 Comme vous pouvez le voir, je ne suis pas Alicia. Alicia est ici, c'est mon étudiant. Nous travaillons dans l'application laboratoire, mais nous faisons une théorique astrophysique pour longtemps pour moi. Comme vous le voyez, c'est un problème très difficile parce que les observations sont très difficiles à obtenir. C'est très difficile. La numérique est très puissante, mais vous pouvez faire ce que vous voulez. Vous devez le mélanger avec tout ce point. Et pour le temps, j'ai envie de présenter un summary de la situation, si vous voulez. En tout cas, j'ai donné à vous des idées, j'ai envie d'avoir une bonne idée de ce qui se passe dans ces systèmes. Au début, je parlerai des factures en astrophysique. Notre problème est concerné par les clusters globales et les galaxies. Ce n'est pas une large-scale structure, comme Michael Joss m'a dit. C'est un autre problème. Vous devez inclurez la relative générale. En ce cas, vous êtes sur une petite scale de l'univers, la scale galactique et la relative générale. Ce n'est pas important. Le deuxième problème que j'ai voulu évoquer, c'est le problème de l'équilibre état, dans lequel l'équilibre est observé par ces systèmes. Ce n'est pas un problème très simple, comme vous le voyez. Après, je présente des instabilités, parce que je pense qu'il y a quelques instabilités qui s'occupercent dans ces systèmes. Et ils sont très intéressants à faire un bon scénario, qui je présente à la maison. Il y a quelques factures. Le modèle standard de la dynamique stelaire est très bien connu. C'est un système avec des n-compagnons en interaction gravitationale. Comme l'n est très large, vous êtes dans un limiter. C'est un problème dans lequel l'équilibre état est associé à une fonction de distribution, qui s'évolue dans le potentiel gravitationnel. Et les équipements dynamiques sont les systèmes de Vlasov Poisson. Il y a une grande densité, qui est l'intégral de la fonction de distribution sur les villes. Et il y a deux constants dans ce problème. Un petit temps, qui est un temps dynamique, qui est obtenu de manière différente. Mais vous avez toujours une diminuie de l'homérité. Un sur la route square de G, le constant de Newton, et la grande densité du système. C'est le temps d'aller par une star, d'aller sur un côté à l'autre côté de la galaxie, si vous voulez. Et il y a un autre temps, qui est un temps long, un temps considéré comme l'infinité dans des systèmes. C'est le temps qui est pris par une collision, pour changer d'une manière ou d'une magnitude, la vitesse dans le système. C'est une très grande calculée par Chandra Sekar, dans le milieu de l'année dernière. Et vous avez beaucoup de controversies sur ce résultat, mais tout le monde est presque dans la connexion avec ce formulaire, il n'y a pas de détails. Et vous voyez que ce temps est N, ce temps-là. Donc, si N est equal à des billions, comme dans la galaxie, puis, comme ce temps-là, il y a des millions, des centaines de millions de années, vous comprendrez que ce temps-là est très, très long, en comparaison entre l'âge de l'univers. Donc, parfois, vous pouvez considérer que ce temps-là est infinit. Mais pour la cluster globale, il n'y a plus de stars, il n'y a plus de stars, il n'y a plus de 1000, 10 000, 100 000 stars. Vous pouvez avoir l'effet de la collision dans la cluster globale. Let's see this point. So, here you have two representative cluster globale. We are a lovely object in the sky. If you've never seen a cluster globale in a telescope, let's try. It's very impressive to see in the reality. You can do this with a small telescope. It is very possible. So, you are here the two kinds of globular cluster. This one, with a large core, and this one with a collapsed core. The difference is very remarkable. So, when you plot the density of stars in function of the radius of the system, in this kind of system you are a large core, and after a low, in this kind of system you are no more a large core, and as we said, a cusp in the core. So, they said, astrophysicists said that this kind of globular cluster are core collapsed. And you have almost 80% of the globular system, the system of the galaxy. In our galaxy, there are almost 100 globular clusters, and there is almost 80 in this state and 20 in this state. And we can say that this effect is an evolution effect. How we can say this? This is very simple to say, to easy to say. You plot all the globular cluster in a diagram, which is, here you are the globular cluster, the alfamast radius, RT, the limit of N, here an estimation of the radius of the core. And when you plot all the globular cluster of the galaxy in this diagram, you see that there are repartition. When you identify what, where there are collapsed core by this symbol, they are all here. And why I said it is an evolution effect is because when you make a color which represents the time, the dynamical time of the system, you see that all the collapsed core are the blue one and the blue one are the one that have the smaller dynamical time. So during the evolution of the galaxy, the evolution of the galaxy during 10 billion years, they are affected by collision and the effect of collision makes this core to collapse as we see in a moment. The model for globular cluster is very old. There is no more explanation but it is very old. There is an explanation but the explanation is not so good as well. This is the king model. The king model it is fine tune heuristic model. You consider an isothermal sphere. You will see why. And you cut it because you have some problem at the end of the distribution. So king said there is no problem. We have to cut and there is no more problem. And you obtain this model. And with three parameters you can fit very well. This is two globular clusters with a core and you can see that you can fit in a very good, with very good accuracy the data and the model. And the collapse core are rather like an isothermal singular sphere. So the density profile is only a low radius to the power minus 2. For the galaxy, the situation is rather different. In fact you have two kind of galaxies. You have LSB which are large, low surface brightness galaxy. We are small galaxy, isolated galaxy. And which are corallaux fashioned. You can see here such of LSB. Which LSB represent in fact 90% of all the galaxy. We can see. It is not very seeable. But when you plot the density if you can of such an object, it is very recent 10 or 20 years we are able to do this. We obtain a corallaux structure. A core with no slope here and minus 2 in the power and the slope. So this is for LSB galaxies and for high surface brightness galaxy. We have such profile for such galaxy or for galaxy form in the large scale structure simulation by cosmologies for example. You obtain such for example, they say this is NFW profile or NSTO profile. And in fact there is a study by Merit some years ago, 10 years ago, which said that NFW is not the same, the better but the best. But rather equivalent profile, it is pruniel simian which is here. And this is a deprojection of the famous Devocouler law. And in this deprojection you see an exponential and you can think that there is some thermodynamics in back in front. The difference between these two kind of galaxies is not an evolution of the system like in global cluster. It's what I think. I am not only the one to think this. The difference between these two systems is between the history and the evolution, the formation of the galaxy. So now what is the problem with equilibrium? If you want to describe a galaxy by thermodynamics, it is a no problem with no solution. A solution by Lindembell in 1767. This is a violent relaxation. So you split the phase space into two energy macro cells. You divide these macro cells into micro cells of the same old. Macro cells contain all the particles with the same energy. And after you compute to maximize the number of complexions and their contents. But there is a lot of problems. They are not distinguishable or not. What are the actual constraints, the energy, the mass, the angular momentum or whatever? Can we use, this is not a simple remark, can we use the steering formula at first order? Like everything do in thermodynamics, statistical thermodynamics. Is it a Vlasov equilibrium and over question I don't know it. So let me present for the person who don't know this, the Lindembell result. So in this calculation, Lindembell suppose that particles are distinguishable. But in fact if you see in detail, he suppose that all the particles are the same mass. So it is a little bit contradictory to use. The constraints imposed are the total mass, the conservation of the total mass and the conservation of the total energy. And as you see, as you know, gravitation impose a classical poly principle because you can put two particles at the same place due to gravitation. So when you compute the number of complexion you obtain this way, which is a Fermi Dirac with distribution with distinguishable particles. And when you compute the distribution, when you look for in this set, so the set of distribution with a finite mass and a finite energy, when you compute the maximum of the entropy, of entropy like, you obtain a Fermi Dirac distribution. As the gravitational system are not degenerate, you can consider that any is very smaller than GE, the degenerate sense of the state, then this distribution reduced to the Maxwell-Boltzmann distribution. So this is the conclusion of Lindembell. But it is good because in the thermodynamic limit when n goes to infinity you obtain Maxwell-Boltzmann distribution. When you integrate this distribution over the velocity you obtain a density exponential of minus an inverse temperature times the potential. And when you plug in into the Vlasov equation you obtain such an equation. And when you use Gidas and Nuremberg theorem of pure mathematical theorem it is said to you that such a system is spherical. So you are happy because everybody thinks that all the gravitating systems are spherical. For example all the globular clusters are spherical. Very few not spherical globular cluster but there is a slow rotation we can explain it. You say to me the spiral galaxies are not spherical but if you have only a disc galaxy, the disc is unstable so around the disc you have a spherical dark material. So in this context all is spherical. So it is a good result but you have a lot of problems with this result. I speak with two problems. First of all this is a classical problem of this kind of calculus. The result is not in the set where you are looking for. Because when you can compute the behavior or the density you obtain in the outer region and you obtain that at the limit when air goes to infinity the density goes like air to the power minus 2. There is a minus here. So when you integrate to obtain the mass over R3, R3 you obtain a divergence. So this system has not a finite mass and you are looking for a system with a finite mass in the basic hypothesis. So it is a problem. A less classic problem perhaps for me you are looking for like a maximum of the entropy in a system which is described as a glass of equation. And glass of equation conserves the entropy. What does it mean to search the maximum of the entropy when it is conserved? For me it is a problem but perhaps it is not a problem. The classical solution like king model we have a cut of this problem. So if you cut you have no problem but you cut after at the end of the story you have to cut at the beginning to be honest. So the cut, the king cut or box there is a large theory in which people put isothermalsphere in a box so you haven't such problems. But what is this box? But if you cut you have no more the problem of infinite mass but you still have the problem of glass of. And there is two modern ideas to solve these two problems simultaneously. The first is a paper by a Brazilian guy and I when this famous trimester when Jihad was in Paris two years ago. We have, I will present to you and our idea six years ago by Orge and Lili Williams. They said that at the end of the system larger the concentration, the number of stars are not so sufficient to use the sterling formula cut at the first order. And when you consider over terms, second order terms and more complicated terms, this gives a finite mass to the system. It is so simple, why they don't think to this before. And I show you what it gives when you do this change. So the distinguishability problem, this is the idea with Beraldo et Tall two years ago. The violent relaxation of self-gravitation, self-gravitating system is not able to produce a full mixed state. It is well known and collision are needed to to to fully mixed the system, but collision takes a long time. So when the system is collapsed, the system is not in a full mixed state. After a few dynamical states, the system only is partially mixed and the phase space contains isolated mixed in Iceland. Particules in mixed regions are indistinguishable among themselves, but distinguishable between the over islands. So in fact you can consider that all particles are the same property in regards to the distinguishability. There are a set of particles we are distinguishable and a set of particles we are indistinguishable. So when you do this, the number of complexes using these remarks becomes this way. And when you do the same calculus, you obtain this distribution. And when it is Maxwellian, you will see it. But this kind of hypothesis assumption breaks the Vlasov symmetry because to obtain Vlasov equation, you know that you have to stop the BBGKY hierarchy. And you have to suppose that all the particles are the same law in this improbability. And in this case, they don't have the same law. So this gives the possibility to the entropy to grow during the violent relaxation. To obtain this kind of distribution function, this kind of density profile, so it is always interesting. You have a core and a halo, but you are still faced to the problem of the mass here. This is the slope of the halo. So you see that including this remark of distinguishability, the slope is always minus 2 at the end of the system. So when you consider the new sterling approximation, as I said in this paper, the situation is changed and this is the same. You are always a Maxwellian, so it is so good for explain what you see. And in this case, you see that the slope is not longer 2 and it is for depends of parameters. So this kind of system has a finite mass, so you don't have any problem and you don't have to cut it before doing the job. Let's me now speak about instabilities quickly. The first instability, which fashion self-gravitating system, is the jeans instability, it is very well known. It is a lot of name, it is very simple. When you consider a new system with velocity dispersion, with constant density and with radius, when the radius of this system is greater than a certain radius, which is the jeans radius, the jeans length, then the system collapses. And what you obtain after the collapse, in fact, you obtain a cor-halostructure. It is always the same thing. And not a general cor-halostructure, in general you obtain a cor with a slope with minus 4. And what we do with Alicia, we remark that this model you obtain is an Isochron model. Let me say a few words about this Isochron model because it is very interesting and fabulous model by Michel Enon, the one of Enon Health. It is in 1968. And the idea of this model, you know that when you have a sphere, I don't resist to write on this marvelous blackboard. You consider a star in this system. And you know that this star is characterized by an energy and a square momentum. You know that the orbit of this star in this system lies in a plane. It is a first-year problem, first-year university problem. And you know that the orbit in this plane. And so you have the distance to the center of this star is periodic. And you can compute the period. And it is very easy between the peri-center to the apocenter. This is very simple. If you don't know, you can find it in classical books. The energy minus the potential minus the square of the momentum. So you can see that in general this period of the star depends on the energy and on the square momentum. Michelinot asks the question, what are the more general potentials for which this period do not depend on L2? And the names of these potentials are the isochron potential. And he obtains the formula. The formula is very simple. It is minus. So you can see that when B is equal to zero, an isochron, it is a Kepler potential. When B tends to go to infinity, you can show that this potential is equal to the harmonic potential. And in between it is an isochron potential. And what you obtain after the genes, the genes instability, you can see here, you can see here, it is this potential. And I think I understand why. It is not in two years ago in the middle of this amphitheater. So here you are the collapse. And in blue and in black you are the density. When you put it into the Blasov equation, you obtain this formula with only one parameter. And when you fit the data with the model, it is okay. So the isochron model is a product of the violent relaxation in this sense. The second instability is well known by the people like you. It is Antonov instability because it is a thermodynamical problem. When you put an isothermal sphere in a box with four parameters, an energy, a temperature, a mass and a radius, we can show that when air goes to infinity, the mass goes to infinity too because it is an isothermal sphere. But when you cut the system, you put it in a box, there is no problem with mass. But you cannot put any isothermal sphere in any box. So you have constraints. If you fix the temperature, then the good parameter is the density contrast. It is the ratio between the value of the density at the center of the system and on the edge of the system. And when you want to put an isothermal sphere fixing the temperature with a density contrast larger than this number, it is impossible. And when you fix the energy, the density contrast for the instability is less, this is well known. This is at the basis of the ensemble in equivalence. And I make the remark that this instability, Antonov instability, the parameter of instability is much more or less, is less, is very, when you add a thermostat to the system, is the problem of caloric curve. I think you know this kind of problems. And the result of an isothermal sphere in a box, it is a Corallo system. And the result, the product of the Gravothermal instability, of the Antonov instability, is the collapse of the core. So we think that it is what happens for globular clusters. Such systems have density contrasts we grow during the revolution. And when density contrasts go up to the limit, the core shrink and collapse and you obtain, this is the common explication. Less known instability, this is radial orbit instability. When you have, we have done a proof some years ago, this is a very strange instability. This is an instability which needs dissipation to develop. This is very interesting, I refer to the paper to see it. And this instability transforms a sphere into a triaxial system. The mechanism is the following. When you are a system which reaches in the velocity space, and when you add a non-radial perturbation with a direction which is non like the overs, there is, you pull one, this is radial, this is pencil, this is radial orbit. And this extends one of the orbits. And there is a torques existing between this orbit and the over. We are close to this. And after, there is a strange. And this is at the end, you obtain a stable equilibrium, which is no more spherical. I can show this mechanism is very, very complicated. What you have to keep in mind is you can produce this instability, not only in simulation but in reality. How to do this instability in reality? You have to make a collapse of something onto something. Let me show something about this. Here you have a plumber sphere, which is a spherical galaxy. And around it, you have a homogeneous system. And this homogeneous system is a gene unstable. And the gene's instability is parametered by the initial virial ratio. When you are far from one, minus one, the instability is very strong. And when you are near one, the instability is not so strong. So when you make a collapse of a system onto another system, you can obtain this instability. This instability is evident on the simulation and you can prove it. In this case, the system receives a lot of radial velocity in this phase space. So it develops the radial orbit instability and develops a bar. In this case, it receives not a sufficient amount of radial velocity. And the instability cannot develop. We think that this kind of instability is at the origin of a primordial elliptical galaxy. In a few years ago, 10 years ago, 20 years ago, when they do simulations, cosmologists, they don't can't explain why there is a primordial elliptical galaxy. They think that all the elliptical galaxies are the product of the merging of two spirals. And so there is no reason to have elliptical galaxies at the beginning of the universe in the simulation. Using this kind of instability, you can produce such a galaxy when you want. But in fact, when you have produced such a galaxy and if it is not isolated, the merging process, something goes on to it, et cetera, et cetera, and it loses its possibility to remain elliptical. But in fact, there is a footprint of this instability in the slope of the density. It is a very interesting paper by Katz, you see it. And this instability is very important to observe what we observe. OK. So let me finish. I finish by putting all these things together. What is the paradigm to explain the formation and evolution of globular system, globular cluster system? At the beginning you have an initial homogeneous cloud. It suffers a violent relaxation, genes instability, it forms an isochron. If the collision on a very few time, dynamical time, on a less, on a more and more longer time, there is a collisions who produced a system like a king. And after the evolution can make the core of the king collapse to form the 20% of the globular cluster you obtain. In the galaxy context, it is different. If you have a hierarchy scenario, so you can idealize by two systems, a smaller into a bigger one. For dynamical reasons, this is a smaller which collapse first. So the first collapse on the gene stem scales to form an isochron. But in this context, the bigger one is a thermostat. And in this case, the critical radius, the critical density contrast is very smaller than 700. And it is equal to 244. So the Antonov instability occurs at the very beginning of the formation of the system. And you have all this part of the system is pushed towards the center. And it is a good hypothesis, a good, a good mechanism to explain why there is object which is in the scope of GAD. Because it is a good possibility to form a supermassive black holes. Which are not present in the over paradigm. We are never seen a big black hole at the center of globular cluster. After the big system collapse onto this and you can produce such thing with a black hole at the center. Or, always with the black hole, but if the collapse is violent, as we see in the simulation, you can obtain a triaxial structure like an elliptical galaxy.