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For spherical fragments of common mass density $\rho$, with $n(D)=K{D}^{-\alpha }$ between specified diameter cutoffs, mass normalization of a power-law size distribution and integration of the geometric cross-section giveIndeed $m=(\pi \rho K/6)\int D^{3-\alpha }dD$ and ${\sigma }_{\mathrm{tot}}=(\pi K/4)\int D^{2-\alpha }dD$; eliminating $K$ proves the formula. If both cutoff ratios raised to $\alpha -3$ and $4-\alpha$ are small, respectively, this reduces to $3m(4-\alpha )D_{\min }^{3-\alpha }{D}_{\max }^{\alpha -4}/[2\rho (\alpha -3)]$. Approaching either endpoint requires the corresponding logarithmic integral.

Assume spherical fragments of the common mass density $\rho$, and let $n(D)=K{D}^{-\alpha }$ count fragments per unit diameter. Mass normalization of a power-law size distribution givesTheir geometric cross-sections are $\pi D^2/4$, hence the cross-section of a power-law fragment population is exactlyFor $3<\alpha <4$, large fragments dominate the mass integral and small fragments dominate the geometric cross-section integral. Neglecting the subdominant endpoints givesThe two actual error parameters are $(D_{\min }/D_{\max }{)}^{\alpha -3}$ and $(D_{\min }/D_{\max }{)}^{4-\alpha }$. They must both be small; the approximation is not uniform as $\alpha$ approaches either endpoint. At $\alpha =3$ or $4$ the corresponding integral is logarithmic, so the endpoint-dominated formula must be replaced by that integral.

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.