Analysis of the secular problem for triple star systems

The long-term dynamics of the three-body problem is studied. The goal is to study the motion of a planet (m1) around a star (m0) that is perturbed by a third-body (m2) (a planet or a brown dwarf star). The gravitational potential is developed in closed form up to the fourth order. Taking into account the triple system, it is shown here the evolution of some orbital parameters of the planet (m1). A comparison considering models with different orders for the disturbing potential is presented. We show that the behavior of the orbit of the inner planet can flip from prograde to retrograde trajectories. This is due to the third-order term, which strongly affects the eccentricity and inclination. We show that the effect of the fourth order term is to change the times when the phenomenon occurs