So far we have been dealing with the propagation of light through media whose refractive indices have been assumed to be constant with position (the assumption of homogeneity). We are now able to embark on the study of the propagation of light through media whose refractive indices – and hence absorptive and dispersive properties – may vary along the ray path; a differential treatment is then in order. More specifically, we shall deal with stratified media whose material properties are constant in planes perpendicular to a given direction. Moreover, our study will not only include passive systems but emission properties of the medium will also be considered (although in the most simplified way).

There are three main hypotheses we should add to proceed with the development that follows:

1. We shall assume that the absorptive, dispersive, and emissive properties of the medium are independent of the light-beam Stokes vector. This is in fact a linear approximation that holds in many astrophysical applications, where, even though the medium may be dependent on the whole radiation field, the angular width of the beam (indeed within the realm of geometrical optics) is so small that its contribution to the physical conditions of the medium can be neglected (e.g., Landi Degl' Innocenti and Landi Degl' Innocenti 1981).

2. […]

With the radiative transfer equation for polarized light to hand, we shall proceed to find solutions and to exploit them both, the equation and its solutions, in order to obtain information about the medium. This chapter is devoted to solutions of the RTE and to the first and simplest diagnostics one can obtain from the observed Stokes profiles. The main emphasis is on concepts rather than numerical details. The latter may be found in the literature (some of the most recent papers are recommended in the bibliography) and in fact are still in continuous evolution and debate. Most of the concepts we describe in this chapter, however, may be said to be well founded nowadays and will help the reader in understanding the topic.

This chapter is devoted to recalling a number of results of importance for development in later chapters. Most of these concepts are assumed to be already known to the reader, and those derivations that are missing will be found in textbooks on optics and electromagnetism. The main aim here is to provide a summary of the polarization properties of the simplest electromagnetic wave one can conceive: the monochromatic, time-harmonic, plane wave.

The terms light and electromagnetic wave will be understood as synonymous throughout the text. More specifically, we will be referring to the visible part of the spectrum and its two nearest neighbors, the ultraviolet and the infrared. Many of the topics discussed are also applicable to other wavelength regions. In particular, it is worth noting that radio observations use most of the concepts we shall be developing here for the optical region, although they are not in principle necessary for that wavelength range.

Polarimetric accuracy is one of the most important goals of modern astronomy. The definition itself of polarimetric accuracy, however, is difficult since we mostly measure polarization differences and are uncertain in establishing the zero level, which is often set by convention. Hence, by “accuracy” we shall understand the sensitivity to variations of the polarization level. Besides the greatest polarimetric accuracy, every astronomical observation should ideally pursue the highest spectral, spatial, and temporal resolution with the widest spatial and spectral coverage. However, all these goals are hard to accomplish at the same time and one always needs to compromise depending on the specific objectives a given observation is aimed at. The amount of available photons from the Sun is never sufficient. In fact, it is equal per resolution element to that from a scarcely resolved star of the same effective temperature. This observational fact is easy to understand (e.g., Mihalas, 1978) if one takes into account the invariance with distance of the specific intensity (energy per unit normal surface, per unit time, per unit frequency interval, per unit solid angle) and its proportionality to the photon distribution function (number of photons per unit volume, per unit frequency interval, per unit solid angle).

Solar polarimetry is, of course, a part of the game and has several limiting factors that govern the final accuracy of the measurements.

In Chapter 6 we spent some time discussing the statistical description of an N-body system in terms of its one-particle distribution function, f. We also introduced an evolution equation for f, the ‘collisionless’ Boltzmann equation. We did not, however, dwell on any solutions of this equation, except to characterise them in terms of Jeans' Theorem. In fact equilibrium solutions have been known and studied for about 100 years, and they are of enduring importance.

The specific choice of f may be made for a variety of reasons. Plummer's model, for instance, is often used for starting a numerical calculation, because of its analytical convenience. The isothermal model is of importance in the study of thermodynamic stability (Chapter 17), while King's model has taken centre stage for many years in the interpretation of observations. Another approach to this particular topic, also based on the phase-space description, deserves a section on its own at the end of the chapter.

By Jeans' Theorem, f is a collisionless equilibrium solution if (but not only if) it depends on the energy, E. Most of the models we mention are of this kind. In what follows we shall usually characterise a model by its distribution function f (E), but that is only part of the story, because f depends on the potential φ (via E), and we need to know how φ depends on the position in space r in order to determine f at any point in phase space.

The previous chapter dealt mostly with a highly idealised model. All stars were single and had the same mass, and the system was isolated. As we saw, even the presence of a spectrum of stellar masses changes the picture, as it is found that gravothermal oscillations set in only for considerably larger values of N. In the present chapter we shall also see that the presence of primordial binaries further weakens their probable relevance. Even when gravothermal oscillations do occur, they seriously affect the structure of only the innermost 1% or so of a cluster. Therefore, in this chapter we concentrate once more on the steady post-collapse evolution of a stellar system. Also, we mainly have in mind a system with a significant population of primordial binaries, and boundary conditions set by the tidal field of the surrounding galaxy. First, however, we consider the simpler case of an isolated cluster.

Isolated clusters

The first thing that is changed in post-collapse evolution when we add primordial binaries is the radius of the core. A similar argument to that of Box 28.1 shows that the ratio rh/rc is now almost independent of N (cf. Problem 1). In fact the ratio depends more on the proportion of binaries (which decreases as the binaries are consumed).

These statements greatly weaken the clues to the occurrence of gravothermal oscillations which we discussed in the case of post-collapse evolution powered by three-body binaries (Chapter 28).

We continue the emphasis on collective effects, i.e. those in which individual interactions between stars are of no importance, but we increasingly focus on those effects that really matter in the million-body problem. Chapter 10 opens with a brief discussion of the notions of equilibrium and stability in this context, but is largely concerned with non-equilibrium phenomena: phase mixing and ‘violent’ relaxation. Another mechanism for evolution of the distribution function, even in static potentials, is diffusion by chaotic motions.

Chapter 11 introduces a variant with a strong astrophysical motivation: the behaviour of N-body systems consisting of particles with time-dependent masses, and how this affects the energy and spatial scale of the system. Much depends on whether the variation is rapid or slow, and in the latter case we can easily study its effect on the distribution function itself.

Again motivated by the astrophysical setting, Chapter 12 introduces the effect of a steady external potential. The problem closely resembles an important idealised version of the motion of the Moon around the Earth under the external perturbing effect of the Sun (Hill's problem). We study the non-integrable motions in this potential, and the important problem of escape. The study is then extended to the case, even more important in applications, of an unsteady external potential.

The following three chapters begin the application of earlier results to the millionbody problem itself. Chapter 16 discusses two effects of two-body gravitational encounters: escape and mass segregation. The first of these actually develops the theory of two-body relaxation further, as we cannot, in this context, approximate encounters by any small-angle scattering approximation. This approach is, how ever, applicable to mass segregation, which is an effect of the tendency to equipartition of energies in two-body encounters. It also has an important influence on the stability of the million-body problem (the ‘mass stratification instability’).

Chapter 17 is also concerned with instability,b ut an instability which even exhibits itself in systems with equal masses. It was first discovered through a remarkable thermodynamic result obtained by Antonov, which helps to explain the relevance of the term ‘gravother modynamics’. This chapter deals with extrema of the entropy, and the stability of linear series of equilibria.

Chapter 18 follows up the previous two chapters by tracing the consequences of the mass stratification and gravothermal instabilities. This is the process referred to as core collapse. In other contexts this would be referred to as an example of “finite-time blow-up” and, in common with other examples of this behaviour, it can be described asymptotically by approximate self-similar solutions of the governing equations.

The present book has its origins in our earlier book Plasma Dynamics published in 1969. Many who used Plasma Dynamics took the trouble to send us comments, corrections and criticism, much of which we intended to incorporate in a new edition. In the event our separate preoccupations so delayed this that we came to the conclusion that we should instead write another book, that might better reflect changes of emphasis in the subject since the original publication. In writing we had two aims. The first was to describe topics that have a place in any core curriculum for plasma physics, regardless of subsequent specialization and to do this in a way that, while keeping physical understanding firmly in mind, did not compromise on a proper mathematical framework for developing the subject. At the same time we felt the need to go a step beyond this and illustrate and extend this basic theory with examples drawn from topics in fusion and space plasma physics.

In developing the subject we have followed the traditional approach that in our experience works best, beginning with particle orbit theory. This combines the relative simplicity of describing the dynamics of a single charged particle, using concepts familiar from classical electrodynamics, before proceeding to a variety of magnetohydrodynamic (MHD) models. Some of the intrinsic difficulties in getting to grips with magnetohydrodynamics stem from the persistent neglect of classical fluid dynamics in most undergraduate physics curricula.

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).