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.

"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, $(n_c,n_e)=(\mathrm{\infty },3/2)$. 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?
    NO!!! This EUREKA moment is WRONG! [EUREKA! In July 2022, I finally realized that, in the context of bipolytropes, the Schönberg-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^st overtone. This very likely indicates that 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.

Polytropes

Isolated n = 1 Polytrope with γ_g = 2

Highlights drawn from our examination of radial oscillations in an isolated, n = 1 polytrope.

Four Modes of Oscillation of an Isolated, $n=1$ Polytrope Assuming ${\gamma }_g=2$
Mode	${\sigma }_c^2$	Neg. Slope $1-({\sigma }_c^2{\pi }^2/12)$	$\mathfrak{𝔉}=\frac{{\sigma }_c^2}{{\gamma }_g}-2\alpha$
Fundamental	2.2405295	3.1287618	-0.879735
1st Overtone	6.340767	-32.06757	1.1703835
2nd Overtone	13.694927	-153.2545	4.8474635
3rd Overtone	28.462829	-665.3074	12.231415
Numerically Determined Eigenfunctions for Various $\mathfrak{𝔉}$

Isolated n = 3 Polytrope with γ_g = 20/13

Highlights drawn from our review of, and successful effort to replicate, the work presented by …

M. Schwarzschild (1941)
Overtone Pulsations for the Standard [Stellar] Model
The Astrophysical Journal, Vol. 94, pp. 245 - 252

Note that, for each identified eigenvector, the square of the eigenfrequency may be obtained via the expression,

${\sigma }_c^2=\left(\frac{3{\gamma }_g}{2}\right){\omega }_{\mathrm{S}\mathrm{c}\mathrm{h}}^2$

$=$

${\gamma }_g[\mathfrak{𝔉}+2\alpha ]={\gamma }_g[\mathfrak{𝔉}+2(3-\frac{4}{{\gamma }_g}\left)\right]$

Four Modes of Oscillation of an Isolated, $n=3$ Polytrope Assuming ${\gamma }_g=\frac{20}{13}$
Mode	$\mathfrak{𝔉}$	${\sigma }_c^2=\frac{20}{13}[\mathfrak{𝔉}+\frac{4}{5}]$
Fundamental	-0.64413	+0.23980
1st Overtone	-0.47014	+0.50748
2nd Overtone	-0.21213	+0.90442
3rd Overtone	+0.12026	+1.41578
Numerically Determined Eigenfunctions for Various $\mathfrak{𝔉}$

Pressure-Truncated n = 5 Polytrope with γ_g = 6/5

Highlights drawn from our examination of radial oscillations in pressure-truncated, n = 5 polytropes.

${\xi }_i$	${\sigma }_c^2$	Numerically Generated Fundamental-Mode Eigenvectors for Various Truncation Radii, ${\xi }_i$
0.75	+13.6915		Excel File: file = Dropbox/WorkFolder/Wiki edits/LinearPerturbation/n5Eigenvectors/n5TruncatedSphere.xlsx --- worksheet = OursPt1 Movie File: file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/n5movie/ --- worksheet = n5Truncated2.gif
1.00	+6.4733
1.25	+3.3666
1.50	+1.8441
1.75	+1.0362
2.00	+0.5835
2.25	+0.3193
2.50	+0.1602
2.75	+0.0619
3.00	$\pm$ 0.0000
3.25	-0.0396
3.50	-0.0653
3.75	-0.0820
4.00	-0.0930
4.50	-0.1048
5.00	-0.1098

Isothermal Sphere

Pressure-Truncated n = ∞ Polytropes with γ_g = 1

Highlights drawn from our review of, and successful effort to replicate, the work presented by …

L. G. Taff, & H. M. van Horn (1974)
Radial Pulsations of Finite Isothermal Gas Spheres
Monthly Notices of the Royal Astronomical Society,
Vol. 168, pp. 427 - 432

Note that, for each identified eigenvector, the square of the eigenfrequency may be obtained via the expression,

${\sigma }_c^2=6[{\lambda }^2{]}_{\mathrm{T}\mathrm{V}\mathrm{H}\mathrm{7}\mathrm{4}}$

$=$

${\gamma }_g[\mathfrak{𝔉}+2\alpha ]={\gamma }_g[\mathfrak{𝔉}+2(3-\frac{4}{{\gamma }_g}\left)\right]$

Eigenfunctions (right) and Corresponding Eigenfrequencies (left) for Isothermal Spheres Truncated at Various Radii
Fundamental
${\xi }_0$ $\mathfrak{𝔉}$ ${\lambda }_0^2=\frac{\gamma (\mathfrak{𝔉}+2\alpha )}{6}$ 2 12.92907 +1.821512 3 5.51614 +0.586023 4 3.29671 +0.216118 5 2.456412 +0.0760687 6 2.092651 +0.0154418 7 1.921062 -0.013156 8 1.8354928 -0.027418 9 1.791388 -0.034769
1st Overtone
${\xi }_1$ $\mathfrak{𝔉}$ ${\lambda }_1^2=\frac{\gamma (\mathfrak{𝔉}+2\alpha )}{6}$ 2 56.87349 +9.14558 3 24.58903 +3.76484 4 13.640525 +1.94009 5 8.798197 +1.13303 6 6.306545 +0.71776 7 4.88991 +0.48165 8 4.024628 +0.33744 9 3.46662 +0.24444

ADDITIONALLY:

• Taking into account our accompanying elaboration regarding the properties of pressure-truncated isothermal spheres, it would be instructive to extend this pair of animations to at least include the model in which the 1^st overtone mode becomes marginally unstable $(\tilde{\xi }\approx 67)$.
• Make a movie that shows in the vicinity of the onset of instability how the eigenfunction that is associated with the 1^st overtone mode varies with $\tilde{\xi }$. As in the context (e.g., above) of our analysis of pressure-truncated n = 5 polytropes, pair it with a movie that shows where each model lies along the equilibrium sequence.

Bipolytropes

Bipolytrope With (n_c, n_e) = (1, 5) and γ_c = γ_e = 5/3

Highlights drawn from our review of, and successful effort to replicate, the work presented by …

J. O. Murphy & R. Fiedler (1985)
Radial Pulsations and Vibrational Stability of a Sequence of Two Zone Polytropic Stellar Models
Proceedings of the Astronomical Society of Australia, Vol. 6, no. 2, pp. 222 - 226

Numerical Values for Two Selected $(n_c,n_e)=(1,5)$ Bipolytropes [to be compared with Table 1 of Murphy & Fiedler (1985)]
MODEL	Source	$\frac{r_i}{R}$	${\mathrm{\Omega }}_0^2$	${\mathrm{\Omega }}_1^2$	$\frac{r}{R}{|}_1$	$1-\frac{M_r}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{|}_1$
10	MF85	0.393	15.9298	21.2310	0.573	1.00E-03
	Here	0.39302	15.93881161	21.24571822	0.5724	3.05E-05
17	MF85	0.933	2.1827	13.9351	0.722	0.232
	Here	0.93277	2.182932207	13.93880866	0.7215	0.24006

Our Determinations for Model 10 $({\xi }_i=2.5646)$
Mode	${\sigma }_c^2$	${\mathrm{\Omega }}^2\equiv \frac{{\sigma }_c^2}{2}\left(\frac{{\rho }_c}{\overline{\rho }}\right)$	$x_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}$	$\frac{d\ln x}{d\ln r^{*}}{|}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}$		$\frac{r}{R}{|}_1$	$1-\frac{M_r}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{|}_1$	$\frac{r}{R}{|}_2$	$1-\frac{M_r}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{|}_2$	$\frac{r}{R}{|}_3$	$1-\frac{M_r}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{|}_3$
				expected	measured
1 (Fundamental)	0.92813095170326	15.93881161	+85.17	8.963286966	8.963085	n/a	n/a	n/a	n/a	n/a	n/a
2	1.237156768978	21.24571822	- 610	12.14743093	12.147337	0.5724	3.05E-05	n/a	n/a	n/a	n/a
3	1.8656033984	32.0380449	+3225	18.62282676	18.6228	0.4845	1.35E-04	0.787	2.05E-07	n/a	n/a
4	2.65901504799	45.66331921	-9410	26.79799153	26.797977	0.4459	2.620E-04	0.7096	1.834E-06	0.8632	1.189E-08

Our Determinations for Model 17 $({\xi }_i=3.0713)$
Mode	${\sigma }_c^2$	${\mathrm{\Omega }}^2\equiv \frac{{\sigma }_c^2}{2}\left(\frac{{\rho }_c}{\overline{\rho }}\right)$	$x_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}$	$\frac{d\ln x}{d\ln r^{*}}{|}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}$		$\frac{r}{R}{|}_1$	$1-\frac{M_r}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{|}_1$	$\frac{r}{R}{|}_2$	$1-\frac{M_r}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{|}_2$	$\frac{r}{R}{|}_3$	$1-\frac{M_r}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{|}_3$
				expected	measured
1 (Fundamental)	1.149837904	2.182932207	+1.275	0.7097593	0.7097550	n/a	n/a	n/a	n/a	n/a	n/a
2	7.34212930615	13.93880866	- 2.491	7.763285	7.763244	0.7215	0.24006	n/a	n/a	n/a	n/a
3	16.345072567	31.03062198	+4.33	18.01837	18.01826	0.5806	0.5027	0.848	0.0541	n/a	n/a
4	27.746934203	52.6767087	-9.1	31.0060	31.0058	0.4859	0.6737	0.7429	0.1974	0.8957	0.0171

Bipolytrope With (n_c, n_e) = (5, 1) and (γ_c, γ_e) = (6/5, 2)

Building upon our review and successful replication of the work presented by …

P. P. Eggleton, J. Faulkner, & R. C. Cannon (1998)
A Small Contribution to the Giant Problem
Monthly Notices of the Royal Astronomical Society, Vol. 298, issue 3, pp. 831 - 834

in which strictly analytical means are used to construct equilibrium models of $(n_c,n_e)=(5,1)$ bipolytropes, here we present highlights drawn from our examination of radial oscillations in these bipolytropes, assuming $({\gamma }_c,{\gamma }_e)=(\frac{6}{5},2)$.

SPREADSHEET:   WorkFolder/Wiki_Edits/BiPolytrope/Faulkner1stOvertone/Stability51.xlsx

• Analytic Specification of n = 5 Core
• Interface Conditions
• Analytic Specification of n = 1 Envelope
• Physical Properties of Core, Normalized to Total Mass

  $\tilde{\rho }$	$=$	${{𝓂}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}^5\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-10}\right(1+\frac{1}{3}{\xi }^2{)}^{-5/2};$
  $\tilde{P}$	$=$	${{𝓂}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}^6\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-12}\right(1+\frac{1}{3}{\xi }^2{)}^{-3};$
  $\tilde{r}$	$=$	${{𝓂}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}^{-2}\left(\frac{{\mu }_e}{{\mu }_c}{)}^4\right(\frac{3}{2\pi }{)}^{1/2}\xi ;$
  ${\tilde{M}}_r\equiv \frac{M_r}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}$	$=$	${{𝓂}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}^{-1}\left(\frac{{\mu }_e}{{\mu }_c}{)}^2\right(\frac{2\cdot 3}{\pi }{)}^{1/2}\left[{\xi }^3\right(1+\frac{1}{3}{\xi }^2{)}^{-3/2}],$

  where,

  ${{𝓂}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}$

  $\equiv$

  $(\frac{2}{\pi }{)}^{1/2}\frac{A{\eta }_s}{{\theta }_i}.$

• Physical Properties of Envelope, Normalized to Total Mass

  $\tilde{\rho }$	$=$	${{𝓂}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}^5(\frac{{\mu }_e}{{\mu }_c}{)}^{-9}{\theta }_i^5\varphi ;$
  $\tilde{P}$	$=$	${{𝓂}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}^6(\frac{{\mu }_e}{{\mu }_c}{)}^{-12}{\theta }_i^6{\varphi }^2;$
  $\tilde{r}$	$=$	${{𝓂}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}^{-2}(\frac{{\mu }_e}{{\mu }_c}{)}^3{\theta }_i^{-2}(2\pi {)}^{-1/2}\eta ;$
  ${\tilde{M}}_r\equiv \frac{M_r}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}$	$=$	${{𝓂}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}^{-1}{\theta }_i^{-1}\left(\frac{2}{\pi }{)}^{1/2}\right(-{\eta }^2\frac{d\varphi }{d\eta }).$

Bipolytrope With (n_c, n_e) = (∞, 3) and (γ_c, γ_e) = (1, 4/3)

As we have pointed out in an accompanying discussion, this is the bipolytropic model that was presented by …

M. Schönberg & S. Chandrasekhar (1942)
On the Evolution of the Main-Sequence Stars
The Astrophysical Journal, Vol. 96, pp. 161 - 172

to suggest that stellar models evolving up the giant branch cannot survive if the helium-core mass tries to climb above some limiting mass ratio.

Toy Model

Let's play with a bipolytropic configuration having $(n_c,n_e)=(5,1)$ and ${\mu }_e/{\mu }_c=\frac{1}{2}$. Evolution will be dictated by the steady increase of the radial location of the core/envelope interface, ${\xi }_i$ from zero (initially) to ${\xi }_i{|}_{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}=2.279258$ which — according to our solution of the bipolytropic LAWE — marks the onset of dynamical instability.

As evolution proceeds along the ${\mu }_e/{\mu }_c=\frac{1}{2}$ sequence, we want to hold $M_{\mathrm{t}\mathrm{o}\mathrm{t}}$ constant. We must choose one additional fixed parameter; it isn't immediately obvious what is the best choice, but we will try setting $K_c$ constant. (Originally, we held the central density, ${\rho }_0$, and $K_c$ constant.) The relevant renormalization is detailed in our chapter tagged SSC/Structure/BiPolytropes/Analytic51Renormalize.

Zero-Age Main Sequence (ZAMS) Configuration

As a check, we note that in the initial (ZAMS) equilibrium configuration $({\xi }_i=0)$ there is no "core" — that is, $M_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}=0$ — so its structure should match that of an isolated $n=1$ polytrope. Specifically, given that, $M_{\odot }=1.99\times 10^{33}\mathrm{g}$, $R_{\odot }=6.96\times 10^{10}\mathrm{c}\mathrm{m}$, and $G=6.674\times 10^{-8}\mathrm{c}{\mathrm{m}}^{\mathrm{3}}{\mathrm{g}}^{\mathrm{-}\mathrm{1}}{\mathrm{s}}^{\mathrm{-}\mathrm{2}}$ …

$M_{\mathrm{t}\mathrm{o}\mathrm{t}}$

$=$

$\frac{4}{\pi }{\rho }_c{R}^3$

$\Rightarrow$

${\rho }_c{|}_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}}$

$=$

$\frac{\pi }{4}\cdot \frac{M_{\odot }}{R_{\odot }^3}=4.64\mathrm{g}\mathrm{c}{\mathrm{m}}^{\mathrm{-}\mathrm{3}}.$

Also,

$\frac{{\rho }_c}{\overline{\rho }}$

$=$

$\frac{{\pi }^2}{3}$

and,

$P_c{|}_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}}$

$=$

$\frac{\pi G}{8}\left(\frac{M_{\odot }^2}{R_{\odot }^4}\right)=4.42\times 10^{15}\mathrm{g}\mathrm{c}{\mathrm{m}}^{\mathrm{-}\mathrm{1}}{\mathrm{s}}^{\mathrm{-}\mathrm{2}}.$

The sun's luminosity is, $L_{\odot }=4\pi R_{\odot }^2{\sigma }_{\mathrm{S}\mathrm{B}}{T}^4=3.846\times 10^{33}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{s}{\mathrm{s}}^{\mathrm{-}\mathrm{1}}$, where, in terms of $a_{\mathrm{r}\mathrm{a}\mathrm{d}}$ and $c$, the Stefan-Boltzmann constant is, ${\sigma }_{\mathrm{S}\mathrm{B}}=\frac{a_{\mathrm{r}\mathrm{a}\mathrm{d}}c}{4}=5.671\times 10^{-5}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{c}{\mathrm{m}}^{\mathrm{-}\mathrm{2}}{\mathrm{s}}^{\mathrm{-}\mathrm{1}}{\mathrm{K}}^{\mathrm{-}\mathrm{4}}.$ Hence, in the context of our ZAMS toy model, the configuration's surface temperature is, $T{|}_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}}$ $=$ $[\frac{L_{\odot }}{4\pi R_{\odot }^2{\sigma }_{\mathrm{S}\mathrm{B}}}{]}^{1/4}=5777\mathrm{K}.$ In the context of our examination of a $1M_{\odot }$ evolutionary track published by 📚 I. Iben, Jr. (1967, ApJ, Vol. 147, pp. 624 - 649) — see below — we can write quite generally that, $L$ $=$ $4\pi R^2{\sigma }_{\mathrm{S}\mathrm{B}}{T}_e^4$ $\Rightarrow (\frac{R}{R_{\odot }}{)}^2$ $=$ $\frac{L}{L_{\odot }}(\frac{T_e}{5777K}{)}^{-4}$ $\Rightarrow \log \left(\frac{R}{R_{\odot }}\right)$ $=$ $\frac{1}{2}\log \left(\frac{L}{L_{\odot }}\right)-2\log \left(\frac{T_e}{5777K}\right)$   $=$ $\frac{1}{2}\log \left(\frac{L}{L_{\odot }}\right)-2\log T_e+7.523.$ Note as well that if an evolving star radiates at an average luminosity, $L_{\mathrm{a}\mathrm{v}\mathrm{g}}/L_{\odot }$, over a specified time interval in units of $10^9\mathrm{y}\mathrm{r}\mathrm{s}$, $(\mathrm{\Delta }t{)}_9$, during the specified interval of time it will have burned through an amount of mass, $\mathrm{\Delta }M/M_{\odot }$, given by the expression, $0.007(\mathrm{\Delta }M)c^2$ $=$ $(\mathrm{\Delta }t)L_{\mathrm{a}\mathrm{v}\mathrm{g}}$ $\Rightarrow \frac{(\mathrm{\Delta }M)}{M_{\odot }}$ $=$ $(\mathrm{\Delta }t{)}_9\cdot \frac{L_{\mathrm{a}\mathrm{v}\mathrm{g}}}{L_{\odot }}[\frac{L_{\odot }}{0.007M_{\odot }{c}^2}\cdot 3.156\times 10^{16}\mathrm{s}]=0.01[(\mathrm{\Delta }t{)}_9\cdot \frac{L_{\mathrm{a}\mathrm{v}\mathrm{g}}}{L_{\odot }}],$ where we have assumed that the efficiency of burning hydrogen to helium is 0.7%.

Most notably, the dimensionless radius that appears in the classic Lane-Emden equation is given by the expression,

$\xi \equiv \frac{r}{a_{\mathrm{n}\mathrm{=}\mathrm{1}}},$

where,

$a_{\mathrm{n}\mathrm{=}\mathrm{1}}\equiv [\frac{(n+1)K_n}{4\pi G}\cdot {\rho }_c^{(1-n)/n}{]}^{1/2}=[\frac{K_{\mathrm{e}\mathrm{n}\mathrm{v}}}{2\pi G}{]}^{1/2}.$

And, given that the surface ($r=R_{\odot }$) of the isolated $n=1$ polytrope occurs at $\xi ={\xi }_1=\pi$, we find that,

$a_{\mathrm{n}\mathrm{=}\mathrm{1}}$	$=$	$\frac{R_{\odot }}{\pi }$
$\Rightarrow K_{\mathrm{e}\mathrm{n}\mathrm{v}}{|}_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}}$	$=$	$\left(\frac{2}{\pi }\right)G{R}_{\odot }^2=2.06\times 10^{14}\mathrm{c}{\mathrm{m}}^{\mathrm{5}}{\mathrm{g}}^{\mathrm{-}\mathrm{1}}{\mathrm{s}}^{\mathrm{-}\mathrm{2}}.$

Double-check:

$P_c{|}_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}}$

$=$

$K_{\mathrm{e}\mathrm{n}\mathrm{v}}{|}_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}}[{\rho }_c{|}_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}}{]}^2=4.42\times 10^{15}\mathrm{g}\mathrm{c}{\mathrm{m}}^{\mathrm{-}\mathrm{1}}{\mathrm{s}}^{\mathrm{-}\mathrm{2}}.$

Yes!

Now, the instant the "core" shows up, we demand that its central density be a factor of $({\mu }_e/{\mu }_c{)}^{-1}$ larger than the density at the inner edge of the envelope; at the same time, we demand that the pressures be the same. Hence the relevant polytropic equation of state for the core must be,

$P$

$=$

$K_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}\left[\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}\rho {]}^{6/5}.$

In our toy model, this means that,

$K_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$

$=$

$P\left[\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}\rho {]}^{-6/5}=(\frac{{\mu }_e}{{\mu }_c}{)}^{6/5}4.42\times 10^{15}\mathrm{g}\mathrm{c}{\mathrm{m}}^{\mathrm{-}\mathrm{1}}{\mathrm{s}}^{\mathrm{-}\mathrm{2}}[4.64\mathrm{g}\mathrm{c}{\mathrm{m}}^{\mathrm{-}\mathrm{3}}{]}^{-6/5}=(\frac{{\mu }_e}{{\mu }_c}{)}^{6/5}7.008\times 10^{14}{\mathrm{g}}^{\mathrm{-}\mathrm{1}/\mathrm{5}}\mathrm{c}{\mathrm{m}}^{\mathrm{1}\mathrm{3}/\mathrm{5}}{\mathrm{s}}^{\mathrm{-}\mathrm{2}}.$

Hence, the ratio,

$\frac{K_{\mathrm{e}\mathrm{n}\mathrm{v}}}{K_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}$

$=$

$2.06\times 10^{14}\mathrm{c}{\mathrm{m}}^{\mathrm{5}}{\mathrm{g}}^{\mathrm{-}\mathrm{1}}{\mathrm{s}}^{\mathrm{-}\mathrm{2}}\left\{\right(\frac{{\mu }_e}{{\mu }_c}{)}^{6/5}7.008\times 10^{14}{\mathrm{g}}^{\mathrm{-}\mathrm{1}/\mathrm{5}}\mathrm{c}{\mathrm{m}}^{\mathrm{1}\mathrm{3}/\mathrm{5}}{\mathrm{s}}^{\mathrm{-}\mathrm{2}}{\}}^{-1}=(\frac{{\mu }_e}{{\mu }_c}{)}^{-6/5}0.293{\mathrm{g}}^{\mathrm{-}\mathrm{4}/\mathrm{5}}\mathrm{c}{\mathrm{m}}^{\mathrm{1}\mathrm{2}/\mathrm{5}},$

which may be rewritten as,

$\frac{K_{\mathrm{e}\mathrm{n}\mathrm{v}}}{K_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}$	$=$	$(\frac{{\mu }_e}{{\mu }_c}{)}^{4/5}0.293[\mathrm{g}\mathrm{c}{\mathrm{m}}^{\mathrm{-}\mathrm{3}}{]}^{-4/5}(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}$
		$=\left[\underset{\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{y}\mathrm{o}\mathrm{f}\mathrm{n}\mathrm{=}\mathrm{5}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}{\underbrace{(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}\frac{\pi }{4}\cdot \frac{M_{\odot }}{R_{\odot }^3}}}{]}^{-4/5}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}.$

This matches the interface condition expression that relates $K_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$ to $K_{\mathrm{e}\mathrm{n}\mathrm{v}}$,

$\frac{K_e}{K_c}$

$=$

${\rho }_c^{-4/5}(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}{\theta }_i^{-4},$

where we appreciate that, in the initial bipolytropic configuration (i.e., when ${\xi }_i=0$), ${\theta }_i=1$.

Adopted Normalizations

We adopt the normalizations in which we hold $M_{\mathrm{t}\mathrm{o}\mathrm{t}}$ and $K_c$ constant, as derived in an accompanying discussion. That is,

$r_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}\equiv \left[\right(\frac{G}{K_c}{)}^{5/2}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^2]$	$=$	$[6.674\times 10^{-8}\mathrm{c}{\mathrm{m}}^{\mathrm{3}}{\mathrm{g}}^{\mathrm{-}\mathrm{1}}{\mathrm{s}}^{\mathrm{-}\mathrm{2}}{]}^{5/2}\times [(\frac{{\mu }_e}{{\mu }_c}{)}^{6/5}7.008\times 10^{14}{\mathrm{g}}^{\mathrm{-}\mathrm{1}/\mathrm{5}}\mathrm{c}{\mathrm{m}}^{\mathrm{1}\mathrm{3}/\mathrm{5}}{\mathrm{s}}^{\mathrm{-}\mathrm{2}}{]}^{-5/2}(1.99\times 10^{33}\mathrm{g}{)}^2$
	$=$	$(\frac{{\mu }_e}{{\mu }_c}{)}^{-3}3.505\times 10^{11}\mathrm{c}\mathrm{m};$
${\rho }_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}\equiv \left[\right(\frac{K_c}{G}{)}^{3/2}\frac{1}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{]}^5$	$=$	$\left\{\right[(\frac{{\mu }_e}{{\mu }_c}{)}^{6/5}7.008\times 10^{14}{\mathrm{g}}^{\mathrm{-}\mathrm{1}/\mathrm{5}}\mathrm{c}{\mathrm{m}}^{\mathrm{1}\mathrm{3}/\mathrm{5}}{\mathrm{s}}^{\mathrm{-}\mathrm{2}}{]}^{3/2}[6.674\times 10^{-8}\mathrm{c}{\mathrm{m}}^{\mathrm{3}}{\mathrm{g}}^{\mathrm{-}\mathrm{1}}{\mathrm{s}}^{\mathrm{-}\mathrm{2}}{]}^{-3/2}[1.99\times 10^{33}\mathrm{g}{]}^{-1}{\}}^5$
	$=$	$(\frac{{\mu }_e}{{\mu }_c}{)}^{9}0.0462\mathrm{g}\mathrm{c}{\mathrm{m}}^{\mathrm{-}\mathrm{3}};$
$P_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}\equiv \left[K_c^{10}{G}^{-9}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{-6}\right]$	$=$	$\left[\right(\frac{{\mu }_e}{{\mu }_c}{)}^{6/5}7.008\times 10^{14}{\mathrm{g}}^{\mathrm{-}\mathrm{1}/\mathrm{5}}\mathrm{c}{\mathrm{m}}^{\mathrm{1}\mathrm{3}/\mathrm{5}}{\mathrm{s}}^{\mathrm{-}\mathrm{2}}{]}^{10}[6.674\times 10^{-8}\mathrm{c}{\mathrm{m}}^{\mathrm{3}}{\mathrm{g}}^{\mathrm{-}\mathrm{1}}{\mathrm{s}}^{\mathrm{-}\mathrm{2}}{]}^{-9}[1.99\times 10^{33}\mathrm{g}{]}^{-6}$
	$=$	$\left(\frac{{\mu }_e}{{\mu }_c}{)}^{12}\right\{[7.008\times 10^{14}][6.674\times 10^{-8}{]}^{-0.9}[1.99\times 10^{33}{]}^{-0.6}{\}}^{10}\mathrm{g}\mathrm{c}{\mathrm{m}}^{\mathrm{-}\mathrm{1}}{\mathrm{s}}^{\mathrm{-}\mathrm{2}}$
	$=$	$(\frac{{\mu }_e}{{\mu }_c}{)}^{12}1.751\times 10^{13}\mathrm{g}\mathrm{c}{\mathrm{m}}^{\mathrm{-}\mathrm{1}}{\mathrm{s}}^{\mathrm{-}\mathrm{2}}.$

Parameter	Core	Envelope
$r=r_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}\tilde{r}=$	$r_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}\left\{{{𝓂}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}^{-2}\right(\frac{{\mu }_e}{{\mu }_c}{)}^4\left(\frac{3}{2\pi }{)}^{1/2}\xi \right\}$	$r_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}\left\{{{𝓂}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}^{-2}\right(\frac{{\mu }_e}{{\mu }_c}{)}^3{\theta }_i^{-2}(2\pi {)}^{-1/2}\eta \}$
$\rho ={\rho }_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}\tilde{\rho }=$	${\rho }_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}\left\{{{𝓂}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}^5\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-10}(1+\frac{1}{3}{\xi }^2{)}^{-5/2}\}$	${\rho }_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}\left\{{{𝓂}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}^5\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-9}{\theta }_i^5\varphi \}$
$P=P_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}\tilde{P}=$	$P_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}\left\{{{𝓂}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}^6\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-12}(1+\frac{1}{3}{\xi }^2{)}^{-3}\}$	$P_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}\left\{{{𝓂}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}^6\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-12}{\theta }_i^6{\varphi }^2\}$
$M_r=M_{\mathrm{t}\mathrm{o}\mathrm{t}}\tilde{M}=$	$M_{\mathrm{t}\mathrm{o}\mathrm{t}}\left\{{{𝓂}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}^{-1}\right(\frac{{\mu }_e}{{\mu }_c}{)}^2\left(\frac{2\cdot 3}{\pi }{)}^{1/2}\right[{\xi }^3(1+\frac{1}{3}{\xi }^2{)}^{-3/2}]\}$	$M_{\mathrm{t}\mathrm{o}\mathrm{t}}\left\{{{𝓂}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}^{-1}{\theta }_i^{-1}\right(\frac{2}{\pi }{)}^{1/2}(-{\eta }^2\frac{d\varphi }{d\eta })\}$

Polytropic Functions and Their Derivatives …

Core:	$\theta (\xi )=[1+\frac{1}{3}{\xi }^2{]}^{-1/2}$		$\frac{d\theta }{d\xi }=-\frac{\xi }{3}[1+\frac{1}{3}{\xi }^2{]}^{-3/2}$
Envelope:	$\varphi =A\left[\frac{\sin (\eta -B)}{\eta }\right]$		$\frac{d\varphi }{d\eta }=\frac{A}{{\eta }^2}[\eta \cos (\eta -B)-\sin (\eta -B)]$

Key Relations; numerical values in parentheses assume $({\mu }_e/{\mu }_c)=\frac{1}{2}$ and ${\xi }_i=2.279258$ … cross-check here:

$\eta$	$=$	$3^{1/2}\left(\frac{{\mu }_e}{{\mu }_c}\right){\theta }_i^2\xi ,$		(0.722596)
${\mathrm{\Lambda }}_i$	$=$	$\frac{{\xi }_i}{\sqrt{3}}\left[\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}\frac{1}{{\theta }_i^2{\xi }_i^2}-1],$		(0.067969)
$A$	$=$	${\eta }_i(1+{\mathrm{\Lambda }}_i^2{)}^{1/2},$		(0.724263)
$B$	$=$	${\eta }_i+{\tan }^{-1}({\mathrm{\Lambda }}_i)-\frac{\pi }{2},$		(-0.780336)
${\eta }_s$	$=$	$\frac{\pi }{2}+{\eta }_i+{\tan }^{-1}({\mathrm{\Lambda }}_i),$		(2.361257)
${{𝓂}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}$	$=$	$(\frac{2}{\pi }{)}^{1/2}\frac{A{\eta }_s}{{\theta }_i},$		(2.255246)
$\frac{{\rho }_c}{\overline{\rho }}$	$=$	$(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}\frac{{\eta }_s^2}{3A{\theta }_i^5},$		(63.29514)
$\nu \equiv \frac{M_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}$	$=$	$\left(\frac{{\mu }_e}{{\mu }_c}{)}^2\sqrt{3}\right[\frac{{\xi }_i^3{\theta }_i^4}{A{\eta }_s}],$		(0.401776)
$q\equiv \frac{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{R}$	$=$	$\left(\frac{{\mu }_e}{{\mu }_c}\right)\sqrt{3}\left[\frac{{\xi }_i{\theta }_i^2}{{\eta }_s}\right].$		(0.306022)
$R_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}$	$=$	$r_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}\left\{{{𝓂}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}^{-2}\right(\frac{{\mu }_e}{{\mu }_c}{)}^3{\theta }_i^{-2}(2\pi {)}^{-1/2}{\eta }_s\}$		$(0.063242r_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}=1.773\times 10^{11}\mathrm{c}\mathrm{m}=2.548R_{\odot })$

Example Evolution

Let's attempt to construct the post-main-sequence evolutionary trajectory for $1M_{\odot }$ stars having a couple of different values of the mean-molecular weight jump, ${\mu }_e/{\mu }_c$. This will be for comparison with the $1M_{\odot }$ evolutionary trajectory published by 📚 Iben (1967); hereafter, Iben67. A remake of this published trajectory is displayed below.

Our toy model does not handle radiation, so we cannot plot luminosity versus effective (surface) temperature, that is, we cannot generate a traditional HR diagram. However, our toy model predicts how the radius of the star should vary with the mass of its core. In the following figure, the solid-green circular markers show how the fractional mass, $\nu$, varies with $R_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}/R_{\odot }$ for the model sequence in which ${\mu }_e/{\mu }_c=1/2$ — note, for example, that among our above numerically evaluated "key relations", $(\nu ,R_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}/R_{\odot })=(0.40178,2.548)$; this point along the curve is marked by the solid-red circular marker. The solid-yellow circular markers show the same information for our toy model sequence in which ${\mu }_e/{\mu }_c=1/4$.

One Solar Mass Evolution

Parameters values deduced from Figure 1 and Table 1 of I. Iben, Jr. (1967) Stellar Evolution.   VI. Evolution from the Main Sequence to the Red-Giant Branch for Stars of Mass $1M_{\odot }$, $1.25M_{\odot }$, and $1.5M_{\odot }$ The Astrophysical Journal, Vol. 147, pp. 624 - 649) Interval $(\mathrm{\Delta }t{)}_9$ $[\log (\frac{L}{L_{\odot }}){]}_{\mathrm{a}\mathrm{v}\mathrm{g}}$ $[\log (\frac{R}{R_{\odot }}){]}_{\mathrm{a}\mathrm{v}\mathrm{g}}$ $\mathrm{\Delta }(M/M_{\odot })$ $\nu$ 1 - 2 3.770 -0.070 -0.05 0.0311 0.0311 2 - 3 2.889 0.06 -0.003 0.0322 0.0633 3 - 4 1.462 0.170 0.044 0.0210 0.0842 4 - 5 1.029 0.275 0.0975 0.0188 0.1030 5 - 6 0.7018 0.38 0.170 0.0163 0.1193 6 - 7 0.292 0.445 0.253 0.0079 0.1272 7 - 10 0.157 0.455 0.333 0.0043 0.1316 10 - 11 0.213 0.51 0.432 0.0067 0.1383 11 - 12 0.185 0.68 0.560 0.0086 0.1468 12 - 13 0.125 0.925 0.708 0.0102 0.1570

The parameter values displayed in the table shown here — immediately on the right of our figure — were determined as follows, referencing the data from Iben67 presented below:

• Interval: —The points displayed along the $1M_{\odot }$ evolutionary trajectory in Figure 1 of Iben67 are numbered in chronological order, from 1 to 13 (except points 8 and 9 are not included). Here, our parameter values reference the time intervals between adjacent points; for example, our table refers to interval "1 - 2," interval "2 - 3," etc.
• $(\mathrm{\Delta }t{)}_9$:   This refers to the amount of evolutionary time that separates the two points of an identified interval. For example, Table 1 of Iben67 states that the star's age at "point 1" along its evolutionary trajectory is $0.0506\times 10^9\mathrm{y}\mathrm{r}\mathrm{s}$ and at "point 2" its age is $3.8209\times 10^9\mathrm{y}\mathrm{r}\mathrm{s}$; hence the interval of time between points 1 and 2 is $(\mathrm{\Delta }t{)}_9=(3.8209-0.0506)\times 10^9\mathrm{y}\mathrm{r}\mathrm{s}=3.770\times 10^9\mathrm{y}\mathrm{r}\mathrm{s}$.
• $[{\log }_{10}(L/L_{\odot }){]}_{\mathrm{a}\mathrm{v}\mathrm{g}}$:  We have performed a linear interpolation in log₁₀ of the luminosity listed for each point along the evolutionary trajectory in order to obtain an average value of the luminosity across each interval. For example, the star's luminosity at "point 1" is approximately $\log (L/L_{\odot })\approx -0.140$ while at "point 2" it is $\log (L/L_{\odot })\approx 0.0$. Hence, while evolving across the 1 - 2 interval, we assume that its luminosity is constant and has the value, $[{\log }_{10}(L/L_{\odot }){]}_{\mathrm{a}\mathrm{v}\mathrm{g}}=\frac{1}{2}[0.0+(-0.140)]=-0.07$.
• $[{\log }_{10}(R/R_{\odot }){]}_{\mathrm{a}\mathrm{v}\mathrm{g}}$:  The average stellar radius over an interval is obtained in the same way. For example, while evolving across the 1 - 2 interval, we assume that its radius is constant and has the value, $[{\log }_{10}(R/R_{\odot }){]}_{\mathrm{a}\mathrm{v}\mathrm{g}}=\frac{1}{2}[-0.073+(-0.027)]=-0.05$.
• $\mathrm{\Delta }(M/M_{\odot })$:   For a given interval, if we multiply the star's average luminosity times the time spent crossing that interval, we obtain a reasonable estimate of the total amount of energy, $E_{\mathrm{\Delta }\mathrm{t}}$, that the star has released over that time interval. Presumably, the same amount of energy was generated via nuclear reactions over that interval. Assuming that the primary reaction is the conversion of hydrogen into helium, the conversion of mass into energy occurs at an efficiency of 0.7%, that is, a reasonable estimate of the amount of mass that gets burned during that interval is,

  $0.007(\mathrm{\Delta }M)c^2$	$\approx$	$\mathrm{\Delta }t\cdot L_{\mathrm{a}\mathrm{v}\mathrm{g}}$
  $\Rightarrow \frac{\mathrm{\Delta }M}{M_{\odot }}$	$\approx$	$\left[\frac{(\mathrm{\Delta }t{)}_9\times 3.156\times 10^{16}\mathrm{s}}{0.007c^2}\right]\cdot 10^{[\log (L/L_{\odot }){]}_{\mathrm{a}\mathrm{v}\mathrm{g}}}\left\{\frac{L_{\odot }}{M_{\odot }}\right\}=0.00970\times (\mathrm{\Delta }t{)}_9\cdot 10^{[\log (L/L_{\odot }){]}_{\mathrm{a}\mathrm{v}\mathrm{g}}}.$

  For example, for interval 1 - 2, $\mathrm{\Delta }M/M_{\odot }\approx (0.0097\times 3.770\times 10^{-0.070})=0.0311$.

• $\nu$:   This is the fraction of the star's total mass that has accumulated in the core by the end of the specified time interval; in our toy model, the total mass is $1M_{\odot }$ so $\nu$ is also the core mass expressed in solar masses. Here the value of $\nu$ is obtained by summing over the mass increments, $\mathrm{\Delta }(M/M_{\odot })$, that have been burned at all previous evolutionary time intervals, including the present one; for example, at the end of time interval 3 - 4, $\nu =0.0311+0.0322+0.0210=0.0842$.

Commentary

Evolutionary Tracks

When we embarked on this investigation, we thought that it would be difficult to quantitatively compare the evolution of our $(n_c,n_e)=(5,1)$ bipolytrope with more realistic stellar evolutionary tracks — as published, for example, by 📚 I. Iben, Jr. (1967, ApJ, Vol. 147, pp. 624 - 649) — because our toy model does not provide a mechanism for assessing the variation in time of the configuration's surface temperature. In the preceding subsection of this discussion, we have demonstrated that a comparison of evolutions is possible if we focus on an examination of how the star's radius varies as the mass of its core steadily increases.

Not surprisingly, we find that the trend is the same in our toy model as it is in Iben67's $1M_{\odot }$ model:   as the core's mass monotonically increases, the star's radius monotonically increases as well. But there appear to be two significant mismatches. (1) In the earliest stage of its evolution, Iben67's model exhibits a steeper rise in the mass of the core for a given radius. (2) If we set ${\mu }_e/{\mu }_c=1/2$ — which is the expected value when considering the fusion of hydrogen into helium — overall, our toy model dumps significantly more mass into the core than does Iben67's evolutionary model. Dropping the mean-molecular-weight ratio from $1/2$ to $1/4$ provides a better match, but it is difficult to come up with an astrophysical argument that would justify such a low ratio.

Given that the trend matches, we will consider this a win! Hopefully a more quantitative match will be obtained when we switch to a bipolytrope that has an isothermal $(n_c=\mathrm{\infty })$ core, instead of an $n_c=5$ core.

Dynamical Instability

We have placed one solid-red circular marker on each of the evolutionary tracks that have been drawn from our toy model. In both cases, this marks the point along the track — at a radius less than $4R_{\odot }$ — where the $(n_c,n_e)=(5,1)$ model becomes dynamically unstable. It is tempting to suggest that this indicates that Iben67's model should also become dynamically unstable before expanding to a radius greater than $4R_{\odot }$. Let's pursue building a bipolytrope that has an isothermal core and see if that system strengthens this argument.

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.."

---

Figure 6-17 (p. 492) in Clayton (1968) displays a reproduction of Figure 1 from 📚 I. Iben, Jr. (1967, ApJ, Vol. 147, pp. 624 - 649). It displays evolutionary tracks of lower-main-sequence population I stars of mass $1M_{\odot },1.25M_{\odot },$ and $1.5M_{\odot }$. Using a pen and ruler to draw scale lines across the figure, the following table catalogues the data from which the $1M_{\odot }$ track in the figure was produced.

Extracted from Published Figure Deduced† Numbered Point Age $(10^9\mathrm{y}\mathrm{r}\mathrm{s})$ $\log \left(\frac{L}{L_{\odot }}\right)$ $\log T_e$ $\log \left(\frac{R}{R_{\odot }}\right)$ 1 0.05060 -0.140 3.763 -0.073 2 3.8209 0.0 3.775 -0.027 3 6.7100 0.120 3.781 0.021 4 8.1719 0.22 3.783 0.067 5 9.2012 0.33 3.780 0.128 6 9.9030 0.43 3.763 0.212 7 10.195 0.46 3.73 0.293 8 … … … … 9 … … … … 10 10.352 0.45 3.688 0.372 11 10.565 0.57 3.658 0.492 12 10.750 0.79 3.645 0.628 13 10.875 1.06 3.633 0.787

^†From above, we know that,

$\log \left(\frac{R}{R_{\odot }}\right)$

$=$

$\frac{1}{2}\log \left(\frac{L}{L_{\odot }}\right)-2\log T_e+7.523.$

It is via this expression that we have deduced how the radius of Iben67's evolving configuration changes over time.

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 $(\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{g}\mathrm{h}\mathrm{l}\mathrm{y}M\le 1.2M_{\odot })$, 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 $(\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{g}\mathrm{h}\mathrm{l}\mathrm{y}M\ge 1.2M_{\odot })$ 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] $2500L_{\odot }$

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."

Padmanabhan (2000)

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 $5M_{\odot }$ 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 $M_{\mathrm{i}\mathrm{c}}\equiv qM$) 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 $\sim 0.1M_{\odot }$, 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 $(\sim 10^4\mathrm{y}\mathrm{r})$, the star may appear as a planetary nebula."

See Also

Twentieth Century

• D. Sugimoto & M. Y. Fujimoto (2000, ApJ, Vol. 538, pp. 837 - 853), Why Stars Become Red Giants. <-- Good list of references in the article.

• R. Bhaskar & A. Nigam (1991, ApJ, Vol. 372, pp. 592 - 596), Qualitative Explanations for Red Giant Formation. <-- §3.3 lists Existing Explanations
  • Eggleton & Faulkner (1981, Proceedings … p. 179 - 182), Why Do Stars Become Red Giants?
  • Iben & Renzini (1984, Physics Reports, Vol. 105, Issue 6, pp. 329 - 406), Single Star Evolution I. Massive Stars and Early Evolution of Low and Intermediate Mass Stars.
  • A. Yahil & L. van den Horn (1985, ApJ, Vol. 296, pp. 554 - 564), Why do Giants Puff Up?
  • J. H. Applegate (1988, ApJ, Vol. 329, pp. 803 - 807), Why Stars Become Red Giants.
  • A. P. Whitworth (1989, MNRAS, Vol. 236, pp. 505 - 544), Why Red Giants are Giant.

• R. Härm & M. Schwarzschild (1975, ApJ, Vol. 200, pp. 324 - 329), Transition from a Red Giant to a Blue Nucleus after Ejection of a Planetary Nebula.
• D. A. Keeley (1970, ApJ, Vol. 161, pp. 657 - 667), Dynamical Models of Long-Period Variable Stars.

(p. 663) "The parameters of the model close to dynamical instability were $M=1.3M_{\odot }$ … 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." Quote from Abell & Goldreich (1966): (p. 239) "We are therefore virtually forced to the conclusion that planetary nebulae are ejected more or less suddenly, although apparently not catastrophically, from stars of extremely large radii."

• B. Paczynski & J. Ziólkowski (1968a, Proceedings from IAU Symposium no. 34: Planetary Nebulae, ed. D. E. Osterbrock and Charles R. O'Dell, p. 396), Dynamical Instability of the Envelopes of Red Supergiants and the Origin of Planetary Nebulae.

(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"

• B. Paczynski & J. Ziólkowski (1968b), Mira Variables.

(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 …"

• R. L. Smith & W. K. Rose (1972, ApJ, Vol. 176, pp. 395 - ), Relaxation Oscillations in the Envelopes of Luminous Red Giants.
• G. S. Kutter & W. M. Sparks (1974, ApJ, Vol. 192, pp. 447 - 456), Studies of Hydrodynamic Events in Stellar Evolution. III. Ejection of Planetary Nebulae.

• I. Iben, Jr. (1974, ARAA, Vol. 12, pp. 215-256), Post Main Sequence Evolution of Single Stars.
• K. De et al. (12 October 2018, Science, Vol. 362, No. 6411, pp. 201 - 206), A Hot and Fast Ultra-stripped Supernova that likely formed a Compact Neutron Star Binary.
• E. Vassiliadis (1994, Astrophysical Journal Supplements Series, Vol. 92, pp. 125 - 144), Post-Asymptotic Giant Branch Evolution of Low- to Intermediate-Mass Stars.
• T. Blöcker (1995b, Astronomy & Astrophysics, Vol. 299, pp. 755 - 769), Stellar Evolution of Low- and Intermediate-Mass Stars. II. Post-AGB Evolution.
• T. Blöcker (1995a, Astronomy & Astrophysics, Vol. 297, pp. 727 - 738), Stellar Evolution of Low- and Intermediate-Mass Stars. I. Mass Loss on the AGB and Its Consequences for Stellar Evolution.
  • (p. 727) "Thus high mass losses must take place during the AGB evolution in order to remove the whole envelope before the core mass reaches the Chandrasekhar limit."
  • (p. 728) "The physical mechanism of mass loss is not well understood up to now, and different approaches are discussed (cf. Lafon & Berruyer 1991)"
• J.-P. J. Lafon & N. Berruyer (1991, Astronomy and Astrophysics Review, Vol. 2, April 1991, pp. 249 - 289), Mass Loss Mechanisms in Evolved Stars
• B. R. Espey & C. Crowley (2008), Proceedings of the Conference on RS Ophiuchi (2006) and the Recurrent Nova Phenomenon, ASP Conference Series, Vol. 401, held 12 - 14 June, 2007 at Keele University, Keele, UK. Mass-Loss from Red Giants. (p. 166)
• Sequence of highly cited papers by Detlev Schönberner:
  • D. Schönberner (1983, ApJ, Vol. 272, pp. 708 - 714), Late Stages of Stellar Evolution. II. - Mass Loss and the Transition of Asymptotic Giant Branch Stars into Hot Remnants.
• L. S. Pilyugin & G. S. Khromov (1981, Astrophysics, Vol. 17, no. 1, pp. 92 - 99), The Origin of Planetary Nebulae.

(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."

• I. Mazzitelli & F. Dantona (1986, ApJ, Vol. 308, pp. 706 - 720), Evolution from the Main Sequence to the White Dwarf Stage for a 3 Solar Mass 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 $10^{-5}{M}_{\odot }{\mathrm{y}\mathrm{r}}^{-1}$. 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."

• A. Renzini (1981, in Physical Processes in Red Giants; Proceedings of the Second Workshop, Erice, Italy, pp. 431 - 446), Red Giants as Precursors of Planetary Nebulae.
• I. Iben, Jr. (1984, ApJ, Vol. 277, pp. 333 - 354), On the Frequency of Planetary Nebula Nuclei Powered by Helium Burning and on the Frequency of White Dwarfs with Hydrogen-Deficient Atmospheres.
• I. Hachisu & M. Kato (1989), in Planetary Nebulae. Proceedings of the 131st. Symposium of the International Astronomical Union, held in Mexico City, Mexico, October 5 - 9, 1987. Editor, Silvia Torres-Peimbert; Publisher, Kluwer Academic Publishers, Dordrecht, The Netherlands, Boston, MA, 1988: Common Envelope Evolutions of Binary System and Formation of Planetary Nebulae

Articles Citing Sun Kwok (2000)

S. Kwok (2000, Cambridge Astrophysics Series, vol. 33), The Origin and Evolution of Planetary Nebulae

Later Edition: S. Kwok (2007, Cambridge Astrophysics Series, vol. 33), The Origin and Evolution of Planetary Nebulae