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How big’s your bow?

Question: What determines the size of a rainbow? They obviously vary as shown
by double rainbows.

Answer: The size of a rainbow is fixed by the way the Sun’s rays go through
the raindrops. When a light ray strikes a raindrop, part of it is reflected and
lost and part is refracted into the drop. When this ray hits the back of the
drop, part of it is refracted out and part is reflected back to the front
surface. Part of this reflected ray is again reflected (see later) and part is
refracted back out.

If the original ray hits near the centre, it will be deflected by 180°
and return along the same path. This is how catseyes work, but you will never
see sunlight reflected this way because of the shadow cast by your head.

But what happens if the original ray hits the raindrop off-centre? As the
point of contact moves away from the centre it reaches a point where many rays
return virtually in the same line, and reinforce each other to make a bright
return at 41° from the sun-line—the line from the Sun to the raindrop.
These returns happen at all points around the sun-line, and combine to form a
bright cone of angle 41° with its axis on the sun-line
(see raindrop B in the diagram).
The light ray is split into its component wavelengths by the
raindrop, and different colours are refracted by different amounts—red
less. blue more. So the bright cone shows rainbow colours, with red on the
outside.

If you look at a sunlit sky, full of raindrops, your eye will be on the
surface of the bright cone of raindrops 41° from your antisun-line—the
line running from your eye to the top of your shadow on the ground. So you will
see the rainbow as a circle that is 41° from the antisun-line, with the red
on the outside. The original rays which hit the drop at the wrong place to form
the rainbow will produce a very faint return, always less than 40° from the
antisun-line, and so inside the rainbow. This makes the sky appear darker above
the bow.

However, a secondary bow can form outside the primary. It is caused by a
double reflection of rays striking raindrops. Some of the lost reflected light
mentioned in the first paragraph can be reflected twice in the raindrop (see
raindrop A in the diagram) and therefore still reaches an observer on the ground
as it finally exits the drop at an angle of around 52° from the
antisun-line. The fact it is reflected twice means the red will now be on the
inside of the cone, and fainter.

Professor A. Black

Winchester, Hampshire

Answer: The variation in apparent size of rainbows is due to several factors.
If the Sun is higher in the sky then less of the rainbow’s arc will be above the
horizon (where it is more visible), and hence it will seem smaller—even
though it is still 41° from the antisolar point. The antisolar point is the
point where an imaginary ray connecting the Sun and the observer meets the
ground, coinciding with the top of the observer’s shadow. If the Sun is above
the horizon, the antisolar point is below the horizon. If the Sun is below the
horizon, the antisolar point will be in the sky.

Similarly, the extent and distance of the water droplets (from the observer)
can give rise to partial arcs, which obviously appear smaller than a full bow.
Finally a rainbow’s relative size is subject to the same optical illusion that
makes the Moon appear larger when it is lower down in the sky—we can more
readily compare its size to the objects on the horizon. So a rainbow behind some
houses may appear smaller than a rainbow spanning the open countryside.

Andy Richmond

London

Answer: The relative position of the Sun, the observer and the bow, lying in
a straight line in that order, was understood by the ancient Greeks. Aristotle
reported that the bow is part of the circumference of the base of a cone with
the Sun at the apex and the eye of the observer on the line from apex to the
centre of the base. This description was extended by medieval philosophers
working between 1200 and 1300 which culminated in the writing of Theodric of
Frieberg in the first decade of the next century. Theodric showed that the
rainbow was formed by individual droplets of water when sunlight falls on rain
or mist.

More than 300 years later René Descartes set out the basis for modern
science in his Discourse on method (1637). Among the appendices is an
extended explanation of the rainbow. In this he identifies 41° 47′ as the
angle between sunlight entering the drops and the maximum intensity of the red
rays leaving to form the primary bow. For the secondary bow, Descartes
calculated the angle as 51° 37′.

J. O. Marsh

Lecturer in History of Science and Technology

Manchester University

This week’s questions

Self-pouring: Recently I saw a demonstration of an amazing liquid that could
pour itself out of a jar. All you had to do was tip the jar slightly so it began
to run out. It then continued until it was all gone, even though the jar was
straightened. What was the liquid and how on Earth does it do this?

Billy Statham

Huddersfield, West Yorkshire

Rise and fall: Last week I had my weight measured. While standing on the
scale I saw that the measuring needle moved slowly up and down. Is it because
I’m breathing in and out, or is there another explanation?

Jane Hanson

London