Abstract

The effect of a concentrated turbid layer on the polarization of skylight is discussed in this paper. Two idealized models of the turbid atmosphere are employed. In one an aerosol layer is situated below a molecular layer and, in the other, above it. The size distribution of the aerosols of refractive index 1.33 is assumed to follow a power law. Inside the aerosol layer, only primary scattering of light is taken into account and polarization effects are neglected. The importance of the interaction between the aerosol and molecular layers is recognized. The resulting computations show broad agreement with available observations. The importance of the location of the turbid layer in characterizing the emergent radiation is revealed. In particular, the computations have shown that the Babinet and Brewster neutral (zero polarization) points can shift towards the sun from their normal positions in a molecular atmosphere when there is low-level turbidity. Such a shift is indicated by observations.

© 1968 Optical Society of America

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References

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  1. Rayleigh, Phil. Mag. XLI, 107 (1871).
  2. S. Chandrasekhar, Radiative Transfer (Oxford University Press, London, 1950), p. 393.
  3. S. Chandrasekhar and D. D. Elbert, Trans. Am. Phil. Soc. (New Series)44, Part 643 (1954).
  4. C. Dorno Veröff, Preuss. Meteor. Inst. (Berlin), No. 303 (1919).
  5. C. Jensen, in Handbuch der Geophysik, F. Linke and F. Möller, Eds. (Gerbrüder Borntraeger, Berlin–Nikolassee, 1942), Vol. 8, pp. 527–621.
  6. Z. Sekera, in Advances in Geophysics, H. E. Landsberg, Ed. (Academic Press Inc., New York, 1956), Vol. III, pp. 43–104.
    [CrossRef]
  7. G. C. Holzwarth and C. R. Nagaraja Rao, J. Opt. Soc. Am. 55, 403 (1965).
    [CrossRef]
  8. Muneyasu Kano, Ph.D. dissertation, Department of Meteorology, University of California, Los Angeles (1964, unpublished). Available at University Microfilm, Ann Arbor, Michigan. A limited number of copies are available for distribution at the Dept. of Meteorology, University of California, Los Angeles 90024.
  9. R. S. Fraser, J. Opt. Soc. Am. 54, 157 (1964).
    [CrossRef]
  10. K. L. Coulson, J. V. Dave, and Z. Sekera, Tables Related to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (University of California Press, Los Angeles, 1960), pp. X + 548
  11. G. Dietze, Z. Meteorol. 5, 86 (1951).

1965 (1)

1964 (1)

1954 (1)

S. Chandrasekhar and D. D. Elbert, Trans. Am. Phil. Soc. (New Series)44, Part 643 (1954).

1951 (1)

G. Dietze, Z. Meteorol. 5, 86 (1951).

1871 (1)

Rayleigh, Phil. Mag. XLI, 107 (1871).

Chandrasekhar, S.

S. Chandrasekhar and D. D. Elbert, Trans. Am. Phil. Soc. (New Series)44, Part 643 (1954).

S. Chandrasekhar, Radiative Transfer (Oxford University Press, London, 1950), p. 393.

Coulson, K. L.

K. L. Coulson, J. V. Dave, and Z. Sekera, Tables Related to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (University of California Press, Los Angeles, 1960), pp. X + 548

Dave, J. V.

K. L. Coulson, J. V. Dave, and Z. Sekera, Tables Related to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (University of California Press, Los Angeles, 1960), pp. X + 548

Dietze, G.

G. Dietze, Z. Meteorol. 5, 86 (1951).

Dorno Veröff, C.

C. Dorno Veröff, Preuss. Meteor. Inst. (Berlin), No. 303 (1919).

Elbert, D. D.

S. Chandrasekhar and D. D. Elbert, Trans. Am. Phil. Soc. (New Series)44, Part 643 (1954).

Fraser, R. S.

Holzwarth, G. C.

Jensen, C.

C. Jensen, in Handbuch der Geophysik, F. Linke and F. Möller, Eds. (Gerbrüder Borntraeger, Berlin–Nikolassee, 1942), Vol. 8, pp. 527–621.

Kano, Muneyasu

Muneyasu Kano, Ph.D. dissertation, Department of Meteorology, University of California, Los Angeles (1964, unpublished). Available at University Microfilm, Ann Arbor, Michigan. A limited number of copies are available for distribution at the Dept. of Meteorology, University of California, Los Angeles 90024.

Nagaraja Rao, C. R.

Rayleigh,

Rayleigh, Phil. Mag. XLI, 107 (1871).

Sekera, Z.

K. L. Coulson, J. V. Dave, and Z. Sekera, Tables Related to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (University of California Press, Los Angeles, 1960), pp. X + 548

Z. Sekera, in Advances in Geophysics, H. E. Landsberg, Ed. (Academic Press Inc., New York, 1956), Vol. III, pp. 43–104.
[CrossRef]

J. Opt. Soc. Am. (2)

Phil. Mag. (1)

Rayleigh, Phil. Mag. XLI, 107 (1871).

Trans. Am. Phil. Soc. (New Series) (1)

S. Chandrasekhar and D. D. Elbert, Trans. Am. Phil. Soc. (New Series)44, Part 643 (1954).

Z. Meteorol. (1)

G. Dietze, Z. Meteorol. 5, 86 (1951).

Other (6)

C. Dorno Veröff, Preuss. Meteor. Inst. (Berlin), No. 303 (1919).

C. Jensen, in Handbuch der Geophysik, F. Linke and F. Möller, Eds. (Gerbrüder Borntraeger, Berlin–Nikolassee, 1942), Vol. 8, pp. 527–621.

Z. Sekera, in Advances in Geophysics, H. E. Landsberg, Ed. (Academic Press Inc., New York, 1956), Vol. III, pp. 43–104.
[CrossRef]

Muneyasu Kano, Ph.D. dissertation, Department of Meteorology, University of California, Los Angeles (1964, unpublished). Available at University Microfilm, Ann Arbor, Michigan. A limited number of copies are available for distribution at the Dept. of Meteorology, University of California, Los Angeles 90024.

K. L. Coulson, J. V. Dave, and Z. Sekera, Tables Related to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (University of California Press, Los Angeles, 1960), pp. X + 548

S. Chandrasekhar, Radiative Transfer (Oxford University Press, London, 1950), p. 393.

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Figures (14)

Fig. 1
Fig. 1

Features of skylight polarization in the sun’s vertical. AR: Arago point; BA: Babinet point; BR: Brewster point; H: Horizon; M: Region of maximum polarization; O: Observer; S: Sun; S: Antisolar point; A: Antisolar distance of Arago point; B1: Solar distance of Babinet point; B2: Solar distance of Brewster point; h: Solar elevation.

Fig. 2
Fig. 2

Scattering geometry. OB: Direction of incidence; OA: Direction of scattering; θ: Scattering angle.

Fig. 3
Fig. 3

Prevalent geometry for a model with a lower aerosol layer and an upper molecular layer (L Model).

Fig. 4
Fig. 4

Prevalent geometry for a model with an upper aerosol layer and a lower molecular layer (U Model).

Fig. 5
Fig. 5

Deviations of the computed positions of the Babinet point in the model turbid atmospheres from those in a molecular atmosphere. – – – – – –: L Model ○○○ 3650 Å; ——: U Model ΔΔΔ 4600 Å; ××× 6250 Å; Turbidity factor T6250 = 3.58.

Fig. 6
Fig. 6

Deviations of the computed positions of the Brewster and Arago neutral points in the model turbid atmospheres from from those in a molecular atmosphere. Notation same as in Fig. 5.

Fig. 7
Fig. 7

Variation of the solar distance of the Babinet point with turbidity. Solar elevation: 6°36. Notation same as in Fig. 5.

Fig. 8
Fig. 8

Variation of the solar distance of the Babinet point with turbidity. Solar elevation: 52°32. Notation same as in Fig. 5.

Fig. 9
Fig. 9

Variation of the solar distance of the Brewster point with turbidity. Solar elevation: 52°32. Notation same as in Fig. 5.

Fig. 10
Fig. 10

Variation of the antisolar distance of the Arago point with turbidity. Solar elevation: 6°36. Notation same as in Fig. 5.

Fig. 11
Fig. 11

Variation of the degree of maximum polarization at 6250 Å with turbidity for typically high (52°32) and low (6°36) solar elevations.

Fig. 12
Fig. 12

Variation of the degree of maximum polarization at 4600 Å with turbidity for typically high (52°32) and low (6°36) solar elevations.

Fig. 13
Fig. 13

Computed values of dispersion of polarization.

Fig. 14
Fig. 14

Measured values of dispersion of polarization. ○○○: Dietze; ●●●: Holzwarth and Rao.

Tables (1)

Tables Icon

Table I The positions of neutral pointsa outside of the sun’s vertical for the solar elevation angle 52.5°.

Equations (17)

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I ( 0 , Ω ) = ( 4 μ ) 1 S ( τ 1 , Ω , Ω 0 ) F ( Ω 0 ) I ( τ 1 , Ω ) = ( 4 μ ) 1 ( τ 1 , Ω , Ω 0 ) F ( Ω 0 ) ,
S L ( τ L , Ω , Ω 0 ) = μ μ 0 μ + μ 0 { 1 exp [ τ L ( 1 μ + 1 μ 0 ) ] } × P L ( Ω , Ω 0 ) , T L ( τ L , Ω , Ω 0 ) = μ μ 0 μ μ 0 [ e τ L / μ e τ L / μ 0 ] P L ( Ω , Ω 0 ) ,
P L ( Ω , Ω 0 ) = ( p 11 0 0 0 0 p 22 p 23 0 0 p 32 p 33 0 0 0 0 p 44 ) ,
p 11 = m = 0 N F m ( μ , μ 0 ) cos m ( ϕ 0 ϕ ) p 22 = m = 0 N E m ( μ , μ 0 ) cos m ( ϕ 0 ϕ ) p 23 = m = 0 N m ( μ , μ 0 ) sin m ( ϕ 0 ϕ ) p 32 = m = 0 N m ( μ , μ 0 ) sin m ( ϕ 0 ϕ ) p 33 = m = 0 N E m ( μ , μ 0 ) cos m ( ϕ 0 ϕ )
p 44 = m = 0 N F m ( μ , μ 0 ) cos m ( ϕ 0 ϕ ) ,
F m ( μ , μ 0 ) π ( 1 + δ 0 m ) = F m * ( μ , μ 0 ) = 0 2 π p 1 ( cos θ ) cos m ( ϕ 0 ϕ ) d ϕ E m ( μ , μ 0 ) π ( 1 + δ 0 m ) = E m * ( μ , μ 0 ) = 0 2 π p 1 ( cos θ ) cos 2 ( β + γ ) × cos ( ϕ 0 ϕ ) d ϕ m ( μ , μ 0 ) π ( 1 δ 0 m ) = m * ( μ , μ 0 ) = 0 2 π p 1 ( cos θ ) sin 2 ( β + γ ) × sin m ( ϕ 0 ϕ ) d ϕ , δ 0 m = { 0 , m 0 1 , m = 0
p 1 ( cos θ ) = 2 π β L K 3 α 0 α 1 ( S 1 S 1 * + S 2 S 2 * ) f ( α ) d α ,
d n / d r = k 1 for 0.03 μ r 0.1 μ
d n / d r = k 2 r 4 for 0.1 μ r 20 μ ,
I ( 0 , Ω ) = 1 4 μ S R ( τ R , Ω , Ω 0 ) F ( Ω 0 ) + I d ( Ω ) exp ( τ R μ ) + 1 4 π μ 2 π T R ( τ R , Ω , Ω ) I d ( Ω ) d Ω
I ( τ t , Ω ) = 1 4 μ T L ( τ L , Ω , Ω 0 ) F ( Ω 0 ) exp ( τ R μ 0 ) + I d ( Ω ) exp ( τ L μ ) + 1 4 π μ 2 π T L ( τ L , Ω , Ω ) × I d ( Ω ) d Ω
I d ( Ω ) = 1 4 μ S L ( τ L , Ω , Ω 0 ) F ( Ω 0 ) exp ( τ R μ 0 ) + 1 4 π μ 2 π S L ( τ L , Ω , Ω ) I d ( Ω ) d Ω
I d ( Ω ) = 1 4 μ T R ( τ R , Ω , Ω 0 ) F ( Ω 0 ) + 1 4 π μ 2 π S R ( τ R , Ω , Ω ) I d ( Ω ) d Ω .
I d ( Ω ) = 1 4 μ T R ( τ R , Ω , Ω 0 ) F ( Ω 0 ) + [ exp ( τ R / μ 0 ) / 16 π μ ] × 2 π S R ( Ω , Ω ) S L ( Ω , Ω 0 ) F d Ω μ + ( 16 π 2 μ ) 1 × 2 π S R ( Ω , Ω ) d Ω μ 2 π S L ( Ω , Ω ) I d ( Ω ) d Ω .
I d ( Ω ) = 1 4 μ T R ( τ R , Ω , Ω 0 ) F ( Ω 0 ) + exp ( τ R / μ ) 16 π μ × 2 π S R ( Ω , Ω ) S L ( Ω , Ω 0 ) F d Ω μ .
I ( τ t , Ω ) = n = 1 5 I n ( Ω ) .
I ( τ t , Ω ) = n = 1 5 I n ( Ω ) .