Introduction

A Galactic Soft X-ray Diffuse Background (SXDB) below ∼2 keV was discovered by Bowyer, Field & Mack (1968). Four soft X-ray all-sky surveys produced maps of this SXDB (McCammon et al. 1983; Marshall & Clark 1984; Garmire et al. 1992; Snowden et al. 1995, 1997) which show complex features, indicating that the SXDB must be made up of several components. However, Snowden et al. (1997) established that the SXDB maps above 0.5 keV are smooth on the south side of the plane, which can be reproduced with only one component: a hot gas in the bulge with a scale height of ∼1.9 kpc. Thus they named this component as the ‘bulge’ emission. The typical temperature was estimated to be ∼0.3 keV.

The ASCA satellite has the capability to observe the SXDB with a reasonable energy resolution (Tanaka et al. 1994), which allows an improved study of line emission. We present here results of our initial analysis of the ASCA spectrum and discuss the bulge emission.

This cycle of lectures presents a self consistent sketch of current understanding about chemical composition of globular clusters and its aftermaths. The first two lectures give basic about nucleosynthesis, chemical models, and abundance determinations. Main results for globular clusters are presented in the next two lectures. In the final lecture I review various indices used to derive abundances from photometry and low dispersion spectroscopy.

Early Nucleosynthesis and models of galactic chemical evolution

In this first lecture I will briefly present the fundamentals of nucleosynthesis and chemical evolution. Owing to lack of time, only few sketches can be given.

The basic observation that we live in an environment rather rich in heavy elements (hereinafter metals) that could not be produced by Big Bang leads us to try to describe the mechanisms of formation of these elements. There is a close interaction between chemical and dynamical evolution of stellar systems; chemical abundances provide then a basic diagnostic for models of galactic evolution.

Figure 1 sketches the most important features to be introduced in this picture. Stars form from condensation of the most dense clouds within the interstellar medium (ISM). Metals are produced by nucleosynthesis processes within the stellar interiors. Stars lose part of their metal-enriched material either through more or less quiescent stellar winds, or through explosive events (SNe) at the end of their lives: the amount of each element produced within stars and returned to the ISM depends on the stellar masses and in some case on the presence of close companions.

Number counts of galaxies in cells on the sky or in space, near neighbor statistics, and distributions of peculiar velocities all enhance our understanding of how galaxies cluster. Even though they contain more information than low-order correlations, we can extend these distributions into highly nonlinear regimes where gravity dominates. General physical principles, rather than detailed orbital dynamics or models, simplify and guide these extensions.

After reviewing some basic mathematical properties of distribution functions, we examine how dynamics describes their linear evolution. As the evolution becomes more and more nonlinear, however, the dynamical arguments give out, just as they did for correlation functions. Mathematical pertubation theory becomes more intractable; its returns diminish. Nonlinearity, spreading from smaller to larger scales, destroys memories of the initial state. Formany initial conditions, nonlinear evolution can lead to quasi-equilibrium. Somewhat unexpectedly, perhaps, this is amenable to a statistical thermodynamic description. Later we will compare the predicted distributions with detailed simulations and observations.

Fractals help characterize the scaling properties of point distributions. Originally astronomers hoped that the galaxy distribution would have simple scaling properties following from the scale-free form of the gravitational potential. It was not to be. Nevertheless, fractals provide useful insights that can be related to correlation and distribution functions.

The main fractal measure of a set of points is its “dimension.” A continuous set of points, such as a line, plane, or spherical ball has an integral topological dimension whose definition goes back at least to Euclid. It refers to the number of independent coordinates needed to locate a point in this set. When Cantor (1883) found his middle-thirds set, however, it became necessary to generalize the concept of dimension. Cantor's set has an uncountably infinite number of points, all disconnected from each other so that there is no continuous interval in the set, even though each point has another which is arbitrarily close. It is simple to construct (Figure 12.1). Start with the unit interval [0, 1] and successively delete the innermost third of each remaining line segment. Cantor's set, C, contains all the points common to the sequence of subsets C0, C1, C2, C3… This set is self-similar since expanding the scale of Cn+1 by a factor 3 gives, for each of the parts of Cn+1 that contains two lines, a shape identical with Cn.

Concepts

Gravity is an extraordinary force and understanding its more profound implications for the cosmological many-body problem requires many strategies. 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 that 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 counterintuitive 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.

For a decade now, the Instituto de Astrofisica de Canrias (IAC) has hosted the Canary Islands Winter School of Astrophysics in which young astrophysicists from all over the world have the opportunity of meeting accredited specialists to study the topics of most active concern in present-day astronomy. During these ten years 80 lecturers and more than 600 students have attended the Winter School, an even higher number not being able to come due to the limited number of places available.

The X Canary Islands Winter School on Astrophysics was dedicated to Globular Clusters, one of the basic sources of our knowledge concerning the lives of the stars and the physics of their evolution.

The School intended to portray a thorough review of research in this field, covering all the relevant disciplines with the aid of the best possible international team of specialists (Canada, Italy, South Africa, Spain, the United Kingdom and the United States), including the theoretical and observational aspects of stellar populations, stellar evolution and chemical abundances, dynamics, variable stars, X-ray sources and the globular clusters of other galaxies.

We take the opportunity to thank local Canarian authorities - Cabildo Insular de La Palma, and Cabildo Insular de Tenerife, as well as the Town Hall of La Laguna, for their continuous support during this and also previous editions of the School.

This tenth Winter School marks a milestone on a long but gratifying journey, in spite of occasional difficulties.

Many motives spur astronomers toward numerical simulations. These computer experiments test well-defined theories, display outcomes of complex interactions, elicit quantitative comparisons with observations, and provoke new insights. Moreover, they are almost always guaranteed to lead to a publishable result. What could be finer and more delightful!?

Streams of simulations have therefore poured forth in abundance. They differ mainly in their assumptions about the amount, nature, and role of dark matter, and in their initial conditions. Most agree with some aspects of observations, but none with all. None, so far, are generally accepted as complete descriptions of galaxy clustering.

As computing power expands, each new generation essentially repeats these simulations with more complicated physical interactions, greater detail, higher resolution, and added parameters. While this development continues, it seems to me wiser not to discuss the latest examples here, for they will soon be as obsolete as their predecessors. Instead, we concentrate on the simplest case: the cosmological many-body problem. Even this reveals a richness of behavior that surpasses current understanding. Understanding is more than simulation, for it embeds the simulations in a much richer conceptual context.

Suppose that cosmological many-body clustering runs on forever. What will happen in the infinite future?

Standard Einstein–Friedmann universes suggest three main possibilities. If the Universe is closed (Ω0 > 1, k = +1) and recollapses into a singularity, all large-scale structure will eventually be destroyed in the big crunch. Whether anything can be resurrected from surviving seeds if the crunch is incomplete (Saslaw, 1991) is unknown. Oscillating universes are possible, though in practice we do not know if the physical requirements for repeated oscillations are consistent with reasonable equations of state. Oscillations whose amplitudes were too small to produce equilibrium would accumulate the debris of previous cycles. Quite apart from the question of increasing entropy, such models would probably require especially fine tuning to produce our observable Universe.

If the Universe is open and expands forever with negative curvature (Ω0 < 1, k = –1), it will expand so rapidly after redshifts z ≲ Ω0–1 (see 30.13) that new larger structures will generally cease to form, and the largest scale patterns at z ≈ Ω0–1 will be essentially frozen. These patterns then tend to expand homologously, becoming increasingly stretched and dilute in physical space: Pascal's nightmare. In models with a cosmological constant, the expansion may pause. But it will have to be carefully tuned, so the quasi-stationary period does not produce overclustering, and also satisfy other constraints.

The study of globular clusters has been and still is essential for furthering our knowledge of such astrophysical phenomena as stellar and galactic evolution, variable and X-ray emission stars, chemical abundances (primordial nucleosynthesis), etc. Globular clusters are ideal laboratories for testing theories of stellar evolution, the chemical evolution of the Universe and the dynamics of N-body systems. They are the oldest known objects whose ages can be independently determined, the closest in proximity to the origin of the Universe and the sole surviving structures of the first stages in the formation of the Galaxy. They provide us with important evidence concerning on the age and formation processes of the Galaxy. Globular Clusters are a fundamental unit of the known Universe, they are also found in all other galaxies within our observational grasp. They are possibly a necessary stage in the formation of galaxies.

Research on Globular Clusters covers a vast amount of territory that was reviewed and collected in the present book. From the photographic plate to the HST most recent results, the field of Globular Clusters was actualised and presented by Ivan R. King, with an interesting Observational Approach to Populations in Globular Clusters, where discusses the observations on which our understanding of globular clusters lies. Steven Majewski, reviews the Stellar Populations and Formation of the Milky Way, with particular emphasis on the role of globular clusters in tracing stellar populations and unravelling the Galactic history.

The search for the structure of our Universe and our position within it never will cease. As we answer each question, others arise with even greater insistence. And the context of our questions is ever changing. From the mythological background of Babylon, to the mechanical clockwork of Newton, through the opening of our minds to prodigous swarms of distant galaxies, to the mathematical models of general relativity and gravitational clustering within them, each new context inspires new questions. Nor is there reason to suppose that the present context will bring this search to a close.

Throughout the roughhewn matrix of our understanding, dark matter weaves threads of uncertainty. Its amount governs the flight of the galaxies and the fate of the Universe. Many models undertake to confine it to various distributions and forms. So far, dark matter has resisted all but gravitational attempts at detection, leaving the models to flicker and shift in the ebb and flow of theoretical fashion.

Nor do observations always provide simple truths. Most are so riddled with selection and filtered with theory that their interpretation is seldom straightforward. Simple ideas like filaments and voids, walls and clusters, become much more complex when closely examined. Their simple grammar often remains suitable mainly for slogans. All good observers know this in their bones. Results, regardless, can still be astounding.

The history of understanding structure in our Universe is older than the story of Odysseus, and has as many twists and turns. Few of these paths remain familiar to most astronomers today, so in the early chapters I have simply collected some essential developments along the way. They are not without surprises. One of which is that many ideas now thought to be novel have really been known for tens, or hundreds, of years. Often they were no more than speculations, but sometimes they captured reality's core.

The past is our future frontier. Distribution functions at high redshift have not yet been observed. Therefore this chapter will be very short.

Here and there we have glimpses of how galaxy clustering may have developed. These are from observations of two-particle correlations, Lyman alpha clouds, merging protogalaxies, and rich clusters, all at great distances. Eventually, when the halfdozen or so high-redshift catalogs now being started accumulate complete and well-chosen samples, they will yield up the distribution functions of the past. Insofar as these agree with the GQED, their evolution is represented by the changing of b. Thus there is time for genuine predictions, such as those of (30.12) and (30.19)–(30.20) shown in Figure 30.5. These catalogs will also test the importance of merging, which would alter N's conservation (see Chapter 36).

At high redshifts the value of b for gravitational quasi-equilibrium evolution depends quite strongly on Ω0. igures 30.5, 31.12, and 31.13 indicate that as we look further into the past, b will decrease more slowly for lower Ω0. This is essentially because the clustering pattern is “frozen” at higher redshifts for lower Ω0. Equivalently, for higher Ω0 most of the evolution occurs more recently. Zhan (1989) and Saslaw and Edgar (1999) give useful diagrams to show how this can help determine Ω0 and the redshift at which galaxies start clustering.

The total mass of the globular cluster system of our Galaxy makes only ∼10−3 the mass of the Galactic disk. It contains, however, ∼20% of all known low-mass binary X-ray sources, about one half of all binary pulsars, and more than a half of the millisecond pulsars in the Galaxy. Close binary systems containing neutron stars should thus form much more easily in the dense stellar environment of globular clusters than elsewhere in the Galaxy. In these lectures we first review the formation mechanism of neutron stars. Then, we present the evolutionary scenarios leading to the formation of binary X-ray sources and binary and millisecond pulsars in the Galactic disk and the Galactic bulge. We later discuss the specific mechanisms to form neutron star binaries in globular clusters. We end by discussing the open issues concerning the origin and evolution of X–ray sources and millisecond pulsars in globular clusters, and their relationship with the structure, dynamics and evolution of the clusters themselves.

Low–mass binary X–ray sources and millisecond pulsars

An early, unexpected result of X–ray astronomy was the discovery of several bright X–ray sources in globular clusters. Later on, searches for radio pulsars have produced many detections, especially of short period pulsars. We begin these lectures with a very schematic presentation of those two kinds of objects.

X–ray binaries

Luminous Galactic binary X–ray sources provided the first evidence of neutron star binaries, that is binary star systems containing neutron stars (Giacconi et al. 1971; Lewin et al. 1971; Schreier et al. 1972; Tananbaum et al. 1972).

Current theoretical predictions concerning the evolution of old, metal-poor stars in galactic globulars are revisited in the light of recent improvements in the input physics. After a short introduction, the role of fundamental physics in constraining stellar structure all along the various evolutionary phases is shortly recalled, together with the additional role played by the evaluation of some macroscopic mechanisms, like convection and diffusion (Sect. 2). Theoretical predictions concerning the CM-diagram location of the best-known evolutionary phases are discussed in some detail, with particular regard to the existing uncertainties in modeling stellar structure as well as in handling observational data (Sect. 3). This discussion is thus extended to the faint stars recently revealed by HST, either at the faint end of the MS or along the WD cooling sequence (Sect. 4). Additional theoretical constraints given by the pulsational properties of RR Lyrae pulsators are recalled (Sect. 5) and the case of extragalactic globulars in the Local Group are briefly discussed (Sect. 6). Some general and methodological considerations close the paper.

The CM diagram: an introduction

The birth of modern physics dates back to the time when Galileo Galilei stated that any attempt to understand the world around us must—first of all—save the phenomena (“salvare i fenomeni”). In modern words, we say that physics is studying relations between observable quantities, so that the identification of suitable “observables” is a first-priority step in any physical investigation.