APOD Mystery Explained

December 5, 2004

I have received about 3000 responses as a result of my "mystery picture" posted as the Astronomy Picture of the Day on September 13, 2004.

Most responses were interesting. Many were funny, charming, and even moving. I answered all messages (it took me six weeks). There were about 50 people (only 2 from all of MIT) who knew what caused the phenomenon. Yet only five of them had a complete understanding of the underlying physics with an accurate explanation of the color sequence, the radius of the outer red circle and even the bright white light coming from inside the colorful bow.

About 500 people believed that the picture shows a glory. Glories are produced by mist, fog and clouds (they can often be seen when your plane is flying over clouds). The physics of glories is very complicated. Since they are the result of diffraction, the colors are always washed out; they are never as crisp and well localized as the colors in my picture. See, e.g., this picture from a web site devoted to atmospheric optics and this Astronomy Picture of the Day. About 100 people believed that the picture was caused by dew. Dew can cause "heiligenschein," which is very different from what you see on my picture; see, e.g., this picture.

My picture is not a glory, and it is not heiligenschein. It is not due to water, dew, fog or mist. It is not due to oil, frost, ice or snow, either. I took it near 1 PM on a warm, sunny, dry day (June 20) in Lincoln, MA. The fact that this was near summer solstice is purely coincidental; a similar picture could have been taken weeks earlier or weeks later.

The unusual image should be called a GLASS BOW. This is what you would see if rain drops were made of glass. Since we all have seen rainbows, I will first say a few words about them. A rainbow results from refraction and reflection of light in spherical water drops. There are, in general, two rainbows: the primary, and the secondary bow. You can see both bows in this picture.

I will only briefly discuss here the primary bow (it's the brighter of the two).

Primary Bow

Sunlight of all colors strikes zillions of water drops. Each water drop reflects a cone of light back into the direction of the sun. This cone is red on the outside and has a cone angle of about 84 degrees (see the figure below). The violet light is further in (cone angle of about 80 degrees). Inside the violet cone, the water drops reflect all colors at roughly equal strength; the light inside the cone is therefore white (reflected sunlight).

When the sun is behind you, and you are looking at rain or water from a fountain or from a waterfall, zillions of water drops will each do their own thing (see the figure), and the result is that you will see a rainbow. If you measure the angle between the shadow of your head (this is the direction 180 degrees away from the sun) and that of any point of the red circle of the primary bow, you will find about 42 degrees (half of 84). You will see the red on the outside of the bow, violet on the inside, and white light inside the bow. The sky is distinctly darker outside the bow than inside. You can see all this very clearly by looking again at this picture.

As an independent but fascinating issue, the light from the colorful bow, and also the white light close to the colorful bow, is strongly linearly polarized. You can recognize linearly polarized light by Haidinger's Brush, but that takes quite some practice. Those of you who are not familiar with this can use a pair of polarized sunglasses. When you look with one eye through one glass and rotate the linear polarizer around in its own plane, you will see that the bow will "come and go."

In my courses on Electricity and Magnetism and on Vibrations and Waves at MIT, I discuss the rainbow in quite some detail (you can watch my 2002 E&M rainbow lecture on the web, see below). My students have to calculate the angles mentioned above (42 degrees for red light and 40 degrees for blue/violet light). They also have to calculate the degree of linear polarization, which is much more complicated. If you are interested, try Problem 10.4 on this problem set. There are also solutions.

Glass Bow

In Problem 10.4, I mention in the last question, "In a world far, far away, rain comes down as small drops of glass (with index of refraction about 1.5). The living souls there talk about a glass bow." I ask my students to calculate the angles for the glass bow. The 42 degree angle for the red in case of water (see the figure), now becomes only about 23 degrees (see the solutions to Problem 10.4(h)). This is due to the fact that the average index of refraction of glass (1.5) is higher than that of water (1.337). I say "average," as the index of refraction differs slightly for different colors. If this were not the case, rainbows and glass bows would exist, but they would be colorless.

My Picture

On June 20, 2004, I visited the sculpture garden of the DeCordova Museum in Lincoln, MA, with my SO, my son, and his SO. Near 1 PM, we walked by the construction site of DeCordova's new visiting center. My son Chuck said, "Dad, look," and there was a "mini" rainbow on the road (I say "mini" because the radius of the bow was only about 20 degrees). I had never seen such a bow before, but it was immediately obvious to me that this bow was caused by spherical transparent beads (I suspected glass). Luckily, my son's SO had a camera, and we took a few pictures. I also took some samples of the beads home as I realized that this would be a great topic for my course at MIT. At that time, we had no idea yet why so many spherical transparent beads were there on the asphalt. I learned only later that spherical glass beads (roughly 1/4 mm in size) are used for sand blasting. Luckily for us, quite a bit had been spilled on the ground. Sand cannot cause the bow; nor can pieces of glass. Only spherical transparent beads can.

Look again at my picture and notice the color sequence: red on the outside, violet on the inside, and a lot of white light inside the bow! There are plenty of glass beads on the asphalt outside the red bow, yet they do not reflect the white sunlight well; only the glass beads located on the asphalt inside the bow do. I always carry with me a linear polarizer, and, of course, we confirmed that the light from the bow was strongly polarized.

The question remains whether there is enough information on my picture for the viewers to be able to definitely exclude water drops as a possible origin of the bow. The answer is: "yes, water can be excluded". If, somehow, fine water drops had been uniformly sprayed in the air over a large area close to the surface on which my shadow is cast, one would have seen an image that is similar to mine. However, there is some key information on my picture and that is the width of the colored bow in comparison with the radius. You can easily measure for yourself with a simple ruler that the width of the colored bow (from the outside of the red to the inside of the violet) is about 16% of the radius of the bow. For a water bow, the width is typically only about 6% of the radius (see the solutions to Problem 10.4(g) - you can make an estimate for yourself by once more looking at this picture. There is no question, water can be excluded! Independently, if this were a water bow, the picture could only have been taken with a wide-angle lens with a field of view of about 90 degrees as the radius of a water bow is about 42 degrees. Pictures taken with such lenses show distortions (see, e.g., this picture) which are absent in my picture.

One can derive from my picture that the linear dimension of the radius of the bow is about 65 cm, by using the shadow of my head as a yardstick. The angular radius of the bow is about 23 degrees. Thus I must have held the camera about 5ft above the ground.

The glass bow will be shown during my lecture on December 7 (in MIT's lecture hall 6-120). The demo guru, Markos Hankin, has prepared about three square meters of black paper on which he glued countless glass beads that we purchased. A strong flood light will take the place of the sun. We have worked very hard to optimize the effect; our indoor glass bow is spectacular! All students have been given linear polarizers early in the course so they will also be able to see that the bow is strongly polarized; they will be thrilled!

75 of my lectures can be viewed on the web (71 on MIT's OpenCourseWare and 4 on MITWorld). You can watch one of my lectures on rainbows (Lecture #31 on this page). In the spring next year, my 23 lectures on Waves and Vibrations will be added, and you will be able to see the glass bow in the lecture hall for yourself (it's Lecture #22). Enjoy it!

\\/\///////@lter Lewin