Learning from a simple model

There are three equations that define this system – the energy balance at the surface, in the atmosphere and for the planet as a whole (only two of which are independent). We can write the equations in terms of the energy fluxes (instead of the temperatures) since it makes the algebra a little clearer.

Surface:  S + \lambda A = G

Atmosphere: \lambda G = 2 \lambda A

Planet: S  = \lambda A + (1-\lambda) G

The factor of two for A (the radiation emitted from the atmosphere) comes in because the atmosphere radiates both up and down. From those equations you can derive the surface temperature as a function of the incoming solar and the atmospheric emissivity as:

G=\sigma T_s^4= {S\over(1 - 0.5\lambda) }

If you want to put some vaguely realistic numbers to it, then with S=240 W/m2 and \lambda=0.769, you get a ground temperature of 288 K – roughly corresponding to Earth. So far, so good.

Point 1: It’s easy to see that the G (and hence T_s) increases from S to 2S as the emissivity goes from 0 (no greenhouse effect) to 1 (maximum greenhouse effect) i.e. increasing the greenhouse effect warms the surface.

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