Editing SSC/Structure/BiPolytropes/FreeEnergy51 (section)

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

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  <td align="right">
<math>\Rightarrow ~~~~ 1-p_c(x) = \frac{P_\mathrm{core}(x)}{P_0} </math>
  </td>
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<math>~=</math>
  </td>
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<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/BiPolytropes/FreeEnergy51#Mass_Profile|derivation of the mass profile]].
The desired integral over this pressure distribution therefore gives,
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<table border="0" cellpadding="8" align="center">

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  <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>

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&nbsp;
  </td>
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<math>~=</math>
  </td>
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<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>

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  <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,
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<math>P_{ic} = K_c \rho_0^{6/5} \biggl(1 + \frac{1}{3}\xi_i^2 \biggr)^{-3} \, ,</math>
</div>
and,
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<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,
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<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>

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  <td align="right">
&nbsp;
  </td>
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<math>~=</math>
  </td>
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<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>

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&nbsp;
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<math>~=</math>
  </td>
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<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,
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<table border="0" cellpadding="5" align="center">

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<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]].