( 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 \leq 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 .