Growth of Intermediate-Mass Black Holes in Globular Clusters

We present results of numerical simulations of sequences of binary-single scattering events of black holes in dense stellar environments. The simulations cover a wide range of mass ratios from equal mass objects to 1000:10:10 solar masses and compare purely Newtonian simulations to simulations in which Newtonian encounters are interspersed with gravitational wave emission from the binary. In both cases, the sequence is terminated when the binary's merger time due to gravitational radiation is less than the arrival time of the next interloper. We find that black hole binaries typically merge with a very high eccentricity (0.93 < e < 0.95 pure Newtonian; 0.85 < e < 0.90 with gravitational wave emission) and that adding gravitational wave emission decreases the time to harden a binary until merger by ~ 30% to 40%. We discuss the implications of this work for the formation of intermediate-mass black holes and gravitational wave detection.Comment: 28 pages including 9 figures, submitted to Ap

Star Clusters with Primordial Binaries: II. Dynamical Evolution of Models in a Tidal Field

[abridged] We extend our analysis of the dynamical evolution of simple star cluster models, in order to provide comparison standards that will aid in interpreting the results of more complex realistic simulations. We augment our previous primordial-binary simulations by introducing a tidal field, and starting with King models of different central concentrations. We present the results of N-body calculations of the evolution of equal-mass models, starting with primordial binary fractions of 0 - 100 %, and N values from 512 to 16384. We also attempt to extrapolate some of our results to the larger number of particles that are necessary to model globular clusters. We characterize the steady-state `deuterium main sequence' phase in which primordial binaries are depleted in the core in the process of `gravitationally burning'. In this phase we find that the ratio of the core to half-mass radius, r_c/r_h, is similar to that measured for isolated systems. In addition to the generation of energy due to hardening and depletion of the primordial binary population, the overall evolution of the star clusters is driven by a competing process: the tidal disruption of the system. We find that the depletion of primordial binaries before tidal dissolution of the system is possible only if the initial number is below 0.05 N, in the case of a King model with W_0=7 and N=4096 (which is one of our longest living models). We compare our findings, obtained by means of direct N-body simulations but scaled, where possible, to larger N, with similar studies carried out by means of Monte Carlo methods.Comment: 15 pages, 18 figures, matches MNRAS accepted version, some sections reorganized but no major change

Star Clusters with Primordial Binaries: I. Dynamical Evolution of Isolated Models

In order to interpret the results of complex realistic star cluster simulations, which rely on many simplifying approximations and assumptions, it is essential to study the behavior of even more idealized models, which can highlight the essential physical effects and are amenable to more exact methods. With this aim, we present the results of N-body calculations of the evolution of equal-mass models, starting with primordial binary fractions of 0 - 100 %, with values of N ranging from 256 to 16384. This allows us to extrapolate the main features of the evolution to systems comparable in particle number with globular clusters. In this range, we find that the steady-state `deuterium main sequence' is characterized by a ratio of the core radius to half-mass radius that follows qualitatively the analytical estimate by Vesperini & Chernoff (1994), although the N dependence is steeper than expected. Interestingly, for an initial binary fraction f greater than 10%, the binary heating in the core during the post collapse phase almost saturates (becoming nearly independent of f), and so little variation in the structural properties is observed. Thus, although we observe a significantly lower binary abundance in the core with respect to the Fokker-Planck simulations by Gao et al. (1991), this is of little dynamical consequence. At variance with the study of Gao et al. (1991), we see no sign of gravothermal oscillations before 150 halfmass relaxation times. At later times, however, oscillations become prominent. We demonstrate the gravothermal nature of these oscillations.Comment: 14 pages, 22 figures, MNRAS accepte

The M/L ratio of massive young clusters

We point out a strong time-evolution of the mass-to-light conversion factor \eta commonly used to estimate masses of dense star clusters from observed cluster radii and stellar velocity dispersions. We use a gas-dynamical model coupled with the Cambridge stellar evolution tracks to compute line-of-sight velocity dispersions and half-light radii weighted by the luminosity. Stars at birth are assumed to follow the Salpeter mass function in the range [0.15--17 M_\sun]. We find that $\eta$, and hence the estimated cluster mass, increases by factors as large as 3 over time-scales of 20 million years. Increasing the upper mass limit to 50 M_\sun leads to a sharp rise of similar amplitude but in as little as 10 million years. Fitting truncated isothermal (Michie-King) models to the projected light profile leads to over-estimates of the concentration par ameter c of $\delta c\approx 0.3$ compared to the same functional fit applied to the proj ected mass density.Comment: Draft version of an ApJ lette

On the effects of dynamical evolution on the initial mass function of globular clusters

In this paper we show the results of a large set of N-body simulations modelling the evolution of globular clusters driven by relaxation,stellar evolution,disk shocking and including the effects of the tidal field of the Galaxy. We investigate the evolution of multi-mass models with a power-law initial mass function (IMF) starting with different initial masses, concentrations, slopes of the IMF and located at different galactocentric distances. We show to what extent the effects of the various evolutionary processes alter the shape of the IMF and to what extent these changes depend on the position of the cluster in the Galaxy. Both the changes in the global mass function and in the local one (measured at different distances from the cluster center) are investigated showing whether and where the local mass function keeps memory of the IMF and where it provides a good indication of the current global mass function. The evolution of the population of white dwarfs is also followed in detail and we supply an estimate of the fraction of the current value of the total mass expected to be in white dwarfs depending on the main initial conditions for the cluster (mass and position in the Galaxy).Simple analytical expression by which it is possible to calculate the main quantities of interest (total mass, fraction of white dwarfs, slope of the mass function) at any time t for a larger number of different initial conditions than those investigated numerically have been derived.Comment: 20 pages LaTeX (MNRAS style), 23 figures, MNRAS (in press

Solvable model of a self-gravitating system

We introduce and discuss an effective model of a self-gravitating system whose equilibrium thermodynamics can be solved in both the microcanonical and the canonical ensemble, up to a maximization with respect to a single variable. Such a model can be derived from a model of self-gravitating particles confined on a ring, referred to as the self-gravitating ring (SGR) model, allowing a quantitative comparison between the thermodynamics of the two models. Despite the rather crude approximations involved in its derivation, the effective model compares quite well with the SGR model. Moreover, we discuss the relation between the effective model presented here and another model introduced by Thirring forty years ago. The two models are very similar and can be considered as examples of a class of minimal models of self-gravitating systems.Comment: 21 pages, 6 figures; submitted to JSTAT for the special issue on long-range interaction