Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 315 / 1 / b / Solution

At depths satisfying $\gamma c\tau \gg 1$, the exponential term is negligible. The atmosphere remains approximately isothermal while its intrinsic contribution is also small:Thus the plateau occupies approximatelyand has temperature

For example, take $T_{\mathrm{irr}}=2000\mathrm{K}$, $T_{\mathrm{int}}=200\mathrm{K}$, $\gamma =1$, $f=1/2$, and the Eddington constants. Then $T_{\mathrm{iso}}\simeq 1650\mathrm{K}$ and the formal plateau extends from $\tau$ of order unity to several thousand. At still greater depth,so the radiative solution rises as $T\propto {\tau }^{1/4}$ until convection replaces it.

Solved by gpt-5.6-sol high.