Raining rocks: an analytical formulation for collision time-scales in planetary systems

The dynamical interaction of minor bodies (such as comets or asteroids) with planets plays an essential role in the planetary system's architecture and evolution. As a result of these interactions, structures such as the Kuiper belt and the Oort cloud can be created. In particular, the collision of minor bodies with planets can drastically change the planet's internal and orbital evolution. We present an analytical formulation to determine the collision time-scale for a minor body to impact a planet for arbitrary geometry. By comparing with a suite of detailed N-body simulations and an analytical method for collision time-scales in the Solar system, we confirmed the accuracy of our analytical formulation. As a proof of concept, we focused on the collision time-scales of minor bodies similar to the Jupiter-family comets and the long-period comets with a Jupiter-like planet. We show that our analytical method yields in good agreement with the numerical simulations. The formalism presented here thus provides a succinct and accurate alternative to numerical calculations.