Evolution of the orbit in the double-averaged planar restricted three-body problem

The double-averaged planar three-body problem is brought to one degree of freedom. For the restricted case, the double-average Hamiltonian is independent of mean anomalies and, therefore, represents an integral of motion. This integral gives the complicated implicit relation between eccentricity e and the distance d of pericenters. The evolution of the orbit is best expressed in the x, y plane for which e, d are polar coordinates. The explicit calculations are carried out up to the fourth degree in eccentricities while the ratio sigma of the semi-major axes is taken into account in all degrees, asuming sigma less than one but not necessarily much less than one. The trajectories in the x, y plane are then ellipses. The center of each ellipse depends on energy. The ellipse degenerating into one point represents the libration point whose x coordinate may be interpreted as the forced eccentricity. The dependence of the parameters on sigma is investigated.