Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 315 / 2 / b / i

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- Solution i

Solution

Apply the formal solution of the radiative transfer equation successively to the two isothermal layers at normal incidence. In local thermodynamic equilibrium,

I_{ν, 2} = I_{ν, 0} e^{- (τ_{1} + τ_{2})} + B_{ν} (T_{1}) (1 - e^{- τ_{1}}) e^{- τ_{2}} + B_{ν} (T_{2}) (1 - e^{- τ_{2}}) .

For optically thin layers this becomes

I_{ν, 2} = I_{ν, 0} + τ_{1} [B_{ν} (T_{1}) - I_{ν, 0}] + τ_{2} [B_{ν} (T_{2}) - I_{ν, 0}] + O (τ^{2}) .

This assumes no scattering and negligible reflection between layers.

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