Practical SPH models for major planets

Modelling planets is done for two main reasons – the first to further understanding of their internal structure and the second to provide models to explore astrophysical situations in which planets play a role. For the latter reason, the requirements on accuracy are less severe, although the planet must be realistic in its major features. A numerical model of a layered giant planet is developed with an iron core, a silicate mantle, an ice region and a hydrogen–helium atmosphere. The Tillotson equation of state is used and examples of two model planets are given, one reproducing the mass and radius of Jupiter quite closely and the other with two Jupiter masses. Transferring these results into a smoothed particle hydrodynamics (SPH) model presents two main difficulties. A uniform distribution of SPH points leads to too few points representing the non-atmospheric component. It is shown that using a distorted lattice enables the core + silicate + ice to be represented by several hundred points so that the evolution of these regions can be followed in detail. Another difficulty concerns the density discontinuities attendant on a layered structure. Density estimates of SPH points are either too large or too small near material interfaces leading to unrealistic pressure gradients and, consequently, to large and unphysical local forces. Algorithms are described for avoiding this difficulty both at material interfaces and near the surface of the planet. In some astrophysical situations involving SPH-modelled planets, the main bulk of the planet is so opaque that internal heat transfer can be neglected. However, surface regions should radiate and a convenient way for including radiation from a planetary surface is described.