We have already mentioned that different combinations of electric and magnetic field vectors can yield the same Poynting vector. This means that forming the vector product of the electric and magnetic field vectors results in a quantity that does not carry unique information about the participating fields. In particular, the Poynting vector contains no information about the polarization state of a transverse electromagnetic field. Thus, by its very deinition, the Poynting vector cannot be used to describe the phenomenon of electromagnetic scattering by, for example, expressing the Poynting vector of the scattered field in that of the incident field.

We have seen in the preceding chapter that the standard descriptor of polarization is the Stokes column vector (7.3). However, this quantity contains no explicit information on the direction of the Poynting vector and can be defined only for a transverse (i.e., plane or spherical) electromagnetic wave, whereas the total electromagnetic field in the near zone of any object (e.g., at any observation point inside a cloud of particles) is never a transverse wave. It is, therefore, highly desirable to introduce an alternative quantity that:

1. • can be defined for any electromagnetic field;

2. • has the dimension of electromagnetic energy flux; and

3. • enables a complete and self-contained description of electromagnetic scattering in the context of practical optical analysis.

This book is addressed to a readership of beginning graduate students and up. In contrast with other monographs on related subjects, it emphasizes some physical and methodological aspects. Two distinctive features are the emphasis on the similarities between the dynamics of stellar systems and the collective behavior of electromagnetic plasmas and the focus on a few important dynamical issues that are raised by observations. In the adopted semiempirical approach, the study of dynamics thus develops under the guidance and inspiration of empirical facts. Therefore, the book, although rich in quantitative analysis, reduces the mathematical discussion to its essentials. The level of presentation is not excessively technical. There is no intention to give a full account of all aspects (many of which are of fundamental importance) of interest in this broad research area; the reader is referred to the many authoritative books already available. Instead, this book tries to capture a lively synthesis to arouse curiosity in a reader who is not already professionally involved in this area of celestial physics.

From the methodological point of view, the book focuses on the general use of asymptotic methods, and it gives in detail the mathematical structure of some derivations when they may be useful for more general purposes. Asymptotic methods are flexible tools to obtain approximate solutions that give priority to the richness of ingredients in a physical problem; technically speaking, these methods recognize the importance of singular perturbations in the realization of physical processes.

A piece of writing that best captures the beauty and mystery surrounding the dynamics of galaxies can be found in chapter 7 of the first volume of The Feynman Lectures on Physics. There, in a few simple sentences and by means of a picture of a globular cluster and a spiral galaxy, we are rapidly brought to the scales and structure of the systems involved and to the underlying limits that our physical knowledge must recognize. We are told that gravitation is the prime actor, stars rarely collide, and angular momentum leads to a contraction in a plane, but we are also reminded of the facts that “of course we cannot prove that the law here is precisely inverse square,” “what determines the shape of these galaxies has not been worked out,” and that we are dealing with systems that are enormously complex.

Approximately four decades earlier, through a decisive distance measurement, some nebulae had been recognized as huge systems millions of light years away; those were nearby galaxies, and galaxies were thus found to be the visible building blocks of the Universe. If we examine that discovery, we see that modern cosmology and quantum mechanics were developed at approximately the same time and well after the formulation of general relativity; they were also developed well after the basic equations that govern the motion of a self-gravitating stellar system were studied by Jeans.

From what has been described so far, it should be clear that the structure and evolution of spiral galaxies depend largely on the amount and distribution of dark matter actually present in these systems. (This statement would apply in even stronger terms to evolutionary scenarios in which galaxy-galaxy interactions play a major role.) In modeling of the basic state (e.g., see Sections 14.4 and 14.5) and discussion of several dynamical mechanisms (e.g., see Section 18.2 or Chapter 19), we have recognized that, in general, a dark halo may be present. However, we have left open the possibility that in some galaxies the halo may dominate, whereas in others it may be relatively small or even insignificant from the dynamical point of view. In this chapter we briefly review the arguments that have brought us to believe that dark matter is indeed present in spiral galaxies. This will also show that many issues remain far from being fully understood.

Starting with the studies by Kapteyn and Oort of the stellar motions in the solar neighborhood (see Section 14.1.2) and the studies by Zwicky and Smith of the galaxy motions inside clusters, it has long been suspected that the Universe contains much more matter than meets the eye, that is, through the telescopes at our disposal.

The theory of orbits is important in the study of the dynamics of galaxies. Because the relaxation times for the relevant processes are very long (see Chapter 7), much of the interest lies in the study of orbits in a mean field. What is learned from these studies applies beyond the case of single particles, for the evolution operator in a continuum description (either in a fluid or in a stellar dynamical description; see Chapter 8) can be essentially identified with the Hamiltonian that governs single-particle orbits.

Galaxy disks are relatively cool systems; that is, most star (or gas cloud) orbits are very close to being circular, so a typical star (or gas cloud) velocity at a given location is very close to the average velocity of rotation of the disk. A proper description of a collection of orbiting particles should be given in terms of a distribution function (see Chapter 14). However, the physically intuitive properties of such cool disks can be easily traced back to the characteristics of quasi-circular orbits of individual particles, which are outlined in this chapter. The deviations from circular orbits are called epicycles. To lowest order, they correspond to a harmonic oscillator in the radial direction characterized by the epicyclic frequency k.

As described in Chapter 9, the concept of the basic state is a key step in the modeling process. As in other contexts, for galaxy disks, identification of an appropriate basic state is done for several purposes. One interesting aspect is that by studying the internal structure of the basic state, we can clarify the overall constraints imposed by self-consistency; in this respect, the two examples that follow, dealing with the vertical and horizontal equilibrium of a rotation-supported disk, are rather simple (in collisionless stellar dynamics, self-consistency imposes much stronger and less intuitive constraints in pressure-supported systems) but very instructive. The models that are set up can then be used to fit the data and thus derive information on the structure of galaxies. Furthermore, a broad choice of equilibrium models can form the basis for systematic stability analyses. In this case, the basic state is meant to be a tool to study evolution, but it should not be assumed to represent necessarily the actual conditions of the galaxy in the distant past.

From the point of view of stellar dynamics, the case of axisymmetric galaxy disks is relatively simple, because for realistic models most orbits are quasi-circular and the disk is rather thin. The related description can then be decoupled at two different levels: (1) the vertical dynamics can be treated separately from the horizontal dynamics; and (2) for the horizontal structure, because the angular momentum essentially identifies the radial coordinate, the problem of self-consistency can be worked out in two separate steps: (2.1) choose a density-potential pair, and (2.2) find a (quasi-Maxwellian) distribution function that supports the assumed density. In contrast, this procedure of going from ρ to f would generally lead to nonphysical models in the spherical case.

Chapter 13 started out by noting the importance of the study of orbits as a key step in the study of the dynamics of galaxies. Here the reason why it is important is briefly summarized. Even better than other stellar systems, elliptical galaxies may be modeled as purely collisionless collections of stars. It is well known that for the distribution function in phase space that is used to describe these systems, the evolution operator dictated by the mean field is the same as that for singleparticle orbits. The resulting Jeans theorem (see Chapter 9) thus reduces the problem of finding the most general stationary solution to the collisionless Boltzmann equation to identification of the relevant integrals of the motion. Furthermore, the frequencies characterizing single-particle orbits play an important role in the stability of the stellar system with respect to internal or driven perturbations. Finally, we should recall that closed periodic orbits are often used to model relatively smooth and cold flow patterns that are sometimes observed around galaxies, thus offering a very interesting diagnostic tool for the underlying potential (see also Subsection 13.6.1 and the discussion of dark matter in Chapters 20 and 24). Unfortunately, when we deal with the dynamics of elliptical galaxies and with the related orbits, we have to face two very difficult issues.

The first important point is that ellipticals are dynamically hot stellar systems. Therefore, even if we focus our attention on axisymmetric basic states, approximations such as those that have led to the discussion of quasi-circular orbits in the context of spiral galaxies are bound to be of little use.

When we make observations, we rely on a number of spectral windows for which we have instruments to study the radiation that reaches the Earth. Obviously, astronomy is centered around optical radiation. After a long period of work with photographic techniques, most of the current optical studies are based on charge-coupled-device (CCD) detectors, which are solid-state devices based on two-dimensional arrays of detection elements with a wide range of capabilities. The visible light extends from the ultraviolet (UV) region (UV radiation is associated with wavelengths in the range 100 to 4,000 Å) to the near-infrared (near-IR) region (IR radiation has wavelengths between 7,500 Å and the millimeter range). Observations from the ground are limited by the way the light is transmitted by the atmosphere. For example, much of the near-IR radiation is absorbed by the atmosphere, and observations beyond 1 μm are best performed at high altitude (where significant transmission occurs in correspondence with the standard IR filters J, H, and K) or directly from space.

New observational windows have been opened, especially in the second part of past century, and these have dramatically changed our view of the Universe (Fig. 2.1). A major step forward has been identification of the 21-cm line (at a frequency of 1,420 MHz; a hyperfine transition in atomic hydrogen associated with the spin-flip in the electron-proton pair), which has given an enormous boost to radioastronomy. Studies of the kinematics of atomic hydrogen, especially those of the 1970s and the early 1980s, have provided the decisive evidence for the existence of dark matter.

Density waves are thought to be at the basis of the explanation of spiral structures in galaxies, especially of the so-called grand-design structure whose extent is on the global scale. A physical discussion of the problem of spiral structure in galaxies will be given in Chapter 18. In this and Chapters 16 and 17 we focus instead on some relevant dynamical mechanisms. As anticipated earlier (Chapter 9), density waves are one of the two main classes of natural perturbations that are expected in a disk (the other class, bending waves, will be addressed in Chapter 19). They leave the equatorial symmetry of the galaxy disk unchanged and are associated with density enhancements and rarefactions that usually break the axisymmetry of the basic state. A special class of density waves (m=0) leaves the disk axisymmetric.

The concept of density waves is a general one because the phenomenon simply reflects the oscillatory character of the disk that has been described in terms of single-star orbits. In practice, the detailed properties of density waves depend on the model considered. In early investigations, the galaxy disk was thought of mainly as a stellar disk, and thus the studies focused on the equations of stellar dynamics.

In Part I we saw that galaxy disks can be thought of as basically comprising two components, Population I and Population II. One component is dominated by cold gas, in atomic or molecular form, but contains significant amounts of stars recently born in the interstellar medium. This component is in a thin layer (at least within the bright optical disk) and is characterized by very low velocity dispersion (often below 10 km s−1) with respect to the circular motions associated with the differential rotation. The other component, Population II, is dominated by relatively old stars, in a thicker layer, and is characterized by higher velocity dispersions; that is, it is warmer from the dynamical point of view. Such separation is only a simplifying tool for dynamical investigations, whereas, in reality, continuous changes of dynamical properties are associated with the many components of a galaxy disk. (The properties of the extraplanar gas were briefly summarized at the end of Chapter 14.)

The gas-dominated component is characterized by small epicycles. Thus, from the dynamical point of view, it is naturally responsible for small-scale features in the disk, which are probably rapidly evolving. (The large-scale spiral structure that will be described in the following chapters then must draw its main support from the stars.) Furthermore, being cold, the gaseous disk is expected to provide an important contribution to the Jeans instability of the disk and, in this sense, to be an important source of excitation also for large-scale spiral modes.

Jeans wrote: “The great nebulae exhibit an enormous difference of structural detail, but Hubble, who has devoted much skill and care to their classification, finds that most of the observed forms can be reduced to law and order.” This general statement applies in even stronger terms today, now that several decades of work have confirmed Hubble's intuition. In fact, “The conclusion is that the modern classification indeed describes a true order among the galaxies, an order not imposed by the classifier.”

The efforts to sharpen the empirical morphological classification are extremely important for our knowledge of the dynamics of normal galaxies. Indeed, “The ultimate purpose of the classification is to understand galaxy formation and evolution.” Thus the very existence of the morphological classification scheme proves that the observed morphology reflects a few intrinsic characteristics that vary with continuity along the Hubble sequence and that the overall structure is likely to be quasi-stationary. This plays a central role in the development of a dynamical framework for the classification of disk galaxies, for which the spiral structure is the most spectacular morphological property considered, as is outlined in Part III.

It has been noted that morphology changes significantly with the wave band of observation, with redder images generally found to be characterized by a higher degree of smoothness and regularity. For a dynamical theory aimed at bringing out the role of gravity in the classification, the recent near-infrared (near-IR) studies that probe the underlying evolved stellar disk thus become of primary importance, especially when they may show a contrast with the morphology based on standard optical images.

Empirical evidence proves that stellar systems do not obey the simple laws of equilibrium thermodynamics. For example, in the solar neighborhood, the data show that the distribution of stellar orbits is such that the associated pressure tensor is definitely anisotropic, which is a primary indication of insufficient relaxation; the axis ratios of the velocity ellipsoid for old disk stars are approximately cr: cθ: cz = 39: 23: 20 (the pressure ratios go with the square of these figures).

In this chapter an elementary discussion is given of why large stellar systems are characterized by a very low degree of collisionality, and then the issue of how such collisionless stellar systems can still be subject to relaxation processes that play a significant role in their evolution is briefly addressed. Thus, even if stellar encounters act on a very long time scale, arbitrary distributions of stellar orbits are probably not realized in nature. It appears that, also with the help of subtle mechanisms of collisionless relaxation, most normal galaxies are in a slowly evolving state of incomplete relaxation.

Two-Star Relaxation Times

The following discussion summarizes well-known results that apply to stellar systems and ionized gases. The two-star relaxation times quantify the effects of star-star collisions in changing the orbit of a star with respect to that determined by the smooth mean field generated by the whole stellar system. Here the stars are considered as point masses interacting with each other according to Newton's law. The relaxation times thus measure the effects of the discreteness of the mass distribution in a stellar system.