Energetics of X-ray Cavities and Radio Lobes in Galaxy Clusters
aa r X i v : . [ a s t r o - ph ] M a y Draft version November 2, 2018
Preprint typeset using L A TEX style emulateapj v. 08/22/09
ENERGETICS OF X-RAY CAVITIES AND RADIO LOBES IN GALAXY CLUSTERS
William G. Mathews and Fabrizio Brighenti , Draft version November 2, 2018
ABSTRACTWe describe the formation and evolution of X-ray cavities in the hot gas of galaxy clusters. Thecavities are formed only with relativistic cosmic rays that eventually diffuse into the surrounding gas.We explore the evolution of cavities formed with a wide range of diffusion rates but which are otherwisesimilar. In previous numerical simulations, in which cavities are formed by injecting ultra-hot butnon-relativistic gas, cavity formation contributes thermal energy that may offset radiative losses in thegas, thereby helping to solve the cooling flow problem. Contrary to these results, we find that X-raycavities formed solely from cosmic rays have a global cooling effect. Most cosmic rays in our cavityevolutions do not move beyond the cooling radius and, as the cluster gas is displaced, contribute toa global expansion of the cluster gas. As cosmic rays diffuse away from the cavities, the nearby gasbecomes buoyant, resulting in a significant outward mass transfer within the cooling radius, carryingrelatively low entropy gas containing cosmic rays to outer regions in the cluster where it remains fortimes exceeding the local cooling time in the hot gas. This post-cavity mass outflow due to cosmicray buoyancy may contribute significantly toward solving the cooling flow problem. For example themass inflow in the Virgo cluster due to radiative cooling can be balanced by buoyant outflow if onlya fraction ∼ . Subject headings:
X-rays: galaxies – galaxies: clusters: general – X-rays: galaxies: clusters – galaxies:cooling flows INTRODUCTION
The hot gas in galaxy clusters loses energy by X-rayemission but does not cool to low temperatures. In re-cent attempts to solve this cooling flow problem it hasbeen almost exclusively assumed that feedback energyfrom accretion onto cluster-centered massive black holescan be delivered to the hot gas in a manner that main-tains the observed temperature and density profiles inspite of radiation losses. The solution to this problemis not straightforward. Even when heat is supplied tothe cluster gas in an ad hoc , optimized, ideally fine-tuned manner, either concentrated or distributed overmany tens of kiloparsecs, it is found that the temperatureand density profiles deviate strongly from those observed(Brighenti & Mathews 2002; 2003). For example, highlyidealized flows in which radiative cooling is perfectly bal-anced by local heating at every radius are inconsistentwith the secular increase in gas density associated withstellar mass loss in the cluster-centered galaxy. In mostgalaxy groups and clusters a positive temperature gradi-ent ( dT /dr >
0) is observed in the inner regions. Sincethe coolest gas is closest to the central source of AGNheating, this central gas must be heated with exquisiteprecision (on short timescales) to maintain the low tem-peratures observed.In spite of these difficulties, a variety of heating mech-anisms continue to be investigated to understand the en-ergetics of hot cluster gas. Heating is usually assumed tobe associated with the formation of X-ray cavities by jet UCO/Lick Observatory, Dept. of Astronomy and Astro-physics, University of California, Santa Cruz, CA 95064 Dipartimento di Astronomia, Universit`a di Bologna, via Ran-zani 1, Bologna 40127, Italy feedback energy proceeding from the central black hole.Weak shock waves emerge when the cluster gas is dis-placed as cavities form (see McNamara & Nulsen 2007 fora review). Weak shocks, observed in a few clusters (e.g.Perseus: Fabian et al 2003; Virgo: Forman et al. 2005;2007), have the desirable capability of dissipating AGNfeedback energy over large regions of the cluster gas, asemphasized by Fabian et al (2003). However, energy dis-sipation by outwardly propagating shock or sound wavesis disproportionally concentrated in the central regionsof clusters where the gas density gradient is smallest.Over time the cumulative wave dissipation in this cen-tral gas (through which all the waves must pass) causesits temperature to become hotter than observed in anycluster (Fujita & Suzuki 2005; Mathews, Faltenbacher &Brighenti 2006). It may be possible to forestall coolingin a limited region of a particular cluster with a properchoice of shock Mach numbers (or sound wave frequen-cies and amplitudes), but this level of fine-tuning seemscontrived.X-ray cavities are thought to provide a convenient mea-sure of the amount of feedback power delivered to the hotgas. Cavities are observed in ∼ −
25% of X-ray brightclusters (Birzan et al. 2004; Rafferty et al. 2006), but theincidence of cavities increases as the cooling time of thecentral gas decreases (Dunn & Fabian 2006). The workdone in displacing a volume V of cluster gas at pressure P is P V and the energy of the material inside the cavityis
P V / ( γ − E cav = [ γ/ ( γ − P V = 4
P V where γ = 4 / t buoy in thecluster gas is a measure of the cavity lifetime, Raffertyet al. (2006) estimate the “cavity jet power” P cav = E cav t buoy . Rafferty et al. find that P cav exceeds the total X-ray lu-minosity L x within the cooling radius (defined where thecooling time is 7.7 Gyrs) in about 60% of their sampleclusters with cavities. They suggest that clusters with P cav < L x – and the many clusters with no known cavi-ties – are in a phase of their feedback energy cycle that isjust now recovering from a recent heating episode. Raf-ferty et al. suggest that all the energy within the cavities P V / ( γ −
1) may be available to heat the cluster gas, notjust the
P V work done by the expanding cavity. Thisimplies that the (thermal) energy of material within thecavity is ultimately shared with the cluster gas. Fur-thermore, for those clusters in which the total cavityjet power is favorable ( P cav > L x ), it is assumed thatthis energy is distributed throughout the cluster gas in amanner that preserves the characteristic gas temperatureand density profiles observed in galaxy clusters. Implicitin this assumed energy distribution is the requirementthat the cavity-forming energy is delivered to the clustergas in an approximately isotropic manner relative to thecentral black hole.Another less often considered possible resolution tothe cooling flow problem is the hypothesis that gas isonly heated at or near the central black hole and thenis buoyantly transported far out into the cluster gas, i.e.a circulation flow (Mathews et al. 2003; 2004). Oneof the initial motivations for mass circulation was thetwo-temperature (and therefore buoyant) flow observedin the galaxy group NGC 5044 (Buote et al. 2003). How-ever, Temi, Brighenti & Mathews (2007a,b) have recentlyfound far-infrared emission from dust extending out to5-10 kpc around many group-centered, X-ray luminouselliptical galaxies. Since the dust lifetime to sputteringdestruction in the hot gas is only ∼ yrs, this obser-vation provides additional strong evidence for ongoingbuoyant outflow from dust-rich cores in the central galax-ies. Additional support for buoyant outflow is providedby the large regions of iron enrichment in the hot gas sur-rounding cluster-centered elliptical galaxies (De Grandiet al. 2004). Since the 50-100 kpc size of these regionsenriched by Type Ia supernovae greatly exceeds that ofthe central galaxy where they occur, an outward masstransfer is essential for their formation. Buoyant outflowis desirable since it can also preserve the observed gasdensity and temperature profiles (Mathews et al. 2004).Alternatively, these profiles can be preserved if mass cir-culates outward in momentum-driven, mass-carrying jets(Brighenti & Mathews 2006).In most theoretical studies of X-ray cavity evolution,it is assumed that the cavities are inflated with ultra-hot gas (occasionally but not always with γ = 4 /
3) (e.g.Br¨uggen & Kaiser 2002; Reynolds et al. 2005; Gar-dini 2007; Pavlovski et al. 2007). If this heated gasis transported in jets from the central black hole, thenthis type of solution implicitly requires an outward masscirculation which in some simulations can be quite large. For example, in the recent double jet 3D calculation ofHeinz et al. (2006), jets with power W j = 10 ergss − enter the cluster gas from the origin with velocity v j = 30 ,
000 km s − . This translates into a mass flux of˙ M j = 2 W/v = 35 M ⊙ yr − and a total injected mass of M j = 3 . × M ⊙ over their 10 year computation. Ifambient cluster gas is entrained in the jets, the outflow-ing mass could be increased further.The jet mass flux may be very much less if most of theirenergy density is in the form of cosmic rays. While it ispossible that jets do transport substantial masses of gasfrom the center (as in Brighenti & Mathews 2006), weexplore here the energetics in the limiting case in whichX-ray cavities are formed only with relativistic cosmicrays that can also diffuse into the cluster gas. For sim-plicity we assume that most of the cosmic rays in cavitiesarrive in jets and are not produced in local shocks. Inthis limit the cavities are formed with pure energy withno appreciable component of rest mass or momentum asin previous numerical simulations. Because of their dif-fusive nature, cosmic rays can eventually penetrate thecluster gas and contribute to the local pressure support.As a consequence the local cluster gas density is reducedby cosmic rays and becomes buoyant. Inhomogeneouslydistributed cosmic rays are a natural driver of buoyantmass outflow. Gas carried out by cosmic ray buoyancymay never return to the cluster center.However, we find that few cosmic rays move beyondthe cooling radius, so the cluster gas expands and coolsglobally. Although some heating is expected from shockwaves, this heating is offset by the global cooling. Con-sequently, X-ray cavities containing cosmic rays result ina net cooling of the cluster gas, not heating as generallyassumed. Buoyant mass outflow resulting from inhomo-geneous cosmic rays and global expansion of the clustergas may help to resolve the cooling flow puzzle.Although it is generally believed that X-ray cavitiesare formed with cosmic rays, this conjecture has only re-cently been tested with detailed calculations (Mathews &Brighenti 2007a,b). However, spherical steady state clus-ter flows including cosmic rays (e.g. B¨ohringer & Mor-fill 1988; Loewenstein et al. 1991) have been developedto explore the possible large scale dynamical influence ofcosmic ray pressure gradients and the dissipation of theirenergy into the thermal gas.For simplicity in this initial treatment of the globaltime-dependent energetics of cluster cavity formationwith cosmic rays, we do not specify the physical natureof the relativistic fluid (electrons or protons) nor do wecalculate radio or inverse Compton fluxes/luminositieswhich would require additional assumed parameters –these important details will be considered in future pa-pers. Nevertheless, the spatial distribution of cosmicrays we calculate defines the region where radio emissioncould be expected. Since we do not consider Coulombheating, the cluster gas responds adiabatically to cosmicray pressure gradients. EQUATIONS AND COMPUTATIONAL PROCEDURE
The combined Eulerian evolution of (relativistic) cos-mic rays (CRs) and thermal gas can be described withthe following four equations: ∂ρ∂t + ∇ · ρ u = 0 (1) ρ (cid:18) ∂ u ∂t + ( u · ∇ ) u (cid:19) = −∇ ( P + P c ) − ρ g (2) ∂e∂t + ∇ · u e = − P ( ∇ · u ) (3) ∂e c ∂t + ∇ · u e c = − P c ( ∇ · u ) + ∇· ( κ ∇ e c ) + ˙ S c (4)where we suppress artificial viscosity terms. Pressuresand thermal energy densities in the plasma and cosmicrays are related respectively by P = ( γ − e and P c =( γ c − e c where we assume γ = 5 / γ c = 4 /
3. Thecosmic ray dynamics are described by e c , the integratedenergy density over the cosmic ray energy or momentumdistribution, e c ∝ R EN ( E ) dE ∝ R p f ( p )(1+ p ) − / dp .The first three equations are the usual equations forconservation of mass, momentum and thermal energyin the hot thermal cluster gas. We do not include op-tically thin radiative losses since our intention here isto study the energetics of cavity creation alone withoutthe complicating effects of a secular radiative energy lossand central cooling. Note that the CR pressure gradientin equation 2 contributes to the motion of the thermalgas. This exchange of momentum between CRs and gasarises as the CRs diffuse through magnetic irregularities(Alfven waves) that are nearly frozen into the hot ther-mal gas. However, magnetic terms do not explicitly enterin the equations because typical magnetic fields in clustergas ∼ − µ G (Govoni & Feretti 2004) are too small,i.e. the magnetic energy densities ∼ B / π < ∼ − ergcm − are generally much less than the thermal energydensity in the hot gas. In addition, the Alfven velocity v A = B/ (4 πρ ) / = 2 n − / e B ( µ G) km s − is typicallymuch less than the sound or flow speeds in cluster gasso the Alfven velocity of the magnetic scatterers can beignored (e.g. Drury & Falle 1986, Jones & Kang, 1990).Equation 4 above describes both the advection of CRswith the gas and their diffusion through the gas. A massconservation equation for the CRs is unnecessary becauseof their negligible rest mass. The CR diffusion coefficient κ is difficult or impossible to calculate in the absenceof detailed information about the magnetic field topol-ogy which is currently unknown. However, we expectthat κ may vary inversely with the density of the ther-mal gas, assuming that the magnetic field strength alsoscales with density. For simplicity we ignore for nowany dependence of κ on CR particle momentum. Sinceobserved radio lobes are very approximately spherical,we assume that κ is isotropic, consistent with a highlyirregular global magnetic field. For these preliminarycalculations it is not necessary to specify the CR compo-sition, either electrons or protons can dominate as longas they are relativistic. Finally, we assume that the totalCR energy density is not substantially reduced by lossesdue to synchrotron emission or interactions with ambientphotons or thermal particles during the cavity evolutiontime. Dissipation of cosmic ray energy into the thermalgas is probably not important in X-ray cavities since the gas temperature in the cavity rims is in fact relativelycooler.The set of equations above are solved in ( r, z ) cylin-drical coordinates using a 2D code very similar to ZEUS2D (Stone & Norman 1992). To be specific, we studythe evolution of X-ray cavities in the well-observed Virgocluster using the analytic fits to the observed gas temper-ature and density profiles suggested by Ghizzardi et al.(2004). The computational grid consists of 100 equallyspaced zones in both coordinates out to 50 kpc plus anadditional 100 zones in both coordinates that increasein size logarithmically out to ∼ g = ( g z , g r ) that establishesexact initial hydrostatic equilibrium for the Virgo clustergas pressure gradient. The cosmic ray diffusion term inequation 4 is solved using implicit Crank-Nicolson dif-ferencing. While this differencing scheme is stable forall time steps, we restrict each time step by the stabil-ity condition for explicitly-differenced diffusion as well asthe Courant condition for numerical stability. Shocks aretreated with a standard artificial viscosity.We assume that the X-ray cavity is formed by CRs thatpropagate in a non-thermal jet from the central blackhole (AGN) to some fixed radius. The CRs are depositedin a gaussian-shaped sphere of characteristic radius r s =2 kpc located at r cav = ( r, z ) = (0 ,
10 kpc), i.e. 10 kpcalong the z -axis. The CR source term in equation 4 istherefore˙ S c = E cav t cav e − (( r − r cav ) /r s ) π / r s erg cm − s − (5)when t < t cav . The integral of ( r s π / ) − e − ( r/r s ) overspace is unity.In the calculations described below our principal ob-jective is to explore the unique diffusive effects of cos-mic rays on cavity formation and energetics. We there-fore restrict the total CR energy of all calculations to E cav = 1 × ergs and adopt t cav = 2 × yrs asthe CR injection time. We choose t cav to be consistentwith X-ray cavity observations; it is shorter than the lo-cal buoyancy time in the cluster gas but sufficiently longnot to produce strong shocks which are not commonlyobserved. At times t > t cav when ˙ S c = 0, the total CRenergy E cr = R e c dV over the grid volume remains verynearly constant but changes slightly due to advection inadiabatic compressions or rarefactions.Since the CR diffusion coefficient is poorly known atpresent – and may vary from one AGN event to another– we consider a wide range of density-dependent coeffi-cients, κ = (cid:26) cm s − : n e ≤ n e cm − ( n e /n e ) cm s − : n e > n e cm − In general, for reasons discussed above, we assume thatthe CR diffusion κ varies inversely with cluster gas den-sity. But κ must be sufficiently large so that the CRdensity is approximately uniform in the X-ray cavitieswhere the cluster gas density is lowest, i.e. we assumethat CRs also diffuse inside the cavity. For example when E cav = 1 × ergs we find that the maximum cavityradius is ∼ κ in the cavitiesbe at least (5 kpc) / t cav ≈ × cm s − and thiscondition is ensured by our adopted κ ( n e ) above. In re-gions of higher n e the density parameter n e determinesthe CR diffusion coefficient. In the following we consider n e = 6 × − − × − cm − . The largest CR diffusioncoefficient κ (corresponding to n e = 6 × − ) is sim-ilar to that required by Mathews & Brighenti (2007b)to explain a common age (10 yrs) for the large radiolobes and the cavity jet (thermal filament) observed byForman et al. (2007) in the Virgo cluster. CAVITY EVOLUTION WITH DIFFERENTCOSMIC RAY DIFFUSIVITIES
Figure 1 shows the gas density ρ ( r, z ) and cosmic rayenergy density e c ( r, z ) at six times during the evolutionof an X-ray cavity and its cosmic ray (radio) lobe. InFigure 1 n e = 6 × − cm − so the diffusion coeffi-cient is rather large, but identical to that used by Math-ews & Brighenti (2007b) to describe the evolution of thecavity jet (thermal filament) in M87/Virgo. The cavityin Figure 1 is formed 10 kpc along the (horizontal) z -axis during time t cav = 0 .
02 Gyr. At time 0 .
024 Gyr,shortly following t cav , the cosmic rays (dotted contours)are tightly confined inside the cavity. By time 0 .
066 Gyrthe cosmic rays have diffused through the cavity wallsforming a small radio lobe and the cavity has just disap-peared – we define the cavity as that region where theplasma density is lower than the original Virgo densityby at least a factor 1/3. A small vortex at ( r, z ) = (5 , z -axis. Theevolution of this feature is described in detail in Mathews& Brighenti (2007b) and we will not repeat that discus-sion again here. During the three later times in Figure 1the denser parts of the cavity jet have fallen back towardthe center of Virgo and the hot gas dynamics becomemore quiescent. However, the cosmic rays are seen todiffuse into a progressively larger region elongated alongthe z -axis. Although not apparent in the gas density con-tours, there is a net outward buoyant migration of thegas in the region occupied by the cosmic rays (the radiolobe).Figure 2 shows the cavity evolution when the cosmicray diffusion is 1000 times lower (when n e > n e =6 × − cm − ) but with all other parameters unchanged.Now the cosmic rays are seen to be very tightly confinedto the cavity region until at least time 0.1 Gyr whenthe cavity is still visible. In this evolution the outerparts of the cavity break away forming a vortex thatmigrates away from the z -axis, carrying its own cosmicrays. By time 0.3 Gyr the brightest parts of the radioimage (corresponding to the largest cosmic ray energydensity) should consist of two separate regions, a featurealong the z -axis extending out to 40 kpc and the vor-tex at ( r, z ) = (14 ,
38) kpc. At this time (0.3 Gyr) wesee an enhanced gas density that accompanies the cos-mic rays along the z -axis. At still later times the cosmicrays continue to reside mostly along the z -axis but bytime 0.9 Gyr a region of low cosmic ray energy densitybecomes visible in the r -direction along the trajectory ofthe receding vortex (now at ( r, z ) = (25 ,
28) kpc).When X-ray cavities are formed in an atmosphere ini- tially at rest, as we assume here, the vortex region at time0.3 Gyr in Figure 2 would appear as a ring when viewedalong the z -axis. To our knowledge no ring-shaped radiofeatures have been observed. This could simply be dueto the faintness of such regions since radio-synchrotronelectrons tend to not to produce observable GHz emissionafter about 10 yrs. Nevertheless, it is clear from Figures1 and 2 that the radio lobe morphology can in princi-ple provide valuable information concerning the cosmicray diffusion coefficient about which very little is knownat present. Finally, since gas phase metal abundancestend to increase toward the centers of clusters, regionsof enhanced abundance are expected to accompany theoutward buoyant migration of cosmic rays.As a further aid in interpreting Figures 1 and 2, Fig-ure 3 shows the pressure profiles along the z -axis at fourtimes for both evolutionary calculations. At early timesthe cavity is visible as the region where P c > P . Withinthe cavity the pressure gradient ( ∼ dP c /dr ) is nearly flatbecause of the enormous pressure scale height of the rel-ativistic fluid. However, except near and within the cavi-ties, the total pressure P + P c (dotted lines) deviates verylittle from the pressure profile in the cluster before thecavity was introduced (dot-dashed lines). This is a con-sequence of the largely subsonic character of the flows.After about 0.1 Gyrs in both calculations the cosmic raysdiffuse sufficiently so that the pressure ratio P c /P ≪ . γ -ray detectability of cosmic rayprotons if nearby clusters are completely filled with cos-mic rays (Pfrommer & Ensslin et al. 2004). Althoughthe cosmic ray pressure and energy density are negligi-ble at these late times, we show below that cosmic rayscontinue to displace about the same volume of gas. Thisresults in a very long-lasting global expansion of clustergas that can be seen by the small discrepancies in the z -axis gas pressure profile (solid lines) relative to that inthe initial cluster (dash-dotted lines) at times 0.3 Gyrs inFigure 3. Nevertheless, the close similarity of the initialand final gas pressure along the z -axis (and elsewhere) atlate times is consistent with our finding that the gas den-sity and temperature (and therefore entropy ) gradientsare also only slightly affected by the cavity evolution. ENERGETICS AND EVOLUTIONOF X-RAY CAVITIES
Figure 4 shows the evolution of cluster gas energies re-sulting from cavity-lobe formation with cosmic rays. Thefour panels show the evolution as the cosmic ray diffu-sion coefficient κ decreases with n e (in cm − ) over awide range: n e = 6 × − (panel a ), 6 × − (panel b ), 6 × − (panel c ), and 6 × − (panel d ). The totalcosmic ray energy integrated over the entire computa-tional region E cr = R e c dV is shown with long dashedlines. E cr ( t ) is seen to rise until t cav = 2 × yrs asthe cavities form, then remain approximately constantafter the cosmic ray source is turned off. E cr is not astrictly conserved energy. The small decrease in E cr ( t ) Figure 8 shows (for the evolution in Figure 2) an approxi-mate z -axis gas entropy profile at 0.9 Gyrs relative to the initialatmosphere. When plotted, s ( z ) at 0.9 Gyrs is almost identicalto that in the original atmosphere from the origin to z ≈
25 kpcand beyond z ≈
42 kpc. In the intermediate region 25 < z < visible at times t > t cav can occur if the cosmic ray en-ergy density is reduced by a secular advective expan-sion with the cluster gas. This decrease in E cr ( t ) after t cav is stronger when κ is smaller (panels a → d ) sincethe cosmic rays are more confined near the z -axis wheremost of the gas expansion occurs. The (small) total ki-netic energy in the cluster gas E kin = 0 . R ρ ( u x + u y ) dV is shown with dotted lines in each panel. The changein potential energy compared to the initial Virgo atmo-sphere, ∆ E pot = R φ [ ρ ( t = 0) − ρ ] dV is shown withshort dashed lines. The gravitational potential φ ( R ) isfound from the initial M87/Virgo atmosphere by inte-grating dφ/dR = − (1 /ρ ) dP/dR where R = ( r + z ) / is the radial coordinate. The change in the total gas ther-mal energy relative to the original atmosphere ∆ E th = R [ e − e ( t = 0)] dV is shown with (the lower) solid lines ineach panel. Finally the dash-dotted line shows the totalenergy E tot = ∆ E th + ∆ E pot + E kin + E cr which is con-stant after time t = t cav to an excellent approximationand is equal to E cav = 10 ergs as expected.Also shown in each panel of Figure 4 is an approx-imate evaluation of the quantity 4 P V where V is thevolume of the X-ray cavity at any time (arbitrarily de-fined as the sum over all grid zone volumes in whichthe gas density is less than ρ ( t = 0) /
3) and P is anestimate of the average pressure in the (sometimes non-contiguous) zones containing the cavity. When cosmicrays are strongly trapped within the cavity, we expect4 P V = E cav = E tot . In panel a of Figure 4 we see that4 P V < E cav which can be expected for larger κ whencosmic rays diffuse through the cavity walls. However,4 P V > E cav is apparent in panels c and d , although 4 P V never exceeds E cav = 10 ergs by more than about 20%.This may be due to our rather approximate estimate of P and V or it may be a real inertial overshoot just afterthe cavity is formed. In any case, we include 4 P V ( t ) inFigure 4 because this is the cluster gas heating energyproposed by Rafferty et al. (2006) and McNamara &Nulsen (2007) in their discussions of cavity energetics.Figure 5 shows the approximate evolution of the cav-ity radius r cav ( t ) and its mean radius in the cluster R cav ( t ). The cavity radius is found from the estimatedcavity volume V by assuming that the cavity is spherical, r cav = (3 V / π ) / . The four lines for r cav and R cav inFigure 5 correspond to the four cosmic ray diffusivities κ as it decreases in each panel in Figure 4: a , dottedline; b , short dashed line; c , long dashed line; d , solidline. The buoyant trajectory of the cavities R cav ( t ) isindependent of κ , but the cavities progress further along R cav ( t ) when κ is smaller. The cavity lifetime is deter-mined when r cav →
0. Notice that the cavity lifetimesare similar for n e = 6 × − (long dashed line), and n e = 6 × − (solid line), suggesting that the cavityevolution would not change much if n e (and therefore κ ) were reduced further.The main result we wish to emphasize is the evolutionof the cluster gas thermal energy ∆ E th ( t ). At early timesduring cavity formation when t < ∼ t cav , ∆ E th increasesbecause of heating due to the weak shock that propa-gates away from the expanding cavity. This initial shockheating increases as κ decreases (panels a → d in Figure4) because (for fixed E cav and t cav ) the shock strengthincreases when cosmic rays are more confined within the cavity. This is consistent with the 1D cavity evolutiondescribed by Mathews & Brighenti (2007a). However,after the cavity is formed ( t > ∼ t cav ) ∆ E th ( t ) decreasesand becomes negative after t ≈ − × yrs. Wesee in Figure 4 that the final energy separation between∆ E th and ∆ E pot is independent of κ . Also notice thatin each panel of Figure 4 the average final value of theenergy change 0 . E th + ∆ E pot ) is not zero, but is ap-proximately equal to the peak thermal energy acquiredfrom the shock, ∼ ∆ E th ( t cav ). Although the total en-ergy change in the cluster ∆ E pot + ∆ E th + E kin increasesas κ decreases, the cluster gas experiences a net cooling(∆ E th <
0) when cavities are formed with cosmic rays.This global cooling is exactly opposite to the results dis-cussed by other authors in which cavity formation is re-garded as an important source of AGN heating requiredto balance radiative losses in the cluster gas and reducecentral cooling in cooling flows.The energy evolutions shown in Figure 4 are not qual-itatively altered when the total cavity energy E cav is in-creased. For example when E cav = 10 ergs (depositedat ( r, z ) = (0 ,
10) kpc in time t cav = 2 × yrs with n e = 6 × − cm − ) all energies are larger but theproportions are similar to those in Figure 4. Althoughthe shock is stronger in this case, so is the buoyant cool-ing and ∆ E th still becomes negative after ∼ yrs.The maximum equivalent spherical radius of this highenergy cavity is about r cav = 15 kpc at log t ∼ .
2, butthe cavity disappears soon afterward at log t = 8 . R cav = 55 kpc. More energetic cavities are bigger andmore buoyant, but don’t last proportionally longer. GLOBAL EXPANSION AND MASS OUTFLOW
To verify that the cluster gas has in fact undergone anet expansion as a result of cavity formation, we com-puted the evolution of the cumulative gas mass distri-bution M ( R ). First we sorted the computational zonesin cylindrical coordinates in order of increasing spheri-cal radius R = ( r + z ) / then we integrated over the7855 sorted zones within R = 50 kpc to determine M ( R ).Figure 6 shows M ( R ) in (one hemisphere of) the initialcluster ( t = 0) and after t = 0 . κ ( n e = 6 × − and 6 × − cm − ). Tomore clearly illustrate the evolution of M ( R ) most ofthe R -variation in Figure 6 has been removed by plotting M ( R )( R/
10 kpc) − . The noise in these plots of M ( R )arises because of the finite number of computational gridzones. Figure 6 clearly shows that mass has been re-moved from the inner regions of the cluster as a result ofcavity formation, i.e. the cluster gas has experienced anet expansion. This expansion is remarkably insensitiveto the cosmic ray diffusivity κ , particularly at R < ∼ M ( R ) is expected to depend bothon the cavity energy E cav and the location of the initialcavity in the cluster gas (10 kpc in our case).The largest radius plotted in Figure 6, R = 50 kpc,is essentially the cooling radius for the cluster, i.e. theradius where the local cooling time is 7.4 Gyrs, whichis comparable to the cluster age. From this plot weestimate that ∆ M = 1 . × M ⊙ has been re-moved from within 10 kpc for both values of n e . At R = 50 kpc the mass outflow is ∆ M = 3 . × M ⊙ (for n e = 6 × − ) and ∆ M = 5 . × M ⊙ (for n e = 6 × − ). Values of ∆ M refer to the total clustermass flow in both hemispheres, assuming that a pair ofcavities are produced by symmetric double jets.Using the Ghizzardi et al. (2004) density and temper-ature profiles for Virgo, we estimate a feedback-free gascooling rate of ˙ M cf ≈ M ⊙ / yr. Therefore if cavities sim-ilar to the one we calculate here are formed with cosmicrays every 6 × yrs, this could result in a mass out-flow at R = 50 kpc equal to the cooling inflow at R = 0from radiation losses. When E cav = 10 ergs the totalmass flowing beyond 50 kpc after time 0.9 Gyr is muchlarger, ∆ M = 4 . × M ⊙ . (Note that the buoyantmass outflow scales with E cav , but the cavity lifetimedoes not.) Successive cavities in Virgo with E cav = 10 erg appearing every 6 × years could balance the massflow from radiative cooling ˙ M cf ≈ M ⊙ / yr and also beconsistent with the current absence of large cavities inM87/Virgo. Of course an expansion outflow at the cool-ing radius comparable to the steady state cooling rate˙ M cf does not in itself shut down radiative cooling nearthe cluster center, but the cooling radius will becomelarger and the cooling time at every radius will increasedue to the slightly lower gas density resulting from theglobal expansion. WHAT INCREASES THE POTENTIAL ENERGYAND DRIVES THE MASS OUTFLOW?
The post-cavity mass flow has two aspects: (1) an in-crease in ∆ E pot in Figure 4 and non-zero ∆ M in Figure6 that result from the global lifting that occurs when anew cavity is formed, and (2) a mass circulation drivennear the z -axis by buoyant gas containing cosmic rays.We discuss each in turn. Global Lifting
Although gas certainly flows beyond R = 50 kpc inall our cavity evolutions, the fraction of total cosmic rayenergy within this radius E cr, /E cr varies. For example,at t = 0 . κ ( n e = 6 × − cm − ),the cosmic ray energy in R <
50 kpc, E cr, = 66 . × ergs, is less than the total cosmic ray energy on thecomputational grid, E cr = 81 . × ergs. In this caseabout 20% of the cosmic rays have moved beyond 50 kpc.For smaller κ ( n e = 6 × − cm − ) after this same timewe find that E cr, = 41 . × ergs is identical to thetotal energy E cr , i.e. almost all cosmic rays are confinedwithin 50 kpc. (Both E cr, and E cr refer to a singlehemisphere.) However, the mass ∆ M flowing across R = 50 kpc is positive for both cases involving large andsmall diffusivities κ .Evidently mass can flow across the cooling radius evenin the absence of local buoyant transport. This mustarise because of the quasi-adiabatic expansion of the en-tire cluster gas when cavities are formed. To understandthis better, we begin by showing that the volume of gasdisplaced by cosmic rays V c depends on the local pres-sure P a in the cluster atmosphere and the total energy E cr of the cosmic rays, but not on the ratio of cosmicray to gas partial pressures, P c /P , which changes as thecosmic rays diffuse into the gas. Imagine a uniform gascontaining a spherical region of volume V that containsa total cosmic ray energy E cr . In pressure equilibrium P a = P c + P where P c = ( γ c − E cr /V and therefore V = ( γ c − E cr /P c . But the partial volume V c contain-ing gas that is actually displaced by the cosmic rays is V c = ( P c /P a ) V = ( γ c − E cr /P a . Therefore as cosmicrays diffuse into larger volumes V , the total volume of gasthat is displaced V c remains unchanged and independentof κ . In a non-uniform cluster environment the volumeof gas displaced by a given total energy of cosmic rays E cr varies with cluster radius as P a ( R ) − .∆ M can be estimated by assuming that all cosmicrays in the original cavity do not diffuse beyond theiroriginal cavity volume V cav and remain at their initialcluster radius R cav = 10 kpc. In this limit, and afterseveral cluster sound crossing times, we expect that themass flow across R = 50 kpc will be equal to the gas massdisplaced by the original cavity, ∆ M ≈ V cav ρ ( R cav ) =6 . × M ⊙ where V cav = (4 π/ r cav , r cav ≈ ρ ( R cav ) = 4 . × − gm s − is the original clusterdensity at R cav . This simple estimate of the outflowingmass is very similar to that found from Figure 6, ∆ M =5 . × M ⊙ .However, discrepancies between computed and esti-mated masses ∆ M are expected because of several ad-ditional details. The cosmic ray energy inside the esti-mated cavity volume is E cr = 3 P V cav = 101 × ergs, where P = P a ( R = 10 kpc) = 1 . × − dynecm − . However, this energy is about twice as large as thevalue of E cr ( t cav ) in the low- κ evolution shown in panel d of Figure 4, so the estimated value of ∆ M aboveshould be reduced by a factor of ∼
2. However, ∆ M isexpected to differ from V cav ρ ( R cav ) because of changesin the mean volume V c of gas displaced by cosmic raysafter their outward motion during time 0.9 Gyr. To es-timate this suppose that the e c -weighted mean radius ofcosmic rays at time 0.9 Gyr is R = 30 kpc (see Figure2) where the gas pressure is P = 0 . × − dynecm − . For constant total cosmic ray energy we expectthe volume of gas displaced to vary as V c ∝ P a − . There-fore if most of the cosmic rays are transported to ∼ M must be increased by ∼ P /P = 2 .
1. The combination of both correctionsleaves our original estimate nearly unchanged. Consider-ing the uncertainties in these estimates, we conclude thatthe total mass flow past R = 50 kpc found from Figure6 for the low- κ evolution, ∆ M = 5 . × M ⊙ , isconsistent with a quasi-adiabatic expansion of the Virgoatmosphere past radius 50 kpc due to gas displaced bycosmic rays when the cavity formed and no buoyant massflow across R = 50 kpc is required. Buoyant Gas Circulation
In addition to this atmospheric lifting, buoyant massflow largely within the cooling radius also contributesto the post-cavity cluster evolution. Buoyant gas in-creases ∆ E pot and as it expands we expect ∆ E th to de-crease. But every buoyant element must be accompaniedelsewhere in the hot gas atmosphere by a downward re-placement flow, largely balancing the buoyant changes in∆ E pot and ∆ E th . If cavities are created by feedback fol-lowing a small accretion onto the central AGN, a buoyantmass circulation flow is a natural response to the inflowexpected from radiative cooling. The low- κ cavity evo-lution ( n e = 6 × − cm − ) in Figure 2 is particularlyinteresting because all buoyant motion occurs within thecooling radius at 50 kpc.Figure 7 shows contours of the radial velocity v r ( r, z )at four times during the low- κ cavity evolution in Figure2 ( n e = 6 × − cm − ). (A plot of the radial velocity inthe high- κ cavity evolution shown in Figure 1 is almostidentical.) We plot v r ( r, z ) only near the z -axis since theradial velocities elsewhere in the atmosphere are verymuch smaller. Figure 7 shows a strong and sustainedsubsonic outflow until at least 0.1 Gyr. At t = 0 . z -axis as the densest partsof the thermal “cavity jet” fall back toward the clustercenter (see Mathews & Brighenti 2007b for more details).Later at t = 0 . t = 1Gyr.Because the entropy decreases monotonically with ra-dius in the initial cluster atmosphere, we can use theentropy as a tracer to confirm that there has been a netmass outflow within 50 kpc during the low- κ evolution(which is essentially adiabatic apart from the weak shockwave). We define the gas entropy as s = 10 − ( e/ρ / )in cgs units. In addition we expect that the buoyancyand outward flow of each gas element will increase withthe partial pressure of the cosmic rays, P c / ( P + P c ). Inthe upper panel of Figure 8 we plot P c / ( P + P c ) against∆ s = s ( t = 0 . − s atm , the change between the entropy s at 0.9 Gyr and the entropy at the same location in theoriginal Virgo atmosphere, s atm . Each point correspondsto a computational grid zone. In those regions where P c / ( P + P c ) is largest we see that ∆ s decreases system-atically with P c / ( P + P c ), indicating that gas containingmore cosmic rays at time 0.9 Gyr originally came fromhigh entropy regions closer to the cluster center.The lower panel of Figure 8 shows (with small points)the location in the cluster at time 0.9 Gyr that containsgas with P c / ( P + P c ) > . s < − . κ cavity evolution. Gas with the highest cos-mic ray content is also the gas with the lowest entropycompared to that at the same radius in the original clus-ter. The spatial distribution of this low entropy gas issimilar to the cosmic ray contours in the final panel ofFigure 2. Low entropy regions in the lower panel of Fig-ure 8 also have a slightly smaller gas density than ad-jacent gas at the same cluster radius R without cosmicrays. During times of interest for the cluster evolution,the gas in these post-buoyant regions containing cosmicrays will never return to the cluster core where it initiallyresided. Figure 8 shows conclusively that low entropy gasoriginally at small cluster radii has been buoyantly trans-ported outward during the low- κ post-cavity evolution,but very little has moved beyond 50 kpc by time 0.9 Gyr.Cosmic rays from cavities are driving a mass circulationin the cluster gas out to a large fraction of the coolingradius.Evidently the long-term decrease in cluster gas mass∆ M = 1 . × M ⊙ within R = 10 kpc where thecavities formed arises due to the combined effects of thejet of thermal gas along the z -axis and the cosmic raybuoyancy that prevents much of this gas from return-ing back to R = 10 kpc. Recall that ∆ M is nearlyindependent of the cosmic ray diffusion coefficient. The cosmic ray energy required to remove a solar mass of gasfrom R = 10 kpc is about e ⊙ = E cav / ∆ M = 8 . × erg M ⊙− . If ˙ M δt = 8 M ⊙ would cool each year inVirgo due to radiation losses alone, the cosmic ray lu-minosity required for a buoyant outflow to balance ˙ M is L cr = e ⊙ ˙ M = 7 . × erg yr − or 2 . × erg s − .However, the rate that energy is created by accretiononto the ∼ × M ⊙ central black hole in M87/Virgois L acc ≈ . M c = 4 . × erg s − . The cosmic ray lu-minosity L cr required to balance a radiating cooling massinflow in Virgo is ∼ L acc . Therefore,if only a fraction ǫ cr ≈ × − of the accretion energyconverts to cosmic rays that inflate X-ray cavities, thismight be sufficient to shut down the cooling flow. A sub-sequent calculation with multiple cavities and radiativecooling will be required to verify this. DISCUSSION
In previous numerical simulations of X-ray cavity evo-lution (e.g. Br¨uggen & Kaiser 2002; Reynolds et al.2005; Gardini 2007; Pavlovski et al. 2007) as well asin the recent review of McNamara & Nulsen (2007) ithas been concluded that the cavities have an importantnet heating effect on the cluster gas. This result can beunderstood because the simulated cavities were inflatedwith ultra-hot (but non-relativistic) gas. As these cav-ities are initially inflated, they perform an amount ofheating ∼ P V on the surrounding cluster gas, but thethermal energy inside the cavities also contributes to thecluster thermal energy budget. When we perform calcu-lations in which cavities are inflated with ultra-hot gas,we also find a (small) net heating when the total clusterthermal energy includes the energy injected inside thecavity. In this case the total energy introduced by thecavity is [ γ/ ( γ − P V = (2 . − P V , depending on theassumed value of γ inside the cavity. (After times > ∼ yr most of the energy of hot-gas cavities is stored in thepotential energy of the cluster.) The approximate valuesof 4 P V plotted in Figure 4 show the estimated cavityheating when cavities are formed with ultra-hot gas (e.g.Birzan et al. 2004; Rafferty et al. 2007; McNamara &Nulsen 2007).But ultra-hot gas is not an appropriate substitute forcosmic rays. When the diffusion κ is sufficiently smalland cosmic rays are strongly trapped within the cavi-ties, we find here that the expanding cavities also heatthe gas by ∼ P V . This can be seen in Figure 4 where∆ E th ( t cav ) ≈ P V . But the cosmic ray contents of ourcavities never contribute to the cluster thermal energy.Instead, the displacement of cluster gas by the diffusing,buoyant cosmic rays results in an overall expansion andcooling of the cluster gas.It should also be recognized that the introduction ofultra-hot gas in previous numerical simulations of cavityevolution implicitly suggests that mass as well as cosmicray energy is transported out from the central AGN, i.e.this conventional means of cavity formation must be re-garded as a (mass) circulation flow similar to those wehave considered (Mathews et al. 2003; 2004). This as-sumed non-relativistic mass outflow can be quite large aswe discussed in the Introduction. While we assume herethat the AGN produces jets of pure relativistic energy, itis possible that real jets do in fact carry non-relativisticmass away from the central AGN (as in Brighenti &Mathews 2006). Nevertheless, current X-ray observa-tions have been unable to detect thermal emission fromultra-hot gas in the cavities (e.g. McNamara & Nulsen2007, but see Mazzotta et al. 2002). CONCLUSIONS
It is generally agreed that the dominant source of feed-back energy in galaxy clusters is accretion onto centralblack holes (AGNs). Some of this accretion energy isthought to be transported out into the cluster gas byjets, forming X-ray cavities. In previous numerical simu-lations of X-ray cavity evolution the cavities were formedby introducing ultra-hot, non-relativistic gas at some ra-dius in the cluster gas.For the first time we consider cavities that are formedexclusively by cosmic rays that can both diffuse throughthe cluster gas and be advected by it. We discuss the en-ergetics of X-ray cavity and radio lobe evolution in thehot gas within the Virgo galaxy cluster. For simplicity weconsider cavities that are formed in time t cav = 2 × yrs by cosmic rays with total energy E cav = 10 ergsbut which have a wide range of cosmic ray diffusion co-efficients κ since this is currently the most uncertain pa-rameter in cavity-lobe evolution. For simplicity, we donot consider here secular energy losses in the cosmic raysdue to synchrotron emission, Coulomb heating, inverseCompton losses, spallation etc., so the total cosmic rayenergy can only change due to local expansion or com-pression by the thermal gas. In this approximation cos-mic ray pressure gradients interact with the cluster gasin a perfectly adiabatic fashion.We conclude that:(1) As cavities expand during formation, they generateweak shock waves that propagate into the surroundingcluster gas where the shock energy is dissipated. Whenthe cosmic ray diffusion κ is low, the maximum shockheating ∼ P V ∼ E cav / κ increasesthe cosmic ray pressure gradients are lower and the shockheating is considerably less. In general, 4 P V , whichvaries with time, is an inaccurate measure of E cav . Theseresults are consistent with our earlier studies of 1D cavityevolution in a uniform gas (Mathews & Brighenti 2007a).(2) The longevity of visible X-ray cavities and the radialdistance that they move out in the cluster gas duringtheir lifetimes both increase as the cosmic ray diffusion κ decreases. Unlike cavities formed with hot gas (e.g.Gardini 2007), cosmic ray cavities remain coherent for > ∼ yrs.(3) In spite of shock heating, cavities formed by cos-mic rays have a net cooling effect on the cluster gas.This is unlike the cluster heating when cavities are in-flated largely with ultra-hot but non-relativistic gas, ascommonly assumed, that later contributes to the clusterthermal energy.(4) The qualitative character of post-cavity energetics remains similar when the cosmic ray diffusion coefficient κ varies over a wide range. After several 10 yrs thetotal energy change in the cluster ∆ E pot + ∆ E th + E kin increases as κ decreases.(5) As cosmic rays displace cluster gas, the entire clus-ter atmosphere expands and cools, increasing the clusterpotential energy and decreasing its thermal energy. Suc-cessive cavities will result in an irreversible expansion ofthe cluster gas as long as most of the cosmic rays remaintrapped within or near the cooling radius, as in our com-puted post-cavity evolutions.(6) Inhomogeneously distributed cosmic rays are an im-portant source of buoyancy and mass flow in the clustergas. As the cluster gas surrounding cavities is penetratedby diffusing cosmic rays, this gas rises in the cluster po-tential, driving a net mass outflow in the cluster gas.Radial mass circulation occurs even when no gas flowsout in the AGN jets.(7) After buoyant gas containing cosmic rays flows fromthe cluster core into distant, low-density regions, it re-mains there never to return during times comparable tothe cluster age. Because of the necessarily positive en-tropy gradient in the pre-cavity cluster, the entropy inthe post-buoyant gas in these distant regions is slightlylower than elsewhere at the same cluster radius.(8) To quantify the previous conclusion, suppose a mass∆ M R within the cluster radius R where the X-ray cav-ity forms is irrevocably transported by cosmic ray buoy-ancy to distant regions in the cluster gas. We find thata cosmic ray energy of e ⊙ = E cav / ∆ M R ≈ ergsremoves a solar mass of gas from the cavity site. Forexample in the Virgo cluster a cosmic ray luminosity ofonly L cr = e ⊙ ˙ M = 2 × erg s − can remove gas at thecooling flow rate ˙ M ≈ M ⊙ yr − that would otherwiseoccur in the absence of feedback energy. Only a fraction ǫ cr ≈ × − of the accretion energy onto the centralblack hole in M87/Virgo, L acc ≈ . M c , if convertedto cosmic rays that inflate X-ray cavities, is in principlesufficient to shut down the cooling flow.Our conclusions suggest that cosmic ray buoyancy pro-vides an important new means of understanding or pos-sibly solving the cooling flow problem with an outwardmass circulation that may balance the inexorable inflowdue to radiation losses. This outward mass flow dueto cosmic ray buoyancy has the added attraction of notheating the gas too much – in fact cavity formation withcosmic rays produces a net cooling.It remains to be determined if the radio emission or π gamma emission from galaxy clusters are at levelsthat are consistent with the buoyant activity required toarrest the cooling flow rate. After a few 10 years mostof the cosmic rays will be in very low density gas wheresuch emission will be greatly reduced.Studies of the evolution of hot cluster gas at UC SantaCruz are supported by NASA and NSF grants for whichwe are very grateful. REFERENCES
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Evolution of an X-ray cavity in Virgo with cosmic ray diffusion parameter n e = 6 × − cm − at six times shown in Gyrsat the upper right of each panel. Solid lines show the gas density contours ρ ( r, z ) in units of 10 − g cm − . Dotted lines show with sixcontours the cosmic ray energy density e c ( r, z ) in units of 10 − erg cm − . Two adjacent contours are labeled and others can be found byextending the same additive variation. Fig. 2.—
Evolution of an X-ray cavity in Virgo with cosmic ray diffusion parameter n e = 6 × − cm − at six times shown in Gyrsat the upper right of each panel. Solid lines show the gas density contours ρ ( r, z ) in units of 10 − g cm − . Dotted lines show with sixcontours the cosmic ray energy density e c ( r, z ) in units of 10 − erg cm − . Two adjacent contours are labeled and others can be found byextending the same additive variation. Due to crowding at early times the cosmic ray contours are difficult to distinguish in the first threepanels. For the final three panels the outer two cosmic ray contours are 14.29 & 31.43 (panel 0.300), 10.0 & 20.0 (panel 0.500), and 4.29& 11.43 (panel 0.900). Fig. 3.—
Pressure profiles along the z -axis in units of 10 − dynes cm − . Each panel contains four (often overlapping) profiles: thegas pressure P ( r ) (solid lines), cosmic ray pressure P c ( r ) (long dashed lines), the total pressure P + P c (dotted lines), and the initial gaspressure in the cluster before the cavity is introduced (dash-dotted lines). Upper four panels: show pressure profiles at four times for theevolution in Fig. 1 with n e = 6 × − cm − . Lower four panels: show pressure profiles at four times for the evolution in Fig. 2 with n e = 6 × − cm − . Each panel is labeled with the time in Gyrs. -50050100-50050100-500501006.5 7 7.5 8 8.5 9-50050100 Fig. 4.—
Global energy evolution in four cavities with decreasing cosmic ray diffusion coefficient κ ∝ n e characterized by densityparameters n e : 6 × − (panel a ), 6 × − (panel b ), 6 × − (panel c ), and 6 × − (panel d ). The energies are labeled as follows:cosmic ray energy E cr (long dashed lines); change in potential energy ∆ E pot (short dashed lines); kinetic energy E kin (dotted lines);change in thermal energy ∆ E th (lower solid lines); and the total energy E tot (dash-dotted lines). The total energy associated with theapproximate cavity volume 4 P V is shown in the upper solid lines. All energies are in units of 10 ergs and are those in the hemispherecontaining our computational grid. Fig. 5.—
Approximate evolution of the cavity radius r cav ( t ) and its mean radius in the cluster R cav ( t ). The cavity radius is foundfrom the estimated cavity volume V by assuming that the cavity is spherical, r cav = [3 V/ π ] / . The four lines correspond to the fourdecreasing cosmic ray diffusivities κ ( n e , n e ) in each panel of Figure 4: a , dotted line; b , short dashed line; c , long dashed line; d , solid line.Each overlapping trajectory R cav ( t ) ends at the time when r cav → Fig. 6.—
Variation of the cumulative spherical mass gas distribution M ( R ) in one hemisphere of the cluster at times t = 0 and 0.9 Gyrfor two values of the cosmic ray diffusion parameter n e shown in parentheses. The cumulative mass is multiplied by ( R/
10 kpc) − toremove most of the radial variation between R = 6 . Fig. 7.—
Evolution of the radial velocity v r ( r, z ) near the z -axis with cosmic ray diffusion parameter n e = 6 × − cm − at four timesshown in Gyrs at the upper right of each panel. Contours are labeled with values of v r in km s − . Fig. 8.—
Top panel : Correlation of the cosmic ray partial pressure P c / ( P + P c ) with the entropy change ∆ s in each computationalzone for the low- κ post-cavity flow over time t = 0 . R = 50 kpc. Bottom panel : Location of regions in the cluster with high cosmic ray partial pressures ( P c / ( P + P c ) > .
3) (points) and regions of largenegative entropy change (∆ s < − .
1) (open squares).1) (open squares).