Difference between revisions of "User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain"

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</div>
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<font color="red"><b>
January 13, 2015:
</b></font>
As is noted in our [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#Third_Effort|accompanying outline of work]], I no longer believe that <math>~\mathfrak{f}_W</math> and <math>~\mathfrak{f}_A</math> have the same expressions as in isolated polytropes.  Hence, all of the material that follows is suspect and needs to be reworked.


After plugging these nontrivial expressions for <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math> into the righthand sides of the above equations for <math>~\Pi_\mathrm{ad}</math> and <math>~\Chi_\mathrm{ad}</math> and, simultaneously, using Horedt's detailed force-balanced expressions for <math>~r_a</math> and <math>~p_a</math> to specify, respectively, <math>~\chi_\mathrm{eq}</math> and <math>~P_e/P_\mathrm{norm}</math> in these same equations &#8212; see [[User:Tohline/SSC/Virial/PolytropesEmbedded/FirstEffortAgain#Summary_2|our accompanying discussion]] &#8212; we find that,
After plugging these nontrivial expressions for <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math> into the righthand sides of the above equations for <math>~\Pi_\mathrm{ad}</math> and <math>~\Chi_\mathrm{ad}</math> and, simultaneously, using Horedt's detailed force-balanced expressions for <math>~r_a</math> and <math>~p_a</math> to specify, respectively, <math>~\chi_\mathrm{eq}</math> and <math>~P_e/P_\mathrm{norm}</math> in these same equations &#8212; see [[User:Tohline/SSC/Virial/PolytropesEmbedded/FirstEffortAgain#Summary_2|our accompanying discussion]] &#8212; we find that,
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It is straightforward to show that this more compact pair of expressions for <math>~\Pi_\mathrm{ad}</math> and <math>~\Chi_\mathrm{ad}</math> satisfy the [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#ConciseVirial|virial theorem presented above]].
It is straightforward to show that this more compact pair of expressions for <math>~\Pi_\mathrm{ad}</math> and <math>~\Chi_\mathrm{ad}</math> satisfy the [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#ConciseVirial|virial theorem presented above]].


{{LSU_WorkInProgress}}
===Physical Meaning of Parameter <math>~\eta_\mathrm{ad}</math>===


===Physical Meaning of Parameter <math>~\eta_\mathrm{ad}</math>===
In association with our [[User:Tohline/SSC/Virial/PolytropesEmbedded/FirstEffortAgain#Summary_2|accompanying derivation of a concise expression for the virial theorem]], we see that the structural form factor associated with the thermal energy reservoir of our configuration is the sum of two terms, specifically,
<div align="center">
<math>~\tilde\mathfrak{f}_A = a_\mathrm{ad} + b_\mathrm{ad} \, ,</math>
</div>
while, as defined in our above discussion, <math>~\eta_\mathrm{ad}</math> is the ratio of these same two terms, specifically, 
<div align="center">
<math>~\eta_\mathrm{ad} = \frac{b_\mathrm{ad}}{a_\mathrm{ad}} \, .</math>
</div>
It is worth pointing out what physical quantities are associated with these two terms. 


As defined in our above discussion, <math>~\eta_\mathrm{ad}</math> is the ratio of the two terms that are summed together in the definition of the structural form factor, <math>~\tilde\mathfrak{f}_A</math>.  It is worth pointing out what physical quantities are associated with these two terms. 


At any radial location within a polytropic configuration, the [[User:Tohline/SSC/Structure/Polytropes#Lane-Emden_Equation|Lane-Emden function]], <math>~\theta</math>, is defined in terms of a ratio of the local density to the configuration's central density, specifically,
At any radial location within a polytropic configuration, the [[User:Tohline/SSC/Structure/Polytropes#Lane-Emden_Equation|Lane-Emden function]], <math>~\theta</math>, is defined in terms of a ratio of the local density to the configuration's central density, specifically,
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Hence, the quantity, <math>~\tilde\theta^{n+1}</math>, which appears as the second term in our definition of <math>~\tilde\mathfrak{f}_A</math>, is the ratio, <math>~(P/P_c)_{\tilde\xi}</math>, evaluated at the surface of the truncated polytropic sphere.  But, by construction, the pressure at this location equals the pressure of the external medium in which the polytrope is embedded, so we can write,
Hence, the quantity, <math>~\tilde\theta^{n+1}</math>, which appears as the second term in our definition of <math>~\tilde\mathfrak{f}_A</math>, is the ratio, <math>~(P/P_c)_{\tilde\xi}</math>, evaluated at the surface of the truncated polytropic sphere.  But, by construction, the pressure at this location equals the pressure of the external medium in which the polytrope is embedded, so we can write,
<div align="center">
<div align="center">
<math>\tilde\theta^{n+1} = \frac{P_e}{P_c}  \, .</math>
<math>b_\mathrm{ad} \equiv \tilde\theta^{n+1} = \frac{P_e}{P_c}  \, .</math>
</div>
</div>


In our [[User:Tohline/SSC/Virial/Polytropes#Strategy2|accompanying detailed analysis]], we have employed the virial theorem expression to demonstrate that the first term in our definition of <math>~\tilde\mathfrak{f}_A</math> provides a measure the configuration's normalized central pressureSpecifically, we show that,
Also, [[User:Tohline/SphericallySymmetricConfigurations/Virial#Structural_Form_Factors|directly from its integral definition]], we have that <math>~\tilde\mathfrak{f}_A = \bar{P}/P_c</math>.  So we can write,
<div align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
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<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\biggl( \frac{4\pi}{3} \biggr) \frac{P_c R_\mathrm{eq}^4}{G M_\mathrm{limit}^2}</math>
<math>~a_\mathrm{ad}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 372: Line 375:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~[3 (n+1) (\tilde\theta^')^2]^{-1} \, .</math>
<math>~\tilde\mathfrak{f}_A - b_\mathrm{ad } = \frac{\bar{P}}{P_c} - \frac{P_e}{P_c}
= \frac{P_e}{P_c}\biggl( \frac{\bar{P}}{P_e} - 1 \biggr) \, .</math>
   </td>
   </td>
</tr>
</tr>
</table>
</table>
</div>
</div>
We conclude, therefore, that quite generally,
We conclude, therefore, that,
<div align="center">
<math>~\eta_\mathrm{ad} = \biggl( \frac{\bar{P}}{P_e} - 1 \biggr)^{-1} \, .</math>
</div>
 
===Desired Pressure-Radius Relation===
 
It is clear from the above discussion that the pair of parametric equations obtained via a [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Detailed_Force-Balanced_Solution|solution of the detailed force-balanced equations]] satisfy our, slightly rearranged,
<div align="center" id="ConciseVirial3">
<font color="#770000">'''Algebraic Expression of the Virial Theorem'''</font><br />
 
<math>
\Pi_\mathrm{ad} \Chi_\mathrm{ad}^4 = \Chi_\mathrm{ad}^{(n-3)/n} - 1 \, .
</math>
</div>
More to the point, it is now clear that this virial theorem expression provides the direct relationship between the configuration's dimensionless equilibrium radius as defined by Horedt, <math>~r_a</math>, and the dimensionless applied external pressure as defined by Horedt, <math>~p_a</math>, that was not apparent from the original pair of parametric relations.  Horedt's parameters, <math>~r_a</math> and <math>~p_a</math>, can be directly associated to our parameters, <math>~\Chi_\mathrm{ad}</math> and <math>~\Pi_\mathrm{ad}</math>, via two new normalizations,  <math>~r_n</math> and <math>~p_n</math>, defined through the relations,
<div align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
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<tr>
<tr>
   <td align="right">
   <td align="right">
<math>(5-n) \tilde\mathfrak{f}_A  </math>
<math>~\Chi_\mathrm{ad} = \frac{r_a}{r_n}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
&nbsp;&nbsp;&nbsp;&nbsp; and &nbsp;&nbsp;&nbsp;&nbsp;
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>~\Pi_\mathrm{ad} = \frac{p_a}{p_n} \, .</math>
\biggl( \frac{3}{4\pi} \biggr) \frac{G M_\mathrm{limit}^2}{P_c R_\mathrm{eq}^4}  + (5-n) \frac{P_e}{P_c}
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
Specifically in terms of the coefficients in the free-energy expression,
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>~r_n^{n-3}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~\equiv</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>~
\biggl( \frac{3}{4\pi} \biggr) \frac{G M_\mathrm{limit}^2}{P_c R_\mathrm{eq}^4} \biggl[1  + \eta_\mathrm{ad} \biggr] \, ,
\frac{(n+1)^n}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{1-n}  
\biggl( \frac{\mathcal{A}}{\mathcal{B}} \biggr)^n \, ,
</math>
</math>
   </td>
   </td>
Line 410: Line 433:
</table>
</table>
</div>
</div>
and that,
and,
<div align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
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<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\eta_\mathrm{ad} </math>
<math>~p_n^{n-3}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~\equiv</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>\biggl[ \frac{4\pi (5-n)}{3} \biggr] \frac{P_e R_\mathrm{eq}^4}{G M_\mathrm{limit}^2} \, .</math>
<math>~
\frac{3^{n-3}}{(4\pi)^4 (n+1)^{3(n+1)}} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{2(n+1)}
\biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr] \, ;
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</table>
</div>
</div>
while, in terms of the structural form factors,


===Desired Pressure-Radius Relation===
It is now clear from our review, above, of [[User:Tohline/SSC/Virial/PolytropesSummary#Detailed_Force-Balanced_Solution|Horedt's detailed force-balanced solution]], that
<div align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
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<tr>
<tr>
   <td align="right">
   <td align="right">
<math>\frac{4\pi (5-n)}{3}\biggl[\frac{P_e R_\mathrm{eq}^4}{G M_\mathrm{limit}^2} \biggr]_\mathrm{Horedt} </math>
<math>~r_n^{n-3}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~\equiv</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\eta_\mathrm{ad} \, .</math>
<math>~
\frac{1}{3} \biggl[ \frac{(n+1)}{5} \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_A} \biggr]^n \mathfrak{f}_M^{1-n}
\, ,
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</table>
</div>
</div>
Hence, the pair of parametric equations obtained via a solution of the detailed force-balanced equations satisfy our, slightly rearranged,
and,
<div align="center" id="ConciseVirial3">
<font color="#770000">'''Algebraic Expression of the Virial Theorem'''</font><br />
 
<math>
\Pi_\mathrm{ad} \Chi_\mathrm{ad}^4 = \Chi_\mathrm{ad}^{(n-3)/n} - 1 \, .
</math>
</div>
More to the point, it is now clear that this virial theorem expression provides the direct relationship between the configuration's dimensionless equilibrium radius as defined by Horedt, <math>~r_a</math>, and the dimensionless applied external pressure as defined by Horedt, <math>~p_a</math>, that was not apparent from the original pair of parametric relations.  Horedt's parameters, <math>~r_a</math> and <math>~p_a</math>, can be directly associated to our parameters, <math>~\Chi_\mathrm{ad}</math> and <math>~\Pi_\mathrm{ad}</math>, via two new normalizations,  <math>~r_n</math> and <math>~p_n</math>, defined through the relations,
<div align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
Line 460: Line 480:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\Chi_\mathrm{ad} = \frac{r_a}{r_n}</math>
<math>~p_n^{n-3}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
&nbsp;&nbsp;&nbsp;&nbsp; and &nbsp;&nbsp;&nbsp;&nbsp;
<math>~\equiv</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\Pi_\mathrm{ad} = \frac{p_a}{p_n} \, .</math>
<math>~
\frac{1}{(4\pi)^8} \biggl[ \frac{3\cdot 5^3}{(n+1)^3} \cdot \frac{\mathfrak{f}_M^2}{\mathfrak{f}_W^3} \biggr]^{n+1} \mathfrak{f}_A^{4n}
\, .
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</table>
</div>
</div>
Specifically in terms of the coefficients in the free-energy expression,
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
 
==Implications Regarding Stability==
 
===Model Sequences===
[[File:AdabaticBoundedSpheres_Virial.jpg|thumb|300px|Equilibrium Adiabatic Pressure-Radius Diagram]]
After choosing a value for the system's adiabatic index (or, equivalently, its polytropic index), <math>~\gamma = (n+1)/n</math>, the functional form of the virial theorem expression, <math>~\Pi_\mathrm{ad}(\chi_\mathrm{ad})</math>, is known and, hence, the equilibrium model sequence can be plotted.  Half-a-dozen such model sequences are shown in the figure near the beginning of this discussion.  Each curve can be viewed as mapping out a single-parameter sequence of equilibrium models; "evolution" along the curve can be accomplished by varying the key parameter, <math>~\eta_\mathrm{ad}</math>, over the physically relevant range, <math>0 \le \eta_\mathrm{ad} < \infty</math>. 
<table border="1" cellpadding="10" width="50%" align="center">
<tr><td align="left">
<font color="maroon">'''ASIDE'''</font> [18 March 2015]:  Many months after I penned the above description of "evolution" along an equilibrium model sequence, I started analyzing in detail the paper by [http://adsabs.harvard.edu/abs/1981PASJ...33..299K Kimura (1981b)].  The following excerpt from &sect;3 of his paper shows that Kimura presented essentially the same description of "evolution along a sequence" several decades ago:
[[File:Kimura1981bExcerpt.png|450px|center|border|Excerpt from section 3 of Kimura (1981b)]]
Kimura uses the subscript "1" to denote the equilibrium value of any physical quantity "Q"; in the figure shown here, on the right, we are holding the equilibrium mass fixed while allowing the external pressure and the configuration volume to vary, so Kimura would say that the figure displays various "M_1 sequences."  And, as is explained more fully in [[User:Tohline/SSC/Structure/PolytropesEmbedded#Kimura.27s_Presentation|an accompanying discussion]], his "truncation parameter" is essentially the same as our truncation radius &#8212; specifically, <math>~\zeta_1 = (n+1)^{1/2}\tilde\xi</math>.  When projected onto our discussion, the physically relevant range of truncation parameter values is, <math>~0 \le \tilde\xi \le \xi_1</math>, where <math>~\xi_1</math> is the [[User:Tohline/SSC/Structure/Polytropes#Lane-Emden_Equation|Lane-Emden radius of an ''isolated'' (unbounded) polytropic sphere]].
</td></tr>
</table>
 
To simplify our discussion, here, we redisplay the above figure and repeat a few key algebraic relations.
<div align="center">
<table border="0" cellpadding="5" align="center">
 
<tr>
   <td align="right">
   <td align="right">
<math>~r_n^{n-3}</math>
<math>~\eta_\mathrm{ad} </math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 483: Line 521:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~\frac{3\cdot 5 ~\tilde\theta^{n+1}}{(n+1) \tilde\xi^2 \tilde\mathfrak{f}_W}
\frac{(n+1)^n}{4\pi} \biggl[ \mathcal{A} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2} \biggr]^n
= \frac{\tilde\theta^{n+1}}{\tilde\theta^{n+1} + 3(\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta}\, ,</math>
\biggl[ \mathcal{B} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-(n+1)/n} \biggr]^{-n} \, ,
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~p_n^{n-3}</math>
<math>~\Pi_\mathrm{ad}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~\equiv</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~\eta_\mathrm{ad} (1 + \eta_\mathrm{ad})^{-4n/(n-3)} \, ,</math>
\frac{3^{n-3}}{(4\pi)^4 (n+1)^{3(n+1)}} \biggl[ \mathcal{A} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2} \biggr]^{-3(n+1)}
\biggl[ \mathcal{B} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-(n+1)/n} \biggr]^{4n} \, ;
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
while, in terms of the structural form factors,
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~r_n^{n-3}</math>
<math>~\Chi_\mathrm{ad}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~\equiv</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~(1 + \eta_\mathrm{ad})^{n/(n-3)} \, .</math>
\frac{1}{3} \biggl[ \frac{(n+1)}{5} \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_A} \biggr]^n \mathfrak{f}_M^{1-n}
   </td>
\, ,
</math>
   </td>
</tr>
</tr>
</table>
</table>
</div>
</div>
and,
Note that the last expression for <math>~\eta_\mathrm{ad}</math> has been obtained after inserting the analytic expression for the structural form-factor, <math>~\tilde\mathfrak{f}_W</math> that &#8212; as has been explained in an [[User:Tohline/SSC/Virial/FormFactors#Viala_and_Horedt_.281974.29_Expressions|accompanying discussion]] &#8212; we derived with the help of [http://adsabs.harvard.edu/abs/1974A%26A....33..195V Viala &amp; Horedt (1974)].
 
===Stability===
 
Analysis of the free-energy function allows us to not only ascertain the equilibrium radius of isolated polytropes and pressure-truncated polytropic configurations, but also the relative stability of these configurations.  We begin by repeating the,
<div align="center" id="RenormalizedFreeEnergyExpression2">
<font color="#770000">'''Renormalized Free-Energy Function'''</font><br />
 
<math>
\mathfrak{G}^{**} = -3 \Chi^{-1} +~ n\Chi^{-3/n} +~ \Pi_\mathrm{ad}\Chi^3 \, .
</math>
</div>
The first and second derivatives of <math>~\mathfrak{G}^{**}</math>, with respect to the dimensionless radius, <math>~\Chi</math>, are, respectively,
<div align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
Line 538: Line 569:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~p_n^{n-3}</math>
<math>~\frac{\partial\mathfrak{G}^{**}}{\partial\Chi}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~\equiv</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~3 \Chi^{-2} -3\Chi^{-(n+3)/n} + 3\Pi_\mathrm{ad} \Chi^2 \, ,</math>
\frac{1}{(4\pi)^8} \biggl[ \frac{3\cdot 5^3}{(n+1)^3} \cdot \frac{\mathfrak{f}_M^2}{\mathfrak{f}_W^3} \biggr]^{n+1} \mathfrak{f}_A^{4n}
\, .
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
<!--  THE FOLLOWING DERIVATION IS CORRECT IN DETAIL, BUT NOT PARTICULARLY USEFUL
Let's plug Horedt's expressions into the virial relation and see how it reduces without inserting specific expressions for the free-energy coefficients, <math>\mathcal{A}</math> and <math>\mathcal{B}</math>.  The lefthand side becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>\Pi_\mathrm{ad} \Chi_\mathrm{ad}^4</math>
<math>~\frac{\partial^2\mathfrak{G}^{**}}{\partial\Chi^2}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 568: Line 587:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~-6 \Chi^{-3} + \frac{3(n+3)}{n} \Chi^{-(2n+3)/n} + 6\Pi_\mathrm{ad} \Chi \, .</math>
\frac{4\pi}{3} \biggl[ \frac{\mathcal{A}^{3(n+1)}}{\mathcal{B}^{4n}} \biggr]^{1/(n-3)} \biggl[ \frac{\mathcal{B}}{\mathcal{A}} \biggr]^{4n/(n-3)}
\biggl(  \frac{P_e}{P_\mathrm{norm} } \biggr)_\mathrm{Horedt} \biggl(  \frac{R_\mathrm{eq}}{R_\mathrm{norm} } \biggr)_\mathrm{Horedt}^4
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
As alluded to, above, equilibrium radii are identified by values of <math>~\Chi</math> that satisfy the equation, <math>\partial\mathfrak{G}^{**}/\partial\Chi = 0</math>.  Specifically, marking equilibrium radii with the subscript "ad", they will satisfy the
<div align="center" id="ConciseVirial2">
<font color="#770000">'''Algebraic Expression of the Virial Theorem'''</font><br />


<tr>
<math>
  <td align="right">
\Pi_\mathrm{ad} = \frac{\Chi_\mathrm{ad}^{(n-3)/n} - 1}{\Chi_\mathrm{ad}^4} \, .
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{4\pi}{3} \biggl[ \frac{1}{\mathcal{A}} \biggr]
\biggl(  \frac{P_e}{P_\mathrm{norm} } \biggr)_\mathrm{Horedt} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm} } \biggr)_\mathrm{Horedt}^4 \, .
</math>
</math>
  </td>
</tr>
</table>
</div>
</div>
While the righthand side becomes,
Dynamical stability then depends on the sign of the second derivative of <math>~\mathfrak{G}^{**}</math>, evaluated at the equilibrium radius; specifically, configurations will be stable if,
<div align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
Line 597: Line 606:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>
<math>~\frac{\partial^2\mathfrak{G}^{**}}{\partial\Chi^2}\biggr|_{\Chi_\mathrm{ad}}</math>
~\Chi_\mathrm{ad}^{(n-3)/n} - 1
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~></math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~0 \, ,</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (stable)
\biggl[ \frac{\mathcal{B}}{\mathcal{A}} \biggr]
\biggl(  \frac{R_\mathrm{eq}}{R_\mathrm{norm} } \biggr)_\mathrm{Horedt}^{(n-3)/n} -1 \, .
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</table>
</div>
</div>
Together, then, we have,
and they will be unstable if, upon evaluation at the equilibrium radius, the sign of the second derivative is less than zero.  Hence, isolated polytropes as well as pressure-truncated polytropic configurations will be stable if,
<div align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
Line 620: Line 623:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>
<math>~0</math>
\frac{4\pi}{3}
\biggl(  \frac{P_e}{P_\mathrm{norm} } \biggr)_\mathrm{Horedt} \biggl(  \frac{R_\mathrm{eq}}{R_\mathrm{norm} } \biggr)_\mathrm{Horedt}^4
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~< </math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~3 \Chi_\mathrm{ad}^{-3} \biggl[ - 2 + \frac{(n+3)}{n} \Chi_\mathrm{ad}^{(n-3)/n} + 2\Pi_\mathrm{ad} \Chi_\mathrm{ad}^4 \biggr]</math>
\mathcal{B} \biggl\frac{R_\mathrm{eq}}{R_\mathrm{norm} } \biggr)_\mathrm{Horedt}^{(n-3)/n} -\mathcal{A}
</math>
   </td>
   </td>
</tr>
</tr>
Line 637: Line 635:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\Rightarrow~~~
&nbsp;
\frac{4\pi}{3} \biggl\{ p_a
\biggl[ \frac{(n+1)^3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2}\biggr]^{(n+1)/(n-3)} \biggr\}
\biggl\{ r_a
\biggl[ \frac{4\pi}{(n+1)^n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{n-1} \biggr]^{1/(n-3)}
\biggr\}^4
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~< </math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~3 \Chi_\mathrm{ad}^{-3} \biggl\{ \frac{(n+3)}{n} \Chi_\mathrm{ad}^{(n-3)/n} + 2[\Chi_\mathrm{ad}^{(n-3)/n} -1] - 2\biggr\}</math>
\mathcal{B} \biggl\{ r_a
\biggl[ \frac{4\pi}{(n+1)^n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{n-1} \biggr]^{1/(n-3)}
\biggr\}^{(n-3)/n} -\mathcal{A}
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
Or, simplifying,
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~
&nbsp;
\frac{1}{3(n+1)} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{2} p_a r_a^4
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~< </math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~r_a^{(n-3)/n}
<math>~3 \Chi_\mathrm{ad}^{-3} \biggl[ \frac{3(n+1)}{n} \Chi_\mathrm{ad}^{(n-3)/n} - 4\biggr]</math>
\mathcal{B} \biggl[ \frac{4\pi}{(n+1)^n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{n-1} \biggr]^{1/n}
-\mathcal{A}
</math>
   </td>
   </td>
</tr>
</tr>
Line 681: Line 659:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\Rightarrow~~~ \frac{1}{3} p_a r_a^4
<math>\Rightarrow~~~~\Chi_\mathrm{ad}</math>
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~> </math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~r_a^{(n-3)/n}
<math>~\biggl[ \frac{4n}{3(n+1)} \biggr]^{n/(n-3)} \, .</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (stable)
\mathcal{B} ( 4\pi )^{1/n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-(n+1)/n}  
-\mathcal{A} (n+1) \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2} \, .
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</table>
</div>
</div>
Derived from the virial theorem, this expression shows, in the most general case, how the equilibrium radius identified by Horedt, <math>~r_a</math>, relates to the dimensionless external pressure, <math>~p_a</math>, as defined by Horedt.  It is somewhat unsatisfactory that this algebraic <math>~p_a - r_a</math> relationship explicitly involves <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>, two of the coefficients found in the free-energy expression.  Unsatisfactory as it may be, its broad applicability can be straightforwardly demonstrated. After plugging in the expressions given above for <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math> in terms of the structural form factors, to obtain,
Reference to this stability condition proves to be simpler if we define the limiting configuration size as,
<div align="center">
<math>~\Chi_\mathrm{min} \equiv \biggl[ \frac{4n}{3(n+1)} \biggr]^{n/(n-3)} \, ,</math>
</div>
and write the stability condition as,
<div align="center">
<math>~\Chi_\mathrm{ad} > \Chi_\mathrm{min} \, .</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (stable)
</div>


When examining the equilibrium sequences found in the upper-righthand quadrant of the figure at the top of this page &#8212; each corresponding to a different value of the polytropic index, <math>~n > 3</math> or <math>~n < 0</math> &#8212; we find that <math>~\Chi_\mathrm{min}</math> corresponds to the location along each sequence where the dimensionless external pressure, <math>~\Pi_\mathrm{ad}</math>, reaches a maximum.  (Keeping in mind that the virial theorem defines each of these sequences, this statement of fact can be checked by identifying where the condition, <math>~\partial\Pi_\mathrm{ad}/\partial\Chi_\mathrm{ad} = 0</math>, occurs according to the [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#ConciseVirial2|algebraic expression of the virial theorem]].)  Hence, we conclude that, along each sequence, no equilibrium configurations exist for values of the dimensionless external pressure that are greater than,
<div align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
Line 703: Line 685:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~p_a r_a^4
<math>~\Pi_\mathrm{max}</math>
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~\equiv</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~r_a^{(n-3)/n} \biggl[3^{(n+1)/n}  \biggr] \frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_M^{(n+1)/n}}
<math>~\Chi_\mathrm{min}^{-4} \biggl[ \Chi_\mathrm{min}^{(n-3)/n} - 1 \biggr] </math>
-\biggl[ \frac{3(n+1)}{5} \biggr] \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} \, ,
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
one need only plug in Horedt's expressions for <math>~r_a</math> and <math>~p_a</math>, and our expressions for the three structural form factors &#8212; all given in terms of <math>~\tilde\theta</math>, <math>~\tilde\theta^'</math>, and <math>~\tilde\xi</math> &#8212; to see that the lefthand side equals the righthand side in precise detail.
END OF BLOCKED-OUT SUBSECTION -->
==Discussion==
===Model Sequences===
[[File:AdabaticBoundedSpheres_Virial.jpg|thumb|300px|Equilibrium Adiabatic Pressure-Radius Diagram]]
After choosing a value for the system's adiabatic index (or, equivalently, its polytropic index), <math>~\gamma = (n+1)/n</math>, the functional form of the virial theorem expression, <math>~\Pi_\mathrm{ad}(\chi_\mathrm{ad})</math>, is known and, hence, the equilibrium model sequence can be plotted.  Half-a-dozen such model sequences are shown in the figure near the beginning of this discussion.  Each curve can be viewed as mapping out a single-parameter sequence of equilibrium models; "evolution" along the curve can be accomplished by varying the key parameter, <math>~\eta_\mathrm{ad}</math>, over the physically relevant range, <math>0 \le \eta_\mathrm{ad} < \infty</math>.  To simplify our discussion, here, we redisplay the above figure and repeat a few key algebraic relations.
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\eta_\mathrm{ad} </math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~\equiv</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\frac{(5-n) \tilde\theta^{n+1}}{3(n+1) (\tilde\theta^')^2} = \biggl[ \frac{4\pi (5-n)}{3} \biggr] \frac{P_e R_\mathrm{eq}^4}{G M_\mathrm{limit}^2}\, ,</math>
<math>~\biggl[ \frac{3(n+1)}{4n} \biggr]^{4n/(n-3)} \biggl[\frac{4n}{3(n+1)} - 1 \biggr]</math>
   </td>
   </td>
</tr>
</tr>
Line 743: Line 709:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\Pi_\mathrm{ad}</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 749: Line 715:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\eta_\mathrm{ad} (1 + \eta_\mathrm{ad})^{-4n/(n-3)} \, ,</math>
<math>~\biggl\{ \biggl[ \frac{3(n+1)}{4n} \biggr]^{4n} \biggl[\frac{n-3}{3(n+1)} \biggr]^{n-3} \biggr\}^{1/(n-3)}</math>
   </td>
   </td>
</tr>
</tr>
Line 755: Line 721:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\Chi_\mathrm{ad}</math>
<math>~\Rightarrow~~~~\Pi_\mathrm{max}^{n-3}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 761: Line 727:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~(1 + \eta_\mathrm{ad})^{n/(n-3)} \, ,</math>
<math>~(4n)^{-4n}~[3(n+1)]^{3(n+1)} ~(n-3)^{n-3} \, .</math>
   </td>
   </td>
</tr>
</tr>
Line 767: Line 733:
</div>
</div>


====When <math>~\eta_\mathrm{ad}</math> is zero====
[In a [[User:Tohline/SSC/Virial/PolytropesEmbedded/FirstEffortAgain#P-V_Diagram|separate, related discussion of the model sequences]] displayed in the above figure, we have actually demonstrated that this same coordinate point was associated with the extremum along each curve.  In that discussion, this special point was identified as <math>~(\Chi_\mathrm{extreme}, \Pi_\mathrm{extreme})</math> instead of as <math>~(\Chi_\mathrm{min}, \Pi_\mathrm{max})</math>.]
For the types of systems that are presently most relevant to astrophysical discussions, the key parameter, <math>~\eta_\mathrm{ad}</math>, can be zero for one of two reasons:  Either <math>~n=5</math>; or <math>~\tilde\theta \rightarrow \theta_{\xi_1} = 0</math>.  In the latter case, all curves converge on the same point, that is, <math>~(\Chi_\mathrm{ad}, \Pi_\mathrm{ad}) = (1, 0)</math>. This corresponds to the case of no external medium <math>~(P_e = 0)</math> and, hence, the associated equilibrium configuration is the familiar [[User:Tohline/SSC/Structure/Polytropes#Polytropic_Spheres|''isolated'' polytropic sphere]].  As can be deduced from our above discussion of the [[User:Tohline/SSC/Virial/PolytropesSummary#ConciseVirial|algebraic expression of the virial theorem]], because <math>~\Chi_\mathrm{ad} = 1</math>, the equilibrium radius of such a configuration is,
 
In the context of a general examination of the free-energy of pressure-truncated polytropes, it is worth noting that this limit on the external pressure also establishes a limit on the coefficient, <math>~\mathcal{D}</math>, that appears in the free energy function.  Specifically, we will not expect to find any extrema in the free energy if,
<div align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
Line 774: Line 741:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\frac{R_\mathrm{eq}}{R_\mathrm{norm}} = \chi_\mathrm{eq}</math>
<math>~\mathcal{D} > \mathcal{D}_\mathrm{max}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~\equiv</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{n/(n-3)} \, .</math>
<math>~(n-3) \biggl\{ \biggl[ \frac{\mathcal{B}}{4n} \biggr]^{4n}~\biggl[ \frac{3(n+1)}{\mathcal{A}} \biggr]^{3(n+1)} ~\biggr\}^{1/(n-3)} \, .</math>
   </td>
   </td>
</tr>
</tr>
</table>
</table>
</div>
</div>
As is demonstrated in an [[User:Tohline/SSC/Virial/Polytropes#Strategy2|accompanying discussion]] and also [[User:Tohline/SSC/Virial/PolytropesSummary#Physical_Meaning_of_Parameter|mentioned above]], after inserting the relevant expressions for the free-energy coefficients, <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>, this provides the key relationship between the mass, equilibrium radius, and central pressure of an isolated polytrope, namely,
 
Finally, it is worth noting that the point along each equilibrium sequence that is identified by the coordinates, <math>~(\Chi_\mathrm{min}, \Pi_\mathrm{max})</math> always corresponds to,
<div align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<math>~\eta_\mathrm{ad} = \eta_\mathrm{crit} \equiv \frac{n-3}{3(n+1)} \, .</math>
</div>


<tr>
<div align="center">
   <td align="right">
<table border="1" align="center" cellpadding="5">
<math>~\frac{P_c R_\mathrm{eq}^4}{G M_\mathrm{limit}^2}</math>
<tr><th align="center" colspan="1">
Summary
</th></tr>
<tr><td align="center">
 
<table border="0" cellpadding="5" align="center">
<tr>
   <td align="right">
<math>~\eta_\mathrm{crit}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~\equiv</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\frac{1}{[4\pi (n+1) (\theta^')^2]_{\xi_1}} \, .</math>
<math>~ \frac{n-3}{3(n+1)} </math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
As we have [[User:Tohline/SSC/Virial/Polytropes#Central_and_Mean_Pressure|reviewed elsewhere]] &#8212; see also our [[User:Tohline/SSC/Structure/Polytropes#Polytropic_Spheres|detailed discussion of isolated polytropes]] &#8212; this is a familiar relationship, appearing prominently in Chapter IV (p. 99, equations 80 and 81) of [[User:Tohline/Appendix/References|Chandrasekhar [C67]]] in association with his discussion of the dimensionless coefficient, <math>~W_n</math>, and the central pressure of polytropes.
In the former case &#8212; that is, in the case where <math>~\eta_\mathrm{ad} \rightarrow 0</math> because the chosen polytropic index is, <math>~n=5</math> &#8212; it must be the case that <math>~\Chi_\mathrm{ad} = 1</math> along the entire sequence (see the green curve labeled <math>~\gamma = (n+1)/n = 6/5</math> in the accompanying figure).  This means that the expression for the central pressure,
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\frac{P_c R_\mathrm{eq}^4}{G M_\mathrm{limit}^2}</math>
<math>~\Pi_\mathrm{max}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~\equiv</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\frac{1}{[4\pi (n+1) (\tilde\theta^')^2]} \, ,</math>
<math>~(n-3) \biggl\{~\frac{ [3(n+1)]^{3(n+1)} }{(4n)^{4n}} \biggr\}^{1/(n-3)} </math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
does not explicitly depend on the size of the applied external pressure.  But the central pressure ''does'' depend on the radial location at which the configuration is truncated, via the parameter <math>~\tilde\theta^'</math>, which is evaluated at <math>~\tilde\xi</math>, rather than at <math>~\xi_1</math>.
===Stability===
Analysis of the free-energy function allows us to not only ascertain the equilibrium radius of isolated polytropes and pressure-truncated polytropic configurations, but also the relative stability of these configurations.  We begin by repeating the,
<div align="center" id="RenormalizedFreeEnergyExpression2">
<font color="#770000">'''Renormalized Free-Energy Function'''</font><br />
<math>
\mathfrak{G}^{**} = -3 \Chi^{-1} +~ n\Chi^{-3/n} +~ \Pi_\mathrm{ad}\Chi^3 \, .
</math>
</div>
The first and second derivatives of <math>~\mathfrak{G}^{**}</math>, with respect to the dimensionless radius, <math>~\Chi</math>, are, respectively,
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\frac{\partial\mathfrak{G}^{**}}{\partial\Chi}</math>
<math>~\Chi_\mathrm{min} </math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~\equiv</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~3 \Chi^{-2} -3\Chi^{-(n+3)/n} + 3\Pi_\mathrm{ad} \Chi^2 \, ,</math>
<math>
\biggl[ \frac{4n}{3(n+1)} \biggr]^{n/(n-3)}
</math>
   </td>
   </td>
</tr>
</tr>
</table>


<tr>
</td>
  <td align="right">
<math>~\frac{\partial^2\mathfrak{G}^{**}}{\partial\Chi^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-6 \Chi^{-3} + \frac{3(n+3)}{n} \Chi^{-(2n+3)/n} + 6\Pi_\mathrm{ad} \Chi \, .</math>
  </td>
</tr>
</tr>
</table>
</table>
</div>
</div>
As alluded to, above, equilibrium radii are identified by values of <math>~\Chi</math> that satisfy the equation, <math>\partial\mathfrak{G}^{**}/\partial\Chi = 0</math>.  Specifically, marking equilibrium radii with the subscript "ad", they will satisfy the
<div align="center" id="ConciseVirial2">
<font color="#770000">'''Algebraic Expression of the Virial Theorem'''</font><br />


<math>
==Mass-Radius Relation==
\Pi_\mathrm{ad} = \frac{\Chi_\mathrm{ad}^{(n-3)/n} - 1}{\Chi_\mathrm{ad}^4} \, .
Up to this point in our discussion, we have focused on an analysis of the pressure-radius relationship that defines the equilibrium configurations of pressure-truncated polytropes.  In effect, we have viewed the problem through the same lens as did [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)] and, separately, [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth (1981)], defining variable normalizations in terms of the polytropic constant, <math>~K</math>, and the configuration mass, <math>~M_\mathrm{tot}</math>, which were both assumed to be held fixed throughout the analysis.  Here we switch to the approach championed by [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)], defining variable normalizations in terms of <math>~K</math> and <math>~P_e</math>, and examining the ''mass-radius'' relationship of pressure-truncated polytropes.
</math>
 
</div>
===Detailed Force-Balanced Solution===
Dynamical stability then depends on the sign of the second derivative of <math>~\mathfrak{G}^{**}</math>, evaluated at the equilibrium radius; specifically, configurations will be stable if,
As has been summarized in our [[User:Tohline/SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|accompanying review]] of detailed force-balanced models of pressure-truncated polytropes, [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)] found that a spherical configuration's equilibrium radius is related to its mass through the following pair of parametric equations:
<div align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="3">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\frac{\partial^2\mathfrak{G}^{**}}{\partial\Chi^2}\biggr|_{\Chi_\mathrm{ad}}</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>~0 \, ,</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (stable)
<math>
\biggl( \frac{n^3}{4\pi} \biggr)^{1/2} \tilde\theta^{(n-3)/2} (- \tilde\xi^2 \tilde\theta^') \, ,
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
and they will be unstable if, upon evaluation at the equilibrium radius, the sign of the second derivative is less than zero.  Hence, isolated polytropes as well as pressure-truncated polytropic configurations will be stable if,
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~0</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">
<math>~3 \Chi_\mathrm{ad}^{-3} \biggl[ - 2 + \frac{(n+3)}{n} \Chi_\mathrm{ad}^{(n-3)/n} + 2\Pi_\mathrm{ad} \Chi_\mathrm{ad}^4 \biggr]</math>
<math>
\biggl( \frac{n}{4\pi} \biggr)^{1/2} \tilde\xi \tilde\theta^{(n-1)/2} \, ,
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
where,
<div align="center">
<math>M_\mathrm{SWS} \equiv
\biggl( \frac{n+1}{nG} \biggr)^{3/2} K^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]} \, ,</math>
</div>
<div align="center">
<math>
R_\mathrm{SWS} \equiv \biggl( \frac{n+1}{nG} \biggr)^{1/2} K^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]} \, .
</math>
</div>


===Mapping from Above Discussion===
====Deriving Concise Virial Theorem Mass-Radius Relation====
Looking back on the definitions of <math>~\Pi_\mathrm{ad}</math> and <math>~\Chi_\mathrm{ad}</math> that we introduced in connection with our initial [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#ConciseVirial|concise algebraic expression of the virial theorem]], we can write,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>~P_e </math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~< </math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~3 \Chi_\mathrm{ad}^{-3} \biggl\{ \frac{(n+3)}{n} \Chi_\mathrm{ad}^{(n-3)/n} + 2[\Chi_\mathrm{ad}^{(n-3)/n} -1] - 2\biggr\}</math>
<math>
~P_\mathrm{norm} \biggl( \frac{3}{4\pi} \biggr) \Pi_\mathrm{ad} \biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr]^{1/(n-3)}
</math>
   </td>
   </td>
</tr>
</tr>
Line 920: Line 886:
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~< </math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~3 \Chi_\mathrm{ad}^{-3} \biggl[ \frac{3(n+1)}{n} \Chi_\mathrm{ad}^{(n-3)/n} - 4\biggr]</math>
<math>
~\biggl( \frac{3}{4\pi} \biggr) \Pi_\mathrm{ad}  \biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr]^{1/(n-3)}
\biggl[ \frac{K^{4n}}{G^{3(n+1)} M_\mathrm{tot}^{2(n+1)}} \biggr]^{1/(n-3)} \, ,
</math>
   </td>
   </td>
</tr>
</tr>
Line 929: Line 898:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>\Rightarrow~~~~\Chi_\mathrm{ad}</math>
<math>~R_\mathrm{eq} </math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~> </math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\biggl[ \frac{4n}{3(n+1)} \biggr]^{n/(n-3)} \, .</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (stable)
<math>
~R_\mathrm{norm} \Chi_\mathrm{ad} \biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{n/(n-3)}  
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
Reference to this stability condition proves to be simpler if we define the limiting configuration size as,
<div align="center">
<math>~\Chi_\mathrm{min} \equiv \biggl[ \frac{4n}{3(n+1)} \biggr]^{n/(n-3)} \, ,</math>
</div>
and write the stability condition as,
<div align="center">
<math>~\Chi_\mathrm{ad} > \Chi_\mathrm{min} \, .</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (stable)
</div>
When examining the equilibrium sequences found in the upper-righthand quadrant of the figure at the top of this page &#8212; each corresponding to a different value of the polytropic index, <math>~n > 3</math> or <math>~n < 0</math> &#8212; we find that <math>~\Chi_\mathrm{min}</math> corresponds to the location along each sequence where the dimensionless external pressure, <math>~\Pi_\mathrm{ad}</math>, reaches a maximum.  (Keeping in mind that the virial theorem defines each of these sequences, this statement of fact can be checked by identifying where the condition, <math>~\partial\Pi_\mathrm{ad}/\partial\Chi_\mathrm{ad} = 0</math>, occurs according to the [[User:Tohline/SSC/Virial/PolytropesSummary#ConciseVirial2|algebraic expression of the virial theorem]].)  Hence, we conclude that, along each sequence, no equilibrium configurations exist for values of the dimensionless external pressure that are greater than,
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\Pi_\mathrm{max}</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~\equiv</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\Chi_\mathrm{min}^{-4} \biggl[ \Chi_\mathrm{min}^{(n-3)/n} - 1 \biggr] </math>
<math>
~\Chi_\mathrm{ad} \biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{n/(n-3)}  
\biggl[ \biggl( \frac{G}{K} \biggr)^n M_\mathrm{tot}^{n-1} \biggr]^{1/(n-3)} \, .
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
The first of these two expressions can be flipped around to give an expression for <math>~M_\mathrm{tot}</math> in terms of <math>~P_e</math> and, then, as normalized to <math>~M_\mathrm{SWS}</math>.  Specifically,
<div align="center">
<table border="0" cellpadding="5">


<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>~ M_\mathrm{tot}^{2(n+1)}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 973: Line 939:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\biggl[ \frac{3(n+1)}{4n} \biggr]^{4n/(n-3)} \biggl[\frac{4n}{3(n+1)} - 1 \biggr]</math>
<math>
~\biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{n-3} \biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr]
\biggl[ \frac{K^{4n}}{G^{3(n+1)}P_e^{n-3} } \biggr]  
</math>
   </td>
   </td>
</tr>
</tr>
Line 985: Line 954:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\biggl\{ \biggl[ \frac{3(n+1)}{4n} \biggr]^{4n} \biggl[\frac{n-3}{3(n+1)} \biggr]^{n-3} \biggr\}^{1/(n-3)}</math>
<math>
~M_\mathrm{SWS}^{2(n+1)} \biggl( \frac{n}{n+1} \biggr)^{3(n+1)}  
\biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{n-3} \biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr]
</math>
   </td>
   </td>
</tr>
</tr>
Line 991: Line 963:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\Rightarrow~~~~\Pi_\mathrm{max}^{n-3}</math>
<math>~ \Rightarrow~~~ \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 997: Line 969:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~(4n)^{-4n}~[3(n+1)]^{3(n+1)} ~(n-3)^{n-3} \, .</math>
<math>
~\biggl( \frac{n}{n+1} \biggr)^{3/2}
\biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(n-3)/[2(n+1)]}  \biggl[ \frac{\mathcal{B}^{2n/(n+1)}}{\mathcal{A}^{3/2}} \biggr] \, .
</math>
   </td>
   </td>
</tr>
</tr>
Line 1,003: Line 978:
</div>
</div>


In the context of a general examination of the free-energy of pressure-truncated polytropes, it is worth noting that this limit on the external pressure also establishes a limit on the coefficient, <math>~\mathcal{D}</math>, that appears in the free energy function.  Specifically, we will not expect to find any extrema in the free energy if,
This means, as well, that we can rewrite the equilibrium radius as,
<div align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\mathcal{D} > \mathcal{D}_\mathrm{max}</math>
<math>~R_\mathrm{eq}^{n-3} </math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~\equiv</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~(n-3) \biggl\{ \biggl[ \frac{\mathcal{B}}{4n} \biggr]^{4n}~\biggl[ \frac{3(n+1)}{\mathcal{A}} \biggr]^{3(n+1)} ~\biggr\}^{1/(n-3)} \, .</math>
<math>
~\Chi_\mathrm{ad}^{n-3} \biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{n}  
\biggl( \frac{G}{K} \biggr)^n M_\mathrm{tot}^{n-1}  
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>


Finally, it is worth noting that the point along each equilibrium sequence that is identified by the coordinates, <math>~(\Chi_\mathrm{min}, \Pi_\mathrm{max})</math> always corresponds to,
<div align="center">
<math>~\eta_\mathrm{ad} = \eta_\mathrm{crit} \equiv \frac{n-3}{3(n+1)} \, .</math>
</div>
<div align="center">
<table border="1" align="center" cellpadding="5">
<tr><th align="center" colspan="1">
Summary
</th></tr>
<tr><td align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\eta_\mathrm{crit}</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~\equiv</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~ \frac{n-3}{3(n+1)} </math>
<math>
~\Chi_\mathrm{ad}^{n-3} \biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{n}
\biggl( \frac{G}{K} \biggr)^n \biggl\{ 
\biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{n-3} \biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr]
\biggl[ \frac{K^{4n}}{G^{3(n+1)}P_e^{n-3} } \biggr]
\biggr\}^{(n-1)/[2(n+1)]}
</math>
   </td>
   </td>
</tr>
</tr>
Line 1,048: Line 1,017:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\Pi_\mathrm{max}</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~\equiv</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~(n-3) \biggl\{~\frac{ [3(n+1)]^{3(n+1)} }{(4n)^{4n}} \biggr\}^{1/(n-3)} </math>
<math>
   </td>
~\Chi_\mathrm{ad}^{n-3} \biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{n}
\biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr]^{(n-1)/[2(n+1)]}
\biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(n-3)(n-1)/[2(n+1)]} \biggl( \frac{G}{K} \biggr)^n
\biggl\{ 
\biggl[ \frac{K^{4n}}{G^{3(n+1)}P_e^{n-3} } \biggr]
\biggr\}^{(n-1)/[2(n+1)]}  
</math>
   </td>
</tr>
</tr>


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\Chi_\mathrm{min} </math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~\equiv</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>
\biggl[ \frac{4n}{3(n+1)} \biggr]^{n/(n-3)}
~\Chi_\mathrm{ad}^{n-3} \biggl\{ \biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{2n(n+1)}
\biggl[ \frac{\mathcal{B}^{4n(n-1)}}{\mathcal{A}^{3(n+1)(n-1)}} \biggr]\biggr\}^{1/[2(n+1)]}
\biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(n-3)(n-1)/[2(n+1)]}
\biggl\{ \biggl( \frac{G}{K} \biggr)^{2n(n+1)}
\biggl[ \frac{K^{4n(n-1)}}{G^{3(n+1)(n-1)}P_e^{(n-3)(n-1)} } \biggr]
\biggr\}^{1/[2(n+1)]}  
</math>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</td>
</tr>
</table>
</div>
===Try Polytropic Index of 4===
====Groundwork====
In an effort to more fully understand what can be learned from an examination of the free-energy, let's play with <math>~n=4</math> polytropic models.  First, let's plot <math>~\mathfrak{G}^{**}(\Chi)</math> using a specific, trial value of the coefficient, <math>~\Pi_\mathrm{ad}</math>, keeping in mind that,
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\eta_\mathrm{crit}\biggr|_{n=4}</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 1,092: Line 1,061:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\frac{1}{15} = 0.066667 \, ;</math>
<math>
~\Chi_\mathrm{ad}^{n-3} \biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(n-3)(n-1)/[2(n+1)]}
\biggl[ \mathcal{A}^{-(n+1)(n-3)} \mathcal{B}^{2n(n-3)} \biggr]^{1/[2(n+1)]}
\biggl[ G^{(3-n)(n+1)} K^{2n(n-3)} P_e^{(n-3)(1-n)} \biggr]^{1/[2(n+1)]}
</math>
   </td>
   </td>
</tr>
</tr>
Line 1,098: Line 1,071:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\Pi_\mathrm{max}\biggr|_{n=4}</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 1,104: Line 1,077:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\frac{15^{15}}{16^{16}} = 0.02373828 \, ;</math>
<math>
   </td>
~R_\mathrm{SWS}^{n-3} \biggl( \frac{n}{n+1} \biggr)^{(n-3)/2}
</tr>
\Chi_\mathrm{ad}^{n-3} \biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(n-3)(n-1)/[2(n+1)]}
\biggl[ \mathcal{A}^{-(n+1)(n-3)} \mathcal{B}^{2n(n-3)} \biggr]^{1/[2(n+1)]}
</math>
   </td>
</tr>


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\Chi_\mathrm{min}\biggr|_{n=4}</math>
<math>~\Rightarrow~~~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} } </math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 1,116: Line 1,093:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\biggl( \frac{16}{15} \biggr)^4 = 1.294538 \, .</math>
<math>
~\biggl( \frac{n}{n+1} \biggr)^{1/2}
\Chi_\mathrm{ad} \biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(n-1)/[2(n+1)]}
\biggl[ \frac{\mathcal{B}^{n/(n+1)}}{\mathcal{A}^{1/2}} \biggr] \, .
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</table>
</div>
</div>
Flipping both of these expressions around, we see that,
<div align="center">
<table border="0" cellpadding="5" align="center">


At the top of the table, shown below, we display a plot of the,
<tr>
<div align="center" id="RenormalizedFreeEnergyExpression2">
  <td align="right">
<font color="#770000">'''Renormalized Free-Energy Function'''</font><br />
<math>~\Pi_\mathrm{ad} </math>
 
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
<math>
\mathfrak{G}^{**} = -3 \Chi^{-1} +~ n\Chi^{-3/n} +~ \Pi_\mathrm{ad}\Chi^3 \, ,
~\frac{4\pi}{3} \biggl\{ \biggl( \frac{n+1}{n} \biggr)^{3(n+1)}  \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{2(n+1)}
\biggl[ \frac{\mathcal{A}^{3(n+1)}}{\mathcal{B}^{4n}} \biggr] \biggr\}^{1/(n-3)} \, ,
</math>
</math>
  </td>
</tr>
</table>
</div>
</div>
where we have set <math>~n = 4</math>, and <math>~\Pi_\mathrm{ad} = 0.01</math>.  Reading quantities off of the plot, the left and right extrema identify equilibria having the following approximate dimensionless radii:  <math>~\Chi_\mathrm{left} \approx 1.03</math> and <math>~\Chi_\mathrm{right} \approx 2.13</math>.  Upon closer examination (plots not shown), we have determined that, <math>~\Chi_\mathrm{left} \approx 1.0494</math> and <math>~\Chi_\mathrm{right} \approx 2.13905</math>.  In accordance with our stability analysis, these values of <math>~\Chi_\mathrm{ad}</math> fall on either side of the demarcation value, <math>~\Chi_\mathrm{min} = (16/15)^4</math>, with the one on the left being a local maximum in the free energy &#8212; indicating an unstable equilibrium &#8212; while the one on the right is a local minimum &#8212; indicating a stable equilibrium.  Next, let's check to see if both extrema satisfy the,
and,
<div align="center" id="ConciseVirial2">
<div align="center">
<font color="#770000">'''Algebraic Expression of the Virial Theorem'''</font><br />
<table border="0" cellpadding="5" align="center">


<math>
<tr>
\Pi_\mathrm{ad} = \frac{\Chi_\mathrm{ad}^{(n-3)/n} - 1}{\Chi_\mathrm{ad}^4} \, .
  <td align="right">
<math>~\Chi_\mathrm{ad} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} }  \biggl( \frac{n+1}{n} \biggr)^{1/2}
\biggl[ \frac{\mathcal{A}^{1/2}}{\mathcal{B}^{n/(n+1)}} \biggr]
\biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(1-n)/[2(n+1)]}
</math>
</math>
</div>
  </td>
For the unstable equilibrium configuration, we calculate,
</tr>
<div align="center">
<math>\Pi_\mathrm{ad} \approx [(1.0494)^{1/4} - 1]/(1.0494)^4 = 1.000024 \times 10^{-2}</math>;
</div>
while, for the stable equilibrium we calculate,
<div align="center">
<math>\Pi_\mathrm{ad} \approx [(2.13905)^{1/4} - 1]/(2.13905)^4 = 1.000018 \times 10^{-2}</math>.
</div>
Because we inserted a value of <math>~\Pi_\mathrm{ad} = 0.01</math> into the free-energy expression, we conclude that, as desired, both identified extrema satisfy the virial relation to the measured accuracy.  These parameter values, and the corresponding values of many other related physical parameters are summarized in the following table, along with the algebraic relations that were used to calculate them.
 
====First Table====


<table border="1" align="center" cellpadding="5">
<tr>
<tr>
   <th align="center" colspan="4">
   <td align="right">
[[File:TryN4Pi0.01.png|450px|Dimensionless Free-Energy Curve]]
&nbsp;
   </th>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} }  \biggl( \frac{n+1}{n} \biggr)^{1/2}
\biggl[ \frac{\mathcal{A}^{1/2}}{\mathcal{B}^{n/(n+1)}} \biggr]
\biggl\{ \biggl( \frac{n+1}{n} \biggr)^{3(n+1)}  \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{2(n+1)}
\biggl[ \frac{\mathcal{A}^{3(n+1)}}{\mathcal{B}^{4n}} \biggr] \biggr\}^{(1-n)/[2(n+1)(n-3)]}
</math>
   </td>
</tr>
</tr>
<tr>
<tr>
   <th align="center" colspan="4">
   <td align="right">
Determined from Plot of Renormalized Free-Energy
&nbsp;
 
  </td>
with <math>~(n, \Pi_\mathrm{ad}) = (4, 0.01)</math>
  <td align="center">
   </th>
<math>~=</math>
  </td>
  <td align="left">
<math>
~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} } \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{(1-n)/(n-3)}  \biggl( \frac{n}{n+1} \biggr)^{n/(n-3)}
\biggl[ \frac{\mathcal{B}}{\mathcal{A}} \biggr]^{n/(n-3)} \, .
</math>
   </td>
</tr>
</tr>
</table>
</div>
Hence, our earlier derived [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#ConciseVirial3|compact expression for the virial theorem]] becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <th align="center">&nbsp;</th>
   <td align="right">
   <th align="center">&nbsp;</th>
<math>~1</math>
   <th align="center" width="25%">Maximum</th>
   </td>
   <th align="center" width="25%">Minimum</th>
   <td align="center">
<math>~=</math>
  </td>
   <td align="left">
<math>
\biggl\{ \frac{R_\mathrm{eq}}{R_\mathrm{SWS} } \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{(1-n)/(n-3)}  \biggl( \frac{n}{n+1} \biggr)^{n/(n-3)}
\biggl[ \frac{\mathcal{B}}{\mathcal{A}} \biggr]^{n/(n-3)} \biggr\}^{(n-3)/n}
</math>
  </td>
</tr>
</tr>
<tr>
<tr>
   <th align="center">&nbsp;</th>
   <td align="right">
   <td align="center"> <math>~\Chi</math> </td>
&nbsp;
   <td align="center"> <math>~1.0494</math> </td>
   </td>
   <td align="center"> <math>~2.13905</math> </td>
   <td align="center">
</tr>
&nbsp;
<tr>
  </td>
  <th align="center" colspan="4">
   <td align="left">
Immediate Implications from Virial Theorem
<math>
   </th>
-~ \frac{4\pi}{3} \biggl\{ \biggl( \frac{n+1}{n} \biggr)^{3(n+1)}  \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{2(n+1)}
\biggl[ \frac{\mathcal{A}^{3(n+1)}}{\mathcal{B}^{4n}} \biggr] \biggr\}^{1/(n-3)}
\biggl\{ \frac{R_\mathrm{eq}}{R_\mathrm{SWS} } \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{(1-n)/(n-3)}  \biggl( \frac{n}{n+1} \biggr)^{n/(n-3)}
\biggl[ \frac{\mathcal{B}}{\mathcal{A}} \biggr]^{n/(n-3)} \biggr\}^4 </math>  
   </td>
</tr>
</tr>
<tr>
<tr>
   <th align="center"><math>~\Chi^{1/4} - 1</math></th>
   <td align="right">
   <td align="center"> <math>~\eta_\mathrm{ad}</math> </td>
&nbsp;
   <td align="center"> <math>~0.012128</math></td>
  </td>
  <td align="center"> <math>~0.20936</math> </td>
   <td align="center">
<math>~=</math>
  </td>
   <td align="left">
<math>
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS} } \biggr)^{(n-3)/n} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{(1-n)/n}  \biggl( \frac{n}{n+1} \biggr)
\biggl[ \frac{\mathcal{B}}{\mathcal{A}} \biggr]
-~ \frac{4\pi}{3} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS} }  \biggr)^4 \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{-2} \biggl( \frac{n}{n+1} \biggr)
\frac{1}{\mathcal{A}} \, .
</math>  
  </td>
</tr>
</tr>
<tr>
</table>
  <th align="center"><math>~(\Chi^{1/4} - 1)\cdot \Chi^{-4}</math></th>
</div>
  <td align="center"> <math>~\Pi_\mathrm{ad}</math> </td>
 
  <td align="center"> <math>~1.000024 \times 10^{-2}</math></td>
Or, rearranged,
  <td align="center"> <math>~1.000018 \times 10^{-2}</math> </td>
 
</tr>
<div align="center" id="CompactStahlerVirial">
<tr>
<table border="1" cellpadding="10" align="center">
  <th align="center" colspan="4">
Associated Detailed Force-Balanced Model Parameters


obtained via interpolation of tabulated numbers on p. 399 of [http://adsabs.harvard.edu/abs/1986Ap%26SS.126..357H Horedt (1986, ApJS, vol. 126)]
  </th>
</tr>
<tr>
<tr>
   <th align="center">&nbsp;</th>
   <td align="right">
  <td align="center"> <math>~\tilde\xi</math> (approx.) </td>
<math>\frac{4\pi}{3} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS} }  \biggr)^4  -
  <td align="center"> <math>~4.81</math></td>
\mathcal{B} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS} } \biggr)^{(n-3)/n} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{(n+1)/n} 
   <td align="center"><math>~1.624</math></td>
+~ \mathcal{A} \biggl( \frac{n+1}{n} \biggr) \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{2} = 0 \, .
</math>  
   </td>
</tr>
</tr>
</table>
</div>
After adopting the modified coefficient definitions,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<tr>
   <th align="center">&nbsp;</th>
   <td align="right">
  <td align="center"> <math>~\tilde\theta</math> (approx.) </td>
<math>~\mathcal{A}_{M_\ell}</math>
   <td align="center"> <math>~0.251</math></td>
  </td>
   <td align="center"><math>~0.709</math></td>
   <td align="center">
<math>~\equiv</math>
  </td>
   <td align="left">
<math>~
\mathcal{A} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2}  =
\frac{1}{5} \cdot \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2}  \, ,</math>
  </td>
</tr>
</tr>
<tr>
<tr>
   <th align="center">&nbsp;</th>
   <td align="right">
  <td align="center"> <math>~- \tilde\theta^'</math> (approx.) </td>
<math>~\mathcal{B}_{M_\ell}</math>
   <td align="center"><math>~0.0727</math></td>
  </td>
   <td align="center"> <math>~0.239</math></td>
   <td align="center">
<math>~\equiv</math>
  </td>
   <td align="left">
<math>~\mathcal{B} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{-(n+1)/n} =
\biggl( \frac{3}{4\pi}\biggr)^{1/n}
\frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_M^{(n+1)/n}} \, ,
</math>
  </td>
</tr>
</tr>
</table>
</div>
<span id="modNormalizations">as well as the modified length- and mass-normalizations,</span> <math>~R_\mathrm{mod}</math> and <math>~M_\mathrm{mod}</math>, such that,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <th align="center"> <math>~\frac{1}{15}\cdot \frac{\tilde\theta^5}{(\tilde\theta^')^2}</math></th>
   <td align="right">
   <td align="center"> <math>~\eta</math> (check) </td>
<math>~\frac{M_\mathrm{SWS}}{M_\mathrm{mod}}</math>
   <td align="center"><math>~0.0126</math></td>
  </td>
   <td align="center"> <math>~0.2091</math></td>
   <td align="center">
</tr>
<math>~\equiv</math>
<tr>
   </td>
  <th align="center" colspan="4">
   <td align="left">
and, hence, Implied Structural Form Factors &amp; Coefficients <math>~\mathcal{B}</math> &amp; <math>~\mathcal{A}</math>
<math>~\biggl( \frac{4\pi}{3} \biggr)^{2n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{3/2} \frac{\mathcal{A}_{M_\ell}^{3/2}}{\mathcal{B}_{M_\ell}^{2n/(n+1)}}  
   </th>
\, ,</math>
   </td>
</tr>
</tr>
<tr>
<tr>
   <th align="center"> <math>~3(-\tilde\theta^')/\tilde\xi</math></th>
   <td align="right">
   <td align="center"> <math>~\mathfrak{f}_M</math></td>
<math>~\frac{R_\mathrm{SWS}}{R_\mathrm{mod}}</math>
   <td align="center"> <math>~0.0453</math></td>
  </td>
   <td align="center"><math>~0.4415</math></td>
   <td align="center">
<math>~\equiv</math>
  </td>
   <td align="left">
<math>~\biggl( \frac{4\pi}{3} \biggr)^{n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{1/2} \frac{\mathcal{A}_{M_\ell}^{1/2}}{\mathcal{B}_{M_\ell}^{n/(n+1)}} \, ,</math>
   </td>
</tr>
</tr>
</table>
</div>
we obtain the
<div align="center" id="ConciseVirialMR">
<font color="#770000">'''Virial Theorem in terms of Mass and Radius'''</font><br />
<math>
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{mod}} \biggr)^4
- \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{mod}} \biggr)^{(n-3)/n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{mod}} \biggr)^{(n+1)/n}
+ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{mod}} \biggr)^2 = 0
\, .
</math>
</div>
For later use we note as well that, with these modified coefficient definitions, we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <th align="center"> <math>~5[3(-\tilde\theta^')/\tilde\xi]^2</math></th>
   <td align="right">
  <td align="center"> <math>~\mathfrak{f}_W</math></td>
<math>~\Pi_\mathrm{ad}^{n-3} </math>
   <td align="center"> <math>~0.01028</math></td>
   </td>
   <td align="center"><math>~0.975</math></td>
   <td align="center">
</tr>
<math>~=</math>
<tr>
   </td>
  <th align="center"> <math>~15(-\tilde\theta^')^2 + \tilde\theta^5</math></th>
   <td align="left">
   <td align="center"> <math>~\mathfrak{f}_A</math></td>
<math>
   <td align="center"> <math>~0.08028</math></td>
~\biggl[ \biggl( \frac{4\pi}{3} \biggr)^{n-3} \biggl( \frac{n+1}{n} \biggr)^{3(n+1)}  
  <td align="center"><math>~1.036</math></td>
\frac{\mathcal{A}_{M_\ell}^{3(n+1)}}{\mathcal{B}_{M_\ell}^{4n}} \biggr] \mathcal{Y}^{2(n+1)} \, ,
</tr>
<tr>
  <th align="center"> <math>~\biggl(\frac{3}{4\pi} \biggr)^{1/4} \mathfrak{f}_M^{-5/4} \cdot \mathfrak{f}_A</math></th>
  <td align="center"> <math>~\mathcal{B}</math></td>
  <td align="center"> <math>~2.682</math></td>
  <td align="center"><math>~2.0122</math></td>
</tr>
<tr>
  <th align="center"> <math>~\frac{\tilde\mathfrak{f}_W}{5 \tilde\mathfrak{f}_M^2} </math></th>
  <td align="center"> <math>~\mathcal{A}</math></td>
  <td align="center"> <math>~1</math></td>
  <td align="center"><math>~1</math></td>
</tr>
<tr>
  <th align="center" colspan="4">
Given <math>~\Pi_\mathrm{ad}</math>, <math>~\Chi</math>, and <math>~\mathcal{B}</math>, we obtain
  </th>
</tr>
<tr>
  <th align="center"> <math>~\frac{3}{4\pi}\mathcal{D} = \frac{3}{4\pi} \Pi_\mathrm{ad} \mathcal{B}^{16} </math></th>
  <td align="center"> <math>~\frac{P_e}{P_\mathrm{norm}}</math></td>
  <td align="center"> <math>~1.71 \times 10^4</math></td>
  <td align="center"><math>~1.72 \times 10^2</math></td>
</tr>
<tr>
  <th align="center"> <math>~\Chi \mathcal{B}^{-4}</math></th>
  <td align="center"> <math>~\chi_\mathrm{eq}</math></td>
  <td align="center"> <math>~0.0203</math></td>
  <td align="center"><math>~0.1305</math></td>
</tr>
<tr>
  <th align="center" colspan="4">
Compare with Horedt's Equilibrium Parameters obtained from DFB Models
  </th>
</tr>
<tr>
  <th align="center"><math>\biggl[  \biggl( \frac{5^3}{4\pi} \biggr)
\tilde\theta( -\tilde\xi^2 \tilde\theta' )^{2} \biggr]^{5}
</math>
</math>
</th>
   </td>
   <td align="center"> <math>~\frac{P_e}{P_\mathrm{norm}}</math></td>
  <td align="center"> <math>~1.76 \times 10^4</math></td>
  <td align="center"><math>~1.73 \times 10^2</math></td>
</tr>
</tr>
<tr>
<tr>
   <th align="center"><math>
   <td align="right">
\biggl( \frac{4\pi}{5^4} \biggr) \tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{-3}
<math>~\Chi_\mathrm{ad}^{n-3} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~
\biggl[ \frac{n}{n+1} \biggl( \frac{\mathcal{B}_{M_\ell}}{\mathcal{A}_{M_\ell}} \biggr)\biggr]^n
\mathcal{X}^{n-3} \mathcal{Y}^{1-n} \, ,
</math>
</math>
</th>
   </td>
   <td align="center"> <math>~\chi_\mathrm{eq}</math></td>
  <td align="center"> <math>~0.0203</math></td>
  <td align="center"><math>~0.130</math></td>
</tr>
</tr>
</table>
</table>
</div>
where <math>~\mathcal{X}</math> and <math>~\mathcal{Y}</math> are defined [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Confirmation|immediately below]].


Now, we are convinced that both extrema identify perfectly valid equilibrium configurations.  However, in the context of astrophysics, the two identified equilibria are not connected to one another in any meaningful way.  In particular, two of the free-energy coefficients, <math>~\mathcal{B}</math> and <math>~\mathcal{D}</math>, have different values in the two cases; and, by inference, the normalized external pressure, <math>~P_e/P_\mathrm{norm}</math>, is different in the two cases.  So the plotted free-energy curve does not represent a "constant pressure" evolutionary trajectory.  How do we identify two equilibria that are associated with the same normalized external pressure?  And how do we identify the free-energy "evolutionary trajectory" that connects the two states?
====Corresponding Concise Free-Energy Expression====
 
Let's also rewrite the [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#FreeEnergyExpression|algebraic free-energy function]] in terms of Stahler's normalized mass and radius variables.  Expressed in terms of the polytropic index, the free-energy function is,
====Second Table====
 
Here, we have decided to look for a stable equilibrium state that is bounded by the same external pressure as the ''unstable'' state that has been identified in the above figure and table.  Rather than going straight to the free-energy expression in search of the desired stable configuration, we cheated a bit.  Using the properties of an <math>~n=4</math> polytrope, as tabulated on p. 399 of [http://adsabs.harvard.edu/abs/1986Ap%26SS.126..357H Horedt (1986, ApJS, vol. 126)], in conjunction with the algebraic expression found in the next-to-last row of the above table, namely,
<div align="center">
<div align="center">
<math>
<math>
\frac{P_e}{P_\mathrm{norm}} = \biggl[  \biggl( \frac{5^3}{4\pi} \biggr) \tilde\theta( -\tilde\xi^2 \tilde\theta' )^{2} \biggr]^{5} \, ,
\mathfrak{G}^* =
-3\mathcal{A} \chi^{-1} +~ n\mathcal{B} \chi^{-3/n} +~ \mathcal{D}\chi^3 \, .
</math>
</math>
</div>
</div>
we examined how <math>~P_e</math> varies with <math>~\tilde\xi</math>.  We found that <math>~P_e/P_\mathrm{norm} = 1.71\times 10^4</math> at <math>~\tilde\xi = 2.6</math>, which is almost identical to the value of the normalized external pressure that we determined was  associated with the unstable equilibrium state (at <math>~\tilde\xi = 4.81</math>) above.  As is illustrated by the figure and table that follows, we determined that the stable equilibrium state associated with this normalized external pressure is the minimum that occurs on the free energy curve having parameters, <math>~(n, \Pi_\mathrm{ad}) = (4, 0.02369)</math>.
First, we recognize that,
<div align="center">
<table border="0" cellpadding="5" align="center">


<table border="1" align="center" cellpadding="5">
<tr>
<tr>
   <th align="center" colspan="4">
   <td align="right">
[[File:TryN4Pi0.0237.png|450px|Dimensionless Free-Energy Curve]]
<math>~\chi \equiv \frac{R}{R_\mathrm{norm}}</math>
   </th>
  </td>
</tr>
  <td align="center">
<tr>
<math>~=</math>
  <th align="center" colspan="4">
  </td>
Determined from Plot of Renormalized Free-Energy
  <td align="left">
<math>~\biggl(\frac{R}{R_\mathrm{SWS}}\biggr) \frac{R_\mathrm{SWS}}{R_\mathrm{norm}} \, .</math>
   </td>
</tr>
</table>
</div>
From the definition of <math>~R_\mathrm{norm}</math> &#8212; reprinted, for example, [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Detailed_Force-Balanced_Solution|here]] &#8212; we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">


with <math>~(n, \Pi_\mathrm{ad}) = (4, 0.02369)</math>
  </th>
</tr>
<tr>
<tr>
   <th align="center">&nbsp;</th>
   <td align="right">
   <th align="center">&nbsp;</th>
<math>~\biggl( \frac{R_\mathrm{SWS}}{R_\mathrm{norm}}\biggr)^{n-3}</math>
   <th align="center" width="25%">Maximum</th>
   </td>
   <th align="center" width="25%">Minimum</th>
   <td align="center">
<math>~=</math>
  </td>
   <td align="left">
<math>~R_\mathrm{SWS}^{n-3} \biggl[ G^{-n} K^n M_\mathrm{tot}^{1-n} \biggr]
</math>
  </td>
</tr>
</tr>
<tr>
<tr>
   <th align="center">&nbsp;</th>
   <td align="right">
   <td align="center"> <math>~\Chi</math> </td>
&nbsp;
   <td align="center"> <math>~1.274</math> </td>
   </td>
   <td align="center"> <math>~1.317</math> </td>
   <td align="center">
<math>~=</math>
  </td>
   <td align="left">
<math>~R_\mathrm{SWS}^{n-3} M_\mathrm{SWS}^{1-n} \biggl[ \biggl(\frac{K}{G}\biggr)^n \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{1-n} \biggr] \, ;
</math>
  </td>
</tr>
</tr>
</table>
</div>
and from the definitions of <math>~R_\mathrm{SWS}</math> and <math>~M_\mathrm{SWS}</math> &#8212; reprinted, for example, [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Detailed_Force-Balanced_Solution_2|here]] &#8212; we have,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <th align="center" colspan="4">
   <td align="right">
Immediate Implications from Virial Theorem
<math>~\biggl( \frac{R_\mathrm{SWS}}{R_\mathrm{norm}}\biggr)^{n-3}</math>
   </th>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \biggl(\frac{K}{G}\biggr)^n \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{1-n} \biggr]
\biggl\{\biggl( \frac{n+1}{nG} \biggr)^{1/2} K^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]} \biggr\}^{n-3}
\biggl\{ \biggl( \frac{n+1}{nG} \biggr)^{3/2} K^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]} \biggr\}^{1-n}
</math>
   </td>
</tr>
</tr>
<tr>
<tr>
   <th align="center"><math>~\Chi^{1/4} - 1</math></th>
   <td align="right">
   <td align="center"> <math>~\eta_\mathrm{ad}</math> </td>
&nbsp;
   <td align="center"> <math>~0.0624</math></td>
  </td>
  <td align="center"> <math>~0.0713</math> </td>
   <td align="center">
<math>~=</math>
  </td>
   <td align="left">
<math>~\biggl[ \biggl(\frac{K}{G}\biggr)^n \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{1-n} \biggr]
\biggl( \frac{n+1}{n} \biggr)^{[(n-3) +3(1-n)]/2} G^{[(3-n) + 3(n-1)]/2} K^{n[(n-3)+2(1-n)]/(n+1)}
</math>
  </td>
</tr>
</tr>
<tr>
<tr>
   <th align="center"><math>~(\Chi^{1/4} - 1)\cdot \Chi^{-4}</math></th>
   <td align="right">
   <td align="center"> <math>~\Pi_\mathrm{ad}</math> </td>
&nbsp;
   <td align="center"> <math>~0.02369</math></td>
   </td>
   <td align="center"> <math>~0.02369 </math> </td>
   <td align="center">
<math>~=</math>
  </td>
   <td align="left">
<math>
~\biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{1-n} \biggl( \frac{n}{n+1} \biggr)^n \, .
</math>
  </td>
</tr>
</tr>
<tr>
</table>
  <th align="center" colspan="4">
</div>
Associated Detailed Force-Balanced Model Parameters
Hence, in each term in the free-energy expression we can make the substitution,
<div align="center">
<table border="0" cellpadding="5" align="center">


obtained via interpolation of tabulated numbers on p. 399 of [http://adsabs.harvard.edu/abs/1986Ap%26SS.126..357H Horedt (1986, ApJS, vol. 126)]
  </th>
</tr>
<tr>
<tr>
   <th align="center">&nbsp;</th>
   <td align="right">
  <td align="center"> <math>~\tilde\xi</math> (approx.) </td>
<math>~\chi </math>
   <td align="center"> ---- </td>
   </td>
   <td align="center"><math>~2.6</math></td>
   <td align="center">
</tr>
<math>~~~\rightarrow~~~</math>
<tr>
   </td>
   <th align="center">&nbsp;</th>
   <td align="left">
   <td align="center"> <math>~\tilde\theta</math> (approx.) </td>
<math>~
  <td align="center">---- </td>
\biggl(\frac{R}{R_\mathrm{SWS}}\biggr) \biggl\{ \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{1-n} \biggl( \frac{n}{n+1} \biggr)^n \biggr\}^{1/(n-3)}
   <td align="center"><math>~0.5048</math></td>
=  
\biggl(\frac{R}{R_\mathrm{SWS}}\biggr)\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}}\biggr)^{(1-n)/(n-3)} 
\biggl\{ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{n-1} \biggl( \frac{n}{n+1} \biggr)^n \biggr\}^{1/(n-3)} \, .
</math>
   </td>
</tr>
</tr>
</table>
</div>
Next, drawing on the definition of <math>~P_\mathrm{norm}</math> &#8212; reprinted, for example, [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Detailed_Force-Balanced_Solution|here]] &#8212; along with the definition of <math>~M_\mathrm{SWS}</math>, we recognize that,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <th align="center">&nbsp;</th>
   <td align="right">
  <td align="center"> <math>~- \tilde\theta^'</math> (approx.) </td>
<math>~\mathcal{D} \equiv \frac{4\pi}{3} \cdot \frac{P_e}{P_\mathrm{norm}}</math>
   <td align="center"> ---- </td>
  </td>
   <td align="center"> <math>~0.175</math></td>
   <td align="center">
<math>~=</math>
  </td>
   <td align="left">
<math>~\frac{4\pi}{3} \cdot P_e \biggl[ K^{-4n} G^{3(n+1)} M_\mathrm{tot}^{2(n+1)} \biggr]^{1/(n-3)} </math>
  </td>
</tr>
</tr>
<tr>
<tr>
   <th align="center"> <math>~\frac{1}{15}\cdot \frac{\tilde\theta^5}{(\tilde\theta^')^2}</math></th>
   <td align="right">
   <td align="center"> <math>~\eta</math> (check) </td>
&nbsp;
   <td align="center"> ---- </td>
  </td>
   <td align="center"> <math>~0.0714</math></td>
   <td align="center">
<math>~=</math>
   </td>
   <td align="left">
<math>~\frac{4\pi}{3} \cdot \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{2(n+1)/(n-3)} P_e \biggl[ K^{-4n} G^{3(n+1)}\biggr]^{1/(n-3)} M_\mathrm{SWS}^{2(n+1)/(n-3)}  </math>
  </td>
</tr>
</tr>
<tr>
<tr>
   <th align="center" colspan="4">
   <td align="right">
and, hence, Implied Structural Form Factors &amp; Coefficients <math>~\mathcal{B}</math> &amp; <math>~\mathcal{A}</math>
&nbsp;
   </th>
   </td>
</tr>
   <td align="center">
<tr>
<math>~=</math>
   <th align="center"> <math>~3(-\tilde\theta^')/\tilde\xi</math></th>
  </td>
   <td align="center"> <math>~\mathfrak{f}_M</math></td>
   <td align="left">
  <td align="center"> ---- </td>
<math>~\frac{4\pi}{3} \cdot \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{2(n+1)/(n-3)}
   <td align="center"><math>~0.2019</math></td>
\biggl\{ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2}\biggl( \frac{n+1}{n}\biggr)^3 \biggr\}^{(n+1)/(n-3)} \, .</math>
   </td>
</tr>
</tr>
</table>
</div>
After making these substitutions into the free-energy function, as well as replacing <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math> with <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math>, respectively, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <th align="center"> <math>~5[3(-\tilde\theta^')/\tilde\xi]^2</math></th>
   <td align="right">
   <td align="center"> <math>~\mathfrak{f}_W</math></td>
<math>~\mathfrak{G}^*</math>
   <td align="center"> ---- </td>
  </td>
  <td align="center"><math>~0.2039</math></td>
   <td align="center">
<math>~=</math>
  </td>
   <td align="left">
<math>~
-3\mathcal{A}_{M_\ell}\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{2}
\biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-1} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}}\biggr)^{-(1-n)/(n-3)} 
\biggl\{ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{n-1} \biggl( \frac{n}{n+1} \biggr)^n \biggr\}^{-1/(n-3)}
</math>
  </td>
</tr>
</tr>
<tr>
<tr>
   <th align="center"> <math>~15(-\tilde\theta^')^2 + \tilde\theta^5</math></th>
   <td align="right">
   <td align="center"> <math>~\mathfrak{f}_A</math></td>
&nbsp;
   <td align="center"> ---- </td>
   </td>
   <td align="center"><math>~0.4922</math></td>
   <td align="center">
&nbsp;
  </td>
   <td align="left">
<math>~
+~ n\mathcal{B}_{M_\ell} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{(n+1)/n}
\biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-3/n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}}\biggr)^{-3(1-n)/[n(n-3)]} 
\biggl\{ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{n-1} \biggl( \frac{n}{n+1} \biggr)^n \biggr\}^{-3/[n(n-3)]}
</math>
  </td>
</tr>
</tr>
<tr>
<tr>
   <th align="center"> <math>~\biggl(\frac{3}{4\pi} \biggr)^{1/4} \mathfrak{f}_M^{-5/4} \cdot \mathfrak{f}_A</math></th>
   <td align="right">
   <td align="center"> <math>~\mathcal{B}</math></td>
&nbsp;
   <td align="center"> ---- </td>
   </td>
   <td align="center"><math>~2.542</math></td>
   <td align="center">
</tr>
&nbsp;
<tr>
  </td>
  <th align="center"> <math>~\frac{\tilde\mathfrak{f}_W}{5 \tilde\mathfrak{f}_M^2} </math></th>
   <td align="left">
  <td align="center"> <math>~\mathcal{A}</math></td>
<math>~
  <td align="center"> <math>~1</math></td>
+~ \frac{4\pi}{3} \cdot \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{2(n+1)/(n-3)}  
  <td align="center"><math>~1</math></td>
\biggl\{ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2}\biggl( \frac{n+1}{n}\biggr)^3 \biggr\}^{(n+1)/(n-3)}
</tr>
\biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^3 \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}}\biggr)^{3(1-n)/(n-3)} 
<tr>
\biggl\{ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{n-1} \biggl( \frac{n}{n+1} \biggr)^n \biggr\}^{3/(n-3)}
  <th align="center" colspan="4">
</math>
Given <math>~\Pi_\mathrm{ad}</math>, <math>~\Chi</math>, and <math>~\mathcal{B}</math>, we obtain
   </td>
   </th>
</tr>
</tr>
<tr>
<tr>
   <th align="center"> <math>~\frac{3}{4\pi}\mathcal{D} = \frac{3}{4\pi} \Pi_\mathrm{ad} \mathcal{B}^{16} </math></th>
   <td align="right">
   <td align="center"> <math>~\frac{P_e}{P_\mathrm{norm}}</math></td>
&nbsp;
   <td align="center"> ---- </td>
  </td>
   <td align="center"><math>~1.71 \times 10^4</math></td>
   <td align="center">
</tr>
<math>~=</math>
<tr>
   </td>
  <th align="center"> <math>~\Chi \mathcal{B}^{-4}</math></th>
   <td align="left">
  <td align="center"> <math>~\chi_\mathrm{eq}</math></td>
<math>~
  <td align="center"> ---- </td>
-3\mathcal{A}_{M_\ell}
  <td align="center"><math>~0.0316</math></td>
\biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-1} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}}\biggr)^{-(1-n)/(n-3)} 
</tr>
\biggl\{ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{[(n-1)-2(n-3)]} \biggl( \frac{n}{n+1} \biggr)^n \biggr\}^{-1/(n-3)}
<tr>
  <th align="center" colspan="4">
Compare with Horedt's Equilibrium Parameters obtained from DFB Models
  </th>
</tr>
<tr>
  <th align="center"><math>\biggl\biggl( \frac{5^3}{4\pi} \biggr)  
\tilde\theta( -\tilde\xi^2 \tilde\theta' )^{2} \biggr]^{5}  
</math>
</math>
</th>
   </td>
   <td align="center"> <math>~\frac{P_e}{P_\mathrm{norm}}</math></td>
  <td align="center"> ---- </td>
  <td align="center"><math>~1.71 \times 10^4</math></td>
</tr>
</tr>
<tr>
<tr>
   <th align="center"><math>
   <td align="right">
\biggl( \frac{4\pi}{5^4} \biggr) \tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{-3}
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+~  n\mathcal{B}_{M_\ell} 
\biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-3/n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}}\biggr)^{-3(1-n)/[n(n-3)]} 
\biggl\{ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{[(n+1)(n-3)-3(n-1)]} \biggl( \frac{n}{n+1} \biggr)^{-3n} \biggr\}^{1/[n(n-3)]}
</math>
</math>
</th>
   </td>
   <td align="center"> <math>~\chi_\mathrm{eq}</math></td>
  <td align="center"> ---- </td>
  <td align="center"><math>~0.0316</math></td>
</tr>
</tr>
</table>


====Summary====
<tr>
 
  <td align="right">
The algebraic free-energy function associated with pressure-truncated <math>~n=4</math> polytropes is,
&nbsp;
<div align="center">
  </td>
<math>
  <td align="center">
\mathfrak{G}^*\biggr|_{n=4} =
&nbsp;
-3\mathcal{A} \chi^{-1} +~ 4\mathcal{B} \chi^{-3/4} +~ \mathcal{D}\chi^3 \, ,
  </td>
  <td align="left">
<math>~
+~ \frac{4\pi}{3} \cdot \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^3 \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}}\biggr)^{[2(n+1)+3(1-n)]/(n-3)}  
\biggl\{ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{[(3(n-1) -2(n+1)]}\biggl( \frac{n}{n+1}\biggr)^{[3n-3(n+1)]} \biggr\}^{1/(n-3)}
</math>
</math>
</div>
  </td>
and the corresponding ''renormalized'' free-energy function is,
</tr>
<div align="center">
<math>
\mathfrak{G}^{**}\biggr|_{n=4} \equiv \mathfrak{G}^* \biggl[ \frac{\mathcal{A}^3}{\mathcal{B}^n} \biggr]^{1/(n-3)} =
-3 \Chi^{-1} +~ 4\Chi^{-3/4} +~ \Pi_\mathrm{ad}\Chi^3 \, .
</math>
</div>
As has been demonstrated, above, the two equilibrium states that are supported by the same external pressure of, <math>~P_e/P_\mathrm{norm} = 1.71 \times 10^4</math>, are associated with extrema found in the following free-energy curves:  The ''unstable'' equilibrium appears as a relative ''maximum'' in the free-energy curves having the coefficient values,
<div align="center">
<math>~\Pi_\mathrm{ad} = 0.01</math> &nbsp;&nbsp;&nbsp;&nbsp; or &nbsp;&nbsp;&nbsp;&nbsp;
<math>(\mathcal{A}, \mathcal{B}, \mathcal{D}) = (1, 2.682, 7.16\times 10^4) \, .</math>
</div>
The ''stable'' equilibrium appears as a relative ''minimum'' in the free-energy curves having the coefficient values,
<div align="center">
<math>~\Pi_\mathrm{ad} = 0.02369</math> &nbsp;&nbsp;&nbsp;&nbsp; or &nbsp;&nbsp;&nbsp;&nbsp;
<math>(\mathcal{A}, \mathcal{B}, \mathcal{D}) = (1, 2.542, 7.16\times 10^4) \, .</math>
</div>


<table border="1" cellpadding="8" align="center" width="75%">
<tr>
  <th align="center">
Configurations Sharing the Same External Pressure
  </th>
</tr>
<tr>
  <td align="left">
'''<font color="maroon">ASIDE:</font>'''  In retrospect, it is obvious that pairs of truncated equilibrium configurations of a given polytropic index that are bounded by the same external pressure &#8212; and, hence, that may share a ''physical'' evolutionary connection &#8212; will share the same value of Horedt's dimensionless pressure,
<div align="center">
<table border="0" cellpadding="3">
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>
&nbsp;
~p_a
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~\equiv</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>~
\tilde\theta^{n+1}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)} \, .
-3\mathcal{A}_{M_\ell}
\biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-1} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}}\biggr)^{-(1-n)/(n-3)} 
\biggl\{ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{(n-5)} \biggl( \frac{n+1}{n} \biggr)^n \biggr\}^{1/(n-3)}
</math>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
 
</div>
<tr>
    
  <td align="right">
&nbsp;
  </td>
   <td align="center">
&nbsp;
   </td>
   </td>
</tr>
</table>
The implication is that a single free-energy curve with ''constant'' coefficients cannot connect the two equilibrium states.  There are certainly two separate equilibrium states that can be supported by the specified external pressure, but these two states exhibit somewhat different values of the structural form factors, which leads to different values of the coefficient, <math>~\mathcal{B}</math>.  The righthand plot in the following figure shows how <math>~\mathcal{B}</math> varies with the applied external pressure in <math>~n=4</math> polytropes. 
<table border="1" align="center" cellpadding="8">
<tr>
  <th align="center">
Variation of Various Physical Parameters along the Sequence of Pressure-Truncated <math>~n=4</math> Polytropes
[Structural data obtained from the table provided on p. 399 of [http://adsabs.harvard.edu/abs/1986Ap%26SS.126..357H Horedt (1986, ApJS, vol. 126)]]
  </th>
</tr>
<tr><td align="center">
[[File:SecondN4Parameters.png|750px|Parameters for n = 4 Embedded Polytropes]]
</td></tr>
<tr>
   <td align="left">
   <td align="left">
'''<font color="maroon">Left:</font>'''  This log-log plot displays the variation with applied external pressure, <math>~p_a</math> (increasing to the right along the horizontal axis), of the renormalized pressure, <math>~\Pi_\mathrm{ad}</math> (light blue diamonds), the renormalized equilibrium radius, <math>~\Chi_\mathrm{ad}</math> (light green triangles), and the key physical parameter, <math>~\eta_\mathrm{ad}</math> (maroon circles).  As the diagram illustrates, each parameter is double-valued, demonstrating that, for any choice of the dimensionless external pressure (as long as the pressure is less than a well-defined limiting value), there are two available equilibrium states.  Along all three curves, parameter values associated with the ''stable'' equilibrium are traced by the ''upper'' portion of the curve.  The red vertical line has been drawn at the value of <math>~{p_a} = 0.176</math>, corresponding to the external pressure <math>~(P_e/P_\mathrm{norm} = 1.71\times 10^4)</math> examined in the above two tables. This red line intersects the <math>~\Pi(p_a)</math> curve at <math>~\Pi = 0.01</math> (unstable state examined above) and at <math>~\Pi = 0.02369</math> (stable state examined above).
<math>~
 
+~ n\mathcal{B}_{M_\ell}
'''<font color="maroon">Right:</font>'''  This plot (linear scale on both axes) shows how <math>~\mathcal{B}</math> (curve outlined by light blue diamonds) varies with the applied external pressure, <math>~P_e/P_\mathrm{norm}</math>, in <math>~n=4</math> polytropes.  The curve bends back on itself, showing that at any value of <math>~P_e</math>, below some limiting value, two equilibrium configurations exist and they have different values of <math>~\mathcal{B}</math>.  The vertical red line identifies the value of the external pressure <math>~(P_e/P_\mathrm{norm} = 1.71\times 10^4)</math> that has been used as an example in the above two tables to illustrate how a pair of ''physically associated'' equilibrium states can be identified.  This red line intersects the displayed curve at <math>~\mathcal{B} = 2.682</math> (unstable state examined above) and at <math>~\mathcal{B} = 2.542</math> (stable state examined above).
\biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-3/n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}}\biggr)^{-3(1-n)/[n(n-3)]}  
\biggl\{ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{(n-5)} \biggl( \frac{n+1}{n} \biggr)^{3} \biggr\}^{1/(n-3)}
</math>
   </td>
   </td>
</tr>
</tr>
</table>
====Curiosity====
[[File:PiVersusPa.png|thumb|300px|Pressure vs. pressure plot]]
The figure displayed here, on the right, is a magnification of a segment of the <math>~\Pi(p_a)</math> curve (light blue diamonds) shown in the lefthand panel of the preceding figure, although here we have used a linear, rather than a log, scale on both axes.  The quantity being plotted along both axes is the external pressure, but normalized in different ways.  The quantity, <math>~p_a</math> (horizontal axis), provides a direct measure of the physical external (hence, also, surface) pressure, while the quantity, <math>~\Pi</math> (vertical axis), is the external pressure ''renormalized'' by a specific combination of the free-energy coefficients.  Our stability analysis has been conducted assuming that the free-energy coefficients &#8212; which are expressible in terms of structural form factors &#8212; are constants, that is, they do not vary with the size of the configuration.  Hence, it is the limiting value of <math>~\Pi_\mathrm{ad}</math>, specifically,
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\Pi_\mathrm{max}\biggr|_{n=4}</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
&nbsp;
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\frac{15^{15}}{16^{16}} = 0.02373828 \, ,</math>
<math>~
+~ \frac{4\pi}{3} \cdot \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^3 \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}}\biggr)^{(5-n)/(n-3)}  
\biggl\{ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{(n-5)}\biggl( \frac{n+1}{n}\biggr)^{3} \biggr\}^{1/(n-3)}
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
that identifies the demarcation between stable and unstable states.  This limiting value is identified by the horizontal red-dashed line in the figure; and the relevant demarcation point appears where this tangent line touches the curve.  According to our stability analysis, equilibrium configurations to the left of this demarcation point are stable while configurations to the right are unstable.
In the context of our discussion of the lefthand diagram in the preceding figure &#8212; see especially the relevant figure caption &#8212; we claimed that, for each physically allowed value of the external pressure, <math>~p_a</math>, the parameter, <math>~\Pi</math>, was double-valued and that configurations along the ''upper'' segment of its curve were stable.  After studying a magnification of this parameter curve near its turning point, a bit of clarification is required.  It appears as though equilibrium models lying along the short ''upper'' segment of the curve that falls between the demarcation/tangent point at <math>~\Pi_\mathrm{max}</math> and the maximum value of <math>~p_a</math> are unstable.  This means that, even though two equilibrium configurations can be constructed at each value of <math>~p_a</math> in this region near and including the turning point, ''both'' configurations are dynamically unstable.  We conclude, therefore, that stable configurations only exist for values of <math>~p_a</math> that are less than the value associated with <math>~\Pi_\mathrm{max}</math>.
==Mass-Radius Relation==
Up to this point in our discussion, we have focused on an analysis of the pressure-radius relationship that defines the equilibrium configurations of pressure-truncated polytropes.  In effect, we have viewed the problem through the same lens as did [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)] and, separately, [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth (1981)], defining variable normalizations in terms of the polytropic constant, <math>~K</math>, and the configuration mass, <math>~M_\mathrm{tot}</math>, which were both assumed to be held fixed throughout the analysis.  Here we switch to the approach championed by [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)], defining variable normalizations in terms of <math>~K</math> and <math>~P_e</math>, and examining the ''mass-radius'' relationship of pressure-truncated polytropes.
===Detailed Force-Balanced Solution===
As has been summarized in our [[User:Tohline/SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|accompanying review]] of detailed force-balanced models of pressure-truncated polytropes, [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)] found that a spherical configuration's equilibrium radius is related to its mass through the following pair of parametric equations:
<div align="center">
<table border="0" cellpadding="3">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>
&nbsp;
~\frac{M_\mathrm{tot}}{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[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{(n-5)}\biggl( \frac{n+1}{n}\biggr)^{3} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}}\biggr)^{(5-n)} \biggr]^{1/(n-3)}
\biggl( \frac{n^3}{4\pi} \biggr)^{1/2} \tilde\theta^{(n-3)/2} (- \tilde\xi^2 \tilde\theta^') \, ,
\biggl\{
-3\mathcal{A}_{M_\ell}\biggl( \frac{n+1}{n}\biggr) \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-1} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}}\biggr)^{2}
+~  n\mathcal{B}_{M_\ell} 
\biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-3/n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}}\biggr)^{(n+1)/n} 
+~ \frac{4\pi}{3} \cdot \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{3} 
\biggr\}\, .
</math>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
Hence, after defining,
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>
<math>~\mathfrak{G}^*_\mathrm{SWS}</math>
~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} }
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=~</math>
<math>~\equiv</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>~
\biggl( \frac{n}{4\pi} \biggr)^{1/2} \tilde\xi \tilde\theta^{(n-1)/2} \, ,
\frac{\mathfrak{G}}{[G^{-3} K^n M_\mathrm{SWS}^{n-5}]^{1/(n-3)}} \biggl( \frac{n}{n+1}\biggr)^{3/(n-3)}
=
\frac{\mathfrak{G}}{[K^{6n} P_e^{5-n}]^{1/[2(n+1)]}} \biggl( \frac{nG}{n+1}\biggr)^{3/2} \, ,
</math>
</math>
   </td>
   </td>
Line 1,614: Line 1,720:
</table>
</table>
</div>
</div>
where,
 
<div align="center">
we can write,
<math>M_\mathrm{SWS} \equiv
<div align="center" id="ConciseFreeEnergyExpression">
\biggl( \frac{n+1}{nG} \biggr)^{3/2} K^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]} \, ,</math>
<table border="0" cellpadding="5" align="center">
</div>
 
<div align="center">
<tr>
<math>
R_\mathrm{SWS} \equiv \biggl( \frac{n+1}{nG} \biggr)^{1/2} K^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]} \, .
</math>
</div>
 
===Mapping from Above Discussion===
Looking back on the definitions of <math>~\Pi_\mathrm{ad}</math> and <math>~\Chi_\mathrm{ad}</math> that we introduced in connection with our initial [[User:Tohline/SSC/Virial/PolytropesSummary#ConciseVirial|concise algebraic expression of the virial theorem]], we can write,
<div align="center">
<table border="0" cellpadding="5">
<tr>
   <td align="right">
   <td align="right">
<math>~P_e </math>
<math>~\mathfrak{G}^*_\mathrm{SWS} </math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 1,637: Line 1,733:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>~
~P_\mathrm{norm} \biggl( \frac{3}{4\pi} \biggr) \Pi_\mathrm{ad} \biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr]^{1/(n-3)}
-3\mathcal{A}_{M_\ell}\biggl( \frac{n+1}{n}\biggr) \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{2} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-1}
+~ n\mathcal{B}_{M_\ell}  \cdot \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{(n+1)/n}  \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-3/n}
+~ \frac{4\pi}{3} \cdot \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{3} \, .
</math>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
Setting the first derivative of this function equal to zero should produce the virial theorem expression.  Let's see &hellip;
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>~\frac{\partial\mathfrak{G}^*_\mathrm{SWS}}{\partial \mathcal{X}}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 1,651: Line 1,754:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>~
~\biggl( \frac{3}{4\pi} \biggr) \Pi_\mathrm{ad} \biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr]^{1/(n-3)}
3\mathcal{A}_{M_\ell}\biggl( \frac{n+1}{n}\biggr) \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{2} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-2}
\biggl[ \frac{K^{4n}}{G^{3(n+1)} M_\mathrm{tot}^{2(n+1)}} \biggr]^{1/(n-3)} \, ,
-~ 3\mathcal{B}_{M_\ell} \cdot \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{(n+1)/n}  \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-(3+n)/n}
+~ 4\pi \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{2}
</math>
</math>
   </td>
   </td>
Line 1,660: Line 1,764:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~R_\mathrm{eq} </math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 1,666: Line 1,770:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>~ 3\biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-2}
~R_\mathrm{norm} \Chi_\mathrm{ad} \biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{n/(n-3)}  
\biggl[ \mathcal{A}_{M_\ell}\biggl( \frac{n+1}{n}\biggr) \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{2}
-~ \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}  
+~ \frac{4\pi}{3} \cdot \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{4} \biggr] \, .
</math>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
Replacing <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math> with <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>, as prescribed by their defined relationships, and setting the expression inside the square brackets equal to zero does, indeed, produce the [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#CompactStahlerVirial|above, boxed-in ''viral theorem'' mass-radius relationship]].


<tr>
====Plotting Concise Mass-Radius Relation====
   <td align="right">
Our derived, [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#ConciseVirialMR|concise analytic expression for the virial theorem]], namely,
&nbsp;
<div align="center">
   </td>
<math>
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{mod}} \biggr)^4
- \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{mod}} \biggr)^{(n-3)/n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{mod}} \biggr)^{(n+1)/n}
+ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{mod}} \biggr)^2 = 0
\, ,
</math>
</div>
 
is plotted for seven different values of the polytropic index, <math>~n</math>, as indicated, in the lefthand diagram of the following composite figure.  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 &#8212; which is, in essence, a statement of the scalar virial theorem &#8212; 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.
 
 
<table border="1" align="center" cellpadding="3">
<tr>
   <td align="center" rowspan="2">
[[File:MassRadiusVirialLabeled.png|350px|Virial Theorem Mass-Radius Relation]]
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
[[File:Stahler1983TitlePage0.png|300px|center|Stahler (1983) Title Page]]
   </td>
   </td>
   <td align="left">
</tr>
<math>
<tr>
~\Chi_\mathrm{ad} \biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{n/(n-3)}
   <td align="center" bgcolor="white">
\biggl[ \biggl( \frac{G}{K} \biggr)^n M_\mathrm{tot}^{n-1} \biggr]^{1/(n-3)} \, .
[[File:Stahler_MRdiagram1.png|300px|center|Stahler (1983) Figure 17 (edited)]]
</math>
   </td>
   </td>
</tr>
</tr>
</table>


</table>
Let's do this again using the mass-radius relation as written explicitly in terms of the normalizations, <math>~M_\mathrm{SWS}</math> and <math>~R_\mathrm{SWS}</math>.  The relevant, generic nonlinear equation is,
</div>
The first of these two expressions can be flipped around to give an expression for <math>~M_\mathrm{tot}</math> in terms of <math>~P_e</math> and, then, as normalized to <math>~M_\mathrm{SWS}</math>.  Specifically,
<div align="center">
<div align="center">
<table border="0" cellpadding="5">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~ M_\mathrm{tot}^{2(n+1)}</math>
<math>~0</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 1,701: Line 1,823:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>~
~\biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{n-3} \biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr]
\biggl( \frac{R}{R_\mathrm{SWS}} \biggr)^4
\biggl[ \frac{K^{4n}}{G^{3(n+1)}P_e^{n-3} } \biggr]
- \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>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>


<tr>
Analytically determined roots:
  <td align="right">
* <math>~n=1</math>
&nbsp;
<div align="center">
  </td>
<math>~\frac{M}{M_\mathrm{SWS}} = \biggl( \frac{10\pi}{3}\biggr)^{1 / 2} \biggl(\frac{R}{R_\mathrm{SWS}} \biggr)^3 \biggl[ \frac{3\cdot 5}{2^3\pi} - \biggl(\frac{R}{R_\mathrm{SWS}} \biggr)^2
  <td align="center">
\biggr]^{-1/2} </math> &nbsp; &nbsp; &nbsp; &nbsp; for, &nbsp; &nbsp; <math>~0 \le \frac{R}{R_\mathrm{SWS}} \le \biggl(\frac{3\cdot 5}{2^3\pi}\biggr)^{1/2} \, .</math>
<math>~=</math>
</div>
  </td>
* <math>~n=3</math>
  <td align="left">
<div align="center">
<math>
<math>~\frac{R}{R_\mathrm{SWS}} = \biggl\{ \biggl[ \biggl(\frac{3}{4\pi}\biggr)\frac{M}{M_\mathrm{SWS}}\biggr]^{4/3}
~M_\mathrm{SWS}^{2(n+1)} \biggl( \frac{n}{n+1} \biggr)^{3(n+1)}  
- \biggl(\frac{1}{5\pi}\biggr) \biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \biggr\}^{1/4} </math> &nbsp; &nbsp; &nbsp; &nbsp; for, &nbsp; &nbsp; <math>~0 \le \frac{M}{M_\mathrm{SWS}} \le \biggl(\frac{3^4\cdot 5^3}{2^8\pi}\biggr)^{1/2} \, .</math>
\biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{n-3} \biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr]
</div>
</math>
* <span id="Isothermal">''Isothermal''</span> (explained [[#IsothermalExplained|immediately below]])
  </td>
<div align="center">
</tr>
<math>~\frac{M}{M_\mathrm{SWS}} = \frac{5}{2} \biggl( \frac{R}{R_\mathrm{SWS}} \biggr) \biggl\{
1 \pm \biggl[ 1 - \frac{16\pi}{15}\biggl( \frac{R}{R_\mathrm{SWS}} \biggr)^2  \biggr]^{1 / 2} \biggr\} </math>
&nbsp; &nbsp; &nbsp; &nbsp; for, &nbsp; &nbsp; <math>~0 \le \frac{R}{R_\mathrm{SWS}} \le \biggl(\frac{3\cdot 5}{2^4\pi}\biggr)^{1/2} \, .</math>
</div>


<tr>
  <td align="right">
<math>~ \Rightarrow~~~ \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\biggl( \frac{n}{n+1} \biggr)^{3/2}
\biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(n-3)/[2(n+1)]}  \biggl[ \frac{\mathcal{B}^{2n/(n+1)}}{\mathcal{A}^{3/2}} \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>


This means, as well, that we can rewrite the equilibrium radius as,
<span id="TabulatedValues">First, we'll create a table of the normalized coordinate values that satisfy this nonlinear expression.</span>
<div align="center">
<div align="center">
<table border="0" cellpadding="5">
<table border="1" align="center" cellpadding="5">
 
<tr>
<tr>
   <td align="right">
   <td align="center" colspan="1"><math>~n =2</math></td>
<math>~R_\mathrm{eq}^{n-3} </math>
  <td align="center" colspan="1"><math>~n =2.8</math></td>
   </td>
  <td align="center" colspan="1"><math>~n =3.5</math></td>
   <td align="center">
   <td align="center" colspan="1"><math>~n = 4</math></td>
<math>~=</math>
   <td align="center" colspan="1"><math>~n = 5</math></td>
  </td>
</tr>
   <td align="left">
<tr>
<math>
<td align="center">
~\Chi_\mathrm{ad}^{n-3} \biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{n}
<table border="0" cellpadding="5" align="center">
\biggl( \frac{G}{K} \biggr)^n M_\mathrm{tot}^{n-1}  
<tr>
</math>
   <td align="center"><math>~ \frac{R}{R_\mathrm{SWS}}</math></td>
   </td>
  <td align="center">&nbsp;</td>
  <td align="center"><math>~ \frac{M}{M_\mathrm{SWS}}</math></td>
</tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
<tr>
  <td align="right">0.3800</td>
  <td align="center">&nbsp;</td>
   <td align="left">0.26562</td>
</tr>
</tr>
<tr>
<tr>
   <td align="right">
   <td align="right">0.4500</td>
&nbsp;
   <td align="center">&nbsp;</td>
  </td>
   <td align="left">0.477153</td>
   <td align="center">
<math>~=</math>
  </td>
   <td align="left">
<math>
~\Chi_\mathrm{ad}^{n-3} \biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{n}
\biggl( \frac{G}{K} \biggr)^n \biggl\{ 
\biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{n-3}  \biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr]
\biggl[ \frac{K^{4n}}{G^{3(n+1)}P_e^{n-3} } \biggr]
\biggr\}^{(n-1)/[2(n+1)]}
</math>
  </td>
</tr>
</tr>
<tr>
<tr>
   <td align="right">
   <td align="right">0.5000</td>
&nbsp;
  <td align="center">&nbsp;</td>
  </td>
   <td align="left">0.70919</td>
   <td align="center">
</tr>
<math>~=</math>
<tr>
   </td>
  <td align="right">0.5500</td>
   <td align="left">
   <td align="center">&nbsp;</td>
<math>
   <td align="left">1.063602</td>
~\Chi_\mathrm{ad}^{n-3} \biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{n}
</tr>
\biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr]^{(n-1)/[2(n+1)]}
<tr>
\biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(n-3)(n-1)/[2(n+1)]} \biggl( \frac{G}{K} \biggr)^n
  <td align="right">0.5800</td>
\biggl\{ 
  <td align="center">&nbsp;</td>
\biggl[ \frac{K^{4n}}{G^{3(n+1)}P_e^{n-3} } \biggr]
   <td align="left">1.39755</td>
\biggr\}^{(n-1)/[2(n+1)]}
</math>
   </td>
</tr>
</tr>
<tr>
<tr>
   <td align="right">
   <td align="right">0.5950</td>
&nbsp;
   <td align="center">&nbsp;</td>
  </td>
   <td align="left">1.64662</td>
   <td align="center">
<math>~=</math>
  </td>
   <td align="left">
<math>
~\Chi_\mathrm{ad}^{n-3} \biggl\{ \biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{2n(n+1)}
\biggl[ \frac{\mathcal{B}^{4n(n-1)}}{\mathcal{A}^{3(n+1)(n-1)}} \biggr]\biggr\}^{1/[2(n+1)]}
\biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(n-3)(n-1)/[2(n+1)]}
\biggl\{ \biggl( \frac{G}{K} \biggr)^{2n(n+1)}
\biggl[ \frac{K^{4n(n-1)}}{G^{3(n+1)(n-1)}P_e^{(n-3)(n-1)} } \biggr]
\biggr\}^{1/[2(n+1)]}
</math>
  </td>
</tr>
</tr>
<tr>
<tr>
   <td align="right">
   <td align="right">0.6050</td>
&nbsp;
  <td align="center">&nbsp;</td>
   </td>
  <td align="left">1.893915</td>
   <td align="center">
</tr>
<math>~=</math>
<tr>
   </td>
   <td align="right">0.6120</td>
   <td align="left">
   <td align="center">&nbsp;</td>
<math>
  <td align="left">2.22372</td>
~\Chi_\mathrm{ad}^{n-3} \biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(n-3)(n-1)/[2(n+1)]}
</tr>
\biggl[ \mathcal{A}^{-(n+1)(n-3)} \mathcal{B}^{2n(n-3)} \biggr]^{1/[2(n+1)]}
<tr>
\biggl[ G^{(3-n)(n+1)} K^{2n(n-3)} P_e^{(n-3)(1-n)} \biggr]^{1/[2(n+1)]}
  <td align="right"><font color="darkgreen">0.6131721</font></td>
</math>
   <td align="center">&nbsp;</td>
   </td>
   <td align="left"><font color="darkgreen">2.433375</font></td>
</tr>
<tr>
  <td align="right">0.6120</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.64923</td>
</tr>
<tr>
  <td align="right">0.6050</td>
  <td align="center">&nbsp;</td>
   <td align="left">3.01688</td>
</tr>
</tr>
<tr>
<tr>
   <td align="right">
   <td align="right">0.5950</td>
&nbsp;
   <td align="center">&nbsp;</td>
  </td>
   <td align="left">3.32037</td>
   <td align="center">
<math>~=</math>
  </td>
   <td align="left">
<math>
~R_\mathrm{SWS}^{n-3} \biggl( \frac{n}{n+1} \biggr)^{(n-3)/2}
\Chi_\mathrm{ad}^{n-3} \biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(n-3)(n-1)/[2(n+1)]}
\biggl[ \mathcal{A}^{-(n+1)(n-3)} \mathcal{B}^{2n(n-3)} \biggr]^{1/[2(n+1)]}
</math>
  </td>
</tr>
</tr>
<tr>
<tr>
   <td align="right">
   <td align="right">0.5800</td>
<math>~\Rightarrow~~~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} } </math>
  <td align="center">&nbsp;</td>
   </td>
  <td align="left">3.658702</td>
   <td align="center">
</tr>
<math>~=</math>
<tr>
   </td>
   <td align="right">0.5500</td>
   <td align="left">
   <td align="center">&nbsp;</td>
<math>
  <td align="left">4.19097</td>
~\biggl( \frac{n}{n+1} \biggr)^{1/2}
</tr>
\Chi_\mathrm{ad} \biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(n-1)/[2(n+1)]}
<tr>
\biggl[ \frac{\mathcal{B}^{n/(n+1)}}{\mathcal{A}^{1/2}} \biggr] \, .
  <td align="right">0.5000</td>
</math>
   <td align="center">&nbsp;</td>
   </td>
   <td align="left">4.94599</td>
</tr>
<tr>
  <td align="right">0.4700</td>
  <td align="center">&nbsp;</td>
   <td align="left">5.38791</td>
</tr>
</tr>
</table>
</div>
Flipping both of these expressions around, we see that,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="right">
   <td align="right">0.4500</td>
<math>~\Pi_\mathrm{ad} </math>
   <td align="center">&nbsp;</td>
  </td>
   <td align="left">5.69164</td>
   <td align="center">
<math>~=</math>
  </td>
   <td align="left">
<math>
~\frac{4\pi}{3} \biggl\{ \biggl( \frac{n+1}{n} \biggr)^{3(n+1)}  \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{2(n+1)}
\biggl[ \frac{\mathcal{A}^{3(n+1)}}{\mathcal{B}^{4n}} \biggr] \biggr\}^{1/(n-3)} \, ,
</math>
  </td>
</tr>
</tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
</table>
</table>
</div>
</td>
and,
<td align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="right">
   <td align="center"><math>~ \frac{R}{R_\mathrm{SWS}}</math></td>
<math>~\Chi_\mathrm{ad} </math>
   <td align="center">&nbsp;</td>
   </td>
   <td align="center"><math>~ \frac{M}{M_\mathrm{SWS}}</math></td>
   <td align="center">
</tr>
<math>~=</math>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
   </td>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
   <td align="left">
<tr><td align="center" colspan="3">&nbsp;</td></tr>
<math>
<tr>
~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} }  \biggl( \frac{n+1}{n} \biggr)^{1/2}
  <td align="right">0.3800</td>
\biggl[ \frac{\mathcal{A}^{1/2}}{\mathcal{B}^{n/(n+1)}} \biggr]
   <td align="center">&nbsp;</td>
\biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(1-n)/[2(n+1)]}
   <td align="left">0.266134</td>
</math>
</tr>
   </td>
<tr>
  <td align="right">0.4500</td>
  <td align="center">&nbsp;</td>
  <td align="left">0.47971</td>
</tr>
<tr>
  <td align="right">0.5000</td>
  <td align="center">&nbsp;</td>
   <td align="left">0.71765</td>
</tr>
</tr>
<tr>
<tr>
   <td align="right">
   <td align="right">0.5250</td>
&nbsp;
   <td align="center">&nbsp;</td>
  </td>
   <td align="left">0.831825</td>
   <td align="center">
<math>~=</math>
  </td>
   <td align="left">
<math>
~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} }  \biggl( \frac{n+1}{n} \biggr)^{1/2}
\biggl[ \frac{\mathcal{A}^{1/2}}{\mathcal{B}^{n/(n+1)}} \biggr]
\biggl\{ \biggl( \frac{n+1}{n} \biggr)^{3(n+1)}  \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{2(n+1)}
\biggl[ \frac{\mathcal{A}^{3(n+1)}}{\mathcal{B}^{4n}} \biggr] \biggr\}^{(1-n)/[2(n+1)(n-3)]}
</math>
  </td>
</tr>
</tr>
<tr>
<tr>
   <td align="right">
   <td align="right">0.5600</td>
&nbsp;
  <td align="center">&nbsp;</td>
  </td>
   <td align="left">1.20977</td>
   <td align="center">
</tr>
<math>~=</math>
<tr>
   </td>
  <td align="right">0.5750</td>
   <td align="left">
   <td align="center">&nbsp;</td>
<math>
   <td align="left">1.427183</td>
~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} } \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{(1-n)/(n-3)}  \biggl( \frac{n}{n+1} \biggr)^{n/(n-3)}
</tr>
\biggl[ \frac{\mathcal{B}}{\mathcal{A}} \biggr]^{n/(n-3)} \, .
<tr>
</math>
  <td align="right">0.5850</td>
   </td>
  <td align="center">&nbsp;</td>
  <td align="left">1.653232</td>
</tr>
<tr>
  <td align="right">0.5900</td>
  <td align="center">&nbsp;</td>
   <td align="left">1.89304</td>
</tr>
</tr>
</table>
</div>
Hence, our earlier derived [[User:Tohline/SSC/Virial/PolytropesSummary#ConciseVirial3|compact expression for the virial theorem]] becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="right">
   <td align="right"><font color="darkgreen">0.5904492</font></td>
<math>~1</math>
   <td align="center">&nbsp;</td>
  </td>
   <td align="left"><font color="darkgreen">1.989927</font></td>
   <td align="center">
<math>~=</math>
  </td>
   <td align="left">
<math>
\biggl\{ \frac{R_\mathrm{eq}}{R_\mathrm{SWS} } \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{(1-n)/(n-3)}  \biggl( \frac{n}{n+1} \biggr)^{n/(n-3)}
\biggl[ \frac{\mathcal{B}}{\mathcal{A}} \biggr]^{n/(n-3)} \biggr\}^{(n-3)/n}
</math>  
  </td>
</tr>
</tr>
<tr>
<tr>
   <td align="right">
   <td align="right">0.5900</td>
&nbsp;
   <td align="center">&nbsp;</td>
  </td>
   <td align="left">2.086584</td>
   <td align="center">
&nbsp;
  </td>
   <td align="left">
<math>
-~ \frac{4\pi}{3} \biggl\{ \biggl( \frac{n+1}{n} \biggr)^{3(n+1)}  \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{2(n+1)}
\biggl[ \frac{\mathcal{A}^{3(n+1)}}{\mathcal{B}^{4n}} \biggr] \biggr\}^{1/(n-3)}
\biggl\{ \frac{R_\mathrm{eq}}{R_\mathrm{SWS} } \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{(1-n)/(n-3)}  \biggl( \frac{n}{n+1} \biggr)^{n/(n-3)}
\biggl[ \frac{\mathcal{B}}{\mathcal{A}} \biggr]^{n/(n-3)} \biggr\}^4 </math>
  </td>
</tr>
</tr>
<tr>
<tr>
   <td align="right">
   <td align="right">0.5850</td>
&nbsp;
   <td align="center">&nbsp;</td>
  </td>
   <td align="left">2.32394</td>
   <td align="center">
<math>~=</math>
  </td>
   <td align="left">
<math>
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS} } \biggr)^{(n-3)/n} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{(1-n)/n}  \biggl( \frac{n}{n+1} \biggr)
\biggl[ \frac{\mathcal{B}}{\mathcal{A}} \biggr]
-~ \frac{4\pi}{3} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS} }  \biggr)^4 \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{-2} \biggl( \frac{n}{n+1} \biggr)
\frac{1}{\mathcal{A}} \, .
</math>
  </td>
</tr>
</tr>
</table>
</div>
Or, rearranged,
<div align="center" id="CompactStahlerVirial">
<table border="1" cellpadding="10" align="center">
<tr>
<tr>
   <td align="right">
   <td align="right">0.5750</td>
<math>\frac{4\pi}{3} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS} }  \biggr)^4  -
  <td align="center">&nbsp;</td>
\mathcal{B} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS} } \biggr)^{(n-3)/n} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{(n+1)/n} 
  <td align="left">2.54527</td>
+~ \mathcal{A} \biggl( \frac{n+1}{n} \biggr) \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{2} = 0 \, .
</tr>
</math>  
<tr>
   </td>
  <td align="right">0.5600</td>
  <td align="center">&nbsp;</td>
   <td align="left">2.75612</td>
</tr>
</tr>
</table>
</div>
After adopting modified length- and mass-normalizations, <math>~R_\mathrm{mod}</math> and <math>~M_\mathrm{mod}</math>, such that,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="right">
   <td align="right">0.5250</td>
<math>~\frac{M_\mathrm{SWS}}{M_\mathrm{mod}}</math>
   <td align="center">&nbsp;</td>
  </td>
   <td align="left">3.07134</td>
   <td align="center">
<math>~\equiv</math>
  </td>
   <td align="left">
<math>~\biggl( \frac{4\pi}{3} \biggr)^{2n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{3/2} \frac{\mathcal{A}^{3/2}}{\mathcal{B}^{2n/(n+1)}} \, ,</math>
  </td>
</tr>
</tr>
<tr>
<tr>
   <td align="right">
   <td align="right">0.4500</td>
<math>~\frac{R_\mathrm{SWS}}{R_\mathrm{mod}}</math>
   <td align="center">&nbsp;</td>
  </td>
   <td align="left">3.460304</td>
   <td align="center">
<math>~\equiv</math>
  </td>
   <td align="left">
<math>~\biggl( \frac{4\pi}{3} \biggr)^{n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{1/2} \frac{\mathcal{A}^{1/2}}{\mathcal{B}^{n/(n+1)}} \, ,</math>
  </td>
</tr>
</tr>
</table>
</div>
we obtain the
<div align="center" id="ConciseVirialMR">
<font color="#770000">'''Virial Theorem in terms of Mass and Radius'''</font><br />
<math>
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{mod}} \biggr)^4
- \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{mod}} \biggr)^{(n-3)/n} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{mod}} \biggr)^{(n+1)/n}
+ \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{mod}} \biggr)^2 = 0
\, .
</math>
</div>
This analytic function is plotted for seven different values of the polytropic index, <math>~n</math>, as indicated, in the lefthand diagram of the following table.
<table border="1" align="center" cellpadding="3">
<tr>
<tr>
   <td align="center" rowspan="2">
  <td align="right">0.3500</td>
[[File:MassRadiusVirialLabeled.png|350px|Virial Theorem Mass-Radius Relation]]
  <td align="center">&nbsp;</td>
   </td>
  <td align="left">3.75881</td>
   <td align="center">
</tr>
[[File:Stahler1983TitlePage0.png|300px|center|Stahler (1983) Title Page]]
<tr>
   </td>
  <td align="right">0.2500</td>
  <td align="center">&nbsp;</td>
  <td align="left">3.97835</td>
</tr>
<tr>
  <td align="right">0.2000</td>
  <td align="center">&nbsp;</td>
  <td align="left">4.09302</td>
</tr>
<tr>
  <td align="right">0.1500</td>
   <td align="center">&nbsp;</td>
  <td align="left">4.232786</td>
</tr>
<tr>
  <td align="right">0.1000</td>
  <td align="center">&nbsp;</td>
  <td align="left">4.430303</td>
</tr>
<tr>
   <td align="right">0.0700</td>
   <td align="center">&nbsp;</td>
  <td align="left">4.60984</td>
</tr>
<tr>
  <td align="right">0.0400</td>
  <td align="center">&nbsp;</td>
   <td align="left">4.9057</td>
</tr>
</tr>
<tr>
<tr>
   <td align="center" bgcolor="white">
  <td align="right">0.0150</td>
[[File:Stahler_MRdiagram1.png|300px|center|Stahler (1983) Figure 17 (edited)]]
   <td align="center">&nbsp;</td>
  </td>
  <td align="left">5.47056</td>
</tr>
</tr>
</table>
</table>
 
</td>
 
<td align="center">
Now that we have this very general, yet concise, algebraic expression for the mass-radius relationship of all pressure-truncated polytropes, let's replace the new "mod" normalizations with Stahler's original normalizations, <math>~R_\mathrm{SWS}</math> and <math>~M_\mathrm{SWS}</math>, to understand more completely how this general expression should be viewed in relation to the parametric relations provided by solutions of the detailed force-balanced models.  We will henceforth use the notation,
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="right">
   <td align="center"><math>~ \frac{R}{R_\mathrm{SWS}}</math></td>
<math>~\mathcal{X}</math>
   <td align="center">&nbsp;</td>
  </td>
   <td align="center"><math>~ \frac{M}{M_\mathrm{SWS}}</math></td>
   <td align="center">
</tr>
<math>~\equiv</math>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
  </td>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
   <td align="left">
<tr><td align="center" colspan="3">&nbsp;</td></tr>
<math>~\frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \, ,</math>
<tr>
   </td>
   <td align="right">0.3800</td>
  <td align="center">&nbsp;</td>
  <td align="left">0.26639</td>
</tr>
</tr>
<tr>
<tr>
   <td align="right">
   <td align="right">0.4500</td>
<math>~\mathcal{Y}</math>
   <td align="center">&nbsp;</td>
  </td>
   <td align="left">0.481072</td>
   <td align="center">
<math>~\equiv</math>
  </td>
   <td align="left">
<math>~\frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \, .</math>
  </td>
</tr>
</tr>
</table>
</div>
In the virial theorem expression we will make the following replacements:
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="right">
   <td align="right">0.5000</td>
<math>~\frac{R_\mathrm{eq}}{R_\mathrm{mod}} ~~~ \rightarrow ~~~ \mathcal{X} \cdot \frac{R_\mathrm{SWS}}{R_\mathrm{mod}}</math>
   <td align="center">&nbsp;</td>
  </td>
   <td align="left">0.722406</td>
   <td align="center">
<math>~=</math>
  </td>
   <td align="left">
<math>~\mathcal{X} \biggl( \frac{4\pi}{3} \biggr)^{n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{1/2} \frac{\mathcal{A}^{1/2}}{\mathcal{B}^{n/(n+1)}} \, ,</math>
  </td>
</tr>
</tr>
<tr>
<tr>
   <td align="right">
   <td align="right">0.5250</td>
<math>~\frac{M_\mathrm{tot}}{M_\mathrm{mod}} ~~~ \rightarrow ~~~ \mathcal{Y} \cdot \frac{M_\mathrm{SWS}}{M_\mathrm{mod}}</math>
   <td align="center">&nbsp;</td>
  </td>
   <td align="left">0.89152</td>
   <td align="center">
<math>~=</math>
  </td>
   <td align="left">
<math>~\mathcal{Y} \biggl( \frac{4\pi}{3} \biggr)^{2n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{3/2} \frac{\mathcal{A}^{3/2}}{\mathcal{B}^{2n/(n+1)}} \, . </math>
  </td>
</tr>
</tr>
</table>
</div>
Next, we recognize that, in order to graphically display the mass-radius relation derived from the virial theorem in the <math>~\mathcal{X}-\mathcal{Y}</math> plane, we must write out the expressions for the free-energy coefficients.  After setting <math>~M_\mathrm{limit}/M_\mathrm{tot} = 1</math> in the [[User:Tohline/SSC/Virial/PolytropesSummary#Structural_Form_Factors|above summary expressions]], we obtain,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<tr>
   <td align="right">
   <td align="right">0.5600</td>
<math>~\mathcal{A}</math>
   <td align="center">&nbsp;</td>
  </td>
   <td align="left">1.246123</td>
   <td align="center">
<math>~=</math>
  </td>
   <td align="left">
<math>\frac{1}{5} \cdot \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2}  = \frac{1}{5-n} \, ,</math>
  </td>
</tr>
</tr>
<tr>
<tr>
   <td align="right">
   <td align="right">0.5650</td>
<math>~\mathcal{B}</math>
   <td align="center">&nbsp;</td>
  </td>
   <td align="left">1.32113</td>
   <td align="center">
<math>~\equiv</math>
  </td>
   <td align="left">
<math>
\biggl( \frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_M^{(n+1)/n}} =
\frac{1}{3(5-n) ( 4\pi )^{1/n}} \biggl[ 3(n+1) (\tilde\theta^')^2 + (5-n)\tilde\theta^{n+1} \biggr]
\biggl( \frac{\tilde\xi}{\tilde\theta^'} \biggr)^{(n+1)/n} \, .
</math>
  </td>
</tr>
</tr>
</table>
</div>
In an effort not to be caught dividing by zero while investigating the specific case of <math>~n=5</math> polytropes, we will adopt as shorthand notation,
<div align="center">
<math>\mathfrak{b}_n \equiv \biggl[ (4\pi)^{1/n} (5-n)\mathcal{B}\biggr]
= \biggl[ (n+1) (\tilde\theta^')^2 + \biggl( \frac{5-n}{3} \biggr)\tilde\theta^{n+1} \biggr]
\biggl( \frac{\tilde\xi}{\tilde\theta^'} \biggr)^{(n+1)/n} \, .
</math>
</div>
Hence, the replacements become,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="right">
   <td align="right">0.5750</td>
<math>~\frac{R_\mathrm{eq}}{R_\mathrm{mod}} </math>
   <td align="center">&nbsp;</td>
  </td>
   <td align="left">1.52651</td>
   <td align="center">
<math>~~~ \rightarrow ~~~</math>
  </td>
   <td align="left">
<math>~\mathcal{X} \biggl( \frac{4\pi}{3} \biggr)^{n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{1/2}
(5-n)^{(n-1)/[2(n+1)]} (4\pi)^{1/(n+1)} \mathfrak{b}_n^{-n/(n+1)}
</math>
  </td>
</tr>
</tr>
<tr>
<tr>
   <td align="right">
   <td align="right">0.5800</td>
&nbsp;
   <td align="center">&nbsp;</td>
  </td>
   <td align="left">1.745165</td>
   <td align="center">
<math>~=</math>
  </td>
   <td align="left">
<math>~\mathcal{X} (5-n)^{(n-1)/[2(n+1)]} \biggl( \frac{n+1}{n} \biggr)^{1/2} (4\pi)^{1/2} \cdot 3^{(1-n)/[2(n+1)]} \cdot
\mathfrak{b}_n^{-n/(n+1)} \, ,
</math>
  </td>
</tr>
</tr>
<tr>
<tr>
   <td align="right">
   <td align="right"><font color="darkgreen">0.5803836</font></td>
<math>~\frac{M_\mathrm{tot}}{M_\mathrm{mod}} </math>
   <td align="center">&nbsp;</td>
   </td>
   <td align="left"><font color="darkgreen">1.823995</font></td>
   <td align="center">
</tr>
<math>~~~ \rightarrow ~~~</math>
<tr>
   </td>
  <td align="right">0.5800</td>
   <td align="left">
   <td align="center">&nbsp;</td>
<math>~\mathcal{Y} \biggl( \frac{4\pi}{3} \biggr)^{2n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{3/2}
   <td align="left">1.90201</td>
(5-n)^{(n-3)/[2(n+1)]} (4\pi)^{2/(n+1)} \mathfrak{b}_n^{-2n/(n+1)} 
</tr>
</math>
<tr>
   </td>
  <td align="right">0.5780</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.01647</td>
</tr>
<tr>
  <td align="right">0.5750</td>
  <td align="center">&nbsp;</td>
   <td align="left">2.11019</td>
</tr>
</tr>
<tr>
<tr>
   <td align="right">
   <td align="right">0.5600</td>
&nbsp;
   <td align="center">&nbsp;</td>
  </td>
   <td align="left">2.35906</td>
   <td align="center">
<math>~=</math>
  </td>
   <td align="left">
<math>~\mathcal{Y} (5-n)^{(n-3)/[2(n+1)]} \biggl( \frac{n+1}{n} \biggr)^{3/2} (4\pi)^{1/2} \cdot 3^{(3-n)/[2(n+1)]} \cdot \mathfrak{b}_n^{-2n/(n+1)}  \, ,
</math>
  </td>
</tr>
</tr>
</table>
</div>
and, in particular, the cross term in the virial theorem expression becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="right">
   <td align="right">0.5400</td>
<math>~\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{mod}}  \biggr)^{(n-3)/n} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{mod}} \biggr)^{(n+1)/n}</math>
  <td align="center">&nbsp;</td>
   </td>
  <td align="left">2.543602</td>
   <td align="center">
</tr>
<math>~=</math>
<tr>
   </td>
   <td align="right">0.5000</td>
   <td align="left">
   <td align="center">&nbsp;</td>
<math>~\mathcal{X}^{(n-3)/n} \mathcal{Y}^{(n+1)/n} \biggl( \frac{5-n}{3} \biggr)^{(n-3)/(n+1)} \biggl( \frac{n+1}{n} \biggr)^{2} (4\pi)^{(n-1)/n}
  <td align="left">2.75557</td>
\cdot \mathfrak{b}_n^{(1-3n)/(n+1)}  \, .
</tr>
</math>
<tr>
   </td>
  <td align="right">0.4500</td>
   <td align="center">&nbsp;</td>
   <td align="left">2.543602</td>
</tr>
<tr>
  <td align="right">0.3800</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.94829</td>
</tr>
<tr>
  <td align="right" bgcolor="yellow"><font color="black">0.3749583</font></td>
  <td align="center">&nbsp;</td>
  <td align="left" bgcolor="yellow"><font color="black">2.948526</font></td>
</tr>
<tr>
  <td align="right">0.3300</td>
  <td align="center">&nbsp;</td>
   <td align="left">2.93161</td>
</tr>
<tr>
  <td align="right">0.2500</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.829401</td>
</tr>
</tr>
</table>
</div>
Via these replacements the [[User:Tohline/SSC/Virial/PolytropesSummary#ConciseVirialMR|concise and general Virial Theorem expression]] derived above morphs into the,
<div align="center" id="ConciseVirialXY">
<font color="#770000">'''Virial Theorem written in terms of <math>~\mathcal{X}</math>, <math>~\mathcal{Y}</math>, and <math>~\mathfrak{b}_n</math>'''</font><br />
<math>
k_\xi \biggl\{
\mathcal{X}^4 \biggl[\frac{4\pi (5-n)}{3} \biggr] - \mathcal{X}^{(n-3)/n} \mathcal{Y}^{(n+1)/n}  (4\pi)^{-1/n} \mathfrak{b}_n + \mathcal{Y}^2 \biggl(\frac{n+1}{n}\biggr)
\biggr\}  = 0 \, ,
</math>
</div>
where the leading coefficient is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="right">
   <td align="right">0.1500</td>
<math>~k_\xi </math>
   <td align="center">&nbsp;</td>
  </td>
   <td align="left">2.578605</td>
   <td align="center">
<math>~\equiv</math>
  </td>
   <td align="left">
<math>~ 4\pi
\biggl( \frac{5-n}{3} \biggr)^{(n-3)/(n+1)} \biggl(\frac{n+1}{n} \biggr)^2 \mathfrak{b}_n^{-4n/(n+1)}  \, .
</math>
  </td>
</tr>
</tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
</table>
</table>
</div>
</td>
 
<td align="center">
In carrying out this last derivation we could be accused of reinventing the wheel, as the expression inside the curly braces is simply <math>~(5-n)</math> times the [[User:Tohline/SSC/Virial/PolytropesSummary#CompactStahlerVirial|virial expression presented inside an outlined box, above]], just before we introduced the modified normalization parameters, <math>~R_\mathrm{mod}</math> and <math>~M_\mathrm{mod}</math>.
 
===Relating and Reconciling Two Mass-Radius Relationships for n = 5 Polytropes===
 
Now, let's examine the case of pressure-truncated, <math>~n=5</math> polytropes.  As we have discussed in the context of [[User:Tohline/SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D5.29|detailed force-balanced models]], [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)] has deduced that all <math>~n=5</math> equilibrium configurations obey the mass-radius relationship,
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="right">
   <td align="center"><math>~ \frac{R}{R_\mathrm{SWS}}</math></td>
<math>~\biggl( \frac{M}{M_\mathrm{SWS}} \biggr)^2 - 5 \biggl( \frac{M}{M_\mathrm{SWS}} \biggr)\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)
  <td align="center">&nbsp;</td>
+ \frac{2^2 \cdot 5 \pi}{3} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^4 </math>
  <td align="center"><math>~ \frac{M}{M_\mathrm{SWS}}</math></td>
  </td>
</tr>
   <td align="center">
<tr>
<math>~=</math>
   <td align="right">0.1000</td>
   </td>
   <td align="center">&nbsp;</td>
   <td align="left">
   <td align="left">0.004224</td>
<math>~0 \, ,</math>
  </td>
</tr>
</tr>
</table>
</div>
where, as [[User:Tohline/SSC/Virial/PolytropesSummary#Detailed_Force-Balanced_Solution_2|reviewed above]], the mass and radius normalizations, <math>~M_\mathrm{SWS}</math> and <math>~R_\mathrm{SWS}</math>, may be treated as constants once the parameters <math>~K</math> and <math>~P_e</math> are specified.  In contrast to this, the mass-radius relationship that we have just derived ''from the virial theorem'' for pressure-truncated, <math>~n=5</math> polytropes is,
<div align="center">
<math>
\biggl( \frac{M_\mathrm{tot}}{M_\mathrm{mod}} \biggr)^2
- \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{mod}} \biggr)^{2/5} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{mod}} \biggr)^{6/5}
+ \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{mod}} \biggr)^4 = 0 \, ,
</math>
</div>
where the mass and radius normalizations,
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="right">
   <td align="right">0.2000</td>
<math>~M_\mathrm{mod}\biggr|_{n=5}</math>
   <td align="center">&nbsp;</td>
  </td>
   <td align="left">0.034709</td>
   <td align="center">
<math>~=</math>
  </td>
   <td align="left">
<math>~M_\mathrm{SWS} \biggl( \frac{3\mathcal{B}}{4\pi} \biggr)^{5/3} \biggl[ \frac{2\cdot 5\pi}{3^2 \mathcal{A}} \biggr]^{3/2}
\, ,</math>
  </td>
</tr>
</tr>
<tr>
<tr>
   <td align="right">
   <td align="right">0.3000</td>
<math>~R_\mathrm{mod}\biggr|_{n=5}</math>
   <td align="center">&nbsp;</td>
   </td>
   <td align="left">0.1230901</td>
   <td align="center">
</tr>
<math>~=</math>
<tr>
   </td>
   <td align="right">0.4000</td>
   <td align="left">
   <td align="center">&nbsp;</td>
<math>~R_\mathrm{SWS} \biggl( \frac{3\mathcal{B}}{4\pi}\biggr)^{5/6} \biggl[ \frac{2\cdot 5\pi}{3^2\mathcal{A}} \biggr]^{1/2} \, ,</math>
   <td align="left">0.31735</td>
   </td>
</tr>
</tr>
</table>
depend, not only on <math>~K</math> and <math>~P_e</math> via the definitions of <math>~M_\mathrm{SWS}</math> and <math>~R_\mathrm{SWS}</math>, but also on the structural form factors via the free-energy coefficients, <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>.  While these two separate mass-radius relationships are similar, they are not identical.  In particular, the middle term involving the cross-product of the mass and radius contains different exponents in the two expressions.  It is not immediately obvious how the two different polynomial expressions can be used to describe the same physical sequence.
This apparent discrepancy is reconciled as follows:  The structural form factors &#8212; and, hence, the free-energy coefficients &#8212; vary from equilibrium configuration to equilibrium configuration.  So it does not make sense to discuss ''evolution along the sequence'' that is defined by the second of the two polynomial expressions.  If you want to know how a given system's equilibrium radius will change ''as its mass changes'', the first of the two polynomials will do the trick.  However, the equilibrium radius of ''a given system'' can be found by looking for extrema in the free-energy function while holding the free-energy coefficients, <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>, constant; more importantly, the relative stability ''of a given equilibrium system'' can be determined by analyzing the behavior of the system's free energy ''while holding the free-energy coefficients constant''.  Dynamically stable versus dynamically unstable configurations can be readily distinguished from one another along the sequence that is defined by the second polynomial expression; they cannot be readily distinguished from one another along the sequence that is defined by the first polynomial expression.  It is useful, therefore, to determine how to map a configuration's position on one of the sequences to the other.
====Plotting Stahler's Relation====
[[File:CorrectedStahlerN5.png|thumb|300px|Pressure vs. pressure plot]]Switching, again, to the shorthand notation,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="right">
   <td align="right">0.4500</td>
<math>~\mathcal{X}</math>
   <td align="center">&nbsp;</td>
  </td>
   <td align="left">0.48177</td>
   <td align="center">
<math>~\equiv</math>
  </td>
   <td align="left">
<math>~\frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \, ,</math>
  </td>
</tr>
</tr>
<tr>
<tr>
   <td align="right">
   <td align="right">0.5000</td>
<math>~\mathcal{Y}</math>
   <td align="center">&nbsp;</td>
  </td>
   <td align="left">0.72493</td>
   <td align="center">
<math>~\equiv</math>
  </td>
   <td align="left">
<math>~\frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \, ,</math>
  </td>
</tr>
</tr>
</table>
</div>
the equilibrium mass-radius relation defined by the first of the two polynomial expressions can be plotted straightforwardly in either of two ways.  One way is to recognize that the polynomial is a quadratic equation whose solution is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="right">
   <td align="right">0.5250</td>
<math>~\mathcal{Y}</math>
  <td align="center">&nbsp;</td>
   </td>
   <td align="left">0.89686</td>
   <td align="center">
</tr>
<math>~=</math>
<tr>
   </td>
  <td align="right">0.5400</td>
   <td align="left">
   <td align="center">&nbsp;</td>
<math>~\frac{5}{2} \mathcal{X} \biggl\{ 1 \pm \biggl[ 1 - \biggl( \frac{2^4\cdot \pi}{3\cdot 5} \biggr) \mathcal{X}^2 \biggr]^{1/2} \biggr\} \, .</math>
  <td align="left">1.028495</td>
   </td>
</tr>
<tr>
  <td align="right">0.5500</td>
   <td align="center">&nbsp;</td>
   <td align="left">1.13574</td>
</tr>
<tr>
  <td align="right">0.5600</td>
  <td align="center">&nbsp;</td>
  <td align="left">1.26965</td>
</tr>
<tr>
  <td align="right">0.5730</td>
   <td align="center">&nbsp;</td>
  <td align="left">1.55527</td>
</tr>
</tr>
</table>
</div>
In the figure shown here on the right &#8212; see also the bottom panel of [[User:Tohline/SSC/Structure/PolytropesEmbedded#Stahler1983Fig17|Figure 2 in our accompanying discussion of detailed force-balance models]] &#8212; Stahler's mass-radius relation has been plotted using the solution to this quadratic equation; the green segment of the displayed curve was derived from the ''positive'' root while the segment derived from the ''negative'' root is shown in orange.  The two curve segments meet at the maximum value of the normalized equilibrium radius, namely, at
<div align="center">
<math>\mathcal{X}_\mathrm{max} \equiv \biggl[ \frac{3\cdot 5}{2^4 \pi} \biggr]^{1/2} \approx 0.54627 \, .</math>
</div>
We note that, when <math>~\mathcal{X} = \mathcal{X}_\mathrm{max}</math>, <math>~\mathcal{Y} = (5\mathcal{X}_\mathrm{max}/2) \approx 1.36569</math>.  Along the entire sequence, the maximum value of <math>~\mathcal{Y}</math> occurs at the location where <math>~d\mathcal{Y}/d\mathcal{X} = 0</math> along the segment of the curve corresponding to the ''positive'' root.  This occurs along the upper segment of the curve where <math>~\mathcal{X}/\mathcal{X}_\mathrm{max} = \sqrt{3}/2</math>, at the location,
<div align="center">
<math>\mathcal{Y}_\mathrm{max} \equiv \biggl[ \frac{3^3 \cdot 5^2}{2^6 } \biggr]^{1/2} \mathcal{X}_\mathrm{max}
= \biggl[ \frac{3^4 \cdot 5^3}{2^{10} \pi } \biggr]^{1/2}  \approx 1.77408 \, .</math>
</div>
The other way is to determine the normalized mass and normalized radius individually through Stahler's pair of parametric relations.  Drawing partly from our [[User:Tohline/SSC/Virial/PolytropesSummary#Detailed_Force-Balanced_Solution_2|above discussion]] and partly from a separate discussion where we provide a [[User:Tohline/SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D5.29|tabular summary of the properties of pressure-truncated <math>~n=5</math> polytropes]], these are,
<div align="center">
<table border="0" cellpadding="3">
<tr>
<tr>
   <td align="right">
   <td align="right"><font color="darkgreen">0.5756189</font></td>
<math>
  <td align="center">&nbsp;</td>
~\mathcal{X}\biggr|_{n=5}
  <td align="left"><font color="darkgreen">1.750930</font></td>
</math>
</tr>
   </td>
<tr>
   <td align="center">
   <td align="right">0.5730</td>
<math>~=~</math>
   <td align="center">&nbsp;</td>
   </td>
  <td align="left">1.93949</td>
   <td align="left">
</tr>
<math>
<tr>
\biggl( \frac{5}{4\pi} \biggr)^{1/2} \tilde\xi \tilde\theta^{2} =
  <td align="right">0.5600</td>
\biggl\{ \frac{3\cdot 5}{2^2 \pi} \biggl[ \frac{\tilde\xi^2/3}{(1+\tilde\xi^2/3)^{2}} \biggr] \biggr\}^{1/2} \, ,
   <td align="center">&nbsp;</td>
</math>
   <td align="left">2.18983</td>
   </td>
</tr>
<tr>
  <td align="right">0.5400</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.376318</td>
</tr>
<tr>
  <td align="right">0.5250</td>
  <td align="center">&nbsp;</td>
   <td align="left">2.46661</td>
</tr>
</tr>
<tr>
<tr>
   <td align="right">
   <td align="right">0.5000</td>
<math>
  <td align="center">&nbsp;</td>
~\mathcal{Y}\biggr|_{n=5}
   <td align="left">2.56895</td>
</math>
</tr>
  </td>
<tr>
   <td align="center">
   <td align="right">0.4600</td>
<math>~=~</math>
   <td align="center">&nbsp;</td>
   </td>
  <td align="left">2.657809</td>
   <td align="left">
<math>
\biggl( \frac{5^3}{4\pi} \biggr)^{1/2} \tilde\theta (- \tilde\xi^2 \tilde\theta^') =
\biggl[  \biggl( \frac{3 \cdot 5^3}{2^2\pi} \biggr) \frac{(\tilde\xi^2/3)^3}{(1+\tilde\xi^2/3)^{4}} \biggr]^{1/2} \, .
</math>
  </td>
</tr>
</tr>
</table>
</div>
The entire sequence will be traversed by varying the Lane-Emden parameter, <math>~\tilde\xi</math>, from zero to infinity.  Using the first of these two expressions, we have determined, for example, that the point along the sequence corresponding to the maximum normalized equilibrium radius, <math>~\mathcal{X}_\mathrm{max}</math>, is associated with an embedded <math>~n=5</math> polytrope whose truncated, dimensionless Lane-Emden radius is,
<div align="center">
<math>
~\tilde\xi \biggr|_{\mathcal{X}_\mathrm{max}} = \frac{1}{5^{1/2}} \biggl[ 2^5\pi - 15 + 2^3\pi^{1/2}(2^4\pi-15)^{1/2} \biggr]^{1/2}
\approx 5.8264 \, .
</math>
</div>
Similarly, we have determined that the point along the sequence that corresponds to the maximum dimensionless mass, <math>~\mathcal{Y}_\mathrm{max}</math>, is associated with an embedded <math>~n=5</math> polytrope whose truncated, dimensionless Lane-Emden radius is, precisely,
<div align="center">
<math>
~\tilde\xi \biggr|_{\mathcal{Y}_\mathrm{max}} = 3 \, .
</math>
</div>
====Plotting the Virial Theorem Relation====
The relevant relation is obtained by plugging <math>~n = 5</math> into the [[User:Tohline/SSC/Virial/PolytropesSummary#ConciseVirialXY|general mass-radius relation derived above]], repeated here for clarity:
<table border="1" cellpadding="8" align="center">
<tr><td>
<div align="center">
<math>
\mathcal{X}^4 \biggl[\frac{4\pi (5-n)}{3} \biggr] - \mathcal{X}^{(n-3)/n} \mathcal{Y}^{(n+1)/n}  (4\pi)^{-1/n} \mathfrak{b}_n + \mathcal{Y}^2 \biggl(\frac{n+1}{n}\biggr)
= 0
</math>
</div>
where,
<div align="center">
<math>\mathfrak{b}_n = \biggl[ (n+1) (-\tilde\theta^')^2 + \biggl( \frac{5-n}{3} \biggr)\tilde\theta^{n+1} \biggr]
\biggl( \frac{\tilde\xi}{-\tilde\theta^'} \biggr)^{(n+1)/n}
</math>
</div>
</td></tr>
</table>
We will begin by plugging <math>~n = 5</math> into these expressions everywhere except for the coefficient <math>~(5-n)</math>, which we will leave unresolved, for the time being, in order to better appreciate the interplay of various terms.  We obtain,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<tr>
   <td align="right">
   <td align="right" bgcolor="yellow"><font color="black">0.41184646</font></td>
<math>~
   <td align="center">&nbsp;</td>
\frac{6}{5} \mathcal{Y}^2 - \biggl( \frac{\mathfrak{b}_I^5 }{4\pi} \cdot \mathcal{X}^{2} \mathcal{Y}^{6}\biggr)^{1/5} 
   <td align="left" bgcolor="yellow"><font color="black">2.688999</font></td>
</math>
</tr>
   </td>
<tr>
   <td align="center">
   <td align="right">0.4100</td>
<math>~=</math>
  <td align="center">&nbsp;</td>
  </td>
   <td align="left">2.68895</td>
   <td align="left">
<math>~
(n-5)\biggl[ \biggl(\frac{4\pi}{3} \biggr) \mathcal{X}^4 -  \biggl( \frac{\mathfrak{b}_{II}^5}{4\pi} \cdot \mathcal{X}^{2} \mathcal{Y}^{6}\biggr)^{1/5} \biggr] \, ,
</math>
   </td>
</tr>
</tr>
</table>
</div>
where, if they are to be assigned values that are actually associated with a particular detailed force-balance model having truncation radius, <math>~\tilde\xi</math>,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<tr>
   <td align="right">
   <td align="right">0.3800</td>
<math>~
   <td align="center">&nbsp;</td>
\mathfrak{b}_I^5 
   <td align="left">2.677703</td>
</math>
  </td>
   <td align="center">
<math>~\equiv</math>
  </td>
   <td align="left">
<math>~
\biggl[ (n+1) (-\tilde\theta^')^2 \biggl( \frac{\tilde\xi}{-\tilde\theta^'} \biggr)^{(n+1)/n} \biggr]^5_{n=5}
= 2^5\cdot 3^5 \biggl[ (-\tilde\theta^')^4 {\tilde\xi}^6 \biggr] 
= 2^5 \cdot 3 \cdot \tilde\xi^{10} (1+\tilde\xi^2/3)^{-6}\, ,
</math>
  </td>
</tr>
</tr>
<tr>
<tr>
   <td align="right">
   <td align="right">0.3000</td>
<math>~
   <td align="center">&nbsp;</td>
\mathfrak{b}_{II}^5
   <td align="left">2.56612</td>
</math>
  </td>
   <td align="center">
<math>~\equiv</math>
  </td>
   <td align="left">
<math>~
\biggl[ \frac{1}{3} ~ \tilde\theta^{n+1}  \biggl( \frac{\tilde\xi}{-\tilde\theta^'} \biggr)^{(n+1)/n} \biggr]^5_{n=5}
= \frac{1}{3^5} ~ \tilde\theta^{30}  \biggl( \frac{\tilde\xi}{-\tilde\theta^'} \biggr)^{6}
= 3(1+\tilde\xi^2/3)^{-6} \, .
</math>
  </td>
</tr>
</tr>
</table>
</div>
Now, if we plug <math>~n=5</math> into the remaining unresolved <math>~(n-5)</math> coefficient, the righthand side goes to zero and the mass-radius relationship provided by the virial theorem becomes,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<tr>
   <td align="right">
   <td align="right">0.2500</td>
<math>~\frac{6}{5}  \mathcal{Y}^2 </math>
   <td align="center">&nbsp;</td>
   </td>
   <td align="left">2.44565</td>
   <td align="center">
</tr>
<math>~=</math>
<tr>
   </td>
   <td align="right">0.2000</td>
   <td align="left">
   <td align="center">&nbsp;</td>
<math>\biggl( \frac{\mathfrak{b}_I^5 }{4\pi} \cdot \mathcal{X}^{2} \mathcal{Y}^{6}\biggr)^{1/5}</math>
   <td align="left">2.28789</td>
   </td>
</tr>
</tr>
<tr>
<tr>
   <td align="right">
   <td align="right">0.1500</td>
<math>~\Rightarrow ~~~~ \mathcal{Y}^4 </math>
   <td align="center">&nbsp;</td>
  </td>
   <td align="left">2.08747</td>
   <td align="center">
<math>~=</math>
  </td>
   <td align="left">
<math>\mathcal{X}^{2} \biggl[ \biggl( \frac{5^5}{2^5\cdot 3^5}\biggr) \frac{\mathfrak{b}^5_{I}}{4\pi} \biggr] 
</math>
  </td>
</tr>
</tr>
<tr>
<tr>
   <td align="right">
   <td align="right">0.1000</td>
<math>~\Rightarrow ~~~~ \mathcal{Y} </math>
   <td align="center">&nbsp;</td>
  </td>
   <td align="left">1.82708</td>
   <td align="center">
<math>~=</math>
  </td>
   <td align="left">
<math>\mathcal{X}^{1/2} \biggl( \frac{5^5 \mathfrak{b}^5_{I}}{2^7\cdot 3^5 \pi}\biggr)^{1/4} \, . </math>
  </td>
</tr>
</tr>
</table>
</table>
</div>
</td>
Hence, for a given value of the structural form factor(s) &#8212; which implies a specific value of the constant coefficient, <math>~\mathfrak{b}_I</math> &#8212; the scalar virial theorem defines a relationship where the normalized mass <math>~(\mathcal{Y})</math> varies as the square root of the normalized radius <math>~(\mathcal{X})</math>.  On the other hand, if we demand that the expression inside the square brackets on the righthand side of the virial theorem relation go to zero on its own &#8212; without relying on the leading coefficient to knock it zero &#8212; the mass-radius relationship provided by the virial theorem becomes,
<td align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5">
<tr>
<tr>
   <td align="right">
   <td align="center"><math>~ \frac{R}{R_\mathrm{SWS}}</math></td>
<math>~\frac{4\pi}{3} ~ \mathcal{X}^4  </math>
   <td align="center">&nbsp;</td>
  </td>
   <td align="center"><math>~ \frac{M}{M_\mathrm{SWS}}</math></td>
   <td align="center">
<math>~=</math>
  </td>
   <td align="left">
<math>\biggl( \frac{\mathfrak{b}_{II}^5}{4\pi} \cdot \mathcal{X}^{2} \mathcal{Y}^{6}\biggr)^{1/5} </math>
  </td>
</tr>
</tr>
<tr>
<tr>
   <td align="right">
   <td align="right">0.1000</td>
<math>~\Rightarrow ~~~~ \mathcal{Y}^6 </math>
   <td align="center">&nbsp;</td>
   </td>
   <td align="left">0.004224</td>
   <td align="center">
</tr>
<math>~=</math>
<tr>
   </td>
   <td align="right">0.2000</td>
   <td align="left">
   <td align="center">&nbsp;</td>
<math>\mathcal{X}^{18} \biggl[ \biggl( \frac{4\pi}{3}\biggr)^5 \frac{4\pi}{\mathfrak{b}^5_{II}} \biggr] 
  <td align="left">0.03471</td>
</math>
  </td>
</tr>
</tr>
<tr>
<tr>
   <td align="right">
   <td align="right">0.3000</td>
<math>~\Rightarrow ~~~~ \mathcal{Y} </math>
   <td align="center">&nbsp;</td>
  </td>
   <td align="left">0.123115</td>
   <td align="center">
<math>~=</math>
  </td>
   <td align="left">
<math>4\pi \mathcal{X}^{3} ( 3^5\mathfrak{b}^5_{II} )^{-1/6} \, . </math>
  </td>
</tr>
</tr>
</table>
</div>
<!-- COMMENT 1 .......
In order to graphically display the mass-radius relation derived from the virial theorem in the <math>~\mathcal{X}-\mathcal{Y}</math> plane, as desired, we must first write out the expressions for the free-energy coefficients.  After setting <math>~M_\mathrm{limit}/M_\mathrm{tot} = 1</math> in the [[User:Tohline/SSC/Virial/PolytropesSummary#Structural_Form_Factors|above summary expressions]], we obtain for all polytropic indexes,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<tr>
   <td align="right">
   <td align="right">0.4000</td>
<math>~\mathcal{A}</math>
   <td align="center">&nbsp;</td>
  </td>
   <td align="left">0.31766</td>
   <td align="center">
<math>~=</math>
  </td>
   <td align="left">
<math>\frac{1}{5} \cdot \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2}  = \frac{1}{5-n} \, ,</math>
  </td>
</tr>
</tr>
<tr>
<tr>
   <td align="right">
   <td align="right">0.4500</td>
<math>~\mathcal{B}</math>
   <td align="center">&nbsp;</td>
  </td>
   <td align="left">0.48278</td>
   <td align="center">
<math>~\equiv</math>
  </td>
   <td align="left">
<math>
\biggl( \frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_M^{(n+1)/n}} =
\frac{1}{3(5-n) ( 4\pi )^{1/n}} \biggl[ 3(n+1) (\tilde\theta^')^2 + (5-n)\tilde\theta^{n+1} \biggr]
\biggl( \frac{\tilde\xi}{\tilde\theta^'} \biggr)^{(n+1)/n} \, .
</math>
  </td>
</tr>
</tr>
</table>
</div>
In an effort not to be caught dividing by zero while investigating the specific case of <math>~n=5</math> polytropes, we will use as shorthand notation,
<div align="center">
<math>\mathfrak{b}_5 \equiv \biggl[ (5-n)\mathcal{B}\biggr]_{n=5}
= \biggl[ \frac{2^3\cdot 3^5}{ \pi} \cdot (-\tilde\theta^')^{4} \tilde\xi^6 \biggr]^{1/5}
= \biggl[ \frac{2^3 \cdot 3}{\pi} \biggl(1 + \frac{\tilde\xi^2}{3} \biggr)^{-6} \tilde\xi^{10} \biggr]^{1/5}
\, ,</math>
</div>
where we have inserted the definition of <math>~\tilde\theta^'</math> as provided for <math>~n=5</math> polytropic structures in another section's [[User:Tohline/SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D5.29|summary table]].  For pressure-truncated <math>~n=5</math> polytropes, we therefore have,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="right">
   <td align="right">0.5000</td>
<math>~\frac{R_\mathrm{eq}}{R_\mathrm{mod}} = \mathcal{X} \cdot \frac{R_\mathrm{SWS}}{R_\mathrm{mod}}</math>
   <td align="center">&nbsp;</td>
  </td>
   <td align="left">0.72866</td>
   <td align="center">
<math>~=</math>
  </td>
   <td align="left">
<math>
~\mathcal{X} \cdot \biggl( \frac{4\pi}{3\mathcal{B}}\biggr)^{5/6} \biggl[ \frac{3^2\mathcal{A}}{2\cdot 5\pi} \biggr]^{1/2}
= (5-n)^{1/3} \mathcal{X} \cdot \biggl( \frac{4\pi}{3\mathfrak{b}_5}\biggr)^{5/6} \biggl[ \frac{3^2}{2\cdot 5\pi} \biggr]^{1/2}
</math>
  </td>
</tr>
</tr>
<tr>
<tr>
   <td align="right">
   <td align="right">0.5250</td>
&nbsp;
   <td align="center">&nbsp;</td>
  </td>
   <td align="left">0.905006</td>
   <td align="center">
<math>~=</math>
  </td>
   <td align="left">
<math>
(5-n)^{1/3} \mathcal{X} \cdot \biggl( \frac{2^{7}\cdot 3\pi^2}{5^3} \biggr)^{1/6} \mathfrak{b}_5^{-5/6} \, ,
</math>
  </td>
</tr>
</tr>
<tr>
<tr>
   <td align="right">
   <td align="right">0.5400</td>
<math>~\frac{M_\mathrm{tot}}{M_\mathrm{mod}} = \mathcal{Y} \cdot \frac{M_\mathrm{SWS}}{M_\mathrm{mod}}</math>
   <td align="center">&nbsp;</td>
  </td>
   <td align="left">1.042907</td>
   <td align="center">
<math>~=</math>
  </td>
   <td align="left">
<math>
~ \mathcal{Y} \cdot \biggl( \frac{4\pi}{3\mathcal{B}} \biggr)^{5/3} \biggl[ \frac{3^2 \mathcal{A}}{2\cdot 5\pi} \biggr]^{3/2}
= (5-n)^{1/6}\mathcal{Y} \cdot \biggl( \frac{4\pi}{3\mathfrak{b}_5} \biggr)^{5/3} \biggl[ \frac{3^2}{2\cdot 5\pi} \biggr]^{3/2}
</math>
  </td>
</tr>
</tr>
<tr>
<tr>
   <td align="right">
   <td align="right">0.5500</td>
&nbsp;
   <td align="center">&nbsp;</td>
  </td>
   <td align="left">1.15886</td>
   <td align="center">
<math>~=</math>
  </td>
   <td align="left">
<math>
(5-n)^{1/6}\mathcal{Y} \cdot \biggl( \frac{2^{11}\cdot 3^8 \pi}{5^9} \biggr)^{1/6}  \mathfrak{b}_5^{-5/3} \, .
</math>
  </td>
</tr>
</tr>
</table>
</div>
So, the cross term that appears in the mass-radius relation obtained from the virial theorem may be written as,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="right">
   <td align="right">0.5600</td>
<math>~\biggl(\frac{R_\mathrm{eq}}{R_\mathrm{mod}}\biggr)^{2/5}\biggl(\frac{M_\mathrm{tot}}{M_\mathrm{mod}}\biggr)^{6/5} </math>
   <td align="center">&nbsp;</td>
  </td>
   <td align="left">1.313712</td>
   <td align="center">
<math>~=</math>
  </td>
   <td align="left">
<math>~\biggl\{\biggl[ 
(5-n)^{1/3} \mathcal{X} \cdot \biggl( \frac{2^{7}\cdot 3\pi^2}{5^3} \biggr)^{1/6} \mathfrak{b}_5^{-5/6}
\biggr]^2 \times \biggl[
(5-n)^{1/6}\mathcal{Y} \cdot \biggl( \frac{2^{11}\cdot 3^8 \pi}{5^9} \biggr)^{1/6}  \mathfrak{b}_5^{-5/3}
\biggr]^6 \biggr\}^{1/5}
</math>
  </td>
</tr>
</tr>
<tr>
<tr>
   <td align="right">
   <td align="right">0.5675</td>
&nbsp;
   <td align="center">&nbsp;</td>
  </td>
   <td align="left">1.511304</td>
   <td align="center">
<math>~=</math>
  </td>
   <td align="left">
<math>~(5-n)^{1/3} \mathcal{X}^{2/5} \mathcal{Y}^{6/5}
\biggl[ \biggl( \frac{2^{7}\cdot 3\pi^2}{5^3} \biggr)^{1/3} \mathfrak{b}_5^{-5/3} \cdot
\biggl( \frac{2^{11}\cdot 3^8 \pi}{5^9} \biggr)  \mathfrak{b}_5^{-10}
\biggr]^{1/5}
</math>
  </td>
</tr>
</tr>
<tr>
<tr>
   <td align="right">
   <td align="right"><font color="darkgreen">0.5692185</font></td>
&nbsp;
  <td align="center">&nbsp;</td>
   </td>
  <td align="left"><font color="darkgreen">1.657839</font></td>
   <td align="center">
</tr>
<math>~=</math>
<tr>
   </td>
  <td align="right">0.5675</td>
   <td align="left">
  <td align="center">&nbsp;</td>
<math>~(5-n)^{1/3} \mathcal{X}^{2/5} \mathcal{Y}^{6/5}
  <td align="left">1.798532</td>
\biggl( \frac{2^{8}\cdot 3^{5} \pi}{5^{6}} \biggr)^{1/3}
</tr>
\mathfrak{b}_5^{-7/3} \, ,
<tr>
</math>
   <td align="right">0.5600</td>
  </td>
   <td align="center">&nbsp;</td>
  <td align="left">1.97061</td>
</tr>
<tr>
  <td align="right">0.5400</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.17282</td>
</tr>
<tr>
  <td align="right">0.5250</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.25888</td>
</tr>
<tr>
  <td align="right">0.5000</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.34793</td>
</tr>
<tr>
  <td align="right">0.4600</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.410374</td>
</tr>
<tr>
  <td align="right" bgcolor="yellow"><font color="black">0.4391754</font></td>
   <td align="center">&nbsp;</td>
   <td align="left" bgcolor="yellow"><font color="black">2.417330</font></td>
</tr>
<tr>
  <td align="right">0.4000</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.396465</td>
</tr>
<tr>
  <td align="right">0.3000</td>
  <td align="center">&nbsp;</td>
  <td align="left">2.19848</td>
</tr>
<tr>
  <td align="right">0.2000</td>
  <td align="center">&nbsp;</td>
  <td align="left">1.84195</td>
</tr>
<tr>
  <td align="right">0.1000</td>
  <td align="center">&nbsp;</td>
  <td align="left">1.31421</td>
</tr>
<tr>
  <td align="right">0.0500</td>
  <td align="center">&nbsp;</td>
  <td align="left">0.930314</td>
</tr>
<tr>
  <td align="right">0.0200</td>
  <td align="center">&nbsp;</td>
  <td align="left">0.58847</td>
</tr>
<tr><td align="center" colspan="3">&nbsp;</td></tr>
</table>
</td>
 
</tr>
</tr>
</table>
</table>
</div>
</div>
and, in entirety, the virial theorem relation becomes,
 
<span id="IsothermalExplained">From a</span> [[User:Tohline/SSC/Virial/Isothermal#Bonnor.27s_.281956.29_Equivalent_Relation|free-energy analysis of isothermal spheres]], we have demonstrated that, when the structural form factors are all set to unity, the statement of virial equilibrium is,
<div align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
Line 2,788: Line 2,504:
   <td align="left">
   <td align="left">
<math>~
<math>~
(5-n)^{1/3}\biggl\{ \mathcal{Y}^2 \cdot \biggl( \frac{2^{11}\cdot 3^8 \pi}{5^9} \biggr)^{1/3} \mathfrak{b}_5^{-10/3}
\biggl( \frac{R}{R_\mathrm{SWS}} \biggr)^4
- \mathcal{X}^{2/5} \mathcal{Y}^{6/5} \biggl( \frac{2^{8}\cdot 3^{5} \pi}{5^{6}} \biggr)^{1/3} \mathfrak{b}_5^{-7/3}
- \frac{3}{4\pi} \biggr( \frac{M}{M_\mathrm{SWS}} \biggr)  \biggl( \frac{R}{R_\mathrm{SWS}} \biggr)
+ (5-n) \mathcal{X}^4 \cdot \biggl( \frac{2^{7}\cdot 3\pi^2}{5^3} \biggr)^{2/3} \mathfrak{b}_5^{-10/3} \biggr\} \, .
+ \frac{3}{20\pi} \biggl( \frac{M}{M_\mathrm{SWS}} \biggr)^2 \, ,
</math>
</math>
   </td>
   </td>
Line 2,796: Line 2,512:
</table>
</table>
</div>
</div>
 
where, in order to be consistent with the above polytropic normalizations, we have adopted the ''isothermal'' normalizations,
A nontrivial solution is obtained by requiring that the terms inside the curly braces sum to zero.  Noting that the third term must be set to zero, on its own, because it retains a leading factor of <math>~(5-n)</math>, the virial theorem relation becomes,
<div align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
Line 2,803: Line 2,518:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\mathcal{Y}^2 \cdot \biggl( \frac{2^{11}\cdot 3^8 \pi}{5^9} \biggr)^{1/3} \mathfrak{b}_5^{-10/3}</math>
<math>~M_\mathrm{SWS}\biggr|_\mathrm{isothermal} \equiv \biggl( \frac{c_s^8}{G^3 P_e} \biggr)^{1 / 2} \, ,</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp;
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\mathcal{X}^{2/5} \mathcal{Y}^{6/5} \biggl( \frac{2^{8}\cdot 3^{5} \pi}{5^{6}} \biggr)^{1/3} \mathfrak{b}_5^{-7/3}
<math>~R_\mathrm{SWS}\biggr|_\mathrm{isothermal} \equiv \biggl( \frac{c_s^4}{G P_e} \biggr)^{1 / 2} \, .</math>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</table>
</div>
</div>
This is a quadratic equation that can be readily solved to provide an analytic expression for the ''isothermal'' mass-radius relation; the relevant expression has already been [[#Isothermal|provided, above]].
====Confirmation====
Rewriting the [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#ConciseVirialMR|just-derived virial theorem expression]] in terms of Stahler's dimensionless radius and mass variables, written in the abbreviated form,
<div align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
Line 2,820: Line 2,539:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\Rightarrow~~~~
<math>~\mathcal{X}</math>
\mathcal{Y}^{2}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~\equiv</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\mathcal{X} \biggl( \frac{5\mathfrak{b}_5}{2 \cdot 3}  \biggr)^{5/2}  
<math>~\frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \, ,</math>
= \mathcal{X} \biggl( \frac{5^5}{2^2 \cdot 3^4 \pi} \biggr)^{1/2} \biggl(1 + \frac{\tilde\xi^2}{3} \biggr)^{-3} \tilde\xi^{5}  \, .
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
.......  END COMMENT -->
<!-- COMMENT 2 ........
Let's go back to the [[User:Tohline/SSC/Virial/PolytropesSummary#CompactStahlerVirial|earlier virial expression]] that still contains the <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math> coefficients and multiply through by <math>~5</math> so that the coefficient of the <math>~\mathcal{X}^4</math> term matches the coefficient found in Stahler's relation.  We have,
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~
<math>~\mathcal{Y}</math>
\biggl(\frac{2^2\cdot 5\pi}{3} \biggr) \mathcal{X}^4  -
5\mathcal{B} \mathcal{X}^{2/5} \mathcal{Y}^{6/5} 
+ 6\mathcal{A} \mathcal{Y}^{2}
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~\equiv</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~0 \, .
<math>~\frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \, ,</math>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</table>
</div>
</div>
Next, let's rewrite the other two terms so that they look more like the terms found in Stahler's expression.
we have,


<div align="center">
<div align="center">
Line 2,867: Line 2,569:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~
<math>~0</math>
\biggl(\frac{2^2\cdot 5\pi}{3} \biggr) \mathcal{X}^4  -
5\mathcal{X}\mathcal{Y} \biggl( \frac{\mathcal{B}^5 \mathcal{Y}}{\mathcal{X}^3} \biggr)^{1/5}
+ 6\mathcal{A} \mathcal{Y}^{2}
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 2,877: Line 2,575:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~0
<math>~\biggl[ \mathcal{X} \cdot \frac{R_\mathrm{SWS}}{R_\mathrm{mod}} \biggr]^4
</math>
- \biggl[ \mathcal{X} \cdot \frac{R_\mathrm{SWS}}{R_\mathrm{mod}} \biggr]^{(n-3)/n}
\biggl[ \mathcal{Y} \cdot \frac{M_\mathrm{SWS}}{M_\mathrm{mod}} \biggr]^{(n+1)/n}
+ \biggl[ \mathcal{Y} \cdot \frac{M_\mathrm{SWS}}{M_\mathrm{mod}} \biggr]^2</math>
   </td>
   </td>
</tr>
</tr>
Line 2,884: Line 2,584:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\Rightarrow~~~
&nbsp;
\biggl(\frac{2^2\cdot 5\pi}{3} \biggr) \mathcal{X}^4  -
5\mathcal{X}\mathcal{Y}
+ \mathcal{Y}^{2}
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 2,894: Line 2,590:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~\mathcal{X}^4 \biggl\{ \biggl( \frac{4\pi}{3} \biggr)^{n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{1/2} \frac{\mathcal{A}_{M_\ell}^{1/2}}{\mathcal{B}_{M_\ell}^{n/(n+1)}} \biggr\}^4
5\mathcal{X}\mathcal{Y} \biggl[\biggl( \frac{\mathcal{B}^5 \mathcal{Y}}{\mathcal{X}^3} \biggr)^{1/5}-1\biggr]
+ \mathcal{Y}^2 \biggl\{ \biggl( \frac{4\pi}{3} \biggr)^{2n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{3/2} \frac{\mathcal{A}_{M_\ell}^{3/2}}{\mathcal{B}_{M_\ell}^{2n/(n+1)}} \biggr\}^2</math>
+(1- 6\mathcal{A} )\mathcal{Y}^{2}
</math>
   </td>
   </td>
</tr>
</tr>
Line 2,906: Line 2,600:
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
&nbsp;
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~ \mathcal{Y} \biggl\{
<math>~
5\mathcal{X}\biggl[\biggl( \frac{\mathcal{B}^5 \mathcal{Y}}{\mathcal{X}^3} \biggr)^{1/5}-1\biggr]
- \mathcal{X}^{(n-3)/n} \mathcal{Y}^{(n+1)/n} \biggl\{ \biggl( \frac{4\pi}{3} \biggr)^{n/(n+1)}  
+(1- 6\mathcal{A} )\mathcal{Y} \biggr\}
\biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{1/2} \frac{\mathcal{A}_{M_\ell}^{1/2}}{\mathcal{B}_{M_\ell}^{n/(n+1)}}\biggr\}^{(n-3)/n}  
\biggl\{ \biggl( \frac{4\pi}{3} \biggr)^{2n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{3/2} \frac{\mathcal{A}_{M_\ell}^{3/2}}{\mathcal{B}_{M_\ell}^{2n/(n+1)}} \biggr\}^{(n+1)/n}  
</math>
</math>
   </td>
   </td>
Line 2,924: Line 2,619:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~ \mathcal{Y} \biggl\{
<math>~\mathcal{X}^4 \biggl\{ \biggl( \frac{4\pi}{3} \biggr)^{4n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{2}
5 \mathcal{B} (\mathcal{Y} \mathcal{X}^2 )^{1/5} - 5 \mathcal{X}
\frac{\mathcal{A}_{M_\ell}^{2}}{\mathcal{B}_{M_\ell}^{4n/(n+1)}} \biggr\}
+(1- 6\mathcal{A} )\mathcal{Y} \biggr\} \, .
+ \mathcal{Y}^2 \biggl\{ \biggl( \frac{4\pi}{3} \biggr)^{4n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{3} \frac{\mathcal{A}_{M_\ell}^{3}}{\mathcal{B}_{M_\ell}^{4n/(n+1)}} \biggr\}</math>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
 
</div>
<tr>
Now, according to Stahler's relation, the lefthand side of our derived expression should be zero when the chosen <math>~(\mathcal{X}, \mathcal{Y})</math> pair identifies an equilibrium configuration.  Therefore, the terms inside the curly brackets on the righthand side of our derived expression should also sum to zero in equilibrium.  Let's see if, indeed, this is the case; as shorthand, we will use,
<div align="center">
<math>\Lambda \equiv \frac{\tilde{\xi}^2}{3} \, .</math>
</div>
From the [[User:Tohline/SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D5.29|tabular summary of detailed force-balanced models]], we have,
<div align="center">
<table border="0" cellpadding="5" align="center">
 
<tr>
   <td align="right">
   <td align="right">
<math>~\tilde\theta</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
&nbsp;
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~  
<math>~
\frac{1}{(1+\Lambda)^{1/2}} \, ,
- \mathcal{X}^{(n-3)/n} \mathcal{Y}^{(n+1)/n} \biggl\{ \biggl( \frac{4\pi}{3} \biggr)^{[(n-3)+2(n+1)]/(n+1)}  
\biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{[(n-3)+3(n+1)]/2n} \frac{\mathcal{A}_{M_\ell}^{[(n-3)+3(n+1)]/2n}}{\mathcal{B}_{M_\ell}^{[(n-3)+2(n+1)]/(n+1)}}\biggr\}
</math>
</math>
   </td>
   </td>
Line 2,956: Line 2,642:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~-~\tilde\theta'</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 2,962: Line 2,648:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~ \frac{1}{3^{1/2}} \cdot
<math>~\biggl\{\mathcal{X}^4 + \biggl[\frac{3(n+1)}{4\pi n}\biggr] \mathcal{A}_{M_\ell} \mathcal{Y}^2 \biggr\}
\frac{\Lambda^{1/2}}{(1+\Lambda)^{3/2}
\biggl\{ \biggl( \frac{4\pi}{3} \biggr)^{4n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{2} \frac{\mathcal{A}_{M_\ell}^{2}}{\mathcal{B}_{M_\ell}^{4n/(n+1)}} \biggr\}</math>
~~~~\Rightarrow~~~~ \biggl( \frac{\tilde{\xi}}{-\tilde\theta'} \biggr) = 3(1+\Lambda)^{3/2} \, ,
</math>
   </td>
   </td>
</tr>
</tr>
Line 2,971: Line 2,655:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\mathcal{X}</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
&nbsp;
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\biggl\{ \frac{3\cdot 5}{2^2\pi} \biggl[
<math>~
\frac{\Lambda}{(1+\Lambda)^2} \biggr] \biggr\}^{1/2}
- \mathcal{X}^{(n-3)/n} \mathcal{Y}^{(n+1)/n} \biggl\{ \biggl( \frac{4\pi}{3} \biggr)^{(3n-1)/(n+1)}  
=
\biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{2} \frac{\mathcal{A}_{M_\ell}^{2}}{\mathcal{B}_{M_\ell}^{(3n-1)/(n+1)}}\biggr\}
\biggl( \frac{3\cdot 5}{2^2\pi} \biggr)^{1/2}  
\frac{\Lambda^{1/2}}{(1+\Lambda)} \, ,
</math>
</math>
   </td>
   </td>
Line 2,988: Line 2,670:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\mathcal{Y}</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 2,994: Line 2,676:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\biggl\{ \frac{3\cdot 5^3}{2^2\pi} \biggl[
<math>~
\frac{\Lambda^3}{(1+\Lambda)^4} \biggr] \biggr\}^{1/2}
\biggl[ \biggl(\frac{4\pi}{3}\biggr) \mathcal{X}^4 - \mathcal{B}_{M_\ell} \cdot \mathcal{X}^{(n-3)/n} \mathcal{Y}^{(n+1)/n}
=
+ \biggl(\frac{n+1}{n}\biggr) \mathcal{A}_{M_\ell} \mathcal{Y}^2 \biggr]  
5\biggl(\frac{3\cdot 5}{2^2\pi}\biggr)^{1/2}  
\biggl\{ \biggl( \frac{4\pi}{3} \biggr)^{(3n-1)/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{2} \frac{\mathcal{A}_{M_\ell}^{2}}{\mathcal{B}_{M_\ell}^{4n/(n+1)}} \biggr\} \, .
\frac{\Lambda^{3/2}}{(1+\Lambda)^2} \, ,
</math>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
Replacing <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math> with <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>, as prescribed by their defined relationships, the expression inside the square brackets becomes the [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#CompactStahlerVirial|above, boxed-in mass-radius relationship]], namely,
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\Rightarrow~~~ \mathcal{Y}\mathcal{X}^2</math>
<math>~
\frac{4\pi}{3} \cdot \mathcal{X}^4 
- \mathcal{B} \cdot \mathcal{X}^{(n-3)/n} \biggl[ \mathcal{Y} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{limit}} \biggr) \biggr]^{(n+1)/n} 
+~ \mathcal{A} \biggl( \frac{n+1}{n} \biggr) \biggl[ \mathcal{Y} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{limit}} \biggr) \biggr]^{2}
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 3,011: Line 2,702:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~0 \, .</math>
5\biggl(\frac{3\cdot 5}{2^2\pi}\biggr)^{3/2}
\frac{\Lambda^{5/2}}{(1+\Lambda)^4} \, ;
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</table>
</div>
</div>
and, from the general definitions given, above,
 
====In Terms of Structural Form-Factors====
 
Alternatively, replacing <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math> by their expressions in terms of the structural form factors gives,  
 
<div align="center">
<div align="center">
<table border="0" cellpadding="5">
<table border="0" cellpadding="5" align="center">
 
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\mathcal{A}</math>
<math>~
\frac{4\pi}{3} \cdot \mathcal{X}^4 
- \mathcal{X}^{(n-3)/n} \mathcal{Y}^{(n+1)/n} \biggl( \frac{4\pi}{3} \biggr)^{-1/n}
\frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_M^{(n+1)/n}}
+ \mathcal{Y}^2\biggl( \frac{n+1}{5n} \biggr) \frac{\tilde\mathfrak{f}_W  }{\tilde\mathfrak{f}_M^2}  
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 3,030: Line 2,728:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>\frac{1}{5-n} \, ,</math>
<math>~0 \, .</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
Finally, inserting into this relation the [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#PTtable|expressions presented above]] for the structural form-factors, <math>~\tilde\mathfrak{f}_M</math> and <math>~\tilde\mathfrak{f}_A</math>, namely,


<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\mathcal{B}</math>
<math>~\tilde\mathfrak{f}_M</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 3,042: Line 2,746:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>~ \biggl[ - \frac{3\Theta^'}{\xi} \biggr]_{\tilde\xi} </math>
\frac{1}{3(5-n) ( 4\pi )^{1/n}} \biggl[ 3(n+1) (\tilde\theta^')^2 + (5-n)\tilde\theta^{n+1} \biggr]
\biggl( \frac{\tilde\xi}{\tilde\theta^'} \biggr)^{(n+1)/n}
</math>
   </td>
   </td>
</tr>
</tr>
Line 3,051: Line 2,752:
<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>~
\tilde\mathfrak{f}_A
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 3,057: Line 2,760:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>~
( 4\pi )^{-1/5} \biggl[ \frac{2\cdot 3}{(5-n)} (\tilde\theta^')^2 + \frac{1}{3}\cdot \tilde\theta^{6} \biggr]
\tilde\theta^{n+1} + \biggl[ \frac{(n+1)}{3\cdot 5} \biggr] \tilde\xi^2 \cdot \tilde\mathfrak{f}_W
\biggl( \frac{\tilde\xi}{\tilde\theta^'} \biggr)^{6/5}  
</math>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
gives us the desired,
<div align="center" id="ConciseVirialXY">
<table border="1" cellpadding="8" align="center">
<tr><td align="center">
<font color="#770000">'''Virial Theorem written in terms of <math>~\mathcal{X}</math>, <math>~\mathcal{Y}</math>, and <math>~\tilde\mathfrak{f}_W</math>'''</font><br />
<math>~
4\pi \cdot \mathcal{X}^4  ~- ~
\mathcal{X}^{(n-3)/n} \mathcal{Y}^{(n+1)/n} ( 4\pi)^{-1/n}
\biggl[\frac{\tilde\xi}{(-\tilde\theta^')}\biggr]^{(n+1)/n} \biggl[\tilde\theta^{n+1} +  \frac{(n+1)\tilde\xi^2}{3\cdot 5} \cdot \tilde\mathfrak{f}_W \biggr]
~+ ~ \mathcal{Y}^2\biggl( \frac{n+1}{3\cdot 5n} \biggr) \frac{\tilde\xi^2}{(- \tilde\theta^')^2} \cdot \tilde\mathfrak{f}_W
= 0 \, .
</math>
</td></tr>
</table>
</div>
===Relating and Reconciling Two Mass-Radius Relationships for n = 5 Polytropes===
Now, let's examine the case of pressure-truncated, <math>~n=5</math> polytropes.  As we have discussed in the context of [[User:Tohline/SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D5.29|detailed force-balanced models]], [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)] has deduced that all <math>~n=5</math> equilibrium configurations obey the mass-radius relationship,
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>~\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^2 - 5 \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)
+ \frac{2^2 \cdot 5 \pi}{3} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^4 </math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 3,072: Line 2,800:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>~0 \, ,</math>
( 4\pi )^{-1/5} \biggl[ \frac{2}{(5-n)} \frac{\Lambda}{(1+\Lambda)^3} + \frac{1}{3}\cdot \frac{1}{(1+\Lambda)^3} \biggr]
3^{6/5} (1+\Lambda)^{9/5}
</math>
   </td>
   </td>
</tr>
</tr>
 
</table>
<tr>
</div>
  <td align="right">
where, as [[User:Tohline/SSC/Virial/PolytropesSummary#Detailed_Force-Balanced_Solution_2|reviewed above]], the mass and radius normalizations, <math>~M_\mathrm{SWS}</math> and <math>~R_\mathrm{SWS}</math>, may be treated as constants once the parameters <math>~K</math> and <math>~P_e</math> are specified.  In contrast to this, the mass-radius relationship that we have just derived ''from the virial theorem'' for pressure-truncated, <math>~n=5</math> polytropes is,
&nbsp;
<div align="center">
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
<math>
\biggl(\frac{3}{4\pi }\biggr)^{1/5} \biggl[ \frac{2\cdot 3}{(5-n)} \Lambda + 1 \biggr]
\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{mod}} \biggr)^2
(1+\Lambda)^{-6/5} \, .
- \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{mod}} \biggr)^{2/5} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{mod}} \biggr)^{6/5}  
+ \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{mod}} \biggr)^4 = 0 \, ,
</math>
</math>
  </td>
</tr>
</table>
</div>
</div>
Hence,
where the mass and radius normalizations,
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~5\mathcal{B} (\mathcal{Y}\mathcal{X}^2)^{1/5}</math>
<math>~M_\mathrm{mod}\biggr|_{n=5}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 3,107: Line 2,824:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~5
<math>~M_\mathrm{SWS} \biggl( \frac{3\mathcal{B}_{M_\ell}}{4\pi} \biggr)^{5/3} \biggl[ \frac{2\cdot 5\pi}{3^2 \mathcal{A}_{M_\ell}} \biggr]^{3/2}  
\biggl(\frac{3}{4\pi }\biggr)^{1/5} \biggl[ \frac{2\cdot 3}{(5-n)} \Lambda + 1 \biggr](1+\Lambda)^{-6/5}
\, ,</math>
\cdot
\biggl\{ 5\biggl(\frac{3\cdot 5}{2^2\pi}\biggr)^{3/2}  
\frac{\Lambda^{5/2}}{(1+\Lambda)^4} \biggr\}^{1/5}
</math>
   </td>
   </td>
</tr>
</tr>
Line 3,118: Line 2,831:
<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>~R_\mathrm{mod}\biggr|_{n=5}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 3,124: Line 2,837:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~R_\mathrm{SWS} \biggl( \frac{3\mathcal{B}_{M_\ell}}{4\pi}\biggr)^{5/6} \biggl[ \frac{2\cdot 5\pi}{3^2\mathcal{A}_{M_\ell}} \biggr]^{1/2} \, ,</math>
\biggl(\frac{3\cdot 5^3}{4\pi }\biggr)^{1/2} \biggl[ \frac{2\cdot 3}{(5-n)} \Lambda + 1 \biggr]\frac{\Lambda^{1/2}}{(1+\Lambda)^2} \, ;
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</table>
</div>
 
while,
depend, not only on <math>~K</math> and <math>~P_e</math> via the definitions of <math>~M_\mathrm{SWS}</math> and <math>~R_\mathrm{SWS}</math>, but also on the structural form factors via the free-energy coefficients, <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math>.  While these two separate mass-radius relationships are similar, they are not identical.  In particular, the middle term involving the cross-product of the mass and radius contains different exponents in the two expressions.  It is not immediately obvious how the two different polynomial expressions can be used to describe the same physical sequence.
 
This apparent discrepancy is reconciled as follows:  The structural form factors &#8212; and, hence, the free-energy coefficients &#8212; vary from equilibrium configuration to equilibrium configuration.  So it does not make sense to discuss ''evolution along the sequence'' that is defined by the second of the two polynomial expressions.  If you want to know how a given system's equilibrium radius will change ''as its mass changes'', the first of the two polynomials will do the trick.  However, the equilibrium radius of ''a given system'' can be found by looking for extrema in the free-energy function while holding the free-energy coefficients, <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math>, constant; more importantly, the relative stability ''of a given equilibrium system'' can be determined by analyzing the behavior of the system's free energy ''while holding the free-energy coefficients constant''.  Dynamically stable versus dynamically unstable configurations can be readily distinguished from one another along the sequence that is defined by the second polynomial expression; they cannot be readily distinguished from one another along the sequence that is defined by the first polynomial expression.  It is useful, therefore, to determine how to map a configuration's position on one of the sequences to the other.
 
====Plotting Stahler's Relation====
 
<span id="MRplot">[[File:CorrectedStahlerN5.png|thumb|300px|Mass (Y) vs. radius (X) plot]]Switching, again, to the shorthand notation,</span>
<div align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
Line 3,137: Line 2,854:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~(1-6\mathcal{A})\mathcal{Y} - 5\mathcal{X}</math>
<math>~\mathcal{X}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~\equiv</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~\frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \, ,</math>
\biggl[ 1- \frac{2\cdot 3}{(5-n)} \biggr]\biggl(\frac{3\cdot 5^3}{2^2\pi}\biggr)^{1/2}
\frac{\Lambda^{3/2}}{(1+\Lambda)^2}
- \biggl( \frac{3\cdot 5^3}{2^2\pi} \biggr)^{1/2}  
\frac{\Lambda^{1/2}}{(1+\Lambda)}
</math>
   </td>
   </td>
</tr>
</tr>
Line 3,154: Line 2,866:
<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>~\mathcal{Y}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~\equiv</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~\frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \, ,</math>
\biggl( \frac{3\cdot 5^3}{2^2\pi} \biggr)^{1/2} \biggl[ \Lambda - \frac{2\cdot 3 }{(5-n)} \Lambda - (1+\Lambda)\biggr]
\frac{\Lambda^{1/2}}{(1+\Lambda)^2}
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
the equilibrium mass-radius relation defined by the first of the two polynomial expressions can be plotted straightforwardly in either of two ways. 
=====Quadratic Equation=====
One way is to recognize that the polynomial is a quadratic equation whose solution is,
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>~\mathcal{Y}_\pm</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 3,175: Line 2,892:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~\frac{5}{2} \mathcal{X} \biggl\{ 1 \pm \biggl[ 1 - \biggl( \frac{2^4\cdot \pi}{3\cdot 5} \biggr) \mathcal{X}^2 \biggr]^{1/2} \biggr\} \, .</math>
- \biggl(\frac{3\cdot 5^3}{4\pi }\biggr)^{1/2}  \biggl[ \frac{2\cdot 3}{(5-n)} \Lambda + 1 \biggr]\frac{\Lambda^{1/2}}{(1+\Lambda)^2} \, .
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</table>
</div>
</div>
So we have the desired result, namely, that these last two expressions have opposite signs but are otherwise identical and, hence, they sum to zero.
In the figure shown here on the right &#8212; see also the bottom panel of [[User:Tohline/SSC/Structure/PolytropesEmbedded#Stahler1983Fig17|Figure 2 in our accompanying discussion of detailed force-balance models]] &#8212; Stahler's mass-radius relation has been plotted using the solution to this quadratic equation; the green segment of the displayed curve was derived from the ''positive'' root while the segment derived from the ''negative'' root is shown in orange.  The two curve segments meet at the maximum value of the normalized equilibrium radius, namely, at
........  END COMMENT 2 -->
<div align="center">
<math>\mathcal{X}_\mathrm{max} \equiv \biggl[ \frac{3\cdot 5}{2^4 \pi} \biggr]^{1/2} \approx 0.54627 \, .</math>
</div>
We note that, when <math>~\mathcal{X} = \mathcal{X}_\mathrm{max}</math>, <math>~\mathcal{Y} = (5\mathcal{X}_\mathrm{max}/2) \approx 1.36569</math>.  Along the entire sequence, the maximum value of <math>~\mathcal{Y}</math> occurs at the location where <math>~d\mathcal{Y}/d\mathcal{X} = 0</math> along the segment of the curve corresponding to the ''positive'' root. This occurs along the upper segment of the curve where <math>~\mathcal{X}/\mathcal{X}_\mathrm{max} = \sqrt{3}/2</math>, at the location,
<div align="center">
<math>\mathcal{Y}_\mathrm{max} \equiv \biggl[ \frac{3^3 \cdot 5^2}{2^6 } \biggr]^{1/2} \mathcal{X}_\mathrm{max}
= \biggl[ \frac{3^4 \cdot 5^3}{2^{10} \pi } \biggr]^{1/2}  \approx 1.77408 \, .</math>
</div>


===Relating and Reconciling Two Mass-Radius Relationships for n = 4 Polytropes===
=====Parametric Relations=====
 
The other way is to determine the normalized mass and normalized radius individually through Stahler's pair of parametric relationsDrawing partly from our [[User:Tohline/SSC/Virial/PolytropesSummary#Detailed_Force-Balanced_Solution_2|above discussion]] and partly from a separate discussion where we provide a [[User:Tohline/SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D5.29|tabular summary of the properties of pressure-truncated <math>~n=5</math> polytropes]], these are,
For pressure-truncated <math>~n=4</math> polytropes, [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)] did not identify a polynomial relationship between the mass and radius of equilibrium configurationsHowever, from his analysis of detailed force-balance models ([[User:Tohline/SSC/Virial/PolytropesSummary#Detailed_Force-Balanced_Solution_2|summarized above]]), we appreciate that the governing pair of parametric relations is,
<div align="center">
<div align="center">
<table border="0" cellpadding="3">
<table border="0" cellpadding="3">
Line 3,194: Line 2,915:
   <td align="right">
   <td align="right">
<math>
<math>
~\mathcal{X}
~\mathcal{X}\biggr|_{n=5}
</math>
</math>
   </td>
   </td>
Line 3,202: Line 2,923:
   <td align="left">
   <td align="left">
<math>
<math>
\biggl( \frac{1}{\pi} \biggr)^{1/2} \tilde\xi \tilde\theta^{3/2} \, ,
\biggl( \frac{5}{4\pi} \biggr)^{1/2} \tilde\xi \tilde\theta^{2} =
\biggl\{ \frac{3\cdot 5}{2^2 \pi} \biggl[ \frac{\tilde\xi^2/3}{(1+\tilde\xi^2/3)^{2}} \biggr] \biggr\}^{1/2} \, ,
</math>
</math>
   </td>
   </td>
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   <td align="right">
   <td align="right">
<math>
<math>
~\mathcal{Y}
~\mathcal{Y}\biggr|_{n=5}
</math>
</math>
   </td>
   </td>
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   <td align="left">
   <td align="left">
<math>
<math>
\biggl( \frac{2^4}{\pi} \biggr)^{1/2} \tilde\theta^{1/2} (- \tilde\xi^2 \tilde\theta^') \, .
\biggl( \frac{5^3}{4\pi} \biggr)^{1/2} \tilde\theta (- \tilde\xi^2 \tilde\theta^') =
\biggl[  \biggl( \frac{3 \cdot 5^3}{2^2\pi} \biggr) \frac{(\tilde\xi^2/3)^3}{(1+\tilde\xi^2/3)^{4}} \biggr]^{1/2} \, .
</math>
</math>
   </td>
   </td>
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</table>
</table>
</div>
</div>
On the other hand, the polynomial that results from plugging <math>~n=4</math> into the [[User:Tohline/SSC/Virial/PolytropesSummary#ConciseVirialXY|general mass-radius relation that is obtained via the virial theorem]] is,
The entire sequence will be traversed by varying the Lane-Emden parameter, <math>~\tilde\xi</math>, from zero to infinity.  Using the first of these two expressions, we have determined, for example, that the point along the sequence corresponding to the maximum normalized equilibrium radius, <math>~\mathcal{X}_\mathrm{max}</math>, is associated with an embedded <math>~n=5</math> polytrope whose truncated, dimensionless Lane-Emden radius is,
 
<div align="center">
<div align="center">
<math>
<math>
\frac{4\pi}{3} \mathcal{X}^4  - \biggl[ \frac{\mathcal{X} \mathcal{Y}^{5}}{4\pi}\biggr]^{1/4} \mathfrak{b}_{n=4} + \frac{5}{4} \mathcal{Y}^2
~\tilde\xi \biggr|_{\mathcal{X}_\mathrm{max}} = 3^{1/2} \, .
= 0 \, ,
</math>
</math>
</div>
</div>
where,
Similarly, we have determined that the point along the sequence that corresponds to the maximum dimensionless mass, <math>~\mathcal{Y}_\mathrm{max}</math>, is associated with an embedded <math>~n=5</math> polytrope whose truncated, dimensionless Lane-Emden radius is, precisely,
<div align="center">
<div align="center">
<math>\mathfrak{b}_{n=4} = \biggl[ 5 (-\tilde\theta^')^2 + \frac{1}{3} \tilde\theta^{5} \biggr]
<math>
\biggl( \frac{\tilde\xi}{-\tilde\theta^'} \biggr)^{5/4} \, .
~\tilde\xi \biggr|_{\mathcal{Y}_\mathrm{max}} = 3 \, .
</math>
</math>
</div>
</div>


[For the record we note that, throughout the structure of an <math>~n=4</math> polytrope, <math>~\mathfrak{b}_{n=4}</math> is a number of order unityIts value is never less than <math>~3^{1/4}</math>, which pertains to the center of the configuration; its maximum value of <math>\approx 5.098</math> occurs at <math>~\tilde\xi \approx 4.0</math>; and <math>~\mathfrak{b}_{n=4} \approx 3.946</math> at its (zero pressure) surface, <math>~\tilde\xi = \xi_1 \approx 14.97</math>.  A plot showing the variation with <math>~P_e</math> of the closely allied parameter, <math>~\mathcal{B}|_{n=4} = (4\pi)^{1/4} \mathfrak{b}_{n=4}</math> is presented in the righthand panel of the [[User:Tohline/SSC/Virial/PolytropesSummary#Summary|above parameter summary figure]].]
<div id="KimuraApplication">
<table border="1" width="90%" align="center" cellpadding="8">
<tr><td align="left">
Referring back to [[User:Tohline/SSC/Structure/PolytropesEmbedded#Turning_Points|our review of turning points]] along equilibrium sequences and, especially, the work of [http://adsabs.harvard.edu/abs/1981PASJ...33..299K Kimura (1981b)], we appreciate that the point that corresponds to the maximum mass, <math>~\mathcal{Y}_\mathrm{max}</math>, is the turning point that Kimura refers to as the "extremum in M<sub>1</sub>" along a p<sub>1</sub> sequence[[User:Tohline/SSC/Structure/PolytropesEmbedded#Location_of_Pressure_Limit|As we have highlighted]], according to Kimura, this point should occur along the sequence where <math>~h_G=0</math>, that is, where the following condition applies:
<div align="center">
<math>~\frac{\tilde\theta^{n+1}}{(\tilde\theta^')^2} = \frac{(n-3)}{2}  \, .</math>
</div>
For the specific case being studied here, namely, <math>~n = 5</math> polytropic configurations, we therefore expect from Kimura's work that <math>~[\tilde\theta^6/(\tilde\theta^')^2] = 1</math> at the "maximum mass" turning point.  Given that,
<div align="center">
<math>~\tilde\xi \biggr|_{\mathcal{Y}_\mathrm{max}} = 3</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>~\Rightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>\tilde\theta_{n=5} = \frac{1}{2}</math>&nbsp;&nbsp;&nbsp;&nbsp; and &nbsp;&nbsp;&nbsp;&nbsp; <math>\tilde\theta_{n=5}^' = -\frac{1}{8} \, ,</math>
</div>
we see that Kimura's condition holds and, hence, that our identification of the location along the sequence of the maximum mass matches Kimura's identification of the location of that turning point.


In both panels of the following figure, the blue curve displays the mass-radius relation for pressure-truncated <math>~n=4</math> polytropes, <math>~\mathcal{Y}(\mathcal{X})</math>, that is generated by Stahler's pair of parametric equations.  The coordinates of discrete points along the curve have been determined from the tabular data provided on p. 399 of [http://adsabs.harvard.edu/abs/1986Ap%26SS.126..357H Horedt (1986, ApJS, vol. 126)]] while Excel has been used to generate the "smooth," continuous blue curve connecting the points; this set of points and accompanying blue curve are identical in both figure panels.  In both figure panels, a set of discrete, triangle-shaped points traces the mass-radius relation, <math>~\mathcal{Y}(\mathcal{X})</math>, that is obtained via the virial theorem, assuming that the coefficient, <math>~\mathfrak{b}_{n=4}</math>, is constant along the sequenceThe "green" sequence in the lefthand panel results from setting <math>~\mathfrak{b}_{n=4} = 3.4205</math>, which is the value of the constant that results from Horedt's tabulated data if the configuration is truncated at <math>~\tilde\xi = 1.4</math>; the "orange" sequence in the righthand panel results from setting <math>~\mathfrak{b}_{n=4} = 4.8926</math>, which is the value of the constant that results from Horedt's tabulated data if the configuration is truncated at <math>~\tilde\xi = 2.8</math>.
We appreciate, as well, that the point corresponding to the maximum normalized equilibrium radius, <math>~\mathcal{X}_\mathrm{max}</math>, is the turning point that Kimura would reference as the "extremum in r<sub>1</sub>" along a p<sub>1</sub> sequence. Following Kimura's analysis [[User:Tohline/SSC/Structure/PolytropesEmbedded#TurningPointXmax|we have shown that this point occurs along the sequence where the following condition applies]]:
 
<div align="center">
<table border="1" cellpadding="8" align="center">
<math>~\frac{\xi (-\theta^')}{\tilde\theta} = \frac{2}{(n-1)} \, ,</math>
<tr>
</div>
  <th align="center">
that is, for the specific case being studied here, we should expect <math>~[\tilde\xi (-\tilde\theta^')/\tilde\theta] = 1/2</math> at the "maximum radius" turning pointGiven that,
Comparing Two Separate Mass-Radius Relations for Pressure-Truncated ''n = 4'' Polytropes
<div align="center">
  </th>
<math>~\tilde\xi \biggr|_{\mathcal{X}_\mathrm{max}} = 3^{1/2}</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</tr>
<math>~\Rightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<tr><td align="center">
<math>\tilde\theta_{n=5} = 2^{-1/2}</math>&nbsp;&nbsp;&nbsp;&nbsp; and &nbsp;&nbsp;&nbsp;&nbsp;
[[File:CompareN4SequencesRevised.png|750px|Comparison of Two Mass-Radius Relations]]
<math>\tilde\theta_{n=5}^' = -(2^3 \cdot 3)^{-1/2} \, ,</math>
</div>
we see that Kimura's condition holds and, hence, that our identification of the location along the sequence of the maximum radius matches Kimura's identification of the location of that turning point.
</td></tr>
</td></tr>
</table>
</table>
</div>


According to [http://adsabs.harvard.edu/abs/1986Ap%26SS.126..357H Horedt's (1986)]] tabulated data, the surface of an isolated <math>~(P_e = 0)</math>, spherically symmetric, <math>~n=4</math> polytrope occurs at the dimensionless (Lane-Emden) radius, <math>~\xi_1 = 14.9715463</math>.  In both panels of the above figure, this ''isolated'' configuration is identified by the discrete (blue diamond) point at the origin, that is, at <math>~(\mathcal{X}, \mathcal{Y}) = (0, 0)</math>.   As we begin to examine pressure-truncated models and <math>~\tilde\xi</math> is steadily decreased from <math>~\xi_1</math>, the mass-radius coordinate of equilibrium configurations "moves" away from the origin, upward along the upper branch of the displayed (blue) mass-radius relation.  A maximum mass of <math>~\mathcal{Y} \approx 2.042</math> (corresponding to a radius of <math>~\mathcal{X} \approx 0.4585</math>) is reached ''from the left'' as <math>~\tilde\xi</math> drops to a value of approximately <math>~3.4</math>.  As <math>~\tilde\xi</math> continues to decrease, the mass-radius coordinates of equilibrium configurations move along the lower branch of the displayed (blue) curve, reaching a maximum radius at <math>~(\mathcal{X}, \mathcal{Y}) \approx (0.555, 1.554)</math> &#8212; corresponding to <math>~\tilde\xi \approx 2.0</math> &#8212; then decreasing in radius until, once again, the origin is reached, but this time because <math>~\tilde\xi</math> drops to zero.
=====Discussion=====
Notice that if the quadratic equation is used to map out the mass-radius relationship, the parameter, <math>~\tilde\xi</math>, never explicitly enters the discussion.  Instead, a radius <math>~0 \le \mathcal{X} \le \mathcal{X}_\mathrm{max}</math> is specified and the ''two'' equilibrium masses associated with <math>~\mathcal{X}</math> &#8212; call them, <math>~\mathcal{Y}_+</math> and <math>~\mathcal{Y}_-</math> &#8212; are determined(The values of the two masses are degenerate at both limiting values of <math>~\mathcal{X}</math>.)  If the pair of parametric relations is used, instead, only ''one'' value of the mass is obtained for each specified value of <math>~\tilde\xi</math>. As <math>~\tilde\xi</math> is increased from <math>~0</math> to <math>~\sqrt{3}</math>, <math>~\mathcal{X}</math> increases monotonically from <math>~0</math> to <math>~\mathcal{X}_\mathrm{max}</math> and the corresponding mass is (only) <math>~\mathcal{Y}_-</math>; that is, as <math>~\tilde\xi</math> is increased from <math>~0</math> to <math>~\sqrt{3}</math>, we move away from the origin in a counter-clockwise direction along the lower segment (colored orange in the [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#MRplot|above figure]]) of the plotted equlibrium sequenceThen, as <math>~\tilde\xi</math> is increased from <math>~\sqrt{3}</math> to <math>~\infty</math>, we continue to move in a counter-clockwise direction along the equilibrium sequence, but now along the upper segment (colored green in the [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#MRplot|above figure]]) of the sequence, back to the origin; that is to say, <math>~\mathcal{X}</math> steadily decreases from <math>~\mathcal{X}_\mathrm{max}</math> back to <math>~0</math> and this time the relevant associated mass is the positive root of the quadratic relation, <math>~\mathcal{Y}_+</math>.  


Clearly, then, each value of <math>~\mathcal{X}</math> is associated with two different values of the parametric parameter, <math>~\tilde\xi</math>.  By inverting the <math>~\mathcal{X}(\tilde\xi)</math> parametric expression we see that, the two values of <math>~\tilde\xi</math> associated with a given equilibrium radius are,
<div align="center">
<table border="0" cellpadding="5" align="center">


If we set <math>~\mathfrak{b}_{n=4} = 3.4205</math> (corresponding to a choice of <math>~\tilde\xi = 1.4</math>), the virial theorem mass-radius relation maps onto the "Stahler" mass-radius coordinate plane as depicted by the set of green, triangle-shaped points in the lefthand panel of the above figure.  While the (green) curve corresponding to this relation does not overlay the blue mass-radius relation, the two curves do intersect.  They intersect precisely at the coordinate location along the blue curve (emphasized by the black filled circle) corresponding to a detailed force-balanced model having <math>~\tilde\xi = 1.4</math>.    In an analogous fashion, in the righthand panel of the figure, the curve delineated by the set of orange triangle-shaped points shows how the virial theorem mass-radius relation maps onto the "Stahler" mass-radius coordinate plane when we set <math>~\mathfrak{b}_{n=4} = 4.8926</math> (corresponding to a choice of <math>~\tilde\xi = 2.8</math>); it intersects the blue mass-radius relation precisely at the coordinate location, <math>~(\mathcal{X}, \mathcal{Y}) \approx (0.5108, 1.965)</math> &#8212; again, emphasized by a black filled circle &#8212; that corresponds to a detailed force-balanced model having <math>~\tilde\xi = 2.8</math>.  Hence, the two relations give the same mass-radius coordinates when the value of <math>~\mathfrak{b}_{n=4}</math> that is plugged into the virial theorem matches the value of <math>~\mathfrak{b}_{n=4}</math> that reflects the structural form factor that is properly associated with a detailed force-balanced model.
<tr>
  <td align="right">
<math>~\tilde\xi_\pm</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{\frac{3}{\alpha} \biggl[ 1 \pm \sqrt{1 - \alpha^2} \biggr]  \biggr\}^{1/2} \, ,</math>
  </td>
</tr>
</table>
</div>


 
where,
When we mapped the virial theorem mass-radius relation onto Stahler's mass-radius coordinate plane using a value of <math>~\mathfrak{b}_{n=4} = 4.8926</math> (as traced by the orange triangle-shaped points in the righthand panel of the above figure), we expected it to intersect the blue curve at the point along the blue sequence where <math>~\tilde\xi = 2.8</math>, for the reason just discussed.  After constructing the plot, it became clear that the two curves also intersect at the coordinate location, <math>~(\mathcal{X}, \mathcal{Y}) \approx (0.255, 1.67)</math> &#8212; also highlighted by a black filled circle &#8212; that corresponds to a detailed force-balanced model having <math>~\tilde\xi \approx 6.0</math>.  This makes it clear that it is the equality of the structural form factors, not the equality of the dimensionless (Lane-Emden) radius, <math>~\tilde\xi</math>, that assures precise agreement between the two different mass-radius expressions.
 
 
As is detailed in our [[User:Tohline/SSC/Virial/PolytropesSummary#Stability|above discussion of the dynamical stability of pressure-truncated polytropes]], an examination of free-energy variations can not only assist us in identifying the properties of equilibrium configurations (via a free-energy derivation of the virial theorem) but also in determining which of these configurations are dynamically stable and which are dynamically unstable.  We showed that, for a certain range of polytropic indexes, there is a critical point along the corresponding model sequence where the transition from stability to instability occurs.  As has been detailed in our [[User:Tohline/SSC/Virial/PolytropesSummary#Try_Polytropic_Index_of_4|above groundwork derivations]], for <math>~n = 4</math> polytropic structures, the critical point is identified by the dimensionless parameters,
<div align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
  <td align="right">
<math>~\alpha</math>
  </td>
   <td align="center">
   <td align="center">
<math>\eta_\mathrm{crit}\biggr|_{n=4}~=~\frac{1}{15} \, ;</math>
<math>~\equiv</math>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
  </td>
<math>\Pi_\mathrm{max}\biggr|_{n=4}~=~\frac{15^{15}}{16^{16}} \, ;</math>
  <td align="left">
&nbsp;&nbsp;&nbsp;&nbsp; and &nbsp;&nbsp;&nbsp;&nbsp;
<math>~\frac{(\mathcal{X}/\mathcal{X}_\mathrm{xmax})^2}{2-(\mathcal{X}/\mathcal{X}_\mathrm{xmax})^2} \, .</math>
<math>\Chi_\mathrm{min}\biggr|_{n=4}~=~\biggl( \frac{16}{15} \biggr)^4 \, .</math>
   </td>
   </td>
</tr>
</tr>
</table>
</table>
</div>
</div>
In the context of the above figure, independent of the chosen value of <math>~\mathfrak{b}_{n=4}</math>, this critical point always corresponds to the maximum mass that occurs along the mass-radius relationship established via the virial theorem.  In both panels of the figure, a horizontal red-dotted line has been drawn tangent to this critical point and identifies the corresponding critical value of <math>~\mathcal{Y}</math>; a vertical red-dashed line drawn through this same point helps identify the corresponding critical value of <math>~\mathcal{X}</math>.  We have deduced (details of the derivation not shown) that, for pressure-truncated <math>~n=4</math> polytropes, the coordinates of this critical point in Stahler's <math>~\mathcal{X}-\mathcal{Y}</math> plane depends on the choice of <math>~\mathfrak{b}_{n=4}</math> as follows:
We note as well that, for a given equilibrium radius, <math>~\mathcal{X}</math>, the ''ratio'' of the two mass solutions is given by a very simple expression, namely,
<div align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
Line 3,284: Line 3,035:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\mathcal{X}_\mathrm{crit}</math>
<math>~\frac{\mathcal{Y}_-}{\mathcal{Y}_+} = \frac{\tilde\xi_-^2}{3}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; or &nbsp; &nbsp; &nbsp; &nbsp;
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\pi^{-1/2} 2^{-16/5} (3\mathfrak{b}_{n=4})^{4/5} \, ,</math>
<math>~\frac{\mathcal{Y}_+}{\mathcal{Y}_-} = \frac{\tilde\xi_+^2}{3} \, .</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
This implies, as well, that,
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\mathcal{Y}_\mathrm{crit}</math>
<math>~\tilde\xi_+ \cdot \tilde\xi_-</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 3,302: Line 3,058:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\pi^{-1/2} 2^{-22/5} (3\mathfrak{b}_{n=4})^{8/5} \, .</math>
<math>~3\, .</math>
   </td>
   </td>
</tr>
</tr>
Line 3,308: Line 3,064:
</div>
</div>


In practice, for a given plot of the type displayed in the above figure &#8212; that is, for a given choice of the structural parameter, <math>\mathfrak{b}_{n=4}</math> &#8212; it only makes sense to compare the location of this critical point to the location of points that have been highlighted by a filled black circle, that is, points that identify the intersection between the two mass-radius relations.  If, in a given figure panel, a filled black circle lies to the right of the vertical dashed line, the equilibrium configuration corresponding to that black circle is dynamically stable.  On the other hand, if the filled black circle lies to the left of the vertical dashed line, its corresponding equilibrium configuration is dynamically unstable.  We conclude, therefore, that the equilibrium configuration marked by a filled black circle in the lefthand panel of the above figure is ''stable''; however, both configurations identified by filled black circles in the righthand panel are ''unstable''.
====Plotting the Virial Theorem Relation====


It is significant that the critical point identified by our free-energy-based stability analysis does not correspond to the equilibrium configuration having the largest mass along "Stahler's" (blue) equilibrium model sequence.  One might naively expect that a configuration of maximum mass along the blue curve is the relevant demarcation point and that, correspondingly, all models along this sequence that fall "to the right" of this maximum-mass point are stable.  But the righthand panel of our above figure contradicts this expectation.  While both of the black filled circles in the righthand panel of the above figure lie to the left of the vertical dashed line and therefore, as just concluded, are both unstable, one of the two configurations lies ''to the right'' of the maximum-mass point along the blue "Stahler" sequence.  This finding is related to [[User:Tohline/SSC/Virial/PolytropesSummary#Curiosity|the curiosity raised earlier]] in our discussion of the structural properties of pressure-truncated, <math>~n=4</math> polytropes.
[[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#ConciseFreeEnergyExpression|Drawing from our above derivations, the concise free-energy expression]] that reflects the properties of pressure-truncated <math>~n = 5</math> polytropic configurations is,
 
===Relating and Reconciling Two Mass-Radius Relationships for n = 3 Polytropes===
 
For pressure-truncated <math>~n=3</math> polytropes, [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)] did not identify a polynomial relationship between the mass and radius of equilibrium configurations.  However, from his analysis of detailed force-balance models ([[User:Tohline/SSC/Virial/PolytropesSummary#Detailed_Force-Balanced_Solution_2|summarized above]]), we appreciate that the governing pair of parametric relations is,
<div align="center">
<div align="center">
<table border="0" cellpadding="3">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>
<math>~\mathfrak{G}^*</math>
~\mathcal{X}
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{18}{5}\cdot \mathcal{A}_{M_\ell}  \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{2} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-1}
+~ 5\mathcal{B}_{M_\ell}  \cdot \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{6/5}  \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-3/5}
+~ \frac{4\pi}{3} \cdot \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{3}  \, ,
</math>
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>~\mathcal{A}_{M_\ell}</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{3}{4\pi} \biggr)^{1/2} \tilde\xi \tilde\theta \, ,
\frac{1}{5} \cdot \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} 
=\frac{1}{3^2\cdot 5} \biggl( \frac{\tilde\xi}{-\tilde\theta^'} \biggr)^2 \tilde\mathfrak{f}_W \, ,
</math>
</math>
   </td>
   </td>
Line 3,336: Line 3,107:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>
<math>~\mathcal{B}_{M_\ell}</math>
~\mathcal{Y}
</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{3^3}{4\pi} \biggr)^{1/2} (- \tilde\xi^2 \tilde\theta^') \, .
\biggl( \frac{3}{4\pi}\biggr)^{1/5} \frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_M^{6/5}}
= \frac{1}{3}( 4\pi)^{-1/5} \biggl( \frac{\tilde\xi}{-\tilde\theta^'} \biggr)^{6/5}
\biggl[ \tilde\theta^6 + \frac{2}{5} \cdot \tilde\xi^2 \tilde\mathfrak{f}_W \biggr] \, .
</math>
</math>
   </td>
   </td>
Line 3,351: Line 3,122:
</table>
</table>
</div>
</div>
On the other hand, the polynomial that results from plugging <math>~n=3</math> into the [[User:Tohline/SSC/Virial/PolytropesSummary#ConciseVirialXY|general mass-radius relation that is obtained via the virial theorem]] is,


The virial theorem which is derived from this free-energy expression provides a mass-radius relationship to be compared with the detailed force-balance relationship presented by Stahler.
Because our intent is to make this comparison, we begin with the virial theorem as written in terms of the variables, <math>~\mathcal{X}</math> and <math>~\mathcal{Y}</math>, and specialized for the case of <math>~n = 5</math> polytropic configurations.  Written in terms of the (constant) coefficients in the free-energy expression, we have
<div align="center">
<div align="center">
<math>
<table border="0" cellpadding="5" align="center">
\frac{2^3 \pi}{3} \mathcal{X}^4  - \biggl[ \frac{\mathcal{Y}^{4}}{4\pi}\biggr]^{1/3} \mathfrak{b}_{n=3} + \frac{4}{3} \mathcal{Y}^2
= 0 \, ,
</math>
</div>
where,
<div align="center">
<math>\mathfrak{b}_{n=3} = \biggl[ 4 (-\tilde\theta^')^2 + \frac{2}{3} \tilde\theta^{4} \biggr]
\biggl( \frac{\tilde\xi}{-\tilde\theta^'} \biggr)^{4/3} \, .
</math>
</div>
 
==Summary Comments==
From our above, detailed analysis of the mass-radius relation for pressure-truncated polytropes, we concluded that configurations along "Stahler's" equilibrium sequence become dynamically unstable at a point that does not coincide with the maximum-mass configuration.  Instead, the onset of dynamical instability is associated with the critical point on the mass-radius relation that arises from the free-energy-based virial theorem.  In drawing this conclusion, we have implicitly assumed that the proper way to analyze an equilibrium configuration's stability is to vary its radius while, not only holding its mass, specific entropy, and surface pressure <math>~(P_e)</math> constant, but also assuming that the configuration's structural form factors are invariable. 
 
 
This seems like a reasonable assumption, given that we're asking how a configuration's characteristics will vary ''dynamically'' when perturbed about an equilibrium state.  While oscillating about an equilibrium state, it seems more reasonable to assume that the system will expand and contract in a nearly homologous fashion than that its internal structure will readily readjust to produce a different ''and'' desirable set of form factors.  In support of this argument, we point to the paper by [http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich &amp; Weber (1980)] which explicitly derives a self-similar solution for the ''homologous'' collapse of stellar cores that can be modeled as <math>~n=3</math> polytropes; an associated [[User:Tohline/Apps/GoldreichWeber80#Homologously_Collapsing_Stellar_Cores|chapter of this H_Book details the Goldreich &amp; Weber derivation]].  Goldreich &amp; Weber use [[User:Tohline/SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29|linear perturbation techniques]] to analyze the stability of their homologously collapsing configurations.  In &sect;IV of their paper, they describe the eigenvalues and eigenfunctions that result from this analysis.  They discovered, for example, that "the lowest radial mode can be found analytically ... [and it] corresponds to a homologous perturbation of the entire core."  Our assumption that the structural form factors remain constant when pressure-truncated polytropic configurations undergo radial size variations therefore appears not to be unreasonable.  (Based on the Goldreich &amp; Weber discussion, we should also look at the published work of [http://adsabs.harvard.edu/abs/1941ApJ....94..245S Schwarzschild (1941, ApJ, 94, 245)], who has evaluated radial modes, and of [http://adsabs.harvard.edu/abs/1941MNRAS.101..367C Cowling (1941, MNRAS, 101, 367)], who has obtained eigenvalues of some low-order nonradial modes.)
 
 
In addition, it would seem that a certain amount of dissipation would be required for the system to readjust to new structural form factors.  In order to test this underlying assumption, following [http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich &amp; Weber (1980)], it would be desirable to carry out a full-blown perturbation analysis that involves looking for, for example, the eigenvector associated with the system's fundamental radial mode of pulsation.  Ideally, we should be using the structural form factors associated with this pulsation-mode eigenfunction in our free-energy analysis of stability.  Better yet, the ''sign'' of the eigenfrequency associated with the system's pulsation-mode eigenvector should signal whether the system is dynamically stable or unstable.


=Serious Concern=
==Statement of Concern==
Throughout our discussion of embedded (pressure-truncated) polytropes &#8212; both on this "summary" page and in an [[User:Tohline/SSC/Virial/Polytropes#Virial_Equilibrium_of_Adiabatic_Spheres|accompanying chapter]] where critical background derivations are presented &#8212;  we have used expressions for the structural form factors that include an overall leading factor of <math>~(5-n)^{-1}</math>.  For clarity, the form factors that we have used [[User:Tohline/SSC/Virial/Polytropes#Summary|for ''isolated'' polytropes]] is reprinted on the lefthand side of the following table while the ones that we have used [[User:Tohline/SSC/Virial/Polytropes#PTtable|for ''pressure-truncated'' polytropes]] is reprinted on the righthand side of the table.
<div align="center">
<table border="1" align="center" cellpadding="5">
<tr>
  <th align="center" colspan="1">
Structural Form Factors for <font color="red">Isolated</font> Polytropes
  </th>
  <th align="center" colspan="1">
Structural Form Factors for <font color="red">Pressure-Truncated</font> Polytropes
  </th>
</tr>
<tr>
  <td align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\mathfrak{f}_M</math>
<math>~
\mathcal{X}^4 
- \frac{3\mathcal{B}_{M_\ell}}{4\pi} \cdot ( \mathcal{X} \mathcal{Y}^3 )^{2/5} 
+~ \frac{9 \mathcal{A}_{M_\ell}}{10\pi}\cdot \mathcal{Y}^{2}
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 3,402: Line 3,140:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~ \biggl[ - \frac{3\Theta^'}{\xi} \biggr]_{\xi_1} </math>
<math>~0 \, ;</math>
   </td>
   </td>
</tr>
</tr>
 
</table>
</div>
or, [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#ConciseVirialXY|from above]], the
<div align="center">
<table border="0" cellpadding="8" align="center">
<tr><td align="center">
<font color="#770000">'''Virial Theorem written in terms of <math>~\mathcal{X}</math>, <math>~\mathcal{Y}</math>, and <math>~\tilde\mathfrak{f}_W</math>'''</font><br />
 
<math>~
\mathcal{X}^4  ~- ~
(\mathcal{X} \mathcal{Y}^3)^{2/5}
\biggl[\frac{\tilde\xi}{4\pi (-\tilde\theta^')}\biggr]^{6/5} \biggl[\tilde\theta^{6} +  \frac{2}{5} \cdot \tilde\xi^2 \tilde\mathfrak{f}_W \biggr]
~+ ~ \mathcal{Y}^2 \biggl[\frac{\tilde\xi}{(- \tilde\theta^')} \biggr]^2 \frac{\tilde\mathfrak{f}_W}{2\cdot 5^2 \pi}
= 0 \, ,
</math>
</td></tr>
</table>
</div>
where, specifically for <math>~n = 5</math> polytropic configurations &#8212; see our [[User:Tohline/SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D5.29|summary of the radial profiles of physical variables]] and our [[User:Tohline/SSC/Virial/FormFactors#Summary_.28n.3D5.29|determination of expressions for the structural form-factors]],
<div align="center">
<table border="0" cellpadding="5" align="center">
 
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>\mathfrak{f}_W </math>
<math>~\tilde\theta</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 3,414: Line 3,173:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\Theta^'}{\xi} \biggr]^2_{\xi_1} </math>
<math>~
( 1+ \ell^2 )^{-1/2} \, ,
</math>
   </td>
   </td>
</tr>
</tr>
Line 3,420: Line 3,181:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>\mathfrak{f}_A  </math>
<math>~\tilde\theta^'</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 3,426: Line 3,187:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>
\frac{3(n+1) }{(5-n)} ~\biggl[ \Theta^' \biggr]^2_{\xi_1} 
- 3^{-1/2} \ell ( 1+ \ell^2 )^{-3/2} \, ,
</math>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
  </td>
  <td align="center">


<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\tilde\mathfrak{f}_M</math>
<math>~\mathfrak{f}_W</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 3,444: Line 3,201:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~ \biggl[ - \frac{3\Theta^'}{\xi} \biggr]_{\tilde\xi} </math>
<math>~
\frac{5}{2^4} \cdot \ell^{-5} 
\biggl[ \ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1 + \ell^2)^{-3} + \tan^{-1}(\ell ) \biggr] \, ,
</math>
   </td>
   </td>
</tr>
</tr>
Line 3,450: Line 3,210:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>\tilde\mathfrak{f}_W </math>
<math>~\ell^2</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~\equiv</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\Theta^'}{\xi} \biggr]^2_{\tilde\xi} </math>
<math>~
\frac{\tilde\xi^2}{3} \, .
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
Once numerical values have been assigned to the free-energy coefficients, <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math>, the mass-radius relationship given by the scalar virial theorem can be compared quantitatively with Stahler's (detailed force-balance) mass-radius relationship.  The simplest, physically reasonable approximation would be to assume uniform-density structures, in which case, <math>~\tilde\mathfrak{f}_M = \tilde\mathfrak{f}_W = \tilde\mathfrak{f}_A = 1</math>, and accordingly, <math>~\mathcal{A}_{M_\ell} = 5^{-1}</math> and <math>~\mathcal{B}_{M_\ell} = (4\pi/3)^{-1/5}</math>.  But a better approximation would be to assign values to the structural form-factors that properly represent the properties of at least one detailed force-balanced model.  By way of illustration, the following table details what the proper values are for the two free-energy coefficients, and other relevant parameters, specifically for the model along Stahler's sequence that sits at <math>~\mathcal{Y}_\mathrm{max}</math> &#8212; that is, the model whose truncation radius is <math>~\tilde\xi = 3</math>.  As is recorded in the table, in this case the precise values of the free-energy coefficients are,
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>\tilde\mathfrak{f}_A  </math>
<math>~\mathcal{A}_{M_\ell}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 3,468: Line 3,236:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>
\frac{3(n+1) }{(5-n)} ~\biggl[ \Theta^' \biggr]^2_{\tilde\xi} + \tilde\Theta^{n+1}
\biggl( \frac{2^4 \pi^2}{3^7} \biggr)^{1/2} \, ,
</math>
</math>
   </td>
   </td>
</tr>
</tr>
</table>


<tr>
  <td align="right">
<math>~\mathcal{B}_{M_\ell}</math>
   </td>
   </td>
</tr>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\biggl( \frac{3}{2^{14}\pi} \biggr)^{1/5} \biggl[  1 +
\biggl( \frac{2^6\pi^2}{3^3} \biggr)^{1/2}\biggr] \, .
</math>
  </td>
</tr>
</table>
</table>
</div>
</div>
This factor seemed destined to become a nuisance in the specific case of <math>~n=5</math> polytropic structures.  But we did not let its appearance in these expressions deter us from using a free-energy analysis to study the equilibrium and stability of spherical polytropes because, after all, the factor of <math>~(5-n)^{-1}</math> appears in [[User:Tohline/Appendix/References|Chandrasekhar's [C67]]] expression for the gravitational potential energy of ''isolated'' polytropes &#8212; see his Equation (90), p. 101. In retrospect, its appearance in the structural form factors for ''isolated'' polytropes did not prove to be a problem because, via a free-energy and virial theorem analysis, the [[User:Tohline/SSC/Virial/Polytropes#Isolated_Nonrotating_Adiabatic_Configuration|derived expression for the configuration's equilibrium radius]] depends on the ratio of <math>~f_W</math> to <math>~f_A</math>, so the awkward factor of <math>~(5-n)^{-1}</math> cancels out.
Notice that, by choosing <math>~\tilde\xi = 3</math>, the evaluation of <math>~\tilde\mathfrak{f}_W</math> is particularly simple, in part, because <math>~\tan^{-1}(\ell) = \tan^{-1} \sqrt{3} = \pi/3</math>, but also because the term <math>~(\ell^4 - 8\ell^2/3 - 1)</math> equals zero.


However, in our discussion of ''pressure-truncated'' <math>~n=5</math> polytropic structures, the factor of <math>~(5-n)^{-1}</math> did not conveniently cancel out at the appropriate time and we were forced to carry out some logical contortions [[User:Tohline/SSC/Virial/PolytropesSummary#Plotting_the_Virial_Theorem_Relation|as we tried to compare the mass-radius relation obtained from the virial theorem]] to Stahler's mass-radius relation, which was derived from detailed force-balance arguments.  This leads us to seriously question whether our, rather casually derived, expressions for the structural form factors in ''pressure-truncated'' polytropes are correct.
==Further Evaluation of n = 5 Polytropic Structures==
Throughout most of this subsection, we will adopt the shorthand notation,
<table align="center" border="1" cellpadding="10">
<tr><td align="center">
<math>~\ell \equiv \frac{\tilde\xi}{\sqrt{3}} ~~~~~\Rightarrow ~~~~~ \ell^2 = \frac{\tilde\xi^2}{3} \, .</math>
</td></tr>
</table>
This will not only simplify the appearance of some expressions, it will facilitate direct comparison with an expression for the free-energy coefficient, <math>~\mathcal{A}</math>, that has been derived in a [[User:Tohline/SSC/Virial/FormFactors#Structural_Form_Factors|companion chapter]] following a different train of logic and with an expression for the normalized gravitational potential energy that has been derived via a brute-force integration in association with our [[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#Free_Energy|discussion of bipolytropic configurations]] where the variable, <math>~x_i</math>, has the same definition as <math>~\ell</math>. 
===Free-Energy Expression===
From our [[User:Tohline/SphericallySymmetricConfigurations/Virial#Free_Energy_Expression|general review of the topic]], to within an additive constant, the free-energy of a nonrotating, pressure-truncated polytrope comes from the sum of three principal energy terms, namely,
<div align="center">
<math>
\mathfrak{G} = W_\mathrm{grav} + \mathfrak{S}_\mathrm{therm} + P_e V \, .
</math>
</div>
Furthermore, as has been shown in our extended [[User:Tohline/SphericallySymmetricConfigurations/Virial#Gathering_it_all_Together|introductory discussion of free energy]], the corresponding ''normalized'' free energy (applied to <math>~n=5</math> or <math>~\gamma_g = 6/5</math> configurations) is,
<div align="center">
<math>
\mathfrak{G}^* \equiv \frac{\mathfrak{G}}{E_\mathrm{norm}} =
-3A\chi^{-1} + 5B \chi^{-3/5} +~ D\chi^3 \, ,
</math>
</div>
where,
<div align="center">
<table border="0" cellpadding="5">


<div align="center" id="ExampleXi3">
<table border="1" align="center" cellpadding="8">
<tr>
  <th align="center" colspan="3">
Determination of Coefficient Values in the Specific Case of <math>~\tilde\xi = 3</math>
  </th>
</tr>
<tr>
<tr>
   <td align="right">
   <td align="center">
<math>~\chi</math>
Quantity
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~\equiv</math>
Analytic Evaluation
   </td>
   </td>
   <td align="left">
   <td align="center">
<math>\frac{R}{R_\mathrm{norm}}  \, ,</math>
Numerical
   </td>
   </td>
</tr>
</tr>
<tr>
<tr>
   <td align="right">
   <td align="center">
<math>~A</math>
<math>~\ell</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~\equiv</math>
<math>~3^{1/2}</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>\frac{1}{5} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2} \, ,</math>
<math>~1.732051</math>
   </td>
   </td>
</tr>
</tr>


<tr>
<tr>
   <td align="right">
   <td align="center">
<math>~B</math>
<math>~\tilde\theta</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~\equiv</math>
<math>~2^{-1}</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>~0.5</math>
\biggl(\frac{3}{4\pi} \biggr)^{1/5} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{6/5}  \cdot \frac{\mathfrak{f}_A}{\mathfrak{f}_M^{6/5}} \, ,
  </td>
</math>
</tr>
 
<tr>
  <td align="center">
<math>~\tilde\theta^'</math>
  </td>
  <td align="center">
<math>~2^{-3}</math>
  </td>
  <td align="left">
<math>~0.125</math>
   </td>
   </td>
</tr>
</tr>


<tr>
<tr>
   <td align="right">
   <td align="center">
<math>~D</math>
<math>~\mathcal{X}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~\equiv</math>
<math>~\biggl( \frac{3^2\cdot 5}{2^6\pi} \biggr)^{1/2}</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>~0.473087</math>
\biggl( \frac{4\pi}{3} \biggr) \frac{P_e}{P_\mathrm{norm}} \, .
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>


Also note that the relevant normalizations are,
<tr>
<div align="center">
  <td align="center">
<math>~E_\mathrm{norm} \equiv \biggl( \frac{K^5}{G^3} \biggr)^{1/2} \, ;</math>&nbsp; &nbsp; &nbsp; &nbsp;
<math>~\mathcal{Y}</math>
<math>~R_\mathrm{norm} \equiv \biggl( \frac{G^5 M_\mathrm{tot}^4}{K^5} \biggr)^{1/2} \, ;</math>&nbsp; &nbsp; &nbsp; &nbsp;
  </td>
<math>~P_\mathrm{norm} \equiv \frac{K^{10}}{G^9 M_\mathrm{tot}^{6}}  \, .</math>&nbsp; &nbsp; &nbsp; &nbsp;
  <td align="center">
</div>
<math>~\biggl( \frac{3^4\cdot 5^3}{2^{10}\pi} \biggr)^{1/2}</math>
  </td>
  <td align="left">
<math>~1.774078</math>
  </td>
</tr>


===Virial Theorem===
<tr>
The traditional expression for the virial theorem in this context is,
  <td align="center">
<div align="center">
<math>~\tilde\mathfrak{f}_W</math>
<math>
  </td>
~2S_\mathrm{therm}^* + W_\mathrm{grav}^* - \frac{3P_e V}{E_\mathrm{norm}} = 0 \, .
  <td align="center">
</math>
<math>~\biggl( \frac{5^2 \pi^2}{2^8\cdot 3^7} \biggr)^{1/2}</math>
</div>
  </td>
From our [[User:Tohline/VE#Adiabatic_Systems|introductory discussion of the thermodynamic energy reservoir]], we know that, for <math>~\gamma_g=6/5</math> configurations,
  <td align="left">
<div align="center">
<math>~0.020993</math>
<math>
  </td>
~S_\mathrm{therm} = \frac{3}{2}(\gamma_g-1) \mathfrak{S}_\mathrm{therm} = \frac{3}{10} \mathfrak{S}_\mathrm{therm}\, .
</tr>
</math>
</div>
So, making this substitution and recognizing that <math>~E_\mathrm{norm} = P_\mathrm{norm}R_\mathrm{norm}^3</math>, the (normalized) virial theorem expression becomes,
<div align="center">
<math>
~\frac{3}{5}\mathfrak{S}_\mathrm{therm}^* + W_\mathrm{grav}^* - \frac{3P_e V}{P_\mathrm{norm}R_\mathrm{norm}^3} = 0 \, .
</math>
</div>


Furthermore, by comparing terms in the first free-energy expression, above, with the second (normalized) free-energy expression, we see that,
<tr>
<div align="center">
  <td align="center">
<math>~W_\mathrm{grav}^* \equiv \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \rightarrow -3A\chi^{-1} \, ;</math>&nbsp; &nbsp; &nbsp; &nbsp;
<math>~\mathcal{A}_{M_\ell}</math>
<math>~\mathfrak{S}_\mathrm{therm}^* \equiv \frac{\mathfrak{S}_\mathrm{therm}}{E_\mathrm{norm}} \rightarrow 5B\chi^{-3/5} \, ;</math>&nbsp; &nbsp; &nbsp; &nbsp;
  </td>
<math>~\frac{P_e V}{E_\mathrm{norm}} \rightarrow D\chi^{3} \, .</math>&nbsp; &nbsp; &nbsp; &nbsp;
  <td align="center">
</div>
<math>~\biggl( \frac{2^4 \pi^2}{3^7} \biggr)^{1/2}</math>
Hence, the normalized virial theorem may be written as,
  </td>
<div align="center">
  <td align="left">
<math>
<math>~0.268711</math>
3B \chi^{-3/5} -3A\chi^{-1}  -~ 3D\chi^3 =0\, .
  </td>
</math>
</tr>
</div>
 
For sake of consistency, let's check this by holding the coefficients <math>~A</math>, <math>~B</math>, and <math>~D</math> constant and setting <math>~d\mathfrak{G}^*/d\chi</math> equal to zero:
<div align="center">
<table border="0" cellpadding="5">


<tr>
<tr>
   <td align="right">
   <td align="center">
<math>~\frac{d\mathfrak{G}^*}{d\chi} = - 3B \chi^{-8/5} + 3A\chi^{-2}  +~ 3D\chi^2 </math>
<math>~\mathcal{B}_{M_\ell}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~\biggl( \frac{3}{2^{14}\pi} \biggr)^{1/5} \biggl[  1 +
\biggl( \frac{2^6\pi^2}{3^3} \biggr)^{1/2}\biggr]</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~0 </math>
<math>~0.830395</math>
   </td>
   </td>
</tr>
</tr>


<tr>
<tr>
   <td align="right">
   <td align="center">
<math>~\Rightarrow ~~~ - \chi^{-1} \biggl[ 3B \chi^{-3/5} - 3A\chi^{-1} -~ 3D\chi^3 \biggr] </math>
<math>~G^*</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~\biggl( \frac{3\cdot 5^3}{2^{12}\pi} \biggr)^{1/2} [  2^3\pi + 3^{5/2} ]</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~0  \, .</math>
<math>~6.951544</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
This, in turn, implies that the expression inside the square brackets sums to zero, which identically matches the (normalized) traditional virial theorem expression.  Excellent!


===Borrowing from Bipolytrope Discussion===
<tr>
In an accompanying chapter that presents the [[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1|detailed force-balanced models of <math>~(n_c, n_e) = (5, 1)</math> bipolytropes]] we explicitly show that, for configurations with the correct equilibrium radius, the virial theorem is satisfied.  In the case of bipolytropes, which are not embedded in an external medium, the relevant normalized virial theorem states that,
  <td align="left">
<div align="center">
Virial:
<math>
~(2{S}_\mathrm{therm}^* + W_\mathrm{grav}^*)_\mathrm{core} + (2{S}_\mathrm{therm}^* + W_\mathrm{grav}^*)_\mathrm{env} = 0 \, .
</math>
</div>


In the bipolytrope, the (truncated) <math>~n=5</math> core is confined by an <math>~n=1</math> envelope; in addition to demanding that the relevant virial theorem be satisfied, there is also a constraint that the pressure at the inner edge of the envelope be equal to the pressure at the (truncated) outer edge of the core.  As we have just discussed, for a (truncated) <math>~n=5</math> polytrope that is confined by a hot, tenuous external medium instead of by an enveloping envelope, the relevant normalized virial theorem is,
<math>~\mathcal{X}^4 - \frac{3\mathcal{B}_{M_\ell}}{4\pi} \cdot (\mathcal{X}\mathcal{Y}^3 )^{2/5}</math>
<div align="center">
<table border="0" cellpadding="5">


<tr>
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <math>~~~~~~~~~~+\frac{9 \mathcal{A}_{M_\ell}}{10\pi} \cdot \mathcal{Y}^2</math>
  <td align="right">
<math>~(2{S}_\mathrm{therm}^* + W_\mathrm{grav}^*)_\mathrm{core} - \frac{3P_e V_\mathrm{eq}}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~0 </math>
<math>~ \frac{3^4\cdot 5^2}{2^{12}\pi} -
  </td>
\frac{3^4\cdot 5^2}{2^{12}\pi} \biggl[ 1 + \biggl( \frac{2^6\pi^2}{3^3} \biggr)^{1/2}\biggr]</math>
</tr>


<tr>
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <math>~~~~~~~~~~+ \biggl( \frac{3^5\cdot 5^4}{2^{18} \pi^2} \biggr)^{1/2}
  <td align="right">
</math>
<math>~\Rightarrow ~~~ (2{S}_\mathrm{therm}^* + W_\mathrm{grav}^*)_\mathrm{core} </math>
  </td>
  <td align="center">
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\frac{3P_e }{(K^5/G^3)^{1/2}} \biggl( \frac{4\pi}{3} R_\mathrm{eq}^3 \biggr) </math>
Sums to zero,
  </td>
</tr>


<tr>
exactly!
  <td align="right">
&nbsp;
</td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi \biggl( \frac{2\cdot 3}{5} \biggr)^{3/2}  \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}}\biggr)^3 \, ,</math>
   </td>
   </td>
</tr>
</tr>
</table>
</table>
</div>
</div>
where,  [[User:Tohline/SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|as discussed/defined in an accompanying chapter of this H_Book]], we have adopted the normalization radius, <math>~R_\mathrm{SWS}</math>, first introduced by [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Steven W. Stahler (1983)].  For <math>~n=5</math> configurations, its definition is,
<div align="center">
<math>
R_\mathrm{SWS}\biggr|_{n=5} = \biggl( \frac{2\cdot 3}{5} \biggr)^{1/2} \biggl[ \frac{(K^{5}/G^3)^{1/2}}{P_\mathrm{e}}\biggr]^{1/3} \, .
</math>
</div>
As has [[User:Tohline/SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|also been discussed in the accompanying chapter]], we can deduce from Stahler's detailed force-balanced models that the equilibrium radius of embedded, <math>~n=5</math> polytropes is given in terms of the dimensionless, ''truncated'' Lane-Emden radius, <math>~\tilde\xi</math> &#8212; and our corresponding variable, <math>\ell</math> &#8212; by the expression,
<div align="center">
<table border="0" cellpadding="3">


<tr>
 
  <td align="right">
The curve traced out by the light-blue diamonds in each panel of the following comparison figure displays Stahler's analytically prescribed mass-radius relation; this curve is identical in all six panels and is the same as the [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Plotting_Stahler.27s_Relation|curve displayed above]] in connection with our description of Stahler's mass-radius relation.  Each point along this "Stahler" curve identifies a model having a different truncation radius, <math>~\tilde\xi</math>; as plotted here, starting near the origin and moving counter-clockwise around the curve, <math>~\tilde\xi</math> is varied from 0.05 to 42.5.  As foreshadowed by the above discussion, the model having the greatest mass <math>~(\mathcal{Y}_\mathrm{max})</math> along the Stahler sequence &#8212; highlighted by the red filled circle in most of the figure panels &#8212;  is defined by <math>~\tilde\xi = 3</math>. 
<math>
 
~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} }\biggr|_{n=5}
In each figure panel, the curve traced out by the orange triangles &#8212; or, in one case, the orange triangles &amp; light purple diamonds &#8212; displays the mass-radius relation defined by the virial theorem.  These "Virial" curves are all defined by the same virial theorem polynomial expression, as just presented, but the coefficient of the <math>~\mathcal{Y}^2</math> term and the coefficient of the <math>~(\mathcal{X}\mathcal{Y}^3)^{2/5}</math> cross term &#8212; essentially, the value of <math>~\mathcal{A}_{M_\ell}</math> and the value of <math>~\mathcal{B}_{M_\ell}</math>, respectively &#8212; have different values in the six separate figure panels.  In each case, a value has been specified for the parameter, <math>~\tilde\xi</math> (as identified in the title of each figure panel), and this, in turn, has determined the values of the two (constant) free-energy coefficients.  For example, in the top-right figure panel whose title indicates <math>~\tilde\xi = 3</math>, the "Virial" curve traces the mass-radius relation prescribed by the virial theorem after the values of the free-energy coefficients have been set to values that correspond to a detailed force-balanced model with this specified truncation radius, that is (see the above table),  <math>~\mathcal{A}_{M_\ell} = 0.268711</math> and <math>~\mathcal{B}_{M_\ell} = 0.830395</math>.  Columns 2 and 3, respectively, of the table affixed to the bottom of the following comparison figure list the values of <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math> that have been used to define the "Virial" curve in each of the six figure panels, in accordance with the value of <math>~\tilde\xi</math> listed in column 1 of the table.
</math>
 
 
<div align="center" id="OverlapPlots">
<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center" colspan="7">
Comparing Two Separate Mass-Radius Relations for Pressure-Truncated ''n = 5'' Polytropes
  </th>
</tr>
<tr>
<td align="center" colspan="3">
[[File:Xi1.26419.png|225px|Comparison of Two Mass-Radius Relations]]
</td>
<td align="center" colspan="2">
[[File:Xi2.0.png|225px|Comparison of Two Mass-Radius Relations]]
</td>
<td align="center" colspan="2">
[[File:Xi3.0.png|225px|Comparison of Two Mass-Radius Relations]]
</td>
</tr>
<tr>
<td align="center" colspan="3">
[[File:Xi3.5.png|225px|Comparison of Two Mass-Radius Relations]]
</td>
<td align="center" colspan="2">
[[File:Xi3.850652.png|225px|Comparison of Two Mass-Radius Relations]]
</td>
<td align="center" colspan="2">
[[File:Xi9.8461.png|225px|Comparison of Two Mass-Radius Relations]]
</td>
</tr>
 
<tr>
  <td align="center" rowspan="2">
<math>~\tilde\xi</math>
  </td>
  <td align="center" colspan="2">
Free-Energy Coefficients
   </td>
   </td>
   <td align="center">
   <td align="center" colspan="2">
<math>~=~</math>
Primary Overlap
   </td>
   </td>
   <td align="left">
   <td align="center" colspan="2">
<math>
Secondary Overlap
\biggl( \frac{5}{4\pi} \biggr)^{1/2} \tilde\xi \tilde\theta^2
 
= \biggl( \frac{5}{4\pi} \biggr)^{1/2} \tilde\xi \biggl(1 + \frac{\tilde\xi^2}{3}\biggr)^{-1}
(see [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#SecondaryOverlap|further elaboration below]])
</math>
   </td>
   </td>
</tr>
</tr>


<tr>
<tr>
   <td align="right">
   <td align="center">
&nbsp;
<math>~\mathcal{A}_{M_\ell}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=~</math>
<math>~\mathcal{B}_{M_\ell}</math>
   </td>
   </td>
   <td align="left">
   <td align="center">
<math>
<math>~\mathcal{X} \equiv \frac{R_\mathrm{eq}}{R_\mathrm{SWS}}</math>
\biggl( \frac{5\cdot 3}{2^2\pi} \biggr)^{1/2} \ell (1 + \ell^2)^{-1} \, .
</math>
   </td>
   </td>
</tr>
   <td align="center">
</table>
<math>~\mathcal{Y} \equiv \frac{M_\mathrm{limit}}{M_\mathrm{SWS}}</math>
</div>
Hence, upon careful evaluation of the thermal energy and gravitational potential energy of truncated <math>~n=5</math> polytropes, we should find that,
<div align="center">
<table border="0" cellpadding="5">
 
<tr>
   <td align="right">
<math>~(2{S}_\mathrm{therm}^* + W_\mathrm{grav}^*)_\mathrm{core} </math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~\mathcal{X}</math>
   </td>
   </td>
   <td align="left">
   <td align="center">
<math>~4\pi \biggl( \frac{2\cdot 3}{5} \biggr)^{3/2}  \biggl[ \biggl( \frac{5\cdot 3}{2^2\pi} \biggr)^{1/2} \ell (1 + \ell^2)^{-1}
<math>~\mathcal{Y}</math>
\biggr]^3 </math>
   </td>
   </td>
</tr>
</tr>


<tr>
<tr>
  <td align="right">
<td align="center">
&nbsp;
1.26419
  </td>
</td>
  <td align="center">
<td align="center">
<math>~=</math>
0.214429
  </td>
</td>
  <td align="left">
<td align="center">
<math>~\biggl( \frac{2\cdot 3^6}{\pi} \biggr)^{1/2} \ell^3 (1 + \ell^2)^{-3} \, .
0.758099
</math>
</td>
  </td>
<td align="center" bgcolor="yellow">
0.520269
</td>
<td align="center" bgcolor="yellow">
0.904142
</td>
<td align="center">
0.388938
</td>
<td align="center">
0.289568
</td>
</tr>
</tr>
</table>
</div>
Well, it just so happens that, in our [[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#twoSplusWcore|accompanying chapter that presents the detailed force-balanced models of <math>~(n_c, n_e) = (5, 1)</math> bipolytropes]], we explicitly carried out the volume integrals defining these two key components of the free energy expression with the results being,
<div align="center">
<table border="0" cellpadding="4">


<tr>
<tr>
  <td align="right">
<td align="center">
<math>~(2S^* + W^*)_\mathrm{core}</math>
2
  </td>
</td>
  <td align="center">
<td align="center">
<math>~=~</math>
0.233842
  </td>
</td>
  <td align="left">
<td align="center">
<math>~\biggl( \frac{2\cdot 3^6}{\pi} \biggr)^{1/2} \biggl[ x_i ^3 (1 + x_i^2)^{-3} \biggr] \, .
0.779836
</math>
</td>
  </td>
<td align="center">
0.540671
</td>
<td align="center">
1.544775
</td>
<td align="center">
0.468918
</td>
<td align="center">
0.570916
</td>
</tr>
</tr>
</table>
</div>
Realizing that the variable, <math>~x_i</math>, in that context is the same as <math>~\ell</math>, in the present context, we see that the two separately derived results are identical to one another.
===Determining Expressions for Free-Energy Coefficients===
We should be able to convert the separately derived expression for <math>~W_\mathrm{grav}^*</math> into an expression for the free-energy coefficient, <math>~A</math>, in equilibrium configurations.  As [[User:Tohline/SSC/Virial/PolytropesSummary#Virial_Theorem|noted above]], for a fixed value of <math>~A</math>,
<div align="center">
<math>~W_\mathrm{grav}^* ~~\rightarrow ~~ -3A\chi^{-1} \, .</math>
</div>
Therefore, in an equilibrium configuration, we can write,
<div align="center">
<table border="0" cellpadding="3">


<tr>
<tr>
  <td align="right">
<td align="center">
<math>
[[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#ExampleXi3|3]]
~W_\mathrm{grav}^*
</td>
</math>
<td align="center">
  </td>
0.268711
  <td align="center">
</td>
<math>~=~</math>
<td align="center">
  </td>
0.830395
  <td align="left">
</td>
<math>
<td align="center">
-3A\chi_\mathrm{eq}^{-1} = -3A \biggl( \frac{R_\mathrm{norm}}{R_\mathrm{eq}}\biggr)
0.473087
= -3A \biggl( \frac{R_\mathrm{SWS}}{R_\mathrm{eq}}\biggr)\biggl( \frac{R_\mathrm{norm}}{R_\mathrm{SWS}}\biggr)</math>
</td>
  </td>
<td align="center">
1.774078
</td>
<td align="center">
0.507387
</td>
<td align="center">
0.798441
</td>
</tr>
</tr>


<tr>
<tr>
  <td align="right">
<td align="center">
<math>
3.5
~\Rightarrow ~~~ A
</td>
</math>
<td align="center">
  </td>
0.288708
  <td align="center">
</td>
<math>~=~</math>
<td align="center">
  </td>
0.861503
  <td align="left">
</td>
<math>
<td align="center">
-\frac{1}{3} W_\mathrm{grav}^* \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}}\biggr)\biggl( \frac{R_\mathrm{SWS}}{R_\mathrm{norm}}\biggr) \, .</math>
0.434310
  </td>
</td>
<td align="center">
1.744359
</td>
<td align="center">
0.515168
</td>
<td align="center">
0.859518
</td>
</tr>
 
<tr>
<td align="center">
3.850652
</td>
<td align="center">
0.303490
</td>
<td align="center">
0.884746
</td>
<td align="center">
0.408738
</td>
<td align="center">
1.699778
</td>
<td align="center">
0.518588
</td>
<td align="center">
0.888969
</td>
</tr>
 
<tr>
<td align="center">
9.8461
</td>
<td align="center">
0.601012
</td>
<td align="center">
1.313904
</td>
<td align="center">
0.186424
</td>
<td align="center">
0.904141
</td>
<td align="center" bgcolor="yellow">
0.520269
</td>
<td align="center" bgcolor="yellow">
0.904143
</td>
</tr>
</tr>
</table>
</table>
</div>
</div>
Now, from immediately above, we know that,
 
 
As is [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Quantitative_Study|discussed more fully, below]], in each of the six panels of the above comparison figure, the "Virial" curve intersects the "Stahler" curve at two locations.  These points of intersection are identified by the black filled circles in each figure panel.  In each case, the intersection point that is farthest along on the Stahler sequence &#8212; as determined by starting at the origin and moving counter-clockwise along the sequence &#8212; identifies the "Primary Overlap" between the two curves.  That is to say, the <math>~(\mathcal{X}, \mathcal{Y})</math> coordinates of this point (see columns 4 and 5 of the table affixed to the bottom of the figure) are the coordinate values that are obtained by plugging the specified value of <math>~\tilde\xi</math> (see the title of the figure panel or column 1 of the affixed table) into [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Parametric_Relations|Stahler's pair of parametric relations]].  The second point of intersection in each panel &#8212; which we will refer to as the "Secondary Overlap" points and whose coordinates are provided in columns 6 and 7 of the affixed table &#8212; appears to be fortuitous and of no particularly significant astrophysical interest.
 
 
The mass-radius diagram displayed in the top-right panel of the [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#OverlapPlots|above comparison figure]] has been reproduced in the upper-left panel of the [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#GraphicalDepictionXi3|following figure]] &#8212; in this case, with a coordinate aspect ratio that is closer to 1:1  &#8212; along with color images of the corresponding free-energy surface, viewed from two different perspectives, and a three-column table listing the 3D coordinates, <math>~(X,Y,Z) = (R_\mathrm{eq}/R_\mathrm{SWS}, M_\mathrm{limit}/M_\mathrm{SWS}, \mathfrak{G}^*)</math>, of the seventeen points that have been used to define the displayed "Virial" curve.  To be more explicit, the rainbow-colored free-energy surface, <math>~\mathfrak{G}^*(R,M_\mathrm{limit})</math>, has been defined by the free-energy function appropriate to pressure-truncated <math>~n=5</math> polytropic configurations [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Plotting_the_Virial_Theorem_Relation|as defined above]], that is,
<div align="center">
<div align="center">
<table border="0" cellpadding="4">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>
<math>~\mathfrak{G}^*</math>
~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} }\biggr|_{n=5}
</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{5\cdot 3}{2^2\pi} \biggr)^{1/2} \ell (1 + \ell^2)^{-1}  \, ;
- \frac{18}{5}\cdot \mathcal{A}_{M_\ell}  \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{2} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-1}  
+~ 5\mathcal{B}_{M_\ell}  \cdot \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{6/5}  \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-3/5}
+~ \frac{4\pi}{3} \cdot \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{3}  \, ,
</math>
</math>
   </td>
   </td>
Line 3,840: Line 3,657:
</table>
</table>
</div>
</div>
and, from our [[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#Free_Energy|accompanying discussion of the free-energy of bipolytropic configurations]], we know that,
<div align="center">
<table border="0" cellpadding="4">


with the values of the two free-energy coefficients set to the values that correspond to a <math>~\tilde\xi = 3</math> virial curve as discussed above, namely, <math>~\mathcal{A}_{M_\ell} = 0.268711</math> and <math>~\mathcal{B}_{M_\ell} = 0.830395</math>.
<div align="center" id="GraphicalDepictionXi3">
<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center" colspan="2">
Free-Energy Surface for ''n = 5'' Polytropic Configurations Truncated at <math>~\tilde\xi = 3</math>
  </th>
  <th align="center" colspan="1">
Radius
  </th>
  <th align="center" colspan="1">
Mass
  </th>
  <th align="center" colspan="1">
Free
Energy
  </th>
</tr>
<tr>
<tr>
  <td align="right">
<td align="center" rowspan="8">
<math>~W^*_\mathrm{core}</math>
[[File:VirialEquilibria.png|150px|Virial Mass-Radius Relation]]
  </td>
</td>
  <td align="center">
<td align="center" rowspan="8" bgcolor="CCFFFF">
<math>~=~</math>
[[File:FreeEnergyFaceOn.png|325px|Virial Mass-Radius Relation on top of Free-Energy Surface]]
  </td>
</td>
  <td align="left">
<td align="right" rowspan="1" bgcolor="lightblue">
<math>~ - \biggl( \frac{3^8}{2^5\pi } \biggr)^{1/2}
0.3152
\biggl[ x_i\biggl(x_i^4 - \frac{8}{3}x_i^2 - 1\biggr) (1 + x_i^2)^{-3}  + \tan^{-1}(x_i) \biggr] \, . </math>
</td>
  </td>
<td align="right" rowspan="1" bgcolor="lightblue">
0.1345
</td>
<td align="right" rowspan="1" bgcolor="lightblue">
0.8231
</td>
</tr>
</tr>
</table>
</div>
So, again realizing that <math>~x_i</math> and <math>~\ell</math> are interchangeable, we have,
<div align="center">
<table border="0" cellpadding="4">


<tr>
<tr>
  <td align="right">
<td align="right" rowspan="1">
<math>~A</math>
0.4159
  </td>
</td>
  <td align="center">
<td align="right" rowspan="1">
<math>~=~</math>
0.3470
  </td>
</td>
  <td align="left">
<td align="right" rowspan="1">
<math>~ \frac{1}{3}
1.9947
\biggl( \frac{3^8}{2^5\pi } \biggr)^{1/2} \biggl( \frac{5\cdot 3}{2^2\pi} \biggr)^{1/2} \ell (1 + \ell^2)^{-1}
</td>
\biggl[ \ell \biggl(\ell^4 - \frac{8}{3}\ell^2 - 1\biggr) (1 + \ell^2)^{-3}  + \tan^{-1}(\ell) \biggr]
\biggl( \frac{R_\mathrm{SWS}}{R_\mathrm{norm}}\biggr)
</math>
  </td>
</tr>
</tr>


<tr>
<tr>
  <td align="right">
<td align="right" rowspan="1" bgcolor="lightblue">
&nbsp;
0.4731
  </td>
</td>
  <td align="center">
<td align="right" rowspan="1" bgcolor="lightblue">
<math>~=~</math>
0.5735
  </td>
</td>
  <td align="left">
<td align="right" rowspan="1" bgcolor="lightblue">
<math>~
3.1095
\biggl( \frac{3^7 \cdot 5}{2^7\pi^2 } \biggr)^{1/2} \ell (1 + \ell^2)^{-1}
</td>
\biggl[ \ell \biggl(\ell^4 - \frac{8}{3}\ell^2 - 1\biggr) (1 + \ell^2)^{-3}  + \tan^{-1}(\ell) \biggr]
\biggl( \frac{R_\mathrm{SWS}}{R_\mathrm{norm}}\biggr) \, .
</math>
  </td>
</tr>
</tr>
</table>
</div>
Finally, we need to determine an expression for the ratio, <math>~R_\mathrm{SWS}/R_\mathrm{norm}</math>.  Drawing the definition of <math>~R_\mathrm{norm}</math> from [[User:Tohline/SphericallySymmetricConfigurations/Virial#Normalizations|our introductory chapter on the virial equilibrium]], we have,
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
  <td align="right">
<td align="right" rowspan="1" bgcolor="white">
<math>~R_\mathrm{norm}\biggr|_{\gamma = 6/5}</math>
0.5035
  </td>
</td>
  <td align="center">
<td align="right" rowspan="1" bgcolor="white">
<math>~=</math>
0.7661
  </td>
</td>
  <td align="left">
<td align="right" rowspan="1" bgcolor="white">
<math>~\biggl[ \biggl( \frac{G}{K}\biggr) M_\mathrm{tot}^{2-6/5} \biggr]^{1/(4-18/5)}
3.9591
= \biggl[ \biggl( \frac{G}{K}\biggr) M_\mathrm{tot}^{4/5} \biggr]^{5/2}
</td>
= \biggl( \frac{G}{K}\biggr)^{5/2} M_\mathrm{tot}^{2} \, .
</math>
  </td>
</tr>
</tr>
</table>
</div>
From our [[User:Tohline/SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D5.29|tabular summary of Stahler's derived mass &amp; radius relationships for truncated, <math>~n=5</math> polytropes]] we have,
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
  <td align="right">
<td align="right" rowspan="1" bgcolor="white">
<math>~M_\mathrm{limit}^2</math>
0.5114
  </td>
</td>
  <td align="center">
<td align="right" rowspan="1" bgcolor="white">
<math>~=</math>
0.8347
  </td>
</td>
  <td align="left">
<td align="right" rowspan="1" bgcolor="white">
<math>~
4.2409
M_\mathrm{SWS}^2 \biggl[ \biggl( \frac{3\cdot 5^3}{2^2\pi} \biggr) \ell^6 (1+\ell^2)^{-4} \biggr] \, .
</td>
</math>
</tr>
  </td>
 
<tr>
<td align="right" rowspan="1" bgcolor="lightblue">
0.5297
</td>
<td align="right" rowspan="1" bgcolor="lightblue">
1.0759
</td>
<td align="right" rowspan="1" bgcolor="lightblue">
5.1453
</td>
</tr>
</tr>
</table>
</div>
In addition, from our [[User:Tohline/SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|review of Stahler's defined normalizations]], we see that,
<div align="center">
<table border="0" cellpadding="5">


<tr>
<tr>
  <td align="right">
<td align="right" rowspan="1" bgcolor="lightblue">
<math>~M_\mathrm{SWS}^2</math>
0.5310
  </td>
</td>
  <td align="center">
<td align="right" rowspan="1" bgcolor="lightblue">
<math>~=</math>
1.4036
  </td>
</td>
  <td align="left">
<td align="right" rowspan="1" bgcolor="lightblue">
<math>~
6.1560
\biggl( \frac{2^3 \cdot 3^3}{5^3} \biggr) G^{-3} K^{10/3} P_e^{-1/3} \, ,
</td>
</math>
  </td>
</tr>
</tr>


<tr>
<tr>
  <td align="right">
<td align="right" rowspan="1" bgcolor="white">
and, &nbsp;&nbsp;&nbsp;<math>~R_\mathrm{SWS}</math>
0.5250
  </td>
</td>
  <td align="center">
<td align="right" rowspan="1" bgcolor="white">
<math>~=</math>
1.5010
  </td>
</td>
  <td align="left">
<td align="right" rowspan="1" bgcolor="white">
<math>~
6.4046
\biggl( \frac{2 \cdot 3}{5} \biggr)^{1/2} G^{-1/2} K^{5/6} P_e^{-1/3} \, ,
</td>
</math>
  </td>
</tr>
</tr>
</table>
</div>
which, when combined to cancel <math>~P_e</math> gives,
<div align="center">
<table border="0" cellpadding="5">


<tr>
<tr>
  <td align="right">
<td align="center" colspan="2" rowspan="9" bgcolor="#CCFFFF">
<math>~M_\mathrm{SWS}^2</math>
[[File:FreeEnergySurface.png|450px|Free-Energy Surface]]
  </td>
</td>
  <td align="center">
<td align="right" rowspan="1" bgcolor="white">
<math>~=</math>
0.5185
  </td>
</td>
  <td align="left">
<td align="right" rowspan="1" bgcolor="white">
<math>~
1.5680
\biggl( \frac{2^6 \cdot 3^6}{5^6} \biggr)^{1/2} G^{-3} K^{10/3}
</td>
\biggl( \frac{5}{2 \cdot 3} \biggr)^{1/2} G^{1/2} K^{-5/6} R_\mathrm{SWS}
<td align="right" rowspan="1" bgcolor="white">
</math>
6.5606
  </td>
</td>
</tr>
</tr>


<tr>
<tr>
  <td align="right">
<td align="right" rowspan="1" bgcolor="lightblue">
&nbsp;
0.5114
  </td>
</td>
  <td align="center">
<td align="right" rowspan="1" bgcolor="lightblue">
<math>~=</math>
1.6206
  </td>
</td>
  <td align="left">
<td align="right" rowspan="1" bgcolor="lightblue">
<math>~
6.6741
\biggl( \frac{2^5 \cdot 3^5}{5^5} \biggr)^{1/2}\biggl( \frac{K}{G}\biggr)^{5/2} R_\mathrm{SWS} \, .
</td>
</math>
</tr>
  </td>
 
<tr>
<td align="right" rowspan="1" bgcolor="black"><font color="white">
0.4731</font>
</td>
<td align="right" rowspan="1" bgcolor="black"><font color="white">
1.7741</font>
</td>
<td align="right" rowspan="1" bgcolor="black"><font color="white">
6.9515</font>
</td>
</tr>
</tr>
</table>
</div>
Hence, we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
  <td align="right">
<td align="right" rowspan="1" bgcolor="lightblue">
<math>~M_\mathrm{limit}^2</math>
0.4343
  </td>
</td>
  <td align="center">
<td align="right" rowspan="1" bgcolor="lightblue">
<math>~=</math>
1.8287
  </td>
</td>
  <td align="left">
<td align="right" rowspan="1" bgcolor="lightblue">
<math>~
7.0247
\biggl( \frac{2^5 \cdot 3^5}{5^5} \biggr)^{1/2}\biggl( \frac{K}{G}\biggr)^{5/2} 
</td>
\biggl[ \biggl( \frac{3\cdot 5^3}{2^2\pi} \biggr) \ell^6 (1+\ell^2)^{-4} \biggr] R_\mathrm{SWS}
</math>
  </td>
</tr>
</tr>


<tr>
<tr>
  <td align="right">
<td align="right" rowspan="1" bgcolor="red"><font color="white">
<math>~\Rightarrow ~~~~ \frac{1}{R_\mathrm{SWS}}</math>
0.3984</font>
  </td>
</td>
  <td align="center">
<td align="right" rowspan="1" bgcolor="red"><font color="white">
<math>~=</math>
1.8333</font>
  </td>
</td>
  <td align="left">
<td align="right" rowspan="1" bgcolor="red"><font color="white">
<math>~
7.0301</font>
\biggl( \frac{2\cdot 3^7 \cdot 5}{\pi^2} \biggr)^{1/2}\biggl( \frac{K}{G}\biggr)^{5/2} 
</td>
\biggl[ \ell^6 (1+\ell^2)^{-4} \biggr] \frac{1}{M_\mathrm{limit}^2} \, .
</math>
  </td>
</tr>
</tr>
</table>
</div>
In combination with the expression for <math>~R_\mathrm{norm}</math>, then, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
  <td align="right">
<td align="right" rowspan="1" bgcolor="lightblue">
<math>~\frac{R_\mathrm{norm}}{R_\mathrm{SWS}}</math>
0.3379
  </td>
</td>
  <td align="center">
<td align="right" rowspan="1" bgcolor="lightblue">
<math>~=</math>
1.7764
  </td>
</td>
  <td align="left">
<td align="right" rowspan="1" bgcolor="lightblue">
<math>~
6.9923
\biggl( \frac{2\cdot 3^7 \cdot 5}{\pi^2} \biggr)^{1/2}
</td>
\biggl[ \ell^6 (1+\ell^2)^{-4} \biggr] \biggl(\frac{M_\mathrm{tot}}{M_\mathrm{limit}}\biggr)^{2} \, ,
</math>
  </td>
</tr>
</tr>
</table>
</div>
which means that, for truncated <math>~n=5</math> polytropes, the expression for the free-energy coefficient is,
<div align="center">
<table border="0" cellpadding="4">


<tr>
<tr>
  <td align="right">
<td align="right" rowspan="1" bgcolor="lightblue">
<math>~A</math>
0.2911
  </td>
</td>
  <td align="center">
<td align="right" rowspan="1" bgcolor="lightblue">
<math>~=~</math>
1.6909
  </td>
</td>
  <td align="left">
<td align="right" rowspan="1" bgcolor="lightblue">
<math>~
6.9541
\biggl( \frac{3^7 \cdot 5}{2^7\pi^2 } \biggr)^{1/2} \biggl( \frac{\pi^2}{2\cdot 3^7 \cdot 5} \biggr)^{1/2}
</td>
\ell (1 + \ell^2)^{-1} \cdot \ell^{-6} (1+\ell^2)^{4} 
\biggl[ \ell \biggl(\ell^4 - \frac{8}{3}\ell^2 - 1\biggr) (1 + \ell^2)^{-3}  + \tan^{-1}(\ell) \biggr]
\biggl(\frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{2}
</math>
  </td>
</tr>
</tr>


<tr>
<tr>
  <td align="right">
<td align="right" rowspan="1" bgcolor="lightblue">
&nbsp;
0.2260
  </td>
</td>
  <td align="center">
<td align="right" rowspan="1" bgcolor="lightblue">
<math>~=~</math>
1.5229
  </td>
</td>
  <td align="left">
<td align="right" rowspan="1" bgcolor="lightblue">
<math>~
6.9091
2^{-4} \ell^{-5} (1+\ell^2)^{3} 
</td>
\biggl[ \ell \biggl(\ell^4 - \frac{8}{3}\ell^2 - 1\biggr) (1 + \ell^2)^{-3}  + \tan^{-1}(\ell) \biggr]
</tr>
\biggl(\frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{2} \, .
 
</math>
<tr>
  </td>
<td align="right" rowspan="1" bgcolor="lightblue">
0.1545
</td>
<td align="right" rowspan="1" bgcolor="lightblue">
1.2749
</td>
<td align="right" rowspan="1" bgcolor="lightblue">
6.8786
</td>
</tr>
</tr>
</table>
</table>
</div>
</div>
Finally, drawing from our [[User:Tohline/SSC/Virial/FormFactors#Gravitational_Potential_Energy|accompanying derivation of expressions for the structural form factors in this case]], we know that,
<div align="center">
<table border="0" cellpadding="3">


In the bottom panel of this figure, the undulating free-energy surface is drawn in three dimensions and viewed from a vantage point that illustrates its "valley of stability" and "ridge of instability;" the surface color correlates with the value of the free energy.  Twelve small colored dots identify extrema &#8212; either the bottom of a valley or the top of a ridge &#8212; in the free-energy function and therefore trace the mass-radius relation defined by the scalar virial theorem.  The 3D coordinates of these twelve points are provided in the three-column table that is affixed to the righthand edge of the figure:  The coordinates of the (only) red dot are provided in row 13 of the table (red has also been assigned as the "bgcolor" of this table row); the equilibrium configuration having the greatest mass along the intersecting Stahler sequence is identified by the (only) black dot (coordinates are provided in row 11 of the table and, correspondingly, bgcolor="black" for that row); bgcolor="lightblue" has been assigned to the other rows of the affixed table that provide coordinates of the other 10, blue dots.
The upper-right panel of this figure presents the two-dimensional projection that results from viewing the identical free-energy surface "from above," along a line of sight that is parallel to the free-energy <math>~(Z)</math> axis and looking directly down onto the radius-mass <math>~(X-Y)</math> plane. From this vantage point, the twelve small colored dots cleanly trace out the <math>~M_\mathrm{limit}(R_\mathrm{eq})</math> equilibrium sequence that is defined by the scalar virial theorem, exactly reproducing the "Virial" curve that is depicted in the mass-radius diagram shown in the upper-left panel of the figure.
==Discussion==
A spherically symmetric, self-gravitating gas cloud whose effective adiabatic exponent is <math>~\gamma < 4/3</math> &#8212; equivalently, <math>~n > 3</math> &#8212; cannot exist in a dynamically stable equilibrium state, in isolation.  Such clouds can be stabilized, however, if they are embedded in a hot, tenuous external medium and effectively confined by an external pressure, <math>~P_e</math>.  The pressure-truncated, <math>~n = 5 ~(\gamma = 6/5)</math> polytropic configurations being discussed here provide examples of such embedded clouds.  A direct analogy can be drawn between this discussion and discussions of pressure-truncated isothermal <math>~(\gamma = 1; n = \infty)</math> clouds &#8212; see, for example, our review of isothermal cloud structures in the context of [[User:Tohline/SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere|Bonnor-Ebert spheres]]. 
===Part I:  Physical Significance of the Two Curves===
The "Stahler" mass-radius relation, plotted as a continuous curve [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Plotting_Stahler.27s_Relation|in the above figure]] and reproduced as a sequence of discrete points in each panel of the subsequent [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Plotting_the_Virial_Theorem_Relation|comparison figure]], identifies the precise mass <math>~(\mathcal{Y})</math> and associated radius <math>~(\mathcal{X})</math> of physically allowed pressure-truncated, <math>~n = 5</math> polytropic configurations over the full range of values of the dimensionless truncation radius, <math>~0 < \tilde\xi < \infty</math>.  Each model along the curve has an internal structure that ensures detailed force balance throughout the configuration; because this internal structure varies from model to model, the values of the structural form-factors &#8212; <math>~\mathfrak{f}_M, \mathfrak{f}_W</math>, and <math>~\mathfrak{f}_A</math> &#8212; and the corresponding values of the coefficients associated with the free-energy function &#8212; <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math> &#8212; will also vary from model to model along the Stahler curve. 
If the values of the coefficients, <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math> (as well as the external pressure and, hence, the additional coefficient, <math>~\mathcal{D}</math>) are  held fixed, the  [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Free_Energy_Function_and_Virial_Theorem|algebraic free-energy function]] defines how a configuration's free energy will change as its overall size is varied.  Extrema in the free energy will identify equilibrium configurations.  Based on this understanding, our derived virial theorem expression for <math>~n = 5</math> polytropic configurations identifies equilibrium radii <math>~(\mathcal{X})</math> associated with various configuration masses <math>~(\mathcal{Y})</math>.  The "Virial" curve that has been plotted in each panel of the above [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Plotting_the_Virial_Theorem_Relation|comparison figure]] shows how the equilibrium radius varies with configuration mass, as dictated by the virial theorem &#8212; and, hence, as identified by extrema in the free-energy function &#8212; assuming that the relevant free-energy coefficients are held fixed.  In each figure panel, this "Virial" curve ''qualitatively'' resembles the quantitatively correct, "Stahler" mass-radius relationship that has been derived from the properties of detailed force-balance models.  The two curves overlap, and cross, wherever the coefficients used to define the "Virial" relation are identical to the coefficient values that are associated with a specific model along the "Stahler" relation.  The two curves do not trace out identical mass-radius relationships simply because the structural form factors vary from model to model along the "Stahler" sequence.
In the context of star formation, the Stahler sequence can be viewed as an ''evolutionary'' sequence for cold protostellar gas clouds that are embedded in a hot, tenuous interstellar medium.  An initially low-mass cloud is represented by an equilibrium configuration that has been truncated at a very small Lane-Emden radius, <math>~\tilde\xi</math>; such clouds will appear near the origin of the displayed <math>~\mathcal{X}-\mathcal{Y}</math> plane, at a point along the "lower" segment of the Stahler mass-radius relation.  Over time, as the cloud grows in mass (through collisions with and accretion of other low-mass clouds, for example), it will slide up the lower segment of the Stahler curve, moving in a counter-clockwise direction further and further away from the plot origin.  The mass-accretion process that drives the cloud's evolution presumably occurs on a time scale that is long compared to the local dynamical-readjustment time of the cloud, allowing the cloud's internal structure time to readjust and establish the properties defined by Stahler's detailed force-balance analysis.
===Part II:  Curve Intersections===
====Early Thoughts====
Notice that, in each frame of the [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#OverlapPlots|above comparison figure]], the "Virial" curve intersects and crosses the "Stahler" curve at two locations.  In each plot these two crossing points are identified by filled black circles and, in each plot, the crossing point that lies farthest along the curve &#8212; again, starting from near the origin and moving around the curve in a counter-clockwise direction &#8212; is associated with the equilibrium model on the Stahler curve that is defined by the same value of <math>~\tilde\xi</math> that was used to define the coefficients of the virial theorem mass-radius relation.  This is not surprising, as the virial theorem ''should'' be precisely satisfied by every one of the equilibrium models along the Stahler sequence, as long as the value of <math>~\tilde\xi</math> that is used to define the coefficients of the free-energy function and, in turn, the virial theorem mass-radius relation is identical to the value of <math>~\tilde\xi</math> that defines the truncation radius of the detailed force-balance model.  For example, the "Virial" curve that appears in the top-right panel of the comparison figure &#8212; a panel whose title includes the notation, <math>~\xi = 3</math> &#8212; intersects the "Stahler" curve at <math>~\mathcal{Y}_\mathrm{max}</math>, that is, at the location of the detailed force-balance model that, as previously explained, has a truncation radius, <math>~\tilde\xi = 3</math>. 
It is not (yet) clear to us what physical significance should be ascribed to the model along the Stahler sequence that is identified by the ''second'' crossing of the "Virial" curve, given that the value of <math>~\tilde\xi</math> associated with the truncation radius of this ''second'' detailed force-balance model is not the same as the value of <math>~\tilde\xi</math> that was used to define the coefficients of the "Virial" curve.  We note that, at least for the range of values of <math>~\tilde\xi</math> sampled in the above figure, this second crossing point seems to hover around the same limited segment of the Stahler sequence. 
By direct analogy with [[User:Tohline/SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere|discussions of Bonnor-Ebert spheres]], the "maximum mass" model associated with <math>~\mathcal{Y}_\mathrm{max}</math> along the Stahler mass-radius relation has important physical significance in astrophysics.  For a given applied external pressure, however, no models exist above some limiting mass &#8212; identified, here, by <math>~\mathcal{Y}_\mathrm{max}</math>.
====Analysis Philosophy====
The mass-radius relationship that derives from detailed force-balanced models is a physically meaningful and reliable statement of how a configuration's equilibrium radius will vary if its mass is changed.  (It must be accepted that the configuration's structural form factors will change as it settles into each new equilibrium state, so such an "evolution" must occur on a ''secular'' time scale.)  From the outset, however, the mass-radius relationship derived via the virial theorem &#8212; which, itself, derives from an analysis of the free energy function &#8212; should not be relied upon for the same physical insight.  Consider, for example, that the scalar virial theorem is obtained from an analysis of the free-energy function by varying a system's size ''while holding constant all coefficients in the free-energy expression''; this means that the system's mass as well as its structural form factors is held fixed while searching for an extremum in the free energy.  The temptation, then, is to use the virial theorem to predict what the configuration's new equilibrium size will be if the system's mass is changed while holding the coefficients in the virial theorem constant.  This means holding the structural form factors constant but not simultaneously holding the mass constant, and this differs from the constraints put on the free-energy function analysis that led to the virial theorem expression in the first place!
But we can combine the two analyses &#8212; the detailed force-balance analysis and the free-energy analysis &#8212; in the following meaningful way.  Use the detailed force-balance analysis to identify the properties of an equilibrium state, specifically, for a given mass, determine the system's equilibrium radius and its accompanying structural form factors.  (The virial theorem will be satisfied by this same set of determined parameter values.)  Then, holding both the mass and the structural form factors constant, see how the free energy of the system varies as the configuration's size changed.  In this manner the system's ''dynamical'' stability can be ascertained.
In summary:  The mass-radius relationship determined from an analysis of detailed force-balanced models defines the physically correct ''secular'' evolutionary track for the system; while, an analysis of the free energy variations about an equilibrium state will answer the question of ''dynamical'' stability.
====Quantitative Study====
The preceding philosophical statements not withstanding, it is still worth understanding the relationship &#8212; if any &#8212; between the pair of models that are identified by the "second crossing" of the Stahler sequence by the "Virial" curve.
<div align="center" id="SecondaryOverlap">
<table border="1" cellpadding="8" align="center">
<tr>
<tr>
  <td align="right">
   <td align="center" colspan="6">
<math>~\frac{M_\mathrm{limit}}{M_\mathrm{tot} } </math>
More Information on Secondary Overlap Points
  </td>
   <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\ell^3 (1+\ell^2)^{-3/2} \, ,
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
which gives,
<div align="center">
<table border="0" cellpadding="4">
<tr>
<tr>
   <td align="right">
   <td align="center" colspan="2">From [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#OverlapPlots|above table]]</td>
<math>~A</math>
  <td align="center" colspan="4">Determined here</td>
</tr>
<tr>
  <td align="center">
<math>~\mathcal{X} \equiv \frac{R_\mathrm{eq}}{R_\mathrm{SWS}}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=~</math>
<math>~\mathcal{Y} \equiv \frac{M_\mathrm{limit}}{M_\mathrm{SWS}}</math>
   </td>
   </td>
   <td align="left">
   <td align="center">
<math>~  
<math>~\tilde\xi_-</math>
\frac{\ell}{2^{4}
  </td>
\biggl[ \ell \biggl(\ell^4 - \frac{8}{3}\ell^2 - 1\biggr) (1 + \ell^2)^{-3} + \tan^{-1}(\ell) \biggr] \, .
  <td align="center">
</math>
<math>~\mathcal{Y}_-</math>
  </td>
  <td align="center">
<math>~\mathcal{A}_{M_\ell}</math>
  </td>
  <td align="center">
<math>~\mathcal{B}_{M_\ell}</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
This exactly matches the expression for the free-energy coefficient, <math>~\mathcal{A}</math>, that we derived separately in conjunction with our [[User:Tohline/SSC/Virial/FormFactors#Gravitational_Potential_Energy|derivation of expressions for the structural form factors]].
==Take Care Comparing Gravitational Potential Energies==
Does this derived relation for the coefficient, <math>~A</math>, make sense?  Well, we've derived the relation by comparing two separate expressions for the gravitational potential energy that were normalized in slightly different ways, so the leading numerical coefficient may not be correct.  We need to repeat the derivation, checking the relative normalizations carefully.  But before doing this, let's determine what we ''expected'' the relation to be, based on the expressions for the structural form factors that we have been using.
From the [[User:Tohline/SSC/Virial/PolytropesSummary#Serious_Concern|lead-in paragraphs of this subsection]], we have previously assumed that,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<tr>
  <td align="right">
<td align="center">
<math>~\mathcal{A}</math>
0.388938
</td>
<td align="center">
0.289568
</td>
  <td align="center">
0.72447
   </td>
   </td>
<td align="center">
0.289568
</td>
   <td align="center">
   <td align="center">
<math>~\equiv</math>
0.20491
   </td>
   </td>
   <td align="left">
   <td align="center">
<math>\frac{1}{5} \cdot \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2}
0.75191
= \frac{1}{5} \cdot \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \biggl\{ \frac{3^2\cdot 5}{5-n} \biggl[ \frac{\tilde\theta^'}{\tilde\xi} \biggr]^2 \biggr\}
\biggl\{ \biggl[ - \frac{3\tilde\theta^'}{\tilde\xi} \biggr] \biggr\}^{-2} = \frac{1}{(5-n)} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \, .
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
According to the line of reasoning presented above, the coefficient, <math>~A</math>, is related to the gravitational potential energy via the expression,
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
  <td align="right">
<td align="center">
<math>~W_\mathrm{grav}</math>
0.468918
</td>
<td align="center">
0.570916
</td>
  <td align="center">
0.98268
   </td>
   </td>
<td align="center">
0.570916
</td>
   <td align="center">
   <td align="center">
<math>~=</math>
0.20889
   </td>
   </td>
   <td align="left">
   <td align="center">
<math>~-3A\chi^{-1} E_\mathrm{norm} </math>
0.75395
   </td>
   </td>
</tr>
</tr>


<tr>
<tr>
  <td align="right">
<td align="center">
&nbsp;
0.507387
  </td>
</td>
  <td align="center">
<td align="center">
<math>~=</math>
0.798441
  </td>
</td>
  <td align="left">
<td align="center">
<math>~-3A \biggl( \frac{K^5}{G^3} \biggr)^{1/2} \biggl( \frac{G^5 M_\mathrm{tot}^4}{K^5} \biggr)^{1/2} \frac{1}{R}</math>
1.17380
  </td>
</td>
<td align="center">
0.798441
</td>
<td align="center">
0.21252
</td>
<td align="center">
0.75652
</td>
</tr>
</tr>


<tr>
<tr>
  <td align="right">
<td align="center">
&nbsp;
0.515168
  </td>
</td>
  <td align="center">
<td align="center">
<math>~=</math>
0.859518
  </td>
</td>
  <td align="left">
<td align="center">
<math>~-3A \biggl(\frac{GM_\mathrm{tot}^2}{R} \biggr) \, .</math>
1.22572
  </td>
</td>
<td align="center">
0.859520
</td>
<td align="center">
0.21360
</td>
<td align="center">
0.75740
</td>
</tr>
</tr>
</table>
 
</div>
From our [[User:Tohline/SphericallySymmetricConfigurations/Virial#Expressions_for_Various_Energy_Terms|introductory layout of the free-energy function for polytropes]] &#8212; see, also, p. 64, Equation (12) of [[User:Tohline/Appendix/References|Chandrasekhar [C67]]] &#8212; the gravitational potential energy is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
  <td align="right">
<td align="center">
<math>~W_\mathrm{grav}</math>
0.518588
  </td>
</td>
  <td align="center">
<td align="center">
<math>~=</math>
0.888969
  </td>
</td>
  <td align="left">
<td align="center">
<math>~ - \int_0^{R_\mathrm{limit}} \biggl( \frac{GM_r}{r} \biggr) 4\pi r^2 \rho dr \, ,</math>
1.25104
  </td>
</td>
<td align="center">
0.888968
</td>
<td align="center">
0.21414
</td>
<td align="center">
0.75785
</td>
</tr>
 
<tr>
<td align="center" bgcolor="yellow">
0.520269
</td>
<td align="center" bgcolor="yellow">
0.904143
</td>
<td align="center">
1.26419
</td>
<td align="center">
0.904142
</td>
<td align="center">
0.21443
</td>
<td align="center">
0.75810
</td>
</tr>
</tr>
</table>
</table>
</div>
</div>
where,
 
 
====Marginal Stability====
As mentioned above, it is widely appreciated that the model having the largest mass &#8212; that is, the model that sits at <math>~\mathcal{Y}_\mathrm{max}</math> &#8212; along the Stahler sequence is of considerable astrophysical significance.  Viewed in terms of a cloud's secular evolution, counter-clockwise along the sequence, something rather catastrophic must happen once the cloud acquires the mass associated with <math>~\mathcal{Y}_\mathrm{max}</math>, because no equilibrium structure is available to the cloud if it gains any additional mass.  It is tempting to associate this point along the Stahler sequence with a dynamical instability, imagining for example that the cloud will begin to dynamically collapse once it reaches this <math>~\mathcal{Y}_\mathrm{max}</math> configuration.  But the "detailed force-balance" technique that is used to define the structure of equilibrium models along the Stahler sequence does not give us any insight regarding a configuration's dynamical stability.
 
Our free-energy analysis ''does'' provide this additional insight.  The mass-radius relationship derived from the scalar virial theorem &#8212; which, itself, was derived via a free-energy analysis &#8212; is qualitatively similar to the mass-radius relationship defined (from a detailed force-balance analysis) by the Stahler sequence; in particular, it also exhibits an upper mass limit.  And our free-energy analysis reveals that this "maximum mass" point associated with the virial theorem separates dynamically stable from dynamically unstable models along the sequence.  This realization fuels the temptation just mentioned; that is, it seems to support the idea that the configuration at <math>~\mathcal{Y}_\mathrm{max}</math> along Stahler's sequence is associated with the onset of a dynamical instability along the sequence.  But this is not the case!  Our free-energy analysis has also shown that, when the structural form-factors &#8212; and, most specifically, the coefficients <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math> &#8212; are assigned the values appropriate to  the configuration at <math>~\mathcal{Y}_\mathrm{max}</math> along Stahler's sequence, the point of maximum mass associated with the corresponding expression for the virial theorem does not coincide with the configuration at <math>~\mathcal{Y}_\mathrm{max}</math>.  The configuration at <math>~\mathcal{Y} = \mathcal{Y}_\mathrm{max} =  1.774078</math> (also identified as the model having <math>~\tilde\xi = 3.0</math>) is found to be dynamically stable.  Both of these realizations are illustrated graphically in the [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#GraphicalDepictionXi3|above figure]].
 
 
Our analysis has shown, instead, that the marginally unstable configuration appears farther along the Stahler sequence when moving in a counter-clockwise direction.  It corresponds to the model having <math>~\tilde\xi = 3.850652</math> instead of <math>~\tilde\xi = 3.0</math>.  While this can be illustrated graphically &#8212; for example, by carefully analyzing and comparing the bottom-center panel with the top-right panel in the [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#OverlapPlots|above figure ensemble]] &#8212; an algebraic demonstration is more definitive.  [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#Stability_4|Our stability analysis has shown]] that, for any pressure-truncated polytropic configuration, the equilibrium structure associated with the point of marginal instability has,
<div align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5">
 
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~M_r</math>
<math>~\biggl( \frac{\mathcal{Y}}{\mathcal{X}^2}\biggr)_\mathrm{crit} </math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~\equiv</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\int_0^r 4\pi r^2 \rho dr \, .</math>
<math>~
\biggl[ \frac{4\pi n}{\mathcal{A}_{M_\ell}(n-3)}\biggr]^{1/2}
\, .
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</table>
</div>
</div>
Now, independent of the chosen normalization, if we use <math>~M_\mathrm{tot}</math> to represent the total mass of an ''isolated'' <math>~n=5</math> polytrope, then from [[User:Tohline/SSC/Structure/PolytropesEmbedded#Review_2|an earlier review]], we have,
For <math>~n=5</math> configurations, this means that the critical model along the equilibrium sequence will have,
<div align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5">
 
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~M_\mathrm{tot}</math>
<math>~\mathcal{X}_\mathrm{crit}^4 </math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 4,245: Line 4,124:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\biggr[ \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr]^{1/2} \rho_c^{-1/5} \, ,</math>
<math>~
   </td>
\biggl[ \frac{\mathcal{A}_{M_\ell}}{10\pi }\biggr] \mathcal{Y}_\mathrm{crit}^2
\, .
</math>
   </td>
</tr>
</tr>
</table>
</table>
</div>
</div>
and we can write, in terms of the Lane-Emden dimensionless radius, <math>~\xi</math>,
But all configurations along Stahler's equilibrium sequence must also obey the [[User:Tohline/SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D5.29|mass-radius relationship]],
<div align="center">
<div align="center">
<math>
<table border="0" cellpadding="5">
\frac{M_r}{M_\mathrm{tot}} = \xi^3 (3 + \xi^2)^{-3/2}  \, .
</math>
</div>
===Virial Chapter===
Now, in our discussion of the virial equilibrium of embedded polytropes, we used the normalizations specified above and wrote,
<div align="center">
<table border="0" cellpadding="5" align="center">
 
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~W_\mathrm{grav}</math>
<math>~\mathcal{Y}^2 - 5\mathcal{Y}\mathcal{X} + \frac{20\pi}{3} \mathcal{X}^4</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 4,269: Line 4,144:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>~
- 4\pi GM_\mathrm{tot} R_\mathrm{norm}^2 \rho_\mathrm{norm} \int_0^{R_\mathrm{limit}/R_\mathrm{norm}} \biggl[\frac{M_r}{M_\mathrm{tot}} \biggr]  r^* \rho^* dr^*
0
\, .
</math>
</math>
   </td>
   </td>
</tr>


</table>
</div>
Combining these two requirements means,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>~\mathcal{Y}_\mathrm{crit}^2 - 5(\mathcal{Y}\mathcal{X})_\mathrm{crit} + \biggl( \frac{2\mathcal{A}_{M_\ell}}{3}\biggr) \mathcal{Y}_\mathrm{crit}^2</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 4,282: Line 4,164:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>~
- E_\mathrm{norm} \int_0^{R_\mathrm{limit}/R_\mathrm{norm}} 3\biggl[\frac{M_r}{M_\mathrm{tot}} \biggr]  r^* \rho^* dr^* \, .
0
</math>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
We can replace <math>~r^* \equiv r/R_\mathrm{norm}</math> with <math>~\xi \equiv r/a_5</math> by recognizing that,
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~M_\mathrm{tot}</math>
<math>~\Rightarrow ~~~~\mathcal{Y}_\mathrm{crit}^2 \biggl[ 1 + \frac{2}{3}\cdot \mathcal{A}_{M_\ell} \biggr] </math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 4,302: Line 4,179:
   <td align="left">
   <td align="left">
<math>~
<math>~
\biggl( \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr)^{1/2} \rho_c^{-1/5}  
5(\mathcal{Y}\mathcal{X})_\mathrm{crit}  
</math>
</math>
   </td>
   </td>
Line 4,309: Line 4,186:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\Rightarrow ~~~ \rho_c^{-2/5} </math>
<math>~\Rightarrow ~~~~\frac{ \mathcal{X}_\mathrm{crit} }{ \mathcal{Y}_\mathrm{crit} }</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 4,316: Line 4,193:
   <td align="left">
   <td align="left">
<math>~
<math>~
\biggl\frac{\pi G^3}{2\cdot 3^4 K^3} \biggr) M_\mathrm{tot}^2 \, ;
\frac{1}{5}\biggl[ 1 + \frac{2}{3}\cdot \mathcal{A}_{M_\ell} \biggr] \, .
</math>
</math>
   </td>
   </td>
</tr>
</tr>
 
</table>
</div>
Now, taking into detailed account the internal structure of pressure-truncated, <math>~n=5</math> polytropic structures as represented in [[User:Tohline/SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D5.29|our summary table of Stahler's equilibrium configurations]], we know that, along Stahler's entire sequence,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~a_5</math>
<math>~\frac{ \mathcal{X} }{ \mathcal{Y} }</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 4,329: Line 4,210:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~\biggl\{ \biggl( \frac{3\cdot 5}{2^2 \pi} \biggr) \frac{\ell^2}{(1+\ell^2)^{2}}  \cdot
\biggl( \frac{3K}{2\pi G} \biggr)^{1/2} \rho_c^{-2/5}  
\biggl( \frac{2^2\pi}{3 \cdot 5^3} \biggr) \frac{(1+\ell^2)^{4}}{(\ell^2)^3}
\biggr\}^{1/2}
</math>
</math>
   </td>
   </td>
Line 4,343: Line 4,225:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~\frac{1 + \ell^2}{5\ell^2}
\biggl(  \frac{3K}{2\pi G} \biggr)^{1/2} \biggl(  \frac{\pi G^3}{2\cdot 3^4 K^3} \biggr)  M_\mathrm{tot}^2
\, ,
</math>
</math>
   </td>
   </td>
</tr>
</tr>


<tr>
</table>
  <td align="right">
</div>
&nbsp;
where we have again adopted the shorthand notation,
  </td>
<div align="center">
  <td align="center">
<math>~\ell^2 \equiv \frac{\tilde\xi^2}{3} \, .</math>
<math>~=</math>
</div>
  </td>
  <td align="left">
<math>~
\biggl(  \frac{\pi}{2^3\cdot 3^7} \biggr)^{1/2\biggl(  \frac{G}{K} \biggr)^{5/2} M_\mathrm{tot}^2 \, ;
</math>
  </td>
</tr>


We conclude, therefore, that in the marginally unstable model along the Stahler equilibrium sequence,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~R_\mathrm{norm}</math>
<math>~1 + \frac{2}{3}\cdot (\mathcal{A}_{M_\ell})_\mathrm{crit}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~=</math>
  </td>
   <td align="left">
   <td align="left">
<math>~
<math>~\frac{1 + \ell_\mathrm{crit}^2}{\ell_\mathrm{crit}^2}
\biggl(  \frac{G}{K} \biggr)^{5/2} M_\mathrm{tot}^{2}  
</math>
</math>
   </td>
   </td>
Line 4,379: Line 4,255:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\Rightarrow~~~ \frac{a_5}{R_\mathrm{norm}}</math>
<math>~\Rightarrow ~~~~~(\mathcal{A}_{M_\ell})_\mathrm{crit} </math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~=</math>
  </td>
   <td align="left">
   <td align="left">
<math>~
<math>~\frac{3}{2} \cdot \ell_\mathrm{crit}^{-2} \, .
\biggl(  \frac{\pi}{2^3\cdot 3^7} \biggr)^{1/2} \, .
</math>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</table>
</div>
</div>
Hence,
Given that the [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Plotting_the_Virial_Theorem_Relation|general expression for]] <math>~\mathcal{A}_{M_\ell}</math> along the Stahler sequence is,
<div align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5">
 
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~r^*</math>
<math>~\mathcal{A}_{M_\ell} </math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~=</math>
  </td>
   <td align="left">
   <td align="left">
<math>~
<math>~\frac{1}{2^4}
\biggl( \frac{\pi}{2^3\cdot 3^7} \biggr)^{1/2} \xi \, .
\biggl[ \ell^{-4} \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr) + \ell^{-5}  (1 + \ell^2)^{3} \tan^{-1}(\ell ) \biggr] \, ,
</math>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</table>
</div>
we deduce that,
<div align="center">
<math>~\ell_\mathrm{crit} = 2.2231751 </math>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
or, equivalently,
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>~\tilde\xi_\mathrm{crit} = 3.850652 \, .</math>
</div>
Hence, also,
<div align="center">
<math>~( \mathcal{X}_\mathrm{crit},  \mathcal{Y}_\mathrm{crit} ) = ( 0.408738, 1.699778 ) \, . </math>
</div>
</div>


Also,
===Part III===
<div align="center">
From our above, detailed analysis of the mass-radius relation for pressure-truncated polytropes, we concluded that configurations along "Stahler's" equilibrium sequence become dynamically unstable at a point that does not coincide with the maximum-mass configuration.  Instead, the onset of dynamical instability is associated with the critical point on the mass-radius relation that arises from the free-energy-based virial theorem.  In drawing this conclusion, we have implicitly assumed that the proper way to analyze an equilibrium configuration's stability is to vary its radius while, not only holding its mass, specific entropy, and surface pressure <math>~(P_e)</math> constant, but also assuming that the configuration's structural form factors are invariable. 
<table border="0" cellpadding="5" align="center">
 


<tr>
This seems like a reasonable assumption, given that we're asking how a configuration's characteristics will vary ''dynamically'' when perturbed about an equilibrium state.  While oscillating about an equilibrium state, it seems more reasonable to assume that the system will expand and contract in a nearly homologous fashion than that its internal structure will readily readjust to produce a different ''and'' desirable set of form factors.  In support of this argument, we point to the paper by [http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich &amp; Weber (1980)] which explicitly derives a self-similar solution for the ''homologous'' collapse of stellar cores that can be modeled as <math>~n=3</math> polytropes; an associated [[User:Tohline/Apps/GoldreichWeber80#Homologously_Collapsing_Stellar_Cores|chapter of this H_Book details the Goldreich &amp; Weber derivation]].  Goldreich &amp; Weber use [[User:Tohline/SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29|linear perturbation techniques]] to analyze the stability of their homologously collapsing configurations.  In &sect;IV of their paper, they describe the eigenvalues and eigenfunctions that result from this analysis.  They discovered, for example, that "the lowest radial mode can be found analytically ... [and it] corresponds to a homologous perturbation of the entire core."  Our assumption that the structural form factors remain constant when pressure-truncated polytropic configurations undergo radial size variations therefore appears not to be unreasonable.  (Based on the Goldreich &amp; Weber discussion, we should also look at the published work of [http://adsabs.harvard.edu/abs/1941ApJ....94..245S Schwarzschild (1941, ApJ, 94, 245)], who has evaluated radial modes, and of [http://adsabs.harvard.edu/abs/1941MNRAS.101..367C Cowling (1941, MNRAS, 101, 367)], who has obtained eigenvalues of some low-order nonradial modes.)
  <td align="right">
 
<math>~\rho_\mathrm{norm}</math>
 
  </td>
In addition, it would seem that a certain amount of dissipation would be required for the system to readjust to new structural form factors.  In order to test this underlying assumption, following [http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich &amp; Weber (1980)], it would be desirable to carry out a full-blown perturbation analysis that involves looking for, for example, the eigenvector associated with the system's fundamental radial mode of pulsation.  Ideally, we should be using the structural form factors associated with this pulsation-mode eigenfunction in our free-energy analysis of stability.  Better yet, the ''sign'' of the eigenfrequency associated with the system's pulsation-mode eigenvector should signal whether the system is dynamically stable or unstable.
  <td align="center">
 
<math>~=</math>
 
  </td>
{{LSU_WorkInProgress}}
  <td align="left">
 
<math>~
==Other Model Sequences==
\frac{3}{4\pi} \biggl[ \frac{K}{G} \biggr]^{15/2} M_\mathrm{tot}^{-5}
===Relating and Reconciling Two Mass-Radius Relationships for n = 4 Polytropes===
</math>
 
  </td>
For pressure-truncated <math>~n=4</math> polytropes, [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)] did not identify a polynomial relationship between the mass and radius of equilibrium configurations.  However, from his analysis of detailed force-balance models ([[User:Tohline/SSC/Virial/PolytropesSummary#Detailed_Force-Balanced_Solution_2|summarized above]]), we appreciate that the governing pair of parametric relations is,
</tr>
<div align="center">
<table border="0" cellpadding="3">


<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>
~\mathcal{X}
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~=~</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>
\frac{3}{4\pi} \biggl[ \frac{K}{G} \biggr]^{15/2} \biggl[ \biggl(  \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr)^{1/2} \rho_c^{-1/5}\biggr]^{-5}
\biggl( \frac{1}{\pi} \biggr)^{1/2} \tilde\xi \tilde\theta^{3/2} \, ,
</math>
</math>
   </td>
   </td>
Line 4,446: Line 4,335:
<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>
~\mathcal{Y}
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~=~</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>
\frac{3}{4\pi} \biggl[ \frac{K}{G} \biggr]^{15/2} \biggl( \frac{\pi G^3}{2\cdot 3^4 K^3} \biggr)^{5/2} \rho_c
\biggl( \frac{2^4}{\pi} \biggr)^{1/2} \tilde\theta^{1/2} (- \tilde\xi^2 \tilde\theta^') \, .
</math>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
On the other hand, the polynomial that results from plugging <math>~n=4</math> into the [[User:Tohline/SSC/Virial/PolytropesSummary#ConciseVirialXY|general mass-radius relation that is obtained via the virial theorem]] is,


<tr>
<div align="center">
  <td align="right">
<math>
&nbsp;
\frac{4\pi}{3} \mathcal{X}^4   - \biggl[ \frac{\mathcal{X} \mathcal{Y}^{5}}{4\pi}\biggr]^{1/4} \mathfrak{b}_{n=4} + \frac{5}{4} \mathcal{Y}^2  
  </td>
= 0 \, ,
  <td align="center">
<math>~=</math>
   </td>
  <td align="left">
<math>~
\biggl(\frac{3^2}{2^4\pi^2} \biggr)^{1/2} \biggl(  \frac{\pi^5 }{2^5 \cdot 3^{20}} \biggr)^{1/2} \rho_c
</math>
</math>
  </td>
</div>
</tr>
where,
 
<div align="center">
<tr>
<math>\mathfrak{b}_{n=4} = \biggl[ 5 (-\tilde\theta^')^2 + \frac{1}{3} \tilde\theta^{5} \biggr]
  <td align="right">
\biggl( \frac{\tilde\xi}{-\tilde\theta^'} \biggr)^{5/4} \, .
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{\pi^3 }{2^9 \cdot 3^{18}} \biggr)^{1/2} \rho_c \, .
</math>
</math>
  </td>
</tr>
</table>
</div>
</div>
Hence,
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
[For the record we note that, throughout the structure of an <math>~n=4</math> polytrope, <math>~\mathfrak{b}_{n=4}</math> is a number of order unity.  Its value is never less than <math>~3^{1/4}</math>, which pertains to the center of the configuration; its maximum value of <math>\approx 5.098</math> occurs at <math>~\tilde\xi \approx 4.0</math>; and <math>~\mathfrak{b}_{n=4} \approx 3.946</math> at its (zero pressure) surface, <math>~\tilde\xi = \xi_1 \approx 14.97</math>.  A plot showing the variation with <math>~P_e</math> of the closely allied parameter, <math>~\mathcal{B}|_{n=4} = (4\pi)^{1/4} \mathfrak{b}_{n=4}</math> is presented in the righthand panel of the [[User:Tohline/SSC/Virial/PolytropesSummary#Summary|above parameter summary figure]].]
  <td align="right">
 
<math>~\rho^* \equiv \frac{\rho}{\rho_\mathrm{norm}}</math>
In both panels of the following figure, the blue curve displays the mass-radius relation for pressure-truncated <math>~n=4</math> polytropes, <math>~\mathcal{Y}(\mathcal{X})</math>, that is generated by Stahler's pair of parametric equations.  The coordinates of discrete points along the curve have been determined from the tabular data provided on p. 399 of [http://adsabs.harvard.edu/abs/1986Ap%26SS.126..357H Horedt (1986, ApJS, vol. 126)]] while Excel has been used to generate the "smooth," continuous blue curve connecting the points; this set of points and accompanying blue curve are identical in both figure panels.  In both figure panels, a set of discrete, triangle-shaped points traces the mass-radius relation, <math>~\mathcal{Y}(\mathcal{X})</math>, that is obtained via the virial theorem, assuming that the coefficient, <math>~\mathfrak{b}_{n=4}</math>, is constant along the sequence.  The "green" sequence in the lefthand panel results from setting <math>~\mathfrak{b}_{n=4} = 3.4205</math>, which is the value of the constant that results from Horedt's tabulated data if the configuration is truncated at <math>~\tilde\xi = 1.4</math>; the "orange" sequence in the righthand panel results from setting <math>~\mathfrak{b}_{n=4} = 4.8926</math>, which is the value of the constant that results from Horedt's tabulated data if the configuration is truncated at <math>~\tilde\xi = 2.8</math>.
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2^9 \cdot 3^{18}}{\pi^3 } \biggr)^{1/2} \frac{\rho}{\rho_c}  
</math>
  </td>
</tr>


<table border="1" cellpadding="8" align="center">
<tr>
<tr>
   <td align="right">
   <th align="center">
&nbsp;
Comparing Two Separate Mass-Radius Relations for Pressure-Truncated ''n = 4'' Polytropes
  </td>
   </th>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(  \frac{2^9 \cdot 3^{18}}{\pi^3 } \biggr)^{1/2} \biggl[ 1 + \frac{\xi^2}{3} \biggr]^{-5/2} \, .
</math>
   </td>
</tr>
</tr>
<tr><td align="center">
[[File:CompareN4SequencesRevised.png|750px|Comparison of Two Mass-Radius Relations]]
</td></tr>
</table>
</table>
</div>
So the energy integral becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
According to [http://adsabs.harvard.edu/abs/1986Ap%26SS.126..357H Horedt's (1986)]] tabulated data, the surface of an isolated <math>~(P_e = 0)</math>, spherically symmetric, <math>~n=4</math> polytrope occurs at the dimensionless (Lane-Emden) radius, <math>~\xi_1 = 14.9715463</math>.  In both panels of the above figure, this ''isolated'' configuration is identified by the discrete (blue diamond) point at the origin, that is, at <math>~(\mathcal{X}, \mathcal{Y}) = (0, 0)</math>.  As we begin to examine pressure-truncated models and <math>~\tilde\xi</math> is steadily decreased from <math>~\xi_1</math>, the mass-radius coordinate of equilibrium configurations "moves" away from the origin, upward along the upper branch of the displayed (blue) mass-radius relation.  A maximum mass of <math>~\mathcal{Y} \approx 2.042</math> (corresponding to a radius of <math>~\mathcal{X} \approx 0.4585</math>) is reached ''from the left'' as <math>~\tilde\xi</math> drops to a value of approximately <math>~3.4</math>.  As <math>~\tilde\xi</math> continues to decrease, the mass-radius coordinates of equilibrium configurations move along the lower branch of the displayed (blue) curve, reaching a maximum radius at <math>~(\mathcal{X}, \mathcal{Y}) \approx (0.555, 1.554)</math> &#8212; corresponding to <math>~\tilde\xi \approx 2.0</math> &#8212; then decreasing in radius until, once again, the origin is reached, but this time because <math>~\tilde\xi</math> drops to zero.
 
 
If we set <math>~\mathfrak{b}_{n=4} = 3.4205</math> (corresponding to a choice of <math>~\tilde\xi = 1.4</math>), the virial theorem mass-radius relation maps onto the "Stahler" mass-radius coordinate plane as depicted by the set of green, triangle-shaped points in the lefthand panel of the above figure.  While the (green) curve corresponding to this relation does not overlay the blue mass-radius relation, the two curves do intersect.  They intersect precisely at the coordinate location along the blue curve (emphasized by the black filled circle) corresponding to a detailed force-balanced model having <math>~\tilde\xi = 1.4</math>.    In an analogous fashion, in the righthand panel of the figure, the curve delineated by the set of orange triangle-shaped points shows how the virial theorem mass-radius relation maps onto the "Stahler" mass-radius coordinate plane when we set <math>~\mathfrak{b}_{n=4} = 4.8926</math> (corresponding to a choice of <math>~\tilde\xi = 2.8</math>); it intersects the blue mass-radius relation precisely at the coordinate location, <math>~(\mathcal{X}, \mathcal{Y}) \approx (0.5108, 1.965)</math> &#8212; again, emphasized by a black filled circle &#8212; that corresponds to a detailed force-balanced model having <math>~\tilde\xi = 2.8</math>.  Hence, the two relations give the same mass-radius coordinates when the value of <math>~\mathfrak{b}_{n=4}</math> that is plugged into the virial theorem matches the value of <math>~\mathfrak{b}_{n=4}</math> that reflects the structural form factor that is properly associated with a detailed force-balanced model.
 
 
When we mapped the virial theorem mass-radius relation onto Stahler's mass-radius coordinate plane using a value of <math>~\mathfrak{b}_{n=4} = 4.8926</math> (as traced by the orange triangle-shaped points in the righthand panel of the above figure), we expected it to intersect the blue curve at the point along the blue sequence where <math>~\tilde\xi = 2.8</math>, for the reason just discussed.  After constructing the plot, it became clear that the two curves also intersect at the coordinate location, <math>~(\mathcal{X}, \mathcal{Y}) \approx (0.255, 1.67)</math> &#8212; also highlighted by a black filled circle &#8212; that corresponds to a detailed force-balanced model having <math>~\tilde\xi \approx 6.0</math>.  This makes it clear that it is the equality of the structural form factors, not the equality of the dimensionless (Lane-Emden) radius, <math>~\tilde\xi</math>, that assures precise agreement between the two different mass-radius expressions.
 
 
As is detailed in our [[User:Tohline/SSC/Virial/PolytropesSummary#Stability|above discussion of the dynamical stability of pressure-truncated polytropes]], an examination of free-energy variations can not only assist us in identifying the properties of equilibrium configurations (via a free-energy derivation of the virial theorem) but also in determining which of these configurations are dynamically stable and which are dynamically unstable.  We showed that, for a certain range of polytropic indexes, there is a critical point along the corresponding model sequence where the transition from stability to instability occurs.  As has been detailed in our [[User:Tohline/SSC/Virial/PolytropesSummary#Try_Polytropic_Index_of_4|above groundwork derivations]], for <math>~n = 4</math> polytropic structures, the critical point is identified by the dimensionless parameters,
<div align="center">
<table border="0" cellpadding="5" align="center">
 
<tr>
  <td align="center">
<math>\eta_\mathrm{crit}\biggr|_{n=4}~=~\frac{1}{15} \, ;</math>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>\Pi_\mathrm{max}\biggr|_{n=4}~=~\frac{15^{15}}{16^{16}} \, ;</math>
&nbsp;&nbsp;&nbsp;&nbsp; and &nbsp;&nbsp;&nbsp;&nbsp;
<math>\Chi_\mathrm{min}\biggr|_{n=4}~=~\biggl( \frac{16}{15} \biggr)^4 \, .</math>
  </td>
</tr>
</table>
</div>
In the context of the above figure, independent of the chosen value of <math>~\mathfrak{b}_{n=4}</math>, this critical point always corresponds to the maximum mass that occurs along the mass-radius relationship established via the virial theorem.  In both panels of the figure, a horizontal red-dotted line has been drawn tangent to this critical point and identifies the corresponding critical value of <math>~\mathcal{Y}</math>; a vertical red-dashed line drawn through this same point helps identify the corresponding critical value of <math>~\mathcal{X}</math>.  We have deduced (details of the derivation not shown) that, for pressure-truncated <math>~n=4</math> polytropes, the coordinates of this critical point in Stahler's <math>~\mathcal{X}-\mathcal{Y}</math> plane depends on the choice of <math>~\mathfrak{b}_{n=4}</math> as follows:
<div align="center">
<table border="0" cellpadding="5" align="center">
 
<tr>
   <td align="right">
   <td align="right">
<math>~\frac{W_\mathrm{grav}}{E_\mathrm{norm} }</math>
<math>~\mathcal{X}_\mathrm{crit}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 4,532: Line 4,416:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>~\pi^{-1/2} 2^{-16/5} (3\mathfrak{b}_{n=4})^{4/5} \, ,</math>
- 3 \biggl(  \frac{\pi}{2^3\cdot 3^7} \biggr) \biggl(  \frac{2^9 \cdot 3^{18}}{\pi^3 } \biggr)^{1/2}  
\int_0^{\tilde\xi} \biggl[\xi^3 (3 + \xi^2)^{-3/2} \biggr]  \xi \biggl[ 1 + \frac{\xi^2}{3} \biggr]^{-5/2} d\xi
</math>
   </td>
   </td>
</tr>
</tr>
Line 4,541: Line 4,422:
<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>~\mathcal{Y}_\mathrm{crit}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 4,547: Line 4,428:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>~\pi^{-1/2} 2^{-22/5} (3\mathfrak{b}_{n=4})^{8/5} \, .</math>
- \biggl(  \frac{2^3 \cdot 3^{6}}{\pi } \biggr)^{1/2}  
\int_0^{\tilde\xi} \xi^4 \biggl[ 1 + \frac{\xi^2}{3} \biggr]^{-4} d\xi \, .
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</table>
</div>
</div>
This needs to be compared with the <math>~W_\mathrm{grav}^*</math> integral that we previously have handled in [[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#Expression_for_Free_Energy|the chapter discussing bipolytrope models]].
 
In practice, for a given plot of the type displayed in the above figure &#8212; that is, for a given choice of the structural parameter, <math>\mathfrak{b}_{n=4}</math> &#8212; it only makes sense to compare the location of this critical point to the location of points that have been highlighted by a filled black circle, that is, points that identify the intersection between the two mass-radius relations.  If, in a given figure panel, a filled black circle lies to the right of the vertical dashed line, the equilibrium configuration corresponding to that black circle is dynamically stable.  On the other hand, if the filled black circle lies to the left of the vertical dashed line, its corresponding equilibrium configuration is dynamically unstable.  We conclude, therefore, that the equilibrium configuration marked by a filled black circle in the lefthand panel of the above figure is ''stable''; however, both configurations identified by filled black circles in the righthand panel are ''unstable''.
 
It is significant that the critical point identified by our free-energy-based stability analysis does not correspond to the equilibrium configuration having the largest mass along "Stahler's" (blue) equilibrium model sequence.  One might naively expect that a configuration of maximum mass along the blue curve is the relevant demarcation point and that, correspondingly, all models along this sequence that fall "to the right" of this maximum-mass point are stable.  But the righthand panel of our above figure contradicts this expectation.  While both of the black filled circles in the righthand panel of the above figure lie to the left of the vertical dashed line and therefore, as just concluded, are both unstable, one of the two configurations lies ''to the right'' of the maximum-mass point along the blue "Stahler" sequence.  This finding is related to [[User:Tohline/SSC/Virial/PolytropesSummary#Curiosity|the curiosity raised earlier]] in our discussion of the structural properties of pressure-truncated, <math>~n=4</math> polytropes.
 
===Relating and Reconciling Two Mass-Radius Relationships for n = 3 Polytropes===
 
For pressure-truncated <math>~n=3</math> polytropes, [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)] did not identify a polynomial relationship between the mass and radius of equilibrium configurations.  However, from his analysis of detailed force-balance models ([[User:Tohline/SSC/Virial/PolytropesSummary#Detailed_Force-Balanced_Solution_2|summarized above]]), we appreciate that the governing pair of parametric relations is,
<div align="center">
<table border="0" cellpadding="3">
 
<tr>
  <td align="right">
<math>
~\mathcal{X}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{3}{4\pi} \biggr)^{1/2} \tilde\xi \tilde\theta \, ,
</math>
  </td>
</tr>
 
<tr>
  <td align="right">
<math>
~\mathcal{Y}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{3^3}{4\pi} \biggr)^{1/2} (- \tilde\xi^2 \tilde\theta^') \, .
</math>
  </td>
</tr>
</table>
</div>
On the other hand, the polynomial that results from plugging <math>~n=3</math> into the [[User:Tohline/SSC/Virial/PolytropesSummary#ConciseVirialXY|general mass-radius relation that is obtained via the virial theorem]] is,
 
<div align="center">
<math>
\frac{2^3 \pi}{3} \mathcal{X}^4  - \biggl[ \frac{\mathcal{Y}^{4}}{4\pi}\biggr]^{1/3} \mathfrak{b}_{n=3} + \frac{4}{3} \mathcal{Y}^2
= 0 \, ,
</math>
</div>
where,
<div align="center">
<math>\mathfrak{b}_{n=3} = \biggl[ 4 (-\tilde\theta^')^2 + \frac{2}{3} \tilde\theta^{4} \biggr]
\biggl( \frac{\tilde\xi}{-\tilde\theta^'} \biggr)^{4/3} \, .
</math>
</div>


{{LSU_HBook_footer}}
{{LSU_HBook_footer}}

Latest revision as of 22:08, 10 February 2019


Virial Equilibrium of Adiabatic Spheres (Summary)

The summary presented here has been drawn from our accompanying detailed analysis of the structure of pressure-truncated polytropes.

Whitworth's (1981) Isothermal Free-Energy Surface
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Detailed Force-Balanced Solution

As has been discussed in detail in another chapter, Horedt (1970), Whitworth (1981) and Stahler (1983) have separately derived what the equilibrium radius, <math>~R_\mathrm{eq}</math>, is of a polytropic sphere that is embedded in an external medium of pressure, <math>~P_e</math>. Their solution of the detailed force-balanced equations provides a pair of analytic expressions for <math>~R_\mathrm{eq}</math> and <math>~P_e</math> that are parametrically related to one another through the Lane-Emden function, <math>~\theta</math>, and its radial derivative. For example — see our related discussion for more details — from Horedt's work we obtain the following pair of equations:

<math> ~\frac{R_\mathrm{eq}}{R_\mathrm{norm}} = r_a \cdot \biggl( \frac{R_\mathrm{Horedt}}{R_\mathrm{norm}} \biggr) </math>

<math>~=~</math>

<math> \tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)} \biggl[ \frac{4\pi}{(n+1)^n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{n-1} \biggr]^{1/(n-3)} \, , </math>

<math> ~\frac{P_\mathrm{e}}{P_\mathrm{norm}} = p_a \cdot \biggl( \frac{P_\mathrm{Horedt}}{P_\mathrm{norm}} \biggr) </math>

<math>~=~</math>

<math> \tilde\theta^{n+1}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)} \biggl[ \frac{(n+1)^3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2}\biggr]^{(n+1)/(n-3)} \, , </math>

where we have introduced the normalizations,

<math>~R_\mathrm{norm}</math>

<math>~\equiv</math>

<math>~\biggl[ \biggl( \frac{G}{K} \biggr)^n M_\mathrm{tot}^{n-1} \biggr]^{1/(n-3)} \, ,</math>

<math>~P_\mathrm{norm}</math>

<math>~\equiv</math>

<math>~\biggl[ \frac{K^{4n}}{G^{3(n+1)} M_\mathrm{tot}^{2(n+1)}} \biggr]^{1/(n-3)} \, .</math>

In the expressions for <math>~r_a</math> and <math>~p_a</math>, the tilde indicates that the Lane-Emden function and its derivative are to be evaluated, not at the radial coordinate, <math>~\xi_1</math>, that is traditionally associated with the "first zero" of the Lane-Emden function and therefore with the surface of the isolated polytrope, but at the radial coordinate, <math>~\tilde\xi</math>, where the internal pressure of the isolated polytrope equals <math>~P_e</math> and at which the embedded polytrope is to be truncated. The coordinate, <math>~\tilde\xi</math>, therefore identifies the surface of the embedded — or, pressure-truncated — polytrope. We also have taken the liberty of attaching the subscript "limit" to <math>~M</math> in both defining relations because it is clear that Horedt intended for the normalization mass to be the mass of the pressure-truncated object, not the mass of the associated isolated (and untruncated) polytrope.

From these previously published works, it is not obvious how — or even whether — this pair of parametric equations can be combined to directly show how the equilibrium radius depends on the value of the external pressure. Our examination of the free-energy of these configurations and, especially, an application of the viral theorem shows this direct relationship. Foreshadowing these results, we note that,

<math>~\biggl[ \biggl(\frac{P_e}{P_\mathrm{norm}}\biggr) \biggl(\frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^4\biggr]_\mathrm{Horedt} </math>

<math>~=</math>

<math> \biggl[ \frac{\tilde\theta^{n+1} }{(4\pi)(n+1)( -\tilde\theta' )^{2}} \biggr] \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{2} \, ; </math>

or, given that <math>~P_\mathrm{norm}R_\mathrm{norm}^4 = GM_\mathrm{tot}^2</math>, this can be rewritten as,

<math>~\biggl[ \frac{P_e R_\mathrm{eq}^4}{G M_\mathrm{limit}^2} \biggr]_\mathrm{Horedt} </math>

<math>~=</math>

<math> \frac{\tilde\theta^{n+1} }{(4\pi)(n+1)( -\tilde\theta' )^{2}} \, . </math>


Free Energy Function and Virial Theorem

The variation with size of the normalized free energy, <math>~\mathfrak{G}^*</math>, of pressure-truncated adiabatic spheres is described by the following,

Algebraic Free-Energy Function

<math> \mathfrak{G}^* = -3\mathcal{A} \chi^{-1} +~ \frac{1}{(\gamma - 1)} \mathcal{B} \chi^{3-3\gamma} +~ \mathcal{D}\chi^3 \, . </math>

In this expression, the size of the configuration is set by the value of the dimensionless radius, <math>~\chi \equiv R/R_\mathrm{norm}</math>; as is clarified, below, the values of the coefficients, <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>, characterize the relative importance, respectively, of the gravitational potential energy and the internal thermal energy of the configuration; <math>~\gamma</math> is the exponent (from the adopted equation of state) that identifies the adiabat along which the configuration heats or cools upon expansion or contraction; and the relative importance of the imposed external pressure is expressed through the free-energy expression's third constant coefficient, specifically,

<math>~\mathcal{D} \equiv \frac{4\pi}{3} \biggl( \frac{P_e}{P_\mathrm{norm}} \biggr) \, .</math>

When examining a range of physically reasonable configuration sizes for a given choice of the constants <math>~(\gamma, \mathcal{A}, \mathcal{B}, \mathcal{D})</math>, a plot of <math>~\mathfrak{G}^*</math> versus <math>~\chi</math> will often reveal one or two extrema. Each extremum is associated with an equilibrium radius, <math>~\chi_\mathrm{eq} \equiv R_\mathrm{eq}/R_\mathrm{norm}</math>.

Equilibrium Adiabatic Pressure-Radius Diagram

Equilibrium radii may also be identified through an algebraic relation that originates from the scalar virial theorem — a theorem that, itself, is derivable from the free-energy expression by setting <math>~\partial\mathfrak{G}^*/\partial\chi = 0</math>. In our accompanying detailed analysis of the structure of pressure-truncated polytropes, we use the virial theorem to show that the equilibrium radii that are identified by extrema in the free-energy function always satisfy the following,

Algebraic Expression of the Virial Theorem

<math> \Pi_\mathrm{ad} = \frac{(\Chi_\mathrm{ad}^{4-3\gamma} - 1)}{\Chi_\mathrm{ad}^4} \, , </math>

where, after setting <math>~\gamma = (n+1)/n</math>,

<math>~\Pi_\mathrm{ad}</math>

<math>~=</math>

<math> ~\mathcal{D} \biggl[ \frac{\mathcal{A}^{3(n+1)}}{\mathcal{B}^{4n}} \biggr]^{1/(n-3)} \, , </math>         and,

<math>~\Chi_\mathrm{ad}</math>

<math>~=</math>

<math> ~\chi_\mathrm{eq} \biggl[ \frac{\mathcal{B}}{\mathcal{A}} \biggr]^{n/(n-3)} \, . </math>

The curves shown in the accompanying "pressure-radius" diagram trace out this derived virial-theorem function for six different values of the adiabatic exponent, <math>~\gamma</math>, as labeled. They show the dimensionless external pressure, <math>~\Pi_\mathrm{ad}</math>, that is required to construct a nonrotating, self-gravitating, adiabatic sphere with a dimensionless equilibrium radius <math>~\Chi_\mathrm{ad}</math>. The mathematical solution becomes unphysical wherever the pressure becomes negative.

If we multiply the above free=energy function through by an appropriate combination of the coefficients, <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>, and make the substitution, <math>~\gamma \rightarrow (n+1)/n</math>, it also takes on a particularly simple form featuring the newly defined dimensionless external pressure, <math>~\Pi_\mathrm{ad}</math>, and the newly identified dimensionless radius, <math>~\Chi \equiv \chi(\mathcal{B}/\mathcal{A})^{n/(n-3)}</math>. Specifically, we obtain the,

Renormalized Free-Energy Function

<math> \mathfrak{G}^{**} \equiv \mathfrak{G}^* \biggl[ \frac{\mathcal{A}^3}{\mathcal{B}^n} \biggr]^{1/(n-3)} = -3 \Chi^{-1} +~ n\Chi^{-3/n} +~ \Pi_\mathrm{ad}\Chi^3 \, . </math>

Relationship to Detailed Force-Balanced Models

Structural Form Factors

In our accompanying detailed analysis, we demonstrate that the expressions given above for the free-energy function and the virial theorem are correct in sufficiently strict detail that they can be used to precisely match — and assist in understanding — the equilibrium of embedded polytropes whose structures have been determined from the set of detailed force-balance equations. In order to draw this association, it is only necessary to realize that, very broadly, the constant coefficients, <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>, in the above algebraic free-energy expression are expressible in terms of three structural form factors, <math>~\tilde\mathfrak{f}_M</math>, <math>~\tilde\mathfrak{f}_W</math>, and <math>~\tilde\mathfrak{f}_A</math>, as follows:

<math>~\mathcal{A}</math>

<math>~\equiv</math>

<math>\frac{1}{5} \cdot \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\tilde\mathfrak{f}_M} \biggr]^2 \cdot \tilde\mathfrak{f}_W \, ,</math>

<math>~\mathcal{B}</math>

<math>~\equiv</math>

<math> \frac{4\pi}{3} \biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr) \frac{1}{\tilde\mathfrak{f}_M} \biggr]_\mathrm{eq}^{(n+1)/n} \cdot \tilde\mathfrak{f}_A = \frac{4\pi}{3} \biggl[ \biggl( \frac{P_c}{P_\mathrm{norm}} \biggr)\chi^{3(n+1)/n} \biggr]_\mathrm{eq} \cdot \tilde\mathfrak{f}_A \, ; </math>

and that, specifically in the context of spherically symmetric, pressure-truncated polytropes, we can write …

Structural Form Factors for Pressure-Truncated Polytropes

<math>~\tilde\mathfrak{f}_M</math>

<math>~=</math>

<math>~ \biggl[ - \frac{3\Theta^'}{\xi} \biggr]_{\tilde\xi} </math>

<math>~ ~\frac{\tilde\mathfrak{f}_A - \tilde\theta^{n+1} }{\tilde\mathfrak{f}_W} </math>

<math>~=</math>

<math>~ \biggl[ \frac{(n+1)}{3\cdot 5} \biggr] \tilde\xi^2 </math>


After plugging these nontrivial expressions for <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math> into the righthand sides of the above equations for <math>~\Pi_\mathrm{ad}</math> and <math>~\Chi_\mathrm{ad}</math> and, simultaneously, using Horedt's detailed force-balanced expressions for <math>~r_a</math> and <math>~p_a</math> to specify, respectively, <math>~\chi_\mathrm{eq}</math> and <math>~P_e/P_\mathrm{norm}</math> in these same equations — see our accompanying discussion — we find that,

<math>~\Pi_\mathrm{ad}</math>

<math>~=</math>

<math>~\eta_\mathrm{ad} (1 + \eta_\mathrm{ad})^{-4n/(n-3)} \, ,</math>

<math>~\Chi_\mathrm{ad}</math>

<math>~=</math>

<math>~(1 + \eta_\mathrm{ad})^{n/(n-3)} \, ,</math>

where the newly identified, key physical parameter,

<math>~\eta_\mathrm{ad} \equiv \frac{b_\mathrm{ad}}{a_\mathrm{ad}}</math>

<math>~=</math>

<math>~\frac{3\cdot 5~ \tilde\theta^{n+1}}{(n+1) \tilde\xi^2 \mathfrak{f}_W} \, .</math>

It is straightforward to show that this more compact pair of expressions for <math>~\Pi_\mathrm{ad}</math> and <math>~\Chi_\mathrm{ad}</math> satisfy the virial theorem presented above.

Physical Meaning of Parameter <math>~\eta_\mathrm{ad}</math>

In association with our accompanying derivation of a concise expression for the virial theorem, we see that the structural form factor associated with the thermal energy reservoir of our configuration is the sum of two terms, specifically,

<math>~\tilde\mathfrak{f}_A = a_\mathrm{ad} + b_\mathrm{ad} \, ,</math>

while, as defined in our above discussion, <math>~\eta_\mathrm{ad}</math> is the ratio of these same two terms, specifically,

<math>~\eta_\mathrm{ad} = \frac{b_\mathrm{ad}}{a_\mathrm{ad}} \, .</math>

It is worth pointing out what physical quantities are associated with these two terms.


At any radial location within a polytropic configuration, the Lane-Emden function, <math>~\theta</math>, is defined in terms of a ratio of the local density to the configuration's central density, specifically,

<math>\theta \equiv \biggl(\frac{\rho}{\rho_c} \biggr)^{1/n} \, .</math>

Remembering that, at any location within the configuration, the pressure is related to the density via the polytropic equation of state,

<math>P = K\rho^{(n+1)/n} \, ,</math>

we see that,

<math>\frac{P}{P_c} = \theta^{n+1} \, .</math>

Hence, the quantity, <math>~\tilde\theta^{n+1}</math>, which appears as the second term in our definition of <math>~\tilde\mathfrak{f}_A</math>, is the ratio, <math>~(P/P_c)_{\tilde\xi}</math>, evaluated at the surface of the truncated polytropic sphere. But, by construction, the pressure at this location equals the pressure of the external medium in which the polytrope is embedded, so we can write,

<math>b_\mathrm{ad} \equiv \tilde\theta^{n+1} = \frac{P_e}{P_c} \, .</math>

Also, directly from its integral definition, we have that <math>~\tilde\mathfrak{f}_A = \bar{P}/P_c</math>. So we can write,

<math>~a_\mathrm{ad}</math>

<math>~=</math>

<math>~\tilde\mathfrak{f}_A - b_\mathrm{ad } = \frac{\bar{P}}{P_c} - \frac{P_e}{P_c} = \frac{P_e}{P_c}\biggl( \frac{\bar{P}}{P_e} - 1 \biggr) \, .</math>

We conclude, therefore, that,

<math>~\eta_\mathrm{ad} = \biggl( \frac{\bar{P}}{P_e} - 1 \biggr)^{-1} \, .</math>

Desired Pressure-Radius Relation

It is clear from the above discussion that the pair of parametric equations obtained via a solution of the detailed force-balanced equations satisfy our, slightly rearranged,

Algebraic Expression of the Virial Theorem

<math> \Pi_\mathrm{ad} \Chi_\mathrm{ad}^4 = \Chi_\mathrm{ad}^{(n-3)/n} - 1 \, . </math>

More to the point, it is now clear that this virial theorem expression provides the direct relationship between the configuration's dimensionless equilibrium radius as defined by Horedt, <math>~r_a</math>, and the dimensionless applied external pressure as defined by Horedt, <math>~p_a</math>, that was not apparent from the original pair of parametric relations. Horedt's parameters, <math>~r_a</math> and <math>~p_a</math>, can be directly associated to our parameters, <math>~\Chi_\mathrm{ad}</math> and <math>~\Pi_\mathrm{ad}</math>, via two new normalizations, <math>~r_n</math> and <math>~p_n</math>, defined through the relations,

<math>~\Chi_\mathrm{ad} = \frac{r_a}{r_n}</math>

     and     

<math>~\Pi_\mathrm{ad} = \frac{p_a}{p_n} \, .</math>

Specifically in terms of the coefficients in the free-energy expression,

<math>~r_n^{n-3}</math>

<math>~\equiv</math>

<math>~ \frac{(n+1)^n}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{1-n} \biggl( \frac{\mathcal{A}}{\mathcal{B}} \biggr)^n \, , </math>

and,

<math>~p_n^{n-3}</math>

<math>~\equiv</math>

<math>~ \frac{3^{n-3}}{(4\pi)^4 (n+1)^{3(n+1)}} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{2(n+1)} \biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr] \, ; </math>

while, in terms of the structural form factors,

<math>~r_n^{n-3}</math>

<math>~\equiv</math>

<math>~ \frac{1}{3} \biggl[ \frac{(n+1)}{5} \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_A} \biggr]^n \mathfrak{f}_M^{1-n} \, , </math>

and,

<math>~p_n^{n-3}</math>

<math>~\equiv</math>

<math>~ \frac{1}{(4\pi)^8} \biggl[ \frac{3\cdot 5^3}{(n+1)^3} \cdot \frac{\mathfrak{f}_M^2}{\mathfrak{f}_W^3} \biggr]^{n+1} \mathfrak{f}_A^{4n} \, . </math>


Implications Regarding Stability

Model Sequences

Equilibrium Adiabatic Pressure-Radius Diagram

After choosing a value for the system's adiabatic index (or, equivalently, its polytropic index), <math>~\gamma = (n+1)/n</math>, the functional form of the virial theorem expression, <math>~\Pi_\mathrm{ad}(\chi_\mathrm{ad})</math>, is known and, hence, the equilibrium model sequence can be plotted. Half-a-dozen such model sequences are shown in the figure near the beginning of this discussion. Each curve can be viewed as mapping out a single-parameter sequence of equilibrium models; "evolution" along the curve can be accomplished by varying the key parameter, <math>~\eta_\mathrm{ad}</math>, over the physically relevant range, <math>0 \le \eta_\mathrm{ad} < \infty</math>.

ASIDE [18 March 2015]: Many months after I penned the above description of "evolution" along an equilibrium model sequence, I started analyzing in detail the paper by Kimura (1981b). The following excerpt from §3 of his paper shows that Kimura presented essentially the same description of "evolution along a sequence" several decades ago:

Excerpt from section 3 of Kimura (1981b)

Kimura uses the subscript "1" to denote the equilibrium value of any physical quantity "Q"; in the figure shown here, on the right, we are holding the equilibrium mass fixed while allowing the external pressure and the configuration volume to vary, so Kimura would say that the figure displays various "M_1 sequences." And, as is explained more fully in an accompanying discussion, his "truncation parameter" is essentially the same as our truncation radius — specifically, <math>~\zeta_1 = (n+1)^{1/2}\tilde\xi</math>. When projected onto our discussion, the physically relevant range of truncation parameter values is, <math>~0 \le \tilde\xi \le \xi_1</math>, where <math>~\xi_1</math> is the Lane-Emden radius of an isolated (unbounded) polytropic sphere.

To simplify our discussion, here, we redisplay the above figure and repeat a few key algebraic relations.

<math>~\eta_\mathrm{ad} </math>

<math>~\equiv</math>

<math>~\frac{3\cdot 5 ~\tilde\theta^{n+1}}{(n+1) \tilde\xi^2 \tilde\mathfrak{f}_W} = \frac{\tilde\theta^{n+1}}{\tilde\theta^{n+1} + 3(\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta}\, ,</math>

<math>~\Pi_\mathrm{ad}</math>

<math>~=</math>

<math>~\eta_\mathrm{ad} (1 + \eta_\mathrm{ad})^{-4n/(n-3)} \, ,</math>

<math>~\Chi_\mathrm{ad}</math>

<math>~=</math>

<math>~(1 + \eta_\mathrm{ad})^{n/(n-3)} \, .</math>

Note that the last expression for <math>~\eta_\mathrm{ad}</math> has been obtained after inserting the analytic expression for the structural form-factor, <math>~\tilde\mathfrak{f}_W</math> that — as has been explained in an accompanying discussion — we derived with the help of Viala & Horedt (1974).

Stability

Analysis of the free-energy function allows us to not only ascertain the equilibrium radius of isolated polytropes and pressure-truncated polytropic configurations, but also the relative stability of these configurations. We begin by repeating the,

Renormalized Free-Energy Function

<math> \mathfrak{G}^{**} = -3 \Chi^{-1} +~ n\Chi^{-3/n} +~ \Pi_\mathrm{ad}\Chi^3 \, . </math>

The first and second derivatives of <math>~\mathfrak{G}^{**}</math>, with respect to the dimensionless radius, <math>~\Chi</math>, are, respectively,

<math>~\frac{\partial\mathfrak{G}^{**}}{\partial\Chi}</math>

<math>~=</math>

<math>~3 \Chi^{-2} -3\Chi^{-(n+3)/n} + 3\Pi_\mathrm{ad} \Chi^2 \, ,</math>

<math>~\frac{\partial^2\mathfrak{G}^{**}}{\partial\Chi^2}</math>

<math>~=</math>

<math>~-6 \Chi^{-3} + \frac{3(n+3)}{n} \Chi^{-(2n+3)/n} + 6\Pi_\mathrm{ad} \Chi \, .</math>

As alluded to, above, equilibrium radii are identified by values of <math>~\Chi</math> that satisfy the equation, <math>\partial\mathfrak{G}^{**}/\partial\Chi = 0</math>. Specifically, marking equilibrium radii with the subscript "ad", they will satisfy the

Algebraic Expression of the Virial Theorem

<math> \Pi_\mathrm{ad} = \frac{\Chi_\mathrm{ad}^{(n-3)/n} - 1}{\Chi_\mathrm{ad}^4} \, . </math>

Dynamical stability then depends on the sign of the second derivative of <math>~\mathfrak{G}^{**}</math>, evaluated at the equilibrium radius; specifically, configurations will be stable if,

<math>~\frac{\partial^2\mathfrak{G}^{**}}{\partial\Chi^2}\biggr|_{\Chi_\mathrm{ad}}</math>

<math>~></math>

<math>~0 \, ,</math>      (stable)

and they will be unstable if, upon evaluation at the equilibrium radius, the sign of the second derivative is less than zero. Hence, isolated polytropes as well as pressure-truncated polytropic configurations will be stable if,

<math>~0</math>

<math>~< </math>

<math>~3 \Chi_\mathrm{ad}^{-3} \biggl[ - 2 + \frac{(n+3)}{n} \Chi_\mathrm{ad}^{(n-3)/n} + 2\Pi_\mathrm{ad} \Chi_\mathrm{ad}^4 \biggr]</math>

 

<math>~< </math>

<math>~3 \Chi_\mathrm{ad}^{-3} \biggl\{ \frac{(n+3)}{n} \Chi_\mathrm{ad}^{(n-3)/n} + 2[\Chi_\mathrm{ad}^{(n-3)/n} -1] - 2\biggr\}</math>

 

<math>~< </math>

<math>~3 \Chi_\mathrm{ad}^{-3} \biggl[ \frac{3(n+1)}{n} \Chi_\mathrm{ad}^{(n-3)/n} - 4\biggr]</math>

<math>\Rightarrow~~~~\Chi_\mathrm{ad}</math>

<math>~> </math>

<math>~\biggl[ \frac{4n}{3(n+1)} \biggr]^{n/(n-3)} \, .</math>      (stable)

Reference to this stability condition proves to be simpler if we define the limiting configuration size as,

<math>~\Chi_\mathrm{min} \equiv \biggl[ \frac{4n}{3(n+1)} \biggr]^{n/(n-3)} \, ,</math>

and write the stability condition as,

<math>~\Chi_\mathrm{ad} > \Chi_\mathrm{min} \, .</math>      (stable)

When examining the equilibrium sequences found in the upper-righthand quadrant of the figure at the top of this page — each corresponding to a different value of the polytropic index, <math>~n > 3</math> or <math>~n < 0</math> — we find that <math>~\Chi_\mathrm{min}</math> corresponds to the location along each sequence where the dimensionless external pressure, <math>~\Pi_\mathrm{ad}</math>, reaches a maximum. (Keeping in mind that the virial theorem defines each of these sequences, this statement of fact can be checked by identifying where the condition, <math>~\partial\Pi_\mathrm{ad}/\partial\Chi_\mathrm{ad} = 0</math>, occurs according to the algebraic expression of the virial theorem.) Hence, we conclude that, along each sequence, no equilibrium configurations exist for values of the dimensionless external pressure that are greater than,

<math>~\Pi_\mathrm{max}</math>

<math>~\equiv</math>

<math>~\Chi_\mathrm{min}^{-4} \biggl[ \Chi_\mathrm{min}^{(n-3)/n} - 1 \biggr] </math>

 

<math>~=</math>

<math>~\biggl[ \frac{3(n+1)}{4n} \biggr]^{4n/(n-3)} \biggl[\frac{4n}{3(n+1)} - 1 \biggr]</math>

 

<math>~=</math>

<math>~\biggl\{ \biggl[ \frac{3(n+1)}{4n} \biggr]^{4n} \biggl[\frac{n-3}{3(n+1)} \biggr]^{n-3} \biggr\}^{1/(n-3)}</math>

<math>~\Rightarrow~~~~\Pi_\mathrm{max}^{n-3}</math>

<math>~=</math>

<math>~(4n)^{-4n}~[3(n+1)]^{3(n+1)} ~(n-3)^{n-3} \, .</math>

[In a separate, related discussion of the model sequences displayed in the above figure, we have actually demonstrated that this same coordinate point was associated with the extremum along each curve. In that discussion, this special point was identified as <math>~(\Chi_\mathrm{extreme}, \Pi_\mathrm{extreme})</math> instead of as <math>~(\Chi_\mathrm{min}, \Pi_\mathrm{max})</math>.]

In the context of a general examination of the free-energy of pressure-truncated polytropes, it is worth noting that this limit on the external pressure also establishes a limit on the coefficient, <math>~\mathcal{D}</math>, that appears in the free energy function. Specifically, we will not expect to find any extrema in the free energy if,

<math>~\mathcal{D} > \mathcal{D}_\mathrm{max}</math>

<math>~\equiv</math>

<math>~(n-3) \biggl\{ \biggl[ \frac{\mathcal{B}}{4n} \biggr]^{4n}~\biggl[ \frac{3(n+1)}{\mathcal{A}} \biggr]^{3(n+1)} ~\biggr\}^{1/(n-3)} \, .</math>

Finally, it is worth noting that the point along each equilibrium sequence that is identified by the coordinates, <math>~(\Chi_\mathrm{min}, \Pi_\mathrm{max})</math> always corresponds to,

<math>~\eta_\mathrm{ad} = \eta_\mathrm{crit} \equiv \frac{n-3}{3(n+1)} \, .</math>

Summary

<math>~\eta_\mathrm{crit}</math>

<math>~\equiv</math>

<math>~ \frac{n-3}{3(n+1)} </math>

<math>~\Pi_\mathrm{max}</math>

<math>~\equiv</math>

<math>~(n-3) \biggl\{~\frac{ [3(n+1)]^{3(n+1)} }{(4n)^{4n}} \biggr\}^{1/(n-3)} </math>

<math>~\Chi_\mathrm{min} </math>

<math>~\equiv</math>

<math> \biggl[ \frac{4n}{3(n+1)} \biggr]^{n/(n-3)} </math>

Mass-Radius Relation

Up to this point in our discussion, we have focused on an analysis of the pressure-radius relationship that defines the equilibrium configurations of pressure-truncated polytropes. In effect, we have viewed the problem through the same lens as did Horedt (1970) and, separately, Whitworth (1981), defining variable normalizations in terms of the polytropic constant, <math>~K</math>, and the configuration mass, <math>~M_\mathrm{tot}</math>, which were both assumed to be held fixed throughout the analysis. Here we switch to the approach championed by Stahler (1983), defining variable normalizations in terms of <math>~K</math> and <math>~P_e</math>, and examining the mass-radius relationship of pressure-truncated polytropes.

Detailed Force-Balanced Solution

As has been summarized in our accompanying review of detailed force-balanced models of pressure-truncated polytropes, Stahler (1983) found that a spherical configuration's equilibrium radius is related to its mass through the following pair of parametric equations:

<math> ~\frac{M_\mathrm{limit}}{M_\mathrm{SWS} } </math>

<math>~=~</math>

<math> \biggl( \frac{n^3}{4\pi} \biggr)^{1/2} \tilde\theta^{(n-3)/2} (- \tilde\xi^2 \tilde\theta^') \, , </math>

<math> ~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} } </math>

<math>~=~</math>

<math> \biggl( \frac{n}{4\pi} \biggr)^{1/2} \tilde\xi \tilde\theta^{(n-1)/2} \, , </math>

where,

<math>M_\mathrm{SWS} \equiv \biggl( \frac{n+1}{nG} \biggr)^{3/2} K^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]} \, ,</math>

<math> R_\mathrm{SWS} \equiv \biggl( \frac{n+1}{nG} \biggr)^{1/2} K^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]} \, . </math>

Mapping from Above Discussion

Deriving Concise Virial Theorem Mass-Radius Relation

Looking back on the definitions of <math>~\Pi_\mathrm{ad}</math> and <math>~\Chi_\mathrm{ad}</math> that we introduced in connection with our initial concise algebraic expression of the virial theorem, we can write,

<math>~P_e </math>

<math>~=</math>

<math> ~P_\mathrm{norm} \biggl( \frac{3}{4\pi} \biggr) \Pi_\mathrm{ad} \biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr]^{1/(n-3)} </math>

 

<math>~=</math>

<math> ~\biggl( \frac{3}{4\pi} \biggr) \Pi_\mathrm{ad} \biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr]^{1/(n-3)} \biggl[ \frac{K^{4n}}{G^{3(n+1)} M_\mathrm{tot}^{2(n+1)}} \biggr]^{1/(n-3)} \, , </math>

<math>~R_\mathrm{eq} </math>

<math>~=</math>

<math> ~R_\mathrm{norm} \Chi_\mathrm{ad} \biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{n/(n-3)} </math>

 

<math>~=</math>

<math> ~\Chi_\mathrm{ad} \biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{n/(n-3)} \biggl[ \biggl( \frac{G}{K} \biggr)^n M_\mathrm{tot}^{n-1} \biggr]^{1/(n-3)} \, . </math>

The first of these two expressions can be flipped around to give an expression for <math>~M_\mathrm{tot}</math> in terms of <math>~P_e</math> and, then, as normalized to <math>~M_\mathrm{SWS}</math>. Specifically,

<math>~ M_\mathrm{tot}^{2(n+1)}</math>

<math>~=</math>

<math> ~\biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{n-3} \biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr] \biggl[ \frac{K^{4n}}{G^{3(n+1)}P_e^{n-3} } \biggr] </math>

 

<math>~=</math>

<math> ~M_\mathrm{SWS}^{2(n+1)} \biggl( \frac{n}{n+1} \biggr)^{3(n+1)} \biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{n-3} \biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr] </math>

<math>~ \Rightarrow~~~ \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}</math>

<math>~=</math>

<math> ~\biggl( \frac{n}{n+1} \biggr)^{3/2} \biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(n-3)/[2(n+1)]} \biggl[ \frac{\mathcal{B}^{2n/(n+1)}}{\mathcal{A}^{3/2}} \biggr] \, . </math>

This means, as well, that we can rewrite the equilibrium radius as,

<math>~R_\mathrm{eq}^{n-3} </math>

<math>~=</math>

<math> ~\Chi_\mathrm{ad}^{n-3} \biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{n} \biggl( \frac{G}{K} \biggr)^n M_\mathrm{tot}^{n-1} </math>

 

<math>~=</math>

<math> ~\Chi_\mathrm{ad}^{n-3} \biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{n} \biggl( \frac{G}{K} \biggr)^n \biggl\{ \biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{n-3} \biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr] \biggl[ \frac{K^{4n}}{G^{3(n+1)}P_e^{n-3} } \biggr] \biggr\}^{(n-1)/[2(n+1)]} </math>

 

<math>~=</math>

<math> ~\Chi_\mathrm{ad}^{n-3} \biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{n} \biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr]^{(n-1)/[2(n+1)]} \biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(n-3)(n-1)/[2(n+1)]} \biggl( \frac{G}{K} \biggr)^n \biggl\{ \biggl[ \frac{K^{4n}}{G^{3(n+1)}P_e^{n-3} } \biggr] \biggr\}^{(n-1)/[2(n+1)]} </math>

 

<math>~=</math>

<math> ~\Chi_\mathrm{ad}^{n-3} \biggl\{ \biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{2n(n+1)} \biggl[ \frac{\mathcal{B}^{4n(n-1)}}{\mathcal{A}^{3(n+1)(n-1)}} \biggr]\biggr\}^{1/[2(n+1)]} \biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(n-3)(n-1)/[2(n+1)]} \biggl\{ \biggl( \frac{G}{K} \biggr)^{2n(n+1)} \biggl[ \frac{K^{4n(n-1)}}{G^{3(n+1)(n-1)}P_e^{(n-3)(n-1)} } \biggr] \biggr\}^{1/[2(n+1)]} </math>

 

<math>~=</math>

<math> ~\Chi_\mathrm{ad}^{n-3} \biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(n-3)(n-1)/[2(n+1)]} \biggl[ \mathcal{A}^{-(n+1)(n-3)} \mathcal{B}^{2n(n-3)} \biggr]^{1/[2(n+1)]} \biggl[ G^{(3-n)(n+1)} K^{2n(n-3)} P_e^{(n-3)(1-n)} \biggr]^{1/[2(n+1)]} </math>

 

<math>~=</math>

<math> ~R_\mathrm{SWS}^{n-3} \biggl( \frac{n}{n+1} \biggr)^{(n-3)/2} \Chi_\mathrm{ad}^{n-3} \biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(n-3)(n-1)/[2(n+1)]} \biggl[ \mathcal{A}^{-(n+1)(n-3)} \mathcal{B}^{2n(n-3)} \biggr]^{1/[2(n+1)]} </math>

<math>~\Rightarrow~~~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} } </math>

<math>~=</math>

<math> ~\biggl( \frac{n}{n+1} \biggr)^{1/2} \Chi_\mathrm{ad} \biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(n-1)/[2(n+1)]} \biggl[ \frac{\mathcal{B}^{n/(n+1)}}{\mathcal{A}^{1/2}} \biggr] \, . </math>

Flipping both of these expressions around, we see that,

<math>~\Pi_\mathrm{ad} </math>

<math>~=</math>

<math> ~\frac{4\pi}{3} \biggl\{ \biggl( \frac{n+1}{n} \biggr)^{3(n+1)} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{2(n+1)} \biggl[ \frac{\mathcal{A}^{3(n+1)}}{\mathcal{B}^{4n}} \biggr] \biggr\}^{1/(n-3)} \, , </math>

and,

<math>~\Chi_\mathrm{ad} </math>

<math>~=</math>

<math> ~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} } \biggl( \frac{n+1}{n} \biggr)^{1/2} \biggl[ \frac{\mathcal{A}^{1/2}}{\mathcal{B}^{n/(n+1)}} \biggr] \biggl( \frac{3\Pi_\mathrm{ad} }{4\pi} \biggr)^{(1-n)/[2(n+1)]} </math>

 

<math>~=</math>

<math> ~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} } \biggl( \frac{n+1}{n} \biggr)^{1/2} \biggl[ \frac{\mathcal{A}^{1/2}}{\mathcal{B}^{n/(n+1)}} \biggr] \biggl\{ \biggl( \frac{n+1}{n} \biggr)^{3(n+1)} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{2(n+1)} \biggl[ \frac{\mathcal{A}^{3(n+1)}}{\mathcal{B}^{4n}} \biggr] \biggr\}^{(1-n)/[2(n+1)(n-3)]} </math>

 

<math>~=</math>

<math> ~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} } \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{(1-n)/(n-3)} \biggl( \frac{n}{n+1} \biggr)^{n/(n-3)} \biggl[ \frac{\mathcal{B}}{\mathcal{A}} \biggr]^{n/(n-3)} \, . </math>

Hence, our earlier derived compact expression for the virial theorem becomes,

<math>~1</math>

<math>~=</math>

<math> \biggl\{ \frac{R_\mathrm{eq}}{R_\mathrm{SWS} } \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{(1-n)/(n-3)} \biggl( \frac{n}{n+1} \biggr)^{n/(n-3)} \biggl[ \frac{\mathcal{B}}{\mathcal{A}} \biggr]^{n/(n-3)} \biggr\}^{(n-3)/n} </math>

 

 

<math> -~ \frac{4\pi}{3} \biggl\{ \biggl( \frac{n+1}{n} \biggr)^{3(n+1)} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{2(n+1)} \biggl[ \frac{\mathcal{A}^{3(n+1)}}{\mathcal{B}^{4n}} \biggr] \biggr\}^{1/(n-3)} \biggl\{ \frac{R_\mathrm{eq}}{R_\mathrm{SWS} } \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{(1-n)/(n-3)} \biggl( \frac{n}{n+1} \biggr)^{n/(n-3)} \biggl[ \frac{\mathcal{B}}{\mathcal{A}} \biggr]^{n/(n-3)} \biggr\}^4 </math>

 

<math>~=</math>

<math> \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS} } \biggr)^{(n-3)/n} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{(1-n)/n} \biggl( \frac{n}{n+1} \biggr) \biggl[ \frac{\mathcal{B}}{\mathcal{A}} \biggr] -~ \frac{4\pi}{3} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS} } \biggr)^4 \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{-2} \biggl( \frac{n}{n+1} \biggr) \frac{1}{\mathcal{A}} \, . </math>

Or, rearranged,

<math>\frac{4\pi}{3} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS} } \biggr)^4 - \mathcal{B} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS} } \biggr)^{(n-3)/n} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{(n+1)/n} +~ \mathcal{A} \biggl( \frac{n+1}{n} \biggr) \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{2} = 0 \, . </math>

After adopting the modified coefficient definitions,

<math>~\mathcal{A}_{M_\ell}</math>

<math>~\equiv</math>

<math>~ \mathcal{A} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2} = \frac{1}{5} \cdot \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} \, ,</math>

<math>~\mathcal{B}_{M_\ell}</math>

<math>~\equiv</math>

<math>~\mathcal{B} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{-(n+1)/n} = \biggl( \frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_M^{(n+1)/n}} \, , </math>

as well as the modified length- and mass-normalizations, <math>~R_\mathrm{mod}</math> and <math>~M_\mathrm{mod}</math>, such that,

<math>~\frac{M_\mathrm{SWS}}{M_\mathrm{mod}}</math>

<math>~\equiv</math>

<math>~\biggl( \frac{4\pi}{3} \biggr)^{2n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{3/2} \frac{\mathcal{A}_{M_\ell}^{3/2}}{\mathcal{B}_{M_\ell}^{2n/(n+1)}} \, ,</math>

<math>~\frac{R_\mathrm{SWS}}{R_\mathrm{mod}}</math>

<math>~\equiv</math>

<math>~\biggl( \frac{4\pi}{3} \biggr)^{n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{1/2} \frac{\mathcal{A}_{M_\ell}^{1/2}}{\mathcal{B}_{M_\ell}^{n/(n+1)}} \, ,</math>

we obtain the

Virial Theorem in terms of Mass and Radius

<math> \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{mod}} \biggr)^4 - \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{mod}} \biggr)^{(n-3)/n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{mod}} \biggr)^{(n+1)/n} + \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{mod}} \biggr)^2 = 0 \, . </math>


For later use we note as well that, with these modified coefficient definitions, we can write,

<math>~\Pi_\mathrm{ad}^{n-3} </math>

<math>~=</math>

<math> ~\biggl[ \biggl( \frac{4\pi}{3} \biggr)^{n-3} \biggl( \frac{n+1}{n} \biggr)^{3(n+1)} \frac{\mathcal{A}_{M_\ell}^{3(n+1)}}{\mathcal{B}_{M_\ell}^{4n}} \biggr] \mathcal{Y}^{2(n+1)} \, , </math>

<math>~\Chi_\mathrm{ad}^{n-3} </math>

<math>~=</math>

<math> ~ \biggl[ \frac{n}{n+1} \biggl( \frac{\mathcal{B}_{M_\ell}}{\mathcal{A}_{M_\ell}} \biggr)\biggr]^n \mathcal{X}^{n-3} \mathcal{Y}^{1-n} \, , </math>

where <math>~\mathcal{X}</math> and <math>~\mathcal{Y}</math> are defined immediately below.

Corresponding Concise Free-Energy Expression

Let's also rewrite the algebraic free-energy function in terms of Stahler's normalized mass and radius variables. Expressed in terms of the polytropic index, the free-energy function is,

<math> \mathfrak{G}^* = -3\mathcal{A} \chi^{-1} +~ n\mathcal{B} \chi^{-3/n} +~ \mathcal{D}\chi^3 \, . </math>

First, we recognize that,

<math>~\chi \equiv \frac{R}{R_\mathrm{norm}}</math>

<math>~=</math>

<math>~\biggl(\frac{R}{R_\mathrm{SWS}}\biggr) \frac{R_\mathrm{SWS}}{R_\mathrm{norm}} \, .</math>

From the definition of <math>~R_\mathrm{norm}</math> — reprinted, for example, here — we can write,

<math>~\biggl( \frac{R_\mathrm{SWS}}{R_\mathrm{norm}}\biggr)^{n-3}</math>

<math>~=</math>

<math>~R_\mathrm{SWS}^{n-3} \biggl[ G^{-n} K^n M_\mathrm{tot}^{1-n} \biggr] </math>

 

<math>~=</math>

<math>~R_\mathrm{SWS}^{n-3} M_\mathrm{SWS}^{1-n} \biggl[ \biggl(\frac{K}{G}\biggr)^n \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{1-n} \biggr] \, ; </math>

and from the definitions of <math>~R_\mathrm{SWS}</math> and <math>~M_\mathrm{SWS}</math> — reprinted, for example, here — we have,

<math>~\biggl( \frac{R_\mathrm{SWS}}{R_\mathrm{norm}}\biggr)^{n-3}</math>

<math>~=</math>

<math>~\biggl[ \biggl(\frac{K}{G}\biggr)^n \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{1-n} \biggr] \biggl\{\biggl( \frac{n+1}{nG} \biggr)^{1/2} K^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]} \biggr\}^{n-3} \biggl\{ \biggl( \frac{n+1}{nG} \biggr)^{3/2} K^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]} \biggr\}^{1-n} </math>

 

<math>~=</math>

<math>~\biggl[ \biggl(\frac{K}{G}\biggr)^n \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{1-n} \biggr] \biggl( \frac{n+1}{n} \biggr)^{[(n-3) +3(1-n)]/2} G^{[(3-n) + 3(n-1)]/2} K^{n[(n-3)+2(1-n)]/(n+1)} </math>

 

<math>~=</math>

<math> ~\biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{1-n} \biggl( \frac{n}{n+1} \biggr)^n \, . </math>

Hence, in each term in the free-energy expression we can make the substitution,

<math>~\chi </math>

<math>~~~\rightarrow~~~</math>

<math>~ \biggl(\frac{R}{R_\mathrm{SWS}}\biggr) \biggl\{ \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{1-n} \biggl( \frac{n}{n+1} \biggr)^n \biggr\}^{1/(n-3)} = \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}}\biggr)^{(1-n)/(n-3)} \biggl\{ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{n-1} \biggl( \frac{n}{n+1} \biggr)^n \biggr\}^{1/(n-3)} \, . </math>

Next, drawing on the definition of <math>~P_\mathrm{norm}</math> — reprinted, for example, here — along with the definition of <math>~M_\mathrm{SWS}</math>, we recognize that,

<math>~\mathcal{D} \equiv \frac{4\pi}{3} \cdot \frac{P_e}{P_\mathrm{norm}}</math>

<math>~=</math>

<math>~\frac{4\pi}{3} \cdot P_e \biggl[ K^{-4n} G^{3(n+1)} M_\mathrm{tot}^{2(n+1)} \biggr]^{1/(n-3)} </math>

 

<math>~=</math>

<math>~\frac{4\pi}{3} \cdot \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \biggr)^{2(n+1)/(n-3)} P_e \biggl[ K^{-4n} G^{3(n+1)}\biggr]^{1/(n-3)} M_\mathrm{SWS}^{2(n+1)/(n-3)} </math>

 

<math>~=</math>

<math>~\frac{4\pi}{3} \cdot \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{2(n+1)/(n-3)} \biggl\{ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2}\biggl( \frac{n+1}{n}\biggr)^3 \biggr\}^{(n+1)/(n-3)} \, .</math>

After making these substitutions into the free-energy function, as well as replacing <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math> with <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math>, respectively, we have,

<math>~\mathfrak{G}^*</math>

<math>~=</math>

<math>~ -3\mathcal{A}_{M_\ell}\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{2} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-1} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}}\biggr)^{-(1-n)/(n-3)} \biggl\{ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{n-1} \biggl( \frac{n}{n+1} \biggr)^n \biggr\}^{-1/(n-3)} </math>

 

 

<math>~ +~ n\mathcal{B}_{M_\ell} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{(n+1)/n} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-3/n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}}\biggr)^{-3(1-n)/[n(n-3)]} \biggl\{ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{n-1} \biggl( \frac{n}{n+1} \biggr)^n \biggr\}^{-3/[n(n-3)]} </math>

 

 

<math>~ +~ \frac{4\pi}{3} \cdot \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{2(n+1)/(n-3)} \biggl\{ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2}\biggl( \frac{n+1}{n}\biggr)^3 \biggr\}^{(n+1)/(n-3)} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^3 \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}}\biggr)^{3(1-n)/(n-3)} \biggl\{ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{n-1} \biggl( \frac{n}{n+1} \biggr)^n \biggr\}^{3/(n-3)} </math>

 

<math>~=</math>

<math>~ -3\mathcal{A}_{M_\ell} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-1} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}}\biggr)^{-(1-n)/(n-3)} \biggl\{ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{[(n-1)-2(n-3)]} \biggl( \frac{n}{n+1} \biggr)^n \biggr\}^{-1/(n-3)} </math>

 

 

<math>~ +~ n\mathcal{B}_{M_\ell} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-3/n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}}\biggr)^{-3(1-n)/[n(n-3)]} \biggl\{ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{[(n+1)(n-3)-3(n-1)]} \biggl( \frac{n}{n+1} \biggr)^{-3n} \biggr\}^{1/[n(n-3)]} </math>

 

 

<math>~ +~ \frac{4\pi}{3} \cdot \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^3 \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}}\biggr)^{[2(n+1)+3(1-n)]/(n-3)} \biggl\{ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{[(3(n-1) -2(n+1)]}\biggl( \frac{n}{n+1}\biggr)^{[3n-3(n+1)]} \biggr\}^{1/(n-3)} </math>

 

<math>~=</math>

<math>~ -3\mathcal{A}_{M_\ell} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-1} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}}\biggr)^{-(1-n)/(n-3)} \biggl\{ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{(n-5)} \biggl( \frac{n+1}{n} \biggr)^n \biggr\}^{1/(n-3)} </math>

 

 

<math>~ +~ n\mathcal{B}_{M_\ell} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-3/n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}}\biggr)^{-3(1-n)/[n(n-3)]} \biggl\{ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{(n-5)} \biggl( \frac{n+1}{n} \biggr)^{3} \biggr\}^{1/(n-3)} </math>

 

 

<math>~ +~ \frac{4\pi}{3} \cdot \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^3 \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}}\biggr)^{(5-n)/(n-3)} \biggl\{ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{(n-5)}\biggl( \frac{n+1}{n}\biggr)^{3} \biggr\}^{1/(n-3)} </math>

 

<math>~=</math>

<math>~\biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{(n-5)}\biggl( \frac{n+1}{n}\biggr)^{3} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}}\biggr)^{(5-n)} \biggr]^{1/(n-3)} \biggl\{ -3\mathcal{A}_{M_\ell}\biggl( \frac{n+1}{n}\biggr) \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-1} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}}\biggr)^{2} +~ n\mathcal{B}_{M_\ell} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-3/n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}}\biggr)^{(n+1)/n} +~ \frac{4\pi}{3} \cdot \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{3} \biggr\}\, . </math>

Hence, after defining,

<math>~\mathfrak{G}^*_\mathrm{SWS}</math>

<math>~\equiv</math>

<math>~ \frac{\mathfrak{G}}{[G^{-3} K^n M_\mathrm{SWS}^{n-5}]^{1/(n-3)}} \biggl( \frac{n}{n+1}\biggr)^{3/(n-3)} = \frac{\mathfrak{G}}{[K^{6n} P_e^{5-n}]^{1/[2(n+1)]}} \biggl( \frac{nG}{n+1}\biggr)^{3/2} \, , </math>

we can write,

<math>~\mathfrak{G}^*_\mathrm{SWS} </math>

<math>~=</math>

<math>~ -3\mathcal{A}_{M_\ell}\biggl( \frac{n+1}{n}\biggr) \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{2} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-1} +~ n\mathcal{B}_{M_\ell} \cdot \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{(n+1)/n} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-3/n} +~ \frac{4\pi}{3} \cdot \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{3} \, . </math>

Setting the first derivative of this function equal to zero should produce the virial theorem expression. Let's see …

<math>~\frac{\partial\mathfrak{G}^*_\mathrm{SWS}}{\partial \mathcal{X}}</math>

<math>~=</math>

<math>~ 3\mathcal{A}_{M_\ell}\biggl( \frac{n+1}{n}\biggr) \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{2} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-2} -~ 3\mathcal{B}_{M_\ell} \cdot \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{(n+1)/n} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-(3+n)/n} +~ 4\pi \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{2} </math>

 

<math>~=</math>

<math>~ 3\biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-2} \biggl[ \mathcal{A}_{M_\ell}\biggl( \frac{n+1}{n}\biggr) \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{2} -~ \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} +~ \frac{4\pi}{3} \cdot \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{4} \biggr] \, . </math>

Replacing <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math> with <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>, as prescribed by their defined relationships, and setting the expression inside the square brackets equal to zero does, indeed, produce the above, boxed-in viral theorem mass-radius relationship.

Plotting Concise Mass-Radius Relation

Our derived, concise analytic expression for the virial theorem, namely,

<math> \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{mod}} \biggr)^4 - \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{mod}} \biggr)^{(n-3)/n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{mod}} \biggr)^{(n+1)/n} + \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{mod}} \biggr)^2 = 0 \, , </math>

is plotted for seven different values of the polytropic index, <math>~n</math>, as indicated, in the lefthand diagram of the following composite figure. For comparison, the schematic diagram displayed on the righthand side of the figure is a reproduction of Figure 17 from Appendix B of 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.


Virial Theorem Mass-Radius Relation

Stahler (1983) Title Page
Stahler (1983) Figure 17 (edited)

Let's do this again using the mass-radius relation as written explicitly in terms of the normalizations, <math>~M_\mathrm{SWS}</math> and <math>~R_\mathrm{SWS}</math>. The relevant, generic nonlinear equation is,

<math>~0</math>

<math>~=</math>

<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>

Analytically determined roots:

  • <math>~n=1</math>

<math>~\frac{M}{M_\mathrm{SWS}} = \biggl( \frac{10\pi}{3}\biggr)^{1 / 2} \biggl(\frac{R}{R_\mathrm{SWS}} \biggr)^3 \biggl[ \frac{3\cdot 5}{2^3\pi} - \biggl(\frac{R}{R_\mathrm{SWS}} \biggr)^2 \biggr]^{-1/2} </math>         for,     <math>~0 \le \frac{R}{R_\mathrm{SWS}} \le \biggl(\frac{3\cdot 5}{2^3\pi}\biggr)^{1/2} \, .</math>

  • <math>~n=3</math>

<math>~\frac{R}{R_\mathrm{SWS}} = \biggl\{ \biggl[ \biggl(\frac{3}{4\pi}\biggr)\frac{M}{M_\mathrm{SWS}}\biggr]^{4/3} - \biggl(\frac{1}{5\pi}\biggr) \biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \biggr\}^{1/4} </math>         for,     <math>~0 \le \frac{M}{M_\mathrm{SWS}} \le \biggl(\frac{3^4\cdot 5^3}{2^8\pi}\biggr)^{1/2} \, .</math>

<math>~\frac{M}{M_\mathrm{SWS}} = \frac{5}{2} \biggl( \frac{R}{R_\mathrm{SWS}} \biggr) \biggl\{ 1 \pm \biggl[ 1 - \frac{16\pi}{15}\biggl( \frac{R}{R_\mathrm{SWS}} \biggr)^2 \biggr]^{1 / 2} \biggr\} </math>         for,     <math>~0 \le \frac{R}{R_\mathrm{SWS}} \le \biggl(\frac{3\cdot 5}{2^4\pi}\biggr)^{1/2} \, .</math>


First, we'll create a table of the normalized coordinate values that satisfy this nonlinear expression.

<math>~n =2</math> <math>~n =2.8</math> <math>~n =3.5</math> <math>~n = 4</math> <math>~n = 5</math>
<math>~ \frac{R}{R_\mathrm{SWS}}</math>   <math>~ \frac{M}{M_\mathrm{SWS}}</math>
 
 
 
0.3800   0.26562
0.4500   0.477153
0.5000   0.70919
0.5500   1.063602
0.5800   1.39755
0.5950   1.64662
0.6050   1.893915
0.6120   2.22372
0.6131721   2.433375
0.6120   2.64923
0.6050   3.01688
0.5950   3.32037
0.5800   3.658702
0.5500   4.19097
0.5000   4.94599
0.4700   5.38791
0.4500   5.69164
 
 
 
 
 
 
<math>~ \frac{R}{R_\mathrm{SWS}}</math>   <math>~ \frac{M}{M_\mathrm{SWS}}</math>
 
 
 
0.3800   0.266134
0.4500   0.47971
0.5000   0.71765
0.5250   0.831825
0.5600   1.20977
0.5750   1.427183
0.5850   1.653232
0.5900   1.89304
0.5904492   1.989927
0.5900   2.086584
0.5850   2.32394
0.5750   2.54527
0.5600   2.75612
0.5250   3.07134
0.4500   3.460304
0.3500   3.75881
0.2500   3.97835
0.2000   4.09302
0.1500   4.232786
0.1000   4.430303
0.0700   4.60984
0.0400   4.9057
0.0150   5.47056
<math>~ \frac{R}{R_\mathrm{SWS}}</math>   <math>~ \frac{M}{M_\mathrm{SWS}}</math>
 
 
 
0.3800   0.26639
0.4500   0.481072
0.5000   0.722406
0.5250   0.89152
0.5600   1.246123
0.5650   1.32113
0.5750   1.52651
0.5800   1.745165
0.5803836   1.823995
0.5800   1.90201
0.5780   2.01647
0.5750   2.11019
0.5600   2.35906
0.5400   2.543602
0.5000   2.75557
0.4500   2.543602
0.3800   2.94829
0.3749583   2.948526
0.3300   2.93161
0.2500   2.829401
0.1500   2.578605
 
 
<math>~ \frac{R}{R_\mathrm{SWS}}</math>   <math>~ \frac{M}{M_\mathrm{SWS}}</math>
0.1000   0.004224
0.2000   0.034709
0.3000   0.1230901
0.4000   0.31735
0.4500   0.48177
0.5000   0.72493
0.5250   0.89686
0.5400   1.028495
0.5500   1.13574
0.5600   1.26965
0.5730   1.55527
0.5756189   1.750930
0.5730   1.93949
0.5600   2.18983
0.5400   2.376318
0.5250   2.46661
0.5000   2.56895
0.4600   2.657809
0.41184646   2.688999
0.4100   2.68895
0.3800   2.677703
0.3000   2.56612
0.2500   2.44565
0.2000   2.28789
0.1500   2.08747
0.1000   1.82708
<math>~ \frac{R}{R_\mathrm{SWS}}</math>   <math>~ \frac{M}{M_\mathrm{SWS}}</math>
0.1000   0.004224
0.2000   0.03471
0.3000   0.123115
0.4000   0.31766
0.4500   0.48278
0.5000   0.72866
0.5250   0.905006
0.5400   1.042907
0.5500   1.15886
0.5600   1.313712
0.5675   1.511304
0.5692185   1.657839
0.5675   1.798532
0.5600   1.97061
0.5400   2.17282
0.5250   2.25888
0.5000   2.34793
0.4600   2.410374
0.4391754   2.417330
0.4000   2.396465
0.3000   2.19848
0.2000   1.84195
0.1000   1.31421
0.0500   0.930314
0.0200   0.58847
 

From a free-energy analysis of isothermal spheres, we have demonstrated that, when the structural form factors are all set to unity, the statement of virial equilibrium is,

<math>~0</math>

<math>~=</math>

<math>~ \biggl( \frac{R}{R_\mathrm{SWS}} \biggr)^4 - \frac{3}{4\pi} \biggr( \frac{M}{M_\mathrm{SWS}} \biggr) \biggl( \frac{R}{R_\mathrm{SWS}} \biggr) + \frac{3}{20\pi} \biggl( \frac{M}{M_\mathrm{SWS}} \biggr)^2 \, , </math>

where, in order to be consistent with the above polytropic normalizations, we have adopted the isothermal normalizations,

<math>~M_\mathrm{SWS}\biggr|_\mathrm{isothermal} \equiv \biggl( \frac{c_s^8}{G^3 P_e} \biggr)^{1 / 2} \, ,</math>

        and        

<math>~R_\mathrm{SWS}\biggr|_\mathrm{isothermal} \equiv \biggl( \frac{c_s^4}{G P_e} \biggr)^{1 / 2} \, .</math>

This is a quadratic equation that can be readily solved to provide an analytic expression for the isothermal mass-radius relation; the relevant expression has already been provided, above.

Confirmation

Rewriting the just-derived virial theorem expression in terms of Stahler's dimensionless radius and mass variables, written in the abbreviated form,

<math>~\mathcal{X}</math>

<math>~\equiv</math>

<math>~\frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \, ,</math>

<math>~\mathcal{Y}</math>

<math>~\equiv</math>

<math>~\frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \, ,</math>

we have,

<math>~0</math>

<math>~=</math>

<math>~\biggl[ \mathcal{X} \cdot \frac{R_\mathrm{SWS}}{R_\mathrm{mod}} \biggr]^4 - \biggl[ \mathcal{X} \cdot \frac{R_\mathrm{SWS}}{R_\mathrm{mod}} \biggr]^{(n-3)/n} \biggl[ \mathcal{Y} \cdot \frac{M_\mathrm{SWS}}{M_\mathrm{mod}} \biggr]^{(n+1)/n} + \biggl[ \mathcal{Y} \cdot \frac{M_\mathrm{SWS}}{M_\mathrm{mod}} \biggr]^2</math>

 

<math>~=</math>

<math>~\mathcal{X}^4 \biggl\{ \biggl( \frac{4\pi}{3} \biggr)^{n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{1/2} \frac{\mathcal{A}_{M_\ell}^{1/2}}{\mathcal{B}_{M_\ell}^{n/(n+1)}} \biggr\}^4 + \mathcal{Y}^2 \biggl\{ \biggl( \frac{4\pi}{3} \biggr)^{2n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{3/2} \frac{\mathcal{A}_{M_\ell}^{3/2}}{\mathcal{B}_{M_\ell}^{2n/(n+1)}} \biggr\}^2</math>

 

 

<math>~ - \mathcal{X}^{(n-3)/n} \mathcal{Y}^{(n+1)/n} \biggl\{ \biggl( \frac{4\pi}{3} \biggr)^{n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{1/2} \frac{\mathcal{A}_{M_\ell}^{1/2}}{\mathcal{B}_{M_\ell}^{n/(n+1)}}\biggr\}^{(n-3)/n} \biggl\{ \biggl( \frac{4\pi}{3} \biggr)^{2n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{3/2} \frac{\mathcal{A}_{M_\ell}^{3/2}}{\mathcal{B}_{M_\ell}^{2n/(n+1)}} \biggr\}^{(n+1)/n} </math>

 

<math>~=</math>

<math>~\mathcal{X}^4 \biggl\{ \biggl( \frac{4\pi}{3} \biggr)^{4n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{2} \frac{\mathcal{A}_{M_\ell}^{2}}{\mathcal{B}_{M_\ell}^{4n/(n+1)}} \biggr\} + \mathcal{Y}^2 \biggl\{ \biggl( \frac{4\pi}{3} \biggr)^{4n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{3} \frac{\mathcal{A}_{M_\ell}^{3}}{\mathcal{B}_{M_\ell}^{4n/(n+1)}} \biggr\}</math>

 

 

<math>~ - \mathcal{X}^{(n-3)/n} \mathcal{Y}^{(n+1)/n} \biggl\{ \biggl( \frac{4\pi}{3} \biggr)^{[(n-3)+2(n+1)]/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{[(n-3)+3(n+1)]/2n} \frac{\mathcal{A}_{M_\ell}^{[(n-3)+3(n+1)]/2n}}{\mathcal{B}_{M_\ell}^{[(n-3)+2(n+1)]/(n+1)}}\biggr\} </math>

 

<math>~=</math>

<math>~\biggl\{\mathcal{X}^4 + \biggl[\frac{3(n+1)}{4\pi n}\biggr] \mathcal{A}_{M_\ell} \mathcal{Y}^2 \biggr\} \biggl\{ \biggl( \frac{4\pi}{3} \biggr)^{4n/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{2} \frac{\mathcal{A}_{M_\ell}^{2}}{\mathcal{B}_{M_\ell}^{4n/(n+1)}} \biggr\}</math>

 

 

<math>~ - \mathcal{X}^{(n-3)/n} \mathcal{Y}^{(n+1)/n} \biggl\{ \biggl( \frac{4\pi}{3} \biggr)^{(3n-1)/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{2} \frac{\mathcal{A}_{M_\ell}^{2}}{\mathcal{B}_{M_\ell}^{(3n-1)/(n+1)}}\biggr\} </math>

 

<math>~=</math>

<math>~ \biggl[ \biggl(\frac{4\pi}{3}\biggr) \mathcal{X}^4 - \mathcal{B}_{M_\ell} \cdot \mathcal{X}^{(n-3)/n} \mathcal{Y}^{(n+1)/n} + \biggl(\frac{n+1}{n}\biggr) \mathcal{A}_{M_\ell} \mathcal{Y}^2 \biggr] \biggl\{ \biggl( \frac{4\pi}{3} \biggr)^{(3n-1)/(n+1)} \biggl[ \frac{3(n+1)}{4\pi n} \biggr]^{2} \frac{\mathcal{A}_{M_\ell}^{2}}{\mathcal{B}_{M_\ell}^{4n/(n+1)}} \biggr\} \, . </math>

Replacing <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math> with <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>, as prescribed by their defined relationships, the expression inside the square brackets becomes the above, boxed-in mass-radius relationship, namely,

<math>~ \frac{4\pi}{3} \cdot \mathcal{X}^4 - \mathcal{B} \cdot \mathcal{X}^{(n-3)/n} \biggl[ \mathcal{Y} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{limit}} \biggr) \biggr]^{(n+1)/n} +~ \mathcal{A} \biggl( \frac{n+1}{n} \biggr) \biggl[ \mathcal{Y} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{limit}} \biggr) \biggr]^{2} </math>

<math>~=</math>

<math>~0 \, .</math>

In Terms of Structural Form-Factors

Alternatively, replacing <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math> by their expressions in terms of the structural form factors gives,

<math>~ \frac{4\pi}{3} \cdot \mathcal{X}^4 - \mathcal{X}^{(n-3)/n} \mathcal{Y}^{(n+1)/n} \biggl( \frac{4\pi}{3} \biggr)^{-1/n} \frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_M^{(n+1)/n}} + \mathcal{Y}^2\biggl( \frac{n+1}{5n} \biggr) \frac{\tilde\mathfrak{f}_W }{\tilde\mathfrak{f}_M^2} </math>

<math>~=</math>

<math>~0 \, .</math>

Finally, inserting into this relation the expressions presented above for the structural form-factors, <math>~\tilde\mathfrak{f}_M</math> and <math>~\tilde\mathfrak{f}_A</math>, namely,

<math>~\tilde\mathfrak{f}_M</math>

<math>~=</math>

<math>~ \biggl[ - \frac{3\Theta^'}{\xi} \biggr]_{\tilde\xi} </math>

<math>~ \tilde\mathfrak{f}_A </math>

<math>~=</math>

<math>~ \tilde\theta^{n+1} + \biggl[ \frac{(n+1)}{3\cdot 5} \biggr] \tilde\xi^2 \cdot \tilde\mathfrak{f}_W </math>

gives us the desired,

Virial Theorem written in terms of <math>~\mathcal{X}</math>, <math>~\mathcal{Y}</math>, and <math>~\tilde\mathfrak{f}_W</math>

<math>~ 4\pi \cdot \mathcal{X}^4 ~- ~ \mathcal{X}^{(n-3)/n} \mathcal{Y}^{(n+1)/n} ( 4\pi)^{-1/n} \biggl[\frac{\tilde\xi}{(-\tilde\theta^')}\biggr]^{(n+1)/n} \biggl[\tilde\theta^{n+1} + \frac{(n+1)\tilde\xi^2}{3\cdot 5} \cdot \tilde\mathfrak{f}_W \biggr] ~+ ~ \mathcal{Y}^2\biggl( \frac{n+1}{3\cdot 5n} \biggr) \frac{\tilde\xi^2}{(- \tilde\theta^')^2} \cdot \tilde\mathfrak{f}_W = 0 \, . </math>

Relating and Reconciling Two Mass-Radius Relationships for n = 5 Polytropes

Now, let's examine the case of pressure-truncated, <math>~n=5</math> polytropes. As we have discussed in the context of detailed force-balanced models, Stahler (1983) has deduced that all <math>~n=5</math> equilibrium configurations obey the mass-radius relationship,

<math>~\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^2 - 5 \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr) + \frac{2^2 \cdot 5 \pi}{3} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^4 </math>

<math>~=</math>

<math>~0 \, ,</math>

where, as reviewed above, the mass and radius normalizations, <math>~M_\mathrm{SWS}</math> and <math>~R_\mathrm{SWS}</math>, may be treated as constants once the parameters <math>~K</math> and <math>~P_e</math> are specified. In contrast to this, the mass-radius relationship that we have just derived from the virial theorem for pressure-truncated, <math>~n=5</math> polytropes is,

<math> \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{mod}} \biggr)^2 - \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{mod}} \biggr)^{2/5} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{mod}} \biggr)^{6/5} + \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{mod}} \biggr)^4 = 0 \, , </math>

where the mass and radius normalizations,

<math>~M_\mathrm{mod}\biggr|_{n=5}</math>

<math>~=</math>

<math>~M_\mathrm{SWS} \biggl( \frac{3\mathcal{B}_{M_\ell}}{4\pi} \biggr)^{5/3} \biggl[ \frac{2\cdot 5\pi}{3^2 \mathcal{A}_{M_\ell}} \biggr]^{3/2} \, ,</math>

<math>~R_\mathrm{mod}\biggr|_{n=5}</math>

<math>~=</math>

<math>~R_\mathrm{SWS} \biggl( \frac{3\mathcal{B}_{M_\ell}}{4\pi}\biggr)^{5/6} \biggl[ \frac{2\cdot 5\pi}{3^2\mathcal{A}_{M_\ell}} \biggr]^{1/2} \, ,</math>

depend, not only on <math>~K</math> and <math>~P_e</math> via the definitions of <math>~M_\mathrm{SWS}</math> and <math>~R_\mathrm{SWS}</math>, but also on the structural form factors via the free-energy coefficients, <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math>. While these two separate mass-radius relationships are similar, they are not identical. In particular, the middle term involving the cross-product of the mass and radius contains different exponents in the two expressions. It is not immediately obvious how the two different polynomial expressions can be used to describe the same physical sequence.

This apparent discrepancy is reconciled as follows: The structural form factors — and, hence, the free-energy coefficients — vary from equilibrium configuration to equilibrium configuration. So it does not make sense to discuss evolution along the sequence that is defined by the second of the two polynomial expressions. If you want to know how a given system's equilibrium radius will change as its mass changes, the first of the two polynomials will do the trick. However, the equilibrium radius of a given system can be found by looking for extrema in the free-energy function while holding the free-energy coefficients, <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math>, constant; more importantly, the relative stability of a given equilibrium system can be determined by analyzing the behavior of the system's free energy while holding the free-energy coefficients constant. Dynamically stable versus dynamically unstable configurations can be readily distinguished from one another along the sequence that is defined by the second polynomial expression; they cannot be readily distinguished from one another along the sequence that is defined by the first polynomial expression. It is useful, therefore, to determine how to map a configuration's position on one of the sequences to the other.

Plotting Stahler's Relation

Mass (Y) vs. radius (X) plot
Switching, again, to the shorthand notation,

<math>~\mathcal{X}</math>

<math>~\equiv</math>

<math>~\frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \, ,</math>

<math>~\mathcal{Y}</math>

<math>~\equiv</math>

<math>~\frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \, ,</math>

the equilibrium mass-radius relation defined by the first of the two polynomial expressions can be plotted straightforwardly in either of two ways.

Quadratic Equation

One way is to recognize that the polynomial is a quadratic equation whose solution is,

<math>~\mathcal{Y}_\pm</math>

<math>~=</math>

<math>~\frac{5}{2} \mathcal{X} \biggl\{ 1 \pm \biggl[ 1 - \biggl( \frac{2^4\cdot \pi}{3\cdot 5} \biggr) \mathcal{X}^2 \biggr]^{1/2} \biggr\} \, .</math>

In the figure shown here on the right — see also the bottom panel of Figure 2 in our accompanying discussion of detailed force-balance models — Stahler's mass-radius relation has been plotted using the solution to this quadratic equation; the green segment of the displayed curve was derived from the positive root while the segment derived from the negative root is shown in orange. The two curve segments meet at the maximum value of the normalized equilibrium radius, namely, at

<math>\mathcal{X}_\mathrm{max} \equiv \biggl[ \frac{3\cdot 5}{2^4 \pi} \biggr]^{1/2} \approx 0.54627 \, .</math>

We note that, when <math>~\mathcal{X} = \mathcal{X}_\mathrm{max}</math>, <math>~\mathcal{Y} = (5\mathcal{X}_\mathrm{max}/2) \approx 1.36569</math>. Along the entire sequence, the maximum value of <math>~\mathcal{Y}</math> occurs at the location where <math>~d\mathcal{Y}/d\mathcal{X} = 0</math> along the segment of the curve corresponding to the positive root. This occurs along the upper segment of the curve where <math>~\mathcal{X}/\mathcal{X}_\mathrm{max} = \sqrt{3}/2</math>, at the location,

<math>\mathcal{Y}_\mathrm{max} \equiv \biggl[ \frac{3^3 \cdot 5^2}{2^6 } \biggr]^{1/2} \mathcal{X}_\mathrm{max} = \biggl[ \frac{3^4 \cdot 5^3}{2^{10} \pi } \biggr]^{1/2} \approx 1.77408 \, .</math>

Parametric Relations

The other way is to determine the normalized mass and normalized radius individually through Stahler's pair of parametric relations. Drawing partly from our above discussion and partly from a separate discussion where we provide a tabular summary of the properties of pressure-truncated <math>~n=5</math> polytropes, these are,

<math> ~\mathcal{X}\biggr|_{n=5} </math>

<math>~=~</math>

<math> \biggl( \frac{5}{4\pi} \biggr)^{1/2} \tilde\xi \tilde\theta^{2} = \biggl\{ \frac{3\cdot 5}{2^2 \pi} \biggl[ \frac{\tilde\xi^2/3}{(1+\tilde\xi^2/3)^{2}} \biggr] \biggr\}^{1/2} \, , </math>

<math> ~\mathcal{Y}\biggr|_{n=5} </math>

<math>~=~</math>

<math> \biggl( \frac{5^3}{4\pi} \biggr)^{1/2} \tilde\theta (- \tilde\xi^2 \tilde\theta^') = \biggl[ \biggl( \frac{3 \cdot 5^3}{2^2\pi} \biggr) \frac{(\tilde\xi^2/3)^3}{(1+\tilde\xi^2/3)^{4}} \biggr]^{1/2} \, . </math>

The entire sequence will be traversed by varying the Lane-Emden parameter, <math>~\tilde\xi</math>, from zero to infinity. Using the first of these two expressions, we have determined, for example, that the point along the sequence corresponding to the maximum normalized equilibrium radius, <math>~\mathcal{X}_\mathrm{max}</math>, is associated with an embedded <math>~n=5</math> polytrope whose truncated, dimensionless Lane-Emden radius is,

<math> ~\tilde\xi \biggr|_{\mathcal{X}_\mathrm{max}} = 3^{1/2} \, . </math>

Similarly, we have determined that the point along the sequence that corresponds to the maximum dimensionless mass, <math>~\mathcal{Y}_\mathrm{max}</math>, is associated with an embedded <math>~n=5</math> polytrope whose truncated, dimensionless Lane-Emden radius is, precisely,

<math> ~\tilde\xi \biggr|_{\mathcal{Y}_\mathrm{max}} = 3 \, . </math>

Referring back to our review of turning points along equilibrium sequences and, especially, the work of Kimura (1981b), we appreciate that the point that corresponds to the maximum mass, <math>~\mathcal{Y}_\mathrm{max}</math>, is the turning point that Kimura refers to as the "extremum in M1" along a p1 sequence. As we have highlighted, according to Kimura, this point should occur along the sequence where <math>~h_G=0</math>, that is, where the following condition applies:

<math>~\frac{\tilde\theta^{n+1}}{(\tilde\theta^')^2} = \frac{(n-3)}{2} \, .</math>

For the specific case being studied here, namely, <math>~n = 5</math> polytropic configurations, we therefore expect from Kimura's work that <math>~[\tilde\theta^6/(\tilde\theta^')^2] = 1</math> at the "maximum mass" turning point. Given that,

<math>~\tilde\xi \biggr|_{\mathcal{Y}_\mathrm{max}} = 3</math>         <math>~\Rightarrow</math>         <math>\tilde\theta_{n=5} = \frac{1}{2}</math>     and      <math>\tilde\theta_{n=5}^' = -\frac{1}{8} \, ,</math>

we see that Kimura's condition holds and, hence, that our identification of the location along the sequence of the maximum mass matches Kimura's identification of the location of that turning point.

We appreciate, as well, that the point corresponding to the maximum normalized equilibrium radius, <math>~\mathcal{X}_\mathrm{max}</math>, is the turning point that Kimura would reference as the "extremum in r1" along a p1 sequence. Following Kimura's analysis we have shown that this point occurs along the sequence where the following condition applies:

<math>~\frac{\xi (-\theta^')}{\tilde\theta} = \frac{2}{(n-1)} \, ,</math>

that is, for the specific case being studied here, we should expect <math>~[\tilde\xi (-\tilde\theta^')/\tilde\theta] = 1/2</math> at the "maximum radius" turning point. Given that,

<math>~\tilde\xi \biggr|_{\mathcal{X}_\mathrm{max}} = 3^{1/2}</math>         <math>~\Rightarrow</math>         <math>\tilde\theta_{n=5} = 2^{-1/2}</math>     and      <math>\tilde\theta_{n=5}^' = -(2^3 \cdot 3)^{-1/2} \, ,</math>

we see that Kimura's condition holds and, hence, that our identification of the location along the sequence of the maximum radius matches Kimura's identification of the location of that turning point.

Discussion

Notice that if the quadratic equation is used to map out the mass-radius relationship, the parameter, <math>~\tilde\xi</math>, never explicitly enters the discussion. Instead, a radius <math>~0 \le \mathcal{X} \le \mathcal{X}_\mathrm{max}</math> is specified and the two equilibrium masses associated with <math>~\mathcal{X}</math> — call them, <math>~\mathcal{Y}_+</math> and <math>~\mathcal{Y}_-</math> — are determined. (The values of the two masses are degenerate at both limiting values of <math>~\mathcal{X}</math>.) If the pair of parametric relations is used, instead, only one value of the mass is obtained for each specified value of <math>~\tilde\xi</math>. As <math>~\tilde\xi</math> is increased from <math>~0</math> to <math>~\sqrt{3}</math>, <math>~\mathcal{X}</math> increases monotonically from <math>~0</math> to <math>~\mathcal{X}_\mathrm{max}</math> and the corresponding mass is (only) <math>~\mathcal{Y}_-</math>; that is, as <math>~\tilde\xi</math> is increased from <math>~0</math> to <math>~\sqrt{3}</math>, we move away from the origin in a counter-clockwise direction along the lower segment (colored orange in the above figure) of the plotted equlibrium sequence. Then, as <math>~\tilde\xi</math> is increased from <math>~\sqrt{3}</math> to <math>~\infty</math>, we continue to move in a counter-clockwise direction along the equilibrium sequence, but now along the upper segment (colored green in the above figure) of the sequence, back to the origin; that is to say, <math>~\mathcal{X}</math> steadily decreases from <math>~\mathcal{X}_\mathrm{max}</math> back to <math>~0</math> and this time the relevant associated mass is the positive root of the quadratic relation, <math>~\mathcal{Y}_+</math>.

Clearly, then, each value of <math>~\mathcal{X}</math> is associated with two different values of the parametric parameter, <math>~\tilde\xi</math>. By inverting the <math>~\mathcal{X}(\tilde\xi)</math> parametric expression we see that, the two values of <math>~\tilde\xi</math> associated with a given equilibrium radius are,

<math>~\tilde\xi_\pm</math>

<math>~=</math>

<math>~\biggl\{\frac{3}{\alpha} \biggl[ 1 \pm \sqrt{1 - \alpha^2} \biggr] \biggr\}^{1/2} \, ,</math>

where,

<math>~\alpha</math>

<math>~\equiv</math>

<math>~\frac{(\mathcal{X}/\mathcal{X}_\mathrm{xmax})^2}{2-(\mathcal{X}/\mathcal{X}_\mathrm{xmax})^2} \, .</math>

We note as well that, for a given equilibrium radius, <math>~\mathcal{X}</math>, the ratio of the two mass solutions is given by a very simple expression, namely,

<math>~\frac{\mathcal{Y}_-}{\mathcal{Y}_+} = \frac{\tilde\xi_-^2}{3}</math>

          or        

<math>~\frac{\mathcal{Y}_+}{\mathcal{Y}_-} = \frac{\tilde\xi_+^2}{3} \, .</math>

This implies, as well, that,

<math>~\tilde\xi_+ \cdot \tilde\xi_-</math>

<math>~=</math>

<math>~3\, .</math>

Plotting the Virial Theorem Relation

Drawing from our above derivations, the concise free-energy expression that reflects the properties of pressure-truncated <math>~n = 5</math> polytropic configurations is,

<math>~\mathfrak{G}^*</math>

<math>~=</math>

<math>~ - \frac{18}{5}\cdot \mathcal{A}_{M_\ell} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{2} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-1} +~ 5\mathcal{B}_{M_\ell} \cdot \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{6/5} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-3/5} +~ \frac{4\pi}{3} \cdot \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{3} \, , </math>

where,

<math>~\mathcal{A}_{M_\ell}</math>

<math>~=</math>

<math>~ \frac{1}{5} \cdot \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} =\frac{1}{3^2\cdot 5} \biggl( \frac{\tilde\xi}{-\tilde\theta^'} \biggr)^2 \tilde\mathfrak{f}_W \, , </math>

<math>~\mathcal{B}_{M_\ell}</math>

<math>~=</math>

<math>~ \biggl( \frac{3}{4\pi}\biggr)^{1/5} \frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_M^{6/5}} = \frac{1}{3}( 4\pi)^{-1/5} \biggl( \frac{\tilde\xi}{-\tilde\theta^'} \biggr)^{6/5} \biggl[ \tilde\theta^6 + \frac{2}{5} \cdot \tilde\xi^2 \tilde\mathfrak{f}_W \biggr] \, . </math>

The virial theorem which is derived from this free-energy expression provides a mass-radius relationship to be compared with the detailed force-balance relationship presented by Stahler. Because our intent is to make this comparison, we begin with the virial theorem as written in terms of the variables, <math>~\mathcal{X}</math> and <math>~\mathcal{Y}</math>, and specialized for the case of <math>~n = 5</math> polytropic configurations. Written in terms of the (constant) coefficients in the free-energy expression, we have

<math>~ \mathcal{X}^4 - \frac{3\mathcal{B}_{M_\ell}}{4\pi} \cdot ( \mathcal{X} \mathcal{Y}^3 )^{2/5} +~ \frac{9 \mathcal{A}_{M_\ell}}{10\pi}\cdot \mathcal{Y}^{2} </math>

<math>~=</math>

<math>~0 \, ;</math>

or, from above, the

Virial Theorem written in terms of <math>~\mathcal{X}</math>, <math>~\mathcal{Y}</math>, and <math>~\tilde\mathfrak{f}_W</math>

<math>~ \mathcal{X}^4 ~- ~ (\mathcal{X} \mathcal{Y}^3)^{2/5} \biggl[\frac{\tilde\xi}{4\pi (-\tilde\theta^')}\biggr]^{6/5} \biggl[\tilde\theta^{6} + \frac{2}{5} \cdot \tilde\xi^2 \tilde\mathfrak{f}_W \biggr] ~+ ~ \mathcal{Y}^2 \biggl[\frac{\tilde\xi}{(- \tilde\theta^')} \biggr]^2 \frac{\tilde\mathfrak{f}_W}{2\cdot 5^2 \pi} = 0 \, , </math>

where, specifically for <math>~n = 5</math> polytropic configurations — see our summary of the radial profiles of physical variables and our determination of expressions for the structural form-factors,

<math>~\tilde\theta</math>

<math>~=</math>

<math>~ ( 1+ \ell^2 )^{-1/2} \, , </math>

<math>~\tilde\theta^'</math>

<math>~=</math>

<math>~ - 3^{-1/2} \ell ( 1+ \ell^2 )^{-3/2} \, , </math>

<math>~\mathfrak{f}_W</math>

<math>~=</math>

<math>~ \frac{5}{2^4} \cdot \ell^{-5} \biggl[ \ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1 + \ell^2)^{-3} + \tan^{-1}(\ell ) \biggr] \, , </math>

<math>~\ell^2</math>

<math>~\equiv</math>

<math>~ \frac{\tilde\xi^2}{3} \, . </math>

Once numerical values have been assigned to the free-energy coefficients, <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math>, the mass-radius relationship given by the scalar virial theorem can be compared quantitatively with Stahler's (detailed force-balance) mass-radius relationship. The simplest, physically reasonable approximation would be to assume uniform-density structures, in which case, <math>~\tilde\mathfrak{f}_M = \tilde\mathfrak{f}_W = \tilde\mathfrak{f}_A = 1</math>, and accordingly, <math>~\mathcal{A}_{M_\ell} = 5^{-1}</math> and <math>~\mathcal{B}_{M_\ell} = (4\pi/3)^{-1/5}</math>. But a better approximation would be to assign values to the structural form-factors that properly represent the properties of at least one detailed force-balanced model. By way of illustration, the following table details what the proper values are for the two free-energy coefficients, and other relevant parameters, specifically for the model along Stahler's sequence that sits at <math>~\mathcal{Y}_\mathrm{max}</math> — that is, the model whose truncation radius is <math>~\tilde\xi = 3</math>. As is recorded in the table, in this case the precise values of the free-energy coefficients are,

<math>~\mathcal{A}_{M_\ell}</math>

<math>~=</math>

<math>~ \biggl( \frac{2^4 \pi^2}{3^7} \biggr)^{1/2} \, , </math>

<math>~\mathcal{B}_{M_\ell}</math>

<math>~=</math>

<math>~ \biggl( \frac{3}{2^{14}\pi} \biggr)^{1/5} \biggl[ 1 + \biggl( \frac{2^6\pi^2}{3^3} \biggr)^{1/2}\biggr] \, . </math>

Notice that, by choosing <math>~\tilde\xi = 3</math>, the evaluation of <math>~\tilde\mathfrak{f}_W</math> is particularly simple, in part, because <math>~\tan^{-1}(\ell) = \tan^{-1} \sqrt{3} = \pi/3</math>, but also because the term <math>~(\ell^4 - 8\ell^2/3 - 1)</math> equals zero.


Determination of Coefficient Values in the Specific Case of <math>~\tilde\xi = 3</math>

Quantity

Analytic Evaluation

Numerical

<math>~\ell</math>

<math>~3^{1/2}</math>

<math>~1.732051</math>

<math>~\tilde\theta</math>

<math>~2^{-1}</math>

<math>~0.5</math>

<math>~\tilde\theta^'</math>

<math>~2^{-3}</math>

<math>~0.125</math>

<math>~\mathcal{X}</math>

<math>~\biggl( \frac{3^2\cdot 5}{2^6\pi} \biggr)^{1/2}</math>

<math>~0.473087</math>

<math>~\mathcal{Y}</math>

<math>~\biggl( \frac{3^4\cdot 5^3}{2^{10}\pi} \biggr)^{1/2}</math>

<math>~1.774078</math>

<math>~\tilde\mathfrak{f}_W</math>

<math>~\biggl( \frac{5^2 \pi^2}{2^8\cdot 3^7} \biggr)^{1/2}</math>

<math>~0.020993</math>

<math>~\mathcal{A}_{M_\ell}</math>

<math>~\biggl( \frac{2^4 \pi^2}{3^7} \biggr)^{1/2}</math>

<math>~0.268711</math>

<math>~\mathcal{B}_{M_\ell}</math>

<math>~\biggl( \frac{3}{2^{14}\pi} \biggr)^{1/5} \biggl[ 1 + \biggl( \frac{2^6\pi^2}{3^3} \biggr)^{1/2}\biggr]</math>

<math>~0.830395</math>

<math>~G^*</math>

<math>~\biggl( \frac{3\cdot 5^3}{2^{12}\pi} \biggr)^{1/2} [ 2^3\pi + 3^{5/2} ]</math>

<math>~6.951544</math>

Virial:

<math>~\mathcal{X}^4 - \frac{3\mathcal{B}_{M_\ell}}{4\pi} \cdot (\mathcal{X}\mathcal{Y}^3 )^{2/5}</math>

          <math>~~~~~~~~~~+\frac{9 \mathcal{A}_{M_\ell}}{10\pi} \cdot \mathcal{Y}^2</math>

<math>~ \frac{3^4\cdot 5^2}{2^{12}\pi} - \frac{3^4\cdot 5^2}{2^{12}\pi} \biggl[ 1 + \biggl( \frac{2^6\pi^2}{3^3} \biggr)^{1/2}\biggr]</math>

          <math>~~~~~~~~~~+ \biggl( \frac{3^5\cdot 5^4}{2^{18} \pi^2} \biggr)^{1/2} </math>

Sums to zero,

exactly!


The curve traced out by the light-blue diamonds in each panel of the following comparison figure displays Stahler's analytically prescribed mass-radius relation; this curve is identical in all six panels and is the same as the curve displayed above in connection with our description of Stahler's mass-radius relation. Each point along this "Stahler" curve identifies a model having a different truncation radius, <math>~\tilde\xi</math>; as plotted here, starting near the origin and moving counter-clockwise around the curve, <math>~\tilde\xi</math> is varied from 0.05 to 42.5. As foreshadowed by the above discussion, the model having the greatest mass <math>~(\mathcal{Y}_\mathrm{max})</math> along the Stahler sequence — highlighted by the red filled circle in most of the figure panels — is defined by <math>~\tilde\xi = 3</math>.

In each figure panel, the curve traced out by the orange triangles — or, in one case, the orange triangles & light purple diamonds — displays the mass-radius relation defined by the virial theorem. These "Virial" curves are all defined by the same virial theorem polynomial expression, as just presented, but the coefficient of the <math>~\mathcal{Y}^2</math> term and the coefficient of the <math>~(\mathcal{X}\mathcal{Y}^3)^{2/5}</math> cross term — essentially, the value of <math>~\mathcal{A}_{M_\ell}</math> and the value of <math>~\mathcal{B}_{M_\ell}</math>, respectively — have different values in the six separate figure panels. In each case, a value has been specified for the parameter, <math>~\tilde\xi</math> (as identified in the title of each figure panel), and this, in turn, has determined the values of the two (constant) free-energy coefficients. For example, in the top-right figure panel whose title indicates <math>~\tilde\xi = 3</math>, the "Virial" curve traces the mass-radius relation prescribed by the virial theorem after the values of the free-energy coefficients have been set to values that correspond to a detailed force-balanced model with this specified truncation radius, that is (see the above table), <math>~\mathcal{A}_{M_\ell} = 0.268711</math> and <math>~\mathcal{B}_{M_\ell} = 0.830395</math>. Columns 2 and 3, respectively, of the table affixed to the bottom of the following comparison figure list the values of <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math> that have been used to define the "Virial" curve in each of the six figure panels, in accordance with the value of <math>~\tilde\xi</math> listed in column 1 of the table.


Comparing Two Separate Mass-Radius Relations for Pressure-Truncated n = 5 Polytropes

Comparison of Two Mass-Radius Relations

Comparison of Two Mass-Radius Relations

Comparison of Two Mass-Radius Relations

Comparison of Two Mass-Radius Relations

Comparison of Two Mass-Radius Relations

Comparison of Two Mass-Radius Relations

<math>~\tilde\xi</math>

Free-Energy Coefficients

Primary Overlap

Secondary Overlap

(see further elaboration below)

<math>~\mathcal{A}_{M_\ell}</math>

<math>~\mathcal{B}_{M_\ell}</math>

<math>~\mathcal{X} \equiv \frac{R_\mathrm{eq}}{R_\mathrm{SWS}}</math>

<math>~\mathcal{Y} \equiv \frac{M_\mathrm{limit}}{M_\mathrm{SWS}}</math>

<math>~\mathcal{X}</math>

<math>~\mathcal{Y}</math>

1.26419

0.214429

0.758099

0.520269

0.904142

0.388938

0.289568

2

0.233842

0.779836

0.540671

1.544775

0.468918

0.570916

3

0.268711

0.830395

0.473087

1.774078

0.507387

0.798441

3.5

0.288708

0.861503

0.434310

1.744359

0.515168

0.859518

3.850652

0.303490

0.884746

0.408738

1.699778

0.518588

0.888969

9.8461

0.601012

1.313904

0.186424

0.904141

0.520269

0.904143


As is discussed more fully, below, in each of the six panels of the above comparison figure, the "Virial" curve intersects the "Stahler" curve at two locations. These points of intersection are identified by the black filled circles in each figure panel. In each case, the intersection point that is farthest along on the Stahler sequence — as determined by starting at the origin and moving counter-clockwise along the sequence — identifies the "Primary Overlap" between the two curves. That is to say, the <math>~(\mathcal{X}, \mathcal{Y})</math> coordinates of this point (see columns 4 and 5 of the table affixed to the bottom of the figure) are the coordinate values that are obtained by plugging the specified value of <math>~\tilde\xi</math> (see the title of the figure panel or column 1 of the affixed table) into Stahler's pair of parametric relations. The second point of intersection in each panel — which we will refer to as the "Secondary Overlap" points and whose coordinates are provided in columns 6 and 7 of the affixed table — appears to be fortuitous and of no particularly significant astrophysical interest.


The mass-radius diagram displayed in the top-right panel of the above comparison figure has been reproduced in the upper-left panel of the following figure — in this case, with a coordinate aspect ratio that is closer to 1:1 — along with color images of the corresponding free-energy surface, viewed from two different perspectives, and a three-column table listing the 3D coordinates, <math>~(X,Y,Z) = (R_\mathrm{eq}/R_\mathrm{SWS}, M_\mathrm{limit}/M_\mathrm{SWS}, \mathfrak{G}^*)</math>, of the seventeen points that have been used to define the displayed "Virial" curve. To be more explicit, the rainbow-colored free-energy surface, <math>~\mathfrak{G}^*(R,M_\mathrm{limit})</math>, has been defined by the free-energy function appropriate to pressure-truncated <math>~n=5</math> polytropic configurations as defined above, that is,

<math>~\mathfrak{G}^*</math>

<math>~=</math>

<math>~ - \frac{18}{5}\cdot \mathcal{A}_{M_\ell} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{2} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-1} +~ 5\mathcal{B}_{M_\ell} \cdot \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{6/5} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-3/5} +~ \frac{4\pi}{3} \cdot \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{3} \, , </math>

with the values of the two free-energy coefficients set to the values that correspond to a <math>~\tilde\xi = 3</math> virial curve as discussed above, namely, <math>~\mathcal{A}_{M_\ell} = 0.268711</math> and <math>~\mathcal{B}_{M_\ell} = 0.830395</math>.


Free-Energy Surface for n = 5 Polytropic Configurations Truncated at <math>~\tilde\xi = 3</math>

Radius

Mass

Free

Energy

Virial Mass-Radius Relation

Virial Mass-Radius Relation on top of Free-Energy Surface

0.3152

0.1345

0.8231

0.4159

0.3470

1.9947

0.4731

0.5735

3.1095

0.5035

0.7661

3.9591

0.5114

0.8347

4.2409

0.5297

1.0759

5.1453

0.5310

1.4036

6.1560

0.5250

1.5010

6.4046

Free-Energy Surface

0.5185

1.5680

6.5606

0.5114

1.6206

6.6741

0.4731

1.7741

6.9515

0.4343

1.8287

7.0247

0.3984

1.8333

7.0301

0.3379

1.7764

6.9923

0.2911

1.6909

6.9541

0.2260

1.5229

6.9091

0.1545

1.2749

6.8786

In the bottom panel of this figure, the undulating free-energy surface is drawn in three dimensions and viewed from a vantage point that illustrates its "valley of stability" and "ridge of instability;" the surface color correlates with the value of the free energy. Twelve small colored dots identify extrema — either the bottom of a valley or the top of a ridge — in the free-energy function and therefore trace the mass-radius relation defined by the scalar virial theorem. The 3D coordinates of these twelve points are provided in the three-column table that is affixed to the righthand edge of the figure: The coordinates of the (only) red dot are provided in row 13 of the table (red has also been assigned as the "bgcolor" of this table row); the equilibrium configuration having the greatest mass along the intersecting Stahler sequence is identified by the (only) black dot (coordinates are provided in row 11 of the table and, correspondingly, bgcolor="black" for that row); bgcolor="lightblue" has been assigned to the other rows of the affixed table that provide coordinates of the other 10, blue dots.

The upper-right panel of this figure presents the two-dimensional projection that results from viewing the identical free-energy surface "from above," along a line of sight that is parallel to the free-energy <math>~(Z)</math> axis and looking directly down onto the radius-mass <math>~(X-Y)</math> plane. From this vantage point, the twelve small colored dots cleanly trace out the <math>~M_\mathrm{limit}(R_\mathrm{eq})</math> equilibrium sequence that is defined by the scalar virial theorem, exactly reproducing the "Virial" curve that is depicted in the mass-radius diagram shown in the upper-left panel of the figure.

Discussion

A spherically symmetric, self-gravitating gas cloud whose effective adiabatic exponent is <math>~\gamma < 4/3</math> — equivalently, <math>~n > 3</math> — cannot exist in a dynamically stable equilibrium state, in isolation. Such clouds can be stabilized, however, if they are embedded in a hot, tenuous external medium and effectively confined by an external pressure, <math>~P_e</math>. The pressure-truncated, <math>~n = 5 ~(\gamma = 6/5)</math> polytropic configurations being discussed here provide examples of such embedded clouds. A direct analogy can be drawn between this discussion and discussions of pressure-truncated isothermal <math>~(\gamma = 1; n = \infty)</math> clouds — see, for example, our review of isothermal cloud structures in the context of Bonnor-Ebert spheres.

Part I: Physical Significance of the Two Curves

The "Stahler" mass-radius relation, plotted as a continuous curve in the above figure and reproduced as a sequence of discrete points in each panel of the subsequent comparison figure, identifies the precise mass <math>~(\mathcal{Y})</math> and associated radius <math>~(\mathcal{X})</math> of physically allowed pressure-truncated, <math>~n = 5</math> polytropic configurations over the full range of values of the dimensionless truncation radius, <math>~0 < \tilde\xi < \infty</math>. Each model along the curve has an internal structure that ensures detailed force balance throughout the configuration; because this internal structure varies from model to model, the values of the structural form-factors — <math>~\mathfrak{f}_M, \mathfrak{f}_W</math>, and <math>~\mathfrak{f}_A</math> — and the corresponding values of the coefficients associated with the free-energy function — <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math> — will also vary from model to model along the Stahler curve.

If the values of the coefficients, <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math> (as well as the external pressure and, hence, the additional coefficient, <math>~\mathcal{D}</math>) are held fixed, the algebraic free-energy function defines how a configuration's free energy will change as its overall size is varied. Extrema in the free energy will identify equilibrium configurations. Based on this understanding, our derived virial theorem expression for <math>~n = 5</math> polytropic configurations identifies equilibrium radii <math>~(\mathcal{X})</math> associated with various configuration masses <math>~(\mathcal{Y})</math>. The "Virial" curve that has been plotted in each panel of the above comparison figure shows how the equilibrium radius varies with configuration mass, as dictated by the virial theorem — and, hence, as identified by extrema in the free-energy function — assuming that the relevant free-energy coefficients are held fixed. In each figure panel, this "Virial" curve qualitatively resembles the quantitatively correct, "Stahler" mass-radius relationship that has been derived from the properties of detailed force-balance models. The two curves overlap, and cross, wherever the coefficients used to define the "Virial" relation are identical to the coefficient values that are associated with a specific model along the "Stahler" relation. The two curves do not trace out identical mass-radius relationships simply because the structural form factors vary from model to model along the "Stahler" sequence.

In the context of star formation, the Stahler sequence can be viewed as an evolutionary sequence for cold protostellar gas clouds that are embedded in a hot, tenuous interstellar medium. An initially low-mass cloud is represented by an equilibrium configuration that has been truncated at a very small Lane-Emden radius, <math>~\tilde\xi</math>; such clouds will appear near the origin of the displayed <math>~\mathcal{X}-\mathcal{Y}</math> plane, at a point along the "lower" segment of the Stahler mass-radius relation. Over time, as the cloud grows in mass (through collisions with and accretion of other low-mass clouds, for example), it will slide up the lower segment of the Stahler curve, moving in a counter-clockwise direction further and further away from the plot origin. The mass-accretion process that drives the cloud's evolution presumably occurs on a time scale that is long compared to the local dynamical-readjustment time of the cloud, allowing the cloud's internal structure time to readjust and establish the properties defined by Stahler's detailed force-balance analysis.

Part II: Curve Intersections

Early Thoughts

Notice that, in each frame of the above comparison figure, the "Virial" curve intersects and crosses the "Stahler" curve at two locations. In each plot these two crossing points are identified by filled black circles and, in each plot, the crossing point that lies farthest along the curve — again, starting from near the origin and moving around the curve in a counter-clockwise direction — is associated with the equilibrium model on the Stahler curve that is defined by the same value of <math>~\tilde\xi</math> that was used to define the coefficients of the virial theorem mass-radius relation. This is not surprising, as the virial theorem should be precisely satisfied by every one of the equilibrium models along the Stahler sequence, as long as the value of <math>~\tilde\xi</math> that is used to define the coefficients of the free-energy function and, in turn, the virial theorem mass-radius relation is identical to the value of <math>~\tilde\xi</math> that defines the truncation radius of the detailed force-balance model. For example, the "Virial" curve that appears in the top-right panel of the comparison figure — a panel whose title includes the notation, <math>~\xi = 3</math> — intersects the "Stahler" curve at <math>~\mathcal{Y}_\mathrm{max}</math>, that is, at the location of the detailed force-balance model that, as previously explained, has a truncation radius, <math>~\tilde\xi = 3</math>.

It is not (yet) clear to us what physical significance should be ascribed to the model along the Stahler sequence that is identified by the second crossing of the "Virial" curve, given that the value of <math>~\tilde\xi</math> associated with the truncation radius of this second detailed force-balance model is not the same as the value of <math>~\tilde\xi</math> that was used to define the coefficients of the "Virial" curve. We note that, at least for the range of values of <math>~\tilde\xi</math> sampled in the above figure, this second crossing point seems to hover around the same limited segment of the Stahler sequence.

By direct analogy with discussions of Bonnor-Ebert spheres, the "maximum mass" model associated with <math>~\mathcal{Y}_\mathrm{max}</math> along the Stahler mass-radius relation has important physical significance in astrophysics. For a given applied external pressure, however, no models exist above some limiting mass — identified, here, by <math>~\mathcal{Y}_\mathrm{max}</math>.

Analysis Philosophy

The mass-radius relationship that derives from detailed force-balanced models is a physically meaningful and reliable statement of how a configuration's equilibrium radius will vary if its mass is changed. (It must be accepted that the configuration's structural form factors will change as it settles into each new equilibrium state, so such an "evolution" must occur on a secular time scale.) From the outset, however, the mass-radius relationship derived via the virial theorem — which, itself, derives from an analysis of the free energy function — should not be relied upon for the same physical insight. Consider, for example, that the scalar virial theorem is obtained from an analysis of the free-energy function by varying a system's size while holding constant all coefficients in the free-energy expression; this means that the system's mass as well as its structural form factors is held fixed while searching for an extremum in the free energy. The temptation, then, is to use the virial theorem to predict what the configuration's new equilibrium size will be if the system's mass is changed while holding the coefficients in the virial theorem constant. This means holding the structural form factors constant but not simultaneously holding the mass constant, and this differs from the constraints put on the free-energy function analysis that led to the virial theorem expression in the first place!

But we can combine the two analyses — the detailed force-balance analysis and the free-energy analysis — in the following meaningful way. Use the detailed force-balance analysis to identify the properties of an equilibrium state, specifically, for a given mass, determine the system's equilibrium radius and its accompanying structural form factors. (The virial theorem will be satisfied by this same set of determined parameter values.) Then, holding both the mass and the structural form factors constant, see how the free energy of the system varies as the configuration's size changed. In this manner the system's dynamical stability can be ascertained.

In summary: The mass-radius relationship determined from an analysis of detailed force-balanced models defines the physically correct secular evolutionary track for the system; while, an analysis of the free energy variations about an equilibrium state will answer the question of dynamical stability.

Quantitative Study

The preceding philosophical statements not withstanding, it is still worth understanding the relationship — if any — between the pair of models that are identified by the "second crossing" of the Stahler sequence by the "Virial" curve.

More Information on Secondary Overlap Points

From above table Determined here

<math>~\mathcal{X} \equiv \frac{R_\mathrm{eq}}{R_\mathrm{SWS}}</math>

<math>~\mathcal{Y} \equiv \frac{M_\mathrm{limit}}{M_\mathrm{SWS}}</math>

<math>~\tilde\xi_-</math>

<math>~\mathcal{Y}_-</math>

<math>~\mathcal{A}_{M_\ell}</math>

<math>~\mathcal{B}_{M_\ell}</math>

0.388938

0.289568

0.72447

0.289568

0.20491

0.75191

0.468918

0.570916

0.98268

0.570916

0.20889

0.75395

0.507387

0.798441

1.17380

0.798441

0.21252

0.75652

0.515168

0.859518

1.22572

0.859520

0.21360

0.75740

0.518588

0.888969

1.25104

0.888968

0.21414

0.75785

0.520269

0.904143

1.26419

0.904142

0.21443

0.75810


Marginal Stability

As mentioned above, it is widely appreciated that the model having the largest mass — that is, the model that sits at <math>~\mathcal{Y}_\mathrm{max}</math> — along the Stahler sequence is of considerable astrophysical significance. Viewed in terms of a cloud's secular evolution, counter-clockwise along the sequence, something rather catastrophic must happen once the cloud acquires the mass associated with <math>~\mathcal{Y}_\mathrm{max}</math>, because no equilibrium structure is available to the cloud if it gains any additional mass. It is tempting to associate this point along the Stahler sequence with a dynamical instability, imagining for example that the cloud will begin to dynamically collapse once it reaches this <math>~\mathcal{Y}_\mathrm{max}</math> configuration. But the "detailed force-balance" technique that is used to define the structure of equilibrium models along the Stahler sequence does not give us any insight regarding a configuration's dynamical stability.

Our free-energy analysis does provide this additional insight. The mass-radius relationship derived from the scalar virial theorem — which, itself, was derived via a free-energy analysis — is qualitatively similar to the mass-radius relationship defined (from a detailed force-balance analysis) by the Stahler sequence; in particular, it also exhibits an upper mass limit. And our free-energy analysis reveals that this "maximum mass" point associated with the virial theorem separates dynamically stable from dynamically unstable models along the sequence. This realization fuels the temptation just mentioned; that is, it seems to support the idea that the configuration at <math>~\mathcal{Y}_\mathrm{max}</math> along Stahler's sequence is associated with the onset of a dynamical instability along the sequence. But this is not the case! Our free-energy analysis has also shown that, when the structural form-factors — and, most specifically, the coefficients <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math> — are assigned the values appropriate to the configuration at <math>~\mathcal{Y}_\mathrm{max}</math> along Stahler's sequence, the point of maximum mass associated with the corresponding expression for the virial theorem does not coincide with the configuration at <math>~\mathcal{Y}_\mathrm{max}</math>. The configuration at <math>~\mathcal{Y} = \mathcal{Y}_\mathrm{max} = 1.774078</math> (also identified as the model having <math>~\tilde\xi = 3.0</math>) is found to be dynamically stable. Both of these realizations are illustrated graphically in the above figure.


Our analysis has shown, instead, that the marginally unstable configuration appears farther along the Stahler sequence when moving in a counter-clockwise direction. It corresponds to the model having <math>~\tilde\xi = 3.850652</math> instead of <math>~\tilde\xi = 3.0</math>. While this can be illustrated graphically — for example, by carefully analyzing and comparing the bottom-center panel with the top-right panel in the above figure ensemble — an algebraic demonstration is more definitive. Our stability analysis has shown that, for any pressure-truncated polytropic configuration, the equilibrium structure associated with the point of marginal instability has,

<math>~\biggl( \frac{\mathcal{Y}}{\mathcal{X}^2}\biggr)_\mathrm{crit} </math>

<math>~=</math>

<math>~ \biggl[ \frac{4\pi n}{\mathcal{A}_{M_\ell}(n-3)}\biggr]^{1/2} \, . </math>

For <math>~n=5</math> configurations, this means that the critical model along the equilibrium sequence will have,

<math>~\mathcal{X}_\mathrm{crit}^4 </math>

<math>~=</math>

<math>~ \biggl[ \frac{\mathcal{A}_{M_\ell}}{10\pi }\biggr] \mathcal{Y}_\mathrm{crit}^2 \, . </math>

But all configurations along Stahler's equilibrium sequence must also obey the mass-radius relationship,

<math>~\mathcal{Y}^2 - 5\mathcal{Y}\mathcal{X} + \frac{20\pi}{3} \mathcal{X}^4</math>

<math>~=</math>

<math>~ 0 \, . </math>

Combining these two requirements means,

<math>~\mathcal{Y}_\mathrm{crit}^2 - 5(\mathcal{Y}\mathcal{X})_\mathrm{crit} + \biggl( \frac{2\mathcal{A}_{M_\ell}}{3}\biggr) \mathcal{Y}_\mathrm{crit}^2</math>

<math>~=</math>

<math>~ 0 </math>

<math>~\Rightarrow ~~~~\mathcal{Y}_\mathrm{crit}^2 \biggl[ 1 + \frac{2}{3}\cdot \mathcal{A}_{M_\ell} \biggr] </math>

<math>~=</math>

<math>~ 5(\mathcal{Y}\mathcal{X})_\mathrm{crit} </math>

<math>~\Rightarrow ~~~~\frac{ \mathcal{X}_\mathrm{crit} }{ \mathcal{Y}_\mathrm{crit} }</math>

<math>~=</math>

<math>~ \frac{1}{5}\biggl[ 1 + \frac{2}{3}\cdot \mathcal{A}_{M_\ell} \biggr] \, . </math>

Now, taking into detailed account the internal structure of pressure-truncated, <math>~n=5</math> polytropic structures as represented in our summary table of Stahler's equilibrium configurations, we know that, along Stahler's entire sequence,

<math>~\frac{ \mathcal{X} }{ \mathcal{Y} }</math>

<math>~=</math>

<math>~\biggl\{ \biggl( \frac{3\cdot 5}{2^2 \pi} \biggr) \frac{\ell^2}{(1+\ell^2)^{2}} \cdot \biggl( \frac{2^2\pi}{3 \cdot 5^3} \biggr) \frac{(1+\ell^2)^{4}}{(\ell^2)^3} \biggr\}^{1/2} </math>

 

<math>~=</math>

<math>~\frac{1 + \ell^2}{5\ell^2} \, , </math>

where we have again adopted the shorthand notation,

<math>~\ell^2 \equiv \frac{\tilde\xi^2}{3} \, .</math>

We conclude, therefore, that in the marginally unstable model along the Stahler equilibrium sequence,

<math>~1 + \frac{2}{3}\cdot (\mathcal{A}_{M_\ell})_\mathrm{crit}</math>

<math>~=</math>

<math>~\frac{1 + \ell_\mathrm{crit}^2}{\ell_\mathrm{crit}^2} </math>

<math>~\Rightarrow ~~~~~(\mathcal{A}_{M_\ell})_\mathrm{crit} </math>

<math>~=</math>

<math>~\frac{3}{2} \cdot \ell_\mathrm{crit}^{-2} \, . </math>

Given that the general expression for <math>~\mathcal{A}_{M_\ell}</math> along the Stahler sequence is,

<math>~\mathcal{A}_{M_\ell} </math>

<math>~=</math>

<math>~\frac{1}{2^4} \biggl[ \ell^{-4} \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr) + \ell^{-5} (1 + \ell^2)^{3} \tan^{-1}(\ell ) \biggr] \, , </math>

we deduce that,

<math>~\ell_\mathrm{crit} = 2.2231751 </math>       or, equivalently,       <math>~\tilde\xi_\mathrm{crit} = 3.850652 \, .</math>

Hence, also,

<math>~( \mathcal{X}_\mathrm{crit}, \mathcal{Y}_\mathrm{crit} ) = ( 0.408738, 1.699778 ) \, . </math>

Part III

From our above, detailed analysis of the mass-radius relation for pressure-truncated polytropes, we concluded that configurations along "Stahler's" equilibrium sequence become dynamically unstable at a point that does not coincide with the maximum-mass configuration. Instead, the onset of dynamical instability is associated with the critical point on the mass-radius relation that arises from the free-energy-based virial theorem. In drawing this conclusion, we have implicitly assumed that the proper way to analyze an equilibrium configuration's stability is to vary its radius while, not only holding its mass, specific entropy, and surface pressure <math>~(P_e)</math> constant, but also assuming that the configuration's structural form factors are invariable.


This seems like a reasonable assumption, given that we're asking how a configuration's characteristics will vary dynamically when perturbed about an equilibrium state. While oscillating about an equilibrium state, it seems more reasonable to assume that the system will expand and contract in a nearly homologous fashion than that its internal structure will readily readjust to produce a different and desirable set of form factors. In support of this argument, we point to the paper by Goldreich & Weber (1980) which explicitly derives a self-similar solution for the homologous collapse of stellar cores that can be modeled as <math>~n=3</math> polytropes; an associated chapter of this H_Book details the Goldreich & Weber derivation. Goldreich & Weber use linear perturbation techniques to analyze the stability of their homologously collapsing configurations. In §IV of their paper, they describe the eigenvalues and eigenfunctions that result from this analysis. They discovered, for example, that "the lowest radial mode can be found analytically ... [and it] corresponds to a homologous perturbation of the entire core." Our assumption that the structural form factors remain constant when pressure-truncated polytropic configurations undergo radial size variations therefore appears not to be unreasonable. (Based on the Goldreich & Weber discussion, we should also look at the published work of Schwarzschild (1941, ApJ, 94, 245), who has evaluated radial modes, and of Cowling (1941, MNRAS, 101, 367), who has obtained eigenvalues of some low-order nonradial modes.)


In addition, it would seem that a certain amount of dissipation would be required for the system to readjust to new structural form factors. In order to test this underlying assumption, following Goldreich & Weber (1980), it would be desirable to carry out a full-blown perturbation analysis that involves looking for, for example, the eigenvector associated with the system's fundamental radial mode of pulsation. Ideally, we should be using the structural form factors associated with this pulsation-mode eigenfunction in our free-energy analysis of stability. Better yet, the sign of the eigenfrequency associated with the system's pulsation-mode eigenvector should signal whether the system is dynamically stable or unstable.



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Other Model Sequences

Relating and Reconciling Two Mass-Radius Relationships for n = 4 Polytropes

For pressure-truncated <math>~n=4</math> polytropes, Stahler (1983) did not identify a polynomial relationship between the mass and radius of equilibrium configurations. However, from his analysis of detailed force-balance models (summarized above), we appreciate that the governing pair of parametric relations is,

<math> ~\mathcal{X} </math>

<math>~=~</math>

<math> \biggl( \frac{1}{\pi} \biggr)^{1/2} \tilde\xi \tilde\theta^{3/2} \, , </math>

<math> ~\mathcal{Y} </math>

<math>~=~</math>

<math> \biggl( \frac{2^4}{\pi} \biggr)^{1/2} \tilde\theta^{1/2} (- \tilde\xi^2 \tilde\theta^') \, . </math>

On the other hand, the polynomial that results from plugging <math>~n=4</math> into the general mass-radius relation that is obtained via the virial theorem is,

<math> \frac{4\pi}{3} \mathcal{X}^4 - \biggl[ \frac{\mathcal{X} \mathcal{Y}^{5}}{4\pi}\biggr]^{1/4} \mathfrak{b}_{n=4} + \frac{5}{4} \mathcal{Y}^2 = 0 \, , </math>

where,

<math>\mathfrak{b}_{n=4} = \biggl[ 5 (-\tilde\theta^')^2 + \frac{1}{3} \tilde\theta^{5} \biggr] \biggl( \frac{\tilde\xi}{-\tilde\theta^'} \biggr)^{5/4} \, . </math>

[For the record we note that, throughout the structure of an <math>~n=4</math> polytrope, <math>~\mathfrak{b}_{n=4}</math> is a number of order unity. Its value is never less than <math>~3^{1/4}</math>, which pertains to the center of the configuration; its maximum value of <math>\approx 5.098</math> occurs at <math>~\tilde\xi \approx 4.0</math>; and <math>~\mathfrak{b}_{n=4} \approx 3.946</math> at its (zero pressure) surface, <math>~\tilde\xi = \xi_1 \approx 14.97</math>. A plot showing the variation with <math>~P_e</math> of the closely allied parameter, <math>~\mathcal{B}|_{n=4} = (4\pi)^{1/4} \mathfrak{b}_{n=4}</math> is presented in the righthand panel of the above parameter summary figure.]

In both panels of the following figure, the blue curve displays the mass-radius relation for pressure-truncated <math>~n=4</math> polytropes, <math>~\mathcal{Y}(\mathcal{X})</math>, that is generated by Stahler's pair of parametric equations. The coordinates of discrete points along the curve have been determined from the tabular data provided on p. 399 of Horedt (1986, ApJS, vol. 126)] while Excel has been used to generate the "smooth," continuous blue curve connecting the points; this set of points and accompanying blue curve are identical in both figure panels. In both figure panels, a set of discrete, triangle-shaped points traces the mass-radius relation, <math>~\mathcal{Y}(\mathcal{X})</math>, that is obtained via the virial theorem, assuming that the coefficient, <math>~\mathfrak{b}_{n=4}</math>, is constant along the sequence. The "green" sequence in the lefthand panel results from setting <math>~\mathfrak{b}_{n=4} = 3.4205</math>, which is the value of the constant that results from Horedt's tabulated data if the configuration is truncated at <math>~\tilde\xi = 1.4</math>; the "orange" sequence in the righthand panel results from setting <math>~\mathfrak{b}_{n=4} = 4.8926</math>, which is the value of the constant that results from Horedt's tabulated data if the configuration is truncated at <math>~\tilde\xi = 2.8</math>.

Comparing Two Separate Mass-Radius Relations for Pressure-Truncated n = 4 Polytropes

Comparison of Two Mass-Radius Relations

According to Horedt's (1986)] tabulated data, the surface of an isolated <math>~(P_e = 0)</math>, spherically symmetric, <math>~n=4</math> polytrope occurs at the dimensionless (Lane-Emden) radius, <math>~\xi_1 = 14.9715463</math>. In both panels of the above figure, this isolated configuration is identified by the discrete (blue diamond) point at the origin, that is, at <math>~(\mathcal{X}, \mathcal{Y}) = (0, 0)</math>. As we begin to examine pressure-truncated models and <math>~\tilde\xi</math> is steadily decreased from <math>~\xi_1</math>, the mass-radius coordinate of equilibrium configurations "moves" away from the origin, upward along the upper branch of the displayed (blue) mass-radius relation. A maximum mass of <math>~\mathcal{Y} \approx 2.042</math> (corresponding to a radius of <math>~\mathcal{X} \approx 0.4585</math>) is reached from the left as <math>~\tilde\xi</math> drops to a value of approximately <math>~3.4</math>. As <math>~\tilde\xi</math> continues to decrease, the mass-radius coordinates of equilibrium configurations move along the lower branch of the displayed (blue) curve, reaching a maximum radius at <math>~(\mathcal{X}, \mathcal{Y}) \approx (0.555, 1.554)</math> — corresponding to <math>~\tilde\xi \approx 2.0</math> — then decreasing in radius until, once again, the origin is reached, but this time because <math>~\tilde\xi</math> drops to zero.


If we set <math>~\mathfrak{b}_{n=4} = 3.4205</math> (corresponding to a choice of <math>~\tilde\xi = 1.4</math>), the virial theorem mass-radius relation maps onto the "Stahler" mass-radius coordinate plane as depicted by the set of green, triangle-shaped points in the lefthand panel of the above figure. While the (green) curve corresponding to this relation does not overlay the blue mass-radius relation, the two curves do intersect. They intersect precisely at the coordinate location along the blue curve (emphasized by the black filled circle) corresponding to a detailed force-balanced model having <math>~\tilde\xi = 1.4</math>. In an analogous fashion, in the righthand panel of the figure, the curve delineated by the set of orange triangle-shaped points shows how the virial theorem mass-radius relation maps onto the "Stahler" mass-radius coordinate plane when we set <math>~\mathfrak{b}_{n=4} = 4.8926</math> (corresponding to a choice of <math>~\tilde\xi = 2.8</math>); it intersects the blue mass-radius relation precisely at the coordinate location, <math>~(\mathcal{X}, \mathcal{Y}) \approx (0.5108, 1.965)</math> — again, emphasized by a black filled circle — that corresponds to a detailed force-balanced model having <math>~\tilde\xi = 2.8</math>. Hence, the two relations give the same mass-radius coordinates when the value of <math>~\mathfrak{b}_{n=4}</math> that is plugged into the virial theorem matches the value of <math>~\mathfrak{b}_{n=4}</math> that reflects the structural form factor that is properly associated with a detailed force-balanced model.


When we mapped the virial theorem mass-radius relation onto Stahler's mass-radius coordinate plane using a value of <math>~\mathfrak{b}_{n=4} = 4.8926</math> (as traced by the orange triangle-shaped points in the righthand panel of the above figure), we expected it to intersect the blue curve at the point along the blue sequence where <math>~\tilde\xi = 2.8</math>, for the reason just discussed. After constructing the plot, it became clear that the two curves also intersect at the coordinate location, <math>~(\mathcal{X}, \mathcal{Y}) \approx (0.255, 1.67)</math> — also highlighted by a black filled circle — that corresponds to a detailed force-balanced model having <math>~\tilde\xi \approx 6.0</math>. This makes it clear that it is the equality of the structural form factors, not the equality of the dimensionless (Lane-Emden) radius, <math>~\tilde\xi</math>, that assures precise agreement between the two different mass-radius expressions.


As is detailed in our above discussion of the dynamical stability of pressure-truncated polytropes, an examination of free-energy variations can not only assist us in identifying the properties of equilibrium configurations (via a free-energy derivation of the virial theorem) but also in determining which of these configurations are dynamically stable and which are dynamically unstable. We showed that, for a certain range of polytropic indexes, there is a critical point along the corresponding model sequence where the transition from stability to instability occurs. As has been detailed in our above groundwork derivations, for <math>~n = 4</math> polytropic structures, the critical point is identified by the dimensionless parameters,

<math>\eta_\mathrm{crit}\biggr|_{n=4}~=~\frac{1}{15} \, ;</math>          <math>\Pi_\mathrm{max}\biggr|_{n=4}~=~\frac{15^{15}}{16^{16}} \, ;</math>      and      <math>\Chi_\mathrm{min}\biggr|_{n=4}~=~\biggl( \frac{16}{15} \biggr)^4 \, .</math>

In the context of the above figure, independent of the chosen value of <math>~\mathfrak{b}_{n=4}</math>, this critical point always corresponds to the maximum mass that occurs along the mass-radius relationship established via the virial theorem. In both panels of the figure, a horizontal red-dotted line has been drawn tangent to this critical point and identifies the corresponding critical value of <math>~\mathcal{Y}</math>; a vertical red-dashed line drawn through this same point helps identify the corresponding critical value of <math>~\mathcal{X}</math>. We have deduced (details of the derivation not shown) that, for pressure-truncated <math>~n=4</math> polytropes, the coordinates of this critical point in Stahler's <math>~\mathcal{X}-\mathcal{Y}</math> plane depends on the choice of <math>~\mathfrak{b}_{n=4}</math> as follows:

<math>~\mathcal{X}_\mathrm{crit}</math>

<math>~=</math>

<math>~\pi^{-1/2} 2^{-16/5} (3\mathfrak{b}_{n=4})^{4/5} \, ,</math>

<math>~\mathcal{Y}_\mathrm{crit}</math>

<math>~=</math>

<math>~\pi^{-1/2} 2^{-22/5} (3\mathfrak{b}_{n=4})^{8/5} \, .</math>

In practice, for a given plot of the type displayed in the above figure — that is, for a given choice of the structural parameter, <math>\mathfrak{b}_{n=4}</math> — it only makes sense to compare the location of this critical point to the location of points that have been highlighted by a filled black circle, that is, points that identify the intersection between the two mass-radius relations. If, in a given figure panel, a filled black circle lies to the right of the vertical dashed line, the equilibrium configuration corresponding to that black circle is dynamically stable. On the other hand, if the filled black circle lies to the left of the vertical dashed line, its corresponding equilibrium configuration is dynamically unstable. We conclude, therefore, that the equilibrium configuration marked by a filled black circle in the lefthand panel of the above figure is stable; however, both configurations identified by filled black circles in the righthand panel are unstable.

It is significant that the critical point identified by our free-energy-based stability analysis does not correspond to the equilibrium configuration having the largest mass along "Stahler's" (blue) equilibrium model sequence. One might naively expect that a configuration of maximum mass along the blue curve is the relevant demarcation point and that, correspondingly, all models along this sequence that fall "to the right" of this maximum-mass point are stable. But the righthand panel of our above figure contradicts this expectation. While both of the black filled circles in the righthand panel of the above figure lie to the left of the vertical dashed line and therefore, as just concluded, are both unstable, one of the two configurations lies to the right of the maximum-mass point along the blue "Stahler" sequence. This finding is related to the curiosity raised earlier in our discussion of the structural properties of pressure-truncated, <math>~n=4</math> polytropes.

Relating and Reconciling Two Mass-Radius Relationships for n = 3 Polytropes

For pressure-truncated <math>~n=3</math> polytropes, Stahler (1983) did not identify a polynomial relationship between the mass and radius of equilibrium configurations. However, from his analysis of detailed force-balance models (summarized above), we appreciate that the governing pair of parametric relations is,

<math> ~\mathcal{X} </math>

<math>~=~</math>

<math> \biggl( \frac{3}{4\pi} \biggr)^{1/2} \tilde\xi \tilde\theta \, , </math>

<math> ~\mathcal{Y} </math>

<math>~=~</math>

<math> \biggl( \frac{3^3}{4\pi} \biggr)^{1/2} (- \tilde\xi^2 \tilde\theta^') \, . </math>

On the other hand, the polynomial that results from plugging <math>~n=3</math> into the general mass-radius relation that is obtained via the virial theorem is,

<math> \frac{2^3 \pi}{3} \mathcal{X}^4 - \biggl[ \frac{\mathcal{Y}^{4}}{4\pi}\biggr]^{1/3} \mathfrak{b}_{n=3} + \frac{4}{3} \mathcal{Y}^2 = 0 \, , </math>

where,

<math>\mathfrak{b}_{n=3} = \biggl[ 4 (-\tilde\theta^')^2 + \frac{2}{3} \tilde\theta^{4} \biggr] \biggl( \frac{\tilde\xi}{-\tilde\theta^'} \biggr)^{4/3} \, . </math>

Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation