What conditions would make it possible to see a naturally occurring fully 360° circular rainbow? Would it even be possible?
6 Answers
The centre of a rainbow is where your shadow would fall. When you're standing on level ground your shadow is obviously on the ground so you only see part of the arc. If you're in an aeroplane or on a mountaintop or somewhere where your shadow would be above ground, you may be able to see the whole circle.
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Glories come to mind. They are easily seen airborne as mentioned by Vineet.
One can also produce a full rainbow in the lab by reflecting light off tiny glass spheres. Frank Crawford (of Berkeley physics Waves) has described it beautifully in an AJP article. He called it "Rainbow dust".
A friend of mine reproduced this effect in the lab sometime back. Here is the picture.
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Photo by H R Madhusudhan, Bangalore Association for Science Education, Jawaharlal Nehru Planetarium
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Yes it can be seen, rainbow can be considered as a set of points which form an angle of $42^{\circ}$ with the sun and our eyes, which comes out to be a perfect circle with center at our shadow made by the light coming from the sun. Usually we see a rainbow from ground and from there these set of points form a semicircle in the atmosphere. But if we observe a rainbow from high enough altitude we can see a complete circular rainbow.

Image source: Colin Leonhardt (Birdseye View Photography). For more information see NASA's Atronomy Picture of the Day, 2014-09-30.
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You can generate a circular rainbow using a garden sprinkler on a sunny day. The top of the rainbow is usually easy to spot - but, by spraying water drops near your feet, the bottom of the rainbow should also become visible.
Waterfalls are great places to see natural rainbows - and, if you are lucky, you might see a circular rainbow, as captured in the video at http://www.philiplaven.com/p5a.html
Most of the "circular rainbows" shown on the web are NOT rainbows! Remember that rainbows are huge: the red arc of the primary rainbow has an angular radius of 42 degrees, thus a full circle rainbow has a diameter of 84 degrees. To capture an entire circular rainbow, you need a camera with an ultra-wide-angle lens (e.g. a fisheye lens).
If you have a picture of a SMALL circular rainbow taken with a normal lens, it is almost certainly a glory. Glories typically appear as one or more coloured rings on clouds surrounding the shadow of an aircraft. Glories usually have an angular radius of less than 5 degrees.
On the other hand, if you see a LARGE circle around the sun, this is probably a halo caused by ice crystals in the atmosphere - see http://www.atoptics.co.uk/halo/circular.htm. Most halos have an angular radius of 22 degrees, but some have an angular radius of 46 degrees.
Good rainbow hunting .....
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Go to the Niagara falls on a sunny day, and you can see there a beautiful circular rainbow creeated by dispersion at the tiny drops of the hazy mist created by the waterfall. I have seen it and was very impressed.
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Yes it's possible. Although small in size, I captured one during a flight.
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5This is in fact a glory. – Emilio Pisanty Jul 16 '12 at 16:08