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### Surface Temperature of Planets

posted Jan 4, 2014, 9:49 AM by Abel Mendez   [ updated Jan 7, 2014, 6:22 AM ]
The mean global surface temperature of a planet in a circular orbit is given by (adapted from Qiu et al., 2003):

 $\large T_{s} = T_o\left (\frac{fL(1-A)}{a^2(1-g)}\right )^ \frac{1}{4}$ (1)

where Ts = mean global surface temperature (K), L* = star luminosity (solar units), a = semi-major axis, f = atmosphere redistribution factor (e.g. = 1 for fast rotators and f = 2 for tidally locked planets without atmospheres)A = bond albedo, and g = normalized greenhouse effect), and To = 278.5 K. The normalized greenhouse effect is defined as (Raval & Ramanathan, 1989):

 $\large g = \frac{G}{\sigma T{_{s}}^{4}}=1-\left (\frac{T_{eq}}{T_s} \right )^4$ (2)

where G = greenhouse effect (W/m2) or greenhouse forcing, and Teq = equilibrium temperature (K). Both A and g are numbers between 0 and 1 that are necessary to understand the temperature of planets. They do not only depend on the surface and atmospheric properties of the planet but also on the surface temperature. For example, Raval & Ramanathan (1989) determined the terrestrial g for clear-skies globally, but for a particular month (April 1985), as:

 $\large g = -0.658 + 3.42\times 10^{-3}T_s$ (3)

where Ts = sea surface temperature (SST), and only valid for temperatures between 275 K to 300 K. Eq. 1 can be easily extended to elliptical orbits assuming that both A and g are nearly constants as function of eccentricity (i.e. constant with orbital changes of Ts). Table 1 show some approximate values of A and g for Venus, Earth, and Mars.

Table 1. Necessary data to calculate the surface temperature of Venus, Earth, and Mars from Eq. 1. Solar luminosity L = 1.0 and f = 1. This solution can be extended to exoplanets given appropriate estimates of A and g.

 Planet a, Semi-Major Axis (AU) A, bond albedo g, greenhouse Venus 0.723 0.750 0.990 Earth 1.000 0.300 0.397 Mars 1.524 0.250 0.086

References

• Qiu, J., Goode, P. R., Pallé, E., Yurchyshyn, V., Hickey, J., Rodriguez, P. M., ... & Koonin, S. E. (2003). Earthshine and the Earth's albedo: 1. Earthshine observations and measurements of the lunar phase function for accurate measurements of the Earth's Bond albedo. Journal of Geophysical Research,108 (D22), 4709.
• Raval, A., & Ramanathan, V. (1989). Observational determination of the greenhouse effect. Nature342 (6251), 758-761.