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The gravitational N-body problem is to describe the evolution of an isolated system of N point masses interacting only through Newtonian gravitational forces. For N =2 the solution is due to Newton. For N =3 there is no general analytic solution, but the problem has occupied generations of illustrious physicists and mathematicians including Laplace, Lagrange, Gauss and Poincaré, and inspired the modern subjects of nonlinear dynamics and chaos theory. The general gravitational N-body problem remains one of the oldest unsolved problems in physics.Many-body problems can be simpler than few-body problems, and many physicists have attempted to apply the methods of classical equilibrium statistical mechanics to the gravitational N-body problem for N ≫ 1. These applications have had only limited success, partly because the gravitational force is too strong at both small scales (the interparticle potential energy diverges) and large scales (energy is not extensive). Nevertheless, we now understand a rich variety of behaviour in large-N gravitating systems. These include the negative heat capacity of isolated, gravitationally bound systems, which is the basic reason why nuclear burning in the Sun is stable; Antonov's discovery that an isothermal, self-gravitating gas in a container is located at a saddle point, rather than a maximum, of the entropy when the gas is sufficiently dense and hence is unstable (the 'gravothermal catastrophe'); the process of core collapse, in which relaxation induces a self-similar evolution of the central core of the system towards (formally) infinite density in a finite time; and the remarkable phenomenon of gravothermal oscillations, in which the central density undergoes periodic oscillations by factors of a thousand or more on the relaxation timescale - but only if N ≳ 104.The Gravitational Million-Body Problem is a monograph that describes our current understanding of the gravitational N-body problem. The authors have chosen to focus on N = 106 for two main reasons: first, direct numerical integrations of N-body systems are beginning to approach this threshold, and second, globular star clusters provide remarkably accurate physical instantiations of the idealized N-body problem with N = 105 – 106.The authors are distinguished contributors to the study of star-cluster dynamics and the gravitational N-body problem. The book contains lucid and concise descriptions of most of the important tools in the subject, with only a modest bias towards the authors' own interests. These tools include the two-body relaxation approximation, the Vlasov and Fokker-Planck equations, regularization of close encounters, conducting fluid models, Hill's approximation, Heggie's law for binary star evolution, symplectic integration algorithms, Liapunov exponents, and so on. The book also provides an up-to-date description of the principal processes that drive the evolution of idealized N-body systems - two-body relaxation, mass segregation, escape, core collapse and core bounce, binary star hardenin...

In this paper we show the results of a large set of N -body simulations modelling the evolution of globular clusters driven by relaxation, stellar evolution, disk shocking and including the effects of the tidal field of the Galaxy. We investigate the evolution of multi-mass models with a power-law initial mass function (IMF) starting with different initial masses, concentrations, slopes of the IMF and located at different galactocentric distances. We show to what extent the effects of the various evolutionary processes alter the shape of the IMF and to what extent these changes depend on the position of the cluster in the Galaxy. Both the changes in the global mass function and in the local one (measured at different distances from the cluster center) are investigated showing whether and where the local mass function keeps memory of the IMF and where it provides a good indication of the current global mass function.The evolution of the population of white dwarfs is also followed in detail and we supply an estimate of the fraction of the current value of the total mass expected to be in white dwarfs depending on the main initial conditions for the cluster (mass and position in the Galaxy). Simple analytical expression by which it is possible to calculate the main quantities of interest (total mass, fraction of white dwarfs, slope of the mass function) at any time t for a larger number of different initial conditions than those investigated numerically have been derived.

Galactic globular clusters are ancient building blocks of our Galaxy. They represent a very interesting family of stellar systems in which some fundamental dynamical processes have been taking place for more than 10 Gyr, but on time scales shorter than the age of the universe. In contrast with galaxies, these star clusters represent unique laboratories for learning about two-body relaxation, mass segregation from equipartition of energy, stellar collisions, stellar mergers, core collapse, and tidal disruption. This review briefly summarizes some of the tremendous developments that have taken place during the last two decades. It ends with some recent results on tidal tails around galactic globular clusters and on a very massive globular cluster in M31. IntroductionThere are about 150 globulars orbiting in the halo of our Galaxy. They look like huge swarms of stars, characterized by symmetry and apparent smoothness. Fig. 1 below displays an image of NGC 5139 ≡ ω Centauri, the brightest and most massive galactic globular cluster. This 40 ′ by 40 ′ image from the Digital Sky Survey does not reach, in spite of its rather large angular size, the outer parts of the cluster. With its tidal radius of about 40-50 ′ , the apparent diameter of ω Centauri on the plane of the sky is significantly larger than the apparent 30 ′ diameter of the full moon. Globular clusters are old stellar systems, made of one single generation of stars. Although still somewhat uncertain, their individual ages range between about 10 and 15 Gyr, with possible significant differences, up to a few gigayears, from one cluster to the other. Other properties of globular clusters exhibit significant variations: e.g., their integrated absolute magnitudes range from M int V = -1.7 to -10.1 mag; their total masses from M tot = 10 3 to 5 × 10 6 M ⊙ ; their galactocentric distances from 2 to 120 kpc. 2A few dynamical time scalesThe dynamics of any stellar system may be characterized by the following three dynamical time scales: (i) the crossing time t cr , which is the time needed by a star to move across the system; (ii) the relaxation time t rlx , which is the meylanreview:

In this paper a cluster is modelled as a smooth potential (due to the cluster stars) plus the steady tidal field of the Galaxy. In this model there is a minimum energy below which stars cannot escape. Above this energy, however, the time‐scale on which a star escapes varies with the orbital parameters of the star (mainly its energy) in a way which we attempt to quantify, with both theoretical arguments and computer simulations. Within the limitations of the model we show that the time‐scale is long enough to complicate the interpretation of full N‐body simulations of clusters, and that stars above the escape energy may remain bound to the cluster for about a Hubble time.