In this chapter we add one ingredient to the topics discussed in the previous chapter. There we outlined what happens to a stellar system when it loses mass, by whatever mechanism. Implicitly, however, we assumed that the system was isolated. Now we add to the picture the fact that the stars in a stellar system are also affected by surrounding matter, and this is especially true of escaping stars. The picture we have in mind is of a system like a globular cluster, orbiting inside a galaxy, which is simply another stellar system, but much larger and more massive.

The way in which the galaxy affects the cluster depends on such factors as the orbit of the cluster, and the distribution of mass within the galaxy. We begin with the simplest non-trivial idealisation. We assume that the orbit of the barycentre of the cluster is circular of radius R. Clearly, this is possible only for certain types of galaxy, e.g. those with axisymmetric potentials φg. We use an accelerating and rotating frame of reference with origin at the barycentre of the cluster, such that the x-axis points radially outward, and the y-axis points in the direction of motion of the cluster. The acceleration of a star in the cluster has several terms, due to: (i) the field of the galaxy; (ii) the gravitational field, φ, of the cluster; (iii) inertial forces, i.e. Coriolis and centrifugal terms.

The previous two chapters were intended to develop a qualitative understanding of the nature of the interactions between binary and single stars, with no more than order-of-magnitude estimates. The present chapter attempts to sharpen these ideas with some approximate quantitative results. We imagine that binaries and single stars are distributed throughout some region of space, such as a part of a star cluster, and we want to know how frequently three-body interactions are taking place.

Cross sections

What is important in applications (Chapters 23f) is the energetics of these interactions, and that is why such stress was laid on the distinction between soft and hard pairs in Chapter 19. In the present chapter this consideration implies that we may be interested in interactions with binaries of a given energy. Encounters with such binaries, however, are taking place all the time with stars which approach from random directions and random distances. Therefore, besides the energy of the binary, we usually do not know or care about the other properties of the participants, except for their statistical distribution. This is true of the approach path of the third body, and also usually it is true of the other five parameters (besides the energy) which determine the relative motion of the binary components.

A stellar system in dynamic equilibrium loses neither mass nor energy. In fact the stellar systems in nature do both, and the reasons for this are both external and internal. In this chapter we consider the latter; that is, we consider a stellar system isolated from all external influences, including gravitational ones.

We have two processes in mind. One is caused by the internal evolution of the stars. Note that this is the first occasion on which we have abandoned the point mass model on which we have relied so far, at least to the extent that we now consider time-dependent masses. The other is caused by the gravitational interactions of pairs of stars, which is really the topic of Chapter 14, and will be discussed rather briefly in this chapter. We also deal with the effects in two ways. One is the scaling treatment (Chapter 9) and the other uses a phase space description.

Evolution of length scale

A single star evolves at a rate which is a rapidly increasing function of its initial mass. Therefore, if we examine the stars in an old stellar system, we find that only those with a sufficiently low mass are more-or-less unevolved, with masses close to those they were born with. Those which were born with higher masses will have evolved, and in the process will have lost mass, leaving a remnant which may take the form of a black hole, a neutron star, or a white dwarf. Simple prescriptions for these aspects of stellar evolution have been in use in stellar dynamics for a long time (see Terlevich 1987, Chernoff & Weinberg 1990 and Problem 1).

In stellar dynamics we do not really study stellar systems like globular clusters and galaxies. That is the job of astronomers. What we do is study models of these systems. Just as in many branches of applied mathematics, a model is nothing other than a mathematical structure into which we try to incorporate our knowledge of the system at hand. Sometimes “knowledge” is not the right word: it may be nothing better than a hunch about how things might work. Often, however, our knowledge will include physical laws, especially the ones we think are relevant. So far in this book, for example, we have implicitly thrown out almost everything we know about stellar systems except the gravitational dynamics.

In the context of stellar dynamics, a model consists of two kinds of mathematical construct. One is the mathematical object used in the description of the system, and the other is the evolution which determines its evolution. So far, for example, we have introduced the N-body model, where the system is described by N time-dependent vectors, which evolve according to Newton's Law of motion. We have also introduced a statistical model of collisionless stellar dynamics, where the system is described by the one-particle distribution function f, which evolves according to the collisionless Boltzmann equation.

In this final chapter of Part II we give a foretaste of the full variety of models for the dynamics of dense stellar systems.

Equilibrium and stability

Mathematicians classify equilibria in various ways. There are, for example, unstable equilibria, which are rarely found in nature, but are important in the theoretical understanding of a complicated dynamical system. Of greater practical importance are stable equilibria. The definition of this concept amounts to saying that, if the system is disturbed slightly from the equilibrium, then it remains in the vicinity of the equilibrium. In nature, however, stable equilibria often exhibit a still stronger behaviour, which mathematicians classify as asymptotic stability. This means that the disturbed system returns to the equilibrium state from which it was disturbed. This happens commonly in nature because of dissipative forces. The process of returning to equilibrium is often referred to as relaxation, and it is one with which we are all familiar (late at night).

With this background it is astonishing that relaxation plays such a central role in stellar dynamics. Not only is there no dissipation in the gravitational many-body problem, there is no equilibrium either. It is true that one can think of some highly artificial solutions which can be regarded as equilibria. The Euler–Lagrange solutions of the three-body problem, in which the three stars appear to be at rest in a uniformly rotating reference frame, come into this class, and, from a more general point of view it may be fruitful to regard a periodic solution as a generalised equilibrium. But even where these solutions are stable, there is no question of asymptotic stability.

Once thought to be virtually devoid of binaries, globular clusters are now known to contain binaries in abundances not very much less than that of the galactic disk. Binaries in such large numbers, containing at least ten per cent of the stars in a typical globular cluster, cannot have resulted from dynamical interactions, and therefore must have formed at the same time that the bulk of the stars were formed. With so many binary stars around, all kinds of interesting reaction channels are possible, in three-body as well as four-body interactions. Binaries containing pulsars are just one example of the unusual objects that can result.

Even without dynamical interactions with other objects, binary star evolution in isolation is quite complicated enough. Compared to the evolution of single stars, a wide variety of new kinds of binaries and single stars can be created, through mass overflow from one star to the other, or through mass loss from the system, at various stages in their combined evolution. The stars can form a common envelope for a while, or one of the stars can explode as a supernova. Even if the explosion is symmetric the binary might not survive, as in the impulsive loss of mass in any stellar system (Chapter 11). Disruption is even more likely if the remnant receives a ‘kick’ (see Hills 1983), and if you find a neutron star in a binary in a star cluster it is likely to have got there by dynamical interactions (e.g. Kalogera 1996).

The following three chapters complicate the million-body problem for astronomically motivated reasons. Chapter 24 explains these by tracing the history of the discovery of binary stars in star clusters, in numbers which imply that they are primordial, i.e. they were born along with the cluster itself. They are associated with several of the remarkable phenomena which help to explain why globular star clusters are so important to astrophysicists, such as the sources of X-rays within them. We contrast their behaviour in star clusters with the much milder behaviour of binaries in less extreme environments.

In systems with many binaries, four-body encounters between two binaries are common. Chapter 25 discusses in detail one of the commoner outcomes: hierarchical triple systems. They are one class of three-body problem where the motion is both non-trivial and amenable to detailed calculation. Since these systems are stable and very long-lived, but may have tiny orbital time scales, such results are important for efficient computer simulation of N-body systems with many primordial binaries.

Chapter 26 discusses the effect of binary–binary encounters on the rest of the system. In important ways they can dominate the effect of the three-body encounters discussed in earlier chapters, though not forever, as binaries are also destroyed in these encounters. The outcomes of the interactions are also more complicated than in three-body encounters, and we show how to classify these.

Since this book is aimed at a broad audience within the physical sciences, we expect most of our readers not to be experts in either astrophysics or mathematics. For those readers, the title of this book may seem puzzling at least. Why should they be interested in the gravitational attraction between bodies? What is so special about a million-body problem, rather than a billion or a trillion bodies? What kind of bodies do we have in mind? And finally, what is the problem with this whole topic?

In physics, many complex systems can be modelled as an aggregate of a large number of relatively simple entities with relatively simple interactions between them. It is one of the most fascinating aspects of physics that an enormous richness can be found in the collective phenomena that emerge out of the interplay of the much simpler building blocks. Smoke rings and turbulence in air, for example, are complex manifestations of a system of air molecules with relatively simple interactions, strongly repulsive at small scales and weakly attractive at larger scales. From the spectrum of avalanches in sand piles to the instabilities in plasmas of more than a million degrees in labs to study nuclear fusion, we deal with one or a few constituents with simple prescribed forces. What is special about gravitational interactions is the fact that gravity is the only force that is mutually attractive.

The last eight chapters, dealing as they have done with interactions between only three or four stars, might seem a long digression away from the subject implied by the title of this book. Yet we shall see, as we take up the thread of the million-body problem where we broke off at the end of Chapter 18, that an understanding of the behaviour of few-body systems is crucial in following the evolution of the system through core collapse and beyond.

We left the system rushing towards core collapse, its central density rising inexorably, so that it would reach infinite values in finite time. How is this catastrophe averted? In fact there is no shortage of choices, for at least five different mechanisms have been proposed over the years. Admittedly, two are rather out of favour at present: a central black hole (e.g. Marchant & Shapiro 1980), or runaway coalescence and evolution of massive stars (Lee 1987a, and Problem 1). The other three involve binary stars in one guise or another, and it is not hard to see why this is attractive. After all, the mechanism responsible for core collapse is a two-body one (Chapter 14). Therefore it is clear that higher-order interactions, which we have neglected so far, might in principle eventually compete with two-body relaxation when the density becomes high enough. And three-body interactions can create binaries (Chapter 21 and Fig. 27.1).

The following three chapters complete the story of the evolution of a million-body system, in its purely stellar dynamical form. Chapter 27 begins by estimating the rate at which the formation and evolution of (non-primordial) binaries effectively generates energy within the system. The first application is to show that this is sufficient to halt core collapse. Then we consider other ways of generating the energy: binaries formed in dissipative two-body encounters between single stars, and primordial binaries; we quantify the extent to which the effectiveness of primordial binaries depends on their abundance and their energy.

In Chapter 28 we consider how a balance can be struck between the creation of energy (by binary interactions) deep in the core and the large-scale structure of the rest of the cluster. We first describe a standard argument which implies that conditions in the core, where the energy is generated, are governed by the overall structure. We outline the core parameters and overall evolution which this argument implies. Next we give arguments to show that this balance can be unstable, and describe the phenomenon of temperature inversion which is associated with the generation of gravothermal oscillations. The manner in which they depend on N (in idealised models) is an example of a Feigenbaum sequence of ‘period-doubling bifurcations’ in this context.

Chapter 29 rounds off the evolution of a million-body system by focusing on the evolution of gross structural parameters: total mass, and a measure of the overall radius.

Can a million-body system be in equilibrium? More precisely, can a model of a million-body system exhibit equilibrium? The answer depends on the model and other conditions. But we already saw in Chapter 8 that equilibrium models of gravitational many-body systems can be constructed. Thus, if it is modelled as a self-gravitating perfect gas it will be in thermal equilibrium if its temperature is uniform. If it is modelled by a Fokker–Planck or Boltzmann equation, then the equivalent condition is that the single-particle distribution be Maxwellian. In both cases the system is required to have infinite mass and extent.

These isothermal models are, in a strict sense, artificial, but they are of great importance conceptually, and for other reasons. In order to make progress in understanding them we shall replace one form of artificiality with another. Instead of dealing with infinite systems, we shall enclose our isothermal system in a spherical enclosure, which at least has the merit of implying that our systems have finite mass and radius. We shall suppose that the enclosure is rigid and spherical; in the N-body model this means that stars bounce off it without loss of energy, while in the gas or phase-space models, it implies that the enclosure is adiabatic. We assume spherical symmetry, and that stars all have the same fixed individual mass. We work entirely, however, with a perfect gas model.

During the past decade or so, theoretical astrophysics has emerged as one of the most active research areas in physics. This advance has also been reflected in the greater interdisciplinary nature of research that has been carried out in this area in recent years. As a result, those who are learning theoretical astrophysics with the aim of making a research career in this subject need to assimilate a considerable amount of concepts and techniques, in different areas of astrophysics, in a short period of time. Every area of theoretical astrophysics, of course, has excellent textbooks that allow the reader to master that particular area in a well-defined way. Most of these textbooks, however, are written in a traditional style that focusses on one area of astrophysics (say stellar evolution, galactic dynamics, radiative processes, cosmology, etc.). Because different authors have different perspectives regarding their subject matter, it is not very easy for a student to understand the key unifying principles behind several different astrophysical phenomena by studying a plethora of separate textbooks, as they do not link up together as a series of core books in theoretical astrophysics covering everything that a student would need.