Editing SSC/Virial/PolytropesEmbeddedOutline (section)

==Overview==
The free-energy function that is relevant to a discussion of the structure and stability of a pressure-truncated configuration having polytropic index, <math>n</math>,  has the form,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>\mathcal{G}(x)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-ax^{-1} +b x^{-3/n} + c x^3 + \mathcal{G}_0
\, ,
</math>
  </td>
</tr>

</table>
</div>
where <math>x</math> identifies the size of the configuration and <math>\mathcal{G}_0</math> is an arbitrary constant.  (As is explained more fully, below, the lefthand panel of Figure 1 displays a free-energy surface of this form for the case, <math>n=5</math>.)   If the coefficients, <math>a, b</math>, and <math>c</math>, are held constant while varying the configuration's size, we see that,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>\frac{d\mathcal{G}}{dx}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
ax^{-2} - \frac{3b}{n}\cdot x^{-(3+n)/n} + 3c x^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
x^{-2} \biggl[ a - \frac{3b}{n}\cdot x^{(n-3)/n} + 3c x^4 \biggr]
\, ,
</math>
  </td>
</tr>

</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>\frac{d^2\mathcal{G}}{dx^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
x^{-3} \biggl[ -2a + \frac{3(3+n)b}{n^2}\cdot x^{(n-3)/n} + 6c x^4 \biggr]
\, .
</math>
  </td>
</tr>

</table>
</div>

===Equilibrium Configurations===
The size, <math>x_\mathrm{eq}</math>, of each equilibrium configuration is determined by setting, <math>d\mathcal{G}/dx = 0</math>.  Hence, <math>x_\mathrm{eq}</math> is given by the root(s) of the polynomial expression that is often referred to as the,
<div align="center" id="ScalarVT">
<font color="#770000">'''Scalar Virial Theorem'''</font><br />
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>x^{(n-3)/n}_\mathrm{eq} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{n}{b} \biggl[\frac{a}{3} + c\cdot x^4_\mathrm{eq} \biggr] \, .
</math>
  </td>
</tr>

</table>
</div>
(The equilibrium radii of <math>n = 5</math> polytropic configurations having a variety of different masses are identified by the sequence of a dozen, small colored spherical dots shown in the [[SSC/Virial/PolytropesEmbeddedOutline#IntroFigures|lefthand panel of Figure 1]].)

===Stability===
The relative stability of each equilibrium configuration is determined by the sign of the second derivative of the free-energy function, evaluated at the specified equilibrium radius.  Specifically, the systems being considered here are stable if the second derivative is positive, but they are unstable if the second derivative is negative.  Evaluating the second derivative in this manner gives,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>\biggl[ x^{3} \cdot \frac{d^2\mathcal{G}}{dx^2}\biggr]_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-2a + \frac{3(3+n)}{n} \biggl[\frac{a}{3} + c\cdot x^4_\mathrm{eq} \biggr] + 6c x^4_\mathrm{eq} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-2a + \frac{(3+n)a}{n}  + \frac{3(3+n)c}{n} \cdot x^4_\mathrm{eq}  + 6c x^4_\mathrm{eq} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{9(n+1)c}{n}\cdot x^4_\mathrm{eq} - \frac{a(n-3)}{n} \, .
</math>
  </td>
</tr>

</table>
</div>
Defining <math>x_\mathrm{crit}</math> as the equilibrium radius at which this function goes to zero gives,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>x_\mathrm{crit} </math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{a(n-3)}{3^2c(n+1)} \biggr]^{1/4} \, .
</math>
  </td>
</tr>

</table>
</div>
(The small red spherical dot in the [[SSC/Virial/PolytropesEmbeddedOutline#IntroFigures|lefthand panel of Figure 1]] identifies the equilibrium configuration at <math>x_\mathrm{crit} </math>.)  We conclude, therefore, that pressure-truncated, equilibrium polytropic configurations having <math>n > 3</math> are stable if,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>x_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>></math>
  </td>
  <td align="left">
<math>
x_\mathrm{crit}  \, ,
</math>
  </td>
</tr>

</table>
</div>
while they are unstable if,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>x_\mathrm{eq}</math>
  </td>
  <td align="center">
<math><</math>
  </td>
  <td align="left">
<math>
x_\mathrm{crit}  \, .
</math>
  </td>
</tr>

</table>
</div>