Rubblising-to-dispersal collision-rate ratio ($R_{\mathrm{rub}}/R_{\mathrm{cc}}=(Q_D^{\ast }/Q_s{)}^{(\alpha -1)/3}$)

For a common projectile power-law size distribution, the rate of impacts exceeding a threshold $Q$ is proportional to $Q^{(1-\alpha )/3}$. Taking the strength-only threshold $Q_s=Q_a{D}^{-a}$ rather than the full catastrophic disruption threshold gives the displayed ratio. Here the strength-shattering rate counts all impacts above $Q_s$, including dispersive ones; subtracting $R_{mcc}$ gives the bound-remnant-only rubblising collision rate. In a gravity-dominated cascade with $\alpha =3$, $a=1/2$, and $b=3/2$, it scales as $D^{4/3}$. Using the actual strength-gravity disruption transition minimum diameter gives coefficient $3^{-2/3}$ in the asymptotic rate ratio; treating that diameter as the equal-contribution crossover silently loses this factor.