The instability of an elastic planet with a liquid core

Considering a spherical planet with a liquid core surrounded by a solid shell, we developed a quasi-static model to investigate the deformation of the double-layered planet with self-gravity and obtained the boundary value problem about radial equilibrium, which is solved by the numerical methods. The effects of governing parameters about geometry, density and bulk modulus on the deformation of the planet with self-gravity were discussed. In addition, we also developed the incremental equation theory to investigate the stability of the double-layered planet under its own gravity. It is concluded that instability is more likely to occur on the planet with smaller liquid cores when the outer radius and density of the planets are constant. Although we only study special double-layered planets, these methods can be conveniently extended to complex multi-layered planets. [Figure not available: see fulltext.].