BOOK REVIEW: The Gravitational Million-Body Problem: A Multidisciplinary Approach to Star Cluster Dynamics

Physics – Condensed Matter – Statistical Mechanics

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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 gg 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 gtrsim 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 hardening, gravothermal oscillations - as well as additional processes such as stellar collisions and tidal shocks that affect real star clusters but not idealized N-body systems.
In a relatively short (300 pages plus appendices) book such as this, many topics have to be omitted. The reader who is hoping to learn about the phenomenology of star clusters will be disappointed, as the description of their properties is limited to only a page of text; there is also almost no discussion of other, equally interesting N-body systems such as galaxies(N approx 106 - 1012), open clusters (N simeq 102 - 104), planetary systems, or the star clusters surrounding black holes that are found in the centres of most galaxies. All of these omissions are defensible decisions. Less defensible is the uneven set of references in the text; for example, nowhere is the reader informed that the classic predecessor to this work was Spitzer's 1987 monograph, Dynamical Evolution of Globular Clusters, or that the standard reference on the observational properties of stellar systems is Binney and Merrifield's Galactic Astronomy. A minor irritation is that many concepts are discussed several times before they are defined, and the index provides no pointer to the primary discussion; thus, for example, there are ten index entries for 'phase mixing' and no indication that the fourth of these refers to the actual definition.
The book is intended as a graduate textbook but more likely it will be used mainly in other contexts: by theoretical researchers, as an indispensable reference on the dynamics of gravitational N-body systems; by observational astronomers, as a readable summary of the theory of star cluster evolution; and by physicists seeking a well-written and accessible introduction to a simple problem that remains fascinating and incompletely understood after three centuries.
Scott Tremaine

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