Why are rainbows in an arc shape?

image source: Wikipedia

Rainbows are an optical phenomenon caused by the reflection of light in water droplets which results in a spectrum of light. We all know what rainbows are, when and why they appear but the question I was wondering to myself was that why do they seem like a semicircle. Well, the answer is pretty obvious – the Earth is blocking off the other half. That leads to more questions like why aren’t they in a straight band and what actually causes them to look like an arc.

First rational explanation of rainbows

The first attempt to rationally explain the appearance of the rainbow was probably that of Aristotle. He proposed that the rainbow is actually an unusual kind of reflection of sunlight from clouds. The imagelight is reflected at a fixed angle, giving rise to a circular cone of “rainbow rays.” Aristotle thus explained correctly the circular shape of the bow and perceived that it is not a material object with a definite location in the sky but rather a set of directions along which light is strongly scattered into the eyes of the observer. The angle formed by the rainbow rays and the incident sunlight was first measured in 1266 by Roger Bacon. He measured an angle of about 42 degrees; the secondary bow is about eight degrees higher in the sky. Today these angles are customarily measured from the opposite direction, so that we measure the total change in the direction of the Sun’s rays. The angle of the primary bow is therefore 180 min us 42, or 13 8, degrees; this is called the rainbow angle. The angle of the secondary bow is 130 degrees.

After Aristotle’s conjecture some 17 centuries passed before further significant progress was made in the theory of the rainbow. In 1304 the German monk Theodoric of Freiberg rejected Aristotle’s hypothesis that the rainbow results from collective reflection by the raindrops in a cloud. He suggested instead that each drop is individually capable of producing a rainbow. Moreover, he tested his conjecture in experiments with a magnified raindrop: a spherical flask filled with water. He was able to trace the path followed by the light rays that make up the rainbow.

Why are they curved?

The sunlight takes a complicated path through each water droplet. It comes in the side closest to the Sun, bends because the index of refraction in water is bigger than that of air (you can see this effect by putting a pencil in a glass of water so that some of it sticks out and looking at it from different angles — the pencil will appear “broken” at the place it crosses the water surface). The sunlight, passing through the water droplet, bounces off the back surface of the droplet, travels back to the other side, and bends once again on its way out.

The reason why the rainbow is curved is because all the angles in the water drop have to be just right for the drop to send some sunlight to you, standing on the ground. So, with the Sun *behind* you, only those water droplets that have the same angle formed by you, the drop, and the Sun (this angle happens to be approximately 42 degrees) will contribute to the rainbow. Other droplets send their light somewhere else, and if you move to a different location, new droplets are needed to make the rainbow you see in the new location. This is why you can’t go to the end of a rainbow to find the mythical leprechauns and pots of gold; anywhere you stand, the rainbow is formed by faraway drops of water reflecting and bending sunlight. The rainbow is curved because the set of all the raindrops that have the right angle between you, the drop, and the Sun lie on a cone pointing at the Sun with you at one tip. The rainbow may look semicircular if the Sun is setting or rising (a good time to see a rainbow because the sunlight at that time can get under rain clouds because it is traveling horizontally). If the Sun is higher in the sky, the Earth gets in the way and you may see less than a semicircular rainbow.

Why don’t rainbows look like straight band of colours?

Sunlight passing through raindrops causes rainbows via a process called refraction, which is the bending of light as it passes from one medium to another. This is analogous to pushing a shopping cart at the edge of a parking lot: if the wheels on one side roll off the pavement onto an adjacent area of grass, the cart will start to turn toward the grass. This is because the wheels moving on the pavement are able to roll faster than the wheels on the grass.

In the case of a rainbow, when sunlight hits a raindrop it does not move as fast through the water as it does through the atmosphere, so it bends a little. The light then turns again as it moves out of the raindrop and back into the air at its original speed. When light hits the rain at just the right angle, it is refracted through a raindrop and into our eyes, causing us to see a rainbow.

But how does the “white” sunlight produce a multicolored rainbow? Sunlight, or “white” light, is actually made up of continuous bands of different colored light–red, orange, yellow, green, blue, indigo and violet. Each color has a different wavelength, or frequency, which refracts slightly differently when it passes from one medium to another. As a result, white light can be broken up into its component colors by being passed through certain medium. For example, a prism can also create rainbows because the glass, like the raindrop, bends the different colors of light at slightly different angles.

The closer the sun is to the horizon, the more of the circle we see. Right at sunset, we would see a full semicircle of a rainbow with the top of the arch 42 degrees above the horizon. The higher the sun is in the sky, the less of a rainbow is visible above the horizon.

Rainbows are circular from height!

As I wrote above, if the Earth gets out of the way, you can see a complete circular rainbow. Check this video out to see how it actually looks like from a helicopter!

Rainbow from a helicopter!
Via The Theory of Rainbow by H. Moyses Nussenzveig (University of Wisconsin), Scientific American, io9 and Department of Physics, University of Illinois.
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3 comments

  1. Great post, I’ve never seen a rainbow in a full circle before. I remember learning a bunch of this stuff in my atmospheric science class. Thanks for sharing! 🙂

    1. Thank you, Stephanie.
      We have just started this blog. We’ll try to post more interesting articles! 😀

      Cheers!

  2. […] Why are rainbows in an arc shape? (asdfscience.wordpress.com) […]

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