Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 315 / 3 / a / Solution

Assume a spherical blackbody planet, uniform reradiation, constant Bond albedo

A_{B}

, and negligible internal luminosity. Radiative equilibrium gives

σ T_{eq}^{4} = \frac{( 1 - A _{B} ) L _{s}}{16 π a _{H}^{2}} .

With

L_{s} = K M_{s}^{β}

,

a_{H} (M_{s}, T) = [\frac{( 1 - A _{B} ) K}{16 πσ T ^{4}}]^{1/2} M_{s}^{β /2} .

The inner and outer circumstellar habitable zone boundaries are obtained by setting

T = T_{m a x}

and

T = T_{m i n}

respectively:

a_{in} \propto T_{m a x}^{- 2} M_{s}^{β /2}, a_{out} \propto T_{m i n}^{- 2} M_{s}^{β /2} .

Real boundaries require wavelength-dependent albedo, clouds, greenhouse effects, and the stellar spectrum, which this equilibrium-temperature model neglects.

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