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Blog - a quantum of warmth (Rev #6, changes)

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This page is a blog article in progress, written by Tim van Beek.

Introducion The Case of the Missing 33 Kelvin

Last time, when we talked about putting earth into a box , we saw that a simple back-of-the-envelope calculation about the energy balance and the resulting average temperature of the earth is surprisingly close to the real world. But there is some gap, because the temperature predicted by a one zero dimensionalenergy balance model is lower than the actual estimated average surface temperature on earth.

In such a situation, as theoretical physicsists, we congratulate ourselves on a successful first approximation, , and look out for the next most important effect that we need to include in our model.

Most of you will of course heard about the effect that climate scientists talk about, which is often - but confusingly - called “greenhouse effect”, or “back radiation”. The term that is most accurate is downward longwave radiation (DLR), however, so I would like to use that instead.

Radiating Splitting Up Up: The Surface Hill and Atmosphere

Since To some get people a have better voiced approximation, concerns instead that of there treating is something wrong with the whole theory earth of radiation, as it a is black applied body, we’ll have to split up the atmosphere, system I into would earth like itself, to and explain its this atmosphere. phenomenon The starting solid with surface of the simplest earth possible consists thought of experiment: A single blackbody in empty space, kept at a fixed lot temperature by a “heat bath”. A heat bath is, like the perfect blackbody, an idealization of an different infinite materials container with of different heat radiation energy properties, so that can it keep is still a system good coupled approximation to say that it at is a fixed black temperature. body.

We But will use the heat atmosphere bath consists in of the example to simplify the reasoning about the effect that emitting a photon couple may have on the temperature of the gases blackbody: only, In and our we example, know it from does quantum not mechanics have that a gas consisting of, say,anyO 2O_2 effect, molecules, because or we assume that the blackbody is kept at a constant temperature by the coupling to the heat bath, no matter what happens.CO 2CO_2 molecules, has very different properties with regard to photon absorption and emission as a black body. In fact, this is one of the reasons for the invention of quantum mechanics in the first place.

a single blackbody in empty space

Tim van Beek: I would like to relate to math from now and then, but would also like to warn those readers with less interest in mathematics, or with less background knowledge, that they may or should skip such sections. Any suggestions how to do this?

In my thought experiment, I would like to have a blackbody of the shape of a cube that radiates from two surfaces only, like from the one with a 1 on it, which I call math technobabble:A 1A_1, and from the one with a 6 on it, which I call A 2A_2.

I If would you also are like interested to ignore any radiation that is emitted at a different angle than perpendicular to the surface. This may seem odd, since the blackbody is supposed to be a diffusive emitter: So it will emitt into all directions in operator theory, you’ll know the same definition way. But this assumption does not alter anything about the thermodynamics of the thought experiment, I will use it to avoid fancy calculations involving “steradians” and other complicated 3D geometry stuff. You can do all the calculations with a more complex geometry as I do here, it does not change anything about the main point.spectrum of an operator. If you haven’t looked into quantum mechanics, however, you’ll be surprised to hear that “spectrum of an operator” is actally related to the emission and absorption “spectrum” molecules: Simplifying somewhat, an eigenvalue of the Hamiltonian operator that describes a molecule corresponds to one line in the emission spectrum of a gas consisting of such molecules.

With all these assumptions, the energy flux is given given by the Stefan-Boltzmann law:

E(T A)[Wm 2]=σT A 4 E(T_A) [\frac{W}{m^2}] = \sigma T_A^4

and the overall energy per second that the blackbody emits is simply

σT A 4(A 1+A 2) \sigma T_A^4 (A_1 + A_2)

Tim van Beek: I’m running out of letters because I used E for the flux, and would like to use it for energy per second also.

Now let’s assume that we bring in a similar blackbody BB with a lower Temperature T BT_B. Blackbody AA will keep radiating as before, blackbody BB will also radiate, but will emit a lower energy flux:

two radiating black bodies

Note that none of the two will stop radiating or change their radiation because the other one is present. Also, the photons emitted by blackbody B have no qualms to cross empty space and be absorbed by blackbody A, therefore there is an energy flux fom B to A. Of course, the net energy transfer per second, on a macroscopic level is from A to B:

Energy net=σT A 4A 2σT B 4B 1 Energy_net = \sigma T_A^4 A_2 - \sigma T_B^4 B_1

For simplicity, let’s say that A 2=B 1=1m 2A_2 = B_1 = 1 m^2, so that the difference becomes

Energy net=σ(T A 4T B 4) Energy_net = \sigma (T_A^4 - T_B^4)

So, the second law of thermodynamics is not violated, because the net energy flow is from the hotter to the colder body. But individual photons are still allowed to tbe emitted by B (the colder body), travel through empty space and be absorbed by A (the hotter body).

In fact this has to happen if quantum statistical mechanics is correct.

Explaining the 33 K Gap: IR-Backradiation

Tim van Beek: The following is just a random collection of material right now!

Here is a nice overview of the spectrum of electromagnetic radiation:

radiation spectrum visualized

BTW, if you doubt that a colder black body can emit low energy photons that are then absorbed by a hotter black body, increasing its energy in the process, you may ponder the question how a microwave oven works.

From the Planck density, we can determine that sun and earth, as black bodies, emit at different wavelenghts:

Spectra of Sun and Earth

Only some components of the atmosphere emit and absorb radiation in the IR part, the part where earth’s spectrum is. These are called - somewhat misleading - “greenhouse gases”. Two prominent ones are H 2OH_2O and CO 2CO_2:

CO2 and H2O radiation spectra

The “atmospheric window” at 8 to 12μm is quite transparent, which means that this radiation passed from the surface to the atmosphere without much ado. Therefore, this window is used by satellites to estimate the surface temperature.

Tim van Beek: I would like to add radiation measurements, maybe some can be found here:

Devices to measure the infrared radiation of the planetary surface are called pyrgeometer, for pyr = fire and geo = earth.

Also have a look here.

Just to have a number, the flux of DLR (downwards longwave radiation) is about 300 Wm 2W m^{-2}.

From Zero to One Dimension

The zero dimensional model has a homogenous inbound energy flux and an averaged albedo. In the next step to refine our model, we could insert a dependency of both the radiation and the albedo of latitude. This results in a one dimensional energy balance model.

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