A self-similar solution for the propagation of a relativistic shock in an exponential atmosphere

We derive a fully relativistic, self-similar solution to describe the propagation of a shock along an exponentially decreasing atmosphere, in the limit of a very large Lorentz factor. We solve the problem in planar symmetry and compute the acceleration of the shock in terms of the density gradient crossed during its evolution. We apply our solution to the acceleration of shocks within the atmosphere of a hypernova and show that velocities consistent with the requirements of gamma-ray burst models can be achieved with exponential atmospheres spanning a wide density range.