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Rainbow
A rainbow is a meteorological phenomenon that is
caused by reflection, refraction and dispersion of
light in water droplets resulting in a spectrum of light
appearing in the sky. It takes the form of a
multicoloured circular arc. Rainbows caused by
sunlight always appear in the section of sky directly
opposite the sun.
Rainbows can be full circles. However, the observer
normally sees only an arc formed by illuminated
droplets above the ground,
[1] and centered on a line
from the sun to the observer's eye.
In a primary rainbow, the arc shows red on the outer
part and violet on the inner side. This rainbow is
caused by light being refracted when entering a
droplet of water, then reflected inside on the back of the droplet and refracted again when leaving it.
In a double rainbow, a second arc is seen outside the primary arc, and has the order of its colours reversed, with red on the inner side
of the arc. This is caused by the light being reflected twice on the inside of the droplet before leaving it.
Overview
Visibility
Number of colours in spectrum or rainbow
Explanation
Mathematical derivation
Variations
Double rainbows
Twinned rainbow
Full-circle rainbow
Supernumerary rainbows
Reflected rainbow, reflection rainbow
Monochrome rainbow
Higher-order rainbows
Rainbows under moonlight
Fogbow
Circumhorizontal and circumzenithal arcs
Rainbows on Titan
Rainbows with different materials
Double rainbow and supernumerary rainbows on the inside of the
primary arc. The shadow of the photographer's head on the bottom
marks the centre of the rainbow circle (antisolar point).
Contents
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Scientific history
Experiments
Culture
See also
Notes
References
External links
A rainbow is not located at a specific distance from the observer, but comes from an optical
illusion caused by any water droplets viewed from a certain angle relative to a light source.
Thus, a rainbow is not an object and cannot be physically approached. Indeed, it is impossible
for an observer to see a rainbow from water droplets at any angle other than the customary one
of 42 degrees from the direction opposite the light source. Even if an observer sees another
observer who seems "under" or "at the end of" a rainbow, the second observer will see a
different rainbow—farther off—at the same angle as seen by the first observer.
Rainbows span a continuous spectrum of colours. Any distinct bands perceived are an artefact
of human colour vision, and no banding of any type is seen in a black-and-white photo of a
rainbow, only a smooth gradation of intensity to a maximum, then fading towards the other side.
For colours seen by the human eye, the most commonly cited and remembered sequence is
Newton's sevenfold red, orange, yellow, green, blue, indigo and violet,
[2][3]
remembered by the
mnemonic Richard Of York Gave Battle In Vain (ROYGBIV).
Rainbows can be caused by many forms of airborne water. These include not only rain, but also
mist, spray, and airborne dew.
Rainbows can be observed whenever there are water drops in the air and sunlight shining from behind the observer at a low altitude
angle. Because of this, rainbows are usually seen in the western sky during the morning and in the eastern sky during the early
evening. The most spectacular rainbow displays happen when half the sky is still dark with raining clouds and the observer is at a spot
with clear sky in the direction of the sun. The result is a luminous rainbow that contrasts with the darkened background. During such
good visibility conditions, the larger but fainter secondary rainbow is often visible. It appears about 10° outside of the primary
rainbow, with inverse order of colours.
The rainbow effect is also commonly seen near waterfalls or fountains. In addition, the effect can be artificially created by dispersing
water droplets into the air during a sunny day. Rarely, a moonbow, lunar rainbow or nighttime rainbow, can be seen on strongly
moonlit nights. As human visual perception for colour is poor in low light, moonbows are often perceived to be white.
[4]
It is difficult to photograph the complete semicircle of a rainbow in one frame, as this would require an angle of view of 84°. For a
35 mm camera, a wide-angle lens with a focal length of 19 mm or less would be required. Now that software for stitching several
images into a panorama is available, images of the entire arc and even secondary arcs can be created fairly easily from a series of
overlapping frames.
Overview
Image of the end of a
rainbow at Jasper National
Park
Visibility
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From above the earth such as in an aeroplane, it is sometimes possible to see a
rainbow as a full circle. This phenomenon can be confused with the glory
phenomenon, but a glory is usually much smaller, covering only 5–20°.
The sky inside a primary rainbow is brighter than the sky outside of the bow. This is
because each raindrop is a sphere and it scatters light over an entire circular disc in the
sky. The radius of the disc depends on the wavelength of light, with red light being
scattered over a larger angle than blue light. Over most of the disc, scattered light at all
wavelengths overlaps, resulting in white light which brightens the sky. At the edge, the
wavelength dependence of the scattering gives rise to the rainbow.
[5]
Light of primary rainbow arc is 96% polarised tangential to the arch.
[6] Light of
second arc is 90% polarised.
A spectrum obtained using a glass prism and a point source is a continuum of
wavelengths without bands. The number of colours that the human eye is able to
distinguish in a spectrum is in the order of 100.
[7] Accordingly, the Munsell colour
system (a 20th-century system for numerically describing colours, based on equal
steps for human visual perception) distinguishes 100 hues. The apparent discreteness
of main colours is an artefact of human perception and the exact number of main
colours is a somewhat arbitrary choice.
Red Orange Yellow Green Blue Indigo Violet

Newton, who admitted his eyes were not very critical in distinguishing colours,
[8]
originally (1672) divided the spectrum into five main colours: red, yellow, green, blue
and violet. Later he included orange and indigo, giving seven main colours by analogy
to the number of notes in a musical scale.
[2][9] Newton chose to divide the visible
spectrum into seven colours out of a belief derived from the beliefs of the ancient
Greek sophists, who thought there was a connection between the colours, the musical
notes, the known objects in the Solar System, and the days of the week.
[10][11][12]
Scholars have noted that what Newton regarded at the time as "blue" would today be
regarded as cyan, and what Newton called "indigo" would today be considered blue.
[3]
Red Orange Yellow Green Cyan Blue Violet

According to Isaac Asimov, "It is customary to list indigo as a colour lying between blue and violet, but it has never seemed to me that
indigo is worth the dignity of being considered a separate colour. To my eyes it seems merely deep blue."
[13]
The colour pattern of a rainbow is different from a spectrum, and the colours are less saturated. There is spectral smearing in a
rainbow owing to the fact that for any particular wavelength, there is a distribution of exit angles, rather than a single unvarying
angle.
[14]
In addition, a rainbow is a blurred version of the bow obtained from a point source, because the disk diameter of the sun
(0.5°) cannot be neglected compared to the width of a rainbow (2°). Further red of the first supplementary rainbow overlaps the violet
of the primary rainbow, so rather than the final colour being a variant of spectral violet, it is actually a purple. The number of colour
Rainbows can form in the spray of a
waterfall (called spray bows).
Rainbows may form in the spray
created by waves.
Eruption of Castle Geyser,
Yellowstone National Park, with
double rainbow seen in the mist
Number of colours in spectrum or rainbow
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bands of a rainbow may therefore be different from the number of bands in a spectrum,
especially if the droplets are particularly large or small. Therefore, the number of
colours of a rainbow is variable. If, however, the word rainbow is used inaccurately to
mean spectrum, it is the number of main colours in the spectrum.
The question of whether everyone sees seven colours in a rainbow is related to the idea
of linguistic relativity. Suggestions have been made that there is universality in the
way that a rainbow is perceived.
[15][16] However, more recent research suggests that
the number of distinct colours observed and what these are called depend on the
language that one uses, with people whose language has fewer colour words seeing
fewer discrete colour bands.
[17]
When sunlight encounters a raindrop, part of the light is reflected and the rest enters the raindrop. The light is refracted at the surface
of the raindrop. When this light hits the back of the raindrop, some of it is reflected off the back. When the internally reflected light
reaches the surface again, once more some is internally reflected and some is refracted as it exits the drop. (The light that reflects off
the drop, exits from the back, or continues to bounce around inside the drop after the second encounter with the surface, is not relevant
to the formation of the primary rainbow.) The overall effect is that part of the incoming light is reflected back over the range of 0° to
42°, with the most intense light at 42°.
[18] This angle is independent of the size of the drop, but does depend on its refractive index.
Seawater has a higher refractive index than rain water, so the radius of a "rainbow" in sea spray is smaller than a true rainbow. This is
visible to the naked eye by a misalignment of these bows.
[19]
The reason the returning light is most intense at about 42° is that this is a turning point – light hitting the outermost ring of the drop
gets returned at less than 42°, as does the light hitting the drop nearer to its centre. There is a circular band of light that all gets
returned right around 42°. If the sun were a laser emitting parallel, monochromatic rays, then the luminance (brightness) of the bow
would tend toward infinity at this angle (ignoring interference effects). (See Caustic (optics).) But since the sun's luminance is finite
and its rays are not all parallel (it covers about half a degree of the sky) the luminance does not go to infinity. Furthermore, the amount
by which light is refracted depends upon its wavelength, and hence its colour. This effect is called dispersion. Blue light (shorter
wavelength) is refracted at a greater angle than red light, but due to the reflection of light rays from the back of the droplet, the blue
light emerges from the droplet at a smaller angle to the original incident white light ray than the red light. Due to this angle, blue is
seen on the inside of the arc of the primary rainbow, and red on the outside. The result of this is not only to give different colours to
different parts of the rainbow, but also to diminish the brightness. (A "rainbow" formed by droplets of a liquid with no dispersion
would be white, but brighter than a normal rainbow.)
The light at the back of the raindrop does not undergo total internal reflection, and some light does emerge from the back. However,
light coming out the back of the raindrop does not create a rainbow between the observer and the sun because spectra emitted from the
back of the raindrop do not have a maximum of intensity, as the other visible rainbows do, and thus the colours blend together rather
than forming a rainbow.
[20]
A rainbow does not exist at one particular location. Many rainbows exist; however, only one can be seen depending on the particular
observer's viewpoint as droplets of light illuminated by the sun. All raindrops refract and reflect the sunlight in the same way, but only
the light from some raindrops reaches the observer's eye. This light is what constitutes the rainbow for that observer. The whole
system composed by the sun's rays, the observer's head, and the (spherical) water drops has an axial symmetry around the axis through
the observer's head and parallel to the sun's rays. The rainbow is curved because the set of all the raindrops that have the right angle
between the observer, the drop, and the sun, lie on a cone pointing at the sun with the observer at the tip. The base of the cone forms a
circle at an angle of 40–42° to the line between the observer's head and their shadow but 50% or more of the circle is below the
horizon, unless the observer is sufficiently far above the earth's surface to see it all, for example in an aeroplane (see above).
[21][22]
Rainbow (middle: real, bottom:
computed) compared to true
spectrum (top): unsaturated colours
and different colour profile
Explanation
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Alternatively, an observer with the right vantage point may see the full circle in a
fountain or waterfall spray.
[23]
It is possible to determine the perceived angle which the rainbow subtends as
follows.
[24]
Given a spherical raindrop, and defining the perceived angle of the rainbow as 2φ,
and the angle of the internal reflection as 2β, then the angle of incidence of the sun's
rays with respect to the drop's surface normal is 2β − φ. Since the angle of refraction
is β, Snell's law gives us
sin(2β − φ) = n sin β,
where n = 1.333 is the refractive index of water. Solving for φ, we get
φ = 2β − arcsin(n sin β).
The rainbow will occur where the angle φ is maximum with respect to the angle β.
Therefore, from calculus, we can set dφ/dβ = 0, and solve for β, which yields
.
Substituting back into the earlier equation for φ yields 2φmax ≈ 42° as the radius
angle of the rainbow.
The term double rainbow is used when both the primary and secondary rainbows are
visible. In theory, all rainbows are double rainbows, but since the secondary bow is
always fainter than the primary, it may be too weak to spot in practice.
Secondary rainbows are caused by a double reflection of sunlight inside the water
droplets. Technically the secondary bow is centred on the sun itself, but since its
angular size is more than 90° (about 127° for violet to 130° for red), it is seen on the
same side of the sky as the primary rainbow, about 10° outside it at an apparent angle
of 50–53°. As a result of the "inside" of the secondary bow being "up" to the observer,
the colours appear reversed compared to those of the primary bow.
The secondary rainbow is fainter than the primary because more light escapes from
two reflections compared to one and because the rainbow itself is spread over a greater
area of the sky. Each rainbow reflects white light inside its coloured bands, but that is
"down" for the primary and "up" for the secondary.
[26] The dark area of unlit sky lying
between the primary and secondary bows is called Alexander's band, after Alexander
Light rays enter a raindrop from one
direction (typically a straight line
from the sun), reflect off the back of
the raindrop, and fan out as they
leave the raindrop. The light leaving
the rainbow is spread over a wide
angle, with a maximum intensity at
the angles 40.89–42°. (Note:
Between 2 and 100% of the light is
reflected at each of the three
surfaces encountered, depending
on the angle of incidence. This
diagram only shows the paths
relevant to the rainbow.)
White light separates into different
colours on entering the raindrop due
to dispersion, causing red light to be
refracted less than blue light.
Mathematical derivation
Mathematical derivation
Variations
Double rainbows
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of Aphrodisias who first described it.
[27]
Unlike a double rainbow that consists of two separate and concentric rainbow arcs,
the very rare twinned rainbow appears as two rainbow arcs that split from a single
base.
[28] The colours in the second bow, rather than reversing as in a secondary
rainbow, appear in the same order as the primary rainbow. A "normal" secondary
rainbow may be present as well. Twinned rainbows can look similar to, but should
not be confused with supernumerary bands. The two phenomena may be told apart
by their difference in colour profile: supernumerary bands consist of subdued
pastel hues (mainly pink, purple and green), while the twinned rainbow shows the
same spectrum as a regular rainbow. The cause of a twinned rainbow is the
combination of different sizes of water drops falling from the sky. Due to air
resistance, raindrops flatten as they fall, and flattening is more prominent in larger
water drops. When two rain showers with different-sized raindrops combine, they
each produce slightly different rainbows which may combine and form a twinned
rainbow.
[29] A numerical ray tracing study showed that a twinned rainbow on a
photo could be explained by a mixture of 0.40 and 0.45 mm droplets. That small
difference in droplet size resulted in a small difference in flattening of the droplet
shape, and a large difference in flattening of the rainbow top.
[30]
Meanwhile, the even rarer case of a rainbow split into three branches was observed
and photographed in nature.
[31]
In theory, every rainbow is a circle, but from the ground, usually only its upper half
can be seen. Since the rainbow's centre is diametrically opposed to the sun's
position in the sky, more of the circle comes into view as the sun approaches the horizon,
meaning that the largest section of the circle normally seen is about 50% during sunset or
sunrise. Viewing the rainbow's lower half requires the presence of water droplets below the
observer's horizon, as well as sunlight that is able to reach them. These requirements are not
usually met when the viewer is at ground level, either because droplets are absent in the required
position, or because the sunlight is obstructed by the landscape behind the observer. From a high
viewpoint such as a high building or an aircraft, however, the requirements can be met and the
full-circle rainbow can be seen.
[32][33] Like a partial rainbow, the circular rainbow can have a
secondary bow or supernumerary bows as well.
[34]
It is possible to produce the full circle when
standing on the ground, for example by spraying a water mist from a garden hose while facing
away from the sun.
[35]
A circular rainbow should not be confused with the glory, which is much smaller in diameter
and is created by different optical processes. In the right circumstances, a glory and a (circular)
rainbow or fog bow can occur together. Another atmospheric phenomenon that may be mistaken
for a "circular rainbow" is the 22° halo, which is caused by ice crystals rather than liquid water
droplets, and is located around the sun (or moon), not opposite it.
Double rainbow with Alexander's band
visible between the primary and
secondary bows. Also note the
pronounced supernumerary bows
inside the primary bow.
Physics of a primary and secondary
rainbow and Alexander's dark band
[25]
(The image of the sun in the picture is
only conventional; all rays are parallel
to the axis of the rainbow's cone)
Twinned rainbow
Circular rainbow
Full-circle rainbow
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In certain circumstances, one or several narrow, faintly coloured bands can be seen
bordering the violet edge of a rainbow; i.e., inside the primary bow or, much more
rarely, outside the secondary. These extra bands are called supernumerary rainbows or
supernumerary bands; together with the rainbow itself the phenomenon is also known
as a stacker rainbow. The supernumerary bows are slightly detached from the main
bow, become successively fainter along with their distance from it, and have pastel
colours (consisting mainly of pink, purple and green hues) rather than the usual
spectrum pattern.
[36] The effect becomes apparent when water droplets are involved
that have a diameter of about 1 mm or less; the smaller the droplets are, the broader the
supernumerary bands become, and the less saturated their colours.
[37] Due to their
origin in small droplets, supernumerary bands tend to be particularly prominent in
fogbows.
[38]
Supernumerary rainbows cannot be explained using classical geometric optics. The
alternating faint bands are caused by interference between rays of light following
slightly different paths with slightly varying lengths within the raindrops. Some rays are in phase, reinforcing each other through
constructive interference, creating a bright band; others are out of phase by up to half a wavelength, cancelling each other out through
destructive interference, and creating a gap. Given the different angles of refraction for rays of different colours, the patterns of
interference are slightly different for rays of different colours, so each bright band is differentiated in colour, creating a miniature
rainbow. Supernumerary rainbows are clearest when raindrops are small and of uniform size. The very existence of supernumerary
rainbows was historically a first indication of the wave nature of light, and the first explanation was provided by Thomas Young in
1804.
[39]
When a rainbow appears above a body of water, two complementary mirror bows may
be seen below and above the horizon, originating from different light paths. Their
names are slightly different.
A reflected rainbow may appear in the water surface below the horizon.
[40] The
sunlight is first deflected by the raindrops, and then reflected off the body of water,
before reaching the observer. The reflected rainbow is frequently visible, at least
partially, even in small puddles.
A reflection rainbow may be produced where sunlight reflects off a body of water
before reaching the raindrops), if the water body is large, quiet over its entire surface,
and close to the rain curtain. The reflection rainbow appears above the horizon. It intersects the normal rainbow at the horizon, and its
arc reaches higher in the sky, with its centre as high above the horizon as the normal rainbow's centre is below it. Due to the
combination of requirements, a reflection rainbow is rarely visible.
Up to eight separate bows may be distinguished if the reflected and reflection rainbows happen to occur simultaneously: The normal
(non-reflection) primary and secondary bows above the horizon (1, 2) with their reflected counterparts below it (3, 4), and the
reflection primary and secondary bows above the horizon (5, 6) with their reflected counterparts below it (7, 8).
[41][42]
Supernumerary rainbows
Contrast-enhanced photograph of a
rainbow with additional
supernumerary bands inside the
primary bow
Reflected rainbow, reflection rainbow
Reflected rainbow
Monochrome rainbow
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Occasionally a shower may happen at sunrise or
sunset, where the shorter wavelengths like blue
and green have been scattered and essentially
removed from the spectrum. Further scattering
may occur due to the rain, and the result can be
the rare and dramatic monochrome or red
rainbow.
[43]
In addition to the common primary and
secondary rainbows, it is also possible for rainbows of higher orders to form. The order of a
rainbow is determined by the number of light reflections inside the water droplets that create it:
One reflection results in the first-order or primary rainbow; two reflections create the secondorder or secondary rainbow. More internal reflections cause bows of higher orders—
theoretically unto infinity.
[44] As more and more light is lost with each internal reflection,
however, each subsequent bow becomes progressively dimmer and therefore increasingly harder
to spot. An additional challenge in observing the third-order (or tertiary) and fourth-order (quaternary) rainbows is their location in
the direction of the sun (about 40° and 45° from the sun, respectively), causing them to become drowned in its glare.
[45]
For these reasons, naturally occurring rainbows of an order higher than 2 are rarely visible to the naked eye. Nevertheless, sightings of
the third-order bow in nature have been reported, and in 2011 it was photographed definitively for the first time.
[46][47] Shortly after,
the fourth-order rainbow was photographed as well,
[48][49] and in 2014 the first ever pictures of the fifth-order (or quinary) rainbow,
located in between the primary and secondary bows, were published.
[50]
In a laboratory setting, it is possible to create bows of much higher orders. Felix Billet (1808–1882) depicted angular positions up to
the 19th-order rainbow, a pattern he called a "rose of rainbows".
[51][52][53]
In the laboratory, it is possible to observe higher-order
rainbows by using extremely bright and well collimated light produced by lasers. Up to the 200th-order rainbow was reported by Ng et
al. in 1998 using a similar method but an argon ion laser beam.
[54]
Tertiary and quaternary rainbows should not be confused with "triple" and "quadruple" rainbows—terms sometimes erroneously used
to refer to the—much more common—supernumerary bows and reflection rainbows.
Like most atmospheric optical phenomena, rainbows can be caused by light from the Sun, but also from the Moon. In case of the
latter, the rainbow is referred to as a lunar rainbow or moonbow. They are much dimmer and rarer than solar rainbows, requiring the
Moon to be near-full in order for them to be seen. For the same reason, moonbows are often perceived as white and may be thought of
as monochrome. The full spectrum is present, however, but the human eye is not normally sensitive enough to see the colours. Long
exposure photographs will sometimes show the colour in this type of rainbow.
[55]
Fogbows form in the same way as rainbows, but they are formed by much smaller cloud and fog droplets that diffract light
extensively. They are almost white with faint reds on the outside and blues inside; often one or more broad supernumerary bands can
be discerned inside the inner edge. The colours are dim because the bow in each colour is very broad and the colours overlap.
Fogbows are commonly seen over water when air in contact with the cooler water is chilled, but they can be found anywhere if the fog
is thin enough for the sun to shine through and the sun is fairly bright. They are very large—almost as big as a rainbow and much
Reflection rainbow (top)
and normal rainbow
(bottom) at sunset
Unenhanced photo of a red
(monochrome) rainbow Higher-order rainbows
Rainbows under moonlight
Fogbow
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broader. They sometimes appear with a
glory at the bow's centre.
[56]
Fog bows should not be confused with
ice halos, which are very common
around the world and visible much
more often than rainbows (of any
order),
[57] yet are unrelated to
rainbows.
The circumzenithal and circumhorizontal arcs are two related optical phenomena
similar in appearance to a rainbow, but unlike the latter, their origin lies in light
refraction through hexagonal ice crystals rather than liquid water droplets. This means
that they are not rainbows, but members of the large family of halos.
Both arcs are brightly coloured ring segments centred on the zenith, but in different
positions in the sky: The circumzenithal arc is notably curved and located high above
the Sun (or Moon) with its convex side pointing downwards (creating the impression
of an "upside down rainbow"); the circumhorizontal arc runs much closer to the
horizon, is more straight and located at a significant distance below the Sun (or Moon).
Both arcs have their red side pointing towards the sun and their violet part away from
it, meaning the circumzenithal arc is red on the bottom, while the circumhorizontal arc
is red on top.
[58][59]
The circumhorizontal arc is sometimes referred to by the misnomer "fire rainbow". In order to
view it, the Sun or Moon must be at least 58° above the horizon, making it a rare occurrence at
higher latitudes. The circumzenithal arc, visible only at a solar or lunar elevation of less than
32°, is much more common, but often missed since it occurs almost directly overhead.
It has been suggested that rainbows might exist on Saturn's moon Titan, as it has a wet surface
and humid clouds. The radius of a Titan rainbow would be about 49° instead of 42°, because the
fluid in that cold environment is methane instead of water. Although visible rainbows may be
rare due to Titan's hazy skies, infrared rainbows may be more common, but an observer would
need infrared night vision goggles to see them.
[60]
Droplets (or spheres) composed of materials with different refractive indices than plain water produce rainbows with different radius
angles. Since salt water has a higher refractive index, a sea spray bow doesn't perfectly align with the ordinary rainbow, if seen at the
same spot.
[61] Tiny plastic or glass marbles may be used in road marking as a reflectors to enhance its visibility by drivers at night.
Due to a much higher refractive index, rainbows observed on such marbles have a noticeably smaller radius.
[62] One can easily
reproduce such phenomena by sprinkling liquids of different refractive indices in the air, as illustrated in the photo.
Spray moonbow at the Lower
Yosemite Fall
Fogbow and glory.
Circumhorizontal and circumzenithal arcs
A circumhorizontal arc (bottom),
below a circumscribed halo
Circumzenithal arc
Rainbows on Titan
Rainbows with different materials
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The displacement of the rainbow due to different refractive indices can be pushed to a peculiar
limit. For a material with a refractive index larger than 2, there is no angle fulfilling the
requirements for the first order rainbow. For example, the index of refraction of diamond is
about 2.4, so diamond spheres would produce rainbows starting from the second order, omitting
the first order. In general, as the refractive index exceeds a number n+1, where n is a natural
number, the critical incidence angle for n times internally reflected rays escapes the domain
. This results in a rainbow of the n-th order shrinking to the antisolar point and vanishing.
The classical Greek scholar Aristotle (384–322 BC) was first to devote serious attention to the
rainbow.
[63] According to Raymond L. Lee and Alistair B. Fraser, "Despite its many flaws and
its appeal to Pythagorean numerology, Aristotle's qualitative explanation showed an
inventiveness and relative consistency that was unmatched for centuries. After Aristotle's death,
much rainbow theory consisted of reaction to his work, although not all of this was
uncritical."
[64]
In Book I of Naturales Quaestiones (c. 65 AD), the Roman philosopher Seneca the Younger discusses various theories of the
formation of rainbows extensively, including those of Aristotle. He notices that rainbows appear always opposite to the sun, that they
appear in water sprayed by a rower, in the water spat by a fuller on clothes stretched on pegs or by water sprayed through a small hole
in a burst pipe. He even speaks of rainbows produced by small rods (virgulae) of glass, anticipating Newton's experiences with prisms.
He takes into account two theories: one, that the rainbow is produced by the sun reflecting in each water drop, the other, that it is
produced by the sun reflected in a cloud shaped like a concave mirror; he favours the latter. He also discusses other phenomena related
to rainbows: the mysterious "virgae" (rods), halos and parhelia.
[65]
According to Hüseyin Gazi Topdemir, the Arab physicist and polymath Ibn al-Haytham (Alhazen; 965–1039), attempted to provide a
scientific explanation for the rainbow phenomenon. In his Maqala fi al-Hala wa Qaws Quzah (On the Rainbow and Halo), alHaytham "explained the formation of rainbow as an image, which forms at a concave mirror. If the rays of light coming from a farther
light source reflect to any point on axis of the concave mirror, they form concentric circles in that point. When it is supposed that the
sun as a farther light source, the eye of viewer as a point on the axis of mirror and a cloud as a reflecting surface, then it can be
observed the concentric circles are forming on the axis."
[66] He was not able to verify this because his theory that "light from the sun is
reflected by a cloud before reaching the eye" did not allow for a possible experimental verification.
[67] This explanation was repeated
by Averroes,
[66] and, though incorrect, provided the groundwork for the correct explanations later given by Kamāl al-Dīn al-Fārisī in
1309 and, independently, by Theodoric of Freiberg (c. 1250–c. 1311)
[68]—both having studied al-Haytham's Book of Optics.
[69]
Ibn al-Haytham's contemporary, the Persian philosopher and polymath Ibn Sīnā (Avicenna; 980–1037), provided an alternative
explanation, writing "that the bow is not formed in the dark cloud but rather in the very thin mist lying between the cloud and the sun
or observer. The cloud, he thought, serves simply as the background of this thin substance, much as a quicksilver lining is placed upon
the rear surface of the glass in a mirror. Ibn Sīnā would change the place not only of the bow, but also of the colour formation, holding
the iridescence to be merely a subjective sensation in the eye."
[70] This explanation, however, was also incorrect.
[66]
Ibn Sīnā's account
accepts many of Aristotle's arguments on the rainbow.
[71]
In Song Dynasty China (960–1279), a polymath scholar-official named Shen Kuo (1031–1095) hypothesised—as a certain Sun Sikong
(1015–1076) did before him—that rainbows were formed by a phenomenon of sunlight encountering droplets of rain in the air.
[72]
Paul Dong writes that Shen's explanation of the rainbow as a phenomenon of atmospheric refraction "is basically in accord with
modern scientific principles."
[73]
A first order rainbow from
water (left) and a sugar
solution (right).
Scientific history
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According to Nader El-Bizri, the Persian astronomer, Qutb al-Din al-Shirazi (1236–1311), gave a fairly accurate explanation for the
rainbow phenomenon. This was elaborated on by his student, Kamāl al-Dīn al-Fārisī (1267–1319), who gave a more mathematically
satisfactory explanation of the rainbow. He "proposed a model where the ray of light from the sun was refracted twice by a water
droplet, one or more reflections occurring between the two refractions." An experiment with a water-filled glass sphere was conducted
and al-Farisi showed the additional refractions due to the glass could be ignored in his model.
[67] As he noted in his Kitab Tanqih alManazir (The Revision of the Optics), al-Farisi used a large clear vessel of glass in the shape of a sphere, which was filled with water,
in order to have an experimental large-scale model of a rain drop. He then placed this model within a camera obscura that has a
controlled aperture for the introduction of light. He projected light unto the sphere and ultimately deduced through several trials and
detailed observations of reflections and refractions of light that the colours of the rainbow are phenomena of the decomposition of
light.
In Europe, Ibn al-Haytham's Book of Optics was translated into Latin and studied by Robert Grosseteste. His work on light was
continued by Roger Bacon, who wrote in his Opus Majus of 1268 about experiments with light shining through crystals and water
droplets showing the colours of the rainbow.
[74]
In addition, Bacon was the first to calculate the angular size of the rainbow. He stated
that the rainbow summit can not appear higher than 42° above the horizon.
[75] Theodoric of Freiberg is known to have given an
accurate theoretical explanation of both the primary and secondary rainbows in 1307. He explained the primary rainbow, noting that
"when sunlight falls on individual drops of moisture, the rays undergo two refractions (upon ingress and egress) and one reflection (at
the back of the drop) before transmission into the eye of the observer."
[76][77] He explained the secondary rainbow through a similar
analysis involving two refractions and two reflections.
Descartes' 1637 treatise, Discourse on Method, further advanced this explanation.
Knowing that the size of raindrops did not appear to affect the observed rainbow, he
experimented with passing rays of light through a large glass sphere filled with water.
By measuring the angles that the rays emerged, he concluded that the primary bow was
caused by a single internal reflection inside the raindrop and that a secondary bow
could be caused by two internal reflections. He supported this conclusion with a
derivation of the law of refraction (subsequently to, but independently of, Snell) and
correctly calculated the angles for both bows. His explanation of the colours, however,
was based on a mechanical version of the traditional theory that colours were produced
by a modification of white light.
[78][79]
Isaac Newton demonstrated that white light was composed of the light of all the
colours of the rainbow, which a glass prism could separate into the full spectrum of
colours, rejecting the theory that the colours were produced by a modification of white
light. He also showed that red light is refracted less than blue light, which led to the
first scientific explanation of the major features of the rainbow.
[80] Newton's corpuscular theory of light was unable to explain
supernumerary rainbows, and a satisfactory explanation was not found until Thomas Young realised that light behaves as a wave under
certain conditions, and can interfere with itself.
Young's work was refined in the 1820s by George Biddell Airy, who explained the dependence of the strength of the colours of the
rainbow on the size of the water droplets.
[81] Modern physical descriptions of the rainbow are based on Mie scattering, work published
by Gustav Mie in 1908.
[82] Advances in computational methods and optical theory continue to lead to a fuller understanding of
rainbows. For example, Nussenzveig provides a modern overview.
[83]
René Descartes' sketch of how
primary and secondary rainbows are
formed
Experiments
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Experiments on the rainbow phenomenon using artificial raindrops, i.e. water-filled
spherical flasks, go back at least to Theodoric of Freiberg in the 14th century. Later,
also Descartes studied the phenomenon using a Florence flask. A flask experiment
known as Florence's rainbow is still often used today as an imposing and intuitively
accessible demonstration experiment of the rainbow phenomenon.
[84][85][86]
It consists
in illuminating (with parallel white light) a water-filled spherical flask through a hole
in a screen. A rainbow will then appear thrown back / projected on the screen,
provided the screen is large enough. Due to the finite wall thickness and the
macroscopic character of the artificial raindrop, several subtle differences exist as
compared to the natural phenomenon,
[87][88]
including slightly changed rainbow
angles and a splitting of the rainbow orders.
A very similar experiment consists in using a cylindrical glass vessel filled with water
or a solid transparent cylinder and illuminated either parallel to the circular base (i.e.
light rays remaining at a fixed height while they transit the cylinder)
[89][90] or under an
angle to the base. Under these latter conditions the rainbow angles change relative to
the natural phenomenon since the effective index of refraction of water changes (Bravais' index of refraction for inclined rays
applies).
[87][88]
Other experiments use small liquid drops,
[52][53] see text above.
Rainbows occur frequently in mythology, and have been used in the arts. One of the
earliest literary occurrences of a rainbow is in the Book of Genesis chapter 9, as part of
the flood story of Noah, where it is a sign of God's covenant to never destroy all life on
earth with a global flood again. In Norse mythology, the rainbow bridge Bifröst
connects the world of men (Midgard) and the realm of the gods (Asgard). Cuchavira
was the god of the rainbow for the Muisca in present-day Colombia and when the
regular rains on the Bogotá savanna were over, the people thanked him offering gold,
snails and small emeralds. The Irish leprechaun's secret hiding place for his pot of gold
is usually said to be at the end of the rainbow. This place is appropriately impossible to
reach, because the rainbow is an optical effect which cannot be approached.
Rainbows appear in heraldry - in heraldry the rainbow proper consists of 4 bands of
color (Or, Gules, Vert, Argent) with the ends resting on clouds.
[91] Generalised
examples in coat of arms include those of the towns of Regen or Pfreimd, both in Bavaria, Germany; and of Bouffémont, France; and
of the 69th Infantry Regiment (New York) of the Army National Guard (USA).
Rainbow flags have been used for centuries. It was a symbol of the Cooperative movement in the German Peasants' War in the 16th
century, of peace in Italy, and of gay pride and LGBT social movements since the 1970s. In 1994, Archbishop Desmond Tutu and
President Nelson Mandela described newly democratic post-apartheid South Africa as the rainbow nation. The rainbow has also been
used in technology product logos, including the Apple computer logo. Many political alliances spanning multiple political parties have
called themselves a "Rainbow Coalition".
Atmospheric optics
Round bottom flask rainbow
demonstration experiment -
Johnson 1882
Culture
Depiction of the rainbow in the Book
of Genesis
See also

