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We revisit the role that gravitational scattering off stars plays inestablishing the steady-state distribution of collisionless dark matter (DM)around a massive black hole (BH). This is a physically interesting problem thathas potentially observable signatures, such as $\gamma-$rays from DMannihilation in a density spike. The system serves as a laboratory forcomparing two different dynamical approaches, both of which have been widelyused: a Fokker-Planck treatment and a two-component conduction fluid treatment.In our Fokker-Planck analysis we extend a previous analytic model to accountfor a nonzero flux of DM particles into the BH, as well as a cut-off in thedistribution function near the BH due to relativistic effects or, further out,possible DM annihilation. In our two-fluid analysis, following an approximateanalytic treatment, we recast the equations as a "heated Bondi accretion"problem and solve the equations numerically without approximation. While boththe Fokker-Planck and two-fluid methods yield basically the same DM density andvelocity dispersion profiles away from the boundaries in the spike interior,there are other differences, especially the determination of the DM accretionrate. We discuss limitations of the two treatments, including the assumption ofan isotropic velocity dispersion.

The Effect of Stars on the Dark Matter Spike Around a Black Hole: A Tale of Two Treatments