Stability of motion in the Sitnikov 3-body problem

Abstract

We study the stability of motion in the 3-body Sitnikov problem, with the two equal mass primaries (m1 = m2 = 0.5) rotating in the x, y plane and vary the mass of the third particle, 0 ≤ m3 < 10−3, placed initially on the z-axis. We begin by finding for the restricted problem (with m3 = 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 m3 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, m3 ≈ 10−6, the “islands” of bounded motion about the z-axis stability intervals are larger than the ones for m3 = 0. Furthermore, as m3 increases, it is the regions of bounded motion closest to z = 0 that disappear first, while the ones further away “disperse” at larger m3 values, thus providing further evidence of an increasing stability of the motion away from the plane of the two primaries, as observed in the m3 = 0 case.