2: Basic Assumptions, Theorems and Polytropes
- Page ID
- 141595
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 2.1: Basic Assumptions
- This page outlines the fundamental assumptions and axioms for understanding stellar structure, including the spherical shape of self-gravitating plasma and the decreasing density with radius. It uses the ideal-gas law for the equation of state and introduces hydrostatic equilibrium, relating pressure gradients to gravitational forces. Key concepts include the mass function M(r) and conservation laws, essential for analyzing stellar interiors.
- 2.2: Integral Theorems from Hydrostatic Equilibrium
- This page explores the moment of mass distribution \(I_{\sigma, \nu}(r)\) and establishes limits for physical quantities like pressure and temperature in stars through integral inequalities. It introduces the β* theorem, analyzing the significance of radiation pressure versus total pressure, especially in massive stars.
- 2.3: Homology Transformations
- This page explains the concept of homology in stellar structures, indicating a one-to-one correspondence between sets. It relates stellar parameters via five variables dependent on radius and utilizes constraints like the ideal-gas law and hydrostatic equilibrium. Homology transformations reveal proportional relationships among variables such as pressure, temperature, and density, essential for understanding stellar evolution during processes like star formation and collapse.
- 2.4: Polytropes
- This page covers key aspects of polytropic equations of state in stellar structure, highlighting the relationship between pressure and density, and introducing the Lane-Emden equation for density distribution. It explains polytropic solutions related to hydrostatic equilibrium, isothermal spheres, and star modeling, emphasizing continuity across boundaries and the impact of polytropic indices on stellar configurations.
- 2.5: Problems
- This page covers essential concepts in stellar structure and physics, emphasizing Chandrasekhar's integral theorems. It explores estimating central temperature limits of stars based on their mass and identifies the critical mass of white dwarfs impacted by relativistic effects. Additionally, it discusses the Lane-Emden equation, solution conditions, and derives mass equations for isothermal spheres.
- 2.6: References and Supplemental Reading
- This page discusses key works on stellar structure, emphasizing integral theorems, internal pressure, polytropes, and isothermal spheres. Significant contributions from authors like S. Chandrasekhar and E.A. Milne are highlighted, focusing on stellar equilibrium and opacity. It also offers resources for deeper insights into polytropes and research on rotating polytropes, aiming to enhance the understanding of stellar mechanics and evolution.