Why are Rainbows Curved as Semicircles?

Why does the rainbow seem curved as a semicircle?
- Anonymous

Nice question! Rainbows are a product of sunlight passing through small droplets of water suspended in the atmosphere (or even falling through it!). Not only are they beautiful, but they are usually rare because you need a rainy day and a sunny day at the same time to make one appear -- the sun has to be shining from one part of the sky, and the rain in another part of the sky before a rainbow can appear.

The sunlight takes a complicated path through each water droplet. It comes in the side closest to the sun, bends because the index of refraction in water is bigger than that of air (you can see this effect by putting a pencil in a glass of water so that some of it sticks out and looking at it from different angles -- the pencil will apppear "broken" at the place it crosses the water surface). The sunlight, passing through the water droplet, bounces off the back surface of the droplet, travels back to the other side, and bends once again on its way out.

The reason why the rainbow is curved is because all the angles in the water drop have to be just right for the drop to send some sunlight to you, standing on the ground. So, with the sun *behind* you, only those water droplets that have the same angle formed by you, the drop, and the sun (this angle happens to be approximately 42 degrees) will contribute to the rainbow. Other droplets send their light somewhere else, and if you move to a different location, new droplets are needed to make the rainbow you see in the new location. This is why you can't go to the end of a rainbow to find the mythical leprechauns and pots of gold; anywhere you stand, the rainbow is formed by faraway drops of water reflecting and bending sunlight. The rainbow is curved because the set of all the raindrops that have the right angle between you, the drop, and the sun lie on a cone pointing at the sun with you at one tip. The rainbow may look semicircular if the sun is setting or rising (a good time to see a rainbow because the sunlight at that time can get under rain clouds because it is traveling horizontally). If the sun is higher in the sky, the earth gets in the way and you may see less than a semicircular rainbow.

The rainbow is colored because the water drops act like little prisms -- how much the light bends when it enters and exits the drop depends on its color, and light from the sun contains contributions from light of all colors. So that 42 degrees above is a bit different for red light and different still for blue light -- you have to look in a different place to see the red rainbow arc and the blue rainbow arc, so you see them as arcs in the sky of different sizes and the rainbow is striped with colors.

You can make your own rainbows with a lawn sprinkler or even a water spray-bottle that can make a fine mist. On a bright, sunny day with the sun at your back, spray some water in front of you in different directions to see where the best rainbows can be seen. Can you make one go all the way around in a circle? Rainbows crop up in the nicest places -- you can see them sometimes at the bottoms of waterfalls and even briefly in the splashes of divers at the swimming pool.

(published on 10/22/2007)

it is to inform you that the link u have give is no longer available online so kindly remove the reference as it may cause inconvenience to others
- Anonymous

Thanks- notes like this help us keep things up-to-date.

Mike W.

(published on 03/13/2009)

Would a rainbow still appear to arc in the vacuum of outer space? Does our atmosphere or the curvature of the earth have anything to do with it?
- Ben (age 32)
Salt Lake City, UT

Rainbows are made by the way light curves in many small drops of water.  So unless you had a way to get a spray of little raindrops near you in space, you wouldn't see a rainbow. I think it would be hard to do, because any little random or systematic velocity that the drops had relative to you initially would soon carry them away.

Our atmosphere is important for a couple of reasons here. It includes the water vapor that forms rain drops when it cools at high altitude. The friction between the drops and the air then keeps them from falling out of the atmosphere more quickly.

The curvature of the Earth is indirectly connected. If the Earth's surface didn't curve then the Earth would be infinite. Its infinite gravity would make the whole situation impossible.

Mike W.

(published on 07/11/2014)

But how do the water droplets, our position, and the sun form a cone?
- Abrez Jilani (age 16)
Srinagar, J&K, India

There are droplets all over the place, but the ones contributing to the rainbow you see form a cone. As the answer above describes, the light that reaches your eye has bent about 42° (depending on the color of the light) from the path it was taking before it hit the droplet. The Sun is very far away, so all the rays coming in from it to our vicinity are nearly parallel. Take any point (you, for instance) and draw all the rays through it that are 42° away from a fixed direction. They form a cone.

Mike W.

(published on 02/10/2015)

It has nothing to do with any roundness of the eye?Lets say all the right raindrops existed in a volume or area in front of you, from the ground to 100s of feet; and the sun light came in as it does to create rainbow; there can never be seen or made a wall of rainbow in the air?
- Daniel (age 25)
New York

The only way the roundness of the eyes is involved is by helping to make the eyes be lenses, so we can see images. If you take a photograph the same rainbow or rainbows appears.

Since the light bends at special angles in the raindrops, there will always be gaps in your field of view with no rainbow in the gap.

Mike W.

(published on 05/29/2015)

If we need a 42 degree angle to see a rainbow, then how can we see a "double rainboow"? Does a double refraction occur? And if so does it allow us to see a rainbow at 2 different angles?
- Austin (age 21)
Albuquerque New Mexico, USA

Hello Austin,

Yes they do and it's a quite common thing.  I have seen it several times.  The main point is that in the secondary rainbow the light rays make an additional internal reflection inside the raindrop.  Curiously, the second rainbow has its colors reversed with respect to the primary one.   There is a nice web site that describes the phenomenon, complete with nice diagrams.

LeeH

(published on 06/06/2015)

I was thinking today that it had to do with the curve of the earth ? Like if the light is all coming in straight, if the earth were flat, it would not be a curve. But, from your answer it sounds like I am wrong.
- Laurie (age 38)
San Jose, CA

Yes, the curvature of the Earth has no direct connection to a rainbow's curvature. Rainbows on top of a hill or in an upward-curving valley have the same sort of arc.

Mike W.

(published on 01/19/2016)

I read your beautiful article on the rainbow curve. Highly appreciate the simplicity and clarity in your explanations. The last paragraph where it was asked about if we could form a circular rainbow. I think that it might be in the similar lines when we see colors around the moon in concentric circles. Or is it something else?
- Jishu (age 30)
Saint Petersburg, Russia

Hello Jishu,

Rainbows can be circular, you just have to be in a position to see them.    For example, I have seen one myself from an airplane.  Also a high mountaintop is a good lookout point for them.

LeeH

(published on 05/05/2016)

Would the curvature of rainbows have anything to do with the Firmament?
- markus (age 51)
australia

I think that is the same question as earlier ones about whether the curvature of the arc is connected with the Earth's curvature. The answer is still no.

Mike W.

(published on 07/18/2016)