Editing VE (section)

===Illustration===
As is derived in [[SSCpt1/Virial#Virial_Equilibrium|an accompanying discussion]], for a uniform-density, uniformly rotating, spherically symmetric configuration of mass <math>M</math> and radius <math>R</math>, 
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>
W_\mathrm{grav}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
- \frac{3}{5} \frac{GM^2}{R_0} \biggl( \frac{R}{R_0} \biggr)^{-1} \, , 
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>
T_\mathrm{kin}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\frac{5}{4} \frac{J^2}{MR_0^2} \biggl( \frac{R}{R_0} \biggr)^{-2} \, , 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
V
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\frac{4}{3} \pi R_0^3 \biggl( \frac{R}{R_0} \biggr)^{3} \, , 
</math>
  </td>
</tr>

</table>
</div> 
where, <math>J</math> is the system's total angular momentum and <math>R_0</math> is a reference length scale.

'''Adiabatic Systems''':  If, upon compression or expansion, the gaseous configuration behaves adiabatically, the reservoir of thermodynamic energy is,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>
\mathfrak{S}_\mathrm{therm} = U_\mathrm{int} = \frac{M K \rho^{\gamma_g-1}}{(\gamma_g - 1)} 
= \frac{M K }{(\gamma_g - 1)} \biggl( \frac{3M}{4\pi R_0^3} \biggr)^{\gamma_g-1} \biggl( \frac{R}{R_0} \biggr)^{-3(\gamma_g-1)} \, .
</math>
  </td>
</tr>

</table>
</div> 
Hence, the adiabatic free energy can be written as,
<div align="center">
<math>
\mathfrak{G} = -A\biggl( \frac{R}{R_0} \biggr)^{-1} +B\biggl( \frac{R}{R_0} \biggr)^{-3(\gamma_g-1)} + C \biggl( \frac{R}{R_0} \biggr)^{-2} + D\biggl( \frac{R}{R_0} \biggr)^3 \, ,
</math>
</div>
where, 
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>A</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{3}{5} \frac{GM^2}{R_0} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>B</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{K}{(\gamma_g-1)} \biggl( \frac{3}{4\pi R_0^3} \biggr)^{\gamma_g - 1} \biggr] M^{\gamma_g} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>C</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\frac{5J^2}{4MR_0^2} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>D</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\frac{4}{3} \pi R_0^3 P_e \, .
</math>
  </td>
</tr>
</table>
</div>

'''Isothermal Systems''':  If, upon compression or expansion, the configuration remains isothermal &#8212; also see Appendix A of {{ Stahler83full }} &#8212; the reservoir of thermal energy is,

<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>
\mathfrak{S}_\mathrm{therm} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
M c_s^2\ln \biggl( \frac{\rho}{\rho_0} \biggr) = - 3 M c_s^2 \biggl( \frac{R}{R_0} \biggr) \, . 
</math>
  </td>
</tr>

</table>
</div> 
Hence, the isothermal free energy can be written as,
<div align="center">
<math>
\mathfrak{G} = -A \biggl( \frac{R}{R_0} \biggr)^{-1} - B_I \ln \biggl( \frac{R}{R_0} \biggr) + C \biggl( \frac{R}{R_0} \biggr)^{-2} + D\biggl( \frac{R}{R_0} \biggr)^3 \, ,
</math>
</div>
where, aside from the coefficient definitions provided above in association with the adiabatic case,
<div align="center">
<table border="0">
<tr>
  <td align="right">
<math>B_I</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
3Mc_s^2 \, .
</math>
  </td>
</tr>

</table>
</div>

'''Summary''': We can combine the two cases &#8212; adiabatic and isothermal &#8212; into a single expression for <math>\mathfrak{G}</math> through a strategic use of the Kroniker delta function, <math>\delta_{1\gamma_g}</math>, as follows:
<div align="center">
<math>
\mathfrak{G} = -A\biggl( \frac{R}{R_0} \biggr)^{-1} +~ (1-\delta_{1\gamma_g})B\biggl( \frac{R}{R_0} \biggr)^{-3(\gamma_g-1)} -~ \delta_{1\gamma_g} B_I \ln \biggl( \frac{R}{R_0} \biggr) +~ C \biggl( \frac{R}{R_0} \biggr)^{-2} +~ D\biggl( \frac{R}{R_0} \biggr)^3 \, ,
</math>
</div>
Once the pressure exerted by the external medium (<math>P_e</math>), and the configuration's mass (<math>M</math>), angular momentum (<math>J</math>), and specific entropy (via <math>K</math>) &#8212; or, in the isothermal case, sound speed (<math>c_s</math>) &#8212;  have been specified, the values of all of the coefficients are known and this algebraic expression for <math>\mathfrak{G}</math> describes how the free energy of the configuration will vary with the configuration's relative size (<math>R/R_0</math>) for a given choice of <math>\gamma_g</math>.