Orbital evolution of planet around a binary star
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We study the secular dynamics of hierarchical (if there is a clearly defined binary and a third body which stays separate from the binary) triple systems composed by a Sun-like central star and a Jupiter-like planet, which are under the gravitational influence of a further perturbing star (brown dwarf). The main goal is to study the orbital evolution of the planet. In special, we investigate the orientation (inclination) and the shape (eccentricity) of its orbit. One key feature explored is the time needed for the first flip in its orientation (prograde to retrograde). The gravitational potential is developed in closed form up to the third order. We have compared the secular evolution of systems with and without the third order term of the disturbing potential. The R2 (quadrupole) and R3 (octupole) terms of the disturbing potential are developed without using the elimination of nodes. Numerical simulations were also performed to compare with the analytical model using the N-body simulations with the Mercury code. The results show that the analytical model are in agreement with the numeric simulations.