Could you, merely by changing coordinates, argue that the Sun gravitationally orbits the Earth? And could Einstein’s theory of gravity, which works equally well in all coordinate systems, allow you to do that?

Despite some claims to the contrary — that all Copernicus really did was choose better coordinates than the ancient Greek astronomers — the answer is: No Way.

How badly does the Sun’s path, nearly circular in Earth-centered (geocentric) coordinates, violate the Earth’s version of Kepler’s law? (Kepler’s third law is the relation T=R^{3/2} between the period T of a gravitational orbit and the distance R, which is half the long axis of the ellipse that the orbit forms.) Since the Moon takes about a month to orbit the Earth, and the Sun is about 400 = 20^{2} times further from Earth than the Moon, the period of the Sun would be 400^{3/2} = 8000 times longer than the Moon’s, i.e. about 600 years, not 1 year.

But is this statement coordinate-independent? Can it serve to prove, even in Einstein’s theory, that the Earth orbits the Sun and the Sun does not orbit the Earth? Yes, it is, and yes, it does. That’s what I claimed last time, and will argue more carefully today.

Of course the question of “Does X orbit Y?” is already complicated in Newtonian gravity. There are many situations in which the question could be ambiguous (as when X and Y have almost equal mass), or when they form part of a cluster of large mass made from many objects of small mass (as with stars within a galaxy.) But this kind of ambiguity is not what’s in question here. Professor Muller of the University of California Berkeley claimed that what is uncomplicated in Newtonian gravity is ambiguous in Einsteinian gravity. And we’ll see now that this is false.

Back before we encountered Professor Richard Muller’s claim that “According to [Einstein’s] general theory of relativity, the Sun does orbit the Earth. And the Earth orbits the Sun,” I was creating a series of do-it-yourself astronomy posts. (A list of the links is here.) Along the way, we rediscovered for ourselves one of the key laws of the planets: Kepler’s third law, which relates the time T it takes for a planet to orbit the Sun to its distance R from the Sun. Because we’ll be referring to this law and its variants so often, let me call it the “T|R law”. [For elliptical orbits, the correct choice of R is half the longest distance across the ellipse.] From this law we figured out how much acceleration is created by the Sun’s gravity, and concluded that it varies as 1/R^{2}.

That wasn’t all. We also saw that objects that orbit the Earth — the Moon and the vast array of human-built satellites — satisfy their own T|R law, with the same general relationship. The only difference is that the acceleration created by the Earth’s gravity is less at the same distance than is the Sun’s. (We all secretly know that this is because the Earth has a smaller mass, though as avid do-it-yourselfers we admit we didn’t actually prove this yet.)

T|R laws are indeed found among any objects that (in the Newtonian sense) orbit a common planet. For example, this is true of the moons of Jupiter, as well as the rocks that make up Jupiter’s thin ring.

Along the way, we made a very important observation. We hadn’t (and still haven’t) succeeded in figuring out if the Earth goes round the Sun or the Sun goes round the Earth. But we did notice this:

This was all in a pre-Einsteinian context. But now Professor Muller comes along, and tells us Einstein’s conception of gravity implies that the Sun goes round the Earth just as much (or just as little) as the Earth goes round the Sun. And we have to decide whether to believe him.

We’ve been having some fun recently with Sun-centered and Earth-centered coordinate systems, as related to a provocative claim by certain serious scientists, most recently Berkeley professor Richard Muller. They claim that in general relativity (Einstein’s theory of gravity, the same fantastic mathematical invention which predicted black holes and gravitational waves and gravitational lensing) the statement that “The Sun Orbits the Earth” is just as true as the statement that “The Earth Orbits the Sun”… or that perhaps both statements are equally meaningless.

But, uh… sorry. All this fun with coordinates was beside the point. The truth, falsehood, or meaninglessness of “the Earth orbits the Sun” will not be answered with a choice of coordinates. Coordinates are labels. In this context, they are simply ways of labeling points in space and time. Changing how you label a system changes only how you describe that system; it does not change anything physically meaningful about that system. So rather than focusing on coordinates and how they can make things appear, we should spend some time thinking about which things do not depend on our choice of coordinates.

And so our question really needs to be this: does the statement “The Earth Orbits the Sun (and not the other way round)” have coordinate-independent meaning, and if so, is it true?

Because we are dealing with the coordinate-independence of a four-dimensional spacetime, which is not the easiest thing to think about, it’s best to build some intuition by looking at a two-dimensional spatial shape first. Let’s look at what’s coordinate-independent and coordinate-dependent about the surface of the Earth.

When we’re trying to figure out whether a confusing statement is really true or not, we have to speak precisely. Up to this stage, I haven’t been careful enough, and in this post, I’m going to try to improve upon that. There are a few small but significant points of clarification to make first. Then we’ll look in detail at what it means to “change coordinates” in such a way that would put the Sun in orbit around the Earth, instead of the other way round.

We’re all taught in school that the Earth goes round the Sun. But if you look around on the internet, you will find websites that say something quite different. There you will find the argument that Einstein’s great insights imply otherwise — that in fact the statements “The Earth goes round the Sun” and “The … Read more

It’s commonly taught in school that the Earth orbits the Sun. So what? The unique strength of science is that it’s more than mere received wisdom from the past, taught to us by our elders. If some “fact” in science is really true, we can check it ourselves. Recently I’ve shown you how to verify, in … Read more

Having confirmed we live on a spherical, spinning Earth whose circumference, diameter and radius are roughly 25000, 8000, and 4000 miles (40000, 13000, and 6500 km) respectively, it’s time to ask about the properties of the objects that are most obvious in the sky: the Sun and Moon. How big are they, and how far away?

Historically, many peoples thought they were quite close. With our global society, it’s clear that neither can be, because they can be seen everywhere around the world. Even the highest clouds, up to 10 miles high, can only be seen by those within a couple of hundred miles or so. If the Moon were close, only a small fraction of us could see it at any one time, as shown in the figure at right. But in fact, almost everyone in the nighttime half of the Earth can see the full Moon at the same time, so it must be much further away than a couple of Earth diameters. And since the Moon eclipses the Sun periodically by blocking its light, the Sun must be further than the Moon.

The classical Greeks were expert geometers, and used eclipses, both lunar and solar, to figure out how big the Moon is and how far away. (To do this they needed to know the size of the Earth too, which Eratosthenes figured out to within a few percent.) They achieved this and much more by working carefully with the geometry of right-angle triangles and circles, and using trigonometry (or its precursors.)

The method we’ll use here is similar, but much easier, requiring no trigonometry and barely any geometry. We’ll use eclipses in which the Moon goes in front of a distant star or planet, which are also called “occultations”. I’m not aware of evidence that the Greeks used this method, though I don’t know why they wouldn’t have done so. Perhaps a reader has some insight? It may be that the empires they were a part of weren’t quite extensive enough for a good measurement.

Even if you’re working from home, so that you’re spending the day at a fixed location on the Earth’s surface, you’re not at a fixed location relative to the Earth’s center. As the Earth turns daily, it carries you around with it. So where are you headed today? Presumably Earth’s spin takes you around in a big circle, right?

That’s great. Which circle?

Point to it, right now.

Let me ask that again, in case that wasn’t clear. With your feet on the ground, looking whichever direction you choose, please show me the circle you’ll be taking today on your travels.

No idea? In my experience, many people have never even thought about it. Those who are willing to hazard a guess have to think for a moment to figure out that the Earth is rotating west to east — that’s why the Sun appears to rise in the east and set in the west. Once they are clear on that point, many people face east, and then indicate a circle that goes straight ahead, which would be combination of east and then down, as you can see in the figure.

To say that another way, if you imagine the circle of travel as being the edge of a disk, that disk would face east-west and slice directly down into the ground.

For the vast majority of us, it turns out this guess is not correct.

So where are we headed? People located at the equator or the poles can answer this more easily than the rest of us, so let’s start with them.