There is a lot of confusion about emissivity. Emissivity is the tendency to emit; particularly the tendency to emit light after absorbing it. In the infrared part of the spectrum, where the earth emits radiation towards the atmosphere, there is very little scattering. Incident light is either absorbed, or it passes through.
It can be seen that the light transmitted is equal to the light that passes unhindered plus the light that is emitted after absorption. The rectangle above can be thought of as a slab of atmosphere. Transmission is the incident light minus the transmitted light. Transmissivity is the transmitted light divided by the incident light. Emissivity is the proportion of light emitted to the light absorbed. For a perfect blackbody, emissivity is 1.
CO2 is far from a perfect blackbody.
Hottel and Leckner measured the column emissivity of CO2 in the 1940’s and 1970’s, respectively, at .14. Nasif Nahle has calculated a much smaller emissivity, .002, using two different methods. Staley and Jurica (1972) get .19. The range seems to be between poor and spectacularly poor emissivity, .002 to .2.
It turns out that in either case, the extremely high (98% in one meter) absorption of CO2 at 15 microns/wave number 667.4, and the poor transmission (2%), renders emissivity unimportant as regards upward radiation. The range of emissivity simply bounds the amount of light that passes straight through without being absorbed.
Above is the one meter absorption of CO2 at 400 ppm. The Q branch at 667 is 98% absorbed in one meter. This means that only 2% is transmitted. This transmission includes both absorbed and re-emitted light and unhindered light passing straight through.
Where emissivity becomes important is back radiation, as an equal amount to what is absorbed and re-radiated up, must be re-radiated downwards as well. The highest published full column emissivity is ~20%. This would seem to be the high limit of full column back radiation.
What about individual layers?
Staley and Jurica give a value of .08 for CO2 emissivity of a slab of one centimeter optical depth, a value of .14 for 10 centimeters, and a value of .19 for a meter. Optical depth is defined in several different ways by astronomers, chemists, and atmospheric scientists. Astronomers treat optical depth as the mean free path through a slab. In this treatment “mean free path”, “path length”, the distance a photon travels after entering before interaction, the average distance between interactions, the distance between the final interaction and escape, the distance you can “see” into the material, and optical depth; are essentially the same.
Optical depth is also defined as the path length times the absorption coefficient. The one meter absorption coefficient for the CO2 fundamental bend is .98. Path length is defined as partial pressure times the layer thickness. If the layer thickness is one meter, and the partial pressure is .04, the path length also becomes .04. We multiply this by the absorption coefficient to get an optical depth .04*.98*100(centimeters of CoE to a meter)=3.9
Beer’s Law defines optical depth as the negative natural log of transmittance. This is one of those mysterious empirical fits that work with surprising frequency. Transmittance is one minus whatever doesn’t get through (absorptance), so the negative natural log for a meter layer thickness and therefore the optical depth is -Ln(.02)=3.9. What luck.
The difference between the astronomical approach to optical depth and the chemistry and physics approach is that for astronomers optical depth is an actual distance, while the physical/chemical optical depth is dimensionless.
We can do a sanity check above where we plot linear values through a one meter slab by projecting centimeter scale values from measurements and calculations (.02 transmission, .08 emissivity/cm optical depth) We know perfectly well that none of this is linear, but transmission, and transmission minus emission are virtually indistinguishable. What this crude exercise can tell us is that it is very unlikely that any 667.4 photons pass through a one meter slab of atmosphere at current CO2 concentration. What we see transmitted through the slab is absorbed and re-emitted.
We can therefore assume that transmission, emission, and back radiation are all equal.
Back radiation must run the gauntlet of CO2 molecules on its way down as well. What we see radiated back down has the same mean free path to escape as what is transmitted up.
While the optical depth and free path remains the same up and down, the number of photons absorbed coming up, compared with those re-emitted in either direction, is reduced by the factor of emissivity. Back (downwelling) radiation can be no more than 2% of upwelling radiation.
The Schwarzschild equation used in radiative transfer models gets this completely backwards. The logic of this equation includes a “sink function” for absorption and a “source function” for re-radiation. This “source function” is given as the absorption coefficient, .98 in this case.
This would be true according to Kirchhoff’s Law if CO2 was a good blackbody with an emissivity of 1. We have seen that CO2 is a lousy blackbody with full column emissivity somewhere between .002 and .2. The value we have developed above, .02, falls in this range.
The “source function” in the Schwarzschild equation must be very significantly adjusted downward to reflect the real world emissivity of CO2.