Abstract
The stability of the Full Two-Body Problem is studied in the case where both bodies are non-spherical, but are restricted to planar motion. The mutual potential is expanded up to second order in the mass moments, yielding a highly symmetric yet non-trivial dynamical system. For this system we identify all relative equilibria and determine their stability properties, with an emphasis on finding the energetically stable relative equilibria and conditions for Hill stability of the system. The energetically stable relative equilibria always correspond to the classical “gravity gradient” configuration with the long ends of the two bodies pointed at each other, however there always exists a second equilibrium in this configuration at a closer separation that is unstable. For our model system we precisely map out the relations between these different configurations at a given value of angular momentum. This analysis identifies the fundamental physical constraints and limitations that exist on such systems, and has immediate applications to the stability of asteroid systems that are fissioned due to a rapid spin rate. Specifically, we find that all contact binary asteroids which are spun to fission will initially lie in an unstable dynamical state and can always re-impact. If the total system energy is positive, the fissioned system can disrupt directly from this relative equilibrium, while if it is negative the system is bound together.
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Bellerose J.E., Scheeres D.J.: Energy and stability in the full two body problem. Celest. Mech. Dyn. Astron. 100(1), 63–91 (2008)
Cendra H., Marsden J.E.: Geometric mechanics and the dynamics of asteroid pairs. Dyn. Syst. 20(1), 3–21 (2005)
Demura H., Kobayashi S. et al.: Pole and Global Shape of 25143 Itokawa. Science 312, 1347–1349 (2006)
Ferraz-Mello S., Rodriguez A., Hussmann H.: Tidal friction in close-in satellites and exoplanets: The Darwin theory re-visited. Celest. Mech. Dyn. Astron. 101(1–2), 171–201 (2008)
Kinoshita H.: First-order perturbations of the two finite body problem. Publ. Astron. Soc. Jpn. 24, 423–457 (1972)
Koon, W.-S., Marsden, J.E., Ross, S., Lo, M.W., Scheeres, D.J.: Geometric mechanics and the dynamics of asteroid pairs. In: Belbruno, E., Folta, D., Gurfil,~P. (eds.) Astrodynamics, Space Missions, and Chaos, vol. 1017, pp. 11–38. Annals of the New York Academy of Science (2004)
Maciejewski A.J.: Reduction, relative equilibria and potential in the two rigid bodies problem. Celest. Mech. Dyn. Astron. 63, 1–28 (1995)
Ostro S.J., Margot J.-L., Benner L.A.M. et al.: Radar imaging of binary near-earth asteroid (66391) 1999 KW4. Science 314, 1276–1280 (2006)
Scheeres D.J.: Stability in the full two-body problem. Celest. Mech. Dyn. Astron. 83, 155–169 (2002)
Scheeres, D.J.: Stability of relative equilibria in the full two-body problem. In: Belbruno, E., Folta, D., Gurfil, P. (eds.) Astrodynamics, Space Missions, and Chaos, vol. 1017, pp. 81–94. Annals of the New York Academy of Science (2004)
Scheeres D.J.: Relative equilibria for general gravity fields in the sphere-restricted full 2-body problem. Celest. Mech. Dyn. Astron. 94, 317–349 (2006)
Scheeres D.J.: Rotational fission of contact binary asteroids. Icarus 189, 370–385 (2007)
Scheeres, D.J.: Minimum energy asteroid recongurations and catastrophic disruptions. Planetary and Space Science in press (2009). doi:10.1016/j.pss.2008.03.009
Simo J.C., Lewis D., Marsden J.E.: Stability of relative equilibria. I. The reduced energy-momentum method. Arch. Ration. Mech. Anal. 115(1), 15–59 (1991)
Wang L.-I., Krishnaprasad P.S., Maddocks J.H.: Hamiltonian dynamics of a rigid body in a central gravitational field. Celest. Mech. Dyn. Astron. 50, 349–386 (1991)
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Scheeres, D.J. Stability of the planar full 2-body problem. Celest Mech Dyn Astr 104, 103–128 (2009). https://doi.org/10.1007/s10569-009-9184-7
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DOI: https://doi.org/10.1007/s10569-009-9184-7