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.

Dynamical friction is a very interesting physical phenomenon, with important applications in astrophysics. At the simplest level, it can be described as the slowing down (“cooling”) of a test particle moving in a sea of field particles due to the cumulative effects of long-range interactions (no geometric collisions are considered). Several approaches have been devised to understand the underlying physics (which is intriguing, as the final result is an irreversible process produced by a time-reversible dynamics; e.g., see Bertin 2014; Binney and Tremaine 2008; Chandrasekhar 1960; Ogorodnikov 1965; Shu 1999; Spitzer 1987). In this chapter, the dynamical friction time is derived in the Chandrasekhar approach by using the impulsive approximation discussed in Chapter 7.

In this chapter, we discuss the complementary approach to that presented in Chapter 12 for the construction of stationary, multicomponent collisionless stellar systems. The Abel inversion theorem is introduced, and then a selection of density–potential pairs of spherical, axisymmetric, and triaxial shapes commonly used in modeling/observational works are presented. We finally discuss the solution of the Jeans equations for spherical and axisymmetric systems, and among other things we show how to compute the various quantities entering the virial theorem. For illustrative purposes, we use some of the derived results to investigate the possible physical interpretations of the fundamental plane of elliptical galaxies.

This chapter explores observations and properties of quasars, which were first observed in the 1960s as point-like sources that emit over a wide range of energies from the radio through the IR, visible, UV, and even extending to the X-ray and gamma-rays. They are now known to be a type of active galactic nucleus thought to be the result of matter accreting onto a supermassive black hole (SMBH) at the center of the host galaxy.

Earth’s Moon is quite distinct from other moons in the solar system, in being a comparable size to Earth. We explore the theory that a giant impact in the chaotic early solar system led to the Moon’s formation, and bombardment by ice-laden asteroids provided the abundant water we find on our planet. Further, we find that Earth’s magnetic field shields us from solar wind protons, that protect our atmosphere from being stripped away. The icy moons of Jupiter and Saturn are the best targets for exploring if life exists elsewhere in the solar system.

In our everyday experience, there is another way we sometimes infer distance, namely by the change in apparent brightness for objects that emit their own light, with some known power or luminosity. For example, a hundred watt light bulb at close distance appears a lot brighter than the same bulb from far away. Similarly, for a star, what we observe as apparent brightness is really a measure of the flux of light, i.e. energy emitted per unit time per unit area.

It turns out that stellar binary (and even triple and quadruple) systems are quite common. In Chapter 10 we show how we can infer the masses of stars, through the study of stellar binary systems. For some systems, where the inclination of orbits can be determined unambiguously, we can infer the masses of the stellar components, as well as the distance to the system. Together with the observed apparent magnitudes, this also gives the associated luminosities of their component stars.