Theory of the Trojan asteroids Part II

In the previously published (1977a) Part I of the paper, the author has constructed a formal long-periodic solution for the case of 1:1 resonance in the restricted problem of three bodies. Here the accuracy of the solution is carried fromO(m) toO(m ^3/2), wherem is the mass parameter of the system.Asymptotic approximations for the period of the motion are obtained for the case of small oscillations about the Lagrangian pointL ₄, in agreement with the classical theory, and for the vicinity of a logarithmic singularity on themean separatrix, passing throughL ₃. The regularizing function (), which removes the singularities of the Poincaré type, is extended to all orders, and the result is used to prove the periodicity of the solution.