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Revision as of 18:01, 31 January 2020
Contents 
Tiled Menu
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[Preamble by Joel E. Tohline — Circa 2020] In an effort to serve ongoing research activities of the astrophysics community, this mediawikibased hypertext book (H_Book) reviews what is presently understood about the structure, stability, and dynamical evolution of (Newtonian) selfgravitating fluids. In the context of this review, the "dynamical evolution" categorization generally will refer to studies that begin with an equilibrium configuration that has been identified — via a stability analysis — as being dynamically (or, perhaps, secularly) unstable then, using numerical hydrodynamic technique, follow the development to nonlinear amplitude of deformations that spontaneously develop as a result of the identified instability.
As is reflected in our choice of the overarching set of Principle Governing Equations, our focus is on studies of compressible fluid systems — those that obey a barotropic equation of state. But, in addition, chapters are included that review what is known about the structure and stability of selfgravitating incompressible (uniformdensity) fluid systems because: (a) in this special case, the set of governing relations is often amenable to a closedform, analytic solution; and (b) most modern computationally assisted studies of compressible fluid systems — both in terms of their design and in the manner in which results have been interpreted — have been heavily influenced by these, more classical, studies of incompressible fluid systems.
As the layout of the following tiled menu reflects, this review is broken into three major topical areas based primarily on geometrical considerations:
 Studies of configurations that are — at least initially — spherically symmetric.
 Twodimensional configurations — such as rotationally flattened spheroidallike or toroidallike structures; or infinitesimally thin, but nonaxisymmetric, disklike structures.
 Configurations that require a full threedimensional treatment — most notably, "spinning" ellipsoidallike structures; or binary systems.
This is very much a living review. The chosen theme encompasses an enormous field of research that, because of its relevance to the astrophysics community, over time is continuing to expand at a healthy pace. As a consequence the review is incomplete now, and it will always be incomplete, so please bear with me. On any given day/week/month I will turn my attention to a topic that seems particularly interesting to me and I will begin writing a new chapter or I will edit/expand the contents of an existing tiled_menu chapter. This necessarily means that all chapters are incomplete while, in practice, some are much more polished than others. Hopefully steady forward progress is being made and the review will indeed be viewed as providing a service to the community.
Context
Global Energy Considerations 

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

Equation of State (EOS)  Ideal Gas  Total Pressure 

Spherically Symmetric Configurations
(Initially) Spherically Symmetric Configurations 

Structural Form Factors  FreeEnergy of Spherical Systems 

OneDimensional PGEs 

Equilibrium Structures
1D STRUCTURE 

Scalar Virial Theorem 

Hydrostatic Balance Equation  Solution Strategies 

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)  Polytropes  Equilibrium Sequence TurningPoints ♥  TurningPoints (Broader Context) 

Free Energy of Bipolytropes (n_{c}, n_{e}) = (5, 1) 

Composite Polytropes (Bipolytropes)  Schönberg Chandrasekhar Mass (1942)  Analytic (n_{c}, n_{e}) = (5, 1)  Analytic (n_{c}, n_{e}) = (1, 5) 

Stability Analysis
1D STABILITY 

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  PressureTruncated n = 5 Configurations 

Exact Demonstration of BKB74 Conjecture  Exact Demonstration of Variational Principle  PressureTruncated n = 5 Polytropes 

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


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

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) 

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

Rotationally Flattened Polytropes  Example Equilibria 

Rotationally Flattened White Dwarfs  Ostriker Bodenheimer & LyndenBell (1966)  Example Equilibria 

Toroidal & ToroidalLike
Definition: anchor ring 

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  TimeDependent Simulations 

 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
 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) 

Nonlinear Dynamical Evolution
2D DYNAMICS 

FreeFall Collapse of an Homogeneous Spheroid 

TwoDimensional Configurations (Nonaxisymmetric Disks)
Infinitesimally Thin, Nonaxisymmetric Disks 

2D STRUCTURE 

Constructing Infinitesimally Thin Nonaxisymmetric Disks 

ThreeDimensional Configurations
(Initially) ThreeDimensional Configurations 

Equilibrium Structures
3D STRUCTURE 

"One interesting aspect of our models … is the pulsation characteristic of the final central triaxial figure … our interest in the pulsations stems from a general concern about the equilibrium structure of selfgravitating, triaxial objects. In the past, attempts to construct hydrostatic models of any equilibrium, triaxial structure having both a high value and a compressible equation of state have met with very limited success … they have been thwarted by a lack of understanding of how to represent complex internal motions in a physically realistic way… We suggest … that a natural attribute of [such] configurations may be pulsation and that, as a result, a search for simple circulation hydrostatic analogs of such systems may prove to a fruitless endeavor. 
— Drawn from §IVa of Williams & Tohline (1988), ApJ, 334, 449 
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 Ou (2006), ApJ, 639, 549 
Ellipsoidal & EllipsoidalLike
Constructing Ellipsoidal & EllipsoidalLike Configurations 

UniformDensity Incompressible Ellipsoids  Jacobi Ellipsoids  Riemann SType Ellipsoids 

 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 (1993), ApJS, 88, 205: 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, 138, 1182: 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
Binary Systems
 S. Chandrasekhar (1963), ApJ, 138, 1182: 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 

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
© 2014  2020 by Joel E. Tohline 