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 ^ { -7 / 4 } and the velocity dispersion as r ^ { -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 \sigma \sim v ^ { - a } , the gas again forms a power-law cusp , but now the SIDM density scales as r ^ { - \beta } , where \beta = ( a + 3 ) / 4 , while its velocity dispersion again varies as r ^ { -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 .