How Rainbows Work
A rainbow is not an object hanging at a fixed spot in the sky. It is a direction. Sunlight enters each spherical raindrop, bends, reflects off the back, and bends again on the way out, concentrating light into a cone about 42 degrees wide. That cone is centered on the antisolar point: the spot directly opposite the sun, which is the direction of your own head’s shadow.
So three things set the bow: you (the tip of the cone), the sun (which aims the cone), and the rain (the screen it lands on). Each panel below moves just one of them.
Move the sun
The sun's height sets how much of the bow clears the horizon.
The top of the bow sits about 34 degrees up. Raise the sun and the bow sinks; lower it and the bow grows toward a half-circle.
Side view: the bow moves with you
Sunlight comes in from the left. Slide yourself and watch your bow follow.
Sunlight arrives from the left in parallel rays. Your bow always sits 42 degrees from the direction opposite the sun, so when you slide left or right it slides right along with you, rebuilt from whatever drops line up. The other viewer has a separate bow from different drops, and it does not move when you do. Nobody shares a rainbow.
Move yourself
Turn your head or walk forward. The bow is locked to a direction, not a spot.
from above
The bow is pinned to the direction opposite the sun. Turn your head and it slides; the person beside you sees a slightly different bow.
Move the rain
Color appears only where the 42 degree cone meets real drops.
Rain sits in front of the antisolar point, so the cone lights up real drops and you see the arc. Slide it away and the colors vanish.
Why the double bow sits farther out
A side view of the light path inside a single raindrop.
The main bow's light reflects once inside each drop and leaves near 42 degrees from the point opposite the sun. The double reflects twice, which sends it out near 51 degrees and flips the color order. That extra bounce also loses light, so the double looks fainter and sits higher, just outside the first bow.
Why 42 degrees, and where the colors come from
Real refracted rays piling up at the maximum angle, and splitting by color.
At b = 0.86 the red ray leaves near 42.4 degrees and the violet ray near 40.6 degrees. Water bends violet a little more than red, so every color has its own angle and its own pile-up: red maxes out near 42 degrees, violet near 40. The bow is just that thin band of maximum angles, red on the outside and violet on the inside. The colors come straight out of the angle.
A few things to notice. The sun’s height controls how much of the bow clears the horizon: a full half-circle at sunrise or sunset, and nothing once the sun climbs past 42 degrees, which is why rainbows favor mornings and evenings (from a plane you can see the whole circle). Because the sun is so far away its rays arrive parallel, so walking does not change its direction; the bow simply travels with you and you can never reach its end. And the colors only appear where that 42-degree cone meets actual raindrops, so you need sun behind you and rain ahead, lined up just so.
The fainter secondary bow comes from light that reflects twice inside each drop. It sits about 51 degrees out with its colors reversed, and the sky between the two bows is noticeably darker (Alexander’s band).
Why these exact angles?
Every ray that enters a drop gets turned by some total amount, and that amount depends on where on the drop it struck. As you scan across the drop, the turning angle drops to a minimum and then climbs again. Near that minimum a whole band of rays leaves in almost the same direction, so the light piles up there, like a caustic on the bottom of a pool. That pile-up is the bright bow, and for water it lands near 42 degrees from the point opposite the sun. The exact value is set by water’s refractive index (about 1.33), and because that index is slightly different for each color (red bends least, violet most), the bow spreads into a band about two degrees wide with red on the outside.
Light actually leaves the drop at every angle, but it is only bright where the turning angle is stationary. One internal reflection gives that pile-up near 42 degrees; two reflections give another near 51 degrees, with the colors reversed. Nothing concentrates between them, which is why Alexander’s band looks dark. Higher-order bows do exist, but they are far too faint to see and sit on the sunward side of the sky.
Why is the double only sometimes there?
The secondary is almost always present, just dim. Only a fraction of the light reflects at each internal bounce, and the secondary bounces twice, so it carries much less energy, spread over a larger ring. To stand out against the background sky it needs bright, direct sun and dense, large raindrops, which is why a heavy shower with a low sun shows the best doubles while a light mist rarely shows one at all.