Post-Newtonian effects on Lagrange's equilateral triangular solution for the three-body problem

Continuing work initiated in earlier publications [Yamada, Asada, Phys. Rev. D 82, 104019 (2010), 83, 024040 (2011)], we investigate the post-Newtonian effects on Lagrange's equilateral triangular solution for the three-body problem. For three finite masses, it is found that the equilateral triangular configuration satisfies the post-Newtonian equation of motion in general relativity, if and only if all three masses are equal. When a test mass is included, the equilateral configuration is possible for two cases: (1) one mass is finite and the other two are zero, or (2) two of the masses are finite and equal, and the third one is zero, namely a symmetric binary with a test mass. The angular velocity of the post-Newtonian equilateral triangular configuration is always smaller than the Newtonian one, provided that the masses and the side length are the same.