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17:58, 23 December 2023 Joel2 talk contribs created page SSC/Virial/Polytropes/Pt1 (Created page with "__FORCETOC__  =Virial Equilibrium of Adiabatic Spheres= Highlights of the rather detailed discussion presented below have been summarized in an accompanying chapter of this H_Book. ==Review== ===Adopted Normalizations=== In an introductory discussion of the virial equilibrium structure of spherically symmetric configuration...")
17:56, 23 December 2023 Joel2 talk contribs created page SSC/Virial/Polytropes/Pt2 (Created page with " ==Nonrotating Adiabatic Configuration Embedded in an External Medium== For a nonrotating configuration <math>~(C=J=0)</math> that is embedded in, and is influenced by the pressure <math>~P_e</math> of, an external medium, the statement of virial equilibrium is, <div align="center"> <math> 3 B\chi_\mathrm{eq}^{3 -3\gamma_g} -~3A\chi_\mathrm{eq}^{-1} -~ 3D\chi_\mathrm{eq}^3 = 0 \, . </math> </div> ====Solution Expressed in Terms of K and M (Whitworth's 1981 Relation)==...")
00:39, 23 December 2023 Joel2 talk contribs created page SSC/Structure/LimitingMasses (Created page with "__FORCETOC__ =Mass Upper Limits= Spherically symmetric, self-gravitating, equilibrium configurations can be constructed from gases exhibiting a wide variety of degrees of compressibility. When examining how the internal structure of such configurations varies with compressibility, or when examining the relative stability of such structures, it can be instructive to construct models using a polytropic equation of state because the degree of compressibility can be adjust...")
00:09, 23 December 2023 Joel2 talk contribs created page SSC/Virial/PolytropesEmbedded/FirstEffortAgain/Pt2 (Created page with " ==Nonrotating Adiabatic Configuration Embedded in an External Medium== For a nonrotating configuration <math>~(C=J=0)</math> that is embedded in, and is influenced by the pressure <math>~P_e</math> of, an external medium, the statement of virial equilibrium is, <div align="center"> <math> 3 B\chi_\mathrm{eq}^{3 -3\gamma_g} -~3A\chi_\mathrm{eq}^{-1} -~ 3D\chi_\mathrm{eq}^3 = 0 \, . </math> </div> ====Solution Expressed in Terms of K and M (Whitworth's 1981 Relation)==...")
00:08, 23 December 2023 Joel2 talk contribs created page SSC/Virial/PolytropesEmbedded/FirstEffortAgain/Pt1 (Created page with "__FORCETOC__  =Virial Equilibrium of Adiabatic Spheres= Highlights of the rather detailed discussion presented below have been summarized in an accompanying chapter of this H_Book. ==Review== ===Adopted Normalizations=== In an introductory discussion o...")
23:43, 22 December 2023 Joel2 talk contribs created page SSC/Structure/PowerLawDensity (Created page with "__FORCETOC__  =Power-Law Density Distributions= Here we begin with the same second-order, one-dimensional ODE that governs the structure of polytropic spheres, namely, the <div align="center"> <span id="LaneEmdenEquation"><font color="#770000">'''Lane-Emden Equation'''</font></span> {{ Math/EQ_SSLaneEmden01 }} </div> and examine whether or not this governing relation can be satisfied...")
23:32, 22 December 2023 Joel2 talk contribs created page File:FreeEnergySurface.png
23:32, 22 December 2023 Joel2 talk contribs uploaded File:FreeEnergySurface.png
23:32, 22 December 2023 Joel2 talk contribs created page File:FreeEnergyFaceOn.png
23:32, 22 December 2023 Joel2 talk contribs uploaded File:FreeEnergyFaceOn.png
23:31, 22 December 2023 Joel2 talk contribs created page File:VirialEquilibria.png
23:31, 22 December 2023 Joel2 talk contribs uploaded File:VirialEquilibria.png
23:20, 22 December 2023 Joel2 talk contribs created page File:Xi9.8461.png
23:20, 22 December 2023 Joel2 talk contribs uploaded File:Xi9.8461.png
23:19, 22 December 2023 Joel2 talk contribs created page File:Xi3.850652.png
23:19, 22 December 2023 Joel2 talk contribs uploaded File:Xi3.850652.png
23:19, 22 December 2023 Joel2 talk contribs created page File:Xi3.5.png
23:19, 22 December 2023 Joel2 talk contribs uploaded File:Xi3.5.png
23:18, 22 December 2023 Joel2 talk contribs created page File:Xi3.0.png
23:18, 22 December 2023 Joel2 talk contribs uploaded File:Xi3.0.png
23:18, 22 December 2023 Joel2 talk contribs created page File:Xi2.0.png
23:18, 22 December 2023 Joel2 talk contribs uploaded File:Xi2.0.png
23:18, 22 December 2023 Joel2 talk contribs created page File:Xi1.26419.png
23:18, 22 December 2023 Joel2 talk contribs uploaded File:Xi1.26419.png
22:43, 22 December 2023 Joel2 talk contribs created page File:CorrectedStahlerN5.png
22:43, 22 December 2023 Joel2 talk contribs uploaded File:CorrectedStahlerN5.png
22:13, 22 December 2023 Joel2 talk contribs created page File:AdabaticBoundedSpheres Virial.jpg
22:13, 22 December 2023 Joel2 talk contribs uploaded File:AdabaticBoundedSpheres Virial.jpg
21:50, 22 December 2023 Joel2 talk contribs created page SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt2 (Created page with " ==Mass-Radius Relation== Up to this point in our discussion, we have focused on an analysis of the pressure-radius relationship that defines the equilibrium configurations of pressure-truncated polytropes. In effect, we have viewed the problem through the same lens as did {{ Horedt70full }} and, separately, {{ Whitworth81full }}, defining variable normalizations in terms of the polytropic constant, <math>K</math>, and the configuration mass, <math>M_\mathrm{tot}</math>...")
21:49, 22 December 2023 Joel2 talk contribs created page SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt3 (Created page with " ==Discussion== A spherically symmetric, self-gravitating gas cloud whose effective adiabatic exponent is <math>\gamma < 4/3</math> — equivalently, <math>n > 3</math> — cannot exist in a dynamically stable equilibrium state, in isolation. Such clouds can be stabilized, however, if they are embedded in a hot, tenuous external medium and effectively confined by an external pressure, <math>~P_e</math>. The pressure-truncated, <math>n = 5 (\gamma = 6/5)</math>...")
21:43, 22 December 2023 Joel2 talk contribs created page SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt1 (Created page with "__FORCETOC__   =Virial Equilibrium of Adiabatic Spheres (Summary)= The summary presented here has been drawn from our accompanying detailed analysis of the structure of pressure-truncated polytropes. ==Detailed Force-Balanced Solution== As has been SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres|discussed in de...")
19:31, 22 December 2023 Joel2 talk contribs created page SSC/FreeEnergy/PolytropesEmbedded/Pt3C (Created page with "=Free Energy of Embedded Polytropes= <table border="1" align="center" width="100%" colspan="8"> <tr> <td align="center" rowspan="2" bgcolor="lightblue" width="25%"><br />Part I: Synopsis<br /> </td> <td align="center" rowspan="2" bgcolor="lightblue" width="25%"><br />Part II: Truncated Polytropes<br /> </td> <td align="center" rowspan="1" colspan="3"bgcolor="l...")
19:28, 22 December 2023 Joel2 talk contribs created page SSC/FreeEnergy/PolytropesEmbedded/Pt3B (Created page with "=Free Energy of Embedded Polytropes= <table border="1" align="center" width="100%" colspan="8"> <tr> <td align="center" rowspan="2" bgcolor="lightblue" width="25%"><br />Part I: Synopsis<br /> </td> <td align="center" rowspan="2" bgcolor="lightblue" width="25%"><br />Part II: Truncated Polytropes<br /> </td> <td align="center" rowspan="1" colspan="3"bgcolor="l...")
19:26, 22 December 2023 Joel2 talk contribs created page SSC/FreeEnergy/PolytropesEmbedded/Pt3A (Created page with "=Free Energy of Embedded Polytropes= <table border="1" align="center" width="100%" colspan="8"> <tr> <td align="center" rowspan="2" bgcolor="lightblue" width="25%"><br />Part I: Synopsis<br /> </td> <td align="center" rowspan="2" bgcolor="lightblue" width="25%"><br />Part II: Truncated Polytropes<br /> </td> <td align="center" rowspan="1" colspan="3"bgcolor="l...")
19:23, 22 December 2023 Joel2 talk contribs created page SSC/FreeEnergy/PolytropesEmbedded/Pt2 (Created page with "=Free Energy of Embedded Polytropes= <table border="1" align="center" width="100%" colspan="8"> <tr> <td align="center" rowspan="2" bgcolor="lightblue" width="25%"><br />Part I: Synopsis<br /> </td> <td align="center" rowspan="2" bgcolor="lightblue" width="25%"><br />Part II: Truncated Polytropes<br /> </td> <td align="center" rowspan="1" colspan="3"bgcolor="l...")
19:20, 22 December 2023 Joel2 talk contribs created page SSC/FreeEnergy/PolytropesEmbedded/Pt1 (Created page with "=Free Energy of Embedded Polytropes= <table border="1" align="center" width="100%" colspan="8"> <tr> <td align="center" rowspan="2" bgcolor="lightblue" width="25%"><br />Part I: Synopsis<br /> </td> <td align="center" rowspan="2" bgcolor="lightblue" width="25%"><br />Part II: Truncated Polytropes<br /> </td> <td align="center" rowspan="1" colspan="3"bgcolor="l...")
18:17, 22 December 2023 Joel2 talk contribs created page SSC/FreeEnergy/PolytropesEmbedded (Created page with "=Chapter Divisions= In <font color="red">October 2023</font>, this very long chapter was subdivided in order to more effectively accommodate edits. Here is a list of the resulting set of shorter chapters: <ol> <li>Free-Energy Synopsis</li> <li>Free-Energy of Truncated Polytropes</li> <li>Free-Energy of BiPolytropes <ul> <li>SSC/FreeEnergy/PolytropesEmbedded/Pt3A|Focus on Five...")
18:06, 22 December 2023 Joel2 talk contribs created page SSC/Structure/Polytropes/VirialSummary (Created page with "__FORCETOC__ =Virial Equilibrium of Pressure-Truncated Polytropes= {| 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>Virial Equilibrium<br />of<br />Pressure-Truncated<br />Polytropes</b></font> |} Here we will draw heavily from: [1] An accompanying SSC/FreeEnergy/PolytropesEmbedded#...")
20:38, 18 December 2023 Joel2 talk contribs created page Appendix/Ramblings/Radiation/RadHydro (Created page with "__FORCETOC__  =Radiation-Hydrodynamics= ==Governing Equations== ===Hayes et al. (2006) — But Ignoring the Effects of Magnetic Fields=== First, referencing §2 of [http://adsabs.harvard.edu/abs/2006ApJS..165..188H J. C. Hayes et al. (2006, ApJS, 165, 188 - 228)] — alternatively see §2.1 of [http://adsabs.harvard.edu/abs/2012ApJS..199...35M D. C. Marcello & J. E. Tohline (2012, ApJS, 199, id. 35, 29 pp)] &#821...")
20:14, 18 December 2023 Joel2 talk contribs created page File:YabushitaMontageB.png
20:14, 18 December 2023 Joel2 talk contribs uploaded File:YabushitaMontageB.png
20:11, 18 December 2023 Joel2 talk contribs created page File:Yabushita1975Combined.png
20:11, 18 December 2023 Joel2 talk contribs uploaded File:Yabushita1975Combined.png
20:09, 18 December 2023 Joel2 talk contribs created page File:IsothermalTrunc12.png
20:09, 18 December 2023 Joel2 talk contribs uploaded File:IsothermalTrunc12.png
20:09, 18 December 2023 Joel2 talk contribs created page File:IsothermalTrunc11.png
20:09, 18 December 2023 Joel2 talk contribs uploaded File:IsothermalTrunc11.png
20:08, 18 December 2023 Joel2 talk contribs created page File:IsothermalTrunc4.png
20:08, 18 December 2023 Joel2 talk contribs uploaded File:IsothermalTrunc4.png
20:06, 18 December 2023 Joel2 talk contribs created page File:DataFileButton02.png

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