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The collinear equilibrium points in the restricted three body problem with triaxial primaries

Sultan Z. Alamri
/ Sobhy E. Abd El-Bar
• Department of Mathematics, Faculty of Science, Taibah University, Al-Madinah, KSA, Saudi Arabia
• Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt
• Other articles by this author:
• De Gruyter OnlineGoogle Scholar
/ Aly R. Seadawy
• Corresponding author
• Department of Mathematics, Faculty of Science, Taibah University, Al-Madinah, Saudi Arabia
• Mathematics Department, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt
• Email
• Other articles by this author:
• De Gruyter OnlineGoogle Scholar
Published Online: 2018-09-06 | DOI: https://doi.org/10.1515/phys-2018-0069

Abstract

The perturbed restricted three body problem has been reviewed. The mass of the primaries are assumed as triaxial. The locations of the collinear points have been computed. Series forms of these locations are obtained as new analytical results. In order to introduce a semi-analytical view, a Mathematica program has been constructed to graph the locations of collinear points versus the whole range of the mass ratio μ taking into account the triaxiality. The resultant figures have been analyzed

PACS: 95.10.Ce; 45.50.Pk; 98.10.+z

1 Introduction

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 [1] and Murray and Drmott [2]. 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 [3]. 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 [4], Tsirogiannis et al. [5], Kushvah and Ishwar [6], Vishnu Namboori et al. [7], Mital et al. [8], Kumar and Ishwar [9], Rahoma and Abd El-Salam [10] and references therein. Ammar [11] analyzed solar radiation pressure effect on the positions and stability of the libration points in elliptic RTBP. Singh [12] 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 [13] 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 [14] 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. [15], Singh and Bello [16, 17], Abd El-Salam and Abd El-Bar [18], Abd El-Bar et al. [19] and Bello and Singh [20] 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 [1]. 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).

$x¨−2ny˙=∂U∂x−ddt∂U∂x˙,y¨+2nx˙=∂U∂y−ddt∂U∂y˙$(1)

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:

$U=n22(1−μ)r12+μr22+(1−μ)r1+μr2+(1−μ)(2σ1−σ2)2r13−3(1−μ)(σ1−σ2)2r15y2+μ(2γ1−γ2)2r23−3μ(γ1−γ2)2r25y2$

with

$r=x2+y2,r1=(x+μ)2+y2r2=(x+μ−1)2+y2,μ=m2m1+m2≤12,$

m1, m2(m1m2) being the masses of the primaries,

$σ1=a12−c125R2,σ2=b12−c125R2,σ1,σ2≪1,γ1=a22−c225R2,γ2=b22−c225R2,γ1,γ2≪1,$

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

$n2=1+322σ1−σ2+322γ1−γ2$

Remark

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 = ÿ = = = 0. These points represent particular solutions of equations of motion

$∂U∂x=∂U∂y=0,$(2)

The explicit formulas are

$Ux=n2x+(x+μ)−(1−μ)r13−3(1−μ)(2σ1−σ2)2r15+15(1−μ)(σ1−σ2)2r17y2+(x+μ−1)−μr23−3μ(2γ1−γ2)2r25+15μ(γ1−γ2)2r27y2$(3)

and

$Uy=n2y−(1−μ)r13+3(1−μ)(2σ1−σ2)2r15+3(1−μ)(σ1−σ2)2r15y$(4)

3 Location of collinear libration points

Any point of the collinear points must, by definition, have z = y = 0, and the solution of the classical RTBP satisfies the conditions of Abd El-Bar and Abd El-Salam [38]

$B1r1+B2r2=1,r1=B1x+μ,r2=−B2μ+x−1$(5)

where, we have (see Figures 1, 2, 3):

Figure 1

The location of L1 and its corresponding parameters

Figure 2

The location of L2 and its corresponding parameters

Figure 3

The location of L3 and its corresponding parameters

$L1:B1=1,B2=1,L2:B1=1,B2=−1,L3:B1=−1,B2=1$

4 Location of L1

The location geometry of L1 can be visualized as given by Figure 1.

Substituting from (5) with the corresponding values of 𝓑1 = 𝓑2 = 1, we get r1 + r2 = 1, r1 = x + μ, r2 = 1 – μx and noting that $\begin{array}{}\frac{\mathrm{\partial }\phantom{\rule{thinmathspace}{0ex}}{r}_{1}}{\mathrm{\partial }\phantom{\rule{thinmathspace}{0ex}}x}=-\frac{\mathrm{\partial }\phantom{\rule{thinmathspace}{0ex}}{r}_{2}}{\mathrm{\partial }\phantom{\rule{thinmathspace}{0ex}}x}=1\end{array}$ , equation (3) becomes:

$Ux=n2(1−μ−r2)−(1−μ)r13+3(1−μ)(2σ1−σ2)2r15−15(1−μ)(σ1−σ2)2r17y2+μr23+3μ2γ1−γ22r25−15μγ1−γ22r27y2$(6)

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

$r1=a1+ε1,r2=b1−ε1,a1=1−b1$(7)

from which we have

$r1=1−b11+ε11−b1r2=b11−ε1b1$(8)

where a1 and b1 are unperturbed positions of r1 and r2 respectively, and b1 is given after some successive approximation by

$b1=α1−α3−α29+227α3+281α4,α=μ3(1−μ)3$(9)

Substituting from equations (7) into equation (6)

$Ux=n21−μ−b1+ε1−1−μ(1−b1)21−2ε11−b1−3(1−μ)(2σ1−σ2)2(1−b1)4(1−4ε11−b1)+15(1−μ)(σ1−σ2)2(1−b1)61−6ε11−b1y2+μb121+2ε1b1+3μ(2γ1−γ2)2b141+4ε1b1−15μ(γ1−γ2)2b161+6ε1b1y2$

Since y = 0, then the above equation may be written as

$Ux=n2(1−μ−b1+ε1)−1−μ(1−b1)21−2ε11−b1−3(1−μ)(2σ1−σ2)2(1−b1)41−4ε11−b1+15(1−μ)(σ1−σ2)2(1−b1)61−6ε11−b1y2+μb121+2ε1b1+3μ(2γ1−γ2)2b141+4ε1b1=0$

which can be solved for ε1 yielding

$ε1=n2+21−μ1−b13−2μb13−6μ2γ1−γ2b15+61−μ2σ1−σ21−b15−1−n2+n2μ+1−μ1−b12−μb12+n2b1−3μ2γ1−γ22b14+31−μ2σ1−σ221−b14$(10)

Setting

$d1=n2+21−μ1−b13−2μb13−6μ2γ1−γ2b15+61−μ2σ1−σ21−b15−1;e1=1b12,f1=1b14,g1=1(1−b1)2,h1=1(1−b1)4$

Using the above relations, equation (10) can be written in the form

$ε1=d1(n2(−1+μ+b1)+1−μg1−μe1−3μ2γ1−γ22f1+31−μ2σ1−σ22h1)$(11)

where b1, d1, e1, f1, g1 and h1 are function of μ and they are given by

$b1=μ31/3−13μ32/3−19μ3+2927μ34/3−5281μ35/3−13μ32+6227μ37/3−12581μ38/3−μ33+Oμ310/3d1=∑i=09Ni1μ3i/3e1=59+233μ13+3μ23−169μ313−5081μ323−10243μ3−608729μ343−148243μ353−9476561μ32−2204819683μ373−2104419683μ383−87034177147μ33+…f1=−7327+331/3μ4/3+4μ+14331/3μ2/3−68932/3μ1/3−500μ1/324331/3+4μ2/3332/3+1324μ6561−434μ4/31968331/3+35180μ5/35904932/3+13294μ259049+339800μ7/3159432331/3+2601328μ8/3478296932/3+3598016μ314348907g1=1+2μ313+73μ323+169μ3+7627μ343+38281μ353+469μ32+58081μ373+97681μ383+10882729μ33+…h1=1+4μ1/331/3+26μ2/3332/3+116μ27+161μ4/32731/3+740μ5/38132/3+3316μ2729+1540μ7/324331/3+2188μ8/324332/3+84460μ319683$

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

$r2=b1−d1(n2−1+μ+b1+1−μg1−μe1−3μ2γ1−γ22f1+31−μ2σ1−σ22h1)$(12)

since the location of L1 is given by

$ξ0,L1=1−μ+r2$(13)

from equation (12) and (13) we get

$ξ0,L1=1−∑i=−39Ni1∗μ3i/3$(14)

where the coefficients $\begin{array}{}{\mathcal{N}}_{i1}^{\ast }\end{array}$ 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

$r1−r2=1,r1=x+μ,r2=x+μ−1,∂r1∂x=∂r2∂x=1$(15)

Substituting from (15) into (3) we get

$Ux=n2(1−μ+r2)−(1−μ)r12+3(1−μ)(2σ1−σ2)2r14−15(1−μ)(σ1−σ2)2r16y2−μr22+3μ2γ1−γ22r24−15μγ1−γ22r26y2$(16)

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

$r1=a2+ε2,r2=b2+ε2,a2=1+b2.$(17)

where a2 and b2 are the unperturbed positions of r1 and r2 respectively, and b2 is given after some successive approximation by the relation

$b2=α1+α3−α29−227α3+281α4,α=μ3(1−μ)3$(18)

Substituting from equations (17) into equation (16) we get

$Ux=n2(1−μ+b2+ε2)−1−μ(1+b2)21−2ε21+b2−3(1−μ)(2σ1−σ2)2(1+b2)41−4ε21+b2+15(1−μ)(σ1−σ2)2(1+b2)61−6ε21+b2y2−μb221−2ε2b2−3μ(2γ1−γ2)2b241−4ε2b2+15μ(γ1−γ2)2b261−6ε2b2y2$

Since y = 0, then the above equation may be written as

$Ux=n2(1−μ+b2+ε2)−1−μ(1+b2)21−2ε21+b2−3(1−μ)(2σ1−σ2)2(1+b2)41−4ε21+b2−μb221−2ε2b2−3μ(2γ1−γ2)2b241−4ε2b2=0$(19)

Equation (19) can be solved for ε2 yielding

$ε2=n2+2μb23+21−μ1+b23+6μ2γ1−γ2b25+61−μ2σ1−σ21+b25−1×n2(μ−b2−1)+μb22+1−μ(1+b2)2+3μ(2γ1−γ2)2b24+3(1−μ)(2σ1−σ2)2(1+b2)4$(20)

Setting

$d2=n2+2μb23+21−μ1+b23+6μ2γ1−γ2b25+61−μ2σ1−σ21+b25−1,e2=1b22,f2=1b24,g2=1(1+b2)2,h2=1(1+b2)4.$

Using the above relations, equation (20) can be written in the form

$ε=d2[n2(μ−b2−1)+(1−μ)g2+μe2+9(1−μ)A12h2+9μA22f2+3μ(2γ1−γ2)2f2+3(1−μ)(2σ1−σ2)2h2]$(21)

where the coefficients 𝓝 are given appendix (D-L2) and where, d2, e2, f2, g2 and h2 are functions of μ andare given by

$b2=μ313+13μ323−19μ3+2527μ343+5681μ353−13μ32+4627μ373+14581μ383−μ33+…d2=∑i=09Ni2μ3i/3e2=59−233μ1/3+3μ2/3−209μ31/3+5881μ32/3+10243μ3−932729μ34/3+196243μ35/3+6736561μ32−3929619683μ37/3+3079619683μ38/3+42916177147μ33+…f2=14327+3μ4/3−4μ+1493μ2/3−148273μ1/3−1012243μ31/3−10538μ26561μ32−45687253144μ37/3+3600328531441μ37/3−2258600531441μ32g2=1−2μ31/3+73μ32/3−169μ3−3227μ34/3+34781μ35/3−509μ32−5281μ37/3+1049μ38/3−12826729μ33+…h2=1−4μ31/3+263μ32/3−11699μ3+899μ34/3+19627μ35/3−294881μ32+146027μ37/3−53227μ38/3−64588729μ33$

Substituting back into equation (21), r2 is a function of b2, d2, e2, f2, g2 and h2 and can be written in the form

$r2=b2+d2[n2(μ−b2−1)+(1−μ)g2+μe2+3μ(2γ1−γ2)2f2+3(1−μ)(2σ1−σ2)2h2]$(22)

since the location of L2 is given by

$ξ0,L2=1−μ+r2$(23)

from equation (22) and (23) we get

$ξ0,L2=1−∑i=−39Ni2∗μ3i/3$

where the coefficients $\begin{array}{}{\mathcal{N}}_{i2}^{\ast }\end{array}$ 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

$r2−r1=1,r1=−(x+μ),r2=1−μ−x,∂r1∂x=∂r2∂x=−1$(24)

Hence substituting from equation (24) into (5), we get

$Ux=n2(1−μ−r2)+(1−μ)r12+3(1−μ)(2σ1−σ2)2r14−15(1−μ)(σ1−σ2)2r16y2+μr22+3μ2γ1−γ22r24−15μγ1−γ22r26$(25)

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

$r1=a3+ε3,r2=b3+ε3,a3=b3−1.$(26)

where a3 and b3 are the unperturbed values of r1 and r2 respectively, and b3 is given after some successive approximation by the relation

$b3=2−7μ12−161μ31728+…$(27)

Substituting from equations (26) into equation (25) and retaining terms up to the first order in the small quantities ε3 we get

$Ux=n21−μ−b3−ε3+1−μ(b3−1)21−2ε3b3−1+3(1−μ)(2σ1−σ2)2(b3−1)41−4ε3b3−1−15(1−μ)(σ1−σ2)2(b3−1)61−6ε3b3−1y2+μb321−2ε3b3+3μ(2γ1−γ2)2b341−4ε3b3−15μ(γ1−γ2)2b361−6ε3b3y2$

Since y = 0, then the above equation may be written as

$Ux=n21−μ−b3−ε3+1−μ(b3−1)21−2ε2b3−1+3(1−μ)(2σ1−σ2)2(b3−1)41−4ε3b3−1+μb321−2ε3b3+3μ(2γ1−γ2)2b34(1−4ε3b3)=0$(28)

Equation (28) can be solved for ε3 to yield

$ε3=−n2−21−μb3−13−2μb33−6μ2γ1−γ2b35−61−μ2σ1−σ2b3−15−1×−n2+n2μ−1−μ−1+b32−μb32+n2b3−3μ2γ1−γ22b34−31−μ2σ1−σ22−1+b34$(29)

now, letting

$d3=−n2−2(1−μ)b3−13−2μb33−6μ(2γ1−γ2)b35−6(1−μ)(2σ1−σ2)b3−15e3=1b32,f3=1b34,g3=1(b3−1)2,h3=1(b3−1)4$

Using the above relations, equation (29) can be written in the form

$ε3=d3(−n2(1−μ)+n2b3−(1−μ)g3−μe3−3μ(2γ1−γ2)2f3−3(1−μ)(2σ1−σ2)2h3)$(30)

where d3, e3, f3, g3 and h3 are functions of μ. They are given by

$d3=∑i=−29Ni3μ3i/3e3=14+7μ48+49μ2768+665μ313824+…f3=116+7μ96+245μ24608+2359μ355296+…g3=19+7μ162+49μ23888+2135μ3209952+…h3=181+7μ729+245μ252488+791μ3236196+…$

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

$r2=b3+d3(−n2(1−μ)+b3−(1μ)g3−μe3−3μ(2γ1−γ2)2f3−3(1−μ)(2σ1−σ2)2h3)$(31)

since the location of L3 is given by

$ξ0,L3=1−μ−r2$(32)

from equation (31) and (32) we get

$ξ0,L3=1−∑i=−39Ni3∗μi/3$(33)

where the coefficients $\begin{array}{}{\mathcal{N}}_{i3}^{\ast }\end{array}$ 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.

Figure 4a

The location of against μ under the effect of an oblate less massive primary and sphere massive primary

Figure 4b

The location of against μ under the effect of an oblate less massive primary and sphere 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.

Figure 4c

The location of L1 against μ under the effect of triaxial massive primary and sphere less massive primary

Figure 4d

The location of L1 against μ under the effect of an oblate massive primary and sphere less massive primary

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.

Figure 4e

The location of L2 against μ under the effect of an oblate massive primary and sphere less massive primary

Figure 4f

The location of L2 against μ under the effect of an oblate less massive primary and sphere 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.

Figure 4g

The location of L2 against μ under the effect of triaxial massive primary and sphere less massive primary

Figure 4h

The location of L2 against μ under the effect of an oblate massive primary and sphere less massive primary

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.

Figure 4i

The location of L3 against μ under the effect of an oblate massive primary and sphere less massive primary

Figure 4j

The location of L3 against μ under the effect of an oblate less massive primary and sphere massive primary

Figure 4k

The location of L3 against μ under the effect of triaxial massive primary and sphere less massive primary

Figure 4l

The location of L3 ratio μ under the effect of an oblate massive primary and sphere less massive primary

6 Conclusion

We have treated the perturbed RTBP with the triaxiality of both primaries up to the terms of order 𝓞 $\begin{array}{}\left(1/{r}_{1,2}^{7}\right)\end{array}$ 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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Appendix (D-L1)

$N01=2−n24+305γ1−305γ22−3σ1+3σ22+3μ1/3−75γ1+752γ2+3μ2/39γ1−92γ2N11=−15+5n22−2515γ13+2515γ26+15σ1−152σ2N21=59−145n212+15865γ19−15865γ218−35σ1+35σ22N31=−287318+335n29−78860γ127+39430γ227+172σ13−86σ23N41=28849−3154n2+301145γ181−301145γ2163−215σ13+215σ26N51=−857518+108n2−790940γ1243+395470γ2243+502σ19−251σ29N81=−462913729+26615n254−31442824γ16561+15721412γ26561+704σ19−352σ29N91=28068072187−145255n2243+56532350γ159049−28266175γ259049−2320σ1243+1160σ2243$

Appendix (D-L2)

$N02=−1−n24−305γ1+305γ22−3σ1+3σ22+3μ1/3−75γ1+752γ2+3μ2/3−9γ1+92γ2N12=−12−5n22−2245γ13+2245γ26−15σ1+152σ2N22=−59−145n212−10465γ19+10465γ218−35σ1+35σ22N32=−299918−335n29−29450γ127+14725γ227−118σ13+59σ23N42=−27573−335n24−44105γ181+44105γ2163−35σ13+35σ26N52=−2217554−469n23−80030γ1243+40015γ2243+128σ19−64σ29N62=−123127243−3305n212−690895γ1729+690895γ21458−85σ127+85σ254N72=−5975081−4360n29−3651725γ12187+3651725γ24374−215σ19+215σ218N82=−831071729−44755n254−12965156γ16561+6482578γ26561−44σ19+22σ29N92=−36518452187−332240n2243−55392640γ159049+27696320γ259049−2270σ1243+1135σ2243$

Appendix (D-L3)

$N03=142−3n2−12σ1+6σ2N33=164−84+560n2−18γ1+9γ2+2256σ1−1128σ2N63=71536(4032+10000n2−810γ1+405γ2+34680σ1−17340σ2)N93=724576(384932+491120n2−65610γ1+32805γ2+1090560σ1−545280σ2)$

Appendix (A-L1)

$N01∗=2−9n24+n44+494γ1−805n2γ12+5139γ12−247γ2+805n2γ24+75γ2σ22+5139γ224+3σ1+9n2σ14+150γ1σ1−75γ2σ1−9σ12−9σ224−3σ22−9n2σ28−75γ1σ2+9σ1σ2−5139γ1γ2N−11∗=−102γ1+3454n2γ1−1971γ12+51γ2−3458n2γ2+1971γ1γ2−19714γ22−90γ1σ1+45γ2σ1+45γ1σ2−452γ2σ2$ $N−21∗=(9γ1−9n2γ1+567γ12−92γ2+92n2γ2−567γ1γ2+5674γ22+27γ1σ1−272γ2σ1−272γ1σ2+274γ2σ2N−31∗=−81γ12+81γ1γ2−81γ224N11∗=−18+79n24−11n44−4720γ13+15319n2γ112−11556γ12+2360γ23−15319n2γ224+11556γ1γ2−2079γ22−3σ1−272n2σ1−361γ1σ1N21∗=75−268n23+44n43+70511γ118−28270n2γ19+22125γ12−70511γ236+14135n2γ29−22125γ1γ2+221254γ22−2σ1+1534n2σ1+2155γ1σ13N31∗=−407318+1121n23−1831n436−217631γ127+337225n2γ154−36168γ12+217631γ254−337225n2γ2108+36168γ1γ2−9042γ22+9σ12−327n2σ14−8372γ1σ19+4186γ2σ19+53σ12−9σ24+327n2σ28+4186γ1σ29−2093γ2σ29−53σ1σ2+53σ224N41∗=32719−2006n23+4571n436+2253133γ1162−1630015n2γ1162+154879γ123−154σ22−2253133γ2324+1630015n2γ2324−154879γ1γ23+154879γ2212−55σ13+49n2σ1+15σ1σ2$ $N51∗=−2824727+33547n227−24991n4108−4893158γ1243+3138185n2γ1243−572404γ129+2446579γ2243−3138185n2γ2486+572404γ1γ29−143101γ229+641σ118−797n2σ14−130655γ1σ1108+130655γ2σ1162+12σ12−641σ236+797n2σ28+130655γ1σ2162−130655γ2σ2324−12σ1σ2+3σ22N61∗=787937486−854033n2486+22696n481+35047541γ11458−9005530n2γ1729+15905525γ12243−35047541γ22916+4502765n2γ2729−15905525γ1γ2234+15905525γ22972−8557σ1162+589n2σ13+589n2σ13+350242γ1σ1243−175121γ2σ1243−78σ12+8557σ2324−589n2σ218−175121γ1σ2243+175121γ2σ2486+78σ1σ2−14σ22$ $N71∗=−954265486+271853n2162−1273n412−49027529γ12187+59331695n2γ18748−12948929γ12243+49027529γ24374−59331695n2γ217496+12948929γ1γ2243−12948929γ22972+2923σ154−257n2σ12−799070γ1σ1729+399535γ2σ1729+15σ12−2923σ2108+257n2σ24+399535γ1σ2729−399535γ2σ218122−15σ1σ2+15σ224N81∗=24024591458−1313483n22916−11426n427+17992143γ113122+12927712n2γ165561+20836592γ12729−177992143γ226244−6463856n2γ265561−20836592γ1γ2729+5209148γ22729−14753σ1486+461n2σ136−8007568γ1σ12187+4003784γ2σ12187−6σ12+14753σ2972−461n2σ272+4003784γ1σ22187−2001892γ2σ22187+6σ1σ2−3σ222N91∗=−1191427−3940255n22187+1084129n4972−164982886γ119683−626757695n2γ178732+223007333γ126561+82491443γ219683+626757695n2γ2157464−223007333γ1γ26561+223007333γ2226244−913σ11458+5333n2σ181+188329745γ1σ16561−188329745γ2σ113122+913σ22916−5333n2σ2162−188329745γ1σ213122+188329745γ2σ226244$

Appendix (A-L2)

$N02∗=1−3n24−n44+479γ1−727n2γ12+3681γ12−479γ22+727n2γ24−3681γ1γ2+3681γ224+6σ1−9n2σ14+348γ1σ1−174γ2σ1−3σ2+9n2σ28+9σ224−174γ1σ2+87γ2σ2−9σ1σ2+9σ12N−12∗=93γ1−3274n2γ1+1971γ12−932γ2+3278n2γ2−1971γ1γ2+19714γ22+144γ1σ1−72γ2σ1−72γ1σ2+36γ2σ2N−22∗=9γ1−9n2γ1+567γ12−92γ2+92n2γ2−567γ1γ2+5674γ22+27γ1σ1−272γ2σ1−272γ1σ2+274γ2σ2N−32∗=81γ12−81γ1γ2+81γ224N12∗=12−41n24−11n44+4354γ13−11617n2γ112+82832/3γ12−2177γ23+11617n2γ224−2484γ1γ2+621γ22+42σ1−272n2σ1+283γ1σ1−283γ2σ12+274n2σ2−283γ1σ22+9σ12−21σ2+283γ2σ24−9σ1σ2+94σ22$ $N22∗=71−170n23−44n43+48017γ118−14182n2γ19−182331/3γ12−48017γ236+7091n2γ29+5469γ1γ2−54694γ22+110σ1−1534n2σ1−1649γ1σ13+1649γ2σ16+3σ12−55σ2+1538n2σ2+1649γ1σ26−1649γ2σ212−3σ1σ2+34σ22N32∗=413318−6377n236−1777n436+65279γ127−68593n2γ154−16752γ12−65279γ254+68593n2γ2108+16752γ1γ2−4188γ22+757σ19−201n2σ14−16360γ1σ19+8180γ2σ19−55σ12757σ212+201n2σ28+8180γ1σ29−4090γ2σ29+55σ1σ2−55σ224$ $N42∗=27116−1019n23−4229n436−166043γ1162+63197n2γ1162−55721γ123+166043γ2324−63197n2γ2324+55721γ1γ23−55721γ2212−44σ1−3n2σ1−50278γ1σ127+25139γ2σ127−15σ12+22σ2+3n2σ22+25139γ1σ227−25139γ2σ254+15σ1σ2−154σ22N52∗=1511527−10601n227−23497n4108−1449466γ1243+453517n2γ1243−27212γ129+724733γ2243−453517n2γ2486+27212γ1γ29−6803γ229−1951σ16+349n2σ14+15743γ1σ181−15743γ2σ1162−12σ12+1951σ212−349n2σ28−15743γ1σ2162+15743γ2σ2324+12σ1σ2−3σ22$ $N62∗=177809486−122333n2486−28960n481−10534933γ11458+538370n2γ1729+3877025γ12243+10534933γ22916−269185n2γ2729−3877025γ1γ2243+3877025γ22972−62437σ1162+281n2σ13+563392γ1σ1243−281696γ2σ1243+84σ12+62437σ2324−281n2σ26−281696γ1σ2243+140848γ2σ2234−84σ1σ2+21σ22N72∗=8141162−80365n2486−21469n436−6513922γ12187−22227281n2γ18748+4233653γ12243+3256961γ22187+22227281n2γ217496−4233653γ1γ2243+4233653γ22972−19063σ1162−57n2σ12+1420652γ1σ1729−710326γ2σ1729+45σ12+19063σ2324+57n2σ24−710326γ1σ2729+355163γ2σ2729−45σ1σ2+45σ224$ $N82∗=1001051458−1357369n22916−27773n427+13081537γ113122−29852018n2γ16561+3497408γ12729−13081537γ236244+14926009n2γ26561−3497408γ1γ2729+874352γ22729+16265σ1162−4037n2σ136−11987038γ1σ12187+5993519γ2σ12187+6σ12−16265σ2324+4037n2σ272+5993519γ1σ22187−5993519γ2σ24374−6σ1σ2+3σ222N92∗=9693742187−730099n2729−1680113n4972−429534110γ119683−1932138199n2γ173732−269045573γ126561+214767055γ219683+1932138199n2γ2157464+269045573γ1γ26561−269045573γ2226244−2665σ11458−4963n2σ181−282835739γ1σ16561+282835739γ2σ113122+2665σ22916+4963n2σ2162+282835739γ1σ213122−282835739γ2σ226244$

Appendix (A-L3)

$N03∗=2+18−2+3n2+12σ1−6σ2(−2+2n2−6σ1+3σ2)N33∗=54−164(−84+560n2−18γ1+9γ2+2256σ1−1128σ2)−1+n2−3σ1+3σ22−1128(2−3n2−12σ1+6σ2)(−40+40n2−18γ1+9γ2−384σ1+192σ2)N63∗=71536(4032+10000n2−810γ1+405γ2+34680σ1−17340σ2)−1+n2−3σ1+3σ22+637686γ1−3γ2+88σ144σ22−3n2−12σ1+6σ2−12048(−84+560n2−18γ1+9γ2+2256σ1−1128σ2)×(−40+40n2−18γ1+9γ2−384σ1+192σ2)N93∗=−749152(−593128+20824n2+595488n4+174420γ1−17100n2γ1−10692γ12−87210γ2+8550n2γ2+10692γ1γ2−2673γ22−1543128σ1+367008n2σ1+367308γ1σ1−183654γ2σ1+1230336σ12+771564σ2−183504n2σ2−183654γ1σ2+91827γ2σ2−1230336σ1σ2+307584σ22)$

About the article

Received: 2018-01-11

Accepted: 2018-04-22

Published Online: 2018-09-06

Citation Information: Open Physics, Volume 16, Issue 1, Pages 525–538, ISSN (Online) 2391-5471,

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© 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

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