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Global Energy Considerations

Principal Governing Equations (PGEs)		Continuity		Euler		1st Law of Thermodynamics		Poisson

 

Equation of State (EOS)		Total Pressure

Spherically Symmetric Configurations

		Structural Form Factors		Free-Energy of Spherical Systems

One-Dimensional PGEs

Equilibrium Structures

Scalar Virial Theorem

Hydrostatic Balance Equation		<math>~\frac{dP}{dr} = - \frac{GM_r \rho}{r^2}</math>		Solution Strategies

 

Isothermal Sphere		<math>~\frac{1}{\xi^2} \frac{d}{d\xi}\biggl( \xi^2 \frac{d\psi}{d\xi} \biggr) = e^{-\psi}</math>		via Direct Numerical Integration

 

Isolated Polytropes		<math>~\frac{1}{\xi^2} \frac{d}{d\xi}\biggl( \xi^2 \frac{d\Theta_H}{d\xi} \biggr) = - \Theta_H^n</math>		Known Analytic Solutions		via Direct Numerical Integration		via Self-Consistent Field (SCF) Technique

 

Zero-Temperature White Dwarf		Chandrasekhar Limiting Mass (1935)

 

Pressure-Truncated Configurations		Bonnor-Ebert (Isothermal) Spheres (1955 - 56)		Polytropes		Equilibrium Sequence Turning-Points

 

Composite Polytropes (Bipolytropes)		Schönberg- Chandrasekhar Mass (1942)		Analytic <math>~(n_c, n_e)</math> = <math>~(5,1)</math>		Analytic <math>~(n_c, n_e)</math> = <math>~(1,5)</math>		Equilibrium Sequence Turning-Points

 

Stability Analysis

Variational Principle

Radial Pulsation Equation		Example Derivations & Statement of Eigenvalue Problem		Jeans (1928) or Bonnor (1957)		Ledoux & Walraven (1958)		Rosseland (1969)		Relationship to Sound Waves

 

Uniform-Density Configurations		Sterne's Analytic Sol'n of Eigenvalue Problem (1937)

 

Pressure-Truncated Isothermal Spheres		<math>~0 = \frac{d^2x}{d\xi^2} + \biggl[4 - \xi \biggl( \frac{d\psi}{d\xi} \biggr) \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi} + \biggl[ \biggl( \frac{\sigma_c^2}{6\gamma_\mathrm{g}}\biggr)\xi^2 - \alpha \xi \biggl( \frac{d\psi}{d\xi} \biggr) \biggr] \frac{x}{\xi^2} </math> where:    <math>~\sigma_c^2 \equiv \frac{3\omega^2}{2\pi G\rho_c}</math>     and,     <math>~\alpha \equiv \biggl(3 - \frac{4}{\gamma_\mathrm{g}}\biggr)</math>		via Direct Numerical Integration

Yabushita's Analytic Sol'n for Marginally Unstable Configurations (1974)		<math>~\sigma_c^2 = 0 \, , ~~~~\gamma_\mathrm{g} = 1</math>  and   <math>~x = 1 - \biggl( \frac{1}{\xi e^{-\psi}}\biggr) \frac{d\psi}{d\xi} </math>

 

Polytropes		<math>~0 = \frac{d^2x}{d\xi^2} + \biggl[ 4 - (n+1) Q \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi} + (n+1) \biggl[ \biggl( \frac{\sigma_c^2}{6\gamma_g } \biggr) \frac{\xi^2}{\theta} - \alpha Q\biggr] \frac{x}{\xi^2} </math> where:    <math>~Q(\xi) \equiv - \frac{d\ln\theta}{d\ln\xi} \, ,</math>    <math>~\sigma_c^2 \equiv \frac{3\omega^2}{2\pi G\rho_c} \, ,</math>     and,     <math>~\alpha \equiv \biggl(3 - \frac{4}{\gamma_\mathrm{g}}\biggr)</math>		Isolated n = 3 Polytrope				Pressure-Truncated Configurations

Exact Demonstration of B-KB74 Conjecture		Exact Demonstration of Variational Principle		Pressure-Truncated n = 5 Polytropes

Our Analytic Sol'n for Marginally Unstable Configurations (2017)		<math>~\sigma_c^2 = 0 \, , ~~~~\gamma_\mathrm{g} = (n+1)/n</math>  and   <math>~x = \frac{3(n-1)}{2n}\biggl[1 + \biggl(\frac{n-3}{n-1}\biggr) \biggl( \frac{1}{\xi \theta^{n}}\biggr) \frac{d\theta}{d\xi}\biggr] </math>

 

Nonlinear Dynamical Evolution

Free-Fall Collapse

 

Collapse of Isothermal Spheres		via Direct Numerical Integration		Similarity Solution

 

Collapse of an Isolated n = 3 Polytrope

 

Two-Dimensional Configurations

PGEs for Axisymmetric Systems

Vertically Thick, Axisymmetric Equilibrium Structures

Constructing Axisymmetric Equilibrium Configurations		Axisymmetric Instabilities to Avoid		Simple Rotation Profiles

 

Solving the Poisson Equation		Exploring the Use of Toroidal Coordinates to Solve the Poisson Equation

Spheroidal & Spheroidal-Like

Uniform-Density (Maclaurin) Spheroids		Maclaurin's Original Text & Analysis (1742)

 

Rotationally Flattened Isothermal Configurations		Hayashi, Narita & Miyama's Analytic Sol'n (1982)

Toroidal & Toroidal-Like

Massless Polytropic Configurations		Papaloizou-Pringle Tori (1984)

 

Self-Gravitating Incompressible Configurations		Dyson-Wong Tori

Nonaxisymmetric Equilibrium Disk Structures

Constructing Infinitesimally Thin Nonaxisymmetric Disks

 

Stability Analysis

Toroidal & Toroidal-Like

Defining the Eigenvalue Problem

 

(Massless) Papaloizou-Pringle Tori		Analytic Analysis by Blaes (1985)

 

Three-Dimensional Configurations

Equilibrium Structures

Ellipsoidal & Ellipsoidal-Like

Constructing Ellipsoidal & Ellipsoidal-Like Configurations

 

		Free-Energy Evolution from the Maclaurin to the Jacobi Sequence

Jacobi Ellipsoids

 

See Also

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation