What is a rainbow?
Author Donald Ahrens in his text Meteorology Today
as "one of the most spectacular light shows observed on earth". Indeed
the traditional rainbow is sunlight spread out into its spectrum of colors
and diverted to the eye of the observer by water droplets.
The "bow" part of the word describes the fact that the rainbow is a group of
nearly circular arcs of color all having a common center.
Where is the sun when you see a rainbow?
This is a good question to start thinking about the physical process that
gives rise to a rainbow. Most people have never noticed that the sun is
always behind you when you face a rainbow, and that the center of the
circular arc of the rainbow is in the direction opposite to that of the sun.
The rain, of course, is in the direction of the rainbow.
What makes the bow?
A question like this calls for a proper physical answer. We will discuss
the formation of a rainbow by raindrops. It is a problem in optics that
was first clearly discussed by Rene Descartes in
1637. An interesting
historical account of this is to be found in Carl Boyer's book, The
Rainbow From Myth to Mathematics. Descartes simplified the study of
the rainbow by reducing it to a study of one water droplet and how it
interacts with light falling upon it.
He writes:"Considering that this bow appears not only in the sky, but
also in the air near us, whenever there are drops of water illuminated by
the sun, as we can see in certain fountains, I readily decided that it arose
only from the way in which the rays of light act on these drops and pass
from them to our eyes. Further, knowing that the drops are round, as has
been formerly proved, and seeing that whether they are larger or smaller, the
appearance of the bow is not changed in any way, I
had the idea of making a very large one, so that I could examine it better.
Descarte describes how he held up a large sphere in the sunlight and looked at
the sunlight reflected in it. He wrote "I found that if the sunlight
came, for example, from the part of the sky which is marked AFZ
and my eye was at the point E, when I put the globe in position BCD, its part
D appeared all red, and much more brilliant than the rest of it; and that
whether I approached it or receded from it, or put it on my right or my left,
or even turned it round about my head, provided that the line DE always
made an angle of about forty-two degrees with the line EM, which we are to
think of as drawn from the center of the sun to the eye, the part D appeared
always similarly red; but that as soon as I made this angle DEM even a
little larger, the red color disappeared; and if I made the angle a little
smaller, the color did not disappear all at once, but divided itself first as
if into two parts, less brilliant, and in which I could see yellow, blue,
and other colors ... When I examined more particularly, in the globe BCD,
what it was which made the part D appear red, I found that it was the rays of
the sun which, coming from A to B, bend on entering the water at the point B,
and to pass to C, where they are reflected to D, and bending there again as
they pass out of the water, proceed to the point ".
This quotation illustrates how the shape of the rainbow is explained. To
simplify the analysis, consider the path of a ray of monochromatic light
through a single spherical raindrop. Imagine how light is refracted as it
enters the raindrop, then how it is reflected by the internal, curved,
mirror-like surface of the raindrop, and finally how it is refracted as it
emerges from the drop. If we then apply the results for a single raindrop
to a whole collection of raindrops in the sky, we can visualize the shape
of the bow.
The traditional diagram to illustrate this is shown here as adapted from
Humphreys, Physics of the Air.
It represents the path of one light ray incident on a water droplet from the
direction SA. As the light beam enters the surface of the drop at A, it
is bent (refracted)
little and strikes the inside wall of the drop
at B, where it is reflected back to C. As it emerges from the drop it is
refracted (bent) again into the direction CE. The angle D represents a
measure of the deviation of the emergent ray from its original direction.
Descartes calculated this deviation for a ray of red light to be about
180 - 42 or 138 degrees.
The ray drawn here is significant because it represents the ray that has the
smallest angle of deviation of all the rays incident upon the raindrop. It
is called the Descarte or rainbow ray and much of the
sunlight as it is refracted and reflected through the raindrop is focused
along this ray. Thus the reflected light is diffuse and weaker except near
the direction of this rainbow ray. It is this concentration of rays
near the minimum deviation that gives rise to the arc of rainbow.
The sun is so far away that we can, to a good approximation, assume that
sunlight can be represented by a set of parallel rays all falling on the
water globule and being refracted, reflected internally, and refracted again
on emergence from the droplet in a manner like the figure. Descartes writes
I took my pen and made an accurate calculation of the paths of the rays
which fall on the different points of a globe of water to determine at which
angles, after two refractions and one or two reflections they will come to
the eye, and I then found that after one reflection and two refractions
there are many more rays which can be seen at an angle of
from forty-one to forty-two degrees than at any smaller angle; and that there
are none which can be seen at a larger angle" (the angle he is referring
to is 180 - D).
A typical raindrop is spherical and therefore its
effect on sunlight is
symmetrical about an axis through the center of the drop and the source of
light (in this case the sun). Because of this symmetry, the
two-dimensional illustration of the figure serves us well and the complete
picture can be visualized by rotating the two dimensional illustration about
the axis of symmetry. The symmetry of the focusing effect of each drop is
such that whenever we view a raindrop along the line of sight defined by
the rainbow ray, we will see a bright spot of reflected/refracted
sunlight. Referring to the figure, we see that the rainbow ray for
red light makes an angle of 42 degrees between the direction of the incident
sunlight and the line of sight. Therefore, as long as the raindrop is viewed
along a line of sight that makes this angle with the direction of incident
light, we will see a brightening. The rainbow is thus a circle of angular
radius 42 degrees, centered on the antisolar point, as shown schematically
We don't see a full circle because the earth gets in the way. The lower the
sun is to the horizon, the more of the circle we see -right at sunset, we
would see a full semicircle of the rainbow with the top of the arch 42
degrees above the horizon. The higher the sun is in the sky, the smaller is
the arch of the rainbow above the horizon.
What makes the colors in the rainbow?
The traditional description of the rainbow is that it is made up of seven
colors - red, orange, yellow, green, blue, indigo, and violet. Actually,
the rainbow is a whole continuum of colors from red to violet and even
beyond the colors that the eye can see.
The colors of the rainbow arise from
two basic facts:
Descartes and Willebrord Snell had determined how a ray of light is bent, or
refracted, as it traverses regions of different densities, such as air
and water. When the light paths through a raindrop are traced for red and
blue light, one finds that the angle of deviation is different for the two
colors because blue light is bent or refracted more than is the red light.
This implies that when we see a rainbow and its band of colors we are
looking at light refracted and reflected from different raindrops,
some viewed at an angle of 42 degrees; some, at an angle of 40 degrees, and
some in between. This is illustrated in this
adapted from Johnson's Physical Meteorology. This rainbow of two
colors would have a width of almost 2 degrees (about four times larger
than the angular size as the full moon). Note that even though
blue light is refracted more than red light in a single drop, we see the
blue light on the inner part of the arc because we are looking along a
different line of sight that has a smaller angle (40 degrees) for the blue.
Sunlight is made up of the whole range of colors that the eye can detect.
The range of sunlight colors, when combined, looks white to the eye.
This property of sunlight was first demonstrated by Sir Isaac Newton in 1666.
Light of different colors is refracted by different amounts when it passes
from one medium (air, for example) into another (water or glass, for example).
Ana excellent laboratory exercise on the mathematics of rainbows is here,
and F. K. Hwang has produced a fine Java Applet illustrating
this refraction, and Nigel Greenwood has written a program that operates in
MS Excel that
illustrates the way the angles change as a function of the sun's angle.
What makes a double
Sometimes we see two rainbows at once, what causes this? We have followed
the path of a ray of sunlight as it enters and is reflected inside the
raindrop. But not all of the energy of the ray escapes the raindrop after it
is reflected once. A part of the ray is reflected again and travels along
inside the drop to emerge from the drop. The rainbow we normally see is
called the primary rainbow and is produced by one internal
reflection; the secondary rainbow arises from two internal
reflections and the rays exit the drop at an angle of 50 degrees° rather
than the 42°degrees for the red primary bow. Blue light emerges at an
even larger angle of 53 degrees°. his effect produces a secondary rainbow
that has its colors reversed compared to the primary, as illustrated in the
drawing, adapted from the Science Universe Series
Sight, Light, and Color.
It is possible for light to be reflected more than twice within a raindrop,
and one can calculate where the higher order rainbows might be seen; but these
are never seen in normal circumstances.
Why is the sky brighter inside a rainbow?
Notice the contrast between the sky inside the arc and outside it. When one
studies the refraction of sunlight on a raindrop one finds that there are
many rays emerging at angles smaller than the rainbow ray, but
essentially no light from single internal reflections at angles greater
than this ray. Thus there is a lot of light within the bow, and very little
beyond it. Because this light is a mix of all the rainbow colors, it is
white. In the case of the secondary rainbow, the rainbow ray is the
smallest angle and there are many rays emerging at angles greater than this
one. Therefore the two bows combine to define a dark region between them -
called Alexander's Dark Band, in honor of Alexander of Aphrodisias who
discussed it some 1800 years ago!
What are Supernumerary Arcs?
In some rainbows, faint arcs just inside and near the top of the primary
bow can be seen. These are called supernumerary arcs and were explained by
Thomas Young in 1804 as arising from the interference of light along certain
rays within the drop. Young's work had a profound influence on theories of
the physical nature of light and his studies of the rainbow were a fundamental
element of this. Young interpreted light in terms of it being a wave of some
sort and that when two rays are scattered in the same direction within a
raindrop, they may interfere with each other. Depending on how the rays mesh
together, the interference can be constructive, in which case the rays
produce a brightening, or destructive, in which case there is a
reduction in brightness. This phenomenon is clearly described in
Nussenzveig's article "The Theory of the Rainbow" in which he writes:
"At angles very close to the rainbow angle the two paths through the
droplet differ only slightly, and so the two rays interfere constructively.
As the angle increases, the two rays follow paths of substantially different
lengths. When the difference equals half of the wavelength, the interference
is completely destructive; at still greater angles the beams reinforce
again. The result is a periodic variation in the intensity of the scattered
light, a series of alternately bright and dark bands."
Mikolaj and Pawel Sawicki have posted several beautiful
of rainbows showing these arcs.
The "purity" of the colors of the rainbow depends on the size of the
raindrops. Large drops (diameters of a few millimeters) give bright
rainbows with well defined colors; small droplets (diameters of about
0.01 mm) produce rainbows of overlapping colors that appear nearly
white. And remember that the models that predict a rainbow arc all assume
spherical shapes for raindrops.
There is never a single size for water drops in rain
but a mixture of many sizes and shapes. This results in a
composite rainbow. Raindrops generally don't "grow" to radii
larger than about 0.5 cm without breaking up because of
collisions with other raindrops, although occasionally drops
a few millimeters larger in radius have been observed when
there are very few drops (and so few collisions between the
drops) in a rainstorm. Bill Livingston suggests: " If you
are brave enough, look up during a thunder shower at the
falling drops. Some may hit your eye (or glasses), but this
is not fatal. You will actually see that the drops are
distorted and are oscillating."
It is the surface tension of water that moulds raindrops
into spherical shapes, if no other forces are acting on them.
But as a drop falls in the air, the 'drag' causes a
distortion in its shape, making it somewhat flattened.
Deviations from a spherical shape have been measured by
suspending drops in the air stream of a vertical wind tunnel
(Pruppacher and Beard, 1970, and Pruppacher and Pitter,
1971). Small drops of radius less than 140 microns (0.014
cm) remain spherical, but as the size of the drop increases,
the flattening becomes noticeable. For drops with a radius
near 0.14 cm, the height/width ratio is 0.85. This flattening
increases for larger drops.
Spherical drops produce symmetrical rainbows, but rainbows
seen when the sun is near the horizon are often observed to
be brighter at their sides, the vertical part, than at their
Alistair Fraser has explained this phenomenon as resulting
from the complex mixture of size and shape of the raindrops.
The reflection and refraction of light from a flattened water
is not symmetrical. For a flattened drop, some of the rainbow
ray is lost at top and bottom of the drop. Therefore, we see
the rays from these flattened drops only as we view them
horizontally; thus the rainbow produced by the large drops is
is bright at its base. Near the top of the arc only small
spherical drops produce the fainter rainbow.
What does a rainbow look like through dark glasses?
This is a "trick" question because the answer depends on whether or not your
glasses are Polaroid. When light is reflected at certain angles it becomes
polarized (discussed again quite well in Nussenzveig's article), and it has
been found that the rainbow angle is close to that angle of
reflection at which incident, unpolarized light (sunlight) is almost
completely polarized. So if you look at a rainbow with Polaroid sunglasses
and rotate the lenses around the line of sight, part of the rainbow will
Other Questions about the Rainbow
Humphreys (Physics of the Air, p. 478) discusses several "popular" questions
about the rainbow:
"What is the rainbow's distance?" It is nearby or far away, according
to where the raindrops are, extending from the closest to the
farthest illuminated drops along the elements of the rainbow cone.
Why is the rainbow so frequently seen during summer and so seldom during
To see a rainbow, one has to have rain and sunshine. In the winter,
water droplets freeze into ice particles that do not produce a rainbow but
scatter light in other very interesting patterns.
"Why are rainbows so rarely seen at noon?" Remember that the center of the
rainbow's circle is opposite the sun so that it is as far below the level
of the observer as the sun is above it.
"Do two people ever see the same rainbow?"
Humphreys points out that "since the rainbow is a special distribution of
colors (produced in a particular way) with reference to a definite point -
the eye of the observer - and as no single distribution can be the same
for two separate points, it follows that two observers do not, and cannot,
see the same rainbow." In fact, each eye sees its own rainbow!!
Of course, a camera lens will record an image of a rainbow which can
then be seen my many people! (thanks to Tom and Rachel Ludovise for
pointing this out!)
- "Can the same rainbow be seen by reflection as
On the basis of the arguments given in the preceding question, bows
appropriate for two different points are produced by different drops; hence,
a bow seen by reflection is not the same as the one seen directly".
What are Reflection Rainbows?
A reflection rainbow is defined as one produced by the reflection of the
source of incident light (usually the sun). Photographs of them are perhaps
the most impressive of rainbow photographs. The reflected rainbow may be
considered as a combination of two rainbows produced by sunlight coming from
two different directions - one directly from the sun, the other from the
reflected image of the sun. The angles are quite different and therefore
the elevation of the rainbow arcs will be correspondingly different.
This is illustrated in a diagram adapted from
Greenler"s Rainbows, Halos, and Glories. The rainbow produced by
sunlight reflected from the water is higher in the sky than is the one
produced by direct sunlight.
What is a Lunar Rainbow?
A full moon is bright enough to have its light refracted by raindrops
just as is the case for the sun. Moonlight is much fainter, of
course, so the lunar rainbow is not nearly as bright as one
produced by sunlight. Lunar rainbows have infrequently been
observed since the time of Aristotle or before. A graphic description of one was writen by Dr. Mikkelson.
Rainbows and Proverbs
There is a delightful book by Humphreys entitled Weather Proverbs and
Paradoxes. In it, he discusses the meteorological justifications of
some proverbs associated with rainbows, such as "Rainbow at night,
shepherd's delight;Rainbow in morning, shepherds take warning,"If there be a
rainbow in the eve,It will rain and leave; But if there be a rainbow in
the morrow It will neither lend nor borrow", and Rainbow to windward, foul
fall the day; Rainbow to leeward, damp runs away."
The meteorological discussion Humphreys presents is appropriate for the
northern temperate zones that have a prevailing wind, and also for a
normal diurnal change in the weather.
William Livingston, a solar astronomer who has also specialized in
atmospheric optical phenomena suggests the following: "Try a hose spray
yourself. As you produce a fine spray supernumeraries up to order three
nicely visible. "Try to estimate the size of these drops compared to a
raindrop. ..."Another thing to try. View a water droplet on a leaf
close-up - an inch from your eye. At the rainbow angle you may
catch a nice bit of color!"
In Minnaert's excellent book Light and Colour in the Open Air you can
find a number of experiments on how to study the nature of rainbows. Here
is an illustration of one of his suggestions. Other demonstration projects are
listed here .
Meg Beal, while a seventh-grader, prepared a science fair project that
illustrated the nature of rainbows. The Beal family provided a
photograph (1MB) of her excellent demonstration.
For those wanting to try to demonstrate the nature of a rainbow in a
classroom, here are examples.
An informative tutorial on optics can be found here.
I am indebted to William C. Livingston, astronomer at the
Astronomy Observatory in Tucson Arizona for
his expert assistance in preparing this paper, and to Seth Sharpless for his
critical reading of the manuscript. Charles A. Knight, an expert on rain
at the National Center
Atmospheric Physics, provided valuable guidance on the interesting
properties of raindrops.
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