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Some massive stars likely fail to produce core-collapse supernovae, but thesefailed supernovae (FSNe) can generate an electromagnetic outburst prior to thedisappearance of the star, as the mass lost to neutrinos during the stellarcore-collapse results in the formation and breakout of a second shock. We showthat when the mass lost to neutrinos is sufficiently small, there are twoself-similar solutions that describe the propagation of a weak shock into ahydrodynamically expanding envelope that simultaneously yield accretion ontothe black hole. The larger-Mach number solution is unstable and yields theminimum Mach number that a shock must have to strengthen into theenergy-conserving regime. Above a critical mass loss there are no weak-shocksolutions, implying that there are only strong explosions if the neutrino massloss is above a critical value, and this value is a few percent of the mass ofthe star (and is physically achievable) for typical parameters. Our resultsimply that the fate of the explosion from a FSN -- weak with little to no massejection or strong with the expulsion of the majority of the envelope -- is asensitive function of the stellar properties and the neutrino mass loss. Wealso show that there is a second type of self-similar solution for the shockthat results in the ``settling'' of the gas near the compact object, which maybe applicable to non-terminal stellar eruptions and the response of a gaseousdisc to gravitational-wave induced mass loss from a binary black hole merger.

The division between weak and strong explosions from failed supernovae