Analytic Central Orbits and their Transformation Group

Simple item page
dc.creatorLynden-Bell, Donald
dc.creatorJin, Shoko
dc.date2007-11-21
dc.date2008-05-18
dc.date.accessioned2026-07-07T09:39:12Z
dc.date.available2026-07-07T09:39:12Z
dc.descriptionA 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.
dc.description12 pages, 8 figures; updated version with minor typographical corrections; published in MNRAS
dc.identifierhttps://arxiv.org/abs/0711.3491
dc.identifierhttp://arxiv.org/abs/0711.3491
dc.identifier2008MNRAS.386..245L
dc.identifierdoi:10.1111/j.1365-2966.2008.13018.x
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/161084
dc.subjectAstrophysics
dc.titleAnalytic Central Orbits and their Transformation Group
dc.typetext

Files

Collections

arXiv