The mystical planet problem

(As part of Rabett Run's Gerlich and Tscheuschner project, Eli has started drafting parts of a response, which we will gift wrap in Bozo paper and send to some unsuspecting journal, but certainly arXiv. This second part comes from Duae Quartunciae. The Editorial Board expresses its thanks=:> [Rabett Run has an exceedingly small Editorial Board] Suggestions for changes and additions are welcome. Below is what was sent with minor formatting changes. It certainly needs a lead in that sets forth the issue and summarizes the result. The organization of the section should make clear why and how G&T are gone astray. Although this mostly covers Section 3.7 I think it should logically go after the part we just worked on which covered section 3.9 BTW, that is a good model for what I think we want. I'll come back to this tomorrow with some suggestions, but will leave it up as is for comments until then - Eli)

UPDATE: Eli has added a bit in Green) anything further


FW?IW the idiocy du jour is that thermal energy is not heat. Thermal energy is heat. Joule showed that about 150 years ago


On page 65 of their paper, Gerlich and Tscheuschner contrast two methods of calculating a temperature for a hypothetical planet, which they call T_eff and T_phys.

The basis for both numbers is a consideration of solar energy reaching the globe of the planet. This is described in section 3.7.4. The sun's emission can be treated as a 5780K blackbody. Scaling for the distance between the Sun and Earth the solar insolation is 1369 W/m² above the atmosphere.

1. Temperatures for a globe exposed to solar radiation.

Gerlich and Tscheuschner consider the amount of energy reaching each point of the Earth's sphere. This is zero on the night side, and on the day side it is scaled by a cosine to account for the angle at which light reaches different regions. The energy is also scaled by 0.7, to account for the amount of energy is reflected away rather than absorbed. (The Earth has an albedo of about 0.3.). At the surface, on average the solar flux is ~340 W/m².

T_eff is the temperature you must to give to every point on the globe in order to radiate all this energy away, again as a blackbody.

Gerlich and Tscheuschner prefer a model which assumes every point on the globe is in equilibrium with the local solar radiation at that point. This corresponds to a planet with no rotation, and with no heat transport over the surface, and uniform albedo. This model is absurd, especially because it neglects the heat capacity of the surface, the atmosphere (about which they have made a great fuss at the beginning of their paper) and most especially the oceans. All of these do not cool anwhere near night temperatures implied by a local radiative equilibrium even at the poles during their long nights.

They then take an average of the temperature for this hypothetical and unphysical planet; ironically calling it the physical average temperature, Tphys.

T_eff and T_phys correspond to the two extremes of having uniform temperatures over the globe, and having temperatures at each point depending only on the instantaneous solar input.

Comparing equations 81 and 83, it can be seen that T_eff = 1.25*sqrt(2)*T_phys = ((1-α)S/4/σ)^0.25, where S is the solar constant (1369) and α is the albedo (0.3). Plugging in the numbers, one gets T_phys = 144K (-129C) and T_eff = 255K (-18C). These values are shown by Gerlich and Tscheuschner in their table 12.

In practice, of course, the distribution of temperature over a planet will be between these two extremes. If the conventional average temperature is taken by integrating real temperatures over the globe, the value T_mean should be between T_phys and T_eff.

From the first law, the energy emitted has to be the same, no matter how temperatures are distributed. It follows that the fourth power of temperature, integrated over the globe, should be an invariant, since this is proportional to energy. This is why T_eff is a more useful quantity in practice than T_phys. In any case, T_phys must be less than T_eff .

The effect of an atmosphere

These values can only be associated with the surface if there is no atmosphere, and no greenhouse effect, so that the surface radiation is equal to the planet's radiation. If there is an atmosphere that absorbs surface radiation, then this atmosphere will be heated from the surface, and will be cooler than the surface as shown in the previous section. Most of the radiation escaping to space will be emitted from the atmosphere, and this is what must match solar input. The surface must be warmer than the atmosphere, by the second law, because the surface is heating the atmosphere.

T_mean corresponds to a level in the upper atmosphere where most of the energy escapes into space, and the average surface temperature T_surf must be somewhat warmer than this.

In practice, when you integrate temperatures over the surface of the Earth, you get about 15 C. This is indeed much greater than the -18 C of T_eff, and this is called the greenhouse effect; the difference between surface temperatures below the atmosphere, and the effective temperature for radiation escaping into space. Arthur Smith (arXiv) has provided a more sophisticated, and thus mathematically complex, consideration of this issue, reaching the same conclusion as here.

Gerlich and Tscheuschner show how to integrate temperatures over the globe's surface, and they correctly note that the value obtained by such integration should be less than T_eff to balance the solar input. They completely fail to note that if you actually do integrate over the surface, you get a value substantially greater than T_eff. The reason for this difference is the greenhouse effect.

This is a bit like having a blanket on a cold night. You end up warmer than you would be without a blanket, but not because the blanket is a source of energy to heat you up. In fact, you are the source of energy heating the blanket, and this means you have to be warmer than the blanket.

Gerlich and Tscheuschner make this elementary mistake in their section 3.9, when they describe the greenhouse as a violation of the second law. In fact, the second law is what requires the surface of a planet to have a higher temperature when there is an atmosphere that is being heated from the surface.

3. The example of the Moon

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.


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