Gravitational stability of spherical self-gravitating relaxation models

The gravitational stability of spherical, self-gravitating, hydrostatically pre-stressed planetary models remains a subject of active interest. Love ( 1907, 1911) was the first to show that purely elastic models can become uns table when values of rigidity and bulk modulus are insufficient to countera ct self-gravitational collapse. We revisit his calculations and extend his work to show that so-called dilatational (or 'D') modes of a viscoelastic s phere can also become unstable to self-gravitation in a specific region of Lame parameter space. As an example, we derive a marginal stability curve f or the dilatational modes of a homogeneous planetary model at spherical har monic degree two. We demonstrate that the stability conditions are independ ent of viscosity and that the instability will occur only when the homogene ous earth model is already unstable to the elastic instability described by Love (1907, 1911). Finally, we also consider a class of Rayleigh-Taylor (o r 'RT') instabilities related to unstable density stratification in planeta ry models. This convective instability is explored using both a homogeneous Maxwell viscoelastic sphere (which has an unstable layering at all depths) and a suite of Maxwell earth models that adopt the elastic and density str ucture of the seismic model PREM (which has regions of unstable density str atification within the upper mantle). We argue that previous studies have s ignificantly overestimated the potential importance of these modes to Earth evolution. For example, suggestions that the timescale of the RT modes is short relative to the age of the Earth face the fundamental problem that th e ensuing convective instability would have long ago destroyed the unstable layering and produced an adiabatic profile. We predict that at low degrees the RT instabilities for a PREM density profile and realistic viscosity st ratification have timescales comparable to the age of the Earth. It is uncl ear, in any event, whether the unstable density layering in the PREM upper mantle is robust.