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 Incoming links: Circumstellar habitable zone

Potential surface habitability requires a stable liquid-water interface, a temperature and pressure outside runaway-greenhouse and high-pressure-ice regimes, and an

H_{2}

inventory large enough for collision-induced greenhouse warming but not so deep that pressure and temperature sterilize the ocean. Long-term atmospheric retention, tolerable stellar ultraviolet and particle flux, nutrient cycling, and chemical disequilibrium compatible with metabolism are also needed. A location in the circumstellar habitable zone alone is insufficient.

For

M_{p} = 5 M_{\oplus}

and

R_{p} = 2 R_{\oplus}

g = \frac{5}{2 ^{2}} g_{\oplus} ≃ 12.3 m s^{- 2} .

Taking

T = 300 K

μ = 2.3

H = \frac{k _{B} T}{μ m _{H} g} ≃ 88 km .

If a strong feature spans

N_{H} = 5

R_{s} = 0.5 R_{⊙}

Δ D ≃ \frac{2 R _{p} N _{H} H}{R _{s}^{2}} ≃ 9 \times 1 0^{- 5} ≃ 90 ppm .

Clouds, hazes, refraction, and smaller molecular abundance reduce this estimate.

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.