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.

Basic Questions

Early analyses (e.g., Gregory and Thompson, 1978; Kirshner et al., 1981; Fairall et al., 1990; Fairall, 1998) of modern magnitude-limited galaxy catalogs revealed large empty regions, originally called voids. These regions filled in somewhat as more sensitive surveys found fainter galaxies (e.g., Kirshner et al., 1987), but they remained underpopulated. Cosmological many-body simulations gave a probability f0(V) for finding such empty regions (Aarseth and Saslaw, 1982), even before their theoretical distribution function was calculated. Soon it became clear that these voids and underdense regions were part of a more general distribution function description.

In retrospect, we may regard many analyses of observed spatial distribution functions as attempts to answer several basic questions. Although these questions are not yet fully answered, and relations among them are not always apparent, we will use them to guide our discussion here.

1. Is the observed form of f(N, V) generally consistent with gravitational quasiequilibrium clustering? If so, does it rule out other possibilities such as particular dark matter distributions or initial conditions?

2. Do the two-dimensional distribution functions for projections onto the sky give a good estimate of f(N, V) or do we need the full three-dimensional distribution function?

3. How does b(r) depend on spatial scale? Can this dependence restrict possible models significantly?

4. […]

Isaac Newton needs no introduction.

Richard Bentley was one of England's leading theologians, with strong scientific interests and very worldly ambitions. Eventually he became Master of Trinity College, Cambridge, reigning for forty-two contentious years. Tyrannical and over-bearing, Bentley tried to reform the College (as well as the University Press) and spent much of the College's income on new buildings, including a small observatory. To balance the College accounts he reduced its payments to less active Fellows, while increasing his own stipend. After ten years of this, some of the Fellows rebelled and appealed to the Bishop of Ely and Queen Anne, the ultimate College authorities, to eject Bentley from the mastership. Various ruses enabled Bentley to put off the trial for another four years. Finally the Bishop condemned Bentley in a public court. But before he could formally deprive Bentley of his mastership, the Bishop caught a chill and died. Queen Anne died the next day. Bentley now put his theological talents to work to convince his opponents that he had won “victory” by divine intervention. So he retained the mastership and raised his salary still higher.

Not explanation, but prediction is our most stringent test of understanding. Many are the explanations after the fact; few are the predictions that agree with later observation. This holds especially for complex phenomena, whether of astrophysics, cosmology, economics, history, or social behavior.

When statistical thermodynamics first yielded distribution functions for cosmological many-body systems, neither simulations nor observations had been analyzed for comparison. Only Hubble's galaxy counts in the limit of large cells were known, though not to us at the time. Correlation functions then dominated studies of largescale structure.

Old simulations, after their spatial distribution functions were analyzed, first showed that the theory was reasonable. As a result, many new simulations, sketched in Part V, began to explore the range of conditions to which the theory applies; this exploration continues.

It took a couple of years to persuade observers to analyze distribution functions for modern catalogs. Would the distributions created by the great analog computer in the sky agree with those of digital computers here on Earth? Is the pattern of the galaxies dominated by simple gravitational clustering? The answer is still a continuing saga. Although observed spatial distribution functions are now moderately well studied, velocity distributions are just beginning to be determined, and predictions of past evolution remain to be observed in our future.

We start with the simplest form of the cosmological many-body problem: the one Bentley posed to Newton (see Chapter 2). In its modern version it asks “If the universe were filled with an initial Poisson distribution of identical gravitating point masses, how would their distribution evolve?” This is clearly an interesting physical problem in its own right, and it provides the simplest model for galaxy clustering in the expanding universe. If we can solve it, other complications such as a range of masses, different initial conditions, galaxy merging, and inhomogeneous intergalactic dark matter may be added to account for observed galaxy clustering. Actually, we will find that these complications are secondary. The simple gravitational cosmological manybody problem contains the essential physics of the observed galaxy distributions.

To formulate the problem thermodynamically in a solvable way, we first examine why the mean gravitational field is not dynamically important and why extensivity is a good approximation. This leads to a derivation of the form of the energy and pressure equations of state. In turn, this derivation provides insight into the requirements for quasi-equilibrium. To complete the basic description, equivalent to finding the third equation of state, Chapter 26 develops very general physical properties of these gravitating systems. This closely follows the analyses by Saslaw and Fang (1996).

One dominant stroke transformed thousands of years of increasingly refined speculation on the structure of our Universe into fact. Hubble (1925a,b,c, 1926, 1929a,b) clinched the extragalactic nature of the “white nebulae” by discovering their Cepheid variable stars. This vastly expanded the known distance scale.

Cepheids are unusually bright stars that pulsate with regular periods ranging from about 10 to 30 days. (The first onewas found in the constellation Cepheus in the Milky Way.) Their crucial property is the relation between a Cepheid's period and its peak intrinsic luminosity, recognized in 1908 (Leavitt, 1912; Hertzsprung, 1913). Brighter Cepheids have longer periods. From the observed periods of Cepheids in nebulae, Hubble could obtain their intrinsic luminosity and thus find the distance from their observed apparent luminosity. The main uncertainty was in calibrating their period–luminosity relation from the independently known distances of Cepheids in our Galaxy. Early calibrations turned out to be wrong, mainly because there are different types of Cepheids, which give somewhat different period–luminosity relations. (Occam's Razor fails again.)

How to observe the dark matter of our Universe remains a primary puzzle of astronomy. Many searches over the radiative spectrum from radio to Xrays have uncovered much of great interest, but not enough to close the Universe. Many laboratory experiments have searched directly for exotic weakly interacting massive particles (WIMPS) streaming through space, but to no avail. Gravitational lensing of more distant sources reveals dark matter in clusters along the line of sight; its amount and detailed location are quite model dependent. So far, the main direct evidence for dark matter comes from the gravitational motions of stars within galaxies and galaxies within clusters. All of it adds up to only Ω0 ≲ 0.3. Its nature, form, and distribution still are unknown.

Close agreement between the form of the GQED and the observed spatial and velocity distributions of galaxies suggests methods for constraining dark matter. It is relatively easy to start with the cosmological many-body model and formulate dark matter variations around it. These variations should not destroy the observed agreement unduly, nor attract epicycles.

As an illustration (Fang and Saslaw, 1999), consider the peculiar velocity distribution function f(v). This is especially sensitive to local dark matter. In the simplest case, the dark matter is closely associated with each galaxy (e.g., in its halo) as it clusters, and (29.4) describes the velocities. This is consistent with observations.

In hierarchical models of galaxy production, many protogalaxies merge at high redshifts. Each merger results in a spectacular wreck, which gradually restructures itself into a more unified system. Collisions engender vast conflagrations of stars as unstable gas clouds collapse and ignite thermonuclear fires. This process repeats and repeats until galaxies form as we know them today.

We see many galaxies still merging at present. In hierarchical models these late mergers are all that remain of earlier more active combining, or they result from encounters in recent dense groups. The long history of merging changes the observed galaxy distribution by destroying the conservation of galaxy numbers and by modifying the luminosity function. The first of these is easier to model; the second, at present, is really a guess. Galaxy luminosities depend on their unknown stellar initial mass functions, their subsequent star formation by merging or other violent activity, their nonthermal radiation, their production and distribution of dust, and their stellar evolution. Of these factors, only stellar evolution is reasonably well understood. On the other hand, number nonconservation depends on galaxy velocities, densities, and collision cross sections. These too must be modeled, but they seem more straightforward.

By describing the evolution of both the luminosity and the spatial distribution functions with one common formalism, we can use their mutual self-consistency to help constrain free parameters.

In stars with a convection zone just below the photosphere, the convective motions might create acoustic waves which propagate outwards through the photosphere. These sound-waves produce an extra pressure, i.e. ‘wave pressure’ in the atmosphere. This pressure will depend on the density and on the amplitudes of the waves. The gradient of the wave pressure results in a force that can drive a stellar wind. If a stellar wind is driven by acoustic wave pressure it is called a ‘sound wave driven wind’.

In this chapter we will first explain the concept of wave pressure by studying the motion of a particle in the presence of an oscillating force. This simple case, first developed by Landau and Lifshitz (1959) shows that oscillations may result in a net force in the direction of the oscillations. In § 6.1 we discuss the motions of particles in an oscillatory field, such as in a sound wave, and we show that this produces a ‘wave pressure’. In § 6.2 we introduce the concepts of the ‘wave action density’ and the ‘acoustic wave luminosity’. These are useful concepts for describing sound wave driven winds. The pressure due to acoustic waves is described in § 6.3. Section (6.4) descibes sound wave driven wind assuming no dissipation of acoustic energy. This results in estimates of the both the mass loss rate and the wind velocity. In § 6.5 we discuss sound wave driven winds with dissipation of the acoustic energy.