Self-similar solutions for relativistic shocks emerging from stars with polytropic envelopes

We consider a strong ultrarelativistic shock moving through a star whose envelope has a polytrope-like density profile. When the shock is close to the star's outer boundary, its behavior follows the self-similar solution given by Sari for implosions in planar geometry. Here we outline this solution and find the asymptotic solution as the shock reaches the star's edge. We then show that the motion after the shock breaks out of the star is described by a self-similar solution remarkably like the solution for the motion inside the star. In particular, the characteristic Lorentz factor, pressure, and density vary with time according to the same power laws both before and after the shock breaks out of the star. After emergence from the star, however, the self-similar solution's characteristic position corresponds to a point behind the leading edge of the flow rather than at the shock front, and the relevant range of values for the similarity variable changes. Our numerical integrations agree well with the analytic results both before and after the shock reaches the star's edge.