Celestial Mechanics and Dynamical Astronomy

, Volume 119, Issue 2, pp 169–206 | Cite as

Small body surface gravity fields via spherical harmonic expansions

  • Yu Takahashi
  • D. J. Scheeres
Original Article


Conventional gravity field expressions are derived from Laplace’s equation, the result being the spherical harmonic gravity field. This gravity field is said to be the exterior spherical harmonic gravity field, as its convergence region is outside the Brillouin (i.e., circumscribing) sphere of the body. In contrast, there exists its counterpart called the interior spherical harmonic gravity field for which the convergence region lies within the interior Brillouin sphere that is not the same as the exterior Brillouin sphere. Thus, the exterior spherical harmonic gravity field cannot model the gravitation within the exterior Brillouin sphere except in some special cases, and the interior spherical harmonic gravity field cannot model the gravitation outside the interior Brillouin sphere. In this paper, we will discuss two types of other spherical harmonic gravity fields that bridge the null space of the exterior/interior gravity field expressions by solving Poisson’s equation. These two gravity fields are obtained by assuming the form of Helmholtz’s equation to Poisson’s equation. This method renders the gravitational potentials as functions of spherical Bessel functions and spherical harmonic coefficients. We refer to these gravity fields as the interior/exterior spherical Bessel gravity fields and study their characteristics. The interior spherical Bessel gravity field is investigated in detail for proximity operation purposes around small primitive bodies. Particularly, we apply the theory to asteroids Bennu (formerly 1999 RQ36) and Castalia to quantify its performance around both nearly spheroidal and contact-binary asteroids, respectively. Furthermore, comparisons between the exterior gravity field, interior gravity field, interior spherical Bessel gravity field, and polyhedral gravity field are made and recommendations are given in order to aid planning of proximity operations for future small body missions.


Asteroid Gravity field Spherical harmonics Spherical Bessel function Proximity operation Laplace’s equation Poisson’s equation Brillouin sphere Bennu Castalia 



This research was supported by NASA’s OSIRIS-REx New Frontiers mission through grant NNM10AA11C.

Supplementary material

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Supplementary material 1 (pdf 5827 KB)
10569_2014_9552_MOESM2_ESM.pdf (234 kb)
Supplementary material 2 (pdf 233 KB)


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.University of Colorado at BoulderBoulderUSA
  2. 2.Jet Propulsion Laboratory/California Institute of TechnologyPasadenaUSA

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