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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

1D STRUCTURE

 

		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				Turning-Points (Broader Context)

 

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>

 

Stability Analysis

1D STABILITY

 

		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

1D DYNAMICS

 

Free-Fall Collapse

 

Collapse of Isothermal Spheres		via Direct Numerical Integration		Similarity Solution

 

Collapse of an Isolated n = 3 Polytrope

 

Two-Dimensional Configurations (Axisymmetric)

PGEs for Axisymmetric Systems

Axisymmetric Equilibrium Structures

2D STRUCTURE (axisymmetric)

 

Constructing Axisymmetric Equilibrium Configurations		Axisymmetric Instabilities to Avoid		Simple Rotation Profiles		Hachisu Self-Consistent-Field [HSCF] Technique		Solving the Poisson Equation

 

Using Toroidal Coordinates to Determine the Gravitational Potential				Attempt at Simplification		Wong's Analytic Potential (1973)

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)		Review of Stahler's (1983) Technique

Toroidal & Toroidal-Like

Definition: anchor ring

Massless Polytropic Configurations		Papaloizou-Pringle Tori (1984)

 

Self-Gravitating Incompressible Configurations		Dyson (1893)		Dyson-Wong Tori

 

Self-Gravitating Compressible Configurations		Ostriker (1964)

 

Stability Analysis

2D STABILITY (of axisymmetric configurations)

 

Toroidal & Toroidal-Like

Defining the Eigenvalue Problem

 

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

 

Two-Dimensional Configurations (Nonaxisymmetric Disks)

2D STRUCTURE (infinitesimally thin, nonaxisymmetric disks)

 

Constructing Infinitesimally Thin Nonaxisymmetric Disks

 

Three-Dimensional Configurations

Equilibrium Structures

3D STRUCTURE

 

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