Editing SSCpt1/Virial (section)

=====Our Choices=====
It is appropriate for us to define some characteristic scales against which various physical parameters can be normalized &#8212; and, hence, their relative significance can be specified or measured &#8212; as the free energy of various systems is examined.  As the system size is varied in search of extrema in the free energy, we generally will hold constant the total system mass and the specific entropy of each fluid element.  (When isothermal rather than adiabatic variations are considered, the sound speed rather than the specific entropy will be held constant.)  Hence, following the lead of both {{ Horedt70full }} and {{ Whitworth81full }}, we will express the various characteristic scales in terms of the constants, <math>G, M_\mathrm{tot},</math> and the polytropic constant, <math>K.</math>  Specifically, we will normalize all length scales, pressures, energies, mass densities, and the square of the speed of sound by, respectively,
<div align="center">
<table border="1" align="center" cellpadding="5">
<tr><th align="center" colspan="2">
Adopted Normalizations
</th></tr>
<tr>
  <td align="center">
Adiabatic Cases
  </td>
  <td align="center">
Isothermal Case
<math>~(\gamma = 1; K = c_s^2)</math>
  </td>
</tr>
<tr><td align="center">

<table border="0" cellpadding="5" align="center">
<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) M_\mathrm{tot}^{2-\gamma} \biggr]^{1/(4-3\gamma)} </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^4}{G^{3\gamma} M_\mathrm{tot}^{2\gamma}} \biggr]^{1/(4-3\gamma)}  </math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">
----
  </td>
</tr>

<tr>
  <td align="right">
<math>E_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>P_\mathrm{norm} R_\mathrm{norm}^3 = 
\biggl[ KG^{3(1-\gamma)}M_\mathrm{tot}^{6-5\gamma} \biggr]^{1/(4-3\gamma)} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\rho_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{3M_\mathrm{tot}}{4\pi R_\mathrm{norm}^3}
= \frac{3}{4\pi} \biggl[ \frac{K^3}{G^3 M_\mathrm{tot}^2} \biggr]^{1/(4-3\gamma )}  </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>c^2_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{P_\mathrm{norm}}{\rho_\mathrm{norm}}
= \frac{4\pi}{3} \biggl[ \frac{K}{(G^3 M_\mathrm{tot}^2)^{\gamma-1}} \biggr]^{1/(4-3\gamma )}  </math>
  </td>
</tr>
</table>

</td>

<td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>R_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{G M_\mathrm{tot}}{c_s^2}   </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>P_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{c_s^8}{G^{3} M_\mathrm{tot}^{2}}   </math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">
----
  </td>
</tr>

<tr>
  <td align="right">
<math>E_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>M_\mathrm{tot} c_s^2 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\rho_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\frac{3}{4\pi} \biggl[ \frac{c_s^6}{G^3 M_\mathrm{tot}^2} \biggr]  </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>c^2_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{4\pi}{3} \biggr) c_s^2 </math>
  </td>
</tr>
</table>

</td>
</tr>

<tr><td align="left" colspan="2">
Note that, given the above definitions, the following relations hold:
<div align="center">
<math>E_\mathrm{norm} = P_\mathrm{norm} R_\mathrm{norm}^3 = \frac{G M_\mathrm{tot}^2}{ R_\mathrm{norm}} 
= \biggl( \frac{3}{4\pi} \biggr) M_\mathrm{tot} c_\mathrm{norm}^2</math>
</div>
</td></tr>
</table>
</div>

It should be emphasized that, as we discuss how a configuration's free energy varies with its size, the variable <math>R_\mathrm{limit}</math> will be used to identify the configuration's size ''whether or not the system is in equilibrium,''  and the parameter,
<div align="center">
<math>\chi \equiv \frac{R_\mathrm{limit}}{R_\mathrm{norm}} \, ,</math>
</div>
will be used to identify the size as referenced to <math>R_\mathrm{norm}</math>.  When an equilibrium configuration is identified <math>(R_\mathrm{limit} \rightarrow R_\mathrm{eq})</math>, we will affix the subscript "eq," specifically,
<div align="center">
<math>\chi_\mathrm{eq} \equiv \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \, .</math>
</div>