Weak Shock Propagation with Accretion. I. Self-similar Solutions and Application to Failed Supernovae

We present solutions for the self-similar propagation of a shock wave in a hydrostatic, adiabatic medium with a point mass gravitational field. In contrast to the well-known, Sedov–Taylor blast wave, these solutions apply to the case when the shock Mach number is of the order of a few, and the energy of the shocked fluid is not conserved but self-consistently modified by the binding energy of the ambient medium that is swept up by the passage of the shock. Furthermore, we show that there is one solution (for a given ambient density profile) that smoothly passes through a sonic point in the post-shock flow and results in accretion onto the central object; in analogy with the Bondi problem, we propose that these solutions are the ones that are most relevant in astrophysical environments. We apply these accreting models to failed supernovae, in which neutron star formation does not unbind the envelope, but a weak shock is still generated in the outer layers of the star from neutrino-induced mass loss. We find excellent agreement between the predictions of our self-similar, shock propagation model and numerical simulations of the collapse of a yellow supergiant; the self-similar solutions reproduce the overall scaling of the shock speed; the time- and space-dependent evolution of the velocity, density, and pressure behind the shock; and the accretion rate onto the black hole. Our results have important implications for the fallback and ejection of material in failed supernovae.