INTERMEDIATE-MASS BLACK HOLES IN EARLY GLOBULAR CLUSTERS

If globular clusters follow the same "M_BH–σ" relation between central black hole mass and effective velocity dispersion as is observed in galaxies (Tremaine et al. 2002), many clusters should host intermediate-mass black holes (IMBHs) with masses in the range 10² ≲ M_BH/M_☉ ≲ 10⁴. Unfortunately, hard observational evidence for IMBHs in globular clusters has proved elusive. Currently, the strongest case is the massive cluster G1 in M31, where dynamical, X-ray, and radio studies are consistent with a mass M_BH ∼ 2 × 10⁴ (Gebhardt et al. 2005; Ulvestad et al. 2007). In our own Galaxy, a measurement of an IMBH mass M_BH ∼ 4 × 10⁴ M_☉ has been reported for ω Centauri (Noyola et al. 2008), although this result has been questioned in a recent study (Anderson & van der Marel 2010; van der Marel & Anderson 2010) suggesting that the IMBH signature becomes much weaker when a more accurate estimate of the cluster center is used. Recently, Ibata et al. (2009) have reported evidence of a density and kinematic cusp in the core of M54, a cluster located at the center of the Sagittarius dwarf galaxy; their models suggest that the cusp could be due to a ∼10⁴ M_☉ IMBH.

For most Galactic globular clusters, kinematic and structural data currently provide only upper limits on possible IMBH masses (McLaughlin et al. 2006; Pasquato et al. 2009; van der Marel & Anderson 2010). However, these upper limits are larger than the values of M_BH expected from the M_BH–σ relation, and do not exclude the possibility that many globular clusters host IMBHs with masses in the above range (van der Marel & Anderson 2010).

The presence of an IMBH in a globular cluster has important consequences for the cluster's structure, kinematics, internal energetics, and long-term dynamical evolution (see, e.g., Baumgardt et al. 2004; Heggie et al. 2007; Trenti et al. 2007; Miocchi 2007; see also McMillan 2008 for a review). In addition, Eddington accretion onto IMBHs has been proposed as the mechanism powering some ultraluminous X-ray sources (ULXs; e.g., Farrell et al. 2009), and IMBHs have long been recognized as potential rich sources of gravitational waves detectable by LISA (Miller 2009). For these reasons, despite the lack of strong observational constraints, the formation and consequent properties of IMBHs have become a topic of considerable theoretical and observational interest.

One possible IMBH formation mechanism is direct collapse of a very massive (⩾250 M_☉) Population III (Pop. III) star (see, e.g., Bond et al. 1984; Madau & Rees 2001, and references therein; but see Abel et al. 2002 for hydrodynamical simulations suggesting that these stars would form in isolation). For non-primordial or more massive IMBHs, the leading formation mechanisms center on dynamical processes in dense star clusters. These include runaway mergers of massive stars in clusters with high central densities (Portegies Zwart & McMillan 2002; Gürkan et al. 2004; Portegies Zwart et al. 2004) and merging of black holes in binaries (see O'Leary et al. 2006, and references therein). However, numerous potential problems affecting these models have been pointed out in the recent literature, suggesting that the net growth rate and hence the final mass of the resulting IMBH may be much lower than that suggested by earlier dynamical simulations (Yungelson et al. 2008; Glebbeek et al. 2009).

Thus, while there appear to be numerous plausible stellar, binary, and dynamical pathways to the formation of relatively "low-mass" (∼100–200 M_☉) IMBHs in clusters, there is currently no clear consensus on a mechanism for the production of high-mass (10³–10⁴ M_☉) IMBHs.

In this Letter, we propose a new scenario for the growth of IMBHs in early globular clusters. It is a natural consequence of the physical conditions that may have obtained in cluster cores during the formation of the second-generation (SG) stars now observed in many globular clusters. Specifically, we show that accretion by a pre-existing seed black hole of just a small fraction of the gas from which the SG stars are formed can lead to black holes in the high-mass (10³–10⁴ M_☉) IMBH range.

The structure of this Letter is as follows. In Section 2, we briefly review the observational evidence for multiple stellar populations in globular clusters and summarize the main elements of our model for the formation and evolution of multiple populations. In Section 3, we present our scenario for IMBH growth. We discuss some consequences of this scenario in Section 4.

Spectroscopic and photometric observations provide strong evidence for multiple stellar generations in globular clusters.

Spectroscopic studies have shown that, even in clusters whose stars have a common Fe abundance, there are substantial star-to-star differences in the abundances of lighter elements (see Gratton et al. 2004 and references therein, and Carretta et al. 2009a, 2009b for a recent spectroscopic study of 19 Galactic clusters). All the Galactic globular clusters for which spectroscopic studies have been carried out show evidence of multiple stellar populations, and in all cases SG stars make up a significant fraction (50%–80%) of the total (D'Antona & Caloi 2008; Carretta et al. 2009a, 2009b).

Several recent photometric studies have provided further support for these results by showing the photometric fingerprints of multiple populations (multiple main sequences, significant main-sequence broadening, or subgiant branch splits) in the color–magnitude diagrams both of "normal" clusters in the Galaxy (Piotto et al. 2007; Marino et al. 2008; Milone et al. 2008; Anderson et al. 2009), and of more complex systems such as ω Cen (Lee et al. 1999; Pancino et al. 2000; Bedin et al. 2004; Villanova et al. 2007), M22 (Marino et al. 2009; Lee et al. 2009), and probably Terzan 5 (Ferraro et al. 2009).

The differences in light-element abundances suggest that SG stars formed out of gas containing matter processed through high-temperature CNO cycle reactions in first-generation (FG) stars. The main candidates currently suggested as possible sources of gas for SG formation are massive asymptotic giant branch (AGB) stars (Ventura et al. 2001) and rapidly rotating massive stars (Decressin et al. 2007). Although many aspects of these two scenarios are still under investigation, in order to form the large mass of SG stars suggested by observations, both scenarios require that either the initial mass function (IMF) of the FG stars was highly anomalous, with an unusually large fraction of massive stars, or the FG population had a normal IMF but was initially at least 10 times more massive than is now observed.

We have recently carried out hydrodynamical and N-body simulations to explore the formation and dynamical evolution of multiple population clusters formed out of AGB star ejecta (D'Ercole et al. 2008). Our simulations show that the gas lost by FG AGB stars collects in a cooling flow in the cluster core, where most SG stars are formed. We have further demonstrated a dynamical mechanism whereby, after the centrally concentrated SG stars have formed, the cluster's response to early mass loss (of intracluster gas and/or supernova ejecta) leads to the escape of a substantial fraction of FG stars. During the subsequent long-term evolution, the SG and FG populations mix by two-body relaxation, plausibly leading to the multiple-population clusters observed today.

We now explore the implications for the growth of a pre-existing seed black hole of the physical conditions in the central regions of the cluster during the SG formation phase.

In the hydrodynamical simulations presented in D'Ercole et al. (2008), we assumed that the cluster FG stars followed a King (1962) density profile,

$$\begin{equation} \rho =\rho _0 \left[1+\left({r \over r_c}\right)^2\right]^{-3/2}, \end{equation} \tag{ 1 }$$

out to a truncation radius R_t, with stellar masses distributed in the range 0.1 < m/m_☉ < 100 according to a "Kroupa" IMF (Kroupa et al. 1993). We adopted a concentration c = log(R_t/r_c) = 1.5 and followed the evolution of the AGB ejecta in clusters having different values of the total FG mass M and truncation radius R_t.

Figure 1 shows the time evolution of the AGB ejecta density and sound speed within 0.1 pc of the cluster center during the SG formation phase, for four clusters spanning broad ranges in initial mass and tidal radius. In all cases, if the cluster already hosts a seed black hole with mass M_BH ∼ 10² M_☉—the result of one of the "prompt" low-mass IMBH formation mechanisms discussed previously, operating in the FG population—the gas conditions shown in Figure 1 will lead to accretion. The Bondi accretion rate onto a black hole of mass M_BH is

$$\begin{eqnarray} \dot{M} &\simeq & 3 \times 10^{-4} \left(M_{\rm BH}\over 100 \,M_{\odot }\right)^2\left(\rho \over 10^{-18}\, \hbox{g cm}^{-3}\right) \nonumber\\ &&\times \left({3\, \hbox{km s}^{-1}\over c_s}\right)^3 \,M_{\odot }\;\hbox{yr}^{-1}, \end{eqnarray} \tag{ 2 }$$

where the density ρ and the sound speed c_s have been normalized to the typical values found in our simulations and shown in Figure 1. For the physical conditions in our four simulations, the Bondi accretion rate is ∼10⁻⁴–10⁻³ M_☉ yr⁻¹, much larger than the Eddington rate, ∼3 × 10⁻⁹(M_BH/M_☉)/η M_☉ yr⁻¹ (Krolik 1999), where η is the radiative efficiency, to be discussed further below. The black hole will therefore accrete gas at the Eddington rate.

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Under these circumstances, the black hole mass will grow on a Salpeter timescale:

$$\begin{equation} T_S \simeq 3.9\times 10^8 \,\eta \, L_E/L \ \hbox{yr}, \end{equation} \tag{ 3 }$$

where L_E ∼ 1.45 × 10³⁸ M_BH/M_☉ erg s⁻¹ is the Eddington luminosity and a value of the mass per electron μ_e ∼ 1.15 has been adopted (see, e.g., Krolik 1999). The value of η is uncertain and depends on the properties of the accretion disk and on the spin of the accreting black hole. Here we adopt η ∼ 0.05–0.1, a range commonly adopted in studies of black hole growth in galactic nuclei. We note that values of η larger than 0.1, appropriate for rapidly rotating black holes, would reduce the growth of the IMBH, while much smaller values (η ≪ 1) typical of thick accretion disks would significantly increase the IMBH growth rate. For accretion at the Eddington rate with η ∼ 0.05–0.1, T_S ∼ 20–40 Myr.

The duration of the SG star formation episode in the models of D'Ercole et al. (2008) is ΔT ≈ 10² Myr. This timescale is constrained by spectroscopic observations showing little variation in the total CNO of FG and SG stars, implying that only AGB progenitors with M > 4–5 M_☉ must have contributed gas to the SG formation process. Thus, if the seed black hole accretes at close to the Eddington rate during this time, its mass will grow by a factor of ∼10–100, leading, by the end of the SG formation phase, to a final black hole mass of M_BH ∼ 10³–10⁴ M_☉.

Several aspects of the scenario just presented merit further discussion and exploration.

We consider first the effect on the cooling flow of the energy radiated by the accreting black hole. To address this issue, we follow the analysis of black hole outflow presented by King (2003, 2005), in which it is shown that the wind produced by the black hole sweeps up the surrounding gas into an expanding shell. Because of effective energy losses due to inverse Compton scattering, the shell expands in the momentum-conserving regime. Under these conditions (see King 2003, 2005), in order for the momentum flux from a black hole accreting at the Eddington rate to overcome gravity and expel the shell from the cluster, the black hole mass must be larger than

$$\begin{equation} M_{\rm BH,crit} = \frac{f_g\kappa \sigma ^4}{\pi G^2}. \end{equation} \tag{ 4 }$$

Here, κ is the electron scattering opacity, f_g is the fraction of the total cluster mass in the form of gas, and it has been assumed that the gas is embedded in an isothermal system with velocity dispersion σ.

Assuming σ ≳ 20 km s⁻¹ (the smallest value for which the AGB ejecta can be retained) and f_g ≈ 0.05 (D'Ercole et al. 2008), we find M_BH,crit ≳ 10⁴ M_☉, much larger than the assumed mass of our initial seed black hole. Thus, feedback from the accreting black hole is not sufficient to alter significantly the global dynamics of the cooling flow or the SG star formation process.

On the other hand, the effects of feedback may significantly alter the local gas dynamics in the vicinity of the black hole. For example, recent detailed two-dimensional hydrodynamical models of accretion onto a 100 M_☉ seed black hole in a dense protogalactic cloud (Milosavljevic et al. 2009) show intermittent accretion, reducing the net accretion rate to about one-third of Eddington. Such a reduction in the mean accretion rate in our case would reduce the growth of the seed black hole to a factor of ∼2–5 for the range of η values adopted in Section 3.

We note here that during the accretion phase the black hole will be very luminous and might be observed as a (possibly intermittent) ULX. We caution however that, before linking this scenario to the ULXs observed in nearby young massive clusters (see, e.g., Portegies Zwart et al. 2010), one must first verify whether these clusters (1) meet the conditions to form a seed black hole, (2) are sufficiently massive, and (3) have the structural properties required for SG star formation.

A second important issue is the possible relation between the IMBH mass and the current structural properties of the parent cluster.

Any correlation between the mass of the seed black hole and the early properties of the cluster should be preserved during the Eddington growth phase, provided that the duration ΔT of the SG star formation episode is approximately independent of the cluster environment. The mass of the seed black hole depends on its formation mechanism. If it is the result of stellar evolution in a massive Pop. III progenitor, a roughly universal mass seems to be the most likely result (see, e.g., Madau & Rees 2001). Early dynamical simulations of the runaway merger scenario suggested that the mass of the runaway should scale with the mass of the cluster (or the cluster core) in which it forms (Gürkan et al. 2004; Portegies Zwart et al. 2004). However, as mentioned in Section 1, strong stellar winds may well limit the actual mass attained, possibly to as little as a few hundred solar masses, again largely independent of the initial cluster mass.

For the black hole growth phase to occur, the cluster must initially have been massive and/or compact enough to retain the AGB ejecta that subsequently flowed into the cluster core and formed the SG population. As discussed in D'Ercole et al. (2008), multiple population clusters must have undergone an early phase of strong FG mass loss and structural evolution; it is therefore not straightforward to connect the initial properties of a cluster during the growth of the seed black hole with the current properties of the cluster hosting an IMBH. However, given the loss of most of the FG population and the possible near-universality of the seed black hole mass, the most likely outcome of the scenario described here seems to be one in which the present IMBH mass has only a weak dependence on current cluster properties.

E.V. and S.M. were supported in part by NASA grants NNX07AG95G, NNX08AH15G, and NNX10AD86G, and by NSF grant AST-0708299. A.D. acknowledges financial support from Italian MIUR through grant PRIN 2007 (prot. 2007JJC53X). F.D. was supported by PRIN MIUR 2007 "Multiple Stellar Populations in Globular Clusters: Census, Characterization and Origin" (prot. n. 20075TP5K9).