Abstract
Recursive computation of mutual potential, force, and torque between two polyhedra is studied. Based on formulations by Werner and Scheeres (Celest Mech Dyn Astron 91:337–349, 2005) and Fahnestock and Scheeres (Celest Mech Dyn Astron 96:317–339, 2006) who applied the Legendre polynomial expansion to gravity interactions and expressed each order term by a shape-dependent part and a shape-independent part, this paper generalizes the computation of each order term, giving recursive relations of the shape-dependent part. To consider the potential, force, and torque, we introduce three tensors. This method is applicable to any multi-body systems. Finally, we implement this recursive computation to simulate the dynamics of a two rigid-body system that consists of two equal-sized parallelepipeds.
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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)
Borderies, N.: Mutual gravitational potential of n solid bodies. Celest. Mech. 18, 295–307 (1978)
Brouwer, D., Clemence, G.M.: Methods of Celestial Mechanics. Academic Press, New York (1961)
Fahnestock, E.G., Scheeres, D.J.: Simulation of the full two rigid body problem using polyhedral mutual potential and potential derivatives approach. Celest. Mech. Dyn. Astron. 96, 317–339 (2006)
Fahnestock, E.G., Scheeres, D.J.: Simulation and analysis of the dynamics of binary near-earth asteroid (66391) 1999KW4. Icarus 194, 410–435 (2008)
Giacaglia, G.E.O., Jefferys, W.H.: Motion of a space station. I. Celest. Mech. 4, 442–467 (1971)
Hobson, E.W.: The Theory of Spherical and Ellipsoidal Harmonics. Cambridge University Press, Cambridge (1931)
Lien, S., Kajiya, J.T.: A symbolic method for calculating the integral properties of arbitrary nonconvex polyhedra. IEEE Comput. Graph. Appl. 4, 35–41 (1984)
Maciejewski, A.J.: Reduction, relative equilibria and potential in the two rigid bodies problem. Celest. Mech. Dyn. Astron. 63, 1–28 (1995)
Montenbruck, O.: Satellite Orbits, Models, Methods, and Applications. Springer, Berlin (2000)
Ostro, S.J., Margot, J.-L., Benner, L.A.M., Giorgini, J.D., Scheeres, D.J., Fahnestock, E.G., et al.: Radar imaging of binary near-earth asteroid (66391) 1999 KW4. Science 314, 1276–1280 (2006)
Paul, M.K.: An expansion in power series of mutual potential for gravitating bodies with finite sizes. Celest. Mech. 44, 49–59 (1988)
Rambaux, N., Bois, E.: Theory of the mercury’s spin-orbit motion and analysis of its main librations. Astron. Astrophys. 413, 381–393 (2004)
Scheeres, D.J., Fahnestock, E.G., Ostro, S.J., Margot, J.-L., Benner, L.A.M., Broschart, S.B., et al.: Dynamical configration of binary near-Earth asteroid (66391) 1999 KW4. Science 314, 1280–1283 (2006)
Schutz, B.E.: The mutual potential and gravitational torques of two bodies to fourth order. Celest. Mech. Dyn. Astron. 24, 173–181 (1981)
Tricarico, P.: Figure-figure interaction between bodies having arbitrary shapes and mass distributions: a power series expansion approach. Celest. Mech. Dyn. Astron. 100, 319–330 (2008)
von Braun, C.: The gravitational potential of two arbitrary rotating bodies with applications to the Earth–Moon system. PhD thesis, The University of Texas at Austin (1991)
Werner, R.A., Scheeres, D.J.: Mutual potential of homogeneous polyhedra. Celest. Mech. Dyn. Astron. 91, 337–349 (2005)
Wigner, E.P.: Group Theory and its Application to the Quantum Mechanics of Atomic Spectra. Academic Press, New York (1959)
Acknowledgments
M. H. is supported by the fellowship from Japan Student Services Organization. Also the authors thank Dr. Sylvio Ferraz-Mello and the anonymous referees for their detailed reviews that improved the clarity and quality of the manuscript.
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Hirabayashi, M., Scheeres, D.J. Recursive computation of mutual potential between two polyhedra. Celest Mech Dyn Astr 117, 245–262 (2013). https://doi.org/10.1007/s10569-013-9511-x
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DOI: https://doi.org/10.1007/s10569-013-9511-x