# How can a rainbow be so steady, even though the droplets causing it can be in such different states?

The rainbow! What magical "thing". And even if you see the droplets of rain move in a sunlit storm, she's steady. I have been trying to understand but there are so many drops involved! And they are moving in turbulent ways on top.
So what's going on? I know that each droplet sends a "rain circle" cone towards us, and that our eyes are sprayed with these cones. These cones all have the same orientation, no matter how the droplets move. Somehow this must be the key, but I don't see how.

• Statistics????? Raindrops are small and many but the rainbow is huge by the time you can see it. May 23, 2021 at 21:14
• Note also that every rainbow is centred around your head at the same angle above the anti-solar point. So if you and your friend are standing next to each other, you are each observing your own personal rainbow, from an independent set of drops. The same is true in time: the rainbow you see in this instant is created by a different set of drops from the one you see in the next instant. So there is very little static about it, the drops' only job seems to be to refract the light and move on to create someone else's rainbow ;). May 23, 2021 at 21:24
• "When you involve statistics, wouldn't all the contributions cancel?" – No, definitely not. That's a common misconception. Suppose you flip 1,000,000,000,000 coins, and you compare the number of heads to the number of tails. I think a lot of people think that the difference will be very small, something like 10 or 20. Actually, the difference will probably be on the order of 1,000,000, and it's very, very unlikely that the difference will be less than 1,000. May 24, 2021 at 12:21
• No. Like I said, that's a common misconception. When you flip a coin, it has no idea which way is towards 0 and which way is away from 0, so there's a 50% chance it'll go towards 0 and a 50% chance it'll go away from 0 (unless the total is already 0, in which case both directions are "away from 0"). So there's no reason why the relative number would converge to zero. The ratio between the number of heads and the number of tails will approach 1 as time goes on, but the difference will not approach 0. May 24, 2021 at 12:50
• @TannerSwett Assuming a fair coin, the average difference will tend to zero. The count of H-T will cross zero infinitely often, and will just as likely be positive as negative at any given point (ignoring priors). May 25, 2021 at 22:52

The color depends on the relative angle between you, the drop, and the sun.

If you were to track an individual drop it would change color as it falls "through" the rainbow. That would be cool to see!

In more detail:

Consider a cartoon similar to the one in @John Hunter's answer. There is a viewer at left looking right. There is a sun behind the viewer. For every point to the right of the viewer we can draw lines from that point to both the viewer and the sun. So we correspond an angle to EVERY point in space.

We can ask "what are the surfaces of constant angle?" Well, imagine a line through the sun and the viewer. All points along this line will be angle zero. Larger angles will be cones coaxial with this center line. These are the surfaces of constant angle.

A rainbow works because, for certain angles, the droplet will reflect the various spectral components of the sun to the viewer.

So what is fixed in space, given the position of the viewer and sun, are the cones of constant angle. These are present whether or not there is water. A rainbow arises when there are water droplets occupying the appropriate cones.

As a droplet falls, it passes through the different cones, meaning the component of the sun that it reflects changes in time as it passes through the cones of differing angles.

• The "cool to see" part is pretty much doable. Use the garden hose in a sunny day. You can see the colorful droplets passing thru the rainbow. May 24, 2021 at 6:59
• "would be cool to see"here you are ;) May 24, 2021 at 7:05
• Upon further thought, this answer somehow reverses the situation (though I can see what you mean!). It states that the rainbow is already there and the droplets change their color (as seen by us) to conform to the colors of the bow. Obviously, they change color but how do these changing colors constitute the rainbow in the first place? I can see, when all droplets are static in a vertical plane, how the bow appears, and upon falling the drops will "fall through". But what about the horizontal direction? Will not different parallel vertical planes give different rainbows? May 24, 2021 at 15:06
• @DescheleSchilder which vertical planes do you mean? Flat side to the observer, differing by distance? May 24, 2021 at 16:10
• @Ruslan I think that what you describe is what I mean, yes. Say, there are two parallel planes with droplets in front of you. Your distance to the right part of these planes is the same as to the left part. The planes are perpendicular to the Earth. May 24, 2021 at 16:21

This picture was posted on this answer, Rainbows and Clouds might be handy again now

The drops are in continuous motion but the two shown will quickly be replaced by others in the same positions...

(although the image has been regularly used on Stack exchange: Copyright 1999 Rebecca McDowell rebeccapaton.net/rainbows/formatn.htm )

• This was in fact the picture that inspired my question! Or better, re-sparked my question. May 23, 2021 at 21:51
• Interesting how sometimes a double rainbow can be seen, quite rare but it happens sometimes...hope this helps May 23, 2021 at 21:54
• There is a slight error in this image: the reflection at the inside-surface of the drop is most certainly not Total Internal Reflection, but rather just "normal" reflection. We've had extensive discussion about this, see this answer for a "corrected" version of the same image, and the question for a discussion as to why it cannot be Total Internal Reflection. May 23, 2021 at 21:58
• @JohnHunter Whenever I see a rainbow, it is almost always double. The second one has bigger visible radius, reversed color order and less brightness. The second one is created by two internal reflections. May 24, 2021 at 6:25
• @Deschele Schilder: I can see why -- the guy definitely looks nonplussed that the droplets keep moving while the rainbow stays still. May 24, 2021 at 20:19

The apparent colour of each particle depends on the position, but only on “2D position in view”, i.e. angular position relative to the observer. (See other answers and questions for explanation on this.) Thus it doesn't matter how chaotic they move: whereever each particle is, it will have the “correct” colour to fit in the rainbow.

• Not simply position: angular position instead, i.e. azimuth and elevation with respect to observer. Change of distance to the observer won't affect the color. May 25, 2021 at 6:09
• @Ruslan so it is position dependent. I didn't say distance-dependent. May 25, 2021 at 7:30
• Distance is the third coordinate of position (in spherical coordinates). May 25, 2021 at 7:31
• But you're right it should be mentioned, because if it did depend on radius then it would disrupt the picture May 25, 2021 at 7:33