What determines the position of a rainbow?

Rainbow magic

Have you ever seen a rainbow and NOT expressed joy or wonder at it? It’s hard, isn’t it? I saw one the other day and couldn’t help but stop and admire it. There is something magical about it that brings out positive emotions in people. Where else in your life do colorful shapes just appear out of thin air in front of you (and then disappear just as fast)? No wonder people in the past had all kinds of mythical lore associated with rainbows.

These days we know that rainbows are formed when sunlight falls on a raindrop and reflects back the full spectrum of light. However, as I was watching the familiar arc of the rainbow circling the horizon, I started thinking about something else. Why did the rainbow appear exactly *here* and not somewhere else on the horizon? Why is it shaped like an arc, and why isn’t the whole sky colorful (since there is rain all over the sky)? Why are rainbows sometimes so big that they cover the whole sky, but other times seem very small? Can another person from another location see that same rainbow in the same position? And what really is at the end of a rainbow (if not a pot of gold)??

The science of rainbows

To answer these questions, I set out to understand how rainbows are formed in the first place.

As we all know, rainbows appear when sunlight and rain (or water droplets) meet.

Although sunlight appears white, in terms of optics, white is actually a combination of all the colors (or wavelengths) of a rainbow. The sun emits all the colors in more or less equal amounts and we call this combination “white”. (Fun fact: this also means that our Sun is actually white, not yellow)

Now sunlight always appears white as it moves through the air since all the colors reach our eyes equally. However, when light passes through a medium denser than air (e.g., a raindrop), something called refraction occurs. Some of that light changes direction and slows down, and the different colors (wavelengths) start to travel at different speeds. Although the speed of light in vacuum is always constant, the speed of light through a medium is different for each color or wavelength of light. This causes the white light to split into different colors, with each color going in a slightly different direction. 

Red has the longest wavelength of visible light, so it is refracted the least. Violet has the shortest wavelength, and is refracted the most. After the refracted light enters the raindrop, it reaches the other side of the raindrop, and some of it is reflected back, now with each color continuing on a slightly different path.

The light is again refracted once it exits the raindrop and travels back towards the earth, each color at a different angle: the red light at approximately 40 degrees and violet at around 42 degrees, with other colors in between. Red is always at the top (outer part) of the rainbow and violet at the bottom (inner part).

And this isn’t just one raindrop; this happens in every raindrop. All of this is concurrently happening whenever millions of rays of light hit millions of droplets of rain, and the raindrops reflect back beams of colorful light. So why isn’t the sky rainbow-colored every time it rains? And why is spotting a rainbow a relatively rare occasion?

How to see a rainbow?

I don’t know about you, but I rarely see rainbows – maybe once or twice a year. However, there are specific conditions that need to be met to witness this sight. Knowing these conditions can significantly increase your chances of spotting a rainbow, especially if your lifelong dream has been to become a rainbow-chaser (like a tornado chaser, but much more chill and with much less mortal danger).

Here are the three key conditions for seeing a rainbow:

1. The sun behind you: The sun must be behind you, with the shadow of your head directly opposite the sun.
2. Rain in front: There must be rain (or water droplets) in front of you.
3. Sun’s position: The sun should be less than 42 degrees above the horizon.

If the sun is high but still below 42 degrees, the rainbow will appear low on the horizon. As the sun sets lower, any rainbows that form will rise higher in the sky, with the most spectacular ones appearing close to sunset. Typically, you will only see half a circle because the Earth’s surface (the horizon) obstructs the full view of the rainbow. However, you can see a full circular rainbow from an airplane or any high elevation, like a mountain or skyscraper, when all other conditions are met.

Why is a rainbow circular?

This is all well and good, but I still don’t exactly understand why the rainbow is shaped like a circle and why the whole sky isn’t rainbow-colored. I dig even deeper and fall into a hole of optics and geometry, which I don’t fully comprehend (you can read more about it here). But I think I get the gist of it.

It all comes down to two facts: 1) all the sun’s rays are parallel (at the same angle); and 2) all water drops are spherical. Because of this, the light always makes the exact same angles when it reflects and exits a raindrop. This has actually been known for hundreds of years, as illustrated in a drawing by the famous philosopher and mathematician René Descartes in 1637.

Because raindrops are spherical, they reflect light in a cone shape. Thus, the rainbow appears along the circumference of the base of this cone. However, we only see the rainbow where the light rays enter our eyes, forming an imaginary cone with our eye at the tip and the rainbow’s circle at the base. If you move, a different cone forms, and you see a different part of the rainbow.

Unique like a rainbow

Rainbows might look like flat images painted across the sky, but they are actually three-dimensional. We perceive them as flat, much like we do with the sun and the moon. However, unlike the sun and the moon, a rainbow is an optical illusion that doesn’t exist in a specific spot in the sky. It’s both everywhere and nowhere at the same time.

The center of a rainbow is directly opposite the sun. So, when you move, the rainbow moves too, always staying in the same spot relative to the sun. The rainbow will thus shift in response to your motion, like a really clingy toddler. This means every rainbow is unique to the observer. If someone else is standing below the rainbow, they’ll see a different rainbow extending from their horizon. Sometimes, there are gaps in the rainbow, which simply means there aren’t enough raindrops in that part of the sky.

So, next time it’s raining nearby and the sun is out, find a place on the open horizon opposite the sun and look out for that magical pot of gold.