Stability of motion in the Sitnikov 3-body problem

Details Stability of motion in the Sitnikov 3-body problem Stability of motion in the Sitnikov 3-body problem

Oct 2007

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2007cemda..99..129s&link_type=abstract

Celestial Mechanics and Dynamical Astronomy, Volume 99, Issue 2, pp.129-148

Astronomy and Astrophysics

Astronomy

8

Restricted 3-Body Sitnikov Problem, Extended And General Sitnikov Problem, Stability Intervals, &Ldquo, Islands&Rdquo, Of Bounded Motion

Scientific paper

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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