OK, so rainbows are produced when water reflects inside a water droplet at 40-42 degrees, or when reflected twice at 51 degrees (http://www.straightdope.com/classics/a910329.html). I knew the first number as I am currently reading Phillip Ball’s “Bright Earth”, where he mentions it … I searched here to see if the second was available and hurrah! it is. Thank you, Cecil.
Are these figures affected at all by impurities in the water? That is, now that rain is more acid than it was 200 years ago, has the angle changed? Does the amount of light reflected also vary with the acidity of the rain? This must be derivable using the index of refraction of the water (you can see this vary when you pour concentrated acid into water … obviously rain is not - yet! - so acid, but I don’t know the effects of smaller pH changes).
How about changes in rainbow formation as related to the size of the water droplets? Droplets won’t form without some kind of impurity to use as a seed (under normal conditions); have changes in the composition of the atmosphere had an effect on the distribution of sizes and hence on rainbows?
And finally, what is the degree of attenuation of the second rainbow of a pair as related to the first? I have never heard of, let alone seen, a triple rainbow. Presumably there is a rainbow corresponding to each internal reflection inside a water droplet; or are some or all of this series eliminated due to size restrictions on the droplets? If we lived in a space station with a pool and were able to create droplets 1 meter in diameter in a large cloud (a rather large station!) how many rainbows would be visible by sunlight?
Part of the reason I ask is because I moved to London, Ontario for eight years, bracketed by my home-town of Toronto. There were much better rainbows in London, bigger, brighter and a few doubles - and I can only think that either the air is better there, or that relative lack of towers gives more sky to observe.
I’ve given lectures on rainbows for years. For references, you might want to have a look at M. Minnaert’s book The Nature of Light and Color in the Open Air (Dover Publ.), Robert Greenler’s Rainbows, Halos, and Glories, R.A.R. Tricker’s Introduction to Meteorological Optics, or the relevant sections in Jeral D. Walker’s The Flying Circus of Physics. I’m sure there must be plenty of good photos of rainbows on the internet. I just visited halo sites last week in preparation for a lecture I gave on Friday, and the photos of ice crystal halos are gorgeous.
In answer to your queries:
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I think you’d need a lot of dissolved material to significantly alter the index of refraction of your water droplets and affect the rainbow position. Have a look at some of the data on refractive index vs. concentration in the CRC Handbook to see how slow a function this is.
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The size of the raindrop profoundly affects the appearance of the rainbow. In fact, you can tell the size of the droplets making up a rainbow by the relative widths of the bands of color. There’s even a useful table in Minnaert’s book. If you want to know why, there’s a detailed explanation in Tricker’s book. He’s the only one I know who gives a detailed derivation of Airy’s Rainbow Integral. In sjhort, rainbows aren’t really a refractive phenomenon, despite what they’ve told you. It’s a diffractive phenomenon, in which the wavefront interferes with itself. This also explain the supernumerary rainbows (Rainbows don’t stop after Violet – they keep going, alternating in color between green and pink).
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Secondary rainbows always have the same intensity relative to the primary bows, dictated by the intensity of the Fresnel reflection at its particular angle (I don’t know it offhand, and I’m too lazy to calculate it or look it up).(secondary bows are caused by light that enters the drop, reflects twice in the interior – as opposed to once for a primarty rainbow – then exits.) If the light forming the rainbows is too dim, and/or if there aren’t enough droplets at the required angle, you won’t see a secondary rainbow.
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Tertiary, quarternary, and higher order rainbows (in which the light reflects three times or four times inside before exiting) arre indeed formed, but you’re unlikely to see them in nature. Each order is successively weaker in intensity, and the 3rd and 4th order bows are located very close to the sun itself, not near the primary and secondary bows. They were first observed in the Middle Ages in the laboratory of an Arabic scientist whose name escapes me at the moment (see the book The Rainbow fromh toathematics for details)
That last reference got garbled. It should be C. Boyer’s The Rainbow from Myth to Mathematics. The Arab scientist had one of those twenty barreled names (everybody, it seems, had them in the Middle Ages), but it ended in “al Shirazi”. He had a similarlyong-winded-named pupil, whose name ended in “al Farisi”. They did their work about twenty years before a Grench/German monk named Theodoric of Freibourg (or Dietrich of Freibourg, according to others, or Thierry of Freibourg, according to still others) did his remarkably modern work. Theodoric’s diagrams still survive, and they are surprisingly modern. What amazes me is that a.) there is no known connection between these two groups, working 1000 miles apart, and b.) Their parallel work was virtuallty unknown to their contemporaries, or to folks after them. For another four hundred years or so rainbow theory continued to be erroneous and off-the-wall, as if these guys had not existed.