Investigating the properties of AGN feedback in hot atmospheres triggered by cooling-induced gravitational collapse
aa r X i v : . [ a s t r o - ph . C O ] A ug Mon. Not. R. Astron. Soc. , 1–8 (2011) Printed 14 August 2018 (MN L A TEX style file v2.2)
Investigating the properties of AGN feedback in hotatmospheres triggered by cooling-induced gravitationalcollapse
Edward C.D. Pope ⋆ , J. Trevor Mendel , Stanislav S. Shabala Department of Physics & Astronomy, University of Victoria, Victoria, BC, V8P 1A1, Canada School of Mathematics and Physics, University of Tasmania, Private Bag 37, Hobart, Tasmania 7001, Australia
14 August 2018
ABSTRACT
Radiative cooling may plausibly cause hot gas in the centre of a massive galaxy, orgalaxy cluster, to become gravitationally unstable. The subsequent collapse of thisgas on a dynamical timescale can provide an abundant source of fuel for AGN heatingand star formation. Thus, this mechanism provides a way to link the AGN accretionrate to the global properties of an ambient cooling flow, but without the implicit as-sumption that the accreted material must have flowed onto the black hole from 10s ofkiloparsecs away. It is shown that a fuelling mechanism of this sort naturally leads toa close balance between AGN heating and the radiative cooling rate of the hot, X-rayemitting halo. Furthermore, AGN powered by cooling-induced gravitational instabilitywould exhibit characteristic duty cycles ( δ ) which are redolent of recent observationalfindings: δ ∝ L X /σ ∗ , where L X is the X-ray luminosity of the hot atmosphere, and σ ∗ is the central stellar velocity dispersion of the host galaxy. Combining this resultwith well-known scaling relations, we deduce a duty cycle for radio AGN in ellipticalgalaxies that is approximately ∝ M BH1 . , where M BH is the central black hole mass.Outburst durations and Eddington ratios are also given. Based on the results of thisstudy, we conclude that gravitational instability could provide an important mecha-nism for supplying fuel to AGN in massive galaxies and clusters, and warrants furtherinvestigation. Key words:
The cooling time of X-ray emitting gas near the centres ofmany massive galaxies and galaxy clusters is much shorterthan the Hubble time. In the absence of heat sources, sig-nificant quantities of the gas would cool and form stars.However, X-ray spectroscopy has shown that the rate atwhich gas cools to low temperatures is significantly lowerthan first expected (e.g. Peterson et al. 2001; Tamura et al.2001; Xu et al. 2002; Sakelliou et al. 2002; Peterson et al.2003; Kaastra et al. 2004; Peterson & Fabian 2006) suggest-ing that the gas is somehow being reheated.Based on both observational and theoretical evidence, itis generally assumed that energy input by a central AGN ispredominantly responsible for reheating the gas. For exam-ple, elliptical galaxies are commonly the hosts of powerful ra-dio Active Galactic Nuclei (AGN). These sources give rise to ⋆ E-mail:[email protected] lobes of radio emission embedded in the X-ray emitting gaswhich permeates massive galaxies and clusters of galaxies(e.g. Bˆırzan et al. 2004; Best et al. 2005; Dunn et al. 2005;Rafferty et al. 2006; Best et al. 2007; Shabala et al. 2008).Moreover, recent observational studies of Brightest Clus-ter Galaxies (BCGs) suggest that radio AGN activity isrelated to the thermal state of its environment. Systemswith short radiative cooling times, or a low central en-tropy, are more likely to exhibit active star formation, opti-cal line-emission and jet-producing AGN (e.g. Burns 1990;Crawford et al. 1999; Cavagnolo et al. 2008; Mittal et al.2009; Rafferty et al. 2008). This suggests that AGN activ-ity is part of a feedback loop that can prevent the ambienthot gas from cooling, and is likely to have other importantconsequences for its environment.Building on the ideas of early work (e.g.Binney & Tabor 1995; Tucker & David 1997;Ciotti & Ostriker 2001; Silk & Rees 1998), theoreticalstudies have drawn attention to the potential, wide-ranging c (cid:13) E.C.D. Pope, J.T. Mendel, S.S. Shabala impact of AGN feedback. For example, semi-analytic modelsof galaxy formation have demonstrated that, in principle,AGN heating can both reheat cooling flows and explainthe exponential cutoff at the bright end of the galaxyluminosity function (e.g. Benson et al. 2003; Croton et al.2006; Bower et al. 2006), see also (Short & Thomas 2009).More recently, AGN heating has been shown to be crucialin shaping the X-ray luminosity-temperature relation ofmassive galaxies (e.g. Puchwein et al. 2008; Bower et al.2008; Pope 2009).Despite these findings, the fundamental details of AGNfeedback remain poorly constrained. Most significantly,there is no clear consensus on how information about thethermal state of the X-ray emitting atmosphere is transmit-ted to the black hole at the centre of the galaxy, or clus-ter. Understanding this link is of great importance becauseit probably facilitates the observationally-inferred long-termbalance between AGN heating and gas cooling, and almostcertainly governs the duty cycle of AGN activity – that is,the fraction of time an AGN is active.Generally, AGN feedback is assumed to be powered byone of the following: a) Bondi accretion of hot materialin the vicinity of the black hole (e.g. Cattaneo & Teyssier2007; Sijacki et al. 2008; Puchwein et al. 2008; Fabjan et al.2010); b) material directly from the ambient cooling flow(e.g. Pizzolato & Soker 2005; Bower et al. 2008; Pope 2009;Pizzolato & Soker 2010). While there are plausible argu-ments for both, based on our current interpretation of theavailable observations, there are also difficulties. For exam-ple, the power output from Bondi accretion is unlikely to besufficient to balance the gas cooling rates in massive clus-ters (e.g Soker 2006). The biggest problem with models incategory b) is finding a mechanism by which the centralblack hole receives information from the cooling flow aboutthe thermal state of the ICM. For example, it is difficult toconceive of how material might flow all the way from thecluster cooling radius onto a central black hole, such that itprompts an AGN outburst on a useful timescale (though seePizzolato & Soker 2005).The model investigated here describes a mechanismthat can reconcile the problem faced by models of type b).While Pizzolato & Soker (2005, 2010) focused on thermalinstability of the ICM as a mechanism for delivering fuel tothe AGN, we focus on the characteristics of AGN feedbackthat is triggered when the hot gas which resides near thecentre of a galaxy becomes gravitationally unstable. Thatis, gas which was previously self-supporting against gravityis somehow destabilised and falls freely towards the galaxycentre, on a dynamical timescale, where it forms stars andfuels the AGN (e.g. Silk & Rees 1998; Fabian 2009; King2009). Gravitational instabilities can be induced by mergerevents (e.g. Silk & Rees 1998) and the effect of radiativecooling (e.g. Birnboim & Dekel 2003). Of these two possi-bilities, only instabilities induced by gas cooling can lead toself-regulated AGN heating, but it is important to mentionthat merger-driven AGN feedback will delay the onset ofgravitational instability induced by radiative cooling. Theextent to which this happens is unclear, but the effect maybe more significant in group and field environments (e.g.Kaviraj et al. 2010), than in clusters.Perhaps importantly, AGN heating driven by cooling-induced gravitational instabilities has the potential to be pe- riodic, as explained by the following argument. In massivegalaxies and clusters there is a continual inflow of materialfrom the large-scale environment, due to the effects of theambient cooling flow. As the mass of material in the hotatmosphere near the centre of a galaxy builds up, it can be-come gravitationally unstable, meaning that some fractionof the gas will collapse and flow on a dynamical timescaleto the centre of the gravitational potential, thereby fuellingthe AGN. Once this fuel has been consumed, the AGN willbe starved of new material until the nearby hot gas onceagain becomes gravitationally unstable. A second instabil-ity will commence after the external inflow has delivered asufficient quantity of new material. Thus, for this mode offuelling, the typical duration of an AGN outburst is largelygoverned by the local dynamical timescale, while the timebetween the onset of successive outbursts is controlled by theambient cooling flow. This is notable because observationsindicate that AGN are only active for a fraction of time de-termined by their environment and host galaxy properties(e.g. Best et al. 2005; Shabala et al. 2008). In addition, asdemonstrated in Pope (2011), periodic heating can supplythe energy required to balance gas cooling, with the mini-mum effort. Thus, periodic AGN activity appears to be anenergetically favourable heating strategy. Therefore, start-ing from the assumption that AGN fuelling is attributableto gravitational instabilities induced by gas cooling, we de-rive AGN duty cycles, outburst durations, star formationrates and discuss implications for numerical simulations.The outline of the article is as follows. In Section 2we present the main model and place constraints on thephenomenological parameters. In Section 3 we show the ex-pected duty cycles and outburst durations that would be as-sociated with AGN heating that is driven by cooling-inducedgravitational instability. The implications are discussed inSection 4, and the results are summarised in Section 5.
In this investigation, we study the properties of AGN feed-back that is driven by the gravitational collapse of hot gasin the vicinity of the supermassive black hole. Therefore,we must first describe the physical conditions of the gasin massive galaxies and clusters that can justifiably lead togravitational instability. Following this we define the charac-teristic timescales of the system and the AGN power outputexpected from this mode of fuelling. These quantities arethen used to determine the observable features predicted bythis scenario.
A self-gravitating sphere of gas will become gravitationallyunstable when the sound-crossing time is greater than thegravitational free-fall time. This is the well-known Jeans cri-terion. Strictly speaking, the criterion only applies in thelimit that the polytropic index of the gas – γ eff , as de-fined in equation (1) – is also lower than some critical value, γ crit ≈ .
2, (e.g. see Birnboim & Dekel 2003, and referencestherein). Since the actual adiabatic index of an ideal, non-relativistic monatomic gas is γ = 5 /
3, a gas of this typecan only become gravitationally unstable if its behaviour is c (cid:13) , 1–8 GN feedback triggered by gravitational collapse modified by heating, cooling and work. For example, if aparcel of gas is heated by an amount ∆ Q , and then does anequal amount of work on its environment (i.e. ∆ W = ∆ Q ),the gas is defined as isothermal, since it acts to maintain aconstant temperature. In this case, the polytropic index ofthe gas must be γ eff = 1, by definition.In the case of the hot, X-ray emitting gas that resides inmassive galaxies and galaxy clusters, we envisage a centralgas mass that is embedded in a pressurised ambient medium.Consequently, the scenario is somewhat reminiscent of theisothermal sphere which is prone to gravitational instabilityif its mass exceeds the Bonnor-Ebert limit. However, it isimportant to note that in this study we do not make anyassumptions about the density distribution of the hot gas .Observations clearly indicate that this gas must be sub-ject to radiative cooling, heating from AGN and stars, and the work done by its surroundings. Therefore, it seems ex-tremely likely that γ eff = 5 / t cool < t dyn . This is perfectly possible because a gas parcelof temperature T and number density n , radiating via ther-mal bremsstrahlung, has a cooling time that is ∝ T / /n .As the gas cools, the temperature will fall, and the densitywill rise, so that the cooling time shortens. It is this con-dition ( t cool < t dyn ) that explains observations which showthat the temperature of the gas remains lower near the cen-tre of the cluster than further out. Indeed, it is preciselythis argument that predicts a cooling catastrophe, and theneed for AGN feedback. However, as shown below, the inflowcondition also predicts the onset of a cooling-induced gravi-tational instability which provides a potentially informativedescription of AGN fuelling in hot atmospheres.Thermodynamically, the mass of gas described abovebehaves like a parcel that cools down (rather than heatsup) when compressed. The appropriate polytropic relationlinking the temperature, T , and volume, V , of the gas parcelis T V γ eff − = constant. Thus, the hot gas in the centres ofmassive galaxies and clusters acts as if 0 < γ eff <
1. Sincethis is less than all plausible values of the critical polytropicindex, the hot gas is likely to be susceptible to gravitationalinstability .For completeness, we also present a brief derivation to il-lustrate the general phenomena that influence the polytropicindex of a gas. This derivation closely follows the work ofBirnboim & Dekel (2003) who define the polytropic index, γ eff , of a Lagrangian fluid element in terms of the logarith-mic time derivative of its pressure, P , and density, ρ , suchthat γ eff ≡ d ln P/ d t d ln ρ/ d t . (1) The gas exhibits characteristics which are somewhere betweenisothermal ( γ eff = 1) and isobaric ( γ eff = 0) states. Accounting for gas cooling, the polytropic index can be ex-pressed as γ eff = γ − n Λ( T )˙ ρe , (2)where γ is the actual adiabatic index of the gas, ˙ ρ is the rateof change of gas density and e is the internal energy per unitmass. As usual, n is the number density of the gas, T is thegas temperature and Λ( T ) is the cooling function. To calculate the local dynamical timescale, we refer backto the definition of the Jeans instability criterion. A self-gravitating gas mass will become gravitationally unstablewhen its mass exceeds some critical value that is compa-rable to the Jeans/Bonnor-Ebert masses. In the centre ofa galaxy, the self-gravity of the gas will become importantwhen its characteristic velocity dispersion is comparable tothe central stellar velocity dispersion, σ ∗ . Thus, if a gas mass M with radius R located at the centre of a galaxy becomesgravitationally unstable, it will collapse on the dynamicaltimescale, t dyn = R/σ ∗ . The average mass flow rate duringthis time will be ˙ M = ∆ M ∆ t ≈ Mt dyn = σ ∗ MR (3)We note that, for a self-gravitating cloud, the characteris-tic velocity dispersion of the constituent particles is relatedto the gravitational potential energy by the virial theorem: αGM/R = σ , where G is Newton’s gravitational constant.In this description the numerical constant α ∼ M = σ ∗ αG ∼ (cid:18) σ ∗
200 km s − (cid:19) M ⊙ yr − . (4)This is the well-known form of the dynamical mass flow rate(c.f. King 2009), modified slightly to account for arbitrarydensity distributions. From this it is straightforward to showthat the duration of the collapse will be M/ ˙ M = t dyn .Following the collapse, mass will flow towards the centreof the galaxy’s gravitational potential where a fraction islikely to be accreted by the supermassive black hole, andthe remainder presumably forms stars nearby or is expelledfrom the galaxy as a result of feedback.As described in the introduction, the fuelling rate pro-vided by gravitational instability can be periodic, for thefollowing reason. The postulated gas cloud near the galaxycentre acts as a reservoir; this reservoir is depleted when thecloud collapses, but is replenished by the inflow of new ma-terial from the larger-scale environment, which forms a newcloud. When it reaches the critical mass, the second cloudwill also become gravitationally unstable and collapse, andso on.The provenance of the material that replenishes the col-lapsing gas cloud is not clear, except that it must come fromthe ambient hot atmosphere. The simplest possibility is thatthe cloud is built up during two phases: 1) an initial collapseof material from slightly further out in the hot atmosphere;2) the subsequent slow inflow of additional material due tothe ambient cooling flow. If this is the case, the first phase c (cid:13) , 1–8 E.C.D. Pope, J.T. Mendel, S.S. Shabala of growth of the new cloud will occur on a similar dynamicaltimescale to the gravitational collapse of the previous cloud.However, at these early times, the new cloud is likely to behotter and of lower mass than the previous cloud, ensuringthat it can be gravitationally stable. More precisely, as longas γ eff < γ crit ≈ .
2, and the cloud mass is below the crit-ical mass, the cloud will not collapse. The second phase ofgrowth will occur on the timescale required for the coolingflow to build up the cloud mass to its gravitationally unsta-ble limit, at which point a collapse will be initiated. Usingthis argument, we can estimate the average time betweengravitational collapses, as shown below.Overall, the local mass inflow rate, ˙ M ext , from the cool-ing flow will be a slowly varying quantity governed by thedifference between the time-averaged heating and coolingrates. Then, in the limit that the majority of the cloud massis built up during this phase, the characteristic time betweenthe triggering of successive gravitational collapses must tendto τ = M ˙ M ext . (5)Assuming that the AGN is only fuelled while the cloud col-lapses, it will be active for a fraction of time δ = t dyn /τ ,known as the duty cycle. By combining equations (4) and(5), we obtain a simple functional form for the AGN dutycycle without having to explicitly calculate the AGN heatingrate δ = t dyn τ = ˙ M ext RMσ ∗ = αG ˙ M ext σ ∗ . (6)Thus, in the present model, the AGN duty cycle can bestraightforwardly related to the local gravitational poten-tial, through σ ∗ , and the external environment through˙ M ext . This is potentially significant, because obervations(e.g. Best et al. 2005; Best 2007; Shabala et al. 2008) indi-cate that the radio AGN duty cycle is heavily influenced byboth local and environmental effects. Below, we argue that˙ M ext is probably closely related to the mass inflow associ-ated with the ambient cooling flow. If a fraction, β , of the inflowing mass rate, ˙ M , reaches theblack hole, the accretion power output will be H = ηβ ˙ Mc = η βσ ∗ αG c ≈ (cid:18) β/α − (cid:19)(cid:18) σ ∗
200 km s − (cid:19) erg s − , (7)where the assumed accretion efficiency is η ≈ .
1. The char-acteristic value of β/α = 10 − is motivated below, and yieldsfavourable comparisons with the observationally-inferred ra-dio AGN duty cycle.Clearly, the AGN energy injection rates associated withthis fuelling mechanism can be very large. Such values areconsiderably larger than would be expected to arise fromthe accretion of nearby hot gas and, if observed in a realsystem, would more commonly be associated with merger-driven fuelling events. However, the reasoning above indi-cates that such an interpretation is not necessarily exclu-sive – high AGN fuelling rates can also be a consequenceof gravitational instability resulting from gas cooling. In ad-dition, it is important to remember that values estimated from equation (7) represent the instantaneous heating rate– the time-averaged values are much more modest.By definition, the time-averaged AGN power output iswritten ¯ H ≡ δH , which can be expanded using equations(6) and (7) to give ¯ H = δH = ηβ ˙ M ext c . (8)Equation (8) shows that the time-averaged AGN heatingrate depends on the properties of the large scale envi-ronment, through ˙ M ext . From this we also conclude that˙ M ext will evolve until the time-averaged heating rate closelymatches the ambient cooling rate of the gas. In the limitthat the time-averaged AGN heating does balance gas cool-ing, ¯ H = L X , equation (8) shows that˙ M ext = L X ηβc ≈ (cid:18) L X erg s − (cid:19)(cid:18) β − (cid:19) − M ⊙ yr − , (9)where again we have assumed η ≈ . β/α ∼ − , with α ∼
1. Below, we motivate the constraints on β by consid-ering the Eddington ratio of the AGN outbursts fuelled bygravitational collapse. The Eddington ratios predicted by this model provide an-other method for comparison with numerical simulationsand observations. Importantly, they also provide a way tocheck the self-consistency of the model, as outlined below.Broadly speaking, the form of energetic output from anAGN can be predicted from its Eddington ratio, defined by R Edd ≡ ˙ m ˙ M Edd , (10)where ˙ m is the black hole accretion rate. ˙ M Edd is the Ed-dington limited accretion rate determined by the balancebetween gravity and radiation pressure˙ M Edd = L Edd ηc = 4 πGm p M BH ηcσ T , (11)where L Edd is the Eddington luminosity, M BH is the blackhole mass, σ T is the Thompson cross-section, m p is the pro-ton mass and the other symbols are as previously defined.By analogy with stellar mass black holes in X-ray bina-ries (e.g. K¨ording et al. 2008), accretion at rates less than ∼
3% of the Eddington limit is radiatively inefficient so thatthe majority of the power output is in the form of kinetic-energy-dominated outflows of relativistic particles which areprominent radio synchrotron emitters. Outflows of this typeare thought to couple strongly to the ambient gas. Con-versely, accretion above this critical rate is radiatively effi-cient, meaning that the power output is predominantly inthe form of photons, rather than kinetic outflows of par-ticles. In this limit, radio jets may still be observed, butthe efficiency of jet production is much lower than in theradiatively inefficient regime (e.g. Maccarone et al. 2003).Furthermore, in the radiatively efficient mode, it has beenargued that only ∼
5% of the accretion power is available toheat the surrounding gas (e.g. Sijacki & Springel 2006), seeKing (2009) for a possible explanation.We note that the comparison with X-ray binaries is notexact, since it does not account for the existence of radio-loud quasars which produce powerful kinetic outflows at high c (cid:13) , 1–8 GN feedback triggered by gravitational collapse Eddington ratios. However, as explained below, we focus onlow accretion rate objects so the distinction does not matterfor the purposes of this model.To calculate the Eddington ratios, we use the well-known relation between the black hole mass and the stellarvelocity dispersion: M BH ≈ . × ( σ ∗ /
200 km s − ) M ⊙ ,where σ ∗ is the central stellar velocity dispersion (e.g.Tremaine et al. 2002). Substituting for the black hole massinto equation (11) yields the Eddington ratio for an AGNfuelled by gravitational collapse R Edd ≈ . (cid:18) β/α − (cid:19)(cid:18) σ ∗
200 km s − (cid:19) − , (12)where the assumed accretion efficiency is η ≈ .
1. Therefore,if β/α . − , accretion following a gravitational collapsecan be radiatively inefficient and produce kinetic outflows.Since α ∼
1, this implies that β . − is required for kineticoutflow production. For larger values of β/α the accretionwill be radiatively efficient.The value of β is depends on the processes that governhow gas travels from kiloparsec scales onto the black hole.Since these processes are highly uncertain, we proceed byassigning β a single, empirical value that encapsulates theaccretion physics in the full range of AGN environments.While this is a simplification, it ensures that the results areas transparent as possible. As indicated in equation (12), wefind that β ∼ − permits a good agreement between themodel and radio observations (Best et al. 2005; Best 2007;Shabala et al. 2008). Reassuringly, equation (12) also showsthat β ∼ − will lead to low Eddington ratios and, there-fore, kinetic outflows which produce radio emission. From the assumptions and derivations outlined above, wepresent the expected duty cycles of radio AGN fuelled bygravitationally destablised gas, the corresponding durationof individual outbursts, and the associated heating rates re-sulting from shocks generated by the AGN outflow.
In this section, we give a more general derivation for theAGN duty cycle. For a constant pressure cooling flow inwhich external heating is not important, the classical massflow rate, gas temperature, and X-ray luminosity are relatedby L X = γγ − M clas k B Tµm p , (13)where k B is Boltzmann’s constant and µm p is the mean massper particle.More generally, the net inflow of mass is determined bythe difference between the time-averaged cooling and heat-ing rates. Observationally, this mass flow rate is estimatedby fitting models to the X-ray spectra. Thus, we refer tothe mass flow rate by its observational name, ˙ M spec , butcalculate it from the following L X − ¯ H = γγ − M spec k B Tµm p . (14) Rearranging equation (14) for the duty cycle, using equation(7), gives δ = αGL X ηβc σ ∗ (cid:18) − ˙ M spec ˙ M clas (cid:19) . (15)Consequently, in the limit that AGN heating exactly bal-ances gas cooling, ˙ M spec / ˙ M clas →
0, the duty cycle simplifiesto δ → αGL X ηβc σ ∗ ∝ L X σ ∗ . (16)In terms of scaled quantities, the duty cycle can be expressedas δ ≈ . (cid:18) β/α − (cid:19) − (cid:18) L X erg s − (cid:19)(cid:18) σ ∗
200 km s − (cid:19) − . (17)Equation (17) shows that AGN fuelled by the gravitationalcollapse of gas in more X-ray luminous clusters should ex-hibit larger duty cycles. That is, those AGN should be activefor a greater proportion of time, as is inferred from obser-vations (Best et al. 2007). However, the precise scaling ofthe AGN duty cycle in clusters is difficult to determine be-cause it is not clear exactly how the X-ray luminosity of thegas, and the central stellar velocity dispersion are related inclusters. Nevertheless, surveys indicate that the X-ray lumi-nosity of a cluster scales as L X ∝ σ n c , where σ c is the velocitydispersion of the cluster potential, and that n ∼ − L X − σ c relation into equation (17), gives δ ∝ σ n c /σ ∗ . Therefore, ifmore massive BCGs are found in more massive clusters, theAGN duty cycle will tend to increase slowly with σ ∗ , whichwould be in qualitative agreement with observations (e.g.Best et al. 2007). Furthermore, assuming β/α ∼ − , thenormalisation of the duty cycle is entirely consistent withthe observational results presented by Best et al. (2007).In contrast, the scaling relations of field elliptical galax-ies yield a much simpler scaling of δ with σ ∗ . Under theassumption of an isothermal gravitational potential, thevelocity dispersion is constant and independent of radius.Again, surveys indicate that the X-ray luminosity scales as L X ∝ σ m ∗ , where m ∼ −
10 (e.g. O’Sullivan et al. 2003;Mahdavi & Geller 2001). Using this fact, the AGN duty cy-cle would scale as δ ∝ σ m − ∗ ∼ σ ∗ . Since supermassive blackhole mass scales as M BH ∝ σ ∗ , the duty cycle can then besaid to increase as δ ∝ M . , which is consistent with ob-servational findings (Best et al. 2005; Shabala et al. 2008).In addition, for a massive galaxy with an X-ray luminosityof 10 erg s − , and σ ∗ ∼
200 km s − the duty cycle wouldbe δ ≈ − (assuming β/α ∼ − ), which is also consis-tent with Best et al. (2005) and Shabala et al. (2008). Con-sequently, both the scaling and normalisation of the dutycycle are largely consistent with observations of radio AGNin field elliptical galaxies and BCGs. The AGN duty cycle is a useful quantity which providesinformation about how frequently an AGN is triggered inorder for heating to balance the effects of radiative cooling.With additional assumptions, the duty cycle also providessome information about the duration of an individual heat-ing event, t dyn , and its environmental dependence. If the c (cid:13) , 1–8 E.C.D. Pope, J.T. Mendel, S.S. Shabala time between the onset of successive AGN outbursts is τ ,the number of heating events which occur during a galaxylifetime, t age , must be N = t age /τ = δ ( t age /t dyn ). Using theprevious results, the expected number of AGN outbursts canbe written N ≈ (cid:18) β/α − (cid:19) − (cid:18) t dyn /t age . (cid:19) − (cid:18) L X erg s − (cid:19) (18) × (cid:18) σ ∗
200 km s − (cid:19) − . By rearranging equation (18), the duration of an outburstcan be expressed in the form t dyn = δt age /N . Using equa-tion (17) for a cluster with L X ∼ erg s − and σ ∗ ∼
200 km s − , the typical outburst duration must be t dyn ∼ . t age . Taking t age ∼ t dyn ∼
50 Myr, whichis sufficient to explain the features AGN-blown bubbles ob-served in many clusters (e.g. Bˆırzan et al. 2004; Dunn et al.2005).The gravitational instability model of AGN fuellingalso offers an explanation for the unexpectedly large num-ber of compact radio sources (e.g. O’Dea & Baum 1997;Shabala et al. 2008) that differs from the accretion disk vari-ability model proposed by Czerny et al. (2009). As shown byAlexander (2000), the maximum stable length of an AGNjet propagating through an atmosphere depends on its powerand the ambient density. For example, at constant jet power,a higher ambient density leads to a shorter stable jet length.Then, since the inflow of material due to gravitational insta-bility will significantly enhance the density in the vicinity ofthe black hole, this enhanced density may also plausibly con-fine and disrupt the resultant AGN jet on kiloparsec scales.If this is correct, the jet can only propagate further outwardsonce the gas has been sufficiently depleted by continued ac-cretion and star formation.
The model investigated above exhibits several features thatare compatible with key observational characteristics ofAGN feedback in both field elliptical galaxies and BCGs.Below we discuss some additional implications of the modelwhich may be significant, but are harder to quantify.
Since only a very small fraction of the inflowing materialreaches the black hole, the vast majority must either formstars, or be dragged out of the galaxy by the AGN outflowitself. As shown in Pope et al. (2010) the mass of ambientmaterial transported by an AGN-blown bubble is approxi-mately equal the mass of gas initially displaced by the bub-ble. In principle, this process can be extremely effective atremoving material from the galaxy, dramatically reducingthe quantity of gas available for forming stars. As a result,it is difficult to quantify star formation rates resulting fromcooling-induced gravitational instability, because it is un-clear how much of the collapsed gas will be retained by thegalaxy, and for how long. Our current best estimate is thatthe mass of material available for forming stars must fall be-tween two well-defined limits. The upper limit is the case in which all of the collapsed gas mass goes into forming stars;the lower limit is the case in which the vast majority of thecollapsed gas mass is expelled by AGN feedback leaving nomaterial available for forming stars.Given that the collapsing gas flows inwards on a dynam-ical timescale, t dyn , with a mass flow rate ˙ M , the collapsingmass can be written as M = ˙ M t dyn . Then, using the factthat the instantaneous mass flow rate is ˙ M = σ ∗ / ( αG ), themaximum mass of gas available for forming stars during eachcollapse, will be M = ˙ M t dyn ≈ σ ∗ αG t dyn ≈ (cid:18) σ ∗
200 km s − (cid:19) (cid:18) t dyn yr (cid:19) M ⊙ . (19)Since a fraction β is assumed to be accreted by the blackhole, this leaves a maximum fraction of 1 − β available forforming stars, in the unlikely event that no material is re-moved due to feedback.However, we can estimate the mass of material removedby the AGN using simple considerations. The total energy, E , injected by an AGN as a result of accreting a mass βM ,will be E = ηβMc . If this energy does work against thepressure, P , of the ambient gas, it will inflate a bubble withvolume, V , given approximately by E = 4 P V , assumingrelativistic bubble contents. Using the definition of the gaspressure P = ρk B T /µm p , where ρ is the ambient gas densityand T is temperature, we can write E = 4 M dis k B T /µm p ,where M dis ≡ ρV is the mass displaced in inflating the bub-ble. As previously noted, Pope et al. (2010) showed that abuoyant AGN-blown bubble will lift a mass comparable to M dis / .The mass of gas lifted out from the centre of a galaxyby AGN feedback can be related to the mass of materialthat collapsed into the galaxy centre due to cooling-inducedgravitational instability M dis / M ≈ ηβ µm p c k B T ≈ (cid:18) β − (cid:19)(cid:18) T K (cid:19) − . (20)According to equation (20), an individual AGN outburstmay remove more gas than actually flowed into the galaxycentre. While this may seem unphysical, it is not – feedbackcould remove all of the material that collapsed into galaxy and additional ambient gas. The fate of the gas lifted outof the galaxy will depend on the AGN power and the depthof the external gravitational potential. There are two mainpossibilities: 1) if the outflow injects sufficient energy, mate-rial will be permanently expelled from the galaxy; 2) if not,the outflow will temporarily lift material out of the galaxy,later allowing it to fall back inwards. It is difficult to estimate how high this material will be liftedbecause it depends strongly on the details of the bubble, the extramass of material it is carrying, and the properties of the ambientatmosphere. Pope et al. (2010) demonstrated that a bubble lift-ing additional material will rise to a height at which the averagedensity of the bubble, plus the lifted mass, is equal to the am-bient density. At this location the buoyancy force goes to zero,and the bubble cannot rise further unless it sheds some of thematerial. The greater the mass carried by the bubble, the less thebubble will rise. This effect directly limits the amount of energyextracted from the bubble. Because of this, it is not possible tosay that the bubble will change the gravitational potential of thelifted mass by an amount defined by: E = ( M dis / (cid:13) , 1–8 GN feedback triggered by gravitational collapse As shown in Pope (2009), AGN fuelled at a small frac-tion of the ambient cooling flow rate can power outflowsthat are capable of ejecting material from the potential ofan elliptical galaxy. However, for the deeper gravitationalpotentials of galaxy groups and clusters (with virial tem-peratures greater than 1-2 keV) there are no black holesthat are massive enough to sustain outflows that can expelgas from the potential. Despite this, typical AGN outflows ingalaxy groups and clusters do still affect their environmentby gently redistributing the gas within the gravitational po-tential. Furthermore, any material that is lifted out of thecentre of the host galaxy will eventually fall back inwards,thereby becoming part of a fountain-like flow.Inspection of equation (20) leads to the expectation thatAGN feedback should remove more ambient gas, per unit ac-creted mass, from cooler (lower mass) systems. As describedabove, this material can be permanently expelled from lowmass elliptical galaxies. Thus, we conclude that star forma-tion in low mass elliptical galaxies would necessarily have tooccur in a limited window of opportunity: after the gravi-tationally unstable material has collapsed into the centre ofthe galaxy, and before it has been removed by AGN feed-back. In other words, any reservoir of material available forforming stars will be short-lived, meaning that star forma-tion episodes are comparatively rare in such systems.When applied to galaxy clusters, equation (20) indi-cates that the mass of material lifted out of the host galaxyby AGN feedback will approximately equal the mass of ma-terial which collapses into the galaxy due to gravitationalinstability. As a result, AGN feedback may also be able toshut-off star formation in BCGs. However, the fountain-likeflow described above would provide a reservoir of material inand around the galaxy which is available for forming stars.Consequently, is reasonable to expect some on-going starformation in BCGs. Interestingly, this conclusion appearsto be compatible with observations; Rafferty et al. (2006)found BCG star formation rates up to ∼
100 M ⊙ yr − , whichcorresponds to 10 M ⊙ over 10 yr.Finally, considering the mass flow rates, we can showthat the ratio of black hole and stellar mass growth ratesmust be ˙ M BH / ˙ M ∗ = β/ (1 − β ) ≈ β . This suggests that,as the age of the system becomes very large, the ratio ofblack hole mass to bulge mass should also tend towards β .It is encouraging to note that a value of β ∼ − agrees wellwith observations (e.g. H¨aring & Rix 2004) and is consistentwith our earlier constraints obtained by comparing the the-oretical AGN duty cycle with radio observations (Best et al.2005; Shabala et al. 2008). The cooling-induced gravitational instability scenario for fu-elling AGN can only be captured by fluid simulations if thefollowing are true: 1) the self-gravity of the gas is included inthe calculations, and
2) the spatial resolution is fine enoughto track the evolution of the gravitationally unstable region.For a collapsing mass M , with a velocity dispersion σ ∗ , theregion of importance has a size R ∼ GMσ ∗ ≈ (cid:18) M M ⊙ (cid:19)(cid:18) σ ∗
200 km s − (cid:19) − kpc . (21) Thus, for all plausible values of M and σ ∗ , the collapsingregion is likely to be smaller than the resolved spatial scalesin cosmological fluid simulations. We note that the densityof these small, self-gravitating regions is likely to be high( ∼ − g cm − for M ∼ M ⊙ and σ ∗ ∼
200 km s − )and scales as ρ ∼ M/R ∝ σ ∗ /M . The aim of this article has been to explore the properties ofAGN feedback that is driven by the gravitational instabilityof hot gas in the locality of a supermassive black hole. Whileit remains unclear whether gravitational instability itself is responsible for delivering fuel to AGN, we have shown that amechanism which behaves similarly could produce outcomesthat are compatible with several key observations of radioAGN in massive galaxies and clusters. The main findings aresummarised below:(i) Gas in the centre of a galaxy which periodically be-comes gravitationally unstable provides a way of linking theAGN fuelling rate to galaxy’s large scale environment with-out requiring gas to flow 10s of kiloparsecs before reachingthe black hole.(ii) According to this model, the AGN duty cycle scalesas δ ∝ L X /σ ∗ , where L X is the X-ray luminosity of the hotgas in a cluster and σ ∗ is the stellar velocity dispersion atthe centre of the galaxy which hosts the AGN.(iii) Applying simple scaling relations to this model, wefind that the duty cycle of radio AGN in massive galaxiesshould scale as δ ∝ σ ∗ ∝ M . , in reasonable agreementwith observations.(iv) The region which collapses may be very small ( < t dyn = δt age /N ∝ L X / ( σ ∗ N ), where t age is the age of the cluster and N is the number of out-bursts during this time. Plausible values for t age and N implythat radio AGN outbursts fuelled by gravitational collapsecan extend up to ∼ − yrs, which is consistent withobservations of AGN-blown bubbles.(vi) Using simple arguments we have shown that AGNfeedback is likely to remove more gas, per unit accretedmass, from the centre of a lower mass galaxy than a BCG.By completely expelling material from lower mass ellipticalgalaxies, AGN feedback can dramatically reduce their starformation rates. In contrast, AGN feedback cannot com-pletely expel gas from a galaxy cluster. As a result, theremay be more material in and around BCGs which is avail-able for forming stars. Thus, it may be more difficult tocompletely shut-off star formation in BCGs than in lowermass galaxies. We thank the anonymous referee for helpful comments thatimproved this work. ECDP would like to thank CITA forfunding through a National Fellowship. JTM acknowledgesfinancial support from the Canadian National Science andEngineering Research Council (NSERC). SSS acknowledges c (cid:13) , 1–8 E.C.D. Pope, J.T. Mendel, S.S. Shabala the Australian Research Council (ARC) for a Super ScienceFellowship.
REFERENCES
Alexander P., 2000, MNRAS, 319, 8Bˆırzan L., Rafferty D. A., McNamara B. R., Wise M. W.,Nulsen P. E. J., 2004, ApJ, 607, 800Benson A. J., Bower R. G., Frenk C. S., Lacey C. G., BaughC. M., Cole S., 2003, ApJ, 599, 38Best P. N., 2007, New Astronomy Review, 51, 168Best P. N., Kauffmann G., Heckman T. M., BrinchmannJ., Charlot S., Ivezi´c ˇZ., White S. D. M., 2005, MNRAS,362, 25Best P. N., von der Linden A., Kauffmann G., HeckmanT. M., Kaiser C. R., 2007, MNRAS, 379, 894Binney J., Tabor G., 1995, MNRAS, 276, 663Birnboim Y., Dekel A., 2003, MNRAS, 345, 349Bower R. G., Benson A. J., Malbon R., Helly J. C., FrenkC. S., Baugh C. M., Cole S., Lacey C. G., 2006, MNRAS,370, 645Bower R. G., McCarthy I. G., Benson A. J., 2008, MNRAS,390, 1399Burns J. O., 1990, ApJ, 99, 14Cattaneo A., Teyssier R., 2007, MNRAS, 376, 1547Cavagnolo K. W., Donahue M., Voit G. M., Sun M., 2008,ApJ, 683, L107Ciotti L., Ostriker J. P., 2001, ApJ, 551, 131Crawford C. S., Allen S. W., Ebeling H., Edge A. C., FabianA. C., 1999, MNRAS, 306, 857Croton D. J., Springel V., White S. D. M., De Lucia G.,Frenk C. S., Gao L., Jenkins A., Kauffmann G., NavarroJ. F., Yoshida N., 2006, MNRAS, 365, 11Czerny B., Siemiginowska A., Janiuk A., Nikiel-Wroczy´nskiB., Stawarz L., 2009, ApJ, 698, 840Dunn R. J. H., Fabian A. C., Taylor G. B., 2005, MNRAS,364, 1343Fabian A. C., 2009, ArXiv e-printsFabjan D., Borgani S., Tornatore L., Saro A., Murante G.,Dolag K., 2010, MNRAS, 401, 1670H¨aring N., Rix H.-W., 2004, ApJ, 604, L89Kaastra J. S., Tamura T., Peterson J. R., Bleeker J. A. M.,Ferrigno C., Kahn S. M., Paerels F. B. S., Piffaretti R.,Branduardi-Raymont G., B¨ohringer H., 2004, A&A, 413,415Kaviraj S., Schawinski K., Silk J., Shabala S. S., 2010,ArXiv e-printsKing A. R., 2009, MNRAS, pp 1914–+K¨ording E. G., Jester S., Fender R., 2008, MNRAS, 383,277Maccarone T. J., Gallo E., Fender R., 2003, MNRAS, 345,L19Mahdavi A., Geller M. J., 2001, ApJ, 554, L129Mittal R., Hudson D. S., Reiprich T. H., Clarke T., 2009,A&A, 501, 835O’Dea C. P., Baum S. A., 1997, ApJ, 113, 148O’Sullivan E., Ponman T. J., Collins R. S., 2003, MNRAS,340, 1375Peterson J. R., Fabian A. C., 2006, Physical Reports, 427,1 Peterson J. R., Kahn S. M., Paerels F. B. S., Kaastra J. S.,Tamura T., Bleeker J. A. M., Ferrigno C., Jernigan J. G.,2003, ApJ, 590, 207Peterson J. R., Paerels F. B. S., Kaastra J. S., Arnaud M.,Reiprich T. H., Fabian A. C., Mushotzky R. F., JerniganJ. G., Sakelliou I., 2001, A&A, 365, L104Pizzolato F., Soker N., 2005, ApJ, 632, 821Pizzolato F., Soker N., 2010, ArXiv e-printsPope E. C. D., 2009, MNRAS, pp 494–+Pope E. C. D., 2011, ArXiv e-printsPope E. C. D., Babul A., Pavlovski G., Bower R. G., DotterA., 2010, MNRAS, 406, 2023Puchwein E., Sijacki D., Springel V., 2008, ApJ, 687, L53Rafferty D. A., McNamara B. R., Nulsen P. E. J., 2008,ApJ, 687, 899Rafferty D. A., McNamara B. R., Nulsen P. E. J., WiseM. W., 2006, ApJ, 652, 216Sakelliou I., Peterson J. R., Tamura T., Paerels F. B. S.,Kaastra J. S., Belsole E., B¨ohringer H., Branduardi-Raymont G., Ferrigno C., den Herder J. W., Kennea J.,Mushotzky R. F., Vestrand W. T., Worrall D. M., 2002,A&A, 391, 903Shabala S. S., Ash S., Alexander P., Riley J. M., 2008,MNRAS, 388, 625Short C. J., Thomas P. A., 2009, ApJ, 704, 915Sijacki D., Pfrommer C., Springel V., Enßlin T. A., 2008,MNRAS, 387, 1403Sijacki D., Springel V., 2006, MNRAS, 366, 397Silk J., Rees M. J., 1998, A&A, 331, L1Soker N., 2006, New Astronomy, 12, 38Tamura T., Kaastra J. S., Peterson J. R., Paerels F. B. S.,Mittaz J. P. D., Trudolyubov S. P., Stewart G., FabianA. C., Mushotzky R. F., Lumb D. H., Ikebe Y., 2001,A&A, 365, L87Tremaine S., Gebhardt K., Bender R., Bower G., DresslerA., Faber S. M., Filippenko A. V., Green R., GrillmairC., Ho L. C., Kormendy J., Lauer T. R., Magorrian J.,Pinkney J., Richstone D., 2002, ApJ, 574, 740Tucker W., David L. P., 1997, ApJ, 484, 602Xu H., Kahn S. M., Peterson J. R., Behar E., PaerelsF. B. S., Mushotzky R. F., Jernigan J. G., BrinkmanA. C., Makishima K., 2002, ApJ, 579, 600 c (cid:13)000