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Thayer Watkins
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The Elevation Angle, Radius
and Center of Curvature of a Rainbow

Rainbows Formed from Direct Rays from the Sun

When a light ray impacts a water droplet it may be refracted and then be reflected a number of times off the inside of the droplet before being refracted on its exit from the water droplet. What happens to a particular ray depends upon where it impacts the droplet with respect to the center of the droplet. If the ray is dead-on-center it could be reflected back out of the droplet in a direction exactly opposite to the direction from which it entered. But generally the ray will be a bit off center and will be reflected backwards with its direction changed by an angle ρ, called here the refraction angle.

A narrow bundle of rays impacting a particular water droplet will generate a cone of light with the cone angle being equal to the refaction angle ρ. In a vertical cross section of the cone some rays are refracted and reflected backward and downward and some are refracted and reflected backward and upward.

Observers on the ground experience only the rays that are refracted downward, but from an elevated observation point such as in an airplane the rays refracted upwards from rain droplets also enter into the formation of the rainbow, as is shown in the diagram below. From an aircraft one can see such full circle rainbows.

Let φ be the elevation angle of the sun. The top of the rainbow is at an elevation angle of (ρ-φ). The bottom of the full circle rainbow is at an elevation angle of -(ρ+φ). This means that the rainbow has an angular radius of ρ and the center of the circle is at an elevation angle of -φ.

Reflection Rainbows

When a large body of water is near the rain shower the refection of the sun off the water can create a reflection rainbow. The elevation angle of a reflection rainbow is determined as shown in the diagram below. A sun ray coming to the water at an elevation angle of φ is reflected upwards at that same angle φ. When that reflected ray impacts a rain droplet it is refracted backward and downward by an angle ρ. The elevation angle of the top of the refection rainbow is an elevation angle of θ of π-(π-(ρ+φ))=(ρ+φ). If there were a full circle reflection rainbow the bottom would be at an elevation angle of -(ρ+φ). This means that the anggulllar radius of the reflection rainbow is (ρ+φ) and the center of curvature of the reflection rainbow is at an elevation angle of 0.

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