American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Variational approach to second species periodic solutions of Poincaré of the 3 body problem
  • DCDS Home
  • This Issue
  • Next Article
    Non-integrability criterium for normal variational equations around an integrable subsystem and an example: The Wilberforce spring-pendulum
March  2013, 33(3): 987-1008. doi: 10.3934/dcds.2013.33.987

Horseshoe periodic orbits with one symmetry in the general planar three-body problem

Abimael Bengochea 1, , Manuel Falconi 2, and  Ernesto Pérez-Chavela 3,

1. 

Departamento de Matemáticas, Facultad de Ciencias, UNAM, Ciudad Universitaria, México, D.F. 04510, Mexico
2. 

Department of Mathematics, Facultad de Ciencias, UNAM, Ciudad Universitaria, México, D.F. 04510
3. 

Departamento de Matemáticas, Universidad Autónoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco 186, México, D.F. 09340

Received  April 2011 Revised  September 2011 Published  October 2012

Using collinear reversible configurations and some properties of symmetry we obtain horseshoe periodic orbits in the general planar three-body problem with masses $m_1\gg m_2 \geq m_3$, which usually represents a system formed by a planet and two small satellites; for instance, the system Saturn-Janus-Epimetheus. For the numerical analysis we have taken the values $m_2/m_1 = 3.5 \times 10^{-4}$ and $m_3/m_1 = 9.7 \times 10^{-5}$ corresponding to $10^5$ times the mass ratios of Saturn-Janus and Saturn-Epimetheus,
Keywords: periodic orbits., General three-body problem, horseshoe orbits.
Mathematics Subject Classification: Primary: 70F15, 70F07, 70H12; Secondary: 37M0.
Citation: Abimael Bengochea, Manuel Falconi, Ernesto Pérez-Chavela. Horseshoe periodic orbits with one symmetry in the general planar three-body problem. Discrete & Continuous Dynamical Systems, 2013, 33 (3) : 987-1008. doi: 10.3934/dcds.2013.33.987
References:
[1]

E. Barrabés and S. Mikkola, Families of periodic horseshoe orbits in the restricted three-body problem, Astron, Astrophys, 432 (2005), 1115-1129. doi: 10.1051/0004-6361:20041483.  Google Scholar

[2]

A. Bengochea and E. Piña, The Saturn, Janus and Epimetheus dynamics as a gravitational three-body problem in the plane, Rev. Mexicana Fís., 55 (2009), 97-105. Google Scholar

[3]

A. Bengochea, M. Falconi and E. Pérez-Chavela, Symmetric horseshoe periodic orbits in the general planar three-body problem, Astrophys. Space Sci., 333 (2011), 399-408. doi: 10.1007/s10509-011-0641-x.  Google Scholar

[4]

J. M. Cors and G. R. Hall, Coorbital periodic orbits in the three body problem, SIAM J. Appl. Dyn. Syst., 2 (2003), 219-237. doi: 10.1137/S1111111102411304.  Google Scholar

[5]

S. F. Dermott and C. D. Murray, The dynamics of tadpole and horseshoe orbits. I. Theory, Icarus, 48 (1981), 1-11. doi: 10.1016/0019-1035(81)90147-0.  Google Scholar

[6]

S. F. Dermott and C. D. Murray, The dynamics of tadpole and horseshoe orbits. II. The coorbital satellites of Saturn, Icarus, 48 (1981), 12-22. doi: 10.1016/0019-1035(81)90148-2.  Google Scholar

[7]

J. R. Dormand and P. J. Prince, A family of embedded Runge-Kutta formulae, J. Comput. Appl. Math., 6 (1980), 19-26. doi: 10.1016/0771-050X(80)90013-3.  Google Scholar

[8]

M. Hénon and J. M. Petit, Series expansion of encounter-type solutions of Hill's problem, Celest. Mech. Dynam. Astron., 38 (1986), 67-100.  Google Scholar

[9]

X. Y. Hou and L. Liu, The symmetric horseshoe periodic families and the lyapunov planar family around $L_3$, Astron. J., 136 (2008), 67-75. doi: 10.1088/0004-6256/136/1/67.  Google Scholar

[10]

J. S. W. Lamb and J. A. G. Roberts, Time-reversal symmetry in dynamical systems: A survey, Phys. D, 112 (1998), 1-39. doi: 10.1016/S0167-2789(97)00199-1.  Google Scholar

[11]

J. Llibre and M. Ollé, The motion of Saturn coorbital satellites in the restricted three-body problem, Astron. Astrophys, 378 (2001), 1087-1099. doi: 10.1051/0004-6361:20011274.  Google Scholar

[12]

K. R. Meyer and G. R. Hall, "Introduction to Hamiltonian Dynamical Systems and the N-Body Problem,'' $1^{st}$ edition, Springer-Verlag, New York, 1992. Google Scholar

[13]

F. J. Muñoz-Almaraz, J. Galán and E. Freire, Families of symmetric periodic orbits in the three body problem and the figure eight, Monogr. Real Acad. Ci. Exact. Fís.-Quím. Nat. Zaragoza, 25 (2004), 229-240.  Google Scholar

[14]

J. M. Petit and M. Hénon, Satellite encounters, Icarus, 66 (1986), 536-555. doi: 10.1016/0019-1035(86)90089-8.  Google Scholar

[15]

A. E. Roy and M. W. Ovenden, On the occurrence of commensurable mean motions in the solar system. II. The mirror theorem, Mon. Not. R. Astron. Soc., 115 (1955), 296-309. Google Scholar

[16]

F. Spirig and J. Waldvogel, The three-body problem with two small masses: A singular-perturbation approach to the problem of Saturn's coorbiting satellites, in "Stability of the Solar System and its Minor Natural and Artificial Bodies,'' (ed. V. G. Szebehely), Reidel, (1985), 53-63. doi: 10.1007/978-94-009-5398-7_5.  Google Scholar

[17]

C. F. Yoder, G. Colombo, S. P. Synnott and K. A. Yoder, Theory of motion of Saturn's coorbiting satellites, Icarus, 53 (1983), 431-443. doi: 10.1016/0019-1035(83)90207-5.  Google Scholar

[18]

C. F. Yoder, S. P. Synnott and H. Salo, Orbits and masses of Saturn's co-orbiting satellites, Janus and Epimetheus, Astron. J., 98 (1989), 1875-1889. doi: 10.1086/115265.  Google Scholar

[19]

J. Waldvogel and F. Spirig, Co-orbital satellites and hill's lunar problem, in "Long-Term Dynamical Behaviour of Natural and Artificial N-Body Systems'' (ed. A. E. Roy), Kluwer, (1988), 223-234. doi: 10.1007/978-94-009-3053-7_20.  Google Scholar

show all references

References:
[1]

E. Barrabés and S. Mikkola, Families of periodic horseshoe orbits in the restricted three-body problem, Astron, Astrophys, 432 (2005), 1115-1129. doi: 10.1051/0004-6361:20041483.  Google Scholar

[2]

A. Bengochea and E. Piña, The Saturn, Janus and Epimetheus dynamics as a gravitational three-body problem in the plane, Rev. Mexicana Fís., 55 (2009), 97-105. Google Scholar

[3]

A. Bengochea, M. Falconi and E. Pérez-Chavela, Symmetric horseshoe periodic orbits in the general planar three-body problem, Astrophys. Space Sci., 333 (2011), 399-408. doi: 10.1007/s10509-011-0641-x.  Google Scholar

[4]

J. M. Cors and G. R. Hall, Coorbital periodic orbits in the three body problem, SIAM J. Appl. Dyn. Syst., 2 (2003), 219-237. doi: 10.1137/S1111111102411304.  Google Scholar

[5]

S. F. Dermott and C. D. Murray, The dynamics of tadpole and horseshoe orbits. I. Theory, Icarus, 48 (1981), 1-11. doi: 10.1016/0019-1035(81)90147-0.  Google Scholar

[6]

S. F. Dermott and C. D. Murray, The dynamics of tadpole and horseshoe orbits. II. The coorbital satellites of Saturn, Icarus, 48 (1981), 12-22. doi: 10.1016/0019-1035(81)90148-2.  Google Scholar

[7]

J. R. Dormand and P. J. Prince, A family of embedded Runge-Kutta formulae, J. Comput. Appl. Math., 6 (1980), 19-26. doi: 10.1016/0771-050X(80)90013-3.  Google Scholar

[8]

M. Hénon and J. M. Petit, Series expansion of encounter-type solutions of Hill's problem, Celest. Mech. Dynam. Astron., 38 (1986), 67-100.  Google Scholar

[9]

X. Y. Hou and L. Liu, The symmetric horseshoe periodic families and the lyapunov planar family around $L_3$, Astron. J., 136 (2008), 67-75. doi: 10.1088/0004-6256/136/1/67.  Google Scholar

[10]

J. S. W. Lamb and J. A. G. Roberts, Time-reversal symmetry in dynamical systems: A survey, Phys. D, 112 (1998), 1-39. doi: 10.1016/S0167-2789(97)00199-1.  Google Scholar

[11]

J. Llibre and M. Ollé, The motion of Saturn coorbital satellites in the restricted three-body problem, Astron. Astrophys, 378 (2001), 1087-1099. doi: 10.1051/0004-6361:20011274.  Google Scholar

[12]

K. R. Meyer and G. R. Hall, "Introduction to Hamiltonian Dynamical Systems and the N-Body Problem,'' $1^{st}$ edition, Springer-Verlag, New York, 1992. Google Scholar

[13]

F. J. Muñoz-Almaraz, J. Galán and E. Freire, Families of symmetric periodic orbits in the three body problem and the figure eight, Monogr. Real Acad. Ci. Exact. Fís.-Quím. Nat. Zaragoza, 25 (2004), 229-240.  Google Scholar

[14]

J. M. Petit and M. Hénon, Satellite encounters, Icarus, 66 (1986), 536-555. doi: 10.1016/0019-1035(86)90089-8.  Google Scholar

[15]

A. E. Roy and M. W. Ovenden, On the occurrence of commensurable mean motions in the solar system. II. The mirror theorem, Mon. Not. R. Astron. Soc., 115 (1955), 296-309. Google Scholar

[16]

F. Spirig and J. Waldvogel, The three-body problem with two small masses: A singular-perturbation approach to the problem of Saturn's coorbiting satellites, in "Stability of the Solar System and its Minor Natural and Artificial Bodies,'' (ed. V. G. Szebehely), Reidel, (1985), 53-63. doi: 10.1007/978-94-009-5398-7_5.  Google Scholar

[17]

C. F. Yoder, G. Colombo, S. P. Synnott and K. A. Yoder, Theory of motion of Saturn's coorbiting satellites, Icarus, 53 (1983), 431-443. doi: 10.1016/0019-1035(83)90207-5.  Google Scholar

[18]

C. F. Yoder, S. P. Synnott and H. Salo, Orbits and masses of Saturn's co-orbiting satellites, Janus and Epimetheus, Astron. J., 98 (1989), 1875-1889. doi: 10.1086/115265.  Google Scholar

[19]

J. Waldvogel and F. Spirig, Co-orbital satellites and hill's lunar problem, in "Long-Term Dynamical Behaviour of Natural and Artificial N-Body Systems'' (ed. A. E. Roy), Kluwer, (1988), 223-234. doi: 10.1007/978-94-009-3053-7_20.  Google Scholar

[1]

Jungsoo Kang. Some remarks on symmetric periodic orbits in the restricted three-body problem. Discrete & Continuous Dynamical Systems, 2014, 34 (12) : 5229-5245. doi: 10.3934/dcds.2014.34.5229
[2]

Rongchang Liu, Jiangyuan Li, Duokui Yan. New periodic orbits in the planar equal-mass three-body problem. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 2187-2206. doi: 10.3934/dcds.2018090
[3]

Niraj Pathak, V. O. Thomas, Elbaz I. Abouelmagd. The perturbed photogravitational restricted three-body problem: Analysis of resonant periodic orbits. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 849-875. doi: 10.3934/dcdss.2019057
[4]

Richard Moeckel. A topological existence proof for the Schubart orbits in the collinear three-body problem. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 609-620. doi: 10.3934/dcdsb.2008.10.609
[5]

Nai-Chia Chen. Symmetric periodic orbits in three sub-problems of the $N$-body problem. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1523-1548. doi: 10.3934/dcdsb.2014.19.1523
[6]

Daniel Offin, Hildeberto Cabral. Hyperbolicity for symmetric periodic orbits in the isosceles three body problem. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 379-392. doi: 10.3934/dcdss.2009.2.379
[7]

Tiancheng Ouyang, Duokui Yan. Variational properties and linear stabilities of spatial isosceles orbits in the equal-mass three-body problem. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 3989-4018. doi: 10.3934/dcds.2017169
[8]

Marcel Guardia, Tere M. Seara, Pau Martín, Lara Sabbagh. Oscillatory orbits in the restricted elliptic planar three body problem. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 229-256. doi: 10.3934/dcds.2017009
[9]

Gianni Arioli. Branches of periodic orbits for the planar restricted 3-body problem. Discrete & Continuous Dynamical Systems, 2004, 11 (4) : 745-755. doi: 10.3934/dcds.2004.11.745
[10]

Elbaz I. Abouelmagd, Juan Luis García Guirao, Jaume Llibre. Periodic orbits for the perturbed planar circular restricted 3–body problem. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1007-1020. doi: 10.3934/dcdsb.2019003
[11]

Hildeberto E. Cabral, Zhihong Xia. Subharmonic solutions in the restricted three-body problem. Discrete & Continuous Dynamical Systems, 1995, 1 (4) : 463-474. doi: 10.3934/dcds.1995.1.463
[12]

Edward Belbruno. Random walk in the three-body problem and applications. Discrete & Continuous Dynamical Systems - S, 2008, 1 (4) : 519-540. doi: 10.3934/dcdss.2008.1.519
[13]

Mitsuru Shibayama. Non-integrability of the collinear three-body problem. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 299-312. doi: 10.3934/dcds.2011.30.299
[14]

Richard Moeckel. A proof of Saari's conjecture for the three-body problem in $\R^d$. Discrete & Continuous Dynamical Systems - S, 2008, 1 (4) : 631-646. doi: 10.3934/dcdss.2008.1.631
[15]

Hiroshi Ozaki, Hiroshi Fukuda, Toshiaki Fujiwara. Determination of motion from orbit in the three-body problem. Conference Publications, 2011, 2011 (Special) : 1158-1166. doi: 10.3934/proc.2011.2011.1158
[16]

Kuo-Chang Chen. On Chenciner-Montgomery's orbit in the three-body problem. Discrete & Continuous Dynamical Systems, 2001, 7 (1) : 85-90. doi: 10.3934/dcds.2001.7.85
[17]

Duokui Yan, Tiancheng Ouyang, Zhifu Xie. Classification of periodic orbits in the planar equal-mass four-body problem. Conference Publications, 2015, 2015 (special) : 1115-1124. doi: 10.3934/proc.2015.1115
[18]

Corey Shanbrom. Periodic orbits in the Kepler-Heisenberg problem. Journal of Geometric Mechanics, 2014, 6 (2) : 261-278. doi: 10.3934/jgm.2014.6.261
[19]

Victoriano Carmona, Emilio Freire, Soledad Fernández-García. Periodic orbits and invariant cones in three-dimensional piecewise linear systems. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 59-72. doi: 10.3934/dcds.2015.35.59
[20]

Jan Bouwe van den Berg, Elena Queirolo. A general framework for validated continuation of periodic orbits in systems of polynomial ODEs. Journal of Computational Dynamics, 2021, 8 (1) : 59-97. doi: 10.3934/jcd.2021004

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (122)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences