Editing SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt1 (section)

==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>