2007/11/30 by Donald Lynden-Bell, D. Lynden-Bell, S. Jin +1 · 1 citation
Engineering · Physics and Astronomy · #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Spacecraft Dynamics and Control #astro-ph
paper · pdf · doi:10.1111/j.1365-2966.2008.13018.x
published as 2008MNRAS.386..245L · 12 pages, 8 figures; updated version with minor typographical corrections; published in MNRAS
openalex publication_date 2008/03/11 · arxiv created 2008/05/18 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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 (mφ), 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.