Abstract
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 ≈ 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.
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Soulis, P., Bountis, T. & Dvorak, R. Stability of motion in the Sitnikov 3-body problem. Celestial Mech Dyn Astr 99, 129–148 (2007). https://doi.org/10.1007/s10569-007-9093-6
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DOI: https://doi.org/10.1007/s10569-007-9093-6