Rainbows: The Inside Story

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This is from an e-mail Q&A with a physics friend that was inspired by a perfect rainbow I saw yeseterday.

I'll reorder it so you don't have to read from bottom up...

Physics is just damn cool. Unless you're falling off a building. Anyway, maybe it's just me, but this is far more fascinating to me than any myth I've ever heard. And I like myths!

Of course, if any physicists/scientists/whatever disagree with the analysis below, comment away! Any other comment welcome, too, of course.

Question:

Why are rainbows bows? That is, why doesn't every single droplet show the prismatic effect? And those that don't (or don't show it to an observer in a given position -- ?); well, why do those that do arrange themselves in a hemisphere?

Answer:

OK, each water droplet in the sky: Think of each droplet as a sphere. Light from the sun impacts the air/water boundary of the droplet. A significant amount of the white (i.e., "contains all colors") light refracts into the droplet. However, the light ray contains different wavelengths of light (different colors), and a law of optics called Snell's Law tells us that the angle of refraction depends on the wavelength of the light. Simple version of the effect: the different colors refract at slightly different angles into the droplet.

[Harder physics version:] Light strikes the air/water boundary at an angle to the "normal" of the surface, the line that is perpendicular to a plane that is tangent to a surface. In the case of a sphere, at each point on the surface, the "normal" is the line that points directly away from the center. Because the index of refraction (the degree to which light is slowed when traveling through a material) is higher for water than for air, Snell's Law implies that the light will bend upon entering the water, bend toward the normal (tend to pass more closely to the center of the sphere than it would if it had kept traveling in the same direction). That explains why light bends. Different colors bend slightly different amounts by this logic: light of different colors has differing wavelength and frequency. Fact of a light wave: frequency times wavelength in a medium = speed of light in that medium. The energy of a light wave is directly proportional to frequency. The energy of the wave doesn't change when it enters a new medium (which means frequency is constant), however the wavelength of the wave does change. (Why? Argh. Fermat's Principle of Least Time is the short answer.) Frequency x reduced wavelength = reduced speed of light. Light of different wavelengths means different speeds means different refraction angles. [End of harder physics version.]

The diverging colors reflect off the back end of the droplet, strike the front end again (light enters droplet from upper right, refracts through to left side of sphere, and refracts out of sphere on the lower right), but the different colors of light strike in different places, and leave the droplet at different angles.

So, white light enters every droplet in the sky, and light of different colors leaves in different directions.

Now, the "right" geometry happens when the angle between the sun's incoming rays and the rays leaving the droplets (angle ABC, where A = sun, B = droplets, and C = you) is about 42 degrees. ABC for the sun's blue light is 40 degrees. ABC for red light is 42 degrees.

Droplets that are slightly farther away and higher in the sky will reflect the red light into your eyes. Different drops that are slightly closer, and slightly lower, will reflect blue light into your eyes. Intermediate colors will reflect at intermediate angles...and you will see a rainbow.

"That is, why doesn't every single droplet show the prismatic effect?" Every droplet DOES show a prismatic effect. But the sun-droplet-you angle has to be about 42ish degrees, and you'll only be able to see ONE of the colors from that drop. Other droplets provide other colors...the combined effect being the rainbow.

Why the hemisphere?

A rainbow would be the base of a cone, if the Earth didn't get in the way. The vertex of the cone is the sun, the center line of the cone passes from the sun through your eyes to the shadow of your eyes (head) on the ground, and the base of the cone (the rainbow) is formed where the sun(vertex)-droplet (base)-eye angle is between 40 and 42-ish degrees. Technically, there is a red cone with one angle, a yellow cone with a slightly different angle, ..., a blue cone with a slightly different angle. You see all the cone bases as a rainbow.

Question:

Interesting twist: a friend of my bro's saw a circular rainbow in a plane! Also, I've seen double-rainbows -- I assume that's just a situation in which there are two "sets" of bows at the proper angles?

One thing that blew me away was the compound nature of the illusion. First, for a creature that doesn't register our visual spectrum, that bow ain't there. I assume bees see UV bows; and so on. Second, unless I misunderstood, the bow itself does not literally exist one color on top of another in a certain place: each color is separated (by miles, feet...?) and only seems to be "stacked" when seen at the proper angle.

Is this correct?
Answer:
I'm guessing bees don't have the visual acuity to see that far away...unless it's mini-rainbow from a sprinkler. :)

You're correct about the "stacked" nature of rainbows. However -- "Also, I've seen double- rainbows -- I assume that's just a situation in which there are two 'sets' of bows at the proper angles?" -- is interesting.

When light rays strike a boundary between different media (air/water), part of the ray reflects (bounces off) and part of it refracts (traverses the boundary and bends). We see a strong single rainbow with the 40-42 degree big arc because much of the light refracts into the droplet, reflects off the back side, then refracts back into the air. However, some of the light will bounce inside the droplet twice: refract in, reflect off the back, reflect off the front, reflect off the back, then refract out of the droplet.

The light that does this double bounce is lower in intensity (quantum mechanics: individual photons are more likely to refract-reflect-refract than refract-reflect-reflect-reflect-refract). The double bounce causes a different geometry: the sun-droplet-eye angle will be bigger by a few degrees...and that's why the second rainbow appears elsewhere in the sky: the color cones whose bases make up the second rainbow have a different angle (probably 48 or 50 degrees or something like that).