General relativistic effect on Sitnikov Three-Body Problem: Restricted Case

We investigate the effect of general relativity on the Sitnikov problem. The Sitnikov problem is one of the simplest three-body problems, in which the two primary bodies (a binary system) have equal mass m and orbit their barycenter, while the third body is treated as a test particle under Newtonian gravity. The trajectory of the test particle is perpendicular to the orbital plane of the binary and passes through the barycenter of the two primaries. To study the general relativistic contributions, we first derive the equations of motion for both the binary and the test particle based on the first post-Newtonian Einstein–Infeld–Hoffmann equation, and integrate these equations numerically. We examine the behavior of the test particle (third body) as a function of the orbital eccentricity of the central binary e, the dimensionless gravitational radius λ, which characterizes the strength of general relativistic effect, and the initial position of the test particle z0. Our numerical calculations reveal the following: (a) Due to general relativistic effect and increasing eccentricity of the central binary, the test particle tends to be ejected from the system. (b) When the binary’s eccentricity is small, it takes longer time for the test particle to be ejected; however, as the eccentricity increases, the particle tends to be ejected more quickly. (c) The escape velocity of the test particle tends to increase with the binary’s eccentricity. (d) When the binary’s eccentricity is small, the test particle is ejected farther from the orbital plane of binary; as the eccentricity increases, the particle tends to be ejected closer to the plane. And (e) Although the acceleration acting on the test particle can be seen to increase as the eccentricity and dimensionless gravitational radius become larger, overall, the acceleration acting when the test particle escapes from the system is generally zero.