HYPERCOMPACT STELLAR SYSTEMS AROUND RECOILING SUPERMASSIVE BLACK HOLES

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 ∼r_k around the SMBH (Equation (1)). Setting M_BH = 3 × 10⁶M_☉ and V_k = 4000 km s⁻¹ gives r_k ≈ 10⁻³ 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 r_k would probably be associated with HCSSs ejected from the most massive galaxies, containing SMBHs with masses M_BH ≈ 3 × 10⁹M_☉ and traveling with a velocity just above escape, ∼2000 km s⁻¹; this implies r_k≈ several pc—similar to a large GC.

Assuming a power-law density profile before the kick, ρ(r) = ρ(r₀)(r/r₀)^−γ, the stellar mass M_k initially within radius r_k is

$$\begin{eqnarray} M_{\rm k} \equiv M(r\le r_{\rm k}) &=& {4\pi \over 3-\gamma } \rho (r_{\rm k})r_{\rm k}^3 \end{eqnarray} \tag{ 4a }$$

$$\begin{eqnarray} \,\,\,\,\,\,\,\, &=& {4\pi \over 3-\gamma } \rho _0 r_0^\gamma \left({GM_{\rm BH}\over V_{\rm k}^2}\right)^{3-\gamma }\,, \end{eqnarray} \tag{ 4b }$$

where ρ₀ ≡ ρ(r₀). As a fiducial radius at which to normalize the pre-kick density profile, we take r₀ = r_•, defined as the radius containing an integrated mass in stars equal to twice M_BH. (We expect r_• to be of order r_infl; see Section 3 for a further discussion.) Equation (4) then becomes

$$\begin{equation} M_{\rm k} = 2M_{\rm BH}\left({GM_{\rm BH}\over r_{\bullet }V_{\rm k}^2}\right)^{3-\gamma }. \end{equation} \tag{ 5 }$$

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

$$\begin{eqnarray} f_{\rm b}\equiv {M_{\rm b}\over M_{\rm BH}} &=& F_1(\gamma)\left({GM_{\rm BH}\over r_{\bullet }V_{\rm k}^2}\right)^{3-\gamma } \end{eqnarray} \tag{ 6a }$$

$$\begin{eqnarray} &\propto & V_{\rm k}^{-2(3-\gamma)} \,, \end{eqnarray} \tag{ 6b }$$

where

$$\begin{equation} 2M_{\rm b}= F_1(\gamma)M_{\rm k}. \end{equation} \tag{ 7 }$$

Kicks large enough to remove a SMBH from a galaxy core must exceed σ, and escape from the galaxy implies (V_k/σ)² ≫ 1; hence r_k ≪ r_infl 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 ∼GM_BH/c³, the kick is essentially instantaneous as seen by stars at distances r ≳ GM_BH/c² ≪ r_k (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 V_k after the kick, the stars respond as if they had received an impulsive velocity change −V_k 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 r = 8r_k at the moment of the kick acquire positive energies with respect to the SMBH and escape. Some of the stars initially at r_k ≲  r < 8r_k escape while others remain bound. The stellar distribution at r ≲ r_k is almost unchanged.

Appendix A presents a computation of the bound mass under these approximations and gives an expression for F₁(γ) in terms of integrals of simple functions. Figure 1(a) plots this expression, and also the function

$$\begin{equation} F_1(\gamma) = 11.6\gamma ^{-1.75} \,, \end{equation} \tag{ 8 }$$

which is seen to be an excellent approximation for 0.7 ≲ γ ≲ 2.5. (As a check, we computed F₁(γ) 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

$$\begin{equation} f_{\rm b} \approx 11.6 \gamma ^{-1.75} \left({GM_{\rm BH}\over r_{\bullet }V_{\rm k}^2}\right)^{3-\gamma }. \end{equation} \tag{ 9 }$$

Setting γ = 1 in this expression gives

$$\begin{equation} f_{\rm b} \approx 2\times 10^{-4} \left({M_{\rm BH}\over 10^7M_\odot }\right)^{2} \left({r_{\bullet }\over 10\ {\rm pc}}\right)^{-2} \left({V_{\rm k}\over 10^3\ {\rm km\ s}^{-1}}\right)^{-4}, \end{equation} \tag{ 10 }$$

which reproduces reasonably well the values for the bound mass found by Boylan-Kolchin et al. (2004) in their N-body simulations of kicked SMBHs; their galaxy models had central power-law density cusps with γ = 1.

Zoom In Zoom Out Reset image size

Setting γ = 1.75, the value corresponding to a collisional (Bahcall–Wolf) cusp gives

$$\begin{equation} f_{\rm b} \approx 5\times 10^{-3} \left({M_{\rm BH}\over 10^7M_\odot }\right)^{1.25}\! \left({r_{\bullet }\over 10\ {\rm pc}}\right)^{-1.25}\! \left({V_{\rm k}\over 10^3\ {\rm km\ s}^{-1}}\right)^{-2.5}, \end{equation} \tag{ 11 }$$

which will be useful in what follows.

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 t ≈ 10(GM_BH/V³_k) and finally into a reflection-symmetric, elongated spheroid with major axis in the direction of the kick at t ≈ 100(GM_BH/V³_k). The latter time is

$$\begin{equation} t_{\rm sym} \sim 3\times 10^{3} {\rm yr} \left({M_{\rm BH}\over 10^7M_\odot }\right) \left({V_{\rm k}\over 1\times 10^3\ {\rm km\ s}^{-1}}\right)^{-3} \,, \end{equation} \tag{ 12 }$$

during which interval the SMBH would travel a distance

$$\begin{equation} d_{\rm sym} \sim 3 {\rm pc} \left({M_{\rm BH}\over 10^7M_\odot }\right) \left({V_{\rm k}\over 10^3\ {\rm km\ s}^{-1}}\right)^{-2}. \end{equation} \tag{ 13 }$$

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

Zoom In Zoom Out Reset image size

In general, the galactic nucleus might be elongated before the kick, and its major axis will be oriented in some random direction compared with V_k. Since the stellar distribution at r ≲ r_k is nearly unaffected by the kick, the generic result will be a bound population that exhibits a twist in the isophotes at r ≈ r_k 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 Figure 2 until late times). The resultant density profiles are shown in Figure 3 for γ = (1, 1.5, 2). Beyond a few r_k, the spherically symmetrized density falls off as ∼r⁻⁴; 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

$$\begin{equation} \rho (r) = {(3-\gamma)M_{\rm D}\over 4\pi } \xi ^{-\gamma } \left(1 + \xi \right)^{\gamma -4}, \ \ \ \ \xi \equiv r/r_{\rm D} \end{equation} \tag{ 14 }$$

is a good fit to these density profiles for 1 ≲ γ ≲ 2, if r_D is set to 2.0r_k; here M_D 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

$$\begin{equation} {M_{\rm D}\over M_{\rm BH}} = 2^{4-\gamma } \left({GM_{\rm BH}\over r_{\bullet }V_{\rm k}^2}\right)^{3-\gamma } \,, \end{equation} \tag{ 15 }$$

implying F₁ ≈ 2^4−γ. This alternate expression for F₁ is plotted as the dashed line in Figure 1(a). Unless otherwise stated, we will use Equation (8) for F₁ in what follows.

Zoom In Zoom Out Reset image size

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 r_infl ≈ (V_k/σ)²r_k, stars will also feel a significant acceleration from the combined attraction of the other stars, leading to a tidally truncated density profile at r ≫ r_k. We ignore that complication in what follows.

We note that r_• is determined by the density of stars just before the massive binary has coalesced, and may be substantially different from r_infl (Equation (3)). In the next section we discuss predictions for r_• 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 r_• as a free parameter.

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

$$\begin{equation} {M_{\rm b}\over M_{\rm BH}} = F_1(\gamma)\left({r_{\rm k}\over r_{\bullet }}\right)^{3-\gamma }\,. \end{equation} \tag{ 16 }$$

As a measure of the size of the HCSS, the effective radius r_eff, i.e., the radius containing one-half of the stellar mass in projection, is preferable to r_k. We define a second form factor F₂ such that

$$\begin{equation} r_{\rm eff} = F_2(\gamma) r_{\rm k}\,. \end{equation} \tag{ 17 }$$

Figure 1(b) plots F₂(γ). Also shown by the dashed line is the relation corresponding to the Dehnen model approximation described above, for which

$$\begin{equation} F_2(\gamma)\approx 1.5(2^{1/(3-\gamma)} -1)^{-1} \end{equation} \tag{ 18 }$$

(Dehnen 1993). The Dehnen model approximation is reasonably good for all γ in the range 0.5 ⩽ γ ⩽ 2.5 and will be used as the default definition for F₂ in what follows.

Combining Equations (16) and (18) gives the mass–radius (M_b–r_eff) relation for HCSSs, in terms of the (yet unspecified) r_•:

$$\begin{eqnarray} M_{\rm b}&=& K(\gamma) M_{\rm BH}r_{\bullet }^{\gamma -3} r_{\rm eff}^{3-\gamma }, \end{eqnarray} \tag{ 19a }$$

$$\begin{eqnarray} K(\gamma) &\equiv & 11.6\gamma ^{-1.75} [(3/2)(2^{1/(3-\gamma)}-1)^{-1}]^{\gamma -3} \end{eqnarray} \tag{ 19b }$$

for γ = (1, 1.5, 2), K = (0.89, 1.41, 2.32).

Stars bound to a recoiling SMBH move within the point-mass potential of the SMBH, for which the local circular velocity is (GM_BH/r)^1/2. The circular velocity at r = r_k is just V_k, so the characteristic (e.g., rms) speed of stars in the bound cloud scales as V_k, motivating us to define a third form factor F₃ such that

$$\begin{equation} \sigma _{\rm obs} = F_3(\gamma)V_{\rm k}, \end{equation} \tag{ 20 }$$

where σ_obs is the measured velocity dispersion. To the extent that γ is known, and/or the dependence of F₃ on γ is weak, it follows that the amplitude of the initial kick can be empirically determined by measuring the velocity dispersion of the stars.

An integrated spectrum will include stars at all (projected) radii within the spectrograph slit. (For example, at the distance of the Virgo cluster, a 1'' slit corresponds to ∼80 pc, larger than r_eff for even the largest HCSSs.) Since V ∼ r^−1/2, the distribution N(V) of line-of-sight velocities of stars within the slit will contain significant contributions from stars moving both much faster and much slower than V_k and can be significantly non-Gaussian.⁴

Figure 4 shows N(V) for bound clouds with γ = 1 and 2, 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 r_k, the results cited below depend weakly on viewing angle.) Since more than 1/2 of the stars lie at r > 2r_k and are moving with v < V_k, the central core of the distribution has an effective width that is much smaller than V_k; most of the information about the high-velocity stars near the SMBH is contained in the extended wings (e.g., van der Marel 1994).

Zoom In Zoom Out Reset image size

Velocity dispersions of stellar systems are typically measured by comparing an observed, absorption line spectrum with template spectra that have been broadened with Gaussian N(V)'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 ultracompact dwarf galaxies (UCDs) 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 Ca ii triplet region (8400 Å ⩽ λ ⩽ 8800 Å), with two broadening functions: N(V) from the top panel of Figure 4, scaled to V_k = 1000 km s⁻¹, and a Gaussian N(V) with σ = 200 km s⁻¹. The two broadening functions produce similar changes in the template spectrum; the N(V) from the bound cloud generates more `peaked' absorption lines, but this difference would be difficult to see in the absence of very high-quality data.

Zoom In Zoom Out Reset image size

We computed the best-fit, Gaussian σ corresponding to the various broadening functions in Figure 4 as a function of V_k. The stellar template of Figure 5 was convolved with Gaussian N(V)'s having σ in the range 2–2000 km s⁻¹ and a step size of 1 km s⁻¹. Each of the Gaussian-convolved templates was then compared with the simulated HCSS spectrum, and the "observed" velocity dispersion σ_obs was defined as the σ 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 shows the results for γ = (1, 2), 150 km s⁻¹ ⩽V_k ⩽ 4000 km s⁻¹, and for (circular) apertures of various sizes. When the entire HCSS is included in the slit, σ_obs ≈ 0.15V_k (γ = 0.5), ≈0.20V_k (γ = 1), and ≈0.35V_k (γ = 2). These values are well fitted by the ad hoc relation

$$\begin{equation} \ln F_3 = -2.17 + 0.56\gamma, \ \ \ 0.5\,{\lesssim}\,\gamma \,{\lesssim}\,2. \end{equation} \tag{ 21 }$$

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

Zoom In Zoom Out Reset image size

We note that some UCDs have σ_obs as large as 40–50 km s⁻¹ and 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 UCDs might be bound by a central black hole; for instance, an observed σ of 50 km s⁻¹ is consistent with a HCSS produced via a kick of ∼250 km s⁻¹ (γ = 1). Detection of the high-velocity wings in N(V) (Figure 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 γ = 1 N(V) plotted in Figure 4, with V_k = 10³ km s⁻¹. 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 ≈40 permits a reasonably compelling determination of a non-Gaussian N(V). This conclusion is reinforced by the inferred values of the Gauss–Hermite (GH) moments σ₀ and h₄; the former measures the width of the Gaussian term in the GH expansion of N(V) while h₄ measures symmetric deviations from a Gaussian. For S/N = 40, the recovered h₄ = 0.18 ± 0.1 (90%), significantly different from zero. (We note that the velocity dispersion corresponding to the GH expansion is $\sigma _{\rm c}=(1\;+\;\sqrt{6}h_4)\sigma _0$ which is close to σ_obs as defined above.) In Section 7 we discuss the feasibility of obtaining HCSS spectra with such a high S/N.

Zoom In Zoom Out Reset image size

Combining Equations (17) and (20), the (stellar) mass–velocity dispersion (M_b–σ_obs) relation for HCSSs becomes

$$\begin{equation} {M_{\rm b}\over M_\odot } \approx F_1(\gamma)\times F_3(\gamma)^{2(3-\gamma)} \left({M_{\rm BH}\over M_\odot }\right) \left({GM_{\rm BH}\over r_{\bullet }}\right)^{3-\gamma } \sigma _{\rm obs}^{2(\gamma -3)}. \end{equation} \tag{ 22 }$$

It is tempting (though only order-of-magnitude correct) to write r_• ≈ r_infl ≈ GM_BH/σ², which allows Equation (22) to be written as

$$\begin{equation} {M_{\rm b}\over M_{\rm BH}} \approx F_1\times F_3^{2(3-\gamma)} \left({\sigma _{\rm obs}\over \sigma }\right)^{2(\gamma -3)}. \end{equation} \tag{ 23 }$$

The dimensionless coefficient in these two expressions is equal to (3 × 10⁻³, 1.2 × 10⁻², 0.4) for γ = (0.5, 1, 2).

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

$$\begin{equation} a\approx a_{\rm h} \equiv {q\over (1+q)^2} {r_{\rm infl}\over 4}\,, \end{equation} \tag{ 24 }$$

with q ≡ M₂/M₁ ⩽ 1 being the binary mass ratio (e.g., Merritt 2006). 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 M_BH = M₁ + M₂ occurs predominantly from stars on eccentric orbits with semimajor axes ∼r_infl, and the relevant relaxation time is therefore ∼t_R(r_infl). Relaxation times at r = r_infl in real galaxies are found to be well correlated with spheroid luminosities (e.g., Figure 4 of Merritt et al. 2007a), dropping below 10 Gyr only in low-luminosity spheroids—roughly speaking, fainter than the bulge of the Milky Way. Such spheroids have velocity dispersions ≲150 km s⁻¹ and contain SMBHs with masses ≲10⁷M_☉. 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 ∼M_☉, 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 such as giant elliptical galaxies.

Denoting the semimajor axis of the massive binary by a(t), one finds (Merritt et al. 2007b)

$$\begin{equation} {1\over t_{\rm R}(r_{\rm infl})}\left|{a\over \dot{a}}\right| \approx A\ln \left({a_{\rm h}\over a}\right) + B \end{equation} \tag{ 25 }$$

for a_eq ≲ a(t) ≲ a_h, where a_eq ≈ 10⁻³r_infl is the separation at which energy losses due to gravitational-wave emission begin to dominate losses due to interaction with stars; (A, B) ≈ (0.016, 0.08) with only a weak dependence on binary mass ratio. The elapsed time between a = a_h and a = a_eq is of order t_R(r_infl).

Figure 8 shows the evolution of the stellar density around a massive binary as it shrinks from a ≈ a_h to a ≈ a_eq; 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, ρ ∼ r^−7/4, and on the same timescale, ∼t_R(r_infl); as a result, a high density of stars is maintained at radii a(t) ≲ r ≲ 0.2r_infl. In effect, the inner edge of the cusp follows the binary as the binary shrinks.

Zoom In Zoom Out Reset image size

Once a(t) drops below ∼a_eq, the binary "breaks free" of the stars and evolves rapidly toward coalescence, leaving behind a phase-space gap corresponding to orbits with pericenters ≲a_eq. (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 a = a_eq and coalescence (Merritt & Wang 2005). Figure 8 suggests that a_eq ≪ (a_h, r_infl). Merritt et al. (2007b) estimated, based on the same Fokker–Planck model used to construct Figure 8, that

$$\begin{equation} {a_{\rm eq}\over r_{\rm infl}} \approx (0.20,0.67,2.3,7.8)\times 10^{-3} \end{equation} \tag{ 26 }$$

for equal-mass binaries with total mass M₁ + M₂ = (10⁵, 10⁶, 10⁷, 10⁸)M_☉; the numbers in parentheses decrease by ∼25% for binaries with M₂/M₁ = 0.1.

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

$$\begin{equation} {a_{\rm eq}\over r_{\rm k}} \approx 10^{-3} \left({V_{\rm k}\over \sigma }\right)^2. \end{equation} \tag{ 27 }$$

The requirement that a_eq < r_k—i.e., that at least some stars remain bound after the kick—then becomes V_k ≲ 30σ, which is never violated by reasonable (V_k, σ) values. However, the inner cutoff exceeds 0.1r_k for V_k ≳ 10σ, a condition that would be fulfilled for σ = 100 km s⁻¹ and V_k ≳ 10³ km s⁻¹. In what follows we ignore the inner cutoff and assume that the Bahcall–Wolf cusp extends to r = 0.

The pre-kick density can therefore be approximated as

$$\begin{equation} \rho (r) = \left\lbrace \begin{array}{lcc}\!\!\! \rho _0\left({r\over r_0}\right)^{-7/4} & & 0\,{\lesssim}\,r \,{\lesssim}\,r_0 \\ \rho _0 & & \,\,\,\,\, r_0\,{\lesssim}\,r \,{\lesssim}\,r_{\rm infl}\end{array} \right., \end{equation} \tag{ 28 }$$

where r₀ ≈ 0.2r_infl and ρ₀ = ρ(r₀) 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

$$\begin{eqnarray} {M_{\rm b}\over M_{\rm BH}} &=& F_1(\gamma) \left({GM_{\rm BH}\over r_{\bullet }V_{\rm k}^2}\right)^{3-\gamma } \end{eqnarray} \tag{ 29a }$$

$$\begin{eqnarray} &\approx & 1.3 \left({GM_{\rm BH}\over r_{\rm infl}V_{\rm k}^2}\right)^{5/4} \left({\rho _0r_{\rm infl}^3\over M_{\rm BH}}\right) \end{eqnarray} \tag{ 29b }$$

$$\begin{eqnarray} &\approx & \left({\sigma \over V_{\rm k}}\right)^{5/2} \left({M_{\rm core}\over M_{\rm BH}}\right) \,, \end{eqnarray} \tag{ 29c }$$

where M_core ≡ ρ₀r³_infl; the last expression assumes r_k < 0.2r_infl, i.e., V_k ≳ 2σ, which is always satisfied for a HCSS that escapes the galaxy core.

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

$$\begin{eqnarray} &&{M_{\rm b}\over M_{\rm BH}} \approx \left({\sigma \over V_{\rm k}}\right)^{5/2} \end{eqnarray} \tag{ 30a }$$

$$\begin{eqnarray} &&\approx 2\times 10^{-2} \left({\sigma \over 200 \mbox{ km s}^{-1}}\right)^{5/2} \left({V_{\rm k}\over 10^3 \mbox{ km s}^{-1}}\right)^{-5/2}. \end{eqnarray} \tag{ 30b }$$

In the case of ejection from a stellar spheroid like that of the Milky Way (σ ≈ 100 km s⁻¹, M_BH ≈ 4 × 10⁶M_☉), we have

$$\begin{equation} M_{\rm b}\approx 10^4 M_\odot \left({V_{\rm k}\over 10^3\, {\rm km\ s}^{-1}}\right)^{-5/2}, \end{equation} \tag{ 31 }$$

i.e.,

$$\begin{eqnarray} 3\times 10^2 \,{\lesssim}\,&M_{\rm b}/M_\odot & \,{\lesssim}\,5\times 10^4, \end{eqnarray} \tag{ 32a }$$

$$\begin{eqnarray} 4000 \ge &V_{\rm k}/({\rm km\ s}^{-1})& \ge 500. \end{eqnarray} \tag{ 32b }$$

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

$$\begin{eqnarray} M_{\rm b}&&\approx 0.4 G^{-5/4}M_{\rm BH}^{-1/4}\sigma ^{5/2}r_{\rm eff}^{5/4} \end{eqnarray} \tag{ 33a }$$

$$\begin{eqnarray} &&\approx 4\times 10^4 M_\odot \left({M_{\rm BH}\over 10^7M_\odot }\right)^{-1/4} \left({\sigma \over 100\ {\rm km\ s}^{-1}}\right)^{5/2} \left({r_{\rm eff}\over 0.1\ {\rm pc}}\right)^{5/4}\nonumber\\ \end{eqnarray} \tag{ 33b }$$

$$\begin{eqnarray} &&\approx 5\times 10^4 M_\odot \left({\sigma \over 100\ {\rm km\ s}^{-1}}\right)^{1.29} \left({r_{\rm eff}\over 0.1\ {\rm pc}}\right)^{5/4}\,, \end{eqnarray} \tag{ 33c }$$

where the final expression assumes the M_BH–σ relation in Equation (2).

The mass–velocity dispersion (M_b–σ_obs) relation for HCSSs follows from Equation (30) with σ_obs = F₃(γ)V_k ≈ 0.30V_k:

$$\begin{eqnarray} M_{\rm b}&\approx & 0.05 M_{\rm BH}\sigma ^{5/2} \sigma _{\rm obs}^{-5/2} \end{eqnarray} \tag{ 34a }$$

$$\begin{eqnarray} &&\approx 2\times 10^5M_\odot \left({\sigma \over 100\ {\rm km\ s}^{-1}}\right)^{7.4} \left({\sigma _{\rm obs}\over 100\ {\rm km\ s}^{-1}}\right)^{-5/2} \end{eqnarray} \tag{ 34b }$$

where the M_BH–σ relation has again been used.

Figures 9 and 10 plot the relations (33) and (34) for σ = (50, 100, 150) km s⁻¹. Plotted for comparison are samples of GCs and dwarf galaxies from the compilation of Forbes et al. (2008).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

In these plots, the minimum r_eff is presumed to be that associated with a kick of ∼4000 km s⁻¹. This condition (combined with the M_BH–σ relation) gives

$$\begin{equation} r_{\rm eff} \,{\gtrsim}\,3.0\times 10^{-3}\, {\rm pc} \left({\sigma \over 100\ {\rm km\ s}^{-1}}\right)^{4.86}. \end{equation} \tag{ 35 }$$

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

$$\begin{equation} r_{\rm eff} \,{\lesssim}\,4.9\, {\rm pc}\, N^{-2} \left({\sigma \over 100\ {\rm km\ s}^{-1}}\right)^{2.86}. \end{equation} \tag{ 36 }$$

Kicks large enough to eject a SMBH completely from its galaxy have N ≈ 5 (Figure 11). Kicks just large enough to remove a SMBH from the galaxy core have N ≈ 2 (Gualandris & Merritt 2008). Figures 9 and 10 show the limits on r_eff and σ_obs corresponding to N = 5 and N = 2, the latter via dashed lines.

Zoom In Zoom Out Reset image size

According to Figure 9, HCSSs in this "collisional" regime can have effective radii as big as ∼1 pc when V_k ≳ V_esc; on the M_b–r_eff plane their distribution barely overlaps with GCs, and extends to much lower sizes and (stellar) masses. However, their velocity dispersions (Figure 10) would always substantially exceed those of either GCs or compact galaxies of comparable (stellar) mass.

By "collisionless" we mean that the nuclear relaxation time is so long that gravitational scattering cannot 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 (Equation (24) has been reached). The relevant radius at which to evaluate the relaxation time is ∼r_infl, the influence radius of the binary (or of the single black hole that subsequently forms). The relaxation time at r_infl in elliptical galaxies is found to correlate tightly with σ or M_BH (Merritt et al. 2007b):

$$\begin{equation} t_{\rm R}(r_{\rm infl}) \approx 8.0\times 10^9\ {\rm yr} \left({M_{\rm BH}\over 10^6 M_\odot }\right)^{1.54}, \end{equation} \tag{ 37 }$$

where solar-mass stars have been assumed. A mass of order M_BH is scattered into the central sink in a time t_R(r_infl), 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 σ ≳ 200 km s⁻¹ or M_BH ≳ 10⁸M_☉.

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 d of the central object, per unit of time, is found to scale roughly linearly with d (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 ∼70% (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 ≫M_BH 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 2006); 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 ρ ∝ r^−γ, with r_• being the radius at which the enclosed stellar mass equals twice M_BH. We computed γ and r_• for a subset of early-type galaxies in the ACS Virgo sample (Côté et al. 2004) for which σ was known; for some of these galaxies the SMBH mass has been measured dynamically while M_BH in the remaining galaxies was computed from Equation (2). Each galaxy was modeled with a point-spread function (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 R_b. The core-Sersic fits were numerically deprojected, and converted from a luminosity to a mass density as in Ferrarese et al. (2006). The radius r_• was then computed from

$$\begin{equation} r_{\bullet }= \left({3-\gamma \over \pi } {M_{\rm BH}\over \rho _0r_0^\gamma }\right)^{1/(3-\gamma)} \,, \end{equation} \tag{ 38 }$$

where γ is the central power-law index of the deprojected density, r₀ is a fiducial radius smaller than R_b which we chose to be 1 pc, and ρ₀ is the mass density at r = r₀.

Figure 12 shows the relation between r_• and σ and between r_• and M_BH. "Core" galaxies (those with projected profiles flatter than Σ ∝ R^−0.5 near the center) are plotted with open circles and "power-law" galaxies (Σ steeper than R^−0.5) galaxies as filled circles. While this distinction is somewhat arbitrary, Figure 12 confirms that the "core" galaxies have larger r_• at given σ or M_BH 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

$$\begin{eqnarray} \log _{10} (r_{\bullet }/\rm pc) &=& -\!4.84 + 2.84 \log _{10} (\sigma /{\rm km\ s}^{-1}) \end{eqnarray} \tag{ 39a }$$

$$\begin{eqnarray} &=& -\!2.92 + 0.56 \log _{10} (M_{\rm BH}/M_\odot), \end{eqnarray} \tag{ 39b }$$

i.e.

$$\begin{eqnarray} r_{\bullet }&\approx & 50\ {\rm pc} \left({\sigma \over 200\ {\rm km\ s}^{-1}}\right)^{2.8} \end{eqnarray} \tag{ 40a }$$

$$\begin{eqnarray} &\approx & 35\ {\rm pc} \left({M_{\rm BH}\over 10^8M_\odot }\right)^{0.56}. \end{eqnarray} \tag{ 40b }$$

While these relations are fairly tight, the γ values show somewhat more scatter, in the range 0.5 ≲ γ ≲ 1.5, and we leave γ as a free parameter in what follows.

Zoom In Zoom Out Reset image size

Combining the relations (40) with Equations (19b) and (2) gives a mass–radius relation for HCSSs in the "collisionless" paradigm:

$$\begin{eqnarray} &&{M_{\rm b}\over 10^4M_\odot } \approx G_{\rm s}(\gamma)\! \left({\sigma \over 200\ {\rm km\ s}^{-1}}\right)^{2.84\gamma -3.66}\!\! \left({r_{\rm eff}\over 0.1\ {\rm pc}}\right)^{3-\gamma } \end{eqnarray} \tag{ 41a }$$

$$\begin{eqnarray} &\approx & G_{\rm m}(\gamma) \left({M_{\rm BH}\over 10^8M_\odot }\right)^{0.560\gamma -0.680} \left({r_{\rm eff}\over 0.1\ {\rm pc}}\right)^{3-\gamma }, \end{eqnarray} \tag{ 41b }$$

where

$$\begin{eqnarray} G_{\rm s}(\gamma) &=& 1.93\times 10^5 \gamma ^{-1.75} \left(750.\over 2^{1/(3-\gamma)}-1\right)^{\gamma -3}, \end{eqnarray} \tag{ 42a }$$

$$\begin{eqnarray} G_{\rm m}(\gamma) &=& 1.16\times 10^5 \gamma ^{-1.75} \left(525.\over 2^{1/(3-\gamma)}-1\right)^{\gamma -3}. \end{eqnarray} \tag{ 42b }$$

Similarly, combining Equations (40) with Equation (22) gives the mass–velocity dispersion relation

$$\begin{eqnarray} {M_{\rm b}\over 10^4M_\odot } &\approx & H_{\rm s}(\gamma) \left({\sigma \over 200\ {\rm km\ s}^{-1}}\right)^{10.92 - 2.02\gamma } \!\!\left({\sigma _{\rm obs}\over 100\ {\rm km\ s}^{-1}}\right)^{2(\gamma -3)}\nonumber\\ \end{eqnarray} \tag{ 43a }$$

$$\begin{eqnarray} &\approx & H_{\rm m}(\gamma) \left({M_{\rm BH}\over 10^8M_\odot }\right)^{2.32-0.44\gamma } \left({\sigma _{\rm obs}\over 100\ {\rm km\ s}^{-1}}\right)^{2(\gamma -3)} \end{eqnarray} \tag{ 43b }$$

where

$$\begin{eqnarray} H_{\rm s}(\gamma) &=& 1.92\times 10^5 \gamma ^{-1.75}[1.2F_3(\gamma)]^{2(3-\gamma)}, \end{eqnarray} \tag{ 44a }$$

$$\begin{eqnarray} H_{\rm m}(\gamma) &=& 1.16\times 10^5 \gamma ^{-1.75}[1.1F_3(\gamma)]^{2(3-\gamma)} \end{eqnarray} \tag{ 44b }$$

and S(γ) is given by Equation (21).

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

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 the 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 M_BH = 0.002M_gal. We confirmed, for γ = 1 and γ = 2, that the fraction of the mass within r_k 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 Section 2. While this is certainly possible, we note that the anisotropy would have to be appreciable at very small radii, a tenth or hundredth of the SMBH influence radius, in order to substantially affect the fraction of the population that is bound after the kick.

For γ < 0.5, there is no isotropic distribution function that can reproduce the density near the SMBH (or rather, the isotropic distribution function would be negative at highly bound energies). For such flat cores, one would need to assume a tangentially anisotropic velocity distribution. This would have the effect of increasing the mass of the post-kick bound population, since it would increase the number of stars on low-velocity (nearly circular) orbits at every radius.

These uncertainties deserve to be more completely addressed in a future paper. For now, we prefer to encapsulate them all in the value of γ.

Recoils large enough to eject SMBHs completely from galaxies can occur even in the absence of gravitational waves, via Newtonian interactions involving three or more massive objects (e.g., Mikkola & Valtonen 1990; Hoffman and Loeb 2006). The same interactions can also hasten coalescence of a binary SMBH by inducing changes in its orbital eccentricity. If the infalling SMBH is less massive than either of the components of the pre-existing binary, M₃ < (M₁, M₂), the ultimate outcome is likely to be ejection of the smaller hole and recoil of the binary, with the binary eventually returning to the galaxy center. If M₃ > M₁ or M₃ > M₂, there will most often be an exchange interaction, with the lightest SMBH ejected and the two most massive SMBHs forming a binary; further interactions then proceed as in the case M₃ < (M₁, M₂). Whether, and to what extent, the ejected SMBH constitutes a HCSS depends on whether it carries a bound population and can retain it during interaction with the other SMBHs. These questions are amenable to high-accuracy N-body simulations, which we hope to carry out in the future.

The presence of significant amounts of cold gas in galaxy nuclei can also accelerate the evolution of a binary SMBH. However, it is not clear whether the net effect of gas would be to increase, or decrease, the mass of a bound stellar population around the coalesced binary, compared with the purely stellar-dynamical estimates made here. On the one hand, gas-dynamical torques can lead to rapid formation of a tightly bound binary SMBH (Mayer et al. 2007), reducing the time that the binary can deplete the stellar density in the core on scales of the SMBH influence radii. On the other hand, the formation of a steep Bahcall–Wolf cusp around a shrinking binary discussed in Section 3 requires that the binary evolution timescale be of the order of the nuclear relaxation time. Cold gas also implies star formation, which could increase the number of bound stars.

We have shown that stars bound to a recoiling SMBH would appear as very compact stellar clusters with exceptionally high velocity dispersions. The density of HCSSs, and therefore the chances of finding them, will be highest in clusters of galaxies, and nearby galaxy clusters such as Virgo and Fornax are therefore well suited to searching for HCSSs.

In their properties, HCSSs share similarities with GCs. However, they would differ from classical GCs by their much larger velocity dispersions and (possibly) higher metal abundances (the nuclei of elliptical galaxies in the Virgo and Coma clusters often have metal abundances comparable with solar, and the cores of quasars, powered by major mergers, frequently show supersolar metallicities). They would differ from stripped galactic nuclei and UCDs (Phillipps et al. 2001) by their typically greater compactness (Figure 19). They would differ from objects in their local environment by possibly showing a large velocity offset.

How can we find HCSSs and distinguish them from other source populations? Systematic searches would be based on color, compactness, spectral properties, or combinations of these.

Imaging searches for compact stellar systems have been or are currently being carried out, focusing especially on the nearest clusters of galaxies Fornax (e.g., Hilker et al. 1999; Drinkwater et al. 2003; Mieske et al. 2008), Virgo (e.g., Côté et al. 2004; Haşegan et al. 2005; Mieske et al. 2006; Firth et al. 2008), and Coma (Carter et al. 2008), and on a number of nearby groups (Evstigneeva et al. 2007). These studies have resulted in the detection of a large number of GCs, and of several UCDs per galaxy cluster.

Two strategies suggest themselves for identifying HCSSs among existing surveys of compact stellar systems in nearby galaxy clusters. One would focus on the faintest HCSSs which are most abundant; the other would concentrate on the brightest objects, which are rare, but most amenable to follow-up spectroscopic observations.

According to Figure 19, HCSSs separate in r_eff − luminosity space most strongly at the smallest effective radii, smaller than galactic GCs, while Figure 18 suggests that their colors should be comparable to (metal-rich) GCs or (gas-poor, i.e., non-star-forming, and non-accreting) galactic nuclei. PSF-deconvolved HST imaging can achieve a spatial resolution better than 0.1 arcsec, corresponding to spatial scales of ∼10 pc at the distance of the Virgo cluster. However, in order to confirm such HCSS candidates, a laborious spectroscopic multi-fiber follow-up survey would then have to be carried out.

Instead, selecting brighter (even though rarer) objects appears more promising and would substantially reduce the exposure time. A key signature of a HCSS is its large velocity dispersion, which would distinguish it from luminous GCs and most known UCDs. At the same time, the highest velocity dispersions would tend to put the broadened absorption lines below the noise. Therefore, HCSSs with σ_obs below several hundred km s⁻¹ might be easiest to detect. These are in fact expected to be the most common (Figure 16).

In order to estimate exposure times, we simulated spectra with the multi-object spectrograph FLAMES attached to the VLT (Pasquini et al. 2002), using the spectrograph GIRAFFE in MEDUSA mode. This spectrograph allows the observation of up to 130 targets at a time at intermediate (∼30 km s⁻¹) to high (∼10 km s⁻¹) spectral resolution. The simulated spectral deconvolutions in Section 2 suggest that a signal-to-noise ratio (S/N) of ∼10 is sufficient to detect the broadened lines, while S/N ≈30 is desirable in order to probe the non-Gaussianity of the broadening function. In order to reach a S/N of 30 for a cluster of M_V ≈ 21 or fainter (M_V ≳ −10 at Virgo) requires excessive integration times in the high-resolution mode. In lower resolution mode, about 10 hours exposure time are required to reach S/N = 30 for M_V = 20. Simply detecting the high velocity dispersion requires less time, of order an hour or less for M_V ≲ 21.

HCSSs share some properties with UCDs and dwarf-globular transition objects ("DGTOs;" Haşegan et al. 2005). These are compact stellar systems with stellar velocity dispersions as high as ∼50 km s⁻¹, masses between 10^6–8 M_☉, and unusually high mass-to-light ratios (e.g., Hilker 2009). Figures 9 and 19 suggest that UCD sizes are consistent with those of the largest ("collisionless") HCSSs, although Figures 10 and 20 suggest that known UCD velocity dispersions are too low by a factor of at least a few. Furthermore Figures 19 and 20 suggest that HCSSs with properties similar to those of UCDs are likely to be rare. However, there is increasing evidence that UCDs are a "mixed bag" (e.g., Hilker 2006), possibly requiring a number of different formation mechanisms (Oh et al. 1995; Fellhauer & Kroupa 2002; Bekki et al. 2003; Mieske et al. 2004; Martini & Ho 2004; Goerdt et al. 2008). Individual HCSSs might therefore hide among the UCD population, and low-mass UCD and DGTO candidates identified in current and future surveys might sometimes be recoiling HCSSs. Objects such as Z2109 (Zepf et al. 2008) with its very unusual and broad [O iii] emission lines are also possible candidates.

The predicted HCSS colors and luminosities summarized in Figures 18–20 reflect the properties of the old (∼5 Gyr) stellar populations that predominate in our models. A HCSS that was discovered near its birthplace might appear much younger than implied by these plots, particularly if it was born in a starburst galaxy resulting from a merger. The light from such a HCSS might be dominated by young massive stars, at least during the ∼0.1–1 Gyr required for it to exit the galaxy. (We are assuming here that massive stars can form, or at least quickly find their way, well inside the SMBH influence radius, as appears to be the case at the center of the Milky Way. We are further assuming that the recoiling SMBH is not accreting at the time of observation since otherwise the quasar might outshine the starlight.) We implicitly excluded this possibility above by assuming 0.1 Gyr as a minimum lag between star formation and SMBH ejection.

Starting roughly 6 Myr after a starburst episode, luminosities and colors of the burst population are expected to increase rapidly due to red supergiant stars (RSGs), which begin to appear when main-sequence stars of initial mass ∼25M_☉ finish core-hydrogen burning. The absolute magnitude of a burst population with standard IMF spikes roughly ∼10 Myr after the burst due to the RSGs; the increase is greatest in the IR bands. For example, in the K band, integrated luminosity increases to a value ∼100 times greater than for the same population at 1 Gyr (Leitherer et al. 1999) before falling off rapidly after ∼14 Myr. Integrated colors reach their reddest values at the same time, i.e., V − I ≈ 1.5. Single RSG stars can approach luminosities of 10⁵–10⁶L_☉ (Davies et al. 2007), so the luminosity and color of an HCSS containing ≪10⁶ stars could undergo enormous relative changes around this time.

There are both advantages and disadvantages to targeting such young HCSSs as a search strategy. (Super)star clusters are commonly found in the centers of starburst galaxies and merger remnants (e.g., Max et al. 2005; Galliano et al. 2008), so in any search based solely on imaging photometry, HCSSs could not be easily distinguished. However, once a HCSS reaches larger separations from the center of the galaxy, it will already deviate from its surroundings by plausibly showing higher metal abundance. On the other hand, even low-resolution spectroscopy could immediately reveal a high velocity relative to the galaxy core or relative to other clusters in the galaxy.

We note that discovery of a young HCSS that is still uniquely associated with its host galaxy can provide important constraints on the timescale associated with late evolution of the binary SMBH that engendered the kick. If coalescence was rapid, the host galaxy of the HCSS will still show signs of recent interaction (for instance, tidal tails). If the massive binary stalled for ≳ 10⁸–10⁹ yr before coalescing, these signs will be mostly gone. Age dating of the recoil event could be based on (projected) distance from the galaxy core and recoil velocity, and/or on the age of the stellar populations.

HCSSs with accreting central black holes, powered either by stellar tidal disruptions or stellar mass loss, were discussed in Paper I. This subpopulation could be efficiently searched for by combining optical properties with information from multi-wavelength surveys, e.g., in the X-ray, UV, or radio band. We speculate that the unusual optical transient source SCP06F6 (Barbary et al. 2009) might be a tidally detonated white dwarf bound to a recoiling SMBH (as discussed already in Paper I). This scenario would fit the high amplitude of variability of the transient, the absence of an obvious host galaxy, and the possible association with a cluster of galaxies at z = 1.1. However, the observed symmetry of the light curve might suggest a lensing origin (Barbary et al. 2009) even though the authors did not favor lensing for other reasons. The unusual optical spectrum of this source could be caused by the tidal-debris disk illuminating the outer disk or the outflowing part of the detonation debris. That way, the observed extreme, unusually broadened absorption features would be caused if we are looking down stream. Gänsicke et al. (2009) recently reported the detection of an X-ray source co-incident with SCP 06F6 with an X-ray luminosity at the lower end of known tidal disruption flares (Komossa et al. 2004). These authors discuss supernovae-related scenarios but also consider tidal disruption of a star, and suggest a preliminary redshift of 0.14, in which case the source is not associated with the cluster at redshift 1.1.

Finally, some of the oldest surviving HCSSs would consist mostly of stellar end states. They would be quite faint, since only very low-mass stars and WDs would remain, but could possibly be identified by their very unusual colors.