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{{LSU_HBook_header}}
{{LSU_HBook_header}}
<font size="+1">
[Preamble by Joel E. Tohline &#8212; ''Circa'' 2020]  &nbsp;In an effort to serve ongoing research activities of the astrophysics community, this media-wiki-based hypertext book (H_Book) reviews what is presently understood about the structure, stability, and dynamical evolution of (Newtonian) self-gravitating 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 &#8212; via a stability analysis &#8212; 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 &#8212; 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 self-gravitating ''incompressible'' (uniform-density) fluid systems because:  (a) in this special case, the set of governing relations is often amenable to a closed-form, analytic solution; and (b) most modern computationally assisted studies of compressible fluid systems &#8212; both in terms of their design and in the manner in which results have been interpreted &#8212; 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:
<ul>
  <li>Studies of configurations that are &#8212; at least initially &#8212; spherically symmetric.</li>
  <li>Two-dimensional configurations &#8212; such as rotationally flattened spheroidal-like or toroidal-like structures; or infinitesimally thin, but nonaxisymmetric, disk-like structures.</li>
  <li>Configurations that require a full three-dimensional treatment &#8212; most notably, "spinning" ellipsoidal-like structures; or binary systems.</li>
</ul>
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.
</font>


==Context==
==Context==


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{| class="wikitable" style="float:right; margin-left: 150px; border-style: solid; border-width: 3px; border-color:black;"
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! style="height: 150px; width: 150px; background-color:#9390DB;"|[[User:Tohline/VE#Global_Energy_Considerations|Global Energy<br />Considerations]]
|}
{| class="wikitable" style="margin-right: auto; margin-left: 0px; border-style: solid; border-width: 3px; border-color: black;"
<!-- |+ style="text-align:left; height:40px;" | <font size="+2">'''CONTEXT'''</font> -->
<!-- |+ style="text-align:left; height:40px;" | <font size="+2">'''CONTEXT'''</font> -->
|-  
|-  
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! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[User:Tohline/PGE/Euler#Euler_Equation|Euler]]
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[User:Tohline/PGE/Euler#Euler_Equation|Euler]]
|
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |1<sup>st</sup> Law of<br />Thermodynamics
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[User:Tohline/PGE/FirstLawOfThermodynamics|1<sup>st</sup> Law of<br />Thermodynamics]]
|
|
! style="height: 150px; width: 150px;"|Poisson
! style="height: 150px; width: 150px;"|[[User:Tohline/SR/PoissonOrigin#Origin_of_the_Poisson_Equation|Poisson]]
|}
|}
<p>&nbsp;</p>
<p>&nbsp;</p>
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{| class="wikitable" 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;" |[[User:Tohline/SR#Supplemental_Relations|<b>Equation<br />of State</b>]]<br />(EOS)
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black;" |[[User:Tohline/SR#Supplemental_Relations|<b>Equation<br />of State</b>]]<br />(EOS)
|
|
! style="height: 150px; width: 150px; " |[[User:Tohline/SR/PressureCombinations#Total_Pressure|Total<br />Pressure]]  
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[User:Tohline/SR/IdealGas#Ideal_Gas_Equation_of_State|Ideal Gas]]
|
! style="height: 150px; width: 150px; " |[[User:Tohline/SR/PressureCombinations#Total_Pressure|Total Pressure]] <br />&nbsp;<hr /><br />[[User:Tohline/Apps/SMS#Rotating.2C_Supermassive_Stars|Bond, Arnett, &amp; Carr<br />(1984)]]
|}
|}


==Spherically Symmetric Configurations==
==Spherically Symmetric Configurations==
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|-
! style="height: 50px; width: 800px; background-color:lightgrey;"|<font color="navy" size="+2">(Initially) Spherically Symmetric Configurations</font>
|}
<p>&nbsp;</p>


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{| class="wikitable" 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=User:Tohline/SSC/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;"|[[User:Tohline/SSC/Virial/FormFactors#PTtable|Structural<br />Form<br />Factors]]
|
! style="height: 150px; width: 150px; background-color:#9390DB; border-left:2px solid black;"|[[User:Tohline/SphericallySymmetricConfigurations/Virial#Free_Energy_Expression|Free-Energy<br />of<br />Spherical<br />Systems]]
|}
{| class="wikitable" style="margin-right: auto; margin-left: 0px; border-style: solid; border-width: 3px; border-color:black;"
|-  
|-  
! style="height: 150px; width: 150px; background-color:lightgreen;" |[[User:Tohline/SphericallySymmetricConfigurations/PGE#Spherically_Symmetric_Configurations_.28Part_I.29|One-Dimensional<br /> PGEs]]
! style="height: 150px; width: 150px; background-color:lightgreen;border-right:2px; " |[[User:Tohline/SphericallySymmetricConfigurations/PGE#Spherically_Symmetric_Configurations_.28Part_I.29|One-Dimensional<br /> PGEs]]
|}
|}




===Equilibrium Structures===
===Equilibrium Structures===
 
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{| class="wikitable" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 2px border-color: black;"
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! style="height: 30px; width: 800px; background-color:lightgrey;"|<font color="white" size="+1">1D STRUCTURE</font>
|}
<p>&nbsp;</p>
{| class="wikitable" 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=User:Tohline/SSC/Synopsis_StyleSheet#Structure|Spherical Structures Synopsis]]
|
! style="height: 150px; width: 150px; background-color:#9390DB;"|[[User:Tohline/VE#Scalar_Virial_Theorem|Scalar<br />Virial<br />Theorem]]
|}
{| class="wikitable" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black;"
<!-- |+ style="text-align:left; height:40px;" | <font size="+2">'''CONTEXT'''</font> -->
<!-- |+ style="text-align:left; height:40px;" | <font size="+2">'''CONTEXT'''</font> -->
|-  
|-  
! style="height: 150px; width: 150px; background-color:lightgreen; border-right:2px solid black;" |[[User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies#Spherically_Symmetric_Configurations_.28Part_II.29|<b>Hydrostatic<br />Balance<br />Equation</b>]]
! style="height: 150px; width: 150px; background-color:lightgreen; border-right:2px solid black;" |[[User:Tohline/SphericallySymmetricConfigurations/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">{{ User:Tohline/Math/EQ_SShydrostaticBalance01 }}</div>
! style="height: 150px; width: 310px; text-align:center; border-right:2px dashed black;" |<div align="center">{{ User:Tohline/Math/EQ_SShydrostaticBalance01 }}</div>
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|}
|}
<p>&nbsp;</p>
<p>&nbsp;</p>
{| class="wikitable" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px border-color: black"
{| class="wikitable" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|-
! style="height: 150px; width: 150px; background-color:#ffff99; " |[[User:Tohline/SSC/Structure/UniformDensity#Isolated_Uniform-Density_Sphere|<b>Uniform-Density<br />Sphere</b>]]
|}
<p>&nbsp;</p>
{| class="wikitable" 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;" |[[User:Tohline/SSC/Structure/IsothermalSphere#Isothermal_Sphere|<b>Isothermal<br />Sphere</b>]]
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black;" |[[User:Tohline/SSC/Structure/IsothermalSphere#Isothermal_Sphere|<b>Isothermal<br />Sphere</b>]]
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! style="height: 150px; width: 310px; text-align:center; border-right:2px dashed black;" |<div align="center">{{ User:Tohline/Math/EQ_SSLaneEmden02 }}</div>
! style="height: 150px; width: 310px; text-align:center; border-right:2px dashed black;" |<div align="center">{{ User:Tohline/Math/EQ_SSLaneEmden02 }}</div>
|
|
! style="height: 150px; width: 150px;" |[[User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies#Numerical|via<br />Direct<br />Numerical<br />Integration]]
! style="height: 150px; width: 150px;" |[[User:Tohline/SSC/Structure/IsothermalSphere#Our_Numerical_Integration|via<br />Direct<br />Numerical<br />Integration]]
|}
|}
<p>&nbsp;</p>
<p>&nbsp;</p>
{| class="wikitable" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px border-color: black"
{| class="wikitable" 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;" |[[User:Tohline/SSC/Structure/Polytropes#Polytropic_Spheres|<b>Isolated<br />Polytropes</b>]]
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black;" |[[User:Tohline/SSC/Structure/Polytropes#Polytropic_Spheres|<b>Isolated<br />Polytropes</b>]]
|
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right:2px dashed black;" |[[User:Tohline/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">{{ User:Tohline/Math/EQ_SSLaneEmden01 }}</div>
! style="height: 150px; width: 310px; text-align:center; border-right:2px dashed black;" |<div align="center">{{ User:Tohline/Math/EQ_SSLaneEmden01 }}</div>
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! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[User:Tohline/SSC/Structure/Polytropes#Known_Analytic_Solutions|Known<br />Analytic<br />Solutions]]
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[User:Tohline/SSC/Structure/Polytropes#Known_Analytic_Solutions|Known<br />Analytic<br />Solutions]]
|
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies#Numerical|via<br />Direct<br />Numerical<br />Integration]]
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[User:Tohline/SSC/Structure/Polytropes#Straight_Numerical_Integration|via<br />Direct<br />Numerical<br />Integration]]
|
|
! style="height: 150px; width: 150px; " |[[User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies#SCF|via<br />Self-Consistent<br />Field (SCF)<br />Technique]]
! style="height: 150px; width: 150px; " |[[User:Tohline/SSC/Structure/Polytropes#HSCF_Technique|via<br />Self-Consistent<br />Field (SCF)<br />Technique]]
|}
|}
<p>&nbsp;</p>
<p>&nbsp;</p>
{| class="wikitable" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px border-color: black"
{| class="wikitable" 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; " |[[User:Tohline/SSC/Structure/WhiteDwarfs#White_Dwarfs|<b>Zero-Temperature<br />White Dwarf</b>]]
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black; " |[[User:Tohline/SSC/Structure/WhiteDwarfs#White_Dwarfs|<b>Zero-Temperature<br />White Dwarf</b>]]
|
|
! style="height: 150px; width: 150px; background-color:#ffeeee;" |[[User:Tohline/SSC/Structure/WhiteDwarfs#Chandrasekhar_mass|Chandrasekhar<br />Limiting<br />Mass]]
! style="height: 150px; width: 150px; background-color:#ffeeee;" |[[User:Tohline/SSC/Structure/WhiteDwarfs#Chandrasekhar_mass|Chandrasekhar<br />Limiting<br />Mass<br />(1935)]]
|}
|}
<p>&nbsp;</p>
<p>&nbsp;</p>
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{| class="wikitable" style="float:right; margin-left: 150px; border-style: solid; border-width: 3px; border-color:black;"
|-
! style="height: 150px; width: 150px; "|[[User:Tohline/SSC/Structure/Polytropes/VirialSummary|Virial Equilibrium<br />of<br />Pressure-Truncated<br />Polytropes]]
|}
{| class="wikitable" 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:#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;" |[[User:Tohline/SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere|Bonnor-Ebert<br />(Isothermal)<br />Spheres]]
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right:2px dashed black;" |[[User:Tohline/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;" |[[User:Tohline/SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres|Polytropes]]
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[User:Tohline/SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres|Polytropes]]
|
|
! style="height: 150px; width: 150px; border-right:2px dotted black;" |[[User:Tohline/Appendix/Ramblings/Turning_Points#Turning_Points|Equilibrium<br />Sequence<br />Turning-Points]]
! style="height: 150px; width: 150px; border-right:2px dotted black;" |[[User:Tohline/SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|Equilibrium<br />Sequence<br />Turning-Points<br /><br /><font color="green" size="+2">&hearts;</font>]]
|
! style="height: 150px; width: 150px; border-right:2px dashed black; " |[[File:MassVsRadiusCombined02.png|130px|link=User:Tohline/SSC/Stability/InstabilityOnsetOverview#Turning_Points_along_Sequences_of_Pressure-Truncated_Polytropes|Equilibrium sequences of Pressure-Truncated Polytropes]]
|
|
! style="height: 150px; width: 150px; " |[[File:MassVsRadiusCombined02.png|150px|Equilibrium sequences of Pressure-Truncated Polytropes]]
! style="height: 150px; width: 150px; " |[[User:Tohline/Appendix/Ramblings/Turning_Points#Turning_Points|Turning-Points<br />(Broader Context)]]
|}
|}
<p>&nbsp;</p>
<p>&nbsp;</p>
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{| class="wikitable" style="float:right; margin-left: 150px; border-style: solid; border-width: 3px; border-color:black;"
|-
! style="height: 150px; width: 150px; "|[[User:Tohline/SSC/Structure/BiPolytropes/FreeEnergy5_1#Free_Energy_of_BiPolytrope_with|Free Energy<br />of<br />Bipolytropes]]<br /><br />(n<sub>c</sub>, n<sub>e</sub>) = (5, 1)
|}
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|-  
|-  
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black;" |[[User:Tohline/SSC/Structure/BiPolytropes#BiPolytropes|<b>Composite<br />Polytropes</b>]]<br />(Bipolytropes)
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black;" |[[User:Tohline/SSC/Structure/BiPolytropes#BiPolytropes|<b>Composite<br />Polytropes</b>]]<br />(Bipolytropes)
|
|
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right:2px dashed black;" |[[User:Tohline/SSC/Structure/LimitingMasses#Sch.C3.B6nberg-Chandrasekhar_Mass|Sch&ouml;nberg-<br />Chandrasekhar<br />Mass]]
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[User:Tohline/SSC/Structure/BiPolytropes/Analytic1.5_3#BiPolytrope_with__and_ne_.3D_3|Milne<br />(1930)]]
|
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1|Analytic<br /><br /><math>~(n_c, n_e)</math><br /> = <br /><math>~(5,1)</math>]]
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right:2px dashed black;" |[[User:Tohline/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;" |[[User:Tohline/SSC/Structure/BiPolytropes/Analytic1_5#BiPolytrope_with_nc_.3D_1_and_ne_.3D_5|Analytic<br /><br /><math>~(n_c, n_e)</math><br /> = <br /><math>~(1,5)</math>]]
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1|Analytic]]<br /><br />(n<sub>c</sub>, n<sub>e</sub>) = (5, 1)
|
|
! style="height: 150px; width: 150px; " |[[User:Tohline/SSC/Stability/InstabilityOnsetOverview#Turning_Points_along_Sequences_of_Pressure-Truncated_Polytropes|Equilibrium<br />Sequence<br />Turning-Points]]
! style="height: 150px; width: 150px; " |[[User:Tohline/SSC/Structure/BiPolytropes/Analytic1_5#BiPolytrope_with_nc_.3D_1_and_ne_.3D_5|Analytic]]<br /><br />(n<sub>c</sub>, n<sub>e</sub>) = (1, 5)
|}
|}
 
<p>&nbsp;</p>


===Stability Analysis===
===Stability Analysis===


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! style="height: 30px; width: 800px; background-color:lightgrey;"|<font color="white" size="+1">1D STABILITY</font>
|}
<p>&nbsp;</p>
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|-
! style="height: 150px; width: 150px; border-right:2px dotted black;" |[[File:SSC_SynopsisImage2.png|150px|link=User:Tohline/SSC/Synopsis_StyleSheet#Stability|Synopsis: Stability of Spherical Structures]]
|
! style="height: 150px; width: 150px; background-color:#9390DB;"|[[User:Tohline/SSC/VariationalPrinciple#Ledoux.27s_Variational_Principle|Variational<br />Principle]]
|}
{| class="wikitable" 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; background-color:lightgreen; border-right:2px solid black;" |<b>Radial<br />Pulsation<br />Equation</b>
|
|
! style="height: 150px; width: 150px; border-right:2px dotted black;" |[[User:Tohline/SSC/Perspective_Reconciliation#Reconciling_Eulerian_versus_Lagrangian_Perspectives|Example<br />Derivations<br />&amp;<br />Statement of<br />Eigenvalue<br />Problem]]
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[User:Tohline/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;" |[[User:Tohline/SSC/Perspective_Reconciliation#Reconciling_Eulerian_versus_Lagrangian_Perspectives|(poor attempt at)<br />Reconciliation]]
|
|
! style="height: 150px; width: 150px;" |[[User:Tohline/SSC/SoundWaves#Sound_Waves|Relationship<br />to<br />Sound Waves]]
|}
<p>&nbsp;</p>
{| class="wikitable" 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: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; border-right:2px dotted black;" |[[File:ImageOfDerivations07GoodLedouxWalraven.png|120px|thumb|center|Ledoux &amp; Walraven (1958)]]
|
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[File:ImageOfDerivations08GoodRosseland.png|120px|thumb|center|Rosseland (1969)]]
! style="height: 150px; width: 150px; " |[[File:ImageOfDerivations08GoodRosseland.png|120px|thumb|center|Rosseland (1969)]]
|
! style="height: 150px; width: 150px;" |[[User:Tohline/SSC/SoundWaves#Sound_Waves|Relationship<br />to<br />Sound Waves]]
|}
|}
<p>&nbsp;</p>
<p>&nbsp;</p>
{| class="wikitable" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px border-color: black"
{| class="wikitable" 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:#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; " |[[User:Tohline/SSC/UniformDensity#The_Stability_of_Uniform-Density_Spheres|Sterne's<br />Analytic Sol'n<br />of<br />Eigenvalue<br />Problem]]
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right:2px dotted black; " |[[User:Tohline/SSC/UniformDensity#The_Stability_of_Uniform-Density_Spheres|Sterne's<br />Analytic Sol'n<br />of<br />Eigenvalue<br />Problem<br />(1937)]]
|
|
! style="height: 150px; width: 150px; " |[[File:Sterne1937SolutionPlot1.png|150px|Equilibrium sequences of Pressure-Truncated Polytropes]]
! style="height: 150px; width: 150px; " |[[File:Sterne1937SolutionPlot1.png|150px|link=User:Tohline/SSC/UniformDensity#Properties_of_Eigenfunction_Solutions|Sterne's (1937) Solution to the Eigenvalue Problem for Uniform-Density Spheres]]
|}
|}
<p>&nbsp;</p>
<p>&nbsp;</p>
{| class="wikitable" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px border-color: black"
{| class="wikitable" 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;" |[[User:Tohline/SSC/Structure/IsothermalSphere#Isothermal_Sphere|<b>Pressure-Truncated<br />Isothermal<br />Spheres</b>]]
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black;" |[[User:Tohline/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">{{ User:Tohline/Math/EQ_RadialPulsation03 }}</div>
! style="height: 150px; width: 620px; text-align:center; border-right:2px dashed black;" |<div align="center">{{ User:Tohline/Math/EQ_RadialPulsation03 }}</div>
|
|
! style="height: 150px; width: 150px; border-right:2px dotted black;" |[[User:Tohline/SSC/Stability/Isothermal#Radial_Oscillations_of_Pressure-Truncated_Isothermal_Spheres|via<br />Direct<br />Numerical<br />Integration]]
! style="height: 150px; width: 150px; border-right:2px dotted black;" |[[User:Tohline/SSC/Stability/Isothermal#Our_Numerical_Integration|via<br />Direct<br />Numerical<br />Integration]]
|
|
! style="height: 150px; width: 310px;  border-right:2px dashed black;" |[[File:TaffVanHorn1974Fundamental.gif|280px|Fundamental-Mode Eigenvectors]]
! style="height: 150px; width: 310px;" |[[File:TaffVanHorn1974Fundamental.gif|240px|link=User:Tohline/SSC/Stability/Isothermal#CompositeDisplay2|Fundamental-Mode Eigenvectors]]
|}
<br />
{| class="wikitable" 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; " |[[User:Tohline/SSC/Stability/InstabilityOnsetOverview#Yabushita.27s_Insight_Regarding_Stability|Yabushita's<br />Analytic Sol'n<br />for<br />Marginally Unstable<br />Configurations<br />(1974)]]
|
|
! style="height: 150px; width: 150px; background-color:#ffeeee;" |[[User:Tohline/SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere|Yabushita's<br />Analytic Sol'n<br />for<br />Marginally Unstable<br />Configurations]]
! 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>
<p>&nbsp;</p>
{| class="wikitable" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px border-color: black"
{| class="wikitable" 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;" |[[User:Tohline/SSC/Structure/Polytropes#Polytropic_Spheres|<b>Polytropes</b>]]
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black;" |[[User:Tohline/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">{{ User:Tohline/Math/EQ_RadialPulsation02 }}</div>
! style="height: 150px; width: 620px; text-align:center; border-right:2px dashed black;" |<div align="center">{{ User:Tohline/Math/EQ_RadialPulsation02 }}</div>
Line 153: Line 250:
! style="height: 150px; width: 150px; border-right:2px dotted black;" |[[User:Tohline/SSC/Stability/n3PolytropeLAWE#Radial_Oscillations_of_n_.3D_3_Polytropic_Spheres|Isolated<br />n = 3<br />Polytrope]]
! style="height: 150px; width: 150px; border-right:2px dotted black;" |[[User:Tohline/SSC/Stability/n3PolytropeLAWE#Radial_Oscillations_of_n_.3D_3_Polytropic_Spheres|Isolated<br />n = 3<br />Polytrope]]
|
|
! style="height: 150px; width: 150px;  border-right:2px dashed black;" |[[File:Schwarzschild1941movie.gif|140px|Schwarzschild's Modal Analysis]]
! style="height: 150px; width: 150px;  border-right:2px dashed black;" |[[File:Schwarzschild1941movie.gif|140px|link=User:Tohline/SSC/Stability/n3PolytropeLAWE#SchwarzschildMovie|Schwarzschild's Modal Analysis]]
|
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[User:Tohline/SSC/Stability/n3PolytropeLAWE#Radial_Oscillations_of_n_.3D_3_Polytropic_Spheres|Pressure-Truncated<br />Configurations]]
! style="height: 150px; width: 150px;" |[[User:Tohline/SSC/Stability/n5PolytropeLAWE#Radial_Oscillations_of_n_.3D_5_Polytropic_Spheres|Pressure-Truncated<br />n = 5<br />Configurations]]
|}
<br />
{| class="wikitable" style="float:right; margin-left: 150px; border-style: solid; border-width: 3px; border-color:black;"
|-
! style="height: 150px; width: 150px;"|[[User:Tohline/Appendix/Ramblings/Nonlinar_Oscillation#Radial_Oscillations_in_Pressure-Truncated_n_.3D_5_Polytropes|''Exact''<br />Demonstration<br />of<br />B-KB74<br />Conjecture]]
|
|
! style="height: 150px; width: 150px; background-color:#ffeeee;" |[[User:Tohline/SSC/Stability/InstabilityOnsetOverview#Analyses_of_Radial_Oscillations|Our<br />Analytic Sol'n<br />for<br />Marginally Unstable<br />Configurations]]
! style="height: 150px; width: 150px; border-left:2px dashed black;"|[[User:Tohline/SSC/VariationalPrinciple#Directly_to_n_.3D_5_Polytropic_Configurations|''Exact''<br />Demonstration<br />of<br />Variational<br />Principle]]
|
! style="height: 150px; width: 150px; background-color:#ffff99; border-left:2px solid black;"|Pressure-Truncated<br />n = 5<br />Polytropes
|}
{| class="wikitable" 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; " |[[User:Tohline/SSC/Stability/InstabilityOnsetOverview#Analyses_of_Radial_Oscillations|Our Analytic Sol'n<br />for<br />Marginally Unstable<br />Configurations<br />(2017)<br /><font color="green" size="+2">&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>
{| class="wikitable" 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; " |[[User:Tohline/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; " |[[User:Tohline/SSC/Stability/BiPolytropes#Marginally_Unstable_Bipolytropes|Our<br />Broader<br />Analysis]]
|}
|}
<p>&nbsp;</p>
<p>&nbsp;</p>


===Nonlinear Dynamical Evolution===
===Nonlinear Dynamical Evolution===


{| class="wikitable" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px border-color: black"
{| class="wikitable" 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 DYNAMICS</font>
|}
<p>&nbsp;</p>
{| class="wikitable" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|-  
|-  
! style="height: 150px; width: 150px; background-color:#ffff99;;" |[[User:Tohline/SSC/FreeFall#Free-Fall_Collapse|<b>Free-Fall<br />Collapse</b>]]
! style="height: 150px; width: 150px; background-color:#ffff99;;" |[[User:Tohline/SSC/FreeFall#Free-Fall|<b>Free-Fall<br />Collapse</b>]]
|}
|}
<p>&nbsp;</p>
<p>&nbsp;</p>
{| class="wikitable" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px border-color: black"
{| class="wikitable" 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>Collapse of<br />Isothermal<br />Spheres</b>
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black;" |[[User:Tohline/SSC/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; 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;" |Self-Similar<br />Solution
! style="height: 150px; width: 150px;" |[[User:Tohline/SSC/IsothermalSimilaritySolution#Similarity_Solution|Similarity<br />Solution]]
|}
|}
<p>&nbsp;</p>
<p>&nbsp;</p>
{| class="wikitable" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px border-color: black"
{| class="wikitable" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|-  
|-  
! style="height: 150px; width: 150px; background-color:#ffff99;" |[[User:Tohline/Apps/GoldreichWeber80#Homologously_Collapsing_Stellar_Cores|<b>Collapse of<br />an Isolated<br />n = 3<br />Polytrope</b>]]
! style="height: 150px; width: 150px; background-color:#ffff99;" |[[User:Tohline/Apps/GoldreichWeber80#Homologously_Collapsing_Stellar_Cores|<b>Collapse of<br />an Isolated<br />n = 3<br />Polytrope</b>]]
|}
|}
<p>&nbsp;</p>
<p>&nbsp;</p>
==Two-Dimensional Configurations (Axisymmetric)==
{| class="wikitable" 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) Axisymmetric Configurations</font>
|}
<p>&nbsp;</p>
{| class="wikitable" style="margin-right: auto; margin-left: 0px; border-style: solid; border-width: 3px; border-color: black;"
|-
! style="height: 150px; width: 150px; background-color:black;" |[[User:Tohline/AxisymmetricConfigurations/Storyline|<font color="white">Storyline</font>]]
|}
<p>&nbsp;</p>
{| class="wikitable" style="margin-right: auto; margin-left: 0px; border-style: solid; border-width: 3px; border-color: black;"
|-
! style="height: 150px; width: 150px; background-color:lightgreen;" |[[User:Tohline/AxisymmetricConfigurations/PGE#Axisymmetric_Configurations_.28Part_I.29|PGEs<br />for<br />Axisymmetric<br />Systems]]
|}
===Axisymmetric Equilibrium Structures===
{| class="wikitable" 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">2D STRUCTURE</font>
|}
<p>&nbsp;</p>
{| class="wikitable" 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;" |[[User:Tohline/AxisymmetricConfigurations/Equilibria|<b>Constructing<br />Steady-State<br />Axisymmetric<br />Configurations</b>]]
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[User:Tohline/2DStructure/AxisymmetricInstabilities|Axisymmetric<br />Instabilities<br />to Avoid]]
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[User:Tohline/AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential|''Simple''<br />Rotation<br />Profiles]]
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[User:Tohline/AxisymmetricConfigurations/HSCF|Hachisu Self-Consistent-Field<br />[HSCF]<br />Technique]]
|
! style="height: 150px; width: 150px; " |[[User:Tohline/AxisymmetricConfigurations/SolvingPE#Common_Theme:_Determining_the_Gravitational_Potential_for_Axisymmetric_Mass_Distributions|Solving the<br />Poisson Equation]]
|}
<p>&nbsp;</p>
{| class="wikitable" 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;" |[[User:Tohline/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=User:Tohline/2DStructure/ToroidalCoordinateIntegrationLimits#Mapping_from_Cylindrical_to_Toroidal_Coordinates|Apollonian Circles]]
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[User:Tohline/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;" |[[User:Tohline/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=User:Tohline/Apps/DysonWongTori#The_Coulomb_Potential|n = 3 contribution to potential]]
|}
====Spheroidal &amp; Spheroidal-Like====
{| class="wikitable" 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; " |[[User:Tohline/Apps/MaclaurinSpheroids#Maclaurin_Spheroids_.28axisymmetric_structure.29|<b>Uniform-Density<br />(Maclaurin)<br />Spheroids</b>]]
|
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right:2px dotted black;" |[[User:Tohline/Apps/MaclaurinSpheroids/GoogleBooks#Excerpts_from_A_Treatise_of_Fluxions|Maclaurin's<br />Original Text<br />&amp;<br />Analysis<br />(1742)]]
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[File:Maclaurin01.gif|150px|center|link=User:Tohline/Apps/MaclaurinSpheroids/GoogleBooks#Prolate_Spheroid|Our Construction of Maclaurin's Figure 291Pt2]]
|
! style="height: 150px; width: 150px; background-color:maroon;" |[[User:Tohline/Apps/MaclaurinSpheroidSequence|<font color="white">Maclaurin<br />Spheroid<br />Sequence</font>]]
|}
<p>&nbsp;</p>
{| class="wikitable" 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>Rotationally<br />Flattened<br />Isothermal<br />Configurations</b>
|
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right:2px dashed black;" |[[User:Tohline/Apps/HayashiNaritaMiyama82#Rotationally_Flattened_Isothermal_Structures|Hayashi, Narita<br /> &amp; Miyama's<br />Analytic Sol'n<br />(1982)]]
|
! style="height: 150px; width: 150px;" |[[User:Tohline/Apps/ReviewStahler83|Review of<br /> Stahler's (1983)<br />Technique]]
|}
<p>&nbsp;</p>
{| class="wikitable" 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>Rotationally<br />Flattened<br />Polytropes</b>
|
! style="height: 150px; width: 150px;" |[[User:Tohline/Apps/RotatingPolytropes|Example<br />Equilibria]]
|}
<p>&nbsp;</p>
{| class="wikitable" 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>Rotationally<br />Flattened<br />White Dwarfs</b>
|
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right:2px dashed black;" |[[User:Tohline/Apps/OstrikerBodenheimerLyndenBell66|Ostriker<br />Bodenheimer<br />&amp; Lynden-Bell<br />(1966)]]
|
! style="height: 150px; width: 150px;" |[[User:Tohline/Apps/RotatingWhiteDwarfs|Example<br />Equilibria]]
|}
====Toroidal &amp; Toroidal-Like====
{| class="wikitable" style="margin-right: auto; margin-left: 75px;"
|-
! style="height: 25px;" |Definition:  [http://www.mathematicsdictionary.com/english/vmd/full/t/torusanchorring.htm anchor ring]
|}
{| class="wikitable" 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>Massless<br />Polytropic<br />Configurations</b>
|
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right:2px dotted black;" |[[User:Tohline/Apps/PapaloizouPringleTori#Massless_Polytropic_Tori|Papaloizou-Pringle<br />Tori<br />(1984)]]
|
! style="height: 150px; width: 150px; background-color:#D0FFFF;" |[[File:TorusMovie1.gif|150px|link=User:Tohline/Apps/PapaloizouPringleTori#Boundary_Conditions|Pivoting PP Torus]]
|}
<p>&nbsp;</p>
{| class="wikitable" 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>Self-Gravitating<br />Incompressible<br />Configurations</b>
|
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right:2px dashed black;" |[[User:Tohline/Apps/DysonPotential|Dyson<br />(1893)]]
|
! style="height: 150px; width: 150px; " |[[User:Tohline/Apps/DWT#Common_Theme:_Determining_the_Gravitational_Potential_for_Axisymmetric_Mass_Distributions|Dyson-Wong<br />Tori]]
|}
<p>&nbsp;</p>
{| class="wikitable" 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>Self-Gravitating<br />Compressible<br />Configurations</b>
|
! style="height: 150px; width: 150px;" |[[User:Tohline/Apps/Ostriker64|Ostriker<br />(1964)]]
|}
<p>&nbsp;</p>
===Stability Analysis===
{| class="wikitable" 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">2D STABILITY</font>
|}
<p>&nbsp;</p>
====Sheroidal &amp; Spheroidal-Like====
{| class="wikitable" 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>Linear<br />Analysis<br /> of<br />Bar-Mode<br />Instability</b>
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[User:Tohline/Apps/RotatingPolytropes/BarmodeIncompressible|Bifurcation<br />from<br />Maclaurin<br />Sequence]]
|
! style="height: 150px; width: 150px;" |[[User:Tohline/Apps/RotatingPolytropes/BarmodeEigenvector|Traditional<br />Analyses]]
|}
<p>&nbsp;</p>
* [https://ui.adsabs.harvard.edu/abs/1949ApJ...109..149C/abstract T. G. Cowling &amp; R. A. Newing (1949)], ApJ, 109, 149:  ''The Oscillations of a Rotating Star''
* [https://ui.adsabs.harvard.edu/abs/1965ApJ...141..210C/abstract M. J. Clement (1965)], ApJ, 141, 210:  ''The Radial and Non-Radial Oscillations of Slowly Rotating Gaseous Masses''
* [https://ui.adsabs.harvard.edu/abs/1963ApJ...137..777R/abstract P. H. Roberts &amp; K. Stewartson (1963)], ApJ, 137, 777:  ''On the Stability of a Maclaurin spheroid of small viscosity''
* [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..825R/abstract C. E. Rosenkilde (1967)], ApJ, 148, 825:  ''The tensor virial-theorem including viscous stress and the oscillations of a Maclaurin spheroid''
* [https://ui.adsabs.harvard.edu/abs/1968ApJ...152..267C/abstract S. Chandrasekhar &amp; N. R. Lebovitz (1968)], ApJ, 152, 267:  ''The Pulsations and the Dynamical Stability of Gaseous Masses in Uniform Rotation''
* [https://ui.adsabs.harvard.edu/abs/1977ApJ...213..497H/abstract C. Hunter (1977)], ApJ, 213, 497:  ''On Secular Stability, Secular Instability, and Points of Bifurcation of Rotating Gaseous Masses''
* [https://ui.adsabs.harvard.edu/abs/1985ApJ...294..474I/abstract J. N. Imamura, J. L. Friedman &amp; R. H. Durisen (1985)], ApJ, 294, 474:  ''Secular stability limits for rotating polytropic stars''
<table border="0" align="center" width="100%" cellpadding="1"><tr>
<td align="center" width="5%">&nbsp;</td><td align="left">
<font color="green">The equilibrium models are calculated using the polytrope version (Bodenheimer &amp; Ostriker 1973) of the Ostriker and Mark (1968) self-consistent field (SCF) code &hellip; the equilibrium models rotate on cylinders and are completely specified by <math>~n</math>, the total angular momentum, and the specific angular momentum distribution <math>~j(m_\varpi)</math>.  Here <math>~m_\varpi</math> is the mass contained within a cylinder of radius <math>~\varpi</math> centered on the rotation axis.  The angular momentum distribution is prescribed in several ways:  (1) imposing strict uniform rotation;  (2) using the same <math>~j(m_\varpi)</math> as that of a uniformly rotating spherical polybrope of index <math>~n^'</math> (see Bodenheimer and Ostriker 1973); and (3) using <math>~j(m_\varpi) \propto m_\varpi</math>, which we refer to as <math>~n^' = L</math>, <math>~L</math> for "linear."</font>
</td></tr></table>
* [https://ui.adsabs.harvard.edu/abs/1990ApJ...355..226I/abstract J. R. Ipser &amp; L. Lindblom (1990)], ApJ, 355, 226:  ''The Oscillations of Rapidly Rotating Newtonian Stellar Models''
* [https://ui.adsabs.harvard.edu/abs/1991ApJ...373..213I/abstract J. R. Ipser &amp; L. Lindblom (1991)], ApJ, 373, 213:  ''The Oscillations of Rapidly Rotating Newtonian Stellar Models.  II. Dissipative Effects''
* [https://ui.adsabs.harvard.edu/abs/2000ApJ...528..946I/abstract J. N. Imamura, J. L. Friedman &amp; 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''
* [https://ui.adsabs.harvard.edu/abs/2003MNRAS.343..619S/abstract M. Shibata, S. Karino, &amp; Y. Eriguchi (2003)], MNRAS, 343, 619 - 626:  ''Dynamical bar-mode instability of differentially rotating stars: effects of equations of state and velocity profiles''
* [https://ui.adsabs.harvard.edu/abs/2019ApJ...877....9H/abstract G. P. Horedt (2019)], ApJ, 877, 9:  ''On the Instability of Polytropic Maclaurin and Roche ellipsoids''
<p>&nbsp;</p>
====Toroidal &amp; Toroidal-Like====
{| class="wikitable" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black;"
<!-- |+ style="text-align:left; height:40px;" | <font size="+2">'''CONTEXT'''</font> -->
|-
! style="height: 150px; width: 150px; background-color:lightgreen; " |[[User:Tohline/Apps/PapaloizouPringle84#Nonaxisymmetric_Instability_in_Papaloizou-Pringle_Tori|<b>Defining the<br />Eigenvalue Problem</b>]]
|}
<p>&nbsp;</p>
{| class="wikitable" 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>(Massless)<br />Papaloizou-Pringle<br />Tori</b>
|
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right:2px dotted black;" |[[User:Tohline/Apps/ImamuraHadleyCollaboration#Characteristics_of_Unstable_Eigenvectors_in_Self-Gravitating_Tori|Analytic Analysis<br />by<br />Blaes<br />(1985)]]
|
! style="height: 150px; width: 150px; " |[[File:N1.5j2_Combinedsmall.png|450px|center|link=User:Tohline/Apps/ImamuraHadleyCollaboration#Plots_of_a_Few_Example_Eigenvectors|j2 Eigenfunction from Blaes85|]]
|}
<p>&nbsp;</p>
{| class="wikitable" 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>Self-Gravitating<br />Polytropic<br />Rings</b>
|
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right:2px dotted black;" |[https://ui.adsabs.harvard.edu/abs/1990ApJ...361..394T/abstract Numerical Analysis<br />by<br />Tohline &amp; Hachisu<br />(1990)]
|
! style="height: 150px; width: 150px; background-color:black; border-right:2px dashed black; " |[[File:Minitorus.animated.gif|150px|center|PP torus instability]]
|
! style="height: 150px; width: 150px;" |[[User:Tohline/Apps/WoodwardTohlineHachisu94#Online_Movies|Thick<br />Accretion<br />Disks]]<br />(WTH94)
|}
<p>&nbsp;</p>
===Nonlinear Dynamical Evolution===
====Sheroidal &amp; Spheroidal-Like====
{| class="wikitable" 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">2D DYNAMICS</font>
|}
<p>&nbsp;</p>
{| class="wikitable" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|-
! style="height: 150px; width: 150px; background-color:#ffff99;;" |[[User:Tohline/Aps/MaclaurinSpheroidFreeFall|<b>Free-Fall<br />Collapse<br />of an<br />Homogeneous<br />Spheroid</b>]]
|}
<p>&nbsp;</p>
{| class="wikitable" 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>Nonlinear<br />Development of<br />Bar-Mode</b>
|
! style="height: 150px; width: 150px; border-right:2px dotted black;" |[[User:Tohline/Apps/RotatingPolytropes/BarmodeLinearTimeDependent|Initially<br />Axisymmetric<br /> &amp; Differentially<br />Rotating<br />Polytropes]]
|
! style="height: 150px; width: 150px; background-color: black;" |[[File:Dissertation.fig3.jpg|112px|link=User:Tohline/Apps/RotatingPolytropes/BarmodeLinearTimeDependent|Cazes Model A Simulation]]
|}
<p>&nbsp;</p>
==Two-Dimensional Configurations (Nonaxisymmetric Disks)==
{| class="wikitable" 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">Infinitesimally Thin, Nonaxisymmetric Disks</font>
|}
<p>&nbsp;</p>
{| class="wikitable" 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">2D STRUCTURE</font>
|}
<p>&nbsp;</p>
{| class="wikitable" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black;"
<!-- |+ style="text-align:left; height:40px;" | <font size="+2">'''CONTEXT'''</font> -->
|-
! style="height: 150px; width: 150px; background-color:lightgreen; " |[[User:Tohline/Apps/Korycansky_Papaloizou_1996#Korycansky_and_Papaloizou_.281996.29|<b>Constructing<br />Infinitesimally Thin<br />Nonaxisymmetric<br />Disks</b>]]
|}
<p>&nbsp;</p>
==Three-Dimensional Configurations==
{| class="wikitable" 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) Three-Dimensional Configurations</font>
|}
<p>&nbsp;</p>
===Equilibrium Structures===
{| class="wikitable" 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">3D STRUCTURE</font>
|}
<!--
<table border="0" cellpadding="3" align="center" width="60%">
<tr><td align="left">
<font color="darkgreen">"One interesting aspect of our models &hellip; is the pulsation characteristic of the final central triaxial figure &hellip; our interest in the pulsations  stems from a general concern about the equilibrium structure of self-gravitating, triaxial objects.  In the past, attempts to construct hydrostatic models of any equilibrium, triaxial structure having both a high <math>~T/|W|</math> value and a compressible equation of state have met with very limited success &hellip; they have been thwarted by a lack of understanding of how to represent complex internal motions in a physically realistic way&hellip; We suggest &hellip; 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.</font>
</td></tr>
<tr><td align="right">
&#8212; Drawn from &sect;IVa of [https://ui.adsabs.harvard.edu/abs/1988ApJ...334..449W/abstract Williams &amp; Tohline (1988)], ApJ, 334, 449
</td></tr></table>
-->
<table border="0" cellpadding="3" align="center" width="60%">
<tr><td align="left">
Special numerical techniques must be developed <font color="darkgreen">"to build three-dimensional compressible equilibrium models with complicated flows."</font>  To date &hellip; <font color="darkgreen">"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 ([https://ui.adsabs.harvard.edu/abs/1984PASJ...36..239H/abstract Hachisu &amp; Eriguchi 1984]; [https://ui.adsabs.harvard.edu/abs/1986ApJS...62..461H/abstract Hachisu 1986)], irrotational systems ([https://ui.adsabs.harvard.edu/abs/1998ApJS..118..563U/abstract Ury&#x016B; &amp; Eriguchi 1998]), and configurations that are stationary in the inertial frame ([https://ui.adsabs.harvard.edu/abs/1996MNRAS.282..653U/abstract Ury&#x016B; &amp; Eriguchi 1996])."</font>
</td></tr>
<tr><td align="right">
&#8212; Drawn from &sect;1 of [https://ui.adsabs.harvard.edu/abs/2006ApJ...639..549O/abstract Ou (2006)], ApJ, 639, 549
</td></tr></table>
====Ellipsoidal &amp; Ellipsoidal-Like====
{| class="wikitable" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black;"
<!-- |+ style="text-align:left; height:40px;" | <font size="+2">'''CONTEXT'''</font> -->
|-
! style="height: 150px; width: 150px; background-color:lightgreen; " |[https://ui.adsabs.harvard.edu/abs/2006ApJ...639..549O/abstract <b>Constructing<br />Ellipsoidal<br /> &amp; Ellipsoidal-Like<br />Configurations</b>]
|}
<p>&nbsp;</p>
{| class="wikitable" style="float:right; margin-left: 150px; border-style: solid; border-width: 3px; border-color:black;"
|-
! style="height: 150px; width: 150px; background-color:#9390DB;"|[[User:Tohline/VE/RiemannEllipsoids|Steady-State<br />2<sup>nd</sup>-Order<br />Tensor Virial<br />Equations]]
|}
{| class="wikitable" 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 />Incompressible<br />Ellipsoids</b>
|
! style="height: 150px; width: 150px; background-color:white; border-right: 2px dashed black; " |[[User:Tohline/ThreeDimensionalConfigurations/HomogeneousEllipsoids|<b>The<br />Gravitational<br />Potential</b>]]<p></p>(A<sub>i</sub> coefficients)
|
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right: 2px dashed black; " |[[User:Tohline/ThreeDimensionalConfigurations/JacobiEllipsoids#Jacobi_Ellipsoids|<b>Jacobi<br />Ellipsoids</b>]]
|
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right: 2px dashed black;" |[[User:Tohline/ThreeDimensionalConfigurations/RiemannStype#Riemann_S-type_Ellipsoids|<b>Riemann<br />S-Type<br />Ellipsoids</b>]]
|
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right: 2px dashed black;" |[[User:Tohline/ThreeDimensionalConfigurations/RiemannTypeI|<b>Type I<br />Riemann<br />Ellipsoids</b>]]
|
! style="height: 150px; width: 150px; background-color:white;" |[[User:Tohline/ThreeDimensionalConfigurations/MeetsCOLLADAandOculusRiftS|<b>Riemann<br />meets<br />COLLADA<br />&amp;<br /> Oculus Rift S</b>]]
|}
<p>&nbsp;</p>
{| class="wikitable" style="margin-right: auto; margin-left: 550px; border-style: solid; border-width: 3px; border-color: black;"
|-
! style="height: 150px; width: 150px; background-color:white; border-right: 2px dashed black; " |[https://www.aimspress.com/article/10.3934/math.2019.2.215/fulltext.html <b>A Gauge Theory<br />of<br />Riemann Ellipsoids</b>]
|
! style="height: 150px; width: 150px; background-color:white; " |[https://ui.adsabs.harvard.edu/abs/2020PhRvL.124e2501S/abstract <b>Nuclear<br />Wobbling Motion</b>]
|}
<p>&nbsp;</p>
{| class="wikitable" 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>Compressible<br />Analogs of<br />Riemann Ellipsoids</b>
|
! style="height: 150px; width: 150px; background-color:pink; border-right: 2px dashed black;" |[[User:Tohline/ThreeDimensionalConfigurations/FerrersPotential|<b>Ferrers<br />Potential<br />(1877)</b>]]
|
! style="height: 150px; width: 150px; background-color:white;" |[[User:Tohline/ThreeDimensionalConfigurations/CAREs|<b>Thoughts<br />&amp;<br />Challenges</b>]]
|}
<p>&nbsp;</p>
* [https://ui.adsabs.harvard.edu/abs/1985Ap.....23..654K/abstract B. P. Kondrat'ev (1985)], Astrophysics, 23, 654:  ''Irrotational and zero angular momentum ellipsoids in the Dirichlet problem''
* [https://ui.adsabs.harvard.edu/abs/1993ApJS...88..205L/abstract D. Lai, F. A. Rasio &amp; S. L. Shapiro (1993)], ApJS, 88, 205:  ''Ellipsoidal Figures of Equilibrium:  Compressible models''
====Binary Systems====
* [https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..539C/abstract S. Chandrasekhar (1933)], MNRAS, 93, 539:  ''The equilibrium of distorted polytropes.  IV. the rotational and the tidal distortions as functions of the density distribution''
* [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1182C/abstract S. Chandrasekhar (1963)], ApJ, 138, 1182: ''The Equilibrium and the Stability of the Roche Ellipsoids''
<table border="0" align="center" width="100%" cellpadding="1"><tr>
<td align="center" width="5%">&nbsp;</td><td align="left">
<font color="green">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.</font>
</td></tr></table>
<p>&nbsp;</p>
===Stability Analysis===
{| class="wikitable" 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">3D STABILITY</font>
|}
====Ellipsoidal &amp; Ellipsoidal-Like====
<p>&nbsp;</p>
====Binary Systems====
* [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1182C/abstract S. Chandrasekhar (1963)], ApJ, 138, 1182:  ''The Equilibrium and the Stability of the Roche Ellipsoids''
* [https://ui.adsabs.harvard.edu/abs/2019ApJ...877....9H/abstract G. P. Horedt (2019)], ApJ, 877, 9:  ''On the Instability of Polytropic Maclaurin and Roche Ellipsoids''
===Nonlinear Evolution===
{| class="wikitable" 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">3D DYNAMICS</font>
|}
<p>&nbsp;</p>
{| class="wikitable" style="float:right; margin-left: 150px; border-style: solid; border-width: 3px; border-color:black;"
|-
! style="height: 150px; width: 150px; background-color:#D0FFFF;"|[[File:JacobiMaclaurin2.gif|300px|link=User:Tohline/ThreeDimensionalConfigurations/EFE_Energies#Animation|Animation related to Fig. 3 from Christodoulou1995]]
|
! style="height: 150px; width: 150px; background-color:#9390DB; border-left:2px solid black;"|[[User:Tohline/ThreeDimensionalConfigurations/EFE_Energies#Properties_of_Homogeneous_Ellipsoids_.282.29|Free-Energy<br />Evolution<br />from the Maclaurin<br />to the Jacobi<br />Sequence]]
|}
{| class="wikitable" 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; " |[[User:Tohline/ThreeDimensionalConfigurations/BinaryFission#Fission_Hypothesis_of_Binary_Star_Formation|<b>Fission<br />Hypothesis</b>]]
|
! style="height: 150px; width: 150px; background-color:white;" |[http://www.phys.lsu.edu/~tohline/fission.movies.html <b>"Fission"<br />Simulations<br />at LSU</b>]
|}
<p>&nbsp;</p>
====Secular====
* [https://ui.adsabs.harvard.edu/abs/1971ApJ...170..143F/abstract M. Fujimoto (1971)], ApJ, 170, 143:  ''Nonlinear Motions of Rotating Gaseous Ellipsoids''
* [https://ui.adsabs.harvard.edu/abs/1973ApJ...181..513P/abstract W. H. Press &amp; S. A. Teukolsky (1973)], ApJ, 181, 513:  ''On the Evolution of the Secularly Unstable, Viscous Maclaurin Spheroids''
* [https://ui.adsabs.harvard.edu/abs/1977ApJ...213..193D/abstract S. L. Detweiler &amp; L. Lindblom (1977)], ApJ, 213, 193:  ''On the evolution of the homogeneous ellipsoidal figures.''
====Dynamical====


=See Also=
=See Also=

Latest revision as of 17:27, 28 May 2021


Tiled Menu

Whitworth's (1981) Isothermal Free-Energy Surface
|   Tiled Menu   |   Tables of Content   |  Banner Video   |  Tohline Home Page   |

[Preamble by Joel E. Tohline — Circa 2020]  In an effort to serve ongoing research activities of the astrophysics community, this media-wiki-based hypertext book (H_Book) reviews what is presently understood about the structure, stability, and dynamical evolution of (Newtonian) self-gravitating 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 self-gravitating incompressible (uniform-density) fluid systems because: (a) in this special case, the set of governing relations is often amenable to a closed-form, 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.
  • Two-dimensional configurations — such as rotationally flattened spheroidal-like or toroidal-like structures; or infinitesimally thin, but nonaxisymmetric, disk-like structures.
  • Configurations that require a full three-dimensional treatment — most notably, "spinning" ellipsoidal-like 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 1st Law of
Thermodynamics
Poisson

 

Equation
of State

(EOS)
Ideal Gas Total Pressure
 

Bond, Arnett, & Carr
(1984)

Spherically Symmetric Configurations

(Initially) Spherically Symmetric Configurations

 

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


Equilibrium Structures

1D STRUCTURE

 

Spherical Structures Synopsis Scalar
Virial
Theorem
Hydrostatic
Balance
Equation

LSU Key.png

<math>~\frac{dP}{dr} = - \frac{GM_r \rho}{r^2}</math>

Solution
Strategies

 

Uniform-Density
Sphere

 

Isothermal
Sphere

LSU Key.png

<math>~\frac{1}{\xi^2} \frac{d}{d\xi}\biggl( \xi^2 \frac{d\psi}{d\xi} \biggr) = e^{-\psi}</math>

via
Direct
Numerical
Integration

 

Isolated
Polytropes
Lane
(1870)

LSU Key.png

<math>~\frac{1}{\xi^2} \frac{d}{d\xi}\biggl( \xi^2 \frac{d\Theta_H}{d\xi} \biggr) = - \Theta_H^n</math>

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)
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)
Analytic

(nc, ne) = (5, 1)
Analytic

(nc, ne) = (1, 5)

 

Stability Analysis

1D STABILITY

 

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

LSU Key.png

<math>~0 = \frac{d^2x}{d\xi^2} + \biggl[4 - \xi \biggl( \frac{d\psi}{d\xi} \biggr) \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi} + \biggl[ \biggl( \frac{\sigma_c^2}{6\gamma_\mathrm{g}}\biggr)\xi^2 - \alpha \xi \biggl( \frac{d\psi}{d\xi} \biggr) \biggr] \frac{x}{\xi^2} </math>

where:    <math>~\sigma_c^2 \equiv \frac{3\omega^2}{2\pi G\rho_c}</math>     and,     <math>~\alpha \equiv \biggl(3 - \frac{4}{\gamma_\mathrm{g}}\biggr)</math>

via
Direct
Numerical
Integration
Fundamental-Mode Eigenvectors


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

<math>~\sigma_c^2 = 0 \, , ~~~~\gamma_\mathrm{g} = 1</math>

 and  

<math>~x = 1 - \biggl( \frac{1}{\xi e^{-\psi}}\biggr) \frac{d\psi}{d\xi} </math>

 

Polytropes

LSU Key.png

<math>~0 = \frac{d^2x}{d\xi^2} + \biggl[ 4 - (n+1) Q \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi} + (n+1) \biggl[ \biggl( \frac{\sigma_c^2}{6\gamma_g } \biggr) \frac{\xi^2}{\theta} - \alpha Q\biggr] \frac{x}{\xi^2} </math>

where:    <math>~Q(\xi) \equiv - \frac{d\ln\theta}{d\ln\xi} \, ,</math>    <math>~\sigma_c^2 \equiv \frac{3\omega^2}{2\pi G\rho_c} \, ,</math>     and,     <math>~\alpha \equiv \biggl(3 - \frac{4}{\gamma_\mathrm{g}}\biggr)</math>

Isolated
n = 3
Polytrope
Schwarzschild's Modal Analysis Pressure-Truncated
n = 5
Configurations


Exact
Demonstration
of
B-KB74
Conjecture
Exact
Demonstration
of
Variational
Principle
Pressure-Truncated
n = 5
Polytropes
Our Analytic Sol'n
for
Marginally Unstable
Configurations
(2017)

<math>~\sigma_c^2 = 0 \, , ~~~~\gamma_\mathrm{g} = (n+1)/n</math>

 and  

<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>

 

BiPolytropes Murphy & Fiedler
(1985b)


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

 

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)
Technique

 

Rotationally
Flattened
Polytropes
Example
Equilibria

 

Rotationally
Flattened
White Dwarfs
Ostriker
Bodenheimer
& Lynden-Bell
(1966)
Example
Equilibria

Toroidal & Toroidal-Like

Definition: anchor ring
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 <math>~n</math>, the total angular momentum, and the specific angular momentum distribution <math>~j(m_\varpi)</math>. Here <math>~m_\varpi</math> is the mass contained within a cylinder of radius <math>~\varpi</math> centered on the rotation axis. The angular momentum distribution is prescribed in several ways: (1) imposing strict uniform rotation; (2) using the same <math>~j(m_\varpi)</math> as that of a uniformly rotating spherical polybrope of index <math>~n^'</math> (see Bodenheimer and Ostriker 1973); and (3) using <math>~j(m_\varpi) \propto m_\varpi</math>, which we refer to as <math>~n^' = L</math>, <math>~L</math> 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)

 

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 Ou (2006), ApJ, 639, 549

Ellipsoidal & Ellipsoidal-Like

Constructing
Ellipsoidal
& Ellipsoidal-Like
Configurations

 

Steady-State
2nd-Order
Tensor Virial
Equations
Uniform-Density
Incompressible
Ellipsoids
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)
Thoughts
&
Challenges

 

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 & Ellipsoidal-Like

 

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

Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation