Abstract

Although it is usually assumed that solar radiation falls on the earth's atmosphere in the form of plane waves, the finite angular size of the solar disk contradicts this assumption. For most purposes, this finite sun effect on computed or measured radiation quantities is negligible. However, in the region of the solar aureole, which is dominated by aerosol diffraction scattering, measurable effects may be obtained. In this paper, we show that the finite sun effect is related to derivatives of the scattering phase function and that a 1% effect may be obtained close to the sun if enough large particles are present in the atmosphere.

© 1981 Optical Society of America

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References

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  1. A. Deepak, “Inversion of Solar Aureole Measurements for Determining Aerosol Characteristics,” in Inversion Methods in Atmospheric Remote Sounding,A. Deepak, Ed. (Academic, New York, 1978).
  2. A. E. S. Green, “Analytic Model Approach to the Inversion of Scattering Data,” in Inversion Methods in Atmospheric Remote Sounding,A. Deepak, Ed. (Academic, New York, 1978).
  3. A. E. S. Green, A. Deepak, B. J. Lipofsky, Appl. Opt. 10, 1263 (1971).
    [CrossRef] [PubMed]
  4. M. D. King, D. M. Byrne, J. Atmos. Sci. 33, 2242 (1976).
    [CrossRef]
  5. M. A. Box, A. Deepak, Appl. Opt. 18, 1376 (1979).
    [CrossRef] [PubMed]
  6. M. A. Box, G. P. Box, A. Deepak, J. Aerosol Sci. 10, 210 (1979).
    [CrossRef]
  7. M. A. Box, A. Deepak, J. Atmos. Sci. 38, 1037 (1981).
    [CrossRef]
  8. M. A. Box, A. Deepak, Appl. Opt. 18, 1941 (1979).
    [CrossRef] [PubMed]
  9. G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1970).
  10. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  11. C. W. Allen, Astrophysical Quantities (Athlone, New York, 1976).

1981 (1)

M. A. Box, A. Deepak, J. Atmos. Sci. 38, 1037 (1981).
[CrossRef]

1979 (3)

1976 (1)

M. D. King, D. M. Byrne, J. Atmos. Sci. 33, 2242 (1976).
[CrossRef]

1971 (1)

Allen, C. W.

C. W. Allen, Astrophysical Quantities (Athlone, New York, 1976).

Arfken, G.

G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1970).

Box, G. P.

M. A. Box, G. P. Box, A. Deepak, J. Aerosol Sci. 10, 210 (1979).
[CrossRef]

Box, M. A.

M. A. Box, A. Deepak, J. Atmos. Sci. 38, 1037 (1981).
[CrossRef]

M. A. Box, A. Deepak, Appl. Opt. 18, 1376 (1979).
[CrossRef] [PubMed]

M. A. Box, G. P. Box, A. Deepak, J. Aerosol Sci. 10, 210 (1979).
[CrossRef]

M. A. Box, A. Deepak, Appl. Opt. 18, 1941 (1979).
[CrossRef] [PubMed]

Byrne, D. M.

M. D. King, D. M. Byrne, J. Atmos. Sci. 33, 2242 (1976).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

Deepak, A.

M. A. Box, A. Deepak, J. Atmos. Sci. 38, 1037 (1981).
[CrossRef]

M. A. Box, A. Deepak, Appl. Opt. 18, 1941 (1979).
[CrossRef] [PubMed]

M. A. Box, A. Deepak, Appl. Opt. 18, 1376 (1979).
[CrossRef] [PubMed]

M. A. Box, G. P. Box, A. Deepak, J. Aerosol Sci. 10, 210 (1979).
[CrossRef]

A. E. S. Green, A. Deepak, B. J. Lipofsky, Appl. Opt. 10, 1263 (1971).
[CrossRef] [PubMed]

A. Deepak, “Inversion of Solar Aureole Measurements for Determining Aerosol Characteristics,” in Inversion Methods in Atmospheric Remote Sounding,A. Deepak, Ed. (Academic, New York, 1978).

Green, A. E. S.

A. E. S. Green, A. Deepak, B. J. Lipofsky, Appl. Opt. 10, 1263 (1971).
[CrossRef] [PubMed]

A. E. S. Green, “Analytic Model Approach to the Inversion of Scattering Data,” in Inversion Methods in Atmospheric Remote Sounding,A. Deepak, Ed. (Academic, New York, 1978).

King, M. D.

M. D. King, D. M. Byrne, J. Atmos. Sci. 33, 2242 (1976).
[CrossRef]

Lipofsky, B. J.

Appl. Opt. (3)

J. Aerosol Sci. (1)

M. A. Box, G. P. Box, A. Deepak, J. Aerosol Sci. 10, 210 (1979).
[CrossRef]

J. Atmos. Sci. (2)

M. A. Box, A. Deepak, J. Atmos. Sci. 38, 1037 (1981).
[CrossRef]

M. D. King, D. M. Byrne, J. Atmos. Sci. 33, 2242 (1976).
[CrossRef]

Other (5)

G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1970).

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

C. W. Allen, Astrophysical Quantities (Athlone, New York, 1976).

A. Deepak, “Inversion of Solar Aureole Measurements for Determining Aerosol Characteristics,” in Inversion Methods in Atmospheric Remote Sounding,A. Deepak, Ed. (Academic, New York, 1978).

A. E. S. Green, “Analytic Model Approach to the Inversion of Scattering Data,” in Inversion Methods in Atmospheric Remote Sounding,A. Deepak, Ed. (Academic, New York, 1978).

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Figures (2)

Fig. 1
Fig. 1

Scattering geometry

Fig. 2
Fig. 2

Contributions to the fractional deviation from point source radiances due to finite solar diameter.

Equations (46)

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L = Φ 0 sec ϕ exp ( τ sec ϕ ) [ τ p F p P p ( ψ 0 ) + τ M F M P M ( ψ 0 ) ] ,
F p = F p ( D ) = 0 τ p exp ( tD ) d t p / τ p ,
F M = 0 τ M exp ( tD ) d t M / τ M ,
L p = sec ϕ exp ( τ sec ϕ ) 0 τ p Ω 0 I 0 P p ( ψ ) × exp [ t ( sec θ sec ϕ ) ] d Ω d t p ,
cos ψ = cos ψ 0 cos ξ + sin ψ 0 sin ξ cos γ ,
cos θ = cos θ 0 cos ξ + sin θ 0 sin ξ cos γ .
π ξ 0 2 = Ω 0 = 6.8 × 10 5 sr .
Φ 0 = 2 π 0 ξ 0 I 0 ( ξ ) ξ d ξ ,
exp ( t sec θ ) exp ( t sec θ 0 ) × ( 1 ½ ξ 2 t sec θ 0 + g 1 ξ cos γ g 2 ξ 2 cos 2 γ ) ,
g 1 = t tan θ 0 sec θ 0 , g 2 = t tan 2 θ 0 sec θ 0 ( 2 ½ t sec θ 0 ) .
P p ( ψ ) = n = 0 ω n P n ( cos ψ ) ,
P p ( ω ) = n = 0 ω p P n ( cos ψ 0 ) P n ( cos ξ ) + 2 m = 1 n ( n m ) ! ( n + m ) ! P n m ( cos ψ 0 ) P n m ( cos ξ ) cos m γ ,
L p = sec ϕ exp ( τ sec ϕ ) 0 τ p d t p exp ( tD ) × n = 0 ω n 0 ξ 0 d ξ I 0 ( ξ ) ξ 0 2 π d γ * { P n ( cos ψ 0 ) P n ( cos ξ ) + 2 m = 1 n ( n m ) ! ( n + m ) ! × P n m ( cos ψ 0 ) P n m ( cos ξ ) cos m γ } * { 1 ½ ξ 2 t sec θ 0 + g 1 ξ cos γ g 2 ξ 2 cos γ 2 } .
L p = 2 π sec ϕ exp ( τ sec ϕ ) 0 τ p d t p × exp ( tD ) n = 0 ω n 0 ξ 0 d ξ I 0 ( ξ ) ξ * { g 1 ξ ( n 1 ) ! ( n + 1 ) ! P n 1 ( cos ψ 0 ) P n 1 ( cos ξ ) + ½ g 2 ξ 2 ( n 2 ) ! ( n + 2 ) ! P n 2 ( cos ψ 0 ) P n 2 ( cos ξ ) + P n ( cos ψ 0 ) P n ( cos ξ ) [ 1 ½ ξ 2 t sec θ 0 ½ g 2 ξ 2 ] } .
P n ( cos ξ ) 1 ¼ n ( n + 1 ) ξ 2 ;
P n 1 ( cos ξ ) ½ n ( n + 1 ) ξ ;
P n 2 ( cos ξ ) ( n 1 ) n ( n + 1 ) ( n + 2 ) ξ 2 ;
P n 1 ( cos ψ 0 ) = sin ψ 0 P n ( cos ψ 0 ) ,
P n 2 ( cos ψ 0 ) = sin 2 ψ 0 P n ( cos ψ 0 ) .
L p = 2 π sec ϕ exp ( τ sec ϕ ) 0 τ p d t p exp ( tD ) × n = 0 ω n 0 ξ 0 d ξ I 0 ( ξ ) ξ * { ½ g 1 ξ 2 sin ψ 0 P n ( cos ψ 0 ) + g 2 ξ 4 sin 2 ψ 0 P n ( cos ψ 0 ) / 16 + P n ( cos ψ 0 ) [ 1 ½ ξ 2 t sec θ 0 ½ g 2 ξ 2 ¼ n ( n + 1 ) ξ 2 + 0 ( ξ 4 ) ] } .
2 π 0 ξ 0 I 0 ( ξ ) ξ 3 d ξ = ½ f ξ 0 2 Φ 0 .
L p = Φ 0 sec ϕ exp ( τ sec ϕ ) 0 τ p exp ( tD ) n = 0 ω n × ( ¼ g 1 f ξ 0 2 sin ψ 0 P n ( cos ψ 0 ) + P n ( cos ψ 0 ) { 1 ¼ f ξ 0 2 × [ t sec θ 0 + g 2 + ½ n ( n + 1 ) ] } d t p .
n = 0 ω n n ( n + 1 ) P n ( cos ψ 0 ) = n = 0 ω n [ 2 cos ψ 0 P n ( cos ψ 0 ) sin 2 ψ 0 P n ( cos ψ 0 ) ] = 2 cos ψ 0 P p ( cos ψ 0 ) sin 2 ψ p P p ( cos ψ 0 )
= cosec ψ 0 d d ψ 0 ( sin ψ 0 d P p d ψ 0 ) .
L p = Φ 0 sec ϕ exp ( τ sec ϕ ) τ p [ F p ( 0 ) P p ( ψ 0 ) ¼ f ξ 0 2 Δ p ( ψ 0 ) ] ,
Δ p ( ψ 0 ) = ½ ( 2 cos ψ 0 P p sin 2 ψ 0 P p ) F p ( 0 ) ½ tan 2 θ 0 sec 2 θ 0 P p F p ( 2 ) + [ ( sec θ 0 + 2 tan 2 θ 0 sec θ 0 ) P p tan θ 0 sec θ 0 sin ψ 0 P p ] F p ( 1 ) ,
F p ( i ) ( D ) = 0 τ p exp ( tD ) t i d t p / τ p ,
Δ p ( ψ 0 ) ½ ( 2 cos ψ 0 P p sin 2 ψ 0 P p ) F p ( 0 ) tan θ 0 sec θ 0 sin ψ 0 P p F p ( 1 )
= ½ n = 0 ω n n ( n + 1 ) P n ( cos ψ 0 ) F p ( 0 ) tan θ 0 sec θ 0 sin ψ 0 P p F p ( 1 ) .
L M = Φ 0 sec ϕ exp ( τ sec ϕ ) τ M [ F M P M ( ψ 0 ) ¼ f ξ 0 2 Δ m ( ψ 0 ) ] ,
P M ( ψ 0 ) = 3 ( 1 + cos 2 ψ 0 ) / 16 π ,
sin ψ 0 P M = 3 sin 2 ψ 0 / 16 π ,
½ ( 2 cos ψ 0 P n sin 2 ψ 0 P n ) = 3 ( 3 cos 2 ψ 0 1 ) / 16 π .
L = Φ 0 sec ϕ exp ( τ sec ϕ ) [ τ p F p P p ( ψ 0 ) + τ M F M P M ( ψ 0 ) ¼ f ξ 0 2 Δ p ( ψ 0 ) τ p ] .
¼ f ξ 0 2 ( cos ψ 0 P p ½ sin 2 ψ 0 P p ) / P p
¼ f ξ 0 2 sin ψ 0 P p / P p .
Δ p ( 0 ° ) = ω n n ( n + 1 ) F p ( 0 ) .
P ( ψ 0 ) α 2 π 1 exp ( α 2 ψ 0 2 ) .
ψ 0 1 d d ψ 0 [ ψ 0 dP ( ψ 0 ) d ψ 0 ] = 4 α 2 P ( ψ 0 ) [ 1 α 2 ψ 0 2 ] .
Δ p ( 0 ° ) = 4 π [ P ( 0 ° ) ] 2 .
I λ ( χ ) = I λ ( 0 ) [ 1 u ( 1 cos χ ) υ ( 1 cos 2 χ ) ] ,
sin χ = tan ξ / tan ξ 0 ξ / ξ 0 .
I λ ( ξ ) = I λ ( 0 ) { 1 u [ 1 ( 1 ξ 2 / ξ 0 2 ) 1 / 2 ] υ ξ 2 / ξ 0 2 } .
Φ 0 = π I λ ( 0 ) ξ 0 2 ( 1 u ½ υ ) .
2 π 0 ξ 0 I λ ( ξ ) ξ 3 d ξ = ½ π I λ ( 0 ) ξ 0 4 ( 1 7 / 15 u υ ) ,
f = f ( u , υ ) = 1 7 u / 15 2 υ / 3 1 u / 3 υ / 2 .

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