A computational method for rotating, multi-layer spheroids with internal jumps

Abstract

Abstract We discuss the structure of differentially rotating, multilayer spheroids containing mass-density jumps and rotational discontinuities at the interfaces. The study is based upon a scale-free, numerical method. Polytropic equations of state and cylindrical rotation profiles are assumed. The Bernoulli equation and the Poisson equation for the gravitational potential are solved for each layer separately on a common computational grid. The 2-layer (core-envelope) case is first investigated in detail. We find that the core and the envelope are not, in general, homothetical in shape (cores are more than spherical than the envelope). Besides, the occurence of a mass-density jump all along the interface is prone to a rotational discontinuity (unless the polytropic indices are the same). In particular, for given rotation laws, the mass-density jump is not uniform along the interface. Tests, trends and examples (e.g. false bipolytrope, critical rotation, degenerate configurations) are given. Next, we consider the general case of systems made of ${\cal L}>2$ layers. This includes the full equation set, the virial equation, a comprehensive step-by-step procedure and two examples of tripolytropic systems. The properties observed in the 2-layer case hold for any pairs of adjacent layers. In spite of a different internal structure, two multilayer configurations can share the same mass, same axis ratio, same size and same surface velocity (which is measured through a degeneracy parameter). Applications concern the determination of the interior of planets, exoplanets, stars and compact objects.