The locations of triangular equilibrium points in elliptic restricted three-body problem under the oblateness and radiation Effects

The Restricted Three-Body Problem (R3BP) considers motion of a third infinitesimal object under the gravitational influences of the primaries (bigger and smaller massive bodies) whose orbits are around the center of mass. If the orbits are elliptical, this belongs to Elliptic R3BP (ER3BP). In planar case it possesses five equilibrium points consisting of three collinear (L1, L2, and L3) and two triangular (L4 and L5). To mimic a better astrophysical R3BP, such as motion of a satellite in star-planet system, the classical problem can be generalized by considering the effects of radiation pressure and oblate spheroid shape on the primaries. We study analytically the locations and the stability of L4 and L5 equilibrium points in the frame of ER3BP with incorporating the effects of radiation for bigger primary and oblateness for smaller primary. Our study suggests that the oblateness factor (A2) and the radiation factor (q1) shift the positions of L4 and L5 points compared with the classical ones. We also find that there is a stability limit for motion of the third body around these points.


Introduction
The Circular Restricted Three-Body Problem (CR3BP) considers motion of a third infinitesimal object under the gravitational influences of the primaries (bigger and smaller massive bodies) whose orbits are circular around the centre of mass. In planar case it possesses five equilibrium points consisting of three collinear (L1, L2, and L3) and two triangular points (L4 and L5). The classical problem of CR3BP was modified by including additional forces. The effect of radiation pressure in CR3BP was first studied by Radzievskii in 1950 for the solar problem [3]. In both papers, however, Radzievskii did not consider the linear stability. The linear stability of equilibrium point was discussed by Chernikov in 1970 [3] who considered the relativistic Poynting Robertson effect which may be treated as perturbation, and Simmons in 1985 [4] with the effect of radiation pressure for all range of value.
This treatment may also be applicable to Elliptical Restricted Three-Body Problem (ER3BP). Singh and Umar in 2012 [1] considered the motion under the influences of an oblate spheroid in bigger primary and a radiation pressure in smaller primary. Kumar  the bigger primary has radiation pressure. Here we study the locations of L4 and L5 in ER3BP with bigger primary contain the radiation and smaller primary is an oblate spheroid.

Equations of motion
We assume that m1 and m2 are masses of the primaries, f is the true anomaly of system, r is the distance between the primaries, e is the eccentricity, and a is the semi-mayor axis of the primaries. In the case of three-body problem, it is more convenience to introduce the third body in the rotational (,, ζ) rather than in the inertial (x, y, z) coordinate system. The unit of time is chosen to make the gravitational constant G = 1. Introducing a transformation to a dimensionless coordinate system equations of motion of the third object in planar case are The potential function V is defined by where n is mean motion of elliptical orbit, with (1,1) and (2,2) are the primaries coordinate.

Radiation pressure
The radiation pressure force (Fp) changes with distance by the same law as the gravitational attraction force of mi (Fgi) and acts opposite to it. It is possible that this force will lead to a reduction of the effective mass of the massive particle. We use qi to represent radiation pressure coefficients of the primaries. We can write the total force as

Oblate spheroid
We use Ai to represent oblateness coefficients of the primaries respectively, such that 0 < Ai << 1 and where AEi and APi are the dimensional equatorial and polar radii of mi. Ri is radius of mi which assume the primary as spherical object.
To mimic the extrasolar planetary system with single star and planet, we assume that only the bigger primary is the source of radiation (q1 ≠ 1, q2 = 1) and only the smaller primary is oblate spheroid (A1 = 0, A2 ≠ 0).

Location of triangular points
Conditions of triangular equilibrium points are given as follows Following [5] from (1) we obtain the location of equilibrium points ( ̅ 0 ,  ̅ 0 ) When the primary is oblate ri = 1 + i with i << 1. The positif value of  ̅ 0 is the position of L4 and the negative value of  ̅ 0 is the position of L5. Figure 1 shows the displacement of triangular points under the effect of radiation and oblateness. The gray dot represents the positition of m2. The yellow dot represents the positition of m1. The positions of infinitesimal object are represented by the blue dots (for classical problem) and the red dots (for the presence of radiation and oblateness effects) with L4 is located above axis and L5 is located below axis. The triangular points shift drawn toward the bigger primary with increment of A2 and reduction of q1.

Linear stability
Define the displacement of triangular points by: The stability of the system depends on the form of  . If all roots are pure imaginary the system is stable.

Conclusion
The position of triangular equilibrium points in ER3BP under the effect of radiation and oblateness primaries have been studied. We conclude that the positions are shift when it recognizes the radiation factor (q1) and oblateness coefficient (A2). The stability regions are changed due to these effects.