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).

The following three chapters complicate the million-body problem for astronomically motivated reasons. Chapter 24 explains these by tracing the history of the discovery of binary stars in star clusters, in numbers which imply that they are primordial, i.e. they were born along with the cluster itself. They are associated with several of the remarkable phenomena which help to explain why globular star clusters are so important to astrophysicists, such as the sources of X-rays within them. We contrast their behaviour in star clusters with the much milder behaviour of binaries in less extreme environments.

In systems with many binaries, four-body encounters between two binaries are common. Chapter 25 discusses in detail one of the commoner outcomes: hierarchical triple systems. They are one class of three-body problem where the motion is both non-trivial and amenable to detailed calculation. Since these systems are stable and very long-lived, but may have tiny orbital time scales, such results are important for efficient computer simulation of N-body systems with many primordial binaries.

Chapter 26 discusses the effect of binary–binary encounters on the rest of the system. In important ways they can dominate the effect of the three-body encounters discussed in earlier chapters, though not forever, as binaries are also destroyed in these encounters. The outcomes of the interactions are also more complicated than in three-body encounters, and we show how to classify these.

Since this book is aimed at a broad audience within the physical sciences, we expect most of our readers not to be experts in either astrophysics or mathematics. For those readers, the title of this book may seem puzzling at least. Why should they be interested in the gravitational attraction between bodies? What is so special about a million-body problem, rather than a billion or a trillion bodies? What kind of bodies do we have in mind? And finally, what is the problem with this whole topic?

In physics, many complex systems can be modelled as an aggregate of a large number of relatively simple entities with relatively simple interactions between them. It is one of the most fascinating aspects of physics that an enormous richness can be found in the collective phenomena that emerge out of the interplay of the much simpler building blocks. Smoke rings and turbulence in air, for example, are complex manifestations of a system of air molecules with relatively simple interactions, strongly repulsive at small scales and weakly attractive at larger scales. From the spectrum of avalanches in sand piles to the instabilities in plasmas of more than a million degrees in labs to study nuclear fusion, we deal with one or a few constituents with simple prescribed forces. What is special about gravitational interactions is the fact that gravity is the only force that is mutually attractive.

The last eight chapters, dealing as they have done with interactions between only three or four stars, might seem a long digression away from the subject implied by the title of this book. Yet we shall see, as we take up the thread of the million-body problem where we broke off at the end of Chapter 18, that an understanding of the behaviour of few-body systems is crucial in following the evolution of the system through core collapse and beyond.

We left the system rushing towards core collapse, its central density rising inexorably, so that it would reach infinite values in finite time. How is this catastrophe averted? In fact there is no shortage of choices, for at least five different mechanisms have been proposed over the years. Admittedly, two are rather out of favour at present: a central black hole (e.g. Marchant & Shapiro 1980), or runaway coalescence and evolution of massive stars (Lee 1987a, and Problem 1). The other three involve binary stars in one guise or another, and it is not hard to see why this is attractive. After all, the mechanism responsible for core collapse is a two-body one (Chapter 14). Therefore it is clear that higher-order interactions, which we have neglected so far, might in principle eventually compete with two-body relaxation when the density becomes high enough. And three-body interactions can create binaries (Chapter 21 and Fig. 27.1).

The following three chapters complete the story of the evolution of a million-body system, in its purely stellar dynamical form. Chapter 27 begins by estimating the rate at which the formation and evolution of (non-primordial) binaries effectively generates energy within the system. The first application is to show that this is sufficient to halt core collapse. Then we consider other ways of generating the energy: binaries formed in dissipative two-body encounters between single stars, and primordial binaries; we quantify the extent to which the effectiveness of primordial binaries depends on their abundance and their energy.

In Chapter 28 we consider how a balance can be struck between the creation of energy (by binary interactions) deep in the core and the large-scale structure of the rest of the cluster. We first describe a standard argument which implies that conditions in the core, where the energy is generated, are governed by the overall structure. We outline the core parameters and overall evolution which this argument implies. Next we give arguments to show that this balance can be unstable, and describe the phenomenon of temperature inversion which is associated with the generation of gravothermal oscillations. The manner in which they depend on N (in idealised models) is an example of a Feigenbaum sequence of ‘period-doubling bifurcations’ in this context.

Chapter 29 rounds off the evolution of a million-body system by focusing on the evolution of gross structural parameters: total mass, and a measure of the overall radius.

Can a million-body system be in equilibrium? More precisely, can a model of a million-body system exhibit equilibrium? The answer depends on the model and other conditions. But we already saw in Chapter 8 that equilibrium models of gravitational many-body systems can be constructed. Thus, if it is modelled as a self-gravitating perfect gas it will be in thermal equilibrium if its temperature is uniform. If it is modelled by a Fokker–Planck or Boltzmann equation, then the equivalent condition is that the single-particle distribution be Maxwellian. In both cases the system is required to have infinite mass and extent.

These isothermal models are, in a strict sense, artificial, but they are of great importance conceptually, and for other reasons. In order to make progress in understanding them we shall replace one form of artificiality with another. Instead of dealing with infinite systems, we shall enclose our isothermal system in a spherical enclosure, which at least has the merit of implying that our systems have finite mass and radius. We shall suppose that the enclosure is rigid and spherical; in the N-body model this means that stars bounce off it without loss of energy, while in the gas or phase-space models, it implies that the enclosure is adiabatic. We assume spherical symmetry, and that stars all have the same fixed individual mass. We work entirely, however, with a perfect gas model.

During the past decade or so, theoretical astrophysics has emerged as one of the most active research areas in physics. This advance has also been reflected in the greater interdisciplinary nature of research that has been carried out in this area in recent years. As a result, those who are learning theoretical astrophysics with the aim of making a research career in this subject need to assimilate a considerable amount of concepts and techniques, in different areas of astrophysics, in a short period of time. Every area of theoretical astrophysics, of course, has excellent textbooks that allow the reader to master that particular area in a well-defined way. Most of these textbooks, however, are written in a traditional style that focusses on one area of astrophysics (say stellar evolution, galactic dynamics, radiative processes, cosmology, etc.). Because different authors have different perspectives regarding their subject matter, it is not very easy for a student to understand the key unifying principles behind several different astrophysical phenomena by studying a plethora of separate textbooks, as they do not link up together as a series of core books in theoretical astrophysics covering everything that a student would need.

Introduction

In this chapter we deal with the physical processes that take place in he diffuse IGM and their signatures in the quasar spectra in the form of hydrogen absorption lines. It uses material developed in several previous chapters, especially Chap. 7.

Gunn–Peterson Effect

We have seen in Chap. 4 that the formation of neutral matter and the decoupling of radiation occurred at z ≃ 103. Because it is unlikely that the formation of structures at lower redshifts could have been 100% efficient, we would expect at least a fraction of the neutral material, especially hydrogen, to remain in the IGM with nearly uniform density. This neutral hydrogen could, in principle, be detected by an examination of the spectrum of a distant source, like a quasar. Neutral hydrogen absorbs Lyman-α photons, which are photons of wavelength 1216 Å that corresponds to the energy difference between the n = 1 and the n = 2 states of the H atom. Because of the cosmological redshift, the photons that are absorbed will have a shorter wavelength at the source and the signature of the absorption will be seen at longer wavelengths at the observer. We expect the spectrum of the quasar to show a dip (known as the Gunn–Peterson effect) at wavelengths on the blue side (shortwards) of the Lyman-α emission line if neutral hydrogen is present between the source and the observer.

Introduction

To understand the features of the universe today, it is necessary to grasp the past history of the universe. We now tackle this issue and describe the physical processes that occur in the earlier phase of the universe. Section 2 develops the basic thermodynamics needed to understand these processes. In Sections 4.3 and 4.4, we consider the possible existence of a relic background of massless or massive fermions (like the neutrinos) in our universe today. In Section 4.5 we discuss the primordial nucleosynthesis and its observational relevance; we study the decoupling of matter from radiation in Section 4.6. In the last section the very early universe and inflationary models are described.

Distribution Functions in the Early Universe

The analysis in Chap. 3 showed that the universe was dominated by radiation at redshifts higher than zeq ≃ 3.9 × 104(ΩNRh2). In the radiation-dominated phase, the temperature of the radiation will be higher than Teq ≃ 9.2 (ΩNRh2)eV ≃ 1.07 × 105(ΩNRh2)K and will be increasing as T ∝ (1 + z).

The contents of the universe, at these early epochs, will be in a form very different from that in the present-day universe. Atomic and nuclear structures have binding energies of the order of a few tens of electron-volts and mega-electron-volts, respectively. When the temperature of the universe was higher than these values, such systems could not have existed as bound objects.

Introduction

Attempts to understand extragalactic objects and the universe by using the laws of physics lead to difficulties that have no parallel in the application of the laws of physics to systems of a more moderate scale. The key difficulty arises from the fact that our universe exhibits temporal evolution and is not in steady state. Thus different epochs in the past evolutionary history of the universe are unique (and have occurred only once), and the current state of the universe is a direct consequence of the conditions that were prevalent in the past. For example, most of the galaxies in the universe have formed sometime in the past during a particular phase in the evolution of the universe. This is in contrast to star formation within a galaxy that we can observe directly and study by using standard statistical methods.

In principle, we should be able to see the events that took place in the universe in the past because of the finite light travel time. By observing sufficiently far-away regions of the universe, we will be able to observe the universe as it was in the past. Although technological innovation will eventually allow us to directly observe and understand all the past events in the history of the universe (especially when neutrino astronomy and gravitational wave astronomy start complementing photon-based observations), we are far from such a satisfactory state of affairs at present.

Introduction

This chapter deals with the Friedmann model for the universe, which is used throughout the study of extragalactic astronomy. The basic framework needed in all the later chapters is also introduced. Some of the discussion requires concepts from the general theory of relativity developed in Vol. I, Chap. 11. The Latin indices go over i, j = 0, 1, 2, 3, and the Greek indices go over α, β = 1, 2, 3. We shall use units with c = 1 in most sections.

The Friedmann Model

To construct the simplest model of the universe, we begin by assuming that the geometrical properties of three-dimensional space are the same at all spatial locations and that these geometrical properties do not single out any special direction in space. Such a three-dimensional space is called homogeneous and isotropic.

The geometrical properties of the space are determined by the distribution of matter through Einstein's equations. It follows therefore that the matter distribution should also be homogeneous and isotropic. This is certainly not true in the observed universe, in which there exists a significant degree of inhomogeneity in the form of galaxies, clusters, etc. We assume that these inhomogeneities can be ignored and the matter distribution may be described by a smoothed-out average density in studying the large-scale dynamics of the universe.