[Paper Review] Stability analysis of Newtonian polytropes

Christian G. Boehmer, Tiberiu Harko|arXiv (Cornell University)|Feb 9, 2009

Advanced Differential Geometry Research3 references2 citations

TL;DR

This paper investigates the stability of Newtonian polytropic fluid spheres using multiple analytical methods—linear stability analysis, Jacobi stability via Kosambi-Cartan-Chern theory, and the Lyapunov function method. By transforming the Lane-Emden equation into an autonomous system and then into a regular second-order differential equation, the study reveals that stability outcomes vary significantly with the polytropic index, offering a robust framework for constraining physical properties of Newtonian stars.

ABSTRACT

We analyze the stability of Newtonian polytropic static fluid spheres, described by the Lane-Emden equation. In the general case of arbitrary polytropic indices the Lane-Emden equation is a non-linear second order ordinary differential equation. By introducing a set of new variables, the Lane-Emden equation can be reduced to an autonomous system of two ordinary differential equations, which in turn may be transformed to another regular second order differential equation. We perform the study of stability by using linear stability analysis, the Jacobi stability analysis (Kosambi-Cartan-Chern theory) and the Lyapunov function method. Depending on the values of the polytropic index characterizing the fluid, these different methods yield different qualitative results on the stability of the solutions. On the other hand, these techniques offer a powerful method for constraining the physical properties of the Newtonian stars.

Motivation & Objective

• To analyze the stability of Newtonian polytropic static fluid spheres governed by the Lane-Emden equation.
• To investigate how different stability analysis techniques yield varying qualitative results depending on the polytropic index.
• To develop a unified framework using multiple analytical methods to constrain the physical properties of Newtonian stars.
• To transform the non-linear Lane-Emden equation into an autonomous system for enhanced stability analysis.

Proposed method

• Transform the non-linear second-order Lane-Emden equation into an autonomous system of two ordinary differential equations via a change of variables.
• Further reduce the system to a regular second-order differential equation to facilitate stability analysis.
• Apply linear stability analysis to assess small perturbations around equilibrium solutions.
• Employ Jacobi stability analysis based on Kosambi-Cartan-Chern theory to evaluate geodesic deviation in the dynamical system.
• Utilize the Lyapunov function method to determine global stability properties of the fluid configurations.
• Compare results across all three methods to assess consistency and sensitivity to the polytropic index.

Experimental results

Research questions

• RQ1 How does the stability of Newtonian polytropic fluid spheres vary with different values of the polytropic index?
• RQ2 What are the qualitative differences in stability outcomes when using linear stability analysis, Jacobi stability, and the Lyapunov function method?
• RQ3 To what extent do the three stability analysis techniques yield consistent or divergent conclusions for the same physical system?
• RQ4 How does the transformation of the Lane-Emden equation into an autonomous system improve the analysis of stability?
• RQ5 Can the combined use of multiple stability methods provide tighter constraints on the physical properties of Newtonian stars?

Key findings

• The stability of Newtonian polytropic fluid spheres depends critically on the value of the polytropic index, with distinct behaviors emerging across different ranges.
• Linear stability analysis reveals instability for certain polytropic indices, particularly those near the critical value marking the onset of gravitational collapse.
• Jacobi stability analysis identifies regions of instability through curvature-based geodesic deviation, offering a geometric interpretation of dynamical instability.
• The Lyapunov function method confirms global stability for specific polytropic indices, providing a complementary energy-based assessment.
• Discrepancies in stability conclusions across methods highlight the sensitivity of results to the analytical framework used.
• The combined application of all three methods offers a more comprehensive and robust characterization of stellar configurations than any single method alone.