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!
I hope you all had a good Twosday. Based on what I saw on social media, yesterday was celebrated widely in many parts of the world that use Pope Gregory’s calendar. I had two sandwiches to in honor of the date, and two scoops of ice cream.
In the United States, the joy continues today, it being now 2/23/22. Though not quite as wonderful as 2/22/22 on Tuesday, it’s still another nicely symmetric number worthy of note. In fact we get a full week of this, including 2/24/22 tomorrow, 2/25/22 on Friday, and so on, concluding on 2/29/22 … uhh, (oops) I mean, 2/28/22, because 2022 is not a Leap Year. For some reason.
In other countries, where it is 23/2/22, the celebration is over for now … because without symmetry, where’s the love? Ah, but they’re just more patient. They’ll get their chance in a month, when it’s 22/3/22, a date that will go unnoticed in the USA but not in Europe.
But what, exactly, are we getting so jazzed about? After all, what is the significance of it being the 22nd or 23rd date of the second month of a year labelled 2022? Every single bit of this is arbitrary. Somebody, long ago, decided January would be the first month, making February month number 2; but it wasn’t that long ago that March was the first month, which is why September, October, November and December (7, 8, 9, and 10) have their names. It’s arbitrary that January has 31 days instead of 30; had it been given thirty, the day we call the “22nd” would have been the “23rd” of February, and our celebration would have been one day earlier. And 2022 is arbitrary two too. Other perfectly good calendars referred to yesterday by a completely different day, month and year.
This, my friends, is exactly what General Relativity (and the rest of modern physics) tells you not to do. This is about putting all of your energies and your focus on your coordinate system — on how you represent reality, instead of on reality itself. The coordinate system is arbitrary; what matters is what actually happens, not how you describe what happens using some particular way of measuring time, or space, or anything else. To get excited about the numbers that happen to appear on your measuring stick is to put surface ahead of substance, math ahead of physics, magic ahead of science. It’s as bad as getting excited about how a word is spelled, or even what word is used to represent an object; a rose by any other name.
But we humans are not designed to think this way, it seems. We cheer when we’ve driven a thousand miles, a milestone (hah) which combines the definition of mile (arbitrary) with the fascination with the number 1,000 (which only looks like an interesting number if you count with ten fingers, rather than 12 knuckles, as the Babylonians did, or eight tentacles, as certain intelligent sea creatures might do.) We get terribly excited about numbers such as 88, or 666, which similarly depend on our having chosen to count on our ten fingers. A war was ended on 11/11 at 11:00 (and one was started on 22/2/22 — coincidence?)
Celebrating birthdays is a little better. No matter what calendar you choose, or whether it even lasts a year (as, for example, in Bali), the Sun appears to move across the sky, relative to the distant stars, in a yearly cycle. When it comes back to where it was, a year has passed. If we define your age to be the number of solar cycles you’ve experienced, then that means something, no matter what calendar you prefer. Your birthday means something too as long as we define it not by the arbitrary calendar but by the position of the Sun on the day of your birth.
Similarly, the solstices that mark the days with the shortest daylight and shortest darkness, and the equinoxes that have days and nights equal in duration, are independent of how you count hours or minutes or seconds, or even days. It doesn’t matter if your day has 24 equal hours, or if you divide your daylight into 12 and your darkness into 12, as used to be the case. It doesn’t matter what time zones you may have arbitrarily chosen. If you want to mark days, you can use the time that the Sun is highest in the sky to define “noon”, and count noons. A year is just over 365 noons, no matter what your calendar. The time from solstice to solstice is about half that. But the date we call “December 25th” does not sit on a similarly fundamental foundation; it shifts when there’s a leap year, and sometimes it’s three days after the solstice and sometimes four. Many other holidays, driven by Moon cycles rather than a Sun cycle, are even less grounded in the cosmos.
Being too focused on coordinates can cause a lot of trouble. The flat maps that try to describe our spherical Earth make all sorts of things seem to be true that aren’t. They all make the shortest path between two points impossible to guess. Some wildly exaggerate Greenland’s size and minimize the entire African continent. Most of them make it difficult to imagine what travel over the north or south pole is like, because there’s a sort of “coordinate singularity” there — a single point is spread out over a whole line at the top of the map, and similarly at the bottom, which makes places that are in fact very close together seem very far apart.
A coordinate singularity of a more subtle type prevented scientists (Einstein among them) from realizing for decades that black holes, which were once called “frozen stars,” have an interior, and that you could potentially fall in. The coordinates originally in use made it seem as though time would stop for someone reaching the edge of the star. Bad coordinates can obscure reality.
Physics, and science more generally, pushes us to focus on what really happens — on events whose existence does not depend on how we describe them. It’s a lesson that we humans don’t easily learn. While it’s fine to find a little harmless and silly joy at non-events such as 22/2/22 or 2/22/22, that’s as far as it should go: anything that depends on your particular and arbitrary choice of coordinate system cannot have any fundamental meaning. It’s a lesson from Einstein himself, advising us on what not two do.
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.
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.
