Analytic Central Orbits and their Transformation Group

A useful crude approximation for Abelian functions is developed and applied to orbits. The bound orbits in the power-law potentials Ar take the simple form (l/r) = 1 + e cos(mφ), where k = 2 − α > 0 and l and e are generalisations of the semilatus-rectum and the eccentricity. m is given as a function of ‘eccentricity’. For nearly circular orbits m is √ 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 α → 0. For such orbits, r vibrates almost harmonically whatever the ‘eccentricity’. Unbound orbits in powerlaw 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, r, r− 4 3 and r. A degeneracy reduces this to three, which are r, r, and r for the Keplerian case. A generalisation of this group includes the isochrone with the Kepler set.