THE STAR INGESTING LUMINOSITY OF INTERMEDIATE-MASS BLACK HOLES IN GLOBULAR CLUSTERS

Suggestive evidence has accumulated that intermediate-mass black holes (IMBHs) exist in some globular clusters (GCs). There is some dynamical evidence for a mass concentration within the central regions of some GCs. The dynamics of stars in the inner regions of nearby clusters such as M15, G1, and ω Centauri suggest the presence of black holes with masses of about 10³, 10⁴, and 4 × 10⁴ M_☉, respectively (Gerssen et al. 2002, 2003; Gebhardt et al. 2002, 2005; Pooley & Rappaport 2006; Noyola et al. 2008). The peculiarities of an unresolved radio source in G1 indicate some unique object at the center (Ulvestad et al. 2007). Recently, somewhat arguable evidence has arisen for the presence of IMBHs in young star clusters, where ultraluminous, compact X-ray sources (ULXs) have been preferentially found to occur (Zezas et al. 2002). Their high luminosities suggest that they are IMBHs rather than binaries containing a normal stellar mass black hole (Portegies Zwart et al. 2004). But before accepting this conclusion (and dismissing alternative ideas) one would like some independent corroboration of the IMBH hypothesis, or, conversely, some way of ruling it out.

We are used to the idea that black holes are implicated in the most powerful sources in the universe, and can (when accreting) be ultraefficient radiators. But there is, for example, no sign of such activity in M15—the X-ray upper limit is no more than 6 × 10³² erg s⁻¹ (Ho et al. 2003). So could a black hole be so completely starved of fuel that it does not reveal its presence? We do not directly know how much gas there is near the IMBH (Pfahl & Rappaport 2001; Pooley & Rappaport 2006), and there is no a priori reason why this region should be swept clean of gas. Gas that is lost from nearby stars (Baumgardt et al. 2006) as well as mass transfer from potentially bound companions (Blecha et al. 2006; Patruno et al. 2006; Hopman et al. 2004) can produce observable signatures. However, it is uncertain whether gas can be adequately supplied to explain ULX activity. The star density, on the other hand, is much better known—after all, if the stars were not closely packed near the center of the cluster we would not have evidence for the black hole at all. As stellar orbits diffuse in phase space, it therefore seems inevitable that some may wander sufficiently close to the hole that they suffer tidal disruption. When a star is disrupted, there is bound to be some radiation from the sudden release of gas. The flares resulting from a disrupted star could be the clearest diagnostic of an IMBH's presence.

It is an intricate although tractable problem in stellar dynamics to calculate the chance that a star passes within the IMBH's tidal radius (Frank & Rees 1976). In a simple case when the velocities are isotropic, the frequency with which a star would enter the zone of vulnerability is ξ ∼ 10⁻⁷M^4/3_h,3(n_*/10⁶ pc^-3)(σ/10 km s^-1)⁻¹ yr^-1, where n_* is the stellar number density in the cluster nucleus and M_h,3 is the black hole's mass in units of 10³ M_☉. This estimate although simplified, agrees well with the fiducial rates derived from detailed N-body simulations of multimass star clusters containing IMBHs (Baumgardt et al. 2004). It may seem, however, that even the modest rate of stellar disruptions given above could have conspicuous consequences. The debris from a disrupted one solar-mass star per hundred million years, swallowed steadily with 10% radiative efficiency, would yield a luminosity of L_steady ∼ 6 × 10³⁷ erg s⁻¹—higher than is observed for X-ray binaries in quiescence. But in reality, one expects brighter flares with short duty cycles, as the time it takes to digest or expel the debris from one star is much shorter than the mean interval between one stellar disruption and the next. This Letter is concerned with the observational manifestations of such phenomena, with particular reference to GCs where the masses of the black holes (if indeed present) are perhaps of the order of 10³–10⁴ M_☉.

A star interacting with a massive black hole cannot be treated as a point mass if it gets so close to the hole that it becomes vulnerable to tidal distortions. Such effects become important when the pericenter distance becomes as small as the tidal radius: R_T ≃ 5 × 10¹¹M^1/3_h,3(R_*/R_☉)(M_*/M_☉)^−1/3 cm. The gravitational radius, R_g ≃ 1.5 × 10⁸M_h,3 cm, scales with mass, whereas the tidal radius goes only as the cube root. The tidal forces at their "surfaces" are thus more gentle for black holes of larger mass, and solar-type stars would be disrupted only after passing irreversibly inside an ultra-massive hole's horizon: M_h ⩽ 7 × 10⁷ M_☉.

When a rapidly changing tidal force starts to compete with a star's self-gravity, the material of the star responds in a complicated way, being stretched along the orbital direction and squeezed at right angles to the orbit (Carter & Luminet 1983; Rees 1988; Evans & Kochanek 1989). To study this problem, we use a three-dimensional smoothed particle hydrodynamics (SPH) method to solve the equations of hydrodynamics. Due to its Lagrangian nature SPH is perfectly suited to follow tidal disruption processes during which the corresponding geometry, densities, and timescales are changing violently. The SPH formulation that we use in this study is described in Rosswog et al. (2008b, 2008a). Common to all runs is the initiation of the calculations with the star being placed safely outside R_T and set it onto a parabolic orbit so that R_min = R_T/3 for a 10³ M_☉ black hole. We have considered three different initial conditions for the approaching star, constructed here by solving the spherically symmetric Lane–Emden equations: A [1, 1, 0.6] M_☉ solar-type star modeled with a polytropic equation of state with adiabatic index Γ = [5/3, 1.4, 1.4] and R_* = [1, 1, 0.75]R_☉.

The gross quantitative behavior of a solar-type star plunging deeply within the tidal radius is examined here (the reader is referred to Section 3 for a discussion on the role of the stellar density structure in shaping the history of the mass accretion rate). Several snapshots taken from our numerical simulations of a 1 M_☉ solar-type star (modeled with Γ = 5/3) are shown in Figure 1. The tidal bulge raised on the star by the black hole becomes of an order unity distortion near pericenter. The resultant gravitational torque spins it up to a good fraction of its corotation angular velocity by the time it gets disrupted. This takes place on a timescale comparable to the crossing time of the star through periastron, Δt ∼ R_*/v_p ∼ 349.6(M_*/M_☉)^−1/6(R_*/R_☉)^3/2M^−1/3_h,3 s.

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The energy required to tear the star apart (that is the star's self-binding energy) is of the order of M_*v²_*, where v_* = (GM_*/R_*)^1/2. During tidal disruption, this energy is supplied at the expense of the orbital kinetic energy, which at pericenter ∼R_T is larger by ∼(M_h/M_*)^2/3 (Rees 1988). Figure 2 displays the evolution of the differential mass distribution in specific energy for the debris. Disruption reduces the orbital energy by the binding energy of the star _* ∼ (GM_*/R_*) ≈ 10⁻⁵c², which is much smaller than the specific kinetic energy at pericenter. The variation of the specific energy in the released gas is determined mainly by the relative depth of a mass element across the disrupted star in the potential well of the black hole. The spread in this specific energy is of the order of vΔv ∼ 10⁻⁴M^1/3_h,3c², where v ≈ v_p = c[R_g/R_T]^1/2 and Δv ≈ v_* = [GM_*/R_*]^1/2. This is much larger than _* and, as a result, almost half the debris escapes on hyperbolic orbits with speeds ∼3000 M^1/6_h,3 km s^-1; the kinetic energy output being ∼10⁵⁰M^1/3_h,3 erg (comparable to the energy of a supernova). The material would be concentrated in a fan close to the orbital plane. Adiabatic cooling, as the material expands, severely reduces the internal radiative energy content before the debris became optically thin. The escaping radiation from the unbound debris would therefore release much less than the initial energy content of the star. There would therefore be no conspicuous flare, until the bound debris fell back onto the IMBH.

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The returning gas does not immediately produce a flare of activity from the black hole. First material must enter quasi-circular orbits and form an accretion torus (Evans & Kochanek 1989). Only then will viscous effects release enough binding energy to power a flare. The bound orbits are very eccentric, and the range of orbital periods is large. The orbital semimajor axis of the most tightly bound debris is a ∼ 10⁴M^−1/3_h,3(R_*/R_☉)(M_*/M_☉)^−2/3 R_g, and the period is only t_a ∼ 6300(a/1.5 × 10⁴R_g)^3/2M^−1/2_h,3 s. If the gaseous debris suffered no internal dissipation due to high viscosity or shocks, it would, after one or two orbital periods, form a highly elliptical disk with a big spread in apocentric distances between the most and least bound orbits, but where at pericenter the orbits are all squeezed in a range δR/R ∼ R_*/R_τ = 0.1(M_*/M_h,3)^1/3. As the stream approaches pericenter, the radial focusing of the orbits therefore acts as an effective nozzle (Figure 3). After pericenter passage, the outflowing gas is on orbits which collide with the infalling stream near the original orbital plane at apocenter, giving rise to an angular momentum redistributing shock (Figure 3) much like those in cataclysmic variable systems. The debris raining down would, after little more than its free-fall time, settle into a disk.

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This orbiting debris starts forming a disk when the most highly bound debris falls back. The simulation shows that the first material returns at a time ⩽t_a, with a peak infall rate roughly given by 31 M_☉ yr^-1 (Figure 4). Such high infall rates are expected to persist, relative steadily, for at least a few orbital periods, before all the highly bound material rains down. Since the amount of highly bound material (dM/d) depends sensitively on the density structure of the star, we find that more centrally condensed stars produce mass feeding rates that are slightly larger in comparison to less centrally condensed stars of comparable mass although they reach the self-similar ∝t^−5/3 phase after fewer orbital periods. The vicinity of the hole would thereafter be fed solely by injection of the infalling matter at a rate³ that drops off roughly as t^−5/3 for t ⩾ t_fb ≈ 10⁵ s. Once the torus is formed, it will evolve under the influence of viscosity, radiative cooling winds, and time-dependent mass inflow.

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The black hole cannot accept matter, with a radiative efficiency = 0.1_0.1, at a rate exceeding $\dot{M}_{\rm Edd}\approx 2 \times 10^{-5} \epsilon _{0.1} M_{{\rm h},3} M_\odot \;{\rm yr^{-1}}$, without exceeding the black hole's Eddington luminosity. The rate at which the stellar debris returns to the vicinity of the black hole exceeds this limit: $\dot{M}\approx 10^{6}\dot{M}_{\rm Edd}$ for = 0.1. Given that this high rate can only be sustained by infall of the highly bound material for a time t_fb, we infer that, if the viscosity were high enough to process all the materials within the infall timescale, then the luminosity of the hole could not remain as high as the Eddington luminosity for longer than t_Edd ≈ 5 × 10^3−3/5_0.1M^−3/5_h,3 t_fb ∼ 17^−3/5_0.1M^−3/5_h,3 yr. The viscosity would have to be implausibly low (i.e., the usual viscous dissipation time t_d for a thick disk would be α⁻¹; α would have to be below 10⁻⁵ for t_d ⩾ t_Edd) for the bulk of the mass to be stored for longer than t_Edd in a reservoir at R ∼ R_T. A luminosity ∼L_Edd = 10⁴¹M_h,3 erg s^-1 can therefore only be maintained for at most 10 years; thereafter the flare would continue to fade as t^−3/5. It is clear from the behavior of $\dot{M}$ that most of the debris would be fed to the hole far more rapidly than it could be accepted if the radiative efficiency were high; much of the bound debris must either escape in a radiatively driven outflow or be swallowed inefficiently.

4.1. Observability of L_Edd Flares

4.2. ULX Activity

4.3. Relevance to GCs and IMBH Growth

We thank H. Baumgardt, L. Chomiuk, J. Strader, J. Guillochon, P. Hut, E. Noyola, C. Hopman, J. Kalirai, D. Kasen, and M. Rees for useful discussions and the referee for constructive suggestions. E.R. acknowledges support from the DOE (SciDAC; DE-FC02-01ER41176) and NSF (0521566). The simulations presented in this Letter were performed on the JUMP computer of the Höchstleistungsrechenzentrum Jülich.