Their calculations of thermal radiation greatly exaggerate the density of the infrared photon radiation in the atmosphere and the extent of the absorption of infrared radiation by infrared-active molecules, commonly called greenhouse gases, such as water vapor and carbon dioxide. Their theory of the transport of heat energy between the surface of the Earth and the atmosphere and through the atmosphere is very wrong. It exaggerates the role of thermal radiation greatly and minimizes the role of the water evaporation and condensation cycle and the role of thermal convection. It cannot be emphasized enough how harmful their mishandling of thermal radiation calculations is to their understanding of the critical issues pertaining to the Earth's climate and to man's role in changing the climate through the use of carbon-based (fossil) fuels.
In the case in which the two black body parallel plane radiators have the same temperature, the volume between them becomes that of a black body radiator. The fundamental characteristic of a black body radiator is the constant energy density in the cavity. I will show that the so-called settled science treatment, which wrongly takes the primary characteristic of a black body radiator to be that the power of emission of radiant energy is given by the Stefan-Boltzmann Law, clearly violates the real principal characteristic of a black body cavity, namely that its constant energy density, e, is given by Stefan’s Law as
This physics may be reviewed by the reader in an excellent textbook called Thermal Physics by Philip M. Morse, Professor of Physics at MIT, published in 1965 by W.A. Benjamin, Inc., New York. Prof. Morse wrote it as a challenging text for seniors and first-year graduate students. I was fortunate to use it in a Thermodynamics course at Brown University in my Junior year. Alternatively, see my post The Greenhouse Gas Hypothesis and Thermal Radiation -- A Critical Review.
In fact, while it is commonly claimed that photons inside the cavity are being 100% absorbed on the walls and an equal amount of radiant energy is emitted from the absorbing wall, the actual case is that the radiant energy incident upon the walls can be entirely reflected from the walls. Planck had derived the frequency spectrum of a black body cavity from an assumption of complete reflection from the walls.
Thus, when these planes are in one another's presence this consensus viewpoint says that
eH = aTH4 in this case.
Consequently, the energy density between the planes is
Here we see that the emissivity which determines the electromagnetic field energy density at the surface is also playing the role of the absorptivity at the absorbing colder surface. So of course, Kirchhoff’s Law of thermal radiation that the emissivity equals the absorptivity of a material in a steady state process applies. There is really nothing at all to prove if one starts with the primary fact and boundary condition that the energy density is the fundamental driver of the thermal radiation of materials.
Update: The extension to gray bodies and other bodies was made on 14 November 2017.
Update: Explanatory additions on 27 November 2017.
Update: Further explanatory text added on 29 November 2017.
Update: Further efforts to help the reader to understand this post and to provide the reader a reason to read it in the introductory paragraphs on 24 March 2018. I am amazed how difficult many readers find it to alter their viewpoint of thermal radiation even when they have no counter argument to my argument presented here.