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Abstract: Two possible ways of dynamical evolution of the extra-solar planet γ Cephei are studied in dependence on the mutual orientation of the planet and the distant star orbits. The Lyapunov stability of the planet orbit is investigated. The evolution of the orbits when resonances are absent is described by the Hamiltonian without short-periodic terms, eliminated by von Zeipel's method.
The question about Lyapunov's stability of the extra-solar planets is considered in the frame of the general three-body problem, i.e. a planet in a binary system revolves around one of the component, and the distance between the star components is much greather than that between the orbiting star and the planet. The differential equations with regards to the eccentricity and the argument of the perigee of the planet, with this Hamiltonian allow to find conditions of the stability for t > t_{0}. The possible conditions of the stability of the extra-solar planet are presented by its orbital parameters -- the angle of the mutual inclination between the planet and the distant star orbits and the angular momentum of the system.
The initial conditions for the investigation of the dynamical evolution were taken from two different sources. In the first case from data of Hatzes et al. (2003) and Schneider (2005), in the second case from Neuhǎuser et al. (2007).
For the first case we obtained the conditions of the stability of the planetary orbit for some values of the longitude of the node laying in defined limits and the value of the mutual inclination. In the second case the angle of the mutual inclination has such a value that small deviations of the initial elements may become in future large for all values of the node. In this case the motion is unstable with respect to the eccentricity of the planetary orbit e_{1} (Chetaev, 1965). The results of the theory were verified by numerical integration.
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Last update: September 18, 2007