March  2019, 24(3): 1007-1020. doi: 10.3934/dcdsb.2019003

Periodic orbits for the perturbed planar circular restricted 3–body problem

1. 

Nonlinear Analysis and Applied Mathematics Research Group (NAAM), Mathematics Department, King Abdulaziz University, Jeddah, Saudi Arabia

2. 

Celestial Mechanics Unit, Astronomy Department, National Research Institute of Astronomy and Geophysics (NRIAG), Helwan 11421, Cairo, Egypt

3. 

Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, Hospital de Marina, 30203-Cartagena, Región de Murcia, Spain

4. 

Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain

Received  July 2017 Revised  February 2018 Published  January 2019

Fund Project: We thank to both reviewers their comments and suggestions that help us to improve this paper. The first and second author of this work were partially supported by MINECO grant number MTM2014-51891-P and Fundación Séneca de la Región de Murcia grant number 19219/PI/14. The first and third author are partially supported by a MINECO-FEDER grant MTM2016-77278-P, a MINECO grant MTM2013-40998-P, and an AGAUR grant number 2014SGR-568

We characterize when the classical first and second kind of periodic orbits of the planar circular restricted $ 3 $–body problem obtained by Poincaré, can be extended to perturbed planar circular restricted $ 3 $–body problems. We put special emphasis when the perturbed forces are due to zonal harmonic or to a solar sail.

Citation: 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
References:
[1]

E. I. AbouelmagdM. S. AlhothualiJ. L. G. Guirao and H. M. Malaikah, The effect of zonal harmonic coefficients in the framework of the restricted three-body problem, Adv. Space Res., 55 (2015), 1660-1672. doi: 10.1016/j.asr.2014.12.030. Google Scholar

[2]

E. I. Abouelmagd, Existence and stability of triangular points in the restricted three-body problem with numerical applications, Astrophys Space Sci., 342 (2012), 45-53. doi: 10.1007/s10509-012-1162-y. Google Scholar

[3]

E. I. Abouelmagd and J. L. G. Guirao, On the perturbed restricted three-body problem, Applied Mathematics and Nonlinear Sciences, 1 (2016), 123-144. doi: 10.21042/AMNS.2016.1.00010. Google Scholar

[4]

E. I. AbouelmagdJ. L. G. Guirao and A. Mostafa, Numerical integration of the restricted three-body problem with Lie series, Astrophys Space Sci., 354 (2014), 369-378. doi: 10.1007/s10509-014-2107-4. Google Scholar

[5]

M. Alvarez and J. Llibre, Heteroclinic orbits and Bernoulli shift for the elliptic collision restricted three–body problem, Archive for Rational Mechanics and Analysis, 156 (2001), 317-357. doi: 10.1007/s002050100116. Google Scholar

[6]

D. BancelinD. Hestroffer and W. Thuillot, Numerical integration of dynamical systems with Lie series relativistic acceleration and nongravitational forces, Celest. Mech. Dyn. Astron., 112 (2012), 221-234. doi: 10.1007/s10569-011-9393-8. Google Scholar

[7]

R. B. Barrar, Existence of periodic orbits of second kind in the restricted problem of three bodies, Astro. J., 70 (1965), 3-4. doi: 10.1086/109672. Google Scholar

[8]

E. BelbrunoJ. Llibre and M. Ollé, On the families of periodic orbits of the circular restricted three–body problem which bifurcate from the Sitnikov problem, Celest. Mech. Dyn. Astron., 60 (1994), 99-129. doi: 10.1007/BF00693095. Google Scholar

[9]

G. D. Birkhoff, Dynamical System, With an addendum by Jurgen Moser. American Mathematical Society Colloquium Publications, Vol. IX American Mathematical Society, Providence, R.I., 1966. Google Scholar

[10]

J. Candy and W. Rozmus, A symplectic integration algorithm for separable Hamiltonian functions, J. Comp. Physiol., 92 (1991), 230-256. doi: 10.1016/0021-9991(91)90299-Z. Google Scholar

[11]

J. R. Cash and A. H. Karp, A variable order Runge-Kutta method for initial value problems with rapidly varying right-hand sides, ACM Trans. Math. Softw., 16 (1990), 201-222. doi: 10.1145/79505.79507. Google Scholar

[12]

J. E. Chambers, A hybrid symplectic integrator that permits close encounters between massive bodies, Mon. Not. R. Astron. Soc., 304 (1999), 793-799. doi: 10.1046/j.1365-8711.1999.02379.x. Google Scholar

[13]

M. Corbera and J. Llibre, Periodic orbits of the Sitnikov problem via a Poincaré map, Celest. Mech. Dyn. Astron., 77 (2000), 273-303. doi: 10.1023/A:1011117003713. Google Scholar

[14]

M. Corbera and J. Llibre, On symmetric periodic orbits of the elliptic Sitnikov problem via the analytic continuation method, Contemporary Mathematics, 292 (2002), 91-127. doi: 10.1090/conm/292/04918. Google Scholar

[15]

M. Corbera and J. Llibre, Symmetric periodic orbits for the elliptical 3–body problem via continution method, Developments in Mathematical and Experimental Physics, Klauaer, Acod., 2003, 113–137.Google Scholar

[16]

M. Corbera and J. Llibre, Periodic orbits of a collinear restricted three body problem, Celest. Mech. Dyn. Astron., 86 (2003), 163-183. doi: 10.1023/A:1024183003251. Google Scholar

[17]

J. M. Cors and J. Llibre, The global flow of the hyperbolic restricted three–body problem, Archive for Rational Mech. and Anal., 131 (1995), 335-358. doi: 10.1007/BF00380914. Google Scholar

[18]

J. M. Cors and J. Llibre, Qualitative study of the hyperbolic collision restricted three–body problem, Nonlinearity, 9 (1996), 1299-1316. doi: 10.1088/0951-7715/9/5/011. Google Scholar

[19]

J. M. Cors and J. Llibre, Qualitative study of the parabolic collision restricted three–body problem, Contemporary Math., 198 (1996), 1-19. doi: 10.1090/conm/198/02485. Google Scholar

[20]

J. M. Cors and J. Llibre, The global flow of the parabolic restricted three–body problem, Celest. Mech. Dyn. Astron., 90 (2004), 13-33. doi: 10.1007/s10569-004-4917-0. Google Scholar

[21]

M. Cuntz, S-type and p-type habitability in stellar binary systems: A comprehensive approach. i. method and applications, Astrophysical Journal, 780 (2014), 14. doi: 10.1088/0004-637X/780/1/14. Google Scholar

[22]

R. DvorakC. Froeschle and C. Froeschle, Stability of outer planetary orbits (p-types) in binaries, Astrophysics, 226 (1989), 335-342. Google Scholar

[23]

S. Eggl and R. Dvorak, An introduction to common numerical integration codes used in dynamical astronomy, In: Souchay, J., Dvorak, R. (eds.) Lecture Notes in Physics, vol. 790, Springer, Berlin, 2010,431–480. doi: 10.1007/978-3-642-04458-8_9. Google Scholar

[24]

S. Eggl, E. Pilat-Lohinger, N. Georgakarakos, M. Gyergyovits and B. Funk, An analytic method to determine habitable zones for s-type planetary orbits in binary star systems, Astrophysical J., 752 (2012), 74. doi: 10.1088/0004-637X/752/1/74. Google Scholar

[25]

S. M. Elshaboury, E. I. Abouelmagd, V. S. Kalantonis and E. A. Perdios, The planar restricted three–body problem when both primaries are triaxial rigid bodies: Equilibrium points and periodic orbits, Astrophys. Space Sci., 361 (2016), Paper No. 315, 18 pp. doi: 10.1007/s10509-016-2894-x. Google Scholar

[26]

G. Gómez, J. Llibre, R. Martínez and C. Simó, 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. doi: 10.1142/9789812810632_bmatter. Google Scholar

[27]

G. Gómez, J. Llibre, R. Martínez and C. Simó, 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

[28]

J. D. Hadjidemetriou, The continuation of Periodic orbits form the restricted to the general three–body problem, Celest. Mech. Dyn. Astron., 12 (1975), 155-174. doi: 10.1007/BF01230209. Google Scholar

[29]

A. Hanslmeier and R. Dvorak, Numerical integration with Lie series, Astron. Astrophys., 132 (1984), 203-207. Google Scholar

[30]

L. Kaltenegger and N. Haghighipour, Calculating the habitable zone of binary star systems. Ⅰ. s-type binaries, Astrophysical J., 777 (2013), 165. doi: 10.1088/0004-637X/777/2/165. Google Scholar

[31]

J. Llibre, On the restricted three–body problem when the mass parameter is small, Celestial Mechanics, 28 (1982), 83-105. doi: 10.1007/BF01230662. Google Scholar

[32]

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

[33]

J. Llibre and R. Ortega, On the families of periodic orbits of the Sitnikov problem, SIAM J. of Applied Dynamical Systems, 7 (2008), 561-576. doi: 10.1137/070695253. Google Scholar

[34]

J. Llibre and C. Piñol, On the elliptic restricted three–body problem, Celest. Mech. Dyn. Astron., 48 (1990), 319-345. doi: 10.1007/BF00049388. Google Scholar

[35]

J. Llibre and C. Stoica, Comet- and Hill-type periodic orbits in restricted (N+1)-body problems, J. Differential Equations, 250 (2010), 1747-1766. doi: 10.1016/j.jde.2010.08.005. Google Scholar

[36]

J. Llibre and D. G. Saari, Periodic orbits for the planar Newtonian three–body problem coming from the elliptical restricted three–body problems, Trans. Amer. Math. Soc., 347 (1995), 3017-3030. doi: 10.1090/S0002-9947-1995-1297534-X. Google Scholar

[37]

J. Llibre and C. Simó, Oscillatory solution in the planar restricted three body problem, Math. Annalen, 248 (1980), 153-184. doi: 10.1007/BF01421955. Google Scholar

[38]

C. R. McInnesA. J. C. McDonaldJ. F. L. Simmons and E. W. MacDonaldg, Solar sail parking in restricted three-body systems, J. Guid. Control Dyn., 17 (1994), 399-406. doi: 10.2514/3.21211. Google Scholar

[39]

K. R. Meyer, G. R. Hall and D. Offin, Introduction to Hamiltonian Dynamical Systems and the N-body Problem, Second edition. Applied Mathematical Sciences, 90, Springer, New York, 2009. doi: 10.1007/978-0-387-09724-4. Google Scholar

[40] C. D. Murray and S. F. Dermott, Solar System Dynamics, Cambridge University Press, 1999. Google Scholar
[41]

H. Poincaré, Leçons de Mécanique Céleste, Librairie Scientifique et Technique Albert Blanchard, Paris, 1987. Google Scholar

[42]

H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste, Vol. 1, 1892; Vol. 2, 1983; Vol. 3, 1899; Gauthier-Villars, Paris. Reprinted by Dover, New York, 1957.Google Scholar

[43]

F. J. T. SalazarC. R. McInnes and O. C. Winter, Periodic orbits for space-based reflectors in the circular restricted three–body problem, Celest. Mech. Dyn. Astron., 128 (2017), 95-113. doi: 10.1007/s10569-016-9739-3. Google Scholar

[44]

R. K. Sharma, The linear stability of libration points of the photogravitational restricted three-body problem when the smaller primary is an oblate spheroid, Astrophys. Space Sci., 135 (1987), 271-281. Google Scholar

[45]

S. Sternberg, Celestial mechanics, Part Ⅱ, W. A. Benjamin, Inc, New York. 1969.Google Scholar

[46]

P. Verrier and T. Waters, Families of periodic orbits for solar sails in the CRBTP, M. Macdonald(ed), Advances in Solar Sailing, Springer Praxis Books, Springer–Verlag Berlin Heidelberg, 2014, 871–884. doi: 10.1007/978-3-642-34907-2_52. Google Scholar

show all references

References:
[1]

E. I. AbouelmagdM. S. AlhothualiJ. L. G. Guirao and H. M. Malaikah, The effect of zonal harmonic coefficients in the framework of the restricted three-body problem, Adv. Space Res., 55 (2015), 1660-1672. doi: 10.1016/j.asr.2014.12.030. Google Scholar

[2]

E. I. Abouelmagd, Existence and stability of triangular points in the restricted three-body problem with numerical applications, Astrophys Space Sci., 342 (2012), 45-53. doi: 10.1007/s10509-012-1162-y. Google Scholar

[3]

E. I. Abouelmagd and J. L. G. Guirao, On the perturbed restricted three-body problem, Applied Mathematics and Nonlinear Sciences, 1 (2016), 123-144. doi: 10.21042/AMNS.2016.1.00010. Google Scholar

[4]

E. I. AbouelmagdJ. L. G. Guirao and A. Mostafa, Numerical integration of the restricted three-body problem with Lie series, Astrophys Space Sci., 354 (2014), 369-378. doi: 10.1007/s10509-014-2107-4. Google Scholar

[5]

M. Alvarez and J. Llibre, Heteroclinic orbits and Bernoulli shift for the elliptic collision restricted three–body problem, Archive for Rational Mechanics and Analysis, 156 (2001), 317-357. doi: 10.1007/s002050100116. Google Scholar

[6]

D. BancelinD. Hestroffer and W. Thuillot, Numerical integration of dynamical systems with Lie series relativistic acceleration and nongravitational forces, Celest. Mech. Dyn. Astron., 112 (2012), 221-234. doi: 10.1007/s10569-011-9393-8. Google Scholar

[7]

R. B. Barrar, Existence of periodic orbits of second kind in the restricted problem of three bodies, Astro. J., 70 (1965), 3-4. doi: 10.1086/109672. Google Scholar

[8]

E. BelbrunoJ. Llibre and M. Ollé, On the families of periodic orbits of the circular restricted three–body problem which bifurcate from the Sitnikov problem, Celest. Mech. Dyn. Astron., 60 (1994), 99-129. doi: 10.1007/BF00693095. Google Scholar

[9]

G. D. Birkhoff, Dynamical System, With an addendum by Jurgen Moser. American Mathematical Society Colloquium Publications, Vol. IX American Mathematical Society, Providence, R.I., 1966. Google Scholar

[10]

J. Candy and W. Rozmus, A symplectic integration algorithm for separable Hamiltonian functions, J. Comp. Physiol., 92 (1991), 230-256. doi: 10.1016/0021-9991(91)90299-Z. Google Scholar

[11]

J. R. Cash and A. H. Karp, A variable order Runge-Kutta method for initial value problems with rapidly varying right-hand sides, ACM Trans. Math. Softw., 16 (1990), 201-222. doi: 10.1145/79505.79507. Google Scholar

[12]

J. E. Chambers, A hybrid symplectic integrator that permits close encounters between massive bodies, Mon. Not. R. Astron. Soc., 304 (1999), 793-799. doi: 10.1046/j.1365-8711.1999.02379.x. Google Scholar

[13]

M. Corbera and J. Llibre, Periodic orbits of the Sitnikov problem via a Poincaré map, Celest. Mech. Dyn. Astron., 77 (2000), 273-303. doi: 10.1023/A:1011117003713. Google Scholar

[14]

M. Corbera and J. Llibre, On symmetric periodic orbits of the elliptic Sitnikov problem via the analytic continuation method, Contemporary Mathematics, 292 (2002), 91-127. doi: 10.1090/conm/292/04918. Google Scholar

[15]

M. Corbera and J. Llibre, Symmetric periodic orbits for the elliptical 3–body problem via continution method, Developments in Mathematical and Experimental Physics, Klauaer, Acod., 2003, 113–137.Google Scholar

[16]

M. Corbera and J. Llibre, Periodic orbits of a collinear restricted three body problem, Celest. Mech. Dyn. Astron., 86 (2003), 163-183. doi: 10.1023/A:1024183003251. Google Scholar

[17]

J. M. Cors and J. Llibre, The global flow of the hyperbolic restricted three–body problem, Archive for Rational Mech. and Anal., 131 (1995), 335-358. doi: 10.1007/BF00380914. Google Scholar

[18]

J. M. Cors and J. Llibre, Qualitative study of the hyperbolic collision restricted three–body problem, Nonlinearity, 9 (1996), 1299-1316. doi: 10.1088/0951-7715/9/5/011. Google Scholar

[19]

J. M. Cors and J. Llibre, Qualitative study of the parabolic collision restricted three–body problem, Contemporary Math., 198 (1996), 1-19. doi: 10.1090/conm/198/02485. Google Scholar

[20]

J. M. Cors and J. Llibre, The global flow of the parabolic restricted three–body problem, Celest. Mech. Dyn. Astron., 90 (2004), 13-33. doi: 10.1007/s10569-004-4917-0. Google Scholar

[21]

M. Cuntz, S-type and p-type habitability in stellar binary systems: A comprehensive approach. i. method and applications, Astrophysical Journal, 780 (2014), 14. doi: 10.1088/0004-637X/780/1/14. Google Scholar

[22]

R. DvorakC. Froeschle and C. Froeschle, Stability of outer planetary orbits (p-types) in binaries, Astrophysics, 226 (1989), 335-342. Google Scholar

[23]

S. Eggl and R. Dvorak, An introduction to common numerical integration codes used in dynamical astronomy, In: Souchay, J., Dvorak, R. (eds.) Lecture Notes in Physics, vol. 790, Springer, Berlin, 2010,431–480. doi: 10.1007/978-3-642-04458-8_9. Google Scholar

[24]

S. Eggl, E. Pilat-Lohinger, N. Georgakarakos, M. Gyergyovits and B. Funk, An analytic method to determine habitable zones for s-type planetary orbits in binary star systems, Astrophysical J., 752 (2012), 74. doi: 10.1088/0004-637X/752/1/74. Google Scholar

[25]

S. M. Elshaboury, E. I. Abouelmagd, V. S. Kalantonis and E. A. Perdios, The planar restricted three–body problem when both primaries are triaxial rigid bodies: Equilibrium points and periodic orbits, Astrophys. Space Sci., 361 (2016), Paper No. 315, 18 pp. doi: 10.1007/s10509-016-2894-x. Google Scholar

[26]

G. Gómez, J. Llibre, R. Martínez and C. Simó, 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. doi: 10.1142/9789812810632_bmatter. Google Scholar

[27]

G. Gómez, J. Llibre, R. Martínez and C. Simó, 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

[28]

J. D. Hadjidemetriou, The continuation of Periodic orbits form the restricted to the general three–body problem, Celest. Mech. Dyn. Astron., 12 (1975), 155-174. doi: 10.1007/BF01230209. Google Scholar

[29]

A. Hanslmeier and R. Dvorak, Numerical integration with Lie series, Astron. Astrophys., 132 (1984), 203-207. Google Scholar

[30]

L. Kaltenegger and N. Haghighipour, Calculating the habitable zone of binary star systems. Ⅰ. s-type binaries, Astrophysical J., 777 (2013), 165. doi: 10.1088/0004-637X/777/2/165. Google Scholar

[31]

J. Llibre, On the restricted three–body problem when the mass parameter is small, Celestial Mechanics, 28 (1982), 83-105. doi: 10.1007/BF01230662. Google Scholar

[32]

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

[33]

J. Llibre and R. Ortega, On the families of periodic orbits of the Sitnikov problem, SIAM J. of Applied Dynamical Systems, 7 (2008), 561-576. doi: 10.1137/070695253. Google Scholar

[34]

J. Llibre and C. Piñol, On the elliptic restricted three–body problem, Celest. Mech. Dyn. Astron., 48 (1990), 319-345. doi: 10.1007/BF00049388. Google Scholar

[35]

J. Llibre and C. Stoica, Comet- and Hill-type periodic orbits in restricted (N+1)-body problems, J. Differential Equations, 250 (2010), 1747-1766. doi: 10.1016/j.jde.2010.08.005. Google Scholar

[36]

J. Llibre and D. G. Saari, Periodic orbits for the planar Newtonian three–body problem coming from the elliptical restricted three–body problems, Trans. Amer. Math. Soc., 347 (1995), 3017-3030. doi: 10.1090/S0002-9947-1995-1297534-X. Google Scholar

[37]

J. Llibre and C. Simó, Oscillatory solution in the planar restricted three body problem, Math. Annalen, 248 (1980), 153-184. doi: 10.1007/BF01421955. Google Scholar

[38]

C. R. McInnesA. J. C. McDonaldJ. F. L. Simmons and E. W. MacDonaldg, Solar sail parking in restricted three-body systems, J. Guid. Control Dyn., 17 (1994), 399-406. doi: 10.2514/3.21211. Google Scholar

[39]

K. R. Meyer, G. R. Hall and D. Offin, Introduction to Hamiltonian Dynamical Systems and the N-body Problem, Second edition. Applied Mathematical Sciences, 90, Springer, New York, 2009. doi: 10.1007/978-0-387-09724-4. Google Scholar

[40] C. D. Murray and S. F. Dermott, Solar System Dynamics, Cambridge University Press, 1999. Google Scholar
[41]

H. Poincaré, Leçons de Mécanique Céleste, Librairie Scientifique et Technique Albert Blanchard, Paris, 1987. Google Scholar

[42]

H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste, Vol. 1, 1892; Vol. 2, 1983; Vol. 3, 1899; Gauthier-Villars, Paris. Reprinted by Dover, New York, 1957.Google Scholar

[43]

F. J. T. SalazarC. R. McInnes and O. C. Winter, Periodic orbits for space-based reflectors in the circular restricted three–body problem, Celest. Mech. Dyn. Astron., 128 (2017), 95-113. doi: 10.1007/s10569-016-9739-3. Google Scholar

[44]

R. K. Sharma, The linear stability of libration points of the photogravitational restricted three-body problem when the smaller primary is an oblate spheroid, Astrophys. Space Sci., 135 (1987), 271-281. Google Scholar

[45]

S. Sternberg, Celestial mechanics, Part Ⅱ, W. A. Benjamin, Inc, New York. 1969.Google Scholar

[46]

P. Verrier and T. Waters, Families of periodic orbits for solar sails in the CRBTP, M. Macdonald(ed), Advances in Solar Sailing, Springer Praxis Books, Springer–Verlag Berlin Heidelberg, 2014, 871–884. doi: 10.1007/978-3-642-34907-2_52. Google Scholar

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