Re: [Phys-L] the color of the sky

[John Denker <jsd@av8n.com>]

On 08/19/2013 05:43 AM, Bob Sciamanda wrote:
> A Nasa site, http://spaceplace.nasa.gov/blue-sky/ , says this:
>
> "Closer to the horizon, the sky fades to a lighter blue or white. The
> sunlight reaching us from low in the sky has passed through even more
> air than the sunlight reaching us from overhead. As the sunlight has
> passed through all this air, the air molecules have scattered and
> rescattered the blue light many times in many directions. Also, the
> surface of Earth has reflected and scattered the light. All this
> scattering mixes the colors together again so we see more white and
> less blue."

That web page leaves out an important part of the physics.

Consider what would happen in a hypothetical atmosphere where
the density was uniform, as in an ideal solid or even an ideal
liquid.  All the molecules would scatter coherently.  This
would lead to 100% forward scattering at each and every
wavelength, no matter how strong or weak the scattering from
any individual molecule.  There would be no scattering to the
side.  The sky would not be blue.  Stars would be visible in
the daytime.

  At this point one may be tempted to explain the previous 
  paragraph in terms of the Huygens construction.  The general
  idea of reconstructing the wave from scattered wavelets is 
  correct, but Huygens missed some important details.  For a 
  description of the problem, and a way to fix it, see
    David A. B. Miller
    "Huygens's wave propagation principle corrected"
    http://www-ee.stanford.edu/~dabm/146.pdf

Not counting clouds, dust, and other impurities, the only
reason we see any light at all coming from the sky is
spatial /fluctuations/ in the density.  The number of
molecules per unit volume in "this" parcel is different 
from "that" parcel.  The reconstructed wavelets from the
average density cancel (in all non-forward directions)
but the wavelets from the fluctuations do not cancel.
For an ideal gas, the particles are distributed randomly, 
so if there are N molecules, the fluctuations scale 
like √N.

So ... the brightness and the color of the sky provide
direct evidence for the size of atoms.  If the molecules
were smaller, we would have a larger number of weaker 
scatterers, so the relative fluctuations √N/N would be 
less, and the amount of scattered light would be less.
The daytime sky would be almost black.  At the other
extreme, a smaller number of strong scatterers would
produce multiple scattering, and the daytime sky would
be almost white.

Some guy worked this out in 1910:
  "Theorie der Opaleszenz von homogenen Flüssigkeiten und
  Flüssigkeitsgemischen in der Nähe des kritischen Zustandes"
  http://www.physik.uni-augsburg.de/annalen/history/einstein-papers/1910_33_1275-1298.pdf
This was a big deal at the time.  Certainly there had 
been previous results that gave evidence for the size 
of atoms (e.g. Loschmidt 1865) ... but not very many.
In particular, at the time of Avogadro death (1856),
nobody had a reasonable estimate for Avogadro's number, 
not even within an order of magnitude.

It is always nice to have good answers for the perennial 
question:  How do we know atoms exist?  Saying that the
fact that the sky is blue gives evidence for the size of
atoms has some limited pedagogical value, limited only
by the fact that the details of the calculation are too 
complicated for the introductory course.  However, the
basic idea of scattering from /fluctuations/ in the density
is correct.  This shows the unity of physics, tying together
ideas about probability, atoms, and waves.

For the next level of detail, see
  http://www.av8n.com/physics/blue-sky.htm