 Solution

Let the shattering threshold be $Q_s=Q_a{D}^{-a}$ and the full catastrophic disruption threshold be $Q_D^{\ast }$. 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 thereforeExcluding the final catastrophic event, the number of rubblising collisions is approximatelyFor the expression printed in the paper, if the gravity-regime term $Q_b{D}^{-b}$ dominates, this becomesThis 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.