Abstract:
In this work we performed a study about the dynamical stability of co-orbital hypothetical planets (Trojans), around the Lagrangian points $L_{4}$ of known planets which form the compact extrasolar planetary systems Kepler-9 and Kepler-56. Using second order symplectic integrators, included in the package of subroutines Swift, and considering the Trojan planets as test particles (massless), we studied a vast grid of initial conditions for the Trojans in the semimajor axis versus eccentricity space of orbital parameters, in order to obtain conclusions of statistical character about its possible existence. The data analysis of the integrations is carried out by determining the survival time of the particles, their maximum eccentricities achieved and the deviation registered in both the semimajor axis and the eccentricity of the orbits of the Trojans, allowing to identify the regions of stable motion. Our work shows that the existence of Trojan planets is possible in both systems, at least from the dynamical point of view, although the size of the stable co-orbital zone is extremely dependent on the initial orbital parameters of the Trojans, the presence of the other planets forming the system and, especially, the occurrence of or near 2:1 mean motion resonances between some of the planets which make up the systems.
