The definition of $X_c$ gives
$$
\frac Q{Q_D^*}=\left(\frac{D_{\rm im}}{X_cD}\right)^3.
$$
The <cratering collision> prescription therefore removes the target-mass fraction
$$
\boxed{1-f_{\rm lr}=\frac12\frac Q{Q_D^*}
=\frac12\left(\frac{D_{\rm im}}{X_cD}\right)^3}.
$$

Solution

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

 Solution

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