A benchmark for gravitational potential up to its third-order derivatives of a vertical cylindrical shell discretized into cylindrical prisms

Studying the gravitational effects of the Earth's topography and crustal layers is a fundamental topic in gravity field modeling in geodesy and geophysics. The introduction of gravitational curvatures (GC), which are the third-order derivatives of the gravitational potential (GP), has recently broadened theoretical research on gravitational effects. Using tensor analysis, this paper comes up with a general formula for the physical parts of the third-order tensor of the potential in cylindrical coordinates. Then, the expressions for the GC of a vertical cylindrical prism are accordingly derived in cylindrical coordinates. Based on the relation among the vertical cylindrical prism, cylindrical shell, and cylinder, the analytical expressions for gravitational effects up to the GC of a vertical cylindrical shell and a cylinder are derived when the computation point is located on the Z-axis no matter whether it is situated below, inside, or above the cylindrical shell and cylinder. Laplace's equation has been adopted to confirm the correctness of the newly derived formulas of the GC. In addition, a benchmark of a cylindrical shell discretized into cylindrical prisms is proposed to reveal the numerical properties of derived GC formulas with the computation point located on the Z-axis. Numerical results reveal that when the computation point's height increases, the relative and absolute errors of the GP, gravitational vector (GV), gravitational gradient tensor (GGT), and GC decrease, in which the relative errors in log₁₀ scale of the nonzero GP, GV, GGT, and GC components are approximately less than -2 when the computation is located below, inside, and above the cylindrical shell. These newly derived formulas lay the theoretical foundation for the GC in cylindrical coordinates and help to investigate the potential applications of the GC in geodesy and geophysics. This new benchmark can become the standard for testing the correctness of the gravitational effects of the cylindrical prism using different numerical algorithms in cylindrical coordinates in practical applications.