Abstract

We performed a calculation of the corona colors that employed Mie theory to obtain the scattered light intensity. The scattered intensity was integrated over the visible spectrum for a number of different cloud droplet size distriubtions. The results were converted to chromaticity coordinates, convolved with the angular size of the sun, and plotted on the 1931 CIE chromaticity diagram. The results were compared to observations of multiple-ring coronas. It was found that, when using Mie theory to estimate cloud droplet sizes, water droplets with diameters in the 7-μm ≲ D ≲ 15-μm range produced the 13 multiple-ring coronas that were observed.

© 1991 Optical Society of America

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References

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  1. M. Minnaert, Tne Nature of Light and Colour in the Open Air (Dover, New York, 1954), articles 160 and 161.
  2. R. A. R. Tricker, Introduction to Meteorological Optics (American Elsevier, New York, 1970), Chap. 5.
  3. R. Greenler, Rainbows, Halos, and Glories (Cambridge U. Press, Cambridge, England, 1980), Chap. 6.
  4. W. J. Humphreys, Physics of the Air (Dover, New York, 1964), Chap. 6.
  5. C. Bohren, “A serendipitous iridescent cloud,” Weatherwise 38, 268–274 (1985).
    [CrossRef]
  6. G. C. Simpson, “Coronas and iridescent clouds,” Q. J. R. Meteorol. Soc. 38, 291–301 (1912).
    [CrossRef]
  7. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), p. 425.
  8. Ref. 7, Sections 11.22, 11.3, and 13.41.
  9. J. G. Wilson, “Note on optical methods of measuring the size of small water drops,” Proc. Cambridge Philos. Soc. 32, 493–498 (1936).
    [CrossRef]
  10. J. Bricard, “Reflexion, refraction et diffraction de la lumiere par une goutte d’eau spherique,” Ann. Geophys. 2, 231–248 (1946).
  11. J. A. Prins, J. J. M. Reesnick, “Buigingstheorie en trichromatische specificatie van de Regenboogkleuren,” Physica 11, 49–60 (1944).
    [CrossRef]
  12. J. D. Walker, “The bright colors in a soap film are a lesson in wave interference,” Sci. Am. 239(3), 232–237 (1978).
    [CrossRef]
  13. Ref. 7, p. 425, appears to be in error on this point. The calculations of Prins and Reesinck and of Buchwald are for the rainbow, not for the corona.
  14. Y. G. Naik, “Correlation between optical and dynamic methods of measuring size of water drops in a cloud,” J. Colloid Sci. 9, 393–399 (1954).
    [CrossRef]
  15. Ref. 7, Fig. 52.
  16. K. Sassen, “Iridescence in an aircraft contrail,” J. Opt. Soc. Am. 69, 1080–1083 (1979).
    [CrossRef]
  17. R. W. Burnham, R. M. Hanes, C. J. Bartleson, Color: A Guide to Basic Facts and Concepts (Wiley, New York, 1963), pp. 130–133.
  18. W. D. Wright, The Measurement of Colour (Hilger, London, 1969), Appendix 2, Table 4.
  19. D. Falk, D. Brill, D. Stork, Seeing the Light (Harper & Row, New York, 1986), p. 244.
  20. D. L. MacAdam, “Visual sensitivities to color differences in daylight,” J. Opt. Soc. Am. 32, 247–274 (1942).
    [CrossRef]
  21. Ref. 18, Fig. 6.3.
  22. C. F. Bohren, “Understanding colors in nature,” Pigment Cell Res. 1, 214–222 (1988).
    [CrossRef] [PubMed]
  23. Ref. 1, p. 215.
  24. Ref. 1, p. 214.
  25. Ref. 2, p. 146.
  26. E. J. McCartney, Optics of the Atmosphere (Wiley, New York, 1976), Table 5.3.
  27. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Appendix A.
  28. A. B. Fraser, “Why can the supernumerary bows be seen in rain showers,” J. Opt. Soc. Am. 73, 1626–1628 (1983).
    [CrossRef]
  29. J. A. Lock, “Observability of atmospheric glories and supernumerary rainbows,” J. Opt. Soc. Am. A 6, 1924–1930 (1989).
    [CrossRef]
  30. S. D. Gedzelman, “In praise of altocumulus,” Weatherwise 41, 143–149 (1988).
    [CrossRef]
  31. For a colorimetric analysis of rainbow photographs see R. L. Lee, “What are ‘all the colors of the rainbow’?” Appl. Opt. 30, 3401–3407 (1991).
    [CrossRef] [PubMed]
  32. H. Lansford, “Shooting the sky,” Weatherwise 35, 73–81 (1982).
    [CrossRef]
  33. S. J. O’Meara, “Yellowstone: a natural phenomenon,” Sky Telesc. 70, 369 (1985).
  34. The data of Fig. 6, where different diameter droplets correspond to different red rings in a single corona, resulted from the corona occurring at the edge of the cloud where the droplet size distribution was not homogeneous.
  35. One additional corona photograph provided by K. Sassen was analyzed for which Fig. 6 predicts a droplet diameter of 21 μm. This corona, however, occurs in a cirrostratus cloud and possesses a number of the features particular to ice crystal coronas. see for example, K. Sassen, “Corona-producing cirrus cloud properties derived from polarization lidar and photographic analyses,” Appl. Opt. 30, 3421–3428 (1991).
    [CrossRef] [PubMed]
  36. Ref. 26, Table 3.6.

1991 (2)

1989 (1)

1988 (2)

S. D. Gedzelman, “In praise of altocumulus,” Weatherwise 41, 143–149 (1988).
[CrossRef]

C. F. Bohren, “Understanding colors in nature,” Pigment Cell Res. 1, 214–222 (1988).
[CrossRef] [PubMed]

1985 (2)

S. J. O’Meara, “Yellowstone: a natural phenomenon,” Sky Telesc. 70, 369 (1985).

C. Bohren, “A serendipitous iridescent cloud,” Weatherwise 38, 268–274 (1985).
[CrossRef]

1983 (1)

1982 (1)

H. Lansford, “Shooting the sky,” Weatherwise 35, 73–81 (1982).
[CrossRef]

1979 (1)

1978 (1)

J. D. Walker, “The bright colors in a soap film are a lesson in wave interference,” Sci. Am. 239(3), 232–237 (1978).
[CrossRef]

1954 (1)

Y. G. Naik, “Correlation between optical and dynamic methods of measuring size of water drops in a cloud,” J. Colloid Sci. 9, 393–399 (1954).
[CrossRef]

1946 (1)

J. Bricard, “Reflexion, refraction et diffraction de la lumiere par une goutte d’eau spherique,” Ann. Geophys. 2, 231–248 (1946).

1944 (1)

J. A. Prins, J. J. M. Reesnick, “Buigingstheorie en trichromatische specificatie van de Regenboogkleuren,” Physica 11, 49–60 (1944).
[CrossRef]

1942 (1)

1936 (1)

J. G. Wilson, “Note on optical methods of measuring the size of small water drops,” Proc. Cambridge Philos. Soc. 32, 493–498 (1936).
[CrossRef]

1912 (1)

G. C. Simpson, “Coronas and iridescent clouds,” Q. J. R. Meteorol. Soc. 38, 291–301 (1912).
[CrossRef]

Bartleson, C. J.

R. W. Burnham, R. M. Hanes, C. J. Bartleson, Color: A Guide to Basic Facts and Concepts (Wiley, New York, 1963), pp. 130–133.

Bohren, C.

C. Bohren, “A serendipitous iridescent cloud,” Weatherwise 38, 268–274 (1985).
[CrossRef]

Bohren, C. F.

C. F. Bohren, “Understanding colors in nature,” Pigment Cell Res. 1, 214–222 (1988).
[CrossRef] [PubMed]

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Appendix A.

Bricard, J.

J. Bricard, “Reflexion, refraction et diffraction de la lumiere par une goutte d’eau spherique,” Ann. Geophys. 2, 231–248 (1946).

Brill, D.

D. Falk, D. Brill, D. Stork, Seeing the Light (Harper & Row, New York, 1986), p. 244.

Burnham, R. W.

R. W. Burnham, R. M. Hanes, C. J. Bartleson, Color: A Guide to Basic Facts and Concepts (Wiley, New York, 1963), pp. 130–133.

Falk, D.

D. Falk, D. Brill, D. Stork, Seeing the Light (Harper & Row, New York, 1986), p. 244.

Fraser, A. B.

Gedzelman, S. D.

S. D. Gedzelman, “In praise of altocumulus,” Weatherwise 41, 143–149 (1988).
[CrossRef]

Greenler, R.

R. Greenler, Rainbows, Halos, and Glories (Cambridge U. Press, Cambridge, England, 1980), Chap. 6.

Hanes, R. M.

R. W. Burnham, R. M. Hanes, C. J. Bartleson, Color: A Guide to Basic Facts and Concepts (Wiley, New York, 1963), pp. 130–133.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Appendix A.

Humphreys, W. J.

W. J. Humphreys, Physics of the Air (Dover, New York, 1964), Chap. 6.

Lansford, H.

H. Lansford, “Shooting the sky,” Weatherwise 35, 73–81 (1982).
[CrossRef]

Lee, R. L.

Lock, J. A.

MacAdam, D. L.

McCartney, E. J.

E. J. McCartney, Optics of the Atmosphere (Wiley, New York, 1976), Table 5.3.

Minnaert, M.

M. Minnaert, Tne Nature of Light and Colour in the Open Air (Dover, New York, 1954), articles 160 and 161.

Naik, Y. G.

Y. G. Naik, “Correlation between optical and dynamic methods of measuring size of water drops in a cloud,” J. Colloid Sci. 9, 393–399 (1954).
[CrossRef]

O’Meara, S. J.

S. J. O’Meara, “Yellowstone: a natural phenomenon,” Sky Telesc. 70, 369 (1985).

Prins, J. A.

J. A. Prins, J. J. M. Reesnick, “Buigingstheorie en trichromatische specificatie van de Regenboogkleuren,” Physica 11, 49–60 (1944).
[CrossRef]

Reesnick, J. J. M.

J. A. Prins, J. J. M. Reesnick, “Buigingstheorie en trichromatische specificatie van de Regenboogkleuren,” Physica 11, 49–60 (1944).
[CrossRef]

Sassen, K.

Simpson, G. C.

G. C. Simpson, “Coronas and iridescent clouds,” Q. J. R. Meteorol. Soc. 38, 291–301 (1912).
[CrossRef]

Stork, D.

D. Falk, D. Brill, D. Stork, Seeing the Light (Harper & Row, New York, 1986), p. 244.

Tricker, R. A. R.

R. A. R. Tricker, Introduction to Meteorological Optics (American Elsevier, New York, 1970), Chap. 5.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), p. 425.

Walker, J. D.

J. D. Walker, “The bright colors in a soap film are a lesson in wave interference,” Sci. Am. 239(3), 232–237 (1978).
[CrossRef]

Wilson, J. G.

J. G. Wilson, “Note on optical methods of measuring the size of small water drops,” Proc. Cambridge Philos. Soc. 32, 493–498 (1936).
[CrossRef]

Wright, W. D.

W. D. Wright, The Measurement of Colour (Hilger, London, 1969), Appendix 2, Table 4.

Ann. Geophys. (1)

J. Bricard, “Reflexion, refraction et diffraction de la lumiere par une goutte d’eau spherique,” Ann. Geophys. 2, 231–248 (1946).

Appl. Opt. (2)

J. Colloid Sci. (1)

Y. G. Naik, “Correlation between optical and dynamic methods of measuring size of water drops in a cloud,” J. Colloid Sci. 9, 393–399 (1954).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (1)

Physica (1)

J. A. Prins, J. J. M. Reesnick, “Buigingstheorie en trichromatische specificatie van de Regenboogkleuren,” Physica 11, 49–60 (1944).
[CrossRef]

Pigment Cell Res. (1)

C. F. Bohren, “Understanding colors in nature,” Pigment Cell Res. 1, 214–222 (1988).
[CrossRef] [PubMed]

Proc. Cambridge Philos. Soc. (1)

J. G. Wilson, “Note on optical methods of measuring the size of small water drops,” Proc. Cambridge Philos. Soc. 32, 493–498 (1936).
[CrossRef]

Q. J. R. Meteorol. Soc. (1)

G. C. Simpson, “Coronas and iridescent clouds,” Q. J. R. Meteorol. Soc. 38, 291–301 (1912).
[CrossRef]

Sci. Am. (1)

J. D. Walker, “The bright colors in a soap film are a lesson in wave interference,” Sci. Am. 239(3), 232–237 (1978).
[CrossRef]

Sky Telesc. (1)

S. J. O’Meara, “Yellowstone: a natural phenomenon,” Sky Telesc. 70, 369 (1985).

Weatherwise (3)

S. D. Gedzelman, “In praise of altocumulus,” Weatherwise 41, 143–149 (1988).
[CrossRef]

H. Lansford, “Shooting the sky,” Weatherwise 35, 73–81 (1982).
[CrossRef]

C. Bohren, “A serendipitous iridescent cloud,” Weatherwise 38, 268–274 (1985).
[CrossRef]

Other (19)

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), p. 425.

Ref. 7, Sections 11.22, 11.3, and 13.41.

M. Minnaert, Tne Nature of Light and Colour in the Open Air (Dover, New York, 1954), articles 160 and 161.

R. A. R. Tricker, Introduction to Meteorological Optics (American Elsevier, New York, 1970), Chap. 5.

R. Greenler, Rainbows, Halos, and Glories (Cambridge U. Press, Cambridge, England, 1980), Chap. 6.

W. J. Humphreys, Physics of the Air (Dover, New York, 1964), Chap. 6.

Ref. 7, p. 425, appears to be in error on this point. The calculations of Prins and Reesinck and of Buchwald are for the rainbow, not for the corona.

R. W. Burnham, R. M. Hanes, C. J. Bartleson, Color: A Guide to Basic Facts and Concepts (Wiley, New York, 1963), pp. 130–133.

W. D. Wright, The Measurement of Colour (Hilger, London, 1969), Appendix 2, Table 4.

D. Falk, D. Brill, D. Stork, Seeing the Light (Harper & Row, New York, 1986), p. 244.

The data of Fig. 6, where different diameter droplets correspond to different red rings in a single corona, resulted from the corona occurring at the edge of the cloud where the droplet size distribution was not homogeneous.

Ref. 26, Table 3.6.

Ref. 1, p. 215.

Ref. 1, p. 214.

Ref. 2, p. 146.

E. J. McCartney, Optics of the Atmosphere (Wiley, New York, 1976), Table 5.3.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Appendix A.

Ref. 18, Fig. 6.3.

Ref. 7, Fig. 52.

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Figures (6)

Fig. 1
Fig. 1

Chromaticity curve for the corona of Eq. (5) in the diffraction model. Point W = (0.333, 0.333) is the equal-energy white point and points R and G are primary spectral red (0.65 μm) and green (0.53 μm). The lines joining R to W and G to W denote various saturations of red and green, respectively. We take the intersections of these lines and the chromaticity curve to represent the red and green corona rings. The grid marks on the chromaticity curve are at intervals of Δv = 10.0 μm deg.

Fig. 2
Fig. 2

Convolved chromaticity curve for the corona in the Mie theory model for (a) D = 6.0 μm, (b) D = 14.0 μm, and (c) D = 21.0 μm. Points W, R, and G are as in Fig. 1. The grid marks on the chromaticity curve are at intervals of Δθ = 1.0° in (a) and Δθ = 0.5° in (b) and (c).

Fig. 3
Fig. 3

Purity of the color of the first four red corona rings, R1 through R4, as a function of water droplet diameter in the Mie theory model.

Fig. 4
Fig. 4

Convolved chromaticity curve for the corona in the Mie theory model for D0 = 14.0 μm and w = 4.0 μm. Points W, R, and G are as in Fig. 1. The grid marks on the chromaticity curve are at intervals of Δθ = 0.5°.

Fig. 5
Fig. 5

Purity of the color of the second through fourth red corona rings, R2 through R4, as a function of the size distribution width w. The solid curve is D0 = 21.0 μm, the dashed curve is D0 = 14.0 μm, and the dot–dash curve is D0 = 8.0 μm.

Fig. 6
Fig. 6

Water droplet diameter as a function of the angular radius θ of the first three red corona rings. The solid curve is the prediction of the Mie theory model and the dashed curve is the prediction of the diffraction model with λG = 0.57 μm. Open circle data points, 22 Aug. 1989; solid circle, 20 Dec. 1989; solid square, 14 Jan. 1990; solid triangle, Ref. 30; open triangle, Ref. 31. The ×’s denote Fraser’s photographic observations of 6 Dec. 1969; open squares, 20 Jan. 1971; pluses, 23 Jan. 1971; stars, 19 Apr. 1971. The a points are Sassen’s photographic observations of 24 Mar. 1976; b, Nov. 1974; c, Feb. 1978; d, June 1977.

Equations (12)

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sin θ N = ( N + 0.22 ) λ G / D ,
r j ( θ ) = d λ I ( D , λ , θ ) r ¯ j ( λ ) b ( λ ) k = 1 3 d λ I ( D , λ , θ ) r ¯ k ( λ ) b ( λ ) ,
I ( D , λ , θ ) = I 0 X 2 D 2 4 R 2 [ J 1 ( X θ ) X θ ] 2 ,
X π D λ ,
r j ( v ) = d λ r ¯ j ( λ ) b ( λ ) [ J 1 ( π v / λ ) π v / λ ] 2 k = 1 3 d λ r ¯ k ( λ ) b ( λ ) [ J 1 ( π v / λ ) π v / λ ] 2 ,
v D θ .
Δ r = [ ( Δ x ) 2 + ( Δ y ) 2 ] 1 / 2
n ( λ ) = 1.3271 + 0.0019 / λ 2 ,
I Mic ( D , λ , θ ) = I 0 D 2 8 X 2 R 2 [ S 1 ( X , θ ) 2 + S 2 ( X , θ ) 2 ] ,
r j conv ( θ ) = - θ max / 2 θ max / 2 d θ ( 1 - θ 2 θ max 2 ) 1 / 2 r j ( θ - θ ) ,
r j ( θ ) = d D d λ N ( D ) I ( D , λ , θ ) r ¯ j ( λ ) b ( λ ) k = 1 3 d D d λ N ( D ) I ( D , λ , θ ) r ¯ k ( λ ) b ( λ ) .
N ( D ) = { N 0 if             D 0 - w / 2 D D 0 + w / 2 , 0 otherwise

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