174.64.8.247: /* Directly to n = 5 Polytropic Configurations */

2021-08-01T20:16:49Z

Directly to n = 5 Polytropic Configurations

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	===Directly to n = 5 Polytropic Configurations===		===Directly to n = 5 Polytropic Configurations===

			{\| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
			\|-
			! style="height: 125px; width: 125px; background-color:white;" \|
			<font size="-1">[[H_BookTiledMenu#MoreStabilityAnalyses\|<b>''Exact''<br />Demonstration<br />of<br />Variational<br />Principle</b>]]</font>
			\|}

	<div align="center">		<div align="center">

174.64.8.247: /* Generalized Governing Integral Relation */

2021-08-01T19:59:03Z

Generalized Governing Integral Relation

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	</div>		</div>

	Now, from an [[~~StabilityVariationalPrincipal~~#Test_Virial_Equilibrium_Condition\|accompanying discussion]], we know that, in equilibrium,		Now, from an [[StabilityVariationalPrinciple#Test_Virial_Equilibrium_Condition\|accompanying discussion]], we know that, in equilibrium,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="3">		<table border="0" cellpadding="3">

174.64.8.247: /* Via Generalized Normalization */

2021-08-01T19:58:14Z

Via Generalized Normalization

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	===Via Generalized Normalization===		===Via Generalized Normalization===
	Next, we'll divide through by the [[~~StabilityVariationalPrincipal~~#Energies_and_Structural_Form_Factors\|normalization energy, as defined in an accompanying discussion]],		Next, we'll divide through by the [[StabilityVariationalPrinciple#Energies_and_Structural_Form_Factors\|normalization energy, as defined in an accompanying discussion]],
	<div align="center">		<div align="center">
	<math>~E_\mathrm{norm} = P_\mathrm{norm}R_\mathrm{norm}^3 = \frac{GM_\mathrm{tot}^2}{R_\mathrm{norm}} \, ,</math>		<math>~E_\mathrm{norm} = P_\mathrm{norm}R_\mathrm{norm}^3 = \frac{GM_\mathrm{tot}^2}{R_\mathrm{norm}} \, ,</math>

Jet53man at 17:36, 1 August 2021

2021-08-01T17:36:43Z

← Older revision		Revision as of 13:36, 1 August 2021
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	<font size="-1">[[H_BookTiledMenu#Stability_Analysis\|<b>Variational<br />Principle</b>]]</font>		<font size="-1">[[H_BookTiledMenu#Stability_Analysis\|<b>Variational<br />Principle</b>]]</font>
	\|}		\|}
	All of the discussion in this chapter will build upon our [[~~User:Tohline/~~SSC/Perturbations#2ndOrderODE\|derivation elsewhere]] of the,		All of the discussion in this chapter will build upon our [[SSC/Perturbations#2ndOrderODE\|derivation elsewhere]] of the,
	<div align="center" id="2ndOrderODE">		<div align="center" id="2ndOrderODE">
	<font color="#770000">'''LAWE:   Linear Adiabatic Wave''' (or ''Radial Pulsation'') '''Equation'''</font><br />		<font color="#770000">'''LAWE:   Linear Adiabatic Wave''' (or ''Radial Pulsation'') '''Equation'''</font><br />

	{{~~User:Tohline/~~Math/EQ_RadialPulsation01}}		{{Math/EQ_RadialPulsation01}}
	</div>		</div>

	We will draw heavily from the papers published by [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux & Pekeris (1941)] and by [http://adsabs.harvard.edu/abs/1964ApJ...139..664C S. Chandrasekhar (1964)], as well as from pp. 458-474 of the review by [http://adsabs.harvard.edu/abs/1958HDP....51..353L P. Ledoux & Th. Walraven (1958)] in explaining how the ''variational principle'' can be used to identify the eigenvector of the fundamental mode of radial oscillation in spherically symmetric configurations. In an associated "Ramblings" appendix, we provide [[~~User:Tohline/~~Appendix/Ramblings/LedouxVariationalPrinciple#Ledoux.27s_Variational_Principle_.28Supporting_Derivations.29\|various derivations that support]] this chapter's relatively abbreviated presentation.		We will draw heavily from the papers published by [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux & Pekeris (1941)] and by [http://adsabs.harvard.edu/abs/1964ApJ...139..664C S. Chandrasekhar (1964)], as well as from pp. 458-474 of the review by [http://adsabs.harvard.edu/abs/1958HDP....51..353L P. Ledoux & Th. Walraven (1958)] in explaining how the ''variational principle'' can be used to identify the eigenvector of the fundamental mode of radial oscillation in spherically symmetric configurations. In an associated "Ramblings" appendix, we provide [[Appendix/Ramblings/LedouxVariationalPrinciple#Ledoux.27s_Variational_Principle_.28Supporting_Derivations.29\|various derivations that support]] this chapter's relatively abbreviated presentation.

	==Ledoux and Pekeris (1941)==		==Ledoux and Pekeris (1941)==
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	</div>		</div>

	(An [[~~User:Tohline/~~SSC/Perturbations#Boundary_Conditions\|accompanying chapter]] provides a broader discussion of this and other astrophysically reasonable boundary conditions that are associated with solutions to the LAWE.)		(An [[SSC/Perturbations#Boundary_Conditions\|accompanying chapter]] provides a broader discussion of this and other astrophysically reasonable boundary conditions that are associated with solutions to the LAWE.)

	===Ledoux & Walraven Approach===		===Ledoux & Walraven Approach===
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	</div>		</div>

	Now, as we have [[~~User:Tohline/SphericallySymmetricConfigurations~~/Virial#Wgrav\|discussed separately]] — see, also, p. 64, Equation (12) of [<b>[[~~User:Tohline/~~Appendix/References#C67\|<font color="red">C67</font>]]</b>] — the gravitational potential energy of the unperturbed configuration is given by the integral,		Now, as we have [[SSCpt1/Virial#Wgrav\|discussed separately]] — see, also, p. 64, Equation (12) of [<b>[[Appendix/References#C67\|<font color="red">C67</font>]]</b>] — the gravitational potential energy of the unperturbed configuration is given by the integral,
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	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</div>		</div>

	for adiabatic systems, the [[~~User:Tohline/SphericallySymmetricConfigurations~~/Virial#Reservoir\|internal energy]] is,		for adiabatic systems, the [[SSCpt1/Virial#Reservoir\|internal energy]] is,
	<div align="center">		<div align="center">
	<math>		<math>
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	\, .</math>		\, .</math>
	</div>		</div>
	(Note that, defined in this way, <math>~I</math> is the same as [[~~User:Tohline/~~VE#Standard_Presentation_.5Bthe_Virial_of_Clausius_.281870.29.5D\|what we have referred to elsewhere]] as the ''scalar moment of inertia'', which is obtained by taking the trace of the [[~~User:Tohline/~~VE#MOItensor\|moment of inertia tensor]], <math>~I_{ij}</math>.)		(Note that, defined in this way, <math>~I</math> is the same as [[VE#Standard_Presentation_.5Bthe_Virial_of_Clausius_.281870.29.5D\|what we have referred to elsewhere]] as the ''scalar moment of inertia'', which is obtained by taking the trace of the [[VE#MOItensor\|moment of inertia tensor]], <math>~I_{ij}</math>.)
	<span id="GoverningIntegral">After inserting these expressions, we have what will henceforth be referred to as the,</span>		<span id="GoverningIntegral">After inserting these expressions, we have what will henceforth be referred to as the,</span>

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	===Via Generalized Normalization===		===Via Generalized Normalization===
	Next, we'll divide through by the [[~~User:Tohline/~~StabilityVariationalPrincipal#Energies_and_Structural_Form_Factors\|normalization energy, as defined in an accompanying discussion]],		Next, we'll divide through by the [[StabilityVariationalPrincipal#Energies_and_Structural_Form_Factors\|normalization energy, as defined in an accompanying discussion]],
	<div align="center">		<div align="center">
	<math>~E_\mathrm{norm} = P_\mathrm{norm}R_\mathrm{norm}^3 = \frac{GM_\mathrm{tot}^2}{R_\mathrm{norm}} \, ,</math>		<math>~E_\mathrm{norm} = P_\mathrm{norm}R_\mathrm{norm}^3 = \frac{GM_\mathrm{tot}^2}{R_\mathrm{norm}} \, ,</math>
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	</div>		</div>

	Now, let's examine how these terms combine if we ''guess'' the [[~~User:Tohline/~~SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds\|analytically defined eigenfunction that applies to marginally unstable, pressure-truncated polytropic configurations]], namely,		Now, let's examine how these terms combine if we ''guess'' the [[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds\|analytically defined eigenfunction that applies to marginally unstable, pressure-truncated polytropic configurations]], namely,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</div>		</div>

	Now, from an [[~~User:Tohline/~~StabilityVariationalPrincipal#Test_Virial_Equilibrium_Condition\|accompanying discussion]], we know that, in equilibrium,		Now, from an [[StabilityVariationalPrincipal#Test_Virial_Equilibrium_Condition\|accompanying discussion]], we know that, in equilibrium,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="3">		<table border="0" cellpadding="3">
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	====Plug in Known Marginally Unstable Solution====		====Plug in Known Marginally Unstable Solution====

	As has been summarized in an [[~~User:Tohline/~~SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds\|accompanying discussion]], we have found that, for marginally unstable pressure-truncated polytropic configurations, the eigenvector associated with the fundamental mode of radial oscillation is prescribed analytically by the following eigenfrequency-eigenfunction pair:		As has been summarized in an [[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds\|accompanying discussion]], we have found that, for marginally unstable pressure-truncated polytropic configurations, the eigenvector associated with the fundamental mode of radial oscillation is prescribed analytically by the following eigenfrequency-eigenfunction pair:
	<div align="center">		<div align="center">
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	=See Also=		=See Also=

	* [[~~User:Tohline/~~Appendix/Ramblings/LedouxVariationalPrinciple#Ledoux.27s_Variational_Principle_.28Supporting_Derivations.29\|Derivations that support this chapter's discussion of the Ledoux Variational Principle]]		* [[Appendix/Ramblings/LedouxVariationalPrinciple#Ledoux.27s_Variational_Principle_.28Supporting_Derivations.29\|Derivations that support this chapter's discussion of the Ledoux Variational Principle]]
	* [https://ui.adsabs.harvard.edu/abs/1967MNRAS.136..293L/abstract D. Lynden-Bell & J. P. Ostriker (1967)], MNRAS, 136, 293: ''On the stability of differentially rotating bodies''		* [https://ui.adsabs.harvard.edu/abs/1967MNRAS.136..293L/abstract D. Lynden-Bell & J. P. Ostriker (1967)], MNRAS, 136, 293: ''On the stability of differentially rotating bodies''
	<table border="0" align="center" width="100%" cellpadding="1"><tr>		<table border="0" align="center" width="100%" cellpadding="1"><tr>

Jet53man: /* Ledoux's Variational Principle */

2021-08-01T17:31:33Z

Ledoux's Variational Principle

← Older revision		Revision as of 13:31, 1 August 2021
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	=Ledoux's Variational Principle=		=Ledoux's Variational Principle=
			{\| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
			\|-
			! style="height: 125px; width: 125px; background-color:#9390DB;" \|
			<font size="-1">[[H_BookTiledMenu#Stability_Analysis\|<b>Variational<br />Principle</b>]]</font>
			\|}
	All of the discussion in this chapter will build upon our [[User:Tohline/SSC/Perturbations#2ndOrderODE\|derivation elsewhere]] of the,		All of the discussion in this chapter will build upon our [[User:Tohline/SSC/Perturbations#2ndOrderODE\|derivation elsewhere]] of the,
	<div align="center" id="2ndOrderODE">		<div align="center" id="2ndOrderODE">

Jet53man: Created page with "FORCETOC =Ledoux's Variational Principle= All of the discussion in this chapter will build upon our User:Tohline/SSC/Perturbatio..."

2021-08-01T17:27:21Z

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