An idealized spherical planet receives stellar irradiance 1300 W m⁻² on a surface normal to the rays. Its albedo is 0.25 and effective thermal emissivity is 0.90. Assume a single uniform temperature, no internal heating, and steady balance between absorbed stellar radiation and thermal emission; use σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴. Derive its equilibrium temperature and recalculate it if albedo decreases by 0.05 with everything else unchanged. Explain the factor of four in the model and one limitation for estimating a real planet's surface temperature.

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Initial temperature 262.9 K; with lower albedo 267.2 K, a rise of 4.276 K. A planet intercepts radiation across its projected disk πR², not its full spherical surface. Only the fraction 1 − a is absorbed; the rest is reflected according to the albedo. Thermal emission comes from surface area 4πR² at power εσT⁴ per unit area. Equating powers gives (1 − a)SπR² = 4πR²εσT⁴; the radius cancels. Thus T = [(1 − a)S/(4εσ)]^(1/4) = 262.9 K. The factor of four reflects the ratio of emitting area to intercepting area. With albedo reduced to 0.2, the absorbed fraction increases. Substitution gives T = 267.2 K, higher by 4.276 K. A real atmosphere absorbs and re-emits radiation at different altitudes, and surface temperature is not necessarily the effective emission temperature. Uneven heating, cloud feedback and spatially varying albedo also limit this deliberately simple global model.

Albedo and a planet's radiation balance · set 5

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