Scattered and Rescattered Sunlight

A. Hammad

doi:10.1364/JOSA.39.000157

Journal of the Optical Society of America
Vol. 39,
Issue 2,
pp. 157-161
(1949)
•https://doi.org/10.1364/JOSA.39.000157

Scattered and Rescattered Sunlight

A. Hammad

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A. Hammad, "Scattered and Rescattered Sunlight," J. Opt. Soc. Am. 39, 157-161 (1949)

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Abstract

In this paper the functions involved in the emission and reception of secondary scattered (rescattered) sunlight are discussed. Index-sum approximations of the form

\sum_{r = 1}^{r = 3} B_{n r} exp (- β_{n r} x)

are obtained for the integrals Ej_n(x), defined by

E j_{n} (x) = \int_{1}^{\infty} exp (- x t) t^{- n} d t .

The factors β_nr are determined numerically by Prony’s method of interpolation by exponentials, and the coefficients B_nr by the method of least squares and Seidel’s method of iteration. The functions Ej_n(x) occur as factors in the integrands of the integrals

L_{n} (c, z, \bar{m}) = \int_{0}^{1} exp (- Z {\bar{m}}^{'}) E j_{n} (c \bar{m} ~ c {\bar{m}}^{'}) d {\bar{m}}^{'}

involved in the emission of secondary scattered light. By means of the index-sum approximations for the integrals Ej_n(x), approximate formulas for the functions L_n are obtained. Numerical tables are given for Ej_n and L_n, showing the high degree of accuracy in our approximations. Likewise, the functions L_n occur as factors in the integrands of the integrals Lu_n and Ld_n involved in the reception of secondary scattered light. These integrals are of the type

\int_{0}^{X} exp (b x) L_{n} (c, z, x) d x .

Hence, approximate formulas for such integrals can be obtained in which the degree of accuracy is higher than that obtained for the functions L_n.

According to the approximate formulas given in this paper, an estimate can be made for secondary scattered sunlight in which the error is less than one percent of the exact value.

The formulas apply to all solar altitudes and to a wide range of wave-lengths in the solar spectrum, in contrast with previously obtained approximate formulas which apply to weakly attenuated radiations and high solar altitudes.

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Equations (22)

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n=3			n=5
r=1	r=2	r=3	r=1	r=2	r=3
B_nr=0.359687	0.124907	0.015417	0.184253	0.062977	0.002769
β_nr=1.251289	2.914563	9.859218	1.118589	1.818283	4.551294
exp(−0.2β_nr)=0.778600	0.558270	0.139200	0.799540	0.695130	0.402420

x	=0	0.2	0.4	0.6	0.8	1.0
Ej₃(x)	Exact=0.50000	0.35195	0.25729	0.19155	0.14432	0.10969
Ej₃(x)	Approx.=0.50001	0.35193	0.25728	0.19155	0.14432	0.10970
Ej₅(x)	Exact=0.25000	0.19221	0.14867	0.11551	0.09007	0.07045
Ej₅(x)	Approx.=0.25000	0.19221	0.14867	0.11551	0.09007	0.07045

x	=0.01	0.2	0.4	0.6	0.8	1.0
Ej₁(x)	Exact=4.03793	1.22265	0.70238	0.45438	0.31060	0.21938
Ej₁(x)	Approx.=4.03793	1.22250	0.70269	0.45642	0.31297	0.21938

Values of L₁
	c=0.1			c=1
sec z	$\bar{m}$ =0	$\bar{m}$ =0.4	$\bar{m}$ =1	$\bar{m}$ =0	$\bar{m}$ =1
1	2.66801	3.24146	2.62655	0.66219	0.43278
1	2.66349	3.26745	2.61746	0.66134	0.43173
6	2.19164	2.58776	1.99599	0.32349	0.05275
6	2.19873	2.61145	1.98154	0.32425	0.05250
Values of L₃
	c=0.1			c=1
sec z	$\bar{m}$ =0	$\bar{m}$ =1		$\bar{m}$ =0	$\bar{m}$ =1

1	0.43484	0.43351		0.17564	0.13810
1	0.43459	0.43327		0.17562	0.13810
6	0.34621	0.33998		0.06455	0.02359
6	0.34601	0.33978		0.06453	0.02359
Values of L₅
	c=0.1			c=1
sec z	$\bar{m}$ =0	$\bar{m}$ =1		$\bar{m}$ =0	$\bar{m}$ =1

1	0.22305	0.22256		0.09803	0.08002
1	0.22305	0.22257		0.09803	0.08002
6	0.17722	0.17491		0.03418	0.01466
6	0.17722	0.17490		0.03418	0.01466

n=3			n=5
r=1	r=2	r=3	r=1	r=2	r=3
B_nr=0.359687	0.124907	0.015417	0.184253	0.062977	0.002769
β_nr=1.251289	2.914563	9.859218	1.118589	1.818283	4.551294
exp(−0.2β_nr)=0.778600	0.558270	0.139200	0.799540	0.695130	0.402420

Abstract

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Tables (4)

Equations (22)

Journal of the Optical Society of America