Iteration Stability for Simple Newtonian Stellar Systems

TL;DR

This paper analyzes the convergence properties of iterative methods for solving Newtonian stellar structure equations, revealing conditions under which the iteration converges or diverges based on the spectral properties of the updating operator.

Contribution

It provides a mathematical analysis of the iteration stability in stellar models, especially for polytropic equations of state, using spectral analysis of the updating operator.

Findings

Fixed-$$ operator is self-adjoint with eigenvalues greater than one.

For fixed central density, the operator has a single zero eigenvalue and is nilpotent.

Convergence depends on the spectral properties of the updating operator.

Abstract

For an equation of state in which pressure is a function only of density, the analysis of Newtonian stellar structure is simple in principle if the system is axisymmetric, or consists of a corotating binary. It is then required only to solve two equations: one stating that the "injection energy", $\kappa$, a potential, is constant throughout the stellar fluid, and the other being the integral over the stellar fluid to give the gravitational potential. An iterative solution of these equations generally diverges if $\kappa$ is held fixed, but converges with other choices. We investigate the mathematical reason for this convergence/divergence by starting the iteration from an approximation that is perturbatively different from the actual solution. A cycle of iteration is then treated as a linear "updating" operator, and the properties of the linear operator, especially its spectrum, determine the convergence properties. For simplicity, we confine ourselves to spherically symmetric models in which we analyze updating operators both in the finite dimensional space corresponding to a finite difference representation of the problem, and in the continuum, and we find that the fixed-$\kappa$ operator is self-adjoint and generally has an eigenvalue greater than unity; in the particularly important case of a polytropic equation of state with index greater than unity, we prove that there must be such an eigenvalue. For fixed central density, on the other hand, we find that the updating operator has only a single eigenvector, with zero eigenvalue, and is nilpotent in finite dimension, thereby giving a convergent solution.