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Let the belt contain $n(D)dD=C{D}^{-\alpha }dD$ bodies. Its total mass fixesFor equal mass density, an impactor of diameter $D$ supplies specific impact energyThus catastrophic disruption requires $D\ge D_{\mathrm{cc}}=X_c{D}_t$, whereThe impactor number density per diameter is $C{D}^{-\alpha }/V$. Neglecting gravitational focusing, the geometric collision cross-section is $\pi (D_t+D{)}^2/4$, so the exact catastrophic planetesimal collision rate in this model isEquivalently, with $A=\pi C{v}_{\mathrm{rel}}/(4V)$,If $D_{\min }\ll X_c{D}_t\ll D_t\ll D_{\max }$, the steep distribution makes impactors just above $X_c{D}_t$ dominate and their collision cross-section is approximately $\pi D_t^2/4$. HenceThis approximation also assumes a common $v_{\mathrm{rel}}$ and size-independent catastrophic disruption threshold $Q_D^{\ast }$.

For equal mass density, the specific impact energy isThus the smallest catastrophic impactor has $D_{\mathrm{im}}=X_{c}D$, whereIntegrating the collision rate givesFor $X_c\ll 1$, the last term is the dominant lower-limit contribution, so the catastrophic planetesimal collision rate is

Cratering impactors satisfy $D_{\mathrm{im}}<X_{c}D$. For $X_c\ll 1$, they also satisfy $D_{\mathrm{im}}\ll D$, so $R_{\mathrm{col}}\simeq A{D}^2{D}_{\mathrm{im}}^{-\alpha }$. The fractional mass-loss rate is thenUsing the catastrophic planetesimal collision rate found above,