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The collision-ejection lifetime of a comet combines two ways to remove it from a specified orbital population: impact and gravitational ejection. If these can be approximated as constant competing Poisson process rates ${\lambda }_c$ and ${\lambda }_e$, the survival probability and mean lifetime areEquivalently, $1/\tau =1/{\tau }_c+1/{\tau }_e$. The probability that collision is the first removal is ${\lambda }_c/({\lambda }_c+{\lambda }_e)$.

Coplanar comet collision time and planetary ejection of a comet estimate the constituent processes under different encounter assumptions. Their reciprocal times should only be added when they refer to a consistent population and approximately stationary rates. Repeated planetary scattering can change the orbit and correlate encounters, invalidating a single exponential lifetime.

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 ${\Gamma }_{\mathrm{col}}=2R_p{\Sigma }_c(a_p)v_{\mathrm{rel}}$.

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 isFor $q\ll a_p$, this becomes ${\pi }^2{Q}^{3/2}{a}_p/(2\sqrt{3}{R}_p\sqrt{G{M}_{\star }})$, so $t_{\mathrm{col}}\propto Q^{3/2}{M}_p^{-1/3}{a}_p$ 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 $t_{\mathrm{col}}/N_c$. It assumes rare encounters and orbital phase mixing, not a Poisson process derived from a single deterministic trajectory.