Emissivity of Atmosphere?

Does the presence of the atmospheric trace gas CO2 give the atmosphere emissivity = 1 for wave number around 667 where CO2 is fully absorbing/emitting, or is the emissivity of the atmosphere there small because CO2 is a trace gas? This is the central question as concerns the impact of the trace gas CO2 on the energy balance of the Earth with atmosphere. We now give an argument suggesting that the emissivity is small, with further more precise evidence given here.

Let us seek an answer in the analysis of blackbody radiation exposed on Computational Blackbody Radiation.  The basic element is the analysis of damped harmonic oscillators under forcing, which shows two basic cases described by damping parameters d and D and oscillator velocities v and V:

1. Resonance: Emission = d x v^2
2. Not Resonance: Emission = D^2 x V^2.

Coupling oscillator 1 and 2 in parallel so that v = V, we have

• Emission of 1+2 = d x v^2 + D^2 V^2 = (d + D^2) V^2 

Assuming that 2 has mass M much bigger than the mass m of 1, it follows that most of the internal energy representing temperature T, is given as carried by 2 because

• T = (m + M) V^2 

which gives

• Emission of 1 + 2 = (d + D^2)/(m + M) x T
• Emissivity of 1 + 2 = (d + D^2)/(m + M)

to be compared with case 1 alone:

• Emission of 1 = d/m x T
• Emissivity of 1 = d/m (representing maximal emissivity = 1 with normalization)

Clearly, if D^2 < d and M >> m, we have that the emissivity of 1 + 2 is comparatively small.

In other words, if CO2 is a small fraction of the atmosphere, the emissivity of the atmosphere in the band where CO2 is absorbing/emitting, is small.

PS For a more relevant alternative analysis see following post.