At temperatures near 300K, they claim that both the warmer and the cooler bodies absorb all of these infra-red photons when they are incident upon a body. Thus, the surface of the Earth absorbs all of the photons emitted by infra-red active gases (commonly called greenhouse gases) in the atmosphere, even though the emitting gas molecules are usually cooler than the surface of the Earth is. The energy of these absorbed photons heats the Earth’s surface and makes it warmer than it otherwise would be. In addition, the surface of the Earth emits photons as though it were interfaced with vacuum and as though it were in radiative equilibrium only, with no other energy transport mechanisms acting on it. This claim is essential to the claim that an increasing concentration of CO2 molecules in the atmosphere will cause a catastrophic increase in the temperature of the Earth’s surface.
Let us take a look at the physics of thermal radiation to examine these claims. This examination will call upon significant knowledge of mathematics, but the explanations are complete with little left as exercises for the reader. We will use thermodynamics concepts mathematically and refer to qualitative concepts of electromagnetic radiation.
Let us examine an enclosure whose walls are at a constant temperature. The radiation within this enclosure is in thermal equilibrium with the walls of the enclosure. In such a case, the energy density of the radiation, e(T), is dependent on the temperature, but independent of the volume enclosed. If the volume enclosed is V, then the total radiation energy, U, is Ve(T). For our enclosure in which the thermal properties are described only in terms of the intensive variables T and P, the radiation pressure on the walls, and the extensive variables V and S, the entropy, we have a dU = T dS – P dV. Or, T dS = dU + P dV.
The radiation pressure on the walls of the uniform temperature enclosure is equal to twice the momentum, p, component perpendicular to the wall of photons reflected off the wall times half the perpendicular velocity component, where the perpendicular velocity component tells us how many photons strike the wall per second and only half have a perpendicular velocity component toward the wall. The pressure P for a photon gas exerted in the x-direction on area A of the wall will be summed over all i = 1 to N photons:
When the cavity above is at a constant temperature Tf with the walls everywhere at that temperature, we know that the emission of photon power from the interior walls equals the absorption of photon power by the walls. If the cavity was at an initial lower temperature Ti and then was heated until it came to a new higher equilibrium temperature of Tf, the emissivity of the walls had to be greater than the absorptivity of the walls during that heating process in order to increase the energy density in the cavity from aTi4 to aTf4. Because the energy density depends on the fourth power of the temperature, a doubling of the temperature requires a factor of 16 times greater energy density. The emissivity of the walls during such a heating process must be much greater than the absorptivity to allow that 16-fold photon energy density increase. The emissivity and the absorptivity of the walls of the cavity are not just a function of the material. They are a function of whether the material is in radiative thermal equilibrium or not as well. It is very important to realize this. When the cavity was at Ti in a radiative equilibrium condition, the emissivity and the absorptivity were equal, but when the cavity was further heated to the new equilibrium temperature of Tf, they were not equal when Ti < T < Tf. Conversely, during a cooling from a higher equilibrium temperature to a new cooler equilibrium temperature, the absorptivity of the walls would have to be greater than the emissivity of the walls.
Among the properties of the cavity with volume V in radiative thermal equilibrium at temperature T is that:
Applying Bose-Einstein statistics for a boson gas with a chemical potential of zero and integrating over the photon standing wave states available in the cavity in radiative thermal equilibrium, one finds that
When the thermal radiative properties of a cavity in radiative equilibrium are applied to a surface which is not in thermal radiative equilibrium in a closed cavity condition, we must be very aware of the differences. When such a surface is surrounded by vacuum and all of its thermal conditions are described by its radiative properties, there will be an infinitesimal volume in the vacuum arbitrarily close to the surface which has an energy density e as described above from the equilibrium cavity condition. Let us imagine two infinite planar surfaces which are very close compared to their dimensions, with only vacuum between them and beyond them. If these two surfaces are at the same temperature T and in equilibrium, the energy density e is clearly aT4 everywhere in the volume between the two planes.
Now let us suppose that one of the two planes is radiatively heated from the back side and reaches a new higher temperature Th, at which the temperature becomes stable. As this hotter plane radiates energy at its higher temperature, it will increase the temperature of the nearby plane, but not to a temperature as high as Th. Let the lower temperature plane have the temperature Tc. Now the energy density in the vacuum arbitrarily close to either side of the hotter plane will be aTh4 and that arbitrarily close to the cooler plane on either side will be aTc4. This is a condition for a plane at a temperature T with a vacuum interface that the energy density e = a T4. Said in a different fashion, the perfectly thermally conducting plane would not be at a temperature T if the energy density of photons in the space immediately at the surface of the plane was not aT4. In the vacuum space in between the two planes the energy density will be a continuously decreasing function from a high value of aTh4 at the hotter plane surface to a lower value of aTc4 at the cooler plane surface. The hotter plane emits and absorbs photons as a surface at an energy density of aTh4. The lower temperature plane absorbs and emits photons as a surface would with an energy density of aTc4 in the vacuum immediately adjacent to the surface.
The energy density between the two planes is not a( Th4 + Tc4), which is what it would have to be if it were true that the photons incident upon the warmer plane from the cooler plane was that due to an energy density of aTc4 even as the emission energy density of photons from that warmer surface was that due to an energy density of aTh4. There has to be a photon energy density gradient between these two planes and this is not consistent with all of the emitted photons from the cooler plane surface being incident upon the warmer plane surface as is usually hypothesized by the advocates of catastrophic man-made global warming. They deny the existence of an electromagnetic field gradient with such rhetorical whimsy as “The photons emitted by the cooler surface must be absorbed by the warmer surface because the warmer surface cannot poll incident photons to see if they came from a warmer or a colder surface.” The electromagnetic field in effect does just that by controlling what photons are created by the field.
Even more can be deduced for this steady-state system. The outside surface of the cooler plane is emitting photons into vacuum in accordance with the energy density e = aTc4. Because its temperature is not changing, this outflow of energy has to be matched by the inflow of energy from the side of the cooler plane facing the warmer plane. The energy density of photons on the vacuum only side is the same as that on the side of the cooler plane facing the warmer plane. Therefor the entire energy density of photons in the vacuum immediately at the surface on the side facing the warmer plane is composed of photons emitted by the warmer plane. The entire energy density of photons at the cooler plane surface interface with the vacuum on the side away from the warmer plane is due to photons emitted from the cooler plane. No photons emitted from the cooler plane are to be found in the vacuum between the planes. If they were, there would be a clear violation of the Law of Energy Conservation, at least given that there is reason to believe that the one lesson that one takes from the physics of a cavity in thermal equilibrium is that the internal energy density is given by e = a T4 and that in general, surfaces at a temperature of T will have an energy density in vacuum immediately adjacent to them at the same energy density such a surface would have in the cavity at thermal equilibrium condition. In this case, not only do no thermally-emitted photons from the cooler plane become absorbed by the warmer plane surface, but none are even incident upon that surface.
Now let us assume that we have the same radiative heat applied to the outside of the warmer plane as before and we introduce a perfect gas between the two planes which has no molecules in it capable of absorbing infra-red energy or any electromagnetic wave energy of any sort. Heat transport between the two planes is now both direct radiative transport and a conductive transport due to gas molecules colliding with the warming wall and picking up heat as an increase in molecular kinetic energy and carrying that energy to the cooler wall. What is the response of the system? Clearly, the temperature difference between the walls is reduced. The hotter wall temperature drops and the cooler wall temperature increases. As the energy flow due to the gas molecule transport of energy increases, the radiative heat flow decreases as the temperature difference between the surfaces decreases.
There is yet another difference that sets in. The electromagnetic field in the volume just outside the warmer plane is determined by the oscillating dipoles in the surface. The amplitude of the oscillations is determined by the temperature. But when a gas molecule collides with this surface and picks up energy from the surface, it does so by decreasing the kinetic energy of one or more of these oscillating dipoles. Those oscillating dipoles are no longer able to pour all of their energy into creating as high an electromagnetic field as they did in the absence of the gas molecule collisions with the surface. The reduced field outside the surface must result in a reduced photon energy density outside the plane surface. Thus, what would have been a photon energy density of aTh4, must now be less than that. We know that this results in a decrease in the number of wave nodes, so the value of Stefan’s constant a is now too large. The energy density of photons is now bTh4, where b is less than a. The emission of photons is accordingly reduced.
Let us now add a film of water to the inside surface of the warmer plane. If the plane is not below the freezing temperature of the water, some of the water will evaporate. This will cool the warmer surface, taking energy out of the oscillation of some of the dipoles in the surface. Thermal radiation emissions will decrease once again, both due to the temperature of the plane being decreased and because the constant in the photon energy density, bT4, just outside the surface will decrease once again. The difference in temperatures between the warmer and the cooler planes will once again decrease. Ah, but you say that water vapor is a greenhouse gas. It is capable of absorbing infra-red radiation. Perhaps that causes a warming effect.
The latent heat of evaporation for water is a large heat which is not easily offset by other effects. If the water vapor concentration is similar to that in air and if the air is near STP conditions, then each water vapor molecule absorbing infra-red radiation is about 4 times (10 times according to Prof. Happer) more likely to lose that infra-red absorbed energy to the surrounding air molecules than it is to re-emit the infra-red energy due to the very high frequency of gas molecule collisions. Half of what it re-emits is on its way toward the cooler plane again in any case. The 80% transferred to heating the non-infra-red active molecules of the air just speeds up their transport of energy to the cooler plane. So only about 10% of the energy absorbed heads back to the hotter plane, which just equals the 10% of the energy radiated toward the cooler plane. This leaves the net effect of the photon absorption by the water vapor molecule to be a heating of the air further from the surface of the warm plane, which increases the air molecule transport of energy between the planes. It is very unlikely that the 10% of the radiant energy absorbed by water vapor and sent back to the hotter surface is a very big effect compared to the cooling effect caused by the evaporation of water on the warmer surface and the speed up of the conductive heat transfer in the gas.
What is more, the excitable water vapor molecule holds more energy due to its additional modes of internal excitation compared to perfect gas molecules. Thus, it can transport more energy to the cooler plane and transfer that greater energy to the plane upon colliding with it. Those internal modes of excitation cause the water molecule to have a higher heat capacity. The water molecule is also lighter than N2 and O2, so at a given temperature, its velocity is greater and it speeds up the transport of heat to the cooler plane relative to the heavier gas molecules.
Complex molecules have been used to fill the space between double glass windows in experiments. It was thought that doing so would impede heat flow as the complex molecule absorbed infra-red energy emissions from the warmer pane of glass. What is found is that the increased heat capacity of the complex molecules simply allows them to transport more energy and transfer it to the cooler glass pane upon colliding with it. These molecules do not succeed in slowing the heat transport at all. The fill gases that work the best are both heavy and simple. Being heavy, their velocity at a given temperature is less than that of a lighter gas molecule. Being simple, such as the monatomic inert gases argon and xenon are, they have a low heat capacity. Both are heavier than N2 and O2, the principal gases in air. In particular, CO2 performs badly as an insulating fill gas in such a case. Dry air is actually a fairly good insulating gas.
There are other important differences between the Earth’s surface and the walls of a cavity. Water covers about 71% of the Earth’s surface. Solar insolation incident on a water surface is not absorbed almost entirely within a micrometer of the water surface. It is absorbed in the first few tens of meters instead. Once that energy is absorbed and distributed over that considerable depth, heat flows to still deeper depths due to the thermal gradient that usually exists with the deeper water being colder. While the absorption of radiant energy on the land surface does occur much, much closer to the surface, heat also flows through soil, sand, and rock there to sub-surface depths. The heat flow from the near surface to deeper depths on either land or water varies throughout a day as the thermal gradient in these zones changes quite a bit through the daily cycle. So, the surface is subject to still another cooling mechanism during the day and often a warming mechanism at night, causing a constantly varying effect in the daily cycle.
In this context, the evaporation of water occurs at a rate which increases exponentially with temperature. Therefore, the water evaporation rate is commonly much higher in the afternoon period of a day and is varying considerably through the course of the day. As is the case with the conduction of heat through water, soil, sand, and rock; the conduction, convection (thermals), and advection (wind) of heat through the air away from the surface also depends upon the commonly varying temperature differentials between the surface and the lower atmosphere through the daily cycle.
Consequently, the Earth’s surface is subject to not just varying thermal radiation absorption and emission in the daily cycle, but it is also subject to varying water evaporation cooling, varying air conduction, convection, and advection cooling, and varying rates of heat flow to deeper depths below the surface or even from deeper depths toward the surface. The kind of radiative thermal equilibrium in the cavity which leads to a photon chemical potential of zero, to emissivity and absorptivity of the cavity wall being equal, and to a surface emitting a number of photons and a total photon energy dependent only on the temperature of the surface generally does not exist during the daily cycle of the Earth’s surface. It is very important that this be borne in mind when discussing the radiant energy transport based on cavities with walls at a constant temperature and only in radiative equilibrium.
Advocates of the catastrophic man-made global warming hypothesis tend to make many ill-considered assumptions about radiative heat flow. They do not understand cavity radiation in thermal equilibrium, they do not understand radiative heat flow between surfaces of differing temperatures, and they do not understand how radiative transport of heat works when other mechanisms are present that also transport heat. I have taken pains to point out a number of these misconceptions in my post Why Greenhouse Gas Theory is Wrong -- An Examination of the Theoretical Basis. Among the consequences specifically of the competing energy transport mechanisms is the fact that the Earth’s surface as a whole has an effective emissivity of a bit less than 0.5a. This commonly drives advocates of the common theory for catastrophic man-made carbon dioxide emissions global warming bananas. They keep pointing at measurements that claim that water, soil, and organic plant materials have emissivities near 0.90 to 0.95, meaning that the value of b in bT4 is 0.90a to 0.95a. When other heat transport mechanisms are prevented in an experimental measurement, the emissivity due to radiation alone will increase relative to its value when other heat transport mechanisms are available. In addition, as discussed above, a surface emissivity or absorptivity is not just a function of its material, but is also a function of whether the surface is changing its temperature. The surface of the Earth is locally changing its temperature all the time.
I hope this post will help those interested to gain a better understanding of these critical thermal radiation issues.
21 May 2015 Update: Important additional deductions were made showing that photons from a cooler body are not absorbed by a warmer body, at least in one simple system.