Stationary spherical accretion into black holes - II. Theory of optically thick accretion

In this paper, the problem of spherical, steady-state, optically thick accretion into black holes is solved. We analyse the integral curves of the differential equations describing the problem. We find a one-parameter family of critical points, where the inflow velocity equals the isothermal sound speed. Physical solutions must pass through one of these critical points. We obtain a complete set of boundary conditions which the solution must satisfy at the horizon of the black hole, and show that these, plus the requirement that the solution pass through a critical point, determine a unique solution to the problem. This analysis leads to a generalization of the well-known Bondi critical point constraint, which arises in the adiabatic accretion problem and which is effective at the point where the inflow velocity equals the adiabatic sound speed. We show that this point can be regarded as a ‘diffused critical point’ in our problem. The analysis also yields a simple expression for the diffusive luminosity at radial infinity. Finally, we find a satisfying explanation for the rather peculiar critical point structure of this problem in an analysis of the characteristics and subcharacteristics present in the problem and in a ‘hierarchical’ analysis of the waves which propagate along them.