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

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