AN ANALYTIC MODEL OF THE CORE OF A SUPERNOVA

We have formulated the problem of the interior of a spherically symmetric star that is collapsing and radiating photons or neutrinos to infinity. The fully relativistic equations are used. This is a starting point for a complete description of a supernova. Boundary conditions are derived by the requirement that the interior metric and its normal derivative match continuously across the boundary to a metric describing the exterior. The metric used for the exterior is the one first derived by Vaidya. The boundary conditions impose severe constraints upon any solution of the equations of motion. We have analytically solved the equations of motion. Various parameters that arise due to initial conditions are fixed by requiring that the pressure versus density curve at the center be as close as possible to a known equation of state. We used the one obtained by Baym, Pethick, and Sutherland for this purpose. We obtained analytic solutions both for the motion of the star and its luminosity as functions of time. To obtain the analytic solution we linearized the two equations derived from the boundary conditions at the edge of the star. This approximation is accurate to one part in 10(4) if the luminosity is less than 10(52) ergs s-1. We also show that the boundary conditions introduce a ''runaway'' solution, i.e. one that increases exponentially with time. We excluded this solution by adjusting the initial conditions.