Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 315 / 4 / c

Solution

Let

F_{abs} = σ_{SB} T_{eq}^{4}

be the globally averaged absorbed stellar flux. The single-layer greenhouse model has an atmosphere transparent to incoming visible light and with infrared absorptivity and emissivity

α

. Treat the surface as a blackbody and neglect internal heating. The layer emits

α σ_{SB} T_{a}^{4}

both upward and downward.

Atmospheric radiative balance gives

α σ_{SB} T_{s}^{4} = 2 α σ_{SB} T_{a}^{4}, T_{a}^{4} = T_{s}^{4} /2 (α > 0) .

Surface radiative balance gives

σ_{SB} T_{s}^{4} = σ_{SB} T_{eq}^{4} + α σ_{SB} T_{a}^{4} .

Eliminating

T_{a}

yields

T_{s} = T_{eq} (1 - \frac{α}{2})^{- 1/4} .

The transparent limit has

T_{s} = T_{eq}

, and a fully infrared-opaque layer has

T_{s} = 2^{1/4} T_{eq}

. The model omits convection, wavelength-dependent opacity, and atmospheric absorption of starlight, so it captures the simplest greenhouse energy balance rather than a full surface-temperature prediction.

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