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20:1720:17, 23 January 2025 diff hist +1 SSC/Structure/BiPolytropes/Analytic15 →Step 6: Envelope Solution current
20:1420:14, 23 January 2025 diff hist −1 SSC/Structure/BiPolytropes/Analytic15 →Step 5: Interface Conditions
20:1420:14, 23 January 2025 diff hist −13 SSC/Structure/BiPolytropes →Solution Steps current

19:5919:59, 10 July 2024 diff hist −5 MediaWiki:Sidebar No edit summary current
19:5819:58, 10 July 2024 diff hist +12 MediaWiki:Sidebar No edit summary
19:5719:57, 10 July 2024 diff hist +40 MediaWiki:Sidebar No edit summary

17:3417:34, 30 June 2024 diff hist 0 MediaWiki:Sidebar No edit summary
17:3417:34, 30 June 2024 diff hist −29 MediaWiki:Sidebar No edit summary
17:3317:33, 30 June 2024 diff hist +37 MediaWiki:Sidebar No edit summary

19:5019:50, 28 June 2024 diff hist +115,199 N Appendix/Ramblings/PowerSeriesExpressions Created page with "__FORCETOC__  =Approximate Power-Series Expressions= ==Broadly Used Mathematical Expressions (shown here without proof)== ===Binomial=== <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~(1 \pm x)^n</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1 ~\pm ~nx + \biggl[\frac{n(n-1)}{2!}\biggr]x^2 ~\pm~ \biggl[\frac{n(n-1)(n-2)}{3!}\biggr]x^3..." current
19:2919:29, 28 June 2024 diff hist 0 N File:WolframAlphaResult.png No edit summary current
19:2619:26, 28 June 2024 diff hist 0 N File:Ostriker64PaperIIFig1.png No edit summary current
19:2119:21, 28 June 2024 diff hist +64,784 N Apps/Ostriker64 Created page with "__FORCETOC__  =Polytropic & Isothermal Tori= {| class="Ostriker64" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 125px; width: 125px; background-color:white;" |<b>Ostriker<br />(1964)</b> |} Here we will focus on the analysis of the structure self-gravitating tori that are composed of compressible — speci..." current
19:0319:03, 28 June 2024 diff hist 0 N File:EH85ten.png No edit summary current
19:0319:03, 28 June 2024 diff hist 0 N File:EH85nine.png No edit summary current
19:0119:01, 28 June 2024 diff hist 0 N File:MPT77eight.png No edit summary current
19:0119:01, 28 June 2024 diff hist 0 N File:MPT77seven.png No edit summary current
18:5818:58, 28 June 2024 diff hist 0 N File:MPT77sixModified.png No edit summary current
18:5818:58, 28 June 2024 diff hist 0 N File:MPT77fiveModified.png No edit summary current
18:5618:56, 28 June 2024 diff hist 0 N File:OurEFEannotated.png No edit summary current
18:5418:54, 28 June 2024 diff hist 0 N File:HTE78Fig10.png No edit summary current
18:0418:04, 28 June 2024 diff hist +32,370 N Apps/MaclaurinToroid Created page with "__FORCETOC__   =Overview= <table border="0" align="center" width="80%" cellpadding="3"><tr><td align="left"> <font color="darkgreen">"It has been previously stated in the literature that the Maclaurin toroid sequence bifurcates from the Maclaurin spheroidal sequence at the <math>P_4(\eta)</math> bifurcation point ({{ EH85full }}, … Since the <math>P_4(\eta)</math> bifurc..." current
17:5517:55, 28 June 2024 diff hist 0 N File:RoverD5over2.png No edit summary current
17:5417:54, 28 June 2024 diff hist 0 N File:DysonCompare01.png No edit summary current
17:4817:48, 28 June 2024 diff hist −1 Appendix/Ramblings/ToroidalCoordinates →Multipole Moment in Toroidal Coordinates current
17:4817:48, 28 June 2024 diff hist 0 N File:TorusIntegration.png No edit summary current
17:4617:46, 28 June 2024 diff hist 0 N File:TCoordsE.gif No edit summary current

18:0018:00, 14 December 2023 diff hist +30 MediaWiki:Sidebar No edit summary

16:3216:32, 12 December 2023 diff hist −13 Appendix/EquationTemplates →Other Equations with Assigned Templates current
16:2916:29, 12 December 2023 diff hist +554 N Template:Math/EQ TRApproximation Created page with "<table border="0" cellpadding="5" align="center"> <tr> <td align="right"> 25px|link=Appendix/EquationTemplates#Other_Equations_with_Assigned_Templates </td> <td align="right"> <math>\Phi_\mathrm{TR}(\varpi,z)</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>-\biggl[ \frac{2GM}{\pi } \biggr]\frac{K(k)}{\sqrt{(\varpi+a)^2 + z^2}}</math> </td> </tr> <tr> <td align="center" colspan="4"> <math>\mathrm{where:}~~~k..." current
16:2716:27, 12 December 2023 diff hist +66 N Template:Math/EQ Polytrope03 Created page with "<math>\rho = \biggl[ \frac{H}{(n+1)K_\mathrm{n}} \biggr]^n </math>" current
16:2616:26, 12 December 2023 diff hist +45 N Template:Math/EQ Polytrope02 Created page with "<math>H = (n+1)K_\mathrm{n} \rho^{1/n}</math>" current
16:2516:25, 12 December 2023 diff hist +42 N Template:Math/EQ Polytrope01 Created page with "<math>P = K_\mathrm{n} \rho^{1+1/n}</math>" current
16:2416:24, 12 December 2023 diff hist +54 N Template:Math/EQ EOSideal02 Created page with "<math>P = (\gamma_\mathrm{g} - 1)\epsilon \rho </math>" current
16:2316:23, 12 December 2023 diff hist +24 N Template:Math/EQ EOSideal00 Created page with "<math>P = n_g k T</math>" current
16:2216:22, 12 December 2023 diff hist +675 N Template:Math/EQ CT99Axisymmetric Created page with "<table border="0" cellpadding="5" align="center"> <tr> <td align="right"> 25px|link=Appendix/EquationTemplates#Other_Equations_with_Assigned_Templates </td> <td align="right"> <math>\Phi(\varpi,z)\biggr|_\mathrm{axisym}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> - \frac{G}{\pi} \iint\limits_\mathrm{config} \biggl[ \frac{\mu}{(\varpi~ \varpi^')^{1 / 2}} \biggr] K(\mu) \rho(\varpi^', z^') 2\pi \varpi^'~ d\v..." current
16:1916:19, 12 December 2023 diff hist +760 N Template:Math/EQ RadialPulsation04 Created page with "<table border=0 cellpadding=2> <tr> <td align="right"> 25px|link=Appendix/EquationTemplates#Stability:_Radial_Pulsation </td> <td align="center"> <math>0 = \frac{d^2x}{dr_0^2} + \frac{1}{r_0}\biggl[4 + \frac{d\ln P_0}{d\ln r_0} \biggr] \frac{dx}{dr_0} + \biggl[ \biggl(3 - \frac{4}{\gamma_g}\biggr) \frac{d\ln P_0}{d\ln r_0} \biggr] \frac{x}{r_0^2} - \frac{1}{\Delta} \biggl[\frac{d\ln P_0}{d\ln r_0}\cdot \frac{\rho_c}{\rho_0} \biggl( \frac{\sigma_c..." current
16:1816:18, 12 December 2023 diff hist +437 N Template:Math/EQ RadialPulsation01 Created page with "<table border=0 cellpadding=2> <tr> <td align="right"> 25px|link=Appendix/EquationTemplates#Stability:_Radial_Pulsation </td> <td align="center"> <math> \frac{d^2x}{dr_0^2} + \biggl[\frac{4}{r_0} - \biggl(\frac{g_0 \rho_0}{P_0}\biggr) \biggr] \frac{dx}{dr_0} + \biggl(\frac{\rho_0}{\gamma_\mathrm{g} P_0} \biggr)\biggl[\omega^2 + (4 - 3\gamma_\mathrm{g})\frac{g_0}{r_0} \biggr] x = 0 </math> </td> </tr> </table>" current
15:4815:48, 12 December 2023 diff hist +350 N Template:Math/EQ SSradiationTransport01 Created page with "<table border=0 cellpadding=2> <tr> <td align="right"> 25px|link=Appendix/EquationTemplates#Traditional_Equations_of_.28Spherical.29_Stellar_Structure </td> <td align="center"> <math>\frac{dT}{dr} = - \frac{ 3 }{ 4a_\mathrm{rad} c} \biggl(\frac{ \kappa \rho }{ T^3 }\biggr) \frac{ L_r }{ 4\pi r^2 }</math> </td> </tr> </table>" current
15:4715:47, 12 December 2023 diff hist +264 N Template:Math/EQ SSmassConservation01 Created page with "<table border=0 cellpadding=2> <tr> <td align="right"> 25px|link=Appendix/EquationTemplates#Traditional_Equations_of_.28Spherical.29_Stellar_Structure </td> <td align="center"> <math>\frac{dM_r}{dr} = 4\pi r^2 \rho</math> </td> </tr> </table>" current
15:4615:46, 12 December 2023 diff hist +286 N Template:Math/EQ SSenergyConservation01 Created page with "<table border=0 cellpadding=2> <tr> <td align="right"> 25px|link=Appendix/EquationTemplates#Traditional_Equations_of_.28Spherical.29_Stellar_Structure </td> <td align="center"> <math>\frac{dL_r}{dr} = 4\pi r^2 \rho \epsilon_\mathrm{nuc}</math> </td> </tr> </table>" current
15:4415:44, 12 December 2023 diff hist +562 N Template:Math/EQ ZTFG01 Created page with "<table border=0 cellpadding=2> <tr> <td align="right" width="35%"> 25px|link=Appendix/EquationTemplates#Equations_of_State </td> <td align="left"> <font><math>P_\mathrm{deg} = A_\mathrm{F} F(\chi) </math></font> </td> </tr> <tr> <td align="center" colspan="2"> where: <math>F(\chi) \equiv \chi(2\chi^2 - 3)(\chi^2 + 1)^{1/2} + 3\sinh^{-1}\chi</math> </td> </tr> <tr> <td align="right" colspan="1"> and: </td> <td align="left"..." current
15:4315:43, 12 December 2023 diff hist +343 N Template:Math/EQ PressureTotal01 Created page with "<table border=0 cellpadding=2> <tr> <td align="right"> 25px|link=Appendix/EquationTemplates#Equations_of_State </td> <td align="center"> <math>p_\mathrm{total} = \biggl(\frac{\mu_e m_p}{\bar{\mu} m_u} \biggr) 8 \chi^3 \frac{T}{T_e} + F(\chi) + \frac{8\pi^4}{15} \biggl( \frac{T}{T_e} \biggr)^4</math> </td> </tr> </table>" current
15:4315:43, 12 December 2023 diff hist +239 N Template:Math/EQ EOSradiation01 Created page with "<table border=0 cellpadding=2> <tr> <td align="right"> 25px|link=Appendix/EquationTemplates#Equations_of_State </td> <td align="center"> <math>P_\mathrm{rad} = \frac{1}{3} a_\mathrm{rad} T^4</math> </td> </tr> </table>" current
15:4215:42, 12 December 2023 diff hist +238 N Template:Math/EQ EOSideal0A Created page with "<table border=0 cellpadding=2> <tr> <td align="right"> 25px|link=Appendix/EquationTemplates#Equations_of_State </td> <td align="center"> <math>P_\mathrm{gas} = \frac{\Re}{\bar{\mu}} \rho T</math> </td> </tr> </table>" current
15:3915:39, 12 December 2023 diff hist +27,268 N Appendix/EquationTemplates Created page with "__FORCETOC__  =50px<font size="+2" color="black">Key Equations</font>= Each of the equations displayed in the Tables, below, encapsulates a physical concept that is fundamental to our understanding of — and, hence our discussion of — the '''structure, stability, and dynamics of self-gravitating fluids.''' The pervasiveness of these physical concepts throughout astrophysics is reflected in the fact t..."
15:3515:35, 12 December 2023 diff hist +52 N Template:Math/MP PolytropicIndex Created page with "<span title="Polytropic index"><math>n</math></span>" current
15:3515:35, 12 December 2023 diff hist +66 N Template:Math/MP PolytropicConstant Created page with "<span title="Polytropic constant"><math>K_\mathrm{n}</math></span>" current
15:3415:34, 12 December 2023 diff hist +65 N Template:Math/MP MeanMolecularWeight Created page with "<span title="Mean molecular weight"><math>\bar{\mu}</math></span>" current
15:3315:33, 12 December 2023 diff hist +69 N Template:Math/MP ElectronMolecularWeight Created page with "<span title="Molecular weight of electrons"><math>\mu_e</math></span>" current

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