English: Considering an atmosphere composed of a single isothermal layer. Because the layer is

opaque in the longwave part of the spectrum, the equivalent blackbody temperature of the planet corresponds to the temperature of the atmosphere. Hence, the atmosphere must emit F units radiation to space as a blackbody to balance the F units of incoming solar radiation transmitted downward through the top of the atmosphere. Because the layer is isothermal, it also emits F units of radiation in the downward direction. Hence, the downward radiation at the surface of the planet is F units of incident solar radiation plus F units of longwave radiation emitted from the atmosphere, a total of 2F units, which must be balanced by an upward emission of 2F units of longwave radiation from the surface. Hence, from the Stefan–Boltzmann law the temperature of the surface of the planet is 303 K, i.e., 48 K higher than it would be in the absence of an atmosphere.

If a second isothermal, opaque layer is added, as illustrated below the flux density of downward radiation incident upon the lower layer will be 2F. (F units of solar radiation plus F units of longwave radiation emitted by the upper layer). To balance the incident radiation, the lower layer must emit 2F units of longwave radiation. Because the layer is isothermal, it also emits 2F units of radiation in the downward radiation. Hence, the downward radiation at the surface of the planet is F units of incident solar radiation plus 2F units of longwave radiation emitted from the atmosphere, a total of 3F units, which must be balanced by an upward emission of 3F units of longwave radiation from the surface.

By induction, the aforementioned analysis can be extended to an N-layer atmosphere. The emissions from the atmospheric layers, working downward from the top, are F, 2F, 3F . . . NF and the corresponding radiative equilibrium temperatures are 303, 335 . . . . [(N 1)F/σ]1/4 K.

The figures used to generate this plot and text were obtained from Wallace, John M. og Hobbs, Peter V. (2006): Atmospheric Science – An Introductory Survey (second edition.), page 121-122. Elsevier. ISBN 978-0-12-732951-2.