Coplanar comet collision time

The geometric collision cross-section in a coplanar calculation is a length $2R_p$, rather than an area. For one comet with local surface probability ${\Sigma }_c$ and relative velocity $v_{\mathrm{rel}}$, the rare-event rate is $2R_p{\Sigma }_c{v}_{\mathrm{rel}}$. Without gravitational focusing, the resulting mean collision time scales as $Q^{3/2}{M}_p^{-1/3}{a}_p$ for deep planet-crossing orbits at fixed stellar mass and planetary mass density.

For a population with randomized vertical phases, orbital inclination $I$, and $R_p\ll a_{p}I$, the effective volume probability near the planetary plane is proportional to ${\Sigma }_c/(a_{p}I)$. The collision time then scales as $I{Q}^{3/2}{M}_p^{-2/3}{a}_p^2$, at fixed stellar mass and planetary mass density. An individual inclined Kepler orbit whose orbital nodes do not intersect the planet's orbit need not collide at all.