Universality in the run-up of shock waves to the surface of a star

We investigate the run-up of a shock wave from inside to the surface of a perfect fluid star in equilibrium and bounded by vacuum. Near the surface we approximate the fluid motion as plane-symmetric and the gravitational field as constant. We consider the "hot" equation of state $P=(\Gamma-1)\rho e$ and its "cold" (fixed entropy, barotropic) form $P=K_0\rho^\Gamma$ (the latter does not allow for shock heating). We find numerically that the evolution of generic initial data approaches universal similarity solutions sufficiently near the surface, and we construct these similarity solutions explicitly. The two equations of state show very different behaviour, because shock heating becomes the dominant effect when it is allowed. In the barotropic case, the fluid velocity behind the shock approaches a constant value, while the density behind the shock approaches a power law in space, as the shock approaches the surface. In the hot case with shock heating,the density jumps by a constant factor through the shock, while the sound speed and fluid velocity behind the shock diverge in a whiplash effect. We tabulate the similarity exponents as a function of the equation of state parameter $\Gamma$and the stratification index $n_*$.