Category Archives: Geology

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)

Geometry From Within: Evidence for a Round Earth

It’s a lot easier to map the Earth than it used to be.  Before satellites, you had to do many careful measurements of distances and directions, at many different locations around the world, and combine them all to build a picture of a world you couldn’t see.  That’s part of why maps and globes made in past centuries had so many inaccuracies and distortions; it was a tough business.  

How that changed in the 1960s!   The first full photograph of the Earth that I’m aware of was made in 1967 by the ATS-3 satellite (were there earlier ones?)  So much simpler… the whole planet laid out in front of you.  You just need a few photographs like this, and the era of measuring from one point on the ground to another is mostly over.

But the challenge of trying to measure things when you’re stuck within them, and can’t step outside them, hasn’t gone away.  Just as we could see in telescopes that the Moon and Mars are ball-shaped, long before we could observe the Earth itself, today we can see other galaxies in great detail, but we still struggle to build a complete picture of our own, the Milky Way. The Gaia satellite is trying hard.

To determine the Earth’s two-dimensional surface is really round took some clever thinking. Aristotle, in ancient Greek times, noted that the Earth’s shadow on the Moon during a lunar eclipse is always curved in the same way — it doesn’t matter what time of day or year the eclipse occurs, or whether the shadow is on the north, east, west or south side of the Moon.  This feature is to be expected if the Earth’s a ball, like the Moon and Sun, and very difficult to explain otherwise.  [Try to figure out what you might see if it were cylinder-shaped!]

But there are other tricks you can use if you have a hunch that the place you live on, or in, is of finite size.

One Dimension: the Possibly Circular Canal

Suppose you live on the banks of a canal, a long thin channel extending off to the horizon, like a river without any flow.  And suppose you suspect that this canal forms a loop, surrounding a large island.  How could you check?   Well, if you had a boat, you could row yourself down the canal; or you could walk along the shore. If the canal is really in the shape of a loop, you’ll eventually come back to your starting point.  But maybe you’re worried such a journey would be too long, difficult, risky, expensive. Do you have other options?  

Here’s one: suppose you could make a big wave moving in the clockwise direction around the canal.  The wave, unlike you, wouldn’t need any food and drink or fuel for the journey — so time and money would not be a problem. The wave would move down the canal at a definite speed [I’m assuming here that it maintains its height], and if the canal were really a loop, then after some time T you’d see the wave return, still moving clockwise, and pass by you.  If you waited the same amount of time T again, you’d see the same wave a second time, again clockwise.  After the same amount of time T, you’d see it a third time. 

If instead the canal were a finite strip, then the wave would reflect off the end, and so the wave would return from the opposite direction. If it were infinite in length, it would never return. And if it had a complicated shape — perhaps a P or an R or a B instead of an O — you would get multiple waves in a complex pattern. But the simple pattern in which the waves return again and again, from the same direction, after a time T, is consistent with the canal being a simple loop.

You could try sending a wave counterclockwise too, and you’d expect the same pattern if the canal’s a loop.

As the wave passes you, you can also estimate its speed v. Having also measured T, you can now determine the length L of the canal. It’s the wave speed times the time T for the wave to go round once:

  • L = v T
Figure 1: You live on the shore of a canal, which you suspect is circular. You could find out how big it is by sending a large wave in either direction, and measuring the time T that it takes to return.

Perhaps making such a wave is too difficult for you, but if you’re lucky, someone or something down the canal may make a giant splash. Then you’ll see the ripples from the splash come by in a similar pattern. Now waves will travel both counterclockwise and clockwise around the canal, and they probably won’t arrive at the same time. That doesn’t matter, though. You’ll see the clockwise waves repeat after a time T, and you’ll see the same for the counterclockwise waves. Seeing both of them repeat after the same time T will give you confidence that the canal’s really a simple loop

To be specific, let’s call t1 the time you measure the first wave, t2 the second wave, t3 the third, t4 the fourth, and so on; if the first wave is counterclockwise, then the second is clockwise (see Figure 2), the third counterclockwise, and so on. (This won’t be true if instead of a loop the canal is in the form of a line segment! A reflection off the end could make the first two waves come from the same direction.) As the clockwise waves will repeat after a time T, and the same for the counterclockwise waves, it will be the case, if the canal’s a loop, that

  • t3 – t1 = t4 – t2 = T
  • L = v T

There’s more; if you know the time ts when the splash happened and you know the wave speed, then you can learn how far away the splash was from you:

  • D = v ( t1 – ts )

But even if you don’t know what time the splash happened, you can figure it out; see Figure 2. The distance traveled by the counterclockwise wave to get to you, plus the distance traveled by the clockwise wave to do the same, equals the full distance round the circle (Figure 2), so the time that the counterclockwise wave required to reach you ( t1 – ts ) plus the corresponding time for the clockwise wave ( t2 – ts ) must be equal to T.

  • T = ( t1 – ts ) + ( t2 – ts ) = t1 + t2 – 2ts , which implies ts = 1/2 (t1 + t2 – T)

If you look closely at these four bold-faced equations, they tell you that you can determine T, L, D and ts , properties of the loop and the splash, if you know t1, t2 and t3 and v, which are all things that you can measure without going anywhere. From this point of view t4 is a bonus, a nice check that things are working as expected.

Even better, if you have a friend down the canal who makes the same measurements, that friend won’t get the same answers for t1, t2, t3 and t4 ; the waves arrive at different times for your friend than for you. But when you obtain T and L and ts from the waves you see, and your friend does the same, you’d better get the same answer — because these are properties of the loop and splash, and don’t care where either you or your friend is located.

Figure 2: A large splash occurs at time ts, and waves travel both counterclockwise (green), in which case they reach you at time t1, and clockwise (red), reaching you at time t2.

By themselves, these equations do not prove the canal is round, though they are consistent with it. They only tell you that it’s a loop of length L, with no kinks which could cause extra reflections. Still, it’s a lot of information for a very low price, without taking a boat around the loop, walking all around it, or sending up a drone to take a photograph. The waves have done all the work for you.

Figure 3: After the counterclockwise wave passes you at time t1, it continues round the canal, and passes you again at time t3 = t1 + T.

Two Dimensions: the Possibly Round Surface of the Earth

What would be different if you lived on a sphere?  (A subtlety of language: by “sphere,” I do not mean “ball”, which is three-dimensional; I mean the surface of the ball, which is two-dimensional.  In this terminology, the Earth is a ball, while its surface is a sphere, approximately.)  Again, of course, you always have the option of traveling round the sphere yourself and exploring it, checking that no matter what direction you go in, if you walk in a perfectly straight line, you will always come back to your starting point after you travel the circumference of the sphere.  But that’s expensive and time-consuming and not very practical.  What other options do you have?

You could wait for a big splash in the atmosphere — a natural one like a volcanic eruption, or an artificial one of similar size (fortunately now forbidden by nuclear testing treaties).   This opportunity, if you want to call it that, came this past week, unfortunately near an inhabited area and at the ocean’s surface within the Kingdom of Tonga, with ensuing loss of life, as well as the destruction of crops and homes; the resulting tsunami even took lives far across the Pacific ocean.  It’s not an experiment we would happily have chosen. But nature has carried it out without asking us; we may as well learn what we can from it.

When water hits hot magma and turns to steam, there’s an immense release of energy, especially if the magma is itself packed with compressed gasses. This is partly why some of the largest explosions in the last two hundred years have occurred when volcanic islands self-destructed; Krakatoa is the most famous.  The latest estimate as of the time of writing is that the one in Tonga last week was overall perhaps only 1/20 times as powerful as Krakatoa, but its plume was enormous, and its shock waves were strong enough to be detected multiple times, in many places, as they traveled round and round the Earth.

The shock wave emanated from the explosion in all directions, moving outward as an ever expanding circle, as you can guess by pure reasoning but also as confirmed by satellite.  After traveling 1/4 of the way around the Earth, the wave front reached a maximum extent — the same size and shape as the equator, though with a different orientation — and then shrank again, converging to a point in Africa exactly halfway around the Earth from the explosion’s location. (A nice visualization of this, and of what I’ll say next, can be found here.) Then the shockwave continued onward, again expanding to the Earth’s full extent, and then shrinking and converging on the very spot where it was created in the first place.   And this process repeated, until the shock wave, gradually losing its energy, faded beyond the point of detectability.

This pattern of outward expansion, convergence to the opposite point, return-ward expansion, and convergence to the original point, means that the waves from the explosion passed every point on Earth multiple times, and did so first moving away from the explosion, then returning, then again moving away, and again returning, until finally they were too small to observe.  That this pattern was seen everywhere, in countries widely spread around the globe, by both professional and civilian weather stations, gives some qualitative evidence that the Earth’s a smooth object with a rounded surface of some type.  For example, here is the pattern of multiple waves crossing, returning, re-crossing and re-returning as measured by weather stations in China; we can see three wave passages clearly (the fourth is too dim to measure well).  And here is a similar pattern in the Netherlands; though it’s only at one location, and only the main shock wave is detected, the shock is seen six times. 

What’s nice is that for a sphere — and only for a sphere [see caveat below] — the equations I wrote earlier for a circular canal still hold, and importantly, they hold everywhere, and have to give the same circumnavigation time T and the same splash time ts. That’s because if you are on a sphere, motion away from the volcano (or indeed any point), in any direction, will take you on a circular path of length equal to the sphere’s circumference. On any other shape, this won’t be true.

[To be fair, I am making a couple of assumptions: for instance, that the volcano was located on a random, not special, point on the Earth. (For example, if the Earth’s surface was oblong instead of circular, then the two points at either end of the oblong are special.) To make a long story short, there are still loopholes to the argument I’m giving here, but they are only relevant if there are very special and unlikely coincidences. Additional volcanoes, would quickly close the loopholes.]

In particular, the equations I introduced earlier should hold in China, about 1/4 around the Earth from Tonga. And they should also hold in the Netherlands, much further from Tonga, in a quite different direction. If the Earth had an uneven shape, then the time to go round the Earth in the direction from Tonga to Beijing would be different from the time to go round it from Tonga to the Netherlands; you wouldn’t get the same T. And if the Earth had edges (as in the absurd flat-earth map), you would see reflection waves; you wouldn’t get the same T or the same ts, and the second big wave across China wouldn’t look like the original one retracing its steps (a fact which already gives qualitative evidence for a round Earth.)

Using publicly available data from anywhere in the world, including what I’ve shown you from China and from the Netherlands, we can check ourselves that the Earth’s a ball and measure its circumference. Let’s do it.

So as not to spoil the fun, I’m going to wait until after the weekend to post the results. You are all encouraged to gather your children together and to try to measure:

  • T, the time it took for the waves to travel around the Earth; do this both with the data from the Netherlands and that from China; do you get similar answers?
  • ts, the time when the eruption occurred; use both the data from the Netherlands and the data from China (make sure you’re using UTC time, so you don’t get confused by time zones). Do you get similar and roughly accurate answers? Is it close to the time reported in this article?
  • v, the speed of the waves, which you can determine by watching how long it takes them to cross a part of China and comparing that time with the distance of that path; caution, make sure you trace a path perpendicular to the wave front.
  • C = T v , the circumference of the Earth, equal to the time it took for the waves to circumnavigate the Earth times their speed. Can you get fairly close?

Caution: You’re not going to get exactly the precise scientifically-known answers, nor will your answers be perfectly consistent, because the data I’ve linked to was neither taken nor presented with scientific levels of precision. But you should be able to get within 10-20%, enough to convince you the Earth’s surface pretty darn close to a sphere. If you want more precision, I’m sure precision data is available (anybody have a good link?) [Also note that there are some extra waves seen in the China map, some of them reverberations from the original explosion, and some due to later, smaller explosions; they travel in the same directions as the original ones, showing they come from the same place. For our purpose here, just keep your focus on the biggest waves.]

The point is that we can learn the Earth is ball-shaped without ever stepping off the Earth, and in fact without even traveling; and we can even learn, from the timing, how big the Earth is.  All it takes is a natural explosion, measurements from a few places, some logic, and simple algebra.   The data is now publicly available, and every science teacher in the world ought to encourage their teenage students to do this exercise!  Not only does it confirm we live on a sphere, it shows that one needs neither a photograph taken from outer space, nor a flight around the world, nor specialized map-making skills, to obtain that proof.

Three Dimensions: The Universe

Now what about the universe as whole?  The Earth and Sun are carrying us along as they travel within a three-dimensional surface.  What is its shape?  How can we know?  [There is also the question of the four-dimensional surface that makes up the space and time of the universe.  I’m not addressing that here, that’s even more complex.]

A circle is a one-dimensional sphere; the surface of the Earth (not its interior) is a two-dimensional sphere. Could the universe be a three-dimensional sphere?   We can’t stand outside it to find out.  In fact it’s far from clear there is meaning to “outside” since, after all, it’s the universe, and might be everything there is. Nevertheless, we can imagine, at least, trying to do a similar experiment.  If there were a huge supernova explosion, or a tremendous flare from a distant black hole as it ripped apart a star, maybe we would see the light arrive from one side of us, and then later see it arrive from the other side, and yet again from the first direction, and so on.

Back before we knew the huge scale of the universe and the tiny speed of light, that might have seemed plausible.  We can’t hope to do anything like this, unfortunately.   But it’s not because the question makes no sense.  The natural Tonga volcano experiment worked thanks to the fact that it’s a small world (after all) and the speed of sound is relatively fast, so it all took less than a day or two.  In the universe, it’s the reverse; it’s a big place and the speed of light is relatively slow.  Our own galaxy, the Milky Way, is itself 100,000 light-years across [i.e. it’s so big that it takes 100,000 years for light, traveling at the fastest speed our universe allows, to cross it], so even if our galaxy were the entire cosmos, as was thought until the 1920s, it would take at least 100,000 years to do this experiment.  And of course we now know the universe is immensely larger than our own galaxy; indeed the most recent map of galaxies extends out, for the brightest galaxies, as far as 10,000,000,000 light years.  Hopeless.

Nevertheless, the possibility that the universe has an interesting shape, and though huge might be small enough that we could see some evidence of its shape, remains a topic of research.  The light from events in the distant past might give us clues.  While a blast wave isn’t something we’d be able to see from multiple perspectives, a long-lasting bright spot on the sky could potentially be seen reaching us from different paths around a complex universe.  The fact that the universe has been expanding over the billions of years since the Hot Big Bang began complicates the thinking, but also provides opportunities.

To give insight into how this could be done is beyond the scope of this blog post, but if you’re curious about it, you might try this long-form article from Quanta Magazine (a highly recommended source for interesting articles.)  

The Lesson for Humankind

The big lesson here: geometry can be learned from the inside.  You don’t need to be outside an object to map it and learn its shape and size. That this is possible explains how mapmakers knew the shapes of continents long before satellites, and how one can determine that the universe is expanding while remaining within it (though the story of how scientists did this, without using the methods described in this post, is for another day.) And if the object is finite, so that no wave can travel forever without eventually returning to you, then it’s possible to infer its shape just by learning how waves travel and bounce around the object. That’s how the depth of the ocean’s deepest point was recently measured, as I described in my last post; and that’s how children (of all ages) should prove for themselves, using publicly available data from last weekend and simple algebra, that the Earth is indeed round.

Mapping the Unknown, Times Three

I took a short break from other projects this weekend. Poking around, I found three particularly lovely science stories, which I thought some readers might enjoy as well. They all involve mapping, but the distances involved are amazingly different.

Starbirth Near to Home

The first one has been widely covered in the media (although a lot of the articles were confused as to what is new news and what is old news.) Our galaxy is full of stars, but also of gas (mainly hydrogen and helium) and dust (tiny grains of material made mainly from heavier elements forged in stars, such as silicon). That gas and dust, referred to as the “interstellar medium”, is by no means uniform; it is particularly thin in certain regions which are roughly in the shape of bubbles. These bubbles were presumably “blown” by the force of large stellar explosions, i.e. supernovas, whose blast waves cleared out the gas and dust nearby.

It’s been known for several decades that the Sun sits near the middle of such a bubble. The bubble and the Sun are moving relative to one another, so the Sun’s probably only been inside the bubble for a few million years; since we’re just passing through, it’s an accident that right now we’re near its center. Called simply the “Local Bubble”, it’s an irregularly shaped region where the density of gas and dust is 1% of its average across the galaxy. If you orient the galaxy in your mind so that its disk, where most of the stars lie, is horizontal, then the bubble stretches several hundred light-years across in the horizontal direction, and is elongated vertically. [For scale: a light-second is 186,000 miles or 300,000 km; the Sun is 8 light-minutes from Earth; the next-nearest star is 4 light-years away; and our Milky Way galaxy is about 100,000 light years across.] It’s been thought for some time that this bubble was created some ten to twenty million years ago by the explosion of one or more stars, probably siblings that were born close by in time and in space, and which at their deaths were hundreds of light-years from the Sun, far enough away to do no harm to Earthly life. [For scale: recall the Sun and Earth are about 5 billion years old.]

Meanwhile, it’s long been suggested that explosion debris from such supernovas can sweep up gas and dust like a snowplow, and that the compression of the gas can lead it to start forming stars. It’s a beautiful story; large stars live fast and hot, and die young, but perhaps as they expire they create the conditions for the next generation. [A star with 40 times the raw material as the Sun has will burn so hot that it will last only a million years, and most such stars die by explosion, unlike Sun-like stars.]

If this is true, then in the region near the Sun, most if not all the gas clouds where stars are currently forming should lie on the current edge of this bubble, and moreover, all relatively young stars, less than 10 million years old, should have formed on the past edge of the bubble. Unfortunately, this has been hard to prove, because measuring the locations, motions and ages of all the stars and clouds isn’t easy. But the extraordinary Gaia satellite has made this possible. Using Gaia’s data as well as other observations, a team of researchers (Catherine Zucker, Alyssa A. Goodman, João Alves, Shmuel Bialy, Michael Foley, Joshua S. Speagle, Josefa Groβschedl, Douglas P. Finkbeiner, Andreas Burkert, Diana Khimey & Cameren Swiggum) has claimed here that indeed the star-forming regions lying within a few hundred light-years of the Sun all lie on the Local Bubble’s surface, and that nearby stars younger than ten million years were born on the then-smaller shell of the expanding bubble. Moreover they claim that the bubble probably formed 14-15 million years ago, at a time when the Sun was about 500 light-years distant.

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Beyond Human Visibility

A couple of interesting scientific stories are making the rounds today, and worth a little physics and general science commentary. The first reminds us just how incredibly limited our sensory perceptions are in telling us about the world, by forcing us to imagine how it may look to animals whose perceptions are slightly different. The second reminds us just how little we know about our own planet. Continue reading