Skip to main content
Log in

Recursive computation of mutual potential between two polyhedra

  • Original Article
  • Published:
Celestial Mechanics and Dynamical Astronomy Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  • Ashenberg, J.: Mutual gravitational potential and torque of solid bodies via inertia integrals. Celest. Mech. Dyn. Astron. 99, 149–159 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Borderies, N.: Mutual gravitational potential of n solid bodies. Celest. Mech. 18, 295–307 (1978)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Brouwer, D., Clemence, G.M.: Methods of Celestial Mechanics. Academic Press, New York (1961)

    Google Scholar 

  • 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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Fahnestock, E.G., Scheeres, D.J.: Simulation and analysis of the dynamics of binary near-earth asteroid (66391) 1999KW4. Icarus 194, 410–435 (2008)

    Article  ADS  Google Scholar 

  • Giacaglia, G.E.O., Jefferys, W.H.: Motion of a space station. I. Celest. Mech. 4, 442–467 (1971)

    Article  ADS  MATH  Google Scholar 

  • Hobson, E.W.: The Theory of Spherical and Ellipsoidal Harmonics. Cambridge University Press, Cambridge (1931)

    Google Scholar 

  • 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)

    Article  Google Scholar 

  • Maciejewski, A.J.: Reduction, relative equilibria and potential in the two rigid bodies problem. Celest. Mech. Dyn. Astron. 63, 1–28 (1995)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Montenbruck, O.: Satellite Orbits, Models, Methods, and Applications. Springer, Berlin (2000)

    Book  MATH  Google Scholar 

  • 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)

    Article  ADS  Google Scholar 

  • Paul, M.K.: An expansion in power series of mutual potential for gravitating bodies with finite sizes. Celest. Mech. 44, 49–59 (1988)

    Article  ADS  MATH  Google Scholar 

  • Rambaux, N., Bois, E.: Theory of the mercury’s spin-orbit motion and analysis of its main librations. Astron. Astrophys. 413, 381–393 (2004)

    Article  ADS  Google Scholar 

  • 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)

    Article  ADS  Google Scholar 

  • Schutz, B.E.: The mutual potential and gravitational torques of two bodies to fourth order. Celest. Mech. Dyn. Astron. 24, 173–181 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Wigner, E.P.: Group Theory and its Application to the Quantum Mechanics of Atomic Spectra. Academic Press, New York (1959)

    MATH  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Masatoshi Hirabayashi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

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

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10569-013-9511-x

Keywords

Navigation