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

Principal
Governing
Equations

(PGEs)
Continuity Euler 1st Law of
Thermodynamics
Poisson

 

Equation
of State

(EOS)
Total
Pressure

Spherically Symmetric Configurations

One-Dimensional
PGEs


Equilibrium Structures

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

 

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

 

Composite
Polytropes

(Bipolytropes)
Schönberg-
Chandrasekhar
Mass
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

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
Equilibrium sequences of Pressure-Truncated Polytropes

 

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

 

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
Our
Analytic Sol'n
for
Marginally Unstable
Configurations

 


Nonlinear Dynamical Evolution

Free-Fall
Collapse

 

Collapse of
Isothermal
Spheres
via
Direct
Numerical
Integration
Self-Similar
Solution

 

Collapse of
an Isolated
n = 3
Polytrope

 

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