**DPS 35th Meeting, 1-6 September 2003**

*Session 46. Other Dynamics*

Oral, Chair: N. J. Rappaport, Saturday, September 6, 2003, 11:20am-12:00noon, DeAnza III
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## [46.04] Generalized Lagrangian Points: Studies of Resonance for Highly Eccentric Orbits

*M. Pan, R. Sari (Caltech)*

A number of dynamically interesting aspects of the solar
system---including, for example, the scattered Kuiper Belt
objects and the creation of the Oort cloud---involve small
bodies on eccentric orbits which are perturbed by a dominant
major planet. Due to the sizeable eccentricities involved,
these situations are hard to study using the usual
eccentricity expansion of the disturbing function applied to
the restricted circular three-body problem.

As an alternate approach, we develop a framework based on
energy kicks for the evolution of high-eccentricity
long-period orbits in the restricted circular planar
three-body problem with Jacobi constant close to 3 and with
secondary to primary mass ratio \mu\ll 1. We use this
framework to explore mean-motion resonances between the test
particle and the massive bodies. This approach leads to a
redefinition of resonance orders for the high-eccentricity
regime in which a p:p+q resonance is called `pth order'
instead of the usual `qth order' to reflect the importance
of interactions at periapse. This approach also produces a
pendulum-like equation describing the librations of
resonance orbits about fixed points which correspond to
periodic trajectories in the rotating frame. A striking
analogy exists between these new fixed points and the
Lagrangian points as well as between librations around the
fixed points and the well-known tadpole and horseshoe
orbits; we call the new fixed points the `generalized
Lagrangian points'. Finally, our approach gives a condition
a~\mu^{-2/5} for the onset of chaos at large semimajor
axis a; a range \mu< ~5\times 10^{-6} in secondary
mass for which a test particle initially close to the
secondary cannot escape from the system, at least in the
planar problem; and a simple explanation for the presence of
asymmetric librations in exterior 1:N resonances and the
absence of these librations in other exterior resonances.

The author(s) of this abstract have provided an email address
for comments about the abstract:
mpan@astro.caltech.edu

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Bulletin of the American Astronomical Society, **35** #4

© 2003. The American Astronomical Soceity.