Since we're in equilibrium we simply have to equate the rates of energy loss and gain: σT4 = ¼(1-α)L⊙. Now, it's slightly more interesting to us to write this in terms of the distance r a planet is from the sun (since you've drawn the comparison between the Moon and the Earth, which essentially share a value for r). So we can sub in for the luminosity (using solar and planetary orbital radii and the temperature of the Sun's photosphere, T⊙) and solve for the planet's surface temperature: T = (1/√2)(1-α)¼√(r⊙/r)T⊙.
If you plug in the values of those constants to find the planet's effective surface temperature you'll get about 252K. Lo and behold, that's roughly the average of the night and day temperature averages you posted for the surface of the Moon (107 Celsius or 380 K and -153 Celsius or 120 K; 380 K + 120K / 2 = 250 K). Regardless of whether the planet is in the uniform temperature limit or instead mixes heat as poorly as the Moon, the effective average surface temperature in equilibrium (in the absence of greenhouse gases) is below the freezing point of water. Greenhouse gases do indeed raise the surface temperature of the planet.
1. neither earth nor Moon are radiating theoretical black bodies, but grey bodies, so stefan-boltzman in this form is not applicable.
2. Earth does not exist at mean temperature and does not radiate at it, because the solar heat is distributed unevenly between equatorial and polar zones. Some areas radiate more intensively than others. Some areas are more "black" than others too.
3. Radiation from the earth surface is not the main mechanism of cooling at all: the main one is contact heat transfer from soil to the lowest layers of air, which initiates convection in the atmosphere, so the actual radiation into open space happens in atmosphere (top layers of stratosphere). Any heat transfer goes ONLY from higher temperatures to the lower ones, so the general heat emission always goes to open space.
4. The actual greenhouse is a transparent CLOSED volume in which convection is not allowed by the ROOF, so the hot air does not leave it; "greenhouse effect" makes only a virtual reference to the actual greenhouse and makes sense only LOCALLY when the area has compact cloud coverage (water vapours are the main "greenhouse gas"). Local processes present the mere aberrations in the global heat-exchange, which is always balanced in/out (with some precession which we can see in weather fluctuations).
And many more points ( ) which Copenhagen climatologists failed to substantiate, otherwise the deal would've been struck. Your calculation is an example of a student problem on black body radiation and completely lacks practical value. Meanwhile, AGW is finished, so we are beating a dead donkey.
Edited by marabod, 31 December 2009 - 06:20 AM.