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

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.