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SSC/FreeEnergy/PolytropesEmbedded/Pt3B - Revision history
2026-06-21T07:49:59Z
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Jet53man: /* Free-Energy of Bipolytropes */
2023-10-15T17:51:41Z
<p><span class="autocomment">Free-Energy of Bipolytropes</span></p>
<a href="https://tohline.education/SelfGravitatingFluids/index.php?title=SSC/FreeEnergy/PolytropesEmbedded/Pt3B&diff=69019&oldid=69018">Show changes</a>
Jet53man
https://tohline.education/SelfGravitatingFluids/index.php?title=SSC/FreeEnergy/PolytropesEmbedded/Pt3B&diff=69018&oldid=prev
Jet53man at 17:50, 15 October 2023
2023-10-15T17:50:33Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 13:50, 15 October 2023</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=See Also=</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=See Also=</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">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:</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><ol></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> <li>[[SSC/FreeEnergy/PolytropesEmbedded/Pt1|Free-Energy Synopsis]]</li></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> <li>[[SSC/FreeEnergy/PolytropesEmbedded/Pt2|Free-Energy of Truncated Polytropes]]</li></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> <li>Free-Energy of BiPolytropes</ins></div></td></tr>
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<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><li>[[SSC/FreeEnergy/PolytropesEmbedded/Pt3A|Focus on Five-One Free-Energy Expression]]</li></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><li>[[SSC/FreeEnergy/PolytropesEmbedded/Pt3B|Focus on Zero-Zero Free-Energy Expression]]</li></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><li>[[SSC/FreeEnergy/PolytropesEmbedded/Pt3C|Overview]]</li></ins></div></td></tr>
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Jet53man
https://tohline.education/SelfGravitatingFluids/index.php?title=SSC/FreeEnergy/PolytropesEmbedded/Pt3B&diff=69013&oldid=prev
Jet53man: Created page with "__FORCETOC__ =Background= Index to original, very long chapter =Free-Energy of Bipolytropes= In this case, the Gibbs-like free energy is given by the sum of four separate energies, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\mathfrak{G}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl[W_\mathrm{grav} + \mat..."
2023-10-15T17:41:51Z
<p>Created page with "__FORCETOC__ =Background= <a href="/SelfGravitatingFluids/index.php/SSC/FreeEnergy/PolytropesEmbedded#Chapter_Divisions" title="SSC/FreeEnergy/PolytropesEmbedded">Index to original, very long chapter</a> =Free-Energy of Bipolytropes= In this case, the Gibbs-like free energy is given by the sum of four separate energies, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\mathfrak{G}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl[W_\mathrm{grav} + \mat..."</p>
<p><b>New page</b></p><div>__FORCETOC__<br />
=Background=<br />
[[SSC/FreeEnergy/PolytropesEmbedded#Chapter_Divisions|Index to original, very long chapter]]<br />
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=Free-Energy of Bipolytropes=<br />
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In this case, the Gibbs-like free energy is given by the sum of four separate energies,<br />
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<math>~\mathfrak{G}</math><br />
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<math>~=</math><br />
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<math>~<br />
\biggl[W_\mathrm{grav} + \mathfrak{S}_\mathrm{therm}\biggr]_\mathrm{core} + \biggl[W_\mathrm{grav} + \mathfrak{S}_\mathrm{therm}\biggr]_\mathrm{env} \, .<br />
</math><br />
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In addition to specifying (generally) separate polytropic indexes for the core, <math>~n_c</math>, and envelope, <math>~n_e</math>, and an envelope-to-core mean molecular weight ratio, <math>~\mu_e/\mu_c</math>, we will assume that the system is fully defined via specification of the following five physical parameters:<br />
* Total mass, <math>~M_\mathrm{tot}</math>;<br />
* Total radius, <math>~R</math>; <br />
* Interface radius, <math>~R_i</math>, and associated dimensionless interface marker, <math>~q \equiv R_i/R</math>;<br />
* Core mass, <math>~M_c</math>, and associated dimensionless mass fraction, <math>~\nu \equiv M_c/M_\mathrm{tot}</math>;<br />
* Polytropic constant in the core, <math>~K_c</math>.<br />
<br />
In general, the warped free-energy surface drapes across a five-dimensional parameter "plane" such that,<br />
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<math>~\mathfrak{G}</math><br />
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<math>~=</math><br />
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<math>~\mathfrak{G}(R, K_c, M_\mathrm{tot}, q, \nu) \, .</math><br />
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=See Also=<br />
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Jet53man