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For an equation of state in which pressure is a function only of density, theanalysis of Newtonian stellar structure is simple in principle if the system isaxisymmetric, or consists of a corotating binary. It is then required only tosolve two equations: one stating that the "injection energy", $\kappa$, apotential, is constant throughout the stellar fluid, and the other being theintegral over the stellar fluid to give the gravitational potential. Aniterative solution of these equations generally diverges if $\kappa$ is heldfixed, but converges with other choices. We investigate the mathematical reasonfor this convergence/divergence by starting the iteration from an approximationthat is perturbatively different from the actual solution. A cycle of iterationis then treated as a linear "updating" operator, and the properties of thelinear operator, especially its spectrum, determine the convergence properties.For simplicity, we confine ourselves to spherically symmetric models in whichwe analyze updating operators both in the finite dimensional spacecorresponding to a finite difference representation of the problem, and in thecontinuum, and we find that the fixed-$\kappa$ operator is self-adjoint andgenerally has an eigenvalue greater than unity; in the particularly importantcase of a polytropic equation of state with index greater than unity, we provethat there must be such an eigenvalue. For fixed central density, on the otherhand, we find that the updating operator has only a single eigenvector, withzero eigenvalue, and is nilpotent in finite dimension, thereby giving aconvergent solution.

Iteration Stability for Simple Newtonian Stellar Systems