A collision producing cross-section $\sigma$ has target diameter
$$
\boxed{D_\sigma=\left(\frac{8D_f\sigma}{\pi}\right)^{1/3}}
$$
by part (ii). The production rate of clumps above $\sigma_{\rm tot}$ is the number of targets in each size interval times their per-target catastrophic rate. Therefore, with $D_0=\max(D_{\min},D_{\sigma_{\rm tot}})$,
$$
N(\gt\sigma_{\rm tot})
=\Delta t\int_{D_0}^{D_{\max}}CD_t^{-\alpha}R_{\rm cc}(D_t)\,dD_t.
$$
Using the small-impactor approximation from part (iii) gives
$$
\boxed{
N(\gt\sigma_{\rm tot})=
\frac{\pi C^2v_{\rm rel}\Delta t}
{4V(\alpha-1)(2\alpha-4)}
X_c^{1-\alpha}
\left(D_0^{4-2\alpha}-D_{\max}^{4-2\alpha}\right)}.
$$
When $D_{\min}\ll D_{\sigma_{\rm tot}}\ll D_{\max}$, this reduces to
$$
\boxed{
N(\gt\sigma_{\rm tot})\simeq
\frac{\pi C^2v_{\rm rel}\Delta t}
{4V(\alpha-1)(2\alpha-4)}
X_c^{1-\alpha}
\left(\frac{8D_f\sigma_{\rm tot}}{\pi}\right)^{(4-2\alpha)/3}}.
$$
Assigning each event to its much larger target avoids double-counting collisions in this $X_c\ll1$ approximation.


Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-316/2/iv/solution