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New methods in celestial mechanics and mission design

Authors: Jerrold E. Marsden and Shane D. Ross
Journal: Bull. Amer. Math. Soc. 43 (2006), 43-73
MSC (2000): Primary 70F07, 70F15; Secondary 37J45, 70H33
Published electronically: November 22, 2005
MathSciNet review: 2188175
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Abstract: The title of this paper is inspired by the work of Poincaré [1890, 1892], who introduced many key dynamical systems methods during his research on celestial mechanics and especially the three-body problem. Since then, many researchers have contributed to his legacy by developing and applying these methods to problems in celestial mechanics and, more recently, with the design of space missions.

This paper will give a survey of some of these exciting ideas, and we would especially like to acknowledge the work of Michael Dellnitz, Frederic Gabern, Katalin Grubits, Oliver Junge, Wang-Sang Koon, François Lekien, Martin Lo, Sina Ober-Blöbaum, Kathrin Padberg, Robert Preis, and Bianca Thiere.

One of the purposes of the AMS Current Events session is to discuss work of others. Even though we were involved in the research reported on here, this short paper is intended to survey many ideas due to our collaborators and others.

This survey is by no means complete, and we apologize for not having time or space to do justice to many important and fundamental works. In fact, the results reported on here rely on and were inspired by important preceding work of many others in celestial mechanics, mission design and in dynamical systems. We mention just a few whose work had a positive influence on what is reported here: Brian Barden, Ed Belbruno, Robert Farquhar, Gerard Gómez, George Haller, Charles Jaffé, Kathleen Howell, Linda Petzold, Josep Masdemont, Vered Rom-Kedar, Radu Serban, Carles Simó, Turgay Uzer, Steve Wiggins, and Roby Wilson. In an upcoming monograph (see Koon, Lo, Marsden, and Ross [2005]), the dynamical systems and computational approach and its application to mission design are discussed in detail.

One of the key ideas is that the competing gravitational pull between celestial bodies creates a vast array of passageways that wind around the Sun, planets and moons. The boundaries of these passageways are realized geometrically as invariant manifolds attached to equilibrium points and periodic orbits in interlinked three-body problems. In particular, tube-like structures form an interplanetary transport network which will facilitate the exploration of Mercury, the Moon, the asteroids, and the outer solar system, including a mission to assess the possibility of life on Jupiter's icy moons. The use of these methods in problems in molecular dynamics of interest in chemistry is also briefly discussed.

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  • [Bate et al.(1971)] Bate, R. R., D. D. Mueller, and J. E. White [1971], Fundamentals of Astrodynamics.
    Dover, New York.
  • [Belbruno(2004)] Belbruno, E. [2004], Capture Dynamics and Chaotic Motions in Celestial Mechanics: With Applications to the Construction of Low Energy Transfers, Princeton University Press. MR 2029316 (2004m:70020)
  • [Combes et al.(1999)] Combes, F., S. Leon, and G. Meylan [1999], $ N$-body simulations of globular cluster tides, Astron. Astrophys. 352, 149-162.
  • [Dellnitz et al.(2001)] Dellnitz, M., G. Froyland, and O. Junge [2001], The algorithms behind GAIO-set oriented numerical methods for dynamical systems. In Ergodic theory, analysis, and efficient simulation of dynamical systems, pages 145-807. Springer, Berlin. MR 1850305 (2002k:65217)
  • [Dellnitz et al.(2005)] Dellnitz, M., K. Grubits, J. E. Marsden, K. Padberg, and B. Thiere [2005], Set-oriented computation of transport rates in 3-degree of freedom systems: the Rydberg atom in crossed fields, Regular and Chaotic Dynamics 10, 173-192.
  • [Dellnitz and Junge(2002)] Dellnitz, M. and O. Junge [2002], Set oriented numerical methods for dynamical systems. In Handbook of Dynamical Systems, Vol. 2, pages 221-264. North-Holland, Amsterdam. MR 1900656 (2003f:37161)
  • [Dellnitz et al.(2005)] Dellnitz, M., O. Junge, W. S. Koon, F. Lekien, M. W. Lo, J. E. Marsden, K. Padberg, R. Preis, S. Ross, and B. Thiere [2005], Transport in dynamical astronomy and multibody problems, Intern. J. of Bifurcation and Chaos 15, 699-727. MR 2136742
  • [Dellnitz et al.(2005a)] Dellnitz, M., O. Junge, M. W. Lo, J. E. Marsden, K. Padberg, R. Preis, S. Ross, and B. Thiere [2005a], Transport of Mars-crossing asteroids from the quasi-Hilda region, Physical Review Letters 94, 231102-1-231102-4.
  • [Dellnitz et al.(2005b)] Dellnitz, M., O. Junge, J. E. Marsden, K. Padberg, R. Preis, S. Ross, and B. Thiere [2005b], Almost invariant sets and celestial mechanics (in preparation).
  • [Dunn(1962)] Dunn, G. L. [1962], A high-speed data link for farside lunar communications, General Electric Co. Report 62 SPC-5, March 1962.
  • [Farquhar(1966)] Farquhar, R. W. [1966], Station-Keeping in the Vicinity of Collinear Libration Points with an Application to a Lunar Communications Problem. In Space Flight Mechanics, Science and Technology Series, volume 11, pages 519-535. American Astronautical Society, New York.
  • [Farquhar and Dunham(1981)] Farquhar, R. W. and D. W. Dunham [1981], A new trajectory concept for exploring the Earth's geomagnetic tail, Journal of Guidance and Control, 4, 192-196.
  • [Farquhar et al.(1980)] Farquhar, R. W., D. P. Muhonen, C. Newman, and H. Heuberger [1980], Trajectories and Orbital Maneuvers for the First Libration-Point Satellite, Journal of Guidance and Control, 3, 549-554.
  • [Farquhar et al.(1977)] Farquhar, R. W., D. P. Muhonen, and D. L. Richardson [1977], Mission Design for a Halo Orbiter of the Earth, Journal of Spacecraft and Rockets 14, 170-177.
  • [Farquhar(1968)] Farquhar, R. [1968], The Control and Use of Libration-Point Satellites, PhD thesis, Stanford University.
  • [Fukushige et al.(2000)] Fukushige, T. and D. C. Heggie [2000], The time-scale of escape from star clusters, Mon. Not. R. Astron. Soc. 318, 753-761.
  • [Gabern et al.(2005)] Gabern, F., W.-S. Koon, J. E. Marsden and S. D. Ross [2005], Theory and computation of non-RRKM lifetime distributions and rates in chemical systems with three or more degrees of freedom, Physica D (to appear).
  • [Gómez et al.(2004)] Gómez, G., W. S. Koon, M. W. Lo, J. E. Marsden, J. Masdemont, and S. D. Ross [2004], Connecting orbits and invariant manifolds in the spatial restricted three-body problem, Nonlinearity 17, 1571-1606. MR 2086140 (Review)
  • [Howell et al.(1997)] Howell, K., B. Barden, and M. Lo [1997], Application of dynamical systems theory to trajectory design for a libration point mission, The Journal of the Astronautical Sciences 45, 161-178. MR 1604793
  • [Jaffé et al.(2002)] Jaffé, C., S. D. Ross, M. W. Lo, J. E. Marsden, D. Farrelly, and T. Uzer [2002], Statistical Theory of Asteroid Escape Rates, Phys. Rev. Lett. 89, 011101-1.
  • [Junge et al.(2005)] Junge, O., J. E. Marsden, and S. Ober-Blöbaum [2005], Discrete mechanics and optimal control, IFAC Proceedings (to appear).
  • [Koon, Lo, Marsden, and Ross(1999)] Koon, W. S., M. W. Lo, J. E. Marsden, and S. D. Ross [1999], Constructing a Low Energy Transfer between Jovian Moons. In Celestial Mechanics : an international conference on celestial mechanics, Evanston, Illinois.
  • [Koon, Lo, Marsden, and Ross(2000)] Koon, W. S., M. Lo, J. E. Marsden, and S. Ross [2000], Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics, Chaos 10, 427-469. MR 1765636 (2001f:70009)
  • [Koon, Lo, Marsden, and Ross(2005)] Koon, W. S., M. Lo, J. E. Marsden, and S. Ross [2005], Dynamical Systems, the Three-Body Problem and Space Mission Design (to be published).
  • [Kresák(1979)] Kresák, L. [1979], Dynamical interrelations among comets and asteroids, in Asteroids, Univ. of Arizona Press, Tucson, 289-309.
  • [Lekien(2003)] Lekien, F. [2003], Time-Dependent Dynamical Systems and Geophysical Flows, PhD thesis, California Institute of Technology.
  • [Lew et al.(2004)] Lew, A., J. E. Marsden, M. Ortiz, and M. West [2004], Variational time integrators, Intern. J. Num. Meth. in Engin. 60, 153-212. MR 2073073 (2005g:74054)
  • [Lo et al.(2001)] Lo, M., B. G. Williams, W. E. Bollman, D. Han, Y. Hahn, J. L. Bell, E. A. Hirst, R. A. Corwin, P. E. Hong, K. C. Howell, B. Barden, and R. Wilson [2001], Genesis Mission Design, The Journal of the Astronautical Sciences 49, 169-184.
  • [Lo and Ross(1998)] Lo, M. W. and S. D. Ross [1998], Low energy interplanetary transfers using invariant manifolds of L1, L2 and halo orbits. In AAS/AIAA Space Flight Mechanics Meeting, Monterey, California.
  • [Lo and Ross(2001)] Lo, M. W. and S. D. Ross [2001], The Lunar L1 Gateway: Portal to the stars and beyond. In AIAA Space 2001 Conference, Albuquerque, New Mexico.
  • [Marsden and West(2001)] Marsden, J. E. and M. West [2001], Discrete mechanics and variational integrators, Acta Numerica 10, 357-514. MR 2009697 (2004h:37130)
  • [Meiss(1992)] Meiss, J. D. [1992], Symplectic maps, variational principles, and transport. Rev. Mod. Phys. 64 (3), 795-848. MR 1183196 (93h:58060)
  • [Meyer and Hall(1992)] Meyer, K. R. and R. Hall [1992], Hamiltonian Mechanics and the N-Body Problem. Texts in Applied Mathematics Science. Springer-Verlag, Berlin.
  • [Monien, Preis and Diekmann(2000)] Monien, B., R. Preis, and R. Diekmann [2000], Quality matching and local improvement for multilevel graph-partitioning. Parallel Computing, 26 (12), 1609-1634. MR 1786939 (2001d:68181)
  • [Poincaré(1890)] Poincaré, H. [1890], Sur le problème des trois corps et les équations de la dynamique, Acta Math. 13, 1-27.
  • [Poincaré(1892)] Poincaré, H. [1892], Les Méthodes Nouvelles de la Mécanique Celeste.
    3 volumes. English translation, New Methods of Celestial Mechanics, History of Modern Physics and Astronomy 13, Amer. Inst. Phys., 1993.
  • [Porter and Cvitanovic(2005)] Porter, M.A. and P. Cvitanovic [2005], Ground control to Niels Bohr: Exploring outerspace with atomic physics, Notices of the AMS 52 (October 2005), 1020-1025.
  • [Rom-Kedar(1999)] Rom-Kedar, V. [1999], Transport in a class of $ n$-d.o.f. systems, in Hamiltonian systems with three or more degrees of freedom (S'Agaró, 1995), Vol. 533 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Kluwer Acad. Publ., Dordrecht, 538-543. MR 1720944 (2000k:37088)
  • [Rom-Kedar and Wiggins(1990)] Rom-Kedar, V. and S. Wiggins [1990], Transport in two-dimensional maps. Arch. Rat. Mech. Anal. 109, 239-298. MR 1025172 (90j:58137)
  • [Ross(2003)] Ross, S. D. [2003], Statistical theory of interior-exterior transition and collision probabilities for minor bodies in the solar system, in Libration Point Orbits and Applications, World Scientific, 637-652.
  • [Ross(2005)] Ross, S. D. [2005], A mechanism for capture, escape, and collision in dynamical astronomy (in preparation).
  • [Ross et al.(2003)] Ross, S. D., W. S. Koon, M. W. Lo, and J. E. Marsden [2003], Design of a Multi-Moon Orbiter. In 13th AAS/AIAA Space Flight Mechanics Meeting, Ponce, Puerto Rico. Paper No. AAS 03-143.
  • [Roy(1988)] Roy, A. E. [1988], Orbital Motion. Adam Hilger, Bristol, 3rd edition.
  • [Serban et al.(2002)] Serban, R., W. S. Koon, M. Lo, J. E. Marsden, L. R. Petzold, S. D. Ross, and R. S. Wilson [2002], Halo orbit mission correction maneuvers using optimal control, Automatica 38, 571-583. MR 2131468 (Review)
  • [Wiggins(1992)] Wiggins, S. [1992], Chaotic transport in dynamical systems. Interdisciplinary Appl. Math. 2. Springer, Berlin-Heidelberg-New York. MR 1139113 (93c:58152)

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Additional Information

Jerrold E. Marsden
Affiliation: Control and Dynamical Systems, California Institute of Technology 107-81, Pasa- dena, California 91125

Shane D. Ross
Affiliation: Department of Aerospace and Mechanical Engineering, University of Southern California, RRB 217, Los Angeles, California 90089-1191

Keywords: Three-body problem, mission design, transport, celestial mechanics
Received by editor(s): May 3, 2005
Received by editor(s) in revised form: July 19, 2005
Published electronically: November 22, 2005
Additional Notes: The first author’s research was supported in part by a Max Planck Research Award and NSF-ITR Grant ACI-0204932.
The second author’s research was supported by an NSF Postdoctoral Fellowship, DMS 0402842.
This article is based on a lecture presented January 7, 2005, at the AMS Special Session on Current Events, Joint Mathematics Meetings, Atlanta, GA
Dedicated: To Henri Poincaré on the 150th anniversary of his birth.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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