Copyright © 2003 jsd
Before we ask why the sky is blue, we should ask whether the sky is blue. There are many times and places where the sky is, in fact, not blue. There is some nifty physics that explains the blue part of the story, which is our topic for today, but we should keep in mind that it is not the whole story.
We shall consider the restricted case of standing on the earth’s surface in the daytime under a clear sky, without clouds, dust, or pollution. This is sometimes a decent approximation in real life. As we shall see, under such conditions we expect the sky to be blue.
A crucial sub-goal will be to understand why the scattering depends on the fourth power of the wavelength.
There are lots of pseudo-explanations out there that focus attention on a piece of sky “one wavelength on a side”. However, I believe any alleged explanation of that sort is wrong physics, and misses the right physics, as will become clear below.
Here is an “executive summary” that outlines where the discussion is going. For details see section 3.
Consider the following diagram of the interaction:
. . . . . . . . . . . . . . . . . . . . . . . . . transmitted --> | | | / |/ | / | /| / | | | | | | | | | | | / |/ | / | /| / | | | | | | | | | | | / |/ | / | /| / | | | | | | | | | | | / |/ | / | /| / | | | | | | | | | | | / |/ | / | /| / | | | incident --> / / / / / / / / / / / / / / / / / / / / / / / / / scattered / / / / --> / / / / / /
The lines represent the crests of the waves.
The dots don’t have much physical significance. They are just the backwards extrapolation of the scattered beam. Loosely speaking, they represent the direction the scattered beam “appears” to be coming from. Ignore the dots if you like.
Here is the diagram again, with labels on some points in the interaction region:
\begin{verbatim} . . . . . . . . . . . . . . . . . . . . . . . . . transmitted --> | | | P |/ | N | /| P | | | | | | | | | | | P |/ | N | /| P | | | | | | | | | | | P |/ | N | /| P | | | | | | | | | | | P |/ | N | /| P | | | | | | | | | | | P |/ | N | /| P | | | incident --> / / / / / / / / / / / / / / / / / / / / / / / / / scattered / / / / --> / / / / / /
We have made some mildly arbitrary choices about the relative phases. This results in no loss of generality.
We assume that the interaction region is transparent to zeroth order. This is consistent with (and stronger than) previous assumptions.
This assumes the interaction region is reasonably large relative to lambda. This is consistent with previous assumptions.
This assumes the scattered beam does not coincide with the transmitted beam. In the other case (i.e. forward scattering) this half-and-half property does not hold, which is a good thing; the distinction allows us to uphold the optical theorem, conservation of energy, and other good things.
Note that this part of the argument is independent of wavelength, for over the relevant range of wavelengths. In the atmosphere, the interaction region is vastly larger than wavelength cubed. That means there are a vast number of P regions, covering half of the interaction region. If the wavelength is smaller, each P zone will be smaller, but there will be more of them, and collectively they will still cover 50% of the interaction region.
The amplitude of these excitations should be independent of frequency, since the compressibility of air doesn’t depend on wavelength.
We are approximating the air as an ideal gas. This should be a very good approximation.
Compressibility being independent of wavelength is another line of argument supporting the previously-mentioned point that we should not focus attention on a single region that has volume on the order of wavelength cubed.
Applying this idea to the air: We have every reason to believe that half of the interaction region is P-type, and half is N-type. We have every reason to believe that air is an ideal gas, to a good enough approximation, indeed more than good enough. Therefore it does not matter how big the P-regions and N-regions are.
This point is worth emphasizing because there are widespread misconceptions that somehow the light is being scattered by “small” regions where the density differs from the average.
This assumes the index is not too different from unity. This is a good assumption for gases under ordinary conditions. If you care about liquids, you could easily relax this assumption and redo the following derivation, thereby obtaining a more general result.
v^{2} (∂ / ∂ x)^{2} φ − (∂ / ∂ t)^{2} φ = 0 (1) |
where v is the speed of light in the medium,^{1} x is position, and t is time. We call equation 1 the “unperturbed” wave equation, for reasons that will be obvious in a moment.
There is a one-to-one relationship between the index of refraction and the speed of light in the medium. To first order, the relationship is:
v = v_{0} / (1 + d) (2) |
where (1 + d) is the index, 1 is the average index (not the index of the vacuum) and d is the fractional deviation from the average index, and v_{0} is the propagation speed associated with the average index. We assume d is small compared to unity.
So in a slightly non-uniform medium we have, to first order in d:
v_{0}^{2} (1−2d) (∂ / ∂ x)^{2} φ − (∂ / ∂ t)^{2} φ = 0 (3) |
and by re-arrangement:
v_{0}^{2} (∂ / ∂ x)^{2} φ − (∂ / ∂ t)^{2} φ = 2d v_{0}^{2} (∂ / ∂ x)^{2} φ (4) |
So, the term involving the index-deviation d can be moved to the RHS. This term can be considered a driving force, i.e. a source term added to the RHS of the unperturbed wave equation (equation 1).
We are assuming that the scattering is not too strong, and the interaction region is not too overly huge, so that we imagine that the light gets scattered at most once. We account for light scattering from the incident beam into the scattered beam, while ignoring any possible secondary scattering (out of the scattered beam). This is called the first Born approximation.
For simplicity, we are ignoring polarization. It would introduce some factors that depend on theta (the scattering angle). You can add them in if you like.
For clarity, I left out the y and z variables in equation 4. But you get the idea.
There is only a rather narrow set of conditions that can give a planet a blue sky. An atmosphere with too little depth and/or too little density would have very little scattering of any kind, so the sky would be black with only a slight tinge of deep blue. At the other extreme, an atmosphere with too much depth and/or too much density would have a lot of second-order and higher-order scattering, violating some of the assumptions made in section 3 and leading to a murky white sky.
Also, as Einstein pointed out in reference 1, the color of the sky can be used to pin down the size of air molecules, within a rather narrow range. The argument goes like this: Macroscopic benchtop measurements tell us the overall refractive index of a parcel of air, but they don’t tell us whether that is due to a smallish number of highly refractive molecules, or a larger number of less refractive molecules. However, the fluctuations in the refractive index do depend on how many molecules there are. If there are N molecules in the parcel, the fluctuations in the number will scale like √(N). Meanwhile, the refraction per molecule must scale like 1/N, to be consistent with the macroscopic observations, so the fluctuations in the index scale like √(1/N). That means that if atoms were 100 times smaller than they really are, there would be 10 times less scattering in the atmosphere, and the sky would be almost black in the daytime.
Copyright © 2003 jsd