= Rubblising-to-dispersal collision-rate ratio
{title2=$R_{\rm rub}/R_{\rm cc}=(Q_D^*/Q_s)^{(\alpha-1)/3}$}
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 $Q_s=Q_aD^{-a}$ 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 $R_{
m cc}$ 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.
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