The rainbow: an arc written in every raindrop

Everyone has seen one; no one has ever stood next to one. A rainbow is not an object hanging in the air — it is an event played out between three points: the sun, the raindrops and your eye. The figure below has two views: first look inside a single drop, then raise the sun above the horizon and watch what happens to the arc.

A drop like a prism, only round

A ray of sunlight entering a spherical drop goes through three events: it refracts on entry, reflects off the back wall, and refracts again on the way out. At each refraction, violet light bends more than red — water has a slightly higher refractive index for it. White light enters as one ray and leaves as a fan of colours. Move the slider in the DROPLET view: where the ray strikes decides the angle at which the fan leaves the drop.

Why 42 degrees, of all things

Drops send light back at all sorts of angles — but not evenly. The formula above has a minimum: around 42° from the antisolar direction, rays from many different impact points pile up into one bright direction. You can see it on the slider: near the minimum, the viewing angle almost stops changing even though the impact point keeps moving. This pile-up — a caustic — is what turns the chaos of a billion droplets into a sharp, luminous circle of radius 42°. We see its slice above the horizon: the arc.

A rainbow is not a thing — it is geometry, private to every eye.

The second bow and the dark band

Some of the light reflects inside the drop not once but twice — and leaves at about 51°. That is the secondary bow: fainter (every reflection costs light) and with the colours reversed, because the extra reflection flips the fan. Between the two bows the sky is noticeably darker — Alexander’s dark band: into that range of angles the drops send no light with either one bounce or two.

The SKY view shows the last piece of the puzzle: the arc’s geometry is chained to the sun. The top of the bow stands at 42° minus the sun’s elevation — which is why the grandest rainbows happen in the morning and towards evening, and at a summer noon, with the sun climbing past 42°, the whole arc slips below the horizon. A garden sprinkler lets you cheat: you can make your own rainbow at any hour, as long as the sun is at your back.

A simplificationWe do pure geometric optics: we ignore interference (the supernumerary bows just inside the main arc), the polarisation of rainbow light, and the fact that large drops flatten while a mist of tiny ones makes a white "fogbow". Refraction and reflection are enough to explain the 42° — and that is the heart of it.