I’ve read that pilots have seen a COMPLETE rainbow, not just the arch. I have my doubts! Is it possible? If it is, how come I’ve never seen a photograph?
It is possible.
Rainbows are not particularly easy to photograph well.
Pilots are busy.
Rainbows are formed when light strikes water droplets at an angle of 48° with respect to an observer, coming from behind the observer. Since a person on the ground has the horizon of the earth (to say nothing of hills/mountains, trees, and buildings) reducing the arc through which a raincircle could be seen, the rainbow will always be an arc that is interrupted by the earth.
However, in an airplane, the horizon will be much farther away–often far enough distant that the rainbow can be seen between the observer and the horizon. Due to the narrow range of angles needed to produce a rainbow, seeing one from an airplane is probably still not a frequent occurrence, but it is possible.
This site gives a pretty good description of the physics of rainbow creation (Ul de Rico notwithstanding).
We had a thread about this a month or so back – check the archives. It IS possible to see a omplete rainbow, but the trick is to be high enough to do so – you have to be in an airplane or a mountaintop to do so. I have a science book from the 1940s that shows a sktch f such a situation. But you also need to have suspended waterdroplets in the full 360 degrees.
Much more common is the GLORY, in which you can see a complete 360 degree rainbow-like circle around the shadow of you airplane as it is projected on a cloudbank beneath you. I just saw one last week when I flew cross-country. In fact, I almost always see one when I fly. But a Glory isn’t a Rainbow, although they’re related.
Here’s the thread:
http://boards.straightdope.com/sdmb/showthread.php?threadid=48587
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I remember using a regular garden hose when I was a kid (washing a car or something) and “creating” a rainbow and following it around in a circle with the hose. Try it one day. Its fun 
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I’ve seen that effect, and had no idea what it was. I’ve been watching out for it every time since then, and begun to wonder if I really saw it or if it was an artefact of the window glass, for example. I’m so pleased to be able to put a name on it, especially such an impressive-sounding name.
Hibernicus:
Glad to hear this was of use to you! Congrartulations on noticing it – I’m sure most people don’t! If you want more info on glories, here’s my usual litany of books:
“Rainibows, Haloes, and Glories” by Greenler
“The Physics of Color in the Open Air” by M. Minnaert (sic)
“Intro. to Meteorological Optics” by R.A. R. Tricker
“The Flying Circus of Physics” by Jearl D. Walker
“Physics from your Airplane Window” by ???
[Mr. Rogers]Kids, it’s easy to make a full-circle rainbow with simple things you can find around your house.[/Mr. Rogers]
Although I discovered this by accident, it is pretty simple. All you need is a sunny day, a bathing suit and lawn sprinkler that sprays drops in a full circle.
Step 1. Wait till the sun is high in the sky.
Step 2. Don bathing suit. (Do this indoors, even on a sunny day.)
Step 3. Turn on lawn sprinkler. (Do this outdoors!)
Step 4. Stand in the middle of the circle of water drops.
Step 5. Look down.
The only tricky part is making sure the sun is high enough. The rainbow will be centered on the anti-solar point – that is, the point directly opposite the sun. If the sun is low the rainbow will be off at an angle and won’t make a complete circle.
Have fun. Be sure to wear safety goggles. And have an adult nearby.
Or Light and Color in the Outdoors by M. G. J. Minnaert - highly recommended. I don’t know if these are the same book but the descriptions sound similar.
I got curious and found this picture of a glory: