Tiled Menu

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Context

Global Energy Considerations

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

Equation of State (EOS)		Ideal Gas		Total Pressure Bond, Arnett, & Carr (1984)

Spherically Symmetric Configurations

(Initially) Spherically Symmetric Configurations

Free-Energy Index				Structural Form Factors		Free-Energy of Spherical Systems

One-Dimensional PGEs		SSC Index

Equilibrium Structures

1D STRUCTURE

		Scalar Virial Theorem

Hydrostatic
Balance
Equation

$\frac{d P}{d r} = - \frac{G M_{r} ρ}{r^{2}}$

Solution
Strategies

Uniform-Density Sphere

Isothermal
Sphere

$\frac{1}{ξ^{2}} \frac{d}{d ξ} (ξ^{2} \frac{d ψ}{d ξ}) = e^{- ψ}$

via
Direct
Numerical
Integration

Isolated
Polytropes

Lane
(1870)

$\frac{1}{ξ^{2}} \frac{d}{d ξ} (ξ^{2} \frac{d Θ_{H}}{d ξ}) = - Θ_{H}^{n}$

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 (n_c, n_e) = (5, 1)

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

Stability Analysis

1D STABILITY

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. The following set of menu tiles include links to chapters where this approach has been applied to: (a) uniform-density configurations, (b) pressure-truncated isothermal spheres, (c) an isolated n = 3 polytrope, (d) pressure-truncated n = 5 configurations, and (e) bipolytropes having $(n_{c}, n_{e}) = (1, 5)$ .
Second, one can derive a variational principle from the equations of small oscillations. Below, an appropriately labeled (purple) menu tile links to a chapter in which the foundation for this approach is developed. 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 … One menu tile, below, links to a chapter in which an analytic (exact) demonstration of the variational principle's utility is provided in the context pressure-truncated n = 5 polytropes.
The third approach is what we have already referred to as a free-energy — and associated virial theorem — 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.

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 … 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. Generally in what follows, this will be referred to as the "B-KB74 conjecture;" a menu tile carrying this label is linked to a chapter in which this approach is used to analyze the onset of a dynamical instability along the equilibrium sequence of pressure-truncated n = 5 polytropes.

--- Text in green taken directly from G. S. Bisnovatyi-Kogan & S. I. Blinnikov (1974); B-KB74, for short.

		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

	$0 = \frac{d^{2} x}{d ξ^{2}} + [4 - ξ (\frac{d ψ}{d ξ})] \frac{1}{ξ} \cdot \frac{d x}{d ξ} + [(\frac{σ_{c}^{2}}{6 γ_{g}}) ξ^{2} - α ξ (\frac{d ψ}{d ξ})] \frac{x}{ξ^{2}}$
where: $σ_{c}^{2} \equiv \frac{3 ω^{2}}{2 π G ρ_{c}}$ and, $α \equiv (3 - \frac{4}{γ_{g}})$

via
Direct
Numerical
Integration

Yabushita's
Analytic Sol'n for
Marginally Unstable
Configurations
(1974)

$σ_{c}^{2} = 0, γ_{g} = 1$

and

$x = 1 - (\frac{1}{ξ e^{- ψ}}) \frac{d ψ}{d ξ}$

Polytropes

	$0 = \frac{d^{2} x}{d ξ^{2}} + [4 - (n + 1) Q] \frac{1}{ξ} \cdot \frac{d x}{d ξ} + (n + 1) [(\frac{σ_{c}^{2}}{6 γ_{g}}) \frac{ξ^{2}}{θ} - α Q] \frac{x}{ξ^{2}}$
where: $Q (ξ) \equiv - \frac{d \ln θ}{d \ln ξ},$ $σ_{c}^{2} \equiv \frac{3 ω^{2}}{2 π G ρ_{c}},$ and, $α \equiv (3 - \frac{4}{γ_{g}})$

Isolated
n = 3
Polytrope

Exact Demonstration of B-KB74 Conjecture		Exact Demonstration of Variational Principle

Pressure-Truncated n = 5 Configurations

Our (2017)
Analytic Sol'n for
Marginally Unstable
Configurations
♥

$σ_{c}^{2} = 0, γ_{g} = (n + 1) / n$

and

$x = \frac{3 (n - 1)}{2 n} [1 + (\frac{n - 3}{n - 1}) (\frac{1}{ξ θ^{n}}) \frac{d θ}{d ξ}]$

B-KB74 Conjecture RE: Bipolytrope (n_c, n_e) = (5, 1)

BiPolytropes		Yabushita (1975) (n_c, n_e) = (∞, 3/2)		Murphy & Fiedler (1985b) (n_c, n_e) = (1,5)		Stability Analysis of BiPolytropes with (n_c, n_e) = (5, 1)

Nonlinear Dynamical Evolution

1D DYNAMICS

Free-Fall Collapse

Collapse of Isothermal Spheres		via Direct Numerical Integration		Similarity Solution

Collapse of an Isolated n = 3 Polytrope

Two-Dimensional Configurations (Axisymmetric)

(Initially) Axisymmetric Configurations

Storyline

PGEs for Axisymmetric Systems

Axisymmetric Equilibrium Structures

2D STRUCTURE

Constructing Steady-State Axisymmetric Configurations		Axisymmetric Instabilities to Avoid		Simple Rotation Profiles		Hachisu Self-Consistent-Field [HSCF] Technique		Solving the Poisson Equation

Using Toroidal Coordinates to Determine the Gravitational Potential				Attempt at Simplification ♥		Wong's Analytic Potential (1973)

Spheroidal & Spheroidal-Like

Uniform-Density (Maclaurin) Spheroids		Maclaurin's Original Text & Analysis (1742)				Maclaurin Spheroid Sequence		Eriguchi, Hachisu & Colleagues

Rotationally Flattened Isothermal Configurations		Hayashi, Narita & Miyama's Analytic Sol'n (1982)		Review of Stahler's (1983) Sol'n Technique

Rotationally Flattened Polytropes		Example Equilibria

Rotationally Flattened White Dwarfs		Ostriker Bodenheimer & Lynden-Bell (1966)		Example Equilibria

Toroidal & Toroidal-Like

Massless Polytropic Configurations		Papaloizou-Pringle Tori (1984)

Self-Gravitating Incompressible Configurations		Dyson (1893)		Dyson-Wong Tori		Maclaurin Toroid Sequence MPT77

Self-Gravitating Compressible Configurations		Ostriker (1964)

Stability Analysis

2D STABILITY

Sheroidal & Spheroidal-Like

Linear Analysis of Bar-Mode Instability		Bifurcation from Maclaurin Sequence		Traditional Analyses

T. G. Cowling & R. A. Newing (1949), ApJ, 109, 149: The Oscillations of a Rotating Star
M. J. Clement (1965), ApJ, 141, 210: The Radial and Non-Radial Oscillations of Slowly Rotating Gaseous Masses
P. H. Roberts & K. Stewartson (1963), ApJ, 137, 777: On the Stability of a Maclaurin spheroid of small viscosity
C. E. Rosenkilde (1967), ApJ, 148, 825: The tensor virial-theorem including viscous stress and the oscillations of a Maclaurin spheroid
S. Chandrasekhar & N. R. Lebovitz (1968), ApJ, 152, 267: The Pulsations and the Dynamical Stability of Gaseous Masses in Uniform Rotation
C. Hunter (1977), ApJ, 213, 497: On Secular Stability, Secular Instability, and Points of Bifurcation of Rotating Gaseous Masses
J. N. Imamura, J. L. Friedman & R. H. Durisen (1985), ApJ, 294, 474: Secular stability limits for rotating polytropic stars

The equilibrium models are calculated using the polytrope version (Bodenheimer & Ostriker 1973) of the Ostriker and Mark (1968) self-consistent field (SCF) code … the equilibrium models rotate on cylinders and are completely specified by $n$ , the total angular momentum, and the specific angular momentum distribution $j (m_{ϖ})$ . Here $m_{ϖ}$ is the mass contained within a cylinder of radius $ϖ$ centered on the rotation axis. The angular momentum distribution is prescribed in several ways: (1) imposing strict uniform rotation; (2) using the same $j (m_{ϖ})$ as that of a uniformly rotating spherical polybrope of index $n^{'}$ (see Bodenheimer and Ostriker 1973); and (3) using $j (m_{ϖ}) \propto m_{ϖ}$ , which we refer to as $n^{'} = L$ , $L$ for "linear."

J. R. Ipser & L. Lindblom (1990), ApJ, 355, 226: The Oscillations of Rapidly Rotating Newtonian Stellar Models
J. R. Ipser & L. Lindblom (1991), ApJ, 373, 213: The Oscillations of Rapidly Rotating Newtonian Stellar Models. II. Dissipative Effects
J. N. Imamura, J. L. Friedman & R. H. Durisen (2000), ApJ, 528, 946: Nonaxisymmetric Dynamic Instabilities of Rotating Polytropes. II. Torques, Bars, and Mode Saturation with Applications to Protostars and Fizzlers
M. Shibata, S. Karino, & Y. Eriguchi (2003), MNRAS, 343, 619 - 626: Dynamical bar-mode instability of differentially rotating stars: effects of equations of state and velocity profiles
G. P. Horedt (2019), ApJ, 877, 9: On the Instability of Polytropic Maclaurin and Roche ellipsoids

Toroidal & Toroidal-Like

Defining the Eigenvalue Problem

(Massless) Papaloizou-Pringle Tori		Analytic Analysis by Blaes (1985)

Self-Gravitating Polytropic Rings		Numerical Analysis by Tohline & Hachisu (1990)		PP torus instability		Thick Accretion Disks (WTH94)		Hadley & Imamura Collaboration

Nonlinear Dynamical Evolution

Sheroidal & Spheroidal-Like

2D DYNAMICS

Free-Fall Collapse of an Homogeneous Spheroid

Nonlinear Development of Bar-Mode		Initially Axisymmetric & Differentially Rotating Polytropes

Two-Dimensional Configurations (Nonaxisymmetric Disks)

Infinitesimally Thin, Nonaxisymmetric Disks

2D STRUCTURE

Constructing Infinitesimally Thin Nonaxisymmetric Disks