Shock Emergence in Supernovae: Limiting Cases and Accurate Approximations

TL;DR

This paper analyzes the dynamics of accelerating shocks in stratified atmospheres, providing accurate formulas for key parameters and exploring limiting cases with analytical solutions.

Contribution

It introduces precise fitting formulas for shock velocity scaling and post-shock acceleration in stratified atmospheres, extending understanding of shock behavior in different density profiles.

Findings

Derived accurate fitting formulas for shock velocity scaling

Provided analytical solutions for uniform atmospheres

Matched shock acceleration outcomes in steep density gradients

Abstract

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.