Existence and Stability of Triangular Points in the Relativistic R3bp When the Primaries are Triaxial Rigid Bodies and Sources of Radiation

Some of the important contributions are by Radzievskii [1,2] Simmons et al. [3] Kunitsyn and Tureshbaev [4] Singh and Ishwar [5] Singh [6]. Some of the significant studies, by considering the triaxiality of one or both primaries, are Khanna and Bhatnagar [7] and Singh [8]. Sharma et al. [9,10] have studied the stationary solutions of the planar restricted three-body problem when one or both primaries are sources of radiation and triaxial rigid bodies with one of the axes as the axis of symmetry and the equatorial plane coinciding with the plane of motion.


Introduction
The restricted three-body problem possesses five stationary solutions called Lagrangian points, three of which called collinear equilibria lie on the line joining the primaries and the other two called equilateral equilibria make equilateral triangles with primaries.
In general, the collinear equilibria are unstable while equilateral ones are stable, in the Lyapunov sense, only in a certain region for the mass parameter. Various authors have made studies on Lagrangian points in the restricted three-body problem by considering the more massive primary or both primaries as source of radiation.
Some of the important contributions are by Radzievskii [1,2] Simmons et al. [3] Kunitsyn and Tureshbaev [4] Singh and Ishwar [5] Singh [6]. Some of the significant studies, by considering the triaxiality of one or both primaries, are Khanna and Bhatnagar [7] and Singh [8]. Sharma et al. [9,10] have studied the stationary solutions of the planar restricted three-body problem when one or both primaries are sources of radiation and triaxial rigid bodies with one of the axes as the axis of symmetry and the equatorial plane coinciding with the plane of motion.
The theory of the general relativity is currently the most successful gravitational theory describing the nature of space and time and well confirmed by observations [11]. Brumberg [12,13] studied the problem in more details and collected most of the important results on relativistic celestial mechanics. He did not obtain only the equations of motion for the general problem of the three bodies but also deduced the equation of the motion for the restricted problem of three bodies. Bhatnagar and Hallan [14] investigated the existence and linear stability of the triangular point 4 5 , L in the relativistic R3BP. They concluded that 4 5 , L are always unstable in the whole range Douskos and Perdios [15] investigated the stability of the triangular points in the relativistic R3BP and contrary to the results of Bhatnagar and Hallan [14], they obtained a region of linear stability in the parameter space
Katour et al. [16] obtained new locations of the triangular points in the framework of relativistic R3BP with oblateness and photogravitional corrections to triangular points.
Singh and Bello [17,18] studied the motion of a test particle in the vicinity of the triangular points 4 5 , L by considering one or both primaries as the sources of radiation in the framework of the relativistic restricted three-body problem (R3BP).
In the present work, we study the existence of the triangular points and their linear stability by considering both primaries as triaxial rigid bodies and sources of radiation. This paper is organized as follows: In Sect. 2, the equations governing the motion are presented; Sect. 3 describes the positions of triangular points, while their linear stability is analyzed in Sect.4; The discussion of the results is given in Sect. 5. In Sect. 6, we conclude our work and highlight the differences and similarities between our work and previous works.

Equations of Motion
In our recent papers Singh and Bello [17,18] we have studied the effect of radiation pressure of the primaries in the relativistic R3BP and found that the positions of triangular points and their stability are affected by both relativistic and radiation factors. In this paper we extend this work by considering both primaries as triaxial rigid bodies as well as sources of radiation.
The pertinent equations of motion of an infinitesimal mass in the relativistic R3BP in a barycentric synodic coordinate system ( ) , ξ η and dimensionless variables can be written as Singh and Bello [17,18] is the ratio of the mass of the smaller primary to the total mass of the primaries, 1 ρ and 2 ρ are distances of the infinitesimal mass from the bigger and smaller primary, respectively; n is the mean motion of the primaries; cis the velocity of light.
characterize the triaxiality of the bigger and smaller primary with a, b, c as lengths of the semi-axes of the bigger primary and as those of the smaller primary. The radiation Radzievskii [1] where gi F and pi F are respectively the gravitational and radiation pressure.
Here as Katour et al. [16] we do not include the parameters

Stability of L 4,5
Since the nature of the linear stability about the point L 5 will be similar to that about L 4 , it will be sufficient to consider here the stability only near L 4 .

Discussion
In this section, we discuss the triangular libration points in the relativistic restricted three-body problem, under the assumption that the primaries are luminous and triaxial. The positions of the triangular points in equation (6) are obtained. It can be seen that they are affected by the relativistic, radiation and triaxiality factors. It is important to note that these triangular libration points cease to be classical one i.e., they no longer form equilateral triangles with the primaries. Rather they form scalene triangles with the primaries. Equation (12) gives the critical value of the mass parameter µ c of the system which depends upon relativistic factor, triaxiality parameters , ( 1,2) σ σ′ ε ε′ When the primaries are non triaxial, the results of the present study tally with those of Singh and Bello [18].
It is noticeable from (13) that that radiation and triaxiality all have destabilizing effects, and therefore the size of the range of stability decreases with increase of the values of these parameters. Evidently, it can also seen that the relativistic factor reduces the size of stability region. When the primaries are non-luminous and non-triaxial, the stability results obtained in this study are in accordance with those of Douskos and Perdios [15] and disagree with Bhatnagar and Hallan [14]. In the absence of relativistic factor, the results obtained in this study are in agreement with those of Sharma [14] and those of Singh [8] when the perturbations are absent. When the primaries are oblate spheroids σ σ σ σ ′ ′ = = i e , the results of equation (6) in this study differ from those of katour et al. [16] when the radiation terms are absent in the realistic part of the potential W.

Conclusion
By considering the primaries as triaxial rigid bodies and sources of radiation in the relativistic CR3BP, we have determined the positions of the triangular points and their linear stability. It is found that their positions and stability region are affected by relativistic, triaxiality and radiation factors. It is further observed that the relativistic, triaxiality and radiation factors have destabilizing tendencies resulting in a decrease in the size of the region of stability. We have noticed that the expressions for A, D, A 2 , C 2 in Bhatnagar and Hallan [14] differ from the present study when the radiation pressure factors are absent and the primaries are spherical ( . . 0, 1, 2). δ σ σ′ Consequently, the characteristic equation is also different. This led them Bhatnagar and Hallan [14] to infer that the triangular points are unstable, contrary to Douskos and Perdios and our results. Our results are also in disagreement with those of Katour et al. [16]. One major distinction is that the expression of the mean motion which they used in their study differ from our own. It seems that there is an error in their expression.