Rainbows are one of natures most beautiful and amazing sights, mystifying people since the dawn of time. Inspiring hundreds of stories, myths and fairytales, rainbows are a source of fascination to many, however they are not well understood.
So let us ask ourselves a few questions; How do rainbows work? Why always those colors? And why are they always arcs?

To answer any of these questions, we once more need to familiarise ourselves with the workings of light, by looking 

Light scattering trough water

back at an older article: “Why do wet clothes get darker?“. Just like in that article we describe light as a wave, and will be working with refractive indexes and reflection. Now to get a rainbow you need two things; rain and the Sun. (Or more generally, water and light.) The water needs to be spread across the sky in tiny droplets and the light must be shining on it from behind the observer. This last thing is a little realized fact, but whenever someone sees a rainbow, the light is coming from behind the person watching. Which means that rainbows must be a form of reflection, but why is it so different from other reflections?

To understand this we will first look at a single droplet of water, which at the end of a downpour is usually very small and almost perfectly spherical. These drops get hit by sunlight, which either reflects or moves through them. In the latter case, the light changes medium from air to water, causing it to bend slightly.

Sunlight reflecting trough a waterdroplet.

 However, the angle of this bend,depends on the frequency of the light that is passing through, where light with lower frequencies (redder) gets bent more than light with higher ones (violet), we call this process dispersion.
This light the travels trough the droplet until it reaches the other side, now because it is at a relatively high angle, most of it gets reflected back into the droplet, rather than moving out through the back. (see 2nd picture) This light then again moves through the droplet until it reaches the front again, this time exiting the droplet because the angle is much smaller. This concept is illustrated in the 3rd picture with red and violet light, the rest of the colors are left out to simplify the image.

Careful measurement by various scientists throughout time has shown that the difference in angles between red and violet light is about 2 degrees and the angle between the entering and exiting light is 42 and 40 degrees respectively. Meaning that no matter at what angle you shine a light on one of these drops, the angle of the reflected light will change by about 40 degrees.

Schematic diagram of a rainbow.

Now if we extend this to process to multiple raindrops, it will become easy to imagine how the different colors of light can reach your eye, as can be seen in the picture above. The highest drops will look red because the red rays hit your eyes, while the other colors move over your head due to the difference in angle between the various colors. This means that the lower the drops, the smaller the angle between you and the drop, so lower drops show higher angle reflections. This is why a rainbow always has red on top and violet on the bottom.

If you go even lower than that, looking at an angle lower than 40 degrees, you will no longer see the separate colors of the second reflection of the water drop, but rather the direct reflection of the light which bounced off the front of the drop. This causes you to see a tiny image of the sun in every one of these drops, which in turn cause the area under the rainbow to look brighter than the area above it. A phenomenon called Alexander’s dark band, called after the Greek philosopher who first described the process. (See top picture.)

We can now say that we understand how rainbows are formed, but what about the arc, how does that come to be? Firstly we need to remember is that a rainbow forms around the center of the so called anti-solar point. Which basically is just the place where the sun would be if you drew a line through it across the sky.

Schematic diagram of a rainbow, Sun and a observer.

Which is why the Sun needs to be facing your back in order for you to see a rainbow, as the light reflecting back travels towards the Sun, and can only reach your eyes if you stand between the two.
Now because the light of the top of the rainbow needs to hit your eyes at 42 degrees, and because we can draw a line 42 degrees away from the anti-solar point all around that point, we get a circle. Which would mean that rainbows should be circles, instead of arcs. Now this is true if there is enough water and nothing obscuring the rainbow, but in most cases the ground causes half of the rainbow to be cut off, a second problem is that there number of droplets is a lot less near the ground.

Rainbow trough an airplane window.

It is however, possible to see a rainbow as a full circle, namely by looking at it from a point where you are not bothered by the ground or a lack of droplets.
Planes, for example, are a great place for seeing full rainbows. This process is illustrated beautifully in the final image, where you can see a circular rainbow around the shadow of the plane. Proving that rainbows are both circular and centered around the antisolar point. (As the shadow is cast in this same point.)

In conclusion, rainbows are optical phenomena caused by sunlight hitting small water droplets. The reflected light is then split into separate colors which are seen at separate angles. The arc forms around the anti-solar point and depends on the location of the Sun in the sky. (The higher the Sun, the lower the rainbow arc, because the antisolar point is below the horizon.)

The same process can be seen in tiny rainbows forming around fountains or garden hoses, or made by shining light in a prism.

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