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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.

Exclude collisions from the ejection count. The effective collision width is $2R_p$, while the favorable noncolliding ejection width is $(b_{\mathrm{ej}}-R_p{)}_{+}$. Define the first flyby-ejection threshold and the equal-rate crossover byThere are no direct noncolliding ejections in this weak-deflection model for $Q\le Q_{\mathrm{hit}}$. Above that threshold, comparison of the two rates gives collision is more likely for $Q<Q_{\mathrm{crit}}$, while ejection is more likely for $Q>Q_{\mathrm{crit}}$. In the deep-crossing limit $Q_{\mathrm{crit}}\simeq [9/(2\sqrt{2})](M_{\star }/M_p)R_p$, so its scaling at fixed density is $M_{\star }{M}_p^{-2/3}{\rho }_p^{-1/3}$. Order-one factors depend on encounter geometry; the robust condition compares $(M_p/M_{\star })Q$ with $R_p$.

Adding the mutually exclusive rare-event loss rates gives the collision-ejection lifetime of a comet:For $Q\le Q_{\mathrm{hit}}$ this is $t_{\mathrm{col}}\propto Q^{3/2}$. For $Q\gg Q_{\mathrm{hit}}$ it approaches $t_{\mathrm{ej}}\propto Q^{1/2}$. A logarithmic sketch therefore bends from slope $3/2$ to slope $1/2$, with no maximum. At the equal-rate crossover the lifetime is half either individual loss time.

These asymptotic branches use the corrected residence density from part iii and require $Q\gg a_p$ and a local encounter scale below the Hill radius. They cannot be extrapolated indefinitely to arbitrarily distant weakly bound orbits.