A Graphical Interpretation of Circumgalactic Precipitation
DDraft version February 2, 2021
Typeset using L A TEX twocolumn style in AASTeX63
A Graphical Interpretation of Circumgalactic Precipitation
G. Mark Voit Michigan State UniversityDepartment of Physics and AstronomyEast Lansing, MI 48824, USA (Received 30 Nov 2020; Revised 19 Jan 2021; Accepted 29 Jan 2021)
Submitted to ApJ LettersABSTRACTBoth observations and recent numerical simulations of the circumgalactic medium (CGM) supportthe hypothesis that a self-regulating feedback loop suspends the gas density of the ambient CGMclose to the galaxy in a state with a ratio of cooling time to freefall time (cid:38)
10. This limiting ratiois thought to arise because circumgalactic gas becomes increasingly susceptible to multiphase conden-sation as the ratio declines. If the timescale ratio gets too small, then cold clouds precipitate out ofthe CGM, rain into the galaxy, and fuel energetic feedback that raises the ambient cooling time. Theastrophysical origin of this so-called precipitation limit is not simple but is critical to understandingthe CGM and its role in galaxy evolution. This paper therefore attempts to interpret its origin assimply as possible, relying mainly on conceptual reasoning and schematic diagrams. It illustrates howthe precipitation limit can depend on both the global configuration of a galactic atmosphere and thedegree to which dynamical disturbances drive CGM perturbations. It also frames some tests of theprecipitation hypothesis that can be applied to both CGM observations and numerical simulations ofgalaxy evolution.
Keywords: miscellaneous — galaxies INTRODUCTIONThis
Letter outlines some key concepts underlying thephenomenon sometimes called circumgalactic precipita-tion. The topic’s history stretches back to investiga-tions of thermal instability in both stratified interstellargas (e.g., Spitzer 1956; Field 1965; Defouw 1970; Bin-ney et al. 2009) and the gaseous cores of galaxy clus-ters (e.g., Fabian & Nulsen 1977; Mathews & Breg-man 1978; Cowie et al. 1980; Nulsen 1986; Malagoliet al. 1987; Balbus & Soker 1989; Loewenstein 1989;McCourt et al. 2012; Gaspari et al. 2013; Choudhury &Sharma 2016; Voit et al. 2017; Voit 2018; Choudhuryet al. 2019). Many of those rather technical papers aredifficult for non-experts to interpret and yet have deepimplications for both observations of the circumgalactic
Corresponding author: G. M. [email protected] medium (CGM) and analyses of numerical simulationsdesigned to model galaxy evolution.Several recent analyses of multiphase circumgalacticgas produced in numerical simulations of galaxy evo-lution have underscored the need for a broader under-standing of the astrophysics at play (e.g., Lochhaas et al.2020; Nelson et al. 2020; Fielding et al. 2020; Esmerianet al. 2020). Those simulations show that feedback fromboth supernovae and active galactic nuclei can maintainthe CGM in a multiphase state characterized by radialprofiles of pressure, gas density, temperature, and spe-cific entropy that have a well-defined median. Aroundeach median profile, fluctuations in density, tempera-ture, and entropy exhibit approximately log-normal dis-tributions with long tails toward lower temperature andentropy and greater gas density. A similar tail is gener-ally not present in the pressure fluctuations, indicating The term multiphase describes a medium in which temperaturesand densities of neighboring regions differ by orders of magnitude. a r X i v : . [ a s t r o - ph . GA ] J a n Voit that the tail consists of gas condensing out of the ambi-ent medium and producing cooler clouds, also known asprecipitation, in the CGM.The explanatory illustrations presented here are there-fore intended to help further an intuitive understandingof the CGM conditions that promote or inhibit devel-opment of a multiphase galactic atmosphere. Section 2discusses thermal instability and how it damps in strat-ified galactic atmospheres. Section 3 shows how variousperturbations of those atmospheres can overcome damp-ing and produce multiphase condensation. Section 4 re-lates the rate of multiphase condensation to both themedian state of the galactic atmosphere and the dis-persion of perturbations around the median. Section 5briefly considers the consequences of altering a galacticatmosphere’s entropy gradient. Section 6 summarizesthe paper. DAMPED THERMAL INSTABILITYThermal stability of a galactic atmosphere dependson whether a low-entropy perturbation within it coolsmore quickly (or heats more slowly) than its surround-ings, causing its entropy contrast to increase with time(e.g., Field 1965; Balbus 1986, 1988). In an opticallythin medium, a low-entropy perturbation can grow be-cause its greater particle density usually makes its ra-diative cooling rate greater. If the background mediumis homogeneous and not gravitationally stratified, thenthe entropy contrast of a low-entropy gas parcel con-tinually increases, without damping, until it reaches atemperature that slows or halts radiative cooling. Sucha medium is thermally unstable and tends to developmultiphase structure on a cooling timescale t cool = 32 nkTn e n i Λ( T, Z, n ) (1)where Λ(
T, Z, n ) is the radiative cooling function of as-trophysical plasma with temperature T , metallicity Z ,and particle density n , defined with respect to electrondensity n e and ion density n i .Gravity makes thermal instability more interesting.The equilibrium configuration of a galactic atmospherethat is both hydrostatic and convectively stable in its po-tential well must have a negative pressure gradient anda positive entropy gradient. Otherwise, convection sortsthe gas parcels until specific entropy becomes a mono-tonically rising function of the gravitational potential φ . The result is an atmosphere with an entropy gradi-ent that can be expressed in terms of the logarithmicslope α K ≡ d ln K/d ln r , where K ≡ kT n − / e , becausechanges in ln K are proportional to changes of specificentropy in a monatomic ideal gas. An adiabatic low-entropy perturbation in such an at-mosphere does not remain a low-entropy perturbationbecause of buoyancy. Instead, it accelerates toward thecenter, causing its entropy contrast to decrease. Even-tually it passes through a layer of equivalent specificentropy into a layer of lower specific entropy. It is thena high-entropy perturbation and begins to decelerate.What follows is a series of buoyancy-driven oscillationswith frequency ω buoy ∼ α / K t − , (2)where the freefall time t ff ≡ (2 r/g ) / is based on thelocal gravitational acceleration g (e.g., Voit et al. 2017).More formally, the perturbation has become an internalgravity wave with a Brunt-V¨ais¨al¨a frequency ω buoy .Those buoyant oscillations can prevent thermal insta-bility from producing multiphase structure in an other-wise static atmosphere with ω buoy t cool (cid:29)
1. Figure 1 il-lustrates the reason. The trajectories it shows were com-puted with the heuristic non-linear thermal instabilitymodel of Voit (2018). At the upper right is a trajectorybeginning at the green diamond on the atmosphere’s me-dian entropy profile, ¯ K ( r ). Along that median profile,
20 30 40 50 60 70 803035404550 20 30 40 50 60 70 80r (kpc)3035404550 K ( k e V c m )
20 30 40 50 60 70 803035404550
Heating > CoolingPositive Buoyancy Heating < CoolingNegative Buoyancy t c o o l / t ff = α K = / δr δK Δ K thermal pumping excites instabilitymotion through ambient medium damps instabilitylow-entropy gas descends faster than it coolsbuoyancy damping reduces entropy contrastperturbation eventually shreds and mixes ¯ K ( r ) Figure 1.
Schematic illustrations of damped thermal insta-bility in a stratified galactic atmosphere. A thick charcoalline shows the entropy profile of a background atmospherewith α K = 2 / t cool /t ff = 10. In the blue region be-low that line, cooling exceeds heating, and gravity pulls theperturbation toward smaller r . Above the line heating ex-ceeds cooling, and buoyancy pushes the perturbation towardlarger r . The perturbation starting at the green diamondbegins with a small amplitude and executes buoyant oscilla-tions amplified by thermal pumping and damped by motionthrough the ambient medium. Its amplitude ends up saturat-ing with δr/r ∼ α − / K ( t ff /t cool ) and δK/K ∼ α / K ( t ff /t cool ).The infalling perturbation starting on the lower right andmarked by a green triangle begins with a larger amplitudeand decays to the saturation amplitude as the perturbationsheds kinetic energy into the ambient medium. raphical Interpretation of Precipitation t cool /t ff = 10. Initially, theperturbation is moving to greater altitude, into the partof the diagram where cooling exceeds heating. Its en-tropy therefore starts to decline on a timescale ∼ t cool ,but it accelerates inward on a shorter timescale ∼ ω − .Before condensation can occur, the perturbation crossesthe median entropy line, into the part of the diagramwhere heating exceeds cooling. Its entropy then rises, itaccelerates outward, and it returns to the median line.Buoyant oscillations follow, as shown in the figure.While the perturbation remains small, thermal pump-ing amplifies those oscillations on a timescale ∼ t cool .Each time the perturbation goes below the median en-tropy, cooling drops it further below the median. Eachtime it goes above the median, heating pushes it furtherabove the median. But amplitude growth also increasesthe damping rate of the oscillations, as motion throughthe ambient medium transfers increasing amounts of ki-netic energy away from the perturbation. Growth there-fore saturates at an amplitude δrr ∼ ( ω buoy t cool ) − ∼ α − / K t ff t cool (3)at which energy losses to the ambient medium equal en-ergy gains through thermal pumping (e.g., Nulsen 1986;McCourt et al. 2012; Voit et al. 2017).Voit et al. (2017) called this saturation process “buoy-ancy damping” but misidentified the channel that drainskinetic energy from the perturbation. The Erratum toVoit et al. (2017) shows that internal gravity waves ex-cited by thermal instability damp by coupling to res-onant pairs of lower frequency gravity waves. Thosewave-triad interactions transfer kinetic energy awayfrom the original unstable wave on a timescale ∼ [ ω buoy ( kr )( δr/r )] − , where k is the wavenumber of theoriginal wave. Growth of the original wave thereforesaturates as this dissipation timescale approaches t cool .Long-wavelength disturbances ( kr ∼
1) consequentlysaturate at the amplitude expressed in equation (3), cor-responding to entropy fluctuations with a fractional am-plitude δK/K ∼ α K ( δr/r ) ∼ α / K ( t ff /t cool ), relative tothe background. However, the actual entropy changesexperienced by a moving perturbation have an ampli-tude ∆ K/K ∼ ( ω buoy t cool ) − ( δr/r ) ∼ α − K ( t ff /t cool ) .Perturbations starting with a larger amplitude thanthe saturation amplitude can also fail to result in mul-tiphase condensation. The lower trajectory in Figure 1shows an example. It enters the figure with K ∼ . K at 80 kpc and proceeds toward smaller radii. However,the perturbation it represents is falling faster than it cancool and ultimately descends through a layer of equiva-lent entropy at r ≈
40 kpc. Buoyant oscillations of de-caying amplitude follow until the perturbation reaches the saturation scale. A more complete model would in-clude fluid instabilities that would shred the oscillatingperturbation and mix it with the ambient medium, per-haps before it is able to reach the saturation scale. GRAVITY WAVES & PRECIPITATIONPrecipitation models have recently received consider-able attention because both observations (Voit et al.2015a,b, 2018, 2019; Voit 2019; Hogan et al. 2017; Pulidoet al. 2018; Babyk et al. 2018) and simulations (McCourtet al. 2012; Sharma et al. 2012; Gaspari et al. 2012,2013; Li et al. 2015; Prasad et al. 2015, 2018; Yang &Reynolds 2016; Meece et al. 2017; Fielding et al. 2017;Esmerian et al. 2020) suggest that coupling between en-ergetic feedback and multiphase condensation enablesat least some galactic atmospheres to self-regulate at amedian ratio t cool /t ff ≈ t cool /t ff ratio of the emergent marginalstate is an order of magnitude greater than the value atwhich buoyancy should interfere with thermal instabil-ity and suppress multiphase condensation, according tothe findings outlined in §
2. Something must therefore beoffsetting the damping effects of buoyancy so that mul-tiphase condensation can proceed. This section outlinessome of the possibilities.The upper panel of Figure 2 illustrates two pathwaysthat can lead to condensation in an otherwise staticgalactic atmosphere. As in Figure 1, all of the pertur-bation trajectories it shows were computed using theheuristic non-linear dynamical model from Voit (2018).In fact, the trajectory beginning at 300 kpc and endingup at 40 kpc is identical to the infalling one in Fig-ure 1. Several other infalling trajectories also begin at300 kpc, and their fate depends on the perturbation’sinitial t cool /t ff ratio. If the ratio begins near unity, theperturbation can cool as least as quickly as it falls, al-lowing it to condense. But if an infalling perturbationbegins with a t cool /t ff ratio much greater than unity,buoyancy damping causes it to settle into and mergewith the ambient atmosphere at an entropy level notvery different from its original value. The fate of in-falling gas coming from cosmological accretion thereforedepends on both its initial t cool /t ff ratio and the entropyprofile of the galactic atmosphere it is entering. A com-plementary analysis of this perturbation mode can be Voit K ( k e V c m ) Outflow-Driven Uplift Damped Internal Gravity WavesGalactic Fountain Cosmological Cold Accretion Cosmological Infall t c o o l / t ff = t c o o l / t ff = Heating > CoolingPositive Buoyancy Heating < CoolingNegative Buoyancy K ( k e V c m ) Driven InternalGravity WavesCircumgalacticPrecipitationHeating > CoolingPositive Buoyancy Heating < CoolingNegative Buoyancy(Chaotic) Cold Accretion
Figure 2.
Perturbation trajectories in idealized galactic at-mospheres with a median entropy profile having α K = 2 / t cool /t ff = 10. Lines and shading shared with Figure 1have the same meanings. Additionally, a dotted line traceswhere t cool /t ff = 1. Perturbations entering the purple regionbelow it inevitably condense, as symbolized by blue circles.The upper panel shows trajectories computed with the modelof Voit (2018) for perturbations in a static atmosphere, andmost of them converge to the saturation amplitude for ther-mally unstable but damped internal gravity waves. Line col-ors for the Cosmological Infall and Outflow-Driven Uplifttrajectories, in the order brown, orange, red, magenta, andpurple, represent increasing susceptibility to condensationin a quiet atmosphere. The lower panel shows what hap-pens to those trajectories in a dynamically noisy atmospherein which random momentum impulses buffet the perturba-tions, which can be considered internal gravity waves thathave been driven to non-linear amplitudes. Many of the tra-jectories then end in condensation, as described in § found in Choudhury et al. (2019), who consider how de-velopment of a multiphase medium depends on the am-plitude of isobaric density perturbations and find thatcondensation of those perturbations depends jointly ontheir initial amplitude and the ambient value of t cool /t ff . The second condensation pathway in the upper panelof Figure 2 begins at small radii. Outflows that lift low-entropy gas to greater altitude can stimulate condensa-tion if they are able to make t cool /t ff (cid:46) ∼ t cool /t ff re-quire greater amounts of uplift. The essential featuresof this condensation mode were captured decades ago bygalactic fountain models for the origin of high-velocityclouds in a galactic atmosphere (Shapiro & Field 1976;Bregman 1980) and are inherent in modern simulationsof condensing galactic winds (e.g., Vijayan et al. 2018;Schneider et al. 2018).Trajectories in the lower panel of Figure 2 begin withthe same initial conditions as those in the upper panel.The only difference is the presence of dynamical noisein the ambient atmosphere. Perturbation trajectories inthe Voit (2018) model can be given random momentumimpulses intended to resemble the effects of turbulenceand other forms of kinetic disturbance. That feature ofthe model was inspired by the finding of Gaspari et al.(2013) that driving of turbulence in a galactic atmo-sphere with a median ratio t cool /t ff ≈
10 can interferewith buoyancy and promote multiphase condensation.The perturbations that condense correspond to internalgravity waves that have been driven toward amplitudeslarge enough for the perturbation’s local value of t cool /t ff to approach unity.The lower panel of Figure 2 shows that dynami-cal noise can cause condensation of perturbations thatwould otherwise damp and converge to the saturationamplitude. Two examples are the red trajectories start-ing at the green diamonds located 20 kpc and 60 kpcfrom the center. Those trajectories represent internalgravity waves that saturate in the upper panel but re-sult in condensation near r ≈
40 kpc in the bottompanel. Two more examples are the orange and salmoncolored trajectories starting 3 kpc from the center. Inthe upper panel, uplift alone is not enough to make themcondense, but random momentum impulses in the lowerpanel end up driving those trajectories into condensa-tion near r ≈ r ≈
15 kpc, which are represented with ma- raphical Interpretation of Precipitation t cool /t ff ≈
1. The other comes from uplifted gas origi-nally at r ≈ r ≈ r < THE GLOBAL PRECIPITATION LIMITThe trajectories in Figure 2 demonstrate that t cool /t ff (cid:46) Individually, those trajectoriesdo not account for why a galactic atmosphere would self-regulate near a much larger median ratio t cool /t ff ∼ σ ln K of the fractional entropyperturbation amplitude δ ln K = [ K − ¯ K ( r )] / ¯ K ( r ) rel-ative to the local median. The low-entropy tail of thatdistribution represents perturbations transitioning intocondensation. Consequently, the precipitation rate ofan atmosphere depends on how its typical perturbationamplitude σ ln K compares with the fractional difference∆ ln K cond = | K − ¯ K | / ¯ K between the median entropy ¯ K and the entropy level K at which t cool /t ff ≈ K , except for a flat tail at the low-entropy end, where t cool /t ff (cid:46)
1. The figure presentsa case in which the median entropy ¯ K corresponds to The criterion ω buoy t cool (cid:46) § cool /t ff p ( t c oo l /t ff ) Heating > Cooling
Positive BuoyancyHeating < Cooling
Negative Buoyancy precipitation gravity waves σ l n K = . σ l n K = . σ l n K = . ≈ Δ ln K cond Figure 3.
Schematic probability distributions of t cool /t ff , K/ ¯ K , T / ¯ T , and ¯ n/n for precipitating atmospheres in ap-proximate pressure balance. The cores of the distributionsare log-normal, with σ ln K = 1 . t cool /t ff = 1. In the coreof each distribution, buoyancy causes entropy perturbationsto oscillate around the median as gravity waves. But in thetail, cooling operates faster than buoyancy, allowing pertur-bations that reach the tail to proceed into multiphase con-densation. The thin line beginning at the green diamond andending at the blue circle shows an example. Given such a dis-tribution, the atmosphere’s precipitation rate is proportionalto the height of the tail and depends on how σ ln K compareswith the fractional difference ∆ ln K cond between ¯ K and thelocal entropy level at which precipitation can occur. t cool /t ff = 10. More generally, ¯ K can be consideredan adjustable parameter of the distribution functionthat determines the median t cool /t ff ratio. The rela-tionship shown between t cool /t ff and the ratio K/ ¯ K as-sumes pressure balance and Λ ∝ T − . , which is ap-propriate for 10 . K < T < . K (e.g., Sutherland& Dopita 1993; Schure et al. 2009). Together, thoseassumptions give t cool ∝ K . and the relationships σ ln t cool ≈ . σ ln K and σ ln t cool ≈ . σ ln T (e.g., Voit2019).Evidence for such a perturbation distribution can beobserved in both idealized and cosmological simulationsof feedback-regulated circumgalactic gas. See, for exam-ple, Figure 1 of Lochhaas et al. (2020) and especially Fig-ure 4 of Fielding et al. (2020). The distributions of T / ¯ T appear log-normal in the core and have tails that leveloff near 0 . T / ¯ T . Fielding et al. (2020) show that the Voit dispersions of those distributions differ, depending onfeatures peculiar to each simulation. The cosmologicalsimulations (from Joung et al. 2012; Springel et al. 2018)analyzed by Fielding et al. (2020) show greater disper-sion in CGM conditions, particularly at large radii, pre-sumably because they include dynamical disturbances,such as cosmological infall, mergers, and stirring of theCGM by orbiting subhalos, that are not present in ide-alized simulations. In those cosmological simulations,the peak of the log-normal distribution is roughly twicethe level of the flat tail, similar to the distribution with σ ln K = 1 in Figure 3. Interestingly, the correspond-ing temperature dispersion ( σ ln T ≈ .
6) is consistentwith the one inferred by Voit (2019) from observationsof CGM O VI absorption around galaxies like the MilkyWay.For the idealized distribution functions in Figure 3,the precipitation rate is exponentially sensitive to theratio ∆ ln K cond /σ ln K , because p (cid:18) t cool t ff (cid:19) ∝ exp (cid:34) − (cid:18) ∆ ln K cond σ ln K (cid:19) (cid:35) (4)for t cool /t ff (cid:38)
1. The model represented in the figure has∆ ln K cond = (ln 10) / . ≈ .
35, because t cool /t ff = 10at ¯ K and t cool ∝ K . . Given this difference between¯ K and the value of K at which t cool /t ff = 1, approxi-mately 9% of a log-normal distribution with σ ln K = 1 . t cool /t ff (cid:46)
1. That percentage drops to ∼
1% in thedistribution with σ ln K = 0 .
6. Precipitation depletes thefraction of the ambient atmosphere with t cool /t ff (cid:46) t cool ≈ t ff in thoseperturbations). And dynamical noise presumably re-stores the tail of the distribution function on a similartimescale.That line of reasoning leads to a hypothesis aboutthe precipitation rate: In a given layer of a galactic at-mosphere it should be similar to the gas mass of thelayer, divided by t ff , times the fraction of mass with t cool /t ff (cid:46)
1. Within the virial radius, the expectedprecipitation rate of a galactic atmosphere representedby the solid line in Figure 3 ( σ ln K = 1 .
0) is then sim-ilar to the CGM gas mass divided by a Hubble time.The precipitation rate corresponding to the long-dashedline ( σ ln K = 0 .
6) is an order of magnitude smaller.In a state of long-term balance, entropy fluctuationscannot be much greater than σ ln K = 1 . t cool /t ff (cid:29) σ ln K = 0 . α K ≈ / t cool /t ff ∼ σ t ≈ . v c . Note that the density fluc-tuations shown in Figure 3 can be considerably largerthan the compressive density fluctuations produced byturbulent speeds of this magnitude in a hydrostatic at-mosphere, which have fractional amplitudes δ ln n ∼ ( σ t /v c ) . That is because the fluctuations depicted inFigure 3 arise instead from vertical displacements of gasin an atmosphere with an entropy gradient. Recent nu-merical simulations by Mohapatra et al. (2020, 2021)have shown turbulence in a stratified medium generatesdensity fluctuations of amplitude δ ln n ∼ α K ( σ t /v c ).The critical velocity dispersion implied by the Voit(2018) model is broadly consistent with the findingsof Gaspari et al. (2013) for numerical simulations thatdrive turbulence, as well as with the observed velocitydispersions of galaxy-cluster cores that appear to be pre-cipitating (Gaspari et al. 2018). The turbulence cannotbe much greater without either damping the perturba-tions through mixing with the ambient gas or overheat-ing the ambient medium through turbulent dissipation(Gaspari et al. 2013, 2017; Banerjee & Sharma 2014;Buie et al. 2018, 2020). Around a galaxy like the MilkyWay, the predicted one-dimensional velocity dispersionof hot gas in a precipitating CGM is therefore σ t ≈ − . Around a massive elliptical galaxy the pre-diction rises to σ t ≈ − . TILTING OF THE ENTROPY PROFILESo far, the paper has focused entirely on entropy pro-files with a constant median t cool /t ff ratio, correspond-ing to α K = 2 / α K = 0, which eliminates buoyancy andproduces singularities in the model of §
2. In that case,buoyancy is unable to damp thermal instability, allowinglow-entropy perturbations to condense on a timescale ∼ t cool regardless of the median t cool /t ff ratio. Thismay be how massive galaxies in the Illustris TNG50simulation manage to maintain multiphase circumgalac-tic gas, even though their ambient atmospheres have amedian t cool /t ff ∼ r ∼ t cool /t ff would ap-pear to be unfavorable to development of a multiphasemedium, but the median entropy slope in that radial raphical Interpretation of Precipitation α K ≈ / α K (cid:46)
0, makingthose regions convectively unstable, free from buoyancydamping, and susceptible to precipitation.Tilting the entropy slope the opposite way has conse-quences that may be more dramatic. An entropy pro-file steep enough for the median t cool /t ff ratio to risewith radius tends to focus multiphase condensation ontothe galaxy’s center, potentially supercharging feedbackfrom the galaxy’s central black hole. Voit et al. (2020)have recently demonstrated how such a tilt may linkquenching of star formation by black-hole feedback witha galaxy’s central stellar velocity dispersion. Obser-vations indeed show that central entropy profiles with α K (cid:29) α K ≈ ω buoy t cool is minimized (Voitet al. 2015b). SUMMARYThis paper has attempted to present the following keyconcepts of self-regulating circumgalactic precipitationas simply as possible, primarily through schematic dia-grams.1.
Buoyancy Damping.
Thermal instability instatic galactic atmospheres with α / K ( t cool /t ff ) ≈ ω buoy t cool (cid:29) § δ ln K ∼ α / K ( t ff /t cool ), relative to the backgroundmedium.2. Local Precipitation Threshold.
Because ofbuoyancy damping, a low-entropy perturbationwithin an atmosphere having α / K ( t cool /t ff ) ≈ ω buoy t cool (cid:29) ω buoy t cool (cid:46) § α K ≈ /
3, then the local thresh-old for precipitation is t cool /t ff (cid:46) Uplift.
Galactic outflows that lift ambient gasnearly adiabatically can lower the local t cool /t ff ratio of the uplifted gas by increasing t ff withoutsignificantly changing t cool (see Figure 2, upperpanel). The amount of uplift required to achieve t cool /t ff (cid:46) t cool /t ff ratio. Therefore, the global ratio is a measure ofthe atmosphere’s susceptibility to multiphase con-densation, when it is disturbed. This route to con-densation resembles a classic galactic fountain.4. Infall.
Alternatively, cosmological infall or strip-ping of low-entropy gas from an orbiting subhalocan introduce perturbations of non-linear ampli-tude that are able to condense if they start with ω buoy t cool (cid:46) Dynamical Driving.
Hydrodynamical distur-bances strong enough to interfere with buoyancydamping produce additional opportunities for con-densation (see Figure 2, lower panel). If all CGMperturbations are represented as internal gravitywaves of amplitude δ ln K ≈ α K ( δ ln r ), then con-densation corresponds to driving of those gravity-wave oscillations to amplitudes that locally sat-isfy ω buoy t cool (cid:46)
1. Drivers of CGM fluctuationsmay include galactic winds, turbulence, cosmolog-ical infall, or stirring by orbiting subhalos.6.
Circumgalactic Precipitation.
Given the random-ness produced by multiple sources of dynamicaldriving, it may be difficult to pinpoint the originof a particular multiphase gas cloud in the CGM(see Figure 2, lower panel). However, consideringall routes to condensation to be forms of circum-galatic precipitation helps to link their collectivepresence with the global characteristics of the am-bient galactic atmosphere.7.
Global Precipitation Limit.
The susceptibility toprecipitation of a stratified galactic atmosphere( α K ∼
1) depends on both its median t cool /t ff ratio and the dispersion σ ln K of entropy fluctua-tions within it (see § t cool /t ff ratio determines the fractional entropy dif-ference ∆ ln K cond between the atmosphere’s me-dian entropy ¯ K and the entropy of a perturba-tion in which t cool /t ff (cid:46)
1. The fraction of theCGM that is able to precipitate therefore dependson the ratio ∆ ln K cond /σ ln K . Numerical simula-tions of the CGM indicate that its entropy fluc-tuations have a log-normal distribution aroundthe median, with σ ln K ≈ . K cond /σ ln K . Consequently, the ra-tio f ≡ ∆ ln K cond /σ ln K needs to be large enoughto avert catastrophic precipitation and an over-whelming feedback response. In a typical galacticatmosphere (with t cool ∝ K . ), these considera- Voit tions yield a global precipitation limit t cool /t ff (cid:38) exp (1 . f σ lnK ) (5)on the median ratio that reduces to t cool /t ff (cid:38) f σ ln K ≈ .
35. Self-regulating precipitationconverges to a value of f at which accretion of coldgas into the galaxy fuels just enough feedback tokeep the CGM in approximate thermal balance.That equilibrium value is likely to be in the range1 (cid:46) f (cid:46) Observable Features.
Two distinct observable fea-tures allow tests of this interpretation of circum-galactic precipitation. Entropy fluctuations in ap-proximate pressure balance with their surround-ings correspond to temperature and density fluctu-ations with log-normal dispersion σ ln T ≈ σ ln n ≈ . σ lnK (see Figure 3). Around a galaxy like theMilky Way, with a CGM temperature ∼ K,the O VI absorption-line column densities are sen-sitive to the amplitude of temperature fluctua-tions, and observations are so far consistent with σ ln T ∼ . ≈ . . v c inthe CGM gas, amounting to 50–70 km s − arounda galaxy like the Milky Way and 100–150 km s − around a massive elliptical galaxy. However, addi-tional modeling will be required to obtain predic-tions of correlations between O VI column densityand absorption-line width. 9. Entropy-Profile Tilt.
All of these predictions forprecipitation substantially change if the CGM en-tropy profile becomes nearly flat ( α K (cid:28)
1) be-cause buoyancy damping is eliminated as α K → ∼ t cool , regardless of the median t cool /t ff ratio ( § t cool /t ff with the atmosphere’s median value of ω buoy t cool ,giving a more general global precipitation limit ω buoy t cool (cid:38)
10 for f σ ln K ≈ . Minimum Precipitation Limit.