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

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.

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 just over a dozen steps, the basics of planetary astronomy; you can

and infer from these laws that the same gravity that makes ordinary objects fall creates an inward acceleration, one that follows an inverse square law, holding certain objects in orbit around the Earth and others in orbit around the Sun.

I won’t count high-precision telescopic observations that can reveal tiny effects, such as stellar aberration, stellar parallax, and Doppler shifts in light from other stars. They’re great, but very tough for non-experts to verify. Isn’t there a simpler source of evidence for this very basic claim about nature — something we can personally check?

Your thoughts? Comments are open. [Be careful, when making suggestions, that you are not assuming that gravity is the dominant force between the Earth and the Sun. That’s something you have to prove. Are you sure there are no additional forces pinning the Earth in place, and/or keeping the Sun in motion around the Earth? What’s your evidence that they’re absent?]

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.

In the last three posts (1,2,3) I showed how to establish the spherical nature of the Earth without the use of geography, geometry or trigonometry. All I used was was the timing of pressure spikes seen in barometers around the world as a result of two volcanic explosions — the one earlier this month from the Kingdom of Tonga, and the Krakatoa eruption of 1883 — along with addition and subtraction. This method, unlike any other I’m aware of, is suitable for especially young students; its only difficulties are conceptual, and even these only involve simple demonstrations, such as can be accomplished with a ball and a rubber band.

The timing data showed that it takes 35-36 hours for a pressure wave to circle the globe. (I showed this for this month’s eruption in the first post, pointed out a logical loophole in the second post, and closed the loophole by showing the same was true for the Krakatoa eruption in the third post.) Next, to determine the size of the globe, all we need is to estimate the pressure wave’s velocity. This requires a bit more information; we need some limited amount of local geography, and timing for one pressure spike as it moves across a small region of the Earth. In brief, all we need is to learn how much time X it took the pressure wave to cross a region of known width W; then the speed of the wave is simply v = W/X.

Measuring the Speed of the Wave

Fortunately a number of people made this easy for us, creating animations in which pressure measurements are shown over a brief period while the pressure wave was crossing their home countries. The only hard part is to make sure that we not only measure timing (X) correctly but that we define the width ( W ) correctly. The width has to be measured perpendicular to the direction of the wave (or equivalently, it has to be the shortest distance between the wave as measured at some initial time t and the wave as measured at a later time t+ X). Otherwise, as you can see in the figure, we’ll overestimate W and thus overestimate v. The difficulty of getting this right, along with the intrinsic thickness of the pressure wave, will be our biggest sources of uncertainty in estimating v.

We already have some circumstantial evidence that v varied by less than 5% or so, based on the success of the method I used to check the Earth’s a sphere. (At the end of this post is some satellite evidence that the Tonga volcano’s pressure wave had a nearly constant v; the evidence seems otherwise for Krakatoa, based on the observed timing of pressure spikes.) But still, in order to be certain that v didn’t vary much, and to reduce uncertainties on our measurements, it would be best to estimate v in a few places. I found useful animations of the pressure wave from Germany, China, New Zealand, and the United States. These represent the wave’s motion in four very different directions: north (and over the pole), northwest, southwest and northeast. Here’s the example from New Zealand, which we’ll go through in detail.

Below are two stills from the above animation, which allows us to see the wave as it first enters New Zealand’s north island and as it exits. The time between the two stills is 1 hour and 2 minutes. How far has the wave traveled in that time? The wave is less obvious in the final still, so while the distance across New Zealand from northeast to southwest is about 720 miles, give or take 10 miles (1140-1175 km), the distance the wave has actually traveled is a bit less certain, perhaps as little as 700 miles or as much as 740 (1125-1190 km). So our measurement of the speed across New Zealand is about 700-740 miles per hour (1125-1190 km per hour.) It would be hard to get a more precise measurement.

When I tried to make similar estimates using the other animations from Germany, China and the United States, I found it was challenging if I tried to determine travel distances over times much less than an hour; the uncertainties were too great. But if the time was much longer than that, it became more difficult to determine the wave’s trajectory– remember it’s important to measure the distance in a direction perpendicular to the wave, so as not to overestimate the distance. In the end, using multiple measurements in both China and the United States and one measurement in Germany, I found the following:

Location

Speed Estimate (mph)

Speed Estimate (kph)

New Zealand

700 – 740

1125 – 1190

China

620 – 700

1000 – 1125

United States

720 – 760

1160 – 1225

Germany

720 – 800

1160 – 1285

The significant spread seen here probably reflects the challenges of an imprecise measurement, rather than actual variation in the wave speed; the round trip times found in an earlier post suggested variation in the speed of no more than 5%. It’s not obvious how to combine these statistically if you really wanted to do this with sophistication, but the whole point of this exercise is to see how far you can get without being sophisticated. So let’s eyeball it: you can see there is a preference for the 680-750 mph range (1095-1205 kph), so let’s take that as our most likely range. Of course you are free to draw a different conclusion from these numbers if you prefer, and to repeat the exercise I’m about to do.

Now that we have an estimate of v, we can determine the Earth’s circumference C. If the pressure wave traveled at constant speed v in the range just suggested, the distance C that it covered in a round trip, which required time T = 35–36 hours, is

C = v T = 23800 – 27000 miles = 38300 – 43500 km

The uncertainty of order 15% is not surprising given the difficulty of determining v, and perhaps its small variation from one place to another, combined with the imperfect measurements of the round-trip time.

The true answer for the Earth’s circumference varies slightly; it is 24,901 miles (40,075 kilometers) around the equator and 24,859 miles (40,008 km) around a circle that passes through both the north pole and the south pole. Of course these precise numbers are measured with sophisticated equipment. They lie well within my estimate (and quite close to its central value of 25400 miles, 40880 km). It shows that with this method, someone with no expertise in atmospheric science or surveying techniques, sitting in a chair in his living room, can characterize the planet. The same is true of kids in a science classroom, given a little time and a lot of guidance.

Some Last Thoughts

Admittedly I have used sophisticated equipment too — the computers, servers and communication lines of the internet, barometers with electronic output, software for putting that output into various useful forms, and social media for its distribution. But what I haven’t needed is illumination, travel, or knowledge of anything other than local geography. This method would work even if the Earth were forever in darkness, if international travel was impossible, and if a large fraction of the Earth had never been mapped.

That’s interesting, because all of the other methods I know for showing the Earth’s a sphere and measuring its size rely on light and/or on travel. Aristotle’s method for inferring Earth’s shape, and Eratosthenes’ method for measuring its size, rely on shadows; Eratosthenes needed geometry, too. If you travel off the Earth you can see the Earth from outside, either in visible light or in other invisible forms of light, such as infrared light — but you need the light. Of course you can remain on the Earth and travel around it, and if you’re really very careful you can learn about the planet’s shape and size without doing a complete circuit of it. That, however, requires some sophistication, and in particular trigonometry.

Here we’ve let a pressure wave do all the travel, and whether in sunlight or in darkness it has left its trace in local atmospheric pressure. We just need the data on that pressure in a few places, mostly without even knowing where those places are. All we need, after that, is addition and subtraction (to find T), followed by a brief application of division (to find v) and multiplication (to find C). I don’t know of a simpler method.

We’re done; now what exactly was the point of all this? I’m sure that there are plenty of people wondering why someone with a Ph.D. in theoretical physics and dozens of papers on particle physics and string theory would spend time showing how to measure something that’s been well-understood for thousands of years. My reasons range from an general interest in history, epistemology and volcanology to a vague concern about how science is taught and understood in the modern world. But that’s a subject for a future post.

Postscript on the Wave Speed

By the way, there’s satellite evidence that the wave speed v was very close to a constant, at least on first half-trip around the Earth. Here’s an animation of the pressure wave on its way out from Tonga (I have not been able to find the original clip), and below is an animation of the pressure wave as it converges on the point exactly opposite Tonga, in southern Algeria. If the wave speed were constant, the converging wave would form a shrinking circle. It’s not quite that, but pretty close! The approximation of a constant speed, while not perfect, is really quite good. And that’s why the methods I used worked so well.

Posted onJanuary 25, 2022|Comments Off on The Earth’s Shape and Size? You Can Measure it Yourself — Part 2

In my last post, I showed, using only simple arithmetic, that the observed atmospheric effects from the January 15th volcanic explosion in the Kingdom on Tonga are consistent with a round Earth. From the timing of the observed spikes in pressure, seen around the world, one can work out how long the pressure wave took to do a round-trip of our planet. It’s clear that the pressure wave from the eruption moved outward and circled the Earth, moving in all directions over the same amount of time (35-36 hours, to within 5% or so). This uniformity is what we would expect if the Earth’s approximately a sphere and the pressure wave had a roughly constant speed.

But at the end of the post I pointed out that this isn’t yet a proof that the Earth’s spherical; there are loopholes, involving possibilities such as an ellipsoidal Earth with the Tonga eruption at one end. And there’s even a flat version of Earth that we can’t rule out with this data!

So in this post, we’ll look at why most shapes for the Earth are ruled out, see why there’s a loophole — why a small number of non-spherical shapes are still consistent with the data — and look at how we might close that loophole.

A Square Earth

Flat Earth’s aren’t plausible, but they are easy to draw and visualize, so I’m going to start by showing why most (but not quite all) flat Earth’s are inconsistent with the data I used in the last post. Once you see the origins of the inconsistencies, the same principles will apply to other shapes that would be much harder to visualize if you didn’t already know what to look for.

Let’s start with a square Earth (yes, square, not a cube — though a cube would have similar problems). This Earth has edges, and we have to figure out what happens to the pressure wave when it gets there. Leaving aside the obvious difficulty that we have no idea how a square planet would hold on to its atmosphere at the edges, there are three easy options for what happens to the pressure wave at the edge:

It disappears.

It bounces back (i.e., it reflects).

It somehow goes round to the back side, crosses it, and reappears.

Disappearance is ruled out immediately, because then the pressure wave would pass each point on Earth once, whereas the data shows it appears multiple times. So let’s focus on the second possibility, the reflecting square. The problems we’ll find here will also affect the third possibility.

There’s another question we have to answer: where is the volcano inside this square? Well, let’s start with the simplest case, where the volcano is dead center. After we see what’s wrong with that, it will be easy to see that an off-center volcano is even worse.

On a square with reflections, the pressure wave expands and then bounces back from the walls, rather than going all the way around as on a sphere. In other words, a round trip from the volcano to a chosen location and back to the volcano involves some reflections instead of a continuous trip. That’s okay in principle, but what’s not okay can be seen in the Figure below. Trips north-south and east-west have the same length, but trips northeast-southwest and northwest-southeast are longer by a square root of 2, about 40% longer. We would certainly have seen this in the pressure spike data; if north-south trips took 36 hours, then northeast-southwest trips would have taken almost 51 hours.

And actually it’s worse than this, because the reflections would make a total mess of the pressure wave. You can get a little intuition for this by tracing the path of the bit of wave that moves west-southwest. It bounces around several times before returning to the volcano!

More generally, what is happening is that the wave is becoming very complex as it reflects multiple times. In the animation below I’ve shown what would happen to a pressure wave on a square. There’s no way we would have seen a simple pattern of spikes in the data around the world had it been square.

Is there any way out of this argument? So far I’ve assumed that the wave travels at a constant speed as it moves away from the volcano. What if it didn’t? What if, instead of forming a circle, it formed a square, which could move out uniformly and bounce back uniformly from the edges, so that all round trips were of the same duration? This would require that the wave’s speed heading toward the corners of the square is 40% faster than it’s speed heading north, south, east and west. That’s a clever idea, and so far, what I’ve told you doesn’t exclude it. But in a later post we’ll use pressure spike data to measure the wave’s speed in various directions, and we won’t see such large variation; so we will rule this out soon enough.

The spike patterns would be at least as complicated, and generally worse, if

the volcano were not dead center on the reflecting square (making the pattern of reflections even more complex — see the figure below);

the pressure wave went round the back of the square Earth;

the square was instead a rectangle with sides of different length; or

the square was instead a triangle, hexagon, parallelogram, a five-pointed star, a crescent, or some irregular shape;

In short, a flat Earth is completely excluded — ruled out by the data — except for one very special shape.

The Flat Disk Earth

Imagine the Earth’s a flat disk, and put the volcano at the exact center. Then, you can get exactly the same pressure spike data as we actually observe. Let’s see why.

If a pressure wave moves off at a constant speed from an explosion at the center of a disk, it will form a ring that moves outward, reflects off the walls, and comes right back to the volcano. And it will do this over and over again. In all directions from the volcano, the out-and-back trips all take the same amount of time; and at each location on Earth, the pressure wave will pass twice during this out-and-back trip. You can go further and check that the equations I used to determine the round-trip time on a spherical Earth will work for a disk Earth too, where T is now the out-and-back time. The spike pattern from a volcano centered on a disk looks identical to that of a volcano on a sphere.

This is only if the volcano is dead center, however. For example, in the figure below, the trip to the right is longer than the trip to the left; and yet again, because the volcano’s not in the center, the reflections off the edges will quickly make the wave extremely complex and lead to a highly irregular pattern of spikes around the world. So an off-center volcano is ruled out. (The situation is no better if the waves, rather than reflecting off the edges, somehow go round the back.)

So the only way to interpret our data, if the Earth is flat, is to conclude that Tonga sits in the very middle of a flat disk. But this is quite a loophole! How can we prove the Earth is not flat?

The Flat-Earthers’ Flat Earth

By the way, what I’ve just told you means that the pressure spike data rules out the flat-disk Earth most popular with flat Earthers. That silly model of Earth puts the north pole at the center and stretches the south pole out into a circle tens of thousands of miles around, with the idea that no one ever actually flies over the south pole to check it out.

Well, let’s leave aside the fact that many scientists, including personal friends of mine, have experiments (Ice Cube, BICEP, South Pole Telescope, and many more…) running within a mile or so of the south pole, and they (and the pilots who fly them there) can confirm it is a point, not an arc tens of thousands of miles wide. But we now have an argument that’s not hearsay: given where the Tonga volcano is located on this flat-disk Earth, an explosion there would never have been able to generate the observed regular and simple pattern of pressure spikes. A 12-year-old can prove the flat-earthers’ model of Earth is definitively ruled out.

And these considerations also show us why a flat Earth that puts Tonga dead center is ruled out too, though not from the pressure spike data. Just as the flat-earther’s model of Earth, with the north pole at the center, spreads the south pole into an arc tens of thousands of miles long, one with Tonga at the center would spread southern Algeria, the region exactly opposite, into an arc tens of thousands of miles long. But even though that’s in the desert, people live there. There are a few roads and a few towns. Residents there would certainly know if driving to the nearest town took many weeks instead of a few hours.

So that one remaining flat Earth is dead too. Good-bye, and good riddance.

But I went through this argument carefully for a reason. Once we understand why a Tonga-centered flat disk Earth is consistent with the pressure spike data, we can understand all the other loopholes, such as ellipsoidal Earths — and we’ll also see how to rule them out too.

A Symmetry

Why was it that every flat Earth gave the wrong pattern for the spike timing except for the flat disk with the volcano at dead center? What was special about that case?

The study in my last post showed that any bit of the pressure wave, as it started at and headed out from the volcano, took the same amount of time to travel outward and back to its starting point. In other words, as far as the pressure wave was concerned, all directions leading away from Tonga are equivalent to one another. East, north, northwest, south-southwest — it doesn’t matter, the length of the round-trip path was always the same.

A fancier way to say that is that there is a symmetry, a rotational symmetry in particular. If you were to put a spike straight through the Earth starting right at the volcano and going through the Earth’s center, and then you rotated the Earth around the spike, the Earth’s shape would stay the same as you did so. If that weren’t true, then not all directions would look the same, and not all round-trip times would be equal.

So what my analysis of the data actually shows is not that the Earth’s a sphere, but only that it is symmetrical around the Tonga volcano — all directions are equivalent. That’s true of a sphere. But it’s also true of a flat disk with the volcano at its center — or of a bowl. And it’s also true of an ellipsoid with the volcano at one end, or of a gourd shape, or of half a sphere.

So how are we going to check that the right shape for the Earth is truly a sphere?

Special Points vs. Typical Points

The symmetry that I just described requires that either

the Tonga volcano is at a very special point on a non-spherical shape, or

the Tonga volcano is at a typical point on a sphere.

We already saw this for the flat disk; we could only reproduce the data if the volcano were at the center, and not if it were off-center (as in the flat-earthers’ flat Earth.) And while it’s true for an ellipsoid with a circular cross-section if the volcano is exactly at one end, it wouldn’t be true if the volcano were anywhere else.

That makes all the non-spherical shapes somewhat implausible, because they require that the Tonga volcano be located at a unique, special place — one of at most two on Earth. And what are the chances that the first big volcanic blast of the internet era would occur at such a special location? There are so many other volcanoes — Vesuvius, Mount Rainier, Mount Erebus, Cotopaxi, Taal, Merapi, and hundreds more — any volcano that isn’t on exactly the opposite side of the Earth from the Tonga volcano would have given asymmetric data, with round trip times that vary widely. Only on a sphere is the Tonga volcano at a typical point, with nothing unusual about it.

So a sphere seems much more plausible. But, hey, that’s just a plausibility argument, and coincidences do happen sometimes. If you want to prove the Earth’s a sphere, this argument is not enough.

Fortunately, it’s now clear where proof would come from. We just need to wait for another similarly-sized eruption, from some other volcano, to create another pressure wave that goes round the Earth. Even if the Tonga volcano were somehow located at a special point on Earth, the next big volcanic blast will almost certainly originate from a typical point. It’s very unlikely that it will lie exactly on the opposite side of the Earth from Tonga. If, after this second blast, we do the same measurement of round-trip times using its pressure spikes, and we again find they all show equal round-trip distances in all directions, then we’ll know the Earth is symmetric around that volcano too. And that’s enough, because only a sphere can be rotationally symmetric around two points (unless those two points are exactly at the opposite ends of an ellipsoid or similar shape.)

The only thing that’s too bad (although it’s also quite fortunate) is that explosions this size don’t happen often. We may not be able to close this loophole for quite a few decades to come…

…unless, rather than looking to the future, we look to the past…?

There is, after all, Krakatoa.

(to be continued)

Comments Off on The Earth’s Shape and Size? You Can Measure it Yourself — Part 2

This week, I’ll describe how one can easily use the Jan 15th explosive volcanic eruption in Tonga to obtain strong evidence that the Earth’s a sphere and determine its circumference, using nothing more than simple arithmetic. This illustration of scientific measurement is perfect for any science classroom, because it uses publicly accessible data, is straightforward enough for a 12-year-old to follow, and is meaningful to every human being. Moreover, students can be set free to find their own data sets online, and yet all will get the same answer in the end. It is my hope that science teachers worldwide will begin to include this exercise in their classrooms.

In this first post, I’ll explain how to verify that the Earth’s approximately a sphere. It’s not quite a proof yet, because there are loopholes to close; but before the end of the week the evidence will be conclusive.

Background

Fortunately, volcanic eruptions as powerfully explosive as Tambora (1815), or even Krakatoa (1883), are seen only a few times a millennium. When they do occur, loss of life and destruction of homes and livelihoods can be immense. The full human cost of the tremendous blast ten days ago, at a mostly underwater volcano in the Kingdom of Tonga, is still not fully known; some islands in the archipelago were completely swamped by large tsunami waves, and the toll in lives and houses is not yet clear. Meanwhile, many aspects of the explosion itself are still puzzling scientists. But these are not the stories for today.

The explosion created a (literally) deafening blast of sound, and a wave of pressure so powerful that it could easily be detected by weather stations around the globe, both those of professionals and those in the homes of ordinary people. In fact, many stations detected the wave passage multiple times. Not since the era of thermonuclear weapons tests, prior to the 1963 nuclear test ban treaty, have we (to my knowledge) observed such a crisply defined pressure wave from an explosion of this magnitude. (The explosion, probably a combination of water flashing to steam upon contacting rising magma, along with the release of gas dissolved in that magma, has been estimated as equivalent to at least 10 megatons of TNT, nearly a thousand times larger than the atomic bombs of World War II and comparable to the largest thermonuclear weapons ever tested.) Back in the ’60s, ordinary people had no easy access to precise data from weather stations, and there were fewer stations around the world, too. Because of today’s technology, this explosion, more than any prior, offers us a unique educational opportunity, a silver lining to this disaster that science teachers across the world should take advantage of.

The Method of Great Circles

How can you tell if the surface you live on is a sphere? Easy, if it’s small enough, like the planet of the Little Prince. You start from your home, and start walking in any direction you choose. Just keep walking straight ahead; you will eventually come home again. Let’s say it took you one hour. Well, now that you’re home, pick another direction, and start walking straight ahead at the same steady pace until you again return home. This second trip should also take you one hour. Repeat as desired; every round trip, in every direction, should cover the same distance, and assuming your walking speed is always the same, it will take the same amount of time.

Each of these trips would be on a path called a “great circle”, which is a circle that divides a sphere into two equal halves; these are the longest circles that you can draw on a sphere, and they each have the same length — the circumference of the sphere. Here’s a drawing with three of them. Famous great circles on the Earth are the equator and all lines of longitude (but not lines of non-zero latitude, which don’t divide the world into equal halves.)

Of course, walking around the Earth would be impractical; not only would it take too long, the oceans would get in your way. You could consider taking an airplane on a series of trips, starting from your home airport and traveling straight ahead until you came back home — but expense, politics and weather would interfere, and the technology for a non-stop round-trip tour isn’t in place.

What’s so useful about a blast wave, for this purpose, is that the wave takes all these great circle trips around the world, in all directions, simultaneously, at no cost to you — not to mention that it’s apolitical. The wave spreads out in all directions, forming an expanding circle; that this was true for the Tonga explosion can be confirmed from pressure measurements, but can also be seen in the satellite images below, of water vapor around the Earth in the hours following the explosion.

Such a wave will continue to spread until its size is as large as the Earth’s circumference; then it shrinks down until it converges at a point exactly on the opposite side of the Earth from the volcano. It then passes through itself and retraces its steps, beginning to grow again. Here’s a visualization, showing an entire round-trip, by @StefFun. Note that one round trip has four stages: expanding from the volcano, shrinking down to the opposite point, expanding again from that point, and shrinking down back the volcano’s location. We can call the first half the “outbound” portion, and the second half the “returning” or “inbound” portion. This pattern repeats over and over until the wave has lost too much energy to be detectable any longer.

It might appear, from these animations, that the wave is going halfway round the Earth and then bouncing back. But in fact, the wave is passing through itself! What’s happening in this round trip is that each little part of the pressure wave is making its own great-circle loop of the Earth. All those great-circle trips happen simultaneously, giving the pattern seen above. And like a sedentary Little Prince, you can use that pattern to see if the Earth’s a sphere.

That’s the Theory. Is it True?

Everything that I’ve just described will be true under two assumptions:

The Earth really is almost spherical.

The shock wave really does travel at an almost constant speed in all directions.

These two assumptions can be tested, and if they are (approximately) true, they can be used to measure (approximately) the size of the Earth. [Note: We’re actually also assuming the atmosphere is thin compared to the size of the Earth, so that the wave’s energy stays trapped in a relatively thin region above the ground.]

Here’s the logic. If the Earth’s a sphere and the pressure wave’s a circle moving at constant speed v, then

each little section of the pressure wave travels around the Earth in a “great circle”, whose length is the circumference of the sphere C.

the “round-trip time”, which we’ll call “T”, is the same for every part of the pressure wave, as illustrated in the tweet above, with T = C / v .

From this behavior of the pressure wave, we obtain a prediction: no matter where you are located on the Earth relative to Tonga, the wave as it passes over you is on a round-the-Earth trip that will take a round-trip time T. During that trip one bit of wave will pass you once during its outbound portion, and the opposite bit of wave, going the other direction, will pass you during its inbound portion; so you will see the wave twice each round trip. Because all parts of the wave are moving at the same speed (by assumption) and all are traveling the same distance (by assumption), you should get the same value of T no matter where you live. If you can measure T, and you have fourteen friends in fourteen other countries who can also measure T in an analogous way, the fifteen of you should all get the same answer.

But how can we measure T, the round-trip time, while sitting at home?

Measuring the Round-Trip Time T

The volcano exploded at about 415 UTC on January 15th. (UTC is a 24 hour universal time which is used world-wide to avoid getting confused by time zones, but it corresponds to a time zone used by several nations in far western Europe and in west Africa.) Its pressure wave was strong enough to create sudden spikes and/or drops in the pressure each time the wave passed by (but let me just refer to this Fdisturbance as a “spike” for brevity.) These could be measured by barometers on the ground. In many places, the wave was strong enough, the atmosphere calm enough, and the barometers precise enough that several spikes were seen.

Here’s an example from the Met Office in the United Kingdom, and one (with average pressure removed to make the spikes easier to see) from Iceland.

Let’s imagine you yourself have a barometer which shows as many as four spikes. Let’s call T_{1}the time between the volcano’s explosion and the appearance of first spike. (I used different notation in my last and more detailed post: T_{1}=t_{1}-t_{s} .) We’ll similarly define T_{2}, T_{3}, T_{4} for the second, third and fourth spike. Then from these four time measurements, there are three independent methods you can use to measure T, and they should all give the same answer.

The key thing to remember, before interpreting these disturbances, is that the pressure wave passes you twice on each of its round trips, and so you see the pressure spike twice per round trip. (Remember each round trip involves four stages, two of them the expansion and contraction of the outbound portion, and two of them the expansion and contraction of the inbound portion. You may want to look at the tweet above if you need a reminder.)

What that means is that spike 1 is caused by the shockwave when it is outbound on its first round trip, and spike 3 is caused when it is outbound on its second round trip, so they are separated by the round-trip time. In other words

T_{3} – T_{1} = T

Similarly, spikes 2 and 4 are caused by the shockwave when it is inbound on its first and second round trips, so they too are separated by the round-trip time.

T_{4} – T_{2} = T

Now the last way to measure T is slightly more subtle, although the answer’s very simple. It turns that

T_{1} + T_{2} = T

Why is this true? It is visualized in the Figure below The key is that the speed v (which we don’t know yet) is constant. The bit of the wave that headed from the volcano towards you took a time T_{1} to reach you, during which the wave covered a distance D_{1} = T_{1} v. (Remember T_{1} is the time that elapsed from the volcanic explosion until your observations of the first spike.) But the second spike was caused by the bit of wave that started in the opposite direction, heading away from you; it reached you after going the long way around the Earth. This required a time T_{2}, and during that time the wave covered a distance D_{2} = T_{2} v. But as you can see from the figure, D_{1} + D_{2} is the entire circumference C of the Earth! So if T_{1} is the time it takes to travel a distance D_{1} , and T_{2} is the time it takes to travel a distance D_{2}, then their sum must be the time it takes to travel the distance C — and that, by definition, is the round trip time T.

So if you see four spikes, you get three ways to measure T that should all agree, as long as the shockwave moves at a constant speed and the line from the volcano to you forms a part of a great circle. If you see three spikes you get two measurements, but even with just two spikes — no simple repeats — you still get one measurement of T.

But if the Earth’s a sphere and the wave’s speed is constant, then everyone around the world should agree on the measurement of T, even though each of us will measure a different T_{1}, T_{2}, T_{3}, T_{4} depending on where we live. If all our measurements of T are the same, then the assumptions we started with — that the eruption caused a circular shock wave of constant speed that moved around a spherical Earth — are consistent with the data. If they are slightly off, then our assumptions are only approximately true, but close enough to give us roughly the right idea.

Let’s grab some data from around the world and see what we get.

Data and Measurement

I obtained data from a variety of places, and did my own estimates of the spike arrival times (which can be done to within an accuracy of 30 to 90 minutes, typically). I then converted those to the time elapsed since the volcanic explosion, being careful to account for time zones and convert to UTC. In some cases I could only determine T_{1} and T_{2}, but sometimes I could get T_{3} or even T_{4} . Then, I computed as many estimates of the round-trip time T that I could obtain with the two, three or four spikes from each location. All this information is given in the table below. You are encouraged to find other sources of data and try this yourself.

Remarkably, from these places that lie in wildly different directions and distances from Tonga, all of the values of T that I obtained fall between 34 ^{3}/_{4} hours and 36 ^{3}/_{4} hours, a variation of less than 10%. (I couldn’t find data from Australia, New Zealand or Southern Africa that showed multiple spikes; do you know of any?) My time measurements were often ambiguous at the 5% level, because the pressure wave often consisted of multiple spikes and dips, so just from my measurement uncertainty one would expect to see several percent variation in these values of T.

The close agreement among the values of T then implies that both of our starting assumptions — that the Earth is spherical and that the pressure wave traveled with a constant speed — are consistent with data, to better than 10%.

About the assumptions: Of course I know, from other data, that the Earth is spherical to within 2% — it is slightly squashed, so that a great circle of longitude is 2% shorter than the length of the equator. So I knew beforehand that the first assumption would be okay to 2%. But given that the speed of waves can vary with temperature and perhaps other atmospheric effects, it wasn’t obvious that the second assumption would work out. Since the numbers all agree, apparently it was more or less correct too.

Is The Earth a Sphere? Mmm… We’re Not Quite Done

So there you have it. Within less than 10%, our assumptions of a roughly spherical Earth and a roughly circular pressure wave of roughly constant speed are consistent with data.

Is this a complete proof of a near-spherical Earth? Nope. We’re close, but there are still loopholes. For example, suppose the Earth looked like an ellipsoid, with the volcano placed exactly at one end. We’d all still find equal values of T. Can you see why?

There’s even a flat-earth hypothesis that we haven’t quite excluded yet! Can you identify which one? (It would easily be ruled out for other reasons, but not from this data alone.)

In the next two posts I’ll show you how to identify the origin of the loophole, and then close it for good. And after that, we’ll measure the circumference of the Earth.

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From the CMS experiment at the Large Hadron collider, a proton-proton collision that created a Higgs boson, which subsequently decayed to two particles of light (shown as green rods.)