A size-independent $Q_D^*$ gives the steady <Dohnanyi collisional cascade> exponent $\alpha=7/2$. The preceding results become
$$
t_{\rm cr}\simeq\frac{0.4}{R_{\rm cc}},
\qquad
t_{\rm cat}\simeq\frac{2(6.22)}{9.94}\frac1{R_{\rm cc}}
\simeq\frac{1.25}{R_{\rm cc}}.
$$
Thus, under these idealized prescriptions, <cratering collisions> remove target mass about three times faster than catastrophic collisions. The conclusion concerns mass removed from targets; both processes contribute fragments to the <collisional cascade>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 316 / 1 / vii / Solution

If $Q_D^*\propto D^b$, part (b) gives
$$
R_{\rm cc}(D)\propto
D^{3-\alpha+b(1-\alpha)/3}.
$$
The mass in one logarithmic size interval scales as $D^3[D n(D)]\propto D^{4-\alpha}$. A steady <collisional cascade> requires the mass destroyed per unit time, and hence the mass flux through every logarithmic interval, to be independent of $D$. Its exponent is therefore
$$
4-\alpha+3-\alpha+\frac{b(1-\alpha)}3=0.
$$
Solving gives
$$
\boxed{\alpha=\frac{21+b}{6+b}}.
$$

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 316 / 3 / d / Solution

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A collisional cascade transfers mass from large bodies to progressively smaller fragments. A steady cascade has a size distribution whose mass flux through logarithmic size bins is independent of size.


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