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Journal articles on the topic 'Collinear Libration points'

Author: Grafiati

Published: 5 June 2025

Last updated: 25 June 2025

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1

Muhammad, Amjad, Aslam Mohd., and Kashif Khan Mohd. "A MATHEMATICAL MODEL IN RESTRICTED THREE-BODY PROBLEM WHEN THE SMALLER PRIMARY IS A UNIFORM STRAIGHT ROD." International Journal of Education &Applied Sciences Research 1, no. 6 (2014): 07–20. https://doi.org/10.5281/zenodo.10686151.

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<strong>Abstract</strong>                  <em>In this paper we have shown the existence of libration points in the restricted three-body problem when the smaller primary is a uniform straight rod. It is found that there exist five libration points out of which L<strong><sub>i</sub> </strong>(i=1, 2, 3) are collinear and L<sub>4, 5 </sub>are non-collinear .The collinear libration points are unstable for 0 ≤ µ ≤ ½ and 0 < l <1 while non-collinear libration points are stable for critical value of mass parameter µ<sub>c </sub>= 0.381410285…. and length parameter l<sub>c</sub> =0.4.  </em> <strong><em> </em></strong> <strong><em>Key Words</em></strong><strong><em>: </em></strong><em>Celestial Mechanics</em><em>, Restricted three-body problem, Collinear Libration points,   Non-collinear Libration Points, Stability of Libration points.</em>

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2

Idrisi, M. Javed. "A study of libration points in CR3BP under albedo effect." International Journal of Advanced Astronomy 5, no. 1 (2016): 1. http://dx.doi.org/10.14419/ijaa.v5i1.6852.

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In this paper this is investigated how albedo perturbed the libration points from its original position? It is found that there exist five libration points, three collinear and two non-collinear and all the libration points are affected by Albedo. The non-collinear libration points are stable for a critical value of mass parameter µ ≤ µc, where µc = µo − (0.00891747 + 0.222579k)α (µo is the critical mass parameter for classical case) but collinear libration points are still unstable.

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3

Idrisi, M. Javed, and Mamta Jain. "Restricted three-body problem with stokes drag effect when less massive primary is an ellipsoid." International Journal of Advanced Astronomy 4, no. 1 (2016): 61. http://dx.doi.org/10.14419/ijaa.v4i1.6140.

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The present paper deals with the effect of Stokes drag force on the existence and stability of collinear and non-collinear libration points in circular restricted three-body problem when less massive primary is an ellipsoid. During the investigation, it is found that there exist five libration points Li (i = 1, 2… 5) out of which three are collinear and two are non-collinear. We observed that the Stokes drag force does not affect the collinear libration points while non-collinear libration points are affected by it and all the libration points either collinear or non-collinear are unstable in Lyapunov sense for the given range of dissipative constant k and mass parameter µ.

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4

Idrisi, M. Javed, та Muhammad Amjad. "Effect of elliptic angle φ on the existence and stability of libration points in restricted three-body problem in earth-moon system considering earth as an ellipsoid". International Journal of Advanced Astronomy 3, № 2 (2015): 87. http://dx.doi.org/10.14419/ijaa.v3i2.5313.

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<p>This paper deals with the existence and the stability of the earth-moon libration points in the restricted three-body problem. In this paper we have considered the bigger primary as an ellipsoid while the smaller one as a point-mass. This is observed that the collinear and non-collinear libration points exist only in the interval 0˚&lt;<em>φ </em>&lt; 45˚. There exist three collinear libration points and the non-collinear libration points are forming a right triangle with the primaries. Further observed that the libration points either collinear or non-collinear all are unstable in 0˚&lt;<em>φ </em>&lt; 45˚.</p>

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5

1M., Javed Idrisi, and Amjad Muhammad. "TWO MORE SOLUTIONS OF NON-COLLINEAR LIBRATION POINTS IN A PLANAR RESTRICTED THREE-BODY PROBLEM WHEN LESS MASSIVE PRIMARY IS AN OBLATE SPHEROID." International Journal of Education &Applied Sciences Research 2, no. 8 (2015): 01–14. https://doi.org/10.5281/zenodo.10691042.

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<strong>Abstract</strong> <em>In this paper the stability of non-collinear libration points in circular restricted three-body problem has been analyzed considering less massive primary as an oblate spheroid. We have considered two cases to find out the location and stability of non-collinear libration points. In case I, it is observed that there exist infinite numbers of non-collinear libration points on the unit circle centered at oblate body and out of these libration points only those lying in the interval  57˚ ≤ ψ ≤ 60˚ are stable for different values of critical mass parameter µ<sub>c</sub>, where ψ is the angle between m<sub>3</sub>, m<sub>2</sub> and m<sub>1</sub> in the same plane. In case II, </em><em>the non-collinear libration points exist only in the interval </em><em>0˚ ≤ φ ≤ 140˚ are stable for different values of critical mass parameter µ<sub>c</sub>, during the analysis<sub> </sub>we got a collinear libration point which is stable for µ<sub>c</sub> ≤ </em><em>0.127284… .</em> <em> </em> <strong><em>Key Words:</em></strong><em> Celestial Mechanics, Restricted three-body problem, Libration points, Linear stability.</em>

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6

M., JAVED IDRISI. "EXISTENCE AND STABILITY OF NON-COLLINEAR LIBRATION POINTS IN RESTRICTED THREE-BODY PROBLEM WHEN SMALLER PRIMARY IS A PROLATE SPHEROID." International Journal of Education &Applied Sciences Research 2, no. 3 (2015): 01–08. https://doi.org/10.5281/zenodo.10687004.

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<strong>Abstract</strong> In this paper the existence and stability of  non-collinear libration points in the restricted three-body problem has been discussed when the smaller primary is a prolate spheroid. We have determined the equations of motion of the infinitesimal mass and then we have investigated the non-collinear libration points and their stability in linear sense. This is found that there exist two non-collinear libration points are stable for a critical value of mass parameter     <em> µ</em> = <em>µ<sub>c</sub></em> ≤ 0.0559836.   <strong>Key Words:</strong> Celestial Mechanics, Restricted three-body problem, Prolate Spheroid, Libration points, Linear Stability

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7

Hou, X. Y., and L. Liu. "Bifurcating families around collinear libration points." Celestial Mechanics and Dynamical Astronomy 116, no. 3 (2013): 241–63. http://dx.doi.org/10.1007/s10569-013-9485-8.

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8

Idrisi, M. Javed, and Kumari Shalini. "Non-collinear libration points in CR3BP when less massive primary is an heterogeneous oblate body with N-layers." International Journal of Advanced Astronomy 4, no. 1 (2016): 39. http://dx.doi.org/10.14419/ijaa.v4i1.5928.

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<p>In the present paper, the existence of non-collinear libration points has been shown in circular restricted three-body problem when less massive primary is a heterogeneous oblate body with N-layers. Further, the stability of non-collinear libration points is investigated in linear sense and found that the non-collinear libration points are stable for the critical value of mass parameter <em>µ</em> ≤ <em>µ<sub>crit</sub></em>= <em>µ</em><sub>o</sub> – 3.32792 <em>k</em><sub>1</sub> – 1.16808 <em>k</em><sub>2</sub>.</p>

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9

Abouelmagd, E. I., F. Alzahrani, A. Hobiny, J. L. G. Guirao, and M. Alhothuali. "Periodic orbits around the collinear libration points." Journal of Nonlinear Sciences and Applications 09, no. 04 (2016): 1716–27. http://dx.doi.org/10.22436/jnsa.009.04.27.

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10

Hou, X. Y., and L. Liu. "ON LYAPUNOV FAMILIES AROUND COLLINEAR LIBRATION POINTS." Astronomical Journal 137, no. 6 (2009): 4577–85. http://dx.doi.org/10.1088/0004-6256/137/6/4577.

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11

Kreisman, B. B. "Stable spatial orbits around collinear libration points." Cosmic Research 48, no. 3 (2010): 265–72. http://dx.doi.org/10.1134/s0010952510030081.

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12

Tkhai, V. N. "Global families of periodic orbits adjacent to libration points in the restricted three-body problem." Astronomičeskij žurnal 101, no. 3 (2024): 263–70. http://dx.doi.org/10.31857/s0004629924030074.

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The restricted circular three-body problem is studied. All global families of periodic orbits adjacent to the libration points are found. A scenario for the evolution of orbits in the family is given. Chains of global families will be highlighted; the chain begins at the triangular libration point, contains global families for the triangular and all collinear libration points, and ends with a family whose orbits are pressed against the main bodies. The evolution of global families in the chain associated with changes in the energy of the system is described. Planar and spatial orbits are studied.

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13

Barden, B. T., and K. C. Howell. "Fundamental Motions Near Collinear Libration Points and Their Transitions." Journal of the Astronautical Sciences 46, no. 4 (1998): 361–78. http://dx.doi.org/10.1007/bf03546387.

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14

GIDEA, MARIAN, and JOSEP J. MASDEMONT. "GEOMETRY OF HOMOCLINIC CONNECTIONS IN A PLANAR CIRCULAR RESTRICTED THREE-BODY PROBLEM." International Journal of Bifurcation and Chaos 17, no. 04 (2007): 1151–69. http://dx.doi.org/10.1142/s0218127407017744.

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The stable and unstable invariant manifolds associated with Lyapunov orbits about the libration point L1between the primaries in the planar circular restricted three-body problem with equal masses are considered. The behavior of the intersections of these invariant manifolds for values of the energy between that of L1and the other collinear libration points L2, L3is studied using symbolic dynamics. Homoclinic orbits are classified according to the number of turns about the primaries.

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15

Popescu, Mihai. "Optimal transfer from collinear libration points with limited rotation speed." Journal of Guidance, Control, and Dynamics 12, no. 1 (1989): 119–21. http://dx.doi.org/10.2514/3.20377.

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16

Dzhumabayeva, A. A., A. L. Kunitsyn, and A. T. Tuyakbayev. "Stabilization of collinear libration points in the earth-moon system." Journal of Applied Mathematics and Mechanics 63, no. 2 (1999): 189–96. http://dx.doi.org/10.1016/s0021-8928(99)00026-x.

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17

Qian, Ying-Jing, Xiao-Dong Yang, Guan-Qiao Zhai, and Wei Zhang. "Planar periodic orbits’ construction around libration points with invariant manifold technique." Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering 233, no. 2 (2017): 498–509. http://dx.doi.org/10.1177/0954410017736544.

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This paper revisits the planar periodic motions around libration points in circular restricted three-body problem based on invariant manifold technique. The invariant manifold technique is applied to construct the nonlinear polynomial relations between ξ-direction and η-direction of a small celestial body during its periodic motion. Such direct nonlinear relations reduce the dimension of the dynamical system and facilitate convenient approximate analytical solutions. The nonlinear directional relations also provide terminal constraints for computing periodic motions. The method to construst periodic orbits proposed in this study presents a new point of view to explore the orbital dynamics. As an application in numerical simulations, nonlinear relations are adopted as topological terminal constraints to construct the periodic orbits with differential correction procedure. Numerical examples verify the validity of the proposed method for both collinear and triangular libration cases.

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18

Alzahrani, F., Elbaz I. Abouelmagd, Juan L. G. Guirao, and A. Hobiny. "On the libration collinear points in the restricted three – body problem." Open Physics 15, no. 1 (2017): 58–67. http://dx.doi.org/10.1515/phys-2017-0007.

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AbstractIn the restricted problem of three bodies when the primaries are triaxial rigid bodies, the necessary and sufficient conditions to find the locations of the three libration collinear points are stated. In addition, the Linear stability of these points is studied for the case of the Euler angles of rotational motion being θi = 0, ψi + φi = π/2, i = 1, 2 accordingly. We underline that the model studied in this paper has special importance in space dynamics when the third body moves in gravitational fields of planetary systems and particularly in a Jupiter model or a problem including an irregular asteroid.

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19

Rui, Zhong, and Xu Shijie. "Periodical Motion of a Tethered Satellite System around Collinear Libration Points." Journal of Aerospace Engineering 24, no. 4 (2011): 488–504. http://dx.doi.org/10.1061/(asce)as.1943-5525.0000077.

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20

Virgili Llop, Josep. "Autonomous optical navigation for orbits around Earth–Moon collinear libration points." Acta Astronautica 86 (May 2013): 119–25. http://dx.doi.org/10.1016/j.actaastro.2013.01.007.

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21

Zhang, Han-Qing, and Shuang Li. "Improved semi-analytical computation of center manifolds near collinear libration points." Research in Astronomy and Astrophysics 18, no. 11 (2018): 138. http://dx.doi.org/10.1088/1674-4527/18/11/138.

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22

Howell, K. C. "Families of Orbits in the Vicinity of the Collinear Libration Points." Journal of the Astronautical Sciences 49, no. 1 (2001): 107–25. http://dx.doi.org/10.1007/bf03546339.

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23

Tkhai, N. V. "On stability of the collinear libration points under internal third-order resonance." Automation and Remote Control 72, no. 9 (2011): 1906–10. http://dx.doi.org/10.1134/s0005117911090128.

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24

Kunitsyn, A. L. "The stability of collinear libration points in the photogravitational three-body problem." Journal of Applied Mathematics and Mechanics 65, no. 4 (2001): 703–6. http://dx.doi.org/10.1016/s0021-8928(01)00075-2.

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25

Ren, Yuan, and Jinjun Shan. "A novel algorithm for generating libration point orbits about the collinear points." Celestial Mechanics and Dynamical Astronomy 120, no. 1 (2014): 57–75. http://dx.doi.org/10.1007/s10569-014-9560-9.

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26

Kunitsyn, A. L., and A. T. Tureshbaev. "On the collinear libration points in the photo-gravitational three-body problem." Celestial Mechanics 35, no. 2 (1985): 105–12. http://dx.doi.org/10.1007/bf01227664.

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27

Jin, Yang, and Bo Xu. "A Modified Targeting Strategy for Station-Keeping of Libration Point Orbits in the Real Earth-Moon System." International Journal of Aerospace Engineering 2019 (September 4, 2019): 1–11. http://dx.doi.org/10.1155/2019/3257514.

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In this paper, a modified targeting strategy is developed for missions on libration point orbits (LPOs) in the real Earth-Moon system. In order to simulate a station-keeping procedure in a dynamic model as realistic as possible, LPOs generated in the circular restricted three-body problem (CRTBP) are discretized and reconverged in a geocentric inertial system for later simulations. After that, based on the dynamic property of the state transition matrix, a modified strategy of selecting target points for station-keeping is presented to reduce maneuver costs. By considering both the solar gravity and radiation pressure in a nominal LPO design, station-keeping simulations about fuel consumption for real LPOs around both collinear and triangular libration points are performed in a high-fidelity ephemeris model. Results show the effectivity of the modified strategy with total maneuver costs reduced by greater than 10% for maintaining triangular LPOs.

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28

Aslanov, Vladimir S., and Daria V. Neryadovskaya. "A Tether System at the L1, L2 Collinear Libration Points of the Mars–Phobos System: Analytical Solutions." Aerospace 10, no. 6 (2023): 541. http://dx.doi.org/10.3390/aerospace10060541.

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This paper is dedicated to identifying stable equilibrium positions of the tether systems attached to the L1 or L2 libration points of the Mars–Phobos system. The orbiting spacecraft deploying the tether is at the L1 or L2 libration point and is held at one of these unstable points by the low thrust of its engines. In this paper, the analysis is performed assuming that the tether length is constant. The equation of motion for the system in the polar reference frame is obtained. The stable equilibrium positions are found and the dependence of the tether angular oscillation period on the tether length is determined. An analytical solution in the vicinity of the stable equilibrium positions for small angles of deflection of the tether from the local vertical is obtained in Jacobi elliptic functions. The comparison of the numerical and analytical solutions for small angles of deflection is performed. The results show that the dependencies of the oscillation period on the length of the tether are fundamentally different for L1 and L2 points. Analytical expressions for the tether tension are derived, and the influence of system parameters on this force is investigated for static and dynamic cases.

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29

Pan, Zhengxu, Mai Bando, Zhanxia Zhu, and Shinji Hokamoto. "Design of Equilateral Array Polygonal Gravitational-Wave Observatory Formation near Lagrange Point L1—Equilateral Triangle and Equilateral Tetrahedral Configurations." Aerospace 11, no. 12 (2024): 1048. https://doi.org/10.3390/aerospace11121048.

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To observe lower-frequency gravitational waves (GWs), it is effective to utilize a large spacecraft formation baseline, spanning hundreds of thousands to millions of kilometers. To overcome the limitations of a gravitational-wave observatory (GWO) on specific orbits, a scientific observation mode and a non-scientific observation mode for GWOs are proposed. For the non-scientific observation mode, this paper designs equilateral triangle and equilateral tetrahedral array formations for a space-based GWO near a collinear libration point. A stable configuration is the prerequisite for a GWO; however, the motion near the collinear libration points is highly unstable. Therefore, the output regulation theory is applied. By leveraging the tracking aspect of the theory, the equilateral triangle and equilateral tetrahedral array formations are achieved. For an equilateral triangle array formation, two geometric configuration design methods are proposed, addressing the fuel consumption required for initialization and maintenance. To observe GWs in different directions and avoid configuration/reconfiguration, the multi-layer equilateral tetrahedral array formation is given. Additionally, the control errors are calculated. Finally, the effectiveness of the control method is demonstrated using the Sun–Earth circular-restricted three-body problem (CRTBP) and the ephemeris model located at Lagrange point L1.

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30

HAN, MingLin, Jing ZHOU, and Jun HU. "Differential correction method in low-thrust orbit transfers near the collinear libration points." SCIENTIA SINICA Technologica 51, no. 2 (2020): 207–20. http://dx.doi.org/10.1360/sst-2020-0140.

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31

Shymanchuk, D. V., A. S. Shmyrov, and V. A. Shmyrov. "Orbital Maneuvering in the Vicinity of Collinear Libration Points Using Light Pressure Forces." Astronomy Letters 47, no. 10 (2021): 710–19. http://dx.doi.org/10.1134/s1063773721100066.

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32

Ibrahim, A. H., M. N. Ismail, A. S. Zaghrout, S. H. Younis, and M. O. El Shikh. "Orbital Motion Around the Collinear Libration Points of the Restricted Three-Body Problem." Journal of Advances in Mathematics and Computer Science 29, no. 1 (2018): 1–16. http://dx.doi.org/10.9734/jamcs/2018/43370.

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33

Zhu, Min, Hamid Reza Karimi, Hui Zhang, Qing Gao, and Yong Wang. "Active Disturbance Rejection Station-Keeping Control of Unstable Orbits around Collinear Libration Points." Mathematical Problems in Engineering 2014 (2014): 1–14. http://dx.doi.org/10.1155/2014/410989.

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An active disturbance rejection station-keeping control scheme is derived and analyzed for station-keeping missions of spacecraft along a class of unstable periodic orbits near collinear libration points of the Sun-Earth system. It is an error driven, rather than model-based control law, essentially accounting for the independence of model accuracy and linearization. An extended state observer is designed to estimate the states in real time by setting an extended state, that is, the sum of unmodeled dynamic and external disturbance. This total disturbance is compensated by a nonlinear state error feedback controller based on the extended state observer. A nonlinear tracking differentiator is designed to obtain the velocity of the spacecraft since only position signals are available. In addition, the system contradiction between rapid response and overshoot can be effectively solved via arranging the transient process in tracking differentiator. Simulation results illustrate that the proposed method is adequate for station-keeping of unstable Halo orbits in the presence of system uncertainties, initial injection errors, solar radiation pressure, and perturbations of the eccentric nature of the Earth's orbit. It is also shown that the closed-loop control system performance is improved significantly using our method comparing with the general LQR method.

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34

Salnikova, T., E. Kugushev, and I. Lutkin. "Space debris in vicinity of collinear libration points of the Earth–Moon system." Acta Astronautica 225 (December 2024): 227–33. http://dx.doi.org/10.1016/j.actaastro.2024.09.015.

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35

Petukhov, V. G., and S. W. Yoon. "Optimization of a Low-Thrust Heliocentric Trajectory between the Collinear Libration Points of Different Planets." Космические исследования 61, no. 5 (2023): 406–19. http://dx.doi.org/10.31857/s0023420623700127.

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The aim of this study is to optimize a low-thrust interplanetary trajectory using collinear libration points L1 and L2 as the junction points of the geocentric or planetocentric segments of the trajectory with the heliocentric segment. The problem of optimizing the heliocentric segment of perturbed low-thrust interplanetary transfer is considered in the four-body ephemeris model, which includes the Sun, Earth, target planet, and spacecraft. To optimize the trajectories, an indirect approach is used based on Pontryagin’s maximum principles and the continuation method. The possibility of reducing the characteristic velocity in comparison with the estimates obtained through the method of zero sphere of influence is shown.

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36

Hou, X. Y., and L. Liu. "On motions around the collinear libration points in the elliptic restricted three-body problem." Monthly Notices of the Royal Astronomical Society 415, no. 4 (2011): 3552–60. http://dx.doi.org/10.1111/j.1365-2966.2011.18970.x.

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37

Héritier, Aurélie, and Kathleen C. Howell. "Natural Regions Near the Collinear Libration Points Ideal for Space Observations with Large Formations." Journal of the Astronautical Sciences 60, no. 1 (2013): 87–108. http://dx.doi.org/10.1007/s40295-014-0027-8.

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38

Ghorbani, Mehrdad, and Nima Assadian. "Optimal station-keeping near Earth–Moon collinear libration points using continuous and impulsive maneuvers." Advances in Space Research 52, no. 12 (2013): 2067–79. http://dx.doi.org/10.1016/j.asr.2013.09.021.

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39

Minglibayev, M. Zh, and T. M. Zhumabek. "ON THE RESTRICTED THREE-BODY PROBLEM." PHYSICO-MATHEMATICAL SERIES 2, no. 336 (2021): 138–44. http://dx.doi.org/10.32014/2021.2518-1726.33.

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The paper analytically investigates the classical restricted three-body problem in a special non-inertial central coordinate system, with the origin at center of forces. In this coordinate system, an analytical expression of the invariant of the centre of forces is given. The existence of the invariant of the centre of forces admits the correct division of the problem into two problems. The first is a triangular restricted three-body problem. The second is a collinear restricted three-body problem. In this paper the collinear restricted three-body problem is investigated. Using the properties of the invariant of centre of forces of the restricted three-body problem in the special non- inertial central coordinate system, the basic differential equations of motion for the collinear restricted three-body problem are obtained when three bodies lie on the same line during all motion. Differential equations of the collinear restricted three-body problem in the rotating non-inertial central coordinate system in pulsating variables are derived. New differential equations of motion for the collinear restricted three-body problem in three regions of possible location of the massless body with stationary solutions corresponding to the three Euler libration points have been derived. The circular collinear restricted three-body problem is investigated in detail. The corresponding Jacobi integrals are obtained. New exact non-stationary partial analytical solutions of the obtained new differential equations of motion of the collinear restricted three-body problem have been found for the considered case.

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40

Bekov, A. A. "The Libration Points and Hill Surfaces in the Restricted Problem of Three Variable-Mass Bodies." International Astronomical Union Colloquium 132 (1993): 277–88. http://dx.doi.org/10.1017/s0252921100066173.

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The paper deals with the study of the arising and disappearence of collinear (Eulerian) L1, L2, L3, triangular (Lagrangian) L4, L5, coplanar L6, L7, ring L0 and infinitely distant L±∞ solutions in a restricted problem of three variable-mass bodies for different time dependencies of main bodies masses and for some additional conditions imposed on the systems parameters. In this case it is assumed that the motion of variable-mass main bodies is determined by the Gylden-Mestschersky problem. The Bill surfaces in the restricted three-body problem where main bodies masses variate isotropically according to the Mestschersky law are studied. Certain possibilities of applying the results of investigations to nonstationary double stellar systems are discussed.

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41

Singh, Jagadish, and Joel John Taura. "Collinear Libration Points in the Photogravitational CR3BP with Zonal Harmonics and Potential from a Belt." International Journal of Astronomy and Astrophysics 05, no. 03 (2015): 155–65. http://dx.doi.org/10.4236/ijaa.2015.53020.

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42

Ibrahim, Ahmed H., Mohammed N. Ismail, Afaf S. Zaghrout, Saher H. Younis, and Mayadah O. El Shikh. "Lissajous Orbits at the Collinear Libration Points in the Restricted Three-Body Problem with Oblateness." World Journal of Mechanics 08, no. 06 (2018): 242–52. http://dx.doi.org/10.4236/wjm.2018.86020.

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43

Hou, X. Y., and L. Liu. "On quasi-periodic motions around the collinear libration points in the real Earth–Moon system." Celestial Mechanics and Dynamical Astronomy 110, no. 1 (2011): 71–98. http://dx.doi.org/10.1007/s10569-011-9340-8.

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44

Nazirov, R. R., N. A. Eismont, E. N. Chumachenko, D. W. Dunham, I. V. Logashina, and A. N. Fedorenko. "Control of spacecraft groupings near sun-earth collinear libration points by means of solar sails." Russian Engineering Research 33, no. 5 (2013): 265–68. http://dx.doi.org/10.3103/s1068798x13050109.

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Petukhov, V. G., and S. W. Yoon. "Optimization of a Low-Thrust Heliocentric Trajectory between the Collinear Libration Points of Different Planets." Cosmic Research 61, no. 5 (2023): 418–30. http://dx.doi.org/10.1134/s0010952523700351.

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Singh, Jagadish, and Aishetu Umar. "The Collinear Libration Points in the Elliptic R3BP with a Triaxial Primary and an Oblate Secondary." International Journal of Astronomy and Astrophysics 04, no. 02 (2014): 391–98. http://dx.doi.org/10.4236/ijaa.2014.42034.

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Singh, Jagadish, and Aishetu Umar. "On motion around the collinear libration points in the elliptic R3BP with a bigger triaxial primary." New Astronomy 29 (May 2014): 36–41. http://dx.doi.org/10.1016/j.newast.2013.11.003.

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Gao, Fabao, and Yongqing Wang. "Approximate Analytical Periodic Solutions to the Restricted Three-Body Problem with Perturbation, Oblateness, Radiation and Varying Mass." Universe 6, no. 8 (2020): 110. http://dx.doi.org/10.3390/universe6080110.

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Abstract:

Against the background of a restricted three-body problem consisting of a supergiant eclipsing binary system, the two primaries are composed of a pair of bright oblate stars whose mass changes with time. The zero-velocity surface and curve of the problem are numerically studied to describe the third body’s motion area, and the corresponding five libration points are obtained. Moreover, the effect of small perturbations, Coriolis and centrifugal forces, radiative pressure, and the oblateness and mass parameters of the two primaries on the third body’s dynamic behavior is discussed through the bifurcation diagram. Furthermore, the second- and third-order approximate analytical periodic solutions around the collinear solution point L3 in two-dimensional plane and three-dimensional spaces are presented by using the Lindstedt-Poincaré perturbation method.

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Aslanov, Vladimir S. "A splitting of collinear libration points in circular restricted three-body problem by an artificial electrostatic field." Nonlinear Dynamics 103, no. 3 (2021): 2451–60. http://dx.doi.org/10.1007/s11071-021-06226-4.

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Tureshbaev, A. T., and U. Sh Omarova. "SUSTAINABILITYOF COLLINEAR LIBRATION POINTS IN A SPATIAL GENERALIZED THREE-BODY PHOTOGRAVITATION PROBLEM WITH RESONANCE OF THIRD ORDER." National Association of Scientists 3, no. 50 (2019): 42–44. http://dx.doi.org/10.31618/nas.2413-5291.2019.3.50.124.

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