No rainbows in the
valley of Shangri-La
When I went trekking in the Himalayas
last month, it rained quite a bit on each day – but mostly after
noon. Unfortunately, despite all the rain, we were not lucky enough
to see a rainbow. I decided that the blame – as usual – lay in
the laws of physics (optics and atmospheric science) and on geometry.
After all, the Sun has to be behind you, and the falling rain in
front, and the clouds should not obscure the Sun's rays, etc.
Our camp was located in a valley – so
that is how I imagined a valley in which rainbows are forevere
forbidden to its denizens. Rather depressing, truly, but I was
already feeling deprived – so the only solace was to concieve of
people even more miserable.
According to wikipedia, if the Sun is
more than 42 degrees from the horizontal, you cannot see a rainbow.
This refers to the primary bow, which lies between 137.6 degrees
(red) and 139.4 degrees (blue). The secondary – which is fainter,
and reversed in sequence – bow lies in the range of angles 129.6
(red) and 126.5 degrees (blue). That means that the secondary bow
would still be visible if the Sun is as high in the sky as 53.5
degrees.
One does not get into the gory details
of trigonometry but it is easy to construct many valleys in which no
rainbows - either primary or secondary - could be seen. Of course,
in some valleys primary bows will not be visible – but secondary
bows can still be seen.
For a valley which is a perfect circle,
with a radius of 1000 metres, the encircling mountain range would
need to be 916 metres high to forbid all primary rainbows, and 1356
metres high to forbid even seconday rainbows. For more realistic
shapes, a little trig would be needed.
The secondary rainbow is about 10% of the total intensity of the
primary bow – and its angular range is about 75% greater. Sightings
of seconday rainbows are rarer than primary bows, because the size
distribution of raindrops needs to be more uniform to get good
intensity.
What about higher-order rainbows? They
are so faint as to be almost invisible against solar glare.
Tertiary and quaternary rainbows have
been seen by the most educated and determined rainbow-chasers in
2011:
http://www.sciencedaily.com/releases/2011/10/111005111001.htmhttp://www.atoptics.co.uk/rainbows/ord34.htm
Unfortunately, the third- and fourth-order rainbows will not slake the thirst of our encircled denizens either. Firstly, these two bows are in the direction of the Sun – rather than opposite to the Sun, as are the first- and second-order bows. That makes the glare problem more acute than for the second-order bow. Secondly, although the total intensity of the 3ed order bow is 24% of the 1st order bow, it is spread over a larger range of angles – so it is fainter. And, lastly, the angle of the 3rd order is at about 40 degrees – almost the same angle to the horizontal as the 1st order bow (but in the opposite direction). The 4th order bow has 15% of the total intensity of the 1st order bow, and its angle is ~45 degrees to the horizontal. So encircling mountains of a similar height would impose a quarantine on rainbows in the valley.
References:
Photographic evidence for the third-order rainbow, M Grossmann, E Schmidt, & A Haussmann, Applied Optics, Vol. 50, Issue 28, pp. F134-F141 (2011).
Photographic observation of a natural fourth-order rainbow, M Theusner, Applied Optics, Vol. 50, Issue 28, pp. F129-F133 (2011)
All this is somewhat depressing for the dwellers of Shangri-La. Keats in his 1820 poem Lamia lamented Newton's decomposing white light and deconstructing the rainbow, accusing him of (sic) “unweaving the rainbow”.
Richard Dawkins has written a book with exactly that title as a rather belated riposte that “ Science is - or ought to be – the inspiration for great poetry”.
That debate will undoubtedly carry on – but what I think true is that few poets would have the wit or the patience to chase down the tertiary and quaternary rainbows.
Or to do as Jearl Walker did, do an experiment to see a dozen bows in a single drop of water (suspended on a wire and artificially illuminated)!
However, the knowledge of rainbows
should be more widely known to the general public – especially poets – because even if the vagaries of weather go
against you, you can still see, without too much trouble, rainbows in
fountains. Providing the sun direction is right and as long as you do
not live in an deprived no-rainbow valley!
