August & September  2019, 12(4&5): 703-710. 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, 2019, 12 (4&5) : 703-710. 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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[26]

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

[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. Google Scholar

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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[26]

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

[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. Google Scholar

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