Editing SSC/Virial/PolytropesSummary

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=Virial Equilibrium of Adiabatic Spheres (Summary)=

The summary presented here has been drawn from our 
[[SSC/Virial/Polytropes#Virial_Equilibrium_of_Adiabatic_Spheres|accompanying detailed analysis of the structure of pressure-truncated polytropes]].

==Detailed Force-Balanced Solution==

As has been [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres|discussed in detail in another chapter]], {{ Horedt70full }},   {{ Whitworth81full }} and {{ Stahler83full }} 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 [[SSC/Structure/Polytropes#Lane-Emden_Equation|the Lane-Emden function]], <math>~\theta</math>, and its radial derivative.  For example &#8212; see our [[SSC/Structure/PolytropesEmbedded#Horedt.27s_Presentation|related discussion for more details]] &#8212; from {{ Horedt70 }} we obtain the following pair of equations:
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
~\frac{R_\mathrm{eq}}{R_\mathrm{norm}} = r_a \cdot \biggl( \frac{R_\mathrm{Horedt}}{R_\mathrm{norm}} \biggr)
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\frac{P_\mathrm{e}}{P_\mathrm{norm}} = p_a \cdot \biggl( \frac{P_\mathrm{Horedt}}{P_\mathrm{norm}} \biggr)
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>
</table>
</div>
where we have introduced the normalizations,

<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>~R_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl[ \biggl( \frac{G}{K} \biggr)^n M_\mathrm{tot}^{n-1} \biggr]^{1/(n-3)} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~P_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{K^{4n}}{G^{3(n+1)} M_\mathrm{tot}^{2(n+1)}} \biggr]^{1/(n-3)}  \, .</math>
  </td>
</tr>

</table>
</div>

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 &#8212; or, pressure-truncated &#8212; 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 {{ Horedt70 }} 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 &#8212; or even ''whether'' &#8212; 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,

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

<tr>
  <td align="right">
<math>~\biggl[ \biggl(\frac{P_e}{P_\mathrm{norm}}\biggr) \biggl(\frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^4\biggr]_\mathrm{Horedt} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>
</table>
</div>
or, given that <math>~P_\mathrm{norm}R_\mathrm{norm}^4 = GM_\mathrm{tot}^2</math>, this can be rewritten as,

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

<tr>
  <td align="right">
<math>~\biggl[ \frac{P_e R_\mathrm{eq}^4}{G M_\mathrm{limit}^2} \biggr]_\mathrm{Horedt} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{\tilde\theta^{n+1} }{(4\pi)(n+1)( -\tilde\theta' )^{2}}   \, .
</math>
  </td>
</tr>
</table>
</div>

==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,
<div align="center" id="FreeEnergyExpression">
<font color="#770000">'''Algebraic Free-Energy Function'''</font><br />

<math>
\mathfrak{G}^* = 
-3\mathcal{A} \chi^{-1} +~ \frac{1}{(\gamma - 1)} \mathcal{B} \chi^{3-3\gamma} +~ \mathcal{D}\chi^3 \, .
</math>
</div>
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,
<div align="center">
<math>~\mathcal{D} \equiv \frac{4\pi}{3} \biggl( \frac{P_e}{P_\mathrm{norm}} \biggr) \, .</math>
</div>
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>.

[[File:AdabaticBoundedSpheres_Virial.jpg|thumb|300px|Equilibrium Adiabatic Pressure-Radius Diagram]]
Equilibrium radii may also be identified through an algebraic relation that originates from the scalar virial theorem &#8212; a theorem that, itself, is derivable from the free-energy expression by setting <math>~\partial\mathfrak{G}^*/\partial\chi = 0</math>.  In our 
[[SSC/Virial/Polytropes#Virial_Equilibrium_of_Adiabatic_Spheres|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,
<div align="center" id="ConciseVirial">
<font color="#770000">'''Algebraic Expression of the Virial Theorem'''</font><br />

<math>
\Pi_\mathrm{ad} = \frac{(\Chi_\mathrm{ad}^{4-3\gamma} - 1)}{\Chi_\mathrm{ad}^4} \, ,
</math>
</div>
where, after setting <math>~\gamma = (n+1)/n</math>,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>~\Pi_\mathrm{ad}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\mathcal{D} \biggl[ \frac{\mathcal{A}^{3(n+1)}}{\mathcal{B}^{4n}} \biggr]^{1/(n-3)}
\, ,
</math> &nbsp; &nbsp; &nbsp; &nbsp; and,
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Chi_\mathrm{ad}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\chi_\mathrm{eq} \biggl[ \frac{\mathcal{B}}{\mathcal{A}} \biggr]^{n/(n-3)} \, .
</math>
  </td>
</tr>

</table>
</div>

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,
<div align="center" id="RenormalizedFreeEnergyExpression">
<font color="#770000">'''Renormalized Free-Energy Function'''</font><br />

<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>
</div>

==Relationship to Detailed Force-Balanced Models==
===Structural Form Factors===

In our [[SSC/Virial/Polytropes#Virial_Equilibrium_of_Adiabatic_Spheres|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 &#8212; and assist in understanding &#8212; 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 [[SSC/Virial/PolytropesSummary#FreeEnergyExpression|algebraic free-energy expression]] are expressible in terms of three [[SSCpt1/Virial#Structural_Form_Factors|structural form factors]], <math>~\tilde\mathfrak{f}_M</math>, <math>~\tilde\mathfrak{f}_W</math>, and <math>~\tilde\mathfrak{f}_A</math>, as follows:
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>~\mathcal{A}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{B}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>
</table>
</div>
and that, specifically in the context of spherically symmetric, pressure-truncated polytropes, we can write &hellip;

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\tilde\mathfrak{f}_M</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl[ - \frac{3\tilde\theta^'}{\tilde\xi} \biggr] \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\tilde\mathfrak{f}_W </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\tilde\theta^'}{\tilde\xi} \biggr]^2 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\tilde\mathfrak{f}_A  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{3(n+1) }{(5-n)} ~\biggl[ \tilde\theta^' \biggr]^2  + \tilde\theta^{n+1} \, .
</math>
  </td>
</tr>
</table>
</div>


<font color="red"><b>
January 13, 2015:
</b></font> 
As is noted in our [[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.

{{ SGFworkInProgress }}

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, we find that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Pi_\mathrm{ad}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\eta_\mathrm{ad} (1 + \eta_\mathrm{ad})^{-4n/(n-3)} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Chi_\mathrm{ad}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(1 + \eta_\mathrm{ad})^{n/(n-3)} \, ,</math>
  </td>
</tr>
</table>
</div>
where the newly identified, key physical parameter, 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\eta_\mathrm{ad} </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{(5-n) \tilde\theta^{n+1}}{3(n+1) (\tilde\theta^')^2} \, .</math>
  </td>
</tr>
</table>
</div>

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 [[SSC/Virial/PolytropesSummary#ConciseVirial|virial theorem presented above]].

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

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,
<div align="center">
<math>\theta \equiv \biggl(\frac{\rho}{\rho_c} \biggr)^{1/n} \, .</math>
</div>
Remembering that, at any location within the configuration, the pressure is related to the density via the polytropic equation of state,
<div align="center">
<math>P = K\rho^{(n+1)/n} \, ,</math>
</div>
we see that,
<div align="center">
<math>\frac{P}{P_c} = \theta^{n+1} \, .</math>
</div>
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">
<math>\tilde\theta^{n+1} = \frac{P_e}{P_c}  \, .</math>
</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 pressure.  Specifically, we show that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{4\pi}{3} \biggr) \frac{P_c R_\mathrm{eq}^4}{G M_\mathrm{limit}^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~[3 (n+1) (\tilde\theta^')^2]^{-1} \, .</math>
  </td>
</tr>
</table>
</div>
We conclude, therefore, that quite generally,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>(5-n) \tilde\mathfrak{f}_A  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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] \, ,
</math>
  </td>
</tr>
</table>
</div>
and that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\eta_\mathrm{ad} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <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>
  </td>
</tr>
</table>
</div>

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

<tr>
  <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>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\eta_\mathrm{ad} \, .</math>
  </td>
</tr>
</table>
</div>
Hence, the pair of parametric equations obtained via a 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">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Chi_\mathrm{ad} = \frac{r_a}{r_n}</math>
  </td>
  <td align="center">
&nbsp;&nbsp;&nbsp;&nbsp; and &nbsp;&nbsp;&nbsp;&nbsp;
  </td>
  <td align="left">
<math>~\Pi_\mathrm{ad} = \frac{p_a}{p_n} \, .</math>
  </td>
</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>
  <td align="right">
<math>~r_n^{n-3}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{(n+1)^n}{4\pi} \biggl[ \mathcal{A} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2} \biggr]^n
\biggl[ \mathcal{B} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-(n+1)/n} \biggr]^{-n} \, ,
</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~p_n^{n-3}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<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>
</tr>
</table>
</div>
while, in terms of the structural form factors,

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

<tr>
  <td align="right">
<math>~r_n^{n-3}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<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>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~p_n^{n-3}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<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>
</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>
  <td align="right">
<math>\Pi_\mathrm{ad} \Chi_\mathrm{ad}^4</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
</tr>

<tr>
  <td align="right">
&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>
  </td>
</tr>
</table>
</div>
While the righthand side becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
~\Chi_\mathrm{ad}^{(n-3)/n} - 1
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{\mathcal{B}}{\mathcal{A}} \biggr]
\biggl(  \frac{R_\mathrm{eq}}{R_\mathrm{norm} } \biggr)_\mathrm{Horedt}^{(n-3)/n} -1 \, .
</math>
  </td>
</tr>

</table>
</div>
Together, then, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<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 align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathcal{B} \biggl(  \frac{R_\mathrm{eq}}{R_\mathrm{norm} } \biggr)_\mathrm{Horedt}^{(n-3)/n} -\mathcal{A}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
\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 align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
</tr>
</table>
</div>
Or, simplifying,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\frac{1}{3(n+1)} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{2} p_a r_a^4
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r_a^{(n-3)/n}
\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>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ \frac{1}{3} p_a r_a^4
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r_a^{(n-3)/n}
\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>
</tr>
</table>
</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,

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

<tr>
  <td align="right">
<math>~p_a r_a^4
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <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}}
-\biggl[ \frac{3(n+1)}{5} \biggr] \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} \, ,
</math>
  </td>
</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>
  <td align="right">
<math>~\eta_\mathrm{ad} </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <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>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Pi_\mathrm{ad}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\eta_\mathrm{ad} (1 + \eta_\mathrm{ad})^{-4n/(n-3)} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Chi_\mathrm{ad}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(1 + \eta_\mathrm{ad})^{n/(n-3)} \, ,</math>
  </td>
</tr>
</table>
</div>

====When <math>~\eta_\mathrm{ad}</math> is zero====
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,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{R_\mathrm{eq}}{R_\mathrm{norm}} = \chi_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{n/(n-3)} \, .</math>
  </td>
</tr>
</table>
</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,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{P_c R_\mathrm{eq}^4}{G M_\mathrm{limit}^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{[4\pi (n+1) (\theta^')^2]_{\xi_1}} \, .</math>
  </td>
</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>
  <td align="right">
<math>~\frac{P_c R_\mathrm{eq}^4}{G M_\mathrm{limit}^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{[4\pi (n+1) (\tilde\theta^')^2]} \, ,</math>
  </td>
</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>
  <td align="right">
<math>~\frac{\partial\mathfrak{G}^{**}}{\partial\Chi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~3 \Chi^{-2} -3\Chi^{-(n+3)/n} + 3\Pi_\mathrm{ad} \Chi^2 \, ,</math>
  </td>
</tr>

<tr>
  <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>
</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 />

<math>
\Pi_\mathrm{ad} = \frac{\Chi_\mathrm{ad}^{(n-3)/n} - 1}{\Chi_\mathrm{ad}^4} \, .
</math>
</div>
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">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial^2\mathfrak{G}^{**}}{\partial\Chi^2}\biggr|_{\Chi_\mathrm{ad}}</math>
  </td>
  <td align="center">
<math>~></math>
  </td>
  <td align="left">
<math>~0 \, ,</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (stable)
  </td>
</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>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~< </math>
  </td>
  <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>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~< </math>
  </td>
  <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>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~< </math>
  </td>
  <td align="left">
<math>~3 \Chi_\mathrm{ad}^{-3} \biggl[ \frac{3(n+1)}{n} \Chi_\mathrm{ad}^{(n-3)/n}  - 4\biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~~\Chi_\mathrm{ad}</math>
  </td>
  <td align="center">
<math>~> </math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{4n}{3(n+1)} \biggr]^{n/(n-3)} \, .</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (stable)
  </td>
</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>
  <td align="right">
<math>~\Pi_\mathrm{max}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\Chi_\mathrm{min}^{-4}  \biggl[ \Chi_\mathrm{min}^{(n-3)/n} - 1 \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{3(n+1)}{4n} \biggr]^{4n/(n-3)} \biggl[\frac{4n}{3(n+1)}  - 1 \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <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>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~\Pi_\mathrm{max}^{n-3}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(4n)^{-4n}~[3(n+1)]^{3(n+1)} ~(n-3)^{n-3} \, .</math>
  </td>
</tr>
</table>
</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,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{D} > \mathcal{D}_\mathrm{max}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <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>
  </td>
</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>
  <td align="right">
<math>~\eta_\mathrm{crit}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ \frac{n-3}{3(n+1)}  </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Pi_\mathrm{max}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~(n-3) \biggl\{~\frac{ [3(n+1)]^{3(n+1)} }{(4n)^{4n}} \biggr\}^{1/(n-3)} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Chi_\mathrm{min} </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{4n}{3(n+1)} \biggr]^{n/(n-3)}  
</math>
  </td>
</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>
  <td align="right">
<math>~\eta_\mathrm{crit}\biggr|_{n=4}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{15} = 0.066667 \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Pi_\mathrm{max}\biggr|_{n=4}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{15^{15}}{16^{16}} = 0.02373828 \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Chi_\mathrm{min}\biggr|_{n=4}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{16}{15} \biggr)^4 = 1.294538 \, .</math>
  </td>
</tr>
</table>
</div>

At the top of the table, shown below, we display a plot of 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>
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,
<div align="center" id="ConciseVirial2">
<font color="#770000">'''Algebraic Expression of the Virial Theorem'''</font><br />

<math>
\Pi_\mathrm{ad} = \frac{\Chi_\mathrm{ad}^{(n-3)/n} - 1}{\Chi_\mathrm{ad}^4} \, .
</math>
</div>
For the unstable equilibrium configuration, we calculate,
<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>
  <th align="center" colspan="4">
[[File:TryN4Pi0.01.png|450px|Dimensionless Free-Energy Curve]]
  </th>
</tr>
<tr>
  <th align="center" colspan="4">
Determined from Plot of Renormalized Free-Energy 

with <math>~(n, \Pi_\mathrm{ad}) = (4, 0.01)</math>
  </th>
</tr>
<tr>
  <th align="center">&nbsp;</th>
  <th align="center">&nbsp;</th>
  <th align="center" width="25%">Maximum</th>
  <th align="center" width="25%">Minimum</th>
</tr>
<tr>
  <th align="center">&nbsp;</th>
  <td align="center"> <math>~\Chi</math> </td>
  <td align="center"> <math>~1.0494</math> </td>
  <td align="center"> <math>~2.13905</math> </td>
</tr>
<tr>
  <th align="center" colspan="4">
Immediate Implications from Virial Theorem 
  </th>
</tr>
<tr>
  <th align="center"><math>~\Chi^{1/4} - 1</math></th>
  <td align="center"> <math>~\eta_\mathrm{ad}</math> </td>
  <td align="center"> <math>~0.012128</math></td>
  <td align="center"> <math>~0.20936</math> </td>
</tr>
<tr>
  <th align="center"><math>~(\Chi^{1/4} - 1)\cdot \Chi^{-4}</math></th>
  <td align="center"> <math>~\Pi_\mathrm{ad}</math> </td>
  <td align="center"> <math>~1.000024 \times 10^{-2}</math></td>
  <td align="center"> <math>~1.000018 \times 10^{-2}</math> </td>
</tr>
<tr>
  <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>
  <th align="center">&nbsp;</th>
  <td align="center"> <math>~\tilde\xi</math> (approx.) </td>
  <td align="center"> <math>~4.81</math></td>
  <td align="center"><math>~1.624</math></td>
</tr>
<tr>
  <th align="center">&nbsp;</th>
  <td align="center"> <math>~\tilde\theta</math> (approx.) </td>
  <td align="center"> <math>~0.251</math></td>
  <td align="center"><math>~0.709</math></td>
</tr>
<tr>
  <th align="center">&nbsp;</th>
  <td align="center"> <math>~- \tilde\theta^'</math> (approx.) </td>
  <td align="center"><math>~0.0727</math></td>
  <td align="center"> <math>~0.239</math></td>
</tr>
<tr>
  <th align="center"> <math>~\frac{1}{15}\cdot \frac{\tilde\theta^5}{(\tilde\theta^')^2}</math></th>
  <td align="center"> <math>~\eta</math> (check) </td>
  <td align="center"><math>~0.0126</math></td>
  <td align="center"> <math>~0.2091</math></td>
</tr>
<tr>
  <th align="center" colspan="4">
and, hence, Implied Structural Form Factors &amp; Coefficients <math>~\mathcal{B}</math> &amp; <math>~\mathcal{A}</math>
  </th>
</tr>
<tr>
  <th align="center"> <math>~3(-\tilde\theta^')/\tilde\xi</math></th>
  <td align="center"> <math>~\mathfrak{f}_M</math></td>
  <td align="center"> <math>~0.0453</math></td>
  <td align="center"><math>~0.4415</math></td>
</tr>
<tr>
  <th align="center"> <math>~5[3(-\tilde\theta^')/\tilde\xi]^2</math></th>
  <td align="center"> <math>~\mathfrak{f}_W</math></td>
  <td align="center"> <math>~0.01028</math></td>
  <td align="center"><math>~0.975</math></td>
</tr>
<tr>
  <th align="center"> <math>~15(-\tilde\theta^')^2 + \tilde\theta^5</math></th>
  <td align="center"> <math>~\mathfrak{f}_A</math></td>
  <td align="center"> <math>~0.08028</math></td>
  <td align="center"><math>~1.036</math></td>
</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>
</th>
  <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>
  <th align="center"><math>
\biggl( \frac{4\pi}{5^4} \biggr) \tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{-3}  
</math>
</th>
  <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>
</table>

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?

====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">
<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} \, ,
</math>
</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>.
 

<table border="1" align="center" cellpadding="5">
<tr>
  <th align="center" colspan="4">
[[File:TryN4Pi0.0237.png|450px|Dimensionless Free-Energy Curve]]
  </th>
</tr>
<tr>
  <th align="center" colspan="4">
Determined from Plot of Renormalized Free-Energy 

with <math>~(n, \Pi_\mathrm{ad}) = (4, 0.02369)</math>
  </th>
</tr>
<tr>
  <th align="center">&nbsp;</th>
  <th align="center">&nbsp;</th>
  <th align="center" width="25%">Maximum</th>
  <th align="center" width="25%">Minimum</th>
</tr>
<tr>
  <th align="center">&nbsp;</th>
  <td align="center"> <math>~\Chi</math> </td>
  <td align="center"> <math>~1.274</math> </td>
  <td align="center"> <math>~1.317</math> </td>
</tr>
<tr>
  <th align="center" colspan="4">
Immediate Implications from Virial Theorem 
  </th>
</tr>
<tr>
  <th align="center"><math>~\Chi^{1/4} - 1</math></th>
  <td align="center"> <math>~\eta_\mathrm{ad}</math> </td>
  <td align="center"> <math>~0.0624</math></td>
  <td align="center"> <math>~0.0713</math> </td>
</tr>
<tr>
  <th align="center"><math>~(\Chi^{1/4} - 1)\cdot \Chi^{-4}</math></th>
  <td align="center"> <math>~\Pi_\mathrm{ad}</math> </td>
  <td align="center"> <math>~0.02369</math></td>
  <td align="center"> <math>~0.02369 </math> </td>
</tr>
<tr>
  <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>
  <th align="center">&nbsp;</th>
  <td align="center"> <math>~\tilde\xi</math> (approx.) </td>
  <td align="center"> ---- </td>
  <td align="center"><math>~2.6</math></td>
</tr>
<tr>
  <th align="center">&nbsp;</th>
  <td align="center"> <math>~\tilde\theta</math> (approx.) </td>
  <td align="center">---- </td>
  <td align="center"><math>~0.5048</math></td>
</tr>
<tr>
  <th align="center">&nbsp;</th>
  <td align="center"> <math>~- \tilde\theta^'</math> (approx.) </td>
  <td align="center"> ---- </td>
  <td align="center"> <math>~0.175</math></td>
</tr>
<tr>
  <th align="center"> <math>~\frac{1}{15}\cdot \frac{\tilde\theta^5}{(\tilde\theta^')^2}</math></th>
  <td align="center"> <math>~\eta</math> (check) </td>
  <td align="center"> ---- </td>
  <td align="center"> <math>~0.0714</math></td>
</tr>
<tr>
  <th align="center" colspan="4">
and, hence, Implied Structural Form Factors &amp; Coefficients <math>~\mathcal{B}</math> &amp; <math>~\mathcal{A}</math>
  </th>
</tr>
<tr>
  <th align="center"> <math>~3(-\tilde\theta^')/\tilde\xi</math></th>
  <td align="center"> <math>~\mathfrak{f}_M</math></td>
  <td align="center"> ---- </td>
  <td align="center"><math>~0.2019</math></td>
</tr>
<tr>
  <th align="center"> <math>~5[3(-\tilde\theta^')/\tilde\xi]^2</math></th>
  <td align="center"> <math>~\mathfrak{f}_W</math></td>
  <td align="center"> ---- </td>
  <td align="center"><math>~0.2039</math></td>
</tr>
<tr>
  <th align="center"> <math>~15(-\tilde\theta^')^2 + \tilde\theta^5</math></th>
  <td align="center"> <math>~\mathfrak{f}_A</math></td>
  <td align="center"> ---- </td>
  <td align="center"><math>~0.4922</math></td>
</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"> ---- </td>
  <td align="center"><math>~2.542</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"> ---- </td>
  <td align="center"><math>~1.71 \times 10^4</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"> ---- </td>
  <td align="center"><math>~0.0316</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>
</th>
  <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>
  <th align="center"><math>
\biggl( \frac{4\pi}{5^4} \biggr) \tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{-3}  
</math>
</th>
  <td align="center"> <math>~\chi_\mathrm{eq}</math></td>
  <td align="center"> ---- </td>
  <td align="center"><math>~0.0316</math></td>
</tr>
</table>

====Summary====

The algebraic free-energy function associated with pressure-truncated <math>~n=4</math> polytropes is,
<div align="center">
<math>
\mathfrak{G}^*\biggr|_{n=4} = 
-3\mathcal{A} \chi^{-1} +~ 4\mathcal{B} \chi^{-3/4} +~ \mathcal{D}\chi^3 \, ,
</math>
</div>
and the corresponding ''renormalized'' free-energy function is,
<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>
  <td align="right">
<math>
~p_a 
</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>
\tilde\theta^{n+1}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)}  \, .
</math>
  </td>
</tr>
</table>
</div>
  
  </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">
'''<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).

'''<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).
  </td>
</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>
  <td align="right">
<math>~\Pi_\mathrm{max}\biggr|_{n=4}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{15^{15}}{16^{16}} = 0.02373828 \, ,</math>
  </td>
</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>
  <td align="right">
<math>
~\frac{M_\mathrm{tot}}{M_\mathrm{SWS} }
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{n^3}{4\pi} \biggr)^{1/2} \tilde\theta^{(n-3)/2} (- \tilde\xi^2 \tilde\theta^') \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} }
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{n}{4\pi} \biggr)^{1/2} \tilde\xi \tilde\theta^{(n-1)/2} \, ,
</math>
  </td>
</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===
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">
<math>~P_e </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
</tr>

<tr>
  <td align="right">
<math>~R_\mathrm{eq} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~R_\mathrm{norm} \Chi_\mathrm{ad} \biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{n/(n-3)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
</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>
  <td align="right">
<math>~ M_\mathrm{tot}^{2(n+1)}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
</tr>

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

<tr>
  <td align="right">
<math>~R_\mathrm{eq}^{n-3} </math>
  </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 M_\mathrm{tot}^{n-1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </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>
  <td align="right">
&nbsp;
  </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{\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>
  <td align="right">
&nbsp;
  </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>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
</tr>

<tr>
  <td align="right">
&nbsp;
  </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>
  <td align="right">
<math>~\Rightarrow~~~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} } </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
</tr>

</table>
</div>
Flipping both of these expressions around, we see that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Pi_\mathrm{ad} </math>
  </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>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
&nbsp;
  </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>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<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>
</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>
  <td align="right">
<math>~1</math>
  </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>
  <td align="right">
&nbsp;
  </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>
  <td align="right">
&nbsp;
  </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>
</table>
</div>

Or, rearranged,

<div align="center" id="CompactStahlerVirial">
<table border="1" cellpadding="10" align="center">

<tr>
  <td align="right">
<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> 
  </td>
</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>
  <td align="right">
<math>~\frac{M_\mathrm{SWS}}{M_\mathrm{mod}}</math>
  </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>
  <td align="right">
<math>~\frac{R_\mathrm{SWS}}{R_\mathrm{mod}}</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}^{1/2}}{\mathcal{B}^{n/(n+1)}} \, ,</math>
  </td>
</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.

<!-- COMMENT OUT ORIGINAL FIGURE until reprint permission received

<table border="1" align="center" cellpadding="3">
<tr>
  <td align="center" rowspan="3">
[[File:MassRadiusVirialLabeled.png|350px|Virial Theorem Mass-Radius Relation]]
  </td>
  <td align="center">
[[File:Stahler1983TitlePage0.png|300px|center|Stahler (1983) Title Page]]
  </td>
</tr>
<tr>
  <td align="center" bgcolor="white">
[[File:Stahler_MRdiagram1.png|300px|center|Stahler (1983) Figure 17 (edited)]]
  </td>
</tr>
</table>

-->

<table border="1" align="center" cellpadding="3" width="80%">
<tr>
  <td align="center" rowspan="3">
[[File:MassRadiusVirialLabeled.png|350px|Virial Theorem Mass-Radius Relation]]
  </td>
<td align="center">
Figure 17 extracted<sup>&dagger;</sup>from p. 184 of [http://adsabs.harvard.edu/abs/1983ApJ...268..165S S. W. Stahler (1983)]<p></p>
"''The Equilibria of Rotating Isothermal Clouds. &nbsp; &nbsp; II. Structure and Dynamical Stability''"<p></p>
ApJ, vol. 268, pp. 165-184 &copy; [http://aas.org/ American Astronomical Society]
</td>
</tr>
<tr>
  <td align="center">
<!-- [[File:Stahler_MRdiagram1.png|300px|center|Stahler (1983) Figure 17 (edited)]] -->
[[Image:AAAwaiting01.png|400px|center|Stahler (1983) Figure 17 (edited)]]
  </td>
</tr>
<tr><td align="left"><sup>&dagger;</sup>Figure displayed here, as a digital image, has been modified from the original publication only via the addition of the word "<font color="red">SCHEMATIC</font>".</td></tr>
</table>


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">

<tr>
  <td align="right">
<math>~\mathcal{X}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{Y}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \, .</math>
  </td>
</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>
  <td align="right">
<math>~\frac{R_\mathrm{eq}}{R_\mathrm{mod}} ~~~ \rightarrow ~~~ \mathcal{X} \cdot \frac{R_\mathrm{SWS}}{R_\mathrm{mod}}</math>
  </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>
  <td align="right">
<math>~\frac{M_\mathrm{tot}}{M_\mathrm{mod}} ~~~ \rightarrow ~~~ \mathcal{Y} \cdot \frac{M_\mathrm{SWS}}{M_\mathrm{mod}}</math>
  </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>
</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>
  <td align="right">
<math>~\mathcal{A}</math>
  </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>
  <td align="right">
<math>~\mathcal{B}</math>
  </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>
</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>
  <td align="right">
<math>~\frac{R_\mathrm{eq}}{R_\mathrm{mod}} </math>
  </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>
  <td align="right">
&nbsp;
  </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>
  <td align="right">
<math>~\frac{M_\mathrm{tot}}{M_\mathrm{mod}} </math>
  </td>
  <td align="center">
<math>~~~ \rightarrow ~~~</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} 
(5-n)^{(n-3)/[2(n+1)]} (4\pi)^{2/(n+1)} \mathfrak{b}_n^{-2n/(n+1)}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </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>
</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>
  <td align="right">
<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>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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} 
\cdot \mathfrak{b}_n^{(1-3n)/(n+1)}  \, .
</math>
  </td>
</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>
  <td align="right">
<math>~k_\xi </math>
  </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>
</table>
</div>

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">

<tr>
  <td align="right">
<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)
+ \frac{2^2 \cdot 5 \pi}{3} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^4 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, ,</math>
  </td>
</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>
  <td align="right">
<math>~M_\mathrm{mod}\biggr|_{n=5}</math>
  </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>
  <td align="right">
<math>~R_\mathrm{mod}\biggr|_{n=5}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
</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>
  <td align="right">
<math>~\mathcal{X}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{Y}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{M_\mathrm{tot}}{M_\mathrm{SWS}} \, ,</math>
  </td>
</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>
  <td align="right">
<math>~\mathcal{Y}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
</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>
  <td align="right">
<math>
~\mathcal{X}\biggr|_{n=5}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\mathcal{Y}\biggr|_{n=5}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </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>
</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>
  <td align="right">
<math>~
\frac{6}{5} \mathcal{Y}^2 - \biggl( \frac{\mathfrak{b}_I^5 }{4\pi} \cdot \mathcal{X}^{2} \mathcal{Y}^{6}\biggr)^{1/5}  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </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>
</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>
  <td align="right">
<math>~
\mathfrak{b}_I^5  
</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>
  <td align="right">
<math>~
\mathfrak{b}_{II}^5
</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>
</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>
  <td align="right">
<math>~\frac{6}{5}  \mathcal{Y}^2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{\mathfrak{b}_I^5 }{4\pi} \cdot \mathcal{X}^{2} \mathcal{Y}^{6}\biggr)^{1/5}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ \mathcal{Y}^4 </math>
  </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>
  <td align="right">
<math>~\Rightarrow ~~~~ \mathcal{Y} </math>
  </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>
</table>
</div>
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,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>~\frac{4\pi}{3} ~ \mathcal{X}^4  </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>
  <td align="right">
<math>~\Rightarrow ~~~~ \mathcal{Y}^6 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>\mathcal{X}^{18} \biggl[ \biggl( \frac{4\pi}{3}\biggr)^5 \frac{4\pi}{\mathfrak{b}^5_{II}} \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ \mathcal{Y} </math>
  </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>
</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>
  <td align="right">
<math>~\mathcal{A}</math>
  </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>
  <td align="right">
<math>~\mathcal{B}</math>
  </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>
</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>
  <td align="right">
<math>~\frac{R_\mathrm{eq}}{R_\mathrm{mod}} = \mathcal{X} \cdot \frac{R_\mathrm{SWS}}{R_\mathrm{mod}}</math>
  </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>
  <td align="right">
&nbsp;
  </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>
  <td align="right">
<math>~\frac{M_\mathrm{tot}}{M_\mathrm{mod}} = \mathcal{Y} \cdot \frac{M_\mathrm{SWS}}{M_\mathrm{mod}}</math>
  </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>
  <td align="right">
&nbsp;
  </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>
</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>
  <td align="right">
<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>
  <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>
  <td align="right">
&nbsp;
  </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>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(5-n)^{1/3} \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>
  </td>
</tr>
</table>
</div>
and, in entirety, the virial theorem relation becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(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}
- \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}
+ (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\} \, .
</math>
  </td>
</tr>
</table>
</div>

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

<tr>
  <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>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <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>
  </td>
</tr>
</table>
</div>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Rightarrow~~~~
\mathcal{Y}^{2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mathcal{X} \biggl( \frac{5\mathfrak{b}_5}{2 \cdot 3}  \biggr)^{5/2} 
= \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>
</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>
  <td align="right">
<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 align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .
</math>
  </td>
</tr>
</table>
</div>
Next, let's rewrite the other two terms so that they look more like the terms found in Stahler's expression.

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

<tr>
  <td align="right">
<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 align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
\biggl(\frac{2^2\cdot 5\pi}{3} \biggr) \mathcal{X}^4  - 
5\mathcal{X}\mathcal{Y} 
+ \mathcal{Y}^{2}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
5\mathcal{X}\mathcal{Y} \biggl[\biggl( \frac{\mathcal{B}^5 \mathcal{Y}}{\mathcal{X}^3} \biggr)^{1/5}-1\biggr]
+(1- 6\mathcal{A} )\mathcal{Y}^{2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \mathcal{Y} \biggl\{
5\mathcal{X}\biggl[\biggl( \frac{\mathcal{B}^5 \mathcal{Y}}{\mathcal{X}^3} \biggr)^{1/5}-1\biggr]
+(1- 6\mathcal{A} )\mathcal{Y} \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \mathcal{Y} \biggl\{
5 \mathcal{B} (\mathcal{Y} \mathcal{X}^2 )^{1/5} - 5 \mathcal{X} 
+(1- 6\mathcal{A} )\mathcal{Y} \biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>
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">
<math>~\tilde\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{1}{(1+\Lambda)^{1/2}} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~-~\tilde\theta'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{3^{1/2}} \cdot
\frac{\Lambda^{1/2}}{(1+\Lambda)^{3/2}}  
~~~~\Rightarrow~~~~ \biggl( \frac{\tilde{\xi}}{-\tilde\theta'} \biggr) = 3(1+\Lambda)^{3/2} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{X}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{ \frac{3\cdot 5}{2^2\pi} \biggl[ 
\frac{\Lambda}{(1+\Lambda)^2} \biggr] \biggr\}^{1/2}
=
\biggl( \frac{3\cdot 5}{2^2\pi} \biggr)^{1/2} 
\frac{\Lambda^{1/2}}{(1+\Lambda)}  \, ,
</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\cdot 5^3}{2^2\pi} \biggl[ 
\frac{\Lambda^3}{(1+\Lambda)^4} \biggr] \biggr\}^{1/2}
=
5\biggl(\frac{3\cdot 5}{2^2\pi}\biggr)^{1/2} 
\frac{\Lambda^{3/2}}{(1+\Lambda)^2}  \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ \mathcal{Y}\mathcal{X}^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
5\biggl(\frac{3\cdot 5}{2^2\pi}\biggr)^{3/2} 
\frac{\Lambda^{5/2}}{(1+\Lambda)^4} \, ;
</math>
  </td>
</tr>
</table>
</div>
and, from the general definitions given, above,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>~\mathcal{A}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>\frac{1}{5-n} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{B}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
( 4\pi )^{-1/5} \biggl[ \frac{2\cdot 3}{(5-n)} (\tilde\theta^')^2 + \frac{1}{3}\cdot \tilde\theta^{6} \biggr]
\biggl( \frac{\tilde\xi}{\tilde\theta^'} \biggr)^{6/5} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{3}{4\pi }\biggr)^{1/5} \biggl[ \frac{2\cdot 3}{(5-n)} \Lambda + 1 \biggr]
(1+\Lambda)^{-6/5} \, .
</math>
  </td>
</tr>
</table>
</div>
Hence,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~5\mathcal{B} (\mathcal{Y}\mathcal{X}^2)^{1/5}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~5
\biggl(\frac{3}{4\pi }\biggr)^{1/5} \biggl[ \frac{2\cdot 3}{(5-n)} \Lambda + 1 \biggr](1+\Lambda)^{-6/5}
\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>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
</tr>
</table>
</div>
while,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(1-6\mathcal{A})\mathcal{Y} - 5\mathcal{X}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
</tr>
</table>
</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.
........  END COMMENT 2 -->

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

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,
<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{1}{\pi} \biggr)^{1/2} \tilde\xi \tilde\theta^{3/2} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\mathcal{Y}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{2^4}{\pi} \biggr)^{1/2} \tilde\theta^{1/2} (- \tilde\xi^2 \tilde\theta^') \, .
</math>
  </td>
</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,

<div align="center">
<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>
</div>
where,
<div align="center">
<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>
</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 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]].]

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>. 

<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center">
Comparing Two Separate Mass-Radius Relations for Pressure-Truncated ''n = 4'' Polytropes
  </th>
</tr>
<tr><td align="center">
[[File:CompareN4SequencesRevised.png|750px|Comparison of Two Mass-Radius Relations]]
</td></tr>
</table>

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">
<math>~\mathcal{X}_\mathrm{crit}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pi^{-1/2} 2^{-16/5} (3\mathfrak{b}_{n=4})^{4/5} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{Y}_\mathrm{crit}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pi^{-1/2} 2^{-22/5} (3\mathfrak{b}_{n=4})^{8/5} \, .</math>
  </td>
</tr>
</table>
</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''.

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>

==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>
  <td align="right">
<math>~\mathfrak{f}_M</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl[ - \frac{3\Theta^'}{\xi} \biggr]_{\xi_1} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathfrak{f}_W </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\Theta^'}{\xi} \biggr]^2_{\xi_1} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathfrak{f}_A  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{3(n+1) }{(5-n)} ~\biggl[ \Theta^' \biggr]^2_{\xi_1}  
</math>
  </td>
</tr>
</table>
  </td>
  <td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\tilde\mathfrak{f}_M</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl[ - \frac{3\Theta^'}{\xi} \biggr]_{\tilde\xi} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\tilde\mathfrak{f}_W </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\Theta^'}{\xi} \biggr]^2_{\tilde\xi} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\tilde\mathfrak{f}_A  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{3(n+1) }{(5-n)} ~\biggl[ \Theta^' \biggr]^2_{\tilde\xi}  + \tilde\Theta^{n+1}
</math>
  </td>
</tr>
</table>

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

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">

<tr>
  <td align="right">
<math>~\chi</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>\frac{R}{R_\mathrm{norm}}  \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~A</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <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>
  </td>
</tr>

<tr>
  <td align="right">
<math>~B</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<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}} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~D</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{4\pi}{3} \biggr) \frac{P_e}{P_\mathrm{norm}} \, .
</math>
  </td>
</tr>
</table>
</div>

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

===Virial Theorem===
The traditional expression for the virial theorem in this context is,
<div align="center">
<math>
~2S_\mathrm{therm}^* + W_\mathrm{grav}^* - \frac{3P_e V}{E_\mathrm{norm}} = 0 \, .
</math>
</div>
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,
<div align="center">
<math>
~S_\mathrm{therm} = \frac{3}{2}(\gamma_g-1) \mathfrak{S}_\mathrm{therm} = \frac{3}{10} \mathfrak{S}_\mathrm{therm}\, .
</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,
<div align="center">
<math>~W_\mathrm{grav}^* \equiv \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \rightarrow -3A\chi^{-1} \, ;</math>&nbsp; &nbsp; &nbsp; &nbsp;
<math>~\mathfrak{S}_\mathrm{therm}^* \equiv \frac{\mathfrak{S}_\mathrm{therm}}{E_\mathrm{norm}} \rightarrow 5B\chi^{-3/5} \, ;</math>&nbsp; &nbsp; &nbsp; &nbsp;
<math>~\frac{P_e V}{E_\mathrm{norm}} \rightarrow D\chi^{3}  \, .</math>&nbsp; &nbsp; &nbsp; &nbsp;
</div>
Hence, the normalized virial theorem may be written as,
<div align="center">
<math>
3B \chi^{-3/5} -3A\chi^{-1}  -~ 3D\chi^3 =0\, .
</math>
</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>
  <td align="right">
<math>~\frac{d\mathfrak{G}^*}{d\chi} = - 3B \chi^{-8/5} + 3A\chi^{-2}  +~ 3D\chi^2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0  </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ - \chi^{-1} \biggl[ 3B \chi^{-3/5} - 3A\chi^{-1} -~ 3D\chi^3 \biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0  \, .</math>
  </td>
</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===
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,
<div align="center">
<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,
<div align="center">
<table border="0" cellpadding="5">

<tr>
  <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 align="left">
<math>~0  </math>
  </td>
</tr>

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

<tr>
  <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>
</tr>
</table>
</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">
<math>
~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} }\biggr|_{n=5}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\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} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{5\cdot 3}{2^2\pi} \biggr)^{1/2} \ell (1 + \ell^2)^{-1} \, .
</math>
  </td>
</tr>
</table>
</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 align="center">
<math>~=</math>
  </td>
  <td align="left">
<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}
\biggr]^3 </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{2\cdot 3^6}{\pi} \biggr)^{1/2} \ell^3 (1 + \ell^2)^{-3} \, .
</math>
  </td>
</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>
  <td align="right">
<math>~(2S^* + W^*)_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{2\cdot 3^6}{\pi} \biggr)^{1/2} \biggl[ x_i ^3 (1 + x_i^2)^{-3} \biggr] \, .
</math>
  </td>
</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>
  <td align="right">
<math>
~W_\mathrm{grav}^*
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
-3A\chi_\mathrm{eq}^{-1} = -3A \biggl( \frac{R_\mathrm{norm}}{R_\mathrm{eq}}\biggr)
= -3A \biggl( \frac{R_\mathrm{SWS}}{R_\mathrm{eq}}\biggr)\biggl( \frac{R_\mathrm{norm}}{R_\mathrm{SWS}}\biggr)</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\Rightarrow ~~~ A
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
-\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>
  </td>
</tr>
</table>
</div>
Now, from immediately above, we know that,
<div align="center">
<table border="0" cellpadding="4">

<tr>
  <td align="right">
<math>
~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} }\biggr|_{n=5}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{5\cdot 3}{2^2\pi} \biggr)^{1/2} \ell (1 + \ell^2)^{-1}  \, ;
</math>
  </td>
</tr>
</table>
</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">

<tr>
  <td align="right">
<math>~W^*_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ - \biggl( \frac{3^8}{2^5\pi } \biggr)^{1/2} 
\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>
</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>
  <td align="right">
<math>~A</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ \frac{1}{3}
\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}
\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>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ 
\biggl( \frac{3^7 \cdot 5}{2^7\pi^2 } \biggr)^{1/2} \ell (1 + \ell^2)^{-1}
\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>
</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>
  <td align="right">
<math>~R_\mathrm{norm}\biggr|_{\gamma = 6/5}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \biggl( \frac{G}{K}\biggr) M_\mathrm{tot}^{2-6/5} \biggr]^{1/(4-18/5)}
= \biggl[ \biggl( \frac{G}{K}\biggr) M_\mathrm{tot}^{4/5} \biggr]^{5/2}
= \biggl( \frac{G}{K}\biggr)^{5/2} M_\mathrm{tot}^{2} \, .
</math>
  </td>
</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>
  <td align="right">
<math>~M_\mathrm{limit}^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
M_\mathrm{SWS}^2 \biggl[ \biggl( \frac{3\cdot 5^3}{2^2\pi} \biggr) \ell^6 (1+\ell^2)^{-4} \biggr] \, .
</math>
  </td>
</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>
  <td align="right">
<math>~M_\mathrm{SWS}^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2^3 \cdot 3^3}{5^3} \biggr) G^{-3} K^{10/3} P_e^{-1/3} \, ,
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
<math>~M_\mathrm{SWS}^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2^6 \cdot 3^6}{5^6} \biggr)^{1/2} G^{-3} K^{10/3} 
\biggl( \frac{5}{2 \cdot 3} \biggr)^{1/2} G^{1/2} K^{-5/6} R_\mathrm{SWS} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2^5 \cdot 3^5}{5^5} \biggr)^{1/2}\biggl( \frac{K}{G}\biggr)^{5/2} R_\mathrm{SWS} \, .
</math>
  </td>
</tr>
</table>
</div>
Hence, we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~M_\mathrm{limit}^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2^5 \cdot 3^5}{5^5} \biggr)^{1/2}\biggl( \frac{K}{G}\biggr)^{5/2}  
\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>
  <td align="right">
<math>~\Rightarrow ~~~~ \frac{1}{R_\mathrm{SWS}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2\cdot 3^7 \cdot 5}{\pi^2} \biggr)^{1/2}\biggl( \frac{K}{G}\biggr)^{5/2}  
\biggl[ \ell^6 (1+\ell^2)^{-4} \biggr] \frac{1}{M_\mathrm{limit}^2} \, .
</math>
  </td>
</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>
  <td align="right">
<math>~\frac{R_\mathrm{norm}}{R_\mathrm{SWS}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2\cdot 3^7 \cdot 5}{\pi^2} \biggr)^{1/2} 
\biggl[ \ell^6 (1+\ell^2)^{-4} \biggr] \biggl(\frac{M_\mathrm{tot}}{M_\mathrm{limit}}\biggr)^{2} \, ,
</math>
  </td>
</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>
  <td align="right">
<math>~A</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ 
\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}
\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>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ 
2^{-4} \ell^{-5} (1+\ell^2)^{3}  
\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>
</table>
</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">

<tr>
  <td align="right">
<math>~\frac{M_\mathrm{limit}}{M_\mathrm{tot} } </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\ell^3 (1+\ell^2)^{-3/2} \, ,
</math>
  </td>
</tr>
</table>
</div>
which gives,
<div align="center">
<table border="0" cellpadding="4">

<tr>
  <td align="right">
<math>~A</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ 
\frac{\ell}{2^{4} }  
\biggl[ \ell \biggl(\ell^4 - \frac{8}{3}\ell^2 - 1\biggr) (1 + \ell^2)^{-3}  + \tan^{-1}(\ell) \biggr] \, .
</math>
  </td>
</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>
  <td align="right">
<math>~\mathcal{A}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<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} 
= \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>
</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>
  <td align="right">
<math>~W_\mathrm{grav}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-3A\chi^{-1} E_\mathrm{norm} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-3A \biggl(\frac{GM_\mathrm{tot}^2}{R} \biggr) \, .</math>
  </td>
</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>
  <td align="right">
<math>~W_\mathrm{grav}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - \int_0^{R_\mathrm{limit}} \biggl( \frac{GM_r}{r} \biggr) 4\pi r^2 \rho dr \, ,</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~M_r</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\int_0^r 4\pi r^2 \rho dr \, .</math>
  </td>
</tr>
</table>
</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,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~M_\mathrm{tot}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggr[ \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr]^{1/2}  \rho_c^{-1/5} \, ,</math>
  </td>
</tr>
</table>
</div>
and we can write, in terms of the Lane-Emden dimensionless radius, <math>~\xi</math>,
<div align="center">
<math>
\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>
  <td align="right">
<math>~W_\mathrm{grav}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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^* 
</math>
  </td>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
- E_\mathrm{norm} \int_0^{R_\mathrm{limit}/R_\mathrm{norm}} 3\biggl[\frac{M_r}{M_\mathrm{tot}} \biggr]  r^* \rho^* dr^* \, .
</math>
  </td>
</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>
  <td align="right">
<math>~M_\mathrm{tot}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(  \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr)^{1/2} \rho_c^{-1/5} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \rho_c^{-2/5} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(  \frac{\pi G^3}{2\cdot 3^4 K^3} \biggr)  M_\mathrm{tot}^2 \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~a_5</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(  \frac{3K}{2\pi G} \biggr)^{1/2} \rho_c^{-2/5} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\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>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </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>

<tr>
  <td align="right">
<math>~R_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(  \frac{G}{K} \biggr)^{5/2} M_\mathrm{tot}^{2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ \frac{a_5}{R_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(  \frac{\pi}{2^3\cdot 3^7} \biggr)^{1/2}  \, .
</math>
  </td>
</tr>
</table>
</div>
Hence,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~r^*</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(  \frac{\pi}{2^3\cdot 3^7} \biggr)^{1/2} \xi \, .
</math>
  </td>
</tr>
</table>
</div>

Also,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\rho_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3}{4\pi} \biggl[ \frac{K}{G} \biggr]^{15/2} M_\mathrm{tot}^{-5}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <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>
  </td>
</tr>

<tr>
  <td align="right">
&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>
  </td>
</tr>
</table>
</div>
Hence,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\rho^* \equiv \frac{\rho}{\rho_\mathrm{norm}}</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>

<tr>
  <td align="right">
&nbsp;
  </td>
  <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>
</table>
</div>
So the energy integral becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{W_\mathrm{grav}}{E_\mathrm{norm} }</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
</tr>
</table>
</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]].


=See Also=


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