SSC/Structure/Polytropes/VirialSummary - Revision history

Joel2: /* Fix External Pressure While Varying Mass */

2026-04-05T00:50:28Z

Fix External Pressure While Varying Mass

← Older revision		Revision as of 20:50, 4 April 2026
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	</table>		</table>

	(As has been demonstrated in an [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt2#modNormalizations\|accompanying discussion]], we could alternatively have obtained this same algebraic relation by shifting to a different pair of mass- and radius-normalizations — namely, <math>M_\mathrm{mod}</math> and <math>R_\mathrm{mod}</math> — instead of choosing these specific values of <math>\mathcal{A}</math> and <math>\mathcal{B}.</math>)		<span id="StahlerSchematic">(As has been demonstrated</span> in an [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt2#modNormalizations\|accompanying discussion]], we could alternatively have obtained this same algebraic relation by shifting to a different pair of mass- and radius-normalizations — namely, <math>M_\mathrm{mod}</math> and <math>R_\mathrm{mod}</math> — instead of choosing these specific values of <math>\mathcal{A}</math> and <math>\mathcal{B}.</math>)

	<table border="1" align="center" cellpadding="8">		<table border="1" align="center" cellpadding="8">

Joel2: /* Fix External Pressure While Varying Mass */

2023-12-28T23:38:40Z

Fix External Pressure While Varying Mass

← Older revision		Revision as of 19:38, 28 December 2023
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	</td></tr>		</td></tr>
	</table>		</table>
	An equilibrium model ''sequence'' is thereby defined for each chosen polytropic index in the range, <math>~0 \le n < \infty</math>.		An equilibrium model ''sequence'' is thereby defined for each chosen polytropic index in the range, <math>0 \le n < \infty</math>.

	Along each equilibrium sequence for which <math>~3 \le n < \infty</math>, there is one ''specific'' equilibrium state that marks a transition from dynamically stable to dynamically unstable configurations. For a given mass, the radius of this ''critical'' configuration is identified by the relation,		Along each equilibrium sequence for which <math>3 \le n < \infty</math>, there is one ''specific'' equilibrium state that marks a transition from dynamically stable to dynamically unstable configurations. For a given mass, the radius of this ''critical'' configuration is identified by the relation,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~		<math>
	\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}}\biggr)_\mathrm{crit}^4		\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}}\biggr)_\mathrm{crit}^4
	</math>		</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>
	\frac{\mathcal{A}}{4\pi}\biggl(\frac{n - 3}{n}\biggr) \biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \, .		\frac{\mathcal{A}}{4\pi}\biggl(\frac{n - 3}{n}\biggr) \biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \, .
	</math>		</math>
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	----		----

	Below, we will examine the behavior of individual virial-equilibrium sequences, using various physical arguments to justify our choice of specific expressions for the coefficients, <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>. Here, in an effort to provide a broad overview, we set,		Below, we will examine the behavior of individual virial-equilibrium sequences, using various physical arguments to justify our choice of specific expressions for the coefficients, <math>\mathcal{A}</math> and <math>\mathcal{B}.</math> Here, in an effort to provide a broad overview, we set,
	<div align="center">		<div align="center">
	<math>~\mathcal{A} = \frac{4\pi n}{3(n+1)}</math>       and       <math>~\mathcal{B} = \biggl( \frac{4\pi}{3} \biggr) \, ,</math>		<math>\mathcal{A} = \frac{4\pi n}{3(n+1)}</math>       and       <math>\mathcal{B} = \biggl( \frac{4\pi}{3} \biggr) \, ,</math>
	</div>		</div>
	and display, in a single diagram, the behaviors of equilibrium sequences having seven different polytropic indexes. Specifically, each curve in the left-hand panel of Figure 1 results from the mass-radius expression,		and display, in a single diagram, the behaviors of equilibrium sequences having seven different polytropic indexes. Specifically, each curve in the left-hand panel of Figure 1 results from the mass-radius expression,
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~0</math>		<math>0</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{R}{R_\mathrm{SWS}}\biggr)^4		\biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^4
	- \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^{(n-3)/n} \biggl( \frac{M}{M_\mathrm{SWS}} \biggr)^{(n+1)/n}		- \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^{(n-3)/n} \biggl( \frac{M}{M_\mathrm{SWS}} \biggr)^{(n+1)/n}
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	</table>		</table>

	(As has been demonstrated in an [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt2#modNormalizations\|accompanying discussion]], we could alternatively have obtained this same algebraic relation by shifting to a different pair of mass- and radius-normalizations — namely, <math>~M_\mathrm{mod}</math> and <math>~R_\mathrm{mod}</math> — instead of choosing these specific values of <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>.)		(As has been demonstrated in an [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt2#modNormalizations\|accompanying discussion]], we could alternatively have obtained this same algebraic relation by shifting to a different pair of mass- and radius-normalizations — namely, <math>M_\mathrm{mod}</math> and <math>R_\mathrm{mod}</math> — instead of choosing these specific values of <math>\mathcal{A}</math> and <math>\mathcal{B}.</math>)

	<table border="1" align="center" cellpadding="8">		<table border="1" align="center" cellpadding="8">
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	----		----

	Below, we will examine the behavior of individual virial-equilibrium sequences, using various physical arguments to justify our choice of specific expressions for the coefficients, <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>. Here, in an effort to provide a broad overview, we set all three structural form factors to unity, in which case the pair of coefficients,		Below, we will examine the behavior of individual virial-equilibrium sequences, using various physical arguments to justify our choice of specific expressions for the coefficients, <math>\mathcal{A}</math> and <math>\mathcal{B}.</math> Here, in an effort to provide a broad overview, we set all three structural form factors to unity, in which case the pair of coefficients,
	<div align="center">		<div align="center">
	<math>~\mathcal{A} = \frac{1}{5}</math>       and       <math>~\mathcal{B} = \biggl( \frac{3}{4\pi} \biggr)^{1/n}</math>,		<math>\mathcal{A} = \frac{1}{5}</math>       and       <math>\mathcal{B} = \biggl( \frac{3}{4\pi} \biggr)^{1/n}</math>,
	</div>		</div>
	and the resulting mass-radius relation of equilibrium sequences is governed by the algebraic mass-radius relation,		and the resulting mass-radius relation of equilibrium sequences is governed by the algebraic mass-radius relation,
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~0</math>		<math>0</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{R}{R_\mathrm{SWS}}\biggr)^4		\biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^4
	- \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^{(n-3)/n}\biggl[ \biggl(\frac{3}{4\pi} \biggr) \frac{M}{M_\mathrm{SWS}}\biggr]^{(n+1)/n}		- \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^{(n-3)/n}\biggl[ \biggl(\frac{3}{4\pi} \biggr) \frac{M}{M_\mathrm{SWS}}\biggr]^{(n+1)/n}
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	</table>		</table>

	For comparison, the ''schematic'' diagram displayed on the righthand side of the figure is a reproduction of Figure 17 from Appendix B of ~~[http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)]~~. It seems that our derived, analytically prescribable, mass-radius relationship — which is, in essence, a statement of the scalar virial theorem — embodies most of the attributes of the mass-radius relationship for pressure-truncated polytropes that were already understood, and conveyed schematically, by Stahler in 1983.		For comparison, the ''schematic'' diagram displayed on the righthand side of the figure is a reproduction of Figure 17 from Appendix B of {{ Stahler83 }}. It seems that our derived, analytically prescribable, mass-radius relationship — which is, in essence, a statement of the scalar virial theorem — embodies most of the attributes of the mass-radius relationship for pressure-truncated polytropes that were already understood, and conveyed schematically, by Stahler in 1983.

	=See Also=		=See Also=

Joel2: /* Fix External Pressure While Varying Mass */

2023-12-28T23:33:19Z

Fix External Pressure While Varying Mass

← Older revision		Revision as of 19:33, 28 December 2023
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	</table>		</table>

	For comparison, the ''schematic'' diagram displayed on the righthand side of the figure is a reproduction of Figure 17 from Appendix B of ~~[http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)]~~. It seems that our derived, analytically prescribable, mass-radius relationship — which is, in essence, a statement of the scalar virial theorem — embodies most of the attributes of the mass-radius relationship for pressure-truncated polytropes that were already understood, and conveyed schematically, by Stahler in 1983.		For comparison, the ''schematic'' diagram displayed on the righthand side of the figure is a reproduction of Figure 17 from Appendix B of {{ Stahler83 }}. It seems that our derived, analytically prescribable, mass-radius relationship — which is, in essence, a statement of the scalar virial theorem — embodies most of the attributes of the mass-radius relationship for pressure-truncated polytropes that were already understood, and conveyed schematically, by Stahler in 1983.


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	</tr>		</tr>
	</table>		</table>
	The left-hand panel of Figure 1 displays this behavior for equilibrium sequences having seven different values of the polytropic index, as labeled. As has been demonstrated in an [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Plotting_Concise_Mass-Radius_Relation\|accompanying discussion]], this governing mass-radius relation can be analytically manipulated into a form that provides either an explicit <math>~M(R)</math> or <math>~R(M)</math> relation for the cases labeled, <math>~n = 1, 3,</math> and <math>~\infty</math>; a generic root-finding technique has been used to generate [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#TabulatedValues\|points along each of the other depicted equilibrium sequences]].		The left-hand panel of Figure 1 displays this behavior for equilibrium sequences having seven different values of the polytropic index, as labeled. As has been demonstrated in an [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt2#Plotting_Concise_Mass-Radius_Relation\|accompanying discussion]], this governing mass-radius relation can be analytically manipulated into a form that provides either an explicit <math>M(R)</math> or <math>R(M)</math> relation for the cases labeled, <math>n = 1, 3,</math> and <math>\infty</math>; a generic root-finding technique has been used to generate [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt2#TabulatedValues\|points along each of the other depicted equilibrium sequences]].

	<table border="1" align="center" cellpadding="8">		<table border="1" align="center" cellpadding="8">

Joel2: /* Basic Relation */

2023-12-28T23:26:41Z

Basic Relation

← Older revision		Revision as of 19:26, 28 December 2023
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	</table>		</table>

	where, when written in terms of the trio of ''[[SSC/FreeEnergy/~~PolytropesEmbedded/Pt2~~#~~PTtable~~\|structural form factors]]'', <math>\tilde{\mathfrak{f}}_A,</math> <math>\tilde{\mathfrak{f}}_M,</math> and <math>\tilde{\mathfrak{f}}_W,</math> the pair of constants,		where, when written in terms of the trio of ''[[SSC/FreeEnergy/Powerpoint#Structural_Form_Factors\|structural form factors]]'', <math>\tilde{\mathfrak{f}}_A,</math> <math>\tilde{\mathfrak{f}}_M,</math> and <math>\tilde{\mathfrak{f}}_W,</math> the pair of constants,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5">		<table border="0" cellpadding="5">

Joel2: /* Fix External Pressure While Varying Mass */

2023-12-28T23:26:02Z

Fix External Pressure While Varying Mass

← Older revision		Revision as of 19:26, 28 December 2023
Line 381:		Line 381:
	===Fix External Pressure While Varying Mass===		===Fix External Pressure While Varying Mass===

	In this case, we want to examine undulations of a two-dimensional free-energy surface that results from allowing <math>~R</math> and <math>~M</math> to vary while holding <math>~P_e</math> fixed. In our [[SSC/FreeEnergy/PolytropesEmbedded#Case_P\|accompanying, more detailed discussion]], this is referred to as '''Case P'''. Motivated by ~~[http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler's (1983)]~~ work, here we adopt the normalizations,		In this case, we want to examine undulations of a two-dimensional free-energy surface that results from allowing <math>~R</math> and <math>~M</math> to vary while holding <math>~P_e</math> fixed. In our [[SSC/FreeEnergy/PolytropesEmbedded#Case_P\|accompanying, more detailed discussion]], this is referred to as '''Case P'''. Motivated by the published work of {{ Stahler83full }}, here we adopt the normalizations,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	</table>		</table>

	where the pair of constants, <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>, have the same definitions in terms of the ''[[SSC/FreeEnergy/~~PolytropesEmbedded~~#~~Free-Energy_of_Truncated_Polytropes~~\|structural form factors]]'' as provided above. The relevant dimensionless free-energy surface is, then, given by the expression,		where the pair of constants, <math>\mathcal{A}</math> and <math>\mathcal{B}</math>, have the same definitions in terms of the ''[[SSC/FreeEnergy/Powerpoint#Structural_Form_Factors\|structural form factors]]'' as provided above. The relevant dimensionless free-energy surface is, then, given by the expression,
	<table border="1" cellpadding="8" align="center">		<table border="1" cellpadding="8" align="center">
	<tr>		<tr>
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	</table>		</table>

	(As has been demonstrated in an [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#modNormalizations\|accompanying discussion]], we could alternatively have obtained this same algebraic relation by shifting to a different pair of mass- and radius-normalizations — namely, <math>~M_\mathrm{mod}</math> and <math>~R_\mathrm{mod}</math> — instead of choosing these specific values of <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>.)		(As has been demonstrated in an [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt2#modNormalizations\|accompanying discussion]], we could alternatively have obtained this same algebraic relation by shifting to a different pair of mass- and radius-normalizations — namely, <math>~M_\mathrm{mod}</math> and <math>~R_\mathrm{mod}</math> — instead of choosing these specific values of <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>.)

	<table border="1" align="center" cellpadding="8">		<table border="1" align="center" cellpadding="8">

Joel2: /* Fix Mass While Varying External Pressure */

2023-12-28T23:18:14Z

Fix Mass While Varying External Pressure

← Older revision		Revision as of 19:18, 28 December 2023
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	</tr>		</tr>
	</table>		</table>
	— which, as is detailed in an [[SSCpt1/Virial#Choices_Made_by_Other_Researchers\|accompanying discussion]], are similar but not identical to the normalizations adopted by ~~[http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)]~~ and by ~~[http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth (1981)]~~ — the coefficients in the above-presented [[#Basic_Relations\|Basic Relations]] become,		— which, as is detailed in an [[SSCpt1/Virial#Choices_Made_by_Other_Researchers\|accompanying discussion]], are similar but not identical to the normalizations adopted by {{ Horedt70full }} and by {{ Whitworth81full }} — the coefficients in the above-presented [[#Basic_Relations\|Basic Relations]] become,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Joel2: /* Virial Equilibrium of Pressure-Truncated Polytropes */

2023-12-28T23:15:14Z

Virial Equilibrium of Pressure-Truncated Polytropes

← Older revision		Revision as of 19:15, 28 December 2023
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	\|}		\|}

	Here we will draw heavily from: [1] An accompanying [[SSC/FreeEnergy/PolytropesEmbedded#Pressure-Truncated_Polytropes\|''Free Energy Synopsis'']]; and [2] A detailed analysis of the [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Plotting_Concise_Mass-Radius_Relation\|Virial Equilibrium of Adiabatic Spheres]].		Here we will draw heavily from: [1] An accompanying [[SSC/FreeEnergy/PolytropesEmbedded#Pressure-Truncated_Polytropes\|''Free Energy Synopsis'']]; and [2] A detailed analysis of the [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt2#Plotting_Concise_Mass-Radius_Relation\|Virial Equilibrium of Adiabatic Spheres]].

	<br />		<br />
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	</table>		</table>

	where, when written in terms of the trio of ''[[SSC/FreeEnergy/PolytropesEmbedded/Pt2#PTtable\|structural form factors]]'', <math>~\tilde{\mathfrak{f}}_M,</math> <math>~\tilde{\mathfrak{f}}_M,</math> and <math>~\tilde{\mathfrak{f}}_M,</math> the pair of constants,		where, when written in terms of the trio of ''[[SSC/FreeEnergy/PolytropesEmbedded/Pt2#PTtable\|structural form factors]]'', <math>\tilde{\mathfrak{f}}_A,</math> <math>\tilde{\mathfrak{f}}_M,</math> and <math>\tilde{\mathfrak{f}}_W,</math> the pair of constants,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5">		<table border="0" cellpadding="5">

Joel2: /* Basic Relation */

2023-12-22T23:44:20Z

Basic Relation

← Older revision		Revision as of 19:44, 22 December 2023
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	</table>		</table>

	where, when written in terms of the trio of ''[[SSC/FreeEnergy/PolytropesEmbedded#PTtable\|structural form factors]]'', <math>~\tilde{\mathfrak{f}}_M,</math> <math>~\tilde{\mathfrak{f}}_M,</math> and <math>~\tilde{\mathfrak{f}}_M,</math> the pair of constants,		where, when written in terms of the trio of ''[[SSC/FreeEnergy/PolytropesEmbedded/Pt2#PTtable\|structural form factors]]'', <math>~\tilde{\mathfrak{f}}_M,</math> <math>~\tilde{\mathfrak{f}}_M,</math> and <math>~\tilde{\mathfrak{f}}_M,</math> the pair of constants,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5">		<table border="0" cellpadding="5">

Joel2: Created page with "FORCETOC =Virial Equilibrium of Pressure-Truncated Polytropes= {| 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;" | Virial Equilibrium
of
Pressure-Truncated
Polytropes |} Here we will draw heavily from: [1] An accompanying SSC/FreeEnergy/PolytropesEmbedded#..."

2023-12-22T22:06:18Z

Created page with "__FORCETOC__ =Virial Equilibrium of Pressure-Truncated Polytropes= {| 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"><b>Virial Equilibrium<br />of<br />Pressure-Truncated<br />Polytropes</b></font> |} Here we will draw heavily from: [1] An accompanying SSC/FreeEnergy/PolytropesEmbedded#..."

New page

__FORCETOC__
=Virial Equilibrium of Pressure-Truncated Polytropes=
{| 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#MoreModels|<b>Virial Equilibrium<br />of<br />Pressure-Truncated<br />Polytropes</b>]]</font>
|}

Here we will draw heavily from: [1] An accompanying [[SSC/FreeEnergy/PolytropesEmbedded#Pressure-Truncated_Polytropes|''Free Energy Synopsis'']]; and [2] A detailed analysis of the [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Plotting_Concise_Mass-Radius_Relation|Virial Equilibrium of Adiabatic Spheres]].

 <br />
 <br />
 <br />
 <br />
 <br />

==Groundwork==
===Basic Relation===
In the context of spherically symmetric, pressure-truncated polytropic configurations, the relevant free-energy expression is,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathfrak{G}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~W_\mathrm{grav} + U_\mathrm{int} + P_eV</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- 3\mathcal{A} \biggl[\frac{GM^2}{R} \biggr] + n\mathcal{B} \biggl[ \frac{K_nM^{(n+1)/n}}{R^{3/n}} \biggr] + \frac{4\pi}{3} \cdot P_e R^3 \, ,</math>
</td>
</tr>
</table>

where, when written in terms of the trio of ''[[SSC/FreeEnergy/PolytropesEmbedded#PTtable|structural form factors]]'', <math>~\tilde{\mathfrak{f}}_M,</math> <math>~\tilde{\mathfrak{f}}_M,</math> and <math>~\tilde{\mathfrak{f}}_M,</math> the pair of constants,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>~\mathcal{A} \equiv \frac{1}{5} \cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2}</math>
</td>
<td align="center">
      and     
</td>
<td align="left">
<math>\mathcal{B} \equiv \biggl(\frac{4\pi}{3} \biggr)^{-1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{f}}_M^{(n+1)/n}} \, .</math>
</td>
</tr>
</table>
</div>

===Often-Referenced Dimensionless Expressions===
When rewritten in a suitably dimensionless form — see two useful alternatives, below — this expression becomes,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathfrak{G}^*</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- a x^{-1} + bx^{-3/n} + c x^3 \, ,</math>
</td>
</tr>
</table>
where <math>~x</math> is the configuration's dimensionless radius and <math>~a</math>, <math>~b</math>, and <math>~c</math> are constants. We therefore have,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d\mathfrak{G}^*}{dx}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{x^2} \biggl[ a - \biggl( \frac{3b}{n} \biggr) x^{(n-3)/n} + 3c x^4 \biggr] \, ,</math>
</td>
</tr>
</table>
and,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d^2\mathfrak{G}^*}{dx^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{x^3} \biggl[\biggl(\frac{n+3}{n}\biggr) \biggl( \frac{3b}{n} \biggr) x^{(n-3)/n} + 6c x^4 - 2a \biggr] \, .</math>
</td>
</tr>
</table>

Virial equilibrium is obtained when <math>~d\mathfrak{G}^*/dx = 0</math>, that is, when

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl( \frac{3b}{n} \biggr) x_\mathrm{eq}^{(n-3)/n} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ a + 3c x_\mathrm{eq}^4 \, .</math>
</td>
</tr>
</table>
And along an equilibrium ''sequence'', the ''specific'' equilibrium state that marks a transition from dynamically stable to dynamically unstable configurations — henceforth labeled as having the ''critical'' radius, <math>~x_\mathrm{crit}</math> — is identified by setting <math>~d^2\mathfrak{G}^*/dx^2 = 0</math>, that is, it is the configuration for which,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[\biggl(\frac{n+3}{n}\biggr) \biggl( \frac{3b}{n} \biggr) x^{(n-3)/n} + 6c x^4 - 2a \biggr]_{x = x_\mathrm{eq}}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~
x_\mathrm{crit}^4
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{a}{3^2c}\biggl(\frac{n - 3}{n+1}\biggr) \, .
</math>
</td>
</tr>
</table>

Inserting the adiabatic exponent in place of the polytropic index via the relation, <math>~n = (\gamma - 1)^{-1}</math>, we have equivalently,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
x_\mathrm{crit}^4
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{a}{3^2c}\biggl(\frac{4-3\gamma}{\gamma}\biggr) \, .
</math>
</td>
</tr>
</table>

===Useful Recognition===

By comparing various terms in the first two algebraic ''Setup'' expressions, above, It is clear that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~W^*_\mathrm{grav} = -ax^{-1}</math>
</td>
<td align="center">
      and,      
</td>
<td align="left">
<math>~U^*_\mathrm{int} = bx^{-3/n} \, .</math>
</td>
</tr>
</table>

Notice, then, that in every equilibrium configuration, we should find,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~- \frac{U^*_\mathrm{int}}{W^*_\mathrm{grav}}\biggr|_\mathrm{eq}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl(\frac{b}{a}\biggr) x_\mathrm{eq}^{(n-3)/n}
=
\frac{n}{3a} \biggl[ a + 3cx^4_\mathrm{eq} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{n}{3} \biggl[ 1 + \biggl(\frac{3c}{a}\biggr) x^4_\mathrm{eq} \biggr] \, .
</math>
</td>
</tr>
</table>
And, specifically in the ''critical'' configuration we should find that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~- \frac{U^*_\mathrm{int}}{W^*_\mathrm{grav}}\biggr|_\mathrm{crit}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{3(\gamma-1)} \biggl[ 1 + \frac{1}{3}\biggl(\frac{4-3\gamma}{\gamma}\biggr) \biggr]
=
\frac{4}{3^2\gamma(\gamma-1)}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~\frac{S^*_\mathrm{therm}}{W^*_\mathrm{grav}}\biggr|_\mathrm{crit}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{2}{3\gamma} \, .
</math>
</td>
</tr>
</table>
The equivalent of this last expression also appears at the end of subsection <b><font color="maroon" size="+1">⑦</font></b> of an [[SSC/SynopsisStyleSheet#Stability|accompanying ''Tabular Overview'']].

==Equilibrium Sequences==
In all of the polytropic configurations being considered here, {{ Math/MP_PolytropicConstant }} is a constant — that is, the specific entropy of all fluid elements is assumed to be the same, both spatially and temporally.

===Fix Mass While Varying External Pressure===

In this case, we want to examine undulations of a two-dimensional free-energy surface that results from allowing <math>R</math> and <math>P_e</math> to vary while holding <math>M</math> fixed. In our [[SSC/FreeEnergy/PolytropesEmbedded#Case_M|accompanying, more detailed discussion]], this is referred to as '''Case M'''. Adopting the normalizations,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>R_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\biggl[ \biggl( \frac{G}{K_n} \biggr)^n M^{n-1} \biggr]^{1/(n-3)} \, ,
</math>
</td>
<td align="center">      </td>
<td align="right">
<math>P_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\biggl[ \frac{K_n^{4n}}{G^{3(n+1)} M^{2(n+1)}} \biggr]^{1/(n-3)}
</math>
</td>
<td align="center">      and,       </td>
<td align="right">
<math>E_\mathrm{norm}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
P_\mathrm{norm} R^3_\mathrm{norm} \, ,
</math>
</td>
</tr>
</table>
— which, as is detailed in an [[SSCpt1/Virial#Choices_Made_by_Other_Researchers|accompanying discussion]], are similar but not identical to the normalizations adopted by [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)] and by [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth (1981)] — the coefficients in the above-presented [[#Basic_Relations|Basic Relations]] become,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~a</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~3\mathcal{A} \, ,
</math>
</td>
<td align="center">      </td>
<td align="right">
<math>~b</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~n\mathcal{B}
</math>
</td>
<td align="center">      and,       </td>
<td align="right">
<math>~c</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\frac{4\pi}{3}\biggl( \frac{P_e}{P_\mathrm{norm}} \biggr) \, .
</math>
</td>
</tr>
</table>

The relevant dimensionless free-energy surface is, then, given by the expression,
<table border="1" cellpadding="8" align="center">
<tr>
<th align="center" colspan="1">''Case M'' Free-Energy Surface</th>
</tr>
<tr><td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\mathfrak{G}_{K,M}^* \equiv \frac{\mathfrak{G}_{K,M}}{E_\mathrm{norm}} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-3\mathcal{A} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1} +~ n\mathcal{B} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-3/n}
+~ \biggl( \frac{4\pi}{3} \biggr) \frac{P_e}{P_\mathrm{norm}} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^3
\, .
</math>
</td>
</tr>
</table>

</td></tr>
</table>

===Fix External Pressure While Varying Mass===

In this case, we want to examine undulations of a two-dimensional free-energy surface that results from allowing <math>~R</math> and <math>~M</math> to vary while holding <math>~P_e</math> fixed. In our [[SSC/FreeEnergy/PolytropesEmbedded#Case_P|accompanying, more detailed discussion]], this is referred to as '''Case P'''. Motivated by [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler's (1983)] work, here we adopt the normalizations,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~R_\mathrm{SWS}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\biggl( \frac{n+1}{nG} \biggr)^{1/2} K_n^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]} \, ,</math>
</td>
<td align="center">    </td>
<td align="right">
<math>~M_\mathrm{SWS}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\biggl( \frac{n+1}{nG} \biggr)^{3/2} K_n^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]} </math>
</td>
<td align="center">     and,</td>
</tr>
<tr>
<td align="right">
<math>~E_\mathrm{SWS}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left" colspan="6">
<math>~
\biggl( \frac{n}{n+1} \biggr) \frac{GM_\mathrm{SWS}^2}{R_\mathrm{SWS}}
=
\biggl( \frac{n+1}{n} \biggr)^{3/2} G^{-3/2}K_n^{3n/(n+1)} P_\mathrm{e}^{(5-n)/[2(n+1)]} \, .</math>
</td>
</tr>

</table>
the coefficients in the above-presented [[#Basic_Relations|Basic Relations]] become,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~a</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~3 \mathcal{A} \biggl( \frac{n+1}{n} \biggr)\biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \, ,
</math>
</td>
<td align="center">      </td>
<td align="right">
<math>~b</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~n\mathcal{B} \biggl(\frac{M}{M_\mathrm{SWS}}\biggr)^{(n+1)/n}
</math>
</td>
<td align="center">      and,       </td>
<td align="right">
<math>~c</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\frac{4\pi}{3} \, ,
</math>
</td>
</tr>
</table>

where the pair of constants, <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>, have the same definitions in terms of the ''[[SSC/FreeEnergy/PolytropesEmbedded#Free-Energy_of_Truncated_Polytropes|structural form factors]]'' as provided above. The relevant dimensionless free-energy surface is, then, given by the expression,
<table border="1" cellpadding="8" align="center">
<tr>
<th align="center" colspan="1">''Case P'':   Free-Energy Surfaces</th>
</tr>
<tr><td align="center">

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathfrak{G}_{K,P_e}^* \equiv \frac{\mathfrak{G}_{K,P_e}}{E_\mathrm{SWS}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- 3 \mathcal{A} \biggl( \frac{n+1}{n} \biggr)\biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-1}
+ n\mathcal{B} \biggl(\frac{M}{M_\mathrm{SWS}}\biggr)^{(n+1)/n} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-3/n}
+ \frac{4\pi}{3} \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^3 \, .
</math>
</td>
</tr>
</table>

</td></tr>
</table>

Across this '''Case P''' free-energy surface, extrema — and, hence, equilibrium configurations — will arise wherever the virial-equilibrium condition is met, that is, wherever

<table border="1" cellpadding="8" align="center">
<tr>
<th align="center" colspan="1">''Case P'':   Virial Equilibrium Sequences</th>
</tr>
<tr><td align="center">

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^4
-\frac{3\mathcal{B}}{4\pi} \biggl(\frac{M}{M_\mathrm{SWS}}\biggr)^{(n+1)/n} \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^{(n-3)/n}
+ \frac{3 \mathcal{A}}{4\pi} \biggl( \frac{n+1}{n} \biggr)\biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \, .</math>
</td>
</tr>
</table>
</td></tr>
</table>
An equilibrium model ''sequence'' is thereby defined for each chosen polytropic index in the range, <math>~0 \le n < \infty</math>.

Along each equilibrium sequence for which <math>~3 \le n < \infty</math>, there is one ''specific'' equilibrium state that marks a transition from dynamically stable to dynamically unstable configurations. For a given mass, the radius of this ''critical'' configuration is identified by the relation,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}}\biggr)_\mathrm{crit}^4
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{\mathcal{A}}{4\pi}\biggl(\frac{n - 3}{n}\biggr) \biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \, .
</math>
</td>
</tr>
</table>

----

Below, we will examine the behavior of individual virial-equilibrium sequences, using various physical arguments to justify our choice of specific expressions for the coefficients, <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>. Here, in an effort to provide a broad overview, we set,
<div align="center">
<math>~\mathcal{A} = \frac{4\pi n}{3(n+1)}</math>       and       <math>~\mathcal{B} = \biggl( \frac{4\pi}{3} \biggr) \, ,</math>
</div>
and display, in a single diagram, the behaviors of equilibrium sequences having seven different polytropic indexes. Specifically, each curve in the left-hand panel of Figure 1 results from the mass-radius expression,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^4
- \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^{(n-3)/n} \biggl( \frac{M}{M_\mathrm{SWS}} \biggr)^{(n+1)/n}
+ \biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \, .</math>
</td>
</tr>
</table>

(As has been demonstrated in an [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#modNormalizations|accompanying discussion]], we could alternatively have obtained this same algebraic relation by shifting to a different pair of mass- and radius-normalizations — namely, <math>~M_\mathrm{mod}</math> and <math>~R_\mathrm{mod}</math> — instead of choosing these specific values of <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>.)

<table border="1" align="center" cellpadding="8">
<tr>
<th align="center" colspan="2">Figure 1:   Virial-Analysis-Based Equilibrium Sequences</th>
</tr>
<tr>
<td align="center" rowspan="2">
[[File:MassRadiusVirialLabeled.png|350px|Virial Theorem Mass-Radius Relation]]
</td>
<td align="center">
[[File:Stahler1983TitlePage0.png|300px|center|Stahler (1983) Title Page]]
</td>
</tr>
<tr>
<td align="center" bgcolor="white">
[[File:Stahler_MRdiagram1.png|300px|center|Stahler (1983) Figure 17 (edited)]]
</td>
</tr>
</table>

For comparison, the ''schematic'' diagram displayed on the righthand side of the figure is a reproduction of Figure 17 from Appendix B of [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)]. It seems that our derived, analytically prescribable, mass-radius relationship — which is, in essence, a statement of the scalar virial theorem — embodies most of the attributes of the mass-radius relationship for pressure-truncated polytropes that were already understood, and conveyed schematically, by Stahler in 1983.

----

Below, we will examine the behavior of individual virial-equilibrium sequences, using various physical arguments to justify our choice of specific expressions for the coefficients, <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>. Here, in an effort to provide a broad overview, we set all three structural form factors to unity, in which case the pair of coefficients,
<div align="center">
<math>~\mathcal{A} = \frac{1}{5}</math>       and       <math>~\mathcal{B} = \biggl( \frac{3}{4\pi} \biggr)^{1/n}</math>,
</div>
and the resulting mass-radius relation of equilibrium sequences is governed by the algebraic mass-radius relation,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^4
- \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^{(n-3)/n}\biggl[ \biggl(\frac{3}{4\pi} \biggr) \frac{M}{M_\mathrm{SWS}}\biggr]^{(n+1)/n}
+ ~\frac{3}{20\pi} \biggl( \frac{n+1}{n}\biggr) \biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \, .</math>
</td>
</tr>
</table>
The left-hand panel of Figure 1 displays this behavior for equilibrium sequences having seven different values of the polytropic index, as labeled. As has been demonstrated in an [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Plotting_Concise_Mass-Radius_Relation|accompanying discussion]], this governing mass-radius relation can be analytically manipulated into a form that provides either an explicit <math>~M(R)</math> or <math>~R(M)</math> relation for the cases labeled, <math>~n = 1, 3,</math> and <math>~\infty</math>; a generic root-finding technique has been used to generate [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#TabulatedValues|points along each of the other depicted equilibrium sequences]].

<table border="1" align="center" cellpadding="8">
<tr>
<th align="center" colspan="2">Figure 1:   Virial-Analysis-Based Equilibrium Sequences</th>
</tr>
<tr>
<td align="center" rowspan="2">
[[File:MassRadiusVirialLabeled.png|350px|Virial Theorem Mass-Radius Relation]]
</td>
<td align="center">
[[File:Stahler1983TitlePage0.png|300px|center|Stahler (1983) Title Page]]
</td>
</tr>
<tr>
<td align="center" bgcolor="white">
[[File:Stahler_MRdiagram1.png|300px|center|Stahler (1983) Figure 17 (edited)]]
</td>
</tr>
</table>

For comparison, the ''schematic'' diagram displayed on the righthand side of the figure is a reproduction of Figure 17 from Appendix B of [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)]. It seems that our derived, analytically prescribable, mass-radius relationship — which is, in essence, a statement of the scalar virial theorem — embodies most of the attributes of the mass-radius relationship for pressure-truncated polytropes that were already understood, and conveyed schematically, by Stahler in 1983.

=See Also=
<ul>
<li>[[SphericallySymmetricConfigurations/IndexFreeEnergy#Index_to_Free-Energy_Analyses|Index to a Variety of Free-Energy and/or Virial Analyses]]</li>
</ul>

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