Contributions to the elliptic restricted three-body problem

Abstract

Two contributions are given for the elliptic case of the restricted three-body problem of Celestial Mechanics. The first contribution consists in ascertaining analytically the eigenvalues about a Lagrangian triangular point as a power series of the eccentricity. This has entailed the determination of a coordinate transformation, whose coefficients are functions of the true anomaly, for the purpose of reducing the linear portion of the differential system to diagonal form. For each power of the eccentricity, a linear system of differential equations was found that yielded the coefficients of the coordinate transformation. The eigenvalues were calculated by imposing the absence of pure secular terms. The second contribution is the establishing of series solutions for the equations of motion about a triangular point by appropriately modifying a procedure devised by this author for the circular problem. Solutions are given in the form of double power series, whose coefficients (functions of the true anomaly) can be found recursively by solving linear differential systems. The two variables of the power series can be obtained in a similar way as solutions of an auxiliary nonlinear differential system of appropriate form. It was further required that the distance between the Lagrangian point and the initial point on the orbit be considered a small parameter.