Editing SSC/BipolytropeGeneralization (section)

===Extrema===
Extrema in the free energy occur when,
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<math>~\mathcal{A}</math>
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<math>~=</math>
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<math>~\mathcal{B} \chi_\mathrm{eq}^{4-3\gamma_c} + \mathcal{C} \chi_\mathrm{eq}^{4-3\gamma_e} \, .</math>
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Also, as stated above, because <math>~P_{ie} = P_{ic}</math> in equilibrium, the ratio of coefficients,
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<math>~\frac{\mathcal{C}}{\mathcal{B}}</math>
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<math>~=</math>
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<math>~\chi_\mathrm{eq}^{3(\gamma_e - \gamma_c)}\biggl[ \frac{(1-q^3) s_\mathrm{env}}{q^3 s_\mathrm{core}} \biggr] \, .</math>
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When put together, these two relations imply,
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<math>~\mathcal{A}</math>
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<math>~=</math>
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<math>~\mathcal{B} \chi_\mathrm{eq}^{4-3\gamma_c} + \chi_\mathrm{eq}^{4-3\gamma_c} \mathcal{B} 
\biggl[ \frac{(1-q^3) s_\mathrm{env}}{q^3 s_\mathrm{core}} \biggr] </math>
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&nbsp;
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<math>~=</math>
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<math>~\mathcal{B} \chi_\mathrm{eq}^{4-3\gamma_c} \biggl[ 1+ \frac{(1-q^3) s_\mathrm{env}}{q^3 s_\mathrm{core}} \biggr] \, .</math>
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But the definition of <math>~\mathcal{B}</math> gives,
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<math>~\mathcal{B} \chi_\mathrm{eq}^{4-3\gamma_c}</math>
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<math>~=</math>
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<math>
\biggl( \frac{4\pi }{3} \biggr) q^3 s_\mathrm{core}   
\biggl[ \frac{1}{\lambda_{ic}} \biggr]_\mathrm{eq} \, .
</math>
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Hence, extrema occur when,
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<math>~\mathcal{A}</math>
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<math>~=</math>
  </td>
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<math>~\biggl( \frac{4\pi }{3} \biggr) q^3 s_\mathrm{core}   
\biggl[ \frac{1}{\lambda_{ic}} \biggr]_\mathrm{eq}  \biggl[ 1+ \frac{(1-q^3) s_\mathrm{env}}{q^3 s_\mathrm{core}} \biggr] </math>
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<math>\Rightarrow ~~~~ \biggl( \frac{3}{2^2 \cdot 5\pi } \biggr) \frac{\nu^2}{q} \cdot f</math>
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<math>~=</math>
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<math>  
\biggl[ \frac{1}{\lambda_{ic}} \biggr]_\mathrm{eq}   \biggl[ q^3 s_\mathrm{core}  +  (1-q^3) s_\mathrm{env} \biggr] </math>
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&nbsp;
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<math>~=</math>
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<math>  
\frac{q^3}{[\lambda_i]_\mathrm{eq}} + \biggl( \frac{3}{2^2\cdot 5\pi} \biggr) \frac{\nu^2}{q} 
+ \frac{(1-q^3)}{[\lambda_i]_\mathrm{eq}} + \biggl( \frac{3}{2^2\cdot 5\pi} \biggr) \frac{\nu^2}{q} \cdot \mathfrak{F}
</math>
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<math>\Rightarrow ~~~~\biggl[ \frac{1}{\lambda_{ic}} \biggr]_\mathrm{eq}  </math>
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<math>~=</math>
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<math>  
\biggl( \frac{3}{2^2 \cdot 5\pi } \biggr) \frac{\nu^2}{q} \cdot (f - 1 - \mathfrak{F})
</math>
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&nbsp;
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<math>~=</math>
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<math>  
\biggl( \frac{2\pi}{3 } \biggr) \sigma^{2} q^2 (g^2 - 1 ) \, .
</math>
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In what follows, keep in mind that,
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<math>~\chi_\mathrm{eq}^{4-3\gamma_c}</math>
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<math>~=</math>
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<math>
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^{4-3\gamma_c} 
= R_\mathrm{eq}^{4-3\gamma_c} \biggl( \frac{K_c}{G} \biggr) M_\mathrm{tot}^{\gamma_c-2} \, ;
</math>
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<math>~K_c \rho_c^{\gamma_c}</math>
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<math>~=</math>
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<math>
K_c \biggl( \frac{\rho_c}{\bar\rho} \biggr)^{\gamma_c} \biggl[ \frac{3M_\mathrm{tot}}{4\pi R^3} \biggr]^{\gamma_c}
= K_c \sigma^{\gamma_c} M_\mathrm{tot}^{\gamma_c} R^{-3\gamma_c} \, ;
</math>
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<math>~\Pi</math>
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<math>~=</math>
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<math>
\frac{3}{2^3 \pi} \biggl(  \frac{GM_\mathrm{tot}^2}{R^4} \biggr) \frac{\nu^2}{q^6} 
= \frac{2\pi}{3} \biggl(  \frac{GM_\mathrm{tot}^2}{R^4} \biggr) \sigma^2 \, .
</math>
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OLD DERIVATION
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<math>P_{i} = K_c \rho_c^{\gamma_c}</math>

<math>\Rightarrow ~~~~ K_c  = P_{i} \sigma^{-\gamma_c} M_\mathrm{tot}^{-\gamma_c} R^{+3\gamma_c}  </math></div>
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NEW DERIVATION
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<math>P_0 = K_c \rho_c^{\gamma_c} </math>

<math>\Rightarrow ~~~~ K_c = P_0 \sigma^{-\gamma_c} M_\mathrm{tot}^{-\gamma_c} R^{+3\gamma_c}  </math>
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&hellip; hence, as derived in the above table &hellip;
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<math>~\frac{1}{\lambda_i} \biggr|_\mathrm{eq}</math>
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<math>~=</math>
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<math>
\sigma^{\gamma_c} \chi_\mathrm{eq}^{4 - 3\gamma_c} 
</math>
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<math>~\frac{1}{\lambda_i} \biggr|_\mathrm{eq}</math>
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<math>~=</math>
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<math>
\sigma^{\gamma_c} \chi_\mathrm{eq}^{4 - 3\gamma_c} -  \biggl( \frac{2\pi}{3} \biggr) q^2 \sigma^2 
</math>
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&hellip; which, when combined with the condition that identifies extrema, gives &hellip;
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<math>~\chi_\mathrm{eq}^{4 - 3\gamma_c}</math>
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<math>~=</math>
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<math>
\biggl( \frac{2\pi}{3 } \biggr) \sigma^{2-\gamma_c} q^2 (g^2 - 1 )
</math>
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<math>~\Rightarrow ~~~~ R_\mathrm{eq}^{4-3\gamma_c} \biggl( \frac{K_c}{G} \biggr) M_\mathrm{tot}^{\gamma_c-2} </math>
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<math>~=</math>
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<math>
\biggl( \frac{2\pi}{3 } \biggr) \sigma^{2-\gamma_c} q^2 (g^2 - 1 )
</math>
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<math>\Rightarrow ~~~~ \frac{ R_\mathrm{eq}^{4}  P_i }{GM_\mathrm{tot}^{2} } </math>
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<math>~=</math>
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<math>
\biggl( \frac{2\pi}{3 } \biggr) \sigma^{2} q^2 (g^2-1)
</math>
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</table>
  </td>
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<math>~\sigma^{\gamma_c}\chi_\mathrm{eq}^{4 - 3\gamma_c} -  \biggl( \frac{2\pi}{3} \biggr) q^2 \sigma^2</math>
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<math>~=</math>
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<math>
\biggl( \frac{2\pi}{3 } \biggr) \sigma^2 q^2 (g^2 - 1 )
</math>
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<math>\Rightarrow ~~~~ \chi_\mathrm{eq}^{4 - 3\gamma_c} </math>
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<math>~=</math>
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<math>
\biggl( \frac{2\pi}{3 } \biggr) \sigma^{2-\gamma_c} q^2 g^2
</math>
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<math>\Rightarrow ~~~~ R_\mathrm{eq}^{4-3\gamma_c} \biggl( \frac{K_c}{G} \biggr) M_\mathrm{tot}^{\gamma_c-2} </math>
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<math>~=</math>
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<math>
\biggl( \frac{2\pi}{3 } \biggr) \sigma^{2-\gamma_c} q^2 g^2
</math>
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<math>\Rightarrow ~~~~ \frac{ R_\mathrm{eq}^{4}  P_0 }{GM_\mathrm{tot}^{2} } </math>
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<math>~=</math>
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<math>
\biggl( \frac{2\pi}{3 } \biggr) \sigma^{2} q^2 g^2
</math>
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These are consistent results because they result in the detailed force-balance relation, <math>P_0 - P_i = q^2 \Pi_\mathrm{eq} \, .</math>
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