  Coughlin et al. (2018) (Paper I) derived and analyzed a new regime of
self-similarity that describes weak shocks (Mach number of order unity) in the
gravitational field of a point mass. These solutions are relevant to low energy
explosions, including failed supernovae. In this paper, we develop a formalism
for analyzing the stability of shocks to radial perturbations, and we
demonstrate that the self-similar solutions of Paper I are extremely weakly
unstable to such radial perturbations. Specifically, we show that perturbations
to the shock velocity and post-shock fluid quantities (the velocity, density,
and pressure) grow with time as $t^{\alpha}$, where $\alpha \le 0.12$, implying
that the ten-folding timescale of such perturbations is roughly ten orders of
magnitude in time. We confirm these predictions by performing high-resolution,
time-dependent numerical simulations. Using the same formalism, we also show
that the Sedov-Taylor blastwave is trivially stable to radial perturbations
provided that the self-similar, Sedov-Taylor solutions extend to the origin,
and we derive simple expressions for the perturbations to the post-shock
velocity, density, and pressure. Finally, we show that there is a third,
self-similar solution (in addition to the the solutions in Paper I and the
Sedov-Taylor solution) to the fluid equations that describes a rarefaction
wave, i.e., an outward-propagating sound wave of infinitesimal amplitude. We
interpret the stability of shock propagation in light of these three distinct
self-similar solutions.
