This was a useful thread! I see that mdgnn was asking about where it was, but otherwise is pretty much priding himself on the 'real world' nature of his vision, as against the abstruse 'two plates in vacuum' discussion. Well, you can lead a horse to water...

I agree with the discussion above, where the implication is that his different vision may be due to assuming 'strong coupling' between the two plates. But I wonder if he is not also confusing the present problem, where there is a steady state, with a continuous net flow of heat from the heater to plate 1 to plate 2 to the outer darkness, with an equilibrium state. If you make plate 2 fully isolated on the 'outer' side, give a burst of heat to the system, then turn off the heater, then assuming there are no losses, the two plates will exchange heat until they have the same temperature (if they have the same heat capacity, the average of their two initial temperatures) and thereafter, the IR photons flowing in one direction will be exactly matched by the IR photons flowing in the other direction. In the limit where the amount of heat flowing into plate 1, and out to space from plate 2, is small compared to the amount of heat being exchanged between the two (imagine plate 2 is almost completely isolated on the outside, with just a tiny emissive patch), you will be close to equilibrium, so the two plates will have the same temperature, near enough (equally, this is the strong coupling limit).

Finally, the analogy to the atmosphere has been deliberately soft-pedalled here, which is a good thing as it enabled a focused discussion. Also, it should be realized that the analogy is not all that good. It is true that much of the heat applied to earth is due to incoming visible light hitting the surface (plate 1), and that most of the heat is lost by black-body type radiation from regions fairly high in the atmosphere (plate 2). However, heat exchange between the surface and the atmosphere is not only by radiation - the earth is not in a vacuum, and air can move up and down convectively, and carry heat with it, in the form of heat itself and in the form of water vapour (which, upon condensing higher up to form rain releases lots of heat). This complicated things quite a bit. Also, the earth is a geographically variegated sphere, not a perfect plate, and it is only heated from one side, though it rotates, etc etc. So it is not possible to make any kind of 'back of the envelope' calculation of the precise temperature on earth. That's why you need models, which are basically hugely extended versions of Jonathan's system, which need to be executed on a computer rather than with paper and pencil. They sort of work OK (they tend to get the temperature in the right ballpark), but they are very far from perfect.

TheBigYinJames, mdgnn is again incorrect, I think. The wavelength of emission from the bodies is irrelevant for the problem as posed here. For what it is worth, though, the wavelength distribution of emitted radiation from a perfect black body is well known. It is given by the Planck Formula, which I will not attempt to type here.

Jonathan, you are right, mdgnn seems to have an idea about thermalisation. That may be what TBYJ is referring to earlier. Well, I can't see myself that this is relevant, but thought I'd try to work out what might be going on.

I think what mdgnn is talking about concerns the processes that can happen to a CO2 molecule, high in the atmosphere. For the purposes here, I think there are four important ones:

(a) absorbtion of a photon of IR light, thereby moving CO2 from its ground state to its first vibrationally excited state (let us assume here that this is the only vibrational state we need to worry about), in which it undergoes bending motions. Lets call this CO2*, and write the absorption as:

CO2 + hf --> CO2*

(b) collisional excitation, whereby a CO2 molecule collides with another molecule, and in the process converts into CO2* (the required energy comes from the translational energy of the molecules - they move a bit slower after the process than before - they have effectively slightly cooled down).

CO2 + M --> CO2* + M

(c) collisional cooling, the reverse of (b):

CO2* + M --> CO2 + M + energy

In collisional cooling, the molecule bumps into another one, say of nitrogen (M = N2), and relaxes to its ground state. The energy released is not converted into a photon, but into translational energy of the molecules - by moving a bit faster, they have effectively warmed up a bit.

(d) spontaneous emission, the reverse of the absorption process (a) above. CO2* loses a photon. This can occur in a random direction, e.g. upwards or downwards.

At the top of the atmosphere, the sequence of steps (b) then (d) allows energy to be emitted into space. As described eloquently by Nullius in Verba on another recent thread, the heat gets to the top of the atmosphere through convection processes, largely.

I think mdgnn is talking about the relative rate of processes (c) and (d). If (c) is faster, then you get an equilibrium between processes (b) and (c), such that the populations of CO2 and CO2* are defined by the local temperature. The gas is thermalised. If however (d) is faster than (c), then upon formation of CO2* by process (a), due to a photon coming from lower down in the atmosphere, then the photon could be re-emitted (presumably upwards only in mdgnn-land - in reality it could occur in any direction) before collisional cooling (c), and the system would not be thermalized. I suppose this could be not accounted for in climate models.

I've tried to estimate the ratio of the rates of these two processes, and for this, I've used data I found in this review on CO2 properties, which I assume to be reliable for my present purposes.

The rate for the 'reaction' CO2* + N2 --> CO2 + N2 (Ns is the main collision partner) is given in Fig. 11, p. 18 of the document. At 0 C, right-hand side of the graph, k is ca. 10^-15 cm^3 s^-1). I estimate that the concentration of N2 molecules in the upper atmosphere is about 10^19 per cubic centimeter. So that would yield a relaxation rate for (c) of about 10^4 s^-1 - most molecules of CO2* would be cooled in less than a millisecond.

Spontaneous emission (d) from the lowest excited vibrational state of CO2 is shown in that document to occur with a rate constant (Einstein coefficient) of 1.6 s-1 (p. 33, Table 5, entry 7). I.e. emission would only occur on a timescale of about a second - much slower than (c).

OK, there are lots of other numbers in there, but most of the processes similar to (d) do seem to have slow rate coefficients, whereas vibrational cooling similar to (c) is quite fast. This is actually an area related to some of the research I do, and my back-of-the-envelope estimate here fits with what I expected: for small molecules, cooling (c) is pretty fast at most pressures. So I think we can assume that photons absorbed by CO2 do get thermalised.

Note that in a metal, the equivalent of process (c) is even faster, so it usually is a pretty good approximation to assume that solids acting as black bodies do get thermalized.