A Moderate Cooling Flow Phase at Galaxy Formation
aa r X i v : . [ a s t r o - ph . C O ] M a y A MODERATE COOLING FLOW PHASE AT GALAXYFORMATION
Noam Soker ABSTRACT
I study the possibility that a cooling flow (CF) exists at the main phase ofsuper massive black hole (SMBH) growth during galaxy formation. To ensurethat jets launched by the SMBH efficiently expel gas from the galaxy, as isrequired by recent results, the gas should be in the hot phase, rather than in coldclouds. The short radiative cooling time of the hot gas leads to the formationof a CF, but heating by the active galactic nucleus (AGN) prevents catastrophiccooling. Cold blobs that start as instabilities in the hot phase feed the SMBHfrom an extended region, form an accretion disk, and lead to the formation ofjets. These jets can expel large quantities of gas out of the galaxy. This cycle,that is termed cold feedback mechanism in CFs in clusters of galaxies, mightexplain the correlation of SMBH to bulge masses. Stars are formed, but at alower rate than what is expected when heating is not included. Such a CF istermed a moderate CF.
1. INTRODUCTION
The tight correlation between the supermassive black hole (SMBH) mass, M BH , and thevelocity dispersion, σ , of the hot component of the host galaxy (e.g., Merritt & Ferrarese2001; Gebhardt et al. 2000; G¨ultekin et al. 2009 ) is well established, and so is the correlationbetween the SMBH mass and the bulge mass, M bulge (e.g., Kormendy & Richstone 1995; Laor2001). It is possible that the M BH − M bulge correlation is determined by a feedback processof the active galactic nucleus (AGN) of the host bulge, powered by the accreting SMBH.The feedback mechanism where AGN jets (outflow; wind) suppress gas from cooling to lowtemperatures and from forming stars was discussed for both cooling flows (CFs) in galaxiesand clusters of galaxies (e.g., Binney & Tabor 1995; Nulsen & Fabian 2000; Reynolds et al.2002; Omma & Binney 2004; Soker & Pizzolato 2005), and in galaxy formation (e.g., Silk& Rees 1998; Fabian 1999; King 2003; Croton et al. 2006; Bower et al. 2008; Shabala & Dept. of Physics, Technion, Haifa 32000, Israel; [email protected] M ∼ M ⊙ ) clumps, Elmegreen et al. (2008b)considered feedback by supernovae (SNe), but not by AGN activity. In their model, theSMBH at the center of the galaxy is formed from the merger of the intermediate mass BHsthat reside in each clump (Elmegreen et al. 2008a). The ratio of SMBH to bulge mass M BH /M bulge ∼ − originates in the massive clumps. In the present paper I assume thatthe SMBH forms by accreting mass from the ISM, and that the M BH − M bulge correlation isdetermined by an AGN feedback mechanism. It is determined in the sense that the AGNactivity limits the ISM mass that is eventually converted to stars by expelling it out of thegalaxy. The ejection of large quantities of gas out of the galaxy during its formation wasfound to be a necessary process in recent studies (e.g., Bower et al. 2008).In a previous paper I tried to account for the SMBH − bulge masses correlation witha feedback mechanism based on jets launched by the SMBH. This feedback is based onnarrow jets that are launched by the central SMBH, and expel large amounts of mass tolarge distances. The condition for an efficient expelling process is that the jets do notpenetrate through the inflowing gas, such that they can deposit their energy in the innerregion where the bulge is formed. A relation between the mass accreted by the SMBH andthe mass that is not expelled, and is assumed to form the bulge, was derived (Soker 2009;this derivation is repeated in section 4.3). It was noted that the same mechanism couldoperate in suppressing star formation in CF clusters, making a tight connection between thefeedback in galaxy formation and CF clusters. In the present paper I extend the comparisonbetween the feedback mechanism operating in CF clusters and during galaxy formation. Myfundamental assumption is that the correlation between the SMBH mass and the host galaxybulge mass is determined by an AGN feedback process that expel large quantities of gas outfrom the galaxy.In section 2 and 3 I argue that for the feedback mechanism to be efficient, as requiredfrom my fundamental assumption, most of the gas should pass through the hot phase, unlikethe case in the model studied by Binney (2004, 2005). In section 4 I discuss a model wherethe feedback between the SMBH and star formation in the bulge occurs through a processsimilar to that operating in CFs in clusters of galaxies. A comparison of CF in clusters tothe process at galaxy formation was done in the past (e.g., Croton et al. 2006). In themodel discussed by Binney (2004; 2005), for example, heating by the AGN activity and theejection of large quantities of gas out from the galaxy are crucial processes, as they are inthe present model. However, Binney (2004) proposed a model where the gas that forms star 3 –has never been heated to the virial temperature, while that gas that was heated to the virialtemperature has negligible contribution to star formation. In the model proposed in section4.2, on the other hand, large quantities of gas in the hot phase cool to form stars. Theadvantage in the presence of large amount of mass in the hot phase is that the hot phaseis more susceptible to AGN heating (Binney 2004; Dekel & Birnboim 2006; Cattaneo et al.2006; Hopkins & Elvis 2009). I summarize the proposed model in section 5.
2. PROBLEMS WITH FEEDBACK DRIVEN BY BONDI ACCRETION
I define the Bondi accretion radius by R B = 2 GM BH C s = 65 GM BH µm H kT (1)where M BH is the SMBH mass , C s = (5 kT / µm H ) / is the ISM sound sped, and allother symbols have their usual meaning. The Bondi accretion rate for an adiabatic index of γ = 5 / M Bondi = 0 . πR B ρC S , (2)where ρ is the density at a large distance from the accreting body. The characteristic inflowtime is τ flow ≡ R B C s . (3)(The inflow velocity at R B is ≪ C s , and the practical inflow time scale is longer than τ flow .)A condition for the existence of a Bondi type accretion flow is that the radiative coolingtime is longer than the inflow time τ cool > τ flow . Otherwise the ISM rapidly cools, the valueof C s rapidly decreases, and the assumptions that lead to the Bondi accretion flow in a hotISM break down. Namely, we are in the regime of accreting cold gas, the problems of whichare discussed in section 3. This condition reads32 nkT Λ n e n p > R B C s , (4)where n , n e , and n p are the total, electron, and proton number density, respectively. Condi-tion (4) can be cast in the form5 . µm H kT Λ R B C s & . πR B ρC s . (5)For the cooling function Λ I assume a zero metallicity composition and take the temper-ature range T & K. This gives Λ ( T ) ≃ × − ( T / K) / erg cm s − (Sutherland & 4 –Dopita 1993). For lower temperatures and higher metallicity the conclusions will be strongereven, as cooling time is shorter. Substituting equation (5) into equation (2) with the aboveapproximation for Λ, gives˙ M Bondi < . × − (cid:18) T K (cid:19) / M BH M ⊙ M ⊙ yr − . (6)The actual limit is lower even for three reasons: (1) Some fraction of metals will be presence,increasing the cooling rate. (2) The inflow time should be shorter than the cooling time notonly from the Bondi radius, but also from a distance of several Bondi radii. (3) In galaxiesthat do not sit at the center of groups and clusters, the virial temperature of the gas is < K. For a temperature of T = 2 × K and zero metallicity, the cooling function islower by a factor of 1.6, and the cooling time shorter by a factor of 5 / . ∼ × − M ⊙ yr − .We can compare the limit from the Bondi accretion as given in equation (6) with theEddington limit ˙ M Edd = 2 . × − (cid:16) ǫ . (cid:17) − M BH M ⊙ M ⊙ yr − , (7)where ǫ is the efficiency of converting mass to radiation in the accretion process. We findthat ˙ M Bondi < . M Edd . It seems there is no time for the BH to grow if it is limited by theBondi accretion, as the e-folding time is τ f = M BH ˙ M Bondi > × (cid:18) T K (cid:19) − / yr . (8)To grow by a factor of 1000, the minimum time required is ∼ > T . K ≪ T virial phase, for which the Bondiaccretion radius is much higher. The problem with cold clouds is that they are very dense.It is very hard to remove such clouds from the galaxy. More over, cold clouds are more likelyto form stars. This solution makes it very hard to explain the termination of star formation.This is the subject of the next section. In any case, models that include Bondi accretionconsider the hot T & K phase.(2) Most of the mass is in the hot phase with T ≃ T virial . However, the SMBH is fead by coldclumps that are embedded in the hot ISM. The clumps as a group contain a small fractionof the total ISM mass. This solution to maintain feedback heating in cluster CFs is termedthe cold feedback mechanism (Pizzolato & Soker 2005, 2010; Soker 2006, 2008b). In section 5 –4 I compare this mechanism as it operates in galaxy formation at high redshifts to its rolein low redshift cluster CFs.
3. PROBLEMS IN MAINTAINING FEEDBACK WITH MASSIVE COLDISM
It is possible that most of the ISM mass resides in cold ( ∼ K) clouds, ratherthan being close to the virial temperature and in a hydrostatic equilibrium. The problemin this case is that the AGN feedback efficiency is too low, because it is extremely difficultto expel dense clouds from the inner regions of the galaxy. In a recent attempt Hopkins &Elvis (2009) note that AGN feedback works more efficiently on the hot phase, and consider achain of processes to overcome this problem. In their model the clouds contain ∼
90% of themass, while ∼
10% of the ISM mass is in the hot phase at hydrostatic equilibrium. A shockpropagating through the hot phase causes the clouds to expand, such that radiation pressurefrom the AGN is more efficient in expelling the bloated clouds. Even in this more efficientscenario, the mass expelled in their model is not sufficient for the feedback mechanism studiedhere. Hopkins & Elvis (2009) take the bulge mass to be 10 M ⊙ , while the ISM mass is only10 M ⊙ . After a time of t = 10 yr from the start of their calculation, the mass expelledin their model is ∆ M e = 5 × M ⊙ . During the same period, for an efficiency of 10%, theSMBH mass has grown by ∆ M SMBH ≃ M ⊙ . This ratio of ∆ M e / ∆ M SMBH ≃
50, is lowerby more than an order of magnitude from the one required in feedback models that accountfor the bulge to SMBH masses ratio, if the feedback is to determine the final mass of thebulge. Had they include radiative cooling, the efficiency of expelling the ISM would be lowereven.The conclusion is that for an AGN feedback to work, most of the gas in the inner regions, r .
10 kpc ∼ . v , must reside in the hot phase; R v is the virial radius.Although the AGN feedback requires the inner regions to be mostly in the hot phase,the gas feeding the galaxy at larger regions can be cold. In the cosmological cold streamsmodel (Dekel et al. 2009a, b and references therein) three cold, low entropy, streams of gaspenetrate the virial radius without being shocked, and reach the central region. Shocks nearthe center are not resolved in the numerical simulations presented by Dekel et al. (2009a),as their resolution is 1 . m ∼ M ⊙ yr − rad .At a distance r from the center the post-shock total (protons, electrons and nuclei) number 6 –density is n ≃ (cid:18) ˙ m M ⊙ yr − rad (cid:19) (cid:16) v
300 km s − (cid:17) − (cid:18) r (cid:19) − cm − , (9)where v is the preshock inflow velocity, scaled with the inflow velocity at r ∼ −
10 kpcfrom the center according to Dekel et al. (2009a). The postshock temperature is ∼ K.For a temperature of ∼ K, and zero metallicity, the cooling function is Λ ≃ − erg cm s − (Sutherland & Dopita 1993). The cooling time of the post-shock gasis τ cs ≃ × (cid:18) ˙ m M ⊙ yr − rad (cid:19) − (cid:16) v
300 km s − (cid:17) / (cid:18) r (cid:19) − (cid:18) Λ − erg cm s − (cid:19) yr , (10)where the dependence of the post shock temperature on velocity has been used. The halfopening angle of a stream is ∼ − ◦ ; I scale with 12 . ◦ . The pressure of the post-shockgas is larger than that of its surroundings, and it expands to the sides in a typical time of τ f ≃ r sin 12 . ◦ v = 3 . × (cid:16) v
300 km s − (cid:17) − (cid:18) r (cid:19) yr . (11)At r ≃ τ cs ≃ τ f . Heating by the AGN will make the formation of a shock wave morelikely. Cantalupo (2010) showed that ionization by star forming regions can remove coolingagents (ions) from the gas and by that efficiently reduces radiative cooling and transforms acold mode accretion into a hot one. The effect is larger even when radiation from the AGNis considered. The effect of removing cooling ions comes in addition to the heating discussednext. Let the gas mass in the inner region be about equal to the stellar mass ∼ M ⊙ .With a total number density of n ≃ − the mass resides within a radius of ∼
10 kpc.To maintain this mass that has a radiative cooling time of ∼ × yr at a temperature of T ≃ K, requires a heating power of ˙ E H ≃ × erg s − . With an efficiency of 10% inconverting mass to energy, the accretion rate onto the SMBH is ∼ . M ⊙ yr − . In 10 yr theSMBH gains a mass of ∼ M ⊙ . In reality, the amount of ISM mass will be lower, as partof it is continuously forming stars, while some of the incoming gas is expelled back to largedistances. This crude calculation and the reduction in radiative cooling rate as discussedby Cantalupo (2010) show that heating by the AGN can maintain a shock wave at a radiusof r & R v ≃ −
100 kpc, as in the simulations presented by Dekel et al. (2009a).However, the model here does require that the streams eventually are shocked at r & . R V , 7 –and a hot pseudo static atmosphere is formed. The presence of hot static atmosphere aroundan already grown SMBH was assumed before, e.g., Short & Thomas (2009).
4. A COOLING FLOW PHASE4.1. The cold feedback mechanism in low redshift clusters
In the cold-feedback model for clusters of galaxies (Pizzolato & Soker 2005, 2010; Soker2006; 2008a) mass accreted by the central SMBH originates in non-linear over-dense blobsof gas residing in an extended region of r ∼ −
50 kpc; these blobs are originally hot, butthen cool faster than their environment and sink toward the center (see also Revaz et al.2008). The mass accretion rate by the central black hole is determined by the cooling timeof the ICM, the entropy profile, and the presence of inhomogeneities (Soker 2006). Mostimportant, the ICM entropy profile must be shallow for the blobs to reach the center as coldblobs. Wilman et al. (2008) suggest that the behavior and properties of the cold clumpsthey observe in the cluster A1664 support the cold feedback mechanism.The cold feedback mechanism in clusters has the following consequences.(1) Cooling flows do exist, but at moderate mass cooling rates: the moderate CF model (Soker et al. 2001). Indeed, in many CF clusters the heating cannot completely offset cool-ing (e.g., Clarke et al. 2004; Hicks & Mushotzky 2005; Bregman et al. 2006; Salome et al.2008; Hudson et al. 2010) and some gas cools to low temperatures and flows inward (e.g.,Peterson & Fabian 2006) and forms stars (Wise et al. 2004; McNamara et al. 2004). Starformation is prominent when the radiative cooling time of the hot gas is short (Rafferty etal. 2008). These observations suggest that indeed cooling of gas from the hot phase to lowtemperatures does take place, including star formation. It is important to note that starformation and the feeding of the SMBH occur simultaneously.(2) The cold feedback mechanism explains why real clusters depart from an ‘ideal’ feedbackloop that is 100% efficient in suppressing cooling and star formation. Simply, the feedbackrequires that non-negligible quantities of mass cool to low temperatures. Part of the massfalls to small radii. Part of this mass forms star, another part is ejected back, and a smallfraction is accreted by the central SMBH.(3) Part (likely most) of the inflowing cold gas is ejected back from the very inner region.This is done by the original jets blown by the SMBH (Soker 2008b). The ejection of this gasis done in a slow massive wide (SMW) bipolar outflow, which are actually two jets. Thebasic mechanism is that the jet does not puncture a hole in the ICM, but rather depositsits energy in the inner region. Wide enough jets deposit their energy in the inner regionsrather than puncturing a hole and expanding to large distances. Rapidly precessing jets or a 8 –relative motion of the medium also prevent such a puncturing. In the case of the formationof low density bubbles in the ICM, it has been shown that in addition to SMW jets, precess-ing jets (Sternberg & Soker 2008a; Falceta-Goncalves et al. 2010) and a relative motion ofthe medium (Morsony et al. 2010) also lead to the inflation of bubbles close to the center.The most striking example of a SMW outflow is presented in the seminal work of Moe et al.(2009; also Dunn et al. 2009). By conducting a thorough analysis, Moe et al. (2009) findthe outflow from the quasar SDSS J0838+2955 to have a velocity of ∼ − , and amass outflow rate of ∼ M ⊙ yr − , assuming a covering fraction of δ ≃ .
2. The coolingand ejection back to the ISM of large quantities of mass, make the feedback process not onlyof energy (heating), but also of mass (Pizzolato & Soker 2005).(4) Such SMW jets can inflate the ‘fat’ bubbles that are observed in many CFs, in clusters,groups of galaxies, and in elliptical galaxies (Sternberg et al. 2007; Sternberg & Soker 2008a,b). The same holds for precessing jets (Sternberg & Soker 2008a; Falceta-Goncalves et al.2010) or a relative motion of the medium (Morsony et al. 2010).(5) The same mechanism that form SMW jets in CF clusters and galaxies, can expel largequantities of mass during galaxy formation, and might explain the SMBH-bulge mass corre-lation (Soker 2009; section 4.3 here).Following the problems with maintaining AGN feedback at galaxy formation by theBondi accretion (sec. 2), and the discussion in section 3, I turn to elaborate on the possibilitythat the feedback at galaxy formation is similar to that in CF clusters.
In their study of galaxy formation with AGN feedback, Croton et al. (2006) considerthe formation of an extended (up to the virial radius of the dark halo) hot atmosphere, andthat gas from the central region of this atmosphere might be accreted onto a central objectthrough a CF. In this section I take upon this, and consider the moderate CF model witha cold feedback. Namely, the gas that feeds the SMBH originates in an extended region ascooling blobs, and not as a Bondi type accretion. Heating of the inner regions by the AGN isa significant process, and the AGN activity level is regulated by the accretion of cold blobsfrom an extended region.I differ from Croton et al. (2006) in a crucial manner. In their model the main growthof the SMBH occurs in the ‘quasar mode’ at high redshifts ( z > z > z ∼ − ∼ . −
10% of its final mass, and the bulge/galaxyis already in the formation process. However, most of the stars in the bulge (or ellipticalgalaxy) have yet to be formed, and so does the SMBH mass. Cold streams that feed thegalaxy (Dekel et al. 2009a, b) are assumed to be shocked at r &
10 kpc, such that most ofthe mass resides in the hot phase. The cooling time of the hot phase is short, and thermalinstabilities lead to the formation of cool blobs that fall inward. These cool blobs are notmuch cooler than their surrounding for most of their journey inward (Pizzolato & Soker2010). Namely, they are not as cold as the gas discussed in section 3. The cool blobs arenot much denser than their hot surroundings, and they can be expel relatively efficiently bythe jets launched by the AGN when AGN activity is high (see section 4.3). A fraction ofthese blobs feed the SMBH. Heating by the SMBH activity, mainly by jets, facilitated theformation of this structure. The jets launched by the accreting SMBH not only heat thegas, but as is the case in the cold feedback mechanism (Pizzolato & Soker 2005), the jetsaccelerate large quantities of gas outward. A structure similar to that in cluster CF at lowredshifts has been formed. Table 1 compares the properties of the two types of the moderateCF models.There are, however, two prominent qualitative differences between the proposed CFmodel at galaxy formation and that in CF in clusters of galaxies.(1) In the case of galaxy formation the cooling time in the inner ∼ (1) Property (2 , Low z Clusters Galaxy formationCentral e − density ( n ce ) 0 . −
10 cm − Central temperature ( T c ) 3 × K 10 KSystem age ( τ age ) 10 yr 10 yrCentral cooling time (4) ( τ c ) 5 × yr 10 yrCooling radius (5) ( r c ) 100 kpc 30 kpcDynamical time at0 . r c ( τ d ) 10 yr 5 × yr τ c /τ d
500 0 . (6) ( ˙ M x ) 10 − M ⊙ yr − − M ⊙ yr − Star formation rate ( ˙ M ∗ ) 0 − . M x ∼ . − . M x Source of gas feeding Cold blobs from an Cold blobs from anthe SMBH extended region. extended region+ inner supersonic inflow.Fate of most gas expelled Inflating large Expelled from the galaxy.from the inner region low-density bubbles(a) Depart from 100% (a) Most of the ISM isefficiency in suppressing susceptible to SMBH jets.Results and implications star formation. (b) Expelling huge amountsof the cold feedback (b) Shallow entropy profile. of mass from the galaxy.mechanism (c) Massive outflows that (c) Can account for thecan inflate ’fat bubbles’. M BH − M bulge correlation(d) Can operate without (Soker 2009; § . § Table 2: (1) In the
Moderate CF model heating is important, but cooling of gas to low temperatures doesoccur, although at a much lower rate than that expected if no heating exists.(2) The values are crude, and most are given to an order of magnitude.(3) Some of the time scales are calculated by using equations from section 2.(4) The cooling time in galaxy formation is lower even, as a zero metallicity was assumed here, but somemetals will be present at an age of ∼ yr.(5) The cooling radius r c is the radius at which the radiative cooling time (no heating included) equals theage of the system.(6) Raw cooling rate is the mass cooling rate if no heating was presence.
11 –(2) In clusters, the region where cooling takes place is ∼ −
50 kpc, or about a fraction ∼ . − . & times the accretion to theSMBH, a ratio that might account for the SMBH − bulge masses correlation (Soker 2009;section 4.3 below).It is interesting to compare the model proposed here with that discussed by Binney(2004). In both models heating, aided by reduction in radiative cooling rate (Cantalupo2010), by the AGN activity is crucial, and in both models the ejection of large quantitiesof gas out from the galaxy takes place. The main difference is that Binney (2004) attributesall star formation to gas that was never heated to about the virial temperature. Instead, Isuggest that a large fraction of the gas is shocked to the virial temperature, and is furtherheated by the AGN to prolong its hot phase. Most of the gas stays in the hot phase for atime longer than the dynamical time, and forms a (pseudo) static medium. Large quantitiesof this gas later cool, and form stars, as observed in cluster CF (but at lower efficiency). Asthe hot gas is much more susceptible to AGN activity, it allows for a feedback process fromthe AGN to work, and determine the masses ratio of the SMBH and bulge (Soker 2009). In this section I briefly summarize the derivation of the correlation between the SMBHmass and the host galaxy bulge mass (Soker 2009). I will not repeat all steps and will notexplain all assumptions, as they are in that paper. I will, however, make some modificationsto account for the present proposal of a CF phase at galaxy formation, and to incorporatethe new results of Soker & Meiron (2010).The basic assumptions are as follows.(1) The feedback mechanism is driven by jets.(2) The properties of jets launched by SMBH have some universal properties. As is shownin a new paper (Soker & Meiron 2010), the basic property is the the ratio of the totalmomentum discharge in the jets to the quantity ˙ M acc c : ǫ p ≡ ˙ M f v f / ( ˙ M acc c ), where ˙ M acc is 12 –the accretion rate onto the SMBH, ˙ M f is the mass flow rate into the two jets, and v f is thejets’ speed. Soker & Meiron (2010) find from statistical analysis of tens of galaxies that inthe feedback model for the correlation ǫ p = 0 . ± . v rel ≃ σ .(5) The cooling surrounding mass M s that resides at a typical distance r s and having adensity ρ s (see below), is flowing inward at a velocity of ∼ σ . Thus, ˙ M s ≃ πr s σρ s , andit is resupplied on a time scale of ∼ r s /σ . This mass will mainly form stars if it is notexpelled by the jets. This assumption is in the heart of the proposed CF phase at galaxyformation. Namely, that the inflowing gas, be it a cooling flow that reaches a fast speed at r ∼ M s ≫ ˙ M acc , as only a small fraction of the inflowing gas at scales of ∼ . −
10 kpc is accreted by the SMBH.If the jets penetrate through the surrounding gas they will be collimated by that gas,and two narrow collimated fast jets will be formed, similar to the flow structure in thesimulations of Sutherland & Bicknell (2007). By fast it is understood that the jet’s velocityis not much below its original velocity. If, on the other hand, the jets cannot penetrate thesurrounding gas they will accelerate the surrounding gas and form SMW (slow-massive-wide)outflow (Soker 2008).I now derive (Soker 2009) the conditions for the jets not to penetrate the surroundinggas, but rather form a SMW outflow. Let the jets from the inner disk zone have a massoutflow rate in both directions of ˙ M f , a velocity v f , and let the two jets cover a solid angleof 4 πδ (on both sides of the disk together). The density of the outflow at radius r is ρ f = ˙ M f πδr v f . (12)Let the jets encounter the surrounding gas residing within a distance r s and having a typicaldensity ρ s ; this is the inflowing cooling gas, that if it is not expelled will form stars. Thehead of each jet proceeds at a speed v h given by the balance of pressures on its two sides. 13 –Assuming supersonic motion this equality reads ρ s v h = ρ f ( v f − v h ) , which can be solved for v h v f v h − πδr s v f ρ s ˙ M f ! / ≃ δ ˙ M s v f ˙ M f σ ! / = 1225 ˙ M s / ˙ M f ! / (cid:18) δ . (cid:19) / (cid:16) v f c (cid:17) / (cid:16) σ
200 km s − (cid:17) − / . (13)where in the second equality the mass inflow rate ˙ M s ≃ πρ s σr s (by assumption 5), hasbeen substituted. The time required for the jets to cross the surrounding gas and break outof it is given by t p ≃ r s v h ≃ r s v f δ ˙ M s v f ˙ M f σ ! / = 4 × (cid:18) r s (cid:19) yr , (14)where in the last equality the same values as in equation (13) have been used.If there are no changes in the relative geometry of the SMBH and inflowing mass, thejets will rapidly penetrate the surrounding gas and expand to large distances. In this casethe jets will not deposit their energy in the inflowing gas. For an efficient deposition ofenergy to the inflowing gas, we require that there will be a relative transverse (azimuthal)motion between the SMBH and the inflowing gas, such that the jets continuously encounterfresh mass. The relevant time is the time that the transverse motion of the jet crosses itwidth τ s ≡ D j /v rel ≃ D j /σ , as by our assumption 4 the relative velocity is v rel ≃ σ . Thewidth of the jet at a distance r s from its source is D j = 2 r s sin α , where α is the half openingangle of the jet. For a narrow jet sin α ≃ α ≃ (2 δ ) / , and τ s = 2(2 δ ) / r s v rel = 4 . × (cid:18) r s (cid:19) (cid:16) v rel
200 km s − (cid:17) − (cid:18) δ . (cid:19) yr . (15)The demand for efficient energy deposition, τ s . t p , reads then˙ M s ˙ M f & v f σv . (16)This result can be understood as follows. The ratio v f σ/v comes from the ratio of theram pressure of the narrow jet to that of the ambient gas which disturbs the jet, and fromthe relative transverse motion of the jet and the abient gas. The number 8 comes from thegeometry of a narrow jet with a relative transverse velocity to that of the ambient gas. Usingthe definition ǫ p ≡ ˙ M f v f / ( ˙ M acc c ) from assumption 2, we derive˙ M s ˙ M acc & ǫ p σcv = 480 (cid:16) ǫ p . (cid:17) (cid:16) σ
200 km s − (cid:17) − (cid:18) σv rel (cid:19) . (17) 14 –Again, it is expected that in its formation phase the galaxy will not be fully relaxed, andthat the relative transverse velocity of the AGN and the inflowing gas will be of the orderof the stellar dispersion velocity, i.e., v rel ≃ σ The accretion rate ˙ M acc is the accretion rate onto the SMBH, and the inflow rate of thesurrounding gas is assumed to form stars in the bulge (if it is not expelled by the jets). Ifthe inflow rate is above the value given by equation (17), the deposition of energy by thejets is efficient enough to expel the mass back to large distances and heat it (Soker 2008b).The interaction of the (narrow or wide) jets blown by the SMBH with the inflowing gas willform a wide outflow (SMW jets), that will expel more of the hot gas that is vulnerable tothe jets. Namely, the jets blown by the SMBH will not allow the bulge to form stars at arate larger than the value of ˙ M s given by equation (17). Following Soker (2009) then, theSMBH to bulge mass ratio is about equal to ˙ M acc / ˙ M s . Equation (17) yields M BH ≃ . M bulge (cid:16) ǫ p . (cid:17) − (cid:16) σ
200 km s − (cid:17) (cid:18) σv rel (cid:19) − . (18)The last equation closes the feedback cycle, in showing that a correlation can be driven byjets blown by the SMBH into the hot ISM. A key issue is that the medium is in the hotphase such that its density is not too high, and therefore it is vulnerable to the action of thejets. This hot phase feeds the SMBH via the process of a moderate cooling flow.
5. SUMMARY
Results from recent years show that the process of galaxy formation requires non-gravitational energy source not only to heat the gas, but also to expel large quantitiesof gas out from the galaxy (Bower et al. 2008). To efficiently eject the ISM from the galaxyby AGN activity the gas must be in the hot phase, namely, its temperature must be aboutthe virial temperature (e.g., Hopkins & Elvis 2009). The conclusion from these studies isthat most of the ISM during galaxy formation must evolves through the hot phase. This gashas a short cooling time, and a cooling flow (CF) is formed in the still-forming galaxy.In the present paper I assumed that the M BH − M bulge correlation is determined by anAGN feedback mechanism that operates during a cooling flow phase at galaxy formation(section 4.3). It is determined by the sense that the AGN activity limits the ISM mass thatis eventually converted to stars by expelling it out of the galaxy. I showed that the Bondiaccretion cannot operate during that phase (section 2), and discussed the requirement thatthe ISM be in the hot phase (section 3). As the radiative cooling time of the hot phase isrelatively short (eq. 10), to maintain a hot phase the infalling gas should be shocked at a 15 –radius of R & cold feedbackmechanism in clusters of galaxies (Pizzolato & Soker 2005, 2010; Soker 2006, 2008b), coldblobs are falling from an extended region. The second channel is a cold supersonic inflowin the inner . moderate CF model (Soker et al. 2001, Soker & David 2003). In section 4.2 the moderate CF model in clustersof galaxies and at galaxy formation are compared, and summarized in Table 1.The moderate CF model proposed here at galaxy formation can be applicable to thesample of obscured AGN studied by Brusa et al. (2009). The median value of L AGN /L Edd in their sample is ∼ − ∼ yr, similar to thegrowth time of the stellar population due to star formation. I propose that a moderate CFexists in these galaxies during the high star formation rate.I thank an anonymous referee for helpful comments. This research was supported bythe Asher Fund for Space Research at the Technion, and the Israel Science foundation. REFERENCES
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