Simple, Accurate, Approximate Orbits in the Logarithmic and a Range of Power-Law Galactic Potentials

Abstract

Curves in a family derived from powers of the polar coordinate formula for ellipses are found to provide good fits to bound orbits in a range of power-law potentials. This range includes the well-known $1/r$ (Keplerian) and logarithmic potentials. These approximate orbits, called p-ellipses, retain some of the basic geometric properties of ellipses. They satisfy and generalize Newton's apsidal precession formula, which is one of the reasons for their surprising accuracy. Because of their simplicity the p-ellipses make very useful tools for studying trends among power-law potentials, and especially the occurence of closed orbits. The occurence of closed or nearly closed orbits in different potentials highlights the possibility of period resonances between precessing, eccentric orbits and circular orbits, or between the precession period of multi-lobed closed orbits and satellite periods. These orbits and their resonances promise to help illuminate a number of problems in galaxy and accretion disk dynamics.