Stellar density cusp around a massive black hole

Abstract

This paper is concerned with the distribution of stars around a massive black hole in a dense star cluster. Spherical symmetry and isotropic distribution functions are assumed along with a power-law mass spectrum of stars to obtain a one-parameter family of equilibrium ''zero-flow'' solutions.The analysis is performed using a Fokker-Planck equation to describe the stellar encounters in their Coulomb potentials in an imposed gravitational potential well of the form phi (r) =phi/sub a/(r/sub a//r)/sup 1/alpha/(..cap alpha..=1 for a massive black hole). We find equilibrium solutions f (E) =KE/sup p/, where p= (3..cap alpha.. -5/2)/2, for the single-mass population case, in agreement with previous results, and for a power law mass spectrum f (E, m) =f/sub 0/m/sup x/ exp(: mE/sup -Y/), where Y=1/2(2X+3) for ..cap alpha..=1 and X>-1. This leads to the density scalings rho (r, m) proportionalm/sup x/ exp (-mr/sup Y/) r/sup -3/2/ and rho (r) proportionalr/sup -3/2-z/, where Z= (X+1)/2(2X+3), which are close to that for a single-mass population. It is shown how this solution may be matched smoothly onto an isothermal sphere configuration with a similar powerlaw mass spectrum.Assuming a stellar mass-luminosity relation Lproportional..mu../sup ..gamma../the luminous flux from the cusp scales as L (r) proportionalr/sup -3/2-z prime/where Z'=more » (X+..gamma..)/2(2X+3) (in the case ..cap alpha..=1). In projection onto the sky this becomes L/sub p/(s) proportionals/sup -1/2-z prime/which is steeper than that due to a single-mass population where L/sub p/(s) proportionals/sup -3/4/.« less