Look Up In the Sky…!

Yesterday’s graduate class in electromagnetism had a bit of extra fun. We did a particular computation in some detail, and arrived at a pair of results. We thought about the main features of the equations we’d derived and I then asked the class if they could think of an example. An example with those equations essentially written all over it. It was the sky. Not just the blueness of the sky (for which the result supplies a partial answer) but the pattern of blueness on the sky, especially when looking through your polarised sunglasses. (You know how you tilt your head when wearing them and you can darken or lighten the sky a bit? Well, that effect is way more effective if you are looking in a direction at right angles to the sun as opposed to either toward or away from the sun.)

So I took the class outside to gaze upon the sky in person, rather than just sit and talk about it. Actually, a little bit of knowledge about the pattern of blue in the sky is useful in a lot of ways. For example it is amusing to me to see how often architects and their artist collaborators get the sky wrong in renderings of things like artist’s impressions of buildings or cityscapes… the cast shadows tell you the direction of the sun, and the sky’s blue patterning thats shown is often wrong when compared to that. See here and here.

It is all to do with how light interacts with (“scatters off”) the various particles that make up the atmosphere. To a good approximation you can describe that as a process of them getting excited by the light and then re-radiating the light in all directions in a particular pattern. You can compute the pattern. The resulting light depends a lot on the polarisation of the light (the angle the electric and magnetic fields that make it up have as they propagate in a particular direction), and what direction you’re looking at and your sunglasses are sensitive to that. Just in case you want to know, the formulae that are “written on the sky” are:
[tex]
\frac{d\sigma}{d\Omega}\sim \frac{1}{\lambda^4}(1+\cos^2\theta)\qquad \Pi(\theta)=\frac{\sin^2\theta}{1+\cos^2\theta}\ ,
[/tex]
and I recommend at this point that you read a good text about the details, and/or see the footnote!**

We had a lot of fun with this (only slightly dampened by going back into class to spend another hour wading through long expressions of what are called “multipole expansions” of scattering results, involving Bessel functions, Hankel functions, Legendre polynomials and vector spherical harmonics… all of which I try to make fun, but sometimes I might fall short of the mark…)

As a bonus we had two comedians in the class! No, I don’t mean any actually registered students (this time!) but two visitors from The Upright Citizen’s Brigade – Alex/Berg and Dax (actually, Darwin and Inspector Gadget) from some nights back when I was judging the Tournament of Nerds (you may recall from here). After the show they were keen to come and see a physics lecture and so I invited them along. Quite a successful cultural exchange, I think!

-cvj

**Roughly, if you draw a line from the sun to you, then [tex]\theta[/tex] is a measure of the angle between that and the direction you’re looking off into. Towards the sun would be zero, and directly away would be 180 degrees. The second function, which tells you about the relative amounts of scattering due of two orthogonal choices of polarised light. It peaks at 90 degrees where the contribution is all from one polarisation… that’s when your sunglasses are at their most effective since they handle a particular polarisation at a given angle… And so forth…