Integrability of motion around galactic razor-thin disks

TL;DR

This paper investigates the motion of particles around galactic razor-thin disks, deriving an approximate third integral of motion based on adiabatic invariance, which predicts the vertical amplitude envelope as a function of surface density.

Contribution

It introduces a new approximate third integral of motion for particles around razor-thin disks, extending previous flattened configuration results to thin-disk models.

Findings

The vertical amplitude follows a $Z(R) \\propto [\\Sigma(R)]^{-1/3}$ relation.

The third integral accurately predicts orbit envelopes in various thin-disk models.

The approach confirms the applicability of adiabatic invariance in galactic disk dynamics.

Abstract

We consider the three-dimensional bounded motion of a test particle around razor-thin disk configurations, by focusing on the adiabatic invariance of the vertical action associated with disk-crossing orbits. We find that it leads to an approximate third integral of motion predicting envelopes of the form $Z(R)\propto[\Sigma(R)]^{-1/3}$, where $R$ is the radial galactocentric coordinate, $Z$ is the z-amplitude (vertical amplitude) of the orbit and $\Sigma$ represents the surface mass density of the thin disk. This third integral, which was previously formulated for the case of flattened 3D configurations, is tested for a variety of trajectories in different thin-disk models.