Of Particular Significance

Tag: astronomy

Black Holes, Mercury, and Einstein: The Role of Dimensional Analysis

In last week’s posts we looked at basic astronomy and Einstein’s famous E=mc² through the lens of the secret weapon of theoretical physicists, “dimensional analysis”, which imposes a simple consistency check on any known or proposed physics equation. For instance, E=mc² (with E being some kind of energy, m some kind of mass, and c the cosmic speed limit [also the speed of light]) passes this consistency condition.

But what about E=mc or E=mc⁴ or E=m²c³ ? These equations are obviously impossible! Energy has dimensions of mass * length² / time². If an equation sets energy equal to something, that something has to have the same dimensions as energy. That rules out m²c³, which has dimensions of mass² * length³ / time³. In fact it rules out anything other than E = # mc² (where # represents an ordinary number, which is not necessarily 1). All other relations fail to be consistent.

That’s why physicists were thinking about equations like E = # mc² even before Einstein was born.

The same kind of reasoning can teach us (as it did Einstein) about his theory of gravity, “general relativity”, and one of its children, black holes. But again, Einstein’s era wasn’t first to ask the question. It goes back to the late 18th century. And why not? It’s just a matter of dimensional analysis.

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POSTED BY Matt Strassler

ON June 28, 2022

Dimensional Analysis: A Secret Weapon in Physics

It’s not widely appreciated how often physicists can guess the answer to a problem before they even start calculating. By combining a basic consistency requirement with scientific reasoning, they can often use a heuristic approach to solving problems that allows them to derive most of a formula without doing any work at all. This week I want to introduce this to you, and show you some of its power.

The trick, called “dimensional analysis” or “unit analysis” or “dimensional reasoning”, involves requiring consistency among units, sometimes called “dimensions.” For instance, the distance from the Earth to the Sun is, obviously, a length. We can state the length in kilometers, or in miles, or in inches; each is a unit of length. But for today’s purposes, it’s irrelevant which one we use. What’s important is this: the Earth-Sun distance has to be expressed in some unit of length, because, well, it’s a length! Or in physics-speak, it has the “dimensions of length.”

For any equation in physics of the form X = Y, the two sides of the equation have to be consistent with one another. If X has dimensions of length, then Y must also have dimensions of length. If X has dimensions of mass, then Y must also. Just as you can’t meaningfully say “I weigh twelve meters” or “I am seventy kilograms old”, physics equations have to make sense, relating weights to weights, or lengths to lengths, or energies to energies. If you see an equation X=Y where X is in meters and Y is in Joules (a measure of energy), then you know there’s a typo or a conceptual mistake in the equation.

In fact, looking for this type of inconsistency is a powerful tool, used by students and professionals alike, in checking calculations for errors. I use it both in my own research and when trying to figure out, when grading, where a student went wrong.

That’s nice, but why is it useful beyond checking for mistakes?

Sometimes, when you have a problem to solve involving a few physical quantities, there might be only one consistent equation relating them — only one way to set an X equal to a Y. And you can guess that equation without doing any work.

Well, that’s pretty abstract; let’s see how it works in a couple of examples.

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POSTED BY Matt Strassler

ON June 21, 2022

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=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 = 20² 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.

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POSTED BY Matt Strassler

ON June 1, 2022

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:

If the Earth goes round the Sun, then its path satisfies the Sun’s T|R law, just like the other planets do.
If the Sun goes round the Earth, then its path does not satisfy the Earth’s T|R law, although the Moon and various human-built satellites do so.

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.

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POSTED BY Matt Strassler

ON May 24, 2022

Earth Orbits the Sun, or Not? Why Coordinates Can’t Be Relevant to the Question.

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.

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POSTED BY Matt Strassler

ON May 18, 2022

Earth Around the Sun, or Not? The Earth-Centered Coordinates You Should Worry About

We’re more than a week into a discussion of Professor Richard Muller’s claim that “According to the general theory of relativity, the Sun does orbit the Earth. And the Earth orbits the Sun. And they both orbit together around a place in between. And both the Sun and the Earth are orbiting the Moon.” Though many readers have made interesting and compelling attempts to prove the Earth orbits the Sun, none have yet been able to say why Muller is wrong.

A number of readers suggested, in one way or another, that we go far from the Sun and Earth and use the fact that out there, far from any complications, Newtonian physics should be good. From there, we can look back at the Sun and Earth, and see what’s going on in an unbiased way. Although Muller would say that you could still claim the Sun orbits the Earth by using “geocentric” coordinates centered on the Earth, these readers argued that such coordinates would not make sense in this distant, Newtonian region.

Are they correct about this?

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POSTED BY Matt Strassler

ON May 16, 2022

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.

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POSTED BY Matt Strassler

ON May 5, 2022

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 Sun goes round the Earth” are equally true, or equally false, or equally meaningless.

Here, for example, is this statement as written in Forbes by professor Richard Muller at the University of California, Berkeley. It opens as follows: “According to the general theory of relativity, the Sun does orbit the Earth. And the Earth orbits the Sun.” I invite you to read the rest of it; it’s not long.

What’s his point? In Einstein’s theory of gravity (“general relativity”), time and three-dimensional space combine together to form a four-dimensional shape, called “space-time”, which is complex and curved. And in general relativity, you can choose whatever coordinates you want on this space-time.

So you are perfectly free to choose a set of coordinates, according to this point of view, in which the Earth is at the center of the solar system. In these coordinates, the Earth does not move, and the Sun goes round the Earth. The heliocentric picture of the planets and the Sun merely represents the simplest choice of coordinates; but there’s nothing wrong with choosing something else, as you like.

This is very much like saying that to use latitude and longitude on the Earth is just a choice. I could use whatever coordinates I want. The equator is special in the latitude-longitude system, since it lies at latitude=0; the poles are special too, at latitude +90 degrees and -90 degrees. But I could just as well choose a coordinate system in which the equator and poles don’t look special at all.

And so, after Einstein, the whole Copernican question — “is the solar system geocentric or heliocentric?” — is a complete red herring… much ado about nothing. As Muller argues in his article, “the revolution of Copernicus was actually a revolution in finding a simpler way to depict the motion, not a more correct way.“

Well? Is this true? If not, why not? Comments are open.

POSTED BY Matt Strassler

ON May 4, 2022

From Kepler’s Law to Newton’s Gravity, Yourself — Part 2

Sometimes, when you’re doing physics, you have to make a wild guess, do a little calculating, and see how things turn out.

In a recent post, you were able to see how Kepler’s law for the planets’ motions (R³=T² , where R the distance from a planet to the Sun in Earth-Sun distances, and T is the planet’s orbital time in Earth-years), leads to the conclusion that each planet is subject to an acceleration a toward the Sun, by an amount that follows an inverse square law

a = (2π)² / R²

where acceleration is measured in Earth-Sun distances and in Earth-Years.

That is, a planet at the Earth’s distance from the Sun accelerates (2π)² Earth-distances per Earth-year per Earth-year, which in more familiar units works out (as we saw earlier) to about 6 millimeters per second per second. That’s slow in human terms; a car with that acceleration would take more than an hour to go from stationary to highway speeds.

What about the Moon’s acceleration as it orbits the Earth? Could it be given by exactly the same formula? No, because Kepler’s law doesn’t work for the Moon and Earth. We can see this with just a rough estimate. The time it takes the Moon to orbit the Earth is about a month, so T is roughly 1/12 Earth-years. If Kepler’s law were right, then R=T^2/3 would be 1/5 of the Earth-Sun distance. But we convinced ourselves, using the relation between a first-quarter Moon and a half Moon, that the Moon-Earth distance is less than 1/10 othe Earth-Sun distance. So Kepler’s formula doesn’t work for the Moon around the Earth.

A Guess

But perhaps objects that are orbiting the Earth satisfy a similar law,

R³=T² for Earth-orbiting objects

except that now T should be measured not in years but in Moon-orbits (27.3 days, the period of the Moon’s orbit around the Earth) and R should be measured not in Earth-Sun distances but in Moon-Earth distances? That was Newton’s guess, in fact.

Newton had a problem though: the only object he knew that orbits the Earth was the Moon. How could he check if this law was true? We have an advantage, living in an age of artificial satellites, which we can use to check this Kepler-like law for Earth-orbiting objects, just the way Kepler checked it for the Sun-orbiting planets. But, still there was something else Newton knew that Kepler didn’t. Galileo had determined that all objects for which air resistance is unimportant will accelerate downward at 32 feet (9.8 meters) per second per second (which is to say that, as each second ticks by, an object’s speed will increase by 32 feet [9.8 meters] per second.) So Newton suspected that if he converted the Kepler-like law for the Moon to an acceleration, as we did for the planets last time, he could relate the acceleration of the Moon as it orbits the Earth to the acceleration of ordinary falling objects in daily life.

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POSTED BY Matt Strassler

ON April 15, 2022

From Kepler’s Law to Newton’s Gravity, Yourself — Part 1

Now that you’ve discovered Kepler’s third law — that T, the orbital time of a planet in Earth years, and R, the radius of the planet’s orbit relative to the Earth-Sun distance, are related by

R³=T²

the question naturally arises: where does this wondrous regularity comes from?

We have been assuming that planets travel on near-circular orbits, and we’ll continue with that assumption to see what we can learn from it. So let’s look in more detail at what happens when any object, not just a planet, travels in a circle at a constant speed.

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POSTED BY Matt Strassler

ON April 11, 2022

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Tag: astronomy

POSTED BY Matt Strassler

POSTED BY Matt Strassler

POSTED BY Matt Strassler

POSTED BY Matt Strassler

POSTED BY Matt Strassler

POSTED BY Matt Strassler

POSTED BY Matt Strassler

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A Guess

POSTED BY Matt Strassler

POSTED BY Matt Strassler

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