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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)
Ideal Gas Total
Pressure

Spherically Symmetric Configurations

(Initially) Spherically Symmetric Configurations

 

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


Equilibrium Structures

1D STRUCTURE

 

Spherical Structures Synopsis 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
Lane
(1870)

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)

 

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

Equilibrium sequences of Pressure-Truncated Polytropes Turning-Points
(Broader Context)

 

Free Energy
of
Bipolytropes


(nc, ne) = (5, 1)
Composite
Polytropes

(Bipolytropes)
Schönberg-
Chandrasekhar
Mass
(1942)
Analytic

(nc, ne) = (5, 1)
Analytic

(nc, ne) = (1, 5)

 

Stability Analysis

1D STABILITY

 

Synopsis: Stability of Spherical Structures Variational
Principle
Radial
Pulsation
Equation
Example
Derivations
&
Statement of
Eigenvalue
Problem
(poor attempt at)
Reconciliation
Relationship
to
Sound Waves

 

Jeans (1928) or Bonnor (1957)
Ledoux & Walraven (1958)
Rosseland (1969)

 

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
n = 5
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>

 

BiPolytropes Murphy & Fiedler
(1985b)


(nc, ne) = (1,5)
Our
Broader
Analysis

 

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)

(Initially) Axisymmetric Configurations

 

PGEs
for
Axisymmetric
Systems


Axisymmetric Equilibrium Structures

2D STRUCTURE

 

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
Apollonian Circles Attempt at
Simplification

Wong's
Analytic Potential
(1973)
n = 3 contribution to potential

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)
Review of
Stahler's (1983)
Technique

 

Rotationally
Flattened
Polytropes
Example
Equilibria

 

Rotationally
Flattened
White Dwarfs
Ostriker
Bodenheimer
& Lynden-Bell
(1966)
Example
Equilibria

Toroidal & Toroidal-Like

Definition: anchor ring
Massless
Polytropic
Configurations
Papaloizou-Pringle
Tori
(1984)
Pivoting PP Torus

 

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

 

Self-Gravitating
Compressible
Configurations
Ostriker
(1964)

 

Stability Analysis

2D STABILITY

 

Toroidal & Toroidal-Like

Defining the
Eigenvalue Problem

 

(Massless)
Papaloizou-Pringle
Tori
Analytic Analysis
by
Blaes
(1985)
N1.5j2 Combinedsmall.png

 

Nonlinear Dynamical Evolution

2D DYNAMICS

 

Two-Dimensional Configurations (Nonaxisymmetric Disks)

Infinitesimally Thin, Nonaxisymmetric Disks

 

2D STRUCTURE

 

Constructing
Infinitesimally Thin
Nonaxisymmetric
Disks

 

Three-Dimensional Configurations

(Initially) Three-Dimensional Configurations

 

Equilibrium Structures

3D STRUCTURE

 

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

 

Fission
Hypothesis

 

Binary Systems

 

Stability Analysis

3D STABILITY

 

Nonlinear Dynamical Evolution

3D DYNAMICS

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