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Observations and theory suggest that core-collapse supernovae can span a range of explosion energies, and when sub-energetic the shockwave initiating the explosion can decelerate to speeds comparable to the escape speed of the progenitor. In these cases, gravity will complicate the explosion hydrodynamics and conceivably cause the shock to stall at large radii within the progenitor star. To understand these unique properties of weak explosions, we develop a perturbative approach for modeling the propagation of an initially strong shock into a time-steady, infalling medium in the gravitational field of a compact object. This method writes the shock position and the post-shock velocity, density, and pressure as series solutions in the (time-dependent) ratio of the freefall speed to the shock speed, and predicts that the shock stalls within the progenitor if the explosion energy is below a critical value. We show that our model agrees very well with hydrodynamic simulations, and accurately predicts (for example) the time-dependent shock position and velocity and the radius at which the shock stalls. Our results have implications for black hole formation and the newly detected class of fast X-ray transients (FXTs). In particular, we propose that a "phantom shock breakout"—where the outer edge of the star falls through a stalled shock—can yield a burst of X-rays without a subsequent optical/UV signature, similar to FXTs. This model predicts the rise time of the X-ray burst, t d, and the mean photon energy, kT, are anticorrelated, approximately as T ∝ t d − 5 / 8 . [ABSTRACT FROM AUTHOR]

Astrophysical explosions that contain dense and ram-pressure-dominated ejecta evolve through an interaction phase, during which a forward shock (FS), contact discontinuity (CD), and reverse shock (RS) form and expand with time. We describe new self-similar solutions that apply to this phase and are most accurate in the limit that the ejecta density is large compared to the ambient density. These solutions predict that the FS, CD, and RS expand at different rates in time and not as single temporal power laws, are valid for explosions driven by steady winds and homologously expanding ejecta, and exist when the ambient density profile is a power law with a power-law index shallower than ∼3 (specifically when the FS does not accelerate). We find excellent agreement between the predictions of these solutions and hydrodynamical simulations, both for the temporal behavior of the discontinuities and for the variation of the fluid quantities. The self-similar solutions are applicable to a wide range of astrophysical phenomena and—although the details are described in future work—can be generalized to incorporate relativistic speeds with arbitrary Lorentz factors. We suggest that these solutions accurately interpolate between the initial “coasting” phase of the explosion and the later, energy-conserving phase (or, if the ejecta is homologous and the density profile is sufficiently steep, the self-similar phase described in R. A. Chevalier).

Some massive stars likely fail to produce core-collapse supernovae, but these failed supernovae (FSNe) can generate an electromagnetic outburst prior to the disappearance of the star, as the mass lost to neutrinos during the stellar core collapse results in the formation and breakout of a second shock. We show that, when the mass lost to neutrinos is sufficiently small, there are two self-similar solutions that describe the propagation of a weak shock into a hydrodynamically expanding envelope that simultaneously yield accretion onto the black hole. The larger Mach number solution is unstable and yields the minimum Mach number that a shock must have to strengthen into the energy-conserving regime. Above a critical mass loss, there are no weak-shock solutions, implying that there are only strong explosions if the neutrino mass loss is above a critical value, and this value is a few percent of the mass of the star (and is physically achievable) for typical parameters. Our results imply that the fate of the explosion from an FSN—weak with little to no mass ejection or strong with the expulsion of the majority of the envelope—is a sensitive function of the stellar properties and the neutrino mass loss. We also show that there is a second type of self-similar solution for the shock that results in the settling of the gas near the compact object, which may be applicable to nonterminal stellar eruptions and the response of a gaseous disk to gravitational-wave induced mass loss from a binary black hole merger.