A new perturbative solution to the motion around triangular Lagrangian points in the elliptic restricted three-body problem

The equations of motion of planar elliptic restricted three body problem are transformed to four decoupled Hill’s equations. By using the Floquet theorem analytic solution to the oscillator equations with time dependent periodic coefficients are presented. We show that the new analytic approach is valid for system parameters 0 < e ≤ 0.05 and 0 < μ ≤ 0.01 where e denotes the eccentricity of primaries while μ is the mass parameter, respectively. We also clarify the transformation details that provide the applicability of the method

A minimal dynamical model for tidal synchronization and orbit circularization

International audienceWe study tidal synchronization and orbit circularization in a minimal model that takes into account only the essential ingredients of tidal deformation and dissipation in the secondary body. In previous work we introduced the model (Escribano et al. in Phys. Rev. E, 78:036216, 2008); here we investigate in depth the complex dynamics that can arise from this simplest model of tidal synchronization and orbit circularization. We model an extended secondary body of mass by two point masses of mass /2 connected with a damped spring. This composite body moves in the gravitational field of a primary of mass ≫ located at the origin. In this simplest case oscillation and rotation of the secondary are assumed to take place in the plane of the Keplerian orbit. The gravitational interactions of both point masses with the primary are taken into account, but that between the point masses is neglected. We perform a Taylor expansion on the exact equations of motion to isolate and identify the different effects of tidal interactions. We compare both sets of equations and study the applicability of the approximations, in the presence of chaos. We introduce the resonance function as a resource to identify resonant states. The approximate equations of motion can account for both synchronization into the 1:1 spin-orbit resonance and the circularization of the orbit as the only true asymptotic attractors, together with the existence of relatively long-lived metastable orbits with the secondary in : ( and being co-prime integers) synchronous rotation