Evaluation of the spherical harmonic coefficients for the external potential of a polyhedral body with linearly varying density

  • Cheng ChenEmail author
  • Yongbing Chen
  • Shaofeng Bian
Original Article


The gravitational potential and its derivatives of a polyhedral body with linearly varying density can be expressed as closed analytical expressions in terms of spherical harmonics. In this paper, a recursive algorithm using linear integrals is proposed by the Gauss divergence theorem and the Stokes theorem to evaluate the spherical harmonic coefficients for the external potential. The formulas can handle a polyhedron with general polygonal faces instead of triangular faces. It is shown that the algorithm is stable. The recursive algorithms for computing the spherical harmonic coefficients of the potentials of a right rectangular prism with linearly varying density and of a prism with density varying with depth following a cubic polynomial are also given. The potential of the prisms can be directly expressed as the spherical harmonics with respect to the whole prism that does not need division, as well as no influence on the stabilities of the recursive algorithms. To verify the correctness of the algorithm in this paper, the numerical experiments for the two actual polyhedral bodies with linearly varying density including a right rectangular prism and the asteroid 433 EROS are implemented.


Gravitational potential Spherical harmonics Polyhedron Normalization Stability 



This work was supported by National Natural Science Foundation of China under Grant Nos. 41474061, 41631072 and 41774021.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of NavigationNaval University of EngineeringWuhanChina

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