On the Orbit Structure of the Logarithmic Potential

TL;DR

This paper analytically explores the phase-space structure of the logarithmic galactic potential, focusing on the stability and bifurcations of key orbit families, and compares analytical predictions with numerical results.

Contribution

It introduces a resonant detuned normal form method based on Lie transforms to analyze orbit stability and bifurcations in the logarithmic potential.

Findings

Analytical expressions for stability thresholds and bifurcations.

Phase-space fractions of orbit families derived and validated.

Comparison with numerical results confirms analytical approach's accuracy.

Abstract

We investigate the dynamics in the logarithmic galactic potential with an analytical approach. The phase-space structure of the real system is approximated with resonant detuned normal forms constructed with the method based on the Lie transform. Attention is focused on the properties of the axial periodic orbits and of low order `boxlets' that play an important role in galactic models. Using energy and ellipticity as parameters, we find analytical expressions of several useful indicators, such as stability-instability thresholds, bifurcations and phase-space fractions of some orbit families and compare them with numerical results available in the literature.