Editing SSC/BipolytropeGeneralization/Pt3

=Bipolytrope Generalization (Pt 3)=
<table border="1" align="center" width="100%" colspan="8">
<tr>
  <td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/BipolytropeGeneralization|Part I:&nbsp; Bipolytrope Generalization]]
&nbsp;
  </td>
  <td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/BipolytropeGeneralization/Pt2|Part II:&nbsp; Derivations]]
&nbsp;
  </td>
  <td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/BipolytropeGeneralization/Pt3|Part III:&nbsp; Examples]]
&nbsp;
  </td>
  <td align="center" bgcolor="lightblue"><br />[[SSC/BipolytropeGeneralization/Pt4|Part IV:&nbsp; Best of the Best]]
&nbsp;
  </td>
</tr>
</table>

==Examples==

===(0, 0) Bipolytropes===
====Review====
In an [[SSC/Structure/BiPolytropes/Analytic00#BiPolytrope_with_nc_.3D_0_and_ne_.3D_0|accompanying discussion]] we have derived analytic expressions describing the equilibrium structure and the stability of bipolytropes in which both the core and the envelope have uniform densities, that is, bipolytropes with <math>~(n_c, n_e) = (0, 0)</math>.  From this work, we find that integrals over the mass and pressure distributions give:
<div align="center">
<table border="0">
<tr>
  <td align="right">
<math>~ \frac{W}{R_\mathrm{eq}^3 P_i} = - \frac{A}{R_\mathrm{eq}^4 P_i} </math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
&nbsp;<math>- ~ \frac{3}{5} \biggl[ \frac{GM_\mathrm{tot}^2}{R^4P_i} \biggr] \biggl( \frac{\nu^2}{q} \biggr) f </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
&nbsp;<math>- ~4\pi q^3 \Lambda f \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{S_\mathrm{core}}{R_\mathrm{eq}^3 P_i} = B_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
~2\pi q^3 (1 + \Lambda) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{S_\mathrm{env}}{R_\mathrm{eq}^3 P_i} = B_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
2\pi \biggl[ (1-q^3) + \frac{5}{2} \Lambda \biggl( \frac{\rho_e}{\rho_0}\biggr) (-2 + 3q - q^3) +
\frac{3}{2q^2} \Lambda \biggl( \frac{\rho_e}{\rho_0}\biggr)^2 (-1 +5q^2 - 5q^3 + q^5) \biggr] \, ,</math>
  </td>
</tr>

</table>
</div>
where,
<div align="center">
<table border="0">

<tr>
  <td align="right">
<math>~\Lambda</math> 
  </td>
  <td align="center">
<math>~\equiv~</math>
  </td>
  <td align="left">
<math>
 \frac{3}{2^2 \cdot 5\pi} \biggl( \frac{GM_\mathrm{tot}^2}{R_\mathrm{eq}^4 P_i} \biggr) \frac{\nu^2}{q^4} \, ,</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
<math>~g^2(q,\rho_e/\rho_c)</math> 
  </td>
  <td align="center">
<math>~\equiv~</math>
  </td>
  <td align="left">
<math>1  + 
\biggl(\frac{\rho_e}{\rho_0}\biggr)  \biggl[ 2 \biggl(1 - \frac{\rho_e}{\rho_0} \biggr) \biggl( 1-q \biggr) + 
\frac{\rho_e}{\rho_0} \biggl(\frac{1}{q^2} - 1\biggr) \biggr] </math>
  </td>
</tr>

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

====Renormalize====
Let's renormalize these energy terms in order to more readily relate them to the [[#Setup|generalized expressions derived above]].
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~R^3 P_i</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~R^3 K_c \biggl(\frac{\rho_{ic}}{\bar\rho} \biggr)^{\gamma_c} \biggl[ \biggl( \frac{3}{4\pi}\biggr) \frac{M_\mathrm{tot}}{R^3} \biggr]^{\gamma_c}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{R}{R_\mathrm{norm}} \biggr]^{3-3\gamma_c} \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\rho_{ic}}{\bar\rho} \biggr]^{\gamma_c} 
K_c M_\mathrm{tot}^{\gamma_c} R_\mathrm{norm}^{3-3\gamma_c}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{R}{R_\mathrm{norm}} \biggr]^{3-3\gamma_c} \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\rho_{ic}}{\bar\rho} \biggr]^{\gamma_c} 
\biggl\{ K_c^{3\gamma_c -4} M_\mathrm{tot}^{\gamma_c(3\gamma_c -4)} \biggl[ \biggl( \frac{K_c}{G} \biggr) M_\mathrm{tot}^{\gamma_c-2} \biggr]^{3-3\gamma_c} \biggr\}^{1/(3\gamma_c -4)}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{R}{R_\mathrm{norm}} \biggr]^{3-3\gamma_c} \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\rho_{ic}}{\bar\rho} \biggr]^{\gamma_c} 
\biggl\{ \frac{G^{3\gamma_c-3} M_\mathrm{tot}^{5\gamma_c-6} }{K_c} \biggr\}^{1/(3\gamma_c -4)}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{R}{R_\mathrm{norm}} \biggr]^{3-3\gamma_c} \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\rho_{ic}}{\bar\rho} \biggr]^{\gamma_c} 
E_\mathrm{norm} \, .</math>
  </td>
</tr>
</table>
</div>

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

<tr>
  <td align="right">
<math>~\biggl[ \frac{GM_\mathrm{tot}^2}{R} \biggr]^{3\gamma_c -4}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\biggl( \frac{R}{R_\mathrm{norm}} \biggr)^{-(3\gamma_c-4)} G^{3\gamma_c -4} M_\mathrm{tot}^{6\gamma_c -8}
\biggl( \frac{G}{K_c} \biggr) M_\mathrm{tot}^{2-\gamma_c}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\biggl( \frac{R}{R_\mathrm{norm}} \biggr)^{-(3\gamma_c-4)} \biggl[ \frac{ G^{3\gamma_c -3} M_\mathrm{tot}^{5\gamma_c -6} }{K_c} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\biggl( \frac{R}{R_\mathrm{norm}} \biggr)^{-(3\gamma_c-4)} E_\mathrm{norm}^{3\gamma_c-4} \, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~~\frac{GM_\mathrm{tot}^2}{R^4 P_i} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\biggl( \frac{R}{R_\mathrm{norm}} \biggr)^{-1} 
\biggl[ \frac{R}{R_\mathrm{norm}} \biggr]^{3\gamma_c - 3} \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\rho_{ic}}{\bar\rho} \biggr]^{-\gamma_c} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{R}{R_\mathrm{norm}} \biggr)^{3\gamma_c - 4} \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\rho_{ic}}{\bar\rho} \biggr]^{-\gamma_c} \, .
</math>
  </td>
</tr>
</table>
</div>

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

<tr>
  <td align="right">
<math>~\Lambda</math> 
  </td>
  <td align="center">
<math>~\equiv~</math>
  </td>
  <td align="left">
<math>
 \frac{3}{2^2 \cdot 5\pi} \frac{\nu^2}{q^4} \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\rho_{ic}}{\bar\rho} \biggr]^{-\gamma_c} 
\biggl( \frac{R}{R_\mathrm{norm}} \biggr)^{3\gamma_c - 4} 
\, .</math>
  </td>
</tr>
</table>
</div>

Given that <math>~\rho_{ic}/\bar\rho = \nu/q^3</math> for the <math>~(n_c, n_e) = (0, 0)</math> bipolytrope,  we can finally write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{R^3 P_i}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\nu}{q^3} \biggr]^{\gamma_c} \biggl( \frac{R}{R_\mathrm{norm}} \biggr)^{3-3\gamma_c} 
\, ,</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Lambda</math> 
  </td>
  <td align="center">
<math>~\equiv~</math>
  </td>
  <td align="left">
<math>
\frac{3}{2^2 \cdot 5\pi} \frac{\nu^2}{q^4} \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\nu}{q^3} \biggr]^{-\gamma_c} 
\biggl( \frac{R}{R_\mathrm{norm}} \biggr)^{3\gamma_c - 4} =
\frac{1}{5} \frac{\nu}{q} \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\nu}{q^3} \biggr]^{1-\gamma_c} 
\biggl( \frac{R}{R_\mathrm{norm}} \biggr)^{3\gamma_c - 4} 
\, .</math>
  </td>
</tr>
</table>
</div>
Hence the renormalized gravitational potential energy becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\frac{W_\mathrm{grav}}{E_\mathrm{norm}}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
- \biggl( \frac{3}{5} \biggr) \frac{\nu^2}{q} \biggl( \frac{R}{R_\mathrm{norm}} \biggr)^{-1} \cdot f
\, ;</math>
  </td>
</tr>
</table>
</div>
and the two, renormalized contributions to the thermal energy become,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{U_\mathrm{core}}{ E_\mathrm{norm} } = \frac{2}{3(\gamma_c-1)} \biggl[ \frac{S_\mathrm{core}}{ E_\mathrm{norm} } \biggr]</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\frac{4\pi q^3 (1 + \Lambda) }{3(\gamma_c-1)} 
\biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\nu}{q^3} \biggr]^{\gamma_c} \biggl( \frac{R}{R_\mathrm{norm}} \biggr)^{3-3\gamma_c}
\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{U_\mathrm{env}}{ E_\mathrm{norm} } = \frac{2}{3(\gamma_e-1)} \biggl[ \frac{S_\mathrm{env}}{ E_\mathrm{norm} } \biggr]</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\frac{4\pi}{3(\gamma_e-1)} \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{(1-\nu)}{(1-q^3)} \biggr]^{\gamma_e} \biggl( \frac{K_e}{K_c} \biggr) 
\biggl[ \frac{K_c^3}{G^3 M_\mathrm{tot}^2} \biggr]^{(3\gamma_c-3\gamma_e)/(3\gamma_c-4)} 
\biggl( \frac{R}{R_\mathrm{norm}} \biggr)^{3-3\gamma_e}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
\times \biggl[ (1-q^3) + \frac{5}{2} \Lambda \biggl( \frac{\rho_e}{\rho_0}\biggr) (-2 + 3q - q^3) +
\frac{3}{2q^2} \Lambda \biggl( \frac{\rho_e}{\rho_0}\biggr)^2 (-1 +5q^2 - 5q^3 + q^5) \biggr] 
\, ,</math>
  </td>
</tr>

</table>
</div>

Finally, then, we can state that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{f}_{WM}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{\nu^2}{q} \cdot f \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~s_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>
1 + \Lambda \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~(1-q^3) s_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>
(1-q^3) + \Lambda\biggl[ \frac{5}{2} \biggl( \frac{\rho_e}{\rho_0}\biggr) (-2 + 3q - q^3) +
\frac{3}{2q^2} \biggl( \frac{\rho_e}{\rho_0}\biggr)^2 (-1 +5q^2 - 5q^3 + q^5) \biggr] 
\, .</math>
  </td>
</tr>
</table>
</div>

====Virial Equilibrium and Stability Evaluation====
With these expressions in hand, we can deduce the equilibrium radius and relativity stability of <math>~(n_c, n_e) = (0, 0)</math> bipolytropes using the generalized expressions provided above.  For example, from the statement of virial equilibrium <math>~(2S_\mathrm{tot} = - W )</math> we obtain,
<div align="center">
<table border="0">
<tr>
  <td align="right">
<math>~q^3 (1 + \Lambda) + (1-q^3) + \frac{5}{2} \Lambda \biggl( \frac{\rho_e}{\rho_0}\biggr) (-2 + 3q - q^3) +
\frac{3}{2q^2} \Lambda \biggl( \frac{\rho_e}{\rho_0}\biggr)^2 (-1 +5q^2 - 5q^3 + q^5)  </math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~q^3 \Lambda \biggl[ 1 + \frac{5}{2q^2} \biggl( \frac{\rho_e}{\rho_c} \biggr) (1-q^2)
+ \frac{1}{2q^5} \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 (2 - 5q^3 + 3q^5) \biggr] </math>
  </td>
</tr>
</table>
</div>

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

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ \frac{1}{\Lambda}</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>\frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) (q-q^3)
+ \frac{1}{2q^2} \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 (2 - 5q^3 + 3q^5) 
- \biggl[ \frac{5}{2} \biggl( \frac{\rho_e}{\rho_0}\biggr) (-2 + 3q - q^3) +
\frac{3}{2q^2} \biggl( \frac{\rho_e}{\rho_0}\biggr)^2 (-1 +5q^2 - 5q^3 + q^5) \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>\frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) (q-q^3 + 2 -3q +q^3)
+ \frac{1}{2q^2} \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 (2 - 5q^3 + 3q^5 +3 - 15q^2+15q^3 -3q^5)  
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>\frac{5}{2}(g^2-1) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ \biggl[ \frac{P_i}{GM_\mathrm{tot}^2} \biggr] R_\mathrm{eq}^4</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>\biggl( \frac{3}{2^3 \pi } \biggr) \frac{\nu^2}{q^4} (g^2-1) \, .
</math>
  </td>
</tr>
</table>
</div>

Or, given the above renormalization, this expression can be written as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\biggl( \frac{R}{R_\mathrm{norm}} \biggr)^{4-3\gamma_c } \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\rho_{ic}}{\bar\rho} \biggr]^{\gamma_c} 
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>\biggl( \frac{3}{2^3 \pi } \biggr) \frac{\nu^2}{q^4} (g^2-1) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~~ \frac{R}{R_\mathrm{norm}} 
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\biggl\{ \biggl( \frac{3}{2^3 \pi } \biggr) \frac{\nu^2}{q^4} (g^2-1) 
\biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\rho_{ic}}{\bar\rho} \biggr]^{\gamma_c} \biggr\}^{1/(4-3\gamma_c)} \, .
</math>
  </td>
</tr>
</table>
</div>


And the condition for dynamical stability is,
<div align="center">
<table border="0">

<tr>
  <td align="right">
<math>-\frac{W}{2}\biggl( \gamma_e - \frac{4}{3}\biggr) - 
(\gamma_e-\gamma_c) S_\mathrm{core} </math>
  </td>
  <td align="center">
&nbsp; <math>~>~</math>&nbsp;
  </td>
  <td align="left">
<math>~0 \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ 2\pi q^3 \Lambda \biggl[ \biggl( \gamma_e - \frac{4}{3}\biggr) f - 
(\gamma_e-\gamma_c) \biggl( 1 + \frac{1}{\Lambda}\biggr) \biggr] </math>
  </td>
  <td align="center">
&nbsp; <math>~>~</math>&nbsp;
  </td>
  <td align="left">
<math>~0 \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl( \gamma_e - \frac{4}{3} \biggr)f - (\gamma_e - \gamma_c) \biggl[1 + \frac{5}{2}(g^2-1) \biggr]</math>
  </td>
  <td align="center">
&nbsp; <math>~>~</math>&nbsp;
  </td>
  <td align="left">
<math>~0 \, .</math>
  </td>
</tr>
</table>
</div>

===(5, 1) Bipolytropes===
In another [[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1|accompanying discussion]] we have derived analytic expressions describing the equilibrium structure of bipolytropes with <math>(n_c, n_e) = (5, 1)</math>.  Can we perform a similar stability analysis of these configurations?
Work in progress!

=Related Discussions=
* [[SSC/BipolytropeGeneralizationVersion2|Newer, Version2 of ''Bipolytrope Generalization'' derivations]].
* [[SSC/Structure/BiPolytropes/Analytic00#BiPolytrope_with_nc_.3D_0_and_ne_.3D_0|Analytic solution]] 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]] with <math>n_c = 5</math> and <math>n_e=1</math>.

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