Rainbow
Why is a rainbow always about 42° from the point opposite the sun, and why is red on the outside? Follow sunlight through a raindrop, see how high the bow will stand in the sky for any sun height, and find the fainter second bow above it.
How Does a Rainbow Form?
Sunlight enters a raindrop, is refracted, reflects off the back of the drop and is refracted again on the way out. Rays hitting the drop at different heights leave at different angles — but they pile up near one angle, the minimum deviation. That concentration of light is the rainbow, about 42° from the antisolar point.
Because the refractive index depends on colour (dispersion), each colour has its own angle: red at 42.3°, violet at 40.6°. Two reflections inside the drop give a fainter secondary bow at about 51°, with the colours reversed. The sky between the bows is darker: Alexander’s dark band.
What You See
sun 15°Inside a Raindrop
1 reflectionRainbow Formulas
k = reflections inside the drop| Quantity | Formula |
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Worked Techniques
Each ray is turned by a total angle D. Near the minimum of D, many rays leave in almost the same direction, so the light there is much brighter.
The bow is a circle around the antisolar point, which lies as far below the horizon as the sun is above it. A low sun gives a tall, almost half-circle bow; a high sun pushes it below the horizon.
A higher index bends light more, so the bow is smaller. Tiny glass beads in road paint make a bright small bow when the sun is low behind you; above n = 2 there is no primary bow at all.
Sources & Verification
- J. D. Walker, “Multiple rainbows from single drops of water and other liquids”, American Journal of Physics 44(5), 421–433 (1976).