Figure 57.3: Impact parameter and the resulting scattered light rays. (From Nussenzveig,Scientific American,
April 1977.)
The rainbow is in the shape of a partial circle, being a larger part of the circle the lower the Sun is in
the sky. The rainbow would, in fact, be a complete circle if the ground weren’t in the way; such complete
rainbows may sometimes be seen from airplanes. The center of the rainbow’s circle is directly opposite the
direction of the Sun in the sky, so if the Sun is setting in the west and it’s raining, look for a rainbow in the
east.
57.3 The Secondary Rainbow
A fainter secondary rainbow appears above (outside) the primary rainbow, and its colors are reversed (violet
on the outside edge and red on the inside edge). It is also a bit wider than the primary bow. The secondary
bow is due to light reflectingtwiceinside the raindrop due to total internal reflection. Some light is lost during
each reflection, so the secondary bow will be fainter than the primary bow.
57.4 Location of the Rainbow
What determines the location of the rainbow in the sky? The center of curvature of the rainbow is opposite
the direction of the Sun, but what determines the angle from the sunline to the rainbow? By convention, we
measure the angle between the Sun and the rainbow, as seen by the observer; this is called therainbow angle.
The primary rainbow has a rainbow angle of about 138 ı, while the for the secondary bow the rainbow angle
is 130 ı.
What determines these angles? Figure 57.3 shows the path of a light ray through a single (spherical)
raindrop. The perpendicular distance between the light ray and the center of the drop is called theimpact
parameter, as shown in the figure. Of course, light rays are hitting the many raindrops at all different im-
pact parameters, so the outgoing light rays are scattered over a range of angles. But from the principles of
geometrical optics, we can calculate the angle of the outgoing light ray (thescattering angleas a function of
impact parameter (Fig. 57.4). In the figure, you can see that the curve for the primary bow (upper curve) has
a minimum for an impact parameter that is about 0.86 times the drop radius. Around this impact parameter,
significant changes in the impact parameter result in nearly the same scattering angle — in essence, many