Rather than combine color and luminance differences in a single
color-difference measure, I show them separately here. Although rainbow
observers cannot make this separation, it does let me address more readily
the issues raised in Refs. 6, 7, 9, and 13. Optically
speaking, cloudbows and fogbows differ very little, so the terms can be used
See Ref. 9, p. 247.
Similar Airy underestimates of intensities in Alexander’s dark band
are evident in Ref. 7, p. 1073 (Fig.
G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data
and Formulae, 2nd ed.
York, 1982), pp.
Ref. 26, pp. 306–309.
D. K. Lynch, W. Livingston, Color and Light in Nature
(Cambridge U. Press,
Cambridge, 1995), p.
119 (Fig. 4.10A). Also see R. A. Anthes, J. J. Cahir, A. B. Fraser, H. A. Panofsky, The Atmosphere, 3rd ed.
Ohio, 1981), Plate 19b, opposite p.
Eliminating deviation angles outside the primary where the Mie and the Airy
150-μm chromaticities diverge noticeably (θ <
137.8°) reduces Δu′, v′¯ to only 0.008901. That still exceeds the
50-μm cloud drop’s Δu′, v′¯ of 0.00571.
Perpendicular (⊥) and parallel (|) directions here are
measured with respect to the scattering plane defined by Sun, water drop,
and observer. This plane’s orientation changes around the rainbow
H. C. van de Hulst, Light Scattering by Small Particles
York, 1981; reprint of 1957 Wiley edition), p.
W. J. Humphreys, Physics of the Air
York, 1964; reprint of 1940 McGraw-Hill
edition), pp. 491–494.
R. A. R. Tricker, Introduction to Meteorological Optics
(American Elsevier, New
York, 1970), pp.
Ref. 11, p. 1963.
C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small
New York, 1983), pp.
Ref. 7, p. 1073 (Fig. 3). Note that Fig. 1’s Mie curve
includes small-scale structure due to external reflections, whereas
Nussenzveig’s figure does not.
Ref. 17, pp. 300–304.
Exceptions include bows seen from mountains, hills, and airplanes in
E. J. McCartney, Optics of the Atmosphere: Scattering by Molecules and
New York, 1976), pp.
163, 170 (Figs. 3.19 and
Although Nussenzveig notes in passing that monodisperse ripples will be
averaged out “over a range of size parameters,” he does not
dwell on the point (Ref. 7, p.
R. L. Lee, A. B. Fraser, The Rainbow Bridge: Rainbows in Art, Myth, and
Science (Penn State Press,
University Park, Pa., to be published), Figs.
8-22 and 8-23.
Consistent with my definition of the natural rainbow, naturalistic here means
“as seen in naturally occurring polydisperse bows.”
Ref. 6, p. 70.
H. M. Nussenzveig, “The theory of the
rainbow,” in Atmospheric Phenomena
Francisco, 1980), pp.
G. B. Airy, “On the intensity of light in the
neighbourhood of a caustic,” Trans.
Cambridge Philos. Soc.6, 379–403
(1838). Airy read his paper before the Society in
May 1836 and March 1838. Remarkably (at least from the biased standpoint of
atmospheric optics), Airy makes no mention of this signal accomplishment in his
Autobiography (Cambridge U.
1896). Instead, Airy’s memorable events from
1836–1838 include his improved filing system for Greenwich
Observatory’s astronomical papers!
C. B. Boyer, The Rainbow: From Myth to Mathematics
(Princeton U. Press,
Princeton, N.J., 1987; reprint
of 1959 Thomas Yoseloff edition), pp.
Ref. 2, p. 313.
All chromaticity and luminance differences are calculated with the
real-number data that underlie Figs. 14–17.
In a personal communication,
G. P. Können (Royal Netherlands Meteorological Institute, De
Bilt, The Netherlands) kindly extended Ref. 11’s
mathematics to include both polarizations of the Airy
Können and de Boer clearly show this phase relationship (Ref. 11, p. 1964).
Ref. 6, p. 70.