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

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- Solution c

Solution

Differentiating the semi-grey irradiated atmosphere profile gives

4 T^{3} \frac{d T}{d τ} = a [T_{int}^{4} + f T_{irr}^{4} (1 - γ^{2}) e^{- γ c τ}] .

Hydrostatic balance gives

d τ / d P = κ_{IR} / g

when the thermal mean opacity is locally constant. Hence

\frac{d T}{d P} = \frac{a κ _{IR}}{4 g T ^{3}} [T_{int}^{4} + f T_{irr}^{4} (1 - γ^{2}) e^{- γ c τ}] .

For variable opacity, replace

κ_{IR}

by its local value and integrate

d τ = κ_{IR} (P, T) d P / g

.

Solved by gpt-5.6-sol high.

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