Self-interacting dark matter cusps around massive black holes
We adopt the conduction fluid approximation to model the steady-state distribution of matter around a massive black hole at the center of a weakly collisional cluster of particles. By ``weakly collisional'' we mean a cluster in which the mean free time between particle collisions is much longer than the characteristic particle crossing (dynamical) time scale, but shorter than the cluster lifetime. When applied to a star cluster, we reproduce the familiar Bahcall-Wolf power-law cusp solution for the stars bound to the black hole. Here the star density scales with radius as ${r}^{\ensuremath{-}7/4}$ and the velocity dispersion as ${r}^{\ensuremath{-}1/2}$ throughout most of the gravitational well of the black hole. When applied to a relaxed, self-interacting dark matter (SIDM) halo with a velocity-dependent cross section $\ensuremath{\sigma}\ensuremath{\sim}{v}^{\ensuremath{-}a}$, the gas again forms a power-law cusp, but now the SIDM density scales as ${r}^{\ensuremath{-}\ensuremath{\beta}}$, where $\ensuremath{\beta}=(a+3)/4$, while its velocity dispersion again varies as ${r}^{\ensuremath{-}1/2}$. Results are obtained first in Newtonian theory and then in full general relativity. Although the conduction fluid model is a simplification, it provides a reasonable first approximation to the matter profiles and is much easier to implement than a full Fokker-Planck treatment or an $N$-body simulation of the Boltzmann equation with collisional perturbations.