Equilibria, stability and chaos in photogravitational Bi-Circular restricted four-body problem.

A model to investigate the influence of gravitational force and radiation pressure on interstellar dust found within the vicinities of certain stellar systems is presented in this study with the inclusion of potential due to the belt. Semi-analytic approach is adopted to examine the dynamical behaviour of the motion of a test particle in the neighbourhood of the radiating stars, namely, Wolf 630, Formalhaut, Omicron Eradani and 36 Ophiuchi respectively. The Libration points were found dependent on the mass ratios of the systems and the radiation pressure exerted by the star(s) and the motion of the dust particle around them is linearly unstable. In each of the four case studies, two Lyapunov Characteristic Exponents were seen positive which validate the chaotic nature of the system, also the Poincare Surface Section revealed the sensitivity of the dynamical system to change in initial conditions. Several scientific questions of great importance in astrophysics and astronomy; such as the motions of interstellar clouds, proto-nebulae, planetary rings and micrometeorites can be answered by engaging this model.

Introduction

Over and again the existence of strong radiation sources in space environment has been reported by various scientists and researchers. Part of the subjects of interest in stellar dynamics is the study of effect of radiating bodies on small particles such as a particle of dust or a particle in a gas cloud. Outside the solar system, it is observed that radiation pressure has greater influence on the motion of infinitesimal particle than the gravitational force. For instance, Lamy and Perrin (1997) reported that depending on a star's spectral type, the size and the material composing the dust particle, the radiation force can be as big as the force of gravity. It is understood that the sun is the radiation source in our solar system meanwhile, there are various instances among the extrasolar systems in which we find some systems having more than one star as the source of radiation with intense exertion of force on the motion of test particle.

Following the report of Maxwell (1873) that electromagnetic radiation exerts pressure upon any surface exposed to it, many scientists such as Bartoli (1883), Heaviside (1893) and Lorentz (1895) have validated the claim both experimentally and theoretically. In fact, Lebedev (1891) propounded that the force of light pressure is inversely proportional to the square of the distance between the particle and the illuminating body. Thus it is imperative to take into cognizance that a combination of the radiation force with gravitational force from system of two or n-bodies can notably alter the dynamic behaviour of any infinitesimal body within the vicinity of that system.

The field of astrodynamics is concerned with the analysis of the motion of natural and artificial objects in space, subject to environmental and artificial forces. So in order to formulate simplified models for complex planetary system, the study of few-bodies problem is adopted as a standard tool in astronomy, astrophysics, solar, stellar and galactic dynamics. In the photogravitational restricted body problem, the motion of an infinitesimal particle in the neighbourhood of body is investigated when the infinitestimal particle is under the gravitational influence and the radiation pressure exerted by some or all of the participating luminous primary bodies. More so, to validate the photogravitationalmodel set up, it is always assumed that the radiation emitted from the participating massive bodies (called primaries) does not affect the motion of the primaries but influences the motion of the test particle only. The test particle is also taken to be spherical and homogenous with non-varying density and its surface has a uniform reflection coefficient with the size not exceeding the wavelength of the incident rays. Screening or shadowing effect from other bodies is made negligible and there are no fluctuations as the light propagates in straight lines.

It was Poyting (1903) who first gave a description of the effect of radiation in the frame of relativity. Robertson (1937) took a cue from this to give an analysis of the effect total radiation forces on a particle. Afterward, Burns et al. (1979) reported a more comprehensive relation between radiation pressure and Poyting-Robertson drag forces. Radzievskii (1950, 1953) was the first to tackle the photogravitational problem in the restricted three-body problem. By investigating the Sun-Planet-particle and Galaxy Kernel-Sun-particle, he observed the existence of two equilibrium points on the plane in symmetry to the plane. Schuerman (1980) later added what he called the particle which means a very small particles or radiant body that is very close to a star. He stated the possibility of radiation force being greater than gravitational force in the vicinity of two radiant primaries.

The restricted four-body problem (R4BP) is an extension of the R3BP. R4BP describes the motion of a body having a mass which is negligible compared to the masses of the three primaries which are set in motion under their mutual gravitational attraction. The three finite masses can assume two main configurations; the collinear configuration as studied by Kalvouridis et al. (2006, 2007); Arribas et al., 2016; Barrabes et al., 2017. The equilateral triangle configuration which is studied by Baltagiannis and Papadakis (2011), Papadouris and Papadakis (2013), Singh and Vincent (2015), and Singh and Omale (2019). In examining of Photogravitational R4BP with Stokes drag by Vincent et al. (2019), showed that under constant dissipative force, collinear equilibrium points do not analytically or numerically exist. Chakraborty and Narayan (2018) reported the effect of Stellar wind and P-R drag on Photogravitational Elliptic Three-Body Problem. Singh and Omale (2019) further emphasized that in the CR4BP, a combination of Stokes drag force and oblateness of the primaries leads to a decaying effect in the orbital elements. In Singh et al. (2020), the investigation revealed that circumstellar dust and radiation pressure in CR4BP in Manve's field cause dissipative effect on the motion of a test particle.

Moreover, in recent times the Bi-Circular Restricted Four-Body Problem (BCR4P) has been investigated with the inclusion of some characterizations by a few authors because of its advantage in applicability to our Solar system and other stellar systems that assume different configurations in which the classical R4BP may not be suitable in applications. Bi-Circular Restricted Four-Body Problem (BCR4P) describes a model where two of the primaries are rotating around their center of mass, which is also revolving together with the third mass around the barycenter of the system. The massless particle is moving under the gravitational influence of the primaries and does not affect their motion. It is assumed that the motion of the primaries, as like as the motion of the test particle are co-planar (see Cronin et al. (1964)). Abouelmagd and Ansari (2019) revealed the motion properties of the infinitesimal body in the framework of Bi-Circular Sun perturbed Earth-Moon System. Singh and Omale (2020) did a study on the Bi-Circular R4BP with dissipative forces and they pointed that the P-R drag exert greater drag effect on the motion of a test particle than the Stokes drag. The Bi-Bircular restricted four body problem (BR4BP) is very useful for analytic and application purpose, for example, The low eccentricity of the Earth's and Moon's orbit and the small inclination of the Moon's orbital plane, make the Bi-Circular R4BP an accurate model to describe the dynamics of a spacecraft in the Sun-Earth-Moon system; see Yagasaki (2004) and Mingotti et al. (2009). Others such as Castella and Jorba (2000), Castella and Jorba (2003), Jorba (2000), Assadian and Pourtakdoust (2010) and Chakraborty and Narayan (2019) have also given their contributions in the area of Bicircular and Bi-Elliptic restricted four body problem.

Our motivation in this study is to investigate within the framework of BCR4BP the dynamical behaviour of a photogravitational model that captures the motion of a dust grain in the vicinity of someunique tripple stellar systems. This work is organized thus; the derivation of the governing equation of motion is presented in section 2. Section 3 and 4 are for the location of libration points and investigation of stability of motion while section 5 is for the examination of chaos by numerical approach and the work is capped with conclusion is section 6.

Model formulation and equations of motion

Potential function for BCR4BP

We formulate this model by taking cue from Huang (1960) and Koon et al. (2000). Let three bodies of finite masses and undergo their motions on the same plane along with fourth body whose mass is negligible to the masses of the three bodies, and its motion is under the impact of the gravitational forces of the three finite masses. Figure 1 below is a schematic of the model set up. We constrain the configuration of the system in such a manner that is taken as the centre of mass of and which revolves around , the centre of mass of the entire system. Moreover, let , and be the distance between and , and , and , and respectively.

Figure 1

The schematic diagram of the Bi-Circular R4BP.

Now, if a rectangular coordinate system is given with its origin at and its three axes being subjected in space so that the axis is perpendicular to the plane of the masses. Then, the four bodies have the coordinates , , and respectively with the equations of motion for given as

Given that and are the angular velocities of the system as it moves about , and as moves about . Then;bearing that

Now, from Kepler's law

A Transformation of to a new coordinate system centred at such that the axis revolves with and and the axis parallel to axis yields;m1, m2 and m3 thus have a new coordinate system as

Substituting equations in 2 and (6), 7 and (8) into system (1) and making use of (9) gives system (10)

With

The system (10) becomeswhere

Taking

Since and is the angle subtended by and at . So that by using (4), 5 and (14), (13) becomes

Noting that is nearly a unity since and in order for the motion of to remain valid.

So that

Thus by utilizing the first three terms of the expansion of the spherical harmonics above (15) becomes

We non-dimensionalized by taking the sum of the masses and and the distance as the units of mass and length, so that the mass parameter with and . Also taking the unit of time so that the gravitational constant , and . We have,

Restricting the motion to the plane, then . Where and are the respective angles between the vectors and with the positive axis. By considering as a constant in a scale of time for few hours, and simplifying (19) we have

Eq. (20) is the gravitational potential for the BCR4BP.

Potential of the material cluster

Miyamoto and Nagai (1975) give the analytic disk potential otherwise recognized as the potential due to the material cluster aswhere is the total mass of the belt, and are parameters that quantify the density profile of the disk. The parameter is called the flatness parameter which determines the flatness of the profile, and is the core parameter which controls the size of the core of the density profile. Suppose , the potential becomes that of a point mass. Confining to the , we take and define then

Mass reduction factor

The mass reduction factor gives the relationship between the force due to radiation pressure and the force due to gravitational force.With where Fp and Fg are the radiation pressure and gravitational force, respectively. It is observed that at constant gravitational force, an increase in the radiation pressure leads to a decrease in the mass reduction factor. Moreover, according to Xuetang and Lizhong (1993).where M and L are the mass and the luminousity of a star respectively, and are the radius and density of the test particle, in the C.G.S system, and is the radiation pressure efficiency factor of a star, and it is taken as unity according to Stefan-Boltzmann's law.

The governing equations of motion

By combining the descriptions offered by Eqs. (20), (22), and (23) we have the equations that govern the motion of a dust particle under the influence of the forces of gravity and the radiation of the three bodies asWhere the potential is given aswhere , , and are the radiation pressure coefficients for the three primaries respectively.

Location of libration points

In this section we shall investigate the location of equilibrium points of the material dust in the sampled triple stellar systems under the following four cases;

when only one star is a radiating source and all the stars are of unequal masses. The practical application in this case is the motion of a test particle in the vicinity of Wolf 630 Triple stellar system.

when two stars are both radiating with all the three stars having unequal masses. The practical application here is the motion of a test particle in the vicinity of Formalhaut Triple stellar system.

when all the three stars are radiating with all the three stars having unequal masses. The practical application in this case is motion of a test particle in the vicinity of Omicron Eradani Triple stellar system.

when all the three stars are radiating with two of stars having equal masses. The practical application here is the motion of a test particle in the vicinity of 36 Ophiuchi Triple stellar system.

It is important to locate the positions of the libration points of the system because it avails us the avenue to investigate the dynamic behavior of the system. These points correspond to the positions in the rotating frame at which the gravitational force and the centrifugal force associated with the rotation of the synodic reference frame are equal, this implies that a dust grain placed at one of these points remains stationary in the synodic frame. Therefore, at the libration points both the velocity and the acceleration components of the test particle vanish i.e. . The data presented in Tables 1, 2, 3, 4 (Johnston 2018) are used to compute the Libration points shown in Tables 5, 6, 7, 8. The stellar distance between the primariesis converted from Light Years to Austronomical Unit. Taking note that 1 Light Year = 63239.7263 AU.

Table 1

Astronomical data for Case 1.

	Stellar System	Distance (Light Year)	VolumetricLuminosity	BorometricLuminosity	Mass (Solar Mass)	Mass Reduction Parameter (q)	Mass Parameter
M1	Wolf 630 D	21.205	0.00000628	0.00076	0.09	0.99999	0.10714
M2	Wolf 630A	21.190			0.56		0.66667
M3	Wolf 630 B	21.190			0.28		0.33333

Table 2

Astronomical data for Case 2.

	Stellar System	Distance (Light Year)	Volumetric Luminosity	Borometric Luminosity	Mass (Solar Mass)	Mass Reduction Parameter (q)	Mass Parameter
M1	Formalhaut B	24.818	0.129	0.189	0.73	0.997425	0.379635
M2	Formalhaut A	25.126	17.8	16.6	1.92	0.914009	0.998492
M3	Formalhaut Ab	25.126			0.0029		0.001508

Table 3

Astronomical data for Case 3.

	Stellar System	Distance (Light Year)	Volumetric Luminosity	Borometric Luminosity	Mass (Solar Mass)	Mass Reduction Parameter (q)	Mass Parameter
M1	Omicron Eradani A	16.255	0.366	0.410	0.78	0.994772	1.27869
M2	Omicron Eradani B	16.352	0.00341	0.010	0.43	0.999769	0.704918
M3	Omicron Eradani C	16.352	0.000699	0.00686	0.18	0.999621	0.295082

Table 4

Astronomical data for Case 4.

	Stellar System	Distance (Light Year)	Volumetric Luminosity	Borometric Luminosity	Mass (Solar Mass)	Mass Reduction Parameter (q)	Mass Parameter
M1	36 Ophiuchi C	19.406	0.0914	0.173	0.71	0.997577	0.417647
M2	36 Ophiuchi A	19.435	0.290	0.336	0.85	0.996068	0.5
M3	36 Ophiuchi B	19.435	0.287	0.333	0.85	0.996104	0.5

Table 5

Libration points for Wolf 630 Triple stellar system.

	α_0=0	α_0=π2	α_0=π
L1	(-0.246837, 0)	(-0.246837, -1.47081∗10−7)	(-0.246837, 0)
L2		(1.13059, -3.13925∗10−6)	(1.13059, 0)
L3	(-1.24364, 0)	(-1.24364, -1.25611∗10−6)	(-1.24364, 0)
L4	(0.000529317, 0)	(0.000529317, -1.11396∗10−10)	(0.000529317, 0)
L5	(0.0359977, 0)	(0.0359977, -5.12002∗10−9)	(0.0359977, 0)
L6	(-0.166668, 0.861781)	(-0.16667, 0.861782)	(-0.166672, 0.861781)
L7	(-0.166668, -0.861781)	(-0.16667, -0.861781)	(-0.166672, -0.861781)

Table 6

Libration points for Formalhaut Triple stellar system.

	α_0=0	α_0=π2	α_0=π
L1	(-0.907965, 0)	(-0.907965, -1.18993∗10−8)	(-0.907965, 0)
L2	(0.966378, 0)	(0.966124, -0.0000148248)	(0.966378, 0)
L3	(-1.07008, 0)	(-1.07008, -6.93004∗10−9)	(-1.07008, 0)
L4	(-0.469678, 0.842999)	(-0.469681, 0.842997)	(-0.46969, 0.842992)
L5	(-0.469678, -0.842999)	(-0.469687, -0.842994)	(-0.46969, -0.842992)

Table 7

Libration points for Omicron Eradani Triple stellar system.

	α_0=0	α_0=π2	α_0=π
L1	(-0.299668, 0)	(-0.299668, -3.72874∗10−9)	(-0.299668, 0)
L2	(1.11542, 0)	(1.11542, -8.66163∗10−8)	(1.11542, 0)
L3	(-1.25239, 0)	(-1.25239, -2.74827∗10−8)	(-1.25239, 0)
L4	(0.000760979, 0)	(0.000760979, 0)	(0.000760979, 0)
L5	(0.0299558, 0)	(0.0299558, 0)	(0.0299558, 0)
L6	(-0.204967, -0.861517)	(-0.204967, -0.861517)	(-0.204967, -0.861517)
L7	(-0.204967, 0.861517)	(-0.204967, 0.861517)	(-0.204967, 0.861517)

Table 8

Libration points for 36 OphiuchiTriple stellar system.

	α_0=0	α_0=π2	α_0=π
L1	(7.18663∗10ˆ-9, 0)	(7.18781∗10ˆ-9, 0)	(7.18515∗10ˆ-9, 0)
L2	(0.0818256, 0)	(0.0818256, -1.01568∗10−9)	(0.0818256, 0)
L3	(-0.0818227, 0)	(-0.0818227, -1.01561∗10−9)	(-0.0818227, 0)
L4	(-1.19158, 0)	(-1.19158, -4.51463∗10−8)	(-1.19158, 0)
L5		(1.19157, -4.51464∗10−8)	(1.19157, 0)
L6	(0.0000119694, 0.860618)	(0.0000119347, 0.860618)	(1.18999∗10ˆ-5, -0.860618)
L7	(0.0000119694, -0.860618)	(0.0000119347, -0.860618)	(1.18999∗10ˆ-5, 0.860618)

The Figures 2 and 3 show the distributions of the libration points on the plane of motion as well as the Contours in the case when only one star is a radiating source and all the stars are of unequal masses, the suitable stellar system in this case is the Wolf 630 system. When the triple stellar system has seven libration points, while at the libration points are six but the shape of the Contours remain the same.

Figure 2

Contours of the fixed/libration points for Wolf 630 system at .

Figure 3

Contours of the fixed/libration points for Wolf 630 system at .

The Figures 4 and 5 above show the distributions of the libration points on the plane of motion as well as the Contours in the case when two stars are both radiating with all the three stars having unequal masses, the suitable stellar system in this case is the Formalhaut system. When and the triple stellar system has five libration pointswith L4 and L5 being symmetrical to each other.

Figure 4

Contours of the fixed/libration points for Formalhaut system at .

Figure 5

Contours of the fixed/libration points for Formalhault system at .

In case three when all the three stars are radiating with all the three stars having unequal masses, the Figures 6 and 7 above show the distributions of the libration points on the plane of motion as well as the Contours, the suitable stellar system in this case is the Omicron Eradani system. This system as well has seven libration points when and .

Figure 6

Contours of the fixed/libration points for Omicron Eradani system at .

Figure 7

Contours of the fixed/libration points for Omicron Eradani system at .

Finally, when all the three stars are radiating with two of stars having equal masses, we considered the Ophiuchi system as the best suitable candidate. It has seven libration points when and . The distributions of the libration points are unique with a circular trajectory at the centre of the plane. These are shown in Figures 8 and 9 respectively.

Figure 8

Contours of the fixed/libration points for 36 Ophiuchi system at .

Figure 9

Contours of the fixed/libration points for 36 Ophiuchi system at .

Linear stability of the equilibrium points

Consider the motion of the infinitesimal body when the coordinates of a particular equilibrium point (x0, y0) is given small displacements. Assuming and are the small displacements in the coordinates such that and . Then Eq. (25) possess the variational equations of motion as given by Eq. (29)where the dots are the derivatives with respect to the actual time t, the subscripts depict the second partial derivatives and superscripts 0 indicate that the values are evaluated at the equilibrium point (x0, y0). We neglect the terms of second and higher powers of and . We take Eq. (30) as the trial solutions of Eq. (29).with as constants and a parameter. Then the characteristic equation of system (29) is derived asWith

Chaos in the system

It is of dynamical interest to investigate a system's sensitivity to change in initial conditions. If a dynamical system is very sensitive to change in initial conditions, then it is chaotic and it becomes difficult to predict its future behavior. That means, if the neighboring orbits separate exponentially fast, then the system possesses irregularity and chaos can be quantified with the aid of Lyapunov characteristic exponents. Assuming some initial condition , and is a nearby point with the initial separation being very small. Also, Let be the separation after iterates. If

Then is called Lyapunov exponent.

A positive Lyapunov exponent confirms that the system is chaotic (see Strogatz, 1994). The first order LCEs areComputed with the help of Mathematica package developed by Sandri (1995). For 100 iterations, we found the LCEs as shown in Table 9. More so, the Figures 10, 11, 12, and 13 below show the exponential rate of divergence of the orbits for the four triple stellar systems respectively. In all the cases, two of the LCEs are seen to be positive we therefore conclude that the system is chaotic which implies that the system is very sensitive to change in initial condition.

Table 9

LCEs for the Triple stellar system in the four cases at .

Wolf 630	Formalhaut	Omicron Eradani	36 Ophiuchi
1.00955	1.00955	1.00955	1.00955
1.00955	1.00955	1.00955	1.00955
-0.00954959	-0.00955075	-0.0095496	-0.00954992
-0.00955327	-0.0095515	-0.00955324	-0.00955289

Figure 10

Divergent orbits for Wolf 630 Triple Stellar System.

Figure 11

Divergent orbits for Formalhaut Triple Stellar System.

Figure 12

Divergent orbits for Omicron Eradani Triple Stellar System Omicron.

Figure 13

Divergent orbits for 36 Ophiuchi Triple Stellar System.

Poincare Surface section

Further, in addition to LCEs being used as a tool to numerically verify chaos, we examine the sensitivity of the dynamical system to change in initial conditions by constructing the Poincare Surface of Section (PSS) in each of the four cases. The phase space is four dimensional with the conditions In each of the cases, we alter the initial conditions from to with just this little change, we noticed significant deviations in the phase space of the Poincare Surface Sections as shown in the Figures 14, 15, 16, 17, 18, 19, 20, 21, especially the sharp contrast between Figures 14 and 15, and, Figures 20 and 21 respectively. Figures 14 and 15 is the PSS of the Formahault system while those of Wolf 630, Omicron Eradani and 36 Ophiuchi are presented in Figures 16, 17, 18, 19, 20, 21 respectively.

Figure 14

PSS for Wolf 630 system when initial conditions is .

Figure 15

PSS for Wolf 630 system when initial conditions is .

Figure 16

PSS for Formalhault system when initial conditions is .

Figure 17

PSS for Formalhault system when initial conditions is .

Figure 18

PSS for Omicron Eradani system when initial conditions is .

Figure 19

PSS for Omicron Eradani system when initial conditions is .

Figure 20

PSS for 36 Ophiuchi system when initial conditions is .

Figure 21

PSS for 36 Ophiuchi system when initial conditions is .

Discussion and conclusion

Considering the fact that radiation pressure from luminous bodies can have significant impact on the motion of a test particle within their sphere of influence, in this study we have numerically explored a model that can be used to generalized the existence of such occurrence. The mathematical formulation of the problem took a different route than the conventional restricted four-body problem so that it can be nearly coherent in modeling the real phenomenon. The Bi-circular version of the problem with the use of real mass value of the primaries and inclusion of the potential from the dust cloud is more realistic and generalizing. With the use of parameters for specific triple stellar systems, we have examined the dynamical behaviour of the system with respect to different values of mass ratio and the nature of radiation from the primaries. We found that the number of the equilibrium points, their positioning on the plane of motion, as well as the orientation of the orbits depend largely on the value of the mass ratio and the radiation coefficients of the systems. This difference is clearly seen in Figures 2, 3, 4, 5, 6, 7, 8, 9. The analysis for the motion stability of the test particle around the equilibrium points in all the four cases revealed that none of the equilibrium points is a stable neighbourhood as shown in Tables 10, 11, 12 and 13. More so, using a numerical approach, the Lyapunov Characteristic Exponents of the system showed that the system is very sensitive to change in initial conditions. The exponential divergence of the Contoursfor each of the systems is shown in Figures 10, 11, 12 and 13 respectively. More so, the Poincare Surface of section for the system shows that the system is sensitive to change in initial conditions as shown in Figures 14, 15, 16, 17, 18, 19, 20, 21. This made us to establish the presence of chaos as a central dynamical behaviour of the system. This model can be adopted to study other phenomenon of interests in celestial mechanics, astronomy, aerospace engineering and astrophysics.

Table 10

Linear Stability of the Libration points for Wolf 630 Triple stellar system.

	α_0=π	λ1,2	λ3,4
L1	(-0.246837, 0)	±3.91969	±2.99058i
L2	(1.13059, 0)	±0.912268	±1.26899i
L3	(-1.24364, 0)	±1.39675	±1.48266i
L4	(0.000529317, 0)	±98.4469i	±100.618i
L5	(0.0359977, 0)	±19.8307	±14.7834i
L6	(-0.166672, 0.861781)	-0.586069 ± 0.953696i	0.586069 ± 0.953696i
L7	(-0.166672, -0.861781)	-0.586069 ± 0.953696i	0.586069 ± 0.953696i

Table 11

Linear Stability of the Libration points for Formalhaut Triple stellar system.

	α_0=π	λ1,2	λ3,4
L1	(-0.907965, 0)	±6.81105	±4.94572i
L2	(0.966378, 0)	±1.89563	±1.74239i
L3	(-1.07008, 0)	±2.97432	±2.38283i
L4	(-0.46969, 0.842992)	-0.578001 ± 0.969233i	0.578001 ± 0.969233i
L5	(-0.46969, -0.842992)	-0.578001 ± 0.969233i	0.578001 ± 0.969233i

Table 12

Linear Stability of the Libration points for Omicron Eradani Triple stellar system.

	α_0=π	λ1,2	λ3,4
L1	(-0.299668, 0)	±4.20594	±3.17907i
L2	(1.11542, 0)	±1.00245	±1.30507i
L3	(-1.25239, 0)	±1.33656	±1.4537i
L4	(0.000760979, 0)	±97.9095i	±100.264i
L5	(0.0299558, 0)	±24.2398	±18.5168i
L6	(-0.204967, -0.861517)	-0.591066 ± 0.961625i	0.591066 ± 0.961625i
L7	(-0.204967, 0.861517)	-0.591066 ± 0.961625i	0.591066 ± 0.961625i

Table 13

Linear Stability of the Libration points for 36 Ophiuchi Triple stellar system.

	α_0=π	λ1,2	λ3,4
L1	(7.18515∗10ˆ-9, 0)	±98.9163i	±100.988i
L2	(0.0818256, 0)	±10.8876	±7.806i
L3	(-0.0818227, 0)	±7.42385	±5.40582i
L4	(-1.19158, 0)	±1.77583	±1.67717i
L5	(1.19157, 0)	±1.77591	±1.67722i
L6	(0.0000118999, -0.860618)	-0.529049 ± 0.893217i	0.529049 ± 0.893217i
L7	(0.0000118999, 0.860618)	-0.529049 ± 0.893217i	0.529049 ± 0.893217i

Declarations

Author contribution statement

Jagadish Singh: Conceived and designed the experiments; Performed the experiments.

Solomon Okpanachi Omale: Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or 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

No data was used for the research described in the article.

Declaration of interests statement

The authors declare no conflict of interest.

Additional information

No additional information is available for this paper.