Binaries in clusters act as energy sources and sinks. As sinks of gravitational potential energy, they can absorb more than one-half the total binding energy of a cluster. As corresponding sources of kinetic energy they heat the cluster.

How fast will binaries form? How important can they be? An analytic theory of these processes, even in the approximations for which it is available, is quite complex. N-body simulations are sometimes the only way to answer detailed questions, especially for a broad spectrum of masses. Nevertheless, many useful results are available, and this section describes some. Following discussions of how binaries form by interactions of single stars, we consider how they can affect a cluster's evolution. From a more general point of view, these can be regarded as examples of energy transfer between different levels of hierarchial structure in a cluster.

Formation by few-body interactions

Binaries may be born directly when stars or galaxies form. We do not really understand how this occurs, so all we can do is add such initial binaries into the cluster in an ad hoc way, and then examine their development. Even if there are no initial binaries, however, the fluctuating gravitational field may chance to bring a few stars so close together, with such low relative velocities, that they remain temporarily bound. Usually these subclusters lead a fleeting existence, for later fluctuations may inject enough kinetic energy to make them decay.

Section 56:

Arp, H.C., 1966. ‘Atlas of peculiar galaxies’, Ap. J. Supp., 123, 1.

Disney, M.J., 1976. ‘Visibility of galaxies’, Nature, 263, 573.

Hubble, E.P., 1930. ‘Distribution of luminosity in elliptical nebulae’, Ap. J., 71, 231.

Kormendy, J., 1982. ‘Observations of galaxy structure and dynamics’, in Morphology and Dynamics of Galaxies, L. Martinet & M. Mayor (eds.) (Saverny: Geneva Observatory).

Reynolds, J.H., 1913. ‘The light curve of the Andromeda Nebula (NGC 224)’, MNRAS, 74, 132.

Roberts, M.S., 1972. ‘Galaxies: landmarks of the universe’, in The Emerging Universe, W.C. Saslaw & K.C. Jacobs (eds.) (Charlottesville: Virginia UP), p. 90.

Rubin, V.C., 1982. ‘Systematics of HII rotation curves’, in Internal Kinematics and Dynamics of Galaxies, IAU Symposium No. 100. E. Athanassoula (ed.) (Dordrecht: Reidel), p. 3 (see also the papers following this one).

Sandage, A., 1961. The Hubble Atlas of Galaxies. (Washington: Carnegie Institution).

Saslaw, W.C., 1970. ‘On the rotation curves of Sb and Sc galaxies’, Ap. J., 160, 11.

Sersic, J.L., 1982. Extragalactic Astronomy (Dordrecht: Reidel).

Thuan, T.X. & Romanishin, W. 1981. ‘The structure of giant elliptical galaxies in poor clusters of galaxies’, Ap. J., 248, 439.

de Vaucouleurs, G., 1948. ‘Researches on the extragalactic nebulae’, Ann. d' Ap., 11, 247.

de Vaucouleurs, G., 1959. ‘Classification and morphology of external galaxies’, in Hand. d. Phys., Vol. 53 (Berlin: Springer-Verlag) pp. 275–310.

de Vaucouleurs, G., de Vaucouleurs, A. & Crowin, H.G., Jr, 1976, Second Reference Catalogue of Bright Galaxies (Austin: Texas UP).

Physical description

If you look for a particular type of response mode in analyzing a mechanical system you might find it. But, if you do not look for it you will never find it. Lack of attention to this truism has often held back progress by promoting the feeling that a system is understood when its Fourier modes have been examined. Thus, it came as a surprise in plasma physics when Landau (1946) discovered that waves in a collisionless plasma, described by the collisionless Boltzmann equation, can damp and decay. At the time it was especially unexpected because damping had been associated with viscosity, and there is no viscosity in such plasmas. Similar behavior occurs in gravitating systems.

The physical reason for collisionless damping arises from the detailed interaction of a wave with the orbits of background stars which are not part of the wave. Thus, this process would not show up in a continuum approximation where the waves could merrily propagate without growth or decay.

To see how this works, consider the idealized triangular (rather than sinusoidal) wave shown in Figure 16. We look at it in a co-moving frame, traveling along with the wave's phase velocity v = ω/k. This ripple has been set up at t = 0 by suddenly imposing a periodic triangular perturbation in the density which gives a similar perturbation to the gravitational potential φ throughout the entire volume.

General description

One of the major mysteries of the Universe is the inhomogeneous distribution of matter throughout space. Another major mystery is the homogeneous distribution of matter throughout space. This apparent contradiction–puzzle–paradox typifies our uncertainty about the origin of structure in the Universe. Both statements are true. This may make them a ‘great truth’ in Niels Bohr's sense of the phrase: a statement whose opposite is also true.

To measure the inhomogeneities of matter we need only look around us. Indeed, the closer we look the greater the density contrasts usually seem to be. We, ourselves, mainly made of water, are about as dense as our planet and a little denser on average than our sun. Each of these is about 1023 times as dense as the average smoothed out density in the solar neighborhood, which is a typical average density within large galaxies. In turn, the average density within large galaxies is about 1000 times that within large clusters of galaxies. If we average over regions containing many large clusters the density drops by another factor of 1000. These relationships are illustrated by Table 2 and Figure 22, both from excellent surveys by de Vaucouleurs (1970, 1971). Although the masses and radii astronomers obtain for large systems are often uncertain by factors of at least two, such uncertainty is hardly visible over the enormous range of these logarithmic scales.

Concepts

So far, we have followed two broad avenues of insight into the instability and clustering of infinite gravitating systems: linear kinetic theory and numerical N-body simulations. Now we turn onto a third avenue: thermodynamics. Classical thermodynamics is a theory of great scope and generality. It survived the relativity and quantum mechanical revolutions of physics nearly intact. In part, this was because among all theories of physics thermodynamics has the least physical content. Its statements relate very general quantities which must be defined anew, through equations of state, for each specific application. With this view, it is natural to ask whether thermodynamics also subsumes gravitating systems.

The answer is yes, with certain caveats and qualifications. Results of gravitational thermodynamics – gravithermodynamics, or GTD for short – are often surprising and counter-intuitive compared to the thermodynamics of ordinary gases. Specific heats, for example, can be negative and equilibrium is a more distant ideal. Basically, these differences are caused by the long-range, unsaturated (unshielded) nature of gravitational forces. As a result, rigorous understanding of GTD is less certain than for ordinary thermodynamics. The present situation is a bit similar to the early thermodynamic gropings of Watt, Carnot, Kelvin and Joule.

Straightforward introduction of gravity into thermodynamics leads again to the Jeans instability from a new point of view. It links up with linear kinetic theory and provides new insight into non-linear clustering.

As groups of gravitating objects condense together they become more and more detached from their surroundings. Growing inhomogeneity breaks the local translational and rotational symmetry more and more strongly. Eventually systems become isolated – no longer influenced significantly by the surrounding distribution of galaxies or stars.

In Part III we consider how systems become isolated, and their resulting properties. These are spherical systems so they retain as much local symmetry as possible. Their study makes it possible to understand the basic physical processes of finite clusters. Complexities of rotation and overall asymmetry will be introduced in Part IV.

The astronomical systems which can be represented approximately as spherical clusters range from globular clusters and galactic nuclei, on a scale of parsecs, to clusters of galaxies, on a scale of megaparsecs. After setting up a general theoretical framework to describe these systems, we briefly review some of their observed dynamical properties and then go on to discuss their internal interactions and evolution in more detail.

Like Robert Louis Stevenson, the great affair of a star in a cluster is to move. When gravithermal instability sets in we want to understand how the distribution of stars evolves. Some of the earliest N-body simulations with 50 or more stars (e.g., Aarseth, 1963) already showed a clear tendency for stellar systems to form three distinct regions. At the center there is a nearly isothermal core. Its radius shrinks and its mass decreases as the system evolves. Its average density, however, increases. The stars which remain in the core bind more tightly together. Beyond the core is the halo. It contains most of the stars which diffuse from the core. Computer models of spherical systems which use the equations of stellar hydrodynamics (see Section 8) show that the halo develops a power law density distribution when all stars have the same mass (Larson, 1970). More detailed models, for isotropic velocities, which use the Fokker–Planck equation (Sections 4 and 5) confirm this power law structure (e.g., Cohn, 1980). The density in the halo ρ∝r-α, where 2.1 ≲ α ≲ 2.4, depending on details of the model. At the outer edge of the halo an evaporation zone develops. Here the density departs from its power law and plummets to zero.

Few orbits in smooth axisymmetric galaxies are exactly circular, and no orbits in unsymmetric force fields are regular. Some orbits, like the denizens of Arcadia, move ergodically hither and thither, described by no special function. Others, more constrained, are partly predictable.

In this section we explore some general properties of orbits in the smooth force field of a flattened galaxy. After a galaxy has formed, the two-body relaxation timescale τR usually becomes very long compared to the dynamical crossing time τc or even to the Hubble time. Therefore, the collisionless Boltzmann equation provides a good description of the internal dynamics. We saw from (7.15) that this description is equivalent to knowing the orbits of stars in the smooth mean field. Thus it is possible to invert the approach of Section 7 by starting from the orbits and building up a self-consistent distribution function (see Section 63). Orbits in unsymmetric systems are naturally very complicated. Finding them usually requires extensive numerical integrations which are highly specific to individual models. Some types of orbits, however, are common to a reasonable range of idealized models, so we may consider them here.

After circular orbits, the next simplest are motions across the symmetry plane z = 0. If these orbits do not stray far above or below the plane, they respond primarily to the local density ρ(r,z) rather than to the overall galactic field.

What types of structure arise from the instability of homogeneous gravitating systems? This is the fundamental question we can now tackle by extending the techniques of Part 1. Eventually we will compare the results with the observed features described in Section 20, and see how these simple initial conditions can account for the main properties of galaxy clustering, but not for the origin of the galaxies themselves.

Many detailed calculations, starting in the mid-1960s but based on the pioneering work of Lifshitz (1946) and Bonnor (1957), have examined the growth of local gaseous instabilities in expanding cosmological models. They treat interactions between gas and radiation, showing mainly how electron scattering damps the growth of large amplitude isothermal perturbations before the era of decoupling, and how radiation viscosity damps small adiabatic perturbations during decoupling. They treat interactions of gas with massive collisionless particles such as heavy neutrinos, and with seeds of finite amplitude gravitational instabilities such as black holes or other condensed pregalactic objects.

The cosmic energy equation

A more useful approach to non-linear correlations would be to derive results from the BBGKY hierarchy which do not depend either on g(1, 2) being small, or on the absence of higher order correlations. In the limit as all correlations become non-linear such results would follow from Liouville's equation itself. Since Liouville's equation is equivalent to the equations of motion such results should also follow from orbit theory.

Conservation of energy applies to non-linear correlations over large regions just as it applies locally. However, in an expanding system we must allow for two modifications. First, expansion causes the kinetic energy of peculiar motions within a given region to decrease. This opposes the growth of irregularities which increase peculiar velocities. Second, correlations extend across the boundaries of any volume. What happens within the volume depends also on clustering around the region. Unlike a perfect gas, an arbitrary subvolume is not self-contained.

The cosmic energy equation describes this situation. It follows directly from the galaxies' equations of motion (Irvine, 1961; Layzer, 1963). Having set up the apparatus of the BBGKY hierarchy in the previous sections we can use it to derive this cosmic energy equation by following the approach of Fall & Severne (1976). From the two lowest equations of the hierarchy we obtain a powerful result valid to all orders for any irregularity in the distribution of galaxies, whether linear or not.

In listing references, I have only included those most directly related to the text. The subject advances so fast that, at time t, there is not much point in listing a bibliography which is up-to-date as of a time t - τ, for it will be out of date by about t + τ. And these days the most informative complete bibliography is found in Astronomy and Astrophysics Abstracts. So with apologies to everyone, past and present, who is not mentioned, here are the references on which parts of the text are based, or to which the text refers directly. Other related references can be found in the sections on problems and extensions. They provide an entry into the literature rather than a summary of it.

There are a number of classical texts on stellar dynamics, orbit theory and celestial mechanics. Representative examples are:

Arnold, V.I., 1978. Mathematical Methods of Classical Mechanics (New York: Springer-Verlag).

Chandrasekhar, S., 1960. Principles of Stellar Dynamics (New York: Dover).

Chandrasekhar, S., 1961. Hydrodynamic and Hydromagnetic Stability (London: Oxford UP).

Chandrasekhar, S., 1969. Ellipsoidal Figures of Equilibrium (New Haven: Yale UP).

Hagihara, Y., 1970, 1972. Celestial Mechanics (Cambridge Mass: MIT Press).

Hamilton, W.R., 1834. On a general method in dynamics Phil. Trans. Roy. Soc., Pt. 11, 124, 247.

Leaving behind the properties of a single isolated system we now begin to examine what happens when two systems interact. Sections 48.2 and 58 already showed how tides can transfer energy and angular momentum. Now we explore some effects of tidal transfer on the internal structure of stellar systems. Since stellar systems have many more degrees of freedom than the interior of an individual star they may respond to external forces in more varied ways. This section describes the basic response of a spherical cluster which passes quickly through the plane of a flat galaxy. Section 65 describes the distortions of two flat systems as they attempt to move slowly past one another.

‘Quick’ and ‘slow’ are relative terms. We saw in Section 47 that what counts is the ratio of the orbital period (or quasi-period) to the timescale for external forces to change. If this ratio is small we can use adiabatic invariants or, more generally, multiple timescale expansions to examine such phenomena as orbit segregation. If this ratio is large we can use the impulsive approximation. A sudden impulsive change of the external field produces a gravitational shock.

it is certainly possible, and probably even common, for a system to contain some orbits which respond smoothly, and others which are shocked, by the same external change. Different parts of the system may respond differently, leading to complicated internal behavior.