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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 (Axisymmetric)

PGEs
for
Axisymmetric
Systems


Axisymmetric Equilibrium Structures

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

Toroidal & Toroidal-Like

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

 

Self-Gravitating
Incompressible
Configurations 		Dyson-Wong
Tori

 

Stability Analysis

Toroidal & Toroidal-Like

Defining the
Eigenvalue Problem

 

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

 

Two-Dimensional Configurations (Nonaxisymmetric Disks)

Constructing
Infinitesimally Thin
Nonaxisymmetric
Disks

 

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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