Abstract
Binary systems are quite common within the populations of near-Earth asteroids, main-belt asteroids, and Kuiper belt asteroids. The dynamics of binary systems, which can be modeled as the full two-body problem, is a fundamental problem for their evolution and the design of relevant space missions. This paper proposes a new shape-based model for the mutual gravitational potential of binary asteroids, differing from prior approaches such as inertia integrals, spherical harmonics, or symmetric trace-free tensors. One asteroid is modeled as a homogeneous polyhedron, while the other is modeled as an extended rigid body with arbitrary mass distribution. Since the potential of the polyhedron is precisely described in a closed form, the mutual gravitational potential can be formulated as a volume integral over the extended body. By using Taylor expansion, the mutual potential is then derived in terms of inertia integrals of the extended body, derivatives of the polyhedron’s potential, and the relative location and orientation between the two bodies. The gravitational forces and torques acting on the two bodies described in the body-fixed frame of the polyhedron are derived in the form of a second-order expansion. The gravitational model is then used to simulate the evolution of the binary asteroid (66391) 1999 KW4, and compared with previous results in the literature.
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References
Ashenberg, J.: Mutual gravitational potential and torque of solid bodies via inertia integrals. Celest. Mech. Dyn. Astron. 99, 149–159 (2007)
Bellerose, J., Scheeres, D.J.: Energy and stability in the full two body problem. Celest. Mech. Dyn. Astron. 100, 63–91 (2008a)
Bellerose, J., Scheeres, D.J.: General dynamics in the restricted full three body problem. Acta Astronaut. 62, 563–576 (2008b)
Borderies, N.: Mutual gravitational potential of N solid bodies. Celest. Mech. 18, 173–181 (1978)
Boué, G., Laskar, J.: Spin axis evolution of two interacting bodies. Icarus 201, 750–767 (2009)
Brouwer, D., Clemence, G.M.: Method of Celestial Mechanics. Academic Press, New York (1961)
Cady, J.W.: Calculation of gravity and magnetic anomalies of finite-length right polygonal prisms. Geophysics 45(10), 1507–1512 (1980)
Chapman, C.R., et al.: Discovery and physical properties of Dactyl, a satellite of Asteroid 243 Ida. Nature 374, 783–785 (1995)
Compère, A., Lemaître, A.: The two-body interaction potential in the STF tensor formalism: an application to binary asteroids. Celest. Mech. Dyn. Astron. 119(3–4), 313–330 (2014)
Ćuk, M., Nesvorný, D.: Orbital evolution of small binary asteroids. Icarus 207, 732–743 (2010)
Fahnestock, E.G.: The full two-body-problem: simulation, analysis, and application to the dynamics, characteristics, and evolution of binary asteroid systems. Ph.D. Dissertation. The University of Michigan (2009)
Fahnestock, E.G., Scheeres, D.J.: Simulation and analysis of the dynamics of binary near-Earth Asteroid (66391) 1999 KW4. Icarus 194, 410–435 (2008)
Giacaglia, C.E., Jefferys, W.H.: Motion of a space station. Celest. Mech. 4, 442–467 (1971)
Golizdra, G.Y.: Calculation of the gravitational field of a polyhedron Izvestiya. Phys. Solid Earth 17(8), 625 (1981)
Hirabayashi, M., Scheeres, D.J.: Recursive computation of mutual potential between two polyhedra. Celest. Mech. Dyn. Astron. 117, 245–262 (2013)
Hou, X., Scheeres, D.J., Xin, X.: Mutual potential between two rigid bodies with arbitrary shapes and mass distributions. Celest. Mech. Dyn. Astron. 127(3), 369–395 (2017)
Kwok, Y.K.: Gravity gradient tensors due to a polyhedron with polygonal FACETS1. Geophys. Prospect. 39(3), 435–443 (1991)
Macmillan, W.D.: The Theory of the Potential. McGraw-Hill Book Company, Inc, New York (1930)
Maciejewski, A.J.: Reduction, relative equilibria and potential in the two rigid bodies problem. Celest. Mech. Dyn. Astron. 63(1), 1–28 (1995)
Marchis, F., Lainey, V., Descamps, P., Berthier, J., Van Dam, M., de Pater, I., et al.: A dynamical solution of the triple asteroid system (45) Eugenia. Icarus 210, 635–643 (2010)
Margot, J.L., Nolan, M.C., Benner, L.A.M., Ostro, S.J., Jurgens, R.F., Giorgini, J.D., et al.: Binary asteroids in the near-earth object population. Science 296(5572), 1445–1448 (2002)
Meirovitch, L.: On the effect of higher-order inertia integrals on the attitude stability of earth-pointing satellites. J. Astronaut. Sci. XV(1), 14–18 (1968)
Nagy, D.: The gravitational attraction of a right rectangular prism. Geophysics 31(2), 362 (1966)
Plouff, D.: Gravity and magnetic fields of polygonal prisms and application to magnetic terrain corrections. Geophysics 41, 727–741 (1976)
Pravec, P., Harris, A.W.: Binary asteroid population. 1. Angular momentum content. Icarus 190(1), 250–259 (2007)
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. Ann. N. Y. Acad. Sci. 1017, 81–94 (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.: Stability of the planar full 2-body problem. Celest. Mech. Dyn. Astron. 104, 103–128 (2009)
Schutz, B.E.: The mutual potential and gravitational torque of two bodies to fourth order. Celest. Mech. 24, 173–181 (1981)
Telford, W.M., Geldart, L.P., Sheriff, R.E.: Applied Geophysics. Cambridge University Press, Cambridge (1976)
Tricarico, P.: Figure–figure interaction between bodies having arbitrary shapes and mass distributions: a power series expansion approach. Celest. Mech. Dyn. Astron. 100(4), 319–330 (2008)
Waldvogel, J.: The Newtonian potential of a homogeneous cube. J. Appl. Math. Phys. (ZAMP) 27, 867 (1976)
Waldvogel, J.: The Newtonian potential of homogeneous polyhedra. J. Appl. Math. Phys. (ZAMP) 30, 388 (1979)
Wang, Y., Xu, S.: Hamiltonian structures of dynamics of a gyrostat in a gravitational field. Nonlinear Dyn. 70(1), 231–247 (2012)
Werner, R.A.: The gravitational potential of a homogeneous polyhedron or don’t cut corners. Celest. Mech. Dyn. Astron. 59(3), 253–278 (1994)
Werner, R.A., Scheeres, D.J.: Exterior gravitation of a polyhedron derived and compared with harmonic and mascon gravitation representations of asteroid 4769 Castalia. Celest. Mech. Dyn. Astron. 65, 313–344 (1997)
Werner, R.A., Scheeres, D.J.: Mutual potential of homogeneous polyhedra. Celest. Mech. Dyn. Astron. 91, 337–349 (2005)
Wisdom, J.: Rotational dynamics of irregularly shaped natural satellites. Astron. J. 94, 1350–1360 (1987)
Wisdom, J., Peale, S.J., Mignard, F.: The chaotic rotation of Hyperion. Icarus 58, 137–152 (1984)
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This work has been supported by the National Natural Science Foundation of China under Grant Nos. 11432001 and 11602009, and the Fundamental Research Funds for the Central Universities.
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Shi, Y., Wang, Y. & Xu, S. Mutual gravitational potential, force, and torque of a homogeneous polyhedron and an extended body: an application to binary asteroids. Celest Mech Dyn Astr 129, 307–320 (2017). https://doi.org/10.1007/s10569-017-9776-6
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DOI: https://doi.org/10.1007/s10569-017-9776-6