In the Tropopause and Second law of Thermodynamics Paradox I discussed a simple dilemma using just temperatures between thermodynamic disks. Radiant energy isn’t as simple as temperature. There is a fourth power relationship between energy flux and temperature. The fourth power relationship creates an envelope of energy related to temperature of the radiant spectrum of an object at a given temperature with a given concentration of elements. Every element has a unique radiant spectrum and elements in combination extend that spectrum. A perfect black body would have a uniform spectrum with no gaps or missing spectral lines.
The Stefan-Boltzmann law, E=sigma(T^4) is the formula to determine the ideal energy intensity of an object at a given temperature. That energy is based on one side of a flat disk, not a sphere. In order to use on a sphere, the geometry of the sphere has to be considered which produces very accurate results for the observable face. The opposite face of the object has to be assumed to follow the same rules, which is perfectly fine if the object is sufficiently massive.
With gray bodies, less than perfect black bodies, those assumptions have to be applied which appear adequate, again if the object is sufficiently massive. Gray bodies with gaseous atmospheres do not always behave completely as expected. How much they deviate from expectations is a major issue.
As with the temperature only example linked above, the same problem can be posed with energy flux instead of temperature with the absorption/emission spectrum of the object considered.
If we let T be 150K, then the flux values for the problem using the full Stefan-Boltzmann constant, σ = 5.670373(21)×10−8 W m−2 K−4, yields F(2T)=459.3Wm-2, F(T)=28.7W/m-2. So in order to have 1/2 the temperature, the cooler disk would only have to absorb 28.7/459.3 or 0.22% of the radiation emitted by the warmer object.
When we place another object between the two, we determined its final temperature would be approximately 3/2T which would be 225K @ 145.3Wm-2. So the simple temperature ratio used is vastly different when using energy flux.
Since the warmer body for some instant in time will continue to emit at 459.3Wm-2 and the center disk only absorbs 145.3, 313Wm-2 would pass through the newly inserted object outside its absorption envelope. The 145.3Wm-2 the warmer object receives from the center disk would be toward the lower energy range of its emission spectrum. If the energy received is within the envelope of the warmer body spectrum, that energy would be absorbed. If not, that energy would pass through the warmer body just as energy outside of the emission spectrum of the cooler disks would pass through.
Since it is likely that the warmer body spectral envelope includes the envelope of the cooler body, we can assume that all the energy is absorbed. In such a case, the warmer body would return approximately one half of that energy, 72.6Wm-2 would be returned to the center disk. It would be incorrect to assume that all of this energy would be absorbed, since the spectral envelop of the center disk is much smaller than the warmer disk.
If the warmer body emits the newly absorbed energy uniformly across its spectrum, the center disk would absorb on the order of 72.6/459.3 or 15.8 percent of that radiation, 13.5Wm-2. We would have a series based on 15.8 percent absorbed which is based on the original spectral envelope of the center disk.
72.6+13.5+1.9+0.36+0.11… or approximately 90Wm-2. The envelope of the center disk would of course increase with temperature changing this relationship, but the law of conservation of energy has to also be considered.
The total energy of the warmer body cannot increase more than the decrease in energy of the other two disks. So the opposite face of the warmer body must decrease as the common face increases.
That changes our series by 50 percent or instead of 90Wm-2 to 45Wm-2.
So the warmer disk would increase to approximately 504.3Wm-2 while its opposite face decreased 45Wm-2 to 414.3Wm-2. The total apparent energy of that object would become 918.6Wm-2 which would have an average apparent temperature of 459.3Wm-2 or 300K degrees. The apparent temperature of the common face would be 307.1 degrees K
What I propose is that is the condition of the atmospheric effect prior to the addition of more carbon dioxide at a location on the surface with an apparent temperature of 307 degrees. Or that the radiant portion of the atmospheric effect is approximate 7K degrees at regions with surface temperature of 307K degrees and that the impact would vary with the surface temperature and the effective radiant layer of the atmosphere. In another post I will add a new disk representing carbon dioxide.
Note: This is a little different that Kirchhoff's law. With longwave radiation there is essentially no reflection. What is not absorbed passes through the disk or wall, and of what is absorbed, 50% is returned instead of reflected. The amount returned is increased by the number of disks returning 50% of their absorbed which either passes through or is absorbed returning 50% and the amount absorbed is dependent of the absorption spectrum, i.e. temperature and/or composition of the disk.