When Exactly Will the Eclipse Happen? A Multimillennium Tale of Computation

Across the US but when exactly will the eclipse eclipse occur at a given location? Being able to predict astronomical events has historically been one of the great triumphs of exact science. But how well can it actually be done now? The answer is well enough that even though the edge of totality moves at just over a thousand miles an hour, it's possible to predict when it will arrive at a given location to within perhaps a second. And as a demonstration of this, the total eclipse, back in 2017, we created a website to let anyone enter their geolocation or address and then immediately compute when the eclipse would reach them, as well as generate many pages of information. And let me just show you that. This is our website, precisioneclipse.com, you type into the input field where you are and it produces a picture that shows the kind of time lapse of what the sun will look like as the moon bites into it and, and contains other information. And then there's a whole giant notebook of further detailed information about the eclipse. Well, predicting eclipse is an old business. These days it's easy to find out when the next solar eclipse will be. Indeed, it's built right into orphan language. It's just a function that tells you in the function solar eclipse that looks like this and you just, it tells you the time of greatest eclipse of the next eclipse. Type in solar eclipse, you get a date out. It's also easy to find out and plot where the region of totality will be. And just have a geolis plot of solar eclipse of total phase polygon and you've got this big stripe going across the US in the case of the upcoming eclipse, for example, determine the total area of totality, including, well, lots of ocean and some of Canada, that the total area of the eclipse will be about a third the area of the US Just can compute that with geo area of the solar eclipse polygon divided by Geo area of USA and the answer is 0.33. But computing eclipses is not exactly a new business. In fact, the antikythera device from 2000 years ago even tried to do it. Using 37 metal gears to approximate the motion of the sun and moon. Yes, with the earth at center, as was the theory of the time. To me, there's something unsettling and cautionary about the fact that the Antikythera device stands as such a solitary piece of technology, forgotten but not surpassed for more than 1600 years. But right there on the bottom of the device is. Let's see, I think we have a picture, a better picture of the device. But the, the. There's a. There's an arm that moves around. Here's the. Here's the device. It's an arrangement of gears and so on. There's an arm that moves around, and when it points to. Well, it's written as an h, but it's actually a capital eta or a capital sigma marking it indicates a possible sun or moon, actually. Yes, that's right. Capital H is an eta. Possible sun or moon eclipse. So the way of setting dates on this device is a bit funky. After all, modern calendar would not be invented for more than 1500 years after this device was produced. But if one takes, for example, the simulation from our orphan demonstrations project, which was calibrated back in 2012 when the demonstration was created, and turns the crank to set the device for April 8, 2024, one gets this picture that I'm showing here of this kind of clockwork thing with arms pointing to a place where the sun and the moon line up, corresponding to an eclipse. So the arm points right at an eta, just as it should for a solar eclipse. And sort of amazing to see this computation successfully happen on a device that was designed 2,000 years ago. By the way, that's a picture of the device as it exists in a museum in Athens. And actually, a few years ago, I happened to drop in on that device, and I think that's a picture of me with. With the device. So you can see roughly how large it is. Well, the results of predicting eclipses are a lot more accurate today, though. Strangely, despite all the theoretical science that's been done, the way we actually compute the position of the sun and moon is conceptually very much like the gears and effectively epicycles of the antikythera device. It's just that now we have the digital equivalent of hundreds of thousands of gears. Okay, so first of all, why do eclipses happen? So a total solar eclipse occurs when the moon gets in front of the sun from the point of view of a particular location on the earth. And it so happens that at this point in the Earth's history, the moon can just block the sun because it has almost exactly the same angular diameter in the sky as the sun, about 0.5 of a degree, or 30 arc minutes. So when does the moon get between the Earth and the sun? Well, basically every time there's a new moon, I. E. Once every lunar month. And this is sort of the. This is sort of the. The rough configuration. But we know that there isn't an eclipse every month. So how come? Well, actually, in the analogous situation, there's a picture, by the way, here, people who are just listening to this. Sun on the left, then Moon in the middle, then Earth on the right. In the analogous situation of Ganymede and Jupiter, there is an eclipse every time Ganymede goes around Jupiter, which happens to be about once a week. Like the Earth, Jupiter's orbit around the sun lies in a particular plane, the plane of the ecliptic. And it turns out that Ganymede's orbit around Jupiter also lies in essentially the same plane. So every time Ganymede reaches the New Moon position, or an official astronomy parlance, when it's aligned in syzygy and spelled syzygy, it's in the right place to cast its shadow onto Jupiter and to eclipse the sun wherever that shadow lands. I mean, from Jupiter. Ganymede appears about three times the size of the Sun. But our Moon is different. Its orbit doesn't lie in the plane of the ecliptic. Instead, it's inclined at about 5 degrees. How it got that way is unknown, but it's presumably related to how the Moon was formed. But that 5 degrees is what makes eclipses so comparatively rare. They can only happen when there's a new Moon configuration of syzygy right at a time when the Moon's orbit passes through the plane of the ecliptic. So let's make a picture of how that works. Here's the Moon going around the Earth. It's kind of a spoke, a series of spokes shown here. And it's colored red whenever it goes through the plane of the ecliptic. It's like a wheel that's been tipped a bit. And when the. When the. When the wheel. When the spokes on the wheel are horizontal, they're going through the plane of the ecliptic. Well, now let's look at what happens over the course of about a year. So here we are looking at the animate, but we don't. We're looking at the dot for where the Moon is each day. And the dot is redder if the Moon is closer to the plane of the ecliptic that day. So if this picture. So what you see in this picture is a kind of a set of curlicues of the Moon going around the more or less circular orbit of the Earth. If the picture was drawn to scale, you'd barely be able to see the Moon's orbit, and it wouldn't ever seem to go backwards like it does in the picture here. But this gives the idea that there are places where in that orbit, There are places where the moon Is in the plane of the ecliptic, so it's color red here and out of the plane of the ecliptic, so it's lighter colored. So now we can start to see how eclipses work. The basic point is that there's a solar eclipse Whenever the moon is both positioned between the earth and the sun, and it's in the plane of the ecliptic. So in this picture, those two conditions correspond to the moon being as far towards the center and as red as possible. Well, so we're only so far showing the position of the exaggerated moon once a day. But let's make things a bit clearer. Show it four times a day. And now prune out cases where the moon Isn't even roughly aligned with the sun. So here's the picture that we get, and we see in this picture Kind of a bunch of places where the moon is sort of Inside the orbit of the earth. And there are just a couple of places in the year where it's shown red, Corresponding to it being in the plane of the ecliptic. So, in fact, in this particular case, There are two points where the moon is lined up and in the plane of the ecliptic, so that it's shown in red here. And these points will correspond to solar eclipses. So in different years, the picture looks slightly different, Essentially because the moon is starting At a different place in its orbit at the beginning of the year. So here are some sort of schematic pictures For a few successive years with different kind of positions for where the moon is. And it's every lunar month, Being between the earth and the sun, and sometimes being shown red here, Meaning that the moon happens to be in the plane of the ecliptic at that particular new moon position. So in this picture, it's not easy to see exactly when the eclipses occur. It's also not possible to tell which are total eclipses, where the moon is exactly lined up, and which are only partial eclipses. But there is at least an indication that. And for example, we can tell here that there are eclipse seasons in different parts of the year when eclipses tend to happen. Okay, so what does the real data look like? So here's a plot for 20 years in the past and 20 years in the future, Showing the actual days in each year when total and partial eclipses occur. So there's a whole grid of new moons here shown in gray. And then there are sort of places where there are total eclipses Shown in red and sort of a Somewhat orderly but somewhat random arrangement of places where there are total and partial eclipses. So there's kind of a successive drift between successive years, which is just that the lunar month, which is 29.53 days, doesn't line up with the year. So the moon doesn't go through a whole number of orbits in the course of the year, with the result that at the beginning of a new year, the moon is in a different phase. But we can see from the picture there's quite a lot of regularity in the general times at which eclipses occur. For example, there are usually two eclipses in any given year, though there can be more. And in 0.2% of years, there can be as many as five eclipses as there were last in 1935. Okay, so let's look at more detail. Let's plot the time differences in fractional years between all successive solar eclipses for 100 years in the past and 100 years in the future. And we see that there's kind of a collection that are very much close to one half a year. There's another at 0.4 times the year, another at 0.1 times the year. But sort of random. Which one of those three choices actually occurs for each successive eclipse. Let's plot the same thing for total solar eclipses. And there's a certain amount of regularity, but there's also lots of fine structure and irregularities in this picture. And being able to correctly predict all these details has basically taken science the better part of a few thousand years. All right, well, let's start with the ancient history. It's hard not to notice an eclipse, and presumably even from the earliest times, people did. But were eclipses just reflections or omens associated with random goings on in the heavens? Perhaps in some kind of soap opera among the gods? Or were they things that could somehow be predicted? A few thousand years ago, it wouldn't have been clear what people like astrologers could conceivably predict. When will the moon be at a certain place in the sky? Will it rain tomorrow? What will the price of barley be? Who will win a battle? Even now, we're not sure how predictable all of these are. But the one clear case where prediction and exact science have triumphed is astronomy, at least as far as the western tradition is concerned. It all seems to have started in ancient Babylon, where, for many hundreds of years, careful observations were made. And in keeping with the ways of that civilization, detailed records were kept. And even today, we still have thousands of daily official diary entries written on what look like tiny chicken scratches. That are preserved on little clay tablets. So it says things like night of the 14th, cold north wind. Moon was in front of Alpha Leones from 15th to 20th. River Rose half a cubit, Barley was 1 Kerr 5 seat 25th, last part of night. Moon was one cubit, eight fingers behind Epsilon Leonis 28th, 74 degrees after sunrise, solar eclipse and so on. So if one looks at what happens on a particular day, one probably can't tell much. But by putting observations together over years or even hundreds of years, it's possible to see all sorts of repetitions and regularities. And back in Babylonian times, the idea arose of using these to construct an ephemeris, a systematic table that said where a particular heavenly body such as the moon was expected to be at any particular time. Needless to say, reconstructing Babylonian astronomy is a complicated exercise in decoding what's by now basically an alien culture. A key figure in that effort was a certain Otto Neugebauer, who happened to work down the hall from me at the Institute for Advanced Study in Princeton in the early 1980s. I would see him almost every day, a kind of quiet white haired chap with a sort of twinkle in his eye. And just sometimes I glimpse his huge filing system of index cards, which I now realize was at the center of understanding Babylonian astronomy. Well, one thing the Babylonians did was to measure surprisingly accurately the repetition period for the phases of the Moon, the so called synodic month or lunation period of, of about 29.53 days. And they noticed that 235 synodic months was very close to 19 years. So that about every 19 years dates and phases of the moon repeat their alignment, forming a so called metonic cycle, named after Meton of Athens, who described it in about 432 BC. It probably helps that the random constellations in the sky form a good pattern against which to measure the precise motion of the Moon. It kind of reminds one of the modern fashion of kind of wearing fractals to make motion capture for movies easier. But the Babylonians noticed all sorts of details of the motion of the Moon. They knew about its anomaly, its periodic speeding up and slowing down in the sky, known to be a consequence of its slightly elliptical orbit. And they measured the average period of this, the so called anomalistic month, to be about 27.55 days. They also noticed that the Moon went above and below the plane of the ecliptic, now known to be because of the inclination of its orbit, with an average period the so called Draconic month that they measured to be about 27.21 days. And by 400 BC they'd notice that every so called saros of about 18 years and 11 days, all these different periods essentially line up. 223 synodic months, 239 anomalistic months and 242 draconic months, with the result that the Moon ends up at the same position relative to the Sun. And this means that if there was an accuracy one Saros, then one can make the prediction that there's going to be an eclipse at the next Saros as well. Well, one was absolutely precise about it. There are all sorts of effects that prevent precise repetition at each Saros. But over timescales of more than 1300 years, there are in fact still strings of eclipses that separated from each other by one Saros. So over the course of such a Saros series, the locations of the eclipses effectively scan across the Earth. The upcoming eclipse that we're going to see on April 8th is number 30 in a series of 71 that began in 1501 AD with an eclipse near the North Pole and will end in 2763 with an eclipse near the South Pole. Any given moment in time will be in the middle of quite a few Saros series. Right now it's about 40. And successive eclipses will always come from different series. But knowing about the Saros cycle is a great first step in predicting eclipses. And it's, for example, what the Antikythera device uses. In a sense, it's a quintessential piece of science. Take many observations, then synthesize a theory from them, or at least a scheme for computation. It's not clear what the Babylonians thought about sort of abstract formal systems, but the Greeks were definitely into them. And by 300 BC, Euclid had defined his abstract system for geometry. So when someone like Ptolemy did astronomy, they did it a bit like Euclid, effectively taking things like the Saros cycle as axioms and then proving from them often surprisingly elaborate geometrical theorems, such as that there must be at least two solar eclipses in a given year. So Ptolemy's almages from around 150 A.D. is an impressive piece of work, containing, among other things, many quite elaborate procedures and explicit tables for predicting eclipses. And yes, even in later printed versions, numbers are still representing, rather confusingly, by letters, as they were in ancient Greek. And you can see sort of tables here and geometrical diagrams and then later printed versions kind of the table of when the eclipses should occur with these kind of Greek numbers. So in Ptolemy's astronomy, Earth was assumed to be at the center of everything. But in modern terms, that just meant he was choosing to use a different coordinate system, which didn't affect most of the things he wanted to do, like working out the geometry of eclipses. And unlike the mainline Greek philosophers, he wasn't trying to make a fundamental theory of the world. He just wanted whatever epicycles and so on, he needed to fit what he observed. The dawn of modern science. So for more than a thousand years, Ptolemy's theory of the Moon defined the state of the art. In the 1300s, Ibam Al Shatir revised Ptolemy's models, achieving somewhat better accuracy. And in 1472, Regiomontanus Johannes Muller, systematizer of trigonometry, published more complete tables of as part of his launch of what was essentially the first ever scientific publishing company. But even in 1543, when Nicholas Copernicus introduced his sun centered model of the solar system, the results he got were basically the same as Ptolemy's, even though his underlying description of what was going on was quite different. It's said that Tycho Brahe got interested in astronomy in 1560 at age 13, when he saw a solar eclipse that had been predicted. And over the next several decades, his careful observation uncovered several effects in the motion of the moon, such as speeding up just before a new moon, before a full moon rather, that eventually resulted in perhaps a factor of five improvement in the prediction of the position of the Moon. To Tycho, eclipses were key tests, and he measured them carefully and worked hard to be able to predict their timing more accurately than to within a few hours. He himself never saw a total solar eclipse, only partial ones. That's a page from Ptolemy's, from Tycho's tables. Eclipses Lunae unum etbajinti solis novam. So I guess 21 lunar eclipses, with my knowledge of Latin probably isn't sufficient to do real time translation of this. In any case, a table of eclipses for a series of years like 1612 and so on. Well, armed with Tycho's observations, Johannes Kepler developed his description of orbits as ellipses and introducing concepts like inclination and the eccentric anomaly. And in 1627 he finally produced his Rudolfwein tables, which got a lot of things right and that had been got wrong before, and included all sorts of detailed tables of lunar positions as well as vastly better predictions for eclipses. So this is Tabularum rudolphi astronomy astronomicarium Horace Tertia, part three. And this is De Ecliptibus Solis at Lunae on Eclipses of the sun and Moon. And there's a table of them there. Well, using Kepler's Rudolfine tables and a couple of other calculations, the first known actual map of a solar eclipse was published in 1654. And while there are sort of some charming inaccuracies in overall geometry, the geometry of the eclipse isn't bad. This shows an eclipse going kind of across Europe and through towards India. And in the original picture, the continent of Africa is very deformed relative to what we now know it is. And somehow the Mediterranean looks a bit smaller in the rendering of geography from 1654. So whether it was Ptolemy's epicycles or Kepler's ellipses, there were plenty of calculations to do in determining the motions of heavenly bodies. And indeed the first known mechanical calculator, accepting the Antikythera device, was developed by a friend of Kepler's, presumably for that purpose. But there wasn't really a coherent underlying theory. It was more a matter of describing effects in ways that could be used to make predictions. So it was a big step forward in 1687 when Isaac Newton published his Principia and claimed that with his laws of motion and gravity it should be possible essentially from first principles, to calculate everything about the motion of the Moon. Charmingly, in his Theory of the World section of that book, he simply asserts as his proposition 22 that all motions of the Moon follow from the principles which we have laid down. Newton was proud of the fact that he could explain all sorts of effects on the basis of his new theory. And it reminds me of our physics project. But when it actually came to actually calculating the detailed motion of the Moon, he had a frustrating time. And even after several years he still couldn't get the right answer. Later editions of the Principia add in the admission that actually the apse of the Moon is about twice as swift. In other words, his answer was wrong by a factor of two. Still, in 1702, Newton was happy enough with his results that he allowed them to be published in the form of 20 page booklet on the Theory of the Moon, which proclaimed that by this theory what by all astronomers was thought most difficult and almost impossible to be done. The excellent Mr. Newton hath now affected, viz. To determine the Moon's place even in her quadratures, in, in all other parts of her orbit besides the syzygies, so accurately by calculation that the difference between that and her true place in the Heavens shall be scarce two minutes. So Newton didn't actually explain his methods, and still not clear precisely what he did or how mathematically rigorous it was or wasn't. But his booklet effectively gave a step by step algorithm to compute the position of the Moon. He didn't claim it worked at the syzygies, that is when the Sun, Moon and Earth are lined up for an eclipse. Though his advertised error of 2 arcminutes was still much smaller than the angular size of the Moon in the sky, which is about 30 arcminutes. But it wasn't the eclipses that were the focus then. It was a very practical problem of Newton's day. Knowing the location of a ship out in the open ocean. It's possible to determine what latitude you're at just by measuring how high the sun gets in the sky. But to determine longitude, you have to correct for the rotation of the Earth, and to do that you have to accurately keep track of time. But back in Newton's days, the clocks that existed simply weren't accurate enough, especially when they were being tossed around on a ship. But particularly after various naval accidents, the problem of longitude was deemed important enough that the British government in 1714 established a board of Longitude to offer prizes to help get it solved. One early suggestion was to use the regularity of the moons of Jupiter, discovered by Galileo, as a way to tell time. But it seemed that a simpler solution, not requiring a powerful telescope, might just be to measure the position of our moon, say relative to certain fixed stars, and then to back compute the time from this. But to do this, one had to have an accurate way to predict the motion of the Moon, which is what Newton was trying to provide. In reality though, it took until the 1760s before tables were produced that were accurate enough to be able to determine time to within a minute arc minute and thus to within a distance of 15 miles or so. And it so happens that right around that time a marine chronometer was invented that directly was able to keep good time. Well, the three body problem, One of Newton's great achievements in the Principia was to solve the so called two body problem and to show that with an inverse square law of gravity, the orbit of one body around another must always be what Kepler had said, an ellipse. In a first approximation, one can think of the Moon as just orbiting the Earth in a simple elliptical orbit. What makes everything difficult is, is that that's just an approximation. Because in reality there's also a gravitational pull on the Moon from the sun and because of this, the Moon's orbit is no longer a simple fixed ellipse. And in fact it ends up being much more complicated. There are a few definite effects one can describe and reason about. The eclipse gets stretched when the ellipse gets stretched. When the Earth is closer to the sun in its own orbit. The orientation of the ellipse precesses like a top as a result of the influence of the sun that there's no way in the end to work out the orbit by pure reasoning. So there's no choice but to go into the mathematics and start solving the equations of the three body problem. In many ways, this represents a new situation from science for science. In the past, one hadn't ever been able to go far without having to figure out new laws of nature. But here the underlying laws were supposedly known, courtesy of Newton. Yet even given these laws, there was difficult mathematics involved in working out what behavior they implied. Over the course of the 1700s and 1800s, the effort to try to solve the three body problem and determine the orbit of the Moon was at the center of mathematical physics and attracted a veritable who's who of mathematicians and physicists. An early entrant was Leonard Euler, who developed methods based on trigonometric series, including much of our current notation for that show you here. And Euler's works contain many immediately recognizable formulas. We can see here, well, lots of cosines and so on. One over one plus k, cosine V or u. The only thing that looks weird is that in the printing, the what is an s looks more like an F in the actual font that's used there. So in the mid-1740s, there was a brief flap, also involving Euler's competitors Clairau and Dalembert, about the possibility the inverse square law for gravity might be wrong. But the problem turned out to be the calculations. And by 1748, Euler was using sums of about 20 trigonometric terms and proudly proclaiming that that the tables he'd produced for the three body problem had predicted the time of a total solar eclipse to within minutes. Actually, he had said there'd be five minutes of totality, whereas in reality there was only one. But he blamed this error on incorrect coordinates he'd been given for the city of Berlin. Mathematical physics moved rapidly over the next few decades, with all sorts of now famous methods being developed, notably by people like Lagrange. And by the 1770s, for example, Lagrange's work was looking, well, kind of just like it would from a modern calculus book. Or from a Walton alpha step by step solution. Well, particularly in the hands of Laplace, there were increasingly obvious success in deriving the observed phenomena of what he called celestial mechanics from mathematics and in establishing the idea that mathematics alone could indeed generate new results in science. At a practical level, measurements of things like the position of the Moon had always been much more accurate than calculations. But now they were becoming comparable driving advances in both. Meanwhile, there was increasing systematization in the production of ephemeris tables. And in 1767 the annual publication began of what was for many years the standard, the British Nautical Almanac. The almanac quoted the position of the moon to the arc second and systematically achieved at least arc minute accuracy. The primary use of the almanac was for navigation and it was what started the convention of using Greenwich as the prime meridian for measuring time. But right at the front of each year's edition of the almanac were the predicted times for eclipses in that year. In 1767, just two. Just two solar eclipses. The math gets more serious. So at a mathematical level, the three body problem is about solving a system of three ordinary differential equations that give the positions of the three bodies as a function of time. If the positions are represented in standard 3D Cartesian coordinates, R sub I equals the list Xi yi zi equations can be stated in a definite form. Here it's Mr. Double dot or R double prime. Let's say R double dot because that's Newton's Fluxion notation. Mr. Double Dot as a function of time. Let's say for the R being the position of the Moon equals minus capital g M sub the Moon, M sub the Moon. The let's say the sun and then the inverse square term, basically one over the distance between the sun and the moon with a vector in the direction between those two and then minus capital g times the product of the mass of the moon, the mass of the. Well actually I got it the wrong way around the way I've written out this equation. But let's say the earth there and then this one over inverse square law between them. So this is basically stating Newton's laws that mass times acceleration is equal to force. The force is the gravitational force and the left hand side Mr. Double Dot is the mass times acceleration. And we have three equations, one for each of the bodies describing the acceleration associated with each body and two terms in the force corresponding to the gravitational pull from each of the other bodies on that body. So the XYZ coordinates that we're using in these equations aren't what traditionally show up in astronomy? For example, in describing the position of the Moon, one might use longitude and latitude on a sphere around the Earth. Or given that one knows that the Moon has a roughly elliptical orbit, one might instead choose to describe its motions by variables that are based on deviations from such an orbit. In principle, it's just a matter of outer break manipulation to restate the equations with any given choice of variables. But in practice, what comes out is often long and complex and can lead to formulas that fill many pages. But okay, so what are the best kind of variables to use for the three body problems? Maybe they should involve relative positions of pairs of bodies or relative angles, or maybe positions in various kinds of rotating coordinate systems. Or maybe quantities that will be constant in a pure two body problem. Over the course of the 1700s and 1800s, many treatises were written exploring different possibilities. But in essentially all cases, the ultimate approach to the three body problem was the same. Set up the problem with the chosen variables, identify parameters that if set to zero, would make the problem collapse to some easy to solve form. Then do a series expansion in powers of these parameters, keeping just some of the terms. The calculations were difficult and people's results often didn't agree. And for example, in 1843, Ada Lovelace noticed that quotes in the solution of the famous problem of the three bodies, there are out of about 295 coefficients of lunar perturbations that have then recently been computed, only 101 that agree precisely both in signs and amounts with previous works. In other words, it was really hard to get those 295 coefficients right. And people, different people who'd computed them, had gotten different answers. By the way, Ada went on to say, rather farsightedly that the calculation of the position of the Moon was something that the analytical engine would be able to solve. The analytical engine being the. The conceptual machine from Charles Babbage that was a mechanical version of what we would now call the universal computer. Well, by the 1860s, Charles Delaunay had spent 20 years developing the most extensive theory of the Moon. Using series expansions, he identified five parameters. Oh, this is actually. I'm sorry, this is something I wanted to mention. This is what Ada actually wrote in her memoir about the analytical engine about the three body problem. But let's go on to Charles Delaunay. Here's what he spent 20 years developing. He identified five parameters with respect to which to do series expansions, eccentricities, inclinations, ratios, orbit sizes. In the end, he generated about 1800 pages full of algebraic formulas. And yes, he really needed Mathematica, but sorry, that was another 130 years before it arrived. But the sad fact for Delaunay was that despite all this effort, he didn't get terribly good answers. Eventually it became clear why. The basic problem was that Delaunay wanted to represent his results in terms of functions like sine and cosine. But in his computations he often wanted to do series expansions with respect to the frequencies of those functions. So here's kind of a minimal case. A series of sine omega plus delta T expanded with respect to delta. What we see is we get sine t omega plus t cosine T omega times delta minus the term of delta squared and the term of delta cubed and so on. Here's the problem. Take a look at the second term. It's T times delta times cosine T omega. The delta may be very small, but the t parameter standing for time. Well, if that delta times T, if T gets big enough, will eventually be big, even if delta is small. So if you want to make predictions very far out, if the time is small, then delta times time will still be small. But if you want to figure out what happens further in the future, that will be a problem, because eventually the delta times T term will get big, because the T will get big, and the higher order terms, which involve t squared and so on, will get even bigger. But unless the Moon is going to escape its orbit or something, the final mathematical expressions that represent its position can't have values that are too big. So in these expressions, the so called secular terms that increase with t with time must somehow cancel out. But the problem is that at any given order in the series expansion, there's no guarantee that will happen in a numerically useful way. And in Delaunay's case, even though with immense effort he went to the seventh order and beyond, it didn't. One nice feature of Delaunay's computation was that it was, in a sense entirely algebraic. Everything was done symbolically, and only at the very end were actual number numerical values of parameters substituted in. But even before Delaunay, Peter Hanson had taken a different approach, substituting numbers as soon as he could and dropping terms based on their numerical size rather than their symbolic form. So his presentations look a lot less pure. You can see in this picture there are a lot of T minus 1800 pieces where T is the time in years and 1800 is the year 1800. More difficult to tell what's going on. But as a practical matter, his results were much Better and in fact were used for many national almanacs from about 1862 to 1922, achieving errors as small as 1 or 2 arc seconds, at least over periods of a decade or so. Over longer periods, the errors could rapidly increase because of the lack of terms that have been dropped as a result of what amounted to numerical accidents. Well, both Delaunay and Hanson tried to represent orbits as series of powers and trigonometric functions, so called Poisson series. But in the 1870s, George Hill in the US National Almanac office proposed instead using as a basis numerically computed functions that came from solving an equation for two body motion with a periodic driving force roughly of the kind the sun exerts on the Moon's orbit. So a large scale effort was mounted and started in 1892. Ernest W. Brown, who had moved to the US but had been a student of George Darwin, Charles Darwin's physicist son, took charge of the project and in 1918 produced what would stand for many years as the definitive tables of the motion of the Moon. So Brown's tables consists of hundreds of pages like the one I'm showing that just have little pieces of formula on them. So ultimately representing the position of the Moon is a combination of about 1400 terms with very precise coefficients. He says right at the beginning of the tables that they aren't particularly intended for unique events like eclipses. But then he goes on and does a worked example of computing an eclipse from 381 BC that have been reported by Ptolemy from the Babylonian tablets, by the way. So it was an impressive indication of how far things had come. But ironically enough, the final presentation of Brown's tables had the same sum of trigonometric functions form that one would get from having lots of epicycles as Ptolemy had at some level. It's not surprising because any function can ultimately be represented by epicycles, just as it can be represented by Fourier or other series. But it's a strange quirk of history that such similar forms were used. Can the three body problem be solved? It's all well and good that one can find approximations to the three body problem, but what about just finding an outright solution like as a mathematical formula? Even in the 1700s there have been some specific solutions found, like Euler's code collinear configuration, three objects in a line, and Lagrange's equilateral triangle configuration that's very famous for the Lagrange points in the Earth Moon sun system, for example. But a century later, no further solutions have been found. And finding a Complete solution to the three body problem was beginning to seem as hopeless as trisecting an angle, solving the quintic or making a perpetual motion machine. And that sentiment was reflected, for example, in a letter Charles Babbage wrote to ada Lovelace in 1843 mentioning the horrible problem of the three bodies. Even though this letter was later misinterpreted by Ada's biographers to be about a romantic triangle, not the three body problem of celestial mechanics. Well, in contrast to the three body problem, what seemed to make the two body problem tractable was that its solutions could be completely characterized by constants of the motion quantities that stay constant with time, in that case notably the direction of the axis of the ellipse. So for many years, one of the big goals of the three body problem was to find constants of the motion. But in 1887, Heinrich Bronze showed that there couldn't be any such constants of the motion, at least expressible as algebraic functions of the standard XYZ, position and velocity coordinates of the three bodies. Then in the 1890s, Henri Poincare showed that there actually couldn't be any constants of the motion that were expressible as any analytic functions of the positions, velocities and mass ratios. One reason that was particularly disappointing at the time was that it had been hoped that somehow constants of the motion would be found in n body problems that would lead to a mathematical proof of the long term stability of the solar system. And as part of Poincare's part of his work, Poincare also saw something else, that at least in particular cases of the three body problem, there was arbitrarily sensory dependence on initial conditions, implying that even tiny errors in measurement could be amplified to arbitrarily large changes in predictive behavior, the classic chaos theory phenomenon. But having discovered that particular solutions of the three body problem could have this kind of instability, Poincare took a different approach that would actually be characteristic of much of pure mathematics. Going forward, he decided to look not at individual solutions, but at the space of all possible solutions. And needless to say, he found that for the three body problem this was very complicated. Though in his efforts to analyze it, he invented the fields of topology. Panquere's work all but ended efforts to find complete solutions to the three body problem. It also seemed to explain why the series expansions of Delaunay and others hadn't worked out. Though in 1912 Carl Sundman did show that, at least in principle, the three body problem could be solved in terms of an Infinite series, albeit one that converges outrageously slowly. But what does it mean to say that there can't be a solution to the three body problem? Galois had shown that there couldn't be a solution to the generic Quincy equation, at least in terms of radicals. But actually it's still perfectly possible to express the solution of the quantic equation in terms of elliptical hypergeometric functions. So why can't there be some more sophisticated class of functions that can be used to solve the three body problem? Well, here are some pictures of what actually happens to the three body problem with various initial conditions. Some of them are very straightforward. Some of them involve little sort of old fashioned phone cable like helices. Some of them look more like, some of them look more scribbly and so on. Well, looking at these gives some indication of why it's not easy to just solve the three body problem. Yes, there are cases where what happens is fairly simple, but there are also cases where it's not and where the trajectories of the three bodies continue to be complicated and tangled for a long time. Just recently wrote about using neural nets in AI to try and make these predictions, and it too can't do it. But, so what fundamentally is going on here? I don't think traditional mathematics is the place to look, but I think what we're seeing is actually an example of a general phenomenon I call computational irreducibility that I discovered in the 1980s in studying the computational universe of possible programs. Many programs, like many instances of the three body problem, behave in quite simple ways. But if you just start looking at all possible programs, it doesn't take long before you start seeing behavior like this, where you just start from a single sort of square at the top of the screen and then you evolve down. You maybe make a sort of a whole triangle of different behavior. And within that triangle, there are things that look very complicated and sometimes seemingly random. How can one tell what's going to happen? Well, one can just keep explicitly running each program and seeing what it does. But, but the question is, is there some systematic way to jump ahead and to predict what will happen without tracing through all the steps? The answer is that in general there isn't. And what I call the principle of computational equivalence suggests that pretty much whenever one sees complex behavior, there won't be. Here's the way to think about this. The system one's studying is effectively doing a computation to work out what its behavior will be. So to jump ahead, we'd in a sense have to do a more Sophisticated computation. But what the principle of computational equivalence says is that actually we can't. And that whether we're using our brains or our mathematics or a Turing machine or anything else, we're always stuck with computations of the same sophistication. So what about the three body problem? Well, I strongly suspect it's an example of computational irreducibility, that in effect the computations it's doing are as sophisticated as any computations that we can do. So there's no way we can ever expect to systematically jump ahead and solve the problem. We also can't expect to just define some new finite class of functions that can just be evaluated to give the solution. So I'm hoping that one day someone will rigorously prove this. There's some technical difficulty because the three body problem is usually formulated in terms of real numbers that immediately have an infinite number of digits. But to compare with ordinary computation, one has to require finite processes to set up initial conditions. Ultimately, one wants to show, for example, that there's a compiler that can go from any program, say for a Turing machine, and can generate instructions to set up initial conditions for a three body problem. So the evolution of the three body problem will give the same results as running that program, implying that three body problem is capable of universal computation. I have to say that I consider Newton, in a sense, very lucky. It could have been that it wouldn't have been possible to work out anything interesting from his theory without encountering the kind of difficulties he had with the motion of the moon. Moon, because one would always be running into computational irreducibility. But in fact there was enough computational reusability and enough that can be computed easily that one could see that the theory was useful in predicting features of the world and not getting wrong answers, like for the apps of the moon. Even if there were some parts that might take two centuries to work out or never be possible at all. Newton himself was certainly aware of the potential issue, saying that least if one was dealing with gravitational interactions between many planets, then to define these motions by exact laws, admitting of any easy of easy calculation exceeds, if I am not mistaken, the force of any human mind. And even today it's extremely difficult to know what the long term evolution of the solar system will be. Not particularly that there's sensory dependence on initial conditions. We actually have measurements that should be precise enough to to determine what will happen for a long time. The problem is we just have to do the computation, a bit like computing the digits of PI, to work out the behavior of the anybody Problem, that is our solar system. Existing simulations show that perhaps a few tens of millions of years in the future, nothing too dramatic can happen. But after that, we don't know. Planets could change their order. Maybe they could even collide or be ejected from the solar system. Computational irreducibility implies that at least after an infinite time, it's actually formally undecidable in the sense of Godel's theorem or the halting problem. What can happen? One of my children actually, when they were very young, asked me whether dinosaurs, when dinosaurs existed, the Earth could have had two moons for years. When I ran into celestial mechanics experts, I would ask them that question, and it was notable how difficult they found it. Most now say that at least at the time of the dinosaurs, we couldn't have had an extra moon. Though a billion years earlier, it's not clear. We used to only have one system of planets to study. And the fact that there were then nine of them used to be a classic philosopher's example of a truth about the world that just happens to be the way it is and isn't necessarily true. Like two plus two equals four. But now, of course, we know a lot about lots of exoplanets, and it's beginning to look as if there might be a theory that for things like how many planets a solar system with, for example, a star of the sun's size is likely to have at some level, there's presumably a process like natural selection, some configuration of planets that aren't fit enough to be stable, some configurations aren't fit enough to be stable, and only those that are survive. In biology, it's traditionally been assumed that natural selection adaptation may somehow what's led to the complexity we see. But actually, I suspect much of it is instead just a reflection of what happens in general in the computational universe, both in biology and in celestial mechanics. Now, in celestial mechanics, we haven't yet seen in the wild any particularly complex forms beyond a few complicated gap structures and rings and tumbling moons and asteroids. But perhaps elsewhere, we'll see things that, like those obviously tangled solutions to the three body problem, become closer to what we're used to in biology. It's remarkable how similar the issues are across so many different fields. For example, the whole idea of using perturbation theory and series expansions that has existed since the 1700s in celestial mechanics is now also core to quantum field theory. And just like in celestial mechanics, there's trouble with convergence. Maybe one should try renormalization or resumation in celestial mechanics. Those are ideas from Quantum field theory. And in the end one begins to realize that there are phenomena, no doubt like turbulence or the three body problem, that are inevitably involve more sophisticated computations and that need to be studied not with traditional mathematics of the kind that was so successful for Newton and his followers, but with the kind of science that comes from exploring the computational universe approaching modern times. Let's get back to the story of the motion of the Moon between Brown's tables and Poincare's theoretical work. By the beginning of the 1900s, the general impression was that whatever could reasonably be computed about the motion of the Moon had been computed. Occasionally there were tests like for example, in 1925 there was a total eclipse visible in New York City. The New York Times perhaps over dramatically said that scientists were tense, wondering whether they or the moon is wrong as eclipse lags five seconds behind. The fact is that a prediction accurate to 5 seconds was remarkably good and we can't do all that much better even today. By the way, the actual article talks extensively about Professor Brown as well as how about the about how the eclipse might disprove Einstein and corroborate the existence of coronium, but doesn't elaborate on the supposed prediction error. Just one of those early examples of how the clickbait headline isn't necessarily followed by the content of the article. Well, as a practical matter, Brown's tables were not exactly easy to use. To find the position of the Moon from them required lots of mechanical desk calculator work as well as careful transcription of numbers. And this led Leslie Comrie in 1932 to propose using punch card based IBM Holorith automatic tabulator. And with the help of Thomas Watson, CEO of IBM, what was probably the first scientific computing laboratory was established to automate computations from Brown's tables. Actually, when I was in elementary school in England in the late 1960s, before electronic calculators always carried around along with my slide rule, a little book of four figure mathematical tables. And I think I found it odd that such a book would have an author. And perhaps for that reason I still remember the name L.J. comrie. That was the person who originally suggested using an automatic tabulator to compute motion of the Moon. Well, by the 1950s, the calculations in Brown's tables were slowly being rearranged and improved to make them more suitable for computers. And then with John F. Kennedy's 1962 we choose to go to the Moon speech, there was suddenly urgent interest in getting the most accurate computations of the Moon's position. As it turned out, Though it was basically a tweaked version of Brown's tables running on a mainframe computer that did the computations for the Apollo program. At first, computers were used in celestial mechanics purely for numerical computation. But by the mid-1960s, there were also experiments in using them for algebraic computation and particularly to automate the generation of series expansions. Wallace Eckert at IBM started using Formak to redo Brown's tables while in Cambridge. David Barton and Steve Bourne. Steve Bourne was later the creator of the Bourne Shell SH in Unix. Built their own Camel C A M A L computer algebra system to try extending the kind of thing Delaunay had done. And by 1970, Delaunay's seventh order calculations had been extended to 20th order. When I myself started to work on computer algebra in 1976, primarily for computations and particle physics, I'd certainly heard about camel, but I didn't know what it had been used for beyond vaguely celestial mechanics. And as a practical theoretical physicist, practicing theoretical physicist in the late 1970s, I have to say that the problem of the Moon that had been so prominent in the 1700s and 1800s and by then fallen into complete obscurity. I remember, for example, in 1984 asking a certain Martin Gutsweiler, who was talking about quantum chaos, what his main interest actually was. And when he said the problem of the Moon, I was flawed. I didn't know there still was any problem with the Moon. As it turns out, in writing this piece, I found out that Gustweiler was actually the person who took over from Eckart and spent nearly two decades working on trying to improve the computations of the position of the Moon. Well, why not just solve it? Traditional approaches to the three body problem come very much from a mathematical way of thinking. But modern computational thinking immediately suggests a different approach. Given the differential equations for the three body problem, we not just directly solve them. And indeed, in the WOLM language there's a built in function ND solve for numerically solving systems of differential equations. So what happens if one just feeds in the equations to the three body problem? Here are the equations written out. I already mentioned roughly what's in these now. Okay, let's set the masses to random values here. M1, M2 M3 equals random real of something. Let's define the initial position and velocity for each body to be random as well. Now we can just use nd solve to get the solutions and it'll just say R1 goes to an interpolating function, gives them as implicit approximate numerical functions of t and now we can just plot them and we've got a solution to the three body problem. Just like that, wiggling around. Well, obviously this is using the Wolfram language in a huge tower of modern technology, but, but would it have been possible, even right from the beginning, for people to generate direct numerical solutions to the three body problem rather than doing all that algebra? Back in the 1700s, Euler already knew what's now called Euler's method for finding approximate numerical solutions to differential equations. So what if he just used that method to to calculate the motion of the moon? The method relies on taking a sequence of discrete steps in time. And if he'd use a step size of a minute, then he would have had to take 40,000 steps to get results for a month. But he should have been able to successfully reproduce the position of the moon to about a percent. If he tried to extend to three months, however, then he would already have had at least a 10% error. Any numerical scheme for solving differential equations in practice eventually builds up some kind of error. But the more one knows about the equations when solving their expected solutions, the more one's able to pre process and adapt things to minimize the error. ND solve has enough automatic adaptivity built into it that it'll do pretty well for a surprisingly long time on a typical three volley problem. It helps that the Wolfram language and NDSOLVE can handle numbers with arbitrary precision, not just machine precision. But if one looks, say, the total energy of the three body system, which one can prove from the equations, should stay constant, then one will typically see an error that slowly builds up in it. One can avoid this if one effectively does a change of variables in the equations to factor out energy. And one can imagine doing a whole hierarchy of algebraic transformations, in a sense, give the numerical scheme as much help as possible. And Indeed, since the 1980s, that's exactly what's been done in practical work on the three body problem and the Earth Moon sun system. So in effect, it's a mixture of the traditional algebraic approach from the 1700s and 1800s together with modern numerical computation. A real Earth, Moon, sun problem. Okay, so what's involved in solving the real problem of the Earth Moon sun system? The standard three body problem gives a remarkably good approximation to the physics of what's happening, but it's obviously not the whole story. For a start, the Earth isn't the only planet in the solar system. And if one's trying to get sufficiently accurate answers, one's going to have to take into account the gravitational effect of Other planets. The most important is Jupiter. And its typical effect on the orbit of the Moon is about at the 10 to the minus 5 level. One part in 100,000, sufficiently large that, for example, Brown had to take it into account in his tables. The next effect is that the Earth isn't just a point mass or even a precise sphere. Its rotation makes it bulge at the equator, and that affects the orbit of the Moon at about the one part in a million level. Orbits around the Earth ultimately depend on the full mass distribution and gravitational field of the Earth, which is, by the way, what Sputnik 1 was nominally launched to map. And both this and the reverse effect from the Moon come in at about the 1 part in 100 million level. At the 1 part in a billion level. There are then effects from tidal deformation, solid tides on the Earth and Moon, as well as from gravitational redshift and other general relativistic phenomena. To predict the position of the Moon as accurately as possible, one ultimately has to have at least some model for these various effects. But there's a much more immediate issue to deal with. One has to know the initial conditions for the Earth, sun and Moon. Or in other words, one has to know as accurately as possible what their positions and velocities were at some particular time. And conveniently enough, there's now a really good way to do that. Because Apollo 11, 14 and 15 all lunar left laser retroreflectors on the Moon. And by precisely timing how long it takes a laser pulse to get from the Earth to round trip to these retroreflectors, it's now possible in effect, to measure the position of the Moon to, to millimeter accuracy. Okay, so how do modern analogs of the Babylonian ephemerides actually work? Internally, they're dealing with the equations for all the significant bodies in the solar system. They do symbolic pre processing to make their numerical work as easy as possible. And then they directly solve the differential equations of the system, appropriately inserting models for things like the mass distribution of the Earth. They start from particular measured initial conditions, then they repeatedly insert new measurements, trying to correct the parameters of the model so as to optimally reproduce all the measurements they have. It's very much like a typical machine learning task, with the training data here being the observations of the solar system and typically fitting just being least squares. But okay, so there's a model one can run to figure out something like the position of the Moon. But one doesn't have to explicitly do that every time one needs to get a result. Instead, one wants in Effect just to store a big table of pre computed results and then to do something like interpolation to get any particular result one needs. And indeed that's how it's done today, how it's really done. Back in the 1960s, NASA started directly solving differential equations for the motion of planets. The Moon was more difficult to deal with. But by the 1980s that too was being handled in a similar way. Ongoing data from things like the lunar retroreflectors was added and all the available historical data was inserted as well. The result of all this was that the JPL development ephemerides jplde. And in addition to new observations being used, the underlying system gets updated every few years, for example, to get what's needed for some spacecraft that's going to some new place in the solar system. The latest is DE441. That follows DE432, which was built for going to Pluto. But so how is the actual ephemeris delivered? Well, for every thousand years covered, the ephemeris has about 100 megabytes of results given us coefficients of Chebyshev polynomials, which are convenient for interpolation and for any given quantity in any given coordinate system over a particular period of time. When access is the appropriate part of these results, and in Wolfram language one can see if one's using it on the desktop, one can see kind of it pulling in the appropriate part of the ephemeris to be able to do some computation. In Wolfram language, it's all packaged up in the function astroposition, which gives the position of the Moon in as an example, we just say astroposition of moon equatorial. And there's the astro position with right ascension and declination. So this is position of the Moon in coordinates relative to the equator of the Earth right now, or at least when I typed this in. Okay, but so how does one find an eclipse? Well, it's an iterative process. Start with an approximation perhaps on the Saros cycle, and interpolate the ephemeris and look at the result. Then keep iterating until one finds just when the Moon will be in the appropriate position. But actually there's some more to do, because what's originally computed are the positions of the barycenters, centers of mass of the various bodies. But now one has to figure out how the bodies are oriented. The Earth rotates and we know its rate quite precisely, but the Moon is basically locked with the same face pointing to the Earth, except that in practice there are small librations where the Moon Wobbles a little bit back and forth, and these turn out to be particularly troublesome to predict. Where will the eclipse be? Okay, so let's say one knows where the earth, moon and sun are. How does one then figure out what type of eclipse will happen and where on the earth the eclipse will actually hit? Well, there's some further geometry to do. And the beginning. What's involved? There's a big picture of the sun and then this kind of cone that sort of has the moon nestled inside it, coming to a point that hits on the surface of the Earth. Basically, the moon generates a cone of shadow. And then the question is how this cone intersects the Earth. If the tip of the cone is inside the Earth, that means there'll be a region of total shadow, an umbra on the earth, and a total eclipse if the tip is above the earth, There'll be an annular eclipse in which there's a ring of sun visible around the moon. So the slightly more complete geometry has kind of the umbra coming to a point and then sort of going out again and a penumbra of sort of partial shadow made by the moon on the Earth that will appear on the earth as a piece bitten out of the sun. Well, okay, so here we're including the penumbra where only part of the sun is shadowed by the moon. And particular case I'm showing here, the umbra cone misses the earth. So there's no total eclipse at all. But there's still a partial eclipse where part of the sun is shadowed. But okay, let's say there's going to be a total eclipse. To see where on the earth the region of totality will be, we have to work out where the cone of total shadow, that is the umbra, will intersect the surface of the Earth. It's a somewhat complicated 3D geometry problem. It's easiest to understand what happens by looking at things from the position of the sun. So here's a picture of the Earth from the position of the sun. And the sort of light gray region, which in this case covers about North America and Central America, is the penumbra. And the little dot in the center is the region of totality, that is the umbra. As the Earth and moon move in their orbits, the region of shadow will move relative to the Earth. But now there's another part of the story, which is the rotation of the Earth. And if we include that, we'll see that the region of totality, at least in this case, traces out a kind of S shaped curve on the surface of the earth. All of this tricky geometry Got figured out in 1824 by Friedrich Bessel, who introduced what are now called the Bessalian elements, Eight variables that specify the location, orientation, and aperture of the umber and penumbra cones, as well as the orientation of the earth As a function of time. And for any given eclipse, the whole story of its appearance and trajectory on the surface of the earth Is determined by. By its basalian elements. When will the eclipse arrive? Okay, so now we know what the trajectory of an eclipse will be. But how do we figure out at what time the eclipse Will actually reach a given point on the earth? Well, first we have to be clear on our definition of time. And there's an immediate issue with the speed of light in special relativity. What does it mean to say that the positions of the earth and sun Are such and such at such and such a time? Because it takes light about eight minutes to get to the earth from the sun? Well, we only get to see where the sun was eight minutes ago, not where it is. Quotes now, and what we really need Is a classic special relativity setup. We essentially imagine that the solar system Is filled with a grid of clocks that have been synchronized by light pulses. And what a modern ephemerist does Is to quote the results for positions of bodies in the solar system Relative to the client times on those clocks. General relativity implies that in different gravitational fields, the clocks will run at different rates. But for our purposes, this is a tiny effect. But what isn't a tiny effect Is including retardation in the equations for the n body problem, Making them become delay differential equations. Retardation because of effects from the finite speed of light. But now there's another issue. If one's observing the eclipse, One's going to be using some timepiece, A phone or something, to figure out what time it is. And if it's working properly, that timepiece should show official civil time. That's based on utc, which is what NTP Internet time is synchronized to. But the issue is that UTC has a complicated relationship to the time used in the astronomical ephemeris. The starting point is what's called UT1, a definition of time in which one day is the average time it takes the earth to rotate once relative to the sun. But the point is that this average time isn't constant because the rotation of the earth is gradually slowing down, Primarily as a result of interactions with the moon. But meanwhile, UTC is defined by an atomic clock whose timekeeping is independent of any issues about the rotation of the earth. There's a convention for keeping UT1 aligned with UTC. If UT1 is going to get more than 0.9 seconds away from UTC, then a leap second is added to UTC. Well, one might think that this would be a tiny effect, but actually, since 1972, a total of 27 leap seconds have been added, as specified in Wolfram language by the function geo orientation data. And exactly when a new leap second will be added is unpredictable. As needed, at least will be unpredictable. It depends on things like what earthquakes have occurred. But we need to account for leap seconds if we're going to get the time of the eclipse correct to the second relative to UCTC or Internet time. There are a few other effects that are also important in the precise observed timing of the eclipse. The most obvious is geoelevation. In doing astronomical computations, the Earth is assumed to be an ellipsoid. There are many different definitions corresponding to different geodetic datums. And that's an issue in defining things like sea level, but it's not relevant here. But if you're at a different height above the ellipsoid, the cone of shadow from the eclipse will reach you at a different time. And the size of this effect can be as much as 0.3 seconds for every thousand feet of height. So all of the effects we've talked about we're readily able to account for. But there's one remaining effect that's a bit more difficult. Right at the beginning or end of totality, one typically sees points of light on the rim of the Moon known as Baily's beads. These are the results of rays of light that make it to us between mountains on the Moon. And figuring out exactly when all these rays are extinguished requires taking GEO elevation data for the Moon and effectively doing full 3D ray tracing. And doing this, one ends up with a rather bizarre conclusion, that the region of shadow on the Earth isn't a perfect circle. Instead, it's roughly a polygon, each of whose edges is associated with a particular Baily's bead. The effect of this can last as long as a second and can cause the precise edge of totality to move by as much as a mile. And one can also imagine effects having to do with the corona of the sun, which is constantly changing. But in the end, even though the shadow of the Moon on the Earth moves at more than 1,000 miles an hour, modern science successfully makes it possible to compute when the shadow will reach a particular point on the Earth to an accuracy of perhaps a second. And that's what our PrecisionEclipse.com website is set up to do. Well, I also wrote a little bit about some experiences that I've had with eclipses. First eclipse I saw was when I was six years old in 1966. But that's a different story, which I mention in another piece that I put out today about sort of technical computation of eclipses. But here's something I wrote in August of 2017. So I said I first saw my first partial eclipse more than 50 years ago. I'd seen one total solar eclipse before in my life. Before the 2017 eclipse, which was in 1991. It was the longest eclipse, 6 minutes and 53 seconds that'll happen for more than a century. There was, I have to say, a certain irony to my experience, especially in view of our efforts now to predict the exact arrival time of the upcoming eclipse. I charted a plane and flown to a small airport in Mexico. I have a picture here taken at that time. That's me on the left with the silly hat. And my friends and I had walked to a beautiful deserted beach and were waiting under a cloudless sky for the eclipse to begin. And I felt very proud of how prepared I was with maps marking for the minute when the eclipse should arrive. But then I realized, there we were, out on a beach with no obvious signs of modern civilization, and nobody had brought any properly set timekeeping device. And in those days, my cell phone was just a phone and didn't even have a signal there. And so it was that I missed seeing a demonstration of an impressive achievement in science. Instead, I got to experience the eclipse pretty much the way people throughout history have experience experienced eclipses. Even if I did know that the moon would continually could gradually eating into the sun and eventually cover it, and that it wouldn't make the world end. There's always something sobering about astronomical events and about realizing just how tiny human scales are compared to them. Billions of eclipses have happened over the course of the earth's history. Recorded history has covered only a few thousand of them. On average, there's an eclipse at any place on the earth roughly every 400 years. In Jackson, Wyoming, which is where I saw the 2017 eclipse. It turns out the next total Eclipse will be 727 years from now, in 2744. In earlier times, civilization built giant monuments to celebrate the motions of the sun and moon. Today, well, back in, in 2017, we're making a website. That website, though, builds on one of the great epics of human intellectual history, stretching back to the earliest times. Of systematic science and encompassing contributions from a remarkable cross section of the most celebrated scientists and mathematicians from past centuries. So at the time of the 2017 eclipse, it was 9,538 days since the eclipse I saw in 1991, the Moon had traveled some 500 million miles around the Earth, and the Earth some 15 billion miles around the sun. But now, in a remarkable triumph of science, it was possible to compute to the second when these will be lined up again. So thinking ahead to April 8, 2024, I had a few other comments. In the days leading up to April 20, August 21, 2017, millions of people accessed our PrecisionEclipse.com website with their guips, kind of increasingly concentrating near the path of totality. I traveled to Wyoming and with a couple of hours to spare, found a place with a clear view across the valley. And so that being by then 2017, I tweeted a picture of myself with silly sun viewing glasses on and the comment the sun is still all there, but not for long. But would our carefully done computations actually be accurate? The eclipse was going to make landfall on the Oregon coast and conveniently we had a spotter right there. From where I was, the partial eclipse was well underway. And then I got a text. Yes, totality in Oregon had come at the predicted second. It would take the shadow of the moon a little under 20 minutes to reach me. And unlike in 1991, I and everyone else had a cell phone with a precise clock and we were able to access our PrecisionEclipse.com site. And here's the picture. So where I was, the eclipse was expected to begin at 11:30 5:02am While I was waiting, I made little images of the crescent from the sun with my fingers repeating something I first noticed more than 50 years earlier when I saw my first partial eclipse at the age of six. At that time I could see through sort of dappled light, through falling through the leaves of a tree, that the patches of light on the ground all were crescents with little pieces bitten out of them, Kind of a pinhole camera effect on the. On the image of the sun. Well, a few minutes before the eclipse, I started to see a strange, strange shimmering, invisible on any video I took shadow bands, a strange and poorly understood eclipse phenomenon. And then there it came, sweeping across the valley. I was looking at totality, arriving right at the predicted second. And I have a probably isn't. I think this is not playable as a video of the eclipse anyway. I'd set up a camera to capture a Video of the eclipse. A little later that day, I did an analysis of it. And since earlier in 2017 I'd started routinely doing live streams, I live streamed that analysis. And in the end I published my notebook to the Wolfram Cloud and Modern Times it's still there and showed at that time that looked at the kind of light curve from the video and saw that the eclipse did indeed arrive at the predicted time. Anyway, six years have passed. Much has happened in our human world since then, but the moon has just inexorably continued in its orbit, and 2,422 days later, it will once again line up to create a total eclipse. And that's what I posted today. I also posted another piece which is about a slightly more technical piece about computing the eclipse astronomy in the Wolfram language that goes through kind of how you compute features of the eclipse and the analysis one can do. And maybe I'll just quickly run through a few things there that fun and then we'll turn it over to some questions. So let's see. So this is my more technical piece about eclipse computation and showing the function solar eclipse that produces the shows location of the eclipse and so on. And then astrographics, showing the actual configuration of the sun and moon, Talking about the pattern of eclipses. These are eclipses over the next. That's eclipses over the next 50 years, eclipses over the next 500 years. You can see eclipses gradually sort of COVID up much of the Earth. And this is sort of sequences of eclipses in a single Saros cycle. And we can see kind of what happens as we go through these Saros cycles of what fraction of the sun is covered as the, as we go through each sort of the moment of eclipse in each Saros cycle. And then talking about how one derives the Saros cycle. And it's a kind of a little piece of number theory, a little piece of rational approximations to ratios of the synodic month, the time between new moons and the draconic month, the time between the moon going through the plane of the ecliptic. Kind of a nice piece of number theory. We can derive the Saros cycle. We can also derive a more subtle cycle called the annex cycle that was identified in the 20th century. And then we can make these kind of charts of sort of all eclipses with respect to their Saros number and their annex number. And that's fairly revealing. We can also just start computing the eclipse from first principles, knowing the positions of the sun and moon and so on, and figuring out for ourselves Sort of when they'll line up. Then I talked a little bit about the analysis of the data that I got from the 2017 eclipse and some interesting pictures of kind of what the terrain looked like and the actual shadow of the moon on that terrain and the kind of 3D shadow, the cone of shadow relative to the terrain. And that's me with my rather unsophisticated equipment or observing the eclipse. And then there's the light curve that I got from that unsophisticated iPad. Looking at the eclipse and we see a big dip, we can try and start computing. Does the dip occur in the right place? Yes, it does. But when we look in more detail we see these weird little ratchet moves here and that. Why is that? Well, it might have been shadow bands or something physicsy like that, but actually it's the camera desperately trying to adjust its aperture to keep the light level constant even as in reality the, the light level is, is going down like this. And it's, it's, it's really just an effect of the, of the instrument there. I just mentioned at the end here, the very first eclipse I saw in 1966. I kind of tell the story of walking to school one day and noticing these crescent shaped patches of light under a tree, going to school, pointing out to other kids, look, there's an eclipse. And basically nobody believed me, which is perhaps a life lesson of interest. But actually it turns out that's, that's the calculation and that was this. I happen to have a picture that I of the, the, the young scientist, so to speak. This was a selfie that I took with a film camera in 1966 with a very definitely look of concentration on my face. And I have to say I make the same weird expression today when I'm concentrating. That's the way it works with us humans. We don't change. All right, let's turn this over to some questions here. Let me see what I have. Let's see, what would eclipses be like on say Europa? Will it be a constant partial eclipses or what? Let's see. Most the inclination of the moon of 5 degrees is somewhat high in the spectrum of inclinations of moons in the solar system. It's, I think. Well, let's see, we can, Let me just something here, let me. Hold on, start sharing this. Let me see what I can do here to, to answer that question. All right, let's see, Let's use a bit of technology here and let's ask for Europa inclination. So this hopefully will give us the inclination of Europa, okay, 0.466 degrees. So it's actually very close to the plane of the ecliptic. And if we look at. Well, we'd have to work out the. Okay, let's do this. Let's live very dangerously. And let us. Okay, let's look at Europa. Europa is a moon of Jupiter. And if we. Let's show you how to actually compute something here. Let's see, we go and start up a language instance here. There we go. Okay, let us compute as they said and as Leibniz would say, calculamus. Let us compute. All right, so we want to know the. Well, let's see this. Let's look at the. Let's start off with Europa. And let's say bigger here, let's say the astral position of Europa. And this is going to give us the current position of Europa from the Earth. And it's got a load in ephemeris. There it is. Now we want to know the position of ephemeris of Europa not from the Earth but from. Let's see. Coordinate system. So let's take a look at the position of the. Of Europa from. I wonder if this is going to work from Jupiter. I'm not sure. That might be the center. Yeah, there we go. Okay, so that's showing us the position of Europa from the center of Jupiter relative to the center of Jupiter. And so we could now say, let us say astrographics of. I wonder if this will work. Maybe how it's going to work. Let's see. Where is Europa in the sky of Jupiter seen from the center of Jupiter. I know. Take a look how this works. Well, let's first of all do Europa in the sky relative to my currently believed geoposition. Okay, so there's Jupiter. It's in the constellation Aries. And Europa will be very close to Jupiter here. I'm sure if I say astro range, let's say 1 arc minute, that will zoom in right around the position of Europa. Not too exciting. It's all black there. And Europa is probably a very small dot. Let's see if I. If I give it a. If I make it a point. Oops, go here. I say. I think point this. And I. Say, for example, style that green. I think there's a good chance. Yeah, there we go. There's a little green dot there. Okay, let's increase the range from 1 arc minute to let's say 50 arc minutes. And let's recompute this. Okay, there we go. So there's. There's A little dot for Europa. There's Jupiter and there's a bunch of stars, fixed stars around it. And that's probably the plane of the ecliptic right there. So anyway, we could compute from this the appearance of Europa from Jupiter. Now, what does it mean? The surface of Jupiter? You've got to be hanging out Jupiter as a gas giant. And the surface, if there is one, is a piece of solid hydrogen deep inside Jupiter. So you have to be sort of floating in the top of the clouds. And it's going to be a little bit more work to compute what that looks like. Now you could ask from the position of Europa. Actually, that's a good question. From the position of Europa. Actually, we could do that. Let's do that. From the position of Europa. So this is from the position that's from the Earth. Now I have to admit I have to look up the documentation here to remind myself how I do this. From the position of. Okay, Astro reference frame, that will tell us the position, I think. So if I say Astro reference frame is. Well, I'm going to do. I'm going to do relative to Europa. Okay, but this is unfortunately relative to the center of Europa. I'd have to choose coordinates. I'd have to choose my attitude longitude on the surface of Europa. Okay, let's go ahead and put in Jupiter here. And let me forget about all this styling and greening and so on because what I'm going to do is look at. From the center of Jupiter. Sorry. From the center of Europa. I hope that was very interesting. Let's forget about this range. Let's just see this. That does this. Okay. Oh, wow, there's Jupiter. Okay, that is from the. From the center of Europa. That's the position of Jupiter. So that's what Jupiter looks like in the sky of Europa. And again, we'd have to be a little bit of fancy footwork to decide this is not. We're not on the center of Europa. We shouldn't be sitting at the center of Europa. We should be trying to work out the what a position on the surface of Europa, which means we have to make an offset by the radius of Europa from the position that we're defining here. And that offset is going to be a little tricky because we have to decide what direction we're going in relative to like the north pole of Europa or whatever. And I don't know the surface features. Let's see if we actually, I don't know if we have. Let's see. We may or may not have. I don't think we have the geographics geomodel Europa. Let me just see if we have that. I don't think we have that, but we'll find out in a moment. Oh, we do. Oh, my gosh. That's the surface features of Europa. Wow. Okay, so we could work out. Let's do that in a different projection. So let's say. Let's do that in geoprojection. So if we were in a particular position on Europa, let's take. What's a good one. Winkle Triple is a nice one. So this is like a map of Europa. So wherever, you know, if we were hanging out at that particular spot there, Europa, some city on Europa or something, we could, we could then work out. Let me. Let me just see. See what it's called. Geogrid lines, arrow automatic. Okay. So that now just added grid lines here. And so we could. We could figure out what the. Relative to this coordinate system. Let's say we go. How about we do this? Let's do something really crazy. Let's put the position of New York City the. The same lat long as New York City is on the Earth. So let's do that. Let's say geoposition. This is going to be a hack, but geoposition of New York City on the Earth. Right. So that's the geoposition. 40.6. Whatever it is there. So now let's do this. Let's see how this will work. Let us try doing this. We'll say geographics of that geoposition. And we're going to use the Europa geomodel. And this is going to be in. Is this going to work? I don't know. Unknown geodetic datum. Ah, I see. We have to. What we have to do is we have to say, so the geomodel outside is Europa. See if we can do this. So let's say this is a point and let's make it red. And this is kind of the analog of New York City in the coordinate system. I hope in the coordinate system that we're using for Europa here. Let's see if we get that. Oh, yeah, there we go. Little dot there. Make it bigger. Let's see. Let's say we say point size 0.05. That thing. All right, so this is a dot at the position on Europa that corresponds to the position of New York City on the Earth. Of course, by corresponding. The latitude is pretty easy to define because that's basically relative to the poles. But the longitude is pretty arbitrary because Europa is rotating. The Earth is rotating. There hasn't been the nautical almanac produced for Europa where somebody said Greenwich will be the prime meridian. Nobody yet staked a claim to the prime meridian of Europa in anything beyond the sort of convention of what was done by this NASA photography. In any case, we could then if we wanted to, we could work out from the New York City analog on Europa, we could work out what Jupiter looks like in the night sky. And probably if we zoom out of it, we'll probably see some other things here. If we zoom out, I don't know what angle this is in the night sky we'll probably see the sun. If we zoom out a bit further, Astro range. Well actually I might be able to. Let me live dangerously. And let's say in Astro range, let's say quantity arc minutes. Let's say P arc minutes. And let me do this. This may not work, but we'll try it anyway. Manipulate this. Oops, manipulate in arc minutes here P goes from let's say 1/8 minute to well, 300 and let's say 90 times 60 arc minutes. Okay, let's see. Okay, so right there we're just staring right at Jupiter. Okay, let's move out, move out, move out. Oh, this is really cool. Okay, so there's Jupiter. We're zooming out, we're seeing more away from Jupiter. Away from Jupiter eventually. Oh, there's the sun. Right, so this, this changed its, its orientation and there's the sun relative to Jupiter from Europa. So this is now as a function. Okay so now as we zoom out, I don't know what, what size is that there? Let's take a Look. That is 3080. What is that? I don't know. That's. How many degrees is that? Well we can, we can just, let's, let's set this up with. Let's say a thousand or 100 arc minutes. Still looking all at Jupiter. A thousand arc minutes. Okay, let's say, well, 3,000 arc minutes. Okay there we've got the sun. Now we've got a picture with the sun in it. So now what we want to do is let me save this away somewhere. Where am I going to put it here? Now let's, let's do this as a function of time and we should see the actual, some, you know, the actual position relative Jupiter. We should see Jupiter covering the sun from Europa. And so we could go ahead and actually what we could do is we could say what's the angular separation? Astro. Angular separation there. What's the angular separation between Jupiter and. Okay, angular separation Between Jupiter and the Sun. By the way, notice that in this picture we can plainly see Jupiter from the position of Europa is much bigger than the sun. Okay, so the angular separation between Jupiter and the sun from the reference frame. Astro reference frame from the reference frame of Europa. And we also want to specify the time. And how do we do that? We have to say. Oh gosh. Okay, I need to read the documentation here to know that. Okay, here we go. Ah, yes. Date object. All right, so let's go date object and let's do this as a function of time over the next month. Let's say date object of T. And let's say make a table of those astro angular separations as a function of time. It's a bad sign that that went red. What did I do wrong here? Astro angular separation. See Jupiter, the Sun. Date object of T. Why is that? Confused? I am. Okay, let's not worry about that for a second. Let's say time from now until now plus one month, for example. Actually, let me, let me look at the period of Europa around this. So let's say Europa period. That will be its period around Jupiter. So orbital period. Oh, 85 hours. So let's, let's do it. Okay, let's go ahead and do it for a month, for example, and let's do it in increments of one hour. And so now if I get this right, let's see if this can compute. Okay, that was wrong. Okay, well I should have paid attention to the fact that it made this thing in red. Okay, let's take a look. What did I do wrong? Observer. I have to combine the observer from. Okay, hold on, hold on, hold on. How do I do that? Wonder whether it works to just say this for the reference frame position and a time in the astro reference frame. Let's take a. If we're lucky, that will work. And it observed Jupiter. Humphty. Humph. What went wrong here? Well, what I was hoping was to plot. Let's just see here. Okay, that's planets from particular data object. How do I get that relative to a particular position? Ah, here we go. Okay, it's a little trickier, right? I've got to say that it's in the equatorial coordinate system. And then I've got to give the time and then the position location. And so now I want to say equatorial. What's wrong? Is this failing again? Humph. What went wrong here? Oh, oh, silly me. I don't want to do that. Silly me. It probably would have worked without any of this. I just need to say I specified the time. See if this works. Oh, bad sign here. Cannot observe Jupiter from. Well, I am confused. Let's just see what happens if I put that there. That's nothing like that. Let's put equatorial in here. Well, I'm a little confused. We're not able to do this. What I should have been able to do is to compute from. This is an example of the angular separation. We're really close here. We should be able to do this. This is an example of the angular separation of Jupiter and Saturn as viewed from the Earth as a function of time from the year 2000 to the year 2040. But the thing that I'm failing here is, is. Oh, that was bad. I did something wrong here. What I was failing on was getting it to know. To do the location relative to the. Relative to Europa. Anyway, sorry, I can't quite work that out in real time. All right, other questions. So that gives some sense, by the way, of how you can do some astronomical computations in Wolfram language. It's very powerful and flexible. And by the way, just to be clear, you know, if we say we can't yet do. I mean, this is solar eclipses and that's the solar eclipse. And if we say solar eclipse of. To remind myself, Sorry, I've got to read documentation again to know how to do this. Let's see, we could say solar eclipse. Gosh, there are many, many, many things one can compute for the solar eclipse. And here, for example. Well, here, let's just do this. This is the upcoming eclipse. We can compute this eclipse map. There it is. Showing different places I'm planning to go. Where is it? I don't know my geography well enough. To Austin, Texas. Okay, I'm just. I'm just going to. I could mark that on the. If I wanted to. I could go. Let's just do that. I'm just going to. Just for fun, let me look up what the appropriate thing is for the. So let's see. This is the umbra polygon. See? Now this is the. Sorry, this is the. I want the. I think I might want graphics data. So I want geographics of solar eclipse affix data. Is that going to give me the right thing? Let's take a look. Okay, that's the eclipse there. So now let's go and say. Now let's zoom in on the location of. So geocenter is Austin, Texas. And let's say georange is 50 miles. Hopefully this will now show us. Okay, there it is. So that's the line of. That's the region of totality and how it intersects. So Austin is right at the edge of the region of totality. That's the center line of the region of totality a little bit further away. So that's. So that's showing us. It's about how far away is it? Well, we can work out the. For this eclipse, we could work out the light as a function of time at that position easily with this. Okay, Let's see. Oh, someone is suggesting what I did wrong here. What did I do wrong? I gave. Now, now plus one hours. Gosh. Maybe somebody can work out what's wrong in this calculation. Because I do want to work out what is the eclipse of the sun from the surface of Europa. If we were hanging out there, what it would look like. And when the eclipses happen, I'm guessing they happen every 85 hours because, look, Jupiter is really big compared to the sun. And every 85 hours, Europa will be in a position relative to. Will be rotating around Jupiter. And so the sun will be obscured by Jupiter at that time. But it'd be nice to actually see that. Okay, let me see. All right, some questions here. Question here. Would the Earth be shielded if there was a major solar storm during an eclipse? Well, I think the solar storm. I think the answer is yes, because I think most of the charged particles streaming from the sun in a solar storm, I mean, the. You know, you can go. Let me share this again. You can go here to WolframAlpha, for example, and we can ask. You can always ask. If you're concerned about these things. You can ask for space weather in Northern Alpha, I think. There we go. So that's the sunspot number as a function of time. Looks like we've got about, well, 79 sunspots as of today on the surface of the sun. And we see there, there's the famous. Back there in 1650, there's the famous Maunder Minimum where it was really cold on the Earth for reasons that are not very well understood. But we see the Aluminia sunspot cycle and we see here information on the amount of flux from the sun of material we can probably look at. In fact, we could go here. I wonder if we can compute this. The X ray, solar X ray flux, long X rays, for example. Let's see whether that has a computation as a function of time. Oh, yes, there we go. So that's showing those. I think those are coronal mass ejection events that are like spewing X rays at the Earth. And if that happened during a solar eclipse, I think the moon is a pretty good shield for those kinds of things. Let's see. Philip Rice asks, could it be the case that the experience of eclipses was a seed in the development of mathematics and hence modern technological society? You know, there's a famous story from probably the 5th century BC of Thales, who was supposedly, you know, was an early scientist who thought that everything was made of water. Not a bad idea that everything is made of the same thing but arranged in different ways. I mean, that's kind of the idea of the atomists and ultimately the idea of our physics project where for us it's made of a hypergraph with atoms of space. But for Thales, everything was made of water. But one of Thale's achievements was that he supposedly predicted the time of an eclipse. And it was just when the Greeks and the Persians were having a big punch up. And in the middle of a battle, the predicted eclipse happens and the sun is covered. And everybody says that's something more significant than us, you know, bashing each other up. And everybody ran away from the battle, or so it's said. I think Herodotus tells that story. That story is probably pretty apocryphal, but that was a time when that was sort of an early eclipse prediction time. I think what is to be said is not necessarily eclipses, but certainly the motion of planets was a major driving force. That was one of the things that the Babylonians first sort of recognized was a regularity of the world. I mean, the heavens. The domain of the gods, so to speak, is where the, the. There was sort of a question of could you predict what was happening in the domain of the gods, in the, you know, the positions of the planets and so on. And that was sort of a big achievement of, of Babylonian work was the realization from the tables that they accumulated over the course of about 800 years every day, recording the positions of planets and so on. They noticed these regularities. And I think at first in an empirical way they said, and now we can predict what's going to happen. And yes, that was a big thing in empirical science. I mean, the actual sort of idea of experimental science and really it's kind of interpretation as experiments, hypotheses, kind of validation and so on. That was a Francis Bacon story from the 1500s, early 1600s. But the, the idea in Babylonian times of we can get these sort of fits of things. I don't think they really understood the difference between the mathematical, the precise Pythagoras theorem Type thing. And it just happens to work that way of astronomy. And indeed, when Ptolemy was writing about things in 150ad, he was presenting his work in very much the way that Euclid had presented his work in what, 200 BC or something of propositions about geometry. These are the ways that triangles work, these are the ways that circles work. Theorem, theorem, theorem, proof, proof, proof. Ptolemy presented his astronomy in the same kind of way of theorem that can be at most five eclipses in a given year. Proof, a bunch of geometry. So I think the notion that of sort of empirical science was one that arose in its full form much later, but the Babylonians were sort of onto that idea and that was certainly a driving force. The fact that there could be any regularities, there could be anything predictable in the world. Back at that time in antiquity, a lot of the way people thought things worked was it's just fate. What happens and will happen in the future is just fate. We can't know it. And it is the capricious activity of the gods that determines what will happen, rather than there is a way that we can predict what will happen. And yes, I think astronomy was sort of the earliest example of where people could say, yes, we can predict what will happen. And that's kind of a driver for the whole idea of science. And yes, leads to kind of the notions of science in time of Copernicus 1600ish, the late 1500s, the, the, the, the, the idea that you could just do science and predict things, even though the things you were talking about seemed in, in, in seem to sort of violate one's common intuition. I mean, yes, prediction of science as the Earth is moving, but yet we're standing still, we're just standing here on the Earth. It doesn't seem to be moving. But that was the moment of just trust science and believe in the science. I would say in modern times, with the whole idea of computational irreducibility, we're kind of seeing the edges of that just trust science idea. Because with computational irreducibility, even though we might know the scientific laws to work out their consequences for any particular thing can be irreducibly difficult. And that's kind of the border, the boundary of this kind of just trust science. But that whole thing started with the Babylonians noticing the regularity of planets. Let's see. There. Predictably, a cycle for these, like every cycle, every eight years. Is it more variable? I'm not sure what this is here. Well, let's see. Candid asks, is the moon 400 times as large as it is distant from the sun at eclipse time. Well, let's see. That's one of those things where we can just compute it. So let's just do that. I don't know is the answer. The initial answer is I don't know, but the next answer is I think I can compute it. So we're asking the. The. Well, we can. We can work out the angular diameter of the moon. We know the radius of the moon. Let's see, the radius of the moon is a thousand miles. The moon, the surface of the moon is about the size of the us. The. So that's the radius of the moon. Now the distance of the moon distance to moon. And it's probably going to give us the exact distance right now to the Moon. It's the radius and the distance from the moon. And then we want to know the. That's going to give us the angular diameter of the moon in the sky. We work that out and then we can work out. Let me just see here, how to do this. We have got. Well, the radius of the sun. Okay, so let's just do this. So it's the radius of the moon, the, the distance from the Earth. Okay, so it's this, the distance from the Earth of the moon. The. The average radius of the moon divided by the distance to the Earth of the Moon. Let's do this and we get a certain result. 0.004, whatever it is. Okay, so now let's do the corresponding thing for the sun. It is the sun. Let's say, let's do this. Distance to the sun. Ta da. Look at that. So this is saying that right now, right now the distance to the. So we take the ratio of these two things, we'll get. So let's take one over that and that's 214 times the. So that's. That's for the radius. So for the diameter it will be. That number will be half that. So I got that right. Diameter is twice this. So it's. So it's. It's. Well, it's two over line 36. No, one over two times line 36 is what the right calculation is. So that's. Okay, so that's 100 times. 107 times. Okay, so right here, this is saying that the average radius of the sun divided by distance from The Earth is 0.0046 at this time. If we did that for the time of the eclipse, we should find that these two things line up and we could, we could do that. But A little bit of typing to just tell it to use the time for each of these that corresponds to the time of the eclipse. Let's see, Elsie is asking to what extent would analog computation be better suited to some of these calculations of N bodies and so on? Well, clearly if you just have the N bodies and you look at them self gravitating, you'll be able to compute what the N bodies do, although you won't have some effects. You'd have to have things that were scaled down as sort of a scaled down orrery of the solar system. Somewhere out in space you could probably recapitulate the solar system with a bunch of satellites out somewhere in the solar system where there isn't. Where? Well, maybe at the Lagrange point where there's equal pulls of gravity from different things or away from the solar system, you could recreate the solar system just a bunch of little satellites. That's sort of an analog computation, but in terms of doing computations with voltage and current and kind of doing it that way where the analog is voltage and current as opposed to digital data where there's ones and zeros. I'm not optimistic that that would give good results, particularly for celestial mechanics. Because celestial mechanics is a place where small changes can lead to large effects. If you make a small perturbation, you'll get very different results. I mean, in the three body problem there are classic cases. Maybe I can even find one very quickly if I can. Oh, maybe I can. Maybe I can. Really capable, I might be able to do this. Let me see, let me see, let's see. Really have my act together here. Yes, I do. Amazing. Okay, so let me show you this. So this is an example of. This is about to be something from my book New Kind of Science. Let me see. Okay, let me throw this up there and that. So this picture is a picture of two sort of idealized stars orbiting around and a third zero mass object, kind of a little planet going up and down in the between these stars. And it turns out if you change the initial conditions for that one planet that's bouncing up and down just a tiny bit, you eventually get these very different trajectories of its height and so on. And that's an example of kind of phenomenon that Henri Poincare talked about. Maxwell had talked about it earlier, but people had talked about it that you can have in these mechanical situations. You can have just a small perturbation leads to a large effect. It's kind of a minimal case of that. Well, a simple case of that is something like a coin or something where you're rolling it and depending on the exact velocity, it will come to rest in a different place. But this is more of a progressive amplification. So for example, here you might drop some disk through a collection, this array of disks, and depending on exactly how the disks hit, it'll bounce back at a. If they, as they hit more and more off center, it'll bounce back at an increasingly increasing rate, so to speak. So that phenomenon of that sort of sense of dependence on initial conditions I think will be very hard to capture an analog computer because in the end you're dealing with effects in the 1 part in 10 to the 15 kind of range. And you just can't measure voltages that accurately. And even if the voltage is sort of made from electrons, you're, you're dealing with, well, just how many electrons were there? You know, you, you won't be able to count accurately enough. So I'm not optimistic that analog computing will be helpful in this case. Let's see. Rob Ryan asks, how do parallel rays intercepted by a 2000 mile diameter object results in an umbra that's 70 miles wide? So in other words, you're asking. Okay, so, so the, the picture that's relevant there. Let me show you the picture that's relevant there to understand what's going on. Let me see. Just have to get the right picture. I could probably remake this picture, but it's easier too. Just find this. Okay, so let's show you this again. All right, So I think the point here is that the rays from the sun are not parallel from a given point on the Earth as you look towards the Moon, it is not the case that if the sun were truly at infinity, the rays from its two sides would be exactly parallel. But from where you're looking on the Earth, you're looking there is an angular. The whole point of an eclipse is that there is an angular separation between the two sides of the sun and there's an angular separation between the two sides of the Moon. And so what's happening is that you're getting a cone of shadow. You're not getting a parallel tube of shadow, you're getting a cone of shadow. And it's the fact that that cone of shadow just lines up with the tip of the cone either intersecting the Earth so that you actually get a region of totality or being above the Earth, in which case you don't see a total eclipse. So that's kind of the geometry of it. It's very Tricky geometry. Very, very tricky geometry. It's, you know, you've got all these cones and, and spheres and ellipsoids and things are intersecting things. But that's the basic, that's the basic idea there. Could there be any perception of the topography of the Moon's surface, its peaks and valleys, and the light observed on the Earth from a location of that experiences totality? King of Pulleys asks Yes. The answer is yes. You see these Baily's beads, which are these little, you know, if you look at a picture of an eclipse, you will see these places as the moon, the disk of the moon covers the sun. There'll be this place where rays of light from the sun are poking through and they're poking through in the valleys on the moon between mountains. It's very tricky because the Moon is librating and it's kind of which things are right at the limb. The edge of the Moon isn't so easy to work out. But if you work that out and you work out the lunar topography, I'm sure we can work out, just for fun, just to see roughly what this is like, let's take a look here. We could say, Okay, let's say we say geoelevation data. I mean, we can go ahead and just look on the moon at the GEO elevation data. Let's say we say Apollo 11 landing site, that will be a position on the moon. And let's say we want to, oh, wow, there it's doing something very elaborate. That's not quite what we want. We just want the lunar landing location. Complicated. Let's see, okay, let's just pick up Apollo 11 lunar landing location and let's just use that rather than the whole fancy thing with markers which we don't need here. And let's say georange arrow, let's say, I don't know, 50 miles or something. And this is now going to give us the topographic data for the moon around the landing. Oh, that's just telling us the actual GEO elevation for that position is relative to sea level, so to speak, on the moon, which is something you have to define. You have to define the zero of elevation on the Moon. This is giving you that, oh, let me go ahead and do this. Let's say instead of that, let me just do this. Let's make a disk around that point. That's a 50 mile radius disk. And let's look at every piece of elevation in that area. And then we're going to do the same thing for the edge of the moon. Second let's get this out. There we go. And now let's say geolist. Let's just make a picture of that elevation data that's going to show us the topography of the Moon around the Apollo 11 landing site. Here we go. Okay, so I guess that was somewhere there. Oops. We see a bunch of craters and so on. Okay, now let's look at the, at the limb of the Moon. So what we want to do is we want to take the geo elevation on the Moon at a geoposition that goes around the rim of the Moon. So where is the rim of the moon we are looking. The coordinate system of the Moon, I believe, has zero, zero being the point that we see in the middle of the, of the Moon as we're looking at the Moon. So if that's the case, 90 degrees off is the limb of the Moon because that's this plus 90 and minus 90 will be the two edges of the Moon. So what we would want to do then is we want to say, let's see, lat long. So it would be. Latitude would be undetermined. So we'd say latitude and the longitude would be 90 degrees. And then we would want to say the geoposition there. So we say oops. And then we want to say it's on the Moon. An important, important thing to do. And then we want to say. So let's, first of all, let's just check that this works. Zero degrees. Actually, I don't want 90 degrees. I just want 90 there. Zero, 90. This would be on the equator of the Moon. Okay, let's see if we can compute the geoelevation of that position on the equator of the Moon. Great. Okay, so now what we're going to do is we're going to compute as a function of where you are. So this is, remember, this is the limb of the Moon. So this is at 90 degrees away from where we're looking straight at the center of the Moon. We're going to compute the as a function of latitude. We're going to compute, make a table of as a function of those latitudes. Let's say latitude from minus 90 to plus 90 in steps of one. Let's say, let's make, see if we can do this. Let's, let's make. So this is going to be the, this is going to be the position of. Oh boy. Sorry. This is taking a while. Why it's taking so long? Anyway, this is going to find us. So relative to quote sea level on the Moon, the position on the equator is a minus 12,000ft. And I'm kind of going to be interested to see how much the Moon bulges. The Moon has a big lump on one side, presumably coming from the formation of the Moon, unlike the Earth, which is rotating and it has a bulge in the middle for the equator from centrifugal force, so to speak. I don't think the Moon works that way, but let's find out. So let's say list, line, plot. And this is now going to be the profile of the limb of the Moon. Oh, wow, what a mess. Okay, so this goes from minus 15,000ft to plus 10,000ft. And so what's happening is, think about this as the edge of the Moon. What's going to happen is at these places here, you're going to see a little, little dot of light as the. Because that's the last place when the, when the limb of the Moon covers the Sun. And when you trace that out and you ray trace it, you find that in most places the shadow is such that the last thing you. You'll see just one of these little Bailey's B points of light on the Moon. But when you look at the umbra, it seems like there are these edges. And you get. Let's see if I can pull up a picture of this. Let me see. Find this. Let me see if I can pull this up quickly. Just looking for something here. Maybe this is a thing. Yes, I think this is a thing. Okay, so this is a post on Wolfram Community recently that this, that there is the profile of the Moon. That's the thing I was computing, just wrapped around the Moon. And I think. Let's see, this should be a way of seeing. Yes, this is a way of using ray tracing to see the actual border of the region of these arrays that come from different parts of the Moon. And this is showing how the region of shadow is formed from those rays. And, and you can see that it's not precisely a circle, and it's much more not a circle than the Moon is not a circle because of the kind of complicated geometry of this. And I think here, where is it? I don't know that one of the points is that if you are on the surface of the Earth at a place which is at one of the corners in this kind of polygonal structure, then right at the moment when the eclipse happens, you get to see two Baily's beads. If you're on one of these edges, you see just one Baily's bead as the last spot of light as the Moon covers The sun. But if you're one of these corners, you get to see two of those. So I'm looking out for that because I didn't really know about that phenomenon before. Let's see. Katie asks can we get super eclipses where the Moon, Venus and Mercury all line up the right way? I mean, the fact is the Moon is a lot bigger in the sky. Let's see if we ask for the. If we say, Ask for. Let's say Venus angular diameter. I think, I think it's, it's quite small. I mean, so there. Okay, so it's 10.3 arc seconds and relative to the Moon. Yeah, it's only 0.5% of the angular diameter. So by the time Venus, I mean, you know, if, if Venus was lined up behind the Moon, we'd never see it. The. There's this whole idea of conjunctions. I wonder whether we actually compute those conjunction are Venus and Mercury. That was called. Oh yeah, there we go. Nearest conjunction of Venus and Mercury, 20 hours in the future. And that would be the time when those line up. I don't know if we say, I bet we can't do this. But let's just see. I suspect we can't do this. I think this is a hard computation. But see what. No, we can't do that. Sorry. I think that's really hard. But that's computing when those two will be lined up. So I think that happens quite soon. So let's see. But that's obviously. And so then the question is when does that happen relative to eclipses? And then when is that lining up with the Moon? We could certainly compute that. It would take me a little while to get everything straightened out to be able to compute that. But we certainly have, we have the technology, so to speak, to compute that. It'll take a couple of lines of orphan language code, I think. All right, I, I think I'm where questions are drying up here. But, but let's. I think there was just one last question here. Would solar panels be more or less efficient on the Moon? I think a little bit more efficient because the atmosphere, Earth's atmosphere cuts out some of the, some light. And in particular it really cuts out. I mean the, the, the cuts out. Well, lots of ultraviolet light, for example. And I don't actually know precisely what the response curves of photovoltaics are. I think probably what you design actually. The ones that go on satellites, I think are pretty similar to the ones that you can put on your roof, whatever. But, but I would think that there would be things that, where you could make use of ultraviolet photons, which don't reach the earth's surface, they're blocked by the ozone layer. I would think you could make more use of those because ultraviolet photons are higher energy. One would think that that would give you something good, but I don't know for sure. All right, well, I think that's it for now here. And I think I may be doing a few more live streams related to the eclipse as we approach the eclipse, maybe for particular groups of people we'll see. But for now, that's what I have to say about the eclipse. They're cool. Go see it if you can. And one of the questions I'm trying to answer is how to pick possible locations that have this correlation in terms of cloud cover. So it's like the three places and you're going to be able to get to any one of those three, but at least one of them, they're uncorrelated in cloud cover. So the probability that all three are covered with cloud is considerably lower. And hopefully by the day before one can figure out where to go and hopefully everybody else won't be going to the exact same place. I have to say that my two total solar eclipses, one of which I saw from a beach in Mexico and one of which I saw from a nice ridge in Jackson Hole, Wyoming. On the Wyoming one, the roads were really jammed with cars even a few hours before the eclipse. I guess more people were there getting to the stripe of totality for the eclipse. Everyone wants to go there. And the roads weren't built to have that many people there. By 1991, Eclipse Experience was simpler because I charged a plane. And that was. I thought there was going to be. Actually I was concerned because I charged a plane from Los Angeles to, to go down to this place in Mexico that was fairly off the. Off the beaten path. But I was concerned that there would be an air traffic control traffic jam of planes. But turns out three dimensional space is very big and the number of planes is not so big. And so that didn't turn out to happen. All right, well, thanks very much. And until another livestream, I should just advertise since we're here. There's my book version of this that talks about what I've talked about here as well as has lots of fun maps of future eclipses. So you can go check out if you miss the eclipse in April or if you're in a different part of the world, then you can go look up in the back here. The details of some future eclipses. I think there's one nice one that goes through Iceland on August 12, 2026, which I might make an effort to go to. Anyway, thanks very much and bye for now.