Some of the grandest encounters in the Universe occur when galaxies pass by one another. In the jetsam of their tidal debris, we often find circumgalactic evidence for close encounters. Sometimes the outer parts of a galaxy pair are distorted into long bridges and tails. We also find large irregular galaxies which have multiple nuclei, perhaps the recent result of a merger.

For a long time, interest in the subject of galaxy merging was quenched by a straightforward nσvT calculation which gave a very small merger probability over the Hubble age T of the Universe. This result does not depend much on the actual value of the Hubble constant. If the observed value of H were larger the volume would decrease, so the observed number density n ∝ H3. Supposing the merger cross section to be roughly proportional to the observed visible cross section gives σ ∝ H-2. (Increasing H means the galaxy is closer and has a smaller linear visible diameter.) The Hubble age T≈H-1. Therefore the resulting probability remains small, regardless of the uncertain current value of H.

The apparent conclusiveness of this argument, however, resulted from the unjustified simplicity of its assumptions. Merging encounters of galaxies are not like random collisions in a gas of particles. For one thing, most mergers within groups and clusters of galaxies occur among the more massive galaxies which tend to collect in the center through mass segregation (Section 46).

One can imagine the fun Mark Twain would have had with the concept of black holes and their influence. Unfortunately, he died in 1910, six years before K. Schwarzschild discovered these singular solutions of general relativity. Although this book is concerned with Newtonian systems, in which the ratio GM/Rc2 of gravitational energy to rest mass energy is very small, the presence of a central singularity which destroys or absorbs stars can have a significant effect on the surrounding Newtonian dynamics. Actually a black hole is just one example of such a singularity. Others are a supermassive star or spinar, and even just a region of such high density that stars disrupt inside it by physically colliding (Section 52).

We have already seen (Section 40.2) how a massive gravitating point modifies the distribution of surrounding stars when this distribution obeys a simple polytropic equation of state. The large point mass induces a central density cusp, in contrast to the normal flat distribution. We did not approach the central mass very closely in Section 40.2 because it destroys the polytropic behavior. Here we venture further in to see how the stellar orbits are distorted.

Let us suppose that the mass of the hole Mh is much greater than the mass of any individual star m*, but much less than the total mass of all stars in the cluster m*N.

Because the classical interaction between gravitating point masses is totally understood – the subject has had its Newton – it is possible to solve the clustering problem completely, in principle. All that is necessary is to integrate the equations of motion with the desired initial conditions and read off the answer. Non-linear regimes are dealt with as easily as linear ones. From the time fast computers became available, about 1960, to the present day, a considerable industry has grown up around these numerical solutions. Beginning with the earliest computations for a couple of dozen particles, the power of computers and techniques has improved by more than an order of magnitude each decade. Now it is possible to follow several thousand galaxies for tens of relaxation times with direct integrations.

New numerical techniques are always being developed. Their general goal is more information about increased numbers of particles for longer times. Compromises are often worked, usually sacrificing information for more particles. Since direct integrations retain the most information, and computers always get better, this approach will continue to flourish.

The direct integration method integrates the equations of motion explicitly to retain complete information about all the galaxy positions and velocities at any time.

Galactic winds

Stellar collisions, novae, supernovae, planetary nebulae, stellar winds and flares all dump gas into a star cluster or galaxy. How much of this gas cools and remains in the cluster, and how much escapes is a fundamental question. The answer is difficult because it depends completely on details of the mass loss process and on the subsequent interactions of gas lost from different stars. Clouds may collide and the heat generated by shocks may cause evaporation. Wandering stars may pass through the clouds, heating them as described in Section 17. The results are clearly very model dependent, so we will just consider a very simple example to illustrate the nature of galactic winds. (See Mathews & Baker, 1971, for a detailed analysis.)

A necessary condition for an atom or ion to escape from the cluster is that its energy exceed roughly twice the average energy per atom of the stars' random motions. (The collisional mean free path determines sufficient escape conditions.) If we can average very crudely over the cluster, then supposing that the star and gas motions are in rough energy equipartition (per particle) gives an equivalent escape temperature Tesc≈μν2*/3κ≈2 x 107(μ/0.5)(ν*/1000 km s-1)2 where μ is the mean particle mass in proton units. From Table 6 we see that, in typical systems where the stellar collision time is much less than a Hubble time, gas must be heated to ~ 5 × 107 K.

In Section 21 we were able to get some analytical understanding of linear perturbation growth. Subsequent sections used a combination of techniques, involving graininess, energy principles, numerical simulations and thermodynamics, to determine non-linear properties of clustering. In this section, we extend the general approach of Section 21 into the non-linear regime. Our aim is to see how long it takes a perturbation to begin contracting, to break away from the expansion of the surrounding universe.

If we were to extend the detailed linear Fourier perturbation technique of Section 21 into the non-linear regime, in a brute force fashion, it would rapidly become too complicated to provide much insight. Instead of that Eulerian technique it becomes simpler to take a Lagrangian point of view, as in the discussion of ‘pancakes’ in Section 35.

The Lagrangian technique follows the motion of a particular object, or set of objects, in the perturbation. (By contrast the Eulerian technique describes what happens as a function of spatial position). A simple inhomogeneity, whose growth typifies many more complicated inhomogeneities to order of magnitude, is just a spherical region of constant density in an otherwise uniformly expanding universe. If the inhomogeneity is to contract, and not merely expand more slowly than the rest of the universe, its density must be greater than the critical density ρc, just necessary to close the universe.

Some non-linear descriptions, such as the cosmic virial theorem, can be derived directly from first principles, but so far this has not been done for clustering efficiency. Here the technique is to construct a model, simple enough to solve analytically, which retains the essential physics of the problem.

Let us suppose there is a hierarchy of clustering on different well-defined levels. Consider the interaction of two adjacent levels of this hierarchy. For example, these levels could represent the clustering of globular clusters (or subgalaxies if these are formed from many globular clusters) to form galaxies. They could also represent the clustering of galaxies to form clusters of galaxies. Objects at the lower level are presumed to be tightly bound; we shall call them particles and represent them by point masses. Objects at the higher level are more loosely bound; they are the clusters.

Our analysis starts in a region which has already undergone its phase transition. The density of a cluster is large compared to the average background density. Not all the particles are in clusters. We shall consider the fate of an arbitrary vagrant particle moving through a field of clusters, and ask: ‘What is the probability that vagrant particles will be captured by clusters?’ This probability is a measure of the clustering efficiency.

The vagrant particle may be one which has been ejected from a forming cluster, or it may never have been bound.

To understand some of the most important properties of gravitating systems we temporarily put aside the effects of their density and velocity gradients, their components of different masses and sizes, and any external forces which may act upon them. Imagine an idealized, isolated, homogeneous gravitating system of particles. Usually in discussing the physics of these systems we will call their particles stars for brevity, although when discussing many astronomical contexts they will often be galaxies, or even clusters of galaxies. These idealized systems will sometimes be finite, sometimes infinite. In later sections we will find that both sizes of homogeneous clusters turn out to be unstable. But never mind that for now; there are more basic properties to consider.

Introduction and basic physical ideas

Is it paradoxical that relaxed nearly spherical galaxies and clusters of galaxies exist? Zwicky (1960) thought so and proclaimed this a major problem of extragalactic astronomy. The difficulty comes from noticing that the two-body relaxation time τR, given by Equation (2.11), for these systems is orders of magnitude longer than the Hubble age of the Universe. If clusters or galaxies did not form near equilibrium, two-body relaxation would not have led them to their present state. And it would seem rather contrived for their initial state to be their final state, like Athena springing from the head of Zeus.

The solution of this difficulty comes from noticing that other relaxation mechanisms exist. Indeed, the further the initial state lies from the relaxed system, the more powerful its relaxation is. King (1962) suggested qualitatively that such a relaxation process must exist and Hénon (1964) confirmed it for a restricted spherical case using numerical simulations. Lynden-Bell's (1967) work pioneered our quantitative understanding of the subject and Shu (1978) later clarified some aspects.

The basic idea behind violent relaxation is that the initial state is so far from equilibrium that large scale collective modes govern its early evolution. Since the initial distributions of masses and velocities are very irregular, individual objects (galaxies or stars) are scattered mainly by groups of objects, rather than by other individual objects.

The spirit beneath the surface of nearly any astronomical phenomenon is gravitation. The reason why gravity is the motive force for much of the Universe is not hard to see. What primarily interests us about the Universe is its structure, including ourselves. And the physical reason for the existence of this structure is gravity. Even in the case of ourselves, it is the force of gravity in massive stars which drives their nuclear reactions to produce heavy elements, then eventually causes the star to explode and spew these elements throughout the galaxy. Some of them collect into new stars and planets, partly through the more gentle ministrations of gravity – and here we are! Of course, the mass of humanity, though important to ourselves, is only about 10-41 of the mass of the visible Universe. To put it another way, we contribute about 10-19 kms-1 Mpc-1 to the Hubble constant.

As our own origin, through star formation and evolution, was driven by gravity, so even more directly does gravity govern the dynamics of other astronomical structure: stellar clusters, the shapes and evolution of galaxies, and the motions of the entire system of galaxies.

Langevin's equation, the Fokker–Planck equation, the master equation, and Boltzmann's equation are all just partial descriptions of gravitating systems. Each is based on different assumptions, suited to different conditions. They all arise from physical, rather intuitive, approaches to the problem. But there is also a more general description from which our previous ones emerge as special cases. We know this must be true because Newton's equations of motion provide a complete description of all the orbits. The trouble with Newton's equations is that they are not very compact: N objects generate 6N equations. True, the total angular and linear momenta, and energy, are conserved, at least for isolated systems, but this is not usually a great simplification.

By extending our imagination, we can cope with the problem. We previously imagined a six-dimensional phase space for the collisionless Boltzmann equation. Each point in this phase space represented the three position and three velocity (or momentum) coordinates of a single particle. It was a slight generalization of the twodimensional phase plane whose coordinates are values of a quantity and its first derivative resulting from a second order differential equation for that quantity. The terminology probably arose from the case of the harmonic oscillator where this plane gave the particular stage or phase in the recurring sequence of movement of the oscillator.