Physics of the circumzenithal arc

Seen from Marple, near the Peak District National Park, on July 13, 2008. The camera looks up. The circumzenithal arc, or 'smile in the sky' is fairly common but is rarely seen. The first impression is that of an upside down rainbow. It is however brighter and more colourful than the rainbow, and is considered the most beautiful of the ice halos. More pictures

Origin

For a description of the circumzenithal arc see the Atmospheric Optics web site. Rainbows occur when sunlight shines on water droplets in the air. Similarly, halos are caused by sunlight shining on ice cystals. Clouds above some 7km in altitude, especially cirrus, will consist of ice. The most common halos are the ring around Sun, and the sundogs (parhelia), which can be seen in the UK in most weeks. The circumzenithal arc forms in plate-like ice crystals. The crystal acts as a prism, causing very pure colours.
Fig.1: Refraction through a plate-like ice crystal


In Fig. 1, the light comes in under an angle iin (measured with respect to the perpendicular, i.e. iin=0 means vertical). It is refracted at the first surface and again at the second surface, and finally leaves at an angle iout. (again iout=0 means perpendicular to the surface.) The index of refraction of ice is n=1.31. The following equation can be derived (see Minnaert: Light and Color in the Outdoors):

The total change in direction is iout + iin -90 degrees. The height of the circumzenithal arc in the sky is given by 90 - iout. (It is also equal to the the height of the Sun above the horizon, plus the change of direction gives above.)
For iin =57.5 degrees, the arc is located at the zenith. For smaller angles (higher Sun), there is no solution: instead the light is reflected at the second surface, travels back into the crystal and exits at the bottom side. This occurs when the altitude of the Sun is above 32.5 degrees. The circumzenithal arc can thus only ever occur when the Sun is lower than this.
The dotted lines in Fig. 2 show the range of angles over which the circumzenithal arc is at its brightest: this is when the arc is at a height of 60 to 75 degrees, and the Sun at 15 to 25 degrees above the horizon.

Colour

The index of refraction varies slightly with wavelength. For ice, it ranges from 1.315 in the blue (450 nm) to 1.308 in the red (650 nm) (see the on-line spectral calculator). Because of the difference, the red arc is slightly lower in the sky than the blue one, and the colour separation makes the arc is brightly coloured.
Figure 2, below, shows the effect. On the left, it shows the height above the horizon where the arc can be found, depending on altitude of the Sun. The location for the different colours are shown by the red and blue line.

Fig. 2: Height on the sky, width of the arc and its brightness, against the altitude of the Sun.


The panel at the upper right of Fig. 2 shows the width of the arc. The red and blue light are separated by typical 1.5 degrees, but the arc gets much wider close to the zenith.

Brightness

The brightness depends on the angle of incidence and on the width of the arc. For very low Sun, the horizontal face of the ice plates intercept very little light: the effective (projected) surface area declines as cos i. At the other extreme, for a 'high' Sun (30 degrees), when the arc is very close to the zenith, is becomes very wide. This divides the light over a larger area and therefore it also becomes fainter. The circumzenithal arc is brightest at intermediate heights.
The panel at the bottom left of the shows how the brightness changes, on a linear scale.

Plate alignment

We have sofar assumed that all ice crystals are identical and pefectly horizontal. The fact that two sides of the plates are perpendicular to each other is a general characteristic of water ice, and this assumption should hold well. Whether all plates are horizontal is a different matter. The plates tend to float (or fall) like leaves. Air pressure keeps them close to horizontal, but not perfectly so.

Fig. 3: Change in location (angle difference between the incoming and outgoing rays) against the horizontal tilt of the refracting crystal


Figure 3 shows the effect of a horizontal misalignment. The horizontal axis shows the angle of the incoming light: this will vary as the plates tilt. The vertical axis shows the change of angle in the plate: the arc will form at this angle above the Sun.
For iin around 65-70 degrees, the change in angle varies very little with tilt. A change in title of 10 degrees will cause not more than 1 degree change in position, and thus broaden the arc by this amount. As the Sun is itself 0.5 degrees across, we can assume that a widening by the tilt of less than this will not have much effect.
We therefore expect that a bright circumzenithal arc will form most easily when the Sun is between 15 and 25 degrees above the horizon, and the tilt of the ice plates should be no more than 5-10 degrees.
A change in angle of 10 degrees puts the arc at markedly different positions if the Sun is near the upper or lower position for a circumzenital arc to occur. Here, one may expect significant broadening, which will make the arc much fainter, and wipe out its colours.

Length of the arc

The light travels through the crystal in the forward direction only, for the circumzenithal arc to occur. It follows that the arc can only be seen between the Sun and the zenith, and not on the opposite side. It can therefore not extend more than 180 degrees. In practice, it is rarely more than 90 degrees, or a quarter circle.
When the vertical side of the ice crystal are not perpendicular to the direction of the light, the light will be refracted not only downward (always by the same amount, which is why the arc is everywhere at the same distance from the zenith) but also horizontally. This causes the extend of the arc and turns the spot into an arc.

Where and when to look

The circumzenithal arc is seen through cirrus clouds. In the image above, it occurs not in the contrail, but in the cold cirrus cloud above it. In the UK, a northwesterly airflow is best, bringing in cold upper air and showers. The Sun must be fairly low in the sky, below 32 degrees: it is brightest when the Sun is between 15 and 25 degrees. In the UK, in mid-summer this is the case in early morning or in the evening. Best are early spring/late winter or early autumn. In mid-winter, the Sun is too low in the sky. The ice crystals which cause the circumzenithal arc have the same shape as those which cause sundogs. If the Sun is at the right height above the horizon, and sundogs are visible, once the cirrus which cause the sundogs reaches 45-50 degrees above th horizon, there is a good chance to see the circumzenithal arc. The most important thing to remember is to look up. The arc is not particularly rare, but is often missed because few people look at the zenith! Note that it is difficult to estimate angles while looking up: the arc will appear to be much closer to the zenith than it actually is.

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