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Gravitational field modelling near irregularly shaped bodies using spherical harmonics: a case study for the asteroid (101955) Bennu

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Abstract

We apply three spherical-harmonic-based techniques to deliver external gravitational field models of the asteroid (101955) Bennu within its circumscribing sphere. This region is known to be peculiar for external spherical harmonic expansions, because it may lead to a divergent series. The studied approaches are (i) spectral gravity forward modelling via external spherical harmonics, (ii) the least-squares estimation from surface gravitational data using external spherical harmonics and (iii) the combination of internal and external series expansions. While the first method diverges beyond any reasonable doubts, we show that the other two methods may ensure relative accuracy from \({\sim }10^{-6}\) to \(10^{-8}\) in the vicinity of Bennu. This is possible even with the second method, despite the fact that it relies on a single series of external spherical harmonics. Our main motivation was to study conceptual differences between spherical harmonic coefficients from satellite data (analogy to the first method) and from surface gravitational data (the second method). Such coefficients are available through the popular spherical-harmonic-based models of the Earth’s gravitational field and often are combined together. We show that the coefficients from terrestrial data may lead to a divergence effect of partial sums, though excellent accuracy can be achieved when such model is used in full. Under (presently) extreme but realistic conditions, the divergence effect of partial sums may affect many near-surface geoscientific applications, such as the geoid/quasigeoid computation or residual terrain modelling. Computer codes (Fortran, MATLAB) and data produced within the study are made freely available at http://edisk.cvt.stuba.sk/~xbuchab/.

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Data Availability:

The input polyhedral model of Bennu’s shape is available at https://sbn.psi.edu/pds/resource/bennushape.html. The data produced within this study are available at http://edisk.cvt.stuba.sk/~xbuchab/. These include: (i) the \({\bar{r}}_{nm}^{\mathrm {S}}\) coefficients from Eq. (1), (ii) spherical harmonic coefficients of \(V^{\mathrm {SGFM}}\) from Eq. (2) (including the effect of the \(V_{\mathrm {ball}}\) term), (iii) the \({\bar{V}}_{nm}^{\mathrm {LS}}(N_{\mathrm {LS}})\) coefficients from Eq. (13) for \(N_{\mathrm {LS}} = 0, 5, 10, \dots , 65\), (iv) the \({\bar{V}}_{nm}^{\mathrm {Ext}}\) and \({\bar{V}}_{nm}^{\mathrm {Int}}\) coefficients computed with \(p_{\mathrm {max}}^{\mathrm {Int}} = p_{\mathrm {max}}^{\mathrm {Ext}} = 1000\), (v) the three grids of the reference gravitational potential from Table 2 and (vi) the GFM_DE_rule package for spatial-domain gravity forward modelling (Fortran and MATLAB) after Fukushima (2017). Due to their size (\({\sim }17.5\) GB), the scalar products from Eq. (10) were not uploaded, but are available on request from BB. Other (already published) routines that were used in this study, particularly for ultra-high-degree spherical harmonic analysis and synthesis (Fortran, C and MATLAB), are also available at the latter link.

Notes

  1. For the Earth, the first radius is taken from Sjöberg (1977) and the other one is equal to the semi-minor axis of GRS80 (Moritz 2000).

  2. Gradshteyn and Ryzhik (2007) acknowledge on asymptotic series that (Page 21) despite the fact that these series diverge, the values of the functions that they represent can be calculated with a high degree of accuracy if we take the sum of a suitable number of terms of such series.

  3. In our case, Bennu’s surface S given by Eq. (1) is sufficiently regular. Therefore, one can refer to the Keldysh–Lavrentiev theorem (e.g. Moritz 1980), implying that the \(\varepsilon \)-accuracy can theoretically be achieved not only above Bennu, but also on its surface.

  4. The integral can be weakly singular.

  5. In our study, \(N_{\mathrm {LS}}\) plays the role of N from Sacerdote and Sansò (2010).

  6. For flattened gravitating bodies such as Bennu or the Earth, advantages of spheroidal/ellipsoidal harmonics over spherical harmonics are well documented in the literature (e.g. Garmier and Barriot 2001; Hu and Jekeli 2015; Reimond and Baur 2016; Sebera et al. 2016).

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Acknowledgements

BB was supported by the project VEGA 1/0809/21. We thank three reviewers for their valuable comments on the manuscript. The computations were performed at the HPC centres at the Slovak University of Technology in Bratislava, the Slovak Academy of Sciences and the University of Žilina, which are parts of the Slovak Infrastructure of High Performance Computing (SIVVP project, ITMS code 26230120002, funded by the European region development funds, ERDF). The maps were produced using the Generic Mapping Tools (Wessel and Smith 1998) and the plots were prepared with the Python’s Matplotlib module (Hunter 2007).

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Authors and Affiliations

  1. Department of Theoretical Geodesy and Geoinformatics, Slovak University of Technology in Bratislava, Radlinského 11, 81005, Bratislava, Slovak Republic

    Blažej Bucha

  2. Department of Civil and Environmental Engineering, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milan, Italy

    Fernando Sansò

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  1. Blažej Bucha
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BB designed the study, conducted all numerical experiments and drafted the manuscript. FS improved the design of the numerical experiments and refined the theoretical background of the study. BB and FS discussed and commented on the manuscript.

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Correspondence to Blažej Bucha.

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Bucha, B., Sansò, F. Gravitational field modelling near irregularly shaped bodies using spherical harmonics: a case study for the asteroid (101955) Bennu. J Geod 95, 56 (2021). https://doi.org/10.1007/s00190-021-01493-w

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