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We derive a fully relativistic, self-similar solution to describe thepropagation of a shock along an exponentially decreasing atmosphere, in thelimit of very large Lorentz factor. We solve the problem in planar symmetry andcompute the acceleration of the shock in terms of the density gradient crossedduring its evolution. We apply our solution to the acceleration of shockswithin the atmosphere of a HyperNova, and show that velocities consistent withthe requirements of GRB models can be achieved with exponential atmospheresspanning a wide density range.Comment: ApJL in pres

A Self-similar Solution for the Propagation of a Relativistic Shock in an Exponential Atmosphere