The theory of the Trojan asteroids, part V

Abstract

E.W. Brown conjectured (1911) that the family of the long-periodic orbits in the Troian case of the restricted problem of three bodies terminates in an asymptotic orbit passing through the Lagrangian point L₃ at t=±∞. In 1977 the author showed that such an orbit deviates from L₃ by the epicyclic term mg (±∞). It is shown here that

$$g\left( { \pm \infty } \right) = 0,$$

so that the Brown conjecture regarding L₃ is false.

Contrary to what Brown believed, there is an entire family ofhomoclinic orbits, doubly asymptotic to short-periodic orbits around L₃. In the complex z-plane of the Poincaré eccentric variables, the latter orbits are circles of radius mR, with R bounded away from zero. The kinematics of the homoclinic family is investigated here in some detail.