We now turn to calculating the effect of CO on radiation, using a one-layer model, i.e. we treat the whole troposphere as a single layer:
Looking at a particular wavelength &lambda; up to &lambda;+d&lambda;, the whole atmosphere has an optical depth OD, while the tropopause has an optical depth 0.12*OD; the troposphere has an optical depth of 0.88*OD.
Thus,  
of the radiation from below the tropopause is transmitted out, but this includes  of the radiation that originates from the ground.
Thus, the weight of the troposphere in determining the radiation that is emitted to outer space is:
A relative increase in the CO concentration means an equal relative increase in the total CO content of the atmosphere, dN/N where N is the number of CO molecules. 
Adding a minute number of such molecules dN will increase the troposphere's weight in determining the radiation for the relevant wavelengths, approximately by the relative amount dN/N, and thus by:
Since CO hardly influences sunlight absorption by Earth, the radiative forcing due to an increase in CO content is equal to the difference in the flux radiated by Earth due to such an increase. To calculate this, one must multiply the above by the difference in radiation due to the difference in temperature. According to Planck's law, this is:
The ground is at temperature T = 288 K, and for the troposphere we will take a typical temperature, the one at the average height of molecules, 6.3 km, where the temperature is T247 K.
Therefore, dI, the change in Earth's emitted radiation is, in a rough approximation, is:
Since dN/N = d(ln N), this can be written as:
The function  is maximal for x = 2.41, with a maximal value of 0.66, and it drops to half this value at x=0.5 and x = 9.2. Thus we look at wavelengths for which the OD is between 0.5 and 9.2: This gives a wavelength band at the width of approximately 1 micron around 17 microns, and less than 1 micron around 13.5 microns. We therefore take:
:&lambda; = 13.5 microns and again 17 microns (summing contributions from both)
:d&lambda; = 0.5 micron for the 13.5 microns band, and 1 micron for the 17 microns band.
Which gives -2.3 W/m for the 13.5 microns band, and -2.7 W/m for the 17 microns band, for a total of 5 W/m.
A 2-fold increase in CO content changes the wavelengths ranges only slightly, and so this derivative is approximately constant along such an increase. Thus, a 2-fold increase in CO content will reduce the radiation emitted by Earth by approximately:
:ln(2)*5 W/m = 3.4 W/m.
More generally, an increase by a factor c/c gives:
:ln(c/c)*5 W/m
These results are close to the approximation of a more elaborate yet simplified model giving 
:ln(c/c)*5.35 W/m, and the radiative forcing due to CO doubling with much more complicated models giving 3.1 W/m.