Somewhere inside a rainbow
Q: I saw a lovely rainbow recently. The sky just inside the bow seemed
brighter than the sky outside the bow. Why? Tom, Bo'ness, Scotland
 Note how the sky brightens across the bow
and on inside. Photo courtesy of NASA.
A:
The sky inside a primary bow is brighter than the sky immediately outside the
bow, because raindrops scatter sunlight into the opposite part of the sky to
form a circular disc of light there. The bright disc is the entire rainbow.
We usually think of the rainbow as an arc of color, but it's much more than
that. Someone as observant as you sees the brightened sky inside,
too. In fact, the colored arc is only the brighter outer rim of the disc.
Raindrops scatter light primarily in the direction of the arc,
but also inside. The earth often cuts off the
rest of the circular disc so we only see a part-of-the-circle rainbow.
Click
here to see a rainbow from an airplane that shows more of the whole
circle.
In fact, "what we call the rainbow is actually the (coloured) edge of a disc
of light," emails
Alistair Fraser, coauthor of The Rainbow Bridge: Rainbows in Art,
Myth, and Science. The raindrops scatter sunlight to form a
many-layered stack of colored discs. "Each colour has a slightly
different sized disc", and, since they overlap except for the edge, the
overlapping colors give white, which brightens the sky on the inside of the
circle. On the edge, however, the different-sized colored discs don't
overlap and display their respective colors — a
rainbow
arc.
You ask why; the answer is part of a winding path through many theories, as our
understanding has deepened to explain the bow's subtleties. Let's start
with how the bow gets into the opposite part of the sky. When we stand
facing the bow, the Sun is behind us and the center of the bow's circular disc
is at our head's shadow. See figure. How does the distorted Sun's image get in
front of us?
The Sun is behind the gentleman.
A ray of light shining from the Sun enters a nearly spherical
raindrop (the blue circle in the figure) in front of him. The light's
path (shown in yellow) then bends twice and reflects once to come into his eye
in front of him although the Sun is behind him. The point C in the figure
is the shadow of his head, which is also the center of the rainbow's circular
disc.
Descartes depicts the falling curtain of rain by the many dots in the picture,
and enlarges one such drop into the blue circle.
 A
rainbow as depicted by Descartes (modified by the author). A sun's ray bends
twice (at B1 and B2) and
reflects once (at R) to form
the bow on the other side of the Sun. Point C shows the gentleman's head's
shadow, which is also the center of the bow's circular disc. Drawing courtesy of Wikipedia.
Now for the main question: why is the inside of the bow bright also?
Most of the light coming into a raindrop follows the yellow path shown in the
figure (and forms the bright colorful arc of the primary bow) but not all
light follow this path. A great deal of light follows paths that bend
even more than the 138 degrees shown, and that light reflects to our eyes from
throughout the raindrop, brightening the inside of the bow in the same way as it
brightens the colorful arc.
Descartes was clever enough to figure this out.
That's it for why the inside of the primary bow is brighter. We could
elaborate on more rainbow theories each encompassing and refining an earlier
theory to explain yet another bow subtlety. We, however, have done the
main job: why the inside is bright.
Basically, the rainbow is a circular disc
of light — bright inside, too — because raindrops,
which are essentially tiny symmetric
spheres, scatter light over an entire circular
disc in the sky.
Further Reading:
What causes a rainbow?
I
saw a lovely rainbow recently. The sky just inside the bow seemed brighter
than the sky outside the bow. Why?
How high do you have to be to see a full-circle rainbow?
How a full-circle rainbow looks
Why
are the colors in the second rainbow backwards?
What a rainbow looks like to a dinosaur, WonderQuest
The Rainbow Bridge: Rainbows in Art, Myth,
and Science by Raymond L. Lee Jr and Alistair B. Fraser
The
rainbow cone (and circle) by Les Cowley, Atmospheric Optics
The optics of a
water drop by Philip Laven
Rainbow,
Wikipedia
What is a
rainbow?
The National Center for Atmospheric
Research
(Answered Jan. 8, 2007, Updated July 22, 2007)
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