p s remnants: rotational and magnetic e their progenitors ......yevgeni kissin doctor of philosophy...

Properties of Stellar Remnants: Rotational and MagneticEvolution of Their Progenitors

by

Yevgeni Kissin

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of Astronomy & AstrophysicsUniversity of Toronto

Copyright c© 2017 by Yevgeni Kissin

Abstract

Properties of Stellar Remnants: Rotational and Magnetic Evolution of Their

Progenitors

Yevgeni Kissin

Doctor of Philosophy

Graduate Department of Astronomy & Astrophysics

University of Toronto

2017

The rotation and magnetic evolution of stars is examined in detail. Advection of

angular momentum by convective plumes is a source of differential rotation, which

both reduces the angular momentum loss to stellar winds and spins up the inner

regions. This spin up not only affects the rotation of the stellar remnant, but also

facilitates the winding of magnetic field lines necessary to amplify a seed field. In

radiative regions we demonstrate that an embedded magnetic field can effectively mix

angular momentum, as well as sustain coupling with adjacent convective zones.

In addition to the resulting angular momentum transport model, we consider a

mechanism for rotationally-induced winding of a seed magnetic field. When this

mechanism operates at a convective-radiative boundary which moves in a Lagrangian

sense towards the convection zone, net helicity is left behind in the newly radiative

material.

We employ this model of angular momentum transport and helicity deposition

in stellar models of a wide range of masses. Our aim is to gauge the effect on the

evolution of rotation and the impact on the properties of stellar remnants. In low-

and intermediate-mass stars we find a quantitative agreement between the predicted

and observed core and surface rotation rates, during both the sub-giant and helium-

burning phases. Continuing the evolution to the ejection of the envelope, we are able

ii

to reproduce the spin periods and magnetic fields of highly magnetic white dwarf

remnants.

We find similar success in massive stellar models, by being able to reproduce both

the spin periods and magnetic fields of pulsars. And in progenitors of black holes

with sufficiently high angular momentum pre-core-collapse, a portion of the infalling

material can reach rotational support. This quenches the accretion onto the newly

formed black hole, and may result in observable phenomena, such as high energy jets.

iii

Dedication

I would like to dedicate this dissertation to my soon-to-be-born first child. May it

serve as an inspiration to aim high and never give up!

iv

Acknowledgements

I would like to thank several people who played a key role in my achievement:

Christopher Thompson for his dedicated supervision over the past six years. I learned

to pay close attention to details, to be thorough, and that always "physics works".

My parents, Olga and Yakov Kissin, for all the sacrifices they made so their children can

have a better and easier life than they did (as exemplified in their dual immigrations).

And my wife, Anna Kissin, for supporting me both physically and emotionally

throughout the entire duration of my PhD program. Me being in graduate school

wasn’t easy for either of us, and it wasn’t her choice, but I couldn’t have done it

without her.

v

Contents

1 Introduction 1

1.1 Rotation and Angular Momentum in Stars . . . . . . . . . . . . . . . . . 3

1.2 Angular Momentum Transport . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Low- and Intermediate-Mass Stars . . . . . . . . . . . . . . . . . . 7

1.2.2 Angular Momentum Pumping by Convective Plumes and Rota-

tional Coupling via Maxwell Stress . . . . . . . . . . . . . . . . . 9

1.2.3 Massive Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2.4 Angular Momentum Transport by Convection and Maxwell

Stresses in High-Mass Stars . . . . . . . . . . . . . . . . . . . . . . 15

1.3 Binary Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.3.1 Low- and Intermediate-Mass Primaries . . . . . . . . . . . . . . . 18

1.3.2 Massive Primaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.4 Rotation and Magnetism of Stellar Remnants . . . . . . . . . . . . . . . . 22

1.4.1 White Dwarfs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.4.2 Neutron Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

1.4.3 Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

1.5 Plan of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2 Rotation of Giant Stars 38

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.1.1 Plan of the Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

vi

2.2 Pumping of Angular Momentum in Deep Convective Envelopes: Back-

reaction from the Coriolis Force . . . . . . . . . . . . . . . . . . . . . . . . 40

2.2.1 Extended Upflows and Downflows with Quasi-Geostrophic Bal-

ance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.2.2 Rapid Mixing Between Upflows and Downflows . . . . . . . . . 43

2.2.3 Model Ω̄(r) Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.2.4 Latitudinal Differential Rotation . . . . . . . . . . . . . . . . . . . 45

2.2.5 Matching of Rotation between Core and Convective Envelope . . 47

2.3 Comparison of 1.5 M� Model Star with Kepler Data . . . . . . . . . . . . 49

2.3.1 Transition from Co & 1 to Co . 1 throughout Bulk of Convective

Envelope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.3.2 Surface Rotation of Subgiants: Angular Momentum Pumping

Versus Magnetized Winds . . . . . . . . . . . . . . . . . . . . . . . 58

2.4 Evolving Rotation Profile with Mass Loss and Interaction with a Planet 60

2.4.1 Loss of Angular Momentum on the RGB and AGB . . . . . . . . 61

2.4.2 Rotation Rate of the Inner Envelope and Core . . . . . . . . . . . 61

2.5 Orbital Evolution of a Planetary Companion to a 1M� Star . . . . . . . 68

2.5.1 Jupiter-mass Planet with ai = 1 AU . . . . . . . . . . . . . . . . . 70

2.5.2 Other Planetary Configurations . . . . . . . . . . . . . . . . . . . 70

2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

2.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

2.7.1 Normalization of Convective Quadrupole During Giant Evolution 77

3 Spin and Magnetism of White Dwarfs 79

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79


3.2 Rotation and Magnetism in a Rapidly Evolving Giant Star . . . . . . . . 82

3.2.1 Combined Effect of Convection and Poloidal Magnetic Field on

Angular Momentum Redistribution . . . . . . . . . . . . . . . . . 82

vii

3.2.2 Magnetic Helicity Growth in Giant Cores . . . . . . . . . . . . . . 84

3.2.3 Relative Importance of Different Convective Episodes . . . . . . 88

3.3 Magnetic Field Amplification Near Core-Envelope Boundary . . . . . . 93

3.3.1 Critical Co for Dynamo Action: Effect of an Intense Radiation

Flux in the Tachocline . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.3.2 Magnetic Buoyancy in the Tachocline . . . . . . . . . . . . . . . . 95

3.3.3 Seeding a Toroidal Magnetic Field in the Tachocline . . . . . . . 97

3.4 White Dwarf Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . 101

3.4.1 Accumulation of Magnetic Helicity in a Radiative Core Relaxing

to Solid Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

3.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

3.5 White Dwarf Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

3.5.1 Dependence of White Dwarf Spin on Coriolis Parameter for

Core-Envelope Decoupling . . . . . . . . . . . . . . . . . . . . . . 114

3.5.2 White Dwarf Spin in Case of Continued Core-Envelope Coupling114

3.5.3 Minimal Magnetic Flux Enforcing Strong Core-Envelope Coupling118

3.6 Magnetic Field Emergence and Decay . . . . . . . . . . . . . . . . . . . . 122

3.7 Summary and Comparison with Alternative Theoretical Approaches . . 128

3.7.1 Some Outstanding Issues . . . . . . . . . . . . . . . . . . . . . . . 131

3.7.2 Competing effects of a collapsing low-mass (∼ 10−4 M�) AGB

envelope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

3.7.3 Impulsive Dynamo Amplification in a Merger? . . . . . . . . . . 136

3.8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

3.8.1 Twisting of a Radial Magnetic Field . . . . . . . . . . . . . . . . . 138

4 Rotation and Magnetism of Massive Stellar Cores 141

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141


viii

4.2 Stellar Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

4.2.1 Building the Star . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

4.2.2 Prescription for Numerical Evolution . . . . . . . . . . . . . . . . 148

4.2.3 Angular Momentum Transport and Magnetic Field Evolution . . 150

4.3 Rotational Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

4.3.1 Initialization of the Rotation . . . . . . . . . . . . . . . . . . . . . 151

4.3.2 Inward Pumping of Angular Momentum by Convection . . . . . 151

4.3.3 Magnetic Angular Momentum Transport Limited by Kink Insta-

bility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

4.3.4 Rotational Evolution between Model Snapshots . . . . . . . . . . 154

4.3.5 Some Basic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

4.3.6 Comparison with Rotationally Induced Mixing . . . . . . . . . . 157

4.4 Magnetic Helicity Accumulation . . . . . . . . . . . . . . . . . . . . . . . 158

4.4.1 Successive Convective Structures . . . . . . . . . . . . . . . . . . . 162

4.5 Combined Magnetic and Rotational Evolution to Collapse . . . . . . . . 165

4.5.1 Dependence on Effective Temperature . . . . . . . . . . . . . . . . 165

4.5.2 Dependence on Stellar Mass . . . . . . . . . . . . . . . . . . . . . 167

4.5.3 Neutron Star Remnants: Rotation and Dipole Magnetic Flux . . 167

4.5.4 Black Hole Remnants: Mass and Spin . . . . . . . . . . . . . . . . 171

4.6 Angular Momentum Injection from a Binary Companion . . . . . . . . . 173

4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

4.7.1 Implications for Growth or Decay of the Magnetic Field Post-

Collapse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

4.8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

4.8.1 MESA inlist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

4.8.2 Mass loss prescription . . . . . . . . . . . . . . . . . . . . . . . . . 184

4.8.3 Angular Momentum Transport by Winding a Poloidal Magnetic

Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

ix

5 Conclusions & Future Work 188

5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

5.2 Predictions and Observational Constraints . . . . . . . . . . . . . . . . . 191

5.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

Bibliography 195

x

List of Figures

2.1 Illustration of the two rotation models proposed in this work. . . . . . . 47

2.2 Flow speed of stellar material with respect to the core-envelope bound-

ary, compared with convective speed in the inner envelope. . . . . . . . 50

2.3 Rotation period at the base of the convective envelope, in a MZAMS =

1.5M� model star, as a function of time during the RGB and early core

He burning phases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.4 Rotation period at the base of the convective envelope of 1.5 M� and

M� models, for different rotation profiles. . . . . . . . . . . . . . . . . . 53

2.5 Rotation period at the base of the convective envelope of 5 M� model,

for different rotation rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.6 Profile of Coriolis parameter with radius during two phases of the

evolution of our 1.5 M� model with initial vrot = 50 km s−1. . . . . . . . 56

2.7 Profile of convective envelope during RGB expansion and AGB expan-

sion in rotation model I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.8 Evolution of surface rotation rate post-MS of a 1.5M� model and com-

parison with observations compiled by Schrijver & Pols (1993). . . . . . 59

2.9 Evolution of spin angular momentum of 1 M� model star during RGB,

core He burning and final AGB phases. . . . . . . . . . . . . . . . . . . . 64

2.10 Evolution of the Coriolis parameter at the base of the convective envelope. 65

2.11 Effect on the 5M� model star of the engulfment of a substellar companion. 66

xi

2.12 Same as right panel of Figure 2.11, but now with steeper inner rotation

profile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

2.13 Stellar radius of a 1M� model post-MS, and the orbital semimajor axis

of a planetary companion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

2.14 Evolution of semimajor axis and eccentricity of a planetary companion

around a 1M� model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

2.15 Probability of engulfment of planets of various masses around a 1M�

model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.1 Evolution of the base of the convective envelope of a 1M� during the

tips of the RGB and AGB. . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.2 Evolution of the base of the convective envelope of a 5M� during the

tips of the RGB and AGB. . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.3 Illustration of the buoyancy of a magnetic flux tube in a tachocline layer. 97

3.4 Relative buoyant speed of a magnetic flux tube in the tachocline layer

during the tips of the RGB and AGB of a 1M� model. . . . . . . . . . . 98

3.5 Thickness of a magnetized shell below the core-envelope boundary

which remains in contact with the active dynamo region. . . . . . . . . . 99

3.6 Minimal seed poloidal field for magnetic buoyancy to overcome the

doward drift of material. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

3.7 Magnetic helicity accumulation as a function of time in the 1M�. . . . . 107

3.8 Comparison of the magnetic helicity generated via the two channels we

consider, in the 5M� model. . . . . . . . . . . . . . . . . . . . . . . . . . . 108

3.9 Dipole magnetic field left behind in the WD remnants of 1M� and 5M�

models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

3.10 Dipole magnetic field left behind in the WD remnants of a 5M� model

as a function of companion mass. . . . . . . . . . . . . . . . . . . . . . . . 110

3.11 Evolution of spin angular momentum of the 1M� model during the tip

of the AGB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

xii

3.12 Evolution of spin angular momentum of the 5M� model during the tip

of the AGB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

3.13 Rotation period of the WD remnants of the 1M� and 5M� models. . . . 115

3.14 Comparison of the minimum angular momentum to maintain cou-

pling during the AGB wind phase of a 1M� and the stellar angular

momentum in cases with planetary interaction. . . . . . . . . . . . . . . 116

3.15 Total and core spin angular momentum of an AGB star during the con-

traction of its hydrogen envelope and the dimensionless core moment

of inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

3.16 Comparison between different estimates of magnetic flux in order to

motivate decoupling of core at end of AGB of 1M� model . . . . . . . . 121

3.17 Comparison between different estimates of magnetic flux in order to

decoupling of core at end of AGB of 5M� model . . . . . . . . . . . . . 122

3.18 Ohmic diffusion timescale in magnetized layer of WD remnant of 1M�

model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

3.19 Ohmic diffusion timescale in magnetized layer of WD remnant of 5M�

model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

3.20 Ohmic diffusion and decay of magnetic flux . . . . . . . . . . . . . . . . 127

3.21 Radius of surface and base of convective envelope versus residual

hydrogen mass at the end of evolution of 1M� model . . . . . . . . . . . 135

4.1 Determination of type of stellar remnant of our models based on com-

pactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

4.2 Convective structure of the 13M� model during both entire evolution

and final year . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

4.3 Convective structure of the 25M� model during final year of evolution 147

4.4 Convective structure of the 25M� model during the pre-MS accretion

phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

xiii

4.5 Kippenhahn diagram of zones which transport angular momentum and

those in which it is frozen, during both entire evolution and final year

of the 13M� model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

4.6 Evolution of the internal angular momentum during the entire evolution

of the 25M� model, using period of collapsed NS as a proxy . . . . . . 156

4.7 Evolution of the internal angular momentum during the entire evolution

of the 25M� model, using period of collapsed NS as a proxy, assuming

no magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

4.8 Comparison of timescales for angular momentum transport by different

mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

4.9 Magnetic flux contribution by different phases of the evolution of the

13M� model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164


25M� model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164


two 40M� models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

4.12 Comparison of the angular momentum loss to a wind for the four stellar

models we consider . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

4.13 Specific angular momentum profiles in the 13 M� and 25 M� models

at four separate times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

4.14 Evolution of the internal angular momentum during the final year of

evolution of the 13M� and 25M� models, using period of collapsed NS

as a proxy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

4.15 Evolution of the internal angular momentum during the final year of

evolution of the two 40M� models, using period of collapsed NS as a

proxy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

xiv

4.16 Specific angular momentum of the two isolated 40 M� models right

before core collapse, compared with the corresponding specific angular

of a stable orbit around a collapse BH . . . . . . . . . . . . . . . . . . . . 172

4.17 Stellar radius and mass as a function of age for both 40 M� models . . . 175

4.18 Impact of external torque from a companion on the angular momentum

stored in the evolving core of the 13M� and 25M� models . . . . . . . . 176

4.19 Impact of external torque from a companion on the angular momentum

stored in the evolving core of the low metallicity 40M� model . . . . . . 177

4.20 Specific angular momentum of the low metallicity 40 M� model right

before core collapse, after experiencing an external torque from a com-

panion, compared with the corresponding specific angular of a stable

orbit around a collapse BH . . . . . . . . . . . . . . . . . . . . . . . . . . 178

4.21 Spin angular momentum of the collapsed BH in the low metallicity

40M� model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

xv

Chapter 1

Introduction

Stars are the building blocks of the universe. They produce the majority of radiative

energy and are responsible for the chemical enrichment of the universe. It is therefore

vital to understand the properties of stars, to be able to explain their impact on their

surroundings. Stars illuminate and heat up the gas around them, and those with

effective temperatures Teff & 30, 000K can even impact the formation of other stars in

the same giant molecular cloud in which they formed. Once stars exhaust their main

source of fuel, hydrogen, they expand to giant dimensions (∼ 102 − 103 times their

original radius) and illuminate at a rate ∼ 10− 103 times higher. (Early-type stars

cool dramatically as they expand, which compensates the increase in radius.) It is

at this phase of evolution that stars lose substantial amounts of mass, which further

intensifies the impact on their surroundings.

The basic features of stellar evolution have been mostly a solved astrophysical

problem for several decades. We are able to describe the evolving properties of

stars quite well, by using simple conservation equations (mass and momentum), as

well as accounting for energy generation from nuclear reactions, and radiative and

convective energy transport (e.g. Eggleton 1971). Then the initial position of a given

star on the Hertzsprung-Russell (HR) diagram, and its subsequent evolution, is largely

determined by its zero age Main Sequence (ZAMS) mass (MZAMS), and metallicity

1

Chapter 1. Introduction 2

(Z).

However, some details of the physics of stars remain poorly understood. These

include internal rotation and its influence on stellar evolution. Stars are commonly

assumed to rotate as solid bodies on the MS, with weak differential rotation imposed

by convection, as is observed in the Sun. This assumption is more questionable

post-MS. Here the specifics of the rotation profile are important, because rotation

helps to induce mixing of chemical elements within stars, which affects the amount

of nuclear fuel available at different stages of evolution, and increases mass loss at

the surface by reducing the effective gravity. In addition, we have observations of

the rotation periods of stellar remnants, which span multiple orders of magnitude in

range (Prot ∼ 0.1− 103 days in white dwarfs and Prot ∼ 10− 103 ms in neutron stars).

Another important piece of physics not tackled in the early stellar evolution models

is the strength and configuration of magnetic fields. Magnetic fields are observed on

the surfaces of stars at various evolutionary stages (from the pre-MS T-Tauri stars, to

the MS Ap and Bp stars, to the magnetic remnants of stellar evolution, white dwarfs

and neutron stars). Magnetic fields are not simply an observable which deserves

an explanation, but are also able to influence the stellar evolution (primarily by

influencing rotation, a connection which will be discussed heavily in this work). The

combined modelling of internal stellar rotation and magnetism in evolving stars is

still in its infancy.

Observations continue to become more detailed, providing us with not only

basic surface properties of stars (mass, radius, temperature, luminosity, rotation rate,

and surface metallicity), but also information about their internal structure. One

important observational development is the detection of propagation of waves within

stars, obtained using asteroseismology (Unno et al. 1989; Christensen-Dalsgaard &

Daeppen 1992). This offers fairy direct probes for the internal density, rotation, and

energy transport (convective vs. radiative). Finally the compact remnant of a star


offers a more indirect probe of its internal workings throughout its "life". Our goal

here is to formulate a realistic picture of the internal rotation and magnetism of stars,

with the minimal accuracy needed to predict the rotation and magnetic properties of

their remnants.

1.1 Rotation and Angular Momentum in Stars

Up until a mere few decades ago, the only direct information regarding the angular

momentum of stars was from observations of their surface rotation velocities (and even

that was only the projected component - v sin i: Bernacca & Perinotto 1974, Conti &

Ebbets 1977, and Stauffer & Hartmann 1987). Low-mass stars, with MZAMS . 1.3M�,

lose a significant fraction of their natal angular momentum while on the MS. This is in

contrast to more massive stars which usually retain the majority of their natal angular

momentum. The difference arises due to the presence of a convective envelope in the

lower mass stars, in which a dynamo is able to amplify a magnetic field. This magnetic

field then causes particles ejected in the stellar wind to continue to co-rotate with the

surface out to a certain distance, which effectively drains angular momentum from the

star. This interaction between magnetic fields and ejected particles was first proposed

by Schatzman (1962). Mestel (1968) later derived rigorous expressions to specify the

loss of angular momentum from stars as a result of the interaction between stellar

wind and an anchored magnetic field, and Skumanich (1972) discovered a dependence

of surface rotation rate on age, v sin i ∝ t−1/2, known now as the Skumanich relation

(which was calibrated in part using the solar rotation rate).

Observations of rotation rates of evolved stars provide a weaker constraint on stellar

models, due to the difficulty of measuring the stellar mass. An indirect measurement

is possible in a star cluster, where the mass of the giant stars roughly corresponds

to the mass of the MS turn-off stars. Schrijver & Pols (1993) compiled observations

of sub-giants and giants from thirty publications. They found that stars spin-down


much faster as they evolve off the MS, than can be expected if they conserve angular

momentum. This effect may be partly due to the redistribution of angular momentum

within the star, and/or to enhanced angular momentum loss to a magnetized wind.

More precise measurements of stellar rotation became possible through astero-

seismology (Unno et al. 1989; Christensen-Dalsgaard & Daeppen 1992). Waves

propagating through stellar matter are influenced by internal rotation and density

gradients, and produce detectable perturbations to the surface brightness. Therefore,

using long cadence observations, we are able to gain a wealth of information about

the internal properties of stars. The first star for which we obtained this insight

was, as is usually the case, our Sun (a field specifically known as helioseismology).

Leighton et al. (1962) were the first to definitively observe the surface perturbations

in the Sun caused by internal acoustic waves. Later observations, made especially rich

by the Solar and Heliospheric Observatory space telescope, provided details regarding

the internal solar rotation (see Christensen-Dalsgaard 2002 for a detailed review of

helioseismology and its results). We now know the Sun rotates approximately close

to a solid body down to ∼ 0.1− 0.2R�, and particularly so in the radiative zone

(below ∼ 0.7R�). In the convective envelope the Sun possesses a latitudinal gradient

in rotation, with faster rotation near the equator, and slower towards the poles.

The roughly rigid solar rotation has led many to postulate a similar rotation profile

in all stars. And for many years this was the most reasonable choice to make, due to

the lack of direct evidence of the internal rotation of stars. This too changed when

instruments sensitive enough to detect the minute brightness fluctuations on the

surfaces of distant stars became available, and especially when space telescopes with

missions which required long cadence observations of stars were launched. CoRoT and

Kepler have produced a wealth of information about the internal rotation of stars over

a range of evolutionary phases. These observations clearly show that stars develop

differential rotation as they grow towards the giant branch post-MS (Beck et al. 2012;


Mosser et al. 2012), coinciding with the development of a deep convective envelope.

We are therefore led to explore the possibility that deep convection zones produce

significant radial differential rotation.

1.2 Angular Momentum Transport

Some of the earliest models of stellar evolution to look at internal rotation were those

of Kippenhahn et al. (1970) and Endal & Sofia (1976). These models did not account

for internal angular momentum transport. Starting from a solid rotation profile on the

ZAMS, these models conserved specific angular momentum throughout. This led to

the unphysical result that internal rotation rates surpassed the Keplerian rate (caused

by the spin-up of the contracting core, due to Ωrot ∝ R−2 whereas ΩKep ∝ R−3/2).

This is obviously problematic. Once angular momentum transport processes were

incorporated into stellar models, it was possible to maintain a coupling of the core to

the envelope, as stars expand on the giant branch (Endal & Sofia 1979).

Angular momentum transport processes generally fit into two categories: dynamic,

and secular. Dynamic processes occur on the rotation or convective timescales. In

radiative layers of the star, both the magnetorotational instability (MRI - Balbus &

Hawley 1998) and the dynamic shear instability work efficiently to dissipate shear on

horizontal layers and enforce "shellular" rotation (as opposed to cylindrical). Secular

processes operate on a slower timescale, comparable to the Kelvin-Helmholtz timescale

(τKH ∼ GM2/RL) or longer. Commonly employed secular processes are meridional

circulation and the secular shear instability, which feeds off a radial velocity gradient.

Meridional circulation (MC), on the other hand, transports angular momentum by

the bulk movement of matter (Maeder & Zahn 1998). MC can either occur by flow

away from the equator along the main rotation axis, then down the surface towards

the equator, and then back down towards the center. Or the flow can be exactly


reverse. The former scenario transports angular momentum from the surface to the

center, the latter scenario from the center to the surface.

In a star, the Prandtl number is low (Pr ∼ νv/νT, with νv the viscous diffusivity

and νT the thermal diffusivity) meaning that the thermal diffusion timescale is much

shorter than the viscous diffusion timescale (the timescale on which angular momen-

tum diffuses microscopically). The radiative diffusion can de-stabilize a radial angular

velocity gradient in a stably stratified layer (Goldreich & Schubert 1967; Fricke 1968).

In the non-axisymmetric case, this process is commonly called rotationally induced

mixing (Zahn 1974).

The dynamic transport processes operate on a shorter timescale, which suggests

they dominate over the secular processes. This has made it a common practice to

presume a "shellular" rotation profile in stellar models, an approximation we use in

our work as well.

The majority of models did not include the effects of magnetic torques on redis-

tributing angular momentum, something which has recently been shown to play an

important role. Spruit & Phinney (1998) considered it in their analysis of massive

stars, in the context of reproducing pulsar rotation rates. They estimated the magnetic

field strength as the minimum required to maintain solid rotation in the core, and

posed that decoupling occurs when the rotation period becomes longer than the evo-

lutionary timescale. Spruit (2002) formulated a more rigorous dynamo model, which

relies on a pinch-type (Tayler) instability of an azimuthal field in a stably stratified

region of the star. The small-scale radial field generated by the instability is then

stretched by differential rotation. The energy lost from differential rotation pushes

the region of the star to rigid rotation. This mechanism, commonly referred to as a

Tayler-Spruit (TS) dynamo, has been implemented numerous times since. Heger et

al. (2005) employed it in the Kepler stellar evolution code (Weaver et al. 1978) to

study the effects of angular momentum transport in massive stars, with a particular


interest on the angular momentum and magnetic fields of remnants. Cantiello et

al. (2014) used the MESA 1-D stellar evolution code (Modules for Experiments in

Stellar Astrophysics, Paxton et al. 2011), which includes the TS dynamo as an angular

momentum transport mechanism. They focused on intermediate-mass stars to study

the evolution of angular momentum as they evolve off the MS.

Very recently, the effects of asteroseismic instabilities on angular momentum

transport have started to be investigated. For example, Fuller et al. (2015) studied the

effects of internal gravity waves (IGWs), and found that they are able to transport

angular momentum quite effectively. We will not consider these effects in our work,

as the sign of the effect varies between authors.

1.2.1 Low- and Intermediate-Mass Stars

Schrijver & Pols (1993) analyzed the rapid spin-down of surface rotation, observed in

sub-giant stars as they evolve towards the giant phase. They assumed the convective

envelope (which develops during the sub-giant phase for stars of all masses) consis-

tently maintains a solid rotation profile, and considered rotational changes to arise

primarily in the radiative zone below. They focused on three possible scenarios: i) the

radiative zone rotates solidly and synchronously with the envelope; ii) it conserves

specific angular momentum at each mass shell; or iii) it rotates rigidly but some shear

is permitted at the convective boundary. They found that none of these three scenarios

is able to account for the spin-down observed, and used this as evidence that strong

magnetic wind breaking must occur to drain angular momentum from the stars. A

similar conclusion was reached by Rutten & Pylyser (1988).

Endal & Sofia (1979) previously considered the effect of both dynamic and secular

transport in model stars of several masses (10, 7, 5, 3, and 1.5M�). They used the

prescription they developed in Endal & Sofia (1976), which assumed convection zones

maintain solid-body rotation. In this early work, they mainly focused on reproducing


surface rotation rates, and specifically on comparing their results with two limiting

cases: perfectly rigid rotation, and conservation of specific angular momentum by

every shell. They found that "realistic" redistribution of internal angular momentum

results in slower surface rotation in evolved stars than both of the above extreme

scenarios (mainly as a result of an internal redistribution of mass by convection,

leading to an increase in the moment of inertia). Their comparison with observed

rotation rates of giant stars is very vague, due to poor observations available at the

time (mere upper limits on rotation rates of giants, whose masses were unknown).

Cantiello et al. (2014) attempted to reproduce the observations of Beck et al.

(2012) and Mosser et al. (2012) by evolving numerical models using the MESA stellar

evolution code. They used the built-in implementation of RIM in MESA to affect both

angular momentum transport and chemical mixing, as well as dynamo-generated

fields and their effect on angular momentum transport (using the TS dynamo pre-

scription of Spruit 2002). Convection zones were treated using the mixing length

theory (MLT), and they claimed to find that the resulting turbulent diffusivity was

so high as to result in rigid rotation within convection zones. (They did consider a

constant specific angular momentum profile in convection zones, but claimed it didn’t

affect their results.) They focused their analysis on a 1.5M� star, rotating initially

with an equatorial rate of 50km/s. They attempted several prescriptions for angular

momentum transport (both with and without the TS dynamo, and assuming local

conservation of specific angular momentum). However, they were unable to reproduce

the observations of strong differential rotation of about a factor 10 between the surface

and core of sub-giants (Beck et al. 2012), nor the dependence of core rotation period

on stellar radius of sub-giant stars (Mosser et al. 2012).


1.2.2 Angular Momentum Pumping by Convective Plumes and Ro-

tational Coupling via Maxwell Stress

We focus on two major gaps in the present understanding of angular momentum

transport in stars. First, we consider the advection of angular momentum by radially

extended plumes in deep convective envelopes. Second, we re-examine the role of

Maxwell stress in redistributing angular momentum in radiative layers of a star. In

particular, we are interested in the case where the layer is threaded by a significant

polar magnetic flux, which is stabilized by finite magnetic helicity.

One of the biggest modifications to internal rotation models brought on by direct

observations, such as asteroseismology, is the ability of convection zones to generate

differential rotation (Beck et al. 2012; Mosser et al. 2012). This is supported by fits

of 1D stellar models to color-magnitude diagrams, and by the results of numeri-

cal simulations of geometrically thick convection zones. Sackmann & Boothroyd

(1991) confirmed the need for a large mixing length in giant star models. The 3D

anelastic simulations of Brun & Palacios (2009) found that such plumes are able to

approximately conserve angular momentum as they rise/fall.

It is common practice to assume rigid rotation in convection zones. This is inspired

partly by the nearly rigid rotation in the solar convection zone, and the assumption

that the high turbulent diffusivity in convective regions is likely to efficiently smooth

rotation gradients. We also assume that convection zones are able to transport angular

momentum efficiently, and maintain a smooth rotation profile within them. However,

we are guided by the numerical simulations of Brun & Palacios (2009) to allow for

the development of differential rotation in geometrically thick convection zones (even

reaching a constant specific angular momentum profile, Ω ∝ r−2). In geometrically

thin convection zones we revert to a solid body rotation profile. For simplicity, we

assume an aspect ratio Rtop/Rbottom = 2 as the boundary between thin and thick

convection zones.


Low- and intermediate-mass stars, those with MZAMS . 8M�, maintain a fairly

simply structure post-MS, with a radiative core and convective envelope. We find that

when these stars reach the tips of the red giant branch (RGB) and asymptotic giant

branch (AGB) a constant specific angular momentum profile develops in the envelope.

Earlier in the expansion, one must account for the backreaction of the Coriolis force

on the inward pumping of angular momentum. When the rotation rate at the base

of the convection zone is sufficiently fast, which we parameterize using the Coriolis

parameter Co ≡ Ωτcon ≥ 1 (Ω is the rotation rate and τcon ≡ lP/vcon is the convective

overturn time, defined as the time it takes a convective plume, moving at vcon, to

traverse a pressure scaleheight lP), the Coriolis effect flattens the rotation profile.

We derive a specific dependence of the rotation profile on the slope of gravitational

acceleration (β, where g(r) ∝ r−β), using the "thermal-wind" approximation of the

vorticity equation (see Section 2.2.1).

We end up with a two-layer rotation profile in convection zones, made up of a

slowly spinning (Co < 1) yet very steep (Ω ∝ r−2) upper part, and a rapidly spinning

(Co ≥ 1) yet shallower lower part. We label the boundary radius between these two

zones Rc. The rotation profile, describing the mean rotation rate over equipotential

shells Ω̄, in the lower part of the envelope (between the base of the convective envelope

Rbenv and Rc) is given by

Ω̄(r) = Ω̄(Rc)(

rRc

)−(1+β)/2(1.1)

=1

τcon(Rc)

(r

Rc

)−α; Rbenv < r < Rc.

In the radiative part we find that the presence of a relatively weak magnetic field

(with magnetic energy density ∼ 10−16 − 10−15 of the convective energy density on

the MS, with the limit rising to ∼ 10−8 − 10−6 at later stages of evolution) is enough

to enforce solid rotation and rotational coupling with the convective envelope. We

therefore obtain a simple rotation profile for post-MS stars, with solid rotation in


the radiative core, rotational coupling across the mantle, and the above discussed

two-layer rotation profile in the convective envelope. Due to the dependence of

rotational coupling on magnetic flux, we estimate that below a critical Co, radial

differential rotation develops in the mantle. This is of particular interest in the rapidly

evolving late AGB phase. In our massive star models we develop a more direct

criterion for enforcement of solid rotation in radiative zones. In all of our models, we

consider angular momentum transport in post-processing, thereby ignoring the effect

of rotation on the stellar structure.

In Section 2.3 we compare the results of our rotation model with observations

of the core rotation rates in sub-giant stars, with a specific focus on a 1.5M� model

rotating at 50km/s on the ZAMS (evolved using MESA and chosen to match the model

analyzed by Cantiello et al. 2014). Such a star experiences weak angular momentum

loss on the MS, and we assume that the same is true after it crosses the Hertzsprung

gap. We find that our model reproduces the rotational difference between surface

and core observed using Kepler (Beck et al. 2012), as well as the evolution of core

rotation rates with stellar radius (Mosser et al. 2012). This comparison demonstrates

that the best fit is obtained for α = 1, which matches our model well, because at

this early stage of evolution, when the core is still relatively large, the dependence

of gravitational acceleration on radius gives β ∼ 1, and hence α ∼ 1. In addition,

we are able to reproduce the core rotation rates of helium clump stars (stars which

are stably burning helium), and are able to tackle the drastic spin-down of surface

rotation of stars as they evolve off the MS. By forgoing the dependence on a solid

rotation profile, as was done by Schrijver & Pols (1993), in favour of what we consider

a more realistic, differential, rotation profile in the deepening convective envelope we

find the majority of the observed spin-down can be accomplished while conserving

total angular momentum. The deepening convective envelope produces increasingly

differential rotation, which naturally spins down the surface. However, there still


remains a need for moderate angular momentum loss, which could occur via magnetic

wind torques (as proposed by Schrijver & Pols 1993).

Asteroseismology will continue to provide observational constraints on internal

angular momentum distribution at different evolutionary stages. For example, Klion

& Quataert (2017) considered the relative rotational splittings of both pressure and

gravity dominated modes, which act as a probe of differential rotation in a star (Beck

et al. 2012). They focused on sub-giant stars with masses in the range M? ∼ 1− 1.5M�,

and radii R? ∼ 4− 5R�, and used appropriate MESA models with different rotation

profiles applied in post-processing. They found that for many stars the rotational

splittings are most consistent with differential rotation concentrated in the mantle

and solid rotation in the envelope, instead of vice versa. Although, a combination of

differential in both mantle and envelope (with α ∼ 1) is moderately consistent. We

would like to point out three key points to consider when making a comparison of our

rotation model and observations: i) It is plausible for strong radial differential rotation

to exist in the mantle of our model stars. We assumed that a minimal magnetic field

will maintain solid rotation in the entire radiative region of the star, however, there

remains a possibility that magnetic fields could result in large-scale buoyant plumes,

which could drive the mantle to substantial differential rotation (Nordhaus et al. 2008;

Section 3.3); ii) The stars considered are sub-giants in which the Coriolis parameter is

substantially greater than unity. These conditions are ripe for the development of MRI,

which would substantially modify the rotational profile. We did not account for this

effect in our models; and iii) So far there exists little to no observational constraints of

the internal rotation of highly expanded giant stars. However, this is the phase along

the giant branch that our model should be most consistent, given Co < 1 in most of

the envelope, and the phase that is of most interest to our dynamo model (due to the

deep penetration of the convective envelope).

Chapter 2 discusses our rotation model in detail. In addition to the 1.5M� model


discussed above, we consider in more detail a solar-type star (given the solar angular

momentum when it reaches a radius R? = 10.9R� as a sub-giant), and a 5M� model

(set to rotate at 50 km/s on the ZAMS) to test the complete evolution up to ejection of

the envelope and emergence of the white dwarf (WD) remnant. These two masses

were chosen to test the effects of our model in a star which contains a convective

envelope on the MS, as well as one which contains a convective core.

It is important to note here the codependence that exists between rotation and

magnetic fields, each affecting the other, and both important to account for.

1.2.3 Massive Stars

Today there are very few direct measurements of internal rotation of massive stars,

especially rapidly rotating ones (Pápics et al. 2012). Some indirect constraints on

angular momentum transport come from the observable properties of their remnants,

especially neutron stars (NSs). We will introduce the relevant properties of NSs in

Section 1.4.2. For the immediate discussion it is only crucial to note that NSs without

close binary companions have spin periods in the range P ∼ 10− 1000ms (Popov &

Turolla 2012), and magnetic fields Br ∼ 3× 1011− 3× 1013G (Bhattacharya & van den

Heuvel 1991). There are other indirect tests of internal angular momentum transport,

such as mixing of chemical elements induced by rotation, which affects the amount

of nuclear fuel available for burning and can contaminate the surface with heavy

elements (Meynet & Maeder 2000), but we will not discuss these observations here.

A comprehensive analysis of angular transport within massive stars is given by

Heger et al. (2000). They considered several non-magnetic angular momentum

transport mechanisms, such as dynamic and secular shear instabilities, and convective

transport (which they assumed forces rigid rotation). They evolved stellar models of

masses 8− 25M� from the ZAMS up to shortly before core collapse. The predicted

NS rotation periods were significantly shorter than is observed, P ∼ 1ms in a 20M�


model.

This result may be due to an underestimate of magnetic torques within radiative

zones, which would cause a net outward transport of angular momentum from the

core. Spruit & Phinney (1998) considered a very simple model for estimating the

magnetic flux threading the material in an evolving massive star, as was mentioned

at the beginning of Section 1.2. They obtained extremely slow rotation in their

remnants, P ∼ 100s, which caused them to propose supernova kicks as a prevalent

source of remnant angular momentum. On the other hand, Spruit & Phinney (1998)

also assumed rigid rotation in convection zones, thereby neglecting inward angular

momentum pumping by convective plumes.

Heger et al. (2005) incorporated a self consistent model for magnetic field

amplification and angular momentum transport, combined with a variety of non-

magnetic instabilities, into massive stellar models. Their dynamo prescription was

based on Spruit (2002). They found that the magnetic field successfully enforced

solid rotation in the inner region of their stellar models, and drastically reduced

the angular momentum stored there (by as much as a factor of ∼ 30-50 compared

to non-magnetic models). The relatively weak poloidal fields they obtained were

unable to maintain core-envelope coupling beyond carbon ignition, resulting in a

fast spinning NS (P ∼ 10ms). In contrast, Spruit & Phinney (1998) assumed perfect

core-envelope coupling and hence obtained very slow rotation.

Fuller et al. (2015) studied the angular momentum transport by internal gravity

waves (IGWs). They found that IGWs can efficiently spin-down the core, relative to

conservation of angular momentum, and obtained pulsar rotation periods & 3ms. On

the other hand, if the core experienced significant spin-down, say due to the presence

of magnetic torques, IGWs are likely to spin-up the core and produce an upper limit

on the rotation period of pulsars ∼ 500ms.


1.2.4 Angular Momentum Transport by Convection and Maxwell

Stresses in High-Mass Stars

Massive stars have far more complicated structures than do low- and intermediate-

mass stars. Post-MS low- and intermediate-mass stars have a very simple structure,

containing a convective envelope and a radiative core, surrounded by one or two

burning shells. In contrast, massive stars posses multiple burning shells, with a

relatively thick convection zone within each one. In fact, there could be multiple

convection zones capable of effectively transporting angular momentum within a

single shell. In addition, the later stages of nuclear burning are so brief that we

must compare their duration with the timescale for angular momentum transport by

magnetic stress. We are therefore forced to consider a more detailed version of our

angular momentum transport mechanism.

In our massive stellar models, the angular momentum transport timescale is

computed within every individual mass shell. This is then compared with the

evolutionary timescale, τev ≡ min[lP/|vr|, (tcc− t)/3] (where vr is the Eulerian velocity

of the mass shell, and the second condition is to account for the time remaining before

core collapse), to determine whether angular momentum is frozen or transported

across this shell.

Transport via convection occurs on the timescale τcon ≡ lP/vcon, where vcon is

the convective speed. Within transporting convection zones (where τcon < τev) a

similar rotation profile to what we introduced in Section 1.2.2 is used, with a slight

modification. We change the strict separation between a rapidly rotating inner region

and a slowly rotating outer region, and instead base the choice of slope of rotation

profile on the specific value of the Coriolis parameter in each shell. The resulting

rotation profile is

Ω(r) ∝

r−2 : Co(r) ≤ 1

r−α = r−(1+β)/2 : Co(r) > 1(1.2)


In radiative zones, angular momentum is transported via Maxwell stresses, due to an

embedded magnetic field. Nominally, this occurs on the Alfvén timescale, given by

τA,r = lP/vA,r, where vA,r ≡ Br/√

4πρ is the standard Alfvén speed. However, above

a threshold magnetic field, the hydromagnetic kink instability suppresses the rate at

which magnetic waves are able to propagate. In Section 4.3.3 we describe the angular

momentum transport while the kink instability grows, and in Appendix 4.8.3 we

provide a detailed derivation. Effectively, the kink instability increases the minimum

magnetic field required to transport angular momentum. Expressing the condition

using the Alfvén speed, the kink instability raises the minimum speed required to

transport angular momentum to

vA,r

∣∣∣∣min

= max

[lPτev

(r

2lPΩτev

)1/4,

lPτev

], (1.3)

where the second condition applies deep within the star in the late stages of evolution,

when the evolutionary time is shorter than the rotation time (τev . Ω−1).

We estimate the magnetic flux threading the material via a self-consistent model

which leaves behind helicity at a receding convective boundary, as described in Section

1.4.2.

To apply our angular momentum transport model to massive stars, we consider

four models (simulated using MESA). 13M� and 25M� models are taken as NS

progenitors, and two 40M� models, one with solar metallicity and one with 30% solar

metallicity, as black hole (BH) progenitors. The remnant of each model is determined

based on the core compactness shortly before core collapse (O’Connor & Ott 2011;

Ertl et al. 2016). We find two important conditions that determine the overall angular

momentum distribution and evolution in our models. One is whether the star expands

to a red or blue supergiant. The main difference between these two states is their

surface temperature. The lower temperature in red supergiants results in a thicker

hydrogen convection zone. This makes a significant difference to the fraction of

angular momentum stored in the surface layers, due to increased inward pumping,


and as a result reduces the angular momentum lost to the stellar wind. We find

that out of our four models, the 13M� and 25M� models become red supergiants,

whereas the two higher mass models remain blue supergiants, probably due to the

greater mass loss (in general it remains unclear what physical properties determine

the surface temperature of a supergiant: see Woosley & Weaver 1995; Ekström et al.

2012). This has a significant influence on the total angular momentum available to a

NS or BH remnant.

The second difference between models is the quantity and thickness of convection

zones within the burning shells of the star. A large number of thick convection

zones implies stronger inward angular momentum pumping. We generally find

that higher luminosities (roughly correlated with total mass) result in more effective

angular momentum pumping. Therefore, although the total angular momentum

of our 13M� and 25M� models is roughly the same right before core collapse, the

angular momentum in the inner ∼ 2M� is much higher in the more massive of the

two models.

In Section 1.4.2 we will introduce our analysis of NS remnant rotation, as well as

magnetic fields. And in Section 1.4.3 we will introduce our analysis of BH spins and a

rotationally imposed cutoff to the mass collapsing to form a BH.

1.3 Binary Interactions

It is important to consider binary interactions when studying the evolving properties

of stars, because most stars form in multiple systems (with the majority being binaries).

The companions can be stellar, sub-stellar, or planetary in mass. Labelling the more

massive of the two objects the primary, and the other one the secondary (or simply

the companion), there are a large variety of interactions that can occur between a

primary and secondary, depending on the physical properties of each object and the


initial orbital configuration. The damping of tides is typically the dominant force-

at-a-distance, and therefore tends to initiate binary interactions. These interactions

can be mild, for example involving the transfer of angular momentum to (from) the

orbit from (to) the rotation of either one or both companions. In the other extreme,

the interactions could involve mass transfer, the formation of a common envelope

surrounding both companions, and/or the eventual merger of the cores of both objects.

Here we are primarily interested in the transfer of angular momentum from the

binary orbit to the primary. Even seemingly negligible low-mass planetary companions

can make a major contribution to the rotation of the primary, due to the relatively

high specific orbital angular momentum.

1.3.1 Low- and Intermediate-Mass Primaries

Recent observational studies have found that ∼ 46% of solar-type stars are primaries in

a multiple system, mostly binaries, with either a stellar or sub-stellar companion (see

Duquennoy & Mayor 1991; Raghavan et al. 2010). However, the binary interaction

with such massive companions could result in significant physical changes in the

primary. We wish to avoid complicated scenarios and therefore focus on planetary

companions to our solar-type model.

The space telescope Kepler, whose main purpose was the detection of extrasolar

planets, has found a plethora of planets (3,498 confirmed planets and 9,564 Objects

of Interest as of July 20171). Focusing on planets around solar-type stars, it is found

that ∼ 15% of such stars harbour Earth-like planets (radius ∼ 1− 1.5R⊕, see Petigura

et al. 2013; Silburt et al. 2015; and the most recent analysis done by Fulton et al.

2017), with orbital periods ranging from Porb ∼ 20− 200 days (the distribution of

planet occurrences appears to be relatively flat with orbital periods). Extrapolated to

longer periods, the prevalence seems to be ∼ 5% for Porb ∼ 200− 400 days. Larger

1See the NASA Exoplanet Archive - http://exoplanetarchive.ipac.caltech.edu


planets appear to be rarer, with a prevalence of ∼ 2% for planets with radii ∼ 4− 6R⊕

(which we might consider as Neptune-like), and a prevalence of ∼ 0.14% is observed

for planets with radii ∼ 9− 11R⊕ (which we might consider as Jupiter-like). These

numbers for the larger planets are estimated for orbital periods shorter than ∼ 100

days, but extrapolated to ∼ 300− 400 days they might only change by a factor of a

few.

Planets orbiting MS stars may be on stable, practically unchanging, orbits through-

out the entire MS evolution. However, as the host star grows on the giant branch, the

tidal interaction drastically increases in strength and may significantly perturb the

orbit of the planet. It is during this phase that planets are most likely to get engulfed

by their host stars. This phenomenon was investigated in several contexts. Kunitomo

et al. (2011) followed up on work by Sato et al. (2008) and Villaver & Livio (2009) to

investigate the dearth of planets within 0.6AU of clump stars, even though plenty of

such planets exist around MS stars. They proposed that tidal interactions between

the growing giant star and companion planets result in the engulfment of the closest

planets. Peterson et al. (1983) considered the engulfment of planets to explain rapid

rotation of clump stars. Livio & Soker (2002) studied the effect of the added angular

momentum to the mass loss and magnetism of the host star. And Privitera (2016)

considered whether planet interactions can explain the fast rotation observed in the

surfaces of some red giant stars.

We have considered the injection of angular momentum from a planet in combi-

nation with the rotation model detailed in Section 1.2.2. Low-mass stars (MZAMS .

1.3M�) lose a significant fraction of their natal angular momentum to magnetic winds

on the MS. Therefore, they are unable to sustain the necessary spin required for an

envelope dynamo while on the giant branch, even with the strong inward pumping of

angular momentum in the convective envelope. These stars require an injection of

angular momentum, which can easily be supplied by a planetary companion.


In addition to tidal dissipation on the orbit of a planet (Hut 1981), we consider a

gravitational perturbation induced by the time dependent density fluctuations in the

convective envelope. We envisage that large scale convective plumes result in a time

dependent external gravitational potential, dominated by the quadruple component.

The magnitude of the quadrupole moment is estimated by integrating the density per-

turbation caused by convection over the entire convective envelope. We then calculate

the external gravitational potential of the star by randomly rotating and modifying

the magnitude of the quadrupole moment once every global convective turnover time

(τcon ∼ (MenvR2?/L?)1/3). Further details are given in Section 2.5. Simply put, tides

tend to circularize and shrink the orbit, whereas the varying quadrupole stochastically

perturbs the orbit and tends, on average, to increase its eccentricity. Therefore, the

main effect of this varying quadrupole is to increase the range of semimajor axes over

which tides are able to drag a companion in. See Section 2.5 for more details.

To investigate the effect of planetary companions on a solar-type primary, we

consider a range of planetary masses (Earth, Neptune, and Jupiter), as well as initial

semimajor axes and eccentricities. We find that incorporating the quadrupole forcing

increases the probability of engulfment, especially for Earth-like planets.

We consider a simpler companion interaction in the case of the 5M� model. This

model reaches a radius ∼ 10 times larger on the AGB, than on the RGB, which

means that is when engulfment is most likely to take place. Therefore, to avoid

evolving orbits of planets throughout the entire post-MS evolution, we estimate the

tidal timescale which would result in an engulfment on the AGB. We consider the

engulfment of planetary and sub-stellar companions with several different masses

M ∼ 3− 100MJupiter. Even though this model does not require injection of angular

momentum for a dynamo to be active, the increased angular momentum positively

affects the magnetic flux amplified by the dynamo, as well as the rotation rate of the

WD remnant.


1.3.2 Massive Primaries

Massive stars, the progenitors of NSs and BHs, form predominantly in multiple

systems. Kobulnicky & Fryer (2007) found that the fraction of massive stars with com-

panions is greater than 80%, suggesting that binary interactions are nearly ubiquitous

(Sana et al. 2012). Much work has been done on the binary interactions of massive

stars, typically focusing on strong interactions and end-of-life properties (core collapse,

remnants, etc.). Binary interactions have been invoked to explain the existence of

Type Ib/c supernovae, thought to be caused by exploding helium stars, exposed due

to the loss of the hydrogen envelope through mass transfer onto a companion (see

Podsiadlowski et al. 1992; Yoon et al. 2010). Binaries were also evolved beyond first

core collapse to study the formation of double compact-star binaries. These systems

must form after common-envelope interactions (either single or double) which cause

the shrinking of the orbital separation between the two massive cores (see Portegies

Zwart & Yungelson 1998; Belczynski et al. 2002). They are of interest for their

eventual coalescence via the emission of gravitational waves (Belczynski et al. 2002;

Abadie et al. 2010).

In order to avoid such complicated interactions, we focus on widely separated

binaries, which only interact via tides and never experience Roche lobe overflow. This

process, although likely to have little effect on the stellar structure of the primary star,

may still significantly augment its total angular momentum and affect the rotation of

its remnant. To test the effect of this tidal interaction, we consider a simple scenario.

For each massive model, we take a companion with half its mass, and place it in

circular orbits of varying sizes. We evolve the orbit using the orbit-averaged tidal force

(Hut 1981).

We evolve the orbit concurrently with the rotational evolution of the primary,

assuming that total angular momentum is conserved. The secondary star is assumed

to remain unevolved, and is treated as a point object. In Sections 1.4.2 and 1.4.3 we


discuss the binary interactions of our massive star models in more detail and present

the result of such an interaction on the pre-core-collapse angular momentum of the

primary.

1.4 Rotation and Magnetism of Stellar Remnants

After the complete exhaustion of nuclear fuel, all stars leave behind a compact remnant.

Low- and intermediate-mass stars, those with MZAMS . 8M�, leave behind a WD

after the ejection of their outer envelope during double shell burning. High-mass

stars, those with MZAMS & 8M�, leave behind either a NS or a BH after their cores

experience loss of pressure support and collapse. The type of remnant left behind

is determined primarily by the compactness of the core shortly before core collapse

(O’Connor & Ott 2011; Ertl et al. 2016). Below a threshold compactness, neutron

degeneracy pressure and neutrino heating, which ramp up as the core reaches nuclear

densities, stall the collapse and send a shockwave which causes the ejection of the

outer layers. This results in a supernova (SN) and leaves behind a NS. At high enough

compactness, the inner layers collapse directly into a BH. Whether this produces an

observable electromagnetic signal depends on the angular momentum of the infalling

material. At low angular momentum the entire star collapses into the event horizon

(Kochanek et al. 2008; O’Connor & Ott 2011), whereas at high angular momentum a

part of the star may become centrifugally supported, producing a relativistic outflow

that blows up the remainder of the star (Woosley 1993).

Even though the formation mechanism of stellar remnants is fairly well understood,

as well as some of their basic properties (such as the equation of state of WDs), there is

still much that is unclear. For example, the origin of rotation and magnetism observed

in these remnants is unknown. We will now discuss each individual remnant type,

summarize the previous work done on their rotation and magnetism, and introduce

our model.


1.4.1 White Dwarfs

WDs are responsible for a variety of interesting phenomena. They are the central

sources of illumination in planetary nebulae, are involved in binary interactions

which create cataclysmic variables, explode as type Ia SNe (which are considered a

standardizable candle for cosmological distance measurements, Branch & Tammann

1992), and more. WDs are fairly well understood, due in part to their prevalence

(Holberg et al. 2016 found 232 WDs within 25pc of the Solar System, in a sample

which they estimated to be 68% complete), and in part to their relatively simple

equation of state. We also understand the mechanism of formation of WDs. They

are the inert remnant cores of stars with MZAMS . 8M�, which are exposed after the

progenitors reach the phase of double-shell burning and eject their outer envelopes

(producing an object observed as a planetary nebula).

However, there are still some mysteries regarding the properties of WDs, specif-

ically concerning their rotation and magnetism. A significant fraction of WDs, up

to ∼ 10% by some estimates (Liebert et al. 2003; Kepler et al. 2013), are found to

possess strong magnetic fields (B ∼ 106 − 109G). Rotation of such strongly magnetic

WDs is observed within a wide range, from as short as one hour to as long as 10

years (Ferrario & Wickramasinghe 2005; Brinkworth et al. 2013), with some evidence

to suggest a positive correlation between the strength of the magnetic field and the

rotation period (Ferrario & Wickramasinghe 2005). The origin of these fields has

been debated for decades, with several hypotheses proposed. Fossil fields and binary

interactions are the main two.

In the fossil field hypothesis it is proposed that magnetic MS stars, specifically Ap

and Bp stars, conserve magnetic flux as they evolve all the way through to envelope

ejection (Angel et al. 1981; Ferrario & Wickramasinghe 2005). Ap and Bp stars have

surface magnetic fields B ∼ 102 − 104G, which translates to a polar magnetic flux

Φ ∼ B · πR2 ∼ 4× 1025− 1027Mx. This is approximately within the range observed in


magnetic WDs. Therefore, if the magnetic flux observed at the surfaces of these stars

also threads their cores, and it is approximately conserved throughout the evolution of

the progenitor, strongly magnetic WDs would be a natural product. This hypothesis is

further supported by the observation that magnetic WDs tend to be more massive than

their non-magnetic counterparts (Kepler et al. 2013), which suggests their progenitors

are on average more massive than those of non-magnetic WDs.

However, strong evidence exists that magnetic WDs form exclusively as a result

of binary interactions (Tout et al. 2010). Magnetic WDs are observed to either be

completely isolated, or in a binary system with a close companion. In the former

group the binary interaction could have involved the disappearance of the companion

by a merger. In the latter group there is strong evidence that a binary interaction has

already taken place. On the other hand, few systems have been discovered containing

a magnetic WD with a binary companion far enough away to conclusively rule out

any past physical interaction.

Tout et al. (2008) suggested that progenitor systems of magnetic WDs undergo

a common envelope interaction once the primary grows on the giant branch, which

causes the in-spiral of the companion (either a low mass MS star or a substellar object).

The injection of angular momentum induces differential rotation in the convective

envelope of the giant, whence the combination of differential rotation and convection

results in a dynamo in the envelope. In this model the strength of the magnetic field

depends on the amount of angular momentum transferred, meaning that smaller final

separations after the ejection of the envelope result in stronger fields as well as shorter

rotation periods. A similar mechanism was proposed by Nordhaus et al. (2011),

wherein the secondary (be it a planet, a Brown Dwarf or a low-mass MS star) spirals

in close to the core where it gets shredded by the strong tidal force. This material then

forms a Keplerian disk around the core in which a magnetic field is generated by the

magnetorotational instability.


We outline the issues with the above scenarios in Section 3.7.3, but briefly: in the

former scenario convection is likely to smooth out the enhanced differential rotation

on a fairly short timescale, and in the latter scenario a question arises about how a

magnetic field amplified in such a transient manner can anchor in the CO core.

Helicity Accumulation Throughout Post-MS Evolution

We propose a mechanism by which a long term dynamo, operating at the boundary

between the convective envelope and radiative core, can source a stable magnetic

field in the core of a post-MS star. Previous treatments of such a dynamo have not

accounted for long term helicity accumulation (Blackman et al. 2001; Nordhaus et

al. 2008). The continuous mass flow across the lower boundary of the convective

envelope, caused by single or double shell burning, gradually delivers magnetized

material to the core. This involves a net transfer of magnetic helicity to the radiative

material. A related configuration is encountered in a pre-MS star, and during the

post-MS retreat of the convective core of a massive star.

Differential rotation is the principal driver of the helicity flux. The mass flow across

the convective-radiative boundary is itself a source of radial differential rotation. The

strength of the radial differential rotation is enhanced by inhomogeneous rotation

in the convection zone. A compensating Maxwell stress is then excited to enforce

solid rotation on the radiative side. The dynamo relies on fast rotation near the

boundary, which we parameterize using the Coriolis parameter Co. A lower bound

on Co comes both from mean-field dynamo theory, and observations of M-dwarfs (the

closest MS analogues of giants), which show that magnetic flux saturates at Rosby

number Ro ≡ 2π/Co . 0.1, corresponding to Co & 60, and declines as Co2 at lower

values. The lower threshold value of Co we use (Co . 1) is chosen due to the different

conditions in giants compared with dwarfs, for example the much higher radiative

flux emerging from the core (see Section 3.3.1).


In order to quantify the remnant magnetic field in a radiative layer, we focus on the

topological invariant H =∫

A · BdV. Here A is the magnetic vector potential from

which the magnetic field B is derived. H evolves only on the long ohmic timescale,

unless an exchange of magnetic twist is possible with the exterior of the medium.

Magnetic helicity has been shown to be an essential ingredient for maintaining

magnetostatic stability in a static and fluid star (Braithwaite & Spruit 2004). In that

respect, the focus on the lower boundary of a convective envelope as a source of net

helicity is justified, as it facilitates the necessary drainage of excess helicity by the

connection to the surface via rapid convective motions.

We envisage a toroidally twisted magnetic field experiencing a MHD flow towards

the core. The growth of H is proportional to the Maxwell torque (∼ BrBφr3) acting be-

low the convective-radiative boundary. It turns out that the dominant source of such a

torque comes from a latitudinal gradient in the Reynolds stress, imposed by convective

motions (as seen in the solar rotation profile and in simulations of convection under

rapid rotation, for example by Brun & Palacios 2009). This imposes a radial offset ∆Ω

in the rotation frequency in the tachochline layer below the convective boundary. The

winding up of the radial magnetic field is concentrated in the tachocline.

The Maxwell stress can be estimated by using the linear winding term in the

induction equation, to obtain a relationship between the radial (Br) and toroidal (Bφ)

fields

2πPdyn

Bφ ∼r∆Ω∆r

Br. (1.4)

Here Pdyn is the dynamo period, which we approximate as Ndyn∆Ω−1. The toroidal

magnetic field is in approximate equipartition with the rotational motions at the base

of the convective envelope,

B2φ ∼ 4πρ(

r∆Ω2π

)2. (1.5)


Therefore, the Maxwell stress works out to

BrBφ4π∼ ρr∆r(∆Ω)

2

2πNdyn. (1.6)

The time derivative of H, obtained by an integral over a sphere displaced below

the convective boundary, is expected to be non-vanishing due to a lack of perfect

reflection symmetry about the rotational equator. There is no reason to expect that the

distribution of magnetic flux in one hemisphere to exactly mirror the distribution in

the other hemisphere; we only expect the total magnetic flux to vanish. More details

regarding this general scheme are presented in Section 3.2.2.

Another reason to focus on the stably stratified tachocline layer below the convec-

tive envelope, is the ability of a magnetic field to anchor to this material. Magnetic

flux tubes then rise buoyantly to retain contact with the dynamo active region, closing

the feedback loop between toroidal and poloidal fields. The buoyancy of a flux tube is

greatly enhanced by the strong radiative flux in an AGB star (or a massive star). It

can be estimated from the temperature deficit induced in it by the added magnetic

pressure (which supplements the thermal pressure). The resulting buoyant speed is

vBr ∼(

B2φ8πP

)Frad

(∆r)2ρN2. (1.7)

P is the total pressure, ∆r is the radial distance from the convective-radiative boundary,

and N is the Brunt-Väisälä frequency. We find that vBr is significantly faster than

the radial speed at which mass shells sink due to nuclear processing vnucr over an

appreciable layer below the convective boundary. This buoyancy has the effect of

extending the active dynamo region and widening the layer with a strong differential

rotation, the tachocline, which remains in contact with the convection zone. In Chapter

3 we present a detailed estimate of seed field required to sustain this effect.

We now compare two types of differential rotation as sources of the Maxwell stress

near the convective-radiative boundary. In addition to the latitudinal gradient at the

base of the convective envelope, driven by convection, there is the radial inflow itself,


which drives differential rotation by the tendency of mass shells to conserve specific

angular momentum as they sink. We label the former mechanism "Ω(θ)", and the

latter "inflow". For both of these we estimate the growth of helicity as follows

dHdt Ω(θ)

= f (∆Mcore) · πεBṀcoreRbenv(ΩbenvRbenv)2, (1.8)

and

dHdt inflow

= f (∆Mcore) · πṀcoreΩbenvR2benvδvr(Rbenv). (1.9)

εB ∼ 10−3 represents the normalization of the Maxwell stress (1.6), Ṁcore is the rate

of mass growth of the core, and δvr(Rbenv) is the speed of mass shells relative to

the convective-radiative boundary. The factor f (∆Mcore) is meant to account for the

cancellation effect induced by the mixing of helicity of opposite sign from the two

hemispheres. One observes that the "Ω(θ)" contribution dominates.

In Section 3.4.2 we present the resulting magnetic flux distribution in the cores of

our two stellar models (solar-type and 5M�). The polar WD magnetic field lies in

the range of BWD ∼ 106 − 108G, with the magnitude increasing proportionally to the

injected angular momentum. The 5M� progenitor does not require an injection of

angular momentum in order to sustain a dynamo and produce a strongly magnetized

WD, but tidal interactions do affect the strength of the embedded field.

We find that we are not able to reproduce magnetic fields higher than∼ 108G. In the

solar-type star, an engulfment of a companion much more massive than Jupiter would

require exceptional circumstances; therefore we can consider the fields obtained as

approximate upper limits. In the 5M� star, an engulfment of a companion as massive

as ∼ 0.1M� may even be more common, therefore fields as high as 108G can be

expected. The extremely magnetic WDs (with fields ∼ 108 − 109G) may form from

the merger of two WDs (García-Berro et al. 2012).


Magnetic Field Ohmic Diffusion

The dynamo process described so far gradually deposits magnetized material into a

shell of limited mass forming the outer boundary of the growing core (destined to

become a WD). In the solar-type model this occurs during both the RGB and AGB,

when the convective envelope penetrates deep enough so as to affect layers destined to

end up in the WD. In the 5M� model only the AGB contributes to the magnetic field

embedded in the WD, due the shallow depth of the base of the convective envelope

during the RGB.

Towards the end of the evolution, as the star loses its envelope, it is also likely to

lose enough angular momentum such that Co at the base of the envelope (Cobenv)

drops below a critical value needed to sustain the dynamo. This means that a thin

non-magnetic layer will prevent the magnetic field from being detectable immediately

after the ejection of the envelope. The buried field will emerge by ohmic diffusion.

We generalize previous calculations of ohmic decay of magnetic fields in WDs

(Cumming 2002) to address the case of a magnetized outer shell. We find that the

magnetic field located at the top of the magnetized layer will emerge in ∼ 107 years,

in both mass models, suggesting a modest lag between the appearance of the WD and

the emergence of the field. The transport time at base of the magnetized layer is ∼ 109

years in the lower mass remnant, compared with ∼ 2× 108 years in the higher mass

remnant. At later times, the field will decay approximately as (t/tdiff)−1/2, where tdiff

is the diffusion timescale.

These results suggest that in low mass WDs magnetic fields are likely to remain

detectable until a very old age. On the other hand, the remnants of nearly all

intermediate-mass stars will experience some field decay at ages exceeding ∼ 109

years.


Rotation of White Dwarfs

Turning now to the rotation period in the remnant WD, we emphasize that it depends

on the angular momentum retained by the core past the phase of envelope ejection.

As the star loses mass it also loses angular momentum, and as long as the core

remains rotationally coupled to the envelope, the core will lose angular momentum as

well. Given that rotational coupling requires a minimum magnetic flux, the strong

dependence of the dynamo-generated field on Cobenv (as suggested by observations

of MS fields, e.g. Reiners et al. 2009) means that that when Cobenv drops below a

critical value, the magnetic flux amplified by a dynamo becomes too weak and the core

decouples. The resulting rotation period depends on the choice of this critical value;

we envision that decoupling should be possible for Cobenv < Cobenv,crit ∼ 0.1− 0.3.

Under the assumption of sustained rotational coupling, the core angular momen-

tum, Jcore, in the 5M� model remains roughly constant as the star loses its envelope

at the end of the AGB, due to the deep convective envelope maintaining a constant

specific angular momentum profile in the envelope. This means that the angular

momentum in the remnant (which we set equal to Jcore) is approximately independent

of when the core decouples. Therefore, the remnant spin is directly proportional

to the initial angular momentum of the star (except in the case of significant angu-

lar momentum injection, such as by the absorption of a massive companion, say

& 3MJupiter).

An interesting change in rotation results from a re-expansion of the envelope

caused by a late helium flash, which is seen in our solar-mass model. During this

flash ∼ 10−3M� of radiative material expands into a large convective envelope, which

results in a final loss of mass, and a significant spin-down of the star. During the main

AGB wind phase the stellar angular momentum drops by a factor ∼ 103, whereas the

core angular momentum drops by a factor ∼ 5. But during the late re-expansion, both

stellar and c

p s remnants: rotational and magnetic e their progenitors ......yevgeni kissin doctor of philosophy...

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