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.

(Left) On a square Earth with reflective edges and a central volcano, round trip times are different for diagonal paths than for horizontal or vertical paths. (Right) In fact round trips can be extremely long at certain angles.

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.

On a square Earth with a reflective edge, a circular blast from a centrally located volcano would lead to a very complicated pattern of pressure spikes that do not correspond to what is observed.

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

  1. the volcano were not dead center on the reflecting square (making the pattern of reflections even more complex — see the figure below);
  2. the pressure wave went round the back of the square Earth;
  3. the square was instead a rectangle with sides of different length; or
  4. 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.

On a square Earth with a reflective edge, a circular blast from a non-centrally located volcano would lead to an even more complicated pattern of pressure spikes.

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.

On a disk-shaped Earth with a reflective edge, a blast from a central volcano would reproduce the pressure spikes observed following this month’s explosion.

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.)

A disk Earth with a volcanic blast away from the center will not have equal round-trip times; the full pattern is very complex, but just the two paths shown are enough to give different round-trip times, not seen in the actual pressure spike data.

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.

The flat earth with north pole at center, popular with a certain set, with a red dot showing the far-off-center location of the Tonga volcano. Pressure waves from that location could not have created the observed pressure spikes.

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.

A flat Earth centered on region of Tonga would spread southern Algeria across tens of thousands of miles; residents of that country would beg to differ.

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?

A pressure wave starting from any point on a sphere, or from a point at either end of an ellipsoid, or from the center of a disk or hemisphere (with reflecting edges), will give a similar pattern to the one observed after the Tonga volcanic explosion. Note this is not true at any other point on the non-spherical surfaces. (Image made with Mathematica 12.)

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 are closed.