3: The Variational Form of the Virial Theorem
- Page ID
- 141454
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 3.1: Variations and Perturbations and their Implications for the Virial Theorem
- This page discusses perturbation analysis, which explores how changes in a system's independent variables affect its behavior. It utilizes governing equations to derive perturbed equations of motion, revealing insights into dynamics near equilibrium states. The analysis results in differential equations that can uncover key macroscopic properties, such as the pulsational period of stars, as shown by researchers like Ledoux, Chandrasekhar, and Fermi.
- 3.2: Radial Pulsations for Self-Gravitating Systems- Stars
- This page applies the virial theorem to derive an expression for radial pulsation frequency in gas spheres, considering energy changes and making assumptions of uniformity. It focuses on Classical Cepheids, establishing a relationship between pulsation periods and stellar characteristics like mass and density. A period formula is introduced that is inversely proportional to the square root of mean density, with estimated Cepheid variable periods ranging from 0.
- 3.3: The Influence of Magnetic and Rotational Energy upon a Pulsating System
- This page examines the influence of magnetic and rotational energies on the pulsation frequency of stellar systems. It covers the approximation of parameters, the derivation of Lagrange's identity, and the virial theorem reflecting energy contributions. The discussion includes the derivation of rotational kinetic energy and variations in magnetic energy, connecting them to pulsational frequency (\(\sigma^2\)).
- 3.4: Variational Form of the Surface Terms
- This page covers the virial theorem and its relevance to stars, focusing on surface integrals and contributions from magnetic fields. It explains gas motion without resistivity and the impact of pressure and magnetic effects on surface terms during pulsations. The page also addresses magnetic stability, showing how surface pressure can enhance stability and the dependence of magnetic terms on geometry, particularly in spherical stars.
- 3.5: The Virial Theorem and Stability
- This page covers the stability of dynamical systems, emphasizing stable equilibria in non-equilibrium contexts like stars. It reviews the virial theorem, Jacobi's stability criterion, and conditions for instability, particularly in relation to total energy and magnetic fields. The impact of pulsation frequencies and the effects of rotation on stellar configurations are explored, noting that while rotation can enhance stability, its effect is limited.
- 3.6: Summary
- This page explores the variational approach to the virial theorem, particularly related to self-gravitating systems and their pulsational stability. It links oscillation modes to density and outlines Jean's stability criterion. Additionally, it examines how magnetic fields and rotation impact stability and pulsation frequencies.
- 3N: Notes to Chapter 3
- This page covers key concepts of energy dynamics in gaseous systems, focusing on gravitational potential and kinetic energy linked to pulsations, density, and temperature. It discusses fluid dynamics equations detailing density variations and boundary conditions, leading to conservation laws. Further, it explains mathematical relationships between density variations and magnetic fields, culminating in simplified expressions for magnetic field variation in relation to perturbations.
- 3.8: References
- This page provides a curated list of references and publications in astrophysics, highlighting contributions from key scientists like Chandrasekhar, Ledoux, and Ostriker. It covers a variety of topics such as stellar structure, hydrodynamic stability, and gravitational interactions, showcasing significant research in the field. Notable journals referenced include the Astrophysical Journal (Ap. J.) and Monthly Notices of the Royal Astronomical Society (M.N.R.A.S.).