  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.
