We present a mathematical model for the propagation of the shock waves that occur during planetary collisions. Such collisions are thought to occur during the formation of terrestrial planets, and they have the potential to erode the planet's atmosphere. We show that under certain assumptions, this evolution of the shock wave can be determined using the method of self similar solutions. This self similar solution is of type II, which means that it only applies to a finite region behind the shock front. This region is bounded by the shock front and the sonic point. Energy and matter continuously flow through the sonic point, so that energy in the self similar region is not conserved, as is the case for type I solutions. Instead, the evolution of the shock wave is determined by boundary conditions at the shock front and at the sonic point. We show how the evolution can be determined for different equations of state, allowing these results to be readily used to calculate the atmospheric mass loss from planetary cores made of different materials.

Previous works developed an analytic model for the propagation of shock waves into atmospheres with a uniform density. In this work we generalized this formalism to account for graded density profiles. These waves can occur in a wide range of astrophysical events, such as collisions in planetary and stellar atmospheres, common envelope explosions and peculiar type Ia supernovae. The behaviour of the shock wave and its evolution can be modelled using type II self similar solutions. In such solutions the evolution of the shock wave is determined by boundary conditions at the shock front and a singular point in the shocked region. We show how the evolution can be determined for different equations of state and density profiles, and compare these results to numerical simulations. We also demonstrate how these results can be applied to a wide range of problems in astrophysics.

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