Joel2 at 04:55, 16 January 2024

2024-01-16T04:55:21Z

← Older revision		Revision as of 00:55, 16 January 2024
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	=Best of the Best=		==Best of the Best==

	==One Derivation of Free Energy==		===One Derivation of Free Energy===

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	==Another Derivation of Free Energy==		===Another Derivation of Free Energy===
	Hence the renormalized gravitational potential energy becomes,		Hence the renormalized gravitational potential energy becomes,
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	==Summary==		===Summary===
	===Understanding Free-Energy Behavior===		====Understanding Free-Energy Behavior====
	'''<font color="#997700">Step 1:</font>''' Pick values for the separate coefficients, <math>\mathcal{A}, \mathcal{B},</math> and <math>\mathcal{C},</math> of the three terms in the normalized free-energy expression,		'''<font color="#997700">Step 1:</font>''' Pick values for the separate coefficients, <math>\mathcal{A}, \mathcal{B},</math> and <math>\mathcal{C},</math> of the three terms in the normalized free-energy expression,
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	===Check It===		====Check It====
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	=Playing With One Example=		==Playing With One Example==
	By setting,		By setting,
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	Combining this expression with the one for <math>~\mathcal{B}</math> gives us the desired result — although, strictly speaking, it is cheating! We can now methodically choose <math>~(\nu, q)</math> pairs and map them into the corresponding values of <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>. And, via an analogous "cheat," the choice of <math>~(\nu, q)</math> also gives us the self-consistent value of <math>~\mathcal{C}</math>. In this manner, we should be able to map out the free-energy surface for any desired set of physical parameters.		Combining this expression with the one for <math>~\mathcal{B}</math> gives us the desired result — although, strictly speaking, it is cheating! We can now methodically choose <math>~(\nu, q)</math> pairs and map them into the corresponding values of <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>. And, via an analogous "cheat," the choice of <math>~(\nu, q)</math> also gives us the self-consistent value of <math>~\mathcal{C}</math>. In this manner, we should be able to map out the free-energy surface for any desired set of physical parameters.

	=Second Example=		==Second Example==
	==Explain Logic==		===Explain Logic===
	[[Image:FreeEnergyExample.jpg\|right\|400px]] The figure presented here, on the right, shows a plot of the free energy, as a function of the dimensionless radius, <math>~\mathfrak{G}^*(\chi)</math>, where,		[[Image:FreeEnergyExample.jpg\|right\|400px]] The figure presented here, on the right, shows a plot of the free energy, as a function of the dimensionless radius, <math>~\mathfrak{G}^*(\chi)</math>, where,
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	==Construction Multiple Curves to Define a Free-Energy Surface==		===Construction Multiple Curves to Define a Free-Energy Surface===
	Okay. Now that we have the hang of this, let's construct a sequence of curves that represent physical evolution at a fixed interface-density ratio, <math>~\rho_e/\rho_c</math>, but for steadily increasing core-to-total mass ratio, <math>~\nu</math>. Specifically, we choose,		Okay. Now that we have the hang of this, let's construct a sequence of curves that represent physical evolution at a fixed interface-density ratio, <math>~\rho_e/\rho_c</math>, but for steadily increasing core-to-total mass ratio, <math>~\nu</math>. Specifically, we choose,
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Joel2 at 04:38, 16 January 2024

2024-01-16T04:38:15Z

Show changes

Joel2: Created page with "=Bipolytrope Generalization (Pt 4)=

Part I: Bipolytrope Generalization
Part II: Derivations
SSC/BipolytropeGeneralizat..."
2024-01-16T04:31:22Z

Created page with "=Bipolytrope Generalization (Pt 4)= <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 4)=
<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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SSC/BipolytropeGeneralization/Pt4 - Revision history

Joel2 at 04:55, 16 January 2024

Joel2 at 04:38, 16 January 2024