Self-similar ultra-relativistic jetted blast wave

TL;DR

This paper derives a universal self-similar solution for ultra-relativistic jetted blast waves, revealing a three-region structure and suggesting it acts as an attractor for directed relativistic explosions like gamma-ray bursts.

Contribution

The paper introduces a new self-similar model for ultra-relativistic jetted blast waves, extending understanding beyond spherical symmetry and proposing it as an attractor solution.

Findings

The solution features three distinct regions: head, envelope, and axial core.

Most energy (~80%) resides outside the head region.

The solution approximately matches previous simulations, indicating universality.

Abstract

Following a suggestion that a directed relativistic explosion may have a universal intermediate asymptotic, we derive a self-similar solution for an ultra-relativistic jetted blast wave. The solution involves three distinct regions: an approximately paraboloid head where the Lorentz factor $\gamma$ exceeds $\sim1/2$ of its maximal, nose value; a geometrically self-similar, expanding envelope slightly narrower than a paraboloid; and an axial core in which the (cylindrically, henceforth) radial flow $u$ converges inward towards the axis. Most ($\sim 80\%$) of the energy lies well beyond the leading, head region. Here, a radial cross section shows a maximal $\gamma$ (separating the core and the envelope), a sign reversal in $u$, and a minimal $\gamma$, at respectively $\sim 1/6$, $\sim1/4$, and $\sim3/4$ of the shock radius. The solution is apparently unique, and approximately agrees with previous simulations, of different initial conditions, that resolved the head. This suggests that unlike a spherical relativistic blast wave, our solution is an attractor, and may thus describe directed blast waves such as in the external shock phase of a $\gamma$-ray burst.