On the Origin of Mass Ejection in Failed Supernovae

Some high-mass stars likely end their lives in underluminous implosions that leave behind a black hole, known as failed supernovae (FSNe). However, neutrinos radiated during proto-neutron star formation generate a weak (Mach $\gtrsim 1$) shockwave in the outer layers of the star, which produces a unique transient as it breaks out of the dying star and signals its imminent disappearance. It was recently shown that there are two self-similar solutions that describe the propagation of this weak shockwave, and these solutions simultaneously contain outward-moving ejecta and fallback accretion onto the black hole. Here we show that the larger Mach number solutions are unstable, such that the Mach number of the shock grows with time $t$ and deviates from the self-similar prediction as $\propto t^{\alpha}$, with $\alpha \lesssim 0.1$, whereas the smaller Mach number solutions are stable. We also show that, above a critical mass loss that is readily achievable in core-collapse supernovae, the shock asymptotically strengthens and approaches the strong limit. Our results imply that it is the mass lost to neutrinos \textit{relative} to the mass enclosed by the shockwave, as well as the stellar density gradient where the shock forms, that primarily dictate its strength and the amount of material it ejects. These criteria explain why red supergiants, which have relative mass losses well in excess of the critical value at the time of shock formation, more readily eject material and create more luminous explosions compared to more compact progenitors.