Gravitational gradients by tensor analysis with application to spherical coordinates

Abstract

This contribution deals with the derivation of explicit expressions of the gradients of first, second and third order of the gravitational potential. This is accomplished in the framework of tensor analysis which naturally allows to apply general formulae to the specific coordinate systems in use in geodesy. In particular it is recalled here that when the potential field is expressed in general coordinates on a 3D manifold, the gradient operation leads to the definition of the covariant derivative and that the covariant derivative of a tensor can be obtained by application of a simple rule. When applied to the gravitational potential or to any of its gradients, the rule straightforwardly provides the expressions of the higher-order gradients. It is also shown that the tensor approach offers a clear distinction between natural and physical components of the gradients. Two fundamental reference systems—a global, bodycentric system and a local, topocentric system, both body-fixed—are introduced and transformation rules are derived to convert quantities between the two systems. The results include explicit expressions for the gradients of the first three orders in both reference systems.