Editing SSC/FreeEnergy/PolytropesEmbedded/Pt3B (section)

====Internal Energy Components====

=====First View=====

Before writing out the expressions for the internal energy of the core and of the envelope, we [[SSC/Structure/BiPolytropes/FreeEnergy00#Virial_Theorem|note from our separate detailed derivation]] that, in either case,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl[\frac{P_i \chi^{3\gamma}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \chi^{3-3\gamma}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\biggl(\frac{P_i }{P_0} \biggr) \biggl(\frac{P_0 }{P_\mathrm{norm}} \biggr)\chi^{3}\biggr]_\mathrm{eq} \biggl[\frac{\chi}{\chi_\mathrm{eq}}\biggr]^{3-3\gamma}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{\biggl(\frac{P_i }{P_0} \biggr) \biggl[ \frac{3}{4\pi } \biggl( \frac{\nu}{q^3} \biggr)\biggr]^{\gamma_c}\chi^{3-3\gamma_c}\biggr\}_\mathrm{eq} \Chi^{3-3\gamma} \, ,</math>
  </td>
</tr>
</table>
</div>
where, in equilibrium,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl(\frac{P_i }{P_0} \biggr)_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 - b_\xi q^2</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~b_\xi q^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{\frac{2}{5}q^3 f + \biggl[1 - \frac{2}{5} q^3( 1+\mathfrak{F} ) \biggr]\biggr\}^{-1} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[1 + \frac{2}{5}q^3 (f - 1-\mathfrak{F} ) \biggr]^{-1} </math>
  </td>
</tr>
</table>
</div>

So, copying from our [[SSC/Structure/BiPolytropes/FreeEnergy00#InternalEnergies|accompanying detailed derivation]], we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{\mathfrak{S}_A}{E_\mathrm{norm}} \biggr)_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{4\pi/3 }{({\gamma_c}-1)}  
\biggl\{\biggl(\frac{P_i }{P_0} \biggr) \biggl[ \frac{3}{4\pi } \biggl( \frac{\nu}{q^3} \biggr)\biggr]^{\gamma_c}\chi^{3-3\gamma_c}\biggr\}_\mathrm{eq} \Chi^{3-3\gamma_c} 
\biggl\{ \biggl( \frac{P_0}{P_{ic}} \biggr) \biggl[ q^3 - \biggl( \frac{3b_\xi}{5} \biggr) q^5 \biggr] \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{1 }{({\gamma_c}-1)}  \biggl[\biggl(\frac{4\pi}{3}\biggr)^{1-\gamma_c}
\biggl( \frac{\nu}{q^3} \biggr)^{\gamma_c}\chi_\mathrm{eq}^{3-3\gamma_c}\biggr] \Chi^{3-3\gamma_c} 
q^3\biggl[ 1 - \biggl( \frac{3}{5} \biggr) b_\xi q^2 \biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl( \frac{\mathfrak{S}_A}{E_\mathrm{norm}} \biggr)_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{4\pi/3 }{({\gamma_e}-1)}  
\biggl\{\biggl(\frac{P_i }{P_0} \biggr) \biggl[ \frac{3}{4\pi } \biggl( \frac{\nu}{q^3} \biggr)\biggr]^{\gamma_c}\chi^{3-3\gamma_c}\biggr\}_\mathrm{eq} \Chi^{3-3\gamma_e} 
\biggl\{ (1-q^3) +  b_\xi \biggl(\frac{P_0}{P_{ie} } \biggr) \biggl[\frac{2}{5} q^5 \mathfrak{F}
\biggr]  \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{1}{({\gamma_e}-1)}  \biggl[\biggl(\frac{4\pi}{3}\biggr)^{1-\gamma_c}
\biggl( \frac{\nu}{q^3} \biggr)^{\gamma_c}\chi_\mathrm{eq}^{3-3\gamma_c}\biggr]  
\Chi^{3-3\gamma_e} \biggl(\frac{P_i }{P_0} \biggr)
\biggl\{ (1-q^3) +  b_\xi \biggl(\frac{P_0}{P_{ie} } \biggr) \biggl[\frac{2}{5} q^5 \mathfrak{F}
\biggr]  \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{1}{({\gamma_e}-1)}  \biggl[\biggl(\frac{4\pi}{3}\biggr)^{1-\gamma_c}
\biggl( \frac{\nu}{q^3} \biggr)^{\gamma_c}\chi_\mathrm{eq}^{3-3\gamma_c}\biggr]  
\Chi^{3-3\gamma_e} 
\biggl\{ (1-b_\xi q^2)(1-q^3) +  b_\xi  \biggl[\frac{2}{5} q^5 \mathfrak{F} \biggr]  \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{1}{({\gamma_e}-1)}  \biggl[\biggl(\frac{4\pi}{3}\biggr)^{1-\gamma_c}
\biggl( \frac{\nu}{q^3} \biggr)^{\gamma_c}\chi_\mathrm{eq}^{3-3\gamma_c}\biggr]  
\Chi^{3-3\gamma_e} (1-q^3)
\biggl\{ 1- \biggl[ 1 - \frac{2}{5} \biggl(\frac{q^3}{1-q^3}\biggr) \mathfrak{F} \biggr]b_\xi q^2   \biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

Furthermore,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl[\biggl(\frac{4\pi}{3}\biggr)^{1-\gamma_c}
\biggl( \frac{\nu}{q^3} \biggr)^{\gamma_c}\chi_\mathrm{eq}^{3-3\gamma_c}\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{3}{4\pi}\biggr)^{\gamma_c - 1}
\biggl( \frac{\nu}{q^3} \biggr)^{\gamma_c}
\biggl\{\chi_\mathrm{eq}^{4-3\gamma_c}\biggr\}^{(3-3\gamma_c)/(4-3\gamma_c)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{3}{4\pi}\biggr)^{\gamma_c - 1}
\biggl( \frac{\nu}{q^3} \biggr)^{\gamma_c}
\biggl\{\frac{1}{2}\biggl(\frac{3}{4\pi} \biggr)^{1-\gamma_c}
\biggl( \frac{\nu}{q^3}\biggr)^{2-\gamma_c}
\biggl[ q^2 + \frac{2}{5} q^5( f - 1-\mathfrak{F} )\biggr] \biggr\}^{(3-3\gamma_c)/(4-3\gamma_c)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{3}{4\pi}\biggr)^{(\gamma_c - 1)/(4-3\gamma_c)}
\biggl( \frac{\nu}{q^3} \biggr)^{(6-5\gamma_c)(4-3\gamma_c)}
\biggl\{\frac{q^2}{2}
\biggl[ 1 + \frac{2}{5} q^3( f - 1-\mathfrak{F} )\biggr] \biggr\}^{(3-3\gamma_c)/(4-3\gamma_c)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{3}{4\pi}\biggr)^{1/(n_c-3)}
\biggl( \frac{\nu}{q^3} \biggr)^{(n_c-5)(n_c-3)}
\biggl\{\frac{q^2}{2}
\biggl[ 1 + \frac{2}{5} q^3( f - 1-\mathfrak{F} )\biggr] \biggr\}^{-3/(n_c-3)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \biggl(\frac{2\cdot 3}{\pi}\biggr) \biggl( \frac{\nu}{q^3} \biggr)^{(n_c-5)} b_\xi^3\biggr]^{1/(n_c-3)} 
\, .
</math>
  </td>
</tr>
</table>
</div>

Hence, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{\mathfrak{S}_A}{E_\mathrm{norm}} \biggr)_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
n_c  \biggl[\biggl(\frac{4\pi}{3}\biggr)^{1-\gamma_c}
\biggl( \frac{\nu}{q^3} \biggr)^{\gamma_c}\chi_\mathrm{eq}^{3-3\gamma_c}\biggr] \Chi^{-3/n_c} 
q^3\biggl[ 1 - \biggl( \frac{3}{5} \biggr) b_\xi q^2 \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
n_c  
\biggl[ \biggl(\frac{2\cdot 3}{\pi}\biggr) \biggl( \frac{\nu}{q^3} \biggr)^{(n_c-5)} b_\xi^3\biggr]^{1/(n_c-3)} 
\Chi^{-3/n_c} 
q^3\biggl[ 1 - \biggl( \frac{3}{5} \biggr) b_\xi q^2 \biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl( \frac{\mathfrak{S}_A}{E_\mathrm{norm}} \biggr)_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
n_e 
\biggl[ \biggl(\frac{2\cdot 3}{\pi}\biggr) \biggl( \frac{\nu}{q^3} \biggr)^{(n_c-5)} b_\xi^3\biggr]^{1/(n_c-3)} 
\Chi^{-3/n_e} (1-q^3)
\biggl\{ 1- \biggl[ 1 - \frac{2}{5} \biggl(\frac{q^3}{1-q^3}\biggr) \mathfrak{F} \biggr]b_\xi q^2   \biggr\} \, .
</math>
  </td>
</tr>

</table>
</div>


=====Second View=====
In our [[SSC/Structure/BiPolytropes/Analytic00#PiDefinition|accompanying discussion of energies associated with detailed force balance models]], we used the notation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Pi</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{3}{2^3\pi}\biggr) \frac{GM_\mathrm{tot}^2}{R^4} \biggl(\frac{\nu}{q^3}\biggr)^2
= P_\mathrm{norm} \chi^{-4}\biggl(\frac{3}{2^3\pi}\biggr) \biggl(\frac{\nu}{q^3}\biggr)^2 \, ,
</math>
  </td>
</tr>
</table>
</div>
which allows us to rewrite the [[#Second_View|above quoted relationship]] between the central pressure and the radius of the bipolytrope as,
<div align="center">
<math>~P_0 = \Pi (qg)^2 \, .</math>
</div>
We [[SSC/Structure/BiPolytropes/Analytic00#Virial_Equilibrium|also showed]] that, in equilibrium, the relationship between the central pressure and the interface pressure is,
<div align="center">
<math>~P_0 =P_i + \Pi_\mathrm{eq} q^2 \, .</math>
</div>
This means that, in equilibrium, the ratio of the interface pressure to the central pressure is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl(\frac{P_i}{P_0}\biggr)_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 - \frac{\Pi_\mathrm{eq} q^2}{P_0}
= 1- \frac{1}{g^2} \, ,
</math>
  </td>
</tr>
</table>
</div>
or given that (see [[#Second_View|above]]),

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
 \frac{5}{2q^3} \biggl[g^2-1\biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
f - 1-\mathfrak{F} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~
g^2
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 1+\frac{2}{5} q^3 ( f - 1-\mathfrak{F} ) \, ,
</math>
  </td>
</tr>
</table>
</div>
we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl(\frac{P_i}{P_0}\biggr)_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 - \frac{\Pi_\mathrm{eq} q^2}{P_0}
= 1- \biggl[ 1+\frac{2}{5} q^3 ( f - 1-\mathfrak{F} )  \biggr]^{-1} \, .
</math>
  </td>
</tr>
</table>
</div>
This is exactly the pressure-ratio expression presented in our "first view" and unveils the notation association,
<div align="center">
<table border="0">

<tr>
  <td align="right">
<math>~b_\xi q^2</math>
  </td>
  <td align="center">
<math>~\leftrightarrow~</math>
  </td>
  <td align="left">
<math>
\frac{1}{g^2} \, .
</math>
  </td>
</tr>
</table>
</div>

From [[SSC/Structure/BiPolytropes/Analytic00#Thermal_Energy_Content|our separate derivation]], we have, in equilibrium,
<div align="center">
<table border="0">
<tr>
  <td align="right">
<math>~\mathfrak{G}_\mathrm{core} = \biggl(\frac{2n_c}{3}\biggr) S_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>\biggl(\frac{2n_c}{3}\biggr) \biggl( \frac{4\pi}{5} \biggr) R_\mathrm{eq}^3 q^5 \biggl (\frac{5P_i}{2q^2} + \Pi \biggr)_\mathrm{eq} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>\biggl( \frac{ q^5n_c}{5} \biggr) R_\mathrm{eq}^3  \biggl( \frac{2^3\pi}{3} \biggr) \Pi_\mathrm{eq} \biggl[\frac{5}{2q^2} \biggl( \frac{P_i}{\Pi} \biggr)_\mathrm{eq} + 1 \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>\biggl( \frac{ n_c}{5} \biggr) \biggl[ R_\mathrm{norm}^3  
P_\mathrm{norm} \biggr] \chi_\mathrm{eq}^{-1} \biggl(\frac{\nu^2}{q}\biggr)  
\biggl[\frac{5}{2q^2} \biggl( \frac{P_i}{P_0} \biggr)_\mathrm{eq}\biggl( \frac{P_0}{\Pi} \biggr)_\mathrm{eq} + 1 \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\biggl[ \frac{\mathfrak{G}_\mathrm{core} }{E_\mathrm{norm}}\biggr]_\mathrm{eq} </math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{ n_c}{5} \biggr) \biggl(\frac{\nu^2}{q}\biggr)  
\biggl[\frac{5}{2q^2} \biggl( 1-\frac{1}{g^2} \biggr)\biggl( q^2g^2\biggr) + 1 \biggr] \chi_\mathrm{eq}^{-1} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{ n_c}{2} \biggr) \biggl(\frac{\nu^2}{q}\biggr)  
\biggl[ g^2-\frac{3}{5}  \biggr] 
\biggl\{\frac{1}{2^{n_c}}\biggl(\frac{4\pi}{3} \biggr)
\biggl( \frac{\nu}{q^3}\biggr)^{n_c-1} (q g)^{2n_c} 
\biggr\}^{-1/(n_c-3)}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~n_c \biggl[ 1- \biggl(\frac{3}{5}\biggr) \frac{1}{g^2}  \biggr]
\biggl( \frac{ 1}{2} \biggr) \biggl(\frac{\nu^2}{q}\biggr)  g^2 
\biggl\{2^{n_c}\biggl(\frac{3}{4\pi} \biggr)
\biggl( \frac{\nu}{q^3}\biggr)^{1-n_c} (q g)^{-2n_c} 
\biggr\}^{1/(n_c-3)}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~n_c \biggl[ 1- \biggl(\frac{3}{5}\biggr) \frac{1}{g^2}  \biggr]
\biggl\{2^{n_c}\cdot 2^{(3-n_c)}\biggl(\frac{3}{4\pi} \biggr)
\biggl( \frac{\nu}{q^3}\biggr)^{1-n_c} \biggl(\frac{\nu}{q^3}\biggr)^{2(n_c-3)} q^{5(n_c-3)} q^{-2n_c} g^{-2n_c} g^{2(n_c-3)}
\biggr\}^{1/(n_c-3)}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~n_c \biggl[ 1- \biggl(\frac{3}{5}\biggr) \frac{1}{g^2}  \biggr]
\biggl\{\biggl(\frac{2\cdot 3}{\pi} \biggr)
\biggl( \frac{\nu}{q^3}\biggr)^{n_c-5} q^{3n_c-15}  g^{-6} 
\biggr\}^{1/(n_c-3)} \, .</math>
  </td>
</tr>
</table>
</div>
Finally, switching from the <math>~g</math> notation to the <math>~b_\xi</math> notation gives,

<div align="center">
<table border="0">
<tr>
  <td align="right">
<math>~\biggl[ \frac{\mathfrak{G}_\mathrm{core} }{E_\mathrm{norm}}\biggr]_\mathrm{eq} </math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~n_c \biggl[ 1- \biggl(\frac{3}{5}\biggr) b_\xi q^2  \biggr]
\biggl\{\biggl(\frac{2\cdot 3}{\pi} \biggr)
\biggl( \frac{\nu}{q^3}\biggr)^{n_c-5} q^{3n_c-15}  b_\xi^3 q^{6} 
\biggr\}^{1/(n_c-3)} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~n_c q^3 \biggl[ 1- \biggl(\frac{3}{5}\biggr) b_\xi q^2  \biggr]
\biggl\{\biggl(\frac{2\cdot 3}{\pi} \biggr)
\biggl( \frac{\nu}{q^3}\biggr)^{n_c-5} b_\xi^3 \biggr\}^{1/(n_c-3)} \, ,</math>
  </td>
</tr>
</table>
</div>
which, after setting <math>~\Chi = 1</math>, precisely matches the above, "first view" expression.  Also from our [[SSC/Structure/BiPolytropes/Analytic00#Thermal_Energy_Content|previous derivation]], we can write,
<div align="center">
<table border="0">
<tr>
  <td align="right">
<math>~\mathfrak{G}_\mathrm{env} = \biggl(\frac{2n_e}{3}\biggr) S_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ 2\pi\biggl(\frac{2n_e}{3}\biggr)
R_\mathrm{eq}^3 \Pi_\mathrm{eq} \biggl\{ (1-q^3) \biggl(\frac{P_i  }{\Pi}\biggr)_\mathrm{eq}  
+ \biggl( \frac{\rho_e}{\rho_0} \biggr)\biggl[ (-2q^2 + 3q^3 - q^5 )
  + \frac{3}{5} \biggl( \frac{\rho_e}{\rho_0} \biggr) ( -1 + 5q^2 -5q^3 +  q^5 )\biggr]\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ 2\pi\biggl(\frac{2n_e}{3}\biggr)
R_\mathrm{eq}^3 \biggl[  P_\mathrm{norm} \chi^{-4}\biggl(\frac{3}{2^3\pi}\biggr) \biggl(\frac{\nu}{q^3}\biggr)^2\biggr]_\mathrm{eq}
\biggl\{ (1-q^3) q^2(g^2-1)  + \biggl(\frac{2}{5}\biggr) q^5 \mathfrak{F} \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ \biggl[  P_\mathrm{norm} R_\mathrm{norm}^3 \biggr] \frac{n_e}{2}
\biggl(\frac{\nu^2}{q^4}\biggr)(1-q^3)
\biggl\{  (g^2-1)  + \frac{2}{5} \biggl(\frac{q^3}{1-q^3} \biggr)\mathfrak{F} \biggr\} 
\chi^{-1}_\mathrm{eq} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~\biggl[ \frac{\mathfrak{G}_\mathrm{env} }{E_\mathrm{norm}}\biggr]_\mathrm{eq} </math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ n_e (1-q^3)
\biggl\{  (g^2-1)  + \frac{2}{5} \biggl(\frac{q^3}{1-q^3} \biggr)\mathfrak{F} \biggr\} \frac{q^2}{2}\biggl(\frac{\nu}{q^3}\biggr)^2
\biggl[\frac{1}{2^{n_c}}\biggl(\frac{4\pi}{3} \biggr)
\biggl( \frac{\nu}{q^3}\biggr)^{n_c-1} (q g)^{2n_c}\biggr]^{-1/(n_c-3)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ n_e (1-q^3)
\biggl\{  (g^2-1)  + \frac{2}{5} \biggl(\frac{q^3}{1-q^3} \biggr)\mathfrak{F} \biggr\} 
\biggl[2^{[n_c-(n_c-3)]} \biggl(\frac{3}{4\pi} \biggr)
\biggl( \frac{\nu}{q^3}\biggr)^{(1-n_c)+2(n_c-3)} q^{2(n_c-3)-2n_c} g^{-2n_c} \biggr]^{1/(n_c-3)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ n_e (1-q^3)
\biggl\{  (g^2-1)  + \frac{2}{5} \biggl(\frac{q^3}{1-q^3} \biggr)\mathfrak{F} \biggr\} 
\biggl[\biggl(\frac{2\cdot 3}{\pi} \biggr)
\biggl( \frac{\nu}{q^3}\biggr)^{n_c-5} q^{-6} g^{-2n_c} \biggr]^{1/(n_c-3)} \, .
</math>
  </td>
</tr>
</table>
</div>
And, finally, switching from the <math>~g</math> notation to the <math>~b_\xi</math> notation gives,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl[ \frac{\mathfrak{G}_\mathrm{env} }{E_\mathrm{norm}}\biggr]_\mathrm{eq} </math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ n_e (1-q^3) (b_\xi q^2)^{-1}
\biggl\{  1 - \biggl[1 - \frac{2}{5} \biggl(\frac{q^3}{1-q^3} \biggr)\mathfrak{F} \biggr]b_\xi q^2\biggr\} 
\biggl[\biggl(\frac{2\cdot 3}{\pi} \biggr)
\biggl( \frac{\nu}{q^3}\biggr)^{n_c-5} q^{-6} (b_\xi q^2)^{n_c} \biggr]^{1/(n_c-3)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ n_e (1-q^3) 
\biggl\{  1 - \biggl[1 - \frac{2}{5} \biggl(\frac{q^3}{1-q^3} \biggr)\mathfrak{F} \biggr]b_\xi q^2\biggr\} 
\biggl[\biggl(\frac{2\cdot 3}{\pi} \biggr)
\biggl( \frac{\nu}{q^3}\biggr)^{n_c-5} q^{-6-2(n_c-3)+2n_c}  b_\xi^{3-n_c+n_c} \biggr]^{1/(n_c-3)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ n_e\biggl[\biggl(\frac{2\cdot 3}{\pi} \biggr)\biggl( \frac{\nu}{q^3}\biggr)^{n_c-5} b_\xi^{3} \biggr]^{1/(n_c-3)} (1-q^3) 
\biggl\{  1 - \biggl[1 - \frac{2}{5} \biggl(\frac{q^3}{1-q^3} \biggr)\mathfrak{F} \biggr]b_\xi q^2\biggr\} \, ,
</math>
  </td>
</tr>
</table>
</div>
which, after setting <math>~\Chi = 1</math>, precisely matches the above, "first view" expression.