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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 

FreeEnergy Index 
Structural Form Factors 
FreeEnergy of Spherical Systems 

OneDimensional PGEs 
SSC Index 

Equilibrium Structures
1D STRUCTURE 

Scalar Virial Theorem 

Hydrostatic Balance Equation 
Solution Strategies 

UniformDensity Sphere 

Isothermal Sphere 
via Direct Numerical Integration 

Isolated Polytropes 
Lane (1870) 
Known Analytic Solutions 
via Direct Numerical Integration 
via SelfConsistent Field (SCF) Technique 

ZeroTemperature White Dwarf 
Chandrasekhar Limiting Mass (1935) 

Virial Equilibrium of PressureTruncated Polytropes 

PressureTruncated Configurations 
BonnorEbert (Isothermal) Spheres (1955  56) 
Embedded Polytropes 
Equilibrium Sequence TurningPoints ♥ 
TurningPoints (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 boundaryvalue problem of the StourmeLiuville 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) uniformdensity configurations, (b) pressuretruncated isothermal spheres, (c) an isolated n = 3 polytrope, (d) pressuretruncated n = 5 configurations, and (e) bipolytropes having .
 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 pressuretruncated n = 5 polytropes.
 The third approach is what we have already referred to as a freeenergy — 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 "BKB74 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 pressuretruncated n = 5 polytropes.
Variational Principle 

Radial Pulsation Equation 
Example Derivations & Statement of Eigenvalue Problem 
(poor attempt at) Reconciliation 
Relationship to Sound Waves 

UniformDensity Configurations 
Sterne's Analytic Sol'n of Eigenvalue Problem (1937) 

PressureTruncated Isothermal Spheres 
via Direct Numerical Integration 

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


Polytropes  Isolated n = 3 Polytrope 

Exact Demonstration of BKB74 Conjecture 
Exact Demonstration of Variational Principle 

PressureTruncated n = 5 Configurations 

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


BKB74 Conjecture RE: Bipolytrope (n_{c}, n_{e}) = (5, 1) 

BiPolytropes  Murphy & Fiedler (1985b) (n_{c}, n_{e}) = (1,5) 
Our Broader Analysis 
Succinct Discussion 

Nonlinear Dynamical Evolution
1D DYNAMICS 

FreeFall Collapse 

Collapse of Isothermal Spheres 
via Direct Numerical Integration 
Similarity Solution 

Collapse of an Isolated n = 3 Polytrope 

TwoDimensional Configurations (Axisymmetric)
(Initially) Axisymmetric Configurations 

Storyline 

PGEs for Axisymmetric Systems 

Axisymmetric Equilibrium Structures
2D STRUCTURE 

Constructing SteadyState Axisymmetric Configurations 
Axisymmetric Instabilities to Avoid 
Simple Rotation Profiles 
Hachisu SelfConsistentField [HSCF] Technique 
Solving the Poisson Equation 

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

Spheroidal & SpheroidalLike
UniformDensity (Maclaurin) Spheroids 
Maclaurin's Original Text & Analysis (1742) 
Maclaurin Spheroid Sequence 

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 & LyndenBell (1966) 
Example Equilibria 

Toroidal & ToroidalLike
Massless Polytropic Configurations 
PapaloizouPringle Tori (1984) 

SelfGravitating Incompressible Configurations 
Dyson (1893) 
DysonWong Tori 

SelfGravitating Compressible Configurations 
Ostriker (1964) 

Stability Analysis
2D STABILITY 

Sheroidal & SpheroidalLike
Linear Analysis of BarMode 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 NonRadial 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 virialtheorem 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) selfconsistent field (SCF) code … the equilibrium models rotate on cylinders and are completely specified by , the total angular momentum, and the specific angular momentum distribution . Here is the mass contained within a cylinder of radius centered on the rotation axis. The angular momentum distribution is prescribed in several ways: (1) imposing strict uniform rotation; (2) using the same as that of a uniformly rotating spherical polybrope of index (see Bodenheimer and Ostriker 1973); and (3) using , which we refer to as , 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 barmode 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 & ToroidalLike
Defining the Eigenvalue Problem 

(Massless) PapaloizouPringle Tori 
Analytic Analysis by Blaes (1985) 

SelfGravitating Polytropic Rings 
Numerical Analysis by Tohline & Hachisu (1990) 
Thick Accretion Disks (WTH94) 
Hadley & Imamura Collaboration 

Nonlinear Dynamical Evolution
Sheroidal & SpheroidalLike
2D DYNAMICS 

FreeFall Collapse of an Homogeneous Spheroid 

Nonlinear Development of BarMode 
Initially Axisymmetric & Differentially Rotating Polytropes 

TwoDimensional Configurations (Nonaxisymmetric Disks)
Infinitesimally Thin, Nonaxisymmetric Disks 

2D STRUCTURE 

Constructing Infinitesimally Thin Nonaxisymmetric Disks 

ThreeDimensional Configurations
(Initially) ThreeDimensional Configurations 

Equilibrium Structures
3D STRUCTURE 

Special numerical techniques must be developed "to build threedimensional compressible equilibrium models with complicated flows." To date … "techniques have only been developed to build compressible equilibrium models of nonaxisymmetric configurations for a few systems with simplified rotational profiles, e.g., rigidly rotating systems (Hachisu & Eriguchi 1984; Hachisu 1986), irrotational systems (Uryū & Eriguchi 1998), and configurations that are stationary in the inertial frame (Uryū & Eriguchi 1996)." 
— Drawn from §1 of 📚 S. Ou (2006, ApJ, Vol. 639, pp. 549  558) 
Ellipsoidal & EllipsoidalLike
SteadyState 2^{nd}Order Tensor Virial Equations 

UniformDensity Incompressible Ellipsoids 
Bernhard Riemann (1861) 

The Gravitational Potential (A_{i} coefficients) 
Jacobi Ellipsoids 
Riemann SType Ellipsoids 
Type I Riemann Ellipsoids 
Riemann meets COLLADA & Oculus Rift S 

A Gauge Theory of Riemann Ellipsoids 
Nuclear Wobbling Motion 

Compressible Analogs of Riemann Ellipsoids 
Ferrers Potential (1877) 
Constructing Ellipsoidal & EllipsoidalLike Configurations 
Thoughts & Challenges 

 B. P. Kondrat'ev (1985), Astrophysics, 23, 654: Irrotational and zero angular momentum ellipsoids in the Dirichlet problem
 📚 D. Lai, F. A. Rasio, & S. L. Shapiro (1993b, ApJ Suppl., Vol. 88, pp. 205  252): Ellipsoidal Figures of Equilibrium: Compressible models
Binary Systems
 S. Chandrasekhar (1933), MNRAS, 93, 539: The equilibrium of distorted polytropes. IV. the rotational and the tidal distortions as functions of the density distribution
 📚 S. Chandrasekhar (1963, ApJ, Vol. 138, pp. 1182  1213): The Equilibrium and the Stability of the Roche Ellipsoids
Roche's problem is concerned with the equilibrium and the stability of rotating homogeneous masses which are, further, distorted by the constant tidal action of an attendant rigid spherical mass. 
Stability Analysis
3D STABILITY 

Ellipsoidal & EllipsoidalLike
Lebovitz & Lifschitz (1996) 

Binary Systems
 📚 S. Chandrasekhar (1963, ApJ, Vol. 138, pp. 1182  1213): The Equilibrium and the Stability of the Roche Ellipsoids
 G. P. Horedt (2019), ApJ, 877, 9: On the Instability of Polytropic Maclaurin and Roche Ellipsoids
Nonlinear Evolution
3D DYNAMICS 

FreeEnergy Evolution from the Maclaurin to the Jacobi Sequence 

Fission Hypothesis 
"Fission" Simulations at LSU 

Secular
 M. Fujimoto (1971), ApJ, 170, 143: Nonlinear Motions of Rotating Gaseous Ellipsoids
 W. H. Press & S. A. Teukolsky (1973), ApJ, 181, 513: On the Evolution of the Secularly Unstable, Viscous Maclaurin Spheroids
 S. L. Detweiler & L. Lindblom (1977), ApJ, 213, 193: On the evolution of the homogeneous ellipsoidal figures.
Dynamical
See Also
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