Krammies

Eli has been mining Gerhard Krimms contribution to arXiv and has come up with yet another Krammi. We had a recent discussion of the surface temperature of the moon here at Rabett Run,

The Moon is a good example to contrast with the Earth. It rotates much more slowly, and therefore has a temperature distribution that approaches what is used by Gerlich and Tscheuschner to derive their "T_phys". Each point on the Moon's surface is tolerably close to radiative balance with the solar input at that point.

The Moon has an albedo of about 0.12. It therefore absorbs more of the incoming solar energy than Earth. Using the solar constant of 1369 W/m², the absorbed radiation for the surface facing the Sun is about 1205 W/m². Hence T_eff for the Moon is (1205/4/σ)^0.25 = 270K, or -3C. This is the temperature that would radiate back the solar energy, if evenly distributed over the moon. But directly facing the Sun, the temperature will be more like (1205/σ)^0.25 = 382K, or 109 C. Albedo is not uniform. In any particularly dark patches, the temperature could even get up to (1369/σ)^0.25 = 394K, or 121C. On the night side, however, temperatures will fall toward absolute zero. Bear in mind that as temperatures fall, so too does the rate of emission of energy. Hence it takes a long time to fall all the way to zero. Say rather that temperatures should fall far enough for the emission of energy to be small.

Now consider data on the Moon from http://www.solarviews.com/eng/moon.htm

Average day temperature is 107 C. Maximum day temperature is 123 C. These are close to theoretical expectation, to within a couple of percent.

The mean night temperature is -153C. This about 120K, and radiates a bit less than 12 W/m². That's less than 1/100 of the solar constant, so the temperature has indeed fallen close to zero, using radiated energy as the basis for comparison.

There's no average temperature given, but the mid point of mean day and mean night temperatures is in the ballpark. This is -23C. And, just as should be expected, it is somewhere between T_phys (-120C) and T_eff (-3C). But it is closer to T_eff, because it is the cool side of the moon that is most different, in absolute temperature, from the unphysical extreme that is the basis of Gerlich and Tscheuschner's T_phys

On Earth, fortunately, we have an atmosphere that has to be heated from the surface. By basic thermodynamics, the Earth's average surface temperature is therefore substantially warmer than our airless moon. where surface radiation escapes directly to space.

but Gerhard shows a contradictory example

Figure 5: Moon's disk temperature at 2.77cm wavelength versus moon phase angle φ during two complete cycles from twice new moon via full moon to new moon again (adopted from Monstein.)

with a comment below of

Our Moon nearly satisfies the requirement of a planet without an atmosphere. It is well known that the Moon has no uniform temperature. There is not only a variation of the temperature from the lunar day to the lunar night, but also from the Moon equator to its poles.

Using Eqs. (1.4) would provide T 270 K e ≈ when the albedo, 0.12 M α = , and the emissivity, 1 M ε = (black body), are considered. However, as illustrated in Figure 5, the mean disk temperature of the Moon observed at 2.77cm wavelength by Monstein (2001) is much lower than this equilibrium temperature

There is nothing necessarily wrong with the Krammigraph (Eli has not checked, but it compares reasonably to similar ones that he has seen. What's up? Work out your answer and send it with to box tops (we are not particular about the boxes) to Rabett Run Krimmi Kontest @ yahno.con. Extra prizes for those who figure out how to send the boxtops.

ANSWER: Look carefully at the wavelength that the brightness temperature was measured at little bunnies. Hmm, 2.77 cm. That's pretty long wave. It turns out that long wavelength radiation (and 2.77 cm is pretty long wave compared to IR even which is micrometers) penetrates through about 4 to 8 times the wavelength. Thus if you use 2.77 cm, what you are measuring is the temperature of the moon 10 - 20 cm below the surface and that is a lot cooler and has a lot less variation with the diurnal cycle. What about if we measured at the surface. Well Apollo 15 left a temperature measuring instrument and this is what it found

Fig. 2. Lunar surface temperature time series from the landing site of the Apollo 15 mission. The series are part of the historical data archive PSPG-00093 of the US National Space Science Data Center. The color-coding of the temperature series used in this diagram is also used in Figs. 3 and 4.

The answer is that the Krammi did not show surface temperature, but below surface temperature.