Stability of the triangular points in the elliptic restricted problem of three bodies

Abstract

A perturbation scheme is used to study the stability of infinitesimal motions about the triangular points in the elliptic restricted problem of three bodies. Fourth-order analytical expressions for the transition curves that separate stable from unstable orbits in the /z-e plane are given. These power series are recast into rational fractions to extend their validity to larger values of eccentricity, e. ANBY3 studied the linear stability of the triangular points numerically using Floquet theory. He presented transition curves that separate the stable from the unstable orbits in the ju-e plane (JJL is the ratio of the smaller primary to the sum of the masses of the two primaries, and e is the eccentricity of the primaries7 orbit) . These curves intersect the JJL axis at AIO and /z&, where jua = 0.03852 is the limiting value of IJL for stable orbits in the circular case, and Hb — 0.02859 is the value of IJL such that one of the periods of motion about the triangular points is exactly twice the period of the orbit of the primaries in the circular case. Bennet2 obtained a first-order analytical expression for the transition curves near jjib using an analytical technique for determination of characteristic exponents. Alfriend and Rand1 obtained second-order analytical expressions for the transition curves at ju a and /z& using the method of multiple scales.57 In this paper, we use a perturbation technique to determine fourth-order analytical expressions for the transition curves. We use this technique rather than the method of multiple scales because we are interested in determining the transition curves only. The amount of algebra involved is considerably less than that required if we use the method of multiple scales because the latter provides the solution in the whole n~e plane. The expansions obtained here are recast into rational fractions8 to extend their validity to larger values of e.