Cool core cycles: Cold gas and AGN jet feedback in cluster cores
aa r X i v : . [ a s t r o - ph . GA ] S e p Draft version August 15, 2018
Preprint typeset using L A TEX style emulateapj v. 5/2/11
COOL CORE CYCLES: COLD GAS AND AGN JET FEEDBACK IN CLUSTER CORES
Deovrat Prasad and Prateek Sharma Joint Astronomy Program and Department of Physics, Indian Institute of Science, Bangalore, India 560012
Arif Babul Department of Physics and Astronomy, University of Victoria, Victoria, BC V8P 1A1, Canada
Draft version August 15, 2018
ABSTRACTUsing high-resolution 3-D and 2-D (axisymmetric) hydrodynamic simulations in spherical geometry,we study the evolution of cool cluster cores heated by feedback-driven bipolar active galactic nuclei(AGN) jets. Condensation of cold gas, and the consequent enhanced accretion, is required for AGNfeedback to balance radiative cooling with reasonable efficiencies, and to match the observed coolcore properties. A feedback efficiency (mechanical luminosity ≈ ǫ ˙ M acc c ; where ˙ M acc is the massaccretion rate at 1 kpc) as small as 6 × − is sufficient to reduce the cooling/accretion rate by ∼ . × M ⊙ ). This value is much smallercompared to the ones considered earlier, and is consistent with the jet efficiency and the fact that onlya small fraction of gas at 1 kpc is accreted on to the supermassive black hole (SMBH). The feedbackefficiency in earlier works was so high that the cluster core reached equilibrium in a hot state withoutmuch precipitation, unlike what is observed in cool-core clusters. We find hysteresis cycles in all oursimulations with cold mode feedback: condensation of cold gas when the ratio of the cooling-timeto the free-fall time ( t cool /t ff ) is .
10 leads to a sudden enhancement in the accretion rate; a largeaccretion rate causes strong jets and overheating of the hot ICM such that t cool /t ff >
10; furthercondensation of cold gas is suppressed and the accretion rate falls, leading to slow cooling of the coreand condensation of cold gas, restarting the cycle. Therefore, there is a spread in core properties,such as the jet power, accretion rate, for the same value of core entropy or t cool /t ff . A fewer numberof cycles are observed for higher efficiencies and for lower mass halos because the core is overheatedto a longer cooling time. The 3-D simulations show the formation of a few-kpc scale, rotationally-supported, massive ( ∼ M ⊙ ) cold gas torus. Since the torus gas is not accreted on to the SMBH,it is largely decoupled from the feedback cycle. The radially dominant cold gas ( T < × K; | v r | > | v φ | ) consists of fast cold gas uplifted by AGN jets and freely-infalling cold gas condensing outof the core. The radially dominant cold gas extends out to 25 kpc for the fiducial run (halo mass7 × M ⊙ and feedback efficiency 6 × − ), with the average mass inflow rate dominating theoutflow rate by a factor of ≈
2. We compare our simulation results with recent observations.
Subject headings: galaxies: clusters: intracluster medium – galaxies: halos – galaxies: jets INTRODUCTION
Majority of baryons in galaxy clusters are in the formof a hot plasma known as the intracluster medium (ICM).In absence of cooling and heating, the ICM is expected tofollow self-similar profiles for density, temperature, etc.,irrespective of the halo mass (Kaiser 1986, 1991; seealso the review by Voit 2005). However, self-similarityis not observed in either groups or clusters (e.g., Ponmanet al. 1999; Balogh, Babul, & Patton 1999; Babul etal. 2002). Moreover, the core cooling times in about athird of clusters is shorter than 1 Gyr, much shorter thantheir age ( ∼ Hubble time; e.g., Cavagnolo et al. 2009;Pratt et al. 2009). Thus, we expect cooling to shapethe distribution of baryons in these cool-core clusters.The existence of cool cores with short cooling times ina good fraction of galaxy clusters is a long-standing puz-zle. According to the classical cooling flow model, clustercores with such short cooling times were expected to cool [email protected]@[email protected] catastrophically and to fuel star formation at a rate of100 − ⊙ yr − (e.g., Fabian 1994; Lewis et al.2000). However, cooling, dropout, and star formationat these high rates are never seen in cluster cores (e.g.,Edge 2001; Peterson et al. 2003; O’Dea et al. 2008).This means that some source(s) of heating is(are) ableto replenish the core cooling losses, thereby preventingrunaway cooling and star-formation.While there are potential heat sources, such as thekinetic energy of in-falling galaxies and sub-halos (e.g.,Dekel & Birnboim 2008), thermal conduction from thehotter outskirts (e.g., Voigt & Fabian 2004; Voit 2011),a globally stable mechanism, which increases rapidly withan increasing hot gas density in the core, is requiredto prevent catastrophic cooling. Observations of severalcool-core clusters by Chandra and
XMM-Newton haveuncovered AGN-jet-driven X-ray cavities, whose mechan-ical power is enough to balance radiative cooling in thecore (e.g., B¨ohringer et al. 2002; Bˆırzan et al. 2004;McNamara & Nulsen 2007). The AGN jets are poweredby the accretion of the cooling ICM on to the supermas- D. Prasad, P. Sharma, A. Babulsive black hole (SMBH) at the center of the dominantcluster galaxy. Thus, more cooling/accretion leads to anenhanced jet power and ICM heating, closing a feedbackloop that prevents runaway cooling in the core.AGN feedback has been long-suspected to play a rolein self-regulating the ICM (e.g., Binney & Tabor 1995;Ciotti & Ostriker 2001; Soker et al. 2001; Babul et al.2002; McCarthy et al. 2008), but a clear picture hasemerged only recently. While AGN feedback should pro-vide feedback heating in cluster cores (as it is enhancedwith ICM cooling), it is not obvious if, for reasonableparameters, AGN heating can keep pace with coolingthat increases rapidly with an increasing core density.Moreover, the dense core gas is expected to be highlysusceptible to fragmentation, leading to the formationof a multiphase medium consisting of cold dense cloudscondensing from the hot diffuse ICM itself. Pizzolato &Soker (2005) suggest that AGN outbursts that result inthe heating of the cluster cores are due to the infall andaccretion of these cold clumps.The importance of cold gas precipitation/feedback hasalso been been highlighted by several recent observations.The fact that there is some multiphase-cooling/star-formation, albeit at a much smaller rate than predictedby the cooling flow estimate (Soker et al. 2001), ties wellwith the idea of a small fraction of the thermally un-stable core gas cooling to the stable atomic and molec-ular temperatures. A lot of this cold gas is expected toform stars, but some should be accreted on to the cen-tral SMBH. Reservoirs of atomic (e.g., Crawford et al.1999; McDonald et al. 2011a; Werner et al. 2014) andmolecular gas (e.g., Donahue et al. 2000; Edge 2001; Sa-lom´e et al. 2006; Russell et al. 2014; O’Sullivan et al.2015), both extended and centrally concentrated, andongoing star formation (e.g., Bildfell et al. 2008; Hicks,Mushotzky, & Donahue 2010; McDonald et al. 2011b)are observed in a lot of cool-core clusters. Additionally,powerful radio jets/bubbles observed in most cool-coreclusters (Cavagnolo et al. 2008; Mittal et al. 2009) canbe interpreted as a signature of kinetic feedback due tocold gas accretion on to the SMBH.Since cool cores are in rough global thermal balance(i.e., the cooling rate minus the heating rate is smallerthan just the radiative cooling rate), the existence of coldgas in cluster cores can be understood as a consequenceof local thermal instability in a weakly stratified atmo-sphere (McCourt et al. 2012; Sharma et al. 2012a; Singh& Sharma 2015). The idealized simulations, which im-pose global thermal equilibrium in the ICM, show thatthe nonlinear evolution of local thermal instability leadsto in-situ condensation of cold gas only if the ratio ofthe cooling time and the free-fall time ( t cool /t ff ) is . ∼ −
100 (Gaspari et al. 2013; Sharmaet al. 2012a). This enhanced accretion rate in the coldphase can explain both the global thermal balance incluster cores and the general lack of massive cooling flows in almost all cool-core clusters whereas, the hot-mode(Bondi) accretion rate appears inadequate by orders ofmagnitude (e.g., McNamara, Rohanizadegan, & Nulsen2011).In detail, the precipitation of the cold gas, followed bya sudden increase in the accretion rate onto the SMBHs,leads to an increase in jet/cavity power and (slight) over-heating of the core. The core expands and as the ratio t cool /t ff rises above the threshold value of t cool /t ff =10,the gas is no longer prone to condensation. The accretionrate drops, as does the jet power. The core cools slowlyand the whole cycle starts again when t cool /t ff .
10. Thefrequency of heating/cooling cycles depend on jet effi-ciency and the halo mass. These features of the coldfeedback model are verified in our numerical simulations.In fact, the simple criterion of t cool /t ff .
10 for the on-set of local thermal instability is expected to be generic –applicable not only to the intracluster medium (ICM) butalso the intragroup medium (IGrM) and the circumgalac-tic medium (CGM) of all galaxies, including the Milkyway (Sharma et al. 2012b; Voit et al. 2015b). This, inturn, has far-reaching implications for providing a com-mon framework for understanding the the breaking ofself-similarity in the properties of hot gas across the hi-erarchy, from galaxies to groups to clusters, the presenceof multi-phase gas in group and clusters cores, and thedetection of cold gas in galaxies at distances of ∼
100 kpc(e.g., Werk et al. 2014). In fact, recent more realisticAGN jet feedback simulations show that cold gas con-densation begins when the t cool /t ff .
10 condition is met,and two distinct cold gas structures emerge: extendedcold filaments which go out 10s of kpc; and a few-kpcrotationally-supported cold torus (Gaspari et al. 2012;Li & Bryan 2014a,b). This dichotomy in cold gas dis-tribution is also seen in observations (e.g., McDonald etal. 2011a).Now that the theoretical models are satisfactorily ableto describe the basic state of the ICM in cool clustercores, and since observations of cold gas and jets/cavitiesare rapidly accumulating, it is ripe to make detailedcomparisons between observations and numerical simula-tions. We also aim to investigate the similarities and dif-ferences in cold gas and jet/bubble properties as a func-tion of the halo mass and feedback efficiency.In this paper, we focus on cool-core clusters and havecarried out 3-D and 2-D (axisymmetric) simulations ofthe interaction of feedback-driven AGN jets with theICM over cosmological timescales, varying the halo massand the feedback efficiency. The 3-D simulations, whichshould correspond more closely to reality, show the for-mation of a cold, massive, angular-momentum-supportedtorus, as seen in previous works (Gaspari et al. 2012; Li& Bryan 2014a,b). This massive cold torus is decoupledfrom the AGN feedback cycle, which is governed by thelow angular momentum, radially-dominant ( | v r | > | v φ | , v r /v φ is the radial/azimuthal component of the velocity)in-falling cold gas. Angular-momentum-supported gas isabsent in 2-D simulations because of axisymmetry andthe absence of rotation in the initial state (stochasticangular momentum can be generated in 3-D because of ∂/∂φ terms in the angular momentum equation). How-ever, 2-D simulations are useful for two reasons: first,they show similar behavior to 3-D simulations, if we onlyconsider the radially-dominant ( | v r | > | v φ | ) cold gas; sec-old gas and AGN jet feedback in cluster cores 3ond, they are much cheaper to run for long timescales,and thus are useful to do parameter scans in halo massand accretion efficiency.Compared to previous works (Gaspari et al. 2012; Li& Bryan 2014a,b), we have carried out simulations withsmaller feedback jet efficiencies. We find that a feedbackefficiency as low as 6 × − (ratio of the input jet powerand ˙ M acc c , where ˙ M acc is the accretion rate measured at1 kpc) is sufficient to reduce the mass accretion/coolingrate by a factor of about 10 compared to the cooling flowvalue in groups and clusters. Such a low feedback effi-ciency fits in nicely with the observations which suggestthat only a small fraction ( ∼ .
01) of the available gas isaccreted by the SMBH (e.g., Loewenstein et al. 2001),and with the estimate of jet efficiency ( ∼ . − . . .
01 the Ed-dington value (e.g., Narayan & Yi 1995; Merloni et al.2003); i.e., . .
22 M ⊙ yr − for a 10 M ⊙ SMBH. Theexpected mass accretion rate on to the SMBH in oursimulations ( ∼ M acc in Table 1) satisfies thisconstraint.We have analyzed the velocity-radius distribution ofthe cold gas in our simulations to compare with re-cent ALMA and
Herschel observations of cold gas struc-ture and kinematics in galaxy/cluster cores (e.g., Mc-Namara et al. 2014; David et al. 2014; Werner etal. 2014). Our simulations help in interpreting obser-vations of cold gas outflows and inflows at scales & . &
500 km s − )atomic/molecular outflows are uplifted by the outgoingAGN jet. The slower ( .
300 km s − ) infall of cold gas isdue to condensation in the dense core. The cold gas inthe rotationally supported torus is at the local circularvelocity ( ∼
200 km s − ).Our paper is organized as follows. In section 2 wepresent the numerical setup, in particular our implemen-tation of mass and kinetic energy injection due to AGNjets. Section 3 presents the key results from our 3-D and2-D simulations, a comparison of 3-D vs. 2-D, and theimpact of parameters such as feedback efficiency and halomass on our results. In section 4 we discuss our resultsand compare with previous simulations and observations,and we conclude with a brief summary in section 5. NUMERICAL SETUP & GOVERNING EQUATIONS
We modify the ZEUS-MP code, a widely-used finite-difference MHD code (Hayes et al. 2006), to simulatecooling and AGN feedback cycles in galaxy clusters. Wesolve the standard hydrodynamic equations using spher-ical ( r, θ, φ ) coordinates, with cooling, external gravity,and mass and momentum source terms due to AGN feed-back: ∂ρ∂t + ∇· ( ρ v ) = S ρ , (1) ∂ρ v ∂t + ∇ . ( ρ v ⊗ v ) = −∇ p − ρ ∇ Φ + S ρ v jet ˆ r , (2) e ddt ln( p/ρ γ ) = − n e n i Λ( T ) , (3) where ρ is the mass density, v is the fluid velocity, p = ( γ − e is the pressure ( e is the internal energydensity and γ = 5 / T ) isthe temperature-dependent cooling function, n e ( n i ) isthe electron (ion) number density given by ρ/ [ µ e ( i ) m p ]( µ e = 1 .
18 and µ i = 1 . K.In addition to the terms shown in Eqs. 1-3, the codeuses the standard explicit artificial viscosity, and hasimplicit diffusion associated with the numerical scheme(Stone & Norman 1992). In addition to the standardnon-linear viscosity, we use the linear viscosity, as rec-ommended by Hayes et al. (2006) for strong shocks (seetheir Appendix B3.2).We use a fixed external NFW gravitational potentialΦ( r ) due to the dark matter halo (Navarro et al. 1996);Φ( r ) = − GM r ln(1 + c r/r )[ln(1 + c ) − c / (1 + c )] , (4)where M ( r ) is the characteristic halo mass (radius)and c ≡ r /r s is the concentration parameter; thedark matter density within r is 200 times the criticaldensity of the universe and r s is the scale radius. In thispaper we focus on cluster and massive cluster runs with M = 7 × M ⊙ and 1 . × M ⊙ , respectively,and adopt c = 4 . S ρ for mass and S ρ v jet ˆ r for the radial momentum to drive AGN jets ( v jet is thevelocity which the jet matter is put in). These sourceterms and the cooling term (in Eq. 3) are applied in anoperator-split fashion. The mass and momentum sourceterms are approximated forward in time and centered inspace. The cooling term is applied using a semi-implicitmethod described in Eq. 7 of McCourt et al. (2012).Our simulations do not include physical processes likestar formation and supernova feedback. Star formationmay deplete some of the cold gas available in the cores(see Li et al. 2015), but this is unlikely to change ourresults for a realistic model of star formation. Super-nova feedback is energetically subdominant compared toAGN feedback, and cannot realistically suppress clustercooling flows (e.g., Saro et al. 2006). We only includethe most relevant physical processes, namely cooling andAGN jet feedback, in our present simulations.
Jet Implementation
Jets are implemented in the active domain by addingmass and momentum source terms as shown in Eqs. 1& 2. The source terms are negligible outside a smallbiconical region centered at the origin around θ = 0 , π ,mimicking mass and momentum injection by fast bipolarAGN jets.The density source term is implemented as S ρ ( r, θ ) = N ˙ M jet ψ ( r, θ ) , We have also carried out narrow-jet simulations with momen-tum injection in the vertical [ ˆ z ] direction, but do not find muchdifference from our runs with momentum injection in the radial[ ˆ r ; see Eq .
2] direction.
D. Prasad, P. Sharma, A. Babulwhere ˙ M jet is the single-jet mass loading rate, ψ ( r, θ ) = (cid:20) (cid:18) θ jet − θσ θ (cid:19) + tanh (cid:18) θ jet + θ − πσ θ (cid:19)(cid:21) × (cid:20) (cid:18) r jet − rσ r (cid:19)(cid:21) ×
14 (5)describes the spatial distribution of the source termwhich falls smoothly to zero outside the small biconicaljet region of radius r jet and half-opening angle θ jet . Wesmooth the jet source terms in space because the Kelvin-Helmholtz instability is known to be suppressed due tonumerical diffusion in a fast flow if the shear layer is unre-solved (e.g., Robertson et al. 2010). The normalizationfactor N = 32 πr (1 − cos θ jet )ensures that the total mass added due to jets per unittime is 2 ˙ M jet . All our simulations use the following jetparameters: σ r = 0 .
05 kpc, θ jet = π/
6, and σ θ = 0 . r jet is scaled with the halo mass; i.e., r jet = 2 kpc (cid:18) M × M ⊙ (cid:19) / . The jet mass-loading rate is calculated from the currentmass accretion rate ( ˙ M acc ) evaluated at the inner radialboundary such that the increase in the jet kinetic energyis a fixed fraction of the energy released via accretion;i.e., ˙ M jet v = ǫ ˙ M acc c . (6)We choose the jet velocity v jet = 3 × km s − (0 . c ; c is the speed of light); such fast velocities are seen in X-ray observations of small-scale outflows in radio galaxies(Tombesi et al. 2010). The jet efficiency ( ǫ ; our fiducialvalue is 6 × − ) accounts for both the fraction of thein-falling mass at the inner boundary (at 1 kpc for thecluster runs) that is accreted by the SMBH and for thefraction of accretion energy that is channeled into the jetkinetic energy. Our results are insensitive to a reasonablevariation in jet parameters ( v jet , r jet , θ jet , σ r , σ θ ), butdepend on the jet efficiency ( ǫ ).Like Gaspari et al. (2012), the jet energy is injectedonly in the form of kinetic energy; we do not add a ther-mal energy source term corresponding to the jet. We notethat Li & Bryan (2014b) have shown that the core evolu-tion does not depend sensitively on the manner in whichthe feedback energy is partitioned into kinetic or ther-mal form. Another difference from previous approaches,which use few grid points to inject jet mass/energy, isthat our jet injection region is well-resolved. Grid, Initial & boundary conditions
Most AGN feedback simulations evolved for cosmolog-ical timescales (e.g., Gaspari et al. 2012; Li & Bryan2014a) use Cartesian grids with mesh refinement. How-ever, we use spherical coordinates with a logarithmically spaced grid in radius, and equal spacing in θ and φ . Theadvantage of a spherical coordinate system is that it givesfine resolution at smaller scales without a complex al-gorithm. Perhaps more importantly, a spherical set upallows for 2-D axisymmetric simulations which are muchfaster and capture a lot (but not all) of essential physics.We perform our simulations in spherical coordinateswith 0 ≤ θ ≤ π , 0 ≤ φ ≤ π , and r min ≤ r ≤ r max , with r [min , max] = [1 , (cid:18) M × M ⊙ (cid:19) / . According to self similar scaling, we have scaled all lengthscales in our simulations (inner/outer radii r min /r max , r , jet radius r jet ) as M / .We apply outflow boundary conditions (gas is allowedto leave the computational domain but prevented fromentering it) at the inner radial boundary. We fix thedensity and pressure at the outer radial boundary to theinitial value and prevent gas from leaving or enteringthrough the outer boundary. Reflective boundary condi-tions are applied in θ (with the sign of v φ flipped) andperiodic boundary conditions are used in φ . We noticedthat cold gas has a tendency to artificially ‘stick’ at the θ boundaries (mainly in 2-D axisymmetric simulations)for our reflective boundary conditions. This cold gas canlead to an unphysically large accretion rate close to thepoles, and hence artificially enhanced feedback heating(Eq. 6). Therefore, we exclude 8 grid-points at eachpole when calculating the mass accretion rate; these ex-cluded angles correspond to only 0.5% of the total solidangle for 128 grid points in the θ direction. All our diag-nostics ( ˙ M acc , entropy profiles, etc.) also exclude thesesmall solid angles close to the poles.The resolution for 3 − D runs is 256 × ×
32 andfor 2 − D runs is 512 × r/r = 0 .
02 (0 . ≈ .
02 (0 .
01) kpc. For such a resolution our integratedquantities (mass accretion rate, jet power, cold gas mass,etc.) are converged.We focus on simulations of a galaxy cluster with M = 7 × M ⊙ but with different parameterssuch as feedback efficiency. For comparison we also car-ried out simulations for a massive cluster with M =1 . × M ⊙ . The initial conditions are the same as inSharma et al. (2012a); i.e., we assume the initial entropyprofile ( K ≡ T keV /n / e ; T keV is the ICM temperature inkeV and n e is the electron number density) of the form K ( r ) = K + K (cid:18) r
100 kpc (cid:19) . , (7)as suggested by Cavagnolo et al. (2009). For our clusterruns, we set K = 10 keV cm and K = 110 keV cm at the start (as in Sharma et al. 2012a). We assume self-similar behavior scaling with M (Kaiser 1986) to set Whether an entropy core exists is debated (Panagoulia, Fabian,& Sanders 2014), but our results are insensitive to our initialconditions. Our ICM profiles change with time and reach a quasi-steady state which may or may not have an entropy core. old gas and AGN jet feedback in cluster cores 5the initial entropy profile for our massive cluster runs (i.e.we assume K = 19 keV cm and K = 210 keV cm ;c.f. McCarthy et al. 2008). Except for early transients,our results are independent of the precise choice of theinitial values of K and K .The outer electron number density is fixed to be n e =0 . − . Given the entropy profile and the densityat the outer radius, we can solve for the hydrostatic den-sity and pressure profiles in an NFW potential (Eq. 4).We introduce small (maximum overdensity is 0.3) iso-baric density perturbations on top of the smooth density(for details, see Sharma et al. 2012a). RESULTS
In this section we describe the key results from oursimulations. Table 1 lists our runs. We begin with theresults from our fiducial 3-D cluster run (C6m5D3 in Ta-ble 1). We show that the 1-D profiles of density, entropy,etc. are consistent with observations. We highlight thecycles of cooling and AGN jet feedback, and the spa-tial and velocity distribution of the cold gas. We showthat there are three components in cold (
T < × K) gas distribution: a massive, centrally-concentrated,rotationally-supported torus; spatially extended and fast( &
500 km s − ) outflows correlated with jets; and slower( .
300 km s − ) in-falling cold gas that condenses outbecause of local thermal instability. Then we comparethe results from our 3-D and 2-D axisymmetric simula-tions. We also explore the dependence of our results onthe halo mass and the jet efficiency. The fiducial 3-D run
We experimented with different values of jet efficiencies( ǫ ; Eq. 6) in our 3-D cluster ( M = 7 × M ⊙ )simulations, and found that the average mass accretionrate for ǫ = 6 × − was about 10% of a pure cooling flow(see Table 1). Therefore, we choose this as our fiducialvalue, which is smaller compared to the values chosenby some recent works (Gaspari et al. 2012; Li & Bryan2014a,b), but is consistent with observational constraints(e.g., O’Dea et al. 2008). Our fiducial value should beconsidered as the smallest efficiency that is required toprevent a cooling flow in a cluster (this critical efficiencydepends on the halo mass, as we shall see later).The minimum ratio of the cooling time ( t cool ≡ nk B T / [2 n e n i Λ]) and the local free-fall time ( t ff ≡ [2 r/g ] / = [2 r /GM ( < r )] / ) is 7 for the initial ICM;this ratio ( t cool /t ff ) is a good diagnostic of the state of thecluster core in rough thermal balance. Since the initialcondition is in hydrostatic equilibrium, there is negligi-ble accretion through the inner boundary, and thereforethere is no jet injection. However, after a cooling timein the core ( ≈
200 Myr) there is a rise in the accretionrate across the inner boundary ( ˙ M acc ), and hence in jetmomentum injection (Eq. 6). The jet powers a bubblethat heats the core and raises t cool /t ff , keeping the massaccretion rate well below the cooling flow value (c.f. toppanel of Fig. 9). After this time the cluster core is in astate of average global thermal balance between radiativecooling and feedback heating via AGN jets. Jets, bubbles & multiphase gas
Figures 1 show the snapshots ( r − θ plane at φ = 0) ofpressure, density, and temperature at different times forour fiducial 3-D run. The X-ray emitting ICM plasma isquite distinct from the dense cold (10 K) gas and fromthe low-density jet/bubble. The cold gas accreting onto SMBH gives rise to AGN jets. Before a cooling time(0.2 Gyr) there are no signs of cooling and jets. After acooling time, accretion rate through the inner boundary(at 1 kpc) increases and bipolar jets are launched (0.29Gyr). The jets are not perfectly symmetric, as they areshaped by the presence of cold gas in their way. Theinhomogeneities in the ICM enhances mixing with (andstirring of) the ICM core, resulting in effectiveness of ourjets even with a low efficiency.Jets are fast in the injection region but become slow,buoyant, and almost in pressure balance with the ICM(compare the upper and middle panels of Fig. 1 ) becauseof turbulent drag and sweeping up of the ICM. In absenceof further power injection, the bubbles are detached fromthe jets and rise buoyantly and mix with the ICM at 10sof kpc scales (3.15 Gyr in Fig. 1). Most of the cold gasis very centrally concentrated (within 10 kpc), but doescondense out at larger radii, although never beyond 30kpc.As jets plough through the dense cold gas clouds, for-ward shock moves ahead of these clouds after partiallydisrupting them. The collision results in a reverse shockand a huge back-flow of hot jet material which mixes withthe cooler ICM, driving the core entropy to higher val-ues. These back-flows and mixing are mainly responsiblefor heating the cluster core.
Radial profiles
Before discussing the detailed kinematics of cold gasand jet cycles, we show in Figure 2 the 1-D profiles of im-portant thermodynamic quantities (entropy [ T keV /n / e ], t cool /t ff , n e , T keV ) as a function of radius for the fiducial3-D run. In addition to the instantaneous profiles (at 1,2, 3, 4 Gyr), the median profile and spread about it areshown. The median is calculated for the entropy mea-sured at 20 kpc (roughly the core size) and all the profileswith entropy within one standard deviation at the sameradius are shown in grey.The spread in quantities outside ∼
20 kpc is quitesmall, but increases toward the center because mul-tiphase cooling (leading to density spikes) and strongjet feedback (leading to overheating) are most effectivewithin the core. The density at 1 Gyr is peaked towardthe center, indicating that the cluster core is in a coolingphase. The spikes in density at 3 Gyr have correspondingspikes in entropy and t cool /t ff profiles, but not as promi-nent in the temperature profile. The temperature fluc-tuations are rather modest compared to fluctuations inother quantities because of dropout and adiabatic cool-ing. Temperature profiles show a general increase withradius, as seen in observations.There is a large spread in entropy toward lower valuesabout the median at radii <
10 kpc (top-left panel inFigure 2). This is because there are short-lived coolingevents during which the entropy in the core decreases sig-nificantly (simultaneously, density increases and t cool /t ff decreases). On the other hand, the increase in the coreentropy is smaller but lasts for a cooling time, which is D. Prasad, P. Sharma, A. Babul Fig. 1.—
The pressure (upper panel), electron number density (middle panel), and temperature (lower panel) contour plots ( R − z planeat φ = 0) in the core at different times for the 3-D fiducial run. The density is cut-off at the maximum and the minimum contour levelshown. The low-density bubbles/cavities are not symmetric and there are signatures of mixing in the core. The left panel corresponds toa time just before a cooling time in the core. The second panel from left shows cold gas dredged up by the outgoing jets. The rightmostpanel shows in-falling extended cold clouds. The pressure maps show the weak outer shock, but the bubbles/cavities so prominent inthe density/temperature plot are indiscernible in the pressure map, implying that the bubbles are in pressure equilibrium and buoyant.Also notice the outward-propagating sound waves in the two middle pressure panels in which the jet is active. The in-falling/rotationally-supported cold gas has a much lower temperature and pressure than the hot phase. The arrows in the temperature plots denote theprojected gas velocity unit vectors. old gas and AGN jet feedback in cluster cores 7 TABLE 1Table of runs
Label dim. N r × N θ × N φ min. resolution M jet efficiency ( ǫ ) h ˙ M acc i ‡ h ˙ M acc , hot i / h ˙ M acc , cold i jet duty(kpc) ( M ⊙ ) ( M ⊙ yr − ) cycle †† (%)C6m5D3 † × ×
32 0.02 7 × × − ‡ × ×
32 0.02 7 × × − × ×
32 0.02 7 × .
01 1.9 1.3 99.8C6m5D2 † × × × × − ‡ × × × × −
153 0.27 47.3C1m2-D2 2 512 × × × × × × − × × × × − × × . × × −
293 (299) 0.3 (0.5) 0.0M6m5D2 2 512 × × . × × − × × . × − × × . × × − × × . × .
01 8.1 3 . × Notes ‘C’ in the label stands for a cluster ( M = 7 × M ⊙ ) and ‘M’ for a massive cluster ( M = 1 . × M ⊙ ). Label C6m5D3 indicatesthat it is a cluster run in 3-D with an efficiency ǫ = 6 × − (Eq. 6). † The fiducial 3-D and 2-D runs. ‡ Angular brackets denote time average over the full run. The quantities in brackets denote values for a pure cooling flow ( ˙ M cf ). Note that8 grid points close to the poles are excluded when calculating the accretion rates. †† Jet duty cycle is defined as the fraction of total time for which the jet power is > erg s − . Fig. 2.—
X-ray emissivity-weighted (considering only 0.5-8 keV gas) 1-D profiles of important thermodynamic variables as a function ofradius. Snapshots at 1, 2, 3, 4 Gyr are shown. Various quantities are obtained by combining 1-D profiles of density and pressure. Themedian and standard deviation ( σ ) of entropy ( K ≡ T keV /n / e ) at 20 kpc are calculated. Various profiles corresponding to the medianentropy at 20 kpc (14 keV cm ) are shown in different panels (black lines with ‘+’). Thick grey lines show the profiles for which the entropyat 20 kpc is within 1 − σ of its median value. D. Prasad, P. Sharma, A. Babullonger in this state. This behavior is generic, fairly insen-sitive to parameters such as the feedback efficiency andthe halo mass.
The cold torus
While Figure 1 shows that cold gas can be dredged upby AGN jets (second panel; see also Revaz, Combes, &Salom´e 2008; Pope et al. 2010) and can also condenseout of the ICM at large scales (fourth panel), majorityof cold gas is at very small scales ( < θ = π/
2) plane at different times; the arrows show theprojection of velocity unit vectors. As the cluster evolvesthe cold gas, condensing out of the hot ICM, gains an-gular momentum from jet-driven turbulence. Becauseof a significant angular velocity, an angular momentumbarrier forms and cold gas circularizes at small radii.Unlike Li & Bryan (2014b), our cold torus is dynamicin nature as AGN jets disrupt it time and again, but itreforms due to cooling. Figure 3 shows the evolution ofthe torus at various stages of the simulation. The top-left panel of Figure 3 shows the cluster center at 0.5 Gyr.Small cold gas clouds are accumulating in the core afterthe first active AGN phase. At 1.3 Gyr, cold gas accret-ing through the inner boundary has an anti-clockwise ro-tational sense. At 1.98 Gyr, cold gas (and the hot gas outof which it condenses) is rotating clockwise. Jet activityleading up to this phase has reversed the azimuthal ve-locity of the cold gas. At all times after this the dynamiccold gas torus rotates in a clockwise sense, essentially be-cause the mass (and angular momentum) in the rotatingtorus is much larger than the newly condensing cold gas.The torus gets disrupted due to jet activity but formsagain quickly. The snapshots at 2.4 and 2.45 Gyr showthat the inner region is covered by the very hot/dilutejet material. If the jets were rapidly changing directionas argued by Babul et al. (2013), we would in factexpect the cold gas torus to be occasionally disruptedby the jets. In the present simulations, however, thisbehavior is an artifact of our feedback prescription; wescale the jet power with the instantaneous mass inflowrate through the inner boundary (see Eq. 6). Even smalloscillations of the cold torus can sometimes lead to a largeinstantaneous mass inflow through the inner boundaryand hence an explosive jet event. The reassuring fact isthat these explosive ‘events’ are rare and the jet materialis quickly mixed with the ICM after these. In reality,most of the cold gas in the torus will be consumed bystar-formation. Only the low angular momentum coldgas that circularizes closer in ( .
100 pc) can be accretedby the SMBH at a short enough timescale.A cold torus forms in all our 3D cluster simulationswith different efficiencies. However extended cold gas islacking at late times in simulations with high jet efficien-cies. Li & Bryan (2014b) show that after 3 Gyr the coldgas settles down in form of a stable torus, with no furthercondensation of extended cold gas. This is inconsistentwith observations which show that about a third of cool-core clusters show H α filaments extending out to 10s ofkpc from the center (McDonald et al. 2010). The bot-tom panels in Figure 3 from our fiducial run show thatthe torus is unsteady even at late times with extendedcold gas condensing out till the end of our run. We com- pare our results in detail with Li & Bryan (2014b) insection 4.1. Velocity and space distribution of cold gas
We find it very instructive to classify the cold gas intotwo components: most of the mass is in the rotationally-dominant gas at . radially dominant component spread over 20 kpc.Figure 4 shows the velocity and space distribution ofrotationally (the left two panels; d M/d ln | v φ | dr and d M/d ln | v r | dr ; | v φ | > | v r | ) and radially (the right twopanels; d M/d ln | v φ | dr and d M/d ln | v r | dr ; | v φ | < | v r | )dominant cold ( T < × K) gas, averaged from 1to 4 Gyr. The rotationally dominant gas distribution( | v φ | > | v r | ; two left panels in Fig. 4) shows two peaksat v φ ≈ ± −
200 km s − and r ≈ . − .
100 km s − .The distribution of the radially dominant cold gas inFigure 4 is quite different from the rotationally dominantgas. In addition to a larger radial extent, the radial ve-locity of the radially dominant component is much larger,going up to ±
600 km s − , much larger than the maxi-mum azimuthal speed. The radial velocity of the closerin gas ( . v r > v jet = 0 . c . The mass in the in-falling radially-dominant cold gas is ≈ twice that of the outgoing coldgas.Figure 5 shows the 1-D velocity distribution of the coldgas averaged from 1 to 4 Gyr. The two large, sharppeaks correspond to the massive clockwise rotating coldtorus. The radially-dominant component ( | v r | > | v φ | )shows a prominent high velocity tail in the positive di-rection. The negative velocity component for velocitieslarger than 300 km s − is also dominated by the radiallyin-falling (rather than rotationally dominant) gas, some-times affected by the fast jet back-flows. The maximumvelocity peak of the radially and rotationally dominantcold gas coincide at ≈ −
200 km s − , correspondingto the circular velocity at ∼ Cooling & heating cycles old gas and AGN jet feedback in cluster cores 9
Fig. 3.—
The 2-D ( z = 0) contour plots of electron number density (in cm − ) in the mid-plane of the very inner region at different timesfor the fiducial 3-D run, with the projection of the velocity unit-vector represented by arrows. The top-left panel shows the beginning ofthe infall of cold gas with random angular momentum. The top-second left panel shows an anti-clockwise transient torus. All times afterthis show a clockwise torus in the mid-plane which waxes and wanes because of cooling and AGN heating cycles. Even at late times thecold torus is not stable and gets disrupted by jets. Fig. 4.—
The velocity-radius distribution of the cold gas (
T < × K) mass averaged from 1 to 4 Gyr. The top-left panel shows the v φ − r mass distribution ( d Md ln | v φ | d ln r ; ∆ v φ = ∆ v r = 20 km s − , ∆ r = 0 . | v φ | > | v r | )gas and the bottom-left panel shows the v r − r distribution for the same gas. The top-right panel shows the v φ − r distribution for theradially-dominant ( | v r | > | v φ | ) cold gas and the bottom-right panel shows the v r − r ( d Md ln | v r | d ln r ) distribution for the same gas. Someof the salient features are: the rotationally-dominant cold gas, which is concentrated mainly within 5 kpc, is more abundant by a factor ∼
40 than the radially-dominant gas; the dominant rotationally-supported clockwise cold torus with a negligible radial velocity (see Fig.3) is clearly visible in the two left panels; the radially-dominant cold gas (with | v r | > | v φ | ) is much more radially extended, going out to25 kpc; the bottom-right panel shows that the in-falling ( v r <
0) cold gas dominates over the outgoing cold gas (by a factor ≈
2) and thatthe outgoing cold gas at . −1000 −800 −600 −400 −200 0 200 400 600 800 100010 velocity (kms − ) m a ss i n c o l dph a s e ( M ⊙ ) dM/d ln | v r | ; 3D dM/d ln | v φ | ; 3D dM/d ln | v r | ; | v r | > | v φ | ; 3D dM/d ln | v φ | ; | v φ | > | v r | ; 3D dM/d ln | v r | ; 2D Fig. 5.—
The velocity distribution of cold gas for the 3-D fidu-cial run with respect to the radial and azimuthal velocities. Alsoshown are the rotationally ( | v φ | > | v r | ) and radially ( | v r | > | v φ | )dominant components. At large velocities the total and radi-ally/rotationally dominant components coincide but at small ve-locities they do not, as expected (at low velocities the other compo-nent of velocity dominates the mass budget). Also shown (dashedline) is the radial velocity distribution for the 2-D fiducial run; theazimuthal velocity is zero for 2-D axisymmetric runs. PSfrag replacements time(Gyr) n o r m a li ze d q u a n t i t i e s ˙ M in , cold , ( M ⊙ yr − )˙ M out , cold , ( M ⊙ yr − ) ˙ M acc ( M ⊙ yr − )jet power (10 erg s − ) Fig. 6.—
The mass inflow (green short-dashed line) and outflow(red solid line) rates in the cold phase measured at 5 kpc as afunction of time. Also shown are the jet power (normalized to10 erg s − ; how jet power is calculated is described in section3.1.5) and the mass accretion rate at 1 kpc (Eq. 6). Note thatthe largest spikes in the cold outflow rates are mostly associatedwith a sudden rise in jet energy, indicating cold gas uplifted byjets. The difference between the mass inflow rate at 5 kpc and at1 kpc (clearly noticeable at ∼ ∼ .
75 Gyr.
One of the distinct features of the cold feedbackparadigm is that we expect correlations in jet power,cold gas mass, mass accretion rate, min( t cool /t ff ), coreentropy, etc. The observations indeed show such corre-lations (e.g., Figs. 1, 2 in Cavagnolo et al. 2008; seealso Voit & Donahue 2014; Sun 2009; McDonald etal. 2011a). In Figure 7 we make ‘phase-space’ plotsof jet power and cold-gas mass (total and the radially-dominant component) as a function of min( t cool /t ff ) forour fiducial 3-D run. Evaluating min( t cool /t ff ): The ratio t cool /t ff is calcu-lated by making radial profiles of emissivity-weighted(only including plasma in the range of 0.5 to 8 keV) in-ternal energy and mass densities. They are combined tocalculate t cool ≡ (3 / nk B T / ( n e n i Λ), and t cool /t ff is cal-culated by taking its ratio with the free-fall time basedon the NFW potential (Eq. 4; t ff ≡ [2 r/g ] / where g ≡ d Φ /dr ). The broad local minimum in t cool /t ff pro-file is searched going in from the outer radius and is usedas min( t cool /t ff ). Evaluating jet power:
The jet power is also calculatedin a novel way, which is close to what is done in obser-vations. We consider the grids with mass density lowerthan a threshold value (chosen to be 0.17 times the initialminimum density in the computational volume; resultsare insensitive to the exact value of the threshold den-sity) to belong to the jet/bubble material, and we simplyvolume-integrate the internal energy density of all suchcells to calculate the jet energy (only considering thermalenergy; we use γ = 5 / γ = 4 / . × ˙ M acc ( M ⊙ yr − ) erg s − . Assumingthis conversion, Figure 6 shows that the two estimates ofjet power are comparable in magnitude but vary ratherdifferently with time. This is because, while ˙ M acc is aninstantaneous quantity varying on a dynamical timescale,our jet power is based on the jet thermal energy whichis an integrated quantity.We anticipate cycles in the evolution of min( t cool /t ff )and jet power or the radially-dominant cold gas. Imag-ine that there is no accreting cold gas at the center; inthis state without heating the core is expected to coolbelow t cool /t ff ∼
10 (because accretion rate in the hotmode is small). The t cool /t ff .
10 state is prone to cold-gas condensation and enhanced feedback heating if ǫ issufficiently high. Energy injection leads to overheatingof the core and an increase in t cool /t ff ; since condensa-tion/accretion is suppressed in this state, both jet power(because of adiabatic/drag losses) and radially-dominantcold gas mass are reduced in this state of t cool /t ff > t cool /t ff ) evolves in form of clock-wise cycles of various widths (a measure of the range ofmin[ t cool /t ff ] before and after the jet event) and heights(jet power). Generally, a smaller t cool /t ff leads to a larger Observers calculate the bubble/cavity mechanical power byassuming it to be in pressure balance with the background ICMand by using a size and an age estimate for the bubble (e.g., seeBˆırzan et al. 2004). Indeed, our bubbles are in pressure balancewith the ICM, as seen in Fig. 1. old gas and AGN jet feedback in cluster cores 11 min(t cool / t ff ) j e t p o w e r ( e r g s − ) min(t cool / t ff ) m a ss i n c o l dph a s e ( M ⊙ ) min(t cool / t ff ) m a ss i n c o l dph a s e ( M ⊙ ; | v r | > | v φ | ) time (Gyr) Fig. 7.—
The variation of jet power, total cold gas mass and the radially-dominant cold gas mass as a function of min( t cool /t ff ) for thefiducial 3-D run from 2.2 to 3.2 Gyr. Color shows the evolution of cluster in time. While jet power (left panel) and radially-dominant coldgas mass (right panel) show clockwise cycles with min( t cool /t ff ), the total cold gas mass (middle panel) simply builds up in time. Noticethe linear scale for the total cold gas mass instead of a log scale for the other two cases. mass accretion rate and a larger jet energy, and thereforelarger overheating and a larger min( t cool /t ff ). Since theefficiency of our fiducial run is rather small ( ǫ = 6 × − ),the cluster core remains with t cool /t ff <
20 at most times.In section 3.2.2 we discuss the dependence of our resultson jet efficiency ( ǫ ).The middle panel of Figure 7 shows the total mass incold gas (most of which is in the cold rotating torus)as a function of min( t cool /t ff ). We see the mass in thecold torus building up in time. We can easily see thatthe total cold gas mass simply builds up in time (see thegreen dashed line in the upper panel of Fig. 9), andis uncorrelated with min( t cool /t ff ). The right panel ofFigure 7 shows the mass in the radially-dominant coldgas (with | v r | > | v φ | ) as a function of min( t cool /t ff ). Thispanel also shows clockwise cycle like jet power shown inthe left panel. A larger radially-dominant cold gas massgenerally implies a higher accretion rate and a larger jetpower, but the features in jet and cold gas cycle are notalways varying in an identical fashion. While the globalevolution in phase space is clockwise, there is haphazardevolution at smaller timescales (e.g., between 2.5 to 2.9Gyr). The 2-D runs
The 3-D simulations are very expensive compared tothe 2-D ones, not only because the number of grid cellsis larger but also because the CFL time step is muchsmaller. The CFL time step in 3-D is dominated by cellsclose to the polar regions ( θ = 0 , π ) and ∝ r sin θ ∆ φ ≈ r ∆ θ ∆ φ/
2, much smaller than in 2-D ( ∝ r ∆ θ ). Our256 × ×
32 (3-D) runs have 8 times more grid cellscompared to our 512 ×
256 (2-D) runs and the CFL timestep is ≈ . Comparison with 3-D
Since 3-D simulations are substantially more time con-suming compared to the 2-D axisymmetric ones, it willbe very useful if some robust inferences can be drawnfrom these faster 2-D computations. To compare the 3-Dsimulations with their 2-D counterparts, we have carriedout the fiducial 3-D simulation in 2-D with identical pa-rameters (the initial density perturbations in 2-D runsare the same as the perturbation for the φ = 0 plane in3-D).Figure 5 compares the time-averaged velocity distribu-tion of cold gas in 2-D and 3-D simulations. Since theazimuthal velocity vanishes in 2-D axisymmetric simu-lations, we compare the radial velocity distribution ofcold gas in 2-D simulations with the radially-dominant( | v r | > | v φ | ) component in 3-D. While the outflowingcold gas has a similar distribution in 2-D and 3-D, theinflowing gas is more dominant in 2-D relative to 3-D be-cause in 3-D a lot of this in-falling cold gas slows downand becomes a part of the rotating cold torus.Table 1 shows that the mass accretion rate throughthe inner boundary for the fiducial runs in 2-D and 3-D are comparable. Unlike in 3-D, we note that there is2 D. Prasad, P. Sharma, A. Babul −6 −5 −4 −3 −2 −3 −2 −1 ǫ ˙ M a cc / ˙ M c f
2D − massive cluster2D − cluster3D − cluster
Fig. 8.—
The mass accretion rate relative to the cooling flowvalue as a function of jet efficiency (Eq. 6) for cluster and massivecluster runs (both 2-D and 3-D). The accretion rate is suppressedmore for a lower halo mass at a fixed ǫ . substantial cold gas sticking to the poles in 2-D due tonumerical reasons. Similarly, in 3-D there is a physicalaccumulation of cold gas in form of a rotating torus.Figure 8 shows the average mass accretion rate throughthe inner radius of our simulation volume ( ˙ M acc ) relativeto the cooling flow rate ( ˙ M cf ). The suppression factor( ˙ M acc / ˙ M cf ) for 3-D cluster simulations (with ǫ = 6 × − , × − , .
01) is comparable to 2-D.Figure 9 shows various important quantities, such asjet power, cold gas mass, mass accretion rate through theinner boundary, as a function of time for the fiducial 3-D(upper panel) and 2-D (bottom panel) runs. Encourag-ingly, various quantities, except the total cold gas mass,show similar trends with time in 2-D and 3-D. The to-tal cold gas mass is much larger in 3-D because of theformation of a massive cold torus which is absent in ax-isymmetry.In both 2-D and 3-D runs min( t cool /t ff ) varies in therange 1 to 10, and is roughly anti-correlated with ˙ M acc and jet power. The maximum jet power goes up to ∼ erg s − in both cases. The mass accretion rateand hence feedback power injection (Eq. 6) is more spikyin 2-D (can go above 100 M ⊙ yr − for some times) be-cause, unlike in 3-D, the cold gas that is accreted in 2-Dcovers full angle 2 π in φ because of axisymmetry. Thejet power, which is calculated by measuring the instan-taneous jet thermal energy, depends on the average massaccretion rate over .
100 Myr rather than the instanta-neous value. Another difference between 2-D and 3-D isthat cold gas can be totally removed (through the innerboundary) after strong feedback jet events in 2-D butthis never happens in 3-D; cold gas (even the radially-dominant component) is present at all times because itis very difficult to evaporate/accrete the massive rotat-ing cold torus. There definitely is a depletion in theamount of radially-dominant cold gas after a strong feed-back event in 3-D (at ∼ . PSfrag replacements ˙ M acc ( M ⊙ yr − )cold mass (10 M ⊙ )cold mass ( | v r | > | v φ | ; 10 M ⊙ ) min( t cool /t ff )jet power (10 erg s − ) time (Gyr) n o r m a li ze d q u a n t i t i e s , - D ˙ M acc ( M ⊙ yr − )cold mass (10 M ⊙ )min( t cool /t ff )jet power (10 erg s − )time (Gyr)normalized quantities, 2-D PSfrag replacements˙ M acc ( M ⊙ yr − )cold mass (10 M ⊙ )cold mass ( | v r | > | v φ | ; 10 M ⊙ )min( t cool /t ff )jet power (10 erg s − )time (Gyr)normalized quantities, 3-D ˙ M acc ( M ⊙ yr − )cold mass (10 M ⊙ ) min( t cool /t ff )jet power (10 erg s − ) time (Gyr) n o r m a li ze d q u a n t i t i e s , - D Fig. 9.—
Various quantities (jet power, cold gas mass, radially-dominant cold gas mass, min[ t cool /t ff ], ˙ M acc ) as a function of timein the fiducial 3-D (top panel) and 2-D (bottom panel) clustersimulations. The data are sampled every 10 Myr. The cycle shownin Figure 7 are based on the top panel. All quantities, except totalcold gas mass, are statistically similar in 3-D and 2-D. The totalcold gas mass, which is dominated by the cold torus, is much largerin 3-D and builds up in time. However, the radially-dominant coldgas ( | v r | > | v φ | ) mass in 3-D is similar to the total cold gas massin 2-D. cycles in jet energy and cold gas mass as a function ofmin( t cool /t ff ). These cycles just reflect the sudden risein the accretion rate ( ˙ M acc ) and jet power due to coldgas condensation and slow relaxation to equilibrium af-ter overheating (notice the fast rise and slow decline injet energy for individual jet events in both panels of Fig.9). Dependence on jet efficiency & halo mass
Till now we have discussed the fiducial cluster simula-tion with a small feedback efficiency ǫ = 6 × − . Inthis section we study the influence of jet efficiency ( ǫ )and halo mass ( M ) on various properties of the clus-ter core. Overall, we find that the effect of an increasinghalo mass is similar to that of a decreasing feedback ef-ficiency. We compare the efficiencies ( ǫ ) ranging from6 × − to 0 .
01. We consider two halo masses: a clusterwith M = 7 × M ⊙ and a massive cluster with M = 1 . × M ⊙ .Table 1 and Figure 8 show that the feedback efficiencyof ǫ = 6 × − is able to suppress the cooling flow byold gas and AGN jet feedback in cluster cores 13 PSfrag replacements cluster, ǫ = 6 × − cluster, ǫ = 6 × − cluster, ǫ = 5 × − massive cluster, ǫ = 6 × − massive cluster, ǫ = 5 × − ˙ M a cc ( M ⊙ y r − ) time(Gyr) Fig. 10.—
The mass accretion rate through the inner radius(smoothed over 50 Myr) as a function of time for different 2-Druns. A lower efficiency and a more massive halo lead to a largeraccretion rate. about a factor of 10 for a cluster but only by a factorof 4 for a massive cluster. This implies that a larger ef-ficiency is required to suppress a cooling flow in a moremassive halo. We note that the pure cooling flow accre-tion rate decreases with a decreasing halo mass becauseof a smaller amount of gas in lower mass halos (see thevalues enclosed in brackets in Table 1).Figure 8 shows that the suppression factor ( ˙ M acc / ˙ M cf )is smaller for the massive cluster, and scales as ǫ − / forboth cluster and massive cluster runs (see also Table 1).A decrease in the accretion rate with an increasing ǫ isnot a surprise; a higher feedback efficiency heats the coremore and maintains t cool /t ff &
10 at most times, resultingin only a few cooling/feedback events. While the averagejet power ( ∼ ǫ ˙ M acc c ∝ ǫ / ) increases with an increasing ǫ , the core X-ray luminosity decreases. This implies thatfeedback heating and cooling do not balance each otherat all times. Heating dominates cooling just after jetoutbursts and cooling dominates in absence of infallingcold gas when t cool /t ff slowly decreases from a value &
10. Thus, for a larger ǫ , for which a cluster spendsmore time in a hot/dilute state, the X-ray emission fromthe core is expected to be smaller (c.f., Fig. 12).Figure 10 shows the mass accretion rate (averaged over50 Myr bins) as a function of time for our 2-D cluster andmassive cluster runs with different feedback efficiencies.The solid red line corresponds to the fiducial 2-D clusterrun (with ǫ = 6 × − ). The green dotted line with amarker, which corresponds to a ten times lower efficiency( ǫ = 6 × − ), shows an accretion rate comparable to acooling flow at most times (see also Table 1). The clus-ter run with ten times higher efficiency ( ǫ = 5 × − ),indicated by black dot-dashed line, shows an average ac-cretion rate of 5.1 M ⊙ yr − (about a fifth of the fiducial2-D run; see Table 1); there are far fewer spikes in ˙ M acc compared to the fiducial run. Similar trends are observedfor the massive cluster runs with ǫ = 6 × − (magentadotted line) and 5 × − (blue double-dot-dashed line).The number of ˙ M spikes in Figure 10 are smaller forlower halo mass and higher feedback efficiency becauseof larger overheating and a longer recovery time after a precipitation-induced jet event.Figure 11 shows the time averaged (from 4 to 5Gyr) and emissivity weighted (0.5-8 keV) 1-D profilesof several key quantities for 2-D cluster and massivecluster runs with different efficiencies: entropy ( K ≡ T keV /n / e ), t cool /t ff , number density and temperature.All profiles look similar to what is seen in observations.The entropy profile flattens toward the center but the en-tropy core is prominent only for the higher efficiency ( ǫ )runs; a ‘core’ with a constant t cool /t ff is more prominentfor lower ǫ and for the massive cluster. As expected, thedensity is lower and the temperature is higher for a largerfeedback heating efficiency. For all efficiencies tempera-ture increases with the radius (except for ǫ = 5 × − which is almost isothermal just after a jet outburst; seethe top-right panel of Fig. 12), as seen in the observa-tions of cool-core clusters.Compared to the cluster runs, the entropy for the mas-sive cluster is higher at larger radii in Figure 11 be-cause the initial entropy was scaled with the halo mass( K ∝ M / ; see section 2.2). The entropy profiles for themassive cluster runs for the two efficiencies are similar;entropy keeps on decreasing as we go toward the center(forming a ‘core’ in t cool /t ff ), more so for ǫ = 6 × − .As we saw with the mass accretion rate in Figure 10,the effect of increasing the efficiency is similar to that ofdecreasing the halo mass. This is expected, as the massaccretion rate for lower mass halos is smaller, and theincrease in jet efficiency and the consequent higher jetpower suppresses accretion. Another point to note in Figure 11 is that the pro-files are rather similar for the massive cluster runs with ǫ = 6 × − and ǫ = 5 × − . The bottom panels ofFigure 12 show that jet events between 4 to 5 Gyr are notable to raise min( t cool /t ff ) much above 10 for these cases.However, top panels of Figure 12 and the bottom panelof Figure 9 show that between 4 to 5 Gyr min( t cool /t ff )increases with an increasing ǫ . Therefore, the core en-tropy (density) for the cluster runs increases (decreases)with an increasing ǫ . Note that the core entropy for thecluster runs with a larger efficiency are not always higher;its only when the core is in the part of the heating cyclewith min( t cool /t ff ) > t cool /t ff ]) as a function of time for 2-D clus-ter ( ǫ = 10 − and 5 × − ; also see the 2-D cluster runwith ǫ = 6 × − in the bottom panel of Fig. 9) andmassive cluster ( ǫ = 6 × − and 5 × − ) runs. Thefirst point to note is that the number of jet events (andhence the number of cycles; e.g., see Fig. 7) is smaller fora higher efficiency and a lower halo mass. Another is thatthe peaks in jet power and min( t cool /t ff ) for a higher effi-ciency are larger, resulting in overheating and longer du-rations for which cold gas and jet power are suppressed.Stronger overheating after jets in higher efficiency (andlower halo mass) runs results because, while the numberof cold accretion events are smaller (compared to lowerefficiency or a larger halo mass), the mass accretion rateduring the multiphase cooling phase is similar (see Fig.10), generally giving larger heating (Eq. 6).As with the mass accretion rate (see the spikes in Fig. We thank the referee for the suggestion to highlight this point.
PSfrag replacementscluster, ǫ = 6 × − cluster, ǫ = 10 − cluster, ǫ = 5 × − massive cluster, ǫ = 6 × − massive cluster, ǫ = 5 × − r/r max T (keV) t cool /t ff n e (cm − ) K ( T k e V / n / e ) PSfrag replacements cluster, ǫ = 6 × − cluster, ǫ = 10 − cluster, ǫ = 5 × − massive cluster, ǫ = 6 × − massive cluster, ǫ = 5 × − r/r max T (keV) t c oo l / t ff n e (cm − ) K ( T keV /n / e ) PSfrag replacementscluster, ǫ = 6 × − cluster, ǫ = 10 − cluster, ǫ = 5 × − massive cluster, ǫ = 6 × − massive cluster, ǫ = 5 × − r/r max T (keV) t cool /t ff n e ( c m − ) K ( T keV /n / e ) PSfrag replacementscluster, ǫ = 6 × − cluster, ǫ = 10 − cluster, ǫ = 5 × − massive cluster, ǫ = 6 × − massive cluster, ǫ = 5 × − r/r max T ( k e V ) t cool /t ff n e (cm − ) K ( T keV /n / e ) Fig. 11.—
Emissivity-weighted (considering plasma in the range 0 . − r max ): entropy ( K ≡ T keV /n / e ; top-left panel); t cool /t ff (top-rightpanel); electron density ( n e ; bottom-left panel); and temperature (in KeV; bottom-right panel). Both min( t cool /t ff ) and core entropydecrease for a lower efficiency or a larger halo mass. Temperature is higher for a higher efficiency, but a cool core (a temperature increasingwith radius) is preserved for all cases, except the highest efficiency run. Spikes in the cluster run with ǫ = 5 × − signify that there iscool, low entropy gas present in the core from 4 to 5 Gyr. ǫ the number of cooling/jet events arelarger for a massive cluster. While t cool /t ff is .
10 andcold gas is present at most times for the massive clus-ter run with ǫ = 6 × − , there are longer periods withmin[ t cool /t ff ] &
10 and lack of cold gas for the cluster run(see bottom panel of Fig. 9). The jet events are moredisruptive (as measured by the rise in min[ t cool /t ff ] af-ter a jet event) in the lower mass halo because the jetpower is relatively large but the hot gas mass is smaller(compare the right panels of Fig. 12). DISCUSSION & ASTROPHYSICAL IMPLICATIONS
The cold mode accretion model, in which local ther-mal instability leads to the condensation and precipita-tion of cold gas and enhanced accretion on to the SMBH,has emerged as a useful framework to interpret variousproperties in cores of elliptical galaxies, groups, and clus-ters (e.g., Pizzolato & Soker 2005; Sharma et al. 2012a;Gaspari et al. 2012; Li & Bryan 2014b). However, thereare several unresolved problems: e.g., the role of angular Here we use the label “cold mode accretion” to refer to thecapture and accretion of cold clouds by SMBH, and not the cos-mological accretion of cold gas sometimes invoked in halos lessmassive than 10 M ⊙ (Birnboim & Dekel 2003). momentum transport, self-gravity and cloud-cloud col-lisions in accretion on to the SMBH (e.g., Pizzolato &Soker 2010; Hobbs et al. 2011; Babul et al. 2013;Gaspari et al. 2014); relative contribution of cold gasat ∼ t cool /t ff becomes smaller than a critical value closeto 10. This leads to a strong feedback heating, whichtemporarily overheats the cluster core. The hot modefeedback, in form of Bondi accretion onto the SMBH, onthe other hand, is not an abrupt switch and increasessmoothly with an increasing (decreasing) core density(temperature). In section 4.1 we discuss the successof the cold accretion model and compare with previoussimulations. In section 4.2 we compare with the recentold gas and AGN jet feedback in cluster cores 15 −1 cluster, ǫ = 10 − n o r m a li ze d q u a n t i t i e s cluster, ǫ = 5 × − −1 massive cluster, ǫ = 6 × − time (Gyr) n o r m a li ze d q u a n t i t i e s massive cluster, ǫ = 5 × − time (Gyr)jet power (10 erg s − )cold gas mass (10 M ⊙ )min(t cool / t ff ) Fig. 12.—
Jet energy, cold gas mass, and min( t cool /t ff ) as a function of time for 2-D runs with different efficiencies ( ǫ = 10 − , × − , × − ) and halo masses ( M = 7 × , . × M ⊙ ). Note that the jet energy and cold gas mass are scaled differently indifferent panels. A smaller efficiency or a larger halo mass leads to many accretion and jet feedback cycles. exquisite cold-gas observations and with statistical anal-yses of X-ray and radio observations of cluster cores. Comparison with previous simulations
There are two broad categories of jet implementationsdescribed in the literature: first, where the jet mass, mo-mentum and energy are injected via source terms (e.g.,Omma et al. 2004; Cattaneo & Teyssier 2007; Li &Bryan 2014a; Gaspari et al. 2012); second, where massand energy are injected as flux through an inner bound-ary (e.g., Vernaleo & Reynolds 2006; Sternberg, Pizzo-lato, & Soker 2007). We use the former approach, whichhas generally been more successful. In this approach, thesudden injection of kinetic energy after cold gas precipi-tation leads to a shock which not only expands verticallybut also laterally, perpendicular to the direction of mo-mentum injection. This lateral spread of jet energy andvorticity generation due to interaction with cold clumpshelp in coupling the jet energy with the equatorial ICM.In the flux-driven approach the jet pressure is usuallytaken to be the same as ICM pressure and the jet drillsa cavity without expanding laterally in the core. Thus,coupling of the jet power is not very effective, unless thejet angle is very broad (Sternberg, Pizzolato, & Soker2007).Our jet modeling is similar to the earliest works suchas Omma et al. (2004); Omma & Binney (2004), whichinject jet mass, momentum and kinetic energy via sourceterms. However, this work focussed on the effect of a sin-gle jet outburst with a fixed power and did not includecooling; the simulations were run for short times ( . ∼
100 in order to match feedback heating withcooling. Bondi accretion is only applicable for a smooth,non-rotating gas distribution, and not for clumpy multi-phase gas which can accrete at a much higher rate (e.g.,Gaspari et al. 2013; Sharma et al. 2012a).Cielo et al. (2014) have studied the detailed structureand thermodynamics of source-term driven cylindricaljets, of different densities and temperatures, interactingwith the ICM but they run for less than 10 Myr. Likeus, they also highlight the importance of hot back flowsin regulating the central ICM.Another set of simulations inflate cavities using jetsdriven by fluxes of mass and momentum at the inner ra-dial boundary (rather than using source terms like us;in cluster context, see Vernaleo & Reynolds 2006; Stern-berg, Pizzolato, & Soker 2007; for MHD modeling ofthe Crab nebula jet, see Mignone et al. 2013). Vernaleo& Reynolds (2006) injected momentum (and kinetic en-ergy) via the inner radial boundary, with an opening an-gle of 15 , in form of 100 times hotter gas but in pressureequilibrium with the ICM. Their jets just drill through anarrow channel without coupling to the catastrophicallycooling core.6 D. Prasad, P. Sharma, A. BabulSternberg, Pizzolato, & Soker (2007) advocated wide(with opening angle & ) boundary-driven jets, suchthat the jet is not as fast, and can lead to vortices andsubstantial mixing in cluster cores. However, since theirsimulations are not run for many cooling times, its un-clear if wide jets and can indeed balance cooling for cos-mological times. Moreover, the fat jets may not repro-duce the observed morphologies of thin jets and fat bub-bles. Using the boundary injection approach, Heinz etal. (2006) emphasize the importance of the dynamicICM in redistributing jet energy but they also run forless than a cooling time.Recent numerical simulations of AGN-driven jets (Gas-pari et al. 2012; Li & Bryan 2014a,b) have been quitesuccessful in producing several observed features such as,the lack of plasma cooling below a third of the ICM tem-perature (Fig. 11 in Li & Bryan 2014b), suppressionof cooling and accretion in the core (by a factor of 10-100 relative to a cooling flow), maintenance of cool-corestructure even with strong intermittent jet events, forma-tion of an angular momentum supported cold-gas torus,viability of AGN feedback from elliptical galaxies to mas-sive clusters. Our simulations are different from theserecent works, which use mesh refinement in a cartesiangeometry, in that we use a spherical coordinate system.We have also tried to push the AGN feedback efficiencytoward the lower limit which is still able to suppress acooling flow. We find that an efficiency ǫ = 6 × − isable to suppress a cluster cooling flow by a factor of 10.Like us, Gaspari et al. (2012) and Li & Bryan (2014b)also make an estimate of the mass accretion rate on tothe SMBH. Gaspari et al. (2012) consider a sphericalshell of radius 0.5 kpc and calculate the mass accretionrate ( ˙ M acc ) due to infalling gas. Li & Bryan (2014b); Liet al. (2015) calculate the mass accretion rate ( ˙ M acc ) bydividing the cold gas mass within 0.5 kpc by 5 Myr (oforder the dynamical timescale). Our estimate of ˙ M acc issimilar to Gaspari et al. (2012), except that we calculateit at 1 kpc. Only a small fraction of ˙ M acc is expected tobe accreted onto the SMBH; thus, the efficiency factor ( ǫ )in Eq. 6 takes into account both the fraction of ˙ M acc thatis accreted by the SMBH and the efficiency of convertingSMBH accretion into jet mechanical energy.While our jet feedback implementation is very similarto Gaspari et al. (2012), our results differ in some key re-spects. The main difference is that we see extended coldgas and jet/cold-gas cycles even at late times (see Figs.1, 3, 7, 9). Like Li & Bryan (2014b), in Gaspari et al.(2012) there is a long-lived rotationally supported torusat few kpc and the extended multiphase gas is lacking atlater times (see their Figs. 10 & 11). The main reasonfor the absence of extended cold gas and strong jets atlate times in previous simulations is a large feedback ef-ficiency. A larger feedback efficiency leads to very strongfeedback heating at early times, and the core reachesrough thermal balance in a state of t cool /t ff >
10 with nofresh extended (radially dominant) cold gas condensingat late times. Since a large fraction of cool core clus-ters show extended cold gas (McDonald et al. 2011a), asmaller value of feedback efficiency seems more consistentwith observations.To solve the problem of a steady cold torus present at late times, very recently, Li et al. (2015) have incor-porated the depletion of cold gas via star formation inthe core, but they adopt the same feedback prescriptionas in their previous works. Star formation exhausts theamount of cold gas within 0.5 kpc, suppresses AGN heat-ing, and leads to a cooling event after which cold gas con-denses again. Thus, they obtain three cooling-feedbackcycles in their fiducial run. In their AGN feedback pre-scription, star formation efficiency primarily determinesthe frequency of cooling/heating cycles. In contrast, ourcycles are determined by the AGN feedback efficiencyand the halo mass (more cycles for a massive halo and asmaller feedback efficiency). While there is some ongo-ing star formation in cool cluster cores, Li et al. (2015)form 3 × M ⊙ in stars over 6.5 Gyr for their fidu-cial run corresponding to the Perseus cluster, a signifi-cant fraction of the mass of the BCG (brightest clustergalaxy; 8 × M ⊙ ; Lim et al. 2008). This does notagree with semi-analytic models which suggest that 80%of the stars of BCGs are assembled before z = 3 (i.e.,only 2 × M ⊙ are expected to form over the past 12Gyr; De Lucia & Blaizot 2007). Moreover, the currentstar formation rate even in the most extreme cool coreclusters is typically .
10 M ⊙ yr − (Hicks, Mushotzky,& Donahue 2010; McDonald et al. 2011b); the averagestar formation rate of Li et al. (2015) would correspondto an unacceptably high value, ≈
46 M ⊙ yr − .We can directly compare our results with Gaspari etal. (2012) as our feedback prescription is similar. Theytried jet efficiency factors of ǫ = 6 × − , .
01. Witha much higher accretion efficiency ( ǫ = 0 .
01) comparedto ours ( ∼ − − − ), Gaspari et al. (2012) get alarger suppression factor ( ˙ M acc / ˙ M cf ∼ − M ∼ M ⊙ . Figure 8 shows how our resultscompare with Gaspari et al. (2012). The suppressionfactor of a massive cluster ( M = 1 . × M ⊙ ) forour fiducial ǫ = 6 × − is 20%, larger than their work.Suppression factor in our massive cluster (cluster) runfor ǫ = 0 .
01, as seen in Figure 8, is 3% (0.8%), in roughagreement with the results of Gaspari et al. (2012).
Comparison with observations
Now that we have done some comparisons with previ-ous simulations, in this section we compare our resultswith observations. We note that the observational com-parison may not be perfect because our simulations lacksome physical processes such as magnetic fields and ther-mal conduction. These effects will be considered later.Moreover, observations suffer from projections effects,and our cluster parameters do not span as broad a range(of halo masses, entropy profiles at large radii, etc.) asencountered in observations.One of the most commonly studied ICM property isits entropy profile (e.g., Cavagnolo et al. 2009). Figure11 shows the time averaged (4-5 Gyr), X-ray emissivityweighted profiles of t cool /t ff and entropy as a functionof radius for our various runs. We see that an entropycore (with a prominent local minimum in t cool /t ff ) is agood approximation for systems in which t cool /t ff & t cool /t ff <
10, in which there isold gas and AGN jet feedback in cluster cores 17 K (keV cm ) M c o l d ( M ⊙ ) M cold (M ⊙ ) j e t p o w e r ( e r g s − ) K ( k e V c m ) min(t cool / t ff ) ǫ = 6 × − ǫ = 10 − ǫ = 5 × − Fig. 13.—
Various important quantities measured at the same time (jet energy, cold gas mass, core entropy, and min[ t cool /t ff ]) plottedagainst each other from our 2-D cluster runs with different efficiencies. The data is sampled every 10 Myr. There is a strong correlationbetween the core entropy and min( t cool /t ff ), especially at larger values of min( t cool /t ff ). There is also a positive correlation between K -jetenergy and min( t cool /t ff )-jet power. Larger efficiency runs lead to a larger value of min( t cool /t ff ) and K . Notice that cold gas is absent ifmin( t cool /t ff ) & M c o l d ( M ⊙ ) K ( k e V c m ) K (keV cm ) M cold (M ⊙ ) j e t p o w e r ( e r g s − ) min(t cool / t ff ) time(Gyr) Fig. 14.—
Important quantities measured at the same time (jet energy, cold gas mass, core entropy, and min[ t cool /t ff ]) plotted againsteach other in our fiducial 3-D cluster run. As in 2-D runs (see Fig. 13), there is a strong correlation between K -min( t cool /t ff ), K − jetpower, and min( t cool /t ff )-jet energy. The cold gas mass is high and becomes almost constant at later times as seen in the top panel of Fig.9. The color-coding corresponds to time. t cool /t ff and a decreasing en-tropy toward the cluster center, albeit with a shallowerslope. This is consistent with recent reanalysis of coreentropy profiles (Panagoulia, Fabian, & Sanders 2014),which suggests that a double power-law entropy profile,with a shallower entropy in the center, better describesthe ICM core. It will be useful to compare the behav-ior of central entropy as a function of min( t cool /t ff ); weexpect entropy cores for min( t cool /t ff ) >
10 and slowlyincreasing entropy profiles for min( t cool /t ff ) . ǫ = 6 × − , − , × − ) and the 3-D fiducialrun, respectively. Data points sampled every 10 Myrare shown. The core entropy ( K ) is obtained by using aleast squares fit to the emissivity-weighted 1-D entropyprofile of gas in 0.5-8 keV range. In both these figures thestrongest correlation is between K and min( t cool /t ff ), asexpected, because both these quantities depend on den-sity and temperature in a similar way ( K ∝ T /n / and t cool /t ff ∝ T / /n ; see Eq. 35 in McCourt et al. 2012);the relation is not one-to-one because K is determinedby entropy near the center and min( t cool /t ff ) by the be-havior at the core radius (beyond which density decreasessharply).The spread in K − min( t cool /t ff ) correlation is largerfor a lower K (or equivalently, min[ t cool /t ff ]; this is alsoseen in observational data shown in Fig. 4 in Voit &Donahue 2014) because a core with constant entropyis not a good description in that case and the entropydecreases inward (see top-left panel of Fig. 11).The correlation between various quantities in Figures13 and 14 are not particularly strong because of the hys-teresis behavior of various quantities (e.g., jet power,radially dominant cold gas mass) with respect to thecore properties (Fig. 7). Figures 13 & 14 show that,in general, the jet power increases for a larger K (ormin[ t cool /t ff ]), particularly for a larger core entropy. Thisis because a large jet power overheats the cluster core andraises its entropy. Other quantities do not show as strongcorrelations in these plots; cold gas mass increases witha lower entropy or a shorter cooling time, but jet energyand cold gas mass show a large spread relative to eachother (since cold gas leads to increase in jet energy, whichin turn suppresses cold gas mass).The 3-D run shown in Figure 14 prominently showsthe sign of the massive torus at late times. Apart fromthis, there are no major differences in 2-D and 3-D. Alsonote that cold gas is missing in ǫ = 5 × − t cool /t ff &
20 (Fig. 13; the same is expected forthe radially dominant cold gas in 3-D). This is consistentwith the observations of Cavagnolo et al. (2008), whofind that H α luminosity is suppressed for a core entropy >
30 keV cm (corresponding to min[ t cool /t ff ] of about20; see the top-right panel of Fig. 13 & Fig. 4 in Voit& Donahue 2014). The onset of star formation in clus-ter cores also happens sharply below the same entropythreshold (Rafferty, McNamara, & Nulsen 2008). Figure14 shows that the core entropy and min( t cool /t ff ) remainbelow 20 even if the instantaneous jet power is as highas 10 erg s − ; core entropy can be much higher (up to100 keV cm ) for higher efficiency (see ǫ = 5 × − in Fig. 13).While Figures 13 & 14 show correlations between var-ious quantities at a given time, we are also interestedin understanding causal relationships between variousquantities such as cold gas mass, min( t cool /t ff ), jet en-ergy. Figure 15 shows the temporal cross-covariance be-tween various important parameters (we take log beforecalculating cross-covariance). Various 2-D and 3-D sim-ulations with different efficiencies are plotted together.The trends which are common to all simulations are likelyto be robust. Robust correlations among various quan-tities occur only with a time lag < t cool /t ff ) decreases ∼ . M acc ). The interpretation is thatthe cooling time (and hence min[ t cool /t ff ]) decreases asthe core cools during the cooling leg of the cycle. Thisleads to the condensation of radially dominant cold gasafter a cooling time (few 100 Myr) and an enhancementof the mass accretion rate and the jet power. Suddenincrease in jet power overheats the core and min( t cool /t ff )increases after a lag of few 100 Myr; cold gas mass and˙ M acc also decrease consequently.The bottom three panels in Figure 15 show that coldgas mass (radially dominant), mass accretion rate, andjet power are positively correlated. A slight skew towarda negative time lag shows that the cold gas mass and themass accretion rate increases first, and that gives riseto an increase in jet power. Thus, the cross-covariancebehavior of different variables is similar to that seen incold gas-jet cycles in Figure 7. Also note in Figure 15that there are smaller number of oscillations for higherfeedback efficiency (or smaller halo mass); this is a re-flection of smaller number of cooling/feedback cycles inthese cases (see Fig. 12).In our 3-D simulations we see the build-up of a massiverotationally-supported torus. A part of this torus shouldcool further and lead to star formation, as argued byLi et al. (2015). While the cold gas (mostly in thetorus) builds up in time and saturates after 2 Gyr, thejet energy shows fluctuations in time even after that (seethe top panel of Fig. 9 and Fig. 14). Therefore, in ourmodels there is no correlation between total cold gas massand jet energy. However, there is a correlation betweenthe radially-dominant cold gas mass and the jet energy(compare green-dotted and black dot-dashed lines in thetop panel of Fig. 9). Thus, although most cold gas isdecoupled from jet feedback, it is the subdominant in-falling cold gas which is powering AGN. This is in linewith the observations of McNamara, Rohanizadegan, &Nulsen (2011), who find no correlation between the jetpower and the available molecular gas. They, therefore,argue that most of the cold gas is converted into starsrather than being accreted by the SMBH.Finally, we compare our simulations with recent obser-vational studies of cold gas kinematics and star forma-tion. These have been studied in unprecedented detailin some elliptical galaxies and clusters, thanks mainly to ALMA and
Herschel telescopes (e.g., McNamara et al.2014; Russell et al. 2014; David et al. 2014; Edge etal. 2010; Werner et al. 2014; Tremblay et al. 2012;old gas and AGN jet feedback in cluster cores 19 −0.5−0.200.20.5 min(t cool / t ff ) - jet power −0.8−0.400.20.5 min(t cool / t ff ) - mass(cold phase) −0.6−0.300.20.4 min(t cool / t ff ) - ˙M acc −0.4−0.200.30.6 mass (cold phase) - jet power −0.4−0.200.30.6 ˙M acc - mass (cold phase) −5 −4 −3 −2 −1 0 1 2 3 4 5−0.4−0.200.20.4 ˙M acc - jet power time (Gyr) cluster; 2D ǫ = 6 × − cluster; 2D ǫ = 10 − cluster; 2D ǫ = 5 × − massive cluster; 2D ǫ = 6 × − massive cluster; 2D ǫ = 5 × − cluster; 3D ǫ = 6 × − Fig. 15.—
Cross-covariance of various quantities (min[ t cool /t ff ], jet energy, mass in cold phase, ˙ M acc ) as a function of time lag to showtemporal relationship between these various quantities. Cross covariance between two quantities as a function of time, as used here, isdefined as: cov( a, b ; τ ) = R T −| τ | [ δa ( t + τ ) δb ( t ) dt ] / (cid:20)qR T | δa ( t ) | dt R T | δb ( t ) | dt (cid:21) , where − T ≤ τ ≤ T is the time lag and δa and δb are mean-subtracted quantities. Since there is a large variation in various quantities (see Figs. 13 & 14), we take log before evaluatingcross-covariance. For the 3-D cluster run we have used the radially-dominant cold gas mass; the cross-covariance is much weaker if we usetotal cold gas mass. Rawle et al. 2012). In this paper we have mainly fo-cussed time-averaged kinematics, as shown in Figures 4and 5. We can clearly see three kinematically distinctcomponents of cold gas: a rotationally-supported mas-sive torus, ballistically infalling cold gas, and jet-upliftedfast cold gas.Observations of different clusters are snapshots at aparticular instant, at which a particular component (e.g.,the rotating torus, a fast outflow, or a radially distributedinflow; see Fig. 6) of the cold gas distribution may bemore prominent. We will present the details of cold gaskinematics in various states of the ICM (with cold in-flows, outflows, and the rotating torus) in a future work.Some of the salient properties of the cold gas distri-bution in the fiducial run are: the rotating cold gastorus, when present, is more massive compared to in-falling cold gas (this component may be exaggerated inour simulations as we do not include star formation thatwould quickly consume some of the cold gas); the ro-tating disk rotates at the almost constant local circularvelocity (100-200 km s − for our fiducial cluster run; theactual value may be larger because we have ignored thegravitational potential due to the BCG) in form of a mas-sive torus within 5 kpc; the radially-dominant cold gasis much more spatially extended (out to few 10s of kpc) compared to the rotating torus, and the majority of thiscomponent also has a velocity close to the circular veloc-ity; some (about 10 M ⊙ ) radially dominant outflowinggas has a radial velocity as high as 1000 km s − (a fastcomponent is seen in the observations of Russell et al.2014 and McNamara et al. 2014); the in-falling cold gas(on average) is about twice as much as the outflowingcomponent.While the accretion rate through the inner radius(dominated by cold gas) is smaller than 100 M ⊙ yr − (the accretion rate on to the SMBH is .
1% of this) atall times (see Fig. 9), the cooling/accretion and outflowrates in the cold gas can be much larger instantaneouslybecause of the massive cold torus buffer (Fig. 6).The observations show varying cold gas kinematicsin different systems: radially in-falling cold molecularclouds of 3 × to 10 M ⊙ in a galaxy group NGC 5044(David et al. 2014); 5 × M ⊙ molecular gas pre-dominantly in a rotating disk, and about 10 M ⊙ in afast (line of sight velocity up to 500 km s − ) outflow inAbell 1835 (McNamara et al. 2014); 10 M ⊙ of molec-ular gas roughly equally divided between as rotating disk(velocity ∼
250 km s − ) and a faster (570 km s − ) in-falling/outflowing component in Abell 1664 (Russell et0 D. Prasad, P. Sharma, A. Babulal. 2014). In our simulations we observe similar compo-nents of the cold gas distribution, as shown in Figures 4and 5. CONCLUSIONS
Cold-mode feedback, due to condensation of cold gasfrom the hot ICM when the local density is higher thana critical value (see below), has emerged as an attractiveparadigm to interpret observations in cluster cool cores.In this paper we have carried out simulations of clustersof halo masses 7 × M ⊙ and 1 . × M ⊙ withfeedback driven AGN jets, varying the feedback efficiencyover a large range (10 − − . ∼
10) even for a feedback efficiency as low as6 × − . This is the major difference from previous jetsimulations, which use a much larger feedback efficiency( & − ; Gaspari et al. 2012; Li & Bryan 2014b; Li etal. 2015). Because of the high feedback efficiency, theprevious simulations attain thermal equilibrium in a hot,low-density core (with t cool /t ff >
10) which does not showcold gas and jet cycles at late times. In contrast, our lowefficiency simulations show cooling/jet cycles even at latetimes.The core undergoes cooling and feedback heating cy-cles because of cold gas precipitation and enhanced ac-cretion on to the SMBH. There are more cycles for alower efficiency and a larger halo mass. The cool-coreappearance is preserved even during strong jet events.Even with large efficiencies, jet feedback raises the coreentropy to several tens of keV cm , and therefore cannotexplain the non-cool-core clusters with large cores andentropies greater than 100 keV cm . The origin of thesenon-cool-core clusters is still poorly understood (see, e.g.,Poole et al. 2008).In this paper we highlight some results that were notemphasized in previous simulations of AGN jet feedbackin clusters; in particular, we compare our results withseveral recent observations. Following are our major con-clusions: • First and most importantly, the results from differ-ent codes, different setups, and different implemen-tation of jet feedback (as long as condensation andaccretion of cold gas is accounted for; e.g., Gaspariet al. 2012; Li et al. 2015) give qualitativelysimilar results. This indicates the robustness ofthe cold feedback mechanism, and the importanceof precipitation (which occurs when t cool /t ff . • We find that a feedback efficiency (defined as theratio of jet mechanical luminosity and the rest massaccretion rate [ ˙ M acc c ] at ∼ × − is sufficient to suppress thecooling/star-formation rate in cluster cores by afactor of about 10 (see Fig. 8). An even smallerefficiency is sufficient for lower mass halos becausethe thermal energy of the ICM is smaller comparedto the rest mass energy. Our fiducial efficiency is atleast 20 times lower than the models of Li & Bryan(2014b) and Gaspari et al. (2012). Our values areconsistent with the expectation that the mass ac-cretion rate on to the SMBH is much smaller than the accretion rate estimated at ∼ ǫ req ˙ M acc c ∼ L X ( ǫ req is the required feedback efficiency, ˙ M acc is the accretion rate estimated at 1 kpc, and L X isthe X-ray luminosity of the cooling core). If we as-sume that the mass accretion rate is a fixed factor f ( ≪
1) of the cooling flow value then, ǫ req f M c c t cool ≈ . k B T M c µm p t cool ( M c is the core mass) implies that ǫ req ≈ c s f c ≈ × − (cid:18) f . (cid:19) − (cid:18) M × M ⊙ (cid:19) / , normalizing to the parameters of our fiducial clus-ter ( T = 2 keV; see the bottom right panel of Fig.11), where c s = k B T /µm p is the sound speed ofthe core ICM. This estimate agrees with our fidu-cial efficiency ǫ = 6 × − . • We observe cycles in jet energy, radially-dominantcold gas mass, and mass accretion rate whichare governed by t cool /t ff measured in the hotphase. If t cool /t ff .
10 cold gas precipitates, andleads to multiphase cooling and enhanced accre-tion on to the SMBH. Sudden rise in the accre-tion rate, for a sufficiently high feedback efficiency,leads to overheating of the core and an increase in t cool /t ff above the threshold for cold gas conden-sation. We emphasize that thermal equilibrium incluster cores only holds in a time-averaged sense.There are cooling/heating cycles during which thecore slowly cools/heats up. The core spends alonger time in the hot state for a larger feedback ef-ficiency and a lower halo mass, leading to a smallernumber of cooling/heating cycles (see Fig. 12).Several observations hint at cycles in jet powerand cooling of the hot gas (see Fig. 7). We donot expect such cycles if feedback occurs via thesmooth hot/Bondi mode as we do not have sud-den cooling/feedback events. The hysteresis be-havior observed in the core X-ray properties ( K ,min[ t cool /t ff ]) and the mass of cold gas, jet power,etc. leads to a large dispersion in the correlationbetween these quantities (see Figs. 13 & 14). Inparticular, the mass accretion rate (at 1 kpc) is in-dependent of the total cold gas mass, which is dom-inated by the rotating cold torus, most of which isconsumed by star formation (rather than accretionon to SMBH; e.g., McNamara, Rohanizadegan, &Nulsen 2011). • We can classify the cold gas in our 3-D simulationsinto two spatially and kinematically distinct com-ponents: a centrally concentrated (within 5 kpc),rotationally supported ( | v φ | ≫ | v r | ) torus (Fig. 3);and extended (both infalling and outgoing) coldold gas and AGN jet feedback in cluster cores 21gas going out to 30 kpc (Fig. 4). The massive,rotationally-supported disk is decoupled from thefeedback loop; the radially-dominant infalling coldgas is what closes the feedback cycle. The coldtorus rotates at the local circular speed (200-300km s − ). The infalling cold gas can be fast ( . − ), but the uplifted cold gas from the rotatingtorus can sometimes reach speeds larger than 1000km s − as it is accelerated by the fast jet (Fig. 5).The mass of the radially-dominant infalling coldgas is about a factor of two times the outflowingcold gas. The massive cold torus is expected to besubstantially depleted by star formation, which wedo not take into account in our simulations. • The minimum in the ratio of the cooling time andthe free fall time (min[ t cool /t ff ]) seems better thanthe core entropy ( K ) for characterizing the coolcores. First of all it is a dimensionless parame- ter which applies for all halo masses (Voit et al.2015a), and secondly it is not sensitive to strongcooling or heating in the very center (unlike K ).The entropy and t cool /t ff panels in Figure 11 showthat a constant t cool /t ff ‘core’, which correspondsto a double power law for the entropy profile (witha slow increase with radius in the core; as arguedin Panagoulia, Fabian, & Sanders 2014), is a bet-ter approximation to clusters in the very cool statewith min( t cool /t ff ) . REFERENCESBabul, A., Balogh, M. L., Lewis, G. F., & Poole, G. B. 2002,MNRAS, 330, 329Babul, A., Sharma, P., & Reynolds, C. S. 2013, ApJ, 768, 11Balogh, M. L., Babul, A., & Patton, D. R. 1999, MNRAS, 307,463Banerjee, N. & Sharma, P. 2014, MNRAS, 443, 687Benson, A. J. & Babul, A. 2009, MNRAS, 397, 1302Bildfell, C., Hoekstra, H., Babul, A., & Mahdavi, A. 2008,MNRAS, 389, 1637Binney, J. & Tabor, G. 1995, MNRAS, 276, 663Birnboim, Y. & Dekel, A. 2003, MNRAS, 345, 349Bˆırzan, L., Rafferty, D. A., McNamara, B. R., Wise, M. W., &Nulsen, P. E. J. 2004, ApJ, 607, 800B¨ohringer, H., Matsushita, K., Churazov, E., Ikebe, Y., & Chen,Y. 2002, A&A, 382, 804Cattaneo A. & Teyssier R., 2007, MNRAS, 376,1547Cavagnolo K. W., Donahue M., Voit M. & Sun M., 2008, ApJ,683,107Cavagnolo, K. W., Donahue, M., Voit, G. M., & Sun, M. 2009,ApJS, 182, 12Cielo, S., Antonuccio-Delogu, V., Macci´o, A. V., Romeo, A. D., &Silk, J. 2014, MNRAS, 439, 2903Ciotti, L., & Ostriker J. 2001, ApJ, 551, 131Crawford, C. S., Allen, S. W., Ebeling, H., Edge, A. C., &Fabian, A. C. 1999, MNRAS, 306, 857David, L. P. et al. 2014, ApJ, 792, 94Dekel, A. & Birnboim, Y. 2008, MNRAS, 383, 119De Lucia, G. & Blaizot, J. 2007, MNRAS, 375, 2Dennis, T. J. & Chandran, B. D. G. 2006, ApJ, 622, 205Donahue, M. et al. 2000, ApJ, 545, 670Dubois, Y., Devriendt, A., Slyz, A., & Teyssier, R. 2010,MNRAS, 409, 985Edge, A. C. 2001, MNRAS, 328, 762Edge, A. C., Oonk, J. B. R., Mittal, R. et al. 2010, A&A, 518, L46Fabian A. C., 1994, ARAA, 32, 277FFabian, A. C. et al. 2003, MNRAS, 344, L43Gaspari, M., Ruszkowski, M., & Sharma, P., 2012, ApJ, 746, 94Gaspari M., Ruszkowski M., & Oh S. P., 2013, MNRAS, 432, 3401Gaspari, M., Ruszkowski, M., Oh, S. P., Brighenti, F., & Temi, P.2014, arXiv:1407.7531Hayes, J. C. et al. 2006, ApJS, 165, 188Heinz, S., Br¨uggen, M., Young, A., & Levesque, E. 2006,MNRAS, 373, L65Hicks, A. K., Mushotzky, R., & Donahue, M. 2010, ApJ, 719, 1844Hobbs, A., Nayakshin, S., Power, C., & King, A. 2011, MNRAS,413, 2633Kaiser, N. 1986, MNRAS222, 323Kaiser, N. 1991, ApJ, 383, 104Lewis, G. F., Babul, A., Katz, N., Quinn, T., Hernquist, L., &Weinberg, D. H. 2000, ApJ, 536, 623Li, Y. & Bryan, G. L. 2014a, ApJ, 789, 153 Li, Y. & Bryan, G. L. 2014b, ApJ, 789, 54Li, Y., Bryan, G., Ruszkowski, M., Voit, G. M., O’Shea, B. W., &Donahue, M. 2015, arXiv:1503.02660Lim, J., Ao, Y., & Dinh-V-Trung 2008, ApJ, 672, 252Loewenstein, M., Mushotzky, R. F., Angelini, L., Arnaud, K. A.,& Quataert, E. 2001, ApJ, 555, L21McCarthy, I. G., Babul, A., Bower, R. G., & Balogh, M. L. 2008,MNRAS, 386, 1309McCourt, M., Sharma, P., Quataert, E. & Parrish, I. J. 2012,MNRAS, 419. 3319McDonald, M., Veilleux, S., Rupke, D. S. N., & Mushotzky, R.2010, ApJ, 721, 1262McDonald M., Veilleux S., & Mushotzky R., 2011, ApJ, 731,33McConald, M., Veilleux S., Rupke, D. S. N., Mushotzky R., &Reynolds, C. S. 2011, ApJ, 734, 95McNamara B. R., & Nulsen P. E. J., 2007, ARA&A, 45, 117McNamara, B. R., Rohanizadegan, M., & Nulsen, P. E. J. 2011,ApJ, 727, 39McNamara, B. R. et al. 2014, ApJ, 785, 44Merloni, A. Heinz, S., & Di Matteo, T. 2003, MNRAS, 345, 1057Mignone, A., Striani, E., Tavani, M., & Ferrari, A. 2013,MNRAS, 436, 1102Mittal, R., Hudson, D. S., Reiprich, T. H., & Clarke, T. 2009,A&A, 501, 835Navarro J. F., Frenk C. S., & White S. D., 1996, ApJ, 462, 563Narayan, R. & Yi, I. 1995, ApJ, 444, 231O’Dea, C. P. et al. 2008, ApJ, 681, 1035Omma H., Binney J., Bryan G., & Slyz A., 2004, MNRAS, 348,1105Omma H., & Binney J., 2004, MNRAS, 350, L13O’Sullivan, E., Combes, F., Hamer, S., Salom´e, P., Babul, A., &Raychaudhury, S. 2015, A&A, 573, A111Panagoulia, E. K., Fabian, A. C., & Sanders, J. S. 2014, MNRAS,438, 2341Peterson, J. R., Kahn, S. M., Faerels, F. B. S. et al. 2003, ApJ,590, 207Pizzolato, F. & Soker, N. 2005, ApJ, 632, 821Pizzolato, F. & Soker, N. 2010, MNRAS, 408, 961Ponman, T. J., Cannon, D. B., & Navarro, J. F. 1999, Nature,397, 135Poole, G. B., Babul, A., McCarthy, I. G., Sanderson, A. J. R., &Fardal, M. A. 2008, MNRAS, 391, 1163Pope, E. C. D., Babul, A., Pavlovski, G., Bower, R. G., Dotter,A. 2010, MNRAS, 406, 2023Pratt, G. W., Croston, J. H., Arnaud, M., & B¨ohringer, H. 2009,A&A, 498, 361Rafferty, D. A., McNamara, B. R., & Nulsen, P. E. J. 2008, ApJ,687, 899Rawle, T. D., Edge, A. C., Egami, E. et al. 2012, 747, 29Revaz, Y., Combes, F., & Salom´e 2008, A&A, 477, L332 D. Prasad, P. Sharma, A. Babul