Still Chasing That Rainbow.In the last issue we began our journey to find the rainbow, and that pot of gold. We had concluded that it would be a long trip, beyond the cloud of water drops that are responsible for the rainbow image.
But, you may say, "I can form a rainbow with a water sprinkler. The water drops are only a few feet away, and I can see the rainbow between myself and a bush. Surely the rainbow must be nearer than the bush."
Think again of the image formed by a mirror. It appears farther away than the mirror. If we use a rangefinder to triangulate the water sprinkler image, we find it is much farther away than that bush. This, like many optical illusions, produces cognitive dissonance, for several clues we use for determining distances seem to be giving contradictory information. Here's where people get into heated discussions and disagreements about what is the "proper" way to determine distances in a situation such as this.
Is the cloud of water drops acting like a mirror? No, not exactly. The drops capture light from the sun, reflect it internally, and it emerges going in the opposite direction, but deviated 42° (red rays). This is the reason the rainbow arc has a radius of 42°, centered on the anti-solar point, a point determined by a straight line from the sun through your eye.
In fact, the light from any small portion of the rainbow is coming from drops over a wide range of distances, all the way through the cloud, perhaps several miles, not from a simple plane mirror surface.
Now you might suppose that the size and shape of the individual water drops, each acting as a tiny lens, might determine the apparent distance to the rainbow image. Surprisingly, they do not. The drops are effectively acting as tiny mirrors. Each one captures a very narrow bundle of light from the sun and simply redirects it back into your eye in a narrow bundle spreading (dispersing) only about 2° into a spectrum of colors. Our eye sees only the reflected light from those drops located in space where they happen to direct light into our eyes.
The sun is so far away that its light reaching the earth diverges very little. Rays from any point on the sun are nearly parallel when they reach the earth. So all water drops receive rays from nearly the same direction, and redirect them back to our eyes, still all nearly parallel. A small caveat: the sun is not a point source of light. It subtends an angle of about 0.5° as seen from earth, but this doesn't compromise our argument, it only adds a slight blurring to the rainbow image.
The bottom line is that the rainbow is as far away as an image of the sun would be if reflected by a huge plane mirror at the distance of the cloud of water drops. So "as far away as the sun" is a good answer. But other answers are close enough for practical purposes—see below.
If you accept the rangefinder definition of distance, this result would be confirmed by direct experiment, by sighting the rainbow simultaneously from two widely separated vantage points.
Now that you understand how the rainbow is formed, you know the answer to the question "Why is the center of any rainbow formed by sunlight always below the horizon?" It is simply because the sun only visible when it is above the horizon. The rainbow's arc is always centered on the anti-solar point—on the shadow of your head. But that doesn't mean, "It's all in your head". You can easily apply the same reasoning to moonbows.
In fact, other optical sky phenomena formed by sunlight appear at the same distance as the sun. These include parhelia, and sun dogs, formed by sunlight refracting through clouds of ice crystals.
You now also know why no one has ever found that pot of gold. There wouldn't be enough gold in it to pay for the trip. And the rainbow would disappear as you passed through the cloud of water drops forming it.
Some nit-picky readers may note that the two lines from sun through your two eyes diverge slightly, since the sun is not infinitely far away. This would put the rainbow's image just a smidgen beyond infinity. Now we get down to the psychology of vision. Binocular vision can tolerate a slight divergence of the two eyes (at least for most people). The divergence here isn't large. How large?
The sun is 5.89 × 1012 inches from earth. Human eyes are spaced about 2.5 inches apart. The divergence angle is therefore 2.43×10-11 degree, approximately. You'd never notice such a small divergence. So the rainbow is at the same distance as the sun, an answer good enough for practical purposes. Infinity is a good answer, too. Even with a large baseline rangefinder, of perhaps a mile, the rainbow image is still stuck pretty near infinity—infinity being defined here as greater than any measurable distance. One could also make a case for the fact that the horizon is usually so far away as to be indistinguishable from infinity seen with the naked eyes. But whenever you use the word "infinity" as if it were a number, you are on a slippery mathematical slope.
Correction.An alert reader, Carl Reiff, noticed my answer to puzzle 2 in the Fall 2016 issue had a misprint in the last sentence. It should have read, "The earth is closer to the sun at full moon. At noon at full moon, the center of the earth is 2×4671 = 9342 km closer to the sun than it is at noon at new moon. So you are also that much closer to the sun at noon at the time of full moon."
Another sharp eyed reader, Patrick Mulvey, spotted an error in the illustration that slipped by when we proofread this. The diagram showed the wrong side of the full moon illuminated. To make amends, here's the proper diagram.