Accretion of Dark Matter onto a Moving Schwarzschild Black Hole: An Exact Solution

We investigate accretion of dark matter onto a moving Schwarzschild black hole. The dark matter is modeled by the collisionless Vlasov gas, assumed to be in thermal equilibrium at infinity. We derive an exact stationary solution and provide a compact formula for the mass accretion rate. In general, the mass accretion rate is a nonmonotonic function of the black hole velocity. A monotonic relation (the accretion rate proportional to the Lorentz factor associated with the velocity of the black hole) is obtained for high asymptotic temperatures of the gas. The derived accretion rates are relevant for the growth of primordial black holes in the early Universe.