The Eccentric Frame Decomposition of Central Force Fields

TL;DR

This paper introduces a new eccentric frame decomposition method for analyzing particle motion in central force fields, providing analytical solutions and applying it to galaxy dark matter models.

Contribution

It presents a novel eccentric frame approach that simplifies the analysis of central force motion, including non-integrable cases, and applies it to astrophysical potentials.

Findings

Analytical description of rosette-shaped orbits in eccentric frames

Classification of motion as circulation or libration in the eccentric frame

Application to Hernquist-Newton potential for galaxy modeling

Abstract

The rosette-shaped motion of a particle in a central force field is known to be classically solvable by quadratures. We present a new approach of describing and characterizing such motion based on the eccentricity vector of the two body problem. In general, this vector is not an integral of motion. However, the orbital motion, when viewed from the nonuniformly rotating frame defined by the orientation of the eccentricity vector, can be solved analytically and will either be a closed periodic circulation or libration. The motion with respect to inertial space is then given by integrating the argument of periapsis with respect to time. Finally we will apply the decomposition to a modern central potential, the spherical Hernquist-Newton potential, which models dark matter halos of galaxies with central black holes.