Rainbow, Double Rainbow, And Alexander’s (Dark) Band

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The reflection and refractions that cause a primary rainbow. Image courtesy of NASA

A rainbow almost seems like a magical phenomenon: Colors splayed across the sky in infinite subtlety and dazzling luminosity — it would be spectacular even without the pot of gold. But what causes the rainbow?

The Physics Of a Rainbow

A rainbow is caused by the prism effect of a water droplet. When light travels from one medium into another in which its speed is different, the light ray is bent; the amount of bending is inversely proportional to the wavelength.

The speed of light in a vacuum is about 3.00 x 108 meters per second. In air, it is a tad slower, but the approximation still holds. In water, on the other hand, the speed of light is much slower — about 2.25 x 108 meters per second.

When a ray of light encounters a water droplet, light entering the droplet is bent (refracted), and the colors are separated. The light then strikes the rear of the droplet and is reflected, then is refracted again when it exits the drop. An observer in the right place sees that fabulous arc of color.

The separation of colors is reversed upon reflection. The first refraction puts the long wavelength red on the inside, but the reflection reverses the spectrum and the second refraction separates the colors even more. The result is that we see red on the outside of the one-reflection rainbow, called the primary rainbow.

How Does A Double Rainbow Work?

If the observer is in the right place with respect to the water droplets, a light ray that bounces twice within the drop  can create a secondary rainbow outside the first. Since each reflection causes the colors to be reversed, the second rainbow appears to have red on the inside and blue on the outside.

Each time light hits the raindrop-air interface, some of the beam is refracted and some reflected. So the secondary rainbow appears dimmer, as additional light is lost in the second reflection.

Alexander’s Dark Band

Approximately 1800 years ago, Alexander of Aphrodisias (the Greeks had a way with names) noticed that when there was a double rainbow, the band between them appeared darker than the rest of the sky. We appropriately call this Alexander’s Dark Band, though Alexander had no idea what caused it.

A double rainbow over NOAA. Photo courtesy of NOAA

The angle made by the line from an observer to the center of a primary rainbow and a line from the observer to the rainbow’s arc is 42°. The comparable angle of the secondary rainbow is 50°.

Between these angles, no light can be scattered to the observer by primary or secondary reflections. Since light can be scattered from this region by other reflections and refractions, the band is not completely dark.

Higher Order Rainbows

Theoretically rainbows could be produced by any number of reflections inside a raindrop. However, the three and four reflection rainbows occur between the observer and the sun, which makes them virtually impossible to see. After that, they are so faint as to be undetectable by the human eye under any circumstances.

Best Conditions For Viewing A Rainbow

Rainbows are fairly rare because they require atmospheric conditions that don’t commonly exist. There must be sun shining on raindrops; there can be no obstructions between the sun and the raindrops, or between the raindrops and the observer.

In the middle latitudes, rainbows most commonly occur in the evening, after rain has passed from west to east and the weather has started to clear. The sunlight has an unobstructed path to the rain, and the observer has a clear view.

Can We See A Full Circle Rainbow?

A rainbow could make a full circle if the earth didn’t get in the way. From an airplane it is fairly common to see the full circle of a rainbow.

Other Applications Of The Rainbow Phenomenon

One of the most powerful tools astronomers have is the prism, with which they separate the color spectra of stars and galaxies. Of course, water is not the preferred medium. Solid prisms are much easier to handle; any transparent material such as glass works well.

By separating starlight into its constituent wavelengths, astronomers can determine, among other things, how fast a star or galaxy is moving and what trace substances it contains.

But even without practical value, a rainbow would still be as exquisite as it is ephemeral.

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