MERGERS OF STELLAR-MASS BLACK HOLES IN NUCLEAR STAR CLUSTERS

Our approach is similar to that of O'Leary et al. (2006), who focus on globular clusters with velocity dispersions σ_1D ⩽ 20 km s⁻¹. Here, however, we concentrate on the more massive and tightly bound nuclear star clusters. Our departure point is the relation found by Ferrarese et al. (2006) between the masses and velocity dispersions of such clusters:

$$\begin{equation} M_{\rm nuc}=10^{6.91\pm 0.11} (\sigma _{\rm 1D,bulge}/54\;{\rm km\;s}^{-1})^{4.27\pm 0.61}\,M_\odot \;. \end{equation} \tag{ 1 }$$

Assuming that there is no massive central BH for these low velocity dispersions, the half-mass relaxation time for the system is (see Binney & Tremaine 1987) $t_{\rm rlx}\approx {N/2\over {8\ln N}}t_{\rm cross}$, where N ≈ M_nuc/0.5 M_☉ is the number of stars in the system (assuming an average mass of 0.5 M_☉) and t_cross ≈ R/σ_3D is the crossing time. Here, R = GM_nuc/σ²_3D is the radius of the cluster. If we assume that $\sigma _{\rm 3D}=\sqrt{3}\sigma _{\rm 1D}$ and that typically σ_1D,bulge ≈ 2σ_1D ≈ σ_3D, this gives

$$\begin{equation} t_{\rm rlx}\approx 1.3\times 10^{10}\;{\rm yr} (\sigma _{\rm 3D}/54\;{\rm km\;s}^{-1})^{5.5}\;. \end{equation} \tag{ 2 }$$

The relaxation time scales inversely with the mass of an individual star (Binney & Tremaine 1987), so a 10 M_☉ BH will settle in roughly 1/20 of this time. Also note that large N-body simulations with broad mass functions evolve to core collapse within roughly 0.2 half-mass relaxation times (Portegies Zwart & McMillan 2002; Gürkan et al. 2004); hence, in the current universe, clusters with velocity dispersions σ_3D < 60 km s⁻¹ will have had their central potentials deepened significantly.

The amount of deepening of the potential, and thus the escape speed from the center of the cluster, depends on uncertain details such as the initial radial dependence of the density and the binary fraction. Given that the timescale for segregation of the BHs in the center is much less than a Hubble time, we will assume that the escape speed is roughly 5σ_1D, as is the case for relatively rich globular clusters (Webbink 1985). This may well be somewhat conservative, because the higher velocity dispersion here than in globulars suggests that a larger fraction of binaries will be destroyed in nuclear star clusters. This could lead to less efficient central energy production and hence deeper core collapse than is typical in globulars.

With this setup, our task is to follow the interactions of BHs in the central regions of nuclear star clusters, where we will scale by stellar number densities of n ∼ 10⁶ pc⁻³ (a characteristic value near the center of the Milky Way; see Genzel et al. 2003b) because of density enhancements caused by relaxation and mass segregation. Our hypothesis is then that binary-single interactions will (1) allow BHs to swap into binaries even if they began as single objects, and (2) harden BH binaries to the extent that they can merge while still in the nuclear star cluster. If a BH starts its life with a binary companion, then the interaction time is short, because every interaction it has will be a binary-single encounter that has a high cross-section. If instead the BH begins as a single object, the binary-single interaction rate is much less because it relies on the BH encountering comparatively rare binaries. This is the case we will consider, because if there is enough time for a BH to capture into a binary and then harden, there is certainly enough time for a BH that is born into a binary to harden.

All binaries in the cluster will be hard, that is, will have internal energies greater than the average kinetic energy of a field star, because otherwise they will be softened and ionized quickly (e.g., Binney & Tremaine 1987). If, for example, we consider binaries of two 1 M_☉ stars in a system with σ_3D = 50 km s⁻¹, then for the binary to be hard, the semimajor axis has to be less than a_max ∼ 1 AU. Studies of main-sequence binaries in globular clusters, which have σ_1D ∼ 10 km s⁻¹, suggest that after billions of years, roughly 5%–20% of them survive, with the rest falling victim to ionization or collisions (Ivanova et al. 2005). The binary fraction will be lower in nuclear star clusters due to their enhanced velocity dispersion, but since when binaries are born, they appear to have a constant distribution across the log of the semimajor axis from ∼10⁻²–10³ AU (e.g., Abt 1983; Duquennoy & Mayor 1991), the reduction is not necessarily by a large factor. We will scale by a binary fraction f_bin = 0.01, which is likely to be somewhat low and thus we will slightly overestimate the time needed for a BH to be captured into a binary.

If a BH with mass M_BH gets within a couple of semimajor axes of a main-sequence binary, the binary will tidally separate and the BH will acquire a companion. The timescale on which this happens is t_bin = (nΣσ_3D)⁻¹, where Σ = πr²_p[1 + 2GM_tot/(σ_3D²r_p)] is the interaction cross-section for pericenter distances ⩽r_p when gravitational focusing is included. Here, M_tot is the mass of the BH plus the mass of the binary. If we assume that M_BH = 10 M_☉ and it interacts with a binary with two 1 M_☉ members and an a = 1 AU semimajor axis, then the typical timescale on which a three-body interaction and capture of one of the stars occur is

$$\begin{eqnarray} t_{\rm 3-bod}&=&(n\Sigma \sigma _{\rm 3D})^{-1}\approx 3\times 10^9\;{\rm yr} (n/10^6\;{\rm pc}^{-3})^{-1}\nonumber\\ &&\times(f_{\rm bin}/0.01)^{-1}(\sigma _{\rm 3D}/50\;{\rm km\;s^{-1}})(a/1\;{\rm AU})^{-1}. \end{eqnarray} \tag{ 3 }$$

With rapid sinking, BHs can form a subcluster in the galaxy core. This will decrease the number density of main-sequence stars in the core and, hence, of main-sequence binaries (although binaries, being heavier than single stars, will be overrepresented). The exchange process might thus take somewhat longer. The timescale in Equation (3) is, however, small enough compared to a Hubble time that we start our simulations by assuming that each BH has exchanged into a hard binary, and follow its evolution from there.

Another important question is whether, after a three-body interaction, a BH binary will shed the kinetic energy of its center of mass via dynamical friction and sink to the center of the cluster before another three-body encounter. If not, the kick speeds will add in a random walk, thus increasing the ejection fraction.

To compute this we note that the local relaxation time of a binary is

$$\begin{equation} t_{\rm rlx}={0.34\over {\ln \Lambda }}{{\sigma _{\rm 3D}}^3\over { G^2\langle m\rangle M_{\rm bin}n}} \end{equation} \tag{ 4 }$$

(Spitzer 1987), where ln Λ ∼ 10 is the Coulomb logarithm, 〈m〉 is the average mass of interloping stars, n is their number density, and M_bin is the mass of the binary. The timescale for a three-body interaction is t_3-bod=(nΣσ_3D)⁻¹ as above. Note, however, that for this calculation we assume that the BH has already captured into a binary. Therefore, it can interact with every star instead of just those in binaries, and thus the factor f_bin is no longer applicable and the timescale is typically 100 times less than indicated by the numerical factor in Equation (3). For a gravitationally focused binary, which is of greatest interest because only these could in principle produce three-body recoil sufficient to eject binaries or singles, r_p < GM_bin/σ_3D². If we also assume that the total mass M_tot of the three-body system is close to M_bin because most of the interlopers have much less mass than the BH, then Σ ≈ 2πr_pGM_bin/σ_3D² and

$$\begin{equation} t_{\rm 3-bod}\approx {\sigma _{\rm 3D}\over {2\pi nr_p GM_{\rm bin}}}\;. \end{equation} \tag{ 5 }$$

If we let r_p = qGM_bin/σ_3D², with q < 1, then

$$\begin{equation} t_{\rm 3-bod}\approx {{\sigma _{\rm 3D}}^3\over {2\pi qG^2M_{\rm bin}^2 n}} \end{equation} \tag{ 6 }$$

so that

$$\begin{equation} t_{\rm rlx}/t_{\rm 3-bod}\approx {2q\over {\ln \Lambda }}{M_{\rm bin}\over { \langle m\rangle }}\;. \end{equation} \tag{ 7 }$$

The encounters most likely to deliver strong kicks to the binary occur when the binary is very hard, q ≪ 1; hence, this quantity is typically less than unity, meaning that after a three-body encounter, a binary has an opportunity to share its excess kinetic energy via two-body encounters and thus settle back to the center of the cluster. We, therefore, treat the encounters separately rather than adding the kick speeds in a random walk.

In a given encounter, suppose that a binary of total mass M_bin = M₁ + M₂, a reduced mass μ = M₁M₂/M_bin, and a semimajor axis a_init interacts with an interloper of mass m_int, and that the kinetic energy of the interloper at infinity is much less than the binding energy of the binary (i.e., this is a very hard interaction). If after the interaction the semimajor axis is a_fin < a_init, then energy and momentum conservation mean that the recoil speed of the binary is given by $v_{\rm bin}^2=G\mu {m_{\rm int}\over {M_{\rm bin}+m_{\rm int}}} \left(1/a_{\rm fin}-1/a_{\rm init}\right)$, and the recoil speed of the interloper is v_int = (M_bin/m_int)v_bin. For example, suppose that M₁ = M₂ = 10 M_☉, M_int = 1 M_☉, a_init = 0.1 AU, and a_fin = 0.09 AU. The binary then recoils at v_bin = 15 km s⁻¹ and stays in the cluster, whereas the interloper recoils at v_int = 300 km s⁻¹ and is ejected.

We treat all three objects as point masses, but in fact main-sequence stars are extended enough that they have a good chance of being tidally disrupted in an encounter with a BH binary. Almost all of the disrupted mass is eventually ejected at speeds comparable to the binary orbital speed; hence, we assume that tidal disruptions of main-sequence stars have the same effect on the binary energetically as a normal three-body ejection. Note, however, that since unlike for interactions with point sources, the ejected mass will not go in a single direction, it is likely that tidal disruptions will not cause the binary to recoil as much and thus such interactions are likely to result in a somewhat greater retention fraction than we calculate. We also find that close approach distances are great enough that post-Newtonian corrections are not necessary during interactions, and that a small enough fraction of stars are involved in these encounters that the effect on the mass distribution due to mergers and ejections is negligible. In addition, we assume throughout our calculations that nuclear star clusters do not have massive BHs at their centers.

The central regions of the clusters undergo significant mass segregation, and thus the mass function will be at least flattened, and possibly inverted. This has been observed for globular clusters (see Table 3 of Sosin 1997 or Table 1 of De Marchi et al. 2007) and is also seen in numerical simulations (e.g., Baumgardt et al. 2008 or Gill et al. 2008). To include this effect, when we consider the mass of a BH, its companion, or the interloping third object in a binary-single encounter, we go through two steps. First, we select a zero-age main-sequence (ZAMS) mass between 0.2 M_☉ and 100 M_☉ using a simple power-law distribution dN/dM ∝ M^−α. While there is evidence that the upper limit of the ZAMS might be greater than 120 M_☉ (Oey & Clarke 2005), we chose the more traditional value of 100 M_☉ (Kroupa & Weidner 2003) to be conservative. We allow α to range anywhere from 2.35 (the unmodified Salpeter distribution) to −1.0, where smaller values indicate the effects of mass segregation. Second, we evolve the ZAMS mass to a current mass. Our mapping is that for M_ZAMS < M_ms,max, where M_ms,max is 1 M_☉ or 3 M_☉ depending on the model used, the star is still on the main sequence and retains its original mass; for 1 M_☉ < M_ZAMS < 8 M_☉, the star has evolved to a white dwarf, with mass M_WD = 0.6 M_☉ + 0.4 M_☉(M_ZAMS/M_☉ − 0.6)^1/3; for 8 M_☉ < M_ZAMS < 25 M_☉, the star has evolved to a neutron star, with mass M_NS = 1.5 M_☉ + 0.5 M_☉(M_ZAMS − 8 M_☉)/17 M_☉; and for M_ZAMS > 25 M_☉, the star has evolved to a BH with mass M_BH = 3 M_☉ + 17 M_☉(M_ZAMS − 25 M_☉)/75 M_☉. Therefore, we assume that BH masses range from 3 M_☉ to 20 M_☉.

These prescriptions are overly simplified in many ways. We, therefore, explore different mass function slopes, main-sequence cutoffs, and so on, and find that our general picture is robust against specific assumptions. Note that, consistent with O'Leary et al. (2006), we find that there is a strong tendency for the merged BHs to be biased toward high masses. Therefore, if BHs with masses greater than 20 M_☉ are common, these will dominate the merger rates. This is important for data analysis strategies, because the low-frequency cutoff of ground-based gravitational wave detectors implies that higher-mass BHs will have proportionally more of their signal in the late inspiral, merger, and ringdown.

The three-body interactions themselves are assumed to be Newtonian interactions between point masses and are computed using the hierarchical N-body code HNBody (K. Rauch and D. Hamilton 2008, in preparation), using the driver IABL developed by Kayhan Gültekin (see Gültekin et al. 2004, 2006 for a detailed description). These codes use a number of high-accuracy techniques to follow the evolution of gravitating point masses. Between interactions, we use the Peters equations (Peters 1964) to follow the gradual inspiral and circularization of the binary via emission of gravitational radiation. This is negligible except near the end of any given evolution.

We begin by selecting the mass of the BH and of its companion (which does not need to be a BH) from the evolved mass function. We also begin with a semimajor axis that is 1/4 of the value needed to ensure that the binary is hard. We do this because soft binaries are likely to be ionized and thus become single stars rather than merge. We also select an eccentricity from a thermal distribution P(e)de = 2ede. We then allow the binary to interact with single field stars drawn from the evolved mass function, one at a time, until either (1) the binary merges due to gravitational radiation, (2) the binary is split apart and thus ionized (this is exceedingly rare given our initial conditions), or (3) the binary is ejected from the cluster. The entire set of interactions until merger typically takes millions to tens of millions of years, and only rarely more than a hundred million years, so it finishes in much less than a Hubble time. This implies that the total time for an individual, initially single, BH to merge with another object is dominated by the few billion years needed to capture into a binary rather than by subsequent interactions. In the course of these interactions there are typically a number of exchanges, which usually swap in more massive for less massive members of the binary. This is the cause of the bias toward high-mass mergers that was also found by O'Leary et al. (2006). As shown in Table 1, for α < 1, most BHs acquire a BH companion in the process of exchanges, and for α ⩽ 0.5 virtually all do.

The results in Table 1 are focused on different mass function slopes and escape speeds. As expected, we find that for V_esc > 150 km s⁻¹, the overwhelming majority of BH binaries merge in the nuclear star cluster rather than being ejected (see Figure 1). This is the difference from lower-σ globular clusters, where the mergers happen outside the cluster. Note also that in addition to few binaries being ejected, there are typically only 1–2 single BHs ejected per merger, suggesting that greater than 50% of holes will merge. In contrast, at the 50 km s⁻¹ escape speed typical of globulars, greater than 20 single BHs are ejected per merger, suggesting an efficiency of less than 10%. For well-segregated clusters (with α ⩽ 0), the average mass of BHs that merge, binary ejection fraction and number of singles ejected, and number of BHs that merge with each other instead of other objects are all insensitive to the particular mass function slope. For less segregated clusters with α>0, the retention fraction of BHs rises rapidly to unity because most of the objects that interact with the holes are less massive stars. For example, in clusters with α ⩾ 1.0, about 10% of BH mergers occur with neutron stars, in contrast to a few percent or less for more segregated clusters. In such clusters, there might be a channel by which the mass of the holes increases via accretion of main-sequence stars, but we expect α>0 to be rare for nuclear star clusters because of the shortness of the segregation times of BHs. Overall, there appears to be a wide range of realistic parameters in which fewer than 10% of binary BHs are ejected before merging.

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