Editing Appendix/Ramblings/OriginOfPlanetaryNebulae (section)

==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?