Category Archives: Science and Modern Society

News Flash: Has a New Axial Higgs Boson (Possibly Dark Matter) Been Discovered?

No.

No, no, no.

I was tempted to blame the science journalists for the incredibly wrong articles about this, but in fact it seems entirely the fault of the scientists involved.

Continue reading

A Big Think Made of Straw: Bad Arguments Against Future Colliders

Here’s a tip.  If you read an argument either for or against a successor to the Large Hadron Collider (LHC) in which the words “string theory” or “string theorists” form a central part of the argument, then you can conclude that the author (a) doesn’t understand the science of particle physics, and (b) has an absurd caricature in mind concerning the community of high energy physicists.  String theory and string theorists have nothing to do with whether such a collider should or should not be built.

Such an article has appeared on Big Think. It’s written by a certain Thomas Hartsfield.  My impression, from his writing and from what I can find online, is that most of what he knows about particle physics comes from reading people like Ethan Siegel and Sabine Hossenfelder. I think Dr. Hartsfield would have done better to leave the argument to them. 

An Army Made of Straw

Dr. Hartsfield’s article sets up one straw person after another. 

  • The “100 billion” cost is just the first.  (No one is going to propose, much less build, a machine that costs 100 billion in today’s dollars.)  
  • It refers to “string theorists” as though they form the core of high-energy theoretical physics; you’d think that everyone who does theoretical particle physics is a slavish, mindless believer in the string theory god and its demigod assistant, supersymmetry.  (Many theoretical particle physicists don’t work on either one, and very few ever do string theory. Among those who do some supersymmetry research, it’s often just one in a wide variety of topics that they study. Supersymmetry zealots do exist, but they aren’t as central to the field as some would like you to believe.)
  • It makes loud but tired claims, such as “A giant particle collider cannot truly test supersymmetry, which can evolve to fit nearly anything.”  (Is this supposed to be shocking? It’s obvious to any expert. The same is true of dark matter, the origin of neutrino masses, and a whole host of other topics. Its not unusual for an idea to come with a parameter which can be made extremely small. Such an idea can be discovered, or made obsolete by other discoveries, but excluding it may take centuries. In fact this is pretty typical; so deal with it!)
  • “$100 billion could fund (quite literally) 100,000 smaller physics experiments.”  (Aside from the fact that this plays sleight-of-hand, mixing future dollars with present dollars, the argument is crude. When the Superconducting Supercollider was cancelled, did the money that was saved flow into thousands of physics experiments, or other scientific experiments?  No.  Congress sent it all over the place.)  
  • And then it concludes with my favorite, a true laugher: “The only good argument for the [machine] might be employment for smart people. And for string theorists.”  (Honestly, employment for string theorists!?!  What bu… rubbish. It might have been a good idea to do some research into how funding actually works in the field, before saying something so patently silly.)

Meanwhile, the article never once mentions the particle physics experimentalists and accelerator physicists.  Remember them?  The ones who actually build and run these machines, and actually discover things?  The ones without whom the whole enterprise is all just math?

Although they mostly don’t appear in the article, there are strong arguments both for and against building such a machine; see below.  Keep in mind, though, that any decision is still years off, and we may have quite a different perspective by the time we get to that point, depending on whether discoveries are made at the LHC or at other experimental facilities.  No one actually needs to be making this decision at the moment, so I’m not sure why Dr. Hartsfield feels it’s so crucial to take an indefensible position now.

Continue reading

Celebrating 2/22/22 (or was it 22/2/22)?

I hope you all had a good Twosday. Based on what I saw on social media, yesterday was celebrated widely in many parts of the world that use Pope Gregory’s calendar. I had two sandwiches to in honor of the date, and two scoops of ice cream.

In the United States, the joy continues today, it being now 2/23/22. Though not quite as wonderful as 2/22/22 on Tuesday, it’s still another nicely symmetric number worthy of note. In fact we get a full week of this, including 2/24/22 tomorrow, 2/25/22 on Friday, and so on, concluding on 2/29/22 … uhh, (oops) I mean, 2/28/22, because 2022 is not a Leap Year. For some reason.

In other countries, where it is 23/2/22, the celebration is over for now … because without symmetry, where’s the love? Ah, but they’re just more patient. They’ll get their chance in a month, when it’s 22/3/22, a date that will go unnoticed in the USA but not in Europe.

But what, exactly, are we getting so jazzed about? After all, what is the significance of it being the 22nd or 23rd date of the second month of a year labelled 2022? Every single bit of this is arbitrary. Somebody, long ago, decided January would be the first month, making February month number 2; but it wasn’t that long ago that March was the first month, which is why September, October, November and December (7, 8, 9, and 10) have their names. It’s arbitrary that January has 31 days instead of 30; had it been given thirty, the day we call the “22nd” would have been the “23rd” of February, and our celebration would have been one day earlier. And 2022 is arbitrary two too. Other perfectly good calendars referred to yesterday by a completely different day, month and year.

This, my friends, is exactly what General Relativity (and the rest of modern physics) tells you not to do. This is about putting all of your energies and your focus on your coordinate system — on how you represent reality, instead of on reality itself. The coordinate system is arbitrary; what matters is what actually happens, not how you describe what happens using some particular way of measuring time, or space, or anything else. To get excited about the numbers that happen to appear on your measuring stick is to put surface ahead of substance, math ahead of physics, magic ahead of science. It’s as bad as getting excited about how a word is spelled, or even what word is used to represent an object; a rose by any other name.

But we humans are not designed to think this way, it seems. We cheer when we’ve driven a thousand miles, a milestone (hah) which combines the definition of mile (arbitrary) with the fascination with the number 1,000 (which only looks like an interesting number if you count with ten fingers, rather than 12 knuckles, as the Babylonians did, or eight tentacles, as certain intelligent sea creatures might do.) We get terribly excited about numbers such as 88, or 666, which similarly depend on our having chosen to count on our ten fingers. A war was ended on 11/11 at 11:00 (and one was started on 22/2/22 — coincidence?)

Celebrating birthdays is a little better. No matter what calendar you choose, or whether it even lasts a year (as, for example, in Bali), the Sun appears to move across the sky, relative to the distant stars, in a yearly cycle. When it comes back to where it was, a year has passed. If we define your age to be the number of solar cycles you’ve experienced, then that means something, no matter what calendar you prefer. Your birthday means something too as long as we define it not by the arbitrary calendar but by the position of the Sun on the day of your birth.

Similarly, the solstices that mark the days with the shortest daylight and shortest darkness, and the equinoxes that have days and nights equal in duration, are independent of how you count hours or minutes or seconds, or even days. It doesn’t matter if your day has 24 equal hours, or if you divide your daylight into 12 and your darkness into 12, as used to be the case. It doesn’t matter what time zones you may have arbitrarily chosen. If you want to mark days, you can use the time that the Sun is highest in the sky to define “noon”, and count noons. A year is just over 365 noons, no matter what your calendar. The time from solstice to solstice is about half that. But the date we call “December 25th” does not sit on a similarly fundamental foundation; it shifts when there’s a leap year, and sometimes it’s three days after the solstice and sometimes four. Many other holidays, driven by Moon cycles rather than a Sun cycle, are even less grounded in the cosmos.

Being too focused on coordinates can cause a lot of trouble. The flat maps that try to describe our spherical Earth make all sorts of things seem to be true that aren’t. They all make the shortest path between two points impossible to guess. Some wildly exaggerate Greenland’s size and minimize the entire African continent. Most of them make it difficult to imagine what travel over the north or south pole is like, because there’s a sort of “coordinate singularity” there — a single point is spread out over a whole line at the top of the map, and similarly at the bottom, which makes places that are in fact very close together seem very far apart.

A coordinate singularity of a more subtle type prevented scientists (Einstein among them) from realizing for decades that black holes, which were once called “frozen stars,” have an interior, and that you could potentially fall in. The coordinates originally in use made it seem as though time would stop for someone reaching the edge of the star. Bad coordinates can obscure reality.

Physics, and science more generally, pushes us to focus on what really happens — on events whose existence does not depend on how we describe them. It’s a lesson that we humans don’t easily learn. While it’s fine to find a little harmless and silly joy at non-events such as 22/2/22 or 2/22/22, that’s as far as it should go: anything that depends on your particular and arbitrary choice of coordinate system cannot have any fundamental meaning. It’s a lesson from Einstein himself, advising us on what not two do.

Why Simple Explanations of Established Facts Have Value

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.

Continue reading

The Best Proof that the Earth Spins

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!

Continue reading

How Can You Check that the Earth Spins?

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.

Continue reading

The Earth’s Shape and Size? You Can Measure it Yourself — Part 4

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 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:

LocationSpeed Estimate (mph)Speed Estimate (kph)
New Zealand700 – 7401125 – 1190
China620 – 7001000 – 1125
United States720 – 7601160 – 1225
Germany720 – 8001160 – 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.

The Earth’s Shape and Size? You Can Measure it Yourself — Part 1

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:

  1. The Earth really is almost spherical.
  2. 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.

The UK’s Met office observed two disturbances in first two days hours following the Tonga Eruption. The timing of these spikes will be our focus.
Spikes obtained from pressure measurements in Iceland, with slow pressure variations removed to make the spikes clearer (credit Halldór Björnsson @halbjo via Prof. Evgenia Ilyinskaya @EIlyinskaya and volcanodiscovery.com)

Let’s imagine you yourself have a barometer which shows as many as four spikes.  Let’s call T1 the time between the volcano’s explosion and the appearance of first spike.  (I used different notation in my last and more detailed post: T1=t1-ts .) We’ll similarly define T2, T3, T4 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

  • T3 – T1 = 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.

  • T4 – T2 = T

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

  • T1 + T2 = 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 T1 to reach you, during which the wave covered a distance D1 = T1 v. (Remember T1 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 T2, and during that time the wave covered a distance D2 = T2 v.  But as you can see from the figure, D1 + D2 is the entire circumference C of the Earth! So if T1 is the time it takes to travel a distance D1 , and T2 is the time it takes to travel a distance D2, then their sum must be the time it takes to travel the distance C — and that, by definition, is the round trip time T.

From the volcano at bottom left to the observer at right, there are two paths that one can take on a great circle. The shorter one, of length D1 , can be covered at speed v in time T1 ; the longer, of length D2 can be covered at the same speed in time T2 . As the circumference of the great circle is D1 + D2, the time required to traverse it entirely is T1 + T2 .

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 T1, T2, T3, T4 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 T1 and T2, but sometimes I could get T3 or even T4 . 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.

LocationT1T2T3T4T1 + T2T3T1T4T2
Iceland131523004915590035hr 15min36hr 00min36hr 00min
Beijing09102635452035hr 45min36hr 10min
Netherlands150021305015561536hr 30min35hr 15min34hr 45min
Hawaii, USA044531153945674536hr 00min35hr 00min36hr 30min
New Jersey, USA1115243535hr 40min
Switzerland1545210036hr 45min
Seattle, USA083027454415634536hr 15min35hr 45min36hr 00min
Southern Chile0845280536hr 50min
Miami10152530451535hr 45min35hr 00min

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?

If the Earth were shaped like an ellipsoid, and the Tonga volcano were exactly at one end, the measurement we just performed would also have given a universal value for the round-trip time. (Image produced with Mathematica 11.3).

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.

(to be continued)