Black holes in radiation-dominated gas: an analogue of the Bondi accretion problem

Black holes, unlike other compact objects, are able to accrete matter more rapidly than their Eddington rate, $$\dot M_\text E=L_\text E/c^2$$. Nevertheless, at such a high $$\dot M$$, radiation will probably be emitted by the in-falling gas in copious enough quantities to have a profound influence on the flow. To aid in understanding the nature of this influence, we study the steady flow, on to a stationary Schwarzschild black hole, of a uniform, non-relativistic gas in which radiation pressure swamps thermal pressure at infinity, and in which Thomson scattering provides the only radiation–gas couple. Asymptotic radiation pressure p_∞ and matter density ρ_∞ determine an asymptotic sound speed c_∞, from which one can derive an accretion rate $$\dot M_\text B$$ corresponding to the adiabatic flow of a $$\gamma=4/3$$ gas. The actual accretion rate depends on the optical depth τ_B of a column of unperturbed gas spanning the Bondi radius, $$r_B=GM/c^2_\infty$$. If $$\tau_B\gt(\sqrt{2}/3)(c/c_\infty)$$, then the flow is adiabatic, and $$\dot M=\dot M_\text B$$. For a somewhat smaller τ_B, diffusion is efficient enough for the radiation to leak out of the gas as it moves towards the trans-sonic point. As a result, the sound speed decreases inwards in the subsonic region, while the density must increase steeply to maintain pressure balance. $$\dot M$$ may then exceed $$\dot M_\text B$$ by a factor of up to $$(\sqrt{2}/3)(c/\tau_\text B c_\infty)$$, although this effect can be limited by thermal pressure. Finally, for small enough τ_B the diffusion approximation breaks down, and radiation drag limits an otherwise thermally-determined $$\dot M$$. Our boundary conditions occur within super-massive $$(M/M_\odot\gtrsim10^2)$$ stars, and in the pre- and post-recombination universe. If a super-massive star of $$M/M_\odot\lesssim5\times10^5$$ happens to have a small $$(M_\text {BH}\lesssim1 M_\odot)$$ black hole passing through it on a bound orbit, it will capture and be swallowed by the hole, before it has a chance to ignite its nuclear fuel and blow itself apart. In any milieu where both small black holes and super-massive stars are likely to form (e.g. a dense star cluster), this may provide a natural mechanism for forming black holes in the mass range $$10^2-10^5M_\odot$$.