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Re: Sunsets

The thread on sunsets (and the blue color of the sky) has caused me enough
cognitive discord to unpack a few references in an attempt to scatter some
light on the subject.
Dan's original attempt to explain Rayleigh scattering in terms of a single
dipole oscillator
was dissonant with Max Born's (in _Atomic Physics_, 6th ed., pp. 21-22 and
pp. 333-335) assertion that the observed scattering of light by the atmosphere
was a fluctuation phenomena. He claims that if the density of air were uniform
throughout that the waves scattered by individual molecules would annul each
other and that the sky would appear black. An earlier post by John seemed to
support this approach, and his
recent suggestion as to how to solve the problem seemed to be in line with
this approach. Dan's attempt to solve this problem following John's suggestion
seemed to be in agreement with Born's assertion that all phenomena involving
deviation from the mean depend on the fact that the mean square deviation of
the number n is equal to the mean value of the number. In this case n is the
number of molecules in a fixed volume v, part of a larger volume V. Born
proves his assertion for the case that v is much smaller than V (pp. 333 -
335). He states that spontaneous deviations of the density of molecules from
the mean value are associated with a change in almost all physical properties
of the gas (such as index of refraction). If the deviation in the property can
be mathematically related to mean square deviation, then the measurement of
the deviation in the property leads to a value of the mean square deviation,
and hence to a good approximation to the number n of molecules in volume v.
Along with the ideal gas law, p*V=(#moles)*N(A)*k*T, Avogadro's number N(A)
could be determined, I would think.

If I interpret John's most recent message correctly, I get the impression that
scattering is not completely annulled by a homogeneous atmosphere, the
intensity going as 1/lambda^2, and that a calculation of the scattering ,
taking fluctuation into account would lead to the scattered intensity going
somewhere between 1/lambda^2 and1/lambda^4.

The only reference that I have access to with a discussion of how single
scattering relates to observed Rayleigh scattering is _Radiation and Optics_
by John M. Stone, McGraw_Hill, 1963, Chap. 14. He claims that, although
fluctuation in density gives rise to the general scattering from optically
dense media (media in which there are many atoms or molecules in a wavelength
cube including air for the visible wavelength range), single scattering leads
to the same result for optically dense gases as long as the ideal gas laws are
quite well obeyed and the index of refraction is close to unity (as in the
case of air and other highly transparent gases). On this basis, Stone chooses
to develop Rayleigh scattering in terms of the single scattering model of
independent dipole oscillators with no mathematical treatment of the
statistical approach. He cites the accuracy with which Avogadro's number has
been obtained by this method is an experimental check.

Stone admits (p. 337) that single scattering does not hold for an optically
dense medium
without fluctuation. He states that in an optically dense medium at any given
atom the secondary electric fields from the other atoms is not negligible. The
dipole responds to the resultant internal field, not just the applied field.
He states, "The problem is therefore no less than solving simultaneously for
the motions of all charges in the body, taking into account the interactions
through secondary waves, or the coupling, as it is called." The problem was
solved by Ewald, Oseen, and others. One of the consequences is the Oseen
extinction theorem, which, I presume, is the basis for Born's assertion that
the sky would appear black in the absence of fluctuation. For the Ewald-Oseen
theory, Stone refers the reader to _Optics_ by Born and Wolfe for a partial
treatment, and to the earlier _Optik_ by Born for a more complete treatment --
also to _Theory of Electrons_ by Rosenfeld, cautioning the reader that these
are very difficult reading.

Regarding the experimental verification of Rayleigh's law, Stone mentions (p.
348) that Rayleigh himself compared the intensity of light from the zenith to
that of direct sunlight weakened by reflection by white paper at several
frequencies. He found quite good agreement with the fourth-power (of the
frequency) law. Stone further states that careful measurements show that the
law is only approximately true for air, attributing the discrepancy to
resonances in the IR and UV in agreement with what John has already pointed
out.

Regarding Cliff's request for an elementary explanation, a few websites had
discussions which I found interesting. One at
http://www.weburbia.com/physics/blue_sky.html has some historical points of
interest including the assertion that Tyndall and Rayleigh originally
thought that dust particles and small water droplets were responsible for the
blue color of the sky. The color of the Martian sky is also discussed. Another
by Matt McIrvin at http://world.std.com/~mmcirvin/bluesky.html (with a link to
a newer version) attempts to explain the model of scattering by a single
dipole oscillator in a non-mathematical way, but including the idea of
retarded potentials. He even
arrives at the dependence on the fourth power of the frequency with little
mathematics. A short note at
http://www.faqs.org/faqs/astronomy/faq/part2/section-17.html indicates how
larger particles in the atmosphere have led to unusual colors of the sky --
including conditions leading to a blue or green moon.

If memory serves me correctly, Max Born discussed the blue sky in his little
popular paperback, _The Restless Universe_ published by Dover, but now out of
print. This book had "movies" about physical phenomena such as dipole
radiation that one observed by quickly thumbing the pages -- in the style of
the "Mickey Mouse" books of the 1930's. I believe he discussed it as a
fluctuation phenomena as in his _Atomic Physics_.

Regarding the history of Rayleigh's law (1871 according to Born), I am not
sure that Rayleigh went beyond the single scattering model in terms of dipole
radiation. Born attributes the explanation in terms of fluctuation to Marian
Smoluchowski (1908). Others attribute it to Einstein. I believe that they
arrived at it independently. There is some biographical information about
Smoluchowski at
http://hermes.umcs.lublin.pl/users/Kosmulsk/michal/smolucho.htm. He was
awarded a prize for the theoretical explanation of the Brownian Motion by the
Vienna Academy of Sciences in 1908. (I am conscious of the name Smoluchowski,
because I heard his son, Roman Smoluchowski, a distinguished solid-state
physicist and astrophysicist who died in 1996, lecture in a public series on
Cosmogony and Cosmology in 1967. According to the latter's obituary at
http://www.informatics.sunysb.edu/apap/archives/1996/0025.html , Marian
Smoluchowski knew Einstein quite well.)

Hugh Logan
Retired physics teacher


John Denker wrote:

At 07:21 PM 8/30/00 -0700, Daniel Schroeder wrote:
However, for shorter-wavelength
light, we need to consider smaller cells, for which the number of
molecules per cell will fluctuate by a greater percentage.

Right.

This effect should further enhance the blueness of the scattered light,
so the detected spectrum should be the sun's spectrum, times 1/lambda^4,
times another factor that decreases with increasing lambda. I conclude
that the best-fit "color temperature" of the sky should be even hotter
than my original, naive model predicted.

Well, how did I do?

Not bad, but you missed a factor of lambda^2.

Recall that N scatterers working independently produce an intensity
proportional to N (square before you add) whereas N scatterers working
together produce a voltage proportional to N and an intensity proportional
to N squared (add before you square). If they cooperate in cohorts of M
then the intensity goes like MN.

For air molecules, if you do all the geometry and all the counting, I
believe that you will find that in the absence of fluctuations, the
scattered voltage goes like 1/lambda (not 1/lambda^2) and the scattered
intensity goes like 1/lambda^2 (not 1/lambda^4).

When we consider fluctuations, the smaller cells fluctuate more, so the
final spectrum is somewhat hotter than 1/lambda^2 (but less hot than
1/lambda^4).

I haven't checked this carefully, but it smells right to me, and it seems
to be in reasonable agreement with the photographers' "9000 K" data
point. (Note that a pure 1/lambda^4 spectrum corresponds to an infinite
color temperature; anything steeper than that would be "hotter than
infinity".)

===========

Note that the 1/lambda^4 per-molecule result applies only to electrons
bound to molecules, and only to frequencies below resonance. This can be
generalized to a wider range of frequencies, namely: w^4 / (w^2 - w0^2)
where w (omega) is the actual frequency (2 pi c / lambda) and w0 is the
resonant frequency.

Note that the 1/lambda^2 collective result given above is consistent with
the familiar result that the index of refraction for ordinary substances
(solids, liquids, and reasonably dense gasses) is constant except near
resonances:
n = 1 + s /(w^2 - w0^2)
where s is some strength factor.

If/when the cohort-counting argument does not apply (e.g. for molecules
that are far apart relative to the wavelength) then we expect an extremely
dispersive dispersion relation:
n = 1 + s' w^2/(w^2 - w0^2).
with a probably rather small strength factor s'.