1.6: Summary

In this chapter, I have tried to lay the groundwork for the classical virial theorem by first demonstrating its utility, then deriving it in several ways and lastly, examining an important premise of its application. An underlying thread of continuity can be seen in all that follows comes from the Boltzmann transport Equation. It is a theme that will return again and again throughout this book. In section 1, we sketched how the Boltzmann transport Equation yields a set of conservation laws which in turn supply the basic structure Equations for stars. This sketch was far from exhaustive and intended primarily to show the informational complexity of this form of derivation. Being suitably impressed with this complexity, the reader should be in an agreeable frame of mind to consider alternative approaches to solving the vector differential Equations of structure in order to glean insight into the behavior of the system. The next two sections were concerned with a highly classical derivation of the virial theorem with section 2 being basically the derivation as it might have been presented a century ago. Section 3 merely updated this presentation so that the formalism may be used within the context of more contemporary field theory. The only 'tricky' part of these derivations involves the 'pairing' of forces. The reader should make every effort to understand or conceptualize how this occurs in order to understand the meaning of the virial itself. The assumption that the forces are derivable from a potential which is described by a power law of the distance alone, dates back at least to Jacobi and is often described as a homogeneous function of the distance.

In the last section, I attempted to provide some insight into the meaning of a very important theorem generally known as the ergodic theorem. Its importance for the application of the virial theorem cannot be too strongly emphasized. Although almost all systems of interest in stellar astrophysics can truly be regarded as ergodic, many systems in stellar dynamics cannot. If they are not, one cannot replace averages over time by averages over phase or the ensemble of particles without further justification.