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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.