Spheroidal and ellipsoidal harmonic expansions of the gravitational potential of small Solar System bodies. Case study: Comet 67P/Churyumov‐Gerasimenko

Abstract

Gravitational features are a fundamental source of information to learn more about the interior structure and composition of planets, moons, asteroids, and comets. Gravitational field modeling typically approximates the target body with a sphere, leading to a representation in spherical harmonics. However, small celestial bodies are often irregular in shape and hence poorly approximated by a sphere. A much better suited geometrical fit is achieved by a triaxial ellipsoid. This is also mirrored in the fact that the associated harmonic expansion (ellipsoidal harmonics) shows a significantly better convergence behavior as opposed to spherical harmonics. Unfortunately, complex mathematics and numerical problems (arithmetic overflow) so far severely limited the applicability of ellipsoidal harmonics. In this paper, we present a method that allows expanding ellipsoidal harmonics to a considerably higher degree compared to existing techniques. We apply this novel approach to model the gravitational field of comet 67P, the final target of the Rosetta mission. The comparison of results based on the ellipsoidal parameterization with those based on the spheroidal and spherical approximations reveals that the latter is clearly inferior; the spheroidal solution, on the other hand, is virtually just as accurate as the ellipsoidal one. Finally, in order to generalize our findings, we assess the gravitational field modeling performance for some 400 small bodies in the Solar System. From this investigation we generally conclude that the spheroidal representation is an attractive alternative to the complex ellipsoidal parameterization, on the one hand, and the inadequate spherical representation, on the other hand.

Forward Gravity Modelling to Augment High-Resolution Combined Gravity Field Models

“…The debate on how well we can model the very high frequencies is ongoing. It is discussed in the literature that using ellipsoidal harmonics, which are used in the present study, instead of spherical harmonics can improve the convergence regions of the gravity field series expansion (Hu and Jekeli 2015;Reimond and Baur 2016). Accordingly, divergence may start at a higher degree in our approach than in an approach based on spherical harmonics and in our validation we do not see any evidence of divergence at degree 3660.…”

Convergence and divergence in spherical harmonic series of the gravitational field generated by high‐resolution planetary topography—A case study for the Moon

“…Our tests conclusively showed that numerical problems cannot explain the results of this study. Furthermore, our results are in line with other studies on the divergence problem [e.g., Hu and Jekeli , ; Reimond and Baur , ] that attribute series divergence as cause for invalid values produced by the spectral technique inside the Brillouin sphere.…”

“…In recent time, the divergence behavior of low-degree harmonic series expansions of gravity field functionals has been intensively studied for irregularly shaped bodies such as the Martian moons Deimos and Phobos [Hu and Jekeli, 2015]; asteroids, e.g., 433 Eros [Hu, 2012], Castalia, and Bennu [Takahashi et al, 2013;Takahashi and Scheeres, 2014]; and comets, e.g., 67P/Churyumov-Gerasimenko [Reimond and Baur, 2016]. All studies demonstrated substantial divergence for evaluation points inside the Brillouin sphere, occurring already at low spectral resolution, showing the SH series unable to model the near-surface exterior gravity field of irregularly shaped bodies with adequate precision and detail.…”

“…Because of their widespread application in planetary sciences, the present study focuses on exterior spherical harmonic series, in our case, of gravity values. We acknowledge that some studies discuss spheroidal (ellipsoidal) instead of spherical harmonics to improve the convergence region of the gravity field series expansions [ Hu and Jekeli , ; Reimond and Baur , ]. The ellipsoidal approach, however, does not appear promising for the Moon, given its negligible flattening.…”

Ejection of large particles from cometary nuclei in the shape of prolate ellipsoids

“…Images taken by Osiris (optical, spectroscopic, and infrared remote imaging system) on board the Rosetta spacecraft allowed astronomers to construct a map of the surface of the 67P/C‐G nucleus. Therefore, in the last 2 years, a number of valuable and very advanced papers on the 67P/C‐G gravitational field have been published (Groussin et al ; Jorda et al ; Lhotka et al ; Pötzold et al ; Preusker et al ; Reimond & Baur, ; Sierks et al ). Summarizing the results of these all papers, it can be stated that the nucleus of 67P/C‐G has a conspicuous bilobate shape with the following overall dimensions along its main axes: 4.34 × 2.60 × 2.12 km.…”