Stability of motion in the Sitnikov 3-body problem

We study the stability of motion in the 3-body Sitnikov problem, with the two equal mass primaries (m (1) = m (2) = 0.5) rotating in the x, y plane and vary the mass of the third particle, 0 <= m (3) < 10(-3), placed initially on the z-axis. We begin by finding for the restricted problem (with m (3) = 0) an apparently infinite sequence of stability intervals on the z-axis, whose width grows and tends to a fixed non-zero value, as we move away from z = 0. We then estimate the extent of "islands" of bounded motion in x, y, z space about these intervals and show that it also increases as |z| grows. Turning to the so-called extended Sitnikov problem, where the third particle moves only along the z-axis, we find that, as m (3) increases, the domain of allowed motion grows significantly and chaotic regions in phase space appear through a series of saddle-node bifurcations. Finally, we concentrate on the general 3-body problem and demonstrate that, for very small masses, m (3) approximate to 10(-6), the "islands" of bounded motion about the z-axis stability intervals are larger than the ones for m (3) = 0. Furthermore, as m (3) increases, it is the regions of bounded motion closest to z = 0 that disappear first, while the ones further away "disperse" at larger m (3) values, thus providing further evidence of an increasing stability of the motion away from the plane of the two primaries, as observed in the m (3) = 0 case.