The Earth’s Shape and Size? You Can Measure it Yourself — Part 3

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.

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