Energy-conserving Relativistic Corrections to Strong-shock Propagation

Abstract

Astrophysical explosions are accompanied by the propagation of a shockwave through an ambient medium. Depending on the mass and energy involved in the explosion, the shock velocity V can be nonrelativistic ( V  ≪  c , where c is the speed of light), ultrarelativistic ( V  ≃  c ), or moderately relativistic ( V  ∼ few × 0.1 c ). While self-similar energy-conserving solutions to the fluid equations that describe the shock propagation are known in the nonrelativistic (the Sedov–Taylor blastwave) and ultrarelativistic (the Blandford–McKee blastwave) regimes, the finite speed of light violates scale invariance and self-similarity when the flow is only mildly relativistic. By treating relativistic terms as perturbations to the fluid equations, here we derive the , energy-conserving corrections to the nonrelativistic Sedov–Taylor solution for the propagation of a strong shock. We show that relativistic terms modify the post-shock fluid velocity, density, pressure, and the shock speed itself, the latter being constrained by global energy conservation. We derive these corrections for a range of post-shock adiabatic indices γ (which we set as a fixed number for the post-shock gas) and ambient power-law indices n , where the density of the ambient medium ρ a into which the shock advances declines with spherical radius r as ρ a  ∝  r − n . For Sedov–Taylor blastwaves that terminate in a contact discontinuity with diverging density, we find that there is no relativistic correction to the Sedov–Taylor solution that simultaneously satisfies the fluid equations and conserves energy. These solutions have implications for relativistic supernovae, the transition from ultra- to subrelativistic velocities in gamma-ray bursts, and other high-energy phenomena.