arXiv:astro-ph/0307084v1 4 Jul 2003 o.Nt .Ato.Soc. Astron. R. Not. Mon. (1) uiLevin Yuri in waves band. gravitational in LISA and holes discs, black AGN and self-gravitating stars massive of Formation (2) rne oebr2018 November 1 printed bandfo bevtoso tla eoiisi u gal our in velocities stellar of observations from obtained Schlosman likely 1980, AGN is an Sunyaev 1987). of It Begelman and parts and hole. outer (Kolykhalov the black disc understanding in central occur in accretion the will issue formation to star major delivered that a is is gas self-gravit discs how timescale; accretion dynamical AGN a of on unsta fragmentation are to discs Self-gravitating ble 2002). 1999, Goodman Kumar and 1999, 1987, Kolykhalov Jure Begelman 1978, and (Paczynski Schlosman 1980, hole away Sunyaev parsec black a central of a fraction a from beyond self- extend become they must if discs gravitating accretion AGN Theoretical the AGNs. that show in models holes black disc supermassive accretion around self-gravitating form that believed widely is It INTRODUCTION 1 c 0 apelHl,Uiest fClfri,Bree,Cal Berkeley, California, of University Hall, Campbell 601 00RAS 0000 aainIsiuefrTertclAtohsc,Univers Astrophysics, Theoretical for Institute Canadian eety oespotn vdnefrti rcs was process this for evidence supporting some Recently, 1 , 2 000 0–0 00)Pitd1Nvme 08(NL (MN 2018 November 1 Printed (0000) 000–000 , egdcupi e est e udeso oa as h bigg The mass. solar of hundreds few up a evolution. few-tens-solar-m the their produce to clumps; which of tens larger stars, end few massive into form a th merge and is collapse of and clump density disc that merged argue surface remaining We su a the clumps. the a dense from once into around fragments turbulen material that disc forming by edge find the transported outer disc value, We is critical the a self-gravity. momentum at angular of disc’s the form dynamics the which stars the in massive hole, analyze black which We in disc. scenario accretion a propose We ABSTRACT ii,te h xr etn sisffiin osaiietedisc. the be stabilize is to holes insufficient embedded is the mo heating onto the extra accretion of st the of range then may rate interesting limit, holes the an if black for contrast, embedded instability By Toomre the t the onto at against accrete accretion disc can from holes black feedback embedded the the If holes. black embedded fgaiainlwvssol eacmaidb eetbechan detectable r a accretion engine. by and central accompanied masses the be of hole luminosity should black waves plausible t gravitational from some of distinct detect for is be LISA Also, mergers will the the by these scenarios. mergers from such of observable limit, signal parameters dynamics gravitational-wave be Eddington of the the should range the significantly reasonable waves to a alter gravitational for comparable not the rate will wav and gravitational disc a merger, of accretion dis at burst a the accreting a of from produce is Merger will hole. hole hole black black black central central the the towards with dragged be and disc ecntutamdlo h G icwihicue xr etsour heat extra includes which disc AGN the of model a construct We h meddbakhlswl neatgaiainlywt h massive the with gravitationally interact will holes black embedded The can s t fTrno 0S.Gog tet oot,O 5 H,Can 3H8, M5S ON Toronto, Street, George St. 60 Toronto, of ity ac- fri,94720 ifornia, y - tr hc ol uvv vnatraceinot h hole and the gas onto of accretion ring after even a survive hole would black which the stars fragmenta- around “The produce would abstract: tion Sunyaev’s and qu Kolykhalov (we galaxies from other the in for was discs 1980 accretion in what circum-black-hole Sunyaev like and Kolykhalov much of by remant disc, proposed or- a already accretion is nearly self-gravitating disc is stellar dense discs the a disc two that the the argued have of of than LB disc orientation thickness thogonal. relative second and the radius a Genzel LB; larger for (2003). in a evidence al. has found et. have which also Ott who stars have from (2003), (2003) LB’s data al. al. hole. recent et. et. black most SgrA* Genzel the have the by used around and confirmed disc (2000), was thin finding al. a in et. move Genzel ple of that data found velocity analyzed have LB) 3-D (2003, the Beloborodov and Levin center. tic ∼ 0preto asv tr rmtesam- the from stars massive of percent 80 A T E tl l v1.4) file style X s lc oe tthe at holes black ass dmnhy n that and monthly, ed a fohrmerger other of hat eBnirt,then rate, Bondi he o h Eddington the low h lmsaccrete clumps the -onbakhole black c-born ei h optical the in ge icecesa exceeds disc e blz h AGN the abilize eagethat argue We . s ftecentral the If es. e parameters. del ts h burst the ates, eidcdby induced ce fa AGN an of s nlya of year final e rmthe from ces e asof mass per permassive s clumps est accretion a drag gas ada ote 2 Levin has ceased”.) The idea for star formation in the disc around higher than the Eddington limit, than the AGN disc cannot SgrA* was originally proposed by Morris et. al. (1999); in be stabilized against its own selfgravity, and one would need their scenario the disc’s self-gravity does not play a central some other source of heating to make the disc stable. role. Instead, the compression of the disc material is achieved We then argue (Section V) that the disc-born black hole by radiation and wind pressure from the central engine. Mor- interacts gravitationally with the accretion disc, and mi- ris et. al. (1999) have identified the circumnuclear ring as the grates inward on the timescale of 107 years. The merger of ∼ most likely source of supply for the disc material. However, the migrating black hole with the central black hole will pro- the circumnuclear ring is strongly misaligned with the disc duce gravitational waves. We show that for a broad range found by LB and by Genzel et. al. (2003). Thus, it seems of AGN accretion rates the final inspiral is unaffected by more likely that the gas in the disc would come from a dis- gas drag, and therefore the gravitational-wave signal should rupted molecular cloud on a plunging trajectory (Sanders be detectable by LISA. The rate of these mergers is uncer- 1998). tain, but if even a fraction of a percent of the disc mass is A disc of stars could also be the remnant of a tidally converted into black holes which later merge with the cen- disrupted young stellar cluster (Gerhard 2001, Portegiese tral black hole, then LISA should detect monthly a signal Zwart, McMillan, and Gerhard 2003, McMillan and Porte- from such a merger. The final inspiral may occur close to giese Zwart 2003). This scenario becomes more credible if the equatorial plane of the central supermassive hole and the dense cluster core contains an intermediate-mass black is likely to follow a quasi-circular orbit, which would make hole (Hansen and Milosavlevic, 2003). The gravitational pull the gravitational-wave signal distinct from those in other as- from the latter would allow the core to stay intact until it trophysical merger scenarios. If the disc-born black hole is comes within 0.1pc from the SgrA*, where the massive stars sufficiently massive, it will disrupt accretion flow in the disc are observed to reside. during the final year of its inpiral, thus making an optical Both the disc fragmentation and disrupted cluster sce- counterpart to the gravitational-wave signal. narios have implications for the dynamics of the galactic nuclei and may provide channels for producing bursts of gravitational waves detectable by LISA. In this paper we investigate the consequences of disc fragmentation. 2 PHYSICS OF AN ACCRETING The dynamics of the fragmenting disc is strongly af- SELF-GRAVITATING DISC fected by the feedback energy input from the starburst. The importance of self-gravitating accretion discs in astro- Colliln and Zahn (1999) have conjectured that the feed- physics has long been understood (Pazynsky 1978, Lin and back from this star formation will prevent the accretion disc Pringle 1987). It was conjectured that the turbulence gen- from becoming strongly self-gravitating. However, Goodman erated by the gravitational (Toomre) instability may act (2003) has used general energy arguments to show that the as a source of viscosity in the disc. This viscosity would feedback from star formation is insufficient to prevent an both drive accretion and keep the disc hot; the latter would AGN disc with the near-Eddington accretion rate from be- act as a negative feedback for the Toomre instability and coming strongly self-gravitating at a distance of pc from ∼ would keep the disc only marginally unstable. Recently, the central black hole. This is distinct from the case of galac- there has been big progress in our understanding of the self- tic gas discs, for which there is evidence that the feedback gravitating discs, due to a range of new and sophisticated from star formation protects them from their self-gravitaty. numerical simulations (Gammie 2001, Mayer et. al. 2002, In this paper we concentrate on the physics of the self- Rice et. al. 2003). In our analysis, we shall rely extensively gravitating disc and make a semi-analytical estimate of the on these recent numerical results. possible mass range of stars formed in such discs (Sections Consider an accretion disc which is supplied by a gas II and III). Our principal conclusion is that the stars can be infall. This situation may arise when a merger or some other very massive, up to hundreds of solar masses. These massive major event in a galaxy delivers gas to the proximity of a stars evolve quickly and produce stellar-mass black holes as supermassive black hole residing in the galactic bulge. Let the end product of their evolution. Σ(r) be the surface density of the disc. We follow the evolu- Goodman (2003) has mentioned the possibility that the tion of the disc as Σ(r) gradually increases due to the infall. AGN disc is stabilized by feedback from accretion onto black We begin by assuming that initially there is no viscosity ⋆ holes embedded in the disc . In Section IV, we investigate mechanism, like Magneto-Rotational Instability (MRI), to Goodman’s conjecture and study whether the feedback from transport the angular momentum and keep the disc hot . accretion onto these black holes may indeed stabilize a disc † This assumption is valid when the ionization fraction of the around a supermassive black hole, thus helping to fuel bright gas in the disc is low, i.e. when the gas is far enough from AGNs. We find that if the embedded black holes are accret- a central source (about a thousand Schwarzchild radii from ing at a super-Eddington rate set by the Bondi-Hoyle for- the supermassive black hole). We also, for the time being, mula, then the extra heating they provide may be able to neglect irradiation from the central source; this may be a stabilize the AGN disc. However, if their acretion rate is no good assumption if a disrupted molecular cloud forms a disc but the accretion onto the hole has not yet begun. As will ⋆ Previously, Kolykhalov and Sunyaev (1980) have suggested that the x-rays produced by accretion onto the embedded black holes would be absorbed by the disc and reradiated in the in- † When the disc begins to fragment, the viscousity due to self- frared. The luminocity and spectra of the infrared radiation of gravity-driven shocks exceeds the one due to MRI; see below. the internally heated AGN disc was recently computed by Sirko Therefore, while computing the disc parameters at fragmentation, and Goodman (2003). it is reasonable to ignore MRI
c 0000 RAS, MNRAS 000, 000–000 On the formation of black holes 3 be discussed below, irradiation is important for some regions use the analytical fit to the opacity (in cm2/gm) from the in the AGN discs we are considering. However, as shown in appendix of Bell and Lin, 1994. the following subsection, inclusion of irradiation or other 2 source of heating will only strengthen the case for formation κ = 0.0002 TK for T < 166K, −16× −7 of massive stars. κ = 2 10 TK for 166K
2 8 We now find the critical surface density and midplane 4 4 2mp πGΣ Teff T f(τ)= f(τ) ., (5) temperature as a function of Ω. We use Eq. (3) to express Σ ∼ kB Ω as a function of T and Ω, then substitute this function into where τ = κΣ/2 is the optical depth of the disc; here κ(T ) Eqs. (6) and (8), and set α = αcrit. After simple algebra, we is the opacity of the disc. The function f(τ) = τ for an obtain optically thin disc, and f(τ) = 1/τ for an optically thick 3 disc. We combine these two cases in our model by taking Ω + pΩ= q, (10) τ where f(τ)= . (6) τ 2 + 1 2 πG m p = 2 p , We have used Eq. (3) in the last step of Eq. (5). The flux κ(Tcrit) kB Tcrit from the disc is powered by the accretion energy: 2 32σ (πG) 2 2 3 q = (mp/kB ) Tcrit. (11) 3 2 9 2 9 2 Σ 9α κ(T ) F = Ω M˙ = αΩc Σ= α(πG) . (7) crit crit 8π 8 s 8 Ω There is an analytical solution to Eq. (10): Here α parametrizes viscous dissipation due to self-gravity (Gammie 2001); we have used Eq. (2) in the last step. Using Ω= w p/(3w), (12) − Eqs (4), (7), and (5), we can express the viscosity parameter where α as 2 3 1/2 1/3 8σ 2m 4 Σ5 w = q/2+[(q/2) +(p/3) ] . (13) α = p (πG)6f(τ) . (8) { } 9 k Ω7 B In Figure 1 we make a plot of Tcrit as a function of the It is very important to emphasize that in this model α is only orbital period, for concreteness we set αcrit = 0.3. The criti- a function of Σ and Ω: the temperature in the midplane is cally self-gravitating disc is optically thin if the second term determined by Eq. (3), and this temperature sets the opacity of the LHS of Eq. (10) is dominant, and optically thick oth- which in turn determines the optical depth τ = κΣ/2. The erwise. This can be expressed as a condition on the critical opacity in the range of temperatures and densities of interest temperature: the disc is optically thin if to us is set by light scattering off ice grains and, in some 2/15 Tcrit < 14Kα , (14) cases, by scattering off metal dust. The relevant regimes crit are worked out in the literature on protoplanetary discs; we and optically thick for higher critical temperatures. The an-
c 0000 RAS, MNRAS 000, 000–000 4 Levin gular frequency above which the critically unfragmented disc Here, M is the black-hole mass, and r and h are the radius becomes optically thick is and the scaleheight of the disc. Thus, even if the clumps do
−11 −1 not merge with each other and stop their growth at the gap- Ωtransition 16.3 10 sec . (15) ≃ × opening mass, the mass of the formed stars will be biased We use Eqs. (2) and (3) to find the critical surface den- towards the high-mass end. In the next section we discuss sity Σcrit, which is plotted in Fig. 2, and the scaleheight the effects of clump mergers and the mass range of stars hcrit = cs/Ω of a marginally fragmenting disc. The Toomre born after the disc fragments. ¯ 2 mass Mcl = Σcrithcrit is the mass scale of the first clumps which form in the first stage of fragmentation. In Fig. 3, we plot the Toomre mass of the critically fragmenting disc as a 3 EVOLUTION OF THE FRAGMENTED DISC function of the orbital period. The value of M¯ cl is not large enough for the initial clump Gammie’s simulations show that once the disc fragments, to open a gap in the accretion disc. The newly-born clump the clumps merge and form significantly larger objects. In will therefore accrete from the disc. The Bondi-Hoyle esti- fact, his razor-thin shearing box turned into a single gas mate of the accretion rate gives M˙ cl ΩM¯ cl, i.e. we ex- lump at the end of his simulation. ∼ pect the mass of the new clump to grow on the dynamical For merger to be possible, the clumps should not col- timescale until it becomes large enough to open a gap in lapse into individual stars before they can coalesce with each the gas disc. The upper limit M˜ cl of this value is the mass other. Let’s check if this is the case. which opens a gap in the original gas disc with Σ = Σcrit Consider a spherical nonrotating clump of radius R and just before it fragments: mass Mcl. First, assume that the clump is optically thin. The
1/2 1/2 energy radiated from the clump per unit time is M˜ cl M¯ cl (40παcrit) (r/hcrit) ; (16) ≃ 4 2 4 Wcool σT R κ(T )Σ MclσT κ(T ). (19) see Eq. (4) of Lin and Papaloizou (1986). Once the gas is ∼ ∼ depleted from the disc, we expect the initial distribution of This radiated power cannot exceed the clumps gravitational 1.5 2.5 −2.5 the clump masses to be concentrated between M¯ cl and M˜ cl. binding energy released in free-fall time, G M R . To- The clump masses will evolve when clumps begin to merge gether with κ(T ) T 2 [since icegrains dominate opacity for ∝ with each other; this is addressed in section III. the optically-thin marginally fragmenting disc–see Eq. (14)], this condition implies that 2.0.0.1 Effect of irradiation and other sources of −5/12 T < T˜ = T0R , (20) heating . So far in determining the structure of the self- gravitating disc, we have neglected external or internal heat- where T0 is a constant for the collapsing optically thin ing of the disc. This is certainly a poor approximation in clump. The temperature T˜ in Eq. (20) is less than the virial −1 many cases. Irradiation from AGN or surrounding stars, or temperature, which scales as R . Therefore, after the col- feedback from star formation inside the disc can be the dom- lapse commenses, the clump is not virialized while it is opti- inant source of heating of the outer parts of AGN discs (eg, cally thin. The temperature cannot be much smaller than T˜, Shlosman and Begelman 1987, 1989). For example, iradia- since otherwise the cooling rate would be much smaller than tion from circumnuclear stars will keep the disc temperature the rate of release of the gravitational binding energy, and at a few tens of Kelvin, which is higher than the critical tem- the gas would heat up by quasi-adiabatic compression. The perature we obtained for a self-gravitating disc beyond 0.1pc inequality in Eq. (20) should be substituted by an approx- 7 away from 10 M⊙ black hole. imate equality, and therefore we have during optically-thin However, extra heating will always work to increase the collapse critical temperature at which the disc fragmentation occurs. −5/12 T R . (21) Therefore, the values of the critical surface density Σcr, ∝ scaleheight hcr, and the mass of the initial fragment Mcl The optical depth scales as obtained above should be treated as lower bounds of what τ R−11/3. (22) might be expected around real AGNs or protostars. Higher ∝ values of these quantities would only strengthen main con- and hence rises sharply as the clump’s radius decreases; as clusions of this paper. LB have found that when the rate the clump shrinks it becomes optically thick . It is possible ˙ ‡ of accretion M is constant throughout the disc, the Toomre to show that once the clump is optically thick, it virializes ¯ mass Mcl is given by quickly with it’s temperature T R−1. For κ T 2 (ice- ∝ ∝ −1 grains), the cooling time of an optically thick clump scales ¯ α Mcl = 1.8M⊙ with the clump radius as 0.3 × ˙ 2 0.5 1.5 −3 Mc M r tcool R . (23) 6 , (17) ∝ Ledd 3 10 M⊙ 0.2pc × and that the gap-opening mass is ‡ The contraction of an optically thin clump may be compli- α −0.5 cated by subfragmentation, since the Jean’s mass for such clump Mgap = 62M⊙ (18) scales as R7/8 ∝ τ −0.23. We suspect that in most cases the clump 0.3 × 0.5 1.5 becomes optically thick before it subfragments, since the Jean’s 2 0.5 Mc˙ M r r mass has a slow dependence on the optical depth. However, only 6 . Ledd 3 10 M⊙ 0.2pc 10h detailed numerical simulations can resolve these issues. × c 0000 RAS, MNRAS 000, 000–000 On the formation of black holes 5
The characteristic timescale for the clump to collide with gap-opening mass of the clump Mgap is given by Eq. (25) of another clump scales with the clump radius as Rafikov (2001):
−2 1/3 1/2 tcollision R . (24) Mgap I Mfr M ∝ = . (26) M 27/6π1/2 Σr2 Σr2 From Eqs. (23) and (24), we see that the collision rate de- is creases less steeply than the coolling rate as a function of the We use the numerical factor 2−7/6π−1/2I = 1.5 appro- radius of an optically thick clump. Therefore, merger can be priate for thin discs. By taking Q = 1 we get an efficient way of increasing the clump’s mass. 1/3 Mgap 14Mfr(Mfr/M¯ cl) (r/hcrit). (27) This conclusion is no longer valid when the tempera- ≃ ture of the clump becomes larger than 200K; then the In Figure 4 the masses Mis and Mgap are plotted as a func- ∼ 1/2 6 opacity is dominated by metal dust with κ T . In this tion of radius for a 3 10 M⊙ black hole; when we calculate −1.5∝ × case the cooling time scales as tcool R . The collision Mgap we conservatively set Mfr = M¯ cl and not to the larger ∝ timescale increases faster than the cooling time as the clump value M˜ cl. It is likely that the most massive clumps will shrinks, and naively one would expect that mergers may not reach Mgap, but it will be more difficult to form a clump be efficient in growing the clump masses. with the mass Mis. However, we have neglected the rotational support From Fig. 4 we see that the most massive clumps can within a clump. Each clump is initially rotating with an- reach tens hundereds of solar masses. The maximum mass gular frequency comparable to the clump’s inverse dynami- would be even larger if we included the heating of the disc cal timescale; for example, in a Keplerian disc each clump’s by external irradiation or internal starburst. It is plausi- initial angular velocity is Ω/2. Therefore each clump will ble that these very massive clumps will form massive stars; ∼ shrink and collapse into a rotationally supported disc, and the masses of the stars may be comparable to the masses the size of this disc is comparable to the size of the original of the original clumps; see McKee and Tan (2002) and ref- clump (this picture seems to be in agreement with Gammie’s erences therein. Stars with masses of a few tens of solar simulations). Thus rotational support generally slows down masses will produce black holes as the end product of their the collapse of an individual fragment and makes mergers rapid (< 106yr) evolution; the characteristic mass of these between diferent fragments to be efficient. black holes is believed to be around 10M⊙. A recent work Magnetic braking is one of the ways for the clump to lose by Mirabel and Rodrigues (2003) shows that the black holes its rotational support§ (see, e.g., Spitzer 1978). One gener- whose projenitors have masses > 40M⊙ do not recieve a ve- ally expects a horizontal magnetic field to be present in a dif- locity kick at their birth. Thus, the disc-born black holes are ferentially rotating disc due to the MRI (Balbus and Hawley, likely to remain embedded in the disc. 1991). Ionization fraction in the disc is expected to be small, so the magnetic field is saturated at a subequipartition value B = βBeq, with β << 1. Horizontal magnetic field will cou- 4 AGN DISC WITH EMBEDDED ple inner and outer parts of the differentially rotating clump STELLAR-MASS BLACK HOLES on the Alfven crossing timescale talfven tdynamical/β, and ∼ Once formed, the stellar-mass black holes will dramatically the collapse will proceed on this timescale as well. affect the dynamics of the AGN disc. We assume that the What is the maximum mass that the clump can achieve? disc is replenished due to continuous infall, and in this sec- This issue has been analyzed for the similar situation of a tion we analyze a steady-state AGN disc which is heated protoplanetary core accreting from a disc of planetesimals by the energy released due to the the accretion onto stellar- (Rafikov 2001 and references therein). The growing clump mass black holes embedded in the disc. cannot accrete more mass than is present in it’s “feeding We make an anzatz that the heated disc is optically annulus”. This gives the maximum “isolation” mass of a thick and is supported by the radiation pressure. Once we clump: obtain the solution for the structure of the disc, we will 2 3/2 (2πr Σcrit) 3/2 derive the regime of validity for our anzats. We thus have Mis = 2π√2M¯ cl(r/hcrit) , (25) ∼ M 1/2 for the sound speed in the disc midplane: 2 4 4 where, as above, M¯ = Σ h is the mass scale of the aT 2aT cs cl crit crit c2 = , (28) first clumps to form from a disc; see, e.g., Eq. (2) of Rafikov s 3ρ ∼ 3Σ Ω (2001). However, numerical work of Ida and Makino (1993) and hence and analytical calculations of Rafikov (2001) indicate that 4κF the isolation mass may be hard to reach. The consider a cs . (29) massive body moving on a circular orbit through a disc of ∼ 3Ωc gravitationally interacting particles, and they find that when Here we have used the expression for the flux through the disc face, F caT 4/(2Σκ). the mass of the body exceeds some critical value, an annular ∼ gap is opened in the particle disc around the body’s orbit. For simplicity, we assume that there are Nbh black holes We can idealize a disc consisting of fragments as a disc of of mass Mbh embedded in a disc within radius r from the particles of a typical fragment mass Mfr. Once a growing central hole of mass M. For this “planetary” system to be 1/3 clump opens a gap in a disc of gravitationally interacting stable, Nbh cannot be much bigger than (M/Mbh) , i.e. the fragments, the clump’s growth may become quenched. This embedded holes cannot be within each other’s Hill spheres. To make further progress, we need to know the rate of accretion onto the black holes embedded in the disc. Be- § Another way is via collisions with other clumps. cause the density of the disc material is high, the well-known
c 0000 RAS, MNRAS 000, 000–000 6 Levin
3/7 3/7 2/7 Bondi-Hoyle formula gives a super-Eddington accretion rate; −3 α0.3 ǫ (Mbh/10M⊙) rbondi/h = 2 10 1, (36) it is not clear at this moment how feasible this is (but see ∗ 2/7 3/7 ≪ M7 m˙ e.g. Begelman 2002). We therefore consider below two sep- arate models: the “Bondi-Hoyle” model, in which the em- which gives some credibility to the Bondi-Hoyle estimate of bedded black holes accrete at the Bondi-Hoyle rate, and the the accretion rate onto the disc-born black hole. “Eddington” model, in which the accretion onto the embed- The expressions for other useful quantities characteriz- ded holes occurs at the Eddington limit. ing the disc are given below: 5/21 1/7 (M /10M⊙) (m/ǫ ˙ ) h/r 0.08 bh (37) ∼ 1/7 5/21 4.1 The “Bondi-Hoyle” model α0.3 M7 5/21 11/42 7 (Mbh/10M⊙) M7 . In this model, the accretion rate for the embedded black cs 10 cm/s , (38) ∼ 1/7 1/7 1/2 home of mass Mbh is given by the Bondi-Hoyle formula, (ǫ/m˙ ) α0.3 (r/0.1pc) ˙ 5/21 ˙ 2 2 NrmbhMbh (Mbh/M⊙) Mbh 4πrbondiρcs 2πrbondiΣΩ. (30) 0.08 . (39) ≃ ∼ M˙ ∼ α1/7ǫ1/7m˙ 6/7M 5/21 For the latter expression to be true, the Bondi radius 0.3 7 2 The last quantity is the characteristic fraction of the disc rbondi = GMbh/cs must be less than the scaleheight of the disc; this condition will be checked once the structure of the mass which gets converted into embedded black holes. disc is worked out. We assume that the fraction ǫ of the rest-mass energy 4.2 The “Eddington” model of the accreted material goes into heating of the disc¶, is thermalized, and is eventually radiated as flux F from the The Eddington acretion rate for the embedded black hole is disc surface: 4πGMbh M˙ bh = , (40) Nbh ˙ 2 ǫaκ0c F = ǫ 2 Mbhc . (31) 2πr where ǫa is the radiative efficiency of an accretion flow. We Finally, the equation for the central hole’s accretion rate analyze the structure of the disc by repeating the steps out- lined in the previous section, and by adopting the upper ˙ 2 M = 3παΣcs/Ω, (32) 1/3 limit (M/Mbh) for the number of the black holes embed- together with the Equations (29), (30), and (31) allow us ded in the disc. We find to find the structure of the disc once the quantities Mbh, 5 2 3 Q 3 10 α0.3(Mbh/M) (κ/κ0) , (41) Nbh, M, M˙ =m ˙ M˙ edd, r, α, κ, and ǫ are fixed. After some ∼ ∗ algebra, we find the expression for the Toomre parameter of and the disc 2/3 h/r 2(κ/κ0)(Mbh/M) . (42) 4/7 4/7 5/7 ∼ α (κ/κ )(ǫ/m˙ ) (M /10M⊙) Q 10 0.3 0 bh . (33) However an embedded black hole will open a gap if its mass ∼ 3/14 3/2 M7 (r/0.1pc) is bigger than 2.5 We see that a typical AGN disc can be stabilized by feedback Mbhgap √40α(h/r) M, (43) from embedded black holes out to a radius of about a parsec. ≃ see Eqs. (4) of Lin and Papaloizou (1986). Using Eq. (42), Here κ0 is the opacity for Thompson scattering, and Nbh was 1/3 we see that set to its upper limit, (M/Mbh) . Recently, Sirko and Goodman (2002) analyzed SEDs 2/3 Mbhgap/Mbh 20(Mbh/M) 1. (44) from AGN discs in which Q = 1 is enforced by the unspec- ∼ ≪ ified heating sources. They have shown that current obser- The inequality above holds unless we allow the individual vations limit such self-gravitating discs to be no larger than masses of the embedded holes be of order of tens of thou- 5 7 10 Schwartzchild radii, about 0.1pc for 10 Modot black sands solar masses. Therefore it is impossible to maintain ∼ hole. This motivates our normalization for the radius used a stable disc by using the feedback from Eddington-limited in Eq. (33). black holes: either these holes are not massive enough to We now check the assumptions which were used in cal- provide sufficient feedback, or they are so massive that they culating the disc structure: open gaps in and become decoupled from the disc. An unsup- ported disc would fragment completely and the fuel supply 1/4 5/56 15/28 prad 2 (κ/κ0) M7 (Mbh/10M⊙) through the disc to the central engine would be stopped. 1.5 10 1/14 1,(34) pgas ∼ ∗ 1/14 3/8 ≫ α0.3 (ǫ/m˙ ) (r/0.1pc) so the disc is radiation-pressure dominated; 5 MERGER OF THE CENTRAL BLACK HOLE (m/ǫ ˙ )5/7M 41/42 AND THE DISC-BORN BLACK HOLE. τ 102 7 1, (35) ∼ 10/21 1/2 5/7 ≫ (Mbh/10Modot) (r/0.1pc) α0.3 It is likely that the newly-born black hole in the disc inspirals hence the disc is optically thick; and towards the central black hole. We imagine that the stellar- mass black hole is embedded into a massive accretion disc which forms due to continuing infall of gas from the galactic ¶ We envisage that each hole powers a minijet which carries a bulge, after the black hole is born. If the black hole opens a significant fraction of the accretion energy. gap in the disc, it will move towards the central black hole
c 0000 RAS, MNRAS 000, 000–000 On the formation of black holes 7 together with the disc (type-II migration; Gould and Rix The issue of gas-drag influence on the LISA signal was 2000, and Armitage and Natarajan 2001). The timescale for first addressed by Narayan (2000); see also Chakrabarti such inspiral is the accretion timescale, (1996). Narayan’s analysis is directly applicable to low-
−1/2 −3/2 2 luminocity non-radiative quasi-spherical accretion flows, 6 M7 0.1pc r tinspiral 10 yr . (45) which might exist around supermassive black holes when ∼ α0.1 r 30h the accretion rate is < 0.01 of the Eddington limit. Narayan If on the other hand, the black hole is not massive enough concluded that non-radiative flows will not have any ob- to open the gap, it will migrate inwards by exciting den- servable influence on the gravitational-wave signals seen by sity waves in the disc (type-I migration). The speed of this LISA. Below we extend Narayan’s analysis to the case of inward drift is given by radiative disc-like flows with high accretion rate, which are 2 likely to be present in high-luminocity AGNs. G MbhΣh vin 2β 3 ; (46) Consider a non-rotating central black hole of mass ∼ csr 6 M6 10 M⊙, accreting at a significant fractionm ˙ of the Ed- × cf. Eqs. (9), (B4) and (B5) of Rafikov (2002), and Ward dington rate, M˙ =m ˙ M˙ edd. The accretion disc in the region (1986). Here β is a numerical factor of order 5 for Q 1, and ≫ of interest (< 10RS , where Rs is the Schwartzchild radius of can be significantly larger for Q 1. For a self-gravitating ∼ the central black hole) is radiation-pressure dominated, and disc with Q 1, we find 2 ∼ the opacity κ = 0.4cm /g is due to the Thompson scatter- 1.5 7 5 30h 10M⊙ r 0.5 ing. By following the standard thin-disc theoryk (Shakura tinspiral 10 yr M7 . (47) and Sunyaev, 1973), we get ∼ β r Mbh 0.1pc 64πc2 ǫ r 3/2 What determines whether the gap is open or not is Σ(r)= 4g/cm2 , (48) whether a stellar-mass black hole has time to accrete a few 27αΩκ2M˙ ∼ αm˙ rs thousand solar masses of gas, which would put it above where ǫ is the efficiency with which the accreted mass con- the gap-opening threshold for the typical disc parameters. verts into radiation, and rs is the Schwartzchild radius of The Eddington-limited accretion occurs on a timescale of the central black hole; and 108yr, much longer than the characteristic type-I inspiral ∼ 3M˙ Ωκ time. Thus, in this case the black holes don’t accrete much cs = . (49) on their way in. On the other hand, the Bondi-Hoyle for- 8πc mula predicts the mass e-folding timescale of a few hundred The disc black hole in orbit around the central black years. Thus, if the black hole is allowed to accrete at the hole excites density waves in the disc (Goldreich and Bondi-Hoyle rate, its mass increases rapidly until it opens Tremaine, 1980); these waves carry angular momentum flux a gap in the disc. Then, the inspiral proceeds via type-II 2 ΣrΩ F0 (GMbh) 3 ; (50) migration. ∼ cs From Eqs. (45) and (47) we see that the embedded here, as before, M is the mass of the orbiting disc black black hole experiencing type-I or type-II migration would bh hole. Ward (1987) has argued that the torque acting on the merge with the central black hole on the timescale 106—107 orbiting body is dLdw/dt (h/r)F0. We can compute the years, shorter than the typical timescale of an AGN activity. ∼ characteristic timescale for the orbit evolution due to the Thus it is plausible that the daughter disc-born black hole is density-waves torque: brought towards the parent central black hole; the mass of the daughter black hole might grow significantly on the way L 1 M M h 2 tdw = = 2 . (51) in. Gravitational radiation will eventually become the dom- dLdw/dt Ω Mbh Σr r inant mechanism driving the inspiral, and the final merger One must compare tdw to the timescale tgw of orbital evo- will produce copious amount of gravitational waves. In the lution due to gravitational-radiation-reaction tourque: next subsection we show that these waves are detectable by 2 4 LISA for a broad range of the black hole masses and the disc 5 crs r tgw = 8tm = . (52) accretion rate. 8 GMbh rs Here, tm is the time left before the disc and central holes merge, i.e. the integration time for the LISA signal. Opti- 5.1 Influence of the accretion disc on the inspiral mistically we could expect to follow the LISA signal to 0.1 signal as seen by LISA of a cycle (Thorne 1999). Therefore, if q = 10nmtgw/tdw It is realistic to expect that LISA would follow the last year is less than unity, the disc drag does not impact detection of the inspiral of the disc-born black hole into the central of the final inspiral; cf. Eq. (16) of Narayan (2000). Here black hole. Generally, one must develop a set of templates nm = Ωtgw/(5π) is the number of cycles the disc black hole which densely span the parameter space of possible inspiral will make before merging with the central black hole. Using signals. In order for the final inspiral to be detectable, one Eqs. (48), (49), (51), and (52), we get of the templates must follow the signal with the phaseshift 3 13/8 ǫ (M /10M⊙) between the two not exceeding a fraction of a cycle. There- −7 0.1 bh 21/8 q 2 10 3 13/4 tm,yr, (53) ≃ × m˙ α0.1 fore, if the drag from the disc will alter the waveform by a M6 fraction of a cycle over the signal integration time (e.g., 1 year), detection of the signal with high signal-to-noise ratio k The disc is no longer thin close to the central black hole, how- will become problematic. Below we address the influence of ever our estimates of the disc structure should be correct to an the accretion disc on the final inpiral waveform. order of magnitude.
c 0000 RAS, MNRAS 000, 000–000 8 Levin where ǫ = 0.1ǫ0.1, α = 0.1α0.1 , and tm = 1yr tm,yr. We see 5.2 The merger event rate as seen by LISA ∗ that for a large range of parameters q < 1, and the disc drag does not influence the inpiral signal. However, note that q The details of black-hole formation and evolution in the disc are uncertain, and it seems impossible to make a reliable is a very sensitive function ofm ˙ and Mbh. For instance, the estimate of an event rate for LISA from this channel of black- disc will influence significantly an inspiralling 100M⊙ black hole when the accretion rate is down to a few percent of hole mergers. Nonetheless, from Eq (39) we see that in order the eddington limit. One then needs to reduce the influence for an extended accretion disc in a bright AGN to be stable, of the disc on the LISA signal by choosing to observe the a significant mass fraction of the material acreted by the smaller portion of the final inspiral, i.e. by choosing a smaller cental black hole must go into the stellar-mass black holes embedded in the disc. It is worth working through a simple integration time tm. There is another important source of drag experi- example to show that the merger of disc-born holes with the enced by the inpiralling black hole; it was first analyzed central hole may be an important source for LISA. by Chakrabarti (1993, 1996). Generally, there is a radial Studies of integrated light coming from Galactic Nuclei pressure gradient in an accretion disc; this pressure gradient show that the supermassive black holes acquire significant makes the asimuthal velosity of the disc gas slightly differ- part and perhaps almost all of their mass via accretion of ent from a velocity of the test particle on a circular orbit at gas; see, e.g., Yu and Tremaine (2002) and references therein. the same radius. Therefore the inspiralling black hole will Assume, for the sake of our example, that a mass fraction experience a head wind from a gas in the accretion disc; by η of this gas is converted into 100M⊙ black holes on the accreting gas from the disc the black hole will experience the way in, and that all of these black holes eventually merge with the central black hole. LISA can detect such mergers braking torque which will make it lose it’s specific angular 5 7 momentum. This tourque is given by to z = 1 if the central black hole is between 10 and 10 solar masses (one needs to assume that the central black ˙ τwind = Mbh∆vr, (54) hole is rapidly rotating at the high-mass end of this range). The mass density of such black holes in the local universe where M˙ bh is rate of accretion onto the inspiraling black hole 5 3 2 ⊙ from the disc, and ∆v c /vo is the speed of the headwind is 10 M /Mpc (Salucci et. al. 1999). We can estimate, ∼ s ∼ experienced by the black hole moving with the orbital speed using Figure 3 of Pei et. al. (1995), that about 10 percent of integrated radiation from Galactic Nuclei comes from red- vo. We assume that the inspiraling hole accretes with the Bondi-Hoyle rate, shifts accessible to LISA, z < 1. This implies that supermas- 2 3 sive black holes acquired 10 percent of their mass at z < 1. M˙ bh πρ(GMbh) /c , (55) ≃ s We shall therefore assume that 10 percent of the mass of the where ρ = Σ/h = ΣΩ/cs is the density of the ambient disc black holes in LISA mass range was accreted at z < 1. This 3 implies that there were 100η = η0.01 mergers per mpc gas. By using the last equation in Eq. (54), we obtain the ∼ expression for the tourque τwind, which turns out to be the which are potentially detectable by LISA. When we multi- same as the tourque from the density waves, up to the nu- ply this by volume out to z = 1 and divide it by the Hubble merical factor between 1 and 10. We see therefore that in- time, we get an estimate of the LISA event rate from such clusion of Chakrabarti’s “accretion” torque is important for mergers, the detailed analysis, but does not qualitatively change our conclusions. dN/dttoymodel 10η0.01 /yr. (59) Can the orbiting black hole open a gap in the disc during ∼ it’s final inspiral? In order to overcome the viscous stresses The estimate above should be treated as an illustration of which oppose opening the gap, the mass of the orbiting black importance for our channel of the mergers, rather than as a hole should be greater than the threshold value given by concrete prediction for LISA. Eq. (43). The radius rin from which the inspiral begins is Currently, it is not known how to compute well the grav- given by itational waveform produced by an inpiral with an arbitrary 1/4 −1/2 1/4 rin = 4(Mbh/10M⊙) M6 tm,yrrs, (56) eccentricity and inclination relative to the central black hole. This might pose a great challenge to the LISA data analysis. and the scaleheight of the disc is radius-independent in the Indeed, in the currently popular astrophysical scenario, the radiation-dominated inner region (this is true only if you stellar-mass black hole gets captured on a highly eccentric treat the central black hole as a Newtonian object): orbit with arbitrary inclination relative to the central black 3m ˙ hole (Siggudrson and Rees, 1997). By contrast, in our sce- h = rs. (57) 4ǫ nario the inpiral occurs in the equatorial plane, and the sig- The gap will be open in the disc, therefore, if the mass of nal is well understood (Hughes 2001 and references therein). the inspiraling black hole exceeds Hence the template for detection is readily available, and the 20/13 −2/13 4 merger channel we consider has a clear observational sig- mgap m˙ M 10 M⊙. (58) ∼ 6 nature which distinguishes it from other channels. It is an During the final year of merger of the 10M⊙ disc-born black open question whether the inpiraling hole can acquire high 6 hole and the 10 Modot central black hole, the gap will be eccentricity by ineracting with the disc or with other orbit- open if the accretion rate onto the central black hole is ing masses (Goldreich and Sari 2003, Chiang et. al. 2002); about five percent of the Eddington limit. Such gap-opening it seems likely that at least in some cases interaction with merger can result in sudden changes in the AGN luminos- the disc will act to circularize the orbit. Finn and Thorne ity and produce an optical counterpart to the gravitational- (2000) have performed a detailed census of parameter space wave signal. for circular-orbit inspirals as seen by LISA.
c 0000 RAS, MNRAS 000, 000–000 On the formation of black holes 9
6 CONCLUSIONS Bate, M. R., Bonnel, I. A, and Bromm, V. 2002, MNRAS, 332, 65 In this paper, we rely on numerical simulations by Gammie Bell, K. R., and Lin, D. N. C. 1994, ApJ, 427, 987 (2001) to develop a formalism for self-gravitating thin discs Begelman, M. C. 2002, ApJ, 568, L97-100 which are gaining mass by continuous infall. We present a Binney, J., and Tremaine, S. 1987, Galactic Dynamics (Prince- way to calculate the critical temperature, surface density, ton University Press) and scaleheigh of the disc just prior to fragmentation. Our Chakrabarti, S. K 1993, ApJ, 411, 610 formalism naturally includes both optically thin and opti- Chakrabarti, S. K 1996, Phys. Rev. D, 53, 2901 cally thick discs. Chiang, E. I., Fisher, D., and Thommes, E. 2002, ApJ, 564, We then speculate on the outcome of the nonlinear L105 Collin, S., and Zahn, J.-P. 1999, A&A, 344, 433 physical processes which follow fragmentation: accretion and Finn, L. S., and Thorne, K. S. 2000, Phys. Rev. D, 62, 124021 merger of smaller fragments into bigger ones. We find an up- Gammie, C. F. 2001, ApJ, 553, 174 per bound on the mass of final, merged fragments, and we Genzel, R., et. al. 2000, MNRAS, 317, 348 give a plausibility argument that some fragments will indeed Genzel, R., et. al. 2003, astro-ph/0305423 reach this upper bound. We thus predict that very massive Gerhard, O. 2001, ApJ, 546, L39 stars of tens or even hundreds of solar masses will be pro- Goldreich, P., and Sari, R. 2003, ApJ, 585, 1024 duced in self-gravitating discs around supermassive black Goldreich, P., and Tremaine, S. 1980, 241, 425 holes. The end product of fast evolution of these massive Goodman, J. 2003, MNRAS, 339, 937 stars will be stellar-mass black holes. Gould, A., and Rix, H.-W. 2000, ApJL, 532, 29 Finally, we make an argument that the disc-born black Hansen, B., and Milosavljevic, M. 2003, submitted to ApJ Letters (astro-ph/0306074) holes in AGNs find a way to merge with central black holes. Hughes, S. A. 2001, Class. Quant. Grav., 18, 4067, and refer- We consider a purely toy-model example of what the rate ences herein of such mergers might be, as seen by LISA; we illustrate Ida, S.,& Makino, J., Icarus, 106, 210 this rate might be high enough to be interesting for future Kolykhalov, P. I., and Sunyaev, R. A. 1980, SvAL, 6, 357 gravitational-wave (GW) observations. The GW signal from Kumar, P. 1999, ApJ, 519, 599 this merger channel is distinct from that of other channels, Levin, Y., and Beloborodov, A. M. 2003, ApJ, 590, 33L and can be readily modeled using our current theoretical Levin, Y, and Matzner, C. 2003, in preparation. understanding of the final stages of the inspiral driven by Lin, D. N. C., and Papaloizou, J. 1986, ApJ, 309, 846 gravitational-radiation reaction. We show that for a broad Lin, D. N. C., and Pringle, J. E. 1987, MNRAS, 225, 607 range of accretion rates and black-hole masses the drag from Mayer, L., et. al. 2002, Science, 298, 1756 McKee, C. F., and Tan, J. C. 2002, Nature, 416, 59 accretion disc will not be large enough to pollute the signal McMillan, S., and Portegies Zwart, S. 2003, submitted to ApJ and make inspiral template invalid. In some cases, the in- (astro-ph/0304022) spiralling hole will open a gap in the accretion disc close Mirabel, F., and Rodrigues, I. 2003, Science, 300, 1119 to the central black hole, thus producing a possible optical Morris, M., Ghez, A. M., and Becklin, E. E. 1989, Advances in counterpart for the gravitational-wave burst generated by Space Research, 23, 959 the merger. Narayan, R. 2000, ApJ, 536, 663 Ott, T. et. al. 2003, in preparation. Paczysnki, B. 1978, AcA, 28, 91 Pei, Y. 1995, ApJ, 438, 623 7 ACKNOWLEDGEMENTS Portegies Zwart, S., McMillan, S., and Gerhard, O. 2003, ac- cepted to ApJ (astro-ph/0303599) I have greatly benefited from discussions with Chris Matzner Rafikov, R. R. 2001, AJ, 122, 2713 throughout this project. He has suggested the model for the Rafikov, R. R. 2002, ApJ, 572, 566 heated disc in section IV, and has independently verified Rice, W. K., et. al. 2003, MNRAS, 339, 1025 the formulae in that section. Prof. Matzner has graciously Salucci, P., Szuszkiewicz, E., Monaco, P.,& Danese, L. 1999, declined to be a co-author of this paper. I have learned a lot MNRAS, 307, 637 Sanders, R. H. 1998, MNRAS, 294, 35 about accretion discs from Eliot Quataert. After the bulk of Shakura, N. I, and Sunyaev, R. A. 1973, A&A, 24, 337 this project was completed, I have found out that Jeremy Shlosman, I., and Begelman, M. 1987, Nature, 329, 810 Goodman and Jonathan Tan have pursued similar line of Shlosman, I., and Begelman, M. 1989, ApJ, 341, 685 research; our open exchange of ideas is much appreciated. Siggurdsson, S., and Rees, M. J. 1997, MNRAS, 284, 318 I also thank Andrei Beloborodov, Andrew Melatos, Norm Sirko, E., and Goodman, J. 2002, MNRAS, 341, 501 Murray, Frank Shu, Anatoly Spitkovsky, Kip Thorne, and Thorne, K. S. 1999, private communications Andrew Youdin for discussions and insights. Sarah Levin Ward, W. R. 1986, Icarus, 67, 164 has contributed to the clarity of the prose. This research was Yu, Q., and Tremaine, S. 2002, MNRAS, 335, 965 supported by TAC at UC Berkeley, by NSERC at CITA, and by the School of Physics at the University of Melbourne.
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c 0000 RAS, MNRAS 000, 000–000 10 Levin
Figure 1. The temparature of the critically fragmenting disc as a function of the orbital period.
Figure 2. The surface density of the critically fragmenting disc as a function of the orbital period.
Figure 3. The Toomre mass of the critically fragmenting disc as a function of the orbital period.
Figure 4. The isolation and gap opening masses plotted as a function of radius for the critically fragmenting disc without ex- ternal sources of heating. The black hole mass is taken to be 6 3 × 10 M⊙.
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