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==Two-Dimensional Configurations (Axisymmetric)== {| class="Chap7A" 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> </p> {| class="Chap7B" style="margin-right: auto; margin-left: 0px; border-style: solid; border-width: 3px; border-color: black;" |- ! style="height: 150px; width: 150px; background-color:black;" |[[AxisymmetricConfigurations/Storyline|<font color="white">Storyline</font>]] |} <p> </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=== {| class="Chap8A" 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> </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> </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">♥</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 & Spheroidal-Like==== <br /> {| class="Chap9A" 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; " |[[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;" |[[Apps/MaclaurinSpheroids/GoogleBooks#Excerpts_from_A_Treatise_of_Fluxions|Maclaurin's<br />Original Text<br />&<br />Analysis<br />(1742)]] | ! style="height: 200px; width: 200px; border-right:2px dashed black;" |[[File:Maclaurin01.gif|282px|link=Apps/MaclaurinSpheroids/GoogleBooks#Prolate_Spheroid|Our Construction of Maclaurin's Figure 291Pt2]] | ! style="height: 150px; width: 150px; border-right:2px solid black; background-color:maroon;" |[[Apps/MaclaurinSpheroidSequence|<font color="white">Maclaurin<br />Spheroid<br />Sequence</font>]] |} <p> </p> {| class="Chap9B" 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;" |[[Apps/HayashiNaritaMiyama82#Rotationally_Flattened_Isothermal_Structures|Hayashi, Narita<br /> & Miyama's<br />Analytic Sol'n<br />(1982)]] | ! style="height: 150px; width: 150px;" |[[Apps/ReviewStahler83|Review of<br /> Stahler's (1983)<br />Sol'n Technique]] |} <p> </p> {| class="Chap9C" 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;" |[[Apps/RotatingPolytropes|Example<br />Equilibria]] |} <p> </p> {| class="Chap9D" 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;" |[[Apps/OstrikerBodenheimerLyndenBell66|Ostriker<br />Bodenheimer<br />& Lynden-Bell<br />(1966)]] | ! style="height: 150px; width: 150px;" |[[Apps/RotatingWhiteDwarfs|Example<br />Equilibria]] |} ====Toroidal & Toroidal-Like==== <br /> {| class="Chap10B" 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;" |[[Apps/PapaloizouPringleTori#Massless_Polytropic_Tori|Papaloizou-Pringle<br />Tori<br />(1984)]] | ! style="height: 150px; width: 250px; background-color:#D0FFFF;" |[[File:TorusMovie1.gif|250px|link=Apps/PapaloizouPringleTori#Boundary_Conditions|Pivoting PP Torus]] |} <p> </p> {| class="Chap10C" 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;" |[[Apps/DysonPotential|Dyson<br />(1893)]] | ! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[Apps/DWT#Common_Theme:_Determining_the_Gravitational_Potential_for_Axisymmetric_Mass_Distributions|Dyson-Wong<br />Tori]] | ! style="height: 150px; width: 150px; " |[[Apps/MaclaurinToroid|Maclaurin<br />Toroid Sequence]]<br />{{ MPT77hereafter }} |} <p> </p> {| class="Chap10D" 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;" |[[Apps/Ostriker64|Ostriker<br />(1964)]] |} <p> </p> ===Stability Analysis=== {| class="Chap11A" 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> </p> ====Sheroidal & Spheroidal-Like==== {| class="Chap11B" 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>Incompressible<br />Sequences</b> | ! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[Appendix/Ramblings/MacSphCriticalPoints|Critical Points<br />Along<br />Maclaurin Spheroid<br />Sequence]] | ! style="height: 150px; width: 150px;" |[[Apps/EriguchiHachisu/Models|Eriguchi, Hachisu<br />& Colleagues]] |} <p> </p> {| class="Chap11B" 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;" |[[Apps/RotatingPolytropes/BarmodeIncompressible|Bifurcation<br />from<br />Maclaurin<br />Sequence]] | ! style="height: 150px; width: 150px;" |[[Apps/RotatingPolytropes/BarmodeEigenvector|Traditional<br />Analyses]] |} <p> </p> * [https://ui.adsabs.harvard.edu/abs/1949ApJ...109..149C/abstract T. G. Cowling & 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 & 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 & 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 & 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%"> </td><td align="left"> <font color="green">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."</font> </td></tr></table> * [https://ui.adsabs.harvard.edu/abs/1990ApJ...355..226I/abstract J. R. Ipser & 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 & 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 & 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, & 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> </p> ====Toroidal & Toroidal-Like==== {| class="Chap11C" 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; " |[[Apps/PapaloizouPringle84#Formulation_of_Eigenvalue_Problem|<b>Defining the<br />Eigenvalue Problem</b>]] |} <p> </p> {| class="Chap11D" 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;" |[[Apps/ImamuraHadleyCollaboration#Analytic_Solution|Analytic Analysis<br />by<br />Blaes<br />(1985)]] | ! style="height: 150px; width: 150px; " |[[File:N1.5j2_Combinedsmall.png|450px|center|link=Apps/ImamuraHadleyCollaboration#Plots_of_a_Few_Example_Eigenvectors|j2 Eigenfunction from Blaes85|]] |} <p> </p> {| class="Chap11E" 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 & 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; background-color:white; border-right:2px dashed black; " |[[Apps/WoodwardTohlineHachisu94#Online_Movies|Thick<br />Accretion<br />Disks]]<br />(WTH94) | ! style="height: 150px; width: 150px;" |[[Apps/ImamuraHadleyCollaboration|Hadley &<br />Imamura<br />Collaboration]] |} <p> </p> ===Nonlinear Dynamical Evolution=== ====Sheroidal & Spheroidal-Like==== {| class="Chap12A" 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> </p> {| class="Chap12B" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black" |- ! style="height: 150px; width: 150px; background-color:#ffff99;" |[[Aps/MaclaurinSpheroidFreeFall|<b>Free-Fall<br />Collapse<br />of an<br />Homogeneous<br />Spheroid</b>]] |} <p> </p> {| class="Chap12C" 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;" |[[Apps/RotatingPolytropes/BarmodeLinearTimeDependent|Initially<br />Axisymmetric<br /> & Differentially<br />Rotating<br />Polytropes]] | ! style="height: 150px; width: 150px; background-color: black;" |[[File:Dissertation.fig3cropped.png|112px|link=Apps/RotatingPolytropes/BarmodeLinearTimeDependent|Cazes Model A Simulation]] |} <p> </p>
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