The continuation of periodic orbits from the restricted to the general three-body problem

Abstract

It is proved that a symmetric periodic orbit of the circular planar restricted three-body problem can be continued analytically, when the mass of the third body is small but not negligible, to a periodic motion of the general three-body problem in a rotating frame of reference whose origin coincides with the center of mass of the two bodies with large masses and itsx axis always contains these bodies. The two bodies with the large masses describe periodic motion on thex axis of the rotating frame while the third body, with the small mass, describes a symmetric periodic orbit in this frame. The motion of the two bodies lying on thex axis is always stable, whereas the periodic orbit of the third body in the rotating frame is stable or unstable depending on whether or not the nonzero characteristic exponents of the original periodic orbit of the restricted problem are of the stable or unstable type, respectively. It is also shown that for a fixed value of the small mass of the third body, a family of symmetric periodic orbits of the restricted problem can be continued analytically to a family of periodic motions of the general three-body problem.