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PostThe Sundial Experiment, Part II: Cones and Straight Lines (Istvan Simon, USA, 12/30/16 3:01 pm)
In my December 28 post I promised that there is a truly beautiful and simple explanation that involves birthday party hats to the puzzle offered by my sundial observation made by me more than two decades ago. I wanted to give the opportunity for WAISers to try to solve the puzzle by themselves, so I delayed this response. For those who are intrigued by the puzzle but were just waiting for the promised explanation, I will provide the solution in this post.
To recall, I had observed the astonishing fact on my sundial that day that the tip of the shadow had moved in a straight line, from say 10 AM to 2 PM. This I had found too difficult to explain. I knew that the explanation had to come from the basic motion of the earth in space.
The earth orbits the sun during the year, and while doing this it also rotates daily around its axis. Furthermore the earth's axis is tilted 23.5 degrees from the vertical to the ecliptic (the plane on which the earth orbits around the sun during a year). I knew all this, as does every elementary school child and educated adult today. Yet the geometry of working all of this out is still complicated enough so that even mathematicians might be a bit stumped.
The key insight for finding a simple explanation was provided by Alfred Kaufman, who sadly passed away on January 4, 2016, and with whom I had played chamber music for decades. Al was a brilliant physicist who worked at the Lawrence Livermore National Laboratory. What Al did also shows how a brilliant scientist approaches a difficult problem. His insight is a magical and marvelous simplification of the problem. Al suggested moving the experiment to the North Pole.
This is a huge simplification, because on the North Pole the vertical coincides with the axis of the earth. Thus the complicated compound motion of the earth described above becomes much simpler at the North (or South) Pole.
Let's imagine that the experiment is performed on June 21 at the height of the summer in the Northern hemisphere. A person standing on the North Pole observing his shadow would see the sun off the vertical by an angle of 66.5 degrees, the complement of the tilt of the earth's axis. Since standing on the axis of the earth greatly simplifies his motion in space, the observer would be simply turning around with the earth's rotation while the sun was fixed off at an angle. It is immediately clear that his shadow would move in a circle. An immediate interesting consequence of all this is that the sun is up 24 hours on the North Pole that day. This is a fact that all of us knew already, but what's interesting here is that it comes naturally out of our imagined sundial thought experiment.
I gave another clue in my December 28th post: that the explanation had to do with birthday party hats, so let me deliver on this next. Imagine that we marked with a visible red thread the path of the sun's ray that hit the horizontal ground at 10:00 AM immediately at the tip of the shadow. Now imagine we marked with a blue thread the path of the sun's ray at the tip of the shadow at 10:05 AM. Imagine doing the same with a yellow thread at 10:10 AM and so on. Since the shadow describes a circle, as we have just seen, all these multicolored threads would lie on the surface of a cone. The vertex of the cone would be the observer's head, and the cone would end where it hits the ground. Hence the circular path of the shadow is the intersection of this cone with the horizontal plane of the ground. The cone is the birthday party hat that I talked about.
The next step in the explanation is a fact that Eratosthenes already knew about 2300 years ago. He knew that the sun is very very far from the earth. A ray of the sun that hits the ground on the North pole at 10:00 am on June 21 and a ray of the sun that hits Pleasanton at the same time both started their journey to the earth at the Sun simultaneously. So these two sun-rays meet at the sun. But because the Sun is so far away, for all practical purposes these two sun-rays are parallel.
So if we move the observer to Pleasanton from the North Pole, the rays of the Sun at different times in Pleasanton would also form a cone exactly parallel to the cone at the North pole. The red thread in the North pole would be parallel to the red thread in Pleasanton, the blue thread in the North pole parallel to the blue thread in Pleasanton, and so on. The difference is that the party hat in Pleasanton would be off the vertical, sort of like if the observer were a bit drunk. The path of the shadow in Pleasanton is still the intersection of this cone with the horizontal plane of the ground, but the axis of the cone is parallel to the axis of the earth, and so no longer coincides with the vertical in Pleasanton. Therefore the path of the shadow would describe the intersection of the cone with a tilted plane rather than a plane perpendicular to the axis of the cone.
A little to the south of the North Pole we would get an ellipse instead of a circle, further south we would get a hyperbola, and if the intersecting plane were parallel to one of the cone's "sides" it would be a parabola. In other words our sundial leads naturally to the discovery of conic sections.
WAISers are invited to cut a birthday party hat at various angles with scissors to confirm what I just said.
So on June 21 the path of the shadow in Pleasanton is not a straight line but a hyperbola. But this still did not explain the astonishing mystery I stumbled on with my real experiment in Pleasanton. Why did I observe a straight line?
To understand this we must return to the North Pole and observe what happens in successive days. The angle of the sun with the axis of the earth changes as the earth progresses in its yearly orbit around the sun. To see this it is best to get rid of the earth entirely and just move the axis around the sun. Get a pencil, tilt it and then move it around a central point where the sun is supposed to be. The orbit of the earth is very nearly circular.
As we have seen at the North Pole, the angle of the sun with the axis of the earth is 66.5 degrees on about June 21. But as the axis of the earth moves around the sun, this angle increases. On about December 21 it is 180 degrees minus 66.5--that is, it is now an obtuse angle of 113.5 degrees. At some point midway through these two dates the angle is 90 degrees. This is the autumnal equinox. It happens around September 21.
So what happens to the party hat on the North Pole? As the earth progresses through its orbit the party hat gradually opens up, and the radius of the circle of the shadow becomes bigger and bigger. When the equinox is reached, the radius has become infinite and the party hat degenerated into a plane. It is no longer a cone, but a horizontal plane at the North Pole. The intersection of this plane with the horizontal plane in Pleasanton is a straight line! This is what I observed in my real experiment in my backyard. By a phenomenal coincidence I had performed the experiment near the date of the autumnal equinox. The intersection of two planes is of course always a straight line, so our mystery is solved.
What happens after the autumnal equinox? On the North Pole the sun is now below the horizon, so we have 24 hours of night rather than 24 hours of day. The cone is now inverted, it is upside down because the sun is now in the southern hemisphere. After the equinox the path of the shadow becomes again a conic section but its curvature is now inverted.
The astute observer of the photographs that I sent with my December 28 post can clearly see the curvature of the path of the shadow shown in the markings we made on the sundial on Christmas day.
I believe that the story that I told in my December 28 post about playing chamber music happened in David Haupt's home in Livermore. Muriel Haupt was the excellent cellist of our string quartet.
I hope that WAISers that had the patience to read through this long post will agree with me that this is a truly beautiful and simple explanation of a complicated phenomenon that I discovered by accident with an improvised sundial in my backyard more than 20 years ago. I hope that my description of the explanation was clear enough that anyone could follow it, even a 9-year-old child, though the length of it probably exceeds the attention span of a 9-year-old, and perhaps many adults as well. And I hope that WAISers will also agree that the hero of my story are two scientists, Eratosthenes and my friend Alfred Kaufman. Both Eratosthenes and Al Kaufman were great men, and they were both scientists.
Our string quartet also mentioned in my story was called the Wolf Gang, a pun on the name of another great man, perhaps the greatest of all men who ever lived. All of us in the Wolf Gang loved the music that this greatest of all men, Wolfgang Amadeus Mozart wrote and played his music often. I would like to lovingly dedicate this post to the memory of these three great men, a long dead Greek scientist, an American scientist who died earlier this year, and an Austrian composer, who died 225 years ago on December 5, 1791. In a few days it will be the first anniversary of the death of Al Kaufman. Rest in peace my good friend.
JE comments: A wonderful tribute to all three men. I think I understand Istvan Simon's careful explanation, although I'm not sure I could replicate the experiment...yet. When necessary, it's useful to "plead the Fifth" and say I'm a Humanist.
A very happy 2017 to you and your family, Istvan.