Abstract
In this analysis, resonant flybys were explored within the context of the circular restricted three-body problem using dynamical systems theory. The first step in this process involved the construction of a flyby trajectory continuously transiting between 3:4 and 5:6 resonances in the Jupiter-Europa circular restricted three-body problem. An examination of this trajectory revealed that it followed the invariant manifolds of resonant orbits during these transitions. It was discovered that these transitions occurred for specific energies where the invariant manifolds of the 3:4 and 5:6 resonant orbits were closely related. The potential of the information obtained from this analysis for use in mission design was demonstrated by developing resonance transition trajectories using resonant orbit homoclinic and heteroclinic connections.
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LEXELL, A.J. and EULER, L. Recherches et calculs sur la vraie orbite elliptique de la comète de l’an 1769. St Pétersbourg: De l’impr. de l’Academie imperiale des sciences, 1770.
LEXELL, A.J. Reflexions sur le temps périodique des comètes en général et principalement sur celui de la cométe observée en 1770. 1772.
LEXELL, A.J. “Recherches sur la période de la comète, observée en 1770, d’après les observations de M. Messier,” Histoire de l’Académie Royale des Sciences, mémoires, année 1776, 1779, pp. 638–651.
LAPLACE, P.S. Traité de Mécanique Céleste, Vol. 4. Paris: Courcier, 1805.
LE VERRIER, U.J. “Theorie de la Comete Periodique de 1770,” Annales de l’Observatoire de Paris, Vol. 3, 1857, pp. 203–270.
D’ALEMBERT, J. Opuscules Mathématiques, Vol. 6, Chez Briasson, Paris, 1773.
TISSERAND, F. “Sur la Théorie de la Capture des Comètes Périodiques,” Bulletin Astronomique, Vol. 6, June 1889, pp. 241–257.
TISSERAND, F. Traité de la Mécanique Céleste: Théories des Satellites de Jupiter et de Saturne Perturbations des Petites Planétes, Vol. 4, Gauthier-Villars, Paris, 1896.
MINOVITCH, M.A. “A Method for Determining Interplanetary Free-Fall Reconnaissance Trajectories,” Technical Memo 312-130, Jet Propulsion Laboratory, August 1961.
MINOVITCH, M.A. “The Determination and Characteristics of Ballistic Interplanetary Trajectories under the Influence of Multiple Planetary Attractions,” Technical Report 32-464, Jet Propulsion Laboratory, October 1963.
STURMS, F. M. and CUTTING, E. “Trajectory Analysis of a 1970 Mission to Mercury via a Close Encounter with Venus,” Technical Report 32-943, Jet Propulsion Laboratory, May 1966.
FLANDRO, G.A. “Fast Reconnaissance Missions to the Outer Solar System Utilizing Energy Gained from the Gravitational Field of Jupiter,” Astronautica Acta, Vol. 12, No. 4, 1966.
STRANGE, N.J. and LONGUSKI, J.M. “Graphical Method for Gravity-Assist Trajectory Design,” Journal of Spacecraft and Rockets, Vol. 39, January–February 2002, pp. 9–16.
OKUTSU, M., DEBBAN, T.J., and LONGUSKI, J.M. “Tour Design Strategies for the Europa Orbiter Mission,” presented as paper AAS 01-463 at the AAS/AIAA Astrodynamics Specialist Conference, July 30–August 2 2001.
BOLLT, E. and MEISS, J.D. “Targeting Chaotic Orbits to the Moon through Recurrence,” Physics Letters A, Vol. 204, August 28 1995, pp. 373–378.
SCHROER, C.G. and OTT, E. “Targeting in Hamiltonian Systems that have Mixed Regular/Chaotic Phase Spaces,” Chaos, Vol. 7, December 1997, pp. 512–519.
BELBRUNO, E. and MARSDEN, B. “Resonance Hopping in Comets,” Astronomical Journal, Vol. 113, No. 4, 1997, pp. 1433–1444.
LO, M. and ROSS, S. “Low Energy Interplanetary Transfers Using Invariant Manifolds of LI, L2, and Halo Orbits,” Proceedings of the AAS/AIAA Space Flight Mechanics Meeting, Monterey, California, February 9–11 1998.
KOON, W.S., LO, M.W., MARSDEN, J.E., and ROSS, S.D. “Heteroclinic Connections between Periodic Orbits and Resonance Transitions in Celestial Mechanics,” Chaos, Vol. 10, June 2000, pp. 427–469.
HOWELL, K.C., MARCHAND, B., and LO, M.W. “Temporary Satellite Capture of Short-Period Jupiter Family Comets from the Perspective of Dynamical Systems,” The Journal of the Astronautical Sciences, Vol. 49, October–December 2001, pp. 539–557.
LO, M.W., ANDERSON, R.L., WHIFFEN, G., and ROMANS, L. “The Role of Invariant Manifolds in Low Thrust Trajectory Design (Part I),” presented as paper AAS 04-288 at the AAS/AIAA Spaceflight Dynamics Conference, Maui, Hawaii, February 8–12, 2004.
ANDERSON, R.L. and LO, M.W. “The Role of Invariant Manifolds in Low Thrust Trajectory Design (Part II),” presented as paper AIAA 2004-5305 at the AIAA/AAS Astrodynamics Specialist Conference, Providence, Rhode Island, August 16–19, 2004.
LO, M.W., ANDERSON, R.L., LAM, T., and WHIFFEN, G. “The Role of Invariant Manifolds in Low Thrust Trajectory Design (Part III),” presented as paper AAS 06-190 at the AAS/AIAA Astrodynamics Specialist Conference, Tampa, Florida, January 22–26, 2006.
ANDERSON, R.L. and LO, M.W. “Role of Invariant Manifolds in Low-Thrust Trajectory Design,” Journal of Guidance, Control, and Dynamics, Vol. 32, November–December 2009, pp. 1921–1930.
ANDERSON, R.L. and LO, M.W. “Dynamical Systems Analysis of Planetary Flybys and Approach: Planar Europa Orbiter,” Journal of Guidance, Control, and Dynamics, Vol. 33, November–December 2010, pp. 1899–1912.
ANDERSON, R.L. Low Thrust Trajectory Design for Resonant Flybys and Captures Using Invariant Manifolds, Ph.D. Dissertation, University of Colorado at Boulder, ccar.colorado.edu/∼rla/papers/andersonphd.pdf, 2005.
SZEBEHELY, V. Theory of Orbits: The Restricted Problem of Three Bodies, Academic Press, New York, 1967.
ROY, A.E. Orbital Motion, Adam Hilger, Philadelphia, Pennsylvania, Third Ed., 1988.
MURRAY, CD. and DERMOTT, S.F. Solar System Dynamics, Cambridge University Press, Cambridge, United Kingdom, 1999.
BARRABÉS, E. and GÓMEZ, G. “Three-Dimensional p-q Resonant Orbits Close to Second Species Solution,” Celestial Mechanics and Dynamical Astronomy, Vol. 85, 2003, pp. 145–174.
HOWELL, K.C. and BREAKWELL, J.V. “Three-Dimensional, Periodic, ‘Halo’ Orbits,” Celestial Mechanics, Vol. 32, 1984, pp. 53–71.
MASDEMONT, J. and MONDELO, J.M. “Notes for the Numerical and Analytical Techniques Lectures (draft version),” Advanced Topics in Astrodynamics Summer Course, Barcelona, July, 2004, July 2004.
WIGGINS, S. “Introduction to Applied Nonlinear Dynamical Systems and Chaos,” Vol. 2 of Texts in Applied Mathematics. Second Ed., New York: Springer-Vertag, 2003.
PARKER, T. and CHUA, L.O. Practical Numerical Algorithms for Chaotic Systems, Springer-Veriag, New York, 1989.
JOHANNESEN, J.R. and D’AMARIO, L.A. “Europa Orbiter Mission Trajectory Design,” presented as paper AAS 99-360 at the AAS/AIAA Astrodynamics Specialist Conference, Girdwood, Alaska, August 16–19, 1999.
WILSON, R.S. “Derivation of Differential Correctors Used in GENESIS Mission Design,” IOM 312.1-03-002, Jet Propulsion Laboratory, 2003.
PERNICKA, H.J. “The Numerical Determination of Lissajous Orbits in the Circular Restricted Three-Body Problem,” Master’s Thesis, Purdue University, December 1986.
ANDERSON, R.L., LO, M.W., and BORN, G.H. “Application of Local Lyapunov Exponents to Maneuver Design and Navigation in the Three-Body Problem,” presented as paper AAS 03-569 at the AAS/AIAA Astrodynamics Specialist Conference, Big Sky, Montana, August 3–7, 2003.
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Anderson, R.L., Lo, M.W. A Dynamical Systems Analysis of Resonant Flybys: Ballistic Case. J of Astronaut Sci 58, 167–194 (2011). https://doi.org/10.1007/BF03321164
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DOI: https://doi.org/10.1007/BF03321164