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.