{
  "id": "rainbow",
  "title": "Rainbow",
  "lang": "en",
  "source_lang": "en",
  "summary": "A rainbow is a multicoloured arc of light formed when sunlight is refracted, dispersed and reflected inside airborne water droplets, seen opposite the Sun.",
  "category": "science",
  "subcategory": "physics",
  "type": "concept",
  "aliases": [],
  "generated_at": "2026-10-08",
  "model": "pro/claude-haiku-5-5",
  "url": "https://aiwiki.page/en/rainbow/",
  "translations": {
    "zh-Hans": {
      "url": "https://aiwiki.page/zh/rainbow/",
      "json": "https://aiwiki.page/zh/rainbow.json"
    }
  },
  "sections": [
    {
      "heading": "",
      "text": "A rainbow is a multicoloured arc of light that appears in the sky when sunlight enters airborne water droplets, is refracted and dispersed, and is reflected back toward the observer. It is seen opposite the Sun, most often while rain is falling in the sunlit part of the sky. Because the arc depends on the viewer's line of sight, each observer sees a personal rainbow, and it has no fixed physical location. The phenomenon is a common everyday example of visible light being separated into its colours, and it follows the same Snell's law that governs a glass prism.\n\nRainbows matter to science because they tested and shaped optical theory. Early explanations by Aristotle's successors and by René Descartes used geometric ray tracing, Isaac Newton used them to argue that white light is a mixture of colours, and later wave-optics and scattering theories explained faint extra bands that ray tracing could not account for. The rainbow has also been a long-standing subject in art, literature and religious symbolism across many cultures.\n\nThis entry first describes the geometry and appearance of rainbows, then the optics of the primary and secondary bows, the wave effects that produce supernumerary bows, and related variants such as fogbows and moonbows. It then covers the question of how many colours a rainbow contains, the history of its explanation, its cultural and symbolic role, and its use in measurement and optical research."
    },
    {
      "heading": "Appearance and geometry",
      "text": "Viewed from the ground, the rainbow is the visible upper part of a ring of light centred on the antisolar point, the spot in the sky directly opposite the Sun as seen by the observer. The primary bow is seen between about 40° and 42° from the antisolar point, and its angular radius is usually quoted as about 42°. Its colour order runs in a fixed direction: reds are found on the outside of the bow, closest to the Sun, with the blues on the inside. The Sun's position therefore fixes which way the bow curves, and the shadow of the observer's head marks the centre of the circle.\n\nThe bow has no fixed place in the landscape. The centre of the curve is always on a line from the Sun to the observer's eye. Only droplets lying on the cone of 42° around that line return concentrated light to a particular viewer, and these drops may be at any distance, but must be on the 42° circle. Two people standing side by side therefore see two different arcs made by different droplets, and moving a few metres shifts the bow in the sky. This is why a rainbow is better understood as a geometric construction tied to the observer than as an object like a cloud or a building.\n\nWhether a full circle is visible depends on the observer's height. On the ground, the lower part of the ring is hidden by the surface, so only the upper arc is seen. Mountain climbers and pilots sometimes see near or full-circle rainbows, as the viewer must be above the observable horizon so that the bottom of the circle doesn't get blocked from view. From an aircraft the situation is different because rain or mist exists both above and below the aeroplane, so it is possible to see the entire circle of the rainbow from an airplane since there can be falling droplets both above and below you. In February 2023, for example, a helicopter crew filmed a complete ring over Waimea Canyon on Kaua'i. A full ring also requires enough suspended droplets to form a continuous backdrop; when flying, if there is enough mist they can be seen as full circles.\n\nThe sky inside the bow is brighter than the sky outside it. Light from droplets that do not lie exactly on the deviation angle still adds to the general illumination, so the region within the 42° circle receives extra scattered light. All rays other than the minimum deviation ray simply add to the general brightness within this 42°-radius circle. As a result, the sky inside a primary rainbow is brighter than the sky outside of the bow. This contrast is most noticeable when the bow is viewed against a dark background.\n\nThe bow itself is not uniformly bright. Its intensity rises steeply as the angle approaches the bow's boundary from inside, so the most luminous part of the colour band is generally the portion nearest the inner side, although the precise profile depends on drop size and the wavelength of light. The primary bow is usually the brightest of the bows. The primary bow is usually brighter than any of the other bows. Its colours also become weaker where droplets are very small, a point developed in the sections on droplet optics and fogbows."
    },
    {
      "heading": "Refraction, dispersion and internal reflection in a droplet",
      "text": "A raindrop in free fall is close to spherical and behaves like a small lens. When sunlight meets the drop, it first crosses the air–water boundary and bends according to Snell's law, \\(\\sin i = n \\sin r\\), where \\(i\\) is the angle of incidence, \\(r\\) the angle of refraction inside the water, and \\(n\\) the refractive index. Water's index is about 1.33 for visible light. Because \\(n\\) changes slightly with wavelength, red, orange, yellow, green, blue and violet are each bent by a slightly different amount. The beam is therefore spread into a fan of colours at the first surface, and this wavelength-dependent spreading is what physicists call dispersion.\n\nThe light does not simply pass through the drop. At the far side, part of the beam is reflected back into the water, while the rest leaves the drop. The reflection happens at the water–air boundary itself, not from a metal coating as in an ordinary mirror. The reflected light then reaches the front surface, crosses back into air, and is refracted a second time, which widens the colour separation further. For a single internal reflection, the total angle through which the ray is turned can be written as \\(D = \\pi + 2i - 4r\\), where \\(i\\) and \\(r\\) are linked by Snell's law. Each ray is therefore deflected by a definite amount that depends on how far from the drop's centre it entered.\n\nThe key feature of this relation is that \\(D\\) is not monotonic. A ray entering near the axis of the drop is turned almost through 180°. As the entry point moves outward, the deflection decreases until it reaches a minimum of about 138° for yellow light, and then it increases again. Since the observer's line of sight is measured from the antisolar point, the visible angle is \\(180^\\circ - 138^\\circ = 42^\\circ\\). The rays near this minimum deviation crowd together. A narrow band of angles around 138° receives more emerging light than any other interval of the same width, so the bow is brightest there. This concentration is a caustic in the optical sense, and it is comparable to the bright curved patterns that a cup of tea can cast on its own surface.\n\nThe colour order follows from the same geometry. Red light has the smallest refractive index and is deviated least, so its minimum deviation is closest to 138°, and it emerges at about 42° from the antisolar point. Violet is deviated most, with a minimum near 140°, which places it at roughly 40°. Red therefore lies on the outer edge of the primary bow and violet on the inner edge, and the intermediate colours fall between them. The angular difference between red and violet is only about 2°, which is why the whole bow occupies a narrow band of sky even though the colour sequence is clearly visible."
    },
    {
      "heading": "Secondary bows, higher-order bows and the reversed sequence",
      "text": "The secondary bow\n\nThe secondary bow comes from light that bounces twice inside a droplet before leaving it. The extra internal reflection pushes the emerging rays outward, so this bow sits at about 51° from the anti-solar point, outside the primary bow, roughly nine degrees beyond it. Published values vary slightly: one source gives a range of 50-53 degrees, and the figure depends on wavelength, since in one calculation the blue edge lies at a radius of 53° and the red edge at 50°. The second path is also much dimmer. At each encounter with the water–air boundary some light escapes, so less light remains to follow the double-bounce route, and the bow is usually much fainter than the primary.\n\nReversed colour order\n\nThe colour sequence flips because the total angular turning along the secondary path is larger. The minimum deviation for the secondary is about 230°, which exceeds the 180° that would send light straight back toward the source. Once the total deviation passes that threshold, the ordering inverts. Red, which is refracted least, emerges on the inner edge, and violet lies on the outer edge, the reverse of the primary. Secondary: red on inner edge and violet on outer edge, primary: red on outer edge and violet on inner. The mechanism is the same refraction and internal reflection described for the primary, applied to a longer path through the same kind of drop.\n\nAlexander's dark band\n\nThe unlit gap between the two arcs is called Alexander's dark band. It exists because scattered light from the primary path stops at about 42°, while light from the secondary path does not reach inward past roughly 51°. Nothing from the drops fills the angles in between, so the sky there looks dim by comparison with the brighter sky inside the primary and outside the secondary. The band is named after Alexander of Aphrodisias, the 2nd-century philosopher who first described it, although the date is sometimes given as around 200 AD.\n\nHigher-order bows\n\nEach additional internal reflection defines a higher order. Primary (first order) rainbows have only one reflection, secondary (second order) rainbows have two reflections, and third order rainbows would have three reflections within the droplet. Theory allows the sequence to continue indefinitely, since higher orders theoretically continue to infinity. In practice the losses at every reflection make these bows progressively harder to detect, so that higher order bows become increasingly faint and their colors spread over larger areas of the sky.\n\nThe third and fourth orders lie on the Sun's side of the sky, which makes them especially difficult to observe. Third and fourth order rainbows appear on the same side as the sun, obscured by its brightness. The fifth and sixth orders are hidden within the much brighter primary bow. Beyond the fourth order, the bows are generally invisible to the naked eye, although ordinary cameras have managed to capture third and fourth order rainbows in single images, and laboratory setups using intense monochromatic sources have detected much higher orders, with a 200th-order rainbow recorded in the lab."
    },
    {
      "heading": "Wave effects and supernumerary bows",
      "text": "Ray-based geometric optics predicts a primary bow whose colours shade smoothly into the sky, with nothing structured inside it. Real primary rainbows often show a set of faint, pastel arcs on the inner side of the main bow, separated by narrow darker bands. These are called supernumerary bows. They cannot be explained by refraction and dispersion alone, and they depend on the wave nature of light.\n\nInterference inside the droplet\n\nThe cause is interference. Rays that strike a droplet at slightly different distances from its axis leave it at nearly the same angle, but they have travelled paths of slightly different length through the water. Where the path difference is a whole number of wavelengths, the waves reinforce each other; where it is an odd multiple of half a wavelength, they cancel. The scattered intensity therefore varies periodically with angle, producing alternating bright and dark bands, and the bright maxima form the supernumerary arcs. Because the path difference depends on the droplet's radius, so does the angular position of each arc. Larger drops give narrower spacing between arcs, while smaller drops give wider spacing, and the arcs can rarely be distinguished in drops larger than about a millimetre across.\n\nThe fringes are fragile. If the drops vary in size, the maxima produced by different drops fall at slightly different angles and blur together, and the overlapping colours of the dispersed light further wash out the pattern. Clear supernumeraries therefore require small droplets of nearly uniform size. This also helps explain why the arcs are often most visible near the top of the bow, where, as one analysis notes, raindrops tend to be more uniform in size than at other points of the bow.\n\nFrom Young to Airy\n\nThomas Young was the first to link these arcs to interference. In 1803 he proposed that the dark gaps between them were produced by destructive interference between the two rays that emerge from a drop in the same direction. His simple two-ray model, however, predicted an unphysical infinite intensity at the main bow, and its predicted supernumerary positions were not accurate in all respects.\n\nGeorge Airy's treatment, published in the 1830s (one source gives 1836, another 1838), replaced this picture. Instead of two rays, Airy summed the contributions from all points on the incoming wavefront, travelling in all directions. This gave a finite intensity at the caustic edge, a maximum that does not coincide with the geometric-optics prediction, and supernumerary positions that correctly depend on droplet radius. Airy's theory did have a limitation: the amplitude distribution across the incoming wavefront had to be assumed rather than derived. Later work extended the approach to polarised light and to higher-order bows.\n\nMie theory and the exact solution\n\nGustav Mie's 1908 work gave a rigorous solution of Maxwell's equations for the scattering of light by homogeneous spheres, which is the complete electromagnetic description of a droplet. Mie theory reproduces the full intensity profile of the rainbow, including features that Airy's approximation only partly captures. Its drawback is that it provides little physical insight into which ray paths cause a given feature, and its calculations were laborious until computers became available. Comparisons show that the Airy approximation matches Mie theory closely for drops about a millimetre across, including the positions and intensities of the supernumerary maxima, and still fairly well for drops of about 0.1 mm. For much smaller drops, where the arcs overlap strongly, Airy's theory becomes less reliable."
    },
    {
      "heading": "Variants: fogbows, moonbows and other rainbow forms",
      "text": "A fogbow, often called a white rainbow, forms in fog or mist rather than rain. It appears as a bow in fog rather than rain. The droplets involved are far smaller than raindrops, typically under 0.05 millimetres across. At that scale, bending around a droplet outweighs bending through it, so the colours are smeared rather than split. The UK Met Office attributes this to diffraction becoming the dominant effect, rather than refraction. The result is a very weak bow, with a red outer edge and bluish inner edge, and in many cases it looks almost entirely white. The first recorded sighting dates to between August 1737 and July 1739 in modern-day Ecuador, reported by Antonio de Ulloa. Fogbows are discussed in more detail in fogbow.\n\nA moonbow is the same kind of bow produced by moonlight. Other than the difference in the light source, its formation is the same as for a solar rainbow: light is refracted and reflected by water droplets, and the arc sits opposite the Moon in the sky. The difficulty is brightness. Moonbows are much fainter than solar rainbows, due to the smaller amount of light reflected from the surface of the Moon. Estimates often put moonlight at roughly 400,000 times fainter than sunlight. Because the light is so dim, it is difficult for the human eye to discern colours in a moonbow. Vision at low light depends on rod cells, which carry no colour information, so observers usually report a pale white or grey arc. Long-exposure photography, which allows a camera sensor to collect light over time, can reveal the vibrant colours hidden within the moonbow. Viewing conditions are also demanding: the Moon must be near full, positioned low in the sky, ideally less than 42 degrees above the horizon, and the surroundings must be dark.\n\nA reflection rainbow is a different effect and is easy to confuse with a double rainbow. It is not a mirror image of the primary bow. Instead, rays of light pass through raindrops, reflect off the water surface, and then travel upwards, which produces an arc pointing toward the sky. The water surface is typically a lake or bay, and the effect has been photographed over a bay, where the light bounces off the water and passes back through the droplets.\n\nA true double rainbow, by contrast, is a primary bow with a fainter secondary bow outside it. A double rainbow appears when some light undergoes two internal reflections inside a raindrop, producing a secondary arc outside the primary rainbow. The phrase therefore does not refer to two primary bows seen together. The secondary bow's reversed colour order is treated in the section on higher-order bows.\n\nThe three variants share the same basic droplet optics, but they are not identical in physics. Fogbows depend on diffraction, a wave effect, to a much greater degree than ordinary rainbows, and moonbows depend on the same geometric refraction as solar rainbows while being limited by light intensity and by human vision. Changes in droplet size, illumination and viewing geometry therefore determine which effect dominates in each case."
    },
    {
      "heading": "Colour perception and the number of colours",
      "text": "The light in a rainbow forms a continuous gradient across the electromagnetic spectrum. Wavelengths shade into one another with no physical break where one hue ends and the next begins. The sharp bands people see are therefore largely a result of how observers classify what they see, not a feature of the light itself. The familiar count of colours comes mainly from a historical choice, and from the way human perception and language sort a smooth range of hues into named categories.\n\nThe standard list of seven colours traces back to Isaac Newton. Around 1665 he passed white light through a prism and identified seven constituent colours (red, orange, yellow, green, blue, indigo and violet), not necessarily because that's how many hues he saw, but because he thought that the colors of the rainbow were analogous to the notes of the musical scale. Accounts differ on how his count developed. One history holds that in his early experiments he recorded only five colours, red, yellow, green, blue and violet, and added orange and indigo only after later, repeated experiments. Newton himself, however, acknowledged in the Opticks that the spectrum was continuous, and that the named colours were only the principal points within it. The musical analogy has no basis in measurement, a point that historians of science have made repeatedly, yet the seven-colour list spread through textbooks and popular accounts and remains in common use.\n\nLanguages divide the colour space in different ways too. In 1969 cognitive researchers Brent Berlin and Paul Kay examined basic colour terms in 20 different languages, and their work suggested that the number of basic colour terms across languages is limited, with a proposed ceiling of a maximum of 11 basic color terms. Real languages vary considerably within that range. Russian, for example, obligatorily distinguishes two kinds of blue: lighter blues (\"goluboy\") and darker blues (\"siniy\"), whereas English uses a single basic term. Estimates for remote communities also differ. The Himba of Namibia were found to use a 5-colour-term language, by which is meant that the same word is used for green and blue regions of colour space when tested in 2004, but a later study using a wider sample of colours concluded that the Himba now have seven terms, or more properly categories, that are independent of other colour terms.\n\nThe clear bands in the rainbow also reflect a perceptual effect called categorical perception. Under this effect, hues within a category look more similar than would be predicted by their perceptual distance. Experiments with Russian and English speakers illustrate the point. In a study by Winawer and colleagues in 2007, Russian speakers were faster than English speakers at discriminating between light blue and blue in a reaction time task, but their advantage disappeared when they had to simultaneously do a verbal task. This suggests that active use of language can shape fast discrimination. Later work has complicated the picture. A 2020 study found that the Russian boundary was sharper than the English one, but it was also affected by frequency and order biases, demonstrating that \"Russian blues\" are less well-structured than previously thought.\n\nThe debate over these effects links the question of rainbow colours to wider questions in cognitive science and neuroscience. Colour vision begins with three types of cone cells in the retina, whose signals are combined and compared in the visual system before reaching the brain, where they are interpreted as hues. Research on how culture, language and attention alter this processing, often summarised under the heading of linguistic relativity, is still ongoing. The consensus is that the seven bands of a rainbow are a convenient and culturally shaped description of a continuous physical spectrum, not a fixed property of the light."
    },
    {
      "heading": "History of explanation",
      "text": "Attempts to explain the rainbow began in antiquity, but for centuries they treated the bow as a feature of the clouds rather than of the individual drops. Aristotle was the first to give a full account of the phenomenon, which he placed among the topics of physics. In his Meteorology he described the bow as sunlight reflected from clouds arranged on a hemisphere around the observer. He recognised three colours and held that the secondary bow repeated the primary in reverse order and with weaker colours. Later observers found this model unsatisfactory. Ibn Sina (Avicenna) noted that rainbows appear even without dark clouds to act as mirrors, which pointed toward water droplets as the true source.\n\nIn the early fourteenth century, two thinkers working independently reached a correct qualitative account. Theodoric of Freiberg (c. 1250–c. 1310) wrote De iride around 1304–1311, and he argued that a globe of water could serve as a magnified raindrop rather than a miniature cloud. He tested this with glass spheres, crystals and mirrors. Kamal al-Din al-Farisi (c. 1267–1319), working in Persia, reached a similar conclusion in his Tanqih al-manazir. He used a transparent sphere filled with water and a camera obscura to trace the light paths, and he proposed that sunlight is refracted twice by a droplet with one or more internal reflections between. Both men built on the optics of Ibn al-Haytham. Their models explained the order of the bows and the reflections involved, but they could not explain why light concentrates at a particular angle, so the bright band of the bow remained unaccounted for.\n\nDescartes supplied that missing geometry. In the Meteors appended to his Discours de la méthode (1637), René Descartes traced rays through a spherical drop using refraction and reflection and calculated the angles at which they emerge. He tested the model with water-filled glass spheres. Historians of science have described this as an early example of computational theoretical physics. Descartes still lacked an explanation for the colours, because he treated light as a uniform substance and did not account for why the different hues separate. Isaac Newton addressed this problem with prism experiments in the 1660s and 1670s, which showed that white sunlight is a mixture of colours that are refracted by different amounts. He then incorporated the dispersion of water into Descartes' theory, which gave a quantitative account of both the position and the colour sequence of the bows. He published these results in Opticks in 1704.\n\nThe nineteenth and twentieth centuries added the wave picture. Thomas Young explained the faint fringes inside the primary bow through interference, and George Airy later developed a wave treatment that accounted for the bow's brightness distribution and its small shift from the geometric angle. Gustav Mie's 1908 solution of the electromagnetic scattering problem for spheres provided an exact framework that reproduced the observed intensity of rainbows in full. Together these theories moved the explanation from geometric rays in a droplet to a complete physical description, linking the rainbow to refraction, interference and scattering."
    },
    {
      "heading": "Mythology, religion and art",
      "text": "Many cultures treated the rainbow as a link between the human world and the divine. In Greek myth, Iris, the personified rainbow, was a messenger of the gods, and Hesiod's Theogony names her as a daughter of Thaumas and Electra. She was imagined as travelling swiftly between Olympus and the mortal world. In Norse tradition, Bifröst is a bridge that joins Midgard, the realm of humans, to Asgard, the home of the gods. The Prose Edda describes it as a bridge guarded by the god Heimdallr, and it is often pictured as a band of brilliant colour.\n\nIn the Abrahamic traditions, the rainbow is most closely tied to the Genesis account of the Flood. In Genesis 9, God sets a bow in the clouds as a sign of a covenant with Noah and with every living creature, promising that the earth will not again be destroyed by a flood. The Hebrew word qeshet also means a weapon, and some commentators have read the rainbow as a bow that God hangs up and sets aside. Jewish and Christian theological writing has returned to this passage for centuries, discussing the rainbow as a reminder of divine promise and a sign of mercy. The covenant motif is specific to this narrative and is not equally central in every Abrahamic tradition.\n\nIn East Asia, rainbows appear in both official records and literary texts. Chinese dynastic histories grouped rainbows among celestial phenomena in treatises on omens, where an unusual rainbow could be read as a sign of disorder at court or of a disturbance in the natural order. Other accounts treated auspicious rainbows as reflections of a ruler's virtue. The Chinese character 虹 carries the insect radical 虫, which reflects older popular views of the rainbow as a serpent-like creature drinking from the sky or the earth. In Japan, rainbows are recorded in court diaries and described in poetry, often as striking, transient sights that carry seasonal or emotional meaning.\n\nIn European painting and literature, the rainbow became a motif of hope, transience and divine order. Dutch landscape painters of the seventeenth century, including Jacob van Ruisdael, placed rainbows over stormy skies and open country, using them to heighten contrast between darkness and light. Romantic poets took up the image as a symbol of wonder and of the child's capacity for awe; William Wordsworth's \"My Heart Leaps Up\" is among the best-known examples. In the twentieth century, the rainbow became a widely recognised emblem of unity and diversity. The rainbow flag, designed by Gilbert Baker for San Francisco Pride in 1978, shows how a natural image can be adapted to a new social meaning. The rainbow also appears routinely in science communication, in textbook diagrams and in advertising, where it carries associations of optimism and colour rather than of any particular doctrine."
    },
    {
      "heading": "Measurement and applied optics",
      "text": "The bow's geometry carries quantitative information about the droplet that produces it. The position of the first-order bow depends on both the droplet's refractive index and its diameter, so measuring where the bow falls and how its intensity is distributed can yield those two quantities. This principle underlies rainbow refractometry, a family of light-scattering techniques for transparent spherical droplets. In a typical arrangement, a laser beam crosses a spray and a detector, often a CCD array, records the angular intensity profile of the scattered light. Fitting that measured profile against a theoretical model, usually computed with Lorenz–Mie theory, returns the refractive index and size of the droplet. The fine ripples inside the bow add information on diameter, and their phase shift can detect changes of only a few nanometres in droplet size.\n\nBecause the refractive index of a liquid varies with temperature, the same measurement can be used as a thermometer. Work on the sprays of falling droplets has shown that the angular position of the bow can be converted into a droplet temperature. The French nuclear safety institute IRSN used this approach in its TOSQAN facility to study heat and mass transfer between a spray and the surrounding gas, in a setting representative of a severe reactor accident. In the global variant, which uses the bow formed by a whole group of droplets rather than a single drop, the technique has been applied to cryogenic sprays of liquid nitrogen, oxygen and methane. For those sprays, the temperature inferred from the signal agreed with thermocouple readings to within about 7.5 percent across the range of 80 to 110 K. Because the refractive index also depends on composition, the same method can in principle detect changes in mixture ratio, a capability that has been explored for binary mixtures and turbulent spray flames.\n\nPractical use is constrained by signal quality. Dense sprays, unusual refractive indices and long working distances often prevent the full bow from being recorded, and researchers have therefore developed methods that work with partial bows. One long-range system, which combines signals from several crossing beams through a synthetic aperture, reported refractive-index uncertainties of about \\(8\\times10^{-4}\\) and size uncertainties of about 3 µm for a monodisperse droplet stream measured at 1.65 m. Extensions to planar and three-dimensional arrangements allow droplets to be located as well as characterised. For colloidal suspensions and emulsions, the amplitude of the bow can be used to estimate the concentration of dispersed inclusions, since those inclusions attenuate the light that forms the bow.\n\nRainbow theory is also used outside direct measurement. In teaching laboratories, glass spheres and water droplets illuminated by a laser or a lamp reproduce the primary and secondary bows, allowing students to compare observed angles with calculated values. In computational optics, the rainbow serves as a stringent benchmark: a code that reproduces the bow's angle, width and intensity structure over a range of droplet sizes is generally considered to handle scattering correctly. Such codes are then applied to atmospheric aerosols, fuel sprays and industrial spray systems, where the droplet size distribution and refractive index may be known only approximately."
    },
    {
      "heading": "Atmospheric conditions and occurrence",
      "text": "A rainbow needs sunlight arriving from behind the observer and airborne water droplets in the part of the sky the observer faces. The droplets must be large enough to refract and reflect light efficiently, so rainbows are most often seen when rain falls from a cloud while the Sun still lights the drops. The classic setting is a partly sunny shower: the cloud blocks the Sun on the observer's side, while the Sun breaks through behind them and illuminates the falling rain. Garden sprinklers, fountains, waterfalls and sea spray can produce the same effect, because the droplets are the only requirement, not the weather itself.\n\nThe Sun's elevation decides whether a bow is visible from the ground. The primary bow's apex lies about 42° above the anti-solar point, so when the Sun is at elevation \\(h_\\odot\\) below 42°, the apex sits at roughly \\(42^\\circ - h_\\odot\\) above the horizon. When \\(h_\\odot\\) exceeds about 42°, the whole bow lies below the horizon and cannot be seen from the ground. This is why rainbows are most common in the early morning and late afternoon in many places: the Sun is low, and more of the arc lies above the horizon. Midday bows are not impossible, though. Near the December solstice at 50° N, the noon Sun reaches only about 17° above the horizon, so a bow with its apex near 25° can appear around midday in winter.\n\nThe observer's height also changes what can be seen. From ground level, the lower part of the circle is hidden by the land, because the rays aimed at it strike the surface before reaching any droplets. From a mountain summit or an aircraft, those downward rays can pass over the ground and reach falling rain, so more of the circle becomes visible. The same geometry means that the Sun's height and the observer's altitude together set how much of the full circle appears, with the lower-altitude observer seeing the least.\n\nOccurrence depends on climate and season. Rainbows need frequent showers combined with sunshine, so they tend to appear in regions and seasons with convective afternoon storms, where a shower can pass while the Sun is still shining on the far side of the sky. Persistent cloud or steady rain that never clears can prevent them, and in dry regions the scarcity of droplets makes rainbows rare. Seasonal changes in rainfall, humidity and the Sun's angle alter the chance of seeing one at a given place. In tropical regions, where the Sun is high for much of the day, bows are seen mainly in the early morning and late afternoon, whereas at high latitudes the low winter Sun extends the window in which bows can form. Because the bow is always centred on the anti-solar point, an observer who moves changes the bow's apparent position, so a rainbow seen from one spot may not be seen from a nearby one at the same moment."
    }
  ],
  "links": [
    "refraction",
    "electromagnetic-spectrum",
    "rene-descartes",
    "isaac-newton",
    "art",
    "literature",
    "religion",
    "sun",
    "cloud",
    "refractive-index",
    "dispersion-optics",
    "total-internal-reflection",
    "optics",
    "interference",
    "electromagnetic-radiation",
    "moon",
    "brain",
    "cognitive-science",
    "neuroscience",
    "ibn-al-haytham",
    "scattering",
    "laser"
  ],
  "sources": [
    {
      "title": "primary rainbow - Glossary of Meteorology",
      "url": "https://glossary.ametsoc.org/wiki/primary-rainbow/"
    },
    {
      "title": "Rainbow",
      "url": "https://en.wikipedia.org/wiki/Rainbow"
    },
    {
      "title": "RAINBOWS",
      "url": "https://earthsci.org/processes/weather/rainbow/rainbow.html"
    },
    {
      "title": "A circular rainbow? See the stunning spectacle caught on camera in Hawaii",
      "url": "https://www.fox5dc.com/news/a-circular-rainbow-see-stunning-spectacle-caught-on-camera-in-hawaii"
    },
    {
      "title": "Primary and Secondary Rainbows",
      "url": "http://hyperphysics.phy-astr.gsu.edu/hbase/atmos/rbowpri.html"
    },
    {
      "title": "Can You See A Full Circular Rainbow From A Plane?",
      "url": "https://infinitylearn.com/question-answer/can-you-see-a-full-circular-rainbow-from-a-plane"
    },
    {
      "title": "Physics Tutorial - Refraction and Lenses - Refraction Phenomena - Rainbow Formation",
      "url": "https://www.physicsclassroom.com/tutorial/refraction-and-lenses/refraction-phenomena/rainbow-formation"
    },
    {
      "title": "Rainbow Formation: Light, Water, and the 42-Degree Arc",
      "url": "https://www.theperpetuallycurious.org/articles/rainbow-formation/"
    },
    {
      "title": "Physics of Rainbows Guide",
      "url": "https://www.physicstheories.com/physics-of-rainbows"
    },
    {
      "title": "Rainbow Science: How Rainbows Form and How to Find Them",
      "url": "https://inlightofnature.com/how-rainbows-form/"
    },
    {
      "title": "The mathematical physics of rainbows and glories - ScienceDirect",
      "url": "https://www.sciencedirect.com/science/article/abs/pii/S037015730100076X"
    },
    {
      "title": "Refraction and the Formation of Rainbows: A Detailed Explanation",
      "url": "https://thestemprofessor.com/article/refraction-and-the-formation-of-rainbows-a-detailed-explanation/"
    },
    {
      "title": "7 Rare Sky Phenomena and Types of Rainbows You Can See",
      "url": "https://www.farmersalmanac.com/types-of-rainbows"
    },
    {
      "title": "Double, Twin, and Triple Rainbows",
      "url": "https://www.crystalinks.com/doublerainbows.html"
    },
    {
      "title": "Chromatic dispersion, rainbows and Alexander's dark band",
      "url": "https://www.animations.physics.unsw.edu.au/jw/light/chromatic_dispersion_rainbows_Alexanders_dark_band.htm"
    },
    {
      "title": "Alexander%27s band",
      "url": "https://en.wikipedia.org/wiki/Alexander%27s_band"
    },
    {
      "title": "Secondary Rainbows - RainbowSpec",
      "url": "https://rainbowspec.observer/secondary/"
    },
    {
      "title": "The Physics of Rainbows: Why You Can Never Walk to the End of One",
      "url": "https://world-of-physics.com/blog/physics-of-rainbows-light-refraction-dispersion/"
    },
    {
      "title": "Invisible Rainbows: Secrets of the Sky’s Most Colorful Phenomena - AMS Headlines",
      "url": "https://headlines.ametsoc.org/2023/07/13/invisible-rainbows-secrets-of-the-skys-most-colorful-phenomena/"
    },
    {
      "title": "Higher Order Rainbows - OPOD - Atmospheric Optics",
      "url": "https://atoptics.co.uk/blog/higher-order-rainbows-opod/"
    },
    {
      "title": "Rainbow",
      "url": "https://education.nationalgeographic.org/resource/rainbow/"
    },
    {
      "title": "Secondary Rainbow - Atmospheric Optics",
      "url": "https://atoptics.co.uk/blog/secondary-rainbow-2/"
    },
    {
      "title": "From Alexander of Aphrodisias to Young and Airy",
      "url": "https://www.dmf.unisalento.it/~manca/lab3/Jackson-Rainbow.pdf"
    },
    {
      "title": "The Theory of the Rainbow",
      "url": "http://bionics.seas.ucla.edu/education/MAE_182A/Airy_Eq_Rainbows.pdf"
    },
    {
      "title": "Common principles behind rainbows and boat wakes",
      "url": "https://pubs.aip.org/aapt/ajp/article/93/10/781/3363823/Common-principles-behind-rainbows-and-boat-wakes"
    },
    {
      "title": "Common principles behind rainbows and boat wakes",
      "url": "https://arxiv.org/pdf/2508.05644"
    },
    {
      "title": "The mathematical physics of rainbows and glories",
      "url": "https://tlakoba.w3.uvm.edu/AppliedUGMath/auxpaper_rainbow_glory_review.pdf"
    },
    {
      "title": "Rainbows: Mie computations and the Airy approximation",
      "url": "https://opg.optica.org/ao/abstract.cfm?uri=ao-30-1-106"
    },
    {
      "title": "Simulation of rainbows, coronas and glories using Mie theory and the Debye series - ScienceDirect",
      "url": "https://www.sciencedirect.com/science/article/abs/pii/S0022407304001840"
    },
    {
      "title": "Simulation of rainbows, coronas, and glories by use of Mie theory Philip Laven",
      "url": "https://patarnott.com/atms749/pdf/SimulationsRainbowGlories.pdf"
    },
    {
      "title": "Introduction",
      "url": "http://www.philiplaven.com/p1.html"
    },
    {
      "title": "Fog bow",
      "url": "https://en.wikipedia.org/wiki/Fog_bow"
    },
    {
      "title": "Mysterious \"fogbow\" weather phenomenon appears over California",
      "url": "https://www.newsweek.com/fogbow-rainbow-photographed-san-francisco-1751231"
    },
    {
      "title": "White rainbows are real—but very rare. Here's what causes them.",
      "url": "https://www.nationalgeographic.com/environment/article/white-rainbow-fogbow"
    },
    {
      "title": "Moonbow",
      "url": "https://en.wikipedia.org/wiki/Moonbow"
    },
    {
      "title": "Can There Be a Rainbow at Night? - Biology Insights",
      "url": "https://biologyinsights.com/can-there-be-a-rainbow-at-night/"
    },
    {
      "title": "Have you ever seen a reflection rainbow? Here's how they work - ABC News",
      "url": "https://www.abc.net.au/news/2024-07-26/reflection-rainbow-explained/104136612"
    },
    {
      "title": "The Science Behind the Triple Rainbow",
      "url": "https://www.nbcbayarea.com/weather/the-science-behind-the-triple-rainbow/2250318/"
    },
    {
      "title": "The Science Behind Fully Double Rainbows",
      "url": "https://www.rainbowsymphony.com/blogs/blog/the-science-behind-fully-double-rainbows"
    },
    {
      "title": "Newton’s Color Theory, ca. 1665",
      "url": "https://www.the-scientist.com/newtons-color-theory-ca-1665-31931"
    },
    {
      "title": "Why Does a Rainbow Have Seven Colors? Newton, Music, and the Cultural Definition of Color - Origin Trace",
      "url": "https://origin-trace.com/article/why-rainbow-has-seven-colors/"
    },
    {
      "title": "a Cross Cultural Study on Color Perception and Memory",
      "url": "https://digitalcommons.usf.edu/cgi/viewcontent.cgi?article=6568&context=etd"
    },
    {
      "title": "Russian blues reveal the limits of language influencing colour discrimination",
      "url": "https://www.researchgate.net/publication/340502008_Russian_blues_reveal_the_limits_of_language_influencing_colour_discrimination"
    },
    {
      "title": "(PDF) The Russian Blues Reveal Effects of Language on Color Discrimination",
      "url": "https://www.researchgate.net/publication/6358875_The_Russian_Blues_Reveal_Effects_of_Language_on_Color_Discrimination"
    },
    {
      "title": "Augmenting a colour lexicon",
      "url": "https://www.nature.com/articles/s41599-022-01045-3"
    },
    {
      "title": "(PDF) Augmenting a colour lexicon",
      "url": "https://www.academia.edu/69454197/Augmenting_a_colour_lexicon"
    },
    {
      "title": "Russian blues reveal the limits of language influencing colour discrimination - ScienceDirect",
      "url": "https://www.sciencedirect.com/science/article/abs/pii/S0010027720301001"
    },
    {
      "title": "Indigo",
      "url": "https://en.wikipedia.org/wiki/Indigo"
    },
    {
      "title": "Color Vision: Color Categories Vary with Language after All - ScienceDirect",
      "url": "https://www.sciencedirect.com/science/article/pii/S0960982207014819"
    },
    {
      "title": "Theodoric of Freiberg and Kamal al-Din al-Farisi Independently Formulate the Correct Qualitative Description of the Rainbow",
      "url": "https://www.encyclopedia.com/science/encyclopedias-almanacs-transcripts-and-maps/theodoric-freiberg-and-kamal-al-din-al-farisi-independently-formulate-correct-qualitative"
    },
    {
      "title": "Theodoric of Freiberg",
      "url": "https://www.encyclopedia.com/science/encyclopedias-almanacs-transcripts-and-maps/theodoric-freiberg"
    },
    {
      "title": "Theodoric of Freiberg",
      "url": "https://en.wikipedia.org/wiki/Theodoric_of_Freiberg"
    },
    {
      "title": "Kamāl al-Dīn al-Fārisī",
      "url": "https://en.wikipedia.org/wiki/Kamal_al-Din_al-Farisi"
    },
    {
      "title": "FĀRESĪ, KAMĀL-AL-DĪN ABU’L-ḤASAN MOḤAMMAD - Encyclopaedia Iranica",
      "url": "https://www.iranicaonline.org/articles/faresi-2/"
    },
    {
      "title": "(PDF) A short history of the rainbow",
      "url": "https://www.researchgate.net/publication/303594800_A_short_history_of_the_rainbow"
    },
    {
      "title": "Aristotle, Meteorology (Arist.Met.)",
      "url": "https://topostext.org/work/865"
    },
    {
      "title": "Deconstructing rainbows",
      "url": "https://physicsworld.com/a/deconstructing-rainbows/"
    },
    {
      "title": "Descartes’ Method (Stanford Encyclopedia of Philosophy)",
      "url": "https://plato.stanford.edu/entries/descartes-method/"
    },
    {
      "title": "Synthetic aperture rainbow refractometry for droplet refractive index and size measurement with long range: Standard and global modes - ScienceDirect",
      "url": "https://www.sciencedirect.com/science/article/abs/pii/S0032591022007549"
    },
    {
      "title": "Planar Rainbow Refractometry For Size, Refractive Index And Position Measurement Of Droplets In A Plane",
      "url": "https://www.researchgate.net/publication/372753411_Planar_Rainbow_Refractometry_For_Size_Refractive_Index_And_Position_Measurement_Of_Droplets_In_A_Plane"
    },
    {
      "title": "(PDF) Development of the global Rainbow refractometry to measure spray droplets temperature in a large containment vessel",
      "url": "https://www.researchgate.net/publication/228485722_Development_of_the_global_Rainbow_refractometry_to_measure_spray_droplets_temperature_in_a_large_containment_vessel"
    },
    {
      "title": "Temperature and Size Measurements of Cryogenic Spray Droplets with Global Rainbow Refractometry",
      "url": "https://arc.aiaa.org/doi/10.2514/1.B36750"
    },
    {
      "title": "Full article: Theoretical investigation of global rainbow refractometry in spray combustion of polydisperse microdroplets",
      "url": "https://www.tandfonline.com/doi/full/10.1080/09500340.2021.1890251"
    },
    {
      "title": "Secondary rainbow refractometry for droplet refractive index and size measurement",
      "url": "https://www.researchgate.net/publication/355350931_Secondary_rainbow_refractometry_for_droplet_refractive_index_and_size_measurement"
    },
    {
      "title": "Simultaneous measurement of refractive index, diameter and colloid concentration of a droplet using rainbow refractometry - ScienceDirect",
      "url": "https://www.sciencedirect.com/science/article/abs/pii/S0022407319307216"
    },
    {
      "title": "www.frontiersin.org",
      "url": "https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2023.1326332/pdf"
    },
    {
      "title": "Development and application of the global rainbow refractometry for the study of heat and mass transfers in a spray; Developpement et application de la refractometrie arc-en-ciel global pour l'etude des transferts massique et thermique dans un spray (Thesis/Dissertation)",
      "url": "https://www.osti.gov/etdeweb/biblio/20972270"
    }
  ],
  "license": "https://creativecommons.org/licenses/by/4.0/"
}