Divergence-free spherical harmonic gravity field modelling based on the Runge–Krarup theorem: a case study for the Moon

Recent numerical studies on external gravity field modelling show that external spherical harmonic series may diverge near or on planetary surfaces. This paper investigates an alternative solution that is still based on external spherical harmonic series, but capable of avoiding the divergence effect. The approach relies on the Runge–Krarup theorem and the iterative downward continuation. In theory, Runge–Krarup-type solutions are able to approximate the true potential within the entire space external to the masses with an arbitrary ε -accuracy, ε >0 . Using gravity implied by the lunar topography, we show numerically that this technique avoids indeed the divergence effect, at least at the studied 5 arc-min resolution. In the context of the iterative scheme, we show that a function expressed as a truncated solid spherical harmonic expansion on a general star-shaped surface possesses an infinite surface spherical harmonic spectrum after it is mapped onto a sphere. We also study the convergence of the gradient approach, which is a technique for efficient grid-wise synthesis on irregular surfaces. We show that the resulting Taylor series may converge slowly when analytically upward continuing from points inside the masses. The continuation from the mass-free space should therefore be preferred. As an underlying topic of the paper, spherical harmonic coefficients from spectral gravity forward modelling and their Runge–Krarup counterpart are numerically studied. Regarding their different nature, we formulate some research topics that might contribute to a deeper understanding of the current methodologies used to develop combined high-degree spherical harmonic gravity models.