06 Fakultät Luft- und Raumfahrttechnik und Geodäsie
Permanent URI for this collectionhttps://elib.uni-stuttgart.de/handle/11682/7
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Item Open Access Analytical solutions for gravitational potential up to its third-order derivatives of a tesseroid, spherical zonal band, and spherical shell(2023) Deng, Xiao-Le; Sneeuw, NicoThe spherical shell and spherical zonal band are two elemental geometries that are often used as benchmarks for gravity field modeling. When applying the spherical shell and spherical zonal band discretized into tesseroids, the errors may be reduced or cancelled for the superposition of the tesseroids due to the spherical symmetry of the spherical shell and spherical zonal band. In previous studies, this superposition error elimination effect (SEEE) of the spherical shell and spherical zonal band has not been taken seriously, and it needs to be investigated carefully. In this contribution, the analytical formulas of the signal of derivatives of the gravitational potential up to third order (e.g., V , Vz, Vzz, Vxx, Vyy, Vzzz, Vxxz, and Vyyz) of a tesseroid are derived when the computation point is situated on the polar axis. In comparison with prior research, simpler analytical expressions of the gravitational effects of a spherical zonal band are derived from these novel expressions of a tesseroid. In the numerical experiments, the relative errors of the gravitational effects of the individual tesseroid are compared to those of the spherical zonal band and spherical shell not only with different 3D Gauss–Legendre quadrature orders ranging from (1,1,1) to (7,7,7) but also with different grid sizes (i.e., 5∘×5∘, 2∘×2∘, 1∘×1∘, 30′×30′, and 15′×15′) at a satellite altitude of 260 km. Numerical results reveal that the SEEE does not occur for the gravitational components V , Vz, Vzz, and Vzzzof a spherical zonal band discretized into tesseroids. The SEEE can be found for the Vxxand Vyy, whereas the superposition error effect exists for the Vxxzand Vyyzof a spherical zonal band discretized into tesseroids on the overall average. In most instances, the SEEE occurs for a spherical shell discretized into tesseroids. In summary, numerical experiments demonstrate the existence of the SEEE of a spherical zonal band and a spherical shell, and the analytical solutions for a tesseroid can benefit the investigation of the SEEE. The single tesseroid benchmark can be proposed in comparison to the spherical shell and spherical zonal band benchmarks in gravity field modeling based on these new analytical formulas of a tesseroid.Item Open Access Self-consistent transformation of first-, second-, and third-order potential gradients among Cartesian, cylindrical, and spherical coordinates(2026) Deng, Xiao-Le; Sneeuw, NicoKnowing how to transform potential gradients between different coordinate systems is of fundamental importance in potential field theory. For first- and second-order gradients, such transformations are conventionally dealt with in terms of vector-matrix notation. However, matrix notation is not helpful for deriving the expressions for transformation of third-order potential gradients. In this contribution, we derive the general detailed expressions for transformation of first-, second-, and third-order potential gradients between two coordinate systems by using the direct expansion method. As examples of these general expressions, we derive detailed expressions for forward and inverse transformations of physical components of first-, second-, and third-order potential gradients among Cartesian, cylindrical, and spherical coordinates. Laplace’s equation has been applied for a first validation of partial expressions. However, to validate all newly derived expressions in a systematic way, we propose the closed-loop transformation cycle method that presents a full-fledged commutative diagram of forward and backward transformations among all three coordinate systems, i.e., potential gradients can become themselves after the closed round-trip transformation cycle among Cartesian, cylindrical, and spherical coordinates. Results reveal that this transformation cycle method confirms the correctness of all derived expressions. These general expressions for transformation of first-, second-, and third-order potential gradients can be applied under arbitrary two coordinate systems, and their detailed expressions can be systematically validated by the proposed transformation cycle method.