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

Main Article Content

Jagadish Singh
Tyokyaa K. Richard

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.

Keywords:
Out-of-Plane, stability, libration points, ER3BP, Oblateness, Zonal Harmonics J4.

Article Details

How to Cite
Singh, J., & Richard, T. K. (2020). The Motion of a Test Particle around the Out-of-Plane Libration Points in the ER3BP with Oblateness up to Zonal Harmonics J_4. International Astronomy and Astrophysics Research Journal, 2(4), 1-15. Retrieved from https://journaliaarj.com/index.php/IAARJ/article/view/28
Section
Original Research Article

References

Singh J, Umar A, On the stability of triangular libration points in the elliptic R3BP under radiating and oblate primaries, Astrophys. Space Sci. 2012;341:349-358.

Singh J, Umar A. On out of plane libration points in the Elliptic restricted three-body problem with radiation and oblate primaries, Astrophys. Space Sci. 2013;344:13-19.

Singh J, Tyokyaa KR. Stability of triangular points in the elliptic restricted three- body problem with oblateness up to zonal harmonic J_4 of both primaries, Eur. Phys. J. Plus. 2016;131:365.

Singh J, Tyokyaa KR, Stability of collinear points in the elliptic restricted three- body problem with oblateness up to zonal harmonic J_4 of both primaries, Eur. Phys. J. Plus. 2017;132:330.

Bhatnagar KB, Hallan PP. Effect of perturbed potentials on the stability of libration points in the restricted problem, Celest. Mech. 1979;20:95-103.

Abdulraheem A, Singh J. Combined effects of perturbations, radiation and oblateness on the periodic orbits in the restricted three-body problem, Astrophys. Space Sci. 2006;317:9-13

Singh J, Mohammed HI. Robe’s circular restricted three-body problem under oblate and triaxial primaries EMP. 2012;109:1-11.

Singh J, Leke O. Libration points and stability in the restricted three-body problem with oblateness and variable masses, Astrophys. Space Sci. 2012;340:27-41.

Singh J, Leke O. “Effects of oblateness, perturbations, radiation and varying masses on the stability of libration points in the restricted three-body problem” Astrophysics and Space Science. 2013;344:51.

Singh J, Leke O. Analytic and numerical treatment of motion of dust grain particle around triangular libration points with post-AGB binary star and disc, Advances in Space research. 2014;54:1659-1677.

Subbarao PV, Sharma RK. A note on the stability of the triangular points of libration in the restricted three-body problem, Astron & Astrophys. 1975;43:381-383.

Ammar MK. Third-order secular solution of the variational equations of motion of a satellite in orbit around a non-spherical planet, Astrophys. Space Sci. 212;340:43.

Tsirogiannis GA, Douskos CN, Perdios EA. Computation of the liapunov orbits in the photogravitational RTBP with oblateness, Astrophys. Space Sci. 2006;305:389.

Sharma RK. The linear stability of libration points of the photogravitational restricted three-body problem when the smaller primary is an oblate spheroid, Astrophys. Space Sci. 1987;135:271.

Javed Idrisi M, Shahbaz Ullah M. Non-collinear libration points in ER3BP with albedo effect and oblateness. J. Astrophys. Astr. 2018;39:28.

Javed Idrisi M, Shahbaz Ullah M. A study of Albedo Effects on Libration points in the Elliptic Restricted Three-Body Problem. Astronautical Sciences; 2020.

Javed Idrisi M, Shahbaz Ullah M, Vinay Kumar. Elliptic restricted synchronous three-body problem (ERS3BP) with a mass dipole model, New Astronomy; 2020.

Radzievskii VV. Out-of-plane points in the Sun-Planet-Particle and Galaxy Kernel-Sun-Particle cases, Astron. Zh. 1950;27:250.

Das MK, Narang P, Mahajan S, Yuasa M. On out of plane equilibrium points in Photo-Gravitational restricted three-body problem, Astrophys. Astr. 2009;30:177-185.

Abouelmagd EI. Existence and stability of triangular points in the restricted three-body problem, Astrophys. Space Sci. 2012;342:45-53.

Singh J, Amuda TO. Out-of-Plane equilibrium points in the photogravitational circular restricted three-body problem with oblateness and P-R Drag, Astron. and Astrophy. 2015;36:291-305.

Singh J. Effects of the out-of-plane equilibrium points in the Coriolis and Centrifugal forces, Astrophys. Space Sci. 2012;342:303.

Ershkov SV. Yarkovsky effects in a modified photogravitational three-bodies, IJPAM. 2012;78:3.

Szebehely VG. Theory of Orbits. Academic Press, New York; 1967.

Danby JMA. Fundamentals of Celestial Mechanics. Second Edition, Revised & Enlarged. Willmann-Bell, Inc Publisher. 1988;115.

Douskos CN, Markellos VV. Out-of-Plane equilibrium points in the restricted three-body problem with oblateness, Astron. and Astrophy. 2006;466:357-362.