Re: blue sky

[John Denker <jsd@MONMOUTH.COM>]

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.

> 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.

Perhaps this is the source of the confusion:  I wrote about
 -- index of refraction, and
 -- scattering phase shift
in the same message.  Note that when light passes through a plate of a given
thickness and a given frequency-independent index, it picks up a _time
delay_ that is independent of frequency.  However it picks up a _scattering
phase shift_ that is proportional to frequency.  Phase is frequency*time.

I'm sorry if I didn't adequately delineate the discussion of phase from the
discussion of index.


At 02:39 AM 9/2/00 -0400, Hugh Logan wrote:
> 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

Statements like that crop up a lot in discussions of the blue sky.  However,
I believe that misses the real physics, and I regret having fallen for that
trap earlier this week.  ISTM the real physics is as follows:

Consider the following diagram of the interaction:

   .
  .    .
 .    .    .
.    .    .    .
    .    .    .    .
        .    .    .    .                     transmitted -->
|   |   |   |   |/  | / |  /|   |   |   |   |   |   |   |   |
|   |   |   |   /   |/  | / |  /|   |   |   |   |   |   |   |
|   |   |   |   |   /   |/  | / |  /|   |   |   |   |   |   |
|   |   |   |   |   |   /   |/  | / |  /|   |   |   |   |   |
  incident -->              /    /    /    /
                                /    /    /    /
                                    /    /    /    /
                                        /    /    /    /
                                            /    /    /    /
                                scattered       /    /    /    /
                                         -->        /    /    /
                                                        /    /
                                                            /


Here it is again with labels on some points in the interaction region:

   .
  .    .
 .    .    .
.    .    .    .
    .    .    .    .
        .    .    .    .                     transmitted -->
|   |   |   |   |/  | / |  /|   |   |   |   |   |   |   |   |
|   |   |   |   /   |/  | / |  /|   |   |   |   |   |   |   |
|   |   |   |   |   /   |/  | N |  /|   |   |   |   |   |   |
|   |   |   |   |   |   P   |/  | / |  /|   |   |   |   |   |
  incident -->              /    /    /    /
                                /    /    /    /
                                    /    /    /    /
                                        /    /    /    /
                                            /    /    /    /
                                scattered       /    /    /    /
                                         -->        /    /    /
                                                        /    /
                                                            /

Suppose we are trying to identify a pattern in the index-deviations that
will result in strong scattering.
  -- At point P, a crest lines up with a crest.  A positive deviation in the
index will make a positive contribution to the overall interaction.
  -- At point N, a crest lines up with a trough.  A _negative_ deviation in
the index will make a positive contribution to the overall interaction.


In fact, we can classify all the zones in the interaction region as to
whether a positive or negative index-deviation results in a positive
contribution to the desired interaction.  Call them the "P zones" and "N
zones" respectively.  (This assumes we have made some arbitrary choices
about the relative phases;  this can be done without loss of generality.)
(This also assumes that the interaction region is transparent to zeroth
order, which is consistent since we considering the limiting case of pure
air without clouds, dust, or pollution -- but this is not generally a good
assumption in real life.)

Note that the P zones collectively cover half the interaction region, while
the N zones cover the other half.  (This statement is obviously independent
of lambda.  This assumes the interaction region is reasonably large, which
is consistent with previous assumptions.)

Essentially we have figured out what sort of diffraction grating (or
hologram) would be ideal for creating the desired interaction.  According to
this analysis, we are getting coherent contributions from throughout the
interaction region -- which is vastly larger than wavelength cubed!

Now the statistical question reduces to this:  what is the chance that the
air will fluctuate into a configuration that has an extra-large number of
molecules in the P zones, and an extra-small number of molecules in the N
zones?  Essentially we are talking about thermally-excited sound modes.

As far as I can tell, the amplitude of these excitations should be
independent of frequency, since the compressibility of air doesn't depend on
wavelength.  That means I'm skeptical of any argument that says that we
should consider fluctuations in a volume on the order of wavelength cubed,
implying that short-wavelength fluctuations should be more prominent.

There _are_ things in the calculation that depend on wavelength.  In
particular, if you look at _The Feynman Lectures on Physics_ volume 1 figure
30-10, and do a little work, you will discover that for his problem, one
dimension of a typical N zone scales like the square root of lambda.  It
also scales like the square root of the distance (r) to the observation
point.  This affects how many atoms contribute to the integral, and
therefore affects the total scattering amplitude.

And be warned that the physics of our problem (random medium) is not the
same as the physics Feynman was doing (uniform medium).

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


===================

The next step is to put on my "scholar" hat and track down some primary
references.

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. :-)

I also note that in the last 25 years there have been hundreds of papers on
"atmospheric scattering" from the planetary science community.  There's an
entire journal devoted to "Quantitative Spectroscopy and Radiative Transfer".

A tangentially interesting online reference is
  http://www.agu.org/revgeophys/stephe01/stephe01.html