Model of Atmosphere with CO2 shows Small Emissivity

Here is a another argument indicating that the effect of the atmospheric trace gas CO2 on the radiation balance of the Earth is small.

Our model of blackbody radiation consists of collection of oscillators with small damping with equal oscillator internal energy T representing temperature, with oscillator resonance frequencies n varying from 1 to a cut-off set at T and each oscillator radiating

• $E_n = \gamma T n^2$

where $\gamma$ is a universal constant, which is Planck’s Law.  Summing over n from 1 to T, we obtain the total radiance

• $E =\sum_n \gamma T n^2 = \sigma T^4$

which is Stefan-Boltzmann’s law with $\sigma = \gamma /3$.  In the case of only one resonance frequency $n = T$, the radiance would be reduced to

• $e = \gamma T T^2 =\gamma T^3 = 3 E/T$

with the reduction factor $1/T$.

The radiance of an atmosphere which is fully opaque over the entire spectrum would radiate $E$, while an atmosphere opaque only for a specific frequency near cut-off $T$, would radiate $e\approx 3E/T$ with a reduction factor  $1/T$.

We conclude that the emissivity of transparent atmosphere with a trace gas like CO2 with only a few isolated resonances, would scale like 1/T and thus be small as soon as T is bigger than say 100 K.

We thus find theoretical evidence from a basic model that the emissivity of the Earth’s atmosphere with the trace gas CO2 would be small, and thus that CO2 would have little effect on the Earth’s radiation balance.

We thus find theoretical evidence from a basic model that the emissivity of the Earth’s atmosphere with the trace gas CO2 would be small, and thus that CO2 would have little effect on the Earth’s radiation balance.