74. METEOROLOGICAL OPTICS 
the height of the layer responsible for the purple light. 
By graphical approximation he arrived at values be- 
tween 25 and 31 km for the upper boundary of the 
layer, and 18 km for the thickness of the generating ray 
beam, that is, a thickness of the layer of roughly the 
same magnitude. Smosarski [c. 14, 50] estimated the 
lower boundary at 8 km and the upper boundary at 
17 km. The relationship between purple lights and 
high dust layers of volcanic origin seems to confirm at 
least the order of magnitude of these values. The pos- 
sibility of a connection with the ozone layer is an un- 
explored question. Incidentally, Gruner [14], assuming 
the purple light to be a corona, also determined the 
order of magnitude of the particle diameter as between 
1 and 1.5 yp. For these small particles, however, the 
classical diffraction theory fails [45] as was shown in the 
preceding section. 
ANGULAR ELEVATION 
DARK 
SEGMENT 
PIZ 
aerate 
234 56 7 8 9 10 II 12 13 14 15 
SUN'S DEPRESSION (°) 
Fig. 12—Average course of dark and bright segments at two 
Swiss stations. 
Several series of geometric measurements of the ele- 
vation of the upper boundary and the width of the anti- 
twilight arch have been made. The course of its angular 
elevation is similar to that of the dark segment (Fig. 
12). Mendelssohn and Dember [33] made a few spectral 
measurements by photographic photometry, but their re- 
sults confuse, rather than elucidate, the visual observa- 
tion. The annual and secular variations of antitwilight- 
arch occurrence were determined by Smosarski [50] who 
found, in Poland, a maximum frequency in autumn and 
winter, a minimum in summer, and a good correlation 
with volcanic activities. A direct connection with the 
occurrence of purple lights does not seem to exist. Ac- 
cording to computations by Smosarski [c. 14], as the 
sun’s depression increases, the antitwilight arch is pro- 
duced by the rearward scattering of sunlight by smaller 
particles at higher altitudes. However, according to 
Mendelssohn and Dember [83], the antitwilight arch 
is supposedly fixed in a layer between 4 and 9 km. This 
problem will be discussed below. 
Of the many observations on the movement of the 
dark and bright segments, only the averages of two 
series obtained at Piz Languard (8280 m) and Steck- 
born (430 m) in Switzerland, as reported by Gruner 
[14], are shown as examples in Fig. 12. As the sun sinks 
below the horizon, the dark segment rises more rapidly 
than does the antisolar point; the ascent rate increases 
with the sun’s depression, until the segment fades from 
view between 5° and 6° depression. Between 7° and 8° 
depression the bright segment appears at an elevation of 
roughly 25° above the solar point and descends, first 
rapidly then more slowly, to the western horizon. Ac- 
cording to the interpolated (dashed) portions of the 
curves, the invisible transit through the zenith occurs 
at a sun’s depression between 6° and 7°. This agrees 
well with observations of the zenith brightness by 
Brunner [c. 14] and Hulburt [18], and of global illumi- 
nation by Siedentopf and Holl [48]; these authors pre- 
sent curves of brightness and illumination, respectively, 
versus sun’s depression, that show a definite inflection 
point between 6° and 7° sun’s depressions. This fact is 
involved in the problem of the height at which this 
phenomenon occurs. In this connection the change in 
relative variability of the dark segment’s elevation at 
various sun’s depressions deserves attention. It has 
been shown [26, 40] that the variability decreases rap- 
idly until the sun’s depression is about 2.5°, then more 
slowly, although the opposite trend was to be expected 
in view of the decreased accuracy of measurement at 
greater sun’s depressions. This fact was interpreted as 
being caused by a transition of the dark segment at 
about 2.5° sun’s depression into the stratosphere, where 
marked changes in turbidity from day to day are less 
frequent [40]. 
The sun’s depressions at which the last traces of the 
bright segment disappear below the horizon have been 
variously used to compute the height at which the 
density of the atmosphere is sufficiently great to produce 
visible scattering of the direct sunlight. However, the 
resulting values of 50-65 km [14] are still quite prob- 
lematic because of subjective factors involved in the 
perception of faint light and of the effects of attenua- 
tion of the direct light rays at various altitudes. Ex- 
cept for a slight increase in elevation of the dark seg- 
ment in summer and with diminished transparency of 
the lower layers of the atmosphere, no clear-cut rela- 
tionships between the turbidity of the air and the 
bright segment, nor a definite seasonal variation of 
either segment have been established [14, 16, 18, 26, 
40]. 
Theories and Problems. Although many theoretical 
approaches to the problems of twilight phenomena 
have been made, no complete theory exists as yet. The 
theories of the dark and bright segments and of the 
antitwilight arch [15, 16] agree qualitatively with, but 
differ quantitatively from, the observations. For exam- 
ple, the elevations of the dark segment, computed from 
the spectral intensities of light scattered by a Rayleigh 
atmosphere (disregarding multiple scattering), were 
considerably smaller than the observed ones [14, 15]. 
The principal ideas underlying the explanations of 
