Spherical Bessel Harmonic Gravity Expansions and Their Properties Related to Sir Isaac Newton's Shell Theorem via the Riemann Zeta Function

The conventional spherical harmonics used to describe the gravity field surrounding a non-spherical mass only converge exterior to the body’s Brillouin (circumscribing) sphere. The complementary gravity field interior to this sphere is provided by spherical Bessel harmonics. We present an algorithm for calculating the Bessel harmonic coefficients directly from the surface integral of an arbitrary shape with homogenous density. Exploring the special case of a homogenous sphere, we show that the Bessel harmonic expansion is a generalization of Newton's Shell theorem. In this special case, the non-zero Bessel coefficients are shown to be related to the Riemann zeta function.