Even if you’re working from home, so that you’re spending the day at a fixed location on the Earth’s surface, you’re not at a fixed location relative to the Earth’s center. As the Earth turns daily, it carries you around with it. So where are you headed today? Presumably Earth’s spin takes you around in a big circle, right?
That’s great. Which circle?
Point to it, right now.
Let me ask that again, in case that wasn’t clear. With your feet on the ground, looking whichever direction you choose, please show me the circle you’ll be taking today on your travels.
Most people who hazard a guess imagine that if they face east (toward the rising Sun, which here is into the plane of your screen), they are traveling on a circle that cuts vertically into the ground. But this is true for very few of us.
No idea? In my experience, many people have never even thought about it. Those who are willing to hazard a guess have to think for a moment to figure out that the Earth is rotating west to east — that’s why the Sun appears to rise in the east and set in the west. Once they are clear on that point, many people face east, and then indicate a circle that goes straight ahead, which would be combination of east and then down, as you can see in the figure.
To say that another way, if you imagine the circle of travel as being the edge of a disk, that disk would face east-west and slice directly down into the ground.
For the vast majority of us, it turns out this guess is not correct.
So where are we headed? People located at the equator or the poles can answer this more easily than the rest of us, so let’s start with them.
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.
