Analytical study of a four-body configuration in exoplanet scenarios

Lately, more and more exoplanets are being discovered and the number is expected to increase thanks to the excellent programs dedicated to this task. Thus it is important to study the dynamical behavior of planetary systems. Among all the problems that may be considered, we are interested in four-body planetary systems. In this article, we first examine the possible cases in four-body planetary systems including satellites, in both single and multiple stars. Step decomposition is a methodology explained by A. Abad and J.A. Docobo to deal with n -body problems with hierarchy. The use of this technique permits the reduction of the different cases to five configurations. The analytical integration of one of these cases, a double star with a planet and a satellite orbiting one of the components of the stellar system, is the objective. First the Hamiltonian of the problem is given and then we formulate it in terms of three small parameters which can be reduced to two in this case. After that, using Hori’s biparametric method, the dependence on the angular variables is removed, thereby allowing the integration of the problem.