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.

Mass segregation was one of the early important results to emerge from computer N-body simulations of clusters with a few dozen members. The heavier stars would gradually settle towards the center, increasing their negative binding energy. Lighter stars would preferentially populate the halo, with reduced binding energy. Later, direct integrations using many hundreds of stars showed the same tendency, as did models which integrated the Fokker–Planck equation for many thousands of stars.

We would expect this behavior from the basic properties of gentle relaxation by two-body encounters. Equation (14.65) shows that the timescale for dynamical friction to significantly decrease the energy of a massive star of mass M0 is less than the relaxation timescale ρR for lighter stars of mass m(moving with the same velocity) by a factor m/M0. As massive stars in the outer regions of a cluster lose energy to the lighter ones, they fall toward the center and increase their velocity. The massive stars continue to lose the kinetic energy they gain by falling, and continue to fall. The lighter stars, on the other hand, increase their average total energy and move into the halo. As light stars rise through the system, their velocity decreases, altering the local relaxation time for remaining massive stars. The net relaxation for any individual star will therefore involve an average of τR over its orbit.

A collapsing stock market, astronomers jumping aboard the bandwagon of the latest ‘hot topic’, rowboat crews, and galaxy formation – all are examples of collective effects. If the strokes of all the rowers in a boat have randomly distributed phases, some pulling upstream while others pull downstream, the boat at best performs a linear random walk. But if all the rowers row in phase, their strengths add nearly linearly (there is some turbulence) and away glides the boat. If buyers and sellers of stocks are about evenly balanced, the price of an average share records little change. A mild selling imbalance somewhat decreases prices. If the decrease deepens and spreads, people begin to worry that they may lose their fortunes. ‘Sell out fast!,’ the word goes 'round. Panic strikes. Everyone sells. The market collapses. In this case, not only do all the phases act together, but the action reinforces itself. It is highly non-linear and the result is catastrophe. Astrophysical bandwagons illustrate still other aspects of collective phenomena (left as an exercise for the reader).

Gravitating systems are prone to a variety of collective effects. Previously, whenever we examined the motion of a star in a system, we have always asked what the system can do to the star, not what the star can do to the system. Obviously a heavyweight star will do more to a system than a lightweight one will, and this change in the system in turn will react back on the motion of the star.

History has shown that the Universe is complex. The more closely we examine astronomical systems the more branches of physical theory we need to understand their behavior and origin, and the more we must specialize the general theory of the last three parts to apply to particular systems. Passing through this book we have successively examined systems of decreasing symmetry. In Part II we dropped the homogeneity that dominated Part I. In Part III we dropped the infinite boundaries that dominated Part II. Now, in Part IV, we are about to drop the spherical symmetry that dominated Part III. As a result there is less to say of a general nature. To compensate, there arises a richness of detail which can often be compared with observations. Although such comparisons are not within the scope of this book, a sketch of some basic observed dynamical properties of non-spherical systems helps give a useful perspective.

In forming impressions of galactic structure we must be especially careful of selection effects. These influence what we observe in subtle ways. Rates of extragalactic information flow are slow, and the natural desire to get interesting results from long nights spent observing often subconsciously affects the choice of objects and places to examine. Versatile and sensitive detectors continue to improve this situation.

On a more fundamental level the brightness of the night sky limits our vision.

Hitherto in Part III, we have considered the properties of stationary relaxed stellar systems. We know such systems cannot be realistic because they are not stable. Their most obvious instability is the evaporation of high energy stars. These carry net energy away from the system; remaining stars swarm inside a deeper gravitational well and cluster more strongly together.

Even if a cluster did not evaporate, something must clearly be wrong with the assumption that it could be stable. The fact that, in a Newtonian system, more and more kinetic energy can be produced by continued gravitational concentration suggests that instability may occur wholly internally through a redistribution of the stars. In a spherical system we would expect the center to become denser, losing energy by ejecting high energy stars into the outer parts, and forming a steeper gravitational well.

A simple physical argument suggests how this instability arises. Systems satisfying the virial theorem (9.30) have 2T + W = 0 or since the total energy E = T + W, they satisfy E = – T. Although systems, or parts of systems, may not satisfy the virial theorem exactly, they are close enough that this relation describes the essential part of the physics needed to understand gravithermal instability. The kinetic energy of random motions T is proportional to the temperature. (A useful confusion of nomenclature?) Thus decreasing the energy of a gravitationally bound system, increases its temperature.

Section 37:

Gunn, J.E. & Gott, J.R. iii, 1972. ‘On the infall of matter into clusters of galaxies and some effects on their evolution’, Ap. J., 176, 1.

Roxburgh, I.W. & Saffman, P.G., 1965. ‘The growth of condensations in a Newtonian model of the steady-state universe’, MNRAS, 129, 181.

Sandage, A.R., 1963. ‘The ability of the 200-inch telescope to discriminate between selected world models’, Ap. J., 133, 355.

Section 38:

Aarseth, S.J. & Lecar, M., 1975. ‘Computer simulations of stellar systems’, Ann. Rev. Astron. and Astrophys., 13, 1.

Aarseth, S.J. and Saslaw, W.C., 1972. ‘Virial mass determinations of bound and unstable groups of galaxies’, Ap. J., 172, 17.

Hénon, M., 1964. ‘L'evolution initiale d'un amas spherique’, Annals. Astrophys, 27, 83.

Kadomtsev, B.B. & Pogutse, O.P., 1970. ‘Collisionless relaxation in systems with Coulomb interactions’, Phys. Rev. Lett., 25, 1155.

King, I.R., 1962. ‘The structure of star clusters, i. An empirical density law’, A. J., 67, 471.

Lynden-Bell, D., 1967. ‘Statistical mechanics of violent relaxation in stellar systems’, MNRAS, 136, 101.

Saslaw, W.C., 1969. ‘Gravithermodynamics. ii. Generalized statistical mechanics of violent agitation’, MNRAS, 143, 437.

Saslaw, W.C., 1970. ‘Violent relaxation in galaxies and stellar systems’, MNRAS, 150, 299.

Shu, F.H., 1978. ‘On the statistical mechanics of violent relaxation’, Ap. J., 225, 83.

Zwicky, F., 1960. ‘The age of large globular clusters of galaxies’, PASP, 72, 365.

Section 40:

Bailey, M.E. & MacDonald, J., 1981. ‘A comparison between velocity dispersion profiles of de Vaucouleurs and King galaxy models’, MNRAS, 194, 195.

Mass is one of the most basic properties of stellar and galactic systems, so it is natural that astronomers have developed several methods to determine it from observations. Unfortunately, results of different methods often disagree. The most notorious examples are the estimates of masses of clusters of galaxies. A voluminous, if not massive, literature has grown to surround this controversy. On the one hand, the mass can be estimated by independently finding the mass–luminosity ratio of nearby galaxies of each morphological type – spiral, elliptical, or irregular. Counting galaxies of each type in a distant cluster and measuring their luminosities then gives the total mass of the cluster galaxies. On the other hand, the mass of a cluster can be estimated by measuring the galaxy separations and velocities and applying the virial theorem. This dynamical mass usually turns out to be greater than the luminosity mass, often by a factor ~ 10 but occasionally by a factor ~ 100.

Several possible causes may contribute to this discrepancy. The mass–luminosity ratio of nearby galaxies may be intrinsically different from those in more distant groups. There, accretion, tidal interactions, or gaseous ablation, for example, may have altered galaxies even if they were similar to begin with. The mass–luminosity ratio of nearby galaxies may have been underestimated, particularly if they contain extended haloes of dark matter – gas, dust, low luminosity stars, black holes, massive neutrinos or other ʾinos, pebbles, etc.