Reflection symmetries of Isolated Self-consistent Stellar Systems

TL;DR

This paper proves that certain symmetry properties of galaxy models depend on their distribution functions, showing spherical symmetry for energy-only models and reflection symmetry for models depending on energy and angular momentum.

Contribution

It provides a new proof of Lichtenstein's Theorem and introduces a general Symmetry Theorem linking distribution function symmetries to galaxy reflection symmetries.

Findings

Energy-only distribution functions imply spherical symmetry.

Distribution functions depending on energy and angular momentum imply reflection symmetry.

The results are derived using elliptic PDE symmetry methods.

Abstract

Isolated, steady-state galaxies correspond to equilibrium solutions of the Poisson--Vlasov system. We show that (i) all galaxies with a distribution function depending on energy alone must be spherically symmetric and (ii) all axisymmetric galaxies with a distribution function depending on energy and the angular momentum component parallel to the symmetry axis must also be reflection-symmetric about the plane $z=0$. The former result is Lichtenstein's Theorem, derived here by a method exploiting symmetries of solutions of elliptic partial differential equations, while the latter result is new. These results are subsumed into the Symmetry Theorem, which specifies how the symmetries of the distribution function in configuration or velocity space can control the planes of reflection symmetries of the ensuing stellar system.