Jet53man: /* Old Approach */

2022-04-26T20:15:25Z

Old Approach

← Older revision		Revision as of 16:15, 26 April 2022
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	====Old Approach====		====Old Approach====

	As has been developed in, for example, our [[~~SSCpt1~~/Virial/PolytropesEmbedded/FirstEffortAgain#Review\|accompanying review]], we adopt the following normalizations:		As has been developed in, for example, our [[SSC/Virial/PolytropesEmbedded/FirstEffortAgain#Review\|accompanying review]], we adopt the following normalizations:
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174.64.8.247: /* Old Approach */

2021-08-01T20:10:16Z

Old Approach

← Older revision		Revision as of 16:10, 1 August 2021
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	====Old Approach====		====Old Approach====

	As has been developed in, for example, our [[~~SSC~~/Virial/PolytropesEmbedded/FirstEffortAgain#Review\|accompanying review]], we adopt the following normalizations:		As has been developed in, for example, our [[SSCpt1/Virial/PolytropesEmbedded/FirstEffortAgain#Review\|accompanying review]], we adopt the following normalizations:
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	Then, from separate summaries — both [[~~SSCpt1Virial~~#Summary_of_Normalized_Expressions\|here]] and [[SSCpt1/Virial/FormFactors#Implication_for_Structural_Form_Factors\|here]] — we can write,		Then, from separate summaries — both [[SSCpt1/Virial#Summary_of_Normalized_Expressions\|here]] and [[SSCpt1/Virial/FormFactors#Implication_for_Structural_Form_Factors\|here]] — we can write,
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174.64.8.247 at 20:06, 1 August 2021

2021-08-01T20:06:01Z

← Older revision		Revision as of 16:06, 1 August 2021
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	====Old Approach====		====Old Approach====

	As has been developed in, for example, our [[~~User:Tohline/~~SSC/Virial/PolytropesEmbedded/FirstEffortAgain#Review\|accompanying review]], we adopt the following normalizations:		As has been developed in, for example, our [[SSC/Virial/PolytropesEmbedded/FirstEffortAgain#Review\|accompanying review]], we adopt the following normalizations:
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	Then, from separate summaries — both [[~~User:Tohline/SphericallySymmetricConfigurations/Virial~~#Summary_of_Normalized_Expressions\|here]] and [[~~User:Tohline/SSC~~/Virial/FormFactors#Implication_for_Structural_Form_Factors\|here]] — we can write,		Then, from separate summaries — both [[SSCpt1Virial#Summary_of_Normalized_Expressions\|here]] and [[SSCpt1/Virial/FormFactors#Implication_for_Structural_Form_Factors\|here]] — we can write,
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	where the [[~~User:Tohline/SphericallySymmetricConfigurations~~/Virial#Structural_Form_Factors\|structural form factors are defined]] as follows:		where the [[SSCpt1/Virial#Structural_Form_Factors\|structural form factors are defined]] as follows:

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	This gives, specifically for [[~~User:Tohline/SSC~~/Virial/FormFactors#PTtable\|specifically for pressure-truncated polytropic configurations]],		This gives, specifically for [[SSCpt1/Virial/FormFactors#PTtable\|specifically for pressure-truncated polytropic configurations]],
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	====New Approach====		====New Approach====
	In order to accommodate the structural integrals required by the Ledoux variational principle, let's re-derive some of these key energy and form-factor expressions. Basically, we will be repeating some [[~~User:Tohline/SphericallySymmetricConfigurations~~/Virial#Expressions_for_Various_Energy_Terms\|earlier derivations]].		In order to accommodate the structural integrals required by the Ledoux variational principle, let's re-derive some of these key energy and form-factor expressions. Basically, we will be repeating some [[SSCpt1/Virial#Expressions_for_Various_Energy_Terms\|earlier derivations]].

	=====Mass=====		=====Mass=====
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	Polytropic Lane-Emden Equation<p></p>		Polytropic Lane-Emden Equation<p></p>
	{{ ~~User:Tohline/~~Math/EQ_SSLaneEmden01 }}		{{ Math/EQ_SSLaneEmden01 }}
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	we realize that,		we realize that,
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	When <math>~\Upsilon_U(\xi) = 1</math>, then according to [[~~User:Tohline/SSC~~/Virial/FormFactors#Viala_and_Horedt_.281974.29_Expressions\|Viala & Horedt (1974)]], this integral over polytropic functions becomes,		When <math>~\Upsilon_U(\xi) = 1</math>, then according to [[SSCpt1/Virial/FormFactors#Viala_and_Horedt_.281974.29_Expressions\|Viala & Horedt (1974)]], this integral over polytropic functions becomes,
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	which matches the expression for <math>~{\tilde\mathfrak{f}}_A</math> [[~~User:Tohline/SSC~~/Virial/FormFactors#PTtable\|derived earlier]].		which matches the expression for <math>~{\tilde\mathfrak{f}}_A</math> [[SSCpt1/Virial/FormFactors#PTtable\|derived earlier]].

	=====Modified Gravitational Potential Energy=====		=====Modified Gravitational Potential Energy=====
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	Now, according to [[~~User:Tohline/SSC~~/Virial/FormFactors#Viala_and_Horedt_.281974.29_Expressions\|Viala & Horedt (1974)]], when <math>~\Upsilon_W(\xi) = 1</math>, this integral over polytropic functions becomes,		Now, according to [[SSCpt1/Virial/FormFactors#Viala_and_Horedt_.281974.29_Expressions\|Viala & Horedt (1974)]], when <math>~\Upsilon_W(\xi) = 1</math>, this integral over polytropic functions becomes,

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	As we have [[~~User:Tohline/SSC~~/Virial/FormFactors#Implication_for_Structural_Form_Factors\|detailed elsewhere]], from this, we have deduced that, for polytropic configurations,		As we have [[SSCpt1/Virial/FormFactors#Implication_for_Structural_Form_Factors\|detailed elsewhere]], from this, we have deduced that, for polytropic configurations,
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	Let's see if this expression is valid when we plug in the expressions for the equilibrium parameter pair — <math>~R_\mathrm{eq}</math> and <math>~P_e</math> — that has been given by [[~~User:Tohline/~~SSC/Structure/PolytropesEmbedded#Horedt.27s_Presentation\|Horedt (1970)]], namely,		Let's see if this expression is valid when we plug in the expressions for the equilibrium parameter pair — <math>~R_\mathrm{eq}</math> and <math>~P_e</math> — that has been given by [[SSC/Structure/PolytropesEmbedded#Horedt.27s_Presentation\|Horedt (1970)]], namely,
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	where we have taken into account the [[~~User:Tohline/SphericallySymmetricConfigurations~~/Virial#Choices_Made_by_Other_Researchers\|shift in normalization factors]],		where we have taken into account the [[SSCpt1/Virial#Choices_Made_by_Other_Researchers\|shift in normalization factors]],
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	<tr><th colspan="3" align="center">Switch from [[~~User:Tohline/~~SSC/Structure/PolytropesEmbedded#Horedt.27s_Presentation\|Hoerdt's (1970)]] Normalization</th><tr>		<tr><th colspan="3" align="center">Switch from [[SSC/Structure/PolytropesEmbedded#Horedt.27s_Presentation\|Hoerdt's (1970)]] Normalization</th><tr>

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	==The Ledoux Variational Principle==		==The Ledoux Variational Principle==
	Drawing from a [[~~User:Tohline/~~SSC/VariationalPrinciple#Ledoux.27s_Expression\|separate presentation of Ledoux's variational principle]], let's normalize his Lagrangian using the same normalizations that have been used, above. His expression is …		Drawing from a [[SSC/VariationalPrinciple#Ledoux.27s_Expression\|separate presentation of Ledoux's variational principle]], let's normalize his Lagrangian using the same normalizations that have been used, above. His expression is …

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174.64.8.247: Created page with "FORCETOC =Free-Energy Stability Analysis= ==Most General Case== Consider a free-energy function of the form,
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2021-08-01T20:00:48Z

Created page with "__FORCETOC__  =Free-Energy Stability Analysis= ==Most General Case== Consider a free-energy function of the form, <div align="center">..."

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Jet53man: /* Old Approach */

174.64.8.247: /* Old Approach */

174.64.8.247 at 20:06, 1 August 2021

174.64.8.247: Created page with "__FORCETOC__ =Free-Energy Stability Analysis= ==Most General Case== Consider a free-energy function of the form, ..."

174.64.8.247: Created page with "FORCETOC =Free-Energy Stability Analysis= ==Most General Case== Consider a free-energy function of the form,
..."