On the amplitudes of spherical harmonics of gravitational potencial and generalised products of inertia

Abstract

The vector field of the force of gravitational attraction due to an extended rigid body (of arbitrary irregular geometrical shape, and with an arbitrary internal mass distribution inside it) at any point outside the body can be derived from the gradient of a scalar field, its gravitational potential. In terms of spherical polar coordinates (distance from the origin, colatitude or latitude, and longitude) that potential can be expanded as an absolutely convergent series of spherical harmonics, involving Legendre polynomials and associated Legendre functions of the first kind depending on the colatitude (or the latitude) and circular functions depending on the longitude. In the present contributed paper we establish, in terms of the so–called “integrals of inertia” (or “generalised products of inertia”) of the body, general formulae for the amplitudes (i.e., for the coefficients) of the different zonal, tesseral, and sectorial harmonics of any degree and order in the said series expansion of the gravitational potential outside the body.