why is rainbow always an arc?
Because the water drops are round.
Har har. No.*
A rainbow is an arc because its shape is entirely defined by the geometry between the sun's location and your eye. You will find that a rainbow's centre point is always exactly opposite the sky from where the sun is, offset by an angle of about 42 degrees. There are an infinite number of points that form this angle between the sun and your eyes - they all lie on a circle. If part of that circle is filled with raindrops, they will all refract the sun at the same angle back to you.
Google 'physics of a rainbow' for more detail.
* seriously, you can get rainbow effects off non-round objects, such as sun halos off ice crystals. So no.
And why is it 42 degrees in every direction from anti-solar point? Where is that symmetry coming from? Would the rainbow be still a circular arc, if the raindrops where ellipsoids, scaled by the factor 2 vertically?
It's not, exactly. Each color is refracted at a slightly different angle, depending on the wavelength (and size of rain drop but all raindrops are rougly the same size). But the angle for all wavelengths of visible light are about 42 degrees.
In fact raindrops are not spherical - they are oblate ellipsoids, flattened vertically by air resistance. More accurately, they are said to resemble the top half of a hamburger bun.
I think it's save to assume that the OP is aware that there are multiple arcs, not just one. He seems to be asking about the reason for the circular shape.
Nothing it perfectly spherical in nature. But how would the rainbow look like, if they were strongly flatten, say by a factor of 0.5?
I can't be sure but I don't think it would make a difference. Any elliptical object will still have spot on its inner surface that reflects at the proper angle.
I lie. I suspect that it would make a difference but not in the way one might think. It would still form a circle - and at ~42 degrees - but it would be weaker and the colours would blur together more (producing more white, like a sun halo). The differing refraction would be what makes the diff.
I can't back this up though; it is an educated guess.
Of course it does, as A.T said. the arc nature of rainbow is caused by circular symmetry of a droplet.
Lecture on rainbows:
Soooo how do you explain rainbows from ice crystals?
Ice rainbows = round
Ice crystals = not round
Ice crystals have some symmetry too (hexagonal). But due to their asymmetries they also produce "sun dogs" (bright spots) or "light pillars":
Water drops don't produce such effects, due to their spherical symmetry. That's why the (water) rainbow is always just an arc.
Well, my point here was that the illustrated sundog still has a circular shape (though parts are faded out of existence - pretty much exactly what I was trying to say in post #10. OK, except the fact that the sundog is adjacent to - rather than opposite - the sun.)
The idea here is to discourage the possible erroneous thought that somehow oblate shaped raindrops might form an oblate rainbow.
Ask yourself question if it is possible to generate a rainbow from only one droplet and then you will know why arc shape is caused by the circular symmetry (or just watch the lecture).
Answering your question regarding crystals the explanation is imo quite simple: the same rule applies to crystals, they also must have circular symmetry. This is also why you will not see rainbow on snow surface at shinny day (snowflakes have irregular shapes).
The circular shape in the case of the ice crystals comes from their random orientation, not their shape
- A spherical water drop diverts the light by 138° in all possible directions that have this angle to the sun-rays. So all the water drops that are on the 42° cone between your eye and the anti-solar point are sending light into your eye.
- A water crystal diverts the light by 22°, but only in a few directions. So you only see a tiny fraction of the crystals on the 22° cone between your eye and the sun. And this only if they are randomly oriented. If they start falling and align themselves vertically, the arc disappears and the sun dogs appear.
The OP's question:
Why is rainbow always an arc?
Can be interpreted in two ways:
1) Why isn't it sometimes something else than an arc?
Answer: The spherical symmetry of the drops makes sure that orientation is irrelevant, and the effect is always an arc.
2) Why are arcs formed?
Answer: All the objects (drops or crystals) divert the light by the same angle. So with random orientation there will be a brighter circle at that certain angle from the sun.
All right. For the sake of argument, I'll concede for the moment.
So, if a cloud of water droplets were able to individually have some non-spherical shape, what effect are you proposing that would have on the shape of the rainbow? Would we see a big squashed, oblate rainbow?
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