Abstract
We examine the stability of the triangular Lagrange points L 4 and L 5 for secondary masses larger than the Gascheau’s value \({\mu_{\rm G}= (1-\sqrt{23/27}/2)= 0.0385208\ldots}\) (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 \({\mu \approx 0.039}\), 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 \({\mu \ge \mu_G}\). We show that μ G is actually the first value of a series \({\mu_0 (=\mu_G), \mu_1,\ldots, \mu_i,\ldots}\) 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 \({\mu_\infty = 0.0463004\ldots}\) which generalizes Gascheau’s work.
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Sicardy, B. Stability of the triangular Lagrange points beyond Gascheau’s value. Celest Mech Dyn Astr 107, 145–155 (2010). https://doi.org/10.1007/s10569-010-9259-5
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DOI: https://doi.org/10.1007/s10569-010-9259-5