existence and stability of triangular points in the restricted three-body problem with numerical applications

in this paper, we prove that the locations of the triangular points and their linear stability are affected by the oblateness of the more massive primary in the planar circular restricted three-body problem, considering the effect of oblateness for j₂ and j₄. after that, we show that the triangular points are stable for 0 < μ < μ_c and unstable when μ_c ≤ μ ≤ 1/2 , where μc is the critical mass parameter which depends on the coefficients of oblateness. on the other hand, we produce some numerical values for the positions of the triangular points, μ and μ_c using planets systems in our solar system which emphasis that the range of stability will decrease; however this range sometimes is not affected by the existence of j₄ for some planets systems as in earth–moon, saturn–phoebe and uranus–caliban systems.