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The process of modeling elliptical galaxies (or of 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. There much 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) in order to identify general and flexible classes of models with realistic properties. The main goal was then to produce a sound physical basis for a dynamical study of some outstanding morphological aspects of galaxy disks, especially spiral structure. Still 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 nice analytical tools and clarify some of the issues related to the support of equilibrium configurations.

Some important dynamical questions related to elliptical galaxies that can be addressed by constructing and by studying stellar dynamical equilibrium configurations are the following. Why is there basically one universal luminosity profile on the global scale of elliptical galaxies? What is such universality telling us about the formation and the long term evolution of these stellar systems? Can we "explain" it, i.e. can we propose a simple physical scenario for it? How can we reconcile the variety of observed kinematical profiles with the existence of a relatively fixed photometric structure? To what extent can pressure anisotropy be "blamed" for the variety of observed kinematical profiles? Given the fact that spiral galaxies appear to be embedded in massive dark halos, how would a dark halo influence the dynamics of an elliptical galaxy? Would dark matter just follow the luminous component, so that only negligible M/L gradients occur, or should we expect, by analogy with spiral galaxies, that rather diffuse dark halos coexist with more compact stellar distributions? How can we practically measure the amount and distribution of dark matter around elliptical galaxies? Given the range of observed shapes, can we infer the relevant distribution of intrinsic shapes among galaxies? How are the various shapes dynamically supported? Are the shapes dominated by pressure anisotropy or are they produced with significant contributions from internal streaming and figure rotation? In other words, how frequent and how significant is the presence of pressure anisotropy? Can we have clues on how the shapes of ellipticals may be generated in terms of plausible formation scenarios? On the smaller scale, can we have a satisfactory description of the core and nuclear regions of ellipticals? What can the resulting models tell us about the possible ubiquity of massive black holes at the center of elliptical galaxies?

There are of course other important issues, such as the existence and the role of scaling laws, and, in general, questions related to the evolution of the elliptical galaxies on the Hubble timescale, which most likely escape a pure dynamical discussion, because they depend on more complex physical mechanisms, on the detailed star formation processes, and on stellar evolution. Some of these points will be considered in Part five.

In this chapter we will describe the properties of stellar dynamical models that may help us answer some of the questions raised above. We will make use of the Jeans theorem, considered as a useful working tool. We will try to see whether, based on guidelines dictated by physical arguments, the theorem can lead us to models that match and clarify the most significant observed properties of elliptical galaxies. If the process works, we will consider it as an indication of the plausibility of the physical arguments made. If the process fails, we should try to revise our physical scenarios and we might even question whether the Jeans theorem, in the simple form that is usually adopted, should be applied at all to the physical systems under investigation. In particular, some difficulties that are encountered in the construction of realistic, triaxial dynamical models may be telling us that fully triaxial stellar systems cannot be approximated by exactly integrable models. In practice, for reasons to be illustrated below, we will emphasize a modeling process where one starts from physically justified distribution functions.

Within the same basic framework of stellar dynamical tools (orbits, integrals of the motion, and the Jeans theorem) very different alternative approaches can be taken (see section 22.1). Apart from some technical aspects that are involved, two major lines of thought characterize these efforts. On one side, some studies, in line with those of many other branches of physics, try to be predictive, and thus focus on the consequences of physical assumptions in relation to the observations. In a semi-empirical approach, such process gets constant inspiration from the data. These efforts try to produce a "theory" which may be disproved by existing data or by new observations. In a second line of thinking, which is essentially descriptive, one aims at extracting from the data as much as possible before making physical assumptions, with the hope that the physical aspects can be discussed later when all the structural parameters are secured "directly" from the data. Almost by definition, the descriptive stage is unable to provide explanations. Probably at the root of this other point of view, in contrast with the semi-empirical approach, there lies the dream of deriving the evolution of stellar systems starting from directly measured initial conditions. We have often emphasized that such an inherently deductive approach is bound to suffer from severe limitations. In practice, since the data points sample only limited spatial regions with finite accuracy, projected along the line-of-sight, the "descriptive" inversion process has also to rely on a set of assumptions (in the best cases, on the geometry and on the mass-to-light ratio profile) and it is known to be inherently unstable, thus bound to lead easily to non-physical solutions.

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