In this chapter we derive the basic equations for Jeans instability in the simplest formulation. This is meant to be the first example in the astrophysical context of the concepts introduced in Chapter 11. In reality, the study of stability brings us one step beyond the concept of dispersive waves. Most of what has been described in Chapter 11 is applicable in cases in which the frequency in the relevant dispersion relation is allowed to be complex, ω =ωR + iωI, provided that |ωI| ≪ |ωR|. But we should be aware that in the opposite limit we are no longer talking about waves in the usual sense.

In the simplest case, gravity and pressure forces are the main ingredients in the Jeans instability mechanism. Quite different results are derived from the same ingredients, depending on the model considered. In particular, although the fluid model is characterized by a dispersion relation for which neutral waves (ωI =0) are allowed, a kinetic analysis leads to a dispersion relation in which stable perturbations are generally damped. At the end of the chapter, we briefly examine the linearized equations for perturbations of a plane-parallel slab, demonstrating how, in general, the collisionless Boltzmann equation is solved by integration along the unperturbed characteristics.