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Whitworth's (1981) Isothermal Free-Energy Surface

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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

 

Free-Energy
Index
Whitworth's (1981) Isothermal Free-Energy Surface Structural
Form
Factors
Free-Energy
of
Spherical
Systems
One-Dimensional
PGEs
SSC
Index

Equilibrium Structures

1D STRUCTURE

 

Spherical Structures Synopsis Scalar
Virial
Theorem
Hydrostatic
Balance
Equation

LSUkey.png

Solution
Strategies


Uniform-Density
Sphere

 

Isothermal
Sphere

LSUkey.png

via
Direct
Numerical
Integration


Isolated
Polytropes
Lane
(1870)

LSUkey.png

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

Equilibrium sequences of Pressure-Truncated Polytropes 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)
Equilibrium sequences of (5, 1) Bipolytropes


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 .
  • 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.

 
 

Synopsis: Stability of Spherical Structures 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)
Sterne's (1937) Solution to the Eigenvalue Problem for Uniform-Density Spheres

 

Pressure-Truncated
Isothermal
Spheres

LSUkey.png

where:        and,    

via
Direct
Numerical
Integration
Fundamental-Mode Eigenvectors

 

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

 and  

 

Polytropes

LSUkey.png

where:           and,    

Isolated
n = 3
Polytrope
Schwarzschild's Modal Analysis


Exact
Demonstration
of
B-KB74
Conjecture
Exact
Demonstration
of
Variational
Principle
Pressure-Truncated
n = 5
Configurations
n5 Truncated Movie


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

 and  

 


B-KB74
Conjecture
RE: Bipolytrope


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


(nc, ne) = (1,5)
Our
Broader
Analysis
Succinct
Discussion

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
Apollonian Circles Attempt at
Simplification

Wong's
Analytic Potential
(1973)
n = 3 contribution to potential

Spheroidal & Spheroidal-Like


Uniform-Density
(Maclaurin)
Spheroids
Maclaurin's
Original Text
&
Analysis
(1742)
Our Construction of Maclaurin's Figure 291Pt2 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
& Lynden-Bell
(1966)
Example
Equilibria

Toroidal & Toroidal-Like

 

Massless
Polytropic
Configurations
Papaloizou-Pringle
Tori
(1984)
Pivoting PP Torus

 

Self-Gravitating
Incompressible
Configurations
Dyson
(1893)
Dyson-Wong
Tori

 

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

 

 

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 , 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."

 

Toroidal & Toroidal-Like

Defining the
Eigenvalue Problem

 

(Massless)
Papaloizou-Pringle
Tori
Analytic Analysis
by
Blaes
(1985)
N1.5j2 Combinedsmall.png

 

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
Cazes Model A Simulation

 

Two-Dimensional Configurations (Nonaxisymmetric Disks)

Infinitesimally Thin, Nonaxisymmetric Disks

 

2D STRUCTURE

 

Constructing
Infinitesimally Thin
Nonaxisymmetric
Disks

 

Three-Dimensional Configurations

(Initially) Three-Dimensional Configurations

 

Equilibrium Structures

3D STRUCTURE

Special numerical techniques must be developed "to build three-dimensional 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 & Ellipsoidal-Like

 

Steady-State
2nd-Order
Tensor Virial
Equations
Uniform-Density
Incompressible
Ellipsoids
Bernhard
Riemann
(1861)

 

The
Gravitational
Potential

(Ai coefficients)
Jacobi
Ellipsoids
Riemann
S-Type
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
& Ellipsoidal-Like
Configurations
Thoughts
&
Challenges

 

Binary Systems

 

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 & Ellipsoidal-Like

Lebovitz & Lifschitz (1996)

 

Binary Systems

Nonlinear Evolution

3D DYNAMICS

 

Animation related to Fig. 3 from Christodoulou1995 Free-Energy
Evolution
from the Maclaurin
to the Jacobi
Sequence
Fission
Hypothesis
"Fission"
Simulations
at LSU

 

Secular

Dynamical

See Also

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