9 - The Collisionless Boltzmann Equation and the Jeans Theorem

Summary

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 R^6N 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.