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We consider the steady-state density and velocity dispersion profiles ofcollisionless matter around a Schwarzschild black hole (BH) and its associatedrate of accretion onto the BH. We treat matter, which could be stars or darkmatter particles, whose orbits are {\it unbound} to the BH, but still governedby its gravitational field. We consider two opposite spatial geometries for thematter distributions: an infinite, 3D cluster and a 2D razor-thin disk, bothwith zero net angular momentum. We demonstrate that the results dependcritically on the adopted geometry, even in the absence of angular momentum. Weadopt a simple monoenergetic, isotropic, phase-space distribution function forthe matter as a convenient example to illustrate this dependence. The effect ofbreaking strict isotropy by incorporating an unreplenished loss cone due to BHcapture of low-angular momentum matter is also considered. Calculations are allanalytic and performed in full general relativity, though key results are alsoevaluated in the Newtonian limit. We consider one application to show that therate of BH accretion from an ambient cluster can be significantly less thanthat from a thin disk to which it may collapse, although both rates areconsiderably smaller than Bondi accretion for a (collisional) fluid with asimilar asymptotic particle density and velocity dispersion (i.e., soundspeed).

Spikes and accretion of unbound, collisionless matter around black holes