Tidal Disruption of Stellar Objects by Hard Supermassive Black Hole Binaries
aa r X i v : . [ a s t r o - ph ] D ec Tidal Disruption of Stellar Objects by Hard Supermassive BlackHole Binaries
Xian Chen
Department of Astronomy, Peking University, 100871 Beijing, China [email protected]
F. K. Liu
Department of Astronomy, Peking University, 100871 Beijing, China [email protected] andJohn Magorrian
Rudolf Peierls Center for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford,OX1 3NP, United Kingdom [email protected]
ABSTRACT
Supermassive black hole binaries (SMBHBs) are expected by the hierarchicalgalaxy formation model in ΛCDM cosmology. There is some evidence in theliterature for SMBHBs in AGNs, but there are few observational constraints onthe evolution of SMBHBs in inactive galaxies and gas-poor mergers. On thetheoretical front, it is unclear how long is needed for a SMBHB in a typicalgalaxy to coalesce. In this paper we investigate the tidal interaction betweenstars and binary BHs and calculate the tidal disruption rates of stellar objects bythe BH components of binary. We derive the interaction cross sections betweenSMBHBs and stars from intensive numerical scattering experiments with particlenumber ∼ and calculate the tidal disruption rates by both single and binaryBHs for a sample of realistic galaxy models, taking into account the generalrelativistic effect and the loss cone refilling because of two-body interaction. Weestimate the frequency of tidal flares for different types of galaxies using theBH mass function in the literature. We find that because of the three-bodyslingshot effect, the tidal disruption rate in SMBHB system is more than one 2 –order of magnitude smaller than that in single SMBH system. The difference ismore significant in less massive galaxies and does not depend on detailed stellardynamical processes. Our calculations suggest that comparisons of the calculatedtidal disruption rates for both single and binary BHs and the surveys of X-rayor UV flares at galactic centers could tell us whether most SMBHs in nearbygalaxies are single and whether the SMBHBs formed in gas-poor galaxy mergerscoalesce rapidly. Subject headings: black hole physics — galaxies: kinematics and dynamics —-galaxies: nuclei — methods: numerical — X-ray: galaxies
1. INTRODUCTION
In cold dark matter (CDM) cosmology, galaxies form hierarchically and present-daygalaxies are the products of successive mergers (Kauffmann & Haehnelt 2000; Springel et al.2005). Recent observations show that almost all galaxies harbor supermassive black holes(SMBHs) at their centers (Richstone et al. 1998; Ferrarese & Ford 2005) and the blackhole (BH) masses tightly correlate with the properties of their host galaxies such as themass of stellar bulge (Magorrian et al. 1998; H¨aring & Rix 2004), the bulge luminosity(McLure & Dunlop 2002), and the nuclear stellar velocity dispersion (Ferrarese & Merritt2000; Gebhardt et al. 2000; Tremaine et al. 2002). The correlations between BH mass andgalaxy properties imply that the growth of SMBHs and the formation and evolution of galax-ies are closely linked. The correlation is likely induced by galaxy major mergers (mergingof galaxies with comparable mass) in which both rapid star formation and gas accretiononto SMBHs are triggered and the feedback from the central active galactic nuclei (AGNs)regulates the growth of both SMBHs and galaxy bulges. The coevolution scenario can suc-cessfully explain not only the correlations between SMBHs and their host galaxies but alsomany of the observed evolutions of galaxies and AGNs (Springel et al. 2005; Hopkins et al.2006; Croton et al. 2006).During galaxy mergers part of the gas originally in galactic plane is driven to galaxy cen-ter, triggering starburst and the accretion of central SMBH (Gaskell 1985; Hernquist & Mihos1995). If both galaxies in the merging system harbor SMBHs at their centers, a pair of AGNscould form which is an intriguing object for both observations and theories. Because of dy-namical friction the two SMBHs quickly sink to the common center of the two merginggalaxies, forming a bound, compact ( ∼
10 pc) supermassive black hole binary (SMBHB,Begelman et al. 1980), which would be difficult to resolve in any but the closest galaxies us-ing current telescopes. As the separation of the binary shrinks, dynamical friction becomes 3 –less and less efficient and three-body interaction between SMBHB and the stars passing bybecomes more and more important. At separations ∼ . − ∼ yr (Ivanov et al. 1999; Gould & Rix 2000;Armitage & Natarajan 2002; Escala et al. 2005; Kazantzidis et al. 2005).Although many physical processes have been proposed in the literature to boost thehardening rates of SMBHBs, the abundance of binary BHs in galaxy centers has not yet beenstrongly constrained observationally. Because of their compactness, bound SMBHBs are verydifficult to resolve directly with current telescopes, and only unbound SMBHBs in a couple ofgas-rich (wet) galaxy mergers have been identified (Komossa 2006). So far all the evidencefor bound and hard SMBHBs are indirect and model dependent. The prototype evidencefor SMBHBs in AGNs is the helical morphology of radio jets in many radio galaxies whichmay be due to the precession of jet orientation in a SMBHB system (Begelman et al. 1980).It has been suggested that the periodic outbursts observed in some AGNs (Sillanp¨a¨a et al.1988; Liu et al. 1995, 1997; Raiteri et al. 2001) are due to the orbital motion of the jet-emitting BH in SMBHB (Villata & Raiteri 1999; Ostorero et al. 2004), the precession ofspin axis of the rotating primary BH (Romero et al. 2000), the interaction between SMBHBand a standard accretion disk (Sillanp¨a¨a et al. 1988; Valtaoja et al. 2000) or an advectiondominated accretion flow (ADAF) (Liu & Wu 2002; Liu et al. 2006). X-shaped featuresin a subclass of radio galaxies have been attributed to the interaction and alignment ofSMBHBs and standard accretion disks (Liu 2004, alternatives see Merritt & Ekers 2002; 4 –Zier 2005). There is also some evidence for the coalescence of SMBHBs in AGNs. Liu et al.(2003) suggested that the double-double radio galaxies and the restarting jet formation insome radio galaxies are due to the removal and refilling of inner accretion disk because ofthe interaction with SMBHBs. Although there is much circumstantial evidence for bothactive and coalesced SMBHBs in AGNs, it is still unclear what fraction of SMBHBs wouldcoalesce during or before the AGN phase and how many could survive to the later dormantor inactive phase. Because present observations cannot give constraints on the physicalprocesses which have been proposed to boost the hardening rates of SMBHBs, Liu & Chen(2007) suggested measuring the acceleration of jet precession as a way of determing thehardening rates of SMBHBs in AGNs and to place strong constraints on models of SMBHBevolution. In inactive galaxies or gas-poor mergers the observational evidence for SMBHBsis very rare. One possible way to identify uncoalesced SMBHBs in inactive galaxies is todetect hypervelocity binary stars ejected by SMBHBs with the three-body sling-shot effect(Lu et al. 2007). However, this method cannot be applied to distant galaxies. Therefore, inthis paper we suggest a way to determine statistically whether SMBHBs in nearby galaxieshave coalesced.One inevitable impact of a non-rorating SMBH on its stellar environment is that starspassing by the BH as close as the tidal radius r t ≃ r ∗ (cid:18) M • M ∗ (cid:19) / ≃ − pc (cid:18) r ∗ r ⊙ (cid:19) (cid:18) M ∗ M ⊙ (cid:19) − / (cid:18) M • M ⊙ (cid:19) / (1)would be tidally disrupted (Hills 1975; Rees 1988), where r ∗ and M ∗ are the radius andmass of the stars and M • is the BH mass. Part of the stellar debris will be spewed to highlyeccentric bound orbits and later fall back onto the BH, giving rise to an outburst decayingwithin months to years (Rees 1988). These tidal flares are definitive evidences of SMBHsin inactive galaxies (Komossa & Greiner 1999; Komossa et al. 2004; Halpern et al. 2004;Gezari et al. 2006). Since the frequency of stellar disruption depends on the surroundingstellar distribution (Syer & Ulmer 1999; Magorrian & Tremaine 1999, thereafter MT99),the stellar disruption rate can be used as a probe of the inner structure of galactic nucleus.For nearby early-type galaxies recent theoretical calculation with single BH without takinginto account general relativistic (GR) effects gives an averaged stellar disruption rate of ∼ − yr − per galaxy if the stellar distribution is spherically symmetric (Wang & Merritt2004) or possibly 1 − ∼ − − − . 5 –Merritt & Wang (2005) considered the stellar disruption by single BHs immediately afterSMBHB coalescence. They found that the tidal disruption rate is significantly suppresseddue to the low density core forming during the hard stage of SMBHBs. Since the two stagesinvestigated by Ivanov et al. (2005) and Merritt & Wang (2005) are two short phases in theevolution of SMBHBs and binary BHs with mass ratios & − spend most of their lifetimeon the hard stage (Y02), it would be very interesting to calculate the tidal disruption rateof stellar objects for hard SMBHBs. Therefore, in this paper we calculate tidal disruptionrates for hard and stalling SMBHBs. We suggest that the comparison of expected tidaldisruption rate by single SMBHs or hard SMBHBs and the future survey of X-ray or UVflares at nearby galactic centers could tell us whether most of the SMBHs in nearby galaxiesare single or binary and whether the SMBHBs would coalesce rapidly, as expected by somemodels.The stellar disruption rate in a galaxy harboring a SMBHB depends on: (1) the ratethat stars are fed from large-scale to the vicinity of SMBHB and (2) the probability thata star moving in the vicinity of the SMBHB passes close enough to one of the BHs tobe tidally disrupted. Although there has been much work on the former part, intendedto study the evolution timescale of SMBHB (Q96; Y02; Milosavljevi´c & Merritt 2003), thelatter is largely unaddressed in the literature. Because of the chaotic nature of three-bodyinteraction, numerical scattering experiments are needed to give the probability of tidaldisruption. Previous numerical experiments on star-SMBHB interactions mainly focus onenergy and angular momentum exchanges (Q96; Sesana et al. 2006) and the limited particlenumbers are insufficient to tackle the rare events of very close encounters such as tidaldisruptions. Therefore, we begin with intensive numerical scattering experiments (normally ∼ particles in each run) to calculate the tidal disruption cross sections for hard SMBHBs.Then we apply our results to a sample of nearby galaxies and estimate the tidal disruptionrates of stellar objects both by single and binary BHs in different types of galaxies.This paper is organized as follows. In § § §
4. In § § § §
8. 6 –
2. PROPERTIES OF HARD SMBHB SYSTEMS2.1. Loss Cone Filling Rate
A SMBHB with masses M and M ( M > M ) becomes hard when the semimajor axis a decreases to a . a h = Gµ σ ∗ , (2)(Q96) where a h is the hardening radius, σ ∗ is the one-dimensional stellar velocity disper-sion of background stars, and µ = M M /M is the reduced mass with M = M + M .Observations of nearby galaxies show that BH mass tightly correlates with bulge velocitydispersion (Ferrarese & Merritt 2000; Gebhardt et al. 2000). If the total BH mass M ofSMBHB also follows the empirical M • − σ ∗ relation: M = 10 . M ⊙ ( σ ∗ /
200 km s − ) . (Tremaine et al. 2002), equation (2) reduces to a h ≃ .
13 pc q (1 + q ) M / , (3)where M = M / M ⊙ and q = M /M is the mass ratio of the binary. In this paper weonly consider q > .
01 because the dynamical friction timescale for SMBHBs with q ≪ . ∼ a about the mass center of thebinary (if not tidally disrupted or swallowed by the BHs) will be eventually expelled withan average energy gain ∆ E = − GM M ∗ ∆ a a ≃ KGµM ∗ /a , (4)where K is a dimensionless factor about 1 . J and specific binding energy E . We define the losscone to consist of all orbits with pericenters less than a , which results in a conical regionwith the boundary given by J ( E , a ) = 2 a [ GM /a − E ]. In a non-spherical galaxy angularmomentum is no longer conserved and there can be more than one orbit family. Nevertheless,instead of a “loss cone” one can still define a “loss zone” inside which stars will be deliveredto the SMBHB.Mechanisms such as two-body relaxation tend to refill the loss cone so the decay ofSMBHB continues. The stars refilled into loss cone are originally far from the SMBHB so 7 –that E ≪ GM /a (5)and J ( a ) ≃ GM a . (6)If the characteristic change of J during one orbital period of a star is much smaller than J lc ,the loss cone refilling behaves like a diffusion process in which the loss cone filling rate is notsensitive to J or a (diffusive regime, Cohn & Kulsrud 1978). Otherwise, the stars enteringthe loss cone have a uniform distribution with respect to J so that the loss cone filling rateis proportional J or a (pinhole regime).In inactive galaxies gas dynamics is probably not important, so a hard binary losesenergy mainly through interacting with loss cone stars. The evolution timescale t h at thishard stage depends on the loss cone filling rate F lc ( a ), the number of stars fed into a sphereof radius a about the mass center of the binary per unit time. From equation (4) and d ( GM / a ) /dt = F lc ( a )∆ E we have t h = | a/ ˙ a | = M / [2 KF lc ( a ) m ∗ ] (7)(see also Y02), which implies that about an amount of F lc ( a ) t h ≃ M / K of stars shouldbe consumed to reduce a by a single e -fold, in agreement with recent simulations (Merritt2006). A fraction of the stars fed into the loss cone of SMBHB would be scattered to thevicinity of a BH and get tidally disrupted. The ratio of the tidal radius of the primary BHand the separation of the binary is about r t a h ≃ . × − M − / (cid:18) r ∗ r ⊙ (cid:19) (cid:18) M ∗ M ⊙ (cid:19) − / (1 + q ) / q − (cid:16) σ
200 km s − (cid:17) ≃ . × − M − / (cid:18) r ∗ r ⊙ (cid:19) (cid:18) M ∗ M ⊙ (cid:19) − / (1 + q ) / q − (8)(from eq. [1] and [3]), which is small for typical hard SMBHBs This implies that most starsin the loss cone would be expelled instead of being tidally disrupted.To further quantify the probability of tidal disruption, we introduce Σ i ( r ) ( i = 1 , r from the i th BH. The 8 –probability that a loss cone star is disrupted by the i th BH estimates Σ i ( r ti ) / Σ i ( a ), where r ti is the tidal radius of the i th BH. Then the stellar disruption rate is related to the losscone filling rate and the tidal disruption cross section as˙ N di ≃ F lc ( a ) Σ i ( r ti )Σ i ( a ) . (9)For a single BH the geometrical cross section can be written out analytically asΣ( r ) = πr (cid:18) GM • /rv / (cid:19) , (10)which increases with the gravitational potential GM • /r and decreases with stars’ kineticenergy v / v is the velocity at apocenter. Whengravitational focusing is important ( GM • /r ≫ v /
2) the cross section scales as r . In SMBHBsystems the presence of a companion BH tends to increase the term GM • /r by deepeningthe gravitational potential and also increase the term v / GM /r is GM /r and theequivalent increase in v / q GM /a . So the cross section of the primary BH isapproximately ˜Σ ( r ) = πr (cid:18) GM /rv / ηq GM /a (cid:19) (11)where η is a correction factor of order unity. We will see below that for different q s the bestfit is obtained when η = 1 / (1 + q ). For the secondary, the roles of M and M exchange,so the cross section can be obtained by replacing q in equation (11) with 1 /q . Quinlan hasseparately explained the behavior of the cross sections at r/a ≫ q and r/a ≪ q from anotherway of understanding (Q96), while our explanation and empirical formula are general for any r/a . Because of the effect of GR equation (11) is no longer valid when r ∼ r S , where r S =2 GM • /c is the Schwarzschild radius. Since the ratio of r S and r t is r S r t ≃ . M / (cid:18) r ∗ r ⊙ (cid:19) − (cid:18) M ∗ M ⊙ (cid:19) / , (12)GR effect is important when M • > . × M ⊙ ( r t < r S for solar type stars). In practice,one can use the pseudo-Newtonian potential ψ ( r ) = − GM • r − r S (13) 9 –(Paczy´nski & Witta 1980) to simulate the GR effect so equation (11) becomes˜Σ ( r ) = πr (cid:18) GM / ( r − r S1 ) v / ηq GM /a (cid:19) (14)(notice that a ≫ r S i ), equivalent to multiplying the non-relativistic cross section by a correc-tion factor r/ ( r − r S ). Equation (14) diverges when r → r S , but since a star plunging into thesphere of marginally bound radius 2 r S about a BH eventually falls into to the event horizon,the cross section at [ r S , r S ] should be constant, equal to the cross section at r = 2 r S .Both equation (11) and (14) are evaluated at the first time that a star passes thebinary. In some cases a star could be scatted onto temporally bound orbit and encounterswith the binary many times before expelled or disrupted (Hills 1983; Q96). During suchmulti-encounters the probability of tidal disruption is expected to be enhanced. Since it isvery difficult to derive analytical cross sections for these complicated encounters, scatteringexperiments are needed. Stars on near-radial orbits have v ≪ GM /a , then equations (8),(11), and (14) suggest that the probability of tidally disruption scales as r ti /a , which is oforder 10 − − − . Therefore, to get statistically meaningful tidal disruption cross sectionsa large number of particles should be used in the scattering experiments.
3. METHOD3.1. Scattering Experiments
In this section we describe our numerical scattering experiments of the restricted three-body problem. The method adopted here is similar to those in Q96 and Sesana et al. (2006)but the number of test particles in our experiments is much larger, typically ∼ , to givestatistically meaningful cross sections at 10 − < r/a < − .In the scattering experiments, instead of using E , the specific binding energy of a particlein the combined potential of stars and BHs, we use a more convenient parameter E = GM /r − v /
2, the specific binding energy about the BH binary, where r is the radiusabout the mass center and v is the velocity of particle. We denote the initial specific bindingenergy with E and initial velocity with v . In our problem the relevant energy range is − qGM / a . E ≪ GM /a (from eq. [2] and eq. [5]).In each scattering experiment, the origin is chosen at the mass center of the binary.In the case − qGM / a . E < r = ( x, y, z ) = ( b, , ∞ ) with asymptotic velocity v = ( v x , v y , v z ) = (0 , , − v ), where b is the impact parameter, the minimum separation between particle and mass center if the 10 –particle feels no gravitational field. The fiducial value for v is 10 − p GM /a , to reproduceFig. 3 in Q96. Given v , b is uniformly sampled in the range [0 , b ], where b max = √ J lc ( a ) /v . Particles with b > b max hardly reach r < a therefore contribute little to thetidal disruption cross section (see § E ≪ GM /a (bound case),initially we put particles at r = ( x, y, z ) = (0 , , r ) with an isotropic velocity distribution( v = p GM /r − E )). r satisfies a ≪ r < GM /E because in realistic SMBHBsystems most stars enter the loss cone on near-radial orbits.Another six parameters (four if the binary is circular) are set to fix the initial conditionsof the binary: (1) the mass ratio q ; (2) the eccentricity of binary orbit e ; (3) the inclinationof orbital plane; (4) the argument of pericenter; (5) the longitude of ascending node; (6) theinitial binary phase. In each set of experiments with fixed q and e , the cosine of the orbitalinclination angle is evenly sampled in [ − ,
1] and the other three angular parameters areuniformly distributed in [0 , π ], resulting in an isotropically filling of the loss cone.Given the above initial conditions, the motion of a massless particle is governed by thefollowing coupled first-order differential equations:˙ r = v ˙ v = − G X i =1 M i ( r − r i ) | r − r i | , (15)where r i is the position of the i th BH. In each scattering experiment we first move a particlefrom its initial position to r k = a (10 q ) / along a Keplerian orbit about a point mass M at the origin. At r k the quadrupole force from the binary is eight orders of magnitudesmaller than GM /a . Then we integrate the particle’s orbit with the subroutine dopri8 (Hairer et al. 1987), an explicit Runge-Kutta method of order (7)8. We set the thresholdof fractional error per step in r and v to be 10 − . Raising this threshold up to 10 − doesnot significantly change our results. The integration stops if one of the following conditionsis satisfied (1) the particle leaves the sphere of radius r k with negative binding energy; (2)the physical integration timescale exceeds 10 yr; (3) the integration timestep reaches 10 .A small fraction ( . . v = − G X i =1 M i ( r − r i ) | r − r i | ( | r − r i | − r S i ) .
11 –and we stop the integration once a particle reaches 1 .
01 Schwarzschild radius about eitherof the BHs. In the GR experiment the ratio of r S1 and a h , r S1 a h ≃ . × − M / qq − , (16)should be set in priori. For the illustrative purpose we always set M = 1. We will discussthe effect of changing M on tidal disruption cross sections in § After each scattering experiment we record the minimum separation between the par-ticle and each BH. At the end of all experiments we count the number N i ( r ) of parti-cles whose minimum separations from the i th BH are less than r . Then the normalizedmulti-encounter cross section (particle are allowed to encounter the binary as many timesas they could until they are expelled) is calculated with Σ i ( r ) / Σ i ( a ) = N i ( r ) /N i ( a ). ThePoissonian error in the counts is p N i ( r ), so the error for the normalized cross sectionis N i ( r ) /N i ( a ) p /N i ( r ) + 1 /N i ( a ) and the corresponding fractional error in statistics is σ stat ( r ) = p /N i ( r ) + 1 /N i ( a ).Following Q96, we also record the minimum separation between a particle and eachBH during their first encounter and derive the single-encounter cross sections. In §
4. RESULTS FROM THE SCATTERING EXPERIMENTS4.1. Non-relativistic Cross Sections
First we consider the unbound case. We set v = 10 − p GM /a (after Fig. 3 inQ96), q = 0 .
01, and use the Newtonian potential. The normalized mullti- and single-encounter cross sections from N ≃ particles are presented as solid lines in the toppanel of Figure 1. The cross sections are plotted as a function of r/a so that they can beeasily scaled. The perturbations at the lower ends are due to statistical fluctuation. Inthe top panel of Figure 1 we also plot the empirical single-encounter cross sections (seeeq. [11]) as dashed lines. We estimate the fractional error of the empirical formula with σ app ≡ | Σ i ( r ) / Σ i ( a ) − ˜Σ i ( r ) / ˜Σ i ( a ) | / [Σ i ( r ) / Σ i ( a )], where Σ i ( r ) / Σ i ( a ) is from scattering ex-periments and ˜Σ i ( r ) from equation (11). σ app and σ stat for the primary and secondary BHs 12 –are shown in the middle and bottom panels of Figure 1. For both BHs we found that at r ≪ a σ app is almost always below σ stat , indicating that the empirical cross sections agreesvery well with the numerical results. At r > qa , although σ app is larger than σ stat , thedifference does not exceed 10%.For q = 0 . q = 1, the numerical (solid lines) and empirical approximations (dashedlines) cross sections are presented in the top left and top right panels of Figure 2. σ stat (solid lines) and σ app (dotted lines) for the single-encounters are presented in the middle andbottom panels. Although the approximations become worse when q is large, the differencebetween approximation error and statistic fluctuation is always below 10%.Figure 1 and Figure 2 show that at r . q a both the multi- and single-encounter crosssections scales linearly as r/a , implying that gravitational focusing is important for veryclose encounters. The cross sections at r/a < − could be obtained by using this scaling.It is also clear that the normalized multi-encounter cross sections are greater than the single-encounter ones as is predicted in § M = 1) about either BH. This is because the probabilityfor a particle to enter the sphere of tidal radius about the i th BH n times roughly scalesas [Σ i ( r ti ) / Σ i ( a )] n or ( r ti /a ) n , which becomes extremely low for r ti ≪ a and n >
1. Forthe similar reason, the event that a particle successively approaches the tidal radii of theprimary and secondary BHs is also extremely rare.
To investigate the effect of binding energy, we carried out two test experiments for E = − .
01 and E = 0 .
01 (in unit of GM /a ), each with ∼ particles and q = 0 .
01. Inthe unbound case the asymptotic velocity of the intruding particles is v ≃ . p GM /a ,slightly greater than the limit 0 . p GM q/a (from the fitting formula eq. 17 in Q96) abovewhich the binary becomes soft. In the bound case, initially the particles are at r = 50 a with an isotropic velocity of v ≃ . p GM /a . We do not consider E = 0 . r a .Results from these test experiments are presented in Figure 3 as solid lines. Crosssections for the fiducial value E ∼ − − GM /a (from § E which is expected byequation 11, Figure 3 shows that in both bound and unbound cases the normalized crosssections for both multi- and single-encounters seem not varying significantly with E . This 13 –Fig. 1.— Top : Normalized multi- and single-encounter cross sections from 10 particles for q = 0 .
01. From top to bottom, the first two solid lines, respectively, refer to the multi-and single-encounter cross sections of the primary BH, and the third and fourth ones areof the secondary. Empirical cross sections (dashed lines) and positions of tidal radii andSchwarzschild radii for M = 10 M ⊙ (short vertical lines ) are also plotted. Middle : Frac-tional errors of statistical fluctuation (solid) and empirical cross sections (dotted), for theprimary BH.
Bottom : The same as the middle panel but for the secondary BH. 14 –Fig. 2.— Close-encounter cross sections and fractional errors for q = 0 . particles) and q = 1 (the right panels, 10 particles). Lines have the same meanings asthose in Figure 1. 15 –Fig. 3.— Normalized multi- and single-encounter cross sections from ∼ particles (solidlines, q = 0 .
01) for E = 0 .
01 ( top ) and E = − .
01 ( bottom ). E is in unit of GM /a .Dotted lines are from the fiducial experiments ( § | E | ≪ GM /a , the velocity of the particles passing r . a from the binaryis not sensitive to E but to the depth of the gravitational potential at r . Therefore, differentparticles passing by the same BH binary feel the similar strength of gravitational focusingand have similar interaction cross sections with the BH components of binary. Since thenormalized cross sections do not vary significantly with E as long as | E | ≪ GM /a , itis reasonable to apply the fiducial cross sections to various binaries with a wide range ofsemimajor axis. For example, according to equation (2), varying E from − − GM /a to − − GM /a corresponds to increasing a from ∼ − q − − a h to ∼ . q − − a h . In both experiments for E > E < J and the loss cone filling rate is isotropic, i.e., the cross sections obtained in § § d Σ i ( r ti ) /dJ / Σ i ( a )with respect to J . The triangles and squares, respectively, refer to the primary and sec-ondary BHs. d Σ i ( r ti ) /dJ / Σ i ( a ) is obtained as follows. For each bin of ∆ J we countthe number ∆ N i ( r ti ) of particles whose initial angular momenta fall in this bin and min-imum separations from the i th BH are less than r ti . Here r ti is from equation (8) with M = 1 and we only use the data for multi-encounters. Then d Σ /dJ is calculated with∆ N i ( r ti ) / ∆ J /N i ( a ) so that the integration over J results in the normalized cross sectionof tidal disruption. The Poissonian errors in the counts have been indicated by the error barsin Figure 4. Note that the differential cross section [or ∆ N i ( r ti ) alone] derived in this way isproportional to the probability of tidal disrupting a particle with initial angular momentum J . Generally speaking, both differential cross sections for the primary and secondary BHsare flat at J < J ≃ GM a . The differential cross sections start to cut off above J andparticles with initial angular momentum greater than 2 J contribute little to the cross sec-tions. When q ≪ J at J ≪ J lc but even in this case the contribution to the total cross section from particleswith larger J is still significant.The cross sections also depend on the direction of the initial angular momentum vector J . We define θ as the relative angle between J and binary’s orbital angular momentum.Figure 5 shows the differential tidal disruption cross sections with respect to cos θ . Thedifferential cross sections are obtained following the same described above. When q ≪ GM a ) for q = 0 .
01 ( top ), q = 0 . middle ), and q =1 ( bottom ). Triangles and squares, respectively, refer to the differential cross sections ofprimary and secondary BHs. Error bars indicate the Poissonian errors in the counts of tidaldisruption events. When differential cross section is low, large perturbation occurs due tothe noise in the counts. 18 –particles on corotating orbits (cos θ >
0) are more likely to be disrupted, while for secondaryBHs the dependence of the cross sections on θ is weak. Unless the stellar distribution isextremely flattened or the star cluster around SMBHB is significantly rotating the totalcross section would not differ much from those presented in Figures 1 and 2. To see clearly the effect of the eccentricity of binary’s orbit, we set an extreme eccen-tricity e = 0 . q = 0 .
01 and 0 .
1, each with 10 particles. The results (solid lines) are presented in Figure 6. Cross sections for e = 0 (dottedlines, from § To study the effect of GR we set the gravitational potential to be pseudo-Newtonianwith equation [13] and M = 1 and then repeated the scattering experiments. For each q theresult from 10 particles is shown in Figure 7 as solid lines. Dotted lines are results from thenon-relativistic experiments. Below the marginally bound radius 2 r S both the cross sectionsfor multi- and single-encounters become constant, representing the relativistic effect.The dashed lines are empirical cross section in the GR case. For the single-encounterswe have used equation (14) and set the cross sections constant between r = r S and 2 r S . Theresulting cross sections agree well with the numerical ones. The approximate cross sectionsto the multi-encounters are obtained by multiplying the non-relativistic cross sections bythe correction factor r/ ( r − r S ) and keeping the cross sections constant below 2 r S . Theseresulting cross sections well agree with the numerical ones as long as r S ≪ q a , i.e., in thegravitational focusing dominated regime. Due to GR effect the cross sections at r = r S arefour times greater than the non-relativistic ones.For other values of M the location of Schwarzschild radius moves along the r/a axis.Instead of repeating the whole simulation we use non-GR cross sections to calculate theapproximate relativistic cross sections, following the procedure described above. 19 –Fig. 5.— Differential tidal disruption cross sections with respect to cos θ for q = 0 .
01 ( top ), q = 0 . middle ), and q = 1 ( bottom ). Triangles and squares, respectively, refer to theprimary and secondary BHs. 20 –Fig. 6.— Normalized multi- and single-encounter cross sections from 10 particles (solidlines) for elliptical binaries with e = 0 . q = 0 .
01 ( top ) or q = 0 . bottom ). Dotted lines areresults from the circular binary case. Short vertical lines have the same meanings as thosein Fig. 1. 21 –Fig. 7.— Relativistic cross sections from 10 particles (solid lines) for q = 0 .
01 ( top left ), q = 0 . bottom left ), and q = 1 ( bottom right ). The empirical cross sections in GR case(dashed lines) and the cross sections from non-GR experiments (dotted lines) are also plotted.Short vertical lines have the same meanings as those in Fig. 1. 22 –
5. STELLAR DISRUPTION RATES IN NEARBY GALAXIES5.1. Galaxy Sample
In this section we estimate the tidal disruption rates (eq. 9) of stellar objects for asample of nearby galaxies. The largest uncertainty in our calculation comes from the losscone refilling rate F lc ( a ); for simplicity, and to ease comparison with earlier work, we assumethat two-body relaxation is the only process contributing to F lc ( a ).Our galaxy sample consists of 51 nearby elliptical galaxies which are listed in Table 1.For each galaxy the BH mass is obtained with H¨aring & Rix ’s M • − M bulge relation, wherethe bulge mass M bulge is from the scaling relation between stellar mass and luminosity(Magorrian et al. 1998). The assumed mass to light ratios and the resulting BH massesare given by columns 2 and 3 in Table 1. Table 1. Galaxy sample Υ V log M • log F lcsingle log a stall log t evol log F lcbinary log ˙ N d log ˙ N d log ˙ N d log ˙ N d log ˙ N d log ˙ N d Name ( M ⊙ /L ⊙ ) ( M ⊙ ) (yr − ) (pc) (yr) (yr − ) (yr − ) (yr − ) (yr − ) (yr − ) (yr − ) (yr − )(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)A2052 7.41 9.17 -5.89 1.76 13.55 -4.88 -9.75 -12.54 -10.33 -11.87 -11.11 -11.41NGC 1023 4.88 7.84 -4.13 0.25 9.98 -2.65 -6.47 -9.71 -7.03 -8.54 -7.59 -7.61NGC 1172 5.39 8.16 -4.21 1.09 10.50 -2.85 -7.38 -10.20 -7.97 -9.49 -8.76 -9.06NGC 1316 7.60 9.25 -4.64 0.89 12.46 -3.71 -7.72 -10.47 -8.25 -9.80 -9.04 -8.94NGC 1399 6.33 8.67 -5.28 0.87 12.73 -4.57 -8.73 -11.60 -9.27 -10.83 -10.10 -10.10NGC 1400 5.69 8.33 -4.79 0.97 11.49 -3.67 -8.05 -11.33 -8.59 -10.17 -9.57 -9.57NGC 1426 5.04 7.95 -4.44 0.95 10.38 -2.94 -7.45 -10.29 -7.97 -9.51 -8.78 -8.78NGC 1600 7.44 9.19 -5.80 1.33 13.81 -5.13 -9.57 -12.49 -10.13 -11.70 -10.97 -10.91NGC 1700 6.25 8.63 -4.37 1.18 10.84 -2.71 -7.22 -10.06 -7.74 -9.24 -8.55 -8.55NGC 221 2.71 5.96 -3.68 -3.34 8.52 -3.16 -4.96 -6.48 -4.84 -6.07 -5.37 -5.37NGC 224 4.62 7.67 -4.54 -0.47 10.91 -3.75 -7.00 -9.81 -7.49 -8.98 -8.04 -8.10NGC 2636 3.93 7.15 -5.05 0.88 9.95 -3.30 -7.99 -11.30 -8.53 -10.10 -9.49 -9.49NGC 2832 7.76 9.32 -5.53 1.57 13.57 -4.75 -9.41 -12.69 -9.94 -11.44 -10.95 -10.95NGC 2841 4.66 7.69 -4.59 0.74 10.21 -3.02 -7.36 -10.68 -7.92 -9.46 -8.92 -8.92NGC 3115 5.39 8.16 -4.03 0.56 10.19 -2.53 -6.56 -9.59 -7.12 -8.62 -7.87 -7.80NGC 3377 4.53 7.60 -3.85 -0.40 10.04 -2.94 -6.28 -9.07 -6.77 -8.26 -7.37 -7.36NGC 3379 5.21 8.05 -4.93 0.63 11.48 -3.93 -8.07 -10.89 -8.62 -10.13 -9.31 -9.31NGC 3599 4.55 7.61 -4.61 0.92 10.16 -3.05 -7.63 -10.41 -8.18 -9.70 -9.04 -9.26NGC 3605 4.13 7.31 -4.78 0.44 10.00 -3.19 -7.36 -10.26 -7.94 -9.42 -8.79 -8.87NGC 3608 5.48 8.21 -4.79 0.94 11.12 -3.41 -7.80 -11.07 -8.35 -9.91 -9.31 -9.31NGC 4168 6.39 8.70 -5.56 1.20 12.77 -4.58 -9.09 -11.93 -9.61 -11.11 -10.45 -10.39NGC 4239 3.50 6.78 -5.24 -0.02 10.02 -3.74 -7.67 -10.47 -8.19 -9.70 -8.90 -8.90NGC 4365 6.71 8.86 -5.24 0.96 12.82 -4.46 -8.65 -11.50 -9.21 -10.69 -10.06 -10.14NGC 4387 3.95 7.17 -4.83 0.06 10.02 -3.36 -7.26 -10.09 -7.78 -9.25 -8.40 -8.49NGC 4434 4.01 7.22 -4.48 -0.37 9.98 -3.27 -6.74 -9.42 -7.25 -8.74 -7.86 -7.85NGC 4458 4.01 7.22 -4.31 -0.39 9.99 -3.28 -6.73 -9.43 -7.25 -8.73 -7.86 -7.84NGC 4464 3.54 6.82 -3.99 -2.38 9.78 -3.56 -5.73 -7.62 -5.79 -7.13 -6.40 -6.40NGC 4467 2.92 6.20 -4.52 -2.89 9.59 -3.99 -6.01 -7.73 -5.96 -7.27 -6.55 -6.55NGC 4472 7.29 9.12 -5.30 0.83 13.29 -4.67 -8.66 -11.43 -9.20 -10.72 -9.95 -9.89NGC 4478 4.49 7.58 -4.78 0.80 10.27 -3.20 -7.66 -10.55 -8.20 -9.76 -9.03 -8.97NGC 4486 7.05 9.02 -5.57 0.83 13.40 -4.89 -7.27 -10.08 -7.76 -9.25 -8.34 -8.40NGC 4486b 3.18 6.48 -5.11 -0.89 10.00 -4.03 -8.91 -11.65 -9.45 -10.98 -10.24 -10.14NGC 4551 4.10 7.28 -4.70 0.01 10.00 -3.23 -7.03 -10.27 -7.56 -9.09 -8.18 -8.19
24 –
For each galaxy we built simple spherical models using the same Nuker law parametersas those in Wang & Merritt (2004) and separately calculated the loss cone filling rates dueto two-body interaction in single and binary BH cases (see MT99 and Y02 for detaileddescription of the calculation).For single BHs we did two sets of calculations to investigate the effect GR on loss conefilling rate F lcsingle . In the non-GR case we calculated J lc with r t from equation (1) while inthe GR case we substituted r t with 2 r S if r t < r S . In both cases the resulting loss conefilling rates are similar, within a factor of 1 . r S ≫ r t . This is because the losscones of the most massive BHs are diffusive so that F lcsingle is insensitive to the size of losscone. In Table 1 we only give F lcsingle for the non-GR case. The mean of log F lcsingle is − . − .
44 for galaxies with M • < M ⊙ , which is within a factor of 2of the rates obtained by Wang & Merritt (2004), the differences being completely accountedfor by our different assumed BH masses.For binary BHs GR effect is not important to loss cone filling rate because we areinterested in the case a ≫ r S . For each galaxy we have calculated the loss cone filling ratesfor binary BHs with a set of a and derived the evolution timescale t evol ( a ) with1 t evol ( a ) = 1 t h ( a ) + 1 t gr ( a ) (17)(Y02), where t gr is the evolution timescale due to gravitational wave radiation (Peters 1964).Then the binary’s stalling radius a stall is determined by locating the peak of the t evol ( a ) curve.If t evol ( a stall ) > yr, which is the case for most of our model galaxies, we increase a stall until (1) t evol ( a stall ) < yr or (2) a stall > . . a stall , t evol ( a stall ), and F lc ( a stall ) given by either criteria (1)or (2) are independent of q . The results for q = 0 .
1, the typical mass ratio in the literaturefor the present-day SMBHBs, are given in Table 1, while the results for q = 0 .
01 and 1 areessentially identical. Although the loss cone of SMBHB in E − J space could be morethan four orders of magnitude wider than that of a single BH (eq.s [6] and [8]), the loss conefilling rate for binary BH increases only by a factor of 10, which implies that the loss coneof SMBHB is also in the diffusive regime. Table 1—Continued Υ V log M • log F lcsingle log a stall log t evol log F lcbinary log ˙ N d log ˙ N d log ˙ N d log ˙ N d log ˙ N d log ˙ N d Name ( M ⊙ /L ⊙ ) ( M ⊙ ) (yr − ) (pc) (yr) (yr − ) (yr − ) (yr − ) (yr − ) (yr − ) (yr − ) (yr − )(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)NGC 4552 5.66 8.32 -4.81 0.82 11.79 -3.98 -8.19 -11.03 -8.77 -10.32 -9.58 -9.88NGC 4564 4.72 7.73 -4.24 0.23 10.00 -2.77 -6.61 -9.86 -7.16 -8.66 -7.79 -7.88NGC 4570 4.80 7.79 -4.11 -0.20 9.97 -2.69 -6.16 -8.93 -6.67 -8.14 -7.25 -7.23NGC 4621 5.88 8.43 -4.11 0.82 10.53 -2.60 -6.78 -9.65 -7.35 -8.83 -8.14 -8.14NGC 4636 6.28 8.65 -5.52 0.80 12.95 -4.81 -8.93 -11.85 -9.49 -11.00 -10.14 -10.14NGC 4649 6.80 8.90 -5.37 0.82 13.09 -4.70 -8.75 -11.76 -9.30 -10.79 -10.04 -9.97NGC 4697 5.64 8.30 -4.69 0.66 11.05 -3.25 -7.34 -10.61 -7.90 -9.45 -8.59 -8.55NGC 4742 4.18 7.35 -3.50 -1.98 9.80 -3.05 -5.35 -7.34 -5.48 -6.85 -6.11 -6.10NGC 4874 8.57 9.64 -6.07 1.61 14.50 -5.37 -9.92 -12.73 -10.51 -12.06 -11.29 -11.59NGC 4889 8.33 9.54 -5.77 1.60 14.13 -5.09 -9.67 -12.45 -10.24 -11.78 -11.07 -11.30NGC 524 6.13 8.57 -5.10 1.00 12.13 -4.06 -8.38 -11.73 -8.94 -10.45 -9.84 -9.97NGC 5813 6.44 8.73 -5.14 1.10 12.51 -4.29 -8.67 -11.95 -9.21 -10.79 -10.09 -10.21NGC 5845 4.66 7.69 -4.41 0.78 10.00 -2.81 -7.20 -10.48 -7.77 -9.31 -8.73 -8.73NGC 596 5.52 8.24 -4.61 0.95 10.80 -3.07 -7.46 -10.73 -8.02 -9.57 -8.97 -8.97NGC 6166 8.46 9.60 -6.24 1.69 14.68 -5.59 -10.26 -13.55 -10.81 -12.39 -11.79 -11.79NGC 720 6.23 8.62 -5.56 0.96 13.00 -4.89 -9.17 -11.93 -9.72 -11.23 -10.57 -10.79NGC 7332 4.70 7.72 -3.73 -0.56 10.01 -2.80 -5.96 -8.73 -6.44 -7.90 -6.98 -7.02NGC 7768 7.73 9.31 -5.42 1.62 13.36 -4.56 -9.27 -12.19 -9.82 -11.36 -10.76 -10.76Note. — Column 1 is the galaxy name. Column 2 is the V -band mass-to-light ratio and Column 3 gives the corresponding BH mass according to the M • − M bulge relation (H¨aring & Rix 2004). Columns 4 is the loss cone filling rates for single BHs in non-GR case. Columns 5, 6, and 7 list the stallingradius, evolution timescale, and loss cone filling rate for SMBHBs with mass ratio q = 0 .
1, the results for q = 0 .
01 and 1 being essentially identical. Both t evol and F lcbinary are evaluated at a stall , given by the minimum of equation 17. The tidal disruption rates for q = 0 .
01 are given in Columns 8 and 9,respectively for the primary and secondary BHs. Columns 10 and 11 have the same meanings as those in Columns 9 and 10, but for q = 0 .
1, while Columns12 and 13 are for q = 1.
26 –
First, we ignore GR effect and only consider tidal disruption of solar type stars. In thiscase the stellar disruption rates for single BHs have been given by column 4 in Table 1. Forbinaries the rates are calculated according to equation (9) with the loss cone filling rateslisted in Table 1 and the non-relativistic cross sections from § N d binary =˙ N d + ˙ N d ) is significantly lower than that for single BH ( ˙ N d single ). The contrast between˙ N d binary and ˙ N d single is more than one order of magnitude and increases with q . The significantreduction of tidal disruption rates in binary BH systems originates in the depletion of loss-cone stars and the suppression of loss cone filling rate by the SMBHB.Mechanisms other than two-body relaxation would increase the loss cone filling ratesin both single and binary SMBH systems (MT99; Y02; Merritt & Milosavljevi´c 2005). Theenhancement in loss cone filling rate is uncertain because it depends on processes whichare not well understood. However, equation (7) implies that the lifetime of hard SMBHBdecreases with loss cone filling rate. So for long-life SMBHB with t h & yr the loss conefilling rate should not significantly exceed F lccri ≡ M / (2 Km ∗ yr). Correspondingly thestellar disruption rate in SMBHB system is unlikely to be much greater than˙ N d cri ≡ F lccri (cid:20) Σ ( r t )Σ ( a h ) + Σ ( r t )Σ ( a h ) (cid:21) . (18)These upper limits of stellar disruption rates for binary BHs have been presented in Figure 8as dashed lines. In 48 out of 51 cases ˙ N d binary is below ˙ N d cri , reflecting that two-body relaxationis inefficient in refilling the loss cone. For galaxies with M • . M ⊙ ˙ N d cri is always lowerthan ˙ N d single , so long-life SMBHBs in these galaxies would manifest themselves by prominentlysuppressing stellar disruption rates no matter what mechanism is responsible for the losscone replenishing. At M • & M ⊙ ˙ N d cri could be higher than ˙ N d single if M • ≫ M ⊙ or q ≪
1, implying that in these galaxies SMBHBs are not so distinct in the efficiency of stellardisruption.
In the relativistic case there is a critical mass M cri above which a BH would swallowthe whole star without tidal disruption (Hills 1975). For solar type stars recent relativisticsimulation (Ivanov & Chernyakova 2006) gave that M cri ∼ (4 − × M ⊙ if the BH isnon-spinning and M cri ∼ M ⊙ if the BH is maximally spinning. Due to the ambiguity of 27 –Fig. 8.— Tidal disruption rates of solar-type stellar objects as a function of total BH massin Newtonian theory. Dots and circles, respectively, refer to the disruption rates for primaryand secondary BHs. Triangle are for single BHs. Dashed lines are thresholds above whichSMBHBs would coalesce within 10 yr. The tidal disruption rates are derived under theassumptions that each galaxy is spherical and that the loss cone refilling is due to two-bodyrelaxation. 28 –BH spin it is not clear whether a star approaching a BH more massive than 10 M ⊙ wouldnecessarily be disrupted and produce a flare. However, it is expected that stellar disruptionsby these massive BHs are rare because only those stars approaching the spinning BH alonga corotating orbit close to the equatorial plane could get disrupted. For less massive BHs( M • . M ⊙ ) although equation (1) suggests that tidal disruption is inevitable ( r t & r S ),part of the stellar debris with low angular momentum may directly plunge into the eventhorizon without producing a flare (Nolthenus & Katz 1983). The proportion of such debrisis sensitive to the spin of the star just before tidal disruption and is uncertain so far, butthe proportion should be low when r t ≪ r S . Because of all these uncertainties, we adopteda rough assumption that flares are produced only in the condition r t > r S .Figure 9 shows the stellar disruption rates when GR effect is taken into account. Thetidal disruption rates for binary BHs are calculated with the loss cone filling rates fromTable 1 and the relativistic tidal disruption cross sections presented in § r t / ( r t − r S ) times greater than the non-relativisticones if r t > r S , or 4 r S /r t times greater if r S < r t < r S . The dashed lines show thethresholds ˙ N d cri above which SMBHBs would coalesce within 10 yr. They are derivedaccording to equation (18) but with relativistic cross sections. To study the frequency oftidal flares we have set the stellar disruption rates zero when r t < r S , thus the sudden cutoffat ∼ (1 + q )10 M ⊙ and ∼ q − (1 + q )10 M ⊙ is artificial.At M • ≪ M ⊙ the stellar disruption rates are similar to those presented in Figure 8and again we find that the stellar disruption rates for SMBHBs are considerably lower thanthose for single BHs. While at M • ≫ M ⊙ the only sources producing tidal flares are thesecondary BHs, though the rates are low. The results of previous sections are based on the assumption that each galaxy is com-posed of stars with solar mass and radius. Since none of our sample galaxies exhibits recentnuclear star formation, taking into account a mass spectrum of main-sequence stars wouldresult in numerous low mass stellar objects which have smaller mean tidal disruption crosssections and are less efficient to relax via two-body interaction. But these effects would notqualitatively change the tidal disruption rates since: (a) the tidal radius is not sensitive tothe stellar mass because r ∗ ∼ m . ∗ (Bond, Arnett & Carr 1984) so that r t ∼ m . ∗ ; (b)the increment in stellar number density to reproduce the mass distribution inferred fromobservations compensates for the decrease in the efficiency of two-body relaxation. 29 –Fig. 9.— Tidal disruption rates of solar-type stellar objects as a function of total BH mass,including general relativisitic effects. Symbols and dashed lines have the same meanings asthose in Figure 8. 30 –However, tidal disruption of off-main-sequence giant stars becomes important whenBHs are more massive than 10 M ⊙ because in this case the tidal radius is greater than theSchwarzschild radius. Following MT99 we assume that giant stars with time-averaged radius r g = 15 r ⊙ contribute g = 1% to the total stellar population. These values are consistent withthe approximate stellar evolution model given in Syer & Ulmer 1999. For single BHs thedisruption rates of giants in the diffusive loss cone limit are obtained with gF lcsingle (MT99),where the loss cone filling rates F lcsingle in relativistic case are from Table 1. For binary BHsthe disruption rates are calculated in the same way as in § r t < r S .Figure 10 shows the stellar disruption rates and the thresholds ˙ N d cri when both solartype and giant stars are taken into account. At M • . M ⊙ the stellar disruption ratesare dominated by disruption of solar type stars therefore similar to those in Figure 9. At M • > M ⊙ disruption of giant stars take over in both single and binary BH systems,resulting in low but no longer zero flaring rates. The suppression of stellar disruption rateby SMBHB is still obvious if the loss cone refilling is dominated by two-body relaxation, butother mechanisms could potentially enhance the stellar disruption rates for massive SMBHBs( M • > M ⊙ ) to a level indistinguishable from those for single BHs.
6. RATES OF TIDAL FLARES IN LOCAL UNIVERSE
To estimate the density of tidal flares in local universe we adopted the mass functionsof SMBH for early (E+S0) and late (Sabcd) type galaxies from Marconi et al. (2004), whichare converted from the distribution of stellar velocity dispersion using the empirical M • − σ ∗ relation (Tremaine et al. 2002). The mass functions are presented in Figure 11 and inthe following we assume that the total BH mass ( M ) in SMBHB systems follows thesame distribution. At M • < M ⊙ the mass distribution of BHs is unclear because thedemographics of the BHs in dwarf galaxies is not well established. The total BH massdensity according to Figure 11 is 2 . × M ⊙ Mpc − ( H = 70 km s − Mpc − throughoutthis paper), consistent with the value given by Yu & Tremaine (2002).The flaring rates per unit volume are calculated by integrating the BH mass functionsweighted by the flaring rates. The flaring rates for single and binary BHs are from Fig-ures 9 (we have smoothed the flaring rates) and the fiducial field of integration is M • ( M ) ∈ [10 M ⊙ , M ⊙ ]. In single BH case we do a second set of calculations with M • ∈ [10 M ⊙ , M ⊙ ]to account for the flaring rates from the intermediate-mass BHs (IMBHs, 10 M ⊙ < M • < M ⊙ ) at the centers of dwarf galaxies, while for binary BHs we skip such calculations 31 –Fig. 10.— Total tidal disruption rates both of stellar-type stellar objects and red giants as afunction of total BH mass, including general relativistic effect and assuming that red giantscontribute 1% to the loss cone filling rate. Symbols and dashed lines have the same meaningsas those in Figure 8. 32 –Fig. 11.— number density of SMBHs as a function of stellar velocity dispersion or BH mass(Marconi et al. 2004). 33 –because most dwarf galaxies have not experienced mergers (Haehnelt & Kauffmann 2002).The flaring rates at M • M ⊙ are from the linear extrapolation of the logarithmic flaringrate at M • > M ⊙ . We do not consider flares produced by giants because we will see belowthat these flares differ from those produced by normal stars. Our result is not sensitive tothis treatment because the mass function of SMBHs cuts off steeply above 10 M ⊙ whereflares from giants become dominant.The flaring rates per unit volume for different types of galaxies and for both single andbinary BHs are presented in Table 2. Note that taking into account asymmetric stellar dis-tributions and relaxation mechanisms other than two-body interaction would significantlyincrease the loss cone filling rates in both single and binary BH systems, therefore dramat-ically increase the flaring rates. According to Table 2, it is obvious that if SMBHBs areubiquitous in the centers of both early and late type galaxies the flaring rate in local uni-verse would be more than one order of magnitude lower than that in single BH case. It isalso clear that dwarf galaxies, if they follow the same mass distribution as that of massivegalaxies and harbor IMBHs at their centers, would contribute the majority to the flares inlate type galaxies but have little effect on the flaring rate in early type galaxies.If a fraction, f bin , of SMBHs in the investigated galaxies are in binary and a fraction1 − f bin are single, the expected tidal disruption rate is˙ N f = (1 − f bin ) ˙ N f single + f bin ( ˙ N f + ˙ N f ) . (19)˙ N f as a function of f bin is presented in Figure 12. It shows that ˙ N f is not sensitive to q unless f bin ≃
1. In § f bin − ˙ N f diagrams to constrain f bin for differenttypes of galaxies.
7. DISCUSSION
The hierarchical structure formation model in CDM cosmology predicts that SMBHBsare continuously formed across the merging history of galaxies. Constraining the abundanceof SMBHBs among inactive galaxies is essential to test the theoretical models of the dynamicevolution of binary BHs. In this paper we have studied the effect of hard binary BHs onstellar disruption rates, trying to find distinct observational signatures of SMBHBs in galaxycenters. We focus our attention on inactive galaxies which contain the final products of BHmergers so we do not consider the effect of gas on SMBHB evolution.We have carried out numerical scattering experiments designed for hard SMBHBs toinvestigate the probability of stellar disruption. Since r t ≪ a h , in each set of experimentsusually a large number of test particles ( N ≃ ) were used to give statistical meaningful 34 –Fig. 12.— Expected tidal disruption rates of stellar objects by black holes as a function ofthe fraction of binaries for all (upper), early (lower left), and late (lower right) type galaxies.Solid lines are our calculated flaring rates for spherical galaxies with two-body relaxation.Dotted, short-dashed, and dot-dashed lines are calculated by simply multiplying the solidlines with, respectively, 2, 10, and 100. The long-dashed horizontal lines indicates the bestestimates of the flaring rates in real universe and the shaded regions corresponding to anassumed statistical uncertainty of one order magnitude. 35 –results. For solar type stars the test particle approximation should be valid outside r t as longas E ≪ GM /a . In this situation the self-binding energy of a star ∼ Gm ∗ /r ∗ is smaller bya factor ( M • /M ∗ ) / than its kinetic energy at r t so the loss of orbital energy due to tidaldissipation is not important. The test particle approximation may be problematic for giantstars because they are much less concentrated and may suffer from mass stripping duringtheir encounters with BHs. This uncertainty could be incorporated into the uncertainty in r g . In single BH case the stellar disruption rate is not sensitive to r g if the loss cone isin the diffusive regime but would be proportional to r g if the loss cone is in the pinholeregime (MT99). For binary BHs the loss cone filling rate does not depend on r g but the tidaldisruption cross section scales with r g when gravitational focusing is important. Thereforeunless the effective r g deviates significantly from our choice r g = 15 r ⊙ our results for giantstars would not be qualitatively different.We find that in binary BH systems the multi-encounter cross sections are greater thanthe single-encounter ones. This is because binary BHs tend to scatter particles onto tem-porary bound orbits which enhances the number of star-binary encounters. If a star is onlypartly disrupted during each close passage from the BHs, multi-flares would be producedoccur due to the multi-encounters, which provides a distinct observational signature of SMB-HBs. However, we have seen in § r t ≪ a . Even if r t ∼ a equation (12) implies that a/r S1 ∼ M − / so grav-itational wave radiation would quickly drive the BHs to coalesce, making the multi-flaresunlikely. A secondary flare could also be produced if the secondary member of SMBHBcaptures sufficient debris from the star disrupted by the primary BH, or vise versa. Theprobability of these events depends on the opening angle of all the spewed debris about thestellar disrupting BH, which needs to be determined by hydrodynamic simulations.We have shown that the normalized cross sections are not sensitive to the star’s initialbinding energy or to the eccentricity of the BH binary, but do depend on the amplitude andorientation of the star’s initial angular momentum. The cross sections presented in Figures 1and 7, which have been equally averaged over the whole loss cone and over all directions,can be directly used if the loss cone of SMBHB is isotropic and is in the pinhole regime. Inthe diffusive loss cone limit the stars falling to the binary BHs satisfy J & J lc . According toFigure 4 the differential tidal disruption cross sections at J & J lc are lower than those insidethe loss cone, so the mean tidal disruption cross section in the diffusive limit is smaller thanthose presented in Figures 1 and 7.We find that the tidal disruption rates in binary BH systems are more than one order ofmagnitude lower than those in single BH systems. The suppression of stellar disruption rateby SMBHB originates in the low loss cone filling rate in binary BH systems, which results 36 –from the stellar cusp destruction due to the hardening of SMBHBs. Even efficient losscone filling mechanisms are taken into account the contrast between the stellar disruptionrates in single and binary BH cases are still prominent in less massive galaxies with M • . M ⊙ , because the survival of SMBHBs requires that the loss cone filling rate should notexceed a critical value which is proportional to the total BH mass of SMBHB. Our result isqualitatively different from that of Ivanov et al. (2005), who suggested that a SMBHB willenhance the stellar disruption rates. The difference is due to that we are studying a muchlater evolution stage at which the dense galactic cusps considered by Ivanov et al. have beendestroyed during the hardening of SMBHBs. According to the calculation in Y02 binaryBHs reach the hard radius in about a dynamic friction timescale of the merging galaxy. Sothe situation considered by Ivanov et al. (2005) applies for a short period relative to thelifetime of a galaxy. Merritt & Wang (2005) suggested that because loss cone refilling takestime the stellar disruption rate immediately after SMBHB coalescence is significantly belowthe final steady disruption rate. But this effect is prominent only in massive galaxies with M • > M ⊙ , where the relaxation timescale is long. Our result implies that in less massivegalaxies the flaring rate could give stringent constraint to the abundance of SMBHB.In larger galaxies with M • > M ⊙ tidal flares are dominated by disruption of giants.However, their spectral and variable properties may be different from those of solar typestars. A tidal flare is expected to be produced at a scale about r t , initially radiating atthe Eddington limit L Edd ≃ . × ( M • / M ⊙ ) erg s − with a thermal spectrum ofeffective temperature T eff ≃ [ L Edd / (4 πr t σ )] / = 2 . × ( r ∗ /r ⊙ ) − / ( m ∗ /M ⊙ ) / M / Kand decaying on a timescale t flare ∼ πGM • (2∆ E ) − / ≃ . r ∗ /r ⊙ ) / ( m ∗ /M ⊙ ) − M / ,where ∆ E ∼ GM • r ∗ /r t characterizes the span of the binding energy of the debris (Rees1988). For solar type stars the spectrum of tidal flare peaks at UV or soft X-ray and theluminosity decays in about M / year. While for giant stars with r ∗ = 15 r ⊙ and m ∗ = m ⊙ the spectrum peaks at optical or UV band and the flare dims on a timescale of about 60 M / years. Therefore if a UV/X-ray outburst decaying within months to one year is detected ina galaxy with total BH mass much greater than 10 M ⊙ (determined by other observations),it is likely that a less massive secondary SMBH resides in the galactic center. Such event isanother signature of SMBHB, though Figure 10 suggests that the frequency of such event islow. So far six candidate tidal disruption events have been observed in nearby inactive galax-ies. Among them, one was observed at z = 0 .
37 by the UV telescope
GALEX (Gezari et al.2006) and the rest were discovered in galaxies at z . .
15 during the one year ROSAT All SkySurvey (
RASS ) (Komossa 2002). The Hubble types for the host galaxies of the five
RASS flares are not well determined: NGC 5905 is of a SB galaxy and the other four galaxies looklike ellipticals (Komossa 2002). From the observations of
RASS
Donley et al. (2002) inferred 37 –a total flaring rate of ˙ N f obs ∼ × − yr − Mpc − for all types of galaxies. From the RASS results we can also give a rough estimate of the flaring rates for different types of galaxies:˙ N f obs ∼ . × − yr − Mpc − for early type galaxies and ˙ N f obs ∼ × − yr − Mpc − for latetypes. We plot ˙ N f obs as long-dashed horizontal lines in Figure 12 and the corresponding un-certainties with shaded areas. Because of the poor statistics in ˙ N f obs and the ill-determinationof Hubble types of the host galaxies, the results are highly uncertain.We did the calculation based on the assumptions that galaxies are spherical and two-body scattering relaxation dominates the loss-cone refilling. However, taking into accountmore realistic non-spherical stellar distributions and other loss-cone refilling mechanismsbesides two-body interaction will probably significantly increase the loss-cone refilling ratesin both single and binary SMBH systems. MT99 and Y02 showed that taking into accountaxisymmetric stellar distribution will increase the loss cone refilling rates by a factor of 2 forboth single SMBHs and hard SMBHBs. The loss cone refilling rates in triaxial galaxies areslightly higher than those in axisymmetric ones if the triaxiality is less than 0 . −
10 relativeto those in the case of two-body relaxation, depending on the eccentricities of stellar orbits(Rauch & Tremaine 1996; Rauch & Ingalls 1998; G¨urkan & Hopman 2007). To calculate thetidal disruption rates by taking into account the above physical processes is out of the scopeof this paper. To illustrate the effects of increased loss-cone refilling rates and tidal disruptionrates on the estimated fraction of binary BHs, f bin , we simply multiply our calculated stellardisruption rates ˙ N f by 2, 10, and 100 and plot the results in Figure 12. In Figure 12, theintersections of long-dashed and solid lines give the lower limit to f bin , while the intersectionsof long-dashed and dot-dashed lines roughly give the upper limits. When all types of galaxiesare considered, binary fraction f bin is between 0 . .
0. For early type galaxies, the solidlines do not intersect the long-dashed line, implying that other loss-cone filling mechanisms inaddition to two-body relaxation should also be important. For late type galaxies, Figure 12suggests a binary fraction f bin ∼
75% and that extreme triaxiality may not be common,otherwise the current observation would imply an extremely high fraction of SMBHBs. Wewould like to note that the constraints on f bin are very uncertain due to the large uncertaintyin the current ˙ N f obs . Detection of more candidate tidal disruption events is needed to givebetter constraints on f bin and on the dominant loss-cone filling mechanisms in different typesof galaxies.The non-detection of X-ray or UV flares in dwarf galaxies seems inconsistent with thecalculated high flaring rates. The discrepancy may be due to (1) a lack of sufficient sensitivityin current surveys, (2) the continuous and steady accretion of stellar debris onto IMBHsduring successive tidal disruption events (Milosavljevi´c et al. 2006), or (3) a lack of IMBHs 38 –in the centers of dwarf galaxies. Future X-ray and UV surveys with improved sensitivity andangular resolution would help constraining the fraction of IMBHs in galaxy centers and giveinteresting implications to the formation and the merging history of IMBHs (Volonteri et al.2003; Volonteri 2007).
8. CONCLUSIONS
To investigate whether SMBHBs are ubiquitous in nearby inactive galaxies or coalescerapidly in galaxy mergers, we have studied the tidal disruption rates of stellar objects inboth single and binary SMBH systems. We have calculated the interaction cross sectionsbetween hard SMBHBs and intruding stars by carrying out intensive numerical scatteringexperiments with typically 10 particles, taking into account the initial binding energy andangular momentum of particle, the eccentricity of the orbit of SMBHB, and including generalrelativistic effects. We have also derived empirical formulae for the relativistic cross sections,which can be applied to SMBHBs with a wide range of semimajor axis and todal BH mass.We have calculated the rate of loss cone refilling due to two-body relaxation for asample of 51 nearby galaxies, assuming that each galaxy is spherical. The steady loss conefilling rates in binary BH systems would be significantly suppressed due to the three-bodyinteraction between SMBHBs and stars passing by. We have calculated the tidal disruptionrates respectively for single and binary SMBHs by combining the loss cone filling rates andthe tidal disruption cross sections. We find that the tidal disruption rate in SMBHB systemsis more than one order of magnitude lower than that in single SMBH systems. For galaxieswith BHs more massive than 10 M ⊙ , a UV/X-ray flare at galactic center decaying withinone year provides strong evidence of a secondary BH, although the probability of such eventsis low.Finally we have calculated the flaring rates in local universe using the BH mass functiongiven in the literature. The comparison of the calculated flaring rates and the preliminaryresults from current X-ray surveys could not yet tell whether SMBHBs are ubiquitous in localuniverse. Future UV/X-ray surveys with improved sensitivity and duration are needed.We are grateful to Dr. Xue-Bing Wu, Dr. Stefanie Komossa, and Dr. Vladimir Karas formany constructive comments. We thank the referee for helpful comments and suggestions.Many thanks are due to Bing-Xiao Xu, Ran Wang, Lei Qian and Zhao-Yu Li for fruitfuldiscussions. During this work we have used the SGI Altix 330 system at the Departmentof Astronomy, Peking University (PKU) and CCSE-I HP Cluster of PKU. This work issupported by the National Natural Science Foundation of China (No. 10573001) and by 39 –the national 973 program (No. 2007CB815405). JM thanks the Royal Society for financialsupport. REFERENCES
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43 –Table 2. Flaring rate per unit volume in yr − Mpc − Galaxy type ˙ N f single ˙ N f single q ˙ N f ˙ N f (1) (2) (3) (4) (5) (6)Early 6 . × − . × − . × − . × − Early 6 . × − . × − . × − . × − Early 6 . × − . × − . × − . × − Late 3 . × − . × − . × − . × − Late 3 . × − . × − . × − . × − Late 3 . × − . × − . × − . × − Note. — Column 1 gives the galaxy type: ’Early’=E+S0, ’Late’=Sabcd.Columns 2 and 3 are the flaring rates per unit volume in single BH case, theformer being calculated for M • > M ⊙ while the latter for M • > M ⊙ . q is the assumed mass ratio of SMBHB, and ˙ N f and ˙ N f2