For the <Dohnanyi collisional cascade> exponent $\alpha=7/2$ and $D_{\max}\gg D_{\min}$,
$$
C\simeq\frac{3M}{\pi\rho D_{\max}^{1/2}},
\qquad
X_c^{-5/2}=\left(\frac{2Q_D^*}{v_{\rm rel}^2}\right)^{-5/6}.
$$
Substitution in part (iv) yields
$$
\boxed{
N(\gt\sigma_{\rm tot})\simeq
\frac{3}{80}
\frac{M^2v_{\rm rel}\Delta t}
{\rho^2VD_{\max}D_f\sigma_{\rm tot}}
\left(\frac{2Q_D^*}{v_{\rm rel}^2}\right)^{-5/6}}.
$$
Thus
$$
N\propto
M^2\Delta t\,V^{-1}\rho^{-2}D_{\max}^{-1}D_f^{-1}
(Q_D^*)^{-5/6}v_{\rm rel}^{8/3}\sigma_{\rm tot}^{-1}.
$$
The $M^2/V$ dependence is the pair-collision scaling. Longer-lived clumps are more numerous, stronger bodies disrupt less often, and smaller fragments put more <geometric cross-section> into each event. At fixed total mass, increasing $D_{\max}$ lowers the normalization of the <power-law size distribution> and therefore lowers the event rate.

Solution

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

 Solution

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