On the stability of circular orbits in galactic dynamics: Newtonian thin disks

TL;DR

This paper establishes a new vertical stability criterion for circular orbits in galactic thin disk models, correcting previous misconceptions and providing analytical tools for orbit analysis in such systems.

Contribution

It introduces a rigorous vertical stability criterion based on surface density and develops an approximate third integral for nearly equatorial orbits.

Findings

Vertical stability depends on positive surface density.

The criterion corrects previous assumptions based on epicyclic frequency.

An approximate third integral describes orbit shapes in the meridional plane.

Abstract

The study of off-equatorial orbits in razor-thin disks is still in its beginnings. Contrary to what was presented in the literature in recent publications, the vertical stability criterion for equatorial circular orbits cannot be based on the vertical epicyclic frequency, because of the discontinuity in the gravitational field on the equatorial plane. We present a rigorous criterion for the vertical stability of circular orbits in systems composed by a razor-thin disk surrounded by a smooth axially symmetric distribution of matter, the latter representing additional structures such as thick disk, bulge and (dark matter) halo. This criterion is satisfied once the mass surface density of the thin disk is positive. Qualitative and quantitative analyses of nearly equatorial orbits are presented. In particular, the analysis of nearly equatorial orbits allows us to construct an approximate analytical third integral of motion in this region of phase-space, which describes the shape of these orbits in the meridional plane.