**35th Meeting of the AAS Division on Dynamical Astronomy, April 2004**

*Session 8 Techniques*

Oral, Friday, April 23, 2004, 9:30am-12:55pm,
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## [8.07] On the Orbital Evolution of Satellites near the Equator of a Precessing Planet

*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.