Escape and accretion by cratering impacts: Formulation of scaling relations for high-speed ejecta

Numerous small bodies inevitably lead to cratering impacts on large planetary bodies during planet formation and evolution. As a consequence of these small impacts, a fraction of the target material escapes from the gravity of the large body, and a fraction of the impactor material accretes onto the target surface, depending on the impact velocities and angles. Here, we study the mass of the high-speed ejecta that escapes from the target gravity by cratering impacts when material strength is neglected. We perform a large number of cratering impact simulations onto a planar rocky target using the smoothed particle hydrodynamics method. We show that the escape mass of the target material obtained from our numerical simulations agrees with the prediction of a scaling law under a point-source assumption when $v_{\rm imp} \gtrsim 12 v_{\rm esc}$, where $v_{\rm imp}$ is the impact velocity and $v_{\rm esc}$ is the escape velocity of the target. However, we find that the point-source scaling law overestimates the escape mass up to a factor of $\sim 70$, depending on the impact angle, when $v_{\rm imp} \lesssim 12 v_{\rm esc}$. Using data obtained from numerical simulations, we derive a new scaling law for the escape mass of the target material for $v_{\rm imp} \lesssim 12 v_{\rm esc}$. We also derive a scaling law that predicts the accretion mass of the impactor material onto the target surface upon cratering impacts by numerically evaluating the escape mass of the impactor material. Our newly derived scaling laws are useful for predicting the escape mass of the target material and the accretion mass of the impactor material for a variety of cratering impacts that would occur on large planetary bodies during planet formation.