Abstract
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.
Highlights
In the literature, several authors considered the three-body problem to study the dynamics of triple systems, such as [1]-[7]
We investigate the dynamics of the three-body problem considering a triple system
We developed the disturbing potential in closed form up to the fourth-order in a small parameter
Summary
Several authors considered the three-body problem to study the dynamics of triple systems, such as [1]-[7]. Reference [6] presented a study where the authors took into account the octupole term in the disturbing potential. They showed that the inclination of the inner planet varies from prograde (i < 90◦) to retrograde (i > 90◦) for a specific problem, where i is the mutual inclination between the two orbits. After some algebraic manipulations we can show that the potential given by Eq (3) is in agreement with the ones shown in references [4, 9, 13, 17]; the potential given by Eq (4) is in agreement with the references [9] and [17] and the potential given by Eq (5) is in agreement with the reference [17]
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