Convergence and divergence in spherical harmonic series of the gravitational field generated by high‐resolution planetary topography—A case study for the Moon. Issue 8 (18th August 2017) - Hirt, Christian; Kuhn, Michael
Abstract: Theoretically, spherical harmonic (SH) series expansions of the external gravitational potential are guaranteed to converge outside the Brillouin sphere enclosing all field‐generating masses. Inside that sphere, the series may be convergent or may be divergent. The series convergence behavior is a highly unstable quantity that is little studied for high‐resolution mass distributions. Here we shed light on the behavior of SH series expansions of the gravitational potential of the Moon. We present a set of systematic numerical experiments where the gravity field generated by the topographic masses is forward‐modeled in spherical harmonics and with numerical integration techniques at various heights and different levels of resolution, increasing from harmonic degree 90 to 2160 ( 61 to 2.5 km scales). The numerical integration is free from any divergence issues and therefore suitable to reliably assess convergence versus divergence of the SH series. Our experiments provide unprecedented detailed insights into the divergence issue. We show that the SH gravity field of degree‐180 topography is convergent anywhere in free space. When the resolution of the topographic mass model is increased to degree 360, divergence starts to affect very high degree gravity signals over regions deep inside the Brillouin sphere. For degree 2160 topography/gravity models, severe divergence (with several 1000 mGal amplitudes) prohibits accurate gravity modeling over most of the topography. As a key result, we formulate a new hypothesis to predict divergence: if the potential degree variances show a minimum, then the SH series expansions diverge somewhere inside the Brillouin sphere and modeling of the internal potential becomes relevant. Key Points: New systematic experiments reveal behavior of spherical harmonic series near the Moon's topography as function of resolution and altitude Gravity from degree‐180 models free of divergence, degree‐360 models partially divergent, and degree‐2160 models severely divergent New hypothesis: local minimum in potential degree variances foreshadows series divergence inside the Brillouin sphere Plain Language Summary: Planetary scientists often use series expansions to represent gravity field variations. However, this mathematical technique may produce grossly incorrect results when applied with very high resolution on rugged planetary bodies such as the Moon. This is also called series divergence. This study presents new systematic experiments that compare the series expansions with a second mathematical technique called numerical integration. The experiments provide for the first time insights into the behavior of series expansions for modeling of the Moon's gravity field. As main result, the series expansions—as commonly used in planetary sciences—cannot be used to accurately represent km to 10 km details of the Moon's gravity field near the surface of the Moon. The paper identifies the chances and limitations of frequently used modeling techniques for the Moon. It presents a new hypothesis that uses a new, simple indicator for occurrence or absence of series divergence. The methods presented can be applied to other terrestrial planets, e.g., Venus and Mars, to obtain similarly detailed insight into the series expansions.