Editing SSC/Structure/BiPolytropes/FreeEnergy51/Pt3

=Free Energy of BiPolytrope with (n<sub>c</sub>, n<sub>e</sub>) = (5, 1)=
<table border="1" align="center" width="100%" colspan="8">
  <td align="center" bgcolor="lightblue" width="33%"><br />[[SSC/Structure/BiiPolytropes/FreeEnergy51|Part I:&nbsp; Mass Profile]]
&nbsp;
  </td>
  <td align="center" bgcolor="lightblue"3 width="33%"><br />[[SSC/Structure/BiPolytropes/FreeEnergy51/Pt2|Part II:&nbsp; Gravitational Potential Energy]]
&nbsp;
  </td>
  <td align="center" bgcolor="lightblue"><br />[[SSC/Structure/BiPolytropes/FreeEnergy51/Pt3|Part III:&nbsp; Thermal Energy Reservoir]]
&nbsp;
  </td>
</tr>
</table>

==Thermodynamic Energy Reservoir==

===The Core===

From our [[SSC/BipolytropeGeneralizationVersion2#Separate_Thermodynamic_Energy_Reservoirs|introductory discussion of the free energy of bipolytropes]], the energy contained in the core's thermodynamic reservoir may be written as,
<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{2}{3({\gamma_c}-1)} \biggl( \frac{\chi}{\chi_\mathrm{eq}} \biggr)^{3-3\gamma_c} \biggl[ \frac{2\pi P_{ic} \chi^3}{P_\mathrm{norm}} \biggr]_\mathrm{eq}  
\biggl[ q^3 s_\mathrm{core} \biggr] \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
  <td align="right">
<math>~q^3 s_\mathrm{core} </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ 
\int_0^q  3\biggl[\frac{1 - p_c(x)}{1-p_c(q)} \biggr]  x^2 dx \, ,
</math>
  </td>
</tr>
</table>
</div>
defines the relevant integral over the core's pressure distribution.  According to our [[SSC/Structure/BiPolytropes/Analytic51#Profile|derivation of the properties of detailed force-balance <math>~(n_c, n_e) = (5, 1)</math> bipolytropes]] &#8212; see also the relevant derivations [[SSC/BipolytropeGeneralizationVersion2#Bipolytrope_Generalization|in our accompanying overview]] &#8212;  in this case the pressure throughout the core is defined by the dimensionless function,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~P^* \equiv \frac{P_\mathrm{core}(\xi)}{P_0} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ 1-p_c(x) = \frac{P_\mathrm{core}(x)}{P_0} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( 1 + a_\xi x^2 \biggr)^{-3} \, ,</math>
  </td>
</tr>
</table>
</div>
where, <math>~a_\xi</math> is defined above in connection with our [[SSC/Structure/BiiPolytropes/FreeEnergy51#Mass_Profile|derivation of the mass profile]].
The desired integral over this pressure distribution therefore gives,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
  <td align="right">
<math>~q^3 s_\mathrm{core} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
3\biggl( 1 + a_\xi q^2 \biggr)^{3} \int_0^q \frac{x^2 dx}{(1+a_\xi x^2)^3}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
3\biggl( 1 + a_\xi q^2 \biggr)^{3} \biggl\{ 
\frac{\tan^{-1}[a_\xi^{1/2}q]}{2^3 a_\xi^{3/2}} 
+ \frac{q}{2^3 a_\xi (a_\xi q^2 +1)}
- \frac{q}{2^2 a_\xi (a_\xi q^2 +1)^2}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{3}{2^3 a_\xi^{3/2}} \biggl( 1 + a_\xi q^2 \biggr)^{3} \biggl\{ 
\tan^{-1}[a_\xi^{1/2}q]
+ \frac{a_\xi^{1/2}q}{(a_\xi q^2 +1)}
- \frac{2a_\xi^{1/2}q}{(a_\xi q^2 +1)^2}
\biggr\}
</math>
  </td>
</tr>

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

Next, let's examine the factor in square brackets with an "eq" subscript.   From our [[SSC/Structure/BiPolytropes/Analytic51#Profile|derivation of the properties of detailed force-balance <math>~(n_c, n_e) = (5, 1)</math> bipolytropes]], we know that,
<div align="center">
<math>P_{ic} = K_c \rho_0^{6/5} \biggl(1 + \frac{1}{3}\xi_i^2 \biggr)^{-3} \, ,</math>
</div>
and,
<div align="center">
<math>
\chi_\mathrm{eq} = \biggl(\frac{R_\mathrm{edge}}{R_\mathrm{norm}} \biggr)_\mathrm{eq} 
= \frac{1}{q}\biggl(\frac{r_i}{R_\mathrm{norm}}\biggr)_\mathrm{eq}
= \frac{1}{q}\biggl[ \frac{K_c^{1/2} G^{-1/2} \rho_0^{-2/5}}{R_\mathrm{norm}}\biggr] \biggl( \frac{3}{2\pi} \biggr)^{1/2} \xi_i \, .
</math>
</div>

Hence, the relevant factor may be rewritten as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{2\pi P_{ic} \chi_\mathrm{eq}^3}{P_\mathrm{norm}} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
2\pi \biggl[ \frac{K_c \rho_0^{6/5}}{P_\mathrm{norm}}\biggr] \biggl(1 + \frac{1}{3}\xi_i^2 \biggr)^{-3} 
\biggl\{ \frac{1}{q}\biggl[ \frac{K_c^{1/2} G^{-1/2} \rho_0^{-2/5}}{R_\mathrm{norm}}\biggr] \biggl( \frac{3}{2\pi} \biggr)^{1/2} \xi_i \biggr\}^3
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{3^3}{2\pi} \biggr)^{1/2} \biggl[ \frac{K_c^{5/2} G^{-3/2}}{P_\mathrm{norm} R_\mathrm{norm}^3 }\biggr] \biggl(1 + \frac{1}{3}\xi_i^2 \biggr)^{-3} 
\biggl( \frac{\xi_i }{q}\biggr)^3
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{3^6}{2\pi} \biggr)^{1/2} \biggl(1 + a_\xi q^2 \biggr)^{-3} a_\xi^{3/2} \, ,
</math>
  </td>
</tr>
</table>
</div>
where, the last expression has been obtained by employing the substitution, defined above, <math>~\xi_i = (3a_\xi)^{1/2}q</math>.  Finally, then, 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{2}{3({\gamma_c}-1)} \biggl( \frac{\chi}{\chi_\mathrm{eq}} \biggr)^{3-3\gamma_c} 
\biggl\{\biggl( \frac{3^8}{2^7\pi} \biggr)^{1/2} \biggl[ 
\tan^{-1}[a_\xi^{1/2}q] - a_\xi^{1/2}q ~\frac{(1 - a_\xi q^2)}{(1 + a_\xi q^2)^2} \biggr]
\biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>
As it should, the term inside the curly brackets precisely matches the analytic expression for the dimensionless thermal energy of the core, <math>~S^*_\mathrm{core}</math>, that has been derived elsewhere in conjunction with our [[SSC/Structure/BiPolytropes/Analytic51#Expression_for_Free_Energy|discussion of the detailed force-balanced structure of this bipolytrope]].

===The Envelope===

Similarly, the energy contained in the envelope's thermodynamic reservoir may be written as,
<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{env}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{2}{3({\gamma_e}-1)} \biggl( \frac{\chi}{\chi_\mathrm{eq}} \biggr)^{3-3\gamma_e} \biggl( \frac{P_{ie}}{P_{ic}} \biggr)
\biggl[ \frac{2\pi P_{ic} \chi^3}{P_\mathrm{norm}} \biggr]_\mathrm{eq}  
\biggl[ (1-q^3) s_\mathrm{env} \biggr] \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
  <td align="right">
<math>~(1-q^3) s_\mathrm{env} </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ 
\int_q^1  3\biggl[ 1 - p_e(x) \biggr]  x^2 dx \, ,
</math>
  </td>
</tr>
</table>
</div>
defines the relevant integral over the envelope's pressure distribution.  According to our [[SSC/Structure/BiPolytropes/Analytic51#Profile|derivation of the properties of detailed force-balance <math>~(n_c, n_e) = (5, 1)</math> bipolytropes]] &#8212; see also the relevant derivations [[SSC/BipolytropeGeneralizationVersion2#Bipolytrope_Generalization|in our accompanying overview]] &#8212;  the pressure throughout the envelope is defined by the dimensionless function,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~P^* \equiv \frac{P_\mathrm{env}(\eta)}{P_0} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\theta_i^6 \phi^2(\eta) = \theta_i^6 \biggl( \frac{A}{\eta} \biggr)^2 \sin^2(\eta-B) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ 1-p_e(x) \equiv \frac{P_\mathrm{env}(x)}{P_{ie}} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{P_{ic}}{P_{ie}}  \biggr) \biggl( \frac{P_0}{P_{ic}}  \biggr) \frac{P_\mathrm{env}(x)}{P_0} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{P_{ic}}{P_{ie}}  \biggr) \biggl( \theta_i^{-6} \biggr) \theta_i^6 \biggl( \frac{A}{b_\eta x} \biggr)^2 \sin^2(b_\eta x-B) \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{P_{ic}}{P_{ie}}  \biggr) \biggl( \frac{A}{b_\eta} \biggr)^2 \frac{\sin^2(b_\eta x-B)}{x^2} \, ,
</math>
  </td>
</tr>
</table>
</div>
where, <math>~b_\eta</math> has been defined above in connection with our [[SSC/Structure/BiPolytropes/FreeEnergy51#The_Envelope|derivation of the envelope's mass profile]].  The desired integral over this pressure distribution therefore gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(1-q^3)s_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
3 \biggl( \frac{P_{ic}}{P_{ie}}  \biggr) \biggl( \frac{A}{b_\eta} \biggr)^2 \int_q^1 \sin^2(b_\eta x-B) dx
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3}{4} \biggl( \frac{P_{ic}}{P_{ie}}  \biggr) \biggl( \frac{A^2}{b_\eta^3} \biggr) \biggl[ 2b_\eta x -\sin[2(b_\eta x - B)] \biggr]_q^1 \, ,
</math>
  </td>
</tr>
</table>
</div>
where, as before, we have dropped the integration constant because it cancels upon insertion of the specified integration limits.  Therefore, 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{env}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{2}{3({\gamma_e}-1)} \biggl( \frac{\chi}{\chi_\mathrm{eq}} \biggr)^{3-3\gamma_e} 
\biggl[ \frac{2\pi P_{ic} \chi^3}{P_\mathrm{norm}} \biggr]_\mathrm{eq}  
\frac{3}{4} \biggl( \frac{A^2}{b_\eta^3} \biggr) \biggl[ 2b_\eta x -\sin[2(b_\eta x - B)] \biggr]_q^1  \, .
</math>
  </td>
</tr>
</table>
</div>
Now, drawing from our above derivation steps and discussion, we know that,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~b_\eta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~3^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr) \frac{\theta_i^2 \xi_i}{q} \, ,</math>
  </td>
</tr>
</table>
</div>
and
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{2\pi P_{ic} \chi_\mathrm{eq}^3}{P_\mathrm{norm}} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{3^6}{2\pi} \biggr)^{1/2} \biggl(1 + a_\xi q^2 \biggr)^{-3} a_\xi^{3/2} = \biggl( \frac{3^3}{2\pi} \biggr)^{1/2} \biggl( \frac{\theta_i^2\xi_i }{q}\biggr)^3 \, .
</math>
  </td>
</tr>
</table>
</div>

Finally, then, we can write,

<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{env}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{2}{3({\gamma_e}-1)} \biggl( \frac{\chi}{\chi_\mathrm{eq}} \biggr)^{3-3\gamma_e} \biggl\{
\frac{3}{4} A^2\biggl[ \biggl( \frac{3^3}{2\pi} \biggr)^{1/2} \biggl( \frac{\theta_i^2\xi_i }{q}\biggr)^3 \biggr] 
\biggl[ 3^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr) \frac{\theta_i^2 \xi_i}{q} \biggr]^{-3} \biggl[ 2b_\eta x -\sin[2(b_\eta x - B)] \biggr]_q^1  \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{2}{3({\gamma_e}-1)} \biggl( \frac{\chi}{\chi_\mathrm{eq}} \biggr)^{3-3\gamma_e} \biggl\{
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} A^2  \biggl( \frac{3^2}{2^5\pi} \biggr)^{1/2}  \biggl[ 2b_\eta x -\sin[2(b_\eta x - B)] \biggr]_q^1  \biggr\} \ .
</math>
  </td>
</tr>
</table>
</div>
The term inside the curly brackets precisely matches the analytic expression for the dimensionless thermal energy of the envelope, <math>~S^*_\mathrm{env}</math>, that has been derived elsewhere in conjunction with our [[SSC/Structure/BiPolytropes/Analytic51#Expression_for_Free_Energy|discussion of the detailed force-balanced structure of this bipolytrope]].

==Virial Theorem==
As has been shown in our [[SSC/BipolytropeGeneralizationVersion2#More_Utilitarian_Form|accompanying overview]], the condition for equilibrium based on a free-energy analysis &#8212; that is, the virial theorem &#8212; is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\mathcal{A}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mathcal{B}_\mathrm{core} \chi_\mathrm{eq}^{4-3\gamma_c} + \mathcal{B}_\mathrm{env} \chi_\mathrm{eq}^{4-3\gamma_e} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4\pi}{3} \biggl[ \frac{P_i R_\mathrm{edge}^4}{GM_\mathrm{tot}^2} \biggr]_\mathrm{eq} [
q^3 s_\mathrm{core} + (1-q^3) s_\mathrm{env} ] \, .
</math>
  </td>
</tr>

</table>
</div>

For <math>~(n_c, n_e) = (0, 0) </math> bipolytropes, the relevant coefficient functions are,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{A}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{5} \biggl(\frac{\nu^2}{q}\biggr) f \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~q^3 s_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
q^3 \biggl(\frac{P_0}{P_{ic}} \biggr) \biggl[ 1 - \frac{3}{5}q^2 b_\xi\biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~(1-q^3) s_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(1-q^3) +  \biggl(\frac{P_0}{P_{ie}} \biggr) \frac{2}{5} q^5 \mathfrak{F} b_\xi \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~f</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
 <td align="left">
<math>
1+
\frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr)  \biggl(\frac{1}{q^2}-1 \biggr) +\biggl( \frac{\rho_e}{\rho_c}  
\biggr)^2 \biggl[ \frac{1}{q^5}-1 + \frac{5}{2} \biggl( 1-\frac{1}{q^2}\biggr)\biggr]  \, ,
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
<math>~\frac{P_{ic}}{P_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1- p_c(q) = 1 - b_\xi q^2 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~b_\xi</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{3}{2^3 \pi} \biggr) 
\frac{G M_\mathrm{tot}^2 }{P_0 R_\mathrm{edge}^4} \biggl( \frac{\nu}{q^3}\biggr)^2\, .</math>
  </td>
</tr>

</table>
</div>

Plugging these expressions into the equilibrium condition shown above, and setting the interface pressures equal to one another, gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{5} \biggl(\frac{\nu^2}{q}\biggr) f</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4\pi}{3} \biggl[ \frac{P_i R_\mathrm{edge}^4}{GM_\mathrm{tot}^2} \biggr]_\mathrm{eq} \biggl\{
q^3 \biggl(\frac{P_0}{P_{i}} \biggr) \biggl[ 1 - \frac{3}{5}q^2 b_\xi\biggr]  
+ (1-q^3) +  \biggl(\frac{P_0}{P_{i}} \biggr) \frac{2}{5} q^5 \mathfrak{F} b_\xi \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4\pi}{3} \biggl[ \frac{P_0 R_\mathrm{edge}^4}{GM_\mathrm{tot}^2} \biggr]_\mathrm{eq} \biggl\{
q^3 \biggl[ 1 - \frac{3}{5}q^2 b_\xi\biggr]  
+ (1-q^3)( 1- b_\xi q^2)  +  \frac{2}{5} q^5 \mathfrak{F} b_\xi \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4\pi}{3} \biggl[ \frac{P_0 R_\mathrm{edge}^4}{GM_\mathrm{tot}^2} \biggr]_\mathrm{eq} \biggl\{
1 - b_\xi \biggl[ \frac{3}{5}q^5  + q^2(1-q^3) - \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{4\pi}{3} \biggl[ \frac{P_0 R_\mathrm{edge}^4}{GM_\mathrm{tot}^2} \biggr]_\mathrm{eq}  \biggl[
\frac{1}{b_\xi} - q^2 + \frac{2}{5} q^5( 1+\mathfrak{F} ) \biggr] b_\xi
</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
<math>\Rightarrow~~~~\biggl( \frac{2^3 \pi}{3} \biggr) 
\frac{P_0 R_\mathrm{edge}^4}{G M_\mathrm{tot}^2 } \biggl( \frac{q^3}{\nu}\biggr)^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
q^2 + \frac{2}{5} q^5( f - 1-\mathfrak{F} )
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~
\frac{P_0 R_\mathrm{edge}^4}{G M_\mathrm{tot}^2 } 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{3}{2^3 \pi} \biggr) \biggl( \frac{\nu}{q^3}\biggr)^2 \biggl\{ q^2 + \biggl( \frac{\rho_e}{\rho_c} \biggr) \biggl[
2q^2(1-q) + \biggl( \frac{\rho_e}{\rho_c} \biggr) (1-3q^2 + 2q^3) \biggr] \biggr\} \, .</math>
  </td>
</tr>
</table>
</div>

This exactly matches the equilibrium relation that was derived from our [[SSC/Structure/BiPolytropes/Analytic00#CentralPressure|detailed force-balance analysis of]] <math>(n_c, n_e) = (0, 0)</math> bipolytropes.

=Related Discussions=
* [[SSC/Structure/BiPolytropes/FreeEnergy00|Free-energy determination of equilibrium configurations for BiPolytropes]] with <math>~n_c = 0</math> and <math>n_e=0</math>.
* [[SSC/Structure/BiPolytropes/FreeEnergy51#Free_Energy_of_BiPolytrope_with|Free-energy determination of equilibrium configurations for BiPolytropes]] with <math>n_c = 5</math> and <math>~n_e=1</math>.
* [[SSC/Structure/BiPolytropes/Analytic00#BiPolytrope_with_nc_.3D_0_and_ne_.3D_0|Analytic solution of Detailed-Force-Balance BiPolytrope]] with <math>n_c = 0</math> and <math>~n_e=0</math>.
* [[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1|Analytic solution of Detailed-Force-Balance BiPolytrope]] with <math>n_c = 5</math> and <math>~n_e=1</math>.
* [[SSC/BipolytropeGeneralization|Old ''Bipolytrope Generalization'' derivations]].

=See Also=

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