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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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

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

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

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

[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.

[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.

[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.

[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.

[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.

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