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20:24, 21 June 2024 Joel2 talk contribs created page File:Wong73VaryingRoverd.gif
20:24, 21 June 2024 Joel2 talk contribs uploaded File:Wong73VaryingRoverd.gif
20:23, 21 June 2024 Joel2 talk contribs created page File:MovieWongComposite.gif
20:23, 21 June 2024 Joel2 talk contribs uploaded File:MovieWongComposite.gif
20:20, 21 June 2024 Joel2 talk contribs created page File:FlatColorContoursCropped.png
20:20, 21 June 2024 Joel2 talk contribs uploaded File:FlatColorContoursCropped.png
20:09, 21 June 2024 Joel2 talk contribs created page 2DStructure/ToroidalGreenFunction (Created page with "__FORCETOC__  =Using Toroidal Coordinates to Determine the Gravitational Potential= NOTE: An earlier version of this chapter has been shifted to our "Ramblings" Appendix. Here we build upon our accompanying review of...")
20:05, 21 June 2024 Joel2 talk contribs created page 2DStructure/ToroidalCoordinates (Created page with "__FORCETOC__  =Using Toroidal Coordinates to Determine the Gravitational Potential= ==Preface== <table border="1" width="90%" align="center" cellpadding="15"> <tr><td align="left"> <table border="0" cellpadding="8" align="center" width="60%"> <tr><td align="center"> Abstract Joel E. Tohline (September, 2017) </td></tr> </table> <table border="0" cellpadding="3" align="center" width="60%"> <tr><td...")
19:57, 21 June 2024 Joel2 talk contribs created page AxisymmetricConfigurations/PoissonEq (Created page with "__FORCETOC__  =Solving the (Multi-dimensional) Poisson Equation Numerically= ==Overview== The set of Principal Governing Equations that serves as the foundation of our study of the structure, stability, and dynamical evolution of self-gravitating fluids contains an equation of motion (the ''Euler'' equation) that includes an acceleration due to local gradients in the (Newtonian) gravitational p...")
19:54, 21 June 2024 Joel2 talk contribs created page Apps/DysonWongTori (Created page with "__FORCETOC__  =Self-Gravitating, Incompressible (Dyson-Wong) Tori=  <table border="1" cellpadding="8" align="center"> <tr> <th algin="center" colspan="5"> Schematics Illustrating ''Anch...")
19:49, 21 June 2024 Joel2 talk contribs created page Template:LSU CT99CommonTheme3B (Created page with "Beginning with ''his'' integral expression for <math>\Phi(\eta,\theta)|_\mathrm{axisym}</math>, {{ Wong73 }} was able to complete the integrals in ''both'' coordinate directions and obtain an analytic expression for the potential both inside and outside of a uniformly charged (equivalently, uniform-density), circular torus. This is a remarkable result that has been largely unnoticed and unappreciated by the astrophysics community. We detail...")
19:48, 21 June 2024 Joel2 talk contribs created page Template:LSU CT99CommonTheme3A (Created page with "Referencing [MF53], {{ Wong73full }} developed an integral expression for the Coulomb (equivalently, gravitational) potential whose foundation is the single summation, Green's function expression for the reciprocal distance between two points in toroidal coordinates. For axisymmetric mass distributions‡, Wong obtained, <table border="0" cellpadding="5" align="center"> <tr> <td align="righ...")
19:47, 21 June 2024 Joel2 talk contribs created page Template:LSU CT99CommonTheme3 (Created page with "Now, beginning with Version 2 of our expression for the ''Gravitational Potential of Axisymmetric Mass Distributions'', let's also map the (unprimed) cylindrical coordinate pair, <math>~(\varpi, z)</math>, to the same (but, unprimed) toroidal coordinate system, <math>~(\eta,\theta)</math>, and place the toroidal coordinate system's ''anchor ring'' in the equatorial plane of the cylindrical-coordinate system such that, <math>~(\varpi_a,z_a) = (a,0)</math>. This gives, wh...")
19:46, 21 June 2024 Joel2 talk contribs created page Template:LSU CT99CommonTheme2A (Created page with "Referencing (equivalently, Version 1 of) the above-identified integral expression for the ''Gravitational Potential of an Axisymmetric Mass Distribution,'' [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, Huré & Hersant (2012)] offer the following assessment in §6 of their paper: <table border="0" cellpadding="3" align="center" width="60%"> <tr><td align="left"> "The important question we have tried to clarify...")
19:45, 21 June 2024 Joel2 talk contribs created page Template:LSU CT99CommonTheme2 (Created page with "Suppose we rewrite (Version 1 of) the above-highlighted Key integral expression such that the (primed) coordinate location of each mass element is mapped from cylindrical coordinates <math>~(\varpi^', z^')</math> to a toroidal-coordinate system <math>~(\eta^',\theta^')</math> whose ''anchor ring'' cuts through the meridional plane at the cylindrical-coordinate location, <math>~(\varpi_a,z_a)</math>. This desired mapping is handled via the pair of relations,...")
19:44, 21 June 2024 Joel2 talk contribs created page Template:LSU CT99CommonTheme1B (Created page with "[http://adsabs.harvard.edu/abs/1974ApJ...194..393D Deupree (1974)] and, separately, [http://adsabs.harvard.edu/abs/1983ApJ...268..155S Stahler (1983a)] have argued that a reasonably good approximation to the gravitational potential due to any extended axisymmetric mass distribution can be obtained by adding up the contributions due to many ''thin rings'' — with <math>~\delta M(\varpi^', z^')</math> being the appropriate differential mass contributed by each ring e...")
19:43, 21 June 2024 Joel2 talk contribs created page Template:LSU CT99CommonTheme1Awhite (Created page with "In §102 of a book titled, [https://www.amazon.com/Theory-Potential-W-D-Macmillan/dp/0486604861/ref=sr_1_2?s=books&ie=UTF8&qid=1503444466&sr=1-2&keywords=the+theory+of+the+potential ''The Theory of the Potential'', MacMillan (1958; originally, 1930)] derives an analytic expression for the gravitational potential of a uniform, infinitesimally thin, circular "hoop" of radius, <math>~a</math>. Throughout our related discussions, we generally will refer to this addition...")
19:41, 21 June 2024 Joel2 talk contribs created page Template:LSU CT99CommonTheme1 (Created page with "The gravitational potential (both inside and outside) of any axisymmetric mass distribution may be determined from the following integral expression that we will refer to as the, <div align="center"> '''Gravitational Potential of an Axisymmetric Mass Distribution (Version 1)''' {{ Math/EQ_CT99Axisymmetric }} </div> and, <math>~K(\mu)</math> is the complete elliptic integral of the first kind. This Key Equation† may be strai...")
19:38, 21 June 2024 Joel2 talk contribs created page AxisymmetricConfigurations/SolvingPE (Created page with "__FORCETOC__  =Common Theme: Determining the Gravitational Potential for Axisymmetric Mass Distributions= You have arrived at this page from our ''Tiled Menu'' by clicking on the chapter title that is also identified, below, by the light-blue-colored icon. You may proceed directly to that chapter by clicking (again) on that highlighted chapter icon. However, we have brought y...")
19:37, 21 June 2024 Joel2 talk contribs created page AxisymmetricConfigurations/HSCF (Created page with "__FORCETOC__  =Introduction= {| class="AxisymmetricConfigurations" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 150px; width: 150px; background-color:white;" |Hachisu Self- Consistent-Field [HSCF] Technique |} Text colored dark green has been extracted ''verbatim'' from...")
19:34, 21 June 2024 Joel2 talk contribs created page AxisymmetricConfigurations/Equilibria (Created page with "__FORCETOC__  =Axisymmetric Configurations (Steady-State Structures)= {| class="AxisymmetricConfigurations" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 150px; width: 150px; background-color:lightgreen;" |Constructing Steady-State Axisymmetric Configurations |} Equilibr...")
19:33, 21 June 2024 Joel2 talk contribs created page AxisymmetricConfigurations/PGE (Created page with "__FORCETOC__  =Axisymmetric Configurations (Governing Equations)= {| class="AxisymmetricConfigurationsPGE" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 125px; width: 125px; background-color:lightgreen;" |PGEs for Axisymmetric Systems |} If the self-gravitating configura...")
19:32, 21 June 2024 Joel2 talk contribs created page AxisymmetricConfigurations/Storyline (Created page with "__FORCETOC__  =(Initially) Axisymmetric Configurations= {| class="AxisymmetricConfigurations" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 150px; width: 150px; background-color:black;" |Storyline |} <table border="0" cellpadding="3" align="center" width="60%"> <t...")
19:31, 21 June 2024 Joel2 talk contribs created page Apps/GoldreichWeber80 (Created page with "__FORCETOC__  =Homologously Collapsing Stellar Cores= {| class="n3Dynamics" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 125px; width: 125px; background-color:#ffff99;;" |Collapse of an Isolated n = 3Polytrope |} ==Review of Goldreich and Weber (1980)== This is principally a review of the...")
19:30, 21 June 2024 Joel2 talk contribs created page SSC/Dynamics/IsothermalSimilaritySolution (Created page with "__FORCETOC__  =Similarity Solution= {| class="IsothermalCollapse" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 150px; width: 150px; background-color:#ffff99;;" |Similarity Solution |} Much of the material in this chapter has been drawn from §4.1 of a [http://adsabs.harvard.edu/abs/1982FCPh.......")
19:28, 21 June 2024 Joel2 talk contribs created page SSC/Dynamics/IsothermalCollapse (Created page with "__FORCETOC__  =Collapse of Isothermal Spheres= {| class="IsothermalCollapse" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 150px; width: 150px; background-color:#ffff99;;" |Collapse of Isothermal Spheres |} <br /...")
19:23, 21 June 2024 Joel2 talk contribs created page SSC/Dynamics/FreeFall (Created page with "__FORCETOC__  =Free-Fall= {| class="PolytropeStability" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 125px; width: 125px; background-color:#ffff99;;" |Free-Fall Collapse |} In our broad study of the "dynamics of self-gravitating fluids," we are interested in examining how, in a wide variety of phys...")
19:19, 21 June 2024 Joel2 talk contribs created page SSC/StabilityConjecture/Bipolytrope51 (Created page with "__FORCETOC__ =Examine B-KB74 Conjecture in the Context of <math>(n_c, n_e) = (5 , 1)</math> Bipolytropes= {| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 125px; width: 125px; background-color:white;" |B-KB74 Conjecture RE: Bipolytrope (nc, ne) = (5, 1) |} In §6 of their paper, [...")
19:15, 21 June 2024 Joel2 talk contribs created page SSC/Stability/Yabushita75 (Created page with "__FORCETOC__  =Stability of a BiPolytrope with an Isothermal Core= This analysis pulls largely from {{ Yabushita75full }}; the focus is on bipolytropes having <math>(n_c, n_e) = (\infty, \tfrac{3}{2})</math>. In an accompanying discussion, we summarize the steps that Yabushita took — from 1968 and 1974, to 1975 — that led up to his discovery of an analytic description of the...")
23:24, 21 January 2024 Joel2 talk contribs created page SSC/Stability/Polytropes/Pt3 (Created page with "__FORCETOC__  =Radial Oscillations of Polytropic Spheres= <table border="1" align="center" width="100%" colspan="8"> <tr> <td align="center" rowspan="1" bgcolor="lightblue" width="33%"> Part I: Wave Equation </td> <td align="center" rowspan="1" bgcolor="lightblue" width="33%"> Part II: Boundary Conditions </td> <td align="...")
23:23, 21 January 2024 Joel2 talk contribs created page SSC/Stability/Polytropes/Pt2 (Created page with "__FORCETOC__  =Radial Oscillations of Polytropic Spheres= <table border="1" align="center" width="100%" colspan="8"> <tr> <td align="center" rowspan="1" bgcolor="lightblue" width="33%"> Part I: Wave Equation </td> <td align="center" rowspan="1" bgcolor="lightblue" width="33%"> Part II: Boundary Conditions </td> <td align="...")
23:18, 21 January 2024 Joel2 talk contribs created page SSC/Stability/Polytropes (Created page with "=Radial Oscillations of Polytropic Spheres= <table border="1" align="center" width="100%" colspan="8"> <tr> <td align="center" rowspan="1" bgcolor="lightblue" width="33%"> Part I: Wave Equation </td> <td align="center" rowspan="1" bgcolor="lightblue" width="33%"> Part II: Boundary Conditions </td> <td align="center" rowspan="1" bgcolor="lightblue"> SSC/...")
23:05, 21 January 2024 Joel2 talk contribs created page File:BruteForceWhiteBoardsmall.png
23:05, 21 January 2024 Joel2 talk contribs uploaded File:BruteForceWhiteBoardsmall.png
22:57, 21 January 2024 Joel2 talk contribs created page File:TrialN1Eigenfunction.png
22:57, 21 January 2024 Joel2 talk contribs uploaded File:TrialN1Eigenfunction.png
21:53, 20 January 2024 Joel2 talk contribs created page SSC/Stability/n1PolytropeLAWE/Pt3 (Created page with "__FORCETOC__ =Radial Oscillations of n = 1 Polytropic Spheres (Pt 3)= <table border="1" align="center" width="100%" colspan="8"> <tr> <td align="center" rowspan="1" bgcolor="lightblue" width="33%"> Part I: Search for Analytic Solutions </td> <td align="center" rowspan="1" bgcolor="lightblue" width="33%"> Part II: New Ideas </td> <td align="center" ro...")
21:52, 20 January 2024 Joel2 talk contribs created page SSC/Stability/n1PolytropeLAWE/Pt2 (Created page with "__FORCETOC__ =Radial Oscillations of n = 1 Polytropic Spheres (Pt 2)= <table border="1" align="center" width="100%" colspan="8"> <tr> <td align="center" rowspan="1" bgcolor="lightblue" width="33%"> Part I: Search for Analytic Solutions </td> <td align="center" rowspan="1" bgcolor="lightblue" width="33%"> Part II: New Ideas </td> <td align="center" ro...")
21:52, 20 January 2024 Joel2 talk contribs created page SSC/Stability/n1PolytropeLAWE (Created page with "__FORCETOC__ =Radial Oscillations of n = 1 Polytropic Spheres (Pt 1)= <table border="1" align="center" width="100%" colspan="8"> <tr> <td align="center" rowspan="1" bgcolor="lightblue" width="33%"> Part I: Search for Analytic Solutions </td> <td align="center" rowspan="1" bgcolor="lightblue" width="33%"> Part II: New Ideas </td> <td align="center" ro...")
17:51, 18 January 2024 Joel2 talk contribs created page File:Chatterji51Summary2.png
17:51, 18 January 2024 Joel2 talk contribs uploaded File:Chatterji51Summary2.png
15:36, 18 January 2024 Joel2 talk contribs created page SSC/VariationalPrinciple/Pt2 (Created page with "__FORCETOC__  =Ledoux's Variational Principle= <table border="1" align="center" width="100%" colspan="8"> <tr> <td align="center" bgcolor="lightblue" width="50%"> Part I: Ledoux & Pekeris (1941) </td> <td align="center" bgcolor="lightblue"> Part II: Exploratory Ideas </td> </tr> </table> =Related, Exploratory Ideas= <!-- OM...")
15:34, 18 January 2024 Joel2 talk contribs created page SSC/VariationalPrinciple (Created page with "__FORCETOC__  =Ledoux's Variational Principle= <table border="1" align="center" width="100%" colspan="8"> <tr> <td align="center" bgcolor="lightblue" width="50%"> Part I: Ledoux & Pekeris (1941) </td> <td align="center" bgcolor="lightblue"> Part II: Exploratory Ideas </td> </tr> </table> {| class="PGEclass" style="float:lef...")
15:18, 18 January 2024 Joel2 talk contribs created page SSC/SoundWaves (Created page with "__FORCETOC__   =Sound Waves= {| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 125px; width: 125px; background-color:white;" | Relationship to Sound Waves |} A standard technique that is used throughout astrophysic...")
14:48, 18 January 2024 Joel2 talk contribs created page SSC/Structure/BiPolytropes/FreeEnergy51/Pt3 (Created page with "=Free Energy of BiPolytrope with (nc, ne) = (5, 1)= <table border="1" align="center" width="100%" colspan="8"> <td align="center" bgcolor="lightblue" width="33%"> Part I: Mass Profile </td> <td align="center" bgcolor="lightblue"3 width="33%"> Part II: Gravitational Potential Energy </td> <td align="center" bgcolo...")
14:45, 18 January 2024 Joel2 talk contribs created page SSC/Structure/BiPolytropes/FreeEnergy51/Pt2 (Created page with "__FORCETOC__ =Free Energy of BiPolytrope with (nc, ne) = (5, 1)= <table border="1" align="center" width="100%" colspan="8"> <td align="center" bgcolor="lightblue" width="33%"> Part I: Mass Profile </td> <td align="center" bgcolor="lightblue"3 width="33%"> Part II: Gravitational Potential Energy </td> <td align="c...")
14:40, 18 January 2024 Joel2 talk contribs created page SSC/Structure/BiiPolytropes/FreeEnergy51 (Created page with "__FORCETOC__ =Free Energy of BiPolytrope with (nc, ne) = (5, 1)= <table border="1" align="center" width="100%" colspan="8"> <td align="center" bgcolor="lightblue" width="33%"> Part I: Mass Profile </td> <td align="center" bgcolor="lightblue"3 width="33%"> Part II: Gravitational Potential Energy </td> <td align="ce...")
19:27, 17 January 2024 Joel2 talk contribs created page SSC/Virial/PolytropesEmbedded/SecondEffortAgain (Created page with "__FORCETOC__   =Virial Equilibrium of Adiabatic Spheres (Summary)= <table border="1" align="center" width="100%" colspan="8"> <tr> <td align="center" bgcolor="lightblue" width="33%"> Part I: Groundwork </td> <td align="center" bgcolor="lightblue" width="33%"> SSC/Virial/PolytropesEmbedded/SecondEf...")
16:19, 17 January 2024 Joel2 talk contribs created page Template:PK2007hereafter (Created page with "PK2007")
16:19, 17 January 2024 Joel2 talk contribs created page Template:PK2007full (Created page with "📚 [https://www.sciencedirect.com/science/article/pii/S0021999106002555 A. Y. Poludnenko & A. M. Khokhlov (2007, J. Comp. Phys., Vol. 220, Issue 2, pp. 678 - 711)]")

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