Modes of a stellar system II: non-ergodic systems

TL;DR

This paper extends the theory of van Kampen modes to non-ergodic stellar systems with distribution functions depending on actions, providing a framework to analyze their disturbances and dynamics.

Contribution

It derives an energy equation for disturbances in non-ergodic systems and generalizes van Kampen mode theory to such systems, enabling better understanding of galaxy and star cluster dynamics.

Findings

Van Kampen modes with different frequencies are orthogonal.

Energies of van Kampen modes are additive.

Insights from ergodic systems apply to real clusters and galaxies.

Abstract

An equation is derived for the energy of a small disturbance in a system that is generated by a distribution function (DF) of the form $f(\vJ)$ -- most galaxies and star clusters can be closely approximated by such a DF. The theory of van Kampen modes is extended to such general systems. An inner product on the space of DFs is defined such that the energy of a disturbance is its norm under this product. It is shown that van Kampen modes that differ in frequency are then orthogonal, with the consequence that the energies of van Kampen modes are additive. Consequently, most of the insight into the dynamics of ergodic systems that was gained in a recent paper on the van Kampen modes of ergodic systems applies to real clusters and galaxies.