In recent years, the importance and significance of the libration points for space applications has increased within the scientific community. This is because these points are natural equilibrium solutions of the restricted three-body problem and offer the unique possibility to obtain a fixed configuration with respect to two primaries. Thereby, the solution to libration points could alleviate a lot of mission constraints which are not realizable with the classical Keplerian two-body orbits. Moreover, exploiting the stable and unstable part of the dynamics regarding these points, low-energy interplanetary, moon-to-moon transfers of practical interest can be obtained. Around each of the three collinear equilibrium points a family of unstable orbits exists, see Abouelmagd et al. . These orbits are useful for many space applications requiring a fixed configuration with respect to two primary bodies. The orbits are also useful when the calculation of planar Lyapunov orbits that emerge from these points is necessary and the ballistically captured transfers are needed, for more details see Koon et al. 
The model of the three-body problem is used to determine the possible motions of three bodies which attract each other according to Newton’s law of inverse squares. It started with Newton’s perturbative studies on the inequalities of the lunar motion. In physics and classical mechanics the problem has two conspicuous meanings:
In its conventional sense, the problem yields an initial set of data that characterize the positions and velocities of three bodies at a specified time. In accordance with the laws of classical mechanics, the motions of the three bodies can be determined.
In an extended modern sense, the three-body problem is a class of problems in classical or quantum mechanics that model the motion of three particles.
Historically, the first specific three-body problem receiving extended study was the one involving the lunar theory, the motion of the Moon under the gravitational influence of the Earth and the Sun. Improving the accuracy of the lunar theory came to be of topical interest at the end of the eighteenth century. This interest mainly arose from the belief that the lunar theory could be applicable to navigation at sea in the development of a method for determining geographical longitude. Following Newton’s work it was appreciated that a major part of the problem in lunar theory consisted in evaluating the perturbing effect of the Sun on the motion of the Moon around the Earth.
Some significance of the three-body problem comes from two links. For the first, it is considered a special case of the n-body problem, which describes how n objects will move under one of the physical forces, such as gravity. These problems have an analytical solution in the form of a convergent power series. For the second link, the problem can be reduced to a perturbed two body problem. Some applications are conveniently treated by perturbation theory, in which the system is considered as a two-body problem plus additional forces causing deviations from a hypothetical unperturbed two-body trajectory.
The restricted three-body problem under the effects of perturbed forces has been receiving considerable attention from researchers. For instance, the existence of libration points, their stability and the periodic orbits in the proximity of these points under the oblateness, triaxialty of the primaries or the effect of photogravitational force or combination of them are studied by Sharma , Singh and Ishwar , Sharma et al. [5, 6], Singh and Mohammed , Abouelmagd and El-Shaboury , Abouelmagd , Abouelmagd [10, 11], Abouelmagd et al. , Abouelmagd and Sharaf , Abouelmagd et al. [14, 15], Abouelmagd et al. [1, 16, 17, 18], Abouelmagd and Mostafa , Abouelmagd et al. , Abouelmagd and Guirao .
In the framework of the restricted three-body problem, for the case of rigid bodies having an axis and plane of symmetry, the conditions for existence of collinear and equilateral equilibrium solutions have been studied by Vidyakin  and Duboshin .
Sharma et al. [5, 6] studied the existence and stability of libration points in the restricted three-body problem when both the primaries are triaxial rigid bodies in the case of stationary rotational motion (θi = ψi = φi = 0). While  studies the stability of infinitesimal motions about the triangular equilibrium points in the elliptic restricted three body problem when the bigger primary is radiating and the smaller is a triaxial rigid body. They used a technique based on Floquetï£¡s Theory for determination of characteristic exponents in the system with periodic coefficients.
The basic dynamical features of the restricted three–body problem when the primaries are triaxial rigid bodies are studied by . The equilibrium libration points are identified and their stability is determined in the special cases when the Euler’s angles of rotational motion are θi = ψi = φi = π/2 and θi = ψi = π/2, φi = 0, i = 1, 2 accordingly. They proved that there are three unstable collinear equilibrium points and two triangular such points which may be stable. Special attention has also been paid to the study of simple symmetric periodic orbits and 31 families consisting of such orbits have been determined. It has been found that only one of these families consists entirely of unstable members while the remaining families contain stable parts indicating that other families bifurcate from them. Finally, using the grid search technique, a global solution in the space of initial conditions is obtained and which is comprised by simple and of higher multiplicities symmetric periodic orbits as well as escape and collision orbits.
In this paper, we consider the restricted three-body problem when both primaries are triaxial rigid bodies in two cases of stationary rotational motion θi = 0, ψi + φi = π/2 and θi = 0, ψi + φi = 0 where i = 1, 2,. The necessary and sufficient conditions to find the locations of the three collinear points are found in five cases. The linear stability of motion in the proximity of these points is also studied. This work is organized as follows. An overview of the significance, some applications of the three-body problem and the aim of the present work have been discussed in this section. A background on the restricted three-body problem when the primaries are triaxial rigid bodies in the general case of Euler angles of rotational motion are given in Section 2. In Section 3 the equations of motion are found when Euler angles of rotational motion are θi = 0, ψi + φi = π/2. In Section 4 the conditions of existence of the three libration collinear points are studied in five different cases, while in Section 5 the linear stability of motion around the libration collinear points are investigated. Finally, a conclusion is sketched in Section 6 and we refer to how one can obtain the corresponding results in the case of Euler angles of rotational motion being θi = 0 and ψi + φi = 0, i = 1, 2.
Let (X1, Y1, Z1), (X2, Y2, Z2) and (X, Y, Z) be the coordinates of the masses m1, m2 and m in a sidereal frame respectively. m1 and m2 are the primaries moving in a circular orbit around their center of mass and m is the mass of the infinitesimal body that moves in the same plane of the primaries under their gravitational field without affecting their motion. Now we assume that the distance between the primaries and the sum of their masses are taken equal to unity, while the unit of time is chosen so as to make the gravitational constant unity too. The principal axes of the primaries are oriented to the synodic axes by Euler angles θi, ψi and φi, i = 1, 2. In addition we assume that r1 and r2 are the distances of m from m1 and m2 respectively, where (1)
furthermore we also suppose that μ1 = m1 = 1 − μ and μ2 = m2 = μ where μ ∈ (0, 1/2] denotes the mass ratio. Therefore, the coordinates of the three masses m1, m2 and m can be written in a synodic frame as (μ, 0, 0), (μ − 1, 0, 0) and (x, y, z), correspondingly.
Since the principal axes are supposed to rotate with the same angular velocity as that of the rigid bodies and the bodies are moving around their center of mass without rotation, the Euler angles remain constant throughout the motion. Thereby the equations of motion of the infinitesimal mass m in a synodic coordinate system with dimensionless variables are governed in the form [see 5] (2)
where Ω is the potential function which is given by [see also 26] (3)
and n is the perturbed mean motion. Also, (I1i, I2i, I3i) are the principal moments of inertia of the triaxial rigid body of mass mi at its center of mass with (ai, bi, ci) as its axes, while Ii is the moment of inertia about the connected line between the rigid body mi with the center of the infinitesimal body of mass m, and i = 1, 2 such that Iij ≠ Iji and Ii are controlled by (4)
where (li, mi, ni) are the direction cosines of the connected line with respect to the principal axes of mi.
Now we shall adopt the notation and terminology of  and follow his procedure, then we denote the unit vectors along the principal axes at m1 or m2 by (i, j, k) and the unit vectors parallel to the synodic coordinates axes by (e1, e2, e3), and with the help of Euler angles (θi, ψi, φi), the relation between vectors can be expressed as: (5)
the elements of matrix A can be determined by the following vectors (7)
Now the potential function Ω in Eq. (3) can be rewritten in the form (8)
and the mean motion n is governed by (10)
where (11) (12)
and R is the separation distance between the primaries, i = 1, 2, see .
3 Equations of motion when θi = 0, ψi + φi = π/2
In the case of the Euler angels of rotational motion being θi = 0, ψi + φi = π/2, we obtain a2i = c3i = 1, b1i = − 1 and the other parameters are equal to zero. The equation of motion Eq. (2) can be rewritten in the form (13)
and the mean motion n is governed by (16)
simply the mean motion will take the form: (18)
Regarding Eq. (18), the mean motion when the rotational motion of Euler angles are θi = 0, ψi + φi = π/2 can be analyzed for several cases:
If the primaries are spherical bodies (classical problem), see the book  (A1i = A2i = A3i = 0, i = 1,2), thereby one obtains unperturbed mean motion n = 1.
If the bigger primary is an oblate spheroid and the smaller is a spherical body  (A11 = A21, A31 ≠ 0, A12 = A22 = A32 = 0), then the perturbed mean motion is faster than the unperturbed motion (Keplerian motion) for the oblate body with A11 > A31. However, If A31 > A11 (when the bigger primary is a prolate body) the perturbed mean motion is slower than the Keplerian motion.
If the bigger primary is a spherical body and the smaller is an oblate spheroid  (A11 = A21 = A31 = 0, A12 = A22, A32 ≠ 0), then the perturbed mean motion is faster than the unperturbed motion for the oblate body with A11 > A31. However, If A32 > A12 (when the smaller primary is a prolate body) the perturbed mean motion is slower than the Keplerian motion.
If both primaries are oblate spheroids (A1i = A2i, A3i ≠ 0, i = 1, 2), +(A12 − A32)] , then the perturbed mean motion is faster than the unperturbed motion for oblate bodies when A1i > A3i. However, If A3i > A1i (when the primaries are prolate bodies) the perturbed mean motion is slower than the Keplerian motion. If one of the primaries is oblate and the other is prolate then the perturbed mean motion will be faster or slower than the unperturbed mean motion depending on whether the sign of (A11 − A31) + A12 − A32) is positive or negative.
In the case of the rotational motion of Euler angles being (θi = 0, ψi + φi = π/2), the perturbed mean motion will be faster or slower than the Keplerian motion in the following cases
If one of the primaries is spherical and the other is a triaxial body, then the perturbed mean motion is faster or slower than the Keplerian motion, according to whether the sign of (2A2i − A1i − A3i) is positive or negative.
If both primaries are triaxial rigid bodies, then the perturbed mean motion is faster than the Keplerian motion when or (2A21 − A11 − A31) + (2A22 − A12 − A32) > 0, otherwise it will be slower.
4 Existence of libration collinear points when θi = 0, ψi + φi = π/2
In general the libration points are the equilibria solutions of the dynamical system, which describes the motion of an infinitesimal body.
Since Ω = Ω(x, y), then (19)
The positions of the equilibrium points are the solutions of the following equations (21)
Since the principal axes are different in triaxial rigid bodies we can assume that the triaxial rigid body of mass mi, i = 1, 2, be nearly a sphere with radius R0i and thereby one obtains (24)
and the mean motion will take the form (28)
where r1 = |x − μ| and r2 = |x − μ + 1|.
Hence we can rewrite Eq. (29) as (30)
To investigate the existence of libration collinear points and determine their locations we have to study the behavior of the function f. In this context the derivative of the function f will be controlled by (34)
Now Eqs. (46–48) show that the signs of the fifth and seventh terms of the function f′ will be effected by the values of (2σ2i − σ1i − σ3i), (i = 1, 2), according to whether both of these values are positive or negative or with different signs, while the sum of the first three terms is not affected and will positive all the time because σ1i, σ2i, σ3i ≪ 1 and δi < 1. Thereby we will investigate under which conditions f′(x) > 0 in the open intervals (−∞, μ − 1), (μ − 1, μ) and (μ, ∞) and we need to find the the necessary and sufficient conditions which makes (35)
In this context we may analyze five cases, which lead to f'(x) > 0, these cases are:
(σ21 ≥ σ11 ≥ σ31) and (σ22 ≥ σ12 ≥ σ32),
(2σ21 − σ11 − σ31) ≥ 0 and (2σ22 − σ12 − σ32) ≥ 0,
(2σ21 − σ11 − σ31) ≤ 0 and (2σ22 − σ12 − σ32) ≤ 0,
(2σ21 − σ11 − σ31) ≥ 0 and (2σ22 − σ12 − σ32) ≤ 0,
(2σ21 − σ11 − σ31) ≤ 0 and (2σ22 − σ12 − σ32) ≥ 0.
Since the parameters μ, δi, σ1i, σ2i, σ3i as well as the quantity 1 − μ are positive, with regard to the previous five cases the necessary and sufficient conditions can be translated to the unequal equations (36–40) respectively.
(36) (37) (38) (39) (40)
where the derivatives f′1, f′2 and f′3 are given in Appendix for every case.
Hence f′(x) > 0 in each of the open intervals (−∞, μ − 1), (μ − 1, μ) and (μ, ∞) and it follows that f(x) is strictly increasing in these intervals too. In addition Eq. (30) shows that
Hence, we deduce that, f(μ − 2) < 0, f[(μ − 1)−] > 0;f[(μ − 1)+] < 0, f(0) > 0 and f(μ+) < 0, f(μ + 1) > 0 and we concludes that there are only three zeros for f(x), when y = 0, which lie in the intervals (μ − 2, μ − 1), (μ − 1, 0) and (μ, μ + 1). Hence, there are three collinear libration points that lie in the intervals(μ − 2, μ − 1), (μ − 1, 0) and (μ, μ + 1), respectively. We will denote these by the symbols L1, L2 and L3
Now we can establish that there exists one and only one real value for x in each of the open intervals (−∞, μ − 1), (μ − 1, μ) and (μ, ∞) such that f(x) = 0 and the function f(x) is strictly increasing in these intervals, when one of the five conditions is achieved, otherwise we may have more than three collinear points.
5 Stability of the libration collinear points
To study the stability of motion around the libration collinear points we assume that (x0, y0) is one of the coordinates of these points and ξ and η are the variation which describe the possible motion of the infinitesimal body around one libration collinear points, where this variation is defined as (41)
where the partial derivatives of the second order of Ω are denoted by the subscripts x, y and the superscript 0 indicates that such derivative is evaluated at one of the libration collinear points. Hence the associated characteristic equation to Eqs. (42) is (43)
The character of the solution of the variational dynamical system depends on the character of the solution for ω2 from the quadratic of Eq. (43). The solution is stable only if the quadratic has two unequal negative roots for ω2, see  for more details.
For the case of (σ21 ≥ σ11 ≥ σ31) and (σ22 ≥ σ12 ≥ σ32) and additionally L1, L2 and L3 laying in the intervals (μ − 2, μ − 1), (μ − 1, 0) and (μ, μ + 1), respectively, we can assume that the coordinates of L1 is (μ − 1 − ξ, 0), then r1 = 1 + ξ and r2 = ξ with 0 < ξ ≪ 1. Using Eqs. (44), we can write where F(ξ) ≅ F(0+) = ∞ and G(ξ) ≅ G(0+) = −∞, therefore and we have then at L1. By the same way we can prove that for L2 and L3.
Now we can say that the discriminant of Eq. (43) is positive under the conditions which are stated in (36–40)for every case. Therefore, the four roots of the characteristic Eq. (43) are real where two of them are positive and the other two are negative. Hence the four roots are controlled by ω1,2 = ±ω11, ω3, 4 = ±iω12 where ω11, ω12 are reals and i is the imaginary unit. Thereby the general solution of Eq. (42) can be written in the form (45)
The solution in Eqs. (45) is constructed under the first condition but we can construct this solution the same way for the other four cases using the associated condition in every case.
Finally Eqs. (45) shows that the motion in the proximity of libration collinear points is unbounded because ω1,2 are real and the trajectory of the infinitesimal body will include some terms, which will grow without limit. Hence the case of instability of the libration collinear points does not change regard to the rotational motion when Euler angles are θi = 0, ψi + φi = π/2. Therefore the motion is unstable.
It is worth mentioning that in the case that the primaries are triaxial rigid bodies when the Euler angles of the rotational motion are θi = 0, ψi + φi = 0, the corresponding results can be obtained by interchanging the parameters σ11 BY σ12 and σ21 by σ22 in the corresponding results given in the case of Euler angles are θi = 0, ψi + φi = π/2.
In this paper the existence of libration collinear points and their linear stability were studied in the restricted three-body problem. This study established the setting of the primaries as being triaxial bodies in the case when the Euler angles of rotational motion are θi = 0, ψi + φi = π/2. The necessary and sufficient conditions to determine the locations of the three collinear points are found. In addition we show that the motion in the proximity of these points is unstable. It is worth mentioning that in the setting of the rotational motion when the primaries are triaxial rigid bodies with θi = 0, ψi + φi = 0, i = 1, 2, the corresponding results can be obtained by interchanging the parameters σ11 and σ21; σ21 and σ22. Finally, we refer to one of the significants of the collinear points in space missions, they are considered the optimal placement to transfer a spacecraft to an associated stable manifold.
Case 1 and 2:
When (σ21 ≥ σ11 ≥σ31) and (σ22 ≥ σ12 ≥ σ32) or (2σ21 − σ11 − σ31) ≥ 0 and (2σ22 − σ12 − σ32) ≥ 0, (46) (47) (48)
When (2σ21 − σ11 − σ31) ≤ 0 and (2σ22 − σ12 − σ32) ≤ 0 (49) (50) (51)
When (2σ21 − σ11 − σ31) ≥ 0 and (2σ22 − σ12 − σ32) ≤ 0 (52) (53) (54)
When (2σ21 − σ11 − σ31) ≤ 0 and (2σ22 − σ12 − σ32) ≥ 0 (55) (56) (57)
This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant no. (36 - 130 - 36 RG). The authors, therefore, acknowledge with thanks DSR technical and financial support. Furthermore, this work was partially supported by MICINN/FEDER: grant number MTM2011-22587; MINECO: grant number MTM2014-51891-P, and Fundación Séneca de la Región de Murcia: grant number 19219/PI/14.
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Published Online: 2017-03-15