Alright, let me make some comments on the contents of the recent paper by Chilingar and co-authors in Environmental Geology: Greenhouse gases and greenhouse effect.
First of all, the title is rather misleading. The paper hardly deals with the greenhouse effect, as we will see shortly. Secondly, the main "theory" presented in the paper, which the authors call the "adiabatic theory of greenhouse effect" is nothing new and has already been discussed by Eli Rabett.
The essence of the theory is that atmospheric temperature can be related to pressure (i.e., altitude in the atmosphere). The fact that the temperature profile in the Earth's troposphere is dominated by the adiabatic gradient, i.e. the effect that air cools as it ascends and expands, is of course textbook knowledge, although Chilingar et al. do not acknowledge this. If you would like to know how it works, for example, follow this link to lecture notes of my colleagues in atmospheric physics at our institute. Using this fact and a strange version of a global radiation balance, Chilingar et al. construct their equation (1) to describe temperature as a function of pressure with two empirical factors b ("scaling factor") and alpha ("adiabatic exponent").
I do not have a clue why the "precession angle" of the Earth enters the radiation balance term. One might have to follow the Sorokhtin and Chilingar references given at the outset of this discussion, but these aren't easy to find (maybe the one in Energy Sources would be available, but we already know that these authors like to duplicate their papers in that journal and Environ. Geol.). I do know, however, that the angle of 23.44° to which the authors refer is usually called the inclination angle or axial tilt, whereas precession refers to the change in the direction of the tilted rotation axis. Anyway, if you are interested in a more conventional radiation balance, see these lecture notes.
Chilingar's equation (1) has a form that fits the temperature profile in the troposphere, they only need to adjust their two empirical parameters. Adjusting b sets the temperature at the surface - this is of course where a proper discussion of the radiation balance and greenhouse effect would be needed. But the authors circumvent this by simply setting b to fit the current mean surface temperature of the Earth. The parameter alpha is indeed a well-known exponent in adiabatic gas theory. For a dry atmosphere it would be given by c_p/c_v - 1 = R/c_p = 0.4. For a humid atmosphere it has to be adjusted for the latent heat in the condensation process. Chilingar et al. introduce two further fit parameters to adjust alpha, C_w and C_r, which they refer to as accounting for the effects of water vapor and - yes, indeed - absorption of heat by greenhouse gases.
I am not sure if the radiation correction term C_r makes much sense, but in any case this seems to be the parameter by which Chilingar et al. describe the greenhouse effect. To summarise the theory thus far, it has three adjustable fit parameters: b to adjust the surface temperature, C_w and C_r to adjust the adiabatic lapse rate. Values of these parameters are found by fitting equation (1) to the current temperature profile of Earth and Venus for comparison. They are quite different for the two planets, which is not further explained. However, an essential assumption of the paper then appears to be that these parameters are somehow characteristic for the two planets, INDEPENDENT OF THE COMPOSITION OF THEIR ATMOSPHERES. This is a rather strong assumption.
The theory and the above assumption are then taken to the extreme by calculating the temperature of Earth for a hypothetical pure carbon-dioxide atmosphere. The authors explicitly state that b is kept fixed and with a bit of calculation one can verify that they also do so for C_w and C_r. Given that C_r is meant to reflect the heat absorption by greenhouse gases, I find it slightly surprising that a pure CO2 atmosphere should have the same value as the current N2/O2 atmosphere. Wouldn't it be here where the authors should discuss the greenhouse effect as they announced so loudly in their title?
Well, they don't but rather find that (given their theory and assumption) a world with a pure CO2 atmosphere would in fact be 6.5°C cooler than the present day world. Great result, isn't it? Funny, though, that Venus with its CO2 atmosphere is so much hotter than Earth. This is due to the much higher factor b and a different C_r. But for Earth, these parameters would of course stay constant even if we were to convert all oxygen to CO2. No problem with global warming there. Just a bit tough to breathe, maybe...
In the event you are not convinced by Chilingar et al.'s theory, they have a backup argument as well, very typical for the rethoric of climate sceptics. This time, towards the end of the article, they dig up the old argument that the climate and CO2 reconstructions from ice cores have shown that temperature changes lead fluctuations of the greenhouse gases, not vice versa. This is correct, but not "indisputable evidence to the fact that the changes in CO2 concentrations of the atmosphere are the effect of global temperature changes, and not their cause." This is a frequent logical mistake. Yes, temperature increases in the past have caused CO2 to rise. This does, however, not exclude the (quite probable) possibility that rising CO2 in turn contributed to warming, and does so today.
There are some other mistakes in the paper as well, of course. One of the more obvious is the list of partial pressures in the Earth's atmosphere (after equation (2), cited literally here): "pN2 = 0.7551; pCO2 = 0.00046 pN2 = 0.7551 and pAr = 0.0128 atm are the partial pressures of the
corresponding gases (Voitkevich et al. 1990)".
Apart from the typo of repeating the N2 value twice, I wonder about the values and the Voitkevich reference. Compare the above values with the standard composition of the atmosphere in any textbook...