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The restricted (equilateral) four-body problem consists of three bodies of masses m 1 , m 2 and m 3 (called primaries) lying in a Lagrangian configuration of the three-body problem i.e., they remain fixed at the apices of an equilateral triangle in a rotating coordinate system. A massless fourth body moves under the Newtonian gravitation law due to the three primaries; as in the restricted three-body problem (R3BP), the fourth mass does not affect the motion of the three primaries. In this paper we explore symmetric periodic orbits of the restricted four-body problem (R4BP) for the case of two equal masses where they satisfy approximately the Routh's critical value. We will classify them in nine families of periodic orbits. We offer an exhaustive study of each family and the stability of each of them.

We consider a restricted four-body problem on the dynamics of a massless particle under the gravitational force produced by three mass points forming an equilateral triangle configuration. We assume that the mass m 3 of one primary is very small compared with the other two, m 1 and m 2 , and we study the Hamiltonian system describing the motion of the massless particle in a neighborhood of m 3 . In a similar way to Hill's approximation of the lunar problem, we perform a symplectic scaling, sending the two massive bodies to infinity, expanding the potential as a power series in m 1/3 3 , and taking the limit case when m 3 → 0. We show that the limiting Hamiltonian inherits dynamical features from both the restricted three-body problem and the restricted four-body problem. In particular, it extends the classical lunar Hill problem. We investigate the geometry of the Poincaré sections, direct and retrograde periodic orbits about m 3 , libration points, periodic orbits near libration points, their stable and unstable manifolds, and the corresponding homoclinic intersections. The motivation for this model is the study of the motion of a satellite near a jovian Trojan asteroid.

In this work we perform a numerical exploration of the families of planar periodic orbits in the Hill's approximation in the restricted four body problem, that is, after a symplectic scaling, two massive bodies are sent to infinity, by mean of expanding the potential as a power series in m 1/3 3 , (the mass of the third small primary) and taking the limit case when m 3 → 0. The limiting Hamiltonian depends on a parameter µ (the mass of the second primary) and possesses some dynamical features from both the classical restricted threebody problem and the restricted four-body problem. We explore the families of periodic orbits of the infinitesimal particle for some values of the mass parameter, these explorations show interesting properties regarding the periodic orbits for this problem, in particular for the Sun-Jupiter-asteroid case. We also offer details on the horizontal and vertical stability of each family.2010 Mathematics Subject Classification. 70F15, 70F16.

The restricted three-body problem posses the property that some classes of doubly asymptotic orbits are limits members of families of periodic orbits, this phenomena has been known as the "Blue Sky Catastrophe" termination. A similar case occurs in the restricted four body problem for the collinear equilibrium point named L2. We make an analytical and numerical study of the stable and unstable manifolds to verify that the hypothesis under which this phenomena occurs are satisfied.

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