An Infrared Photon’s Life in The Atmosphere

Did you know that the fraction of the atmosphere that is CO₂ is 0.0004 by volume? By the atmosphere I really mean the troposphere, which is the layer of atmosphere between the Earth’s surface and approximately 17 km (11 miles) above the Earth in the middle latitudes. It is approximately 20 km (12 miles) deep in the tropics, and in the polar region the troposphere is only about 7 km (4.3 miles) thick in winter. About 75% of the total atmospheric mass and 99% of its water vapor and aerosols reside in the troposphere. This is the layer where all the Earth’s weather resides and where all green house warming takes place.  

The number for the CO₂ fraction comes from measured values of approximately 400 parts per million (ppm) by volume. That means that if you count the number of molecules in a given volume of air, for every 1 million molecules of all types you count, you will find 400 CO₂ molecules. Its fraction of the atmosphere is then

.

The actual number density of CO₂ at the Earth’s surface is n = 9.8×10²¹ m^-3 = 1/L³, where L is the average distance between CO₂ molecules.Therefore, even though carbon dioxide forms a very small fraction of the atmosphere, there is still a huge number of carbon dioxide molecules in a cubic meter. The density gives us the distance L as L ≈ 4.7×10^-8 m, which is a very small distance indeed.

At first you might think that L is the average distance an IR photon must travel between collisions, but you would be wrong. The time between collisions comes in two parts. The first part is the time of flight of the photon between the CO₂ molecules which is

τ_f = L/c =1.6×10^-16 sec where c = 3.00×10⁸ m/sec is the speed of

light. The second part is the time in which the photon remains captured by the carbon dioxide molecule, which is the time the molecule takes to decay back to its original state and remit the photon. Depending on the mechanism that stimulates the decay, I have seen estimates of decay time that vary from τ_c = 1 μs = 1.0×10^-6 sec to τ_c = 0.45 sec. Whatever value we take for the capture time τ_c, we will have in any case a total time between collisions of τ = τ_f + τ_c ≈ τ_c.

Another important fact about the CO₂ molecule is that it has only four main peaks in its absorption spectrum in the infrared region at wavelengths of 1437, 1955, 2013 and 2060 nanometers. This means that the molecule can absorb only four photons, and only one photon in a narrow band about each absorption peak. Any photon not within one of those four bands will not be absorbed under any situation. All infrared photons outside the absorption bands will almost immediately be lost to space without affecting the atmosphere. Because the photon’s average time of flight between CO₂ molecules,  τ_f , is much less than its capture time τ_c , we can expect a constant flux of infrared photons passing totally saturated molecules at the speed of light at the lowest levels of the atmosphere. The flux of photons in the appropriate frequency bands for absorption simply pass the saturated molecules faster than the molecules can give up the photons absorbed to make energy levels available to allow absorption of more photons! How thick, Δ, is this layer of totally or almost totally saturated molecules? That is a problem that I have not worked out yet, but at the very least we can give it a lower limit of cτ_c < Δ.

Whatever this length is, it is the effective diffusion step size between collisions of the photons with the CO₂ molecules. The bigger this step size is, the sooner these photons will diffuse to the top of the troposphere to be lost to space, and the fewer collisions they would make with unsaturated molecules. The fewer such collisions, the fewer the momentum transfers between the photons and the molecules, and therefore the less heating of the atmosphere there would be.

Let us see if we can make an estimate of the maximum time a photon will stay in the atmosphere. The shorter the diffusion step size is, the longer it will take for the photons to diffuse to the top of the atmosphere. Therefore, if we make the approximation

Δ ≥ Δ₁ = cτ_c

we will maximize the estimate of the average lifetime of an infrared photon in the atmosphere. Since the smallest Δ₁ gives the longest average photon stay, we will take the smallest value for τ_c=1 μs = 1.0×10^-6 sec. Then our estimate for the step size becomes

Δ₁ = cτ_c = 300 m.

This is a huge number! The reason it is so large is that once a CO₂ molecule becomes saturated, it stays saturated for a relatively long time, allowing photons to go a long distance before they suffer a collision with a molecule.

After the photons go through a finite number of collisions, they will start to dribble out of the atmosphere. Which will be the first photons lost? They will be the ones always emitted by the interacting molecules in a vertical direction, i.e. always forward-scattered. Let us say that this starts to happen after N collisions, and let us take the troposphere thickness as its largest value over the equator, 20 km. Then

NΔ₁ = cNτ_c = 300N m = 2×10⁴ m or N ≈ 66.

The first photons will be lost to space after 66 collisions and a time of Nτ_c = 66τ_c = 6.6×10^-5 sec = 66 μsec after having been reflected by the Earth’s surface. Of course these photons are only the first ones out of the total set of photons that started at the Earth’s surface at the same time. Even if it takes a hundred times the 66 μsec for the rest of the diffusing photons to dribble out, the time is shockingly small. And recall that these numbers result from a systematic set of assumptions that minimize the diffusion step size and maximize the time the photons stay within the atmosphere. Also, we implicitly made the simplifying  assumption that the CO₂ molecule density was constant and not decreasing with altitude as in reality. Taking this into account would give us a diffusion step size that increased with altitude, therefore decreasing the time required for photon escape, the number of collisions the photons experience with carbon dioxide molecules, and the amount of atmospheric heating. The actual times, number of collisions, and atmospheric heating are probably considerably smaller. Of course, we have yet to calculate the rate of heating. To do that we really should treat this theoretical problem in a more systematic and careful manner.

It should be apparent that infrared photons are not in any way “trapped” by greenhouse gases; their trip back to space is merely delayed by diffusion and not by much. All those infrared photons that do not have energies that would cause a molecular transition to a higher energy state, probably the vast majority, would not interact with the carbon dioxide at all. Although our theoretical estimates are rather rough, a more careful treatment would almost certainly cast the Anthropogenic Global Warming (AGW) theory in a much worse light.

In our next post on this subject, we will take a look at some of the evidence that the AGW proponents cite.

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