Why is a rainbow round?

The usual picture is of a coloured object hanging in the air at some place, like a bridge. That picture is wrong. A rainbow is not a thing at a place; it is a direction. It is the set of directions, measured from your own eye, in which raindrops happen to send you a lot of sunlight. Every drop that lies in one of those directions flashes at you, whether it is 100 metres or 2 kilometres away. Move, and a different set of drops does the job.

To see why those directions form a circle, start with a single drop and a single ray of sunlight.

One ray, one drop

A falling raindrop of ordinary size (a millimetre or two across) is very nearly a sphere. When sunlight reaches it, three things happen to the light that makes a rainbow. It is refracted (bent) as it crosses from air into water, because light travels more slowly in water. It is reflected once from the inside of the back of the drop. And it is refracted again on the way out. The deviation is the total angle through which the ray has been turned, from its original direction to its final one. A ray that went straight back towards the Sun would have a deviation of 180°.

How much a ray is turned depends on where it enters. Call that the impact parameter: how far from the drop’s centre line the ray strikes, as a fraction of the drop’s radius. Zero means dead centre; one means grazing the edge.

A ray through the centre comes straight back: deviation 180°. Rays further out are turned less, down to about 138° for a ray entering at 0.86 of the radius. Beyond that, the deviation rises again. The ray at the bottom of that valley is called the Descartes ray, after René Descartes, who worked it out by tracing rays through a large glass globe filled with water and published the result in Les Météores in 1637. (He was not the first to try a water-filled glass globe: Theodoric of Freiberg and Kamal al-Din al-Farisi did so around 1304–1309. Descartes added the law of refraction and the numbers.)

Turn it around: the outgoing ray makes an angle of 180° − 138° = 42° with the line pointing straight away from the Sun. Stand with your back to the Sun and look at the shadow of your own head. That point, exactly opposite the Sun, is the antisolar point. No ray that reflects once inside a drop can leave at more than 42° from it.

A whole beam

The bottom of a valley is flat. Near the minimum, a big change in where the ray enters makes almost no change to where it leaves. So when a whole beam of sunlight falls on the drop, a large share of it leaves in nearly the same direction, about 42° from the antisolar point. That pile-up is the rainbow.

The figure below sends 4,000 evenly spaced rays into the upper half of a drop and counts how many leave in each half-degree of angle from the antisolar point. The light piles up against a hard edge and nothing at all comes out beyond it.

About one ray in eight leaves within the last degree before the edge, so a narrow ring of directions gets far more light than any other. (The figure counts rays in one flat slice through the drop, evenly spaced. In a real round drop each ray stands for a ring of light, and the outer rays stand for more of it; weighted that way the share is about one-fifth.) The edge itself depends on colour. Water bends violet light a little more than red: its refractive index (the ratio of the speed of light in vacuum to its speed in water) is about 1.331 for red light at 700 nm and 1.344 for violet at 400 nm. That small difference moves the edge from 42.4° for red to 40.6° for violet. Isaac Newton, in Opticks (1704), was the first to explain the bow’s colours this way: each colour has its own amount of bending, and so its own rainbow, nested inside one another. Red is on the outside, violet on the inside.

From one drop to the sky

Now step back. You stand with the Sun behind you. Sunlight falls on every drop in a shower, all in the same direction. A drop sends you its bright flash only if the line from your eye to that drop makes an angle of about 42° with the line from your eye to the antisolar point.

Which drops satisfy that? All the ones lying on a cone with its tip at your eye, its axis pointing at the shadow of your head, and a half-angle of 42°. Any drop on that cone, near or far, works; any drop off it doesn’t. Looked at from its tip, a cone is a circle. That is the whole answer: the rainbow is round because the drops that can send you their flash all sit at the same angle from one point, and a set of directions all at the same angle from one direction is a circle.

It also explains the rest. The bow is always opposite the Sun, because its centre is the antisolar point. Your neighbour has their own antisolar point and their own cone, so they see light from mostly different drops, at least among the nearby rain shown in Figure 3. That changes with distance. The ring of lit drops is about 3 cm thick for every metre of distance, while your friend’s cone is displaced by only a metre or two (it depends on the Sun’s height); so beyond roughly 50 to 100 m the two rings overlap, and for a shower a kilometre away you share nearly all the same drops. Each of you sees such a drop at a very slightly different angle, so the colour it sends you can differ (clearly for rain a few tens of metres away, hardly at all at a kilometre). And the ground usually hides part of the circle.

The antisolar point is as far below the horizon as the Sun is above it. With the Sun at 20°, the top of the bow stands 42° − 20° = 22° up. With the Sun on the horizon, the centre of the bow is on the horizon too, and you see a full half-circle. Once the Sun climbs above about 42°, the whole primary circle is below the horizon and there is no primary rainbow to be seen from the ground (the fainter second bow, 51° out, hangs on until the Sun is about 50° high; tick the box to see it). That is why you don’t see rainbows around midday in summer: in London the midsummer Sun stands 62° high at noon. From an aeroplane or a mountain top, where you can look down into the rain, you can sometimes see the full circle.

Notice too that the sky is brighter inside the bow than outside it. Figure 2 showed why: every ray leaves at 42° or less from the antisolar point, so the drops inside the circle all send you some light, while the primary bow sends nothing at all from outside it.

The second bow and the dark band

Some light reflects twice inside the drop before leaving. Its deviation also has a minimum, and that gives a fainter second bow at about 51° from the antisolar point (50.2° for red, 53.6° for violet in this model). The extra reflection turns the light the other way round, so the colours are reversed: red on the inside. The second bow is fainter because some light escapes the drop at each reflection. And twice-reflected light leaves only at angles outside 51°.

So between about 42° and 51° neither kind of light arrives: the primary sends its light inside its circle, the secondary outside its own. That strip of noticeably darker sky between the two bows is called Alexander’s dark band, after Alexander of Aphrodisias, who described it around 200 AD, about eleven centuries before Theodoric and al-Farisi first explained the two bows.