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 methodologies of 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, allowing these results to be readily used to calculate the atmospheric mass loss from planetary cores made of different materials. We demonstrate that, as a planetary shock converges to the self similar solution, it loses information about the collision that created it, including the impact angle for oblique collisions.
Highlights
The formation of terrestrial planets is thought to proceed in two phases
One of the interesting features of this map is that, for small impactors, the offset has very little influence on the outcome. This is because small enough impactors deposit all their energy in the target regardless of obliquity. This trend is in agreement with the findings of Yalinewich and Schlichting [15], who showed that, when the shock radius is considerably larger than the size of the impactor, the shock wave tends to the self similar solution found in the previous section, and in the process loses some of the information about the initial conditions
Planet formation involves a phase of giant collisions, when planets and planetary cores smash into each other
Summary
The formation of terrestrial planets is thought to proceed in two phases. In the first phase, rocky cores form by accretion in the protoplanetary disc [1], and, in the second stage, multiple planetary cores merge to form planets after the disc evaporates [2,3]. It is difficult to observe planetary collisions directly, as they are expected, in the most optimistic scenario, to produce a faint, short (few hours) and hard (X-ray and EUV) transient [8] Study of this phase is based on theoretical or computational models, and indirect observational evidence, such as isotope ratios on Earth [9] or debris rings around exoplanetary systems [10]. The shock waves in planetary collisions are Type II solutions. In such cases, conservation laws cannot be used and the motion is determined by the behaviour near a singularity [24].
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