A coplanar target sweeps a strip of width $2R_p$ through the <comet>'s local position probability. Hence the relevant <geometric collision cross-section> is a length, and the rare collision rate per <comet> is $\Gamma_{\rm col}=2R_p\Sigma_c(a_p)v_{\rm rel}$.

Using the corrected <phase-mixed comet surface density> and the notation $s=q/a_p$, $\gamma=\sqrt{3-2\sqrt{2s}}$, the <coplanar comet collision time> is
$$
\boxed{t_{\rm col}\simeq\frac{\pi^2Q^{3/2}a_p\sqrt{1-s}}{2R_p\gamma\sqrt{GM_\star}}.}
$$
For $q\ll a_p$, this becomes $\pi^2Q^{3/2}a_p/(2\sqrt3R_p\sqrt{GM_\star})$, so $t_{\rm col}\propto Q^{3/2}M_p^{-1/3}a_p$ at fixed stellar mass and planetary <mass density>. This is the mean waiting time for one orbiting <comet>; the mean interval between impacts from $N_c$ such independent <comets> is $t_{\rm col}/N_c$. It assumes rare encounters and orbital <phase mixing>, not a <Poisson process> derived from a single deterministic trajectory.

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 316 / 1 / v / Solution

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> $v_{\rm rel}$, the rare-event rate is $2R_p\Sigma_c v_{\rm rel}$. 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>.

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