Dispersion. Sunlight is a mixture of colours. Water has a slightly bigger refractive index for violet (about 1.343) than for red (about 1.331), so at each surface violet is bent a little more than red. The single white ray therefore splits into a fan of colours inside the drop.

Primary bow (one reflection). The light is refracted as it enters, reflected once at the back of the drop and refracted again as it leaves. Rays that hit the drop near 0.86 of the radius all leave in almost the same direction (the minimum deviation), so the light is bright there: red leaves at about 42.5° and violet at about 40.6° from the line pointing away from the Sun. The red light reaching your eye comes from higher drops, so red is on the outside of the primary bow.

sin i = n sin r; one reflection: D = 180° + 2i − 4r; two reflections: D = 360° + 2i − 6r

Secondary bow (two reflections). The light enters near the bottom of the drop and reflects twice. It leaves at about 50.2° (red) to 53.5° (violet), so the secondary bow is higher and wider. The extra reflection turns the fan of colours over, so the colours are reversed: red on the inside. Some light is lost at every reflection, so the secondary bow is fainter. Between the bows no light is sent to you: this is Alexander’s dark band.

Why a bow. All the drops at the same angle from the line pointing away from the Sun send you the same colour, and together they form a circle centred on the anti-solar point (the shadow of your head). The ground hides the lower part, so you see an arc. The higher the Sun, the lower the bow.

What is left out. The real spread of the colours is small (about 1.8° in the primary bow), so it is spread out (5 times in the raindrop, 2 times in the sky) in the pictures unless you untick the box. The drop is much bigger than in real life, and the brightness lost at each surface is not shown. The refractive index uses n = 1.3239 + 0.0031/λ² (λ in micrometres).

Things to try. (1) Press Aim at the rainbow ray, then move the slider a little either side. Why is the light bright here? (2) Which colour comes out at the bigger angle in each bow? (3) Why are the colours reversed in the secondary bow? (4) Raise the Sun. At what height does the primary bow disappear?