 So we'll start the second lecture of this morning. So it's quite a different subject. Le son sur les hypothèses cosmogonyques. Poincaré's view of solar system formation. What remains valid today by Alessandro Morbidelli. Please. Thank you and thank you very much for this invitation. It's an honor for me to be here in this meeting and honor the memory of Poincaré. So I will speak about this book, Le son sur les hypothèses cosmogonyques. So essentially it's origin on hypotheses and models, lectures on hypotheses and models on origins. Poincaré gave a series of lectures on origins at the University of La Sorbonne. And actually the book is not written by him. It's written by Henri Verne on the basis of Poincaré's lecture notes. But Verne is essentially just a scriber and when you read the book it's essentially Poincaré speaking. And Verne is not at all showing up. So Poincaré was a mathematician, was not an astronomer. So he doesn't really have his own theories. So in this book it's like a broader view of the models and the theories of the time that Poincaré describes with his mathematician touch. So he describes all these models with some uniform level of formalism, which sometimes goes well beyond what the original authors have done. And it's nice to see some puns, like for instance this one on Maxwell, where Poincaré writes that his work, his ideas, monk de rigueur, mm de clarté, so lack of rigor, and even of clarity. And so he writes everything with mathematical formalism so that the ideas are properly laid down. So although Poincaré doesn't have, oops, his own model and theory to defend, when he describes the theories of the others, of course he says explicitly where he agrees and where he doesn't agree, he doesn't hesitate to criticize some of these models so we can get through the book what actually Poincaré was thinking at the time. So the book is on origins, which means a little bit on everything. So the first part is on the origin of the solar system and the planets. And then it goes to discuss the interior of the planets, the origin of the heat inside the stars and the planets, the origin of stars, and the origin of stellar clusters, and even at the end something towards the galaxy. So because I'm a planetary scientist and I work in planet formation theories, I'm biased. And so here I will only describe the first part of the book where Poincaré discusses about the models for the origin of the solar system. So the basic observations from which astronomers at the time were starting from was that the eight planets of the solar system orbit the sun all in the same direction and its direction is the same as the direction of rotation of the sun. So the orbits are all progred. The orbits of the planets of the solar system are basically circular and they are basically co-plated. And so these three facts, the progred orbital rotation of co-planarity and circularity, suggested that maybe these planets formed from a flat circular structure like a disk. And of course astronomers knew from Saturn's rings from very early on. And it's always said, even today, when you read the review papers and books, that the idea of the protoplanetary disk, the disk surrounding the sun from which the planets formed, comes from Kant and Laplace. And so indeed Poincaré devotes the first three chapters of the book to Kant and Laplace. So Kant, what did Kant think? Kant think thought that in the beginning there was an ebule at rest, no rotation, basically uniform. And with some slight density fluctuations. And by gravity, the gas started to accumulate around one of these density fluctuations, reaching so higher and higher density there. And by coming together towards the center, the gas was colliding with each other. And these random collisions eventually gave to the nebula some rotation. This is of course wrong because there is angular momentum conservation. So if the nebula is at rest at the beginning, it should be at rest always. And there is no way that collisions, random collisions, gives a global rotational motion to the nebula. So this is a fatal flaw in the thought of Kant, which completely disqualifies Kant in the eyes of Poincaré, who is very, very critical in his right sentences like this. No mathematician would ever buy this stuff. And obviously Kant is a philosopher. He doesn't belong to the mathematician club. So it's not that very well respected. In just six pages, Poincaré closes the chapter and moves on. Oh, that's very funny to read. Laplace, instead, is a totally different story. Laplace is a mathematician. He's highly respected by Poincaré. So it's Poincaré champion, Poincaré hero. And his theory is what clearly Poincaré likes the most. He describes it in two chapters, 60 pages total. The most developed description. And he also proposes some amendments to Laplace's theory but always moves in the framework of Laplace's ideas as a sort of mentor for him. And as we will see, actually the original Laplace idea are quite different from a modern view. This was discovered for myself as well because typically people say Laplace got it all. And actually as we will see in a second, the ideas of Laplace are relatively different from the current ones. So what did Laplace find? Laplace's theory was based on two assumptions. So that there is a nebula. Here represents spherical at the beginning. Here represented by this green circle. And this nebula has a central condensation whose boundary is here represented by this red circle. And there is a lot of mass here in the central condensation compared to the rest of the nebula. This is the first assumption, central condensation. The second key assumption is that the nebula is rotating because Laplace knew angular momentum conservation. But Laplace did not know about the work of Kant. And when he quotes some precursors of this idea he puts other people and not Kant at all. So the second assumption is that the nebula is using rotation and it's in a uniform rotation. So the rotational frequency is the same wherever you are from the spin axis. And this top plot here is on the same scale as the bottom drawing. So this is the radius. Zero is at the center of the central condensation and one is the initial radius, outer radius of the nebula. And the white line shows the frequency of rotation. As you can see it's the same, whatever the radius, whatever the distance from the spin axis. And I also report for reference because it will become important this red curve which is the centrifugal equilibrium. That means that if the rotation rate is that of the red curve, a particle is in orbit because the gravitational force is counterbalanced by the centrifugal force. Okay, so this is the start of Laplace idea. And then Laplace imagine that the central condensation contracts because of its gravity and by contract it spins up. That's again angular momentum conservation. For those who don't know, I don't know if anybody, somebody cannot know this, but angular momentum conservation is that law so that for instance when a skater spins with the arms open and then closes the arms, it spins faster. And so this is exactly the same thing. When the central condensation contracts, it's like for the skater to close the arms so we start to spin faster. And so as you can see here, this rotation rate goes up compared to beginning, beginning was here and now it goes up. But it's instantaneously propagated by viscosity flow out to the nebula. So not only the central condensation spins faster, but the entire nebula spins faster. That's key in Laplace model that at any time the nebula rotation rate is uniform. So then the central condensation contracts even further so it spins up faster. And we reach a point where at the extreme of the nebula, the rotational rate is exactly the centrifugal equilibrium rate. At this point, the outer surface of the nebula, which is an equipotential surface, starts to be distorted. It's not a circle anymore. It starts to become ellipsoidal. As the central condensation contracts even further, again the frequency rate goes up and now it intersects the centrifugal equilibrium at some radius, which is smaller than one. That means that the gas beyond this location is now at centrifugal equilibrium, is detached from the nebula, is in fact in orbit around the central condensation. The equipotential surface of the nebula now stops at this point where the centrifugal equilibrium starts and all the gas that, so it becomes more contracted and all the gas that is now outside of this equipotential level has to slide along the equipotential level and come down on the plane. And as the central condensation contracts more and more, as you can see this intersection point between the rotation rate and the centrifugal equilibrium curve moves inward and inward. That means that the surface of the nebula shrinks, this point moves to the center and more and more gas is left in orbit beyond the limit of the nebula. And so you can continue like that and you get the central condensation more and more dense, which eventually forms the sun and accumulates a lot of heat by contraction and the nebula becomes small and the disk becomes extended. So this is basically the idea of Laplace that Poincare describes in mathematical terms using these equipotential levels and the equations for rotating nebula and so on in a very, very clear way. So is this all? Is this the solution of the problem? No, because for Laplace and I think also for Poincare, it was inconceivable that you can form a limited number of planets, eight planets out of a uniform nebula. Why something uniform should give origin to a discrete set of objects? So Laplace and Poincare were after an idea, a different idea that not to form a continuous homogeneous disk but rather to form a set of rings with gaps in between rings so that each ring would then give you origin to one planet and so if you have eight rings you get eight planets at the end. How to get these rings? The original description of Laplace is a little bit hand-waving and so it's formalized in Poincare's book following actually the formalization provided by Mr. Rosch, which I'm trying to explain here. So the idea is like, the beginning is like what I showed before for Poincare, for Laplace. So the nebula, the central condensation contracts at some point the rotational frequency becomes equal to the centrifugal equilibrium frequency at the border of the nebula. So the next step, the contraction of the central core we start to form the ring. Now the idea is that when the ring forms and the gas outside of the equipotential surface slides down to go to the plane then the gas that is now at the surface was inside the nebula before and now it's exposed for the first time to vacuum. So because it's exposed for the first time to vacuum it's not blanketed by the gas that was originally beyond it now it starts to irradiate heat away into the vacuum and by irradiating heat it cools down and of course by cooling down the pressure inside the nebula decreases and the pressure is what keeps the nebula inflated. So by cooling down and decreasing the temperature then the envelope of the nebula shrinks. But when the envelope of the nebula shrinks the rotational frequency doesn't change because essentially the angular momentum of the entire nebula is due to the central condensation and it's propagated to the entire nebula by friction, by viscosity. So even if the nebula would like to speed up by shrinking essentially it has to stay at the same rotation frequency of the central condensation. So this shrinking without changing the rotational frequency doesn't leave any gas behind and that's how a gap is produced. And then the central condensation will keep contracting and when it contracts it speeds up the whole thing and so new gas is left behind beyond the new surface of the nebula. But then new gas is exposed to vacuum so it irrigates heat, decreases the pressure the atmosphere of the nebula contracts again without changing the rotational frequency which leaves behind another gap. And then the central condensation contracts and this leaves behind gas in orbit and then the nebula contracts and it's another gap and so on and so forth. This is really bright. But, and so this forms a discrete number of rings depending on how many times the nebula cools relative to the central condensation shrinking. Starting from the Titus Baudelot that describes geometrically the separation of the planets Poincaré even computed the times at which the atmosphere had to contract to leave behind the rings and gaps. And he concluded that the times at which the atmosphere contracted had to increase with a geometric progression. So if T0 is the time at which first the atmosphere progresses, the atmosphere compresses then the next one T1, T2 will be two times T0 T3 will be three times T0 and so on and so forth. So in Laplace model even in this variant to form the rings as you have seen it's crucial that the nebula is in uniform rotation and that this uniform rotation is the rotation frequency of the central condensation. So for Laplace the reason for that is viscosity is friction and so the nebula is forced by friction to rotate like a rigid sphere essentially. Cannot have any shearing because shearing exerts friction. Now Poincaré in this chapter mentions that Helmholtz computed that if one uses for the nebula the same kind of molecular viscosity that characterizes the atmosphere of the earth it would take something like 10 to the 22 years for a nebula extended up to Saturn to Neptune sorry to acquire a uniform rotation rate which is the rotation rate of the Sun. So this is sort of enormous number and so Poincaré says that but for him is not really a fatal stop to the theory essentially thinks well then probably there is some other mechanism to make the uniform, the nebula rotating uniformly because this is really required in order to form the disk and the rings and in this so the fact that Poincaré in some sense ignores this problem is actually very modern because we still have this problem today. Now we observe a protoplanetary disks and we observe even the accretion of material onto the stars that we can measure. And so from these two observations one can deduce the viscosity in these disks and the viscosity in the protoplanetary disks is many, many, many orders of magnitude larger than the molecular viscosities. And so we still have this discrepancy of timescale even today in modern times and the solution of astronomers has been to invent a concept of turbulent viscosity so it's not molecular viscosity that actually drives the evolution of the disk it's this turbulent viscosity due to the turbulence in the gas. The viscosity is very, very enhanced but it's still a very open subject of research to understand what generates turbulence and what actual viscosity turbulence can generate. So this is an open problem even today and doesn't stop us from believing in accretional disks and so it's very conceivable that for Poincaré it was about the same. So as I think you have understood the Laplace idea in particular is very different from the modern idea of a protoplanetary disk. And actually there was someone not very known in the history of science to Viscont de l'Igonde to which Poincaré devote chapter five and pretty developed chapter of 33 pages who actually imagined the formation of a protoplanetary disk in a much, much more modern way compared to Laplace and actually went very, very close to what we think today is happening. So and it's a curious that this person is essentially disappeared from the history of science. So for instance if you do a search on Wikipedia of all the names I'm quoting this is the only one that doesn't have a page of his own. So and I really found out about this person reading Poincaré book so that's very interesting. So L'Igonde said that, okay, initially starting thinking that at the beginning the nebula is a uniform rotation doesn't look like very realistic but probably a very rarefied nebula a just random motion in the gas but if you do the sum of all the velocities you will get some total angular momentum which would not be exactly zero. Why should it be zero? Zero is a very specific number would be very non-generic number to get. So the nebula despite it's chaotically moving in all directions has some non-zero angular momentum. Due to the inelastic collisions of gas particles and that was also a concept that he introduced that the molecules lose energy and by losing energy they fall towards the center of the nebula and start to form a central condensation. This is not exactly right the reality is a little bit more complicated because when two molecules collide they lose energy so they emit photons but these photons can be reabsorbed by other molecules which are accelerated. So actually in a modern concept we should worry about the gas opacity and the optical depth. But okay, the idea is about right if you think that the nebula is optically thin. And so by the gas falling to the center by angular momentum condensation it starts to speed up. Also this nebula because of this inelastic collisions has to become flatter and flatter and damp on a plane orthogonal to the total angular momentum of the system. And this is easy to understand by this is the angular momentum vector and this is the orthogonal plane by definition of angular momentum and orthogonal plane it means that if you take all the velocities of all the molecules along the z direction the arithmetical sum of all these velocities is zero. And so by colliding and damping the velocities between elastic collisions of course the z component of the velocities decreases, decreases, decreases until it becomes zero which means that the nebula becomes flat on the plane orthogonal to the angular momentum. So by this concept of inelastic collisions Ligunde could explain from an originally rarefied chaotic nebula the formation of the central condensation which eventually forms the sun the formation of a flat structure around it everything conserving angular momentum. And then again that was the issue why how to form eight planets out of the disk and Ligunde thought that rings form but rings form because of sort of gravitational instability not because of this game of cooling and contraction that I showed before from Laplace. And this is actually extremely modern. What I'm a little bit surprised about is that Poincare describes all these ideas does not criticize all of them because I think cannot be criticized but is not particularly enthused by that it presents this like a model among others and maybe sentimentally is more still more attached to Laplace uniformly rotating nebula and ring model maybe because it's more mathematically defined and you can do more you can really write equations to describe the evolution of the neighboring Laplace model and these equations are those from which I did the movies before the animation before whereas this is more qualitative you know and so in the chapter from five most of the chapter is devoted to dynamics of a medium where collisions are inelastic and comparing the evolution so the contraction to the conservation of entropy and to what the conservation of entropy predicts. Okay, so what we know now about protoplanetary disks before going to the formation of the planets well disks exist and now we can see with the most powerful telescope stars forming in the star forming regions and the young stars are all surrounded by protoplanetary disks some disks are seen phase on and so they appear like dark circles on a bright background which is the giant molecular cloud which is ionized and bright other disks are seen from the side so they appear like a black horizontal band over again a bright background some disks are bright themselves because they are undergoing foot evaporation so they are highly ionized so it's quite amazing that actually basically the idea of the protoplanetary disk is right and that the people from Laplace even can't could just by imagination imagine something like that imagine that the star forms by central condensation central accumulation of gas so the basic lines are definitely correct and we now have the proof these are not models anymore this is reality on how the nebula actually contracts and speeds up and forms a star and forms the disk things are a little bit different certainly the more much different from what Laplace envisioned similar to what Ligon de envisioned now we can do this kind of calculations using numerical simulations obviously numerical simulation made by Matthew Bate so in the simulation you will see the nebula forming so the density is represented by this color scale at the beginning the density is very very low so it's all black and this image is on the plane orthogonal to the total angular momentum of the initial nebula okay so the disk will be flat so let's see the movie so the gas starts to go towards the center that's why the density increases and increases because of angular momentum conservation this thing at the center spins faster and faster until it becomes rotationally unstable it develops a bar and then this bar launches a spiral density waves the spiral density waves extract a transport angular momentum and that's the reason for which new gas can still go and feed the star and we can observe the gas being accreted by the star using like an alpha emission lines and at the same time a disk is formed due to the gas that still has too much angular momentum to go to the center and if you wait longer and longer most of the gas accretes eventually onto the center by viscosity so as you can see there are no rings and there is also no uniform rotation really except at the very beginning when the bar forms now let's go back to Laplace idea of the rings because this is what von Karay buys and let's see how the planet could form the idea is that the planet with the gas cooling the oops sorry with the gas cooling these rings contract both radially and vertically so that the density increases and eventually the planet is formed and in Laplace he doesn't question himself of why one ring should give one object and not several on the same orbit this is something that von Karay details and Laplace didn't now the idea that the disk uh... by cooling down becomes narrower and narrower is actually wrong because we know now that if one takes into account the viscosity of the gas then actually you have viscous spreading not contraction the idea that the disk contracts radially is unfortunately not correct the big issue of the time was to understand not really why a planet forms because that was considered uh... conceivable if the density becomes high enough but why the prior planets have a prerogative rotation this was a key question at the time and for Laplace again friction tends to make uniform the angular velocity of the ring for Kepler laws the angular velocity of the inner part of the ring is larger than the angular velocity of the outer part of the ring but because of friction eventually these two rotational velocities have to become the same like in the original uniform rotation rate of the nebula and when the two rotation angular rotation rates are the same the linear velocities of course larger here than here because the radius is larger and so the planet that eventually forms by this uniform rotating gas will have a prerogative rotation because the outer velocity is larger than the inner velocity will actually be synchronous relative to the sun so we have one rotation in one orbital period but the synchronous rotation is indeed a prerogative rotation so this is the only idea of Laplace that Poincare criticizes explicitly saying l'explication de Laplace est un suffisant the explanation of Laplace is insufficient and the reason is that Laplace didn't know this but Poincare knows this after the work of Maxwell which was not clear and not rigorous but he wrote it again that if a disk has an intensity which is large enough larger than this quantity where omega is the rotational frequency of the disk then the disk becomes unstable gravitationally unstable and so it has to clump and form a self-gravitating clumps of gas and also not from just a unique planet but a number of of self-gravitating clouds of gas now modern astronomers use this criterion which is called Tumorist criterion for gravitational instability and when you read books the Tumorist criterion is given like this so it's not the volume density it's the surface density that has to be bigger than omega Cs is the sound speed but actually if one thinks about the sound speed is the product between omega and the scale height of the disk and sigma divided by the scale height of the disk is just two times the density so if you do the transformation you'll find exactly the same thing so the modern celebrated Tumorist criterion actually was worked out well before by Maxwell so because of this then as I said the ring will not become infinitely thin and infinitely narrow and to form only one planet unlike Laplace and Vision it will eventually start to break down in self-gravitating sphere of gas and so you will have several self-gravitating sphere of gas on similar orbits but not quite identical and because the inner guy, the inner self-gravitating sphere of gas will orbit faster than the outer self-gravitating sphere of gas when these things will collide they will merge forming eventually one unique planet but because the inner one's orbit always faster than the outer one the final planet will turn like this not like this so a planet formed by this mechanism should always have a retrograde rotation and not a prograde rotation unlike what is observed so this issue of the prograde rotation was the real big question of the time but first before going further and see the explanation for the prograde rotation let me review the observations of the time what did the people believe so for the astronomers of the time they thought that all the inner planets are all prograde including Venus that's which is retrograde and probably because Venus is quite a uniform atmosphere they couldn't observe the clouds and so they could not figure out that Venus was retrograde so all the planets Mercury, Venus, Earth, Mars, Jupiter and Saturn were all prograde but then they believed that Uranus and Neptune are retrograde now Uranus in reality the spin axis is essentially horizontal the obliquity of Uranus is 98 degrees technically it's retrograde but not really spin down like Venus but Pankare never mentions that actually so he considers it like a retrograde planet Neptune in reality is perfectly prograde but why the astronomers thought it was retrograde at the time because the surface of Neptune is very uniform so you can't detect any rotation just by looking at Neptune but in 1846 the major satellite of Neptune Triton was discovered a Triton is a retrograde satellite it turns in a retrograde in the opposite direction relative to the spin of Neptune and at the time the astronomers knew that all the major satellites of the giant planet Jupiter and Saturn orbit the planet in the same direction of the spin of the planet so for Neptune they could not measure the spin of the planet but they could measure the orbit of Triton and by analogy if Triton turns clockwise they thought that Neptune was turning clockwise too so in a retrograde direction and so this thing first six planets are prograde the last two are retrograde is very very important in the conceptions of the theories of the time so the first explanation of why the planets are prograde I have a prograde rotation comes from Phi who actually was the director of the institute the mechanics analyst here in Paris and Phi proposed that the planets form still inside the nebula when the nebula is still uniform before it develops the central condensation why? because if you are inside the uniform sphere of material then the orbital speed instead of decreasing like one over square root of R increases like R and so that means that in the scheme the outer planet the outer self-gravitating sphere of gas orbits faster than the inner one and so upon emerging this will give a unique planet on a prograde rotation and so the central condensation and the formation of the star has to occur later after planet formation at least after that Jupiter and the other planets form so Poincare dismisses this idea as crazy because when the center condensation forms that means that the mass inside the orbit of a planet increases enormously and if you increase the mass of that attracts that changes the gravity that the planet feels then the orbit of the planet migrates inwards and so Poincare computed with an easy math that in Phi's theory the Earth should have formed that 13 AU is the current distance between the Sun and the Earth so 13 times further than the Earth actually is now and Saturn 22 times further Saturn now is at 9 AU and he considered this crazy of course so this is the reason for dismissing Phi's theory in Poincare's book chapter 4 there is actually a positive side in Phi's theory that Poincare mentions is that in Phi's model the planets form before the Sun and this was appealing because at the time the age of the Sun evaluated from cooling times was only 100 million years but the geologist by looking at sediments and fossils and so on had deduced that the Earth at least had 200 million years so the Earth looked older than the Sun and this theory would explain why but Poincare doesn't really believe any of this dating that the activity was being discovered it was obvious that anything about heating and generation of heating and cooling was going to change so he thinks this is all nothing to think about so and this argument is much more solid so Phi is disqualified so the idea of Poincare solution of Poincare and this is really his own solution actually written in the book this is my thought and for the progressive rotation of the planet is based on tides the theory of tides was very well developed in those years by this person George Darwin who was the Sun or Charles Darwin also died 100 years ago and a really great man understood a lot of things in particular the evolution of the Moon and the origin of the Moon and his ideas are now being reconsidered more and more so then Poincare says okay tides are important and tides are particularly extremely efficient when the protoplanet is still an extended sphere of gas for these are the tides exerted by the Sun on the planet so the planets form like hot and extended clouds of gas they are retrograde because of the way they form but they are very affected by the tides and as George Darwin showed the tides have the effect of slowing down the retrograde rotation of the planet until it is synchronous so that it has one spin in one orbit but the synchronous rotation is actually a progressive rotation and then the cloud the cloud that is the precursor of the planet contracts because it cools down and this has two effects first it frees the planet from the tidal grip because the narrower is the planet the less sensitive it is to the tides raised by the Sun so eventually the tides from the Sun weakens and the planet can become asynchronous and also by contraction due to angular momentum conservation and understood it it speeds up so the rotation frequency will become faster and faster so it will be asynchronous but because it starts from a synchronous rotation which is prograde it will be a fast and prograde the rotation this is very clever Juan Carré brings forward three observational evidences for this theory being right so the first one is that the inner planets the first six are prograde and the last two are retrograde and Juan Carré says great this is because tides becomes weaker and weaker with the distance from the Sun so Uranus and Neptune felt tides which were too weak to ever become synchronous and so the reason to turn from retrograde to prograde there's something to do with something that has to become more and more prominent with the distance decreasing distance from the star from the Sun exactly like tides do then the fact that the satellites of the planets have orbit have prograde orbits so they spin around the planet in the same direction of the rotation of the planet of course Laplace and Juan Carré and everybody thought that the formation of the satellite is just repeats the same process as the formation of the planets around the Sun when the planet contracts it leaves behind the disk and from this disk the satellites form and of course the disk around the planet has to turn in the same direction of the rotation of the planet but the most distant satellites are actually retrograde and Juan Carré says this on the basis of one observation which is Phoebe Phoebe was discovered in 1899 is the biggest irregular satellite of Saturn we call irregular satellite the satellites that instead of having a coplanar circular orbits have eccentric inclined orbits and some of them are retrograde and Phoebe the biggest one has a retrograde orbit and it is the most distant the irregular satellites are much much more distant than the regular satellites which are very close to the planet so Juan Carré says great this is the proof that the most distant satellite formed early when the planet was still retrograde before becoming synchronous and before becoming prograde and so this sequence the outer satellites are retrograde the inner satellites are prograde is the proof that the planets form retrograde and then eventually they are turned into prograde okay what do we know today about planet formation actually the idea the Juan Carré and the others were after is actually very similar to one modern concept of planet formation which is gravitational instability a mechanism for planet formation which has been resurrected by the works of Cameron or Alan Boss and others so there is this criterion this gravitational stability criterion that says that when the density is bigger than the rotation frequency squared then the disk is gravitationally unstable so in the outer part of the disk where omega is small indeed a disk can fragment in self-gravitating clumps of gas but these clumps typically are very massive much more massive than Jupiter and this happens only far away when omega is small actually what people at the time of Juan Carré did not understand is that it's not enough to wait long time so that the disk becomes thinner and thinner and raw increases so that eventually raw can be larger than the rotational frequency whatever whatever is the rotational frequency so at any distance from the star the reason is that the major source of heating in the disk is viscous heating and that depends on their density itself so you can't increase arbitrarily the density without increasing internally heating which makes the disk thicker and decreases the density so the density cannot become arbitrarily large in the disk but it's self-regulated by the viscosity so this relationship can only be true farther out in the disk when omega is small and so the theory of planet formation by gravitational instability actually seems to work but predates only very massive planets very far away from the star we might see this kind of planets for instance the system HR8799 which is a system imaged with telescopes show this so this is the central star which has been cancelled using modern coronography techniques and the star is surrounded by three planets the masses are not very well known but from their brightness it's about from five to ten Jupiter masses and they are very far away 24 AU, 38 AU, 68 AU from the central star remind you that Jupiter is at five and Saturn is at nine so it can work but for other worlds not really ours for our solar system we think things are different and the modern concept of planet formation envision of course a protoplanetary disk but consider the parallel evolution of the gas and the dust and the solid component so inside the gas component there is the solid component that sediments in the midplane of the disk the inner part is dominated by rocks because it's hot the outer part is dominated by ices because the temperature is low and then the solids accrete with each other forming solid bodies which are bigger and bigger and eventually if a solid body becomes massive enough and there is still gas around then it can start to attract the gas and form the gaseous atmospheres that characterizes the giant planets like Jupiter and Saturn so this curves around from Pollack's model so this is the mass of the solid component of the planet which increases up to about 10 Earth masses and then once 10 Earth masses are reached the planet starts to attract the gas so this is the total mass of the planet gas plus solid so the difference is the total mass amount of mass in gas and this eventually goes into a runaway accretion fashion and the planet of hundreds of Earth masses like Saturn and Jupiter can be formed in this model so first formation of the solid core and then accretion of the gas the progress rotation of the planet is no mystery it's enough to look at the dynamics of the gas in the vicinity of the core of the planet so this is the direction to the Sun this is the orbital direction of the planet and these are the streamlines of the gas so you can see the gas comes in like this and turns like this something you can easily understand by solving the equations of the heat problem and so the accretion of the gas necessarily gives to the gaseous planet a progress rotation for the terrestrial planet is a little bit different because there is no gas and the rotation of the terrestrial planet is due to the collisions between the rocky bodies and because the rocky bodies can become eccentric actually it's expected that the final rotation of the terrestrial planet is a multirangle so in Poincaré books actually there is no mention of the physical differences between terrestrial planets and giant planets and actually people at the time thought that even Jupiter was solid that Jupiter was understood to be gaseous planet mostly gaseous planet only 1933 and they knew the differences in densities but they just thought that these were differences in heat and so there is no distinction that terrestrial planets and giant planets somehow must form differently or take two different evolutionary paths there is no distinction between the dynamics of gas and solids the idea is that everything forms like a cloud of gas and then upon cooling any gas eventually would become solid and so there is no need, no discussion the need about to remove eventually hydrogen and helium because hydrogen and helium would never become solid at the conditions of temperature and pressure all this old acid so to conclude three take home lessons by reading this book the first one is that science really evolves by standing on the shoulders of giants it's quite impressive to see how these people like Laplace, Poincaré and so on were the brightest minds at the time were still relatively far from the modern concepts that we have and science has evolved astronomy planetary science has evolved enormously since and this is not because there was a super genius after Poincaré where it would revolutionize the entire field just because of a community effort and the piece after piece, brick after brick the edifice has been built and also a lot of observations have come in in planetary science it's an observationally driven science there is no doubt the second one is a lesson of humility of course even if because of the standing on the shoulders of giants and building an edifice brick after brick by definition I would say the modern models are the best models ever made in the history of science doesn't mean that they are good doesn't mean they are correct we still have a lot of problems in understanding some things and we are aware of that but I would even go further maybe something that we think we understand is not correct Laplace fought to understand the origin of his rings and of course he was not correct so it's probably the same thing and given that I do planet formation models myself I question myself what will be my models in 100 years from now and I'm lucky if they can last 100 years they will be eventually be looked like naive and superficial I'm sure of that and the last take home message is that even the brightest minds can be misled by bogus observations so this is a pun for our friends or enemies the observers but this is actually interesting so I'm referring of course to the fact that Neptune was fought to be retrograde and Venus was fought to be prograde and Feebe was fought to be satellite like the others and this is something actually we should keep in mind today because with the discovery of extrasolar planets we are bombarded by new data every day and there is a tendency in the community to try to write down and publish theories to explain the data as they come in forgetting that these data are still very confusing very partial, very dominated by observational biases in some cases even wrong and so we should learn these lessons and I mean a good theoretician should not start thinking until he really feels that the data are solid and robust and of course it's a skill to understand when this occurs because if you wait too long then the others do everything you have nothing else to do so but there is one thing on which Poincaré is really modern and I would like to end on this is this text which is actually not in any chapter it's in the introduction of the book that Poincaré wrote and it says this which I translate in English I can read it anyway it's too small it says Poincaré says that Laplace did not mention explicitly the possibility that other of other planetary systems but clearly implicitly thought that they should exist and they should all be more or less the same because they come out of a universal formation process which is the formation of the disk and the formation of the rings and then Poincaré says that the recent progresses of stellar astronomy does not allow us to still think the same and because telescope has discovered such diversity of stellar systems of course extraterrestrial planets did not well not know that we did not expect and then Poincaré goes on saying if the stellar systems are so diverse on each other there are individual stars, double stars, triple stars massive stars, dwarf stars different temperatures are different there is no way the other planets will be like our own the same kind of diversity that we see in stars will be also in the planetary system and this is of course perfectly right in 1995 the first extraterrestrial planet has been discovered since we've found a thousand there is a huge diversity of planetary system this is shortly summarized in this diagram which shows the distance from the central star and the eccentricity of all the giant planets discovered today and as you can see giant planets can be at any distance from the sun not just very far from the star like our Jupiter and Saturn can have some, some have even orbits much much smaller than that of Mercury some have circular orbits like our own planets but others have very eccentric orbits like the comets in our solar system and in terms of masses everything has been discovered from 10 Jupiter masses down to one Earth masses with a continuous mass distribution so really there is a huge diversity Poincaré predicted it it was totally forgotten and that's why astronomers were sort of shocked when this diversity of planetary systems came out from telescope surveys but this is what Poincaré already expected thank you very much this beautiful lecture and Shakespearean conclusion so maybe there are questions yes, please thank you very much for this lecture it was extremely interesting I have a question regarding the magneto-hydrodynamic aspect of this problem the fundamental equation existed we arrived at the same period and Poincaré doesn't mention any of these effects no, exactly and do you have any comments on that? I think so, of course Maxwell equations were known but the idea that magneto-rotational, magneto-dynamics could give rights to turbulence and solve this problem of the viscosity is much much much more modern and I think they did not have really the tools at the time to figure that out maybe not even yeah, I mean there was this realization that the molecular viscosity was not enough to keep the Nebula uniform but they did not really try to go further than that but it's true so when I said that we have the same problem today and so the molecular viscosity is not the cause of viscosity of the disk we think it's turbulent viscosity then the question is what gives origin to turbulence in the disk and magneto-rotational instability is considered by most of the people in the community today the main driver of turbulence in the disk which explains the viscosity that is indirectly observed by the Stellar-Akrician way so Alessandro, so thanks first for this the magnificent lecture because I had the book for a while in my shelf and I've started to read it a little bit but never dare to really go through and I should thank you for that my question is, did you look to what was the following of this book did it have any impact or not or was it totally forgotten did you look to that or not really but my impression is that so Laplace ideas and remained in the community certainly and at least the idea of the disk and things started to progress later when star formation was revisited and with the first computer simulations and this was in the 1950s more or less but the idea of the disk was never a question so something that actually went into the common culture of the astronomers at any time and then the fact that Jupiter is gaseous was discovered in 1933 and I think boosted a little bit the idea that some planets at least can start from the gas component and also pointed out that there are two clearly different categories of planets in the solar system and so everything has to be much more complex than what was explained at the time and sorry, perhaps it's a stupid question but what for the asteroid belt I'm not sure, was it already discovered by this time and if yes, well the asteroid is small so I'm not sure, what did they think of them and what do we think of them now for the information? Okay, so the first few asteroids had been discovered in 1901 was the discovery of Ceres, so here we are in 1911 so I think 10, 15 asteroids were known thinking if Juan Carreira mentions them I think he mentions them in passing and he says that there was this law, the Tizus Bode law describing the distance of the planets from the Sun and they had a gap so the fact that they found four or five bodies at this place was considered like what the proof that something was there and probably exploded and I think he mentions that in passing in the introduction and not in the chapters anymore and now we believe that asteroids are more the leftover of planetary formation they never formed the planet probably because Jupiter formed earlier and disturbed the region preventing further acquisition of the region. So thank you again. Thank you.