Editing VE (section)

==Free Energy Expression==
Associated with any isolated, self-gravitating, gaseous configuration we can identify a total Gibbs-like free energy, <math>\mathfrak{G}</math>, given by the sum of the relevant contributions to the total energy of the configuration,
<div align="center">
<math>
\mathfrak{G} = W_\mathrm{grav} + \mathfrak{S}_\mathrm{therm} + T_\mathrm{kin} + P_e V + \cdots
</math>
</div> 
Here, we have explicitly included the gravitational potential energy, <math>W_\mathrm{grav}</math>, the ordered kinetic energy, <math>T_\mathrm{kin}</math>, a term that accounts for surface effects if the configuration of volume <math>V</math> is embedded in an external medium of pressure <math>P_e</math>, and <math>\mathfrak{S}_\mathrm{therm}</math>, the reservoir of thermodynamic energy that is available to perform work as the system expands or contracts.  Our above discussion of the [[#Scalar_Virial_Theorem|scalar virial theorem]] provides mathematical definitions of each of these energy terms, except for <math>\mathfrak{S}_\mathrm{therm}</math>, which we discuss now.

===Reservoir of Thermodynamic Energy===
<math>\mathfrak{S}_\mathrm{therm}</math> derives from the differential, "PdV" work that is often discussed in the context of thermodynamic systems.  It should be made clear that, here, "dV" refers to the differential volume ''per unit mass,''  so it should be written as "<math>~d(\rho^{-1})</math>", to be consistent with the notation used throughout this H_Book.  Therefore, the differential thermodynamic work is,
<div align="center">
<math>d\mathfrak{w} = Pd(1/\rho) = -  \biggl( \frac{P}{\rho^2} \biggr) d\rho \, .</math>
</div>
After an ''evolutionary'' equation of state has been adopted, this differential relationship can be integrated to give an expression for the energy per unit mass, <math>\mathfrak{w}</math>, that is potentially available for work.  Then we define the thermodynamic energy reservoir as,
<div align="center">
<math>\mathfrak{S}_\mathrm{therm} \equiv - \int \mathfrak{w} ~dm \, .</math>
</div>

====Isothermal Systems====
If each element of gas maintains its temperature when the system undergoes compression or expansion &#8212; that is, if the compression/expansion is isothermal &#8212; then, the relevant evolutionary equation of state is,
<div align="center">
<math>P = c_s^2 \rho \, ,</math>
</div>
where the constant, <math>c_s</math>, is the isothermal sound speed.  In this case, the expression for the differential thermodynamic work becomes,
<div align="center">
<math>d\mathfrak{w} =  -  \biggl( \frac{c_s^2}{\rho} \biggr) d\rho =  -  c_s^2 d\ln\rho \, .</math>
</div>
Hence, to within an additive constant, we have,
<div align="center">
<math>\mathfrak{w} = -  c_s^2 \ln \biggl(\frac{\rho}{\rho_0}\biggr) \, ,</math>
</div>
where, <math>\rho_0</math> is a (as yet unspecified) reference density, and integration throughout the configuration gives (for the isothermal case),
<div align="center">
<math>\mathfrak{S}_\mathrm{therm} = +  \int c_s^2 \ln \biggl(\frac{\rho}{\rho_0}\biggr) dm \, .</math>
</div>

====Adiabatic Systems====
If, upon compression or expansion, the gaseous configuration evolves adiabatically, the pressure will vary with density as,
<div align="center">
<math>P = K \rho^{\gamma_g} \, ,</math> 
</div>
where, <math>K</math> specifies the specific entropy of the gas and {{ Template:Math/MP_AdiabaticIndex }} is the ratio of specific heats that is relevant to the phase of compression/expansion.  In this case, the expression for the differential thermodynamic work becomes,
<div align="center">
<math>d\mathfrak{w} =  -  K \rho^{{\gamma_g}-2} d\rho =  -  \frac{K}{({\gamma_g}-1)} d\rho^{{\gamma_g}-1} \, .</math>
</div>
Hence, to within an additive constant, we have,
<div align="center">
<math>\mathfrak{w} = -  \frac{1}{({\gamma_g}-1)} \biggl( \frac{P}{\rho} \biggr) \, ,</math>
</div>
and integration throughout the configuration gives (for the adiabatic case),
<div align="center">
<math>\mathfrak{S}_\mathrm{therm} = +  \int \frac{1}{({\gamma_g}-1)} \biggl( \frac{P}{\rho} \biggr)  dm 
= \frac{2}{3({\gamma_g}-1)}  \int \frac{3}{2} \biggl( \frac{P}{\rho} \biggr)  dm
= \frac{2}{3({\gamma_g}-1)} S_\mathrm{therm} \, ,</math>
</div>
where, as introduced in our above discussion of the [[#Scalar_Virial_Theorem|scalar virial theorem]], <math>S_\mathrm{therm}</math> is the system's total thermal (''i.e.,'' random kinetic) energy.

====Relationship to the System's Internal Energy====
It is instructive to tie this introductory material to the classic discussion of thermodynamic systems, which relates a change in the system's internal energy per unit mass, <math>\Delta u_\mathrm{int}</math>, to the differential work, <math>\Delta \mathfrak{w}</math>, via the expression,
<div align="center">
<math>\Delta u_\mathrm{int} = \Delta Q - \Delta \mathfrak{w} \, ,</math>
</div>
where, <math>\Delta Q</math> is the change in heat content of the system.  

'''Isothermal Evolutions''':  Because the internal energy is only a function of the temperature, we can set <math>\Delta u_\mathrm{int} = 0</math> for expansions or contractions that occur isothermally.  Hence, for isothermal evolutions the change in heat content can immediately be deduced from the expression derived for the differential work; specifically, <math>\Delta Q = \Delta \mathfrak{w}</math>.

'''Adiabatic Evolutions''':  By definition, <math>\Delta Q = 0</math> for adiabatic evolutions, in which case we find <math>\Delta u_\mathrm{int} = - \Delta \mathfrak{w}</math>.  The definition of the thermodynamic energy reservoir can therefore be rewritten as,
<div align="center">
<math>\mathfrak{S}_\mathrm{therm} = - \int \mathfrak{w} ~dm = + \int u_\mathrm{int} ~dm = U_\mathrm{int} \, .</math>
</div>
Quite generally, then &#8212; in sync with the above derivation &#8212; we can replace <math>\mathfrak{S}_\mathrm{therm}</math> by,
<div align="center">
<math>~U_\mathrm{int} = \frac{2}{3(\gamma_g-1)} S_\mathrm{therm} \, ,</math> 
</div>
in the expression for the free energy when analyzing adiabatic evolutions.

===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>.