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 (along z-axis) 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 z¯0. Our numerical calculations reveal the following; as general relativistic effects λ increase and the eccentricity e of the binary orbit grows, the distance r¯ between the test particle and the primary star undergoes complicated oscillations over time. Consequently, the gravitational force acting on the test particle also varies in a complex manner. This leads to a resonance state between the position z¯ of the test particle and the distance r¯, causing the energy E of the test particle to become E≥0. This triggers the effective ejection of the test particle due to the gravitational slingshot effect. In this paper, we shall refer to this ejection mechanism of test particle as the “Sitnikov mechanism.” As a concrete phenomenon that becomes noticeable, the increase in general relativistic effects and the eccentricity of the binary orbit leads to the following: (a) ejection of test particles from the system in a shorter time, and (b) increasing escape velocity of the test particle from the system. As an astrophysical application, we point out that the high-velocity ejection of test particles induced by the Sitnikov mechanism could contribute to elucidating the formation processes of astrophysical jets and hyper-velocity stars.