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Stability of motion in the Sitnikov 3-body problem. (English) Zbl 1162.70312

Summary: 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, \(0 \leq m_3 < 10^{-3}\), 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, \(m_3 \approx 10^{-6}\), 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.

MSC:

70F07 Three-body problems
70K20 Stability for nonlinear problems in mechanics
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