Editing VE (section)

==Whitworth (1981) and Stahler (1983)==
The above formulation of a [[#Free_Energy_Expression|Gibbs-like free energy]] has been motivated by the {{ Stahler83 }} analysis of stability of isothermal gas clouds, and it closely parallels the discussion by {{ Whitworth81full }} of "global gravitational stability for one-dimensional polytropes."  Whitworth introduces a "global potential function," <math>\mathfrak{u}</math>, that is the sum of three "internal conserved energy modes,"
<div align="center">
<table border="0">
<tr>
  <td align="right">
<math>
\mathfrak{u}
</math>
  </td>
  <td align="center">
<math>
= 
</math>
  </td>
  <td align="left">
<math>
\mathfrak{g} + \mathfrak{B}_\mathrm{in} + \mathfrak{B}_\mathrm{ex} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
&nbsp;
<math>
- \frac{3}{5} \frac{GM_0^2}{R_0} \biggl(\frac{R}{R_0} \biggr)^{-1} 
+ (1-\delta_{1\eta})\biggl[ \frac{KM_0^\eta}{(\eta - 1)} V_0^{(1-\eta)} \biggr] \biggl(\frac{R}{R_0}\biggr)^{3(1-\eta)}
- \delta_{1\eta} \biggl[ 3KM_0 \ln\biggl(\frac{R}{R_0} \biggr) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp; 
  </td>
  <td align="left">
&nbsp;
<math>
+ P_\mathrm{ex} V_0 \biggl( \frac{R}{R_0} \biggr)^{3}
</math>
  </td>
</tr>

</table>
</div>
Clearly Whitworth's global potential function, <math>\mathfrak{u}</math>, is what we have referred to as the configuration's Gibbs-like free energy, with <math>\eta</math> being used rather than <math>\gamma_g</math> to represent the ratio of specific heats in the adiabatic case.  Our expression for <math>\mathfrak{G}</math> would precisely match his expression for <math>\mathfrak{u}</math> if we chose to examine the free energy of a nonrotating configuration, that is, if we set <math>C=J=0</math>.