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=Tiled Menu=
{| class="TopBanner" width=100% style="margin-right: auto; margin-left: 0px; border-style: solid; border-width: 3px; border-color:black;"
|-
! style="height: 50px; background-color:black;"|[[File:HBook_title_Fluids2.png|780px|link=User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#Virial_Equilibrium_of_Embedded_Polytropic_Spheres|Whitworth's (1981) Isothermal Free-Energy Surface]]
|}

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==Context==

{| class="Chap1B" style="float:right; margin-left: 150px; border-style: solid; border-color:black;"
|-
! style="height: 150px; width: 150px; background-color:#9390DB;"|[[VE|Global Energy<br />Considerations]]
|}
{| class="Chap1A" style="border-style: solid; border-width: 3px; border-color: black;"
|- 
! style="height: 150px; width: 150px; background-color:lightgreen; border-right:solid black;" |[[PGE|<b>Principal<br />Governing<br />Equations</b>]]<br />(PGEs)
|
! style="height: 150px; width: 150px; background-color:white; border-right:dashed black;" |[[PGE/ConservingMass|Continuity]]
|
! style="height: 150px; width: 150px; border-right:dashed black;" |[[PGE/Euler|Euler]]
|
! style="height: 150px; width: 150px; border-right:dashed black;" |[[PGE/FirstLawOfThermodynamics|1<sup>st</sup> Law of<br />Thermodynamics]]
|
! style="height: 150px; width: 150px;"|[[PGE/PoissonOrigin|Poisson]]
|}
<p>&nbsp;</p>
{| class="Chap1C" style="margin-right: auto; margin-left: 0px; border-style: solid; border-width: 3px; border-color: black"
|- 
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black;" |[[SR|<b>Equation<br />of State</b>]]<br />(EOS)
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[SR/IdealGas|Ideal Gas]]
|
! style="height: 150px; width: 150px; " |[[SR/PressureCombinations|Total Pressure]] <br />&nbsp;<hr /><br />[[Apps/SMS|Bond, Arnett, &amp; Carr<br />(1984)]]
|}


==Spherically Symmetric Configurations==
{| class="Chap2A" width=100% style="margin-right: auto; margin-left: 0px; border-style: solid; border-width: 3px; border-color:navy;"
|-
! style="height: 50px; width: 800px; background-color:lightgrey;"|<font color="navy" size="+2">(Initially) Spherically Symmetric Configurations</font>
|}
<p>&nbsp;</p>

{| class="Chap2B" style="float:right; margin-left: 150px; border-style: solid; border-width: 3px; border-color:black;"
|-
! style="height: 150px; width: 150px;"|[[File:FreeNRGpressureRadiusIsothermal.png|150px|link=SSCpt1/Virial/PolytropesEmbeddedOutline#Virial_Equilibrium_of_Embedded_Polytropic_Spheres|Whitworth's (1981) Isothermal Free-Energy Surface]]
|
! style="height: 150px; width: 150px; border-left:2px dashed black;"|[[SSCpt1/Virial/FormFactors#Synopsis|Structural<br />Form<br />Factors]]
|
! style="height: 150px; width: 150px; background-color:#9390DB; border-left:2px solid black;"|[[SSCpt1/Virial#Free_Energy_Expression|Free-Energy<br />of<br />Spherical<br />Systems]]
|}
{| class="Chap2C" style="margin-right: auto; margin-left: 0px; border-style: solid; border-width: 3px; border-color:black;"
|- 
! style="height: 150px; width: 150px; background-color:lightgreen;border-right:2px; " |[[SSCpt1/PGE|One-Dimensional<br /> PGEs]]
|}

===Equilibrium Structures===
{| class="Chap3A" width=100% style="margin-right: auto; margin-left: 0px; border-style: solid; border-width: 3px; border-color:darkgrey;"
|-
! style="height: 30px; width: 800px; background-color:lightgrey;"|<font color="white" size="+1">1D STRUCTURE</font>
|}
<p>&nbsp;</p>
{| class="Chap3B" style="float:right; margin-left: 150px; border-style: solid; border-width: 3px; border-color:black;"
|-
! style="height: 150px; width: 150px; border-right:2px dotted black;" |[[File:SSC_SynopsisImage1.png|150px|link=SSC/SynopsisStyleSheet#Structure|Spherical Structures Synopsis]]
|
! style="height: 150px; width: 150px; background-color:#9390DB;"|[[VE#Scalar_Virial_Theorem|Scalar<br />Virial<br />Theorem]]
|}
{| class="Chap3C" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black;"
|- 
! style="height: 150px; width: 150px; background-color:lightgreen; border-right:2px solid black;" |[[SSCpt2/IntroductorySummary#Applications|<b>Hydrostatic<br />Balance<br />Equation</b>]]
|
! style="height: 150px; width: 310px; text-align:center; border-right:2px dashed black;" |<div align="center">{{ Template:Math/EQ_SShydrostaticBalance01 }}</div>
|
! style="height: 150px; width: 150px;" |[[SSCpt2/SolutionStrategies#Solution_Strategies|Solution<br />Strategies]]
|}


{| class="Chap3D" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|- 
! style="height: 150px; width: 150px; background-color:#ffff99; " |[[SSC/Structure/UniformDensity#Isolated_Uniform-Density_Sphere|<b>Uniform-Density<br />Sphere</b>]]
|}
<p>&nbsp;</p>
{| class="Chap3E" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|- 
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black;" |[[SSC/Structure/IsothermalSphere#Isothermal_Sphere|<b>Isothermal<br />Sphere</b>]]
|
! style="height: 150px; width: 310px; text-align:center; border-right:2px dashed black;" |<div align="center">{{ Math/EQ_SSLaneEmden02 }}</div>
|
! style="height: 150px; width: 150px;" |[[SSC/Structure/IsothermalSphere#Our_Numerical_Integration|via<br />Direct<br />Numerical<br />Integration]]
|}


{| class="Chap3F" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|- 
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black;" |[[SSC/Structure/Polytropes#Polytropic_Spheres|<b>Isolated<br />Polytropes</b>]]
|
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right:2px dashed black;" |[[SSC/Structure/Lane1870#Lane.27s_1870_Work|<b>Lane<br />(1870)</b>]]
|
! style="height: 150px; width: 310px; text-align:center; border-right:2px dashed black;" |<div align="center">{{ Math/EQ_SSLaneEmden01 }}</div>
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[SSC/Structure/Polytropes#Known_Analytic_Solutions|Known<br />Analytic<br />Solutions]]
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[SSC/Structure/Polytropes#Straight_Numerical_Integration|via<br />Direct<br />Numerical<br />Integration]]
|
! style="height: 150px; width: 150px; " |[[SSC/Structure/Polytropes#HSCF_Technique|via<br />Self-Consistent<br />Field (SCF)<br />Technique]]
|}

<span id="MoreModels">&nbsp;</span>
{| class="Chap3G" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|- 
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black; " |[[SSC/Structure/WhiteDwarfs#White_Dwarfs|<b>Zero-Temperature<br />White Dwarf</b>]]
|
! style="height: 150px; width: 150px; background-color:#ffeeee;" |[[SSC/Structure/WhiteDwarfs#Chandrasekhar_mass|Chandrasekhar<br />Limiting<br />Mass<br />(1935)]]
|}


{| class="Chap3H" style="float:right; margin-left: 150px; border-style: solid; border-width: 3px; border-color:black;"
|-
! style="height: 150px; width: 150px; "|[[SSC/Structure/Polytropes/VirialSummary|Virial Equilibrium<br />of<br />Pressure-Truncated<br />Polytropes]]
|}
{| class="Chap3I" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|- 
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black;" |<b>Pressure-Truncated<br />Configurations</b>
|
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right:2px dashed black;" |[[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere|Bonnor-Ebert<br />(Isothermal)<br />Spheres<br />(1955 - 56)]]
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres|Embedded<br />Polytropes]]
|
! style="height: 150px; width: 150px; border-right:2px dotted black;" |[[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|Equilibrium<br />Sequence<br />Turning-Points]]<br /><font color="green">&hearts;</font>
|
! style="height: 150px; width: 150px; border-right:2px dashed black; " |[[File:MassVsRadiusCombined02.png|130px|link=SSC/Stability/InstabilityOnsetOverview#Turning_Points_along_Sequences_of_Pressure-Truncated_Polytropes|Equilibrium sequences of Pressure-Truncated Polytropes]]
|
! style="height: 150px; width: 150px; " |[[Appendix/Ramblings/TurningPoints#Turning_Points|Turning-Points<br />(Broader Context)]]
|}


{| class="Chap3J" style="float:right; margin-left: 150px; border-style: solid; border-width: 3px; border-color:black;"
|-
! style="height: 150px; width: 150px; "|[[SSC/Structure/BiPolytropes/FreeEnergy51#Free_Energy_of_BiPolytrope_with|Free Energy<br />of<br />Bipolytropes]]<br /><br />(n<sub>c</sub>, n<sub>e</sub>) = (5, 1)
|}
{| class="Chap3K" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|- 
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black;" |[[SSC/Structure/BiPolytropes#BiPolytropes|<b>Composite<br />Polytropes</b>]]<br />(Bipolytropes)
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[SSC/Structure/BiPolytropes/Analytic1.53#BiPolytrope_with__and_ne_.3D_3|Milne<br />(1930)]]
|
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right:2px dashed black;" |[[SSC/Structure/LimitingMasses#Sch.C3.B6nberg-Chandrasekhar_Mass|Sch&ouml;nberg-<br />Chandrasekhar<br />Mass<br />(1942)]]
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[SSC/Structure/BiPolytropes/Analytic15#BiPolytrope_with_nc_.3D_1_and_ne_.3D_5|Murphy (1983)<br /><br /><br />Analytic]]<br />(n<sub>c</sub>, n<sub>e</sub>) = (1, 5)
|
! style="height: 150px; width: 150px; border-right:2px dashed black; " |[[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1|Eggleton, Faulkner<br />&amp; Cannon (1998)<br /><br />Analytic]]<br />(n<sub>c</sub>, n<sub>e</sub>) = (5, 1)
|
! style="height: 150px; width: 150px; " |[[Image:TurningPoints51Bipolytropes.png|150px|link=SSC/Stability/BiPolytropes#Planned_Approach|Equilibrium sequences of (5, 1) Bipolytropes]]
|}
<br />

===Stability Analysis===

{| class="Chap4A" width=100% style="margin-right: auto; margin-left: 0px; border-style: solid; border-width: 3px; border-color:darkgrey;"
|-
! style="height: 30px; width: 800px; background-color:lightgrey;"|<font color="white" size="+1">1D STABILITY</font>
|}

<font color="darkgreen"><span id="BKB74pt1">Three different approaches are used in the study of hydrodynamical stability of stars</span> and other gravitating objects &hellip; &nbsp; 
<ul><li>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.</font> 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 <math>(n_c, n_e) = (1, 5)</math>.</li>

<li>Second, one can derive <font color="darkgreen">a variational principle from the equations of small oscillations.</font>  Below, an appropriately labeled (purple) menu tile links to a chapter in which the foundation for this approach is developed.  <!-- 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> 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.</li>

<li>The third approach is what we have already referred to as a free-energy &#8212; and associated virial theorem &#8212; 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> </li>
</ul>

<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 &hellip;  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>  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.
<div align="right">--- Text in ''green'' taken directly from [http://adsabs.harvard.edu/abs/1974A%26A....31..391B G. S. Bisnovatyi-Kogan &amp; S. I. Blinnikov (1974)]; B-KB74, for short.</div>  
&nbsp;<br />
&nbsp;<br />

{| class="Chap4B" style="float:right; margin-left: 150px; border-style: solid; border-width: 3px; border-color:black;"
|-
! style="height: 150px; width: 150px; border-right:2px dotted black;" |[[File:SSC_SynopsisImage2.png|150px|link=SSC/SynopsisStyleSheet#Stability|Synopsis: Stability of Spherical Structures]]
|
! style="height: 150px; width: 150px; background-color:#9390DB;"|[[SSC/VariationalPrinciple#Ledoux.27s_Variational_Principle|Variational<br />Principle]]
|}
{| class="Chap4C" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|- 
! style="height: 150px; width: 150px; background-color:lightgreen; border-right:2px solid black;" |<b>Radial<br />Pulsation<br />Equation</b>
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29|Example<br />Derivations<br />&amp;<br />Statement of<br />Eigenvalue<br />Problem]]
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[SSC/PerspectiveReconciliation#Reconciling_Eulerian_versus_Lagrangian_Perspectives|(poor attempt at)<br />Reconciliation]]
|
! style="height: 150px; width: 150px;" |[[SSC/SoundWaves#Sound_Waves|Relationship<br />to<br />Sound Waves]]
|}
<p>&nbsp;</p>
{| class="Chap4D" style="margin-right: auto; margin-left: 230px; border-style: solid; border-width: 3px; border-color: black"
|- 
! style="height: 150px; width: 150px; border-right:2px dotted black;" |[[File:ImageOfDerivations06GoodJeansBonnor.png|120px|thumb|center|Jeans (1928) or Bonnor (1957)]]
|
! style="height: 150px; width: 150px; border-right:2px dotted black;" |[[File:ImageOfDerivations07GoodLedouxWalraven.png|120px|thumb|center|Ledoux &amp; Walraven (1958)]]
|
! style="height: 150px; width: 150px; " |[[File:ImageOfDerivations08GoodRosseland.png|120px|thumb|center|Rosseland (1969)]]
|}
<p>&nbsp;</p>
{| class="Chap4E" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|- 
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black; " |<b>Uniform-Density<br />Configurations</b>
|
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right:2px dotted black; " |[[SSC/Stability/UniformDensity#The_Stability_of_Uniform-Density_Spheres|Sterne's<br />Analytic Sol'n<br />of Eigenvalue<br />Problem<br />(1937)]]
|
! style="height: 150px; width: 150px; " |[[File:Sterne1937SolutionPlot1.png|150px|link=SSC/Stability/UniformDensity#Properties_of_Eigenfunction_Solutions|Sterne's (1937) Solution to the Eigenvalue Problem for Uniform-Density Spheres]]
|}
<p>&nbsp;</p>
{| class="Chap4F" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|- 
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black;" |[[SSC/Stability/Isothermal|<b>Pressure-Truncated<br />Isothermal<br />Spheres</b>]]
|
! style="height: 150px; width: 620px; text-align:center; border-right:2px dashed black;" |<div align="center">{{ Math/EQ_RadialPulsation03 }}</div>
|
! style="height: 150px; width: 150px; border-right:2px dotted black;" |[[SSC/Stability/Isothermal#Our_Numerical_Integration|via<br />Direct<br />Numerical<br />Integration]]
|
! style="height: 150px; width: 310px;" |[[File:TaffVanHorn1974Fundamental.gif|400px|Fundamental-Mode Eigenvectors]]
|}
<span id="MoreStabilityAnalyses">&nbsp;</span>

{| class="Chap5A" style="margin-right: auto; margin-left: 230px; border-style: solid; border-width: 3px; border-color: black"
|- 
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right:2px dotted black; " |[[SSC/Stability/InstabilityOnsetOverview#Yabushita.27s_Insight_Regarding_Stability|Yabushita's<br />Analytic Sol'n for<br />Marginally Unstable<br />Configurations<br />(1974)]]
|
! style="height: 150px; width: 310px;"|<table border="0" cellpadding="2" align="center">
<tr>
  <td align="center">
<math>\sigma_c^2 = 0 \, , ~~~~\gamma_\mathrm{g} = 1</math>
  </td>
</tr>
<tr>
  <td align="center">
&nbsp;and &nbsp;
  </td>
</tr>
<tr>
  <td align="center">
<math>x = 1 - \biggl( \frac{1}{\xi e^{-\psi}}\biggr) \frac{d\psi}{d\xi} </math>
  </td>
</tr>
</table>
|}
<p>&nbsp;</p>
{| class="Chap5B" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|- 
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black;" |[[SSC/Stability/Polytropes#Radial_Oscillations_of_Polytropic_Spheres|<b>Polytropes</b>]]
|
! style="height: 150px; width: 620px; text-align:center; border-right:2px dashed black;" |<div align="center">{{ Math/EQ_RadialPulsation02 }}</div>
|
! style="height: 150px; width: 150px; border-right:2px dotted black;" |[[SSC/Stability/n3PolytropeLAWE#Radial_Oscillations_of_n_.3D_3_Polytropic_Spheres|Isolated<br />n = 3<br />Polytrope]]
|
! style="height: 150px; width: 150px; " |[[File:Schwarzschild1941movie.gif|300px|link=SSC/Stability/n3PolytropeLAWE#SchwarzschildMovie|Schwarzschild's Modal Analysis]]
|}
<br />
{| class="Chap5C" style="float:right; margin-left: 150px; border-style: solid; border-width: 3px; border-color:black;"
|-
! style="height: 150px; width: 150px;"|[[Appendix/Ramblings/NonlinarOscillation#Radial_Oscillations_in_Pressure-Truncated_n_.3D_5_Polytropes|''Exact''<br />Demonstration<br />of<br />B-KB74<br />Conjecture]]
|
! style="height: 150px; width: 150px; border-left:2px dashed black;"|[[SSC/VariationalPrinciple#Directly_to_n_.3D_5_Polytropic_Configurations|''Exact''<br />Demonstration<br />of<br />Variational<br />Principle]]
|}
{| class="Chap5D" style="margin-right: auto; margin-left: 230px; border-style: solid; border-width: 3px; border-color: black"
|- 
! style="height: 150px; width: 150px; border-right:2px dotted black;" |[[SSC/Stability/n5PolytropeLAWE#Radial_Oscillations_of_n_.3D_5_Polytropic_Spheres|Pressure-Truncated<br />n = 5<br />Configurations]]
|
! style="height: 250px; width: 500px; " |[[File:N5Truncated2.gif|600px|n5 Truncated Movie]]
|}
<br />
{| class="Chap5E" style="margin-right: auto; margin-left: 230px; border-style: solid; border-width: 3px; border-color: black"
|- 
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right:2px dotted black; " |[[SSC/Stability/InstabilityOnsetOverview#Polytropic_Stability|Our (2017)<br />Analytic Sol'n for<br />Marginally Unstable<br />Configurations<br /><font color="green">&hearts;</font>]]
|
! style="height: 150px; width: 465px;"|<table border="0" cellpadding="2" align="center">
<tr>
  <td align="center">
<math>~\sigma_c^2 = 0 \, , ~~~~\gamma_\mathrm{g} = (n+1)/n</math>
  </td>
</tr>
<tr>
  <td align="center">
&nbsp;and &nbsp;
  </td>
</tr>
<tr>
  <td align="center">
<math>~x = \frac{3(n-1)}{2n}\biggl[1 + \biggl(\frac{n-3}{n-1}\biggr) \biggl( \frac{1}{\xi \theta^{n}}\biggr) \frac{d\theta}{d\xi}\biggr] </math>
  </td>
</tr>
</table>
|}
<p>&nbsp;</p>
<br />
{| class="Chap5F" style="float:right; margin-left: 150px; border-style: solid; border-width: 3px; border-color:black;"
|-
! style="height: 150px; width: 150px;"|[[SSC/StabilityConjecture/Bipolytrope51|B-KB74<br />Conjecture<br />RE: Bipolytrope]]<br /><br />(n<sub>c</sub>, n<sub>e</sub>) = (5, 1)
|}
{| class="Chap5G" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|- 
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black; " |<b>BiPolytropes</b>
|
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right:2px dashed black; " |[[SSC/Stability/MurphyFiedler85|Murphy &amp; Fiedler<br />(1985b)]]<br /><br />(n<sub>c</sub>, n<sub>e</sub>) = (1,5)
|
! style="height: 150px; width: 150px; " |[[SSC/Stability/BiPolytropes/HeadScratching|Our<br />Broader<br />Analysis]]
|}

===Nonlinear Dynamical Evolution===

{| class="Chap6A" width=100% style="margin-right: auto; margin-left: 0px; border-style: solid; border-width: 3px; border-color:darkgrey;"
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! style="height: 30px; width: 800px; background-color:lightgrey;"|<font color="white" size="+1">1D DYNAMICS</font>
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<p>&nbsp;</p>
{| class="Chap6B" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
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! style="height: 150px; width: 150px; background-color:#ffff99;;" |[[SSC/Dynamics/FreeFall#Free-Fall|<b>Free-Fall<br />Collapse</b>]]
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<p>&nbsp;</p>
{| class="Chap6C" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
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! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black;" |[[SSC/Dynamics/IsothermalCollapse#Collapse_of_Isothermal_Spheres|<b>Collapse of<br />Isothermal<br />Spheres</b>]]
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[http://adsabs.harvard.edu/abs/1993ApJ...416..303F via<br/>Direct<br />Numerical<br />Integration]
|
! style="height: 150px; width: 150px;" |[[SSC/Dynamics/IsothermalSimilaritySolution#Similarity_Solution|Similarity<br />Solution]]
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<p>&nbsp;</p>
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! style="height: 150px; width: 150px; background-color:#ffff99;" |[[Apps/GoldreichWeber80#Homologously_Collapsing_Stellar_Cores|<b>Collapse of<br />an Isolated<br />n = 3<br />Polytrope</b>]]
|}
<p>&nbsp;</p>

==Two-Dimensional Configurations (Axisymmetric)==
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! style="height: 50px; width: 800px; background-color:lightgrey;"|<font color="navy" size="+2">(Initially) Axisymmetric Configurations</font>
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<p>&nbsp;</p>
{| class="Chap7B" style="margin-right: auto; margin-left: 0px; border-style: solid; border-width: 3px; border-color: black;"
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! style="height: 150px; width: 150px; background-color:black;" |[[AxisymmetricConfigurations/Storyline|<font color="white">Storyline</font>]]
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<p>&nbsp;</p>
{| class="Chap7C" style="margin-right: auto; margin-left: 0px; border-style: solid; border-width: 3px; border-color: black;"
|- 
! style="height: 150px; width: 150px; background-color:lightgreen;" |[[AxisymmetricConfigurations/PGE#Axisymmetric_Configurations_.28Part_I.29|PGEs<br />for<br />Axisymmetric<br />Systems]]
|}


===Axisymmetric Equilibrium Structures===

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! style="height: 30px; width: 800px; background-color:lightgrey;"|<font color="white" size="+1">2D STRUCTURE</font>
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<p>&nbsp;</p>
{| class="Chap8B" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black;"
|- 
! style="height: 150px; width: 150px; background-color:lightgreen; border-right:2px solid black;" |[[AxisymmetricConfigurations/Equilibria|<b>Constructing<br />Steady-State<br />Axisymmetric<br />Configurations</b>]]
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[2DStructure/AxisymmetricInstabilities|Axisymmetric<br />Instabilities<br />to Avoid]]
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential|''Simple''<br />Rotation<br />Profiles]]
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[AxisymmetricConfigurations/HSCF|Hachisu Self-Consistent-Field<br />[HSCF]<br />Technique]]
|
! style="height: 150px; width: 150px; " |[[AxisymmetricConfigurations/SolvingPE#Common_Theme:_Determining_the_Gravitational_Potential_for_Axisymmetric_Mass_Distributions|Solving the<br />Poisson Equation]]
|}
<p>&nbsp;</p>
{| class="Chap8C" style="margin-right: auto; margin-left: 230px; border-style: solid; border-width: 3px; border-color: black"
|-
! style="height: 150px; width: 150px; border-right:2px dotted black;" |[[2DStructure/UsingTC#Common_Theme:_Determining_the_Gravitational_Potential_for_Axisymmetric_Mass_Distributions|Using<br />Toroidal Coordinates<br />to Determine the<br />Gravitational<br /> Potential]]
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[File:Apollonian_myway4.png|150px|link=2DStructure/ToroidalCoordinateIntegrationLimits#Mapping_from_Cylindrical_to_Toroidal_Coordinates|Apollonian Circles]]
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[2DStructure/TCsimplification#Common_Theme:_Determining_the_Gravitational_Potential_for_Axisymmetric_Mass_Distributions|Attempt at<br />Simplification<br /><br /><font color="green" size="+2">&hearts;</font>]]
|
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right:2px dotted black;" |[[Apps/WongAP#Common_Theme:_Determining_the_Gravitational_Potential_for_Axisymmetric_Mass_Distributions|Wong's<br />Analytic Potential<br />(1973)]]
|
! style="height: 150px; width: 150px; background-color:#D0FFFF;" |[[File:MovieWongN4.gif|130px|link=Apps/DysonWongTori#The_Coulomb_Potential|n = 3 contribution to potential]]
|}

====Spheroidal &amp; Spheroidal-Like====

=See Also=


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