Why Is a Rainbow Curved?

The shower has just passed, the garden is dripping, and the late sun comes out behind you. And there it is over the rooftops: a great coloured arch, as neat as if it had been drawn with a compass. Which raises a fair question. Rain falls in straight lines and sunlight travels in straight lines, so who is drawing the curve?

The short answer

Nobody is drawing an arch at all. A rainbow is a circle, and the ground is hiding the bottom of it. The US National Weather Service describes a rainbow as a group of nearly circular arcs of colour sharing one centre, and puts the rest plainly: we do not see the full circle because the earth gets in the way. The centre of that circle is the point exactly opposite the sun, which is where the shadow of your own head falls.

How a raindrop builds a circle

Each raindrop works like a tiny round mirror with a prism attached. Sunlight bends as it enters the drop, bounces off the curved inside of the far wall, and bends again on the way out. That bending spreads white light into colours. The light does not leave evenly in all directions, though. A great deal of it is concentrated along one path, which comes back at about 42 degrees from the direction the sunlight was travelling.

So a drop only sends you its colour if it sits at just that angle from your eye. Now ask where all the drops with the right angle are. As the physicists at the University of Illinois explain, they lie on a cone with you at its tip, pointing away from the sun. Look along a cone from its tip and what you see is a circle. Red light leaves the drop at a slightly wider angle than blue, which is why red makes the outside edge of the bow and blue the inside.

• 42° from the point opposite the sun to the main bow
• ≈ 50° where the second, fainter bow appears
• 2 reflections inside each drop for the second bow

What people often get wrong

The common picture is of an arch standing somewhere in a field, with two ends you could walk to. In 17th-century Ireland, the National Weather Service notes, people said you were as likely to find a pot of gold as the end of a rainbow, and they had the odds about right. The bow is not an object in a place. It is a set of directions measured from your own eye, so when you walk, it walks with you. The person standing next to you is at the tip of a slightly different cone, and sees a bow made by different drops.

The size of the arch changes through the day as well. The lower the sun, the more of the circle rises above the horizon, and at sunset you get a full half-circle whose top is 42 degrees up. If the sun climbs higher than 42 degrees, the whole circle sinks below the horizon and there is nothing to see from the ground. That is why the grandest rainbows turn up when the sun is low.

How we know

You can check it by going up. The Met Office notes that from a mountain or an aeroplane the ground no longer cuts the picture off, and you may see the whole ring. The geometry also makes a prediction that is easy to test on a lucky day. Light that reflects twice inside a drop comes out at about 50 degrees, so a second, fainter bow should sit outside the first one with its colours in reverse order. It does. The strip of sky between the two bows looks darker than the rest, and it even has a name, Alexander's band. Not bad for a few million drops of water, each doing the same simple thing and each sending a little colour your way.