In this chapter, we show how the (infinite) set of equations known as the Jeans equations is derived by considering velocity moments of the collisionless Boltzmann equation (CBE) discussed in Chapter 9. The Jeans equations are very important for physically intuitive modeling of stellar systems, and they are some of the most useful tools in stellar dynamics. In fact, while the natural domain of existence of the solution of the CBE is the six-dimensional phase space, the Jeans equations are defined over three-dimensional configuration space, allowing us to achieve more intuitive modeling of directly observable quantities. The physical meaning of the quantities entering the Jeans equations is also illustrated by comparison with the formally analogous equations of fluid dynamics. Finally, by taking the spatial moments of the Jeans equations over the configuration space, the virial theorem in tensorial form is derived, complementing the more elementary discussion in Chapter 6.

With this chapter, the final part of the book, dedicated to collisionless stellar systems, begins. As should be clear, in order to extract information from the N-body problem, we need to move to a different approach than direct integration of the differential equations of motion, and a first (unfruitful) attempt will be based on the Liouville equation. In fact, the basic reason for the “failure” of the Liouville approach is that, despite its apparent statistical nature, the dimensionality of the phase space Γ where the function f(6N) is defined is the same as that of the original N-body problem. Suppose instead we find a way to replace the 6N-dimensional R6N phase space Γ with the six-dimensional one-particle phase space γ: we can reasonably expect that the problem would be simplified enormously, and in fact along these lines we will finally obtain the collisionless Boltzmann equation, one of the conceptual pillars of stellar dynamics.

Armed with the power of the Jeans theorem, we now proceed to formulate and discuss the so-called direct problem of collisionless stationary stellar dynamics. This approach is best suited for systems where empirical/dynamical arguments can lead to a plausible ansatz for the form of the underlying distribution function, expressed in terms of the relevant integrals of motion. In the absence of such an ansatz and in the presence of specific requirements (in general motivated by observations) for the density and velocity dispersion profiles, a different and complementary approach based on the use of the Jeans equations is often followed, which is the subject of Chapter 13.

In astronomy in general, and in the study of stellar systems in particular, one is often led to consider the effects of an “external” gravitational field on a body of some spatial extension: examples are satellites around planets, binary stars, open and globular clusters in galaxies, and galaxies in clusters of galaxies. The general problem can be mathematically very difficult; however, when the extension of the body of interest is small compared to the characteristic length scale of the external gravitational field (i.e., when the system is in the tidal regime), the problem becomes more tractable. In this chapter, we provide the basic ideas and tools that can be used in stellar dynamics when dealing with tidal fields. Among other things, we will find that tidal fields are not always expansive (as in the familiar case of the Earth–Moon system), as they can be also compressive (e.g., for stellar systems inside galaxies or for galaxies in galaxy clusters).

In this last chapter, we discuss a final theoretical step of the moments approach illustrated in Chapter 13: under the assumption that the macroscopic profiles (e.g., density and velocity dispersion) of each component are known, there is a possibility of recovering the phase-space distribution function (DF) of a model and checking its positivity (i.e., verifying the model consistency). The problem of recovering the DF is in general a technically difficult inverse problem, and even when it is doable, unicity of the recovered DF is not guaranteed, so that a simple consistency analysis is quite problematic. Fortunately, there are special cases when (in principle) the DF can be obtained analytically (generally in integral form), and in these cases a few general and useful consistency conditions can be proved, such as the so-called global density slope–anisotropy inequality. The student is warned that this chapter is somewhat more technical than the others; however, the additional effort needed for its study will be well repaid by the understanding of some nontrivial results allowing for the construction of phase-space consistent collisionless stellar systems.

The N-body problem, the study of the motion of N point masses (e.g., stars) under the mutual influence of their gravitational field, is one of the central problems of classical physics, and the literature on the subject is immense, starting withNewton’s Principia (e.g., see Chandrasekhar 1995). Conceptually, it is the natural subject of celestial mechanics more than of stellar dynamics; however, experience suggests that some space should be devoted to an overview of the exact results of the N-body problemin a book like this. In fact, due to the very large number of stars in stellar systems, stellar dynamics must rely on specific techniques and assumptions, and one may legitimately ask which of the obtained results hold true in the generic N-body problem; for example, these exact results represent invaluable tests for validating numerical simulations of stellar systems. The virial theorem is onesuch result, and in this chapter we present a first derivation of it, while an alternative derivation in the framework of stellar dynamics will be discussed in Chapter 10.

In this chapter, we introduce and solve (by means of the Laplace–Runge–Lenz vector) the two-body problem, with an emphasis on the properties of hyperbolic orbits, and specifically on the so-called slingshot effect. The obtained results will be used in Chapters 7 and 8 for the derivation of two fundamental timescales characterizing the dynamical evolution of stellar systems (i.e., the two-body relaxation time and dynamical friction time).

We summarize here the most important mathematical results used in the text (and useful for solving some of the proposed exercises). Overall, this appendix is intended more as a selection of mathematical results relevant to stellar dynamics than an organic presentation of definitions and theorems. A good knowledge of linear algebra and calculus at the undergraduate level is assumed, and the interested reader is encouraged to refer to some of the excellent classical treatises of mathematical physics, such as Arfken and Weber (2005), Bender and Orszag (1978), Courant and Hilbert (1989), Dennery and Krzywicki (1967), Ince (1927), Jeffrey and Jeffrey (1950), Kahn (2004), and Morse and Feshbach (1953). To help with further study, in the following sections more specific references are sometimes also provided.

This chapter is aimed at introducing in an elementary yet rigorous way the mathematical properties of the Newtonian gravitational field together with the Second Law of Thermodynamics, upon which stellar dynamics is founded. We begin from the gravitational field of a point mass, and then we move to consider the field of extended mass distributions by using the superposition principle. A direct proof of Newton’s First and Second theorems for homogeneous shells is worked out, followed by a different derivation based on the Gauss theorem.

When observed as astronomical objects, stellar systems appear projected on the plane of the sky. As a consequence, it is important to set the framework for a correct understanding of the relation between intrinsic dynamics and projected properties. Unfortunately, while it is always possible (at least in principle) to project a model and then compare the results with observational data, the operation of inversion (i.e., the recovery of three-dimensional information starting from projected properties) is generally impossible due to obvious geometric degeneracies. Spherical and ellipsoidal geometries are among the few exceptional cases that will be discussed in some depth in Chapter 13. Here, instead, the reader is provided with some of the general concepts and tools needed for projecting the most important properties of stellar systems on a projection plane.

In this chapter, we introduce one of the fundamental and most far-reaching concepts of stellar dynamics (and of plasma physics for the case of electric forces): that of “gravitational collisions." As an application of the framework developed, the two-body relaxation time is derived (in the Chandrasekhar approach) by using the so-called impulsive approximation. The concepts of the Coulomb logarithm and of infrared and ultraviolet divergence are elucidated, with an emphasis on the importance of the correct treatment of angular momentum for collisions with small impact parameters, an aspect that is sometimes puzzling for students due to presentations in which the minimum impact parameter appears as something to be put into the theory “by hand." On the basis of the quantitative tools devised in this chapter, we will show that large stellar systems, such as elliptical galaxies, should be considered primarily as collisionless, while smaller systems, such as small globular clusters and open clusters, exhibit collisional behavior. These different regimes are rich in astrophysical consequences, both from the observational and the theoretical points of view.