Coordinate Independence, Kepler, and Planetary Orbits

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=R3/2 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 = 202 times further from Earth than the Moon, the period of the Sun would be 4003/2 = 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.

Read more

Even in Einstein’s General Relativity, the Earth Orbits the Sun (& the Sun Does Not Orbit the Earth)

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/R2.

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.

Read more

Is it Meaningful to Say that Earth Goes Round the Sun, or Not? (And Why Is This So Hard…?)

Is the statement “The Sun Orbits the Earth” false? Not according to professor Richard Muller of the University of California, Berkeley, as I discussed yesterday. Muller argues that Einstein’s theory of general relativity implies that you can view the Sun as orbiting the Earth if you like, or that both the Sun and Earth orbit Venus, or a random point in space, or anything else for that matter. Meanwhile, every science textbook in our kids’ classrooms says that “The Earth Orbits the Sun“. But for all of our discussions yesterday on this subject, we did not yet collectively come to any conclusions as to whether Muller is right or wrong. And we can’t hope to find evidence that the Earth orbits the Sun if the reverse is equally true!

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.

Read more

Sun Around the Earth, or Earth Around the Sun?  Did Einstein Say “It’s all the same”?

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

Earth Goes Around the Sun? What’s Your Best Evidence?

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

Which is Bigger, the Sun or the Earth?  Check it Yourself!

Once you’ve convinced yourself the Earth’s a spinning sphere of diameter about 8000 miles (13000 km), and you’ve estimated the Moon’s size and distance (diameter about 1/4 Earth’s, and distance about 30 times Earth’s diameter), it’s easy to convince yourself the Sun’s bigger than the Earth, and much further than the Moon.  It just takes a couple of triangles, and a bit of Moon-gazing.

Since that’s all there is to it, you can guess that the ancient Greek astronomers, masters of geometry, already knew the Sun’s the larger of the two.  That said, they never did quite figure out how big and far the Sun actually is; we need modern methods for that.

It’s Just a Phase

The Moon goes through a monthly cycle of phases, lasting about 291/2 Earth days, in which the part that glows brightly with reflected sunlight grows and shrinks, from crescent to full and back again.  The phases arise because there are two simple ways of dividing the Moon in half:

  • At any moment, the half of the Moon that faces Earth — let’s call it the near half of the Moon — is the only half that we can potentially see. (We’d only be able to see the far half, facing away from Earth, if the Moon were transparent, or a big mirror was sitting beyond the Moon.)
  • At any moment, the half of the Moon that faces the Sun is brightly lit — let’s call it the lit half.  The other half is dark, and its presence can only be detected by the fact that it can block stars that it moves in front of, and through a very dim glow in which it reflects sunlight that first reflected from the Earth (called “Earthshine.”)  

The phases arise because the lit half and the near half aren’t the same, and the relationship between them changes from night to night.   See the diagram below. When the Moon is more or less between the Sun and the Earth (it rarely passes exactly between, because its orbit is tilted by a few degrees out of the plane of the drawing below) then the Moon’s lit half is its far half, and the near half is unlit. We call this dark view of the Moon the “New Moon” because it is traditionally viewed as the start of the Moon’s monthly cycle. 

Figure 1: The Moon’s phases, assuming the Sun’s much further than the Moon. When the Moon is roughly between the Earth and Sun, its near half coincides with the unlit half, making it invisible (New Moon). As the cycle proceeds, more of the near half intersects with the lit half; after 1/4 or the cycle, the Moon’s near half is half lit and half unlit, giving us a “half Moon.” At the cycle’s midpoint, the near side coincides with the lit half and the Moon appears full. The cycle then reverses, with the other half Moon occurring after 3/4 of the cycle.

When the Moon is on the opposite side of the Earth from the Sun (but again, rarely eclipsed by Earth’s shadow because of its tilted orbit), then its near side is its lit side, and that creates the “Full Moon”, a complete white disk in the sky. 

At any other time, the near side of the Moon is partly lit and partly unlit. When the line between the Moon and Earth is perpendicular to the Earth-Sun line, then the lit side and unlit side slice the near side in half, and the Moon appears as a half-disk cut down the middle.

When I was a child, I wondered why half this half-lit phase of the Moon, midway between New Moon (invisible) and Full Moon (the bright full disk), was called “First Quarter”, when in fact the Moon at that time is half lit.  Why not “First Half?”  Two weeks later, the other half of the near-side of the Moon is lit, and why is that called “Third Quarter” and not, say, “Other Half”?

This turns out to have been an excellent question. The fact that a Half Moon is also a First Quarter Moon tells us that the Sun is large and far away!

Read more

An Experience of a Lifetime: My 1999 Eclipse Adventure

Back in 1999 I saw a total solar eclipse in Europe, and it was a life-altering experience.  I wrote about it back then, but was never entirely happy with the article.  This week I’ve revised it.  It could still benefit from some editing and revision (comments welcome), but I think it’s now a good read.  It’s full of intellectual observations, but there are powerful emotions too.

If you’re interested, you can read it as a pdf, or just scroll down.

 

 

A Luminescent Darkness: My 1999 Eclipse Adventure

© Matt Strassler 1999

After two years of dreaming, two months of planning, and two hours of packing, I drove to John F. Kennedy airport, took the shuttle to the Air France terminal, and checked in.  I was brimming with excitement. In three days time, with a bit of luck, I would witness one the great spectacles that a human being can experience: a complete, utter and total eclipse of the Sun.

Read more

A Solar Eclipse Tomorrow (Sunday)

Appropriate for General Readership Tomorrow there will be a solar eclipse (i.e., the moon will pass between the earth and the sun, blocking the sun’s light.) Those of us on the east coast of the United States who wake up to a clear sky at dawn will see the rising sun partially eclipsed, as much … Read more

%d bloggers like this: