Editing Appendix/Ramblings/OriginOfPlanetaryNebulae (section)

==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>"