Editing SSC/Structure/PolytropesEmbedded/n5 (section)

====Overlap with Whitworth's Presentation====
The curve labeled <math>~n=5</math> in the top two panels of Figure 1 shows how <math>R_\mathrm{eq}</math> varies with the applied external pressure <math>P_e</math>; as shown, the curve has two segments &#8212; configurations that are stable (blue diamonds) and configurations that are unstable (red squares).  Following the lead of [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth] (1981, MNRAS, 195, 967) &#8212; for clarification, read the [[SSC/Structure/PolytropesASIDE1|accompanying ASIDE]] &#8212; these two quantities have been respectively normalized (or, "referenced") to,
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<math>
R_\mathrm{rf}\biggr|_\mathrm{n=5} \equiv \frac{2^6}{3^3} \biggl( \frac{\pi}{5^5} \biggr)^{1/2} \biggl[ \frac{G^5 M^4}{K^5} \biggr]^{1/2} ~~~\Rightarrow ~~~ \frac{R_\mathrm{eq}}{R_\mathrm{rf}} = \biggl( \frac{5^5}{2^{15}\cdot 3} \biggr)^{1/2} \frac{(3+\xi_e^2)^3}{\xi_e^5} \, ;
</math>
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and,
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<math>
P_\mathrm{rf}\biggr|_\mathrm{n=5} \equiv \frac{3^{12} 5^9}{2^{26} \pi^3} \biggl( \frac{K^{10}}{G^9 M^6} \biggr) ~~~\Rightarrow ~~~ \frac{P_e}{P_\mathrm{rf}} = \biggl( \frac{2^{29}\cdot 3^{3} }{5^9}  \biggr) \frac{\xi_e^{18}}{(3 + \xi_e^2)^{12}}  \, .
</math>
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We see that this <math>~n=5</math> model sequence bends back on itself.  That is to say, for this polytropic index there is an externally applied pressure above which no equilibrium configuration exists.  This limiting pressure arises along the curve where, 
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<math>\frac{dP_e}{dR_\mathrm{eq}} = \biggl( \frac{dP_e}{d\xi_e} \biggr) \biggl( \frac{dR_\mathrm{eq}}{d\xi_e} \biggr)^{-1} = 0 \, .</math>
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Evaluation of this expression shows that the limiting pressure occurs precisely at <math>\xi_e = 3</math>,  that is,
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<math>
\biggl( \frac{P_e}{P_\mathrm{rf}} \biggr)_\mathrm{max} = \biggl( \frac{2^{29}\cdot 3^{3} }{5^9}  \biggr) \frac{3^{18}}{12^{12}} = \frac{2^5 \cdot 3^9}{5^9} \, ,
</math>
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and the radius of this limiting configuration is,
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<math>
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr) = \biggl( \frac{5^5}{2^{15}\cdot 3} \biggr)^{1/2} \frac{12^3}{3^5} = \biggl( \frac{5^5}{2^3 \cdot 3^5} \biggr)^{1/2} \, .
</math>
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On the log-log plot displayed in the top-right panel of Figure 1, the location of this special point is <math>[ \log(P_e/P_\mathrm{rf}) , \log(R_\mathrm{eq}/R_\mathrm{rf}) ] \approx [ -0.49149, +0.10308 ] \, .</math>

We note as well that a conversion from Whitworth's normalizations to the normalizations adopted by Horedt produce the following coordinates for the limiting model configuration:
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<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
~p_a|_\mathrm{max}
</math>
  </td>
  <td align="center">
<math>~=~</math>
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  <td align="left">
<math>
~\frac{3^{12}}{2^{24}} \, ,
</math>
  </td>
</tr>
</table>
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and, at this bounding pressure, the model has an equilibrium radius,
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<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
~r_a
</math>
  </td>
  <td align="center">
<math>~=~</math>
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  <td align="left">
<math>
\frac{2^6}{3^3} \, .
</math>
  </td>
</tr>
</table>
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