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5-22-2006 Mass Deficits, Stalling Radii, and the Merger Histories of Elliptical David Merritt Rochester Institute of Technology

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Recommended Citation David Merritt 2006 ApJ 648 976

This Article is brought to you for free and open access by RIT Scholar Works. It has been accepted for inclusion in Articles by an authorized administrator of RIT Scholar Works. For more information, please contact [email protected]. arXiv:astro-ph/0603439v3 13 Jun 2007 ikl atnn19;Qiln19) asv binary massive rate A a rate 1983 at 1996). hardens to (Hills Quinlan 1992; reference experiments Valtonen & with scattering Mikkola discussed from often derived is equations nucleus intra- galactic including a galaxies, 2005) from al. ap- et ejected (Holley-Bockelmann that nebulae been populations planetary other cluster have for Tremaine to & and (Yu Way 2006), pear Milky al. high-velocity the et the of Haardt halo for 2003; the responsible in be observed also stars 200 may Graham 2002; SBHs brigh al. Binary in et cores (Milosavljevic of galaxies presence the elliptical and 2003) al. et Gopal-Krishna n rcsino ai es(ose l 93 oeoe al. et Romero 1993; al. et (Roos jets including 2000), bendin radio the properties of 2003), 2003), precession Xie (Komossa the 2000; and al. galaxies explain et (Valtaoja active variability to AGN and binary invoked normal and increasingly of 1980), are Rees SBHs & Blandford, (Begelman, together ino tr nances hc stetpco h current the of topic the is which nucleus, distribu- a the in in stars change of binary-induced tion the computing when 2005). Wang & Merritt 2002; host Yu the 2001; of hybrid Biermann et model many & a Haardt for (Zier into basis binary 2005; a the imbed also which 2004) by are al. schemes equations galaxy et These al. the Holey-Bockelmann 2006). from et al. ejected Sesana (e.g. stars b; in binary mass 2003a, galactic the the in compute al. SBHs to et binary and Volonteri of been (e.g. rates have evolution nuclei (2) estimate and to (1) used Equations coefficient. dimensionless where where n ades h asi tr jce ytebnr satisfie binary the by ejected stars in mass eccentric The ratio, mass hardness. binary and the on depends that coefficient est n eoiydseso,and dispersion, velocity and density h yaia neato famsiebnr ihsasin stars with binary massive a of interaction dynamical The aaymresbigsprasv lc oe (SBHs) holes black supermassive bring mergers Galaxy qain 1 n 2 r o atclryueu however useful particularly not are (2) and (1) Equations rpittpstuigL using typeset Preprint D ATVERSION RAFT a M X stebnr eimjraxis, semi-major binary the is n h itiuino akmte ttecneso galaxie of centers the headings: at Subject matter dark of bi distribution for the Implications and models. formation galaxy hierarchical with si rnil eae oteglx’ egrhsoy u t but by history, stars merger galaxy’s of ejection the to to accuracy due related nucleus principle the in of is mass the in decrease lc oems.We oprdwt bevdms ect,t deficits, mass observed with compared When mass. hole black h aayspeeitn est rfie ec,after Hence, profile. density pre-existing galaxy’s the 12 and - ASDFCT,SALN AI,ADTEMRE ITRE FE OF HISTORIES MERGER THE AND RADII, STALLING DEFICITS, MASS iaysprasv lc oelae nipito galac a on imprint an leaves hole black supermassive binary A ≡ M M Z 12 1 F sae ai oe Mrit&Ees2002; Ekers & (Merritt lobes radio -shaped + EBRUARY A N T h oa aso h iay h ofceti hsrelation this in coefficient the binary; the of mass total the E bd iuain r sdt airt hsrlto.Mass relation. this calibrate to used are simulations -body M tl mltajv 10/09/06 v. emulateapj style X d 1. dt 2 d ln dM h iayms,and mass, binary the , INTRODUCTION  ( 1 ,2008 5, j e a 1 / a  ) = = eateto hsc,RcetrIsiueo Technology, of Institute Rochester Physics, of Department H JM G H σ 12 ρ sadmninesrate dimensionless a is ρ , and σ J r h stellar the are sasecond a is rf eso eray5 2008 5, February version Draft D AVID ity, (2) (1) . 4). Abstract g s ; t N M ERRITT mergers, aysaln ai,teoii fhprvlct stars, hyper-velocity of origin the radii, stalling nary ilycag h tla est nasottm,o re th order of time, short less, a or in time density crossing galaxy stellar the change tially paper. esa ag s10 num- as particle large direc with as high-accuracy, feasible bers now that are such integrations years, summation recent in simulation of regime collisionless essentially galaxies. the to scaled be not can 200 (Yu binary the to N stars t of long supply too the much affect are significantly times relaxation o central effects lum – “scouring” binary the the massive rea of in evidence in especially show that – effects galaxies but nous discreteness relaxation, these two-body Spurzem of is & galaxies counterpart Merritt circumstances (Berczik, The in stall would even 2005). binary evolving real a continue where e to an cause binary which bedded effects, discreteness other and relaxation but problems; profile. density aeycmue rma qainlk 2,i ol tl be still would accu- it be (2), could relate like time to equation this difficult an before the the from if binary about computed Even the at rately 1996). by stall (Valtonen ejected hard to becomes expected mass it be that would e time binary same galaxies, many the in of in be- Furthermore, place lution to taken start distribution. already (2) stellar has and the change (1) considerable like a othe equations valid, In that come 2001). time Merritt the & by Milosavljevic of words, 4 Figure (e.g. pair uino h iaysice rmrpd–i.e. evo- – the rapid which from at switches time binary The the binary. of inspiralling mas lution tight two second, a the form until a particles continued sive are and integrations object con- The galaxy mass. massive a point central of evolution a the follow taining to used are simulations by driven are repopulation. that loss-cone stages collisional slow like process late, of the which independent from – be – evolution should binary normalized, of appropriately phases when early rapid, the separate 04 uuhg ta.20;Brzk ert Spurzem & Merritt of Berczik, values high 2005; such Given al. 2005). et Fukushige 2004; r discussed. are s erlto a ee enqatfid ee high- Here, quantified. been never has relation he bd iuain htaedmntdb iceeeseffec discreteness by dominated are that simulations -body oee,teehsbe uhpors nteatof art the in progress much been has there However, N hsi h prahaotdi h rsn paper. present the in adopted approach the is This bd ehiuswudse ob h ouint these to solution the be to seem would techniques -body h iay h antd ftems deficit mass the of magnitude The binary. the N i eutipis1 implies result his i ulu ntefr fa“asdfii, a deficit,” “mass a of form the in nucleus tic M bd xeiet hwta h w Bssubstan- SBHs two the that show experiments -body ef de ect r hw obe to shown are deficits ohse,N 14623 NY Rochester, eed nywal on weakly only depends ≈ N 0 bd nertosaepaudb spurious by plagued are integrations -body . M 5 N 6 j e Drade l 03 ulnrse al. et Gualandris 2003; al. et (Dorband M oosral hne ntenuclear the in changes observable to • LPIA GALAXIES LLIPTICAL with ∼ < before N M ∼ < • h nl(current) final the ,i codwith accord in 3, N hyfr tightly-bound a form they M tbcmspsil to possible becomes it , ef de M 2 / ≈ M 0 1 . 5 N ron or M -independent N 12 -dependent , N N -body -body real vo- m- 2). a f N ts t- i- o e - r l , 2 David Merritt

– to slow – i.e. N-dependent – is identified, and the properties a ≈ ah, where ah is the semi-major axis of a “hard” binary, of the galaxy are recorded at that time. The degree to which a sometimes defined as clear separation of the two regimes is possible, for a given N, Gµ a = . (6) depends on the binary mass ratio, becoming more difficult as h σ2 the mass ratio becomes more extreme. Results are presented 4 2 here for mass ratios in the range 0.025 ≤ M2/M1 ≤ 0.5. If one writes rh = GM1/σ – equivalent to the definition (3) −2 We find (§4) that the mass deficit – the difference in inte- in a nucleus with ρ ∼ r – then ah can also be written grated mass between initial and final nuclear density profiles q r – at the end of the rapid evolutionary phase is proportional to a = h . (7) h (1 + q)2 4 the total mass of the binary, Mde f ≈ 0.5M12, with only a weak dependence on M2/M1 or on the initial density profile. This The exact definition of a “hard” binary varies from authorto result can be motivated by simple arguments (§2): the smaller author, and this vagueness reflects the difficulty of relating the M2, the tighter the binary which it forms before stalling. evolution of an isolated binary to one embedded in a galaxy. A mass deficit of ∼ 0.5M• is at least a factor two An isolated binary in a fixed background begins to harden at smaller than the typical mass deficits observed in bright an approximately constant rate, (d/dt)(1/a) ≈ const., when elliptical galaxies (Milosavljevic et al. 2002; Graham 2004; a ∼< ah (Quinlan 1996). But in a real galaxy, a hard binary Ferrarese et al. 2006) but we show via an additional set of would efficiently eject all stars on intersecting orbits, and its N-body simulations (§5) that the effect of binary SBHs on hardening rate would suddenly drop at a ≈ ah. This effect a nucleus is cumulative, scaling roughly in proportion both has been observed in N-body experiments with sufficiently to the number of mergers and to the final mass of the SBH. large N (Makino & Funato 2004; Berczik, Merritt & Spurzem Hence, observed values of Mde f /M• can be used to constrain 2005). the number N of mergers that have taken place since the era Supposing that the binary “stalls” at a ≈ ah, it will have at which the SBHs first formed. We find (§6) that 1 ∼< N ∼< 3, given up an energy consistent with expectations from hierarchical structure for- GM1M2 GM1M2 mation theory. Finally, in §7 and §8, the implications for ∆E ≈− + (8a) binary stalling radii, ejection of high-velocity stars, and the 2rh 2ah distribution of dark matter are discussed. 1 2 2 ≈− M2σ + 2M12σ (8b) 2. STAGES OF BINARY EVOLUTION 2 ≈ 2M σ2 (8c) Here we review the stages of binary SBH evolution and dis- 12 cuss the connection between evolution of a binary in N-body to the stars in the nucleus. In other words, the energy trans- simulations and in real galaxies. ferred from the binary to the stars is roughly proportional to Let M1 and M2 to be the masses of the two components of the combined mass of the two SBHs. This result suggests that the binary, and write M12 ≡ M1 + M2, q ≡ M2/M1 ≤ 1, and the binary will displace a mass in stars of order its own mass, µ ≡ M1M2/M12. In what follows we assume that the larger independent of the mass of the infalling hole. This prediction SBH is located initially at the center of the galaxy and that the is verified in the N-body simulations presented below. smaller SBH spirals in. Since the “hard” binary separation ah is ill-defined for bi- The evolution of the massive binary is customarily divided naries embedded in galaxies, we define here a more useful into three phases. quantity. The stalling radius astall is defined as the separation 1. At early times, the orbit of the smaller SBH decays due at which evolution of the binary would halt, d(1/a)/dt = 0, in to dynamical friction from the stars. This phase ends when the absence of any mechanism to re-supply the stellar orbits. the separation R12 between the two SBHs is ∼ rh, the gravita- For instance, if the galaxy potential were completely smooth, tional influence radius of the larger hole. We define rh in the the binary’s decay would halt once all the stars on orbits in- usual way as the radius of a sphere around M1 that encloses a tersecting the binary had been ejected or otherwise (e.g. due stellar mass of 2M1: to shrinkage of the binary) stopped interacting with it. The M (r )= 2M . (3) motivation for this definition is the very long time scales, in ⋆ h 1 luminous elliptical galaxies, for orbital re-population by two- 2. When R12 falls below ∼ rh, the two SBHs form a body encounters. Even in galaxies with relatively short cen- bound pair. The separation between the two SBHs drops tral relaxation times, the binary’s hardening rate would drop rapidly in this phase, due both to dynamical friction act- drastically at a = astall and the likelihood of finding the binary ing on M2, and later to ejection of stars by the binary at a separation near astall would be high. (Milosavljevic & Merritt 2001). The motion of the smaller This definition implies a dependence of astall on the mass SBH around the larger is approximately Keplerian in this ratio of the binary, but (unlike Equation 6) it also implies a phase; we denote the semi-major axis by a and the eccen- dependence on the initial density profile and shape (spherical, tricity by e. The relative velocity of the two SBHs for e = 0 axisymmetric, triaxial) of the galaxy, since these factors may is influence the mass in stars that can interact with the binary. GM12 V = (4) The definition is operational: it can only be applied by “doing bin r a the experiment,” i.e. imbedding the binary in a galaxy model, and the binary’s energy is turning off gravitational perturbations (aside from those due to the binary itself), and observing when the decay halts. GM M E = − 1 2 . (5) In a numerical simulation with finite N, gravitational en- 2a counters will continue supplying stars to the binary at rates 3. The rapid phase of binary evolution comes to an end much higher than those in real galaxies. One can still esti- 2 when the binary’s binding energy reaches ∼ M12σ , i.e. when mate astall if the particle number is large enough that a clear Mass Deficits in Galaxies 3

generated from the unique isotropic phase-space distribution TABLE 1 ρ PARAMETERS OF THE N-BODY INTEGRATIONS function that reproduces Dehnen’s (r) in the combined grav- itational potential of the stars and the central point mass. The γ ′ ′ second SBH was introduced into this model at t = 0, and M2/M1 rh rh astall astall/rh Mdef/M12 −2 −2 given a velocity roughly 1/2 of the circular orbital velocity. 0.5 0.5 0.264 0.37 1.49 × 10 4.08 × 10 0.60 γ γ 0.5 0.25 0.264 0.32 1.18 × 10−2 3.69 × 10−2 0.46 In the models with = 0.5 and = 1.0, the initial separa- 0.5 0.1 0.264 0.30 5.38 × 10−3 1.82 × 10−2 0.33 tion of the two massive particles was 1.6, while in the mod- 0.5 0.05 0.264 0.29 3.60 × 10−3 1.26 × 10−2 0.27 els with γ = 1.5 the initial separation was 0.5. Each realiza- 0.5 0.025 0.264 0.28 1.91 × 10−3 6.82 × 10−3 0.24 tion was then integrated forward using the N-body integrator

−3 −2 described in Merritt, Mikkola & Szell (2005), an adaptation 1.0 0.5 0.165 0.24 7.75 × 10 3.23 × 10 0.62 NBODY1 1.0 0.25 0.165 0.22 4.83 × 10−3 2.25 × 10−2 0.54 of Aarseth (1999) to the GRAPE-6 special-purpose 1.0 0.1 0.165 0.19 2.37 × 10−3 1.23 × 10−2 0.45 computer. Close encounters between the two SBH particles, 1.0 0.05 0.165 0.19 1.36 × 10−3 7.27 × 10−3 0.39 and between the SBHs and stars, were regularized using the 1.0 0.025 0.165 0.18 6.21 × 10−4 3.36 × 10−3 0.34 algorithm of Mikkola and Aarseth (Mikkola & Aarseth 1990, 1993). 3 2 1.5 0.5 0.0795 0.13 3.66 × 10− 2.77 × 10− 0.63 Three parameters suffice to define the initial models: the 1.5 0.25 0.0795 0.11 2.18 × 10−3 1.95 × 10−2 0.56 −3 −2 binary mass ratio q, the central density slope of the Dehnen 1.5 0.1 0.0795 0.10 1.26 × 10 1.29 × 10 0.41 γ ρ 1.5 0.05 0.0795 0.10 7.78 × 10−4 8.19 × 10−3 0.38 model ≡−d log /d logr, and the number of particles N. 1.5 0.025 0.0795 0.09 4.60 × 10−4 4.84 × 10−3 0.39 Five values of the binary mass ratio were considered: q = (0.5,0.25,0.1,0.05,0.025). For each choice of q, three val- ues of γ were used: 0.5,1.0, and 1.5. Finally, each of these change takes place in the hardening rate at some value of a. 15 initial models was integrated with two different values of Better still, by repeating the experiment with different values N: 1.2 × 105 and 2.0 × 105. Mass deficits were computed as of N, one can hope to show that the rapid evolutionary phase described in Paper I. comes to an end at a well-defined time and that subsequent The parameters of the N-body integrations are summarized evolution occurs at a rate that is a decreasing function of N. in Table 1. Columns three and four give two different esti- One advantage of defining astall in this operational way is mates of the binary’s influence radius. The first, rh, is the that it eliminates the need for many of the qualitative distinc- radius containing a mass in stars equal to twice M1 in the tions that have been made in the past between the different initial model. The second is the radius containing a mass regimes of binary evolution, e.g. “soft binaries” vs. “hard bi- in stars equal to twice M1 + M2 at a time t = tstall , the esti- naries,” hardening via “dynamical friction” vs. hardening via ′ mated stalling time. The second definition, rh, is the relevant “loss-cone draining,” etc. (Yu 2002). It also allows for mech- one when comparing the results of the N-body integrations anisms that are not reproduced in the scattering experiments to a real galaxy which has already experienced the scouring on which Equations (1) and (2) are based, e.g. the “secondary effects of a binary SBH. The other columns in Table 1 give slingshot” (Milosavljevic & Meritt 2003), or the effect of the estimates of the stalling radius astall ≡ a(tstall ) and the mass changing nuclear density on the rate of supply of stars to the deficit Mde f (tstall ), derived as described below. binary. It would seem natural to compute astall using so-called “collisionless” N-body codes that approximate the gravita- tional potential via smooth basis functions (Clutton-Brock 4. RESULTS 1973; van Albada & van Gorkom 1977; Hernquist & Ostriker Figure 1 illustrates the evolution of the relative orbit in the 1992). Such algorithmshave in fact been applied to the binary five integrations with γ = 1.5 and N = 200K. The three phases SBH problem (Quinlan & Hernquist 1997; Hemsendorf et al. discussed above are evident. (1) The separation R12 between 2002; Chatterjee et al. 2003) but with results that are not al- the two massive particles gradually decreases as dynamical ways consistent with those of direct-summation codes or with friction extracts angular momentum from the orbit of M2. (2) the predictions of loss-cone theory (compare Chatterjee, Hen- When R12 ≈ rh, the rapid phase of binary evolution begins. quist & Loeb 2003 with Berczik, Merritt & Spurzem 2005). (3) When the separation drops to ∼ ah, evolution again slows. These inconsistencies may be due to difficulties associated Figure 1 shows clearly that the separation of the two massive with incorporatinga binary into a collisionless code, ambigui- particles at the start of the final phase is smaller for smaller ties about the best choice of origin for the potential expansion, M2. etc. Until these issues can be resolved, direct-summation N- Figure 1 suggests that the stalling radius is of order ah, but body codes seem a safer choice. better estimates of astall can be made by comparing integra- tions of the same model carried out with different N. Fig- 3. MODELS AND METHODS ure 2 makes the comparison for the model with γ = 1.0 and The galaxy models and N-body techniques used here are q = 0.1. After ∼ 15 orbits, the mean separation between the similar to those described in Merritt & Szell (2005) (hereafter two massive particles has dropped from ∼ 1 to ∼ 0.2 ≈ rh and Paper I). In brief, Monte-Carlo realizations of steady-state, the second, rapid phase of binary hardening begins. Prior to spherical galaxy models were constructed using Dehnen’s this time, the two integrations with different N find essentially (1993) density law. The models contained an additional, cen- identical evolution of R12. As discussed above, the evolution tral point mass representing the more massive of the two during this phase is due to some combination of dynamical SBHs. The mass of this particle, M1, was alwaysset to 0.01in friction acting on M2 and ejection of stars by the increasingly- units where the total mass in stars Mgal was one. (The Dehnen hard binary, both of which are N-independent processes. The scale length rD and the gravitational constant G are also set to contribution of dynamical friction to the evolution can be es- unity in what follows.) Stellar positions and velocities were timated using Chandrasekhar’s (1943) expression for the dy- 4 David Merritt

FIG. 1.— Evolution of the binary separation in five N-body integra- tions with γ = 1.5 and N = 200K; binary mass ratios are, from left to right, 0.5,0.25,0.1,0.05,0.025. Vertical lines show the times identified as t = tstall . The upper horizontal line indicates rh, the influence radius of the more mas- sive hole in the initial model. Lower horizontal lines show ah as defined in Equation (7). The rapid phase of decay continues until a ≈ ah, with the result that the binary’s binding energy at the end of this phase is nearly independent of M2. namical friction force: ∞ 2 ∆ π π 2 ρ v f h vki = −4 (4 G M2 ) dv f f f (v f )H1(v,v f(9a)), Z0  v  Λ ln if v > v f , FIG. 2.— Upper panel: Thick (black) line shows evolution of the binary H1 = (9b) γ 0 ifv < v f . separation in the N-body integration with = 1.0, M2/M1 = 0.1, N = 200K. Thin (red) line is the evolution predicted by the dynamical friction equation This is the standard approximation in which lnΛ is “taken (9) assuming a fixed galaxy. Horizontal lines indicate rh and ah, the latter as out of the integral”; v is the velocity of the massive object, defined in Equation (7). The inset shows the evolution of the inverse semi- major axis of the binary for this integration, and for a second integration with v f is the velocity of a field star, and f f (v f ) is the field-star N = 120K. Lower panel: The mass deficit, as defined in the text, for the same velocity distribution, normalized to unit total number and as- two N-body integrations. Lines show least-squares fits to the time intervals sumed fixed in time. The result of integrating equation (9), 80 ≤ t ≤ 110 and 110 ≤ t ≤ 200. with lnΛ = 5.7, is shown in the upper panel of Figure 2 as In order to make still more accurate estimates of tstall and the red (thin) line. Chandrasekhar’s formula accurately repro- astall, the hardening rate, duces the evolution of the relative orbit until a time t ≈ 80, af- d 1 ter which it predicts that the separation should drop to zero ata s(t) ≡ , (10) finite time. However as shown in the lower panel of Figure 2, dt a by t = 80 – roughly at the start of the second evolutionary was estimated from the N-body data by fitting smoothing −1 phase – the infall of M2 has begun to displace stars and lower splines to the measured values of a vs. t and differenti- the central density, and the dynamical friction force must drop ating. In a collisionless galaxy, s(t) would reach a peak value below the value predicted by equation (9) with fixed ρ and f f . during the rapid phase of binary evolution, then drop rapidly The inset in Figure 2 shows the inverse semi-major axis of to zero as the stars on intersecting orbits are ejected by the the binary, 1/a, versus time in the two integrations of this binary; tstall would be the time at which s ≈ 0. In the N-body model with different N. The two curves are coincident un- integrations, s will never fall completely to zero since gravi- til a ≈ ah but diverge at later times. This is the expected tational encounters continue to scatter stars into the binary’s result (Berczik, Merritt & Spurzem 2005): once the hard bi- loss cone. nary has ejected most of the stars on intersecting orbits, con- Figure 3 shows s(t) as extracted from the N-body integra- tinued hardening requires orbital repopulation which occurs tions with γ = 0.5. In each case, s reaches a peak value during on a time scale that is roughly proportional to N. The dif- the rapid phase of evolution and then declines. In the mod- ference between the two integrations is also apparent in the els with largest M2, loss-cone repopulation is least efficient, lower panel of Figure 2, which shows mass deficits in the two and the hardening rates in the two integrations with different integrations. Based on this comparison, the stalling radius– N “track” each other well past the peak. As M2 is decreased, the value of a at which the evolution ceases to be independent loss cone repopulation becomes more efficient and the two −3 of particle number – is astall = a(t ≈ 110) ≈ 2×10 , and the s(t) curves deviate from each other at progressively earlier mass deficit when a = astall is Mde f ≈ 0.5M1. times with respect to the peak. Mass Deficits in Galaxies 5

FIG. 3.— Evolution of the binary hardening rate s ≡ (d/dt)(1/a) in N- body integrations with γ = 0.5; binary mass ratios are, from left to right, 0.5,0.25,0.1,0.05,0.025. Solid curves are for N = 200K and dashed curves are for N = 120K. Vertical solid lines are the estimates of tstall . Nearly- vertical dashed lines show the approximate, asymptotic behavior of s(t) in a purely collisionless galaxy, assuming t(ah)=(40,50,60). Even in the absence of loss cone repopulation, a binary in a collisionless galaxy would continue to evolve at late times as stars with progressively longer orbital periods reach pericen- ter and interact with it. The asymptotic behavior of s(t) in a spherical non-evolving galaxy in the absence of encounters is roughly FIG. 4.— Stalling radius (a) and mass deficit at t = tstall (b) as functions of the binary mass ratio. Curve in (a) is Equation (12); curve in (b) is Equation d 1 2 (13). s(t)= ≈ 16π K fm(E)dE (11) dt a Z ues. For instance, Figure 2 shows that the mass deficit in the where fm(E) is the phase-space mass density of stars and K N = 200K integration of the (γ = 1.0,q = 0.1) model varies is a constant that defines the mean, dimensionless change of only between 0.50 ∼< Mde f /M1 ∼< 0.65 for 110 ≤ t ≤ 140. In- energy of the binary in one interaction with a star (Yu 2002). tegrations with larger N will ultimately be required in order to Equation (11) equates the rate of change of the binary’s en- improve on these estimates. ergy with the rate at which stars, moving along their unper- Theresults are presentedin Table 1 and Figure4. The upper ′ turbed orbits, enter the binary’s influence sphere and extract panel of Figure 4 shows astall as a fraction of rh, the binary’s energy from it. Depopulation of the orbits as stars are ejected influence radius (the radius containing a mass in stars equal to is represented by progressively restricting the range of the en- M1 + M2) at tstall . The measured points are compared with ergy integral; the lowest allowed energy at any time t is that astall q corresponding to an orbit with radial period P(E)= t. Fol- ′ = 0.2 2 ; (12) lowing Yu (2002), we set K = 1.6, the approximate value rh (1 + q) for a “hard” binary, and assume that the integration starts at this functional form was motivated by Equation (7). Given a = ah. Figure 3 compares the solution to Equation (11) with the likely uncertainties in the estimated astall values, Figure 4 the N-body hardening rates for (γ = 0.5,q = 0.5). Clearly, the ′ suggests that there is no significant dependence of astall/rh “draining” of long-period orbits contributes only minimally on the initial nuclear density profile. This result will be used to the late evolution seen in the N-body integrations, i.e. the below to estimate stalling radii in observed galaxies. evolution in phase three is driven almost entirely by loss-cone The lower panel of Figure 4 shows mass deficits at t = tstall repopulation. as a fraction of M12. The line in this figure is Plots like those in Figure 3 were used to estimate tstall for γ Mde f each ( ,M2/M1). Unavoidably, these estimates are somewhat = 0.70q0.2. (13) subjective. For the larger values of M2, tstall was taken to be M12 the time when the s t curves for the two different N-values ( ) This relation is a good fit to the γ = 1.0 and γ = 1.5 models, begin to separate. For the integrations with smaller M , col- 2 though it overestimates M for γ = 0.5. In any case, the lisional effects are more important, and the two s(t) curves de f dependence of Mde f on binary mass ratio is weak, consistent separate even before the peak value is reached. In these inte- with the prediction made above. grations, tstall was taken to be the time at which s(t) in the N = In galaxies with initially steep nuclear density profiles, 200K integration reached its first minimum after the peak. ρ −γ γ ∼ r , 1.0 ∼< ∼< 1.5, Figure 4 suggests that mass deficits These estimated tstall values are likely to be systematically generated by “stalled” binaries should lie in the narrow range larger than the true values in a collisionless galaxy. However the uncertainties in the estimated values of astall ≡ a(tstall ) and Mde f 0.4 ∼< ∼< 0.6, 0.05 ∼< q ∼< 0.5. (14) Mde f (tstall ) are probably modest, at least for the larger M2 val- M1 + M2 6 David Merritt

FIG. 5.— Density profiles at t = tstall . Thickness of curves decreases from q = 0.5 to q = 0.025. Dotted lines are the initial density profiles.

To a good approximation, M 0 5M . de f ≈ . • TABLE 2 Density profiles at t = tstall are shown in Figure 5. It is in- MULTI-STAGE N-BODY INTEGRATIONS teresting that none of the profiles exhibits the very flat, nearly constant density cores seen in some bright elliptical galaxies M2 N M• Mdef Mdef /M• Mdef /(N M•) 0.0050 1 0.0150 0.0095 0.63 0.63 (Kormendy 1985; Lauer et al. 2002; Ferrarese et al. 2006). 2 0.0200 0.024 1.20 0.60 3 0.0250 0.041 1.64 0.55 4 0.0300 0.057 1.90 0.48 5. MULTI-STAGE MERGERS The weak dependence of M /M on initial density pro- 0.0025 1 0.0125 0.0070 0.56 0.56 de f 12 2 0.0150 0.017 1.13 0.57 file and on q found above has an obvious implication: in re- 3 0.0175 0.027 1.54 0.51 peated mergers, mass deficits should increase cumulatively, 4 0.0200 0.035 1.75 0.44 even if expressed as a multiple of M•, the combined mass of the two SBHs at the end of each merger. If the stellar 0.0010 1 0.0110 0.004 0.41 0.41 2 0.0120 0.010 0.79 0.40 mass displaced in a single merger is ∼ 0.5M12, then (assum- 3 0.0130 0.014 1.10 0.37 ing that the two SBHs always coalesce before the next SBH 4 0.0140 0.018 1.29 0.32 falls in) the mass deficit following N mergers with M2 ≪ M1 is ∼ 0.5N M• with M• the accumulated mass of the SBH. linearly with both N and M• for small N . Equation (13) could be used to derive a more precise pre- Of course, these results are only meaningful under the as- diction for the dependence of the mass deficit on N . But di- sumption that the binary manages to coalesce between succes- rect simulation is a better approach. To this end, multi-stage sive merger events. Infall of a SBH into a nucleus containing N-body integrations were carried out. The starting point for an uncoalesced binary would almost certainly result in larger γ each set of integrations was one of the = 1.5, M1 = 0.01 values of Mde f /M• than those given in Table 2, for reasons N-body models described above, extracted at t = tstall . The discussed below. two massive particles were replaced by a single particle with mass equal to their combined mass, and with position and 6. COMPARISON WITH OBSERVED MASS DEFICITS velocity equal to the center-of-mass values for the binary. Mass deficits, computed from observed luminosity pro- A second massive particle, with mass equal to the original files, have been published for a number of “core” galaxies value of M2, was then added and the model integrated for- (Milosavljevic et al. 2002; Ravindranath, Ho & Filippenko ward until the new stalling radius was reached. The process 2002; Graham 2004; Ferrarese et al. 2006). As first empha- was then repeated. In each set of experiments, the mass M2 sized by Milosavljevic et al. (2002), computing Mde f is prob- of the second SBH particle was kept fixed; in other words, lematic due to the unknown form of the galaxy’s luminosity M•, the accumulated central mass, increased linearly with profile before it was modified by the binary. Graham (2004) the number N of mergers, while the binary mass ratio de- noted that Sersic’s law provides the best global fit to the lumi- creased with N . Three different values of M2 were tried: nosity profiles of early-type galaxies and bulges and proposed M2 = (0.005,0.0025,0.001). that mass deficits be defined in terms of the deviation of the Figure 6 and Table 2 give the results. The cumulative ef- inner profile from the best-fitting Sersic law. This procedure fect of multiple mergers is clear from the figure: Mde f /M• was followed also by Ferrarese et al. (2006) in their study of increases roughly linearly with N for the first few mergers, Virgo galaxies using HST/ACS data. reaching values of ∼ 1 for N = 2 and ∼ 1.5 for N = 3. For Figure 7 summarizes the results from the Graham (2004) larger N and for smaller M2, the increase of Mde f /M• with N and Ferrareseetal. (2006) studies. Mass deficits from beginsto drop below a linear relation. That this shouldbe so is Graham (2004) were increased by 0.072 in the logarithm to clear from the results of the single-stage mergers (Figure 4). correct an error in the mass-to-light ratios (A. Graham, pri- Nevertheless, Figure 6 verifies that (at least for q ∼> 0.1) hi- vate communication). Graham (2004) gives two estimates erarchical mergers produce mass deficits that scale roughly of M• for each galaxy, based on the empirical correlation Mass Deficits in Galaxies 7

FIG. 6.— Density profiles and mass deficits in the multi-stage integrations. (a) M2 = 0.005; (b) M2 = 0.0025; (c) M2 = 0.001. In the upper panels, dotted lines show the initial density profile and solid lines show ρ(r) at the end of each merger, i.e. at t = tstall . In the lower panels, points show Mdef /M• at tstall and dotted lines show Mdef /M• = 0.5N , where M• is the accumulated SBH mass.

FIG. 7.— (a) Observed mass deficits from Graham (2004) (filled circles) and Ferrarese et al. (2006) (stars). Thick, thin, dashed and dotted lines show Mdef /M• = 0.5,1,2 and 4 respectively. (b) Histogram of Mdef /M•. 8 David Merritt with concentration parameter (Graham et al. 2001) and on the or more of the SBHs could eventually be ejected by the grav- Gebhardt et al. (2000) version of the M• − σ relation. We itational slingshot (Mikkola & Valtonen 1990); and to larger recomputed SBH masses for Graham’s galaxies using the values of Mde f , since multiple SBHs are more efficient than most current version of the M• −σ relation (Ferrarese & Ford a binary at displacing stars (Merritt et al. 2004b). (The for- 2005). Ferrarese et al. (2006) computed M• for their sample mer point is only relevant to galaxies – unlike NGC 5903 – galaxies in the same way; we have replaced M• for three of for which M• has actually been measured.) The chaotic inter- their galaxies (NGC4486, NGC4374, NGC4649) with values action between three SBHs would probably also assist in the derived from detailed kinematical modelling (Macchetto et al. coalescence of the two most massive holes 1997; Bower et al. 1998; Gebhardt et al. 2003). One “core” by inducingrandom changesin their relative orbit (Blaes et al. galaxy from Ferrareseet al. (2006) – NGC 798 – was ex- 2002). (d) The gravitational-wave rocket effect is believed cluded since it contains a large-scale stellar disk. capable of delivering kicks to a coalescing binary as large −1 The histogram of Mde f /M• values is shown in Figure 7b. as ∼ 200 km s (Favata et al. 2004; Blanchet et al. 2005; There is a clear peak at Mde f /M• ≈ 1. Herrmann et al. 2006; Baker et al. 2006). The stellar density Sersic’s law – which describes the projected, or surface, drops impulsively when the SBH is kicked out, and again luminosity density of early-type galaxies – implies a space when its orbit decays via dynamical friction. Mass deficits −γ density that varies as j(r) ∼ r at small radii, where γ ≈ produced in this way can be as large as ∼ M• (Merritt et al. (n − 1)/n and n is the Sersic index. The galaxies plotted in 2004a; Boylan-Kolchin et al. 2004). (e) Stars bound to the γ Figure 7 have n ∼> 5, hence 0.8 ∼< ∼< 1.0; in other words, infalling SBH – which were neglected in the N-body simu- the mass deficits for these galaxies were computed under the lations presented here – might affect Mde f , although it is not assumption that the pre-existing nuclear density was a power clear what the direction or magnitude of the change would be. law with γ ≈ 1. This is approximately the same inner depen- Focussing again on the majority of galaxies in Figure 7 γ dence as in our = 1 Dehnen-model galaxies. While the cu- with 0.5 ∼< Mde f /M• ∼< 1.5, we can ask whether values of mulative N-body mass deficits shown in Figure 6 and Table 2 N in the range 1 ≤ N ≤ 3 are consistent with hierarchi- were based on initial models with γ = 1.5, Figure 4 suggests cal models of galaxy formation. If the seeds of the cur- that the results would have been almost identical for γ = 1, at rent SBHs were present at large redshift, the ancestry of least during the first few stages of the merger hierarchy. a bright galaxy could include dozens of mergers involv- Comparing Figures 6 and 7, we therefore conclude that ing binary SBHs (Volonteri et al. 2003a; Sesana et al. 2004). most “core” galaxies have experienced 1 ∼< N ∼< 3 mergers However the more relevant quantity is probably the num- (i.e. 0.5 ∼< Mde f /M• ∼< 1.5), with N = 2 the most common ber of mergers since the era at which most of the gas was value. depleted, since star formation from ambient gas could re- A few galaxies in Figure 7 have significantly larger generate a density cusp after its destruction by a binary SBH mass deficits; the most extreme object is NGC 5903 with (Graham 2004). Haehnelt & Kauffmann (2002) calculate just Mde f /M• ≈ 4.5. Of course the uncertaintities associated this quantity, based on semi-analytic models for galaxy merg- with the Mde f and M• values in Figure 7 may be large; ers that include prescriptions for star- and SBH-formation. the former due to uncertain M/L corrections, the latter due Their Figure 2 shows probability distributions for N as a to various difficulties associated with SBH mass estimation function of galaxy luminosity for mergers with q > 0.3. For (e.g. Maciejewski & Binney 2001; Valluri, Merritt & Em- galaxies like those in Figure 7 (MV ∼< − 21,MB ∼< − 20), sellem 2004). In the case of NGC 5903, for which M• Haehnelt & Kauffmann (2002) find a median N of ∼ 1, but was computed from the M• − σ relation, we note that pub- particularly among the brightest galaxies, they find that val- lished velocity dispersions for this galaxy range from 192 km ues of N as large as 3 or 4 are also likely. This prediction is s−1 (Smith et al. 2000) to 245 km s−1 (Davies et al. 1987); consistent with Figure 7. the corresponding range in the inferred mass is Probably the biggest uncertainty in this analysis is the un- 8 < < 8 1.36×10 ∼ M•/M⊙ ∼ 4.45×10 and the range in Mde f /M• known behavior of the binary after it reaches a ≈ astall and be- σ is 1.9 ∼< Mde f /M• ∼< 6.2. Even if were known precisely, forethe next SBH falls in. If mergersare “dry,” gas-dynamical there will always be a substantial uncertainty associated with torques can not be invoked to accelerate the coalescence, im- any M• value derived from an empirical scaling relation. plying a complicated interaction between three SBHs when But assuming for the moment that the numbers plot- the next merger event occurs. As discussed above, the net ef- ted in Figure 7 are accurate, how might the galaxies with fect would be an increase in Mde f /M•. On the other hand, if Mde f /M• > 2 be explained? There are several possibilities. gas is present in sufficient quantities to assist in coalescence, (a) These galaxies are the product of N ∼> 4 mergers. Two it may also form new stars – decreasing Mde f – and/or accrete of the largest Mde f /M• values in Figure 7 are associated with onto the SBH – increasing M•; in either case, Mde f /M• would M87 and M49, both extremely luminous galaxies that could be smaller than computed here. have experienced multiple mergers. However, Figure 6 sug- 7. STALLED BINARY SEPARATIONS gests that Mde f /M• values ∼> 3 might be difficult to produce via repeated mergers. (b) The primordial density profiles in Equation (12) gives an estimate for the stalling radius of ′ these galaxies were flatter than Sersic’s law at small radii. a binary SBH in terms of its influence radius rh; the latter (c) The two SBHs failed to efficiently coalesce during one or is defined as the radius of a sphere containing a stellar mass more of the merger events, so that a binary was present when equal to twice M• ≡ M1 + M2, after the binary has reached a third SBH fell in. Indeed this is likely to be the case in the astall, i.e. after it has finished modifying the stellar density largest (lowest density) galaxies in Figure 7, which have little profile. One can use Equation (12) to estimate binary separa- gas and extremely long timescales for loss-cone repopulation tions in galaxies where M• and ρ(r) are known, under the as- by two-body relaxation. Infall of a third SBH onto an uncoa- sumption (discussed in more detail below) that no additional lesced binary is conducive to smaller values of M•, since one mechanism has induced the binary to evolve beyond astall. Table 3 gives the results for the seven Virgo “core” galax- Mass Deficits in Galaxies 9

TABLE 3 astall depend on geometry? (b) If a binary continues to harden VIRGO “CORE”GALAXIES below astall , what would be the effect on Mde f ? The answer to the latter question is complex. Con- sider a galaxy containing a stalled binary, and suppose that astall astall vesc vesc ′ some mechanism – gravitational encounters, time-dependent Galaxy BT M• rh q = 0.5 q = 0.1 vej q = 0.5 q = 0.1 (1) (2) (3) (4) (5) (6) (7)torques (8) (9)from a passing galaxy, perturbations from a third NGC 4472 -21.8 5.94 130. (1.6) 5.6 (0.070) 2.1 (0.026) 562. 1395SBH,. new1865 star. formation, etc. – has the net effect of plac- NGC 4486 -21.5 35.7 460. (5.7) 20. (0.25) 7.6 (0.095) 733. 1480ing. additional2175. stars on orbits that intersect the binary. These NGC 4649 -21.3 20.0 230. (2.9) 10. (0.13) 3.8 (0.047) 776. 1590. 2325. NGC 4406 -21.0 4.54 90. (1.1) 4.0 (0.050) 1.5 (0.019) 590. 1255stars. will1790 be. ejected, and the binary will shrink. However NGC 4374 -20.8 17.0 170. (2.1) 7.6 (0.094) 2.8 (0.035) 832. 1635there. need2435. not be any net change in the mass deficit, since NGC 4365 -20.6 4.72 115. (1.4) 5.0 (0.063) 1.9 (0.023) 533. 1115the. new1615 stars. are first added, then subtracted, from the nuclear NGC 4552 -20.3 6.05 73. (0.91) 3.2 (0.040) 1.2 (0.015) 757. 1500density.. 2230 (Second-order. changes in ρ due to the time depen- Notes. – Col. (1): New General Catalog (NGC) numbers. Col. (2): Absolute dence of the gravitational potential are being ignored.) This 8 argument suggests that the good correlation found here (Fig- B-band galaxy magnitudes. Col. (3): Black hole masses in 10 M⊙. Col. (4): Black hole influence radii, defined as the radii containing a mass in stars ure 4) between astall and Mde f need not apply more generally, equal to 2M•, in pc (arcsec). Col. (5): Binary stalling radii for q = 0.5, in pc in time-dependent or non-spherical situations. (arcsec). Col. (6): Binary stalling radii for q = 0.1, in pc (arcsec). Col. (7): −1 An example of the latter is a binary embedded in a triax- Typical ejection velocity from a binary SBH with a = astall (km s ). Cols. ial galaxy which contains centrophilic (box or chaotic) or- (8), (9): Escape velocity (km s−1). bits. The mass in stars on centrophilic orbits can be extremely ies shown in Figure 7. Influence radii were computed using large, ≫ M•, in a triaxial galaxy, although many orbital pe- parametric (“core-Sersic”) fits to the luminosity profiles, as riods will generally be required before any given star passes described in Ferrarese et al. (2006); that paper also gives the near enough to the binary to interact with it (Merritt & Poon algorithm which was used for converting luminosity densities ′ 2004; Holley-Bockelmann & Sigurdsson 2006). Nevertheless into mass densities. When discussing real galaxies, rh is just the binary need not stall (Berczik et al. 2006). Self-consistent the currently-observed influence radius rh and henceforth the calculations of the effect of a binary on the nuclear density prime is dropped. profile have not been carried out in the triaxial geometry, but Stalling radii are given in Table 3 assuming q = 0.5 and it is clear that mass deficits might be smaller than in the spher- q = 0.1; astall for other values of q can be computed from ical geometry since some stars ejected by the binary will be Equation (12). Table 3 also gives angular sizes of the binaries on orbits with very large (≫ r ) characteristic radii. assuming a distance to Virgo of 16.52 Mpc. Typical sepa- h The influence of different geometries on astall and Mde f will rations between components in a stalled binary are found to be investigated in future papers. be ∼ 10 pc (0.1′′) (q = 0.5) and ∼ 3 pc (0.03′′) (q = 0.1). Of course these numbers should be interpreted as upper lim- 8.2. Mass Deficits and Core Radio Power its; nevertheless they are large enough to suggest that binary An intriguing relation was discovered by SBHs might be resolvable in the brighter Virgo galaxies if Balmaverde & Capetti (2006) and Capetti & Balmaverde both SBHs are luminous. (2006) between the radio and morphological properties of active galaxies. Radio-loud AGN – active nuclei with a large ratio of radio to optical or radio to X-ray luminosities – are 8. DISCUSSION uniquely associated with “core” galaxies. In fact, the slope of the inner brightness profile appears to be the only quantity We have shown that the mass deficit produced by a binary that reliably predicts whether an AGN is radio-loud or SBH at the center of a spherical galaxy is “quantized” in units radio-quiet. Capetti & Balmaverde (2006) also showed that of ∼ 0.5(M1 + M2), with only a weak dependence on binary when the radio-loud and radio-quiet galaxies are considered mass ratio, and that Mde f /M• grows approximately linearly separately, there is no dependence of radio loudness on the with the number of merger events. These results were com- nuclear density profile within either class. In other words, pared with observed mass deficits to concludethat most bright the fact that an active galaxy is morphologically a “core” elliptical galaxies have experienced 1 − 3 mergers since the galaxy predicts that it will be radio-loud but does not predict era at which gas was depleted and/or star formation became how loud. While the origin of this connection is unclear, it inefficient. might reflect the influence of SBH rotation on radio power In this section, we explore some further implications of the (Wilson & Colbert 1995): “core” galaxies have experienced N-body results, and discuss their generality. a recent merger, and coalescence of the two SBHs resulted in a rapidly-spinning remnant. 8.1. Loss-Cone Refilling and Non-Spherical Geometries Two results from the N-body work presented here may The approach adopted in this paper was motivated by the be relevant to the Capetti & Balmaverde correlation. First, fact that mass deficits, or cores, are only observed at the cen- as shown above, the mass deficit is a weak function of the ters of galaxieswith very long relaxation times. In such galax- mass ratio of the binary that produced it, i.e., the shape ies, collisional repopulation of orbits depleted by the binary of the nuclear density profile is poorly correlated with the would act too slowly to significantly influence the binary’s mass of the infalling hole. Since the degree of spin-up evolution (Valtonen 1996), and the binary would stop evolv- is also weakly correlated with M2/M1 (Wilson & Colbert ing once it had interacted with all stars on intersecting orbits. 1995; Merritt & Ekers 2002; Hughes & Blandford 2003) – But even in the completely collisionless case, evolution of the even very small infalling holes can spin up the larger hole binary, and its effect on the local distribution of stars, could to near-maximal spins – it follows that SBH rotation should be different in nonspherical (axisymmetric, triaxial, ...) ge- be poorly correlated with mass deficit. This may explain the ometries. There are really two questions here: (a) How does weak dependence of radio loudness on profile slope found by 10 David Merritt

Capetti & Balmaverde (2006) for the core galaxies. value of M•. Ejection velocities are seen to be much smaller Second, as argued above, core galaxies may contain uncoa- than vesc in all cases, implying that essentially no stars would lesced binaries. Perhaps the presence of a second SBH is the be ejected into the intracluster medium. main factor that determines radio loudness. 8.4. Dark-Matter Cores 8.3. High-Velocity Stars The specific energy change of a dark matter particle inter- Gravitational slingshot ejections by a binary SBH produce acting with a binary SBH is identical to that of a star. At least a population of high-velocity stars with trajectories directed in the case of bright elliptical galaxies like those in Table 3, it away from the center of the galaxy. A few hyper-velocitystars is unlikely that dark matter was ever a dominant component have been detected in the halo of the Milky Way (Brown et al. near the center or that it significantly influenced the evolu- 2005; Hirsch et al. 2005; Brown et al. 2006), and a binary tion of the binary. But ejection of dark matter particles by a SBH at the Galactic center is a possible model for their ori- massive binary would produce a core in the dark matter distri- gin. The short nuclear relaxation time would accelerate the bution similar in size to the luminous-matter core, rcore ≈ rh. hardening of a massive binary by ensuring that stars were A more definite statement about the variation with radius of scattered into its sphere of influence (Yu & Tremaine 2003). ρDM/ρ⋆ near the center of a galaxy containing a binary SBH In this model, stars could have been ejected during the late is probably impossible to make without N-body simulations stages of the binary’s evolution, when a ≪ ah, at high enough that contain all three components. However Table 3 suggests velocities and large enough numbers to explain the observed dark matter core radii of hundreds of parsecs in bright ellipti- hyper-velocity stars. cal galaxies. Binary SBHs have also been invoked to explain other popu- It follows that the rates of self-interaction of supersymmet- lations, e.g. intergalactic planetary nebulae in the Virgo clus- ric particles at the centers of galaxies like M87 (Baltz et al. ter (Arnaboldi et al. 2004). If every Virgo galaxy harbored a 2000) would be much lower than computed under the assump- binary SBH with mass ratio q = 0.1 which evolved, via sling- tion that the dark matter still retains its “primordial” density shot ejection of stars, to the gravitational-radiation regime, the profile (Navarro et al. 1996; Moore et al. 1998). This fact has total mass ejected would have been ∼ 2% of the luminous implications for so-called “indirect” dark matter searches, in mass of the cluster (Holley-Bockelmann et al. 2005). But at which inferences are drawn about the properties of particle least in the brighter Virgo galaxies – including the galaxies dark matter based on measurements of its self-annihilation by- in Table 3 – two-body relaxation times are much too long for products (Bertone et al. 2004). One of the proposed search collisional resupply of binary loss cones, and there is no com- strategies is to identify a component of the diffuse gamma- pelling reason to assume that the binary SBHs in these galax- ray background that is generated by dark matter annhila- ies would have continued interacting with stars beyond astall. tions in halos at all redshifts (Ullio et al. 2002; Taylor & Silk At these separations, velocities between the two components 2003). Calculations of the background flux (Ando 2005; of the binary are relatively modest, implying a much lower Elsässer & Mannheim 2005) have so far always assumed that probability of ejection of stars at velocities large enough to the dark matter distribution does not change with time. Since escape the galaxy. the annihilation flux in a smooth dark matter halo is dom- The mean specific energy changeof a star that interacts with inated by the the center, these calculations may substantially a hard binary is ∼ 3Gµ/2a (Hills 1983; Quinlan 1996). Com- over-estimate the contribution of galactic halos to the gamma- bining this with Equation (12) for astall gives the typical ejec- ray background. tion velocity from a stalled binary: GM 1/2 M 1/2 r −1/2 12 −1 12 h I thank Pat Cote, Laura Ferrarese and Alister Graham ve j ≈ 4.0 ≈ 830 kms 8 ,  rh  10 M⊙   10 pc for help with the galaxy data that were analyzed in §6 and (15) for illuminating discussions. This work was supported by independent of mass ratio. Of course this is an upper limit in grants AST-0071099, AST-0206031, AST-0420920 and AST- the sense that ejection velocities are lower when t < tstall . Ta- 0437519 from the NSF, grant NNG04GJ48G from NASA, ble 3 gives ve j for the Virgo “core” galaxies and also vesc, the and grant HST-AR-09519.01-A from STScI. The N-body cal- escape velocity, defined as −2Φ(astall) with Φ(r) the grav- culations presented here were carried out at the Center for itational potential includingp the contribution from the binary, the Advancement of the Study of Cyberinfrastructure at RIT modelled as a point with mass equal to the currently-observed whose support is gratefully acknowledged.

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