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If a spherical parent and its fragments have the same mass density, substitute $m=\pi \rho D_c^3/6$ into the endpoint-dominated cross-section of a power-law fragment population. The diameter obeysThis inversion requires specified fragment cutoffs. If $D_{\max }=\eta D_c$ instead, solving that dependence changes the power of $D_c$ to $\alpha -1$. It is useful for estimating the parent comet needed to supply an observed debris disk, subject to the assumed fragment model.

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.