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

Solution

At depths satisfying

γ c τ ≫ 1

, the exponential term is negligible. The atmosphere remains approximately isothermal while its intrinsic contribution is also small:

T_{int}^{4} (b + τ) ≪ f T_{irr}^{4} (b + \frac{1}{γ c}) .

Thus the plateau occupies approximately

\frac{1}{γ c} ≪ τ ≪ f (\frac{T _{irr}}{T _{int}})^{4} (b + \frac{1}{γ c}) - b

T_{iso} ≃ T_{irr} [a f (b + \frac{1}{γ c})]^{1/4} .

For example, take

T_{irr} = 2000 K

T_{int} = 200 K

γ = 1

f = 1/2

, and the Eddington constants. Then

T_{iso} ≃ 1650 K

and the formal plateau extends from

τ

of order unity to several thousand. At still greater depth,

T^{4} ≃ a T_{int}^{4} τ + e x t co n s t an t,

so the radiative solution rises as

T \propto τ^{1/4}

until convection replaces it.

Solved by gpt-5.6-sol high.

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