Optimized formulas of the gravitational field of a vertical cylindrical prism

Modeling the gravitational effects of the topography and other layers of the Earth’s interior is one of the fundamental topics in geodesy and geophysics. Previous formulas of the Gravitational Potential (GP), Gravitational Vector (GV), and Gravitational Gradient Tensor (GGT) of a vertical cylindrical prism were derived from complex conversion relations and were relatively complicated. In this contribution, we derive the optimized formulas through a particular geometrical relation between the computation point and integration point, which are consistent with the previous expressions. The analytical formulas of the GP, GV, and GGT of a cylindrical shell are presented when the computation point is located on the polar axis. Based on these formulas, a cylindrical shell benchmark is put forward to evaluate the numerical properties of the cylindrical prism, that is, to discretize a whole cylindrical shell into cylindrical prisms. Beyond the improved simplicity, our optimized formulas with a second-order 3D Taylor series expansion help to save computation time (particularly for the GGT up to 20%). Numerical results reveal that when the computation point's vertical distance changes, the relative and absolute errors are symmetric with respect to the center vertical distance of the cylindrical shell. Accompanying codes in Python for a vertical cylindrical prism and a cylindrical shell are provided at https://www.github.com/xiaoledeng/optimized-formulas-of-gp-gv-ggt.