Editing VisTrails/H Book (section)

===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 second-order ODE &#8212; for example, the [[User:Tohline/SSC/Perturbations#2ndOrderODE|''Linear Adiabatic Wave Equation (LAWE)'']] &#8212; comes directly from ''perturbing'' the equilibrium model and a ''linearization'' of the overarching set of [[User:Tohline/PGE#Principal_Governing_Equations|principal governing equations]].  In the set of chapters that is identified in the upper-left quadrant of the following table of content, we explain how the relevant wave equation for self-gravitating 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 upper-right quadrant of the table, builds on a free-energy analysis of self-gravitating 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 &hellip;" 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. Bisnovatyi-Kogan &amp; S. I. Blinnikov (1974)].  <font color="darkgreen">Three different approaches are used in the study of hydrodynamical stability of stars and other gravitating objects &hellip; &nbsp; 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.  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_Walraven|Ledoux &amp; 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 &amp; Lebovitz (1968)], [http://adsabs.harvard.edu/abs/1967MNRAS.136..293L Lynden-Bell 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 &hellip;</font>  The third approach is what we have referred to as a free-energy 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 &amp; 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 &#8212; whether the model is stable or not &#8212; then the most straightforward procedure is to use the third, static method (Zeldovich 1963; Dmitrie &amp; 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.

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[[User:Tohline/SSC/SoundWaves#Sound_Waves|Background:  Sound Waves]]

<font color="darkblue">'''Solution Strategy Assuming Spherical Symmetry:'''</font>
* [[User:Tohline/SSC/Stability_Eulerian_Perspective#Stability_of_Spherically_Symmetric_Configurations_.28Eulerian_Perspective.29|Eulerian Perspective]]
* [[User:Tohline/SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29|Lagrangian Perturbations]]
* [[User:Tohline/SSC/Perspective_Reconciliation#Reconciling_Eulerian_versus_Lagrangian_Perspectives|Reconciliation]]
<p></p>
<table align="left" border="1" cellpadding="3">
<tr><th align="center" colspan="3"><font color="darkblue">Summary of Various ''Adiabatic Wave Equation'' Derivations</font></th></tr>
<tr>
<td align="center">[[File:ImageOfDerivations06GoodJeansBonnor.png|100px|thumb|center|Jeans (1928) or Bonnor (1957)]]</td>
<td align="center">[[File:ImageOfDerivations07GoodLedouxWalraven.png|100px|thumb|center|Ledoux &amp; Walraven (1958)]]</td>
<td align="center">[[File:ImageOfDerivations08GoodRosseland.png|100px|thumb|center|Rosseland (1969)]]</td>
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</table>
  </td>
  <td align="right">
[[User:Tohline/SSC/VariationalPrinciple|Ledoux's Variational Principle]]<p></p>
[[User:Tohline/SSC/VirialStability|Early Chatting with Kundan Kadam]]<p></p>
[[User:Tohline/SSC/BipolytropeGeneralization#Bipolytrope_Generalization|Bipolytrope Generalization]]
  </td>
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  <td align="left" colspan="2">
Example Solutions:
* [[User:Tohline/SSC/UniformDensity#The_Stability_of_Uniform-Density_Spheres|Uniform-density sphere]]
* Isolated Polytropes
** [[User:Tohline/SSC/Stability/Polytropes#Radial_Oscillations_of_Polytropic_Spheres|Setup]]
** n = 1:&nbsp; [[User:Tohline/SSC/Stability/n1PolytropeLAWE#Radial_Oscillations_of_n_.3D_1_Polytropic_Spheres|Attempt at Formulating an Analytic Solution]]
** n = 3:&nbsp; [[User:Tohline/SSC/Stability/n3PolytropeLAWE|Numerical Solution]] to compare with [http://adsabs.harvard.edu/abs/1941ApJ....94..245S M. Schwarzschild (1941)]
** n = 5:&nbsp; [[User:Tohline/SSC/Stability/n5PolytropeLAWE#Radial_Oscillations_of_n_.3D_5_Polytropic_Spheres|Attempt at Formulating an Analytic Solution]]
* Radial Oscillations in, and Stability of, Pressure-Truncated Configurations ([[User:Tohline/SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|Overview]])
** [[User:Tohline/SSC/Stability/Isothermal|Pressure-Truncated Isothermal Spheres]]
* [[User:Tohline/SSC/Stability_BoundedCompositePolytropes|Bounded and Composite Polytropes]]
* Bipolytropes having <math>~(n_c, n_e) = (0, 0)</math>
** [[User:Tohline/SSC/Stability/BiPolytrope0_0|Overview]] 
** Initial set of chapters that used incorrect interface matching condition
*** [[User:Tohline/SSC/Stability/BiPolytrope0_0Old|One Analytically Specified Eigenvector for a Bipolytrope]] having <math>~(n_c, n_e) = (0, 0)</math>
*** [[User:Tohline/SSC/Stability/BiPolytrope0_0Details|Initial Detailed Derivation]] 
*** [[User:Tohline/Appendix/Ramblings/Additional_Analytically_Specified_Eigenvectors_for_Zero-Zero_Bipolytropes#Searching_for_Additional_Eigenvectors_of_Zero-Zero_Bipolytropes|Additional Analytically Specified Eigenvectors]]
*** [[User:Tohline/Appendix/Ramblings/NumericallyDeterminedEigenvectors#Numerically_Determined_Eigenvectors_of_a_Zero-Zero_Bipolytrope|Numerically Determined Eigenvectors]]
**[[User:Tohline/SSC/Structure/BiPolytropes/Analytic0_0#Stability_Condition|Free-Energy Stability Analysis]]
**[[User:Tohline/SSC/Stability/BiPolytrope0_0CompareApproaches|Compare Eigenvector and Free-Energy Analyses]]
  </td>
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<tr>
  <td align="left" colspan="2">Also to be studied: [http://adsabs.harvard.edu/abs/1981PASJ...33..299K Kimura] (1981, PASJapan, 33, 299)</td>
</tr>
</table>