SOLAR RADIANT ENERGY 25 
lated to m’NK,, or K, times their geometrical cross 
section, so that 
Ih = In exp(—2rr’NK,), (23) 
where r is the radius of the drops and JN is their number. 
Figure 7 shows A, as a function of the parameter X 
= 2nr/. Equation (23) holds for nonabsorbing par- 
ticles and is valid at wave lengths where water ab- 
sorption is negligibly small; the summation is required 
over drop radii when the drop sizes are not uniform. 
For particles where n, ~ 1.33, A, will differ from the 
values of Fig. 7; van de Hulst [76] shows some varia- 
tions. From Fig. 7 it is important to note that for some 
ratios of particle size to wave length the scattering does 
not always increase with decreasing wave length. 
4.0 
3.0 
(0) 10 20 30 40 50 
X=2mr/d 
Fie. 7.—Seattering-area coefficient K, for liquid-water 
drops in nonabsorbed spectral regions as a function of \ and 
of drop radius 7. (After Houghton and Chalker [46].) 
Multiple Scattering. In the case of pure Rayleigh scat- 
tering (if it ever can be said to exist in the atmosphere) 
the scattering is symmetrical about the particle so that, 
for example, the forward- and backward-scattered 
energy are equal. But as the particles become larger the 
scattering mcreases in the forward direction [384; 55, p. 
161]. Calculations of the effect of such particles, es- 
pecially for multiple scattering, become rather complex; 
a they have been undertaken by several authors 
76). 
The energy which is scattered from the original solar 
beam will in general be scattered more than once on its 
way down to the earth’s surface or else out to space. 
From an empirical viewpoint, regarding the contribu- 
tion of the cloudless sky radiation to the total radiation 
on a horizontal surface, Kimball [52] states that of the 
radiation scattered from the direct solar beam, half 
will be scattered down and half up. In a pure Rayleigh 
atmosphere this would be the case. But for the actual 
conditions of the atmosphere, it serves only as a useful 
approximation for average daily values of cloudless- 
_ sky radiation after the total scattering from the direct 
beam has been evaluated. 
For instantaneous values, Kimball found the ratio of 
the direct sunlight component on a horizontal surface 
I, to the total solar and sky radiation on a horizontal 
surface Q to be a function of the zenith distance of the 
sun (Table IV). 
Tasie IV. Vatues or /;/Q as A Function or Sun’s ZENITH 
Distance Z 
(After Kimball [51]) 
Z 30.0 /48.3 |60.0 |66.5 |70.7 (73.6 (75.7 (77.4 |78.7 |79.8 
(deg) | | 
T,/Q 0.84) 0.84 0.80| 0.78| 0.76] 0.72 ale 0.65) 0.63 
Similar values have also been found by other obsery- 
ers [55, p. 356]. If we designate the diffuse sky radiation 
by D, then D = Q — T,. As might be expected, the 
ratio D/@ increases where the atmospheric turbidity is 
high [55]. Measurements illustrating this point (e.g. 
illumination measurements) have often been made with 
instruments for relatively narrow spectral bands. It 
would be desirable to measure the ratio for nonspectral 
radiation with varying atmospheric transmissions or 
turbidity factors. 
Albedo. By the albedo of a body we mean the fraction 
of the incident energy which is reflected by the body. 
Thus the albedo A of the planet Harth is the fraction cf 
the energy incident at the “outer limits” of the at- 
mosphere which is returned to space; the albedo of a 
cloud ‘‘surface” is the fraction of the energy incident 
upon the cloud which is reflected by the cloud. 
The terrestrial entities which reflect solar energy are 
(1) the cloudless atmosphere, (2) clouds, and (3) the 
earth’s surface. The sum of these three reflections is the 
total energy reflected by the Earth. Expressed as a 
fraction of the extraterrestrial solar energy intercepted 
by the Earth, this sum determines A. 
The Albedo of the Cloudless Almosphere. 1. Pure Dry 
Air. We have seen that within a few per cent the frac- 
tional spectral depletion of solar energy by molecular 
scattering is represented by the Rayleigh scattering 
law, and the summation over wave length is readily 
given as a function of air mass from curve (1) of Fig. 6. 
The earth as seen from the sun can be considered as 
made up of narrow concentric circular rings centered 
about the subsolar point. To an observer in a particular 
ring the sun is at a specified zenith distance, and hence 
in each ring the optical air mass m is known. From m 
and the area of the rings, together with the above- 
mentioned relation between scattering and air mass, 
the amount of energy scattered by the entire atmo- 
sphere can readily be calculated and found to be about 
15 per cent of the incident energy [30]. This was calcu- 
lated using Fowle’s data; Fig. 6 gives a slightly smaller 
value. However, this energy is scattered in all direc- 
tions. To determine the fraction scattered upward 
(away from the earth’s surface) it is necessary to assume 
something about the angular scattering by each par- 
ticle. Scattering by small particles, such as molecules, 
is symmetrical about the particle [34]. Hence as much 
energy is scattered up as is scattered down. Even for 
