by Ben Wouters
The concept of Effective Temperature (Te) originates in astronomy, and is a rule of thumb calculation to estimate the radiative temperatures of planetary bodies in a solar system. For planets without atmosphere, no internal heat source and surface emissivity equal to 1 the Te is the whole story. All mentioned factors increase the “base” Te. See eg this site
for more details.
Surprisingly the Te for the moon using albedo 0,11 is ~270K, but the actual average temperature is much LOWER (~197K). This means that either the moons albedo is massively higher (~0,75 iso 0,11) or there is a serious flaw in the way the Te’s are calculated. Let me show you that the Te for the moon is ~161K iso ~270K and for Earth ~151K iso the well-known 255K
Basically Te is arrived at by taking the surface temperature of a sun and calculating the remaining radiation (Total Solar Irradiance (TSI)) at the distance of the body under consideration. The amount of energy intercepted by a body is equal to the TSI times the cross-sectional area of the body. Since a sphere has 4 times the area of a circle, the TSI is divided by 4 to arrive at the average RADIATION/m2 on the body. A reduction for reflected radiation is also applied. So far so good. To arrive at the Te, the Stefan-Boltzmann formula (SB) is used on this number, assuming “black body” (BB) behaviour of the body. This last step is where things go wrong: using the average radiation to arrive at a temperature iso calculating different temperatures and then averaging them. You cannot average the input for a non-linear equation like the SB formula (fourth power) and expect a meaningful result. The following spread sheet demonstrates this nicely. It shows the radiation heating two identical BB plates, the resulting temperature using SB and the average temperature. Notice that in all cases the AVERAGE radiation is 240 W/m2.
The last example (480/0) represents the situation for heavenly bodies with only one sun, like our own planet Earth. A better way to calculate Te is to calculate the average radiation only for the sunny side of a body, calculate the average temperature for that side and THEN average with the dark side which has a radiative temperature of 0K, so dividing by 2 of the result for the sunny side will do. Since the original method didn’t use it, I also ignore the ~2,77K resulting from the cosmic background radiation.
Here are the Te calculations for the moon and earth:
I assume a TSI for both of 1364 W/m2, the moon reflects 11% and Earth 30%.
Moon: (1364 W/m2 x 0,89)/2 = 607 W/m2 SB => 322K => Te = (322K + 0K)/2 = 161K
Earth: (1364 W/m2 x 0,70)/2 = 477 W/m2 SB => 303K => Te = (303K + 0K)/2 = 151K
All this means that a non-rotating grey body at our distance from the sun, without atmosphere, no internal heat and in radiative balance will have this average surface temperature. Change in any of the mentioned factors will give a higher ACTUAL temperature.
I’ll show that this reasoning is correct by looking at some temperature plots of the moon.
Some observations: - the temperature nicely follows radiation values during daytime, but doesn’t drop to 0K at night - apparently some heat storage takes place during the day that re-radiates during the night, given enough time most probably converging to the same temperature as Latitude 890 Winter - temperature at Latitude 890 Winter seems to converge to 30-40K given enough time without sunshine
My conclusion is that for some reason the moon has a “base” temperature of ~30K, most probably caused by internal heat. On deep crater floors near the poles temperatures as low as 25K are found. A heat flow of ~100mW/m2 would be enough to explain this temperature. At the poles Earthshine is an unlikely candidate.
So the moon behaves reasonably well like a BB, except: - it reflects some of the radiation (=> making it a “grey body”) - it rotates and some heat storage is taking place - its “no radiation” temperature is not 0K, but ~25-40K
With its “base” temperature of ~25K, some leftover heat from the previous “day” and the sun adding the rest the average measured moon temperature of ~197K can be easily explained.
Back to Earth: The correct Te for Earth of 151K and consequently the Greenhouse Effect being ~139K iso ~35K leaves us with the problem of explaining Earths average surface temperature of ~290K. Is CO2 even more powerful than previously thought? Or is there a much more plausible explanation?
Ben Wouters, Zuid Scharwoude, Netherlands.