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

Write the power-law size distribution as $n(D)=k{D}^{-\alpha }$. The total geometric cross-section of the optically thin debris disk isBecause blackbody grains at radius $r$ intercept the fraction ${\sigma }_{\mathrm{tot}}/(4\pi r^2)$ of the stellar luminosity, the fractional luminosity of a debris disk givesAnother integral then yieldsfor $D_{\min }\ll D\ll D_{\max }$.

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For equal material densities, a projectile of diameter $D_{\mathrm{cc}}$ catastrophically disperses a target of diameter $D$ whenAssume $D_{\min }\ll D_{\mathrm{cc}}\ll D$, neglect gravitational focusing, and approximate the collision cross-section by $\pi D^2/4$. The collision rate per target isSubstitution giveswhere

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For size-independent $Q_D^{\ast }$, the total rate of destructive events whose target exceeds $D$ isSince the integrand is proportional to $D^{\prime 3-2\alpha }$ and $\alpha >3$, the lower limit dominates. With $n(D)=k{D}^{-\alpha }$,The mean interval between such events is therefore

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If $Q_D^{\ast }\propto D^b$, part (b) givesThe mass in one logarithmic size interval scales as $D^3[Dn(D)]\propto D^{4-\alpha }$. 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 thereforeSolving gives

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The total destructive-event rate above $D$ scales asThe steady-state relation from part (d) impliesso the exponent is exactly $-3$. ConsequentlyThe power $3$ is independent of the disruption-law index $b$: changing $b$ changes the equilibrium size-distribution index $\alpha$ in precisely the compensating way.

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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.

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