In the original Star Wars movie there is a brief but memorable scene of two stars shining down from the afternoon sky. We find it striking because we are so used to seeing just a single star in the sky, but in fact it is the Sun that is unusual. In the neighbourhood of the Sun most stars are binaries, and some belong to triple systems or even little groups with still larger numbers of stars (Duquennoy & Mayor 1991). Our nearest neighbours, for example, are a binary (α Centauri) with a distant third companion (Proxima Centauri, see Matthews & Gilmore 1993). Such systems are unlikely to arise by chance encounters (Problem 1), and so the abundance of binaries and triples suggests that most stars are born that way. Indeed binaries and other multiples are most common among the youngest stars (see Kroupa 1995).

A brief observational history

With this background, astronomers were perplexed by how difficult it was to find any binaries in globular star clusters. Admittedly, wide visual pairs were not expected, as such binaries would be destroyed by encounters with other stars in the dense environment of a globular cluster (Problem 21.1). Indeed, there is observational evidence that this has happened (Côté et al. 1996). But it should have been possible to detect closer binaries by observing periodic variations in their radial velocity, or else through the discovery of eclipsing variable stars.

A single coupling constant

The gravitational N-body problem can be defined as the challenge to understand the motion of N point masses, acted upon by their mutual gravitational forces (Eq. (1.1)). From the physical point of view, a fundamental feature of these equations is the presence of only one coupling constant: the constant of gravitation, G = 6.67 × 10-8 cm3 g-1 s-2 (see Seife 2000 for recent measurements). It is even possible to remove this altogether by making a choice of units in which G = 1. Matters would be more complicated if there existed some length scale at which the gravitational interaction departed from the inverse square dependence on distance. Despite continuing efforts, no such behaviour has been found (Schwarzschild 2000).

The fact that a self-gravitating system of point masses is governed by a law with only one coupling constant (or none, after scaling) has important consequences. In contrast to most macroscopic systems, there is no decoupling of scales. We do not have at our disposal separate dials that can be set in order to study the behaviour of local and global aspects separately. As a consequence, the only real freedom we have, when modelling a self-gravitating system of point masses, is our choice of the value of the dimensionless number N, the number of particles in the system.

Introduction

On the face of it, solving an equation of motion to determine the orbit of a single charged particle in prescribed electric and magnetic fields may not seem like the best way of going about developing the physics of plasmas. Given the central role of collective interactions hinted at in Chapter 1 and the subtle interplay of currents and fields that will be explored in the chapters on MHD that follow, it is at least worth asking “Why bother with orbit theory?”. One attraction is its relative simplicity. Beyond that, key concepts in orbit theory prove useful throughout plasma physics, sometimes shedding light on other plasma models.

Before developing particle orbit theory it is as well to be clear about conditions under which this description might be valid. Intuitively we expect orbit theory to be useful in describing the motion of high energy particles in low density plasmas where particle collisions are infrequent. More specifically, we need to make sure that the effect of self-consistent fields from neighbouring charges is small compared with applied fields. Then if we want to solve the equation of motion analytically the fields in question need to show a degree of symmetry. We shall find that scaling associated with an applied magnetic field is one reason – indeed the principal reason – for the success of orbit theory. Particle orbits in a magnetic field define both a natural length, rL, the particle Larmor radius, and frequency, ω, the cyclotron frequency.

Triple systems are very familiar. The motion of the Earth and Moon around the Sun is lightly perturbed by the other planets, and if such effects are neglected it is a nice example of a triple system. Furthermore, the distance between the Earth and Moon is much smaller than their distance from the Sun, and so it is an example of what is called a hierarchical triple system. The dynamics of such a system can be understood, to a satisfactory first approximation, as two Keplerian motions. In the case of the Earth–Moon–Sun system, one of these is the familiar motion of the Moon relative to the Earth, and the other is the motion of the barycentre of the Earth–Moon system around the Sun. The barycentre lies within the Earth, in fact, and we are more familiar with the picture that the Earth orbits around the Sun, but it is more accurate to say that it is the motion of the barycentre that is approximately Keplerian. This was realised by Newton (Principia, Book I, Prop. LXV), and it was he who really originated the study of hierarchical triples.

The mass ratios in the Earth–Moon–Sun system are rather extreme. Even though the Sun is so distant, its mass is so great that it exerts a much greater force on the Moon than the Earth does.

This chapter deals with two effects of two-body encounters. In a general way this process was discussed in Chapter 14, but now we begin to study the effects on the system itself. Furthermore, the theory described there is applicable only to one of the two topics of this chapter. That theory describes the cumulative effects of many weak scatterings, and is perfectly adequate for an understanding of mass segregation. The escape of stars from an isolated stellar system, however, is controlled by single, more energetic encounters, and a better theory is necessary. The theory we shall describe is illustrative of a whole body of theory which improves on that of Chapter 14, though for most purposes (e.g. mass segregation) the improvements are unimportant.

Escape

We consider the case of an isolated stellar system. For this case, a star with speed ν will almost certainly escape if ν2/2 + φ > 0, where φ is the smoothed potential of the system at the location of the star, with the convention that φ → 0 at infinity. The exceptions are binary components (for which the true potential differs significantly from the smoothed potential), and an escaper which, on its way out, interacts with another star in such a way that its energy once again becomes negative. The latter possibility is rare in large systems (King 1959), precisely because two-body relaxation takes place on a much longer time scale than orbital motions (Chapter 14).

Even though this is a book about dense stellar systems (i.e. what is often called ‘collisional’ stellar dynamics, though no physical collisions need take place), it rests on a foundation of ‘collisionless’ stellar dynamics, and the relevant aspects are surveyed in these five chapters. In addition, we outline the various ways in which the effects of gravitational encounters can be incorporated, though the details are deferred to later sections of the book.

Chapter 5 begins with a discussion of the main aspects of the thermodynamic behaviour of N-body systems: how a stellar system responds to being put in contact with a ‘heat bath’, for instance. In fact, stellar systems tend to cool down if heat is added; paradoxical though this might seem, it helps us to understand even the motion of an Earth satellite. A toy model helps to explain what is happening.

Chapter 6 introduces the basic tools used for describing large numbers of gravitating particles: phase space, the distribution function f, the gravitational potential, and the equation governing the evolution of f (the ‘collisionless Boltzmann equation’). We outline some of its solutions, and aspects of the manner in which they evolve, especially phase mixing. We also look at the development of Jeans' instability.

For our purpose the most important distribution functions are those exhibiting spherical spatial symmetry. Therefore Chapter 7 is devoted to the motion of stars in spherical potentials, including constants of the motion and their link with symmetry.

Introduction

In this final chapter an attempt is made to sketch the classical mathematical structure underlying the various theoretical models which have been used throughout the book. The knowledge of where a particular model fits within the overall picture helps us both to understand the relationship to other models and to appreciate its limitations. Of course, we have touched upon these relationships and limitations already so the task remaining is to construct the framework of classical plasma theory and show how it all fits together.

Since collisional kinetic theory is the most comprehensive of the models that we have discussed we could begin with it as the foundation of the structure we wish to build. Indeed, we shall demonstrate its pivotal position. This would, however, be less than satisfactory for two reasons. The first and basic objection is that, so far, we have merely assumed a physically appropriate model for collisions. We have not carried out a mathematical derivation of the collision term. In fact, enormous effort has gone into this task though we shall present only a brief resumé. In doing so, we shall show how the separation of the effects of the Coulomb force into a macroscopic component (self-consistent field) and a microscopic component (collisions) appears quite naturally in the mathematical derivation of the collisional kinetic equation. This is the second reason for starting at a more fundamental level than the collisional kinetic equation itself.

To lighten the burden of the mathematical analysis we have, wherever convenient, restricted calculations to a one-component (electron) plasma. The ions, however, are not ignored but treated as a uniform background of positive charge.

As first introduced by Maxwell, the term ‘relaxation’ meant the process by which a deformed elastic body returned to equilibrium. It was then extended to the dynamical theory of gases, where equilibrium is a statistical equilibrium characterised by the particular form of the distribution of energies, and then transferred by Jeans to stellar dynamics. In stellar dynamics equilibrium is never achieved, because particles escape, but one can still think of a ‘quasi-equilibrium’ on the time scale of many crossing times. Even so, in due course the term ‘relaxation’ gradually became applied to several mechanisms which alter such properties as the energies of the stars, whether or not they have anything to do with the approach to equilibrium. More recently it has been argued (Merritt 1999) that the term should be extended further to apply to any one of a variety of mechanisms which cause evolution of the distribution function, whether or not the quantities like energy or angular momentum are altered. The history of the word reflects the development of the subject, from its initial concern with equilibrium models to its modern concern with dynamical evolution.

This chapter deals with one mechanism of relaxation, in which the energy of one star is altered by its interaction with one other. It is often called ‘collisional’ relaxation, though the interaction is entirely gravitational; real collisions we do not discuss until Chapter 31.

Introduction

Ideal MHD is used to describe macroscopic behaviour across a wide range of plasmas and in this chapter we consider some of the most important applications. Being dissipationless the ideal MHD equations are conservative and this leads to some powerful theorems and simple physical properties. We begin our discussion by proving the most important theorem, due to Alfvén (1951), that the magnetic field is ‘frozen’ into the plasma so that one carries the other along with it as it moves. This kinematic effect arises entirely from the evolution equation for the magnetic field and represents the conservation of magnetic flux through a fluid element. Of course, any finite resistivity allows some slippage between plasma and field lines but discussion of these effects entails non-ideal behaviour and is postponed until the next chapter.

The concept of field lines frozen into the plasma leads to very useful analogies which aid our understanding of the physics of ideal MHD. It also suggests that one might be able to contain a thermonuclear plasma by suitably configured magnetic fields, although research has shown that this is no easily attainable goal. Further, since the ideal MHD equations are so much more amenable to mathematical analysis they can be used to investigate realistic geometries. The theory has thereby provided a useful and surprisingly accurate description of the macroscopic behaviour of fusion plasmas showing why certain field configurations are more favourable to containment than others.

Introduction

Linearization gives rise to such simplification that in many cases it is pushed to its limits and sometimes beyond in the hope that by understanding the linear problem we may gain some insight into the non-linear physics. Perhaps the clearest example of the progress that can be made by analysing linearized equations is in cold plasma wave theory, but linearization, in one form or another, is almost universally applied. For instance, the drift velocities of particle orbit theory are of first order in the ratio of Larmor radius to inhomogeneity scale length. In kinetic theory it is invariably assumed that the distribution function is close to a local equilibrium distribution.

A question of fundamental importance is then, ‘How realistic and relevant are linear theories?’ Some problems are essentially non-linear in that there is no useful small parameter to allow linearization. Examples of these are sheaths, discussed in Chapter 11, and shock waves. Primarily, our intention is to address the subsidiary question: ‘Given that there is a valid linear regime, to what extent need we concern ourselves with non-linear effects?’

Of course, if the linear solution predicts instability then we know that, in time, it will become invalid because the approximation on which the linearization is based no longer holds good. In such cases the aim might be to identify and investigate non-linear processes that come into play and quench the instability. However, an unstable linear regime is emphatically not a pre-requisite for an interest in nonlinear phenomena. There are many situations in which the linear equations give only stable solutions but the non-linear equations are secular, i.e. under certain conditions some solutions grow with time.

Newton's equations for the gravitational N-body problem are the starting point for all four chapters in Part I, but each time seen in a different light. To the astrophysicist (Chapter 1) they represent an accurate model for the dynamical aspects of systems of stars, which is the subject known as stellar dynamics. We distinguish this from celestial mechanics, and sketch the distinction between the two main flavours of stellar dynamics. This book is largely devoted to what is often (but maybe misleadingly) called collisional stellar dynamics. This does not refer to actual physical collisions, though these can happen, but to the dominant role of gravitational encounters of pairs of stars. In dense stellar systems their role is a major one. In collisionless stellar dynamics, by contrast, motions are dominated by the average gravitational force exerted by great numbers of stars.

We lay particular emphasis on the stellar systems known as globular star clusters. We survey the gross features of their dynamics, and also the reasons for their importance within the wider field of astrophysics. Though understanding the million-body problem is not among the most urgent problems in astrophysics, through globular clusters it has close connections with several areas which are. Another practical topic we deal with here is that of units, which may be elementary, but is one area where the numbers can easily get out of hand.

Chapter 2 looks at the N-body equations from the point of view of theoretical physicists.

Following the evolution of a star cluster is among the most computer-intensive and delicate problems in science, let alone stellar dynamics. The main challenges are to deal with the extreme discrepancy of length and time scales, the need to resolve the very small deviations from thermal equilibrium that drive the evolution of the system, and the sheer number of computations involved. Though numerical algorithms of many kinds are used, this is not an exercise in numerical analysis: the choice of algorithm and accuracy are dictated by the need to simulate the physics faithfully rather than to solve the equations of motion as exactly as possible.

Length/time scale problem

Simultaneous close encounters between three or more stars have to be modelled accurately, since they determine the exchange of energy and angular momentum between internal and external degrees of freedom (Chapter 23). Especially the energy flow is important, since the generation of energy by double stars provides the heat input needed to drive the evolution of the whole system, at least in its later stages (Chapter 27). Unfortunately, the size of the stars is a factor 109 smaller than the size of a typical star cluster. If neutron stars are taken into account, the problem is worse, and we have a factor of 1014 instead, for the discrepancy in length scales.

The time scales involved are even worse, a close passage between two stars taking place on a time scale of hours for normal stars, milliseconds for neutron stars (Table 3.1).

Introduction

The plasma state is often referred to as the fourth state of matter, an identification that resonates with the element of fire, which along with earth, water and air made up the elements of Greek cosmology according to Empedocles.† Fire may indeed result in a transition from the gaseous to the plasma state, in which a gas may be fully or, more likely, partially ionized. For the present we identify as plasma any state of matter that contains enough free charged particles for its dynamics to be dominated by electromagnetic forces. In practice quite modest degrees of ionization are sufficient for a gas to exhibit electromagnetic properties. Even at 0.1 per cent ionization a gas already has an electrical conductivity almost half the maximum possible, which is reached at about 1 per cent ionization.

The outer layers of the Sun and stars in general are made up of matter in an ionized state and from these regions winds blow through interstellar space contributing, along with stellar radiation, to the ionized state of the interstellar gas. Thus, much of the matter in the Universe exists in the plasma state. The Earth and its lower atmosphere is an exception, forming a plasma-free oasis in a plasma universe. The upper atmosphere on the other hand, stretching into the ionosphere and beyond to the magnetosphere, is rich in plasma effects.

Solar physics and in a wider sense cosmic electrodynamics make up one of the roots from which the physics of plasmas has grown; in particular, that part of the subject known as magnetohydrodynamics – MHD for short - was established largely through the work of Alfvén.

Introduction

Although ideal MHD is often a good model for astrophysical and space plasmas and is widely employed in fusion research it is never universally valid, for the reasons discussed in Section 4.1. In this chapter we consider some of the most important effects which arise when allowance is made for finite resistivity and, in the case of shock waves, other dissipative mechanisms. Even though the dissipation may be very weak the changes it introduces are fundamental. For example, finite resistivity enables the plasma to move across field lines, a motion forbidden in ideal MHD. Usually, the effects of this diffusion are concentrated in a boundary layer so that mathematically the problem is one of matching solutions, of the non-ideal equations in the boundary layer and ideal MHD elsewhere. On the length scale of the plasma the boundary layer may be treated as a discontinuity in plasma and field variables and, depending on the strength of the flow velocity, this discontinuity may appear as a shock wave.

A comparison of Tables 3.1 and 3.2 reveals that the difference between resistive and ideal MHD is the appearance of extra terms proportional to the plasma resistivity, η ≡ σ−1, in the evolution equations for P and B.

Introduction

Historically studies of wave propagation in plasmas have provided one of the keystones in the development of plasma physics and they remain a focus in contemporary research. Much was already known about plasma waves long before the subject itself had any standing, early studies being prompted by practical concerns. The need to allow for the effect of the geomagnetic field in determining propagation characteristics of radio waves led to the development, by Hartree in 1931, of what has become known as Appleton–Hartree theory. About the same time another basic plasma mode, electron plasma oscillations, had been identified. In 1926 Penning suggested that oscillations of electrons in a gas discharge could account for the anomalously rapid scattering of electron beams, observed over distances much shorter than a collisional mean free path. These oscillations were studied in detail by Langmuir and were identified theoretically by Tonks and Langmuir in 1928.

Alfvén's pioneering work in the development of magnetohydrodynamics led him to the realization in 1942 that magnetic field lines, pictured as elastic strings under tension, should support a class of magnetohydrodynamic waves. The shear Alfvén wave, identified in Section 4.8, first appeared in Alfvén's work on cosmical electrodynamics. Following the development of space physics we now know that Alfvén (and other) waves pervade the whole range of plasmas in space from the Earth's ionosphere and magnetosphere to the solar wind and the Earth's bow shock and beyond.

There is a bewildering collection of plasma waves and schemes for classifying the various modes are called for. Plasma waves whether in laboratory plasmas or in space are in general non-linear features.

It often happens in science that progress is made, not so much by the discovery of new facts, but by organising knownfacts in a new way. An illustration from chemistry is the periodic table, and in astronomy the HR (Hertzsprung–Russell)diagram is a perfect example. In its modern form this diagram, now called a CM (colour–magnitude) diagram, is a scatter plot of the luminosity or absolute magnitude of a sample of stars plotted against their colour (see Fig. 30.2). It is an immensely powerful tool, so familiar that its power is taken for granted, and it is invaluable for studying the evolution of stars. In this chapter our aim is to provide something comparable for star clusters, though our goal is the much more modest one of helping the reader to grasp in a few pictures the essentials of what has been described in greater detail in earlier chapters of this book.

The links between many of the dynamical processes we have discussed, and one or two others, are summarised in Fig. 33.1. Centre stage is mass loss, which occurs through the agency of several processes. Meanwhile, as the system is losing mass and heading towards oblivion, various processes are also causing its internal structure to evolve. Chief among these is two-body relaxation, which causes more massive stars to segregate inwards, and the core to collapse. Both mechanisms enhance the importance of interactions between primordial binaries, which eventually bring the collapse to a halt.