“Professor, look at the rainbow in the sky!”

I was walking across campus one beautiful (and apparently clear) November morning. Two days prior, I had been explaining to a class the calculus-based mathematics underlying rainbow formation. A student* in that class greeted me with the above remark that morning, and I followed her outstretched arm until I noticed the multicolored smile near the zenith—probably at the most about 10° of arc from the zenith. It was a circumzenithal arc (CZA) [Fig. 1(a)], formed by specific light ray paths through hexagonal plate ice crystals populating thin cirrus clouds. Crystals giving rise to the CZAs are predominately horizontally oriented as they fall due to the aerodynamic forces around them. They are only visible in favorable conditions (i.e., cirrus clouds being present) when the Sun’s altitude φ is about 32° of arc or less.

Fig. 1.

(a) CZA (courtesy of Tom Bean). (b) and (c) CHAs (courtesy of Ardy Lyon and Judy Smith, respectively).

Fig. 1.

(a) CZA (courtesy of Tom Bean). (b) and (c) CHAs (courtesy of Ardy Lyon and Judy Smith, respectively).

Close modal

Question 1: Use Fig. 2 to explain this restriction on the altitude of the Sun. The incoming ray enters the upper horizontal face of an ice crystal and exits through a vertical side. You may assume the (relative) refractive index of ice is ∼1.31. (Hint: examine the conditions for total internal reflection of rays inside a crystal.)

Fig. 2.

Diagram explaining the formation of a CZA.

Fig. 2.

Diagram explaining the formation of a CZA.

Close modal

Question 2(a): By rotating Fig. 2 counterclockwise by 90°, explain why circumhorizontal arcs (CHAs) [Figs. 1(b) and (c)] form (under favorable conditions) only when the solar altitude exceeds 58° of arc. (In this case, the incoming ray enters a vertical face of an ice crystal and exits via the lower horizontal face.)

Question 2(b): By considering the tilt of Earth’s imaginary N–S axis toward the plane of its orbit around the Sun (∼23.5°), explain why in the northern hemisphere CHAs cannot be seen above about latitude 56° N.

I thank Tom Bean, Ardy Lyon, and Judy Smith for permission to use their photographs.

Look for the answers online at tpt.aapt.org under “Browse,” at the very end of the current issue.

Fermi Questions are brief questions with answers and back-of-the-envelope estimation techniques. To submit ideas, please email John Adam (jadam@odu.edu).