Please help me welcome now Professor Alain Garrieli. Thank you very much, Christine, and thank you all for coming, and hello for all people who are watching online. So, geometry of nature, we've looked at small things. Today we're going to look at a very big thing, the shape of planets. We take for granted that planets are mostly spherical, and today I want to challenge this belief and show you this simple question has been hotly debated for the last 300 years and indeed has shaped modern science and mathematics. So, if the earth is not spherical, what is the shape of the earth? Well, the best science tells us that the shape of the earth is the geoid, which is another way to say that it is the shape of the earth. So, where do we start? Well, like everything in science, we start with the ancient Greek. And since the 6th century BC, people, learned people, believed that the earth was spherical, mostly because it was a perfect form of nature, according to the Greek geometry, but also they realized by looking at the lunar eclipse that they were seeing the shadow of the earth on the moon, and that the most likely shape was a sphere. And already in the third century BC, Aristotle in 240 BC managed to give an estimate of the radius of the Earth that was actually extremely close to the exact one. And that belief was held up to the 17th century, right at the age of enlightenment. So what happened then? Well, there were two main observations that challenged this idea of perfect sphericity. And the first one was related to time, measure of time, and the second one was related to measure of distance. So if it's not perfectly spherical, it's somewhat flat. So I brought you a little uh model of the earth right here. It's a model, we call it a globe. So if the earth is not flat, it can either be, according to the time, a prolate, meaning it's shape like this, or oblate, like so. And the big difference here is not the position of the pole, is that the prolate shape is a surface of revolution, meaning the all the intersection with the with these planes makes circle. The oblate shape is also a surface of revolution, but in the first case, the long axis is longer than the radius of the larger circle, and in the second case, the pole-to-pole is shorter. Thank you. So if we define the flatness as being 1 minus b over a, flatness is zero when it's perfectly spherical, flatness is negative for prolate, uh, which would look like a lemon, and flatness is positive for oblate, which would look more like a squash. So what was the great debate? Well, was the earth prolate or oblate? There were two sides of the argument: the site of the Cassinis and the site of Newton, more or less English and French. And this is the debate that lasted for about 50 years. And to help us with the problem, we will not only travel the world today, but we also go around Paris to find all the great contributors to the problem. So you have the Cassinis, you have the Bernoulli, who were following Descartes in this uh belief of how the planets were organized. And on the other side, you had Newton, Voltaire, and Eugenes. So, on what did they base their opinion? The first main observation was the major development, major technological development, was about time measure and the discovery or the development by Ergens of the pendulum clock. And up to Eugene, the best clock were making about maybe 10-15 minutes error a day, and the pendulum clock, when it was introduced, reduced that to 10 to 15 seconds a day. And within 10 years, the technology developed to one to two seconds a day. And when Jean Richer was asked by the French Academy to go and do some astronomy, so he went to Paris all the way to Cayenne, in French Guyana, in 1673, he brought with him a very well-calibrated pendulum clock. And the first thing he realized when he arrived there is that his pendulum clock was losing two and a half minutes a day. So the period of the pendulum clock, the period in Guyana, was slower than the period in Paris. This was a very anomalous uh observation, but Newton seized on that and saw that as a way to prove that gravitation was indeed universal, that it applies to everything. Indeed, if the period of a pendulum is longer in Guyana than in Paris, well, we know from the standard uh laws of physics of the pendulum that the period is related to the length of the bob divided by the gravity, the square root of both. So longer length, longer period, but smaller, smaller period for the same length means that the gravity in Guyana was less than the gravity of Paris. And Newton then came with an extremely clever idea. He says, let's think of the globe, and let's imagine that I have a canal drawing from the pole all the way to the center of the earth, connected to another canal going from the center of the earth to the equator, at any point in the equator. And I fill this with fluid. What he computed then, he says the T fluid has to be balanced, and if I introduce the centrifugal acceleration at the equator, so purely due to the motion of the earth, divided by the gravitational acceleration, and call it M, Newton deduced that the flatness must be 5 over 4 times M. And already at that time he had some estimate of what M was. It was 1 over 290, and so he approximates, he says the flatness is about 1 over 230, which is positive, hence the earth is oblate. Also oblate. So for us, if you look at his computation and understand what he's doing, and you can look at the write-up in the appendix, I show what the computation is explicit is explicitly, uh, it's a very, it's a very good argument, and we'll naturally conclude that the earth was oblate. But there are two problems with that at the time. First, nobody really at the time understood what Newton was writing. It was very difficult, and if you try to read Newton, you'll have the same problem that the people in the time. Even the great Bernoulli, he was the greatest, one of the greatest mathematicians in the early uh 18th century, says that it was completely gibberish. He read it, talking about his problem. I read it and read it and I don't understand it. You had to wait for Émilie Duchât to not only translate Newton, but really annotate and correct many of Newton's mistakes. But that comes only in the 1750s. The second problem is that there were conflicting observations. There was another set of observations, not related to time, but to length. So, what is the idea? Another way to think about measuring the flatness of the earth is to measure length along the longitude, uh the latitude. So you take the earth, assume that it's an oblate, you look at the north star, and you start at the pole. So the north star would be right on the horizon, and you keep going all the way to the point where the north star is at 10 degrees, and you measure that distance, that in red here. And so you have that distance in red. And now you go to the pole and you do the same the same type of measurement. You start where a north star is right at 90 degrees over your head, and you measure the distance along the latitude all the way to the point, so you go full south all the way to the point where the north star is not at 80 degrees. And if the earth is oblate, the distance of that meridian is longer at the pole that it's at the equator. Of course, you're not gonna actually measure 10 degrees because that would be about more than a thousand kilometers, but let's just say one degree. So if you measure one degree, and if the degree you measure at the equator is less than the one you measure at the pole, then the earth should be oblate, otherwise it's prolate. How do you measure that? Big development in the 17th century again. The idea is quite remarkable. So suppose that you want to measure this distance from latitude n to n plus one, zero to one or ninety or close to the pole. How do you do that? Well, you're not gonna measure with ropes and things all the way to uh 100 kilometers, 110 kilometers. What you're gonna do is you're gonna measure one distance. This is called the base. So you measure that very carefully, you measure that base. And then after that, that's the only measure, only distance that you're gonna measure, only lengths you're gonna measure. After that, it's all looking at angles. What you look at is the tip of a mountain, a big tree, a church, or something like that. And you look at the two angles here, alpha 1 and alpha 2, to that, to that point. Very precisely, and they developed wonderful instruments at the time to do that. Now, if I know two angles in a triangle, I know the third one because the sum of the angle is pi, 180 degrees. So I know all the angles. I also know the law of sine that tells me that the length of an edge of a triangle divided by the sine of the opposite angle is constant in a triangle. So if I have the law of sine and I know all the angle, I can write, I can find a1 as a3 sine alpha one divided by sine of alpha three that I know now. And so I know a1 and I also know a2 by the same law of sine, I can explicitly find. Okay, now I know the two sides of a triangle, all I need is to find another point of reference, let's say this guy here, and find the precisely the angle to this point. And now I know I repeat the same process, I know the length of this triangle, and I continue doing that with a lot of different triangles, and there is a lot of possible errors and all that, but eventually I can do that. This is the meridian de Picard, Jean Picard with Snellius was one of the first were the one to develop the technology, going to the observatory of Paris, and here is the base length. And this is a method that's further developed in the 18th century. Here's the the meridian of Paris going to France, and right there, I don't know if you can see it as a bunch of little triangles going there. And what the Cassini showed, what the Cassini showed is that the length in the south of France appears to be longer than in the north of France. And so they concluded that the earth was uh was prolate. But the debate kept going on for years and years, way after the death of Newton in 1727, and eventually the French Academy of Science decided to settle the problem with two major expeditions. That if we want to do it properly, measuring meridian in France is too small. There are too many errors that can that can propagate to have good measurement. You have to go to the equator and to the pole. And so two expeditions, first one in 1735, went to Peru, which is now Equator, around Quito. And the second one left a year later, 1736, to Laplan. So the Laplan expedition is well known because it was led by Maupertuis, who became famous after that. It also involved Claireau, young prodigy, mathematician who could do amazing computation and was very helpful. And a certain Celsius, Swedish scientist, who must have been very cold because later on he came up with a new scale of temperature, the one that we use. And within a year they came back with the result that the length of a meridian was 70 57,437 toises. So I remind you that a toise, there was no meter at the time, is six French uh feet, and the French have longer feet than the British, it's about 1.949 meters, compared to the one that people knew at the time in France, the one in Paris, which was 57 and 60 toise, and clearly showed that the earth was oblate, it settled the debate. What about the Peru expedition? Well, they waited and they waited and they waited. It's amazing. It's one of the greatest uh scientific adventures of all time. I think it has all the elements that would make great uh Hollywood movies. I mean, I'm talking pirates and private privateers and earthquake, volcanic eruption, mysterious disease, robbery, murder, and yes, even romance. But it took them almost nine years to come back in 1744, and they came back with very good measurement, 56,734 twice, was only 20 meters away from the exact measurement that we know today. But by the time the debate was completely settled, though they did bring the knowledge of rubber that became a big technology after that. But going back to Maupertui, already, when all the other guys were in Peru, he became a superstar in French circle. And Voltaire said that not only he flattened the earth, as you see in his engraving, he also flattened the Cassini in the problem. Later on in life, Voltaire got a little bit sour with Maupertuis, Maupertui left for Berlin and all that, and wrote, You have confirmed in the lands of boredom what Newton knew without leaving home. So it's quite remarkable because all this technology not only gave the shape of the Earth quite with good accuracy, but also developed the method that was used in 1790 for the Paris meridian, going from the equator all the way to the pole to define the meter as one divided by 10 million of that meridian arc. It was not quite precise enough, but the meter that will later be in Paris was defined as such based on the length of the Earth. Now, fast forward 200 years, we know a lot more about the geoid. We know it very precisely, and in great part due to Gladys West, who passed two months ago. She modeled the shape of the Earth and also developed a satellite geodesy model and the global positioning system. So probably all of you today have a piece of Gladys West mathematic in your pocket. And I thought that her contribution was so remarkable that the French should really have a street name after her, but of course the mairie is a little slow in doing that. So I did it for them, and I put her in the fifth arrondissement, which is my favorite place. So what do we know now, thanks to her work and her husband, who also worked on the problem? Well, we know very precisely what A and B uh are, and that the flatness is 1 over 298.257, very close to Newton's estimate, but Newton was lucky in a way he he had two errors in his computation that somewhat cancel. What's remarkable is that the earth or the potential, the zero potential for gravitation, is almost a perfect ellipsoid of revolution. And here is the 96 uh geoid uh standard model, and it shows you, the color shows you the deviation from the ellipsoid of the potential, and the deviation are less than 100 meters in both directions. So we're talking about the zero potential, not the mountain. So that's the mean average of the earth. So the earth is very exactly, very extremely close to this ellipsoid. So we know a lot about the uh Earth, but let me go back to the problem that Newton started. Now, people at the time say, okay, the Earth is important, but what about other planets? What about what if we assume that the planet is a fluid? What is the solution there? What are the possibilities if the rotation is different, and so on? So, what is the mathematical problem associated with that? You take a fluid planet, you assume a density rho, you assume it's subject to its own gravity, and you make it rotate at an angular velocity omega. And the question now is to find the shape as a function of g, omega, and rho. And in order to do that, you define one number, like you usually do in physics, that really characterize the entire problem. Because going faster or having uh uh denser planets or changing the density, the effect are the same. The only important things that matter is the ratio of this parameter, omega square divided by pi g rho, the energy associated with uh rotation divided by the gravitational energy. So you really have one planet, and the best way to think about it, you fix a planet and you make it rotate faster and faster, and then what would be the shape of that planet? This problem was first solved by uh MacLaurin, and what he did, he generalized, he fully generalized the argument of uh Newton. Uh and let me explain you this very important result that stood for at least a hundred years. On the x-axis, you have the eccentricity, which is another measure of the flatness, it's a more common one used in mathematics. So it's the square root of one minus a square over b square. E equals zero is the sphere, and the deviation away from that e increase. And on the y-axis, you have this parameter m. So what that means is that no velocity, zero velocity here, you have a sphere, zero eccentricity, and if you increase, so if you increase the velocity, you get a point on this curve, and you get the eccentricity corresponding to that. So let's start spinning our earth and seeing what the solution are from Lac MacLaurin. Here is I start here and I increase the angular velocity. And you go up, as you go up, the eccentricity increases, so it gets more and more flatter, more and more ablaze. So up to a point where you reach this bifurcation point, I'll come back to the bifurcation point in a minute. You keep increasing, and you get all the way to the top of this curve. And then you can decrease, and then you get very strange solutions. The eccentricity is extremely flat earth. But these are all possible solutions as shown by MacLaurin. Up to the point where you have like almost a pancake. So, of course, that started an entire research program in the 18th century. One of the questions, well, if I take one, I have two possible solutions, right? I have this one and this one. So which one is going to be selected? Uh so Euler started looking at the problem. Legendre shows that close to the sphere there is only one solution. Gauss worked on the problem, Cauchy developed the entire theory of continuum mechanics with application of this type in mind. Poisson developed the theory of potential that is central in uh any kind of field theory. Monge worked on the problem, Laplace worked on the possibility of having tri axial ellipse. But Lagrange, by the end of the 18th century, concluded that only ellipsoid of revolution can exist. So that the only solution that are ellipsoids are ellipsoid of revolution, where one of the sections has a perfect circle, like the MacLaurin solution. However, in the 19th century, Iacobi looked back at the problem of the solution of Lagrange and did not believe in Lagrange. He said, I don't think that's right. And indeed, he proved that at this point there is another branch of solution which are not ellipsoid of revolution. So what do this look like? Well, now you have an ellipsoid which three axes will have different lengths. So every section that is normal to any of these axes would give an ellipse and not a disk. So triaxial and show that this is a possible solution also. Liouville shows that these two branches meet at this point. So what do they look like? Now we're talking about strange solutions, and we'll see even stranger one. Well, this is what they look like. At the bifurcation point, instead of following the previous curve, we can follow the curve of Jacobi. And what you have is this more like cigar shape, which don't have uh which which are not ellipsoid of revolution. And here is an example of a known one, which is Omea, which is in the Kuiper belt, so past uh uh Neptune. It's a dwarf planet that has a period of roughly 284 years, and it is a Jacobi ellipsoid with a very good uh very good approximation. And this is due to the fact that it rotates very quickly, so it's actually very close to this point of bifurcation, and it's on the blue curve, not on the red curve. But the problem of which curve is stable and which curve would be selected started yet another research program due to Dirichlet that was later studied by Dedekin and Riemann, and Dedekin and Riemann both showed the existence of different types of ellipsoid with different assumptions on the fluid inside. And Riemann was the first to propose a criterion for stability of this different solution. However, his criterion of stability would later found to be wrong also a century later. So much work was done by the end of the 19th century, and that problem that in 1873 Isaac Todd Hunter, the big historian of science of the 19th century, wrote a summary of the work, and it's a summary of two volumes of more about a thousand pages. So it was really a central problem, but the story would yet take another turn with the work of the great Poincaré. And Poincaré said, well, if MacLaurin ellipsoids ferry can turn into the Jacobi ellipsoid, so surface of revolution to triaxial, maybe there are other solutions than the Jacobi ellipsoid. And he calls this solution piriform for shape of pairs, pois. And he says that there should be a bifurcation point. That's also when Poincaré introduced the notion of bifurcation. When you have two solutions, they can bifurcate and you can follow one versus the other one. So that's also going to the foundation of dynamical systems, the modern way of thinking about this problem. But what Poincare shows is that Ts can evolve into strange shape, like this one. That's what he called piriform solution. Is the third harmony can can can be uh can become a solution. And what do they look like? No, that's even stranger. Here is what this solution looks like. Now I'm following Poincare's branch, and a solution starts looking very bizarre indeed. Nothing like you would ever think a planet would look like. And this program of research got so popular, so fashionable, that people loved it. And that was another, for 40 years, people started looking at this problem because Poincaré had his idea that it would explain, for instance, satellite formation, like the formation of the moon, that maybe as they form, as they deform, they elongate and they finally pinch up, and that's how you create a satellite. James worked on the problem. Lyapunov worked on the problem all the way to Cartan. And Cartan in 1924 comes and does a very precise mathematical computation and shows that all these solutions starting from Poincaré that people have been looking for 40 years are all unstable. They can never be seen in reality. They do not exist. They are possible equilibrium solutions, but not solutions that you would actually see. And up to that point, more or less people start looking at the problem all the way to Subrahmaya and Chandrasekar, who wrote, who looked again in the problem in the 1960s and wrote the definite treatise on this problem, correcting past mistakes and finding all the solutions. He went on to look at the evolution of star, theoretical astrophysics, and won the Nobel Prize in 1983 for all his work. And that's really the full solution or the best of what we know about fluid planet. But you tell me we're not on a fluid planet, right? We know that they're free planets, but there are other planets that are not fluid. So are these solid planets spherical without without angular velocity at a stable in the same sense that the fluid planets are. What this is called the Lame problem. It was studied starting in 1854 by Gabriel Lame. And by 1917, James Jeans concluded that based on his result and analysis, the spherical configuration can never become unstable. So as far as solid planet was, there is essentially no places to go. This is really the end of the road. Except things are always a little bit more complicated. There is the little problem of Earth tide. We know that the formation of planets that are due to moon and other bodies. We know that there are tides, like the tides you go on the seaside, but the earth itself is moving with much lower amplitude, maybe a meter or something like that, depending on the different place. And that's a problem that both Calvin and Rayleigh look at. But the one who made the most important contribution to this problem is really Augustus Love. And Augustus Love won the Adams Prize, young Augustus Love in 1890, for his uh for his treatise called Some Problem of Geodynamics. And the problem he looked at is the following. Here is the Earth, and here is the Moon. And he says, what is the effect on the of the Moon on changing the shape of the Earth? Right? So if I know the length and the size of this and I know the property of a solid earth, how is it going to be deformed? Right? And it's an important part of uh planetary science. He comes up with this number called H2 that describes the effects of the effect of what is called a tide raiser, and it's commonly used to compute the earth. But in the late 1990s, when I was still in the US, uh there was some interest in planetary science, and somebody from planetary science came to us, and us is uh myself and Michael Taber. He says we have this uh interesting problem. We try to understand um the shape of Europa, and we have this altimetric data from satellite. And the reason that they're interested in is that they believe that Europa has a very thick layer of ice, but they don't know how big the layer of ice is. But they have models for that, and they want to relate the deformation during the day of Europa to the size. So if you assume different size, you can predict how much is going to be deformed and check that against your data. And why is that important? Because depending on the size of the ice and the temperature below the ice, that might be just the right environment for life to emerge. And so whether or not there is life on Europa. And so that that was a big question. And so, with a student, Sarah Frey, we looked at the problem and we went back and revisited the problem of the type raiser, the theory of love. And we found that there was a problem with this theory, is that for certain value of the parameter of elasticity, the theory allowed for singularities. That means that according to the theory, the if you write it there, whatever the size of the tide razor, you would get an arbitrarily large deformation of the Earth. Right? And so you have a singularity there, and when you have a singularity, you always have to pause a second, because uh, as Sherlock Holmes said, singularity is inevitably a clue. You do not expect this planet to explode or anything like that, because clearly it is theory breakdown at some point, and so something else must be happening, but the singularity themselves might be explaining something that's going on below it, might be capturing something. So I didn't think too much about the problem. I moved I moved to the UK in Oxford, and uh a few years later I had the visiting professor, Feija, from Harbin Technology, now he's a professor in Harbin Technology, uh University in China, uh, and he was looking for fun problem because he's interested in fun problem and has this idea that I'm doing these kinds of things. And I said, Well, we have this, I have this problem with the planet that I want to revisit, and I think we have to do it correctly, meaning not the way Love did it, because the theory or the tools that they had a century ago were not sufficient, and they were very open to the fact that they were making a series of assumptions that might not be valid. But now we have better tools, and I said, let's look at the problem. Let's look at the pure mathematical problem again of uh elastic planet. So my entire universe now has one planet, no tide raiser. It has density rho, it's a solid planet, so it has elasticity mu. So meaning small mu means it's easily deformable like rubber, and large mu it's hard to deform like steel. So you have a parameter to play depending on what you want. And these planets initially start with zero gravity, and then you turn on gravity, so it's under the effect of self-gravity. And we said first thing we want to know, it was the radius of this planet as a function of density, elasticity, and g. Very much like the original problem with the fluid planet, but now with a solid planet. Except this we don't even have rotation yet. Then the problem is the following whether or not a planet can become unstable under its own gravity field. Well, we know that if we press on a rubbery object, eventually they can deform in different ways, and that's an instability. And would gravity be sufficient to deform the planet in itself? So there are two steps to that. You start with an initial planet without gravity, you turn out gravity, because you're the master of your equation, so you in in your little world you can do every everything that you want. And you get a spherically but stress configuration, right? So that means that at different places you have different compression. And the problem now is that is this configuration becoming unstable according to different mode harmonic? Can it be squash or like like the piriform solution that would be the n equal three solution and so on? Okay. So let's take a pause and let's try to do that. The first step is very easy to think about it. You take a radius, a planet, a spherical planet of radius one of given mass and material. Radius is one because there is nothing else in your universe, so you might as well measure the length with respect to your planet. No gravity and no and constant density. Now you turn on gravity and you obtain a compressed planet. And so the question is to find the deformation, little r, which is the position or the radius of a point that was initially at capital R, as a function of all the parameters. And again, the key parameter here is a combination of all these parameters I show you. It's a it's 4π, forget about the pi, rho squared, density squared divided by mu. It gives you the relative effect of gravity and elasticity, and it has no dimension, which is very convenient. And so the problem is to find the radius according to different size or different elasticity. The easiest way to think about it, since you can change parameters in multiple ways, is to think of the same material, same elasticity, and that's how we're going to think about it, but planets of increasing size, made of the same material and density, right? So we're going to grow a planet, and as you grow, gravity gets more and more important because you have more mass. So think of the key parameter eta as increasing the size of the planet. Now, the equation that governed that spherical deformation we derived, and this equation is this one. I'm not gonna go, don't worry, not gonna go try to explain you all the detail of it. What I want to show you is that it's both very simple, simple equation that can be written with multiple terms but on two lines. It's also a very difficult equation. It's a highly non-linear second order differential equation with boundary values, and I'm not even showing the boundary values. This is the second order. It's the second derivative of the function little r with respect to capital R. But if you can solve this equation, then you can find R1 as a function of eta. You see eta is a parameter appearing in the equation. So all you need to do is now you reduce the problem of solving a second order uh ordinary differential equation and find the solution as a function of the parameter. Exactly like one did when we do F equals Ma, because A is the second derivative of position, that's another second order uh differential equation. But due to the multiple effects that gravity uh plays, this equation and elasticity, this equation is much more complicated. But we have tools to do that, both computationally and uh and analytically. So computationally, uh Usman Kudio, who is also on the paper, and John Chapman, uh, will look at this equation and solve it. Usman is now was a student in Oxford, and he he worked on that with us, and now he's a professor at UCSD, and John Chapman is the professor of mathematics and its application uh in Oxford. So, what did we find? Well, again, the solution are going to be quite interesting. We found many planets. So let me explain you this graph. Now I have the mass on the x-axis, so when you increase on the x-axis, increase like that for you, eta increase, the mass increase, and I have the initial radius on the on the y-axis. Since I started with the radius one, if I have no mass, there is no gravity effect, so the radius is one. So on this curve, for a given mass, I can find the radius of the current planet. So I increase the mass, it compresses itself, and I keep increasing the mass. The solution that Love proposed is the one here, and it's only valid in this small region, is that line there that you can barely see. And soon after, Love computation is not valid anymore. But you reach a point, interesting point, for a given mass, you reach a point here in the solution where you find that there is no solution for mass behind that. And what that means is that you do have gravitational collapse. That means the elasticity of the planet is not enough to go against the force of gravity and the planet fully compressed to a point. And we're not talking relative effect or black holes or anything like that. This is just traditional uh uh mechanics that already have this type of singularity and collapse. But if you go along this curve, you find two solutions in that neighborhood. So there are two possible radius. That's interesting. If you keep going, you find that there are masses for which not only there are one or two solutions, but three solutions. Hmm. Now you have region where you have one, two, and three. So which way next? Well, if you keep going on this curve, this curve is curling like a little spiral, and you can prove that you have n solutions possible with n equal one, two, zero, one, two, three, four, and arbitrary. So these equations support planets of different radii, and right in that little, very little window of parameters, these radii could be a lot of different things. Right? You can choose to have exactly n solution. So, okay, now you have the main the main the main solution. Remember step two, and what James says, that symmetral symmetrical configuration can never become unstable. So the step two is that you compress the planet, and does it become unstable to T's perturbation? Well, you can do the computation, that's even more involved. There are different ways of doing it, but what you find, the result is also interesting, is that the curve there, the curve with all the spiral, all these points, all these earth, all these planets are unstable. Right? So that's an interesting result. But what about the collapse? Okay. So when I told you that there is one parameter of elasticity, I didn't tell you the whole story. There is another way to characterize elasticity, is the resistance to compression. So a body can be easily compressible or not, or can resist compression. Like if you have a form of rubber, they can have the same parameter eta, but one is easier to compress fully than the other one. And so, depending on the material, and we don't know exactly what the materials are, you can penalize compression with another parameter that tells you that in large deformation it gets harder and harder to further compress. So you start pushing and it gets harder and harder to compress. This is the parameter alpha. So what that means is that alpha close to zero, you don't have much resistance to compression. But alpha close to one, you have high resistance to compression. So when alpha is large and the planet gets compressed, it resists us more and prevents collapse. And indeed, for alpha large enough, like 0.6, you start compressing, mass gets bigger and bigger, compress, but then you always end up with the same planet size. So you can get as big as you want, you always end up on the same side. What happened with moderate alpha, where things get a little bit more interesting, that with this problem you have zoo of solutions and you have to sort them out. So for moderate alpha, uh what you have is that you have a stable that is called bistable. That means the two planets on the upper branch and the lower branch are both stable. You can have a highly compressed planet that can exist and stable, or you get there is another problem, or you can have a big one for the same parameter. And now you have quite a uh complex overall picture. Depending on the compression, you can have either one and only one spherical solution, that's the one in red. You can have one to n spherical solution, so depending on the parameter, you can go up to n. Or you can have zero to infinitely many spherical solutions. And you can find, depending on whether it's highly it's easily compressible or it resists highly to compression, you can find a different uh different solution. So, okay, you're going to ask me, of course. If that was your question, too bad. What about real planets? Well, it's very hard, right? Because reality is always hard. Uh let's take Mercury, for instance, let's do very rough approximation. I'm not saying my theory is applicable directly to uh Mercury, but just to get a ballpark idea of where we are in terms of stability. Where we know that Mercury is about 80 gigapascal in elasticity, we know its density, we know its mass and its radius. And if you assume that it does not resist compression, then in that case, before collapse, its maximal radius would be about the same order. So it's not that far from the instability limit. We don't know if the range is correct, but that's that we're not that far. So the the theory tells you that there may be some interesting things that happen for real planets. What about other planets? Well, we had a very nice talk a few months ago by Chris Lintot here in Gresham as a Gresham lecture about exoplanets. And I remember when I was a kid, uh there was barely the time when I got interested in science as a teenager, that people for the first time thought they could prove that they were exoplanets, they were other planets orbiting other stars. Well, with the technology we have now, we have more than six stars and exoplanets and all, planets outside our solar system. And they can be very big, like gas giants, which are fluid planets, or they can be smaller. And it turns out that uh a rocky planet, the one that has solid, are much smaller than the big one. There's not really not that many very big planets. And it's something that's that's well known in uh in in astrophysics, is that the typical rocky planet may be 1.6, 1.8 times the size of the Earth, and after that you really have a valley, you don't really see many of them. So there is a discrepancy about the size of the planet. So maybe the fact that we don't find bigger planets might be related to the ability to sustain uh gravitational stress during the evolution. I don't know. But there may be something interesting there. So what about the other effect? Well, there's plenty of things we don't know. There's the behavior in high compression, we don't know exactly how a material like a solid planet would behave in high compression. Does it become plastic? That means it yields without supporting stress. At the phase transition, that turns rock into magma and so on, and change the type of system that you have, then you have to worry about temperature effect. Uh if you have uh fluid inside, like on Earth with a crust, then you have to take into account the fluid pressure between the two, and we've done that partly. Uh, what about if you have anisotropic layering, you have different layers that have different properties, and rotation, I didn't even look at the effect of rotation on this solution, or the effect of other bodies in the universe, or the possibility of seismic waves. So these are all valid lines of inquiry, and mostly I have no idea. And uh on that, I'd like to thank you very much.