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For spherical planetesimals with mass density $\rho$ and total-number distribution $n(D)=K{D}^{-\alpha }$ between $D_{\min }$ and $D_{\max }$,Thus $K=6M_{\mathrm{tot}}(4-\alpha )/[\pi \rho (D_{\max }^{4-\alpha }-D_{\min }^{4-\alpha })]$ for $\alpha \ne 4$. At $\alpha =4$, use $K=6M_{\mathrm{tot}}/[\pi \rho \log (D_{\max }/D_{\min })]$. Divide $n(D)$ by belt volume to obtain the number-density distribution used in a planetesimal collision rate.

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.

The impactors' number density per diameter is $K{D}_{\mathrm{im}}^{-\alpha }/V$. With gravitational focusing neglected, the geometric collision cross-section is $\pi (D+D_{\mathrm{im}}{)}^2/4$. The planetesimal collision rate is consequentlywhere