 Solution

Let $d$ be an impactor diameter and assume it is much smaller than target diameter $D$, with equal material mass density. Neglect gravitational focusing and take the encounter speed to be the stated target-size law $v_{\mathrm{rel}}=v_p{D}^p$. The specific impact energy isA catastrophic planetesimal collision requires $Q\ge Q_D^{\ast }=Q_b{D}^b$. Therefore the catastrophic projectile diameter isUse $\max (D_{\mathrm{cc}},D_{\min })$ as the actual lower size cutoff when integrating a rate. If this exceeds $D_{\max }$, the specified population contains no destructive impactors. The small-impactor approximation itself requires $D_{\mathrm{cc}}\ll D$.