Composite self-similar solutions for relativistic shocks: the transition to cold fluid temperatures

TL;DR

This paper develops a new self-similar analytical solution describing the transition of relativistic shock-driven flows from relativistic to subrelativistic temperatures during stellar envelope breakout, supported by numerical validation.

Contribution

It introduces a novel self-similar solution for post-shock breakout flows with cooling from relativistic to subrelativistic temperatures, extending previous models.

Findings

Numerical results agree with the new self-similar solution.

Corrections for spherical geometry are significant only near the stellar surface.

The solution relates terminal Lorentz factors to post-shock Lorentz factors.

Abstract

The flow resulting from a strong ultrarelativistic shock moving through a stellar envelope with a polytrope-like density profile has been studied analytically and numerically at early times while the fluid temperature is relativistic--that is, just before and just after the shock breaks out of the star. Such a flow should expand and accelerate as its internal energy is converted to bulk kinetic energy; at late enough times, the assumption of relativistic temperatures becomes invalid. Here we present a new self-similar solution for the post-breakout flow when the accelerating fluid has bulk kinetic Lorentz factors much larger than unity but is cooling through $p/n$ of order unity to subrelativistic temperatures. This solution gives a relation between a fluid element's terminal Lorentz factor and that element's Lorentz factor just after it is shocked. Our numerical integrations agree well with the solution. While our solution assumes a planar flow, we show that corrections due to spherical geometry are important only for extremely fast ejecta originating in a region very close to the stellar surface. This region grows if the shock becomes relativistic deeper in the star.