Geometric mechanics and the dynamics of asteroid pairs

Annals of the New York Academy of Sciences

Volumn 1017, Issue , 2004, Pages 11-38

  Koon, Wang Sang ^a   Marsden, Jerrold E ^a   Ross, Shane D ^a   Lo, Martin ^b   Scheeres, Daniel J ^c  

^a CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

^b CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

^c UNIVERSITY OF MICHIGAN   (United States)

Author keywords

Asteroid pairs; Dynamics; Full body problem; Geometric mechanics

Indexed keywords

ASTRONOMY; CONFERENCE PAPER; ENERGY; EVOLUTION; GEOMETRY; GRAVITY; MATHEMATICAL ANALYSIS; MATHEMATICAL MODEL; SIMULATION; SPACE; THEORETICAL MODEL;

ASTEROIDEA;

EID: 3042831489     PISSN: 00778923     EISSN: None     Source Type: Book Series    
DOI: 10.1196/annals.1311.002     Document Type: Conference Paper

References (111)

1. KRISHNAPRASAD, P.S. & J.E. MARSDEN. 1987. Hamiltonian structure and stability for rigid bodies with flexible attachments. Arch. Rat. Mech. Anal. 98: 137-158.
2. KRUPA, M., M. SCHAGERL, A. STEINDL, et al. 2001. Relative equilibria of tethered satellite systems and their stability for very stiff tethers. Dyn. Syst. 16: 253-278.
3. SCHEERES, D.J. 2004. Stability of relative equilibria in the full two-body problem. Ann. N.Y. Acad. Sci. 1017: this volume.
4. WISDOM, J. 1987. Rotational dynamics of irregularly shaped natural satellites. Astron. J. 94: 1350-1360.
5. TOUMA, J. & J. WISDOM. 1994. Evolution of the Earth-Moon system. Astron. J. 108: 1943-1961.
6. BECK, J.A. & C.D. HALL. 1998. Relative equilibria of a rigid satellite in a circular keplerian orbit. J. Astronaut. Sci. 46: 215-247.
7. WANG, L.S., P.S. KRISHNAPRASAD & J.H. MADDOCKS. 1991. Hamiltonian dynamics of a rigid body in a central gravitational field. Celest. Mech. Dyn. Astron. 50: 349-386.
8. WANG, L.S., P.S. KRISHNAPRASAD & J.H. MADDOCKS. 1992. Steady rigid-body motions in a central gravity field. J. Astronaut. Sci. 40: 449-478.
9. WANG, L.S., K.-Y. LIAN & P.-T. CHEN. 1995. Steady motions of gyrostat satellites and their stability. IEEE Trans. Auto. Cont. 40: 1732-1743.
10. RUI, C., I. KOLMANOVSKY & N.H. MCCLAMROCH. 2000. Nonlinear attitude and shape control of spacecraft with articulated appendages and reaction wheels. IEEE Trans. Auto. Cont. 45: 1455-1469.
11. SHEN, J. & N.H, MCCLAMROCH. 2001. Translational and rotational spacecraft manuevers via shape change actuators. In Proc. 2001 American Control Conference, 3961-3966.
12. BLOCK, A.M., P. CROUCH, J.E. MARSDEN & T.S. RATIU. 2002. The symmetric representation of the rigid body equations and their discretization. Nonlinearity 15: 1309-1341.
13. DOBROVOLSKIS, A.R. & J.A. BURNS. 1980. Life near the roche limit: behavior of ejecta from satellites close to planets. Icarus 42: 422-441.
14. CHAUVINEAU, B., P. FARINELLA & F. MIGNARD. 1993. Planar orbits about a triaxial body: applications to asteroidal satellites. Icarus 105: 370-384.
15. SCHEERES, D.J. 1994. Dynamics about uniformly rotating tri-axial ellipsoids. Applications to asteroids. Icarus 110: 225-238.
16. SCHEERES, D.J., S.J. OSTRO, R.S. HUDSON & R.A. WERNER. 1996. Orbits close to asteroid 4769 castalia. Icarus 121: 67-87.
17. SCHEERES, D.J., S.J. OSTRO, R.S. HUDSON, et al. 1998. Dynamics of orbits close to asteroid 4179 toutatis. Icarus 132: 53-79.
18. SCHEERES, D.J., D.D. DURDA & P.E. GEISSLER. 2002. The fate of asteroid ejecta. In Asteroids III. W.M. Bottke, et al., Eds.: University of Arizona Press, Tuscon.
19. SCHEERES, D.J., B.G. WILLIAMS & J.K. MILLER. 2000. Evaluation of the dynamic environment of an asteroid: applications to 433 eros. J. Guidance Cont. Dyn. 23: 466-475.
20. DUBOSHIN, G.N. 1958. The differential equations of translational - rotational motion of mutally attracting rigid bodies. Astronomicheskii Zhurnal 35: 265.
21. MACIEJEWSKI, A.J. 1995. Reduction, relative equilibria and potential in the two rigid bodies problem. Celest. Mech. Dyn. Astron. 63: 1-28.
22. SCHEERES, D.J. 2002. Stability in the full two-body problem. Celest. Mech. Dyn. Astron. 83: 155-169.
23. SCHEERES, D.J., S.J. OSTRO, R.A. WERNER, et al. 2000. Effects of gravitational interactions on asteroid spin states. Icarus 147: 106-118.
24. SCHEERES, D.J. 2001. Changes in rotational angular momentum due to gravitational interactions between two finite bodies. Celest. Mech. Dyn. Astron. 81: 39-44.
25. SCHEERES, D.J. 2002. Stability of binary asteroids. Icarus 159: 271-283.
26. BLOCK, A.M. 2003. Nonhonomomic Mechanics and Control. Applied Mathematical Sciences Series. Springer-Verlag.
27. MARSDEN, J.E. & T.S. RATIU. 1999. Introduction to Mechanics and Symmetry. Texts in Applied Mathematics, 17: Springer-Verlag.
28. MARSDEN, J.E. 1992. Lectures on Mechanics, Vol. 174. London Math. Society, Lecture Note Series. Cambridge University Press.
29. BLOCH, A.M., P.S. KRISHNAPRASAD, J.E. MARSDEN & T.S. RATIU. 1994. Dissipation induced instabilities. Ann. Inst. H. Poincaré, Analyse Nonlineéaire 11: 37-90.
30. BLOCH, A.M., P.S. KRISHNAPRASAD, J.E. MARSDEN & T.S. RATIU. 1996. The Euler-Poincaré equations and double bracket dissipation. Comm. Math. Phys. 175: 1-42.
31. DUNCAN, M., T. QUINN & S. TREMAINE. 1989. The long-term evolution of orbits in the solar system - a mapping approach. Icarus 82: 402-418.
32. WISDOM, J. & M. HOLMAN. 1991. Symplectic maps for the n-body problem. Astron. J. 102: 1528-1538.
33. LEW, A., J.E. MARSDEN, M. ORTIZ & M. WEST. 2004. Variational time integration for mechanical systems. Intern. J. Num. Meth. Engin. To appear.
34. LEW, A., J.E. MARSDEN, M. ORTIZ & M. WEST. 2004. Asynchronous variational integrators. Arch. Rat. Mech. Anal. 167: 85-146.
35. JAFFÉ, C., S.D. ROSS, M.W. LO, et al. 2002. Statistical theory of asteroid escape rates. Phys. Rev. Lett. 89: 011101-1.
36. DELLNITZ, M., O. JUNGE, W.-S. KOON, et al. 2003. Transport in dynamical astronomy and multibody problems. Preprint.
37. NÉRON DE SURGY, O. & J. LASKAR. 1997. On the long term evolution of the spin of the earth. Astron. Astrophys. 318: 975-989.
38. GREENBERG, R., G.V. HOPPA, P. GEISSLER, et al. 2002. The rotation of europa. Celest. Mech. Dyn. Astron. 83: 35-47.
39. BOTTKE, W.F., R. JEDICKE, A. MORBIDELLI, et al. 2000. Understanding the distribution of near-Earth asteroids. Science 288: 2190-2194.
40. BOTTKE, W.F., D. VOKROUHLICK , M. BROŽ, et al. 2001. Dynamical spreading of asteroid families by the yarkovsky effect. Science 294: 1693-1696.
41. SPITALE, J. & R. GREENBERG. 2001. Numerical evaluation of the general yarkovsky effect: effects on semimajor axis. Icarus 149: 222-234.
42. RUBINCAM, D. 2000. Radiative spin-up and spin-down of small asteroids. Icarus 148: 2-11.
43. NEISHTADT, A.I., D.J. SCHEERES, V.V. SIDORENKO & A.A. VASILIEV. 2002. Evolution of comet nucleus rotation. Icarus 157: 205-218.
44. YEOMANS, D.K. & P.W. CHODAS. 1989. An asymmetric outgassing model for cometary nongravitational accelerations. Astron. J. 98: 1083-1093.
45. MARGOT, J.L., M.C. NOLAN, L.A.M. BENNER, et al. 2002. Binary asteroids in the near-earth object population. Science 296: 1445-1448.
46. VEILLET, C., J.W. PARKER, I. GRIFFIN, et al. 2002. The binary kuiper-belt object 1998 ww31. Nature 416: 711.
47. MERLINE, W.J., S.J. WEIDENSCHILLING, D.D. DURDA, et al. 2002. Asteroids do have satellites. In Asteroids III. W.M. Bottke, A. Cellino, P. Paolicchi & R.P. Binzel, Eds.: 289-312. University of Arizona Press, Tucson.
48. BELTON, M.J.S., B.E.A. MUELLER, L.A. D'AMARIO, et al. 1996. The discovery and orbit of 1993 (243)1 dactyl. Icarus 120: 185-199.
49. BENNER, L.A.M., S.J. OSTRO, J.D. GIORGINI, et al. 2001. 1999 kw4. IAU Circular 7730.
50. HARRIS, A.W. 2002. On the slow rotation of asteroids. Icarus 156: 184-190.
51. 22 on orbit energy and angular momentum. Celest. Mech. Dyn. Astron. 73: 339-348.
52. SIMO, J.C., D.R. LEWIS & J.E. MARSDEN. 1991. Stability of relative equilibria I: the reduced energy momentum method. Arch. Rat. Mech. Anal. 115: 15-59.
53. MARSDEN, J.E. & M. WEST. 2001. Discrete mechanics and variational integrators. Acta Numerica 10: 357-514.
54. KOON, W.S., M.W. LO, J.E. MARSDEN & S.D. ROSS. 2000. Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics. Chaos 10: 427-469.
55. CENDRA, H., J.E. MARSDEN & T.S. RATIU. 2001. Lagrangian reduction by stages, volume 152 of Memoirs. American Mathematical Society, Providence.
56. MARSDEN, J.E., T.S. RATIU & J. SCHEURLE. 2000. Reduction theory and the lagrange-routh equations. J. Math. Phys. 41: 3379-3429.
57. MACIEJEWSKI, A.J. 1999. The two rigid bodies problem, reduction and relative equilibria. In Hamiltonian Systems with Three or More Degrees of Freedom (S'Agaró, 1995). 475-479.
58. GOŹDZIEWSKI, K. & A.J. MACIEJEWSKI. 1999. Unrestricted planar prolbme of a symmetric body and a point mass, triangular libration points and their stability. Celest. Mech. Dyn. Astron. 75: 251-285.
59. LEONARD, N.B. & J.E. MARSDEN. 1997. Stability and drift of underwater vehicle dynamics: mechanical systems with rigid motion symmetry. Physica D 105: 130-162.
60. MARSDEN, J., G. MISIOLEK, M. PERLMUTTER & T. RATIU. 1998. Symplectic reduction for semidirect products and central extensions. Diff. Geom. Appl. 9: 173-212.
61. MARSDEN, J.E., R. MONTGOMERY & T.S. RATIU. 1990. Reduction, symmetry and phases in mechanics. Memoirs of the AMS, Vol. 436. Amer. Math. Society, Providence.
62. CLERC, M. & J.E. MARSDEN. 2001. Dissipation-induced instabilities in an optical cavity laser: a mechanical analog near the 1:1 resonance. Phys. Rev. E 64: 067603.
63. LARA, M. & D.J. SCHEERES. 2002. Stability bounds for three-dimensional motion close to asteroids. J. Astronaut. Sci. 50(4): 389-409.
64. ALBER, M.S., G.G. LUTHER, J.E. MARSDEN & J.M. ROBBINS. 1998. Geometric phases, reduction and Lie-Poisson structure for the resonant three-wave interaction. Physica D 123: 271-290.
65. HERNANDEZ, A. & J.E. MARSDEN. 2004. Bifurcation of singular relative equilibria by the blowing up method. SIAM J. Appl. Dyn. Systems. To appear.
66. STERN, S.A., W.F. BOTTKE & H.F. LEVISON. 2003. Regarding the putative eccentricity of charon's orbit. Astron. J. 125: 902-905.
67. SAWAI, S., D.J. SCHEERES & S. BROSCHART. 2002. Control of hovering spacecraft using altimetry. J. Guidance Cont. Dyn. 25: 786-795.
68. CHANG, D., A.M. BLOCH, N. LEONARD, et al. 2002. The equivalence of controlled lagrangian and controlled hamiltonian systems. Cont. Calc. Variations (special issue dedicated to J.L. Lions) 8: 393-422.
69. ROM-KEDAR, V. & S. WIGGINS. 1990. Transport in two-dimensional maps. Arch. Rat. Mech. Anal. 109: 239-298.
70. LEKIEN, F. & J.E. MARSDEN. 2004. Separatrices in high-dimensional phase spaces: application to van der waals dissociation. In preparation.
71. LEKIEN, F. & C. COULLIETTE. 2004. MANGEN: computation of hyperbolic trajectories, invariant manifolds and lobes of dynamical systems defined as 2d+1 data sets. In preparation.
72. LEKIEN, F. 2003. Time-Dependent Dynamical Systems and Geophysical Flows. Ph.D. Thesis, California Institute of Technology.
73. WIGGINS, S. 1992. Chaotic Transport in Dynamical Systems. Interdisciplinary Appl. Math., Vol. 2. Springer, Berlin-Heidelberg-New York.
74. ROM-KEDAR, V., A. LEONARD & S. WIGGINS. 1990. An analytical study of transport, mixing and chaos in an unsteady vortical flow. J. Fluid Mech. 214: 347-394.
75. ROM-KEDAR, V. & S. WIGGINS. 1991. Transport in two-dimensional maps: concepts, examples, and a comparison of the theory of rom-kedar and wiggins with the markov model of mackay, meiss, ott, and percival. Physica D 51: 248-266.
76. COULLIETTE, C. & S. WIGGINS. 2001. Intergyre transport in a wind-driven, quasigeostrophic double gyre: an application of lobe dynamics. Nonlinear Process. Geophys. 8: 69-94.
77. LEKIEN, F., C. COULLIETTE & J.E. MARSDEN. 2004. Lagrangian structures in very high frequency radar data and optimal pollution timing. In Proc. 7th Experimental Chaos Conference. American Institute of Physics. 162-168.
78. TIYAPAN, A. & C. JAFFÉ. 1995. Chaotic scattering: an invariant fractal tiling of phase space. J. Chem. Phys. 103: 5499-5511.
79. JALNAPURKAR, S.M., M. LEOK, J.E. MARSDEN & M. WEST. 2004. Discrete routh reduction. Found. Comput. Math. To appear.
80. SAHA, P. & S. TREMAINE. 1992. Symplectic integrators for solar system dynamics. Astron. J. 104: 1633-1640.
81. MARSDEN, J.E., G.W. PATRICK & S. SHKOLLER. 1998. Multisymplectic geometry, variational integrators and nonlinear PDEs. Comm. Math. Phys. 199: 351-395.
82. KANE, C., J.E. MARSDEN, M. ORTIZ & M. WEST. 2000. Variational integrators and the newmark algorithm for conservative and dissipative mechanical systems. Int. J. Num. Meth. Eng. 49: 1295-1325.
83. ROWLEY, C.W. & J.E. MARSDEN. 2002. Variational integrators for point vortices. Proc. CDC 40: 1521-1527.
84. SCHLICK, T. 2002. Molecular Modeling and Simulation. Interdisciplinary Applied Mathematics, Vol. 21. Springer-Verlag, New York.
85. MARSDEN, J.E., S. PEKARSKY & S. SHKOLLER. 2000. Symmetry reduction of discrete lagrangian mechanics on lie groups. J. Geom. Phys. 36: 140-151.
86. DELLNITZ, M. & O. JUNGE. 2002. Set oriented numerical methods for dynamical systems. In Handbook of Dynamical Systems, Vol. 2. 221-264. North-Holland, Amsterdam.
87. MURRAY, C.D. & S.F. DERMOTT. 1999. Solar System Dynamics. Cambridge.
88. CHO, S.N., H. MCCLAMROCH & M. REYHANOGLU. 2000. Dynamics of multi-body vehicles and their formulation as nonlinear control systems. In Proc. 2000 American Control Conference. 3908-3912.
89. HAGERTY, P., A.M. BLOCH & M. WEINSTEIN. 1999. Radiation induced instability in interconnected systems. In Proc. 38th Conf. Decisions Control. IEEE 651-656.
90. WASHABAUGH, P.D. & D.J. SCHEERES. 2002. Energy and stress distributions in ellipsoids. Icarus 159: 314-321.
91. ASPHAUG, E. & W. BENZ. 1996. Size, density, and structure of comet shoemaker-levy 9 inferred from the physics of tidal breakup. Icarus 121: 225-248.
92. RICHARDSON, D.C., W.F. BOTTKE & S.G. LOVE. 1998. Tidal distortion and disruption of Earth-crossing asteroids. Icarus 134: 47-76.
93. HARRIS, A.W. 1994. Tumbling asteroids. Icarus 107: 209-211.
94. BOTTKE, W.F. & H.J. MELOSH. 1996. The formation of binary asteroids and doublet craters. Icarus 124: 372-391.
95. DURDA, D.D. 1996. The formation of asteroidal satellites in catastrophic collisions. Icarus 120: 212-219.
96. DORESSOUNDIRAM, A., P. PAOLICCHI, A. VERLICCHI & A. CELLINO. 1997. The formation of binary asteroids as outcomes of catastrophic collisions. Planet. Space. Sci. 45: 757-770.
97. BOTTKE, W.F., D.C. RICHARDSON, P. MICHEL & S.G. LOVE. 1999. 1620 geographos and 433 eros: shaped by planetary tides? Astron. J. 117: 1921-1928.
98. MICHEL, P., W. BENZ, P. TANGA & D.C. RICHARDSON. 2001. Collisions and gravitational reaccumulation: forming asteroid families and satellites. Science 294: 1696-1700.
99. GOLDREICH, P., Y. LITHWICK & R. SARI. 2002. Formation of kuiper-belt binaries by dynamical friction and three-body encounters. Nature 240: 643-646.
100. CHANDRASEKHAR, S. 1969. Ellipsoidal Figures of Equilibrium. Yale University Press.
101. BLOCH, A.M., D. CHANG, N. LEONARD & J.E. MARSDEN. 2001. Controlled lagrangians and the stabilization of mechanical systems ii: potential shaping. IEEE Trans. Auto. Cont. 46: 1556-1571.
102. SHEN, J., N.H. MCCLAMROCH & A.M. BLOCH. 2004. Local equilibrium controllability of multibody systems controlled via shape change. IEEE Trans. Auto. Cont. To appear.
103. MEISS, J.D. 1992. Symplectic maps, variational principles, and transport. Rev. Mod. Phys. 64: 795-848.
104. 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). 538-543.
105. BEIGIE, D., A. LEONARD & S. WIGGINS. 1991. A global study of enhanced stretching and diffusion in chaotic tangles. Phys. Fluids A 3: 1039-1049.
106. MALHOTRA, N. & S. WIGGINS. 1998. Geometric structures, lobe dynamics, and lagrangian transport in flows with aperiodic time dependence, with applications to rossby wave flow. J. Nonlinear Sci. 8: 401-456.
107. POJE, A.C. & G. HALLER. 1999. Geometry of cross-stream mixing in a double-gyre ocean model. Phys. Oceanogr. 29: 1649-1665.
108. HALLER, G. & G. YUAN. 2000. Lagrangian coherent structures and mixing in two-dimensional turbulence. Physica D 147: 352-370.
109. HALLER, G. 2001. Distinguished material surfaces and coherent structures in 3d fluid flows. Physica D 149: 248-277.
110. JAFFÉ, C., D. FARRELLY & T. UZER. 2000. Transition state theory without time-reversal symmetry: chaotic ionization of the hydrogen atom. Phys. Rev. Lett. 84: 610-613.
111. ROSS, S.D., W.S. KOON, M.W. LO & J.E. MARSDEN. 2003. Design of a multi-moon orbiter. In 13th AAS/AIAA Space Flight Mechanics Meeting, Ponce, Puerto Rico, AAS 03-143.