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Linearized Analysis of Adiabatic Oscillations of Rotating Gaseous Stars

Tetu Makino

TL;DR

This paper tackles the linear adiabatic oscillations of rotating self-gravitating gaseous stars by formulating the perturbation problem in Lagrangian coordinates and deriving the ELASO system $\\partial_t^2\\bm{u}+\\bm{B}\\partial_t\\bm{u}+\\mathcal{L}\\bm{u}=0$ together with the quadratic pencil $\\mathfrak{L}(\\lambda)=\\lambda^2+\\lambda\\bm{B}+\\bm{L}$. It constructs admissible stationary backgrounds via a distorted Lane-Emden function and proves a basic well-posedness result for ELASO using Friedrichs extensions and Hille–Yosida theory, with separate treatments for barotropic and baloclinic cases. The paper also analyzes the eigenvalue problem in the quadratic pencil framework, discusses possible eigenmode expansions, and introduces a general seminorm-based notion of stability, including special cases such as Cowling's approximation. Overall, it provides a mathematically rigorous foundation for the linear dynamics of rotating stellar oscillations and clarifies spectral structure and stability criteria for both barotropic and non-barotropic backgrounds.

Abstract

We study adiabatic oscillations of rotating self-gravitating gaseous stars in mathematically rigorous manner. The internal motion of the star is supposed to be governed by the Euler-Poisson equations with rotation of constant angular velocity under the equation of state of the ideal gas. The motion is supposed to be adiabatic, but not to be barotropic in general. This causes a free boundary problem to gas-vacuum interface. Existence of solutions to the linearized equation in the Lagrange coordinates of the perturbations around a fixed stationary solution, the eigenvalue problem with concept of quadratic pencil of operators, and the stability problem with a new concept of stability introduced in this article are discussed.

Linearized Analysis of Adiabatic Oscillations of Rotating Gaseous Stars

TL;DR

This paper tackles the linear adiabatic oscillations of rotating self-gravitating gaseous stars by formulating the perturbation problem in Lagrangian coordinates and deriving the ELASO system together with the quadratic pencil . It constructs admissible stationary backgrounds via a distorted Lane-Emden function and proves a basic well-posedness result for ELASO using Friedrichs extensions and Hille–Yosida theory, with separate treatments for barotropic and baloclinic cases. The paper also analyzes the eigenvalue problem in the quadratic pencil framework, discusses possible eigenmode expansions, and introduces a general seminorm-based notion of stability, including special cases such as Cowling's approximation. Overall, it provides a mathematically rigorous foundation for the linear dynamics of rotating stellar oscillations and clarifies spectral structure and stability criteria for both barotropic and non-barotropic backgrounds.

Abstract

We study adiabatic oscillations of rotating self-gravitating gaseous stars in mathematically rigorous manner. The internal motion of the star is supposed to be governed by the Euler-Poisson equations with rotation of constant angular velocity under the equation of state of the ideal gas. The motion is supposed to be adiabatic, but not to be barotropic in general. This causes a free boundary problem to gas-vacuum interface. Existence of solutions to the linearized equation in the Lagrange coordinates of the perturbations around a fixed stationary solution, the eigenvalue problem with concept of quadratic pencil of operators, and the stability problem with a new concept of stability introduced in this article are discussed.
Paper Structure (14 sections, 28 theorems, 772 equations)

This paper contains 14 sections, 28 theorems, 772 equations.

Table of Contents

  1. Introduction
  2. Existence of stationary solutions --Distorted Lane-Emden functions
  3. Derivatin of ELASO
  4. Basic existence theorem for ELASO --Application of the Hille-Yosida theory
  5. Case of barotropic backgrounds
  6. Case of baloclinic backgrounds
  7. Eigenvalue problem --Possibility of eigenmode expansions
  8. Stability problem --New concepts of stability
  9. ELASO($\blacklozenge$) of astrophysicists
  10. Analysis of ELASO($\blacklozenge$)
  11. Trivial displacements
  12. Functional analysis of ELASO($\blacklozenge$)
  13. Splitting ELASO($\blacklozenge$) with respect to tha azimuthal numbers
  14. Axially symmetric perturbations and Rayleigh's instability

Key Result

Theorem 1

Let a smooth function $\omega_b$ on $[0,+\infty[$, a smooth function $\Sigma$ on $\mathbb{R}$ and a positive number $\rho_O$ be given. Supoose that If $\frac{6}{5}<\gamma <2$, and $\frac{1}{\rho_O} \|\Omega +\omega_b\|_{L^{\infty}}^2$ is sufficiently small, then there uniquely exists a barotropic admissible stationary solution $(\rho_b, S_b, \bm{v}_b))$ such that $S_b=\Sigma(\rho_b^{\gamma-1})$$\

Theorems & Definitions (48)

  • Definition 1
  • Definition 2
  • Remark 1
  • Definition 3
  • Remark 2
  • Remark 3
  • Theorem 1
  • Remark 4
  • Remark 5
  • Remark 6
  • ...and 38 more