Editing SSC/BipolytropeGeneralization (section)

===Shift to Central Pressure Normalization===
Let's rework the definition of <math>~\lambda_i</math> in two ways:  (1) Normalize <math>~R_\mathrm{eq}</math> to <math>~R_\mathrm{norm}</math> and normalize the pressure to <math>~P_\mathrm{norm}</math>; (2) shift the referenced pressure from the pressure at the interface <math>~(P_i)</math> to the central pressure <math>~(P_0)</math>, because it is <math>~P_0</math> that is directly related to <math>~K_c</math> and <math>~\rho_c</math>; specifically, <math>P_0 = K_c \rho_c^{\gamma_c}</math>.  Appreciating that, in equilibrium,
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<math>~P_i</math>
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<math>~=</math>
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<math>~P_0 - q^2 \Pi_\mathrm{eq}
= K_c \rho_c^{\gamma_c} - \frac{3}{2^3 \pi} 
\biggl( \frac{GM_\mathrm{tot}^2}{R_\mathrm{eq}^4} \biggr) \biggl( \frac{\nu^2}{q^6} \biggr) q^2 \, ,</math>
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the left-hand-side of the last expression, above, can be rewritten as,
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<math>~\frac{1}{\lambda_i} \biggr|_\mathrm{eq}</math>
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<math>~\equiv</math>
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<math>~\frac{R_\mathrm{eq}^4 P_i}{GM_\mathrm{tot}^2} </math>
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&nbsp;
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<math>~=</math>
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<math>~\frac{R_\mathrm{eq}^4}{GM_\mathrm{tot}^2} 
\biggl[ P_0 - \frac{3}{2^3 \pi} 
\biggl( \frac{GM_\mathrm{tot}^2}{R_\mathrm{eq}^4} \biggr) \biggl( \frac{\nu^2}{q^6} \biggr) q^2\biggr] </math>
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&nbsp;
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<math>~=</math>
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<math>~\frac{R_\mathrm{eq}^4 P_0}{GM_\mathrm{tot}^2} - \frac{2\pi}{3} \biggl[ \biggl( \frac{3}{4\pi} \biggr)
\frac{\nu}{q^3} \biggr]^2 q^2 \, .</math>
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Hence, the virial equilibrium condition gives,
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<math>
\frac{R_\mathrm{eq}^4 P_0}{GM_\mathrm{tot}^2} - \frac{2\pi}{3} \biggl[ \biggl( \frac{3}{4\pi} \biggr)
\frac{\nu}{q^3} \biggr]^2 q^2
</math>
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<math>~=</math>
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<math>
\frac{2\pi}{3} \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\nu}{q^3} \biggr]^2 q^2 (g^2-1)  
</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>
\frac{2\pi}{3} \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\nu}{q^3} \biggr]^{2} q^2 g^2  \, .
</math>
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This result precisely matches [[SSC/Structure/BiPolytropes/Analytic00#CentralPressure|the result obtained via the detailed force-balanced conditions]] imposed through hydrostatic equilibrium.

Adopting our new variable normalizations and realizing, in particular, that,
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<math>~R_\mathrm{norm}^4 P_\mathrm{norm}</math>
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<math>~=</math>
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<math>~GM_\mathrm{tot}^2 \, ,</math>
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the expression alternatively can be rewritten as,
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<math>~\frac{1}{\lambda_i} \biggr|_\mathrm{eq}</math>
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<math>~\equiv</math>
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<math>~\frac{R_\mathrm{eq}^4 P_i}{GM_\mathrm{tot}^2} 
= \chi_\mathrm{eq}^4 \biggl( \frac{P_i}{P_\mathrm{norm}} \biggr) </math>
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&nbsp;
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<math>~=</math>
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<math>
\chi_\mathrm{eq}^4 \biggl\{  \frac{K_c \rho_c^{\gamma_c}}{P_\mathrm{norm}} - \frac{2\pi}{3} 
\biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\nu}{q^3}\biggr]^2 q^2 \biggl( \frac{GM_\mathrm{tot}^2}{R_\mathrm{eq}^4P_\mathrm{norm}} \biggr) \biggr\}
</math>
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&nbsp;
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<math>~=</math>
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<math>
\chi_\mathrm{eq}^4 \biggl\{  \frac{K_c }{P_\mathrm{norm}} \biggl[ \frac{\rho_c}{\bar\rho} \biggl( \frac{3M_\mathrm{tot}}{4\pi R_\mathrm{norm}^3} \biggr) \chi_\mathrm{eq}^{-3} \biggr]^{\gamma_c} 
- \frac{2\pi}{3} \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\nu}{q^3}\biggr]^2 q^2 \chi_\mathrm{eq}^{-4} \biggr\}
</math>
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&nbsp;
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<math>~=</math>
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<math>
\chi_\mathrm{eq}^{4-3\gamma_c} \biggl[\biggl( \frac{3}{4\pi} \biggr) \frac{\nu}{q^3} \biggr]^{\gamma_c} \frac{K_c }{P_\mathrm{norm}} \biggl( \frac{M_\mathrm{tot}^{\gamma_c}}{R_\mathrm{norm}^{3\gamma_c}} \biggr) 
- \frac{2\pi}{3} \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\nu}{q^3}\biggr]^2 q^2
</math>
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&nbsp;
  </td>
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<math>~=</math>
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<math>
\chi_\mathrm{eq}^{4-3\gamma_c} \biggl[\biggl( \frac{3}{4\pi} \biggr) \frac{\nu}{q^3} \biggr]^{\gamma_c} 
- \frac{2\pi}{3} \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\nu}{q^3}\biggr]^2 q^2 \, .
</math>
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Normalized in this manner, the virial equilibrium (as well as the hydrostatic balance) condition gives,
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<math>
\chi_\mathrm{eq}^{4-3\gamma_c} \biggl[\biggl( \frac{3}{4\pi} \biggr) \frac{\nu}{q^3} \biggr]^{\gamma_c} 
- \frac{2\pi}{3} \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\nu}{q^3}\biggr]^2 q^2
</math>
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<math>~=</math>
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<math>
\frac{2\pi}{3} \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\nu}{q^3} \biggr]^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>
\frac{2\pi}{3} \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{\nu}{q^3} \biggr]^{2-\gamma_c} q^2 g^2  \, .
</math>
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