The restricted three-body problem (RTBP) is defined as a system where an infinitesimal mass, m3, is attracted gravitationally by two finite arbitrary masses called primaries, m1 and m2, but their motion is not influenced. The term restricted comes from the fact that all masses are assumed to move in the same plane defined by the two revolving primaries which revolve around their center of mass in circular orbits. This definition is widely used in almost all classical books of celestial mechanics, e.g. Szebehely  and Murray and Drmott . For a more generalized version of the problem many authors have amended the potential function with some relevant perturbing forces; e.g. considering oblate primaries instead of spherical masses. Or even more generalized as triaxial bodies, inclusion of the relativistic effects, assuming the primaries are emitters, and/or move in a resisting medium. Even if the RTBP is not integrable, a number of special solutions can be found in the rotating frame where the third body has zero velocity and zero acceleration. These solutions correspond to equilibrium positions in the rotating frame at which the gravitational forces and the centrifugal force associated with the rotation of the synodic frame all cancel, with the result that a particle located at one of these points appears stationary in the synodic frame. There are five equilibrium points in the circular RTBP, three of them are collinear points, namely L1, L2, L3 and the another two are triangular points, namely L4, L5. The position of the infinitesimal body is displaced a little from the equilibrium point due to the some perturbations. If the resultant motion of the infinitesimal mass is a rapid departure from the vicinity of the point, we can call such a position of equilibrium point an “unstable” one, if however the body merely oscillates about the equilibrium point, it is said to be a “stable position” (in the sense of Lyapunov), Abd El-Salam . In general, the dynamics of a circular and/or elliptical three-body problem is widely applicable toastrophysics, for example stellar/solar system dynamics and Earth-Moon system.This problem consequently received more attention from astronomers and dynamical system scientists. In spite of it, the solutions of this problem has been developed over the past centuries.
The literature concerning RTBPis extensive and it is worth highlighting here some relevant and recent studies dealingwith RTBP, with and without considering different perturbations: Sharma , Tsirogiannis et al. , Kushvah and Ishwar , Vishnu Namboori et al. , Mital et al. , Kumar and Ishwar , Rahoma and Abd El-Salam  and references therein. Ammar  analyzed solar radiation pressure effect on the positions and stability of the libration points in elliptic RTBP. Singh  formulated the triangular librationpoints nonlinear stability under the Coriolis effect and centrifugal forces as small perturbations in addition to the effect of primaries oblateness and radiation pressures. Singh and Umar  investigated the effect of luminous and oblate spheroids primaries on the locations and stability of the collinear libration points. In another work, Singh and Umar  studied the effect of the big primary’s triaxial and spherical shape of the companion on the locations and stability of the collinear libration points. They found that the position of collinear libration points and their stability are affected with their considered perturbations in addition to the eccentricity and the semi-major axis of the primaries orbit.
Katour et al. , Singh and Bello [16, 17], Abd El-Salam and Abd El-Bar , Abd El-Bar et al.  and Bello and Singh  were concerned with the relativistic RTBP in addition to some different perturbations—the primaries oblateness andradiation from one of the primaries—upon the equilibrium points locations and stability. They noticed that the stability regions of the concerned equilibrium points are varied (expanding or shrinking) related to the critical mass value and depending upon the value of their considered perturbations.
The fundamental structure in nature and science such as RTBP, plasma containment in tokamaks and stellarators for energy generation, population ecology, chaoticbehavior in biological systems, neural networks and solitonicfibre optical communication devices can be expound by nonlinear partial differential equations [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37].
The aim of this study is the determination of the locations of the collinear equilibrium points, taking into consideration the fact that both primaries are triaxial. This paper will be organized as follows: Section 1 is an historical introduction, the equations of motion in an RTBP with triaxial primaries are formulated in Section 2, the computations related to the location of equilibrium points with the considered perturbations are introduced in section 3. Section 4 highlightssomenumerical simulations with a discussion and forthcoming works. The paper finishes with a conclusion in Section 5.
2 Equations of motion
We shall adopt the notation and terminology of Szebehely . As a consequence, the distance between the primaries m1 and m2 does not change and is taken equal to one; the sum of the masses of the primaries is also taken as one. The unit of time is also chosen as to make the gravitational constant unity.
The equations of motion of the infinitesimal mass m3 in the RTBP in a synodic co-ordinate system(x, y)in dimensionless variables in which the primary coordinates on the x-axis (–μ, 0), (1 – μ, 0) are given by Brumberg (1972).
where U is the potential–like function of the RTBP, which can be written as composed of two components (compound), namely the potential of the classical RTBP potential U are given by:
m1, m2(m1 ≥ m2) being the masses of the primaries,
ai, bi, ci, i = 1, 2 are the semi-axes of the triaxial of two primaries respectively and R the distance between the primaries.
The mean motion n of the primaries is given by
The oblateness case of the primaries can be deduced directly from applying the condition ai = bi, i = 1, 2 or alternatively σ1 = σ2.
The libration points are obtained from equations of motion after setting ẍ = ÿ = ẋ = ẏ = 0. These points represent particular solutions of equations of motion
The explicit formulas are
3 Location of collinear libration points
4 Location of L1
The location geometry of L1 can be visualized as given by Figure 1.
Then it may be reasonable in our case to assume that positions of the equilibrium points L1 are the same as given by classical RTBP but perturbed due to the triaxial primaries
from which we have
where a1 and b1 are unperturbed positions of r1 and r2 respectively, and b1 is given after some successive approximation by
Since y = 0, then the above equation may be written as
which can be solved for ε1 yielding
Using the above relations, equation (10) can be written in the form
where b1, d1, e1, f1, g1 and h1 are function of μ and they are given by
where the coefficients 𝓝i1 are given appendix (D-L1)
Substituting back into equation (11), r2 is being a function of b1, d1, e1, f1, g1 and h1. It can be written in the form
since the location of L1 is given by
where the coefficients are given appendix (B-L1)
4.1 Location of L2
The geometry of L2 can be visualized as given by Figure 2.
Follow the same procedure as with L1, with the corresponding values of𝓑1 = 1, 𝓑2 = −1. Substituting into (5) we get
Then it may be reasonable in our case to assume that position of the equilibrium point L2 is the same as given by classical (RTBP) but perturbed due to the triaxial primaries
where a2 and b2 are the unperturbed positions of r1 and r2 respectively, and b2 is given after some successive approximation by the relation
Since y = 0, then the above equation may be written as
Equation (19) can be solved for ε2 yielding
Using the above relations, equation (20) can be written in the form
where the coefficients 𝓝 are given appendix (D-L2) and where, d2, e2, f2, g2 and h2 are functions of μ andare given by
Substituting back into equation (21), r2 is a function of b2, d2, e2, f2, g2 and h2 and can be written in the form
since the location of L2 is given by
where the coefficients are given appendix (A-L2)
4.2 Location of L3
The geometry of L3 can be visualized as given by Figure 2.
Follow the same procedure as in L1, with the corresponding values of 𝓑1 = – 1, 𝓑2 = 1, substituting into (5) we get
Then it may be reasonable in our case to assume that position of the equilibrium point L3 is the same as given by classical restricted three-body problem but perturbed due tothe triaxial primaries
where a3 and b3 are the unperturbed values of r1 and r2 respectively, and b3 is given after some successive approximation by the relation
Since y = 0, then the above equation may be written as
Equation (28) can be solved for ε3 to yield
Using the above relations, equation (29) can be written in the form
where d3, e3, f3, g3 and h3 are functions of μ. They are given by
where the coefficients 𝓝i1 are given appendix (D-L3).
Substituting back into equation (29) to yield r2 as a function of b3, d3, e3, f3, g3 and h3. It can be rewritten in the form
since the location of L3 is given by
where the coefficients are given Appendix (A-L3)
5 Numerical representations
In the following section, we will draw the locations of collinear points Li, i = 1, 2, 3 versus the mass ratio μ ∈ (0, 0.1) taking into account the effect oftrixial primaries.In all casesthe black curve represents the classical unperturbed RTBP in which the primaries are considered spheres. The blue and the red curves represent the increasing magnitude in the perturbing parameters.
5.1 Analysis of L1 location
Considering the massive primary as oblate, it can be seen in Figure 4a that the location of is shifted towards the massive primary, i.e. towards the barycenter. Since any equilibrium point emerges from the balance between the gravitational field and the rotational field of the primaries, we can conclude that the resultant of these forces is to perturb the location of L1 towards the massive primary. This is very logical dynamical effect, since the additional mass bulge due to oblateness of the massive primarycauses gravitational attraction towards the center. The effect is noticeable for a relatively higher mass ratio. In view of Figure 4b, and considering the less massive primary as oblate, the location of L1 is largely shifted towards the less massive primary, i.e. away from the barycenter. The effect of perturbation is much bigger than that in Figure 4a due to the close proximity of the point to the less massive primary.
In view of the Figure 4c and Figure 4d, the dynamical effects are towards and away from the barycenter, respectively. The size of perturbation is small comparedto the effects of oblateness due to the magnitude of the perturbing parameter.
5.2 Analysis of L2 location
From Figure 4e, considering the massive primary as oblate, the location of L2 is shifted towards the barycenter for mass ratios larger that the classical critical mass ratio, namely the Routian value μc = 0.03841. In Figure 4f, considering the less massive primary as oblate, the location of L2 is also largely shifted towards to thebarycenter. The effect of perturbation is much bigger than that shown in Figure 4e due to the close proximity of the point to the less massive primary.
Referring to Figure 4g and Figure 4h, the dynamical effects are towards and away from the barycenter respectively. These effects could be easily interpreted by considering balancing between the gravitational and rotational fields. The size of perturbation is small comparedto the effects of oblateness due to the magnitude of the perturbing parameter.
5.3 Analysis of L3 location
The Figures 4i-4l show the locations of L3 considering the massive primary as oblate in 4i and the less massive primary as oblate in Figure 4j. Figure 4k and 4l show the perturbed location of L3 due to the triaxiality of the primaries. It is very clear that the dynamics are dominated by the massive primary.
We have treated the perturbed RTBP with the triaxiality of both primaries up to the terms of order 𝓞 perturbations. As expected, these perturbations bring deviations of the locations of the equilibrium points from classical RTBP. In this work, we computed and illustrated these deviations in collinear points explicitly as functions in the mass ratio. We analyzed the oblate RTBP as special cases of a triaxialproblem. All the dynamical effects could be properly interpreted in view of balancing between the gravitational and rotational fields. It is observed that the dynamics of L3 are clearly dominated by the massive primary, while the dynamics of L1 and L2 are dominatedby the less massive primary due to their close proximity to it. When investigating the triaxiality effects, it is noticed that the size of perturbation is small compared to the effects of oblateness due to the magnitude of the perturbing parameter.
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About the article
Published Online: 2018-09-06
Citation Information: Open Physics, Volume 16, Issue 1, Pages 525–538, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2018-0069.
© 2018 S. Z. Alamri et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0