Limiting Eccentricities for Stable Trojan Librations.
Eugene K. Rabe · The Astronomical Journal · 1965
None of the known Troj an planets has an orbital eccentricity e exceeding 0.15, in contrast to a corresponding limit of about 0.54 for "ordinary" minor planets. Since some numerical integrations of Troj an traj ectories, in the rotating frame of the restricted problem of three bodies, also suggest the existence of certain stability limits for the magnitude of the free short-period terms and thus of the heliocentric eccentricities, a high-order stability analysis seemed desirable. The earlier first-order results from Hill's equation are valid only for very small oscillations about the basic reference librations of long period (Rabe, Astron. J. 66, 500; 67, 382). If the third-order powers and products of the displacements from the periodic reference orbit are considered in the differential equations for these superimposed fluctuations, the characteristic exponent ic becomes a function of the arbitrary amplitude a of the principal short-period term and thus of the nearly proportional eccentricity e. Complex values of c and thus instability are obtained when e exceeds a certain limiting value emax, which is found to be approximately 0.21 for very small reference librations, and to decrease with the increasing size of the basic long-period libration, as illustrated in the following tabulation for three such reference solutions: d0 A e~ax 1.0001 0?22 0.21 1.01 21.35 0.16 1.02 43.06 0.08 Here d0 denotes a parameter (Rabe, loc. cit.) closely related to the total longitudinal amplitude A of the reference libration. Only two known Trojans have A values exceeding 430 If the more realistic elliptic restricted problem is considered, the listed values of emax may still be increased by the approximate amount 0.05 of Jupiter's orbital eccentricity, to the first order of accuracy. While fourth- and higher-order terms have been omitted in this analysis, it is remarkable how well the here established theoretical limits for stable eccentricities account for the observed maximum of 0.15. This work was supported by the National Aeronautics and Space Administration.