### Abstract

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 σ∼v-a, the gas again forms a power-law cusp, but now the SIDM density scales as r-β, where β=(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.

Original language | English (US) |
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Article number | 023506 |

Journal | Physical Review D - Particles, Fields, Gravitation and Cosmology |

Volume | 89 |

Issue number | 2 |

DOIs | |

State | Published - Jan 16 2014 |

### ASJC Scopus subject areas

- Nuclear and High Energy Physics
- Physics and Astronomy (miscellaneous)

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## Cite this

*Physical Review D - Particles, Fields, Gravitation and Cosmology*,

*89*(2), [023506]. https://doi.org/10.1103/PhysRevD.89.023506