Computation of the Locations of the Libration Points in the Relativistic Restricted Three-Body Problem

Problem statement: In this study, the effects of Relativistic Restric ted three-Body Problem (in brief RRTBP) on the equilibrium points of both tria ngular and collinear is considered. The approximate locations of the collinear and triangular points ar e determined. Series expansions are used to develop in μ and 1/c as small parameters. To check the validity of our s lution, when ignoring 1/c 2 terms we get directly the classical results of the restricted three-body problem. Conclusion/Recommendations: A MATHEMATICA program is constructed to give a numeri cal application of the relativistic perturbations i the locations of the equilibrium points of the thre e body problem.


INTRODUCTION
The three-body problem considers three mutually interacting masses m 1 , m 2 and m 3 . In the restricted three-body problem, m 3 is taken to be small enough so that it does not influence the motion of m 1 and m 2 , which are assumed to be in circular orbits about their center of mass. The orbits of the three masses are further assumed to all lie in a common plane. If m 1 and m 2 are in elliptical instead of circular orbits, the problem is variously known as the "elliptic restricted problem" or "pseudo restricted problem" Szebehely (1967).
The history of the restricted three-body problem began with Euler and Lagrange continues with Jacobi (1836), Hill (1878), Poincare (1957) and Birkhoff (1915). Euler and Courvoisier (1980) first introduced a synodic (rotating) coordinate system, the use of which led to an integral of the equations of motion, known today as the Jacobian integral. Euler himself did not discover the Jacobi integral which was first given by Jacobi (1836) who, as Wintner remarks, "rediscovered" the synodic system. The actual situation is somewhat complex since Jacobi published his integral in a sidereal (fixed) system in which its significance is definitely less than in the synodic system.
Many authors hope to investigate the relativistic effects in this problem. But unfortunately, the Einstein field equations are nonlinear and therefore cannot in general be solved exactly. By imposing the symmetry requirements of time independence and spatial isotropy we are able to find one useful exact solution, the Schwarzschild metric, but we cannot actually make use of the full content of this solution, because in fact the solar system is not static and isotropic.
Indeed, the Newtonian effects of the planet's gravitational fields are an order of magnitude greater than the first corrections due to general relativity and completely swamp the higher corrections that are in principle provided by the exact Schwarzschild solution. It is worth noting to highlight some important articles in this field.
Computed the post-Newtonian deviations of the triangular Lagrangian points from their classical positions in a fixed frame of reference for the first time, but without explicitly stating the equations of motion. Treated the relativistic (RTBP) in rotating coordinates. He derived the Lagrangian of the system and the deviations of the triangular points as well. Weinberg (1972) calculated the components of the metric tensor by using the post-Newtonian approximation in order to obtain the (RTBP) problem equations of motion. Soffel (1989) obtained the angular frequency ω of the rotating frame for the relativistic two-body problem. Brumberg (1972;1991) studied the problem in more details and collected most of the important results on relativistic celestial mechanics.
In this study, the approximate positions of the collinear and triangular points are determined using the equations of motion of the relativistic restricted threebody problem. The formulas are expanded in terms of µ and 2 1 c as small parameters

Equations of motion:
The equations of motion of the infinitesimal mass in the (RRTBP) in a synodic frame of reference (ξ, η), in which the primary coordinates on the ξ -axis (-µ, 0), (1-µ, 0) are kept fixed and the origin at the center of mass, is given by Brumberg (1991), From Bhatnagar and Hallan (1997) Eq. 1: where, U is the potential-like function of the (RRTBP), which can be written as composed of two components, namely the potential of the classical (RTBP) potential U c and the relativistic correction U r Eq. 2: where, U c and U r are given by Eq. 3:    (1), it is apparent that an equilibrium solution exists relative to the rotating frame when the partial derivatives of the pseudopotential functions (U ξ , U η , U ζ ) are all zero, i.e., U = const. These points correspond to the positions in the rotating frame at which the gravitational and the centrifugal forces associated with the rotation of the synodic frame all cancel, with the result that a particle located at one of these points appears stationary in the synodic frame. There are five equilibrium points in the circular (RTBP), also known as Lagrange points or libration points. Three of them (the collinear points) lie along the ξ-axis: one interior point between the two primaries and one point on the far side of each primary with respect to the barycenter. The other two libration points (the triangular points) are each located at the apex of an equilateral triangle formed with the primaries. One of the most commonly used nomenclatures (and the one that we will use here) defines the interior points as L 1 , the point exterior to the small primary (the planet) as L 2 and the point L 3 as exterior to the large primary (the Sun), see Fig. 1. The triangular points are designated L 4 and L 5 , with L 4 moving ahead of the ξ-axis and L 5 trailing ξ-axis, along the orbit of the small primary as the synodic frame rotates uniformly relative to the inertial frame (as shown in Fig. 1) Remark: All five libration points lie in the ξ-η plane, i.e., in the plane of motion The libration points are obtained from equations of motion after setting 0 ξ = η = ζ = ɺɺ ɺɺ ɺɺ . These points represent particular solutions of equations of motion Eq. 5: The explicit formulas are (Remembering that U = U c + U r ) Eq. 6: And Eq. 7: Location of collinear libration points: Location of L 1 .
The collinear points must, by definition, have ξ = η = 0, since 2 1 / c 1 << and the solution of the classical (RTBP) is (Fig. 2  Using (8) Eq. 6, can be written explicitly in terms of r 1 and r 2 as Eq. 9: Then it may be reasonable in our case to assume that position of the equilibrium points L 1 are the same as given by classical (RTBP) but perturbed due to the inclusion of the relativistic correction by quantities ( ( ) 2 1 O 1 / c ε ≡ Eq. 10: r 1 = a 1 +ε 1 , r 2 = b 1 -ε 1 , a 1 = 1-b 1 where, a 1 and b 1 are unperturbed positions of r 1 and r 2 respectively and b 1 is given after some successive approximation by: Substituting from Eq. 10 into Eq. 9 and retaining the terms up to the first order in the small parameter ε 1 , we get: Setting, 1 1 1 3 3 1 1 1 Eq. 11 -13 can be solved for ε 1 to yield:   Location of L 2 : The L 2 point locates outside the small massive primary of mass µ. We now drive an approximate location for this point. At the L 2 point we have (see Fig. 3) Eq. 14: 1 1 2 1 2 2 r r 1, r , r r r 1, 1 The procedure is similar to Eq. 10 with little modification according to the location of ξ o , L2 as: With a 2 and b 2 are the unperturbed positions of r 1 and r 2 respectively, where b 2 is by: As is done in the calculation of the location of ξ o , L1 we can similarly calculate the location of ξ o , L2 as Eq. 15: The procedure is similar to Eq. 10 with little modification according to the location of ξ o , L2 as: where a 3 and b 3 are the unperturbed values of r 1 and r 2 respectively and b 3 is given after some successive approximation by: Substituting in the second set of (7) and (8) and solving for ξ and η up to the first order in the involved small quantities ε 1 and ε 2 , we get: Substituting the values of r 1 , r 2 , ξ and η into equations: Evaluating the included partial derivatives and retaining the terms up to the first order in the small parameters ε 1 , ε 2 and also the first order terms in the relativistic correction, we obtained Eq. 20: (1 2 ) 8c (1 ) 0,(1 ) Which represent two simultaneous equations in ε 1 and ε 2 . Their solutions are being Eq. 21: Substituting the values of ε 1 and ε 2 into Eq. 19, yields the coordinates of the triangular points Eq. 22: (1 2 ) 5 (1 ) 2 4c 3 1 (1 (6 6 5)) 2 12c The classical limit: To check our solution, when ignoring 2 1 / c terms we get directly the classical results of the restricted three-body problem as:   In Fig. 5 the difference L iR -L 1c seems to be constant beyond µ ≥ 0.05. In Fig. 6 the difference L 2R -L 2c increases with increasing the mass ratioµ. In Fig. 7 the difference L 3R -L 3c increases with increasing the mass ratio µ within the domain 0 ≤ µ ≤ 0.35 and decreases with increasing µ within the domain 0.35 ≤ µ ≤ 0.5. Fig.  8 represents an assembly plot of the all collinear equilibrium points.
We may see that, in most of these cases, the positions of L 3 and L 2 are much more affected by the influence of the relativistic terms than that of the equilibrium point, L 1

ACKNOWLEDGMENT
The researchers are deeply indebted to the Professor Dr. M. K. Ahmed, the professor of space dynamics at Cairo university, faculty of Science, department of Astronomy for his valuable discussions and critical comments and ideas that help us to finalize this study. This research work was supported by a grant No (164/1432) from the deanship of the scientific research at Taibah university, Al-Madinah Al-Munawwarah, K.S.A.