Editing Appendix/Ramblings/OriginOfPlanetaryNebulae (section)

==Sub-Projects Undertaken==
In order for the above proposed numerical investigation to provide informative results, it is important that we establish a firm understanding of a variety of related, but less complicated, concepts and problems.  An emphasis has been placed on tackling problems that can described as fully as possible using analytic, rather than purely numerical, techniques.  The following subsections provide a list, along with brief description, of related sub-projects that we have studied, to date. 

===Polytropes===

====Isolated n = 1 Polytrope with &gamma;<sub>g</sub> = 2====
Highlights drawn from [[SSC/Structure/BiPolytropes/Analytic51Renormalize#Isolated_n_=_1_Polytrope|our examination of radial oscillations in an isolated, n = 1 polytrope]].

<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center" colspan="4"><b>Four Modes of Oscillation of an Isolated, <math>n=1</math> Polytrope Assuming <math>\gamma_g = 2</math></b></td>
</tr>
<tr>
  <td align="center">Mode</td>
  <td align="center"><math>\sigma_c^2</math></td>
  <td align="center">Neg. Slope<br><math>1 - (\sigma_c^2\pi^2/12)</math></td>
  <td align="center"><math>\mathfrak{F} = \frac{\sigma_c^2}{\gamma_g} - 2\alpha</math></td>
</tr>
<tr>
  <td align="center">Fundamental</td>
  <td align="center">2.2405295</td>
  <td align="center">3.1287618</td>
  <td align="center">-0.879735</td>
</tr>
<tr>
  <td align="center">1<sup>st</sup> Overtone</td>
  <td align="center">6.340767</td>
  <td align="center">-32.06757</td>
  <td align="center">1.1703835</td>
</tr>
<tr>
  <td align="center">2<sup>nd</sup> Overtone</td>
  <td align="center">13.694927</td>
  <td align="center">-153.2545</td>
  <td align="center">4.8474635</td>
</tr>
<tr>
  <td align="center">3<sup>rd</sup> Overtone</td>
  <td align="center">28.462829</td>
  <td align="center">-665.3074</td>
  <td align="center">12.231415</td>
</tr>

<tr><td align="center" colspan="4">
'''Numerically Determined Eigenfunctions for Various <math>~\mathfrak{F}</math><br />
[[File:N1osc06.gif|700px|Animated gif showing oscillation modes for n = 1 polytrope]]
</td></tr></table>


====Isolated n = 3 Polytrope with &gamma;<sub>g</sub> = 20/13====
Highlights drawn from [[SSC/Stability/n3PolytropeLAWE|our review of, and successful effort to replicate,]] the work presented by &hellip; <div align="center">{{ Schwarzschild41figure }}</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 = \biggl( \frac{3\gamma_g}{2} \biggr)\omega^2_\mathrm{Sch}</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 border="1" align="center" cellpadding="8">
<tr>
  <td colspan="3" align="center"><b>Four Modes of Oscillation of an Isolated, <math>n=3</math> Polytrope Assuming <math>\gamma_g = \frac{20}{13}</math></b></td>
</tr>
<tr>
  <td align="center">Mode</td>
  <td align="center"><math>\mathfrak{F}</math></td>
  <td align="center"><math>\sigma_c^2 = \frac{20}{13}\biggl[ \mathfrak{F} + \frac{4}{5}\biggr]</math></td>
</tr>
<tr>
  <td align="center">Fundamental</td>
  <td align="center">-0.64413</td>
  <td align="center">+0.23980</td>
</tr>
<tr>
  <td align="center">1<sup>st</sup> Overtone</td>
  <td align="center">-0.47014</td>
  <td align="center">+0.50748</td>
</tr>
<tr>
  <td align="center">2<sup>nd</sup> Overtone</td>
  <td align="center">-0.21213</td>
  <td align="center">+0.90442</td>
</tr>
<tr>
  <td align="center">3<sup>rd</sup> Overtone</td>
  <td align="center">+0.12026</td>
  <td align="center">+1.41578</td>
</tr>

<tr>
  <td align="center" colspan="3">
'''Numerically Determined Eigenfunctions for Various <math>~\mathfrak{F}</math><br />
[[File:Schwarzschild1941movie.gif|400px|Eigenfunctions for Standard Model]]
  </td>
</tr>
</table>

====Pressure-Truncated n = 5 Polytrope with &gamma;<sub>g</sub> = 6/5====
Highlights drawn from [[SSC/Stability/n5PolytropeLAWE|our examination of radial oscillations in pressure-truncated, n = 5 polytropes]].

<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center"><math>\xi_i</math></td>
  <td align="center"><math>\sigma_c^2</math></td>
  <th align="center" colspan="2">Numerically Generated Fundamental-Mode Eigenvectors<br />for Various Truncation Radii, <math>\xi_i</math></th>
</tr>
<tr>
  <td align="left">0.75</td>
  <td align="left">+13.6915</td>
  <td align="center" rowspan="16">
[[File:N5Truncated2.gif|650px|n5 Truncated movie]]
  </td>
  <td align="center" rowspan="16">
Excel File:<br />
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/LinearPerturbation/n5Eigenvectors/n5TruncatedSphere.xlsx --- worksheet = OursPt1]]
Movie File:<br />
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/n5movie/ --- worksheet = n5Truncated2.gif]]
  </td>
</tr>
<tr>
  <td align="left">1.00</td>
  <td align="left">+6.4733</td>
</tr>
<tr>
  <td align="left">1.25</td>
  <td align="left">+3.3666</td>
</tr>
<tr>
  <td align="left">1.50</td>
  <td align="left">+1.8441</td>
</tr>
<tr>
  <td align="left">1.75</td>
  <td align="left">+1.0362</td>
</tr>
<tr>
  <td align="left">2.00</td>
  <td align="left">+0.5835</td>
</tr>
<tr>
  <td align="left">2.25</td>
  <td align="left">+0.3193</td>
</tr>
<tr>
  <td align="left">2.50</td>
  <td align="left">+0.1602</td>
</tr>
<tr>
  <td align="left">2.75</td>
  <td align="left">+0.0619</td>
</tr>
<tr>
  <td align="left">3.00</td>
  <td align="left" bgcolor="pink"><math>\pm</math> 0.0000</td>
</tr>
<tr>
  <td align="left">3.25</td>
  <td align="left" bgcolor="pink">-0.0396</td>
</tr>
<tr>
  <td align="left">3.50</td>
  <td align="left" bgcolor="pink">-0.0653</td>
</tr>
<tr>
  <td align="left">3.75</td>
  <td align="left" bgcolor="pink">-0.0820</td>
</tr>
<tr>
  <td align="left">4.00</td>
  <td align="left" bgcolor="pink">-0.0930</td>
</tr>
<tr>
  <td align="left">4.50</td>
  <td align="left" bgcolor="pink">-0.1048</td>
</tr>
<tr>
  <td align="left">5.00</td>
  <td align="left" bgcolor="pink">-0.1098</td>
</tr>
</table>

===Isothermal Sphere===
====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">
<tr>
  <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">
<tr>
  <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>

===Bipolytropes===

====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>

====Bipolytrope With (n<sub>c</sub>, n<sub>e</sub>) = (5, 1) and (&gamma;<sub>c</sub>, &gamma;<sub>e</sub>) = (6/5, 2)====
Building upon [[SSC/Structure/BiPolytropes/Analytic51|our review and successful replication]] of the work presented by &hellip; <div align="center">{{ EFC98figure }}</div>
in which strictly analytical means are used to construct ''equilibrium'' models of <math>(n_c, n_e) = (5, 1)</math> bipolytropes, here we present highlights drawn from [[SSC/Stability/BiPolytropes/HeadScratching|our examination of radial oscillations in these bipolytropes]], assuming <math>(\gamma_c, \gamma_e) = (\tfrac{6}{5}, 2)</math>.

<font color="red">SPREADSHEET:</font> &nbsp; WorkFolder/Wiki_Edits/BiPolytrope/Faulkner1stOvertone/Stability51.xlsx
<ul>
  <li>[[SSC/Structure/BiPolytropes/Analytic51#Steps_2_&_3|Analytic Specification of n = 5 Core]]
  </li>
  <li>[[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|Interface Conditions]]</li>
  <li>[[SSC/Structure/BiPolytropes/Analytic51#Step_6:_Envelope_Solution|Analytic Specification of n = 1 Envelope]]</li>
  <li>[[SSC/Structure/BiPolytropes/Analytic51Renormalize#Step_4:_Throughout_the_core|Physical Properties of Core, Normalized to Total Mass]]

<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>\tilde{\rho}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\mathcal{m}_\mathrm{surf}^5 \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-10} 
\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2} \, ;</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\tilde{P}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\mathcal{m}_\mathrm{surf}^6 \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} 
\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3} \, ;</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\tilde{r}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\mathcal{m}_\mathrm{surf}^{-2} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{4} 
\biggl(\frac{3}{2\pi}\biggr)^{1/2} \xi \, ;</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\tilde{M}_r \equiv \frac{M_r}{M_\mathrm{tot}}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\mathcal{m}_\mathrm{surf}^{-1} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{2} 
\biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} \biggl[ \xi^3 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} \biggr] \, ,</math>
  </td>
</tr>
</table>
where,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>\mathcal{m}_\mathrm{surf}</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left"><math>\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i} \, .</math></td>
</tr>
</table>

  </li>
  <li>[[SSC/Structure/BiPolytropes/Analytic51Renormalize#Step_8:_Throughout_the_envelope|Physical Properties of Envelope, Normalized to Total Mass]]

<table align="center" border="0" cellpadding="3">
<tr>
  <td align="right">
<math>\tilde\rho</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\mathcal{m}_\mathrm{surf}^5 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-9} \theta^{5}_i \phi \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\tilde{P}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\mathcal{m}_\mathrm{surf}^6 \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \theta^{6}_i \phi^{2} \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\tilde{r}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\mathcal{m}_\mathrm{surf}^{-2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{3} \theta^{-2}_i (2\pi)^{-1/2}\eta \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\tilde{M}_r \equiv \frac{M_r}{M_\mathrm{tot}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\mathcal{m}_\mathrm{surf}^{-1}~ \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr) \, .</math>
  </td>
</tr>
</table>

  </li>
</ul>

====Bipolytrope With (n<sub>c</sub>, n<sub>e</sub>) = (&infin;, 3) and (&gamma;<sub>c</sub>, &gamma;<sub>e</sub>) = (1, 4/3)====
As we have pointed out in an [[SSC/Structure/LimitingMasses#Schönberg-Chandrasekhar_Mass|accompanying discussion]], this is the bipolytropic model that was presented by &hellip; <div align="center">{{ SC42figure }}</div>
to suggest that stellar models evolving up the giant branch cannot survive if the helium-core mass tries to climb above some limiting mass ratio.