This post was begun some time ago and we experienced so much trouble replicating the MODTRAN Planck curves that two rambling subsequent posts transpired recounting the misadventure. Here, and here. We eventually concluded that the Planck curves in the MODTRAN output graphic can be satisfactorily replicated if the MODTRAN line radiances are scaled by a factor of 10,000. No mention of this tidbit is made in the MODTRAN documentation or many preceding graphics from IRIS satellite measurements plotted against Planck curves.
Having replicated the curves vs radiances, it should now be possible to use brightness temperature to determine altitude by lapse, after similarly scaling the radiances. Brightness temperature is simply an inversion of the Planck formula to give temperature rather than radiance.
We resume the post below as originally begun. Hold on tight, nothing has been smooth sailing so far.
We left off a prior post with the graphic shown again below. It plots MODTRAN CO2 only (the other absorbing gasses zeroed out) 410ppm CO2 total upward IR radiation in one kilometer increments from the surface (1 meter) to 70 km.
This is well and good to establish that several watts of CO2 radiance to space is coming from the stratosphere, but what altitudes and which CO2 absorption bands are involved?
We established from the graphic above that the deepest part of the CO2 deviation from the surface Planck curve conforms to and radiates at the 220K Planck curve.
This portion of the CO2 radiance deviation conforming to the 220K Planck curve corresponds to the rotations linked to the fundamental bending mode of the CO2 molecule.
But what is MODTRAN actually seeing from 70 km? The 70 km line of sight crosses 220K on the lapse curve three times! This is where the ability to calculate the emission temperature from the radiance might be helpful, particularly at temperatures that don’t cross the lapse curve three times.
In the graphic above we have labelled the significant absorption lines of the CO2 molecule in a CO2 only greenhouse world. MODTRAN has plenty of resolution to distinguish these according to the table below.
If we use our eyechrometer on our densified Planck curves below, we can get approximate radiative temperatures for the significant lines.
If CO2 were a good blackbody, we would be able to check these radiative temperatures using the Stephan-Boltzmann equation. We tried this but it yielded temperatures far out of range for our atmosphere. CO2 is a lousy blackbody with a column emissivity of about .2. Applying this correction was no help.
In the next post we will check these eyeball temperatures against similarly scaled brightness temperatures.