Journal Article

An analytical solution for Kepler's problem

András Pál

in Monthly Notices of the Royal Astronomical Society

Published on behalf of The Royal Astronomical Society

Volume 396, issue 3, pages 1737-1742
Published in print July 2009 | ISSN: 0035-8711
Published online June 2009 | e-ISSN: 1365-2966 | DOI: http://dx.doi.org/10.1111/j.1365-2966.2009.14853.x
An analytical solution for Kepler's problem

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In this paper, we present a framework which provides an analytical (i.e. infinitely differentiable) transformation between spatial coordinates and orbital elements for the solution of the gravitational two–body problem. The formalism omits all singular variables which otherwise would yield discontinuities. This method is based on two simple real functions for which the derivative rules are only required to be known, all other applications – e.g. calculating the orbital velocities, obtaining the partial derivatives of radial velocity curves with respect to the orbital elements – are thereafter straightforward. As it is shown, the presented formalism can be applied to find optimal instants for radial velocity measurements in transiting explanatory systems to constrain the orbital eccentricity as well as to detect secular variations in the eccentricity or in the longitude of periastron.

Keywords: methods: analytical; techniques: radial velocities; celestial mechanics; ephemerides

Journal Article.  3820 words.  Illustrated.

Subjects: Astronomy and Astrophysics

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