Self-similarity of the third type in ultra relativistic blastwaves

A new type of self-similarity is found in the problem of a plane-parallel, ultra-relativistic blast wave, propagating in a powerlaw density profile of the form $\rho \propto z^{-k}$. Self-similar solutions of the first kind can be found for $k<7/4$ using dimensional considerations. For steeper density gradients with $k>2$, second type solutions are obtained by eliminating a singularity from the equations. However, for intermediate powerlaw indices $7/4<k<2$ the flow does not obey any of the known types of self-similarity. Instead, the solutions belong to a new class in which the self-similar dynamics are dictated by the non self-similar part of the flow. We obtain an exact solution to the ultra-relativistic fluid equations and find that the non self-similar flow is described by relativistic expansion into vacuum, composed of (1) an accelerating piston that contains most of the energy and (2) a leading edge of fast material that coincides with the interiors of the blastwave and terminates at the shock. The dynamics of the piston itself are self-similar and universal, and do not depend on the external medium. The exact solution of the non self-similar flow is used to solve for the shock in the new class of solutions.