6: Relativistic Stellar Structure
- Page ID
- 141630
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 6.1: Introduction
- This page discusses the limitations of traditional stellar structure models that assume spherical symmetry and Newtonian gravity, emphasizing the necessity of general relativity for understanding certain stellar phenomena. It reviews the historical context of applying general relativity to stars, especially in relation to pulsars and neutron stars.
- 6.2: Field Equations of the General Theory of Relativity
- This page discusses the general theory of relativity, which connects space-time geometry to matter-energy through Einstein's field equations. It highlights key concepts such as the metric tensor, Einstein tensor, stress-energy tensor, and curvature tensors. The equations reflect the relationship between geometric properties and physical aspects, demonstrating the interplay between geometry and physics in gravitational phenomena.
- 6.3: Oppenheimer-Volkoff Equation of Hydrostatic Equilibrium
- This page covers the Schwarzschild metric in general relativity, emphasizing spherical symmetry. It presents the metric in spherical coordinates while deriving the Einstein field equations to show solutions yielding the Schwarzschild metric for regions outside stars. Additionally, it discusses gravitational potential, hydrostatic equilibrium, and the Oppenheimer-Volkoff equation, which defines the structure of relativistic stars by accounting for pressure and energy density.
- 6.4: Equations of Relativistic Stellar Structure and Their Solutions
- This page explores the construction of stellar models for relativistic stars, demonstrating that extreme gravity simplifies modeling compared to Newtonian frameworks. It highlights the impact of nuclear forces and outlines a basic constant-density model for mass and pressure calculations. Additionally, it discusses the mass limits of neutron stars, noting a maximum mass near 1 solar mass due to neutrino losses and a secondary limit over 2 solar masses.
- 6.5: Relativistic Polytrope of Index 3
- This page covers the equation of state for relativistic degenerate gases, emphasizing polytropes of index n = 3 relevant to white dwarfs and massive stars. It introduces the Virial theorem and discusses general relativity's impact on gravitational stability, particularly for objects nearing limiting configurations like white dwarfs and super-massive stars.
- 6.6: Fate of Super-massive Stars
- This page examines the characteristics and evolution of super-massive stars and their relationship with white dwarfs, focusing on Eddington luminosity, energy balance, and mass-radius relationships. It discusses how radiation pressure stabilizes stars up to a point, after which massive stars face collapse, potentially forming black holes if energy production fails.
- 6.7: Problems
- This page covers stellar physics, emphasizing polytropes and their indices. It modifies the hydrostatic equilibrium equation to derive the Oppenheimer-Volkoff equation and explores the mass-radius law for super-massive stars linked to the proton-proton cycle. The stability mass limit for white dwarfs under general relativity is assessed, alongside relativistic integrals for polytropes and minimum radius calculations.
- 6.8: References and Supplemental Reading
- This page provides an overview of significant references on stellar structure, emphasizing neutron stars, supermassive stars, and high-density matter dynamics. It highlights contributions from key physicists like Landau, Oppenheimer, Salpeter, and Chandrasekhar, covering crucial topics such as gravitational contraction and massive neutron cores. The author acknowledges E. R. Capriotti's insights and underscores the importance of historical context and recent advancements in the field.