User:Tohline/Appendix/Ramblings/OriginOfPlanetaryNebulae
On the Origin of Planetary Nebulae
This chapter — initially created by J. E. Tohline on 19 November 2016 — is intended primarily to provide a summary of the research that has been undertaken following a discussion that took place on 3 July 2013 with Kundan Kadam (an LSU graduate student, at the time) regarding the stability of bipolytropes.
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Context
Why do stars become red giants? In particular, why does a star on the main sequence — whose internal density profile is only moderately centrally concentrated — readjust its internal structure to become a red giant — which has a highly centrally condensed structure — 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 Schönberg-Chandrasekhar mass limit.
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. Henrich & Chandraskhar (1941) and Schönberg & 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 Schönberg-Chandrasekhar mass limit was initially identified as a "turning point" along a sequence of equilibrium 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 — toward the turning point — 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 — as emphasized by, for example, Eggleton, Faulkner, and Cannon (1998) — evolution of a star from the main sequence toward the red giant branch may be naturally associated with evolution toward the Schönberg-Chandrasekhar mass limit.
From a structural or stability standpoint, this is not good because self-gravitating, isothermal structures have a tendency toward instability.
Treating as bipolytrope with <math>~(n_c, n_e) = (\infty, 3/2)</math> SC showed that this evolution is accompanied by a structural change toward a more centrally condensed structure. Faulkner et al. showed this more cleanly with analytic bipolytropic structures with <math>~(n_c, n_e) = (5, 1)</math>. We wondered whether mass-transfer in a binary system might accelerate the process. What type of instability results from exceeding the SC mass limit? Is it secular, or might it be dynamical? And what is the consequence; does the core collapse on a dynamical time scale; or does the envelope get "kicked off" to form a planetary nebula; or both?
See Also
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