Convergence of spherical harmonic series expansion of the Earth’s gravitational potential

Abstract

Given a continuous boundary value on the boundary of a “simply closed surface” ∂S that encloses the whole Earth, a regular harmonic fictitious field V*(P) in the domain outside an inner sphere Ki that lies inside the Earth could be determined, and it is proved that V*(P) coincides with the Earth’s real field V(P) in the whole domain outside the Earth. Since in the domain outside the inner sphere Ki and the fictitious regular harmonic function V*(P) could be expressed as a uniformly convergent spherical harmonic series, it is concluded that the Earth’s potential field could be expressed as a uniformly convergent spherical harmonic expansion series in the whole domain outside the Earth.