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SSC/Structure/BiiPolytropes/FreeEnergy51 - Revision history
2026-04-22T04:15:05Z
Revision history for this page on the wiki
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https://selfgravitatingfluids.education/JETohline/index.php?title=SSC/Structure/BiiPolytropes/FreeEnergy51&diff=1322&oldid=prev
Joel2 at 19:04, 18 January 2024
2024-01-18T19:04:36Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 15:04, 18 January 2024</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>and note that the expression for the coefficient, <math>~b_\eta</math>, becomes simpler as well, specifically,</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>and note that the expression for the coefficient, <math>~b_\eta</math>, becomes simpler as well, specifically,</div></td></tr>
<tr><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: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><div align="center"></div></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><div align="center<ins style="font-weight: bold; text-decoration: none;">" id="EquilibriumRadius</ins>"></div></td></tr>
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Joel2
https://selfgravitatingfluids.education/JETohline/index.php?title=SSC/Structure/BiiPolytropes/FreeEnergy51&diff=1321&oldid=prev
Joel2: /* The Envelope */
2024-01-18T18:58:39Z
<p><span class="autocomment">The Envelope</span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 14:58, 18 January 2024</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l697">Line 697:</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></div></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></div></div></td></tr>
<tr><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: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>But, from the [[<del style="font-weight: bold; text-decoration: none;">SSC/Structure/BiiPolytropes/FreeEnergy51</del>#<del style="font-weight: bold; text-decoration: none;">The_Core</del>|above derivation of the mass profile in the core]], we know that,</div></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>But, from the [[#<ins style="font-weight: bold; text-decoration: none;">DensityProfile</ins>|above derivation of the mass profile in the core]], we know that,</div></td></tr>
<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><div align="center"></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><div align="center"></div></td></tr>
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Joel2
https://selfgravitatingfluids.education/JETohline/index.php?title=SSC/Structure/BiiPolytropes/FreeEnergy51&diff=1320&oldid=prev
Joel2: /* The Core */
2024-01-18T18:57:19Z
<p><span class="autocomment">The Core</span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 14:57, 18 January 2024</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>in which case,</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>in which case,</div></td></tr>
<tr><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: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><div align="center"></div></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><div align="center<ins style="font-weight: bold; text-decoration: none;">" id="DensityProfile</ins>"></div></td></tr>
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Joel2
https://selfgravitatingfluids.education/JETohline/index.php?title=SSC/Structure/BiiPolytropes/FreeEnergy51&diff=1319&oldid=prev
Joel2: /* The Envelope */
2024-01-18T18:53:16Z
<p><span class="autocomment">The Envelope</span></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 14:53, 18 January 2024</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></div></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></div></div></td></tr>
<tr><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: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>But, from the [[SSC/Structure/<del style="font-weight: bold; text-decoration: none;">BiPolytropes</del>/FreeEnergy51#The_Core|above derivation of the mass profile in the core]], we know that,</div></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>But, from the [[SSC/Structure/<ins style="font-weight: bold; text-decoration: none;">BiiPolytropes</ins>/FreeEnergy51#The_Core|above derivation of the mass profile in the core]], we know that,</div></td></tr>
<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><div align="center"></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><div align="center"></div></td></tr>
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Joel2
https://selfgravitatingfluids.education/JETohline/index.php?title=SSC/Structure/BiiPolytropes/FreeEnergy51&diff=1315&oldid=prev
Joel2 at 18:41, 18 January 2024
2024-01-18T18:41:58Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 14:41, 18 January 2024</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l2">Line 2:</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>=Free Energy of BiPolytrope with (n<sub>c</sub>, n<sub>e</sub>) = (5, 1)=</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>=Free Energy of BiPolytrope with (n<sub>c</sub>, n<sub>e</sub>) = (5, 1)=</div></td></tr>
<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><table border="1" align="center" width="100%" colspan="8"></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><table border="1" align="center" width="100%" colspan="8"></div></td></tr>
<tr><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: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> <td align="center" bgcolor="lightblue" width="33%"><br />[[SSC/Structure/<del style="font-weight: bold; text-decoration: none;">BiPolytropes</del>/FreeEnergy51|Part I:&nbsp; Mass Profile]]</div></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> <td align="center" bgcolor="lightblue" width="33%"><br />[[SSC/Structure/<ins style="font-weight: bold; text-decoration: none;">BiiPolytropes</ins>/FreeEnergy51|Part I:&nbsp; Mass Profile]]</div></td></tr>
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Joel2
https://selfgravitatingfluids.education/JETohline/index.php?title=SSC/Structure/BiiPolytropes/FreeEnergy51&diff=1314&oldid=prev
Joel2: Created page with "__FORCETOC__ =Free Energy of BiPolytrope with (n<sub>c</sub>, n<sub>e</sub>) = (5, 1)= <table border="1" align="center" width="100%" colspan="8"> <td align="center" bgcolor="lightblue" width="33%"><br />Part I: Mass Profile </td> <td align="center" bgcolor="lightblue"3 width="33%"><br />Part II: Gravitational Potential Energy </td> <td align="ce..."
2024-01-18T18:40:37Z
<p>Created page with "__FORCETOC__ =Free Energy of BiPolytrope with (n<sub>c</sub>, n<sub>e</sub>) = (5, 1)= <table border="1" align="center" width="100%" colspan="8"> <td align="center" bgcolor="lightblue" width="33%"><br /><a href="/JETohline/index.php/SSC/Structure/BiPolytropes/FreeEnergy51" title="SSC/Structure/BiPolytropes/FreeEnergy51">Part I: Mass Profile</a> </td> <td align="center" bgcolor="lightblue"3 width="33%"><br /><a href="/JETohline/index.php/SSC/Structure/BiPolytropes/FreeEnergy51/Pt2" title="SSC/Structure/BiPolytropes/FreeEnergy51/Pt2">Part II: Gravitational Potential Energy</a> </td> <td align="ce..."</p>
<p><b>New page</b></p><div>__FORCETOC__<br />
=Free Energy of BiPolytrope with (n<sub>c</sub>, n<sub>e</sub>) = (5, 1)=<br />
<table border="1" align="center" width="100%" colspan="8"><br />
<td align="center" bgcolor="lightblue" width="33%"><br />[[SSC/Structure/BiPolytropes/FreeEnergy51|Part I:&nbsp; Mass Profile]]<br />
&nbsp;<br />
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<td align="center" bgcolor="lightblue"3 width="33%"><br />[[SSC/Structure/BiPolytropes/FreeEnergy51/Pt2|Part II:&nbsp; Gravitational Potential Energy]]<br />
&nbsp;<br />
</td><br />
<td align="center" bgcolor="lightblue"><br />[[SSC/Structure/BiPolytropes/FreeEnergy51/Pt3|Part III:&nbsp; Thermal Energy Reservoir]]<br />
&nbsp;<br />
</td><br />
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</table><br />
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{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"<br />
|- <br />
! style="height: 125px; width: 125px; background-color:white;" |<font size="-1">[[H_BookTiledMenu#MoreModels|<b>Free Energy<br />of<br />Bipolytropes</b>]]<br />(n<sub>c</sub>, n<sub>e</sub>) = (5, 1)</font><br />
|}<br />
Here we present a specific example of the equilibrium structure of a bipolytrope as determined from a free-energy analysis. The example is a bipolytrope whose core has a polytropic index, <math>n_c = 5</math>, and whose envelope has a polytropic index, <math>n_e = 1</math>. The details presented here build upon an [[SSC/BipolytropeGeneralizationVersion2|overview of the free energy of bipolytropes that has been presented elsewhere]].<br />
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<br />
==Preliminaries==<br />
<br />
==Mass Profile==<br />
<br />
===The Core===<br />
The core has <math>~n_c = 5 \Rightarrow \gamma_c = 1+1/n_c = 6/5</math>. Referring to the general relation [[SSC/BipolytropeGeneralizationVersion2#Partitioning_the_Mass|as established in our accompanying overview]], and using <math>~\rho_0</math> to represent the central density, we can write,<br />
<div align="center"><br />
<table border="0" cellpadding="5" align="center"><br />
<br />
<tr><br />
<td align="right"><br />
<math>(\mathrm{For}~0 \leq x \leq q)</math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br />
<math>~M_r </math><br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math><br />
M_\mathrm{tot} \biggl( \frac{\nu}{q^3} \biggr) \biggl( \frac{\rho_0} {{\bar\rho}_\mathrm{core}}\biggr)_\mathrm{eq} \int_0^{x} 3<br />
\biggl[ \frac{\rho(x)}{\rho_0} \biggr]_\mathrm{core} <br />
x^2 dx \, . <br />
</math><br />
</td><br />
</tr><br />
<br />
</table><br />
</div><br />
<br />
Drawing on the derivation of [[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1|detailed force-balance models of <math>~(n_c, n_e) = (5, 1)</math> bipolytropes]], the density profile [[SSC/Structure/BiPolytropes/Analytic51#Step_4:__Throughout_the_core_.28.29|throughout the core]] is,<br />
<div align="center"><br />
<table border="0" cellpadding="5" align="center"><br />
<br />
<tr><br />
<td align="right"><br />
<math>~\biggl[ \frac{\rho(\xi)}{\rho_0} \biggr]_\mathrm{core}</math><br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math>~\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2} \, ,</math><br />
</td><br />
</tr><br />
</table><br />
</div><br />
where the dimensionless radial coordinate is,<br />
<div align="center"><br />
<table border="0" cellpadding="5" align="center"><br />
<tr><br />
<td align="right"><br />
<math>~\xi</math><br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math>~\biggl[ \frac{G \rho_0^{4/5}}{K_c} \biggr]^{1/2} \biggl( \frac{2\pi}{3} \biggr)^{1/2} r \, .</math><br />
</td><br />
</tr><br />
</table><br />
</div><br />
Switching to the [[SSCpt1/Virial#Normalizations|normalizations that have been adopted in the broad context of our discussion of configurations in virial equilibrium]] and inserting the adiabatic index of the core <math>~(\gamma_c = 6/5)</math> into all normalization parameters, we have,<br />
<div align="center"><br />
<table border="0" cellpadding="5" align="center"><br />
<br />
<tr><br />
<td align="right"><br />
<math>~R_\mathrm{norm} = \biggl[ \biggl(\frac{G}{K_c} \biggr) M_\mathrm{tot}^{2-\gamma} \biggr]^{1/(4-3\gamma)}</math><br />
</td><br />
<td align="center"><br />
<math>~\Rightarrow</math><br />
</td><br />
<td align="left"><br />
<math>~R_\mathrm{norm} = \biggl( \frac{G^5 M_\mathrm{tot}^4}{K_c^5} \biggr)^{1/2} \, ,</math><br />
</td><br />
</tr><br />
<br />
<tr><br />
<td align="right"><br />
<math>~\rho_\mathrm{norm} = \frac{3}{4\pi} \biggl[ \frac{K_c^3}{G^3 M_\mathrm{tot}^2} \biggr]^{1/(4-3\gamma)}</math><br />
</td><br />
<td align="center"><br />
<math>~\Rightarrow</math><br />
</td><br />
<td align="left"><br />
<math>~\rho_\mathrm{norm} = \frac{3}{4\pi} \biggl( \frac{K_c^{3}}{G^3 M_\mathrm{tot}^2} \biggr)^{5/2} \, .</math><br />
</td><br />
</tr><br />
</table><br />
</div><br />
Hence, we can rewrite,<br />
<br />
<div align="center"><br />
<table border="0" cellpadding="5" align="center"><br />
<tr><br />
<td align="right"><br />
<math>~\xi</math><br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math>~\biggl( \frac{r}{R_\mathrm{norm}} \biggr) \biggl( \frac{\rho_0}{\rho_\mathrm{norm}} \biggr)^{2/5} \biggl[ \frac{G }{K_c} \biggr]^{1/2} <br />
\biggl( \frac{2\pi}{3} \biggr)^{1/2} R_\mathrm{norm} \rho_\mathrm{norm}^{2/5}</math><br />
</td><br />
</tr><br />
<br />
<tr><br />
<td align="right"><br />
&nbsp;<br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math>~r^* (\rho_0^*)^{2/5} \biggl[ \frac{G }{K_c} \biggr]^{1/2} \biggl( \frac{2\pi}{3} \biggr)^{1/2} <br />
\biggl( \frac{G^5 M_\mathrm{tot}^4}{K_c^5} \biggr)^{1/2} \biggl( \frac{3}{4\pi} \biggr)^{2/5} \biggl( \frac{K_c^{3}}{G^3 M_\mathrm{tot}^2} \biggr)</math><br />
</td><br />
</tr><br />
<br />
<tr><br />
<td align="right"><br />
&nbsp;<br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math><br />
~r^* (\rho_0^*)^{2/5} \biggl[ \biggl( \frac{2\pi}{3} \biggr)^{5} \biggl( \frac{3}{4\pi} \biggr)^{4} \biggr]^{1/10}<br />
= r^* (\rho_0^*)^{2/5} \biggl[ \frac{\pi}{2^3 \cdot 3}\biggr]^{1/10} \, .<br />
</math><br />
</td><br />
</tr><br />
</table><br />
</div><br />
Now, following the same approach as was used in our [[SSCpt1/Virial#Separate_Time_.26_Space|introductory discussion]] and appreciating that our aim here is to redefine the coordinate, <math>~\xi</math>, in terms of normalized parameters evaluated ''in the equilibrium configuration,'' we will set,<br />
<div align="center"><br />
<table border="0" cellpadding="5" align="center"><br />
<br />
<tr><br />
<td align="right"><br />
<math>~r^*</math><br />
</td><br />
<td align="center"><br />
<math>~\rightarrow~</math><br />
</td><br />
<td align="left"><br />
<math><br />
~ x \chi_\mathrm{eq} \, ;<br />
</math><br />
</td><br />
</tr><br />
<br />
<tr><br />
<td align="right"><br />
<math>~\rho_0^*</math><br />
</td><br />
<td align="center"><br />
<math>~\rightarrow~</math><br />
</td><br />
<td align="left"><br />
<math><br />
\biggl[ \frac{\rho_0}{\bar\rho} \biggr]_\mathrm{core} \biggl( \frac{{\bar\rho}_\mathrm{core}}{\rho_\mathrm{norm}} \biggr)<br />
= \biggl[ \frac{\rho_0}{\bar\rho} \biggr]_\mathrm{core} \frac{\nu M_\mathrm{tot}/(q^3 R_\mathrm{edge}^3)_\mathrm{eq}}{M_\mathrm{tot}/R_\mathrm{norm}^3} <br />
= \frac{\nu}{q^3} \biggl[ \frac{\rho_0}{\bar\rho} \biggr]_\mathrm{core} \chi_\mathrm{eq}^{-3} \, .<br />
</math><br />
</td><br />
</tr><br />
<br />
</table><br />
</div><br />
Then we can set,<br />
<div align="center"><br />
<table border="0" cellpadding="5" align="center"><br />
<tr><br />
<td align="right"><br />
<math>~\xi</math><br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math>~(3a_\xi)^{1/2} x \, ,</math><br />
</td><br />
</tr><br />
</table><br />
</div><br />
in which case,<br />
<div align="center"><br />
<table border="0" cellpadding="5" align="center"><br />
<br />
<tr><br />
<td align="right"><br />
<math>~\biggl[ \frac{\rho(x)}{\rho_0} \biggr]_\mathrm{core}</math><br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math>~\biggl( 1 + a_\xi x^2 \biggr)^{-5/2} \, ,</math><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br />
where the coefficient,<br />
<div align="center"><br />
<table border="0" cellpadding="5" align="center"><br />
<br />
<tr><br />
<td align="right"><br />
<math>~(3a_\xi)^{1/2}</math><br />
</td><br />
<td align="center"><br />
<math>~\equiv</math><br />
</td><br />
<td align="left"><br />
<math>~<br />
\chi_\mathrm{eq} \biggl[ \frac{\nu}{q^3} \biggl( \frac{\rho_0}{\bar\rho} \biggr)_\mathrm{core} \chi_\mathrm{eq}^{-3} \biggr]^{2/5} <br />
\biggl( \frac{\pi}{2^3 \cdot 3}\biggr)^{1/10} <br />
=\chi_\mathrm{eq}^{-1/5} \biggl[ \frac{\nu}{q^3} \biggl( \frac{\rho_0}{\bar\rho} \biggr)_\mathrm{core} \biggr]_\mathrm{eq}^{2/5} <br />
\biggl( \frac{\pi}{2^3 \cdot 3}\biggr)^{1/10} <br />
</math><br />
</td><br />
</tr><br />
<br />
<tr><br />
<td align="right"><br />
<math>\Rightarrow~~~~a_\xi</math><br />
</td><br />
<td align="center"><br />
<math>~\equiv</math><br />
</td><br />
<td align="left"><br />
<math>~<br />
\frac{1}{3} \biggl\{ \chi_\mathrm{eq}^{-1/5} \biggl[ \frac{\nu}{q^3} \biggl( \frac{\rho_0}{\bar\rho} \biggr)_\mathrm{core} \biggr]_\mathrm{eq}^{2/5} <br />
\biggl( \frac{\pi}{2^3 \cdot 3}\biggr)^{1/10} \biggr\}^2<br />
= \chi_\mathrm{eq}^{-2/5} \biggl[ \frac{\nu}{q^3} \biggl( \frac{\rho_0}{\bar\rho} \biggr)_\mathrm{core} \biggr]_\mathrm{eq}^{4/5} <br />
\biggl( \frac{\pi}{2^3 \cdot 3^6}\biggr)^{1/5} \, .<br />
</math><br />
</td><br />
</tr><br />
</table><br />
</div><br />
We therefore have,<br />
<div align="center"><br />
<table border="0" cellpadding="5" align="center"><br />
<br />
<tr><br />
<td align="right"><br />
<math>~M_r\biggr|_\mathrm{core} </math><br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math><br />
M_\mathrm{tot} \biggl[ \frac{\nu}{q^3} \biggl( \frac{\rho_0} {{\bar\rho}}\biggr)_\mathrm{core} \biggr]_\mathrm{eq} \int_0^{x} 3<br />
\biggl( 1 + a_\xi x^2 \biggr)^{-5/2} x^2 dx <br />
</math><br />
</td><br />
</tr><br />
<br />
<tr><br />
<td align="right"><br />
&nbsp;<br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math><br />
M_\mathrm{tot} \biggl[ \frac{\nu}{q^3} \biggl( \frac{\rho_0} {{\bar\rho}}\biggr)_\mathrm{core} \biggr]_\mathrm{eq} <br />
\biggl[ x^3\biggl( 1 + a_\xi x^2 \biggr)^{-3/2} \biggr] \, .<br />
</math><br />
</td><br />
</tr><br />
<br />
</table><br />
</div><br />
Note that, when <math>~x \rightarrow q</math>, <math>~M_r|_\mathrm{core} \rightarrow M_\mathrm{core} = \nu M_\mathrm{tot}</math>. Hence, this last expression gives,<br />
<div align="center"><br />
<table border="0" cellpadding="5" align="center"><br />
<br />
<tr><br />
<td align="right"><br />
<math>~\nu M_\mathrm{tot}</math><br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math><br />
M_\mathrm{tot} \biggl[ \frac{\nu}{q^3} \biggl( \frac{\rho_0} {{\bar\rho}}\biggr)_\mathrm{core} \biggr]_\mathrm{eq} <br />
\biggl[ q^3\biggl( 1 + a_\xi q^2 \biggr)^{-3/2} \biggr] <br />
</math><br />
</td><br />
</tr><br />
<br />
<tr><br />
<td align="right"><br />
<math>\Rightarrow~~~~\biggl[\biggl( \frac{\rho_0} {{\bar\rho}}\biggr)_\mathrm{core} \biggr]_\mathrm{eq}</math><br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math><br />
\biggl( 1 + a_\xi q^2 \biggr)^{3/2} \, .<br />
</math><br />
</td><br />
</tr><br />
</table><br />
</div><br />
Hence, finally,<br />
<div align="center"><br />
<table border="0" cellpadding="5" align="center"><br />
<br />
<tr><br />
<td align="right"><br />
<math>~M_r\biggr|_\mathrm{core} </math><br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math><br />
\nu M_\mathrm{tot} \biggl( \frac{x^3}{q^3} \biggr) <br />
\biggl[ \frac{ 1 + a_\xi x^2 }{ 1 + a_\xi q^2 } \biggr]^{-3/2} \, ;<br />
</math><br />
</td><br />
</tr><br />
<br />
</table><br />
</div><br />
and, after the equilibrium radius, <math>~\chi_\mathrm{eq}</math>, has been determined from the free-energy analysis, the coefficient, <math>~a_\xi</math>, can be determined via the relation,<br />
<div align="center"><br />
<table border="0" cellpadding="5" align="center"><br />
<tr><br />
<td align="right"><br />
<math>~\chi_\mathrm{eq}^{2} </math><br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math>~\biggl( \frac{\pi}{2^3 \cdot 3^6}\biggr)<br />
\biggl( \frac{\nu}{q^3} \biggr)^{4} \biggl( 1 + a_\xi q^2 \biggr)^{6} a_\xi^{-5} \, .<br />
</math><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br />
<table border="1" cellpadding="5" align="center" width="90%"><br />
<tr><td align="left"><br />
<br />
MORE USEFUL:<br />
<br />
Letting, <math>\ell \equiv \xi/\sqrt{3}</math>,<br />
<div align="center"><br />
<table border="0" cellpadding="5" align="center"><br />
<br />
<tr><br />
<td align="right"><br />
<math>~a_\xi</math><br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math>~\biggl( \frac{\ell_i}{q} \biggr)^2 \, ,</math><br />
</td><br />
</tr><br />
<br />
<tr><br />
<td align="right"><br />
<math>~\tilde\mathfrak{f}_\mathrm{Mcore}</math><br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math>~\biggl( \frac{\bar\rho}{\rho_c} \biggr)_\mathrm{core}<br />
=<br />
(1 + \ell_i^2)^{-3/2} \, .<br />
</math><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br />
</td></tr><br />
</table><br />
<br />
===The Envelope===<br />
The envelope has <math>~n_e = 1 \Rightarrow \gamma_e = 1+1/n_e = 2</math>. Again, referring to the general relation [[SSC/BipolytropeGeneralizationVersion2#Partitioning_the_Mass|as established in our accompanying overview]], and continuing to use <math>~\rho_0</math> to represent the central density, we can write,<br />
<div align="center"><br />
<table border="0" cellpadding="5" align="center"><br />
<br />
<tr><br />
<td align="right"><br />
<math>(\mathrm{For}~q \leq x \leq 1)</math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br />
<math>~M_r </math><br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math><br />
M_\mathrm{tot} \biggl\{\nu + \biggl( \frac{1-\nu}{1-q^3} \biggr) \int_{x_i}^{x} 3<br />
\biggl[ \frac{\rho(x)}{\bar\rho} \biggr]_\mathrm{env} <br />
x^2 dx \biggr\} \, .<br />
</math><br />
</td><br />
</tr><br />
<br />
</table><br />
</div><br />
<br />
Drawing on the derivation of [[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1|detailed force-balance models of <math>~(n_c, n_e) = (5, 1)</math> bipolytropes]], the density profile [[SSC/Structure/BiPolytropes/Analytic51#Step_8:__Throughout_the_envelope_.28.2|throughout the envelope]] is,<br />
<div align="center"><br />
<table border="0" cellpadding="5" align="center"><br />
<br />
<tr><br />
<td align="right"><br />
<math>~\biggl[ \frac{\rho(\eta)}{\rho_0} \biggr]_\mathrm{env}</math><br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math>~A \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta_i^5 \biggl[ \frac{\sin(\eta - B)}{\eta} \biggr] \, ,</math><br />
</td><br />
</tr><br />
</table><br />
</div><br />
where definitions of the constants <math>~A</math> and <math>~B</math> are given in an [[SSC/Structure/BiPolytropes/Analytic51#Parameter_Values|accompanying table of parameter values]], and the dimensionless radial coordinate is,<br />
<div align="center"><br />
<table border="0" cellpadding="5" align="center"><br />
<tr><br />
<td align="right"><br />
<math>~\eta</math><br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math>~\biggl[ \frac{G \rho_0^{4/5}}{K_c} \biggr]^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta_i^2 (2\pi)^{1/2} r \, .</math><br />
</td><br />
</tr><br />
</table><br />
</div><br />
Using the same radial and mass-density normalizations as defined, above, for the core, we can write,<br />
<div align="center"><br />
<table border="0" cellpadding="5" align="center"><br />
<tr><br />
<td align="right"><br />
<math>~\eta</math><br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math>~r^* (\rho_0^*)^{2/5}\biggl[ \frac{G}{K_c} \biggr]^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta_i^2 (2\pi)^{1/2} <br />
R_\mathrm{norm} \rho_\mathrm{norm}^{2/5}</math><br />
</td><br />
</tr><br />
<br />
<tr><br />
<td align="right"><br />
&nbsp;<br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math><br />
~r^* (\rho_0^*)^{2/5} \biggl( \frac{3}{4\pi} \biggr)^{2/5} \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta_i^2 (2\pi)^{1/2} \, .<br />
</math><br />
</td><br />
</tr><br />
</table><br />
</div><br />
Next, we set,<br />
<div align="center"><br />
<table border="0" cellpadding="5" align="center"><br />
<br />
<tr><br />
<td align="right"><br />
<math>~r^*</math><br />
</td><br />
<td align="center"><br />
<math>~\rightarrow~</math><br />
</td><br />
<td align="left"><br />
<math><br />
~ x \chi_\mathrm{eq} \, ;<br />
</math><br />
</td><br />
</tr><br />
<br />
<tr><br />
<td align="right"><br />
<math>~\rho_0^*</math><br />
</td><br />
<td align="center"><br />
<math>~\rightarrow~</math><br />
</td><br />
<td align="left"><br />
<math><br />
\biggl[ \frac{\rho_0}{\bar\rho} \biggr]_\mathrm{env} \biggl( \frac{{\bar\rho}_\mathrm{env}}{\rho_\mathrm{norm}} \biggr)<br />
= \biggl[ \frac{\rho_0}{\bar\rho} \biggr]_\mathrm{env} \frac{(1-\nu) M_\mathrm{tot}/[(1-q^3) R_\mathrm{edge}^3]_\mathrm{eq}}{M_\mathrm{tot}/R_\mathrm{norm}^3} <br />
= \frac{1-\nu}{1-q^3} \biggl( \frac{\rho_0}{\bar\rho} \biggr)_\mathrm{env} \chi_\mathrm{eq}^{-3} \, .<br />
</math><br />
</td><br />
</tr><br />
<br />
</table><br />
</div><br />
Hence, we can write,<br />
<div align="center"><br />
<math>~\eta = b_\eta x \, ,</math><br />
</div><br />
where,<br />
<div align="center"><br />
<table border="0" cellpadding="5" align="center"><br />
<tr><br />
<td align="right"><br />
<math>~b_\eta</math><br />
</td><br />
<td align="center"><br />
<math>~\equiv</math><br />
</td><br />
<td align="left"><br />
<math><br />
~\chi_\mathrm{eq}^{-1/5} \biggl[ \frac{1-\nu}{1-q^3} \biggl( \frac{\rho_0}{\bar\rho} \biggr)_\mathrm{env} \biggr]^{2/5} <br />
\biggl( \frac{3}{4\pi} \biggr)^{2/5} \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta_i^2 (2\pi)^{1/2} \, .<br />
</math><br />
</td><br />
</tr><br />
</table><br />
</div><br />
In which case,<br />
<div align="center"><br />
<table border="0" cellpadding="5" align="center"><br />
<br />
<tr><br />
<td align="right"><br />
<math>~\biggl[ \frac{\rho(x)}{\rho_0} \biggr]_\mathrm{env}</math><br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math>~A \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta_i^5 \biggl[ \frac{\sin(b_\eta x - B)}{b_\eta x} \biggr] \, ,</math><br />
</td><br />
</tr><br />
</table><br />
</div><br />
so,<br />
<div align="center"><br />
<table border="0" cellpadding="5" align="center"><br />
<br />
<tr><br />
<td align="right"><br />
<math>~M_r \biggr|_\mathrm{env}</math><br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math><br />
M_\mathrm{tot} \biggl\{\nu + \biggl[ \biggl( \frac{1-\nu}{1-q^3} \biggr) \biggl( \frac{\rho_0}{\bar\rho}\biggr)_\mathrm{env} \biggr]_\mathrm{eq} \int_{q}^{x} 3<br />
A \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta_i^5 \biggl[ \frac{\sin(b_\eta x - B)}{b_\eta x} \biggr] <br />
x^2 dx \biggr\} <br />
</math><br />
</td><br />
</tr><br />
<br />
<tr><br />
<td align="right"><br />
&nbsp;<br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math>\nu M_\mathrm{tot} + <br />
M_\mathrm{tot} \biggl[ \biggl( \frac{1-\nu}{1-q^3} \biggr) \biggl( \frac{\rho_0}{\bar\rho}\biggr)_\mathrm{env} \biggr]_\mathrm{eq} \biggl( \frac{\mu_e}{\mu_c} \biggr) <br />
\frac{3A\theta_i^5 }{b_\eta} <br />
\int_{q}^{x} \sin(b_\eta x - B) x dx <br />
</math><br />
</td><br />
</tr><br />
<br />
<tr><br />
<td align="right"><br />
&nbsp;<br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math>\nu M_\mathrm{tot} + <br />
M_\mathrm{tot} \biggl[ \biggl( \frac{1-\nu}{1-q^3} \biggr) \biggl( \frac{\rho_0}{\bar\rho}\biggr)_\mathrm{env} \biggr]_\mathrm{eq} \biggl( \frac{\mu_e}{\mu_c} \biggr) <br />
\frac{3A\theta_i^5 }{b_\eta} <br />
\biggl[ - \frac{\sin(B-b_\eta x) + b_\eta x \cos(B - b_\eta x)}{b_\eta^2} \biggr]_q^x <br />
</math><br />
</td><br />
</tr><br />
<br />
<tr><br />
<td align="right"><br />
&nbsp;<br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math>\nu M_\mathrm{tot} + <br />
M_\mathrm{tot} \biggl[ \biggl( \frac{1-\nu}{1-q^3} \biggr) \biggl( \frac{\rho_0}{\bar\rho}\biggr)_\mathrm{env} \biggr]_\mathrm{eq} \biggl( \frac{\mu_e}{\mu_c} \biggr) <br />
\frac{3A\theta_i^5 }{b_\eta^3} <br />
\biggl[ C_1-\sin(B-b_\eta x) -xb_\eta \cos(B - b_\eta x) \biggr] \, ,<br />
</math><br />
</td><br />
</tr><br />
<br />
</table><br />
</div><br />
where, <math>~C_1</math> is a constant obtained by evaluating the integral at the interface <math>~(x = x_i = q)</math>, specifically,<br />
<div align="center"><br />
<math><br />
C_1 \equiv \sin(B-b_\eta q)+ b_\eta q \cos(B - b_\eta q) \, .<br />
</math><br />
</div><br />
<br />
Now, this expression can be significantly simplified by drawing on earlier results of this section as well as on attributes of the corresponding detailed force-balanced model. First, independent of the specific density profiles that define the structure of a bipolytrope, the ratio of the ''mean'' densities of the two structural regions is,<br />
<div align="center"><br />
<table border="0" cellpadding="5" align="center"><br />
<tr><br />
<td align="right"><br />
<math>~\frac{\bar\rho_e}{\bar\rho_c}</math><br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math>~\frac{q^3(1-\nu)}{\nu(1-q^3)} \, .</math><br />
</td><br />
</tr><br />
</table><br />
</div><br />
Hence the bracketed pre-factor of the second term of the expression for <math>~M_r|_\mathrm{env}</math> may be rewritten as,<br />
<div align="center"><br />
<table border="0" cellpadding="5" align="center"><br />
<tr><br />
<td align="right"><br />
<math>~\biggl[ \biggl( \frac{1-\nu}{1-q^3} \biggr) \biggl( \frac{\rho_0}{\bar\rho}\biggr)_\mathrm{env} \biggr]_\mathrm{eq} </math><br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math>~\biggl[ \frac{\nu}{q^3} \biggl( \frac{\rho_0}{\bar\rho}\biggr)_\mathrm{core} \biggr]_\mathrm{eq} \, .</math><br />
</td><br />
</tr><br />
</table><br />
</div><br />
But, from the [[SSC/Structure/BiPolytropes/FreeEnergy51#The_Core|above derivation of the mass profile in the core]], we know that,<br />
<div align="center"><br />
<table border="0" cellpadding="5" align="center"><br />
<br />
<tr><br />
<td align="right"><br />
<math>\biggl[\biggl( \frac{\rho_0} {{\bar\rho}}\biggr)_\mathrm{core} \biggr]_\mathrm{eq}</math><br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math><br />
\biggl( 1 + a_\xi q^2 \biggr)^{3/2} = \biggl( 1 + \frac{1}{3} \xi_i^2 \biggr)^{3/2} = \theta_i^{-3} \, ,<br />
</math><br />
</td><br />
</tr><br />
</table><br />
</div><br />
where the final step comes from knowledge of the expression for <math>~\theta_i</math> drawn from the detailed force-balanced model (see, for example, the associated [[SSC/Structure/BiPolytropes/Analytic51#Parameter_Values|''Parameter Values'' table]]). Hence, we can write,<br />
<div align="center"><br />
<table border="0" cellpadding="5" align="center"><br />
<br />
<tr><br />
<td align="right"><br />
<math>~M_r \biggr|_\mathrm{env}</math><br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math>\nu M_\mathrm{tot} + <br />
M_\mathrm{tot} \biggl( \frac{\mu_e}{\mu_c} \biggr) <br />
\frac{3\nu A\theta_i^2 }{(b_\eta q)^3} <br />
\biggl[ C_1-\sin(B-b_\eta x) -xb_\eta \cos(B - b_\eta x) \biggr] \, ,<br />
</math><br />
</td><br />
</tr><br />
<br />
</table><br />
</div><br />
and note that the expression for the coefficient, <math>~b_\eta</math>, becomes simpler as well, specifically,<br />
<div align="center"><br />
<table border="0" cellpadding="5" align="center"><br />
<br />
<tr><br />
<td align="right"><br />
<math>~b_\eta</math><br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math><br />
~\chi_\mathrm{eq}^{-1/5} \biggl( \frac{3}{4\pi} \biggr)^{2/5} \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta_i^2 (2\pi)^{1/2} \biggl( \frac{\nu}{q^3 \theta_i^3} \biggr)^{2/5} <br />
</math><br />
</td><br />
</tr><br />
<br />
<tr><br />
<td align="right"><br />
<math>~\Rightarrow ~~~~\chi_\mathrm{eq} </math><br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math><br />
~b_\eta^{-5} \biggl( \frac{3^4 \pi}{2^3} \biggr)^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^5 \frac{\nu^2 \theta_i^4}{q^6} \, .<br />
</math><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br />
Next &#8212; and, again, drawing from knowledge of the [[SSC/Structure/BiPolytropes/Analytic51#Step_6:__Envelope_Solution|internal structure of the detailed force-balanced model]], in particular, realizing that,<br />
<div align="center"><br />
<math>b_\eta q = \eta_i = 3^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta_i^2 \xi_i \, ,</math><br />
</div><br />
&#8212; note that the constant, <math>~C_1</math>, can be rewritten as,<br />
<br />
<div align="center"><br />
<table border="0" cellpadding="5" align="center"><br />
<br />
<tr><br />
<td align="right"><br />
<math>~C_1</math><br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math>~b_\eta q \cos(b_\eta q - B) - \sin(b_\eta q- B) </math><br />
</td><br />
</tr><br />
<br />
<tr><br />
<td align="right"><br />
&nbsp;<br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math>~\eta_i \cos(\eta_i - B) - \sin(\eta_i - B) </math><br />
</td><br />
</tr><br />
<br />
<tr><br />
<td align="right"><br />
&nbsp;<br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math>~\frac{\eta_i^2}{A} \biggl( \frac{d\phi}{d\eta} \biggr)_i = - \frac{1}{A} \biggl( \frac{\mu_e}{\mu_c} \biggr)^2 3^{1/2} \theta_i^4 \xi_i^3</math><br />
</td><br />
</tr><br />
<br />
<tr><br />
<td align="right"><br />
&nbsp;<br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math><br />
~- \frac{1}{A} \biggl( \frac{\mu_e}{\mu_c} \biggr)^2 3^{1/2} \theta_i^4 \biggl[ 3^{-1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta_i^{-2} b_\eta q\biggr]^3 <br />
</math><br />
</td><br />
</tr><br />
<br />
<tr><br />
<td align="right"><br />
&nbsp;<br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math><br />
~- \frac{1}{3A\theta_i^2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} (b_\eta q)^3 \, ,<br />
</math><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br />
which means,<br />
<div align="center"><br />
<table border="0" cellpadding="5" align="center"><br />
<br />
<tr><br />
<td align="right"><br />
<math>~M_r \biggr|_\mathrm{env}</math><br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math>\nu M_\mathrm{tot} - <br />
\nu M_\mathrm{tot} \biggl\{ 1 - \frac{1}{C_1} \biggl[ \sin(B-b_\eta x) + xb_\eta \cos(B - b_\eta x) \biggr] \biggr\} <br />
</math><br />
</td><br />
</tr><br />
<br />
<tr><br />
<td align="right"><br />
&nbsp;<br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math> <br />
\frac{\nu M_\mathrm{tot}}{C_1} \biggl[ \sin(B-b_\eta x) + xb_\eta \cos(B - b_\eta x) \biggr] \, .<br />
</math><br />
</td><br />
</tr><br />
<br />
</table><br />
</div><br />
<br />
<table border="1" cellpadding="5" align="center" width="90%"><br />
<tr><td align="left"><br />
<br />
MORE USEFUL:<br />
<br />
Letting, <math>\ell \equiv \xi/\sqrt{3}</math>,<br />
<div align="center"><br />
<table border="0" cellpadding="5" align="center"><br />
<br />
<tr><br />
<td align="right"><br />
<math>~b_\eta = \eta_s</math><br />
</td><br />
<td align="center"><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br />
and<br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br />
</td><br />
<td align="left"><br />
<math>~b_\eta q = \eta_i = 3\biggl( \frac{\mu_e}{\mu_c} \biggr) \ell_i (1 + \ell_i^2)^{-1} \, ,</math><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br />
<div align="center"><br />
<table border="0" cellpadding="5" align="center"><br />
<tr><br />
<td align="right"><br />
<math>~\tilde\mathfrak{f}_\mathrm{Menv} \equiv \biggl( \frac{\bar\rho}{\rho_c}\biggr)_\mathrm{env}</math><br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math>~ \frac{q^3(1-\nu)}{\nu(1-q^3)} \cdot \tilde\mathfrak{f}_\mathrm{Mcore}<br />
</math><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br />
</td></tr><br />
</table><br />
<br />
=See Also=<br />
<br />
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Joel2