Periodic Orbits of Quantised Restricted Three-Body Problem
Abstract
:1. Introduction
2. Model Description
2.1. Equations of Motion for PCQRTBP
2.2. Hamiltonian of PCQRTBP
3. Periodic Orbits of First Kind
4. Periodic Orbits of the Second Kind
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Dvorak, R.; Froeschlé, C.; Froeschle, C. Stability of outer planetary orbits (P-types) in binaries. Astron. Astrophys. 1989, 226, 335–342. [Google Scholar]
- Cuntz, M. S-type and P-type habitability in stellar binary systems: A comprehensive approach. I. Method and applications. Astrophys. J. 2013, 780, 14. [Google Scholar] [CrossRef] [Green Version]
- Eggl, S.; Pilat-Lohinger, E.; Georgakarakos, N.; Gyergyovits, M.; Funk, B. An analytic method to determine habitable zones for S-type planetary orbits in binary star systems. Astrophys. J. 2012, 752, 74. [Google Scholar] [CrossRef] [Green Version]
- Kaltenegger, L.; Haghighipour, N. Calculating the habitable zone of binary star systems. I. S-type binaries. Astrophys. J. 2013, 777, 165. [Google Scholar] [CrossRef]
- Gómez, G.; Llibre, J.; Martínez, R.; Simó, C. Dynamics and Mission Design Near Libration Points. Vol. I Fundamentals: The Case of Collinear Libration Points (World Scientific Monograph Series in Mathematics, Vol. 2); World Scientific: Singapore, 2001. [Google Scholar]
- Gómez, G.; Llibre, J.; Martínez, R.; Simó, C. Dynamics and Mission Design Near Libration Points. Vol. II Fundamentals: The Case of Triangular Libration Points (World Scientific Monograph Series in Mathematics, Vol. 3); World Scientific: Singapore, 2001. [Google Scholar]
- Abouelmagd, E.I.; Alhothuali, M.; Guirao, J.L.; Malaikah, H. The effect of zonal harmonic coefficients in the framework of the restricted three-body problem. Adv. Space Res. 2015, 55, 1660–1672. [Google Scholar] [CrossRef]
- Abouelmagd, E.I. Existence and stability of triangular points in the restricted three–body problem with numerical applications. Astrophys. Space Sci. 2012, 342, 45–53. [Google Scholar] [CrossRef]
- Abouelmagd, E.I.; Guirao, J.L.; Mostafa, A. Numerical integration of the restricted three-body problem with Lie series. Astrophys. Space Sci. 2014, 354, 369–378. [Google Scholar] [CrossRef]
- Bancelin, D.; Hestroffer, D.; Thuillot, W. Numerical integration of dynamical systems with Lie series: Relativistic acceleration and non-gravitational forces. Celest. Mech. Dyn. Astron. 2012, 112, 221–234. [Google Scholar] [CrossRef] [Green Version]
- Candy, J.; Rozmus, W. A symplectic integration algorithm for separable Hamiltonian functions. J. Comput. Phys. 1991, 92, 230–256. [Google Scholar] [CrossRef]
- Cash, J.R.; Karp, A.H. A variable order Runge-Kutta method for initial value problems with rapidly varying right-hand sides. ACM Trans. Math. Softw. (TOMS) 1990, 16, 201–222. [Google Scholar] [CrossRef]
- Chambers, J.E. A hybrid symplectic integrator that permits close encounters between massive bodies. Mon. Not. R. Astron. Soc. 1999, 304, 793–799. [Google Scholar] [CrossRef]
- Eggl, S.; Dvorak, R. An introduction to common numerical integration codes used in dynamical astronomy. In Dynamics of Small Solar System Bodies and Exoplanets; Springer: Berlin/Heidelberg, Germany, 2010; pp. 431–480. [Google Scholar]
- Elshaboury, S.; Abouelmagd, E.I.; Kalantonis, V.; Perdios, E. The planar restricted three-body problem when both primaries are triaxial rigid bodies: Equilibrium points and periodic orbits. Astrophys. Space Sci. 2016, 361, 315. [Google Scholar] [CrossRef]
- Hanslmeier, A.; Dvorak, R. Numerical integration with Lie series. Astron. Astrophys. 1984, 132, 203–207. [Google Scholar]
- Sharma, R.K. 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, 271–281. [Google Scholar] [CrossRef]
- Hadjidemetriou, J.D. The continuation of periodic orbits from the restricted to the general three-body problem. Celest. Mech. 1975, 12, 155–174. [Google Scholar] [CrossRef]
- Llibre, J.; Saari, D.G. Periodic orbits for the planar newtonian three-body problem coming from the elliptic restricted three–body problems. Trans. Am. Math. Soc. 1995, 347, 3017–3030. [Google Scholar] [CrossRef]
- Alshaery, A.A.; Abouelmagd, E.I. Analysis of the spatial quantized three-body problem. Results Phys. 2020, 17, 103067. [Google Scholar] [CrossRef]
- Battista, E.; Esposito, G. Restricted three-body problem in effective–field-theory models of gravity. Phys. Rev. D 2014, 89, 084030. [Google Scholar] [CrossRef] [Green Version]
- Donoghue, J.F. Leading quantum correction to the Newtonian potential. Phys. Rev. Lett. 1994, 72, 2996. [Google Scholar] [CrossRef] [Green Version]
- Poincaré, H. Les Méthodes Nouvelles de la Mécanique Céleste; Gauthier-Villars: Paris, France; 1892; Volume 1; 1983; Volume 2; 1899; Volume 3; Reprinted by Dover: New York, NY, USA, 1957. [Google Scholar]
- Poincaré, H. Leçons de Mécanique Céleste; Gauthier-Villars: Paris, France, 1905; Volume 1, 1907; Volume 2, Part I; 1909; Volume 2, Part II; 1910; Volume 3. [Google Scholar]
- Ragos, O.; Perdios, E.; Kalantonis, V.; Vrahatis, M. On the equilibrium points of the relativistic restricted three-body problem. Nonlinear Anal. Theory Methods Appl. 2001, 47, 3413–3418. [Google Scholar] [CrossRef]
- Douskos, C.; Kalantonis, V.; Markellos, P.; Perdios, E. On Sitnikov-like motions generating new kinds of 3D periodic orbits in the R3BP with prolate primaries. Astrophys. Space Sci. 2012, 337, 99–106. [Google Scholar] [CrossRef]
- Zotos, E.E. Fractal basins of attraction in the planar circular restricted three–body problem with oblateness and radiation pressure. Astrophys. Space Sci. 2016, 361, 181. [Google Scholar] [CrossRef] [Green Version]
- Zotos, E.E. Basins of convergence of equilibrium points in the pseudo–Newtonian planar circular restricted three-body problem. Astrophys. Space Sci. 2017, 362, 195. [Google Scholar] [CrossRef] [Green Version]
- Bjerrum-Bohr, N.E.J.; Donoghue, J.F.; Holstein, B.R. Quantum gravitational corrections to the nonrelativistic scattering potential of two masses. Phys. Rev. D 2003, 67, 084033. [Google Scholar] [CrossRef] [Green Version]
- Yamada, K.; Asada, H. Post-Newtonian effects of planetary gravity on the perihelion shift. Mon. Not. R. Astron. Soc. 2012, 423, 3540–3544. [Google Scholar] [CrossRef] [Green Version]
- Zhou, T.-Y.; Cao, W.-G.; Xie, Y. Collinear solution to the three-body problem under a scalar–tensor gravity. Phys. Rev. D 2016, 93, 064065. [Google Scholar] [CrossRef]
- Cao, W.G.; Zhou, T.Y.; Xie, Y. Uniqueness of First Order Post-Newtonian Collinear Solutions for Three–Body Problem under a Scalar-Tensor Theory. Commun. Theor. Phys. 2017, 68, 455. [Google Scholar] [CrossRef]
- Strominger, A.; Trivedi, S.P. Information consumption by Reissner-Nordström black holes. Phys. Rev. D 1993, 48, 5778. [Google Scholar] [CrossRef] [Green Version]
- Kazakov, D.; Solodukhin, S. On quantum deformation of the Schwarzschild solution. Nucl. Phys. B 1994, 429, 153–176. [Google Scholar] [CrossRef] [Green Version]
- Lu, X.; Xie, Y. Gravitational lensing by a quantum deformed Schwarzschild black hole. Eur. Phys. J. C 2021, 81, 627. [Google Scholar] [CrossRef]
- Gao, B.; Deng, X.M. Dynamics of charged test particles around quantum-corrected Schwarzschild black holes. Eur. Phys. J. C 2021, 81, 983. [Google Scholar] [CrossRef]
- Lin, H.Y.; Deng, X.M. Bound orbits and epicyclic motions around renormalization group improved Schwarzschild black holes. Universe 2022, 8, 278. [Google Scholar] [CrossRef]
- Lathrop, D.P.; Kostelich, E.J. Characterization of an experimental strange attractor by periodic orbits. Phys. Rev. A 1989, 40, 4028. [Google Scholar] [CrossRef] [PubMed]
- Sternberg, S. Celestial Mechanics. Part II; W. A. Benjamin, Inc.: New York, NY, USA, 1969. [Google Scholar]
- Meyer, K.R.; Offin, D.C. Introduction to Hamiltonian Dynamical Systems and the N-Body Problem; Springer: Cham, Switzerland, 2009. [Google Scholar]
- Llibre, J.; Simó, C. Oscillatory solutions in the planar restricted three-body problem. Math. Ann. 1980, 248, 153–184. [Google Scholar] [CrossRef]
- Llibre, J.; Piñol, C. On the elliptic restricted three-body problem. Celest. Mech. Dyn. Astron. 1990, 48, 319–345. [Google Scholar] [CrossRef]
- Cors, J.M.; Llibre, J. The global flow of the hyperbolic restricted three-body problem. Arch. Ration. Mech. Anal. 1995, 131, 335–358. [Google Scholar] [CrossRef]
- Corbera, M.; Llibre, J. Periodic orbits of the Sitnikov problem via a Poincaré map. Celest. Mech. Dyn. Astron. 2000, 77, 273–303. [Google Scholar] [CrossRef] [Green Version]
- Corbera, M.; Llibre, J. Periodic orbits of a collinear restricted three-body problem. Celest. Mech. Dyn. Astron. 2003, 86, 163–183. [Google Scholar] [CrossRef] [Green Version]
- Llibre, J.; Ortega, R. On the families of periodic orbits of the Sitnikov problem. SIAM J. Appl. Dyn. Syst. 2008, 7, 561–576. [Google Scholar] [CrossRef]
- Llibre, J.; Stoica, C. Comet-and Hill-type periodic orbits in restricted (N + 1)-body problems. J. Differ. Equ. 2011, 250, 1747–1766. [Google Scholar] [CrossRef] [Green Version]
- Yamada, K.; Asada, H. Collinear solution to the general relativistic three–body problem. Phys. Rev. D 2010, 82, 104019. [Google Scholar] [CrossRef] [Green Version]
- Yamada, K.; Asada, H. Uniqueness of collinear solutions for the relativistic three–body problem. Phys. Rev. D 2011, 83, 024040. [Google Scholar] [CrossRef] [Green Version]
- Szebehely, V. Theory of Orbits: The Restricted Problem of Three Bodies; Technical Report; Yale University: New Haven, CT, USA, 1967. [Google Scholar]
- Murray, C.D.; Dermott, S.F. Solar System Dynamics; Cambridge University Press: Cambridge, UK, 1999. [Google Scholar]
- Arenstorf, R.F. Periodic Solutions of the Restricted Three Body Problem Representing Analytic Continuations of Keplerian Elliptic Motions; National Aeronautics and Space Administration: Washington, DC, USA, 1963. [Google Scholar]
- Barrar, R. Existence of periodic orbits of the second kind in the restricted problems of three bodies. Astron. J. 1965, 70, 3. [Google Scholar] [CrossRef]
- Morbidelli, A. Modern Celestial Mechanics: Aspects of Solar System Dynamics; Taylor & Francis: London, UK, 2002. [Google Scholar]
- Birkhoff, G.D. Dynamical Systems; American Mathematical Soc.: Providence, RI, USA, 1927; Volume 9. [Google Scholar]
- Battista, E.; Dell’Agnello, S.; Esposito, G.; Di Fiore, L.; Simo, J.; Grado, A. Earth-Moon Lagrangian points as a test bed for general relativity and effective field theories of gravity. Phys. Rev. D 2015, 92, 064045. [Google Scholar] [CrossRef] [Green Version]
- Zhao, S.S.; Xie, Y. Solar System and stellar tests of a quantum-corrected gravity. Phys. Rev. D 2015, 92, 064033. [Google Scholar] [CrossRef]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Abouelmagd, E.I.; García Guirao, J.L.; Llibre, J. Periodic Orbits of Quantised Restricted Three-Body Problem. Universe 2023, 9, 149. https://doi.org/10.3390/universe9030149
Abouelmagd EI, García Guirao JL, Llibre J. Periodic Orbits of Quantised Restricted Three-Body Problem. Universe. 2023; 9(3):149. https://doi.org/10.3390/universe9030149
Chicago/Turabian StyleAbouelmagd, Elbaz I., Juan Luis García Guirao, and Jaume Llibre. 2023. "Periodic Orbits of Quantised Restricted Three-Body Problem" Universe 9, no. 3: 149. https://doi.org/10.3390/universe9030149