Captain Meteo: Rainbows

Q (Banerjee asks): Rainbow phenomena are predicted very accurately assuming spherical raindrops. But falling raindrops do not remain spherical because of aerodynamic effects. What is the shape of a falling raindrop and what are the effects of that shape on rainbows?

First: about raindrop shapes. The smaller ones (1mm or less) are pretty spherical. Bigger ones tend to a top-half of the hamburger bun shape, see e.g. here. Those bigger than about 4 mm tend to get ripped apart by aerodynamic forces.

Let’s start with ordinary rainbows from spherical drops. We have mostly noticed that the Sun is always at our back when we see rainbows, and if have played around a bit with sprayers on hoses. that we can get complete circular rainbows under the right circumstance. Descartes seems to have been the first to study rainbows systematically, so let’s quote from him, via this marvelous NCAR document: About Rainbows. [You will want to look at the picture in the link]

He writes:"Considering that this bow appears not only in the sky, but also in the air near us, whenever there are drops of water illuminated by the sun, as we can see in certain fountains, I readily decided that it arose only from the way in which the rays of light act on these drops and pass from them to our eyes. Further, knowing that the drops are round, as has been formerly proved, and seeing that whether they are larger or smaller, the appearance of the bow is not changed in any way, I had the idea of making a very large one, so that I could examine it better.

Descartes describes how he held up a large sphere in the sunlight and looked at the sunlight reflected in it. He wrote "I found that if the sunlight came, for example, from the part of the sky which is marked AFZ
and my eye was at the point E, when I put the globe in position BCD, its part D appeared all red, and much more brilliant than the rest of it; and that whether I approached it or receded from it, or put it on my right or my left, or even turned it round about my head, provided that the line DE always made an angle of about forty-two degrees with the line EM, which we are to think of as drawn from the center of the sun to the eye, the part D appeared always similarly red; but that as soon as I made this angle DEM even a little larger, the red color disappeared; and if I made the angle a little smaller, the color did not disappear all at once, but divided itself first as if into two parts, less brilliant, and in which I could see yellow, blue, and other colors ...

When I examined more particularly, in the globe BCD, what it was which made the part D appear red, I found that it was the rays of the sun which, coming from A to B, bend on entering the water at the point B, and to pass to C, where they are reflected to D, and bending there again as they pass out of the water, proceed to the point ".

What happens when see that bit of color from a drop is that a ray of light, coming from more or less behind us, has entered the spherical drop, been bent slightly by refraction as it enters, been reflected from the back of the droplet, been refracted again as it leaves, and ultimately entered our eye. The important point is that 42 degree angle that Descartes observed – the reflected rays come out at an angle of 42 degrees to the incident rays. Imagine, if you will, a straight line drawn from the Sun, through your head, and out into the beyond. You will noticed that the observed position of the rainbow is along the 42 degree arc about that axis from Sun through head.

The other important thing Descartes noticed was that slightly different angles correspond to different colors of light. Some colors are refracted more than others, and it’s that color separation that produces the rainbow of colors. It’s also possible for rays of light to bounce twice inside the droplet before exiting – such double reflections produce a fainter, color reversed rainbow at about 50 degrees.

Banerjee asked what effect the aerodynamic distortion of the drops has on the rainbow. The most common such distortion observed is a flattening of the drops. That flattening is believed to be responsible for the fact that for low Sun angles the edges of the rainbow are sometimes brighter than the top. The flattening of the big drops (little drops remain nearly spherical) causes the rainbows rays to disperse and leak out of the drops.

Another point worth mentioning is that you should notice that the region inside the arc is noticeably brighter than the outside. This is another rainbow effect, due to the fact that:

When one studies the refraction of sunlight on a raindrop one finds that there are many rays emerging at angles smaller than the rainbow ray, but essentially no light from single internal reflections at angles greater than this ray. Thus there is a lot of light within the bow, and very little beyond it. Because this light is a mix of all the rainbow colors, it is white. In the case of the secondary rainbow, the rainbow ray is the smallest angle and there are many rays emerging at angles greater than this one. Therefore the two bows combine to define a dark region between them - called Alexander's Dark Band, in honor of Alexander of Aphrodisias who discussed it some 1800 years ago!

This quote, and most of the content of the present post, are taken or adapted from the NCAR reference cited above.