64 METEOROLOGICAL OPTICS 
unknown as yet. For example, the practical problem 
of the mutual influence of sky shape and cloudiness es- 
timation needs further exploration. Simultaneous photo- 
graphic records of the sky, estimation of cloudiness, 
and determination of the shape of the sky (under all 
possible states of sky) may furnish an individual cor- 
rection factor for adjustment of cloudiness estimates. 
In particular, the quality of cloudiness estimations for 
individual cloud layers in the presence of other cloud 
decks needs special attention, as does the effect of the 
configuration of the terrestrial horizon on such estima- 
tions. More observations are also needed on the follow- 
ing effects: possible seasonal variations; the terrestrial 
horizon distance in conjunction with the configuration 
of the visual field between horizon and observer; the 
measured brightness level and brightness distribution 
over the sky. 
Wholly unexplored is the possible effect of an obsery- 
er’s altitude above the earth’s surface on his impression 
of the sky shape and consequently the accuracy of his 
cloudiness estimation; this phase is especially of m- 
terest for meteorological observers on mountains or in 
airplanes. In this connection, the apparent shapes of 
the terrestrial:surface and of cloud layers, as seen from 
above, deserve attention, because the quality of es- 
timations of cloudiness below an aerial observer hinges 
on this problem. 
REFRACTION PHENOMENA IN THE 
CLOUD-FREE ATMOSPHERE 
In the subsequent discussions reference is made only 
to the visible portion of the electromagnetic spectrum, 
although analogous phenomena occur with other wave 
lengths. The refractive index mn) of air for various wave 
lengths and its dependence on the air density is well 
known from laboratory determinations [81, 42]. An 
example of the magnitude of the dispersion and tem- 
perature effect on the refractive index in air at normal 
pressure (760 mm Hg) is given in Table V. Obviously, 
TaBLE V. VARIATION OF (7)-1)10° with Wave LENeTH 
AND TEMPERATURE 
(After Meggers and Peters [31]) 
(A) 0c 15C 30C 
3983 297 282 268 
5069 293, 278 263 
7664 290 275 261 
the effect of dispersion is small in comparison with that 
of temperature. The refractive indices of the various 
gaseous constituents of air differ even more significantly 
from each other, but sce the composition of air varies 
only slightly, this effect is negligible for all practical 
purposes. 
Phenomena due to Average Density Gradient. When 
penetrating the atmosphere, which is assumed to con- 
sist of concentric, equidistant isopyenic surfaces, an 
extraterrestrial light ray describes a curve whose well- 
known equation is 
mr sini = k = const, (1) 
where r is the distance of an isopyenic surface from the 
earth’s center, 7 the angle of incidence, and the refrac- 
tive index n (for white light) is a function of r, n(r). In 
Fig. 4, 7, = CO is the earth’s radius; an observer at O 
sees a light ray L, that enters the atmosphere at P 
with the angle of imcidence 7, at the apparent zenith 
distance 6, instead of the true zenith distance 6. The 
difference 0 — 0, = R, the astronomical refraction. In 
its general form, the equation of the ray curve PO in 
polar coordinates is 
a ee Ne 
Oi i = = ip ? (2) 
or expressed in terms of astronomical refraction: 
. : 
No? SIN O, 
R= || ee dn (3) 
Dy nvV/ nr = (iris Sur i, 
The integrals, to be taken between the upper limit of the 
atmosphere (where n = nz) and the earth’s surface 
. 
ZENITH v4 
/ iL 
/ 
if 
/ 
/ Le 
DOK 
</ 
Lox KV) 
(© 
Fie. 4—Schematie diagram of astronomical refraction. 
(index 0), have no general solutions, because the change 
of n with altitude must first be known. 
Various authors made different assumptions regard- 
ing the function n(r) and computed the astronomical re- 
fraction for various apparent zenith distances. Wunsch- 
mann [59], who compared several of the results, prefers 
Gylden’s refraction data, because they are based on 
a hypothesis which agrees closely with data obtained 
from aerological ascents. Recently Link and Sekera 
[28] computed tables of various characteristics of the 
ray path for zenith distances from 75° to 90° and alti- 
tudes up to 60 km. They considered the vertical density 
distribution separately for summer and winter, using 
average density conditions as revealed by balloon 
soundings, sound propagation, and twilight phenomena. 
Correction tables for the effect of deviations from nor- 
mal in local conditions of pressure, temperature, and 
humidity have also been variously computed [31, 42]. 
These corrections are important chiefly for apparent 
zenith distances up to 70°, for which the values of & 
