Is the 'Greenhouse Effect' based on a 'cool' Sun?

        Most descriptions of the 'greenhouse effect' are similar to that offered by Wikipedia: "If an ideal thermally conductive blackbody was the same distance from the Sun as the Earth, it would have an expected blackbody temperature of 5.3°C. However, since the Earth reflects about 30%... of the incoming sunlight, the planet's actual blackbody temperature is about -18 or -19°C, about 33°C below the actual surface temperature of about 14 °C or 15 °C. The mechanism that produces this difference between the actual temperature and the blackbody temperature is due to the atmosphere and is known as the greenhouse effect."

        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:

where

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

We see that the wavelength at maximum intensity is in the visible spectrum at about .475 microns or 475 nanometers, which is what the Handbook of Chemistry and Physics, 82nd Edition, CRC Press, 2001 also shows as the maximum 'Solar Spectral Irradiance.' The Wein displacement law can then be used to calculate the Sun surface temperature based upon the maximum irradiance wavelength, resulting in a Sun surface temperature of 6101°K (5828°C). Another source gives an even higher figure, "The Sun's outer visible layer is called the photosphere and has a temperature of 6000°C," which is equivalent to 6273°K.

        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

, 

        If we now use the Sun surface temperature calculated from Wein's displacement law as the input to the equation above, we find the Earth surface temperature without an atmosphere should be 294.26°K or 21.11°C. Since the 'average' Earth surface temperature is commonly cited as 15°C, that means the Earth with a 'greenhouse gas' atmosphere along with the additional factors of convection, evapo-transpiration, and adiabatic effects is colder than an Earth without!

see also the peer-reviewed Chilingar et al paper Cooling of the atmosphere due to CO2

h/t comments by Gord, graph source