# 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.

 "This approximation is as dangerously crude as it is computationally economic." — Drawn from §I of Härm & Schwarzschild (1975)

## 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 — 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.

Rationale:   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, $~(n_c, n_e) = (\infty, 3/2)$. 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 has been emphasized by, for example, Eggleton, Faulkner, and Cannon (1998) — 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 Schönberg-Chandrasekhar mass limit.

Motivating Questions:   With this general scenario in mind, we began to wonder — as, almost certainly, other astrophysicists before us have wondered:

• What type of instability — dynamical or secular (?) — is associated with the equilibrium sequence turning point that is synonymous with the Schönberg-Chandrasekhar mass limit?
• If it is a secular instability, do stars normally find a way — via one or more secular mechanisms — 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 — presumably catastrophic — result of encountering the Schönberg-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?
• Might evolution toward the Schönberg-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 Schönberg-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 — and the consequential development of wonderfully photogenic planetary nebulae — is a natural outcome of evolving stars encountering the Schönberg-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:   The LSU astrophysics group ought to employ its three-dimensional hydrodynamic code to investigate what happens when a bipolyropic star (the donor) — with an isothermal (or nearly isothermal) core and a core-to-total mass ratio that is near the Schönberg-Chandrasekhar mass limit — 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 Schönberg-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?

## Sub-Projects Undertaken

In order for the above proposed numerical investigation to provide informative results, it is important that we establish a firm understanding of a variety of related, but less complicated, concepts and problems. An emphasis has been placed on tackling problems that can described as fully as possible using analytic, rather than purely numerical, techniques. The following subsections provide a list, along with brief description, of related sub-projects that we have studied, to date.

# Textbook Explanations

## Clayton (1968)

Here, we consider the descriptions presented by D. D. Clayton (1968).

Evolution to the Red-Giant Branch: (§6-7, p. 485) "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 …" As the star's evolution proceeds, the temperature of the inert (isothermal) core will continue to increase, as will "the temperature of a shell of hydrogen surrounding the core … 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."

Stellar Pulsation: (§6-10, p. 504) "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." Analyses that attempt to explain the existence and properties of regular variable stars — such as the Cepheids and RR Lyrae variables — 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 "… physical effects contributing to the status of the stability of the zone."

• Γ mechanism: "The first term always contributes to stability … [but its] influence is diminished in ionization zones."
• κ mechanism: "The second term reflects the way in which the opacity varies during the pulsation. Positive values of $~\kappa_T$ and $~\kappa_P$ would imply that the opacity increases upon contraction, which would remove energy from the radiation flux … at the proper time to drive mechanical work."

Mass Loss: (§6-9, p. 501) "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 … 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.."

## Hansen & Kawaler (1994)

Here, we consider the descriptions presented by [HK94].

Evolution to the Red-Giant Branch & the SC Limit: (§2.3, pp. 53-55) "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 … that the structure of the star begins to change radically and the main sequence phase ends. … evolution to … the red giant branch (RGB) is characterized by a continual expansion and reddening of the star to lower [surface] temperatures. "

"In an early 1942 study Schönberg 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 … The envelope … however, responds by expanding rapidly … This signals the end of the main sequence phase of evolution for the star …"

Stellar Pulsation:

Mass Loss & Formation of Planetary Nebula: (§2.4.1, pp. 60-61) "The ignition of helium under electron degenerate conditions …" occurs via "… an explosive runaway or helium flash. … 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."

(§2.4.1, p. 62) "The critical point about the [observationally determined] masses … 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."

## Rose (1998)

Here, we consider the descriptions presented by W. K. Rose (1998).

Evolution to the Red-Giant Branch & the SC Limit: (§8.2, p. 267) "… after hydrogen depletion has occurred in their cores main-sequence stars evolve onto the red-giant branch. Low-mass stars $~(\mathrm{roughly}~M \leq 1.2 M_\odot)$, which burn hydrogen by means of the proton-proton chain on the main sequence evolve gradually from main sequence to red-giant evolutionary stages … The cores of stars that are sufficiently massive $~(\mathrm{roughly}~M \geq 1.2 M_\odot)$ 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 …"

(§8.2, p. 268) "Because the thermonuclear energy release that results from hydrogen burning is very large … the time-derivative term in Equation (2.131) can be neglected in calculating main-sequence stellar models. If [this same] 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 $~\approx 0.1 - 0.15$ times the mass of the star. These limiting isothermal core masses are referred to collectively as the Schönberg-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."

(§8.2, p. 269) "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."

Stellar Pulsation: (§8.1, p. 260) "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° as compared to the phase of minimum radius … Extensive hydrogen ionization zones cause [long-period variables] to become unstable to radial pulsations … The pulsations of … Cepheids result from both hydrogen and helium ionization zones."

(§1.5, p. 24) "… asymptotic-giant-branch stars become pulsationally unstable after their luminosities become [greater or on the order of] $~2500 L_\odot$

Mass Loss & Formation of Planetary Nebula: (§8.1, p. 260) "The [pulsation] amplitudes" of long-period variables "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."

(§1.5, p. 24) "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."

Here, we consider the descriptions presented by [P00]; note that in Chapter 3 of Volume II, subsection 3.4 is titled, Evolution of High-Mass Stars while subsection 3.5 is titled, Evolution of Low-Mass Stars.

Evolution to the Red-Giant Branch & the SC Limit: (Vol. II, §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, "Further evolution depends on the structural changes that take place in the [inert] helium core … the helium core is fairly homogeneous [in, for example, a $~5 M_\odot$ star] because of the mixing that is due to the original convective transport … 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 $~M_\mathrm{ic} \equiv qM$) to support the envelope of mass $~(1-q)M$. It turns out that this is possible only if the fraction of the mass in the core is below a critical value called the" Schönberg-Chandrasekhar (SC) limit.

For the remainder of §3.4.3, [P00] discusses in considerable detail — relying heavily on virial-theorem-based arguments — 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 §3.5.1 (p. 152), he re-emphasizes that "The effect of shell burning is … 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 $~\sim 0.1 M_\odot$, 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."

He also emphasizes the following. (Vol. II, §3.4.3, p. 148) "During" evolution from the main sequence to the red-giant branch, "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."

Stellar Pulsation: (Vol. II, §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 "net amount of work done by each layer of the star during one cycle of oscillation … 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," for example, the opacity in a layer of the envelope increases when the layer is compressed. "(Under normal circumstances, opacity actually decreases with compression.) … [This] exception occurs in the layers of the star that are partially ionized … This mechanism is called the κ mechanism."

Mass Loss & Formation of Planetary Nebula: (Vol. II, §3.6, p. 163) "The rapid expansion of the star … 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. … Modelling the resulting stellar wind from fundamental considerations is extremely difficult and no reliable theory exists at present."

(Vol. II, §3.6, p. 165) For sufficiently low mass stars, "… carbon ignition does not take place … 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."

## Maoz (2016)

Here, we consider the descriptions presented by D. Maoz (2016).

Evolution to the Red-Giant Branch: (§4.1, p. 65) "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 … This is the red-giant 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."

Mass Loss & Formation of Planetary Nebula: (§4.1, p. 67) "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 thermal pulses — roughly periodic flashes of enhanced helium shell burning."

(§4.1, p. 68) "… the remaining outer envelopes of the star expand to the point that they are completely blown off and dispersed. During this very brief stage $~(\sim 10^4~\mathrm{yr})$, the star may appear as a planetary nebula."

 (p. 663) "The parameters of the model close to dynamical instability were $~M = 1.3 M_\odot$ … and core mass-fraction $~=0.20$. This model is the same as model 2.5 in Paper I = D. A. Keeley (1970, ApJ, Vol. 161, p. 643) … This model was calculated to test the possibility that an extreme red giant could eject its envelope and form a planetary nebula. G. O. Abell & P. Goldreich (1966, PASP, Vol. 78, P. 232) presented arguments in favor of this hypothesis. The problem was subsequently pursued by B. Paczynski (1968, Acta Astr., Vol. 18, p. 511), B. Paczynski & J. Ziólkowski (1968a) and (1968b) and the writer."
 (p. 398) "We propose the following scheme for the late phases of stellar evolution. After the exhaustion of helium [sic] in the core the star evolves into the region of red supergiants and moves up on the H-R diagram very close to the Hayashi border … The mass of the helium and carbon core and the luminosity due to the helium and hydrogen-shell sources increase. A star with total mass smaller than about 4 $~M_\odot$ will terminate this type of evolution with an outflow of hydrogen-rich matter as a result of the dynamical instability of the extended envelope. We suggest that planetary nebulae are formed in this way"
 (p. 265) "We suggest that a planetary nebula is formed in this way. This process seems to be impossible for a star with larger mass. In the latter case the mass of the core in which all the nuclear energy sources are exhausted will finally exceed the Chandrasekhar limit for degenerate configurations. The dynamical instability of the core, followed by a supernova explosion may be expected …"
 (p. 98) "Of the three mechanisms considered earlier for the separation of the planetary nebula, preference must clearly be given to the ejection of a massive shell at a single time; for it is free of many of the difficulties inherent in the hypothesis of gradual accumulation of matter in the neighborhood of the star that is the precursor of the planetary nebula and its central star."
 (§ VI, p. 711) "… we summarize the major concepts which are currently accepted for the evolution through the PN stage." i) "Renzini (1981) first considered that the very existence of PNs implies that, at a given time, the mass loss rate from asymptotic giant branch (AGB) stars must become much larger than the normal 'wind' rate (Reimers 1975). This 'superwind' must be at least several times $~10^{-5} M_\odot~\mathrm{yr}^{-1}$. ii) "The precise phase during the AGB evolution at which the super wind sets in is also a fundamental parameter for the following evolution, as the helium shell burning is more or less efficient at different stages. Iben (1984) followed numerically a number of cases, showing, for instance, how the complete loss of the hydrogen layer may be achieved if, after the PN ejection, there is the possibility of igniting a final helium shell flash." iii) "It is not clear at all whether PN ejection is a hydrodynamical event or not. The few existing hydrodynamic computations (e.g., Kutter and Sparks 1974) do not consider the resulting behavior of the hydrostatic remnant, whose further evolution to the blue is crucial for the appearance of the PN. many of the non hydrodynamical computations — starting from a fundamental paper by Härm and Schwarzschild (1975) and end with Iben (1984) — simulate the dynamics ejection by 'scaling' the models before ejection to smaller masses. It is not proven that this procedure is physically meaningful."

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