Relativistic Flows after Shock Emergence

Abstract

We investigate relativistic flows that occur after a shock wave generated inside a star reaches the surface. First, the effect of sphericity is included through successive approximations by adding correction terms to an already known self-similar ultrarelativistic solution for plane-parallel geometry. The inclusion of sphericity increases the early acceleration compared with the original plane-parallel flow. Second, we obtain semianalytic solutions for a mildly relativistic flow in which the rest-mass energy density is nonnegligible in the equation of state (EOS). To take this into account, we use enthalpy and pressure instead of density and pressure as thermodynamic variables. These solutions assume self-similar evolution except for the initial conditions. Third, we carry out numerical calculations with a special relativistic hydrocode based on Godunov's method to check the applicability of the sphericity corrections and semianalytic solutions. The EOS used in our calculations includes the rest-mass energy density. Comparisons with the numerical calculations support the validity of the sphericity corrections. The evolution of the pressure and Lorentz factor of each fluid element in the semianalytic solution for mildly relativistic flows matches the numerical results, at least for early phases. We also numerically investigate the final free-expansion phase. For spherically symmetric or plane-parallel ultrarelativistic flows, we could not observe this phase even when the self-similar variables grew to 106. However, flows in which the ratio of pressure to density at shock breakout is less than unity did reach free expansion. We derive the final energy distributions for these flows and compare with previous work.