Linear Stability of the Elliptic Lagrangian Triangle Solutions in the Three-Body Problem

This paper concerns the linear stability of the well-known periodic orbits of Lagrange in the three-body problem. Given any three masses, there exists a family of periodic solutions for which each body is at the vertex of an equilateral triangle and travels along an elliptic Kepler orbit. Reductions are performed to derive equations which determine the linear stability of the periodic solutions. These equations depend on two parameters – the eccentricity e of the orbit and the mass parameter β=27(m1m2+m1m3+m2m3)/(m1+m2+m3)2. A combination of numerical and analytic methods is used to find the regions of stability in the βe-plane. In particular, using perturbation techniques it is rigorously proven that there are mass values where the truly elliptic orbits are linearly stable even though the circular orbits are not.