Perturbed locations and linear stability of collinear Lagrangian points in the elliptic restricted three body problem with triaxial primaries

This paper aims to study the effect of the triaxiality and the oblateness as a special case of primaries on the locations and stability of the collinear equilibrium points of the elliptic restricted three body problem (in brief ERTBP). The locations of the perturbed collinear equilibrium points are first determined in terms of mass ratio of the problem (the smallest mass divided by the total mass of the system) and different concerned perturbing factors. The difference between the locations of collinear points in the classical case of circular restricted three body problem and those in the perturbed case is represented versus mass ratio over its range. The linear stability of the collinear points is discussed. It is observed that the stability regions for our model depend mainly on the eccentricity of the orbits in addition to the considered perturbations.


Introduction
The general three-body problem is described as a third body of infinitesimal mass m 3 that is attracted by two finite masses called primaries but not influencing their motion, moves in the plane defined by the two revolving primaries. The primaries are of arbitrary masses m 1 and m 2 and revolve around their center of mass in circular orbits (this system denoted briefly, CRTBP), see for instance, the fundamental celestial mechanics books, e.g. Szebehely (1976) and Murray and Drmott (1999). For more generalization, it is supposed that the primaries revolve around common center of mass in elliptical orbits (this system denoted briefly, ERTBP). ERTBP appears a more difficulty involved in handling it. Even if the CRTBP is not integrable, a number of special solutions can be found. The points where the third body has zero velocity and zero acceleration in the rotating frame are special solutions. Such points are called equilibrium points. 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 . In general case, the dynamics of circular and/or elliptical three-body problem has a wide applications in astrophysics for example but not all stellar/solar system dynamics and Earth-Moon system. So this problem received more attention from the astronomers and dynamical system scientists and can't be enumerated. in spite of it, the solutions of this problem has been developed over the past centuries.
It seems good to present some of the important related works focused on the ERTBP, Sharma et al. (2001) analyzed the equilibrium points stability retrieved that the triangular quilibrium points are stable conditionally but collinear points are always unstable. Ammar (2008) analyzed solar radiation pressure effect on the positions and stability of the libration points in ERTBP. Singh (2011) formulated the triangular libration points nonlinear stability under the effect Coriolis and centrifugal forces as small perturbations in addition to the effect of primaries oblateness and radiation pressures.  investigated the stability of triangular equilibrium points in the ERTBP, considering both primaries are oblate and emit light energy simultaneously.  investigated the luminous and oblate spheroids of primaries effect on the locations and stability of the collinear libration points. In another work, Singh and Umar (2014) studied the effect of the big primary's triaxial and a 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 as well. Katour et al. (2014); Singh andBello (2014a,b, 2015a,b); Abd El-Salam and Abd El-Bar (2015); Abd El-  and Bello and Singh (2016) concerned with the relativistic R3BP in addition to some different perturbations; the primaries oblateness, radiation from one of the primaries, upon the equilibrium points locations and stability. They noticed that the stability regions of the their concerned equilibrium points are varied (expanding or shrinking) related to the critical mass value and depending upon the value of their considered perturbations.
In recent works, the calculations of the effect of different perturbations like oblateness, ellipsoidal primaries and photogravitational relativistic on the equilibrium points locations and its stability in the restricted three-body problem are investigated, Wang et al. (2018); Wu et al. (2018); Abd El-Salam and Abd El-Bar (2018) and Xin and Hou (2017). They concluded that the collinear points stability is highly affected at a whole range of mass, contrary to the stability regions of the triangular points which are affected differently depend on the perturbations kind.
The aim of this study is the determination of the locations of the collinear equilibrium points and investigating their linear stability in the ERTBP taking into consideration both primaries triaxial. The oblateness is discussed as a special case of triaxiality.This aim comes as a continuation study for a previously started work by Abd . The equations of ERTBP is not adequate easy to dealing with because the Hamiltonian of the problem be-comes time-dependent. Utilizing from the pulsating and non-uniformly rotating coordinate reference frame, the system can be brought to a form where the primaries positions become at fixed and the infinitesimal body motion can be analyzed relative to their locations. This paper will be organized after this introduction section as follows: the equation of motion in ERTBP with oblate and triaxial primaries will be formulated in section 2, the computations related to the location of equilibrium points in ERTBP with the considered perturbations will be introduced in section 3. Sec. 4 highlighted the small displacement effect of a test particle located in collinear points. In sec. 5 all the computations related to the stability of collinear points are introduced. Finally a numerical simulations are presented in sec. 6 with discussion and a work conclusion is shown in sec. 7.

Equations of motion
The motion of an infinitesimal particle in the ERTBP with oblate and triaxial primaries in a dimensionless, barycentric and pulsating rotating coordinate system (ξ , η) is described by Abd El-Salam (2015) With the potential-like function U given by where f is the true anomaly of the more massive primary, m 1 , e is the eccentricity of any primary's orbit, µ is the ratio of the mass of smaller primary to the total mass of the primaries and it satisfies 0 < µ ≤ 1/2, n is the mean motion, Aσ = A 1 + (2σ1 − σ 2) and Aγ = A 2 + (2γ1 − γ 2) are the total averaged oblateness and triaxiality coefficients for larger and smaller primaries respectively. A 1 , A 2 , (2σ1 − σ 2) and (2γ1 − γ 2) are the oblateness and triaxiality factors for larger and smaller primaries respectively. r 1 and r 2 , the distances of the infinitesimal mass from larger and smaller primaries respectively are given by.
Finally the mean motion of the system is given by

Existence of the collinear equilibrium points
Since the equilibrium points have zero relative velocities and zero relative accelerations, then the locations of them is determined by equating the partial derivatives of the potential-like function to zero, The solutions of the Eqs. (5-6) with η ≠ 0 give the location of the triangular equilibrium points L 4 and L 5 whereas the trivial solution η = 0 gives the location of the collinear equilibrium points L 1 , L 2 and L 3 which are the cases of our study. Ignoring the perturbing coefficients in Eqs. (5-6), oblateness, triaxiality and ellipticity, The system will be retrieved to CRTBP system.

Linear stability analysis
To study the stability of the equilibrium points, a small displacement is occurred to a test particle located in those points due to the included perturbations. We will call the point a "stable position" if the body simply oscillates around the equilibrium point, also the point can be called as an "unstable one" if the body leavs rapidly from the point neighborhood; in the Lyapunov sense. In order to study the effect of a small displacement on a test particle subjected on the equilibrium collinear points, the equations of motion must be linearized around these points. Similarity transformation technique is used in order to solve the system of linear differential equation resulted from the linearization process. Consider a small displacement (X, Y) from the collinear equilibrium points (ξ i , η i ) , i = 1, 2, 3 such that ξ = ξ i + X and η = η i + Y, the linearized equation can be expressed as Y where λ j are the characteristic roots, which can be evaluated from the corresponding characteristic equation where Also α j and β j are dependent constants and they are defined as By meditation at Eq. (12), the positive eigenvalue gives rise to exponential growth that lead to instability case whereas the imaginary eigenvalue gives rise to oscillation terms that lead to stability case. The exponential decay is resulted from negative eigenvalue. However, from Eq. (13) the negative eigenvalue is resulted together with positive one from the equation.
It is clear that λ 1 = −λ 2 and λ 3 = −λ 4 , then at the stable points, the roots of the characteristic equation must satisfy λ 2 1 = λ 2 2 < 0 and λ 2 3 = λ 2 4 < 0. This implies that for stability case the following sufficient and necessary conditions should be satisfied simultaneously

Perturbed locations of Lagrangian points
The location of collinear Lagrangian points in the case of CRTBP is determined in many textbooks of celestial mechanics, e.g. ?. In this section, the required computation is to determine the locations of these collinear libration points in ERTBP with oblate and triaxial primaries. These new locations will be obtained as a perturbation of CRTBP Lagrangian points' locations as follows.

Location of L 1
The location of L 1 in the case of CRTBP is determined by the following power series, with where R 1 and R 2 are the positions of L 1 with respect to m 1 and m 2 , respectively. In the case of ERTBP with oblatness and triaxiality case, R 1 and R 2 will be slightly changed due to the perturbations in addition to ellipticity of the primaries orbit, where ε 1 and ε 2 are small perturbations factors. For the case of perturbed first Lagrangian point L 1 (ξ1, 0) we have Denote ε 1 = ε and R 1 = R then using Eq. (20) and Eq. (21) yields Substitute Eq. (22) into Eq. (5) yields This equation can be written as Since ε R << 1, ε 1−R << 1 then we can use the Maclaurin formula (1 ± M) m 1 ± mM The solution will be ε = (23) By looking at Eq. (23), we find that ε equals to the negative ratio of the first partial derivative of the potential-like function with respect to ξ at r 1 = R 1 = R of non perturbed location of L 1 to that the second partial derivative of the potential-like function with respect to the same variable at Then the perturbed location of L 1 can be expressed as, Removing all oblatness and triaxility factors for both Larger and smaller primaries (Aσ , Aγ → 0) also remove the ellipticity factor (n → 1), the perturbation factor will approach to zero, i.e., Which confirms the unperturbed location obtained, see Eq. (19).

Location of L 2
The location of L 2 in the case of CRTBP can be determined as with R 1 − R 2 = 1, α = ( µ 3(1−µ) ) 1/3 , where R 1 and R 2 are the positions of L 2 with respect to m 1 and m 2 . Similar to L 1 , the perturbation factor can be expressed as The formal coordinate of L 2 will be formulated as: Removing all perturbations, the location of L 2 will be retrieved to its position in CRTBP lim ( n,Aσ ,Aγ ) →(1,0,0) Which confirms again the unperturbed location obtained, see Eq. (28).

Location of L 3
The location of L 3 in the case of CRTBP can be determined as with R 2 − R 1 = 1, α = µ (1−µ) where R 1 and R 2 are the positions of L 3 with respect to m 1 and m 2 . Similar to the previous cases, The coordinates of L 3 will be Which confirms again the unperturbed location obtained, see Eq. (32).

Graphical representation and analysis
In this section we will use the numerical values of the included parameters listed in Table 1 to introduce some graphical representations illustrating the effect of the considered perturbation on: 1. The locations of the collinear points. 2. The fate of a small displacement on a test particle located at each point. 3. Stability region versus the mass ratio for each point.
The initial conditions of the orbit of the primaries can be found in Abd El- . Eccentricity = 0.2, semimajor axis =1.1, square of mean motion = 0.96714015, true anomaly = 0 degree and F = (︀ 1/1.2 )︀ . In addition to the following domains of the perturbing parameters tabulated in the following Table 1.
In what follows, we gave some plots to illustrate and interpret the difference between the locations of L i , i = 1, 2, 3 in the classical CRTBP and in the perturbed ERTBP. In Figures 1, 4, 7 we plotted the locations of L i , i = 1, 2, 3 versus the whole mass ratio µ ∈ ]0, 0.5].
In Figure 1 the location decreases with increasing the mass ratio, i.e. the equilibrium point L 1 moves left towards the barycenter. In Figure 7 the location decreases with increasing the mass ratio, i.e. the equilibrium point L 1 moves right towards the barycenter. This may be attributed mainly to the gravitational field of the massive primary. The size of perturbation between the classical CRTBP and in the perturbed ERTBP in determination of the location of L 1 and L 3 become significant and noticeable for larger mass ratios, approximately µ > 0.25. The gravitational fields of the primaries in both case illustrated in Figure 1 and in Figure 7 have only one line of action towards the left at L 1 and towards the right at L 3 . This may interpret well the decreasing nature of the curves obtained in Figure 1, 7.
In Fig. 4 the location increases with increasing the mass ratio, i.e. the equilibrium point L 2 moves right towards the les massive object away from the barycenter, which is very difficult to be interpreted but it may partially attributed due to close vicinity of L 2 to the less massive primary and the gravitational fields of the primaries have two inverse lines of action at L 2 , to the left towards the massive body and to the right towards the less massive one. Therefore the gravitational field of the less massive body will be more effective at L 2 than the gravitational field of the massive primary. Also the rotating frame may give a contribution to the effect.
The size of perturbation between the classical CRTBP and in the perturbed ERTBP in determination of the location of L 2 is very tiny. This may support our interpretation that we are very colse to the less massive primary.
In Figures 2, 5, 8 small displacement leads to departure which enhances the instability of L i , i = 1, 2, 3 illustrated in Figures 3, 6, 9. The only difference is the velocity of departure from equilibrium point.
According to Eq. (13, 14, 15), the characteristic equations of L i , i = 1, 2, 3 of ERTBP for µ = 0.01 are respectively given by   Figure 2. The growth of a small displacement for a test particle placed at L1 vs. f .
For L 3 Figure 5. The growth of a small displacement for a test particle placed at L2 vs. f .
It is clear that the first term is a decay and the third and fourth is an oscillation while the second term represents a growth that is the source of unstable case for L 1 and it dominates all other terms, see Figures 2, 5, 8. The  value of the constant term in the characteristic equation is always negative overall 0 < µ ≤ 1 2 . This implies that the first condition of stability is not achieved. See Figures 3, 6, 9.

Conclusion
We have considered the ellipticity, oblateness and triaxiality perturbations as a modification to the classical CRTBP. These perturbations as expected bring deviations of the locations of the equilibrium points from classical CRTBP. In this work, we computed and illustrated these deviations in collinear points explicitly as functions in the mass ration. We investigated the stability of the collinear equilibrium points. The absolute deviation in the location of all collinear points is the ratio between the first partial derivative of the potential-like function to the second partial derivative of the same function with respect to ξ axis at the solution of classical CRTBP. In all collinear points, the size of perturbations coming from the different perturbing sources are almost comparable at small and moderate mass ratios, But this becomes distinguishable for late moderate and high mass ratios. Also in all cases, the equilibrium points undergo instabilities in the whole domain of mass ratio.