Analytical Solutions for Gravitational Potential up to Its Third-order Derivatives of a Tesseroid, Spherical Zonal Band, and Spherical Shell

Abstract

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, \(V_{z}\), \(V_{zz}\), \(V_{xx}\), \(V_{yy}\), \(V_{zzz}\), \(V_{xxz}\), and \(V_{yyz}\)) 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., \(5^{\circ }\times 5^{\circ }\), \(2^{\circ }\times 2^{\circ }\), \(1^{\circ }\times 1^{\circ }\), \(30^{\prime }\times 30^{\prime }\), and \(15^{\prime }\times 15^{\prime }\)) at a satellite altitude of 260 km. Numerical results reveal that the SEEE does not occur for the gravitational components V, \(V_{z}\), \(V_{zz}\), and \(V_{zzz}\) of a spherical zonal band discretized into tesseroids. The SEEE can be found for the \(V_{xx}\) and \(V_{yy}\), whereas the superposition error effect exists for the \(V_{xxz}\) and \(V_{yyz}\) 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.