Editing SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt1

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=Virial Equilibrium of Adiabatic Spheres (Summary)=

The summary presented here has been drawn from our 
[[SSC/Virial/PolytropesEmbedded/FirstEffortAgain|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>.

<table border="1" cellpadding="8" align="right"><tr>
<td align="center">Figure 1</td></tr><tr>
<td align="center">[[File:AdabaticBoundedSpheres_Virial.jpg|300px|Equilibrium Adiabatic Pressure-Radius Diagram]]</td></tr></table>
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/PolytropesEmbedded/FirstEffortAgain#Renormalization|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 our Figure 1 "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" id="PTtable">
<table border="1" align="center" cellpadding="5">
<tr><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>~\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>~
~\frac{\tilde\mathfrak{f}_A - \tilde\theta^{n+1} }{\tilde\mathfrak{f}_W}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{(n+1)}{3\cdot 5} \biggr] \tilde\xi^2 
</math>
  </td>
</tr>
</table>

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


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 the {{ Horedt70 }} detailed force-balanced expressions for <math>r_a</math> and <math>p_a</math> to specify, respectively, <math>~\chi_\mathrm{eq}</math> and <math>P_e/P_\mathrm{norm}</math> in these same equations &#8212; see [[SSC/Virial/PolytropesEmbedded/FirstEffortAgain#Summary_2|our accompanying discussion]] &#8212; 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} \equiv \frac{b_\mathrm{ad}}{a_\mathrm{ad}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3\cdot 5~ \tilde\theta^{n+1}}{(n+1) \tilde\xi^2 \mathfrak{f}_W} \, .</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 [[#ConciseVirial|virial theorem presented above]].

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

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


At any radial location within a polytropic configuration, the [[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>b_\mathrm{ad} \equiv \tilde\theta^{n+1} = \frac{P_e}{P_c}  \, .</math>
</div>

Also, [[SSCpt1/Virial#Structural_Form_Factors|directly from its integral definition]], we have that <math>\tilde\mathfrak{f}_A = \bar{P}/P_c</math>.  So we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a_\mathrm{ad}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tilde\mathfrak{f}_A - b_\mathrm{ad } = \frac{\bar{P}}{P_c} - \frac{P_e}{P_c}
= \frac{P_e}{P_c}\biggl( \frac{\bar{P}}{P_e} - 1 \biggr) \, .</math>
  </td>
</tr>
</table>
</div>
We conclude, therefore, that,
<div align="center">
<math>~\eta_\mathrm{ad} = \biggl( \frac{\bar{P}}{P_e} - 1 \biggr)^{-1} \, .</math>
</div>

===Desired Pressure-Radius Relation===

It is clear from the above discussion that the pair of parametric equations obtained via a [[#Detailed_Force-Balanced_Solution|solution of the detailed force-balanced equations]] satisfy our, slightly rearranged, 
<div align="center" id="ConciseVirial3">
<font color="#770000">'''Algebraic Expression of the Virial Theorem'''</font><br />

<math>
\Pi_\mathrm{ad} \Chi_\mathrm{ad}^4 = \Chi_\mathrm{ad}^{(n-3)/n} - 1 \, .
</math>
</div>
More to the point, it is now clear that this virial theorem expression provides the direct relationship between the configuration's dimensionless equilibrium radius as defined by {{ Horedt70 }}, <math>r_a</math>, and the dimensionless applied external pressure as defined by {{ Horedt70 }}, <math>p_a</math>, that was not apparent from the original pair of parametric relations.  The {{ Horedt70 }} 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( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{1-n} 
\biggl( \frac{\mathcal{A}}{\mathcal{B}} \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( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{2(n+1)} 
\biggl[ \frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}} \biggr] \, ;
</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>

==Implications Regarding Stability== 

===Model Sequences===
<!-- [[File:AdabaticBoundedSpheres_Virial.jpg|thumb|300px|Equilibrium Adiabatic Pressure-Radius Diagram]]-->
After choosing a value for the system's adiabatic index (or, equivalently, its polytropic index), <math>\gamma = (n+1)/n</math>, the functional form of the virial theorem expression, <math>\Pi_\mathrm{ad}(\chi_\mathrm{ad})</math>, is known and, hence, the equilibrium model sequence can be plotted.  Half-a-dozen such model sequences are shown in Figure 1, above.  Each curve can be viewed as mapping out a single-parameter sequence of equilibrium models; "evolution" along the curve can be accomplished by varying the key parameter, <math>\eta_\mathrm{ad}</math>, over the physically relevant range, <math>0 \le \eta_\mathrm{ad} < \infty</math>.  
<table border="1" cellpadding="10" width="90%" align="center">
<tr><td align="left">
<font color="maroon">'''ASIDE'''</font> [18 March 2015]:  Many months after I penned the above description of "evolution" along an equilibrium model sequence, I started analyzing in detail the paper by  {{ Kimura81bfull }}.  The following excerpt from &sect;3 of his paper shows that Kimura presented essentially the same description of "evolution along a sequence" several decades ago:
<!-- [[File:Kimura1981bExcerpt.png|450px|center|border|Excerpt from section 3 of Kimura (1981b)]] -->
<table border="0" cellpadding="3" align="center" width="80%">
<tr><td align="left">
<font color="darkgreen">
"It can be seen that if a certain quantity, say <math>Q_1</math>, is fixed, then a sequence of bounded polytropes is constructed by varying a truncation parameter <math>\zeta_1</math> in the range <math>\zeta_0 < \zeta_1 < \zeta_f</math>, where <math>\zeta_0</math> is the dimensionless radius of the inner boundary, and <math>\zeta_f</math> that of the free surface.  Such a sequence will be termed a '<math>Q_1</math>-sequence'."
</font>
</td></tr>
<tr><td align="right">
— Drawn from {{ Kimura81bfull }} 
</td></tr></table>

Kimura uses the subscript "1" to denote the equilibrium value of any physical quantity "<math>Q</math>"; in Figure 1 above, we are holding the equilibrium mass fixed while allowing the external pressure and the configuration volume to vary, so Kimura would say that the figure displays various "M_1 sequences."  And, as is explained more fully in [[SSC/Structure/PolytropesEmbedded#Kimura.27s_Presentation|an accompanying discussion]], his "truncation parameter" is essentially the same as our truncation radius &#8212; specifically, <math>\zeta_1 = (n+1)^{1/2}\tilde\xi</math>.  When projected onto our discussion, the physically relevant range of truncation parameter values is, <math>0 \le \tilde\xi \le \xi_1</math>, where <math>\xi_1</math> is the [[SSC/Structure/Polytropes#Lane-Emden_Equation|Lane-Emden radius of an ''isolated'' (unbounded) polytropic sphere]].
</td></tr>
</table>

To simplify our discussion, here, we redisplay the above figure and repeat a few key algebraic relations.
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\eta_\mathrm{ad} </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{3\cdot 5 ~\tilde\theta^{n+1}}{(n+1) \tilde\xi^2 \tilde\mathfrak{f}_W} 
= \frac{\tilde\theta^{n+1}}{\tilde\theta^{n+1} + 3(\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta}\, ,</math>
  </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>
Note that the last expression for <math>\eta_\mathrm{ad}</math> has been obtained after inserting the analytic expression for the structural form-factor, <math>\tilde\mathfrak{f}_W</math> that &#8212; as has been explained in an [[SSCpt1/Virial/FormFactors#Viala_and_Horedt_.281974.29_Expressions|accompanying discussion]] &#8212; we derived with the help of {{ VH74full }}.

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

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>