Energy-conserving Relativistic Corrections to Strong-shock Propagation

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 × 0.1c). 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 { \mathcal O }({V}²/{c}²), 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.