The Best Proof that the Earth Spins

In my last post I gave you a way to check for yourself, using observations that are easy but were unavailable to ancient scientists, that the Earth is rotating from west to east. The clue comes from the artificial satellites and space junk overhead. You can look for them next time you have an hour or so under a dark night sky, and if you watch carefully, you’ll see none of them are heading west. Why is that? Because of the Earth’s rotation. It is much more expensive to launch rockets westward than eastward, so both government agencies and private companies avoid it.

In this post I want to describe the best proof I know of that the Earth rotates daily, using something else our ancestors didn’t have. Unlike the demonstration furnished by a Foucault pendulum, this proof is clear and intuitive, involving no trigonometry, no complicated diagrams, and no mind-bending arguments.

The Magic Star-Pointing Wand

Let’s start by imagining we owned something perfect (almost) for demonstrating that the Earth is spinning daily. Suppose we are given a magic wand, with an amazing occult power: if you point it at a distant star, any star (excepting the Sun), from any location on the Earth, it will forever stay pointed at that star. Just think of all the wonderful things you could do with this device!

As a simple example, imagine you go to the equator near Quito, Ecuador, on the day when Mintaka, the right-most star in the belt of the constellation Orion, is directly overhead at midnight. Right at midnight, you point the magic wand straight up at Mintaka. By 6 am, Mintaka is setting in the west, and the wand points west too. At noon, with the Sun overhead, Mintaka is directly opposite the Sun, on the opposite side of the Earth; and so your wand points straight down. At 6pm Mintaka rises, and the wand points toward it in the east. And when Mintaka is again overhead, the wand points straight up. From your perspective, it has rotated once during the 24 hour period.

[Actually, it takes 23 hours and 56 minutes, not 24 hours. That’s a subtle point, which we can return to in a later post.]

And from then on, through this second night and into the third and fourth, the wand follows Mintaka on its path around the sky. It doesn’t matter if it’s cloudy or stormy or clear; it doesn’t matter if it’s day or night. Always, like magic, the wand points at Mintaka.

Of course, if the Earth is indeed rotating, the right way to understand the wand’s motion is that it is an illusion, and instead the star is fixed on the sky, and the wand is fixed in space. It is you who are being rotated, carried along as the Earth rotates. This is illustrated in the figure below.

There’s nothing special about the location, the star, or the timing. You could have first pointed the wand at 3:16 am instead of midnight. You could have pointed the wand at Mintaka from, say, Oslo or Buenos Aires; the wand will again rotate in a plane once a day, though the orientation of the plane, which is perpendicular to the Earth’s spin axis, will change depending on your latitude.

Or you could have aimed at other bright stars, such as Sirius, or Vega, or Arcturus, or at dim stars with no special names. You could have pointed the wand at the star Betelgeuse at the moment it rises above the trees outside your home on December 14th, or at the Large Magellenic Cloud of stars on February 2nd at 11:15 p.m. in your favorite southern vacation spot. It matters not at all: the wand, once aimed, never lets go of its target.

And that means, as each night and day go by, the wand will seem to perform a little pirouette, coming back to its starting point about 24 hours later. Rather than rotating in a plane, the end of the wand pointed at a random star will go around in a smaller circle, while the motion of the wand as a whole outlines a cone. The center of that cone will point along the Earth’s rotation axis, right at the North Star (called Polaris), if the target star is north of the equator, or at the South Star (called Polaris Australis) if the target star is south of the equator. If pointed straight at the North or South Stars, it will not budge, for those stars’ apparent locations in the sky remain fixed even as the Earth rotates.

Since this works at every location on Earth and with every star in the sky, would this prove the Earth is rotating? No. Our magic wand only proves that either the Earth is rotating, moving you around in a circle with the wand standing still, or that the stars are rotating, carrying the wand with them. This wand gives us an opportunity to measure how we are moving relative to the stars, or vice versa, but by itself doesn’t determine who is moving.

But maybe, with a little more knowledge of magic, we could learn which is which. If only we had such a magic wand.

The Wand Exists?

In reality, we don’t have a wand that works by such magic. We have something better: a wand that works by physics. Rather than something occult created by a wizard, it’s a concrete object created by one or more engineers, and it’s quite easy to understand and use, at least in principle. And when we understand how it works, it becomes clear that the stars are standing still, and it is the Earth that is rotating.

This wand, called a gyroscope, is a wonderful device. It was invented in 1812 by German physicist Johann Bohnenberger, though Foucault, the inventor of the famous pendulum, gave it its name in 1852. Foucault became interested in gyroscopes for the same reason we are discussing them here: he wasn’t satisfied with his pendulum as a demonstration of the Earth’s rotation, because it is somewhat counter-intuitive. In particular, it’s not easy to describe to a non-expert (or even a physics student) why it rotates every 24 hours at the north or south pole, every 32 hours in Paris, and every five and a half days in Caracas. Foucault wanted something simpler and more obvious. The behavior of a gyroscope, as a mathematician friend apparently suggested to him, would fit the bill.

The simplest gyroscopes are simply spinning tops: a piece of metal or other material, whose shape is not essential except that it must be carefully engineered to be as round and balanced as possible around a central axis. To become a gyroscope, it simply needs to be set spinning, and the axis around which it spins plays the role of our magic wand. It’s therefore essential that the axis be free to rotate in any direction, which requires a nested set of gimbals. It has to spin for 24 hours if we’re to use it as I’ve described, and it has to be absolutely free from any stray forces that might cause it to lose its sense of direction.

You can buy a cheap toy gyroscope at a store, but that won’t do at all; it’s not nearly good enough. Better ones are available at physics departments, where they serve for important demonstrations and experiments; later on I’ll link to a video that shows one. But even these aren’t good enough for our task. Foucault, using the best techniques he had available back then, only managed to design a small one that could spin for 10 minutes… enough to demonstrate that the Earth rotates, but not enough to make a spectacle on the same level as the majestic swing of a pendulum hung high in a cavernous room.

But a gyroscope of extreme tenacity and stability has all the properties of our magic wand. No matter where you live and no matter what time of day you choose and no matter where you choose to point it, its axis seems to move in a circle every 24 hours (except when pointed right along the Earth’s spin axis, in which case it remains fixed). That means 360 degrees rotation per day, which is 15 degrees per hour, or one degree every four minutes. (Almost exactly.) Unlike Foucault’s pendulum, its apparent rotation occurs at the same rate as the Earth’s real rotation, no matter where it is on Earth.

Like the imaginary wand, this device would demonstrate either that the Earth spins every 24 hours, or the stars zip around the Earth every 24 hours. But unlike the mythical wand that operates by magic, we know exactly how a gyroscope works. There’s no magic: any connection with the stars is irrelevant. You can point it at a gap right between two stars, and it will still point at that gap indefinitely.

Why the apparent magic if there’s no special relationship between this terrestrial gadget and the stars? It is a coincidence, though a dramatic one.

1. Because of a principle of physics, (conservation of angular momentum, which I’ll describe in a moment), a well-constructed stable isolated gyroscope always rotates around a fixed direction.
2. Because the stars are so incredibly far away, their rapid motion relative to one another and to our Sun is too slow for us to notice over periods of years or even lifetimes.

In short, the gyroscope’s fixed by principle, the stars by mere accident; and the combination of the two explains why a gyroscope tracks whatever star (or absence of star) its axis is aimed at.

Importantly, you can check that a gyroscope’s axis is stable even without knowing why! Before you point the gyroscope at a star and leave it alone, you should play with it a few minutes — a time short enough that the Earth’s rotation (if any) would be unnoticeable. Try lifting it, or turning its housing upside-down or sideways, or tilting the table it might be sitting on; you will see the axis of its spin does not budge. Beautiful demonstrations of this can be seen in this video, where high quality gyroscopes at the Caltech physics department are demonstrated. See especially 00:45-01:00, and again, perhaps more visibly, at 02:15-03:45. Of course if you strike the gyroscope, or shake it hard enough, or you unbalance it in any way, you’ll cause it to lose its aim. Indeed, you’ll see evidence that even the gyroscopes in the video, while good for demonstrations, aren’t stable enough for the task of showing the Earth rotates. But as long as you are gentle with it, a perfectly-engineered gyroscope will successfully maintain its orientation.

In space, it’s easier to get a gyroscope to point reliably. In this European Space Agency video from the international space station, you can see that even a toy gyroscope will do a pretty good job in the apparently weightless environment. But of course we want a gyroscope that can be stable on Earth, and a toy won’t do.

Now that you’ve played with this perfect gyroscope for a while and convinced yourself that it maintains its spin axis, you can point it at a star. There are three things that can happen.

• The gyroscope is not as well made as you need, and so its axis wanders due to its imperfections. If you point it at another star, it will again wander, but in a different way, since the imperfections have nothing to do with the motion of the stars.
• The gyroscope is high-enough quality, and the Earth does not rotate. Then the spin axis will remain pointed at exactly the same spot in your room — at your ceiling, or your window, or the blotch on the floor, or the dirty sock under the bed — all day long. It will not follow the stars, but neither will not wander; it will seem to you to remain fixed.
• The gyroscope is high-enough quality to remain stable, and the Earth does rotate. Then the gyroscope will behave like our imaginary magic wand, and appear to track the stars and, from your point of view, to rotate daily.

So just try out the gyroscope for a few days and draw your own conclusion. If you do this experiment and find behavior that mimics the imaginary magic wand we started with, you may safely conclude that the Earth rotates once a day, around an axis that points at the North Star in the north and at the South Star in the south. Its once-a-day rotation explains in an intuitive way that the Earth spins, and that the stars only seem to rotate around the Earth daily.

This is the clear demonstration of Earth’s spin that Foucault was seeking. Why does it work?

Conservation of Angular Momentum

A balanced, isolated gyroscope won’t lose its orientation because of the principle of nature called the “conservation of angular momentum”. Here “angular momentum” is a measure of spinning that accounts not only how fast the object spins but also the axis around which it spins, and “conservation” means “never changing”. You can see this principle at work in this video from MIT.

Conservation of angular momentum has many consequences — it explains why spinning figure skaters can spin faster when they pull their arms in toward their bodies — but for our purposes, the only consequence we care about is this: an isolated spinning object will maintain its spin axis and spin rate indefinitely.

In fact, the Earth itself is such an object — it’s one of the best gyroscopes we know. That’s why the axis of the Earth has pointed for centuries at the two stars Polaris (the North Star) and Polaris Australis (the corresponding South Star), which are the stars that appear to be essentially stationary throughout the night. The other planets also are natural gyroscopes, as are the Sun, other stars, black holes, and indeed most large objects in the universe.

But reality is never perfect. No gyroscope is ever perfectly isolated, and interactions with other objects tend to alter its spin rate and gradually destabilize its spin axis. Even the Earth’s axis wobbles, though slowly compared to a human lifetime. For the gyroscope we want, sitting on the Earth, there will be friction at the gyroscope’s supports which will slow its spin. Even if these are countered by a motor that keeps it spinning, there are usually other small effects which, over a day’s time, can cause this wand to lose its magic. To design a mechanical gyroscope which is stable enough to perform a complete daily rotation is not an easy task at all. Foucault, in 1852, did quite well to make one which was stable for ten minutes!

State of the Art?

The fact that gyroscopes have still not replaced Foucault’s pendulum in science museums around the world reflects the technological challenge involved, and it has remained a topic of research. Consider Andre Geim, who after first winning an Ig Nobel prize in 2000 for levitating frogs using powerful magnets, later shared the 2010 Nobel prize for the discovery of the wonder material called graphene. In 2001 he wrote a paper (unfortunately behind a paywall) entitled “Detection of Earth Rotation with a Diamagnetically Levitating Gyroscope”. This paper was co-authored with his hamster, Tisha, who apparently participated in the levitation experiments. Despite the co-author, the paper is written in a serious vein. In it Geim claims that he built a magnetically levitated gyroscope that operated for two hours (limited only by the cost of running his powerful 15 Tesla magnet) and rotated at the expected 15 degrees per hour. He then discusses the technical advantages and limitations of this approach and how it might be improved. That said, I have not yet found a single citation of this paper that reproduces or directly uses his result, nor can I find any talk in which he or anyone else describes this phenomenon, so I’m still not sure how seriously to take this paper. Let me know if you know more than I do.

In any case we are beyond the current limits of my knowledge and far beyond my expertise. There are a number of modern gyroscope-imitators, some of which operate on different principles from the mechanical gyroscopes I’ve been discussing, and I don’t understand all of them. It would be great if there’s one whose operation was intuitively clear, which could do the job of maintaining its orientation for more than a day, and whose cost might someday drop enough to make it affordable, at least to museums; but I don’t see one yet. If anyone reading this is more knowledgeable about this subject that I am, please chime in in the comments.

The Ideal Spinning World

Here’s my dream for the future. Someday, I hope that every major science museum — maybe even many universities — will own three sophisticated, stable, affordable gyroscopes. Why three? Because there are many lovely things you can do with three. You could, for example, pick three stars at random — that is, point the three gyroscopes in any directions you choose — and then watch how the gyroscopes rotate during the day. The three cones that the gyroscopes’ axes sweep out over the day will all be centered on the Earth’s axis. You’ll also notice that the angles between the gyroscopes never change, as you’d expect if they are truly at fixed orientations and it’s you who are rotating.

A second option would be to point one gyroscope at the North Star (Polaris), a second at a bright star elsewhere, such as Aldebaran, and the third at the Sun. When you check the next day, the first gyroscope won’t have moved, the second will still be pointing at Aldebaran, but the third will no longer be pointing quite at the Sun; it will be about 1 degree off. This change per day will continue. Half a year later, the gyroscope would point opposite the Sun, out into deep space. And if you could watch it all year, the Sun would come back into alignment with the third gyroscope only after the full year had passed. This, of course, is a sign that the Sun orbits the Earth once a year — or is it the other way round?

A gyroscope pointed at the North Star won’t seem to move; one pointed at a random star, such as Aldebaran, will rotate in a cone and remain pointed at that star; but one pointed at the Sun, though it will also rotate in a cone, will fail to follow the Sun by about 1 degree per day, or a full 360 degrees per year.

A last option I’ll mention is that you could arrange the three gyroscopes so that they point at right angles, setting up what’s known as a Cartesian coordinate system. The simplest way to do this would be to point one at Polaris and place the other two in the plane perpendicular to the first one and to each other. Then, over the day, the first gyroscope wouldn’t budge, but the other two would seem to rotate smoothly around it.

The Best Proofs Come from the Most Unlikely Sources

Until and unless these kinds of mechanical gyroscopes become possible, we may have to satisfy ourselves with laser gyroscopes, which are far less intuitive than mechanical ones but have the same magical ability to know where they point. (I won’t try to describe how they work here, since it’s quite subtle, and I’d rather see someone build a mechanical one anyway.) I haven’t found any free videos that clearly show these in action, but they are definitely capable of demonstrating that the Earth rotates. This was done on camera in Netflix’ movie Behind the Curve, in which one of these laser gyroscopes is seen to rotate 15 degrees per hour and thus 360 degrees per day.

There is resonance, in this story, with the 17th century scientists who laughed at spin-earthers because cannonballs on a rotating Earth wouldn’t fly straight; we laugh back, because in fact they do not fly straight (i.e. the Coriolis effect, known long before Coriolis but clarified by him in 1835.) In the case at hand, the experiment that proves the Earth rotates was performed by someone who didn’t believe the Earth rotates. The fellow in question, a flat earther, was confident that if he bought a good enough gyroscope, reportedly at a cost of $20,000, it wouldn’t rotate at all, and thus he fully expected he would prove, on camera, that anyone who claims the Earth spins is ridiculous. Of course he was wrong — the gyroscope rotated at 15 degrees per hour. We thank him for his efforts.

But naturally, being a flat earther, rather than recognizing that he’d perfectly reproduced the prediction of the competing theory and admitting defeat, he set about to make reality fit his beliefs. He described his discovery and his reaction in this video, where he claimed the sky must be causing the gyroscope’s rotation. To test this notion, he tried putting the gyroscope in metal containers in an attempt to shield it from these unknown sky effects, but, hey, the gyroscope just kept rotating. Sensible people will be giggling about this for a long time, assuming the whole thing wasn’t staged. (It probably wasn’t; you can’t make this stuff up, can you?)

To see the full Netflix video requires that you pay. At the moment, I can’t find any free videos that allow you to observe the Earth’s rotation convincingly. But perhaps a company that sells ring laser gyroscopes would like to set one up as a combination public service/marketing opportunity? Just point a camera at it and let it sit, recording its orientation hour by hour, day by day, as the Earth turns?