User:Tohline/H Book
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(→Structure:) 

(16 intermediate revisions not shown)  
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*** [[User:Tohline/PGE/Hybrid_Scheme#Hybrid_Advection_SchemeJay Call's Hybrid Scheme]]  *** [[User:Tohline/PGE/Hybrid_Scheme#Hybrid_Advection_SchemeJay Call's Hybrid Scheme]]  
** <math>1^\mathrm{st}</math> Law of Thermodynamics  ** <math>1^\mathrm{st}</math> Law of Thermodynamics  
  ** Poisson Equation  +  ** [[User:Tohline/SR/PoissonOrigin#Origin_of_the_Poisson_EquationPoisson Equation]] 
* [[User:Tohline/SR#Supplemental_RelationsSupplemental Relations]]  * [[User:Tohline/SR#Supplemental_RelationsSupplemental Relations]]  
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*** [[User:Tohline/SR/Ptot_QuarticSolution#Determining_Temperature_from_Density_and_PressureQuartic Temperature Solution]]  *** [[User:Tohline/SR/Ptot_QuarticSolution#Determining_Temperature_from_Density_and_PressureQuartic Temperature Solution]]  
  * [[User:Tohline/VE#Global_Energy_Considerations  +  * Global Energy Considerations 
+  ** [[User:Tohline/VE#Global_Energy_ConsiderationsTraditional Presentation Based on Virial Theorem]]  
+  ** [[User:Tohline/StabilityVariationalPrincipalVariational Principle Approach]]  
+  ** [[User:Tohline/SSC/SynopsisSynopsis]] ([[User:Tohline/SSC/Synopsis_StyleSheetAlternate Style Sheet]])  
=Applications=  =Applications=  
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===Stability:===  ===Stability:===  
  Suppose we want to study the '''stability''' of one of the spherically symmetric, equilibrium structures that has been described in the above "Structure" chapters. One approach is to solve an eigenvalue problem whose governing secondorder ODE — for example, the [[User:Tohline/SSC/Perturbations#2ndOrderODE''Adiabatic Wave Equation'']] — comes directly from ''perturbing'' the equilibrium model and a ''linearization'' of the overarching set of [[User:Tohline/PGE#Principal_Governing_Equationsprincipal governing equations]]. In the set of chapters that is identified in the upperleft quadrant of the following table of content, we explain how the relevant wave equation for selfgravitating systems is derived from either a ''Lagrangian'' perspective or an ''Eulerian'' perspective, and how it resembles the equation that describes the propagation of sound waves in terrestrial fluids. Another approach, which is described by the chapter(s) in the upperright quadrant of the table, builds on a freeenergy analysis of selfgravitating systems: If an extremum in the free energy is identified as a local energy ''minimum'', then the identified equilibrium configuration is stable; but if the extremum is a local energy ''maximum'', then the equilibrium configuration is unstable. In the lower portion of the following table of content, links are provided to chapters in which one of these  +  Suppose we want to study the '''stability''' of one of the spherically symmetric, equilibrium structures that has been described in the above "Structure" chapters. One approach is to solve an eigenvalue problem whose governing secondorder ODE — for example, the [[User:Tohline/SSC/Perturbations#2ndOrderODE''Linear Adiabatic Wave Equation (LAWE)'']] — comes directly from ''perturbing'' the equilibrium model and a ''linearization'' of the overarching set of [[User:Tohline/PGE#Principal_Governing_Equationsprincipal governing equations]]. In the set of chapters that is identified in the upperleft quadrant of the following table of content, we explain how the relevant wave equation for selfgravitating systems is derived from either a ''Lagrangian'' perspective or an ''Eulerian'' perspective, and how it resembles the equation that describes the propagation of sound waves in terrestrial fluids. Another approach, which is described by the chapter(s) in the upperright quadrant of the table, builds on a freeenergy analysis of selfgravitating systems: If an extremum in the free energy is identified as a local energy ''minimum'', then the identified equilibrium configuration is stable; but if the extremum is a local energy ''maximum'', then the equilibrium configuration is unstable. 
+  
+  [This paragraph ultimately should be moved to a "more …" page] <span id="BKB74pt1">Text written in ''green'' has been taken directly from the introductory paragraphs of</span> [http://adsabs.harvard.edu/abs/1974A%26A....31..391B G. S. BisnovatyiKogan & S. I. Blinnikov (1974)]. <font color="darkgreen">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. This approach is subject to great numerical difficulties and has been carried through successfully only for the simplest case</font>[s]. For spherically symmetric, Newtonian systems, see, for example, [[User:Tohline/SSC/Stability/n3PolytropeLAWE#Schwarzschild_.281941.29 Schwarzschild (1941)]] or [[User:Tohline/SSC/Perspective_Reconciliation#Approach_by_Ledoux_and_WalravenLedoux & Walraven (1958)]]. Second, one can derive <font color="darkgreen">a variational principle from the equations of small oscillations. This principle replaces the straightforward solution of these equations:</font> In the context of rotating Newtonian systems, see, for example, [http://adsabs.harvard.edu/abs/1964ApJ...140.1045C Clement (1964)], [http://adsabs.harvard.edu/abs/1968ApJ...152..267C Chandrasekhar & Lebovitz (1968)], [http://adsabs.harvard.edu/abs/1967MNRAS.136..293L LyndenBell and Ostriker (1967)], or [http://adsabs.harvard.edu/abs/1972ApJS...24..319S Schutz (1972)]. <font color="darkgreen">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 …</font> The third approach is what we have referred to as a freeenergy analysis. <font color="darkgreen">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.</font> For non rotating systems having <math>~\gamma_g</math> near 4/3, see, for example, [http://adsabs.harvard.edu/abs/1966ApJ...144..180F Fowler (1964)] or Zeldovich & Novikov (Soviet journal, 1965).  
+  
+  <span id="BKB74pt2"><font color="darkgreen">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.</font></span>  
+  
+  In the lower portion of the following table of content, links are provided to chapters in which one of these primary solution strategies is employed to analyze the stability of ''specific'' equilibrium models.  
<table border="1" cellspacing="2" cellpadding="8" width="100%">  <table border="1" cellspacing="2" cellpadding="8" width="100%">  
<tr>  <tr>  
<td align="left">  <td align="left">  
  [[User:Tohline/SSC/SoundWavesBackground: Sound Waves]]  +  [[User:Tohline/SSC/SoundWaves#Sound_WavesBackground: Sound Waves]] 
<font color="darkblue">'''Solution Strategy Assuming Spherical Symmetry:'''</font>  <font color="darkblue">'''Solution Strategy Assuming Spherical Symmetry:'''</font>  
  * [[User:Tohline/SSC/Stability_Eulerian_PerspectiveEulerian Perspective]]  +  * [[User:Tohline/SSC/Stability_Eulerian_Perspective#Stability_of_Spherically_Symmetric_Configurations_.28Eulerian_Perspective.29Eulerian Perspective]] 
* [[User:Tohline/SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29Lagrangian Perturbations]]  * [[User:Tohline/SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29Lagrangian Perturbations]]  
* [[User:Tohline/SSC/Perspective_Reconciliation#Reconciling_Eulerian_versus_Lagrangian_PerspectivesReconciliation]]  * [[User:Tohline/SSC/Perspective_Reconciliation#Reconciling_Eulerian_versus_Lagrangian_PerspectivesReconciliation]]  
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</td>  </td>  
<td align="right">  <td align="right">  
  <  +  [[User:Tohline/SSC/VariationalPrincipleLedoux's Variational Principle]]<p></p> 
  +  [[User:Tohline/SSC/VirialStabilityEarly Chatting with Kundan Kadam]]<p></p>  
[[User:Tohline/SSC/BipolytropeGeneralization#Bipolytrope_GeneralizationBipolytrope Generalization]]  [[User:Tohline/SSC/BipolytropeGeneralization#Bipolytrope_GeneralizationBipolytrope Generalization]]  
</td>  </td>  
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Example Solutions:  Example Solutions:  
* [[User:Tohline/SSC/UniformDensity#The_Stability_of_UniformDensity_SpheresUniformdensity sphere]]  * [[User:Tohline/SSC/UniformDensity#The_Stability_of_UniformDensity_SpheresUniformdensity sphere]]  
  * [[User:Tohline/SSC/Stability/Polytropes#Radial_Oscillations_of_Polytropic_Spheres  +  * Isolated Polytropes 
  * [[User:Tohline/SSC/Stability/IsothermalPressureTruncated Isothermal Spheres]]  +  ** [[User:Tohline/SSC/Stability/Polytropes#Radial_Oscillations_of_Polytropic_SpheresSetup]] 
+  ** n = 1: [[User:Tohline/SSC/Stability/n1PolytropeLAWE#Radial_Oscillations_of_n_.3D_1_Polytropic_SpheresAttempt at Formulating an Analytic Solution]]  
+  ** n = 3: [[User:Tohline/SSC/Stability/n3PolytropeLAWENumerical Solution]] to compare with [http://adsabs.harvard.edu/abs/1941ApJ....94..245S M. Schwarzschild (1941)]  
+  ** n = 5: [[User:Tohline/SSC/Stability/n5PolytropeLAWE#Radial_Oscillations_of_n_.3D_5_Polytropic_SpheresAttempt at Formulating an Analytic Solution]]  
+  * Radial Oscillations in, and Stability of, PressureTruncated Configurations ([[User:Tohline/SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_PressureTruncated_Gas_CloudsOverview]])  
+  ** [[User:Tohline/SSC/Stability/IsothermalPressureTruncated Isothermal Spheres]]  
* [[User:Tohline/SSC/Stability_BoundedCompositePolytropesBounded and Composite Polytropes]]  * [[User:Tohline/SSC/Stability_BoundedCompositePolytropesBounded and Composite Polytropes]]  
* Bipolytropes having <math>~(n_c, n_e) = (0, 0)</math>  * Bipolytropes having <math>~(n_c, n_e) = (0, 0)</math>  
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*** [[User:Tohline/SSC/Stability/BiPolytrope0_0DetailsInitial Detailed Derivation]]  *** [[User:Tohline/SSC/Stability/BiPolytrope0_0DetailsInitial Detailed Derivation]]  
*** [[User:Tohline/Appendix/Ramblings/Additional_Analytically_Specified_Eigenvectors_for_ZeroZero_Bipolytropes#Searching_for_Additional_Eigenvectors_of_ZeroZero_BipolytropesAdditional Analytically Specified Eigenvectors]]  *** [[User:Tohline/Appendix/Ramblings/Additional_Analytically_Specified_Eigenvectors_for_ZeroZero_Bipolytropes#Searching_for_Additional_Eigenvectors_of_ZeroZero_BipolytropesAdditional Analytically Specified Eigenvectors]]  
+  *** [[User:Tohline/Appendix/Ramblings/NumericallyDeterminedEigenvectors#Numerically_Determined_Eigenvectors_of_a_ZeroZero_BipolytropeNumerically Determined Eigenvectors]]  
**[[User:Tohline/SSC/Structure/BiPolytropes/Analytic0_0#Stability_ConditionFreeEnergy Stability Analysis]]  **[[User:Tohline/SSC/Structure/BiPolytropes/Analytic0_0#Stability_ConditionFreeEnergy Stability Analysis]]  
**[[User:Tohline/SSC/Stability/BiPolytrope0_0CompareApproachesCompare Eigenvector and FreeEnergy Analyses]]  **[[User:Tohline/SSC/Stability/BiPolytrope0_0CompareApproachesCompare Eigenvector and FreeEnergy Analyses]]  
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* [[User:Tohline/Apps/PapaloizouPringleToriPapaloizouPringle (massless) Tori]] [[User:Tohline/Apps/PapaloizouPringleTori#Boundary_Conditions<font size="+1">🎦</font>]]  * [[User:Tohline/Apps/PapaloizouPringleToriPapaloizouPringle (massless) Tori]] [[User:Tohline/Apps/PapaloizouPringleTori#Boundary_Conditions<font size="+1">🎦</font>]]  
* Selfgravitating, Isothermal Rings  * Selfgravitating, Isothermal Rings  
  * Selfgravitating, Polytropic Tori  +  * [[User:Tohline/Apps/Tori/PolytropicSelfgravitating, Polytropic Tori]] 
* [[User:Tohline/2DStructure/ToroidalCoordinates#Using_Toroidal_Coordinates_to_Determine_the_Gravitational_PotentialUsing Toroidal Coordinates to Determine the Gravitational Potential]] [[User:Tohline/2DStructure/ToroidalCoordinates#Perform_Angular_Integration_Over_Test_Mass_Distribution<font size="+1">🎦</font>]]  * [[User:Tohline/2DStructure/ToroidalCoordinates#Using_Toroidal_Coordinates_to_Determine_the_Gravitational_PotentialUsing Toroidal Coordinates to Determine the Gravitational Potential]] [[User:Tohline/2DStructure/ToroidalCoordinates#Perform_Angular_Integration_Over_Test_Mass_Distribution<font size="+1">🎦</font>]]  
Current revision as of 16:14, 9 August 2017
Preface from the original version of this HyperText Book (H_Book):
November 18, 1994
Much of our present, basic understanding of the structure, stability, and dynamical evolution of individual stars, shortperiod binary star systems, and the gaseous disks that are associated with numerous types of stellar systems (including galaxies) is derived from an examination of the behavior of a specific set of coupled, partial differential equations. These equations — most of which also are heavily utilized in studies of continuum flows in terrestrial environments — are thought to govern the underlying physics of all macroscopic "fluid" systems in astronomy. Although relatively simple in form, they prove to be very rich in nature... <more>
 Tiled Menu  Tables of Content  Banner Video  Tohline Home Page  
Pictorial Table of Contents
Context
 Principal Governing Equations
 Continuity Equation
 Euler Equation
 1^{st} Law of Thermodynamics
 Poisson Equation
 Global Energy Considerations
Applications
Spherically Symmetric Configurations
Introduction (Alternate Introduction)
Structure:
🎦 ⚜
Here we show how the set of principal governing equations (PGEs) can be solved to determine the equilibrium structure of spherically symmetric fluid configurations — such as individual, nonrotating stars or protostellar gas clouds. After supplementing the PGEs by specifying an equation of state of the fluid, the system of equations is usually solved by employing one of three techniques to obtain a "detailed forcebalanced" model that provides the radius, , of the equilibrium configuration — given its mass, , and central pressure, , for example — as well as details regarding the internal radial profiles of the fluid density and fluid pressure. As our various discussions illustrate (see the table of contents, below), simply varying the powerlaw index in a polytropic equation of state gives rise to equilibrium configurations that have a wide variety of internal structural profiles.
If one is not particularly concerned about details regarding the distribution of matter within the equilibrium configuration, a good estimate of the size of the equilibrium system can be determined by assuming a uniformdensity structure then identifying local extrema in the system's global free energy. In the astrophysics community, the mathematical relation that serves to define the properties of configurations that are associated with such freeenergy extrema is often referred to as the scalar virial theorem … <more>
In the following table, each green check mark identifies and provides a link to an H_Book chapter that presents a detailed discussion of the topic that is identified on the left — for example, the equilibirum structure of "isolated polytropes" or an "isothermal sphere embedded in an external medium." Mathematical models that provide full solutions to the PGEs, including details regarding the internal structural profiles of equilibrium configurations, are derived in chapters whose check marks fall under the column labeled "Detailed ForceBalance." Insight into the properties of equilibrium systems that is revealed via an analysis of a system's freeenergy and the corresponding scalar virial theorem is presented in chapters whose check marks fall under the column labeled "Virial Equilibrium."
Solution Strategies: 
Detailed ForceBalance 
Virial Equilibrium 



Stability:
Suppose we want to study the stability of one of the spherically symmetric, equilibrium structures that has been described in the above "Structure" chapters. One approach is to solve an eigenvalue problem whose governing secondorder ODE — for example, the Linear Adiabatic Wave Equation (LAWE) — comes directly from perturbing the equilibrium model and a linearization of the overarching set of principal governing equations. In the set of chapters that is identified in the upperleft quadrant of the following table of content, we explain how the relevant wave equation for selfgravitating systems is derived from either a Lagrangian perspective or an Eulerian perspective, and how it resembles the equation that describes the propagation of sound waves in terrestrial fluids. Another approach, which is described by the chapter(s) in the upperright quadrant of the table, builds on a freeenergy analysis of selfgravitating systems: If an extremum in the free energy is identified as a local energy minimum, then the identified equilibrium configuration is stable; but if the extremum is a local energy maximum, then the equilibrium configuration is unstable.
[This paragraph ultimately should be moved to a "more …" page] Text written in green has been taken directly from the introductory paragraphs of G. S. BisnovatyiKogan & S. I. Blinnikov (1974). 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. 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), LyndenBell 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 freeenergy 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 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.
In the lower portion of the following table of content, links are provided to chapters in which one of these primary solution strategies is employed to analyze the stability of specific equilibrium models.
Solution Strategy Assuming Spherical Symmetry:

Ledoux's Variational Principle Early Chatting with Kundan Kadam  
Example Solutions:


Also to be studied: Kimura (1981, PASJapan, 33, 299) 
Dynamics:

TwoDimensional Configurations
Structure:
Solution Strategies


UniformDensity & Centrally Condensed Structures:
Toroidal Structures:
Nonaxisymmetric Disks & Annuli: 
Stability:
Solution Strategy


Centrallycondensed Structures: Toroidal Structures:

Dynamics:
ThreeDimensional Configurations
 Introduction
 The Foundation Provided by Chandrasekhar's EFE
 Fission Hypothesis of Binary Star Formation 🎦
Structure:
Solution Strategies 
Example Solutions:

Stability:
 Lou & Bai (2011, MNRAS, 415, 925) — 3D perturbations in an isothermal selfsimilar flow
Dynamics:
Related Projects Underway
Appendices
See Also
© 2014  2021 by Joel E. Tohline 