Dr. Christopher S. Baird

# Why does a rainbow exist only in a narrow band?

Category: Physics      Published: January 15, 2014

By: Christopher S. Baird, author of The Top 50 Science Questions with Surprising Answers and Associate Professor of Physics at West Texas A&M University

Public Domain Image, source: U.S. National Park Service.

A rainbow does not exist only in a narrow band. A primary rainbow is the light pattern resulting from sunlight entering a spherical water drop, bouncing once off the back of the drop, and exiting. Defined this way, a primary rainbow exists at all viewing angles from 0° to 42° and therefore forms a wide disc. The viewing angle is the angle formed between the point exactly opposite the sun and the direction you are looking. It's true that the most vivid and spectrally pure part of the rainbow exists only in a narrow band from about 41° to 42°, but this is only one part of the rainbow. There are three main parts to a primary rainbow:

1. The bright and nearly spectrally pure spread of colors existing in a narrow band from about 41° to 42° (consisting of red, orange, yellow, green, blue, violet).
2. The faint and odd-looking, partially-mixed colors existing in a narrow band from about 39° to 41° (purple, teal, gray).
3. The faint white color existing in a broad disc from about 0° to 39°. This faint white color just makes the background look brighter.

All of these parts of the rainbow arise from the exact same thing: light rays refracting (bending) upon entering a spherical water drop, reflecting once off the drop's back surface, and then refracting again upon exiting the drop. The different parts of a rainbow look so different simply based on the way the colors are mixing. Using the laws of reflection and refraction, it is easy to derive a relationship between the viewing angle at which you see a light ray, and the associated height of the corresponding original light ray with respect to the drop center's. The result is shown in the image for the colors red (640 nm), yellow (585 nm), green (530 nm), blue (475 nm), and violet (420 nm).

Plot of the viewing angle of the rainbow's light rays as a function of the height of the corresponding original ray of sunlight relative to the drop's center. A viewing angle of 0° is looking directly away from the sun. A height of 0 is an incident ray hitting the dead center of the drop and a height of 1 is an incident ray just glancing the side of the drop. Public Domain Image, source: Christopher S. Baird.

The x-axis plots the height at which the original rays of sunlight hit the spherical water drop relative to the drop's center. A height of 0 corresponds to a direct, dead center hit on the drop and a height of 1 corresponds to an incident ray just glancing off the side. The y-axis plots the viewing angle at which we see light exit the water drop and is measured relative to the antisolar point. A viewing angle of 0° corresponds to looking directly away from the sun, and a viewing angle of 180° corresponds to looking directly towards the sun. The curves on the plot relate the incident ray height and final viewing angle for rays of different colors.

White sunlight contains all colors. After the white sunlight scatters off the water drops in the air, different colors end up at different viewing angles. Analyzing this graph, we find that at a viewing angle between 41° and 42°, the colors are well separated. An observer therefore sees this part of the rainbow as a vivid, virtually pure spread of colors. Between 39° and 41°, these colors begin to blend incompletely, leading to odd colors. Finally, for viewing angles between 0° and 39°, there is almost an equal amount of all colors. When all colors of light are mixed together evenly, the color you get is white. The part of the rainbow extending from 0° and 39° is therefore white. As this graph should make obvious, the color red is scattered into all angles from 0° to 42°. But because of the mixing in of other colors, the only angle at which we see almost pure red is at 42°.