For my money, rainbows are the best thing in the sky, day or night. It's much more than an atmospheric optical phenomenon: it's the freshness of the air after a shower or the dramatic, low angle of the light. All of it awesome. If the weather fails to cooperate, you can see a rainbow in the fine mist of the garden hose (and wash the car at the same time), or failing all else, generate one on your computer's screen.

Interestingly the word "rainbow" doesn't refer to its most important characteristic: color. Although we often call it a rainbow, the spectrum produced by a prism isn't the same one seen in the sky. True, reflection and refraction are involved, and it is dispersion or the difference in refraction for different wavelengths that spreads out the colors. But the colored arcs in the sky are the edges of all the light rays bounced and concentrated through the drop and back to your eye. Red light is bent less and so is scattered wider resulting in the red outer edge. The sky under the bow is brighter because light of all colors (albeit at much lower intensity) is scattered there, but none is above. This produces the dark feature known as Alexander's band, and if you're fortunate, you'll see a secondary bow--with wider, reversed color banding--higher still in the sky. Those rays are from light twice reflected inside the drops.

Ray theory explains all this and is the basis of the computer simulation shown at top. All that is really needed is trig tables or a good pocket calculator and a lot of free time. René Descartes first followed light ray's paths through the drop and calculated the angles of the primary and secondary bows, about 42° and 51°, respectively, from the point directly opposite the Sun. Ray theory predicts dozens of rainbows, a spectacular sky full of arcs of colors. Because so little light energy is left after the first two internal reflections, higher order bows have never been seen in nature. But in the laboratory, light reflected and refracted in "bows" of up to 13th order has been observed!

Below the primary, faint supernumary bows may occasionally be seen. Ray theory fails here. Supernumary bows result from the interference of light traveling slightly different angles and thus distances and phase shifts on its path through the raindrop. George Airy derived the rainbow integral as part of this theory. Mathematically, it is just a tad too complicated to code easily in a spreadsheet. Airy theory also takes into account rain drop size as well as the spread of droplet sizes, explaining why some bows are more colored or saturated, while others, particularly fogbows of especially fine droplets, appear ghostly and almost colorless.

Light from rainbows is polarized: a fact that you can demonstrate for yourself by putting on polarized sunglasses in the rain and staring away from the Sun. But be prepared for strange looks from passersby. Just explain to them that Gustav Mie's theory explains this feature. It's very complicated but also very cool: Mie theory is the solution to Maxwell's equations for the electric and magnetic field of the light waves as they interact with a sphere of water in air, boundary conditions included. The solutions are in terms of power series, complex numbers and Legendre polynomials. Numerically, typical plots show curves with tens to hundreds of peaks and troughs. They look more like combs than bows.

Here, we've run out of theories. As far as I know, no one has a quantum mechanical theory of the rainbow. And I wouldn’t want to see it if they did. Existing models explain most everything optical one can see and appreciate. Sprinkles of mist on the eyeglasses, distant thunder and a delightful play of light off the clouds won't come out those equations no matter how hard you study them. Keep you eye on the late-day spring shower or get out the garden hose and wash the winter grime from the car. A rainbow could be yours to enjoy.

Moondark is written by Doug Miller and published on the web, in the Delmarva Star Gazers'Star Gazer News and in the Delaware Astronomical Society's FOCUS. Please address comments and suggestions to dmiller@udel.edu. This document was last revised on 24 March '00. All text and images copyright © 2000 Douglas C. Miller, All Rights Reserved. This material may not be reproduced in any form without prior permission.