Kepler’s third law is so simple to state that (as shown last time) it is something that any grade school kid, armed with Copernicus’s data and a calculator, can verify. Yet it was 75 years from Copernicus’s publication til Kepler discovered this formula! Why did it take Kepler until 1618, nearly 50 years of age, to recognize such a simple relationship? Were people just dumber than high-school students back then?
Here’s a clue. We take all sorts of math for granted that didn’t exist four hundred years ago, and calculations which take an instant now could easily take an hour or even all day. (Imagine computing the cube root of 4972.64 to part-per-million accuracy by hand.) In particular, one thing that did not exist in Copernicus’ time, and not even through much of Kepler’s, was the modern notion of a logarithm.
Much of this work was done by Nicolai Copernicus himself, the most famous of those philosophers who argued for a Sun-centered universe rather than an Earth-centered universe during the millennia before modern science. He had all the ingredients we have, minus knowledge of Uranus and Neptune, and minus the clues we obtain from telescopes, which would have confirmed he was correct.
Copernicus knew, therefore, that although the planetary distances from the Sun and their cycles in the sky (which astrologers [not astronomers] have focused on for centuries) don’t seem to be related, the distances and their orbital times around the Sun are much more closely related. That’s what we saw in the last post.
Let me put these distances and times, relative to the Earth-Sun distance and the Earth year, onto a two-dimensional plot. [Here the labels are for Mercury (Me), Venus (V), Mars (Ma), Jupiter (J), Saturn (S), Uranus (U) and Neptune (N).] The first figure shows the planets out to Saturn (the ones known to Copernicus).
Figure 1: The planets’ distances from the Sun (in units of the Earth-Sun distance) versus the time it takes them to complete an orbit (in Earth-years). Shown are the planets known since antiquity: Mercury (Me), Venus (V), Mars (M), Jupiter (J) and Saturn (S).
The second shows them out to Neptune, though it bunches up the inner planets to the point that you can’t really see them well.
Figure 2: Same as Figure 1, but now showing Uranus (U) and Neptune (N) as well.
You can see the planets all lie along a curve that steadily bends down and to the right.
Copernicus knew all of the numbers that go into Figure 1, with pretty moderate precision. But there’s something he didn’t recognize, which becomes obvious if we use the right trick. In the last post, we sometimes used a logarithmic axis to look at the distances and the times. Now let’s replot Figure 2 using a logarithmic axis for both the distances and the times.
Figure 3: Same as Figure 2, but now with both axes in logarithmic form.
Oh wow. (I’m sure that’s the equivalent of what Kepler said in 1618, when he first painstakingly calculated the equivalent of this plot.)
It looks like a straight line. Is it as straight as it looks?
Figure 4: Same as Figure 3, but with a blue line added to show how well a straight line describes the distance/time relationship of the planets, and with a grid added that passes through 1, 10, 100, 1000 Earth-years and distances equal to 1, 10, 100 Earth-Sun distances.
And now we see three truly remarkable things about this graph:
First, the planet’s distances to the Sun and orbital times lie on a very straight line on a logarithmic plot.
Second, the slope of the line is 2/3 (2 grid steps up for every 3 steps right) rather than, say, 7.248193 .
Third, the line goes right through the point (1,1), where the first horizontal and first vertical lines cross.
What else can we learn just with simple observations? Since the stars’ daily motion is an illusion from the Earth’s spin, and since the stars do not visibly move relative to one another, our attention is drawn next to the motion of the objects that move dramatically relative to the stars: the Sun and the planets. Exactly once each year, the Sun appears to go around the Earth, such that the stars that are overhead at midnight, and thus opposite the Sun, change slightly each day. The question of whether the Earth goes round the Sun or vice versa is one we’ll return to.
Let’s focus today on the planets (other than Earth) — the wanderers, as the classical Greeks called them. Do some of them go round the Earth? Others around the Sun? Which ones have small orbits, and which ones have big orbits? In answering these questions, we’ll start to build up a clearer picture of the “Solar System” (in which we include the Sun, the planets and their moons, as well as asteroids and comets, but not the stars of the night sky.)
The Basic Patterns
If we make the assumption (whose validity we will check later) that the planets are moving in near-circles around whatever they orbit, then it’s not hard to figure out who orbits who. For each possible type of orbit, a planet will exhibit a different pattern of sizes and phases across its “cycle“ when seen through binoculars or a small telescope. Even with the naked eye, a planet’s locations in the sky and changes in brightness during its cycle give us strong clues. Simply by looking at these patterns, we can figure out who orbits who.
Once you’ve convinced yourself the Earth’s a spinningsphere of diameter about 8000 miles (13000 km), and you’ve estimated the Moon’s size and distance(diameter about 1/4 Earth’s, and distance about 30 times Earth’s diameter), it’s easy to convince yourself the Sun’s bigger than the Earth, and much further than the Moon. It just takes a couple of triangles, and a bit of Moon-gazing.
Since that’s all there is to it, you can guess that the ancient Greek astronomers, masters of geometry, already knew the Sun’s the larger of the two. That said, they never did quite figure out how big and far the Sun actually is; we need modern methods for that.
It’s Just a Phase
The Moon goes through a monthly cycle of phases, lasting about 291/2 Earth days, in which the part that glows brightly with reflected sunlight grows and shrinks, from crescent to full and back again. The phases arise because there are two simple ways of dividing the Moon in half:
At any moment, the half of the Moon that faces Earth — let’s call it the near half of the Moon — is the only half that we can potentially see. (We’d only be able to see the far half, facing away from Earth, if the Moon were transparent, or a big mirror was sitting beyond the Moon.)
At any moment, the half of the Moon that faces the Sun is brightly lit — let’s call it the lit half. The other half is dark, and its presence can only be detected by the fact that it can block stars that it moves in front of, and through a very dim glow in which it reflects sunlight that first reflected from the Earth (called “Earthshine.”)
The phases arise because the lit half and the near half aren’t the same, and the relationship between them changes from night to night. See the diagram below. When the Moon is more or less between the Sun and the Earth (it rarely passes exactly between, because its orbit is tilted by a few degrees out of the plane of the drawing below) then the Moon’s lit half is its far half, and the near half is unlit. We call this dark view of the Moon the “New Moon” because it is traditionally viewed as the start of the Moon’s monthly cycle.
Figure 1: The Moon’s phases, assuming the Sun’s much further than the Moon. When the Moon is roughly between the Earth and Sun, its near half coincides with the unlit half, making it invisible (New Moon). As the cycle proceeds, more of the near half intersects with the lit half; after 1/4 or the cycle, the Moon’s near half is half lit and half unlit, giving us a “half Moon.” At the cycle’s midpoint, the near side coincides with the lit half and the Moon appears full. The cycle then reverses, with the other half Moon occurring after 3/4 of the cycle.
When the Moon is on the opposite side of the Earth from the Sun (but again, rarely eclipsed by Earth’s shadow because of its tilted orbit), then its near side is its lit side, and that creates the “Full Moon”, a complete white disk in the sky.
At any other time, the near side of the Moon is partly lit and partly unlit. When the line between the Moon and Earth is perpendicular to the Earth-Sun line, then the lit side and unlit side slice the near side in half, and the Moon appears as a half-disk cut down the middle.
When I was a child, I wondered why half this half-lit phase of the Moon, midway between New Moon (invisible) and Full Moon (the bright full disk), was called “First Quarter”, when in fact the Moon at that time is half lit. Why not “First Half?” Two weeks later, the other half of the near-side of the Moon is lit, and why is that called “Third Quarter” and not, say, “Other Half”?
This turns out to have been an excellent question. The fact that a Half Moon is also a First Quarter Moon tells us that the Sun is large and far away!
Last time I described an easy way for you to determine the size of the Moon — easier than the famous techniques used by the classical Greeks. (We don’t need to know the Earth’s circumference, as they did, if we’re ok with a moderately precise estimate.) Once you’ve done that, there’s an simple method, well known since classical times, for figuring out how far away the Earth’s companion is. That’s what I’ll describe in this short post.
(What’s not so easy is to determine the distance and size of the Sun. The classical Greeks failed in their efforts. We’ll need a more modern approach… but that’s for next week.)
Size Versus Distance
Even the early classical Greeks knew something about the Sun, just from the fact that the Moon and Sun appear roughly the same size to our eyes — that is, they occupy about the same amount of sky. If the Sun is twice as far away as the Moon, its diameter must be twice as big, in order that it appear the same size. That’s illustrated in the figure below. If it is ten times as far away, its diameter must be ten times as big. If it’s four hundred times as far away, its diameter is four hundred times as big. (Spoiler: that last one’s the truth; but we’ll get to it later.)
If the Moon is a distance L away from you, and another object twice as far away appears to be the same size in the sky, then that object’s diameter must be twice the Moon’s diameter D. This logic applies more generally to objects further and nearer than the Moon.
You can run this logic in the other direction; if something perfectly blocks the Moon, then if it’s ten times closer than the Moon its diameter must be ten times smaller. If it’s a billion times closer than the Moon, it must be a billion times smaller.
Having confirmed we live on a spherical, spinning Earth whose circumference, diameter and radius are roughly 25000, 8000, and 4000 miles (40000, 13000, and 6500 km) respectively, it’s time to ask about the properties of the objects that are most obvious in the sky: the Sun and Moon. How big are they, and how far away?
If the Moon were close to Earth, then at any one time it would only be visible over a small part of the Earth, as indicated in light blue. But in fact (except at new moon) about half the Earth can see it at a time.
Historically, many peoples thought they were quite close. With our global society, it’s clear that neither can be, because they can be seen everywhere around the world. Even the highest clouds, up to 10 miles high, can only be seen by those within a couple of hundred miles or so. If the Moon were close, only a small fraction of us could see it at any one time, as shown in the figure at right. But in fact, almost everyone in the nighttime half of the Earth can see the full Moon at the same time, so it must be much further away than a couple of Earth diameters. And since the Moon eclipses the Sun periodically by blocking its light, the Sun must be further than the Moon.
The classical Greeks were expert geometers, and used eclipses, both lunar and solar, to figure out how big the Moon is and how far away. (To do this they needed to know the size of the Earth too, which Eratosthenes figured out to within a few percent.) They achieved this and much more by working carefully with the geometry of right-angle triangles and circles, and using trigonometry (or its precursors.)
The method we’ll use here is similar, but much easier, requiring no trigonometry and barely any geometry. We’ll use eclipses in which the Moon goes in front of a distant star or planet, which are also called “occultations”. I’m not aware of evidence that the Greeks used this method, though I don’t know why they wouldn’t have done so. Perhaps a reader has some insight? It may be that the empires they were a part of weren’t quite extensive enough for a good measurement.
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.
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.
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.
