The temporal variation of the spherical and Cartesian multipoles of the gravity field: the generalized MacCullagh representation

The Cartesian moments of the mass density of a gravitating body and the sph erical harmonic coefficients of its gravitational field are related in a pe culiar way. In particular, the products of inertia can be expressed by the spherical harmonic coefficients of the gravitational potential as was deriv ed by MacCullagh for a rigid body. Here the MacCullagh formulae are extende d to a deformable body which is restricted to radial symmetry in order to a pply the Love-Shida hypothesis. The mass conservation law allows a represen tation of the incremental mass density by the respective excitation functio n. A representation of an arbitrary Cartesian monome is always possible by sums of solid spherical harmonics multiplied by powers of the radius. Intro ducing these representations into the definition of the Cartesian moments, an extension of the MacCullagh formulae is obtained. In particular, for exc itation functions with a vanishing harmonic coeffcient of degree zero, the (diagonal) incremental moments of inertia also can be represented by the ex citation coefficients. Four types of excitation functions are considered, n amely: (1) tidal excitation; (2) loading potential, (3) centrifugal potenti al; and (4) transverse surface stress. One application of the results could be model computation of the length-of-day variations and polar motion, whi ch depend on the moments of inertia.