  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 like 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 analytic formulation to determine the collision
timescale 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 timescales in the solar system, we confirmed the accuracy of our
analytic formulation. As a proof of concept, we focused on the collision rate
of minor bodies randomly distributed around a Jupiter-like planet, emulating a
Kuiper belt-like disk. 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.
