On the elastodynamics of rotating planets

TL;DR

This paper derives comprehensive equations of motion for rotating, self-gravitating planets, separating elastic deformation from rigid motions, and extends the framework to layered fluid-solid bodies with potential applications in planetary dynamics.

Contribution

It introduces a novel decomposition of planetary motion equations, incorporating a Tisserand frame, and extends the approach to layered fluid-solid planetary models with new conservation laws.

Findings

Derived equations for elastic and rigid motions in rotating planets.

Extended the framework to layered fluid-solid planetary models.

Discussed equilibria, linearization, and applications to planetary dynamics.

Abstract

Equations of motion are derived for (visco)elastic, self-gravitating, and variably-rotating planets. The equations are written using a decomposition of the elastic motion that separates the body's elastic deformation from its net translational and rotational motion as far as possible. This separation is achieved by introducing degrees of freedom that represent the body's rigid motions; it is made precise by imposing constraints that are physically motivated and should be practically useful. In essence, a Tisserand frame is introduced exactly into the equations of solid mechanics. The necessary concepts are first introduced in the context of a solid body, motivated by symmetries and conservation laws, and the corresponding equations of motion are derived. Next, it is shown how those ideas and equations of motion can readily be extended to describe a layered fluid--solid body. A possibly new conservation law concerning inviscid fluids is then stated. Thereafter the equilibria and linearisation of the fluid--solid equations of motion are discussed, along with new equations for use within normal-mode coupling calculations and other Galerkin methods. Finally, the extension of these ideas to the description of multiple, interacting fluid--solid planets is qualitatively discussed.