SSCpt1/Virial/PolytropesEmbeddedOutline - Revision history

174.64.8.247: /* See Also */

2021-07-16T23:13:26Z

174.64.8.247: /* Second Effort */

2021-07-16T23:12:15Z

Second Effort

← Older revision		Revision as of 19:12, 16 July 2021
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	===Second Effort===		===Second Effort===
	My [[~~User:Tohline/~~SSC/Virial/PolytropesSummary#Virial_Equilibrium_of_Adiabatic_Spheres_.28Summary.29\|second attempt to analytically define the free energy, and then the virial equilibrium, of pressure-truncated (embedded) polytropic spheres]] built upon my ''first effort'' and, for a couple of different polytropic indexes, focused on comparing the mass-radius relationship embodied in detailed force-balanced models against the mass-radius relationship implied by the virial theorem. A lot of reasonable results seem to have arisen from a discussion of [[~~User:Tohline/~~SSC/Virial/PolytropesSummary#Relating_and_Reconciling_Two_Mass-Radius_Relationships_for_n_.3D_4_Polytropes\|models (done numerically using Excel) with <math>~n=4</math> polytropic index]]. And there are some nice aspects of [[~~User:Tohline/~~SSC/Virial/PolytropesSummary#Relating_and_Reconciling_Two_Mass-Radius_Relationships_for_n_.3D_5_Polytropes\|models with an <math>~n=5</math> index]], but these models raise some [[~~User:Tohline/~~SSC/Virial/PolytropesSummary#Serious_Concern\|serious concerns]] related to the fact that two of our "derived" form-factor expressions involve division by the factor, <math>~(5-n)</math>, that is, division by zero.		My [[SSC/Virial/PolytropesSummary#Virial_Equilibrium_of_Adiabatic_Spheres_.28Summary.29\|second attempt to analytically define the free energy, and then the virial equilibrium, of pressure-truncated (embedded) polytropic spheres]] built upon my ''first effort'' and, for a couple of different polytropic indexes, focused on comparing the mass-radius relationship embodied in detailed force-balanced models against the mass-radius relationship implied by the virial theorem. A lot of reasonable results seem to have arisen from a discussion of [[SSC/Virial/PolytropesSummary#Relating_and_Reconciling_Two_Mass-Radius_Relationships_for_n_.3D_4_Polytropes\|models (done numerically using Excel) with <math>~n=4</math> polytropic index]]. And there are some nice aspects of [[SSC/Virial/PolytropesSummary#Relating_and_Reconciling_Two_Mass-Radius_Relationships_for_n_.3D_5_Polytropes\|models with an <math>~n=5</math> index]], but these models raise some [[SSC/Virial/PolytropesSummary#Serious_Concern\|serious concerns]] related to the fact that two of our "derived" form-factor expressions involve division by the factor, <math>~(5-n)</math>, that is, division by zero.

	===Third Effort===		===Third Effort===

174.64.8.247: /* First Effort */

2021-07-16T23:11:03Z

First Effort

← Older revision		Revision as of 19:11, 16 July 2021
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	==Outline of Detailed Investigations Leading to Above Summary==		==Outline of Detailed Investigations Leading to Above Summary==
	===First Effort===		===First Effort===
	My [[~~User:Tohline/~~SSC/Virial/Polytropes#Virial_Equilibrium_of_Adiabatic_Spheres\|first attempt to analytically define the free energy, and then the virial equilibrium, of pressure-truncated (embedded) polytropic spheres]] was developed as a direct extension of my description of the virial equilibrium of ''isolated'' polytropes. An important outcome of this "first effort" was the unveiling of analytic expressions for the key structural form factors, both [[~~User:Tohline/~~SSC/Virial/Polytropes#Summary\|for ''isolated'' polytropes]] and, separately, [[~~User:Tohline/~~SSC/Virial/Polytropes#PTtable\|for ''pressure-truncated'' polytropic structures]].		My [[SSC/Virial/Polytropes#Virial_Equilibrium_of_Adiabatic_Spheres\|first attempt to analytically define the free energy, and then the virial equilibrium, of pressure-truncated (embedded) polytropic spheres]] was developed as a direct extension of my description of the virial equilibrium of ''isolated'' polytropes. An important outcome of this "first effort" was the unveiling of analytic expressions for the key structural form factors, both [[SSC/Virial/Polytropes#Summary\|for ''isolated'' polytropes]] and, separately, [[SSC/Virial/Polytropes#PTtable\|for ''pressure-truncated'' polytropic structures]].

	I am very confident that the form-factor expressions presented for isolated polytropes are all correct because they have been cross-checked with expressions for closely related "integral" parameters discussed by Chandrasekhar [C67]. Although the form-factor expressions derived for pressure-truncated polytropes make some sense — they look very similar to the ones presented for isolated polytropes and seem to behave properly for models which, based on detailed force-balanced analysis, are known to be in equilibrium — I have much less confidence that they are correct. A couple of strategies were developed in an effort to demonstrate the validity and utility of these more general form-factor expressions, resulting in the derivation of a ''concise virial equilibrium'' relation,		I am very confident that the form-factor expressions presented for isolated polytropes are all correct because they have been cross-checked with expressions for closely related "integral" parameters discussed by Chandrasekhar [C67]. Although the form-factor expressions derived for pressure-truncated polytropes make some sense — they look very similar to the ones presented for isolated polytropes and seem to behave properly for models which, based on detailed force-balanced analysis, are known to be in equilibrium — I have much less confidence that they are correct. A couple of strategies were developed in an effort to demonstrate the validity and utility of these more general form-factor expressions, resulting in the derivation of a ''concise virial equilibrium'' relation,

174.64.8.247: /* Second Effort, Second Time Around */

2021-07-16T23:04:17Z

Second Effort, Second Time Around

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	===Second Effort, Second Time Around===		===Second Effort, Second Time Around===
	[[~~User:Tohline/~~SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Virial_Equilibrium_of_Adiabatic_Spheres_.28Summary.29\|In an accompanying chapter]], we reproduce the discussion associated with our [[~~User:Tohline/SSC~~/Virial/PolytropesEmbeddedOutline#Second_Effort\|"Second Effort", as referenced above]], but revise key sections to incorporate corrected expressions for the structural form factors.		[[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Virial_Equilibrium_of_Adiabatic_Spheres_.28Summary.29\|In an accompanying chapter]], we reproduce the discussion associated with our [[SSCpt1/Virial/PolytropesEmbeddedOutline#Second_Effort\|"Second Effort", as referenced above]], but revise key sections to incorporate corrected expressions for the structural form factors.

	=See Also=		=See Also=

174.64.8.247: /* First Effort, Second Time Around */

2021-07-16T23:03:24Z

First Effort, Second Time Around

← Older revision		Revision as of 19:03, 16 July 2021
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	===First Effort, Second Time Around===		===First Effort, Second Time Around===
	[[~~User:Tohline/~~SSC/Virial/PolytropesEmbedded/FirstEffortAgain\|In an accompanying chapter]], we reproduce the discussion associated with our [[~~User:Tohline/SSC~~/Virial/PolytropesEmbeddedOutline#First_Effort\|"First Effort", as referenced above]], but correct expressions for <math>~\mathfrak{f}_W</math> and <math>~\mathfrak{f}_A</math>, as identified in our "Third Effort" and, accordingly, re-derive various affected expressions that follow.		[[SSC/Virial/PolytropesEmbedded/FirstEffortAgain\|In an accompanying chapter]], we reproduce the discussion associated with our [[SSCpt1/Virial/PolytropesEmbeddedOutline#First_Effort\|"First Effort", as referenced above]], but correct expressions for <math>~\mathfrak{f}_W</math> and <math>~\mathfrak{f}_A</math>, as identified in our "Third Effort" and, accordingly, re-derive various affected expressions that follow.

	===Second Effort, Second Time Around===		===Second Effort, Second Time Around===

174.64.8.247: /* Third Effort */

2021-07-16T22:59:16Z

Third Effort

← Older revision		Revision as of 18:59, 16 July 2021
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	===Third Effort===		===Third Effort===
	In an attempt to answer the serious concern(s) raised during our first two efforts, we finally buckled down and [[~~User:Tohline/SSC~~/Virial/FormFactors#Structural_Form_Factors\|performed the integrals necessary to determine expressions for key structural form factors in the cases where the internal structure is known analytically]], specifically, for indexes <math>~n=5</math> and <math>~n=1</math>. The result is that the individual expressions derived by direct integration for <math>~\mathfrak{f}_W</math> and for <math>~\mathfrak{f}_A</math> ''do not match'' the general form-factor expressions that were rather cavalierly "derived" during our first effort. Oddly enough, as we discovered while [[~~User:Tohline/SSC~~/Virial/FormFactors#Fiddling_Around\|fiddling around with the new results]], the ''ratio'' of these form factors appears to be the same as before, namely,		In an attempt to answer the serious concern(s) raised during our first two efforts, we finally buckled down and [[SSCpt1/Virial/FormFactors#Structural_Form_Factors\|performed the integrals necessary to determine expressions for key structural form factors in the cases where the internal structure is known analytically]], specifically, for indexes <math>~n=5</math> and <math>~n=1</math>. The result is that the individual expressions derived by direct integration for <math>~\mathfrak{f}_W</math> and for <math>~\mathfrak{f}_A</math> ''do not match'' the general form-factor expressions that were rather cavalierly "derived" during our first effort. Oddly enough, as we discovered while [[SSCpt1/Virial/FormFactors#Fiddling_Around\|fiddling around with the new results]], the ''ratio'' of these form factors appears to be the same as before, namely,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Jet53man: /* See Also */

2021-07-16T17:19:16Z

Jet53man: /* See Also */

2021-07-16T16:42:20Z

Jet53man: /* Stability (M) */

2021-07-15T21:18:48Z

Stability (M)

← Older revision		Revision as of 17:18, 15 July 2021
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	====Stability (M)====		====Stability (M)====
	Similarly, according to the [[~~User:Tohline/SSC/Virial/PolytropesEmbeddedOutline~~#Stability\|above-derived stability criterion]], pressure-truncated polytropic configurations will only be stable if,		Similarly, according to the [[#Stability\|above-derived stability criterion]], pressure-truncated polytropic configurations will only be stable if,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5">		<table border="0" cellpadding="5">
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\frac{R_\mathrm{eq}}{R_\mathrm{SWS}} </math>		<math>\frac{R_\mathrm{eq}}{R_\mathrm{SWS}} </math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~></math>		<math>></math>
	</td>		</td>
	<td align="left">		<td align="left">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\frac{M_\mathrm{limit}}{M_\mathrm{SWS}} </math>		<math>\frac{M_\mathrm{limit}}{M_\mathrm{SWS}} </math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~<</math>		<math><</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>
	\biggl[ \frac{4\pi n}{\mathcal{A}_{M_\ell}(n-3)}\biggr]^{1/2} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^2		\biggl[ \frac{4\pi n}{\mathcal{A}_{M_\ell}(n-3)}\biggr]^{1/2} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^2
	\, .		\, .
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	{{ ~~WorkInProgress~~ }}		{{ SGFworkInProgress }}

	==Outline of Detailed Investigations Leading to Above Summary==		==Outline of Detailed Investigations Leading to Above Summary==

Jet53man: /* Virial Equilibrium (M) */

2021-07-15T21:17:37Z

Virial Equilibrium (M)

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	====Virial Equilibrium (M)====		====Virial Equilibrium (M)====
	Plugging these coefficient assignments into the [[~~User:Tohline/SSC/Virial/PolytropesEmbeddedOutline~~#Equilibrium_Configurations\|above mathematical prescription of the virial theorem]] gives the mass-radius relationship for pressure-truncated, polytropic equilibrium configurations, namely,		Plugging these coefficient assignments into the [[#Equilibrium_Configurations\|above mathematical prescription of the virial theorem]] gives the mass-radius relationship for pressure-truncated, polytropic equilibrium configurations, namely,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5">		<table border="0" cellpadding="5">
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\mathcal{B}_{M_\ell} \cdot \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{(n+1)/n} \biggl( \frac{R}{R_\mathrm{SWS}} \biggr)^{(n-3)/n} </math>		<math>\mathcal{B}_{M_\ell} \cdot \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{(n+1)/n} \biggl( \frac{R}{R_\mathrm{SWS}} \biggr)^{(n-3)/n} </math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
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	</table>		</table>
	</div>		</div>
	which matches the [[~~User:Tohline/~~SSC/Virial/PolytropesEmbedded/SecondEffortAgain#ConciseFreeEnergyExpression\|statement of virial equilibrium presented in our accompanying, more detailed analysis]].		which matches the [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#ConciseFreeEnergyExpression\|statement of virial equilibrium presented in our accompanying, more detailed analysis]].

	====Stability (M)====		====Stability (M)====

← Older revision		Revision as of 19:13, 16 July 2021
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	=See Also=		=See Also=
	<ul>		<ul>
	<li>[[~~User:Tohline/~~SphericallySymmetricConfigurations/IndexFreeEnergy#Index_to_Free-Energy_Analyses\|Index to a Variety of Free-Energy and/or Virial Analyses]]</li>		<li>[[SphericallySymmetricConfigurations/IndexFreeEnergy#Index_to_Free-Energy_Analyses\|Index to a Variety of Free-Energy and/or Virial Analyses]]</li>
	<li>[[SSC/Index\|Spherically Symmetric Configurations (SSC) Index]]</li>		<li>[[SSC/Index\|Spherically Symmetric Configurations (SSC) Index]]</li>
	</ul>		</ul>

← Older revision		Revision as of 13:19, 16 July 2021
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	<ul>		<ul>
	<li>[[User:Tohline/SphericallySymmetricConfigurations/IndexFreeEnergy#Index_to_Free-Energy_Analyses\|Index to a Variety of Free-Energy and/or Virial Analyses]]</li>		<li>[[User:Tohline/SphericallySymmetricConfigurations/IndexFreeEnergy#Index_to_Free-Energy_Analyses\|Index to a Variety of Free-Energy and/or Virial Analyses]]</li>
	<li>[[SSC/Index\|Spherically Symmetric Configurations (SSC) Index]]		<li>[[SSC/Index\|Spherically Symmetric Configurations (SSC) Index]]</li>
	</ul>		</ul>


	{{ SGFfooter }}		{{ SGFfooter }}

← Older revision		Revision as of 12:42, 16 July 2021
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	<ul>		<ul>
	<li>[[User:Tohline/SphericallySymmetricConfigurations/IndexFreeEnergy#Index_to_Free-Energy_Analyses\|Index to a Variety of Free-Energy and/or Virial Analyses]]</li>		<li>[[User:Tohline/SphericallySymmetricConfigurations/IndexFreeEnergy#Index_to_Free-Energy_Analyses\|Index to a Variety of Free-Energy and/or Virial Analyses]]</li>
			<li>[[SSC/Index\|Spherically Symmetric Configurations (SSC) Index]]
	</ul>		</ul>


	{{ SGFfooter }}		{{ SGFfooter }}