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.

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).