The simplest examples of the gravitational slingshot are a binary which scatters a single star and the breakup of an initially bound three-body system. Resulting recoil may eject both the single body and the binary from the system. If galactic nuclei contain supermassive objects which form binaries, these processes may have especially dramatic astronomical consequences (Saslaw, Valtonen & Aarseth, 1974). They are also important in ordinary star clusters with N ≲ 102, and in the very center of richer clusters.

Performing large numbers of three-body experiments shows how scattering behaves. There is an amusing contrast with high energy physics. Our (inexpensive) gravitational accelerator is a computer. We experiment not to find the microscopic law describing the interaction – we've known that since Newton – but to understand realistic applications of this law. Even our relatively low energy particles, typical stars moving at ~ 102 km s-1, have kinetic energies exceeding ~ 1050GeV!

Although there are a great many exact mathematical results on the analytical dynamics of three-body systems, they usually apply only under very restricted circumstances. Normally they are asymptotic or perturbation calculations (see Arnold, 1978; Pars, 1965). These are not sufficient to understand the rich range of important phenomena occurring in realistic physical scattering, so we must resort to numerical simulations and approximate non-linear analyses (see Heggie, 1975(a, b), reference in Section 48).

Introduction

Spiral structure is the frosting on the cake of galactic rotation. Like frosting, it is very alluring and has been greatly admired. Self-gravitating stellar disks are very lively objects; many processes contribute to the observed spiral patterns in galaxies. Our knowledge of this subject, greatly extended in the last two decades, still changes quickly. This section just describes some of the basic gravitational principles and questions common to many discussions.

What are the forces that drive spiral structure? To answer ‘differential rotation’ is true, but not much more informative than the ancient physicians who replied that morphine produces sleep because it contains a ‘dormative principle’. Spiral instabilities resulting from differential rotation are familiar in both magnetic and fluid systems, with or without gravitation.

The patterns of rotating magnetic field lines are so suggestive that for many years they were popular as an explanation for spiral structure, either directly or through their influence on star formation. The decline of magnetic explanations occurred when more accurate measurements of the galactic magnetic field strength showed it was significantly less than expected. Even today these measurements are uncertain because the topology of the field is obscure and we are not really sure how to average it over large distances.

Previous sections have made it plausible that an object in a gravitating system near equilibrium can be considered to be immersed in a bath of fluctuating forces, along with an average mean field force. We now consider a simple mathematical model for the time evolution of orbits. We use this intuitive physical picture to try to capture the essence of the problem in a fairly simple way. An advantage of this procedure is that it readily suggests modifications of the description for an improved physical picture. The results can always be checked against N-body computer experiments, and we will discuss their more exact derivation in Section 10.

At first sight, the simplest model might seem to represent the motion of each star by Newton's equation of motion with a stochastic force β(t) which fluctuates in time, i.e.,. But this turns out to be too simple. It makes the velocity undergo Brownian motion (for a Gaussian distribution of fluctuations) with an everincreasing root mean square value vrms ∝t½. Correspondingly, the root mean square position of an average star also departs monotonically from its initial value. These two properties are inconsistent with conservation of total energy, for the increase in kinetic energy must be compensated by a contraction of the system to decrease the potential energy. But the Brownian increase of every star's root mean square position from its initial value prevents the system from becoming very small.

Gas dynamical processes

Many phenomena occur when stars plunge through clouds of gas. Among the most dramatic is the formation of shocks and ionized wakes, especially around stars which emit strongly in the ultraviolet. The ionizing radiation can be produced either by the star directly, or from the bow shock accompanying supersonic motion. Although we will not usually include gas dynamic and radiative processes, this one is an exception since it is important. So we give a brief general discussion of the phenomenon. Then we will describe collisionless accretion, the slowing down of stars by gas, and modifications of the Jeans and two-stream instabilities.

Suppose a star moves supersonically through a cloud of hydrogen. (The role of heavier atoms is mainly to increase and complicate the radiation processes.) In the direction of motion there is a bow shock which embraces the star more tightly at high Mach numbers (v*/vsound). To estimate the temperature immediately behind the shock front, equate the thermal energy 3kT/2 to the kinetic energy of an atom, giving, where v⊥100 is the inflow velocity (in units of 100kms-1) normal to the shock surface. As this shock-heated gas flows behind the star, it expands and cools. Part of the cooling will be from free–free radiation, and part from the expansion itself. From standard formulae for free-free emission by an ionized gas, one learns that the timescale for substantial radiation is τff ≈ 106v100n-1 yr, where n is the gas number density cm-3.

As we move from the idealizations of mathematical physics to the messy realities of astronomical systems, the very nature of our understanding is transformed. From basic principles we move to models, and from general models to special examples. Observed examples seldom do more than whisper the approximations we need to decode them. All we have left for guidance is a jumble of parametrized ideas, and a hope of relations among them. We understand through the inspired guess, or by accident.

Introduction – spinning polytropes

In our continued progression toward decreasing symmetry, we now arrive at the most individualistic systems of all. Previously we discussed flattened systems which were mainly thin disks. But those cannot represent the large observed class of elliptical galaxies. Ellipticals show moderate rotation, and as a result they have less global coherence than either spirals or spherical non-rotating systems. Pressure, rotation and internal currents all combine to determine a variety of structures and shapes.

Historically, some of the great eighteenth, nineteenth- and early-twentieth century mathematicians began to develop this subject by rigorously analyzing certain simple, highly specialized cases. Magnificent though these achievements were – and they often led to important mathematical insights – they also raised the stature of certain models so high that their astronomical importance became greatly exaggerated. All these models were built from uniform, incompressible fluids. The first of these non-spherical self-gravitating models were Maclaurin's (mid-eighteenth- century) spheroids. They have a stately uniform rotation, with no internal motions. Gravity is balanced by incompressibility and rotation. This balance can exist only for certain values of the spin ratio (58.1), nearly equal to unity. Two types of Maclaurin spheroids, with small and large eccentricities, are possible equilibrium figures.

So simple and satisfying seemed Maclaurin's results, that nearly a century went by before Jacobi realized that other equilibrium figures exist.

Ask a fundamental astronomical question, and the chance of an accurate answer is small. Throughout this account of gravitating systems, nearly each section ends on an incomplete note. We are just beginning to understand the richness and subtlety implicit in a system whose components interact with a force as simple as an inverse square. On all scales, from the sun and its planets to cosmic clusters of galaxies, flock insistent but unanswered questions.

For how long will the solar system be stable? Secular perturbations that grow as the planets follow their courses may eventually end in a resonance that forces ejection. Analytic theory gives us some reassurance, but not a definite answer. Computer simulations do not have the accuracy needed. And if we think of our solar system itself as an analog computer, then the calculation has not yet been done. All we know for certain is that the orbits of planets remaining today have avoided ejection for billions of years. Someday the sun may not rise tomorrow.

The uncertainties of planets orbiting the sun are relatively tame compared to the few-body problem whose masses nearly are equal. More opportunities for resonance flourish. A star hardly knows which way to go. Are computer experiments the only method for predicting the outcome from given initial conditions?

Parallel to the comparison developed in Sections 15 and 16, we next turn to new phenomena which graininess introduces into the growth of perturbations. In a major application of this theory, the grains are galaxies. Despite our ignorance of their formation, we can ask how gravity causes galaxies to cluster. Does the clustering we expect explain what we see?

The simplest result can even be anticipated from our analysis in the last section. Consider a gas of galaxies. Let it be uniform except for √N fluctuations. If we treat it, for a moment, as a fluid, then the growth rate of perturbations in a standard cosmology, ρ1(t)∝ R(t)∝t⅔, tells us that observed clusters with ~ 104 galaxies should be able to form easily starting at redshifts of 102 - 103. So galaxy clustering promises to be more understandable than galaxy formation, although it still has its mysteries.

Realizing the relative ease of galaxy clustering, it becomes natural to push the process back a step. Could galaxies themselves be the result of an earlier clustering? Suppose that isothermal perturbations of the Jeans mass at decoupling, ~ 106 M⊙, started (somehow) with large amplitudes and formed bound systems. It would take about 104 of these to form a small galaxy.

The purpose of this section is to give a very brief guide to the major dynamical properties of some real astronomical systems. It is not meant to be a review of the latest observations, for these change almost daily and often a long time must pass before they can be put into proper perspective. Specific astronomical observations, for understandable reasons, are seldom repeated. Often the pressures of time on large telescopes, or astronomical careers, are too great. The result is that, unlike laboratory physics experiments, it is difficult to gauge the real uncertainties (in contrast to the formal error limits) which surround particular observations. Another feature of astronomical observations is that a good deal of theory is usually needed to make the observation itself. This is because the systems we observe are complicated and seldom yield a basic physical quantity in a straightforward manner. The measurements we want must usually be strained through a network of interpretation.

Steady advances in observing technology mean that, at any given time, the most exciting frontier observations will often be near the limits of available instruments. Frequently these observations will just be able to rise above the ‘noise level’, and astronomers then speak of ‘a two- or three-sigma effect’, sigma being the standard error added by noise.

The plan of this monograph is divided into four main parts. These parts develop in order of decreasing symmetry, from idealized infinite homogeneous systems to finite flattened irregular systems. Along this sequence, the ratio of model applications to fundamental physical ideas and techniques increases. Even so, I have tended to emphasize the basic physics over detailed applications. Specific astronomical models wax and wane as data and fashions change, but the principles on which they are built have much longer lifetimes. Thus the degree to which various topics are discussed does not always reflect their popularity in today's, or yesterday's, literature.

Nearly all the theory described in this book is based on classical Newtonian gravity. Relativistic generalizations of almost every aspect are possible, and there was a flurry of these generalizations in the 1960s and early 1970s. It was greatly stimulated by possible applications to quasars. Although quasars still are unsolved, no evidence has developed that relativistic star clusters are needed to explain them. That, plus the fact that there are enough fascinating things to say about observed non-relativistic systems, persuaded me to restrict this book to classical gravity.

The book is reasonably self-contained in that most of its results can be obtained directly from preceding ones, sometimes with two or three intermediate algebraic steps to be added by the reader. Usually these steps are straightforward and they are outlined in the text.

Many N-body simulations have shown how galaxies disperse and cluster as the universe expands. We shall just examine some representative examples. The 4000-body simulations of Aarseth and his collaborators (see bibliography for references) were designed to see how different initial conditions and cosmological models, within reasonable ranges, alter the clustering of galaxies. Experiments whose initial velocities had a pure Hubble flow behaved much like those which also had peculiar Maxwellian velocities smaller than the expansion velocity. Gravitational graininess built up the self-consistent velocity field after about one initial Hubble expansion in either case.

Somewhat more important is the galaxies' mass distribution. In the computation this does not change with time, although small groups which form can mimic the distant effects of larger mass galaxies at later stages of clustering. Various simulations have shown that the more massive galaxies have a greater tendency to cluster. Their two-point correlation amplitude, for example, increases faster than for less massive galaxies. Linear analysis (Equation (24.1)) leads us to expect this, at least during the initial clustering when the velocity distribution does not depend strongly on mass. Experiments also show that many results, especially regarding spatial distribution, do not depend strongly on detailed properties of the mass spectrum, so long as it continuously spans a range of at least an order of magnitude.

Equation of state

Imagine life as it may have been a million years ago. You are in the jungle, being stalked by a tiger. Your ability to survive depends on pattern recognition. If you can only see the stripes on the tiger (small scale correlations), but not the overall effect of the tiger itself, you will be at some disadvantage. Perhaps this is how the ability of our eyes to recognize high order correlation functions developed. Similarly, restricting our understanding of galaxy clustering to just the two- or three-particle correlation functions means we miss a lot of the action. We need a simple measure of high order clustering which can also be related to basic gravitational physics.

In Section 27 we saw that gravitational clustering can be characterized by the distribution of voids. These, in turn, are related to the high order correlations which describe the galaxies which should have been in the region of the void but are not. We may generalize the idea of a void by working in terms of distribution functions f(N) which give the probability of finding any number of galaxies in a volume V of arbitrary size and shape. For N = 0, the distribution of voids is f(0), which is calculated in (27.7) for a Poisson distribution.