Editing SSC/Stability/BiPolytropes/RedGiantToPN (section)

==Truncated n = 5 Polytrope==

In [[SSC/Structure/PolytropesEmbedded/n5#Fig3|Figure 3 of an accompanying discussion]], we show where various turning points lie along the equilibrium sequence of truncated <math>n=5</math> polytropes.



<div align="center" id="Fig3">
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<b>Figure 3:</b> &nbsp; Equilibrium Sequences of Pressure-Truncated, n = 5 Polytropic Spheres<br />(viewed from several different astrophysical perspectives)
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  <td align="center"><font color="black" size="+2">&#x25CF;</font></td><td align="center"><math>~\xi_e</math></td>
  <td align="center" width="300px"><sup>&dagger;</sup>External Pressure vs. Volume<br /><font size="-1">(Fixed Mass)</font></td>
  <td align="center" width="300px">Mass vs. Radius<br /><font size="-1">(Fixed External Pressure)</font></td>
  <td align="center" width="300px"><sup>&Dagger;</sup>Mass vs. Central Density<br /><font size="-1">(Fixed External Pressure)</font></td>
  <td align="center" width="300px">Mass vs. Central Density<br /><font size="-1">(Fixed Radius)</font></td>
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  <td align="center" colspan="1"><font color="yellow" size="+2">&#x25CF;</font></td>  <td align="center" colspan="1">&radic;3</td>
  <td align="center" colspan="1" rowspan="4">(a)<br />
[[File:N5Sequence01B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
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  <td align="center" colspan="1" rowspan="4">(b)<br />
[[File:N5Sequence02B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
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  <td align="center" colspan="1" rowspan="4">(c)<br />
[[File:N5Sequence03B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
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  <td align="center" colspan="1" rowspan="4">(d)<br />
[[File:N5Sequence04B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
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  <td align="center" colspan="1"><font color="darkgreen" size="+2">&#x25CF;</font></td>  <td align="center" colspan="1">3</td>
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  <td align="center" colspan="1"><font color="purple" size="+2">&#x25CF;</font></td>  <td align="center" colspan="1">&radic;15</td>
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  <td align="center" colspan="1"><font color="red" size="+2">&#x25CF;</font></td>  <td align="center" colspan="1">9.01</td>
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  <td align="center" colspan="2">&nbsp;</td>
  <td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^3 \biggl[ \xi^{18} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-12} \biggr]_\tilde\xi</math><br /> vs. <br />
<math>\biggl(\frac{\pi}{2\cdot 3}\biggr)^{5/2} \biggl[ \xi^{-15} \biggl(1 + \frac{\xi^2}{3} \biggr)^{9}\biggr]_\tilde\xi</math>
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  <td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2}\biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs. 
<br /> <math>\biggl(\frac{3}{2\pi}\biggr)^{1 / 2} \biggl[ \xi \biggl(1 + \frac{\xi^2}{3} \biggr)^{-1} \biggr]_\tilde\xi</math></td>
  <td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2} \biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \biggl(1 + \frac{\xi^2}{3} \biggr)^{5/2}\biggr]_\tilde\xi</math>
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  <td align="center" colspan="1"><math>\biggl[ \frac{2^3\cdot 3}{\pi} \biggr]^{1 / 4} \biggl[  \xi^{5/2}\biggl(1 + \frac{\xi^2}{3}\biggr)^{-3 / 2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \frac{3}{2\pi} \biggr]^{5 / 4} \tilde\xi^{5 / 2}</math>
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<ul>
<li>
<font color="red">KEY RESULT:</font>
<ul>
  <li>The maximum "Bonnor-Ebert type" mass and external pressure occurs along the sequence precisely at <math>\tilde\xi = 3</math>.</li>
  <li>It is precisely at this turning point that the equilibrium model is marginally (dynamically) unstable; the eigenfunction is parabolic.</li>
  <li>[[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|For all <math>3 < n < \infty</math>]], the location along the relevant sequence presents an analogous turning point whose location and whose eigenfunction is known analytically.</li>
</ul>