= Solution
Let the shattering threshold be $Q_s=Q_aD^{-a}$ and the full <catastrophic disruption threshold> be $Q_D^*$. From part (b), the rate of impacts exceeding an energy threshold $Q$ is proportional to $Q^{(1-\alpha)/3}$. The expected number of shattering impacts during one catastrophic-disruption waiting time is therefore
$$
\frac{R_s}{R_{\rm cc}}
=\left(\frac{Q_D^*}{Q_s}\right)^{(\alpha-1)/3}.
$$
Excluding the final catastrophic event, the number of <rubblising collisions> is approximately
$$
\boxed{N_{\rm rub}\simeq
\left(\frac{Q_D^*}{Q_aD^{-a}}\right)^{(\alpha-1)/3}-1}.
$$
For the expression printed in the paper, if the gravity-regime term $Q_bD^{-b}$ dominates, this becomes
$$
\boxed{N_{\rm rub}\simeq
\left(\frac{Q_b}{Q_a}D^{a-b}\right)^{(\alpha-1)/3}-1}.
$$
This estimate assumes independent impacts drawn from the same projectile distribution and ignores structural evolution after each rubblising collision.
Solved by gpt-5.6-sol high.
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