This post reports on some progress on resolving our difficulty replicating MODTRAN Planck curves in a prior post. In a personal communication, Dr. David Archer kindly provided the spreadsheet used to produce the MODTRAN curves.

The standard wave number Planck formula is:

Dr. Archer’s MODTRAN formula is:

We have reddened the factors where Dr. Archer’s intensity differs from Planck radiance. Assumptions can be dangerous, but we assume that when MODTRAN “Raw Model Output” is in units of radiance, the graphic is also in units of radiance.

The problem we encountered was that in consistent units of radiance, we could not reproduce the MODTRAN graphic. We got the result below:

Which differs significantly from this:

Our intention had been to increase the density of Planck curves for better resolution of the radiative temperatures in the CO2 deviation.

Remarkably, we discovered that when the factor of Pi is removed from Dr. Archer’s equation, it yields identical values to the standard Planck equation we have used.

What this must mean is that Dr. Archer’s factors of **10^8** in the dividend and **100** in the divisor are symmetrical. If so, why bother?

We discovered that the MODTRAN graphic can be acceptably reproduced in units of radiance on the same axis if the Total Radiance output is scaled by 10^4.

That’s a lot of scaling. We have done our share of scaling to make things comparable, but always with far more modest scaling factors. At 10^4 you approach a realm where scaling matters for stuff like quantum effects.

We also discovered that an eerily visually similar result can be achieved on separate axes without scaling if another .2 increment (of 10^-5 W/m2) is added to the Total Radiance axis. This essentially stretches the axis, thereby compressing the CO2 deviation.

Wow. What does all this have to say about the reliability of temperatures implied by the relationship between the Planck curves and the CO2 deviation? Ultimately the scaling (or compressing) exercise is about getting the Total Radiance away from the deviation close to surface temperature. Nobody believes that over a wide area like the tropics, anything is radiating 10-15 degrees above surface temperature, as our first efforts implied.

Once you take the plunge of setting the top of the deviation reasonably below surface temperature, things appear to fall into place nicely. The shape of the P and R rotations on either side of the fundamental bend, and the top corners of the deviation match the Planck curves too well to deny a relationship.

The graphic below appears on the University of Chicago MODTRAN website.

It is a validation of the program against measurements over the Sahara by the IRIS satellite. It is in units of radiance. (Having the N from radiance decide it needed to appear in the middle of the graphic is something that would happen to me!)

Most of the old satellite radiances are plotted against Planck curves. It is dissapointing that no mention is ever made that the measured radiances are grossly scaled or massaged to fall in line.

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The Planck curves illustrated are given by the following formula,

B(Ns, T)= 2hc^2(Ns^3/(e^((hcNs/kT)-1)

The intensity is B as a function of wavenumber Ns and temperature T.

This is the version commonly used in spectroscopy and differs from that used in physics because in physics Ns= 2πν/c as opposed to Ns=ν/c.

The formula you have is not in the wavenumber domain (as you have quoted) as you don’t have wavenumber but frequency in the equation.

Regards.

Thank you Geoff. I have since discovered the provenance of Dr. Archer’s adjustments. The spectracalc website gives the wave number formula as:

Wavenumbers are the number of waves per cm, units of length rather than time. I think this explains why when Dr. Archer’s Pi is removed the wavelength and wavenumber curves become identical.

I still don’t see why one would add 10^8 and 100 factors that cancel and have no effect. I sort of understand Dr. Archer’s Pi as the radians (radiance) of half a circle, but radiance is already defined by solid angle. Perhaps he is trying to bridge the 2Pi v/c and v/c definitions, but wouldn’t he need to adjust “raw model output” in radiance for Pi as well?

The silliest part of all is that the version 5 MODTRAN tape 7sc files contain BBODY_T[K] output that is exactly what I am trying to get at. The University of Chicago output does not include this and I can find no way to access it. I would gladly have simply used this, but maybe it is better to learn the hard way that there is considerable murkiness lurking in the apparent certainty of published numbers.

I agree that the ‘10^8’ and ‘100’ factors appear without explanation. And explanation is required.

You have said,

“I think this explains why when Dr. Archer’s Pi is removed the wavelength and wavenumber curves become identical”

But with respect, the wavelength and wavenumber curves differ by a factor of 1.76 in the position of the peak with conversion between optical accountancy methods. The peak for 294K in wavenumber accountancy units is 576cm-1, and the Wien peak for 294K expressed in wavenumber is 1014cm-1.

In the wavelength domain the Planck function has 1/λ^5 as opposed to ν^3 or σ^3.

The wavenumber domain is preferred because it is analogous to frequency and can be made to be the same curve as the frequency domain which is directly related then to photon energy except that you don’t need factors of 10E13 or 14 to think about.

This logic does render the Sun (peaking at 837nm converted from wavenumber) an infrared star which explains why 51% of its energy is infrared not visible light (37%) as is popular believed, with then the bulk of its spectral intensity in the physical domain where it excites protons not electrons.

I think you have a better understanding of the different units than I. I’m aware the choice is not trivial, but gravitated to wave numbers, figuring since MODTRAN used them I could just stay consistent and everything would work out.

I assume your factor of 1.76 in the position of the peak applies on the wavelength/number axis. If the radiances shift correspondingly the relationship to temperature remains the same? If not, I despair of ever knowing temperature from wavenumber.

I believe the confusion we have both felt is because of the nature of the wavenumber domain. The units in cm-1 are arbitrary as the standard formula you have for intensity would give m-1 not cm-1. That is where the factors of 10^8 and 100 come from. The shape of the spectrum in ν, cm-1, or in m-1 is exactly the same shape but requires scaling between standard SI intervals in metres and the chosen cm-1.

Spectralcalc specify that their wavenumber relationship considers,

σ=ν/100c,

which gives units of cm-1 from,

N=ν/c,

which would remain as m-1.

As ν is to the power of three in the intensity equation then a factor of 10^6 arises when converted to wavenumber cm-1 from frequency.

Further, Spectralcalc also state that intensity in wavenumber (cm-1), Lσ,

Lσ=(100c)Lν

So a further factor of 100 is involved to scale the total intensity per steradian to the wavenumber domain from frequency. This factor of 100 appears to provide the 10^8 scaling factor when multiplied by the 10^6 noted earlier.

As I have indicated earlier the 10^8 is compensated exactly by the additional factor of 100 in the divisor as the sole requirement of this is to allow values to be in cm-1 and not in m-1, as a purely arbitrary drift from SI units.

I cannot state this for certain, but it could be that the 10^4 scaling factor you have noted is still a result of using the arbitrary units of cm, as the intensity, in SI units, is through a square metre area which obviously differs when expressed in square cm by a factor of 10^4.

The standard formulae produce intensity or radiance in W/m^2/sr which, when multiplied by π then gives the total intensity or radiant emittance in W/m-2.

I would have thought that the IRIS comparison to Modtran should not have a factor of π in the Modtran calculations as the output is per steradian.

In response to your questions about the factor of 1.76, this only applies to conversion between wavenumber/frequency conversion to wavelength and it does apparently move the peak intensity leaving all else the same. So yes the relationship to temperature remains exactly the same, with only the fact that you cannot assume an emissivity of unity with gases, as you have sensibly noted. So you can only be reserved about specifying a temperature when you can only do this by taking ε=1 as lower emissivity versions require higher corresponding emission temperatures which come from lower altitudes to ε=0 which is the surface radiation transmitted without attenuation. At most wavenumbers the intensity is the product of attenuated transmission from a higher temperature source below and emission from the effective surface you are attempting to identify. At some wavenumbers with very high opacity, such that several optical depths over short distances exclude transmission, then this temperature sensing technique has value but has to be used with caution elsewhere.

If you can specify the opacity as high then this is of value.

In the upper atmosphere the required broadening mechanism should not be present and should lead to very low opacity due to fine line width. This ‘should’ render the upper atmosphere at 220K incapable of significant emissive power in the same way that Mars’ atmosphere has near zero emissivity because it is too cold for thermal broadening and too low a pressure for significant collisionally induced broadening.