Cratering impactors satisfy $D_{\rm im}<X_cD$. For $X_c\ll1$, they also satisfy $D_{\rm im}\ll D$, so $R_{\rm col}\simeq AD^2D_{\rm im}^{-\alpha}$. The fractional mass-loss rate is then
$$
\frac{\dot m_{\rm cr}}m
\simeq\int_{D_{\min}}^{X_cD}
\frac12\left(\frac{D_{\rm im}}{X_cD}\right)^3
AD^2D_{\rm im}^{-\alpha}\,dD_{\rm im}
\simeq\frac{A}{2(4-\alpha)}X_c^{1-\alpha}D^{3-\alpha}.
$$
Using the <catastrophic planetesimal collision rate> found above,
$$
\boxed{\frac m{\dot m_{\rm cr}}
\simeq\frac{2(4-\alpha)}{\alpha-1}\frac1{R_{\rm cc}(D)}}.
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-316/1/v/solution