Characterization of instabilities in the problem of elastic planetary tides
In 1911, A. E. H. Love published a linear elastic model for the tidal deformation of planetary bodies. Using numerical techniques that were unavailable to Love, surprising behaviors of the tidal solution have been found: tides of finite, even substantial, height are possible in the presence of an infinitesimal tide raiser, thus indicating some sort of instability.
The Love tidal model was for the deformation of a homogeneous sphere. In order to better understand the nature of the instabilities in this model, I consider the effect of adding a radially dependent density profile to the model. For a given singularity, an increase in the initial density gradient causes the singularity to change locations in parameter space. For steep enough density gradient, the singularity is pushed outside the realm of physically meaningful parameter space for certain initial radial density profiles.
Self-gravitation appears to be the likely mechanism for the driving of the tidal instability. The nature of the behavior of self-gravitation will be studied by considering an exact elastic formulation of the problem. In this way, a more complete view of the processes involved in the tidal deformation of a body can be explored. I find that each of the curves of singularity loci observed in the tidal problem correspond to instabilities in different modes for the exact elastic self-gravitation problem.