A case of commensurability induced by oblateness

In the three-dimensional restricted three-body problem, it is known that there exists a near one-to-one commensurability ratio between the planar angular frequencies (s _{1, 2, 3}) and the corresponding angular frequency (S ₂) in thez-direction at the three collinear equilibria (L _{1, 2, 3}), which is significant for small and practically important values of the mass parameter (). When the more massive primary is treated as an oblate spheroid with its equatorial plane coincident with the plane of motion of the primaries, it is established that oblateness induces a one-to-one commensurability at the exterior pointL ₃ (to the right of the more massive primary) and at the interior pointL ₂ for 01/2 and that atL ₁ no such commensurability exists. However, the values of the oblateness coefficient (A ₁) involved atL ₂ are too high to have any practical significance, while those atL ₃ being small for small values of may be useful for generating periodic orbits of the third kind.