Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 315 / 1 / a / Solution

Let ${\tau }_{\nu }$ increase downward and let $\mu >0$ be the outward direction cosine. Assume a plane-parallel, static, nonscattering atmosphere in local thermodynamic equilibrium, so the source function is $B_{\nu }[T(P)]$. Hydrostatic balance givesThe formal solution of the radiative transfer equation between $P_1>P_2$, corresponding to ${\tau }_1>{\tau }_2$, isFor an isothermal atmosphere, $B_{\nu }(T)$ is constant andIf the lower boundary is itself thermalized, $I_{\nu ,1}=B_{\nu }(T)$ and the upward intensity remains the blackbody value.