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.

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.