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

A useful first estimate places the radiative-convective boundary where the intrinsic term becomes comparable to the deep irradiation plateau:ThusWith the assumptions of part (b), $g=10\mathrm{m}{\mathrm{s}}^{-2}$ and ${\kappa }_{\mathrm{IR}}=10^{-3}{\mathrm{m}}^2\mathrm{k}{\mathrm{g}}^{-1}$, one finds ${\tau }_{\mathrm{RCB}}\sim 6.2\times 10^3$ and $P_{\mathrm{RCB}}\sim 6.2\times 10^7\mathrm{Pa}\simeq 620\mathrm{bar}$.

More precisely one equates the local radiative logarithmic gradient to the adiabatic gradient. A constant-opacity grey profile approaches ${\nabla }_{\mathrm{rad}}=1/4$, so a diatomic adiabat with ${\nabla }_{\mathrm{ad}}=2/7$ requires the realistic increase of opacity with depth to become convective. The numerical pressure is therefore an order-of-magnitude estimate, sensitive mainly to $T_{\mathrm{int}}$ and deep opacity.

Solved by gpt-5.6-sol high.