Third-order development of shape, gravity, and moment of inertia for highly flattened celestial bodies. Application to Ceres

¹ IMCCE, Observatoire de Paris – PSL Research University, Sorbonne Universités – UPMC Univ. Paris 06, Univ. Lille 1, CNRS, 77 Avenue Denfert-Rochereau, 75014 Paris, France
e-mail: Nicolas.Rambaux@obspm.fr
² LGLTPE, CNRS UMR5276, ENS de Lyon, Site Monod, 15 parvis René Descartes, 69007 Lyon, France
³ Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA

Received: 20 July 2015
Accepted: 4 November 2015

Abstract

Context. We investigate the hydrostatic shape and gravitational potential coefficients of self-gravitating and rotating bodies large enough to have undergone internal differentiation and chemical stratification. Quantifying these properties under the assumption of hydrostatic equilibrium forms the basis for interpreting shape and gravity data in terms of interior structure and infer deviations from hydrostaticity that can bring information on the thermal and chemical history of the objects.

Aims. The main purpose is to show the importance of developing the reference hydrostatic shape for relatively fast rotating bodies up to third order to reach an accuracy of a few tens of meters. This paper especially focuses on Ceres, for which high-resolution shape data are being obtained by the Dawn spacecraft, with a projected accuracy better than 200 m/pixel.

Methods. To improve the accuracy on the determination of geodetic parameters, we numerically integrated Clairaut’s equations of rotational equilibrium expanded up to third order in a small parameter m, the geodetic parameter.

Results. Previous studies of Ceres have been based on shape models developed to first order. However, we show that the first-order theory underestimates (a−c) (where a and c are the equatorial and polar radii) by 1.8 km, which leads to underestimating the extent of mass concentration and is insufficient to interpret the upcoming observations by Dawn space mission. Instead, by using the third-order theory, we obtain an accuracy of 25 meters that is better than the accuracy expected from Dawn. Then, we derive the following geodetical quantities: flattening and other shape parameters, gravitational potential coefficients, and moments of inertia, by using the Ceres models constrained by observations obtained with the Hubble Space Telescope and ground-based adaptive optics telescopes. The difference in equatorial and polar radii for a large parametric space of interior models is investigated, and the large (a−c) corresponds to a model with a low density contrast.

Conclusions. This type of modeling will also prove instrumental to infer non-hydrostatic contributions to Ceres’ shape that are to be measured by Dawn.