Editing SSC/Structure/BiPolytropes/Analytic51/Pt4 (section)

===Equilibrium Condition===
====Global====

Recognizing from the above [[SSC/Structure/BiPolytropes/Analytic51#Parameter_Values|Table of Parameters]] that,
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<math>~A</math>
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<math>~=~</math>
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<math>~\frac{\eta_i}{\sin(\eta_i - B)} \, ,</math>
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[because <math>~\phi_i = 1</math>]
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  <td align="right">
<math>~(\eta_s - B)</math>
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<math>~=~</math>
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<math>~\pi \, ,</math>
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[hence, <math>~\sin^2(\eta_s - B) = 0</math>]
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</tr>

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  <td align="right">
<math>~\eta_i</math>
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<math>~=~</math>
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<math>~3^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr) \xi_i \biggl( 1 + \frac{1}{3} \xi_i^2 \biggr)^{-1}\, ,</math>
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&nbsp;
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we can rewrite this last "envelope virial" expression as,

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<math>~\biggl( 2S + W \biggr)_\mathrm{env}</math>
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<math>~=~</math>
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<math>- ~ \biggl( \frac{2}{\pi} \biggr)^{1/2}  \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \eta_i^3</math>
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</tr>

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&nbsp;
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<math>~=~</math>
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<math>- ~ \biggl( \frac{2}{\pi} \biggr)^{1/2} 3^{3/2}  \xi_i^3 \biggl( 1 + \frac{1}{3} \xi_i^2 \biggr)^{-3} \, .</math>
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This expression is equal in magnitude, but opposite in sign to the "core virial" expression derived earlier.   Hence, putting the core and envelope contributions together, we find,
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<math>~\biggl( 2S + W \biggr)_\mathrm{tot} ~=~ 2(S_\mathrm{core} + S_\mathrm{env}) + (W_\mathrm{core} + W_\mathrm{env})</math>
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<math>~=~</math>
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<math>~ 0 \, .</math>
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</tr>
</table>
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This demonstrates that the detailed force-balanced models of <math>~(n_c, n_e) = (5,1)</math> bipolytropes derived above are also all in virial equilibrium, as should be the case.  More importantly, showing that these four separate energy integrals sum to zero helps provide confirmation that the four energy integrals have been derived correctly.  This allows us to confidently proceed to an evaluation of the relative dynamical stability of the models.

====In Parts====
In section <b><font color="maroon" size="+1">&#x2469;</font></b> of our ''[[SSC/SynopsisStyleSheet#Bipolytropes|Tabular Overview]]'', we speculated that, in bipolytropic equilibrium structures, the statements

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<math>~2S_\mathrm{core} + W_\mathrm{core} = 3P_i V_\mathrm{core}</math>
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&nbsp; &nbsp; and &nbsp; &nbsp;
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<math>~2S_\mathrm{env} + W_\mathrm{env} = - 3P_i V_\mathrm{core}  \, ,</math>
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hold separately.  Let's evaluate the "PV" term.  We find that,
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<math>~
3P_i V_\mathrm{core} = 4\pi P_i r_i^3
</math>
  </td>
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<math>~=</math>
  </td>
  <td align="left">
<math>~
4\pi \biggl( 1 + \frac{\xi_i^2}{3} \biggr)^{- 3}\biggl(\frac{3}{2\pi}\biggr)^{3 / 2} \xi_i^3 
</math>
  </td>
</tr>

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&nbsp;
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<math>~=</math>
  </td>
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<math>~
\biggl( 1 + \frac{\xi_i^2}{3} \biggr)^{- 3}\biggl(\frac{2 \cdot 3^3 }{\pi} \biggr)^{1 / 2} \xi_i^3 \, .
</math>
  </td>
</tr>
</table>
This is '''precisely''' the "extra term" that shows up (with opposite signs) in the above-derived expressions for the separate quantities, <math>~(2S + W)_\mathrm{core}</math> and <math>~(2S + W)_\mathrm{env}</math>.  Hence our speculation has been shown to be correct, at least for the case of bipolytropes with, <math>~(\gamma_c, \gamma_e) = (\tfrac{6}{5}, 2)</math>.