**DDA2001, April2001**

*Session 4. Posters*

Monday, 8:00pm
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## [4.15] (n:1) Resonances and Stability of Restricted Three-Body Systems

*N. Haghighipour (Northwestern University)*

Numerical integrations of P-type binary planetary systems
have shown instabilities at (n:1), 3 < n < 9,
commensurabilities between the orbital period of the planet
and the period of the binary (Holman and Wiegert, 1999). It
has also been shown that when some dissipative force is
introduced to the system, the system can be temporarily
captured in an (n:1) resonance during which the planet
migrates outwards and its orbital eccentricity undergoes
drastic changes (Haghighipour, 2000). Presentation of an
analytical treatment of the dynamical evolution of such
systems is the purpose of this article.

In a quest for an analytical proof of stability or
instability of the P-type binary planetary systems above,
dynamics of a restricted three-body system at resonance is
studied. The method of partial averaging near a resonance is
employed to show how the system evolves while captured in a
resonance. The first order partially averaged system at
resonance is shown to be a pendulum-like equation whose
dynamics resembles the long term evolution of the main
planetary system. Studying this averaged system, I will
present analytical arguments on how (n:1) resonances are
established and will examine the stability of the main
planetary system at (1:1), (2:1) and (3:1) resonances for
different values of the masses of the bodies and their
initial velocities and distances. Also, the relationship
between the orbital parameters of the planet and the binary
for establishing a stable system at those resonances are
discussed.

Holman M.J. and Wiegert P.A., 1999, AJ, 117, 621.

Haghighipour N., 2000, MNRAS, 316, 845.

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