Comparison of three methods for computing the gravitational attraction of tesseroids at satellite altitude
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Cited by 3 publications
References 26 publications
In recent years, there are research trends from constant to variable density and low-order to high-order gravitational potential gradients in gravity field modeling. Under the research circumstances, this paper focuses on the variable density model for gravitational curvatures (or gravity curvatures, third-order derivatives of gravitational potential) of a tesseroid and spherical shell in the spatial domain of gravity field modeling. In this contribution, the general formula of the gravitational curvatures of a tesseroid with arbitrary-order polynomial density is derived. The general expressions for gravitational effects up to the gravitational curvatures of a spherical shell with arbitrary-order polynomial density are derived when the computation point is located above, inside, and below the spherical shell. When the computation point is located above the spherical shell, the general expressions for the mass of a spherical shell and the relation between the radial gravitational effects up to arbitrary-order and the mass of a spherical shell with arbitrary-order polynomial density are derived. The influence of the computation point’s height and latitude on gravitational curvatures with the polynomial density up to fourth-order is numerically investigated using tesseroids to discretize a spherical shell. Numerical results reveal that the near-zone problem exists for the fourth-order polynomial density of the gravitational curvatures, i.e., relative errors in $$\log _{10}$$ log 10 scale of gravitational curvatures are large than 0 below the height of about 50 km by a grid size of $$15'\times 15'$$ 15 ′ × 15 ′ . The polar-singularity problem does not occur for the gravitational curvatures with polynomial density up to fourth-order because of the Cartesian integral kernels of the tesseroid. The density variation can be revealed in the absolute errors as the superposition effects of Laplace parameters of gravitational curvatures other than the relative errors. The derived expressions are examples of the high-order gravitational potential gradients of the mass body with variable density in the spatial domain, which will provide the theoretical basis for future applications of gravity field modeling in geodesy and geophysics.
In recent years, there are research trends from constant to variable density and low-order to high-order gravitational potential gradients in gravity field modeling. Under the research circumstances, this paper focuses on the variable density model for gravitational curvatures (or gravity curvatures, third-order derivatives of gravitational potential) of a tesseroid and spherical shell in the spatial domain of gravity field modeling. In this contribution, the general formula of the gravitational curvatures of a tesseroid with arbitrary-order polynomial density is derived. The general expressions for gravitational effects up to the gravitational curvatures of a spherical shell with arbitrary-order polynomial density are derived when the computation point is located above, inside, and below the spherical shell. When the computation point is located above the spherical shell, the general expressions for the mass of a spherical shell and the relation between the radial gravitational effects up to arbitrary-order and the mass of a spherical shell with arbitrary-order polynomial density are derived. The influence of the computation point’s height and latitude on gravitational curvatures with the polynomial density up to fourth-order is numerically investigated using tesseroids to discretize a spherical shell. Numerical results reveal that the near-zone problem exists for the fourth-order polynomial density of the gravitational curvatures, i.e., relative errors in $$\log _{10}$$ log 10 scale of gravitational curvatures are large than 0 below the height of about 50 km by a grid size of $$15'\times 15'$$ 15 ′ × 15 ′ . The polar-singularity problem does not occur for the gravitational curvatures with polynomial density up to fourth-order because of the Cartesian integral kernels of the tesseroid. The density variation can be revealed in the absolute errors as the superposition effects of Laplace parameters of gravitational curvatures other than the relative errors. The derived expressions are examples of the high-order gravitational potential gradients of the mass body with variable density in the spatial domain, which will provide the theoretical basis for future applications of gravity field modeling in geodesy and geophysics.
Analytical Solutions for Gravitational Potential up to Its Third-order Derivatives of a Tesseroid, Spherical Zonal Band, and Spherical Shell
The 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, $$V_{z}$$ V z , $$V_{zz}$$ V zz , $$V_{xx}$$ V xx , $$V_{yy}$$ V yy , $$V_{zzz}$$ V zzz , $$V_{xxz}$$ V xxz , and $$V_{yyz}$$ V yyz ) 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., $$5^{\circ }\times 5^{\circ }$$ 5 ∘ × 5 ∘ , $$2^{\circ }\times 2^{\circ }$$ 2 ∘ × 2 ∘ , $$1^{\circ }\times 1^{\circ }$$ 1 ∘ × 1 ∘ , $$30^{\prime }\times 30^{\prime }$$ 30 ′ × 30 ′ , and $$15^{\prime }\times 15^{\prime }$$ 15 ′ × 15 ′ ) at a satellite altitude of 260 km. Numerical results reveal that the SEEE does not occur for the gravitational components V, $$V_{z}$$ V z , $$V_{zz}$$ V zz , and $$V_{zzz}$$ V zzz of a spherical zonal band discretized into tesseroids. The SEEE can be found for the $$V_{xx}$$ V xx and $$V_{yy}$$ V yy , whereas the superposition error effect exists for the $$V_{xxz}$$ V xxz and $$V_{yyz}$$ V yyz of 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.
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