Abstract
In the present paper, a study on the beyond-Newtonian collinear circular restricted \((3+1)\)-body problem with spinning primaries is presented. In the setup of the collinear circular restricted four-body problem (CCR4BP), the three primaries are situated in the collinear central configuration where the masses of two peripheral primaries are equal. We have first derived the equations of motion for a system composed of three Kerr-like primaries. Further, we have unveiled how the existence, evolution, and stability of the libration points in the CCR4BP with beyond-Newtonian first-order correction terms and spin of the primaries depend on the mass parameter \(\beta \) and control parameter \(\varepsilon \) intending to manage the contribution of the beyond-Newtonian terms. It is observed that equilibria of the system are always even and varying from six to eighteen.
Similar content being viewed by others
Data Availability
All data generated or analysed during this study are included in this published article (and its supplementary information files).
References
Abbott, B.P., Abbott, R., Abbott, T.D., et al.: Astrophysical implications of the binary black hole merger gw150914. Astrophys. J. 818, L22 (2016)
Arribas, M., Abad, A., Elipe, A., Palacios, M.: Equilibria of the symmetric collinear restricted four-body problem with radiation pressure. Astrophys. Space Sci. 361, 84 (2016a). https://link.springer.com/article/10.1007/s10509-016-2671-x
Arribas, M., Abad, A., Elipe, A., Palacios, M.: Out-of-plane equilibria in the symmetric collinear restricted four-body problem with radiation pressure. Astrophys. Space Sci. 361, 270 (2016b). https://link.springer.com/article/10.1007/s10509-016-2858-1
Asada, H.: Gravitational wave forms for a three-body system in Lagrange s orbit: parameter determinations and a binary source test. Phys. Rev. D 80, 064021 (2009). https://journals.aps.org/prd/abstract/10.1103/PhysRevD.80.064021
Baltagiannis, A.N., Papadakis, K.E.: Equilibrium points and their stability in the restricted four-body problem. Int. J. Bifurc. Chaos 21, 2179–2193 (2011a). https://doi.org/10.1142/S0218127411029707
Baltagiannis, A.N., Papadakis, K.E.: Families of periodic orbits in the restricted four-body problem. Astrophys. Space Sci. 336, 357–367 (2011b). https://doi.org/10.1007/s10509-011-0778-7
Baltagiannis, A.N., Papadakis, K.E.: Periodic solutions in the Sun-Jupiter-Trojan Asteroid-Spacecraft system. Planet. Space Sci. 75, 148–157 (2013). https://doi.org/10.1016/j.pss.2012.11.006
Barnes, S.A.: Ages for illustrative field stars using gyrochronology: viability, limitations, and errors. Astrophys. J. 669(2), 1167 (2007)
De, S., Roychowdhury, S., Banerjee, R.: Beyond-Newtonian dynamics of a planar circular restricted three-body problem with Kerr-like primaries. Mon. Not. R. Astron. Soc. 501(1), 713–729 (2021). https://doi.org/10.1093/mnras/staa3733
Dubeibe, F.L., Lora-Clavijo, F.D., González, G.A.: Pseudo-Newtonian planar circular restricted 3-body problem. Phys. Lett. A 381(6), 563–567 (2017). https://doi.org/10.1016/j.physleta.2016.12.024
Ernst, F.J.: New formulation of the axially symmetric gravitational field problem. Phys. Rev. 167, 1175–1177 (1968)
Everitt, C.W.F., et al.: The Gravity Probe B test of general relativity. Class. Quantum Gravity 32(22), 224001 (2015)
Farr, W.M., Stevenson, S., Miller, M.C., et al.: Nature 548, 426 (2017)
Gardner, J.P., Mather, J.C., Clampin, M., et al.: The James Webb Space Telescope. Space Sci. Rev. 123(4), 485–606 (2006). https://doi.org/10.1007/s11214-006-8315-7
Hamilton, D.P.: Celestial mechanics: fresh solutions to the four-body problem. Nature 533, 187 (2016). https://doi.org/10.1038/nature17896
Imai, T., Chiba, T., Asada, H.: Choreographic solution to the general-relativistic three-body problem. Phys. Rev. Lett. 98, 201102 (2007). https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.98.201102
Leandro, E.S.: On the central configurations of the planar restricted four-body problem. J. Differ. Equ. 226, 323–351 (2006). https://doi.org/10.1016/j.jde.2005.10.015
Li, D., Wu, X., Liang, E.: Figure-eight orbits in three post-Newtonian formulations of triple black holes. Phys. Rev. D 104, 044039 (2021). https://journals.aps.org/prd/abstract/10.1103/PhysRevD.104.044039
Michalodimitrakis, M.: The circular restricted four-body problem. Astrophys. Space Sci. 75, 289–305 (1981). https://doi.org/10.1007/BF00648643
Moore, C.: Braids in classical dynamics. Phys. Rev. Lett. 70, 3675 (1993)
Moore, C., Nauenberg, M.: New periodic orbits for the \(n\)-body problem. J. Comput. Nonlinear Dyn. 1(4), 307–311 (2006). https://doi.org/10.1115/1.2338323
Muhammad, S., Duraihem, F.Z., Zotos, E.E.: On the equilibria of the restricted four-body problem with triaxial rigid primaries-I. Oblate bodies. Chaos Solitons Fractals 142, 110500 (2021)
Papadouris, J.P., Papadakis, K.E.: Equilibrium points in the photogravitational restricted four-body problem. Astrophys. Space Sci. 344, 21–38 (2013)
Scholz, A., et al.: A universal spin-mass relation for Brown dwarfs and planets. Astrophys. J. 859(2), 153 (2018). https://doi.org/10.3847/1538-4357/aabfbe
Singh, J., Omale, S.O.: Combined effect of Stokes drag, oblateness and radiation pressure on the existence and stability of equilibrium points in the restricted four-body problem. Astrophys. Space Sci. 364(6), 1–10 (2019). https://doi.org/10.1007/s10509-019-3494-3
Sotiriou, T.P., Apostolatos, T.A.: Corrections and comments on the multipole moments of axisymmetric electrovacuum spacetimes. Class. Quantum Gravity 21(24), 5727 (2004). https://doi.org/10.1088/0264-9381/21/24/003
Suraj, M.S., Aggarwal, R., Arora, M.: On the restricted four-body problem with the effect of small perturbations in the Coriolis and centrifugal forces. Astrophys. Space Sci. 362, 159 (2017)
Suraj, M.S., Mittal, A., Arora, M., et al.: Exploring the fractal basins of convergence in the restricted four-body problem with oblateness. Int. J. Non-Linear Mech. 102, 62–71 (2018a). https://www.sciencedirect.com/science/article/abs/pii/S0020746217308417
Suraj, M.S., Aggarwal, R., Mittal, A., Asique, M.C.: The effect of radiation pressure on the basins of convergence in the restricted four-body problem. Chaos Solitons Fractals 141 110347 (2020a). https://www.sciencedirect.com/science/article/abs/pii/S0960077920307426
Suraj, M.S., Aggarwal, R., Mittal, A., Meena, O.P., Asique, M.C.: On the spatial collinear restricted four-body problem with non-spherical primaries. Chaos Solitons Fractals 133 109609 (2020b). https://doi.org/10.1016/j.chaos.2020.109609
Šuvakov, M., Dmitrašinović, V.: Three classes of Newtonian three-body planar periodic orbits. Phys. Rev. Lett. 110(11), 114301 (2013)
Tokovinin, A.: The updated multiple star catalog. Astrophys. J. Suppl. Ser. 235, 6 (2018)
Valtonen, M., Karttunen, H.: The Three-Body Problem. Cambridge University Press, Cambridge (2006)
Valtonen, M., Mikkola, S.: The few-body problem in astrophysics. Annu. Rev. Astron. Astrophys. 29(1), 9–29 (1991)
Yamada, K., Asada, H.: Collinear solution to the general relativistic three-body problem. Phys. Rev. D 82, 104019 (2010)
Yamada, K., Asada, H.: Uniqueness of collinear solutions for the relativistic three-body problem. Phys. Rev. D 83, 024040 (2011). https://journals.aps.org/prd/abstract/10.1103/PhysRevD.83.024040
Zotos, E.E.: Revealing the basins of convergence in the planar equilateral restricted four-body problem. Astrophys. Space Sci. 362, 2 (2017). https://doi.org/10.1007/s10509-016-2973-z
Funding
F.L.D was partially supported by MinCiencias (Colombia) Grant 8863 and by Universidad de los Llanos Grant C09-F04-010-2019. Dr. Rajiv Aggarwal was partially supported by Department of Science and Technology, India, under scheme MATRICS (MTR/2018/000442).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Suraj, M.S., Dubeibe, F.L., Aggarwal, R. et al. On the beyond-Newtonian collinear circular restricted \((3 + 1)\)-body problem with spinning primaries. Astrophys Space Sci 367, 55 (2022). https://doi.org/10.1007/s10509-022-04081-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10509-022-04081-2