# How are Rainbows Formed? The Physics of Rainbows

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## Somewhere over the Rainbow

Rainbows are some of the most magical natural phenomena. They are bright, colorful and have contributed to folklore for thousands of years. How, then, are these endless visions formed?

## What is a Rainbow

Rainbows are formed through the processes of light reflection and refraction. Sunlight enters the raindrop and is broken into its spectrum of colors. Depending on the angle of entry, some wavelengths of light are then reflected off the internal curve of the raindrop and sent back out toward the ground. If the angle of refraction is between 40 and 42 degrees, only certain wavelengths of light are sent back toward the Earth. Raindrops that refract the light at 40 degrees send only blue wavelengths toward the ground, while those with a 42 degree angle send red wavelengths (41 degrees produces yellow light). Between 40 and 42 there can be combinations so that any one particular color in the spectrum could be sent toward the observer. This means that when we see a rainbow, we are seeing different refractions and reflections from millions of raindrops. Again, any light that is refracted at an angle greater than 42 degrees ends up in the atmosphere, while rays that refract at less than 40 degrees don’t come back to Earth in a single wavelength, but in a combination of all wavelengths. The end result in this case is white light.

Thus, the rainbow is an optical effect based on observer position and light reflection and refraction, not an actual physical object.

## Why is the Rainbow Always an Arc Shape?

The reason a rainbow is always an arc shape was first discussed at length by Rene Descartes way back in 1637. He based his observations on two major assumptions; all raindrops are spherical and the distance to the sun is large enough that all sun rays can be assumed to be parallel to each other as they hit the raindrops. He set up an experiment with a small globe which stood in for a single raindrop. By varying the angle of light entry, he calculated the entry and exit angles for all possible visible light wavelengths. Here he determined that for the optimal angle of the visible light spectrum to produce a visible rainbow to a viewer at ground level, the angular radius of the raindrop must be 42 degrees.

Once this angle was determined, he checked it against a real rainbow and just as he had found in his experiment, the angular radius of the arc of his observed rainbow was 42 degrees with the apex directly opposite the sun.

Even though this is true, some rainbows seem to have steeper arcs than others. How can this be? Simple. As an observer, you can only see some of the rainbow most of the time. If you were to take an angular measurement, you would find that the arc of the rainbow is consistent; it is just the reference points that you are using that causes the visual anomaly (just like how the moon looks smaller the higher it is in the sky although it is always the same size). Just as a short footnote, if the Earth wasn’t in our way, we would be able to see a full rainbow circle.

## Double and Lunar Rainbows

Occasionally, if the angle of the sun hitting the water droplets is just right, a double rainbow can form. This happens when the light reflects inside of the rain droplet two extra times and exits at an angle between 50 and 53 degrees. These secondary rainbows are seen above the primaries and their color pattern is reversed with blue being at the bottom and red at the top. They are less brilliant in color because of the extra diffusion caused by the two extra reflections. The space between the two rainbows is called “Alexander’s Dark Band” as the light between the rainbows is generally darker than the surrounding sky. The interplay of light waves in this region often causes complimentary colors to eliminate each other.

Lunar rainbows are a very rare occurrence and have been recorded only a handful of times since Greek times. When the light of the full moon is reflected at the perfect angle to produce a rainbow, there is often no good backdrop to see it against. Therefore, even though the conditions may be favorable, if there isn’t a lightly-colored band of clouds in the atmosphere, then the rainbow won’t be visible to observers on the ground.

## When is the Best Time to See a Rainbow?

The best time to see a rainbow is in the mid-morning or the mid to late evening. This is because the angle of the sun’s rays at these times are closer to the 42 degree angle needed to produce the optical effect. The noon sun is too high to produce many rainbows, not that they can’t happen at noon, it is just far less likely. It is also good to note that rainbows often appear in the morning just before it is going to rain (because the sun is in the East and rain clouds generally travel west to east) and after the rain has stopped in the evening (as the sun is in the West). You, the observer, must be between the sun and the rain with the sun to your back to see a rainbow. Remember, the reflection comes from the backside of the raindrop so you must be on the same side of the rain as the sun.

Of course, the sun isn’t the only source of light that can be used to create a rainbow nor is the rain the only place to get the water droplets needed. Some of the more interesting places to see a rainbow are in your sprinkler at home, near a geyser or even in the spray from a waterfall (this one is quite spectacular).

## References

The National Center for Atmospheric Research and the UCAR Office of Programs, https://eo.ucar.edu/rainbows/

Finding and Photographing Rainbows, https://www.photocentric.net/rainbow_physics.htm

The Rainbow, https://mysite.du.edu/~jcalvert/astro/bow.htm

Image Courtesy of Martyn Gorman and licensed for reuse under this Creative Commons Licence, https://www.geograph.org.uk/photo/426517