Apparently this article of mine on the basic physics and mathematics of radiation on a planet relating to the Greenhouse effect, became somewhat the subject of discussion on a German climate science thread. Given the language difference, I didn't quite follow what they were on about, and so asked for a summary in a related discussion. Note that my article has been discussed at length previously and Google shows lots more references around the 'net.
It's always interesting to see how somebody becomes confused, but it's odd to see them insistently persist in that confusion. Gerhard Kramm seems to have completely ignored the main focus of my article, jumped a couple of paragraphs in, and found something he thinks is critically wrong in the definitions I wrote down in equations 7 and 8. He seems to be absolutely certain that when I described "effective" temperature and emissivity, that I meant "average". This despite the fact that I separately defined average temperature elsewhere with its obvious meaning (equation 11), so that "effective" temperature is clearly quite different from "average".
Kramm even seems to have missed equation 4, where I define an "effective" albedo in a way that doesn't have any direct reference to averaging. His selective misunderstanding of my use of the word "effective" is very strange. To be absolutely clear, "average" has an operational meaning that is in essence the same for every quantity it is applied to: the mean value of a quantity across all instances of it. In contrast, the word "effective", used as I do it, means simply a particular value for a quantity that is in some way representative for the particular discussion at hand. For albedo, the "effective" value relates actual reflection of energy from the planet's surface to the actual total input energy from the Sun, so as an average it would have to be weighted by the level of that input solar energy. Similarly for the effective radiative temperature in this discussion, the particularly useful value corresponds to the uniform temperature that would given the same Stefan-Boltzmann emissions as the given temperature distribution under uniform emissivity. And the effective emissivity, as I defined it, is a similar value that, with the effective radiative temperature, corresponds to the actual distribution of emissivities and temperatures across the planet associated with Stefan-Boltzmann radiation.
Nevertheless, Kramm persists, and despite my and several other people's attempts at explanation, today he
has posted this complete misrepresentation of my paper, suggesting that I don't understand calculus. This sort of arrogant attack seems typical of Kramm - as far as I can tell on the German language thread he has repeatedly told various people they don't understand basic physics, they need to go back to school, and so forth.
In any case, the clear confusion he exhibits in his "basic rules of calculus" post is that he interprets the subscript "eff" in the equations in my paper (7, 8, and 9, but ) as somehow the simple averaging operator, despite my use of a different subscript and explicit use of the word "effective", rather than "average, in the text. In the specific cases of equations 7 and 8 he claims to reproduce them in his post, but replaces my "eff" subscript with angle brackets - <> - indicating a pure average, and so not a reproduction of my argument at all.
He also seems to have trouble with the normal representation of surface integrals which enter into the equations I used. Many of them refer to total energy and so are not averages at all, but indeed need the actual (r squared) area involved. For example, equations 1, 3-6, and 9-10 all refer to totals, not averages. Using a common framework for equations 7 and 8 was only natural; for a spherical planet it does reduce to the same averaging process over solid angle that Kramm describes, but that's hardly a necessary step in calculating the quantities involved. In fact, there's a loss of generality doing it Kramm's way, as surface area is only proportional to solid angle if the radius is normal to the area element in question - true for a sphere, but not for anything else.
To a PhD physicist this stuff is pretty trivial math; I'm surprised at Kramm's continued misunderstanding. Various posters on Rabett's blog seem to think it's deliberate in some fashion; I'll grant that anybody who gets something wrong does try to put the best face on it, but at some point you just have to admit you were wrong and move on. Will Kramm? I think I'm done with trying to explain it to him. Nobody else seems to have had this particular problem with the argument in my paper!