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• 18:15, 29 June 2024 Joel2 talk contribs created page Appendix/Mathematics/ToroidalSynopsis01 (Created page with "<!-- __FORCETOC__ will force the creation of a Table of Contents --> <!-- __NOTOC__ will force TOC off --> =Synopsis of Toroidal Coordinate Approach= ==Basics== Here we attempt to bring together — in as succinct a manner as possible — our approach and [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W C.-Y. Wong's (1973)] approach to determining the gravi...")
• 18:14, 29 June 2024 Joel2 talk contribs created page Template:Math/EQ Q1minusHalf01 (Created page with "<table border="0" cellpadding="5" align="center"> <tr> <td align="right"> 25px|link=Appendix/SpecialFunctions#Toroidal_Function_Evaluations </td> <td align="right"> <math>Q^1_{-\frac{1}{2}}(z)</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> - \biggl[\frac{1}{2(z-1)} \biggr]^{1 / 2} E(k) </math> </td> </tr> <tr> <td align="center" colspan="4"> where:   <math>k = \sqrt{ \frac{2}{z+1}} \, .</math>...")
• 18:12, 29 June 2024 Joel2 talk contribs created page Template:Math/EQ QplusHalf01 (Created page with "<table border="0" cellpadding="5" align="center"> <tr> <td align="right"> 25px|link=Appendix/SpecialFunctions#Toroidal_Function_Evaluations </td> <td align="right"> <math>Q_{+\frac{1}{2}}(z)</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> z \sqrt{ \frac{2}{z+1} }~K\biggl( \sqrt{ \frac{2}{z+1} } \biggr) ~-~ [2(z+1)]^{1 / 2} E\biggl( \sqrt{ \frac{2}{z+1} } \biggr) </math> </td> </tr> <tr> <td align="cente...")
• 18:07, 29 June 2024 Joel2 talk contribs created page Template:Math/EQ PplusHalf01 (Created page with "<table border="0" cellpadding="5" align="center"> <tr> <td align="right"> 25px|link=Appendix/SpecialFunctions#Toroidal_Function_Evaluations </td> <td align="right"> <math>P_{+ \frac{1}{2}}(z)</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{2}{\pi} \biggl[z + \sqrt{ z^2-1} \biggr]^{1 / 2} ~E\biggl( \sqrt{ \frac{2(z^2-1)^{1 / 2}}{z + (z^2-1)^{1 / 2}}} \biggr) </math> </td> <td align="center" rowspa...")
• 18:02, 29 June 2024 Joel2 talk contribs created page Template:Math/EQ QminusHalf01 (Created page with "<table border="0" cellpadding="5" align="center"> <tr> <td align="right"> 25px|link=Appendix/SpecialFunctions#Toroidal_Function_Evaluations </td> <td align="right"> <math>Q_{-\frac{1}{2}}(z)</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \sqrt{ \frac{2}{z+1} } ~K\biggl( \sqrt{ \frac{2}{z+1}} \biggr) </math> </td> <td align="center" rowspan="2" width="20%">      for example …</td>...")
• 18:01, 29 June 2024 Joel2 talk contribs created page Template:Math/EQ PminusHalf01 (Created page with "<table border="0" cellpadding="5" align="center"> <tr> <td align="right"> 25px|link=Appendix/SpecialFunctions#Toroidal_Function_Evaluations </td> <td align="right"> <math>P_{-\frac{1}{2}}(z)</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{2}{\pi} \biggl[\frac{2}{z+1}\biggr]^{1 / 2} ~K\biggl( \sqrt{ \frac{z-1}{z+1}} \biggr) </math> </td> <td align="center" rowspan="2" width="20%">    &nb...")
• 17:59, 29 June 2024 Joel2 talk contribs created page Template:Math/EQ Toroidal00 (Created page with "<table border="0" cellpadding="5" align="center"> <tr> <td align="right"> 25px|link=http:Appendix/Equation_templates#Special_Function_Relationships </td> <td align="right"> <math>~ P_\nu^n(z) </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{\Gamma(\nu + n + 1)}{\Gamma(\nu - n + 1)} P_\nu^{-n}(z) </math> </td> </tr> <tr> <td align="center" colspan=4> [https://authors.library.caltech.edu/43491/1/V...")
• 17:57, 29 June 2024 Joel2 talk contribs created page Template:Math/EQ EllipticIntegral04 (Created page with "<table border="0" cellpadding="5" align="center"> <tr> <td align="right"> 25px|link=Appendix/SpecialFunctions#Complete_Elliptic_Integrals </td> <td align="right"> <math>E(\mu)</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> 1 ~+~ \frac{1}{2}\biggl( \ln \frac{4}{k'} - \frac{1}{1\cdot 2}\biggr)(k')^2 ~+~ \frac{1^2\cdot 3}{2^2\cdot 4}\biggl( \ln \frac{4}{k'} - \frac{2}{1\cdot 2} - \frac{1}{3\cdot 4}\biggr)(k')...")
• 17:55, 29 June 2024 Joel2 talk contribs created page Template:Math/EQ EllipticIntegral02 (Created page with "<table border="0" cellpadding="5" align="center"> <tr> <td align="right"> 25px|link=Appendix/SpecialFunctions#Complete_Elliptic_Integrals </td> <td align="right"> <math>\frac{2E(k)}{\pi}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> 1 - \frac{1}{2^2} ~k^2 - \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4 - \biggl(\frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2~\frac{ k^6 }{5} - \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^...")
• 17:54, 29 June 2024 Joel2 talk contribs created page Template:Math/EQ EllipticIntegral03 (Created page with "<table border="0" cellpadding="5" align="center"> <tr> <td align="right"> 25px|link=Appendix/SpecialFunctions#Complete_Elliptic_Integrals </td> <td align="right"> <math>K(\mu)</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \ln \frac{4}{k^'} + \frac{1}{2^2}\biggl( \ln\frac{4}{k^'} - \frac{2}{1\cdot 2} \biggr)(k')^2 + \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 \biggl( \ln\frac{4}{k^'} - \frac{2}{1\cdot 2} -...")
• 17:53, 29 June 2024 Joel2 talk contribs created page Template:Math/EQ EllipticIntegral01 (Created page with "<table border="0" cellpadding="5" align="center"> <tr> <td align="right"> 25px|link=Appendix/SpecialFunctions#Complete_Elliptic_Integrals </td> <td align="right"> <math>\frac{2K(k)}{\pi}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> 1 + \biggl( \frac{1}{2} \biggr)^2k^2 + \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4 + \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k^6 + \biggl( \frac{1\cdot 3\cdot...")
• 17:50, 29 June 2024 Joel2 talk contribs created page Template:Math/EQ Gamma01 (Created page with "<table border="0" cellpadding="5" align="center"> <tr> <td align="right"> 25px|link=Appendix/SpecialFunctions#Gamma_Function </td> <td align="right" colspan="2"> <math>~ \Gamma(z) ~\Gamma(1-z) </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{\pi}{\sin(\pi z)} </math> </td> <td align="center" rowspan="1"><math>~\biggl|</math></td> <td align="center" rowspan="3"> for example, if<br /> <math>~z \...")
• 17:48, 29 June 2024 Joel2 talk contribs created page Appendix/SpecialFunctions (Created page with "__FORCETOC__ <!-- __NOTOC__ will force TOC off --> =50pxSpecial Functions= ===Gamma Function=== <div align="center"> <table border=3 cellpadding=5 cellspacing=1 width="95%" bordercolor="darkblue"> <tr> <th colspan=3 align="center"> <font size="+1" color="darkblue">Gamma Function</font> </th> </tr> <tr> <td colspan=2> To insert a given equation into any Wiki document, type ...<br /><center> {{ Math/<i><font color="red">Template_Name</fo...")
• 17:43, 29 June 2024 Joel2 talk contribs created page Template:Math/EQ Toroidal01 (Created page with "<table border="0" cellpadding="5" align="center"> <tr> <td align="right"> 25px|link=Appendix/Equation_templates#Relationships_Between_Various_Associated_Legendre_Functions </td> <td align="right"> <math> Q_\nu[t t^' - (t^2-1)^{1 / 2} (t^{'2} - 1)^{1 / 2} \cos\psi] </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> Q_\nu(t) P_\nu(t^') + 2\sum_{n=1}^\infty (-1)^n Q^n_\nu(t) P^{-n}_\nu(t^') \cos(n\psi) </math>...")
• 15:12, 29 June 2024 Joel2 talk contribs created page File:DysonTorusIllustration03.png
• 15:12, 29 June 2024 Joel2 talk contribs uploaded File:DysonTorusIllustration03.png
• 15:00, 29 June 2024 Joel2 talk contribs created page Apps/DysonPotential (Created page with "__FORCETOC__ <!-- will force the creation of a Table of Contents --> <!-- __NOTOC__ will force TOC off --> =Dyson (1893)= {| class="HNM82" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 125px; width: 125px; background-color:#ffeeee;" |<b>Dyson<br />(1893)</b> |} Our focus, here, is on the pioneering work of [http://adsabs.harvard.edu/abs/1893RSPTA.184....")
• 14:57, 29 June 2024 Joel2 talk contribs created page File:FukushimaTorusIllustration02.png
• 14:57, 29 June 2024 Joel2 talk contribs uploaded File:FukushimaTorusIllustration02.png
• 14:56, 29 June 2024 Joel2 talk contribs created page File:WongTorusIllustration02.png
• 14:56, 29 June 2024 Joel2 talk contribs uploaded File:WongTorusIllustration02.png
• 14:55, 29 June 2024 Joel2 talk contribs created page File:OstrikerTorusIllustration01.png
• 14:55, 29 June 2024 Joel2 talk contribs uploaded File:OstrikerTorusIllustration01.png
• 14:55, 29 June 2024 Joel2 talk contribs created page File:MacMillanTorusIllustration02.png
• 14:55, 29 June 2024 Joel2 talk contribs uploaded File:MacMillanTorusIllustration02.png
• 14:53, 29 June 2024 Joel2 talk contribs created page File:DysonTorusIllustration04.png
• 14:53, 29 June 2024 Joel2 talk contribs uploaded File:DysonTorusIllustration04.png
• 14:49, 29 June 2024 Joel2 talk contribs created page Appendix/Ramblings/CCGF (Created page with "__FORCETOC__ <!-- will force the creation of a Table of Contents --> <!-- __NOTOC__ will force TOC off --> =Compact Cylindrical Green Function (CCGF)= ==Preface by Tohline== [http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl & Tohline (1999)] present an expression for the Newtonian gravitational potential in terms of a ''Compact Cylindrical Green's Function'' expansion. Over a professional career that dates back to 1976, this has turned out to be one of my mos...")
• 14:47, 29 June 2024 Joel2 talk contribs created page Apps/DWT (Created page with "__FORCETOC__ <!-- __NOTOC__ will force TOC off --> =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...")
• 14:18, 28 June 2024 Joel2 talk contribs created page File:Figure4THH2012.png
• 14:18, 28 June 2024 Joel2 talk contribs uploaded File:Figure4THH2012.png
• 14:17, 28 June 2024 Joel2 talk contribs created page File:DiagramToroidalCoordinates.png
• 14:17, 28 June 2024 Joel2 talk contribs uploaded File:DiagramToroidalCoordinates.png
• 14:15, 28 June 2024 Joel2 talk contribs created page File:LSU CombinedTori.jpg
• 14:15, 28 June 2024 Joel2 talk contribs uploaded File:LSU CombinedTori.jpg
• 14:12, 28 June 2024 Joel2 talk contribs created page Appendix/Ramblings/ToroidalCoordinates (Created page with "__FORCETOC__ <!-- will force the creation of a Table of Contents --> <!-- __NOTOC__ will force TOC off --> =Toroidal Configurations and Related Coordinate Systems= This rather long and rambling chapter reveals how my train of thought progressed as I effectively taught myself how to use a toroidal coordinate system to determine the gravitational potential of an axisymmetric configuration. A summary of the key results from this chapter can be found in a primary chapter o...")
• 13:53, 28 June 2024 Joel2 talk contribs created page File:PickettDurisenDavis96Fig1.png
• 13:53, 28 June 2024 Joel2 talk contribs uploaded File:PickettDurisenDavis96Fig1.png
• 13:41, 28 June 2024 Joel2 talk contribs created page File:PP84Tori.jpg
• 13:41, 28 June 2024 Joel2 talk contribs uploaded File:PP84Tori.jpg
• 13:39, 28 June 2024 Joel2 talk contribs created page Apps/PapaloizouPringleTori (Created page with "__FORCETOC__ <!-- __NOTOC__ will force TOC off --> =Massless Polytropic Tori= {| class="HNM82" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 150px; width: 150px; background-color:#ffeeee;" |<b>Paploizou-Pringle<br />Tori<br />(1984)</b> |} (''aka'' Papaloizou-Pringle Tori) In a seminal paper that focused on an analysis of nonaxisymmetric instabiliti...")
• 13:35, 28 June 2024 Joel2 talk contribs created page Apps/RotatingWhiteDwarfs (Created page with "__FORCETOC__ <!-- __NOTOC__ will force TOC off --> =Rotationally Flattened White Dwarfs= {| class="HNM82" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 150px; width: 150px; background-color:white;" |<b>Example<br />Equilibria</b> |}  <br />  <br />  <br />  <br />  <br />  <br />  <br /> ==Example Equilibrium Co...")
• 13:34, 28 June 2024 Joel2 talk contribs created page Apps/OstrikerBodenheimerLyndenBell66 (Created page with "__FORCETOC__ <!-- __NOTOC__ will force TOC off --> =Rotationally Flattened White Dwarfs= {| class="HNM82" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 150px; width: 150px; background-color:#ffeeee;" |<b>Ostriker<br />Bodenheimer<br />& Lynden-Bell<br />(1966)</b> |} As we have reviewed in an SSC/Structure/WhiteDwarfs#Chandrasekhar_mass|acco...")
• 13:32, 28 June 2024 Joel2 talk contribs created page Apps/RotatingPolytropes (Created page with "__FORCETOC__ <!-- __NOTOC__ will force TOC off --> =Rotationally Flattened Polytropes= {| class="HNM82" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 150px; width: 150px; background-color:white;" |<b>Example<br />Equilibria</b> |} Here we review what has been learned over the past century+ regarding the structural properties of rotationally flatt...")
• 13:29, 28 June 2024 Joel2 talk contribs created page Apps/ReviewStahler83 (Created page with "__FORCETOC__ <!-- __NOTOC__ will force TOC off --> =Stahler's (1983) Rotationally Flattened Isothermal Configurations= {| class="HNM82" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 150px; width: 150px; background-color:white;" |<b>Review of<br />Stahler's (1983)<br />Sol'n Technique</b> |} Consider the collapse of an isothermal cloud (characteriz...")
• 13:23, 28 June 2024 Joel2 talk contribs created page File:HNM82ThreeDisksAB.png
• 13:23, 28 June 2024 Joel2 talk contribs uploaded File:HNM82ThreeDisksAB.png
• 13:21, 28 June 2024 Joel2 talk contribs created page Apps/HayashiNaritaMiyama82 (Created page with "__FORCETOC__ <!-- __NOTOC__ will force TOC off --> =Rotationally Flattened Isothermal Structures= {| class="HNM82" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 150px; width: 150px; background-color:#ffeeee;" |<b>Hayashi, Narita<br />& Miyama's<br />Analytic Sol'n<br />(1982)</b> |} [https://ui.adsabs.harvard.edu/abs/1982PThPh..68.1949H/abstra...")
• 13:11, 28 June 2024 Joel2 talk contribs created page File:Q0.5P G1 triangleD.png
• 13:11, 28 June 2024 Joel2 talk contribs uploaded File:Q0.5P G1 triangleD.png