THE CLOSE STELLAR COMPANIONS TO INTERMEDIATE-MASS BLACK HOLES

In this work we use the direct-summation N-body star cluster dynamics code NBODY6 (Aarseth 1999, 2003) in a version enabled for graphics processing unit-accelerated calculations (Nitadori & Aarseth 2012)³ to follow the evolution of star clusters in a realistic tidal field following the implementation of Heggie et al. (2006) and subsequent papers (Trenti et al. 2007a, 2007b, 2008, 2010; Gill et al. 2008; Trenti 2008; Pasquato et al. 2009; Vesperini & Trenti 2010; Trenti & van der Marel 2013). Simulations containing N stellar particles are performed in N-body units where G = M_T = −4 E_T = 1, in which M_T and E_T are the total mass and energy of the cluster initial conditions (Heggie & Mathieu 1986). The corresponding dynamical time unit is

$$\begin{eqnarray}&&{t}_{{\rm{d}}}={{GM}}_{{\rm{T}}}^{5/2}/{(-4{E}_{{\rm{T}}})}^{3/2}.\end{eqnarray} \tag{ 1 }$$

The stellar cluster's dynamical state relaxes over approximately a relaxation time:

$$\begin{eqnarray}&&{t}_{{\rm{rh}}}=\displaystyle \frac{0.138\;{{Nr}}_{{\rm{h}}}^{3/2}}{\mathrm{ln}(0.11N)},\end{eqnarray} \tag{ 2 }$$

defined using the half-mass radius, r_h (Spitzer 1987; Trenti et al. 2010). The external tidal field of the host galaxy is implemented following Trenti et al. (2007b), and the reader is directed to Section 2 of that paper for details.

The NBODY6 distribution embeds the single stellar evolution (SSE) and binary stellar evolution (BSE) codes of Hurley et al. (2000, 2002) as a simplified, on-the-fly implementation of stellar evolution.⁴ The binaries in our clusters evolve with the standard BSE implementations of binary evolution, including tidal circularization, magnetic braking, Roche lobe overflow, and binary coalescence (Hurley et al. 2002).

We initialize our models with an IMBH of mass M_bh at rest at the system center of mass. This IMBH particle then evolves identically to the remaining particles. In this sense, our simulations do not self-consistently account for the formation of such an IMBH; instead, we aim to simulate its subsequent dynamical evolution in the cluster context. Because the IMBH is more massive than any of the stellar particles, it sinks toward the cluster center, motivating our initial condition. We find within the first core relaxation time the BH acquires nonzero energy relative to the cluster center of mass through strong interactions with cluster-core stars, rapidly erasing any imprint of this initial condition. We include gravitational wave energy and angular momentum losses from any very compact binaries following the Peters (1964) formulation, which is particularly relevant for the IMBH and its companion (e.g., Miller & Hamilton 2002; Gültekin et al. 2004, 2006; Leigh et al. 2014).

Close interactions with the IMBH can lead to the tidal destruction of a star or the inspiral and merger of a compact object. We handle those events as follows in our simulations. For stars, we delete any star passing within one tidal radius,

$$\begin{eqnarray}&&{r}_{{\rm{t}}}={({M}_{\mathrm{bh}}/{M}_{*})}^{1/3}{R}_{*},\end{eqnarray} \tag{ 3 }$$

of the BH at pericenter. In a tidal disruption event, the star is often on a weakly bound or near-parabolic orbit (Frank & Rees 1976; Lightman & Shapiro 1977). In this case, approximately half of the tidal debris remains bound to the BH, while the other half is ejected (Rees 1988). We model this behavior by adding half of the disrupted star's mass to the IMBH's mass following a disruption event. An ideal implementation of the stellar tidal disruption process would account for the stellar structure in determining the degree of mass loss and would also allow for partial tidal disruptions (Guillochon & Ramirez-Ruiz 2013). Partial tidal stripping is particularly relevant to evolved stars that contain a dense, tidally invulnerable core (MacLeod et al. 2012, 2013; Bogdanović et al. 2014). However, in this work we adopt the simplified, full-disruption model described above. We implement gravitational wave inspiral and merger events through the inclusion of gravitational wave corrections to the two-body equation of motion of tight pairs of compact objects (Peters 1964). Rather than allowing merging particles to spiral to arbitrarily small separations, we instead merge pairs of compact objects whose inspiral time becomes shorter than one N-body time unit, as in Equation (1). We do not apply a kick to the remnant M_bh, although such a kick may be relevant for some mergers (Holley-Bockelmann et al. 2008; Konstantinidis et al. 2013).

We have implemented several diagnostics to allow us to better study close companions to the IMBH. We record all three-body encounters in which a third object approaches with a pericenter distance within three times the semimajor axis of the most-bound companion to the IMBH. We also include the parameters of all tidal-disruption and merger events described above. For close tidal passages that do not lead to disruption, those with pericenter distances r_t < r_p < 10³ r_t, we also record the system parameters. While we record full-cluster mass, position, and velocity snapshots every 10 N-body time units, we record the IMBH's position and velocity every 0.1N-body time unit. Every N-body time unit we record the parameters of the most tightly bound partner to the IMBH.

In this paper, we present four groups of simulations with differing initial parameters. Simulations within each group have statistically different realizations of the initial conditions. Table 1 defines the properties of these simulation groups. We use the McLuster code (Küpper et al. 2011) to generate the initial stellar distribution following the King (1966) models with W₀ = 7. Our simulations include an IMBH that is initialized at the center of mass with zero velocity and N = (1–2) × 10⁵ stars. The size of our clusters is constrained by the computational expense of the direct N-body method. These clusters are thus at the low-mass end of the spectrum of GC masses and a factor of 10–100 less massive than typical nuclear star clusters. In later sections we consider how our results might scale to larger N clusters.

The stars in our initial conditions follow a Kroupa (2001) initial mass function, within the mass range 0.1 M_⊙–30 M_⊙, with no binaries. The metallicity is one-tenth solar. We initialize our clusters as tidally underfilling by a factor of 1.5. In N-body units, the initial tidal radius is 10.455 rather than 6.95 for W₀ = 7 in our simulations (compare to Table 1 of Trenti et al. 2007b). This allows for some cluster expansion due to mass loss before stars begin to escape. We consider cases of different IMBH mass, varied between 0.13% and 0.26% of the initial cluster mass.

Finally, we consider two different cases for the velocity kicks imparted to stellar-remnant neutron stars (NSs) and BHs (for a recent discussion, see Contenta et al. 2015). Rather than choosing absolute kick velocities based on observational constraints, here we choose velocities relative to the cluster velocity dispersion in order to produce different retention fractions of stellar remnants. This strategy is needed because the clusters we are able to simulate with the direct N-body method are small relative to realistic dense star clusters. In both cases, remnants are given a kick drawn from a Maxwellian distribution with sigma of either 1 or 2.5 times the initial cluster velocity dispersion, ${\sigma }_{*}=\sqrt{{{GM}}_{{\rm{T}}}/{r}_{{\rm{h}}}}$. These kick distributions imply that 19.8% or 1.6% of kicked objects have kick velocities less than σ_*, which, in the absence of binaries, is proportional to the fraction of retained remnants.

Some previous work has mentioned or studied close companion stars to IMBHs in GCs. Here we describe the methodological differences in our new simulations as they compare to this work. Baumgardt et al. (2004b) run direct N-body simulations of clusters containing IMBHs and up to N = 131 K stars. They plot several sequences of the semimajor axis of tightly bound stars as a function of time from their simulations containing IMBHs (their Figures 14 and 15). These sequences show a long-lived most-bound companion in several cases with a semimajor axis considerably smaller than the other stars and hint at the possible universality of IMBH companions. Baumgardt et al. (2006) focus on the production of mass-transferring binary pairs containing the IMBH and a star. They run N-body simulations including approximate tidal dissipation and gravitational wave radiation from stellar orbits and study the statistics of tidally captured stars in young (<12 Myr) clusters as possible ULX progenitors. Konstantinidis et al. (2013) also model small, young (10 Myr age) clusters initially containing N = 32k stars using direct N-body integration. The clusters have an IMBH with M_bh = 500–1000 M_⊙. They study one simulation in detail, in which the IMBH captures a stellar BH companion, goes through several cycles of exchange, and eventually inspirals and merges. Konstantinidis et al. (2013) argue that the merger generates sufficient gravitational wave recoil to eject the IMBH from their low-mass simulation cluster. Mapelli et al. (2013c), Mapelli & Zampieri (2014), and Ziosi et al. (2014) study the dynamics of massive stellar-mass BHs (up to 10–100 M_⊙) in a large sample of N-body calculations of young star clusters with a range of metallicities and N = 5500 stars.

Leigh et al. (2014) also run N-body simulations containing IMBHs with between N = 32k and N = 131k stars, but they do not include gravitational radiation from compact binaries. After the formation of an IMBH–BH compact binary, the eccentricity of this innermost binary varies over time. The binary passes many times through orbital configurations that should have driven it to merger. As a result, the simulations of Leigh et al. (2014) show artificially long-lived companion objects and a single companion persisting throughout the cluster evolution. Recently, Giersz et al. (2015) have used the MOCCA Monte Carlo Cluster simulation code to study accretion-fed IMBH growth over GC lifetimes. Their simulations, particularly their "slow" scenario with a single massive BH, show the cycles of replacement of the innermost object missing from earlier N-body work. However, as Giersz et al. (2015) describe, the Monte Carlo code is not well suited to studying the detailed dynamics of a small number of stars tightly bound to the IMBH, nor is it well suited to the inclusion of a single, massive object.

Blecha et al. (2006) adopt a slightly different approach. Rather than simulate an entire GC, they use a binary evolution code to evolve the IMBH and any companions with a fixed background of scatterers drawn from a GC core distribution function. While this approach facilitates modeling a larger number of IMBH pairs and a detailed treatment of IMBH–star interactions, it does not capture the dynamical feedback of the IMBH and its partner on the host cluster.

The simulations presented in this paper were designed specifically to study close companion objects to the IMBH in realistic GCs. This involved combining some of the best aspects of the previous work described above. In particular, we take advantage of the accurate dynamics of the N-body methods and implement specific interaction outcomes for close interactions between the IMBH and stellar objects like tidal disruptions and gravitational wave inspirals. We model an IMBH that is a realistically small fraction (<0.3%) of the total cluster mass. This approach limits our study to relatively small GCs with N₀ ∼ 10⁵, with their proportionately low-mass IMBHs, M_bh ∼ 10² M_⊙, but it allows us to follow their long-term evolution over 6–10 Gyr without approximation. In Section 4, we discuss some considerations in scaling the lessons learned from our simulations to more massive clusters or IMBHs.

Our simulated clusters exhibit a global evolution similar to those of others studied in detail combining IMBHs and a tidal field (e.g., Trenti et al. 2007a, 2010; Gill et al. 2008; Lützgendorf et al. 2013a; Giersz et al. 2015). Thus, we comment only briefly on the generic properties of our model clusters to set the stage for our consideration of the dynamics of their IMBH-hosting cores. Clusters in a tidal field slowly dissolve as (typically low-mass) stars escape from the system. Our models of simulation groups A and B are run for 6 Gyr, during which time the total number of stars decreases to ≈2 × 10⁴. Simulation groups C and D are run to the point of dissolution in the tidal field, which occurs after 9 and 10.4 Gyr, respectively. Mass loss from stellar evolution rapidly decreases the cluster mass by ∼30% in the first 50 Myr, then proceeds increasingly gradually as time goes on. The combination of these effects dictates that the mass range of typical cluster stars narrows over the course of the GC's evolution. At early times, stars are the most massive objects other than the IMBH, while at later times, for example ≳1 Gyr, the turnoff mass has decreased to <3 M_⊙, and stellar remnants are the most massive objects. Figure 1 in Lützgendorf et al. (2013a)  gives an excellent overview of the time evolution of N-body models of GCs containing IMBHs.

The IMBH sinks nearly to the cluster center of mass, as for example can be seen in Figure 2 of Konstantinidis et al. (2013). Once in the core, dynamical interactions with the IMBH and its companion object act as a heat source for the cluster, stalling core collapse or causing the system to expand (e.g., Quinlan 1996; Yu & Tremaine 2003; Trenti et al. 2007a; Lützgendorf et al. 2013a). Figure 1 shows an example cluster evolution from simulation group A. The half-mass and core radii expand initially in response to stellar evolution mass loss and mass loss from kicked supernova-remnant objects. These characteristic radii then stabilize for the remainder of the simulation, and the cluster neither expands nor contracts as energy loss balances dynamical heating. Over time, the IMBH grows by accreting the debris of tidally disrupted stars and by swallowing inspiraling compact objects. The individual sequences are, of course, variable, but the IMBHs in our clusters grow by ∼30% over the evolutionary span of the cluster lifetime.

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The process of capture of a close companion begins with the two-body relaxation of the cluster. The stellar cluster's dynamical state relaxes over approximately a relaxation time, t_rh (Spitzer 1987). This relaxation drives the stellar distribution surrounding the IMBH to higher and higher densities until a steady-state distribution is reached (Peebles 1972; Bahcall & Wolf 1976; Frank & Rees 1976; Lightman & Shapiro 1977). Because the IMBH mass is much greater than the average stellar mass, many stars can be gravitationally bound to the IMBH at a given time. For these stars,

$$\begin{eqnarray}&&\varepsilon =-\displaystyle \frac{{{GM}}_{\mathrm{bh}}}{{r}_{{\rm{bh}}}}+\displaystyle \frac{1}{2}{v}_{{\rm{bh}}}^{2}\lt 0,\end{eqnarray} \tag{ 4 }$$

where we have written their orbital energy per unit mass ε with respect to the IMBH in terms of their instantaneous separation r_bh and velocity v_bh relative to the IMBH. As such, these are the stars that occupy a phase space where the IMBH is energetically dominant in their orbits. In general, these stars satisfy

$$\begin{eqnarray}&&a\lesssim {r}_{{\rm{h}}}={{GM}}_{\mathrm{bh}}/{\sigma }_{{\rm{c}}}^{2},\end{eqnarray} \tag{ 5 }$$

where a = GM_bh/2ε is the semimajor axis of the bound star, and σ_c is the core velocity dispersion. While many stars may exist in such bound orbits, we would not describe each of these as a companion to the IMBH.

Instead, we will consider the IMBH to have a companion when its most-bound star is significantly more tightly bound than the second-most-bound. One way to express this is by comparison of those stars' semimajor axes (or equivalently orbital energies) with respect to the IMBH. The condition can be written a₀/a₁ ≪ 1, which implies ε₀/ε₁ ≫ 1. Here the coefficient 0 denotes the most bound, and the coefficient i denotes the (i + 1)th most tightly bound object. In these cases, the most-bound object has hierarchically separated from the rest of the cluster and sphere-of-influence stars. The remaining stars then orbit in the combined potential of the IMBH and its companion. The hierarchically isolated companion can exist relatively unperturbed until it suffers a strong encounter.

How does the most-bound star end up in an orbit hierarchically isolated from the other cluster stars? We explore this question visually in Figure 1. This diagram shows characteristic radii during a time series of one of the case A simulations. From top to bottom, the radii plotted are the cluster half-mass radius, r_h, the core radius, r_c, and the radius enclosing an IMBH mass of stars, r_m. Next we show the characteristic radii of some of the most-bound companions to the IMBH. These include the semimajor axis of the fifth-most-bound star a₄, the second-most-bound star, a₁, and the most-bound, a₀. We also show the pericenter distance of the most-bound object, r_p,0. We color the most-bound object depending on its stellar evolutionary type.

The larger radii, those that describe the bulk of the cluster, like the core and the half-mass radii, are nearly constant in time. As the radii get smaller and describe fewer and fewer particles, a higher degree of variability is seen. The semimajor axis of the most-bound object, however, exhibits qualitatively different behavior than the other radii or even the other tightly bound objects (as shown by a₁ and a₄). The most-bound star goes through a cyclic behavior of tightening semimajor axis followed by replacement with a new object. This cycle is seen to repeat throughout the time series of Figure 1.

The objects energetically bound to the IMBH all represent pairings that are "hard" relative to the cluster background energy (Heggie 1975; Hut & Bahcall 1983). In clusters, binaries that are hard tend to get harder through interactions with remaining cluster stars (Heggie 1975). This process can be understood as a statistical trend toward energy equipartition between the binary and the background because kinetic energy tends to transfer from the binary components to the perturber, ejecting it faster than it entered (Hills 1975a, 1975b). In this picture, each encounter leaves the IMBH plus companion pair (on average) somewhat more tightly bound and ejects the perturbing object with high velocity (Saslaw et al. 1974).

Among the multiple stars bound to the IMBH, only the most tightly bound object is "hard" relative to the other bound stars. The orbits of less-bound objects cannot easily perturb the innermost orbit. The converse implies that the less-bound stars, despite being hard relative to the cluster core energy, are not hard relative to the more tightly bound companion to the IMBH. These less-bound stars exchange energy with the innermost binary in a manner that prevents their orbits from tightening significantly. Because strong perturbations drive the binary hardening process, the continuous hardening of the most-bound star's orbit can be approximated relative to the properties of the cluster background as

$$\begin{eqnarray}&&\displaystyle \frac{{da}}{{dt}}\propto \displaystyle \frac{G{\rho }_{{\rm{c}}}}{{\sigma }_{{\rm{c}}}}{a}_{0}^{2},\end{eqnarray} \tag{ 6 }$$

where ρ_c is the mass density of the cluster core environment surrounding the binary (Quinlan 1996; Heggie & Hut 2003). This hardening rate implies that the timescale for change of the semimajor axis, ${a}_{0}/{\dot{a}}_{0}$, scales as ${a}_{0}^{-1}$ because ${\dot{a}}_{0}\propto {a}_{0}^{2}$. These scalings for the time evolution of the most-bound object's semimajor axis suggest the qualitative behavior seen in Figure 1, where a₀ first decreases rapidly while a₀ is large, then decreases more slowly as a₀ decreases.

In this section, we examine some of the orbital properties that define the population of most-bound companions to the IMBH. Figure 2 highlights some of the key orbital demographics of these companions. The lower left panel of Figure 2 displays the cumulative distribution function of the log of the ratio of the semimajor axis of the most-bound star to the second-most-bound, $\mathrm{log}({a}_{0}/{a}_{1})$. This diagram reveals that, in our simulations, the IMBH has a companion that is hierarchically isolated from the rest of the cluster stars a large fraction of the time: 90% of the time, the companion has ${a}_{0}\lt \frac{1}{3}{a}_{1}$. Approximately 25% of the time, a₀ < 10⁻²a₁.

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The other panels of Figure 2 examine the distributions of orbital characteristics in physical units. In the upper left panel, we show the orbital semimajor axis distribution of most-bound companions. The largest separations are around 10³ au. This distance becomes similar to the average distance of the less-bound stars within the IMBH's sphere of influence (see, for example, Figure 1) and, as a result, determines the maximum separation at which an object is able to persist as most bound without exchange. The peak of the distribution lies in the range 10¹–10² au, and the minimum separation observed in our simulations is a few astronomical units. The upper right panel shows the eccentricity distribution of companion objects. It is nearly thermal for most of the range of e, with some decrement observed near e ∼ 1, because objects that enter into orbits that are too eccentric interact directly with the IMBH at the orbital pericenter. The orbital separation distribution, along with the object masses, map to an orbital period distribution, shown in the lower right panel of Figure 2.

As can be inferred from an inspection of Figure 1, companions to the IMBH do not persist indefinitely. Instead, these objects undergo cycles of replacement following their termination—either through exchange or direct interaction with the IMBH. In Figure 3, we show the residence time of a given object as the companion to the IMBH. In the gray histogram, we measure the persistence time of each individual partnership with the IMBH. With this diagnostic, we can examine how long a typical close partnership between the IMBH and a companion lasts. Figure 3 reveals a bimodal distribution of persistence times of partnerships between the IMBH and other objects. One peak of this distribution lies around 10²–10⁴ years, and the second peaks around 10⁷ years. A tail toward longer residence times connects the two peaks. The origin of the first peak in this distribution can be traced back directly to the orbital period of the typical most-bound companion, as shown in the right-hand panel of Figure 2. The longest orbital periods of the most-bound companion occur when a₀ ∼ a₁ and are 10² years for the parameters used here. The short-residence-time branch can thus be associated with objects that persist as a partner to the IMBH for only one orbit. Near the peak of the distribution, the gray histogram in Figure 3 records objects that last for a small integer number of orbits. Associating the peak at ∼10³ years in Figure 3 with N orbits of an object near the maximum period for most-bound companions, we find that ∼10 orbits is a typical residence time before an exchange of hierarchy. A tail to longer residence times represents objects whose orbits harden and thus persist for many more orbital periods.

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The origin of the secondary peak at ∼10⁶–10⁸ years arises from objects whose orbits harden sufficiently to become significantly hierarchically isolated from the remaining cluster. This timescale thus becomes comparable to the cluster relaxation timescale of ∼0.6 Gyr. In the blue histogram of Figure 3, we show the distribution of residence times for most-bound companion objects sampled at even time intervals (therefore it is possible to count the same long-lived companion at multiple sampling times in this case). If we observe the cluster at any given time, we are very likely to see a companion with a residence time of 10⁶–10⁸ years. This impression is confirmed visually in Figure 1, where the populations of objects that we see in the time series last, on average, for timescales of fractions of a gigayear or more.

The color coding of stellar evolutionary state in Figure 1 shows that the demographics of the most-bound objects are remarkably diverse. This outcome is similar to the results from Mapelli & Zampieri (2014) for systems undergoing Roche lobe overflow. Companion objects include BHs (1–2 Gyr), as well as main sequence (MS) and giant-branch stars, and white dwarfs (WD) of various masses. In some cases, a single object will persist as the closest companion for long enough that it evolves and its stellar evolutionary state changes. One example of this is seen between ∼2.2 and 2.4 Gyr, where a single most-bound star evolves from the MS to the giant branch and eventually tidally interacts with the IMBH. In this section, we examine the statistical distributions of stellar evolutionary type, mass, and radius of the most-bound companion objects.

The diversity of companion types seen in Figure 1 is reflected quantitatively in Figure 4. The upper left panel of this figure shows the companions' stellar evolutionary type. The most prevalent companions to the IMBH are MS stars, but compact objects like WDs, NSs, and BHs all represent significant contributions to the total demographics. The companion's radius distribution, which we show in the lower left panel of Figure 4, directly reflects the stellar evolutionary status of the companions. We show the range of radii encompassing WDs to giant stars in this panel. The primary peaks of the radius distribution lie at 10⁻² R_⊙ and ≳1 R_⊙, reflecting the WD and massive MS populations; a smaller third peak is seen between 10¹–10² R_⊙ that includes stars at various phases along the giant branch.

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The objects' mass distribution, shown in the right-hand panels of Figure 4, exhibits a broad main peak centered at slightly more than a solar mass that is made up of MS stars, giants, massive WDs, and NSs. This peak lies substantially above the mean mass of cluster stars, which is initially ∼0.58 M_⊙. We also observe a tail toward much higher companion masses (up to ∼10 M_⊙) made up mostly of MS stars, giants, and stellar-mass BHs. That the mass of most of the IMBH's companions is significantly larger than that of a typical cluster star is not surprising. The trend toward energy equipartition in two-body interactions causes stars more massive than the mean mass to sink toward orbits deep in the cluster core, where they are very likely to come into direct interaction (and potentially partnership) with the IMBH (e.g., Gürkan et al. 2004; Goswami et al. 2012; Leigh et al. 2014). Further, more massive secondary objects are likely to replace an existing companion if they do undergo a three-body interaction (Fullerton & Hills 1982; Sigurdsson & Phinney 1993; Mapelli & Zampieri 2014; Samsing et al. 2014). In the lower right panel of Figure 4, we show the unnormalized distribution of companion masses in order to demonstrate that there is strong evolution in the companion mass distribution with stellar population age. Young ages, <2 Gyr, where the turnoff mass is still high, contribute the bulk of the companions ≫1 M_⊙. This temporal dependence in the companion mass can be contrasted to the approximately time-independent companion orbital parameter distributions described in the previous subsection.

Close partnerships to the IMBH can be terminated through one of several channels. In this section, we describe the possible channels and explore their relative likelihoods.

3.5.1. Channels

Three possible channels of termination of a close partnership with the IMBH exist. They are shown in the diagram of Figure 5 and explained below.

• 1.  
  Exchange: Exchanges can occur when a third body's pericenter distance from the IMBH becomes similar to the semimajor axis of the binary companion a₀. When this occurs, the system undergoes a strong three-body interaction, during which the gravitational attraction between both objects and the IMBH is similar. Either star may emerge from the three-body interaction as the new most-bound object. Statistically, there is also a chance that the IMBH could be the ejected object, leaving behind a binary of two stars. However, since the mass ratio is large, this exchange outcome is extremely unlikely (Heggie et al. 1996). Throughout the duration of our simulations, we record all perturbers to the central binary with pericenter distance less than three times a₀. This allows us to track the statistics of these exchange interactions. We find that the bulk of exchanges are due to strong three-body interactions, where the previous perturber emerges from the encounter as the new companion. A small fraction, of order of a few percent, are the result of more complex multibody strong interactions where an object other than the strongest perturber emerges as the companion immediately following the encounter.
• 2.  
  Tidal Disruption: An examination of Figure 1 reveals that, despite having the highest binding energy to the IMBH, the orbit of the most-bound star is still subject to perturbations. Not only does its orbit tighten through interactions with other cluster-core stars, but its orbital angular momentum wanders in magnitude and direction, as can be seen through the orbital pericenter distance, r_p,0. As a result, the most-bound star will be occasionally perturbed into an orbit that leads it to pass within a tidal radius, r_t, of the IMBH. In our simulations, we record these tidal disruption events and delete the particle from the simulation, as described in Section 2. In reality, some stars, especially those with condensed cores, can lose part of their envelope gas without being completely disrupted (MacLeod et al. 2012, 2013). The remnants of these partial tidal stripping events could then go on to have further interaction with the IMBH (Hopman & Portegies Zwart 2005; Bogdanović et al. 2014).
• 3.  
  Gravitational Wave Inspiral: Just as the orbital wandering of the most-bound object permits the occasional tidal-disruption event, if the companion to the IMBH is a compact object, it may also undergo an intermediate-mass ratio gravitational-wave inspiral (IMRI) (Amaro-Seoane et al. 2007; Mandel et al. 2008; Sesana et al. 2008; Mandel & Gair 2009; Konstantinidis et al. 2013). In our simulations, these are flagged as occurring when the gravitational-wave inspiral time becomes less than one N-body time unit, and the particle is removed from the simulation (see Section 2). Holley-Bockelmann et al. (2008) and Konstantinidis et al. (2013) explore the retention of IMBHs suffering a gravitational wave recoil following an IMRI event. IMBHs that merge with stellar-mass BHs with mass ratio q ≳ 0.1 have a significant probability of being ejected from the cluster. Smaller kicks would temporarily offset the IMBH from the cluster core. These inspirals typically occur when the companion is in a very eccentric orbit and are driven primarily by the close pericenter distance, rather than, for example, a close semimajor axis, much like the tidal disruptions (and in agreement with the findings of Gültekin et al. 2004). The N-body simulations of Baumgardt et al. (2004b) and Leigh et al. (2014) lack this merger channel, and Leigh et al. (2014) demonstrate that this leads to unrealistically long-lived central partnerships. If the companion is a BH or NS, it will be swallowed as it merges with the IMBH. However, in some cases where the companion is a WD, it will tidally disrupt before merging with the IMBH (e.g., Casalvieri et al. 2006; Ivanov & Papaloizou 2007; Sesana et al. 2008; Zalamea et al. 2010; Krolik & Piran 2011; MacLeod et al. 2014). This can proceed in one of two ways. In most cases, tidal heating likely leads to the disruption of the WD while it is still in an eccentric orbit. However, in some cases the orbit may fully circularize before the onset of Roche lobe overflow.

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3.5.2. Termination Demographics

The termination of partnerships with the IMBH, as well as other close encounters, is displayed in Figure 6. In this figure, we show the same time series plotted in Figure 1. Here, we extend the range of the plot to include radii very close to the IMBH, including the IMBH's Schwarzschild radius, r_s. We also include tidal radii of typical WDs, MS stars, and giant stars. Overset on this time series we highlight the orbital semimajor axis (solid symbol) and pericenter distance (open symbol) of encounters leading to either tidal disruption or gravitational wave inspiraling. Tidal disruptions are shown with circular points colored blue for MS star disruptions and red for giant star disruptions. Gravitational wave inspirals are marked with green stars.

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Figure 6 reveals the broad demographics of changes of partnership with the IMBH. Many changes of partnership appear to arise from strong interactions between the companion to the IMBH and other cluster stars—often including the second-most-bound star, whose semimajor axis, a₁, is occasionally similar to a₀. When a₀ ∼ a₁, changes of partnership are frequent, and as a result the constantly changing companion's orbit does not have the opportunity to tighten significantly. An example of this scenario is seen in the 2.5–4.5 Gyr span of Figure 6. Close encounters with the IMBH often accompany strong encounters: many tidal-disruption events either involve the most-bound companion or stars scattered into a similar orbit. The chaotic three-body dynamics of the IMBH, companion, and perturber system then cause a fraction of stars to be scattered directly toward the IMBH. An exception to this requirement for scatterings comes in the form of giant star disruptions, which can occasionally occur when the star evolves to become tidally vulnerable to the IMBH rather than being strongly scattered ("spoon-feeding" the IMBH: MacLeod et al. 2013).

In Table 3, we examine the termination of close partnerships with the IMBH, as well as the rates of close encounters with the IMBH. The most common, by far, means for changes of partnership with the IMBH is the exchange interaction, contributing approximately 98% of the changes in partnership. These exchanges dominate the large peak of IMBH companions that have residence times in the range of 1–10⁶ years, as shown in Figure 3. The mean exchange rate is (2.5–5) × 10⁻⁷ yr⁻¹. Of these exchanges, ∼98% are simple three-body exchanges in which the strongest perturber becomes the new most-bound companion. The remaining 2% are more complex multibody interactions. These may involve more than three objects, or they may involve the tidal disruption or inspiral of one or more objects. Gravitational-wave inspirals and tidal-disruption events contribute to the remaining termination of partnerships with the IMBH. The tidal-disruption rate is of order 10⁻⁸ yr⁻¹, and the inspiral rate is of order 10⁻⁹ yr⁻¹ in our simulations. In Section 5, we compare these event rates to previous predictions and mention their potential significance for electromagnetic and gravitational transients.

As can be inferred from Figure 6, the longer a companion object persists, the more likely it is that the pair will be split by direct interaction with the IMBH rather than exchange. Longer-lived companion pairs segregate to smaller orbital semimajor axes, reducing the cross section for encounters or exchange in proportion to a₀. Meanwhile, interaction with the IMBH becomes progressively more likely at smaller a₀ because the orbital eccentricity needed to produce a close pericenter passage becomes smaller. Thus, many long-lived companions appear to only be replaced when they are scattered to their disruption or when they undergo a gravitational wave inspiral. These terminations contribute to the large fraction of tidal disruption events and gravitational wave inspirals arising from the most-bound companion in Table 3.

In Figure 7, we plot comparative time series, analogous to that shown in Figure 1, for examples from simulation groups B–D (which may be compared directly to Figure 1 for simulation group A). Even looking at the examples in this figure, dynamical and demographic differences and similarities distinguish themselves. The pattern of orbital tightening and replacement of a most-bound companion is repeated throughout the panels, despite their varying cluster and IMBH parameters. The global evolution of the clusters is largely similar as well: cluster core and half-mass radii are relatively constant in time, just as with simulation group A.

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Differences in the stellar dynamics and demographics of the companion objects also become apparent on closer inspection of Figure 7. Simulation group B, with a larger population of retained BHs, shows a larger quantity of BHs joining the IMBH as most-bound companions. The cycles of orbital hardening experienced by these BHs are longer lived than those of most of their stellar counterparts. By contrast, simulation group D, with a larger total number of stars than the fiducial case, shows more frequent exchanges of partnership but a diversity of companion-object stellar evolutionary types.

These BHs preferentially enter into partnerships with the IMBH because of their significantly higher-than-average mass, which causes them to sink in the cluster potential due to dynamical friction. When these stellar-remnant BHs interact with the IMBH and its companion, their high mass makes them very likely to replace an existing companion. Heggie et al. (1996) have shown that the exchange cross section depends quite strongly on the mass of the tertiary, perturbing body, even in the case where the mass ratio between the binary components is large. In Figure 8, we plot the fraction of encounters that result in an exchange given a particular perturber-to-companion mass ratio, M_p/M₀. This fraction is computed by counting the fraction of encounters in which the perturber comes within a₀ of the binary center of mass at the pericenter that result in an exchange. While the exchange fraction depends strongly on the perturber mass ratio, it does not depend nearly as strongly on other cluster properties. This plot implies that massive objects can exchange into the inner binary with relative ease and that, as companions, they can undergo many encounters, each hardening their orbit before one leads to an exchange.

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Figure 9 further traces how companion demographics propagate to orbital distributions. The histogram of orbital semimajor axis is shown for each of the simulation groups A–D and is broken down by companion type. Visually from Figure 7, and here quantitatively, we can infer that types of companions have different subdistributions. When combined in varying percentages, these demographic differences lead to variations in the overall semimajor axis distribution of IMBH companions. The most obvious example is BHs. Based on the histograms of Figure 9, BHs exist in orbits that are preferentially among the most tightly bound. The distribution is broad, and over the course of the simulation there were BHs at all separations. We can understand this distinction because BHs are preferentially difficult to replace in exchange interactions because of their high mass and are not prone to tidal disruption when the companion has an orbit with e ∼ 1.

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By contrast, giant stars tend to exist in the least-bound orbits in Figure 9. These objects are especially tidally vulnerable and so are subject to tidal disruption by the IMBH at larger orbital separations for a given eccentricity than are more-compact MS stars or WDs. The cases of MS stars and WDs lie intermediate to that of giants and BHs. MS stars' innermost semimajor axes are also limited by tidal disruption as their orbits cycle through high-eccentricity phases (note the rapidly varying eccentricity of the innermost orbit in Figure 1). WDs are less tidally vulnerable but are not as massive as the BHs, so their distribution peaks at somewhat larger separations. The case of the NSs is interesting and somewhat more subtle. Although these remnants are ∼3× the average stellar mass in the cluster, they are subject to kicks on formation, and they do not resegregate to the cluster core as efficiently as the more-massive BHs.

Figure 10 quantitatively explores how these demographic differences propagate to the other properties of the most-bound companions to the IMBH. The upper left panel shows companion-object demographics, which are also tabulated in Table 2. Qualitatively, the demographics remain unchanged in that the companions include stellar remnants like WDs, BHs, and NSs, along with MS and giant stars. As seen in Figures 7 and 9, the most dramatic difference arises with simulation group B and the retention of a higher fraction of BHs in the cluster's central regions. Simulation group D, with its larger total particle number, also exhibits a mild enhancement in the fraction of BH companions. This difference can be traced to the larger total number of BHs retained in these simulations (despite the small retention fraction). Companion demographics shape the companion mass distribution, shown in the upper right panel of Figure 10. The enhanced BH fraction is reflected in the higher peak at companion masses ≳3 M_⊙ for simulation groups B and D.

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In addition to demographic changes in the typical properties of IMBH companions, there are also dynamical differences we can observe across our simulations. We begin by examining how the long-lasting tail of the residence-time distribution of IMBH companions is affected by changing cluster properties in the central panel of Figure 10 (the residence time plotted is counted per partnership, as is the gray distribution in Figure 3). This tail of long-lived companion objects is shaped by processes of companion exchange and disruption. As a result, it is influenced by a combination of cluster, IMBH, and companion demographic properties. An increased fraction of BH companions (as in simulation case B) implies more long-lasting partnerships because the IMBH's companion is considerably more immune to exchange interactions (see Figure 8) and to tidal disruption. In fact, we can observe this directly in the termination statistics of Table 3, where we see that exchanges are significantly less frequent in simulation group B (because of the mass sensitivity of the exchange cross section; see Figure 8). Tidal disruptions are also nearly a factor of two less common than in case A, with a smaller fraction arising from the most-bound companion. With lower IMBH mass, as evaluated in simulation group C, the weaker gravitational focusing of the surrounding stars leads to the lower exchange rate tabulated in Table 3. This propagates to more objects with long residence times in the 10⁶–10⁸ years range. The larger number of stars present in simulation group D, by contrast, implies a larger encounter and exchange rate and shorter typical residence times.

Table 3.  Close Encounter and Termination Demographics of Most-bound Companions to the IMBH

Name	${\dot{N}}_{{\rm{ex}}}$	${\dot{N}}_{{\rm{insp}}}$	${\dot{N}}_{{\rm{tde}}}$	fex,3b	finsp,0	ftde,0
	(Gyr−1)	(Gyr−1)	(Gyr−1)	(%)	(%)	(%)
A	530.86	1.77	10.11	98.62	80.65	43.50
B	253.01	1.76	4.54	98.32	83.87	33.75
C	318.70	1.76	5.22	99.16	83.33	47.89
D	884.06	3.65	11.68	98.78	78.95	30.14

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The lower panels of Figure 10 explore differences in companion periods, P₀, and hierarchies, a₀/a₁. Here distinctions arise both in the form of companion demographics and IMBH mass. All of simulation groups B–D exhibit shorter typical orbital periods than does simulation group A. The peak in the orbital period distribution shifts by about 1 dex from ∼10 years to ∼1 year, but the distribution remains broad in all cases. The companion hierarchy, as measured by a₀/a₁, is also affected both by companion demographics and IMBH mass. The fiducial case, simulation group A, shows companions that are less segregated than in groups B–D. In the case of simulation group B, the tighter companion distribution can be traced to the increased BH fraction and correspondingly longer companion residence time (which allows the IMBH–companion pair more time to harden prior to disruption). Simulation group D's tightly bound tail can be similarly explained by a mildly increased fraction of BH companions.

The IMBH mass effect on companion hierarchy can also be observed in the lower left panel of Figure 10. The median a₀/a₁ shifts by a factor of approximately three between simulation groups A and C, which have a factor of two difference in IMBH mass. The origin of this distinction can be understood through a consideration of the dynamics of orbital hardening and exchange for IMBH companions. A typical close passage of a perturbing body of mass M_p through the IMBH–companion pair carries away an energy of the order of

$$\begin{eqnarray}&&{\rm{\Delta }}{E}_{{\rm{p}}}\approx \displaystyle \frac{{{GM}}_{{\rm{0}}}{M}_{{\rm{p}}}}{2{a}_{0}},\end{eqnarray} \tag{ 7 }$$

when it passes with orbital pericenter within the semimajor axis of the inner binary (Yu & Tremaine 2003). As compared to the orbital energy, E₀ = GM_bh M₀/2 a₀, this typical per-encounter change of energy is

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}{E}_{{\rm{p}}}}{{E}_{0}}\approx \displaystyle \frac{{M}_{{\rm{p}}}}{{M}_{\mathrm{bh}}}.\end{eqnarray} \tag{ 8 }$$

So the fractional, per-encounter hardening rate of the IMBH and companion pair scales inversely with the IMBH mass. But, the rate of encounters also changes with IMBH mass. The encounter cross section is dictated by gravitational focusing onto the inner binary and is Σ_enc ≈ 2πGM_bha₀/v². Associating a typical velocity with the cluster core velocity dispersion, the encounter rate, ${\dot{N}}_{{\rm{enc}}}\propto {n}_{{\rm{c}}}{\sigma }_{{\rm{c}}}{{\rm{\Sigma }}}_{{\rm{enc}}}$, where n_c is the core number density, thus scales as

$$\begin{eqnarray}&&{\dot{N}}_{{\rm{enc}}}\propto {n}_{{\rm{c}}}{\sigma }_{{\rm{c}}}^{-1}{M}_{\mathrm{bh}}{a}_{{\rm{0}}},\end{eqnarray} \tag{ 9 }$$

proportional to both M_bh and a₀, as long as the low-angular-momentum phase space of encountering orbits is efficiently replenished (e.g., Frank & Rees 1976). As a result, although each encounter with a perturber carries away less energy ∝M_bh⁻¹, the encounters are more common ∝M_bh, and the innermost binary hardens at an identical rate regardless of IMBH mass.

Although the hardening rate remains unchanged with differing IMBH masses, the residence time, as shown in the central panel of Figure 10, does change. In particular, when the encounter rate goes up, so does the exchange rate of IMBH companions. This is because the exchange fraction in our simulations does not appear to depend strongly on IMBH mass, as shown in Figure 8; instead it is sensitive primarily to the perturber–companion mass ratio. As IMBH masses go up, their companions harden at the same rate but are exchanged with increasing frequency, restarting a new hardening cycle. This result implies, when extrapolated to larger IMBH masses, that only IMBHs of moderate mass ratio to their surrounding stars (perhaps those in the range ∼10²–10⁴ M_⊙) will ever exhibit tightly segregated companions.

We have focused so far on IMBH companions across a range of stellar system ages. However, most GCs are very old stellar systems, with ages >10 Gyr (e.g., Marín-Franch et al. 2009; Forbes et al. 2015; Trenti et al. 2015). How might the companions to IMBHs in these very old stellar systems differ? To examine this question, we look at several evolutionary sequences of the last several gigayears of cluster evolution of the models in simulation group D. These evolutionary tracks are shown in Figure 11. We find that by these late epochs interactions with the IMBH (either mergers or ejections) have exhausted the supply of stellar-mass BHs that might fall into partnership with the IMBH (Leigh et al. 2014). If realistic GCs begin with much larger masses and N than their present-day properties (or our smaller-still simulated systems), they would retain a similar fraction but a larger number of BHs to late times (Morscher et al. 2015). If these BHs remain in the cluster, we would expect them to mass-segregate toward the cluster core and, perhaps, play a role in shaping the time dependence of the companion mass function, as shown in Figure 4. At 10 Gyr, the turnoff mass is now ∼1 M_⊙ and is therefore less than the mass of typical stellar remnants like WDs and NSs. As a result, the most common companions for the IMBH in our simulations at these times are massive WDs, those with masses ≳1 M_⊙, which are therefore more massive than the MS stars in the cluster core. NSs similarly form a fraction of late-time companions because those that were retained in the cluster (but perhaps kicked to the outskirts) have had sufficient time to resegregate into the cluster core.

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One motivation for performing this study was to access whether close companions to the IMBH are ever driven into multiple-passage tidal interactions. Such multipassage, tidal-disruption flaring interactions can be difficult to achieve. For multiple passages of similar strength to be sustained, the star must remain bound to the IMBH with its orbit relatively unchanged (in energy or, more importantly, angular momentum). Further, the object undergoing the passages must expand upon mass loss, either through its adiabatic response to mass loss or heating due to the violent interaction. These requirements suggest that orbits leading to multipassage disruptions must exist in a phase space where the per-orbit scatter due to the remaining cluster stars is small (MacLeod et al. 2013). For MS stars or WDs in clusters with as few stars as we consider here, the only star in such an orbit is the most-bound companion to the IMBH. Thus, if this most-bound star enters into a weak tidal encounter with the IMBH, the stripping may persist for several passages.

In our simulations, we find that a reasonable fraction (30%–50%) of tidal-disruption events involve the most-bound star (Table 3). Although we track only full disruptions dynamically in our simulations, many of these would, in fact, lead to partial tidal disruptions over multiple orbits. Stars with condensed cores relative to their envelopes, in particular evolved stars and those with radiative envelopes, are particularly able to retain a bound core despite losing part of their envelope (MacLeod et al. 2012; Guillochon & Ramirez-Ruiz 2013). We find that some giant stars evolve to the point of mass transfer with the IMBH (MacLeod et al. 2013). WDs that undergo gravitational-wave inspiraling are also very likely to disrupt before their orbits circularize, leading to episodic mass transfer and repeated flaring (MacLeod et al. 2014). Many of the disrupted most-bound stars reach the tidal radius through a secular interaction with other tightly bound objects, opening the possibility of a large number of passages with similar pericenter distance. In the ensuing repeated tidal stripping episodes, the individual flares would be qualitatively similar to other tidal-disruption flares, albeit with less total fluence. The repetition time between flares would be governed by the orbital period distribution of the most-bound companion.

This leads us to the case of the extraordinary repeated-flaring transient HLX-1 (Farrell et al. 2009) by a putative IMBH with M_bh ∼ 10⁴ M_⊙ (e.g., Davis et al. 2011). If this object is powered by the partial tidal stripping of a star in an eccentric orbit, the object must have gone through ≳10 passages based on the light curve, with an orbital period near a year. The IMBH mass predicted for HLX-1 is nearly a factor of 100 larger than those we are able to simulate. However, with the scalings of Section 4 as guidance, we can speculate whether these properties appear plausible given our results.

First, having tightly bound stars in orbit around the IMBH appears likely given the IMBH (rather than supermassive) nature of the putative HLX-1 BH. Although scalings to a more massive system would modify them, our distributions of orbital periods easily extend to orbital periods in the range of years to tens of years. From a pericenter distance slightly outside the tidal radius, the donor star could have been tidally excited until it started transferring mass to the IMBH (Hopman et al. 2004; Baumgardt et al. 2006; Blecha et al. 2006). Second, a large fraction of tidal-disruption events in our low-density, small-N clusters originate from the most-bound companion to the IMBH (as compared to the supermassive BH in a galactic center context upon which most tidal-disruption literature is defined). We observe that the most-bound stars that tend to interact with the IMBH in our simulations arrive in disruptive orbits through secular interaction, rather than single, strong scattering events. Thus, it would not be entirely surprising if a tightly bound star in the HLX-1 system ended up in a grazing, rather than fully disruptive, interaction with the IMBH. In conclusion, a scenario involving a hierarchically isolated companion star being driven to interact with the IMBH is a scenario that qualitatively explains the properties of a repeated-flaring source like HLX-1, and an extrapolation of our results indicates that such a configuration can be established through dynamical interactions of stars with the IMBH.

5.3.1. Luminous Companions to a Dark IMBH

The calculations of this paper have demonstrated that IMBHs residing in clusters acquire and retain close companions. In rare periods in which the IMBH disrupts and accretes its companion, a luminous accretion signature will mark the presence of the IMBH. We discuss this possibility further in the following section, but here we consider the ability of companion objects to reveal the IMBH in periods of relative quiescence. Among the companion objects, approximately half are luminous stars like MS stars and giants, while the other half are WDs, NSs, and smaller BHs. Can this luminous population of stars serve to reveal the presence of otherwise dark IMBHs in nearby clusters?

First, the binary nature of the IMBH plus companion pair could potentially reveal the IMBH's presence. The typical companion orbits found in our simulations (semimajor axes of ∼10² au and periods of ∼10 years, shown in Figure 2) imply typical orbital velocities of hundreds of km s⁻¹. If spectroscopically detected, this orbital motion would strongly point to a massive companion because of the high velocity given combined with relatively long orbital period (compared to a stellar-mass binary). This approach extends work already being done to explore the radial velocities of stars bound to potential IMBHs in cluster cores (e.g., Noyola et al. 2010; Lützgendorf et al. 2013b). But we argue that, among the tightly bound stars, it would be reasonable to expect a single, very compact companion orbit in a large fraction of clusters hosting an IMBH.

Luminous stars also shed substantial stellar winds. With the star tightly bound to the IMBH, we examine whether and how these winds might contribute to fueling accretion activity onto the IMBH. The relevant scale for comparison here is the Bondi accretion rate at which the IMBH could accrete gas from the cluster-core interstellar medium, ${\dot{M}}_{{\rm{bondi}}}$ = 4π (GM_bh)²ρ/c_s³ (Volonteri et al. 2011). This interstellar medium is shaped by the interacting stellar winds of all the stars in the cluster core. Taking typical values of 10⁻²⁵ g cm⁻³ and sound speed c_s = 10⁶–10⁷ cm s⁻¹, we find ${\dot{M}}_{{\rm{bondi}}}\approx {10}^{-11}\mbox{--}{10}^{-15}{M}_{\odot }$ yr⁻¹ (e.g., Naiman et al. 2012, 2013).

In Figure 12, we compare this nominal accretion rate from the background of stars to that fed by the companion. Here we plot the duty cycle (fraction of time) spent accreting above a given level. The upper series of lines in this figure shows ${\dot{M}}_{0}$, the mass-loss rate of the companion star, estimated using Reimers' (1975) mass-loss formula, $\dot{M}$ = 4 × 10⁻¹³η_R (L₀M₀/R₀)/(L_⊙M_⊙/R_⊙) with η_R = 0.5 (McDonald & Zijlstra 2015). We then estimate the fraction of this mass lost that is bound to the IMBH using the formulae of Miller et al. (2014, described in their Section 4.1). The bound fraction depends on the orbital properties through the semimajor axis and eccentricity, which determine the velocity of the wind-shedding star relative to the IMBH. It also depends on the terminal velocity of the winds themselves, which we take to be the star's escape speed, $\sqrt{2{{GM}}_{0}/{R}_{0}}$. The slow, massive winds of giant stars, for example, are particularly easy to capture. We plot the portion of companion wind material that is bound to the IMBH as ${\dot{M}}_{{\rm{bh}}}$ in Figure 12. Most of these captured winds come when the IMBH hosts a giant star companion. These cumulative distributions show that the IMBH captures the winds of its companion at rates comparable to ${\dot{M}}_{{\rm{bondi}}}$ of order 5% of the time in our clusters.

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As a result, we can expect that the ≳50% of stellar IMBH companions should illuminate their host BH through their winds, providing a minimum accretion rate onto the IMBH. However, the fact that the IMBH has a companion does not imply that most IMBHs should accrete substantially above the Bondi accretion rate from the ambient medium. Of course, how these alternative fuel sources transfer into accretion luminosity represents a significant uncertainty. Media captured from the cluster core has low net angular momentum and likely does not efficiently form a disk (e.g., Cuadra et al. 2008), perhaps contributing to a low radiative efficiency. There is still more uncertainty in how much of the wind-deposited gas might reach an IMBH in scenarios with realistic feedback (Naiman et al. 2009; Park & Ricotti 2011, 2012, 2013) and what fraction of the resultant accretion energy would contribute to radio and X-ray luminosities (Maccarone 2004; Maccarone et al. 2005; Strader et al. 2012; Wrobel et al. 2015). Given these uncertainties, wind feeding from the companion implies a minimum activity level for the fraction of IMBHs hosting giant star or MS companions.

5.3.2. Electromagnetic and Gravitational Transients

We have explored the question of close companion objects to IMBHs in dense cluster environments using direct N-body numerical simulations in this paper. This chosen approach carries the advantage of direct integration of each particle's equation of motion, ensuring accurate dynamics. However, it necessitates some approximations because of the computational expense of each individual simulation. In particular, in our analysis we consider clusters with initial mass of ∼5 × 10⁴ M_⊙ or 10⁵ M_⊙. By the time these clusters evolve, they are less massive (and thus less dense) than the typical Milky Way GC. We similarly include IMBHs of relatively low initial mass, 75 or 150 M_⊙. More massive clusters might harbor more-massive-still IMBHs, changing the IMBH-to-star mass ratio. While these constraints are shared by every direct N-body simulation method, they necessitate a consideration of how the stellar dynamics will scale and extrapolate to more massive systems found in Nature. We have qualitatively explored the response and dependence of our results to the dependencies in cluster mass and density and IMBH mass in Section 4. To extend our simulated dynamics into the regimes that are found in more massive IMBH environments, an N-body algorithm particularly well suited to studying more extreme mass ratios, like that recently published by Karl et al. (2015), may offer a promising numerical approach.

We have included no primordial binaries in the simulations presented in this work. This simplification is not realistic; binary fractions are thought to be of order a few percent in typical GCs (Heggie & Hut 2003). In future work, we intend to include a realistic binary fraction and study the role these primordial binaries may play in interacting with the IMBH and its close companions (Pfahl 2005). Before addressing this question with numerical simulations, we can make a few analytic estimates of their potential importance. In a cluster environment, only hard binaries survive the interactive cluster-core environment (Heggie 1975). To an order of magnitude, these binaries have separation ${{GM}}_{{\rm{bin}}}/{\sigma }_{{\rm{c}}}^{2}$. If such a binary passes close to the IMBH and is split by the Hills (1988) mechanism, then one star can remain bound to the IMBH while the other is ejected. The bound star has a minimum typical semimajor axis of

$$\begin{eqnarray}&&{a}_{{\rm{min}}}\sim {\left(\displaystyle \frac{{M}_{\mathrm{bh}}}{{M}_{{\rm{bin}}}}\right)}^{2/3}{a}_{{\rm{bin}}},\end{eqnarray} \tag{ 10 }$$

and plugging in a few typical numbers for our simulation clusters, this might indicate a_min ∼ 3 × 10¹⁵ cm for marginally hard source binaries. This is near the peak of the semimajor axis distribution of IMBH companions in our case A simulations (as seen, for example, in Figure 2). Thus, in some cases, these split binaries would leave behind stars with semimajor axes similar to the most-bound IMBH companion. Additional strong interactions between the newly bound star and the companion would ensue with the possibility of an exchange of hierarchy.

Trenti et al. (2007a) run direct N-body simulations of star clusters with primordial binaries and a central IMBH under idealized conditions (N = 16384 equal-mass particles) and find that the IMBH presence increases the binary disruption rate, in qualitative agreement with the theoretical predictions by Pfahl (2005). However, the large majority of the binaries are split by interactions with other stars in the dense central stellar cusp surrounding the IMBH, rather than by the IMBH tidal field (Trenti et al. 2007a, Figure 6). This hints that a majority of binaries in relatively low angular momentum orbits may be split before reaching the IMBH and that those that survive as bound pairs and experience a close encounter with the IMBH are likely to be preferentially harder binaries. The earlier idealized experiments of IMBH–binary interactions also highlight the importance of resolving the dynamics of the sphere of influence exactly through direct N-body simulations. We plan to explore these effects more fully with the aid of future simulations.