SSC/FreeEnergy/PolytropesEmbedded - Revision history

Jet53man: /* Chapter Divisions */

2023-10-15T17:31:04Z

Chapter Divisions

← Older revision		Revision as of 13:31, 15 October 2023
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	<ul>		<ul>
	<li>[[SSC/FreeEnergy/PolytropesEmbedded/Pt3A\|Focus on Five-One Free-Energy Expression]]</li>		<li>[[SSC/FreeEnergy/PolytropesEmbedded/Pt3A\|Focus on Five-One Free-Energy Expression]]</li>
	<li>[[SSC/FreeEnergy/PolytropesEmbedded/Pt3B\|Focus on ~~One~~-~~One~~ Free-Energy Expression]]</li>		<li>[[SSC/FreeEnergy/PolytropesEmbedded/Pt3B\|Focus on Zero-Zero Free-Energy Expression]]</li>
	<li>[[SSC/FreeEnergy/PolytropesEmbedded/Pt3C\|Overview]]</li>		<li>[[SSC/FreeEnergy/PolytropesEmbedded/Pt3C\|Overview]]</li>
	</ul>		</ul>

Jet53man: /* Chapter Divisions */

2023-10-15T17:29:38Z

Chapter Divisions

← Older revision		Revision as of 13:29, 15 October 2023
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	<li>Free-Energy of BiPolytropes		<li>Free-Energy of BiPolytropes
	<ul>		<ul>
	<li>[[SSC/FreeEnergy/PolytropesEmbedded/Pt3A\|Free-Energy ~~of BiPolytropes~~]]</li>		<li>[[SSC/FreeEnergy/PolytropesEmbedded/Pt3A\|Focus on Five-One Free-Energy Expression]]</li>
			<li>[[SSC/FreeEnergy/PolytropesEmbedded/Pt3B\|Focus on One-One Free-Energy Expression]]</li>
			<li>[[SSC/FreeEnergy/PolytropesEmbedded/Pt3C\|Overview]]</li>
	</ul>		</ul>
	</li>		</li>

Jet53man: /* Chapter Divisions */

2023-10-15T17:27:54Z

Chapter Divisions

← Older revision		Revision as of 13:27, 15 October 2023
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	<li>[[SSC/FreeEnergy/PolytropesEmbedded/Pt1\|Free-Energy Synopsis]]</li>		<li>[[SSC/FreeEnergy/PolytropesEmbedded/Pt1\|Free-Energy Synopsis]]</li>
	<li>[[SSC/FreeEnergy/PolytropesEmbedded/Pt2\|Free-Energy of Truncated Polytropes]]</li>		<li>[[SSC/FreeEnergy/PolytropesEmbedded/Pt2\|Free-Energy of Truncated Polytropes]]</li>
			<li>Free-Energy of BiPolytropes
			<ul>
			<li>[[SSC/FreeEnergy/PolytropesEmbedded/Pt3A\|Free-Energy of BiPolytropes]]</li>
			</ul>
			</li>
	</ol>		</ol>

Jet53man: /* Chapter Divisions */

2023-10-15T17:17:46Z

Chapter Divisions

← Older revision		Revision as of 13:17, 15 October 2023
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	<ol>		<ol>
	<li>[[SSC/FreeEnergy/PolytropesEmbedded/Pt1\|Free-Energy Synopsis]]</li>		<li>[[SSC/FreeEnergy/PolytropesEmbedded/Pt1\|Free-Energy Synopsis]]</li>
			<li>[[SSC/FreeEnergy/PolytropesEmbedded/Pt2\|Free-Energy of Truncated Polytropes]]</li>
	</ol>		</ol>


	=Free-Energy Synopsis=		=Free-Energy Synopsis=

Jet53man: /* Free-Energy Synopsis */

2023-10-15T17:07:53Z

Free-Energy Synopsis

← Older revision		Revision as of 13:07, 15 October 2023
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	__FORCETOC__ <!-- will force the creation of a Table of Contents -->		__FORCETOC__ <!-- will force the creation of a Table of Contents -->
	<!-- __NOTOC__ will force TOC off -->		<!-- __NOTOC__ will force TOC off -->
			=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>[[SSC/FreeEnergy/PolytropesEmbedded/Pt1\|Free-Energy Synopsis]]</li>
			</ol>


	=Free-Energy Synopsis=		=Free-Energy Synopsis=

Jet53man: /* From Detailed Force-Balance Models */

2023-10-14T22:18:02Z

From Detailed Force-Balance Models

Show changes

Jet53man: /* Reconcile With Known Analytic Expression */

2023-10-14T22:14:25Z

Reconcile With Known Analytic Expression

← Older revision		Revision as of 18:14, 14 October 2023
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	===Reconcile With Known Analytic Expression===		===Reconcile With Known Analytic Expression===
	From our [[SSC/Structure/BiPolytropes/FreeEnergy51#The_Core_2\|earlier derivations]], it appears as though,		From our [[SSC/Structure/BiPolytropes/FreeEnergy51#The_Core_2\|earlier derivations]], it appears as though,
	~~<div align="center">~~
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\chi_\mathrm{eq} \equiv \frac{R_\mathrm{eq}}{R_\mathrm{norm}}</math>		<math>\chi_\mathrm{eq} \equiv \frac{R_\mathrm{eq}}{R_\mathrm{norm}}</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\biggl( \frac{3^8}{2^5\pi} \biggr)^{-1/2}		<math>\biggl( \frac{3^8}{2^5\pi} \biggr)^{-1/2}
	\biggl(\frac{3}{2^4}\biggr) \biggl( \frac{q}{\ell_i}\biggr)^{5}\biggl(\frac{\nu}{q^3} \biggr)^2 \biggl( 1 + \ell_i^2 \biggr)^{3}		\biggl(\frac{3}{2^4}\biggr) \biggl( \frac{q}{\ell_i}\biggr)^{5}\biggl(\frac{\nu}{q^3} \biggr)^2 \biggl( 1 + \ell_i^2 \biggr)^{3}
	</math>		</math>
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	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\biggl(\frac{2\cdot 5}{3}\biggr)^{5/2} \biggl(\frac{q}{\nu^2} \biggr)^{3/2}		<math>\biggl(\frac{2\cdot 5}{3}\biggr)^{5/2} \biggl(\frac{q}{\nu^2} \biggr)^{3/2}
	\biggl[\biggl( \frac{\pi}{2^8 \cdot 3 \cdot 5^5} \biggr)^{1/2} \biggl(\frac{\nu^2}{q} \biggr)^{5/2}		\biggl[\biggl( \frac{\pi}{2^8 \cdot 3 \cdot 5^5} \biggr)^{1/2} \biggl(\frac{\nu^2}{q} \biggr)^{5/2}
	\frac{(1 + \ell_i^2)^3}{\ell_i^5} \biggr] \, .		\frac{(1 + \ell_i^2)^3}{\ell_i^5} \biggr] \, .
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	</tr>		</tr>
	</table>		</table>
	~~</div>~~

	This implies that,		This implies that,
	~~<div align="center">~~
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\frac{\mathcal{A}_\mathrm{biP} }{\mathcal{B}_\mathrm{biP}}</math>		<math>\frac{\mathcal{A}_\mathrm{biP} }{\mathcal{B}_\mathrm{biP}}</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~\approx</math>		<math>\approx</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>
	\biggl[\biggl( \frac{\pi}{2^8 \cdot 3 \cdot 5^5} \biggr)^{1/2} \biggl(\frac{\nu^2}{q} \biggr)^{5/2}		\biggl[\biggl( \frac{\pi}{2^8 \cdot 3 \cdot 5^5} \biggr)^{1/2} \biggl(\frac{\nu^2}{q} \biggr)^{5/2}
	\frac{(1 + \ell_i^2)^3}{\ell_i^5} \biggr]^{2/5}		\frac{(1 + \ell_i^2)^3}{\ell_i^5} \biggr]^{2/5}
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	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\biggl(\frac{\nu^2}{q} \biggr)		<math>\biggl(\frac{\nu^2}{q} \biggr)
	\biggl( \frac{\pi}{2^8 \cdot 3 \cdot 5^5} \biggr)^{1/5} \frac{(1 + \ell_i^2)^{6/5}}{\ell_i^2}		\biggl( \frac{\pi}{2^8 \cdot 3 \cdot 5^5} \biggr)^{1/5} \frac{(1 + \ell_i^2)^{6/5}}{\ell_i^2}
	</math>		</math>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\Rightarrow ~~~~ \biggl[ \mathfrak{a}_c\biggl(\frac{\nu}{q}\biggr) + \mathfrak{a}_e \biggl(\frac{1-\nu}{1+q}\biggr) \biggr] </math>		<math>\Rightarrow ~~~~ \biggl[ \mathfrak{a}_c\biggl(\frac{\nu}{q}\biggr) + \mathfrak{a}_e \biggl(\frac{1-\nu}{1+q}\biggr) \biggr] </math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~\approx</math>		<math>\approx</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{1}{2^2\cdot 5}\biggl(\frac{\nu^2}{q} \biggr)		<math>\frac{1}{2^2\cdot 5}\biggl(\frac{\nu^2}{q} \biggr)
	\frac{(1 + \ell_i^2)^{6/5}}{\ell_i^2} \biggl[5\mathfrak{b}_c		\frac{(1 + \ell_i^2)^{6/5}}{\ell_i^2} \biggl[5\mathfrak{b}_c
	+ \mathfrak{b}_e \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \frac{q^3(1-\nu)^2}{\nu^2(1-q^3)} \biggr]		+ \mathfrak{b}_e \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \frac{q^3(1-\nu)^2}{\nu^2(1-q^3)} \biggr]
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\Rightarrow ~~~~ \biggl[ \mathfrak{a}_c + \mathfrak{a}_e \cdot \frac{q(1-\nu)}{\nu(1+q)} \biggr] </math>		<math>\Rightarrow ~~~~ \biggl[ \mathfrak{a}_c + \mathfrak{a}_e \cdot \frac{q(1-\nu)}{\nu(1+q)} \biggr] </math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~\approx</math>		<math>\approx</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{\nu}{2^2\cdot 5}		<math>\frac{\nu}{2^2\cdot 5}
	\frac{(1 + \ell_i^2)^{6/5}}{\ell_i^2} \biggl[5\mathfrak{b}_c		\frac{(1 + \ell_i^2)^{6/5}}{\ell_i^2} \biggl[5\mathfrak{b}_c
	+ \mathfrak{b}_e \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \frac{q^3(1-\nu)^2}{\nu^2(1-q^3)} \biggr]		+ \mathfrak{b}_e \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \frac{q^3(1-\nu)^2}{\nu^2(1-q^3)} \biggr]
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	</tr>		</tr>
	</table>		</table>
	~~</div>~~

	==Focus on Five-One Free-Energy Expression==		==Focus on Five-One Free-Energy Expression==

Jet53man: /* Five-One Bipolytropes */

2023-10-14T22:10:28Z

Five-One Bipolytropes

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Jet53man: /* Reconcile With Known Analytic Expression */

2023-10-14T22:03:40Z

Reconcile With Known Analytic Expression

← Older revision		Revision as of 18:03, 14 October 2023
Line 6,329:		Line 6,329:

	===Reconcile With Known Analytic Expression===		===Reconcile With Known Analytic Expression===
	From our [[SSC/Structure/BiPolytropes/~~FreeEnergy5_1~~#The_Core_2\|earlier derivations]], it appears as though,		From our [[SSC/Structure/BiPolytropes/FreeEnergy51#The_Core_2\|earlier derivations]], it appears as though,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Jet53man: /* Free-Energy of Bipolytropes */

2021-10-14T21:43:40Z

Free-Energy of Bipolytropes

Show changes