Re: Sunsets

[John Denker <jsd@MONMOUTH.COM>]

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'.