The dynamics of galaxies is to some extent similar to that of electromagnetic plasmas. This is not at all a new way of looking at stellar systems. In fact, the work of Jeans, whose goal was to describe the motion of a large collection of stars under the influence of their mutual gravitational attractions (see Chapter 8), and the work of Chandrasekhar, in setting out the equations for the effects of collisions in a stellar system (see Chapter 7), precede much of the work related to the Vlasov equation and the various transport processes in electromagnetic plasmas. Thus it is no surprise that many scientists have contributed, either explicitly or indirectly, to both fields of electromagnetic and gravitational plasmas. Several articles emphasize the common concepts at the roots of the two fields. To be sure, the field of stellar dynamics is much less diverse than that of electromagnetic plasmas. This is probably more the result of the several cancellations related to the fact that the charge-to-mass ratio equals unity for the gravitational charge, which limits the number of relevant frequency windows, rather than the effect of the absence of the complexity of the full set of Maxwell's equations (with classical gravity, we need to keep track of only the Poisson equation).

What are the key analogies and differences? To answer this question, we distinguish among fundamental aspects, basic modeling tools, specific mechanisms, and general approach.

From the discussion of Chapter 20 it should be clear that spiral galaxies possess significant amounts of dark matter in the form of a diffuse halo. Thus we should naturally expect that elliptical galaxies also possess sizable dark halos. Can we prove the validity of this expectation? How can we measure the amount and distribution of dark matter in elliptical galaxies? Note that in contrast to the case of spirals, we may easily imagine here that the halo has the same spatial distribution as the luminous matter because both components can be thought to be the result of collisionless collapse.

In general, elliptical galaxies lack a straightforward, radially extended kinematic tracer, such as the atomic hydrogen emission used to derive the rotation curves of spiral galaxies. Furthermore, although for spirals we can produce convincing tests to judge the overall spatial symmetry of the underlying gravitational potential, for ellipticals, such symmetry is a major open question not only in general but also especially for individual objects; in particular, it is difficult to determine whether the system is basically axisymmetric or triaxial. Thus it is not surprising to find that the data of many objects can in principle be fitted by models that assume a constant mass-to-light ratio. However, in many cases, models of this type are found to be difficult to justify from the physical point of view; for example, if the velocity-dispersion profile is rather flat, as for NGC 4472, a significant bias of the pressure tensor in the tangential direction would be required.

Much of the study of the dynamics of collective systems proceeds through investigation of the properties of equilibrium configurations (often referred to as basic states) and analysis of their stability with respect to a variety of perturbations. The choice of the equilibriumstate is primarily a choice of symmetry. The study of how perturbations evolve on such a basic state is thus a study of the mechanisms that lead to a breaking of the assumed symmetry (although we may consider perturbations that preserve the original symmetry). Obviously, the simplest case that can be considered is that of the linear limit, in which perturbations have vanishingly small amplitudes, much as in the case of small oscillations for a system of point masses in classical mechanics.

In this chapter, general remarks are made about this common framework, and some interesting consequences for the internal structure of collisionless stellar systems are noted. By means of a nontrivial example from plasma physics and an elementary example from classical mechanics, some key concepts are demonstrated and a few important issues are raised that will be found later to be a common thread in the study of the dynamics of normal galaxies. In Chapter 10, the main results and some technical aspects of the process of studying equilibrium and stability for one relatively simple class of self-gravitating fluids, that is, the classical ellipsoids, are summarized.

Within the above-mentioned framework it is implicitly assumed that in many observed systems, some simple symmetries are approximately realized, which is intuitively true.

How can we formulate appropriate equations that are able to provide a realistic and quantitative description of the processes involved in the dynamics of galaxies? How should we model specific phenomena, such as the occurrence of spiral structure or the equilibrium configuration that underlies the observed photometric properties of elliptical galaxies? The modeling process can be articulated in two steps: (1) choice of equations and (2) identification of the relevant basic state and parameter characterizations. Because we are dealing with extremely complex and composite systems (which include different populations of stars, gas in various forms, magnetic fields, etc.), we should recognize from the beginning that no matter how detailed we try to be, we are introducing only model equations, that is, equations for ideal systems that have only some of the ingredients of the real systems we would like to describe. Once the equations are chosen, a second major aspect of the modeling process is that of choosing a good match to the parameters, boundary conditions, and overall characteristics that define the astrophysical phenomena to be studied.

There is a tradeoff between the number of ingredients that can be included in the equations and the model and what can be practically calculated. Special cases with fewer ingredients may be amenable to exact analytical solutions or to efficient numerical calculations, with the obvious advantages implied by such a possibility.

This final chapter is meant to be a bridge between the dynamics of disk galaxies and the dynamics of disks on much smaller scales. The paradigm of mass accretion on a central object by means of a dissipative disk has long been studied in various contexts. The general framework was initially considered as a tool to describe the formation stages of the solar nebula to clarify the mechanisms that generate stars and planets. Later, the picture was reexplored in the context of high-energy astrophysics. There are thus two interesting aspects: the formation and dynamical evolution of a disk during collapse and the possibility of extracting gravitational energy from systems dominated by rotation. In the general picture, key roles are played by viscous dissipation and angular-momentum transport.

These concepts bring us to consider some of the topics that are currently at the frontier of astrophysical investigations. At the smallest scale, for masses of the order of a few solar masses and lengths of the order of a few astronomical units, much of the current interest addresses the processes of star and planet formation; here great progress comes from the ongoing discovery and study of extrasolar planets and detailed observations of star-forming regions with dedicated telescopes from the ground and from space. At similarly small scales, high-energy astrophysics phenomena are produced when accretion occurs onto compact stellar objects and stellar-mass black holes; these processes generally affect binary systems, in which material from a relatively normal star is captured by the compact star.

The process of modeling elliptical galaxies (or constructing models of interest for elliptical galaxies) can be interpreted in very different ways depending on the goals we have in mind. Some of this variety of approaches was already implicit in Chapter 14 in the context of the modeling of galaxy disks. Theremuch of the emphasis was on the construction of realistic basic states as a prerequisite for an appropriate stability analysis; thus the discussion focused on a number of physical arguments (see especially Section 14.4) aimed at identifying general and flexible classes of models with realistic properties. The main goal was to produce a sound physical basis for a dynamical study of some outstanding morphological aspects of galaxy disks, especially spiral structure. Nevertheless, in that chapter we found it instructive to describe a few different models; for example, in Section 14.1 we took a close look at the vertical equilibrium of the disk, which might have been explored even further if we had in mind a deeper analysis of the problem of dark matter in the solar neighborhood. In addition, we also introduced a few exact models (Section 14.3) that offer useful analytical tools and clarify some of the issues related to the support of equilibrium configurations.

There are a number of important dynamical questions related to elliptical galaxies:

For bright elliptical galaxies, why is there basically one universal luminosity profile (the R1/4 law) on the global scale?

What is such universality telling us about the formation and the long-term evolution of these stellar systems?

The themes introduced in Chapter 10 are demonstrated in one class of dynamical systems, which, because of its simplicity, serves as a bridge from the case of single-particle dynamics to the more complex collective behavior of continuous media. The study of incompressible self-gravitating fluids with ellipsoidal shape focuses on an environment with a limited number of degrees of freedom. Still it opens the way to several subtle phenomena that have attracted the interest of many eminent mathematicians and physicists, starting with Newton.

The classical ellipsoids were originally studied as models, as a starting point, to interpret the properties of rotating celestial bodies, such as planets and stars. Curiously, similar concepts have recently found applications in completely different fields, such as the physics of atomic nuclei and black holes. Tools related to the simple properties of ellipsoidal mass distributions have found applications in the study of the formation of structures by collapse (“pancakes” or protogalaxies) in the cosmological context.

In galactic dynamics, classical ellipsoids might be thought of as the natural representation of elliptical galaxies. Unfortunately, spectroscopic observations in the mid-1970s proved that bright ellipticals are not flattened by rotation and thus do not fit this simple picture (see Subsection 4.2.2 and Part IV).

The galaxies we see today owe their current structure to a combination of initial conditions, set by the processes that led to their formation, and a number of mechanisms that have shaped their evolution. Some regularities that we observe, in their morphology (such as the regularities captured by the Hubble classification scheme), in their overall luminosity profiles (exponential or R1/4), or in the existence of well-defined scaling laws (such as the luminosity-velocity relation for spirals and the fundamental plane for elliptical galaxies), demand a physical explanation in terms of formation and evolution.

The discussion here becomes necessarily speculative, especially because we lack strong and direct empirical constraints. It is true that distant quasars and gas-rich absorption systems detected at relatively high redshifts provide clues about the conditions under which galaxies were formed, and relatively normal galaxies have been observed out to z ≈ 5 (i.e., up to a lookback time greater than 90 percent of the age of the Universe) and beyond. Yet we lack the type of detailed quantitative structural information that has allowed us to develop a satisfactory picture of the dynamics of normal galaxies, as observed at z ≈ 0. Furthermore, even when we manage to obtain some data on their internal constitution, for example, in galaxies at z ≈ 1, we do not have direct information on the way such objects have evolved from their progenitors and are going to evolve into the systems that we see in the nearby Universe.

In this chapter we derive the basic equations for Jeans instability in the simplest formulation. This is meant to be the first example in the astrophysical context of the concepts introduced in Chapter 11. In reality, the study of stability brings us one step beyond the concept of dispersive waves. Most of what has been described in Chapter 11 is applicable in cases in which the frequency in the relevant dispersion relation is allowed to be complex, ω =ωR + iωI, provided that |ωI| ≪ |ωR|. But we should be aware that in the opposite limit we are no longer talking about waves in the usual sense.

In the simplest case, gravity and pressure forces are the main ingredients in the Jeans instability mechanism. Quite different results are derived from the same ingredients, depending on the model considered. In particular, although the fluid model is characterized by a dispersion relation for which neutral waves (ωI =0) are allowed, a kinetic analysis leads to a dispersion relation in which stable perturbations are generally damped. At the end of the chapter, we briefly examine the linearized equations for perturbations of a plane-parallel slab, demonstrating how, in general, the collisionless Boltzmann equation is solved by integration along the unperturbed characteristics.

The discussion of density waves given in Chapter 15 is only one important step in the study of the dynamics of even perturbations on an axisymmetric self-gravitating disk. As already emphasized at the end of that chapter, we should complete the analysis by taking into account the inhomogeneity of the disk and the relevant physical conditions at the boundaries. However, the focus on the study of waves is given priority because of its simplicity and its usefulness in identifying the relevant dynamical mechanisms (see Section 15.5) and, historically, because of its immediate application in testing the density-wave theory against observations of spiral structures in galaxies within a semiempirical approach.

Given the symmetry of the basic state and the radial inhomogeneity of the disk, we consider disturbances for which the perturbed density of the disk is given by σ1(r)exp[i(ωt−mθ)] and, for given values of ω and m, look for appropriate functions σ1(r) that satisfy the dynamical equations and the relevant physical conditions at the boundaries. In general, this is an eigenvalue problem, and the associated spectrum is not continuous in the sense that the problem admits solutions for only selected values of the frequency. These solutions are often called global modes or simply modes. For a given model, once we have formulated the (linearized) mathematical problem, we can use numerical integration for the calculation of such modes with no need for the approximate description in terms of waves contained in a local dispersion relation [see Eq. (15.1)].

This chapter is devoted to a very simple summary of the main concepts at the basis of gravitational lensing and its astrophysical applications, with special attention to the frequent case in which galaxies act either as lenses or as sources affected by the phenomenon of lensing. After the discovery of the first clear macroscopic evidence of gravitational lensing, that is, the observation of a double image of a single quasar produced by an intervening galaxy acting as a lens, enormous progress has been made in terms of observations and theoretical results, which has produced a vast literature dealing with a number of distinct and extremely interesting related topics. As we will see, the physical basis of gravitational lenses is such that the most spectacular phenomena are produced by distant lenses on very distant sources; this is why the rapid progress and growing interest in this subject largely coincide with the systematic exploration of the distant Universe at redshifts of cosmological interest started at the end of the past century.

The main reason to present such a summary here is that one of the most interesting themes developed in the past ten to fifteen years in relation to the general problem of mass diagnostics for galaxies and clusters of galaxies is that of the combined use of gravitational lensing and dynamics. The use of dynamics as a diagnostic tool to probe the mass distribution in galaxies and other systems is generally based on the assumption that the object under investigation has reached a state of approximate dynamical equilibrium.

Since the year of publication of the first edition of Dynamics of Galaxies, the field has developed significantly. This book keeps the structure of the first edition but includes much new material. In general terms, it contains an up-to-date view of the basic phenomenology (in the eyes of a theorist), a new discussion of the properties of dark halos in galaxies in the light of recent results and projects, an extended description of the dynamics of quasi-relaxed stellar systems in connection with the dynamics of globular clusters, an introduction to gravitational lensing, and a primer on the properties of self-gravitating accretion disks. Motivation to undertake the writing of the new edition has come not only from personal research in the above-mentioned topics but also especially from the teaching experience in graduate and undergraduate courses (in Milan and Pisa); therefore, in this new edition the set of problems has increased considerably, especially in relation to Parts I and II.

The details of additions that characterize this edition are the following. Two new chapters have been placed at the end of the book, in Part V. Chapter 26, “Galaxies and Gravitational Lensing,” provides the basic elements of the theory of gravitational lensing and a summary of some interesting astrophysical applications; an incentive to write such a chapter was given by the fact that in the last few years the combined use of stellar dynamics and gravitational lensing has proved to be an excellent diagnostic of the mass distribution in distant self-gravitating systems.

With most of the necessary dynamical tools already introduced, we now turn to the astrophysical problem. In this chapter we will try to highlight only some important points. The reader is referred to the monograph by Bertin and Lin (1996) for the full definition of the astrophysical issues involved, a physical outline of the modal theory in the semiempirical context, and the description of many important observational tests not mentioned here.

Spiral structure in galaxies has been the focus of a vast set of observational and theoretical investigations during the entire twentieth century. A decisive turning point in the course of these studies was the realization, in the early 1960s, that density waves are at the basis of the most spectacular phenomena associated with spiral structure. In particular, this realization came as the conclusion of the pioneering work of Bertil Lindblad, who first formulated the hypothesis that the large-scale spiral structure in galaxies is quasi-stationary despite the presence of differential rotation in the disk. Because of differential rotation, any material structure could not persist but would be stretched and would wrap up on a very short time scale. This is commonly known as the winding dilemma. In contrast, as the manifestation of waves, the density concentrations that define the spiral arms could move in the disk in a manner that is, to a large extent, decoupled from the motion of the individual particles (stars or gas clouds) that support it, much like a sound wave in an ordinary gas. Thus spiral arms, if associated with a wave phenomenon, could survive differential rotation as quasi-stationary patterns.

The gas layer of our Galaxy is known to be warped in a rather asymmetric way. Outside the solar circle, the HI disk reaches ≈ 3 kpc above the plane on one side (in the North), whereas it extends down to only 1 kpc below the plane on the other side (in the South); in the outermost regions, at a galactocentric radius of ≈ 30 kpc, the warp is found to reach the height of 6 kpc. Many galaxies are affected by similar global distortions of the disk (Fig. 19.1; see also Fig. 3.4 in Part I). In a survey of twenty-six edge-on galaxies, twenty were found to be warped in HI, often in an asymmetric way, with the warp usually starting at the edge of the optical disk. Warps also have been noted (although often only as marginal evidence) for the stellar disk. The fact that stars and gas both participate in the warp is an indication that the phenomenon is probably dominated by gravitational forces. The presence of significant warps only in the outer regions, where the stellar disk ends, may simply be due to the small inertia of the disk in the outer parts compared with the heavier inner regions.

Besides the frequent integral-sign appearance (i.e., dominance of an m = 1 relatively open distortion of the disk), the morphology of warps is not as well documented.

The discovery of dark halos, the spectroscopic evidence that elliptical galaxies are dominated by collisionless dynamics (which rules out the applicability of simple fluid models for their description) and the opening of new observational windows (especially in the near infrared) able to provide direct information on the underlying stellar mass distribution in galaxies have significantly changed, in the third quarter of last century, our perception of the internal structure of galaxies. As will be pointed out in general terms in Part II, the modeling tools and the theories developed to explain many interesting observations (from the study of global spiral and bar modes of galaxy disks to the construction of self-consistent anisotropic collisionless models to explain the universality of the luminosity profile of elliptical galaxies) present many analogies to parallel work in the physics of electromagnetic plasmas. At the frontier of current research in extragalactic astrophysics, the Hubble Space Telescope and new large telescopes from the ground are giving us a view of the early dynamical stages of galaxies, providing interesting insights into aspects of galactic formation and evolution, and, on the small scale for relatively nearby galaxies, are offering unprecedented accurate data on their structure and kinematics.

Within the same basic framework of dynamical studies, very different approaches can be taken. Dynamics is a powerful tool that allows us not only to interpret the data and to form a better picture of the internal structure of galaxies but also to develop and test physical scenarios. As such, its role is not limited to the descriptive stage. Two extremes are often taken in the study of the dynamics of galaxies.