Unweaving the rainbow
It’s not often that something so pretty and fascinating has some more beautiful and easy (!) maths and physics behind it. But with rainbows, that definitely the case!
When Sir Isaac Newton described the colors of the rainbow through the refraction of light, the poet John Keats was horrified. Keats complained (through a poem, of course) that a mathematical explanation robbed these marvels of nature of their magic. I highly disagree with Keats and I am sure you will too!
Refraction
Refraction is the phenomenon where light’s path is changed (refracted) due to the medium it is traveling in. A medium can be anything from the vacuum of space to the glass in your window. In our case, it is between a water droplet and the air. Light travels at different speeds in different mediums and yes that means that things can kinda be faster than light, they just can’t be faster than c the speed of light in a perfect vacuum. Now imagine light as a wave moving through space and think of a wavefront moving towards you at the beach. You can maybe figure why it is bent when it goes from a medium where it travels quicker to a slower medium. Hopefully, the simple gif here helps you get an intuition.
We most commonly encounter refraction when we look at objects in water (like a straw in a glass) or when we get mini rainbows from glass in our windows. At which angle light is refracted is determined by the refraction index n, where n = c/c’ (lightspeed in medium 1 divided by lightspeed in medium 2). The famous formula, Snell’s law, determines the incoming angle α and the outgoing angle α’.
Snell’s law: n * sin(α) = n’ * sin(α’)
Droplets
So back to the rainbow. When sunlight enters a droplet, some light will be reflected off and some refracted into the raindrop.
a_1 is the incidental angle. The secondary angle is a bit smaller because the refractive index of water is around 1,34. From here some of the sunlight reflects off the backside of the raindrop and then leaves the raindrop through the “bottom” where the light is refracted again, with the same angle change as when it entered the raindrop.
Now, depending on where the light enters the raindrop it will exit the raindrop at different angles. We can calculate the total deflection D by adding up all grey angles:
We then calculate the derivative for the incidental angle to find the minimum. This minimum angle has the most intense light and creates what we can see as a rainbow. Remember that we are plotting the total deflection, the angle between the sunlight and you looking up will be 180°-D. This ends up to be around 42° and does not depend on the size of the water droplet.
Supposedly Descartes (mostly known for his philosophy) figured all this out graphically, but he did not understand why the rainbow showed different colors. He didn’t know that every medium has a different refractive index for each color which is a wavelength in the electromagnetic spectrum, Blue’s refractive index is around 1,342 while red is around 1,331 resulting in different deflection angles.
There are so many further interesting facts to point out but I will try to keep the list short:
In theory, there are many more orders of the rainbow, each one reflects light once more inside of the rain drop. The second-order rainbow which has its colors reversed is the only one you can frequently see with bare eyes and is located about 10° above the first-order rainbow.
Rainbows seen from the ground can only occur in the morning or the evening due to the 42°-degree angle between you and the sun. If the sun is higher than 42° -degrees, the rainbow will be below the horizon (unless you are up high and looking down). Seen from an airplane, rainbows are full circles directly opposite of the sun!
Last but not least, seawater has a higher refractive index than rainwater, so the radius of the “seabow” is a bit smaller, making for some crazy photos!
So next time you see one of these colorful arcs appear in the sky, try to remember the elegant math behind what you’re seeing!
And as always, stay curious!
