Self Similar Shocks in Atmospheric Mass Loss Due to Planetary Collisions

Abstract

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.

Lunar and Martian Silica

“…In the second stage, the shock begins to decelerate as it sweeps up more mass. As the shock expands, it loses information about the impactor's incidence angle, and converges to the same universal solution (Yalinewich & Remorov 2020). For this reason, we can just consider head on collisions.…”

“…Energy and momentum conservation provide limits on the value of this power law index (Zeldovich 1956) 1 > 𝛽 > 1/2. The value of 𝛽 depends on the equation of state of the soil (Housen et al 1983;Yalinewich & Remorov 2020), but is typically close to 2/3. As the shock excavates a crater basin, material is expelled out of the mouth of the basin.…”

The Propagation of Strong Shocks into Planetary and Stellar Atmospheres

“…As was done in a previous paper (Yalinewich & Remorov 2020), instead of considering a three dimensional problem, we consider the one-dimensional analogue called the impulsive piston problem (Adamskii 1956;Zel'dovich & Raizer 1967). In this scenario, a thin wafer hits a much thicker slab of material (with a certain density distribution) and both are perfectly cold prior to the collision.…”

“…It does so by reduce the hydrodynamic equations, which are partial differential equations, to ordinary differential equations. We also describe how these equation can be numerically integrated, and how the value of α can be obtained, as was done for the case ω = 0 in (Yalinewich & Remorov 2020).…”

“…Our model is a novel generalisation to the classical impulsive piston problem (Adamskii 1956;Yalinewich & Remorov 2020). In the classical problem, a thin wafer collides with a semi infinite space filled with a cold gas with a uniform density.…”