Velocities are the current frontier, as spectrographs with fiber optic cameras scramble to produce larger catalogs. Some galaxies will have good secondary distance indicators and be useful for peculiar velocities. Catalogs whose peculiar velocities are homogeneously sampled can provide direct comparisons with theoretical predictions.

So far, most catalogs of peculiar velocities are for restricted samples such as spirals in clusters, or for isolated field galaxies. This again involves finding a suitable definition of a cluster or field galaxy. Relating convenient morphological definitions of clusters to the underlying physics of their formation is often quite difficult. It is simpler, and perhaps less model dependent, to consider the combined peculiar velocity distribution for all clustering scales and degrees from isolated field galaxies and small groups to the richest dense clusters. The corresponding predictions of gravitational quasi-equilibrium clustering are given for f(v) by (29.4) and for the observed radial velocity distribution f(vr) by (29.16). Chapter 32 describes their agreement with cosmological many-body simulations.

The GQED predictions are for local peculiar velocities averaged over a large and varied system. They do not include the motion of the system itself, which corresponds to regional bulk flow produced by distant, rare, and massive attractors. There is general agreement that such bulk flows exist, though their detailed nature is subtle and still controversial.

Despite difficulties in solving the BBGKY kinetic equations directly, judicious approximations give valuable insights into their physical properties. As soon as the system leaves the linear regime, these approximations replace a more rigorous analysis. Computer simulations, although they require further approximations, can tell if these insights are genuine and provide a useful description. In this chapter we discuss several physical approximations; following chapters summarize simulations and observations.

Scaling

Scaling has acquired many meanings in different contexts of large-scale structure and thereby caused some confusion. One form of scaling is the simple geometric relation between angular correlation functions W(θ) in catalogs of various limiting magnitudes, given by (14.38). This is closely related to the manufacture of catalogs rather than to the underlying dynamics of clustering. We mention it further in Chapter 20. A more physical form of scaling results from noticing that the Ω0 = 1 Einstein–de Sitter cosmology contains no scale length. Nor is there any scale length in the Newtonian gravitational force between galaxies. Therefore, the argument goes, correlation functions should not contain any scale lengths either and should be power laws.

Unfortunately this argument neglects the actual dynamical development of correlations. Initially, if the positions of newly formed galaxies are not correlated in a gravitationally self-consistent way on all scales, gravitational interactions of galaxies will tend to produce self-consistent correlations.

Stubbornness, stamina, boldness, and luck enabled William Herschel to connect our Universe with Newton's and Kant's speculations. Leaving Hanover in 1757 after the French occupation, he settled in England as an itinerant teacher, copier, and composer of music, becoming organist of the Octagon Chapel at Bath in 1766. But his real interest from childhood was astronomy. He privately built a succession of larger and larger reflecting telescopes and systematically swept the heavens. His sister, Caroline, emigrating in 1772, helped with these nightly observations, to the eventual destruction of her own singing career. In 1781, Herschel had the great luck to find Uranus, the first planet discovered since the days of the ancients, although he originally thought it was just a comet. Fame followed quickly, and fortune soon after when George III granted him a pension for life. He supplemented this by building small telescopes for sale (until his wealthy marriage in 1788) and became a full-time astronomer. Career paths, like the subject itself, have changed considerably since then.

For twenty years, starting in 1783, Herschel searched for nebulae with his 20-foot telescope and its 18 7/10 inch speculum mirror. Messier's catalog, available in 1781, had inspired him first to try to resolve known nebulae with his superior telescope, and then to discover more.

Ideas of connectivity join with those of shape to describe the topology of the galaxy distribution. This addresses the question much discussed in the 1950s (see Chapter 7) of whether clusters are condensations in an otherwise uniform sea of galaxies or whether clusters are just the edges of voids and underdense regions. The question resurfaced in the 1980s when astronomers noticed fairly large three-dimensional volumes containing relatively few galaxies (Tifft & Gregory, 1976; Kirshner et al., 1981; de Lapparent et al.,1986; Kauffmann & Fairall, 1991). Consequently, much high-energy speculation arose over the origin of voids and cellular structure in the early universe. The main question was: Are clusters or voids the fundamental entities of the galaxy distribution? The answer is: both or neither.

It all depends on how you look at the distribution. If galaxies are the fundamental entities, then clusters and voids are just derivative configurations. If clusters and voids are fundamental, imposed by conditions in the early universe, then galaxies are just derivative markers. If dark matter dominates the universe, the situation becomes even more murky. In any case, topology helps quantify the conditions where relatively underdense or overdense regions dominate. It is most useful, so far, on scales at least several times that of the two-point correlation function (∼ 5 h–1 Mpc where h is the Hubble constant in units of 100 km s–1 Mpc–1).

About Pulsating Stars

Introduction

The discussions in the following sections are limited to pulsating variables and thus omit such objects as eclipsing and cataclysmic variables. Rather than try to cover every conceivable aspect of the subject an attempt is made to discuss in detail a few problems of current interest. This has meant that some types of pulsating variable are not dealt with at all, e.g. Type II Cepheids (including RV Tau stars), SX Phe variables and the recently discovered pulsating K giants in globular clusters (Edmonds and Gilliland 1996). In several of the areas covered strongly divergent views are held by different workers. In such cases an attempt is made to summarize the arguments of the various groups whilst at the same time indicating what in the present writer's opinion seems most likely to be the correct interpretation.

Pulsating stars are of importance for a variety of reasons. First, a study of their light, colour, and radial velocity, changes through the pulsation cycle tell us a great deal about the stars themselves – about their structure – which we cannot easily learn in other ways. Secondly, because pulsating variables are rather easily classified into groups with homogeneous properties it is possible to use them, provided their absolute magnitudes can be calibrated, to derive distances. Pulsating stars are at the basis of the galactic and extragalactic distance scales and are important in determining the distances and ages of classical, old, globular clusters.

Direct Methods

For inspiring new insights into galaxy clustering, for testing our understanding of gravitational many-body physics, and for detailed comparisons with observation, nothing works better than computer experiments. But they also have trade-offs and dangers. The trade-offs are among detailed physical information, computational speed, and number of physical particles. The dangers are lack of uniqueness and a tendency to examine only a small range of models based just on different parameters rather than on different basic ideas.

A variety of numerical techniques, all compromises, have been developed for different types of problems. The simplest problem considers the evolution of a distribution of N points with the same or different masses in the background of an expanding universe. We shall call this the cosmological many-body problem. Each point mass represents a galaxy and its associated halo. This is a good approximation if we are not concerned with galaxies' tidal interactions and mergers, or with inhomogeneous dynamically important intergalactic matter. Such complications can be added using other techniques to determine their significance.

Cosmological many-body problems are usually solved by integrating all the N particles' equations of motion. This is the direct method. Since only particle–particle interactions occur, it is also called the particle–particle (PP) method. Though direct, it is not straightforward. There are N equations, each with N terms, leading to ∼N2 operations. Moreover, when particles come close together, their high accelerations require short time steps.

Globular clusters provide ideal laboratories for studying the dynamical behaviour of N-body systems. If one includes in one's definition of ‘globular cluster’ the young and intermediate age rich star clusters in the Magellanic Clouds, then one has at hand a set of objects that can serve as a testing ground for theories that describe self-gravitating systems of point masses at any stage in their evolution. These different stages include violent relaxation, a gradual approach to quasi-static equilibrium through two-body relaxation, the dramatic collapse, probably followed by oscillations, of the cluster core, and ultimately dissolution of the cluster as it contributes its stars to the parent/host galaxy's field (usually halo) population. Understanding the mechanisms that hasten the dissolution of a cluster can help us reconstruct the original population of clusters in a given galaxy. This in turn can guide theories of globular cluster formation, and, to the extent that globular clusters trace the early stages of galaxy evolution, the formation of galaxies themselves. This chapter provides an overview of the life of a globular cluster (Section 1), derives the time scales relevant to various stages of cluster evolution (Section 2), and discusses the main observable qualities of clusters relevant to their dynamical evolution: their surface brightness profiles (Section 3) and their internal velocity dispersions (Section 4). In Section 5 some recent results from a large HST project to study the formation and evolution of rich star clusters in the Large Magellanic Cloud are described.

Two of the three main ingredients for understanding the universe during the first half of the twentieth century were observational: its immense size and its expansion. The third was Einstein's general theory of relativity. It related the force of gravity to the structure of spacetime. Two years after his definitive account of the theory, Einstein (1917) applied it to cosmology. His first model, introducing the cosmological constant,was static – matter without motion. Shortly afterward deSitter (1917a,b) discovered an expanding but empty solution of Einstein's equations – motion without matter. Then Friedmann (1922) found the intermediate solutions with both expansion and matter, which Lemaître (1927) independently rediscovered. Eddington (1930, 1931a) was about to publish them independently yet again when Lemaître, who had formerly been his student, gently reminded him that they were already known. So Eddington publicized these solutions more widely and also showed that Einstein's static universe would become unstable if condensations formed within it.

A small fraction of cosmological thought during this period strayed from the homogeneous models to the nature and origin of structure in the universe.

This introductory chapter discusses the observations on which our understanding of globular clusters lies. Successive sections deal with photometry, chemical abundances, the details of color-magnitude diagrams, the distance scale, luminosity and mass functions, and the lower end of the main sequence. An appendix treats the dynamical role of binaries in globular clusters.

Astronomy aims at an understanding of the facts and phenomena that we see, and the processes by which they came about—and in the best of possible cases, the recognition of why they had to be this way and could not have been otherwise. The first stage in this endeavor is to see what is there, and, to the extent that we can, how it became that way.

Globular clusters can in many ways be considered the crossroads of astronomy. They have played a central role in the unfolding of our astronomical understanding, to which they bring two singular advantages: first, each cluster (with a possible rare exception) is a single and specific stellar population, stars born at the same time, in the same place, out of the same material, and differing only in the rate at which each star has evolved. Such a group is much easier to study than the hodgepodge that makes up the field stars of the Milky Way. Second, globular clusters are made up of nearly the oldest—perhaps the very oldest—stars of the Universe, and as such they give us an unparalleled opportunity to probe the depths of time that are the remotest to reach.

Finally, four vignettes of the future. Some basic questions whose true understanding awaits new ideas and new observations. They follow in order of their solution's remoteness.