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This is a climate model:
T = [(1-a)S/(4es)]1/4
(T is temperature, a is the albedo, S is the incoming solar radiation, e is the emissivity, and s is the Stefan-Boltzmann constant)

Lindy, the 1/4 that appears should be an exponent, as in

T = [(1-a)S/(4es)]^(1/4)

It is conventional to use the greek letters epsilon for emissivity (e) and sigma (s) for the Stefan-Boltzmann constant.

So you would usually see the equation written as

T = [(1-a)S/(4es)]^(1/4)

This equation just expresses the balance between incoming solar radiation (1-a)S/4 and outgoing emission from the Earth: esT^4 . When these quantities are equal, as they must be in radiative equilibrium, then the above relation for T is obtained.

The temperature T here is the temperature of the emitting surface, which would be the surface of the Earth if there were no greenhouse gases in the atmosphere. For Earth, T= 255 K or about -18 C.

Actually, of course, CO2, H2O and CH4 (methane) are present, and make the atmosphere opaque to various degrees in these molecular bands. Then, the effective "emitting surface" is high in the atmosphere at the wavelengths of the greenhouse bands, and the temperature T is therefore the temperature at this location. Since temperature decreases with height, that means the surface must be warmer than the radiative equilibrium temperature T, and it is. For the Earth, the mean global surface temperature is about 288 K= 15 C, or some 33 C warmer because of the greenhouse effect.

Phil

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