Gravitational Brownian motion as inhomogeneous diffusion: black hole populations in globular clusters

Zacharias Roupas

Published 2020-06-23Version 1

Recent theoretical, numerical developments supported by observational evidence strongly suggest that many Globular Clusters host a black hole (BH) population in their centers. This is in contrast to previous long-standing belief that a BH subcluster would evaporate after undergoing core collapse and decoupling from the cluster. We propose that one mechanism which may add a stabilizing pressure to a BH population is the inhomogeneous Brownian motion generated by fluctuations of the stellar gravitational field. We argue that the diffusion equation for Brownian motion in inhomogeneous medium with spatially varying diffusion coefficient and temperature, discovered firstly by Van Kampen, applies to self-gravitating systems. Applying the stationary phase space probability distribution to a single BH immersed in a Plummer Globular Cluster we infer it may wanderer as far as $\sim 0.05,\,0.1,\,0.5{\rm pc}$ for mass $m_{\rm b} \sim 10^3, \,10^2,\,10{\rm M}_\odot$, respectively. We further find that the fluctuations of a fixed stellar mean gravitational field are sufficient to stabilize a BH population above the Spitzer instability threshold. We, nevertheless, identify an instability, whose onset depends on the Spitzer parameter $S = (M_{\rm b}/M_\star) (m_{\rm b}/m_\star)^{3/2} $ and parameter $B = \rho_{\rm b}(0) (4\pi r_c^3/M_b)(m_\star/m_{\rm b})^{3/2} $, where $\rho_{\rm b}(0)$ is the Brownian population central density. For a Plummer sphere the instability occurs at $(B,S) = (140,0.25)$. For $B > 140$ we get very cuspy BH subcluster profiles, unstable under the support of fluctuations alone. For $S > 0.25$ there do not exist stationary states of the BH population inhomogeneous diffusion equation.