The collinear equilibrium points in the elliptic restricted synchronous three-body problem under an oblate primary and a dipole secondary

Abstract

The study investigates the collinear positions and stability in the elliptic restricted synchronous three-body problem under an oblate primary and a dipole secondary for Luhman 16 and HD188753 systems. Our study has established four collinear equilibrium points $\left(L_{1,2,3,6}\right)$ which are greatly affected by the parameters under review. The collinear position $L_1$ move away and closer as the parameters increase and decrease respectively. For the collinear positions $L_{2}and{L}_3$, we witnessed a uniform space movement away from the origin in the negative direction while $L_6$ seems to be moving closer to the origin from the negative part of the origin. We observed changes in the movements of the collinear positions $\left(L_{1,2,3,6}\right)$ as a result of the half distance between the mass dipoles and the oblateness of the primary for the problem under review. The movements away and closer to the origin from collinear positions do not change the status of the collinear points as they remain unstable and unchanged. It is also found that as the half distance between mass dipoles and oblateness of the primary increase, the region of stability of the collinear positions decreases for the aforementioned binary systems. The collinear equilibrium point $\left(L_3\right)$ is stable for the characteristic roots $\left({\lambda }_{\mathrm{1,2}}\right)$ for Luhman 16 system. This is evidenced by at least one characteristic root, a positive real part and a complex root. The stability of collinear points in most cases are unstable for the stated binary systems in Lyapunov.

1. Introduction

There has been a significant increase in the study of minor celestial bodies like comets and asteroids in the space community. Since the first successful mission of Galileo, which involved close flybys of 951 Gaspra and 243 Ida, several missions involving the study of small bodies have been conducted by NASA, ESA, and JAXA, including NEAR, ROSETTA, and HAYABUSA. These missions' explorations advanced our understanding of the solar system's formation and served as the starting point for deeper space explorations. Closed asteroid (or comet) orbital dynamics provide the greatest difficulties when the orbit is designed in the vicinity of minor celestial bodies, except for flyby missions, because of the combined effects of the asteroids' rotation and their irregular gravitational field on vicinal objects by Ref. [1]. The term “heliostationary flight” refers to the placement of a spacecraft at one of the Sun-asteroid collinear libration sites. During this process, the asteroid is typically considered a point mass with a spherically symmetric gravitational potential, according to Williams and Abate [2].

Zeng et al. [3] examined the connection between the rotating mass dipole and naturally elongated bodies. According to their research, the elongated celestial body can be approximated by five dipole model parameters (but only four independently), including the mass ratio, system mass, spinning period, characteristic distance, and the ratio of gravitational and centrifugal forces. Seven example asteroids and comets—1620 Geographos, 216 Kleopatra, 951 Gaspra, 1996 HWI, 2063 Bacchus, 25,143 Itokawa, and 103P/Hartley-2—were used to show the relevant parameters of their simplified models.

For the past several decades, researchers have studied the dynamics around some simple bodies to understand the common characteristics of irregularly shaped bodies, such as ellipsoids, disks, dumbbell-shaped bodies, homogeneous cubes, and material segments [[4], [5], [6], [7], [8], [9]].

Zeng et al. [10] noted that it might be possible to approximate the potential distribution of nearly axisymmetrical elongated celestial bodies using the spinning mass dipole. They obtained four novel equilibrium points in addition to the five typical equilibrium points around the main of a prolate spheroid by using the boundary values of the oblateness as (−0.05 to 0.05). The triangle's equilibrium points $E_1$ and $E_7$, which are independent of mass ratio, are present when the force ratio is between 0.37 and 2.07. When the force ratio is low, the collinear equilibrium point $E_1$ is conditionally stable. Additionally, conditionally stable are the triangular equilibrium points $E_4$ and $E_5$, whose stable zone grows as the oblateness decreases.

A binary asteroid's dynamical model is constructed with a patched three-body method. A cost-effective landing trajectory can be found for manifolds interaction between patched three-body problems using the Poincare analysis on the Surface Of Equivalence (SOE). The Circular Restricted Three-Body Problem (CR3BP) can be exploited using the patched three-body technique, and cost-effective trajectories can be generated beginning from periodic or quasi-periodic orbits and associated invariant manifolds by Ref. [11]. Barbosa Torres dos Santos et al. [12] considered a motion of negligible mass travelling in a system composed of other massive bodies. While one of the bodies is modeled as an irregularly shaped and rotating mass dipole in both collinear and non-collinear scenarios, the other is believed to have a spherical shape. According to their findings, with smaller values of the mass parameter, the collinear points are unstable, whereas the non-collinear ones are linearly stable.

Zotos et al. [13] examined equilibrium points associated with convergence basins in the frame work of the restricted problem with a modifield gravitational potential. They found the basin boundaries on the plane (x, y) always fractal for all possible values of the involved parameters.

Alshaery and Abouelmagd [14] studied spatial quantized restricted three-body problem as the new version of the restricted three-body problem. They explored permissible regions both the inside and outside planes of motion. Their results show that the regions of permissible motion are decreasing with increasing Jacobian value while the forbidden regions are increasing with increasing Jacobian value.

Zotos et al. [15] carried out a quantitative orbit classification of the planar restricted three-body problem with application to the motion of a satellite around Jupiter. Their results show that the classic Poincare Surface of sections are useful to qualitative classification of phase-space regions with periodic, quasi-periodic or chaotic motion.

Ershkov et al. [16] considered an additional well-known effect of differential rotation which obviously takes place in the gaseous of fluid convection zone of primary giant planet. Their study revealed that a combined contribution to the tidal dissipation Uranus + from satellite yields a magnitude od decelerations for semi-major axis of all the satellites of Uranus.

Singh and Tyokyaa [17] recently investigated the stability and velocity sensitivity of the libration points in the elliptic restricted synchronous three-body problem for the Luhman16 and HD188753 systems under an oblate primary and a dipole secondary. Following their findings, the triangular points for the binary systems HD188753 and Luhman 16 exhibit stable and unstable behavior for specific values of oblateness, particularly when the parameters are varied differently with an order of commensurability k. Their results show stable behavior for the triangular points for the HD188753 system and unstable behavior for the Luhman 16 system.

Collinear libration points have a long-standing interest in the area of celestial mechanics. Numerous studies have been conducted to determine the location and stability of collinear equilibrium points in the circular or elliptic restricted three-body problem. In the axisymmetric restricted three-body problem with both primary acting as radiation sources, some analysis has revealed that the collinear libration point remains unstable, according to Abouelmagd and El-Shaboury [18]. Both Kunitsyn [19] and Kunitsyn et al. [20] investigated the properties of collinear equilibrium points. Their findings supported the collinear points' stability in exceptional conditions. In the case of a fourth-order resonance taking into account the radiation of both primaries, the collinear points can be stable in the sense of Lyapunov [21].

Singh and Leke [22] recorded stable collinear equilibrium points with the gravitational constant k (kappa). But even with the addition of the constant k, the out-of-plane equilibrium points are still unstable. Singh and Tyokyaa [23] examined the positions and stability of collinear equilibrium points in the elliptic restricted three-body problem with oblateness of the primaries up to zonal harmonic $J_4$. They stated that the collinear points remain unstable for the binary systems: HD188753 and Gliese 667.

Our present work is to study the collinear positions and stability in the elliptic restricted synchronous three-body problem under an oblate primary and a dipole secondary for Luhman 16 and HD188753 systems.

The paper spans five sections: the equations of motion are presented in section 2; in section 3, the location of collinear libration points is analyzed; section 4 presents the stability of the collinear equilibrium points and the conclusions are drawn from section 5.

2. Equations of motion

The equations of motion of the infinitesimal mass (third body) under the influence of an oblate primary and a dipole secondary using dimensionless variables and a barycentric synodic coordinate system $\left(\xi ,\eta ,\zeta \right)$ according to Ref. [17] can be written as follows;

$${\xi }^{″}-2{\eta }^{\prime }=\frac{\partial \Omega }{\partial \xi },{\eta }^{″}+2{\xi }^{\prime }=\frac{\partial \Omega }{\partial \eta },{\zeta }^{″}=\frac{\partial \Omega }{\partial \zeta }$$ (1)

with the force function

$$\Omega ={\left(1-e^2\right)}^{-\frac{1}{2}}\left[\frac{1}{2}\left({\xi }^2+{\eta }^2\right)+\frac{1}{n^2}\left\{\frac{\left(1-2\nu \right)}{r_1}+\frac{\left(1-2\nu \right)A}{2{r_1}^3}+\frac{\nu }{r_{21}}+\frac{\nu }{r_{22}}\right\}\right]$$ (2)

The mean motion, $n\text{,}$ is given as

$${r_1}^2={\left(\xi -{\xi }_1\right)}^2+{\eta }^2+{\zeta }^2$$

$${r_{21}}^2={\left(\xi -{\xi }_{21}\right)}^2+{\eta }^2+{\zeta }^2$$ (3)

$${r_{22}}^2={\left(\xi -{\xi }_{22}\right)}^2+{\eta }^2+{\zeta }^2$$

$$n^2=\frac{{\left(1+e^2\right)}^{\frac{1}{2}}}{a\left(1-e^2\right)}\left[1+\frac{3A}{2}\right]$$ (4)

Where $\nu =\frac{M_{21}}{M_1+M_{21}+M_{22}},0<\nu <\frac{1}{4},M_1>M_{21}=M_{22}$ and that, $M_1$ is the mass of the oblate primary, $M_2=M_{21}+M_{22}$ is the mass of the dipole secondary composed by the two equal masses $M_{21}and{M}_{22}$. ${\xi }_1=-2\nu ,{\xi }_{21}=1-2\nu -d,{\xi }_{22}=1-2\nu +d,M_1>M_{21}=M_{22}$.

$2d$ is the distance between $M_{21}and{M}_{22}$ in canonical units or the length of the largest axis of the dipole secondary. $r_1$, $r_{21}$ and $r_{22}$ are the respective distances of $M_1,M_{21}and{M}_{22}$ from the infinitesimal body; $a$ and $e$ are, respectively, the semi-major axis and eccentricity of the orbits, and $A$ is the oblateness coefficient of the primary.

3. Locations of collinear equilibrium points

We consider the points; $L_1\left(\xi >{\xi }_{22}\right),L_2\left({\xi }_1<\xi <{\xi }_{21}\right),L_3\left({\xi }_{21}<\xi <{\xi }_{22}\right)and{L}_6\left({\xi }_1>\xi \right)$

The collinear equilibrium points are the solutions of equations; ${\mathrm{\Omega }}_{\xi }={\mathrm{\Omega }}_{\eta }={\mathrm{\Omega }}_{\zeta }=0$ together with $\eta =\zeta =0$. Now, considering equations (1), (2), (3), (4) we have;

$${\left(1-e^2\right)}^{-\frac{1}{2}}\left[\xi -\frac{1}{n^2}\left\{\frac{\left(1-2\nu \right)\left(\xi -{\xi }_1\right)}{{r_1}^3}+\frac{3\left(1-2\nu \right)\left(\xi -{\xi }_1\right)A}{2{r_1}^5}+\frac{\nu \left(\xi -{\xi }_{21}\right)}{{r_{21}}^3}+\frac{\nu \left(\xi -{\xi }_{22}\right)}{{r_{22}}^3}\right\}\right]=0$$ (5)

$${\left(1-e^2\right)}^{-\frac{1}{2}}\left[\eta \left(1-\frac{1}{n^2}\left(\frac{\left(1-2\nu \right)}{{r_1}^3}+\frac{3\left(1-2\nu \right)A}{2{r_1}^5}+\frac{\nu }{{r_{21}}^3}+\frac{\nu }{{r_{22}}^3}\right)\right)\right]=0$$ (6)

$$\frac{{\left(1-e^2\right)}^{-\frac{1}{2}}}{n^2}\left[-\zeta \left\{\frac{\left(1-2\nu \right)}{{r_1}^3}+\frac{3\left(1-2\nu \right)A}{2{r_1}^5}+\frac{\nu }{{r_{21}}^3}+\frac{\nu }{{r_{22}}^3}\right\}\right]=0$$ (7)

To locate equilibrium points on the $\xi -axis$, we put $\eta =\zeta =0$ into equation (3) and substituting the values of ${r_1}^2$, ${r_{21}}^2$ and ${r_{22}}^2$ into equation (5), we have

$$n^2\xi -\left[\frac{\left(1-2\nu \right)\left(\xi -{\xi }_1\right)}{{r_1}^3}+\frac{3\left(1-2\nu \right)\left(\xi -{\xi }_1\right)A}{2{r_1}^5}+\frac{\nu \left(\xi -{\xi }_{21}\right)}{{r_{21}}^3}+\frac{\nu \left(\xi -{\xi }_{22}\right)}{{r_{22}}^3}\right]=0$$ (8)

To obtain the equilibrium point of the collinear point on the $\xi -axis$, we divide the orbital plane into four points as; $\xi >{\xi }_{22},{\xi }_1<\xi <{\xi }_{21},{\xi }_{21}<\xi <{\xi }_{22}and{\xi }_1>\xi$.

The four points are considered in the following Cases: I, II, III and IV.

Case I

Let the collinear point $L_1$ be on the right side of the smaller primary $M_{22}$ at a distance $\rho$ from it on the $\xi -axis$ (i.e. $\xi >{\xi }_{22}$)

where ${\xi }_1=-2\nu ,{\xi }_{21}=1-2\nu -dand{\xi }_{22}=1-2\nu +d$.

Then.

$\xi ={\xi }_{22}+\rho =1-2\nu +d+\rho$ that is $\xi -1+2\nu -d=\rho$

$$\left(\xi -{\xi }_1\right)=\left(1-2\nu +d+\rho \right)-\left(-2\nu \right)=1-2\nu +d+\rho +2\nu =1+d+\rho whichimplies{r}_1=\left|1+d+\rho \right|$$

$$\left(\xi -{\xi }_{21}\right)=\left(1-2\nu +d+\rho \right)-\left(1-2\nu -d\right)=1-2\nu +d+\rho -1+2\nu +d=\rho +2dwhichimplies{r}_{21}=\left|\rho +2d\right|$$

$$\left(\xi -{\xi }_{22}\right)=\left(1-2\nu +d+\rho \right)-\left(1-2\nu +d\right)=1-2\nu +d+\rho -1+2\nu -d=\rho whichimplies{r}_{22}=\left|\rho \right|$$

Substituting the values of $\xi ,\left(\xi -{\xi }_1\right),\left(\xi -{\xi }_{21}\right),\left(\xi -{\xi }_{22}\right)r_1,r_{21}and{r}_{22}$ into equation (8), we get

$$2n^2{\rho }^2{\left(\rho +2d\right)}^2{\left(1+d+\rho \right)}^4\left(1-2\nu +d+\rho \right)-2\left(1-2\nu \right){\rho }^2{\left(\rho +2d\right)}^2{\left(1+d+\rho \right)}^2-3\left(1-2\nu \right){\rho }^2{\left(\rho +2d\right)}^{2}A-2\nu {\rho }^2{\left(1+d+\rho \right)}^4-2\nu {\left(\rho +2d\right)}^2{\left(1+d+\rho \right)}^4=0$$ (9)

where $n^2=\frac{1}{a}\left(1+\frac{3A}{2}+\frac{3e^2}{2}\right)$.

Case II

Let the collinear point $L_2$ be on the left side of the smaller primary $M_{21}$ at a distance $\rho$ from it on the $\xi -axis$ (i.e ${\xi }_1<\xi <{\xi }_{21}$)

where ${\xi }_1=-2\nu ,{\xi }_{21}=1-2\nu -dand{\xi }_{22}=1-2\nu +d$.

Then

$$\xi ={\xi }_{21}-\rho =\left(1-2\nu -d\right)-\left(\rho \right)=1-2\nu -d-\rho$$

$$r_1=\left(\xi -{\xi }_1\right)=\left(1-2\nu -d-\rho \right)-\left(-2\nu \right)=1-2\nu -d-\rho +2\nu =1-d-\rho whichimplies{r}_1=\left|1-d-\rho \right|$$

$$r_{21}=\left(\xi -{\xi }_{21}\right)=\left(1-2\nu -d-\rho \right)-\left(1-2\nu -d\right)=1-2\nu -d-\rho -1+2\nu +d=-\rho whichimplies{r}_{21}=\left|\rho \right|$$

$$r_{22}=\left(\xi -{\xi }_{22}\right)=\left(1-2\nu -d-\rho \right)-\left(1-2\nu +d\right)=1-2\nu -d-\rho -1+2\nu -d=-\left(\rho +2d\right)whichimplies{r}_{22}=\left|\rho +2d\right|$$

Substituting the values of $\xi ,\left(\xi -{\xi }_1\right),\left(\xi -{\xi }_{21}\right),\left(\xi -{\xi }_{22}\right)r_1,r_{21}and{r}_{22}$ into equation (8), we obtain

$$2n^2{\rho }^2{\left(\rho +2d\right)}^2{\left(1-d-\rho \right)}^4\left(1-2\nu -d-\rho \right)-2\left(1-2\nu \right){\rho }^2{\left(\rho +2d\right)}^2{\left(1-d-\rho \right)}^2-3\left(1-2\nu \right){\rho }^2{\left(\rho +2d\right)}^{2}A+2\nu {\left(\rho +2d\right)}^2{\left(1-d-\rho \right)}^4+2\nu {\rho }^2{\left(1-d-\rho \right)}^4=0$$ (10)

where $n^2=\frac{1}{a}\left(1+\frac{3A}{2}+\frac{3e^2}{2}\right)$.

Case III

Let the collinear point $L_3$ be on the left side of the smaller primary $M_{22}$ at a distance $\rho$ from it on the $\xi -axis$ (i.e ${\xi }_{21}<\xi <{\xi }_{22}$)

where ${\xi }_1=-2\nu ,{\xi }_{21}=1-2\nu -dand{\xi }_{22}=1-2\nu +d$.

Then

$$\xi ={\xi }_{22}-\rho =\left(1-2\nu +d\right)-\left(\rho \right)=1-2\nu +d-\rho$$

$$r_1=\left(\xi -{\xi }_1\right)=\left(1-2\nu +d-\rho \right)-\left(-2\nu \right)=1-2\nu +d-\rho +2\nu =1+d-\rho whichimplies{r}_1=\left|1+d-\rho \right|$$

$$r_{21}=\left(\xi -{\xi }_{21}\right)=\left(1-2\nu +d-\rho \right)-\left(1-2\nu -d\right)=1-2\nu +d-\rho -1+2\nu +d=2d-\rho whichimplies{r}_{21}=\left|2d-\rho \right|$$

$$r_{22}=\left(\xi -{\xi }_{22}\right)=\left(1-2\nu +d-\rho \right)-\left(1-2\nu +d\right)=1-2\nu +d-\rho -1+2\nu -d=-\rho whichimplies{r}_{22}=\left|\rho \right|$$

Substituting the values of $\xi ,\left(\xi -{\xi }_1\right),\left(\xi -{\xi }_{21}\right),\left(\xi -{\xi }_{22}\right)r_1,r_{21}and{r}_{22}$ into equation (8), we get

$$2n^2{\rho }^2{\left(\rho -2d\right)}^2{\left(1+d-\rho \right)}^4\left(1-2\nu +d-\rho \right)-2\left(1-2\nu \right){\rho }^2{\left(\rho -2d\right)}^2{\left(1+d-\rho \right)}^2-3\left(1-2\nu \right){\rho }^2{\left(\rho -2d\right)}^{2}A+2\nu {\rho }^2{\left(1+d-\rho \right)}^4+2\nu {\left(\rho -2d\right)}^2{\left(1+d-\rho \right)}^4=0$$ (11)

Where $n^2=\frac{1}{a}\left(1+\frac{3A}{2}+\frac{3e^2}{2}\right)$.

Case IV

Let the collinear point $L_6$ be on the left side of the bigger primary $M_1$ at a distance $\rho$ from it on the $\xi -axis$ (i.e ${\xi }_1>\xi$)

where ${\xi }_1=-2\nu ,{\xi }_{21}=1-2\nu -dand{\xi }_{22}=1-2\nu +d$.

Then

$$\xi =-\rho +{\xi }_1=\left(-\rho \right)+\left(-2\nu \right)=-\rho -2\nu$$

$$r_1=\left(\xi -{\xi }_1\right)=\left(-\rho -2\nu \right)-\left(-2\nu \right)=-\rho -2\nu +2\nu =-\rho whichimplies{r}_1=\left|\rho \right|$$

$$r_{21}=\left(\xi -{\xi }_{21}\right)=\left(-\rho -2\nu \right)-\left(1-2\nu -d\right)=-\rho -2\nu -1+2\nu +d=-\rho -1+d=-\left(1-d+\rho \right)whichimplies{r}_{21}=\left|1-d+\rho \right|$$

$$r_{22}=\left(\xi -{\xi }_{22}\right)=\left(-\rho -2\nu \right)-\left(1-2\nu +d\right)=-\rho -2\nu -1+2\nu -d=-\rho -1-d=-\left(1+d+\rho \right)whichimplies{r}_{22}=\left|1+d+\rho \right|$$

Substituting the values of $\xi ,\left(\xi -{\xi }_1\right),\left(\xi -{\xi }_{21}\right),\left(\xi -{\xi }_{22}\right)r_1,r_{21}and{r}_{22}$ into equation (8), we get

$$2n^2{\rho }^4{\left(1-d+\rho \right)}^2{\left(1+d+\rho \right)}^2\left(-\rho -2\nu \right)+2\left(1-2\nu \right){\rho }^2{\left(1-d+\rho \right)}^2{\left(1+d+\rho \right)}^2+3\left(1-2\nu \right){\left(1-d+\rho \right)}^2{\left(1+d+\rho \right)}^{2}A+2\nu {\rho }^4{\left(1+d+\rho \right)}^2+2\nu {\rho }^4{\left(1-d+\rho \right)}^2=0$$ (12)

where $n^2=\frac{1}{a}\left(1+\frac{3A}{2}+\frac{3e^2}{2}\right)$.

The collinear points denoted by $L_1,L_2,L_{3}and{L}_6$ are evidenced by cases I, II, III, and IV respectively. Using Equations (9), (10), (11), (12) for various oblateness $\left(A\right)$, mass ratio $\left(\nu \right)$, the half distance between mass dipoles $\left(d\right)$, and mean motion ($n$), we compute numerically using MATHEMATICA software with the help of Table 1, Table 2. We find the value of $\rho$ and substitute it into the following: $\xi =1-2\nu +d+\rho =L_1\left(CaseI\right),\xi =1-2\nu -d-\rho =L_2\left(CaseII\right),\xi =1-2\nu +d-\rho =L_3\left(CaseIII\right)and\xi =-2\nu -\rho =L_6\left(CaseIV\right)$. The effects of the parameters mentioned above on the positions of the collinear points are shown in Table 3 and Fig. 1, Fig. 2, Fig. 3, Fig. 4 for Luhman 16 binary system and Table 4 and Fig. 5, Fig. 6, Fig. 7, Fig. 8 for HD188753 binary system.

The figures below show the effects of the half dipole mass and oblateness on the collinear equilibrium points positions for the systems: Luhman-16 and HD188753.

As shown in Table 3 and Fig. 2, Fig. 3, Fig. 4, Fig. 5, the collinear position $L_1$ move away from the origin while the collinear positions $L_2,L_{3}and{L}_6$ both move closer to the origin. The movement of $L_{2}and{L}_3$ is in the positive direction while $L_6$ is from the negative direction. It is found that the effects on the positions $L_1,L_2,L_{3}and{L}_6$ occurred owing to an increase in the half distance between the mass dipoles and the oblateness of the primary for the binary system Luhman 16.

As for the HD188753 binary system, we witnessed the effects of the aforementioned parameters on the collinear position in Table 4 and Fig. 6, Fig. 7, Fig. 8, Fig. 9. The collinear position $L_1$ move away from the origin as the parameters under study increase to some points and start moving backward as the parameters keep increasing. For the collinear positions $L_{2}and{L}_3$, we witnessed the same space movement away from the origin in the negative direction while $L_6$ seems to be moving closer to the origin from the negative part of the origin. We observed changes in the movements of the collinear positions $\left(L_{1,2,3,6}\right)$ as a result of the half distance between the mass dipoles and the oblateness of the primary for the problem under review. The movements away and closer to the origin from collinear positions do not change the status of the collinear points as they remain unstable and unchanged.

Table 1.

Numerical data for the binary systems: Luhman 16 and HD188753.

Binary system	Masses			Semi-major axis $\left(\mathit{a}\right)$	Eccentricity $\left(\mathit{e}\right)$	Spectral Type
	${\mathit{M}}_{\mathbf{1}}$	${\mathit{M}}_{\mathbf{21}}$	${\mathit{M}}_{\mathbf{22}}$
Luhman 16	$33.20$	$0.032$	$0.032$	$2.63\times {10}^{-6}$	$0.338$	$L7.5$
HD188753	$0.99$	$0.62$	$0.62$	$0.0058$	$0.502$	$G8V$

Table 2.

Numerical data in a dimensionless form for the binary systems: Luhman 16 and HD188753.

Binary system	Mass ratio $\left(\mathit{\nu }\right)$	Semi-major axis $\left(\mathit{\alpha }=\mathbf{1}-\mathit{a}\right)$	Eccentricity $\left(\mathit{e}\right)$	Half Distance between mass dipoles $\left(\mathit{d}\right)$	Oblateness $\left(\mathit{A}\right)$
Luhman 16	$0.000962$	$0.999997$	$0.338$	$0.010$	$0.00000$
				$0.015$	$0.00150$
				$0.020$	$0.00300$
				$0.025$	$0.00450$
				$0.030$	$0.00600$
				$0.035$	$0.00750$
				$0.040$	$0.00900$
				$0.045$	$0.01050$
HD188753	$0.2780$	$0.9942$	$0.502$	$0.010$	$0.00000$
				$0.015$	$0.00150$
				$0.020$	$0.00300$
				$0.025$	$0.00450$
				$0.030$	$0.00600$
				$0.035$	$0.00750$
				$0.040$	$0.00900$
				$0.045$	$0.01050$

Table 3.

Effects of half mass dipole distance, oblateness, and other constant parameters on the collinear equilibrium points positions for Luhman-16 systems: $\nu =0.000962,\alpha =1-a=0.999997,e=0.338$.

Half Distance between mass dipoles $\left(\mathit{d}\right)$	Oblateness $\left(\mathit{A}\right)$	Collinear Equilibrium points positions
		${\mathit{L}}_{\mathbf{1}}$	${\mathit{L}}_{\mathbf{2}}$	${\mathit{L}}_{\mathbf{3}}$	${\mathit{L}}_{\mathbf{6}}$
$0.010$	$0.00000$	$1.0723979$	$0.8975113$	$0.8975116$	$-0.9494600$
$0.015$	$0.00150$	$1.0742521$	$0.8968232$	$0.8968236$	$-0.9479124$
$0.020$	$0.00300$	$1.0766762$	$0.8958037$	$0.8958036$	$-0.9480944$
$0.025$	$0.00450$	$1.0795474$	$0.8944725$	$0.8944726$	$-0.9482764$
$0.030$	$0.00600$	$1.0827621$	$0.8928525$	$0.8928526$	$-0.9484554$
$0.035$	$0.00750$	$1.0862394$	$0.8909677$	$0.8909676$	$-0.9486344$
$0.040$	$0.00900$	$1.0899183$	$0.8888424$	$0.8888426$	$-0.9488114$
$0.045$	$0.01050$	$1.0937539$	$0.8864999$	$0.8880156$	$-0.9489874$

Fig. 1.

Configuration of the model.

Fig. 2.

Effects of half dipole mass (d) and Oblateness (A) on Location (${\mathit{L}}_{\mathbf{1}}$) for Luhman-16 system with $\nu =0.000962,\alpha =1-a=0.999997,e=0.338$.

Fig. 3.

Effects of half dipole mass (d) and Oblateness (A) on Location (${\mathit{L}}_{\mathbf{2}}$) for Luhman-16 system with $\nu =0.000962,\alpha =1-a=0.999997,e=0.338$.

Fig. 4.

Effects of half dipole mass (d) and Oblateness (A) on Location (${\mathit{L}}_{\mathbf{3}}$) for Luhman-16 system with $\nu =0.000962,\alpha =1-a=0.999997,e=0.338$.

Table 4.

Effects of half mass dipole distance, oblateness, and other constant parameters on the collinear equilibrium points positions for HD188753 systems: $\nu =0.2780,\alpha =1-a=0.9942,e=0.502$.

Half Distance between mass dipoles $\left(\mathit{d}\right)$	Oblateness $\left(\mathit{A}\right)$	Collinear Equilibrium points positions
		${\mathit{L}}_{\mathbf{1}}$	${\mathit{L}}_{\mathbf{2}}$	${\mathit{L}}_{\mathbf{3}}$	${\mathit{L}}_{\mathbf{6}}$
$0.010$	$0.00000$	$1.087625$	$-0.077503$	$-0.077503$	$-1.130915$
$0.015$	$0.00150$	$1.087434$	$-0.076567$	$-0.076567$	$-1.131949$
$0.020$	$0.00300$	$1.087327$	$-0.075715$	$-0.075715$	$-1.132969$
$0.025$	$0.00450$	$1.087306$	$-0.074946$	$-0.074946$	$-1.133975$
$0.030$	$0.00600$	$1.087369$	$-0.074259$	$-0.074259$	$-1.134967$
$0.035$	$0.00750$	$1.087518$	$-0.073653$	$-0.073653$	$-1.135945$
$0.040$	$0.00900$	$1.087750$	$-0.073127$	$-0.073127$	$-1.136911$
$0.045$	$0.01050$	$1.088067$	$-0.072679$	$-0.072679$	$-1.137864$

Fig. 5.

Effects of half dipole mass (d) and Oblateness (A) on Location (${\mathit{L}}_{\mathbf{6}}$) for Luhman-16 system with $\nu =0.000962,\alpha =1-a=0.999997,e=0.338$.

Fig. 6.

Effects of half dipole mass (d) and Oblateness (A) on Location (${\mathit{L}}_{\mathbf{1}}$) for HD188753 system with $\nu =0.2780,\alpha =1-a=0.9942,e=0.502$.

Fig. 7.

Effects of half dipole mass (d) and Oblateness (A) on Location (${\mathit{L}}_{\mathbf{2}}$) for HD188753 system with $\nu =0.2780,\alpha =1-a=0.9942,e=0.502$.

Fig. 8.

Effects of half dipole mass (d) and Oblateness (A) on Location (${\mathit{L}}_{\mathbf{3}}$) for HD188753 system with $\nu =0.2780,\alpha =1-a=0.9942,e=0.502$.

Fig. 9.

Effects of half dipole mass (d) and Oblateness (A) on Location (${\mathit{L}}_{\mathbf{6}}$) for HD188753 system with $\nu =0.2780,\alpha =1-a=0.9942,e=0.502$.

4. Stability of collinear points

Szebehely [24] stated that the motion that remains in the equilibrium point's small neighborhood after it has been disturbed is termed “stable."

To examine the stability of the collinear points, we consider the points lying in $\xi >{\xi }_{22},{\xi }_1<\xi <{\xi }_{21},{\xi }_{21}<\xi <{\xi }_{22}and{\xi }_1>\xi$ respectively.

Considering the stability of a collinear point for which $\xi >{\xi }_{22}$ we have that

$$r_1=\left|\xi +2\nu \right|,r_{21}=\left|\xi -1+2\nu +d\right|,r_{22}=\left|\xi -1+2\nu -d\right|$$ (13)

Given the second partial derivatives as;

$${\mathrm{\Omega }}_{\xi \xi }={\left(1-e^2\right)}^{-\frac{1}{2}}\left[1-\frac{1}{n^2}\left\{\frac{\left(1-2\nu \right)}{{r_1}^3}+\frac{3\left(1-2\nu \right)A}{2{r_1}^5}+\frac{\nu }{{r_{21}}^3}+\frac{\nu }{{r_{22}}^3}\right\}+\frac{1}{n^2}\left\{\frac{3\left(1-2\nu \right){\left(\xi +2\nu \right)}^2}{{r_1}^5}+\frac{15\left(1-2\nu \right){\left(\xi +2\nu \right)}^{2}A}{2{r_1}^7}+\frac{3\nu {\left(\xi -1+2\nu +d\right)}^2}{{r_{21}}^5}+\frac{3{\nu \left(\xi -1+2\nu -d\right)}^2}{{r_{22}}^5}\right\}\right]$$ (14)

$${\mathrm{\Omega }}_{\eta \eta }={\left(1-e^2\right)}^{-\frac{1}{2}}\left[1-\frac{1}{n^2}\left(\frac{\left(1-2\nu \right)}{{r_1}^3}+\frac{3\left(1-2\nu \right)A}{2{r_1}^5}+\frac{\nu }{{r_{21}}^3}+\frac{\nu }{{r_{22}}^3}\right)+\frac{{\eta }^2}{n^2}\left(\frac{3\left(1-2\nu \right)}{{r_1}^5}+\frac{15\left(1-2\nu \right)A}{2{r_1}^7}+\frac{3\nu }{{r_{21}}^5}+\frac{3\nu }{{r_{22}}^5}\right)\right]$$ (15)

$${\mathrm{\Omega }}_{\xi \eta }={\left(1-e^2\right)}^{-\frac{1}{2}}\left[\frac{\eta }{n^2}\left\{\frac{3\left(1-2\nu \right)\left(\xi +2\nu \right)}{{r_1}^5}+\frac{15\left(1-2\nu \right)\left(\xi +2\nu \right)A}{2{r_1}^7}+\frac{3\nu \left(\xi -1+2\nu +d\right)}{{r_{21}}^5}+\frac{3\nu \left(\xi -1+2\nu -d\right)}{{r_{22}}^5}\right\}\right]$$ (16)

Substituting equation (13) into equation (14), we have

$${\mathrm{\Omega }}_{\xi \xi }^0={\left(1-e^2\right)}^{-\frac{1}{2}}\left[1+\frac{1}{n^2}\left(\frac{\left(1-2\nu \right)}{{\left|\xi +2\nu \right|}^3}+\frac{3\left(1-2\nu \right)A}{{\left|\xi +2\nu \right|}^5}+\frac{\nu }{{\left|\xi -1+2\nu +d\right|}^3}+\frac{\nu }{{\left|\xi -1+2\nu -d\right|}^3}\right)\right]>0$$ (17)

From equation (8)

$$n^2\xi -\left[\frac{\left(1-2\nu \right)\left(\xi -{\xi }_1\right)}{{r_1}^3}+\frac{3\left(1-2\nu \right)\left(\xi -{\xi }_1\right)A}{2{r_1}^5}+\frac{\nu \left(\xi -{\xi }_{21}\right)}{{r_{21}}^3}+\frac{\nu \left(\xi -{\xi }_{22}\right)}{{r_{22}}^3}\right]=0$$

But $r_1=\left(\xi -{\xi }_1\right),r_{21}=\left(\xi -{\xi }_{21}\right)and{r}_{22}=\left(\xi -{\xi }_{22}\right)$ for collinear equilibrium points $\left(i.e\eta =\zeta =0fromequation\left(3\right)\right)$

Then

$$\frac{\left(1-2\nu \right)}{{r_1}^2}=n^2\xi -\frac{3\left(1-2\nu \right)A}{2{r_1}^4}-\frac{\nu }{{r_{21}}^2}-\frac{\nu }{{r_{22}}^2}$$ (18)

Substituting equation (18) into equation (15) with $\eta =\zeta =0$ we obtain

$${\mathrm{\Omega }}_{\eta \eta }={\left(1-e^2\right)}^{-\frac{1}{2}}\left[1-\frac{\xi }{r_1}+\frac{1}{n^2}\left(\frac{\nu }{r_1.{r_{21}}^2}+\frac{\nu }{r_1.{r_{22}}^2}-\frac{\nu }{{r_{21}}^3}-\frac{\nu }{{r_{22}}^3}\right)\right]$$ (19)

Given that $n^2=\frac{1}{a}\left(1+\frac{3A}{2}+\frac{3e^2}{2}\right)$, then $\frac{1}{n^2}=a\left(1-\frac{3A}{2}-\frac{3e^2}{2}\right)$. But from equation (13), ${\xi =r}_1-2\nu$. Now, substituting the values of $\frac{1}{n^2}$ and $\xi$ into equation (19) and by $\nu <\frac{1}{4},A,e^2\ll 1,r_1>1,r_{21}<r_{22}<1$ we have

$${\mathrm{\Omega }}_{\eta \eta }^0=\left[\frac{\nu }{r_1}\left(2+e^2\right)+\left[\frac{a\nu }{r_1.{r_{21}}^2}+\frac{a\nu }{r_1.{r_{22}}^2}-\frac{a\nu }{{r_{21}}^3}-\frac{a\nu }{{r_{22}}^3}\right]\left(1-\frac{3A}{2}-e^2\right)\right]<0$$ (20)

Now, for equation (16), we have

$${\mathrm{\Omega }}_{\xi \eta }^0=0\eta =0$$ (21)

Likewise, for the collinear points lying in the interval ${\xi }_1<\xi <{\xi }_{21},{\xi }_{21}<\xi <{\xi }_{22}and{\xi }_1>\xi$ we have that,

$${\mathrm{\Omega }}_{\xi \xi }^0>0,{\mathrm{\Omega }}_{\eta \eta }^0<0and{\mathrm{\Omega }}_{\xi \eta }^0=0$$

Now, considering the characteristics equation of the system given by Singh and Tyokyaa [17];

$${\lambda }^4-\left({\mathrm{\Omega }}_{\xi \xi }^0+{\mathrm{\Omega }}_{\eta \eta }^0-4\right){\lambda }^2+{\mathrm{\Omega }}_{\xi \xi }^0{\mathrm{\Omega }}_{\eta \eta \eta }^0-{\left({\mathrm{\Omega }}_{xy}^0\right)}^2=0$$ (22)

From equation (22) ${\mathrm{\Omega }}_{\xi \xi }^0{\mathrm{\Omega }}_{\eta \eta \eta }^0-{\left({\mathrm{\Omega }}_{xy}^0\right)}^2<0$ therefore, its discriminant is positive, and the roots can be expressed as as ${\lambda }_{\mathrm{1,2}}=\pm b$ and ${\lambda }_{\mathrm{3,4}}=\pm ic$ where $b$ and $c$ are real. This shows that the motion in the neighborhood of the collinear points is unstable since it is not bounded.

To ascertain the status of stability of the collinear equilibrium positions, we compute numerical values using equations (17), (20), (21) into equation (22) with the help of the software MATHEMATICA. These values are tabulated in Table 5, Table 6, Table 7, Table 8 for the binary systems: Luhman 16 and HD188753. We have observed in Table 5, Table 6, Table 7, Table 8 that, as the half distance between mass dipoles and oblateness of the primary increase, the region of stability of the collinear positions decreases for both binary systems.

Table 5.

The characteristic roots $\left({\lambda }_{\mathrm{1,2}};{\lambda }_{\mathrm{3,4}}\right)$ of collinear point $\left(L_{1}and{L}_2\right)$ for Luhman-16 systems: $\nu =0.000962,\alpha =1-a=0.999997,e=0.338$.

Half Distance between mass dipoles $\left(\mathit{d}\right)$	Oblateness $\left(\mathit{A}\right)$	Location	Characteristic roots
		${\mathit{L}}_{\mathbf{1}}$	${\mathit{\lambda }}_{\mathbf{1,2}}$	${\mathit{\lambda }}_{\mathbf{3,4}}$
$0.010$	$0.00000$	$1.0723979$	$\pm 3.16491$	$0\pm 2.43674i$
$0.015$	$0.00150$	$1.0742521$	$\pm 3.24412$	$0\pm 2.4863i$
$0.020$	$0.00300$	$1.0766762$	$\pm 3.3403$	$0\pm 2.54691i$
$0.025$	$0.00450$	$1.0795474$	$\pm 3.44582$	$0\pm 2.61377i$
$0.030$	$0.00600$	$1.0827621$	$\pm 3.55536$	$0\pm 2.68347i$
$0.035$	$0.00750$	$1.0862394$	$\pm 3.66562$	$0\pm 2.7539i$
$0.040$	$0.00900$	$1.0899183$	$\pm 3.77483$	$0\pm 2.82387i$
$0.045$	$0.01050$	$1.0937539$	$\pm 3.88207$	$0\pm 2.89276i$
$\mathit{d}$	$\mathit{A}$	${\mathit{L}}_{\mathbf{2}}$	${\mathit{\lambda }}_{\mathbf{1,2}}$	${\mathit{\lambda }}_{\mathbf{3,4}}$
$0.010$	$0.00000$	$0.8975113$	$\pm 2.02219$	$0\pm 1.60256i$
$0.015$	$0.00150$	$0.8968232$	$\pm 2.06743$	$0\pm 1.62742i$
$0.020$	$0.00300$	$0.8958037$	$\pm 2.12708$	$0\pm 1.66086i$
$0.025$	$0.00450$	$0.8944725$	$\pm 2.19954$	$0\pm 1.70205i$
$0.030$	$0.00600$	$0.8928525$	$\pm 2.28314$	$0\pm 1.75008i$
$0.035$	$0.00750$	$0.8909677$	$\pm 2.37623$	$0\pm 1.80409i$
$0.040$	$0.00900$	$0.8888424$	$\pm 2.47737$	$0\pm 1.86329i$
$0.045$	$0.01050$	$0.8864999$	$\pm 2.58531$	$0\pm 1.92697i$

Table 6.

The characteristic roots $\left({\lambda }_{\mathrm{1,2}};{\lambda }_{\mathrm{3,4}}\right)$ of collinear points $\left(L_{3}and{L}_6\right)$ for Luhman-16 systems: $\nu =0.000962,\alpha =1-a=0.999997,e=0.338$.

Half Distance between mass dipoles $\left(\mathit{d}\right)$	Oblateness $\left(\mathit{A}\right)$	Location	Characteristic roots
		${\mathit{L}}_{\mathbf{3}}$	${\mathit{\lambda }}_{\mathbf{1,2}}$	${\mathit{\lambda }}_{\mathbf{3,4}}$
$0.010$	$0.00000$	$0.8975116$	$-0.685506\pm 0.919434i$	$0.685506\pm 0.919434i$
$0.015$	$0.00150$	$0.8968236$	$-0.66714\pm 0.970443i$	$0.66714\pm 0.970443i$
$0.020$	$0.00300$	$0.8958036$	$-0.610931\pm 0.996466i$	$0.610931\pm 0.996466i$
$0.025$	$0.00450$	$0.8944726$	$-0.496661\pm 0.996028i$	$0.496661\pm 0.996028i$
$0.030$	$0.00600$	$0.8928526$	$-0.1893\pm 0.954177i$	$0.1893\pm 0.954177i$
$0.035$	$0.00750$	$0.8909676$	$\pm 0.209013$	$0\pm 1.43531i$
$0.040$	$0.00900$	$0.8888426$	$\pm 0.722371$	$0\pm 1.67807i$
$0.045$	$0.01050$	$0.8880156$	$\pm 1.08045$	$0\pm 1.98554i$
$\mathit{d}$	$\mathit{A}$	${\mathit{L}}_{\mathbf{6}}$	${\mathit{\lambda }}_{\mathbf{1,2}}$	${\mathit{\lambda }}_{\mathbf{3,4}}$
$0.010$	$0.00000$	$-0.9494600$	0 $\pm 0.0978511i$	$0\pm 0.892534i$
$0.015$	$0.00150$	$-0.9479124$	0 $\pm 0.0982423i$	$0\pm 0.889507i$
$0.020$	$0.00300$	$-0.9480944$	0 $\pm 0.0986352i$	$0\pm 0.886485i$
$0.025$	$0.00450$	$-0.9482764$	0 $\pm 0.0990288i$	$0\pm 0.883474i$
$0.030$	$0.00600$	$-0.9484554$	0 $\pm 0.099425i$	$0\pm 0.880462i$
$0.035$	$0.00750$	$-0.9486344$	0 $\pm 0.0998218i$	$0\pm 0.877463i$
$0.040$	$0.00900$	$-0.9488114$	0 $\pm 0.100221i$	$0\pm 0.874466i$
$0.045$	$0.01050$	$-0.9489874$	0 $\pm 0.100621i$	$0\pm 0.871477i$

Table 7.

The characteristic roots $\left({\lambda }_{\mathrm{1,2}};{\lambda }_{\mathrm{3,4}}\right)$ of collinear point $\left(L_{1}and{L}_2\right)$ for HD188753 systems: $\nu =0.2780,\alpha =1-a=0.9942,e=0.502$.

Half Distance between mass dipoles $\left(\mathit{d}\right)$	Oblateness $\left(\mathit{A}\right)$	Location	Characteristic roots
		${\mathit{L}}_{\mathbf{1}}$	${\mathit{\lambda }}_{\mathbf{1,2}}$	${\mathit{\lambda }}_{\mathbf{3,4}}$
$0.010$	$0.00000$	$1.087625$	$\pm 1.33159$	$0\pm 1.23703i$
$0.015$	$0.00150$	$1.087434$	$\pm 1.33271$	$0\pm 1.23683i$
$0.020$	$0.00300$	$1.087327$	$\pm 1.33431$	$0\pm 1.23688i$
$0.025$	$0.00450$	$1.087306$	$\pm 1.33636$	$0\pm 1.23718i$
$0.030$	$0.00600$	$1.087369$	$\pm 1.33889$	$0\pm 1.23771i$
$0.035$	$0.00750$	$1.087518$	$\pm 1.34185$	$0\pm 1.23849i$
$0.040$	$0.00900$	$1.087750$	$\pm 1.34528$	$0\pm 1.23951i$
$0.045$	$0.01050$	$1.088067$	$\pm 1.34914$	$0\pm 1.24075i$
$\mathit{d}$	$\mathit{A}$	${\mathit{L}}_{\mathbf{2}}$	${\mathit{\lambda }}_{\mathbf{1,2}}$	${\mathit{\lambda }}_{\mathbf{3,4}}$
$0.010$	$0.00000$	$-0.077503$	$\pm 3.11479$	$0\pm 1.52913$
$0.015$	$0.00150$	$-0.076567$	$\pm 3.13841$	$0\pm 1.51553$
$0.020$	$0.00300$	$-0.075715$	$\pm 3.16274$	$0\pm 1.50174$
$0.025$	$0.00450$	$-0.074946$	$\pm 3.18783$	$0\pm 1.48773$
$0.030$	$0.00600$	$-0.074259$	$\pm 3.21373$	$0\pm 1.47348$
$0.035$	$0.00750$	$-0.073653$	$\pm 3.24047$	$0\pm 1.45897$
$0.040$	$0.00900$	$-0.073127$	$\pm 3.26809$	$0\pm 1.44415$
$0.045$	$0.01050$	$-0.072679$	$\pm 3.29663$	$0\pm 1.42901$

Table 8.

The characteristic roots $\left({\lambda }_{\mathrm{1,2}};{\lambda }_{\mathrm{3,4}}\right)$ of collinear point $\left(L_{3}and{L}_6\right)$ for HD188753 systems: $\nu =0.2780,\alpha =1-a=0.9942,e=0.502$.

Half Distance between mass dipoles $\left(\mathit{d}\right)$	Oblateness $\left(\mathit{A}\right)$	Location	Characteristic roots
		${\mathit{L}}_{\mathbf{3}}$	${\mathit{\lambda }}_{\mathbf{1,2}}$	${\mathit{\lambda }}_{\mathbf{3,4}}$
$0.010$	$0.00000$	$-0.077503$	$-2.23889\pm 0.955792i$	$2.23889\pm 0.955792i$
$0.015$	$0.00150$	$-0.076567$	$-2.23821\pm 0.961645i$	$2.23821\pm 0.961645i$
$0.020$	$0.00300$	$-0.075715$	$-2.23751\pm 0.96717i$	$2.23751\pm 0.96717i$
$0.025$	$0.00450$	$-0.074946$	$-2.23681\pm 0.972342i$	$2.23681\pm 0.972342i$
$0.030$	$0.00600$	$-0.074259$	$-2.23613\pm 0.977138i$	$2.23613\pm 0.977138i$
$0.035$	$0.00750$	$-0.073653$	$-2.2547\pm 0.981536i$	$2.2547\pm 0.981536i$
$0.040$	$0.00900$	$-0.073127$	$-2.23483\pm 0.985514i$	$2.23483\pm 0.985514i$
$0.045$	$0.01050$	$-0.072679$	$-2.23423\pm 0.989053i$	$2.23423\pm 0.989053i$
$\mathit{d}$	$\mathit{A}$	${\mathit{L}}_{\mathbf{6}}$	${\mathit{\lambda }}_{\mathbf{1,2}}$	${\mathit{\lambda }}_{\mathbf{3,4}}$
$0.010$	$0.00000$	$-1.130915$	$-1.39251\pm 0.810259i$	$1.39251\pm 0.810259i$
$0.015$	$0.00150$	$-1.131949$	$-1.39491\pm 0.807817i$	$1.39491\pm 0.807817i$
$0.020$	$0.00300$	$-1.132969$	$-1.39724\pm 0.80541i$	$1.39724\pm 0.80541i$
$0.025$	$0.00450$	$-1.133975$	$-1.39951\pm 0.803035i$	$1.39951\pm 0.803035i$
$0.030$	$0.00600$	$-1.134967$	$-1.40172\pm 0.800692i$	$1.40172\pm 0.800692i$
$0.035$	$0.00750$	$-1.135945$	$-1.40387\pm 0.798378i$	$1.40387\pm 0.798378i$
$0.040$	$0.00900$	$-1.136911$	$-1.40597\pm 0.796096i$	$1.40597\pm 0.796096i$
$0.045$	$0.01050$	$-1.137864$	$-1.40802\pm 0.793842i$	$1.40802\pm 0.793842i$

As represented in Table 6, the collinear equilibrium point $\left(L_3\right)$ is stable for some values of half distance between mass dipole $\left(d\right)$ and oblateness $\left(A\right)$ for Luhman 16 system for the characteristic roots $\left({\lambda }_{\mathrm{1,2}}\right)$. This is evidenced by ${\lambda }_{\mathrm{1,2}}$ having a negative real part with a complex root for a collinear point $\left(L_3:0.8975116to0.8928526\right)$ see Table 6. Other collinear points or values considered in the Luhman 16 system are unstable as the characteristic root is a positive real part or at least one characteristic root is a positive real part with a complex root (see Table 5, Table 6).

We have also observed stable collinear equilibrium points $\left(L_{3}and{L}_6\right)$ for values of half distance between mass dipole $\left(d\right)$ and oblateness $\left(A\right)$ for the system HD188753 considering the characteristic roots $\left({\lambda }_{\mathrm{1,2}}\right)$ as shown in Table 8. The collinear $\left(L_{3}and{L}_6\right)$ are unstable for the characteristic roots (${\lambda }_{\mathrm{3,4}}$) in the sense of Lyapunov as at least a positive real part and a complex root is evidenced (see Table 8). In Table 7, the collinear equilibrium points $\left(L_{1}and{L}_2\right)$ are unstable for the HD188753 system as at least a positive real part and or a positive real part with a complex root is recorded.

Our study establishes four collinear equilibrium points $\left(L_{1,2,3,6}\right)$ unlike those of [23,[25], [26], [27]] whose collinear equilibrium points are $L_{1,2,3}$. The unstable stability behavior of this study agrees with those of [23,[25], [26], [27]].

5. Conclusion

We have studied the Collinear Equilibrium points in the Elliptic Restricted Synchronous Three-Body Problem under an oblate primary and a dipole secondary. Analytical solutions are drawn from equations (9), (10), (11), (12), (17), (20), (21), (22). Using the software MATHEMATICA, we computed numerical values from equations (9), (10), (11), (12), (22), which are presented in Table 3, Table 4, Table 5, Table 6, Table 7, Table 8 and Fig. 2, Fig. 3, Fig. 4, Fig. 5. We observed changes in the movements of the collinear positions $\left(L_{1,2,3,6}\right)$ as a result of the half distance between the mass dipoles and the oblateness of the primary for the problem under review. The movement away and closer to the origin from collinear positions changes the stability behaviour of the collinear points as they are stable and unstable for some characteristic roots depending on the values of the parameters under study. Our study reviewed that for both binary systems, the collinear points are stable for some values of the half distance between mass dipole $\left(d\right)$ and oblateness $\left(A\right)$ for the characteristic roots $\left({\lambda }_{\mathrm{1,2}}\right)$ and are unstable for some values of $d$ and $A$ for the characteristic roots $\left({\lambda }_{\mathrm{3,4}}\right)$.

It is further observed that, in each of Table 5, Table 6, Table 7, Table 8, as the half distance between mass dipoles and oblateness of the primary increase, the region of stability of the collinear positions decreases for both binary systems. The unstable stability behavior of this study agrees with those of [23,[25], [26], [27]].

Declarations

Author contribution statement

Jagadish Singh: Conceived and designed the experiments; contributed reagents, materials, analysis tools or data.Tyokyaa K. Richard: Performed the experiments; Analyzed and interpreted the data; Wrote the paper.

Funding statement

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Data availability statement

Data will be made available on request.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.