AIAA-46400-479

Engineering Notes Analysis of Periodic and Quasi-Periodic Orbits in the Earth–Moon System Pooja Dutt∗ and R. K. Sharma∗ Indian Space Research Organisation, Thiruvananthapuram, 695 022, India DOI: 10.2514/1.46400 I. Introduction O NE of the earliest investigations of the Poincaré surfaces ofsection method [1] to discover periodic orbits in the restricted three-body problemwas byHenon [2–5] in a number of publications. Jefferys [6] also used this method to evolve a large number of periodic orbits in the restricted three-body problem. After that, this technique has been used by many researchers to find periodic orbits in the restricted three-body problem. Some of the important contributions are by Smith [7], Smith and Szebehely [8], Tuckness [9,10], Scott and Spencer [11], Howell and Kakoi [12], Demeyer and Gurfil [13], Guilera [14], and Kolemen et al. [15]. The motivation for the present study comes from the work of Winter [16], which deals with a family of simply symmetric retrograde periodic orbits around the moon in the rotating Earth–moon–particle system in the framework of the planar circular restricted three-body problem for the Earth–moon mass ratio (0.01215). This family of periodic orbits is one particular case of the family of periodic orbits classified by Broucke [17]. II. Planar Circular Restricted Three-Body Problem In this model, two bodies (known as the primaries) of masses m1 and m2 are assumed to rotate about their center of mass in circular orbits in a plane under the influence of their mutual gravitational attraction [18,19]. The objective is to describe the motion of a third infinitesimal mass, typically representing a spacecraft. To simplify the analysis for this study, the infinitesimal mass is further restricted in the plane of motion of the two primaries. The resulting problem is commonly referred to as the planar circular restricted three-body problem. The unit of length is the distance between the primaries and the unit of time is chosen in order to set the gravitational parameter as unity. We use a rotating coordinate system �x; y�, which rotates uniformlywith unit velocity, where the x axis is along the line joining the two primaries. In the present study, we simulate the motion of a spacecraft placed in the Earth–moon system. The coordinates are scaled so that the Earth is at the location ���; 0�, and the moon is at �1 � �; 0�, where� is the mass ratiom2=�m1 �m2�, andm1 andm2 are themasses of Earth andmoon, respectively. The equations for the motion of the spacecraft in the rotating frame of reference are �x � 2_y � x���1 � �� �x� �� r31 � � �x � 1� �� r32 (1) �y� 2_x � y�� � 1 � � r31 � � r32 � y (2) where r21 � �x� ��2 � y2; r22 � �x � 1� ��2 � y2 (3) The dynamical system has an integral of the motion, known as Jacobi’s constant, given by C� x2 � y2 � 2 � 1 � � r1 � � r2 � � _x2 � _y2 (4) III. Poincaré Surface of Section The orbital elements of the test particle at any instant can be determined from its position �x; y� and velocity � _x; _y�, which correspond to a point in a four-dimensional phase space. To visualize this four-dimensional manifold (which includes the periodic orbits, quasi-periodic orbits, and the nonperiodic trajectories on a two- dimensional figure), we need to constrain the manifold by two dimensions. A convenient way of achieving this is to choose the trajectories that have the same value of C, and to take a Poincaré sectionwhen these orbits cross a plane (say, y� 0). This technique is good at identifying periodic and quasi-periodic orbits, determining the regular or chaotic nature of the trajectory, and determining the extent of the particle’s motion throughout the phase space. As per Kolmogorov–Arnold–Moser (KAM) theory, the point represents a periodic orbit in the rotating frame, and the closed curves around the point correspond to the quasi-periodic orbits. IV. Numerical Results As inWinter [16], the starting conditions in our computationswere chosen such that for each value of the Jacobi constantC, the values of x were selected by taking y� _x� 0 and _y > 0. This meant that the integrationwas usually started at the pericenter of the particle’s orbit. We have used the fourth-order Runge–Kutta–Gill method for numerical integration of the equations of motion. The integration step size of 0.01 in time was found to be adequate for our numerical studies. Over 1500 starting conditions in the range of Jacobi constant C between 1.5 and 4.24, which covers the region that contains the periodic orbits considered, were used. Experimentation for the distance interval �x between two consecutive starting conditions with the sameCwas done for�x� 0:01 to 0.0001. It was found that �x� 0:001 ismore suitable for our studies. In the present dynamical system, the largest of the quasi-periodic orbits (KAM tori), which correspond to the maximum amplitude oscillation, can be taken as a parameter tomeasure the degree of stability of the periodic orbit with respect to the region around it in the phase space. With the aim to obtain periodic orbits near the moon, we have generated Poincaré surfaces of section up toC� 4:24.We divide this family of orbits into two categories. Category I is orbits consisting of oscillations near the moon, and category II is orbits consisting of oscillations near the Earth. The evolution of the category I orbits, in terms of increasing value of the Jacobi constant, can be further divided into four distinct stages. In the first stage, from C� 2:4 to 2.84, the maximum amplitude of oscillation decreases uniformly until it disappears at C� 2:84895. Then, in the second stage, the islands reappear again atC� 2:85 and the stability grows, reaching a peak at C� 2:9 and then decreasing until they disappear completely around C� 2:955. The periodic orbit at C� 2:954 corresponds to the collinear Lagrangian equilibrium point L1 (x� 0:83689). In the above two zones, our Received 19 July 2009; revision received 9 February 2010; accepted for publication 9 February 2010. Copyright © 2010 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 0731-5090/10 and $10.00 in correspondence with the CCC. ∗Vikram Sarabhai Space Centre, Applied Mathematics Division. JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS Vol. 33, No. 3, May–June 2010 1010 Fig. 2 Poincaré surface of section for Jacobi constant C� 2:84895. Fig. 1 Poincaré surface of section for Jacobi constant C� 2:84. J. GUIDANCE, VOL. 33, NO. 3: ENGINEERING NOTES 1011 Fig. 3 Poincaré surface of section for Jacobi constant C� 2:86. Fig. 4 Poincaré surface of section for Jacobi constant C� 2:95. 1012 J. GUIDANCE, VOL. 33, NO. 3: ENGINEERING NOTES Fig. 5 Poincaré surface of section for Jacobi constant C� 2:955. Fig. 6 Poincaré surface of section for Jacobi constant C� 2:98. J. GUIDANCE, VOL. 33, NO. 3: ENGINEERING NOTES 1013 Fig. 7 Location of the periodic orbit as a function of Jacobi constant for the two categories of periodic orbits. Fig. 8 Size ofKAMtori as a function of Jacobi constantC. The upper line corresponds to the leftmost tip and the lower line corresponds to the rightmost tip of the island obtained by Poincaré surface of section, at the line of conjunction. 1014 J. GUIDANCE, VOL. 33, NO. 3: ENGINEERING NOTES Fig. 9 Poincaré surface of section for Jacobi constant C� 3:05. Fig. 10 Poincaré surface of section for Jacobi constant C� 3:15. J. GUIDANCE, VOL. 33, NO. 3: ENGINEERING NOTES 1015 results and figures do not match reasonably with Winter [16]. The differencesmay be attributed to the finer step size of�x� 0:001 that we have used in our study. The step size used byWinter is 0.01. In the third stage, the size of the KAM tori increases up to C� 3:066 and then it decreases as C increases up to 4.24, where the periodic orbits are very close to the moon [17]. The disappearance of the region of stability is caused by the intersection of the central periodic orbit and the unstable periodic orbit lying at the three corners of the triangular stability region [20]. This transition atC� 2:84895 and 2.955 is shown in Figs. 1–6. This category of periodic orbits slowly traverses toward the moon. Figure 7 gives the location of the periodic orbit as a function of the Jacobi constant. These results were generated from the Poincaré surface of section, considering, for each Jacobi constant, the size of the largest island (quasi-periodic orbit) in the line of conjunction (values of xwhen _x� 0). Figure 8 provides the size of the KAM tori from C� 2:6 to 4.3. Meanwhile, another set of islands is created. They attain peak stability, merge with other sets of islands, and mostly remain near Earth. Figure 7 clearly shows the evolution of category II periodic orbits at C� 2:7, whose stability keeps increasing up to C� 3:05. Meanwhile, another set appears at C� 2:85. Its stability increases and then merges with the bigger island. The merged orbit keeps gaining stability thereafter with increase in the value of C. Figures 9 and 10 give the representatives of this family. One of the interesting surfaces of section is forC� 2:7, which has been expanded in Fig. 11. The center of the very first (segregated) island, which belongs to category II is at x� 0:163639 and _x� 0:0. With this initial conditionwhen the system of equations ofmotion (1) and (2) is integrated, a stable periodic orbit around Earth is obtained. The central point of the second island is at x� 0:460175 and _x� 0:0, which yields a kidney-bean-shaped stable periodic orbit around themoon. Similarly shaped quasi-periodic orbits are obtained from the other points of the surface of section. It is clearly seen that as wemove away from the center of the island, the periodic orbit slowly changes into a quasi-periodic orbit, preserving the shape. Sensitivity to initial conditions was also analyzed in some of the cases. If the trajectory is started from the point at the lower end of the bigger island, with x� 0:5919 and _x��0:4224, the particle no longer remains in the same region. Its trajectory moves toward the moon but maintains a distance. But if the initial condition is perturbed in the fifth decimal place and integrated, the trajectory moves and is captured at the moon near t� 3:5. Such sensitivity to initial conditions is suggestive of chaos. V. Conclusions We have used the technique of Poincaré surface of section to study a family of periodic and quasi-periodic orbits. Using this technique we have determined the location of the periodic orbits and their stability in terms of the maximum amplitude of oscillation. We have divided this family of orbits into two categories: namely, the category of islands near moon (category I) and those near the Earth (category II). We found that the category I orbits slowly traverse toward the moon and that there is a kind of separatrix of two different types of quasi-periodic orbit around the periodic orbits at values of the Jacobi constant equal to 2.84895 and 2.955. The category II orbits remain mostly near Earth and their stability increases with increasing value of the Jacobi constant. Acknowledgments The authors would like to thank S. V. Sharma, Deputy Director, Vikram Sarabhai Space Centre (Aeronautics Entity), and S. Swaminathan, Group Director, Aerospace Flight Dynamics Group, for their kind support. The authors are highly thankful to the Associate Editor, Robert Melton, and the three reviewers for their very constructive and motivating comments, which helped in bringing the paper to the present form. Fig. 11 Various orbits obtained from the Poincaré surface of section for C� 2:7. 1016 J. GUIDANCE, VOL. 33, NO. 3: ENGINEERING NOTES References [1] Poincaré, H., Les Méthodes Nouvelles de la Mécanique Celeste, Vol. 1, Gauthier-Villas, Paris, 1892, p. 82. [2] Henon, M., “Exploration Numerique du Probleme Restreint: I,” Annales d’Astrophysique, Vol. 28, Feb. 1965, pp. 499–511. [3] Henon, M., “Exploration Numerique du Probleme Restreint: II,” Annales d’Astrophysique, Vol. 28, Feb. 1965, pp. 992–1007. [4] Henon, M., “Exploration Numerique du Probleme Restreint: III,” Bulletin Astronomique, Ser. 3, Vol. 1, 1965, pp. 57–79. [5] Henon, M., “Exploration Numerique du Probleme Restreint: IV,” Bulletin Astronomique, Ser. 3, Vol. 2, 1965, pp. 49–66. [6] Jefferys, W. 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V., “Stationary Solutions and Their Characteristic Exponents in the Restricted Three-Body ProblemWhen theMoreMassive Primary is anOblate Spheroid,”CelestialMechanics, Vol. 13, No. 2, 1976, pp. 137–149. doi:10.1007/BF01232721 [20] Henon, M., “Numerical Exploration of the Restricted Problem, VI: Hill’s Case: Non-Periodic Orbits,”Astronomy and Astrophysics, Vol. 9, Nov. 1970, pp. 24–36. J. GUIDANCE, VOL. 33, NO. 3: ENGINEERING NOTES 1017