Abstract

The specific intensities of nonparallel light in the direction of a stellar source, such as the sun, resulting from multiple scattering and diffuse ground reflection of a unit flux of parallel radiation incident on a plane parallel, Rayleigh atmosphere, equivalent to the Earth’s atmosphere in composition and density, are evaluated following S. Chandrasekhar’s extension of the Rayleigh theory. Values are given as a function of direction of the source and normal optical thickness covering the visible and near ultra violet spectrum.

It is shown that the flux resulting from such nonparallel radiation, for a sufficiently small solid angle (10−3 radian) around the source, when the latter is within 50° or less of the vertical, is of the order of 10−5 of the reduced flux of the direct solar beam, for radiation within the visible range. The relative importance of the scattered light increases with normal optical thickness and zenith distance of the source.

It is pointed out that certain observations of apparent atmospheric transmissions of solar radiation in the blue end of the visible spectrum exceed in magnitude significantly the values obtained from the exact Rayleigh scattering theory, and it is suggested that this anomaly may be due to non-Rayleigh particles existing in the high atmosphere, with strong forward scattering characteristics.

© 1953 Optical Society of America

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References

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  1. J. W. Strutt, Phil. Mag. 41, 107, 274 (1871).
  2. Rayleigh, Phil. Mag. 47, 375 (1899).
  3. C. G. Abbot and et al., Annals of the Astrophysical Observatory of the Smithsonian Institutian., (U. S. Government Printing Office, Washington, D. C., 1902–1932). Vols. II to V, inclusive.
  4. S. Chandrasekhar, Radiative Transfer (Clarendon Press, Oxford, England, 1950).
  5. “Tables relating to Rayleigh scattering of light in the atmosphere,” , Contr. AF 19 (122)–239, Air Force Cambridge Research Center (1952).
  6. S. Chandrasekhar, Nature 167, 51 (1951).
    [CrossRef]
  7. D. Deirmendjian, , Contr. AF 19(122)–239, Air Force Cambridge Research Center (mimeographed). Also M. A. thesis, University of California, Los Angeles (1952).
  8. C. Dorno, Monthly Weather Rev. 53, 519 (1925).
    [CrossRef]
  9. H. S. Stewart and T. A. Curcio, J. Opt. Soc. Am. 42, 801 (1952).
    [CrossRef]

1952 (1)

1951 (1)

S. Chandrasekhar, Nature 167, 51 (1951).
[CrossRef]

1925 (1)

C. Dorno, Monthly Weather Rev. 53, 519 (1925).
[CrossRef]

1899 (1)

Rayleigh, Phil. Mag. 47, 375 (1899).

1871 (1)

J. W. Strutt, Phil. Mag. 41, 107, 274 (1871).

Abbot, C. G.

C. G. Abbot and et al., Annals of the Astrophysical Observatory of the Smithsonian Institutian., (U. S. Government Printing Office, Washington, D. C., 1902–1932). Vols. II to V, inclusive.

Chandrasekhar, S.

S. Chandrasekhar, Nature 167, 51 (1951).
[CrossRef]

S. Chandrasekhar, Radiative Transfer (Clarendon Press, Oxford, England, 1950).

Curcio, T. A.

Deirmendjian, D.

D. Deirmendjian, , Contr. AF 19(122)–239, Air Force Cambridge Research Center (mimeographed). Also M. A. thesis, University of California, Los Angeles (1952).

Dorno, C.

C. Dorno, Monthly Weather Rev. 53, 519 (1925).
[CrossRef]

Rayleigh,

Rayleigh, Phil. Mag. 47, 375 (1899).

Stewart, H. S.

Strutt, J. W.

J. W. Strutt, Phil. Mag. 41, 107, 274 (1871).

J. Opt. Soc. Am. (1)

Monthly Weather Rev. (1)

C. Dorno, Monthly Weather Rev. 53, 519 (1925).
[CrossRef]

Nature (1)

S. Chandrasekhar, Nature 167, 51 (1951).
[CrossRef]

Phil. Mag. (2)

J. W. Strutt, Phil. Mag. 41, 107, 274 (1871).

Rayleigh, Phil. Mag. 47, 375 (1899).

Other (4)

C. G. Abbot and et al., Annals of the Astrophysical Observatory of the Smithsonian Institutian., (U. S. Government Printing Office, Washington, D. C., 1902–1932). Vols. II to V, inclusive.

S. Chandrasekhar, Radiative Transfer (Clarendon Press, Oxford, England, 1950).

“Tables relating to Rayleigh scattering of light in the atmosphere,” , Contr. AF 19 (122)–239, Air Force Cambridge Research Center (1952).

D. Deirmendjian, , Contr. AF 19(122)–239, Air Force Cambridge Research Center (mimeographed). Also M. A. thesis, University of California, Los Angeles (1952).

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Figures (4)

Fig. 1
Fig. 1

Schematic representation of elementary scattering processes.

Fig. 2(a)
Fig. 2(a)

Variation of the specific intensity (including the factor π) of diffuse light in the direction of the source with the air mass (sec Θ0), for five normal optical thicknesses. Albedo=0.

Fig. 2(b)
Fig. 2(b)

Variation of the specific intensity (including the factor π) of reflected light in the direction of the source on the same basis as for Fig. 2(a), corresponding to Lambert reflection with albedo 0.80.

Fig. 3
Fig. 3

Variation of the ratio specific intensity (including the factor π) of diffuse light to the reduced flux, both in the direction of the source, for albedo=0. The zenith distance Θ0 of the source is shown on the top scale for reference.

Tables (9)

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Table I The wavelengths λ corresponding to selected normal optical thicknesses τ at sea level, computed from an up-to-date model of the terrestrial molecular atmosphere.

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Table II Derived parameters from the independent variable μ=cosΘ, where Θ0 represents zenith distance of the source, and secΘ0 or the “air mass” is an index of the optical path length in a plane-parallel, stratified medium neglecting astronomical refraction.

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Table III Reduced flux πFR of parallel radiation due to Rayleigh attenuation, as a function of normal optical thickness and μ0, for a surface normal to the incoming radiation.

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Table IV Computed values of the specific intensity Im of the nonparallel radiation due to multiple Rayleigh scattering, in the direction of the source (μ=μ0) in a plane parallel atmosphere, based on an incident flux of πF0=1 at the outer boundary, and no reflection from the lower boundary.

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Table V Values of the ratio Im/πFR of the specific intensity of diffuse light to reduced flux, resulting from the values of Tables III and IV.

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Table VI Values of the ratio IΛ/πFR of the specific intensity of reflected light to reduced flux, for albedos of 0.25 and 0.80, respectively, using the values of Tables III and VIII.

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Table VII Theoretical flux (Im+IΛω of nonparallel radiation including the effect of Lambert reflection with albedo 0.25, originating from a cone equivalent to a solid angle Δω=109.1×10−5 around the source, for a surface oriented normal to the direction of the source.

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Table VIII Computed values of the specific intensity IΛ of nonparallel radiation from rescattered light after reflection from the lower boundary with albedos of 0.25 and 0.80 respectively, according to the Lambert law. The assumptions are the same as for Table IV.

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Table IX Ratio (Im+IΛω/πFR, of the flux of nonparallel radiation to the reduced flux as provided by Tables VII and III, respectively, for a surface oriented normal to the direction of the source.

Equations (13)

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π F λ = π F R + π F m + π F Λ
π F R = exp ( - τ / μ 0 ) ,
I l + I r = ( 3 / 32 π ) ( μ 0 / μ - μ 0 ) [ ( ψ 0 + χ 0 ) ( ξ + σ ) + 2 ( ϕ 0 + ζ 0 ) ( η + θ ) - ( ξ 0 + σ 0 ) ( ψ + χ ) - 2 ( θ 0 + η 0 ) ( ϕ + ζ ) + 4 μ μ 0 ( 1 - μ 2 ) 1 2 × ( 1 - μ 0 2 ) 1 2 ( X 0 ( 1 ) Y ( 1 ) - Y 0 ( 1 ) X ( 1 ) ) × cos ( φ 0 - φ ) + ( 1 - μ 0 2 ) ( X 0 ( 2 ) Y ( 2 ) - Y 0 ( 2 ) X ( 2 ) ) ( 1 - μ 2 ) cos 2 ( φ 0 - φ ) ] ,
I m = lim μ μ 0 φ φ 0 ( I l + I r )
I m = ( 3 μ 0 / 32 π ) [ ( ψ 0 + χ 0 ) ( ξ 0 + σ 0 ) - ( ξ 0 + σ 0 ) ( ψ 0 + χ 0 ) + 2 { ( ϕ 0 + ζ 0 ) ( η 0 + θ 0 ) - ( θ 0 + η 0 ) ( ϕ 0 + ζ 0 ) } + 4 μ 0 2 ( 1 - μ 0 2 ) × ( X 0 ( 1 ) Y 0 ( 1 ) - Y 0 ( 1 ) X 0 ( 1 ) ) + ( 1 - μ 0 2 ) 2 ( X 0 ( 2 ) Y 0 ( 2 ) - Y 0 ( 2 ) X 0 ( 2 ) ) ] ,
I Λ = μ 0 Λ 4 ( 1 - Λ s ¯ ) π ( γ l + γ r ) [ 2 - ( γ l + γ r ) ] ,
π F λ = exp ( - τ / μ 0 ) + ( I l + I r ) cos ( Θ - Θ 0 ) d ω + ( I l * + I r * ) cos ( Θ - Θ 0 ) d ω ,
π F λ = exp ( - τ / μ 0 ) + ( I m + I Λ ) Δ ω ,
π F R q R = exp ( - τ / μ 0 ) ,
π F λ = π F 0 q 1 sec Θ 0 ; or , log π F λ = log π F 0 + log q 1 · sec Θ 0 .
q 1 sec Θ 0 = π F R + ( I m + I Λ ) Δ ω ,
sec Θ 0 log q 1 = log π F R + log [ 1 + [ ( I m + I Λ ) Δ ω ] / π F R ]
log q 1 = - τ + μ 0 ( I m + I Λ ) Δ ω exp ( - τ / μ 0 )