This calculation is based upon the Stefan-Boltzmann law, radius of the Sun and Earth, distance between the Sun and Earth, and albedo (reflection, mostly from clouds), from which the following equation is derived:
Te = Temperature of the Earth surface
Ts = Temperature of the Sun surface
Rs = Radius of the Sun = 6.96x10^8 m
a = Albedo (reflection of incoming solar radiation mostly from clouds) = 0.306
E = IR emissivity of the Earth (assumed to be ~1)
D = the Astronomical Unit = distance between the Sun and Earth = 1.496x10^11 m
If we plug in the value for the temperature of the Sun (5778°K) used by the Wikipedia article, we find that the temperature of the Earth without an atmosphere (setting albedo to 0) should be 278.68°K (5.53°C). This is very close to the 5.3°C ideal blackbody temperature as stated above. If we then set albedo to the commonly used value of 0.306 for Earth with an atmosphere, we find the Te calculation drops to -18.8°C. The 'greenhouse effect' is then calculated from the globally 'averaged' earth surface temperature of 15°C - -18.8 = 33.8°C. Note also the G&T paper has shown that it is not possible to calculate an 'average' Earth surface temperature (p.70-71), which also renders the 'greenhouse effect' calculation moot. In addition, all of these calculations are based on the incorrect assumption that the Earth, atmosphere, and Sun can each behave as ideal blackbodies.
The value for Sun surface temperature used in these calculations (5778°K or 5505°C) is a low estimate and thus biases the calculation to show an enhanced 'greenhouse effect.' The reason why is shown in the graph below of the observed distribution of solar radiation reaching the top of the atmosphere as a function of wavelength (solid black line).
|Ts based on wavelength at Max Intensity|
If we had used the Sun surface temperature of 5778°K assumed by the Wikipedia article, we would find that the Sun emitted more energy than an ideal blackbody throughout the visible spectrum, which is impossible:
|Solar flux exceeds blackbody Wikipedia Ts||,|
see also the peer-reviewed Chilingar et al paper Cooling of the atmosphere due to CO2
h/t comments by Gord, graph source