= Solution
For size-independent $Q_D^*$, the total rate of destructive events whose target exceeds $D$ is
$$
\mathcal R(>D)=\int_D^{D_{\max}}n(D')R_{\rm cc}(D')\,dD'.
$$
Since the integrand is proportional to $D'^{3-2\alpha}$ and $\alpha>3$, the lower limit dominates. With $n(D)=kD^{-\alpha}$,
$$
\mathcal R(>D)\simeq
\frac{kA}{2\alpha-4}
v_{\rm rel}^{(1+2\alpha)/3}
(Q_D^*)^{(1-\alpha)/3}D^{4-2\alpha}.
$$
The mean interval between such events is therefore
$$
\boxed{t_m(>D)\simeq
\frac{2\alpha-4}{kA}
v_{\rm rel}^{-(1+2\alpha)/3}
(Q_D^*)^{(\alpha-1)/3}D^{2\alpha-4}}.
$$
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