= Parent-body size from a fragment cross-section
{title2=$D_c$}
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 obeys
$$
D_c^3=\frac{4(\alpha-3)}{\pi(4-\alpha)}
\sigma_{\rm tot}D_{\min}^{\alpha-3}D_{\max}^{4-\alpha}.
$$
This 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.
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