### 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,” , 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 is held fixed, but converges with other choices. To understand the mathematical reasons for this, we start the iteration from an approximation that is perturbatively different from the actual solution and, for the current study, confine ourselves to spherical symmetry. A cycle of iteration is treated as a linear “updating” operator, and the properties of the linear operator, especially its spectrum, determine the convergence properties. 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- 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.

Received 10 March 2009
Accepted 06 June 2009
Published online 13 July 2009

Acknowledgments:
We gratefully acknowledge support for this work under NSF Grant Nos. PHY-0554367 and PHY-0503366, NASA Grant No. NNG05GB99G, by the Greek State Scholarships Foundation, and by the Center for Gravitational Wave Astronomy. We thank Alan Farrell for supplying some computational results.

Article outline:

I. INTRODUCTION
A. Background
B. Spherically symmetric equations
C. Outline
II. FIXED-ITERATION
A. Self-adjoint linearized updating operator
B. Polytropes and eigenvalue bounds
C. Numerical investigations
III. FIXED DENSITY
A. Fixed central density and finite difference computations
B. Fixed central density and the continuum
C. Fixed intermediate density and finite difference computations
D. Fixed intermediate density and continuum models
IV. SUMMARY AND CONCLUSIONS

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