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Rainbows

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Several times in my life I've seen rainbows following or during rain storms. A few times, I have actually seen two rainbows near each other. During my recent vacation in Israel, I saw two rainbows but with an unusual relationship. The rainbows appeared to be optically connected: the right and left rainbows were concentric with the colors of one side a mirror image of the colors of the other side. Between the two rainbows was a dark band that appeared to be dark violet. Attached below is a picture I took through the car window.

Any physicists care to comment on this phenomenon? Is it usual? I had never seen anything like it before.

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Funny you should mention this aspect of rainbows. I had an interesting discussion at work about them on Friday.

An senior engineer I work with was involved with judging some national science competition where the students ran their own experiments on topics of their choosing. Two of the students had work that was far above that of the others. One of them did a study of refraction and reflection in rainbows. He worked out the math behind seven orders of rainbows!

Anyway, to your question:

Essentially the white light of the sun reflects once within the rain drop, and the water acts like a prism to spread the spectrum through refraction. The light emerges at angles that differ slightly according to the wavelength, so red comes at a different angle then blue.

As we look at a rainbow, we see light from different drops, some higher above the ground than others. The height above the ground means we view these drops at different angles, and hence we see different colors of light as a function of height. (Actually it is radial distance not height that matters, which is why rainbows follow circular arcs.)

Now usually we see just one rainbow, but it is possible to see multiple spectra under certain conditions. What happens is that the light doesn't make just one reflection inside the drop, but several, each time with less intensity than the previous one. Each successive reflection increases the value of the angle at which the light emerges (so if red light emerges at a certain angle for the primary rainbow, it emerges at a larger angle in the secondary rainbow, and so on).

This is why we sometimes see secondary rainbows, why they follow arcs of larger radii, and why they are fainter. Also it is possible to see more than two, but as each increasing order is fainter, it is a rarer event.

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One other aspect I just realized from looking at the photos: the spectra is reversed for the secondary rainbow. Notice how red is on the "outside" on the primary but on the "inside" on the secondary.

This is an instance of left-right inversion due to reflection. To see the effect, look at your reflection in a mirror, then use a second mirror to look at your image in the primary mirror. The first reflection is a left-right inverted image of yourself, and the second mirror inverts your image again.

I should add, too, that I looked up the issue of rainbows in Colour: Why the World Isn't Grey by Hazel Rossotti. This is a nice book with qualitative descriptions of a variety of color phenomena in the world.

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One other aspect I just realized from looking at the photos: the spectra is reversed for the secondary rainbow. Notice how red is on the "outside" on the primary but on the "inside" on the secondary.

Yes. That is exactly what really spurred my question. I'm not sure if there's always a specific orientation to the spectra (red on inside and violet on outside). I'm not sure if it's clear in the photo, but the inner rainbow was much brighter than the outer rainbow. Most "double" rainbows I've seen have not been concentric and have never been "related" to each other. The radius of curvature was always different and were in a different spatial orientation.

This is an instance of left-right inversion due to reflection. To see the effect, look at your reflection in a mirror, then use a second mirror to look at your image in the primary mirror. The first reflection is a left-right inverted image of yourself, and the second mirror inverts your image again.

I should add, too, that I looked up the issue of rainbows in Colour: Why the World Isn't Grey by Hazel Rossotti. This is a nice book with qualitative descriptions of a variety of color phenomena in the world.

I can see where reflection would reverse the order of the spectal colors. If the second rainbow is due to reflection, then what explains the large "violet" boundary between the two rainbows. Do you think the distance is related to wavelength? Would the second rainbow be a "real" or primary rainbow in that it is not produced by light from the sun but by relfected light from an adacent rainbow?

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I can see where reflection would reverse the order of the spectal colors. If the second rainbow is due to reflection, then what explains the large "violet" boundary between the two rainbows. Do you think the distance is related to wavelength? Would the second rainbow be a "real" or primary rainbow in that it is not produced by light from the sun but by relfected light from an adacent rainbow?

Each rainbow is produced by a combination of reflection and refraction with the particular water drops that show color. Light in the primary rainbow reflects just once within the drop. Light in the secondary rainbow reflects twice within the drop. It is not light from the first rainbow reflecting into or otherwise affecting the secondary rainbow. Different particular drops are at different locations and operate independent of each other; the rainbow is an effect of the collective action of many drops, white light, and other things.

Think of it another way, if it helps: White light hits a drop of water vapor. It internally reflects once, with refraction spreading out the colors as light propogates through the drop. This first internal reflection has some light escape. The rest of the light continues within the drop, internally reflecting a second time (and inverting the color order) before having some of the light escape. Of course, there's less light emitted the second time, but it is still there. This process continues, each time changing angles and inverting and losing intensity. So, the odd-numbered rainbows would have the color sequence ROYGBIV (outside to inside) and the even-numbered ones would be the reverse: VIBGYOR.

The light that you see, qua observer, depends on the viewing angle. At most angles you don't see any light. At another angle you see blue light in the primary; at another, red light in the primary; at another, red in the secondary; and so on. Now, if you change location, the viewing angle changes, and the rainbow will change. (Often we are far from the rainbow, so small changes in the location lead to angular changes that are too small the make a difference.)

Also, the secondary rainbow is no less real than the primary. It's just less intense, so usually harder to see.

Re: the violet boundary: I don't see it in the photos. I'm not sure what you mean.

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Re: the violet boundary: I don't see it in the photos. I'm not sure what you mean.

Thanks for your analysis. If you look carefully at the area between the rainbows, (it's about 3 inches wide after you open the lower picture), you'll see a dark band connecting the violet colors. The band is darker than the sky that is to the left or right of the left or right rainbow. It's not a boundary, but a violet band between the two rainbows.

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If you look carefully at the area between the rainbows ...

Even though yours is less pronounced than ideal, you are fortunate to have captured an image of this band. In the literature this is generally known as Alexander's dark band, or Alexander's dark space. It is named after Alexander of Aphrodisias, who in the 2nd century AD introduced a paradox based upon Aristotle's explanation for rainbows. According to the reflection principles outlined by Aristotle, the intensity of the reflected light should produce a brightened sky between the double rainbows, but observation clearly indicated that region was darker, not lighter than the surrounding sky. Hence the "Aphrodisian paradox" of Alexander and the eventual labeling of the region with his name.

It was Descartes who was actually first to explain the Alexander dark space by reference to the laws of refraction. Starting with an idealized single drop of water, Descartes used the refraction laws to actually calculate the path of many different light rays through the drop. By carefully studying the consequences of fine differences in the angle of incident light, Descartes identified a peak at a particular angle of incidence, which he associated with the visual brightening of the primary rainbow that Ed described in his post. Descartes also realized that beyond this peak value no light rays are refracted, which explains the existence of Alexander's dark space and resolves the "Aphrodisian paradox."

Most texts will explain the Alexander dark space through the idealization of geometric optics, but, of course, in real life the dark area is not devoid of all light. The idealization of geometric optics does not account for higher order processes, and there is some interesting work, like accounting for the deformation of the water drop, that explains in a limited manner how internal reflections creep into Alexander's dark space, brightening it somewhat beyond the idealization.

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Even though yours is less pronounced than ideal, you are fortunate to have captured an image of this band. In the literature this is generally known as Alexander's dark band, or Alexander's dark space. It is named after Alexander of Aphrodisias, who in the 2nd century AD introduced a paradox based upon Aristotle's explanation for rainbows.

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WOW. I'm excited at having discovered something that has intrigued physicists for millennia!!! :D

It is also of interest that the Alexander's dark space did change intensities as we drove. Visually, at times, it was a little darker than what I captured in the photo. The rainbows "travelled" with us for several minutes. They jumped from one side of the road to the other. At times, I could see the entire arc. When they were on the left side, they were much closer to the road; maybe 50 yards away. I could actually see it end on the ground (no pot of gold was noted :D ). When on the right side of the road, the rainbow was much farther away.

Any idea on what determines the width of the dark space? It was fairly large. Any idea on how often this phenomenon is observed?

PS. Since this happened in Israel, should I attach any religious significance to this? A message maybe? An epiphany? :D;)

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WOW. I'm excited at having discovered something that has intrigued physicists for millennia!!! :D

It is not that easy to capture the dark space on film. The classic photos that are shown in texts are culled from many many lesser attempts. As they say, you were in the right spot, at the right time.

It is also of interest that the Alexander's dark space did change intensities as we drove. Visually, at times, it was a little darker than what I captured in the photo. The rainbows "travelled" with us for several minutes. They jumped from one side of the road to the other. At times, I could see the entire arc. When they were on the left side, they were much closer to the road; maybe 50 yards away. I could actually see it end on the ground (no pot of gold was noted :D ). When on the right side of the road, the rainbow was much farther away.

Yes, these are all common descriptions. Rainbows are very much a frame of reference phenomena, hence the old sayings about chasing the rainbow and then later the rainbow is chasing you.

Any idea on what determines the width of the dark space? It was fairly large.

I do not think there is any really good explanation for the width, other than some dependence on the index of refraction and the viewing angle.

Any idea on how often this phenomenon is observed?

Of this I am not sure, but photos of Alexander's dark space are sparse as compared to photos of more ordinary rainbows.

PS. Since this happened in Israel, should I attach any religious significance to this? A message maybe? An epiphany? :D;)

Yes, it is undoubtedly very significant; Iran better watch out lest they become another example of Alexander's dark space!

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Yes, it is undoubtedly very significant; Iran better watch out lest they become another example of Alexander's dark space!

I shall gather the troops forthwith!!

Actually, I'll take the easier path and just push a few buttons! What does a nuclear rainbow look like?

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Rainbows ;) happy sigh! My favorite memories are from Hawaii, where every day is a rainbow day.

An senior engineer I work with was involved with judging some national science competition where the students ran their own experiments on topics of their choosing. Two of the students had work that was far above that of the others. One of them did a study of refraction and reflection in rainbows. He worked out the math behind seven orders of rainbows!

Ed, how terrific! Do they post the results and the students' work?

I found this link that shows a beautiful rainbow and Alexander's dark space.

The photographer of that great rainbow sells prints, but too large for my apartment. But I can't resist showing this one of him with an advertising set:

pepsiPrints_675.jpg

Man with his work. Hmm, but they're upside down to him. Oh well, I still like it.

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Ed, how terrific! Do they post the results and the students' work?
It was a science fair some 20+ years ago.

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