Effect of oblateness and radiation pressure on angular frequencies at collinear points

In the three-dimensional restricted three-body problem, by considering the more massive primary as an oblate spheroid with its equatorial plane coincident with the plane of motion as well as source of radiation, it is found that the collinear point L ₁ comes nearer to the primaries with the increase in oblateness and radiation pressure, while L ₂ and L ₃ move away from the more massive primary with the increase in oblateness and come nearer to it with the increase in radiation pressure. It is noted that the angular frequency s ₁ at L ₁ increases with oblateness as well as with radiation pressure. s ₂ increases with oblateness and decreases with radiation pressure and s ₃ decreases with oblateness and increases with radiation pressure. A study on the norms of the characteristic roots λ and s at L ₁, L ₂ and L ₃ is carried out. It is established that for certain oblateness and radiation pressure parameters there is a one-to-one commensurability at the collinear points L ₂, L ₃ between the planar angular frequencies (s _2,3) and the corresponding angular frequency (s _z ) in the z-direction, and that at L ₁ no such commensurability exists. At L ₂ and L ₃, the value of oblateness parameter providing the commensurability decreases with the increase in the radiation pressure. However, the commensurable angular frequencies and eccentricity of the periodic orbits decrease at L ₂ and increase at L ₃, with the increase in the radiation pressure.