Editing Appendix/Ramblings/OriginOfPlanetaryNebulae (section)

====Pressure-Truncated n = &infin; Polytropes with &gamma;<sub>g</sub> = 1====
Highlights drawn from [[SSC/Stability/Isothermal#Previously_Published_Eigenvalues_and_Eigenfunctions|our review of, and successful effort to replicate,]] the work presented by &hellip; <div align="center">{{ TVH74figure }}</div>

Note that, for each identified eigenvector, the square of the eigenfrequency may be obtained via the expression,
<table align="center" cellpadding="5" border="0">
<tr>
  <td align="right"><math>\sigma_c^2 = 6\biggl[\lambda^2\biggr]_\mathrm{TVH74}</math></td>
  <td align="right"><math>=</math></td>
  <td align="left">
<math>
\gamma_g \biggl[ \mathfrak{F} + 2\alpha\biggr]
=
\gamma_g \biggl[ \mathfrak{F} + 2\biggl(3 - \frac{4}{\gamma_g}  \biggr)\biggr]
</math>
  </td>
</tr>
</table>

<table align="center" border="1" cellpadding="5" width="75%">
<tr>
  <td align="center" colspan="2">
<b>Eigenfunctions (right) and Corresponding Eigenfrequencies (left)<br />for Isothermal Spheres Truncated at Various Radii</b>
  </td>
</tr>
<tr>
  <td align="center">Fundamental</td>
  <td align="left" rowspan="2">
<div align="center">
[[File:TaffVanHorn1974Fundamental.gif|600px|Fundamental mode animation]]
</div>
  </td>
</tr>
<tr>
  <td align="center">
<table border="0" align="center" cellpadding="6">
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  <td align="center"><math>~\xi_0</math></td>
  <td align="center"><math>~\mathfrak{F}</math></td>
  <td align="center"><math>~\lambda_0^2 = \frac{\gamma(\mathfrak{F}+2\alpha)}{6}</math></td>
</tr>
<tr>
  <td align="center">2</td>
  <td align="center">12.92907</td>
  <td align="center" bgcolor="white">+1.821512</td>
</tr>
<tr>
  <td align="center">3</td>
  <td align="center">5.51614</td>
  <td align="center" bgcolor="white">+0.586023</td>
</tr>
<tr>
  <td align="center">4</td>
  <td align="center">3.29671</td>
  <td align="center" bgcolor="white">+0.216118</td>
</tr>
<tr>
  <td align="center">5</td>
  <td align="center">2.456412</td>
  <td align="center" bgcolor="white">+0.0760687</td>
</tr>
<tr>
  <td align="center">6</td>
  <td align="center">2.092651</td>
  <td align="center" bgcolor="white">+0.0154418</td>
</tr>
<tr>
  <td align="center">7</td>
  <td align="center">1.921062</td>
  <td align="center" bgcolor="pink">-0.013156</td>
</tr>
<tr>
  <td align="center">8</td>
  <td align="center">1.8354928</td>
  <td align="center" bgcolor="pink">-0.027418</td>
</tr>
<tr>
  <td align="center">9</td>
  <td align="center">1.791388</td>
  <td align="center" bgcolor="pink">-0.034769</td>
</tr>
</table>
  </td>
</tr>
<tr>
  <td align="center">1<sup>st</sup> Overtone</td>
  <td align="left" rowspan="2">
<div align="center">
[[File:TaffVanHorn1974Harmonic.gif|600px|First Harmonic mode animation]]
</div>
  </td>
</tr>
<tr>
  <td align="center">
<table border="0" align="center" cellpadding="6">
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  <td align="center"><math>~\xi_1</math></td>
  <td align="center"><math>~\mathfrak{F}</math></td>
  <td align="center"><math>~\lambda_1^2 = \frac{\gamma(\mathfrak{F}+2\alpha)}{6}</math></td>
</tr>
<tr>
  <td align="center">2</td>
  <td align="center">56.87349</td>
  <td align="center" bgcolor="white">+9.14558</td>
</tr>
<tr>
  <td align="center">3</td>
  <td align="center">24.58903</td>
  <td align="center" bgcolor="white">+3.76484</td>
</tr>
<tr>
  <td align="center">4</td>
  <td align="center">13.640525</td>
  <td align="center" bgcolor="white">+1.94009</td>
</tr>
<tr>
  <td align="center">5</td>
  <td align="center">8.798197</td>
  <td align="center" bgcolor="white">+1.13303</td>
</tr>
<tr>
  <td align="center">6</td>
  <td align="center">6.306545</td>
  <td align="center" bgcolor="white">+0.71776</td>
</tr>
<tr>
  <td align="center">7</td>
  <td align="center">4.88991</td>
  <td align="center" bgcolor="white">+0.48165</td>
</tr>
<tr>
  <td align="center">8</td>
  <td align="center">4.024628</td>
  <td align="center" bgcolor="white">+0.33744</td>
</tr>
<tr>
  <td align="center">9</td>
  <td align="center">3.46662</td>
  <td align="center" bgcolor="white">+0.24444</td>
</tr>
</table>
  </td>
</tr>
</table>

<font color="red">ADDITIONALLY:</font><br />
<ul>
  <li>
Taking into account our [[SSC/Stability/InstabilityOnsetOverview#Elaboration|accompanying elaboration]] regarding the properties of pressure-truncated isothermal spheres, it would be instructive to extend this pair of animations to at least include the model in which the 1<sup>st</sup> overtone mode becomes marginally unstable <math>( {\tilde\xi} \approx 67)</math>.
  </li>
  <li>
Make a movie that shows ''in the vicinity of the onset of instability'' how the eigenfunction that is associated with the 1<sup>st</sup> overtone mode varies with <math>\tilde\xi</math>.  As in the context (''e.g.,'' above) of our analysis of pressure-truncated n = 5 polytropes, pair it with a movie that shows where each model lies along the equilibrium sequence.
  </li>
</ul>