# Periodic Orbits of Quantised Restricted Three-Body Problem

^{1}

^{2}

^{3}

^{4}

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## Abstract

**:**

## 1. Introduction

## 2. Model Description

#### 2.1. Equations of Motion for PCQRTBP

#### 2.2. Hamiltonian of PCQRTBP

## 3. Periodic Orbits of First Kind

**Theorem**

**1.**

## 4. Periodic Orbits of the Second Kind

**Theorem**

**2.**

## 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]

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**MDPI and ACS Style**

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

**AMA Style**

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 Style**

Abouelmagd, 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