Iteration stability for simple Newtonian stellar systems

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,” $κ$⁠, 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 $κ$ 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-$κ$ 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.