Originally Posted by alastair
The scheme used today is based on the Schwarzschild's equation, which he proposed for the radiation in the Sun. Emden then used it for the Earth's atmosphere but that behaves very differently. The Sun's atmophere radiates as a black body, but the Earth's atmosphere only emits lines. It is the Earth's surface that radiates as a blackbody.
With the Sun, each layer radiates based on its temperture, but in the Earth's atmosphere the greenhouse gases, which are the only radiators, radiate based on their excitation. They are excited by the blackbody radiation from the surface of the Earth, but are then relaxed by collisions with other air molecules, Thus their warming effect is confined to a layer of the atmosphere only 30m high.
It has been argued that this means that increasing CO2 would have little effect on global temperatures, but using Beer's Law if you double CO2 then the layer heated would only be 15 m high. Thus the greenhouse effect increases linearly with gas concentration, not logarithimetically as proposed at present. Moreover, the thin layer of the atmosphere which touches the earth's surface will warm the most, and cause mountain and sea ice to melt. This change to the Earth's albdeo has been a major driving force in climates of the past.
If you look at spectra of the Sahara, Arctic, or Mars you will find that CO2 always emits with a Plackian temperature of 218 K. Rather than basing the radiation from CO2 on the Planckian temperature of the air layer, it would be more correct to base it on 218 K, ie a fixed value. Water vapour would require a higher value, and a also a small component for layer temperature.
You have misunderstood a lot of things. Here, I only outline one of them:
You have misunderstood Eq. (6.9) in Schwarzschild's book, Structure and Evolution of the Stars. Although that equation look exactly as the energy density of black body radiation, E = a T^4, you have mis-interpreted this equation. You should NOT just read the face value of this equation. You should read the derviation of this equation (i.e. how to get this equation from Eq. 6.5). The problem Schwarzschild taking about is NOT on the emission itself, but on the radiative transfer through a gas. This is also the problem people talking about in solar physics and meterology. Notice that the Sun's atmosphere does NOT really radiates as a black body.
Please read the derivation carefully. The derivation is given on page 37 to 41 (Chapter 2). Notice that the emission is represented by j in this derviation. Hence, the equation you referred to is NOT about radiation emission.
The line emissions of air molecules have also considered in solar physics and meterology because they are PART of the radiation calculation. Have you read page 620 of Hansen's paper that come with EdGCM?.