Determination of the Earth’s disturbing potential and its functionals as series of spherical harmonics, without using the Laplace equation.

In this work we present a new method of determining the disturbing potential T and its functionals without using the Laplace equation.

The first step of this method is to solve two Dirichlet boundary value problems on the surface of the geoid related to a new partial differential equation (created by the author) named as G – modified Helmholtz equation. The solutions are formed after spherical approximation and describe gravity anomaly Δg and gravity disturbance δg as series of spherical harmonics.

In the second step we determine the disturbing potential T as a solution of a Dirichlet boundary value problem on the same boundary surface related to a very simple partial differential equation. Its Dirichlet boundary condition is formed with the aid of gravity anomaly, gravity disturbance and the fundamental boundary condition in spherical approximation. The solution is expressed as a series of spherical harmonics. As an epilogue to this work we present some new formulae for normal gravity γ, gravity g, vertical gradient of gravity, and mean curvature for actual equipotential surfaces as series of spherical harmonics.

The difference between known spherical harmonics and the introduced spherical harmonics is that the latter has the polar distance r in irrational powers. The advantage of this method is the simple determination of gravity anomaly, gravity disturbance and disturbing potential since it involves only Dirichlet boundary value problems. Finally this method shows that the determination of gravity anomaly and gravity disturbance can be made without determining the disturbing potential.