Editing Appendix/Ramblings/OriginOfPlanetaryNebulae (section)

====Bipolytrope With (n<sub>c</sub>, n<sub>e</sub>) = (1, 5) and &gamma;<sub>c</sub> = &gamma;<sub>e</sub> = 5/3====
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">{{ MF85bfigure }}</div>



<table border="1" align="center" cellpadding="8" >
<tr>
  <td align="center" colspan="7">
'''Numerical Values for Two Selected <math>(n_c, n_e) = (1, 5)</math> Bipolytropes'''<br />
[to be compared with Table 1 of [http://adsabs.harvard.edu/abs/1985PASAu...6..219M Murphy &amp; Fiedler (1985)]]
  </td>
</tr>
<tr>
  <td align="center">MODEL</td>
  <td align="center">Source</td>
  <td align="center"><math>~\frac{r_i}{R}</math></td>
  <td align="center"><math>~\Omega_0^2</math></td>
  <td align="center"><math>~\Omega_1^2</math></td>
  <td align="center"><math>~\frac{r}{R}\biggr|_1</math></td>
  <td align="center"><math>~1-\frac{M_r}{M_\mathrm{tot}}\biggr|_1</math></td>
</tr>
<tr>
  <td align="center" rowspan="2">10</td>
  <td align="center" bgcolor="pink">MF85</td>
  <td align="left">0.393</td>
  <td align="left">15.9298</td>
  <td align="left">21.2310</td>
  <td align="left">0.573</td>
  <td align="left">1.00E-03</td>
</tr>
<tr>
  <td align="center">Here</td>
  <td align="right">0.39302</td>
  <td align="right">15.93881161</td>
  <td align="right">21.24571822</td>
  <td align="right">0.5724</td>
  <td align="left">3.05E-05</td>
</tr>
<tr>
  <td align="center" rowspan="2">17</td>
  <td align="center" bgcolor="pink">MF85</td>
  <td align="left">0.933</td>
  <td align="left">2.1827</td>
  <td align="left">13.9351</td>
  <td align="left">0.722</td>
  <td align="left">0.232</td>
</tr>
<tr>
  <td align="center">Here</td>
  <td align="left">0.93277</td>
  <td align="left">2.182932207</td>
  <td align="left">13.93880866</td>
  <td align="left">0.7215</td>
  <td align="left">0.24006</td>
</tr>
</table>


<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center" colspan="12">
'''Our Determinations for Model 10''' <math>~(\xi_i = 2.5646)</math>
  </td>
</tr>
<tr>
  <td align="center" rowspan="2">Mode</td>
  <td align="center" rowspan="2"><math>~\sigma_c^2</math></td>
  <td align="center" rowspan="2"><math>~\Omega^2 \equiv \frac{\sigma_c^2}{2} \biggl( \frac{\rho_c}{\bar\rho}\biggr)</math></td>
  <td align="center" rowspan="2"><math>~x_\mathrm{surf}</math></td>
  <td align="center" colspan="2"><math>~\frac{d\ln x}{d\ln r^*}\biggr|_\mathrm{surf}</math></td>
  <td align="center" rowspan="2"><math>~\frac{r}{R}\biggr|_1</math></td>
  <td align="center" rowspan="2"><math>~1 - \frac{M_r}{M_\mathrm{tot}}\biggr|_1</math></td>
  <td align="center" rowspan="2"><math>~\frac{r}{R}\biggr|_2</math></td>
  <td align="center" rowspan="2"><math>~1 - \frac{M_r}{M_\mathrm{tot}}\biggr|_2</math></td>
  <td align="center" rowspan="2"><math>~\frac{r}{R}\biggr|_3</math></td>
  <td align="center" rowspan="2"><math>~1 - \frac{M_r}{M_\mathrm{tot}}\biggr|_3</math></td>
</tr>
<tr>
  <td align="center">''expected''</td>
  <td align="center">measured</td>
</tr>
<tr>
  <td align="center">1<br /><font size="-1">(Fundamental)</font></td>
  <td align="right">0.92813095170326</td>
  <td align="right">15.93881161</td>
  <td align="right">+85.17</td>
  <td align="right">8.963286966</td>
  <td align="right">8.963085</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
</tr>
<tr>
  <td align="center">2</td>
  <td align="right">1.237156768978</td>
  <td align="right">21.24571822</td>
  <td align="right">- 610</td>
  <td align="right">12.14743093</td>
  <td align="right">12.147337</td>
  <td align="right">0.5724</td>
  <td align="right">3.05E-05</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
</tr>
<tr>
  <td align="center">3</td>
  <td align="right">1.8656033984</td>
  <td align="right">32.0380449</td>
  <td align="right">+3225</td>
  <td align="right">18.62282676</td>
  <td align="right">18.6228</td>
  <td align="right">0.4845</td>
  <td align="right">1.35E-04</td>
  <td align="right">0.787</td>
  <td align="right">2.05E-07</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
</tr>
<tr>
  <td align="center">4</td>
  <td align="right">2.65901504799</td>
  <td align="right">45.66331921</td>
  <td align="right">-9410</td>
  <td align="right">26.79799153</td>
  <td align="right">26.797977</td>
  <td align="right">0.4459</td>
  <td align="right">2.620E-04</td>
  <td align="right">0.7096</td>
  <td align="right">1.834E-06</td>
  <td align="center">0.8632</td>
  <td align="center">1.189E-08</td>
</tr>
<tr>
  <td align="center" colspan="12">[[File:MF85Figure2B.png|800px|Match Figure 2 from MF85]]</td>
</tr>
</table>


<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center" colspan="12">
'''Our Determinations for Model 17''' <math>(\xi_i = 3.0713)</math>
  </td>
</tr>
<tr>
  <td align="center" rowspan="2">Mode</td>
  <td align="center" rowspan="2"><math>~\sigma_c^2</math></td>
  <td align="center" rowspan="2"><math>~\Omega^2 \equiv \frac{\sigma_c^2}{2} \biggl( \frac{\rho_c}{\bar\rho}\biggr)</math></td>
  <td align="center" rowspan="2"><math>~x_\mathrm{surf}</math></td>
  <td align="center" colspan="2"><math>~\frac{d\ln x}{d\ln r^*}\biggr|_\mathrm{surf}</math></td>
  <td align="center" rowspan="2"><math>~\frac{r}{R}\biggr|_1</math></td>
  <td align="center" rowspan="2"><math>~1 - \frac{M_r}{M_\mathrm{tot}}\biggr|_1</math></td>
  <td align="center" rowspan="2"><math>~\frac{r}{R}\biggr|_2</math></td>
  <td align="center" rowspan="2"><math>~1 - \frac{M_r}{M_\mathrm{tot}}\biggr|_2</math></td>
  <td align="center" rowspan="2"><math>~\frac{r}{R}\biggr|_3</math></td>
  <td align="center" rowspan="2"><math>~1 - \frac{M_r}{M_\mathrm{tot}}\biggr|_3</math></td>
</tr>
<tr>
  <td align="center">''expected''</td>
  <td align="center">measured</td>
</tr>
<tr>
  <td align="center">1<br /><font size="-1">(Fundamental)</font></td>
  <td align="left">1.149837904</td>
  <td align="left">2.182932207</td>
  <td align="right">+1.275</td>
  <td align="right">0.7097593</td>
  <td align="right">0.7097550</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
</tr>
<tr>
  <td align="center">2</td>
  <td align="left">7.34212930615</td>
  <td align="left">13.93880866</td>
  <td align="right">- 2.491</td>
  <td align="right">7.763285</td>
  <td align="right">7.763244</td>
  <td align="right">0.7215</td>
  <td align="right">0.24006</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
</tr>
<tr>
  <td align="center">3</td>
  <td align="left">16.345072567</td>
  <td align="left">31.03062198</td>
  <td align="right">+4.33</td>
  <td align="right">18.01837</td>
  <td align="right">18.01826</td>
  <td align="right">0.5806</td>
  <td align="right">0.5027</td>
  <td align="right">0.848</td>
  <td align="right">0.0541</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
</tr>
<tr>
  <td align="center">4</td>
  <td align="left">27.746934203</td>
  <td align="left">52.6767087</td>
  <td align="right">-9.1</td>
  <td align="right">31.0060</td>
  <td align="right">31.0058</td>
  <td align="right">0.4859</td>
  <td align="right">0.6737</td>
  <td align="right">0.7429</td>
  <td align="right">0.1974</td>
  <td align="center">0.8957</td>
  <td align="center">0.0171</td>
</tr>
<tr>
  <td align="center" colspan="12">[[File:MF85Figure3.png|800px|Match Figure 3 from MF85]]</td>
</tr>
</table>