SSC/FreeEnergy/PowerPoint - Revision history

Joel2: /* Case M Equilibrium Conditions */

2024-07-17T18:45:58Z

Case M Equilibrium Conditions

← Older revision		Revision as of 14:45, 17 July 2024
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	=====Case M Equilibrium Conditions=====		=====Case M Equilibrium Conditions=====
	Employing the [[~~SphericallySymmetricConfigurations~~/Virial#Choices_Made_by_Other_Researchers\|renormalization factors]],		Employing the [[SSCpt1/Virial#Choices_Made_by_Other_Researchers\|renormalization factors]],

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Joel2: /* Basic Properties */

2024-07-17T18:34:47Z

Basic Properties

← Older revision		Revision as of 14:34, 17 July 2024
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	Consistent with our [[SSC/BipolytropeGeneralization#Free_Energy_and_Its_Derivatives\|generic discussion of the stability of bipolytropes]] and the ''specific'' discussion of [[SSC/Structure/BiPolytropes/~~Analytic5_1~~#Stability_Condition\|the stability of bipolytropes having]] <math>(n_c, n_e) = (5, 1)</math>, it can straightforwardly be shown that <math>\partial \mathfrak{G}/\partial \chi = 0</math> is satisfied by setting <math>\Chi = 1</math>; that is, the equilibrium condition is,		Consistent with our [[SSC/BipolytropeGeneralization#Free_Energy_and_Its_Derivatives\|generic discussion of the stability of bipolytropes]] and the ''specific'' discussion of [[SSC/Structure/BiPolytropes/Analytic51#Stability_Condition\|the stability of bipolytropes having]] <math>(n_c, n_e) = (5, 1)</math>, it can straightforwardly be shown that <math>\partial \mathfrak{G}/\partial \chi = 0</math> is satisfied by setting <math>\Chi = 1</math>; that is, the equilibrium condition is,

	Furthermore, the equilibrium configuration is unstable whenever <math>~\partial^2 \mathfrak{G}/\partial \chi^2 < 0</math>, that is, the transition from stable to unstable configurations whenever,		Furthermore, the equilibrium configuration is unstable whenever <math>~\partial^2 \mathfrak{G}/\partial \chi^2 < 0</math>, that is, the transition from stable to unstable configurations whenever,

Joel2: /* Coincidence Between Points of Secular and Dynamical Instability */

2024-07-17T18:33:39Z

Coincidence Between Points of Secular and Dynamical Instability

← Older revision		Revision as of 14:33, 17 July 2024
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	====Coincidence Between Points of Secular and Dynamical Instability====		====Coincidence Between Points of Secular and Dynamical Instability====
	From the [[~~User:Tohline/~~SSC/Structure/BiPolytropes/~~Analytic5_1~~#Model_Sequences\|accompanying graphical display of equilibrium sequences]], it seems that a <math>~\nu_\mathrm{max}</math> turning point will only exist in five-one bipolytropes for <math>~\mu_e/\mu_c</math> less than some value — call it, <math>~(\mu_e/\mu_c)_\mathrm{begin}</math> — which is less than but approximately equal to <math>~\tfrac{1}{3}</math>. As we move along any sequence for which <math>~\mu_e/\mu_c < (\mu_e/\mu_c)_\mathrm{begin}</math>, in the direction of increasing <math>~\ell_i</math>, it is fair to ask whether the system becomes dynamically unstable (at <math>~[x_\mathrm{eq}]_\mathrm{crit}</math>) before or after it encounters the point of secular instability marked by <math>~\nu_\mathrm{max}</math>.		From the [[SSC/Structure/BiPolytropes/Analytic51#Model_Sequences\|accompanying graphical display of equilibrium sequences]], it seems that a <math>~\nu_\mathrm{max}</math> turning point will only exist in five-one bipolytropes for <math>~\mu_e/\mu_c</math> less than some value — call it, <math>~(\mu_e/\mu_c)_\mathrm{begin}</math> — which is less than but approximately equal to <math>~\tfrac{1}{3}</math>. As we move along any sequence for which <math>~\mu_e/\mu_c < (\mu_e/\mu_c)_\mathrm{begin}</math>, in the direction of increasing <math>~\ell_i</math>, it is fair to ask whether the system becomes dynamically unstable (at <math>~[x_\mathrm{eq}]_\mathrm{crit}</math>) before or after it encounters the point of secular instability marked by <math>~\nu_\mathrm{max}</math>.


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Joel2: /* Basic Properties */

2024-07-17T18:33:08Z

Basic Properties

← Older revision		Revision as of 14:33, 17 July 2024
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	Comparing the general free-energy expression at the beginning of this chapter with the free-energy expression provided via our [[~~User:Tohline/~~SSC/FreeEnergy/PolytropesEmbedded#BiPolytropeFreeEnergy\|accompanying summary discussion of five-one bipolytropes]], we find that,		Comparing the general free-energy expression at the beginning of this chapter with the free-energy expression provided via our [[SSC/FreeEnergy/PolytropesEmbedded#BiPolytropeFreeEnergy\|accompanying summary discussion of five-one bipolytropes]], we find that,

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	Consistent with our [[~~User:Tohline/~~SSC/BipolytropeGeneralization#Free_Energy_and_Its_Derivatives\|generic discussion of the stability of bipolytropes]] and the ''specific'' discussion of [[~~User:Tohline/~~SSC/Structure/BiPolytropes/Analytic5_1#Stability_Condition\|the stability of bipolytropes having]] <math>~(n_c, n_e) = (5, 1)</math>, it can straightforwardly be shown that <math>~\partial \mathfrak{G}/\partial \chi = 0</math> is satisfied by setting <math>~\Chi = 1</math>; that is, the equilibrium condition is,		Consistent with our [[SSC/BipolytropeGeneralization#Free_Energy_and_Its_Derivatives\|generic discussion of the stability of bipolytropes]] and the ''specific'' discussion of [[SSC/Structure/BiPolytropes/Analytic5_1#Stability_Condition\|the stability of bipolytropes having]] <math>(n_c, n_e) = (5, 1)</math>, it can straightforwardly be shown that <math>\partial \mathfrak{G}/\partial \chi = 0</math> is satisfied by setting <math>\Chi = 1</math>; that is, the equilibrium condition is,

	Furthermore, the equilibrium configuration is unstable whenever <math>~\partial^2 \mathfrak{G}/\partial \chi^2 < 0</math>, that is, the transition from stable to unstable configurations whenever,		Furthermore, the equilibrium configuration is unstable whenever <math>~\partial^2 \mathfrak{G}/\partial \chi^2 < 0</math>, that is, the transition from stable to unstable configurations whenever,
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	[[~~User:Tohline/~~SSC/Structure/BiPolytropes/~~Analytic5_1~~#Stability_Condition\|Table 1 of an accompanying chapter]] — and the red-dashed curve in the figure adjacent to that table — identifies some key properties of the model that marks the transition from stable to unstable configurations along equilibrium sequences that have various values of the mean-molecular weight ratio, <math>~\mu_e/\mu_c</math>.		[[SSC/Structure/BiPolytropes/Analytic51#Stability_Condition\|Table 1 of an accompanying chapter]] — and the red-dashed curve in the figure adjacent to that table — identifies some key properties of the model that marks the transition from stable to unstable configurations along equilibrium sequences that have various values of the mean-molecular weight ratio, <math>~\mu_e/\mu_c</math>.

	====Coincidence Between Points of Secular and Dynamical Instability====		====Coincidence Between Points of Secular and Dynamical Instability====

Joel2: /* Case P Equilibrium Conditions */

2024-07-17T18:30:31Z

Case P Equilibrium Conditions

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	=====Case P Equilibrium Conditions=====		=====Case P Equilibrium Conditions=====

	The [[~~User:Tohline/~~SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation\|equilibrium radius and corresponding configuration mass]] from a "Case P" analysis are,		The [[SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation\|equilibrium radius and corresponding configuration mass]] from a "Case P" analysis are,
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	According to our [[~~User:Tohline/~~SSC/Structure/PolytropesEmbedded#Other_Limits\|review of, especially, Kimura's work]], the turning point associated with <math>~M_\mathrm{max}</math> occurs where,		According to our [[SSC/Structure/PolytropesEmbedded#Other_Limits\|review of, especially, Kimura's work]], the turning point associated with <math>~M_\mathrm{max}</math> occurs where,

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Joel2: /* Case M Equilibrium Conditions */

2024-07-17T18:26:48Z

Case M Equilibrium Conditions

← Older revision		Revision as of 14:26, 17 July 2024
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	=====Case M Equilibrium Conditions=====		=====Case M Equilibrium Conditions=====
	Employing the [[~~User:Tohline/~~SphericallySymmetricConfigurations/Virial#Choices_Made_by_Other_Researchers\|renormalization factors]],		Employing the [[SphericallySymmetricConfigurations/Virial#Choices_Made_by_Other_Researchers\|renormalization factors]],

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	we find from detailed force-balance analyses that the [[~~User:Tohline/~~SSC/Structure/PolytropesEmbedded#General_Properties\|equilibrium radius and corresponding external pressure]] for "Case M" configurations are,		we find from detailed force-balance analyses that the [[SSC/Structure/PolytropesEmbedded#General_Properties\|equilibrium radius and corresponding external pressure]] for "Case M" configurations are,
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	which matches the expression derived in an [[~~User:Tohline/~~SSC/Structure/Polytropes#Lane-Emden_Equation\|ASIDE box found with our introduction of the Lane-Emden equation]], and		which matches the expression derived in an [[SSC/Structure/Polytropes#Lane-Emden_Equation\|ASIDE box found with our introduction of the Lane-Emden equation]], and
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	There are two turning points: One associated with a maximum in <math>~P_e</math> and one associated with a maximum in <math>~R_\mathrm{eq}</math>. According to [[~~User:Tohline/~~SSC/Structure/PolytropesEmbedded#Turning_Points\|Kimura's discussion]], the first of these occurs in the configuration for which,		There are two turning points: One associated with a maximum in <math>~P_e</math> and one associated with a maximum in <math>~R_\mathrm{eq}</math>. According to [[SSC/Structure/PolytropesEmbedded#Turning_Points\|Kimura's discussion]], the first of these occurs in the configuration for which,

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	This agrees with the [[~~User:Tohline/~~SSC/FreeEnergy/PolytropesEmbedded#Case_M\|expression derived in a separate <font color="red">ASIDE</font>]]; as was pointed out in that context, the root of this equation is: <math>~\ell \approx 2.223175</math>, that is, <math>~\tilde\xi \approx 3.85065</math>. This point along the equilibrium sequence is identified by the red circular dot in Figure 3, below. It is almost — but definitely not — coincident with the configuration along the sequence (marked by the yellow circular dot) that is associated with the minimum-radius turning point.		This agrees with the [[SSC/FreeEnergy/PolytropesEmbedded#Case_M\|expression derived in a separate <font color="red">ASIDE</font>]]; as was pointed out in that context, the root of this equation is: <math>\ell \approx 2.223175</math>, that is, <math>\tilde\xi \approx 3.85065</math>. This point along the equilibrium sequence is identified by the red circular dot in Figure 3, below. It is almost — but definitely not — coincident with the configuration along the sequence (marked by the yellow circular dot) that is associated with the minimum-radius turning point.

	<div align="center" id="Figure3">		<div align="center" id="Figure3">

Joel2 at 18:24, 17 July 2024

2024-07-17T18:24:04Z

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Joel2: Created page with "FORCETOC =Supporting Derivations for Free-Energy PowerPoint Presentation= The derivations presented here are an extension of our accompanying free-energy synopsis. These additional details proved to be helpful while developing an overarching PowerPoint presentation. {{LSU_HBook_header}} ==General Free-Energy Expression== We're considering a free-energy function of the following form:

2024-07-17T18:10:27Z

Created page with "FORCETOC =Supporting Derivations for Free-Energy PowerPoint Presentation= The derivations presented here are an extension of our accompanying free-energy synopsis. These additional details proved to be helpful while developing an overarching PowerPoint presentation. {{LSU_HBook_header}} ==General Free-Energy Expression== We're considering a free-energy function of the following form: <div a..."
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