The Non Relativistic Interiors of Ultra-Relativistic Explosions: Extension to the Blandford McKee Solutions

TL;DR

This paper develops a new self-similar solution extending the Blandford-McKee model to describe the interior, mildly relativistic to Newtonian regions of ultrarelativistic blast waves, improving understanding of their internal structure.

Contribution

It introduces a novel self-similar solution that accurately models the interior of ultrarelativistic blast waves beyond the Blandford-McKee approximation, including the formation of secondary shocks.

Findings

Velocity profile is independent of shock Lorentz factor.

Solution is valid down to R/Γ² behind the shock.

Secondary shock forms under certain density conditions.

Abstract

The hydrodynamics of an ultrarelativistic flow, enclosed by a strong shock wave, are described by the well known Blandford-McKee solutions in spherical geometry. These solutions, however, become inaccurate at a distance $\sim R/2$ behind the shock wave, where $R$ is the shock radius, as the flow approaches Newtonian velocities. In this work we find a new self-similar solution which is an extension to the Blandford-McKee solutions, and which describes the interior part of the blast wave, where the flow reaches mildly relativistic to Newtonian velocities. We find that the velocity profile of the internal part of the flow does not depend on the value of the shock Lorentz factor, $\Gamma$, and is accurate from $r=0$ down to a distance of $R/\Gamma^2$ behind the shock. Despite the fact that the shock wave is in causal contact with the entire flow behind it, a singular point appears in the equations. Nevertheless, the solution is not required to pass through the singular point: for ambient density that decreases slowly enough, $\rho \propto r^{-k}$ with $k<\frac{1}{2}(5-\sqrt{10})\cong0.92$, a secondary shock wave forms with an inflow towards the origin.