As has been seen in Chapter 18, two-body relaxation predicts its own downfall. It leads to the collapse of the core and, at the level of simplified models, infinite central density. Clearly, some new dynamical processes, beyond two-body encounters, must come into play. The very high density is the clue, for it suggests that a third body may, with increasing probability, intervene in the two-body encounters which mediate relaxation. In Chapter 27 it will be seen that three-body encounters do indeed act on a sufficiently short time scale, late in core collapse, to have a decisive influence on events. As we note there, this is not the only mechanism that can work, but we concentrate on it for the time being.

The mechanism is a two-stage one, both stages involving three-body encounters. In the first stage, which we consider in Chapter 21, a three-body interaction leads to the formation of a binary star and a single third body (which acts as a kind of catalyst). In the second stage, this binary interacts with other single stars (again in three-body reactions). In this chapter we shall study three-body encounters in isolation, in order to uncover those properties which allow them to play their crucial role in rescuing the cluster from collapse. Clusters get into this difficulty because of their negative heat capacity, and in fact it is the negative heat capacity of binaries which comes to their rescue.

The following three chapters are devoted to two-body interactions in the context of the million-body problem. Chapter 13 shows that these cause neighbouring orbits to diverge with an approximately exponential time dependence. Beginning with the three-body problem, we go on to investigate the N-dependence of the e-folding time scale for the divergence. Were late the phenomenon to the exponential divergence of geodesics in an alternative formulation of the problem.

Chapter 14 is quintessential collisional stellar dynamics. Here we consider the cumulative effect of many two-body encounters on the motion of a single star: the theory of two-body relaxation. We develop a number of standard formulae for the first and second moments of the cumulative change in its velocity. The first moment corresponds to the phenomenon of dynamical friction. We then go on to incorporate this theory into an evolution equation (the Fokker–Planck equation) for the distribution function. We approximate this equation in a form appropriate to the situation in stellar dynamics, when the time scale of relaxation is much longer than that of orbital motions. This also incorporates the evolution which may result from slow changes in the potential.

Chapter 15 takes a close look at the two-body problem itself. We show, in particular, that the two-body collision singularity is a removable singularity. This introduces a number of topics which might seem surprising in the context of the million-body problem: the Lenz vector, quaternions, the Hopf map, the simple harmonic oscillator, and even a transformation into four dimensions.

A strange black box

Compared to laboratory situations, a self-gravitating star cluster is a very strange object. Imagine that you were handed a star cluster in a closed box, so that you could only measure the temperature at the surface of the box. Imagine also that you could change the conditions of the star cluster from the outside in two ways: (1) you could put the box inside a larger box with a different temperature, as an effective heat bath, in order to change the temperature inside; (2) you could change the size of the box, compressing or expanding its volume.

So far, there is nothing unusual, and we might still pretend that we are about to carry out a textbook thermodynamics experiment. But when we dip our box into a heat bath, something strange may occur: depending on the exact conditions inside the box, the box may exhibit a most bizarre behaviour. When placed in a colder environment, the box may actually heat up, without limit. The only way to cool the box back to its original temperature would be to place it temporarily inside an even hotter environment – but not for too long, otherwise it will cool to below its original temperature.

This contrary tendency of self-gravitating systems corresponds to the fact that such systems can exhibit a negative heat capacity. We will return to this mysterious character later on, when we analyse its effects in detail, both on macroscopic scales, governing the evolution of a star cluster as a whole, and on a microscopic scale, when we deal with few-body systems.

Globular star clusters have an important place in modern astrophysics for several reasons, but let us mention just two here. Firstly, they are a laboratory for the study of gravitational interactions and dynamical evolution, and this is the motivation for much of the research that we have written about in this book. Secondly, however, each cluster is also a sample of stars of very similar age and composition, and are an ideal test-bed for theories of stellar evolution.

Over the years these two aspects of cluster studies built up their own communities of theorists and observers, and their own suites of problems (Fig. 30.1). Now what is remarkable is that, for many years, research in these two areas proceeded in almost total isolation from each other. It was possible to pursue an active and successful research career on one side of this diagram (Fig. 30.1) without even being aware of the existence of the people working on the other side. Whenever one did need something from the other side, the most primitive tool for the job was used. Dynamicists would use mass functions which to any observational astronomer would seem distinctly bizarre, while observers, if they ever needed a theoretical model, would dust off an old one which ignored decades of subsequent theoretical development. Dynamicists were fascinated by the problems of stellar systems with stars of only two different possible masses, while fitting a Fokker–Planck model was something that no non-theorist ever attempted.

In the last chapter we saw that core collapse ends in what is (by stellar dynamics standards) a blaze of energy, which is emitted in interactions involving primordial binaries. Dramatic though that sounds, the real climax of the whole book is reached in the present chapter, where for the first time we catch a glimpse of the entire lifespan of a cluster. Admittedly we concentrate here on a highly idealised cluster, isolated from the rest of the Universe, consisting of stars of equal mass, and totally devoid of primordial binaries. In the next chapter we shall relax some of these idealisations.

Steady post-collapse expansion

What happens to the energy generated at the close of core collapse? To answer this question it is easiest to think of the cluster as a conducting, self-gravitating mass of gas, with its temperature decreasing from the core to the tenuous boundary. Because of the assumed temperature gradient, the heat flux is outwards, and the time scale for the transport of thermal energy is of order the local relaxation time, tr. Therefore, after an interval of order trh, i.e. the value of tr at the radius rh which contains half the mass, the thermal energy generated at core bounce has diffused throughout the bulk of the cluster. In the same time interval M, the mass of the cluster, has hardly altered (cf. Problem 16.1).

As we have seen, primordial binaries exist in some abundance in globular clusters, and so there is good reason to study the behaviour of a million-body problem in which there are many binaries. How do such clusters evolve?

To some extent the binaries may be treated as heavy point masses, provided we are interested in interactions with other stars at distances much greater than the typical semi-major axis of a binary. Thus the binaries behave like a heavy species, and the process of mass segregation (Chapter 16) ensures that they become heavily concentrated towards the core of the cluster. It follows that, even if we start with a cluster in which only ten per cent of the stars are binaries, we quickly find ourselves looking at part of the system (the core) where the abundance of binaries is more like 90%. Incidentally, it has even been argued that young star clusters may consist entirely of binaries. For one reason or another, then, the study of stellar systems in which essentially all stars are binaries is an important one.

Something was already said about the interaction between two binaries in the previous chapter. There we quickly focused on the formation of hierarchical triple systems in such encounters. In the present chapter we focus on the energetics of these interactions. We shall see in the next chapter that this is the main way in which these interactions feed back into the overall dynamics of a stellar system.

While the last chapter roughed out a picture of what happens in an interaction between a binary star and a third body, there are three very different ways in which the picture can be sharpened. One is to develop approximate analytical results on the outcome of an encounter, and that is successful in various limiting cases, e.g. very distant encounters, very hard binaries, and so on. This is the approach of Chapter 21. To cover the middle ground between these extremes, there is no substitute for computational studies (Chapter 22). In the present chapter, however, we push the analytical methods in the opposite direction, and examine minute corners of parameter space which may be of no conceivable value in applications. The merit of this approach is that rigorous statements become possible, at least in expert hands, and the resulting ideas help to develop our intuition of what can happen in more realistic situations. Our approach is quite informal, and places emphasis on the ideas behind the proofs, without any technical details.

Fractals and chaos

We first turn to resonances, those long-lived but temporarily bound triple systems that often arise in scattering events. Our first aim here is to discuss one situation which makes it particularly clear why the outcome of a resonance depends sensitively on the initial data. We can then argue, at least physically, that the system forgets details of the initial conditions of its formation, and that the breakup is determined (in a statistical sense) only by the quantities which are preserved in the evolution, i.e. energy and angular momentum.

Introduction

In this chapter we turn to a consideration of the physics of inhomogeneous plasmas. Since virtually all plasmas whether in the laboratory or in space are to some degree inhomogeneous, all that can be attempted within the limits of a single chapter is to outline some general points and illustrate these with particular examples. Throughout the book we have dealt in places with plasmas which were inhomogeneous in density or temperature and confined by spatially inhomogeneous magnetic fields. In the case of the Z-pinch the high degree of symmetry allowed us to find analytic solutions in studying the equilibrium. By contrast for a tokamak, even with axi-symmetry, solutions to the Grad-Shafranov equation could only be found numerically. Indeed the only general method of dealing theoretically with problems in inhomogeneous plasmas is by numerical analysis.

Nevertheless useful analytic insights may be gained in two limits. In the first, plasma properties change slowly in the sense that for an inhomogeneity scale length L and wavenumber k, kL ≫ 1 and one can appeal to the WKBJ approximation described in Section 11.2. In this limit we shall draw on illustrations from the physics of wave propagation in inhomogeneous plasmas. If we picture a wave propagating in the direction of a density gradient, at some point on the density profile it may encounter a cut-off or a resonance. As we found in Chapter 6, propagation beyond a cut-off is not possible and the wave is reflected, whereas at a resonance, wave energy is absorbed. The WKBJ approximation breaks down in the neighbourhood of both cut-offs and resonances.

The analytical approaches sampled in the previous chapter have some good uses, but providing accurate useful numbers is not always one of them. They may provide suitable scaling laws, showing how the statistics of three-body scattering depends on the masses involved, but there is usually an overall coefficient that must be determined in some other way. We now turn to a technique which can fill in such gaps. Actually it is marvellous how complementary the two techniques are. Numerical methods are not good at determining cross sections of very rare events, e.g. very close triple approaches, but it is often precisely these little corners of parameter space where analytical methods are feasible.

Numerical methods offer astrophysicists a tool quite analogous to the kinds of particle colliders in use by high-energy physicists. There beams of particles are fired at targets (or other particles), and the relative frequencies of different kinds of collision debris can be observed. Using numerical methods we can see what happens when a binary (the target) is fired on by a single particle, and the experiment may be repeated as often as we care.

Numerical studies of three-body encounters go back almost one hundred years. In 1920 L. Becker published results on exchange encounters (see Chapter 19) which were carried out with the aid of ‘mechanical quadrature’. But it was not until the era of electronic computing that the investigation of triple scattering orbits became a sizable industry.

The last two chapters have assembled most of the qualitative arguments by which the evolution of the core of a stellar system can be understood. In summary, the tendency towards equipartition drives the more massive stars to smaller radii. Unless their total mass is sufficiently small, equipartition cannot be reached by the time the heavier stars become essentially self-gravitating. When that happens they are eventually subject to the gravothermal instability. It is the purpose of the present chapter to flesh out this outline, but we shall do so in two passes, as it were. First we shall examine the time scales on which these processes act, and a number of factors which modify the simple picture; and we shall explain the qualitative nature of the resulting evolution. Then we turn to a more detailed description of one case which has been studied in great detail: self-similar collapse in systems with stars of equal mass.

The big picture

The time scale for equipartition, te, was discussed in Chapter 16 (see Eq. (16.14)). It is useful to compare it with the standard relaxation time tr (Eq. (14.12)). For this purpose we evaluate the mean kinetic energy per unit mass for each species by 〈Ei〉 = v2/2, independent of mass, i.e. we assume equipartition of velocities.

Up to this point in the book we have largely turned our back on the microscopic character of the million-body problem. Usually we have approximated the gravitational field by that of a smooth distribution of matter. Now we concentrate on the interactions between small numbers of stars in the system, often only two or three stars at a time. In later parts of the book we shall see how these microscopic processes influence the large-scale behaviour.

The purpose of this chapter is to look at the question of sensitivity to initial conditions in the million-body problem. Much current research in other dynamical problems is devoted to this question, because of its importance for prediction, for the foundations of statistical mechanics, and perhaps even for the survival of life on Earth. What lies behind this remark is the fact that the question of sensitivity is linked to stability, and the stability of the solar system is something we rely on implicitly. But the collision of comet Shoemaker–Levy with Jupiter in 1994 reminded us that the solar system is not the well regulated clock we often take it for. Less well known is the recent realisation that the rotation of the Earth (which itself influences climate strongly) appears to be stabilised by the presence of the Moon (see Laskar 1996).

Consider the one-body problem. The star proceeds with uniform rectilinear motion r1(t). Another single star, not interacting with the first, and starting with a similar initial velocity and position, exhibits similar motion r2(t).

The remaining chapters of the book go beyond the N-body problem as it is understood outside astrophysics. Here the fact that the bodies are stars is essential.

Chapter 30 sets the scene by summarising the various dynamical processes that have been introduced so far, the various kinds of stars and other relevant topics which are of interest in astrophysics (such as colour–magnitude diagrams) and, most importantly, the relations between these two sides of the problem. Special attention is paid to those kinds of stars which are readily observed and where dynamical processes are most immediately relevant: blue stragglers, millisecond pulsars and X-ray sources.

Chapter 31 analyses in some detail the simplest process where the stellar nature of the bodies is vital: collisions and other encounters between two individual stars, where the gravitational interaction is not the whole story. We estimate the rate of collisions, and how it depends on the stellar density and the kinds of stars present. Non-gravitational interactions are also vital in understanding the role and evolution of binary stars, especially when interactions with other (single) stars occur frequently enough. The effects of collisions on the participating stars are outlined, and we consider the dynamics of near-collisions, where non-gravitational effects are important (‘tidal capture’).

The dynamics and evolution of binary stars are taken up in detail in Chapter 32. Special attention is paid to the ways in which blue stragglers can arise from interactions involving binaries.

Introduction

We know from classical electrodynamics that accelerated charged particles are sources of electromagnetic radiation. Particles accelerated in electric or magnetic fields radiate with distinct characteristics. Electric micro-fields present in the plasma result in bremsstrahlung emission by plasma electrons. External radiation fields interacting with the plasma give rise to scattered radiation. Charged particles moving in magnetic fields emit cyclotron or synchrotron radiation, depending on the energy range of the particles.

The interaction of radiation with plasmas in all its aspects – emission, absorption, scattering and transport – is a key to understanding many effects in both laboratory and natural plasmas. Laboratory plasmas in particular do not radiate as black bodies so that an integrated treatment of emission, absorption and transport of radiation is usually needed. Core plasma parameters such as electron and ion temperatures and densities as well as plasma electric and magnetic fields may all be determined spectroscopically, in the most general sense of the term. Rather arbitrarily we shall confine our discussion to radiation from fully ionized plasmas thus excluding line radiation on which many diagnostic procedures are based. To some extent alternative spectroscopic techniques, in particular light scattering, have replaced if not entirely supplanted measurements of line radiation as preferred diagnostics of some key parameters in fusion plasmas (see Hutchinson (1988)). In the course of this chapter we shall outline the basis of some of these diagnostics, notably those that rely on bremsstrahlung and cyclotron radiation as well as those involving light scattering. We shall limit our discussion of radiation to plasmas in thermal equilibrium, with few exceptions. Non-thermal emission, while an important issue in practice, is in many instances still relatively poorly understood.

Introduction

Much of plasma physics can be adequately described by fluid equations, namely, the MHD or wave equations. However, these are derivative descriptions in which some information about the plasma has been suppressed. In situations where that information matters it is necessary to go to a deeper level of physical description.

The information that gets lost in a fluid model is that relating to the distribution of velocities of the particles within a fluid element, since the fluid variables are functions of position and time but not of velocity. Any physical properties of the plasma that depend on this microscopic detail can be discovered only by a description in six-dimensional (r, v) space. Thus, instead of starting with the density of particles, n(r, t), at position r and time t, we begin with the so-called distribution function, f (r, v, t), which is the density of particles in (r, v) space at time t. The evolution of the distribution function is described by kinetic theory.

With the additional information on particle velocities within a volume element introduced by a phase space description we now have microscopic detail that we did not have before. For that reason, kinetic and fluid theories are identified as microscopic and macroscopic, respectively.

Introduction

When the fields induced by the motion of the plasma particles are significant in determining that motion, particle orbit theory is no longer an apt description of plasma behaviour. The problem of solving the Lorentz equation self-consistently, where the fields are the result of the motion of many particles, is no longer practicable and a different approach is required. In this chapter, by treating the plasma as a fluid, we derive various sets of equations which describe both the dynamics of the plasma in electromagnetic fields and the generation of those fields by the plasma.

The fluid equations of neutral gases and liquids are usually derived by treating the fluid as a continuous medium and considering the dynamics of a small volume of the fluid. The aim is to develop a macroscopic model that, as far as possible, is independent of the detail of what happens at the molecular level. In this sense the approach is the opposite of that adopted in particle orbit theory where we seek information about a plasma by examining the motion of individual ions and electrons. In experiments one seldom makes measurements or observations at the microscopic level so we require a macroscopic description of a plasma similar to the fluid description of neutral gases and liquids. This is obtained here by an extension of the methods of fluid dynamics, an approach that conveniently skims over some fundamental difficulties inherent in plasmas. The chief of these is that a plasma is not really one fluid but at least two, one consisting of ions and the other electrons.

In order to progress from qualitative arguments and toy models it is necessary to set up apparatus for describing a gravitational N-body system. There are several ways in which this can be done.

One common approach is to employ the N position vectors ri and the N velocity vectors vi of the stars at some time. Each of these vectors has three components, and so the entire system can be described by a single 6N-dimensional vector, i.e. a single point in a 6N-dimensional space Г. This is a useful description, because it is sufficient to specify uniquely the entire subsequent evolution of the system, as the equations of motion are of second order; they describe the motion of this point through Г. Implicitly, therefore, this is the description adopted in N-body methods, even though it is more natural to think of N particles moving in a six-dimensional phase space.

This description in a 6N-dimensional space can be turned into a statistical one if we imagine a collection of stellar systems, each described by a distinct point in Г. If their distribution is described by a probability density function f, the evolution of f is determined by the equations of motion, and indeed is equivalent to them. This description is almost never used in stellar dynamics.

Another way of describing a stellar system is to represent each star by a single point in a six-dimensional space with coordinates r and v.

The previous chapter was concerned with the consequences of two-body interactions, but made use of nothing more than an approximate solution of the two-body problem. Here we consider the classical two-body problem without approximations. It is one of the oldest solved problems of dynamics, and so, as we mentioned in the preface, it is no longer really a problem. Yet its structure is of enduring interest, and offers new surprises each time we view it from a fresh angle.

Along with the simple harmonic oscillator, the Kepler problem is to dynamics what the Platonic solids are to geometry. And, just as there is a duality among the latter (for example the cube, with six faces and eight vertices, is dual to the octahedron, with six vertices and eight faces), we shall see that there is an intimate link between these two dynamical problems. This chapter may look self-indulgent compared with the serious issues of stellar dynamics in the surrounding chapters, and should perhaps be in a box of its own, but in fact some of the results we shall survey have important applications to the million-body problem. The reason is that we shall be taking a close look at the singularity of the two-body equations, where numerical methods cause a lot of trouble.

Removing the collision singularity

Consider first the one-dimensional Kepler problem. With a suitable scaling, the equation of motion is

This equation is singular, corresponding to a collision in the Kepler problem.

Core collapse leads to high stellar densities, where interactions may involve more than just two stars at a time. The chapters in this section are therefore devoted to three-body interactions, especially interactions between a binary star and a single star. One of our aims in these chapters is to show that important aspects of the three-body problem can be understood from various points of view, even though the problem itself lacks a general mathematical solution.

Chapter 19 takes a phenomenological approach, applying notions of equipartition and energy conservation. This already classifies encounters according to whether the binary is hard or soft. In some interactions with hard binaries the result (temporarily) is like a miniature star cluster of three stars, and our previous knowledge of the behaviour of star clusters can suggest how this evolves.

Chapter 20 takes an informal mathematical view of the same phenomena. We see that the breakup of triple star clusters exhibits a sensitive dependence on initial conditions, partly justifying a statistical treatment. One of the standard examples in which this is most readily understood is Sitnikov's problem, which we use to introduce the Smale horseshoe. Finally we prove informally a theorem which shows that permanent capture into a triple configuration is (practically) impossible, and end with some recent surprising discoveries about permanently bound triple systems.

Chapter 21 takes a course halfway between the previous ones, exploiting a mixture of approximate analytical tools and physical arguments to develop theoretical results on the outcome of three-body interactions.