Analytic Central Orbits and their Transformation Group

TL;DR

This paper develops an approximation for Abelian functions to analyze orbits in power-law and logarithmic potentials, introduces a transformation group linking orbits across different potentials, and explores their properties and degeneracies.

Contribution

It presents a novel approximation for Abelian functions applied to orbital analysis and extends orbit transformations into a group connecting multiple potentials, including the isochrone.

Findings

Derived simple orbit forms for power-law potentials

Established a transformation group linking orbits in different potentials

Identified degeneracies reducing the potential set to three core cases

Abstract

A useful crude approximation for Abelian functions is developed and applied to orbits. The bound orbits in the power-law potentials A*r^{-alpha} take the simple form (l/r)^k = 1 + e cos(m*phi), where k = 2 - alpha > 0 and 'l' and 'e' are generalisations of the semi-latus-rectum and the eccentricity. 'm' is given as a function of 'eccentricity'. For nearly circular orbits 'm' is sqrt{k}, while the above orbit becomes exact at the energy of escape where 'e' is one and 'm' is 'k'. Orbits in the logarithmic potential that gives rise to a constant circular velocity are derived via the limit of small alpha. For such orbits, r^2 vibrates almost harmonically whatever the 'eccentricity'. Unbound orbits in power-law potentials are given in an appendix. The transformation of orbits in one potential to give orbits in a different potential is used to determine orbits in potentials that are positive powers of r. These transformations are extended to form a group which associates orbits in sets of six potentials, e.g. there are corresponding orbits in the potentials proportional to r, r^{-2/3}, r^{-3}, r^{-6}, r^{4/3} and r^{-4}. A degeneracy reduces this to three, which are r^{-1}, r^2 and r^{-4} for the Keplerian case. A generalisation of this group includes the isochrone with the Kepler set.