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I know that we observe rainbows from light being reflected off of water in the atmosphere. So, what angle of incidence must exist between an observer and the sun in order to see the colors of the rainbow?

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  • $\begingroup$ I don't think that really matters, what matters is the angle between the rain drop and the observer which is $42^{\circ}$ $\endgroup$
    – Courage
    Dec 10, 2015 at 5:48
  • $\begingroup$ But you need at least 3 points for there to be an angle. The relevant angle is between the eye and the sun, where between them lies the water vapor, in which the angle begins. $\endgroup$
    – Blake.W
    Dec 10, 2015 at 5:57

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The answer depends on the color you are observing, and whether you are looking at the primary or secondary ("inverted colors") rainbow. This is demonstrated in this diagram from http://eo.ucar.edu/rainbows/ (Found by googling "rainbow angle");

enter image description here

The angles given are for the edges of the rainbow (red and violet) with the colored lines matching the color of the corresponding edge. Note also that the secondary rainbow has an extra internal reflection, making the incident and refracted beams cross over. This explains why the colors look "upside down".

Note - the angles given are between the direction of the observer's own shadow (the sun is coming from behind the observer) and the apparent position of the rainbow. The apparent angle between the sun and the rainbow would be $180-\theta$, or roughly between 125 and 140 degrees. Whatever you do, an angle is defined by three points or two directions. Note that the sun is "really far away" so we treat all its rays as parallel in this diagram; and we ignore the finite size (about 0.5 $^\circ$) of the solar disk

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  • $\begingroup$ I think the OP is asking about the angle between observer and the sun and not between the observer and the rainbow. $\endgroup$
    – Courage
    Dec 10, 2015 at 9:09
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    $\begingroup$ @Vishwaas The angle shown is between the sun ray and the observer's line of sight. If he means the line from the sun to the observer's head, this is always $0^{\circ}$ (sun rays are all parallel). $\endgroup$ Dec 10, 2015 at 13:29
  • $\begingroup$ @Owen Boyle okay, I thought the OP was asking about the angle between the sun and the observer (that is the angle between the line joining the sun and the observer, and the ground) $\endgroup$
    – Courage
    Dec 10, 2015 at 14:58
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The water droplets in the air refract the white light that is given out by the sun, acting like prisms, which splits the whtie light into the continuous spectrum which includes the visible range thereby forming a rainbow.

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  • $\begingroup$ I fully understand how and why rainbows form. I am inquiring the specific angle that exists between the sun and observer. $\endgroup$
    – Blake.W
    Dec 10, 2015 at 5:29
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The angle between your line of sight and the rays of sunlight are given in the diagram posted by @Floris. So $40-42^{\circ}$.

The rainbow is formed in a plane that is perpendicular to the rays of the sun. The Sun's rays are all parallel (it's awfully far away). You see the pattern along a line of sight that is $40-42^{\circ}$ from the sun ray (exactly as in the diagram).

You are perhaps thinking that by moving, you can change your angle to the sun ray? This would be true if it were a single beam from a point source but you have to remember that the sun is much bigger than Earth and much, much further away than the rainclouds. So, for us on planet Earth, the sun rays are essentially a set of infinite, parallel lines.

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