Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 316 / 1 / v / Solution

Cratering impactors satisfy $D_{\mathrm{im}}<X_{c}D$. For $X_c\ll 1$, they also satisfy $D_{\mathrm{im}}\ll D$, so $R_{\mathrm{col}}\simeq A{D}^2{D}_{\mathrm{im}}^{-\alpha }$. The fractional mass-loss rate is thenUsing the catastrophic planetesimal collision rate found above,