Third-order stability of the long-period Trojan librations

Abstract

view Abstract Citations (25) References (1) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Third-order stability of the long-period Trojan librations Rabe, Eugene Abstract Nearly all existing studies of the stability properties of periodic solutions of the restricted problem of three bodies are based on linearized variation equations, so that the results are valid only for infinitesimally small displacements from the given periodic orbit. Since the normally nonperiodic Trojan-type librations in the sun-Jupiter plane, about either one of the two equilateral points of Lagrange, may be treated as pre- dominantly short-period oscillations about a suitable long-period reference solution, the question of the actual dependence of the librational stability on the magnitude of such superposed oscillations is a matter of considerable interest. The second- and third-order terms of the variation equations are therefore included in the present analysis of the stability of the long-period Trojan librations which are available since their earlier determination (Rabe 1961, 1962) by numerical methods. For any such reference libration, the first- order solution from the linearized variation equations involves a purely imaginary characteristic exponent ic, independent of the arbitrary oscillation amplitude. In the second-order increments to this solution, new short-period terms involving 2ic emerge, together with additional terms of the basic long period T, while the "stability constant" c, of order 12, remains unchanged and independent of the constants of integration. At the third-order stage of the solution, however, c becomes a function of the principal amplitude of the short- period oscillations which is nearly proportional to the helicentric eccentricity e of such a nonperiodic Trojan planet (the long-period reference Trojans have nearly circular orbits around the sun). The initial phase of the principal oscillation has a comparatively higher-order effect on c. For sufficiently small e, and for basic long-period libration amplitudes typical of the real Trojans, the oscillating librational motion remains stable, because the properly adjusted c (e) are found to remain real. For e values exceeding a certain critical em depending on the reference libration, however, instability is indicated by the resulting two conjugate complex values of c. This limiting stable eccentricity e is about 0.19 for very small basic librations, but decreases with the increasing size of the reference orbit. When the latter extends over about 430 in longitude, for example, has decreased to roughly 0.08. The stability limits indicated by the present results agree very well with the observed maximum eccentricities and libration amplitudes of the known Trojan planets. Publication: The Astronomical Journal Pub Date: February 1967 DOI: 10.1086/110196 Bibcode: 1967AJ.....72...10R full text sources ADS |