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M. Efroimsky, M.A. Murison (U.S. Naval Observatory)
We can describe an arbitrary trajectory as a continuous family of instantaneous Kepler orbits tangent to this trajectory. Mathematically, such instantaneous Kepler orbits obey the so-called Lagrange constraint, and the six Keplerian parameters of such instantaneous orbits are called osculating elements.
Modeling of a trajectory with a non-tangent family of Kepler orbits violates the Lagrange constraint. Parameters of such non-tangent instantaneous ellipses or hyperbolae are called non-osculating elements. A set of such elements (i.e., a particular family of non-tangent instantaneous orbits describing the disturbed motion) can be singled out through fixing some supplementary condition different from the Lagrange constraint.
A certain set of non-osculating orbital elements was first employed by Goldreich (1965), addressing orbital motion about a precessing oblate planet. Goldreich studied alterations that frame wobble (considered as a perturbation) inflicts upon the Lagrange planetary equations. He demonstrated that the only alteration is the addition of a new term to the disturbing function. Efroimsky and Goldreich (2004) proved that with osculating elements this addition is insufficient and that extra terms will emerge in the planetary equations, terms that will not be parts of the disturbing function. Therefore, by applying Goldreich's result to osculating elements one will miss relevant terms. Hence, several problems involving satellite, ring, and galaxy dynamics may need to be reconsidered.
The problem, though, is in fact more generic. As proven by Efroimsky and Goldreich (2003), whenever the perturbation is described by a velocity-dependent addition to the Lagrangian (i.e., by a momentum-dependent addition to the Hamiltonian), the Hamilton-Jacobi procedure always leads to non-osculating elements. This subtlety is not visible with the naked eye and has remained long unappreciated.
In this communication we touch upon the problem of evolution of satellite orbits initially close to the equatorial plane of a wobbling planet.
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Bulletin of the American Astronomical Society, 36 #2
© 2004. The American Astronomical Soceity.