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 A r^-α take the simple form (l/r)^k = 1 + e cos (m φ), where k = 2 - α > 0 and l and e are generalizations of the semi-latus-rectum and the eccentricity. m is given as a function of `eccentricity'. For nearly circular orbits m is , while the above orbit becomes exact at the energy of escape where e is 1 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, r² 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 r⁴. A degeneracy reduces this to three, which are r^-1, r² and r^-4 for the Keplerian case. A generalization of this group includes the isochrone with the Kepler set.