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.

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. ]]>

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.

]]>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.

]]>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.

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