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20:45, 12 January 2024 Joel2 talk contribs created page Appendix/Ramblings/T6CoordinatesPt2 (Created page with "__FORCETOC__  =Concentric Ellipsoidal (T6) Coordinates (Part 2)= ==Orthogonal Coordinates== ===Speculation5=== ====Spherical Coordinates==== <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~r\cos\theta</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~z \, ,</math> </td> </tr> <tr> <td align="right"> <math>~r\sin\theta</math> </td> <td align="center">...")
20:44, 12 January 2024 Joel2 talk contribs created page Appendix/Ramblings/ConcentricEllipsoidalCoordinates (Created page with "__FORCETOC__  =Concentric Ellipsoidal (T6) Coordinates= ==Background== Building on our general introduction to ''Direction Cosines'' in the context of orthogonal curvilinear coordinate systems, and on our previous development of T3 (concentric oblate-spheroidal) and T5 (concentric e...")
20:42, 12 January 2024 Joel2 talk contribs created page Appendix/Ramblings/EllipticCylinderCoordinates (Created page with "  =Elliptic Cylinder Coordinates= ==Background== Building on our general introduction to ''Direction Cosines'' in the context of orthogonal curvilinear coordinate systems, here we detail the properties of [https://en.wikipedia.org/wiki/Elliptic_cylindrical_coordinates Elliptic Cylinder Coordinates]. First, we pres...")
20:40, 12 January 2024 Joel2 talk contribs created page Appendix/Ramblings/DirectionCosines (Created page with "__FORCETOC__  =Direction Cosines= Following [<b><font color="red">MF53</font></b>], a generalized coordinate system consists of a threefold family of surfaces whose equations in terms of Cartesian coordinates are, <math>~\xi_1(x,y,z) = </math> constant, <math>~\xi_2(x,y,z) = </math> constant, and <math>~\xi_3(x,y,z) =</math> constant. The lines of intersection of these surfaces constitute...")
20:39, 12 January 2024 Joel2 talk contribs created page Appendix/Ramblings/myPhys (Created page with "__FORCETOC__   =Files on LSU P&A Servers= ==Primary Web== public_html = /opt1/usrlocal/apache2/htdocs/graceland/faculty/tohline = [http://www.phys.lsuedu/faculty/tohline http://www.phys.lsu.edu/faculty/tohline] … ===HOMEPAGE RELEVANT1=== <ul> <li> index.html </li> <li> indexhtv.html: Homepage prior to "accessibility statement" added by htv (Hortensia) </li> <li> inde...")
18:07, 11 January 2024 Joel2 talk contribs created page Template:EQ SSLaneEmden01 (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{1}{\xi^2} \frac{d}{d\xi}\biggl( \xi^2 \frac{d\Theta_H}{d\xi} \biggr) = - \Theta_H^n</math> </td> </tr> </table>")
18:05, 11 January 2024 Joel2 talk contribs created page Template:EQ SSLaneEmden02 (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{1}{\xi^2} \frac{d}{d\xi}\biggl( \xi^2 \frac{d\psi}{d\xi} \biggr) = e^{-\psi}</math> </td> </tr> </table>")
22:23, 10 January 2024 Joel2 talk contribs created page SSC/Structure/BiPolytropes/Analytic51 (Created page with "__FORCETOC__ =BiPolytrope with n<sub>c</sub> = 5 and n<sub>e</sub> = 1= {| 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;" |<font size="-1"><b>Eggleton, Faulkner<br />& Cannon (1998)<br /><br />Analytic</b><br />(n<sub>c</sub>, n<sub>e</sub>) = (5, 1)</font> |} File:CommentButton02.png|right|100px|Comm...")
22:19, 10 January 2024 Joel2 talk contribs created page File:Figure s.ps.png
22:19, 10 January 2024 Joel2 talk contribs uploaded File:Figure s.ps.png
22:18, 10 January 2024 Joel2 talk contribs created page SSC/Stability/MurphyFiedler85 (Created page with "__FORCETOC__  =Review of the BiPolytrope Stability Analysis by Murphy & Fiedler (1985b)= ==Overview== In the stability analysis presented by [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy & Fiedler (1985b)], the relevant polytropic indexes are, <math>~(n_c, n_e) = (1,5)</math>. Structural properties of the underlying equilibrium models have been reviewed in SSC/Structure/BiPolytropes/Analytic15#BiPolytrope_with...")
22:12, 10 January 2024 Joel2 talk contribs created page Appendix/Ramblings/PatrickMotl (Created page with "__FORCETOC__  =Discussing Patrick Motl's 2019 - 2020 Simulations= ==May 1 (from Patrick)== <table border="1" cellpadding="10" width="60%" align="center"><tr><td align="left"> Hi Joel, I hope things are well for you and yours. I did finally have a couple research students this semester that were able to slog their way into unix, programming, etc. far enough to do some useful things. They ran two spherical bipolytropes with my ol...")
22:09, 10 January 2024 Joel2 talk contribs created page SSC/Stability/UniformDensity (Created page with "__FORCETOC__   =The Stability of Uniform-Density Spheres= {| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 125px; width: 125px; background-color:#ffeeee;" | <font size="-1"><b>Sterne's<br />Analytic Sol'n<br />of Eigenvalue<br />Problem</b><br />(1937)<...")
18:22, 9 January 2024 Joel2 talk contribs created page File:Sterne37OmegaVsGammaLabeled.png
18:22, 9 January 2024 Joel2 talk contribs uploaded File:Sterne37OmegaVsGammaLabeled.png
13:06, 9 January 2024 Joel2 talk contribs created page Template:TVH74hereafter (Created page with "TVH74")
13:05, 9 January 2024 Joel2 talk contribs created page Template:TVH74full (Created page with "📚 [https://ui.adsabs.harvard.edu/abs/1974MNRAS.168..427T/abstract L. G. Taff, & H. M. van Horn (1974, MNRAS, Vol. 168, pp. 427 - 432)]")
13:04, 9 January 2024 Joel2 talk contribs created page Template:TVH74figure (Created page with "[https://ui.adsabs.harvard.edu/abs/1974MNRAS.168..427T/abstract L. G. Taff, & H. M. van Horn (1974)]<br />''Radial Pulsations of Finite Isothermal Gas Spheres''<br />Monthly Notices of the Royal Astronomical Society,<br />Vol. 168, pp. 427 - 432")
13:02, 9 January 2024 Joel2 talk contribs created page Template:TVH74 (Created page with "📚 [https://ui.adsabs.harvard.edu/abs/1974MNRAS.168..427T/abstract Taff, & van Horn (1974)]")
12:52, 9 January 2024 Joel2 talk contribs created page SSC/Stability/Isothermal (Created page with "__FORCETOC__  =Radial Oscillations of Pressure-Truncated Isothermal Spheres= {| 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;" | <font size="-1"><b>via<br />Direct<br />Numerical<br />Integration</b></font> |} ==Overview== Both {{ Ebert55full }} and {{ Bonnor56f...")
12:46, 9 January 2024 Joel2 talk contribs created page SSC/Stability/n5PolytropeLAWE (Created page with "__FORCETOC__  =Radial Oscillations of n = 5 Polytropic Spheres= ==Background== ===General Form of the LAWE for Spherical Polytropes=== In an accompanying discussion, we derived the so-called, <div align="center" id="2ndOrderODE"> <font color="#770000">'''Adiabatic Wave''' (or ''Radial Pulsation'') '''Equation'''</font><br /> {{Math/EQ_RadialPulsation01}} </div> whose solution gives eigenfunction...")
12:43, 9 January 2024 Joel2 talk contribs created page SSC/Stability/n3PolytropeLAWE (Created page with "__FORCETOC__  =Radial Oscillations of n = 3 Polytropic Spheres= {| 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;" | <font size="-1"><b>Isolated<br />n = 3<br />Polytrope</b></font> |} ==Background== <br /> <br /> <br /> <br /> ===Our...")
12:24, 9 January 2024 Joel2 talk contribs created page SSC/Structure/BiPolytropes/Analytic51Renormalize (Created page with "__FORCETOC__ =BiPolytrope with (n<sub>c</sub>, n<sub>e</sub>) = (5, 1) Renormalized= <table border="1" align="center" width="100%" colspan="8"> <tr> <td align="center" rowspan="1" bgcolor="lightblue" width="33%"><br />Part I: (5, 1) Analytic Renormalize<br /> </td> <td align="center" rowspan="1" bgcolor="lightblue" width="33%"><br />SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt2|Part II: E...")
12:23, 9 January 2024 Joel2 talk contribs created page SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt2 (Created page with "__FORCETOC__ =BiPolytrope with (n<sub>c</sub>, n<sub>e</sub>) = (5, 1) Renormalized= <table border="1" align="center" width="100%" colspan="8"> <tr> <td align="center" rowspan="1" bgcolor="lightblue" width="33%"><br />Part I: (5, 1) Analytic Renormalize<br /> </td> <td align="center" rowspan="1" bgcolor="lightblue" width="33%"><br />SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt2|Part II: E...")
12:08, 9 January 2024 Joel2 talk contribs created page SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt3 (Created page with "__FORCETOC__ =BiPolytrope with (n<sub>c</sub>, n<sub>e</sub>) = (5, 1) Renormalized= <table border="1" align="center" width="100%" colspan="8"> <tr> <td align="center" rowspan="1" bgcolor="lightblue" width="33%"><br />Part I: (5, 1) Analytic Renormalize<br /> </td> <td align="center" rowspan="1" bgcolor="lightblue" width="33%"><br />SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt2|Part II: E...")
11:51, 9 January 2024 Joel2 talk contribs created page Appendix/Ramblings/OriginOfPlanetaryNebulae (Created page with "  =On the Origin of Planetary Nebulae= This chapter — initially created by J. E. Tohline on 19 November 2016 — is intended primarily to provide a summary of the research that has been undertaken following <font color="red">a discussion that took place on 3 July 2013</font> with Kundan Kadam (an LSU graduate student, at the time) regarding the stability of...")
11:47, 9 January 2024 Joel2 talk contribs created page SSC/Stability/BiPolytropes/Index (Created page with "__FORCETOC__  =Organizational Index= This index helps organize and summarize the insight that we have gained from our broad investigation into the relative stability of spherically symmetric bipolytropic configurations. The most relevant summary chapter is named, SSC/Stability/BiPolytropes. ==Motivation== Schönberg-Chandrasekhar mass limit: <ul> <li> SSC/Structure/LimitingMasses#Schönberg-C...")
11:46, 9 January 2024 Joel2 talk contribs created page Template:LAWEsolutions (Created page with "<table border="1" align="center" cellpadding="5"> <tr> <td align="center" colspan="3"><b>Isolated n = 1 Polytrope</b></td> </tr> <tr> <td align="center" rowspan="2"><math>\gamma_g = (n+1)/n = 2</math><br /><p><br /><p><br />250px|n = 1 Eigenvectors</td> <td align="center" rowspan="2"><math>\gamma_g = \tfrac{5}{3}</math><br />250px|n = 1 Eigenvectors</td> <td align="left">Key Reference(s): <ul><li>{{ Chatterj...")
11:45, 9 January 2024 Joel2 talk contribs created page File:N5Truncated2.gif
11:45, 9 January 2024 Joel2 talk contribs uploaded File:N5Truncated2.gif
11:44, 9 January 2024 Joel2 talk contribs created page File:OurImprovedSummary2.png
11:44, 9 January 2024 Joel2 talk contribs uploaded File:OurImprovedSummary2.png
11:43, 9 January 2024 Joel2 talk contribs created page File:N1osc06.gif
11:43, 9 January 2024 Joel2 talk contribs uploaded File:N1osc06.gif
17:54, 8 January 2024 Joel2 talk contribs created page File:VariationOf2Modes.png
17:54, 8 January 2024 Joel2 talk contribs uploaded File:VariationOf2Modes.png
17:48, 8 January 2024 Joel2 talk contribs created page File:FourModesMuRatio100.png
17:48, 8 January 2024 Joel2 talk contribs uploaded File:FourModesMuRatio100.png
17:46, 8 January 2024 Joel2 talk contribs created page File:NEWCompositeDlabeled.png
17:46, 8 January 2024 Joel2 talk contribs uploaded File:NEWCompositeDlabeled.png
17:45, 8 January 2024 Joel2 talk contribs created page File:Mod0MuRatio100.png
17:45, 8 January 2024 Joel2 talk contribs uploaded File:Mod0MuRatio100.png
17:44, 8 January 2024 Joel2 talk contribs created page File:Mode0EigenfunctionsCombinedSmall.png
17:44, 8 January 2024 Joel2 talk contribs uploaded File:Mode0EigenfunctionsCombinedSmall.png
16:15, 8 January 2024 Joel2 talk contribs created page SSC/Stability/BiPolytropes/Pt2 (Created page with "=Review of the Analysis by Murphy & Fiedler (1985b)= As we have detailed separately, the boundary condition at the center of a polytropic configuration is, <div align="center"> <math>\frac{dx}{d\xi} \biggr|_{\xi=0} = 0 \, ;</math> </div> and the boundary condition at the surface of an isolated polytropic configuration is, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{d\ln x}{...")
16:09, 8 January 2024 Joel2 talk contribs created page SSC/Stability/BiPolytropes/Pt3 (Created page with "=Radial Oscillations of (n<sub>c</sub>, n<sub>e</sub>) = (5, 1) Models= ==Foundation== In an accompanying discussion, we derived the so-called, <div align="center" id="2ndOrderODE"> <font color="#770000">'''Adiabatic Wave''' (or ''Radial Pulsation'') '''Equation'''</font><br /> {{Math/EQ_RadialPulsation01}} </div> whose solution gives eigenfunctions that describe various radial modes of oscillation in spherically symmetric, self-gravi...")
15:53, 8 January 2024 Joel2 talk contribs created page SSC/Stability/BiPolytropes/Pt4 (Created page with " =Reconciliation= <table border="0" cellpadding="8" align="right"> <tr> <th align="center">Figure 7: Conflicting Instability Regions</th> </tr> <tr> <td align="center" colspan="10">400px|Marginally unstable models</td> </tr> </table> Figure 7, shown here on the right, is identical to the right-hand panel of Figure 4, as displayed above. From a standard global, free-energy analysis — such as the one #Virial_Sta...")
20:26, 3 January 2024 Joel2 talk contribs created page File:Sterne1937CritGamma1.png
20:26, 3 January 2024 Joel2 talk contribs uploaded File:Sterne1937CritGamma1.png
20:21, 3 January 2024 Joel2 talk contribs created page SSC/Stability/GammaVariation (Created page with "=How Does Stability Change with γ<sub>g</sub>?= ==Isolated Uniform-Density Configuration== =How Does Stability Change with P<sub>e</sub>?=")

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