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Let ${\tau }_1$ and ${\tau }_2$ be the top-down optical depths at $P_1$ and $P_2$, and let $\mu$ be the outward direction cosine. The formal solution of the radiative transfer equation givesFor a semi-infinite lower layer in local thermodynamic equilibrium, $I_{\nu ,0}=B_{\nu }(T_3)$.

DefineThe emergent planetary surface flux isHence the band-centre planet-star ratio isIf $T_1=T_2=T_3=T_p$, the weights telescope to $1/2$, so $F_{p,\nu }=\pi B_{\nu }(T_p)$ and the expression reduces to the blackbody thermal eclipse depth, about $72\mathrm{ppm}$ for $T_p=600\mathrm{K}$.

Solved by gpt-5.6-sol high.

Without scattering and in local thermodynamic equilibrium, the radiative transfer equation has the emergent solutionThe Eddington-Barbier relation gives the useful approximationA molecular band has larger opacity than its neighboring continuum and therefore reaches optical depth unity at lower pressure.

If temperature decreases outward, the band samples cooler gas and appears in absorption. If the atmosphere is isothermal, both levels have the same source function and the feature disappears. If an atmospheric thermal inversion makes the upper layer hotter, the band appears in emission. For a weak separation of formation pressures,which explicitly shows that feature sign and amplitude measure the vertical temperature gradient.

Solved by gpt-5.6-sol high.