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

For the Dohnanyi collisional cascade exponent $\alpha =7/2$ and $D_{\max }\gg D_{\min }$,Substitution in part (iv) yieldsThusThe $M^2/V$ 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.