The Effect of Rotation on Stellar Structure and Evolution

Clemson University TigerPrints

All Theses Theses

May 2020

Jaad Antoun Tannous Clemson University, [email protected]

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Recommended Citation Tannous, Jaad Antoun, "The Effect of Rotation on Stellar Structure and Evolution" (2020). All Theses. 3267. https://tigerprints.clemson.edu/all_theses/3267

This Thesis is brought to you for free and open access by the Theses at TigerPrints. It has been accepted for inclusion in All Theses by an authorized administrator of TigerPrints. For more information, please contact [email protected]. The Effect of Rotation on Stellar Structure and Evolution

A Dissertation Presented to the Graduate School of Clemson University

In Partial Fulﬁllment of the Requirements for the Degree Master of Science Physics

by Jaad A. Tannous May 2020

Accepted by: Prof. Bradley S. Meyer, Committee Chair Prof. Mounib El Eid Prof. Dieter Hartmann Prof. Stephen Kaeppler Abstract

In this thesis, we investigate the eﬀects of rotation on the evolution of stars in the mass range of 2-15 M assuming solar-like initial composition. We have used a well tested hydrodynamical stellar evolution program [El Eid et al., 2009], which has been extended to include a one-dimensional treatment of rotational instabilities.

The calculations for stars in the mass range up to 8 M have been per- formed to include the Asymptotic Giant Branch (AGB) in order to ﬁgure out whether rotational instabilities can eﬀect the so called ”third dredge up”, leading to a neutron source for the s-process nucleosynthesis.

In the case of massive stars, above 8 M , rotational effects are expected to become important during the main sequence evolution, mainly owing to the so called ”Eddington-Sweet” circulation. An indication of the effect of rotation should show up through the surface enrichment of 4He and 14N at the end of core Hydrogen burning. Another important aspect of rotation is the study of its effect on the shell carbon burning phase, where the s-process nucleosynthesis occurs.[El Eid et al., 2000]

ii Acknowledgments

The work discussed in this dissertation is the culmination of eﬀorts from two departments from two diﬀerent universities.

From the faculty at the American University of Beirut, I would like to ex- press my sincere gratitude to my advisor, Prof. Mounib el Eid, for taking me under his wing and his continuous support of my Master’s and PhD studies, his patience, motivation, and immense knowledge. Also, to the department chair, Prof. Jihad

Touma, the graduate advisor, Prof. Michel Kazan, and the department secretary

Joumana Abi Falah, who facilitated the process and provided a good place to vent at times.

From the faculty at Clemson University, I would like to thank Prof. Bradley

Meyer, who’s insight gave added value to the dissertation, and sitting as committee chairman. To Prof. Murray Daw, who suﬀered through my questions about transferring the dissertation from AUB to Clemson. To the remaining committee members, professors Dieter Hartmann and Stephen Kaepler, who’s comments and insights further enhanced the writing of this dissertation.

Other than professional aid, a more personal form of help and sanity checks were always welcome. To Elie Salameh and Rodrique Badr, thank you for always being there to oﬀer insight and advice on how to tackle problems, both numerically and in life.

Last but not the least, I would like to thank my family for their undying patience and unyielding support they have given me throughout the projects duration and beyond.

iii Table of Contents

Title Page ...... i

Abstract ...... ii

Acknowledgments ...... iii

List of Tables ...... v

List of Figures...... vi

1 Introduction...... 1

2 Method of Calculation...... 3 2.1 Stellar Structure Equations ...... 3 2.2 Present Treatment of Rotation ...... 7

3 Main Sequence ...... 14 3.1 Eﬀect of Rotation on the Main Sequence...... 16

4 Eﬀects on Helium Burning ...... 20 4.1 He Flash...... 21 4.2 Blue Loops ...... 25

5 The Phase of the Asymptotic Giant Branch (AGB) ...... 30 5.1 TP-AGB and the Eﬀect of Rotation ...... 30 5.2 The S-Process and Rotation...... 35

6 Conclusion & Future Work...... 38 6.1 Summary ...... 38 6.2 Future Work ...... 39

References...... 41

iv List of Tables

3.1 A table indicating the considered stellar masses. The rotational speeds were taken from[Heger et al., 1999, Langer et al., 1999, Endal and Soﬁa, 1978, Piersanti et al., 2013]. The zero values indicate non-rotating cases . 16

6.1 Table summarizing the incorporated instabilities due to rotation . . 39

v List of Figures

2.1 Mass enclosed between two spherical shells of thickness dr ...... 4

3.1 Reactions involved in the branchings of the proton-proton chain [Guenther and Demarque, 1997] ...... 14 3.2 The CNO cycle(s). Unstable isotopes are represented by dashed circles, and the dashed line shows leakage out of the chains [Arnould et al., 1999] 15 3.3 Energy generation rate by the CNO cycle and the PP-chain . . . . . 15 3.4 HR diagrams for 2, 3, 7, and 15 M during the Main Sequence phase 17 3.5 fp, fT , and fT /fp as a function of mass coordinate during the main sequence for a 3 M star ...... 17 3.6 Extent of the convective core as a function of time during the main sequence ...... 18 3.7 4He and 14N surface abundances during the main sequence for 15 M star ...... 18 3.8 Diﬀusion coeﬃcient magnitudes during the main sequence ...... 19

4.1 The helium flash of a 2M star and its subsequent mini flashes. The time has been zeroed at the start of the flash...... 21 4.2 Convective structure of the core of a 2 M star undergoing the He-Flash 22 4.3 Comparison of the helium flash luminosities between rotating and non-rotating cases for 2M star ...... 23 4.4 Comparison of the core’s convective structure during the He-flash between rotating and non-rotating cases for 2M star ...... 24 4.5 Diffusion coefficients during the He-Flash as a function of mass coordinate ...... 25 4.6 HR-Diagram, integrated luminosities, and Convective structure of a static 7 M star ...... 26 4.7 The HR diagrams for a 7 M comparing the static case with two rotating cases, one having fc = 1 and the other fc = 1/30 ...... 27 4.8 The integrated luminosities and convective structure of a rotating 7 M with an uncalibrated mixing coefficient during the Helium burning phase ...... 28 4.9 Diffusion coefficients for the 7 M during the He burning phase . . . 28 4.10 The variation of 4He and 14N surface abundance as a function of helium central mass fraction during the helium burning phase of a 7 M star ...... 29

5.1 Schematic sketch of the mixing epsiodes during and between thermal pulses [Kippenhahn et al., 1990]...... 31 5.2 Integrated luminosities and convective structure during the TP-AGB phase of a 3 M star ...... 32 5.3 The comparison between integrated helium luminosity for rotating and non rotating cases for 3 M during the TP-AGB phase . . . . . 33

vi th 5.4 Convective structure of 3 M star during the 15 thermal pulse. The blue line represents the extent of the helium core and the orange line represents the extent of the C/O core...... 34 5.5 Diﬀusion coeﬃcients during and between 15th & 16th pulses of a 3 M star going through TP-AGB...... 35 5.6 The s-process path through the Zr, Nb and Mo isotopes.Thick lines show the main path of the s-process, while the thinner lines show the less important branchings[Nicolussi et al., 1998]...... 36 5.7 Hydrogen,13C, and 14N mass fractions between 15th & 16th pulses as a function of mass coordinate ...... 37

vii Chapter 1

Introduction

In the present thesis, we address the evolution of stars after they have been formed in a complicated process and ﬁnally reach the so called ”Zero Age Main

Sequence (ZAMS)”, where the stars start to increase their luminosity while evolving in the Hertzsprung-Russell diagram (HRD); by hydrostatically burning Hydrogen.

The evolution of stars is rather well determined on a computational basis. Given the basic stellar structure equations, see chapter 2, and suitable boundary conditions, a star can be modeled when the mass and initial composition are prescribed. This is the content of the so called ”Vogt-Russell” theorem, which states: The mass and composition structure throughout a star uniquely determine its radius, luminosity and internal structure, as well as its subsequent evolution.

The theorem’s statement is rather incomplete, as the eﬀects of magnetic

ﬁelds and rotation are not included, which motivates the present work. While stellar modeling without rotation is successful in many respects, see [El Eid et al., 2009], the eﬀect of rotation may play a certain role in various phases of the evolution.

We will consider the eﬀect of rotation on the structure and evolution of low- mass stars (LMS), up to 2 M , which undergo the helium ﬂash at core He-ignition.

Stars of higher masses, up to 8 M and are called intermediate mass stars (IMS), avoid the He-ﬂash and evolve to the Asymptotic Giant Branch (AGB) phase. This is also encountered in the case of LMS. It is well known that the thermal pulsations occur during the AGB phase, which leads to severe mass loss, so that the stars end their evolutions as C/O white dwarfs. It is also well known that the main component

1 of the s-process nucleosynthesis is produced during the AGB phase, initiated by a neutron source governed by the reaction 13C(α, n)16O [Busso et al., 1999]. It is intriguing to study the eﬀect of rotation on the production of this neutron source.

The massive stars of masses in excess of 8 M evolve through all the nuclear burning phases from hydrogen burning to silicon burning. They undergo core collapse and end up as supernovae explosions.

The present thesis is organized as follows. In chapter 2 we summarize the basic stellar structure equations and present their modifications (sect 2.2) in a one-dimensional formulation. In chapter 3, the evolution of the stars on the main sequence are presented with and without the effect of rotation. Chapter 4 concerns the He-burning phase of the stars having different masses. Chapter 5 deals with the evolution of the stars on the AGB phase. Finally, chapter 6 presents a summary of this present work.

2 Chapter 2

Method of Calculation

In this chapter, we introduce the necessary modiﬁcations to the stellar structure equations incorporated into the stellar code we have used, see [El Eid et al., 2009], along with other updates. We essentially follow the treatment done by Endal &

Sofia [Endal and Sofia, 1976] concerning the modification of the equations. The angular momentum transport equation and the diffusion coefficients describing the rotationally induced instabilities are similar to Heger et al. [Heger et al., 1999].

2.1 Stellar Structure Equations

In general, stellar structure and evolution are described by ﬁve basic equations, which will be summarized in the following sub-sections. The equations are derived on spherically symmetric and concentric shells. A fully detailed description of the structure equations can be found in the text by Kippenhahn et al.[Kippenhahn et al., 1990] and Icko Iben[Iben, 2013].

2.1.1 Continuity Equation

Let mr be the mass enclosed in a shell of radius r, as shown in Fig. 2.1. The mass enclosed between the radii r and r + dr is:

m(r + dr) m(r) = ρ(r)dV (2.1) −

Where dV = 4πr2dr is the volume of the shell. Combining this with the

3 Figure 2.1: Mass enclosed between two spherical shells of thickness dr deﬁnition of the derivative on the left-hand side of eq. 2.1, we get :

∂r 1 = 2 (2.2) ∂mr 4πρ(r)r

2.1.2 Momentum Equation

As can be found in reference [Kippenhahn et al., 1990], the momentum conservation equation can be written as

∂u 2 ∂Pr Gmr = 4πr 2 (2.3) ∂t − ∂mr − r

During the major burning phases, stars achieve a state of hydrostatic equi-

∂u librium, which means that the acceleration term ∂t can be set to zero. In other words, Eq. (2.3) indicates then that the pressure force encounters the gravity force.

However, in certain phases like the He-ﬂash (see sect. 4.1), the acceleration term stabilizes the calculations and enables a hydrodynamical scheme to solve the stellar structure equations.

2.1.3 Energy Conservation Equation

The luminosity proﬁle in the star is determined by the energy conservation equation which is written as (see ref. [Kippenhahn et al., 1990] for details):

∂Lr ∂r Pr dρr = (n ν ) + 2 (2.4) ∂mr − − ∂t ρr dt

4 In this equation, n is the nuclear energy generation rate, ν is the neutrino energy loss rule, r is the speciﬁc internal energy per gram. The second and third term represent the gravitational energy per gram, g. Therefore, Eq. 2.4 can be rewritten as

∂Lr = n ν + g (2.5) ∂mr −

The energy loss rate ν becomes important only during advanced burning phases, i.e carbon-burning and onward, since they are produced by diverse processes described in chap. 18 of reference [Kippenhahn et al., 1990]

2.1.4 Energy Transport Equation

In general, the energy transport in stars is due to radiation, convection and conduction. Considering the ﬁrst two modes of energy transport, the energy transport equation is written in the form:

∂Tr Tr Gmr = 4 (2.6) ∂mr −Pr 4πr ∇

In case of radiative or convective transport, the temperature gradient = ∇ dlnT dlnP , is given by

3κ LrPr = rad = 4 (2.7) ∇ ∇ 16πacG Tr mr

Where κ is the Rosseland mean opacity. In case of convective transport, = ad. ∇ ∇ P δ This happens in the deep interior, where ad = , with cp the speciﬁc heat at ∇ T ρcp ∂lnρ constant pressure and δ = ( )p, which is obtained from the equation of state. − ∂lnT In the case of envelop convection, is obtained from the solution of the mixing ∇ length theory(details in chapter 7 of [El Eid et al., 2009]).

2.1.5 Convection and Compositional Mixing

The change of chemical species is determined by nuclear transmutation and convective mixing. If rotational mixing is not considered, the variation of chemical

5 species is obtained by solving the following equation:

∂X ∂ 2 ∂X dX n = 4πr2ρ D n + n (2.8) ∂t ∂m ∂m dt nuc

In order to solve this equation, a criterion for convection is needed, and since it is a diffusion equation, the coefficient of diffusion (D) should be given. In case of the Schwarzschild criterion for convection, convective instability is encountered when

rad > ad (2.9) ∇ ∇

Both of which have been deﬁned in the previous section. Using the Schwarzschild criterion neglects the eﬀect of the mean molecular gradient

∂lnρ µ = (2.10) ∇ ∂lnP

If µ is involved, then the criterion for convective instability is more strict being ∇ given by: ϕ rad > ad + µ (2.11) ∇ ∇ δ ∇

Where ∂lnρ ϕ = (2.12) ∂lnµ P,T

The quantities ϕ and δ are determined by the given equation of state.

The diﬀusion coeﬃcient appearing in equation 2.8 above can be determined using the mixing length theory. As given in [Langer et al., 1985] and references therein, it is given by:

c 1/3 DC = 0.44HP gβ(1 β) ad( rad ad) (2.13) κρ − ∇ ∇ − ∇

Where β is the gas pressure component of the total pressure. This expression is derived using DC = (1/3)αvconvHP with α = 2.0 and HP is the pressure scale

P height given by HP = ρg .

Adopting the Schwarzschild criterion encounters the problem that rad ∇ does not match µ at the boundary of the convective zone, for example, the con- ∇

6 vective core boundary. What is needed is an extension of mixing beyond that boundary, but without energy transport.

There is no unique approach to treat this extra mixing. One approach is derived from two-dimensional calculations by [Freytag et al., 1996], suggesting a treatment of overshooting given by:

2z − f.H DOS = DC e P (2.14)

In this equation, z = rb r , where rb is the radius of the convective boundary as | − | determined by the Schwarzschild criterion, and f is a parameter characterizing the extension of the extra mixing in terms of the pressure scale height. The value itself is not well known. A value of f = 0.02 was suggested by [Herwig et al., 1997].

In case of the Ledoux criterion, the extended mixing is described by the process of semiconvection, which is a vibrational instability with a timescale less than the time scale of convection. This diffusion process is described by a parameterized diffusion coefficient as described in [Langer et al., 1985] for example.

2.2 Present Treatment of Rotation

2.2.1 Modiﬁcation of the Stellar Structure Equations

Centrifugal forces acting on the stellar matter affect the shapes of surfaces having constant pressure, density, and temperature, deviating them from spherical symmetry. This leads to a direct modification of the momentum and energy transport equations. The modifications stem from the notion that mass shells correspond to isobars instead of spherical shells. For any quantity f varying on an isobar, the mean value is defined by 1 I f = f dσ (2.15) h i Sp Sp

where dσ is an element of isobaric surface area which encloses a volume

4 3 Vp = 3 πrp. The eﬀective gravitational acceleration, g, is normal to Sp. It is given

7 by:

" #1/2 ∂ψ ∂ψ 2 1 ∂ψ 2 g = = + (2.16) ∂n ∂r r ∂θ

ψ is the Roche potential as described by Kippenhahn & Thomas [Kippenhahn and Thomas, 1970]:

GM 1 ψ = + ω2r2 sin2 θ (2.17) r 2

The first modification concerns the momentum equation, eq.(2.3). The influence of rotation is described by a factor fp given by:

4 4πrp −1 −1 fp = g (2.18) GmpSp

This leads to a modiﬁed momentum equation

∂P Gmp 1 = 4 fp 2 r¨p (2.19) ∂mp − 4πrp − 4πrp

The second modiﬁcation is on the radiative temperature gradient:

−1 " 2 2 # 3κ P Lp fT rp ∂ rp rad = 1 + (2.20) 16πacG T 4 m f Gm f ∂t2 ∇ p p p p mp

and,

2 2 ! 4πrp −1 −1 fT = g g (2.21) Sp h i

The terms g and g−1 are integrated using a 5 point Lagrange-Gaussian h i quadrature in the present models.

2.2.2 Angular Momentum Transport

Inside a star, angular momentum is transported and redistributed due to mass motions induced by convection, circulations, and shears. If the transport is treated as a diﬀusive process, then according to [Heger et al., 1999], [Endal and Soﬁa, 1978], and [Pinsonneault et al., 1989], it can be written as:

∂ω 1 ∂ 2 ∂ω 2ω ∂r = 4πr2ρ r2ν (2.22) ∂t r2 ∂m ∂m − r ∂t

8 Where ν is the turbulent viscosity. Equations (2.8) & (2.22) are solved with reﬂecting boundary conditions, i.e Neumann boundary conditions set to zero.

2.2.3 Diﬀusive Coeﬃcients

The diffusion coefficients used in equations (2.8) & (2.22) are related to multiple instabilities. All the required thermodynamic quantities used in the subsequent sections have been defined in previous sections.

2.2.3.1 Eddington-Sweet Circulation (ES)

A rotating star can not be in hydro-static and radiative thermal equilibrium. This is because surfaces of constant temperature and pressure do not coincide[Von Zeipel, 1924]. This causes large scale circulations. Homogenization on isobars are due to the radial component of the circulation, which will be considered here. According to [Kippenhahn, 1974], the circulation along the radial component is given by:

2 3 2 2 ad ω r l 2nucr 2r 3 ve = ∇ 2 (2.23) δ( ad rad) (Gm) l − m − 4πρr ∇ − ∇ In the presence of µ gradients, meridional circulation has to work against the potential, thus it might be suppressed or even inhibited. The stabilizing velocity is given by: HP ϕ µ vµ = ∇ (2.24) τKH δ( ad rad) ∇ − ∇ Gm2 where, τKH = rl is the Kelvin-Helmholtz timescale, To calculate the diffusion coefficient, the individual direction of the veloci- ties is not important since the stabilizing velocity is always opposite to the meridional flow. This results in a reduction of the effective flow. Following [Endal and Sofia, 1978], the effective velocity in the present models is calculated by:

vES = max ( ve vµ , 0) (2.25) | | − | |

The diﬀusion coeﬃcient is calculated by taking the product of the velocity and length scale of the circulation, which is the minimum of the extent of the

9 dr instability, dinst, and the velocity scale height, HES = . dlnvES

DES = min (dinst,HES) vES (2.26)

2.2.3.2 Secular Shear Instability (SSI)

The secular shear instability arises in fluids when the thermal diffusion timescale is much shorter than the angular momentum diffusion timescale. Gradi- ents in the mean molecular weight may inhibit the instability and must be taken into account. According to Endal & Sofia [Endal and Sofia, 1978], two stability criteria must be violated simultaneously for the instability to set in. The first being:

2 PrRe,c ρδ dlnr Ris,1 = ( ad rad) g > Ri,c (2.27) 8 P ∇ − ∇ dω and the second 2 ρϕ µ dlnr Ris,2 = ∇ g > Ri,c (2.28) P dω

1 Ri,c is the critical Richardson number, taken to be 4 . The critical Reynolds number, Re,c, is assumed to be 2500 [Heger et al., 1999], and the Prandtl number,Pr, is around 10−6 in stellar interiors [Endal and Soﬁa, 1978]. From it, the kinematic

4acT 3 viscosity can be extracted ν = PrK; where K = 3κρ is the thermal conductivity. The circulation velocity associated with the process is computed from the timescale and the length of the turbulent elements

s ν dω vSSI = (2.29) Re,c dlnr

limited to the adiabatic speed of sound cs. For the typical length scale, the velocity scale height of the ﬂow is assumed

dr Hv,SSI = (2.30) dlnvSSI limited to the pressure scale height. The resulting diﬀusion coeﬃcient is given by:

2 max (Ris,1,Ris,2) DSSI = min (vSSI , cs) min (Hv,SSI ,HP ) 1 (2.31) − Ri,c

10 The last term in the diﬀusion coeﬃcient allows us to smoothly turn on the instability with increasing violation of the criteria.

2.2.3.3 Goldreich-Schubert-Fricke Instability (GSF)

In the inviscid limit, Pr 1, two conditions can be derived for secular stability to axisymmetric perturbations [Goldreich and Schubert, 1967, Fricke, 1968].

They are the same as the Taylor-Proudmann and Rayleigh conditions for an incom- pressible ﬂuid:

∂ω ∂j = 0 = 0 (2.32) ∂z ∂r

The first condition, Taylor-Proudmann, indicates that meridional flows will be driven if the rotational profile is not conservative. It is also in contradiction with the ”shellular” rotation law enforced by baroclinic instability. Thus, the

GSF instability will tend to enforce uniform rotation in chemically homogeneous regions[Endal and Soﬁa, 1978].

The GSF instability has a stronger dependence on diﬀerential rotation than the ES circulation, and the large scale circulation velocity in the equatorial plane is estimated by:

2HT dlnω vg = ve (2.33) Hj dlnr

dr dr Here, HT = is the temperature scale height and Hj = is the − dlnT dlnj angular momentum scale height. The instability has the same µ-dependence as the

ES circulation, therefore the circulation velocity is calculated in the same manner:

vGSF = max ( vg vµ , 0) (2.34) | | − | |

And similarly to calculating the DES, DGSF is calculated by:

DGSF = min (dinst,Hv,GSF ) vGSF (2.35)

dr with Hv,GSF = . For small angular velocity gradients, the ES circula- dlnvGSF tion dominates, while the GSF becomes more important as the diﬀerential rotation

11 increases.

2.2.3.4 Calibrating the Mixing Eﬃciencies

The aforementioned rotationally induced instabilities are subject to con- siderable uncertainties because they result from order of magnitude estimates of relevant time and length scales. Therefore, efficiency factors of order unity are introduced in order to calibrate the diffusion coefficients with observational data.

This is based on the treatment done by [Pinsonneault et al., 1989].

The first adjustable parameter introduced is fc. It is responsible for re- ducing the diffusion coefficients generated by rotationally induced instabilities that go into the diffusion coefficient of eq. (2.13), while their full values go into the turbulent viscosity in eq(2.21). Which is to say that:

D = Dconv + fc (DES + DSSI + DGSF ) (2.36) ∗

ν = Dconv + DES + DSSI + DGSF (2.37)

The second adjustable parameter, fµ, describes the sensitivity of the rotationally induced mixing to µ-gradients. It is included by replacing µ with fµ µ. ∇ ∇ The major diﬀerence between this treatment and the one done by [Heger et al., 1999] is that the Dynamical Shear Instability and the Solberg-Høiland Instabilty have been neglected due to their lack of impact as shown by [Endal and Soﬁa, 1978]. Our choice for fµ and fc are 0.05 and 1/30 respectively as is done by [Heger et al., 1999, Chaboyer and Zahn, 1992, Pinsonneault et al., 1989, Piersanti et al., 2013, Langer et al., 1999].

2.2.3.5 Enhanced Mass Loss & Angular Momentum Loss

Mass loss in stars comes from two sources, nuclear burning and surface activity in the form of stellar winds. The mass loss rates used in the code, see

[El Eid et al., 2009], are modiﬁed to account for stellar rotation according to Friend

& Abbott[Friend and Abbott, 1986],

1 ξ M˙ (ω) = M˙ (ω = 0) ξ 0.43 (2.38) ∗ 1 Ω ≈ −

12 where

v 2 Gm κL Ω = vcrit = (1 Γ) Γ = (2.39) vcrit r − 4πcGm

Where Γ is the Eddington factor and Ω is the ratio of the surface equatorial rate to the critical rotation rate.

The loss of angular momentum from the surface due to stellar winds is approximated by removing the angular momentum along the surface layer

J˙ = Mj˙ surf (2.40)

Where jsurf is the speciﬁc angular momentum at the star’s surface.

13 Chapter 3

Main Sequence

The ﬁrst major burning phase in the Hydrogen burning for any formed star. During this phase, Hydrogen is transformed mainly into Helium, which is accomplished by the proton-proton chain (see Fig. 3.1) for stars of initial masses up to 1.5 solar masses. For those of higher initial masses, the H-burning proceeds by the CNO cycle (see Fig. 3.2). A fully detailed description can be found in chapter

6 of [Rolfs et al., 1988]

Figure 3.1: Reactions involved in the branchings of the proton-proton chain [Guenther and Demarque, 1997]

14 Figure 3.2: The CNO cycle(s). Unstable isotopes are represented by dashed circles, and the dashed line shows leakage out of the chains [Arnould et al., 1999]

Figure 3.3 reinforces the idea that the PP-chain energy generation is dom- inating for temperatures between 4 106 and 1.5 107 K, which is the case for × × stars having less than 1.5 M .

Figure 3.3: Energy generation rate by the CNO cycle and the PP-chain

Also from the ﬁgure, it is observed that the CNO cycle is much more temperature sensitive than the PP-Chain at higher temperatures, rendering it the

15 dominant energy generating cycle for hydrogen burning for stars more massive than

1.5M .

3.1 Eﬀect of Rotation on the Main Sequence

Stars spend 90% of their life time on the main sequence. This longevity will allow enough time for the discussed dissipation mechanisms to reduce any rapid core rotation built up during contraction to the main sequence, however, this is phase is not considered in the present work. Thus, rotation will not play a significant role during and early post main sequence stages [Endal and Sofia, 1976]. This holds true for LMS and IMS. For MS however, rotation’s effect is more pronounced during this stage.

Table 3.1 lists the stars taken into study along with their initial ZAMS rotation speeds. The listed LMS and IMS have been evolved all the way till the

TP-AGB phase, while the MS till the end of He burning.

Time on the Main Sequence 8 Mass (M ) Surface Rotation Speed (Km/s) Time (10 yrs) 2 0 10.011231 2 120 10.351031 3 0 3.0536659 3 250 3.1982587 7 0 0.37788084 7 210 0.38469581 15 0 0.10537300 15 300 0.11779796

Table 3.1: A table indicating the considered stellar masses. The rotational speeds were taken from[Heger et al., 1999, Langer et al., 1999, Endal and Soﬁa, 1978, Piersanti et al., 2013]. The zero values indicate non-rotating cases

The different assumptions of different surface rotation speeds is related to observations that the more massive stars have higher rotations. Comparing the times spent on the main sequence between with and without rotation, we see an increase of up to 11 % in the main sequence lifespan. The increased longevity can be explained by a reduced temperature profile, as shown by the HR diagrams in

Figure 3.4.

16 1.6 2.4 3 M non-rotating ⊙

) ) 3 M rotating ⊙ 1.4 ⊙ 2.2 ⊙ L/L L/L ( ( 10 10 1.2 2 M non-rotating 2.0 Log ⊙ Log 2 M rotating ⊙ 1.0 1.8 4.00 3.95 3.90 3.85 3.80 4.15 4.10 4.05 4.00 3.95 3.90 3.85 Log10(Teff ) Log10(Teff )

3.8 4.8

) 3.6 ) 4.6 ⊙ ⊙ L/L L/L

( 3.4 ( 4.4 10 10 7 M non-rotating 15 M non-rotating Log 3.2 ⊙ Log 4.2 ⊙ 7 M rotating 15 M rotating ⊙ ⊙ 3.0 4.0 4.35 4.30 4.25 4.20 4.15 4.10 4.50 4.45 4.40 4.35 4.30 4.25 Log10(Teff ) Log10(Teff )

Figure 3.4: HR diagrams for 2, 3, 7, and 15 M during the Main Sequence phase

fP fT 1.06 fT /fP

1.04

1.02

1.00

Inﬂuence Factors 0.98

0.96

0.94

0.0 0.5 1.0 1.5 2.0 2.5 3.0 Mr/M ⊙

Figure 3.5: fp, fT , and fT /fp as a function of mass coordinate during the main sequence for a 3 M star

Sustained burning at a lower temperatures can be explained by a larger

convective core. Figure 3.5 shows that fT /fp > 1 during the main sequence. Since it factors into the radiative gradient, as shown in eq. 2.20, the Schwarzschild criterion

is satisﬁed for a larger depth. The increase is illustrated in ﬁgure 3.6.

17 0.3 0.6

0.2 0.4 ⊙ ⊙ /M /M r r

M 0.1 M 0.2 2 M non-rotating 3 M non-rotating ⊙ ⊙ 2 M rotating 3 M rotating 0.0 ⊙ 0.0 ⊙ 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 time (109 yrs) time (108 yrs)

1.5 4 ⊙ ⊙ 1.0 /M /M r r

M M 2 0.5 7 M non-rotating 15 M non-rotating ⊙ ⊙ 7 M rotating 15 M rotating 0.0 ⊙ 0 ⊙ 0 1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 time (107 yrs) time (107 yrs)

Figure 3.6: Extent of the convective core as a function of time during the main sequence

non-rotating non-rotating 0.2975 rotating 0.0040 rotating

0.2950 0.0035 0.2925

0.0030 0.2900

0.0025 0.2875 Surface Abundance Surface Abundance N He 14 4 0.2850 0.0020

0.2825 0.0015

0.2800 0.0010 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 time (107 yrs) time (107 yrs)

4 14 Figure 3.7: He and N surface abundances during the main sequence for 15 M star

Since the diﬀerence is more pronounced for 15 M , we expect enhanced surface abundances for 4He and 14N, as they are by-products of the CNO cycle.

Figure 3.7 shows the diﬀerence between surface abundances of 4He and 14N in

18 the rotating and non-rotating cases. The figure indeed shows a steady a steady enhancement over the main sequence time. This can be explained by the ES circulation coming into effect during this stage of the evolution, providing added mixing throughout the star, as discussed by [Heger et al., 1999, Langer et al., 1999] and depicted in figure 3.8. Further analysis of Fig. 3.8 indicates that core mixing is enhanced in all the stars by ES circulation, followed closely by the SSI in LMS and

IMS.

Figure 3.8: Diﬀusion coeﬃcient magnitudes during the main sequence

19 Chapter 4

Eﬀects on Helium Burning

After the core Hydrogen is exhausted, the star’s core starts to contract while shell Hydrogen burning, driven by the CNO cycle in case of the stars considered in this work, forces the outer layers to expand, and the star is said to have entered the

Red Giant Phase. When the central temperature exceeds 108K, the triple-alpha

8 process becomes eﬀective. The temperature 6 10 K is possible at high density as in low mass stars, since the triple-alpha rate depends on the square of the density.

4He +4 He 8 Be + γ (4.1) 2 2 →4

8Be +4 He 12 C + γ (4.2) 4 2 →6

Since 8Be is unstable, the reaction (4.1) decays to two alpha particles. However, the high density achieved in the core of red giant stars enables the capture of a third alpha particle before the decay occurs, so that Carbon is produced according to reaction (4.2). Various effects can be observed during, or just before, core Helium burning depending on initial mass of the star. LMS of solar-like composition having mass 6 2.2M evolve through the core Helium flash, while the IMS of same composition do not encounter the He-flash since the electron degeneracy is not strong enough. After core He ignition, stars of masses in the range 4-12M exhibit blue loops in the HR diagram as shown in Fig. (4.4) in case of the 7M star. Detailed calculations concerning the blue loops were reported by [Halabi et al., 2012] and

20 references therein.

4.1 He Flash

As stated above, LMS of solar-like composition with mass 2.2M expe- 6 rience the core Helium ﬂash. This is a characteristic evolutionary phase of low mass

stars which proceeds under electron degeneracy. Observation has shown that the

ﬂash is not a destructive event, as stars evolve past it and proceed to fuse Helium

under hydrostatic conditions.

8 Lstar LH 6 LHe

4 ) ⊙ 2 L/L ( 10

Log 0

2 −

4 −

0.0000 0.0025 0.0050 0.0075 0.0100 0.0125 0.0150 0.0175 time (108 yrs)

Figure 4.1: The helium flash of a 2M star and its subsequent mini flashes. The time has been zeroed at the start of the flash.

The ﬂash occurs when the highly temperature dependent triple-alpha pro-

cess, described above, operates in a strong electron degenerate environment. When

the electrons are degenerate, the pressure becomes proportional only to the density.

Since the energy generation rate by the triple-alpha process changes like T 41 near

108K, the accumulation of nuclear energy cannot be transported through radiation.

Therefore the star expands violently in order to lift up the stellar layers from the

degenerate conditions, such that the He-burning can proceed quiescently. A con-

vection zone is formed to facilitate the energy transport as illustrated in Fig.(4.2)

21 0.5

0.4

0.3 ⊙ /M r

M 0.2

0.1

0.0 1.2224 1.2226 1.2228 1.2230 1.2232 1.2234 1.2236 1.2238 1.2240 time (109 yrs)

Figure 4.2: Convective structure of the core of a 2 M star undergoing the He-Flash

The fact that the star is not disrupted as the existance of the Horizontal

Branch (HB) clearly indicates is because the produced energy is used up to expand

the outer layers of the star. This expansion weakens the H-burning shell as seen by

the drop in the luminosity during the transition to the HB, where the He-burning

proceeds normally. Figures 4.3 & 4.4 show the comparison between the rotating

and non-rotating cases. The main flash of the rotating case has a luminosity of ≈ 8.8 7.6 10 L vs 10 L for the non-rotating case, which lifts the degeneracy much more effectively. This leads to a smaller number of required mini flashes to completely

lift the degeneracy as shown in ﬁgures 4.4. However, the time required to lift the

degeneracy is nearly the same in both cases, 1.5 106 years. During the ﬂash, ≈ ∗ according to ﬁgure 4.5, the secular shear instability is the dominant rotationally

induced instability in the outer shells while the Eddington-Sweet circulation takes

eﬀect towards the star’s degenerate core. Comparing the behavior with that during

the main sequence, ﬁgure 3.8, the depth and location of these instabilities within

the star have switched. This is to be expected since the ﬂash increases the speed of

mass motion towards the outer shells, stabilizing the meridional ﬂow and encourages

shearing.

22 8 Lstar LH 6 LHe

4 ) ⊙ 2 L/L ( 10

Log 0

2 −

4 −

0.0000 0.0025 0.0050 0.0075 0.0100 0.0125 0.0150 0.0175 time (108 yrs)

(a) Non-Rotating case

Lstar 8 LH LHe

) 4 ⊙ L/L ( 10 2 Log

2 −

0.0000 0.0025 0.0050 0.0075 0.0100 0.0125 0.0150 0.0175 time (108 yrs)

(b) Rotating case

Figure 4.3: Comparison of the helium ﬂash luminosities between rotating and non- rotating cases for 2M star

23 0.5

0.4

0.3 ⊙ /M r

M 0.2

0.1

0.0 1.2224 1.2226 1.2228 1.2230 1.2232 1.2234 1.2236 1.2238 1.2240 time (109 yrs)

(a) Non-Rotating case

0.5

0.4

⊙ 0.3 /M r M 0.2

0.1

0.0 1.23325 1.23350 1.23375 1.23400 1.23425 1.23450 1.23475 1.23500 time (109 yrs)

(b) Rotating case

Figure 4.4: Comparison of the core’s convective structure during the He-ﬂash between rotating and non-rotating cases for 2M star

24 Figure 4.5: Diﬀusion coeﬃcients during the He-Flash as a function of mass coordinate

4.2 Blue Loops

On the onset of Helium burning in stars in the range (4-12) M exhibit what is known as blue loops on the HR diagrams [Halabi et al., 2012, Halabi, 2014], refer to figure 4.6. The blue loop is essentially a crossing to a higher effective temperature and returning back to the RGB. What triggers the crossing is the reinforcement of shell hydrogen burning, via the CNO cycle, during core helium burning. Figure 4.6a displays the blue loop on the HR diagram. This occurs when as hydrogen burning shell burning is triggered, as indicated by the increased hydrogen luminosity in the top part of figure 4.6b. The convective structure, in the bottom part of figure 4.6b, indicates that the blue loop occurs between the first and second dredge up, and the outer layers of the star during this phase are radiative.

Considering that blue loops are important to explain super giants and δ- cepheids, they impose an observational constraint on rotational instabilities. An uncalibrated mixing coeﬃcient will lead to excessive mixing, driving material away from the hydrogen burning shell, preventing it from reaching ignition density.

25 4.2

4.0

) 3.8 ⊙ L/L ( 10 3.6 Log

3.4

3.2

4.3 4.2 4.1 4.0 3.9 3.8 3.7 3.6 Log10(Teff )

(a) HR-Diagram exhibiting a blue loop

0 ) ⊙ 5 − L/L ( 10 10 − Lstar

Log 15 − LH 20 LHe −

⊙ 4 /M r M 2

0 4.0 4.2 4.4 4.6 4.8 time (107 yrs)

(b) Convective Structure and integrated luminosities During Helium Burning

Figure 4.6: HR-Diagram, integrated luminosities, and Convective structure of a static 7 M star

26 Figure 4.7: The HR diagrams for a 7 M comparing the static case with two rotating cases, one having fc = 1 and the other fc = 1/30

Figure 4.7 compares the HR diagrams for three 7 M stars. The uncalibrated case does not experience the blue loop phenomenon. This reinforces the notion of introducing calibration terms to ensure that the observational constraints are met. The convective structure and integrated luminosities of this case suggest that the excess mixing drives the intermediate mass star to behave like a low mass star, see ﬁgures 4.8.

27 5

0 )

⊙ 5 − L/L ( 10 10 − Lstar Log 15 − LH 20 LHe −

⊙ 4 /M r M 2

0 4.2 4.4 4.6 4.8 5.0 5.2 5.4 time (107 yrs)

Figure 4.8: The integrated luminosities and convective structure of a rotating 7 M with an uncalibrated mixing coeﬃcient during the Helium burning phase

Figure 4.9: Diﬀusion coeﬃcients for the 7 M during the He burning phase

Analyzing the calibrated case shows that the blue loop crosses to a lower

surface temperature than the static case, but returns to the RGB at a higher lu-

minosity. This is due to the enhanced surface abundances of 4He, which in turn

28 alters the opacity that factors into the luminosity generation. The enhancement is

a direct result of the Eddington-Sweet circulation as depicted in ﬁgure 4.9. It is also

worth noting the GSF instability has a higher magnitude during this phase than on

the main sequence and operates in the outer shells instead of the core. This added

mixing enhances the 4He and 14N surface abundances as depicted in ﬁgures 4.10.

0.310 0.0036

0.0035 0.305

0.0034

0.300 0.0033

0.0032 Surface Abundance Surface Abundance N

He 0.295 14 4 0.0031

0.0030

0.290 non-rotating non-rotating 0.0029 rotating rotating

4.0 4.2 4.4 4.6 4.8 5.0 4.0 4.2 4.4 4.6 4.8 5.0 time (107 yrs) time (107 yrs)

Figure 4.10: The variation of 4He and 14N surface abundance as a function of helium central mass fraction during the helium burning phase of a 7 M star

A positive result is that calibration of the diﬀusion coeﬃcients is required

in the one-dimensional treatment of rotation to save the blue loops which are an

observational reality.

29 Chapter 5

The Phase of the Asymptotic

Giant Branch (AGB)

5.1 TP-AGB and the Eﬀect of Rotation

The AGB phase is divided into two parts, the early AGB (E-AGB) and the thermally pulsing AGB (TP-ABG). During the E-AGB phase, shell H-burning supplies the star’s luminosity, while the inner stellar layers keep contracting. This structure changes when the TP-AGB is initiated by the ignition of the He-burning shell surrounding the C/O core. This occurs in a thin layer in which the temperature of 108K is required to start the triple-alpha process. As mentioned in chap. 4, the strong dependence of this process on the temperature leads to enormous energy production in a thin layer that cannot be transported by radiation. Consequently, a convective zone is formed during this thermal runaway which enables the energy transport. The eruption of the He-shell leads to the expansion of the upper layers, so that the H-burning shell becomes temporarily ineﬀective (dormant state). After the relaxation of the shell He-burning, the star contracts, leading to the revival of the shell H-burning. This is the mechanism of the thermal pulsation on the AGB that is the interplay between the thermal instability of the shell He-burning on a short time scale, and the stabilizing eﬀect of the shell H-burning on a much longer time scale (see some results below).

What happens is that the erupted convective He-shell mixes the carbon

30 toward the H-burning shell, and after its relaxation, the formed convective envelop rich in Hydrogen extends inward and may mix protons into the He-C layers, so that 13C is created which delivers a neutron source for the main component of the s-process as described below. The mixing process of protons during the interpulse period is what is known as the ”third dredge up (TDU)”. Figure 5.1 illustrates the mixing process.

Figure 5.1: Schematic sketch of the mixing epsiodes during and between thermal pulses [Kippenhahn et al., 1990]

At time t1 the thermal pulse of the He-shell starts and triggers the intershell convection zone (ISCZ), which grows in mass and may reach the H-shell location at time t2. The H-shell, however, has extinguished at this time due to the radial expansion of the intershell. After the pulse terminates, the outer convection zone

(OCZ) can extend deeper than before (at t3) and mixes both protons into the intershell as well as carbon, produced in the He-shell and transported upwards by the

ISCZ, into the envelope. Upon contraction of the intershell, the OCZ recedes, the

H-shell reignites, and the proton-enriched intershell layers heat to suﬃciently high temperatures to allow 12C(p, γ)13C reactions in a radiative environment, forming the 13 C-pocket, which marks the start of s-process nucleosynthesis. At t4 the next pulse cycle starts, eventually leading to dredge-up of s-process elements and carbon

4 to the surface of the AGB star. The interpulse time t4 t1 is of order 10 years, − the timescale t3 t1 (or t6 t4) is a few hundred years − −

31 5.0

) 2.5 ⊙ L/L

( 0.0 10

Lstar

Log 2.5 − LH 5.0 − LHe

0.60

0.58 ⊙ 0.56 /M r

M 0.54

0.52

5.3820 5.3825 5.3830 5.3835 5.3840 5.3845 time (108 yrs)

Figure 5.2: Integrated luminosities and convective structure during the TP-AGB phase of a 3 M star

As depicted in ﬁgure 5.2, thermal pulses increase rapidly in strength after

the ﬁrst few, so third dredge-ups are generally the deepest and most likely to circu-

late core material to the surface. The relevant mass range for this type of evolution

is between 1 & 12 M , and is divided into two groups. The ﬁrst group is the group of normal AGB stars. These stars are the progenitors of C/O white dwarfs and had

masses on the ZAMS ranging from 1-7 M . The second group is called the Super AGB phase. These stars occupy the second part of the mass range and are heavy

enough to evolve through carbon burning to form electron degenerate Oxygen/Neon

(O/Ne) cores[El Eid, 2016].

32 6.0 Non-Rotating Rotating 5.5

5.0

4.5

4.0

3.5

Integrated Helium Luminosity 3.0

2.5

2.0 0.0000 0.0001 0.0002 0.0003 0.0004 0.0005 time (109 yrs)

Figure 5.3: The comparison between integrated helium luminosity for rotating and non rotating cases for 3 M during the TP-AGB phase

From an observational stand point, one can detect two major diﬀerences

between the static and rotating cases as shown in Fig. 5.3. The ﬁgure displays the

comparison between the integrated He luminosities for rotating and non rotating

cases for 3 M ,and the time has been zeroed in both graphs at the ﬁrst thermal pulse for both cases. We observe that the time between pulses is generally shorter

for the rotating case, yet the pulses have higher luminosities.

Analyzing ﬁgure 5.4 allows us to reach the following conclusions. Rotation

shifts up the position where the convective zone occurs during the thermal pulse,

which is a direct result of the modiﬁed Schwarzchild criterion in equation 2.20. It

does not however, extend the range of the zone itself, nor alleviate the need for

convective overshooting.

33 Non-Rotating Case Rotating Case 0.58

0.57

0.56 ⊙

/M 0.55 r M

0.54

0.53

0.52 0 80 160 240 320 400 480 560 640 720 800 3700 3850 4000 4150 4300 4450 4600 4750 time (yrs) +5.38144 108 time (yrs) +5.6352 108 × ×

th Figure 5.4: Convective structure of 3 M star during the 15 thermal pulse. The blue line represents the extent of the helium core and the orange line represents the extent of the C/O core

To further aid the explanation, we determine which of the considered rota-

tionally induced instabilites take eﬀect during the pulse and in between pulses.

34 th th Figure 5.5: Diﬀusion coeﬃcients during and between 15 & 16 pulses of a 3 M star going through TP-AGB

According to ﬁgure 5.5, the secular shear instability is the dominant rotationally induced instability both during and in between pulses, indicating it should be responsible for enhanced mixing while combined with convective overshooting.

5.2 The S-Process and Rotation

The slow neutron capture process, s-process, occurs mostly in TP-AGB stars. It generates more than half the atomic nuclei heavier than iron. During the process, iron remnant from a supernova of a previous generation of stars absorbs neutrons to form heavier isotopes. If the isotope is stable, a series of increases of atomic mass can occur. But if the isotope is unstable, beta minus decay occurs and the next element of higher atomic number is produced. A range of elements can be produced by the s-process due to alpha decay steps along the chain.

The process is slow in the sense that there is suﬃcient time for the beta decay to occur before the next neutron capture; decades may pass between one neutron capture and the next. The extent to which the process can produce isotopes is determined by the degree to which a star can produce neutrons. The quanti- tative yield is proportional to the amount available in the star’s initial abundance

35 Figure 5.6: The s-process path through the Zr, Nb and Mo isotopes.Thick lines show the main path of the s-process, while the thinner lines show the less important branchings[Nicolussi et al., 1998] distribution.The main uncertainty in calculating the s-process in AGB stars is to get the right abundance distribution which is termed the “13C pocket”. Some extra mixing is needed to have the right abundance of this Carbon isotope. The main problem is mixing of protons from the convective envelope is needed to create the

13C pocket. However it is important that not too many protons are mixed into the carbon-rich region to enable the complete operation of the CN-cycle to produce a high abundance of 14N, which is a neutron poison via the reaction 14N(n, p)14C.

It is clear that the s-process will not be eﬃcient when 14N is more abundant than

13C

One distinguishes the main and the weak s-process component. The main component produces heavy elements beyond Sr and Y, and up to Pb in the low- est metallicity stars. The production sites of the main component are low-mass asymptotic giant branch stars[Boothroyd, 2006]. The main component relies on the

13C neutron source through the 13C(α, n)16O reaction [Busso et al., 1999]. The weak component of the s-process, on the other hand, synthesizes s-process isotopes of elements from iron group seed nuclei to 58F e on up to Sr and Y, and takes place at the end of helium- and carbon-burning in massive stars[El Eid et al., 2009,

L.-S et al., 2000]. It employs primarily the 22Ne neutron source through the 22Ne(α, n)25Mg reaction. These stars will become supernovae at their demise and spew those s- process isotopes into interstellar gas.

With rotation, the angular velocity may produce strong shear instability

36 between the contracting core and the expanding envelope. Unlike the partial mixing which seems to be caused by extra mixing due to overshooting during the TDU, this shear instability does not disappear at the end of the TDU. In other words, continuous mixing of protons into the top of the He-intershell may lead to complete operation of the CN-cycle and high abundance of 14N would result. Due to the neutron poison reaction 14N(n, p)14C, the neutron exposure will be severely reduced, leading to ineﬀective production of the s-process elements(see [Piersanti et al., 2013]).

Another point is to include the eﬀect of the magnetic ﬁeld, see [Suijs et al., 2008], which may modify the angular momentum in the star and reduce the mixing between the core and the envelope. This is still an open problem.

Figure 5.7: Hydrogen,13C, and 14N mass fractions between 15th & 16th pulses as a function of mass coordinate

To study the eﬀect of rotation on the s-process, we refer to ﬁgure 5.7. The

ﬁgure compares the logarithm of the mass fractions of hydrogen, 13C, and 14N as a function of the mass coordinate. The mass fractions were taken in between pulses, the stage at which any variation of 13C or 14N would appear. The ﬁgure indicates no change in either mass fractions, indicating no proton mixing at this depth. This is due to the lack of convective overshooting, as mentioned before.

37 Chapter 6

Conclusion & Future Work

6.1 Summary

In the present work, the eﬀect of rotation has been studied on processes and phases during the evolution of stars of mass 2, 3, 7, and 15 M . The stars low and intermediate mass stars were evolved to the AGB phase and the massive star was evolved to the end of helium burning. The comparisons of theoretical predictions with observations allowed us to properly calibrate the instabilities and get more insight to interior modeling. The results are summarized as follows:

1. On the main sequence, chapter 3, we showed that stars shift towards lower

surface temperatures on the HR diagram and live longer on the main sequence.

This was sustained by larger convective cores. The main rotation induced

instability during this phase is the Eddington-Sweet circulation. For massive

stars, the latter lead to enhanced surface abundances of 14N and 4He.

2. For helium burning, chapter 4, we analyzed the eﬀect on blue loops for inter-

mediate masses and the helium ﬂash for low masses. For the helium ﬂashed,

we showed that the ﬂash itself has a higher amplitude for rotating cases than

static cases, lifting the degeneracy more eﬀectively. During the ﬂash itself,

the ES circulation operated in the core while the secular shear instability op-

erated in the outer shells. For blue loops, we showed the importance of proper

mixing calibration as it is possible that rotation, via the ES circulation with

the GSF instability, can prevent the hydrogen shell from igniting, eliminating

38 the blue loop altogether. The two instabilities also lead to enhanced 14N and

4He surface abundances.

3. On the AGB and for the s-process, chapter 5, we showed that rotation leads

to stronger helium shell ﬂashes during the thermal pulses. This in turn lead

to sharper convective zones. The dredge-up following the helium ﬂash also

reaches deeper, mixing more protons with 12C, increasing the 13C supply.

The enhanced 13C supply leads to a higher neutron density thus enhancing

s-process abundance output.

The following table summarizes the rotational instabilities used and during which phase their eﬀect was the most pronounced:

Instability Table Instability Description Phase in which the instability dominates Eddington-Sweet Cir- Large scale circula- Dominant instability culation tions generated by a during the main se- rotating star because quence and He burning it can not be in hy- phases drostatic and radiative thermal equilibrium simultaneously Secular Shear Instabil- Arises when the ther- ity mal diffusion timescale Helium flash for is much shorter than LMS the angular momentum diffusion timescale TP-AGB Phase for LMS and IMS

Goldreich-Schubert- A combination be- Minor eﬀect localized Fricke Instability tween the Rayleigh and in and around the con- Taylor-Proudmann vective core of the star conditions. The in- during all phases stability itself has a stronger dependence on diﬀerential rotation than ES circulation

Table 6.1: Table summarizing the incorporated instabilities due to rotation

6.2 Future Work

This work paved the way for investigations along the following tracks.

Firstly, to properly study the eﬀects on the s-process itself, we need to incorpo-

39 rate a full nuclear network software to properly calculate the yields. This will be done in colaboration with Prof. Meyer, by utilizing his nuclear network found here http://sourceforge.net/projects/nucnet-tools. Secondly, the framework has been laid down to incorporate dynamical instabilities and magnetic ﬁelds to observe their full eﬀect on structure and abundances alike.

This thesis has addressed several aspects of stellar evolution in the mass range under consideration and highlighted some of the present uncertainties as well as the shortcomings of the simpliﬁcations currently employed in stellar models. It has also raised several questions and leaves the door open to further investigations on diﬀerent levels as outlined above.

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