# EQUILIBRIUM POINTS IN THE CR3BP OF THREE OBLATE BODIES UNDER THE EFFECTS OF CIRCUMBINARY DISC AND RADIATING PRIMARIES WITH POYNTING-ROBERTSON DRAG

### Abstract

We study numerically the generalized planar photogravitational circular restricted three-body problem, where an infinitesimal body is moving under the Newtonian gravitational attraction of two bodies which are finite, moving in circles around their center of mass fixed at the origin of the coordinate system, where both bodies are situated on the horizontal -axis. The third body is significantly smaller compared to the masses of the two bodies (primaries) where its influence on them can be neglected. The three participating bodies are modeled as oblate spheroids, under effect of radiation of the two main masses together with effective Poynting-Robertson drag and both of them are enclosed by a belt of homogeneous circular cluster of material points. In this paper, the existence and location of the equilibrium points and their linear stability are explored for various combinations of the model’s parameters. We observe that under constant P–R drag effect, collinear equilibrium solutions cease to exist but there are in the absence of the drag forces. We found that five or seven non-collinear equilibrium points may lie on the plane of primaries motion depends on the particular values of model’s parameters, and it is seen that the perturbing forces have significant effects on their positions and linear stability. In our model, the binary system Kruger 60 is used, and it is found that the positions of the equilibria and their stability are affected by these perturbing forces. In the case where seven critical points exist, all the equilibria are...

### References

Abouelmagd, E. I , Asiri, H. M. and Sharaf , M. A., (2013). Effect of oblateness in the perturbed restricted three-body problem,” Meccanica. 48, 2479–2490.

Capdevila, L.R and Howell, K.C. (2018). A transfer network linking Earth, Moon, and the triangular libration point regions in the Earth-Moon system. Adv. Space. Res. 62, 1826–1852.

Chernikov, J. A. (1970). The photogravitational restricted three-body problem. Sov. Astron. 14, 176–181.

Gao, F. and Wang, R. (2020). Bifurcation analysis and periodic solutions of the HD 191408 system with triaxial and radiative perturbations. Universe 6, 35.

Gyegwe, J. M., Vincent, A. E. and Perdiou, A.E. (2022). On the stability of the triangular equilibrium points in the photogravitational R3BP with an oblate infinitesimal and triaxial primaries for the binary Lalande 21258 system, in Approximation and Computation in Science and Engineering, ed. by Th. M. Rassias and P. Pardalos. Springer Optim. Its Appl. 180 (Springer, Cham,) 397—415.

Jiang, I. G. and Yeh, L. C. (2004). The drag-induced resonant capture for Kuiper Belt objects. Mon. Not. R. Astron. Soc. 355, 29–32.

Jiang, I. G. and Yeh, L. C. (2006). On the Chermnykh-like problem: The equilibrium points, Astrophys. Space Sci. 305, 341–345.

Kalantonis, V. S., Perdios, E. A. and Perdiou, A. E. (2008). The Sitnikov family and the associated families of 3D periodic orbits in the photogravitational RTBP with oblateness. Astrophys. Space Sci. 315, 323–334.

Kalantonis, V. S., Vincent, A. E., Gyegwe, J. M., and Perdios, E. A. (2021). “Periodic Solutions Around the Out-Of-Plane Equilibrium Points in the Restricted Three-Body Problem with Radiation and Angular Velocity Variation” in Nonlinear Analysis and Global Optimization. Editors Th. M. Rassias and P. M. Pardalos (Cham: Springer), 251–275. Springer Optimization and Its Applications. doi:10.1007/978-3-030-61732-5_11

Kishor, R. and Kushvah, B. S. (2013). Linear stability and resonances in the generalized photogravitational Chermnykh-like problem with a disc. Mon. Not. R. Astron. Soc. 436, 1741–1749.

Leke,O. and Singh, J. (2021). Exploring Effect of Perturbing Forces on Periodic Orbits in the Restricted Problem of Three Oblate Spheroids with Cluster of Material Points. Int. Astron. and Astrophy. Research J., 2(4), 48-73.

Lhotka , C. and Celletti, A. (2015).The effect of Poynting-Robertson drag on the triangular Lagrangian points. Icarus. 250, 249–261

Orbeti, P. and Vienne, A. (2003). An upgrade theory for Helene Telesto, and Calypso. Astron. Astrophy. 397: 353—359.

Papadakis, K.E. (1996). Families of periodic orbits in the photogravitational three-body problem, AP&SS, 245, 1—13

Papadakis, K.E. (1995). The geometry of the Roche coordinates and zero-velocity curves in the photogravitational three-body problem, Astrophys. Space Sci. 232, 337–354

Papadakis, K.E. (2006). The planar photogravitational Hill problem. Int. J. Bifurcat. Chaos, 16, 1809–1821

Papadakis, K., Ragos, O., and Litzerinos C. (2009). Asymmetric periodic orbits in the photogravitational Copenhagen problem. Journal of Computational and Applied Mathematics 227, 102-114

Pal, A. K. and Kushvah, B.S. (2015). Geometry of halo and Lissajous orbits in the circular restricted three-body problem with drag forces. Mon Not R Astron Soc..46(1), 959–72.

Perdios, E.A., Kalantonis, V.S., Perdiou, A.E., Nikaki, A.A., (2015). Equilibrium points and related periodic motions in the restricted three- body problem with angular velocity and radiation effects. Adv. Astron. 2015, 473–483.

Poynting, J. H. (1903). Radiation in the solar system: its effect on temperature and its pressure on small bodies. Philos. Trans. Roy. Soc. London 202, 525–552.

Radzievskii , V.V. (1950). The restricted problem of three bodies taking account of light pressure. Astron. Zh. 27, 250–256.

Radzievskii, V.V. (1953). The space photogravitational restricted three-body problem. Astron. Zh.. 30, 25–273.

Ragos, O. and Zafiropoulos, F. A. (1995). A numerical study of the influence of the Poynting-Robertson effect on the equilibrium points of the photogravitational restricted three– body problem. I. Coplanar case. Astron. Astrophys. 300, 568–578.

Robertson, H. P. (1937). Dynamical effects of radiation in the solar system. MNRAS. 97, 423–438,.

Schuerman, D.W. (1980). The restricted three-body problem including radiation pressure. Astrophys. J. 238, 337–342

Simmons, J. F. L, McDonald, A. J. C. and Brown, J. C. (1985). The restricted 3–body problem with radiation pressure. Celest. Mech. 35, 145–187.

Singh, J. and Amuda, T. O. (2017). Effects of Poynting-Robertson (P-R) drag, radiation, and oblateness on motion around the L4,5 equilibrium points in the CR3BP. J. Dyn. Syst. Geom. Theor. 15, 177.

Singh, J. and Taura, J. J. (2013). Motion in the generalized restricted three-body problem. Astrophys. Space Sci. 343, 95–106.

Stenborg ,T. N. (2008). Collinear Lagrange point solutions in the circular restricted three-body problem with radiation pressure using Fortran. In: Argyle, R.W., Bunclark, P.S., Lewis, J.R. (Eds.), Astronomical Data Analysis Software and Systems XVII, Astronomical Society of the Pacific Conference Series. 394,734–737.

Stanley, P. and Whipple, F. L. (1950). The Poynting-Robertson effects on meteor orbits. Am Astron Soc. 111,134–141

Szebehely, V.G., 1967. Theory of Orbits. Academic Press, New York.

Taura and Leke (2022). Derivation of the dynamical equations of motion of the R3BP with variable masses and disk. FUDMA Journal of Sciences (FJS) 6 (4): 125 – 133.

Tyokyaa, K. R. and Atsue, T (2020). Positions and stability of libration points in the radiating and oblating bigger primary of circular restricted three-body problem. FUDMA Journal of Sciences (FJS) 4 (2) :523 – 531

Vincent, A. E. and Perdiou, A. E. (2021a). Poynting-Robertson and oblateness effects on the equilibrium points of the perturbed R3BP: Application on Cen X-4 binary system, in eds. T. M. Rassias, Nonlinear Analysis, Differential Equations, and Applications, Springer Optim. Its App. 173, 131–147.

Vincent, A. E. and Kalantonis, V.S. (2023). Motion around the equilibrium points in the photogravitational R3BP under the effects of Poynting-Robertson drag, circumbinary belt and triaxial primaries with an oblate infinitesimal body: Application on Achird Binary System, in Th. M. Rassias and P. Pardalos, (Eds) Analysis, Geometry, Nonlinear Optimization and Applications, Springer, Cham. https://doi.org/10.1142/9789811261572

Vincent, A. E. and Singh, J. (2022). Out-of-plane equilibria in the perturbed photogravitational restricted three-body problem with Poynting-Robertson drag, Heliyon, 8, e09603 doi.org/10.1016/j.heliyon.2022. e09603.

Vincent, A. E., Perdiou A. E. and Perdios, E.A. (2022). Existence and stability of equilibrium points in the R3BP with triaxial-radiating primaries and an oblate massless body under the effect of the circumbinary disc. Fron. Astron. Space Sci. 9:877459

Vincent, A. E. and Perdiou, A. E. (2021b). Existence and stability of equilibrium points under the influence of Poynting-Robertson and Stokes drags in the restricted three-body problem", in Rassias Th.M., Parasidis, I. N., Providas, E. (Eds) Mathematical Analysis in Interdisciplinary Research, Springer, Cham. DOI: 10.1007/978-3-030-84721-0-37

Vincent, A. E., Tsirogiannis, G. A., Perdiou, A. E., and Kalantonis, V. S. (2024). Equilibrium Points and Lyapunov Families in the Circular Restricted Three-body Problem with an Oblate Primary and a Synchronous Rotating Dipole Secondary: Application to Luhman-16 Binary System. New Astronomy 105 (2024) 102108

Vincent, A. E., Taura, J. J. and Omale, S. O. (2019): Existence and stability of equilibrium points in the photogravitational restricted four- body problem with Stokes drag effect, Astrophysics and Space Science 364 (10)

Yousuf, S. and Kishor, R. (2019). Effects of the albedo and disc on the zero velocity curves and linear stability of equilibrium points in the generalized restricted three-body problem. Monthly Notices of the Royal Astronomical Society. 488(2), 1894–1907.

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