Characterization of Instabilities in the Tidal Deformation of a Plaetary Body

In 1911, A.E.H. Love published a linear elastic model for the tidal amplitude of a uniform, compressible, self-gravitating body. Our recent numerical evaluations of the solution to Love's governing equations reveal portions of parameter space for which an infinitesimal tide raiser can raise a tide of arbitrary height. In addition, using a solution technique different from Love's, investigations have been made into the effect of allowing non-uniform, radially varying density and elasticity in Love's formulation. It has been found that the tidal instabilities persist when the body initially has a radially varying density profile. To further investigate the nature of the tidal instabilities, an exact elastic formulation of the self-gravitational collapse of a uniform sphere is considered. Analysis reveals that the instability curves observed in Love's tidal solution for a uniform sphere and in the tidal solution for a sphere with radially varying density and elasticity correspond to various unstable modes of self-gravitation.