Hamilton, D.P. 1994. A comparison of Lorentz, planetary
gravitational, and satellite gravitational resonances.
Icarus 109, 221-240.
We consider a charged dust grain whose orbital motion is dominated by
a planet's point-source gravity, but perturbed by higher-order terms
in the planet's gravity field as well as by the Lorentz force arising
from an asymmetric planetary magnetic field. Perturbations to
Keplerian orbits due to a non-spherical gravity field are expressed in
the traditional way: in terms of a disturbing function which can be
expanded in a series of spherical harmonics (Kaula 1966). In order to
calculate the electromagnetic perturbation, we first write the Lorentz
force in terms of the orbital elements and then substitute it into
Gauss' perturbation equations. This procedure is analogous to the
derivation of gravitational disturbing functions, except, since the
Lorentz force has no associated potential, the perturbation of each
orbital element must be calculated separately. We use our result to
derive strengths of Lorentz resonances and elucidate their properties.
In particular, we compare Lorentz resonances to two types of
gravitational resonances: those arising from periodic tugs of a
satellite and those due to the attraction of an arbitrarily-shaped
planet.
We find that Lorentz resonances share numerous properties with their
gravitational counterparts and show, using simple physical arguments,
that several of these patterns are fundamental, applying not only to
our expansions, but to all quantities expressed in terms of orbital
elements. Some of these patterns have been previously called
``d'Alembert rules'' for satellite resonances. Other similarities
arise because, to first-order in the perturbing force, the three
problems share an integral of the motion. Yet there are differences
too; for example, first-order inclination resonances exist for
perturbations arising from planetary gravity and from the Lorentz
force, but not for those due to an orbiting satellite. Finally, we
provide a heuristic treatment of a particle's orbital evolution under
the influence of drag and resonant forces. Particles brought into
mean-motion resonances experience either trapping or resonant
``jumps,'' depending on the direction from which the resonance is
approached. We show that this behavior does not depend on the details
of the perturbing force but rather is fundamental to all mean-motion
resonances.
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