Editing Appendix/Ramblings/OriginOfPlanetaryNebulae (section)

==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>"

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<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 {{ Iben67hereafter }}'s evolving configuration changes over time.