A numerical method for reorientation of rotating tidally deformed visco-elastic bodies

Currently the true polar wander (TPW) of a visco-elastic body is studied with three types of approches: (i) a linear dynamic approach which applies the linearised Liouville equation (e.g. Wu and Peltier [1984]); (ii) a non-linear dynamic approach which is based on the quasi-fluid approximation (e.g. Ricard et al. [1993]); (iii) a long-term limit approach which only considers the fluid limit of a reorientation (e.g. Matsuyama and Nimmo [2007]). Several limitations of these approaches have not been studied: the range for which the linear approach is accurate, the validity of the quasi-fluid approximation, and the dynamic solution for TPW of a tidally deformed rotating body. In order to deal with these isssues, we establish a numerical procedure which is able to determine the large angle reorientation of a visco-elastic celestial body that can be both centrifugally and tidally deformed. We show that the linear approach leads to significant errors for loadings near the poles or the equator. For instance, when the loading is placed at 10 degree colatitude on a model representing the Earth, the maximum allowed TPW is just 0.2 degree for the error of the linear method to remain below 1%. Secondly, we show that slow relaxation modes can have a significant effect on large angle TPW of Earth or other planets and ignoring these modes can lead to large error for the transient response of TPW. Finally, we show that reorientation of a tidally deformed body driven by a positive mass anomaly near the poles has a preference for rotating around the tidal axis instead of towards it. At a tidally deformed body, positive mass anomalies are more likely to be found near the equator and the plane perpendicular to the tidal axis, while negative mass anomalies tend to be near the great circle with longitudes 0 and 180 degree.