Abstract

An optical model of the parhelic circle is computed which includes the Stokes parameters of polarization. The model is based on one of two closely related mechanisms in crystals which are known to exist in cirrus clouds. An analysis of the circumstances of occurrence shows that one mechanism—external reflection from the side faces of ice crystal plates with c axes vertical—probably generates most parhelic circles.

© 1979 Optical Society of America

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References

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  1. M. Minnaert, The Nature of Light and Color in the Open Air, (Dover, New York, 1954).
  2. D. K. Lynch, “Atmospheric Halos,” Sci. Am. 238, (4), 144–152 (1978).
    [Crossref]
  3. R. S. McDowell, “The Formation of Parhelia at Higher Solar Elevations,” J. Atmos.Sci. 31, 1876–1884, (1974).
    [Crossref]
  4. E. Kidson, “Halo Complex,” Met. Mag. 66, 17–18 (1931).
  5. J. Findlater, “Remarkable Halo Display at Gibralter,” Weather 2, 247 (1947).
  6. G. A. Jones and K. J. Wiggins, “Halo Phenomena at Odiham,” Weather 19, 289–290 (1964).
    [Crossref]
  7. A. E. Moon, (untitled correspondence), Met. Mag. 69, 121–122 (1934).
  8. A. B. Fraser (private communication at OSA meeting Keystone, Colorado), 1978.
  9. F. G. Maunsell, “Parhelic Circle,” Weather 6, 245 (1951).

1978 (1)

D. K. Lynch, “Atmospheric Halos,” Sci. Am. 238, (4), 144–152 (1978).
[Crossref]

1974 (1)

R. S. McDowell, “The Formation of Parhelia at Higher Solar Elevations,” J. Atmos.Sci. 31, 1876–1884, (1974).
[Crossref]

1964 (1)

G. A. Jones and K. J. Wiggins, “Halo Phenomena at Odiham,” Weather 19, 289–290 (1964).
[Crossref]

1951 (1)

F. G. Maunsell, “Parhelic Circle,” Weather 6, 245 (1951).

1947 (1)

J. Findlater, “Remarkable Halo Display at Gibralter,” Weather 2, 247 (1947).

1934 (1)

A. E. Moon, (untitled correspondence), Met. Mag. 69, 121–122 (1934).

1931 (1)

E. Kidson, “Halo Complex,” Met. Mag. 66, 17–18 (1931).

Findlater, J.

J. Findlater, “Remarkable Halo Display at Gibralter,” Weather 2, 247 (1947).

Fraser, A. B.

A. B. Fraser (private communication at OSA meeting Keystone, Colorado), 1978.

Jones, G. A.

G. A. Jones and K. J. Wiggins, “Halo Phenomena at Odiham,” Weather 19, 289–290 (1964).
[Crossref]

Kidson, E.

E. Kidson, “Halo Complex,” Met. Mag. 66, 17–18 (1931).

Lynch, D. K.

D. K. Lynch, “Atmospheric Halos,” Sci. Am. 238, (4), 144–152 (1978).
[Crossref]

Maunsell, F. G.

F. G. Maunsell, “Parhelic Circle,” Weather 6, 245 (1951).

McDowell, R. S.

R. S. McDowell, “The Formation of Parhelia at Higher Solar Elevations,” J. Atmos.Sci. 31, 1876–1884, (1974).
[Crossref]

Minnaert, M.

M. Minnaert, The Nature of Light and Color in the Open Air, (Dover, New York, 1954).

Moon, A. E.

A. E. Moon, (untitled correspondence), Met. Mag. 69, 121–122 (1934).

Wiggins, K. J.

G. A. Jones and K. J. Wiggins, “Halo Phenomena at Odiham,” Weather 19, 289–290 (1964).
[Crossref]

J. Atmos.Sci. (1)

R. S. McDowell, “The Formation of Parhelia at Higher Solar Elevations,” J. Atmos.Sci. 31, 1876–1884, (1974).
[Crossref]

Met. Mag. (2)

E. Kidson, “Halo Complex,” Met. Mag. 66, 17–18 (1931).

A. E. Moon, (untitled correspondence), Met. Mag. 69, 121–122 (1934).

Sci. Am. (1)

D. K. Lynch, “Atmospheric Halos,” Sci. Am. 238, (4), 144–152 (1978).
[Crossref]

Weather (3)

J. Findlater, “Remarkable Halo Display at Gibralter,” Weather 2, 247 (1947).

G. A. Jones and K. J. Wiggins, “Halo Phenomena at Odiham,” Weather 19, 289–290 (1964).
[Crossref]

F. G. Maunsell, “Parhelic Circle,” Weather 6, 245 (1951).

Other (2)

M. Minnaert, The Nature of Light and Color in the Open Air, (Dover, New York, 1954).

A. B. Fraser (private communication at OSA meeting Keystone, Colorado), 1978.

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Figures (8)

FIG. 1
FIG. 1

Two mechanisms that form the parhelic circle. (a) External reflection from the side faces of plates oriented with their c axes vertical. (b) External reflection from the basal faces of pencil crystals. In each case the orientation about the c axis is arbitrary.

FIG. 2
FIG. 2

Coordinate system used to analyze the parhelic circle. S: vector from the observer O to the sun. P: vector from the observer to an arbitrary point on the parhelic circle P. e: elevation of the sun and parhelic circle. a: azimuth of an arbitrary point on the PHC.

FIG. 3
FIG. 3

Total intensity I of the PHC as a function of azimuth a, with elevation e as a parameter. Note that for low elevations (<30°) the maximum intensity occurs near 22°, in the vicinity of the parhelia.

FIG. 4
FIG. 4

Azimuth of the greatest intensity of the parhetic circle as a function of elevation.

FIG. 5
FIG. 5

Degree D of linear polarization as a function of azimuth. When the elevation is greater than 52.6°, the Brewster angle, the light from the PHC is never totally polarized.

FIG. 6
FIG. 6

Azimuth of total polarization as a function of elevation.

FIG. 7
FIG. 7

Angle θ between the vertical plane at azimuth a and the plane of incidence for various elevations.

FIG. 8
FIG. 8

Ratio of minimum-to-maximum intensity as a function of azimuth.

Equations (14)

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1 = F ( a , e ) * sin ( a / 2 ) * cos ( e ) ,
F ( a , e ) = r p + r s ,
r p = [ tan ( i i ) ] 2 / [ tan ( i + i ) ] 2 ,
r s = [ sin ( i i ) ] 2 / [ sin ( i + i ) ] 2 ,
i = cos 1 [ cos ( e ) * sin ( a / 2 ) ] ,
i = sin 1 [ ( sin ( i ) ] / n .
S 0 = r p + r s ,
S 1 = r p r s ,
S 2 = 2 * ( r p * r s ) 1 / 2 cos ( ϕ ) ,
S 3 = 2 * ( r p * r s ) 1 / 2 sin ( ϕ ) .
S = cos ( e ) î + 0 ĵ + sin ( e ) k ̂ ,
P = cos ( e ) * cos ( a ) î + cos ( e ) * sin ( a ) ĵ + sin ( e ) k ̂ ,
θ = 180 ° cos 1 ( sin ( e ) [ cos ( a ) 1 ] { sin 2 ( a ) + sin 2 ( e ) * [ cos ( a ) 1 ] 2 } 1 / 2 ) ,
D = S 1 / S 0 ,