7: Structure of Distorted Stars
- Page ID
- 141639
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 7.1: Introduction
- This page examines the challenges of modeling stellar structures when stars deviate from spherical shapes due to rotation or companion stars. It outlines the associated physical phenomena and observational difficulties, such as orientation-dependent appearances affecting measurements of luminosity and temperature. The text proposes a theoretical framework emphasizing cases of axial symmetry to simplify complexities while addressing critical issues in understanding distorted stars.
- 7.2: Classical Distortion- The Structure Equations
- This page details the transformation of classical equations in stellar structures due to the loss of spherical symmetry, introducing vector forms and axial symmetry concepts. It covers the scalar potential in fluid dynamics, angular momentum equilibrium, and effects of gravitational and magnetic distortions. The discussion extends to interpreting axis-symmetric perturbations in distortion problems, indicating a systematic approach to solve complex distortions.
- 7.3: Solution of Structure Equations for a Perturbing Force
- This page discusses the need for incorporating essential physical processes in stellar structures, focusing on perturbative analysis and the balance of state variables in hydrostatic equilibrium. It highlights the complexity added by velocity fields and the discrepancies in available equations versus unknowns.
- 7.4: Von Zeipel's Theorem and Eddington-Sweet Circulation Currents
- This page covers three interrelated topics: von Zeipel's theorem, which indicates limitations on energy production in non-spherical stars due to hydrostatic and radiative principles; the relationship between entropy, temperature, and pressure gradients in thermodynamic systems, including energy transport dynamics; and the dynamics of Eddington-Sweet circulation currents, highlighting how these currents affect stellar mixing and evolution depending on rotation speed.
- 7.5: Rotational Stability and Mixing
- This page explores the intricacies of stability in rotating stars, emphasizing shear instabilities caused by velocity gradients and their role in angular momentum redistribution. It examines forces like thermal stratification and chemical composition gradients that influence these instabilities. Additionally, the Goldreich-Schubert-Fricke instability is discussed, noting its minimal impact in most stars but potential significance in slow rotation scenarios.
- 7.6: Problems
- This page examines the intricacies of modeling rapidly rotating magnetic neutron stars, highlighting the importance of a systematic approach and the assumptions required for effective calculations. It presents a specific relationship among key variables and discusses necessary adjustments to the conservation of mass equation to incorporate mechanistic flows from Eddington-Sweet currents, emphasizing the relationship between rotation and magnetic fields in these celestial objects.
- 7.7: References and Supplemental Reading
- This page reviews the advancements in understanding distorted stars, especially rotating ones, over the last century. It emphasizes key literature, including Chandrasekhar's work on distorted polytropes and contemporary reviews by Toussel and Roxburgh. Major themes include the historical evolution of theories and the role of rotation in stellar development. The page also serves as a resource for further reading, featuring essential papers and reviews in stellar astrophysics.