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• 19:50, 28 June 2024 Xxsrm talk contribs created page Appendix/Ramblings/PowerSeriesExpressions (Created page with "__FORCETOC__ <!-- __NOTOC__ will force TOC off --> =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...")
• 19:29, 28 June 2024 Xxsrm talk contribs created page File:WolframAlphaResult.png
• 19:29, 28 June 2024 Xxsrm talk contribs uploaded File:WolframAlphaResult.png
• 19:26, 28 June 2024 Xxsrm talk contribs created page File:Ostriker64PaperIIFig1.png
• 19:26, 28 June 2024 Xxsrm talk contribs uploaded File:Ostriker64PaperIIFig1.png
• 19:21, 28 June 2024 Xxsrm talk contribs created page Apps/Ostriker64 (Created page with "__FORCETOC__ <!-- __NOTOC__ will force TOC off --> =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...")
• 19:03, 28 June 2024 Xxsrm talk contribs created page File:EH85ten.png
• 19:03, 28 June 2024 Xxsrm talk contribs uploaded File:EH85ten.png
• 19:03, 28 June 2024 Xxsrm talk contribs created page File:EH85nine.png
• 19:03, 28 June 2024 Xxsrm talk contribs uploaded File:EH85nine.png
• 19:01, 28 June 2024 Xxsrm talk contribs created page File:MPT77eight.png
• 19:01, 28 June 2024 Xxsrm talk contribs uploaded File:MPT77eight.png
• 19:01, 28 June 2024 Xxsrm talk contribs created page File:MPT77seven.png
• 19:01, 28 June 2024 Xxsrm talk contribs uploaded File:MPT77seven.png
• 18:58, 28 June 2024 Xxsrm talk contribs created page File:MPT77sixModified.png
• 18:58, 28 June 2024 Xxsrm talk contribs uploaded File:MPT77sixModified.png
• 18:58, 28 June 2024 Xxsrm talk contribs created page File:MPT77fiveModified.png
• 18:58, 28 June 2024 Xxsrm talk contribs uploaded File:MPT77fiveModified.png
• 18:56, 28 June 2024 Xxsrm talk contribs created page File:OurEFEannotated.png
• 18:56, 28 June 2024 Xxsrm talk contribs uploaded File:OurEFEannotated.png
• 18:54, 28 June 2024 Xxsrm talk contribs created page File:HTE78Fig10.png
• 18:54, 28 June 2024 Xxsrm talk contribs uploaded File:HTE78Fig10.png
• 18:04, 28 June 2024 Xxsrm talk contribs created page Apps/MaclaurinToroid (Created page with "__FORCETOC__ <!-- will force the creation of a Table of Contents --> <!-- __NOTOC__ will force TOC off --> =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...")
• 17:55, 28 June 2024 Xxsrm talk contribs created page File:RoverD5over2.png
• 17:55, 28 June 2024 Xxsrm talk contribs uploaded File:RoverD5over2.png
• 17:54, 28 June 2024 Xxsrm talk contribs created page File:DysonCompare01.png
• 17:54, 28 June 2024 Xxsrm talk contribs uploaded File:DysonCompare01.png
• 17:48, 28 June 2024 Xxsrm talk contribs created page File:TorusIntegration.png
• 17:48, 28 June 2024 Xxsrm talk contribs uploaded File:TorusIntegration.png
• 17:46, 28 June 2024 Xxsrm talk contribs created page File:TCoordsE.gif
• 17:46, 28 June 2024 Xxsrm talk contribs uploaded File:TCoordsE.gif
• 16:29, 12 December 2023 Xxsrm talk contribs created page 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...")
• 16:27, 12 December 2023 Xxsrm talk contribs created page Template:Math/EQ Polytrope03 (Created page with "<math>\rho = \biggl[ \frac{H}{(n+1)K_\mathrm{n}} \biggr]^n </math>")
• 16:26, 12 December 2023 Xxsrm talk contribs created page Template:Math/EQ Polytrope02 (Created page with "<math>H = (n+1)K_\mathrm{n} \rho^{1/n}</math>")
• 16:25, 12 December 2023 Xxsrm talk contribs created page Template:Math/EQ Polytrope01 (Created page with "<math>P = K_\mathrm{n} \rho^{1+1/n}</math>")
• 16:24, 12 December 2023 Xxsrm talk contribs created page Template:Math/EQ EOSideal02 (Created page with "<math>P = (\gamma_\mathrm{g} - 1)\epsilon \rho </math>")
• 16:23, 12 December 2023 Xxsrm talk contribs created page Template:Math/EQ EOSideal00 (Created page with "<math>P = n_g k T</math>")
• 16:22, 12 December 2023 Xxsrm talk contribs created page 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...")
• 16:19, 12 December 2023 Xxsrm talk contribs created page 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...")
• 16:18, 12 December 2023 Xxsrm talk contribs created page 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>")
• 15:48, 12 December 2023 Xxsrm talk contribs created page 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>")
• 15:47, 12 December 2023 Xxsrm talk contribs created page 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>")
• 15:46, 12 December 2023 Xxsrm talk contribs created page 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>")
• 15:44, 12 December 2023 Xxsrm talk contribs created page 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"...")
• 15:43, 12 December 2023 Xxsrm talk contribs created page 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>")
• 15:43, 12 December 2023 Xxsrm talk contribs created page 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>")
• 15:42, 12 December 2023 Xxsrm talk contribs created page 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>")
• 15:39, 12 December 2023 Xxsrm talk contribs created page Appendix/EquationTemplates (Created page with "__FORCETOC__ <!-- __NOTOC__ will force TOC off --> =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...")
• 15:35, 12 December 2023 Xxsrm talk contribs created page Template:Math/MP PolytropicIndex (Created page with "<span title="Polytropic index"><math>n</math></span>")
• 15:35, 12 December 2023 Xxsrm talk contribs created page Template:Math/MP PolytropicConstant (Created page with "<span title="Polytropic constant"><math>K_\mathrm{n}</math></span>")