Abstract
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.
Similar content being viewed by others
Notes
Figure 7 shows a subset of Figures A.1 and A.2 (in Appendix A). The cross section is the \(xz\)-plane and \(n=10\).
Figure 8 shows a subset of Figures B.1 and B.2 (in Appendix B). The cross section is the \(xz\)-plane and \(n\) \(=\) \(10\).
The acceleration error plot for case 1 for Bennu is shown in Fig. 11. The identical plot is given in Figure A.6.
The acceleration error plot for case 1 for Castalia is shown in Fig. 12. The identical plot is given in Figure B.5.
The acceleration errors are shown in Fig. 13.
The acceleration errors are shown in Fig. 14.
References
Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, chap. 9, 10. U.S. Department of Commerce and Knovel Library (online), 10th ed. (1972)
Allen, A.J., Palmer, P.L., Papaloizou, J.: A conservative numerical technique for collisionless dynamical systems: comparision of the radial and circular orbit instabilities. Mon. Not. R. Astron. Soc. 242, 576–594 (1992)
Arfken, G.: Mathematical Methods for Physicists, chap. 9, 11, 12, 3rd edn. Academic Press Inc, NY (1985)
Brillouin, M.: Equations aux Dériveées partielles du 2e ordre. Domaines à connexion multiple. Fonctions sphériques non antipodes. Annales De L’Institut H. Poincaré 4(2), 173–206 (1933)
Cangahuala, L.A.: Augmentations to the Polyhedral Gravity Model to Facilitate Small Body Navigation. American Astronomical Society, Washington DC (2005)
Cunningham, L.E.: On the computation of the spherical harmonic terms needed during the numerical integration of the orbital motion of an artificial satellite. Celest. Mech. 2, 207–216 (1970)
Gottlieb, P.: Estimation of local lunar gravity features. Radio Sci. 5(2), 301–312 (1970)
Hanna, J.R., Rowland, J.H.: Fourier Series, Transforms, and Boundary Value Problems, chap. 1, 2, 5, 6, 2nd edn. A Wiley-Interscience Publication, London (1990)
Herrera-Sucarrat, E., Palmer, P., Roberts, R.: Modeling the gravitational potential of a nonspherical asteroid. J. Guid. Control Dyn. 36(3), 790–798 (2013)
Hudson, R.S., Ostro, S.J.: Shape of asteroid 4769 castalia, (1989 PB) from inversion of radar images. Science 263(1994), 940–943 (1989)
Jones, B.A., Born, G.H., Beylkin, G.: A Cubed Sphere Gravity Model for Fast Orbit Propagation, pp. 09–137. American Astronomical Society, Washington, DC (2009)
Jones, B.A., Efficient Models for the Evaluation and Estimation of the Gravity Field, Ph.D. thesis, The University of Colorado at Boulder (2010)
Kaula, W.M.: Theory of Satellite Geodesy, chap. 1. Blaisdell Publishing Company, Waltham, Massachusetts (1966)
Kholchevnikov, C.: Le développement du potentiel dans le cas d’une densité analytique. Celest. Mech. 3, 232–240 (1971)
Kholchevnikov, C.: Le développement du potentiel dans le cas d’une densité lisse. Celest. Mech. 6, 214–220 (1972)
Kholshevnikov, C.: On convergence of an asymmetrical body potential expansion in spherical harmonics. Celest. Mech. 16, 45–60 (1977)
Liu, H., Zou, J.: Zeros of the Bessel and spherical Bessel functions and their applications for uniqueness in inverse acoustic obstacle scattering. J. Appl. Math. 72, 817–831 (2007)
Lundberg, J.B., Schutz, B.E.: Recursion formulas of Legendre functions for use with nonsingular geopotential models. J. Guid. Control Dyn. 11, 31–38 (Feb. 1988)
MacRobert, T.M.: Spherical Harmonics: An Elementary Treatise on Harmonic Functions with Applications, chap. 2, 4, 6, 8, 14, 2nd edn. Dover Publications Inc, NY (1948)
Nolan, M., Magri, C., Howell, E., Benner, L.A.M., Giorgini, J.D., Hergenrother, C., Hudson, R., Lauretta, D., Margot, J.-L., Ostro, S., Scheeres, D.: Shape model and surface properties of the OSIRIS-REx target asteroid (101955) Bennu from radar and lightcurve observations. Icarus 226, 629–640 (2013)
Ostro, S.J., Chandler, J.F., Hine, A.A., Rosema, K.D., Shapiro, I.I., Yeomans, D.K.: Radar images of asteroid 1989 PB. Science 248, 1523–1528 (1990)
Palmer, P.L.: Stability of Collisionless Stellar Systems—mechanisms For The Dynamical Structure of Galaxies, chap. 10, pp. 181–190. Kluwer Academic Publishers, Dordrecht (1994)
Park, R.S., Werner, R.A., Bhaskaran, S.: Estimating small-body gravity field from shape model and navigation data. J. Guid. Control Dyn. 33, 212–221 (2010)
Pinsky, M.A.: Introduction to Partial Differential Equations with Applications, chap. 0, 2, 3, 4. MacGraw-Hill Book Company, NY (1984)
Russell, R.P., Arora, N.: Global Point Mascon Models for Simple. Accurate and Parallel Geopotential Computation, pp. 11–158. AAS/AIAA Space Flight Mechanics Meeting (2011)
Takahashi, Y., Scheeres, D.: Generalized density distribution estimation for small bodies. In: 23rd AAS/AIAA Space Flight Mechanics Meeting, No. 13–265 (2013)
Takahashi, Y., Scheeres, D.: Surface Gravity Fields for Asteroids and Comets. In: 22nd AAS/AIAA Space Flight Mechanics Meeting (2012)
Takahashi, Y., Scheeres, D., Werner, R.A.: Surface gravity fields for asteroids and comets. J. Guid. Control Dyn. 36(2), 362–374 (2013)
Tapley, B.D., Schutz, B.E., Born, G.H.: Statistical Orbit Determination. Elsevier Academic Press, Amsterdam (2004)
Watson, C.: A Treatise on the Theory of Bessel Functions, chap. 2, 3, 15, 2nd edn. Cambridge University Press, Cambridge (1944)
Werner, R.A.: Evaluating Descent and Ascent Trajectories Near Non-Spherical Bodies. Tech. Rep. http://www.techbriefs.com/component/content/article/8726, Jet Propulsion Laboratory (2010)
Werner, R.A., Scheeres, D.J.: Exterior gravitation of a polyhedron derived and compared with harmonic and mascon gravitation representations of asteroid 4769 castalia. Celest. Mech. Dyn. Astron. 65, 314–344 (1997)
Werner, R.A.: Spherical Harmonic coefficients for the potential of a constant-density polyhedron. Comput. Geosci. 23(10), 1071–1077 (1997)
Acknowledgments
This research was supported by NASA’s OSIRIS-REx New Frontiers mission through grant NNM10AA11C.
Author information
Authors and Affiliations
Corresponding author
Electronic supplementary material
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
Takahashi, Y., Scheeres, D.J. Small body surface gravity fields via spherical harmonic expansions. Celest Mech Dyn Astr 119, 169–206 (2014). https://doi.org/10.1007/s10569-014-9552-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10569-014-9552-9