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===Equilibrium Structures===
===Vertically Thick, Axisymmetric Equilibrium Structures===
 
====Spheroidal & Spheroidal-Like====
{| class="wikitable" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black;"
{| class="wikitable" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black;"
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! style="height: 150px; width: 150px; background-color:lightgreen; border-right:2px solid black;" |[[User:Tohline/AxisymmetricConfigurations/SolutionStrategies#Axisymmetric_Configurations_.28Structure_.E2.80.94_Part_II.29|<b>Constructing<br />Axisymmetric<br />Equilibrium<br />Configurations</b>]]
! style="height: 150px; width: 150px; background-color:lightgreen; border-right:2px solid black;" |[[User:Tohline/AxisymmetricConfigurations/SolutionStrategies#Axisymmetric_Configurations_.28Structure_.E2.80.94_Part_II.29|<b>Constructing<br />Axisymmetric<br />Equilibrium<br />Configurations</b>]]
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! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[User:Tohline/AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential|''Simple''<br />Rotation<br />Profiles]]
! style="height: 150px; width: 150px; border-right:2px dotted black;" |[[User:Tohline/2DStructure/AxisymmetricInstabilities|Axisymmetric Instabilities<br />to Avoid]]
|
! style="height: 150px; width: 150px; " |[[User:Tohline/AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential|''Simple''<br />Rotation<br />Profiles]]
|}
<p>&nbsp;</p>
{| class="wikitable" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black;"
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! style="height: 150px; width: 150px; background-color:lightgreen; border-right:2px solid black;" |<b>Solving the<br />Poisson Equation</b>
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! style="height: 150px; width: 150px; border-right:2px dotted black;" |[[User:Tohline/2DStructure/ToroidalCoordinates#Using_Toroidal_Coordinates_to_Determine_the_Gravitational_Potential|Exploring the Use of<br />Toroidal Coordinates<br />to Solve the<br />Poisson Equation]]
! style="height: 150px; width: 150px; border-right:2px dotted black;" |[[User:Tohline/2DStructure/ToroidalCoordinates#Using_Toroidal_Coordinates_to_Determine_the_Gravitational_Potential|Exploring the Use of<br />Toroidal Coordinates<br />to Solve the<br />Poisson Equation]]
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! style="height: 150px; width: 150px;" |[[File:Apollonian_myway4.png|150px|link=User:Tohline/2DStructure/ToroidalCoordinateIntegrationLimits#Mapping_from_Cylindrical_to_Toroidal_Coordinates|Apollonian Circles]]
! style="height: 150px; width: 150px;" |[[File:Apollonian_myway4.png|150px|link=User:Tohline/2DStructure/ToroidalCoordinateIntegrationLimits#Mapping_from_Cylindrical_to_Toroidal_Coordinates|Apollonian Circles]]
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<p>&nbsp;</p>
 
====Spheroidal &amp; Spheroidal-Like====
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====Thin, Nonaxisymmetric Disks====
===Nonaxisymmetric Equilibrium Disk Structures===
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{| class="wikitable" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black;"
<!-- |+ style="text-align:left; height:40px;" | <font size="+2">'''CONTEXT'''</font> -->
<!-- |+ style="text-align:left; height:40px;" | <font size="+2">'''CONTEXT'''</font> -->

Revision as of 15:33, 11 August 2017


Tiled Menu

Whitworth's (1981) Isothermal Free-Energy Surface
|   Tiled Menu   |   Tables of Content   |  Banner Video   |  Tohline Home Page   |

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