AI 中文总结
研究由特定欧拉-泊松系统控制的液体莱恩-埃姆登星，通过制定转折点原理，结合气态尾部分析，得出增长模式数量相关结论，恢复强化径向稳定性结果并获非径向稳定性转折点准则。
AI 中文摘要
固定绝热指数后，由具有‘刚性气体’状态方程\(p = \rho^\gamma - 1\)的欧拉-泊松系统控制的球对称液体莱恩-埃姆登星形成一个单参数族，由中心密度\(\kappa=\bar\rho_\kappa(0)\in(1,\infty)\)自然参数化。与气态情况不同，液体自由边界打破了族的自相似性并使质量-半径曲线弯曲，当\(\gamma<2(d - 1)/d\)时产生质量极值。我们按照泽尔多维奇、惠勒和索恩的精神，以及哈季奇、林和赖因作品中严格的相对论对应物，制定了一个转折点原理：径向增长模式的数量等于线性化算子的负莫尔斯指数，沿族局部恒定，且仅在质量-半径曲线的转折点处变化，那里的弯曲方向决定是获得还是失去一个增长模式。我们描述了大中心密度极限，其中气态尾部（从平面动力系统读出）决定质量-半径曲线是否螺旋，从而决定增长模式的数量是保持有界还是趋于无穷。作为推论，我们恢复并强化了已知的液体莱恩-埃姆登星的径向（不）稳定性结果，并结合现有的非径向分析，获得了非径向稳定性的转折点准则。
英文摘要
Upon fixing the adiabatic index, the spherically symmetric liquid Lane-Emden stars governed by the Euler-Poisson system with a "stiffened gas" equation of state $p=ρ^γ-1$ form a one-parameter family, naturally parametrised by the central density $κ=\barρ_κ(0)\in(1,\infty)$. In contrast to the gaseous case, the liquid free boundary breaks the self-similarity of the family and bends the mass-radius curve, creating extrema of the mass when $γ<2(d-1)/d$. We formulate a turning point principle in the spirit of Zel'dovich, Wheeler and Thorne, and of its rigorous relativistic counterpart from the works of Hadžić, Lin and Rein: the number of radial growing modes equals the negative Morse index of the linearised operator, is locally constant along the family, and can change only at the turning points of the mass-radius curve, the bending orientation there dictating whether a growing mode is gained or lost. We describe the large central density limit in which the gaseous tail (read off from a planar dynamical system) determines whether the mass-radius curve spirals, and hence whether the number of growing modes remains bounded or tends to infinity. As corollaries we recover and sharpen the known radial (in)stability results for liquid Lane-Emden stars and, in combination with existing non-radial analysis, obtain a turning point criterion for non-radial stability.
Comments32 pages
