Re: Sunsets

[Daniel Schroeder <DSCHROEDER@CC.WEBER.EDU>]

> Consider a region in the atmosphere so small that coherence is
> guaranteed.  A cell lambda/2pi on a side should be small enough.  Ask
> yourself how many thousands of air molecules there are in such a cell --
> all scattering coherently, guaranteed by construction.
>
> Argue from the most elementary statistical principles that any two such
> cells have "pretty much" the same number of molecules, and that therefore
> the troposphere is more like glass than like a collection of independent
> Rayleigh scatterers.

OK, here goes:  Such a cell is on the order of 100 nm across, which is
about 20 times the average distance between air molecules.  So the cell
contains roughly 10^4 molecules, and this number should fluctuate, from
cell to cell, by its own square root, about 100 or 1%.  Let's say I'm
looking in a direction 90 degrees from the sun.  Along my line of sight
lie two such cells, separated in distance by half a wavelength.  Along
comes a light wave from the sun, which is scattered my way by the
molecules in both cells.  Within each cell, the scattering is coherent,
but we get destructive interference between the two cells because one
is lambda/2 farther away from me than the other.  If the cells contained
the same number of molecules, this interference would be total.  But
instead, one contains a 1% excess of molecules, so the destructive
interference is only 99% complete.  Hence, the brightness of the
scattered light is much less than it would be if the scattering were
completely decoherent.

If we replace the air by a transparent solid or liquid, then each
cell contains 1000 times more molecules, and furthermore, the number
of molecules from cell to cell doesn't really fluctuate at all, so
the destructive interference in my direction is essentially complete.

Of course, the underlying mechanism, the reason why the molecules
scatter in the first place, is the same that I've already given:
the molecules become oscillating dipoles as the wave passes by, and
these dipoles radiate.  At this level, the scattering introduces a
1/lambda^4 dependence on wavelength.  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.  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?

Dan