Gravitational field of a homogeneous spherical longitudinal sector

Abstract

Abstract When investigating the gravitational field (e.g., gravitational potential, gravitational vector, gravitational gradient tensor, and gravitational curvature) in the spherical coordinate system, there are currently three benchmarks: a spherical shell, a spherical zonal band, and a spherical prism. To obtain simple analytical formulas similar to those of a spherical shell and to reduce computation time with respect to a spherical zonal band, we introduce a spherical longitudinal sector as a new benchmark. It is defined geometrically as the part of a spherical shell confined between two meridians. Alternatively, one can consider the spherical longitudinal sector to be a spherical prism without latitude restrictions, i.e., extending from pole to pole. Based on the relationship with a spherical prism, we derive analytical expressions for the gravitational potential, gravitational vector, gravitational gradient tensor, and gravitational curvature of a homogeneous spherical longitudinal sector when the computation point is on the polar axis. The effectiveness of this proposed benchmark of a spherical longitudinal sector discretized into spherical prisms has been validated through numerical experiments. Numerical results indicate that, compared to a spherical zonal band discretized into spherical prisms, a spherical longitudinal sector improves computational efficiency by a factor of 2, corresponding to a 50% reduction in computation time when the longitudinal and colatitudinal grid sizes are identical. A spherical longitudinal sector computes roughly 360/ n times faster than a spherical shell, where n is the longitudinal grid size in degree. The analytical expressions for the gravitational field of a homogeneous spherical longitudinal sector can be an additional benchmark for the gravity forward modeling in geodesy, geophysics, and planetary sciences.