Hypercompact Stellar Systems

Hypercompact Stellar Systems Around Recoiling Supermassive Black Holes

Abstract

A supermassive black hole ejected from the center of a galaxy by gravitational wave recoil carries a retinue of bound stars – a “hypercompact stellar system” (HCSS). The numbers and properties of HCSSs contain information about the merger histories of galaxies, the late evolution of binary black holes, and the distribution of gravitational-wave kicks. We relate the structural properties (size, mass, density profile) of HCSSs to the properties of their host galaxies and to the size of the kick, in two regimes: collisional (), i.e. short nuclear relaxation times; and collisionless (), i.e. long nuclear relaxtion times. HCSSs are expected to be similar in size and luminosity to globular clusters but in extreme cases (large galaxies, kicks just above escape velocity) their stellar mass can approach that of ultra-compact dwarf galaxies. However they differ from all other classes of compact stellar system in having very high internal velocities. We show that the kick velocity is encoded in the velocity dispersion of the bound stars. Given a large enough sample of HCSSs, the distribution of gravitational-wave kicks can therefore be empirically determined. We combine a hierarchical merger algorithm with stellar population models to compute the rate of production of HCSSs over time and the probability of observing HCSSs in the local universe as a function of their apparent magnitude, color, size and velocity dispersion, under two different assumptions about the star formation history prior to the kick. We predict that HCSSs should be detectable within 2 Mpc of the center of the Virgo cluster and that many of these should be bright enough that their kick velocities (i.e. velocity dispersions) could be measured with reasonable exposure times. We discuss other strategies for detecting HCSSs and speculate on some exotic manifestations.

1. Introduction

A natural place to search for supermassive black holes (SMBHs) is at the centers of galaxies, where they presumably are born and spend most of their lives. But it has become increasingly clear that a SMBH can be violently separated from its birthplace as a result of linear momentum imparted by gravitational waves during strong-field interactions with other SMBHs ((Peres 1962); (Bekenstein 1973); (Redmount & Rees 1989)). The largest net recoils are produced from configurations that bring the two holes close enough together to coalesce. Kick velocities following coalescence can be as high as km s in the case of nonspinning holes ((González et al. 2007a); (Sopuerta et al. 2007)); km s for maximally spinning, equal mass BHs on initially circular orbits ((Campanelli et al. 2007); (González et al. 2007b); (Herrmann et al. 2007); (Pollney 2007); (Tichy and Marronetti); (Brügmann et al. 2008); (Dain et al. 2008); (Baker et al. 2008)); and even higher, km s, for black holes that approach on nearly-unbound orbits ((Healy et al. 2008)). Since escape velocities from the centers of even the largest galaxies are km s ((Merritt et al. 2004)), it follows that the kicks can in principle remove SMBHs completely from their host galaxies. While such extreme events may be relatively rare (e.g. (Schnittman & Buonanno 2007); (Schnittman 2007)), recoils large enough to displace SMBHs at least temporarily from galaxy cores – to distances of several hundred to a few thousand parsecs – may be much more common ((Merritt et al. 2004); (Madau and Quataert 2004); (Gualandris & Merritt 2008); (Komossa & Merritt 2008b)).

Komossa et al. (2008) reported the detection of a recoil candidate. This quasar exhibits a kinematically offset broad-line region with a velocity of 2650 km s , and very narrow, restframe, high-excitation emission lines which lack the usual ionization stratification – two key signatures of kicks. In addition to spectroscopic signatures ((Merritt et al. 2006b); (Bonning et al. 2007)), recoiling SMBHs could be detected by their soft X-ray, UV and IR flaring ((Shields & Bonning 2008); (Lippai et al. 2008); (Schnittman and Krolik 2008)) resulting from shocks in the accretion disk surrounding the coalesced SMBH. Detection of recoiling SMBHs in this way is contingent on the presence of gas. But only a small fraction of nuclear SMBHs exhibit signatures associated with gas accretion, and a SMBH that has been displaced from the center of its galaxy will only shine as a quasar until its bound gas has been used up ((Loeb 2007)). The prospect that the SMBH will encounter and capture significant amounts of gas on its way out are small ((Kapoor 1976)).

A SMBH ejected from the center of a galaxy will always carry with it a retinue of bound stars. The stars can reveal themselves via tidal disruption flares or via accretion of gas from stellar winds onto the SMBH ((Komossa & Merritt 2008a), hereafter Paper I). The cluster of stars is itself directly observable, and that is what we discuss in the current work. The linear extent of such a cluster is fixed by the magnitude of the kick velocity and by the mass of the SMBH:

(1a)
(1b)

Reasonable assumptions about the density of stars around the binary SMBH prior to the kick (Paper I) then imply a total luminosity of the bound population comparable to that of a globular star cluster.

In this paper we discuss the properties of these “hyper-compact stellar systems” (HCSSs) and their relation to host galaxy properties. Our emphasis is on the prospects for detecting such objects in the nearby universe at optical wavelengths, and so we focus on the properties that would distinguish HCSSs from other stellar systems of comparable size or luminosity. As noted in Paper I, a key signature is their high internal velocity dispersion: because the gravitational force that binds the cluster comes predominantly from the SMBH, of mass , stellar velocities will be much higher than in ordinary stellar systems of comparable luminosity. Other signatures include the small sizes of HCSSs (unfortunately, too small to be resolved except for the most nearby objects); their high space velocities (due to the kick); and their broad-band colors, which should resemble more closely the colors of galactic nuclei rather than the colors of uniformly old and metal-poor systems like globular clusters.

As we discuss in more detail below (§2), a remarkable property of HCSSs is that they encode, via their internal kinematics, the velocity of the kick that removed them from their host galaxy. A measurement of the velocity dispersion of the stars in a HCSS is tantamount to a measurement of the amplitude of the kick – independent of how long ago the kick occurred; the black hole mass; and the space velocity of the HCSS at the moment of observation. This property of HCSSs opens the door to an empirical determination of the distribution of gravitational-wave kicks.

The outline of the paper is as follows. §2 derives the relations between the structural parameters of HCSSs– mass, radius, and internal velocity dispersion – given assumed values for the slope and density normalization of the stellar population around the SMBH just before the kick. In §3, models for the evolution of binary SMBHs are reviewed and their implications for the pre-kick distribution of stars are described. These results, combined with the relations derived in §2, allow us to relate the structural parameters of HCSSs to the global properties of the galaxies from which they were ejected. §4 discusses the effect of post-kick dynamical evolution of the HCSSs on their observable properties. Stellar evolutionary models are used to predict the luminosities and colors of HCSSs and their post-kick evolution in §5, and in §6, the evolutionary models are combined with models of hierarchical merging to estimate the number of HCSSs to be expected per unit volume in the local universe as a function of their observable properties. §7 discusses search strategies for HCSSs and various other observable signatures that might be uniquely associated with them. In §8 we briefly discuss the inverse problem of reconstructing the distribution of recoil velocities from an observed sample of HCSSs. §9 sums up and suggests topics for further investigation.

2. Structural Relations

In what follows, we adopt the relation in the form given by Ferrarese & Ford (2005):

(2)

with the 1-D velocity dispersion of the galaxy bulge. The influence radius of the SMBH is defined as

(3a)
(3b)

and .

2.1. Bound Population

As discussed in Paper I, a recoiling SMBH carries with it a cloud of stars on bound orbits. Just prior to the kick, most of the stars that will remain bound lie within a sphere of radius around the SMBH (eq. 1). Setting and km sgives pc as an approximate, minimum expected value for the size of a HCSS; such a small size justifies the adjective “hypercompact”. The largest values of would probably be associated with HCSSs ejected from the most massive galaxies, containing SMBHs with masses and travelling with a velocity just above escape, km s; this implies several pc – similar to a large globular cluster.

Assuming a power law density profile before the kick, , the stellar mass initially within radius is

(4a)
(4b)

where . As a fiducial radius at which to normalize the pre-kick density profile, we take , defined as the radius containing an integrated mass in stars equal to twice . (We expect to be of order ; see §3 for a further discussion.) Equation (4) then becomes

(5)

After the kick, the density profile will be nearly unchanged at but will be strongly truncated at larger radii. We define to be the total mass in stars that remain bound to the SMBH after the kick, and write

(6a)
(6b)

where

(7)

Kicks large enough to remove a SMBH from a galaxy core must exceed , and escape from the galaxy implies ; hence to a good approximation. It follows that stars that remain bound following the kick will be moving essentially in the point-mass potential of the SMBH both before and after the kick. To the same order of approximation, the SMBH’s velocity is almost unchanged as it climbs out of the galaxy potential well (at least during the relatively short time required for the stars to reach a new steady state distribution after the kick). Finally, since the bulk of the recoil is imparted to the SMBH in a time , the kick is essentially instantaneous as seen by stars at distances ((Schnittman et al. 2008)).

These three approximations allow the properties of the bound population to be computed uniquely given the initial distribution (Paper I). Transferring to a frame moving with velocity after the kick, the stars respond as if they had received an implusive velocity change at the instant of the kick, causing the elements of their Keplerian orbits about the SMBH to instantaneously change. As a result, all initially-bound stars outside of the sphere at the moment of the kick acquire positive energies with respect to the SMBH and escape. Some of the stars initially at escape while others remain bound. The stellar distribution at is almost unchanged.

Figure 1.— Dimensionless factors that describe (a) the stellar mass bound to a kicked SMBH (eq. 6) and (b) its effective radius (eq. 17). Thick (black) line in the upper panel is the exact expression derived in Appendix A and thin (black) line is the approximation, given in eq. 8. Open circles in (b) were computed using a Monte-Carlo algorithm. Dashed (blue) lines in both panels show the Dehnen-model approximations of eqs. (15) and (18).
Figure 2.— Evolution of the bound population following a kick; the kick was in the direction at . Each frame is centered on the (moving) SMBH. Stars were initially distributed as a power law in density, ; only stars which remain bound following the kick are plotted. Unit of length is and frames (a,b,c,d) correspond to times of () in units of .

Appendix A presents a computation of the bound mass under these approximations and gives an expression for in terms of integrals of simple functions. Figure 1a plots this expression, and also the function

(8)

which is seen to be an excellent approximation for . (As a check, we computed in another way: we generated Monte-Carlo samples of positions and velocities corresponding to an isotropic, power-law distribution of stars around the SMBH prior to the kick and discarded the stars that would be unbound after the kick.) Combining equations (6) and (8), we get

(9)

Setting in this expression gives

(10)

which reproduces reasonably well the values for the bound mass found by Boylan-Kolchin et al. (2004) in their -body simulations of kicked SMBHs; their galaxy models had central power-law density cusps with .

Setting , the value corresponding to a collisional (Bahcall-Wolf) cusp, gives

(11)

which will be useful in what follows.

Figure 3.— Steady state, spherically symmetrized density profiles of the bound population for . Dotted lines show the pre-kick densities; dashed (blue) lines are Dehnen-model fits.

Given the elements of the Keplerian orbits after the kick, the subsequent evolution of the stellar distribution can be computed by simply advancing the positions in time via Kepler’s equation. (Alternately the stellar trajectories can be brute-force integrated; both methods were used as a check.) Figure 2 shows how the bound population evolves from its initially spherical configuration, into a fan-shaped structure at , and finally into a reflection-symmetric, elongated spheroid with major axis in the direction of the kick at . The latter time is

(12)

during which interval the SMBH would travel a distance

(13)

Observing the kick-induced asymmetry would only be possible for a short time after the kick; however the elongation of the bound cloud at would persist indefinitely.

In general, the galactic nucleus might be elongated before the kick, and its major axis will be oriented in some random direction compared with . Since the stellar distribution at is nearly unaffected by the kick, the generic result will be a bound population that exhibits a twist in the isophotes at and a radially-varying ellipticity.

Continuing with the same set of approximations made above, we can compute the steady-state distribution of the bound population by fixing the post-kick elements of the Keplerian orbits and randomizing the orbital phases (or equivalently by continuing the integration of Fig. 2 until late times.) The resultant density profiles are shown in Figure 3 for . Beyond a few , the spherically-symmetrized density falls off as ; the stars in this extended envelope move on eccentric orbits that were created by the kick.

It turns out that Dehnen’s (1993) density law:

(14)

is a good fit to these density profiles for , if is set to ; here is the total (stellar) mass. Figure 3 shows the Dehnen-model fits as dashed lines. Using the expressions in Dehnen (1993), it is easy to show that the Dehnen models so normalized satisfy

(15)

implying . This alternate expression for is plotted as the dashed line in Figure 1a. Unless otherwise stated, we will use equation (8) for in what follows.

So far we have assumed that stars remaining bound to the SMBH experience only its point-mass force. In reality, beyond a radius of order , stars will also feel a significant acceleration from the combined attraction of the other stars, leading to a tidally truncated density profile at . We ignore that complication in what follows.

We note that is determined by the density of stars just before the massive binary has coalesced, and may be substantially different from (eq. 3). In the next section we discuss predictions for based on a number of models for the evolution of the massive binary prior to the kick.

Before doing so, we first present the mass-radius and mass-velocity dispersion relations for the bound population, expressed in terms of as a free parameter.

2.2. Mass-Radius Relation

Combining equations (1) and (6), we get

(16)

As a measure of the size of the HCSS, the effective radius , i.e. the radius containing one-half of the stellar mass in projection, is preferable to . We define a second form factor such that

(17)

Figure 1(b) plots . Also shown by the dashed line is the relation corresponding to the Dehnen-model approximation described above, for which

(18)

(Dehnen 1993). The Dehnen model approximation is reasonably good for all in the range and will be used as the default definition for in what follows.

Combining equations (16) and (18) gives the mass-radius () relation for HCSS’s, in terms of the (yet unspecified) :

(19a)
(19b)

for , .

2.3. Mass-Velocity Dispersion Relation

Stars bound to a recoiling SMBH move within the point-mass potential of the SMBH, for which the local circular velocity is . The circular velocity at is just , so the characteristic (e.g. rms) speed of stars in the bound cloud scales as , motivating us to define a third form factor such that

(20)

where is the measured velocity dispersion. To the extent that is known, and/or the dependence of on is weak, it follows that the amplitude of the initial kick can be empirically determined by measuring the velocity dispersion of the stars.

Figure 4.— Line-of-sight distribution of velocities of stars bound to a recoiling SMBH, as seen from a direction perpendicular to the kick. Initially ; the phase-space distribution following the kick was computed as in Paper I. Solid curves show as defined by all bound stars (thick) and progressively thinner curves show defined by bound stars within a projected distance of from the SMBH. Dashed (blue and red) curves show Gaussian distributions with () and () respectively.

An integrated spectrum will include stars at all (projected) radii within the spectrograph slit. (E.g. at the distance of the Virgo cluster, a slit corresponds to pc, larger than for even the largest HCSS’s.) Since , the distribution of line-of-sight velocities of stars within the slit will contain significant contributions from stars moving both much faster and much slower than and can be significantly non-Gaussian1.

Figure 5.— Absorption line spectrum of the K0III star HR 7615, convolved with two broadening functions. Thick (black) curve: from the top panel of Fig. 4, computed from the entire bound population, assuming km s. Thin (blue) curve: Gaussian with km s.

Figure 4 shows for bound clouds with and , as seen from a direction perpendicular to the kick. (This is the a priori most likely direction for observing a prolate object. Since the HCSS is nearly spherical within a few , the results cited below depend weakly on viewing angle.) Since more than 1/2 of the stars lie at and are moving with , the central core of the distribution has an effective width that is much smaller than ; most of the information about the high velocity stars near the SMBH is contained in the extended wings (e.g. (van der Marel 1994)).

Velocity dispersions of stellar systems are typically measured by comparing an observed, absorption line spectrum with template spectra that have been broadened with Gaussian ’s; the comparison is either made directly in intensity-wavelength space (e.g. (Morton & Chevalier 1973)) or via cross-correlation (e.g. (Simkin 1974)). For example, internal velocities of UCDs (ultra-compact dwarf galaxies) in the Virgo and Fornax clusters have been determined in both ways (e.g. Hilker et al. 2007; Mieske et al. 2008). Figure 5 shows the results of broadening the spectrum of a K0 star in the CaII triplet region (Å Å), with two broadening functions: from the top panel of Figure 4, scaled to km s, and a Gaussian with km s. The two broadening functions produce similar changes in the template spectrum; the from the bound cloud generates more ‘peaked’ absorption lines, but this difference would be difficult to see absent very high quality data.

We computed the best-fit, Gaussian corresponding to the various broadening functions in Figure 4 as a function of . The stellar template of Figure 5 was convolved with Gaussian ’s having in the range to km sand a step size of km s. Each of the Gaussian-convolved templates was then compared with the simulated HCSS  spectrum, and the “observed” velocity dispersion was defined as the for which the Gaussian-convolved template was closest, in a least-squares sense, to the HCSS  spectrum. No noise was added to either the HCSS  or comparison spectra.

Figure 6.— Velocity dispersions that would be inferred from broadened absorption-line spectra of HCSS’s. Solid (black) lines: ; dashed (blue) lines: , where is the power-law index of the stellar density profile before the kick. Thick curves correspond to all bound stars; thinner curves correspond to an observing aperture that includes only bound stars within a projected distance and from the SMBH (as viewed from a direction perpendicular to the kick). Dotted lines show and .
Figure 7.— Recovery of HCSS broadening functions from simulated absorption line spectral data with various amounts of added noise. Blue lines are the input (from Figure 4, with and km s). Solid lines are the recovered s and dash-dotted lines are 90% confidence bands. and are coefficients of the Gauss-Hermite fit to the recovered ; 90% confidence intervals on the parameters are given.

Figure 6 shows the results for , km s km s, and for (circular) apertures of various sizes. When the entire HCSS  is included in the slit, (), (), and (). These values are well fit by the ad hoc relation

(21)

As the aperture is narrowed, increases to values closer to , although as argued above, realistic slits would be expected to include essentially the entire HCSS and we will assume this in what follows.

We note that some ultra-compact dwarf galaxies (UCDs) have as large as km sand that the implied masses are difficult to reconcile with simple stellar population models, which has led to suggestions that the UCDs are dark-matter dominated (Hilker et al. 2007; Mieske et al. 2008). Alternatively, some UCD’s might be bound by a central black hole; for instance, an observed of km sis consistent with an HCSS  produced via a kick of km s(). Detection of the high-velocity wings in (Fig. 4) could distinguish between these two possibilities.

While spectral deconvolution schemes exist that can do this (e.g. (Saha & Williams 1994); (Merritt 1997)), they require high signal-to-noise ratio data. Precisely how high is suggested by Figure 7, which shows the results of simulated recovery of HCSS broadening functions from absorption line spectra. The spectrum of Figure 5 was convolved with the plotted in Figure 4, with km s. Noise was then added to the broadened spectrum (as indicated in the figures by the signal-to-noise ratio S/N) and the broadening function was recovered via a non-parametric algorithm ((Merritt 1997)); confidence bands were constructed via the bootstrap. Figure 7 suggests that S/N permits a reasonably compelling determination of a non-Gaussian . This conclusion is reinforced by the inferred values of the Gauss-Hermite (GH) moments and ; the former measures the width of the Gaussian term in the GH expansion of while measures symmetric deviations from a Gaussian. For S/N, the recovered (90%), significantly different from zero. (We note that the velocity dispersion corresponding to the GH expansion is which is close to as defined above.) In §7 we discuss the feasibility of obtaining HCSS spectra with such high S/N.

Combining equations (17) and (20), the (stellar) mass-velocity dispersion () relation for HCSS’s becomes

(22)

It is tempting (though only order-of-magnitude correct) to write , which allows equation (22) to be written

(23)

The dimensionless coefficient in these two expressions is equal to for .


3. The Pre-Kick Stellar Density

While the linear extent of a HCSS  is determined entirely by and (eq. 1), its luminosity and (stellar) mass depend also on the density of stars around the SMBH (i.e. around the massive binary) just prior to the kick. In this section we discuss likely values for the parameters that determine the pre-kick density of stars near the SMBH and the implications for the mass that remains bound after the kick. In a following section we will relate mass to luminosity and color.

Two inspiralling SMBHs first form a bound pair when their separation falls to , the influence radius of the larger hole. This distance is a few parsecs in a galaxy like the Milky Way. The separation between the two SMBHs then drops very rapidly (on a nuclear crossing time scale) to a fraction of as the binary kicks out stars on intersecting orbits via the gravitational slingshot ((Merritt 2006a)). Because a massive binary tends to lower the density of stars or gas around it, the two SMBHs may stall at this separation, never coming close enough together ( pc) that gravitational wave emission can bring them to full coalescence. This is the “final parsec problem.”

Of course, in order for a kick to occur, the two SMBHs must coalesce, and in a time shorter than Gyr. Roughly speaking, this requires that the density of stars or gas near the binary remain high until shortly before coalescence. This implies, in turn, a relatively large mass in stars that can remain bound to the SMBH after the kick, hence a relatively large luminosity for the HCSS  that results.

Converting these vague statements into quantitative estimates of the stellar density just before the kick requires a detailed model for the joint evolution of stars and gas around the shrinking binary. A number of such models have been discussed (see (Gualandris & Merritt 2008), for a review). Here we focus on the two that are perhaps best understood:

  • Collisional loss-cone repopulation. If the two-body relaxation time in the pre-kick nucleus is sufficiently short, gravitational scattering between stars can continually repopulate orbits that were depleted by the massive binary, allowing it to shrink on a timescale of ((Yu 2002)). This process can be accelerated if the nucleus contains perturbers that are significantly more massive than stars, e.g. giant molecular clouds ((Perets & Alexander 2008)). Repopulation of depleted orbits guarantees that the density of stars near the binary will remain relatively high as the binary shrinks.

  • Collisionless loss-cone repopulation. In non-axisymmetric (barred, triaxial or amorphous) galaxies, some orbits are “centrophilic,” passing near the galaxy center each crossing time. This can imply feeding rates to a central binary as large as even in the absence of collisional loss-cone repopulation ((Merritt & Poon 2004)). Because the total mass on centrophilic orbits can be , interaction of the binary with a mass in stars need not imply a significiant decrease in the local density of stars, again implying a large pre-kick density near the binary.

We now discuss these two pathways in more detail and their implications for the pre-kick stellar density near the SMBH.

3.1. Collisional loss-cone repopulation

At the end of the rapid evolutionary phase described above, the binary forms a bound pair with semi-major axis

(24)

with the binary mass ratio (e.g. (Merritt 2006a)). Stars on “loss cone” orbits that intersect the binary have already been removed via the gravitational slingshot by this time, and continued evolution of the binary is determined by the rate at which these orbits are repopulated – in this model, via gravitational scattering. Scattering onto loss-cone orbits around a central mass occurs predominantly from stars on eccentric orbits with semi-major axes , and the relevant relaxation time is therefore . Relaxation times at in real galaxies are found to be well correlated with spheroid luminosities (e.g. Figure 4 of (Merritt et al. 2007a)), dropping below Gyr only in low-luminosity spheroids – roughly speaking, fainter than the bulge of the Milky Way. Such spheroids have velocity dispersions km sand contain SMBHs with masses . Binary SMBHs in more luminous galaxies might still evolve to coalescence via this mechanism, but only if they contain significant populations of perturbers more massive than , e.g. giant molecular clouds or intermediate mass black holes; Perets & Alexander (2008) have argued that this might generically be the case in the remnants of gas-rich galaxy mergers though this model is unlikely to work in gas-poor, old systems like giant elliptical galaxies.

Denoting the semi-major axis of the massive binary by , one finds ((Merritt et al. 2007b))

(25)

for , where is the separation at which energy losses due to gravitational wave emission begin to dominate losses due to interaction with stars; () () with only a weak dependence on binary mass ratio. The elapsed time between and is of order .

Figure 8.— Evolving stellar density around a binary SMBH of mass in a spherical galaxy containing Solar-mass stars, in the “collisional loss cone repopulation” regime ((Merritt et al. 2007b)). Solid lines show at five different times, between and . The density falls to zero at and smaller values of correspond to later times; total elapsed time is where is the gravitational influence radius of the massive binary. Dotted line shows the initial (pre-binary) galaxy density and dashed line has the Bahcall-Wolf (1976) slope, .

Figure 8 shows the evolution of the stellar density around a massive binary as it shrinks from to ; the evolution was computed using the Fokker-Planck formalism described in Merritt et al. (2007b). The same gravitational encounters that scatter stars into the binary also drive the distribution of stellar energies toward the Bahcall-Wolf (1976) “zero-flux” form, , and on the same time scale, ; as a result, a high density of stars is maintained at radii . In effect, the inner edge of the cusp follows the binary as the binary shrinks.

Once drops below , the binary “breaks free” of the stars and evolves rapidly toward coalescence, leaving behind a phase-space gap corresponding to orbits with pericenters . (In a similar way, evolution of a binary SMBH in response to gravitational waves and gas-dynamical torques leaves behind a gap in the gaseous accretion disk; Milosavljevic & Phinney 2005.) Gravitational scattering will only partially refill this gap in the time between and coalescence ((Merritt & Wang 2005)). Figure 8 suggests that . Merritt et al. (2007b) estimated, based on the same Fokker-Planck model used to construct Figure 8, that

(26)

for equal-mass binaries with total mass ; the numbers in parentheses decrease by for binaries with .

Figure 9.— Effective radius vs. bound stellar mass for HCSSs. Thick solid lines (blue hatched area) are based on the “collisional” loss cone repopulation model and assume a galaxy central velocity dispersion of km s(from left to right). Thin solid lines (red hatched area) are based on the “collisionless” loss cone repopulation model; the three lines in each set assume a galaxy central velocity dispersion of km s(right to left) and the three sets of lines are for (black) , 1.0 (green) and 1.5 (orange). For both models, solid lines extend to a maximum based on the assumption that (escape from the galaxy) while dashed lines correspond to the weaker condition (escape from the galaxy core). HCSSs to the left of the dash-dotted (magenta) line are expected to expand appreciably over their lifetime. Data points are from Forbes et al. (2008). Filled circles: E galaxies. Open circles: Ultra-compact Dwarfs (UCDs) and Dwarf-Globular Transition Objects (DGTOs). Stars: globular clusters.
Figure 10.— Observed velocity dispersion vs. bound stellar mass for HCSSs. Thick solid lines (blue hatched area) are based on the “collisional” loss cone repopulation model and assume a galaxy central velocity dispersion of km s(from left to right). Thin solid lines (red hatched area) are based on the “collisionless” loss cone repopulation model; the three lines in each set assume a galaxy central velocity dispersion of km s(right to left) and the three sets of lines are for (black) , 1.0 (green) and 1.5 (orange). For both models, solid lines extend to a minimum based on the assumption that (escape from the galaxy) while dashed lines correspond to the weaker condition (escape from the galaxy core). Other symbols are as in Fig. 9.

Following the kick, the density profile of Figure 8 will be truncated beyond . The inner cutoff at satisfies

(27)

The requirement that – i.e. that at least some stars remain bound after the kick – then becomes , which is never violated by reasonable () values. However, the inner cutoff exceeds for , a condition that would be fulfilled for km sand km s. In what follows we ignore the inner cutoff and assume that the Bahcall-Wolf cusp extends to .

The pre-kick density can therefore be approximated as

(28)

where and is the density of the galaxy core. Using equations (6) and (28), the mass remaining bound to the coalesced SMBH after a kick is then

(29a)
(29b)
(29c)

where ; the last expression assumes , i.e. , which is always satisfied for a HCSS that escapes the galaxy core.

Figure 11.— Escape velocity from the center of a Sersic-law galaxy as a function of Sersic index , in units of the central, projected, 1d velocity dispersion as measured through a circular aperture. Constant mass-to-light ratio was assumed and the effect of the SMBH on the potential or on the motions of stars was ignored in computing and . The four curves (black, red, green, blue) correspond to aperture radii of (0.01,0.03,0.1,0.3) in units of the half-light radius of the galaxy.

To the extent that the galaxy core was itself created by the massive binary during its rapid phase of evolution, then is of order unity ((Merritt 2006a)). (Following the kick, the core will expand still more; Gualandris & Merritt 2008.) Making this assumption yields

(30a)
(30b)

In the case of ejection from a stellar spheroid like that of the Milky Way ( km s, ), we have

(31)

i.e.

(32a)
(32b)

Combining equations (1), (17) and (30) gives the mass-radius relation in the collisional regime:

(33c)

where the final expression assumes the relation in equation (2).

The mass-velocity dispersion () relation for HCSSs follows from equation (30) with :

(34a)
(34b)

where the relation has again been used.

Figures 9 and 10 plot the relations (33), (34) for km s. Plotted for comparison are samples of globular clusters and dwarf galaxies from the compilation of Forbes et al. (2008).

In these plots, the minimum is presumed to be that associated with a kick of km s. This condition (combined with the relation) gives

(35)

The maximum is associated with the smallest that is of physical interest. We express this value of as which allows us to write

(36)

Kicks large enough to eject a SMBH completely from its galaxy have (Figure 11). Kicks just large enough to remove a SMBH from the galaxy core have ((Gualandris & Merritt 2008)). Figures 9 and 10 show the limits on and corresponding to and , the latter via dashed lines.

According to Figure 9, HCSSs in this “collisional” regime can have effective radii as big as pc when ; on the plane their distribution barely overlaps with globular clusters, and extends to much lower sizes and (stellar) masses. However their velocity dispersions (Fig. 10) would always substantially exceed those of either globular clusters or compact galaxies of comparable (stellar) mass.

3.2. Collisionless loss-cone repopulation

By “collisionless” we mean that the nuclear relaxation time is so long that gravitational scattering can not refill the loss cone of a massive binary at a fast enough rate to significantly affect the binary’s evolution after the hard-binary regime (eq. 24) has been reached. The relevant radius at which to evaluate the relaxation time is , the influence radius of the binary (or of the single black hole that subsequently forms). The relaxation time at in elliptical galaxies is found to correlate tightly with or ((Merritt et al. 2007b)):

(37)

where Solar-mass stars have been assumed. A mass of order is scattered into the central sink in a time , and this is also roughly the mass that must interact with the binary in order for it to shrink by a factor of order unity. Even allowing for variance in the phenomenological relations (37), it follows that collisional loss cone refilling is unlikely to significantly affect the evolution of a binary SMBH in galaxies with km sor .

An alternative pathway exists for stars in these galaxies to interact with a central binary. If the large-scale galaxy potential is non-axisymmetric, a certain fraction of the stellar orbits will have filled centers – these are the (non-resonant) box or centrophilic orbits, which are typically chaotic as well due to the presence of the central point mass ((Merritt & Valluri 1999)). Stars on centrophilic orbits pass near the central object once per crossing time; the number of near-center passages that come within a distance of the central object, per unit of time, is found to scale roughly linearly with ((Gerhard & Binney 1985); (Merritt & Poon 2004)), allowing the rate of supply of stars to a central object to be computed simply given the population of centrophilic orbits. While the latter is not well known for individual galaxies, stable, self-consistent triaxial galaxy models with central black holes can be constructed with chaotic orbit fractions as large as ((Poon & Merritt 2004)). Placing even a few percent of a galaxy’s mass on centrophilic orbits is sufficient to bring two SMBHs to coalescence in 10 Gyr ((Merritt & Poon 2004)). Furthermore, the effect of the binary on the density of stars in the galaxy core is likely to be small, since the mass associated with centrophilic orbits is and stars on these orbits spend most of their time far from the center.

Here, we make the simple assumption that the observed core structure of bright elliptical galaxies is similar to what would result from the decay and coalescence of a binary SMBH in the collisionless loss-cone repopulation model. In other words, we assume that the binary SMBHs that were once present in these galaxies did coalesce, and the cores that we now see are relics of the binary evolution that preceded that coalescence. By making these assumptions, we are probably underestimating the density around a SMBH at the time of a kick, since some observed cores will have been enlarged by the kick itself ((Gualandris & Merritt 2008)). Also, core sizes in local (spatially resolved) galaxies are likely to reflect a series of past merger events ((Merritt 2006a)); SMBHs that recoiled during a previous generation of mergers would probably have carried a higher density of stars than implied by the current central densities of galaxies.

Above we characterized the pre-kick mass density as , with the radius at which the enclosed stellar mass equals twice . We computed and for a subset of early-type galaxies in the ACS Virgo sample ((Côté et al. 2004)) for which was known; for some of these galaxies the SMBH mass has been measured dynamically while in the remaining galaxies was computed from equation (2). Each galaxy was modelled with a PSF-convolved, core-Sersic luminosity profile ((Graham et al. 2003)), which assumes a power law relation between luminosity density and projected radius inside a break radius . The core-Sersic fits were numerically deprojected, and converted from a luminosity to a mass density as in Ferrarese et al. (2006). The radius was then computed from

(38)

with the central power-law index of the deprojected density; is a fiducial radius smaller than which we chose to be pc and is the mass density at .

Figure 12.— Relation between , the radius containing a mass in stars equal to twice , and central velocity dispersion (top) or black hole mass (bottom), for galaxies in the ACS Virgo sample. Open circles are “core” galaxies; dashed lines are fit to just these points while dotted lines are fit to the entire sample.

Figure 12 shows the relation between and and between and . “Core” galaxies (those with projected profiles flatter than near the center) are plotted with open circles and “power-law” galaxies ( steeper than ) galaxies as filled circles. While this distinction is somewhat arbitrary, Figure 12 confirms that the “core” galaxies have larger at given or than the “power-law” galaxies, consistent with the idea that the central densities of “core” galaxies have been most strongly affected by mergers. The best-fit relations defined by the core galaxies alone are

(39a)
(39b)

i.e.

(40a)
(40b)

While these relations are fairly tight, the values show somewhat more scatter, in the range , and we leave as a free parameter in what follows.

Combining the relations (40) with equations (19b) and (2) gives a mass-radius relation for HCSSs in the “collisionless” paradigm:

(41a)
(41b)

where

(42a)
(42b)

Similarly, combining equations (40) with equation (22) gives the mass-velocity dispersion relation:

(43a)
(43b)

where

(44a)
(44b)

and is given by equation (21).

These relations are plotted in Figures 9 and 10. The allowed locus in the diagram (indicated by red vertical lines) now includes the region occupied also by globular clusters and compact E galaxies. However, velocity dispersions remain much higher than those observed so far in these classes of object.

3.3. Unbound stars

The high, pre-kick stellar densities near the binary which are required for coalescence in the stellar-dynamical models do not necessarily imply that these stars are bound to the SMBH. For instance, in a triaxial galaxy populated by radially-anisotropic box orbits, some of the stars that are momentarily near SMBH will be on orbits that make them unbound with respect to the hole, even before the kick. Another example is loss-cone repopulation by “massive perturbers”; in this model stars are “shot” inward to the binary on eccentric orbits, many with high enough velocities that they would be unbound in the absence of the galactic potential.

When deriving the velocity distribution of stars near the SMBH, we assumed isotropy and we neglected the effect of the galaxy’s gravitational potential on the stellar orbits. These assumptions would be violated in some (though not all) of the loss-cone repopulation mechanisms that have been invoked to solve the stalling problem. Here we discuss briefly the consequences of relaxing these assumptions.

We first note that in the isotropic case, unbound stars are negligibly important. We verified this by constructing fully self-consistent models of galaxies containing SMBHs and counting the unbound stars at each radius. We used (isotropic) Dehnen (1993) models, which have an inner power-law density profile; the self-consistent distribution function describing the stars was computed assuming a central point with mass . We confirmed, for and , that the fraction of the mass within that is unbound for kick velocities exceeding the escape velocity is never more than a few percent.

If the pre-kick velocity distribution were radially anisotropic, more stars would be unbound with respect to the massive binary and the post-kick bound population would be smaller than what was computed in §2. While this is certainly possible, we note that the anisotropy would have to be appreci