Discovery of an Equilibrium Circle in the Circular Restricted Three Body Problem

Problem statement: A traial to find equilibrium points in out of plan e of the restricted three body problem. Approach: Solution of the equations of motion at equilibrium points. Linearizing the equations of motion and computing the eigen values to investigate the stability. Results: New triangular equilibrium points in plane perpendicula r to the plane of motion and passing through Lagrange triangular equilibrium points are obtained . A circle of equilibrium points, namely Fawzy equilibrium circle is discovered. Infinite number o f equilibrium points located on the circumference o f this circle are computed. The obtained solutions ar e checked via obtaining some special cases. The stability of Fawzy equilibrium circle is studied. T he oscillatory stable solutions as η-dependent and ξdependent stabilities are derived. Conclusion: The equilibrium points in out of plane of motion o f the restricted three body problem is investigated. We o btained the following very new results:First we obtained Fawzy ξξ-triangular equilibrium points in the plane η = 0. Second we obtained the so called Fawzy equilibrium circle. We checked our solutions via obtaining some special cases. We studied the stability of Fawzy equilibrium circle. We derived t he oscillatory stable solutions as η-dependent and ξdependent stabilities. The stability of Lagrange as well as Fawzy triangular equilibrium points followed directly because they are subsets of the F awzy equilibrium circle.


INTRODUCTION
In any assumed isolated two-body massive orbiting system (such as the Sun and the Earth) there are five equilibrium points, L i , i = 1, 2, 3, 4, 5 these points usually called Lagrangian or libration points. At these points the gravitational pulls are in balance. Any infinitesimal body at any point of the Lagrangian points would be held there without getting pulled closer to either of massive bodies. The points L 1 , L 2 , L 3 are colinear with the line joining the two massive bodies, while the triangular points L 4 , L 5 are found 60° ahead of and behind the less massive body, along its orbit. These two triangular points L 4,5 are forming equilateral triangles with the two massive bodies. The Restricted Three Body Problem (RTBP in brief) is now defined as a system consisting of two massive bodies, called the primaries, revolving in a circular orbits around their center of mass and a third body of infinitesimally small mass which moves in the primaries' orbital plane. The three collinear equilibria at L 1 , L 2 , L 3 are unstable, while the two triangular solutions L 4 , L 5 are stable, in the linear analysis, for only a certain values of the mass ratio in the interval (0, µ) where µ = 0.038521 is the Routhian which modifies when including different perturbations (cf. Szebehely, 1967;Subba and Sharma, 1994;1997;1988;1986;Narayan and Ramesh, 2011). The global stability of these points have been studied by several authors: (Leontovich, 1962;Deprit and Deprit-Bartholome, 1967;Markeev, 1969;Szebehely 1979;Narayan and Ramesh, 2008;Singh, 2011;Kumar and Ishwar, 2011;Douskos, 2011;Shankaran et al., 2011). Their final conclusions are that in the planar case the triangular points L 4,5 are always stable within some domain of mass ratio which is modified when including such kind of different perturbations.
In this study the author is searching for new equilirium points in the peripindicular planes to the plane of motion of the primaries. We follow the same very well known way in RTBP but with the required modifications. We first compute their positions of such equilibrium points and then study their stability.
where n=1 is the mean motion of the rotating system and U is the potential-like (pseudopotential) function of the RTBP is given by Eq.  (1), it is apparent an equilibrium solution exists relative to the rotating frame when the partial derivatives of the pseudopotential function are all zero, i.e., these points correspond to the positions in the rotating frame at which the gravitational forces and the centrifugal force 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. The libration points are obtained from equations of motion (1) after setting These points represent particular solutions of equations of motion Eq. 3: Equilibrium Points in the plane η η η η=0: The author expects a new set of equilibrium points in the ξξ plane. Now we can rename the triangular Lagrange equilibrium points L 4 , L 5 to become 4 L ξη , 5 L ξη this because Lagrange found them in ξη. In this study the author found a new set of equilibrium points in the ξξ plane. This new set will be called in the future works Fawzy ξξ-triangular equilibrium points 4 L ξζ , 5 L ξζ . To calculate the locations of these new equilibrium points, rewrite the potential-like function U in a different form. By definition, this new set has η = 0 and Eq. 4: Thus we can write Eq. 5: The system (3) can be written as Eq. 6: This system is verified when: from which one can easily obtain r 1 = r 2 = 1. Substituting these results back to the Eq. 4 yields: Solving these two equations simultaneously yields Eq. 7: As expected from the symmetry of the problem, we obtained the locations of the new equilibrium points. It looks similar to Lagrange triangular equilibrium points ( ) but except the plane in which they lie: i.e., in the plane η = 0 Eq. 8: Equilibrium points in the plane parallel to ξ ξ ξ ξ = 0 plane: We consider a test particle at a plane parallel to the plane ξ = 0 (except at planes ξ = -µ and ξ = 1-µ because we did not find any more equilibrium points there). According to these constraints we have Eq. 9: Thus we can write: The system (3) can be written as Eq. 10: This system is verified when from which one can easily obtain r 1 = r 2 = 1. Substituting these results back to the Eq. 9 yields Eq. 12: Solving these two equations simultaneously for ξ yields 1 = 2 ξ − µ , substituting back into Eq. 8 we get . Therefore we have infinite number of solutions (equilibrium points), their coordinates are given by Eq. 13: . Also if we set η = 0 Fawzy triangular equilibrium points follow directly as 4 L ξζ , 5 L ξζ on respective at ( )

Stability of fawzy equilibrium circle:
We are going to study the stability of Fawzy equilibrium circle. At each step we will examine our solution using the well known results in RTBP. To examine the stability, an infinitesimal body would be displaced a little from the Fawzy equilibrium circle. If the resultant motion of the particle is a rapid departure from the vicinity of this circle we can call such a circle an unstable circle, if however the particle merely oscillates about the equilibrium points of its circumference, it is said to be a stable positions. If we have a mixed situation then we will determine the stable points and the unstable ones.
To examine the stability of the orbits in the vicinity of Fawzy equilibrium circle the equations of motion are linearized around the circumference of the equilibrium circle with coordinates Eq. 14: . . .
Let us denote to the state vector by ( ) To solve this system we compute first the eigenvalues of the coefficient matrix A as follows Eq. 18: Therefore we get the characteristic equation, with degree n in λ with possible complex roots. After getting the eigenvalues λ's substitute them into (A-λI)X= 0 and solve for the components X. The resulting equations are coupled (different components are appeared in the same equation). Using the similarity transformation Y = BX we can transform this coupled system to uncoupled one as: where, BAB -1 is a diagonal matrix with the eigenvalues of A on the diagonal, the matrix B is constructed from the n eigenvalues the matrix A. Now The coupled system (17) is transformed to uncoupled Eq. 19: The solutions of the transformed system are easily found as Eq. 20: where c i are n constants of integration. We must now transform back to the original variables in components form as Eq. 21 and 22: which are linear differential equations with constant coefficients so long as only first order terms are retained. Let a solution of the problem be Eq. 23: where, A, B, C and λ are constants. To find the expressions for A, B and C Eq. 15 can be rewritten, using the suggested solution, as Eq. 24: which can be written in matrix notation as Eq. 25: This system has nontrivial solution if Eq. 26: where σ is the root of the characteristic determinant and U αβ are the second partial derivatives with respect to the variables mentioned in the subscripts and evaluated at the equilibrium points. Evaluating the partial derivatives included in Eq. 27. Now computing the coefficients of ' s, λ then the characteristic Eq. 27 can be written in the form Eq. 28: The quantity inside the inner square root can be checked for the negative values as follows; setting: which can be written as ( ) As is clear from Eq. 33 and 35 the general solution for the components of the position and velocity around the equilibrium points of on the circumference of the circle involve a linear combinations, linearly independent terms, of that leads by Euler identity to sines and cosines. Thus we obtained oscillatory stable solutions when η or ξ satisfies the critria derived in (33) and (35).