70 METEOROLOGICAL OPTICS 
The frequency of different halos, their diurnal and 
annual or seasonal variations, and their relation to 
weather situations, have been investigated [1, c. 34, 52]. 
The similarities or differences in the results can generally 
be ascribed to climatic characteristics. The relationship 
between halo occurrence and solar activity is, however, 
still quite problematic. Visser [52] found a decrease in 
halo frequency with increasing relative number of sun- 
spots® up to 90 and then an increase for higher sunspot 
numbers. Archenhold [1] arrived at a direct linear 
relationship, and a halo periodicity of 27-28 days, which 
he associates with the rotation of the sun. This periodic- 
rainbow, whose radius to the red inner border is about 
51°. Inside the primary and outside the secondary bow, 
supernumerary bows showing fewer and fainter colors 
are often visible. When the sun’s rays are reflected by a 
smooth water surface’ before striking the suspended 
droplets, primary and secondary reflection rainbows may 
appear whose center lies as much above the horizon as 
that of the regular bows lies below it. Thus, the reflec- 
tion bows intersect the respective regular ones at the 
horizon. When the reflecting water surface is undulated 
by a smooth swell, the reflection bow may deform into 
vertical shafts [57]. Droplets of radii < 30 » may pro- 
Taste VII. PrincipaL Genetic Features or Mayor Hato PHENOMENA 
: Orientation of 
Halo Refracting angle principal crystal Special characteristics (s = sun’s elevation) 
axis 
2? 60° Random Incident ray at 90° to principal axis. 
Parhelia 60° Vertical Intensity rapidly decreases for s > 50°. 
Circumscribed to 22°-halo 60° Horizontal | Upper and lower tangent ares join at s = 30°. 
At s = 55° elliptic with long horizontal axis. 
At s = 90° circular and coincides with 22°-halo. 
Parry-ares 60° Horizontal | Pair of crystal sides vertical for s < 30°, horizontal 
for 30° < s < 50°. 
46° 90° Random Incident ray at 90° to principal axis. 
Cireumzenithal are 90° Vertical Ray entrance at crystal base; limited to s S$ 32°; 
also possible with horizontal axis and pair of 
sides horizontal, but rare. 
Cricumhorizontal are 90° Vertical Ray entrance at crystal side; limited to s = 58°; 
also possible with horizontal axis and pair of 
sides horizontal. 
Lateral tangent ares of 22°-halo (Lowitz- 60° Oscillating | Oscillation < +30° about vertical in plane that is 
ares) parallel to sun’s meridional plane. Limited to 
252° <s < 50°. 
Infralateral tangent arcs of 46°-halo 90° Horizontal | Ray entrance at crystal base. Limited to 0° < s 
< 68°. 
Supralateral tangent ares of 46°-halo 90° Horizontal | Ray entrance at crystal side. Limited to 0° < s 
< 32°. 
ity was shown to be spurious [389]. Although a certain 
degree of correlation with solar activity seems to exist, 
there is hardly a direct relationship (e.g., corpuscular 
solar radiation furnishing sublimation nuclei), consider- 
ing the prerequisites that must be fulfilled for a halo to 
be observable [34, 39]. 
Rainbows. The colorful arcs around the antisolar 
point that appear on sheets of water droplets are called 
rainbows, although these phenomena may be produced 
by dew droplets on the ground or water sprays. The 
primary rainbow has an angular radius to the red outer 
border of roughly 42°; concentric to it is the secondary 
6. According to Schindler [e. 36], a relationship between 
sunspots and halos begins only at a spot number of 35 to 40. 
duce a broad white band with faintly tinted borders, 
the fogbow, between about 37° and 40° distance from 
the antisolar point. Rainbows or fogbows produced by 
drops in a horizontal plane appear in the form of conic 
sections, that is, hyperbolic, elliptic, or parabolic ares, 
depending on whether the sun’s elevation is respectively 
smaller than, larger than, or equal to, the aperture of the 
cone (42° or 51°). 
The well-known explanation by Descartes considered 
geometrical optics alone. Airy, basing his theory on 
wave optics, explained the variation of colors with 
droplet size and the supernumeraries as interference 
rings. His rainbow integral was also solved by others. 
The distribution of intensity, color, and polarization of 
