Abstract

Naturally occurring tertiary rainbows are extraordinarily rare and only a handful of reliable sightings and photographs have been published. Indeed, tertiaries are sometimes assumed to be inherently in visible because of sun glare and strong forward scattering by raindrops. To analyze the natural tertiary’s visibility, we use Lorenz–Mie theory, the Debye series, and a modified geometrical optics model (including both interference and nonspherical drops) to calculate the tertiary’s (1) chromaticity gamuts, (2) luminance contrasts, and (3) color contrasts as seen against dark cloud backgrounds. Results from each model show that natural tertiaries are just visible for some unusual combinations of lighting conditions and raindrop size distributions.

© 2011 Optical Society of America

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  1. T. Bergman, “Von den Erklärungen des Regenbogens,” Der königlich schwedischen Akademie der Wissenschaften Abhandlungen aus der Naturlehre, Haushaltungskunst und Mechanik 21, 231–243 (1762).
  2. C. Hartwell, “Description of a tertiary rainbow,” American Journal of Science and Arts 17 (2nd series), 56–57 (1854).
  3. D. E. Pedgley, “A tertiary rainbow,” Weather 41, 401 (1986).
  4. J. R. Prescott, “Tertiary rainbows,” Phys. World 17, 18 (2004).
  5. J. Heilermann, “Ueber den dritten Regenbogen,” Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht 11, 72–73 (1880).
  6. T. C. Lewis, “A tertiary rainbow,” Nature 32, 523 (1885). Lewis later retracts his identification, noting that the arc’s angular size and color order were consistent with a halo, not a tertiary bow.
    [CrossRef]
  7. J. N. Huyette, “A tertiary rainbow,” Mon. Weather Rev. 32, 325–326 (1904). Huyette in fact does not identify his uncurved “bright streak” (probably a circumhorizontal arc) as a tertiary bow—an editor confidently makes this error in a detailed postscript.
    [CrossRef]
  8. M. Grossmann, E. Schmidt, and A. Haussmann, “Photographic evidence for the third-order rainbow,” Appl. Opt. 50, F134–F141 (2011). Grossman’s photograph was posted online 1 June 2011 at http://atoptics.wordpress.com.
    [CrossRef] [PubMed]
  9. M. Theusner, “Photographic observation of a natural fourth-order rainbow,” Appl. Opt. 50, F129–F133 (2011). Theusner’s photograph of a tertiary-quaternary rainbow pair was posted online 12 June 2011 at http://atoptics.wordpress.com.
    [CrossRef] [PubMed]
  10. R. A. R. Tricker, Introduction to Meteorological Optics(American Elsevier, 1970), p. 57.
  11. J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
    [CrossRef]
  12. W. J. Humphreys, Physics of the Air (McGraw-Hill, 1940), p. 476.
  13. C. B. Boyer, “The tertiary rainbow: An historical account,” Isis 49, 141–154 (1958).
    [CrossRef]
  14. J. D. Walker, “The amateur scientist,” Sci. Am. 239, 185–186 (1978). Here Walker relays Prescott’s possible sighting of a tertiary rainbow and notes that “perhaps under some circumstances rainbows of higher order might be visible.”
  15. R. Greenler, Rainbows, Halos, and Glories (Cambridge, 1980), pp. 6–7.
  16. R. L. Lee, Jr. and A. B. Fraser, The Rainbow Bridge: Rainbows in Art, Myth, and Science (Pennsylvania State U. Press, 2001), pp. 290–291.
  17. C. F. Bohren and A. B. Fraser, “Newton’s zero-order rainbow: Unobservable or nonexistent?,” Am. J. Phys. 59, 325–326(1991).
    [CrossRef]
  18. M. Grossmann, E. Schmidt, and A. Haussmann, “Photographic evidence for the third-order rainbow,” Appl. Opt. 50, F134–F141 (2011). Grossman’s photograph was posted online 1 June 2011 at http://atoptics.wordpress.com.
    [CrossRef] [PubMed]
  19. M. Theusner, “Photographic observation of a natural fourth-order rainbow,” Appl. Opt. 50, F129–F133 (2011). Theusner’s photograph of a tertiary-quaternary rainbow pair was posted online 12 June 2011 at http://atoptics.wordpress.com.
    [CrossRef] [PubMed]
  20. T. Bergman, “Von den Erklärungen des Regenbogens,” Der königlich schwedischen Akademie der Wissenschaften Abhandlungen aus der Naturlehre, Haushaltungskunst und Mechanik 21, 231–243 (1762). Michael Vollmer of the Brandenburg University of Applied Sciences graciously translated this passage.
  21. C. Hartwell, “Description of a tertiary rainbow,” American Journal of Science and Arts 17 (2nd series), 56–57 (1854).
  22. D. E. Pedgley, “A tertiary rainbow,” Weather 41, 401 (1986).
  23. D. E. Pedgley, 35 Thamesmead, Crowmarsh, Oxfordshire OX10 8EY, United Kingdom (personal communication, 2009).
  24. J. R. Prescott, “Tertiary rainbows,” Phys. World 17, 18 (2004).
  25. J. R. Prescott, Physics Department, University of Adelaide, SA 5005, Australia (personal communication, 2010).
  26. D. E. Pedgley, 35 Thamesmead, Crowmarsh, Oxfordshire OX10 8EY, United Kingdom (personal communication, 2009).
  27. J. Heilermann, “Ueber den dritten Regenbogen,” Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht 11, 72–73 (1880). Michael Vollmer of the Brandenburg University of Applied Sciences graciously translated this passage.
  28. L. F. Richardson, “Measurement of water in clouds,” Proceedings of the Royal Society of London, Series A 96, 19–31 (1919).
    [CrossRef]
  29. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981; reprint of 1957 Wiley edition), p. 231. In van de Hulst’s notation (and ours), p is the number of refractions from water to air that occurs before a ray reaches an observer. Thus p=0 indicates light externally reflected and diffracted forward by the droplet. Note that rainbow order n=p−1, where n is the number of internal reflections, so the tertiary with n=3 corresponds to p=4.
  30. K. Sassen, “Angular scattering and rainbow formation in pendant drops,” J. Opt. Soc. Am. 69, 1083–1089 (1979).
    [CrossRef]
  31. A. W. Green, “An approximation for the shapes of large raindrops,” J. Appl. Meteorol. 14, 1578–1583 (1975).
    [CrossRef]
  32. D. S. Langley and P. L. Marston, “Generalized tertiary rainbow of slightly oblate drops: observations with laser illumination,” Appl. Opt. 37, 1520–1526 (1998).
    [CrossRef]
  33. J. A. Lock, C. L. Adler, B. R. Stone, and P. D. Zajak, “Amplification of high-order rainbows of a cylinder with an elliptical cross section,” Appl. Opt. 37, 1527–1533 (1998).
    [CrossRef]
  34. J. Hernández-Andrés, R. L. Lee, Jr., and J. Romero, “Calculating correlated color temperatures across the entire gamut of daylight and skylight chromaticities,” Appl. Opt. 38, 5703–5709 (1999). Data for Fig. 1 are interpolated in h0 from solar-disc irradiance spectra measured by LI-COR Environmental, 4647 Superior Street, Lincoln, Neb. 68504.
    [CrossRef]
  35. D. C. Blanchard, “Raindrop size-distribution in Hawaiian rains,” J. Meteorol. 10, 457–473 (1953). In Fig. 2, the Blanchard DSD data are for his sample number 113, whose maximum drop radius is 0.65 mm.
    [CrossRef]
  36. C. W. Ulbrich, “Natural variations in the analytical form of the raindrop size distribution,” J. Clim. Appl. Meteorol. 22, 1764–1775 (1983). In Fig. 2, the Cb_0 DSD data are for the first thunderstorm rain model listed in Ulbrich’s Table 2.
    [CrossRef]
  37. S. D. Gedzelman, “Simulating rainbows in their atmospheric environment,” Appl. Opt. 47, H176–H181 (2008).
    [CrossRef]
  38. C. F. Bohren and E. E. Clothiaux, Fundamentals of Atmospheric Radiation (Wiley-VCH, 2006), p. 177.
  39. R. L. Lee, Jr., “Mie theory, Airy theory, and the natural rainbow,” Appl. Opt. 37, 1506–1519 (1998).
    [CrossRef]
  40. P. Laven, “Simulation of rainbows, coronas, and glories by use of Mie theory,” Appl. Opt. 42, 436–444 (2003).
    [CrossRef] [PubMed]
  41. For the tertiary rainbow, a local maximum in scattered radiance at θ is equivalent in geometrical optics terms to a minimum deviation angle at 360°−θ.
  42. G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, 1982, 2nd ed.), pp. 158–164.
  43. G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, 1982, 2nd ed.), pp. 306–310.
  44. J. A. Lock, “Theory of the observations made of high-order rainbows from a single water droplet,” Appl. Opt. 26, 5291–5298 (1987).
    [CrossRef] [PubMed]
  45. P. Laven, “Simulation of rainbows, coronas and glories using Mie theory and the Debye series,” J. Quant. Spectrosc. Radiat. Transfer 89, 257–269 (2004). All Debye series calculations were made with Laven’s MiePlot program, which is freely available for download at http://www.philiplaven.com/mieplot.htm.
    [CrossRef]
  46. R. L. Lee, Jr., “Measuring overcast colors with all-sky imaging,” Appl. Opt. 47, H106–H115 (2008).
    [CrossRef] [PubMed]
  47. A quantitative definition of chromaticity gamut is given in R. L. Lee, Jr., “What are ‘all the colors of the rainbow’?,” Appl. Opt. 30, 3401–3407, 3545 (1991).
    [CrossRef] [PubMed]
  48. While light that forms the primary and secondary rainbows (p=2 and 3, respectively) is negligible for θ=37°–43°, we include it here for completeness. Note that Eq.  calculates the C(θ) that results from adding p=4 and 5 rays at a givenθ rather than comparing luminances at adjacent θ. As a result, Fig.  shows the contrast consequences of higher-order scattering.
  49. J. Gorraiz, H. Horvath, and G. Raimann, “Influence of small color differences on the contrast threshold: its application to atmospheric visibility,” Appl. Opt. 25, 2537–2545 (1986).
    [CrossRef] [PubMed]
  50. In order to demonstrate how the tertiary’s visibility separately depends on color and luminance contrast, here we do not use a combined metric for color and luminance differences such as the CIELUV color difference ΔEuv*.
  51. T. Nousiainen and K. Muinonen, “Light scattering by Gaussian, randomly oscillating raindrops,” J. Quant. Spectrosc. Radiat. Transfer 63, 643–666 (1999).
    [CrossRef]
  52. O. N. Ross and S. G. Bradley, “Model for optical forward scattering by nonspherical raindrops,” Appl. Opt. 41, 5130–5141(2002).
    [CrossRef] [PubMed]
  53. A. W. Green, “An approximation for the shapes of large raindrops,” J. Appl. Meteorol. 14, 1578–1583 (1975).
    [CrossRef]
  54. For example, excluding p=0 rays from Fig.  would increase its peak Cb_0 contrast C by <0.2%. Similarly, excluding p=0 rays from Fig.  would increase its peak Cb_0 color difference Δu′v′ by <14% of the JND given there. Both of these changes are negligible visually.
  55. A. B. Fraser, “Inhomogeneities in the color and intensity of the rainbow,” J. Atmos. Sci. 29, 211–212 (1972).
    [CrossRef]
  56. D. E. Pedgley, “A tertiary rainbow,” Weather 41, 401 (1986).
  57. D. E. Pedgley, “A tertiary rainbow,” Weather 41, 401 (1986).
  58. C. Hartwell, “Description of a tertiary rainbow,” American Journal of Science and Arts 17 (2nd series), 56–57 (1854).
  59. J. R. Prescott, “Tertiary rainbows,” Phys. World 17, 18(2004).
  60. G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, 1982, 2nd ed.), pp. 138–139. Figure 11 was developed for a Sony Trinitron display with the following CIE 1976 primaries: u′(red)=0.4289, v′(red)=0.5268, u′(green)=0.1169, v′(green)=0.5594, u′(blue)=0.1721, v′(blue)=0.1744; each color channel has a gray-level gamma of 1.8.
  61. Similar techniques are used in R. J. Kubesh, “Computer display of chromaticity coordinates with the rainbow as an example,” Am. J. Phys. 60, 919–923 (1992).
    [CrossRef]
  62. K. Sassen, “Angular scattering and rainbow formation in pendant drops,” J. Opt. Soc. Am. 69, 1083–1089 (1979).
    [CrossRef]
  63. D. S. Langley and P. L. Marston, “Generalized tertiary rainbow of slightly oblate drops: observations with laser illumination,” Appl. Opt. 37, 1520–1526 (1998).
    [CrossRef]
  64. C. Hartwell, “Description of a tertiary rainbow,” American Journal of Science and Arts 17 (2nd series), 56–57 (1854).
  65. J. R. Prescott, “Tertiary rainbows,” Phys. World 17, 18(2004).

2011 (4)

2008 (2)

2004 (5)

J. R. Prescott, “Tertiary rainbows,” Phys. World 17, 18 (2004).

J. R. Prescott, “Tertiary rainbows,” Phys. World 17, 18 (2004).

P. Laven, “Simulation of rainbows, coronas and glories using Mie theory and the Debye series,” J. Quant. Spectrosc. Radiat. Transfer 89, 257–269 (2004). All Debye series calculations were made with Laven’s MiePlot program, which is freely available for download at http://www.philiplaven.com/mieplot.htm.
[CrossRef]

J. R. Prescott, “Tertiary rainbows,” Phys. World 17, 18(2004).

J. R. Prescott, “Tertiary rainbows,” Phys. World 17, 18(2004).

2003 (1)

2002 (1)

1999 (2)

1998 (4)

1992 (1)

Similar techniques are used in R. J. Kubesh, “Computer display of chromaticity coordinates with the rainbow as an example,” Am. J. Phys. 60, 919–923 (1992).
[CrossRef]

1991 (2)

1987 (1)

1986 (5)

J. Gorraiz, H. Horvath, and G. Raimann, “Influence of small color differences on the contrast threshold: its application to atmospheric visibility,” Appl. Opt. 25, 2537–2545 (1986).
[CrossRef] [PubMed]

D. E. Pedgley, “A tertiary rainbow,” Weather 41, 401 (1986).

D. E. Pedgley, “A tertiary rainbow,” Weather 41, 401 (1986).

D. E. Pedgley, “A tertiary rainbow,” Weather 41, 401 (1986).

D. E. Pedgley, “A tertiary rainbow,” Weather 41, 401 (1986).

1983 (1)

C. W. Ulbrich, “Natural variations in the analytical form of the raindrop size distribution,” J. Clim. Appl. Meteorol. 22, 1764–1775 (1983). In Fig. 2, the Cb_0 DSD data are for the first thunderstorm rain model listed in Ulbrich’s Table 2.
[CrossRef]

1979 (2)

1978 (1)

J. D. Walker, “The amateur scientist,” Sci. Am. 239, 185–186 (1978). Here Walker relays Prescott’s possible sighting of a tertiary rainbow and notes that “perhaps under some circumstances rainbows of higher order might be visible.”

1976 (1)

J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
[CrossRef]

1975 (2)

A. W. Green, “An approximation for the shapes of large raindrops,” J. Appl. Meteorol. 14, 1578–1583 (1975).
[CrossRef]

A. W. Green, “An approximation for the shapes of large raindrops,” J. Appl. Meteorol. 14, 1578–1583 (1975).
[CrossRef]

1972 (1)

A. B. Fraser, “Inhomogeneities in the color and intensity of the rainbow,” J. Atmos. Sci. 29, 211–212 (1972).
[CrossRef]

1958 (1)

C. B. Boyer, “The tertiary rainbow: An historical account,” Isis 49, 141–154 (1958).
[CrossRef]

1953 (1)

D. C. Blanchard, “Raindrop size-distribution in Hawaiian rains,” J. Meteorol. 10, 457–473 (1953). In Fig. 2, the Blanchard DSD data are for his sample number 113, whose maximum drop radius is 0.65 mm.
[CrossRef]

1919 (1)

L. F. Richardson, “Measurement of water in clouds,” Proceedings of the Royal Society of London, Series A 96, 19–31 (1919).
[CrossRef]

1904 (1)

J. N. Huyette, “A tertiary rainbow,” Mon. Weather Rev. 32, 325–326 (1904). Huyette in fact does not identify his uncurved “bright streak” (probably a circumhorizontal arc) as a tertiary bow—an editor confidently makes this error in a detailed postscript.
[CrossRef]

1885 (1)

T. C. Lewis, “A tertiary rainbow,” Nature 32, 523 (1885). Lewis later retracts his identification, noting that the arc’s angular size and color order were consistent with a halo, not a tertiary bow.
[CrossRef]

1880 (2)

J. Heilermann, “Ueber den dritten Regenbogen,” Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht 11, 72–73 (1880).

J. Heilermann, “Ueber den dritten Regenbogen,” Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht 11, 72–73 (1880). Michael Vollmer of the Brandenburg University of Applied Sciences graciously translated this passage.

1854 (4)

C. Hartwell, “Description of a tertiary rainbow,” American Journal of Science and Arts 17 (2nd series), 56–57 (1854).

C. Hartwell, “Description of a tertiary rainbow,” American Journal of Science and Arts 17 (2nd series), 56–57 (1854).

C. Hartwell, “Description of a tertiary rainbow,” American Journal of Science and Arts 17 (2nd series), 56–57 (1854).

C. Hartwell, “Description of a tertiary rainbow,” American Journal of Science and Arts 17 (2nd series), 56–57 (1854).

1762 (2)

T. Bergman, “Von den Erklärungen des Regenbogens,” Der königlich schwedischen Akademie der Wissenschaften Abhandlungen aus der Naturlehre, Haushaltungskunst und Mechanik 21, 231–243 (1762).

T. Bergman, “Von den Erklärungen des Regenbogens,” Der königlich schwedischen Akademie der Wissenschaften Abhandlungen aus der Naturlehre, Haushaltungskunst und Mechanik 21, 231–243 (1762). Michael Vollmer of the Brandenburg University of Applied Sciences graciously translated this passage.

Adler, C. L.

Bergman, T.

T. Bergman, “Von den Erklärungen des Regenbogens,” Der königlich schwedischen Akademie der Wissenschaften Abhandlungen aus der Naturlehre, Haushaltungskunst und Mechanik 21, 231–243 (1762).

T. Bergman, “Von den Erklärungen des Regenbogens,” Der königlich schwedischen Akademie der Wissenschaften Abhandlungen aus der Naturlehre, Haushaltungskunst und Mechanik 21, 231–243 (1762). Michael Vollmer of the Brandenburg University of Applied Sciences graciously translated this passage.

Blanchard, D. C.

D. C. Blanchard, “Raindrop size-distribution in Hawaiian rains,” J. Meteorol. 10, 457–473 (1953). In Fig. 2, the Blanchard DSD data are for his sample number 113, whose maximum drop radius is 0.65 mm.
[CrossRef]

Bohren, C. F.

C. F. Bohren and A. B. Fraser, “Newton’s zero-order rainbow: Unobservable or nonexistent?,” Am. J. Phys. 59, 325–326(1991).
[CrossRef]

C. F. Bohren and E. E. Clothiaux, Fundamentals of Atmospheric Radiation (Wiley-VCH, 2006), p. 177.

Boyer, C. B.

C. B. Boyer, “The tertiary rainbow: An historical account,” Isis 49, 141–154 (1958).
[CrossRef]

Bradley, S. G.

Clothiaux, E. E.

C. F. Bohren and E. E. Clothiaux, Fundamentals of Atmospheric Radiation (Wiley-VCH, 2006), p. 177.

Fraser, A. B.

C. F. Bohren and A. B. Fraser, “Newton’s zero-order rainbow: Unobservable or nonexistent?,” Am. J. Phys. 59, 325–326(1991).
[CrossRef]

A. B. Fraser, “Inhomogeneities in the color and intensity of the rainbow,” J. Atmos. Sci. 29, 211–212 (1972).
[CrossRef]

R. L. Lee, Jr. and A. B. Fraser, The Rainbow Bridge: Rainbows in Art, Myth, and Science (Pennsylvania State U. Press, 2001), pp. 290–291.

Gedzelman, S. D.

Gorraiz, J.

Green, A. W.

A. W. Green, “An approximation for the shapes of large raindrops,” J. Appl. Meteorol. 14, 1578–1583 (1975).
[CrossRef]

A. W. Green, “An approximation for the shapes of large raindrops,” J. Appl. Meteorol. 14, 1578–1583 (1975).
[CrossRef]

Greenler, R.

R. Greenler, Rainbows, Halos, and Glories (Cambridge, 1980), pp. 6–7.

Grossmann, M.

Hartwell, C.

C. Hartwell, “Description of a tertiary rainbow,” American Journal of Science and Arts 17 (2nd series), 56–57 (1854).

C. Hartwell, “Description of a tertiary rainbow,” American Journal of Science and Arts 17 (2nd series), 56–57 (1854).

C. Hartwell, “Description of a tertiary rainbow,” American Journal of Science and Arts 17 (2nd series), 56–57 (1854).

C. Hartwell, “Description of a tertiary rainbow,” American Journal of Science and Arts 17 (2nd series), 56–57 (1854).

Haussmann, A.

Heilermann, J.

J. Heilermann, “Ueber den dritten Regenbogen,” Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht 11, 72–73 (1880). Michael Vollmer of the Brandenburg University of Applied Sciences graciously translated this passage.

J. Heilermann, “Ueber den dritten Regenbogen,” Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht 11, 72–73 (1880).

Hernández-Andrés, J.

Horvath, H.

Humphreys, W. J.

W. J. Humphreys, Physics of the Air (McGraw-Hill, 1940), p. 476.

Huyette, J. N.

J. N. Huyette, “A tertiary rainbow,” Mon. Weather Rev. 32, 325–326 (1904). Huyette in fact does not identify his uncurved “bright streak” (probably a circumhorizontal arc) as a tertiary bow—an editor confidently makes this error in a detailed postscript.
[CrossRef]

Kubesh, R. J.

Similar techniques are used in R. J. Kubesh, “Computer display of chromaticity coordinates with the rainbow as an example,” Am. J. Phys. 60, 919–923 (1992).
[CrossRef]

Langley, D. S.

Laven, P.

P. Laven, “Simulation of rainbows, coronas and glories using Mie theory and the Debye series,” J. Quant. Spectrosc. Radiat. Transfer 89, 257–269 (2004). All Debye series calculations were made with Laven’s MiePlot program, which is freely available for download at http://www.philiplaven.com/mieplot.htm.
[CrossRef]

P. Laven, “Simulation of rainbows, coronas, and glories by use of Mie theory,” Appl. Opt. 42, 436–444 (2003).
[CrossRef] [PubMed]

Lee, R. L.

Lewis, T. C.

T. C. Lewis, “A tertiary rainbow,” Nature 32, 523 (1885). Lewis later retracts his identification, noting that the arc’s angular size and color order were consistent with a halo, not a tertiary bow.
[CrossRef]

Lock, J. A.

Marston, P. L.

Muinonen, K.

T. Nousiainen and K. Muinonen, “Light scattering by Gaussian, randomly oscillating raindrops,” J. Quant. Spectrosc. Radiat. Transfer 63, 643–666 (1999).
[CrossRef]

Nousiainen, T.

T. Nousiainen and K. Muinonen, “Light scattering by Gaussian, randomly oscillating raindrops,” J. Quant. Spectrosc. Radiat. Transfer 63, 643–666 (1999).
[CrossRef]

Pedgley, D. E.

D. E. Pedgley, “A tertiary rainbow,” Weather 41, 401 (1986).

D. E. Pedgley, “A tertiary rainbow,” Weather 41, 401 (1986).

D. E. Pedgley, “A tertiary rainbow,” Weather 41, 401 (1986).

D. E. Pedgley, “A tertiary rainbow,” Weather 41, 401 (1986).

D. E. Pedgley, 35 Thamesmead, Crowmarsh, Oxfordshire OX10 8EY, United Kingdom (personal communication, 2009).

D. E. Pedgley, 35 Thamesmead, Crowmarsh, Oxfordshire OX10 8EY, United Kingdom (personal communication, 2009).

Prescott, J. R.

J. R. Prescott, “Tertiary rainbows,” Phys. World 17, 18(2004).

J. R. Prescott, “Tertiary rainbows,” Phys. World 17, 18 (2004).

J. R. Prescott, “Tertiary rainbows,” Phys. World 17, 18(2004).

J. R. Prescott, “Tertiary rainbows,” Phys. World 17, 18 (2004).

J. R. Prescott, Physics Department, University of Adelaide, SA 5005, Australia (personal communication, 2010).

Raimann, G.

Richardson, L. F.

L. F. Richardson, “Measurement of water in clouds,” Proceedings of the Royal Society of London, Series A 96, 19–31 (1919).
[CrossRef]

Romero, J.

Ross, O. N.

Sassen, K.

Schmidt, E.

Stiles, W. S.

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, 1982, 2nd ed.), pp. 158–164.

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, 1982, 2nd ed.), pp. 306–310.

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, 1982, 2nd ed.), pp. 138–139. Figure 11 was developed for a Sony Trinitron display with the following CIE 1976 primaries: u′(red)=0.4289, v′(red)=0.5268, u′(green)=0.1169, v′(green)=0.5594, u′(blue)=0.1721, v′(blue)=0.1744; each color channel has a gray-level gamma of 1.8.

Stone, B. R.

Theusner, M.

Tricker, R. A. R.

R. A. R. Tricker, Introduction to Meteorological Optics(American Elsevier, 1970), p. 57.

Ulbrich, C. W.

C. W. Ulbrich, “Natural variations in the analytical form of the raindrop size distribution,” J. Clim. Appl. Meteorol. 22, 1764–1775 (1983). In Fig. 2, the Cb_0 DSD data are for the first thunderstorm rain model listed in Ulbrich’s Table 2.
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981; reprint of 1957 Wiley edition), p. 231. In van de Hulst’s notation (and ours), p is the number of refractions from water to air that occurs before a ray reaches an observer. Thus p=0 indicates light externally reflected and diffracted forward by the droplet. Note that rainbow order n=p−1, where n is the number of internal reflections, so the tertiary with n=3 corresponds to p=4.

Walker, J. D.

J. D. Walker, “The amateur scientist,” Sci. Am. 239, 185–186 (1978). Here Walker relays Prescott’s possible sighting of a tertiary rainbow and notes that “perhaps under some circumstances rainbows of higher order might be visible.”

J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
[CrossRef]

Wyszecki, G.

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, 1982, 2nd ed.), pp. 306–310.

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, 1982, 2nd ed.), pp. 138–139. Figure 11 was developed for a Sony Trinitron display with the following CIE 1976 primaries: u′(red)=0.4289, v′(red)=0.5268, u′(green)=0.1169, v′(green)=0.5594, u′(blue)=0.1721, v′(blue)=0.1744; each color channel has a gray-level gamma of 1.8.

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, 1982, 2nd ed.), pp. 158–164.

Zajak, P. D.

Am. J. Phys. (3)

J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
[CrossRef]

C. F. Bohren and A. B. Fraser, “Newton’s zero-order rainbow: Unobservable or nonexistent?,” Am. J. Phys. 59, 325–326(1991).
[CrossRef]

Similar techniques are used in R. J. Kubesh, “Computer display of chromaticity coordinates with the rainbow as an example,” Am. J. Phys. 60, 919–923 (1992).
[CrossRef]

American Journal of Science and Arts (4)

C. Hartwell, “Description of a tertiary rainbow,” American Journal of Science and Arts 17 (2nd series), 56–57 (1854).

C. Hartwell, “Description of a tertiary rainbow,” American Journal of Science and Arts 17 (2nd series), 56–57 (1854).

C. Hartwell, “Description of a tertiary rainbow,” American Journal of Science and Arts 17 (2nd series), 56–57 (1854).

C. Hartwell, “Description of a tertiary rainbow,” American Journal of Science and Arts 17 (2nd series), 56–57 (1854).

Appl. Opt. (16)

J. Gorraiz, H. Horvath, and G. Raimann, “Influence of small color differences on the contrast threshold: its application to atmospheric visibility,” Appl. Opt. 25, 2537–2545 (1986).
[CrossRef] [PubMed]

J. A. Lock, “Theory of the observations made of high-order rainbows from a single water droplet,” Appl. Opt. 26, 5291–5298 (1987).
[CrossRef] [PubMed]

A quantitative definition of chromaticity gamut is given in R. L. Lee, Jr., “What are ‘all the colors of the rainbow’?,” Appl. Opt. 30, 3401–3407, 3545 (1991).
[CrossRef] [PubMed]

R. L. Lee, Jr., “Mie theory, Airy theory, and the natural rainbow,” Appl. Opt. 37, 1506–1519 (1998).
[CrossRef]

D. S. Langley and P. L. Marston, “Generalized tertiary rainbow of slightly oblate drops: observations with laser illumination,” Appl. Opt. 37, 1520–1526 (1998).
[CrossRef]

D. S. Langley and P. L. Marston, “Generalized tertiary rainbow of slightly oblate drops: observations with laser illumination,” Appl. Opt. 37, 1520–1526 (1998).
[CrossRef]

J. Hernández-Andrés, R. L. Lee, Jr., and J. Romero, “Calculating correlated color temperatures across the entire gamut of daylight and skylight chromaticities,” Appl. Opt. 38, 5703–5709 (1999). Data for Fig. 1 are interpolated in h0 from solar-disc irradiance spectra measured by LI-COR Environmental, 4647 Superior Street, Lincoln, Neb. 68504.
[CrossRef]

J. A. Lock, C. L. Adler, B. R. Stone, and P. D. Zajak, “Amplification of high-order rainbows of a cylinder with an elliptical cross section,” Appl. Opt. 37, 1527–1533 (1998).
[CrossRef]

O. N. Ross and S. G. Bradley, “Model for optical forward scattering by nonspherical raindrops,” Appl. Opt. 41, 5130–5141(2002).
[CrossRef] [PubMed]

P. Laven, “Simulation of rainbows, coronas, and glories by use of Mie theory,” Appl. Opt. 42, 436–444 (2003).
[CrossRef] [PubMed]

R. L. Lee, Jr., “Measuring overcast colors with all-sky imaging,” Appl. Opt. 47, H106–H115 (2008).
[CrossRef] [PubMed]

S. D. Gedzelman, “Simulating rainbows in their atmospheric environment,” Appl. Opt. 47, H176–H181 (2008).
[CrossRef]

M. Theusner, “Photographic observation of a natural fourth-order rainbow,” Appl. Opt. 50, F129–F133 (2011). Theusner’s photograph of a tertiary-quaternary rainbow pair was posted online 12 June 2011 at http://atoptics.wordpress.com.
[CrossRef] [PubMed]

M. Theusner, “Photographic observation of a natural fourth-order rainbow,” Appl. Opt. 50, F129–F133 (2011). Theusner’s photograph of a tertiary-quaternary rainbow pair was posted online 12 June 2011 at http://atoptics.wordpress.com.
[CrossRef] [PubMed]

M. Grossmann, E. Schmidt, and A. Haussmann, “Photographic evidence for the third-order rainbow,” Appl. Opt. 50, F134–F141 (2011). Grossman’s photograph was posted online 1 June 2011 at http://atoptics.wordpress.com.
[CrossRef] [PubMed]

M. Grossmann, E. Schmidt, and A. Haussmann, “Photographic evidence for the third-order rainbow,” Appl. Opt. 50, F134–F141 (2011). Grossman’s photograph was posted online 1 June 2011 at http://atoptics.wordpress.com.
[CrossRef] [PubMed]

Der königlich schwedischen Akademie der Wissenschaften Abhandlungen aus der Naturlehre, Haushaltungskunst und Mechanik (2)

T. Bergman, “Von den Erklärungen des Regenbogens,” Der königlich schwedischen Akademie der Wissenschaften Abhandlungen aus der Naturlehre, Haushaltungskunst und Mechanik 21, 231–243 (1762).

T. Bergman, “Von den Erklärungen des Regenbogens,” Der königlich schwedischen Akademie der Wissenschaften Abhandlungen aus der Naturlehre, Haushaltungskunst und Mechanik 21, 231–243 (1762). Michael Vollmer of the Brandenburg University of Applied Sciences graciously translated this passage.

Isis (1)

C. B. Boyer, “The tertiary rainbow: An historical account,” Isis 49, 141–154 (1958).
[CrossRef]

J. Appl. Meteorol. (2)

A. W. Green, “An approximation for the shapes of large raindrops,” J. Appl. Meteorol. 14, 1578–1583 (1975).
[CrossRef]

A. W. Green, “An approximation for the shapes of large raindrops,” J. Appl. Meteorol. 14, 1578–1583 (1975).
[CrossRef]

J. Atmos. Sci. (1)

A. B. Fraser, “Inhomogeneities in the color and intensity of the rainbow,” J. Atmos. Sci. 29, 211–212 (1972).
[CrossRef]

J. Clim. Appl. Meteorol. (1)

C. W. Ulbrich, “Natural variations in the analytical form of the raindrop size distribution,” J. Clim. Appl. Meteorol. 22, 1764–1775 (1983). In Fig. 2, the Cb_0 DSD data are for the first thunderstorm rain model listed in Ulbrich’s Table 2.
[CrossRef]

J. Meteorol. (1)

D. C. Blanchard, “Raindrop size-distribution in Hawaiian rains,” J. Meteorol. 10, 457–473 (1953). In Fig. 2, the Blanchard DSD data are for his sample number 113, whose maximum drop radius is 0.65 mm.
[CrossRef]

J. Opt. Soc. Am. (2)

J. Quant. Spectrosc. Radiat. Transfer (2)

T. Nousiainen and K. Muinonen, “Light scattering by Gaussian, randomly oscillating raindrops,” J. Quant. Spectrosc. Radiat. Transfer 63, 643–666 (1999).
[CrossRef]

P. Laven, “Simulation of rainbows, coronas and glories using Mie theory and the Debye series,” J. Quant. Spectrosc. Radiat. Transfer 89, 257–269 (2004). All Debye series calculations were made with Laven’s MiePlot program, which is freely available for download at http://www.philiplaven.com/mieplot.htm.
[CrossRef]

Mon. Weather Rev. (1)

J. N. Huyette, “A tertiary rainbow,” Mon. Weather Rev. 32, 325–326 (1904). Huyette in fact does not identify his uncurved “bright streak” (probably a circumhorizontal arc) as a tertiary bow—an editor confidently makes this error in a detailed postscript.
[CrossRef]

Nature (1)

T. C. Lewis, “A tertiary rainbow,” Nature 32, 523 (1885). Lewis later retracts his identification, noting that the arc’s angular size and color order were consistent with a halo, not a tertiary bow.
[CrossRef]

Phys. World (4)

J. R. Prescott, “Tertiary rainbows,” Phys. World 17, 18 (2004).

J. R. Prescott, “Tertiary rainbows,” Phys. World 17, 18 (2004).

J. R. Prescott, “Tertiary rainbows,” Phys. World 17, 18(2004).

J. R. Prescott, “Tertiary rainbows,” Phys. World 17, 18(2004).

Proceedings of the Royal Society of London, Series A (1)

L. F. Richardson, “Measurement of water in clouds,” Proceedings of the Royal Society of London, Series A 96, 19–31 (1919).
[CrossRef]

Sci. Am. (1)

J. D. Walker, “The amateur scientist,” Sci. Am. 239, 185–186 (1978). Here Walker relays Prescott’s possible sighting of a tertiary rainbow and notes that “perhaps under some circumstances rainbows of higher order might be visible.”

Weather (4)

D. E. Pedgley, “A tertiary rainbow,” Weather 41, 401 (1986).

D. E. Pedgley, “A tertiary rainbow,” Weather 41, 401 (1986).

D. E. Pedgley, “A tertiary rainbow,” Weather 41, 401 (1986).

D. E. Pedgley, “A tertiary rainbow,” Weather 41, 401 (1986).

Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht (2)

J. Heilermann, “Ueber den dritten Regenbogen,” Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht 11, 72–73 (1880). Michael Vollmer of the Brandenburg University of Applied Sciences graciously translated this passage.

J. Heilermann, “Ueber den dritten Regenbogen,” Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht 11, 72–73 (1880).

Other (16)

R. A. R. Tricker, Introduction to Meteorological Optics(American Elsevier, 1970), p. 57.

R. Greenler, Rainbows, Halos, and Glories (Cambridge, 1980), pp. 6–7.

R. L. Lee, Jr. and A. B. Fraser, The Rainbow Bridge: Rainbows in Art, Myth, and Science (Pennsylvania State U. Press, 2001), pp. 290–291.

W. J. Humphreys, Physics of the Air (McGraw-Hill, 1940), p. 476.

For example, excluding p=0 rays from Fig.  would increase its peak Cb_0 contrast C by <0.2%. Similarly, excluding p=0 rays from Fig.  would increase its peak Cb_0 color difference Δu′v′ by <14% of the JND given there. Both of these changes are negligible visually.

C. F. Bohren and E. E. Clothiaux, Fundamentals of Atmospheric Radiation (Wiley-VCH, 2006), p. 177.

For the tertiary rainbow, a local maximum in scattered radiance at θ is equivalent in geometrical optics terms to a minimum deviation angle at 360°−θ.

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, 1982, 2nd ed.), pp. 158–164.

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, 1982, 2nd ed.), pp. 306–310.

D. E. Pedgley, 35 Thamesmead, Crowmarsh, Oxfordshire OX10 8EY, United Kingdom (personal communication, 2009).

J. R. Prescott, Physics Department, University of Adelaide, SA 5005, Australia (personal communication, 2010).

D. E. Pedgley, 35 Thamesmead, Crowmarsh, Oxfordshire OX10 8EY, United Kingdom (personal communication, 2009).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981; reprint of 1957 Wiley edition), p. 231. In van de Hulst’s notation (and ours), p is the number of refractions from water to air that occurs before a ray reaches an observer. Thus p=0 indicates light externally reflected and diffracted forward by the droplet. Note that rainbow order n=p−1, where n is the number of internal reflections, so the tertiary with n=3 corresponds to p=4.

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, 1982, 2nd ed.), pp. 138–139. Figure 11 was developed for a Sony Trinitron display with the following CIE 1976 primaries: u′(red)=0.4289, v′(red)=0.5268, u′(green)=0.1169, v′(green)=0.5594, u′(blue)=0.1721, v′(blue)=0.1744; each color channel has a gray-level gamma of 1.8.

While light that forms the primary and secondary rainbows (p=2 and 3, respectively) is negligible for θ=37°–43°, we include it here for completeness. Note that Eq.  calculates the C(θ) that results from adding p=4 and 5 rays at a givenθ rather than comparing luminances at adjacent θ. As a result, Fig.  shows the contrast consequences of higher-order scattering.

In order to demonstrate how the tertiary’s visibility separately depends on color and luminance contrast, here we do not use a combined metric for color and luminance differences such as the CIELUV color difference ΔEuv*.

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Figures (11)

Fig. 1
Fig. 1

Normalized spectral irradiances of the sun’s disc measured at solar elevation angle h 0 = 7.4 ° , the same h 0 as for Pedgley’s tertiary rainbow observation. This spectral illuminant is used in calculating Figs. 3, 4, 5, 6, 7, 8, 9, 10, 11, and its CIE 1976 UCS coordinates are u = 0.2532 , v = 0.5288 .

Fig. 2
Fig. 2

Raindrop number densities N ( r EV ) form drop-size distributions (DSDs) as functions of equivalent-volume radius r EV for orographic rain and thunderstorm rain. These two DSDs are used in calculating Figs. 3, 4, 5, 6, 7, 8, 9, 10, 11.

Fig. 3
Fig. 3

Portion of the CIE 1976 UCS diagram, showing the chromaticity coordinates of Fig. 1’s illuminant (marked with an ×) and u ( θ ) , v ( θ ) chromaticity curves as functions of scattering angle θ for tertiary rainbows as predicted by Lorenz–Mie theory for spher ical raindrops. All chromaticities are calculated using (1) Riemann sums from 400 700 nm in 1 nm steps, (2) the DSDs shown in Fig. 2, and (3) a completely black background. The horizontal line at lower left is a typical MacAdam u JND for nearby chromaticities.

Fig. 4
Fig. 4

Chromaticity curves as functions of scattering angle θ for tertiary rainbows as predicted by the Debye series for spherical raindrops. All chromaticities are calculated using (1) Riemann sums from 400 700 nm in 5 nm steps, (2) the DSDs shown in Fig. 2, and (3) a bluish cloud background spectrum L OVC added to the rainbow spectra using a relative weight w ( L OVC ) = 0.025 .

Fig. 5
Fig. 5

Luminance contrast C ( θ ) for tertiary rainbows as predicted by the Debye series for spherical raindrops. All model parameters are as in Fig. 4; the nominal threshold contrast C thresh = 0.02 .

Fig. 6
Fig. 6

Color difference Δ u v ( θ ) for tertiary rainbows against their visual background as predicted by the Debye series for spher ical raindrops. All model parameters are as in Fig. 4; the JND or threshold Δ u v = 0.001285 is a typical MacAdam semimajor axis for nearby chromaticities.

Fig. 7
Fig. 7

Chromaticity curves as functions of scattering angle θ for tertiary rainbows as predicted by our modified geometrical optics (GO) model for oblate spheroidal raindrops. All chromaticities are calculated using (1) Riemann sums from 400 700 nm in 10 nm steps, (2) the DSDs shown in Fig. 2, (3) rainbow clock angle α = 0 ° 15 ° , and (4) a bluish cloud background spectrum L OVC added to the rainbow spectra using a relative weight w ( L OVC ) = 0.025 .

Fig. 8
Fig. 8

All GO model parameters are as in Fig. 7, except that α = 15 ° 30 ° .

Fig. 9
Fig. 9

Luminance contrast C ( θ ) for tertiary rainbows as predicted by our modified GO model for oblate spheroidal raindrops. All model parameters are as in Fig. 8; the nominal threshold contrast C thresh = 0.02 .

Fig. 10
Fig. 10

Color difference Δ u v ( θ ) for tertiary rainbows against their visual background as predicted by our modified GO model for oblate spheroidal raindrops. All model parameters are as in Fig. 8; the threshold Δ u v = 0.001285 .

Fig. 11
Fig. 11

Maps of tertiary rainbow colors versus θ as predicted by (a) Mie and (b) Debye theories for spherical raindrops, and (c) our modified GO model for oblate spheroidal raindrops at α = 0 ° 15 ° . All scattering models use the Cb_0 DSD. In (b) and (c), diffuse light from a bluish cloud background spectrum L OVC is weighted by w ( L OVC ) = 0.025 .

Equations (1)

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C ( θ ) = ( L v ( p = 4 , 5 ) L v ( other ) ) / L v ( other ) ,

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