The Rainbow

1. Emma Vander Ende MAT 194 Northern Kentucky University Decartes René Decartes, the French mathematician and philosopher, was the first to offer an actual theoretical representation of the rainbow. in the form of a short essay. Decartes hypothesized that light enters raindrops, is inflected at least once within the raindrop, thus bending the light, and then exits. His approach compared the sines of the incidence and refraction angles of the rainbow, and he was accused of plagiarism for this method, since a Dutch mathematician by the name of Willebrord Snellius had developed a similar law (now known as Snell’s Law; sometimes the Snell-Decartes Law) using a different trigonometric function, and Decartes could not properly support his formulations. Nevertheless, Decartes’s model remains an extremely important step despite the in studying the science and mathematics of rainbows. Art and Myth Rainbows have fascinated human beings since they were first glimpsed. A beautiful arc of color, it has been the object of mythology and art for ages. Perhaps most well-known for its place in Irish mythology as hiding a Leprechaun’s pot of gold at the end of the rainbow, the rainbow has become both a playful and political symbol. In Greek and Roman mythology, the goddess Iris is known as the goddess of the rainbow, and it is said that the rainbow represents the bridge from this life to the afterlife (Raymond and Fraser, 2001). The rainbow has been mentioned in many religions; this is not surprising, given the mysterious and otherworldly nature of the rainbow and the fascination it holds for human eyes. It is perhaps not a great stretch of the imagination to imagine it as belonging to the divine. Scientific thinking about rainbows began with the ancient Greek scholars, who took great interest in the natural world and sought to explain it. Early philosophers such as Homer and Hesiod thought the rainbow to be a thing of divinity. One of Thales’s (who held the belief that water is central to all natural phenomena) students, Anaximenes, first recognized its relationship to the sun. He was also the first to recognize that bending of light creates the rainbow, although he did not understand why (Boyer, 1987). The Rainbow • The tantalizing relationship between light, color, and mathematics Modern Symbolism Today, the rainbow flag stands for peace, equality, and diversity. It is used not only as a peace flag, but as the symbol chosen by the GLBTQ community to stand for equality and gay rights. It has been used as a symbol for diversity due to it having the full spectrum of color, an analogy for the diversity in human beings and in human culture. René Decartes’s sketch of light refraction involved with rainbows.

2. Airy’s rainbow integral is perhaps the most important mathematical insight into the rainbow to date. Closely related to this integral is the nature of the geometric caustic (figure 3). A optical caustic is an “envelope” of light rays that have been bent by a curved surface; this is what happens inside the water drops in a rainbow (notice in figure 2, the red circle; this is an example of caustic reflection). The results of Airy’s integral showed that the point of greatest intensity along the rainbow did not occur at the geometric caustic, or the point where the angle of incoming light had the least deviation from the ray in Cartesian coordinates, but on the external sides of the caustic. Airy’s integral has been tested experimentally, and the results have bolstered Airy’s arguments. It was later that another mathematician, George Stokes, developed a more efficient way of calculating the position of many more “bands” in the rainbow than Airy’s integral. He derived a “...formula from the wave-front equation, y = kx3, the expression giving the angular deviation θ (between the emergent ray in question and the least deviated ray) in terms of the angle of inclination J of the ray, the wave length λ of the light, and the radius R of the drop, and the quantity which Airy had designated by m” (Boyer, 1987). It is perhaps interesting to note that water drops in rainbow theory have been assumed to be spherical. High speed photography has, however, shown that deviation in rain drop shape does exist, with larger droplets tending to flatten out, while small drops are kept spherical due to the cohesion of the water molecules. (Boyer, 1987) • The Mathematics • The different colors of the rainbow come from the different frequencies of light. Light is, actually, a spectrum of color, ranging from red to violet, with all the major colors in between. The difference in frequency and wavelength is directly related to the color we perceive; with a different wave frequency, the light will refract at a different angle from the raindrop. What we perceive as a band of color is actually a continuous spectrum. However, the precise position of these “bands” is determined by the size of the raindrop, and therefore the appearance varies slightly. Rainbows also do not have fixed positions in the sky (so there is no “end” to follow!); a rainbow’s position in the sky is relative to the observer’s position and the position of the sun. • A major flaw in Decartes’s early model of the rainbow is that it considered only a single ray. It turns out that the interaction of rays of light (wave interference) is important. The search for mathematics which explain the nature of the rainbow has been long, and key players have been among the most famous of mathematicians, including Isaac Newton. Much of the mathematics involved in rainbows has evolved from mathematics discovered for light in general, but some contributions from mathematicians such as Airy have been specifically for the rainbow. Airy’s rainbow integral, written today as: • describes the intensity of light as it relates to a point on the rainbow (m is the parameter that determines the departure angle of a ray) (Boyer, 1987). • Airy’s model does have a flaw, though; it fails when the size of the raindrops are between 10 and 50 microns, the size that is most common in clouds. Recognizing this, W. V. R. Malkus, another mathematician, and his group of research associates, “...substituted for the approximate wave-front equation previously used a more precise parametric form, • Figure 2. wave interference; view the circle as a single • raindrop, and the lines as rays of light from the sun. • Figure 1. reflection of light • Figure 3. optical caustic through a glass of water

3. Lunar Rainbows and White Rainbows The formation of rainbows is not dependent on the light of sun. While not as intense as the light of the sun, the light of the moon (the light of sun reflected off of the moon) is sufficient enough to generate a rainbow. They are much weaker than other rainbows, and thus can only be seen when the moon is full. They are normally pale, and thus have sometimes been confused with the white rainbow, which forms when the raindrops are very small. The white rainbow has the appearance of a single white band which is much thicker than ordinary rainbows and has an orange hue on the inside of the bow, with a blue hue on the outside. This phenomenon agrees with the earlier calculations for small drops. With ordinary rainbows, the key angle (related to the radius) is 42º, but with white rainbows, this angle is less; this explains the positioning of the few colors in the white rainbow. where sin(g) = m sin(q) and H = 4m(1 - cos(q)) - (1 - cos(g)) and m is the the index of refraction, g is the angle of incidence, q the angle of refraction, and a the radius of a spherical drop” (Boyer, 1987.) The Circular Rainbow All rainbows are, in fact, circular; the horizon normally blocks the full circle from our view. It is very rare to see a circular rainbow in nature, however, and they are only seen around the sun (or moon) as in the picture below. They can also be seen from above; sometimes pilots are able to see circular rainbows below their airplanes. They are not halos, although it might be easy to mistake them as such. It is relatively easy to create a circular rainbow artificially. One method is to take a garden hose (with a mist setting) in mid-to-late afternoon sun, and spray the air until a rainbow is visible. You can then follow the circumference of the rainbow by moving the hose. The effect is that of tracing the whole circular rainbow. The Double Rainbow A double rainbow forms when light is reflected multiple times in a water droplet and exits at a different angle from a normal rainbow (50-52º; normal rainbows have an exit angle of 42º); this produces a secondary, fainter rainbow. This phenomenon is fairly common (if you haven’t seen it, drop what you’re doing next time the sun shines after it rains and go look for one!). Interestingly, the order of the colors in the secondary rainbow is the inverse of the first rainbow (see picture below). Additional rainbows are not impossible, but are simply too faint for the eye to see. Above: a picture of a lunar rainbow.

4. Sun Dogs Sun Dogs are bright spots that appear around the sun when light passes through ice crystals (usually in cirrus clouds), They are named Sun Dogs because they are “loyal” to the sun, as dogs are loyal to humans. They form in pairs, with a 22-degree angle halo around the sun. Due to the conditions that Sun Dogs form in, they have been used as an indicator of rain sine the time of the Greeks. Soap Bubbles Rainbows sometimes occur in soap bubbles and other surfaces such as pools of oil and the surfaces of CDs. In the case of soap bubbles, the thickness of the bubble wall changes with its area, and this thickness determines which wavelengths of light are absorbed and reflected. Thus, you can get beautiful images such as the one below. Rainbows on Titan Amazingly, rainbows may be possible on Titan, one of Saturn’s moons. Titan is abundant in both liquid and methane in the atmosphere, and scientists are finding signs that it may be raining on Titan. Since a rainbow is formed by the interaction of light and clear droplets of liquid - on Earth, water, but methane is also transparent - there is no reason that a rainbow could not form on Titan. According to one scientist, Les Crowley, “A methane rainbow would be larger than a water rainbow, with a primary radius of at least 49º for metane vs 42.5º for water...The order of the colors, however, wold be the same: blue on the inside and red on the outside, with an overall hint of orange caused by Titan’s orange sky.”[3] “For poets and fabulists the rainbow has served as a ubiquitous source of inspiration, but mathematics also has given to the bow a “beauty bare” which only the deeply initiated can fully appreciate.” - Boyer, 1987. References and Works Cited: •Boyer, C.B.. (1987). The Rainbow: From Myth to Mathematics. Princeton, New Jersey: Princeton University Press. •Raymond, L. L. Jr., Fraser A.B.. (2001). The Rainbow Bridge: Rainbows in Art, Myth, and Science. University Park, Pennsylvania: Pennsylvania State University Press. •Minneart, M.. (1954). Light & Color. New York, N.Y., Dover Publications. •Field, Tom. "Rainbow Physics: How Rainbows Form (Photography)." Www.PhotoCentric.Net. Web. 30 Apr. 2011. <http://www.photocentric.net/rainbow_physics.htm>. •Sawicki, Pawel, and Mikolaj Sawicki. "Supernumerary Rainbows." 19 July 1999. Web. 30 Apr. 2011. <http://www.jal.cc.il.us/~mikolajsawicki/rainbows.htm>. •"Rainbow Lab: Light." The Geometry Center Welcome Page. Web. 25 Apr. 2011. <http://www.geom.uiuc.edu/education/calc-init/rainbow/light.html>. [1] •Figure 2: http://www.usna.edu/Users/oceano/raylee/RainbowBridge/RB_images/Fig8_7.jpg •"About Rainbows." About Rainbows. Web. 26 Apr. 2011. <http://eo.ucar.edu/rainbows/>. •Google Images •Sun Dogs: http://www.islandnet.com/~see/weather/elements/sundog.htm •Bubbles: http://www.nanonet.go.jp/english/kids/k-make/bubble-t.html •http://www.wikihow.com/Make-a-Rainbow •[3] http://science.nasa.gov/science-news/science-at-nasa/2005/25feb_titan2/ NOTE: Some variables were changed in the quotes due to the nature of the equation-generating tool used in this presentation.