Light-cone geometry: the key to special relativity

WE HAVE SEEN how an index notation is strikingly helpful in the development of physical formulae for flat three-dimensional space. We found it convenient to work with a fixed Cartesian coordinate system, expressing the components of vectors and tensors with respect to that system. We know, nevertheless, as a matter of principle, that the general conclusions we draw are independent of the particular coordinatization chosen for the underlying space.

We now propose to formulate special relativity in essentially the same spirit. We shall regard space-time as a flat four-dimensional continuum with coordinates xa (a = 0,1,2,3). The points of space-time are called ‘events’, and we are interested in the relations of events to one another. Our purpose here is two-fold: first, to review some aspects of special relativity pertinent to that which follows later; and second, to develop further a number of index-calculus tools which are very useful in general relativity as well as special relativity.

Abstract. Data on kinematics, spatial distributions, and galaxy morphology in different density regimes within individual galaxy clusters show that many clusters are not in a stationary state but are still in the process of forming.

INTRODUCTION

Paradigms for galaxy clusters are changing. As in all tearing away from secure positions (Kuhn 1970) the process is controversial, yet continuing. Most papers in this volume suggest directions that will probably lead to even stronger new ideas about cluster cosmogony. We are concerned in this review with physical properties that have relevance for the question of whether clusters of galaxies are generally stationary, changing only slowly in a crossing time or if they are dynamically young. We examine if parts of a cluster may still be forming, falling onto an old dense core that would have been the first part of a density fluctuation to collapse even if all galaxies in a cluster are the same age, having formed before the cluster. During the 1930's the stationary nature of clusters seemed beyond doubt. A suggestion that they are dynamically young would have been too radical even for Zwicky who was the model of prophetic radicals. Rather, Zwicky (1937) took the stationary state to be given in making his calculation of a total mass, following an earlier calculation by Sinclair Smith (1936). The justification was that rich clusters such as Coma (1257 +2812; or Abell A1656), Cor Bor (1520 +2754; A2065), Bootis (1431 +3146; A1930), and Ursa Major No.2 (1055 +5702; A1132), known already to Hubble (1936) and to Humason (1936), appear so regular.

Abstract. Past X-ray surveys have shown that clusters of galaxies contain hot gas. Observations of this hot gas yield measurements of the fundamental properties of clusters. Results from a recent study of the X-ray luminosity function of local Abell clusters is described. Future surveys are discussed and the potential for studying the evolution of clusters is analyzed.

INTRODUCTION

The systematic study of clusters began with the surveys of Abell (1958) and Zwicky et al. (1968) who each created well defined catalogues according to specific definitions of the object class. In particular Abell defined clusters as overdensities of galaxies within a fixed physical radius around a center, classifying such objects as a function of their apparent magnitude (distance) and of their overdensity (“richness”).

The first X-ray survey of the sky by the UHURU X-ray satellite showed that “rich” nearby clusters were powerful X-ray sources (Gursky, et al. 1971, Kellogg et al. 1972). Subsequent spectroscopic studies detected X-ray emission lines of highly ionized iron and demonstrated that the X-ray emission was produced by thermal radiation of a hot gas with temperatures in the range of 30 to 100 million degrees (Mitchell et al. 1976, Serlemitsos, et al. 1977).

With the launch of the HEAO1 and the Einstein Observatories, surveys of significant samples of nearby clusters demonstrated that as a class, clusters of galaxies are bright X-ray sources with luminosities between 1042 and 1045 ergs/sec (Johnson, et al. 1983, Abramopoulos and Ku 1983, and Jones and Forman 1984).

Abstract. This contribution reviews the X-ray properties of clusters of galaxies and includes a brief summary of the X-ray characteristics of early-type galaxies and compact, dense groups. The discussion of clusters of galaxies emphasizes the importance of X-ray observations for determining cluster substructure and the role of central, dominant galaxies. The X-ray images show that substructure is present in at least 30% of rich (Abell) clusters and, hence that many rich clusters whose other properties are those of dynamically young systems, suggests that most cluster classification systems which utilize a property related to dynamical evolution, require a second dimension related to the dominance of the central galaxy. X-ray surveys of rich clusters show that central, dominant galaxies are twice as common as optical classifications suggest. The evidence for mass deposition (“cooling flows”) around central, dominant galaxies is reviewed. Finally, the implications of X-ray gas mass and iron abundance measurements for understanding the origin of the intracluster medium are discussed.

HOT GAS IN GALAXIES, GROUPS, AND CLUSTERS

Hot gas has been been found to be commonly associated with both individual early-type galaxies and with the poor and rich clusters in which they lie. Although this presentation will concentrate on the hot gas in rich clusters, we briefly describe the characteristics of individual galaxies and groups, as well as clusters since their evolution and present epoch properties are interrelated. Recent reviews of X-ray properties of clusters of galaxies include Forman and Jones (1982) and Sarazin (1986).

Abstract. On-going removal of the low density outer interstellar HI gas occurs in galaxies passing through the central regions of clusters with moderately high X-ray luminosity. Although the galaxies currently maintain their spiral morphology, they are HI deficient by as much as a factor of ten relative to their counterparts at larger cluster radii or in the field. The HI distribution in deficient galaxies is truncated well interior to the optical radius as the gas is removed preferentially from the outer portions. In contrast, the molecular hydrogen component, derived from observations of CO, seems undisturbed. Galaxies that are HI poor by a factor of ten may be gas poor by only a factor of three. At the same time, other indicators suggest a reduction in the star formation rate in most H I deficient galaxies, but some objects may suffer an enhanced gas depletion if star formation is actually induced by the interaction. While the intracluster medium is the likely catalyst for gas removal, the exact sweeping mechanism is unclear. Early-type objects seem to be even more HI poor than late-type ones, perhaps supporting the suggestion of a fundamental difference in the orbital anisotropy of early and late type spirals. While it seems possible that after disk fading, stripped spirals would ultimately resemble S0's, it is unlikely that all S0's result from such gas sweeping events since the process seems viable only in the cores of rich clusters.

Abstract. A new, combined N-body and 3D hydrodynamic simulation algorithm is used to study the dynamics of the intracluster medium (ICM) in rich clusters of galaxies. Results of a program to study an ensemble of clusters covering a range of cluster richness within the framework of a cold dark matter (CDM) dominated universe are presented. Comparison with observations for both individual cluster characteristics and properties of the ensemble is emphasized. Predictions arising from the numerical models will be discussed and directions for future work in this area outlined.

INTRODUCTION

The intergalactic space in rich clusters of galaxies is permeated by a hot, ionized plasma which emits a continuum of X-rays generated by the scattering of energetic electrons off protons and ions. This thermal bremsstrahlung emission is observed to distances R ∼ 1 Mpc and spectral fits indicate temperatures T ∼ 108 K, so if the gas is confined by the gravitational potential of the cluster the binding mass must be of order M≃G−1(kT/μmp)R ∼ 3 × 1014 M⊙. The X-rays from the extended intracluster medium thus reflect emission from the largest relaxed, self-gravitating entities known in the universe.

The issues one would like to understand both observationally and theoretically range from the internal and structural—What are the spatial gas density and temperature profiles? Intrinsic shapes? How do these relate to optical properties? How do they evolve with redshift?—to the global and statistical—What is the expected abundance of clusters as a function of luminosity, temperature or any other observable?

Abstract. If the universe has closure density and the spectrum of primordial density fluctuations is a power law, the lack of any preferred scale means that the clustering should evolve in a scale invariant manner. These self-similar models allow one to approximately predict the evolution of the clustering in e.g., the ‘standard’ cold dark matter model. I describe how these models yield predictions for the evolution of the cluster populations. Particular attention is given to the range of spectral indices for which the scaling should be valid. I argue than the allowed range is −3 < n < 1, though quite what happens for spectra near the upper bound is somewhat unclear. The cold dark matter power spectrum has spectral index n ≃ −1 on the mass scale of clusters. For this value of n, I find that the comoving density of clusters classified according to virial temperature Tv or by Abell's richness, should show weak positive density evolution ∂log n(Tv, z)/∂z ≃ +0.3. Clusters classified by total X-ray luminosity should show strong positive density evolution ∂log n(Lx, z)/∂z ≃ +3, but the assumptions used to predict the total X-ray luminosity are somewhat questionable. More robust predictions can be made for the halo emission, and I describe an evolutionary test which should be feasible with ROSAT.

INTRODUCTION

Rich clusters have had much impact on cosmological theory. They give the strongest indication that the universe contains copious amounts of dark matter and give an empirical estimate of the baryon to dark matter ratio.

Abstract. I discuss some issues that arise in the attempt to understand what rich clusters of galaxies might teach us about cosmology. First, the mean mass per galaxy in a cluster, if applied to all bright galaxies, yields a mean mass density ∼ 30 percent of the critical Einstein-de Sitter value. Is this because the mass per galaxy is biased low in clusters, or must we learn to live in a low density universe? Second, what is the sequence of creation? There are theories in which protoclusters form before galaxies, or after, or the two are more or less coeval. Third, can we imagine that clusters formed by gravitational instability out of Gaussian primeval density fluctuations? Or do the observations point to the non-Gaussian perturbations to be expected from cosmic strings, or explosions, or even some variants of inflation? These issues depend on a fourth: do we know the gross physical properties of clusters well enough to use them as constraints on cosmology? I argue that some are too well established to ignore. Their implications for the other issues are not so clear, but one can see signs of progress.

THE STATISTICS OF CLUSTERS OF GALAXIES

To draw lessons for cosmology, we need not only the physical properties of individual clusters but also an understanding of how typical the numbers are. The issue here is whether the Abell catalog or any other now available is adequate for the purpose.

Abstract. This article describes the current status of various methods for determining the dark matter distribution in clusters. Despite a great deal of progress recently we still do not have good mass constraints for even one cluster. The reasons for this are discussed. New observational tools and methods of analysis should however lead to some results in the near future.

INTRODUCTION

One of the many interesting aspects of clusters of galaxies is that they appear to contain large amounts of missing mass. The evidence for this has largely been based on the application of the standard virial theorem. More sophisticated approaches which utilize cluster velocity dispersion profiles came to similar conclusions but assumed that the mass distribution in clusters was the same as that of the light (galaxy) distribution. While this may be true it is definitely at present an assumption. Much recent theoretical work has argued for different distributions for the dark and luminous components of the universe. One of the consequences of this is that we should not assume that the mass distribution in clusters parallels the light distribution. Without this assumption it is very difficult to constrain the mass distribution in clusters, and consequently total cluster masses are not as yet well determined (Bailey 1982, The & White 1986, Merritt 1987).

The cluster mass distribution is an important ‘parameter’ in that it directly influences many of the physical processes that occur in clusters.

Abstract. Various observations indicate that the cluster environment can affect the structure and dynamics of galaxies. This review concentrates on the effect the environment can have on three of the most basic properties of a galaxy; the morphological type, the size, and the distribution of mass. A reexamination of the morphology - density relation suggests that the fundamental driver may be related to some global property of the cluster, such as the distance from the cluster center, rather than some local property, such as membership in a local subclump within the cluster. While there is good evidence that the size of a galaxy can be increased (i.e., cD galaxies) or decreased (i.e., early type galaxies near the centers of clusters) by the cluster environment, it is not clear what physical mechanism is responsible. There is tentative evidence that rotation curves of spiral galaxies near the centers of clusters are falling, perhaps indicating that the dark halo has been stripped off. Rotation curves for spiral galaxies in compact groups are even more bizarre, providing strong evidence that the group environment has affected the kinematics of these galaxies.

INTRODUCTION

Perhaps the three most basic questions an extragalactic astronomer might be asked are:

1. Why are some galaxies flattened into disks while others are elliptical in shape?

2. How big are galaxies?

3. How massive are galaxies?

Although we can fill journals with details about galaxies, an astronomer cannot really answer these three basic questions with any confidence.

Abstract. Recent progress in understanding four processes that play a large role in the evolution of clusters of galaxies is reviewed. These are dynamical friction, mergers, collisional tidal stripping and the cluster mean field tide. Recent estimates for the growth rate of the cD galaxy and its frequency of appearance are discussed. In spherical relaxed clusters the theoretical and observational results for the accretion rate of a central massive galaxy seem to be quite consistent. It appears that a major part of the cD formation must occur in subclusters. Recent work on the formation of clusters containing galaxies and dark matter suggests that considerable mass segregation occurs in small subclusters (provided clusters form in a bottom up manner). This appears to be a result of dynamical friction. It may imply that visible clusters are embedded in large dark matter halos and that cluster M/L's have been underestimated.

INTRODUCTION

Clusters of galaxies represent a fascinating, if formidable, challenge for the theorist. Although relatively young in terms of their crossing times (Tcr = R/v), the galaxies are sufficiently large and massive that they interact with each other and the intracluster medium on a timescale comparable to their ages. A sensible way to organize a report on this complicated subject is, by analogy with stellar structure, to report first on the detailed processes which may occur (the ‘hydrodyamics’ and atomic ‘physics’ of the problem) and then to examine the effects of these processes (the analog of ‘evolution’) on the galaxies and the cluster.

Clusters of galaxies are probably the largest gravitationally bound entities in the universe. They offer a laboratory for studying such diverse astrophysical problems as the form of the initial fluctuation spectrum, the evolution and formation of galaxies, environmental effects on galaxies, and the nature and quantity of dark matter in the universe, as well as providing tracers of the large-scale structure. The view that clusters are dynamically relaxed systems has been challenged by the demonstration of significant substructure in the galaxy and X-ray distribution within clusters (see the chapters herein by Geller, Cavaliere & Colafrancesco, Fitchett, Richstone, and Forman). There is, however, still some dissent on the reality of subclustering (see the discussion in West's chapter). New simulations of the formation and evolution of the dark matter and gas distributions in clusters are giving interesting results—their confrontation with observations may yield information on the nature of the initial density fluctuations required to form galaxies and enable us to solve some of the problems in this field (e.g., the so-called “β–discrepancy”). The simulations should also allow for better comparisons between theory and optical and X-ray observations (see the chapters by Cavaliere & Colafrancesco, Evrard and West). The abundance and velocity dispersions of rich clusters, and measurements of their clustering properties and peculiar motions may provide strong constraints on theories of galaxy formation (see the chapters by Kaiser, Peebles and West).