Effect on L 4,5 in the ER3BP when Both Primaries are Radiating with Oblateness up to Zonal Harmonic J 4

: This study examines the triangular points in the elliptic restricted three-body problem when both primaries are sources of radiation as well as oblate spheroids with oblateness up to zonal harmonic J4. The positions of triangular points and their critical mass ratio are seen to be affected by the eccentricity, semi major axis, radiation and oblateness of both primaries up to zonal harmonic J4. We highlight the effects of the said parameters on the locations of the triangular points of 61 CYGNI and STRUVE 2398. The triangular points of these systems are found to be unstable.


Introduction
If three particles are free to move in space under their mutual gravitational influence and initially move in any given manner, then this is called three-body problem (3BP). If one of these particles is so much smaller (called infinitesimal mass) than the other two particles (called primaries) and has a negligible effect on their motion, then the 3BP reduces to the restricted three-body problem (R3BP). The R3BP is called circular or elliptic restricted three-body problem (CR3BP or ER3BP), if the two particles with dominant masses move around their common centre of mass along circular or elliptic orbits.
A number of communications have considered the primaries to be either point masses or strictly spherical in shape. Generally however, celestial and stellar bodies are axisymmetric (oblate, triaxial or prolate spheroids). Certain planets and their satellites (Earth, Jupiter, Saturn Charon and our Moon) and stars (Achernar, Alfa Arae, Regulus, VFTS 102, Vega and Altair) are sufficiently oblate/triaxial for the departure from sphericity to be very significant in the R3BP.
In the study of stability of equilibrium points, [13] studied the stability of the equilibrium points of the R3BP. He established in the linear sense that the triangular points 4 5 L and L are stable for 0 0.03852... C µ µ < ≤ = Also, [14] established the complete solution of the R3B and discussed the existence and linear stability of the equilibrium points for all the value of radiation pressure of both luminous bodies. They show that, the inner Lagrange point, 1 L , can be stable, but only when both large masses are luminous. Further, [15] studied the existence of libration points for the generalized photogravitational R3BP by considering the infinitesimal mass as an oblate spheroid and both finite masses as a source of radiation. They found that, there was a possibility of nine libration points, in which three are collinear, four are coplanar and two are triangular.
Taking one or both primaries as sources of radiation or oblate spheroids or both, the effect of oblateness and radiation pressure of the primaries on the existence and stability of equilibrium points in the CR3BP were analysed by [2], [16][17][18]. [2] Studied the effect of oblateness and radiation pressure forces of the primaries on the locations and the linear stability of the triangular points in the R3BP. They found that, these points are stable for 0 C µ µ < < and unstable for 1 2 C µ µ < < , and C µ depends on the radiating and oblateness coefficients. The influence of the eccentricity of the orbits of the primary bodies with or without radiation pressures on the existence of the equilibrium points and their stability was touched upon to an extent by [16], [19] and [20].
A zonal harmonic is a spherical harmonic which reduces to a Legendre polynomial [21]. These harmonics are termed "zonal" since the curves on a unit sphere (with centre at the origin) on which vanish are parallels of latitude which divide the surface into zones. Several studies [22][23][24] have included the second zonal harmonics in their investigation of the R3BP. Singh and Taura [22] examined the combined effect of radiation and oblateness up to 4 of both primaries, together with additional gravitational potential from the circular cluster of material points on the motion of an infinitesimal body under the frame-work of circular restricted three-body problem (CR3BP), [23] established that the locations of the triangular points and their linear stability are affected by the oblateness up to 4 of the bigger primary in the planar CR3BP. While [24] investigated the Influence of the Zonal harmonics of the primary on L4,5 in the photographical ER3BP.
Also, [10] investigated the stability of triangular points in the elliptic R3BP under radiating and oblate primaries of the binary systems Achird, Luyten 726-8, Kruger 60, Alpha Centauri AB and Xi Bootis, and binary pulsars respectively, moving in elliptic orbits around their common centre of mass. They found that, the triangular points are affected by the eccentricity oblateness, radiation semi-major axis; they however remain stable. Here, the work of [10] will be extended by including the oblateness up to zonal harmonics J 4 . Thus the aim of the paper is to study the motion of the infinitesimal body in the ER3BP when both primaries are sources of radiation as well as oblate spheroids up to zonal harmonic J 4 of the primaries, using the binary systems 61 CYGNI and STRUVE moving in elliptic orbits around their common centre of mass.
This paper is organised in seven sections; section 1 is the introduction; section 2 deals with the equation of motion; section 3 focuses on the location of triangular points. The linear stability of these points is examined in section 4, the numerical application is also described in section 5, while the discussion and conclusion are represented in section 6 and section 7, respectively.

Equations of Motion
The equations of motion of an infinitesimal mass, in the ER3BP with oblate as well as luminous primaries, can be written in the dimensionless-pulsating coordinate system ( , , ) following [10] and [22] as; The mean motion, n, is given by

ILCPA Volume 83
Here, m 1 , m 2 are the masses of the bigger and smaller primaries positioned at the points ( ,0,0),i=1,2; , q 1 , q 2 are their radiation factors; , are their distances from the infinitesimal mass; respectively; a and e are the semi-major axis and eccentricity of the orbits respectively; characterize the zonal harmonic oblateness of the bigger and smaller primaries whose mean radii are R 1 and R 2 respectively.

Location of Triangular Points
The equilibrium points are the solutions of the equations Ω = Ω = Ω = 0, which yield The last equation yields = 0. This implies the existence of equilibrium points. The triangular points are the solutions of the first two equations of system (5) with ≠ 0 . From which we obtain; In the absence of oblateness of the primaries, system (6) provides When oblateness is considered, the value of 1 2 will change slightly by 1 and 2 ( ), respectively so that; Considering only linear terms in 1 , 1 , 2 , 2 and 2 and neglecting their products, (3) gives; In the case of spherical luminous primaries, (7) and (8) give Using (8) and (9) in (6), we get International Letters of Chemistry, Physics and Astronomy Vol. 83 3 Substituting for 1 2 from (10)in (9) we obtain; Using (4) and (11), we get; The points ( , ± ), obtained by (12) in the − plane are denoted by L 4,5 ( , ± ) and are known as the triangular equilibrium points.

Linear Stability of Triangular Points
The notion of stability can be applied to other types of problems. It is probably the important aspect in sciences as it refers to what we call "reality". Everything should be stable to be observable. For example, in quantum mechanics, energy levels are those that are stable since unstable levels cannot be observed.
Mathematically, if a dynamical system is in a state of equilibrium, it remains in that state for all time. A real system is subjected to perturbations or disturbances. [13] stated that, the motion which remains in the small neighbourhood of the equilibrium point after it has been disturbed is termed "stable". The motion of a particle in the ξη − plane is investigated by giving the triangular points small displacement (θ, ω). Then we write ξ = ξ 0 + θ And η = η 0 + ω In the variational form, we have the equations of motion as Then, their characteristic equation is Where the superscript 0 indicates that the partial derivatives are elevated at the triangular point ( 0 , 0 ) In the case of triangular points, we have Substituting these values in the characteristic equation (13) and considering only the linear terms in (14) is a quadratic equation in 2 , which yields We can conclude from the nature of the solutions = , = that these will be bounded and periodic only if is pure imaginary. Therefore, for stable motion, we choose , 1, 2 , ℎ ℎ 2 < 0 i.e In the case when (16) is not satisfied, the characteristic roots will be either real or complex conjugate. In the case of complex roots, the positive real part leads to instability of the investigated triangular points. From (15), we now have; ∆ = 3�9 + 4 + 2 1 + 2 2 + 12 1 + 12 1 − The necessary conditions for the stability of the triangular points are given by (16) and (18). The solution of the quadratic equation ∆= 0 i.e, when the discriminant vanishes for gives the critical value of the mass parameter as; Equation (19) represents the effect of radiation pressures, oblateness up to J4 of the primaries, the semi-major axis and the eccentricity of the orbits on the critical mass value.

Numerical Applications
The triangular points given by (12) of the problem are obtained numerically for the binary systems 61 Cygni and STRUVE 2398. They all have oblate and radiating primaries. The numerical data about the system is contained in table 1.
Using table 1, we compute numerically using MATHEMATICA software, the locations of the triangular points for the binary systems. Table 2 and 3 show the effect of oblateness, while table 4 and 5 show the effect of radiation pressure of the primaries. From the table it is shown that increasing the oblateness coefficient while keeping radiation pressure constant causes the values of both and to decrease. And increasing the radiation pressure while keeping the oblateness constant causes the value of both and to increase.
We used Equation (19) to calculate the numerical values of the critical mass parameter for different values of oblateness. Table 6 shows the effect of the size of the region of stability while varying oblateness. The condition of stability 0< µ < µ c is not satisfied for this system. This is because µ c < 0; which confirms from this table the instability of the triangular points.

Discussion
The positions of triangular points L4, 5 of the problem are seen to be affected by the introduction of the oblateness, radiation pressure, eccentricity and semi-major axis of both primaries up to zonal harmonics J4. In the case J4=0, this coincides with those of [10] (i.e 2 = 2 = 0), the coordinates reduce to = Taking semi-major axis as unity, disregarding eccentricity and oblateness up to 4 ( . = 1, 1 = 2 = 1 = 2 = = 1 = 2 = 0) the coordinates reduce to = which fully coincide with classical case of [13] The effect of oblateness and radiation pressure on the location of triangular points for the binary system 61 CYGNI and STRUVE 2389 are shown in table 2, 3, 4 and 5; and graphically, in figure 1 and 2. It is found out that there is a shift on both coordinates towards the and axis respectively.
The critical value of the mass parameter µ c of the system (table 6) is used to determine the size of the region of stability and also in analysing the behaviours of the parameters involved therein. The triangular points of the binary systems are found to be unstable, which confirm the result of [10] with J4=0 (i.e A2=B2=0 ), reduce to; = ( 1 + 2 ) − � 14 9√69 � 2 . In the absence of radiation pressure, eccentricity, and potential from the circular cluster of material µ c confirm the result of [22]. However, when the primaries are spherical and they move in circular orbits, it corresponds to the classical case of [13] reduce to; = International Letters of Chemistry, Physics and Astronomy Vol. 83

Conclusion
The positions of triangular points have been determined under the assumption that both primary bodies move in elliptic orbits under their common centre of mass, with both primaries radiating and oblate. Their linear stability has also been examined. It is found that their positions and stability are significantly affected by the eccentricity of the orbit, semi-major axis, oblateness and radiation factor of the primary, all of which have destabilizing tendencies resulting in a decrease in the size of region of stability.