Small Emissivity of Atmosphere with CO2

This is a continuation of the previous post Emissivity of the Atmosphere? asking the question of the emissivity of the atmosphere around wave number 667 where the trace gas CO2 is resonant while the main gases O2 and N2 are not resonant (assuming no water vapor).

We consider a model of the atmosphere as two damped oscillators here coupled in series with for simplicity acoustic damping, one with big mass M and velocity V representing O2 and N2, and one with small mass m and velocity v representing the trace gas CO2, both subject to the forcing F.

The analysis of Computational Blackbody Radiation shows that the emission E of the system as the energy consumed by the damping, is given as

• E = d v^2 + D^2 V^2 with dv^2 = D^2V^2 = F^2   

while the temperature T as the total kinetic energy (x 2) is given by

• T = mv^2 + MV^2. 

We compute assuming m << M so that T ~ MV^2 

• E ~ 2 x D^2/M x T

with thus a coefficient of emissivity 

• epsilon = 2 x D^2/M 

to be compared with the coefficient d/m in case only the small (resonant) oscillator is present. 

The emissivity of the system may thus be much smaller than that of the resonant oscillator, if the resonant oscillator has a small mass compared to the system mass. The reason is the presence of the factor D^2 in the non-resonant case to be compared with the factor d in the resonant case, expressing the essence of the analysis.

In other words, we have in a simple model discovered small emissivity of the system compared with the (maximal) emissivity of the small resonant component, that is small atmospheric emissivity by the presence of the resonant trace gas CO2.