SSC/BipolytropeGeneralization/Pt3 - Revision history

Joel2: /* Examples */

2024-01-16T04:49:28Z

Examples

← Older revision		Revision as of 00:49, 16 January 2024
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	=Examples=		==Examples==

	==(0, 0) Bipolytropes==		===(0, 0) Bipolytropes===
	===Review===		====Review====
	In an [[SSC/Structure/BiPolytropes/Analytic00#BiPolytrope_with_nc_.3D_0_and_ne_.3D_0\|accompanying discussion]] we have derived analytic expressions describing the equilibrium structure and the stability of bipolytropes in which both the core and the envelope have uniform densities, that is, bipolytropes with <math>~(n_c, n_e) = (0, 0)</math>. From this work, we find that integrals over the mass and pressure distributions give:		In an [[SSC/Structure/BiPolytropes/Analytic00#BiPolytrope_with_nc_.3D_0_and_ne_.3D_0\|accompanying discussion]] we have derived analytic expressions describing the equilibrium structure and the stability of bipolytropes in which both the core and the envelope have uniform densities, that is, bipolytropes with <math>~(n_c, n_e) = (0, 0)</math>. From this work, we find that integrals over the mass and pressure distributions give:
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	===Renormalize===		====Renormalize====
	Let's renormalize these energy terms in order to more readily relate them to the [[#Setup\|generalized expressions derived above]].		Let's renormalize these energy terms in order to more readily relate them to the [[#Setup\|generalized expressions derived above]].
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	===Virial Equilibrium and Stability Evaluation===		====Virial Equilibrium and Stability Evaluation====
	With these expressions in hand, we can deduce the equilibrium radius and relativity stability of <math>~(n_c, n_e) = (0, 0)</math> bipolytropes using the generalized expressions provided above. For example, from the statement of virial equilibrium <math>~(2S_\mathrm{tot} = - W )</math> we obtain,		With these expressions in hand, we can deduce the equilibrium radius and relativity stability of <math>~(n_c, n_e) = (0, 0)</math> bipolytropes using the generalized expressions provided above. For example, from the statement of virial equilibrium <math>~(2S_\mathrm{tot} = - W )</math> we obtain,
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	==(5, 1) Bipolytropes==		===(5, 1) Bipolytropes===
	In another [[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|accompanying discussion]] we have derived analytic expressions describing the equilibrium structure of bipolytropes with <math>(n_c, n_e) = (5, 1)</math>. Can we perform a similar stability analysis of these configurations?		In another [[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|accompanying discussion]] we have derived analytic expressions describing the equilibrium structure of bipolytropes with <math>(n_c, n_e) = (5, 1)</math>. Can we perform a similar stability analysis of these configurations?
	Work in progress!		Work in progress!

Joel2 at 04:41, 16 January 2024

2024-01-16T04:41:04Z

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Joel2: Created page with "=Bipolytrope Generalization (Pt 3)=

Part I: Bipolytrope Generalization
Part II: Derivations
SSC/BipolytropeGeneralizat..."
2024-01-16T04:30:56Z

Created page with "=Bipolytrope Generalization (Pt 3)= <table border="1" align="center" width="100%" colspan="8"> <tr> <td align="center" bgcolor="lightblue" width="25%"><br />Part I: Bipolytrope Generalization </td> <td align="center" bgcolor="lightblue" width="25%"><br />Part II: Derivations </td> <td align="center" bgcolor="lightblue" width="25%"><br />SSC/BipolytropeGeneralizat..."

New page
=Bipolytrope Generalization (Pt 3)=
<table border="1" align="center" width="100%" colspan="8">
<tr>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/BipolytropeGeneralization|Part I:  Bipolytrope Generalization]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/BipolytropeGeneralization/Pt2|Part II:  Derivations]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/BipolytropeGeneralization/Pt3|Part III:  Examples]]
 
</td>
<td align="center" bgcolor="lightblue"><br />[[SSC/BipolytropeGeneralization/Pt4|Part IV:  Best of the Best]]
 
</td>
</tr>
</table>

=Related Discussions=
* [[SSC/BipolytropeGeneralizationVersion2|Newer, Version2 of ''Bipolytrope Generalization'' derivations]].
* [[SSC/Structure/BiPolytropes/Analytic00#BiPolytrope_with_nc_.3D_0_and_ne_.3D_0|Analytic solution]] with <math>n_c = 0</math> and <math>n_e=0</math>.
* [[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1|Analytic solution]] with <math>n_c = 5</math> and <math>n_e=1</math>.

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