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What is a rainbow?

Author Donald Ahrens in his text Meteorology Today describes a rainbow as “one of the most spectacular light shows observed on earth”. Indeed the traditional rainbow is sunlight spread out into its spectrum of colors and diverted to the eye of the observer by water droplets. The “bow” part of the word describes the fact that the rainbow is a group of nearly circular arcs of color all having a common center.

Where is the sun when you see a rainbow?

This is a good question to start thinking about the physical process that gives rise to a rainbow. Most people have never noticed that the sun is always behind you when you face a rainbow, and that the center of the circular arc of the rainbow is in the direction opposite to that of the sun. The rain, of course, is in the direction of the rainbow.

What makes the bow?

A question like this calls for a proper physical answer. We will discuss the formation of a rainbow by raindrops. It is a problem in optics that was first clearly discussed by Rene Descartes in 1637. An interesting historical account of this is to be found in Carl Boyer’s book, The Rainbow From Myth to Mathematics. Descartes simplified the study of the rainbow by reducing it to a study of one water droplet and how it interacts with light falling upon it.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.

Descarte 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 “.

This quotation illustrates how the shape of the rainbow is explained. To simplify the analysis, consider the path of a ray of monochromatic light through a single spherical raindrop. Imagine how light is refracted as it enters the raindrop, then how it is reflected by the internal, curved, mirror-like surface of the raindrop, and finally how it is refracted as it emerges from the drop. If we then apply the results for a single raindrop to a whole collection of raindrops in the sky, we can visualize the shape of the bow.

The traditional diagram to illustrate this is shown here as adapted from Humphreys, Physics of the Air. rain.jpg It represents the path of one light ray incident on a water droplet from the direction SA. As the light beam enters the surface of the drop at A, it is bent (refracted) a little and strikes the inside wall of the drop at B, where it is reflected back to C. As it emerges from the drop it is refracted (bent) again into the direction CE. The angle D represents a measure of the deviation of the emergent ray from its original direction. Descartes calculated this deviation for a ray of red light to be about 180 – 42 or 138 degrees.

The ray drawn here is significant because it represents the ray that has the smallest angle of deviation of all the rays incident upon the raindrop. It is called the Descarte or rainbow ray and much of the sunlight as it is refracted and reflected through the raindrop is focused along this ray. Thus the reflected light is diffuse and weaker except near the direction of this rainbow ray. It is this concentration of rays near the minimum deviation that gives rise to the arc of rainbow.

The sun is so far away that we can, to a good approximation, assume that sunlight can be represented by a set of parallel rays all falling on the water globule and being refracted, reflected internally, and refracted again on emergence from the droplet in a manner like the figure. Descartes writes

I took my pen and made an accurate calculation of the paths of the rays which fall on the different points of a globe of water to determine at which angles, after two refractions and one or two reflections they will come to the eye, and I then found that after one reflection and two refractions there are many more rays which can be seen at an angle of from forty-one to forty-two degrees than at any smaller angle; and that there are none which can be seen at a larger angle” (the angle he is referring to is 180 – D).

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