Equilibrium of stellar dynamical systems in the context of the Vlasov-Poisson model
aa r X i v : . [ a s t r o - ph ] J un Equilibrium of stellar dynamical systems inthe context of the Vlasov-Poisson model
J´erˆome Perez
Applied Mathematics Laboratory, Ecole Nationale Sup´erieure de TechniquesAvanc´ees, 32 Bd Victor, 75739 Paris Cedex 15
Abstract
This short review is devoted to the problem of the equilibrium of stellar dynamicalsystems in the context of the Vlasov-Poisson model. In a first part we will reviewsome classical problems posed by the application of the Vlasov-Poisson model tothe astrophysical systems like globular clusters or galaxies. In a second part we willrecall some recent numerical results which may give us some some quantitative hintsabout the equilibrium state associated to those systems.
Key words:
Classical gravitation, Vlasov-Poisson system, Equilibrium
Globular clusters and galaxies are concentration of stars whose physical char-acteristics are such that allow astrophysicists to model them as being in someequilibrium state for not too long time scales. The large number of stars andsome properties of the gravitational interaction − which evidently play a keyrole for their dynamics − made the Vlasov-Poisson model a good candidatefor their modeling. In this short review we present some results and problemsof this field of research. The Vlasov-Poisson model is a mean field approximation of the gravitationalpotential generated by a large assembly of point masses in the context of the
Email address: [email protected] (J´erˆome Perez).
Preprint submitted to Elsevier 20 December 2018 on dissipative kinetic theory. In order to apply this model to self-gravitatingsystems like globular clusters or galaxies, we have to pose three classical hy-potheses : First of all, it suffices to consider only Newtonian gravity for themean field interaction between stars . Second, all stars are statistically equiv-alent, particularly they do have the same mass and they do not evolve. This isa drastic approximation but it could be relevant in a mean sense and duringquite long evolution stages of these objects. Third, the considered systemsmust be non-dissipative on the time scales of interest. Nevertheless, it is wellknown that stellar dynamical systems could be dissipative mainly by two pro-cesses : On the one hand , such systems could contain gas which dissipatestheir total energy by dynamical friction. This feature must be considered formodeling the visible part of spiral galaxies. On the other hand it is well knownthat deviations from the mean-field forces due to the discreteness of the actualstellar distribution will alter the distribution of kinetic versus potential ener-gies as computed from the Vlasov-Poisson equations. In fact, this is a problemof time scales. The dynamical time T d of self-gravitating systems depends onlyon its mean density ˜ ρ via the relation T d ≈ / √ G ˜ ρ where G is the Newtongravitational constant (see (6) for details). It is typically the time taken bya test star to cross the system. Another duration is under interest : the time T r required for the mean velocity of the system to change by of order itself.This time has been estimated by Chandrasekhar for uniform systems to beproportional to the dynamical one via the relation T r ≈ N T d / ln N (see (6)p. 189 for details). It is typically the time for which the dissipative process likeencounters plays a key role in the system’s dynamics. Galaxies are composedof at least N = 10 stars, therefore this time limitation is not relevant in thiscase. The situation is not so clear for globular clusters. The number of theircomponents ranges from 10 to 10 . If initial stages (during a few hundredof dynamical times ...) could be non dissipative and therefore potentially de-scribed by Vlasov equation, during late stages − for billions years clusters − encounters dissipate energy and stellar dynamicists introduce Fokker-Planckformalism. Self-gravitating collisionless systems could be modelized by a phase space dis-tribution function f ( r , p , t ) and a mean field potential ψ ( r , t ). Vectors r and p are the usual conjugated position and momentum which are elements of R d The general theory of relativity could also be considered in the context of thecosmological principle. The system takes then into account the scale factor a ( t ) ofthe Universe. This formalism is often used in the context of formation of the largestructures of the Universe. d dimensional systems. The functions f and ψ are coupled by the Vlasov-Poisson system, which is for d = 3: ∂f∂t + { E , f } = 0 where E = p m + mψψ ( r , t ) = − Gm Z f ( r ′ , p ′ , t ) | r − r ′ | d r ′ d p ′ ⇔ ρ ( r , t )∆ ψ = 4 πG z }| { m Z f ( r , p ′ , t ) d p ′ ψ bounded at ∞ The quantity E denotes the mean field energy of a test star and plays a centralrole in this system. The function ρ ( r , t ) represent the mean mass densitydistribution of the system. Writing the Vlasov equation as above − i.e. usingthe Poisson brackets − makes trivial the well known result that every positiveand normed function f o of the mean field energy is a steady state solutionof this system. A natural question, initially posed by S. Chandrasekhar inthe middle of the last century, could then be : What are the properties ofphysical systems whose distribution function writes f o = f o ( E ) ? Althoughthe answer of the inverse problem was well known by astrophysicists , thenatural direct one was solved only fifty years later (2), mainly by using inthis context a difficult, but classical, mathematical theorem by Gidas, Ni andNirenberg (1). The main ingredients of this result could be sketched as follow: If the distribution function depends only on the mean field energy, then themean mass density depends on the position r only through the mean fieldpotential f o = f ( E ) ⇒ ρ o ( r ) = m Z f p ′ m + mψ o ! d p ′ = ρ o ( ψ o )In this case, Poisson equation writes ∆ ψ o = c ρ o ( ψ o ) where c is a positive con-stant. The physical context allows additional hypotheses : Newtonian gravityimposes that ψ o ( r ) is a negative function bounded at infinity. The classicalcontinuous limit let us consider that ρ o ( r ) is a positive and continuous func-tion. The GNN theorem (1) then allows us to claim that ψ o = ψ o ( | r | ) andtherefore the corresponding system has spherical symmetry in the spatial partof the phase space. Properties of the system in the velocity part of the phasespace are more evident : The dispersion velocity tensor is clearly proportionalto unity, therefore the system is said isotropic in velocity space. Spherical andisotropic steady state solutions of Vlasov-Poisson systems was intensively stud-ied in the context of stellar dynamics as classical models for globular clusters As a matter of fact, a spherical self-gravitating system must be associated to aradial gravitationnal potential which produces naturally a radial density profile byPoisson equation f o = f o ( E, L ), an extension of the GNN theorem showsthat if the mass density is monotonic, then the system is always spherical.However, the tangential velocity dependence of L = r v t makes the systemanisotropic in the velocity space. The asymmetry of such systems is associatedto an imbalance in the ratio of radial over tangential star orbits velocity distri-bution. Since the end of the 80’s, stellar dynamicists have understood that suchanisotropy could be at the origin of an interesting instability. As a matter offact, if anisotropic spherical stellar systems are generally stable against radialperturbations , it could be proved that systems whose cannot be infinitesi-mally perturbed by radial disturbances are intrinsically unstable(4). This isthe fine mechanism of Radial Orbit Instability : For pure radial orbit system,each star orbit extension is exactly the radius of the whole system. Thus,such a system cannot receive infinitesimal radial perturbation which affectsby definition only an infinitesimal part of the system. On the contrary, anynon radial perturbation associated to a given spatial direction could stretchor compress infinitesimally the spatial extension of an associated star orbit.This feature generates a tidal friction which makes an instability to grow andforms a triaxial system from an initial sphere. Radial Orbit Instability triggerswhen a sufficient amount of radial orbits is present in the system − a generalcriterion is given in (4) from distribution function susceptibility to receive ra-dial perturbations. As indicated by numerical analysis (11), this feature couldbe at the origin of triaxiality in some self-gravitating systems like ellipticalgalaxies.Less is known about steady state solutions characterized by distribution func-tions depending, in addition of E, on more complicated integrals. If somespecial models associated to f o = f ( E, L z ) are clearly triaxial (see (6) for de-tails) , there is, up to now, no extensions of the famous GNN theorem whichallows to claim anything in a general way.4 Equilibrium of a stellar system
Taking into account the time problem limitation presented in section 2 andtheir generally quiet physical properties, one can modelize globular clustersand at least elliptical galaxies by steady state solutions of the Vlasov-Poissonsystem. The fundamental question that we have to answer is : What are theassociated distribution functions ?This problem was attacked by stellar dynamicists using three approaches :Thermodynamics, comparison with observational data and numerical simula-tion.The thermodynamical approach, e.g. (13), is based on the classical assumptionof statistical physics which associates the equilibrium state to the maximumBoltzmann entropy one. In the late sixties, (3) reconsider this problem andfailed thoroughly : The classical result of these works is the isothermal spherewhich distribution function is f o = f ( E ) ∝ exp ( βE ). This failure comes fromthe fact that in the 3 − D case, such entropy maximizer no exists. Nevertheless,interesting features come from such an analysis when the system is put in anunphysical box − see (10) for a review of this topic. Another problem is thefact that the Boltzmann entropy S = − Z f ln f d r d p which is extremalized in this approach, is a conserved Casimir functional inthe Vlasov-Poisson context !Observational approach consists in the integration in the models of observedproperties of globular clusters or galaxies. A non exhaustive list of such amodels is presented in classical text books like (6) or more recently (7). Mostfamous ones are the King model for globular clusters which is an arbitrarilytruncated isothermal sphere, and the Navarro-Frenk-White(8) radial profilefor dark matter halos associated to the galaxies. From a theoretical point of A Casimir functional is on the form C [ f ] = R C ( f ) d r d p where C is a smoothfunction. It is well known that such functionals are time conserved quantities if f is a solution of the Vlasov equation: d C [ f ] dt = Z dC ( f ) df ∂f∂t d r d p = − Z dC ( f ) df (cid:26) p m ∂f∂ x − m ∂ψ∂ x ∂f∂ p (cid:27) d r d p = − Z (cid:26) p m ∂C∂ x − m ∂ψ∂ x ∂C∂ p (cid:27) d r d p = 0where the last equality follows by intergrating the first term over x and the secondterm over p , since f → | x | , | p | → ∞ . The idea is to produce an equilibrium state from the gravitational collapse ofan arbitrary physical set of point masses. This is not a recent idea and it waspioneered by Spitzer since the late sixties. The effective realization of such astudy in a general way was the result of a serie of works displayed over morethan twenty years ((11) and reference therein, (12) and reference therein). Inthose works , a general understanding of observed equilibrium properties ofglobular clusters and dark matter halos of galaxies follows from a detailedanalysis of numerical gravitational collapses of a sufficiently large set of N point masses. Gravitational collapse from physical initial state produces gen-erally a sphere. Flatness is possible by Radial Orbit Instability but it requiresstrong and inhomogeneous collapse . The radial mass density profile of suchspheres splits into two distinct classes : Homogeneous initial conditions col-lapses into a core halo-structure. It consists of a constant density region (thecore) which extension can reach the half-mass radius of the system. This coreis surrounded by a radial power law decreasing density region (the halo). Theother class is composed by sufficiently inhomogeneous initial conditions whosecollapse toward a monotonic radial power law decreasing density without no-table central core. The origin of the collapsed core of such systems could beexplained by the effect of the Antonov Instability discovered in the thermo-dynamical context (see (10) for a review). These two classes could be directlylinked to the two classes of Newtonian self-gravitating systems which are glob-ular clusters and galaxies. The classical hierarchical galaxy formation scenariois evidently linked to the inhomogeneous class. It produces collapsed core − which is a favorable process to form the observed and always mysterious supermassive black holes − and potential flattening in violent cases. The smaller,quiet, more isolated and then homogeneous case, could be naturally inter-preted as the generic globular cluster formation process. This could explainin the same operation, their generic spherical shape, their typical core-halodensity profile, and finally their generic lack of intermediate mass black hole An important amount of solid rotation could also be invoked but this property isgenerally not observed in a sufficient way in stellar systems, like elliptical galaxiesin particular.
6n central region . The case of spiral galaxies formation and evolution is morespecific and the role of dissipative process cannot be neglegted. Perhaps Vlasovequation is no more relevant for the description of such structures. Taking into account some physical constraints, the globular clusters and thegalaxies could be suitably modelized by the Vlasov-Poisson model. In addi-tion to this modelization, accurate numerical simulations allows to obtain aglobal understanding of important stages of their evolutions and of their maindifferences. However, a lot of fundamental works are always to be done beforethe study of very particular properties of the self-gravitating systems.
Acknowledgements
The author thanks the referees for their very careful reading of the first versionof the paper, and M. Kiessling for the communication of the fundamentalreference (13).
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