SSC/BipolytropeGeneralization/Pt2 - Revision history

Joel2: /* Detailed Derivations */

2024-01-16T04:47:21Z

Detailed Derivations

← Older revision		Revision as of 00:47, 16 January 2024
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	=Detailed Derivations=		==Detailed Derivations==
	Dividing the free-energy expression through by <math>~E_\mathrm{norm}</math> generates,		Dividing the free-energy expression through by <math>~E_\mathrm{norm}</math> generates,
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	==Free Energy and Its Derivatives==		===Free Energy and Its Derivatives===

	Now, the free energy can be written as,		Now, the free energy can be written as,
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	==Equilibrium==		===Equilibrium===
	The radius, <math>~R_\mathrm{eq}</math>, of the equilibrium configuration(s) is determined by setting the first derivative of the free energy to zero. Hence,		The radius, <math>~R_\mathrm{eq}</math>, of the equilibrium configuration(s) is determined by setting the first derivative of the free energy to zero. Hence,

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	This is the familiar statement of virial equilibrium. From it we should always be able to derive the radius of equilibrium configurations.		This is the familiar statement of virial equilibrium. From it we should always be able to derive the radius of equilibrium configurations.

	==Stability==		===Stability===

	To assess the relative stability of an equilibrium configuration, we need to determine the sign of the second derivative of the free energy, evaluated at the equilibrium radius. If the sign of the second derivative is positive, the system is dynamically stable; if the sign is negative, he system is dynamically unstable. Using the above statement of virial equilibrium, that is, setting,		To assess the relative stability of an equilibrium configuration, we need to determine the sign of the second derivative of the free energy, evaluated at the equilibrium radius. If the sign of the second derivative is positive, the system is dynamically stable; if the sign is negative, he system is dynamically unstable. Using the above statement of virial equilibrium, that is, setting,

Joel2 at 04:43, 16 January 2024

2024-01-16T04:43:12Z

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Joel2: Created page with "=Bipolytrope Generalization (Pt 2)=

Part I: Bipolytrope Generalization
Part II: Derivations
SSC/BipolytropeGeneralizat..."
2024-01-16T04:30:26Z

Created page with "=Bipolytrope Generalization (Pt 2)= <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..."

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=Bipolytrope Generalization (Pt 2)=
<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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