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Register a new userFormation of Massive Black Holes in Dense Star Clusters. II. IMF and Primordial Mass Segregation
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Sanghamitra Goswami
Publication date
2011
fields
Physics
and research's language is
English
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A promising mechanism to form intermediate-mass black holes (IMBHs) is the runaway merger in dense star clusters, where main-sequence stars collide and form a very massive star (VMS), which then collapses to a black hole. In this paper we study the effects of primordial mass segregation and the importance of the stellar initial mass function (IMF) on the runaway growth of VMSs using a dynamical Monte Carlo code for N-body systems with N as high as 10^6 stars. Our code now includes an explicit treatment of all stellar collisions. We place special emphasis on the possibility of top-heavy IMFs, as observed in some very young massive clusters. We find that both primordial mass segregation and the shape of the IMF affect the rate of core collapse of star clusters and thus the time of the runaway. When we include primordial mass segregation we generally see a decrease in core collapse time (tcc). Moreover, primordial mass segregation increases the average mass in the core, thus reducing the central relaxation time, which also decreases tcc. The final mass of the VMS formed is always close to sim 10^-3 of the total cluster mass, in agreement with the previous studies and is reminiscent of the observed correlation between the central black hole mass and the bulge mass of the galaxies. As the degree of primordial mass segregation is increased, the mass of the VMS increases at most by a factor of 3. Flatter IMFs generally increase the average mass in the whole cluster, which increases tcc. For the range of IMFs investigated in this paper, this increase in tcc is to some degree balanced by stellar collisions, which accelerate core collapse. Thus there is no significant change in tcc for the somewhat flatter global IMFs observed in very young massive clusters.
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We study the early dynamical evolution of young, dense star clusters using Monte Carlo simulations for systems with up to N~10^7 stars. Rapid mass segregation of massive main-sequence stars and the development of the Spitzer instability can drive these systems to core collapse in a small fraction of the initial half-mass relaxation time. If the core collapse time is less than the lifetime of the massive stars, all stars in the collapsing core may then undergo a runaway collision process leading to the formation of a massive black hole. Here we study in detail the first step in this process, up to the occurrence of core collapse. We have performed about 100 simulations for clusters with a wide variety of initial conditions, varying systematically the cluster density profile, stellar IMF, and number of stars. We also considered the effects of initial mass segregation and stellar evolution mass loss. Our results show that, for clusters with a moderate initial central concentration and any realistic IMF, the ratio of core collapse time to initial half-mass relaxation time is typically ~0.1, in agreement with the value previously found by direct N-body simulations for much smaller systems. Models with even higher central concentration initially, or with initial mass segregation (from star formation) have even shorter core-collapse times. Remarkably, we find that, for all realistic initial conditions, the mass of the collapsing core is always close to ~10^-3 of the total cluster mass, very similar to the observed correlation between central black hole mass and total cluster mass in a variety of environments. We discuss the implications of our results for the formation of intermediate-mass black holes in globular clusters and super star clusters, ultraluminous X-ray sources, and seed black holes in proto-galactic nuclei.
Collisions were suggested to potentially play a role in the formation of massive stars in present day clusters, and have likely been relevant during the formation of massive stars and intermediate mass black holes within the first star clusters. In the early Universe, the first stellar clusters were particularly dense, as fragmentation typically only occurred at densities above $10^9$cm$^{-3}$, and the radii of the protostars were enhanced due to the larger accretion rates, suggesting a potentially more relevant role of stellar collisions. We present here a detailed parameter study to assess how the number of collisions as well as the mass growth of the most massive object depends on the properties of the cluster, and we characterize the time evolution with three effective parameters, the time when most collisions occur, the duration of the collisions period, as well as the normalization required to obtain the total number of collisions. We apply our results to typical Population III (Pop.III) clusters of about $1000$M$_odot$, finding that a moderate enhancement of the mass of the most massive star by a factor of a few can be expected. For more massive Pop.III clusters as expected in the first atomic cooling halos, we expect a more significant enhancement by a factor of $15-32$. We therefore conclude that collisions in massive Pop.III clusters were likely relevant to form the first intermediate mass black holes.
Observations of young star-forming regions suggest that star clusters are born completely mass segregated. These initial conditions are, however, gradually lost as the star cluster evolves dynamically. For star clusters with single stars only and a canonical initial mass function, it has been suggested that traces of these initial conditions vanish at a time $tau_mathrm{v}$ between 3 and $3.5,t_mathrm{rh}$ (initial half-mass relaxation times). Since a significant fraction of stars are observed in binary systems and it is widely accepted that most stars are born in binary systems, we aim to investigate what role a primordial binary population (even up to $100,%$ binaries) plays in the loss of primordial mass segregation of young star clusters. We used numerical $N$-body models similar in size to the Orion Nebula Cluster (ONC) -- a representative of young open clusters -- integrated over several relaxation times to draw conclusions on the evolution of its mass segregation. We also compared our models to the observed ONC. We found that $tau_mathrm{v}$ depends on the binary star fraction and the distribution of initial binary parameters that include a semi-major axis, eccentricity, and mass ratio. For instance, in the models with $50,%$ binaries, we find $tau_mathrm{v} = (2.7 pm 0.8),t_mathrm{rh}$, while for $100,%$ binary fraction, we find a lower value $tau_mathrm{v} = (2.1 pm 0.6),t_mathrm{rh}$. We also conclude that the initially completely mass segregated clusters, even with binaries, are more compatible with the present-day ONC than the non-segregated ones.
Several dynamical scenarios have been proposed that can lead to prompt mass segregation on the crossing time scale of a young cluster. They generally rely on cool and/or clumpy initial conditions, and are most relevant to small systems. As a counterpoint, we present a novel dynamical mechanism that can operate in relatively large, homogeneous, cool or cold systems. This mechanism may be important in understanding the assembly of large mass-segregated clusters from smaller clumps.
Intermediate-mass black holes (IMBHs) could form via runaway merging of massive stars in a young massive star cluster (YMC). We combine a suite of numerical simulations of YMC formation with a semi-analytic model for dynamical friction and merging of massive stars and evolution of a central quasi-star, to predict how final quasi-star and relic IMBH masses scale with cluster properties (and compare with observations). The simulations argue that inner YMC density profiles at formation are steep (approaching isothermal), producing some efficient merging even in clusters with relatively low effective densities, unlike models which assume flat central profiles resembling those of globular clusters (GCs) {em after} central relaxation. Our results can be approximated by simple analytic scalings, with $M_{rm IMBH} propto v_{rm cl}^{3/2}$ where $v_{rm cl}^{2} = G,M_{rm cl}/r_{rm h}$ is the circular velocity in terms of initial cluster mass $M_{rm cl}$ and half-mass radius $r_{rm h}$. While this suggests IMBH formation is {em possible} even in typical clusters, we show that predicted IMBH masses for these systems are small, $sim 100-1000,M_{odot}$ or $sim 0.0003,M_{rm cl}$, below even the most conservative observational upper limits in all known cases. The IMBH mass could reach $gtrsim 10^{4},M_{odot}$ in the centers nuclear star clusters, ultra-compact dwarfs, or compact ellipticals, but in all these cases the prediction remains far below the present observed supermassive BH masses in these systems.
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