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Whitworth's (1981) Isothermal Free-Energy Surface
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Context

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

 

Equation
of State
(EOS) 		Total
Pressure

Spherically Symmetric Configurations

Whitworth's (1981) Isothermal Free-Energy Surface Structural
Form
Factors 		Free-Energy
of
Spherical
Systems
One-Dimensional
PGEs


Equilibrium Structures

Scalar
Virial
Theorem
Hydrostatic
Balance
Equation

LSU Key.png

<math>~\frac{dP}{dr} = - \frac{GM_r \rho}{r^2}</math>

Solution
Strategies

 

Isothermal
Sphere

LSU Key.png

<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

LSU Key.png

<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 		Equilibrium sequences of Pressure-Truncated Polytropes

 

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) 		Sterne's (1937) Solution to the Eigenvalue Problem for Uniform-Density Spheres

 

Pressure-Truncated
Isothermal
Spheres

LSU Key.png

<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 		Fundamental-Mode Eigenvectors


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

LSU Key.png

<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 		Schwarzschild's Modal Analysis 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 		Apollonian Circles

Spheroidal & Spheroidal-Like

Uniform-Density
(Maclaurin)
Spheroids 		Maclaurin's
Original Text
&
Analysis
(1742)
Our Construction of Maclaurin's Figure 291Pt2

 

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

Toroidal & Toroidal-Like

Massless
Polytropic
Configurations 		Papaloizou-Pringle
Tori
(1984) 		Pivoting PP Torus

 

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)
N1.5j2 Combinedsmall.png

 

Three-Dimensional Configurations

Equilibrium Structures

Ellipsoidal & Ellipsoidal-Like

Constructing
Ellipsoidal
& Ellipsoidal-Like
Configurations

 

Animation related to Fig. 3 from Christodoulou1995 Free-Energy
Evolution
from the Maclaurin
to the Jacobi
Sequence
Jacobi
Ellipsoids

 

See Also

Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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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

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