Relativistic Flows after Shock Emergence

We investigate relativistic flows after a shock wave generated in a star arrives at the surface. First, the sphericity effect is involved through a successive approximation procedure by adding correction terms to an already known self-similar solution in the ultra-relativistic limit assuming the plane parallel geometry. It is found that the lowest order correction terms are of the order of (a/R∗) √ 3−8) for a star with a radiative envelope, where a is the distance from the stellar surface at the moment of shock breakout and R∗ the stellar radius. We also find that the involvement of sphericity increases the acceleration in the early phase as compared with that of the original plane-parallel flow. Second, we obtain semi-analytic solutions for a mildly relativistic flow in which the rest mass energy density is not negligible in the equation of state. To take into account this, we use the enthalpy and the pressure instead of using the density and the pressure as thermodynamic variables. These solutions assume self-similar evolutions except for the initial conditions. Third, we have carried out numerical calculations with a special-relativistic hydrodynamical code based on the Godunov method in order to check the applicability of the above sphericity corrections and the semi-analytic solutions. The equation of state used in our calculations includes the rest mass energy density. Comparisons with results of numerical calculations support the validity of the sphericity correction terms. The evolutions of the pressure and the Lorentz factor of each fluid element of the semi-analytic solution for mildly relativistic flows match the numerical results at least in early phases. We also investigate the final free expansion phases by this code. For either spherical symmetric or plane-parallel ultra-relativistic flows, we could not observe the free expansion phase till the self-similar variables become as large as 10. On the contrary, the flow in which the ratio of the pressure to density at the shock breakout is less than unity reaches the free expansion. We have derived the final energy distributions for these flows and compare them with previous works. Subject headings: hydrodynamics—shock waves—relativity—self-similar