Editing Appendix/Ramblings/OriginOfPlanetaryNebulae (section)

=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 {{ Iben67hereafter }}'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>"