Formation of the Circumzenithal Arc

Circumzenithal arc is produced by the refraction of light on hexagonal ice crystals. If the crystals are plate shaped (their height is relatively small), their largest face is horizontally aligned due to different aerodinamic reasons. We will show how rays of Sun are refracted while passing through such oriented plate crystals.

Assuming parallel rays coming from the Sun, consider the rays that enter a hexagonal prism on its upper horizontal face and leave on one of the vertical sides. (See Figure 1.) The path of the light is made up by three segments: before entering, inside, after leaving the crystal.

The refraction at the boundary of segments and is invariant to any rotation of the prism around the vertical axis. Let denote the elevation of the Sun, the angle between the horizontal plane and ligth segment and the refraction index of ice. According to Snell's law:

 (1)

The refraction / at the vertical face of the prism is more interesting. Note that despite both pairs of segments / and / being coplanar, the two refraction planes are different in general. Instead of the current coordinate system having a fixed light ray segment and plane rotating freely around the vertical axis, we make fixed for the analysis below.

As depicted in Figure 2, let denote the intersection of ray and plane , and a point of ray at some fixed height. In the coordinate system fixed to plane , the possible locations for point form a circle. Let be the center of this circle, and the projection of onto the diameter of the circle that is in . The refraction plane is defined by ray and the normal of at . Since is parallel to , the plane of refraction contains .

The next figure depicts the refraction / . Let and be the images of and after a reflection in , respectively. Let be a point on ray such that the lengths and are equal, furthermore be the projection of onto line . Finnally, let be the intersection of and . Applying Snell's law gives:

 (2)

We are going to determine the possible locations of point while moves on the semicircular arc depicted above. Let denote the horizontal plane in which this circle lies, while is its reflection in . Since , therefore . Beside, (2) states that , which is constant, since all the points of the semicircle are equidistant from . Thus is in the intersection of this sphere and plane , which is a circle. Figure 4 shows plane including , which is the reflection of in .

As a consequence of being on a circular arc in a horizontal plane, the angle of ray to the vertical line is independent of the orientation of the crystal. The circumzenithal arc is therefore produced by the ice crystals that are on the cone, which has a vertical axis and its apex is at the viewer.

Below we express the angle under which the arc is seen and the angular width of the arc in therms of the elevation of the sun (). Consider the vertical plane in which rays and lie. If the side face of the hexagonal crystal is perpendicular to this plane, ray also lies in this plane. This implies and , therefore and . From (1) and (2) recall that

 (3)

Based on these,
 (4)

Since , this is always positive. A necessary condition for the appearance of the circumzenithal arc follows from the fact that , thus
 (5) (6)

Therefore if , the angular radius of the arc is
 (7)

The angular width of the arc can be calculated as follows. The path of the light projected into the horizontal plane is shown in Figure 4 using double stripped lines. The maximum deviation of the light in the horizontal plane is the half of the angular width circumzenithal arc. The maximum deviation angle corresponds to the angle of total reflection, i.e., when , thus

 (8)

so the ratio of the circles' radii in is
 (9)

To obtain the ratio of the radii, consider again the case when all 3 ray segments are in the same vertical plane. Using the notations of Figure 3, having and and using (2) this can be written as
 (10)

With the help of equations 3,
 (11)

Condition for this being positive leads to the same constraints that we got for (4). Obviously, this is less than , therefore for the prevoiusly defined range of , the angular with of the circumzenithal arc is
 (12)