## Ptfi874

déi which is more illuminating when written as

Note that Eq. (8.75) does not give the radiant energy in the twice-refracted radiation. To obtain this would require including the Fresnel transmission coefficients for the two interfaces. Figure 8.27: Deviation (i.e., scattering) of incident light by an angle \$ because of refraction by a 60° prism that is part of a hexagonal plate.

Figure 8.28 shows deviation angle as a function of incidence angle for a 60° ice prism that is part of a hexagonal plate. For angles of incidence less than about 13° the transmitted ray is totally internally reflected at the second interface; for angles of incidence greater than about 70° reflection rises sharply (e.g., see Fig. 7.6) and hence transmission plunges. Thus most incident rays of consequence lie in this range. According to Eq. (8.75) the probability density P(\$) becomes infinite if the deviation angle has a minimum, which it does for « 40°; the corresponding deviation angle is about 22°. The observable manifestation of this singularity, or caustic, at the angle of minimum deviation for a 60° ice prism is a bright spot about 22° from either or both sides of a sun low in the sky. These bright spots are called sun dogs, because they accompany the sun, or parhelia or mock suns.

The minimum deviation angle \$m, and hence the angular position of sun dogs, depends on the prism angle A and refractive index n (see Prob. 8.20):

Because n varies with wavelength, the separation between the angles of minimum deviation for red (650 nm) and violet (430 nm) light is about 0.7° (Fig. 8.28), slightly greater than the angular width of the sun. As a consequence, sun dogs may be tinged with color, most noticeably toward the sun. Because the refractive index of ice is least at the red end of the spectrum, the red components of a sun dog are closest to the sun. Moreover, for any wavelength, except that corresponding to red, a horizontal line tangent at the minimum angle to the curve of deviation angle versus incident angle intersects curves for other wavelengths. Because of this overlap of deviation angles, red is the purest color in sun dogs, which fade into white away from their red inner edges.

ra 45

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Angle of Incidence (Degrees)

Figure 8.28: Deviation (i.e., scattering) angle versus angle of incidence for a 60° ice prism and a 90° ice prism. The solid line is for a wavelength of 650 nm, the dashed line for 430 nm.

With increasing solar elevation, sun dogs move away from the sun. A falling ice plate is approximately equivalent to a prism the angle of which increases with solar elevation. From Eq. (8.76) it follows that the angle of minimum deviation, hence the position of the sun dog, also increases. Why is only the 60° prism portion of a hexagonal plate singled out for attention? According to Fig. 8.27, a hexagonal plate could be considered to be made up of 120° prisms. For a ray to be refracted twice, its angle of incidence at the second interface must be less than the critical angle [see Eq. (4.63)], which imposes limitations on the prism angle. For n « 1.31 (ice at visible wavelengths), all incident rays are totally internally reflected by prisms with angles greater than about 99.5°.

A close relative of the sun dog is the 22° halo, a bright ring approximately 22° from the sun. Lunar halos and moon dogs are also possible, but because of their low brightness may not be noticed as frequently as their solar counterparts. Until Alistair Fraser analyzed halos the conventional wisdom had been that they obviously were the result of randomly oriented crystals, yet another example of jumping to conclusions. By combining optics and aerodynamics, he showed that ice crystals small enough to be randomly oriented by Brownian motion are too small to yield sharp scattering patterns.

But partially oriented larger plates can produce halos, especially ones of non-uniform brightness. Each part of a halo is contributed to by a plate with a different tip angle, angle between the normal to the plate and the vertical. The transition from oriented plates with zero tip angle to randomly oriented plates occurs over a narrow range of sizes. In the transition region, plates can be small enough to be partially oriented yet large enough to give a distinct contribution to the halo. Moreover, the mapping between tip angles and azimuthal angles on the halo depends on solar elevation. When the sun is near the horizon, plates can give a distinct halo over much of its azimuth. When the sun is high in the sky, hexagonal plates cannot give a sharp halo but hexagonal columns - another possible form of atmospheric ice particles -can. The stable position of a falling column is with its long axis horizontal. When the sun is Angle of Incidence (Degrees)

Figure 8.28: Deviation (i.e., scattering) angle versus angle of incidence for a 60° ice prism and a 90° ice prism. The solid line is for a wavelength of 650 nm, the dashed line for 430 nm.

directly overhead, such columns can give a uniform halo even if they all lie in the horizontal plane. When the sun is not overhead but well above the horizon, columns also can give halos.

A corollary of Fraser's analysis is that halos are caused by crystals with sizes in the range 12-40 |m. Larger crystals are oriented, smaller crystals too small to yield distinct scattering patterns. More or less uniformly bright halos with the sun neither high nor low in the sky could be caused by mixtures of hexagonal plates and columns or by clusters of bullets (rosettes). Fraser opines that the latter is more likely.

One of the by-products of his analysis is an understanding of the relative rarity of the 46° halo. Rays can be transmitted through two sides of a hexagonal column (A = 60°) or through one side and an end (A = 90°). For n = 1.31 and A = 90° Eq. (8.76) yields a minimum deviation angle of 46° (see Fig. 8.28). Although 46° halos are possible, they are seen much less frequently than 22° halos. Plates cannot give distinct 46° halos although columns can. But they must be solid and most columns have hollow ends. Moreover, the range of sun elevations is restricted.

Like the green flash, ice-crystal phenomena are not intrinsically rare. Halos and sun dogs can be seen frequently once you know what to look for, where, and when. Hans Neuberger reports that halos were observed in State College, Pennsylvania an average of 74 days a year over a 16-year period, with extremes of 29 and 152 halos a year. Although the 22° halo was by far the most frequently seen display, ice-crystal displays of all kinds were seen, on average, more often than once every four days at a location not especially blessed with clear skies. Although thin clouds are necessary for ice-crystal displays, clouds thick enough to obscure the sun are their bane.