Of Particular Significance

Tag: earth

How “Shooting Stars” Measure the Distance to the Sun

(This is the fourth post in a series, though it can be read independently; here are post #1 , post #2 , and post #3.)

For many years, I thought that measuring the distance to the Sun was quite difficult for a non-astronomer. I had the impression that it requires precision measurements, often involving telescopes or information from satellites, and that it was only easy to obtain a minimum distance and a maximum distance that were still quite far apart, as I explained in my last post.

But it’s not true. As I’ll explain today, it turns out that anyone can estimate the distance to the Sun, at night, with nothing more than the naked eye, basic reasoning, and… meteors.

Just from the fact that a long meteor crosses the sky in a few seconds, you can infer that the Earth-Sun distance is something like 100 million miles (km). If the Sun were only 10 million miles (km) away, the meteors would lazily drift among the stars, only a bit faster than the motions of the space station and other satellites, which take minutes to cross the sky. Meanwhile, if the Sun were a billion miles (km) away, then meteors would flash across the sky in a fraction of a second.

With a little more work and knowledge, you can use meteors to make an estimate of the Sun’s distance that’s well within a factor of 2 of the truth. It’s not even that hard to get a precise measurement that’s good to 10% or so.

It may seem odd that one can use such little specks of dust in the Earth’s atmosphere to determine, without a telescope, how far it is to the Sun. But in fact the method is simple. It’s so simple that it must have been pointed out two centuries ago. Curiously, though, I’ve never seen it written down anywhere. It seems to be little-known, even to scientists.

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POSTED BY Matt Strassler

ON January 23, 2024

How Far Could the Sun Possibly Be?

(This is the third post in a series, though it can be read independently; here are post #1 and post #2; and post #4 blows this one out of the water, so don’t miss it!)

Measuring the distance to the Sun is challenging, for reasons explained in my last post. Long ago, the Greek thinker Aristarchus proposed a geometric method, which involves estimating the Moon’s sunlit fraction on a certain date. Unfortunately, because the Sun is so far away, his approach isn’t powerful enough; Aristarchus himself underestimated the distance. [This last remained true for later astronomers before the 17th century, though they got closer to the truth, presumably by using more precise methods than you or I could easily apply. I doubt anyone truly found a maximum possible distance to the Sun just using geometry.] The best we can do, using Aristarchus’ method and our naked eyes, is determine a minimum possible distance to the Sun: a few million miles.

Figure 1: A simple application of Aristarchus’ method tells us that the minimum distance to the Sun is a few million miles (km), ruling out the red region. But the entire green region is still allowed.

Today we’ll see how to obtain a maximum distance to the Sun, using an approach suggested in the previous post: by measuring speeds. Specifically, we’ll make use of a speed that the ancient astronomers weren’t aware of: the speed of light, also known as the cosmic speed limit c. That’s 186,000 miles (300,000 km) per second, or 5.9 trillion miles (9.5 trillion km) per year. We’ll find the Sun’s distance is less than 12 billion miles… still much larger than its true distance, but a significant improvement on our starting point!

*Figure 2: By the end of this post, we’ll know a maximum possible distance for the Sun, too.*

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POSTED BY Matt Strassler

ON January 19, 2024

What Ocean Tides Teach Us About the Sun and Moon

The Moon has a four-week cycle; it is full every four weeks (actually every 29.5 days). But ocean tides exhibit a two-week cycle; they are large one week and then smaller the next.

Specifically, as in Fig. 1 below, ocean tides are stronger (“spring tides”) around New Moon and Full Moon than they are at First Quarter Moon and Last Quarter Moon (“neap tides”). The pattern is seen, roughly at least, all around the world (though the details are not simple, as they depend on the shape of the coastline and other factors.)

Figure 1: Tides in Anchorage, Alaska, USA, during October 2023; the blue line shows how the water rises and falls about twice a day (note the vertical columns are each **two** days wide!) The pattern of strong and weak tides on alternate weeks is clearly visible. New Moon occurred on October 14 and Full Moon on October 28th, just before the peak tides.

What’s behind this pattern? And what does it tell us about the Sun and Moon that we wouldn’t otherwise easily know? Perhaps surprisingly, it tells us that the average mass density of the Sun — its mass divided by its volume — is only a little smaller than the average mass density of the Moon. Here’s why.

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POSTED BY Matt Strassler

ON January 12, 2024

How to Tell that the Earth Spins

Continuing with the supplementary material for the book, from its Chapter 2. This is in reference to Galileo’s principle of relativity, a central pillar of modern science. This principle states that perfectly steady motion in a straight line is indistinguishable from no motion at all, and thus cannot be felt. This is why we don’t feel our rapid motion around the Earth and Sun; over minutes, that motion is almost steady and straight. I wrote

. . . Our planet rotates and roams the heavens, but our motion is nearly steady. That makes it nearly undetectable, thanks to Galileo’s principle.

To this I added a brief endnote, since the spin of the Earth can be detected, with some difficulty.

As pointed out by the nineteenth-century French physicist Léon Foucault, the Earth’s rotation, the least steady of our motions, is reflected in the motion of a tall pendulum. Many science museums around the world have such a “Foucault pendulum” on exhibit.

But for those who would want to know more, here’s some information about how to measure the Earth’s spin.

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POSTED BY Matt Strassler

ON September 21, 2023

Beyond the Book (and What the Greeks Knew About the Earth)

Since the upcoming book is basically done, it’s time for me to launch the next phase of the project — the supplementary material, which will be placed here, on this website.

Any science book has to leave out many details of the subjects it covers, and omit many important topics. While my book has endnotes that help flesh out the main text, I know that some readers will want even more information. That’s what I’ll be building here over the coming months. I’ll continue to develop this material even after the book is published, as additional readers explore it. For a time, then, this will be a living, growing extension to the written text.

As I create this supplementary material, I’ll first post it on this blog, looking for your feedback in terms of its clarity and accuracy, and hoping to get a sense from you as to whether there are other questions that I ought to address. Let’s try this out today with a first example; I look forward to your comments.

In Chapter 2 of the book, I have written

Over two thousand years ago, Greek thinkers became experts in geometry and found clever tricks for estimating the Earth’s shape and size.

This sentence then refers to an endnote, in which I state

The shadow that the Earth casts on the Moon during a lunar eclipse is always disk-shaped, no matter the time of day, which can be true only for a spherical planet. Earth’s size is revealed by comparing the lengths of shadows of two identical objects, separated by a known north-south distance, measured at noon on the same day.*

Obviously this is very terse, and I’m sure some readers will want an explanation of the endnote. Here’s the explanation that I’ll post on this website:

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POSTED BY Matt Strassler

ON September 19, 2023

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

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

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Tag: earth

POSTED BY Matt Strassler

POSTED BY Matt Strassler

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