The monotonicity of the apsidal angle using the theory of potential oscillators

TL;DR

This paper proves that in certain central force systems, the apsidal angle increases monotonically with energy or eccentricity, using potential oscillator theory, which enhances understanding of orbital dynamics.

Contribution

It establishes the monotonic relationship between the apsidal angle and energy for forces of specific power-law form, applying potential oscillator theory.

Findings

Apsidal angle is monotonic with energy for forces with (r)  r^{-(\u03b1+1)} and \u03b1<2.

The result applies to a class of central force systems relevant in celestial mechanics.

Provides a theoretical proof connecting orbital parameters and force laws.

Abstract

In a central force system the angle between two successive passages of a body through pericenters is called the apsidal angle. In this paper we prove that for central forces of the form $f(r)\sim \lambda r^{-(\alpha+1)}$ with $\alpha<2$ the apsidal angle is a monotonous function of the energy, or equivalently of the orbital eccentricity.