Plate 07 · Light & Sound
Why a Rainbow Is Round
It does not hang anywhere in the sky. It is a ring of light 42 degrees around the shadow of your own head. And all of it happens inside a raindrop.
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Light in a Raindrop
Sunlight entering a raindrop bends as it crosses from air into water, reflects off the back of the drop, and bends again on its way out.
Each color bends by a slightly different amount: violet the most, red the least. A beam of white light is pulled apart into a row of colors.
Cross-section of a raindrop · greatly enlarged
42 Degrees
Light that enters the drop at different places leaves in different directions. But they share a limit: none comes back more than about 42 degrees off the line it came in on, and a great many crowd in just short of it — so that is where the light is brightest.
For red light the angle is about 42.4 degrees; for violet, about 40.8.
Back to the Sun
So to see a rainbow, the Sun must be behind you and the rain in front. The bow is a ring of light about 42 degrees out from the shadow of your head: red on the outside, violet on the inside.
The ground hides the lower part of the ring, which is why we usually see only an arc.
Your Own Rainbow
Every raindrop scatters light in all directions, but only the drops on that 42-degree ring send their concentrated, colored light into your eyes. The person beside you sees a different rainbow, made by different drops.
Walk toward it and the rainbow moves away just as fast. There is no end of the rainbow to reach.
The Second Bow
Light that reflects twice inside the drops makes a second bow, at about 50 to 53 degrees. It is fainter, and its colors run the other way: red inside, violet outside.
Between the two bows, neither once- nor twice-reflected light arrives, so that band of sky looks darker.
No Rainbows at a Summer Noon
The higher the Sun, the lower the rainbow. Once the Sun climbs past 42 degrees, the main bow sinks entirely below the horizon.
In Beijing on the June solstice, the Sun stands higher than that from about 8:40 in the morning until about 3:50 in the afternoon, so the main bow cannot appear. In winter the noon Sun climbs only into the twenties; whenever sun and rain meet, a rainbow is possible all day.
Beijing time · computed from the Sun’s position
Your Turn
Raise and lower the Sun and watch the bow rise and set. Turn on the second bow. Then put someone beside you and compare your rainbows.
Try thisHow high can the Sun climb before your rainbow sinks below the horizon?
What the model leaves out
The model traces rays through round raindrops with geometric optics, counting light that reflects once or twice. The wave nature of light (which adds faint extra bands inside the bow), and the size and shape of the drops, are left out. Water’s refractive index comes from a fit to measured values. The rain, ground and figures in the scene are illustrative and not to scale.
A common misreading
A rainbow is not an object standing somewhere. You cannot walk under it, and it has no end. It is a set of directions: every direction about 42 degrees away from the shadow of your head.
Sources
Follow the question
Plate 13Why the Sky Is BlueThe sky is not reflecting the sea. The air itself is throwing the blue in sunlight across the sky.
Plate 29A Curved Piece of GlassLight bends a little as it passes from air into glass. Shape the glass into a curve and light passing through every part of it can meet at one point — which is how magnifying glasses, cameras and your eyes work.
