Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 316 / 2 / iv / Solution

A collision producing cross-section $\sigma$ has target diameterby part (ii). The production rate of clumps above ${\sigma }_{\mathrm{tot}}$ is the number of targets in each size interval times their per-target catastrophic rate. Therefore, with $D_0=\max (D_{\min },D_{{\sigma }_{\mathrm{tot}}})$,Using the small-impactor approximation from part (iii) givesWhen $D_{\min }\ll D_{{\sigma }_{\mathrm{tot}}}\ll D_{\max }$, this reduces toAssigning each event to its much larger target avoids double-counting collisions in this $X_c\ll 1$ approximation.