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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

Text written in green has been taken directly from the introductory paragraphs of G. S. Bisnovatyi-Kogan & S. I. Blinnikov (1974); referred to, below, as B-KB74. Three different approaches are used in the study of hydrodynamical stability of stars and other gravitating objects …   The first approach is based on the use of the equations of small oscillations. In that case the problem is reduced to a search for the solution of the boundary-value problem of the Stourme-Liuville type for the linearised system of equations of small oscillations. The solutions consist of a set of eigenfrequencies and eigenfunctions. This approach is subject to great numerical difficulties and has been carried through successfully only for the simplest case[s]. For spherically symmetric, Newtonian systems, see, for example, Schwarzschild (1941) or Ledoux & Walraven (1958). Second, one can derive a variational principle from the equations of small oscillations. This principle replaces the straightforward solution of these equations: In the context of rotating Newtonian systems, see, for example, Clement (1964), Chandrasekhar & Lebovitz (1968), Lynden-Bell and Ostriker (1967), or Schutz (1972). With the aid of the variational principle, the problem is reduced to the search of the best trial functions; this leads to approximate eigenvalues of oscillations. In spite of the simplifications introduced by the use of the variational principle and by not solving the equations of motion exactly, the problem still remains complicated … The third approach is what we have referred to as a free-energy analysis. When this method is used, it is not necessary to use the equations of small oscillations but, instead, the functional expression for the total energy of the momentarily stationary (but not necessarily in equilibrium) star is sufficient. The condition that the first variation of the energy vanishes, determines the state of equilibrium of the star and the positiveness of a second variation indicates stability. For non rotating systems having ${\gamma }_g$ near 4/3, see, for example, Fowler (1964) or Zeldovich & Novikov (Soviet journal, 1965).

If one wants to know from a stability analysis the answer to only one question — whether the model is stable or not — then the most straightforward procedure is to use the third, static method (Zeldovich 1963; Dmitrie & Kholin 1963). For the application of this method, one needs to construct only equilibrium, stationary models, with no further calculation. Generally the static analysis gives no information about the shape of the modes of oscillation, but, in the vicinity of critical points, where instability sets in, this method makes it possible to find the eigenfunction of the mode which becomes unstable at the critical point. This method is referred to, below, as the "B-KB74 conjecture."  
 

		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 (2017) Analytic Sol'n for Marginally Unstable Configurations ♥		${\sigma }_c^2=0,{\gamma }_{\mathrm{g}}=(n+1)/n$  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