Copyright. Jean-Philippe Munoz

Transcription

1 Copyright by Jean-Philippe Munoz 2008

2 The Dissertation Committee for Jean-Philippe Munoz certifies that this is the approved version of the following dissertation: Sun-Perturbed Dynamics of a Particle in the Vicinity of the Earth-Moon Triangular Libration Points Committee: Bob E. Schutz, Supervisor David G. Hull Wallace T. Fowler Cesar A. Ocampo Daniel J. Scheeres

3 Sun-Perturbed Dynamics of a Particle in the Vicinity of the Earth-Moon Triangular Libration Points by Jean-Philippe Munoz, B.S, M.S.E. Dissertation Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy The University of Texas at Austin August 2008

4 A mes Parents, A mes Grands-Parents, A ma Marraine.

5 Acknowledgments First and foremost, I would like to thank my parents, for their love and support, so far from home. I also would like to thank Dr. Bob Schutz for his guidance throughout this research, as well as Dr. Cesar Ocampo, for his many contributions to this work, most notably with the Copernicus software. To my friends, Scott Zimmer, Ravi Mathur, Laurent Froideval and Nadege Pie, I would like to express my gratitude, not only for their help and support but also for making feel at home, here in Austin. Finally, last but most certainly not least, I want to thank my girlfriend Lindsay Taylor, for her unconditional love and for putting up with me through the redaction of this dissertation. Jean-Philippe Munoz The University of Texas at Austin August 2008 v

6 Sun-Perturbed Dynamics of a Particle in the Vicinity of the Earth-Moon Triangular Libration Points Publication No. Jean-Philippe Munoz, Ph.D. The University of Texas at Austin, 2008 Supervisor: Bob E. Schutz This study focuses on the Sun s influence on the motion near the triangular libration points of the Earth-Moon system. It is known that there exists a very strong resonant perturbation near those points that produces large deviations from the libration points, with an amplitude of about 250,000 km and a period of 1,500 days. However, it has been shown that it is possible to find initial conditions that negate the effects of that perturbation, even resulting in stable, although very large, periodic orbits. Using two different models, the goal of this research is to determine the initial configurations of the Earth-Moon-Sun system that produce minimal deviations from the libration points, and to provide a better understanding of the vi

7 dynamics of this highly nonlinear problem. First, the Bicircular Problem (BCP) is considered, which is an idealized model of the Earth-Moon-Sun System. The impact of the initial configuration of the Earth-Moon-Sun system is studied for various propagation times and it is found that there exist two initial configurations that produce minimal deviations from L 4 or L 5. The resulting trajectories are very sensitive to the initial configuration, as the mean deviation from the libration points can decrease by 30,000 km with less than a degree change in the initial configuration. Two critical initial configurations of the system were identified that could allow a particle to remain within 30,000 km of the libration points for as long as desired. A more realistic model, based on JPL ephemerides, is also used, and the influence of the initial epoch on the motion near the triangular points is studied. Through the year 2007, 51 epochs are found that produce apparently stable librational motion near L 4, and 60 near L 5. But the motion observed depends greatly on the initial epoch. Some epochs are even found to significantly reduce the deviation from L 4 and L 5, with the spacecraft remaining within at most 90,000 km from the triangular points for upwards of 3,000 days. Similarly to what was observed in the BCP, these trajectories are found to be extremely sensitive to the initial epoch. vii

8 Contents Acknowledgments v Abstract vi List of Tables xi List of Figures xii Chapter 1 Introduction The Circular Restricted Three-Body Problem The Equations of Motion The Libration Points Motivation and Problem Overview Bibliographical Survey The Bicircular Problem Zero-Velocity Curves Equilibrium Points Behavior about L Periodic Orbits about L viii

9 Chapter 2 Simulations in the Bicircular Problem Goal and Method Linearizing the Equations of Motion Power Spectra Study of the Vicinity of L Emergence of a Resonance Phenomenon The 3-1 Resonance Influence of the Initial Configuration of the Earth-Moon-Sun System Tentative Explanations Prolonging Bounded Motion Sensitivity to the Initial Conditions Conclusions Chapter 3 Numerical Study of the Real-World Model Objective and Overview Model and Output Frame Copernicus Numerical Integration Influence of the Initial Epoch Apparently Stable Motion Detailed Examples Minimizing the Deviation from L 4 and L Sensitivity to the Initial Epoch Configurations of the Favorable Initial Epochs Further Refinement and Numerical Integration Settings ix

10 3.6 Conclusions Chapter 4 Conclusions and Recommendations Summary of the Results Discussion and Future Work Bibliography 152 Vita 156 x

11 List of Tables 2.1 Coefficients of de Vries Equations of Motion Epochs of 2007 Producing Apparently Stable Motion Epochs of 2007 Producing Small Deviations from the Libration Points Configuration of the Favorable Epochs of 2007 in the L 4 case Configuration of the Favorable Epochs of 2007 in the L 5 case xi

12 List of Figures 1.1 A representation of the CRTBP Location of the libration points in the Earth-Moon system A representation of the Bicircular Problem Zero-velocity curve in the BCP for H = Zero-velocity curve in the BCP for H = Zero-velocity curve in the BCP for H = Zero-velocity curve in the BCP for H = Zero-velocity curve in the BCP for H = Hamiltonian of a particle at rest at L 4 versus θ Location of the equilibrium points in the BCP for varying positions of the Sun Trajectory of a spacecraft initially at equilibrium point Trajectory of a spacecraft starting at L Distance from the spacecraft to L 4 versus time First stable periodic orbit about L Second stable periodic orbit about L Trajectory obtained by propagating first periodic orbit for 500 days. 28 xii

13 2.1 Trajectory of a spacecraft initially at rest at L4 using the linearized equations Trajectory of a spacecraft initially at rest at L4 using the BCP Equations Distance from the spacecraft to L 4 using the BCP and linearized equations Trajectory of a spacecraft initially at rest at L Distance to L 4 in kilometers for a spacecraft initially at rest at L Power spectrum for a spacecraft initially at rest at L Trajectory of a spacecraft initially at rest at x L km Distance to L 4 in kilometers for a spacecraft initially at rest, x L km Power spectrum for a spacecraft initially at rest at x L km Trajectory of a spacecraft initially at rest at x L km Distance to L 4 in kilometers for a spacecraft initially at rest, x L km Power spectrum for a spacecraft initially at rest at x L km Trajectory of a spacecraft initially at rest at x L km Distance to L 4 in kilometers for a spacecraft initially at rest, x L km Power spectrum for a spacecraft initially at rest at x L km Trajectory of a spacecraft initially at rest at y L km Distance to L 4 in kilometers for a spacecraft initially at rest, y L km Power spectrum for a spacecraft initially at rest at y L km. 50 xiii

14 2.19 Trajectory of a spacecraft initially at L 4, ẋ 0 = Normalized Units Distance to L 4 in kilometers for a spacecraft initially at L 4, ẋ 0 = Normalized Units Power spectrum for a spacecraft initially L 4, ẋ 0 = Normalized Units Trajectory of a spacecraft initially at L 4, ẋ 0 = Normalized Units Distance to L 4 in kilometers for a spacecraft initially at L 4, ẋ 0 = Normalized Units Power spectrum for a spacecraft initially L 4, ẋ 0 = Normalized Units Trajectory of a spacecraft initially at L 4, ẋ 0 = Normalized Units Distance to L 4 in kilometers for a spacecraft initially at L 4, ẋ 0 = Normalized Units Power spectrum for a spacecraft initially L 4, ẋ 0 = Normalized Units Trajectory with no expansion and contraction propagated for 870 days Distance to L 4 in kilometers for a trajectory with no expansion and contraction Power spectrum for a trajectory with no expansion and contraction Trajectory with no expansion and contraction propagated for 2,000 days xiv

15 2.32 Distance to L 4 in kilometers for a trajectory with no expansion and contraction Power spectrum for a trajectory with no expansion and contraction Mean distance to L 4 versus the initial value of θ Mean Distance to L 5 versus the initial value of θ Mean distance to L 4 versus the initial value of θ near the first minimum Mean distance to L 4 versus the initial value of θ near the second minimum Mean distance to L 4 versus the initial value of θ near 129 degrees Trajectory of a spacecraft initially at L 4 with θ = degrees Distance to L 4 versus time for θ = degrees Trajectory of a spacecraft initially at L 4 with θ = degrees Distance to L 4 versus time for θ = degrees Number of days required to reach 30,000 km from L 4 versus the initial value of θ Trajectory of a spacecraft initially at L 4 with θ = degrees Distance to L 4 versus time for θ = degrees Mean distance to L 4 versus the initial value of θ for 520 Time Units Number of days required to reach 30,000 km from L 4 versus the initial value of θ Trajectory of a spacecraft initially at L 4 with θ = degrees Trajectory of a spacecraft relative to L 4 for θ = degrees Trajectory of a spacecraft relative to L 4 for θ = degrees Distance to L 4 in kilometers for Figure Distance to L 4 in kilometers for Figure xv

16 2.53 Power spectrum of the distance to L 4 for θ = degrees Power spectrum of the distance to L 4 for θ = degrees Three-body acceleration in Normalized Units for a spacecraft initially at rest at L 4 versus the initial value of θ Acceleration due to the Sun in Normalized Units for a spacecraft initially at rest at L 4 versus the initial value of θ Mean acceleration in Normalized Units for a spacecraft initially at rest at L 4 versus the initial value of θ Trajectory of a spacecraft initially at rest at L 4 for θ = degrees Time history of the distance to L 4 spacecraft initially at rest at L 4 for θ = degrees Mean distance to L 4 versus the initial x-coordinate Mean distance to L 4 versus the initial y-coordinate Mean distance to L 4 versus the initial value of ẋ Mean distance to L 4 versus the initial value of ẏ Difference between the RK5 and BSE integrators for the April 22 epoch Difference between the RK5 and DLSODE integrators for the April 22 epoch Difference between the RK5 and BSE integrators over 1,500 days for the January 10 07:18:38 ET epoch Difference between the RK5 and DLSODE integrators over 1,500 days for the January 10 07:18:38 ET epoch Difference between the RK5 and BSE integrators over 3,000 days for the January 10 07:18:38 ET epoch xvi

17 3.6 Difference between the RK5 and DLSODE integrators over 3,000 days for the January 10 07:18:38 ET epoch Positions of the Sun producing apparently stable motion about L Positions of the Sun producing apparently stable motion about L Trajectory in the (x, y) plane for April 22 00:00:00 ET at L Trajectory in the (x, z) plane for April 22 00:00:00 ET at L Distance to L 4 in kilometers for April 22 00:00:00 ET Distance to the initial position for April 22 00:00:00 ET propagated for 10,000 days Trajectory in the (x, y) plane for March 31 00:00:00 ET at L Distance to the L 4 in kilometers for March 31 00:00:00 ET Trajectory in the (x, y) plane for December 22 00:00:00 ET at L Distance to L 4 in kilometers for the December 22 00:00:00 ET Distance to L 4 in kilometers for the December 22 Epoch propagated for 6,200 days Trajectory in the (x, y) plane for April 13 00:00:00 ET at L Distance to L 5 in kilometers for April 13 00:00:00 ET Distance to L 5 in kilometers for the April 13 00:00:00 ET epoch propagated for 10,000 days Trajectory in the (x, y) plane for May 4 00:00:00 ET at L Distance to L 5 in kilometers for May 4 00:00:00 ET Trajectory in the (x, y) plane for May 4 00:00:00 ET at L 5 propagated for 1,500 days Distance to L 5 in kilometers for May 4 00:00:00 ET propagated for 1,500 days xvii

18 3.25 Positions of the Sun producing small deviations from L Positions of the Sun producing small deviations from L Trajectory of a spacecraft initially at rest at L 4 for the May 29 14:50:10 ET epoch Time history of the distance to L 4 for a spacecraft initially at rest at L 4 on May 29 14:50:10 ET Trajectory of a spacecraft initially at rest at L 4 on November 9 19:54:50 ET Time history of the distance to L 4 for a spacecraft initially at rest at L 4 on November 9 19:54:50 ET Trajectory of a spacecraft initially at rest at L 5 on December 12 23:37:50 ET Time history of the distance to L 5 for a spacecraft initially at rest at L 5 on December 12 23:37:50 ET Trajectory of a spacecraft initially at rest at L 5 on March 3 05:26:00 ET Time history of the distance to L 5 for a spacecraft initially at rest at L 5 on March 3 05:26:00 ET Trajectory in the (x, z)-plane of a spacecraft initially at rest at L 5 on December 12 23:37:50 ET Number of days elapsed before reaching 90,000 km from L 4 versus initial epoch for May 29 14:50:10 ET Number of days elapsed before reaching 90,000 km from L 5 versus initial epoch for December 12 23:37:50 ET xviii

19 3.38 Number of days elapsed before reaching 100,000 km from L 4 versus initial epoch for November 9 19:54:50 ET Number of days elapsed before reaching 100,000 km from L 5 versus initial epoch for the March 3 05:26:00 ET Number of days elapsed before reaching 90,000 km from L 5 versus initial epoch for August 9 01:00:00 ET Number of days elapsed before reaching 90,000 km from L 4 versus the initial value of ψ for May 29 14:50:10 ET Number of days elapsed before reaching 90,000 km from L 5 versus the initial value of ψ for December 12 23:37:50 ET Time history of the distance to L 4 for May 29 14:50:06.96 ET Time history of the distance to L 4 for August 9 23:03:00 ET xix

20 Chapter 1 Introduction 1.1 The Circular Restricted Three-Body Problem The Circular Restricted Three-Body Problem [1,6] is a particular case of the threebody problem where two gravitating bodies, called primaries (for example the Earth and the Moon) are in a circular orbit about their common center of mass, and the third body, for example a spacecraft, has a sufficiently small mass to have no appreciable influence on the motion of the primaries. In this Dissertation, the Circular Restricted Three-Body Problem will be denoted as CRTBP. The simplest way to represent the CRTBP is in a rotating frame, centered at the center of mass of the primaries, such that both primaries lie on the x-axis, and where the z-axis is given by the angular velocity vector of each the primaries with respect to their center of mass, ω. Figure 1.1 depicts this rotating frame in the Earth-Moon case. In order to simplify the equations of motion for the spacecraft, all physical quantities can be normalized. Both the distance between the Earth and the Moon and ω are set to unity. We also define µ = M Moon M Moon + M Earth (1.1) 1

21 where M Moon is the mass of the Moon and M Earth is the mass of the Earth. A new time unit (TU) can be chosen so that the period of revolution of the Moon and the Earth about their center of mass T p is T p = 2 π(t U) (1.2) With this new time unit, the value of the gravitational constant G can be set to 1. The advantage of normalizing those quantities is that the CRTBP can be fully described using the sole mass parameter µ. Figure 1.1: A representation of the CRTBP In this rotating frame, the coordinates of the Earth are ( µ, 0, 0) and the coordinates of the Moon are (1 µ, 0, 0). The position of a spacecraft will be defined by a vector r, of coordinates (x, y, z), whereas its position relative to the Earth and the Moon will be described by the vectors r 1 of length r 1 and r 2 of length r 2. 2

22 1.2 The Equations of Motion Since the units have been normalized, the sum of the gravitational accelerations F applied to the spacecraft, due to the Earth and the Moon, and expressed in the rotating frame shown on Figure 1.1, can be written as: F = 1 µ r1 3 r 1 µ r2 3 r 2 (1.3) This yields the expression of r, the acceleration of the spacecraft in the rotating frame shown on Figure 1.1 [4,6]: r = F 2ω v ω (ω r) (1.4) Where v is the velocity of the spacecraft relative to the rotating frame. Since ω = 1, the right-hand side of Equation 1.4 is computed by substituting Equations 1.5 and 1.6: 2ẏ 2ω v = 2ẋ 0 x ω (ω r) = y 0 (1.5) (1.6) And the equations of motion for the spacecraft in the rotating frame are [4]: (1 µ)(x + µ) ẍ = ÿ = r 3 1 (1 µ)y r 3 1 yµ r 3 2 µ(x 1 + µ) r x + 2ẏ (1.7) + y 2ẋ (1.8) 3

23 (1 µ)z z = r1 3 zµ r2 3 (1.9) 1.3 The Libration Points The libration points, also called Lagrangian points, are five equilibrium points of the CRTBP, that were initially discovered by Lagrange in 1772 [1,4]. These points are such that their acceleration and velocity is zero. To determine the location of the libration points, consider the rotational potential Ω: Ω = 1 2 (x2 + y 2 ) + 1 µ r 1 + µ r 2 (1.10) From the equations of motion of the CRTBP, it can be verified that: ẍ 2ẏ = Ω x ÿ + 2ẋ = Ω y z = Ω z (1.11) (1.12) (1.13) Where: Ω x Ω y Ω z = x (1 µ)(x + µ) r 3 1 = y (1 µ)y r 3 1 = (1 µ)z r 3 1 yµ r 3 2 zµ r 3 2 µ(x 1 + µ) r 3 2 (1.14) (1.15) (1.16) Since at the equilibrium points, both accelerations and velocities are zero, the libration points are such that Ω x = Ω y = Ω Ω z = 0. First, it is clear that z = 0 if z = 0, therefore all five libration points are in the (x, y) plane. Solving for the x and y 4

24 coordinates shows that three points, L 1, L 2 and L 3 are on the x-axis, which is why they are known as collinear libration points. The other two points, L 4 and L 5, are the result of a particular solution, for which r 1 = r 2 = 1, and lie at the third point of an equilateral triangle whose base is the line between the Earth and the Moon, which is why they are known as triangular libration points. The location of all five Lagrangian points is shown on Figure 1.2. Figure 1.2: Location of the libration points in the Earth-Moon system The collinear points have been shown to be linearly unstable, whereas the triangular points L 4 and L 5 of the Earth-Moon system are stable, as µ is less [4,6]. Of course, once the Sun is added to the system, L 4 and L 5 are no longer equilibrium points. However, the dynamics in their vicinity has been object of interest for the past 50 years, as will be presented in the next Section. 5

25 1.4 Motivation and Problem Overview Since space exploration began in the late 1950 s, the prospect of a permanent base in space has always been considered but often dismissed as science fiction. But as we venture farther and farther into space, the appeal of an outpost outside of the Earth s gravity field is becoming stronger. In 1974, O Neil [7] suggested the use of well-known equilibrium points of the Earth-Moon system, based on the restricted problem of three bodies, called the triangular Lagrangian points, as location for a space colony. Those points are located at the apex of an equilateral triangle whose base is the Earth-Moon line. Another possible location for a permanent outpost would be the far side of Moon, in which case a major issue would be communication. This problem could be solved, at least in part, by placing a satellite at a triangular Lagrangian point, allowing for constant communication between Earth and the outpost. However, the research conducted so far into the behavior of a spacecraft in the vicinity of the Earth-Moon triangular Lagrangian under the perturbation of the Sun shows that it is highly dependent on the initial conditions and configuration of the system, and can lead to very large deviations from the Lagrangian point. The purpose of this work is therefore to perform a detailed numerical investigation of the behavior of a spacecraft in the vicinity of the triangular Lagrangian points (also denoted L 4 and L 5 ), once the solar perturbation is considered, in order to provide a better understanding of the motion. First a simplified model, the Bicircular Problem, will be considered. The influence of the initial conditions on the motion will be studied, as well as the impact the impact of the initial configuration of the Earth-Moon-Sun system, in an attempt to find trajectories that remain in the close vicinity of the triangular libration points for extended periods of time. 6

26 An ephemeris-based model will also be used to determine if, and how, the results obtained in the Bicircular Problem can be used in a more realistic model. 1.5 Bibliographical Survey The existence of the triangular Lagrangian point has been known since 1772 [1], and their existence confirmed by observation in the case of the Sun-Jupiter system for instance, through the discovery of Trojan asteroids. But L 4 and L 5 in the Earth- Moon system had been largely dismissed until 1961, when Kordylewski [8] claimed to have observed dust clouds in the vicinity of L 4 and L 5. Even though those findings could not be confirmed [9], they sparked the interest of some researchers. De Vries [10] published extensively on the subject in the 1960 s. Using the approximation of the Bicircular Problem, where the orbits of the Earth and the Moon about their common center of mass and about the Sun are assumed to be circular and coplanar, de Vries linearized the equations of motions of the spacecraft in the vicinity of L 4 in the Sun-perturbed Earth-Moon system. He was able to identify four fundamental frequencies of the motion, using a first order approximation, and twelve other frequencies using a second order approximation. Two of the four first-order frequencies are due to the Earth-Moon three-body effects and are very close to the frequencies obtained when the influence of the Sun is not considered. The other two frequencies correspond to the frequency of the solar perturbation, and twice that frequency. When the second order solution is considered, only two of the twelve frequencies seem to be of interest. The first one corresponds to a period of about 45 months, and the other is the frequency of the solar perturbation. De Vries also shows that the method converges poorly, however, in that the coefficients of the third order approximation are not significantly smaller than that of the second 7

27 order approximation. In fact the linearized motion only matches the numerically integrated trajectory for a few months. The results of this paper will be studied in more detail in the next Chapter. Wolaver [11] uses the first order solution obtained by de Vries (and therefore the approximation of the Bicircular Problem) to determine the initial conditions and the initial position of the Sun that will minimize the displacement of the spacecraft from L 4 for about 400 days. Three initial positions of the Sun were obtained and refined, and finally the best solution showed a maximum displacement of 5,349 km. According to Szebehely [4], this represents a reduction of 55 percent of the envelope of motion of the spacecraft for the first 200 days and of 30 percent over the next 275 days. Wolaver also points out that this solution does not persist in a more realistic, ephemeris-based model. Tapley and Lewallen [12] used a slightly modified model, since it includes the inclination of the Sun s orbit, and numerically integrated the equations of motion. They found that the motion of the satellite initially at L 4 is very dependent on the initial position of the Sun and that the amplitude of the motion is greater than what was predicted by the linearized equations. They also determined the equilibrium positions of the restricted four-body Earth-Moon-Sun system, and finally that the impulse necessary to maintain a spacecraft at L 4 for an entire year is about 2,360 lb-sec/slug, and that keeping the spacecraft at a point near L 4 requires a larger impulse. Tapley and Schutz [13,14,15], using the same model as Tapley and Lewallen, show that the numerical integration of the trajectory of a spacecraft initially at L 4 does not yield an escaping trajectory, but rather an expanding and contracting trajectory, of period 1500 days and amplitude 220,000 km. After 3,900 days, a 8

28 lunar encounter is observed and the spacecraft escapes the system. As Wolaver [11] had noticed that the constants of the motion (such as µ) seem to have a great influence of the behavior of the spacecraft, Tapley and Schutz also used slightly modified constants and found that a minimal change in the constants resulted in an expanding and contracting trajectory that persisted for 8,000 days, underlying the sensitivity of the dynamics about L 4. The trajectory of a spacecraft originally at L 5 of the elliptic three-body problem was finally computed using the JPL DE3 ephemeris, for an initial epoch corresponding to a solar eclipse in 1967, and the trajectory obtained displayed a similar expansion and contraction, this time with a period of 650 days, and persisted for 2,500 days. This long-term expansion and contraction seems to be a fundamental characteristic of the motion about L 4 or L5 of the Sun-perturbed Earth-Moon system. Despite the wide departures from L 4 described above, it is possible to find closed, periodic orbits about L 4, using simplified models of the Earth-Moon-Sun system. Schechter [16] used a fourth-order development of the Hamiltonian of a three-dimensional system that did not include the eccentricity of the Moon s orbit but did include the regression of the lunar nodes, and was able to draw several conclusions. First, that even small coplanar motions near L 4 will grow, proving that the triangular points are no longer stable. He also showed that the out of plane component is negligible compared to the coplanar motion, and that two periodic orbits exist, one stable, with a semi-major axis of about 97,000 km, and another unstable, and slightly larger than the stable one. Both orbits have the same period as the period of the Sun s perturbation. Independently, Kolenkiewicz and Carpenter [17,18], using a gravitationally consistent model similar to the Bicircular Problem, found not one, but two stable, 9

29 periodic orbits, out of phase by about 180 degrees, but larger than what Schechter predicted (145,000 km). They also found a small, unstable orbit very close to L 4, which was in agreement with Schechter s conclusion that small coplanar motions near L 4 would grow large. By means of Lie transformations, Kamel [3] confirmed the existence of the two stable orbits described by Kolenkiewicz and Carpenter. Kamel used a thirddegree analysis and was able to match Kolenkiewicz and Carpenter s results very closely. He also showed that the inclusion of the eccentricity of the Moon s orbit makes those orbits quasi-periodic. Later, Gomez et al [19] and Jorba [2] confirmed not only the existence of two stable, out of phase periodic orbits in the Bicircular Problem, but also the existence of the unstable smaller orbit, this time using Poincare sections. Additionally, Gomez et al found two larger, unstable periodic orbits whose period is three times that of the solar perturbation. Scheeres [31] used a restricted four-body Hill s model and was able to identify one periodic orbit with a period matching that of the solar perturbation. And while the use of Hill s model could not account for both stable orbits found by Kamel and Kolenkiewicz and Carpenter, the size and shape of the resulting orbit is in very good agreement with what was found by the other authors. When a more realistic model is considered, there remain quasi-periodic orbits in the vicinity of L 4 and L 5, as shown by Companys et al in [24]. By using a Legendre polynomial expansion, the Lagrangian of the system can be expressed so as to include not only the gravitational perturbation of the Sun and other planets, but also the solar radiation pressure and the nonsphericity of the primaries. The polynomials then found are expanded into power series and all the terms greater than adimensional units are kept (with this threshold, solar radiation pressure, 10

30 the perturbations due to other planets and the nonsphericity of the Earth and Moon are neglected). Then, a quasi-periodic can be targeted by imposing the form of the coordinates. However, if the same initial conditions are used in a model built with JPL ephemerides, the orbit starts to grow after 2 years and becomes very large after 5 years (approximately 400,000 km). To remedy this, a parallel shooting method can be used to refine the solution and the quasi-periodic orbit obtained can be potentially refined to last for as long as required. The spacecraft remains within 43,000 km of L 4 in the x-direction and 31,000 km in the y-direction (Figures 4 and 5 in [24]). Since the stable orbits found are fairly large and do not subsist outside of the very simplified model that is the Bicircular Problem, Castella and Jorba [20] considered invariant curves in the Poincare sections of the Bicircular Problem to find three families of three-dimensional tori, two of which are connected. The existence of these families is remarkable in the sense that some of the trajectories seem to persist in the real system. Jorba [2] uses the JPL DE406 ephemeris and shows that several trajectories generated using the initial conditions from the tori described in [20] persist for at least 1,000 years. For the second family, which Jorba calls VF2, he finds 8 trajectories that subsist for 1,000 years about L 4, and 85 about L 5. The number of those trajectories is much greater when considering the third family, VF3, as several thousands of trajectories are found that persist for over 1,000 years, around both L 4 and L 5. Those trajectories, however, are very large (230,000 km in the Earth-Moon plane, and up to 600,000 km out of the plane). As was suggested by Wolaver [11], the initial configuration of the system (the initial position of the Sun) is also a very significant factor. Schutz [5] used the JPL DE19 and DE96 ephemerides and propagated the trajectory of a spacecraft initially 11

31 at L 4 for 37 epochs in October 1967 and April He was able to show that the complex resonance that was observed in the Bicircular model still exists (resulting in an expansion and contraction of the trajectory), but that some epochs do not produce librational motion. The period and amplitude of the motion vary greatly with the epoch, but the typical amplitude is about 60 degrees behind the Earth-L 5 line to 30 degrees ahead of the line. The period of the expansion and contraction may vary between 800 and 1,400 days. Also, several epochs seem to produce stable motion, at least for the integration time of 14 years. 1.6 The Bicircular Problem The Bicircular Problem [2,19] is an idealized model of the Earth-Moon-Sun system. In this model, it is assumed that the orbit of the Earth and the Moon about their common center of mass is circular, and that the center of mass of the Earth-Moon system revolves around the center of mass of the Earth-Moon-Sun system in a circular, coplanar orbit. As in the CRTBP, the mass of the spacecraft is assumed to be sufficiently small not to affect any other gravitating body. We will use the same rotating frame as for the CRTBP (Figure 1.1) and the same normalized units. Figure 1.3 is a representation of the Bicircular Problem. 12

32 Figure 1.3: A representation of the Bicircular Problem 13

33 It should be noted that, in this frame, the Sun revolves clockwise about the center of mass of the Earth-Moon system. Therefore θ, the angle between the x-axis and the position vector of the Sun, is defined negatively. One method to determine the equations of motion for a spacecraft in the BCP is to consider the Hamiltonian of the system. Using superposition [2], the Hamiltonian of the BCP is the sum of the Hamiltonian of the CRTBP and the perturbation due to the inclusion of the Sun in the model. The Hamiltonian for the CRTBP [2,19] is: H CRT BP = 1 2 (p2 x + p 2 y + p 2 z) + yp x xp y 1 µ r 1 µ r 2 (1.17) where p x, p y and p z are the generalized momenta, defined by: p x = ẋ y (1.18) p y = ẏ + x (1.19) p z = ż (1.20) The perturbation due to the Sun can be expressed as follows [2,19]: H Sun = m s r 3 m s a 2 (ysin(θ) xcos(θ)) (1.21) s Where r 3 is the distance from the spacecraft to the Sun, a s is distance from the center of mass of the Earth-Moon system to the Sun, and m s is the mass of the Sun. Here it is important to note that all the quantities are normalized. The mass of the Sun was divided by the sum of the masses of the Earth and the Moon, the radius of the Sun s orbit was divided by the distance between the Earth and the Moon, and the angular velocity ω s of the Sun is defined as ω s = 2 π T s, where T s is the period 14

34 of the Sun s apparent orbit in the rotating frame, expressed in Time Units (TU). Therefore, the expression of the Hamiltonian in the BCP is: H BCP = 1 2 (p2 x+p 2 y+p 2 z)+yp x xp y 1 µ µ m s m s (ysin(θ) xcos(θ)) (1.22) r 1 r 2 r 3 a 2 s To obtain the equations of motion of a spacecraft in the BCP, Hamilton s canonical equations [1] are used, and the resulting equations are [11,19]: (1 µ)(x + µ) µ(x 1 + µ) ẍ = r1 3 r2 3 ÿ = (1 µ)y r 3 1 yµ r x + 2ẏ m s(x x s ) r3 3 + y 2ẋ m s(y y s ) z = r 3 3 (1 µ)z r 3 1 m s a 2 cos(θ) (1.23) s + m s a 2 sin(θ) (1.24) s zµ r 3 2 m sz r 3 3 (1.25) r 1 = (x + µ) 2 + y 2 + z 2 (1.26) r 2 = (x 1 + µ) 2 + y 2 + z 2 (1.27) x s = a s cos(θ) (1.28) y s = a s sin(θ) (1.29) r 3 = (x x s ) 2 + (y y s ) 2 + z 2 (1.30) 15

35 1.7 Zero-Velocity Curves Similarly to what exists in the CRTBP [4,6], zero-velocity curves can be found in the BCP [32], that define the regions where motion is permissible for a given energy. Unlike what occurs in the CRTBP, however, the Hamiltonian of the system changes with time, and therefore the shape of the curves will evolve as the position of the Sun changes. If only planar motion is considered, the zero-velocity curves can be obtained by rewriting Equation 1.22, substituting for p x and p y using Equations 1.18 and The expression of H becomes: H = 1 2 (ẋ2 + ẏ 2 ) 1 2 (x2 + y 2 ) 1 µ r 1 µ r 2 m s r 3 m s a 2 (ysin(θ) xcos(θ)) (1.31) s Defining : U = 1 2 (x2 + y 2 ) + 1 µ r 1 + µ r 2 + m s r 3 + m s a 2 (ysin(θ) xcos(θ)) (1.32) s H can be rewritten as H = 1 2 V 2 U, where V is the norm of the velocity vector of the spacecraft in the rotating frame. It follows that, if θ is held constant, the equation of the zero-velocity curves in the BCP will be H + U = 0. Figures 1.4 to 1.8 below show the zero-velocity curves for increasing values of H between and , for four values of θ: 0, π 4, π 2, and 3π 4. By symmetry, the curves would be identical for values of θ between π and 2π. 16

36 Figure 1.4: Zero-velocity curve in the BCP for H = Figure 1.5: Zero-velocity curve in the BCP for H =

37 Figure 1.6: Zero-velocity curve in the BCP for H = Figure 1.7: Zero-velocity curve in the BCP for H =

38 Figure 1.8: Zero-velocity curve in the BCP for H =

39 As was observed in the CRTBP, as H increases, new regions of permissible motion appear. The zero-velocity curves open up at each of the collinear libration point, although the position of the Sun does have a significant influence, as shown on Figure 1.5 for example. For the same value of H, the region past L 2 may or may not be accessible depending on θ. The same holds for the region past L 3 on Figure 1.7. Also, the regions near L 4 and L 5 are only accessible for sufficiently high values of H. This lower bound can be easily obtained by plotting the Hamiltonian of a particle at rest at L 4 versus θ, as shown on Figure 1.9. Figure 1.9: Hamiltonian of a particle at rest at L 4 versus θ As expected, the perturbation due to the Sun is periodic, with the maximum of H occurs for θ = radians or 29.4 degrees. The minimum occurs for θ = radians or degrees. 20

40 1.8 Equilibrium Points The zero-velocity curves shown in the previous Section also emphasize how the position of the equilibrium points, and especially the triangular points, is affected by the perturbation of the Sun [12]. In order to get the precise location of the equilibrium points of the BCP, one can proceed exactly the same way as for the CRTBP [4]. The rotational potential U is now defined as: U = 1 2 (x2 + y 2 ) + 1 µ r 1 + µ r 2 + m s r 3 + m s a 2 (ysin(θ) xcos(θ)) (1.33) s From the equations of motion of the BCP (Equations 1.23, 1.24 and 1.25), it is verified that: ẍ 2ẏ = U x ÿ + 2ẋ = U y z = U z (1.34) (1.35) (1.36) Where: U x (1 µ)(x + µ) = x r 3 1 U y µ(x 1 + µ) r 3 2 = y (1 µ)y r 3 1 yµ r 3 2 m s(x a s cos(θ)) r3 3 m s(y + a s sin(θ)) U z r 3 3 = (1 µ)z r 3 1 m scos(θ) a 2 s + m ssin(θ) a 2 s zµ r 3 2 m sz r 3 3 (1.37) (1.38) (1.39) Again, the position of the equilibrium points will be such that U x = U y = U z = 0, for any given value of θ. The equation U z = 0 yields that the solutions are all in the (x, y) plane, and by numerically solving the equations for 100 values of θ between 21

41 0 and 2π, the position of the equilibrium points is obtained, and plotted on Figure 1.10, where the x and y coordinates are given in Distance Units (1DU = 384, 400 km). Figure 1.10: Location of the equilibrium points in the BCP for varying positions of the Sun This plot further emphasizes the importance of the Sun s perturbation, especially in the vicinity or the triangular points. Whereas the impact of the fourth body perturbation is minimal near L 1 and very small near L 2, it is undeniably a major factor near L 4 and L 5, and even more so near L 3. Even though the BCP is a very simplified model, it clearly shows that the CRTBP is simply not realistic enough, at least in the Earth-Moon case, to be used to study the motion near the triangular libration points. 22

42 It should also be noted that those equilibrium points are only valid for the particular position of the Sun for which they were computed. A spacecraft placed at one of these points with no initial velocity will not remain at that position. Figure 1.11 shows the trajectory of a spacecraft initially at the upper triangular equilibrium point for θ = 0, propagated for 220 days, with coordinates given in Distance Units. The trajectory obtained is incidentally in very good agreement with what was obtained by Tapley and Lewallen [12], despite the fact that, unlike the model used in [12], the BCP does not account for the inclination of the Sun s orbit with respect to the Earth-Moon rotating frame. Figure 1.11: Trajectory of a spacecraft initially at equilibrium point 23

43 1.9 Behavior about L 4 The numerical propagation of the trajectory of a spacecraft in the vicinity of L 4 of the CRTBP in the BCP shows the existence of a strong perturbation due to the Sun that results in an expansion and contraction of the trajectory [13,14]. Figure 1.12 shows the trajectory of a spacecraft initially at L 4 of the CRTBP with no initial velocity. The plot shows the trajectory in the rotating frame, with a distance unit of 100, 000 km. Initially θ = 0, and the trajectory is integrated for 3,200 days. Figure 1.12: Trajectory of a spacecraft starting at L 4 24

44 As expected the spacecraft experiences large deviations from its initial position, its trajectory expanding and contracting, with an amplitude of about 250,000 km and a period of 1,500 days, as seen on Figure 1.13, which shows the distance from the spacecraft to L 4, its starting point, versus time in days. The period of the expansion-contraction is larger slightly larger the period of 1,330 days obtained using de Vries equations [10], which could be explained by the fact that the linearized equations he obtained are only representative of the motion for a few months, but could also suggest that the expansion and contraction is a phenomenon more complex that a simple resonance between the frequency of the solar perturbation and a frequency associated with the three-body motion. Figure 1.13: Distance from the spacecraft to L 4 versus time 25

45 1.10 Periodic Orbits about L 4 Kamel [3], Kolenkiewicz and Carpenter [17] and Gomez et al [19] have all shown, using different approaches, that in the BCP, there exist two stable, periodic orbits, whose period is the period of the Sun s perturbation. Those orbits are shown in Figures 1.14 and The orbits are shown in an Earth-Moon rotating frame, centered at the center of mass of the Earth-Moon system. Figure 1.14: First stable periodic orbit about L 4 26

46 Figure 1.15: Second stable periodic orbit about L 4 27

47 It is apparent that those orbit are very large, about 230,000 km along the x- axis and 160,000 km along the y-axis, and their stability can be illustrated by Figure 1.16, when the first stable orbit was propagated for 500 days. Mathematically, their stability is established by computing the monodromy matrix associated with the two trajectories [19], and computing their eigenvalues. The norm of the eigenvalues obtained are less than one, confirming the stability of these orbits. Figure 1.16: Trajectory obtained by propagating first periodic orbit for 500 days The results presented in this chapter illustrate the strong influence of the Sun on the dynamics about L 4, and show how the CRTBP cannot be used to adequately describe the motion of a spacecraft near the triangular libration points of the Earth-Moon system. However, not only the existence in the BCP, of stable, periodic orbits about L 4 and L 5, albeit very large, but also the sensitivity to the initial conditions and the nonlinearity of the dynamics involved suggests that the vicinity of the triangular points should be investigated further. 28

48 Chapter 2 Simulations in the Bicircular Problem 2.1 Goal and Method To better understand the dynamics in the proximity of L4 of the CRTBP in the Bicircular Problem (BCP), the nature of the perturbation due to the Sun must first be considered. When determining how this perturbation affects the spacecraft, the goal will be to find regions in which the long period oscillations either vanish or are significantly weakened, and therefore allow the spacecraft to remain in the close vicinity of the libration points. First, the linearized model presented by de Vries [10] will be summarized and its major results presented. By means of a Fast Fourier Transform, the findings of the linearized model will be compared to what is observed using the full BCP equations. Finally, the influence of the initial configuration of the Earth-Moon-Sun system on the librational motion near L 4 and L 5 is investigated, with special attention given to the amplitude of the motion. 2.2 Linearizing the Equations of Motion As mentioned above, this section summarizes and expands on the results obtained by de Vries [10]. We will keep, however, the same notation as in Chapter 1, so as to 29

49 avoid any confusion. As a reminder, the model considered is the Bicircular Problem, in which the Earth and the Moon revolve on a circular orbit about their common center of mass, and that center of mass revolves on a circular, coplanar orbit about the center of mass of the Earth-Moon-Sun system. Using a Legendre Polynomial expansion, de Vries shows that the force function can be expressed as the sum of the contribution due to the Earth and the Moon, and that due to the Sun. A Taylor series expansion is used to compute the equation of motion in the vicinity of L 4. Here it is important to note that the rotating Earth-Moon frame considered is now centered at L 4, instead of the center of mass of the Earth-Moon system, as was the case in Chapter 1. The new coordinates, used by de Vries, will be denoted X, Y, Z. However, for the purpose of this study, the out-of-plane motion will be neglected. First, let us only consider the terms in the force function due to the Earth and the Moon. As the coordinates of L 4 are known and the units have been normalized, the equation of motion near L 4 are: Ẍ 2Ẏ = 3 4 X (1 2µ) 3Y (2.1) Ÿ + 2Ẋ = 9 4 Y (1 2µ) 3X (2.2) These equations are very similar to those obtained in the three-body problem [21]. The inclusion of the solar terms of the force function is more problematic, and only the first two terms of the Taylor series expansion will be considered. The equations of motion obtained are as follows: Ẍ 2Ẏ C 1X C 2 Y = C 4 + C 6 cos(2θ) + C 7 sin(2θ) +C 8 cos(θ) + C 9 sin(θ) + C 11 ν(xcos(2θ) Y sin(2θ)) 30

50 +C 12 ν(3x L cos(θ) Y L sin(θ))x +C 12 ν(y L cos(θ) X L sin(θ))y Ÿ + 2Ẋ C 2X C 3 Y = C 5 + C 7 cos(2θ) C 6 sin(2θ) C 9 cos(θ) + C 10 sin(θ) C 11 ν(xcos(2θ) Y sin(2θ)) +C 12 ν(y L cos(θ) X L sin(θ))x +C 12 ν(x L cos(θ) 3Y L sin(θ))y The expression of the constants and their value can be found in Table 2.1, and θ is defined on Fig 1.3. The numerical values in Table 2.1 were obtained using µ = , ω s = , a s = and m s = These values are consistent with the ones used by Gomez et al [19], but differ slightly from those used by de Vries. 31

51 Table 2.1: Coefficients of de Vries Equations of Motion Coefficient Expression Numerical Value ν C 1 3 m s a 3 s ν C 2 3 3(1 2µ) C ν X C L ν Y C L ν X C L ν C 7 3Y Lν C 8 3ν(3X2 L +Y L 2) 8a s C 9 3X L Y L ν 4a s C 10 3ν(3Y 2 L +X2 L ) 8a s C C a s

52 These equations of motion can now be numerically integrated. Figure 2.1 shows the trajectory of a spacecraft initially at L 4 with θ = π, which was the initial value chosen by de Vries, propagated for 180 days, in a rotating frame centered at L 4. Figure 2.1: Trajectory of a spacecraft initially at rest at L4 using the linearized equations 33

53 This trajectory should be compared to what is obtained by integrating the equations of motion of the BCP. To that end, Figure 2.2 is the trajectory of a spacecraft initially at L 4, with θ = π, propagated for 180 days, this time using the equations of motion of the BCP, as established in Chapter 1 (Equations 1.23 and 1.24). Here, the trajectory is shown in the Earth-Moon rotating frame defined on Figure 1.3. Figure 2.2: Trajectory of a spacecraft initially at rest at L4 using the BCP Equations 34

54 To illustrate the discrepancy between the two cases, Figure 2.3 shows the distance from the spacecraft to L 4 versus time. The solid line corresponds to the full BCP equations, and the dashed line to the linearized equations. The behavior of the spacecraft is similar in both cases, even though there is a clear discrepancy that appears after about 80 days. Here again, in both cases, the initial value of θ is π. Figure 2.3: Distance from the spacecraft to L 4 using the BCP and linearized equations 35

55 The two curves match almost exactly initially, at least for about 70 days, and their behavior remains very similar throughout. But strong discrepancies are apparent around 90, 140 and 165 days. Despite those differences, the linearized equations describe the motion closely of the spacecraft closely enough to be used to isolate the harmonics of the motion. First, let us consider the left-hand side of the linearized equations: Ẍ 2Ẏ C 1X C 2 Y = 0 Ÿ + 2Ẋ C 2X C 3 Y = 0 The characteristic equation [21] associated with them is: λ 4 + (1 ν)λ (1 (1 2µ)2 ) + 3ν 2 + ν2 4 = 0 (2.3) This equation has four imaginary solutions, of the form λ i = ±a j where a and a Those numerical values differ slightly form the the ones determined by de Vries, as we are using different values for µ and ν. One can also note that the introduction of the Sun s perturbation affects the values of those fundamental frequencies through the parameter ν. If only the three-body problem is considered, the fundamental frequencies for the chosen value of µ are and Since the frequencies are given in normalized units, it is useful to calculate the corresponding period: a 1 corresponds to a period of about days and a 2 corresponds to days. In order to determine the frequencies associated with the right-hand side of the linearized equations, de Vries assumes that the solutions of the linearized system 36

56 of equations are of the form: X = ν i X i (2.4) i=0 Y = ν i Y i (2.5) i=0 Where X 0 and Y 0 correspond to a first order solution, X 1 and Y 1 correspond to a second order solution, and so on. By regrouping the terms according to the powers of ν, the first order solution is derived from the linearized equations, discarding all the terms not containing ν. De Vries was able to explicitly solve these equations by using LaPlace transforms, and found that four frequencies were present, a 1, a 2, ω s and 2ω s, where ω s corresponds to a period of about days and 2ω s corresponds to a period of days. It is also found that the contribution of 2ω s is significantly greater than that of ω s, and that the contributions of a 1 and a 2 are larger still, but depend on the initial conditions. Obtaining the second order solutions is more difficult. Out of twelve possible frequencies, de Vries selected two that he deemed worthy of close consideration. The first one is ω s, as is is numerically close to a 1, and therefore may produce a resonance phenomenon, and the second one is a 1 ω s, as it produces a very long perturbation, with a period close to 1,330 days. It was then concluded that the a 1 ω s term was insignificant in the equations of motion and that the resulting amplitude in the solutions was small, but that the ω s term was very significant, almost as much as the first order solution. However, as was shown on Figure 2.3, this linearized system only matches the full BCP equation for a few months, so Fast Fourier Transforms will be used in the next section to determine the exact spectral composition of the solar perturbation and the results will be compared to the de Vries analytical findings. 37

57 2.3 Power Spectra The study of the spectral composition of the perturbation was performed as follows: starting at L 4, and with no initial velocity, the initial values of x was incrementally increased and decreased by DU, then the process was repeated for the initial value of y. Using Matlab, the trajectory of the spacecraft was propagated and the time history of the distance from the spacecraft to L 4 was computed. A Fast Fourier Transform of this distance is generated, and the power spectrum is plotted. For all the simulations, the initial value of θ was set to 0. Since the power spectra obtained are very similar, only a few representative cases are shown in this Dissertation. Also, the goal of this study is to gain a better understanding of the expansion and contraction experienced by the spacecraft. Therefore, for each initial condition, the propagation time was adjusted so that only the expansion and contraction phase was considered. The constants used will be chosen consistent with those used by Gomez et al [19]: µ = , m s = , a s = , ω s = , and the trajectories were integrated using Matlab s ode113 integrator, which is a variable step and variable order Adams-Bashforth-Moulton integrator [27], and checked against an explicit Runge-Kutta 7-8 integrator developed by V. Govorukhin [25] Study of the Vicinity of L 4 Figures 2.4, 2.5 and 2.6 respectively show the trajectory of a spacecraft initially at rest at L4 propagated for 3,200 days, the distance of the spacecraft to L 4 versus time, and then the power spectrum obtained. 38

58 Figure 2.4: Trajectory of a spacecraft initially at rest at L 4 Figure 2.5: Distance to L 4 in kilometers for a spacecraft initially at rest at L 4 39

59 Figure 2.6: Power spectrum for a spacecraft initially at rest at L 4 40

60 Aside from the long-period perturbation, whose period is about 1,600 days and is not clearly discernible on the power spectrum, it appears that there are three additional peaks, corresponding to perturbations of period 29.24, and days respectively. These three periods are extremely close the period of the apparent orbit of the Sun in the system, which is days, as well as the period corresponding to the a 1 frequency. A second, weaker group of peaks, corresponding to periods of about 14.7 days can also be seen, and this corresponds approximately to half the period of the orbit the Sun. The two weaker peaks correspond to and days. The initial position of the spacecraft is slightly changed, as it is displaced by km to the right along the x-axis. Figures 2.7, 2.8 and 2.9, show the trajectory propagated for 3,200 days, the distance to L 4, and finally the power spectrum. Figure 2.7: Trajectory of a spacecraft initially at rest at x L km 41

61 Figure 2.8: Distance to L 4 in kilometers for a spacecraft initially at rest, x L km Figure 2.9: Power spectrum for a spacecraft initially at rest at x L km 42

62 In this case, four more peaks can be found corresponding approximately to the period of the Sun, with periods between and 30.3 days. These periods are again close to the period of the orbit of the Sun in this system, just as close as the periods observed on Figure 2.6. Also, the 15-day period oscillation (peaks between and days) is still identified, although it is slightly weaker than on Figure 2.6. Two relatively strong peaks are observed at and days, which is about half the period corresponding to a 2. Finally, a peak at days is also present. Next, Figure 2.10 shows the trajectory of the spacecraft initially at rest, and displaced from L 4 by km to the left along the x-axis. This trajectory was propagated for 3,200 days. Figures 2.11 and 2.12 respectively show the distance to L 4 and the power spectrum. Figure 2.10: Trajectory of a spacecraft initially at rest at x L km 43

63 Figure 2.11: Distance to L 4 in kilometers for a spacecraft initially at rest, x L km Figure 2.12: Power spectrum for a spacecraft initially at rest at x L km 44

64 Again, significant peaks remain for perturbations with periods between 28.2 and days, and around 15 days. But for the first time, the bi-monthly perturbation seems to be stronger than the monthly effects. Also, the peaks around 21.8 and 45 days are still observed, but weaker than in previous cases. Figures 2.13, 2.14 and 2.15 show the trajectory, propagated for only 2,000 days, of a spacecraft initially at rest and displaced from L4 by km to the right along the x-axis, then the distance to L 4 in kilometers and the power spectrum obtained. 45

65 Figure 2.13: Trajectory of a spacecraft initially at rest at x L km 46

66 Figure 2.14: Distance to L 4 in kilometers for a spacecraft initially at rest, x L km Figure 2.15: Power spectrum for a spacecraft initially at rest at x L km 47

67 In this case, the power spectrum only exhibits two major peaks aside from the long-period oscillations, corresponding to periods of 29.4 days and days respectively. The fact that fewer peaks were found may be attributed to a shorter propagation time. Peaks around 21.9 and 45 days are still identifiable. Unlike Figure 2.12 however, the monthly perturbation is again the strongest. Next, a deviation from L 4 in the y-direction is considered. Since the results are similar to what was obtained for deviations along the x-axis, only one case will be presented. Below are Figures 2.16, 2.17 and 2.18, which show the trajectory of the spacecraft initially displaced from L 4 by km upwards and propagated for 2,500 days, the distance from the spacecraft to L 4 and the power spectrum. 48

68 Figure 2.16: Trajectory of a spacecraft initially at rest at y L km 49

69 Figure 2.17: Distance to L 4 in kilometers for a spacecraft initially at rest, y L km Figure 2.18: Power spectrum for a spacecraft initially at rest at y L km 50

70 The power spectrum exhibits one main peak at 29.4 days, and other peaks at around 15 days. From all the previous cases, it appears that, in the close neighborhood of L 4, the spacecraft is subjected to long-period perturbations - with periods of the order of 1,500 days - and to a lesser extent, to monthly and bi-monthly oscillations. Even though de Vries linearized equations did not predict the long-period oscillations, they accurately predicted that the other main perturbations would be monthly and bi-monthly. However, the results may differ greatly when an initial velocity at L 4 is considered, as presented next Emergence of a Resonance Phenomenon Figures 2.19, 2.20 and 2.21 show the trajectory of spacecraft initially at L 4 with an initial velocity ẋ 0 = Normalized Units (or about m/s), the distance from the spacecraft to L 4 throughout the trajectory and the associated power spectrum. 51

71 Figure 2.19: Trajectory of a spacecraft initially at L 4, ẋ 0 = Normalized Units Figure 2.20: Distance to L 4 in kilometers for a spacecraft initially at L 4, ẋ 0 = Normalized Units 52

72 Figure 2.21: Power spectrum for a spacecraft initially L 4, ẋ 0 = Normalized Units 53

73 This particular case exhibits the same type of behavior that was observed when varying the initial position of the spacecraft, and the period associated with the two main secondary peaks is now and days, with smaller peaks for periods of and days. But there is another sizable family of peaks at about days (or half the period corresponding to a 2 ), similar to what was observed on Figure 2.9, and a weaker family of peaks at about 21.8 days. As we increase the initial velocity of the spacecraft, however, a new behavior starts to emerge, as illustrated on Figures 2.22, 2.23 and Figure 2.22 shows the trajectory of a spacecraft initially at L 4 with an initial velocity ẋ 0 = Normalized Units (or about m/s), propagated for 2,700 days. Figure 2.23 and 2.24 respectively show the distance form the spacecraft to L 4 and the power spectrum generated. Figure 2.22: Trajectory of a spacecraft initially at L 4, ẋ 0 = Normalized Units 54

74 Figure 2.23: Distance to L 4 in kilometers for a spacecraft initially at L 4, ẋ 0 = Normalized Units Figure 2.24: Power spectrum for a spacecraft initially L 4, ẋ 0 = Normalized Units 55

75 As was already seen in previous cases, a peak at about days appears, but this time stronger than the bi-monthly perturbation. This occurs as, on Figure 2.23, the long-period oscillations seem to vanish for about 1,000 days. The spacecraft appears to enter a mode in which the long-period oscillations are damped and this allows the spacecraft to remain much closer to L 4 than in previous cases. A lower initial velocity is chosen, in this case ẋ 0 = Normalized Units (or about 13.3 m/s), and the trajectory is plotted on Figure Figures 2.26 and 2.27 show the distance from the spacecraft to L 4 and the power spectrum. Figure 2.25: Trajectory of a spacecraft initially at L 4, ẋ 0 = Normalized Units 56

76 Figure 2.26: Distance to L 4 in kilometers for a spacecraft initially at L 4, ẋ 0 = Normalized Units Figure 2.27: Power spectrum for a spacecraft initially L 4, ẋ 0 = Normalized Units 57

77 This case shows the lower bound of the initial velocity ẋ 0 for which the spacecraft enters the mode in which the long-period oscillations disappear. Figure 2.24 shows the peaks at 45 and 22 days are still present, but weaker than in the previous case. The periods associated with the five main peaks are 29.4, 14.77, 43.1, and 21.9 days respectively. The perturbations with periods greater than 40 days, which are only dominant in the last two cases presented, seem to be responsible for the mode seen on Figure

78 2.3.3 The 3-1 Resonance Using the results shown on Figure 2.23, the initial position and velocities of the spacecraft were reset in order to study the motion in the region where the longperiod oscillations appear to vanish, between 1,500 and 2,300 days. Figures 2.28, 2.29 and 2.30 show respectively thee trajectory, propagated for 870 days, the distance from the spacecraft to L 4 and the power spectrum. Figure 2.28: Trajectory with no expansion and contraction propagated for 870 days 59

79 Figure 2.29: Distance to L 4 in kilometers for a trajectory with no expansion and contraction Figure 2.30: Power spectrum for a trajectory with no expansion and contraction 60

80 In this particular case only a major peak for a period of about 45 days is found, and even though the amplitude of the oscillations changes, the period remains unchanged, unlike what had been observed for the cases presented in section A numerical procedure was implemented to slightly modify the initial conditions of the trajectory shown on Figure 2.28 so that the spacecraft is not subjected to the expansion and contraction for at least 2,000 days. The resulting trajectory, plotted on Figure 2.31, exhibits a quasi-constant amplitude in the oscillations of the distance from L 4 to the spacecraft (Figure 2.32). Figure 2.33 shows the power spectrum for the distance to L 4 for the trajectory on Figure 2.31, where only one major peak at days can be seen. This type of orbit can be referred to as a 3-1 resonant quasi-periodic orbit [26], since the period of revolution of the spacecraft about L 4 is approximately 3 T S. Figure 2.31: Trajectory with no expansion and contraction propagated for 2,000 days 61

81 Figure 2.32: Distance to L 4 in kilometers for a trajectory with no expansion and contraction Figure 2.33: Power spectrum for a trajectory with no expansion and contraction 62

82 2.4 Influence of the Initial Configuration of the Earth-Moon-Sun System Wolaver [11] had shown, using the linearized equations above, that the amplitude of the librational motion can be significantly reduced by modifying the initial position of the Sun (in our case the initial value of θ). In this section, the goal is to determine the influence of the initial value of θ using the equations of motion of the BCP derived in Chapter 1 (Equations 1.23 and 1.24). The trajectory of a spacecraft initially at rest at L 4 was therefore propagated for 2,000 initial values of θ between 0 and 360 degrees (the initial value of θ being incrementally increased after each propagation). The propagation time was 120 Time Units, which is about 520 days. For each trajectory, the mean distance to L 4 was computed using 100 data points per Time Units, and Figure 2.34 shows the mean distance to L 4, in kilometers, versus the initial value of θ, in degrees. The same process was repeated for L 5, and the resulting plot is shown on Figure

83 Figure 2.34: Mean distance to L 4 versus the initial value of θ Figure 2.35: Mean Distance to L 5 versus the initial value of θ 64

84 Two minima appear in each case, with mean distances of less than 10,000 km. In the case of L 4, the minima occur at 135 degrees and 320 degrees. In the case of L 5, they occur at 82.5 degrees and degrees. The neighborhood of these minima is now considered, by using the same approach but with a longer propagation time (300 Time Units, or about 1,300 days), and 1,000 initial values of θ. Figures 2.36 and 2.37 show the mean distance to L 4 for initial values of θ close to the minima found above. Figure 2.36: Mean distance to L 4 versus the initial value of θ near the first minimum 65

85 Figure 2.37: minimum Mean distance to L 4 versus the initial value of θ near the second 66

86 By increasing the propagation time, the minimal distance also increased slightly, whereas the initial value of θ that produces the minimum decreased (129.3 degrees and degrees). In order to refine the solution even further, the propagation time was increased again, this time up to 520 Time Units, or 2,300 days. Only the case of the first minimum about L 4 will be presented, since the behavior at the second minimum is extremely similar. Figure 2.38 shows the mean distance to L 4 versus the initial value of θ near 129 degrees, using 100 initial values of θ for a spacecraft initially at rest at L 4. Figure 2.38: Mean distance to L 4 versus the initial value of θ near 129 degrees 67

87 The minimum is now clearly visible, for θ = degrees initially. When the same process is conducted on the second minimum, the initial value of θ that produces the minimum is found to be degrees. To illustrate the behavior of a spacecraft initially at rest at L 4 for those two initial values of θ, Figures 2.39, 2.40, 2.41 and 2.42 respectively show the trajectory, propagated for 2,300 days, and the distance to L 4, for both θ = degrees and θ = degrees. Figure 2.39: Trajectory of a spacecraft initially at L 4 with θ = degrees 68

88 Figure 2.40: Distance to L 4 versus time for θ = degrees Figure 2.41: Trajectory of a spacecraft initially at L 4 with θ = degrees 69

89 Figure 2.42: Distance to L 4 versus time for θ = degrees 70

90 In both cases, the shape of the curve is very similar, but in the second case, the amplitude of the motion is a bit smaller, as proven by Figure Either trajectory shows a much smaller deviation from L 4 than any of the trajectories shown so far, with a maximum distance from L 4 of about 28,000 km, as opposed to 250,000 km in Figure 2.5, for instance. However, these small amplitude oscillations only last for a limited propagation time, and ultimately the large oscillations, as seen on Figure 2.4, will reappear. An interesting result is to compute after how long, in the neighborhood of the optimal values of θ, the spacecraft is subjected to the large oscillations. To obtain Figure 2.43, the trajectory of a spacecraft initially at L 4 was propagated for 1,000 Time Units (4,400 days), for 38 initial values of θ around the first minimum (between 128 and 130 degrees). The distance to L 4 was calculated for each case, and the time at which that distance first reaches 30,000 km was stored. Figure 2.43 shows the time necessary to first reach 30,000 km from L 4 versus the initial value of θ. 71

91 Figure 2.43: Number of days required to reach 30,000 km from L 4 versus the initial value of θ 72

92 There appears to be a critical initial of θ that allows the spacecraft to remain within 30,000 km of L 4 for almost 4,000 days, and given the shape of the curve on Figure 2.43, it seems that this number can be further refined. For instance, an initial value of θ of degrees produces a trajectory that stays within 30,0000 km of L 4 for 4,513 days. What is all the more remarkable is that an initial value of θ of 129 degrees results in an trajectory that remains within 30,000 km of L 4 for 1,939 days. This corresponds to an improvement of 2,574 days by modifying the initial value of θ by only degrees, which the Sun covers in a little more than 8.5 minutes. This behavior could be used to remain in the close vicinity of L 4, provided it subsists in a more realistic model. Figures 2.44 and 2.45 show the trajectory of a spacecraft initially at L 4 for an initial value of θ of degrees, and the distance from the spacecraft to L 4 versus time. The propagation time is 1,000 Time Units, or 4,400 days. Figure 2.44: Trajectory of a spacecraft initially at L 4 with θ = degrees 73

93 Figure 2.45: Distance to L 4 versus time for θ = degrees 74

94 The existence of those two critical values of θ is also apparent when plotting the mean distance from the spacecraft to L 4 versus the initial value of θ for a propagation time of 520 Time Units, or about 2,300 days, as shown on Figure 2.46, which was generated using 2,000 initial values of θ. There is a sharp drop in the mean distance for the initial values of θ found above. We can also note that for some initial values of θ (between approximately 15 and 45 degrees and between 195 and 215 degrees) there are great variations in the mean distance, which can be explained by escaping trajectories. Repeating this process around L 5 producing an extremely similar plot, with very sharp minima around 80.5 and degrees. Figure 2.46: Mean distance to L 4 versus the initial value of θ for 520 Time Units 75

95 For completeness sake, the method used to generate Figure 2.43 was also used around the other initial value of θ that minimizes the deviation from L 4. Figures 2.47 shows the number of days required for a spacecraft to reach 30,000 km from L 4 versus the initial value of θ. Once again, a very sharp maximum appears at a critical value of θ, this time about degrees. The initial value of θ was then further refined to degrees, to produce a trajectory that remain within 30,000 km of L 4 for upwards of 4,500 days. This trajectory is shown on Figure Figure 2.47: Number of days required to reach 30,000 km from L 4 versus the initial value of θ 76

96 Figure 2.48: Trajectory of a spacecraft initially at L 4 with θ = degrees 77

97 It appears on both Figures 2.44 and 2.48 that there is a very simple way to decrease the mean distance to L 4 even further, simply by starting the propagation at a later time on the same trajectory. To illustrate this, Figures 2.49 and 2.50 show the trajectories obtained from 2.44 and 2.48, propagated for 300 Time Units (about 1,300 days), but this time the center of the frame has been moved to L 4. Also Figures 2.51 and 2.52 show the distance from the spacecraft to L 4 for both trajectories. Figure 2.49: Trajectory of a spacecraft relative to L 4 for θ = degrees 78

98 Figure 2.50: Trajectory of a spacecraft relative to L 4 for θ = degrees Figure 2.51: Distance to L 4 in kilometers for Figure

99 Figure 2.52: Distance to L 4 in kilometers for Figure

100 The power spectra of the distance to L 4 for the trajectories shown on Figures 2.49 and 2.50 was also obtained, and are shown on Figures 2.53 and There are remarkably few well defined peaks, only 4 in the each case, illustrating that those solutions are in fact quasi-periodic orbits. The periods associated with each harmonic frequency, from strongest to lowest, are 45.87, 17.6, 43.1, 12.72, and days. The similarities between the results are striking, suggesting that the dynamics involved are very similar. The period of the motion about L 4 is close to twice the period associated with the largest peak, or about 90 days, which is very close to the period associated with a 2 and three times the period of the solar perturbation. However, none of the other periods match the fundamental frequencies discussed in section 2.2. Figure 2.53: Power spectrum of the distance to L 4 for θ = degrees 81

101 Figure 2.54: Power spectrum of the distance to L 4 for θ = degrees 82

102 2.5 Tentative Explanations In an attempt to understand why there exists two critical initial values of θ, for which the spacecraft remains in a very close vicinity of the triangular points, the following approach was used. The trajectory of a spacecraft initially at rest at L 4 was propagated for 120 Time Units (or 520 days) for 2,000 initial values of θ between 0 and 2π. First the mean acceleration due to the three-body dynamics was computed and plotted on Figure 2.55 versus the initial value of θ. The mean acceleration due to the solar perturbation was plotted on Figure 2.56, and finally the total mean acceleration was plotted on Figure It is interesting to note that the three-body acceleration is significantly larger than the acceleration due to the Sun. Also, the minima of the total acceleration correspond approximately to the position of the Sun producing small deviations from L 4. However, the minima of the acceleration due to the Sun occur for higher values of θ than the ones producing the small deviations. It would appear that the spacecraft is subjected to a smaller perturbation for the initial values of θ close to the ones producing small deviations, but a more detailed study would be necessary to draw any definite conclusions. 83

103 Figure 2.55: Three-body acceleration in Normalized Units for a spacecraft initially at rest at L 4 versus the initial value of θ Figure 2.56: Acceleration due to the Sun in Normalized Units for a spacecraft initially at rest at L 4 versus the initial value of θ 84

104 Figure 2.57: Mean acceleration in Normalized Units for a spacecraft initially at rest at L 4 versus the initial value of θ 85

105 2.6 Prolonging Bounded Motion As was suggested by Figure 2.43, it appears possible for the smaller oscillations about L 4 to persist for longer periods of time, provided the initial value of θ is chosen properly. To that end, a numerical procedure was developed to refine the initial value of θ so that, for any specified propagation time, the smaller oscillations would persist. As an example, a propagation time of 1,400 Time Units was chosen, and a starting initial value of θ of degrees. The initial value of θ obtained was degrees. Figure 2.58 and 2.59 show the trajectory and the time history of the distance to L 4 of a spacecraft initially at rest at L 4, propagated for 1400 Time Units (or about 6,100 days). This process could be repeated for longer propagation times. However, the difference between the refined value of θ and its starting value is only degrees, for an improvement of 2,451 days (the trajectory shown on Figure 2.58 remains within 30,000 km of L 4 for 6,964 days). It is therefore going to be difficult to refine the initial value of θ further, in order to remain in the close vicinity of L 4 longer, as longer propagation times will require greater and greater numerical accuracy. 86

106 Figure 2.58: Trajectory of a spacecraft initially at rest at L 4 for θ = degrees Figure 2.59: Time history of the distance to L 4 spacecraft initially at rest at L 4 for θ = degrees 87

107 2.7 Sensitivity to the Initial Conditions In this section the influence of the initial position of the spacecraft on its mean deviation from L 4 is studied. The initial value of θ is set to degrees, which has been shown to produce a trajectory that remains within 30,000 km of L 4 for more than 4,500 days if the spacecraft is initially at rest at L 4. The elements of the initial state vector are varied one by one, holding the other three constant, the trajectory is propagated for 520 Time Units, or approximately 2,300 days, and the mean distance to L 4 is computed and stored. Figure 2.60 shows the evolution of the mean distance to L 4 when the initial x-coordinate is varied from x L4 300 km to x L km, Figure 2.61 shows the evolution of the mean distance to L 4 when the initial y-coordinate is varied from y L4 300 km to y L km, and Figures 2.62 and 2.63 show the evolution of the mean distance to L 4 when the initial values of ẋ and ẏ are varied from -10 to 10 m/s. Figure 2.60: Mean distance to L 4 versus the initial x-coordinate 88

108 Figure 2.61: Mean distance to L 4 versus the initial y-coordinate 89

109 Figure 2.62: Mean distance to L 4 versus the initial value of ẋ Figure 2.63: Mean distance to L 4 versus the initial value of ẏ 90

110 Modifying the initial velocity of the spacecraft produces a very sharp minimum of the mean distance to L 4 at 0, suggesting the dynamics around the triangular points are very sensitive to the initial velocity. Interestingly, the minima of the mean distance to L 4 on Figure 2.60 and 2.61 do not occur exactly at L 4, but at x L km in the case of Figure 2.60 and y L km in the case of Figure The improvement in mean distance is actually small (about 110 km), but it illustrates that the mean distance to L 4 can be reduced by slightly modifying the initial position. The trajectories obtained using either the initial value of x or the initial value of y producing the minimum deviation from L 4, however, only remain within 30,000 km of L 4 for about 2,500 days, as opposed to 4,500 days if the spacecraft was initially at L Conclusions First, using de Vries linearization of the equations of motion [10], we established that there exist four fundamental frequencies associated with the motion of a spacecraft in the vicinity of L 4. The periods associated with those frequencies are 14.76, 28.72, and days respectively. Another key frequency was also identified, using a second order solution, corresponding to a resonance between one of the frequencies associated with the three-body dynamics and the frequency of the solar perturbation. This frequency, a 1 ω s, corresponds, using our constants, to a period of 1,330 days, and came closest to explaining the long period oscillations described in Chapter 1. Unfortunately, de Vries showed that long period oscillation was insignificant in the linearized solutions. But since the solutions of the linearized equations only match the full equations of motion for about three months, it is not surprising that the long term effects were lost in the linearization. 91

111 The numerical experiments that were then conducted showed that the predominant frequencies were indeed the long-period oscillation, which corresponds approximately to the a 1 ω s frequency but was not predicted by the linearized model, then a 1, ω s and 2ωs, which was predicted by the analytical solution. However, the a 2 frequency was not observed in the vicinity of L 4 until initial velocities were considered. Displacing the spacecraft from L 4 has a small effect on the frequencies of the motion, but the periods observed tend to increase the farther away from L 4 the spacecraft initially is. The emergence of the 3-1 resonance is another key result. It may be related to the a 2 frequency obtained by de Vries, as the period associated with it is close to 3T s as well. Another possibility is that the trajectory obtained is a continuation of a periodic orbit of the CRTBP around L 4. However, a more detailed numerical and analytical study would be required to draw any definite conclusions. This quasi-periodic solution illustrates that initial conditions can be found in the vicinity of L 4 so that the spacecraft is not subjected to the long-period oscillations. However, this trajectory remains fairly large (about 50,000 km from L 4 ). Finally, the effect of the initial configuration of the Earth-Moon-Sun system was studied. It was found that choosing an appropriate initial value of θ can reduce the deviation from L 4 of a spacecraft initially at rest at L 4 by a factor of 10 over up to 6,700 days. There are actually two initial values of θ producing a minimum deviation from L 4. However, the dynamics was found to be very sensitive to the initial configuration. It was shown that a difference of about degrees in the initial value of θ leads to a trajectory that remains in the close vicinity of L 4 for more that 2,000 days longer. This sensitivity would allow one to refine the initial value of θ even further, which could lead to a trajectory that stays within 30,000 km 92

112 of L 4 for as long as desired, provided sufficient numerical accuracy. The solutions can also be improved, in the sense that they would remain closer to L 4, by starting at a later point on the trajectory. The resulting trajectories are quasi-periodic orbits about L 4, whose amplitude is about 20,000 km and that revolve about L 4 in about 3T s. Through the numerical investigation of the Bicircular Problem, it was established that the large expansion and contraction that occurs in the vicinity of the triangular libration points can be damped or even vanish. In particular, it was found that the initial configuration of the Earth-Moon-Sun system has a great influence on the motion and can allow a spacecraft to remain in the close vicinity of the libration points for extended periods of time. While the characteristics of this motion were studied here numerically, a future investigation could be conducted using analytical methods, and hopefully explain why there exist two such favorable initial positions of the Sun. In the next Chapter, a more realistic model, obtained using JPL DE405 ephemerides, is considered. We will study the nature of the solar perturbation in this model as well, and the influence of the initial epoch, which determines the configuration of the Earth-Moon-Sun system at the initial time. We will search for the epochs that produce a minimal deviation from the libration points, and attempt to replicate the behavior that was observed in Section

113 Chapter 3 Numerical Study of the Real-World Model 3.1 Objective and Overview After studying the fundamental characteristic of the motion of a spacecraft near L 4 in the BCP, we now move on to a more realistic model to investigate the dynamics near the triangular Lagrangian points. In this aspect, the model is based on SPICE ephemerides, which represent the highest accuracy ephemerides now in use, and are available from and maintained by NASA s Navigation and Ancillary Information Facility. SPICE gets its ephemerides data from different ephemeris files depending on the celestial object considered. In the case of the Earth, Moon and Sun, which are the only objects considered in this study, SPICE uses the JPL DE 405 ephemerides. Schutz [5] showed, using an ephemeris-based model as well, how the initial epoch influences the nature of the motion near L 4 and L 5, with some epochs producing apparently stable librational motion and others producing no librational motion at all. One of the goals of this study is the investigation of the influence of the initial configuration of the Earth-Moon-Sun system throughout the year 2007, in order to determine which epochs allow the spacecraft to remain close to L 4 and L 5 for the longest time. Once those epochs are identified, the sensitivity to the initial epoch 94

114 will be considered. 3.2 Model and Output Frame In this model, only the gravitational forces applied to a simulated spacecraft due to the Earth, the Moon and the Sun are considered. Other forces, such as the gravitational forces of other planets, the nonsphericity of the Earth and the Moon and other planets, and solar radiation pressure are neglected. The equations of motion of a spacecraft are numerically integrated in an Earth-centered non-rotating J2000 frame. The position of the Earth, the Moon and the Sun are obtained via the SPICE ephemerides. The numerically integrated trajectories of the simulated spacecraft are then plotted in two slightly different output frames. The first is an Earth-centered rotating frame, such that the Earth-Moon line is the x-axis (with the Moon on the positive side), and the z-axis is given by the angular momentum of the Moon about the center of mass of the Earth-Moon system. The second output frame is an Earth-centered rotating pulsating frame, with the same axes as the first output frame, but such that all distances and velocities are scaled to keep the Earth- Moon distance fixed (384,400 km). The main advantage of using a rotating-pulsating frame is that the positions of L 4 and L 5 are fixed. Since the orbits of the Earth and the Moon are not circular in the ephemerides-based model, the positions of L 4 and L 5 are not constant in the Earth-centered rotating frame, and will have to be computed for each epoch. This is done by converting the coordinates of L 4 and L 5 in the rotating-pulsating frame back to the rotating frame, yielding the coordinates and velocities of the triangular Lagrangian points in the elliptic three-body problem. 95

115 3.3 Copernicus The numerical integration of the trajectories was performed using a software named Copernicus, developed at the University of Texas at Austin by Dr. Cesar Ocampo and his team. Copernicus is a trajectory design and optimization software, written in standard Fortran 90 and using 64-bit arithmetic, that allows to propagate, target and optimize a large variety of trajectories in the solar system [22]. In the current version, Copernicus encompasses eight different optimizers and five integrators. The integrators chosen for this study are ODEINT, which uses either a fifthorder Runge-Kutta-Cash-Karp integrator or a Burlisch-Stoer method [28,29,30], and the DLSODE integration package [23], which uses a variable order Adams-Moulton method (of order 1 to 12) for non-stiff problems, and a Backward Differentiation Formula (BDF) of order 1 to 5 for stiff problems. More details about the integrators and the numerical accuracy are presented in the next Section. 3.4 Numerical Integration As mentioned in the previous Section, the numerical propagation of the trajectories was obtained using three different integrators, two of them driven by ODEINT, which are the 5th order Runge-Kutta-Cash-Karp integrator (RK5) and the Burlisch- Stoer extrapolation integrator (BSE), and the third is the DLSODE (Double precision Livermore Solver for Ordinary Differential Equations) integrator. All three integrators use a variable step size, to achieve the prescribed tolerance. The error tolerance for the BSE and RK5 integrators was set to 10 12, the relative error tolerance for the DLSODE integrator was set to and its absolute error tolerance to Since the goal of this study is to determine the characteristics of the motion 96

116 in the vicinity of the triangular libration points, the choice was made to use three integrators to ensure that the trajectories obtained are indeed representative of the motion of the spacecraft and not an artifact of a particular integrator. DLSODE was used as the primary integrator, and the other two were used to verify the results. To compare the results, let us consider two initial epochs for a spacecraft initially at rest at L 4 in the rotating-pulsating frame: April 22 00:00:00 Ephemeris Time (ET) and January 10 07:18:38 ET. The first epoch was chosen because the resulting trajectory exhibits the expansion and contraction already seen in Chapter 2, and requires long propagation times (5,000 days or more). The second epoch, however, allows the spacecraft to remain in the close vicinity of L 4, without the typical expansion and contraction, and requires a much shorter propagation time (3,000 days or less). More details about both epochs will be presented in the following Sections. In order to illustrate that the spacecraft remains within the close vicinity of L 4, for the January 10 07:18:38 ET epoch, the mean distance from the spacecraft to L 4 throughout the 1,500 propagation is computed using each integrator, and using 5 data points a day. In the case of the RK5 integrator, the mean distance obtained is km, as opposed to km in the BSE case and km in the DLSODE case. This emphasizes the good agreement between the integrators, with a maximum difference of about 50 centimeters over 1,500 days. To further confirm the validity of our results, one can also consider Figures 3.1 to 3.6. For both epochs mentioned above, the trajectory of a spacecraft initially at L 4 was propagated with all three integrators, and the time history of the distance to L 4 was computed in each case, using 5 data points per day. The difference between the time histories obtained was plotted to evaluate the level of agreement between the integrators throughout the propagation. First, for the April 22 epoch, the difference, 97

117 in kilometers, between the time histories of the distance to L 4 obtained with the BSE and RK5 integrators is shown on Figure 3.1 and the difference between the RK5 and DLSODE integrators is shown in Figure 3.2. In both cases the propagation time is 5,000 days. Figure 3.1: Difference between the RK5 and BSE integrators for the April 22 epoch 98

118 Figure 3.2: Difference between the RK5 and DLSODE integrators for the April 22 epoch 99

119 The three integrators appear to be in very good agreement throughout the 5,000 day-propagation, especially considering how long the integration time is. The maximum difference is about 15 to 20 meters in the first case and 25 to 30 meters in the second case, which may seem large in value, but should be compared to the distances considered in the study, and namely the Earth-Moon distance. It corresponds to a level of agreement of about 10 8 between the integrators. These results are all the more remarkable when compared to Figures 3.2 to 3.6, which show the difference between the time histories of the distance to L 4 for the January 10 07:18:38 ET epoch, for 1,500 and 3,000-day propagations. Figure 3.3: Difference between the RK5 and BSE integrators over 1,500 days for the January 10 07:18:38 ET epoch 100

120 Figure 3.4: Difference between the RK5 and DLSODE integrators over 1,500 days for the January 10 07:18:38 ET epoch Figure 3.5: Difference between the RK5 and BSE integrators over 3,000 days for the January 10 07:18:38 ET epoch 101

121 Figure 3.6: Difference between the RK5 and DLSODE integrators over 3,000 days for the January 10 07:18:38 ET epoch 102

122 While the integrators are still in good agreement, especially through 1,500 days, the difference increases significantly at the end of the propagation to reach 4 to 5 km. This case illustrates that the trajectories that do not exhibit the typical expansion-contraction seem to be more sensitive to numerical integrators errors. However, our goal is to determine the characteristics of the motion of spacecraft in the vicinity of L 4 and L 5, and the level of agreement between the integrators shows that their use is appropriate for that purpose. And while the difference between the integrators is not large enough to compromise the validity of the results presented in this chapter, it could definitely become a factor for longer propagation times. 3.5 Influence of the Initial Epoch Apparently Stable Motion It is already known [5] that some epochs produce apparently stable librational motion. The goal in this section is to investigate more thoroughly these epochs and the configurations of the Earth-Moon-Sun system, through the year The study was conducted as follows. For each day in 2007, at 00:00:00 Ephemeris Time (ET), the trajectory of a spacecraft initially at rest in the rotating-pulsating frame, at L 4 and L 5 of the three-body problem is propagated for 2,000 days. If the librational motion persists after 2,000 days, the trajectory is propagated for 3,500 and then 5,000 days. Trajectories for which the librational motion persists for more than 5,000 days will be considered apparently stable. Table 3.1 below shows, for both L 4 and L 5, the epochs that produce apparently stable motions for each month in Interestingly, more epochs were found for L 5 than for L 4. This is in agreement with what had been observed by Schutz, for fewer initial epochs. 103

123 Table 3.1: Epochs of 2007 Producing Apparently Stable Motion Month Number of Epochs L4 Epochs L4 (Calendar Day) Number of Epochs L5 Epochs L5 (Calendar Day) January 6 1, 2, 15, 17, 23, , 14, 22, 29 February 5 8, 15, 16, 21, , 20 March 2 23, , 14, 15, 20, 21 April 2 16, , 13, 14, 18, 19, 20, 28, 29 May 2 14, , 12, 13, 14, 15, 18, 19 June 7 4, 6, 12, 13, 27, 28, , 10, 11, 16, 17, 18 July 6 6, 8, 12, 13, 21, , 3, 5, 10, 17 August 6 3, 4, 11, 25, 26, , 2, 9, 23, 31 September 6 1, 2, 9, 23, 24, , 15, 21, 25, 29 October 3 1, 23, , 22, 23 November 2 22, , 20, 21, 26, 27 December 4 15, 20, 22, , 12, 20, 26,

124 Unlike what happens at L 4, the largest concentration of the apparently stable epochs around L 5 occurs in the spring, with 20 epochs in March, April and May, as opposed to only 6 around L 4 for the same three months. The number of epochs per month is comparable for the rest of the year. Also, the only epochs that produce apparently stable motion about both L 4 and L 5 are May 14, May 15, September 25, October 23 and December 20, even though many of the apparently stable epochs occur within a few days of each other. From Table 3.1 it can be seen that an apparently stable motion can be observed for several consecutive initial epochs, up to 5 days in May about L 5, and therefore there are not only epochs, but regions that produce apparently stable librational motion. The position of the Sun in the Earth-centered rotating frame, for each apparently stable epoch, is shown on Figures 3.7 and 3.8. Figure 3.7: Positions of the Sun producing apparently stable motion about L 4 105

125 Figure 3.8: Positions of the Sun producing apparently stable motion about L 5 106

126 In the case of L 4, the positions of the Sun producing apparently stable motion are a few days before the new Moon, a few days before the full Moon, at approximately -45 degrees with respect to the Earth-Moon line, at approximately 135 degrees ahead of Earth-Moon line, and one epoch was found close to 90 degrees from the Earth-Moon line. In the case of L 5, the epochs obtained are a few days after the full Moon, a few days after the new Moon, with one epoch very close to the new Moon, and at +45 degrees and -135 degrees with respect to the Earth-Moon line. However no epochs were found close to ±90 degrees from the Earth-Moon line. The comparison between Figure 3.7 and Figure 3.8 also shows an approximate symmetry about the x-axis Detailed Examples During the study, it was observed that not only does the epoch influence the duration of the librational motion, but it has a very strong impact on the amplitude and period of the motion as well. To illustrate this, several epochs were chosen and the trajectories associated plotted, this time in the rotating pulsating frame, as well as the distance to the initial position of the spacecraft, which in this frame is L 4 or L 5. They are shown on the figures below. The first case presented is the April 22 epoch at L 4, Figure 3.9 show the trajectory in the (x, y) plane, Figure 3.10 shows the trajectory in the (x, z) plane, and Figure 3.11 shows the distance from the spacecraft to L 4 versus time. 107

127 Figure 3.9: Trajectory in the (x, y) plane for April 22 00:00:00 ET at L 4 Figure 3.10: Trajectory in the (x, z) plane for April 22 00:00:00 ET at L 4 108

128 Figure 3.11: Distance to L 4 in kilometers for April 22 00:00:00 ET 109

129 From Figure 3.11, it appears that the period of the expansion-contraction is approximately 1,000 days, and more interestingly that the amplitude does not seem to increase. This trajectory was then integrated for 15,000 days, and again the distance from the spacecraft to L 4 was plotted, as shown on Figure 3.12, for the first 10,000 days of propagation. The amplitude of the oscillation not only does not increase, but seems to decrease as time elapses, suggesting a very stable behavior. Figure 3.12: Distance to the initial position for April 22 00:00:00 ET propagated for 10,000 days The next case is the March 31 epoch at L 4, Figure 3.13 shows the trajectory in the (x, y) plane propagated for 5,000 days, and Figure 3.14 shows the distance form the spacecraft to L 4. The trajectory in the (x, z) plane for the next cases is not shown, since the amplitude of the out-of-plane motion is much smaller than that of the in-plane motion, as illustrated by Figure

130 Figure 3.13: Trajectory in the (x, y) plane for March 31 00:00:00 ET at L 4 Figure 3.14: Distance to the L 4 in kilometers for March 31 00:00:00 ET 111

131 Here the period of the expansion-contraction seems to change after about 3000 days. At first, the period of the motion is about 1,500 days, but at approximately 3,000 days, the period is reduced to slightly less than 1,000 days. Concurrently, the amplitude of the oscillations tends to increase, indicating that the spacecraft would eventually escape the system. This is confirmed by propagating the trajectory for another 5,000 days, after which the spacecraft enters an elliptic orbit about the Earth. Another interesting epoch for L 4 is December 22, as shown on Figure 3.15 and Figure 3.15: Trajectory in the (x, y) plane for December 22 00:00:00 ET at L 4 112

132 Figure 3.16: Distance to L 4 in kilometers for the December 22 00:00:00 ET In the case of the December 22 epoch, the period is shorter than in the previous cases, about 830 days, and the amplitude does not increase, which would indicate a very stable motion. In order to confirm this, this trajectory was propagated for 10,000 days. As is shown on Figure 3.17, the amplitude of the motion increases greatly after 6,000 days and the spacecraft escapes the system. This emphasizes the fact that even if the motion appears stable for a 16-year propagation time, it is still possible that the spacecraft may escape the vicinity of L

133 Figure 3.17: Distance to L 4 in kilometers for the December 22 Epoch propagated for 6,200 days Consider now the motion at L 5. Several epochs produced very remarkable behaviors, some of which are shown below. First, Figures 3.18 and 3.19 illustrate the trajectory of the spacecraft propagated for 5,000 days and the distance from the spacecraft to its initial position, L 5, for the April 13 epoch. 114

134 Figure 3.18: Trajectory in the (x, y) plane for April 13 00:00:00 ET at L 5 Figure 3.19: Distance to L 5 in kilometers for April 13 00:00:00 ET 115

135 At first, Figure 3.19 shows that the librational motion is close to being periodic, with a period of about 1,200 days. But, between 2,500 and 3,500 days, a new behavior emerges, reminiscent of the behavior found in Chapter 2, with the large oscillations vanishing and much smaller oscillations (approximately 50,000 km in amplitude) appear, with a period near 500 days. After 3,500 days, however, the large oscillations return, with a slightly longer period, about 1,500 days. This trajectory was integrated for 10,000 days, and the distance to the L 4 is shown on Figure This figure shows that there is a second window, about 900 days in length, where the oscillations are smaller (about 75,000 km in amplitude), between 4,700 and 5,600 days. After that, the windows where smaller oscillations are observed appear to get shorter and the amplitude of those oscillations increases. Figure 3.20: Distance to L 5 in kilometers for the April 13 00:00:00 ET epoch propagated for 10,000 days 116

136 Finally, the May 4 epoch about L 5 was selected, as it produces another very interesting trajectory. Figures 3.21 and 3.22 show the trajectory, propagated for 5,000 days, and the distance from the spacecraft to L 5. Figure 3.21: Trajectory in the (x, y) plane for May 4 00:00:00 ET at L 5 117

137 Figure 3.22: Distance to L 5 in kilometers for May 4 00:00:00 ET 118

138 This case exhibits characteristics not observed in other cases. For this epoch, the spacecraft did not experience the typical expansion and contraction for about 1,800 days. As a matter of fact, the spacecraft remained relatively close to its initial position, with only a few peaks above 50,000 km. This behavior is similar to characteristics observed in the BCP in Chapter 2. The first 1,500 days of this trajectory were plotted separately on Figure 3.23 and the distance to the L 5 on Figure It was found that the mean distance to the initial position through 1,500 days was 23,099 km. Figure 3.23: Trajectory in the (x, y) plane for May 4 00:00:00 ET at L 5 propagated for 1,500 days 119

139 Figure 3.24: Distance to L 5 in kilometers for May 4 00:00:00 ET propagated for 1,500 days 120

140 3.5.3 Minimizing the Deviation from L 4 and L 5 The trajectory shown on Figure 3.23 shows that, just as in the BCP, it is possible to choose the initial epoch so that the deviation from the Lagrangian point be significantly reduced. In this section, we will investigate the epochs that produce a motion similar to the May 4 epoch at L 5 and refine the initial epoch to delay the expansion and contraction of the trajectory. To that end, initial epochs producing small oscillations for at least 3,000 days, at both L 4 and L 5 were determined. These epochs were sought using the following method. The trajectory of a spacecraft initially at rest at L 4 or L 5 was propagated for 3,000 days, for two initial epochs a day (00:00:00 ET and 12:00:00 ET) and for each day of the year Using trial and error, the epochs producing the smallest deviations from the libration points were refined so that the trajectory of the spacecraft would not exhibit the typical expansion and contraction for 3,000 days. Table 3.2 lists the initial epochs obtained, as well as the mean distance to the libration point through 1,500 days. Again, the epochs are given in Ephemeris Time. Unlike what was observed with apparently stable motion, the number of epochs producing small deviations from the libration points is very similar in both cases, with 12 epochs at L 4 and 13 at L 5. Interestingly, many favorable epochs at L 4 are very close to those at L 5. For instance the difference between the March 2 epoch at L 4 and the March 3 epoch at L 5 is only 7 hours and 26 minutes, and the difference between the July 8 epoch at L 4 and the July 9 epoch at L 5 is 14 hours and 23 minutes. This was not predicted by the BCP, where a difference of about 50 degrees in the optimal initial value of θ was observed between the L 4 and L 5 case, which the Sun covers in about 4.2 days. 121

141 Table 3.2: Epochs of 2007 Producing Small Deviations from the Libration Points Epoch L4 case Mean Distance to L4 (km) Epoch L5 Case Mean Distance to L5 (km) Jan 10 07:18: Jan 12 05:28: Jan 30 15:31: Jan 29 21:14: Mar 2 22:00: Feb 7 00:26: Apr 2 08:27: Mar 3 05:26: May 1 12:49: May 4 00:30: May 29 14:50: May 28 06:46: Jul 8 09:43: Jun 6 21:57: Aug 9 23:03: Jul 9 00:06: Sep 10 07:51: Jul 23 15:32: Oct 11 01:56: Aug 9 01:00: Nov 9 19:54: Aug 19 23:19: Dec 8 10:01: Sep 13 10:26: X X Dec 12 23:37:

142 The positions of the Sun in the (x, y)-plane of the Earth-centered rotating frame for each favorable epoch is shown on Figure 3.25 for the L 4 case and Figure 3.26 for the L 5 case. The positions corresponding to a mean distance smaller than 25,000 km over 1,500 days (see Table 3.2 above) are represented by a and those corresponding to a mean distance greater than 25,000 km are represented by a square. Figure 3.25: Positions of the Sun producing small deviations from L 4 123

143 Figure 3.26: Positions of the Sun producing small deviations from L 5 124

144 There is a very stark contrast between the two figures. Figure 3.25 is approximately in agreement with the results of the BCP, with the best epochs corresponding to similar configurations of the Earth-Moon-Sun system. There are, however, other favorable epochs (just before the new Moon and just before the full Moon for instance), but they correspond to larger mean distances to L 4. In the L 5 case, there are three epochs corresponding approximately to the predictions of the BCP (the May 4 epoch, the September 13 epoch and the December 12 epoch), and three others, which are just as viable, but were not observed in Chapter 2. They are the July 9 epoch, the July 23 epoch and the August 9 epoch. It should also be noted that, for both libration points, the mean distances observed are about three times as large as those found in the BCP. To illustrate the behavior of a particle for those epochs, the trajectories corresponding to the lowest and highest mean distance are shown for both L 4 and L 5, as well as the time history of the distance to the libration points, on Figures 3.27 to The propagation time is 1,500 days. 125

145 Figure 3.27: Trajectory of a spacecraft initially at rest at L 4 for the May 29 14:50:10 ET epoch Figure 3.28: Time history of the distance to L 4 for a spacecraft initially at rest at L 4 on May 29 14:50:10 ET 126

146 Figure 3.29: Trajectory of a spacecraft initially at rest at L 4 on November 9 19:54:50 ET Figure 3.30: Time history of the distance to L 4 for a spacecraft initially at rest at L 4 on November 9 19:54:50 ET 127

147 Figure 3.31: Trajectory of a spacecraft initially at rest at L 5 on December 12 23:37:50 ET Figure 3.32: Time history of the distance to L 5 for a spacecraft initially at rest at L 5 on December 12 23:37:50 ET 128

148 Figure 3.33: Trajectory of a spacecraft initially at rest at L 5 on March 3 05:26:00 ET Figure 3.34: Time history of the distance to L 5 for a spacecraft initially at rest at L 5 on March 3 05:26:00 ET 129

149 There is an obvious symmetry between the trajectories around L 4 and L 5, for both the largest and smallest mean distance, although the largest mean distance was found at L 5 and the smallest one was found at L 4. But none of the trajectories on Figures 3.27 to 3.34 exhibit the expansion and contraction that was observed for the April 22 00:00:00 ET epoch, for example. The spacecraft remains within 60,000 km of the libration points in the case of figures 3.27 and 3.31, and within 90,000 km of the libration points in Figures 3.29 and Also, the out-of-plane component was not shown, since it is negligible compared to the in-plane motion, as illustrated, in the case of the December 12 23:37:50 ET epoch about L 5, on Figure Figure 3.35: Trajectory in the (x, z)-plane of a spacecraft initially at rest at L 5 on December 12 23:37:50 ET 130

150 3.5.4 Sensitivity to the Initial Epoch As was seen in Chapter 2, the dynamics around the positions of the Sun that produce smaller deviation from the libration points is extremely sensitive to the initial configuration (Figure 2.43). Such sensitivity is therefore expected in this model as well. To illustrate this, a plot similar to Figure 2.43 was obtained for initial epochs producing small deviations from the libration points. This was achieved using the the following method. Starting 10 minutes before the initial epoch considered (obtained from Table 3.2), the trajectory of a spacecraft initially at rest at L 4 or L 5 was propagated and the number of days elapsed before it reached 90,000 km from its initial position was stored. The process was repeated twenty times, increasing the initial epoch by one-minute increment after each propagation. In order to gain better resolution around the epoch that produced the maximum number of days, the experiment was conducted again, this time using 5-second increments, but starting only minute before the initial epoch considered, and stopping one minute after. Figure 3.36 shows the number of days required for a spacecraft to reach 90,000 km from L 4 in the case of the May 29 14:50:10 ET epoch, versus the initial epoch in Julian days, whereas Figure 3.37 depicts the December 12 23:37:50 ET epoch at L

151 Figure 3.36: Number of days elapsed before reaching 90,000 km from L 4 versus initial epoch for May 29 14:50:10 ET Figure 3.37: Number of days elapsed before reaching 90,000 km from L 5 versus initial epoch for December 12 23:37:50 ET 132

152 The parallel between Figures 3.36, 3.37 and 2.43 is clear and it appears that a very similar behavior to the one observed in the BCP has been found using an ephemeris-based model. However, Figures 3.36 and 3.37 correspond to the most favorable epochs found in this study (the mean distances associated with them were the smallest). The epochs corresponding to the largest mean distances were therefore also studied, and the results are shown on Figures 3.38 and Note that since the mean distance in those cases is larger, the number of days required for the spacecraft to reach 100,000 km from L 4 or L 5 was recorded, and not 90,000 km as was done above. Figure 3.38: Number of days elapsed before reaching 100,000 km from L 4 versus initial epoch for November 9 19:54:50 ET 133

153 Figure 3.39: Number of days elapsed before reaching 100,000 km from L 5 versus initial epoch for the March 3 05:26:00 ET 134

154 In the case of L 5, it was also observed that some of the epochs did not match the configuration of the Earth-Moon-Sun system predicted by the BCP. Hence, the August 9 01:00:00 ET epoch at L 5 was also studied to ensure that the results seen in the previous four cases persisted. Again, the number of days required for a spacecraft to reach 90,000 km from L 5 was recorded. Figure 3.40: Number of days elapsed before reaching 90,000 km from L 5 versus initial epoch for August 9 01:00:00 ET 135

155 In every case, the sensitivity to the initial epoch is clear, with smaller deviations from the libration points persisting for up to 1,500 days longer by modifying the initial epoch by less than 30 seconds. In order to compare the results to those obtained in the BCP, the initial epoch was converted to an angle, ψ, defined as the angle between the projection of the Earth-Sun line onto the Earth-Moon plane, and the Earth-Moon line. To match the configuration of the BCP, ψ is defined negatively and ψ = 0 corresponds to a new-moon configuration. Figures 3.41 and 3.42 show the number of days required to reach 90,000 km from L 4 or L 5 versus ψ for the May 29 14:50:10 ET epoch at L 4 and the December 12 23:37:50 ET epoch at L 5. Figure 3.41: Number of days elapsed before reaching 90,000 km from L 4 versus the initial value of ψ for May 29 14:50:10 ET Comparing those results to Figure 2.43, it appears that the sensitivity to the initial value of ψ is even greater than that to the initial value of θ in the BCP. 136

156 Figure 3.42: Number of days elapsed before reaching 90,000 km from L 5 versus the initial value of ψ for December 12 23:37:50 ET However, part of the difference in sensitivity may be due to the fact that, unlike in the case of the BCP, the initial position of the Sun is now out of the (x, y) plane. Also, the angle ψ is measured in a rotating frame centered at the Earth, and not the center of mass of the Earth-Moon system, although this has little impact on the value of ψ. 137

157 3.5.5 Configurations of the Favorable Initial Epochs In this section, the exact configurations of the Earth-Moon-Sun system for the favorable epochs found above are given. The coordinates of the Sun in the Earth-centered Earth-Moon rotating frame as well as the Earth-Moon distance are given in Table 3.3, in the L 4 case, and Table 3.4 for the L 5 case. In Tables 3.3 and 3.4, X Sun is the x-coordinate of the Sun in the Earth-centered Earth-Moon rotating frame, Y sun is its y-coordinate and Z Sun is its z-coordinate. 138

158 Table 3.3: Configuration of the Favorable Epochs of 2007 in the L4 case Epoch Earth-Moon Distance (10 5 km) XSun (10 8 km) YSun (10 8 km) ZSun (10 7 km) Jan 10 07:18: Jan 30 15:31: Mar 2 22:00: Apr 2 08:27: May 1 12:49: May 29 14:50: Jul 8 09:43: Aug 9 23:03: Sep 10 07:51: Oct 11 01:56: Nov 9 19:54: Dec 8 10:01:

159 Table 3.4: Configuration of the Favorable Epochs of 2007 in the L5 case Epoch Earth-Moon Distance (10 5 km) XSun (10 8 km) YSun (10 8 km) ZSun (10 7 km) Jan 12 05:28: Jan 29 21:14: Feb 7 00:26: Mar 3 05:26: May 4 00:30: May 28 06:46: Jun 6 21:57: Jul 9 00:06: Jul 23 15:32: Aug 9 01:00: Aug 19 23:19: Sep 13 10:26: Dec 12 23:37:

160 Comparing Tables 3.3 and 3.4 to Table 3.2, it appears that the initial Earth- Moon distance has very little effect on the resulting value of the mean distance to L 4 and L 5. For instance, the smallest mean distance from L 4 are achieved at the January 30, May 29 and August 9 epochs, for which the initial Earth-Moon distances are approximately 385,000 km, 404,000 km and 378,000 km respectively. Another interesting point is that the last four favorable epochs in the L 4 case occur at similar configuration of the Earth-Moon-Sun system, which can easily be explained by the fact that the initial epochs are approximately 30 days apart, which is the period of the solar motion in the rotating frame. Comparing Tables 3.3 and 3.4 however, one can also notice that if the favorable epochs at L 4 are approximately evenly distributed in time, such is not the case about L 5. In fact, in the L 5 case, two favorable epochs January, two in July and two in August are separated by only 17, 15 and 10 days respectively, whereas the are no favorable epochs in April, October and November Further Refinement and Numerical Integration Settings Similarly to what was done in Chapter 2, the initial epoch can be further refined to remain in the close vicinity of L 4 and L 5 for longer periods of time. Figure 3.36, for instance, suggests that it may be possible to further refine the initial epoch of the May 29 14:50:10 ET case about L 4. Indeed, an initial epoch of May 29 14:50:06.96 ET produces a trajectory that remains within 60,000 km of L 4 for 4,450 days. Figure 3.43 shows the time history of the distance to L 4 for the resulting trajectory. However, as was shown at the beginning of the Chapter, such long propagation times for epochs producing no expansion and contraction of the trajectory can lead to numerical errors. Therefore, further refinement of the initial epoch, so 141

161 Figure 3.43: Time history of the distance to L 4 for May 29 14:50:06.96 ET that the spacecraft would remain close to the libration points for extended periods of time, can pose numerical difficulties that should be investigated and their impact determined before using longer propagation times. 3.6 Conclusions In this Chapter, a model of the Earth-Moon-Sun system was built using SPICE ephemerides, and propagated using a software named COPERNICUS, developed at the University of Texas at Austin by Dr. Cesar Ocampo and his team. The motion near L 4 and L 5 was studied both in an Earth-centered Earth-Moon rotating frame and in an Earth-centered Earth-Moon rotating-pulsating frame. The trajectories were integrated numerically using three integrators: DLSODE and two integrators driven by ODEINT, namely, RK5 and BSE, which are respectively a 5th order Runge-Kutta-Cash-Karp integrator and a Bulirsch-Stoer integrator. 142

162 The first experiments conducted were an extension of the work done by Schutz [5]. The motion of a spacecraft initially at rest at L 4 and L 5 was propagated for each day of the year 2007 at 00:00:00 ET. The initial epochs producing apparently stable librational motion (in the sense that the librational motion persisted for at least 5,000 days) were noted. Through the year 2007, 51 epochs were found that produce apparently stable motion about L 4 and 60 about L 5. The initial configuration of the Earth-Moon-Sun system for each of these epochs were plotted in both the L 4 and L 5 case, and an approximate symmetry between the two cases was found. However, a great variety of behaviors was observed in the librational motion obtained. Some cases experience very large deviations from the libration points, such as the April 22 epoch at L 4, others remain relatively close to the libration points for extended periods of time, such as the May 4 epoch at L 5. This is consistent with what had been observed in Chapter 2, where two initial configurations of the BCP also produced smaller deviations from the libration points. However, the configurations of the Earth-Moon-Sun system producing the smaller deviation can vary greatly, in the ephemeris-based model, and do not necessarily correspond to the configurations of the BCP seen in Chapter 2. Initial epochs producing smaller deviations from L 4 and L 5 for the year 2007 were also determined. Unlike what was found during the first experiment, the number of such epochs was found to be nearly identical for L 4 and L 5 (12 and 13). The epochs were determined by selecting those that allowed the spacecraft to remain in the close vicinity of L 4 and L 5 (less than 90,000 km), and not exhibit the typical expansion-contraction of the trajectory, for at least 3,000 days. The mean distance from the spacecraft to the libration points was also determined through the first 1,500 days of the trajectory. The most favorable initial epochs produced 143

163 trajectories that remained within approximately 21,000 km of L 4 and 22,000 km of L 5 on average. Finally, the sensitivity to the initial epoch was studied. Using the same methodology that was used in Chapter 2, the number of days required for the spacecraft to reach a prescribed distance from L 4 or L 5 (either 90,000 km or 100,000 km, depending on the mean distance from the spacecraft to the libration point for the initial epoch considered) was determined. And as was the case in Chapter 2, there appear to exist critical initial epochs for which the spacecraft never reaches the prescribed distance. As a test case, the May 29 epoch at L 4 was used, the initial epoch was refined, using trial and error, so as to produce a trajectory that remains within 60,000 km of L 4 for 4,500 days. This initial epoch can most likely be refined further. In order to compare these results to those obtained in Chapter 2, the initial epoch was converted to an angle ψ, given by the projection of the position of the Sun on the (x, y)-plane with respect to the Earth-Moon line. The motion appears to be more sensitive to ψ in this model that it was to θ in the BCP, but those results may be skewed, as ψ does not account for the out-of-plane component of the Sun s position. 144

164 Chapter 4 Conclusions and Recommendations 4.1 Summary of the Results This research focused on the Sun-perturbed motion of a particle near the Earth- Moon triangular libration points, with special interest given to the influence of the initial configuration of the Earth-Moon-Sun system. In Chapter 1, a brief overview of the Circular Restricted Three-Body Problem was given, as well as review of previous contributions. Finally, the Bicircular Problem was introduced, and the main characteristics of the motion near L 4 and L 5 were presented. In Chapter 2, the BCP was used to study the nature of the solar perturbation. Fast Fourier Transforms were used to determine that, as was predicted by de Vries [10], there exist monthly and bi-monthly perturbations, of varying amplitude depending on the initial conditions. However, the dominant perturbation was found to have a period of about 1,500 to 1,600 days, and an amplitude of about 250,000 km, and was not predicted by de Vries linearized equations to have such a large influence. These results confirmed earlier works, for instance Tapley and Schutz [13]. It was also shown how, by modifying the initial conditions of the particle, the effect of the resonant perturbation could be negated, and that the resulting trajec- 145

165 tories revolved about the libration points in about three times the period of the solar perturbation. The influence of the initial configuration was studied, through the initial value of the angle θ, which describes the position of the Sun in the Earth- Moon rotating frame. The trajectories of a spacecraft initially at rest at L 4 or L 5 were propagated for up to 2,300 days and the mean distance from the spacecraft to its initial position was computed and stored. Two initial values of θ were found to produce very steep minima of the mean distance. The vicinity of these minima was also considered, and two critical initial configurations of the Earth-Moon-Sun system were identified, for which the spacecraft would potentially never escape the close vicinity of the libration points (less than 30,000 km form L 4 or L 5 ). The first minimum in the L 4 was used as an example, and the initial value of θ was refined so that the spacecraft remained within 30,000 km of L 4 for a propagation time of almost 7,000 days. The resulting trajectory, however, was shown to be extremely sensitive to the initial value of θ, since a difference in θ of the order of degrees produced a trajectory that remained within 30,000 km of L 4 for 2,400 fewer days. Finally, it was observed that the largest deviations from the libration points occurred in the first few hundred days of propagation, and therefore an easy way to allow the spacecraft to remain closer to L 4 or L 5 is to start the propagation at a later time on the same trajectory. This was done in the case of the first minimum about L 4 and the resulting trajectory was a quasi-periodic orbit, that revolved about L 4 in approximately three times the period of the solar perturbation. In Chapter 3, a more realistic model of the Earth-Moon-Sun system was considered. Using the COPERNICUS software, currently in development at the University of Texas at Austin, the motion of a particle near the triangular points of the Earth-Moon system was studied using the SPICE ephemerides. First, the initial 146

166 epochs of the year 2007 producing apparently stable librational motion - in the sense that the libration motion persisted for at least 5,000 days - were determined. It was found that there exist 51 such epochs in the L 4 case and 60 in the L 5 case, and the initial configurations producing apparently stable motion were found to be approximately symmetrical. Through these experiments, it was also determined that the nature of the librational motion varies greatly depending on the epoch. Interestingly, some motions remain comparatively very close to the libration points. Such is the case, for instance for the May 4 epoch in the L 5 case. This led to the second part of the experiment, determining which epochs produce the smallest deviations from the libration points. In the year 2007, 12 epochs were found in the L 4 case and 13 in the L 5 for which the trajectory of the spacecraft would not exhibit the typical expansion and contraction, and remain in the close vicinity of the libration points for at least 3,000 days. This is very reminiscent of what was observed in Chapter 2. This time, however, the mean distance of the spacecraft to the libration points was on the order of 25,000 km, as opposed to 10,000 km in Chapter 2. The sensitivity to the initial epoch was considered, and for initial configurations close to those producing the smallest deviations, the number of days required for the spacecraft to reach 90,000 km from L 4 or L 5 was determined. The results bore a striking resemblance with what was observed in Chapter 2, and suggest that in this model as well, there exist critical initial epochs for which the spacecraft would never leave the close vicinity of the triangular points. This was tested using the May 29 epoch of the L 4 case, and the initial epoch could be refined so that the spacecraft would remain within 60,000 km of L 4 for about 4,500 days. But in this model again, our results clearly show how sensitive the resultant motion is to the initial configuration. 147

167 4.2 Discussion and Future Work The results from Chapter 2 confirmed that it is possible to find initial conditions in the BCP for which the effects of the resonant perturbation vanish, and the spacecraft remains in the close vicinity of the libration points. This had already been observed by other authors, for instance Castella and Jorba [20], who found invariant tori that grew from the periodic orbits initially discovered by Kamel [3] and Kolenkiewicz and Carpenter [17]. Our trajectories however, are planar, and revolve around L 4 and L 5 in about three times the period of the solar perturbation. While Wolaver [7] expected that the initial values of θ that produce the minimal deviations from L 4 would be offset by 180 degrees, and our experiments have shown that such is actually not the case. This is understandable, as the problem is actually not symmetrical with respect to the origin. It was also very interesting to find that, just by choosing an appropriate initial value of θ, the motion of the spacecraft can remain bounded within 30,000 km of the libration points. Some tests were conducted to show that the mean distance from the spacecraft to the triangular points can be further reduced by slightly modifying the initial conditions, but only for limited propagation times, unlike the solution shown on Figure 2.54 that could be prolonged indefinitely, provided that the numerical integration is accurate enough for such long propagation times. Future work in the Bicircular Problem could include the use of stroboscopic maps to study the motion the vicinity of the libration points, similarly to what was done by Scheeres [31]. Such a map would record the position and velocity of a spacecraft for the same position of the Sun every solar period, and possibly provide a better understanding of the dynamics near L 4 and L 5. Another interesting approach would be to generalize the results obtained in this Dissertation to other systems than 148

168 the Earth-Moon-Sun system, by considering the impact of changing the values of µ, m s and a s on the mean deviation from the libration points (Figure 2.46). While they provided some interesting insight into the dynamics of the problem, the results of Chapter 2 are, for practical purposes, of little use. The fact that some of the characteristics of the BCP seem to translate to a more realistic model, however, is very promising. In Chapter 3, the work of Schutz [5] was first expanded by testing, as an initial epoch, every day of the year Our study confirmed that there were indeed more initial epochs producing apparently stable motion at L 5 than L 4-60 to 51. But the sample size used (1 year) is too short to draw any definite conclusions, especially that the lunar nodes precess along the ecliptic with respect to the equinox with a period of 18.6 years. This will undeniably have an effect on the motion near L 4 and L 5. Initial epochs should therefore be tested on a much longer time frame (at least 18.6 years), to determine if the discrepancy between the number of apparently stable epochs persists. The influence of the initial epoch has a very strong influence on the nature of the librational motion as well. The April 22 epoch at L 4, for instance, produces apparently very stable motion but very large deviations from the triangular point. In contrast, the May 4 epoch produces relatively small deviations from L 5 for about 1,800 days. This phenomenon can definitely be useful in planning a mission to the triangular points, and our study found 25 initial epochs (12 at L 4 and 13 at L 5 ) for which the spacecraft remains in a close vicinity of the libration points (approximately 25,000 km). As an example, the Aug 9 23:03:00 ET epoch, produces deviations of, at most, about 50,000 km, and this for well over 2,000 days, except for an isolated peak at the beginning of the propagation. This peak can be averted simply by starting the propagation at a later time on the trajectory. The time history of the 149

169 distance to L 4 is given on Figure 4.1 below. Figure 4.1: Time history of the distance to L 4 for August 9 23:03:00 ET This trajectory can be compared to the one suggested by Companys et al [24], who found a quasi-periodic solution that remains within 43,000 km of L 4 in the x-direction and 31,000 km in the y-direction. Our solution remains within 48,000 km of L 4 in the x-direction (except for the isolated peak mentioned above), and 33,000 km in the y-direction, and this for more than 2,000 days. Our trajectory, however, can likely be improved by modifying the initial conditions, which could be done as part of a future study. But it is interesting to note that simply by adjusting the initial epoch, a solution can be found that is comparable, in terms of deviation from the libration points, to the solution proposed by Companys et al. Finally, even though the model used in Chapter 3 is much more realistic than the Bicircular Problem, it could still be improved. The most significant effect not included in our study is solar radiation pressure, but including it would require some 150