SSC/Virial/PolytropesEmbeddedOutline - Revision history

Jet53man: /* The Physics */

2021-10-28T17:15:42Z

The Physics

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	<font color="maroon"><b>Pressure-Truncated Polytropes</b></font>: But, a [[SSC/Virial/Polytropes#Nonrotating_Adiabatic_Configuration_Embedded_in_an_External_Medium\|configuration embedded in an external medium]] of pressure, <math>~P_e</math>, will have a (pressure-truncated) surface whose radius, <math>~R_\mathrm{limit}</math>, corresponds to the radial location at which the configuration's internal pressure drops to a value that equals <math>~P_e</math>. In this case as well, one might choose to refer to <math>~M_\mathrm{limit}</math> as the total mass; on the other hand, it might be more useful to distinguish <math>~M_\mathrm{limit}</math> from <math>~M_\mathrm{tot}</math>, continuing to rely on <math>~M_\mathrm{tot}</math> to represent the mass of the corresponding ''isolated'' polytrope.		<font color="maroon"><b>Pressure-Truncated Polytropes</b></font>: But, a [[SSC/Virial/Polytropes#Nonrotating_Adiabatic_Configuration_Embedded_in_an_External_Medium\|configuration embedded in an external medium]] of pressure, <math>~P_e</math>, will have a (pressure-truncated) surface whose radius, <math>~R_\mathrm{limit}</math>, corresponds to the radial location at which the configuration's internal pressure drops to a value that equals <math>~P_e</math>. In this case as well, one might choose to refer to <math>~M_\mathrm{limit}</math> as the total mass; on the other hand, it might be more useful to distinguish <math>~M_\mathrm{limit}</math> from <math>~M_\mathrm{tot}</math>, continuing to rely on <math>~M_\mathrm{tot}</math> to represent the mass of the corresponding ''isolated'' polytrope.

	<font color="maroon"><b>BiPolytropes</b></font>: When discussing [[SSC/~~BipolytropeGeneralization_Version2~~#Bipolytrope_Generalizatio\|bipolytropes]], the limit of integration, <math>~R_\mathrm{limit}</math>, will naturally refer to the radial location that defines the outer edge of the configuration's "core" and, at the same time, identifies the radial "interface" between the bipolytrope's core and its envelope. In this case, <math>~M_\mathrm{limit}</math> corresponds to the mass of the core rather than to the total mass of the bipolytropic configuration.		<font color="maroon"><b>BiPolytropes</b></font>: When discussing [[SSC/BipolytropeGeneralizationVersion2#Bipolytrope_Generalizatio\|bipolytropes]], the limit of integration, <math>~R_\mathrm{limit}</math>, will naturally refer to the radial location that defines the outer edge of the configuration's "core" and, at the same time, identifies the radial "interface" between the bipolytrope's core and its envelope. In this case, <math>~M_\mathrm{limit}</math> corresponds to the mass of the core rather than to the total mass of the bipolytropic configuration.
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	====Coefficient Definitions====		====Coefficient Definitions====

	The {{ ~~Whitworth81full~~ }} <font color="red">'''Case P'''</font> analysis of pressure-truncated polytropic spheres produces the following governing free-energy function — referred to by Whitworth as the "global potential function":		The {{ Whitworth81 }} <font color="red">'''Case P'''</font> analysis of pressure-truncated polytropic spheres produces the following governing free-energy function — referred to by Whitworth as the "global potential function":

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Jet53man: /* See Also */

2021-10-14T21:19:56Z

Jet53man: /* Second Effort, Second Time Around */

2021-10-13T19:02:35Z

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 [[SSC/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=

Jet53man: /* First Effort, Second Time Around */

2021-10-13T19:01:28Z

First Effort, Second Time Around

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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 [[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.

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

Jet53man: /* Third Effort */

2021-10-13T19:00:35Z

Third Effort

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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,
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	<math>~\mathfrak{f}_M = \frac{\bar\rho}{\rho_c} = \biggl[- \frac{3\tilde\theta^'}{\tilde\xi}\biggr] \, ,</math>		<math>~\mathfrak{f}_M = \frac{\bar\rho}{\rho_c} = \biggl[- \frac{3\tilde\theta^'}{\tilde\xi}\biggr] \, ,</math>
	</div>		</div>
	~~which~~ is the same as before and the same as for isolated polytropes. We also have determined that,		is the same as before and the same as for isolated polytropes. We also have determined that,
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	<math>~\biggl(\frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr) \frac{1}{\mathfrak{f}_M} =		<math>~\biggl(\frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr) \frac{1}{\mathfrak{f}_M} =

Jet53man: /* Second Effort */

2021-10-13T18:53:06Z

Second Effort

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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===

Jet53man: /* First Effort */

2021-10-13T18:52:14Z

First Effort

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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,

Jet53man: /* Coefficient Definitions */

2021-10-12T20:29:13Z

Coefficient Definitions

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	where, drawing from ~~[http://adsabs.harvard.edu/abs/1983ApJ...268..165S Steven W. Stahler's~~ (~~1983~~)~~] work~~ — see also our [[SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation\|accompanying discussion]],		where, drawing from {{ Stahler83full }} (hereafter, {{ Stahler83hereafter }}) — see also our [[SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation\|accompanying discussion]],
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	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Jet53man: /* ASIDE: Isothermal Configurations */

2021-10-12T18:57:17Z

ASIDE: Isothermal Configurations

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	Our Figure 2 ~~provided~~ graphical depictions of the free-energy surface, <math>\mathcal{G}(R, P_e) = 2\mathcal{U}/3\mathcal{U}_\mathrm{rf}</math>, associated with pressure-truncated, uniform-density isothermal configurations — see equation (10) of {{ Whitworth81 }} or our restatement of this equation, [[SSC/Virial/PolytropesEmbeddedOutline#WhitworthFreeEnergyExpression\|above]]:		Our Figure 2 provides graphical depictions of the free-energy surface, <math>\mathcal{G}(R, P_e) = 2\mathcal{U}/3\mathcal{U}_\mathrm{rf}</math>, associated with pressure-truncated, uniform-density isothermal configurations — see equation (10) of {{ Whitworth81 }} or our restatement of this equation, [[SSC/Virial/PolytropesEmbeddedOutline#WhitworthFreeEnergyExpression\|above]]:
	<ul>		<ul>
	<li>''Lower-right quadrant'': <sup>&dagger;</sup>A reproduction of the black & white Figure 2 from {{ Whitworth81 }} ; as displayed here, the aspect ratio of the figure has been modified slightly in order to facilitate comparison with the images displayed in other quadrants of this figure. Continuous lines display the radial dependence of the free energy for a variety of different values of the external pressure, as labelled. Quoting directly from Whitworth's figure caption, "Filled circles mark stable equilibria; the open circles mark unstable equilibria; and the cross marks the critical neutral equilibrium state" at <math>~(R_\mathrm{eq}/R_\mathrm{rf}, P_e/P_\mathrm{rf}) = (1.0, 1.0)</math>. </li>		<li>''Lower-right quadrant'': <sup>&dagger;</sup>A reproduction of the black & white Figure 2 from {{ Whitworth81 }} ; as displayed here, the aspect ratio of the figure has been modified slightly in order to facilitate comparison with the images displayed in other quadrants of this figure. Continuous lines display the radial dependence of the free energy for a variety of different values of the external pressure, as labelled. Quoting directly from Whitworth's figure caption, "Filled circles mark stable equilibria; the open circles mark unstable equilibria; and the cross marks the critical neutral equilibrium state" at <math>~(R_\mathrm{eq}/R_\mathrm{rf}, P_e/P_\mathrm{rf}) = (1.0, 1.0)</math>. </li>
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	</div>		</div>

	Figure 2 from {{ Whitworth81 }} — reproduced in the lower-right quadrant of ~~the~~ [[SSC/Virial/PolytropesEmbeddedOutline#3DIsothermalSurface\|composite image ~~shown immediately above~~]] — does an excellent job of conveying many of the essential elements of this isothermal free-energy surface within the constraints imposed by a two-dimensional black & white line plot. In order to construct this compact plot, Whitworth employed a different free-energy normalization parameter for each selected value of the external pressure. Specifically, he used,		Figure 2 from {{ Whitworth81 }} — reproduced in the lower-right quadrant of [[SSC/Virial/PolytropesEmbeddedOutline#3DIsothermalSurface\|our Figure 2 composite image]] — does an excellent job of conveying many of the essential elements of this isothermal free-energy surface within the constraints imposed by a two-dimensional black & white line plot. In order to construct this compact plot, Whitworth employed a different free-energy normalization parameter for each selected value of the external pressure. Specifically, he used,
	<div align="center">		<div align="center">
	<math>~\mathcal{G}_0(P_e) = \frac{2}{3}\biggl[ 1 + \ln\biggl( \frac{P_e}{4P_\mathrm{rf}} \biggr)\biggr]		<math>~\mathcal{G}_0(P_e) = \frac{2}{3}\biggl[ 1 + \ln\biggl( \frac{P_e}{4P_\mathrm{rf}} \biggr)\biggr]
	- \frac{3}{2} \biggl( \frac{P_e}{4P_\mathrm{rf}} \biggr)^{1/3} \, .</math>		- \frac{3}{2} \biggl( \frac{P_e}{4P_\mathrm{rf}} \biggr)^{1/3} \, .</math>
	</div>		</div>
	In an effort to quantitatively compare (and check for accuracy) our results with Whitworth's, we have adopted the same <math>~\mathcal{G}_0(P_e)</math> normalization function when generating the multicolored, three-dimensional free-energy surface that is displayed (with three different projections) in ~~the~~ [[SSC/Virial/PolytropesEmbeddedOutline#3DIsothermalSurface\|~~above~~ composite image]], along with the reproduction of Whitworth's Figure 2. Aside from this pressure-dependent normalization parameter, the surface drawn for comparison with Whitworth's Figure 2 is identical to the one displayed in the righthand panel of ~~the~~ [[SSC/Virial/PolytropesEmbeddedOutline#IntroFigures\|~~figure~~ at the top of this page]].		In an effort to quantitatively compare (and check for accuracy) our results with Whitworth's, we have adopted the same <math>~\mathcal{G}_0(P_e)</math> normalization function when generating the multicolored, three-dimensional free-energy surface that is displayed (with three different projections) in [[SSC/Virial/PolytropesEmbeddedOutline#3DIsothermalSurface\|our Figure 2 composite image]], along with the reproduction of Whitworth's Figure 2. Aside from this pressure-dependent normalization parameter, the surface drawn for comparison with Whitworth's Figure 2 is identical to the one displayed in the righthand panel of [[SSC/Virial/PolytropesEmbeddedOutline#IntroFigures\|our Figure]], at the top of this page.

	===Our Case P Analysis Allowing for Nonuniform Density Structures===		===Our Case P Analysis Allowing for Nonuniform Density Structures===

Jet53man: /* Virial Equilibrium of Embedded Polytropic Spheres */

2021-10-12T18:47:36Z

Virial Equilibrium of Embedded Polytropic Spheres

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	[[File:FreeNRGmassRadiusN5.png\|300px\|n = 5 Free-Energy Surface]]		[[File:FreeNRGmassRadiusN5.png\|300px\|n = 5 Free-Energy Surface]]
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	<td align="center" colspan="1">		<td align="center" colspan="1" bgcolor="#C0FFFF">
	[[File:FreeNRGpressureRadiusIsothermal.png\|300px\|Whitworth's (1981) Isothermal Free-Energy Surface]]		[[File:FreeNRGpressureRadiusIsothermal.png\|300px\|Whitworth's (1981) Isothermal Free-Energy Surface]]
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	<td align="center" colspan="1" rowspan="1" bgcolor="#~~D0FFFF~~">		<td align="center" colspan="1" rowspan="1" bgcolor="#C0FFFF">
	[[File:EnergyRadiusViewTop5.png\|250px\|Whitworth's (1981) Isothermal Free-Energy Surface]]		[[File:EnergyRadiusViewTop5.png\|250px\|Whitworth's (1981) Isothermal Free-Energy Surface]]
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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>
	</ul>		</ul>

	{{ SGFfooter }}		{{ SGFfooter }}