Characteristic exponents of the five equilibrium solutions in the elliptically restricted problem

Abstract

The characteristic roots and exponents for the five equilibrium solutions in the elliptically restricted problem of three bodies are found numerically with Floquet theory. The method is essentially the same as that employed by Moulton in his classical study of the eighth satellite of Jupiter and recently by Danby in his study of the stability of the triangular points in the elliptic problem. The full range of mass ratio is considered and eccentricities as large as 0.975 are handled satisfactorily. Numerical difficulties that arose in grossly unstable situations were overcome by employing known symmetries of the characteristic equation. For the triangular points the results provide a refinement and extension of Danby's transition curves. The unstable region is found to be divided into three parts. The roots are of a different type in each of these parts. For the collinear points, as could be expected, no region of variational stability was found. For any value of mass ratio, the magnitudes of the real and imaginary parts of the characteristic exponent always increase with larger eccentricity.