Read 2.3.2 again. Stimulated emission is only significant in the absence of collisions.

Under conditions as found in the troposphere with collision rates between molecules of several 109 s−1, any induced transition rate due to the thermal background radiation is orders of magnitude smaller, and even up to the stratosphere and mesosphere, most of the transitions are caused by collisions, so that above all they determine the population of the states and in any case ensure a fast adjustment of a local thermodynamic equilibrium in the gas.

And it is included in the Swarzschild equation. At LTE, absorption must equal emission whether there is stimulated emission or not.

]]>For the moment, I’m ignoring scattering. The Schwarzschild equation tells us how radiation traveling through a non-scattering atmosphere changes with incremental distance traveled. It is accurate enough to make reasonable estimates of how radiation behaves in the earth’s atmosphere.

At a more fundamental level, however, emission and absorption of photons are described by Einstein coefficients. How is the Schwarzschild equation derived from this perspective? It seems to me that their must be a mathematical relationship between Einstein coefficients (and excited state lifetime) and the absorption cross-section. As best I can remember. the physics presented at SOD hasn’t covered this relationship. Instead, absorption cross-sections are MEASURED in the laboratory and we apply that cross-section to both the absorption AND emission terms of Schwarzschild equation. This paper presumably addresses some of these questions. See equations 16 and 17, connecting Einstein coefficients to the Planck function. Also see the Schwarzschild eqn, which arises from eqns 47 and 48, which connect a single cross-section to the Einstein A coefficient.

http://www.hindawi.com/journals/ijas/2013/503727/

In this paper, a lot of physics (that I don’t understand in detail) is needed to connect the absorption coefficient we measure in the laboratory to the coefficient used in the emission term of the Schwarzschild equation. The connection between absorption and emission is always made by assuming radiation in equilibrium with its surroundings at some temperature, ie that absorption rate = emission rate. And a Boltzmann distribution of energy states is often postulated (LTE), though Section 4.3 of this paper doesn’t. Section 2.3.2 deals with stimulated or induced emission (which isn’t included the Schwarzschild eqn.) I’m surprised to see that stimulated emission rates for some of the strongest CO2 lines may be as big as 4% of the spontaneous emission rate.

]]>The Einstein coefficients do not *define* the Swarzschild equation. It can be derived without any assumptions about the details of the absorptivity/emissivity as a function of wavelength. If instead of absorption, you include scattering from aerosols and call it extinction, the Einstein coefficients are not sufficient. Also, If you are not using a line-by-line program to calculate radiative transfer using the equation, you’ll never see them.

..these MIT physics’ notes are wrong..

Time for you to do the heavy lifting.

Instead of assuming that because someone at MIT may have been lacking precision in their physics notes (I haven’t looked at your link), and “therefore” the whole of climate science may also have copied this same “mistake” – just take a look at the equations of radiative transfer and explain which step is wrong – and why:

Understanding Atmospheric Radiation and the “Greenhouse” Effect – Part Six – The Equations.

I will wait for your update.

]]>Emissivity is fully as well defined for a isothermal layer as it’s for a surface. It’s the ratio of the radiation flux that originates in the layer and exits the layer to the radiation flux of a black body. A thin highly transparent layer has a small emissivity, but that does not make the definition any less clear.

Emittance is much more confusing that emissivity. I would say that emissivity is not confusing at all, while emittance is, and it’s particularly confusing when it’s used as it is those lecture notes. Those notes are extremely concise, thus it’s not surprising that they don’t explain everything, but the second of the pages linked is poor even on the standards of very concise presentation, and not only because of the use of the word emittance. (The first page is OK).

]]>http://www.hindawi.com/journals/ijas/2013/503727/

The same author has a more controversial article in the same journal

http://www.scipublish.com/journals/ACC/papers/download/3001-846.pdf

]]>We should start by including (at least) three alternatives, not two:

– transmission

– emission or absorption

– reflection or (back-)scattering

In case of gas transmission and emission/absorption dominate. There’s also some scattering, but to a good approximation it can often be ignored.

In case of typical solid surfaces we have emission/absorption and reflection/back-scattering.

For a layer of water (or glass) all three must be included. It’s, however, possible to consider oceans (etc.) in a way where transmission is excluded, when the depth of absorption is not of interest. In this case reflection/scattering occurs mostly at surface, but also as scattering from varying depth.

The sum of transmissivity, absorptivity, and reflectivity is always one, whether one of the three is negligible (as reflectivity is typically for gases) or all three are significant.

From this consideration, you can see that there’s nothing strange in the definition of the absorptivity of a layer of gas.

Defining emissivity is not as clear for a thick layer of gas, when the temperature is not constant throughout the layer. Comparing with a blackbody is not possible, when the temperature is not well defined. For layers thin enough to have a single temperature emissivity and absorptivity are both well defined and equal also for gases.

(On the terminology emittance seems to be the most problematic. I think that it’s better to avoid using it as a synonym for emissivity. On that point I agree with AlecM, but that must be almost the only issue related to physics on which I agree with him.)

]]>Einstein coefficients are not directly related to the Swarzschild equation. That’s the differential equation describing how the intensity of a beam of light of a given wavelength varies with distance along a path where the absorptivity and emissivity at that wavelength are between zero and one.

The Einstein coefficients are the probabilities of the spontaneous (A) or stimulated (B) emission of a photon from an excited state. If you know the number of excited states, the volumetric spontaneous emission intensity is the product of the number of states and the corresponding Einstein coefficient for that state.

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