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Observations and theory suggest that core-collapse supernovae can span arange of explosion energies, and when sub-energetic, the shockwave initiatingthe explosion can decelerate to speeds comparable to the escape speed of theprogenitor. In these cases, gravity will complicate the explosion hydrodynamicsand conceivably cause the shock to stall at large radii within the progenitorstar. To understand these unique properties of weak explosions, we develop aperturbative approach for modeling the propagation of an initially strong shockinto a time-steady, infalling medium in the gravitational field of a compactobject. This method writes the shock position and the post-shock velocity,density, and pressure as series solutions in the (time-dependent) ratio of thefreefall speed to the shock speed, and predicts that the shock stalls withinthe progenitor if the explosion energy is below a critical value. We show thatour model agrees very well with hydrodynamic simulations, and accuratelypredicts (e.g.) the time-dependent shock position and velocity and the radiusat which the shock stalls. Our results have implications for black holeformation and the newly detected class of fast X-ray transients (FXTs). Inparticular, we propose that a ``phantom shock breakout'' -- where the outeredge of the star falls through a stalled shock -- can yield a burst of X-rayswithout a subsequent optical/UV signature, similar to FXTs. This model predictsthe rise time of the X-ray burst, $t_{\rm d}$, and the mean photon energy,$kT$, are anti-correlated, approximately as $T \propto t_{\rm d}^{-5/8}$.

Low-energy Explosions in a Gravitational Field: Implications for Sub-energetic Supernovae and Fast X-ray Transients