Shock Emergence in Supernovae: Limiting Cases and Accurate Approximations

We examine the dynamics of accelerating normal shocks in stratified planar atmospheres, providing accurate fitting formulae for the scaling index relating shock velocity to the initial density and for the post-shock acceleration factor as functions of the polytropic and adiabatic indices which parameterize the problem. In the limit of a uniform initial atmosphere there are analytical formulae for these quantities. In the opposite limit of a very steep density gradient the solutions match the outcome of shock acceleration in exponential atmospheres.