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Global Energy Considerations

Principal Governing Equations (PGEs)		Continuity		Euler		1st Law of Thermodynamics		Poisson

 

Equation of State (EOS)		Total Pressure

Spherically Symmetric Configurations

Structural Form Factors		Free Energy of Spherical Systems

One-Dimensional PGEs

Equilibrium Structures

Hydrostatic Balance Equation		<math>~\frac{dP}{dr} = - \frac{GM_r \rho}{r^2}</math>		Solution Strategies

 

Isothermal Sphere		<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		<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

 

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

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)

 

Pressure-Truncated Isothermal Spheres		<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

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		<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				Pressure-Truncated Configurations

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

 

See Also

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | 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