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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

 

Sheroidal & Spheroidal-Like

 

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

 

Jacobi
Ellipsoids

 

Binary Systems

 

Roche's problem is concerned with the equilibrium and the stability of rotating homogeneous masses which are, further, distorted by the constant tidal action of an attendant rigid spherical mass.

 

Stability Analysis

3D STABILITY

Ellipsoidal & Ellipsoidal-Like

 

Binary Systems

Nonlinear Evolution

3D DYNAMICS

 

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

 

Secular

Dynamical

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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