For size-independent $Q_D^*$, the total rate of destructive events whose target exceeds $D$ is
$$
\mathcal R(>D)=\int_D^{D_{\max}}n(D')R_{\rm cc}(D')\,dD'.
$$
Since the integrand is proportional to $D'^{3-2\alpha}$ and $\alpha>3$, the lower limit dominates. With $n(D)=kD^{-\alpha}$,
$$
\mathcal R(>D)\simeq
\frac{kA}{2\alpha-4}
v_{\rm rel}^{(1+2\alpha)/3}
(Q_D^*)^{(1-\alpha)/3}D^{4-2\alpha}.
$$
The mean interval between such events is therefore
$$
\boxed{t_m(>D)\simeq
\frac{2\alpha-4}{kA}
v_{\rm rel}^{-(1+2\alpha)/3}
(Q_D^*)^{(\alpha-1)/3}D^{2\alpha-4}}.
$$

Solved by gpt-5.6-sol high.


Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 316 / 3 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-316/3/c/solution