Abstract
This contribution deals with the derivation of explicit expressions of the gradients of first, second and third order of the gravitational potential. This is accomplished in the framework of tensor analysis which naturally allows to apply general formulae to the specific coordinate systems in use in geodesy. In particular it is recalled here that when the potential field is expressed in general coordinates on a 3D manifold, the gradient operation leads to the definition of the covariant derivative and that the covariant derivative of a tensor can be obtained by application of a simple rule. When applied to the gravitational potential or to any of its gradients, the rule straightforwardly provides the expressions of the higher-order gradients. It is also shown that the tensor approach offers a clear distinction between natural and physical components of the gradients. Two fundamental reference systems—a global, bodycentric system and a local, topocentric system, both body-fixed—are introduced and transformation rules are derived to convert quantities between the two systems. The results include explicit expressions for the gradients of the first three orders in both reference systems.
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References
Albertella A, Migliaccio F, Sansò F, Tscherning CC (2000) The space-wise approach—overall scientific data strategy. In: Suenkel (ed) From Eötvös to Milligal, Final Report, ESA Study, ESA/ESTEC Contract No. 3329/98/NL/GD, 267–298
Albertella A, Migliaccio F, Sansò F (2002) GOCE: The earth gravity field by space gradiometry. Celest Mech Dyn Astron 81: 1–15
Borisenko AI, Tarapov IE (1979) Vector and tensor analysis with applications, revised english edition. Dover Publications, New York
Caola MJ (1978) Solid harmonics and their addition theorems. J Phys A Math Gen 11: L23–L25
Cunningham LE (1970) 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
Fantino E, Casotto S (2008) Methods of harmonic synthesis for global geopotential models and their first-, second- and third-order gradients. J Geod. doi:10.1007/s00190-008-0275-0
Garmier R, Barriot J-P (2001) Ellipsoidal Harmonic Expansions of the Gravitational Potential: Theory and Application. Celest Mech Dyn Astron 79: 235–275
Gelfand IM (1989) Lectures on Linear Algebra. Dover Publ, New York
Grafarend EW (1986) Differential geometry of the gravity field. Manuscr Geod 11: 29–37
Hassani S (1999) Mathematical Physics – A Modern Introduction to its Foundations. Springer, New York
Hopkins J (1973) Mathematical models of geopotential gradients. In: Proceedings of the symposium on Earth’s gravitational field & secular variations in position, pp 93–105
Hotine M (1969) Mathematical geodesy, environmental science services administration (ESSA) monograph no. 2, U.S. Department of Commerce, Washington, DC
Ilk KH (1983) Ein Beitrag zur Dynamik ausgedhenter Körper, Deutsche Geodaetische Kommission, Bayerischen Akademie der Wissenschaften, Heft Nr. 288, München
Keller W, Sharifi MA (2005) Satellite gradiometry using a satellite pair. J Geod 78: 544–557
Koop R (1993) Global Gravity Field Modeling Using Satellite Gravity Gradiometry, Netherlands Geodetic Comission, Publications on Geodesy, New Series, Number 38, Delft, The Netherlands
Koop R, Stelpstra D (1989) On the computation of the gravitational potential and its first and second order derivatives. Manuscr Geod 14: 373–382
MacMillan WD (1930) Theory of the potential. McGraw-Hill, New York
Marussi A (1951) Fondamenti di Geodesia Intrinseca. Pubbl Comm Geodet Ital Terza Ser Mem 7. (1985) Intrinsic geodesy (trans: Marussi A). Springer, Berlin
Moritz H (1967) Kinematical geodesy. Report no. 92, Department of Geodetic Science, Ohio State University, Columbus
Moritz H (1971) Kinematical geodesy II. Report no. 165, Department of Geodetic Science, Ohio State University, Columbus
Moritz H, Hoffman-Wellenhof B (1993) Geometry, relativity, geodesy. Wichmann, Karlsruhe
Magnus JR, Neudecker H (1988) Matrix differential calculus, with applications in statistics and econometrics. Wiley, Chichester
Ostro SJ, Hudson RS, Nolan MC, Margot J-L, Scheeres DJ, Campbell DB, Magri C, Giorgini JD, Yeomans DK (2000) Radar observations of asteroid 216 Kleopatra. Science 288(5467): 836–839
Reed GB (1973) Application of kinematical geodesy for determining the short wave length components of the gravity field by satellite gradiometry. Report no. 201, Department of Geodetic Science, Ohio State University, Columbus
Rummel R (1986) Satellite gradiometry. In: Sünkel (1986) Mathematical and numerical techniques in physical geodesy. Springer, Berlin, pp 317–363
Rummel R, Van Gelderen M, Koop R, Schrama E, Sansò F, Brovelli M, Migliaccio F, Sacerdote F (1993) Spherical harmonic analysis of satellite gradiometry. Publications on Geodesy, New Series, No. 39, Netherlands Geodetic Commission
Schutz B (1980) Geometrical methods of mathematical physics. Cambridge University Press, Cambridge
Spain B (1960) Tensor calculus. Oliver & Boyd, Edinburgh. Reprinted (2003) Dover, Mineola
Thong NC, Grafarend EW (1989) A spheroidal harmonic model of the terrestrial gravitational field. Manuscr Geod 14: 285–304
Tóth Gy, Földvàry L (2005) Effect of geopotential model errors on the projection of GOCE gradiometer observables. In: Jekeli C, Bastos L, Fernandes J (eds) Gravity, Geoid and space missions, IAG symposia 129, Springer, Berlin
Tscherning CC (1976) Computation of the second-order derivatives of the normal potential based on the representation by a legendre series. Manuscr Geod 1: 71–92
Wrede RC (1972) Introduction to vector and tensor analysis. Dover Publications, New York
Zund J (1989) A mathematical appreciation of Antonio Marussi’s contributions to geodesy. In: Sacerdote F, Sansò F (eds) Proc. II Hotine-Marussi symposium on mathematical geodesy, Pisa, 5–8 June 1989, 1–18
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Casotto, S., Fantino, E. Gravitational gradients by tensor analysis with application to spherical coordinates. J Geod 83, 621–634 (2009). https://doi.org/10.1007/s00190-008-0276-z
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DOI: https://doi.org/10.1007/s00190-008-0276-z