Editing SSC/Virial/PolytropesSummary (section)

==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 &eta;<sub>ad</sub> = 0====
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 [[SSC/Structure/Polytropes#Polytropic_Spheres|''isolated'' polytropic sphere]].  As can be deduced from our above discussion of the [[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 [[SSC/Virial/Polytropes#Strategy2|accompanying discussion]] and also [[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 [[SSC/Virial/Polytropes#Central_and_Mean_Pressure|reviewed elsewhere]] &#8212; see also our [[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 [[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>.