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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   Bond, Arnett, & Carr (1984)

Spherically Symmetric Configurations

(Initially) Spherically Symmetric Configurations

 

		Structural Form Factors		Free-Energy of Spherical Systems

One-Dimensional PGEs

Equilibrium Structures

1D STRUCTURE

 

		Scalar Virial Theorem

Hydrostatic Balance Equation		$\frac{dP}{dr}=-\frac{G{M}_r\rho }{r^2}$		Solution Strategies

Uniform-Density Sphere

 

Isothermal Sphere		$\frac{1}{{\xi }^2}\frac{d}{d\xi }\left({\xi }^2\frac{d\psi }{d\xi }\right)=e^{-\psi }$		via Direct Numerical Integration

Isolated Polytropes		Lane (1870)		$\frac{1}{{\xi }^2}\frac{d}{d\xi }\left({\xi }^2\frac{d{\mathrm{\Theta }}_H}{d\xi }\right)=-{\mathrm{\Theta }}_H^n$		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)		Embedded Polytropes		Equilibrium Sequence Turning-Points ♥				Turning-Points (Broader Context)

Free Energy of Bipolytropes (nc, ne) = (5, 1)

Composite Polytropes (Bipolytropes)		Milne (1930)		Schönberg- Chandrasekhar Mass (1942)		Murphy (1983) Analytic (nc, ne) = (1, 5)		Eggleton, Faulkner & Cannon (1998) Analytic (nc, ne) = (5, 1)

Stability Analysis

1D STABILITY

 

		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)

 

Pressure-Truncated Isothermal Spheres		$0=\frac{d^{2}x}{d{\xi }^2}+[4-\xi (\frac{d\psi }{d\xi }\left)\right]\frac{1}{\xi }\cdot \frac{dx}{d\xi }+\left[\right(\frac{{\sigma }_c^2}{6{\gamma }_{\mathrm{g}}}){\xi }^2-\alpha \xi (\frac{d\psi }{d\xi }\left)\right]\frac{x}{{\xi }^2}$ where:    ${\sigma }_c^2\equiv \frac{3{\omega }^2}{2\pi G{\rho }_c}$     and,     $\alpha \equiv (3-\frac{4}{{\gamma }_{\mathrm{g}}})$		via Direct Numerical Integration

 

Yabushita's Analytic Sol'n for Marginally Unstable Configurations (1974)		${\sigma }_c^2=0,{\gamma }_{\mathrm{g}}=1$  and   $x=1-\left(\frac{1}{\xi e^{-\psi }}\right)\frac{d\psi }{d\xi }$

 

Polytropes		$0=\frac{d^{2}x}{d{\xi }^2}+[4-(n+1)Q]\frac{1}{\xi }\cdot \frac{dx}{d\xi }+(n+1)\left[\right(\frac{{\sigma }_c^2}{6{\gamma }_g})\frac{{\xi }^2}{\theta }-\alpha Q]\frac{x}{{\xi }^2}$ where:    $Q(\xi )\equiv -\frac{d\ln \theta }{d\ln \xi },$    ${\sigma }_c^2\equiv \frac{3{\omega }^2}{2\pi G{\rho }_c},$     and,     $\alpha \equiv (3-\frac{4}{{\gamma }_{\mathrm{g}}})$		Isolated n = 3 Polytrope				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) ♥		${\sigma }_c^2=0,{\gamma }_{\mathrm{g}}=(n+1)/n$  and   $x=\frac{3(n-1)}{2n}[1+(\frac{n-3}{n-1}\left)\right(\frac{1}{\xi {\theta }^n}\left)\frac{d\theta }{d\xi }\right]$

 

BiPolytropes		Murphy & Fiedler (1985b) (nc, ne) = (1,5)		Our Broader Analysis

See Also