Rainbows by elliptically deformed drops. II. The appearance of supernumeraries of high-order rainbows in rain showers

Gunther P. Können; James A. Lock

doi:10.1364/AO.56.000G98

Applied Optics
Vol. 56,
Issue 19,
pp. G98-G103
(2017)
•https://doi.org/10.1364/AO.56.000G98

Rainbows by elliptically deformed drops. II. The appearance of supernumeraries of high-order rainbows in rain showers

Gunther P. Können and James A. Lock

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Gunther P. Können and James A. Lock, "Rainbows by elliptically deformed drops. II. The appearance of supernumeraries of high-order rainbows in rain showers," Appl. Opt. 56, G98-G103 (2017)

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Abstract

The appearance of supernumeraries of high-order rainbows in heavy rain showers is explored for rainbows up to order five ( $p = 6$ ). This is done by using a combination of the ray-theory-based first-order Möbius approximation for high-order rainbows with the Airy approximation of the rainbow radiance distribution. We conclude that supernumerary formation of rainbows of order three, four, and five is possible in natural rain showers. Supernumeraries of the third-order and fourth-order rainbows are preferentially formed near the bottom of these rainbows. A strategy for observing supernumeraries of high-order rainbows is proposed.

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Equations (35)

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			Principal Maximum		First Supernumerary		Second Supernumerary
	Descartes Rainbow Scattering Angle	Sun Height Angle with Largest Separation Between Maxima	$a_{eff, 0}$	Angular Separation from Descartes Rainbow Angle	$a_{eff, 1}$	Angular Separation from Principal Maximum	$a_{eff, 2}$	Angular Separation from First Supernumerary
First-order ( $p = 2$ )	137.9°	Rainbow top: 21° (bottom: −21°) ^b	0.14 mm	1.03°	0.22 mm	1.42°	0.25 mm	0.84°
Second-order ( $p = 3$ ) ^c	129.1°	Rainbow top: 50° (bottom: −50°)	0.46 mm	0.78°	0.62 mm	1.21°	0.68 mm	0.76°
Third-order ( $p = 4$ )	41.7°	(Rainbow top: −21°) bottom 21°	0.17 mm	2.29°	0.26 mm	3.18°	0.30 mm	1.88°
Fourth-order ( $p = 5$ )	43.7°	(Rainbow top: −22°) bottom: 22°	0.23 mm	2.35°	0.36 mm	3.25°	0.42 mm	1.93°
Fifth-order ( $p = 6$ )	128.2°	Rainbow top: 26° (bottom: −26°)	0.18 mm	3.37°	0.28 mm	4.67°	0.32 mm	2.77°

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