Nonlinear perturbations and weak shock waves in isentropic atmospheres

Acoustic perturbations to stellar envelopes can lead to the formation of weak shock waves via nonlinear wave-steepening. Close to the stellar surface, the weak shock wave increases in strength and can potentially lead to the expulsion of part of the stellar envelope. While accurate analytic solutions to the fluid equations exist in the limits of low amplitude waves or strong shocks, connecting these phases generally requires simulations. We address this problem using the fact that the plane parallel Euler equations, in the presence of a constant gravitational field, admit exact Riemann invariants when the flow is isentropic. We obtain exact solutions for acoustic perturbations and show that after they steepen into shock waves, Whitham's approximation can be used to solve for the shock's dynamics in the weak to moderately strong regimes, using a simple ordinary differential equation. Numerical simulations show that our analytic shock approximation is accurate up to moderate ($\sim$ few--15) Mach numbers, where the accuracy increases with the adiabatic index.