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

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 $D_{\min }\ll D\ll D_{\max }$.

Solved by gpt-5.6-sol high.