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Re: blue sky



John Denker wrote:

At 05:50 PM 8/31/00 -0400, Hugh Logan wrote:
... Max Born's ... 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.

I hope we can all agree on that! It seems super-obvious based on everything
we know about the optics of homogenous materials.

Apparently, arriving at such results on the basis of the Ewald-Oseen
theory was not
all that obvious. John Stone, in _Radiation and Optics_ (p. 338) agrees
that the
results that he cited, are results already known on empirical grounds,
presumably
for optical media that are actually sufficiently uniform -- like glass.
But a
uniform atmosphere, to which the Ewald-Osteen theory was applied was
hypothetical,
the real atmosphere being a fluctuating one. Presumably, the Ewald-Oseen
theory can
be applied to situations where the results are not so obvious. According
to Stone,
"The Ewald-Oseen theory ... gives as deep an insight into the optics of
dense media
as is possible on a classical basis, but a more useful approach is
through the
macroscopic Maxwell equations ... ." Stone's treatment of optics is
based on the
Lorentz theory of electrons. In Chap. 15 he arrives at the macroscopic
Maxwell
equations from microscopic Lorentz form of these equations. He states
that the
"most correct and satisfying procedure for arriving at the macroscopic
theory of
Maxwell from the microscopic theory of Lorentz, particularly at high
frequencies,
follows the lines of Ewald and Oseen ... . This difficult calculation
can be
avoided ... ." Once the macroscopic Maxwell equations are applied to
homogeneous
media, one can arrive at "super obvious " results such as Snell's law
and the
reflection laws (via the Fresnel equations as I recall), consistent with
the
results of the more difficult Ewald-Oseen theory. It is obvious that
that the
macroscopic Maxwell theory cannot explain scattering by microscopic
particles of
the atmosphere, since it assumes that the granularity of the microscopic
picture
has been removed by an averaging process -- much as the microscopic
details of
bodies are assumed to be averaged out in the mechanics of continuous
bodies.
However, Stone (p. 339) points out that the macroscopic Maxwell theory
applies to
problems in which the scatterers are small bodies of an optically dense
medium.
These bodies would be large compared with the microscopic particles of
the
surrounding medium. As an example, he cites Mie scattering by spheres
(as in the
case of water drops in fog or rain). Similar applications to scatterers
with shapes
other than spheres have been carried out. The "obvious" results of the
Ewald-Oseen
theory as applied to the hypothetical atmosphere of uniform density
(without
fluctuation) are obtained by treating the microscopic particles (I
think) of the
atmosphere and their couplings. Assuming I have stated this correctly,
it is
probably already obvious to the
pros.

(I apologize for so much reference to Stone's text, but it is the only
relevant one
that I have conveniently at hand. I have only skimmed this book, having
picked it
up years ago because of its treatment of the Kirchoff theory of
diffraction.
Whether or not one likes the approach to physical optics based on the
Lorentz
electron theory, I now appreciate that the book is exemplary in
specifying the
models on which the development is based. Stone
does not develop all of the ideas discussed qualitatively in the text --
like the
Ewald-Oseen and Mie theories, presumably because of the intermediate
level of the
text. Nor does he develop scattering by the statistical approach to any
extent. The
text is probably out of print, as it is not listed at amazon.com. Stone
was on the
faculty of U. of Cal., Berkeley.)

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

Eeek? That's not right. I didn't say that.

I was referring to the critique of Dan's solution of the problem:

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).

To summarize:
is as follows:
-- Yes, coherence _depends_ on wavelength.
-- No, you can't blithely set the coherence length _equal_ to the wavelength.

Thanks for the diagram (including the revised one) and the corresponding
explanation, which was interesting -- particularly the idea that the
thermally-excited sound mode
excitation didn't depend on the wavelength; also the idea that the
interaction
region was vastly larger than the wavelength cubed. I don't know if
something like
this is what Stone had in mind when he stipulated that the total volume
of a gas of
low density could not be too large for single scattering to hold. In any
case I
don't think that fluctuation can be ignored in explaining the blue sky
as was
suggested elsewhere.

Thanks for the references. I will try to find my copy of Feynman, Vol.
1.

At 05:50 PM 8/31/00 -0400, Hugh Logan wrote:
Stone refers the reader to ... cautioning the reader that these
are very difficult reading.

Hmmmm. Born. Smoluchowski. Einstein. Anderson. I'm not sure this topic
should be considered low-hanging fruit. :-)

The difficult reading that Stone refers to (p. 340) is the Ewald-Oseen
theory which
happens to be treated in texts by Born and Wolfe, Born, and Rosenfeld.
Everybody
knows that _Optics_ by Born and Wolfe is a difficult text, even for
other topics.
As mentioned, Stone avoids developing the statistical work of
Smoluchowski and
Einstein. I am not sure which Anderson is being referring to. (I recall
an _Optics
of the Electromagnetic Spectrum_ by one of the Andersons.)

Hugh Logan
Retired physics teacher