Runaway Merging of Black Holes: Analytical Constraint on the Timescale

Abstract

Following the discovery of a black hole (BH) with a mass of 10^3-10^6 M(sun) in a starburst galaxy M82, we study formation of such a BH via successive merging of stellar-mass BHs within a star cluster. The merging has a runaway characteristic. This is because massive BHs sink into the cluster core and have a high number density, and because the merging probability is higher for more massive BHs. We use the Smoluchowski equation to study analytically the evolution of the BH mass distribution. Under favorable conditions, which are expected for some star clusters in starburst galaxies, the timescale of the runaway merging is at most of order 10^7 yr. This is short enough to account for the presence of a BH heavier than 10^3 M(sun) in an ongoing starburst region.