The Motion of a Test Particle around the Out-of-Plane Libration Points in the ER3BP with Oblateness up to Zonal Harmonics J_4

Abstract

The study of the Elliptic Restricted Three-Body Problem (ER3BP) in our present work  considered the  coordinates taken oblateness up to zonal harmonics  for Gliese 667 and Sirius systems. The   coordinates considered here is termed; the out-of-plane libration points. The out-of-plane libration points is birthed from the existence of the three-dimsional Restricted Three-Body Problem (R3BP). Its points or positions are denoted by . These positions ( ) lie in the plane almost directly above and below the center of each oblate primary. We have computed numerically the positions of the out-of-plane libration points  and its stability to show the effects of the parameters involved. With the help of the software MATHEMATICA, we have observed the topologies of Zero-Velocity Curves (ZVC) for the stated problem. It is found that, the positions of the out-of-plane libration points for the binary systems: Gliese 667 and Sirius seems to respectively move away and closer in the absence and present of oblateness. For the stability, it is evidenced that, for each set of values, there exist at least one complex root with positive real part and hence in Lyapunov sense, the stability of the out-of-plane libration points are unstable for the binary systems mentioned above.