I’ve received various comments, in public and in private, that suggest that quite a few readers are wondering why a Ph.D. physicist with decades of experience in scientific research is spending time writing blog posts on things that “everybody knows.” Why discuss unfamiliar but intuitive demonstrations of the Earth’s shape and size, and why point out new ways of showing that the Earth rotates? Where’s all the discussion of quantum physics, black holes, Higgs bosons, and the end of the universe?
One thing I’m not doing is trying to convince flat-earthers! A flat-earther’s view of the world is so full of conceptual holes that there’s no chance of filling them. Such an effort would be akin to trying to convince a four-year-old Santa Claus devotee that the jolly fellow can’t actually fly through the air and visit half a billion homes, stopping to eat the cookies left for him in every one, all in one night. Logic has no power on a human whose mind is already made up. (If you’re an adult, don’t be that human.)
Instead my goals are broader, and more contemplative than corrective. Here are a few of them.
In my last post I gave you a way to check for yourself, using observations that are easy but were unavailable to ancient scientists, that the Earth is rotating from west to east. The clue comes from the artificial satellites and space junk overhead. You can look for them next time you have an hour or so under a dark night sky, and if you watch carefully, you’ll see none of them are heading west. Why is that? Because of the Earth’s rotation. It is much more expensive to launch rockets westward than eastward, so both government agencies and private companies avoid it.
In this post I want to describe the best proof I know of that the Earth rotates daily, using something else our ancestors didn’t have. Unlike the demonstration furnished by a Foucault pendulum, this proof is clear and intuitive, involving no trigonometry, no complicated diagrams, and no mind-bending arguments.
The Magic Star-Pointing Wand
Let’s start by imagining we owned something perfect (almost) for demonstrating that the Earth is spinning daily. Suppose we are given a magic wand, with an amazing occult power: if you point it at a distant star, any star (excepting the Sun), from any location on the Earth, it will forever stay pointed at that star. Just think of all the wonderful things you could do with this device!
Well, now that we’ve seen how easily anyone who wants to can show the Earth’s a sphere and measure its size — something the classical Greeks knew how to do, using slightly more subtle methods — it’s time to face a bigger challenge that the classical Greeks never figured out. How can we check, and confirm, that the Earth is spinning daily, around an axis that passes through the north and south poles?
We definitely need techniques and knowledge that the Greeks didn’t have; the centuries of Greek astronomy included many great thinkers who were too smart to be easily fooled. The problem, fundamentally, is that it is not obvious in daily life that the Earth is spinning — we don’t feel it, for reasons worthy of a future discussion — and it’s not obvious in astronomy either, because it is hard to tell the difference between the Earth spinning versus the sky spinning. In fact, if it’s the sky that’s spinning, it’s clear why we don’t feel the motion of the Earth’s spin, whereas if the Earth is spinning then you will need to explain why we don’t feel any sense of motion. Common sense tells us that we, and the Earth, are stationary. So even though many people over the centuries did propose the Earth is spinning, it was very hard for them to convince anyone; they had neither the right technology nor a coherent understanding of basic physics.
Broken Symmetry
One way to differentiate a rotating Earth from a non-rotating one is to focus on the notion of symmetry. On a non-rotating featureless ball, even if we define it to have north and south poles, there’s no difference between East and West. There’s a symmetry: if you look at a mirror image of the ball, West and East are flipped, but there’s nothing about the ball that looks any different.
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.
As the wave moves from lower left to upper right, its speed can be estimated by measuring the distance W traveled during a certain time period. But the line drawn for this measurement must be perpendicular to the wave (black arrow); if we let geometry or geography fool us into measuring in any other direction (as in the red arrows), we will overestimate W and thus the speed.
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.
The massive eruption from Hunga Tonga-Hunga Ha'apai volcano on Saturday sent a pressure wave all around the globe.
It travelled the ~2000km to NZ in around 2 hours and was even picked up on its second pass.
The pressure wave from the Tonga volcano crossing New Zealand.
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.
Two stills from the above animation, 1:02 apart, showing the pressure wave as it enters New Zealand from the northeast and as it exits to the southwest just a few hours after the explosion.
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.
In the first post in this series, I showed, using pressure spikes in barometers from around the world, that the pressure wave from the volcano that exploded in the Kingdom of Tonga earlier this month circled the Earth about once every 36 hours (accurate to within 5% or so, which is about two hours). It required only grade school arithmetic to do it, too! That the round-trip time is the same in many different directions provides evidence that the Earth is a sphere, obtained without the need for photographs, expensive travel or even geography!
But as I showed in my second post, it’s too quick to view it as proof of a spherical Earth. There’s a loophole.
What the direction-independence of the pressure wave’s round-trip time proves is that the Earth has some amount of symmetry: if you are standing at the volcano, then no matter in which direction you look, the Earth has the same shape (to within 5% or so). But there are many shapes that have this symmetry, not just a sphere. For instance, an ellipsoid, a gourd, a flat disk or an inverted bowl would all have this symmetry, as long as the Tonga volcano were centrally located: at one end of the ellipsoid or gourd, or in the center of the flat disk or bowl. Even though it’s unlikely that the Tonga volcano would be at such a very special point on a non-spherical Earth, we can’t prove that it’s not the case without more information.
If the Tonga Volcano were located at the end of an ellipsoid, or at the center of a disk or bowl with reflective edges, it would have created pressure spikes with the same pattern as if it were on a sphere. (Image created with Mathematica 11.3.)
As I pointed out, though, the pressure wave from a second volcanic blast of a similar nature, arising from another point on the Earth, wouldn’t show the same independence of direction unless either
the Earth’s a sphere, or
the Earth’s not a sphere, but the second volcano is located at the exact opposite side of the Earth from the Tonga volcano.
The second possibility is extremely unlikely, especially as the relevant location, southern Algeria, has no volcanoes! So if the round-trip times for a second natural explosion are the same in all directions, that proves the Earth’s a sphere.
Such powerful and dangerous eruptions are rare, fortunately, and so it might seem that we will have to wait a long time to close this loophole. But in fact, we can look to the past, where the famous 1883 explosion of Krakatoa, between the islands of Sumatra and Java in Indonesia, fits the bill. The same types of pressure spikes were observed then as we have observed this month. The only challenge is to find that century-old data.
Pressure spikes, similar to those seen around the world in mid-January, observed in late August 1883 from the Krakatoa blast.
It actually isn’t much of a challenge. The Royal Society, an organization based in London with an outsized role in the history of modern science, spent the years following the blast collecting all the data that we might ever want. And as I realized on Monday night, the full Royal Society report from 1888 is available online, via Google Books and perhaps other sources. It took me five minutes to find the pressure data, and thirty seconds to find the tables that I needed to close the loophole and prove, once and for all, that the Earth’s a nice round ball.
That’s worth thinking about. The Royal Society’s experts had to collect all this data by sending letters to keepers of weather records, located in remote places all around the world. Not only did they need all the details of atmospheric pressure over time following the Krakatoa eruption, they also had to be very careful that they interpreted the timing correctly. In those days, time zones were very new, and weren’t universally adopted, so it would have been very easy to mistake the meaning of any local time marked on the pressure charts. It must have been hard work, prone to errors. On top of this, they couldn’t know exactly when the biggest explosion happened — there were no satellites there to see it, and of the few eyewitnesses, none apparently had a precise clock — so they had to infer the timing of the blast from the pressure data itself.
Meanwhile, while some experts were studying the pressure spikes, other experts were collecting other information about the eruption: the tsunamis, the eruptive history, the materials ejected by the volcano, the optical and electromagnetic effects and the eyewitness reports. By the time everything was collated and ready for public distribution, it was 1888 — over four years later. Copies of the Royal Society report were buried in large public and university libraries, but this 600 page document wasn’t necessarily something you could find at your small town bookstore. Even a few decades ago, it wasn’t the easiest information to obtain quickly.
But that has changed in the era of the internet and of projects such as Google Books. Indeed, what took the Royal Society four years for Krakatoa now takes almost no time at all. For the Tonga volcano, pressure data from many places, including weather stations owned by ordinary people, was reported almost in real time via social media and various websites. That made it easy to show the Earth is probably a sphere within a few days, almost as soon as the data came in. Closing the last loopholes, to really prove the Earth’s a sphere, simply required a short visit to the Great Library in the Cloud. All this can be done by pretty much anyone, including internet-enabled schoolchildren with a science teacher who provides guidance as to what to do and why.
The Krakatoa Report’s Data and the Round Trip Time
So let’s open the pages of the Royal Society report, and see what it contains.
In my first post in this series (and also in the post before that) I pointed out that if you have the pressure data from a certain city and can see the spikes that were generated by the volcano’s pressure wave, then it is simple arithmetic to determine the round-trip travel time T of that bit of pressure wave that traveled from the volcano to that city. If the Earth’s really (approximately) a sphere and the pressure wave moves at an (approximately) constant speed, then the pressure wave will travel uniformly around the Earth, and every location in the world will find the same time T, no matter how far or in what direction relative to the volcano.
More specifically, I pointed out that if you observed, say, four pressure spikes that occurred after the blast by times T1, T2, T3, T4, then there are three ways to measure T. (If you only saw three spikes, then you get two measurements; if only two, as is the most common, then you still get one shot at T.)
T3 – T1 = T
T4 – T2 = T
T1 + T2 = T
The first two relations are easy to understand: T1 is the first pass of the outbound pressure wave, and T3 is the second pass of the outbound wave (while T2 and T4 are the first and second pass of the inbound wave), so the time between T3 and T1 is just the round-trip time T, and the same is true for T4 and T2. The last one is trickier, and I point you to the relevant section of the first post in this series.
For the Tonga volcano explosion, I collected data from nine locations around the world and ended up with about twenty measurements of T, all of which fell between 34 3/4 and 36 3/4 hours. It’s not surprising that there’s some variation. First, it can be hard to say exactly when a pressure spike happened; often each spike is really multiple spikes very close together (for instance, see the second figure here) as the wave goes by, so should you choose the largest spike, or the leading spike, for the timing? The difference can be as big as an hour. The data can also be clouded (heh) by local weather, which can move the pressure around for other reasons, and make the start of the spiking hard to identify. Second, the wave’s speed was surely not exactly constant; it probably varied by a few percent due to temperature variations and other effects that I don’t personally understand. Third, we know the Earth’s not a perfect sphere; it’s slightly squashed at the poles, by about 2 percent — though two percent of 36 hours is about twenty minutes, so that’s relatively small effect. So the fact that the answers are all consistent within a two hour range is actually pretty solid evidence that the Earth’s symmetrical in all directions around Tonga, and probably a sphere.
What about Krakatoa? The Royal Society managed to obtain over forty measurements of pressure readings, most of them with multiple spikes and some with as many as seven. These are arranged in two tables, one showing the odd-numbered spikes (the outgoing pressure wave) and one showing the even-numbered spikes (the returning pressure wave). Careful: the times in their raw data are shown relative to midnight Greenwich Mean Time, not to the Krakatoa blast, which occurred very close to 3:00 Greenwich Mean Time (best estimate being 2:56), so you need to subtract about three hours to obtain T1, T2, etc. That will be important at the end of this section.
Then the authors of this section of the report calculated T3 – T1 (and T5 – T3, etc., which measure later round-trip times) and put that in a table, shown below. (I’ve crossed out a few entries, because the Royal Society questioned the data quality for those cases.) And what did they find for T, the round-trip time for the Krakatoa pressure wave? In location after location, they found something close to 35 – 36 hours — a little more here, a little less there, but essentially the same as what one finds for the Tonga volcano pressure wave.
In the green-boxed columns are the round trip times (for the first, second, and third trips where available) for the outbound pressure wave, as measured in many locations around the world. Note that they show hours with two decimal places, not hours and minutes. I crossed out entries where the Royal Society judged the measurement data (listed elsewhere in the report) to be problematic.
Next the authors calculated T4 – T2 (and T6 – T4, when available) and put that in a table also. Of course they find something close to 35 – 36 hours again, though sometimes a bit less.
Same as the previous figure, but now for the inbound (returning) pressure wave.
The authors then used the data to figure out the timing of the big explosion; if you’re curious how they did that, also just using arithmetic, see this post. We’ll just accept their timing, and with the risk of a small amount of logical circularity, we can calculate T1 + T2, which the Royal Society didn’t do. Let’s look at an example of how this is done from the report’s timing tables.
A small fraction of the timing data tables of the report, showing the times of wave passages relative to midnight Greenwich Mean Time (GMT, or UTC nowadays); the volcano exploded around 2:56 GMT
The Melbourne weather observatory saw the first spike at 8:14 GMT, but since the volcano exploded around 2:56 GMT, we should subtract 2:56 from this number to get T1 = 5:18 . The second spike (the first column of the second table) was at 34:25, and so T2 = 34:25 – 2:56 = 31:29. Adding these two numbers together gives T1 + T2 = 36:47 = 36.78 hours. Repeating this for all the locations with two reliable spikes, we again find 35 – 36 hours, plus or minus an hour or so.
The round-trip times that I obtained by adding the Royal Society’s recorded times for the first and second pressure spikes; locations with data marked as questionable in the report are not shown here, but give similar answers.
Implications
In these results, there is some amount of variation, especially in data from North America. The Royal Society authors noticed this, of course, and spent quite a few pages of their report trying to understand it. Apparently the pressure wave moved a little faster in some directions than others, though with variation no more than 10%. Why did this happen? (And why, so far, have we seen no sign of such a large variation in this month’s pressure wave?) I’m certainly not expert enough to say. In fact, I have the impression that atmospheric scientists have been debating the implications of this variation ever since, at least as recently as 2010.
In fact one of the possible advantages of using T1 + T2 to calculate T, aside from the fact that many sites measured two pressure spikes but not as many measured three or more, is that these variations may have tended to cancel out. (For instance, if a northward-moving part of the wave moved faster than average and the southward-moving part moved slower by an equal amount, that would shift T3 – T1 and T4 – T2 but not T1 + T2.) You can see there’s somewhat greater uniformity in my numbers than in the round-trips as calculated in the Royal Society’s tables; but still, round trip times as measured in North America are longer by a few percent.
Nevertheless, for our current purposes, the differences are small. To within 10%, both Tonga and Krakatoa pressure waves indicate that they are at symmetric points on the Earth — and since they’re not on opposite ends of the Earth, this proves the Earth’s a sphere, to 10% or better. Flat Earths are flat out, as are bowl Earths, gourd Earths and highly elliptical Earths.
Moreover, because the round trip times are essentially the same for both eruptions, the Earth apparently hasn’t grown or shrunk, nor have the speeds of pressure waves significantly changed, during the past 140 years. In all that time, only the speed of information has changed, which is why I can write this post within two weeks of the explosion, before the ash has even settled on the ground.
Looking Ahead
We now know, without any loopholes, the shape of the Earth; but what of its size? Since we know the round-trip time T, all we need to determine the Earth’s circumference C is the speed v with which the pressure waves were traveling:
C = v T.
We could guess the waves were traveling at sound speed, but apparently that’s really not the right way to think about these huge waves; and in any case sound speed varies with pressure and thus with altitude, and so it’s not at all clear which value for sound speed we would want to use. It would be better to actually measure v directly from the pressure data. We can do this, without assuming the Earth’s a sphere, by looking at how quickly the pressure wave crossed small parts of the world. For the recent explosion, that data is available too, and we’ll use it next time to find v.
(to be continued)
As reproduced in the Royal Society report, one of six drawings of the sunset just west of London on November 26th, 1883, three months after Krakatoa’s eruption. Created by Mr. W. Ascroft.
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
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.
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…?
