A Study of the Orbits of the Logarithmic Potential for Galaxies

TL;DR

This paper refines the understanding of galaxy dynamics within the logarithmic potential by correcting previous work, analyzing orbital equations with advanced methods, and exploring gravitational lensing implications.

Contribution

It provides small corrections to prior orbital analyses, applies the p-ellipse method to the radial orbital equation, and discusses gravitational lensing calculations using special functions.

Findings

Refined periodic orbit and bifurcation analysis.

Analytical and numerical determination of the apsidal angle.

Application of Lambert W and Polylogarithm functions to lensing computations.

Abstract

The logarithmic potential is of great interest and relevance in the study of the dynamics of galaxies. Some small corrections to the work of Contopoulos & Seimenis (1990) who used the method of Prendergast (1982) to find periodic orbits and bifurcations within such a potential are presented. The solution of the orbital radial equation for the purely radial logarithmic potential is then considered using the p-ellipse (precessing ellipse) method pioneered by Struck (2006). This differential orbital equation is a special case of the generalized Burgers equation. The apsidal angle is also determined, both numerically as well as analytically by means of the Lambert W and the Polylogarithm functions. The use of these functions in computing the gravitational lensing produced by logarithmic potentials is discussed.