Energy-conserving Relativistic Corrections to Strong-shock Propagation

Abstract

Astrophysical explosions are accompanied by the propagation of a shock wave through an ambient medium. Depending on the mass and energy involved in the explosion, the shock velocity V can be non-relativistic (V ≪ c, where c is the speed of light), ultra-relativistic (V ≃ c), or moderately relativistic (V ∼ f ew × 0.1c). While self-similar, energy-conserving solutions to the fluid equations that describe the shock propagation are known in the non-relativistic (the Sedov-Taylor blastwave) and ultra-relativistic (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 O(V 2 /c 2 ), energy-conserving corrections to the non-relativistic, 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 sub-relativistic velocities in gamma-ray bursts, and other high-energy phenomena.All of these explosion scenarios involve the formation and expansion of a shock wave into its surroundings, and this shock leaves in its wake a "sea" of post-shock fluid (as, of course, do explosion scenarios not initiated by the collapse of a massive star, such as compact object mergers; e.g., Li & Paczyński 1998;Levinson et al. 2002;Nakar & Piran 2011;Abbott et al. 2017). One of the most useful techniques for describing the spatial and temporal evolution of the post-shock gas and of the shock itself is self-similarity. This mathematical technique exploits the scale invariance of the fluid equations and, in the absence of any temporal or spatial scales of the ambient medium, the necessary scale invariance of the solutions to those equations (e.g., Ostriker & McKee 1988).Among the best-known examples of a self-similar solution to the fluid equations is the Sedov-Taylor (ST) blastwave (Sedov 1959;Taylor 1950). The ST blastwave describes the propagation of an energy-conserving, strong (Mach number much greater than one) shock into an ambient medium that possesses a power-law density profile. The conservation of energy implies that there is a unique shock speed V that can be directly related to the initial energy of the explosion, the impulsive injection of which initiated the e...

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