Dynamical evolution of black hole subsystems in idealized star clusters

In this paper, globular star clusters which contain a subsystem of stellar mass black holes (BH) are investigated. This is done by considering two-component models, as these are the simplest approximation of more realistic multimass systems, where one component represents the BH population and the other represents all the other stars. These systems are found to undergo a long phase of evolution where the centre of the system is dominated by a dense BH subsystem. After mass segregation has driven most of the BH into a compact subsystem, the evolution of the BH subsystem is found to be influenced by the cluster in which it is contained. The BH subsystem evolves in such a way as to satisfy the energy demands of the whole cluster, just as the core of a one-component system must satisfy the energy demands of the whole cluster. The BH subsystem is found to exist for a significant amount of time. It takes approximately 10t_rh,i, where t_rh,i is the initial half-mass relaxation time, from the formation of the compact BH subsystem up until the time when 90 per cent of the subsystem total mass is lost (which is 10³ times the half-mass relaxation time of the BH subsystem at its time of formation). Based on theoretical arguments, the rate of mass-loss from the BH subsystem (Mdot₂) is predicted to be -βζM/(αt_rh), where M is the total mass, t_rh is the half-mass relaxation time and α, β, ζ are three dimensionless parameters (see Section 2 of the main text for details). An interesting consequence of this is that the rate of mass-loss from the BH subsystem is approximately independent of the stellar mass ratio (m₂/m₁) and the total mass ratio (M₂/M₁) (in the range m₂/m₁ ≳ 10 and M₂/M₁ ∼ 10^-2, where m₁ and m₂ are the masses of individual low-mass and high-mass particles, respectively, and M₁ and M₂ are the corresponding total masses). The theory is found to be in reasonable agreement with most of the results of a series of N-body simulations, and with all of the models if the value of ζ is suitably adjusted. Predictions based on theoretical arguments are also made about the structure of BH subsystems. Other aspects of the evolution are also considered such as the conditions for the onset of gravothermal oscillation.