View source for Appendix/Ramblings/OriginOfPlanetaryNebulae

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=On the Origin of Planetary Nebulae=
This chapter &#8212; initially created by J. E. Tohline on 19 November 2016 &#8212; is intended primarily to provide a summary of the research that has been undertaken following <font color="red">a discussion that took place on 3 July 2013</font> with Kundan Kadam (an LSU graduate student, at the time) regarding the stability of bipolytropes.

<table border="0" cellpadding="3" align="center" width="60%">
<tr><td align="left">
<font color="darkgreen">
"This approximation is as dangerously crude as it is computationally economic."</font>
</td></tr>
<tr><td align="right">
&#8212; Drawn from &sect;I of [http://adsabs.harvard.edu/abs/1975ApJ...200..324H H&auml;rm &amp; Schwarzschild (1975)] 
</td></tr></table>

==Context==
Why do stars become red giants?  In particular, why does a star on the main sequence &#8212; whose internal density profile is only moderately centrally concentrated &#8212; become a red giant &#8212; which has a highly centrally condensed structure &#8212; at the end of the core-hydrogen-burning phase of its evolution?  It seems likely that this evolutionary transition is triggered by an instability associated with the [[SSC/Structure/LimitingMasses#Sch.C3.B6nberg-Chandrasekhar_Mass|Sch&ouml;nberg-Chandrasekhar mass limit]].  

''Rationale'': &nbsp; As hydrogen fuel is exhausted at the center of the star and burning shifts predominantly to a surrounding, off-center shell location, the helium core that is left behind is inert and approximately isothermal because the helium, itself, is not hot enough to burn.   [http://adsabs.harvard.edu/abs/1941ApJ....94..525H Henrich &amp; Chandraskhar (1941)] and [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Sch&ouml;nberg &amp; Chandrasekhar (1942)] discovered that equilibrium stellar structures with isothermal cores ''can'' be constructed, but only if the fraction of the star's mass that is contained in the core is below a well-defined, limiting value.  This so-called Sch&ouml;nberg-Chandrasekhar mass limit was initially identified as a "[[SSC/FreeEnergy/EquilibriumSequenceInstabilities#Instabilities_Associated_with_Equilibrium_Sequence_Turning_Points|turning point]]" along a sequence of equilibrium [[SSC/Structure/BiPolytropes#BiPolytropes|''bipolytrope'']] models in which the effective polytropic index of the core (c) and envelope (e) were, <math>~(n_c, n_e) = (\infty, 3/2)</math>.  Evolution along this equilibrium sequence &#8212; ''toward'' the turning point &#8212; is naturally associated with stellar evolution off the main sequence.  Specifically, one expects to see a slow (secular) but steady increase in the fraction of the star's mass that is enclosed within the isothermal core as the hydrogen-burning shell slowly works its way outward from the center.  An examination of the bipolytropic models along this sequence also reveals that, as the mass of the isothermal core increases, the star's equilibrium structure becomes more and more centrally condensed.  As a result &#8212; as has been emphasized by, for example, [http://adsabs.harvard.edu/abs/1998MNRAS.298..831E Eggleton, Faulkner, and Cannon (1998)] &#8212; the substantial structural change that occurs in a star as it evolves from the main sequence toward the red giant branch may be simply a natural consequence of evolution toward the Sch&ouml;nberg-Chandrasekhar mass limit.

''Motivating Questions'': &nbsp; With this general scenario in mind, we began to wonder &#8212; as, almost certainly, other astrophysicists before us have wondered:
* What type of instability &#8212; dynamical or secular (?) &#8212; is associated with the equilibrium sequence turning point that is synonymous with the Sch&ouml;nberg-Chandrasekhar mass limit?
** If it is a ''secular'' instability, do stars normally find a way &#8212; via one or more secular mechanisms &#8212; to readjust their structure as they approach the turning point and avoid encountering the mass limit altogether?
** If it is a ''dynamical'' instability, what is the &#8212; presumably catastrophic &#8212; result of encountering the Sch&ouml;nberg-Chandrasekhar mass limit?  Does the core collapse on a free-fall time scale; is the envelope ejected instead?  Or, perhaps envelope ejection occurs in concert with the core's collapse?  <br /><font color="darkgreen">NO!!! This EUREKA moment is WRONG!</font> [<font color="red">EUREKA!  In July 2022, I finally realized that, in the context of bipolytropes, the Sch&ouml;nberg-Chandrasekhar mass limit is precisely associated with the onset of a dynamical instability.  But the unstable eigenfunction is ''not'' the fundamental mode; it is the 1<sup>st</sup> overtone.  This very likely indicates that envelope ejection occurs in concert with the core's collapse!</font>]
* Might evolution toward the Sch&ouml;nberg-Chandrasekhar mass limit be hastened in situations where the hydrogen-shell-burning (bipolytropic) star has a binary companion?  The natural, gradual expansion of the star's envelope as it evolves off of the main sequence may bring its surface into contact with the binary system's Roche lobe and, as a result, some of the star's mass will be transferred to its companion.  This means that, even if the amount of mass contained within the inert helium (isothermal) core does not increase, the ''fraction'' of the star's mass that is contained in the core is destined to increase because the star's total mass is decreasing as a result of mass transfer.  If the mass-transfer rate is high enough, perhaps the secular mechanisms that help an isolated star avoid the Sch&ouml;nberg-Chandrasekhar mass limit will not have sufficient time to operate and, as a result, the evolving star is pushed past the limit.  How catastrophic is this?
* Perhaps envelope ejection &#8212; and the consequential development of wonderfully photogenic planetary nebulae &#8212; is a natural outcome of evolving stars encountering the Sch&ouml;nberg-Chandrasekhar mass limit.  And perhaps a star is more likely to be pushed to/past this limit if it has a binary companion. 

''Proposed Numerical Investigation'': &nbsp; The LSU astrophysics group ought to employ its three-dimensional hydrodynamic code to investigate what happens when a bipolyropic star (the donor) &#8212; with an isothermal (or nearly isothermal) core and a core-to-total mass ratio that is near the Sch&ouml;nberg-Chandrasekhar mass limit &#8212; fills its Roche lobe and transfers mass to its stellar companion (the accretor).  After a fairly predictable amount of (envelope) mass has been transferred from the donor to the accretor, the donor should encounter the Sch&ouml;nberg-Chandrasekhar mass limit.  What will the result be?  Does the initially bipolytropic donor's internal structure readjust on a dynamical time scale in response to this encounter?  Does its core collapse; and/or does its envelope rapidly expand?

==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.

==Toy Model==

Let's play with a bipolytropic configuration having <math>(n_c, n_e) = (5, 1)</math> and <math>\mu_e/\mu_c = \tfrac{1}{2}</math>.  Evolution will be dictated by the steady increase of the radial location of the core/envelope interface, <math>\xi_i</math> from zero (initially) to <math>\xi_i|_\mathrm{final} = 2.279258</math> which &#8212; [[SSC/Stability/BiPolytropes#Fundamental_Modes|according to our solution of the bipolytropic LAWE]] &#8212; marks the onset of dynamical instability.

As evolution proceeds along the <math>\mu_e/\mu_c = \tfrac{1}{2}</math> sequence, we want to hold <math>M_\mathrm{tot}</math> constant.  We must choose one additional fixed parameter; it isn't immediately obvious what is the best choice, but we will try setting <math>K_c</math> constant.  ([[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_=_5_and_ne_=_1|Originally]], we held the central density, <math>\rho_0</math>, and <math>K_c</math> constant.)  The relevant renormalization is [[SSC/Structure/BiPolytropes/Analytic51Renormalize|detailed in our chapter tagged SSC/Structure/BiPolytropes/Analytic51Renormalize]].

===Zero-Age Main Sequence (ZAMS) Configuration===
As a check, we note that in the initial (ZAMS) equilibrium configuration <math>(\xi_i=0)</math> there is no "core" &#8212; that is, <math>M_\mathrm{core} = 0</math> &#8212; so its structure should match that of an [[SSC/Structure/Polytropes#n_=_1_Polytrope|isolated <math>n = 1</math> polytrope]].  Specifically, given that, <math>M_\odot = 1.99\times 10^{33}~\mathrm{g}</math>, <math>R_\odot = 6.96\times 10^{10}~\mathrm{cm}</math>, and <math>G = 6.674\times 10^{-8}~\mathrm{cm^3~g^{-1}~s^{-2}}</math> &hellip; 
<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right"><math>M_\mathrm{tot}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\frac{4}{\pi} \rho_c R^3</math></td>
<td align="center">&nbsp; &nbsp; &nbsp; <math>\Rightarrow</math> &nbsp; &nbsp; &nbsp; </td>
  <td align="right"><math>\rho_c\biggr|_\mathrm{initial}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\frac{\pi}{4} \cdot \frac{M_\odot}{R_\odot^3} = 4.64~\mathrm{g~cm^{-3}} \, .</math></td>
</tr>
</table>
Also,
<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right"><math>\frac{\rho_c}{\bar\rho}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\frac{\pi^2}{3}</math></td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; </td>
  <td align="right"><math>P_c\biggr|_\mathrm{initial}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\frac{\pi G}{8} \biggl( \frac{M_\odot^2}{R_\odot^4} \biggr) = 4.42 \times 10^{15}~\mathrm{g~cm^{-1}~s^{-2}} \, .</math></td>
</tr>
</table>

<table border="1" align="center" cellpadding="5" width="80%"><tr><td align="left">
The sun's luminosity is, <math>L_\odot = 4\pi R_\odot^2 \sigma_\mathrm{SB} T^4 = 3.846\times 10^{33}~\mathrm{ergs~s^{-1}}</math>, where, in terms of {{ Math/C_RadiationConstant }} and {{ Math/C_SpeedOfLight }}, the Stefan-Boltzmann constant is,
<div align="center">
<math>\sigma_\mathrm{SB} = \frac{a_\mathrm{rad} c}{4} = 5.671\times 10^{-5} ~\mathrm{erg~cm^{-2}~s^{-1}~K^{-4}}\, .</math>
</div>
Hence, in the context of our ZAMS toy model, the configuration's surface temperature is,

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>T\biggr|_\mathrm{initial}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\biggl[ \frac{L_\odot}{4\pi R_\odot^2 \sigma_\mathrm{SB} } \biggr]^{1 / 4} = 5777~\mathrm{K} \, .</math>
  </td>
</tr>
</table>

----

In the context of our examination of the [[#IbenFigure|Iben (1967) evolutionary track, below]], we can write quite generally that,

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>L</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>4\pi R^2 \sigma_\mathrm{SB}T_e^4</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ \biggl(\frac{R}{R_\odot}\biggr)^2 </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\frac{L}{L_\odot}\biggl( \frac{T_e}{5777 K}\biggr)^{-4}</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ \log\biggl(\frac{R}{R_\odot}\biggr) </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\tfrac{1}{2}\log\biggl(\frac{L}{L_\odot} \biggr)
- 2 \log\biggl( \frac{T_e}{5777 K}\biggr)</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\tfrac{1}{2}\log\biggl(\frac{L}{L_\odot} \biggr)
- 2 \log T_e + 7.523 \, .</math>
  </td>
</tr>
</table>

----

Note as well that if an evolving star radiates at an average luminosity, <math>L_\mathrm{avg}/L_\odot</math>, over a specified time interval in units of <math>10^9~\mathrm{yrs}</math>, <math>(\Delta t)_9</math>, during the specified interval of time it will have burned through an amount of mass, <math>\Delta M/M_\odot</math>, given by the expression,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>0.007(\Delta M)c^2</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>(\Delta t)L_\mathrm{avg}</math></td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ \frac{(\Delta M)}{M_\odot}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
(\Delta t)_9 \cdot \frac{L_\mathrm{avg}}{L_\odot} \biggl[
\frac{L_\odot}{0.007 M_\odot c^2} 
\cdot 3.156\times 10^{16}~\mathrm{s}
\biggr]
=
0.01 \biggl[ (\Delta t)_9 \cdot \frac{L_\mathrm{avg}}{L_\odot}  \biggr] \, ,
</math>
  </td>
</tr>
</table>
where we have assumed that the efficiency of burning hydrogen to helium is 0.7%.
</td></tr></table>

Most notably, the dimensionless radius that appears in the classic Lane-Emden equation is given by the expression,

<div align="center">
<math>
\xi \equiv \frac{r}{a_\mathrm{n = 1}} ,
</math>
</div>
where,
<div align="center">
<math>
a_\mathrm{n=1} \equiv \biggl[\frac{(n+1)K_n}{4\pi G} \cdot \rho_c^{(1-n)/n} \biggr]^{1/2} 
=
\biggl[\frac{K_\mathrm{env}}{2\pi G} \biggr]^{1/2}
\, .
</math>
</div>
And, given that the surface (<math>r = R_\odot</math>) of the isolated <math>n=1</math> polytrope occurs at <math>\xi = \xi_1 = \pi</math>, we find that,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>a_\mathrm{n=1}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\frac{R_\odot}{\pi}</math></td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ K_\mathrm{env}\biggr|_\mathrm{initial}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\biggl( \frac{2}{\pi} \biggr) G R_\odot^2 = 2.06 \times 10^{14}~\mathrm{cm^5~g^{-1}~s^{-2}} \, .</math></td>
</tr>
</table>
Double-check:
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>P_c\biggr|_\mathrm{initial}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
K_\mathrm{env}\biggr|_\mathrm{initial}\biggl[ \rho_c\biggr|_\mathrm{initial}\biggr]^2
=
4.42\times 10^{15} ~\mathrm{g~cm^{-1}~s^{-2}} \, .
</math>
  </td>
</tr>
</table>
Yes!


Now, the instant the "core" shows up, we demand that its central density be a factor of <math>(\mu_e/\mu_c)^{-1}</math> larger than the density at the inner edge of the envelope; at the same time, we demand that the pressures be the same.  Hence the relevant polytropic equation of state for the core must be,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>P</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
K_\mathrm{core}\biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \rho \biggr]^{6 / 5} \, .
</math>
  </td>
</tr>
</table>
In our toy model, this means that,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>K_\mathrm{core}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
P \biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \rho \biggr]^{-6 / 5} 
=
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{6 / 5} 4.42\times 10^{15} ~\mathrm{g~cm^{-1}~s^{-2}} 
\biggl[ 4.64~\mathrm{g~cm^{-3}} \biggr]^{-6 / 5}
=
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{6 / 5} 7.008\times 10^{14} \mathrm{g^{-1 / 5}~cm^{13 / 5}~s^{-2}}
\, .
</math>
  </td>
</tr>
</table>
Hence, the ratio,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>\frac{K_\mathrm{env}}{K_\mathrm{core}}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2.06 \times 10^{14}~\mathrm{cm^5~g^{-1}~s^{-2}}
\biggl\{\biggl(\frac{\mu_e}{\mu_c}\biggr)^{6 / 5} 7.008\times 10^{14} \mathrm{g^{-1 / 5}~cm^{13 / 5}~s^{-2}}\biggr\}^{-1}
=
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-6 / 5}0.293 ~\mathrm{g^{-4 / 5}~cm^{12 / 5}}
\, ,
</math>
  </td>
</tr>
</table>
which may be rewritten as,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>\frac{K_\mathrm{env}}{K_\mathrm{core}}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{4/ 5}0.293 \biggl[\mathrm{g~cm^{-3}} \biggr]^{- 4 / 5}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
=\biggl[ \underbrace{ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\pi}{4}\cdot \frac{M_\odot}{R_\odot^3} }_\mathrm{central~ density~of~n = 5 ~ core} \biggr]^{- 4 / 5}
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \, .
</math>
  </td>
</tr>
</table>

This matches the [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface condition expression]] that relates <math>K_\mathrm{core}</math> to <math>K_\mathrm{env}</math>,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>\frac{K_e}{K_c}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\rho_c^{-4 / 5} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \theta_i^{-4} \, ,
</math>
  </td>
</tr>
</table>
where we appreciate that, in the initial bipolytropic configuration (i.e., when <math>\xi_i = 0</math>), <math>\theta_i = 1</math>.

===Adopted Normalizations===
We adopt the normalizations in which we hold <math>M_\mathrm{tot}</math> and <math>K_c</math> constant, as [[SSC/Structure/BiPolytropes/Analytic51Renormalize#NewNormalization|derived in an accompanying discussion]].  That is,

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

<tr>
  <td align="right"><math>r_\mathrm{norm} \equiv \biggl[\biggl( \frac{G}{K_c} \biggr)^{5 / 2} M_\mathrm{tot}^{2}  \biggr]</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>
\biggl[ 6.674\times 10^{-8}~\mathrm{cm^3~g^{-1}~s^{-2}}\biggr]^{5 / 2}  \times \biggl[
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{6 / 5} 7.008\times 10^{14} \mathrm{g^{-1 / 5}~cm^{13 / 5}~s^{-2}} \biggr]^{-5 / 2} 
\biggl( 1.99\times 10^{33}~\mathrm{g} \biggr)^{2} 
</math></td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3} 3.505\times 10^{11}~\mathrm{cm} 
\, ;
</math></td>
</tr>

<tr>
  <td align="right"><math>\rho_\mathrm{norm} \equiv \biggl[\biggl( \frac{K_c}{G} \biggr)^{3 / 2} \frac{1}{M_\mathrm{tot}} \biggr]^{5}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl\{
\biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{6 / 5} 7.008\times 10^{14} \mathrm{g^{-1 / 5}~cm^{13 / 5}~s^{-2}} \biggr]^{3 / 2} 
\biggl[ 6.674\times 10^{-8}~\mathrm{cm^3~g^{-1}~s^{-2}} \biggr]^{-3 / 2}
\biggl[1.99\times 10^{33}~\mathrm{g}\biggr]^{-1}
\biggr\}^5 
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{9} 0.0462 ~\mathrm{g}~\mathrm{cm^{-3}} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>P_\mathrm{norm} \equiv \biggl[K_c^{10} G^{-9} M_\mathrm{tot}^{-6}  \biggr]</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>
\biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{6 / 5} 7.008\times 10^{14} \mathrm{g^{-1 / 5}~cm^{13 / 5}~s^{-2}} \biggr]^{10} 
\biggl[ 6.674\times 10^{-8}~\mathrm{cm^3~g^{-1}~s^{-2}} \biggr]^{-9} 
\biggl[ 1.99\times 10^{33}~\mathrm{g} \biggr]^{-6} 
  </math></td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{12} \biggl\{
\biggl[ 7.008\times 10^{14}  \biggr] 
\biggl[ 6.674\times 10^{-8} \biggr]^{-0.9} 
\biggl[ 1.99\times 10^{33}~\biggr]^{-0.6} 
\biggr\}^{10}~\mathrm{g ~cm^{-1}~s^{-2} }
  </math></td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{12} 1.751\times 10^{13} ~\mathrm{g ~cm^{-1}~s^{-2} }\, .
  </math></td>
</tr>
</table>

<table border="1" align="center" cellpadding="5">

<tr>
  <td align="center">Parameter</td>
  <td align="center">[[SSC/Structure/BiPolytropes/Analytic51Renormalize#CoreParameters|Core]]</td>
  <td align="center">[[SSC/Structure/BiPolytropes/Analytic51Renormalize#EnvelopeParameters|Envelope]]</td>
</tr>

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

<tr>
  <td align="right">
<math>\rho = \rho_\mathrm{norm} \tilde{\rho} =</math>
  </td>
  <td align="center">
<math>\rho_\mathrm{norm}
\biggl\{
\mathcal{m}_\mathrm{surf}^5 \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-10} 
\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2} 
\biggr\}
</math>
  </td>
  <td align="center">
<math>\rho_\mathrm{norm} 
\biggl\{
\mathcal{m}_\mathrm{surf}^5 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-9} \theta^{5}_i \phi 
\biggr\}
</math>
  </td>
</tr>

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

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

'''Polytropic Functions and Their Derivatives &hellip;'''
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">Core:</td>
  <td align="center">
<math>
\theta(\xi) = \biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{-1/2}
</math>
  </td>
<td align="center" width="10%">&nbsp;</td>
  <td align="center">
<math>
\frac{d\theta}{d\xi} = - \frac{\xi}{3}\biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{-3/2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">Envelope:</td>
  <td align="center">
<math>
\phi = A \biggl[ \frac{\sin(\eta - B)}{\eta} \biggr]
</math>
  </td>
<td align="center">&nbsp;</td>
  <td align="center">
<math>
\frac{d\phi}{d\eta} = \frac{A}{\eta^2} \biggl[ \eta\cos(\eta-B) - \sin(\eta-B) \biggr] 
</math>
  </td>
</tr>
</table>

'''Key Relations'''; numerical values in parentheses assume <math>(\mu_e/\mu_c) = \tfrac{1}{2}</math> and <math>\xi_i = 2.279258</math> &hellip; [[SSC/Stability/BiPolytropes#Equilibrium_Properties_of_Marginally_Unstable_Models|cross-check here]]:
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>\eta 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
3^{1 / 2}\biggl(\frac{\mu_e}{\mu_c}\biggr) \theta_i^2 \xi \, ,
</math>
  </td>
  <td align="center" width="10%">&nbsp;</td>
  <td align="left">(0.722596)</td>
</tr>

<tr>
  <td align="right"><math>\Lambda_i</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\frac{\xi_i}{\sqrt{3}} 
\biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1}\frac{1}{\theta_i^2 \xi_i^2} - 1\biggr] \, ,</math></td>
  <td align="center" width="10%">&nbsp;</td>
  <td align="left">(0.067969)</td>
</tr>

<tr>
  <td align="right"><math>A</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math> 
\eta_i\biggl(1 + \Lambda_i^2\biggr)^{1 / 2} \, ,</math></td>
  <td align="center" width="10%">&nbsp;</td>
  <td align="left">(0.724263)</td>
</tr>

<tr>
  <td align="right"><math>B</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\eta_i + \tan^{-1}(\Lambda_i) - \frac{\pi}{2} \, ,</math></td>
  <td align="center" width="10%">&nbsp;</td>
  <td align="left">(-0.780336)</td>
</tr>

<tr>
  <td align="right"><math>\eta_s</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math> 
\frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \, ,</math></td>
  <td align="center" width="10%">&nbsp;</td>
  <td align="left">(2.361257)</td>
</tr>

<tr>
  <td align="right"><math>\mathcal{m}_\mathrm{surf}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i} \, ,</math></td>
  <td align="center" width="10%">&nbsp;</td>
  <td align="left">(2.255246)</td>
</tr>

<tr>
  <td align="right"><math>\frac{\rho_c}{\bar\rho}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s^2}{3A\theta_i^5} \, ,</math></td>
  <td align="center" width="10%">&nbsp;</td>
  <td align="left">(63.29514)</td>
</tr>

<tr>
  <td align="right"><math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^2 \sqrt{3} ~ \biggl[ \frac{\xi_i^3 \theta_i^4}{A\eta_s}\biggr] \, ,</math></td>
  <td align="center" width="10%">&nbsp;</td>
  <td align="left">(0.401776)</td>
</tr>

<tr>
  <td align="right"><math>q \equiv \frac{r_\mathrm{core}}{R}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr) \sqrt{3} ~ \biggl[ \frac{\xi_i \theta_i^2}{\eta_s}\biggr] \, .</math></td>
  <td align="center" width="10%">&nbsp;</td>
  <td align="left">(0.306022)</td>
</tr>

<tr>
  <td align="right"><math>R_\mathrm{surf}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>r_\mathrm{norm} 
\biggl\{
\mathcal{m}_\mathrm{surf}^{-2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{3} \theta^{-2}_i (2\pi)^{-1/2}\eta_s 
\biggr\}</math></td>
  <td align="center" width="10%">&nbsp;</td>
  <td align="left"><math>(0.063242 ~r_\mathrm{norm} = 1.773\times 10^{11} ~\mathrm{cm} = 2.548~R_\odot)</math></td>
</tr>
</table>

===Example Evolution===

Let's attempt to construct the post-main-sequence evolutionary trajectory for <math>1 M_\odot</math> stars having a couple of different values of the mean-molecular weight jump, <math>\mu_e/\mu_c</math>.  This will be for comparison with [[#IbenFigure|Iben's (1967a) evolution]] displayed below.  Our toy model  does not handle radiation, so we cannot plot Luminosity versus effective (surface) temperature, that is, we cannot generate a traditional HR diagram.  However, our toy model predicts how the radius of the star should vary with the mass of its core.  In the following figure, the solid-green circular markers show how <math>\nu</math> varies with <math>R_\mathrm{surf}/R_\odot</math> for the model sequence in which <math>\mu_e/\mu_c = 1 / 2</math> &#8212; note, for example, that among our above numerically evaluated "key relations", <math>(\nu, R_\mathrm{surf}/R_\odot) = (0.40178, 2.548)</math>; this point along the curve is marked by the solid-red circular marker.  The solid-yellow circular markers show the same information for the model sequence in which <math>\mu_e/\mu_c = 1 / 4</math>.

<table border="0" align="center" cellpadding="5">
<tr>
  <td align="center">
[[File:EvolveComparison3Labeled.png|center|500px|One Solar Mass Evolution]]
  </td>
  <td align="center">
<table border="1" align="center" cellpadding="5">

<tr>
  <td align="center" colspan="6">Parameters values deduced from<br />Figure 1 and Table 1 of [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..624I/abstract Iben67a]</td>
</tr>

<tr>
  <td align="center">Interval</td>
  <td align="center"><math>(\Delta t)_9</math></td>
  <td align="center"><math>\biggl[\log \biggl(\frac{L}{L_\odot}\biggr)\biggr]_\mathrm{avg}</math></td>
  <td align="center"><math>\biggl[\log \biggl(\frac{R}{R_\odot}\biggr)\biggr]_\mathrm{avg}</math></td>
  <td align="center"><math>\Delta (M/M_\odot)</math></td>
  <td align="center"><math>\nu</math></td>
</tr>

<tr>
  <td align="center">1 - 2</td>
  <td align="center">3.770</td>
  <td align="center">-0.070</td>
  <td align="center">-0.05</td>
  <td align="center">0.0311</td>
  <td align="center">0.0311</td>
</tr>

<tr>
  <td align="center">2 - 3</td>
  <td align="center">2.889</td>
  <td align="center">0.06</td>
  <td align="center">-0.003</td>
  <td align="center">0.0322</td>
  <td align="center">0.0633</td>
</tr>

<tr>
  <td align="center">3 - 4</td>
  <td align="center">1.462</td>
  <td align="center">0.170</td>
  <td align="center">0.044</td>
  <td align="center">0.0210</td>
  <td align="center">0.0842</td>
</tr>

<tr>
  <td align="center">4 - 5</td>
  <td align="center">1.029</td>
  <td align="center">0.275</td>
  <td align="center">0.0975</td>
  <td align="center">0.0188</td>
  <td align="center">0.1030</td>
</tr>

<tr>
  <td align="center">5 - 6</td>
  <td align="center">0.7018</td>
  <td align="center">0.38</td>
  <td align="center">0.170</td>
  <td align="center">0.0163</td>
  <td align="center">0.1193</td>
</tr>

<tr>
  <td align="center">6 - 7</td>
  <td align="center">0.292</td>
  <td align="center">0.445</td>
  <td align="center">0.253</td>
  <td align="center">0.0079</td>
  <td align="center">0.1272</td>
</tr>

<tr>
  <td align="center">7 - 10</td>
  <td align="center">0.157</td>
  <td align="center">0.455</td>
  <td align="center">0.333</td>
  <td align="center">0.0043</td>
  <td align="center">0.1316</td>
</tr>

<tr>
  <td align="center">10 - 11</td>
  <td align="center">0.213</td>
  <td align="center">0.51</td>
  <td align="center">0.432</td>
  <td align="center">0.0067</td>
  <td align="center">0.1383</td>
</tr>

<tr>
  <td align="center">11 - 12</td>
  <td align="center">0.185</td>
  <td align="center">0.68</td>
  <td align="center">0.560</td>
  <td align="center">0.0086</td>
  <td align="center">0.1468</td>
</tr>

<tr>
  <td align="center">12 - 13</td>
  <td align="center">0.125</td>
  <td align="center">0.925</td>
  <td align="center">0.708</td>
  <td align="center">0.0102</td>
  <td align="center">0.1570</td>
</tr>
</table>
  </td>
</tr>
</table>

The parameter values displayed in the table shown here &#8212; immediately on the right of our figure &#8212; were determined as follows, referencing the data from [[#IbenFigure|Iben67a presented below]]:
<ul>
  <li>Interval: &#8212;The points displayed along the <math>1 M_\odot</math> evolutionary trajectory in Figure 1 of Iben67a are numbered in chronological order, from 1 to 13 (except points 8 and 9 are not included).  Here, our parameter values reference the ''time intervals'' between adjacent points; for example, our table refers to interval "1 - 2," interval "2 - 3," etc.</li>
  <li><math>(\Delta t)_9</math>: &nbsp; This refers to the amount of evolutionary time that separates the two points of an identified interval.  For example, Table 1 of Iben67a states that the star's age at "point 1" along its evolutionary trajectory is <math>0.0506\times 10^{9}~\mathrm{yrs}</math> and at "point 2" its age is <math>3.8209\times 10^{9}~\mathrm{yrs}</math>; hence the interval of time between points 1 and 2 is <math>(\Delta t)_9 = (3.8209 - 0.0506)\times 10^{9}~\mathrm{yrs}= 3.770 \times 10^{9}~\mathrm{yrs}</math>.</li>
  <li><math>[\log_{10}(L/L_\odot)]_\mathrm{avg}</math>:&nbsp; We have performed a linear interpolation ''in log<sub>10</sub>'' of the luminosity listed for each point along the evolutionary trajectory in order to obtain an ''average'' value of the luminosity across each interval.  For example, the star's luminosity at "point 1" is approximately <math>\log(L/L_\odot) \approx -0.140</math> while at "point 2" it is <math>\log(L/L_\odot) \approx 0.0</math>.  Hence, while evolving across the 1 - 2 interval, we assume that its luminosity is constant and has the value, <math>[\log_{10}(L/L_\odot)]_\mathrm{avg} = \tfrac{1}{2}[0.0 + (- 0.140)] = -0.07</math>.</li>
  <li><math>[\log_{10}(R/R_\odot)]_\mathrm{avg}</math>:&nbsp; The average stellar radius over an interval is obtained in the same way.  For example, while evolving across the 1 - 2 interval, we assume that its radius is constant and has the value, <math>[\log_{10}(R/R_\odot)]_\mathrm{avg} = \tfrac{1}{2}[-0.073 + (- 0.027)] = -0.05</math>.</li>
  <li><math>\Delta (M/M_\odot)</math>: &nbsp; For a given interval, if we multiply the star's average luminosity times the time spent crossing that interval, we obtain a reasonable estimate of the total amount of energy, <math>E_\mathrm{\Delta t}</math>, that the star has released over that time interval.  Presumably, the same amount of energy was generated via nuclear reactions over that interval.  Assuming that the primary reaction is the conversion of hydrogen into helium, the conversion of mass into energy occurs at an efficiency of 0.7%, that is, a reasonable estimate of the amount of mass that gets burned during that interval is,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>0.007 (\Delta M)c^2</math></td>
  <td align="right"><math>\approx</math></td>
  <td align="left"><math>\Delta t \cdot L_\mathrm{avg}</math></td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ \frac{\Delta M}{M_\odot}</math></td>
  <td align="right"><math>\approx</math></td>
  <td align="left"><math>\biggl[\frac{(\Delta t)_9 \times 3.156\times 10^{16}~\mathrm{s}}{0.007 c^2}\biggr] \cdot 10^{[\log(L/L_\odot)]_\mathrm{avg}} \biggl\{ \frac{L_\odot}{M_\odot} \biggr\}
= 0.00970 \times (\Delta t)_9\cdot 10^{[\log(L/L_\odot)]_\mathrm{avg}} \, .
</math></td>
</tr>
</table>
For example, for interval 1 - 2, <math>\Delta M/M_\odot \approx (0.0097 \times 3.770 \times 10^{-0.070}) = 0.0311</math>.
</li>
  <li><math>\nu</math>: &nbsp; This is fraction of the star's total mass that has accumulated in the core; in our toy model, the total mass is <math>1 M_\odot</math> so <math>\nu</math> is also the core mass expressed in solar masses.  Here the value of <math>\nu</math> is obtained by summing over the mass increments, <math>\Delta(M/M_\odot)</math>, that have been burned at all previous evolutionary time intervals, including the present one; for example, at the end of time interval 3 - 4, <math>\nu = 0.0311+0.0322+0.0210 = 0.0842</math>.</li>
</ul>

=Textbook Explanations=

==Clayton (1968)==

Here, we consider the descriptions presented by [http://adsabs.harvard.edu/abs/1968psen.book.....C D. D. Clayton (1968)].

<font color="red">'''Evolution to the Red-Giant Branch:'''</font> (&sect;6-7, p. 485) "<font color="darkgreen">The core continues to contract as the hydrogen is exhausted, leaving a central region of helium plus heavier trace elements.  This helium core will tend to be isothermal because nuclear energy generation has ceased &hellip;</font>"  As the star's evolution proceeds, the temperature of the inert (isothermal) core will continue to increase, as will "<font color="darkgreen">the temperature of a shell of hydrogen surrounding the core &hellip; The increased internal temperatures require the expansion of the stellar radius to keep the temperature gradient at a consistently low level.  The star therefore reddens at a relatively rapid rate while the hydrogen-burning shell slowly increases the mass of the helium core.</font>"

<font color="red">'''Stellar Pulsation:'''</font> (&sect;6-10, p. 504) "<font color="darkgreen">By 1930 it was clear, thanks largely to the work of Eddington, that a pulsating star must in fact be some type of heat engine, in which some continuously operating mechanism transforms thermal energy into the mechanical energy of the oscillation.</font>"  Analyses that attempt to explain the existence and properties of ''regular variable stars'' &#8212; such as the Cepheids and RR Lyrae variables &#8212; focus on stellar (envelope) structures that are dynamically stable, according to ''adiabatic'' stability analyses, but that harbor a tendency toward growing oscillatory amplitude when non-adiabatic effects are considered.  Specifically referencing the three terms in equation (6-116) on p. 511, we can identify the principal "<font color="darkgreen">&hellip; physical effects contributing to the status of the stability of the zone.</font>" 
* ''&Gamma; mechanism:''  "<font color="darkgreen">The first term always contributes to stability &hellip; [but its] influence is diminished in ionization zones.</font>"
* ''&kappa; mechanism:''  "<font color="darkgreen">The second term reflects the way in which the opacity varies during the pulsation.  Positive values of <math>~\kappa_T</math> and <math>~\kappa_P</math> would imply that the opacity increases upon contraction, which would remove energy from the radiation flux &hellip; at the proper time to drive mechanical work.</font>"

<font color="red">'''Mass Loss:'''</font> (&sect;6-9, p. 501) "<font color="darkgreen">Mass loss is a self-descriptive term that is used to describe any process by which the main body of the star, defined as the gravitationally bound mass, reduces its mass by ejecting surface layers &hellip; Mass loss can occur in a variety of forms and can be initiated by a variety of physical mechanisms.  Any catastrophic event in which a massive outer layer is lifted off into space by some internal instability must result in a drastically new structure for the remaining core.  So special are these circumstances that they will not be discussed here..</font>"

----

<span id="IbenFigure">Figure 6-17 (p. 492)</span> in Clayton (1968) displays a reproduction of Figure 1 from {{ Iben67full }}.  It displays evolutionary tracks of lower-main-sequence population I stars of mass <math>1 M_\odot , 1.25 M_\odot ,</math> and <math>1.5 M_\odot</math>.  Using a pen and ruler to draw scale lines across the figure, the following table catalogues the data from which the <math>1 M_\odot</math> track in the figure was produced.

<table border="0" align="center" cellpadding="0">
<tr><td align="center">
<table border="1" align="center" cellpadding="5">

<tr>
  <td align="center" colspan="4">Extracted from Published Figure</td>
  <td align="center" colspan="1">Deduced<sup>&dagger;</sup></td>
</tr>

<tr>
  <td align="center">Numbered<br />Point</td>
  <td align="center">Age<br /><math>(10^9~\mathrm{yrs})</math></td>
  <td align="center"><math>\log \biggl(\frac{L}{L_\odot}\biggr)</math></td>
  <td align="center"><math>\log T_e</math></td>
  <td align="center"><math>\log \biggl(\frac{R}{R_\odot}\biggr)</math></td>
</tr>

<tr>
  <td align="center">1</td>
  <td align="center">0.05060</td>
  <td align="center">-0.140</td>
  <td align="center">3.763</td>
  <td align="center">-0.073</td>
</tr>

<tr>
  <td align="center">2</td>
  <td align="center">3.8209</td>
  <td align="center">0.0</td>
  <td align="center">3.775</td>
  <td align="center">-0.027</td>
</tr>

<tr>
  <td align="center">3</td>
  <td align="center">6.7100</td>
  <td align="center">0.120</td>
  <td align="center">3.781</td>
  <td align="center">0.021</td>
</tr>

<tr>
  <td align="center">4</td>
  <td align="center">8.1719</td>
  <td align="center">0.22</td>
  <td align="center">3.783</td>
  <td align="center">0.067</td>
</tr>

<tr>
  <td align="center">5</td>
  <td align="center">9.2012</td>
  <td align="center">0.33</td>
  <td align="center">3.780</td>
  <td align="center">0.128</td>
</tr>

<tr>
  <td align="center">6</td>
  <td align="center">9.9030</td>
  <td align="center">0.43</td>
  <td align="center">3.763</td>
  <td align="center">0.212</td>
</tr>

<tr>
  <td align="center">7</td>
  <td align="center">10.195</td>
  <td align="center">0.46</td>
  <td align="center">3.73</td>
  <td align="center">0.293</td>
</tr>

<tr>
  <td align="center">8</td>
  <td align="center">&hellip;</td>
  <td align="center">&hellip;</td>
  <td align="center">&hellip;</td>
  <td align="center">&hellip;</td>
</tr>

<tr>
  <td align="center">9</td>
  <td align="center">&hellip;</td>
  <td align="center">&hellip;</td>
  <td align="center">&hellip;</td>
  <td align="center">&hellip;</td>
</tr>

<tr>
  <td align="center">10</td>
  <td align="center">10.352</td>
  <td align="center">0.45</td>
  <td align="center">3.688</td>
  <td align="center">0.372</td>
</tr>

<tr>
  <td align="center">11</td>
  <td align="center">10.565</td>
  <td align="center">0.57</td>
  <td align="center">3.658</td>
  <td align="center">0.492</td>
</tr>

<tr>
  <td align="center">12</td>
  <td align="center">10.750</td>
  <td align="center">0.79</td>
  <td align="center">3.645</td>
  <td align="center">0.628</td>
</tr>

<tr>
  <td align="center">13</td>
  <td align="center">10.875</td>
  <td align="center">1.06</td>
  <td align="center">3.633</td>
  <td align="center">0.787</td>
</tr>
</table>

</td>
<td align="center" bgcolor="white">[[File:Fig1Labeled.png|450px|Iben67a Figure 1]]</font>
</td>

</tr>
</table>

<sup>&dagger;</sup>From [[#Zero-Age_Main_Sequence_%28ZAMS%29_Configuration|above]], we know that,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>\log\biggl(\frac{R}{R_\odot}\biggr) </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\tfrac{1}{2}\log\biggl(\frac{L}{L_\odot} \biggr)
- 2 \log T_e + 7.523 \, .</math>
  </td>
</tr>
</table>

It is via this expression that we have deduced how the radius of Iben's evolving configuration changes over time.

==Hansen &amp; Kawaler (1994)==

Here, we consider the descriptions presented by [<b>[[Appendix/References#HK94|<font color="red">HK94</font>]]</b>].

<font color="red">'''Evolution to the Red-Giant Branch &amp; the SC Limit:'''</font> (&sect;2.3, pp. 53-55)  "<font color="darkgreen">Following exhaustion of hydrogen in the core, the growing helium remnant is surrounded by an active shell of burning hydrogen, which supplies the power for the star.  The core itself, however, has no energy source of its own (except some input from contraction) and hence it tends to be isothermal since no temperature gradients are required.  It is at this point &hellip; that the structure of the star begins to change radically and the main sequence phase ends. &hellip; evolution to &hellip; the red giant branch (RGB) is characterized by a continual expansion and reddening of the star to lower [surface] temperatures. </font>" 

"<font color="darkgreen">In an early 1942 study Sch&ouml;nberg and Chandrasekhar demonstrated that when an isothermal helium core is built up to a mass corresponding to about 10% of the initial hydrogen mass of the star, it is no longer possible to maintain quasi-hydrostatic equilibrium for the core of the model star if pressure support is due to an ideal gas.  Other studies, including evolutionary calculations, support this by showing that the core contracts and heats rapidly &hellip; The envelope &hellip; however, responds by expanding rapidly &hellip;  This signals the end of the main sequence phase of evolution for the star &hellip;</font>"

<font color="red">'''Stellar Pulsation:'''</font>


<font color="red">'''Mass Loss &amp; Formation of Planetary Nebula:'''</font> (&sect;2.4.1, pp. 60-61)  "<font color="darkgreen">The ignition of helium under electron degenerate conditions &hellip;</font>" occurs via "<font color="darkgreen">&hellip; an explosive runaway or ''helium flash''. &hellip; The major evidence that leads us to believe that the helium flash is not explosive enough to disrupt the star in a serious fashion is that stars in the post-helium flash stage are observed and they constitute the horizontal branch of globular clusters, for example.</font>"

(&sect;2.4.1, p. 62)  "<font color="darkgreen">The critical point about the</font> [observationally determined] <font color="darkgreen">masses &hellip; for RR Lyrae stars, and therefore HB stars, is that they are less than the masses of their progenitor main sequence stars.  If these results are correct, then mass must have been lost during the time elapsed between the main sequence and the HB stages.  Such mass loss from red giants is observed, but the physical mechanism is not well understood.</font>"

==Rose (1998)==

Here, we consider the descriptions presented by [http://adsabs.harvard.edu/abs/1998asa..book.....R W. K. Rose (1998)].

<font color="red">'''Evolution to the Red-Giant Branch &amp; the SC Limit:'''</font> (&sect;8.2, p. 267)  "<font color="darkgreen">&hellip; after hydrogen depletion has occurred in their cores main-sequence stars evolve onto the red-giant branch.  Low-mass stars <math>~(\mathrm{roughly}~M \leq 1.2 M_\odot)</math>, which burn hydrogen by means of the proton-proton chain on the main sequence evolve gradually from main sequence to red-giant evolutionary stages &hellip; The cores of stars that are sufficiently massive <math>~(\mathrm{roughly}~M \geq 1.2 M_\odot)</math> to burn hydrogen by means of the CNO cycle on the main sequence contract rapidly (i.e., in a Kelvin-Helmholtz timescale) after hydrogen core exhaustion, and then evolve more rapidly onto the red-giant branch &hellip;</font>"

(&sect;8.2, p. 268)  "<font color="darkgreen">Because the thermonuclear energy release that results from hydrogen burning is very large &hellip; the time-derivative term in Equation (2.131) can be neglected in calculating main-sequence stellar models.  If</font> [this same] <font color="darkgreen">term is neglected in calculating post-main-sequence evolution then the calculated stellar models have isothermal cores that are surrounded by hydrogen-burning shells.  Numerical calculations show that isothermal cores consisting of a nondegenerate gas surrounded by a hydrogen-burning shell source do not exist if the core mass exceeds <math>~\approx 0.1 - 0.15</math> times the mass of the star.  These limiting isothermal core masses are referred to collectively as the Sch&ouml;nberg-Chandrasekhar limit.   The existence of a limiting isothermal core for a particular initial mass main-sequence star shows that core contraction must occur in post-main-sequence evolution.</font>"

(&sect;8.2, p. 269)  "<font color="darkgreen">Numerical solutions of the equations of stellar interiors show that as the core mass of a red giant increases, the luminosity and radius increase by a large factor but the core radius changes by only a small amount.</font>"

<font color="red">'''Stellar Pulsation:'''</font> (&sect;8.1, p. 260)  "<font color="darkgreen">The instability that drives pulsations in RR Lyrae variables, Cepheids and long-period variables is associated with hydrogen and helium ionization zones.  The large heat capacity of these ionization zones causes the phase of maximum luminosity to be delayed by approximately 90&deg; as compared to the phase of minimum radius &hellip; Extensive hydrogen ionization zones cause</font> [long-period variables] <font color="darkgreen"> to become unstable to radial pulsations &hellip; The pulsations of &hellip; Cepheids result from both hydrogen and helium ionization zones.</font>"

(&sect;1.5, p. 24)  "<font color="darkgreen">&hellip; asymptotic-giant-branch stars become pulsationally unstable after their luminosities become</font> [greater or on the order of] <math>~2500 L_\odot</math>

<font color="red">'''Mass Loss &amp; Formation of Planetary Nebula:'''</font> (&sect;8.1, p. 260)  "<font color="darkgreen">The [pulsation] amplitudes</font>" of long-period variables "<font color="darkgreen">become sufficiently large that shock waves are generated in their atmospheres.  The standard scenario for producing mass loss from these stars is that shock waves eject mass.</font>"

(&sect;1.5, p. 24)  "<font color="darkgreen">Long-period variables experience significant mass loss.  The final phase of mass loss on the red-giant branch leads to the formation of a planetary nebula.  If a luminous red giant ejects a mass shell, and as a consequence the remnant star becomes nearly hydrogen deficient, then the remnant star evolves rapidly off the red-giant branch and into the region of the H-R diagram occupied by the central stars of planetary nebulae.</font>"

==Padmanabhan (2000)==

Here, we consider the descriptions presented by [<b>[[Appendix/References#P00|<font color="red">P00</font>]]</b>]; note that in Chapter 3 of Volume II, subsection 3.4 is titled, ''<b>Evolution of High-Mass Stars</b>'' while subsection 3.5 is titled, ''<b>Evolution of Low-Mass Stars</b>''.

<font color="red">'''Evolution to the Red-Giant Branch &amp; the SC Limit:'''</font> (Vol. II, &sect;3.4.3, p. 142)  Once the hydrogen fuel in the core is nearly exhausted and hydrogen burning occurs primarily in a shell immediately surrounding the core, "<font color="darkgreen">Further evolution depends on the structural changes that take place in the [inert] helium core &hellip; the helium core is fairly homogeneous [in, for example, a <math>~5 M_\odot</math> star] because of the mixing that is due to the original convective transport &hellip; Further, it will be nearly isothermal because the vanishing of luminosity implies the vanishing of the temperature gradient.  The equilibrium of such a star depends on the ability of an isothermal core (with mass <math>~M_\mathrm{ic} \equiv qM</math>) to support the envelope of mass <math>~(1-q)M</math>.  It turns out that this is possible only if the fraction of the mass in the core is below a critical value called the</font>" Sch&ouml;nberg-Chandrasekhar (SC) limit.

For the remainder of &sect;3.4.3, [<b>[[Appendix/References#P00|<font color="red">P00</font>]]</b>] discusses in considerable detail &#8212; relying heavily on virial-theorem-based arguments &#8212; how the SC limit should be viewed in high-mass stars, where the core remains non-degenerate, versus in low-mass stars where electron degeneracy sets in.  Then in &sect;3.5.1 (p. 152), he re-emphasizes that "<font color="darkgreen">The effect of shell burning is &hellip; very different in low-mass stars compared with what we have seen in high-mass stars.  Because the cores are nearly degenerate, the [SC] limit is fairly irrelevant for low-mass stars.  As the burning shell causes the core mass to exceed <math>~\sim 0.1 M_\odot</math>, the core contraction would have produced sufficient degeneracy to circumvent the [SC] constraint.  At this stage, the core is made of degenerate, isothermal helium and no rapid core contraction occurs.</font>"

He also emphasizes the following.  (Vol. II, &sect;3.4.3, p. 148) "<font color="darkgreen">During</font>" evolution from the main sequence to the red-giant branch, "<font color="darkgreen">the core and the envelope regions behave in a very different way.  The study of the trajectories of different mass shells inside the star as functions of time based on numerical integration of equations of stellar evolution shows that the core collapses while the envelope expands.</font>"


<font color="red">'''Stellar Pulsation:'''</font> (Vol. II, &sect;3.7.2, p. 178)  Finite-amplitude, ''sustained'' oscillations in stars can only be explained in terms of ''non-adiabatic'' effects.  Such explanations are usually couched in terms of a measure of the "<font color="darkgreen">net amount of work done by each layer of the star during one cycle of oscillation &hellip; To drive the oscillations, heat must enter the layer during the high-temperature part of the cycle and exit during the low-temperature part.  Different layers of the star may have different phase relations as regards such a process, and whether the oscillations will be sustained or not will depend on the net effect.  Favourable circumstances for sustained oscillations occur if</font>," for example, the opacity in a layer of the envelope increases when the layer is compressed.  "<font color="darkgreen">(Under normal circumstances, opacity actually decreases with compression.) &hellip; [This] exception occurs in the layers of the star that are partially ionized &hellip; This mechanism is called the <b>&kappa; mechanism</b>.</font>"


<font color="red">'''Mass Loss &amp; Formation of Planetary Nebula:'''</font> (Vol. II, &sect;3.6, p. 163) "<font color="darkgreen">The rapid expansion of the star &hellip; implies that the outer regions of the star are very loosely bound.  Hence it is possible for matter to escape from the star in the form of a steady outflow, usually called a ''stellar wind''. &hellip; Modelling the resulting stellar wind from fundamental considerations is extremely difficult and no reliable theory exists at present.''</font>"

(Vol. II, &sect;3.6, p. 165) For sufficiently low mass stars, "<font color="darkgreen">&hellip; carbon ignition does not take place &hellip; and a more gradual ejection of material from the star in the form of shell flashes, winds, and envelope pulsations will lead to an expanding shell of gas around the core.  This expanding shell of gas is called a ''planetary nebula''.</font>"

==Maoz (2016)==

Here, we consider the descriptions presented by [http://adsabs.harvard.edu/abs/1998asa..book.....R D. Maoz (2016)].

<font color="red">'''Evolution to the Red-Giant Branch:'''</font> (&sect;4.1, p. 65)  "<font color="darkgreen">Once most of the hydrogen in the core of a star has been converted into helium, the core contracts and the inner temperatures rise.  As a result, hydrogen in the less-processed regions outside the core starts to burn in a shell surrounding the core.  Stellar models consistently predict that at this stage there is a huge expansion of the outer layers of the star &hellip; This is the <b>red-giant</b> phase.  The huge expansion of the star's envelope is difficult to explain by means of some simple and intuitive argument, but it is well understood and predicted robustly by the equations of stellar structure.</font>"

<font color="red">'''Mass Loss &amp; Formation of Planetary Nebula:'''</font> (&sect;4.1, p. 67)  "<font color="darkgreen">Evolved stars undergo large mass loss, especially on the red-giant branch and on the asymptotic branch, as a result of the low gravity in their extended outer regions and the radiation pressure produced by their large luminosities.  Mass loss is particularly severe on the AGB during so-called <b>thermal pulses</b> &#8212; roughly periodic flashes of enhanced helium shell burning.</font>"

(&sect;4.1, p. 68)  "<font color="darkgreen">&hellip; the remaining outer envelopes of the star expand to the point that they are completely blown off and dispersed.  During this very brief stage <math>~(\sim 10^4~\mathrm{yr})</math>, the star may appear as a <b>planetary nebula</b>.</font>"

=See Also=
==Twentieth Century==
* [http://adsabs.harvard.edu/abs/2000ApJ...538..837S D. Sugimoto &amp; M. Y. Fujimoto (2000, ApJ, Vol. 538, pp. 837 - 853)], ''Why Stars Become Red Giants.'' <-- Good list of references in the article.

* [http://adsabs.harvard.edu/abs/1991ApJ...372..592B R. Bhaskar &amp; A. Nigam (1991, ApJ, Vol. 372, pp. 592 - 596)], ''Qualitative Explanations for Red Giant Formation.'' <-- &sect;3.3 lists ''Existing Explanations''
** [http://adsabs.harvard.edu/abs/1981ASSL...88..179E Eggleton &amp; Faulkner (1981, Proceedings &hellip; p. 179 - 182)], ''Why Do Stars Become Red Giants?''
** [http://adsabs.harvard.edu/abs/1984PhR...105..329I Iben &amp; Renzini (1984, Physics Reports, Vol. 105, Issue 6, pp. 329 - 406)], ''Single Star Evolution I.  Massive Stars and Early Evolution of Low and Intermediate Mass Stars.''
** [http://adsabs.harvard.edu/abs/1985ApJ...296..554Y A. Yahil &amp; L. van den Horn (1985, ApJ, Vol. 296, pp. 554 - 564)], ''Why do Giants Puff Up?''
** [http://adsabs.harvard.edu/abs/1988ApJ...329..803A J. H. Applegate (1988, ApJ, Vol. 329, pp. 803 - 807)], ''Why Stars Become Red Giants.''
** [http://adsabs.harvard.edu/abs/1989MNRAS.236..505W A. P. Whitworth (1989, MNRAS, Vol. 236, pp. 505 - 544)], ''Why Red Giants are Giant.''


* [http://adsabs.harvard.edu/abs/1975ApJ...200..324H R. H&auml;rm &amp; M. Schwarzschild (1975, ApJ, Vol. 200, pp. 324 - 329)], ''Transition from a Red Giant to a Blue Nucleus after Ejection of a Planetary Nebula.''
* [http://adsabs.harvard.edu/abs/1970ApJ...161..657K D. A. Keeley (1970, ApJ, Vol. 161, pp. 657 - 667)], ''Dynamical Models of Long-Period Variable Stars.''
<table border="1" cellpadding="10" align="center" width="90%"><tr><td align="left">
(p. 663) "The parameters of the model close to dynamical instability were <math>~M = 1.3 M_\odot</math> &hellip; and core mass-fraction <math>~=0.20</math>.  This model is the same as model 2.5 in Paper I = [http://adsabs.harvard.edu/abs/1970ApJ...161..643K D. A. Keeley (1970, ApJ, Vol. 161, p. 643)] &hellip; This model was calculated to test the possibility that an extreme red giant could eject its envelope and form a planetary nebula.  [http://adsabs.harvard.edu/abs/1966PASP...78..232A G. O. Abell &amp; P. Goldreich (1966, PASP, Vol. 78, P. 232)] presented arguments in favor of this hypothesis.  The problem was subsequently pursued by [http://adsabs.harvard.edu/abs/1968AcA....18..511P B. Paczynski (1968, Acta Astr., Vol. 18, p. 511)], [http://adsabs.harvard.edu/abs/1968IAUS...34..396P B. Paczynski &amp; J. Zi&oacute;lkowski (1968a)] and [http://adsabs.harvard.edu/abs/1968AcA....18..255P (1968b)] and the writer."
<br />
Quote from Abell &amp; Goldreich (1966): 
* (p. 239) <font color="darkgreen">"We are therefore virtually forced to the conclusion that planetary nebulae are ejected more or less suddenly, although apparently not catastrophically, from stars of extremely large radii."</font>
</td></tr></table>

* [http://adsabs.harvard.edu/abs/1968IAUS...34..396P B. Paczynski &amp; J. Zi&oacute;lkowski (1968a, Proceedings from IAU Symposium no. 34: ''Planetary Nebulae'', ed. D. E. Osterbrock and Charles R. O'Dell, p. 396)], ''Dynamical Instability of the Envelopes of Red Supergiants and the Origin of Planetary Nebulae.''
<table border="1" cellpadding="10" align="center" width="90%"><tr><td align="left">
(p. 398) "We propose the following scheme for the late phases of stellar evolution.  After the exhaustion of helium [sic] in the core the star evolves into the region of red supergiants and moves up on the H-R diagram very close to the Hayashi border &hellip;  The mass of the helium and carbon core and the luminosity due to the helium and hydrogen-shell sources increase.  A star with total mass smaller than about 4 <math>~M_\odot</math> will terminate this type of evolution with an outflow of hydrogen-rich matter as a result of the dynamical instability of the extended envelope.  We suggest that planetary nebulae are formed in this way"
</td></tr></table>

* [http://adsabs.harvard.edu/abs/1968AcA....18..255P B. Paczynski &amp; J. Zi&oacute;lkowski (1968b)], ''Mira Variables.''
<table border="1" cellpadding="10" align="center" width="90%"><tr><td align="left">
(p. 265) "We suggest that a planetary nebula is formed in this way.  This process seems to be impossible for a star with larger mass.  In the latter case the mass of the core in which all the nuclear energy sources are exhausted will finally exceed the Chandrasekhar limit for degenerate configurations.  The dynamical instability of the core, followed by a supernova explosion may be expected &hellip;"
</td></tr></table>

* [http://adsabs.harvard.edu/abs/1972ApJ...176..395S R. L. Smith &amp; W. K. Rose (1972, ApJ, Vol. 176, pp. 395 - )], ''Relaxation Oscillations in the Envelopes of Luminous Red Giants.''
* [http://adsabs.harvard.edu/abs/1974ApJ...192..447K G. S. Kutter &amp; W. M. Sparks (1974, ApJ, Vol. 192, pp. 447 - 456)], ''Studies of Hydrodynamic Events in Stellar Evolution.  III. Ejection of Planetary Nebulae.''

* [https://ui.adsabs.harvard.edu/abs/1974ARA%26A..12..215I/abstract I. Iben, Jr. (1974, ARAA, Vol. 12, pp. 215-256)], ''Post Main Sequence Evolution of Single Stars.''
* [http://adsabs.harvard.edu/abs/2018Sci...362..201D K. De et al. (12 October 2018, Science, Vol. 362, No. 6411, pp. 201 - 206)], ''A Hot and Fast Ultra-stripped Supernova that likely formed a Compact Neutron Star Binary.''
* [http://adsabs.harvard.edu/abs/1994ApJS...92..125V E. Vassiliadis (1994, Astrophysical Journal Supplements Series, Vol. 92, pp. 125 - 144)], ''Post-Asymptotic Giant Branch Evolution of Low- to Intermediate-Mass Stars.''
* [http://adsabs.harvard.edu/abs/1995A%26A...299..755B T. Bl&ouml;cker (1995b, Astronomy &amp; Astrophysics, Vol. 299, pp. 755 - 769)], ''Stellar Evolution of Low- and Intermediate-Mass Stars.  II.  Post-AGB Evolution.''
* [http://adsabs.harvard.edu/abs/1995A%26A...297..727B T. Bl&ouml;cker (1995a, Astronomy &amp; Astrophysics, Vol. 297, pp. 727 - 738)], ''Stellar Evolution of Low- and Intermediate-Mass Stars.  I.  Mass Loss on the AGB and Its Consequences for Stellar Evolution.''
**(p. 727)  "Thus high mass losses must take place during the AGB evolution in order to remove the whole envelope before the core mass reaches the Chandrasekhar limit."
**(p. 728) "The physical mechanism of mass loss is not well understood up to now, and different approaches are discussed (cf. Lafon &amp; Berruyer 1991)"
* [http://adsabs.harvard.edu/abs/1991A%26ARv...2..249L J.-P. J. Lafon &amp; N. Berruyer (1991, Astronomy and Astrophysics Review, Vol. 2, April 1991, pp. 249 - 289)], ''Mass Loss Mechanisms in Evolved Stars''
* [http://adsabs.harvard.edu/abs/2008ASPC..401..166E B. R. Espey &amp; C. Crowley (2008)], Proceedings of the Conference on RS Ophiuchi (2006) and the Recurrent Nova Phenomenon, ASP Conference Series, Vol. 401, held 12 - 14 June, 2007 at Keele University, Keele, UK.  ''Mass-Loss from Red Giants''. (p. 166)
* Sequence of highly cited papers by Detlev Sch&ouml;nberner:
** [http://adsabs.harvard.edu/abs/1983ApJ...272..708S D. Sch&ouml;nberner (1983, ApJ, Vol. 272, pp. 708 - 714)], ''Late Stages of Stellar Evolution. II. -  Mass Loss and the Transition of Asymptotic Giant Branch Stars into Hot Remnants.''
* [http://adsabs.harvard.edu/abs/1981Ap.....17...92P L. S. Pilyugin &amp; G. S. Khromov (1981, Astrophysics, Vol. 17, no. 1, pp. 92 - 99)], ''The Origin of Planetary Nebulae.''
<table border="1" cellpadding="10" align="center" width="90%"><tr><td align="left">
(p. 98) "Of the three mechanisms considered earlier for the separation of the planetary nebula, preference must clearly be given to the ejection of a massive shell at a single time; for it is free of many of the difficulties inherent in the hypothesis of gradual accumulation of matter in the neighborhood of the star that is the precursor of the planetary nebula and its central star."
</td></tr></table>

* [http://adsabs.harvard.edu/abs/1986ApJ...308..706M I. Mazzitelli &amp; F. Dantona (1986, ApJ, Vol. 308, pp. 706 - 720)], ''Evolution from the Main Sequence to the White Dwarf Stage for a 3 Solar Mass Star.''
<table border="1" cellpadding="10" align="center" width="90%"><tr><td align="left">
(&sect; VI, p. 711) "&hellip; we summarize the major concepts which are currently accepted for the evolution through the PN stage." 
* i) "[http://adsabs.harvard.edu/abs/1981ASSL...88..431R Renzini (1981)] first considered that the very existence of PNs implies that, at a given time, the mass loss rate from asymptotic giant branch (AGB) stars must become much larger than the normal 'wind' rate (Reimers 1975).  This 'superwind' must be at least several times <math>~10^{-5} M_\odot~\mathrm{yr}^{-1}</math>.  
* ii) "The precise phase during the AGB evolution at which the super wind sets in is also a fundamental parameter for the following evolution, as the helium shell burning is more or less efficient at different stages. [http://adsabs.harvard.edu/abs/1984ApJ...277..333I Iben (1984)] followed numerically a number of cases, showing, for instance, how the complete loss of the hydrogen layer may be achieved if, after the PN ejection, there is the possibility of igniting a final helium shell flash."
* iii) "It is not clear at all whether PN ejection is a hydrodynamical event or not.  The few existing hydrodynamic computations (e.g., [http://adsabs.harvard.edu/abs/1974ApJ...192..447K Kutter and Sparks 1974]) do not consider the resulting behavior of the hydrostatic remnant, whose further evolution to the blue is crucial for the appearance of the PN.  many of the non hydrodynamical computations &#8212; starting from a fundamental paper by [http://adsabs.harvard.edu/abs/1975ApJ...200..324H H&auml;rm and Schwarzschild (1975)] and end with [http://adsabs.harvard.edu/abs/1984ApJ...277..333I Iben (1984)] &#8212; simulate the dynamics ejection by 'scaling' the models before ejection to smaller masses.  It is not proven that this procedure is physically meaningful."
</td></tr></table>
* [http://adsabs.harvard.edu/abs/1981ASSL...88..431R A. Renzini (1981, in Physical Processes in Red Giants; Proceedings of the Second Workshop, Erice, Italy, pp. 431 - 446)], ''Red Giants as Precursors of Planetary Nebulae.''
* [http://adsabs.harvard.edu/abs/1984ApJ...277..333I I. Iben, Jr. (1984, ApJ, Vol. 277, pp. 333 - 354)], ''On the Frequency of Planetary Nebula Nuclei Powered by Helium Burning and on the Frequency of White Dwarfs with Hydrogen-Deficient Atmospheres.''
* [https://ui.adsabs.harvard.edu/abs/1989IAUS..131..541H/abstract I. Hachisu &amp; M. Kato (1989)], in Planetary Nebulae.  Proceedings of the 131st. Symposium of the International Astronomical Union, held in Mexico City, Mexico, October 5 - 9, 1987.  Editor, Silvia Torres-Peimbert; Publisher, Kluwer Academic Publishers, Dordrecht, The Netherlands, Boston, MA, 1988:  ''Common Envelope Evolutions of Binary System and Formation of Planetary Nebulae''

==Articles Citing Sun Kwok (2000)==
[https://ui.adsabs.harvard.edu/abs/2000oepn.book.....K/abstract S. Kwok (2000, Cambridge Astrophysics Series, vol. 33)], ''The Origin and Evolution of Planetary Nebulae''

Later Edition: [https://www.cambridge.org/us/catalogue/catalogue.asp?isbn=9780521039079 S. Kwok (2007, Cambridge Astrophysics Series, vol. 33)], ''The Origin and Evolution of Planetary Nebulae''

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