The Expulsion of Stellar Envelopes in Core‐Collapse Supernovae

Abstract

We examine the relation between presupernova stellar structure and the distribution of ejecta in core-collapse supernovae of Types Ib, Ic, and II, under the approximations of adiabatic, spherically symmetric Ñow. We develop a simple yet accurate analytical formula for the velocity of the initial forward shock that traverses the stellar envelope. For material that does not later experience a strong reverse shock, the entropy deposited by this forward shock persists into the Ðnal, freely expanding state. We demonstrate that the Ðnal density distribution can be approximated with simple models for the Ðnal pressure distribution , in a way that matches the results of simulations. Our results indicate that the distribution of density and radiation pressure in a starÏs ejecta depends on whether the outer envelope is radiative or convective, and if convective, on the composition structure of the star. Our models are most accurate for the high-velocity ejecta cast away from the periphery of a star. For stellar structures that limit to a common form in this region, the resulting ejecta limit to a common distribution at high velocities because the blast wave forgets its history as it approaches the stellar surface. We present formulae for the Ðnal density distribution of this material as a function of mass, for both radiative and efficiently convective envelopes. These formulae limit to the well-known planar, self-similar solutions for mass shells approaching the stellar surface. However, the assumption of adiabatic Ñow breaks down for shells of low optical depth, so this planar limit need not be attained. The event of shock emergence, which limits adiabatic Ñow, also produces a soft X-ray burst of radiation. Formulae are given for the observable properties of this burst and their dependence on the parameters of the explosion. Motivated by the relativistic expansion recently inferred by Kulkarni et al. for the synchrotron shell around SN 1998bw, we estimate the criterion for relativistic mass ejection and the rest mass of relativistic ejecta. We base our models for the entire ejecta distribution on the high-velocity solution, on our shock-velocity formula, and on realistic radiation pressure distributions. We also present simpler, but less Ñexible, analytical approximations for ejecta distributions. We survey the ejecta of the polytropic hydrogen envelopes of red super-giants. Our models will be useful for studies of the light curves and circumstellar or interstellar interactions of core-collapse supernovae, and of the birth of pulsar nebulae in their ejecta.