Science & Technology Q&A for Kids (and others) [Part 142]

Hi, everyone. Welcome to another episode of Q and A about science and technology for kids and others. And I see a bunch of questions here about eclipses for those who are in the US or nearby. There is a total solar eclipse that's happening on April 8th in about a month and a half, month and a bit. And so people are asking about the science behind the eclipse. So I could talk a bit about that. Actually, I have to hawk something that just came out. This is a book about the eclipse that I wrote that I actually kind of started researching at the time of the 2017 total solar eclipse that was visible in the U.S. but anyway, this just got finished and just got published. So there I get to hawk it here. Okay, so let's talk about eclipses. And a very basic question is, why do eclipses happen? What are eclipses? And so on. So solar eclipse happens when the moon gets between the sun and the Earth, so it blocks the sun. And a total solar eclipse happens when things are lined up so that the disk of the moon just covers the disk of the sun and everything goes dark for a few minutes at the particular place on the Earth where the total eclipse is happening. And that's kind of the thing of the eclipse. It's sort of this coincidence of the orbit of the moon and the Earth and so on around the sun just line up so that the moon just covers the disk of the sun. Now, actually, we have a very weird case here on Earth about eclipses, because first of all, eclipses on Earth are quite rare. They happen. Actually, there could be as many as five in a year. But I think that last happened in 1935, that there were five total solar eclipses in one year. But normally there's one or two total solar eclipses per year, and they're often incredibly obscure. They're only total over some small region in an incredibly obscure part of the Earth. Well, a couple of things are needed to be able to get an exciting eclipse. One is that your moon better be able to cover the disk of the sun. If it doesn't, then, well, yes, something goes in front of the sun, but it doesn't cover it. So, for example, if we look towards the sun, which one shouldn't do with one's bare eyes, so to speak. But, for example, the planet Mercury, which is the planet closest to the sun, it regularly transits the sun. It goes across the sun, but it's a tiny speck on the disk of the sun. And from the Earth, we can see it going in front of the sun, but it's a tiny speck. It's way too small and too far away to significantly cover the disk of the Sun. The Moon happens to be just so that it just covers the disk of the Sun. It happens to be that way in our times, in the time of the so generally, the Moon is progressively moving further away from the Earth. It's essentially what's happening is it's kind of like the Moon is gradually kind of losing energy, losing angular momentum. It's gradually kind of slowing down. And that means it's going further and further away from the Earth. And in the time of the dinosaurs, the Moon was much closer to the Earth. In another few hundred million years or whatever, it'll be significantly further away from the Earth. In the time of the dinosaurs, it was easier to have the moon cover the disk of the sun because the Moon was closer to the Earth and looked bigger in the sky. In the future, sufficiently far in the future, there'll no longer be total solar eclipses on the Earth because the disk of the moon will no longer cover the disk of the sun looked at from the surface of the Earth. Now, for other planets, for example, on Mars, the moons are pretty small. And so on Mars, even though Mars is further away from the sun than the Earth, so the sun looks smaller in the Martian sky than it does in the sky from Earth. The moons of Mars are small enough that they're just little specks. Like Mercury is going across the disk of the Sun. On Jupiter, I think Jupiter is far enough away that the disk of the sun is fairly small by the time you get to Jupiter. And I think Ganymede, the largest moon of Jupiter, I think covers the sun when looked at from the surface of, well, Jupiter. Jupiter doesn't quite have a surface in the same way that the Earth has a surface. Jupiter is a gas giant planet where it's kind of its atmosphere all the way down. Probably maybe the center of Jupiter is a piece of solid hydrogen. But mostly it's not got a solid surface like the Earth. It's mostly just gas of different densities and so on. But any case. So we are kind of special on Earth that we have a situation where our moon just covers the disk of the sun as viewed from where the Earth is Now. Another way that we're special is that, for example, so that we have a planet like Mars, for example, it's orbiting the Sun. It's orbiting around the sun in a certain plane, and the moons of Mars are orbiting around Mars. And the plane in which those moons orbit turns out to Be aligned with the plane in which Mars orbits the sun. So the plane in which the planets orbit the sun, it's called the plane of the ecliptic. It's not coincidental that the word eclipse appears in the term ecliptic. But in any case, when the solar system is, for the most part, kind of pretty flat. As in the sun is in the middle. And the planets are all orbiting in one plane, for example, Pluto orbits a bit out of that plane. And other things that aren't classified as planets Also orbit out of the plane. But as far as the regular eight planets are concerned, they orbit rather accurately in this particular plane. Presumably when the solar system was formed, what happened was that there was a rotating cloud of gas. And that as a result of the rotation, the thing kind of spun out In a particular direction. And formed this kind of flat pancake of material. And as the planets were formed, they sort of condensed out of that pancake of material. The fact that sort of you get these pancakes of material when things are rotating. Is the same reason that galaxies like ours, Big galaxies like ours, Are also quite flat. That's just a feature of what happens to material. When it's attracting different parts through gravity. And when it's spinning around As a result of some angular momentum that it's been given. So, in a first approximation, the planets orbit within this single plane. And, for example, the moons of Mars Orbit within the plane of the ecliptic. Within that same plane that the orbit of Mars is around the sun. So what does that mean? Well, it means that if you're looking from the surface of Mars. And you're looking towards the sun, Every so often, the moons of Mars Will go in between you and the sun. Because everything is in the same plane. So that has to happen that every sort of Martian lunar month, which I think is a week or two. The Phobos and Deimos, the moons of Mars, Will be between you on the surface of Mars and the Sun. And so there'll be a not very convincing eclipse. Because the moon is really small. Compared to the disk of the sun there. But it will be at least kind of structurally like an eclipse. Okay, so Earth is different. Because our moon does not orbit in the plane of the ecliptic. Our moon is 20, about 20 degrees. I should know that precise number. Away from the plane of the ecliptic. Its orbit is kind of. It's off. By the way, that's an independent phenomenon from the fact that the Earth is rotating at the earth is 23 degrees. The axis of rotation of the earth is 23 degrees away from the plane of the ecliptic. That's in the end, why we have seasons and things like this is because it's not the Earth spinning on its axis. That spinning around the axis is not sticking straight out of the plane of the ecliptic. The Earth is not turning around in that same plane. Different planets have different inclinations with respect to that. So, for example, I think Uranus is actually tipped upside down relative to it's sort of rotating backwards, so to speak, relative to the rotation of Uranus around the sun, so to speak. So that's a different phenomenon from the fact that the orbit of the Moon is out of the plane of the ecliptic. So what does it mean that the orbit of the Moon's out of the plane of the ecliptic? That's the reason that if it was within the plane of the ecliptic every lunar month, every time the Moon goes around the Earth, which is about every 28 days, the moon would find itself between the Earth and the sun and there will be a total solar eclipse. But that doesn't happen because the orbit of the Moon is tipped relative to the plane in which the Earth is orbiting the Sun. And so that means that most of the time when the Moon is in its orbit between the Earth and the sun, it's up or down relative to the plane that the Earth is orbiting in. And so when you look towards the sun, you don't see the Moon there. So when an eclipse happens is in that rare case, when the orbit of the Moon, when the Moon in its orbit has got to a point where as it orbits it goes. Sometimes it's above the plane of the ecliptic, sometimes it's below. But once every so called draconic month, it is in the plane of the ecliptic as it orbits. So there are two different things. One is, how often do we get a new Moon? Does it orbit? Where is it between? There's the Earth, there's the Sun. Where is the Moon relative to the Sun? That's one question. And the other is, where is the Moon relative to the plane of the ecliptic? And you can get an eclipse when you will get an eclipse when those two things exactly line up. When the Earth, when the Moon is in its orbit between the Earth and the sun, and it is, and it's between the Earth and the sun, and it's at the place in its orbit where it is in the plane of the ecliptic. So sometimes it's above, sometimes it's below, but there's a moment when it goes through the plane of the ecliptic. And when that moment happens, when the moon is also between the sun and the Earth, then you'll have an eclipse. So that's sort of the basic thing is when, when the Moon is when you have that sort of coincidence of the so called synodic month and the draconic month, when, when you have those things lining up, then you have the, then you will have an eclipse. So when do those line up? Well, it's kind of like you are trying to work out, I don't know, you've got, I don't know the number, the number 11 and the number 13. And you're saying, well, what multiple of 11 is also a multiple of 13? That's kind of the thing that you're trying to work out when will you have it be the case that the orbit of the Moon with respect to this draconic month lines up with the orbit of the Moon with respect to the synodic months? And so it's kind of like saying what multiple of 11 is the same as a multiple of 13? Well, for example, in that particular case it's 11 times 13. That if you have 13 elevens, it's the same as 11 13s. If those things weren't both primes, then you would have a different, and you could have a different arrangement of how many of one number is equal to how many of the other number. But that's sort of the question is how do you line up these, how do you get it to be the case? So let's say an eclipse has happened, the Earth has been in this place where it precisely lines up. And now you're waiting how long until you get the next eclipse? Well, it has to be some number of draconic months because it's got to be back in the place where it is in the plane of the ecliptic and some number of synodic months because it's got to be between the sun and the Earth. And so the question is how many draconic months equal a certain number of synodic months? And so there's this kind of question of numbers of these are both sort of, these are both just sort of numbers with lots of digits. And you're asking when do you get a good approximation to these two things being equal? So it turns out one time you get a good approximation is about every 18 years. And that leads to the thing called the Saros cycle. And the Saros cycle is a way of sort of guessing when there'll be an eclipse, because every Sarosaurus is about 18 years. Every saros, it's. I don't remember how many it is. It's like 480 or something draconic months. I should know these numbers and some number of synodic months. And it's a place where that number of draconic months equals, to a good approximation, that different number of synodic months. And that coincidence happens about every 18 years. There's another coincidence that happens every 200 and something years, the Inex. I think that's called the Inex cycle. But so the Saros cycle is the thing that sort of determines, okay, you had an eclipse, when are you going to get the next one? Now, it's not precise because those numbers don't precisely line up. But that's a good guess for when an eclipse could happen. There are a number of things that can sort of go wrong with predicting when the eclipse happens. One of them is that the Earth that the moon doesn't isn't necessarily its orbit is a bit complicated. And it could be a little bit too close, too far away from the Earth. And if it's too far away from the Earth, it is not going to. As the moon gets further away from the Earth, it will get smaller relative to the disk of the sun when viewed from the Earth. And its orbit has it be a variable distance away from the Earth. If it's a little bit too far away, it won't quite cover the disk of the sun. And so you get what's called an annular eclipse, where the disk of the sun sort of peeks out from behind the moon. So the first sort of thing to know is, given that eclipse has happened, another one could happen about 18 years later in the Saros cycle. Now, things don't line up precisely. So what you'll have is this whole Saros series of eclipses where you roughly get an eclipse every 18 years. But after a couple of thousand years, things don't line up properly and that series will be over because the numbers just don't line up well enough and no longer sort of aligns to get the moon in the right place. So at any given time, there are a bunch of different Saros series that are operating. So different eclipses will be one of the members of a series that's spaced by 18 years. But they may be the different eclipses that happen may be members of different series. So you don't necessarily have to wait 18 years for an eclipse because there may be a different Saros series That is a different sort of way in which the Moon and the Earth align and it again is spaced by 18 years, but it's at a different position, so to speak. So they usually. Oh, I should know this as well, maybe. I think they're like a dozen or so Saros series that are active at a given time, that are sort of producing eclipses at a given time in history. Okay, so then this. So if you want to know sort of more precisely when can the eclipse happen and so on, you can go and look at these other series, you look at these numbers, and there'll be some sort of way in which, if you take the Draconic month and the Cyanoic month, you divide the lengths of time for these two, it'll be some irrational number, and there'll be a rational approximation, an approximation to that number that says it's roughly this whole number divided by this whole number. And that means that approximation will tell you sort of at what point you will have a coincidence of number of draconic months, number of synodic months. But as you go on, you can find successive rational approximations. Let me explain how that works. So, for example, if you have something like PI, 3.14159, et cetera, PI, the first approximation to PI is it's about 3. The next approximation to PI might be it's about 22 over 7. Then there's another approximation to PI. I have to work it out. Let's say it's 312 over 111 or something. It's some other pair of numbers. And you keep going. And you can always get better approximations to PI as a ratio of whole numbers. You'll never precisely get to PI because PI is an irrational number. It can't be represented as a ratio of whole numbers. But you'll get closer and closer approximations by having longer and longer bigger and bigger numbers in the numerator and denominator of that fraction. And so it is with approximating this ratio of draconic to synodic months, which is what determines, once you have one of those rational approximations, you're saying this number of Draconic months is almost equal to that number of synodic months. And that means that there'll be this coincidence that will cause an eclipse to happen or have the potential to have an eclipse happen. So that's sort of the basic mathematics, I suppose, of when eclipses can happen. Now, this Saros cycle was identified a very long time ago. It was known to the Babylonians, and they had this big project the astronomical diaries in which every day they would go and record the state of the sky, you know, what planets were visible, this planet is near those stars, et cetera, et cetera, et cetera. They also record the weather. They'd record the height of the river, the cost of grain, few other things. They did that recording for 700 years. So I think about 2% of the tablets on which those records were committed have survived. But the thing that from those astronomical diaries one could see things like these 18 year cycles, and people knew about the 18 year cycle of eclipses at that time. In fact, one place that was made use of was this thing called the Antikythera device, which is this one example of a computer that we have found from antiquity. It's this little thing about the size of a hand or something, made from a little bit bigger than that, made from a bunch of cogs. And the cogs kind of move around and their teeth and their size is such that they kind of move around, sort of an orrery that simulates the motion of the Moon and the sun and so on. At that time, they were sort of assuming that they were representing the sun going around the Earth, which for the purposes of their device was really no different from saying that the Earth goes around the sun. It's the same sort of mathematics at that level. But anyway, their device represented that, and their device represented the Saros cycle and effectively could tell one when could an eclipse conceivably next happen? Now, the fact that an eclipse is conceivable then doesn't tell you it will actually happen. It doesn't tell you where on the Earth it will happen. So that to work that out is a more complicated kind of issue. And people. So that was just sort of things are going around in cycles. Then you can start asking, well, can we actually compute where the Moon is going to be, where the Earth is going to be to figure out whether there'll be an eclipse or not, rather than just saying, oh, it roughly goes around in this orbit and the orbit is periodic and so on. Can we just work out, using physics, where the Moon is going to be? Well, nothing like that was possible in antiquity. That started to be possible when we started understanding things like how gravity works or the laws for gravity. And that was really a thing that was pretty much Isaac Newton in 1687 was the guy who kind of gave his universal law of gravity, that the force of attraction of gravity is proportional to the one over the square of the distance between two bodies and proportional the mass of one body times the mass of the other body. But importantly, it's this inverse square law of gravity. And what Newton did at the time was to use that law together with his laws of motion, laws of mechanics that say things like force is equal to mass times acceleration. That the law of gravity plus that law of motion, plus the sort of tool of calculus, was what allowed Newton to work out the orbits of planets and more particularly and more impressively in those days, the orbits of comets. It was a time when Halley's comet was set to return and there was sort of a question of, of how that all worked. And the thing which came out was, oh, you can use this inverse square law of gravity together with the laws of motion, together with calculus, and you can compute what the orbit of this comet will be. So one of the things that Newton really wanted to work out was the orbit of the Moon. And he wanted to work out, given the observations of the Moon, the angle that it's at and the length of a lunar month, things like this. He wanted to work out just exactly where will the Moon be so that you could do things like work out eclipses and so on. He didn't succeed. His calculations got the wrong answer. That's not because the underlying theory is wrong. It's because it's actually a pretty hard problem. Because the standard problem that Newton had worked out was the so called two body problem. The problem of given essentially a star and an idealized planet, what is the orbit of that idealized planet around that star? And the answer is, it's always mathematically an ellipse. And then you can work out the properties of that ellipse. The two body problem is one that Newton really solved and showed. The answer is an ellipse. And you can work out sort of everything you want to know about where the various bodies will be at different times and so on. Okay, now add a third body, for example, sun, Earth, Moon. Things become extremely much more complicated. And in fact, you can work out, in Wolfram language, you can easily use the function n body simulation to work out what the actual orbits of three bodies look like. And it's a remarkable thing because if you start them off from sort of random positions with random velocities, chances are you'll get this big snarled mess of kind of these things orbiting these things and twisting around and so on. Sometimes the outcome will be all three bodies just escape from each other. Sometimes the outcome will be two bodies are in orbit around each other. You've got kind of a two body system, and the other One gets ejected. And sometimes you'll end up, rarely you'll end up with sort of a triple body system. And we see that with stars. We see usually like our sun is a star all on its own, but something like Alpha Centauri and nearest other star is a triple star system with three stars in it. There are plenty of other stars which are binary stars where there are two stars orbiting each other. And this three body problem tells one sort of what happens when there are three bodies. And sometimes you end up with one being ejected, et cetera. And just sometimes you end up with all three sort of doing some long time dance orbiting around each other. But the main takeaway is the three body problem is a really complicated problem and it's really complicated to work out what those orbits are. And it's pretty obvious that it's complicated. But if you actually do the computation a few times, you'll just see these orbits are really complicated. Snarly messes Actually this problem of working out the three body problem was a problem that kind of dominated mathematical physics in the 19th century, for example, and some extent the 18th century. People really wanted to work out the answer to the three body problem. They wanted to see can you work out sort of a mathematical formula for the three body problem as you can work out a mathematical formula for the two body problem? They never succeeded. My own guess is that the three body problem is in some sense computationally irreducible. That there really isn't a way to get a kind of oh we know everything that's going to happen formula for the three body problem. That's an interesting outstanding problem in mathematical physics to try and work out whether that's really the case or not. But any case, it's a practical matter to solve the where's the moon going to be? You have to solve the three body problem. Newton for example, in his didn't have really the mathematical apparatus to be able to get a good approximation to the three body problems. We had various techniques but the main technique that one uses is series expansions. So if you know kind of calculus, mathy type stuff, you might know that something like a trigonometric function like sine X, you can say sine x is roughly x plus x cubed over three factorial minus. Am I going to get this wrong? I have to think about this. Minus x to the fifth over five factorial. Is that right? And then you keep going there. But it's an infinite series, it's something times x plus something times x cubed. In the case of sine plus something times x to the fifth and so on. It's this series of terms. And it's a common mathematical trick to say we've got this thing and it's really a complicated function. It's wiggling around in all kinds of ways. But let's decompose that into a sum of pieces in terms of functions that we do understand, like x x squared x cubed or sine x sine 2 x sine 3 x sine 4 x functions we understand and say this thing that we are trying to put together is some sum of terms, each of which is something we understand. And there's some multiple of that thing. And we add all these together. And kind of what you have to hope is that in that series expansion that somehow the importance of successive terms goes down so that most of what you care about is in the first few terms that you can work out. So that's the key approach in celestial mechanics, in the problem of working out the motion of things like planets, is to use series expansions and to basically say, oh, what's important is this particular feature, like this fundamental orbit of the Moon around the Earth. But there's a little correction factor that comes that you deal with in pairs of orbits, let's say, and then another one for triples of orbits and so on. And you hope that that series somehow converges. Problem is that in celestial mechanics is sort of a fundamental problem of celestial mechanics. A little bit mathy why this happens, but the. The fundamental problem is that you don't have that kind of convergence in certain cases. Can I explain why? Roughly, the reason is the following. That. So when you have a sine wave, what is a sine wave wiggles up and down like this. Probably many people know exactly what it is. But if you take a circle and you have a point on the circle and you're rotating the circle around its center, and you look how high is that point? As the circle goes round that point, sometimes that point will be down at the axis, sometimes that point will be below the axis, sometimes it'll be above. You keep rotating the circle at a fixed rate. And you ask how high is that point away from sort of the baseline that goes through the center of the circle? That quantity, the how high is that point is the sign of the angle that you're turning the circle around at. So if you turn the. If the circle is at angle T, if T is sort of the. The time the circle is. Is at angle T, then the height of that point is sine T S. I N t of T. So that. And so, for example, if you're looking at questions about, you know, where is the moon relative to the, the things like the plane of the ecliptic and so on, there's a lot of the story of, you know, there's a lot of sine T, a lot of the sine function that shows up. Here's where things get tricky. What's tricky is that the actual sort of main perturbation to it just being, oh, it's just a sine function, it's wiggling and it keeps wiggling. The same way is not that it is something like, you know, 0.2 times sine T plus 0.1 times sin 2T or something like that. Instead, what happens is that inside the sine function, the frequency of rotation is not what you think it is. There's sort of a change in the frequency of rotation. And when you try and work it out and you do the calculus and you work out this series expansion, you'll discover that when you have that change in the frequency, that it can sort of spray an infinite number of terms into this series. And it's that phenomenon that makes it tricky to do celestial mechanics. It makes it tricky to work out these series expansions and so on. And so what Newton tried to do was to use this series approximation method to work out the motion of the moon. He didn't get it right for actually reasons related to this phenomenon I was describing. Many famous mathematicians worked on this over the course of the next couple of centuries. Euler worked on it, a bunch of other people worked on gets very mathematically complicated and in fact, by the mid-1800s, people were producing practically book long formulas that were all the terms in this series. And it was just a very difficult thing to work out. One of the more famous ones, oh, gosh, what's the person's name? Not particularly well known person, but there was a person who worked out the motion of the moon, supposedly using one of these series approximations. It's a big, long, messy formula that goes on for tens of pages in some printed book. Unfortunately, there was a mistake in one of the terms in this series and sort of everything after that is wrong. And so that wasn't a successful way of computing the position of the moon. By the way, you say, did anybody actually get anything out of all these series approximations? The discovery of the planet Neptune was a result of two people, Adams and Le Verrier, who worked out all these series approximations and said, you know what, the orbit of the planet Uranus is not what we think it, what it should be based on the other planets that we know in the solar system. There must be something else that is wiggling that that is having a gravitational effect on the planet Uranus. It must be another planet. Oh, here's where it should be. And you turn the telescope there and there's Neptune. But if you actually look at the books those guys wrote, they're big thick books and they're full of algebra, they're full of these series expansions trying to work out what really should be the orbit of the planet Uranus based on the orbits of Jupiter and Saturn and all these other planets. So it's a tricky business. It's a difficult thing to work this out. The Earth, Moon, sun system is particularly difficult. It's a place where these series approximations don't converge as rapidly as they do in other cases. It's really a hard case. So. But in. And what ends up getting done when you make sort of, when you present the answer to this is what the motion is going to be, you typically present it as a sum of terms, each of which is kind of a sine wave type term. That's a little bit ironic because back in antiquity when people were trying to work out what is the motion of all the planets and things like this, they had this idea of epicycles. Ptolemy in the first century A.D. i guess, was kind of famous for setting up a very elaborate version of this. The idea was that the planets were sort of moving. People eventually said they were moving on crystal spheres in the sky. And these crystal spheres were kind of cog, like things that had certain sizes and so on and certain rates of rotation. And the planets were kind of moving around on this giant kind of orrery in the sky that was associated with the so called epicycles. Epicycles were basically like sort of conceptually like kind of cogs moving at certain rates. And you would explain the motion of a planet by it's this cog and that cog and that cog. And that's what leads to the particular motion we see. Very tricky issue is the so called retrograde motion of Mars. The fact that if you look at the position of the planet Mars and the sky from the Earth, it goes backwards some of the time. And that happens if you look at the geometry of what happens when you're from the Earth and the Earth is going around the Sun, Mars is going around the sun, the rate at which Mars is going around the sun, the rate at which the Earth is going around the sun, if you kind of do all the geometry, you'll see that in the sky of the Earth, Mars will seem to go backwards some amount of time. It'll have these loops in the sky that go backwards as well as forwards. And that was hard to explain in terms of epicycles. But people made up, oh, there are these cogs and these cogs, and you can kind of make that motion that way. Well, then when the Copernican system came in and people realized, actually, everything's going around the sun, and it became a bit simpler to see how that all worked. But nevertheless, the mathematics of sort of these cogs, cogs and cogs, and sort of lots of circular motions and so on, all connected together, that mathematics is still perfectly good approximation to what we actually see in the motion of planets and the Moon and so on. And even today, where Ptolemy might have talked about, I don't know, two or three epicycles being responsible for the observed motion of planets, today when we have mathematical approximations, we might have the equivalent of 20,000 epicycles. It's kind of like the Antikythera device had a few cogs in it, maybe four or five. And now we have a machine, except it's actually just a piece of software that has the equivalent of 20,000 cogs that are the way that we work out the position of the Moon and so on. Well, this whole question about can you work out where the Moon is going to be? Was sort of a driving force in the development of lots of kinds of computational devices. The very first ever. Well, in modern times, the Antikythera device was an astronomical device back in antiquity, and no doubt there were many copies of it back in those days. But the first known sort of modern device was made by a chap called Shakad, who was a friend of Kepler's and was made for Kepler for his kind of researches into astronomical motions and so on. So there's been sort of a long history of using computational devices and computers to work out things about the motion of planets and the Moon and so on. And people, I guess by the early 1900s, 1920s and so on, people with mechanical calculators and such like had gotten pretty good at working out the motion of the Moon. There was a thing called Brown's tables, which were sort of, for a long time, kind of the standard for working out the motion of the Moon. It should be said that the people who were really sort of published information about the motion of the moon were navies. The Naval Almanac. There was One in the uk, one in the US this was a place where it was like, when's there going to be a new moon? You know, that's a way to tell time and so on. And that was something that was tabulated. And the nautical almanacs also tabulated the times of eclipses and such like. It might be mentioned that, well, there's sort of this fundamental problem that if you want to know where are you on the Earth today, you use the GPS satellites. We know the orbits of the GPS satellites. We measure the time delays of signals coming from different GPS satellites. We can triangulate where we are on the Earth rather accurately at this point. But back in the day, there weren't any GPS satellites. So how did you do it? Well, it was pretty easy to work out the latitude that you were at on the Earth, how sort of far up between the equator and the North Pole you were. That could easily be worked out from the. What is it, the altitude of the sun? I think. Let's think about that. But. Well, okay, you can work that out. That without. You can just work that out by observing the sun. Yes, yes. But if you want to work out your longitude, you need one other piece of information. You need to know what time it is. You need to independently know the position of the sun and what time it is. And the problem was it was pretty hard to keep time. And particularly if you're on a ship that was sort of bouncing up and down all over the place, over the waves and so on. If you had something like a pendulum clock that was swinging back and forth, there was very hard to maintain the pendulum kind of swinging at the right rate, even as you were kind of, you know, having the sort of the thing that the pendulum is attached to, bouncing up and down in the waves and so on. So this was a long time issue of could you make a marine chronometer, an accurate timekeeping device that would work when the ship was kind of bouncing around on the sea? And there were a variety of proposals that were made for how you could avoid having to work out, do this precise timekeeping using an actual timekeeping device. One of the more interesting ones was to use the moons of Jupiter. So the moons of Jupiter were discovered by Galileo in 1608, and the moons of Jupiter go around Jupiter at a certain rate. And so one idea was, if you want to know what time it is, you just look at the configuration of the moons of Jupiter and you look it up in a table. Telescopes weren't quite routinely good enough that that was A possible thing to do. And before the marine chronometer was invented, it seemed like a possible way to tell time accurately was to just look at the positions of the moons of Jupiter, because that would be something that had known kind of periods and so on. So in any case, but the question of sort of when was there, when exactly were the new moons? What was the position of the moon? That was something important for navigation. That something got worked out in these nautical almanacs and so on. The final sort of where is the moon? Question, by the 1930s, people were getting pretty confident of sort of where is the moon? And among other things that allowed one to predict eclipses. It was, I think in the. When was it maybe 1950s, there was an eclipse that was total in New York City. And there had been a prediction about what the edge of totality, of the region of totality would be. And it was off by a few blocks, but it was good enough that that was where it was at. So let me explain a little bit more about the eclipse and what you see on the Earth when there's a total eclipse. So basically the moon is getting between the sun and the Earth. So you're seeing the shadow of the Moon. The moon casts a shadow on the Earth, it doesn't cast a very big shadow. There is a region where there is sort of total shadow where no part of the sun is visible on the Earth. That's the so called umbra region. There's a much larger region where just sort of a piece of the sun but not all of the Sun. The sun has a bite taken out of it, so to speak, when viewed from the Earth, but it is not completely covered by the moon. That's the so called penumbra. And in that region you will see a partial eclipse. You'll see the sun partially covered by the Moon. It's only in this small region where things are exactly lined up, you'll see sort of the shadow of the moon. It's. It's the sort of complete shadow of the moon. And when you look at the sun, it will be completely covered by the moon. And by the way, many eclipses end up being just partial eclipses. There's no place where the shadow of the moon falls on the Earth. So if you imagine the sun is in one place, the moon is in some place, the moon is casting a shadow. But if the Earth isn't in quite the right place, that shadow may be cast somewhere off into space and not on the surface of the Earth. The question is, is that shadow cast Somewhere on the surface of the Earth where that shadow is cast on the surface of the Earth, that will be where there is a total eclipse. Now, things are moving. The Earth is rotating. The Moon is moving in its orbit. That means that the position of that shadow will move. In fact, the position of the shadow moves at, I think, around 1800 miles an hour on the surface of the Earth. Most of that speed comes from the motion of the Moon in its orbit. The Moon is moving in its orbit, and it's kind of getting out of the way or getting into the way. And where the thing is lined up is changing rapidly because the Moon is moving in its orbit, but also the Earth is rotating from underneath. If you're in a particular place on the surface of the Earth and you're rotating with the Earth, then. That's another thing that's moving you out of the way of the shadow. Now, I think the speed of that shadow will vary depending on what latitude you're at on the Earth and various other effects. Because when the Concorde was out and about able to go at what, about a thousand miles an hour, it could. It successfully chased some eclipses, so to speak. So you could stay in the region of totality. You could stay in the place where there was a shadow, because you could go fast enough on the surface of the Earth that you could kind of keep up with the speed at which the shadow of the Moon was moving, at least for a while. But any case. So a typical total eclipse, there will be a swath of shadow that goes across the Earth. It's not very wide. It's. What is it, maybe 20 miles wide? Something like that. In most places, it varies depending on where on the Earth it is. But it's a thin region of totality. And that the spot where there is totality, which is roughly an ellipse on the surface of the Earth, that spot moves at maybe 1800 miles an hour. And that spot is maybe 20 miles across. And so what you see in a total eclipse is you can actually see the two total eclipses I've seen in my life. In both cases, you can actually see the shadow coming towards you rapidly. So I saw one total eclipse in 1991 in Mexico, actually, where I was right on a beach looking out over the ocean, and you could see the shadow coming in over the ocean. I saw another total solar eclipse in 2017 in Montana. No, in Wyoming, sorry, where it was up on some ridge looking over a valley. And you could see the shadow come across the valley as the shadow of the Moon sort of moved on the surface of the earth. And what then happens is there is a brief time in which things are lined up enough that the shadow that you're in this region of complete shadow. So, for example, for the eclipse that I saw In Mexico in 1991, it was about six and a half minutes of totality. I think that's the longest period of totality for a century or so. In the eclipse that I saw in Wyoming, it was maybe three and a bit minutes of totality. It depends on the exact lining up of things, how long the period of totality will be for a particular eclipse. And it depends on how far the moon is away from the earth on the particular time when the eclipse happens. But in any case, so when you look at where is the path of totality, where's the path of the eclipse? It sort of cuts across the earth. And there are different paths for different eclipses. And if you make a picture in this little book of mine, there's pictures of where the eclipse paths will be for the next 500 years. And they kind of cut across in all kinds of places across the earth. So, for example, the eclipse that's about to happen in April 8th will cut across from someplace out in the Pacific Ocean going through Mexico, coming up through Texas, goes up towards the Great Lakes and then Maine and then goes out into the Atlantic Ocean. And it makes that sort of transit over the course of what, a couple of hours or something as the moon is moving in its orbit and the earth is moving, is rotating around. So what do you actually see in one of these eclipses? All kinds of things happen. I mean, it's kind of weird because it's kind of like if there's no cloud cover and so on, it's kind of like somebody switched the lights off and it's fairly dramatic. There are. Well, I remember that the eclipse I saw in 1991, one of the things that's really kind of weird is that the birds think it's nighttime and they stop chirping and so on. And other things that sort of come out at night come out because they say, oh, it's nighttime, we're time for us to get up type thing. That's kind of a weird thing. Another phenomenon that you see with eclipses is. Well, you see in the eclipse in 2017, I saw this really weird thing that right before the eclipse, for maybe a minute or so before the eclipse, this weird shimmering that you could see everywhere. These are things called shadow bands, and they occur in some eclipses, but not others. Nobody really quite knows what they are. They're probably the result of turbulence in the atmosphere causing sort of different as the atmosphere is cooled because it was being heated by the sun, but the sun got covered up and so things start to cool down. And as it cools down, there's motion of air, there's little motion of air that is moving around and that produces potentially this kind of shimmering effect. It's a really weird effect. I tried to take video of it and failed. People typically fail to take videos of this, possibly because it's not really a truly imaged thing. It's a thing that somehow is a feature of kind of the way that light is coming to one and the phase of the light and so on, and how you're actually imaging it in your eyes. And it's not something that is obviously capturable by the sort of way that a camera works as opposed to the way that things work in your eye. Not clear how that exactly works out. There are other things about. The thing that this question about predicting the arrival of the eclipse. For the 2017 eclipse, we made a website to predict the arrival of the eclipse within one second that's very non trivial. It requires both precise knowledge of the orbits of the moon and so on. And it requires knowing things about the topography of the Earth. And it requires knowing things like leap seconds about how the Earth has slowed down over time and exactly which second relative to the clocks that we have, will the eclipse arrive at. But it seems like we got it right so far as we could tell to within the second at that time. Actually the thing that is the final limit is mountains on the moon. So one of the things that happens is as the moon is covering the sun, the question is when is the last ray of light from the sun? When will it get through? And that depends on where there's a valley. If the moon was completely circular and it's viewed from the Earth, then it would just be like, oh well, it crossed the edge of the sun and we're done. But actually the moon has hills and valleys and so on, which are quite big actually compared to the ones on Earth, they're proportionately bigger than the ones on Earth. And so if the moon, if there's sort of a ray of light that can sort of shine through some valley between the mountains on the moon, that ray of light can still get through, even though based on sort of the position of the center of the moon, you would think that the sun would have been covered. And so there are these things called baily's beads, which are kind of the last remaining pieces of light that you see before the whole disk of the sun is covered by the moon. I suspect there are also other effects like the sun has big prominences that splurt out from its outer atmosphere that probably produce a certain amount of light that's even pretty far away from the official position of the sun. So there's trickiness to get below the 1 second level is, I think, quite difficult and depends on these kind of complicated effects. Because even knowing which mountain, the moon is more or less locked with one face towards the earth, more or less, we get to see the same moon face, so to speak, all the time. But there is this effect called libration, where the Moon wiggles a little bit. And that effect has the consequence that if you want to know which mountain is right at the edge of the moon, right on the limb of the moon, so to speak, one side of the moon that will be affected by where you are in this wiggling of the moon. So that's yet another complicated effect that you have to work out. So in any case. Well, what more to say about eclipses? I have to say, when there are many interesting phenomena, I have to say the first eclipse I saw was when I was six years old. And I'm very proud to say that I noticed it for myself because I was walking to school, which one did in those days, and I noticed, you know, there's patches of light under trees. And I noticed that the patches of light looked really weird. And then I realized, gosh, if you looked at the sun, it had a big piece bitten out of it. And these patches of light under the tree were they. Usually they're kind of little images of the sun, effectively. They're little circles that are kind of the pattern. And it's kind of like a pinhole camera. If you have a source of light and you have just this pinhole, and light is coming through the pinhole, and then you're making an image of it somewhere else just by the. By the geometry of the light has to come. If you have some shape of light as it goes through the pinhole that always has to go through that one point, then you'll form an image that is an upside down version, let's say, of that original shape of the source of light. And so it is with the sun going through sort of the gaps between leaves on a tree. But when the sun has a big bite taken out of it, those shapes that you get look like they have bites taken out of them. All lined up, very weird effect. And in fact, for the 2017 eclipse, you just take your fingers and you just look at the. You make something which is kind of like a miniature pinhole and you can see on any object you can see during the time when there's a partial eclipse, you can see this kind of crescent shape that comes about from that effect. But so that was, I have to say, in one of those life forming experiences, having seen this eclipse walking to school, I'm like, I get there and tell people, look, there's this eclipse, isn't that cool? And most of the kids just didn't believe me. So whatever. It's one of those. By the time you can plainly see, okay, Admittedly it's like kids don't look at the sun type thing. Although In England it's 53 degrees north latitude, the sun is not as bright as it is in some other places. So. But it's kind of one of those formative experiences of even if it's so obvious, you just look in the sky and you could see the sun has a big bite taken out of it. And people sort of don't believe that that's what's going on. That's kind of an interesting thing to observe happen and something that I've certainly observed in very different forms throughout my life of even very obvious things that you can point out people don't necessarily believe and understand. But that's a different issue. All right, a few questions here and then we should probably wrap up. Let's see. Aria asks, can we still be wrong about the position of Earth and its motion or is that debate over? It's really over. We know very, very precisely the motion of the Earth and the Moon. In fact, one of the things that was one of the results of the Apollo missions was the Apollo astronauts left a lunar laser retroreflector which is like a big. It has those sort of glass spheres that are kind of like you have a miniature forms on road signs or you're having these cat's eyes that are stuck in roads. They're retroreflectors. So you shine light at one of these beads and it gets reflected back exactly the way it came, kind of like a good mirror type thing. But so by shining pulses of laser light at the moon, seeing it reflect back and timing exactly how long it took for the pulse of light to come back. The position of the Moon is now known to maybe a centimeter, something like that. And that allows one to verify these calculations of the orbit of the Moon very precisely. Also Allows one to verify things like this fundamental feature of gravity. That inertial mass, the mass that is associated with how difficult it is to get things to start to move, is the same as gravitational mass, the mass that's associated with how much gravity a particular object produces. That principle of equivalence, that's a fundamental feature of Einstein's general theory of relativity, for example, that's verified by looking at the motion of the Moon using the laser retroreflector to, like, one part in a quadrillion. So one part billionth, one part in a thousand billion. So, yes, the motion of these things really is known at this point. There are effects, I could have said, when you try to work out motion of the Moon as accurately as we now know it, There are effects from all the planets in the solar system. Jupiter has a big effect. Other planets have smaller effects, but they all have an effect. There are also effects from relativity theory, general relativity and so on. There are a bunch of effects from that. That gets very tricky, Very tricky. Issues about what, Even when you do the calculations of working out what is the time that. The time on the Moon, the time on the Earth, how do you line up those times? That's a whole story of reference frames and simultaneity conventions and so on in relativity and such like, Let's see. So. Oh, by the way, I mean, what one question you might ask. Actually, one question I didn't address is I mentioned that unlike most other moons in the solar system, the Earth's moon is not orbiting in the plane of the ecliptic. The Earth's Moon is kind of off at an angle. And why is that? Most likely, it's because of the way the Moon was formed. The current most favored theory is that in that early primeval formation of all the planets period, that there was a thing, asteroid like thing, maybe almost the size of Mars, that hit the primordial Earth. And that was a big splat that happened. It's often referred to as the big splat. And that collision will have liquefied the Earth and the Earth. And the theory would be that the Earth recondensed, reformed, but a piece of it broke off and formed the Moon. And that the Moon is a mixture of material that was part of the original Earth, together with things that were part of this impactor that hit the Earth. And that there's some sort of evidence for that. Because if you look at the Moon. The Moon, okay, so in the Earth, the Earth is really quite. Well, the Earth is shaped, is very close to a sphere, but also if you look at the gravity of the Earth, the gravity, you know, you learn the acceleration due to gravity is 9.81 meters per second per second. That's a standard high school physics fact. That number varies slightly depending on where you are. On the Earth, it varies. Sometimes it isn't even going straight down towards the center of the Earth. The acceleration vector is slightly off. If you're near mountains, it's a bit different, et cetera, et cetera, et cetera. Depends on how much mass there is in different places. But on the Earth, the acceleration due to gravity is mostly pretty close to being towards the center of the Earth and about the same everywhere on the Earth. That's not true on the Moon. On the Moon, there's a big lump on one side of the Moon where the gravity is different. And that means when you have a satellite orbiting the Moon, its orbit is more complicated than the orbit of a satellite around the Earth. It's more difficult to have a stable orbit around the Moon and so on. And so that sort of supports this idea that there's a piece of the thing that hit the Earth and the Moon, the primordial Earth, that there's a piece of that stuck in the Moon, so to speak, that has a different kind of thing, and that leads to this kind of anomaly. And the gravity of the Moon, by the way. I mean, there are all kinds of questions one can ask, like, was there only one moon of the Earth? Did the Earth have three moons back in the day, you know, a few billion years ago, when this original event happened, did it originally produce three or four moons? Or did we always only have one moon? We don't know the answer to that. There's a question that sort of a fun question one of my kids actually asked when he was quite young was when there were dinosaurs, could the Earth have had two moons? And I tried asking that to a bunch of people who know about celestial mechanics and these kinds of things over the course of many years. And people would usually say, we don't know, we don't know, we don't know. I think by now people are more or less saying it probably couldn't have been true 65 million years ago. Probably the Earth only had one moon 65 million years ago. But if you ask about a billion years ago, it's not clear anymore. And the Earth could have had multiple moons. And what would have happened to the other moons? Well, most likely they escaped. Because if they crashed into the Earth or the Moon, we probably would see evidence of that. Certainly if they Crashed into the Earth. We'd see evidence of that if they were of substantial size. Another question you can ask, if you're just sort of asking questions about how these things work, is, are there moon moons? Like, you can think of the. You know, there's the sun, and the sun has planets going around it, and then there's planets, and they have moons going around them. It's kind of like the planets are the moons of the sun. And then the moons have. Are the kind of analog of planets around the sun. That's what. When Galileo first saw the moons of Jupiter, the thing that he was sort of very impressed with is, oh, that's kind of like the planets must be going around the sun, Just like these moons are going around Jupiter. So then the question is, can there be moon moons, that is moons going around moons? And the answer is, well, there are some small ones, but mostly the orbits of moon moons don't tend to be stable. It tends to be the case that there are enough other things that have an effect on. On the sort of. That are pulling those moon moons away so they don't keep going around in that same orbit for a long time. In fact, the solar system in general is probably the result of sort of a natural selection of different kind of planets, because probably our solar system started with a lot more planets Than we have today. And things happened like planets crashed into each other, or planets got ejected from the solar system and are probably wandering very lonely in interstellar space, Sort of not having reached a star. And, you know, we don't know at all how many planets there originally were in the solar system. I mean, one thing that's become clear by looking at exoplanets, planets around other stars is that our solar system is pretty typical. And, you know, it always used to be the case. People always assumed that the fact that there were, in those days, nine planets, now considered to be eight planets, was sort of an arbitrary fact about the world that just is the way the world happens to be. It's now increasingly clear that for a star of the size of the sun, you can work out, oh, there have to be about this number of planets. There typically will be about this number of planets. So in any case, the reason that we are the result of many coincidences, but the fact that we have eclipses that are sort of just, you know, we only have rare eclipses, is a consequence of the fact that the moon was formed in this strange way, probably different from the way the moons of other planets were formed, where probably just when the planet Condensed out of the kind of original dust and gas and so on that some pieces were sort of left over and didn't form into the planet. I'm not sure that's right actually. But. And that that led to the typical Moon. But that our case is weird because our Moon was the result of a sort of additional impact on the Earth. Let's see, Storin observes, the Earth's Moon is important for the tides. Absolutely. For the climate. I'm not so sure about that. Certainly for the tides. I mean the reason that there are tides on the Earth is that. Well, both the sun and the Moon have an effect on tides. But what's basically happening is there's the Earth, it's pretty much solid. The Earth does squidge a bit. There are so called solid tides that come because the gravity, gravity from the Moon sort of squashes the Earth. And that same thing happens to the Moon. There are solid tides on the Moon, but mostly the Earth is pretty hard to squidge. The thing that isn't hard to squidge is the oceans. And so what happens is the center of the Earth is being attracted to the Moon. The piece of the Earth that is closer to the Moon than the center is more attracted to the Moon than the center is because there's this inverse square law. And when you're closer to the Moon there is a stronger force of attraction by gravity than if you are at the position of the center of the Earth. Similarly, if you're on the opposite side of the Earth, you're further away. And so the force of gravity is less big. So what does that mean? That means the Earth is being overall attracted to the Moon. But the piece of the Earth that's closest to the Moon is being extra attracted to the Moon. And the piece of the Earth that's on the opposite side from the Moon is being less attracted to the Moon. So that means that if you think about if you can have water in the oceans of the Earth that the ocean is sort of pulled up in the direction towards the Moon and is let to go away from the Earth in the direction that is sort of through the Earth from the Moon. So what happens, that's why there are two tides per day roughly is because the water is being sort of pulled up in the direction towards the Moon and is sort of receding away on the direction away from the Moon. So as the Earth turns around, assume the Moon stays more or less in the same place. Cause it takes 28 days to go around the Earth. The Moon is. There's what's called a tidal deformation, as there's sort of the Earth gets kind of the oceans get put well, the Earth as well. But it's more obvious with the liquid ocean that it gets sort of deformed so that the thing is kind of like a squashed shape where it's kind of elongated towards the moon and away from the moon. Now Newton, for example, knew that explanation for the tides. And you might say, oh, that means the tides are solved. Not even close. You still can't compute tides at every place on the Earth. We've been interested in doing that for WolframAlpha for a long time. There are, I don't know, about 20,000 TIDE stations on the Earth that measure the tides in different places. And there are just a zillion different effects that affect the way the tides actually operate on the Earth. There is a cycle. How long is it? Is it an 18 year cycle? I think it may be, but it isn't for the same reason as the Saros cycle. But there's a cycle of a few decades in the tides on the Earth where you'll get the same configuration of tides. But if you look at the actual height of the water, by the way, the sort of naive version of just this tidal deformation would make the height be essentially a sine wave over time. But it isn't that. It's much more complicated. And there are places where there are features of the ocean, like the Bay of Fundy is a famous one, where there are very complicated and large tides. There are other places where there are quite small tides. It's really hard to work out the tides and there still isn't good computation of the tides for every place on Earth. And sometimes people, people really care about tides from time to time. For example, the D day adventure of 1944 in World War II, when British and Americans kind of took back the beaches in France that was critically dependent on tides to make that work out. And in fact, one of the strange consequences of that fact, that that military operation depended a lot on being able to predict the tides, meant that when we were first using tide data in things like WolframAlpha, we had this strange interaction with the British Admiralty which they said, oh no, our tide tables are classified. It's like, that doesn't make any sense. The tides repeat every number of years. So we can just go back to the tide tables from the 1800s that were measured and so on, and we'll have a very good approximation of the tides today. So like, okay, I guess that's true, but the Tides have significant. If you're going to make a, you know, if you're going to invade from the sea, so to speak, then it really matters whether what the tide is likely to be at that time. So people have really cared about that and for other reasons too. But the computation of tides has been really difficult. I don't think that the. I don't know of interactions between the climate and features of the tides. I don't think the tides are. I don't know. Actually, it's a good question. As sea level rises and so on, what effect that has on the tides, I'm not sure. Let's see. Technic asks about experiments reflecting lasers from the Moon. Before the Apollo missions, there was certainly radar, radio bouncing off the Moon. I don't think there were at least good laser experiments because the Moon, it looks pretty bright in the sky. But the albedo of the Moon is I think about 0.1 or something. So albedo is if you shine light on something, how much of that light is reflected off? But on the Moon, even though the Moon on its surface, the rock is quite dark, but it is illuminated against the night sky. It looks bright to us even though only maybe a tenth of the light that's falling on the Moon is actually being reflected from the Moon. But it also has the feature that it is mostly diffuse reflection. So there's nothing that reflects it back like a mirror. You know, when light falls on a surface, there are several different ways in which it gets reflected back. If it's a mirror, then light comes in at a certain angle and it bounces off the mirror and comes back at the sort of complementary angle. The thing that's special about these retroreflectors is that the light comes in and it is reflected back exactly the way it came. And I'm now having to think about how the geometry of that works. But the idea is you have a sphere of glass and how does that work? It just concentrates all that. I mean, it's kind of just like a mirror, but it's more effective than a mirror. And I'm blanking on exactly why that is right now. But there are these sort of two fundamentally different ways in which light gets reflected from things. One is what happens with things like metals where the light comes in and it excites a bunch of electrons in the metal and the light gets re emitted on that sort of complementary angle. The other thing that happens is diffuse reflection where light comes in and then the light is kind of emitted at all angles, so called Lambertian reflectors. Where. And that's what your average kind of random non shiny surface is, a diffuse reflector. And the moon rock on the moon, lunar regolith is pretty much a diffuse reflector like that. So when light comes in, most of the light is scattered off at at all sorts of different angles. It doesn't come back at a specific angle, which means there's much less intensity from the light in the light. You know, if you shine a laser in one direction, there's much less intensity because you don't get a reflection of the laser light right back the way it came. You just get it scattered in all directions. I mean, the thing that happens with light like that is when surfaces are shinier, what's happening is more of the light is being concentrated back in that kind of specular reflection, mirror like reflection direction. So you go from this completely diffuse thing where light goes in all directions to something where it's completely concentrated like a mirror. And you end up getting these kinds of different effects of the amount of light that's concentrated around the direction of the mirror like direction, and that determines the amount of shininess. In computer graphics, there's lots of parameterization of this. It's usually called the specularity is how much kind of the sort of highlight of kind of mirror like reflection there is. And there are often one sort of. One can look at these properties of materials in terms of things like the power laws for the amount of light as a function of angle relative to its precise reflection direction. And that's in modern times with GPUs there are a whole technology of so called shaders, which are little pieces of code that sort of say, given that light in this simulated scene is falling in this direction, you have this spotlight that's shining on this thing. What will be the actual sort of effect of that light when it is reflected from this object? And some objects are really complicated. Like skin is really complicated because the light, it's a little bit translucent. Light goes a little distance into skin and then it's not necessarily scattered or reflected from the surface of the skin, it's reflected from below the surface of the skin. Things like hair are really complicated because that hair is comparable in size to wavelength of light. And all kinds of complicated effects happen where you're not just getting the light from sort of the surface of the hair, so to speak. That's what in sort of early computer graphics you would have sort of hair that looks like plastic because it was just like the light was being kind of reflected from the surface. It's a much more complicated thing to do. Hair and fur and so on. It's a feature of the world of computer graphics. It always used to be the case at the annual Siggraph conference. There would be these kind of. These movies shown and they would have, okay, this year. I remember I was there the year that fur got solved. The year before that, fur always looked terrible in computer graphics. And then finally fur got solved and another year fabric got solved. You know, kind of being able to represent kind of the sort of the folds and so on in fabric and different things. Another year plants got solved. And so there are these different kind of tricks that in some cases are very tied into the physics of what's going on. In other cases, they're more just sort of tricks of computer graphics. But in modern times, that's kind of the technology is to use GPUs to make these shaders to basically simulate what happens to lots of idealized light rays and so on, hitting objects and such like. Okay, let's see. Well, one last question here. Jamie is asking, do other planets experience eclipses? Yes, as I mentioned, in the case of Mars, the geometry is such that the moons of Mars are really small compared to the disk of the sun. But most other planets, their moons orbit in the plane of the ecliptic, which means that every. Every month. By month, I mean, orbital period of their moon. Every month, the moon is between the surface of the planet and the sun. We're special in that. That happens rarely. I believe, on Jupiter, as I was saying before, I believe on Jupiter, Ganymede covers the disk of the sun. From Jupiter, sun looks pretty small in the sky of Jupiter. Jupiter. Jupiter's a lot further away than the Earth. And I think for exoplanets, well, I don't think any exomoons have been discovered yet. I mean, it's unquestionably the case that there are moons around exoplanets. And you can ask kind of what happens in those cases. Something would be fun to simulate. And I don't think I've ever seen this is. When you're a planet orbiting a binary star system, you'll have a bunch of much funkier behavior with respect to eclipses. Because if you've got two stars in your sky, there's sort of the question of, well, you know, maybe there's a time. I mean, most planets in binary star systems, exoplanets, they orbit far away from where the stars are. They're not like weaving in between the stars. But that would be even more exotic. And you can imagine kind of eclipses in that case. I suspect that most of those weaving type planets won't have stable orbits over the course of a billion years. They will have crashed into their star at some time or another or something like that. All right, well, we should wrap up there. As I said, I was yakking about eclipses here because I just put out this little book that is actually mostly things I wrote from the 2017 eclipse and modernized for this eclipse. It's kind of fun. It's got all kinds of cool graphics about, I think, let's see if there's one here that's a cool graphic about the actual sort of the shadow of the moon and that's the topography of the particular place in Wyoming where I happened to see the 2017 eclipse. And anyway, this book was my attempt to kind of trace the history of how it became possible to predict when the eclipse would occur with our kind of final result of the website that is already up for this eclipse. Precisioneclipse.com that should be able to compute the time of the eclipse within one second for anybody who's going to be in the path of totality. By the way, this little book also has a bunch of convenient maps of where the eclipse is going to go in different parts of the US and so on, and exactly at what time. I have to say, in just telling a personal story about this, the 1991 eclipse which went through Mexico, I decided to go see this eclipse. And it was kind of a slightly off the beaten track part of Mexico. And so I charted a plane and went down there with some other people. And we arrived there and this is 1991, before smartphones or any of those kinds of things which get to this lovely beach that had nobody else on it ready to see this eclipse. And I knew what time precisely the eclipse should occur in the particular position because I've been looking at the aviation maps and so on and seeing where could we land and where would it be relative to the maximum period of the eclipse and all that kind of thing. And I realized we're on this beach and nobody has an accurate timekeeping device. So the most accurate way to know the time is to just watch when the eclipse started. And that was obviously by the time of the 2017 eclipse. Everybody had a smartphone that was accurate, that was all aligned to the central clock of the Internet and so on and not to the GPS satellites. I think it is that right. I don't know how the synchronization is done. It's probably done through the NTP Internet time protocol. But in any case, by that point, one knew locally what the time was. One didn't just have to look at the sky and see what, What was happening with the eclipse. I have to say I really find it striking when you see an eclipse that you realize people have been seeing these things throughout human history. And in the early, you know, much earlier times, it must have been really weird to see an eclipse. Not least because I doubt that any human had seen more than one total solar eclipse in their lifetime until quite modern times. I mean, before things were known about where the eclipses were and people could travel around the earth and so on, people will have seen at most one eclipse in their life. And so that, I mean, I'm sort of surprised that there isn't more mythology that developed from, look, I saw this eclipse when I was 11 years old or something and this weird thing happened and then you're going to tell stories about it your whole life. And most people you tell those stories to will never see an eclipse. And so it will be this kind of mysterious thing that happened at one time. And I think it's sort of the fact that eclipses are predictable is kind of a great kind of feather in the cap of science, that it's a thing which we really can predict. There are lots of things about the world that we can't really predict, some of them which I think are fundamentally unpredictable based on things like computational irreducibility and so on. But eclipses are one where we got it. This is a thing where science has been successful. We can, with mathematical science, we can really predict eclipses to within one second now. So. Okay, I should wrap up there. Thanks for joining me. This was a bit of a one topic eclipse session, but we'll chat about other things other times. Thanks, bye for now.