Effect of Oblateness of the Secondary up to JJ44 on LL44,55 in the Photogravitaional ER3BP

In a synodic-pulsating dimensionless coordinate, with a luminous primary and an oblate secondary, we examine the effects of radiation pressure, oblateness (quadruple and octupolar i.e. 2 4 J and J ) and eccentricity of the orbits of the primaries on the triangular points 4,5 L in the ER3BP. 2 4 J and J have been shown to disturb the motion of an infinitesimal body and 4 J particularly has significant effects on a satellite’s secular perturbation and orbital precessions. The influence of these parameters on the triangular points of Zeta Cygni, 54 Piscium and Procyon A/B are highlighted in this study. Triangular points are stable in the range 0 C μ μ < < and their stability is affected by said parameters.

repulsive force of the radiation pressure. The photogravitational restricted three-body problem models adequately the motion of a particle of a gas-dust cloud which is in the field of two gravitating and radiating stars. The summary action of gravitational and light repulsive forces may be characterized by the mass reduction factor . The motion of particles in the stellar system may be of particular interest. Among the various possible motion of the particle, the equilibrium positions around the libration points of a rotating system of coordinates have practical applications. The existence and stability of equilibrium points were studied by Chernikov (1970), Kunitsyn & Perezhogin (1978) and Singh & Umar (2012a) in the case of one luminous body, while Schuerman (1980), Luk'yanov (1984,1988), Simmons et al. (1985), Kunitsyn &Tureshbaev (1985), Kunitsyn (2000Kunitsyn ( , 2001 and Singh & Umar (2012b) in the case when both bodies are sources of radiation.
The quadruple mass moment 2 of an aspherical body disturbs the motion of a satellite both at the Newtonian and Post-Newtonian levels (Soffel et al. 1988), so also does the octupolar mass moment 4 . 4 has significant effects particularly in the satellite's secular perturbation and orbital precessions. These shifts are quite relevant in a number of practical applications including global gravity field determination (Konopliv et al. 2013 andRenzetti 2013) and fundamental physics in space Iorio 2005Iorio , 2006Iorio , 2007a Singh and Umar 2013c, & Umar and Singh 2014, 2015. Taking account of the oblateness of the earth, Ammar et al (2102) have conducted an analytic theory of the motion of a satellite and solved the equations of the secular variations in a closed form, while Abouelmagd (2012) analyzed the effect of oblateness of the more massive primary up to J4 in the planar CR3BP and proved that the positions and stability of the triangular points are affected by this perturbation. This paper investigates the effects of radiation pressure of the primary and the oblateness of the secondary up to 4 on the triangular points in the ER3BP. It can be applied to the Sun-Earth system, Zeta Cygni, 54 Piscium and Procyon A/B. This paper is organized as follows: -sections 2 presents the equations of motion; section 3 finds the locations of triangular equilibrium points of the system; while section 4 examines the linear stability and section 5 provides the numerical applications of the problem. Finally, the conclusions are drawn in section 6.

2.Equations of Motion
In a synodic-pulsating dimensionless coordinate system, with axes that expand and shrink, considering the primary to be luminous and the secondary an oblate spheroid, with oblateness up to J4, we present the equations of motion of the ER3BP following Singh and Umar (2012b); and Singh and Taura (2013b) as where the force function, and the mean motion The distance of the third body from the primary and secondary are: and 0 < = 2 1 + 2 < ½ is the mass ratio with 1 , 2 as the masses of the primaries positioned at the points ( , 0,0) , i= 1,2; 1 , & 2 are their oblateness up to octupolar mass moment ( 4 ) coefficients Bi = J2i 2 2 (i=1,2) characterize the oblateness of the smaller primary of mean radius R2 and quadruple and octupolar mass moments (Zonal Harmonic Co-efficient) J2 and J4 respectively, while and are respectively the semi-major axis and eccentricity of the orbits.

Positions of Triangular Points
The equilibrium solutions of the problem are obtained by equating all velocities and acceleration components of the dynamical systems to zero. That is, the equilibrium points are the solutions of the equations: The positions of the triangular points are obtained from the first two equations of equation (5) above with ≠ 0 and = 0. From which; when oblateness of the smaller primary is absent i.e. B 1 = B 2 = 0, we have = n 2 , (i=1,2), when the oblateness is considered the value of r2 will change slightly by , say and (9), becomes; Using 4 & 11, we get The co-ordinates ( , ± ) obtained in equation (12) are the triangular libration points and are denoted by L 4,5 . Using equation (12), for various oblateness coefficients 1 & 2 ,we compute numerically the positions of the triangular points as given in tables 1-4 to show the effects of 1 & 2 radiation q, eccentricity e and semi-major axis a. These effects are shown graphically in figures 1-8.

Stability of Triangular Libration Points
To examine the linear stability of an infinitesimal body near the triangular point L 4 (ξ 0 , η 0 ) we displace it to a position ξ= ξ 0 + , η = η 0 + , where α, β are small displacements. Substituting these values in the equations of motion (1) and considering only the linear terms, the variational equations of motion corresponding to the system are given as: - The second order partial derivatives of Ω are represented by the subscripts, while the superscript 0 implies that the partial derivatives are to be evaluated at the libration point L 4 (ξ 0 , η 0 ).
Hence, the characteristics equation corresponding to the system is: - Neglecting second and higher order terms of B 1 , B 2 , 2 and their products, the values of the partial derivatives at the triangular point (12) are obtained as 13 Substituting these values into equation (13) above and neglecting product and higher order terms, we get, Where; Equation (14) is a quadratic equation in terms of , which yields; Its roots are where the discriminant = ( 4 From equation (16)  For the stability of the libration points as given in equation (16) above, and equating the discriminant to zero i.e. Δ = 0 and solving for μ, we obtain the critical mass parameter as: - Where, The value of the critical mass parameter to ten decimal places is: - Since > 0, in the interval 0 < < , this implies that the roots of equation (15) are pure imaginary numbers, hence the triangular libration points are stable in this region. In < < ½, < 0, the real parts of the two roots are (15) are positive, therefore the triangular points are unstable. If = , = 0, the roots in (15) are double roots and hence the triangular points are unstable.
Hence, the triangular points are stable for 0 < < where the critical mass parameter depends on the radiation pressure factor, oblateness and the quadruple and octupolar mass moment of the smaller primary, the semi-major axis and eccentricity of the orbits on the critical mass value.

Numerical Application
We compute numerically the locations of the triangular points of Zeta Cygni, Procyon A/B and 54 Piscium. Zeta Cygni (ζ Cyg) belongs to the northern constellation of Cygnus and is the brightest member of the constellation, with an apparent visual magnitude of 3.26. The primary component is a giant star and the secondary component has a 12 th magnitude companion believed to be a white dwarf (Cygnus Constellation 2016). Procyon (Alpha Canis Minoris) is a binary star, consisting of a white main sequence star, a yellowish star brighter than our sun belonging to spectra type of F5IV called Procyon A and a faint white dwarf companion of spectra type DA. It is the eighth brightest star in the night sky, and has an apparent magnitude of 0.4 and absolute magnitude of 2.68 (Fred 2011). While, 54 Piscium is an orange dwarf star of the sixth magnitude, belonging to the Pisces constellation and class KO dwarf star, with a low mass body and apparent visual magnitude of 5.87 (Jim 2013). We assume an eccentricity 0.  Tables 3 and 4 show the effects of increasing the eccentricity and semi-major axis on the locations of the triangular points of Zeta Cygni. Interestingly,

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IJARM Vol. 13 we find that for e>0.5, the triangular points cease to exist. These are shown graphically in figures 4, 5, and 6. Figure 7 is a surface representation of the effect of eccentricity on L4.
Finally, for an arbitrarily system with µ = 0.035, the effect of radiation pressure on the size of the region of stability is investigated, highlighting the effect of eccentricity as presented in table 5 and figure 8.

Conclusions
The positions and linear stability of the triangular libration points have been obtained and are affected by the oblateness (up to J4), eccentricity and semi-major axis of the orbits. These effects are shown numerically and graphically. Figures 1, 2 and 3 (table 2) show the effects of quadruple and octupolar mass moments (B1 and B2) on the triangular points of Zeta Cygni, Procyon A/B and 54 Piscium. It is seen clearly that as eccentricity (table 3)  Our results in the circular case confirms those of Sharma (1987) and Ishwar and Elipe (2001) with J4=0 in ours. They also agree with those of Singh and Ishwar (1999) and AbdulRaheem and Singh