Stability of motions near Lagrange points in the Elliptic Restricted Three Body Problem

Transcription

1 MSC Computational Physics Thesis Stability of motions near Lagrange points in the Elliptic Restricted Three Body Problem Zografos Panagiotis Supervising Professor: Voyatzis George Aristotle University of Thessaloniki Faculty of Science, Department of Physics 1

2 Contents Abstract 3 1. Introduction 4 2. Description of the Elliptic Restricted Three Body Problem 6 A. Equations of Motion in the Inertial Barycentric system 11 B. Equations of Motion in the Rotating system 15 C. The Five Lagrange points of Equilibrium 20 D. The Fast Lyapunov Indicator (FLI) The Stability of The Lagrange points of Equilibrium and Stability range 28 A. Stability of the Lagrange points 28 B. Regular orbits in the neighborhood of the Lagrange points 33 C. Examination of the results 97 D. Appreciation for Exoplanetary systems Conclusions 102 APPENDIX 103 References 143 2

3 Abstract The nature of stability of the Lagrange points has been studied in the two-dimensional, elliptic restricted three body problem. The regular regions near the neighborhoods of Lagrange points L 4 and L 5 have been computed for different values of the mass ratio μ and the orbital eccentricity e of the primaries. It has been found that the volume of stability regions decreases as both μ and e increase. The range of the regular regions near L 4 of 15 extrasolar planetary systems, consisting of one star and one Jupiter-like planet, has been calculated. Given a large enough regular region, there exists the possibility of co-orbital companions or Trojan-like objects in stable orbits around the triangular equilibrium points. A system of rotating-pulsating coordinates has been used. The equations of motion and the corresponding variational equations have been integrated numerically using the Runge- Kutta method of the 4th order. The nature of stability of the Lagrange points has been calculated using the Floquet theory. The range of the regular regions θ has been determined by calculation of the Fast Lyapunov Indicators (FLI) for a grid of initial conditions around L 4 and L 5. For the purposes of this study several computer programs have been written using C and C++ programming language. 3

4 1. Introduction The elliptic restricted three body problem is a special case of the three body problem. This means that one of the three bodies has mass equal to zero, and that the other two bodies move in Keplerian elliptic orbits. The purpose here is to study the motion of the massless body under the gravitational influence of the other two bodies. This study can be carried out in two or three dimensions, but here we chose the former and the simplest. Unlike the circular restricted three body problem which is a well-documented and simple dynamical system, the elliptic problem is considerably more complicated and presents different challenges in its study. This is due to the fact that the elliptic problem is non-conservative, as the absence of the energy integral adds numerous implications. The interest of studying the elliptic problem stems from the fact that it is a better approximation of the motions that occur in the solar system in comparison to the circular problem. For instance, the motion of the moon as it is affected by the Sun and the Earth, or the motion of a satellite under the influence of the Earth and the moon, are better studied by using primaries moving in elliptic orbits. The equations of motion of the elliptic problem differ from those in the circular problem in three important places: 1. The elliptic problem contains two parameters instead of one, the eccentricity e and the mass ratio of the primary bodies μ. 2. The elliptic problem has no energy (Jacobi) integral. 3. The independent variable is present in the equations of motion, even when rotating axes are used. 4

5 The details of the solution of the equations of motion and the variational equations are discussed in Section 2. This work concentrates on the nature of stability of the triangular Lagrange equilibrium points L 4 and L 5, as well as the stability of orbits in their neighboring regions for a large number of μ and e (including cases of e=0) and different initial conditions of the massless body. This work s goal is to determine the regions near the Lagrange points, where if the massless body is left motionless (initial velocity components equal to zero), it will follow a stable or regular orbit in the neighborhood of the Lagrange points and not veer to infinity. In order to accomplish this, the nature of stability of the five Lagrange points - which according to Broucke (1969) falls into seven categories - was calculated for different pairs of μ-e using the Floquet theory. All of the above were made possible by numerical integration of the equations of motion of the elliptic problem in the rotating-pulsating frame of reference using the Runge-Kutta method of the 4th order, as well as integration of the variational equations. The stability of the neighborhood of the Lagrange points in the elliptic three-body problem is of great interest when the problem is applied to exoplanetary systems (EPS). The vast majority of extrasolar planetary systems that have been discovered contain one Jupiter-like gas giant, their masses ranging from 5M up to several Jupiter masses (Schwarz, Dvorak, Süli & Érdi, 2007). The importance of the stable regions near the equilibrium points and especially near the triangular Lagrange points L 4 and L 5 becomes apparent if one poses the question of whether or not a Trojan-like planet may move in a stable orbit around the aforementioned Lagrange points. Even more importantly, can such planets be habitable? When astronomers look for life in EPS, they have to know the location of the so-called habitable zone (HZ), which is defined 5

6 as a region around the star where the planet could receive enough radiation to maintain liquid water on its surface and to be able to build a stable atmosphere (Schwarz, Dvorak, Süli & Érdi, 2007). When the gas giant of an EPS moves into the HZ, a habitable satellite could have a stable orbit (e.g. Titan around Saturn), or a Trojan-like planet could exist in a stable orbit near the L 4 and L 5 Lagrange points (Schwarz, Dvorak, Süli & Érdi, 2007). The results of this work concerning the regular regions near the equilibrium points, depending on the mass ratio μ and the eccentricity e, are cross-examined with known EPS in order to suggest possible candidate systems for investigation of the possible existence of Trojan-like planets in those systems. The computer programs needed for the calculations were written in C programming language. The results, diagrams and data matrices are provided and discussed in Section Description of the Elliptic Restricted Three Body Problem In this section, the equations of motion of the elliptic problem relative to different coordinate systems are presented. The results of this study were obtained by numerical integration of these equations in two dimensions. Suppose a particle called the satellite is moving under the gravitational influence of two massive bodies according to Newton s laws. The massive bodies are called the primaries. All three objects are moving on the same plane. The problem is defined as restricted because the satellite is massless meaning that it can t influence the motion of the primaries, which move in elliptic Keplerian orbits relative to each other 6

7 or relative to their center of mass. Elliptic orbits with various values of eccentricity e including e=0 (circular problem), are considered in this study. In the following, the equations of motion in an inertial barycentric frame of reference will be presented, although the most important calculations use rotating systems of coordinates. At the initial value of the independent variable (t=0), the primaries always lie on the x-axis at an apse, at periapsis at minimum distance or apopsis at maximum distance (Broucke, 1969). Also, a system of regular units is used in such a way that the semi-major axis α and the mean motion n of the motion of the primaries equals 1. In this case, the masses of the primaries (including the gravitational constant) can be written as m 1 = 1 μ, m 2 = μ < 1 2. (1) The distance between the primaries is r = (1 cose) = p 1 + e cosv (2) where E is the eccentric anomaly, v the true anomaly and p = (1 e 2 ) (3) 7

8 is the semilatus rectum. In this barycentric system of coordinates, the coordinates of the primaries are ξ 1 = μ r cosv = μ(cose e) (4a) η 1 = μ r sinv = μ(1 e 2 ) 1 2 sine (4b) ξ 2 = (1 μ) r cosv = (1 μ)(cose e) (4c) η 2 = (1 μ)r sinv = (1 μ)(1 e 2 ) 1 2 sine (4d). Kepler s equation will be used to relate the eccentric anomaly E with the time t: t + x = E e sine (5) Here, the phase constant x is either equal to 0 (periapsis) or π (apopsis). Equation (5) can be differentiated with respect to t to produce the differential equation de dt = 1 1 e cose (6) which is solved numerically in order to acquire E, which in turn is used to calculate the coordinates of the primaries from eq. (4). A few more formulas relating to the motion of the primaries are proven useful. The first and second derivatives of r, v and E are 8

9 r = e sinv p 1 2 = e sine, r = r e cosv r 2 = p r r 3 v = p1 2 r 2, v = E = 1 r, 2e sinv r 3 (7) E = e sinv r 2 p 1. 2 Using eq. (7), the energy integral of the two-body problem of the primaries can be verified: 1 2 (r 2 + r 2 v 2 ) 1 r = 1 2. (8) The above derivatives are notated with a prime to indicate differentiation with respect to the time t. Later, the true anomaly v will be used as the independent variable, and the corresponding derivatives will be indicated by dots. The derivatives of r with respect to v are r = er2 sinv p, r = 2r 2 r + r (1 r ). (9) p The energy integral can then be written as 9

10 p (r 2 2 r + r) 1 r = 1 2. (10) By combining eq. (10) and the second eq. (9), the following differential equation for r is obtained: r = 2 p r3 + 3 p r2 r. (11) Eq. (11) can be used to determine r except when p=0 or e=1, in which case eq. (1) has to be used, after solving Kepler s equation eq. (5) or (6). Also, since changes in the independent variable from t to v will be made, the relation between the t- derivatives and the v-derivatives of any given quantity F are F = p1 2 r 2 F, F = p(rf 2r F ) r 5. (12) In the following sections, a special set of coordinates (ξ, η ) will be introduced. These are called reduced or pulsating coordinates and they introduce a radial scale change such that the elliptic motion of the primaries in the system (ξ, η) is transformed in a circular motion in the system (ξ, η ) (Broucke, 1969).The relation between the two sets is proportional to r: ξ = rξ, η = rη. (13) 10

11 The reduced coordinates of the primaries m 1 and m 2 are then written as ξ 1 = μ cosv, η 1 = μ sinv ξ 2 = (1 μ) cosv, η 2 = (1 μ) sinv (14) and represent circular motion (with non-constant angular momentum (Broucke, 1969).When r=0 or e=1 this transformation cannot be used. A. Equations of Motion in the Inertial Barycentric system As mentioned before, in this work most calculations were made by using a rotating system of coordinates. However, before the equations of motion of the satellite relative to this system are presented, it is important to present the equations of motion in the inertial barycentric system, as the transition to the rotating system will be much easier. Theoretically, the results of this study can be obtained by solely using the inertial system, although the calculations are much less convenient. In the inertial barycentric frame of reference, the equations of motion of the satellite are derived from the Lagrangian function L = 1 2 (ξ 2 + η 2 ) + 1 μ s 1 + μ s 2. (15) 11

12 The distances between the satellite and the primaries m 1 and m 2 respectively are s 1 2 = (ξ ξ 1 ) 2 + (η η 1 ) 2 s 2 2 = (ξ ξ 2 ) 2 + (η η 2 ) 2 (16) If q j and q j denote the generalized positions and generalized velocities respectively such that q 1 = ξ, q 1 = ξ, then by using the equation q 2 = η q 2 = η (17) d dt ( L q j (Hadjidemetriou, 2000) the equations of motion are thus ) L q j = 0 (18) ξ = (1 μ)(ξ ξ 1) 3 μ(ξ ξ 2) 3 s 1 s 2 η = (1 μ)(η η 1) s 1 3 μ(η η 2) s 2 3. (19) Lagrangian eq. (15) is non-conservative because it explicitly contains the independent variable t through s 1 and s 2. According to Broucke (1969) using the true anomaly v as the independent variable, the Lagrangian equation would change to 12

13 L = r2 p 1 2 L = p1 2 2r 2 (ξ2 + η 2 ) + r2 p 1 2 ( 1 μ s 1 + μ s 2 ). (20) If the reduced or pulsating coordinates (ξ, η ) are used, according to eq. (13), and ξ = r ξ + rξ, η = r η + rη, ξ = r ξ + rξ ξ = r η + rη (21) the Lagrangian equations in eqs. (15) and (20) can be transformed accordingly to L = r2 2 (ξ 2 + η 2) + rr (ξ ξ + η η ) + r 2 2 (ξ2 + η ) μ (1 + μ ) (22) r r 1 r 2 1 2r L = p1 2 (ξ 2 + η 2 ) + p 2 r + r p 1 2 (ξ ξ + η η ) + p1 2 2 r 2 r 2 (ξ2 + η ) 2 ( 1 μ r 1 + μ r 2 ) (23) where r 2 1 = (ξ ξ 1 ) 2 + (η η ) 2 1 = s 1 2 r 2 r 2 2 = (ξ ξ 2 ) 2 + (η η 2 ) 2 = s 2 2 r 2. (24) 13

14 Dividing eq. (23) by p 1 2 and substituting r by its value from eq. (9), the Lagrangian transforms into L = 1 (ξ 2 e r sinv + η 2 ) + 2 p (ξ ξ + η η ) + e2 r 2 sin 2 v 2p 2 (ξ 2 + η 2 ) + r p (m 1 r 1 + m 2 r 2 ). (25) Eq. (25) can be replaced by a simpler Lagrangian by subtracting the following exact differential, which by taking into account eq. (2) and eq. (9) is d dv (e r sinv 2p = (ξ 2 + η )) 2 e r sinv p (ξ ξ + η η ) + e2 r 2 sin 2 v (ξ 2 2p 2 + η 2 ) (1 r p ) (ξ 2 + η 2 ), (26) from eq. (25). The above exact differential can be safely omitted from the Lagrangian without change in the equations of motion. Thus, the simplified Lagrangian is L = 1 2 (ξ2 + η 2 ) (r 1) (ξ2 p + η ) 2 + r p (m 1 r 1 + m 2 r 2 ). (27) The equations of motion in the inertial barycentric pulsating frame of reference, with the true anomaly v as independent variable, are thus 14

15 ξ = ( r p 1) ξ r p (m 1(ξ ξ 1 ) 3 + m 2(ξ ξ 2 ) 3 ) r 1 r 2 η = ( r p 1) η r p (m 1(η η ) m 2(η η ) 2 3 ) r 1 r 2 (28) The forces which are present in the last equations of motion, eq. (28), contain three terms: an apparent radial force that comes only from the radial scale change of the coordinate system, and m 1 and m 2, which are, of course, the Newtonian attraction potential from the two primaries. B. Equations of motion in the Rotating system Now, the equations of motion in a rotating barycentric system are presented. The angle of rotation is the true anomaly v. The angular velocity of the axes are time dependent, unless the eccentricity e equals zero (Broucke, 1969). The equations of motion will be written both with time t and the true anomaly v as independent variable, in ordinary and in pulsating coordinates. The equations of motion written in rotating pulsating coordinates and with the true anomaly v as the independent variable provide certain advantages, which will be discussed later, and this is the reason they were chosen for the calculations. The ordinary rotating coordinates (x, y ) are related to the inertial coordinates (ξ, η) by 15

16 ξ = x cosv y sinv η = x sinv + y cosv (29) The rotating coordinates simplify things by keeping the two primaries m 1 and m 2 on the x-axis permanently. Their coordinates in this system are x 1 = μ r, y 1 = 0 x 2 = (1 μ) r, y 2 = 0 (30) These coordinates are not constant; the two primaries are oscillating on the x-axis (Broucke, 1969). Lagrangian eq. (15) combined with eqs. (29) transforms into L = 1 2 (x 2 + y ) 2 + (x y + y x )v (x2 + y 2 )v 2 + ( m 1 s 1 + m 2 s 2 ) (31) and the equations of motion derived from this Lagrangian are 16

17 x 2y v y v x v 2 = (1 μ)(x x ) 1 3 μ(x x ) 2 (32a) 3 s 1 s 2 y 2x v x v y v 2 (1 μ)y = 3 μy 3 s 1 s. (32b) 2 The above equations will be transformed to the desired form by writing them using rotating-pulsating coordinates (x,y). These coordinates are defined in the same way as eq. (13): x = rx, y = ry (33) Lagrangian eq. (31) transforms in the new form L = r2 2 (x 2 + y 2 ) + rr (xx + yy ) + p 1 2(xy yx ) (r 2 + p r 2) (x2 + y 2 ) + 1 r (m 1 r 1 + m 2 r 2 ) (34) and the corresponding equations of motion are written as r 2 x 2y p rr x 1 r x = 1 r (m 1(x x 1 ) 3 + m 2(x x 2 ) 3 ) (35a) r 1 r 2 r 2 y 2x p rr y 1 r y = 1 r (m 1y r m 2y 3 r ) (36b) 2 17

18 The above equations of motion are written with time t as the independent variable, but they can be simplified if the true anomaly v is used as the independent variable instead. This change in variable is done by using the expression of v of eq. (7), and in the same way it was used to obtain the Lagrangian eq. (20). By subtracting the exact derivative, as in eqs. (26), (27), d dv (1 2 r r (x2 + y 2 )) (37) and dividing the Lagrangian equation by p 1 2, then eq. (34) finally transforms into L = 1 2 (x 2 + y 2 ) + (xy yx ) + r p [1 2 (x2 + y 2 ) + m 1 r 1 + m 2 r 2 ] (35). The corresponding equations of motion are thus x 2y = r p (x m 1(x x 1 ) r 1 3 m 2(x x 2 ) r 2 3 ) y + 2x = r p (y m 1y r 1 3 m 2y r 2 3 ). (36) 18

19 The Lagrangian eq. (35) and its equations of motion (36), present the unique advantage that they differ only by a factor of r = 1 from the Lagrangian and classical equations of p 1+e cosv motion of the circular restricted three body problem. They have also been used in the study of orbits near the equilibrium points in this work. It is important to note that since the numerical integration of eqs. (36) was achieved using the Runge-Kutta 4 method, the system of equations (36) was altered slightly in such a way that instead of containing two second order differential equations, it contained four first order differential equation. This was easily done using the following substitutions: u x = x, u x = x u y = y, u y = y (37) The system (36) using eqs. (37) transforms into x = u x u x 2u y = r p (x m 1(x x 1 ) r 1 3 m 2(x x 2 ) r 2 3 ) (38) y = u y u y + 2u x = r p (y m 1y r 1 3 m 2y r 2 3 ) 19

20 where u x, u y are, of course, the satellites velocity components. The systems (36) and (38) of differential equations are equivalent. C. The five Lagrange points of Equilibrium The five Lagrange points (equilibrium points) are present both in the circular and in the elliptic three body problem. At these points, the satellite remains in the same position relative to the primaries if no other forces are applied to it (i.e. the force field at these points is zero). The study of the five Lagrange points, their stability as well as their neighborhoods was made mainly by using the rotating-pulsating coordinates (x, y), but the inertial coordinates (ξ, η) can also be used. In the rotating-pulsating frame of reference, the Lagrange points are fixed (Broucke, 1969). From the equations of motion (36), it is clear that there are five particular solutions with constant coordinates and with x = y = x = y = 0. These constant coordinates are solutions to the equations of motion by setting their right side equal to zero. From eqs. (36) we obtain x m 1(x x 1 ) 3 m 2(x x 2 ) 3 r 1 r 2 y m 1y r 1 3 = x (1 1 μ r 1 3 μ r 2 3 ) + μ(1 μ) ( 1 r r 1 3 ) = 0 (39a) m 2y r μ = y (1 3 μ 3 r 1 r ) = 0 (39b) 2 20

21 In the above equations m 1, m 2 were replaced by m 1 = 1 μ, m 2 = μ (40) and x 1, x 2 were replaced by their values in the rotatingpulsating system: x 1 = μ, x 2 = 1 μ. (41) The case where the eccentricity e = 1 has been excluded because in the rotating-pulsating system of coordinates r must be always different from zero. The eqs. (39) are the same equations one would arrive at in the study of the classical circular three-body problem (Broucke, 1969). The first two equilibrium solutions are easily found from eqs. (39) if r 1 = r 2 = 1. Since the distances between the primaries and the satellite are r 1 2 = (x x 1 ) 2 + y 2 r 2 2 = (x x 2 ) 2 + y 2 (42) solving the system (42) for r 1 = r 2 = 1 provides the coordinates of the L 4 and L 5 Lagrange points which correspond to the equilateral triangle configurations with the two primaries. They are thus: 21

22 L 4 : x 4 = 1 2 (1 2μ), y 4 = (43) L 5 : x 5 = 1 2 (1 2μ), y 5 = The other three solutions of eq. (39) are L 1, L 2 and L 3. These are called collinear equilibrium points because they lie on the syzygy-axis (the line of the primaries) where y=0 (Broucke, 1969). Their abscissa x therefore is root of equation f(x) x + (1 μ) (x x 1) r μ (x x 2) r 2 3 (44) where r 1 3 = x x 1 3 r 2 3 = x x 2 3 (45) Equation (44) has one root in each of the three intervals on both sides of m 1 and m 2 and between them. This is easily verifiable if one looks at the derivative of f(x): 22

23 df(x) dx 2(1 μ) = 1 x x 1 3 2μ < 0. (46) x x 2 3 Since the derivative is negative in each interval, f(x) is monotonously decreasing from + to in each interval thus crossing the x-axis. In order to obtain the abscissae of the L 1, L 2, L 3 Lagrange points, eq. (44) was numerically solved using the Newton-Raphson method. The following figure show the change in x L1, x L2, and x L3 with respect to the mass ratio μ. Figure 1: Change in x L1, x L2, and x L3 with respect to the mass ratio μ. The mass ratio is between < μ < 0. 5 and the eccentricity is e= mu 0.5 mu mu xl xl xl3 As aforementioned, the purpose of this work is to study the neighborhood of the Lagrange points of equilibrium, in order to determine the stability of these points as well as the stability of their neighborhoods. To achieve this, the equations of motion, eqs. (36), must be linearized in the neighborhood of 23

24 the Lagrange points in order to obtain the so-called first order variational equations (Broucke, 1969). Eqs. (36) are written in the form x 2y r p x = r p U x (47) y + 2x r p y = r p U y where U is the potential function U = 1 μ r 1 + μ r 2. (48) The subscripts x and y in U are used to represent the partial derivatives of U. The variational equations derived from eqs. (47) are written as δx 2δy r p δx = r p (U xxδx + U xy δy) δy + 2δx r p δy = r p (U yxδx + U yy δy) (49) 24

25 Again, as with the equations of motion (36), the variational equations (49) were numerically integrated using the Runge- Kutta of the 4th order method. For this reason, the system (49) was transformed to include four first-order differential equations, in a similar way as eqs. (38). This transformed system is written as δx = δu x δu x 2δu y r p δx = r p (U xxδx + U xy δy) (50) δy = δu y δu y + 2δu x r p δy = r p (U yxδx + U yy δy) Equations (49) are a system of linear differential equations with nonconstant periodic coefficients, due to factor r, which depends on the cosine of the true anomaly v. In the circular three-body problem, the respective system would have constant coefficients since r = 1. This principal difference between the circular and the elliptic problem, makes the study of the latter a more difficult proposition, thus a more complex method is required in order to study the variational equations in the elliptic problem. The Floquet theory was used, in order to determine the stability of the Lagrange points. Omitting the details of the theory, a 4x4 monodromy matrix D is obtained by integrating the variational equations for a single period of the true anomaly v (0 < v < 2π) using the following initial conditions for the variational equations: δx(0) = 1, δy(0) = 0, δu x (0) = 0, δu y (0) = 0 (51) 25

26 At v = 2π the integration of the variational equations yields δx(2π) = δx 1, δy(2π) = δy 1, (52) δu x (2π) = δu x1, δu y (2π) = δu y1 The integration of system (50) was repeated three more times, each time changing the place of unity in the initial conditions (51). For instance, the second integration would use δx(0) = 0, δy(0) = 1, δu x (0) = 0, δu y (0) = 0 (53) in order to obtain δx(2π) = δx 2, δy(2π) = δy 2, (54) δu x (2π) = δu x2, δu y (2π) = δu y2 and so forth. The monodromy matrix D is thus constructed in the following way: δx 1 δx 2 δx 3 δx 4 δy 1 δy 2 δy 3. D = ( δu x1 δu x2.. ) (55) δu y1 δu y2.. 26

27 The stability of any point is then determined by numerically calculating the eigenvalues and their magnitudes of the monodromy matrix D using the widely used QR algorithm. The QR algorithm has its basis on the QR decomposition, in which a matrix A is decomposed to a product A=QR of an orthogonal matrix Q and an upper triangular matrix R. Formally, let A be a real matrix of which we want to compute the eigenvalues, and let A 0 :=A. At the k-th step (starting with k = 0), we compute the QR decomposition A k = Q k R k where Q k is an orthogonal matrix (i.e., Q T = Q 1 ) and R k is an upper triangular matrix. We then form A k+1 = R k Q k. Note that A k+1 = R k Q k = Q k 1 Q k R k Q k = Q k 1 A k Q k = Q k T A k Q k so all the A k are similar and hence they have the same eigenvalues. The algorithm is numerically stable because it proceeds by orthogonal similarity transforms. Under certain conditions, the matrices A k converge to a triangular matrix. The eigenvalues of a triangular matrix are listed on the diagonal, and the eigenvalue problem is solved. All the above calculations were made for many different values for the mass ratio μ and the eccentricity e. The results and the kinds of stability of the Lagrange points are discussed in Section 3. D. The fast Lyapunov indicator (FLI) The regions near the Lagrange points of equilibrium are of great interest and part of this work focuses on finding the parts of these regions which produce regular satellite orbits. This is again accomplished by integrating the equations of motion (38) and the variational equations (50) together for initial conditions that correspond to the neighboring region of the Lagrange points, this time calculating for every single orbit the so-called fast Lyapunov indicator or FLI. 27

28 The FLI is a great tool which allows the identification of regular and chaotic orbits. The FLI accomplishes this by measuring the mean exponential distancing of a neighboring orbit to a control orbit (Voyatzis & Meletlidou, 2015). There are many formulas that define FLI, but the following was used: FLI(v) δx(v) 2 + δy(v) 2 + δu x (v) + δu y (v) = log v ( ) (56) For a regular orbit of the system, the FLI increases in value at a slow rate as v. On the other hand, for a chaotic orbit the FLI increases at a much faster rate. Therefore, after a comparatively small interval of the independent variable v, the FLI either has a small value, indicating a regular orbit or it has a large value which indicates a chaotic orbit (Voyatzis & Meletlidou, 2015). 3. The Stability of The Lagrange points of Equilibrium and their Neighborhoods A. Stability of the Lagrange points The stability of the Lagrange points will be discussed. As shown in Section 2C, the type of stability of any point can be calculated by constructing a monodromy matrix D by 28

29 integrating the variational equations (50) as indicated by the Floquet theory. The stability then depends upon the corresponding eigenvalues of the monodromy matrix D, λ 1, λ 2, λ 3, λ 4. Of particular interest are, of course, the Lagrange points of equilibrium and for which combinations of the mass ratio μ and the eccentricity e they present linear stability. It is well known that the collinear Lagrange points L 1, L 2, L 3 found in the circular three body problem, show instability for any combination of the values of μ and e (Szebehely, 1967). This fact persists in the elliptic problem and as a result, only the triangular Lagrange points L 4, L 5 were tested for linear stability. The four eigenvalues, which can be real or complex numbers, of the 4x4 monodromy matrix D are found not to be independent of one another. It is proved that the eigenvalues form reciprocal pairs. If e.g. λ 1 R then λ 2 = 1/λ 1 is also an eigenvalue. Also if λ 1 = a + ib C then λ 2 = a ib is also an eigenvalue. Depending on the place of the four eigenvalues on the unit circle, there exist seven stability regions. The six are unstable regions and one is stable. The stable region corresponds to the eigenvalues that are on the unit circle. The seven stability regions and the properties of their eigenvalues are described in Table 1. 29

30 Region Table 1: Properties of the seven stability regions Properties of eigenvalues Remarks Properties of Orbits λ, 1, μ, 1 λ μ 1 λ = 1/λ, μ = 1/μ (λ and μ complex with λ = μ = 1) 2 λ = μ, μ = λ (λ and μ complex) 3 λ real, μ real λμ < 0 4 λ real >0 μ real >0 5 λ real <0 μ real <0 6 λ real >0 μ complex μ = 1/μ 7 λ complex λ = 1/λ μ real <0 (Broucke, 1969). All four on unit circle Not on unit circle Two positive and two negative Four real positive Four real negative Two real positive and two complex on unit circle Two real negative and two complex on unit circle Stability Complex instability Even-odd instability Even-even instability Odd-odd instability Even-semiinstability Odd-semiinstability 30

31 Figure 2: Eigenvalue configuration for stability regions μ Region 1 y 1.0 λ 1.5 Region 2 y λ /μ x x /λ 1.0 1/μ 1.0 1/λ 1.5 μ Region 3 y y Region /μ μ 1/λ λ x 1/μ 1/λ λ μ x Region y Region 6 y 1.0 μ /μ 1/λ λ μ x λ 1/λ x /μ 31

32 Region 7 y 1.0 λ 0.5 1/μ μ x /λ The results of the analysis concerning the linearized stability of the triangular Lagrange points L 4, L 5 for a wide range of values of the mass ratio μ and the eccentricity e, are presented in Figure 2 Figure 3: Linear stability of the triangular Lagrange points in the elliptic restricted three-body problem The shaded areas in figure 2 represent linear stability. The point denoted by μ on the μ axis corresponds to the value of the mass ratio at which any nonzero eccentricity introduces 32

33 linear instability (Szebehely, 1967, p. 599). The point denoted by μ 0 represents the value at which instability presents itself when e = 0. The numerical calculation of the points (μ, e) that present linear stability, also provide the values of the points μ and μ 0. These points are found to be μ = , μ 0 = As figure 3 shows, the eccentricity of the motion of the two primaries may introduce stability for μ > μ 0. In fact, the orbits around the triangular Lagrange points are stable up to μ = with the proper value of e (Szebehely, 1967). B. Regular orbits in the neighborhood of the Lagrange points In this section, the results of the calculations of the regular orbits around the Lagrange points of equilibrium are presented. These results consist of a number of regions of initial conditions around the equilibrium points for which the satellite follows a regular orbit. The calculations were made for different combinations of the mass ratio μ and the eccentricity e. The range of the regular region is also calculated as the angle θ of the region measured from the axes point of origin. Figure 4 shows exactly how the range of the regular region θ is calculated. 33

34 Figure 4: Range of the regular region θ. The shaded area is the regular region. As mentioned in Section 2D, the fast Lyapunov indicator or FLI was used to distinguish between regular and chaotic orbits. The initial conditions of the satellite that have been used are x 0 = 0, y 0 = 0, which means that the satellite starts motionless, and for x 0, y 0 a grid has been considered with its center at the coordinates of the Lagrange point L 4 (x L4, y L4 )or L 5 (x L5, y L5 ). Thus, the initial conditions x 0 and y 0 are between x Li 0. 9 < x 0 < x Li and y Li 0. 4 < y 0 < y Li It is also important to note that all the calculations were made whit the two primaries at periapsis (minimum elongation). The following figure shows the FLI as a function of the independent variable v for a regular and a chaotic orbit. 34

35 Figure 5: The change in FLI with respect to the true anomaly v for a regular and a chaotic orbit. The orbits have been calculated for μ=0.001 and e=0.05 for 325 orbital periods (v max = 650π). Initial conditions of the regular orbit are x 0 = , y 0 = and for the chaotic orbit x 0 = 0. 5, y 0 = The coordinates of L 4 are (x L4 = , y L4 = ). From figure 5 it can clearly be seen that the value of FLI for a regular orbit stays relatively low whereas it quickly increases for a chaotic orbit. The following two figures show the corresponding orbits of the satellite which are taken for a smaller interval of the independent variable v. 35

36 Figure 6: Regular orbit for μ=0.001 and e=0.05 for 15 orbital periods (v max = 30π). Initial conditions of the satellite are x 0 = , y 0 = The coordinates of L 4 are (x L4 = , y L4 = ). Figure 7: Chaotic orbit for μ=0.001 and e=0.05 for 15 orbital periods (v max = 30π). Initial conditions of the satellite are x , y 0 = The coordinates of L 4 are (x L4 = , y L4 = ). Even though the initial conditions are closer to L 4 in comparison to the regular orbit, a chaotic orbit is produced. 36

37 However, apart from the FLI, as an added measure of accuracy, the actual distance between the satellite and the Lagrange point has also been taken into account. This is calculated by d L = (x x L ) 2 + (y y L ) 2 (57) where x L and y L denote the coordinates of the Lagrange point whose neighborhood is tested. If this distance increases beyond a certain threshold after a certain interval of the independent variable v, then the orbit clearly veers into infinity and thus the initial conditions which produced it are excluded. In addition, orbits which lead to collisions of the satellite with the two primaries are also excluded. The distance between the satellite and the primaries is calculated by d pr1 = (x x 1 ) 2 + (y y 1 ) 2 d pr2 = (x x 2 ) 2 + (y y 2 ) 2 (58) If these distances become zero, a collision has occurred and the corresponding initial conditions are excluded. With all this information the so called stability maps have been constructed around the Lagrange points. These maps consist of the initial condition x 0, y 0 of the satellite, for which it follows a regular orbit. In each such pair of initial conditions 37

38 a color has been added according to the value of FLI that has been calculated for this particular orbit. For the chaotic orbits and the collision orbits a special arbitrary value of 100 is assigned to the FLI for the purposes of visualization. The color ranges from black to yellow with yellow being the region that does not produce regular orbits. It was found that the boundary of the value of the FLI that separates the regular orbits region from the chaotic orbits region is FLI=6.0. The implication here is that initial conditions that produce regular orbits with FLI value close to 6.0 (represented by orange color), could be excluded from the regular region if the integrations were to be made for larger intervals of the independent variable v. For this reason the range of the regular regions have been calculated using initial conditions which produce orbits with a value of FLI close to 1.50 (represented by blue color). The results that follow were obtained for a maximum value of v equal to 650π or 325 orbital periods of the primaries. The interval of μ tested is < μ < 0.04 with step equal to starting from The values of e that have been tested start at 0.0 with step equal to 0.05 and end whenever the range of the regular regions θ reach 0. 38

39 i. Regular regions near equilibrium point L 4 μ=0.001, e=0.0, θ=1.45 rad μ=0.001, e=0.05, θ=1.30 rad 39

40 μ=0.001, e=0.1, θ=1.07rad μ=0.001, e=0.15, θ=0.96 rad 40

41 μ=0.001, e=0.2, θ=0.86 rad μ=0.001, e=0.25, θ=0.80 rad 41

42 μ=0.001, e=0.3, θ=0.74 rad μ=0.001, e=0.35, θ=0.65 rad 42

43 μ=0.001, e=0.4, θ=0.61 rad μ=0.001, e=0.45, θ=0.41 rad 43

44 μ=0.001, e=0.5, θ=0.38 rad μ=0.001, e=0.55, θ=0.18 rad 44

45 μ=0.001, e=0.6, θ=0.12 rad μ=0.001, e=0.65, θ=0.08 rad After e=0.65 there are no initial conditions that produce regular orbits. 45

46 μ=0.0025, e=0.0, θ=1.43 rad μ=0.0025, e=0.05, θ=1.29 rad 46

47 μ=0.0025, e=0.1, θ=1.04 rad μ=0.0025, e=0.15, θ=0.99 rad 47

48 μ=0.0025, e=0.2, θ=0.75 rad μ=0.0025, e=0.25, θ=0.72 rad 48

49 μ=0.0025, e=0.3, θ=0.58 rad μ=0.0025, e=0.35, θ=0.41 rad 49

50 μ=0.0025, e=0.4, θ=0.38 rad μ=0.0025, e=0.45, θ=0.21 rad 50

51 μ=0.0025, e=0.5, θ=0.20 rad μ=0.0025, e=0.55, θ=0.04 rad After e=0.55 there are no initial conditions that produce regular orbits. 51

52 μ=0.005, e=0.0, θ=1.26 rad μ=0.005, e=0.05, θ=1.15 rad 52

53 μ=0.005, e=0.1, θ=0.96 rad μ=0.005, e=0.15, θ=0.50 rad 53

54 μ=0.005, e=0.2, θ=0.41 rad μ=0.005, e=0.25, θ=0.46 rad 54

55 μ=0.005, e=0.3, θ=0.32 rad μ=0.005, e=0.35, θ=0.18 rad 55

56 μ=0.005, e=0.4, θ=0.17 rad μ=0.005, e=0.45, θ=0.08 rad 56

57 μ=0.005, e=0.5, θ=0.0 rad After e=0.5 there are no initial conditions that produce regular orbits. μ=0.0075, e=0.0, θ=1.24 rad 57

58 μ=0.0075, e=0.05, θ=0.99 rad μ=0.0075, e=0.1, θ=0.56 rad 58

59 μ=0.0075, e=0.15, θ=0.28 rad μ=0.0075, e=0.2, θ=0.19 rad 59

60 μ=0.0075, e=0.25, θ=0.27 rad μ=0.0075, e=0.3, θ=0.29 rad 60

61 μ=0.0075, e=0.35, θ=0.03 rad After e=0.35 there are no initial conditions that produce regular orbits. μ=0.01, e=0.0, θ=0.88 rad 61

62 μ=0.01, e=0.05, θ=0.77 rad μ=0.01, e=0.1, θ=0.62 rad 62

63 μ=0.01, e=0.15, θ=0.53 rad μ=0.01, e=0.2, θ=0.25 rad 63

64 μ=0.01, e=0.25, θ=0.06 rad μ=0.01, e=0.3, θ=0.06 rad 64

65 μ=0.01, e=0.35, θ=0.14 rad After e=0.35 there are no initial conditions that produce regular orbits. μ=0.0125, e=0.0, θ=0.65 rad 65

66 μ=0.0125, e=0.05, θ=0.31 rad μ=0.0125, e=0.05, θ=0.16 rad 66

67 μ=0.0125, e=0.1, θ=0.16 rad μ=0.0125, e=0.15, θ=0.13 rad 67

68 μ=0.0125, e=0.2, θ=0.04 rad μ=0.0125, e=0.25, θ=0.09 rad 68

69 μ=0.0125, e=0.3, θ=0.09 rad After e=0.3 there are no initial conditions that produce regular orbits. μ=0.015, e=0.0, θ=0.53 rad 69

70 μ=0.015, e=0.05, θ=0.15 rad μ=0.015, e=0.1, θ=0.18 rad 70

71 μ=0.015, e=0.15, θ=0.18 rad μ=0.015, e=0.2, θ=0.22 rad 71

72 μ=0.015, e=0.25, θ=0.06 rad After e=0.25 there are no initial conditions that produce regular orbits. μ=0.0175, e=0.0, θ=0.74 rad 72

73 μ=0.0175, e=0.05, θ=0.69 rad μ=0.0175, e=0.1, θ=0.48 rad 73

74 μ=0.0175, e=0.15, θ=0.31 rad μ=0.0175, e=0.2, θ= 0.04 rad After e=0.2 there are no initial conditions that produce regular orbits. 74

75 μ=0.02, e=0.0, θ=0.64 rad μ=0.02, e=0.05, θ=0.60 rad 75

76 μ=0.02, e=0.1, θ=0.19 rad After e=0.1 there are no initial conditions that produce regular orbits. μ=0.0225, e=0.0, θ=0.11 rad 76

77 μ=0.0225, e=0.05, θ=0.07 rad μ=0.0225, e=0.1, θ=0.03 rad After e=0.1 there are no initial conditions that produce regular orbits. 77

78 μ=0.025, e=0.0, θ=0.03 rad μ=0.025, e=0.05, θ=0.07 rad After e=0.05 there are no initial conditions that produce regular orbits. 78

79 μ=0.0275, e=0.0, θ=0.12 rad After e=0.05 there are no initial conditions that produce regular orbits. μ=0.03, e=0.0, θ=0.19 rad After e=0.0 there are no initial conditions that produce regular orbits. 79

80 μ=0.0325, e=0.0, θ=0.20 rad μ=0.0325, e=0.05, θ=0.03 rad After e=0.05 there are no initial conditions that produce regular orbits. 80

81 μ=0.035, e=0.0, θ=0.17 rad μ=0.035, e=0.05, θ=0.08 rad 81

82 μ=0.035, e=0.1, θ=0.01 rad After e=0.1 there are no initial conditions that produce regular orbits. μ=0.0375, e=0.0, θ=0.14 rad 82

83 μ=0.0375, e=0.05, θ=0.03 rad μ=0.0375, e=0.1, θ=0.01 rad After e=0.0 there are no initial conditions that produce regular orbits. 83

84 μ=0.04, e=0.0, θ=0.06 rad After μ=0.05, e=0.0 there are no initial conditions that produce regular orbits. ii. Regular regions near equilibrium point L 5 The regular regions near equilibrium point L 5 are almost completely symmetrical to the ones near L 4 the reason being that actually L 5 becomes L 4 as the initial conditions become y y and the independent variable becomes v v. As a result and for the sake of time, less values of the mass ratio μ have been considered here. The results of the calculations are as follows: 84

85 μ=0.005, e=0.0, θ=1.26 rad μ=0.005, e=0.05, θ=1.09 rad 85

86 μ=0.005, e=0.1, θ=1.04 rad μ=0.005, e=0.15, θ=0.54 rad 86

87 μ=0.005, e=0.2, θ=0.46 rad μ=0.005, e=0.25, θ=0.53 rad 87

88 μ=0.005, e=0.3, θ=0.36 rad μ=0.005, e=0.35, θ=0.21 rad 88

89 μ=0.005, e=0.4, θ=0.19 rad μ=0.005, e=0.45, θ=0.10 rad 89

90 μ=0.005, e=0.5, θ=0.0 rad After e=0.5 there are no initial conditions that produce regular orbits. μ=0.01, e=0.0, θ=0.88 rad 90

91 μ=0.01, e=0.05, θ=0.73 rad μ=0.01, e=0.1, θ=0.65 rad 91

92 μ=0.01, e=0.15, θ=0.54 rad μ=0.01, e=0.2, θ=0.28 rad 92

93 μ=0.01, e=0.25, θ=0.06 rad μ=0.01, e=0.3, θ=0.06 rad 93

94 μ=0.01, e=0.35, θ=0.14 rad After e=0.35 there are no initial conditions that produce regular orbits. μ=0.02, e=0.0, θ=0.64 rad 94

95 μ=0.02, e=0.05, θ=0.59 rad μ=0.02, e=0.1, θ=0.19 rad After e=0.1 there are no initial conditions that produce regular orbits. 95

96 μ=0.03, e=0.0, θ=0.19 rad After e=0.0 there are no initial conditions that produce regular orbits. μ=0.04, e=0.0, θ=0.05 rad After e=0.0 there are no initial conditions that produce regular orbits. 96

97 C. Examination of the results This examination of the results focuses on the regular regions near the Lagrange point L 4. The apparent symmetry between the regular regions near L 4 and L 5 means that any conclusions that are drawn for one region can be applied to the other. By examining the range of the regular regions near L 4, it becomes clear that the range θ decreases in value as the mass ratio μ and the eccentricity of the orbits increases as shown in figures 3-5. In fact, the rate of its decrease is rather fast as the regular regions are virtually non-existent after μ=0.03 (save for very low values of e) and e=0.65 in the most extreme cases where μ is closer to zero. This suggests that the chances of the existence of co-orbital companions, such as Trojan-like bodies, increase in dynamical systems with low to moderate orbital eccentricities (< 0.3) (Schwarz, Dvorak, Süli & Érdi, 2007) where the regular area is larger. Figure 8 shows the change in the regular range θ with the mass ratio μ for e = 0.0 and e = 0.1. Since the majority of extrasolar planetary systems that have been discovered resemble a dynamical system not unlike the Sun and Jupiter system (1M = M Jupiter, μ = ), the most interesting range of the mass ratio μ, that warrants further examination, is μ Figure 9 shows the change in the regular range θ with the orbital eccentricity e for μ = and μ =

98 Figure 8: The change in the range of the regular region θ with the mass ratio μ for e=0.0 Figure 9: The change in the range of the regular region θ with the eccentricity e for μ=0.001 and μ=0.005 Figure 10 shows the change of the range θ with both μ and e. While it is clear that θ decreases as e increases, there is a noticeable increase of θ around μ=0.018 as indicated by the bump. 98

99 Figure 10: The change in the range of the regular region θ with the mass ratio μ and the eccentricity e D. Stability Regions of Exoplanetary systems The results of Sections 3B and 3C can be applied to existing exoplanetary systems in an attempt to suggest possible candidate systems in which co-orbital companions could exist in stable orbits near the Lagrange point L 4. By taking into account that the range of the regular region θ decreases as the mass ratio μ and the orbital eccentricity e increase in value, a list of 15 exoplanetary systems has been examined to determine their regular regions. The criteria which the possible candidates must fulfill are a low to moderate eccentricity e (< 0.3), and a mass ratio μ with values between and

100 This would resemble a system in which the planet s mass would be comparable to that of Jupiter, and the host star s mass would be comparable to that of the Sun. This is because, as can be seen in figure 11, the majority of the known exoplanetary systems fulfill these criteria. Figure 11: Distribution of known exoplanetary systems with regards to the mass of the host star (in M ) and the mass of the planet (in M Jupiter ). The color range shows their orbital eccentricity. The distribution is denser around 1M and 1M Jupiter. Table 2 gives the list of the candidate exoplanetary systems for which the range of the regular region θ has been calculated, along with their characteristics. 100

101 Table 2 System M S (M ) M P (M Jup ) μ e θ (rad) HR HD HD WASP HD HD Kepler HD WASP HIP tau Boo Kepler KOI WASP HIP

102 4. Conclusions In this work, the nature of stability of the Lagrange points L 4 and L 5 and of their neighborhoods for different values of the mass ratio μ and the orbital eccentricity e, using the Elliptic Restricted Three Body Problem as a theoretical basis has been studied. The range of the regions near the equilibrium points L 4 and L 5, where a motionless satellite follows regular orbits, has been determined using the Fast Lyapunov Indicators (FLI). The extend of these regions depends on the mass parameter μ and the eccentricity e. The range of the regular regions θ decreases as both the mass ration μ and the eccentricity e increase. It is possible that co-orbital companions such as Trojan-like bodies of small mass exist in the regular regions near L 4 and L 5 of extrasolar planetary systems that consists of a Sun-like star and one Jupiter-like planet. Fifteen exoplanetary systems have been examined and their regular region have been determined in order to propose possible candidates for investigation of the existence of possible coorbital companions in these systems. Further works are necessary to better calculate the range of the regular regions near the equilibrium points considering larger intervals of the independent variable. In addition, the list of exoplanetary systems that could be host to Trojan-like bodies can be greatly expanded, since there are more than 2700 known exoplanetary systems, with more discovered yearly. 102

103 APPENDIX The programs that have been prepared for the this study are presented here. The programming languages C and C++ have been used. Program A: //INTEGRATION OF THE ELLIPCTIC RESTRICTED THREE BODY PROBLEM IN THE BARYCENTRIC INERTIAL FRAME OF REFERENCE #include <iostream> #include <cstdio> #include <cstdlib> #include <cmath> using namespace std; //ODE for eccentric anomaly E double dydx(double x, double y, double e){ double dy; dy=1.0/(1.0-(e*cos(y))); return dy; //ODEs for satellite coordinates in the barycentric inertial frame double dqdt(double t, double q, double p, double vq, double vp, double m, double q1, double q2, double p1, double p2){ double dq; dq=vq; return dq; double dvqdt(double t, double q, double p, double vq, double vp, double m, double q1, double q2, double p1, double p2){ double dvq, s1, s2; s1=sqrt(pow((q-q1), 2.0) + pow((p-p1), 2.0)); s2=sqrt(pow((q-q2), 2.0) + pow((p-p2), 2.0)); dvq=-(1-m)*((q-q1)/(pow(s1, 3.0))) - m*((q-q2)/(pow(s2, 3.0))); return dvq; double dpdt(double t, double q, double p, double vq, double vp, double m, double q1, double q2, double p1, double p2){ double dp; dp=vp; 103

104 return dp; double dvpdt(double t, double q, double p, double vq, double vp, double m, double q1, double q2, double p1, double p2){ double dvp, s1, s2; s1=sqrt(pow((q-q1), 2.0) + pow((p-p1), 2.0)); s2=sqrt(pow((q-q2), 2.0) + pow((p-p2), 2.0)); dvp=-(1-m)*((p-p1)/(pow(s1, 3.0))) - m*((p-p2)/(pow(s2, 3.0))); return dvp; //function f(a) for Newton-Raphson double func(double m, double a){ double fa; fa=(1-a)/(pow(m, 2)) - a/(pow(1-m, 2)) + (m*a)/pow(abs(a), 3) + ((1-m)*(a-1))/pow(abs(a-1), 3); return fa; //function f'(a) for Newton-Raphson (root of a>1) double funcdot(double m, double a){ double fdota; fdota=-(1/pow(1-m, 2)) - (1/pow(m, 2)) + ((1-m)/pow(abs(a-1), 3)) + (m/pow(abs(a), 3)) - ((3*(a-1)*(1-m))/pow(abs(a-1), 4)) - ((3*a*m)/pow(abs(a), 4)); return fdota; //function f'(a) for Newton-Raphson (roots of a<1) double funcdot2(double m, double a){ double fdota2; fdota2=-(1/pow(1-m, 2)) - (1/pow(m, 2)) + ((1-m)/pow(abs(a-1), 3)) + (m/pow(abs(a), 3)) + ((3*(a-1)*(1-m))/pow(abs(a-1), 4)) + ((3*a*m)/pow(abs(a), 4)); return fdota2; int main(int argc, char *argv[]) { double tn,en,k1,k2,k3,k4,e,t,h,t0,e0,e; double m, q1, q2, p1, p2, xi0; double ql1, pl1, ql2, pl2, ql3, pl3; double q4, p4; double q, p, vq, vp, q0, p0, vq0, vp0; double qn, pn, vqn, vpn; double limit; double a1, a2, a3, a4, b1, b2, b3, b4, c1, c2, c3, c4, d1, d2, d3, d4; double a01, a02, a03, an1, an2, an3;//for Newton-Raphson FILE * fp1; 104

105 FILE * fp2; FILE * fp3; FILE * fp4; FILE * fp5; FILE * fp6; FILE * fp7; FILE * fp8; FILE * fp9; /*Initial conditions and step h*/ t0=0.0; // initial time E0=0.0; //initial E //Initial Coordinates q0=-2.0; p0=0.0; //Initial velocity components vq0=-0.0; vp0=-0.6; m=0.2; //mass ratio e=0.9; //eccentricity h=0.005; //RK4 step limit=t0+120; //time limit a01=1.1; //Newton-Raphson starting values a02=-0.01; a03=0.5; tn=t0; En=E0; qn=q0; pn=p0; vqn=vq0; vpn=vp0; /*Newton-Raphson*/ an1=a01; if( func(m, a01)!=0 && funcdot(m, a01)!=0 ){ do{ a01=an1; an1=a01 - ((func(m, a01)/funcdot(m, a01))); while(fabs(an1 - a01) >= ); an2=a02; if( func(m, a02)!=0 && funcdot2(m, a02)!=0 ){ do{ 105

106 a02=an2; an2=a02 - ((func(m, a02)/funcdot2(m, a02))); while(fabs(an2 - a02) >= ); an3=a03; if( func(m, a03)!=0 && funcdot2(m, a03)!=0 ){ do{ a03=an3; an3=a03 - ((func(m, a03)/funcdot2(m, a03))); while(fabs(an3 - a03) >= ); cout << "a1= " << an1 << endl; cout << "a2= " << an2 << endl; cout << "a3= " << an3 << endl; //Coordinates of the Primaries q1=-m*(cos(en)-e); p1=-m*sqrt(1-pow(e,2))*sin(en); q2=(1-m)*(cos(en)-e); p2=(1-m)*sqrt(1-pow(e,2))*sin(en); //Quantity q0 xi0=(1/2.0)-m; //Coordinates of L4/L5 lagrange Point q4=xi0*(cos(en)-e) - (sqrt(3.0)/2.0)*sqrt(1-pow(e,2))*sin(en); p4=xi0*sqrt(1-pow(e,2))*sin(en) + (sqrt(3.0)/2.0)*(cos(en)-e); //Coordinates of L1 (a>1) Lagrange Point ql1=(an1 - m)*(cos(en) - e); pl1=(an1 - m)*(sqrt(1-pow(e,2))*sin(en)); //Coordinates of L2 (a<1) Lagrange Point ql2=(an2 - m)*(cos(en) - e); pl2=(an2 - m)*(sqrt(1-pow(e,2))*sin(en)); //Coordinates of L3 (0<a<1) Lagrange Point ql3=(an3 - m)*(cos(en) - e); pl3=(an3 - m)*(sqrt(1-pow(e,2))*sin(en)); fp1=fopen("eccentricanomaly.txt", "w+"); fp2=fopen("primary1coordinates.txt", "w+"); fp3=fopen("primary2coordinates.txt", "w+"); fp4=fopen("satellitecoordinates.txt", "w+"); fp5=fopen("l4coordinates.txt", "w+"); fp6=fopen("l5coordinates.txt", "w+"); fp7=fopen("l1coordinates.txt", "w+"); fp8=fopen("l2coordinates.txt", "w+"); fp9=fopen("l3coordinates.txt", "w+"); 106