Beating the Rainbow
Moratorium
Dolly Parton said:
“The way I see it, if you want the rainbow, you gotta put up
with the rain.”
Clearly, Dolly never saw a rainbow in a fountain, water
sprinkler or hose. Of course, maybe she was just making a metaphorical point.
Fig.1: 16:24, 8th March 2024, Lodhi
Garden, Delhi
But I know there are people who have spent more than 60
years on Planet Earth and never knew they could see a rainbow in a fountain! My
better half, who is also 60+, recruits (buttonholes?) any nearby children at
the Lodhi Garden to glimpse the lovely rainbows…you have to educate the next
generation! Especially since rainbows are few and far between (even in those
places blessed with a higher number of rainbows), we have to manage our
fountains well.
Obviously, the rainbow you see in a fountain has nothing to
do with the rain…other than the common factor of water droplets. So why not call
it a fountain bow?
Nah, it won’t work. Why? Because you could see one in a
water sprinkler too. You wanna call it a sprinkler-bow? Or you could pinch the
end of a hose so the water exits in a sheet, and, with the sun behind you, you
can see a ‘hose-bow’? It won’t wash.
Back to the wall, you might argue that a fountain is just a
water sprinkler surrounded by water, but it’s probably better to just let
sleeping semanticks lie.
Anyway, it doesn’t matter whether the water drops go up or
down, just so long as the sun is shining at your back, and enough water
droplets are illuminated by the sunlight.
Another point to (partly) refute Dolly is that the rain
could be below a relatively far away cloud, leaving the observer quite dry (as
indicated in the schematic below [1]).
Fig.2 [1]
However, here’s a caveat for people who go fountain-hopping
to catch rainbows on sunny days: the sun needs to be low in the sky, making an
angle with the horizon that is less than 42° [1]. At sunrise (or sunset),
when the angle with the horizon is zero, you see the most that you can get: a
half-circle [2]. As the morning goes on, the rainbow shifts downwards, until
the top disappears below the horizon:
a)
Sunrise: 06:02 am:
b) 07:40:
c)
08:35:
d)
09:20:
The rainbow disappears (because the sun
altitude is higher than 42°).
Fig.3
Fig.3a was taken from the internet [3]. It was cropped to show
what it would have looked like, as the morning proceeds, with the times set as
if the picture had been taken in Delhi (77.1°, 28.7°) on 9th Apr.2024.
In the afternoon, the
same sequence is reversed: the sun reappears, the rainbow shifts upwards, till
it reaches the highest point at sunset (18:40).
Rainbow moratorium:
According to a physics website [1]: ‘rainbows are not seen
in midday’. There’s even a Quora thread [4]: ‘someone tells you they saw a
rainbow at noon; why they’re lying’. Some people replied: “They may not be”.
As you might expect, the period when you can’t see a rainbow
depends on your latitude (see the Table of rainbow times vs latitude at the
end). The duration of the ‘rainbow moratorium’ decreases as you approach the
Arctic Circle (in the Northern Hemisphere, or the Antarctic Circle in the
Southern Hemisphere).
The rainbow moratorium times were calculated using an online
calculator [5] to get the sun altitude vs latitude (on 9th Apr.2024
and on 21st June 2024, the summer solstice).
Red rainbows:
Strictly speaking, you could even see a rainbow before
sunrise (but after dawn) and after sunset (but before dusk), although light
levels will be low. But, interestingly, rainbows around sunrise and sunset
times tend to be mostly red, with traces of yellow [6]. Why? Because the longer
path-length the sun rays have to traverse scatters (by Rayleigh scattering) the
shorter wavelengths of sunlight out of the line of sight: the rays that remain
to strike the raindrops are mostly red, and so are the rainbows. I found a Reddit
user [7] who even specified the time he saw the red rainbow as 6 minutes before
sunrise.
Fig.4: red rainbow: from: Sarah Zielinski https://www.nationalgeographic.com/science/article/151218-rainbow-colour-sunrise-sunset-atmosphere-science
Secondary Rainbow:
However, there’s good news and the bad news.
The good news: there is a secondary rainbow (see the Fig.2) [1]
which is visible as long as the sun makes an angle of less than 51°
with the horizon (i.e. closer to noon).
The bad news? It’s somewhat fainter than the primary bow (43%,
according to [8]), so you have to lucky or savvy (or both) to see it. Also,
it’s at a higher angle, so either the water droplets in the fountain have to go
higher, or you have to get closer (and probably wetter).
Further, the primary rainbow has an angular width of about 1.9°,
while the secondary rainbow has a width of about 3.4°. Since the intensity is spread
over a larger width, this apparent intensity of the secondary rainbow is lowered
even more [9,10]. The primary rainbow forms between about 40.6° and 42.3°,
from the antisolar point (discussed again later).
Couch potatoes (like me) who couldn’t be bothered to physically
go to a fountain/sprinkler or get hold of a hose, can get plenty of rainbows –
all shapes and sizes – on the net (e.g. Google images, Pinterest, etc).
Breaking the moratorium:
Some contrary souls might want to break the rainbow
moratorium. Others are dissatisfied with a mere half-circle rainbow. Why not a
full circle?
There are two ways: you need to gain altitude or latitude:
In the Northern Hemisphere, head North. If you reach a
latitude of 71.5° (i.e.90° + 23.5° – 42°) the Sun basically never gets
an altitude more than 42°. Ergo, no rainbow moratorium on any summer day. However,
there are other days with no rainbow moratorium even at lower latitudes (e.g.
at 60°
on 9th Apr.2024). Similarly, in the Southern Hemisphere, go South.
But you’re still stuck with just a half-circle rainbow at
sunrise/sunset.
Full circle rainbows:
Remember that you get to see the widest arc of the rainbow
at sunset? Well, you can see more of it if you go to a high place (like a
mountain cliff or a high building) - or get in a plane. If you’re close to
sunset (or sunrise, if you’re a lark person), you won’t have to climb too high
[11]
N.B.: ‘altitude’ here means ‘altitude above ground level’
(AGL), not above mean sea level (MSL): going to Srinagar or Denver (Colourado)
will not help break the rainbow moratorium.
Astronomy picture of the day: 30th Sep.2024,
taken from a helicopter near Cottesloe Beach, Perth, Australia [12]:
Fig.5
The exact height you need to go to? That depends upon how
far you are from the water droplets. Of course, the rainbow itself is not
localised to a given distance, since the rays coming to your eye come from a
cone that intersects with the water droplets (see Fig.2 that shows a half-cone).
Say the closest water drops are at a minimum distance dmin from the
observer. Using this distance and the 42° full-angular width of the primary
rainbow, the minimum height hmin is given by:
tan (21°) = hmin/dmin = 1.527
i.e. hmin = 1.527 dmin
This is the height AGL needed to see a full-circle rainbow
at sunrise/sunset. What’s the minimum height at other times? I guess you would
have to go higher. How much? Dunno. Also, the closer the water droplets are,
the smaller the bow appears, for the same angle subtended by the cone.
Another full circle rainbow below [13], except that they
insist on calling it a ‘complete’ bow, a water hose-bow:
Fig.6:
Richard Fleet, webmaster of ‘Glows Bows & Haloes’ produced this hosepipe
bow. The camera field of view was not enough to take in the entire rainbow and
so he carefully took a number of images while keeping the camera in the same
position as far as possible.
Fleet [14] calls it a ‘spraybow’. He used 8 cameras to make
the composite image that covers a field of view (FOV) of about 110°
horizontally. And his website has a lot more…
Applying the equation above and taking Fleet’s height as 1.5
metres would suggest that the water droplets are less than a metre away from
him. (Actually we should take the distance between the cameras that took the
photo and the water drops).
Waterfall rainbows:
Oh, I almost forgot (how could I?): rainbows in waterfalls! Like
Yosemite, Niagara, or your friendly neighbourhood waterfall). Or, just Google a
few images. But, trust me, a real rainbow is the real thing – no matter where
you see it.
Apparently, a video went viral in 2023 of a waterfall in
Yosemite, at a time of ‘unusually heavy water’, taken at 9 am in November 2017 by
photographer Greg Harlow at a time when there were very high winds (making for
a spray of water drops) – leading to an exceptionally strong rainbow along the
entire length of the 1,450 foot long waterfall (or is it 2,400 foot long?) [15].
Check it out!
Effects of Droplet sizes and shapes:
Another problem: not all fountains produce rainbows.
Especially near the central part of the fountain the water drops are quite big.
That should not be a problem, since bigger drops mean a brighter rainbow. But the
spacing between the large raindrops is too much, and it reduces the total
number density of drops or the fill factor. The same fountain will give you
rainbows if the wind is blowing, creating a spray of fine water droplets.
Prisms vs
Spheres? [16]
Fig.7: prisms in (a), sphere in (b)
The shape factor is another point that is a bit unclear.
Almost all explanations of rainbows cite Newton’s prism that disperses white
light into separate colours. Then they claim that water droplets ‘act like
prisms’ [17,18], with ‘like’ as the operative word.
But water drops are spherical, no?
Experiments on light being reflected and refracted by a
glass sphere full of water were done by Kamal Al-Din Al-Farisi and by Rene
Descartes [19], and by the monk Theodoric von Freiberg in 1304 [20]. (A more
recent analysis of the glass sphere full of water, taking into account the
glass thickness [21]). These experiments predate Newton and his prism. But the
advantage of the prism was that the separation of colours was increased by the
refraction of the second surface, making it easier to see the spectrum.
However, almost any shape will create a rainbow – provided
the second surface is not parallel to the first, because that cancels out the
effect of the first surface (so a rectangular shape will not work, but a
cylinder will). I didn’t get this parallelism criterion but I found this on
Quora by a BA student, Shreya Mishra, at Delhi University [22]:
“The objects like rectangular glass
slab cannot scatter light into seven colours because it has the opposite sides
parallel. All the scattering done from first interface is undone when the light
reaches the second interface as the sides are parallel. On the other hand Prism
can scatter light as it's two sides are not parallel so the scattering done by
the first interface is not undone when the light reaches the second interface.
Same is the case with water droplet, no two sides of droplet are parallel so the
scattering is possible in water droplet.”
Do large raindrops contribute much
to rainbows? Probably to a lesser extent, because of the fill factor problem.
But the glass spheres show they could, in principle.
Keats lamented that Newton was responsible for ‘unweaving’
the rainbow, bringing its mystery down to earth. But today we still have
mysteries in elusive rainbows but we want to create, or discover, Rainbow
Machines to democratise access to them!
Higher order rainbows:
I’m not sure if Keats knew about the secondary rainbow. Maybe
he did, maybe he didn’t.
But poetry had little to do with the discovery in 2011 of
tertiary and quaternary rainbows [23], that were predicted earlier, and then
found and photographed in 2011 - under the predicted conditions! You have to
look back towards the sun, and these two bows make 43° with the direction (azimuth,
not elevation) of the sun [24]. The intensity of the tertiary rainbow is about
24% of that of the primary; that of the quaternary is about 15%. Even higher
order bows have been observed in the lab using high-intensity lasers [1].
How come tertiary and quaternary rainbows aren’t often seen?
Apparently, prior to the photographing by Grossmann [24] in 2011 only 5
credible instances were reported in 250 years! The intensity ratios of 24% and
15% are ideal values assuming the raindrops are uniformly the same size
– whereas, in most cases, there is a distribution of sizes from less than 0.1
mm diameter to up to 6 mm dia (above which size the drops break up).
In 2014, the first image of a 5th order rainbow,
the quinary (which hides in Alexander’s dark band!) was taken by Harald Edens
[25].
Droplet shapes and sizes (again):
As raindrops come down, the front ends get a bit flattened (they’re
nowhere near the classic teardrop shape).
We also know that small raindrops (less than 2 mm diameter
says Volpi [26], less than 0.28 mm says Ryomoto [16]; the difference is
probably in the extent of deviation from the spherical shape in the two
analyses) are kept spherical by surface tension [26], while large raindrops,
being more affected by air drag, are not. The latter shape, according to NASA
is like a hamburger: round on top (due to surface tension) and flat on the
lower end of a falling raindrop (due to air pressure) [27]. Further, they
oscillate (shape-changing raindrops), making the math even more messy [28].
The theory of rainbows for elliptical raindrops was started
in 1918 by W.Mobius in the Zeitschrift fur Physik (Do you read German? I don’t.)
leading to what are now called the Mobius shifts (using Mie scattering theory) of
the rainbow angle (by about ± 0.1° in 42°) – which has recently been
extended to rainbows of all orders by J.A.Lock et al [29].
The specifics of the
appearance of the rainbow as a function of droplet size have been detailed by
Ryomoto [16] (see the Table in Appendix lifted from his paper), but one simple
point is that the bigger flattened raindrops tend to scatter sunlight
horizontally, while small spherical raindrops scatter in more directions. So:
the top of the rainbow (that disappears towards noon) is fainter and is
predominantly due to the spherical raindrops.
But a more subtle point is
about the purity of the colours of the rainbow: the reds are pure, but all the
others are overlapped, with violet being the most overlapped. Why? See this website
for details [30a]:
“The
colours of a rainbow are not simple and uniform; rather, they are wonderfully
subtle mixtures. Unlike the colours produced by prisms or circumzenithal arcs,
which consist of almost single wavelengths, rainbow colours are composed of
overlapping disks of light. Each disk corresponds to a specific colour or
wavelength and is brightest at its caustic edge, gradually diminishing towards
its center. Reds create the largest disks, while blues and violets form the
smallest ones.”
This point is clarified by Chris
Baird [31]:
“…although most of the violet
light comes out… at 40.6°, we see that some violet light comes out at all
angles between 0° and 40.6°.”
“Although pure red is mostly bent by a raindrop into a 42.1°
viewing angle to form the outer edge of a rainbow, some of the red is bent into
all angles between 0° and 42.1° because of the curved surface of the raindrop.
Similarly, pure orange is mostly bent into the 41.9° viewing angle, but some
orange is bent into all lower angles as well. The colour in a rainbow at 42.1°
is therefore red, the colour at 41.9° is orange plus a little bit of red, the colour
at 41.7° is yellow plus a little bit of orange and red, etc. The end result is
that the colours in a rainbow tend to blur together and wash each other out.”
“Unlike the spread of colours created by a prism, the spread
of colours created by a spherical raindrop is not a pure spectrum.”
“A prism and a raindrop are in principle very similar. They
both spread white light out into a span of colours through refraction. The main
difference though is that a prism has flat surfaces, leading to a pure
spectrum, while a raindrop has a round surface, leading to an impure spectrum.”
That’s why a spectrograph contains prisms to disperse the
different wavelengths, without any admixture.
One consequence of the dispersion of wavelengths by (mostly)
spherical raindrops [30,31] is that the sky below the rainbow at an angle lower
than 40.6°
is fairly bright: a little bit of all the colours add up to produce white.
This brightness below the bow is discussed by John Hardwick [32]:
“Some of the light scattered by the raindrops can,
however, reach the observer from below the rainbow, which is therefore
brighter than the area above it – but, of course, not nearly as bright as
the rainbow itself.”
Fogbows and dewbows:
If water droplets are too fine, you get fog, and there is a
fogbow. But, like the fog, it’s white,
since the droplet size is so small [16,33].
Droplets in fog typically range from 5 to 15 microns [34].
However, there’s another wrinkle: the dewbow [32]:
“A dew-bow is created when light refracts from water
droplets located on a horizontal plane, such as the grass of a golf course. As
with a standard rainbow, the observer can be pictured as standing at the vertex
of a cone and receiving the rainbow light from angles along the cone’s surface.
The observer interprets the shape of the light source as the cross-section that
is cut through the cone by the plane containing the droplets. So if sunlight
strikes the plane at a very shallow angle, for example in early morning, the
dew-bow has a hyperbolic shape. Later in the day, if the droplets are still
present, the dew-bow will become an ellipse.”
Hardwick [32], asks: if dew is so common, why are
dewbows so rare? He explains: “But I do not think that the density of dewdrops
plays the key role. Dew-bows probably form only when spherical water droplets
are present. As with a standard rainbow, this allows sunrays to be
internally reflected from the back of the drops. However, experiments with a
blade of grass suggest that most drops of water attached to grass are not
spheres but hemispheres. Internal reflection would therefore not be possible
for the range of angles needed for a complete hyperbola to be seen.”
The moral of the story: Size matters…but so does shape.
Possibly, not as much as size, though?
And, experience with higher order bows, indicates that the
size distribution also matters (the traditional Marshall-Palmer or the more
recent Ulbrich) [35].
Alexander’s (dark) band [26]:
“The area
between the primary and secondary bows is noticeably darker than the
surrounding sky. Called “Alexander’s dark band”, it is named for the Greek sage
Alexander of Aphrodisias, who described it in his chronicles in the early 3rd
Century. This geometric band is an area devoid of rays which have passed
through water drops.”
Rainbow capital of the world and the effect of global
warming:
But the rainbow capital of the world is Hawaii, according to
a recent study [36]. There might be a smidgin of doubt because the claimant is
a denizen of Hawaii; but he wouldn’t mislead us, for ulterior motives? The
paper cites meteorological and topographical factors that give Hawaii its edge
(monsoons, mountains). Another paper
from Hawaii examines the influence of climate change on rainbows.
Global warming means less snow and more rain in the Northern
Latitudes, increasing the likelihood of rainbows. Regions with lower rainfall –
like the Mediterranean – will not get any advantage, However, on average, any
point on land on the globe, the number of rainbow days is estimated to increase
by 5% by the year 2100 [37,38].
Newton (again!):
Anyway, it seems that in medieval times only 5 colours were
recognised. Newton added two more…because that seemed to match his mystical Pythagorean
beliefs? [39]. Although, nowadays, a slightly different set is usually used
[17] – and anyway the number of colours is practically (and mathematically)
infinite.
One more point: you can get rainbows in materials other than
water drops e.g. in Newton’s prisms or in glass beads [40]. But apparently not
in diamonds! Ken Ford [8] writes the condition: the refractive index should be
less than 2 for the primary bow, and diamonds have n = 2.42! Consolation: this
restriction applies to primary rainbows, not the higher-order ones, that are
fainter.
12 types of rainbows:
Jean Louis Ricard proposed a scheme to classify rainbows, by
examining ‘hundreds of pictures of primary bows’, into one of 12 types in a
paper presented to the American Geophysical Union in Dec.2015 [41], which is,
sadly, paywalled. Apart from the abstract, all I could access is an interview with
Ricard [42] in which he says ‘four characteristics’ were examined: the primary
bow, the secondary bow, the dark band of Alexander separating the two, and the
additional supernumerary bows: whether all colours are visible, whether there
is a strong Alexander’s band, and whether there are supernumerary bows. The
single most important factor affecting the rainbow’s appearance was the height
of the sun in the sky. The less important factor was the size of the droplets: "wider
drops made rainbows less vivid, with more widely spaced hues.”
To end: the above discussion only scratches the surface. You
can get a much more complete coverage by checking out the references – which
are anyway, a small fraction of the total available on the internet. ‘Conjuring
a rainbow’ gives practical tips for spotting rainbows [43]. If you want to
delve into the maths of rainbows, try Adam for starters [44].
Angular width of rainbows and the critical angle:
About the angular width of the primary rainbow, Adam [44b], gives
the theoretical value as 1.7° (from 40.6° to 42.3°),
but adds that one must add 0.5° to this width, to take into account
the lack of parallelism of the Sun’s rays due to its finite size (as seen by an
observer on the ground). Volpi [9] states that the primary bow is between about 38.72° and 42.86°, from the
antisolar point. The upper value (for red) agrees with the estimates of others,
but the lower one (for violet) does not. It is probably a misprint.
However, a
different point relates to the critical angle θ, defined by:
sin(θ) =
1/n, where n is the refractive index of the water droplet.
“The critical angle for water (which would apply to
raindrops) is 48 degrees (relative to the normal). Therefore, if light strikes
the back of a raindrop at an angle greater than 48 degrees, it will be
reflected back. If the angle is smaller than 48 degrees, the light will simply
pass on through [57]. Since only one colour of light is observed from each
raindrop, an incredible number of raindrops is required to produce the
magnificent spectrum of colours that are characteristic of a rainbow [57].”
This argument is interesting but it ignores 3 things:
i)
the refractive index of water varies with
wavelength from violet to red so the critical angle varies from: 48° (V)
to 48.7°
(R).
ii)
the
finite diameter of the solar disk of 0.5° mentioned above.
iii)
The amount of sunlight incident on the drop is
as high as 1018 photons per second per cm2.
If one incorporates these factors, the ray diagram looks a
bit more like Fig.7b, than that used by [57] that shows a single ray for each
colour being reflected by the back surface of the raindrop in Fig.8:
Fig.8
If you still need something more or different, you could try
the other references [45 - 55]. Ref.45 discusses 12 kinds of rainbows (like
ref.42). Ref.49 suggests the best rainbow locations, including Hawaii, but also
the waterfalls (Niagara, Victoria, Iguazu etc). Yosemite wasn’t included!
The eye of the beholder:
The article [51] by Sara Chodosh is entitled: “Rainbows are
(literally) in the eye of the beholder”. According to the physics [56], someone
standing right next to you sees a different rainbow than you do, since the
reflected rays from a different set of raindrops are just at the right angles
to reach their eyes, to the set of rays that reach yours.
However, Chodosh makes a different point: some people are
colour blind, some are normal trichromats and others are tetrachromats (who can
see more colours than trichromats). Most mammals are trichromats, but dogs see
fewer colours than we do. On the other hand, the mantis shrimp sees thousands. Bees
see ultraviolet light. And even amongst trichromats, the exact distribution of
cones is different; the neural circuits that process vision also differ. So, we
all see different rainbows… for different reasons.
The longest-lasting rainbow:
Most
rainbows last for about an hour. But Ref.54 is a confirmed Guinness Record for
the longest lasting rainbow, seen in Taipei (Taiwan) for almost 9 hours in
Nov.2017! The previous record holder was 6 hours long in Yorkshire. The Yorkshire rainbow was from 09:00 to 15:00 on 14th
Mar.1994 [47].
“Their observations, pictures and video recordings showed the
rainbow lasted from 06:57 until 15:55 – totalling eight hours and 58 minutes.” [47].
Fig.9
The question may
recur: what happened to the rainbow moratorium? The answer is clear: the
rainbow was in the Yangmingshan mountains of Taiwan. Other factors that helped
produce the record: the north-east monsoon traps plenty of moisture in the air
and the winds were slow and steady (2-5 m/s) [55]. The video is available in
many sites (including [55]).
References:
1.
http://hyperphysics.phy-astr.gsu.edu/hbase/atmos/rbowpri.html
2.
https://lightcolourvision.org/diagrams/lower-the-sun-higher-the-rainbow
3.
https://www.flickr.com/photos/amazingsky/28614408407
4.
https://www.quora.com/Someone-tells-you-they-saw-a-rainbow-at-noon-why-do-you-know-they-are-lying.
5.
https://www.omnicalculator.com/physics/sun-angle
6.
https://atoptics.co.uk/blog/red-rainbows/
7.
https://www.reddit.com/r/CasualUK/comments/s0fols/a_rainbow_5_mins_before_sunrise_this_morning/
8.
Ken Ford, Libretext http://www.basic-physics.com/rainbows-figuring-their-angles/
9.
Federica
Volpi: https://inters.org/physics-of-rainbow
10.
Oikofuge: https://oikofuge.com/secondary-rainbows/#:~:text=Firstly%2C%20because%20the%20light%20enters,%C2%BA%20for%20the%20primary%20rainbow.
11.
Rachael Funnell: https://www.iflscience.com/full-circle-rainbows-happen-but-you-ve-gotta-be-at-the-right-elevation-67769
12.
https://apod.nasa.gov/apod/ap140930.html
13.
https://atoptics.co.uk/blog/complete-rainbow/
14.
http://www.dewbow.co.uk/bows/spray15.html
16.
Ryomoto:
https://gpm.nasa.gov/education/articles/shape-of-a-raindrop
17.
https://en.wikipedia.org/wiki/Rainbow
18.
https://physicsworld.com/a/the-subtlety-of-rainbows/
19.
Hüseyin Gazi
Topdemir, “Kamal Al-Din Al-Farisi’s explanation
of the rainbow” Humanity & Social Sciences Journal 2 (1): 75-85, 2007
20.
Roland Stull: https://geo.libretexts.org/Bookshelves/Meteorology_and_Climate_Science/Practical_Meteorology_(Stull)/22%3A_Atmospheric_Optics/22.01%3A_New_Page
21.
Marcus Selmke and Sarah Selmke , “Revisiting the
round-bottom glass flask rainbow experiment ” Am.J.Phys. Dec.2016).
22.
Shreya Mishra:
https://www.quora.com/If-a-drops-shape-is-spherical-how-can-it-act-as-a-prism-while-making-a-rainbow
23.
https://www.livescience.com/16405-quadruple-rainbow-photographed.html
24.
Michael Grossmann et al “Photographic evidence
for third-order rainbow” Applied Optics (Oct.2011) DOI: 10.1364/AO.50.00F134
25.
https://earthsky.org/earth/first-ever-image-of-5th-order-rainbow/
26.
Federica Volpi: https://inters.org/physics-of-rainbow
27.
https://gpm.nasa.gov/education/articles/shape-of-a-raindrop
29. Lock, J.
A., & Können, G. P. (2017). “Rainbows by elliptically deformed drops:
Möbius shift for high-order rainbows.” Applied Optics, 56(19), G88-G9
30. https://atoptics.co.uk/blog/a-perfect-rainbow/
31. Chris Baird: https://www.wtamu.edu/~cbaird/sq/2014/01/30/why-does-a-rainbow-contain-a-pure-spread-of-spectral-colours/
32. John Hardwick: https://physicsworld.com/a/the-subtlety-of-rainbows/
34. Remko
Uijlenhoet Hydrology and Earth System Sciences 5 (2001)615
36. Steven
Businger, “The secrets of the best rainbows on Earth “, Bulletin of American
Meteorological Society (Feb.2021)
37.
https://www.sciencedaily.com/releases/2022/10/221031104444.htm
38.
Kimberly M.Carlson et al ,” Global rainbow
distribution under current and future climates” Global Environmental Change 77
(2022) 102604
39.
Len Fisher, “Music inspired Newton’s rainbow”
(Nature 520(2015) 436)
40.
https://demonstrations.wolfram.com/RainbowsOfDifferentOrderInWaterDropletsAndGlassBeads/
41.
https://ui.adsabs.harvard.edu/abs/2015AGUFM.A53B0378R/abstract
42.
https://www.livescience.com/53149-12-flavors-of-rainbows-identified.html
43.
https://tasmaniangeographic.com/guide-to-rainbows/
44.
a) John A.Adam “An example of Nature’s mathematics: The
Rainbow” “ Digital Commons Virginia
Mathematics Teacher vol. 44, no. 1 19
b)
https://www.sciencedirect.com/science/article/abs/pii/S037015730100076X
45.
https://www.scienceabc.com/nature/are-rainbows-all-the-same.html
46.
https://science.howstuffworks.com/nature/climate-weather/atmospheric/question41.htm
47.
https://www.rmets.org/metmatters/how-are-rainbows-formed
49.
https://www.accuweather.com/en/travel/where-to-find-the-worlds-best-rainbows/632175/amp
50.
https://www.livescience.com/30235-rainbows-formation-explainer.html
51.
Sara Chodosh: https://www.popsci.com/why-rainbows-look-like/
52.
https://www.britannica.com/science/rainbow-atmospheric-phenomenon
53.
https://letstalkscience.ca/educational-resources/stem-explained/whats-in-a-rainbow
54.
Jason Daley: https://www.smithsonianmag.com/smart-news/9-hour-rainbow-sets-new-guinness-record-180968527/
55.
Cindy Sui: https://www.bbc.com/news/world-asia-42219665
56.
https://www.sciencefocus.com/science/why-do-millions-of-raindrops-only-make-one-rainbow
57.
http://ww2010.atmos.uiuc.edu/(Gh)/guides/mtr/opt/wtr/rnbw/frm.rxml
58.
https://personal.math.ubc.ca/~cass/courses/m309-03a/m309-projects/ryomoto/project.html
Fig.10
21st June 2024: moratorium time vs
latitude
|
Latitude |
Start time |
End time |
Duration (hrs
& mins) |
|
0 |
9:09 |
14:55 |
5 hrs 46 mins |
|
10 |
8:48 |
15:16 |
6 hrs 28 mins |
|
20 |
8:35 |
15:24 |
6 hrs 49 mins |
|
30 |
8:27 |
15:37 |
7 hrs 10 mins |
|
40 |
8:26 |
15:38 |
7 hrs 12 mins |
|
50 |
8:35 |
15:29 |
7 hrs 14 mins |
|
60 |
9:02 |
15:02 |
6 hrs |
|
70 |
10:45 |
13:21 |
3 hrs 36 mins |
Moratorium time vs latitude: 9th Apr.2024
|
Latitude |
Start time |
End time |
Duration (hrs
& mins) |
|
0 |
8:52 |
15:12 |
6 hrs 20 mins |
|
10 |
8:47 |
15:16 |
6 hrs 26 mins |
|
20 |
8:49 |
15:14 |
6 hrs 25 mins |
|
30 |
8:59 |
15:04 |
6 hrs 5 mins |
|
40 |
9:22 |
14:42 |
5 hrs 4 mins |
|
50 |
10:11 |
13:52 |
3 hrs 41 mins |
|
55 |
11:15 |
12:45 |
1 hr 30 mins |
April 8th 2024: Delhi latitude:
28.7 Longitude: 77.1
Dawn: 5:40
Sunset:
18:42
Sunrise: 6:04
Dusk: 19:06
Plot elevation vs time
|
time |
Elevation |
azimuth |
|
6:00 |
-1.38 |
80.78 |
|
6:30 |
5.1 |
84.35 |
|
5:30 |
-8.01 |
77.07 |
|
7:00 |
11.35 |
88.19 |
|
8:00 |
24.69 |
95.23 |
|
9:00 |
37.66 |
103.76 |
|
9:30 |
43.97 |
108.96 |
|
10:00 |
50.07 |
115.22 |
|
11:42 |
66.42 |
154.07 |
|
14:30 |
53.38 |
240.73 |
|
15:00 |
47.45 |
247.81 |
|
16:00 |
34.88 |
258.41 |
|
17:00 |
21.86 |
266.6 |
|
18:00 |
8.78 |
273.83 |
|
18:30 |
2.41 |
277.39 |
|
19:00 |
-4.28 |
281.02 |
Fig.11
Ryomoto’s [58] summary of how the rainbow changes as the
droplet size changes:
|
Diameter of
water drop |
Features of the
Rainbow |
|
~ 1-2mm |
The violet is very bright and the green is vivid. The rainbow
contains pure red, but barely any blue. There are many spurious bows,
violet-pink alternating with green without interruption into the primary bow. |
|
~ 0.5mm |
The red is significantly weaker. There are fewer
supernumerary bows, violet-pink and green are again alternating. |
|
~ 0.2-0.3mm |
There is no more red. The bow is broad and well developed
for the rest of the colours. The supernumerary bows become more yellow. If
the diameter of the drops is around 0.2mm, a gap occurs between the
supernumerary bows. If the diameter is less than 0.2mm, a gap is formed
between the primary bow and the first supernumerary bow. |
|
~ 0.08-0.1mm |
The bow is broader and paler, the only vivid colour is
violet. The first supernumerary bow is well seperated from the primary bow
and visibly shows tints of white. |
|
~ 0.06 |
A distinct white stripe is contained in the primary bow. |
|
< 0.05mm |
White Rainbows: Fogbows, Mistbows, Cloudbows |
