Multi-component, rigidly rotating polytropes: improved and extended theory

TL;DR

This paper advances the theory of multi-component, rigidly rotating polytropes by including boundary conditions and intersecting boundaries, providing analytical solutions and a detailed example of complex configurations.

Contribution

It extends existing models to include subsystems with boundary intersections and nonzero boundary densities, and offers analytical solutions for special cases.

Findings

Equilibrium configurations are independent of fluid nature within certain polytropic indices.

Series expansion solutions can be extended beyond the boundary.

A detailed example demonstrates complex intersecting boundary configurations.

Abstract

With respect to earlier investigations, the theory of multi-component, concentric, copolar, axisymmetric, rigidly rotating polytropes is improved and extended, including subsystems with nonzero density on the boundary and subsystems with intersecting boundaries. The formulation is restricted to two subsystems for simplicity but, in principle, can be extended to $N$ subsystems. Equilibrium configurations are independent of the nature of the fluid i.e. collisional or collisionless, provided the polytropic index lies within the range, $1/2\le n\le5$, as in one-component systems. The solution of the equilibrium equations is expanded in power series, which can be continued up to the boundary and outside via starting points placed at increasingly larger distance from the centre of mass. A detailed analysis is devoted to special cases where the solution of the equilibrium equations can be expressed analytically. Finally a guidance example is shown, involving homogeneous subsystems with intersecting boundaries, where a substantially flattened component extends outside a slightly flattened one.