Editing SSC/Virial/Polytropes/Pt2 (section)

=====Strategy2=====
* Pick the desired polytropic index, <math>~n</math>, and a radial coordinate within the isolated polytropic model, <math>~\tilde\xi \leq \xi_1</math>, that will serve as the truncated edge of the embedded polytrope.
* Knowledge of the isolated polytrope's internal structure will give the value of the Lane-Emden function, <math>~\tilde\theta</math>, and its radial derivative, <math>~{\tilde\theta'}</math>, at this truncated edge of the structure.
* According to {{ Horedt70 }} &#8212; see our [[SSC/Structure/PolytropesEmbedded#Horedt.27s_Presentation|accompanying discussion of detailed force-balanced models]] &#8212; the physical radius and external pressure that corresponds to this choice of the truncated edge is given by the expressions,
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<table border="0" cellpadding="3">

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<math>
~\frac{R_\mathrm{eq}}{R_\mathrm{norm}} = r_a \biggl( \frac{R_\mathrm{Horedt}}{R_\mathrm{norm}} \biggr)
</math>
  </td>
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<math>~=~</math>
  </td>
  <td align="left">
<math>
\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)} \biggl[ \frac{4\pi}{(n+1)^n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{n-1} \biggr]^{1/(n-3)} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\frac{P_e}{P_\mathrm{norm}} = p_a \biggl( \frac{P_\mathrm{Horedt}}{P_\mathrm{norm}} \biggr)
</math>
  </td>
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<math>~=~</math>
  </td>
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<math>
\tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)} \biggl[ \frac{(n+1)^3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2} \biggr]^{(n+1)/(n-3)}  \, .
</math>
  </td>
</tr>
</table>
</div>

* Using the chosen value of <math>~\tilde\xi</math> and its associated function values, <math>~\tilde\theta</math> and <math>~\tilde\theta^'</math>, determine the values of the three relevant structural form factors via the following analytic relations:

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<math>~\tilde\mathfrak{f}_M = \frac{\bar\rho}{\rho_c}</math>
  </td>
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<math>~=</math>
  </td>
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<math>~ \biggl[ - \frac{3 \tilde\theta^'}{\tilde\xi} \biggr] \, ,</math>
  </td>
</tr>

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<math>\tilde\mathfrak{f}_W </math>
  </td>
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<math>~=</math>
  </td>
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<math>~\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\tilde\theta^'}{\tilde\xi} \biggr]^2 = \biggl( \frac{5}{5-n} \biggr) \tilde\mathfrak{f}_M^2 \, ,</math>
  </td>
</tr>

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<math>\tilde\mathfrak{f}_A  = \frac{\bar{P}}{P_c}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{3(n+1) }{(5-n)} ~( \tilde\theta^')^2  + \tilde\theta^{n+1} \, .
</math>
  </td>
</tr>
</table>
</div>
* Using these values of the structural form factors, determine the values of the three free-energy coefficients (set <math>~M_\mathrm{limi}/M_\mathrm{tot} = 1</math> for the time being) via the expressions:
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<math>~\mathcal{A}_\mathrm{mod} \equiv (5-n)\mathcal{A}</math>
  </td>
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<math>~=</math>
  </td>
  <td align="left">
<math>\frac{(5-n)}{5} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{2}  \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} 
= \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{2}  \, ,</math>
  </td>
</tr>

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  <td align="right">
<math>~\mathcal{B}_\mathrm{mod} \equiv (5-n)\mathcal{B}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{3}{4\pi} \biggr)^{1/n}
\biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\tilde\mathfrak{f}_M} \biggr]_\mathrm{eq}^{(n+1)/n}
\cdot [(5-n)\tilde\mathfrak{f}_A] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
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<math>~=</math>
  </td>
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<math>
\frac{4\pi}{3} \biggl( \frac{P_c}{P_\mathrm{norm}} \biggr) \chi_\mathrm{eq}^{3(n+1)/n}
\cdot [(5-n)\tilde\mathfrak{f}_A] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{D}_\mathrm{mod} \equiv (5-n) \mathcal{D}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{4\pi}{3} \biggr) (5-n) \frac{P_e}{P_\mathrm{norm}} \, .
</math>
  </td>
</tr>
</table>
</div>
* Plot the following free-energy function and see if the value of <math>~\chi</math> associated with the extremum is equal to the dimensionless equilibrium radius, <math>~R_\mathrm{eq}/R_\mathrm{norm}</math>, as predicted by the {{ Horedt70 }} expression, above:
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<math>~(5-n)\mathfrak{G}^*</math>
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<math>~=</math>
  </td>
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<math>~-3\mathcal{A}_\mathrm{mod} \chi^{-1} -~ \frac{1}{(1-\gamma_g)} \mathcal{B}_\mathrm{mod} \chi^{3-3\gamma_g} +~ \mathcal{D}_\mathrm{mod}\chi^3</math>
  </td>
</tr>

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&nbsp;
  </td>
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<math>~=</math>
  </td>
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<math>~-3\mathcal{A}_\mathrm{mod} \chi^{-1} +n\mathcal{B}_\mathrm{mod} \chi^{-3/n} +~ \mathcal{D}_\mathrm{mod}\chi^3 \, .</math>
  </td>
</tr>
</table>
</div>
* Virial equilbrium &#8212; that is, an extremum in the free energy function &#8212; occurs when <math>~\partial\mathfrak{G}^*/\partial\chi = 0</math>, that is, where,
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<math>~\mathcal{B}_\mathrm{mod} \chi_\mathrm{eq}^{(n-3)/n} - \mathcal{D}_\mathrm{mod}\chi_\mathrm{eq}^4</math>
  </td>
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<math>~=</math>
  </td>
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<math>~\mathcal{A}_\mathrm{mod} \, .</math>
  </td>
</tr>
</table>
</div>

Note that if the coefficient, <math>~\mathcal{B}</math>, is written in terms of the normalized central pressure, the statement of virial equilibrium becomes,
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<math>~\mathcal{A}_\mathrm{mod} </math>
  </td>
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<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{4\pi}{3}\biggl( \frac{P_c}{P_\mathrm{norm}} \biggr) \chi_\mathrm{eq}^{3(n+1)/n} 
\cdot [(5-n)\tilde\mathfrak{f}_A] \biggr] \chi_\mathrm{eq}^{(n-3)/n} - \frac{4\pi}{3} (5-n)\biggl( \frac{P_e}{P_\mathrm{norm}} \biggr) \chi_\mathrm{eq}^4</math>
  </td>
</tr>

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  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4\pi}{3}\chi_\mathrm{eq}^4 \biggl\{\biggl( \frac{P_c}{P_\mathrm{norm}} \biggr) 
\cdot (5-n)\tilde\mathfrak{f}_A  - (5-n)\biggl( \frac{P_e}{P_\mathrm{norm}} \biggr) \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4\pi}{3}\chi_\mathrm{eq}^4 \biggl\{\biggl( \frac{P_c}{P_\mathrm{norm}} \biggr) 
\cdot \biggl[ 3(n+1) (\tilde\theta^')^2 + (5-n)\tilde\theta^{n+1} \biggr]  - (5-n)\biggl( \frac{P_e}{P_\mathrm{norm}} \biggr) \biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>
But, in this situation, <math>~\tilde\theta^{n+1} = P_e/P_c</math>, so virial equilibrium implies,
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<math>~\frac{3}{4\pi}\chi_\mathrm{eq}^{-4} \mathcal{A}_\mathrm{mod} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{P_c}{P_\mathrm{norm}} \biggr) 
\cdot \biggl[ 3(n+1) (\tilde\theta^')^2 + (5-n)\frac{P_e}{P_c} \biggr]  - (5-n)\biggl( \frac{P_e}{P_\mathrm{norm}} \biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~3(n+1) (\tilde\theta^')^2\biggl( \frac{P_c}{P_\mathrm{norm}} \biggr) 
</math>
  </td>
</tr>

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  <td align="right">
<math>~\Rightarrow~~~\frac{P_c}{P_\mathrm{norm}} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}}  \biggr)^4 \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ 4\pi(n+1) (\tilde\theta^')^2 \biggr]^{-1}
</math>
  </td>
</tr>

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  <td align="right">
<math>~\Rightarrow~~~\frac{P_c R_\mathrm{eq}^4}{GM_\mathrm{limit}^2} 
</math>
  </td>
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<math>~=</math>
  </td>
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<math>~\frac{1}{4\pi(n+1) (\tilde\theta^')^2} \, .
</math>
  </td>
</tr>
</table>
</div>