Stability analysis of apsidal alignment in double-averaged restricted elliptic three-body problem

We are dealing with the averaged model used to study the secular effects in the motion of a body of the negligible mass in the context of a spatial restricted elliptic three-body problem. It admits a two-parameter family of equilibria (stationary solutions) corresponding to the motion of the third body in the plane of primaries’ motion, so that the apse line of the orbit of this body is aligned with the apse lines of the primaries’ orbits. The aim of our investigation is to analyze the stability of these equilibria. We show that they are stable in the linear approximation. The Arnold–Moser stability theorem provides sufficient conditions under which this means stability in a nonlinear sense too. These conditions are violated for parameters of the problem that belong to a set formed by a finite number of analytic curves in the parameters’ plane. As it turned out, in the system under consideration, violation of these conditions in some cases actually leads to an instability.