Stability of the triangular Lagrange points beyond Gascheau's value

Details Stability of the triangular Lagrange points beyond Gascheau's value Stability of the triangular Lagrange points beyond Gascheau's value

Jun 2010

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2010cemda.107..145s&link_type=abstract

Celestial Mechanics and Dynamical Astronomy, Volume 107, Issue 1-2, pp. 145-155

Astronomy and Astrophysics

Astronomy

2

Three-Body Problem, Lagrange Points, Global Stability, Routh’S Critical Mass Ratio, Feigenbaum Cascade

Scientific paper

We examine the stability of the triangular Lagrange points L 4 and L 5 for secondary masses larger than the Gascheau’s value {μ_G= (1-sqrt{23/27}/2)= 0.0385208ldots} (also known as the Routh value) in the restricted, planar circular three-body problem. Above that limit the triangular Lagrange points are linearly unstable. Here we show that between μ G and {μ ≈ 0.039}, the L 4 and L 5 points are globally stable in the sense that a particle released at those points at zero velocity (in the corotating frame) remains in the vicinity of those points for an indefinite time. We also show that there exists a family of stable periodic orbits surrounding L 4 or L 5 for {μ ge μ_G}. We show that μ G is actually the first value of a series {μ_0 (=μ_G), μ_1,ldots, μ_i,ldots} corresponding to successive period doublings of the orbits, which exhibit {1, 2, ldots, 2^i,ldots} cycles around L 4 or L 5. Those orbits follow a Feigenbaum cascade leading to disappearance into chaos at a value {μ_infty = 0.0463004ldots} which generalizes Gascheau’s work.

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