NUMERICAL INTEGRATION OF THE RESTRICTED THREE BODY PROBLEM WITH LIE SERIES

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1 NUMERICAL INTEGRATION OF THE RESTRICTED THREE BODY PROBLEM WITH LIE SERIES ELBAZ. I. ABOUELMAGD 1, JUAN L.G. GUIRAO 2 AND A. MOSTAFA 3 Abstract. The aim of this work is to present some recurrence formulas for the equations of motion of an infinitesimal body in the planar restricted three body problem which allow us to integrate numerically this problem via a Lie series approach. For doing this, the equations of motion of the problem are transformed to an origin at one of the libration points and the Lie operator and recurrence formulas for the terms of the Lie series are constructed. In addition, we provide an algorithm that allows us to find any number of Lie series terms and which gives successful calculations for the orbit of the infinitesimal body around one of the libration points. Furthermore, all our calculations are performed under the oblateness effect of the bigger primary up to J 4. Finally, a numerical application of these results is given to the case of the Earth Moon system. 1. Introduction In general terms it is well known that is not possible to find an analytic solution to the set of equations which describe the motion of a gravitational dynamical system regarding three or more celestial bodies. Consequently, numerical methods to obtain an approximation solution at discrete time t i, i {1, 2,..., n}, with step size τ = t i t i 1 considering it constant during the integration process. There are many numerical integration methods that have been used or developed especially in space dynamics to solve systems of ordinary differential equations of a dynamical system. Some of these methods are, for instance: Radau s method, Bulirsch Stoer mehod, Lie series approach, Runge Kutta method which is also called improved Euler s method, Symplectic Integrators, Hybrid integrators, etc. See for a more details analysis of these methods: Everhart [10], Deuflhard [7], Hanslmeier and Dvorak [12], Cash and Karp [4], Candy and Rozmus [3], Chambers [5] respectively as well as the monograph Stoer and Bulirsch [17]. While for an extensive usage of the methods see Sándor et al. [14] and Eggl and Dvorak [9] respectively. Key words and phrases. Numerical integration, Lie series solution, Restricted three body Mathematics Subject Classification. Primary: 70E17, 70E20, 70E40. Secondary: 37C27. 1

2 2 E.I. ABOUELMAGD, J.L.G. GUIRAO, A. MOSTAFA Eggl and Dvorak [9] presents a view of the stay of the art of some of the previous methods and presents integration algorithms of each method to solve the basic Newtonian gravitational N body problem in dynamical astronomy. Also are discussed the main properties of these methods for the Kepler problem. The germ of the original idea of trying to solve a dynamical system via the use of Lie series comes from Gröbner [11]. Hanslmeier and Dvorak [12] develops the method in a real way simplifying the calculation of the Lie terms. Also a recurrence formula for the use of the series expansion is derived and a high speed numerical integration scheme of the classical N body problem is provided. Delva [6] uses the method of Lie series to construct a solution for elliptic restricted three body problem. In this work the the Lie operator for the motion of the infinitesimal body is derived as a function depending on the coordinates, the velocities and the true anomaly of the primaries in a rotating coordinate system. Also the terms of Lie series for the solution are partially computed with recurrence formula. Lie series approach was originally developed for studying the stability and dynamical evolution of Near Earth objects in the inner solar system, see for instance [8]. [13] presents the recurrence formulas for the equations of motion of N body system and states a result which simplifies the derivation of the recurrence formulas for the linearized equations. [15] uses this technique to study the stability of extra solar systems. Moreover, the Lie series obtained in such paper are used to study too the phase space structure of the triangular Lagrangian points, see [16]. Recently, Bancelin et al. [2] generalizes the Lie integrator by considering relativistic acceleration in the frame work of general relativity and a simplified force for the Yarkovsky effect. [2] applies a general iteration procedure to derive the Lie series to any order and precision. Furthermore, an application to integration of the equation of motion for typical Near Earth objects and planet Mercury is given. Our objective in the present paper is to present a numerical integration of the restricted three body problem. In order to do that we transform the equations of motion of the infinitesimal body in a rotating coordinate system in an origin at libration points, see section 2. The equations are constructed when the bigger primary is oblate spheroid considering the effect of oblateness up to J 4. Up to this hypothesis the Lie operator and recurrence formulas for the Lie series terms are constructed, see section 3. Then, we present an algorithm that enables us to find any number of Lie series terms which can be used to approximate the motion of the infinitesimal body around one of the libration points. Furthermore, since the formulas have an easy analytical structure it allows us to program them and we finish our study applying the previous results to the equations of the Earth Moon system, see section 4.

3 NUMERICAL INTEGRATION OF THE RESTRICTED THREE BODY PROBLEM Equations of motion Let m 1 and m 2 be the masses of the primaries such that (m 1 > m 2 ) while m is the mass of the infinitesimal body. The primaries move in circular orbits about their barycenter of mass. The infinitesimal body moves under the gravitational field of the primaries in the same plane, but it is so small that it has no influence on the motion of m 1 and m 2. We assume also that the distance between the primaries is the unit and the sum of masses is equal to one. Let the mass ratio be µ = m 2 /(m 1 +m 2 ). It follows that m 1 = 1 µ and m 2 = µ 1, also the unity of time is chosen, which makes each of the 2 unperturbed mean motion and the universal gravitational constant equal to one. Let the coordinates of m 1 and m 2 and m be designed in inertial frame by ( X 1, Y 1, Z 1 ), (X 2,Y 2, Z 2 ) and (X, Y, Z) respectively, but in a synodic frame are (µ, 0,0), ( 1+µ,0,0) and (x, y,z) respectively. A synodic frame is chosen such that rotates with angular velocity n in positive direction where n = θ and θ is the angle of rotation. The origin of an inertial and synodic frames is taken at the center of mass of the primaries. The orbital plane of m 1 and m 2 are taken as XY plane. We assume that r 1, r 2 are the position vectors of m with respect to m 1 and m 2 respectively, see Figure 1 for details. The equations of motion of the infinitesimal body with the oblateness effect of this problem up to J 4 in a synodic coordinates system xy using dimensionless variables are given below, see [1] for more details. (1) where ẍ 2nẏ = Ω x, ÿ 2nẋ = Ω y, (2) Ω = 1 2 n2 [(1 µ)r µr 2 2] + (1 µ)[ 1 r 1 + A 1 2r 3 1 and n 2 = [ A A 2] 3A 2 8r1 5 ] + µ, r 2 r1 2 = (x µ) + y 2, r2 2 = (x µ + 1) 2 + y 2. Note that n denotes the mean motion, r 1 and r 2 are the magnitudes of the position vectors of the infinitesimal body with respect to m 1 and m 2 respectively, A i = J 2i R1 2i(i = 1, 2) where R 1 is the mean radius of the bigger primary and J 2i represents the dimensionless coefficients of oblateness. The subscripts on the right hand side denote the partial derivation of Ω with respect to the x and y respectively. As we previously stated, in order to formulate the Lie series solution for this problem, first we need to develop explicitly the equations of motion and

4 4 E.I. ABOUELMAGD, J.L.G. GUIRAO, A. MOSTAFA Figure 1. Configuration of an inertial and a synodic frames in the restricted three body problem transform them to an origin at the libration point. For this purpose the equations of motion will be transformed into a system of four first order equations. With the aim of simplifying the notation we define the following functions which are well defined in their natural domain of definition: F (x, y) = 1 (x µ) 2 + y 2 and G(x, y) = 1 (x µ + 1) 2 + y 2. Now substituting equation (2) into (1) and using F (x, y) and G(x, y), the equations of motion of our problem are explicitly controlled by

5 NUMERICAL INTEGRATION OF THE RESTRICTED THREE BODY PROBLEM... 5 (3) (x µ)(1 µ)[n 2 [F (x, y)] 3 3A 1 [F (x, y)]5 ẍ 2nẏ = A 2 8 [F (x, y)]7 ] + µ(x µ + 1)[n 2 [G(x, y)] 3 ] (1 µ)y[n 2 [F (x, y)] 3 3A 1 [F (x, y)]5 ÿ 2nẋ = A 2 8 [F (x, y)]7 ] + µy[n 2 [G(x, y)] 3 ] With the aim of transforming the equation of motion to an origin at the libration point, let the coordinates of the infinitesimal body be (x 1, y 1 ) such that (4) x = x 1 + α, y = y 1 + β where (α, β) are the coordinates of one of the libration points. Hence substituting equations (4) into (3) we obtain (5) ẍ 1 = ÿ 1 = 2ny 1 + (1 µ)(x 1 + α µ)[n 2 [F (x 1 + α, y 1 + β)] 3 3A 1 2 [F (x 1 + α, y 1 + β)] A 2 8 [F (x 1 + α, y 1 + β)] 7 +µ(x 1 + α µ + 1)[n 2 [G(x 1 + α, y 1 + β)] 3 ] 2nx 1 + (1 µ)(y 1 + β)[n 2 [F (x 1 + α, y 1 + β)] 3 3A 1 2 [F (x 1 + α, y 1 + β)] A 2 8 [F (x 1 + α, y 1 + β)] 7 +µ(y 1 + β)[n 2 [G(x 1 + α, y 1 + β)] 3 ] For obtaining a system of four first order equations, we take: hence x 1 = q 1, x 1 = q 2, y 1 = q 3, y 1 = q 4, (6) 1 = q 2, 2 = ẍ 1, 3 = q 4, 4 = ÿ 1. Substituting equations (6) into (5) we obtain

6 6 E.I. ABOUELMAGD, J.L.G. GUIRAO, A. MOSTAFA (7) 1 = q 2, 2nq 4 + (1 µ)(q 1 + α µ)[n 2 [F (q 1 + α, q 3 + β)] 3 2 = 3A 1 2 [F (q 1 + α, q 3 + β)] A 2 8 [F (q 1 + α, q 3 + β)] 7 +µ(q 1 + α µ + 1)[n 2 [G(q 1 + α, q 3 + β)] 3 ] 3 = q 4, 2nq 2 + (1 µ)(q 3 + β)[n 2 [F (q 1 + α, q 3 + β)] 3 4 = 3A 1 2 [F (q 1 + α, q 3 + β)] A 2 8 [F (q 1 + α, q 3 + β)] 7 +µ(q 3 + β)[n 2 [G(q 1 + α, q 3 + β)] 3 ],. 3. The algorithm on the Lie series The aim of this section is to stated an algorithm to construct the linear Lie operator and the recurrence formulas for the Lie series terms for our problem Step 1. The Lie operator over dimension four is given by D = d dt = i=1 q i dq i dt + t. Or if the vector q q(q 1, q 2, q 3, q 4 ) has no explicit time (8) D = i=1 q i dq i dt. See [6] for more details on the definition of this operator. equations (7) into (8), we obtain Substituting

7 NUMERICAL INTEGRATION OF THE RESTRICTED THREE BODY PROBLEM... 7 (9) D = q 2 q 1 2nq 4 + (1 µ)(q 1 + α µ)[n 2 [F (q 1 + α, q 3 + β)] 3 + 3A 1 2 [F (q 1 + α, q 3 + β)] A 2 8 [F (q 1 + α, q 3 + β)] 7 q 2 +µ(q 1 + α µ + 1)[n 2 [G(q 1 + α, q 3 + β)] 3 ] +q 4 q 3 2nq 2 + (1 µ)(q 3 + β)[n 2 [F (q 1 + α, q 3 + β)] 3 + 3A 1 2 [F (q 1 + α, q 3 + β)] A 2 8 [F (q 1 + α, q 3 + β)] 7 q 4 +µ(q 3 + β)[n 2 [G(q 1 + α, q 3 + β)] 3 ] 3.2. Step 2. Construction of the Lie series by applying the Lie operator D to the components of vector function q q(q 1, q 2, q 3, q 4 ) to derive the terms of the Lie series Lie operator acting on q 1. Applying the Lie operator D given by equation (9) to q 1 we obtain Dq 1 = q 2, D 2 q 1 = Dq 2, D 3 q 1 = D 2 q 2,... D n q 1 = D n 1 q 2 Now, we shall construct the recurrence formulas for Dq Lie operator acting on q 2. Let (10) (11) and K 1 = (1 µ)(q 1 + α µ), K 2 = µ(q 1 + α µ + 1) ρ 1 = [F (q 1 + α, q 3 + β)] 3, ρ 2 = [F (q 1 + α, q 3 + β)] 5, ρ 3 = [F (q 1 + α, q 3 + β)] 7, ρ 4 = [G(q 1 + α, q 3 + β)] 3.

8 8 E.I. ABOUELMAGD, J.L.G. GUIRAO, A. MOSTAFA Substituting equations (10) and (11) into the expression of 2 given in (7) we get Also let 2 = 2nq 4 + K 1 [n 2 ρ A 1ρ A 2ρ 3 ] + K 2 [n 2 ρ 4 ]. (12) Then, M 1 = 2nq 4, M 2 = K 1 [n 2 ρ A 1ρ A 2ρ 3 ] + K 2 [n 2 ρ 4 ]. Dq 2 = M 1 + M 2, D 2 q 2 = DM 1 + DM 2, D 3 q 2 = D 2 M 1 + D 2 M 2,... D n q 2 = D n 1 M 1 + D n 1 M 2 We underline that D m M 2 can be computed as follows. Let (13) S 1 = n 2 ρ 1, S 2 = n 2 ρ 4, S 3 = ρ 2, S 4 = ρ 3 and K 3 = 3 2 A 1K 1, (14) K 4 = 15 8 A 2K 1. Substituting equation (13) and (14) into (12) we get while M 2 = K i S i, i=1 In general DM 2 = [K i DS i + S i DK i ]. i=1

9 NUMERICAL INTEGRATION OF THE RESTRICTED THREE BODY PROBLEM... 9 D m M 2 = i=1 j=0 m [D m j K i ][D j S i ], m 0 where D 0 χ χ. Note that DK 1 = a 1 q 2, D n K 1 = a 1 D n 1 q 2 and D n K i = a i D n K 1, i = 2, 3, 4, where a i are constant given by a 1 = 1 µ, a 2 = µ 1 µ, a 3 = 3 2 A 1 and a 4 = 15 8 A 2. On the other hand the recurrence expressions for computing the powers of DS i are{ the following: { T3 i, if i = 1, 2 D n 1 T DS i = and DS n 3 i, if i = 1, 2 i = T i, if i = 3, 4 D n 1 T i, if i = 3, 4 where T 1 = 3[q 2 (q 1 + α µ + 1) + q 4 (q 3 + β)][g(q 1 + α, q 3 + β)] 5, T 2 = 3[q 2 (q 1 + α µ) + q 4 (q 3 + β)][f (q 1 + α, q 3 + β)] 5, T 3 = 5 3 [F (q 1 + α, q 3 + β)] 2 T 2, T 4 = 7 3 [F (q 1 + α, q 3 + β)] 4 T 2 = 7 5 [F (q 1 + α, q 3 + β)] 2 T Lie operator acting on q 3. Applying the Lie operator D given by equation (9) to q 3 we get Dq 3 = q 4, D 2 q 3 = Dq 4, D 3 q 3 = D 2 q 4... D n q 3 = D n 1 q 4 Now, we endeavour to state the recurrence formulas for Dq 4 and its powers Lie operator acting on q 4. Let (15) L 1 = (1 µ)(q 3 + β), L 2 = µ(q 3 + β) Substituting equations (11) and (15) into expression of 4 in (7) we get q 4 = 2nq 2 + L 1 [n 2 ρ A 1ρ A 2ρ 3 ] + L 2 [n 2 ρ 4 ]. Also, let

10 10 E.I. ABOUELMAGD, J.L.G. GUIRAO, A. MOSTAFA (16) then N 1 = 2nq 2, N 2 = L 1 [n 2 p A 1p A 2p 3 ] + L 2 [n 2 p 4 ] Dq 4 = N 1 + N 2, D 2 q 4 = DN 1 + DN 2, D 3 q 4 = D 2 N 1 + D 2 N 2... D n q 4 = D n 1 N 1 + D n 1 N 2 In the last equation D m N 2 can be calculated as follows. Let L 3 = 3 2 A 1L 1, (17) L 4 = 15 8 A 2L 1. Substituting equations (13) and (17) into expression of N 2 of (16) we get while N 2 = L i S i, i=1 In general DN 2 = [L i DS i + S i DL i ]. i=1 D m N 2 = i=1 j=0 m [D m j L i ][D j S i ], m 0; where DL 1 = a 1 q 4, D n L 1 = a 1 D n 1 q 4 and D n L i = a i D n L 1, i = 2, 3, Step 3. Now, we shall computed all terms of the Lie series. Recall that the analytic solution is approximated by the Lie series given by or q = exp[τd](q) q=q0 q = j=0 τ j j! (Dj q) q=q0.

11 NUMERICAL INTEGRATION OF THE RESTRICTED THREE BODY PROBLEM Where q 0 is the initial vector, also for simplicity we can write (D j q) q=q0 as D j q 0. Consequently the components of the vector function q q(q 1, q 2, q 3, q 4 ) are given by where q 2 = Dq 1, q 1 = q 1 (0) + j=1 τ j j! Dj 1 q 2 (0) q 2 = q 2 (0) + j=1 j=1 τ j j! Dj 1 M 1 (0) + j 1 i=1 j=1 m=0 q 3 = q 3 (0) + j=1 where q 4 = Dq 3 and τ j q 4 = q 4 (0) + j! Dj 1 N 1 (0) + τ j j! Dj 1 q 4 (0) j 1 i=1 j=1 m=0 τ j j! [Dj m 1 K i ][D m S i ], τ j j! [Dj m 1 L i ][D m S i ]. Remark 1. Finally, we summarize the steps of the algorithm that we have stated to get a numerical approximation of the solution for the equations of motion of an infinitesimal body in the planar restricted three body problem: (1) The equations of motion must be developed explicitly. (2) The equations motion must be transformed to an origin at the libration points. (3) The equations of motion must be transformed in to a system of four first orders equations. (4) The Lie operator must be constructed. (5) The Lie series terms must be computed. (6) The analytic solution of the problem is approximated by the Lie series expansion. 4. Application to the Earth Moon problem In this section we present some numerical calculations in order to verify the solution obtained in the previous section. Our calculations are made for a satellite (considered a point mass) in the Earth Moon system when (µ = , A 1 = , A2 = 0) The calculations and the curves are made using a commercial symbolic package. First we start with a comparison between the linearized motion about L 4 obtained by the Lie series solution and the exact curve equations obtained in Szebehely [18, pag. 251]. The results show an exact matching over an interval of time of 100 days which is about 3.5 the period of the small primary about the bigger primary which in this case is the period of the Moon about the Earth= day. This is shown in Figure 2 where the calculations have

12 12 E.I. ABOUELMAGD, J.L.G. GUIRAO, A. MOSTAFA Figure 2. The linearized motion obtained by Lie Series solution (dotted curve), and the exact curve (continuous curve). Step size one day and four terms are computed. Figure 3. The linearized motion obtained by Lie Series solution (dotted curve), and the exact curve (continuous curve). Step size one day and two terms are computed. been made using one day step size and four terms calculations. In Figure 3 the calculations are done using only two terms to make a comparison. The initial conditions are taken to be x = y = 0, V x = 0, and V y = 1. Recall that V x is the velocity in x direction, V y is the velocity in y direction. In Figure 4, a curve is plotted for the non linear motion about L 4 using Lie series solution of two hours step with 6 terms calculation. The motion started at L 4 with initial velocity 1 at the y direction. Finally, the conservation of the energy integral c = 2Ω V 2 is tested for both the linear and non linear motions around L 4. A comparison is made at different steps and different orders of Lie series calculations. The constant of energy is calculated at x = y = 0, V x = 0 and V y = 1 in the linear motion

13 NUMERICAL INTEGRATION OF THE RESTRICTED THREE BODY PROBLEM Figure 4. Nonlinear motion about L 4 obtained by Lie Series solution for two hours step size and 6 Lie series terms calculations. which gives c = 1, and at L 4 and V x = 0, V y = 1 for the non linear motion which gives c = 2, see Tables 1 5. Number of days 4 terms 8 terms 12 terms Table 1. Conservation of the energy integral in the linear motion around L 4 for different number of Lie series terms at one day step. Acknowledgements This work has been partially supported by MICINN/FEDER grant number MTM References [1] E.I. Abouelmagd, Existence and stability of triangular points in the restricted three body problem with numerical applications, Astrophys Space Sci. 342 (2012), 45 53

14 14 E.I. ABOUELMAGD, J.L.G. GUIRAO, A. MOSTAFA Number of days 8 terms 12 terms 16 terms Table 2. Conservation of the energy integral in the linear motion around L 4 for different number of Lie series terms at four days step. Number of days 8 terms 12 terms 16 terms Table 3. Conservation of the energy integral in the linear motion around L 4 for different number of Lie series terms at eight days step. [2] D. Bancelin, D. Hestroffer, W. Thuillot, Numerical integration of dynamical systems with Lie series Relativistic acceleration and non-gravitational forces, Celestial Mech. Dyn. Astr. 112 (2012), [3] J. Candy, W. Rozmus, A symplectic integration algorithm for separable Hamiltonian functions, J. Comp. Phys. 92 (1991), [4] J.R. Cash, A.H. Karp, A variable order Runge Kutta Method for initial value problems with rapidly varying right hand sides, ACM Transac Math. Softw. 16(3) (1990), [5] J.E. Chambers, A hybrid symplectic integrator that permits close encounters between massive bodies, Mon. Not. R. Astron. Soc. 304 (1999),

15 NUMERICAL INTEGRATION OF THE RESTRICTED THREE BODY PROBLEM Number of days 4 terms 8 terms Table 4. Conservation of the energy integral in the non linear motion around L 4 for different number of Lie series terms at one day step. Number of days 4 terms 8 terms Table 5. Conservation of the energy integral in the non linear motion around L 4 for different number of Lie series terms at four days step. [6] M. Delva, A Lie integrator program and test for the elliptic restricted three body problem, Astron. Astrophys. 60 (1985), [7] P. Deuflhard, Order and stepsize control in extrapolation methods, Num. Math. 41 (1983), [8] R. Dvorak, E. Pilat Lohinger, On the dynamical evolution of the Atens and the Apollos. Planet., Space Sci. 47 (1999), [9] S. Eggl and R. Dvorak, An introduction to common numerical integration codes used in dynamical astronomy, J. Souchay and R. Dvorak (eds.) Lecture Notes in Physics, vol. 790, Berlin, Springer, (2010) [10] E. Everhart, Implicit single sequence methods for integrating orbits, Celestial Mech. Dyn. Astr. 10 (1974), [11] W. Gröbner, Die Lie-Reihen und ihre Anwendungen, VEB Deutscher Verlag (1967) [12] A. Hanslmeier, R. Dvorak, Numerical integration with Lie series, Astron. Astrophys. 132 (1984), [13] A. Pál, A. Süli, Solving linearized equations of the N body problem using the Lie integration method, MNRAS 381 (2007),

16 16 E.I. ABOUELMAGD, J.L.G. GUIRAO, A. MOSTAFA [14] Z. Sándor, A. Süli, B. Érdi, E. Pilat Lohinger,R. Dvorak, A stability catalogue of the habitable zones in extra solar planetary systems, Monthly Notices of the Royal Astronomical Society 375(4) (2007), [15] R. Schwarz, E. Pilat Lohinger, R. Dvorak, B. Érdi, Z. Sándor, Trojans in habitable zones, Astrobiology. 5 (2005), [16] R. Schwarz, R. Dvorak, A. Süli, B. Érdi, Survey of the stability region of hypothetical habitable Trojan planets,, Astron. Astrophys. 474(3) (2007), [17] J. Stoer, R. Bulirsch, Introduction to numerical analysis, Springer, New York (1980). [18] V. Szebehely, Theory of orbits: the restricted three body problem, Academic Press (1967). 1 Mathematics Department, Faculty of Science and Arts (Khulais), King Abdulaziz University, Jeddah, Saudi Arabia. Corresponding Author address: eabouelmagd@gmail.com, eabouelmagd@kau.edu.sa 2 Departamento de Matemática Aplicada y Estadística. Universidad Politécnica de Cartagena, Hospital de Marina, Cartagena, Región de Murcia, Spain. address: juan.garcia@upct.es 3 Mathematics Department, Faculty of Science, Ain Shams University. Cairo, Egypt