1: Development of the Virial Theorem
- Page ID
- 141428
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 1.1: The Basic Equations of Structure
- This page covers foundational equations in stellar astrophysics, emphasizing conservation principles and the significance of phase-space and the Boltzmann transport equation. It presents simplifications leading to key equations that represent conservation of mass, momentum, and energy, and clarifies relationships among physical quantities like density and pressure.
- 1.2: The Classical Derivation of the Virial Theorem
- This page covers the virial theorem, detailing its derivation from first principles and its application in steady or quasi-steady state systems. It explains the relationship between kinetic and potential energies, emphasizing statistical properties and the significance of averaging over longer time periods to minimize discrepancies.
- 1.3: Velocity Dependent Forces and the Virial Theorem
- This page explains the virial theorem, highlighting that its results are unaffected by velocity-dependent forces. It derives equations of motion for mass points under both velocity-dependent and independent forces. The integration of these equations over time shows that periodic motion leads to the cancellation of certain terms, allowing for the omission of Lorentz or viscosity forces in further discussions on the virial theorem.
- 1.4: Continuum-Field Representation of the Virial Theorem
- This page explores the virial theorem's application in stellar structure, focusing on continuum density fields. It transitions from discrete mass points to volume-integral forms, emphasizing mass conservation and defining total kinetic energy. The relationship between virial and potential energy is established, paralleling previous equations.
- 1.5: The Ergodic Theorem and the Virial Theorem
- This page explores the link between Lagrange's identity and the virial theorem, emphasizing the need for time-averaging in astrophysics and introducing the Ergodic Theorem, which relates ensemble averages to time averages. It discusses the concept of ergodicity, including the quasi-ergodic hypothesis, and the challenges of applying the virial theorem to celestial systems.
- 1.6: Summary
- This page delves into the classical virial theorem, outlining its foundation through the Boltzmann transport equation and its implications for stellar structure. It covers both classical and modern derivations of the theorem, emphasizing the importance of force pairings and potential energy assumptions. Additionally, the ergodic theorem's relevance to the virial theorem in stellar systems is discussed, alongside warnings about its limitations in specific dynamic scenarios.
- 1.7: Notes to Chapter 1
- This page explores mathematical representations of forces and potential energy in physical systems, particularly symmetric interactions among particles. It details the formulation of total force and potential energy through pairwise interactions while avoiding double-counting. Additionally, it introduces force density connected to potential and its distance dependencies.
- 1.8: References
- This page discusses various foundational references in stellar systems and classical mechanics, featuring works by notable figures such as Kurth, Landau, Lifshitz, and Chandrasekhar. It highlights significant contributions from Maxwell and Boltzmann to statistical mechanics and ergodic theory, covering the evolution of ideas from the early 19th to the mid-20th century in this scientific field.