Dynamical evolution of black hole sub-systems in idealised star clusters

Abstract

In this paper, globular star clusters which contain a sub-system 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 multi-mass 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 sub-system. After mass segregation has driven most of the BH into a compact sub-system, the evolution of the BH sub-system is found to be influenced by the cluster in which it is contained. The BH sub-system 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 sub-system 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 sub-system up until the time when 90% of the sub-system total mass is lost (which is of order 10^{3} times the half-mass relaxation time of the BH sub-system at its time of formation). Based on theoretical arguments the rate of mass loss from the BH sub-system (\dot{M}_2) is predicted to be -(beta*zeta*M)/(alpha*t_{rh}), where M is the total mass, t_{rh} is the half-mass relaxation time, and alpha, beta, zeta are three dimensionless parameters (see Section 2 for details). An interesting consequence of this is that the rate of mass loss from the BH sub-system is approximately independent of the stellar mass ratio (m_2/m_1) and the total mass ratio (M_2/M_1) (in the range m_2/m_1 >~ 10 and M_2/M_1 ~ 10^{-2}, where m_1, m_2 are the masses of individual low-mass and high-mass particles respectively, and M_1, M_2 are the corresponding total masses).