Re: Bohren on blue sky and sunsets

[Hugh Logan <hlogan1@BELLSOUTH.NET>]

> And what about the issue of density fluctuations in the atmosphere?
> Irrelevant for an ideal gas, according to Bohren and Fraser.
> They claim that, since the molecular positions in an ideal gas
> are uncorrelated, the total scattering rate is just N (the number
> of molecules) times the molecular scattering rate.  Jackson
> (second edition, page 422) agrees.  Density fluctuations are
> important (they say) in liquids and high-density gases.  I'm not
> sure I fully understand this issue myself, though.

Just how ideal must the gas be, and how do they define a high density
gas?

The reference I cited, _Radiation and Optics_ by Stone agrees that
single scattering (such that the scattered wave for any atom is the same
as though the atom were alone in the incident field) as you have
described above applies to a gas of sufficiently low density that the
atoms are far enough apart that, at a given atom, the field resulting
from the other atoms is weak in comparison with the incident wave. He
does stipulate that the forward direction has to be given special
consideration. He also states that the total volume must not be too
large. He gives the caveat that it is difficult to be very precise about
the conditions under which single scattering holds.

As for the density of a medium in the context of scattering, Stone
emphatically means density in the optical sense. A dense medium in this
sense is one in which the number of atoms in a wavelength cube is
significant -- so that at a given atom, the secondary electric fields
from the other atoms is not negligible.

Stone states that normal air is a dense gas relative to visible light.
There would be at least
10^6 molecules in a wavelength cube at NTP. (Of course, this would be
less at higher altitudes). For such a gas, according to Stone (and Max
Born), fluctuation would have to be taken into account to get the
results of single scattering as if there were no interaction
among the scattering molecules. As I mentioned previously, the index of
refraction of the gas must be close to unity, and the gas should act
quite like an ideal gas. I presume this means the gas should obey the
ideal gas law quite well. But I don't think this means a gas
so ideal that its molecules are mass points with no other attributes.
The molecules must be small compared with the wavelength, but, whether
or not the gas is dense, the molecules must have the capability of
acting like dipole oscillators. (In the single scattering case, perhaps
the gas might be considered more ideal because the interaction -- or
coupling -- among the atoms is negligible.)

Assuming that Born, Einstein, and Smoluchowski were not incorrect, I
would like to see
how the Rayleigh law is derived on the basis of fluctuations and to what
extent the single scattering result for dipole oscillators enters into
this. The elementary explanations I have seen of the fluctuation
approach explain that the relative fluctuation in density is greater
in a wavelength cube at the violet end of the spectrum than at the red
end, resulting in greater scattering at the violet end of the spectrum.
But the fourth power (of the frequency) single oscillator result is not
mentioned. It seems like the obvious wavelength dependence
of the fluctuation would change the already nearly correct single
oscillator result by more than a small correction -- as evident from
recent attempts to solve the problem.

Hugh Logan