Abstract
This work presents an elegant formalism to model the evolution of the full two rigid body problem. The equations of motion, given in a Cartesian coordinate system, are expressed in terms of spherical harmonics and Wigner D-matrices. The algorithm benefits from the numerous recurrence relations satisfied by these functions allowing a fast evaluation of the mutual potential. Moreover, forces and torques are straightforwardly obtained by application of ladder operators taken from the angular momentum theory and commonly used in quantum mechanics. A numerical implementation of this algorithm is made. Tests show that the present code is significantly faster than those currently available in literature.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
If we denote by \(\{C^{(T)}_{lm},S^{(T)}_{lm}\}\) the Stokes coefficients defined in (Tricarico 2008, Eqs. 14, 15), those of the present paper are given by \(\{C_{lm}, S_{lm}\} = N_{lm} \{C^{(T)}_{lm},S^{(T)}_{lm}\}\) with
$$\begin{aligned} N_{lm} = \frac{2}{1+\delta _{m,0}}\frac{(l-m)!}{(l+m)!} . \end{aligned}$$
References
Borderies, N.: Mutual gravitational potential of N solid bodies. Celest. Mech. 18, 295–307 (1978). doi:10.1007/BF01230170
Boué, G., Correia, A.C.M., Laskar, J.: Complete spin and orbital evolution of close-in bodies using a Maxwell viscoelastic rheology. Celest. Mech. Dyn. Astron. 126(1), 31–60 (2016). doi:10.1007/s10569-016-9708-x
Compère, A., Lemaître, A.: The two-body interaction potential in the STF tensor formalism: an application to binary asteroids. Celest. Mech. Dyn. Astron. 119, 313–330 (2014). doi:10.1007/s10569-014-9568-1
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). doi:10.1007/s10569-006-9045-6
Gimbutas, Z., Greengard, L.: A fast and stable method for rotating spherical harmonic expansions. J. Comput. Phys. 228, 5621–5627 (2009)
Hartmann, T., Soffel, M.H., Kioustelidis, T.: On the use of STF-tensors in celestial mechanics. Celest. Mech. Dyn. Astron. 60, 139–159 (1994). doi:10.1007/BF00693097
Hou, X., Scheeres, D.J., Xin, X.: Mutual potential between two rigid bodies with arbitrary shapes and mass distributions. Celest. Mech. Dyn. Astron. (2016). doi:10.1007/s10569-016-9731-y
Maciejewski, A.J.: Reduction, relative equilibria and potential in the two rigid bodies problem. Celest. Mech. Dyn. Astron. 63, 1–28 (1995). doi:10.1007/BF00691912
Mathis, S., Le Poncin-Lafitte, C.: Tidal dynamics of extended bodies in planetary systems and multiple stars. Astron. Astrophys. 497, 889–910 (2009). doi:10.1051/0004-6361/20079054
Paul, M.K.: An expansion in power series of mutual potential for gravitating bodies with finite sizes. Celest. Mech. 44, 49–59 (1988). doi:10.1007/BF01230706
Poincaré, H.: Sur une forme nouvelle des équations de la mécanique. C. R. Acad. Sci. 132, 369–371 (1901)
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). doi:10.1007/s10569-008-9128-7. 0711.2078
Varshalovich, D., Moskalev, A., Khersonskii, V.: Quantum Theory of Angular Momentum. World Scientific, Singapore (1988)
Werner, R.A.: Spherical harmonic coefficients for the potential of a constant-density polyhedron. Comput. Geosci. 23, 1071–1077 (1997). doi:10.1016/S0098-3004(97)00110-6
Werner, R.A., Scheeres, D.J.: Mutual potential of homogeneous polyhedra. Celest. Mech. Dyn. Astron. 91, 337–349 (2005). doi:10.1007/s10569-004-4621-0
Wigner, E.P.: Group theory and its application to the quantum mechanics of atomic spectra. Academic Press, New York (1959)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Boué, G. The two rigid body interaction using angular momentum theory formulae. Celest Mech Dyn Astr 128, 261–273 (2017). https://doi.org/10.1007/s10569-017-9751-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10569-017-9751-2