Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 315 / 3 / a / ii / Solution

Where the spectral components vanish, $a=(R_p/R_{\ast }{)}^2$, soFor extinction cross-section $\sigma \propto {\lambda }^{\gamma }$, the scattering slope of a transmission spectrum obeysSince $y=(R_p/R_{\ast }{)}^2$,After subtracting the Gaussian feature, the model gives $dy/d\log \lambda =\gamma c(\lambda /{\lambda }_1{)}^{\gamma }$. At $\lambda ={\lambda }_1$,or, with $g=G{M}_p/R_p^2$, $T=\bar{m}G{M}_{p}c{R}_{\ast }^2/(2k_B{R}_p^3)$. This estimates the mean isothermal terminator temperature under hydrostatic conditions.

Solved by gpt-5.6-sol high.