback (Figure 9), which may be difficult to “catch”in observations and only occurs when the halo is not invirial equilibrium — giving the expected temperaturefor the wrong reason.A lower-than-expected gas temperature in galaxy ha-los has important implications for analytic CGM mod-els and the initial conditions of idealized CGM simu-lations. If gas is cooler, thermal pressures are lower,radiative cooling is more efficient, expected X-ray sur-face brightnesses are lower, and galaxy halos may beable to maintain higher O VI column densities at largerhalo masses than expected ( § We sug-gest that analytic models and idealized simula-tions adopt the modified virial temperature at afactor of ∼ lower than the standard virial tem-perature for the initial conditions of any modelor simulation. A lower-than-expected halo temperature due to en-ergy contributions from non-thermal motions is not aunique feature in FOGGIE. This phenomenon shouldbe measurable in any self-consistent cosmological simu-lation where gaseous halos are built up along with galax-ies. However, other cosmological simulations with lowerspatial resolution than FOGGIE may not be capturingenough of the energy contained in the small scales of theturbulent cascade in order to make a considerable differ-ence to the overall energy of the halo gas. Indeed, it ispossible that at the resolution of FOGGIE, there is stillsome turbulent energy below the resolution scale that wedo not capture, so the magnitude of the difference be-tween standard and modified virial temperatures maybe even larger than what we find here. A full analysisof the structure of turbulence in the CGM in FOGGIEis forthcoming.0 Lochhaas et al. ACKNOWLEDGMENTSThis study was primarily funded by NASA via anAstrophysics Theory Program grant 80NSSC18K1105.We thank Nicolas Lehner, John O’Meara, Mark Voit,Philipp Grete, David H. Weinberg, and Claire Kopen-hafer for useful discussion. CL and RA were addi-tionally supported by HST GO Enzo (Bryan et al. 2014; Brummel-Smith et al. 2019)and yt Facilities: NASA Pleiades Software: astropy (Astropy Collaboration et al.2013, 2018), Cloudy (Ferland et al. 2017), Enzo (Bryanet al. 2014; Brummel-Smith et al. 2019), grackle (Smithet al. 2017), yt (Turk et al. 2011) APPENDIX A. DEFINITIONS OF VIRIAL RADIUS AND VIRIAL THEOREMThroughout this work, we use the radius enclosing 200 times the critical density of the universe, frequently referredto as R , as the “virial” radius. However, the true “virial radius” is more strictly defined from the collapse of a OGGIE V: Modified Virial Temperature not simply R because the overdensity factor evolves with redshift in a universe with dark energy (Bryan & Norman1998). Instead, the overdensity factor at z = 0 is actually closer to 100, not 200, for the Planck Collaboration et al.(2016) cosmological parameters Ω M = 0 . Λ = 0 . R falls at leastpartially within the zoomed refine box for each of the FOGGIE halos at all redshifts (see § R vir , does not. At z = 0, the difference between these different definitions of “virial radius” for the Tempesthalo is ∼ 50 kpc and this difference is smaller at higher redshift. At z = 0, R ≈ 170 kpc while R vir ≈ 220 kpc, forthe Tempest halo. We show in this appendix that the precise definition of the virial radius does not matter for ourqualitative conclusions.In addition, the virial theorem as presented in Equation (1) is originally defined for the sum of all matter in thesystem, not in thin radial shells as we do in this paper. However, in a galaxy simulation with sources and sinksof energy, such as star formation feedback heating and radiative cooling, a simple sum of the kinetic and potentialenergies of all matter within R or R vir is not expected to satisfy the virial theorem (and R vir is not a hard “edge”to the system in any case). To reduce the impact of energy sources and sinks, we neglected the inner parts of the halonear the stellar component of the galaxy. We calculated the virial energy in radial shells, but we could also calculatethe virial energy (Equation 8) as a cumulative sum (still neglecting the very inner halo near the galaxy r < . R ) tobetter match the original definition of the virial theorem. Figure 10 compares the energies within the halo as functionsof radius, computed either within thin radial shells (top panel) or cumulatively up to r (bottom panel), at z = 0 . z = 0 . 15 because at lower redshift, R vir entirely exits the forced refinement regionof the Tempest halo. We see there are no strong differences in the virial energy calculated either way, although thecumulative virial energy is smoother. The virial radius R vir is also compared to the radius we use throughout thispaper, R , and it is clear there is no strong difference in either the shell virial energy or the cumulative virial energybetween the two radii, as the virial energy becomes fairly flat with radius past ∼ 100 kpc in both cases. The differencebetween the virial energy including non-thermal gas motions and the thermal-only virial energy is far larger than thedifferences in virial energy calculation method or halo radius definition.Figure 11 (top panel) shows the virial energy (Equation 8) of gas within 0 . R < r < R (solid) or within0 . R vir < r < R vir (dashed) as functions of cosmic time, similar to Figure 3. The bottom panel of this figure showsthe same, but the virial energy is computed as the cumulative energy within either R or R vir . We show the virialenergies in this case only down to z = 0 . 15, because at smaller z , R vir entirely exits the zoom-in, high-resolutionbox and is instead located only in the low-resolution simulation domain, and we lose the valuable information aboutsmall-scale non-thermal kinetic energy such as turbulence that is so crucial to the virial energy and temperature ofthe halo gas. 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