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A planet of radius r and albedo alpha (the fraction of incident light it reflects) orbits a star at distance D. The star behaves as a blackbody of temperature T and radius R. Treating the planet as a blackbody radiator (the albedo arising from its atmosphere, oceans, etc.) in thermal equilibrium, what is its equilibrium (blackbody) temperature Tₚ?

1. Tₚ = T * sqrt(R/(2*D)) * (1 - alpha)^(1/4)
2. Tₚ = T * sqrt(R/D) * (1 - alpha)^(1/2)
3. Tₚ = T * (R/(2*D)) * (1 - alpha)^(1/4)
4. Tₚ = T * sqrt(R/(2*D)) * (1 - alpha)

Correct answer: Tₚ = T * sqrt(R/(2*D)) * (1 - alpha)^(1/4)

Solution

The star's luminosity is L = 4*pi*R² * sigma * T⁴. At distance D this gives a flux F = L/(4*pi*D²) = sigma*T⁴ * R²/D². The planet intercepts this over its cross-sectional area pi*r² and absorbs a fraction (1 - alpha): P_abs = (1 - alpha) * F * pi*r². In equilibrium it radiates P_rad = 4*pi*r² * sigma * Tₚ⁴. Equating and cancelling pi*r²*sigma gives Tₚ⁴ = (1 - alpha) * T⁴ * R²/(4*D²), so Tₚ = T * sqrt(R/(2*D)) * (1 - alpha)^(1/4). Note Tₚ is independent of the planet's own radius r.

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