GENERAL METEOROLOGICAL OPTICS 75 
the segments and antitwilight arch are schematically 
demonstrated with the aid of Fig. 13 as follows: Ro to 
R; are sun’s rays passing through the atmosphere whose 
optically effective height may be £;D3. The lowest ray 
Ry touches the earth’s surface at Oo. Owing to scatiter- 
ing and absorption on their way through the air, any 
rays between Fo and & lose part of their short-wave 
components and their intensity is depleted, so that Ko 
may not reach much beyond Oo, the next higher ray a 
little farther, and so on. The envelope of the end points 
of all these rays is represented by the curve OoD,D2D3. 
Consequently, the atmosphere to the right of this curve 
lies in the shadow of the earth. An observer 0O;, for 
whom the sun’s depression is 61, sights the vertex of the 
dark segment at D,, the poimt of tangency of his line 
of sight 01S; to the ray envelope. If he raises his line of 
sight slightly, he perceives light scattered from the still 
illuminated portion of the atmosphere near the enve- 
lope. This prevalently reddish light constitutes the anti- 
twilight arch. Its upper boundary is seen when the line 
of sight at O, is further raised so that the diminishing 
light from the zone of red rays is compensated by the 
increasing light of shorter wave lengths from the higher 
Fig. 13.—Schematic diagram for the dark and bright segments. 
layers of the atmosphere. When the sun sinks further, 
the relative position of the observer shifts to Oo, where 
the line of sight OS. touches the envelope at D». For 
an observer at H3, the passage of the dark segment 
through the zenith is imperceptible, because there is 
not enough contrast between the sky brightness to the 
right and left of the line of sight #3;D;. Finally, the ob- 
server at O3 sees the bright segment at an elevation e. 
The geometric aspects of this problem have been 
studied by various authors [c. 14, 40]. The results of 
computations of the heights of points D at various 
sun’s depressions, although based on different assump- 
tions, agree fairly well as shown by the examples in 
Table IX. According to these heights, pomt D moves 
into the stratosphere at a sun’s depression between 2° 
and 3°, in agreement with the deductions made from 
the variability of the dark segment. However, the prob- 
lem is essentially a photometric one [42]. An attempt at 
a physical solution was made by Dember and Uibe 
[10], who took into consideration the visibility as pro- 
portional to the square root of the measured sky bright- 
ness. Application of this theory by Mendelssohn [33] 
to photometric measurement yielded a constant height 
of the dark segment between 2 and 4 km. However, the 
transit of the dark segment would then have to occur 
not later than at about 3.5° sun’s depression [40], which 
is contrary to observation. Nevertheless, the basic idea 
of including the attenuation of light along the line of 
sight is correct. This was suggested by Exner [42] who, 
however, based his formula on Rayleigh scattering alone 
and disregarded secondary scattering which undoubt- 
edly plays a role in the brightness of the sky below the 
ray envelope [18]. 
TasLE IX. Hetcut or Dark SEGMENT AT VARIOUS 
Sun’s DEPRESSIONS 
6(°) Mohn [c. 14] (km) Neuberger [40] (km) 
1 a 2 
2 11 8 
3 21 17 
4 3l 27 
5 40 38 
The major problematic factors pertaining to the seg- 
ments and antitwilight arch are as follows: Two in- 
fluences on the ray envelope must be considered, that 
of atmospheric refraction which causes a vertical di- 
vergence of the sun’s rays (shown as parallels in Fig. 
13), and that of atmospheric attenuation of the rays 
near the ground. This attenuation tends to counteract 
the effect of refraction by eliminating the lowest, most 
refracted rays [27]. The position and shape of the ray 
envelope is, thus, primarily a function of the trans- 
parency of the atmosphere at and above the point of 
tangency with the earth’s surface. As regards the in- 
tensity and color of the scattered light, most computa- 
tions have been based on an idealized atmosphere in 
which Rayleigh’s theory with its symmetric scattering 
function is valid [14, 15, 18]. However, the real atmos- 
phere contains a large number of particles, especially 
in the lower layers. For this reason, the agreement 
between theory and observations is not satisfactory, 
and, in particular, the variations from day to day ob- 
served in dark and bright segments and antitwilight 
arch remain unexplained. According to Linke [29], the 
rigorous theory by Mie-Debye is more suitable for 
the theoretical approach to the twilight colors. How- 
ever, this theory is difficult to apply to the problems 
at hand, because it still involves the assumption of 
spherical particles. In view of the new theories of the 
anticorona [6, 20], the consideration of rearward dif- 
fraction should be extended to the theory of the anti- 
twilight arch. 
The theory of the purple light which was recognized 
as a diffraction phenomenon by Kiessling [c. 42] was 
established by Gruner [14] along the lines suggested by 
Pernter [42] and others. This theory adequately de- 
scribes the temporal and spatial development of the 
purple light; the basic ideas may briefly be outlined 
with the aid of schematic Fig. 14. The sun’s rays Ro to 
Rs; pass through a dust layer DD’ in which they are 
deprived of their short-wave components. While Fo, 
passing through the dense lower layers, and A, hay- 
ing the longest path through the dust layer, may be 
completely extinguished, the rays around A will emerge 
as a reddish beam of light and enter the lower boundary 
of the dust layer again at P:, where the particles will 
