doi: 10.3934/dcdss.2019044

Libration points in the restricted three-body problem: Euler angles, existence and stability

1. 

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

2. 

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

3. 

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

4. 

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

* Corresponding author: Elbaz I. Abouelmagd

Received  May 2017 Revised  January 2018 Published  November 2018

The objective of the present paper is to study in an analytical way the existence and the stability of the libration points, in the restricted three-body problem, when the primaries are triaxial rigid bodies in the case of the Euler angles of the rotational motion are equal to $ θ_i = π/2, \, ψ_i = 0, \,\varphi_i = π/2 $, $ i = 1, 2 $. We prove that the locations and the stability of the triangular points change according to the effect of the triaxiality of the primaries. Moreover, the solution of long and short periodic orbits for stable motion is presented.

Citation: Hadia H. Selim, Juan L. G. Guirao, Elbaz I. Abouelmagd. Libration points in the restricted three-body problem: Euler angles, existence and stability. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2019044
References:
[1]

E. I. AbouelmagdM. S. AlhothualiJ. L. G. Guirao and H. M. Malaikah, Periodic and secular solutions in the restricted three-body problem under the effect of zonal harmonic parameters, Appl. Math. & Info. Sci., 9 (2015), 1659-1669.

[2]

E. I. AbouelmagdM. S. AlhothualiJ. L. G. Guirao and H. M. Malaikah, On the periodic structure in the planar photogravitational Hill problem, Appl. Math. & Info. Sci., 9 (2015), 2409-2416.

[3]

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.

[4]

E. I. AbouelmagdF. AlzahraniJ. L. G. Guirao and A. Hobiny, Periodic orbits around the collinear libration points, J. Nonlinear Sci. Appl. (JNSA), 9 (2016), 1716-1727. doi: 10.22436/jnsa.009.04.27.

[5]

E. I. AbouelmagdH. M. Asiri and M. A. Sharaf, The effect of oblateness in the perturbed restricted three-body problem, Meccanica, 48 (2013), 2479-2490. doi: 10.1007/s11012-013-9762-3.

[6]

E. I. AbouelmagdM. E. AwadE. M. A. Elzayat and I. A. Abbas, Reduction the secular solution to periodic solution in the generalized restricted three-body problem, Astrophys. Space Sci., 350 (2014), 495-505.

[7]

E. I. Abouelmagd and S. M. El-Shaboury, Periodic orbits under combined effects of oblateness and radiation in the restricted problem of three bodies, Astrophys. Space Sci., 341 (2012), 331-341.

[8]

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.

[9]

E. I. Abouelmagd and M. A. Sharaf, The motion around the libration points in the restricted three-body problem with the effect of radiation and oblateness, Astrophys. Space Sci., 344 (2013), 321-332.

[10]

E. I. Abouelmagd, Stability of the triangular points under combined effects of radiation and oblateness in the restricted three-body problem, Earth Moon Planets, 110 (2013), 143-155.

[11]

E. I. Abouelmagd, The effect of photogravitational force and oblateness in the perturbed restricted three-body problem, Astrophys. Space Sci., 346 (2013), 51-69.

[12]

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.

[13]

E. I. Abouelmagd, A. Mostafa and J. L. G. Guirao, A first order automated Lie transform International Journal of Bifurcation and Chaos, 25 (2015), 1540026, 10pp. doi: 10.1142/S021812741540026X.

[14]

E. I. Abouelmagd and A. Mostafa, Out of plane equilibrium points locations and the forbidden movement regions in the restricted three-body problem with variable mass, Astrophys. Space Sci., 357 (2015), 58-68.

[15]

E. I. Abouelmagd and J. L. G. Guirao, On the perturbed restricted three-body problem, Applied Mathematics and Nonlinear Sciences, 1 (2016), 123-144.

[16]

F. AlzahraniE. I. AbouelmagdJ. L. G. Guirao and A. Hobiny, On the libration collinear points in the restricted three-body problem, Open Physics, 15 (2017), 58-67.

[17]

K. B. Bhatnagar and P. P. Hallan, Effect of perturbed potentials on the stability of libration points in the restricted problem, Celes. Mech. Dyn. Astr., 20 (1979), 95-103. doi: 10.1007/BF01230231.

[18]

R. BrouckeA. Elipe and A. Riaguas, On the figure-8 periodic solutions in the three-body problem, Chaos, Solitons and Fractals, 30 (2006), 513-520. doi: 10.1016/j.chaos.2005.11.082.

[19]

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.

[20]

S. W. McCusky, Introduction to Celestial Mechanics, Addision Wesley, 1963.

[21]

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.

[22]

R. K. SharmaZ. A. Taqvi and K. B. Bhatnagar, Existence of libration Points in the restricted three body problem when both primaries are triaxial rigid bodies, Indian J. Pure Appl. Math., 32 (2001), 125-141.

[23]

R. K. SharmaZ. A. Taqvi and K. B. Bhatnagar, Existence of libration Points in the restricted three body problem when both primaries are triaxial rigid bodies and source of radition, Indian J. Pure Appl. Math., 32 (2001), 981-994.

[24]

J. Singh and B. Ishwar, Stability of triangular points in the photogravitational restricted three body problem, Bull. Astr. Soc. India, 27 (1999), 415-424.

[25]

J. Singh and H. L. Mohammed, Robe's circular restricted three-body problem under oblate and triaxial primaries, Earth Moon Planets, 109 (2012), 1-11. doi: 10.1007/s11038-012-9397-8.

[26]

V. Szebehely, Theory of Orbits: The Restricted Three Body Problem, Academic Press, 1967.

[27]

F. B. Zazzera, F. Topputo and M. Mauro Massari, Assessment of Mission Design Including Utilization of Libration Points and Weak Stability Boundaries, ESA / ESTEC, 2005.

show all references

References:
[1]

E. I. AbouelmagdM. S. AlhothualiJ. L. G. Guirao and H. M. Malaikah, Periodic and secular solutions in the restricted three-body problem under the effect of zonal harmonic parameters, Appl. Math. & Info. Sci., 9 (2015), 1659-1669.

[2]

E. I. AbouelmagdM. S. AlhothualiJ. L. G. Guirao and H. M. Malaikah, On the periodic structure in the planar photogravitational Hill problem, Appl. Math. & Info. Sci., 9 (2015), 2409-2416.

[3]

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.

[4]

E. I. AbouelmagdF. AlzahraniJ. L. G. Guirao and A. Hobiny, Periodic orbits around the collinear libration points, J. Nonlinear Sci. Appl. (JNSA), 9 (2016), 1716-1727. doi: 10.22436/jnsa.009.04.27.

[5]

E. I. AbouelmagdH. M. Asiri and M. A. Sharaf, The effect of oblateness in the perturbed restricted three-body problem, Meccanica, 48 (2013), 2479-2490. doi: 10.1007/s11012-013-9762-3.

[6]

E. I. AbouelmagdM. E. AwadE. M. A. Elzayat and I. A. Abbas, Reduction the secular solution to periodic solution in the generalized restricted three-body problem, Astrophys. Space Sci., 350 (2014), 495-505.

[7]

E. I. Abouelmagd and S. M. El-Shaboury, Periodic orbits under combined effects of oblateness and radiation in the restricted problem of three bodies, Astrophys. Space Sci., 341 (2012), 331-341.

[8]

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.

[9]

E. I. Abouelmagd and M. A. Sharaf, The motion around the libration points in the restricted three-body problem with the effect of radiation and oblateness, Astrophys. Space Sci., 344 (2013), 321-332.

[10]

E. I. Abouelmagd, Stability of the triangular points under combined effects of radiation and oblateness in the restricted three-body problem, Earth Moon Planets, 110 (2013), 143-155.

[11]

E. I. Abouelmagd, The effect of photogravitational force and oblateness in the perturbed restricted three-body problem, Astrophys. Space Sci., 346 (2013), 51-69.

[12]

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.

[13]

E. I. Abouelmagd, A. Mostafa and J. L. G. Guirao, A first order automated Lie transform International Journal of Bifurcation and Chaos, 25 (2015), 1540026, 10pp. doi: 10.1142/S021812741540026X.

[14]

E. I. Abouelmagd and A. Mostafa, Out of plane equilibrium points locations and the forbidden movement regions in the restricted three-body problem with variable mass, Astrophys. Space Sci., 357 (2015), 58-68.

[15]

E. I. Abouelmagd and J. L. G. Guirao, On the perturbed restricted three-body problem, Applied Mathematics and Nonlinear Sciences, 1 (2016), 123-144.

[16]

F. AlzahraniE. I. AbouelmagdJ. L. G. Guirao and A. Hobiny, On the libration collinear points in the restricted three-body problem, Open Physics, 15 (2017), 58-67.

[17]

K. B. Bhatnagar and P. P. Hallan, Effect of perturbed potentials on the stability of libration points in the restricted problem, Celes. Mech. Dyn. Astr., 20 (1979), 95-103. doi: 10.1007/BF01230231.

[18]

R. BrouckeA. Elipe and A. Riaguas, On the figure-8 periodic solutions in the three-body problem, Chaos, Solitons and Fractals, 30 (2006), 513-520. doi: 10.1016/j.chaos.2005.11.082.

[19]

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.

[20]

S. W. McCusky, Introduction to Celestial Mechanics, Addision Wesley, 1963.

[21]

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.

[22]

R. K. SharmaZ. A. Taqvi and K. B. Bhatnagar, Existence of libration Points in the restricted three body problem when both primaries are triaxial rigid bodies, Indian J. Pure Appl. Math., 32 (2001), 125-141.

[23]

R. K. SharmaZ. A. Taqvi and K. B. Bhatnagar, Existence of libration Points in the restricted three body problem when both primaries are triaxial rigid bodies and source of radition, Indian J. Pure Appl. Math., 32 (2001), 981-994.

[24]

J. Singh and B. Ishwar, Stability of triangular points in the photogravitational restricted three body problem, Bull. Astr. Soc. India, 27 (1999), 415-424.

[25]

J. Singh and H. L. Mohammed, Robe's circular restricted three-body problem under oblate and triaxial primaries, Earth Moon Planets, 109 (2012), 1-11. doi: 10.1007/s11038-012-9397-8.

[26]

V. Szebehely, Theory of Orbits: The Restricted Three Body Problem, Academic Press, 1967.

[27]

F. B. Zazzera, F. Topputo and M. Mauro Massari, Assessment of Mission Design Including Utilization of Libration Points and Weak Stability Boundaries, ESA / ESTEC, 2005.

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