Triangular solution to the general relativistic three-body problem for general masses

Continuing work initiated in an earlier publication [T. Ichita, K. Yamada, and H. Asada, Phys. Rev. D 83, 084026 (2011)], we reexamine the post-Newtonian effects on Lagrange’s equilateral triangular solution for the three-body problem. For three finite masses, it is found that a triangular configuration satisfies the post-Newtonian equation of motion in general relativity if and only if it has the relativistic corrections to each side length. This post-Newtonian configuration for three finite masses is not always equilateral, and it recovers previous results for the restricted three-body problem when one mass goes to zero. For the same masses and angular velocity, the post-Newtonian triangular configuration is always smaller than the Newtonian one.

Figure 1

PN triangular configuration. Each mass is located at one of the apexes. $r_I\equiv |{\mathit{r}}_I|(I=1,2,3)$ denotes the orbital radius of each body. Each ${\epsilon }_{IJ}$ denotes the relativistic correction to each side length at the 1PN order. In the equilateral case, ${\epsilon }_{12}={\epsilon }_{23}={\epsilon }_{31}=0$, namely, $r_{12}=r_{23}=r_{31}=a$.