# 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.

## 3 thoughts on “Model of Atmosphere with CO2 shows Small Emissivity”

1. Prof. Johnson

Thank you for the post.

I have long been bothered by the continued use of equilibrium models to describe atmospheric phenomena. I guess it is like using a hammer on a watch when it the only tool you understand.

For example, Pierrehumbert says: “Back radiation” is simply the name given to the downward infrared flux evaluated at the surface. The flux computation (in any direction) proceeds directly from the Planck distribution – – –

Such flux cannot be right for isolated molecules of CO2,interacting kinetically at one temperature, and radiatively with both space and the surface.

2. Does, in your opinion, your reasoning hold for the other gas molecules in the atmosphere as well? If so, would that imply that atmospheres shouldn’t affect a planets climate much at all?