Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 316 / 2 / i / Solution

Assume spherical bodies of common mass density $\rho$, a well-mixed belt, and that $n(D)dD$ counts bodies in the entire belt rather than per unit volume. A body of diameter $D$ has mass $\pi \rho D^3/6$. The mass normalization of a power-law size distribution isConsequentlyThe thin-torus volume is circumference times radial width times full vertical height,The local number-density distribution needed for a planetesimal collision rate is $n(D)/V$. Thus belt geometry does not enter this total-number $K$ but does enter collision rates. This approximation treats the stated height as uniform, independent of size.