Self-similar ultra-relativistic jetted blast wave

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 $γ$ 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 $γ$ (separating the core and the envelope), a sign reversal in $u$, and a minimal $γ$, 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 $γ$-ray burst.

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