SSC/FreeEnergy/PolytropesEmbedded/Pt3A - Revision history

Jet53man: /* See Also */

2023-10-15T17:47:54Z

See Also

← Older revision		Revision as of 13:47, 15 October 2023
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	=See Also=		=See Also=
			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>
			<li>[[SSC/FreeEnergy/PolytropesEmbedded/Pt2\|Free-Energy of Truncated Polytropes]]</li>
			<li>Free-Energy of BiPolytropes
			<ul>
			<li>[[SSC/FreeEnergy/PolytropesEmbedded/Pt3A\|Focus on Five-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>
			</ul>
			</li>
			</ol>


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Jet53man at 17:45, 15 October 2023

2023-10-15T17:45:32Z

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Jet53man: /* Free-Energy of Bipolytropes */

2023-10-15T17:40:07Z

Free-Energy of Bipolytropes

← Older revision		Revision as of 13:40, 15 October 2023
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	=Free-Energy of Bipolytropes=		=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} + \mathfrak{S}_\mathrm{therm}\biggr]_\mathrm{core} + \biggl[W_\mathrm{grav} + \mathfrak{S}_\mathrm{therm}\biggr]_\mathrm{env} \, .
			</math>
			</td>
			</tr>

			</table>
			</div>

			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:
			* Total mass, <math>~M_\mathrm{tot}</math>;
			* Total radius, <math>~R</math>;
			* Interface radius, <math>~R_i</math>, and associated dimensionless interface marker, <math>~q \equiv R_i/R</math>;
			* Core mass, <math>~M_c</math>, and associated dimensionless mass fraction, <math>~\nu \equiv M_c/M_\mathrm{tot}</math>;
			* Polytropic constant in the core, <math>~K_c</math>.

			In general, the warped free-energy surface drapes across a five-dimensional parameter "plane" such that,
			<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>~\mathfrak{G}(R, K_c, M_\mathrm{tot}, q, \nu) \, .</math>
			</td>
			</tr>
			</table>
			</div>

	=See Also=		=See Also=

Jet53man: Created page with "FORCETOC =Background= Index to original, very long chapter =Free-Energy of Bipolytropes= =See Also= {{ SGFfooter }}"

2023-10-15T17:38:19Z

Created page with "__FORCETOC__ =Background= Index to original, very long chapter =Free-Energy of Bipolytropes= =See Also= {{ SGFfooter }}"

New page

__FORCETOC__
=Background=
[[SSC/FreeEnergy/PolytropesEmbedded#Chapter_Divisions|Index to original, very long chapter]]

=Free-Energy of Bipolytropes=

=See Also=

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SSC/FreeEnergy/PolytropesEmbedded/Pt3A - Revision history

Jet53man: /* See Also */

Jet53man at 17:45, 15 October 2023

Jet53man: /* Free-Energy of Bipolytropes */

Jet53man: Created page with "__FORCETOC__ =Background= Index to original, very long chapter =Free-Energy of Bipolytropes= =See Also= {{ SGFfooter }}"

Jet53man: Created page with "FORCETOC =Background= Index to original, very long chapter =Free-Energy of Bipolytropes= =See Also= {{ SGFfooter }}"