简单牛顿恒星系统的迭代稳定性
Iteration Stability for Simple Newtonian Stellar Systems
摘要：
对于一个状态方程，其中压力只是密度的函数，牛顿恒星结构的分析在原则上是简单的，如果系统是轴对称的，或由一个共转双星组成。然后只需要求解两个方程：一个方程说明“注入能量”，$κ$，一个势，在整个恒星流体中是恒定的，另一个是恒星流体的积分，以给出引力势。如果$κ$保持固定，则这些方程的迭代解通常发散，但对于其他选择则收敛。我们调查的数学原因，这种收敛/发散开始迭代从一个近似，是扰动不同的实际解决方案。一个循环的迭代，然后被视为一个线性“更新”算子，和线性算子的性质，特别是它的谱，确定的收敛性能。为了简单起见，我们把自己限制在球对称模型中，我们分析了更新算子在有限维空间对应的有限差分表示的问题，并在连续，我们发现，固定- $κ$运营商是自伴的，一般有一个特征值大于1;在指数大于1的多方状态方程的特殊情况下，我们证明了必有这样一个本征值。另一方面，对于固定的中心密度，我们发现更新算子只有一个特征向量，零特征值，在有限维是幂零的，从而给出收敛解。
英文摘要：
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. 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-$κ$ 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.
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