Editing SSC/FreeEnergy/PolytropesEmbedded/Pt1 (section)

====Reconcile====


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<math>~\biggl[\frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr]^4_\mathrm{crit} </math>
  </td>
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<math>~=</math>
  </td>
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<math>~\biggl[ \frac{(n-3)}{20\pi (n+1)} \biggr] \biggl(\frac{n+1}{n}\biggr)
\biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{2} 
\frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} 
</math>
  </td>
</tr>
</table>

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<math>~\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^4_\mathrm{crit}</math>
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<math>~=</math>
  </td>
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<math>~\frac{1}{20\pi}  \biggl( \frac{n-3}{n+1} \biggr) \biggl( \frac{P_e}{P_\mathrm{norm}} \biggr)^{-1} \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} </math>
  </td>
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</table>

</div>
Taking the ratio, the RHS is,
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<math>~\biggl( \frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{2} \biggl( \frac{P_e}{P_\mathrm{norm}} \biggr)</math>
  </td>
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<math>~=</math>
  </td>
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<math>~P_e M_\mathrm{tot}^2 \biggl[ \frac{G^{3(n+1)} M_\mathrm{tot}^{2(n+1)} }{K^{4n}} \biggr]^{1/(n-3)} 
\biggl[ \biggl( \frac{n+1}{n} \biggr)^{3/2} G^{-3/2} K_n^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]} \biggr]^{-2}
\biggl( \frac{n+1}{n}\biggr)</math>
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</tr>

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&nbsp;
  </td>
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<math>~=</math>
  </td>
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<math>~\biggl( \frac{n+1}{n} \biggr)^{-2}P_e M_\mathrm{tot}^2 \biggl[ G^{3} M_\mathrm{tot}^{2} \biggr]^{(n+1)/(n-3)} K_n^{-4n/(n-3)}
\biggl[  G^{3} K_n^{-4n/(n+1)} P_\mathrm{e}^{(n-3)/(n+1)} \biggr]</math>
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&nbsp;
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<math>~=</math>
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<math>~\biggl( \frac{n+1}{n} \biggr)^{-2}  \biggl[ G^{3} M_\mathrm{tot}^{2} \biggr]^{[(n-3)+(n+1)]/(n-3)} 
\biggl[ K_n^{[(n+1)+(n-3)]/[(n+1)(n-3)] } \biggr]^{-4n} P_\mathrm{e}^{[(n+1)+  (n-3)]/(n+1)} </math>
  </td>
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&nbsp;
  </td>
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<math>~=</math>
  </td>
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<math>~\biggl( \frac{n+1}{n} \biggr)^{-2}  M_\mathrm{tot}^{4(n-1)/(n-3)} G^{[6(n-1)]/(n-3)} 
K_n^{-8(n-1)/[(n+1)(n-3)] }  P_\mathrm{e}^{2(n-1)/(n+1)} \, ;</math>
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while the LHS is,

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<math>~\biggl( \frac{R_\mathrm{norm}}{R_\mathrm{SWS}} \biggr)^{4}</math>
  </td>
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<math>~=</math>
  </td>
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<math>~
\biggl[\biggl( \frac{G}{K} \biggr)^n M_\mathrm{tot}^{(n-1)} \biggr]^{4/(n-3)} 
\biggl\{\biggl( \frac{n+1}{n} \biggr)^{1/2} G^{-1/2} K_n^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]}\biggr\}^{-4}
</math>
  </td>
</tr>

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&nbsp;
  </td>
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<math>~=</math>
  </td>
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<math>~\biggl( \frac{n+1}{n} \biggr)^{-2} 
M_\mathrm{tot}^{4(n-1)/(n-3)} 
G^{[6(n-1)]/(n-3)} 
K^{-8n(n-1)/[(n-3)(n+1)] }  P_\mathrm{e}^{2(n-1)/(n+1)} \, .
</math>
  </td>
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</table>
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Q.E.D.

Now, given that,
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<math>~M_\mathrm{SWS}^{-4(n-1)/(n-3)}</math>
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<math>~=</math>
  </td>
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<math>~\biggl[\biggl( \frac{n+1}{n} \biggr)^{3/2} G^{-3/2} K_n^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]}\biggr]^{-4(n-1)/(n-3)} </math>
  </td>
</tr>

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&nbsp;
  </td>
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<math>~=</math>
  </td>
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<math>~\biggl( \frac{n+1}{n} \biggr)^{-6(n-1)/(n-3)} G^{6(n-1)/(n-3)} K_n^{-8n(n-1)/[(n+1)(n-3)]} P_\mathrm{e}^{2(n-1)/(n+1)} </math>
  </td>
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we have,
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  <td align="right">
<math>~\biggl( \frac{R_\mathrm{norm}}{R_\mathrm{SWS}} \biggr)^{4}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{n+1}{n} \biggr)^{-2} 
\biggl(\frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{4(n-1)/(n-3)} 
\biggl( \frac{n+1}{n} \biggr)^{6(n-1)/(n-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{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{4(n-1)/(n-3)} 
\biggl( \frac{n+1}{n} \biggr)^{4n/(n-3)} 
</math>
  </td>
</tr>

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<math>~\Rightarrow ~~~\biggl( \frac{R_\mathrm{norm}}{R_\mathrm{SWS}} \biggr)^{n-3}</math>
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<math>~=</math>
  </td>
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<math>~
\biggl(\frac{M_\mathrm{tot}}{M_\mathrm{SWS}}\biggr)^{n-1} 
\biggl( \frac{n+1}{n} \biggr)^{n} 
</math>
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By inspection, in the specific case of <math>~n=5</math> (see above), this critical configuration appears to coincide with one of the [[SSC/Structure/PolytropesEmbedded#Other_Limits|"turning points" identified by Kimura]].  Specifically, it appears to coincide with the "extremal in r<sub>1</sub>" along an M<sub>1</sub> sequence, which satisfies the condition,
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<math>~\biggl[ \frac{n-3}{n-1} \biggr]_{n=5}</math>
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<math>~=</math>
  </td>
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<math>~\biggl[ \frac{\tilde\xi \tilde\theta^{n}}{(-\tilde\theta^')}\biggr]_{n=5}</math>
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<math>~\Rightarrow ~~~\frac{1}{2} </math>
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<math>~=</math>
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<math>~3^{1/2}\ell \biggl[ (1 + \ell^2)^{-1/2} \biggr]^5 \biggl[ \frac{\ell}{3^{1/2}} (1+\ell^2 )^{-3/2} \biggr]^{-1}</math>
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&nbsp;
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<math>~=</math>
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<math>~3(1 + \ell^2)^{-1} </math>
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<math>~\Rightarrow~~~ \ell </math>
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<math>~=</math>
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<math>~5^{1/2} \, .</math>
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Hence, according to Kimura, the turning point associated with the configuration with the largest equilibrium radius, corresponds to the equilibrium configuration having,
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<math>~\ell |_\mathrm{R_{max}} = \sqrt{5} \approx 2.2360680 \, .</math>
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This is, indeed, very close to &#8212; but decidedly different from &#8212; the value of <math>~\ell_\mathrm{crit}</math> determined, above!