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

A body of diameter $D$ has mass $m(D)=\pi \rho D^3/6$. Integrating the power-law size distribution therefore givesHenceWhen $D_{\max }\gg D_{\min }$ and $3<\alpha <4$, the belt mass is dominated by its largest bodies and $K\simeq 6M(4-\alpha )/(\pi \rho D_{\max }^{4-\alpha })$.