The Influence of the Size of the Sun on the Sky Light Distribution

Reiner Eiden Meteorologisch-Geophysikalisches Institut, Johannes Guten­berg Universität, Mainz, Germany. Received 1 March 1968.

The equations describing the sky light distribution have been derived on the assumption that the sun is a point source of radiation.1,2 However, the observed sky radiation is the total of the scattered radiation emerging from various points of the surface of the actual sun. Therefore, it seems necessary to in­vestigate whether it is reasonable to compare observed with theoretical values, or to estimate the error of the theoretical values due to the concentration of the solar energy in the center of the sun. A first at tempt in this direction was made by Kos-bahn,3 who got a rough estimation using an observed particular sky light distribution.

For solving this problem in a more general way, the following suppositions are made.

(1) Each surface element dFs of the solar disk has the same radiance. The limb darkening of the sun is neglected.

(2) Within a scattering angle interval φ0 - δs ≤ φ ≤ φ0 + δs, the nonlinear angular distribution of the radiance of sky light. BΛP(φ) originated by a solar point source, is approximated by a straight line (Fig. 1).

The slope a and the intersection φ0 of the straight line with the abscissa are constant within the scattering angle interval under consideration. The angle φ0 is any fixed scattering angle or angular distance from the sun. δs = 4.36 × 1 0 - 3 rad is the average angular radius of the sun. The index p denotes the sun as a point source; λ is the wavelength.

1648 APPLIED OPTICS / Vol. 7, No. 8 / August 1968

Fig. 1. Schematic graph to illustrate the procedure of the linearization of the sky light distribution function BλP(φ0); p denotes the solar point source; φ0 is the intersection of the linearized sky light function with the abscissa outside the interval

of definition.

Fig. 2. Relative error R(φ0) due to an idealized solar radiation source vs scattering angle φ0. Parameter is the upper limiting

radius r2 of the atmospheric aerosol size distribution.

Under these assumptions, the relative error can be calculated by

To outline the magnitude of the relative error, two extreme cases may be considered: (1) a pure Rayleigh atmosphere and (2) a highly disturbed atmosphere. These considerations are restricted to a normal turbid atmosphere and do not include, for example, phenomena like the corona.

(1) The sky radiance of a pure Rayleigh atmosphere has been calculated by Coulson et al., including all scattering processes.2

Using these theoretical values, the relative error caused by sim­plifying the sun as a point source turns out to be R(φ0) < 3 × 10 - 3 %, as close as possible to the sun. A decrease of R(φ) with increasing φ0 can be confirmed, but it is very small.

(2) To analyze the influence of the turbidity and the aerosol size distribution of a turbid atmosphere, theoretical values of the sky radiance given by de Bary et αl.4 are used. The analysis of the theoretical values mentioned reveals that, in the wavelength range 0.45 μ ≤ λ ≤ 1.0 μ, R(φ0) is largest for small wavelength, i.e., λ = 0.45 μ. Yet the differences are very small. The relative air mass of the sun does not affect R(φ0) at all.

I t can be shown that, if the turbidity exceeds T 5 [T = 1 + τD/τR, ΤD, τR = optical thickness of aerosol (D), and Ray­leigh (R) atmosphere], the variations of the relative error R(φ0) by changing turbidity are negligible. In this case, it only de­pends on relative aerosol size distribution and the upper limiting

Fig. 3. Relative error R(φ0) due to an idealized solar radiation source vs scattering angle φ0. Parameter is the exponent ν*

of the exponential atmospheric aerosol size distribution.

radius r2. A variation of the lower limiting radius r1 within the interval 0.04 μ ≤ r1 ≤ 0.08 μ does not noticeably affect R(φ0).

The influence of the variation of the upper limiting radius r2 ≥ 3 μ is demonstrated in Fig. 2, where R(φ0) is plotted vs scattering angle φ0. Near the sun, R(φ0) increases with in­creasing upper limiting radius of the aerosol size distribution. This is identical with the fact that, with increasing upper limiting radius r2, the slope of the sky radiance becomes more and more steep toward the sun. This is the case only until r2 30. Beyond this value and φ0 ≥ 1° the variation of the sky radiance is negligible.5

The same effect can be observed in Fig. 3, where the param­eter ν* of the curves is the exponent of the exponential aerosol size distribution. (dN ~ r-v*+1dr, dN = particles per cm3

in the radius interval dr.) A small exponent ν* means a stronger weighting of the large particles within the aerosol size distribu­tion than in the case of a large ν*. This results in a steeper slope of the sky radiance towards the sun and a higher relative error R(φ0).

To summarize the results, it can be said that the relative error increases with increasing course of the sky radiance to­wards the sun and with increasing radius and number of big aerosol particles in the atmosphere. In any case, the relative error R(φ0) increases with decreasing scattering angle φ, i.e., R(φ0) is greatest close to the sun. But assuming realistic at­mospheric conditions the relative error does not exceed the value of R(φ0) 1% (φ0 ≥ 1°) even in the adverse case of an atmo­spheric aerosol size distribution with an exponent of ν* = 2.5 and an upper limiting radius r2 ≥ 30 μ. If the limb darkening of the sun would be taken into account, the result would be even more favorable. Hence, the theoretical values based on a solar point source of radiation may be used within the measuring accuracy to analyze measured sky light distributions.

The author wishes to thank Kurt Bullrich for his helpful sug­gestions and discussions.

References 1. K. Bullrich, Adv. Geophys. 10, 99 (1964). 2. K. L. Coulson, I. V. Dave, and Z. Sekera, Tables Related to

Radiation Emerging from a Planetary Atmosphere with Ray-leigh Scattering (University of California Press, Berkeley, California, 1960).

3. R. Kosbahn, master's thesis Meteorological Institute, University of Frankfurt/Main (1948).

4. E. de Bary, B. Braun, and K. Bullrich, Tables Related to Light Scattering in a Turbid Atmosphere, Vols. I - I I I , AF Cambridge Research Lab., Special Rep. No. 33, 1965.

5. R. Eiden, Tellus, in publication.

August 1968 / Vol. 7, No. 8 / APPLIED OPTICS 1649