Massive Stars in Low Metallicity Environments

DOCTOR OF PHILOSOPHY

Massive stars in low metallicity environments

Higgins, Erin

Award date: 2020

Awarding institution: Queen's University Belfast

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Download date: 08. Oct. 2021 Massive stars in low metallicity environments

Erin Rose Higgins BSc PGCE

PhD

Queen’s University of Belfast

Dublin Institute for Advanced Studies

Armagh Observatory & Planetarium

June 2020 To the strong women in my life, for their eternal inspiration. Declaration

I declare that the work contained in this thesis has not been submitted for any other award and that it is all my own work. I also conﬁrm that this work fully acknowledges opinions, ideas and contributions from the work of others. The work was done in collaboration with Jorick Vink. I declare that the Word Count of this thesis is less than 80,000 words.

Name: Erin Rose Higgins

Signature:

4 June 2020

External Examiner: Prof. Jose Groh Director of Astrophysics and Tenured Assistant Professor Trinity College Dublin, Ireland

Internal Examiner: Dr. Gavin Ramsay Armagh Observatory and Planetarium Armagh, UK

Principal Supervisor: Prof. Jorick Vink Armagh Observatory and Planetarium Armagh, UK

Ph.D Candidate: Erin Rose Higgins Student identiﬁcation number : 40080963 Armagh Observatory and Planetarium, Dublin Institute for Advanced Studies, Queen’s University Belfast

ii Acknowledgements

To my supervisor Jorick Vink, thank you for your guidance and encouragement throughout my PhD. For moulding me into a better astrophysicist and giving me the best ex- periences, I am indebted to you. To Andreas Sander for your continued motivation and feedback, thank you for the pleasure of your collaboration.

To my childhood sweetheart, Jamie, for our endless discussions on life, the Universe, and even semiconvection. For teaching me to be spontaneous, for growing up beside me, and for supporting me throughout my PhD, I am perpetually thankful for our life together, and for exploring the cosmos with you, my closet-physicist.

To my Mammy, for teaching me how powerful a young woman can be, and for always knowing how to ﬁx anything on our long walks on the beach. You will forever be my inspiration and my best friend, for I am only here because of your immeasurable strength and love.

To my Daddy, for being the origin of my motivation and curiosity, for the years of graphs and for shaping my future, I owe my scientiﬁc nature to you. Thank you for your unwavering determination to get me to where I needed to be and supporting me through everything.

To Kate, watching you become the woman you hope to be brings me so much pride and inspiration that I often forget who the little sister is, thank you for being you.

To my grandparents, for my Grandma’s limitless belief in me and my Granda’s heartfelt pride. Thank you for making me believe I could do anything. For my Nana, who showed me the value of education, though I wish you could be here, a part of you will always remain in me. For my Granda-Jo, who always supported me.

To my PhD-partner Lauren, thank you for making this journey one of laughter and friendship. To all the students and staff at Armagh Observatory, thank you for your support over the years.

iii Abstract

The evolution of massive stars has been a key field of research for many decades, as it has significant effects on a variety of astrophysical disciplines, Langer (2012). Nonetheless, it is still widely uncertain due to a deficit of observational evidence to verify theoretical models. Recent surveys of the galaxy and Magellanic clouds (Evans et al., 2011) in addition to continuous development of stellar evolution codes (Paxton et al., 2015) are, however, replenishing the domain of massive stars so that we may probe the characteristics of their evolution to better understand the early stages of their life.

The evolution of massive stars is highly heterogeneous, sensitive to mass loss, internal mixing, metallicity, rotation, binarity, and magnetic ﬁelds. In this thesis, I have probed the evolution of massive stars for a range of metallicities, including our Milky Way galaxy and the Magellanic Clouds. This work provides an insight into the evolutionary models conducted in an endeavour to understand the dominant physical processes acting on massive stars, chapters 3 and 4. Theoretical predictions of the upper luminosity limit of red supergiants and constraints on internal mixing processes such as convective overshooting and semiconvection are presented in chapter 5. Chapters 1 and 2 provide an introduction to the central research chapters 3-5, with conclusions and future work outlined in chapter 6.

iv List of Publications

A list of publications resulting from the work presented in this thesis are provided below.

Erin R. Higgins & Jorick S. Vink, Massive star evolution: rotation, winds, and overshooting vectors in the mass-luminosity plane. I. A calibrated grid of rotating single star models, 2019, A&A, 622, A50

Erin R. Higgins & Jorick S. Vink, Massive star evolution revealed in the Mass- Luminosity plane, 2019, IAUS, 346, 480H

Erin R. Higgins & Jorick S. Vink, A theoretical investigation of the Humphreys- Davidson limit at high and low metallicity, 2020, A&A, 635

Erin R. Higgins & Jorick S. Vink, Stellar age determination and mass discrepancies in the Mass-Luminosity Plane. II. A grid of rotating single star models with LMC metallicity, 2020, to be submitted to A&A

Erin R. Higgins, Andreas S. Sander, Jorick S. Vink, Raphael Hirschi, Evolution of Wolf-Rayet stars as black hole progenitors, with a new locally consistent mass-loss recipe, 2020, in prep.

v Contents

Declaration ii

Acknowledgements iii

Abstract iv

List of Publications v

1 Introduction 1

1.1 Stellar classiﬁcation and evolution ...... 2 1.1.1 Massive stars ...... 2 1.1.2 Evolutionary channels ...... 5 1.1.3 Final fates ...... 6 1.2 Massive stars throughout the Universe ...... 9 1.2.1 Low metallicity environments ...... 11 1.2.2 Mass Loss as a function of Metallicity ...... 14 1.2.3 Most massive stars in the Universe ...... 14 1.3 Thesis overview ...... 16

2 Theoretical modelling of massive stars 18

2.1 Stellar structure and evolution ...... 18 2.2 1-dimensional stellar evolution ...... 21 2.2.1 MESA v8845 ...... 23 2.2.2 Basic input parameters ...... 23 2.3 Physical processes ...... 25 2.3.1 Convection ...... 25 2.3.2 Rotation ...... 28 2.3.3 Stellar winds ...... 30 2.3.4 Summary ...... 32

3 Galactic evolution in the Mass-Luminosity Plane 34

vi 3.1 Introduction ...... 35 3.2 Method ...... 40 3.2.1 The detached, eclipsing binary HD 166734 : A test-bed for massive star evolution ...... 42 3.3 Mixing and mass loss ...... 46 3.3.1 Envelope stripping and nitrogen enrichment...... 46 3.3.2 Rotationally induced mass loss ...... 49 3.4 Mass - Luminosity Plane ...... 51 3.5 Observational constraints ...... 60 3.5.1 Galactic observational sample ...... 63 3.6 Grid analysis ...... 65 3.6.1 Red supergiant upper luminosity limit ...... 67 3.6.2 Compactness parameter ...... 68 3.7 Discussion and conclusions ...... 71

4 The Magellanic Clouds and determining stellar age 73

4.1 Introduction ...... 74 4.2 Methodology ...... 76 4.2.1 MESA ...... 76 4.2.2 Mass - Luminosity Plane ...... 77 4.3 Evolution models for detached binaries in VFTS ...... 77 4.3.1 R139 : the most massive binary test-bed ...... 79 4.3.2 VFTS063 ...... 80 4.3.3 VFTS116 ...... 80 4.4 Stellar age determination in the M-L plane ...... 81 4.4.1 VFTS age estimates ...... 81 4.5 Grid comparison with observations of O and B supergiants from the VFTS sample ...... 86 4.5.1 Main sequence objects ...... 86 4.5.2 Mass discrepancy ...... 87

vii 4.6 Conclusions ...... 91

5 Post-Main Sequence Evolution to Red Supergiants 93

5.1 Introduction ...... 94 5.2 Method ...... 97 5.2.1 MESA Stellar evolution models ...... 97 5.2.2 Mixing processes ...... 98 5.2.3 Observations in the HRD ...... 99 5.3 Results ...... 100 5.3.1 Evolutionary channels ...... 102 5.3.2 The HD limit ...... 104 5.3.3 Uniﬁed theory of the HD limit at all Z ...... 109 5.3.4 Implications for the B/R ratio ...... 114 5.4 Discussion ...... 117 5.5 Conclusions ...... 119

6 Conclusions and Outlook 123

6.1 Overview ...... 123 6.1.1 Physical processes ...... 123 6.1.2 Evolutionary channels ...... 125 6.1.3 Stellar populations ...... 126 6.1.4 Observations ...... 127 6.1.5 Stellar winds ...... 128 6.1.6 Code comparisons ...... 128 6.1.7 Summary of conclusions ...... 129 6.2 Future work ...... 130 6.3 Wolf-Rayet mass-loss rates to black hole progenitors ...... 132 6.3.1 Milky Way Wolf-Rayet models ...... 133

Appendix A Galactic grid of models 137

viii Appendix B MESA v8845 inlist 144

References 146

List of Figures

1.1 Stellar Structure ...... 3 1.2 Hertzsprung-Russell Diagram ...... 4 1.3 CNO-cycle ...... 5 1.4 Fractions of CCSN types ...... 7 1.5 Stellar remnants with mass and metallicity ...... 8 1.6 Carina Nebula and WR 22 ...... 10 1.7 Tarantula Nebula ...... 12 1.8 Iron-opacity bump ...... 15

2.1 Mass conservation ...... 19 2.2 Uncertainty of an evolutionary track ...... 22 2.3 Temperature and composition gradients ...... 27 2.4 Hunter diagram ...... 29 2.5 Bubble Nebula ...... 32

3.1 HD166734 ...... 44 3.2 Surface nitrogen without rotation ...... 49 3.3 Rotationally-induced mass loss ...... 51 3.4 Evolution plots for HD166734 ...... 53 3.5 Mass-Luminosity Plane ...... 55 3.6 Range of mass-loss rates ...... 56 3.7 Range of rotation rates ...... 58

3.8 Range of overshooting αov ...... 59 3.9 HD166734 M-L plane solution ...... 62 3.10 HD166734 surface nitrogen abundances ...... 63

ix 3.11 Comparison with Bonn and O stars ...... 66 3.12 Spectroscopic HRD ...... 67 3.13 Red supergiants with high overshooting ...... 69 3.14 Compactness parameter of pre-supernova models ...... 70

4.1 VFTS 527 ...... 78 4.2 VFTS 063 ...... 80 4.3 VFTS 116 ...... 82 4.4 Age in the M-L plane ...... 83

4.5 Rotating grid of models (Ω/Ωcrit = 0.1) ...... 84

4.6 Rotating grid of models (Ω/Ωcrit = 0.4) ...... 85

4.7 Grid of models with O and B supergiants (Mevol) ...... 88

4.8 Grid of models with O and B supergiants (Mspec) ...... 89

4.9 Grid of models in HRD with observations for αov = 0.1, 0.5 . . . . . 90

5.1 Kippenhahn diagrams of 60 M with αov= 0.1, 0.5 ...... 101 5.2 Evolutionary stages ...... 103 5.3 Effective temperature as a function of central He abundance, with HRD 107

5.4 Kippenhahn diagrams for 60 M as a function of Z ...... 111 5.5 LMC models with RSG observations ...... 112 5.6 Percentage of core He-burning time ...... 115 5.7 Grid of models in HRD and He-burning timescale ...... 122

6.1 Mass-loss recipe for WR models at Z ...... 133

6.2 Pre-main sequence evolution of 20-30 M WR models ...... 134

6.3 Kippenhahn diagram of a 20 M WR model ...... 135

6.4 Evolution of a 20 M WR star in a HRD ...... 135

A.1 Non-rotating grid ...... 138

−1 A.2 Rotating Z grid for 100 km s ...... 139

−1 A.3 Rotating Z grid for 200 km s ...... 140

−1 A.4 Rotating Z grid for 300 km s ...... 141

x −1 A.5 Rotating Z grid for 400 km s ...... 142

−1 A.6 Rotating Z grid for 500 km s ...... 143

List of Tables

3.1 Grid of models ...... 42 3.2 Observational properties of HD166734 ...... 44 3.3 Galactic O stars ...... 64 3.4 Code comparisons ...... 68

5.1 Core He-burning timescales and B/R ratio ...... 114

xi ”It is reasonable to hope that in a not too distant future we shall be

competent to understand so simple a thing as a star.” - Eddington. Chapter 1

Introduction

Star light is the intrinsic communicator of the local universe, responsible for decoding the composition and evolution of the universe as it alters and expands. It is a col- lection of electromagnetic radiative wavelengths known as a spectrum of light. All astrophysical information about distant cosmic objects is deducted from this spectrum via a method of stellar ﬁngerprinting called spectroscopy. Chemical elements absorb and emit wavelengths of light as they gain energy, providing a unique imprint in the spectrum in the form of absorption lines and emission lines. High resolution spectroscopy allows astronomers to identify key characteristics in the spectrum in order to classify stars in groups based on their mass, abundances, temperature, pressure and luminosity.

Spectroscopic analysis provides the community with stellar parameters with which theoretical models can be compared to, providing evolutionary masses, age estimates, and even their evolutionary history and ﬁnal fates. In order to predict the evolution of stars, we require a large sample of observations at multiple evolutionary phases in order to piece together the history and future of stellar populations.

1 CHAPTER 1. INTRODUCTION

1.1 Stellar classiﬁcation and evolution

Stars are classiﬁed by their spectral type based on their surface temperatures in classes O B A F G K M. They are then categorised into subgroups by their luminosity classi- ﬁcation represented by roman numerals, where I represents supergiants, III represents giants and V represents dwarfs.

It has been argued that the most influential of these spectral factions is the class of O-type stars with the highest luminosities, temperatures and masses, (Langer, 2012). Stars of this spectral class end their lives in violent supernovae or black holes, enriching their surrounding medium with a final burst of synthesised elements, the heaviest in the universe. These stars are expected to be a relation of the first stars, and hence can be seen at the highest redshifts (z≥5) as probes of their locations and the expanse between, Langer (2012). Though O-type stars may be the most luminous of these spectral classes, there is a superior class known as WNh stars which are H-rich and the most massive stars known. This recent classification of young, hydrogen-burning stars displays Wolf-Rayet mass loss features at much higher luminosities with masses as high as 70 - 300 M , Crowther et al. (2010).

1.1.1 Massive stars

The term massive star has accumulated a variety of deﬁnitions, including the mass at which a supernova (SN) progenitor is created. For solar metallicity this is estimated to be in the mass range of 8-12 M , Poelarends et al. (2008). Moreover, research by

Smartt et al. (2009) conﬁrms this lower limit of 8 M since core-collapse supernovae (CCSNe) are an evolutionary channel for stars above this mass. Heger et al. (2003) has estimated an upper mass limit of 65 M where the regime of pair-instability supernovae would be expected, Vink et al. (2015). Beyond this mass physical processes gradually adjust for the stars proximity to the Eddington limit (see 1.2.3), and stars are deﬁned as very massive stars.

A star like our Sun (0.7−1.2 M ) contains a radiative core and convective envelope

2 CHAPTER 1. INTRODUCTION

(Soderblom and King, 1998), whereas at much lower masses of <0.3 M stars are fully convective. Above 1.5 M , stars consist of a convective core surrounded by a radiative envelope. This is due to the change in energy generation in the core and the effect this will have on the temperature gradient, such that a radiative or convective zone is formed. For massive stars the central temperature is high enough to develop a convective core and begin nuclear fusion.

The transition of particles between the core and envelope is theorised such that particles travelling towards the convective boundary do not cease but may continue to move into the radiative zone, e.g. Castellani et al. (1971), just as a car travelling at a given speed does not immediately stop when brakes are applied, Langer (2012). This distance travelled into the radiative envelope is called the overshooting zone, illustrated by the crossed purple lines in Fig. 1.1.

Figure 1.1: Structure of Massive stars in terms of chemical elements (left) and physical processes illustrated in a Kippenhahn diagram (right) where green lines indicate convection, purple crossed lines indicate overshooting, and remaining clear sections represent radiation.

As a high mass star evolves through to the late stages of its life, its envelope may inﬂate due to the iron opacity peak (Fig. 1.8) or diminish due to stellar winds. High temperatures within massive stars lead to the emission of ionising radiation in ultra-violet band, giving rise to these strong stellar winds, Lamers and Cassinelli (1999).

Many massive stars evolve through each stage of nucleosynthesis until a collapsing iron core is formed leading to an unrelenting supernova explosion. Each process of nuclear fusion forms a shell of newly processed chemical elements developing to layers of hydrogen, helium, carbon, oxygen, neon, silicon, and eventually an iron core as seen

3 CHAPTER 1. INTRODUCTION in Fig. 1.1.

Figure 1.2: Hertzsprung-Russell Diagram (HRD) by Ekstrom¨ et al. (2012) correlating spectral type and temperature with luminosity for rotating stellar evolution models. The colour bar indicates the surface abundance of nitrogen.

4 CHAPTER 1. INTRODUCTION

1.1.2 Evolutionary channels

In the ﬁrst instance, a pre-main sequence protostar is formed from a compressed nebula composed primarily of hydrogen and helium, with an inﬁnitesimal fraction of metals (Z). Stars begin their lives on the zero-age main sequence (ZAMS) at various positions dependent on their initial mass and evolve to cooler temperatures as seen in Fig. 1.2. The core of a massive star will begin nucleosynthesis while on the main-sequence (MS), fusing hydrogen to helium via the CNO cycle, Fig. 1.3. At the point of hydrogen depletion, known as the terminal age main sequence (TAMS), the star will ignite a helium burning core with a hydrogen envelope as its exterior.

Figure 1.3: An illustration of CN cycle (hexagon), with two ON loops and an additional branch with reactions, encompassing the CNO cycle of burning Hydrogen to Helium. Adapted from Maeder et al. (1991).

As massive stars evolve through core and shell burning phases, they expand and ma- noeuvre to cooler temperatures in the HRD maturing into a red supergiant (RSG) with a newly-extended radius. Red supergiants may then undergo blue loops where they increase in temperature temporarily, possibly due to mass loss eruptions, or even evolve to hotter temperatures as a blue supergiant (BSG).

Stellar winds lead to signiﬁcant mass loss within massive stars, even for a variety of late evolutionary phases such as in cool supergiants. At a certain mass range (> 40 M ) massive stars may evolve to hotter effective temperatures remaining blue during core

5 CHAPTER 1. INTRODUCTION

He-burning. This stage of evolution denotes the spectroscopically identified object known as the classical Wolf−Rayet (WR) type star, identified by its strong emission lines caused by dense outflows. Primarily helium-burning objects, classical WR’s can be found stripped of their hydrogen envelopes with strong helium emission lines at the surface. Within this class of stars, there are three main subclasses recognised by emission lines of nitrogen (WN), carbon (WC), and oxygen (WO). It has been theorised that a typical O-type star may evolve through one or multiple WR phases before expiring in a supernova explosion, Crowther et al. (1995).

Among the most luminous, hot stars known is the class of luminous blue variables (LBVs), another form of late stage O-type star distinguished as intrinsically bright with variations in both spectra and luminosity on a timescale from days to years.

Massive stars may evolve to one or many of these phases in its life, but since their evolution is sensitive to many factors, it is most probable that each star will have a slightly altered evolution to its neighbour.

Formerly discussed evolutionary channels have considered single star evolution exclusively, yet the possibility of binary stars should not be neglected. In the interest of main-sequence stars, the prominent scenario to consider is the Case A main-sequence mergers as discussed by Langer (2012). For this case, two hydrogen-burning stars in a close binary system will merge to produce a replenished hydrogen-burning star. This case of mergers could affect a population of massive star observations by up to 10%, with mergers appearing younger than their governing cluster age. Sana et al. (2012) conﬁrms the importance of multiplicity in massive star evolution, though it remains debated in its quantitative inﬂuence.

1.1.3 Final fates

In the pre-supernovae stages of evolution, massive stars experience a peak in their stellar winds, though only for ∼ 1% of their lifetime. This means that most of the factors presiding over their fates are dictated during their MS evolution (e.g. mass-loss

6 CHAPTER 1. INTRODUCTION

Figure 1.4: The relative fractions of CCSN types, adapted from Smith et al. (2011), Fig. 1. rates). Nevertheless, the final burst of stellar outflows may influence their surrounding medium and subsequent interactions. Understanding SNe and their progenitors is key to our understanding of the evolving universe as they are responsible for recycling energy and matter throughout the ISM, generation after generation of stars.

Massive stars evolve through core burning phases until they reach iron, where they then have ∼ 0.25 second before their core collapses and their envelope is ejected in a supernova explosion. A variety of supernovae ﬂavours are deﬁned by their presence of H, He and other spectral features. Smartt et al. (2009) stated that the initial mass capable of producing a CCSN is 8± 1 M from RSG progenitors of type II-P SNe, while 60% of all CCSNe come from the mass range 8-15 M due to the shape of the initial mass function (IMF). The relative fractions of SNe types is shown in Fig. 1.4, adapted from Smith et al. (2011).

The review by Smartt et al. (2009) provides a classification of SNe types as follows: a lack of H suggests type I SNe, with a further lack of He and presence of Si classed as type Ia. It is important to note that this class of type Ia CCSN differs from the thermonuclear type Ia SNe from white dwarf binary systems. A presence of He and lack of H in CCSNe defines type Ib SNe, while a lack of H or He defines the type Ic

7 Fig. 1. from How Massive Single Stars End Their Life Heger et al. 2003 ApJ 591 288 doi:10.1086/375341 http://dx.doi.org/10.1086/375341 CHAPTER© 2003. The 1.American INTRODUCTION Astronomical Society. All rights reserved. Printed in U.S.A.

Figure 1.5: Final fates of massive single stars for a range of initial masses and metallicities, adapted from Heger et al. (2003), Fig. 1.

SNe, usually associated with long gamma ray bursts (GRBs). If a strong H presence is noted in the SN spectra, it may be classed as a type II SNe. Sub-classes based on observed light curves include type II-P (with a plateau phase), type II-L (linear decay after peak), and type IIb (begin type II but develop He signatures). Groh et al. (2013) provided an insight into the possible progenitors of these diverse SNe types, conﬁrming type II-P likely have RSG progenitors, type II-L/b have LBV or yellow hyper-giant (YHG) progenitors, and types Ib and Ic having WN and WO progenitors, respectively.

In the wake of a supernova death, the remaining core may condense to form a neutron star (Minit < 25 M ) or a black hole (Minit > 25 M ). Figure 1.5 illustrates the range of stellar remnants as a function of initial mass and metallicity, given a set of stellar evolution prescriptions including mass-loss rates. A neutron star is formed when a massive star has completed nuclear burning and the core compresses under the force of gravity, forcing protons and electrons to merge to form a dense sea of neutrons.

8 CHAPTER 1. INTRODUCTION

A black hole is a more extreme version of these events where the gravitational force is so strong that the star’s core collapses to a singularity in the centre. The range of remnants as a function of metallicity highlight the range of maximum masses of black holes, for our Galaxy this may be ∼ 15 M due to WR mass-loss rates (Vink and de Koter, 2005). At all metallicities, a star with initial mass ∼ 10-25 M will collapse to a neutron star, whereas in the range ∼ 25-40 M the ejected envelope may fall back onto the neutron star, tipping the gravitational force such that a black hole is formed by fallback. Above ∼ 40 M direct collapse black holes are formed at low metallicity, but due to the effects of mass loss at solar metallicity, would only form a neutron star or fallback black hole.

This field of research has become immensely important for the wider community of astrophysics and general physics due to the recent detections of gravitational waves (GWs). These ripples in space-time have confirmed Einsteins theory of general rel- ativity 100 years later to provide the modern world with a better understanding of fundamental physics. The laser interferometer gravitational-wave observatory (LIGO) is a global collaboration with multiple sites around the world which have identified unique patterns of interference corresponding to four categories of GWs. These patterns have been critical in constraining the progenitors of the GW events such as black hole mergers and neutron star mergers. The first detection of GWs was GW150914 with analysis predicting two black holes of ∼ 30 M each merged to form a 62 M black hole. Future advancements in technology will provide detailed information on the location and frequency of these events throughout the universe.

1.2 Massive stars throughout the Universe

Massive stars are signiﬁcant drivers of galaxy evolution and enrichment of the in- terstellar medium (ISM). They are the pro-creators of black holes, neutron stars and supernovae, with the most massive of these stars likely being responsible for the reion- ization of the Universe. Figure 1.6 illustrates these effects of winds on the surrounding

9 CHAPTER 1. INTRODUCTION

Figure 1.6: An image of the Carina Nebula with a centred Wolf-Rayet star WR 22. Credit : ESO’s La Silla Observatory, Chile. medium, with strong outﬂows from the object WR22 within the Carina Nebula. Due to the IMF shape, acquiring a large statistical sample of the most massive stars has been challenging in the past. Without a comprehensive set of observations to constrain the theory of their evolution, massive stars have largely remained a mystery. The VLT-Flames Tarantula Survey (VFTS) of the 30Dor starburst (Evans et al., 2011), has provided a recent multi-epoch, spectroscopic survey of 800 OB stars, critically aiding in our understanding of massive stars (see Sect. 1.2.1).

Very massive stars (VMSs, M > 100 M ), like those of the Arches cluster in the Galac- tic centre, were observed in the R136 cluster of 30Dor. The most massive stars observed to date with masses up to 200-300 M , these observations challenged previous quotes of an upper mass limit ∼ 150 M by Figer (2005a). Schneider et al. (2018)

10 CHAPTER 1. INTRODUCTION

also found that 30Dor yielded an excess in massive stars over 30 M suggesting a top- heavy IMF, meaning massive stars are dominating their host galaxies. Finally, Sana et al. (2013) found a 50% binarity fraction in O stars, greatly affecting the ﬁnal evolutionary fates of massive stars.

We now know that massive stars often evolve in binary or multiple systems and can evolve from higher initial masses than previously predicted, suggesting binary black hole mergers like GW150914 (e.g. Belczynski et al., 2016) may occur frequently throughout the universe. These results highlight the importance of future research on the evolution of massive stars, particularly at the highest mass range. Though efforts have been made in the past to reproduce the evolution of massive stars (e.g. Ekstrom¨ et al., 2012; Brott et al., 2011a), a disconcerting number of theoretical problems remain. The most consequential of issues remains the amount of mass lost during the lives of massive stars since this determines their ﬁnal mass and fate. This greatly affects the ﬁnal masses compared to BH masses estimated from LIGO detections.

1.2.1 Low metallicity environments

In the early universe, the ﬁrst stars would have been over hundreds of times the mass of our Sun, enriching their surrounding neighbourhood with newly forged heavy metals, now essential for our daily lives. To explore the nature of the ﬁrst stars we observe low metallicity environments such as the nearby Magellanic Clouds (e.g. the VFTS, Evans et al., 2011).

The Magellanic Clouds

The Magellanic Clouds are dwarf, satellite galaxies of the Milky Way. They host the nearest star-forming regions and are plentiful with massive stars. With a metallicity of

ZLMC∼ 50% Z and ZSMC∼ 20% Z , the Magellanic Clouds provide the best labora- tory for testing massive star evolution. For this reason, the VLT-FLAMES survey was launched by Evans et al. (2005) as part of an European Southern Observatory (ESO) large programme. A total of 50 O-stars and 500 B-stars were observed in clusters

11 CHAPTER 1. INTRODUCTION

Figure 1.7: The Tarantula Nebula of the Large Magellanic Cloud, hosting the 30 Do- radus region. Credit: ESO/IDA/Danish 1.5 m/R. Gendler, C. C. Thone,¨ C. Feron,´ and J.-E. Ovaldsen

12 CHAPTER 1. INTRODUCTION across the Milky Way and Magellanic Clouds.

Building upon the VLT-FLAMES survey (VFTS), the collaboration by Evans et al. (2011) proposed a multi-epoch spectroscopic survey of over 800 massive stars in the 30 Doradus (30 Dor) region of the Tarantula nebula in the Large Magellanic Cloud (LMC). This region , shown in Fig. 1.7, hosts multiple generations of star formation as well as the most massive stars observed to date in the R136 cluster.

Taylor et al. (2011) found the most massive binary system R139 within the 30 Dor region, suggesting masses of ∼ 65-75 M for both components, later analysed in chapter 4. Sana et al. (2013) studied the multiplicity of the O-star sample finding over 50% in binary or multiple systems, suggesting that binary evolution plays an important role in understanding massive star populations. This study represents a sample 5 times larger than that of the galactic survey of binaries in Sana et al. (2012) where approximately 70% of massive stars were found in binary systems. The rotational velocities of the VFTS were sampled by Ram´ırez-Agudelo et al. (2013) finding a peak at ∼ 80 km s−1 and a high-velocity tail extending to 600 km s−1, with most massive stars rotating at less than 20% of their break-up velocity. Grin et al. (2017) compared these rotational velocities to surface nitrogen enrichment in order to test current rotational mixing theory, finding that only ∼ 70% of stars could be explained by current models while ∼ 30% remained unresolved.

The analysis of observations taken in the VFTS has provided an invaluable insight into massive star evolution through their stellar properties, however, many challenges remain. The discrepancy between spectroscopic masses and evolutionary masses highlight a need for improved stellar atmosphere modelling, stellar evolution modelling or both. The number of SNe compared with their progenitors questions the channels of producing a range of SNe types. The frequency of binary mergers and their impact on stellar populations remains a caveat in single star evolution and population synthesis models. The upper mass limit and IMF have been tested by Schneider et al. (2018) for the 30 Dor region though whether this is representative of other galaxies remains

13 CHAPTER 1. INTRODUCTION unclear.

1.2.2 Mass Loss as a function of Metallicity

Metallicity is a governing parameter of stellar formation and evolution, deﬁned as the amount of chemical elements heavier than helium present in a given environment. Heavy elements (up to iron) are produced in the most massive stars and the heavier elements are created in their supernovae. One can imagine an early universe consisting of mainly hydrogen with few metals, recognised as a low metallicity environment, in comparison to the modern metal-rich universe. Metallicity not only plays a role in the evolution of the universe, but of stars and galaxies as well.

Predictions from Vink et al. (2001) state that metallicity has an effect on the mass loss rates of massive stars. This can be appreciated as the mass loss rates of massive stars are driven by radiative acceleration of spectral lines due to heavy chemical elements such as iron. Photons generated in nuclear fusion reactions as by-products are radiated out towards the surface after millions of years due to the random walk-type motion, absorbed by surface elements which are excited to a higher ionisation state by the energy of the photon, and emitted again as spectral lines driven by surface radiation pressure.

A grid of models was computed for a range of effective temperatures 12,500 - 22,500K and 27,500 - 50,000K, with relation to Γe, the ratio of gravitational acceleration to radiative acceleration due to electron scattering.

1.2.3 Most massive stars in the Universe

Stellar parameters are mainly coexistent in their relations that drive the evolution of stars. The luminosity of a star greatly depends on its mass, as shown in equation 3.2.

However, this relation ﬂattens out above 20 M , leaving the relation L ∝ M.

L M a = (1.1) L M

14 CHAPTER 1. INTRODUCTION

where a varies as a function of mass.

Massive stars have been hypothesised to reach ≈ 150 M , by Figer (2005b) for example, yet the upper mass limit of stars could be much higher, as suggested by Schneider et al. (2018). With a strong mass-luminosity relation, it is expected that higher mass stars would increase respectively in luminosity. However, the ultimate luminosity of a star is anchored by the Eddington limit, the maximum luminosity a star may achieve while remaining in hydrostatic equilibrium, presented in equation 1.2. As stars increase in mass, they approach the Eddington limit, Γedd ≈ 1, causing enhanced mass loss.

grad κL Γedd = = (1.2) g 4πcGM

Figure 1.8: Envelope inﬂation due to the Iron-opacity bump adapted from Grafener¨ et al. (2012) illustrating the fraction of the Eddington factor Γedd in colour.

15 CHAPTER 1. INTRODUCTION

Furthermore, as a result of stellar proximity to the Eddington limit the radiative envelope may inflate due to a physical feature known as the iron-opacity bump. Figure 1.8 illustrates this feature in terms of pressure and temperature, though relates to the optical depth since Teff ∝ τ. The iron-opacity bump occurs at 150 kK, fixing the optical depth at this point, hence constraining the relative radius and density. In order for a star to succeed this bump the density decreases and the corresponding ∆R increases. This increase in radius is known as envelope inflation, and is discussed further by Grafener¨ and Vink (2013).

Since the most massive stars are difﬁcult to calculate in stellar evolution models as they create a density inversion, and mass loss at the surface leads to convergence problems, we focus on the key mass range of observed massive stars for probing physical processes such as internal mixing and mass loss (20-60 M ).

1.3 Thesis overview

While there has been many developments in our understanding of massive star evolution, problems remain between theoretical models and observations. Though surface enrichments have provided insight into the rotational distribution of massive stars, the efficiency of internal mixing has not been calibrated to fit observations. Long-standing problems such as the observed number of blue and red supergiants in different galaxies has caused conflict with theory for decades, while the mechanisms for driving evolution bluewards or redwards also remains unclear. The various evolutionary phases of massive stars have been populated with observations in the recent VFTS sample, though in which timescales and order they occur is still unresolved. Even the main sequence, where stars spend 90% of their lives, consists of uncertainties with regards to the MS length, the amount of internal mixing required to reproduce MS objects, and how the MS phase changes with metallicity. Many of these concerns are highlighted and explored in this work.

In this thesis multiple grids of evolutionary models of massive stars are provided for

16 CHAPTER 1. INTRODUCTION a range of initial masses and metallicities. Physical processes such as mass loss, convective core overshooting and semiconvection are investigated with respect to their in- ﬂuence on core hydrogen and helium burning. A new method of calibrating evolution has been developed in the Mass-Luminosity plane, with an extension to predictions of stellar age. This method has provided constraints on the mass-loss rates and internal mixing of massive stars on the main-sequence. Multiple detached binaries have been analysed as test cases for massive star evolution at various metallicities, providing constraints for larger samples of observations such as the VLT-FLAMES Tarantula Survey.

The one-dimensional stellar evolution code MESA is discussed in chapter 2, including input physical parameters and processes. The Galactic grid of evolution models are presented in chapter 3 with the introduction of the Mass-Luminosity plane. Chapter 4 develops the use of the M-L plane towards estimating stellar age while providing a grid of evolutionary models for the Large Magellanic Cloud. The evolution is extended to red and blue supergiants in chapter 5 with a theoretical investigation of the upper luminosity of red supergiants. Conclusions are presented in chapter 6, with future work and an outlook discussed. Wolf-Rayet evolutionary models are presented in chapter 6 with a new locally-consistent mass-loss recipe for a range of metallicities in an attempt to predict ﬁnal masses which would be comparable to black holes detected by LIGO.

17 Chapter 2

Theoretical modelling of massive stars

2.1 Stellar structure and evolution

Most stars are suspended in a state of equilibrium, seeming constant with time. Yet this state of stability is governed by a set of equations dedicated to understanding the internal structure of stars. Due to gravitational forces and for simplicity, we treat solutions of these equations as spherically symmetric, Kippenhahn and Weigert (1990)1.

If a shell of mass dm is considered with thickness dr at radius r from the centre, as in Figure. 2.1, the mass of this shell can be expressed by equation 2.1. The mass coordinate of this shell is a function of the mass element in the star, and must be written in terms of m rather than r.

Thus the coordinate transformation, equation 2.2, is used leaving the ﬁrst equation of stellar structure (eqn. 2.3), known as the conservation of mass.

dm = 4πr2ρdr (2.1)

1Stellar Structure equations are presented in this section in the form as discussed by Kippenhahn and Weigert (1990)

18 CHAPTER 2. THEORETICAL MODELLING OF MASSIVE STARS

Figure 2.1: Illustration of a shell of mass dm, expressed by equation 2.1, from Kippenhahn and Weigert (1990).

d d dr = · (2.2) dm dr dm

dr 1 = (2.3) dM 4πr2ρ

Maintaining the concept of Figure 2.1, the forces acting on this mass element are shown by P(r) and Fg. As previously stated, lacking observable ﬂuctuation these forces must be balanced as shown in equation 2.4. However, from combining equations 2.3 and 2.4, emerges a second expression in Lagrangian form, known as the equation of hydrostatic equilibrium, (eqn. 2.5).

dP GM = − ρ (2.4) dr r2

dP GM = − (2.5) dM 4πr4

Energy is produced via nucleosynthesis in the core of stars, and may be released by neutrinos at the surface. The amount of energy per second entering the shell of mass dm from the core is denoted l while the amount of energy per second leaving the outer shell m+dm is denoted l+dl. In a stationary case, dl is the amount of energy released

19 CHAPTER 2. THEORETICAL MODELLING OF MASSIVE STARS from nuclear reactions, where is the nuclear energy released per unit mass per second. Equation 2.6 represents the amount of energy dl leaving a shell of mass dm, and if rearranged gives the simplistic equation of energy conservation (eqn. 2.7).

dl = 4πr2ρdr = dm (2.6)

dL = (2.7) dM

Radiative energy is transported via the treatment of diffusion since the mean free path which photons travel to get to the surface of the star is much smaller than the stellar radius. Therefore the ﬂux of photons leaving the star is represented by equation 2.8. Solving this for the temperature gradient and replacing F with the local luminosity l = 4πr2 leaves equation 2.9. Yet, as seen in equation 2.3 a transformation must take place in order to display the equation in terms of the independant variable m, giving equation 2.10, the equation of energy transport.

4ac T3 F = −krad∇T where krad = (2.8) 3 κρ

and ∇ T is the temperature gradient.

∂T 3 κρl = − (2.9) ∂r 16πac r2T 3

∂T 3κl = − (2.10) ∂M 64π2acr4T 3

Coupled with the equation of state and opacity (equations 2.11 and 2.12, respectively), the equations of stellar structure are powerful in their profound ability to explain the

20 CHAPTER 2. THEORETICAL MODELLING OF MASSIVE STARS interior of distant cosmic objects.

ρ P = K( )5/3 (2.11) µe

α β κ = κ0ρ T (2.12)

The interior structure of massive stars plays a signiﬁcant role on their location in the Hertzsprung−Russell diagram (see Fig. 1.2) in conjunction with physical processes acting at the stellar surface.

When using these equations to provide information about observations, we use solar units to provide a more meaningful comparison. M = 1.988x1033 g, L = 3.828x1033 erg

−1 s , R = 6.957x1010 cm, and Teﬀ, = 5772 K.

2.2 1-dimensional stellar evolution

Since the golden age of stellar astrophysics (e.g. Bohm-Vitense,¨ 1958; Schwarzschild, 1958; Henyey et al., 1964; Paczynski, 1971) the ﬁeld has developed substantially, mainly due to technological advances in both computational power and observational surveys. Numerical simulations have allowed for solutions of stellar structure and evolution equations simultaneously. Studies of massive star populations with 1-dimensional stellar evolution codes such as GENEC (Ekstrom¨ et al., 2012), STARS (Eggleton et al., 2006), STERN (Brott et al., 2011a) and others, have led to advanced understanding of stellar interiors and evolutionary phases. These 1-dimensional codes aim to solve the stellar structure and evolution equations described previously with time.

Martins and Palacios (2013) provide a comparison of six modern stellar evolution codes in order to highlight the extent of uncertainties from one grid of models to another. They compared evolutionary tracks in a HRD ﬁnding that they are highly sensitive to their adopted solar abundance composition, convective overshooting and

21 CHAPTER 2. THEORETICAL MODELLING OF MASSIVE STARS

Figure 2.2: Adapted from Martins and Palacios (2013) Fig. 7. The uncertainty of the evolution of a 20 M model with solar metallicity for ﬁve different evolution codes (Geneva, STERN, FRANEC, MESA and Starevol). Rotating models are encompassed by the dark grey envelope, whereas non-rotating models are enclosed in light grey delimited by dashed lines.

22 CHAPTER 2. THEORETICAL MODELLING OF MASSIVE STARS mass-loss rate. Due to the uncertainty of internal mixing efficiencies, the luminosities and subsequently evolutionary mass can be systematically overestimated, see Fig. 2.2. These studies provide evidence for much needed constraints on stellar evolution and input physics such as mixing and mass loss. Until there are sufficient constraints on theoretical modelling, it will be difficult to compare with observations and make assumptions on stellar populations.

2.2.1 MESA v8845

In this thesis the stellar evolution code MESA (modules for experiments in stellar astrophysics) has been utilised (see Paxton et al., 2011, 2013, 2015). MESA is an open source package of modules with active development for its worldwide user network. It is a self-consistent framework with ﬂexible architecture and advanced capabilities. Its thread-safe design allows for parallel computing which greatly reduces computational time such that large grids of models are feasibly calculated for exploring many physical processes.

The module MESAstar is a 1-dimensional stellar evolution code which simultaneously solves the structure and composition equations. The inlist is used for user input to alter basic parameters, while the Fortran module complies the ﬁle run_star_extras.f for new routines added by the user (e.g. Choi et al., 2016).

2.2.2 Basic input parameters

At the beginning of each project, a user may decide to create a pre-MS evolution model towards the MS or begin on the MS, also where to terminate evolution or run until core-collape. The initial mass, metallicity and mixing parameters are set in the inlist_project along with other physical processes such as rotation and mass loss (see Appendix B). The chemical abundances have been selected from Grevesse and Sauval (1998a) though many options are possible. This remains consistent throughout this thesis with the use of scaled-solar abundances in the case of low metallici-

23 CHAPTER 2. THEORETICAL MODELLING OF MASSIVE STARS ties.

Timesteps

The resolution of each model is calculated by the mesh refinement and timesteps, de- fined by the user. A timestep appropriate for each phase of evolution is important for calculating the most reliable and accurate model. The timestep should be sufficiently small to aid convergence while also sufficiently large to calculate the model in a reasonable time. Excessively large changes between each timestep can lead to accuracy issues in subsequent evolutionary phases. At the beginning of each timestep, MESAs- tar checks if spatial mesh adjustment is needed. This involves splitting and merging cells to resolve physical processes locally. Changes from cell to cell can be minimised to resolve variables such as mass, radius and pressure. Cells can also be split to resolve boundary locations such as near the convective core.

Opacities

The equation of state (EOS) tables in MESA are based on the OPAL tables from Rogers and Nayfonov (2002). The OPAL tables are split into two types, where Type I are used for 0 < X < 1-Z and 0< Z < 0.1 for ﬁxed abundance patterns. Type II tables are used for enhanced carbon and oxygen abundances, as well as 0< X < 0.7 and 0< Z < 0.1. Type II tables are important for massive stars and post-main sequence evolution phases. MESA also splits the radiative opacity tables into two temperature regimes; hot (log T > 4.0) and cool ( log T < 4.0). The initial abundances are set by X+Y+Z=1, where solar Z=0.02.

Nuclear reaction rates are imported directly from the JINA REACLIB database (Cyburt et al., 2010). The default nuclear network traces 11 species, but can be expanded for detailed reaction rates in later evolutionary phases.

24 CHAPTER 2. THEORETICAL MODELLING OF MASSIVE STARS

2.3 Physical processes

2.3.1 Convection

The mixing length theory by Bohm-Vitense¨ (1958) denotes the transport of energy in stellar interiors by convection. The distance travelled by a convective cell (lMLT) before dissolving into the surrounding medium is set by a fraction αmlt of the pressure scale height (Hp). The free parameter αmlt is of the order unity and has a default value in MESA of 1.5. While this parameter dictates how efﬁcient convection is, and that it may vary with mass, for the purpose of this study the default value has been assumed to condense the number of free parameters.

Chemical mixing by convection is treated as a time-dependent, diffusive process. The efﬁciency of this mixing is denoted by the diffusion coefﬁcient D.

Convection is located in a stable region where ∇T < ∇ad dictated by the Schwarzschild criterion, where ∇T is the local temperature gradient and ∇ad is the adiabatic temperature gradient. In regions where a composition gradient ∇µ is present, the Ledoux criterion is effective. While test cases have been undertaken to investigate the effects of the Ledoux or Schwarzschild criterion, the former has been implemented in order to test processes such as semiconvection, which would be ommitted in the case of Schwarzschild criterion.

dlnT ∇T = (2.13) dlnP s where s indicates the derivatives of the surrounding material.

P dT ∇ad = (2.14) T dP s

Convective core overshooting

In order to account for the nonzero momentum of the convective cells around convective boundaries, another free parameter has been introduced to account for the distance

25 CHAPTER 2. THEORETICAL MODELLING OF MASSIVE STARS a cell can continue to travel into the radiative region before dissipating. This method is described as convective overshooting and has been implemented in stellar evolution codes to account for the observed MS width and theoretical assumptions of convection.

There are two main types of convective core overshooting, denoted as step and exponential. The former artiﬁcially enhances the core by a fraction of the pressure scale height αov and mixes the material homogeneously as in the core. The resulting diffusion coefﬁcient in the overshooting region for step overshooting is Dov = αovHp.

The latter has been developed to provide a more realistic representation of enhanced mixing with an exponential gradient from the core to a boundary set by fov. However, as there is a second order of uncertainty in the prescription of exponential overshooting (i.e. two free parameters D0 and fov), this work primarily focuses on the step prescription of overshooting.

Semiconvection

The Ledoux criterion for convection incorporates a composition gradient ∇µ, and is important for semiconvective mixing. In convectively unstable regions by the Schwarzschild criterion but stable by the Ledoux criterion, a type of mixing called semiconvection is effective. This is where a thermally unstable medium has a stabilizing, positive composition gradient. It usually takes effect towards the end of core H-burning and during He-burning due to the H-He gradient. Semiconvective regions form in the stellar envelope above the convective core and is implemented in stellar evolution codes as a time-dependent, diffusive process, with a diffusion coefﬁcient Dsc (see equation 2.15).

The range of values of αsemi varies by orders of magnitude throughout literature from 0.1-100 (e.g. Langer, 1991; Yoon et al., 2006; Charbonnel and Zahn, 2007; Cantiello and Langer, 2010).

K ∇T − ∇ad Dsc = αsc (2.15) 6Cpρ ∇L − ∇T

26 CHAPTER 2. THEORETICAL MODELLING OF MASSIVE STARS

Figure 2.3: The ∇µ -(∇rad - ∇ad) stability plane with regimes for convection, radiation, semiconvection and thermohaline mixing, adapted from Salaris and Cassisi (2017). The diagonal line dividing the top left and bottom right diagrams denotes ∇L.

27 CHAPTER 2. THEORETICAL MODELLING OF MASSIVE STARS

where K is the radiative conductivity, Cp is the speciﬁc heat at constant pressure and

αsemi is a free parameter which dictates the efﬁciency of semiconvective mixing.

Figure 2.3 demonstrates where convection, radiation, semiconvection and other mixing processes are active due to the corresponding temperature and composition gradients. For example, when the radiative temperature gradient is larger than the adiabatic temperature gradient and there is a positive composition gradient, then semiconvective mixing occurs. Figure 2.3 also shows thermohaline mixing, a hydrodynamic instability that arises when an unstable composition gradient is stabilised by a temperature gradient.

2.3.2 Rotation

The effects of rotation have been studied throughout stellar astrophysics though remain highly uncertain and challenging to reproduce. Rotation is intrinsically a 3D process and as such remains unrealistic to model in 1D codes. However, the shellular approxi- mation allows rotation to be implemented in 1D, (see Kippenhahn and Thomas, 1970; Endal and Soﬁa, 1976). This is where anisotropic turbulence from differential rotation leads to chemical and velocity gradients spreading out along isobars, (Meynet and Maeder, 1997; Heger et al., 2000a).

The transport of angular momentum and chemical elements is treated as a diffusive process by rotationally-induced instabilities. Heger et al. (2000a) and Maeder and Meynet (2000b) provide a full overview of the ﬁve rotational instabilities: dynamical shear instability (DS), secular shear instability (SSH), Solberg-Høiland instability (SH), Eddington-Sweet circulation (ES) and Goldreich-Schubert-Fricke instability (GSF), where the latter two lead to almost rigid rotation. The Spruit-Taylor dynamo can also be included to induce a magnetic ﬁeld in the presence of differential rotation (Choi et al., 2016).

The angular velocity Ω(r, θ) is developed in spherical harmonics throughout the star

28 CHAPTER 2. THEORETICAL MODELLING OF MASSIVE STARS

Figure 2.4: The Hunter diagram for correlating surface enrichment with rotation rate. Adapted from Brott et al. (2011a) ﬁgure 10. with ﬂuctuations of the angular velocity implemented along isobars.

¯ 1 Ω(r, θ) = Ω(r) + Ω2(r) P2(θ) + (2.16) 5 ¯ where Ω is the average velocity on an isobar, Ω2 is the amplitude and P2 is the Lengen- dre polynomials. The diffusion coefﬁcient D denotes the efﬁciency of chemical mixing and angular momentum transfer, implemented as an arbitrary number in stellar evolution codes. For the purpose of this work, the value D = 0.03 has been implemented from Brott et al. (2011a), for comparable results and without an extensive study on this free parameter.

Rotation has been linked with chemical enrichment at the stellar surface in Hunter et al. (2008) suggesting that it may also be treated as a proxy for estimating the age of populations. Brott et al. (2011a) utilises this method in comparing models with observations of B supergiants to determine a variety of classes such as enriched slowly- rotating B supergiants. Figure 2.4 is adapted from Brott et al. (2011a) to highlight the effect of rotation on surface nitrogen abundance.

29 CHAPTER 2. THEORETICAL MODELLING OF MASSIVE STARS

2.3.3 Stellar winds

Massive stars reveal evidence of strong outflows in their UV and optical spectra. These outflows have been studied for decades through radiative-driven wind theory (e.g. Cas- tor et al., 1975). The effects of these outflows in terms of the mass lost throughout a stars life has been integrated in stellar evolution to predict the possible evolutionary paths a massive star may follow. Mass loss plays a key role in the evolution of massive stars, influencing their evolutionary tracks and surface abundances. The mass-loss rate during the MS lifetime of a star dictates the subsequent evolution leading to either RSGs or WR and LBV phases. The fates of massive stars is also highly impacted by the extent of their winds, determining whether a black hole or neutron star is formed. The magnitude of stellar winds is dominated by the initial mass and metallicity of the star, with higher mass stars having a higher mass-loss rate. Similarly, higher metallicity environments result in stronger mass loss.

A set of stellar wind prescriptions has been collated in the wind routine of MESAstar to include a variety of evolutionary phases. The hot, O star routine is dictated by the Vink et al. (2001) recipe, detailed below, while the cool supergiant phase is inﬂuenced by the Nugis and Lamers (2000) recipe. The combination of these mass-loss rates is known as the Dutch prescription in MESA and prohibits extrapolation of recipes outside of their constrained temperature ranges.

The Vink et al. (2001) mass-loss recipe is implemented as shown for the relevant temperature ranges in equations 2.17 and 2.18. As a result, higher metallicity environments will have a higher mass-loss rate compared to low metallicity environments. This can have many consequences on various evolutionary channels such as evolution towards

30 CHAPTER 2. THEORETICAL MODELLING OF MASSIVE STARS

RSG and WR phases.

log(M˙ ) = − 6.688(±0.080)

5 + 2.210(±0.031) log(L∗/10 )

− 1.339(±0.068) log(M∗/30) (2.17) v /v − 1.601(±0.055) log( ∞ esc ) 2.0

+ 1.07(±0.10) log(Teﬀ /20, 000)

for 12,500

log(M˙ ) = − 6.697(±0.061)

5 + 2.194(±0.021) log(L∗/10 )

− 1.313(±0.046) log(M∗/30) v /v (2.18) − 1.226(±0.037) log( ∞ esc ) 2.0

+ 0.933(±0.064) log(Teﬀ /40, 000)

2 − 10.92(±0.90) (log(Teﬀ /40, 000) )

for 27,500

The mass loss rates for the temperature range 22,500 - 27,500 K remain ambiguous due to the effects of the bi-stability jump. In spite of this, it is possible to infer the appropriate mass loss rate within this range as a function of v∞ / vesc, Vink et al. (2001).

This mass-loss recipe is also dependent on the metallicity of the model, where M˙ (Z) ˙ 0.85 = M × (Zsurf / Z ) .

Figure 2.5 illustrates the effects of radiatively driven winds from an O star creating a stellar wind bubble. The fast wind from a massive star can collide with a previously

31 CHAPTER 2. THEORETICAL MODELLING OF MASSIVE STARS

Figure 2.5: The Bubble Nebula, also known as NGC 7635, observed by the NASA/ESA Hubble Space Telescope. emitted slow wind to create shells around the star travelling through the ISM to form a bow shock.

2.3.4 Summary

The inclusion of the physical processes outlined previously has greatly benefited our understanding of stellar evolution in 1D codes. However, with many uncertainties in their implementation and efficiency of many mixing processes, the reliability of evolutionary tracks remains unclear. Though the implementation of convection and rotation is restricted in 1D, many improvements can be made to benefit our understanding of massive star evolution. By studying the effects of convective core overshooting, we may probe the MS-width at various masses and metallicities, leading to important conclusions about H and He-burning objects. Similarly, by confining the efficiency of

32 CHAPTER 2. THEORETICAL MODELLING OF MASSIVE STARS convection and semiconvection, we may better understand the effect these processes have on later evolutionary phases such as from the MS towards RSGs. The work outlined within this thesis aims to constrain these processes as a function of mass and metallicity.

33 Chapter 3

Galactic evolution in the Mass-Luminosity Plane

In this chapter I present a study of the main sequence evolution of massive stars for the Milky Way. A new method of calibrating stellar evolution is presented in the Mass- Luminosity plane, with analysis of a detached test-bed binary HD166734.

The research outlined in this chapter has been published in Astronomy and Astro- physics as Erin R. Higgins & Jorick S. Vink, Massive star evolution: rotation, winds, and overshooting vectors in the mass-luminosity plane. I. A calibrated grid of rotating single star models, 2019, A&A, 622, A50

Massive star evolution is dominated by various physical effects, including mass loss, overshooting, and rotation, but the prescriptions of their effects are poorly constrained and even affect our understanding of the main sequence. We aim to constrain massive star evolution models using the unique test-bed eclipsing binary HD 166734 with new grids of MESA stellar evolution models, adopting calibrated prescriptions of overshooting, mass loss, and rotation. We introduce a novel tool, called the mass- luminosity plane or M − L plane, as an equivalent to the traditional HR diagram, utilising it to reproduce the test-bed binary HD 166734 with newly calibrated MESA stellar evolution models for single stars. We can only reproduce the Galactic binary

34 CHAPTER 3. GALACTIC EVOLUTION IN THE MASS-LUMINOSITY PLANE

system with an enhanced amount of core overshooting (αov= 0.5), mass loss, and rotational mixing. We can utilise the gradient in the M − L plane to constrain the amount of mass loss to 0.5 - 1.5 times the standard prescription test-bed, and we can exclude extreme reduction or multiplication factors. The extent of the vectors in the M − L plane leads us to conclude that the amount of core overshooting is larger than is normally adopted in contemporary massive star evolution models. We furthermore conclude that rotational mixing is mandatory to obtain the correct nitrogen abundance ratios between the primary and secondary components (3:1) in our test-bed binary system. Our calibrated grid of models, alongside our new M − L plane approach, present the possibility of a widened main sequence due to an increased demand for core overshooting. The increased amount of core overshooting is not only needed to explain the extended main sequence, but the enhanced overshooting is also needed to explain the location of the upper-luminosity limit of the red supergiants. Finally, the increased amount of core overshooting has implications for the compactness of pre-supernova models.

3.1 Introduction

Massive stars with an initial mass above 8 M have a diversity of possible evolutionary channels that are dictated by the dominant processes acting on their structure. The extent of these dependencies are variant with mass, metallicity, and multiplicity. Stel- lar winds have a significant impact on the evolution of O-type stars throughout their lives, leading to evolutionary phases involving Luminous Blue Variables (LBV) and Wolf-Rayet (WR) stars. It is also an important factor for dictating their final masses and determining whether a neutron star or black hole is formed in the final stage of evolution, as extensively reviewed by Chiosi and Maeder (1986a).

On the main sequence (MS), mass loss via stellar winds has the greatest impact at the highest mass ranges. Above '60 M mass loss completely dominates the evolution of

O-type stars (e.g. Vink and Grafener,¨ 2012; Vink, 2015), whilst in the range 30 M <

35 CHAPTER 3. GALACTIC EVOLUTION IN THE MASS-LUMINOSITY PLANE

M < 60 M mass loss is one of the important ingredients (e.g. Langer, 2012; Groh et al., 2014). At lower masses, (i.e. below ∼ 30 M ) the evolution is thought to be heavily inﬂuenced by rotation (e.g. Maeder and Meynet, 2000a). Over the explored mass range within this study, i.e. 8 - 60 M , we will consider the effects of mass loss and rotation, as well as convective overshooting, which may all play a role in the evolution of these stars.

The extension of the convective core by overshooting is a key structural feature which increases the amount of hydrogen (H) dredged into the core, replenishing its supply, thereby extending the MS lifetime. The parameter αov which we explore in this study corresponds to the fraction of the pressure scale height Hp by which particles continue to travel a distance lov beyond the convective core boundary. This form of mixing has been explored for decades, with few constraints on its size (αov) in the high-mass range. It has been argued as essential for reproducing observations, although evidence is lacking for dependencies such as mass (Claret and Torres, 2017). Another process that may potentially affect stellar evolution is the presence of an internal magnetic ﬁeld, however, Grunhut et al. (2012) showed the fraction of surface magnetic O stars to be just on the order of 7%.

Massive star evolution models are currently not able to fully reproduce observations, even the MS (e.g. Vink et al., 2010; Markova et al., 2018), with many physical processes such as rotation and overshooting yet to be fully understood. The MS width and dependencies remain unresolved, while this stage represents 90% of the overall lifetime and sets the stage for later evolutionary phases.

Efforts have been made to map out the evolution of massive stars with systematic grids (e.g. Brott et al., 2011a; Ekstrom¨ et al., 2012) using detailed predictions of chemical abundances, rotation rates, and fundamental parameters such as mass, luminosity, and effective temperature. These models have subsequently been compared to observations for predicting evolutionary stages and characteristics, though due to limitations in both key observations and accurately modelling key physical processes, many as-

36 CHAPTER 3. GALACTIC EVOLUTION IN THE MASS-LUMINOSITY PLANE

sumptions remain, including the amount of core overshooting (αov) that is thought appropriate.

Martins and Palacios (2013) explored a diversity of evolutionary codes (e.g. Ekstrom¨ et al., 2012; Chieffi and Limongi, 2013; Bertelli et al., 2009) in which the implementation of input physics was surveyed, allowing code applicability to be tested, however linear comparisons of physical treatments cannot be drawn due to the variety of prescriptions in different codes. It is clear that all stellar evolution models have a degree of uncertainty, yet to establish a clear comparison between codes, it would be beneficial to examine physical implementations with one and the same code. Therefore, in this work we aim to compute massive star models with both new and existing prescriptions using the same evolutionary code, Modules for Experiments in Stellar Astrophysics (MESA; e.g. Paxton et al., 2011) given its high flexibility and code capabilities, enabling ample comparisons of several key physical processes. Such exploration offers the opportunity for calibrating models with respect to observations.

Due to the variety of possible prescriptions in each code, the evolution of massive O- type stars so far remains model dependent, leaving the MS lifetime ambiguous particularly due to the absence of evidence for objects after the terminal age main sequence (TAMS). Cool B supergiants, the descendants of O-type stars, are less understood, and have yet to be conﬁrmed as core hydrogen or helium burning objects, (Vink et al., 2010). As O-type stars spend the majority of their lifetime on the MS, we would expect a scarcity of B supergiants if they are indeed post-MS objects. However, we observe a (too) large number of these stars (e.g. Garmany and Fitzpatrick, 1988), raising the possibility that these objects are MS, core H-burning stars. The existence of a large number of slow-rotating B supergiants however, (with v sin i< 50 km s−1) is sugges- tive of an evolved star that has completed the MS phase and been spun down, though this depends on initial rotation rates.

Vink et al. (2010) also considered the possibility for bi-stability braking (BSB) as the mechanism by which B supergiants lose their angular momentum (see also Keszthe-

37 CHAPTER 3. GALACTIC EVOLUTION IN THE MASS-LUMINOSITY PLANE lyi et al., 2017). If we consider that B supergiants may not represent the end of the H-burning phase, this could allow for a wider MS, hypothesised by Vink et al. (2010). This would result in a demand for additional mixing of H in the core, which may be fulﬁlled by increased overshooting. Vink et al. (2010) addressed that a higher value of αov would result in a lower critical mass at which BSB would be efﬁcient.

Test models show that BSB occurs in present models with αov=0.335 above a critical mass of 35 M in the Large Magellanic Cloud (LMC), yet the critical mass drops to 20 M for the same metallicity with an increase of overshooting to αov=0.5 (Vink et al., 2010).

The determination of αov for massive stars has been challenging without the aid of astroseismological data for the most massive stars, leading to an array of prescriptions such as the correlation between v sin i and log g (Brott et al., 2011a). Many other estimations of αov have been adapted in stellar evolutionary models leading to a wide variety of potential stellar ages, MS lifetimes, and ﬁnal products (Martins and Palacios, 2013). One of the most straightforward approaches would be to derive it simply from the MS width, which might potentially be possible from the Galactic Hertzsprung- Russell diagram (HRD) in Castro et al. (2014). However, the Galactic data from this study might be biased compared with the unbiased LMC data from the VLT-Flames Surveys of massive stars Evans et al. (2005, 2011) as they do not show a gap between O and B supergiants (Vink et al., 2010), thereby suggesting a more extended MS.

In this study, we attempt to constrain the dominant parameters effecting massive star evolution. For this purpose, we present a grid of evolutionary models for two extreme values of αov= 0.1 and 0.5 to illustrate both lower adopted values and enhanced overshooting, with varying initial masses, rotation rates, and mass-loss rates, thereby highlighting the sensitivity of stellar models in terms of mixing and mass loss. We introduce the Mass-Luminosity Plane as an alternative to the HRD to study the key ingredients in massive star evolution on the MS (see Fig. 3.5). Whilst the fundamental stellar parameters of mass and luminosity have been plotted logarithmically by Maeder (1983) for example, our version of the plot highlights the independent effects of rota-

38 CHAPTER 3. GALACTIC EVOLUTION IN THE MASS-LUMINOSITY PLANE tion, overshooting and mass loss on stellar evolution through vectors, with inverted mass on the x-axis providing a useful comparison to the tracks in the HRD.

Weidner and Vink (2010b) presented an overview of the methods of mass determination for O stars, including the ’mass discrepancy’ often seen between the evolutionary masses and spectroscopic masses. The method of comparing the positions of stars in the HRD with theoretical evolution models (evolutionary masses) has often led to predictions which are systematically higher than the masses derived through stellar spectroscopy (spectroscopic masses) (e.g. Herrero et al., 1992). However, when O stars are found in binary systems, their dynamics can present a model independent mass determination (dynamical masses). Evolutionary masses can present discrepancies amongst themselves when using various theoretical models (e.g. Ekstrom¨ et al., 2012; Brott et al., 2011a) with differing implementations of rotation, convection and mass loss. Although this is not the widely discussed ‘mass discrepancy’ problem, it does highlight the necessity of calibrating stellar evolution models to minimise further discrepancies with spectroscopic and dynamical masses (see e.g. Markova et al., 2018). In the case where dynamical masses agree with the spectroscopic masses, we can have faith in the spectroscopic result, thus allowing for calibration of theoretical evolution models. Similar work has been completed by Southworth et al. (2004); Pavlovski et al. (2018); Tkachenko et al. (2014) for detached eclipsing binaries, however these works utilised lower mass stars (up to ∼15 M ) which did not incorporate the interacting effects of mass loss, overshooting and rotation as we do in this study.

˙ We use constraints relative to αov and M to investigate the possible evolutionary paths of a high mass, detached binary, HD 166734, modelled in this work as a test-bed for single star evolution. As previously mentioned, dominant processes take effect at varying mass ranges, yet with dynamical masses of 39.5 M and 33.5 M for the primary and secondary, respectively, for HD 166734 we may probe the effects of these processes as they interact and overlap. As the spectroscopic masses adeptly agree with the dynamical masses for HD 166734 (Mahy et al., 2017), this system provides a unique opportunity to constrain – and effectively correct – stellar evolution models, whilst for

39 CHAPTER 3. GALACTIC EVOLUTION IN THE MASS-LUMINOSITY PLANE the general case of single massive stars we cannot currently tell if there are issues with spectroscopic masses, leading to mass discrepancies (Markova et al., 2018).

We present a method of producing a calibrated grid of models with an analysis of HD 166734 in section 3.5 and a calibration of mixing processes in section 3.3. We explore a new tool for comparing observations with models in the Mass-Luminosity Plane in section 3.4 and we provide our ﬁnal results for HD 166734 in section 3.5. We present our grid of models alongside a sample of Galactic O-stars in section 3.6, and ﬁnally, we highlight our conclusions in section 3.7. Appendix A provides a complete grid of models for a range of initial masses and rotation rates.

3.2 Method

A set of evolutionary models was calculated for massive MS stars with the 1D, stellar evolution code MESA, for example Paxton et al. (2011), as a comparison for both the primary and secondary of HD 166734 (see Sect. 3.9). The extensive capabilities of this code provide a diverse range of available alterations, enabling the user to compare implementations of physical processes with other code treatments. In this study, we examine the effects of mass loss, convective overshooting, and rotational mixing in terms of fundamental observables such as luminosity, mass, and surface abundances. These models were completed from zero-age main-sequence (ZAMS) to core collapse, unless convergence problems arose in which computations were concluded earlier. We adopt the default metallicity in MESA of Z = 0.02 with the chemical mixture from Grevesse and Sauval (1998a) to provide direct comparisons with chemical abundances in Galactic observations, though other studies have shown a range of initial N abundances (e.g. Brott et al., 2011a).

Convection is treated by the mixing length theory where αMLT = 1.5, with a semiconvection efﬁciency parameter of αsemi = 1. The convective core boundary is deﬁned in these models by the Ledoux criterion1 in order to test the effects of extra convec-

1 φ The Ledoux criterion is denoted by ∇rad < ∇ad + δ ∇µ , but in chemically homogeneous layers

40 CHAPTER 3. GALACTIC EVOLUTION IN THE MASS-LUMINOSITY PLANE tive mixing processes, in which overshooting succeeds convective mixing at the core boundary, increasing the temperature gradient ∇T by implementing a thermal gradient

∇rad (e.g. Choi et al., 2016). This method of extending the core is denoted as step- overshooting, which enhances the core by a factor αov of the pressure scale height Hp. Experiments in the dependencies of this parameter are completed in the following sections. Uncertainties in model parameters such as αov are estimated within the scope of the grid such that a systematic uncertainty of ∼ 0.1 exists.

We then compared our grid with treatments of αov and rotational mixing from Brott et al. (2011a) and Ekstrom¨ et al. (2012) grids since these are used extensively in the community. Brott et al. (2011a) presented a calibration of the overshooting parameter by comparing the TAMS of 16 M models with observations from the FLAMES survey (Evans et al., 2008), suggesting a TAMS at log g = 3.2, since this value represents a drop in v sin i beyond which a large number of slow-rotating B supergiants are located, assumed to be post-MS objects. The model with αov =0.335 corresponded to the log g = 3.2 and has since been used as a static parameter in models to compare against observations of a wide mass range. A lower value of αov=0.1 is applied for models presented by Ekstrom¨ et al. (2012), since calibration was completed with lower mass stars of ∼1.7-2 M where convective mixing plays a dominant role compared to that of rotational mixing. Hence this allowed for a linear calibration of convective overshooting without accounting for the more sophisticated treatment of rotational mixing as prescribed in the GENEC code. We adopt step-overshooting for H-burning phases only, as we aim to better understand the MS width.

Mass-loss rates are adopted from Vink et al. (2001) accounting for metallicity dependencies and the occurrence of the bi-stability jump, an increase of mass loss at 21kK causing effects in the evolution, seen in the HRD. We tested various factors of this mass-loss regime to determine the possibility of extreme rates. We hence explored a range of multiplication factors of Vink et al. (2001) mass-loss rates from 0.1 to 10 times the standard prescription. We later applied rotation in our models through a fully where ∇rad = ∇ad the Schwarzschild criterion is effective.

41 CHAPTER 3. GALACTIC EVOLUTION IN THE MASS-LUMINOSITY PLANE

Table 3.1: Calibrated grid of stellar evolutionary models.

Minitial [ M ] 8, 12, 16, 20, 25, 30, 35, 40, 45, 50, 55, 60 v sin i[ km s−1] 0, 100, 200, 300, 400, 500 αov 0.1, 0.5 diffusive approach with appropriate instabilities such as the Eddington-Sweet circulation, dynamical and secular shear instabilities. We also considered the effects of an internal magnetic field by a Spruit-Taylor dynamo, although we found that this had qualitatively inconsequential effects on our results. The calibration of our single star models are relevant for evolutionary codes which implement rotational mixing in a similar way. If this process is treated as physically different in another code, then the results would differ quantitatively, but qualitatively have the same behaviour, though there may be effects on angular momentum transport (Meynet et al., 2013). A range of diffusion coefficients has not been explored in the scope of this work though may impact the efficiency of mixing.

A systematic grid of models was calculated for comparison with a larger sample, including new prescriptions discussed in Sections 3.3 and 3.4. Table 3.1 shows the range of masses, rotation rates, and overshooting values for which we compose our grid. We choose masses representative for the O-star and early B-star range, with a variety of rotation rates up to break-up speed, and extreme values for αov to explore the extent of extra mixing. We evolved each model to core collapse, unless convergence problems highlighted unlikely solutions. For this purpose, Vink et al. (2001) provided the relevant mass-loss prescription, with a factor of unity for all models in the ﬁrst instance.

3.2.1 The detached, eclipsing binary HD 166734 : A test-bed for

massive star evolution

The eclipsing massive binary HD 166734 (see Table 3.2) provides a unique opportunity to improve physics in stellar evolution models as Mahy et al. (2017) were able to determine the individual stellar parameters, including their positions in the HRD

42 CHAPTER 3. GALACTIC EVOLUTION IN THE MASS-LUMINOSITY PLANE and their dynamical real masses directly. As these dynamical masses were found to be in excellent agreement with their spectroscopic masses, these two stars of this massive binary system, enable us to calibrate and correct the evolutionary masses, thereby constraining the relevant physics in the upper HRD for stars above 30-40 M . Ob- servations of high-mass eclipsing binaries are sparse (on the order ∼10, Tkachenko et al., 2020), and even more extreme for detached, non-interacting stars that may be treated as evolved single stars. Since observations of massive single stars may some- times highlight discrepancies between spectroscopic and evolutionary masses. In this case, we have an ideal opportunity because the dynamical masses are in agreement with spectroscopic masses, providing a tool for calibrating evolutionary masses and thus evolutionary paths of stars that are massive enough for the physics to be heavily inﬂuenced, if not dominated by mass loss via stellar winds.

Though a large fraction of O stars may be present in a binary or multiple system, observations of eclipsing binaries above 30 M are extremely rare (see e.g. Bonanos et al., 2004; de Mink et al., 2009; Pfuhl et al., 2014; Gies, 2003). Hence the stellar parameters derived by Mahy et al. (2017) have provided a unique opportunity to analyse a non- interacting system which can be treated as single stars. The similar values of v sin i for both components may at face value be considered of interest in terms of synchronisation, but Mahy et al. (2017) have argued against synchronisation because the rotation speeds are lower than the orbital period. In addition, we note that the v sin i values are close to the inferred macro-turbulent values of 65 ± 10 km s−1(Mahy et al., 2017) and we therefore urge for caution that the quoted values of v sin i are truly the result of rotation (see Simon-D´ ´ıaz and Herrero, 2014). We thus treat the v sin i values as upper limits, and we consider the similar values of the two components as merely a coincidence, or possibly due to low inclination. We recognise that the estimated v sin i quantities also may be upper limits due to the possibility of macro-turbulence. We can also compare with observed surface N abundances as a secondary assessment of potential rotation rates. Uncertainties in the surface enrichments are adapted from Mahy et al. (2017).

43 CHAPTER 3. GALACTIC EVOLUTION IN THE MASS-LUMINOSITY PLANE

Table 3.2: HD 166734 Properties. Fundamental observational properties of HD 166734, adapted from Mahy et al. (2017). Primary Secondary Teff [K] 32000 ±1000 30500 ±1000 log(L/ L ) 5.840 ± 0.092 5.732 ± 0.104 Mdyn [ M ] 39.5 ± 5.4 33.5 ± 4.6 Mspec [ M ] 37.7 ± 29.2 31.8 ± 26.6 v sin i[ km s−1] 95 ± 10 98 ± 10 [N/H] 8.785 ± 0.706 8.255 ± 0.556 Orbital period 34.5 days

Figure 3.1: An image of the test-case binary HD 166734 as taken by the 2MASS survey and hosted by the Sloan Digital Sky Survey (SDSS).

44 CHAPTER 3. GALACTIC EVOLUTION IN THE MASS-LUMINOSITY PLANE

We utilise this agreement between dynamical and spectroscopic masses, though focus on dynamical masses as the most accurate measurement, allowing HD 166734 to be treated as an excellent test-bed for massive star evolution of the most massive O-type stars. Mahy et al. (2017) analysed the system ﬁnding a composition of two supergiant O-type stars in an eccentric 34.5-day orbital period. The distance to HD 166734 is 1.9kpc estimated from Gaia Collaboration et al. (2018), though due to the low Galactic latitude could have signiﬁcant reddening.

Comparisons to current-day evolutionary sets of models by Brott et al. (2011a) and Ek- strom¨ et al. (2012) by Mahy et al. (2017) revealed that both sets of models over-predict the evolutionary masses, whilst the secondary star appeared to be more evolved than the primary. We consider this latter ﬁnding an artefact of the Mahy et al. (2017) approach rather than an equal-age assumption, and that in reality it is far more likely that both components formed simultaneously. We can therefore use an equal-age assumption in addition to the exact HRD positions and true current day masses to solve the evolutionary mass discrepancy for both components, and at the same time constrain the relevant physics in this mass range.

Our assumption that this binary has evolved from the same initial stage is important for constraints of the MS width and thus for constraining the overshooting parameter and determining the rotation rates and possible evolutionary scenarios. As both stars show limited evidence of an evolved nature, we can exclude extreme events in the past such as eruptive mass loss or binary interactions. Mahy et al. (2017) showed surface nitrogen enrichments with a particle fraction [N/H] ratio of 3:1 between the primary and secondary components respectively. We utilise these abundances as evidence for mixing, as well as constraints for the determination of age.

45 CHAPTER 3. GALACTIC EVOLUTION IN THE MASS-LUMINOSITY PLANE

3.3 Mixing and mass loss

3.3.1 Envelope stripping and nitrogen enrichment.

In developing our initial set of models we aim to minimise interacting physical processes. We start with a set of non-rotating stellar evolution models that exclusively employ mass loss and convective overshooting as mixing processes. In the ﬁrst instance, initial masses were adopted from Mahy et al. (2017) with 56.1 M and 47.4

M for the primary and secondary, respectively; with varying factors of the mass- loss recipe adopted from Vink et al. (2001) for a range of convective overshooting parameters αov. We initially attempted to reproduce characteristics of HD 166734 by following analysis from Mahy et al. (2017) with parameters taken from Brott et al. (2011a) and Ekstrom¨ et al. (2012) grids. We found however that these models do not offer solutions in which sufﬁcient N enrichment is reached. We hence employ greater mixing through increased factors of mass loss and overshooting.

In reproducing the properties of HD 166734, we can constrain the scenarios that display the 3:1 ratio of [N/H] for the primary to secondary by applying a restriction to the model time. As both stars are assumed to be approximately the same age with this ratio of enrichment, we can exclude the vast majority of possible evolutionary scenarios, i.e. those that do not represent these surface chemical enrichments simultaneously. Accordingly, we do not predict the ages of these stars, but we rather allow for constraints such as surface enrichments, rotation rates, and dynamical masses to provide a solution whereby both stars can reproduce the observables concurrently. Analogous to this, isochrones have not been used here as a method of stellar age determination as we have previously highlighted the sensitivity of model dependency on these features, thus leading to a wide range of possible ages.

Massive stars produce surface He on the MS by the CNO cycle, with a rapid increase in 14N by a factor of ∼10 at the surface when CN equilibrium is reached. The occurrence of this observational feature has been reviewed widely by Maeder and Meynet (1987), ﬁnding that increased convective mixing by overshooting has shown to lower the limit

46 CHAPTER 3. GALACTIC EVOLUTION IN THE MASS-LUMINOSITY PLANE for CN equilibrium during the MS.

Maeder and Meynet (1987, 1988, 1991) composed grids of evolutionary models based on inputs of mass-loss rates and convective overshooting αov as the sole mechanisms for chemical mixing. The importance of convective overshooting has been stressed in these early publications as αov leads to a range of stellar ages, due to the dependence of Teﬀ at TAMS on αov, (Maeder and Meynet, 1991). Moreover, the MS luminosity increases by 0.9 dex at the reddest point of the MS when overshooting is accounted for leading to increases in age by factors of 1.5 - 2.7.

Meynet et al. (1994a) presented grids of massive stars with high mass-loss rates since the evolution of the most massive stars is so heavily reliant on the effect of stellar winds. A factor of two enhancement was applied to their mass-loss prescription from Schaller et al. (1992) demonstrating the effects on the evolutionary track presented in a HRD. These results hinted at a metallicity dependency on mass-loss rates, though also show envelope stripping with increased mass loss leading to evolutionary phases such as WR types and quasi-chemical homogeneous evolution (Meynet et al., 1994a). When analysing nitrogen enrichments for these models we ﬁnd that if surface abundances do increase, it is by a sudden step of a factor of ten, representative of CN-equilibrium. This behaviour applies to factors of 1 - 3 of Vink et al. (2001) mass-loss rates, and overshooting αov of 0.1 - 0.8. We also note that models with increased overshooting result in earlier enrichment by up to 1Myr, regardless of mass-loss rates. In ﬁgure 3.2 we present the nitrogen enrichments for a sample of models of primary and secondary masses.

We ﬁnd that chemical mixing of CNO elements by mass loss and overshooting at- tains CN equilibrium before any intermediate enrichment occurs. This demonstrates that a combination of increased mass loss and overshooting results in envelope stripping, whereby fusion products are extensively exposed at the stellar surface. Since this does not provide a solution for reproducing the observed surface enrichments of HD 166734, and moreover any observation with intermediate enrichment, we must ex-

47 CHAPTER 3. GALACTIC EVOLUTION IN THE MASS-LUMINOSITY PLANE plore additional, viable mixing processes, such as rotational mixing.

As we have adopted the method of step-overshooting, physical implications of this may hinder intermediate enrichment since step-overshooting invokes instantaneous homogeneous mixing within the overshooting region, leading to immediate enrichment by a factor of 10 when the envelope is stripped via stellar winds. Therefore, we compared our results with the prescription of exponential overshooting, whereby the length of the scale height is set by a comparable parameter f0, but the overshooting region is mixed by a diffusion gradient.

Nevertheless, these results show similar enrichments, as even though intermediate enrichments may be reached through the overshooting region by altering the diffusion coefﬁcient, elements are not mixed intermittently through the envelope from the overshooting layer. Thus another mixing process capable of mixing the chemical elements from the convective layers through the envelope must be implemented in order to match observed enrichments.

Recent studies of massive star observations (e.g. Brott et al., 2011b; Hunter et al., 2008; Maeder, 2000) suggest that surface enrichments of CNO products may or may not be a result of rotational mixing. Yet, the necessity of rotational mixing has not been stressed with respect to CN equilibrium or observed intermediate enrichments. We therefore tested the effects of rotational mixing as a function of surface enrichment, with a set of rotating models of varied initial rotation rates from 100-500 km s−1. In this set of models we find that a range of intermediate enrichments occurs, also providing solutions for reproducing the 3:1 nitrogen ratios as seen in HD 166734, (Fig.3.2). The comparison in Fig. 3.2 illustrates that rotational mixing is essential in reproducing observational surface enrichments, unless another not yet considered mechanism is identified, since previous mixing processes provide either too little or too much mixing leading to insignificant enrichment or CN equilibrium.

Fig. 3.2 not only demonstrates the necessity of rotational mixing, but also stresses the importance of enhanced overshooting. In the rotating models of Fig. 3.2 we see that

48 CHAPTER 3. GALACTIC EVOLUTION IN THE MASS-LUMINOSITY PLANE

Figure 3.2: Surface nitrogen abundances as a function of stellar age for extreme values of αov and −1 M . The blue lines represent rotating 40 M models with an initial rotation rate of 200 km s , αov=0.1 (dash-dotted), and αov=0.5 (solid). The red lines show the corresponding non-rotating models for the same mass and values of αov respectively.

with an increase in αov from 0.1 to 0.5, we get much larger surface enrichments that may aid our understanding of the unexplained nitrogen enrichments discussed by Grin et al. (2017). As a signiﬁcant fraction of the sample cannot be explained by rotational mixing alone, extended overshooting may help towards resolving this problem.

3.3.2 Rotationally induced mass loss

While analysing a set of rotating models for HD 166734 we discovered a problem with respect to interacting processes such as rotation and mass loss, which consequently have a non-linear effect on the mass and luminosity. We ﬁnd that the initial masses suf- ﬁcient for reproducing the observed luminosities are excessive when aiming to reach the dynamical masses by the time of observed temperatures or evolutionary phases. We therefore calculated a set of lower initial mass models, yet these diminish the luminosity gradient over time so that current data points of HD 166734 remain out of

49 CHAPTER 3. GALACTIC EVOLUTION IN THE MASS-LUMINOSITY PLANE reach. Interpreting an initial mass from the observed luminosity allowed for calcula- tion of a possible mass-loss rate that would enable the current dynamical masses to be reached.

Following this method, we ﬁnd a mass-loss rate of log M˙ = − 5.17, translating to an increase in the mass-loss rate by approximately a factor of 3. We therefore completed further models with increased mass-loss rates of a factor of two and three. We now reached the dynamical masses; this also led to a signiﬁcant drop in luminosity, which correlates to a shallow gradient in the M − L plane (see Fig. 3.6.), suggesting the observed masses and luminosities could not be reproduced simultaneously (see Fig. 3.4). The possibility of rotationally enhanced mass loss started with 1D radiation-driven wind models of Friend and Abbott (1986), who proposed an equatorially enhanced stellar wind and an increased mass-loss rate due to a lower effective gravity at the equator. This result is also included in many massive star evolution models (see Heger et al., 2000b; Brott et al., 2011a). This same implementation is included in the default MESA settings. The mathematical approach in shown in Eq. (3.1), i.e.

˙ ξ ˙ M0 M = 1 where ξ = 0.43. (3.1) 1−Ω

We note that the Geneva group (e.g. Maeder and Meynet, 2015; Ekstrom¨ et al., 2012) employed a slightly different implementation, yet it is based on similar physical principles. Since 1986 there have been many studies of the effects of rotation on radiation- driven wind predictions with several different levels of sophistication, and different results. Recent 2D modelling by Muller¨ and Vink (2014) encountered cases of equato- rial decreases of the mass-loss rate and surface-averaged total mass-loss rates that are lower than for the 1D case. They therefore challenged the implementation of rotationally enhanced mass-loss in stellar evolution modelling, which is still mostly applied; for example it is the default setting in MESA.

Figure 3.3 highlights the change in initial mass-loss rate due to a change in initial

−1 rotation rate from 100 - 300 km s for both a 40 M model. As we consider the

50 CHAPTER 3. GALACTIC EVOLUTION IN THE MASS-LUMINOSITY PLANE

Figure 3.3: Mass-loss rate as a function of stellar age for comparison of rotationally induced mass loss. The red solid lines represent models with default MESA settings of rotationally induced M˙ (see −1 −1 Eq. (3.1)), with an initial mass of 40 M and rotation rates of 100 km s and 300 km s . All other processes have been set to default values to avoid conﬂict in our analysis. The dashed blue lines show the corresponding models with ξ = 0, for the same mass and rotation rates respectively. current enhancement largely as artiﬁcial, we explored the difference between disabling rotationally enhanced mass loss (effectively setting ξ = 0) and enabling it using the default setting (ξ = 0.43). We thus calculated a series of models with various initial masses, rotation rates, mass-loss rates and overshooting parameters.

3.4 Mass - Luminosity Plane

When comparing models in Section 3.3 we ﬁnd that enhanced mass-loss regimes lead to unrealistic luminosities that are too low to reproduce the observed HD 166734 luminosities. We also ﬁnd that an initial mass representative of the observed luminosities is too high to reproduce the much lower dynamical masses with factor unity of Vink et al. (2001) mass-loss rates. If we aim to simultaneously reproduce the mass and luminosity, we must explore all possible dependencies of these properties,

L = µ4M α, (3.2)

51 CHAPTER 3. GALACTIC EVOLUTION IN THE MASS-LUMINOSITY PLANE

where α varies as a function of mass and µ is the mean molecular weight. The most fundamental characteristics of the evolution of a star are its mass and luminosity. As such, when trying to correlate the theoretical evolution of a star with its observables, these properties are essential. Thanks to analysis of HD 166734 by Mahy et al. (2017), we can reliably utilise the luminosities of both stars determined from bolometric magnitudes to calibrate their evolutionary status. This reasoning is also applicable to the masses of HD 166734, as in this circumstance the dynamical masses agree very well with the derived spectroscopic masses, providing a unique opportunity to constrain the mass-loss rates and physical processes during evolution.

The mass and luminosity of a star are reliant on age and mass-loss rate, so we reach a diversity of possible evolutionary scenarios with respect to mass-loss rates and αov. Yet we may constrain these solutions by assuming both objects evolved from the same initial starting point, so we can account for primary and secondary masses to be reached at the same time.

Eq. (3.2) shows that we can increase the luminosity by increased helium abundance. A minor helium enrichment in both the primary and secondary presents the possibility that the initial mass is not required to be insufﬁciently high to reach the dynamical mass. The observed helium enrichment corresponds to an increase in Linit by ≈ 30% or 0.11 dex. This offers a potential scenario that would allow for a higher luminosity and lower Minit, nonetheless it is unlikely that the initial He abundance of HD 166734 is enriched rather than having been exposed as fusion products at the surface during hydrogen burning. We consider this solution unlikely.

Alternatively, the observed luminosity could be higher than would be required for the relevant initial mass due to the evolutionary phase at which these stars are currently undergoing. Beyond the TAMS, we observe an increased luminosity as models evolve to cooler temperatures. If HD 166734 was in fact composed of helium burning objects, the observed luminosity could be explained by this increased post-TAMS. Yet when comparing our models with the observed Teﬀ ’s, we note that both objects remain too

52 CHAPTER 3. GALACTIC EVOLUTION IN THE MASS-LUMINOSITY PLANE

−1 Figure 3.4: Evolution of a 40 M model with an initial rotation rate of 100 km s and αov= 0.1. This is shown for a variety of conventional plots such as the luminosity and mass as a function of stellar age (left upper and lower), as well as in a standard HRD (right upper), and ﬁnally mass as a function of effective temperature.

53 CHAPTER 3. GALACTIC EVOLUTION IN THE MASS-LUMINOSITY PLANE

hot to be post-MS objects, regardless of αov, thus excluding later evolutionary phases as a viable solution.

We observe some models that reach the dynamical mass of the primary due to higher mass-loss rates relative to the higher initial mass, though even these models must be excluded due to the observed Teﬀ since the dynamical mass is only reached during the bi-stability regime at a much cooler temperature than observed. Scaling factors and ˙ dependencies between M, αov and v sin i present a complex situation to break into linear effects.

We constrain our models with HD 166734 observations by utilising a variety of plots for consistency between mass, luminosity, temperature, and age; see ﬁgure 3.4. We explore the HRD position and compare this with the spectroscopic HRD (sHRD), which removes uncertainties with distance and luminosity. Simultaneously, we correlate ages of the primary and secondary with dynamical masses and mass-loss rates. Figure 3.4 illustrates the relevant plots comparing HD 166734 to reproduce the observed masses and luminosities concurrently.

Maeder (1986) discussed the complexity of mixing processes that apply to stellar evolution and the disentanglement required to understand the linear effects of each process fully. Mass loss is thought to dredge up fusion products to the surface while diminish- ing the core mass, extending the MS lifetime at the expense of the He-burning lifetime. In this respect, stellar winds behave similarly to convective overshooting or rotational mixing, even in extreme cases where a star may evolve quasi-chemically homogeneously due to extensive mixing.

Earlier models which solely employ mass loss as the mixing process may present a simpler solution to understanding the full effects of this process. Although it has been stressed that to reproduce observations such as in the 34 open clusters from Maeder and Mermilliod (1981), an extended MS was required leading to conclusions that overshooting is required as an additional mechanism of mixing.

Similarly, in section 3.3 we emphasised the necessity of rotational mixing in repro-

54 CHAPTER 3. GALACTIC EVOLUTION IN THE MASS-LUMINOSITY PLANE

vinit , αov

. M Luminosity

Mass

Figure 3.5: Illustration of the Mass-Luminosity plane with a typical evolutionary track entering the ZAMS at the red dot, evolving along the black arrow. The dotted vector suggests how increased rotation and/or convective overshooting may extend the M − L vector. The curved dashed line represents the gradient at which mass-loss rates affect this M − L vector. The red solid region represents the boundary set by the mass-luminosity relationship, and as such is forbidden.

55 CHAPTER 3. GALACTIC EVOLUTION IN THE MASS-LUMINOSITY PLANE

−1 Figure 3.6: Evolution of both 40 M and 60 M models with initial rotation rates of 100 km s and αov= 0.1 are shown for a variety of factors of Vink et al. (2001) mass-loss rate (0.1 - 3 times), demonstrating the gradients for each model represented by the green and red solid lines in the M − L plane. ducing observed surface abundances. Therefore, since overshooting, mass loss and rotation have similar effects on the MS lifetime and appearance of CNO products at the stellar surface, a method of separating these processes must be developed.

Challenges in reproducing masses and luminosities simultaneously remained while comparing the HRD and mass-age plots. It was consequently thought to be more insightful to compare our models by mass and luminosity directly. Interpreting be- havioural characteristics of physical processes in this way has opened a diversity of information on luminosity and mass, as illustrated in Fig. 3.5.

Figure 3.5 highlights the key features in the Mass-Luminosity Plane. As the star

56 CHAPTER 3. GALACTIC EVOLUTION IN THE MASS-LUMINOSITY PLANE evolves with time, the vector of mass and luminosity increases in length, since MS stars increase in luminosity due to hydrogen burning. In this sense the M − L plot is similar to the HRD in that time can be interchanged with temperature, since we also follow the vector length with respect to temperature, reaching characteristics such as the bi-stability jump. Figure 3.5 demonstrates the evolution of a theoretical model to a particular age or temperature, by which we can compare this point with observations (e.g. an observed effective temperature).

We note that the gradient of this vector is reliant on the mass-loss rate or in this case factors of the mass-loss prescription from Vink et al. (2001). Unsurprisingly, this feature becomes more prominent with higher masses, for example 60 M compared to that of a 20 M model, (see Fig. 3.6). We ﬁnd that the position of the vector at a given evolutionary phase or temperature can only be further extended in length by increased rotation or overshooting αov, since greater mixing leads to higher luminosities (see Fig. 3.7 and 3.8), which have a higher mass-loss rate and subsequently a lower mass.

When analysing our grid of models for the mass range 8-60 M we found a set of features in the M − L plane that provide fundamental boundaries to stellar evolutionary models. Figure 3.5 illustrates one of these boundaries by a red solid forbidden region, by which the mass-luminosity relation (see Eq. 3.2) sets the initial mass and luminosity. As a result of this relationship, stellar evolution models cannot lie within the red forbidden region.

Similarly, if the length of the vector in the M − L plane increases not only with time, but also temperature (as in the HRD) then we can adjust the length of our model based on an observed temperature. Thus we set an initial position and a ﬁnal position in the M − L plane for our models based on observed stellar parameters such as mass, luminosity and temperature. We can then utilise these positions to better understand processes such as rotation, mass loss and overshooting, since these all have an affect on our now ”measured” vector length.

Figures 3.6, 3.7, and 3.8 each illustrate a process which inﬂuences the length or gradi-

57 CHAPTER 3. GALACTIC EVOLUTION IN THE MASS-LUMINOSITY PLANE

Figure 3.7: Evolution of a 40 M model with a factor of unity of the mass-loss prescription and −1 αov= 0.1 shown for a variety of initial rotation rates from 100-500 km s . The length of the M − L vector at a given evolutionary stage can be extended via increased rotation as shown by the blue dots corresponding to TAMS for each model.

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−1 Figure 3.8: Evolution of a 40 M model with an initial rotation rate of 100 km s and a factor of unity for the mass-loss prescription shown for a variety of overshooting αov= 0.1, 0.3, 0.5. The blue dots correspond to TAMS for each model, demonstrating the increase in luminosity or decrease in mass for an increase in overshooting of αov= 0.1 - 0.5. This illustrates the possible further extension of a vector in this plane by extending αov. ent of our vector in the M − L plane. Figure 3.6 demonstrates that the mass-loss rate dictates a steep or shallow gradient, which again must be reached with the initial and final positions determined by the boundaries shown (i.e. the black line representing the forbidden region, and observations illustrating the final position). Figure 3.7 shows the possibility of extending the length of the vector by increasing the initial rotation rate, hence enhancing the luminosity. Finally, we can further extend the vector length by overshooting as represented in figure 3.8 if rotation can be constrained through other methods such as v sin i and surface enrichments.

The range of explored factors of Vink et al. (2001) mass-loss prescription can be seen

59 CHAPTER 3. GALACTIC EVOLUTION IN THE MASS-LUMINOSITY PLANE

in Fig. 3.6 for models with initial masses 40 M and 60 M . As we would expect, the factor of mass-loss rate has a much larger effect at 60 M than the 40 M . We ﬁnd that due to the ’forbidden’ region highlighted in Fig. 3.5, the gradients of models with two-three times the Vink et al. (2001) prescription are much too shallow to reach observed initial luminosities of a 60 M star for example.

Fig. 3.7 illustrates an increase in luminosity by 0.1 dex for an increase in rotation of 200-400 km s−1. We ﬁnd models with initial rotation rates of 100 km s−1and 200 km s−1are indistinguishable in the M − L plane, although a notable increase in luminosity occurs above 200 km s−1. We use the TAMS as a reference point (blue dots) for each model demonstrating the effects of increased mixing by rotation or overshooting.

3.5 Observational constraints

HD 166734 parameter space

To determine the initial parameters of the system HD 166734, we computed a collec- tion of models that adapt our methods from sections 3.3 and 3.4, for a variety of initial masses, mass-loss rates, αov and rotation rates. Due to constraining observations we have reproduced dynamical masses, luminosities, and surface nitrogen abundances based on a selection of parameters.

Since there are multiple solutions to the current evolutionary stage, we present a parameter space in which the system can be reproduced within observational errors. This is necessary as following models with increased rotation or overshooting leads to higher luminosities. For example, models with higher mass-loss rates requires lower initial masses and thus lower rotation rates.

We can reject extreme factors of Vink et al. (2001) mass-loss rates due to the initial mass boundary in ﬁgure 3.5, such that we can reproduce the system with factors 0.5 -

1.5 of the Vink et al. (2001) recipe. For initial masses of 55 - 60 M for the primary and 42 - 47 M for the secondary, we ﬁnd a range of relevant overshooting parameters

60 CHAPTER 3. GALACTIC EVOLUTION IN THE MASS-LUMINOSITY PLANE of 0.3 - 0.5 and 0.1 - 0.4 for the primary and secondary, respectively. We also stress that when calibrating our theoretical models, we ensure that the factor of mass-loss recipe (Vink et al., 2001) remains constant between the two objects to reach the most reliable solution.

When ﬁxing the mass-loss prescription to a factor unity of Vink et al. (2001), we predict initial masses of 55 M and 45 M for the primary and secondary respectively. Initial rotation rates have been selected such that observed surface N abundances are reproduced with 250 km s−1and 120 km s−1 for the primary and secondary, respectively. Having ﬁxed the mass-loss rate and rotation rates of our models, we utilise the M − L plane to measure the necessary overshooting required to reach the observed mass, luminosity and effective temperature of the primary and secondary. We discover greater values of αov required to reproduce these stellar parameters with the primary adopting αov= 0.3 ± 0.1 and the secondary requiring extra mixing of αov= 0.5 ± 0.1 to reach the observed luminosity at their respective effective temperatures.

Figure 3.9 illustrates the evolution of the selected models which simultaneously reproduce observed luminosities and dynamical masses at an age of ∼3 Myrs. Observed nitrogen abundances are reproduced in ﬁgure 3.10, showing the observed 3:1 ratio of the primary to secondary.

Applications from analysis

We can further constrain the evolution of HD 166734 due to constraints provided by v sin i and [N/H] abundances, after we constrain αov in the M − L plane. We seem to require a larger amount of core overshooting for the secondary star than for the more luminous primary. As the primary initial mass is of the order of 60 M , effects such as envelope inﬂation (Grafener¨ et al., 2012; Sanyal et al., 2015) and mass loss may potentially effect the stellar radius of the primary to a larger extent than it would for the secondary. Therefore, instead of arguing for an inverse mass dependence of the

αov, we remain conservative, and consider the αov determination of the secondary star as more secure than that of the primary.

61 CHAPTER 3. GALACTIC EVOLUTION IN THE MASS-LUMINOSITY PLANE

Figure 3.9: HD 166734 constrained models for primary and secondary in the M − L plane (left) and HRD (right).

62 CHAPTER 3. GALACTIC EVOLUTION IN THE MASS-LUMINOSITY PLANE

Figure 3.10: Nitrogen enrichments from models in Fig 3.9. Observations of HD 166734 highlight the desired (3:1) ratio.

3.5.1 Galactic observational sample

We aimed to consolidate our results from sections 3.3-3.5 by overlaying our calibration models for HD 166734 with a sample of 30 Galactic O stars from Markova et al. (2018) to ensure our calibration is representative of a larger sample, and not unique to our selected test-bed HD 166734 only. The analysis by Markova et al. (2018) provided photospheric and wind parameters, including rotation rates, and surface N abundances by applying the model atmosphere code FASTWIND (Puls et al., 2005) to optical spectroscopy. Table 3.3 provides the key parameters we explored. We compared these Galactic data to our grid of models with the aim to constrain treatments of rotation and convection, and we contrast this with treatments from other evolutionary codes.

Over-plotting our models with the Galactic sample from Markova et al. (2018) not only allowed us to contrast evolutionary masses as derived from the spectroscopic HRD with the masses derived from the standard HRD, but also allowed us to compare our model grids with the prescriptions from Brott et al. (2011a). A sample of representative

63 CHAPTER 3. GALACTIC EVOLUTION IN THE MASS-LUMINOSITY PLANE

Table 3.3: Galactic sample of O-stars.Sample of 30 O-type stars analysed by Markova et al. (2018). HD/CPD Teﬀ [kK] log(L/ L ) Mspec [ M ] [N/H] HD 64568a 48.0 ±1.5 5.80 48.5 ±17.9 8.18 HD 46223 43.5±1.5 5.58 38.9 ±14.4 8.58 HD 93204a 40.5±1.0 5.70 60.9 ±22.5 7.78 CPD-59 2600a 40.0 ±1.0 5.40 40.3 ±14.9 7.78 HD 93843a 39.0 ± 1.5 5.91 64.1±23.8 7.98 HD 91572a 38.5 ±1.0 5.35 32.7±12.1 8.37 HD 91824a 39.0±1.0 5.37 32.7±12.1 8.48 HD 63005a 38.5±1.0 5.52 34.4±12.7 8.58 CPD-58 2620a 38.5±1.0 4.99 16.0±5.9 7.98 HD 93222 38.0± 1.0 5.36 35.2±13.0 7.98 CD-47 4551 38.0±1.5 6.19 120.9±44.9 8.08 HD 94963a 36.0±1.0 5.47 23.1±8.6 8.38 HD 94963b 5.62 32.4±12.0 HD 94370a 36.0±1.0 5.36 29.9±11.1 7.78 HD 94370b 5.50 40.5±15.1 HD 92504 35.0±1.0 4.99 19.7±7.3 7.78 HD 75211 34.0±1.0 5.63 43.3±16.1 8.58 HD 46202 34.0±1.0 4.88 22.8±8.4 7.88 HD 152249 31.5±1.0 5.59 25.7±9.5 7.88 HD 151804 30.0±2.0 5.99 62.1±23.9 8.98 CD-44 4865 30.0±1.0 5.26 24.4±9.0 7.98 HD 152003 30.5±1.0 5.66 30.7±11.4 7.78 HD 75222 30.0±1.0 5.56 25.7±9.5 8.38 HD 75222a 5.67 32.8±12.2 HD 78344 30.0±1.0 5.60 33.3±12.3 8.58 HD 169582 37.0±1.0 6.10 86.1±32.1 8.98 CD-43 4690 37.0±1.0 5.53 29.5±10.9 8.38 HD 97848 36.5 ±1.0 5.03 19.6±7.2 8.38 HD 69464 36.0±1.0 5.78 46.9±17.3 8.28 HD 302505 34.0±1.0 5.43 32.4±12.0 8.18 HD148546 31.0±1.0 5.70 35.7±13.2 8.98 HD 76968a 31.0±1.0 5.58 29.8±1.0 8.18 HD 69106 30.0±1.0 5.09 21.8±8.1 8.00

64 CHAPTER 3. GALACTIC EVOLUTION IN THE MASS-LUMINOSITY PLANE

masses 20 M , 40 M and 60 M of our grid, and the applied parameters from Brott ˙ et al. (2011a) models (see Table 3.4) for αovand Mrot, boost that test our new prescriptions discussed in sections 3-4, are denoted in this work as Brott-like models. Note that we make comparisons based on two rotation rates for both Brott-like models and our new grid.

In ﬁgures 3.11 and 3.12, we present our tracks in blue and grey for 100 km s−1and 250 km s−1, respectively, and Brott et al. (2011a) -like tracks in red and pink with 100 km s−1and 250 km s−1, respectively. We ﬁnd a diminished discrepancy between

Mevol, sHRD Mevol, HRD with our new models when compared to that of Brott et al. (2011a) parameters by approximately 0.1dex as a result of increased luminosities with in- ˙ creased αov, as well as the absence of the Mrot, boost. This discrepancy is noted in Markova et al. (2018, pg. 12) as a systematic difference in Ekstrom¨ et al. (2012) models whereas Brott et al. (2011a) models appear 10-20% less massive in the HRD compared to the sHRD for masses above 30 M .

While exploring the possibility of a reduced discrepancy between evolutionary masses derived from the HRD and sHRD, we compared luminosities deduced by Markova et al. (2018) with the recent Gaia DR2 distance estimates (Gaia Collaboration et al., 2018). We found errors in the distances of our sample and therefore luminosities when using the newly calculated distances through Bailer-Jones et al. (2018). However, when considering potential errors due to reddening, we found the reddening error to be larger than the already substantial error in the Gaia distances (2.9±0.3 kpc). Therefore, ﬁnal answers would require a new spectroscopic analysis with proper consideration for reddening parameters and Gaia DR2 distances, which lies beyond the scope of this study.

3.6 Grid analysis

Our systematic grid has been completed for two extreme values of αov = 0.1 and 0.5, since analysis from HD 166734 invokes an argument for increased overshooting of

65 CHAPTER 3. GALACTIC EVOLUTION IN THE MASS-LUMINOSITY PLANE

Figure 3.11: Comparison of parameters taken from Brott et al. (2011a) in red/pink with our new prescription in blue/grey, contrasted alongside our adopted Galactic sample of O stars from Markova et al. (2018).

66 CHAPTER 3. GALACTIC EVOLUTION IN THE MASS-LUMINOSITY PLANE

Figure 3.12: Comparison of models as described in Fig. 3.11. Data from Markova et al. (2018) are shown as black triangles in the form of a sHRD.

αov= 0.5, and αov= 0.1 for allowing comparisons with previously published grids such as Ekstrom¨ et al. (2012) models. For this reason, we show our tracks in ﬁgures 3.11 and 3.12, which are computed with our larger αov prescription of αov= 0.5. Table 5.7. highlights the parameter space in which we compose our grid, compared with models from Ekstrom¨ et al. (2012) and Brott et al. (2011a) in table 3.4, and a full overview of models for a range of initial masses and rotation rates are provided in Appendix

A. We identify the key variances as extra mixing by overshooting of up to αov= 0.5, and decreased mass loss by excluding rotationally induced mass loss. In order for our results to have relevance beyond the MS, we must ensure that observations of later evolutionary phases can be matched with our parameters.

3.6.1 Red supergiant upper luminosity limit

Red supergiants (RSGs) have been observed at luminosities up to log L/ L ≈ 5.5 - 5.8 (Levesque et al., 2005; Humphreys and Davidson, 1994), with an observed cut-off after which RSGs are not created. In analysing our set of models, we compare ﬁnal

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Table 3.4: Comparison of parameter set-up with previously discussed evolutionary grids and this work. Code Bonn parameters GENEC parameters This work : MESA Minitial [ M ] 5 - 60 8 - 120 8-60 −1 vinitial [ km s ] 0 - 550 v/vcrit = 0.4 0, 100, 200, 300, 400, 500 αov 0.335 0.1 0.1- 0.5 Internal B-field Spruit-Tayler - None ˙ Mrot, boost factor 0.43 0 0 stages of evolution to RSG, BSG, or possible WR-type evolution (bluewards). We find that having fixed our mass-loss rates for clarity, overshooting has the dominant effect on the maximum mass/luminosities at which RSGs are formed. We note that the treatment of convection in the outer layer and the adopted mass-loss regime for this evolutionary phase also affects the position of the RSG.

˙ If both αov and M are ﬁxed at lower values such as αov= 0.1, with standard Vink et al.

(2001) mass-loss rates, then RSGs are formed at luminosities of up to log L / L = 6.0, even for masses up to 60 M . Since observations of RSGs suggest a lower cut-off in the range ≈5.5-5.8 dex, a higher value of αov is required to match observations. Models which have adopted αov= 0.5 remain blue above log L/ L = 5.8 without evolving to RSGs, in agreement with the Humphreys-Davidson limit.

We also examine ﬁnal evolutionary phases for models from section 3.4 with factors of ˙ M between 0.5 and 1.5, for αov= 0.1 and 0.5, ﬁnding that regardless of mass-loss rate

(within our accepted parameter range), models which adopt αov of 0.1 result in RSG evolution even at 60 M . Figure 3.13. represents the observed luminosity cut-off for

RSG evolution when implementing an enhanced overshooting of αov= 0.5.

3.6.2 Compactness parameter

Enhancing the overshooting parameter αov has repercussions for the ﬁnal fate of our models. The consequences of our grid results impact ﬁnal mass estimates as well as compactness parameters which may dictate black hole and neutron star formation.

We include our estimates of the compactness parameter ζ2.5 for all ﬁnal models, in

68 CHAPTER 3. GALACTIC EVOLUTION IN THE MASS-LUMINOSITY PLANE

˙ Figure 3.13: Rotating models with αov= 0.5 and a factor of 1.5 M, highlighting the post-MS evolution to the red for log L up to 5.8, but above this limit remaining in the blue at TAMS. Red/green lines denotes different inital masses.

which post-MS evolution is set as standard in MESA, as a function of αov and initial rotation rate. O’Connor and Ott (2011) quantiﬁed the compactness of a presupernova stellar core as seen in Eq. (3.3), where M = 2.5 M is selected as the relevant mass within which the iron core density gradient may be deﬁned. The parameter ζ2.5 thus denotes how easily a presupernova stellar core explodes; with a low value leading to a more likely solution in which the star explodes rather than collapsing to form a black hole. Sukhbold and Woosley (2014) found dependencies in the treatment of convection, including overshooting, with the explodability of presupernova models computed with MESA. The models provided have been calculated until core collapse or convergence problems arose, in which case the compactness parameter may have been estimated from O-burning or earlier evolutionary stages than core collapse. A resolution study was not encompassed within the scope of this study.

M/ M ζM = . (3.3) R(Mbary = M)/1000km

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Figure 3.14: Final value of compactness parameter for the initial mass range 20-60 M with αov= −1 0.1 and 0.5, rotation rates of 0-500 km s . The circles represent models with αov= 0.1 and the triangles −1 represent αov= 0.5. Rotation rates are shown as a fraction of the maximum 500 km s in the relevant colour bar.

We note that with an extended convective core (αov= 0.5), and thus MS lifetime, it is more difﬁcult to form black holes than for αov= 0.1 at the same mass range, as shown in Fig. 3.14. A clear correlation with rotation rate is not reached, but can be compared via the representative colour bar. However, we note that rapidly rotating models with

−1 vinit = 500 km s have a very low ζ2.5 and may explode more easily, regardless of

αov.

Renzo et al. (2017) presented values of ζ2.5 ≥ 0.25 for the mass range 15-30 M with varying wind parameters, which are analogous to our results for a similar mass range with αov= 0.5. However we reach values of ζ2.5 ≥ 0.9 with αov= 0.1 for the mass range 20-30 M , suggesting this level of convective mixing may be less desirable for a high chance of explodability.

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3.7 Discussion and conclusions

We presented a calibrated grid of evolutionary models for the mass range 8 - 60 M ,

−1 with initial rotation rates 0 - 500 km s , and two values of overshooting αov= 0.1 and αov= 0.5. These models have been constrained based on results of our test-bed eclipsing binary HD 166734. We found that rotational mixing is necessary to reproduce the observed intermediate nitrogen enrichments, after ﬁrst having explored the possibility that this could be achieved by overshooting and mass loss alone. In particular, we developed a method of reproducing the eclipsing binary HD 166734 based on the fundamental properties of mass and luminosity, utilising a new tool known as the mass-luminosity plane, the M − L plane. This tool presents extensive information about the dominant physical processes for various mass ranges.

- First of all, the M − L plane allows us to exclude very large increases or reductions in the standard mass-loss rates, via the gradient in the M − L plane. More speciﬁcally, we can exclude mass-loss factors that lie beyond 0.5 - 1.5 times the Vink et al. (2001) prescriptions.

- Secondly, the extension of the data in the M − L plane forces us to conclude that an additional process is required. Therefore, we favour large overshooting values of order αov= 0.5. The reproduced evolution of our test-bed high-mass binary HD 166734 required this enhanced mixing by rotation and overshooting to increase the luminosity to that of the observed primary and secondary luminosities.

- Rotational mixing proves intrinsically necessary as the process whereby nitrogen is dredged to the surface in any intermediate quantity. Even though the process has been widely researched in the last few decades, the importance of reproducing observed surface abundances such as in HD 166734 has not been sufﬁciently emphasised. We conﬁrm that alternative mechanisms such as convection and mass loss cannot alone reproduce observed surface nitrogen abundances.

- Finally, we disfavour the application of rotationally induced mass loss, in agreement with results from 2D computations of Muller¨ and Vink (2014), as interacting processes

71 CHAPTER 3. GALACTIC EVOLUTION IN THE MASS-LUMINOSITY PLANE artiﬁcially altering the initial mass-loss rate, leads to an entangled set of processes that cannot be separately constrained. The evolution of HD 166734 cannot be reproduced with the inclusion of this theory. A follow-up study should explore the effects of lowered initial mass-loss rates with the inclusion of rotationally-induced mass loss, though it is not within the scope of this study.

If we compare observations to our new prescriptions of overshooting, mass loss, and rotation, we now open the possibility of an extended MS width, reinforcing the argument of B supergiants still being core H-burning objects (Vink et al., 2010). This is important for the wider context of evolutionary phases and paths undertaken by massive stars, providing a better understanding of observed populations. Since the main sequence is where stars spend 90% of their lives, it is key to understand the extent of the MS observation sample and whether this includes O, B, and A supergiants.

72 Chapter 4

The Magellanic Clouds and determining stellar age

In this chapter I present a grid of evolutionary models for the Large Magellanic Cloud, comparable to the VFTS observational sample. A development of the Mass-Luminosity plane is included by estimating the age of a test case binary.

The research outlined in this chapter is to be submitted for publication in Astronomy and Astrophysics as Erin R. Higgins & Jorick S. Vink, Stellar age determination and mass discrepancies in the Mass-Luminosity Plane. II. A grid of rotating single star models with LMC metallicity, 2020, to be submitted to A&A.

The theory of massive star evolution has been progressing with the aid of the VLT- FLAMES Tarantula Nebula Survey (VFTS), a homogeneous large sample of O and B supergiants. Yet questions remain about the evolutionary stages and internal mixing efﬁciencies. We aim to further understand the evolution of massive stars by comparisons with a wide ranges of models in the M-L plane. From Chapt. 3 I have used the novel method of the M-L plane to calibrate the evolution of test-case detached binaries which may be used as a stepping stone to better reproduce observations with theoretical models. We develop a more robust method of estimating the age of observations from the M-L plane.We have reproduced the evolution of the most massive binary in

73 CHAPTER 4. THE MAGELLANIC CLOUDS AND DETERMINING STELLAR AGE the VFTS sample, R139, along with two lower mass binaries, VFTS 063 and VFTS 116. The M-L plane has illustrated a discrepancy between the evolutionary and spectroscopic masses of B supergiants from the VFTS sample. I present a grid of LMC metallicity, rotating models as a comparison to the VFTS sample utilised in estimating the age of a large sample of stars.

4.1 Introduction

In this chapter, the main sequence evolution is tested at lower metallicity for a range of test-case binaries and larger samples. Following Chapt.3, the effects of mass loss and overshooting are explored with use of observations from the nearby Large Magellanic

Cloud, a satellite dwarf galaxy of the Milky Way. This galaxy has ∼50% Z and is a useful comparison for how stellar evolution of massive stars changes at different metallicities.

The largest homogeneous sample of massive star observations was obtained by the VLT-FLAMES Tarantula Survey (VFTS) (Evans et al., 2011) providing multi-epoch spectroscopy of over 800 massive stars in the Tarantula Nebula of the Large Magel- lanic Cloud. Previously, due to the lack of observations, massive stars have remained mysterious in their evolutionary channels. Though other surveys have been performed on massive stars, these mainly are photometric or star formation studies and as such the VFTS sample provides a much needed spectropscopic survey of massive stars. The VFTS ESO Large Programme was motivated by the lack of massive star observations and the need to better understand their binarity, evolutionary status and physical processes (such as mass loss and internal mixing). With variations in radial velocities, many massive binary systems were detected, now studied by the Tarantula Massive Binary Monitoring project (TMBM).

Sana et al. (2013) found that the binarity of massive stars in the VFTS sample was ∼ 50% suggesting that multiplicity is an important factor in massive star evolution. Though Sana et al. (2012) found ∼ 70% of massive stars were in binary or multiple

74 CHAPTER 4. THE MAGELLANIC CLOUDS AND DETERMINING STELLAR AGE systems, this sample was much smaller with only 20% of the Sana et al. (2013) sample size. Taylor et al. (2011) suggested that non-interacting binaries would provide a valu- able insight into this evolution, mainly due to reliable estimates of the mass of each component. Moreover, the age and evolutionary traits including chemical mixing and mass loss may be better understood. In Chapt. 3, I utilised this technique in constraining the evolution of the galactic detached binary HD166734 with constraints on the internal mixing processes and mass loss rates.

Efforts have been made by Evans et al. (2011) to better map out the evolution of massive stars, with the aim of constraining the key processes which effect the evolution towards supernovae (SNe) and black hole (BH) progenitors which provide a better understanding of their ﬁnal fates.

Theoretical modelling has improved our understanding of massive stars from previous studies (e.g. Langer, 2012; Chiosi and Maeder, 1986b), though has led to inconsistencies between stellar atmosphere modelling and evolution models. One key discrepancy is between spectroscopic masses and evolutionary masses, now known as the mass discrepancy. Since the masses of M > 20 M stars is highly uncertain, the use of detached binaries provides an important tool in better constraining the mass and evolution of these stars.

In this study we further develop the Mass-Luminosity (M-L, Chapt. 3) Plane (Higgins and Vink, 2019), to estimate the age of massive stars from the VFTS sample. We compare evolutionary and spectroscopic masses determining discrepancies in the M-L plane with observations from Grin et al. (2017); McEvoy et al. (2015). We predict the evolution of the most massive detached binary in the VFTS sample, R139. We provide analysis of two smaller mass binaries (20...50 M ) as well to provide a wide range of evolution in the LMC. This is important in providing a range of evolutionary channels at lower metallicity since the ZLMC is ∼50% Z and the ZSMC is ∼20% Z . The M-L plane tool is extended to calculate the age of the test-case binaries as well as provide examples for a large sample (VFTS), however it has not been extensively

75 CHAPTER 4. THE MAGELLANIC CLOUDS AND DETERMINING STELLAR AGE tested in calculating the age of the entire sample in order to provide overall properties of the VFTS sample.

4.2 Methodology

In this study we utilise the stellar evolution code MESA as well as previously developed methods from Chapt. 3. We compare the observations of detached binaries from the VFTS sample with evolutionary models in the M-L plane. We utilise these archetype binaries in producing a grid of evolutionary models with LMC metallicity as a comparison to the VFTS sample of O and B supergiants, see Sect. 4.5.

4.2.1 MESA

We previously calculated a grid of galactic evolution models in Chapt. 3 for the main sequence, developing the analysis of the MS further for LMC metallicity in this study.

Convection is employed via the mixing length theory developed by Cox and Giuli

(1968) where αmlt= 1.5, while implementing the Ledoux criterion denoted by ∇rad <

φ ∇ad + δ ∇µ , but in chemically homogeneous layers where ∇rad = ∇ad the Schwarzschild criterion is effective. We investigate the efﬁciency of convective core overshooting by including the step overshooting implementation in MESA. This method extends the core by a fraction αov of the pressure scale height Hp. The effects of convective core overshooting are probed with variations of αov= 0.1...0.5. We adopt a scaled-solar metallicity for the LMC with Z = 0.0088, though other studies have utilised a different initial [N/H] abundance (e.g. Brott et al., 2011a).

The Vink et al. (2001) mass loss recipe is implemented for mass loss during the hot, hydrogen-rich phases of evolution (i.e. Teﬀ > 10kK, Xs > 0.7) with a scaling factor of unity as concluded by Higgins and Vink (2019) in our previous work, with de Jager et al. (1988) implemented for cool stars (Teﬀ < 10kK). The effects of rotationally- induced mass loss are discarded as determined in Higgins and Vink (2019). Rotation

76 CHAPTER 4. THE MAGELLANIC CLOUDS AND DETERMINING STELLAR AGE is included for a range of initial rotation rates 0-500 km s−1with rotational instabilities employed in angular momentum transfer and mixing as described by Heger et al. (2000a).

4.2.2 Mass - Luminosity Plane

In Chapt. 3 a unique method of calibrating massive star evolution was developed, comparing the evolutionary masses with observed luminosities we can disentangle physical processes ﬁnding upper limits to v sin i and αov. We calibrated the evolution of a galactic test-bed detached binary, HD166734, using the M-L plane and surface nitrogen abundances as a function of observed Mspec, L, Teﬀ , and v sin i.

In this work we employ a similar method in which surface nitrogen abundances are reproduced simultaneously at an equal age by calibrating initial rotation rates and convective overshooting, while stellar parameters are measured in the M-L plane such that the spectroscopic masses and observed luminosities are reached by the observed effective temperatures. In the ﬁrst instance, we explored variations of increased mass loss and overshooting when reproducing surface nitrogen enrichments, though this resulted in negligible enrichment as seen in Chapt. 3, therefore we also included rotation.

We compare the evolution of three test-case binaries in the M-L plane and HRD, constraining their evolution and stellar parameters. The utility of the M-L plane has been further developed in Sect. 4.4 where the observed effective temperature is compared with the evolutionary track to provide an estimation of the age.

4.3 Evolution models for detached binaries in VFTS

The test-bed binary HD166734 was analysed in Higgins and Vink (2019) through the M-L plane for calibration of a grid of galactic evolutionary models. We found that enhanced core overshooting was required for this test-case at Z . A set of rotating and non-rotating models were then calculated as a comparison of the galactic O supergiant sample. In this study we provide new analysis of three detached binary systems from

77 CHAPTER 4. THE MAGELLANIC CLOUDS AND DETERMINING STELLAR AGE

Figure 4.1: Evolutionary tracks of VFTS 527 with initial masses of 65 M (primary, red) and 60 M (secondary, green), initial rotation of Ω/Ωcrit = 0.4 and αov= 0.1 in both cases. Black triangles represent the observed stellar parameters of VFTS 527 taken from Mahy et al. (2020). the TMBM sample: VFTS 527, VFTS 063, and VFTS 116. The TMBM project has performed analysis of 82 massive binary systems in the LMC, 51 single lined and 31 double lined spectroscopic binaries (Almeida et al., 2017). Spectral disentanglement was undertaken by Mahy et al. (2020) in order to provide estimates of stellar parameters which we utilise in this research to better constrain their evolutionary status and study internal mixing processes (e.g. αov) at multiple mass ranges. Errors in the model parameters are subject to systematic uncertainties due to the grid speciﬁcation, such that αov will have an error of 0.1.

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4.3.1 R139 : the most massive binary test-bed

The most massive O-star binary system R139, VFTS 527, was revealed in the VLT- FLAMES Tarantula Survey (VFTS), an ESO Large programme of multi-epoch optical spectroscopy (Evans et al., 2011). It was discovered to be a massive spectroscopic binary system by Taylor et al. (2011), with similar spectral types for the primary and secondary, classiﬁed as an O6.5Iafc supergiant and O6Iaf supergiant respectively. Further analysed as part of the Tarantula Massive Binary Monitoring (TMBM), the system was found to have slightly lower mass estimates (Almeida et al., 2017), though required a disentangled spectra for a full analysis.

Mahy et al. (2020) have since studied R139 in more detail providing spectroscopic masses, effective temperatures and surface abundances. Utilising these stellar parameters to probe stellar evolution at this high mass range would provide a much better understanding of physical processes in this regime, as well as exploring the location of the TAMS for these masses. Mahy et al. (2020) suggests that R139 may be an equivalent test-bed to that of HD166734, previously analysed in Paper I. Furthermore, it may provide a better understanding of main sequence evolution for two mass ranges at two metallicities.

In this study I have analysed the evolution of R139 with the M-L plane, HRD and surface nitrogen abundances. We ﬁnd that since the surfaces of both objects are enriched, we required an initial rotation of Ω/Ωcrit = 0.4. By ∼2.5x106 yrs the surface rotation rate is approximately 60 km s−1in line with observations due to spin down with mass loss.

When predicting the evolution in the M-L plane based on our assumptions from the surface nitrogen abundances and subsequent rotation rates, we found that minimal overshooting was required in both cases with αov= 0.1. We implement efﬁcient semiconvection of αsemi= 100 as in HV20. Our estimated initial masses for the primary and secondary respectively are 65 M and 60 M . Figure 4.1 illustrates the evolution of the detached binary in both the ML plane and HRD.

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Figure 4.2: Evolutionary models for VFTS 063 with initial masses of 45 M and 22.5 M for the primary (red) and secondary (green) respectively. Increased core overshooting is required of αov= 0.5 and initial rotation of Ω/Ωcrit = 0.4 and 0.1 for the primary and secondary respectively is applied. Black triangles represent the observed stellar parameters of VFTS 063 adapted from Mahy et al. (2020).

4.3.2 VFTS063

We provide estimates of the evolution for the detached binary VFTS063. We ﬁnd that the initial masses of the primary and secondary are 45 M and 22.5 M respectively, from the initial masses selected along the M-L relation and gradient from mass-loss rates. The initial rotation rates are Ω/Ωcrit = 0.4 and 0.1 for the primary and secondary respectively, due to the length of the vector required to reach the observed effective temperature. We also require extra mixing in the form of convective overshooting of

αov= 0.5 for both objects. Figure 4.2 illustrates the evolution of VFTS063 with our estimated evolutionary tracks.

4.3.3 VFTS116

VFTS116 is a much lower mass binary with spectroscopic masses of 16 M and evolutionary masses of 16 M and 13 M for the primary and secondary respectively. We ﬁnd that when utilising the spectroscopic masses in the M-L plane that due to a much lower luminosity for the secondary, this component would be much younger than it’s companion. The evolutionary tracks which best reproduce these spectroscopic masses

80 CHAPTER 4. THE MAGELLANIC CLOUDS AND DETERMINING STELLAR AGE

estimate an initial mass of 15.5 M with initial rotation of Ω/Ωcrit = 0.1 and core overshooting of 0.5, shown in Fig. 4.3.

4.4 Stellar age determination in the M-L plane

The M-L plane requires a degeneracy of up to ﬁve inputs from observations, allowing in depth analysis of the evolution. This allows for more robust evolutionary tracks to be calculated, improving constraints on observations. We utilise as many stellar parameters made available from observations with emphasis on the spectroscopic mass, luminosity, effective temperature and surface abundances. We then make use of theoretical estimates of the rotation rates and convective overshooting from surface enrichments and analysis of the evolutionary tracks.

The M-L plane offers insight into many physical processes acting on the evolution of a star, yet it also provides information on the stage of evolution such as whether a star is on the MS or beyond. While isochrones may estimate the age of a star based on its position in the HRD, in this study we have developed a more robust method of estimating the age of stars, see Fig. 4.4. Our inclusion of up to ﬁve stellar parameters in calibrating evolutionary models via the M-L plane allows us to estimate the age of stars when the observed luminosity, mass and effective temperature are reached simultaneously, in line with our theoretical estimates of rotation and overshooting as a function of surface enrichments. This may provide a more reliable method of estimating the age of stars, even for a large sample as illustrated in ﬁgure 4.5. We explore the full details of this method in section 4.4.1.

4.4.1 VFTS age estimates

Figure 4.4 illustrates an extension of the M-L plane from Fig. 3.5 with the observed effective temperature reached by an evolutionary model at a given age. This comparison of observed stellar parameters in the M-L plane enables a more robust age estimation from our theoretical model. We provide an example of this new method for the

81 CHAPTER 4. THE MAGELLANIC CLOUDS AND DETERMINING STELLAR AGE

Figure 4.3: Evolutionary track of a 16 M model with initial rotation of Ω/Ωcrit = 0.1 and enhanced core overshooting of αov= 0.5. Observational parameters are represented by black (spectroscopic masses) and blue (evolutionary masses) triangles. The M-L plane is shown in the left plot with a corresponding HRD (right). One evolutionary track is shown as both components are expected to have the same initial mass.

82 CHAPTER 4. THE MAGELLANIC CLOUDS AND DETERMINING STELLAR AGE

Age (t)

log Teff = 4.4 Luminosity

log Teff = 4.45

log Teff = 4.5

Mass

Figure 4.4: The Mass-Luminosity plane illustrated with an evolutionary model extending to higher luminosities and lower masses with age until an observed effective temperature is reached (blue star). Hotter effective temperatures are denoted by markers along the evolutionary track. The red forbidden zone marker the initial mass and luminosity of each model based on the mass-luminosity relation.

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Figure 4.5: A grid of evolutionary models with initial masses 8-60 M shown in the M-L plane with initial rotation rates of Ω/Ωcrit = 0.1. The O supergiant sample from Grin et al. (2017) are represented by coloured dots, where the colour is denoted by their v sin i corresponding to the colourbar (right). Dashed lines along the vector/evolution represent effective temperatures ranging from log 4.6-4.0. test-case VFTS 527 from Sect. 4.3.

R139 provides an opportunity as a calibrated test-bed for this method of determining stellar ages since the system has been reproduced with evolutionary models. We note the age at which these models concurrently reach the observed stellar parameters is 2.9 ± 0.1 Myrs. This result lies in agreement with Taylor et al. (2011) though is increased by ∼ 0.5 ± 0.1 Myrs.

The M-L plane provides a new method of determining the age of a large sample of observations by comparing with a calibrated grid of evolutionary models. Figure 4.5 and 4.6 represent rotating models of Ω/Ωcrit = 0.1 and 0.4 respectively for comparison with O supergiants categorised as fast or slow rotators by their associated colour given

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Figure 4.6: A grid of evolutionary models with initial masses 8-60 M shown in the M-L plane with initial rotation rates of Ω/Ωcrit = 0.4. The O supergiant sample from Grin et al. (2017) are represented by coloured dots, where the colour is denoted by their v sin i corresponding to the colourbar (right). Dashed lines along the vector/evolution represent effective temperatures ranging from log 4.6-4.0.

85 CHAPTER 4. THE MAGELLANIC CLOUDS AND DETERMINING STELLAR AGE in the colourbar (0-400 km s−1). The horizontal markers along the evolutionary tracks represent effective temperatures which can be used to equal the observed effective temperature of each data point. This point is then used to provide an estimate of the age at that temperature range in the evolutionary model.

4.5 Grid comparison with observations of O and B su-

pergiants from the VFTS sample

The VFTS survey by Evans et al. (2011) has provided the largest homogeneous sample of O and B stars enabling detailed studies of their evolutionary traits. While much has been discovered about their binarity (Sana et al., 2012), rotation (Ram´ırez-Agudelo et al., 2013), and surface abundances (Grin et al., 2017), many questions remain about their evolutionary stages and internal mechanisms.

In this study we present a grid of evolutionary models in order to reproduce observations of the VFTS O and B supergiant sample. Comparable to Higgins and Vink

(2019), models were calculated for the range of initial masses 8-60 M from pre-MS to core collapse unless convergence problems occurred in ﬁnal evolutionary phases.

We apply overshooting αov of 0.1 and 0.5 to explore the location of the TAMS for

O and B supergiants. Finally, we implement two rotation rates of Ω/Ωcrit = 0.1 and 0.4, as well as a non-rotating set of models in order to best represent our sample of observations.

4.5.1 Main sequence objects

We illustrate the contrast of our evolutionary grid alongside observations in Fig. 4.9 in the form of a HRD with O supergiants represented by blue dots and B supergiants by light blue dots. When comparing the extension of the MS in models with increased overshooting we note that B supergiants would be included as MS objects, suggesting that the TAMS position determined by αov may alter the evolutionary status of the

86 CHAPTER 4. THE MAGELLANIC CLOUDS AND DETERMINING STELLAR AGE

VFTS sample. In Higgins and Vink (2019), we ﬁnd that our test-bed required extra internal mixing in order to reproduce observed luminosities by enhanced core overshooting αov = 0.5. If this conclusion is applied to our LMC grid of models, we must consider B supergiants as H-burning MS objects. Vink et al. (2010) considers the effects of bi-stability braking as a method of reproducing the sample of slow-rotating B supergiants from the VFTS sample, enabling them to be categorised as MS objects which have not spun-down over their MS lifetime. Figure 4.9 demonstrates that B supergiants may be included in MS evolution with αov = 0.5. Due to an unbiased data set from VFTS, we do not observe a gap between O and B supergiants, suggesting B supergiants may in fact remain to be H-burning objects close to the TAMS.

4.5.2 Mass discrepancy

The inconsistencies between spectroscopic masses derived from spectroscopic analysis of observations, and evolutionary masses estimated from evolutionary models has been a problem throughout studies of massive stars. In this study we avail of our systematic grid of models to make comparisons with the VFTS sample of O supergiants from Grin et al. (2017) and B supergiants from McEvoy et al. (2015). While analysing spectroscopic masses of the data in the M-L plane we find that O and B supergiants occupy one population as seen in figure 4.7, indistinguishable of spectral type compared to in a HRD. Since B supergiants are expected to lie beyond O supergiants, we compared O and B supergiants with evolutionary masses from McEvoy et al. (2015) rather than spectroscopic masses finding that indeed the B supergiants now occupied a later evolutionary phase in the M-L plane (see figure 4.8). This suggests that the discrepancy between evolutionary and spectroscopic masses is consequential for B supergiants and their evolutionary stage. In this study we find a systematic overestimation of spectroscopic masses and/or underestimation of luminosities of the B supergiant sample.

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Figure 4.7: Grid of evolutionary models with initial masses 10-60 M alternating colour red/green, compared in the M-L plane with observations of O and B supergiants from Grin et al. (2017) and McEvoy et al. (2015) with evolutionary masses 15-60 M .

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Figure 4.8: Grid of evolutionary models with initial masses 10-60 M alternating colour red/green, compared in the M-L plane with observations of O and B supergiants from Grin et al. (2017) and McEvoy et al. (2015) with spectroscopic masses.

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Figure 4.9: Evolutionary tracks of rotating models (initial masses of 10-60 M ) with αov= 0.1 and 0.5 shown in a HRD and compared with observations of O and B supergiants from Grin et al. (2017) and McEvoy et al. (2015).

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4.6 Conclusions

In this chapter I have provided a new method of calculating stellar age as a function of mass, luminosity and effective temperature in the M-L plane. The development of this tool has provided estimates of age for test-case binaries in the TMBM sample, as well as the possibility of calculating the age of large samples such as the VFTS.

We provide a grid of massive star evolutionary models for a range of masses at LMC metallicity. We compare these models with the VFTS sample of O and B supergiants. In particular we utilise detached binaries from this sample as constraints on evolutionary processes.

We noted from Mahy et al. (2020) that VFTS 527 was in fact the LMC counter-testbed to HD166734 which we have analysed for the galactic grid in Higgins and Vink (2019). Having dynamical masses from Taylor et al. (2011) in agreement with Mahy et al. (2020) analysis we can utilise this binary in the same way to constrain evolution in the LMC. The initial masses of VFTS 527 are such that they lie beyond the mass range of both Galactic and LMC grids (20-60 M ), which results in a variation of relevant physical behaviours such as the mass loss and overshooting.

We ﬁnd that in order to reproduce the mass and luminosities of VFTS 527 at the vector length of the observed temperatures, αov = 0.1 is required, having constrained the rotation rates to Ω/Ωcrit = 0.4 from the surface nitrogen enrichment. We ﬁnd that in a 40 M model the convective core is ∼ 60 % of the entire stellar mass with αov =

0.5 included, whereas in an 80 M model the convective core is already ∼ 70 % of the overall mass with αov = 0.1. Due to such large convective cores in VMSs we require a minor extension of the core by overshooting to reproduce observed abundances and stellar parameters, compared to that of masses below 40-50 M where a larger extension of 0.5 is required.

We utilise the VFTS sample of O supergiants (Grin et al., 2017) and B supergiants (McEvoy et al., 2015) in comparing our grid of models in the M-L plane. When investigating the position of the O and B stars in the M-L plane we discovered a lack

91 CHAPTER 4. THE MAGELLANIC CLOUDS AND DETERMINING STELLAR AGE of distinction between the spectral types, this was a result of employing evolutionary masses taken from McEvoy et al. (2015). When comparing with spectroscopic masses in the M-L plane, we found that B supergiants in fact lie beyond the O star range - though they may still be MS objects if alpha=0.5 is applied.

We present a new and innovative method of determining stellar ages through use of the M-L plane. This method can be utilised for an entire sample of observations, including our testbed VFTS 527.

92 Chapter 5

Post-Main Sequence Evolution to Red Supergiants

In this chapter I present a study of the upper luminosity of red supergiants by probing the effects of semiconvective mixing, convective overshooting mixing and stellar winds for the Milky Way and Magellanic Clouds.

The research outlined in this chapter has been published in Astronomy and Astro- physics as Erin R. Higgins & Jorick S. Vink, A theoretical investigation of the Humphreys-Davidson limit at high and low metallicity, 2020, A&A, 635.

Current massive star evolution grids are not able to simultaneously reproduce the empirical upper luminosity limit of red supergiants, the Humphrey-Davidson (HD) limit, nor the blue-to-red (B/R) supergiant ratio at high and low metallicity. Although previous studies have shown that the treatment of convection and semiconvection play a role in the post-main sequence (MS) evolution to blue/red supergiants, a uniﬁed treatment for all metallicities has not yet been achieved. In this study, I focus on developing a better understanding of what drives massive star evolution to blue and red supergiant phases, with the ultimate aim of reproducing the HD limit at varied metallicities. We discuss the consequences of classifying B and R in the B/R ratio and clarify what is required to quantify a relatable theoretical B/R ratio for comparison with observations.

93 CHAPTER 5. POST-MAIN SEQUENCE EVOLUTION TO RED SUPERGIANTS

For solar, LMC (50% solar), and SMC (20% solar) metallicities, we develop eight grids of MESA models for the mass range 20-60 M to probe the effect of semiconvection and overshooting on the core helium-burning phase. We compare rotating and non- rotating models with efﬁcient (αsemi= 100) and inefﬁcient semi-convection (αsemi=

0.1), with high and low amounts of core overshooting (αov= 0.1 or 0.5). The red and blue supergiant evolutionary phases are investigated by comparing the fraction of core He-burning lifetimes spent in each phase for a range of masses and metallicities. We

ﬁnd that the extension of the convective core by overshooting αov= 0.5 has an effect on the post-MS evolution which can disable semiconvection leading to more RSGs, but a lack of BSGs. We therefore implement αov= 0.1 which switches on semiconvective mixing, though for standard αsemi= 1, would result in an HD limit which is higher than observed at low Z (LMC, SMC). Therefore, we need to implement very efﬁcient semiconvection of αsemi= 100 which reproduces the HD limit at log L/ L ∼ 5.5 for the Magellanic Clouds while simultaneously reproducing the Galactic HD limit of log

L/ L ∼ 5.8 naturally. The effect of semiconvection is not active at high metallicities due to the depletion of the envelope structure by strong mass loss such that semiconvective regions could not form. Metallicity dependent mass loss plays an indirect, yet decisive role in setting the HD limit as a function of Z. For a combination of efﬁcient semiconvection and low overshooting with standard M˙ (Z), we ﬁnd a natural HD limit at all metallicities.

5.1 Introduction

The maximum luminosity (Lmax) of red supergiants (RSGs) is an important tracer for luminous stellar populations of galaxies (Massey, 2003). This limit implies that above a certain luminosity, massive stars do not evolve to the cool supergiant phase but remain compact evolving towards a blue supergiant (BSG) or Wolf-Rayet (WR) star. Humphreys and Davidson (1979) showed that the maximum luminosity of RSGs, now recognised as the HD limit, was log L/ L ∼ 5.8 for the Milky Way, (see also Lamers and Fitzpatrick, 1988; Davies et al., 2018).

94 CHAPTER 5. POST-MAIN SEQUENCE EVOLUTION TO RED SUPERGIANTS

The HD limit actually has two key features within the Hertzsprung-Russell diagram, a cool temperature-independent section represented by Lmax, and a hot temperature- dependent section where the most massive stars stay hot and blue. Mass loss probably plays a direct role on the hot side of the HD limit through the relations with temperature, mass, luminosity, and proximity to the Eddington limit. Yet whether mass loss plays a role in setting the Lmax remains unresolved. Though this will have important consequences for the progenitors of type II-P supernovae and whether the most massive stars produce direct collapse black holes. In this study we probe the effects of mass loss and mixing in setting Lmax.

Studies have shown a metallicity dependence of radiative driven winds (Abbott, 1982; Kudritzki et al., 1987; Vink et al., 2001), which has been thought to inﬂuence the evolution to BSGs and RSGs (e.g. Chiosi and Maeder, 1986b; Massey, 2003; Lamers and Fitzpatrick, 1988). There has often been an expectation that the HD limit shifts to higher luminosities at lower metallicity due to the physics of these metallicity- dependent winds. Perhaps surprisingly, Davies et al. (2018) recently showed the HD limit in the SMC to be similar to or even slightly lower than that of the LMC (log

L/ L ∼ 5.4-5.5), thereby challenging the dominance of line-driven winds in setting

Lmax. Davies et al. (2018) suggest that there is no evidence for metallicity-dependent winds to be the primary factor in setting the HD limit. In this study, we consider the potential indirect effect of stellar winds by probing its effect on internal mixing and its dependency in setting Lmax via the length of time spent as a RSG or BSG as a function of metallicity.

Langer and Maeder (1995) considered when investigating a related issue, known as the blue-to-red supergiant (B/R) ratio, that the treatment of convection played a key role in the timescales of hot and cool supergiant phases during helium-burning (He-burning), particularly mixing processes such as convective overshooting and semiconvection. First studies by van den Bergh and Hagen (1968) showed that the B/R ratio steeply increases with increasing metallicity. Langer and Maeder (1995) scrutinised the B/R ratio in order to determine which physical processes may effect the evolution of O

95 CHAPTER 5. POST-MAIN SEQUENCE EVOLUTION TO RED SUPERGIANTS stars to red or blue supergiant phases. While an appreciation for the sensitivity of the B/R ratio to semiconvection and mass loss was made, a unique treatment of these processes has not been reached in reproducing the B/R ratio at varied metallicities. The B/R supergiant problem has been explored over the years with variations of its deﬁnition used interchangeably. This has caused inconsistencies within theoretical models and the observed number of supergiants. In particular, reproducing the number of BSGs and RSGs over a range of metallicities has proven unattainable, with some input parameters sufﬁciently reproducing B/R at high Z and others at low Z. This poses the question - is there a problem with observations, theory, or both?

Langer and Maeder (1995) presented the problem of predicting the B/R ratio at various Z with theory, while Davies et al. (2018) question the inverse Z-dependency of the observed HD limit. In order to reconcile these issues between theory and observations, we need to better understand the mechanisms which drive the evolution of O stars to BSG an RSG phases. These studies may also have consequences for the red supergiant problem, reviewed by Davies (2017), concerning the number of red supergiants detected by Smartt et al. (2009) as progenitors of supernovae (SNe). Most massive stars above 8 M will evolve as RSGs before exploding as SNe, with the type II-P SNe as the most common. Smartt et al. (2009) studied the pre-supernovae data in order to analyse the progenitors of a range of SNe types. When comparing the observed mass distributions with theoretical predictions of RSG populations, it was found that there was a deﬁciency in SNe from stars with Minit > 17 M often referred to as the ’red supergiant problem’ (see also Kochanek, 2020; Davies and Beasor, 2020).

In this study we compare variations of internal mixing with a focus on the convective core overshooting parameter (αov) and semiconvection efficiency (αsemi). We explore a wide range of model configurations in order to best fit the evolution of RSGs for a range of metallicities. We provide a grid of galactic, LMC, and SMC models which explore the dependencies of each parameter, and discuss the consequences of our results in Sect. 5.4.

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5.2 Method

5.2.1 MESA Stellar evolution models

We avail of the one-dimensional stellar evolution code MESA (Modules for Experi- ments in Stellar Astrophysics) with version 8845 to calculate our models (Paxton et al., 2011, 2013, 2015), adopting physical assumptions from Chapt.3 unless specified oth- erwise. Following our investigation of the MS evolution of massive stars in Chapt.3, we focus here on post-MS evolution. In particular we explore the role of semiconvective mixing in the final stages of H-burning and throughout core He-burning. As the convective core recedes during the MS, semiconvective regions form in the envelope above due to the change in the H/He abundance profile. We also investigate the effect of convective core overshooting during core H-burning as it will affect the size of the He core during later burning phases.

The Ledoux criterion is implemented for convection in order to test extra convective mixing processes such as semiconvection, while employing the mixing length theory as developed by Cox and Giuli (1968) with αmlt= 1.5 (e.g. Jiang et al., 2015; Pax- ton et al., 2015). The use of MLT++ was not included in our calculations. We apply semiconvective mixing in post-MS phases with an efﬁciency, denoted by Langer et al.

(1983), of αsemi varied here from 0.1-100. We include the effect of overshooting by extending the core by a fraction αovof the pressure scale height Hp, known as step overshooting. We vary this fraction from αov= 0.1 - 0.5, since a modest value of 0.1 is adopted by Ekstrom¨ et al. (2012) while a moderate value of 0.335 is adopted by Brott et al. (2011a), but more recently (as in Chapt.3) values of up to 0.5 have been considered (e.g. Higgins and Vink, 2019; Schootemeijer and Langer, 2018), having consequences for an extended MS. We apply the ’Dutch’ wind scheme as the mass loss recipe, with Vink et al. (2001) active in hydrogen-rich (Xs >0.7) hot stars (Teﬀ

> 10kK), and de Jager et al. (1988) for cool stars (Teﬀ < 10kK). We apply rotation of Ω/Ωcrit = 0.4, as in Ekstrom¨ et al. (2012), and compare with non-rotating models. Rotational instabilities are employed in angular momentum transfer and mixing as de-

97 CHAPTER 5. POST-MAIN SEQUENCE EVOLUTION TO RED SUPERGIANTS scribed by Heger et al. (2000a), though we exclude the effects of rotationally-induced mass loss as in Higgins and Vink (2019); Muller¨ and Vink (2014).

For selected initial chemical composition a scaled solar heavy element distribution as provided by Grevesse and Sauval (1998b) has been adopted. We adopt a solar metallicity of Z = 0.02 (Y =0.28), with scaled-solar mass fractions applied to ZLMC =

0.0088 (YLMC =0.26) and ZSMC = 0.004 (YSMC =0.248) for the Magellanic Clouds. We calculate eight grids of models for each metallicity, rotating and non-rotating, high and low overshooting, efficient and inefficient semiconvective mixing. We determine that this grid specification is sufficient to distinguish which effects are predominant as well as determining which processes conflict or coalesce. We provide our full grid of models in Fig. 5.7.

5.2.2 Mixing processes

The effects of overshooting are relevant for the core H-burning phase though will have repercussions for later phases as an increased core size will determine the (Mcc)/

(MT ) ratio. Semiconvection takes affect during the core He-burning phase and will dictate the envelope structure for the ﬁnal phases. Rotational mixing has an effect on internal mixing, though does not inﬂuence our results in reproducing the HD limit.

While we compare both non-rotating models and Ω/Ωcrit = 0.4 rotating models, we provide an illustration of both models throughout though it is important to note that the use of either grid provides qualitatively similar results and as such does not affect our conclusions.

The role of semiconvection applies to slow mixing in a region above the convective core where there is stable convection by the Ledoux criterion but unstable by the Schwarzschild criterion, (Langer et al., 1985). Semiconvective regions are formed when a H/He gradient stabilises the unstable radiative temperature gradient. The efficiency of this mixing is described by a diffusion coefficient which determines how rapidly mixing takes place. Semiconvection affects the hydrogen profile outside the

98 CHAPTER 5. POST-MAIN SEQUENCE EVOLUTION TO RED SUPERGIANTS

He-burning core, changing the H/He abundance gradient which in turn alters the structure of the H envelope causing evolution to either red or blue supergiant phases. Semi- convection is implemented as a time-dependent diffusive process in MESA where the diffusion coefﬁcient, Dsc, is calculated by equation 5.1, as seen in Langer et al. (1983).

K ∇T − ∇ad Dsc = αsc (5.1) 6Cpρ ∇L − ∇T

where K is the radiative conductivity, Cp is the speciﬁc heat at constant pressure and

αsemi is a free parameter which dictates the efﬁciency of semiconvective mixing.

Langer and Maeder (1995) highlight that increased semiconvection leads to a higher number of BSGs whereas less semiconvection leads to more RSGs. This results from tests with the Schwarzschild criterion which increase the ratio of B/R. Langer and Maeder (1995) found that while the Schwarzschild criterion predicts the ratio for galactic metallicity as observed, and the Ledoux criterion can reproduce the ratio at SMC metallicity, none of the treatments tested were capable of simultaneously reconstruct- ing the B/R ratio at both high and low metallicities. Perhaps the indirect effects of mass loss and mixing with metallicity can provide a better understanding of what sets the HD limit, and whether it remains a hard boundary or merely a short-lived phase.

5.2.3 Observations in the HRD

The observed Lmax is set by the most massive RSGs and in the past has been altered due to uncertainties in distances or bolometric corrections (e.g. Davies et al., 2018), but can also be determined by the timescales of the RSG phase at these highest masses. The

Lmax of RSGs was observed to be log L/ L max ∼ 5.4 - 5.5 for both LMC and SMC

(Davies et al., 2018). Probability distributions suggested that the SMC Lmax would be slightly lower than that of the LMC, which suggests that Lmax does not increase with

99 CHAPTER 5. POST-MAIN SEQUENCE EVOLUTION TO RED SUPERGIANTS decreasing metallicity, and in fact it may be the inverse. Luminosity distributions from this study also highlight a cross-over from RSGs to WRs in both Magellanic Clouds at log L/ L ∼ 5.4 - 5.5, suggesting a possible shift in evolutionary channels.

Although most RSGs in Davies et al. (2018) were observed at log L/ L ∼ 5.4 - 5.5 for both LMC and SMC, there was one object in each galaxy observed to be above the HD limit. This poses a question of the HD limit being a hard border which physically should not be passed, or an observational artefact based on short timescales spent above the observed maximum luminosity, suggesting that while the most luminous RSGs are prone to small statistics, they may spend a small fraction of core He-burning as RSGs and can be observed as such, (see Fig. 5.5).

5.3 Results

We explore the effects of semiconvection and overshooting with free parameters αov and αsemi for a range of masses 20-60 M and metallicities Z ,ZLMC, and ZSMC, in order to probe the evolution to RSG or BSG phases. We assume the observed HD limit is determined by the luminosity at which massive stars spend a significant fraction of core He-burning at ∼ log Teff 3.6 (∼ 90% tHe). We note the occurrence of blue loops and thus do not define a model as a RSG if it merely dips into the cooler temperature for a short timescale (∼ 10% tHe). In Sect. 3.1 we explore the evolutionary channels of massive stars which may evolve as RSGs and BSGs. We provide our results for reproducing the HD limit in sections 3.2 and 3.3, before discussing the consequences our results may have on the B/R ratio in section 3.4.

100 CHAPTER 5. POST-MAIN SEQUENCE EVOLUTION TO RED SUPERGIANTS =0.1 ov α 0.5 (left). The colorbar represents the He abundance, whereas = ov α 0.1 (right) or = ov α =0.5 ov α rotating models with LMC metallicity and

M Kippenhahn diagrams of 60 blue circles represent convective regions with hatched blue regions showing an extension of the core by overshooting. White hatched regions illustrate semiconvective mixing. Figure 5.1:

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5.3.1 Evolutionary channels

Chiosi and Maeder (1986b) characterise HRD positions by initial mass, with massive stars at Minit > 60 M , moderate massive stars between 25-60 M , and ’low mass’ massive stars at Minit < 25 M . Massive stars have signiﬁcant winds which shed the outer envelope leaving an exposed core which cannot evolve to form a RSG, but more likely evolve to LBV and WR phases. Moderate massive stars form the basis of our study since these likely evolve to form RSGs since the winds are not strong enough to strip the outer envelope at early evolutionary phases. We later ﬁnd that mixing and mass loss play a combined role in the duration of RSG/BSG phases (see Sect. 5.3.2), though these processes can be separated in the M-L plane as described in Hig- gins and Vink (2019). Lower mass stars may evolve quickly to become a RSG during He-burning, though undergo blue loops before returning to a RSG. The extension of these blue loops is affected by mass loss and mixing. Low mass stars (Minit < 25 M ) evolve to become the dominant population of RSGs which dictates the B/R ratio without reaching the HD limit.

Figure 5.2 illustrates that for massive stars in the mass range 20-60 M we observe MS objects which may include O supergiants, B supergiants and maybe even A supergiants depending on the TAMS position, (see Sect. 5.4). The post-MS follows with higher mass (∼ 40 M ) stars spending a large fraction of He-burning as BSGs before evolving up the Hayashi line as a RSG. These more massive RSGs determine Lmax. For ∼

20 M , the post-MS may be spent mostly as a RSG, populating the majority of RSGs in the B/R ratio.

The main sequence is represented by position 1 with ZAMS-TAMS positions in yellow, followed by BSG and RSG phases. Consequently, the B/R ratio heavily depends on the deﬁnition of a BSG and RSG. Figure 5.2 illustrates that a BSG can be a post-MS object at either pre-RSG or post-RSG phases (positions 2 and 5 respectively). The lower mass RSGs which dominate the population are represented in position 3, while the most luminous RSGs which set the upper luminosity limit are shown in position 4.

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Figure 5.2: Evolutionary models for 20-60 M in HRD form, with TAMS positions shown for αov = 0.1, 0.5. Evolutionary phases are highlighted by number with RSGs at the red dashed line, BSGs at the blue dashed line and the MS before the yellow lines with the HD limit shown in green. Position 1 shows MS objects, positions 2 and 5 are post-MS BSGs which may be pre or post-RSGs. Position 3 represents the majority of observed RSGs, while position 4 illustrates the most luminous RSGs.

103 CHAPTER 5. POST-MAIN SEQUENCE EVOLUTION TO RED SUPERGIANTS

For some intermediate mass models, blue loops are observed, showing post-RSG BSGs represented by position 5. In order to distinguish which models spend a sufﬁcient ratio of He-burning as a RSG, we compare the HRD positions from ZAMS to core He- exhaustion to observe a distinct switching of evolution from RSG to BSG and compare the time spent at hot and cool effective temperatures, see Fig. 5.3.

5.3.2 The HD limit

First comparisons of theoretical models with the observed HRD by Humphreys and Davidson (1979) suggested an empirical boundary for the luminosity of RSGs in the Milky Way and LMC which may be primarily due to the effects of metallicity-dependent mass loss at the highest masses. However, Davies et al. (2018) recently found discrepancies between theory and observations at the highest luminosities suggesting that mass loss may not be primarily responsible for setting the HD limit. Population synthesis models of Davies et al. (2018) with the GENEC code (Ekstrom¨ et al., 2012) showed a decrease in the time spent at luminosities higher than log L/ L ∼ 5.6 during the cool supergiant phase. This suggests that the empirical boundary may be an observable artefact due to the short timescale of RSGs above Lmax. These models not only predict a decrease in the time spent as a RSG above a certain luminosity, but also

ﬁnd a higher Lmax with lower metallicities due to reduced mass loss, with RSGs expected up to log L/ L ∼ 5.7-5.8 in the SMC. Since this is not observed, theory implies that either stars do not evolve to cool supergiants at these higher luminosities, or they spend such a short time in this phase that it is not likely observed.

After H is exhausted at the center, massive stars promptly start burning H in a shell and He in their core. This causes the radius to expand as the effective temperature cools in order to radiate the same nuclear energy from the stellar surface. Since the core mass and nuclear burning rate increases, the luminosity also increases along the Hayashi line before exploding as a SNe in the ﬁnal phase of evolution. The size of the convective core during the MS (Mcc) as a fraction of the total mass (MT ) greatly dictates the post-MS evolution of massive stars since this determines how much mass remains

104 CHAPTER 5. POST-MAIN SEQUENCE EVOLUTION TO RED SUPERGIANTS in the envelope. Further to this, the envelope structure dictates the effectiveness of semiconvection which in turn shapes the H/He gradient.

In Chapt.3 I have provided an analysis of a main-sequence test-bed binary HD166734 which suggested an extension of the core by overshooting of up to αov= 0.5. For this reason, we first attempted to reproduce the HD limit with post-MS models for relatively high values of αov= 0.5. We found that when the core is extended by this fraction of the pressure scale height, the areas required to form semiconvective regions were not sufficient to reproduce any BSGs at all, even at low metallicity. Therefore only RSGs were formed, and a cut-off luminosity did not exist. In other words, we could not properly reproduce the HD limit at any metallicity. Figure 5.1 illustrates the structural changes as a result of enhanced overshooting for a 60 M model, in which semiconvection does not occur for αov= 0.5. This is due to a lack of capacity for semiconvective regions to form leading to evolution of RSGs, even when extremely efficient semiconvection is implemented. Hence we find that increased overshooting of αov= 0.5 prevents evolution to BSGs.

However, models with αov= 0.1 show semiconvective regions forming above the core due to the overall lowered fraction of core to total mass ratio (compared to αov= 0.5) which allows semiconvective regions to form in the envelope. These models allow for a combination of red and blue supergiants which can reproduce the HD limit, depending on the effects of their stellar winds with metallicity.

Similarly, the fraction of core mass to total mass may be altered by the natural effect that mass loss plays in depleting the envelope. In environments where the envelope loses enough mass through Z-driven winds the semiconvective regions, important for dictating RSG/BSG evolution, are prohibited from forming above the core. This is due to the envelope structure which is not large enough to sustain extra mixing in these unstable regions. It is important to note that in these models, it is irrelevant whether very efﬁcient or inefﬁcient semiconvection is assumed since the regions are not developed (Schootemeijer and Langer, 2018, c.f.). Therefore all models with moderate-high

105 CHAPTER 5. POST-MAIN SEQUENCE EVOLUTION TO RED SUPERGIANTS

mass loss, such as Z models, evolve to RSGs during He-burning, unless the envelope is stripped by other means causing WR or LBV-like phases.

In order to constrain the free parameter αsemi and as a result obtain a better understanding of the HD limit over multiple metallicities, we explore each set of models as they evolve through core He-burning, determining the fraction of time spent at hot or cool temperatures (i.e. >10kK or <10kK) such that RSGs or BSGs would be favoured. We investigate the ﬁnal status of each model (BSG/RSG) along with the time spent in each phase in order to determine whether a model would be observed as a RSG or BSG. Rather than implementing the MESA default value of unity for the semiconvection ef-

ficiency parameter αsemi, we here investigate the outcome of increasing and decreasing this by a factor of 10, similar to that of studies by Schootemeijer et al. (2019). We find that this factor is necessary in altering the efficiency to a notable amount.

106 CHAPTER 5. POST-MAIN SEQUENCE EVOLUTION TO RED SUPERGIANTS 1.0 (grey). Left: The = semi α 0.5 with = ov α 1.0 (orange), 0.1 (blue), 100 (red), and ﬁnally = semi α 0.1 and = ov α , LMC rotating models with Z

M Evolution of 40 evolution of the effective temperaturetracks as shown in a a function HRD of highlighting central a He variation of abundance, blue illustrating loops the and fraction RSG evolution. of Increases core in He-burning L spent (orange, at blue RSG tracks) may and be BSG due phases. to H-shell Right: burning. Evolutionary Figure 5.3:

107 CHAPTER 5. POST-MAIN SEQUENCE EVOLUTION TO RED SUPERGIANTS

An illustration of our selection criteria is shown in Fig. 5.3, where the HRD (right) shows whether the model evolves to the red or blue, with the core He fraction mapped out as a function of log Teﬀ (left). Figure 5.3 provides a comparison of core He- burning at cool temperatures (RSG) or hot temperatures (BSG) for a 40 M star of LMC metallicity with various factors of semiconvective efﬁciency. We choose this mass range as it is representative of the Lmax (HD limit) where models may switch from RSG to BSG depending on input parameters.

The main sequence is shown in the upper section of the left plot (Fig. 5.3) with a green box. The models then diverge at the TAMS with the grey model (αov= 0.5) leaving the main sequence later. We ﬁnd that increasing the core size by overshooting αov=

0.5, which may be preferred at lower masses (∼ 20-40 M ) as found in Higgins and Vink (2019), semiconvective regions are unable to form since the envelope mass is insufﬁcient, therefore the grey model in Fig. 5.3 evolves to a RSG and remains so until it explodes as a supernova, regardless of metallicity.

All other models have a value of αov= 0.1 and semiconvection is varied from the default value αsemi= 1 (orange). The orange dashed line remains blue through most of the He-burning phase though it will result in a RSG by core He-exhaustion. Similarly for low semiconvective efficiency, the blue line representing αsemi= 0.1 spends most of core He-burning as a BSG but later dips to cooler temperatures forming a RSG. We confirm that less efficient semiconvection leads to a combination of blue and red supergiants with higher mass models reaching RSG phases but lower mass models remaining blue for most of the evolution, ending the He-burning phase as RSGs. Since observations show the opposite, i.e. more RSGs at lower masses (∼ 20 M ) than higher masses (∼ 60 M ), we find that this is not a solution for αsemi in setting the HD limit.

For more efﬁcient semiconvection (αsemi= 100), the red dashed line shows that while this model begins He-burning as a RSG, it appears merely as a loop back into the BSG temperature range where it spends most of the He-burning timescale. Therefore when

108 CHAPTER 5. POST-MAIN SEQUENCE EVOLUTION TO RED SUPERGIANTS very efﬁcient semiconvection is applied, models evolve to BSGs above a certain mass

(e.g. 55 M at Z ) reproducing the HD limit.

5.3.3 Uniﬁed theory of the HD limit at all Z

Mass loss is decreased at lower metallicity due to its metallicity dependence, the envelope mass is high enough to create larger unstable regions which may be transformed by semiconvection. Therefore semiconvection can be more efﬁcient in lower metallicity environments leading to more BSGs and therefore a lower HD limit. Similarly, at solar metallicity, the envelope structure would be more depleted than at LMC metallicity, prohibiting large unstable regions to develop. This means that semiconvection is overall less prominent for the same αsemi factor in the Galaxy than at lower metallicities.

Figure 5.4 illustrates the extent of mass lost from the envelope of a 60 M model with

∆M ∼ 20 M at solar metallicity, ∆M∼ 10 M at LMC metallicity and only ∆M∼

7 M at SMC metallicity. This creates a variation in Mcc / MT with metallicity leading to a higher Lmax in the Milky Way than in the Magellanic Clouds. The ﬁnal loss of mass during the RSG phase of evolution also has important consequences for the RSG problem and ﬁnal masses which dictate the fate of these stars.

We ﬁnd that in order to reproduce the HD limit at all Z, very efﬁcient semiconvection is needed, coupled with a small convective core or Mcc / MT ratio. This can be achieved by lowering the amount of convective overshooting required at the highest masses where BSGs are expected.

Figure 5.4 demonstrates the impact of metallicity driven mass loss on the envelope mass and subsequently on semiconvective regions and the evolution to RSGs. Each

Kippenhahn diagram in Fig. 5.4 illustrates a rotating 60 M model with αov= 0.1 and αsemi= 100 for respective metallicities. We present a 60 M rotating model as a representation of this effect for each metallicity since mass loss plays a dominant role at this mass range. Note the increase in semiconvective regions with decreasing

109 CHAPTER 5. POST-MAIN SEQUENCE EVOLUTION TO RED SUPERGIANTS metallicity as a result of a larger envelope mass, for one unique set of parameters. The core evolution also becomes more compact with decreasing metallicity, though the overall effect on evolution from the LMC to SMC is dominated by the change in envelope mass.

These structures suggest that metallicity-dependent mass loss plays an important role in setting the HD limit as they explain the previously misunderstood increasing HD limit with increased metallicity which provided conflict between theory and observations prior to this study. We hence find an increased HD limit for increased metallicity in line with observations for our models with αsemi= 100. This suggests that increased semiconvective mixing efficiency dominates the evolution, whereas mass loss plays an important indirect role in the effectiveness of semiconvection and BSG/RSG evolution. Therefore in low metallicity environments where mass loss is weaker, internal mixing is predominant.

We present the most luminous RSGs for the LMC adapted from Davies et al. (2018) compared with our preferred set of parameters in Fig. 5.5 with a rotating set of models of masses 20-60 M for very efﬁcient semiconvection. Black triangles represent the most luminous RSGs with most below log L/ L = 5.4. At this point our models with very efﬁcient semiconvection begin blue loops, with masses higher than 35 M evolving to BSGs rather than RSGs as we would expect.

Our rotating and non-rotating models with αov= 0.1 and highly efﬁcient semiconvection (αsemi= 100) are consistently able to reproduce the HD limit for three metallicities, see Fig. 5.7. Rotating and non-rotating galactic metallicity models result in a HD limit of log L/ L ∼ 5.8-5.9, in agreement with the observed maximum luminosity of RSGs in the Milky Way. Both rotating and non-rotating models of LMC and SMC metallicity

ﬁnd a HD limit of log L/ L ∼ 5.4-5.5 which reproduces the distribution of RSGs in these lower metallicity environments as found by Davies et al. (2018).

Figure 5.6 demonstrates a theoretical HD limit based on the percentage of time spent during core He-burning at cool supergiant effective temperatures. We present our set of

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LMC

SMC

Figure 5.4: Kippenhahn diagrams for 60 M models at galactic, LMC and SMC metallicities with our preferred parameters of very efﬁcient semiconvection and minimal overshooting (αsemi= 100 and αov= 0.1, with rotation). The colorbar represents the He abundance, whereas blue circles represent convective regions with hatched blue regions showing an extension of the core by overshooting. White hatched regions illustrate semiconvective mixing. Note the decrease in envelope mass with increased metallicity, and subsequently decreased semiconvective regions.

111 CHAPTER 5. POST-MAIN SEQUENCE EVOLUTION TO RED SUPERGIANTS

Figure 5.5: Very efﬁcient semiconvective mixing models with minimal overshooting (αov= 0.1 and αsemi= 100, with rotation) for the mass range 20 M to 60 M (in steps of 5 M ) with LMC metallicity (red solid lines). The most luminous RSGs in the LMC adapted from Davies et al. (2018) have been plotted here as black triangles as a comparison with our prescription.

112 CHAPTER 5. POST-MAIN SEQUENCE EVOLUTION TO RED SUPERGIANTS models from the calculated grid which could reproduce the HD limit at all metallicities.

Efﬁcient semiconvection (αsemi= 100) is implemented with minimal core overshooting

(αov= 0.1) for core H and He burning phases, with the indirect effects of stellar winds dominating the post-MS behaviour. Evolutionary models are analysed based on their core He-burning timescale. We compare the time spent above log Teﬀ ∼ 3.6 for BSGs and below log Teﬀ ∼ 3.6 for RSGs. Though this method determines the effective time spent as a RSG in order to determine the likelihood it would be observed above the empirical HD limit, it also provides useful information for the B/R ratio.

We ﬁnd that the galactic metallicity models spend 70-90% of all He-burning as a RSG, leaving the Lmax above log L/ L ∼ 5.8. This is due the the indirect effect of stellar winds on the envelope structure at this metallicity. In the Magellanic Clouds the RSG timescales are bimodal, with lower mass RSGs (25 M ) spending 20-40% of core He- burning as RSG due to their longer lifetimes, reaching 50% at a critical point before decreasing again. The exponentially increasing and decreasing % of time spent as a RSG form the bimodal structure seen in the LMC and SMC models. The critical point corresponds to a theoretical HD limit. The behaviour seen in the Magellanic Clouds is due to the internal mechanisms which drive evolution bluewards/redwards such as mass loss and semiconvection (see Sect. 5.3.2). Above the theoretical HD limit where massive stars spend most of the He-burning phase as a BSG, the RSG timescales are <5% of the overall He-burning time.

For solar metallicity this only occurs at 60-70 M and above log L/ L ∼ 5.9, while for the Magellanic Clouds all models ≥ 40 M spend less than 2% He-burning as a RSG. Table 5.1 provides the analysis of each model shown in Fig. 5.6, with the timescales of core He-burning, RSG and BSG phases. Although this highlights the short timescales RSGs spend above the HD limit, it also provides the B/R ratio for a range of masses and metallicities. The luminosity is taken from the point the model reaches log Teff ∼ 3.6, before the star evolves up the Hayashi line. We ﬁnd that the behaviour seen in the Magellanic clouds models, compared to the Milky Way where the behaviour is not bimodal, is a feature of semiconvective mixing being switched on and off respectively.

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Table 5.1: Timescales for core He-burning models at solar, LMC and SMC metallicity. Comparisons of time spent at hot (log Teﬀ > 3.6) or cool (log Teﬀ < 3.6) supergiant temperatures demonstrates the respective B/R ratio (tBSG / tRSG) for a range of initial masses and metallicities. The luminosity is determined at the bottom of the Hayashi track. Minit ( M ) Z log LHe tHe (yrs) tBSG (yrs) tRSG (yrs) % tRSG B/R 20 Z 5.0622 862176 179694 682481 79.16 0.263 25 Z 5.2978 670013 66537 6569230 90.07 0.110 30 Z 5.4520 435955 22422 176961 94.86 0.054 35 Z 5.5598 305292 15672 603476 94.87 0.055 40 Z 5.6413 189269 12307 176961 93.50 0.070 45 Z 5.6899 116566 10113 106453 91.32 0.095 50 Z 5.7507 29658 9083 20574 69.37 0.442 55 Z 5.8228 45261 7385 37876 83.68 0.195 60 Z 5.8842 25656 7327 18328 71.44 0.399 20 ZLMC 5.1178 917555 774735 142820 15.56 5.425 25 ZLMC 5.3330 6956648 532495 163085 23.45 4.821 30 ZLMC 5.4828 610123 570334 39789 6.52 14.334 35 ZLMC 5.5560 528338 463547 64790 12.26 7.155 40 ZLMC 5.6554 492070 488921 3149 0.64 155.244 45 ZLMC 5.7264 449172 372011 77160 17.18 4.821 50 ZLMC 5.8215 64835 64834 - 0 - 55 ZLMC 5.9589 199940 199939 - 0 - 60 ZLMC 5.9800 57607 57606 - 0 - 20 ZSMC 5.1181 892962 870237 22724 2.55 38.29 25 ZSMC 5.3437 687509 665022 22486 3.27 29.57 30 ZSMC 5.5125 595742 527794 67947 11.41 7.77 35 ZSMC 5.6076 524646 514410 10235 1.95 50.26 40 ZSMC 5.7066 473388 465997 7390 1.56 63.05 45 ZSMC 5.7829 446112 438579 7533 1.68 58.22 50 ZSMC 5.8599 419623 4121689 7454 1.78 55.29 55 ZSMC 5.9071 388815 383790 5025 1.29 76.38 60 ZSMC 5.9477 376330 371621 4708 1.25 78.93

In order to ﬁnd a true B/R ratio, a population synthesis of the current models would need to be completed, though is not within the scope of this study.

5.3.4 Implications for the B/R ratio

In Sect. 5.3 we have presented our results for reproducing the HD limit at multiple metallicities while also comparing the time spent as BSGs and RSGs at the highest luminosities since this is where the HD limit is set. In the following section we discuss the full mass range (20-60 M ) since the RSGs formed at 20 M dominate the B/R

114 CHAPTER 5. POST-MAIN SEQUENCE EVOLUTION TO RED SUPERGIANTS

Figure 5.6: The percentage of time spent during core He-burning as cool RSGs for all 20-60 M models with αsemi=100 and αov= 0.1 (non-rotating). Note the dashed lines represent the observed maximum luminosity of RSGs for each galaxy / our theoretical predictions for the HD limit in each galaxy such that models which lie above these limits must spend less than ∼ 5% of core He-burning at these cool temperatures.

115 CHAPTER 5. POST-MAIN SEQUENCE EVOLUTION TO RED SUPERGIANTS ratio due to the shape of the IMF.

In reproducing the HD limit from our 8 model grid sample, we also analysed the fraction of core He-burning spent as a RSG for a range of masses and metallicities, such that we could determine the likelihood that stars above the HD limit would be observed

(see Fig. 5.6). We ﬁnd that lower mass models (20...30 M ) have a longer He-burning timescale, spending much of the He-burning stage as a BSG before evolving redwards during the later burning phases. This results in a small percentage of He-burning as a RSG before the internal mixing affects the envelope structure dictating the time spent as a RSG or BSG. At a critical point (35-40 M ), models begin to spend 40-50% of the He-burning phase as a RSG due to the effects of internal mixing, in particular efﬁcient semiconvective mixing. The effects are predominately seen in the Magellanic clouds where mass loss is reduced.

The behaviour changes at solar metallicity due to strong stellar winds affecting the envelope structure such that semiconvection does not take place and RSGs are formed at all masses below 60 M (log L/ L ∼ 5.8). The change in behaviour is present between Z and ZLMC due to the absence or presence of semiconvective regions, previously seen in Fig. 5.4. At an intermediate metallicity this feature may switch from a high percentage of RSGs as seen at galactic metallicity to the peak seen at ZLMC depending on whether semiconvective regions form or not due to the indirect effects of mass loss at a given metallicity.

Observations from Eggenberger et al. (2002) illustrated a clear relationship between the blue-to-red supergiant ratio and metallicity for 45 open clusters, finding that the B/R ratio increased with metallicity. This study followed a series of works on the B/R ratio, though Meylan and Maeder (1982) were the first to examine the B/R ratio with stellar clusters in the Milky Way, LMC and SMC finding an increase in B/R ratio by an order of magnitude with an increase in metallicity from ZSMC to Z . Another eval- uation of the B/R ratio is determined by variations in metallicity over galactocentric distance (e.g. van den Bergh and Hagen, 1968). This method has shown that metal-rich

116 CHAPTER 5. POST-MAIN SEQUENCE EVOLUTION TO RED SUPERGIANTS inner sectors of a galaxy have a higher B/R ratio than lower metallicity outer regions. Humphreys and Sandage (1980) demonstrated this for M33 concluding that the B/R ratio decreases with increasing galactocentric distance. It is important to note that while both sets of studies highlight a positive correlation between B/R ratio and metallicity, in this study we are concerned with deﬁning the B/R ratio in different galaxies such as the Milky Way, LMC and SMC.

We ﬁnd that evolutionary models show a decrease in time spent He-burning as a RSG with decreased Z for models 20-30 M which corresponds to an increased B/R with decreasing Z, counter to observations when including O supergiants, (see Table 5.1). Though as we will later discuss, the observed B/R relation with Z may be polluted by MS objects which will increase B with Z due to the varied TAMS position with Z. Results from Eggenberger et al. (2002) show that O supergiants dominate the B sample for at least seven of the 45 clusters studied, suggesting that the observed Z-relation may be a result of categorising B with MS objects as well as post-MS objects.

5.4 Discussion

The deﬁnition of B and R in an observed B/R ratio is important when comparing to theoretical models in order to establish a better understanding of the driving processes which dictate the B/R ratio such as mass loss and mixing. Since the ratio provides information on the post-MS phases we must exclude MS objects, though a lack of clarity on the TAMS position makes this exclusion difﬁcult to determine.

The blue supergiant population is categorised by Eggenberger et al. (2002) and Langer and Maeder (1995) as O, B and A supergiants; or as O, B supergiants by Meylan and Maeder (1982). However, O supergiants are considered H-burning objects while B and A supergiants may be either H or He-burning (Vink et al., 2010). By including MS objects, the overall B sample becomes much larger than R for all metallicities resulting in a high B/R ratio.

The large number of B supergiants found adjacent to O stars in the HRD (without a

117 CHAPTER 5. POST-MAIN SEQUENCE EVOLUTION TO RED SUPERGIANTS gap) may suggest B supergiants are MS objects. Vink et al. (2010) studied the slow rotation rates of B supergiants, which could be explained by bi-stability braking in case the wind timescale is sufﬁciently long, which would require that B supergiants are indeed core hydrogen-burning MS stars. This can be achieved by large core overshooting (Vink et al., 2010). In this case B supergiants (and perhaps even A supergiants) should not be included in the observed B/R ratio. Alternatively, the slow rotation rates of B supergiants could betray their evolved nature (Vink et al., 2010), in which case B supergiants should be included in the B/R ratio. In other words, as long as we have not solved the related issues of the amount of core overshooting, the location of the TAMS, and the origin of the slow rotation rates of BA supergiants, we are not in a position to have a meaningful comparison of the B/R ratio between observations and theoretical model grids.

If we implement increased overshooting of αov =0.5 to allow for B supergiants as

MS objects at moderate masses (20...40 M ) but retain a lower overshooting at higher masses (40...60 M ) to allow for semiconvection, then our B/R ratio can be constrained to post-MS objects which evolve to RSGs then BSGs. Due to the hotter TAMS at

ZLMC,SMC than Z when increased overshooting is applied, the number of BSGs included as MS or post-MS changes with metallicity. This leads to more O supergiants included in B for Z than for ZSMC resulting in an increased B/R with metallicity. If we exclude MS objects from the B sample, we may exclude the Z-dependence of B/R which proves inverse to theory. This requires a detailed analysis of post-MS objects for a range of metallicities with carefully selected criteria for B and R, allowing comparisons with theoretical models.

In Sect. 5.3, we presented a uniﬁed set of input parameters which can reproduce the observed HD limit at varied metallicities. Figure 5.7 provides an overview these models for all three metallicities in the form of a HRD (left) and effective temperature with time (right). The core He abundance is mapped with the colourbar illustrating core He- burning at RSG and BSG effective temperatures. The model conﬁguration included minimal core overshooting (αov = 0.1) which would allow for semiconvective regions

118 CHAPTER 5. POST-MAIN SEQUENCE EVOLUTION TO RED SUPERGIANTS to form in the envelope, promoting BSG evolution. Very efficient semiconvection was also needed (αsemi = 100) for sufficient mixing at all metallicities. We find that by core He-exhaustion, both LMC (middle) and SMC (bottom) models show a maximum luminosity of RSGs at approximately log L/ L ∼ 5.5, while for solar metallicity models (top) RSGs are still formed at log L/ L ∼ 5.8 during core He-burning (see Fig. 5.7).

We established a second order effect in mass loss, as it plays a more signiﬁcant role at

Z , the envelope becomes more depleted than at lower Z, resulting in fewer semiconvective regions, more RSGs and a higher observed maximum L of log L/ L ∼ 5.8

(compared to that of ZLMC,SMC where log L/ L ∼ 5.5). The efficiency of semiconvection in lower z environments is higher due to the lack of mass loss even for αsemi= 0.1, leading to BSGs during most of He-burning but RSGs at final stage of He-burning. The evolution to RSG phases has implications for pre-SNe and final mass estimates due to the strong stellar winds experienced by RSGs. The uncertainty in these mass- loss rates leads to large uncertainties in the final evolution of these stars. Figure 5.1 illustrates the final decrease in envelope mass during the RSG phase, leaving a 10 M variation in the final mass of models with αov= 0.1 or 0.5. Since RSG mass loss rates are increased compared to earlier evolutionary phases, the consequences for increased RSG timescales can be important for determining mass-loss rates for these phases (e.g. Beasor and Davies, 2018) which ultimately affects their final masses, influencing the lack of RSGs as SNe progenitors such as in the ’red supergiant problem’.

5.5 Conclusions

I have developed eight grids of models of masses 20-60 M for solar, LMC and SMC metallicities to probe the effect of semiconvection and overshooting on the core helium- burning phase (see Fig. 5.7). We compare rotating and non-rotating models with high and low semi-convection (αsemi= 0.1...100), and with high and low overshooting

(αov= 0.1...0.5). We conﬁrm that semiconvective mixing alters the envelope structure

119 CHAPTER 5. POST-MAIN SEQUENCE EVOLUTION TO RED SUPERGIANTS such that more blue supergiants (BSGs) are formed with more efﬁcient semiconvection. We ﬁnd that mass loss and overshooting have indirect effects which dictate the effectiveness of semiconvective regions forming, leading to more RSGs with increased mass loss and overshooting. In order to reproduce the HD limit simultaneously at all metallicities we require low αov∼ 0.1 at the mass range where the HD limit is set and above (e.g. at 55 M and above for the Milky Way). This allows for semiconvective regions to form which in turn produce BSGs above the HD limit.

We stress that a consistent efficiency of semiconvection may reproduce observations of the most luminous RSGs. Therefore we may constrain the HD limit solely by the efficiency of semiconvection. At higher masses we note that envelope inflation may play a role in the treatment of overshooting and the ratio of Mcc to MT, (e.g. Grafener¨ et al., 2012). However, since the prescription of core overshooting is uncertain we aim to better constrain the effectiveness of mixing near the core through αov though it may ultimately be attributed to another internal mixing process (such as rotational mixing, perhaps mediated by internal gravity waves or dynamo mechanisms).

We appreciate that although our unique prescription of mixing and mass loss presented in this study is necessary for reproducing the HD limit at various metallicities, it will also have consequences for the B/R ratio. Since we focus on the evolution to RSG and BSG as a ﬁnal stage of He-burning, the dominant mass range under scrutiny is ∼

35...50 M , which sets the HD limit or maximum RSG luminosity. We ﬁnd that the HD limit is an observational artefact based on the likelihood of observing a RSG at such short timescales. Our models spend less than 2% of core He-burning as a RSG above the theoretical HD limit for all metallicities. This suggests that while the HD limit sets a preference for BSG evolution above a certain luminosity range, it is possible to observe RSGs above the HD limit, as in Davies et al. (2018).

We present an estimate of the B/R for a range of masses and metallicities, and disentangle the constraints on B and R so that observational studies may be compared to the theory which drives evolution to BSG and RSG phases. Our ﬁnal set of models are

120 CHAPTER 5. POST-MAIN SEQUENCE EVOLUTION TO RED SUPERGIANTS presented in Fig. 5.7, demonstrating the core He fraction timescales at RSG and BSG phases for Z ,ZLMC and ZSMC.

121 CHAPTER 5. POST-MAIN SEQUENCE EVOLUTION TO RED SUPERGIANTS

Figure 5.7: Left: Evolutionary models for solar metallicity (top), LMC (middle), and SMC (bottom), with core He abundance represented by the colorbar. Right: He-burning timescale as a factor of effective temperature, such that RSGs are formed for all solar models, while only short timescales are spent in the red during He-burning at lower metallicities. Models are calculated for a range of masses 20-60 M with αov = 0.1 and αsemi = 100.

122 Chapter 6

Conclusions and Outlook

6.1 Overview

In this chapter the key results of the work presented in this thesis will be outlined within the context of the field of research. Though many questions remain in the field of massive stars, developments in theoretical modelling have provided clarity in our understanding of their evolutionary channels, and physical processes which influence them. With large-scale observations of massive stars such as the VLT-FLAMES Taran- tula survey (VFTS) (Evans et al., 2011), many comparisons of these theoretical models have been made feasible.

6.1.1 Physical processes

The evolution of massive stars has been studied for decades, leading to an abundance of knowledge on what impacts upon this evolution, such as stellar winds, internal mixing and rotation (e.g. Chiosi and Maeder, 1986b). Nevertheless, our implementation of these processes in theoretical models has been challenging in reproducing observations. Due to limited observational surveys in the past, this has only been made feasible in recent years with studies of the VFTS sample. We now know that massive star evolution is highly inﬂuenced by metallicity, binarity, rotation, mass loss, and

123 CHAPTER 6. CONCLUSIONS AND OUTLOOK internal mixing.

Rotational mixing

In chapter 3, the effects of both internal mixing and mass loss were analysed in reproducing the galactic test-bed binary HD 166734. In this study it was found that simultaneously increasing the mass-loss rate and convective overshooting region resulted in a surface nitrogen enrichment of a factor 10 rather than the observed intermediate (3:1, primary:secondary) enrichment. This resulted in the conclusion that rotational mixing was necessary to mix CNO products such as nitrogen to the surface as is observed by Mahy et al. (2017), and similarly for large scale observations as in Hunter et al. (2008). Clarity has now been provided to the wider modelling community in terms of the inclusion of rotational mixing in all stellar evolution models.

In reproducing HD 166734 a new method of constraining the evolution was established in the M-L plane. Due to the convolution of rotationally-induced mass loss, it was not possible to reproduce the evolution of HD 166734 in the M-L plane. In order to disentangle the processes of rotation and mass loss and ultimately provide analysis of HD 166734, the effects of rotationally-induced mass loss were omitted. Ek- strom¨ et al. (2012) implement a lowered initial mass-loss rate allowing the inclusion of rotationally-induced mass loss, however a full study on the effects of this process is needed for further clariﬁcation.

Convective core overshooting

The extent of convective overshooting was probed for HD 166734 ﬁnding that an increased αov of 0.5 was required for reproducing the observed luminosity and dynamical mass of the primary, simultaneously. While this value has been adopted in the mass range ∼ 40 M , it was found to be much lower in higher mass stars such as VFTS

527 (R139), where αov = 0.1 was needed due to the already increased Mcc/MT. This suggests a possible inverted mass relation with αov, though requires further analysis of test-case binaries at multiple mass ranges. This has consequences for the larger context

124 CHAPTER 6. CONCLUSIONS AND OUTLOOK of internal mixing, as Claret and Torres (2017) have shown in studies of lower mass binaries that they require higher αov and Castro et al. (2018) has suggested an extended

MS. These tests are in agreement with our increase in αov below 50 M .

The reduced value of αov = 0.1 was also required in the analysis of the upper luminosity limit of RSGs in chapter 5 due to the size of the envelope and development of semiconvective regions. Though αov = 0.1 is needed for the mass range where RSG evolution switches to BSG (e.g. above 50 M for Z ), it is possible that αov = 0.5 could be implemented below this threshold.

The possibility of an extended MS due to αov= 0.5 leads to questions on the nature of B supergiants and whether they are H-burning or He-burning objects. As seen in chapter 4, both O and B supergiants are encompassed in the MS-width for models with

αov= 0.5. Though for reduced αov of 0.1, only O supergiants lie in the MS. Further investigation of the MS-width and possibility of B supergiants as H-burning objects is required for a large sample of observations.

6.1.2 Evolutionary channels

The life-cycle of massive stars is heavily inﬂuenced by many processes as previously discussed, leading to many possible evolutionary channels. However these processes can be studied on the MS where stars spend ∼90% of their life, as seen in chapter 3. This study found that processes on the MS can inﬂuence the compactness parameter of a pre-supernova star, such as the choice of αov, where 0.5 leads to more neutron stars and 0.1 leads to more black holes.

Chapter 5 explored the subsequent ∼10% of a stars life where red and blue supergiant evolution is important. It was found that increased semiconvection leads to more BSGs and increased αov leads to decreased envelope mass where semiconvective regions should form, resulting in more RSGs. It was found that for a uniﬁed set of input parameters, αov= 0.1 and αsemi= 100, that the natural effects of mass loss switch on/off the effects of semiconvection such that RSGs are formed up to log L/ L ∼ 5.5

125 CHAPTER 6. CONCLUSIONS AND OUTLOOK

in the Magellanic Clouds as is observed. Due to higher mass-loss rates at Z , the upper luminosity limit of RSGs was found to be much higher log L/ L ∼ 5.8.

At the maximum luminosity of RSGs, or H-D limit, it is thought that post-MS massive stars switch evolutionary channels from RSGs to WRs. The evolution of WR stars is probed in chapter 6.3 for a range of He-core masses (10-70 M ). These models are to be utilised in implementing a new state-of-the-art mass-loss prescription to study the ﬁnal black hole masses, as a comparison to LIGO detections of GW progenitors. These studies have proven key in our understanding of evolutionary channels and how massive stars may end their lives (e.g. Langer, 2012).

6.1.3 Stellar populations

Massive stars are rare, but highly important for stellar populations. It has been shown that the UV ﬂux from a single VMS (∼ 100 M ) can greatly alter the spectrum of an entire galaxy (e.g. Schneider et al., 2018). This means that further analysis of the upper mass limit and IMF shape are required for better constraining population synthesis models. Similarly, since the evolution of VMSs is hindered by their proximity to the Eddington limit (as discussed in Chapt. 5.1), improved theoretical modelling of envelope inﬂation in these stars is required for reproducing their evolution and ultimately their observations (e.g. R136a). In chapter 4 the most massive binary observed, R139, was reproduced in the M-L plane with estimates of the age and stellar parameters.

In chapter 5, the evolution of massive stars to BSGs and RSGs highlighted a discrepancy between the observed B/R ratio and theoretical B/R ratio since the MS-width is yet to be determined. It was concluded that a better representation of B and R is ﬁrst needed in order to make comparisons with observations and implement in population synthesis models.

126 CHAPTER 6. CONCLUSIONS AND OUTLOOK

6.1.4 Observations

The growth in massive star observations is largely due to the VFTS sample (Evans et al., 2011), which has provided the opportunity for critically improving our theoretical understanding of massive stars. The analysis of these stars has proved fruitful in determining the binarity, rotation proﬁle, and surface properties of massive stars (Sana et al., 2013; Ram´ırez-Agudelo et al., 2013; Grin et al., 2017). These characteristics have been critical in providing the research outlined in this thesis.

Surface abundances as history indicators

Hunter et al. (2008) showed that the surface nitrogen abundance of stars could be correlated with the rotation rates to provide a variety of classes (e.g. fast rotating enriched B supergiants). In chapter 3, this method was extended to utilise the observed enrichment of HD 166734 and particularly the 3:1 ratio of the primary to secondary. This ratio acted as a history indicator for the previous evolution of HD 166734 to constrain properties such as the rotation rate, overshooting region and mass-loss rate. These surface enrichments proved key in concluding the necessity of rotational mixing, and in reproducing the test-bed binary in the M-L plane.

Binarity as a tool for single star evolution

Sana et al. (2013) showed that massive stars in the VFTS sample had a binary fraction of ∼50%, proving the importance of binary evolution in population synthesis models and observational analysis. Though many massive stars evolve in higher multiple systems, many are thought to evolve as single stars due to wide orbital separations. This is the case for HD 166734 and R139. Both detached eclipsing binaries have provided essential knowledge of massive stars at various mass ranges and metallicities, as seen in chapters 3 and 4.

127 CHAPTER 6. CONCLUSIONS AND OUTLOOK

6.1.5 Stellar winds

In chapter 3, the Vink et al. (2001) mass-loss recipe was explored through the M-L plane, ranging from 0.1-3 times this recipe in order to distinguish whether extremely low or high factors of mass loss may reproduce surface abundances or stellar parameters of the test-bed binary HD 166734. Due to the shallow gradient of evolution tracks (2-3 times Vink et al. (2001)) in the M-L plane, it was concluded that extremely high mass-loss factors would result in a lowered luminosity which could not reproduce the observed luminosity. Similarly, extremely low factors of the Vink et al. (2001) recipe resulted in very steep gradient in the M-L plane such that the dynamical mass would not be reached. In order for both dynamical masses and observed luminosities to be reproduced, the mass-loss rates were conﬁned to 0.5-1.5 factors of the Vink et al. (2001) recipe.

6.1.6 Code comparisons

Martins and Palacios (2013) provided a comparison of modern stellar evolution codes finding significant discrepancies in stellar parameters and treatments of internal mixing, leading to uncertainties in our theoretical understanding of massive stars. Sys- tematic differences between spectroscopic analysis and evolution models can lead to long-standing problems such as the ’mass discrepancy’ of massive stars since some codes have systematically higher luminosities than others. However, with the widely- developed code MESA, direct comparisons of implementations are now feasible, allowing for better constraints on physical processes. These include studies of mixing and mass loss as seen in this thesis, where the efficiency of semiconvection can be probed as a function of RSG luminosity (chapter 5), and variations in αov can be explored (chapters 3 and 4).

This thesis provides multiple grids of stellar evolution models as a comparison to observations, for a range of masses (20-60 M ), metallicities ( Z ,ZLMCand ZSMC), ro-

−1 tation rates (0-500 km s ), overshooting αov (0.1-0.5), semiconvective efﬁciencies

128 CHAPTER 6. CONCLUSIONS AND OUTLOOK

(0.1-100), and mass-loss rates (1-3 factors of Vink et al. (2001)).

6.1.7 Summary of conclusions

The key conclusions of this thesis are as followed:

• The novel ’mass- luminosity plane’ has proved a useful tool and comparison to the HRD for testing the evolution of massive stars.

• The effects of rotational mixing were proved necessary in reproducing observed intermediate surface nitrogen enrichments.

• Mass-loss rates were reﬁned to 0.5-1.5 times the Vink et al. (2001) recipe due to the gradient in the M-L plane.

• The evolution of test-case binaries HD166734, VFTS 527, VFTS 116 and VFTS 067 have been reproduced in the M-L plane with estimates of initial stellar parameters and mixing prescriptions.

• These test-case binaries have shown a variation in αov from 0.1-0.5, with 0.5

required for the primary component of HD166734 (∼ 40 M ), while the study

of the HD limit has shown that αov= 0.1 is needed for the upper mass range (∼

50 M ) so that semiconvective regions are able to form in the envelope.

• The study of the HD limit also showed that extremely efﬁcient semiconvective mixing was needed to reproduce the maximum luminosity of RSGs at all Z with

αsemi= 100.

• The indirect effects of mass loss in setting the HD limit have proved critical in switching on/off semiconvective mixing leading to red or blue supergiant evolution.

• With a uniﬁed set of input parameters the maximum luminosity of RSGs has

now been reproduced for the Galaxy (log L/ L ∼ 5.8), and Magellanic Clouds

(log L/ L ∼ 5.5).

129 CHAPTER 6. CONCLUSIONS AND OUTLOOK

• The age of stars can be estimated in the M-L plane with comparison of the observed v sin i and effective temperature position in the evolution track.

• A systematic discrepancy of spectroscopic masses of B supergiants was found in the M-L plane for the VFTS sample.

• A range of model grids have been published for comparison with observations

−1 for initial masses 20-60 M , rotation rates 0-500 km s , αov= 0.1-0.5, and

metallicities Z ,ZLMC,ZSMC. (See Appendix A for Z models.)

6.2 Future work

In this thesis many aspects of massive stars have been explored with improvements in our understanding of internal mixing mechanisms and stellar winds. With observations of detached eclipsing binaries and the large sample of massive stars from VFTS (Evans et al., 2011), better constraints on the evolution of these stars has been possible. Yet these studies have also shown the need for further investigation of the remaining problems within the ﬁeld. While constraining main sequence evolution of 20-60 M stars, it was found that αov may vary from 0.1-0.5, leading to questions about the MS-width and possibility of H-burning B supergiants. In order for this problem to be resolved, more systematic studies of detached eclipsing binaries are required. Fur- thermore, the shape and efﬁciency of convective core overshooting must be scrutinised through asteroseismology to provide better implementation in stellar evolution codes (e.g. Aerts et al., 2010). Similarly, the treatment of convection and implementation in 1D stellar evolution may be further improved through 3D models as seen in Jiang et al. (2015).

Schneider et al. (2018) has shown an excess of massive stars in 30Dor with the most massive stars ∼200-300 M . This suggests that the previously estimated upper mass limit of 150 M (Figer, 2005a) may be surpassed by the cluster R136 and that massive stars may evolve closer to the Eddington limit than previously thought. In order to reproduce the evolution of the most massive stars, the effects of envelope inﬂation must

130 CHAPTER 6. CONCLUSIONS AND OUTLOOK be explored and implemented in stellar evolution codes. Moreover, the IMF shape of galaxies must be studied for a range of galaxies and metallicities in order to test the uniqueness of 30Dor and the upper mass limit.

The VFTS community have presented a wide range of results constraining the evolution of massive stars, though problems remain such as the mass discrepancy of O stars (Weidner and Vink, 2010a), the B/R ratio of galaxies (Chapt. 5), mass-loss rates of pre-SNe RSGs (Beasor and Davies, 2018), and a consensus on 1D stellar evolution codes (Martins and Palacios, 2013). These problems may be resolved with additional observational samples and advancements in theory. The Hubble Space Telescope’s Ultraviolet Legacy Library of Young Stars as Essential Standards (ULLYSES) will provide observations of hundreds of massive stars in our local universe, mainly within our Galaxy and the nearby Magellanic Clouds, and will provide the necessary additional observations to test our theoretical models. This legacy program will provide constraints on star formation and evolution, as well as providing the necessary links between evolutionary channels (e.g. towards WRs and BHs), and compare massive star populations with that of 30Dor.

Studies of heavy black hole progenitors have become increasingly important with the development of the LIGO project which detects GW events through black hole mergers. The detections of GWs will provide estimates of the upper masses of stellar mass black holes at varied Z, aiding our constraints on the evolutionary channels which lead to these heavy BHs. Similarly, the statistics of these BH masses provide information on the stellar populations which evolve to BHs. The Advanced LIGO and Advanced Virgo detectors began their third observing run (O3) in April 2019, with hundreds of detections expected, providing a large sample of BH and neutron star masses to con- ﬁne evolutionary channels of massive stars. Improved stellar atmosphere modelling by Sander et al. (2020) has provided new mass-loss rates for black hole progenitors in order to better estimate the ﬁnal masses of these stars and ultimately reproduce the evolution of GW progenitors (see Sect. 6.3).

131 CHAPTER 6. CONCLUSIONS AND OUTLOOK

The key questions remaining in massive star evolution include the final stages of evolution from WR to BH, the final masses of massive stars - in particular their mass-loss rates, and the efficiencies of internal mixing processes. These topics are critical in our understanding of GW and BH progenitors, as well as linking SNe to their progenitors, and in particular the extent of the influence of massive stars on their environment through stellar winds.

6.3 Wolf-Rayet mass-loss rates to black hole

progenitors

In advancing the work conducted in this thesis further to investigate the change in evolutionary channels in the post-MS from RSGs to BSGs above the HD limit, a col- laborative project has been developed to calculate the evolution towards Wolf-Rayet (WR) stars.

Evolved massive stars can reach the hydrogen-free WR stage during core He-burning, where they produce strong stellar winds. Although these stars eventually collapse into heavy black holes (BHs) like those of the LIGO GW sample, we cannot reliably predict their ﬁnal masses since their mass-loss rates are severely uncertain. Higgins et al. (in prep), have developed a project in collaboration with Raphael Hirschi and Jorick Vink which aims to resolve this issue. As a leading expert in stellar atmosphere modelling, A. Sander has composed a set of detailed, comoving-frame radiative transfer models with consistent hydrodynamics for Wolf-Rayet stars (WRs) for a range of masses at different metallicities. This project will provide for the ﬁrst time, a set of locally consistent models with an unprecedented level of detail allowing us to obtain self- consistent mass-loss rates from a given set of WR star parameters.

Figure 6.1 shows the theoretical WR mass-loss rates calculated by A. Sander resulting from the self-consistent PoWR atmosphere models for Z . Implementing mass- loss recipes derived from such models will allow for producing the most accurately

132 CHAPTER 6. CONCLUSIONS AND OUTLOOK

M [M ] 7 8 10 12 15 20 ∗30 50 100 200 400

log M˙ = 12.14 + 1.31 log L −

log M˙ = 39.24 + 6.53 log L 3 − − log M˙ = 13.30 + 1.36 log L −

log M˙ = 4.93 + 4.91 log(log L 4.53) 4 − · − − ]) 1 −

yr 5

− M [ ˙

M 6 − log( 7 −

8 −

4.5 5.0 5.5 6.0 6.5 7.0 7.5 log(L [L ])

Figure 6.1: Mass-loss recipe for WR models at Z with the stellar atmosphere code PoWR, for masses 7-400 M . constrained WR models to date, vital for comparisons with LIGO detections. As a ﬁrst outlook test case, I have written a new subroutine in the stellar evolution code

MESA, implementing the new mass-loss rates for WR models at Z . In the future, this study will be repeated for lower metallicity environments and compared to observations.

6.3.1 Milky Way Wolf-Rayet models

The initial WR models were set up from the pre-MS stage of evolution as seen in Fig. 6.2. These models were set up with multiple inlists for each stage of evolution to change the H-burning core to a mixed He-core by increasing Y to 0.96, see Fig. 6.3. With convergence problems in the higher mass models of these tests (∼

30-40 M ) another set of WR models were tested. In order to create a He-core the

αov region was increased to 0.9 to encompass the entire mass of the star. Figure 6.4 illustrates the evolution of a 20 M model in a HRD with core He abundance shown in

133 CHAPTER 6. CONCLUSIONS AND OUTLOOK

Figure 6.2: The pre-MS evolution of WR models for initial masses 20-30 M . the colour bar. The development of these models will continue for higher masses and lower metallicities, ultimately incorporating the mass-loss recipe mentioned previously in Figure 6.1.

The future of massive star evolution requires a comprehensive analysis of the most massive stars with state-of-the-art stellar evolution codes (MESA, GENEC). I will provide a detailed grid of stellar models with a wide range of masses and metallicities, implementing next-generation mass-loss recipes and internal mixing prescriptions. These models will be crucial in providing the necessary theoretical comparisons to the ﬁrst uniform UV spectroscopic survey of massive stars, ULLYSES (the Hubble UV Legacy programme). Moreover, current efforts in predicting BH masses and linking these to progenitor evolutionary channels are arguably futile without a fundamental understanding of the very stars which lead to their existence. I will apply my grid of

134 CHAPTER 6. CONCLUSIONS AND OUTLOOK

Figure 6.3: Structure diagram or Kippenhahn diagram of a 20 M WR model with time. The Helium abundance is shown by the colour bar and green crossed lines represent the convective core.

Figure 6.4: The evolution of a 20 M WR star in a HRD with the core He abundance shown in colour (red Y=0.96, blue Y=0.1).

135 CHAPTER 6. CONCLUSIONS AND OUTLOOK models in predicting the ﬁnal stellar mass at the highest mass range and compare with LIGO detections to classify BH progenitors.

The gravitational wave era is upon us before our understanding of massive star evolution has progressed to a state where we can classify the progenitors. While estimates of the BH masses have been predicted, we cannot yet establish how massive their progenitor stars were initially. The LIGO Collaboration predicts weekly detections of GW sources, though the statistics will not aid our understanding of the progenitors since we need further studies on VMSs and their dominant processes. From Chapt. 5 we ﬁnd that semiconvection and mass loss play an important role and may be critical at the upper mass limit. In order to better prepare for interpreting GW detections, we must examine the evolution of their progenitors. ULLYSES data will also provide a large sample of massive star observations to aid constraints on the evolution at the most opportune time.

136 Appendix A

Galactic grid of models

A calibrated grid of stellar evolution models is provided for a range of initial masses and rotation rates, for Z , as part of the research outlined in chapter 3. Two extreme values of αov = 0.1 and 0.5 are shown in sold and dashed lines respectively, since analysis from HD 166734 suggests increased overshooting of αov= 0.5, while studies of the upper mass range suggest overshooting of αov= 0.1.

137 APPENDIX A. GALACTIC GRID OF MODELS

Figure A.1: Grid results for the non-rotating models in the mass range 8 -60 M employing αov = 0.1 (solid lines) and αov = 0.5 (dashed lines).

138 APPENDIX A. GALACTIC GRID OF MODELS

Figure A.2: Grid results for the rotating models with initial rotation rates of 100 km s−1 for the mass range as mentioned in Fig. A.2, αov = 0.1 (solid lines) and αov = 0.5 (dashed lines).

139 APPENDIX A. GALACTIC GRID OF MODELS

Figure A.3: Grid results for the rotating models with initial rotation rates of 200 km s−1 for the mass range as mentioned in Fig. A.2, αov = 0.1 (solid lines) and αov = 0.5 (dashed lines).

140 APPENDIX A. GALACTIC GRID OF MODELS

Figure A.4: Grid results for the rotating models with initial rotation rates of 300 km s−1 for the mass range as mentioned in Fig. A.2, αov = 0.1 (solid lines) and αov = 0.5 (dashed lines).

141 APPENDIX A. GALACTIC GRID OF MODELS

Figure A.5: Grid results for the rotating models with initial rotation rates of 400 km s−1 for the mass range as mentioned in Fig. A.2, αov = 0.1 (solid lines) and αov = 0.5 (dashed lines).

142 APPENDIX A. GALACTIC GRID OF MODELS

Figure A.6: Grid results for the rotating models with initial rotation rates of 500 km s−1 for the mass range as mentioned in Fig. A.2, αov = 0.1 (solid lines) and αov = 0.5 (dashed lines).

143 Appendix B

MESA v8845 inlist

−1 The following inlist was used to create a 40 M model with vinit = 100 km s , and

αov= 0.1 for tests in chapter 3.

&star_job

show_log_description_at_start = .true.

eos_file_prefix = ’mesa’

kappa_file_prefix = ’gs98’

set_initial_age = .true. ! begin without pre-MS

initial_age = 0 ! starting model age in years

create_pre_main_sequence_model = .false. ! begin with a pre-main sequence model

pgstar_flag = .true. ! display on-screen plots

pause_before_terminate = .true. ! allow plots to remain on screen

!------ROTATION------!

new_rotation_flag = .true.

change_rotation_flag = .true.

change_initial_rotation_flag = .true.

new_surface_rotation_v = 100 ! in km/s

set_surface_rotation_v = .true.

set_initial_surface_rotation_v = .true.

change_v_flag = .true.

change_initial_v_flag = .true.

new_v_flag = .true.

/ ! end of star_job namelist

&controls

144 APPENDIX B. MESA V8845 INLIST

!------BASIC PARAMETERS------!

initial_mass = 40 ! Mass in Msun units

history_interval = 1

profile_interval = 20

xa_central_lower_limit(1) = 1E-4

xa_central_lower_limit_species(1) = ’he4’

mesh_delta_coeff = 1.5 ! Larger values increase the max deltas,decreases the no. of grid points

mesh_delta_coeff_for_highT = 2.5 ! ˆ for high T

varcontrol_target = 1d-3 ! Target value for relative variation in the structure between models

use_Type2_opacities = .true. ! Default opacities for Massive stars

Zbase = 0.02 ! Base Metallicity: Galacticz=0.014, SMCz=0.004, LMC/Bonn=0.0088

!------MIXING PARAMETERS------!

mixing_length_alpha = 1.5 ! Geneva use 1.6 <40 Msol, and 1.0 >40 Msol. Bonn/MESA default use 1.5

MLT_option = ’Henyey’ ! Options: Cox, ML1, Ml2, Mihalas, Henyey, none

okay_to_reduce_gradT_excess = .false. ! MLT++ on=true/off=false

gradT_excess_lambda1 = -1.0 ! Full MLT++ on

use_Ledoux_criterion = .true. ! Schwarzschild criterion if false

alpha_semiconvection = 1 ! Determines efficiency of semiconvective mixing

!------OVERSHOOTING PARAMETERS------!

overshoot_f0_above_burn_h_core = 0.005 ! overshoot distance in the expense of the core: MS

overshoot_f0_above_burn_h_shell = 0.005

overshoot_f0_below_burn_h_shell = 0.005

step_overshoot_D0_coeff = 1 ! Diffusion coefficient D at point r0

step_overshoot_f_above_burn_h_core = 0.1 ! Step overshooting values:

step_overshoot_f_above_burn_h_shell = 0.1 ! alpha_ov= 0.335 -> Bonn Model

step_overshoot_f_below_burn_h_shell = 0.1 ! alpha_ov= 0.1 -> Geneva Model

!------MASS LOSS------!

hot_wind_scheme = ’Vink’!Vink et al 2001 treatment of mass loss

hot_wind_full_on_T = 1.2d4 !T limits

hot_wind_full_off_T = 1.0d4 !T limits

Vink_scaling_factor = 1d0 !"Mass-loss predictions for O and B stars as a function of metallicity"

mdot_omega_power = 0d0

!------ROTATION------!

! Chemical Mixing ! 1 -> on, 0 -> off

D_DSI_factor = 1 ! dynamical shear instability

D_SH_factor = 0 ! Solberg-Hoiland

D_SSI_factor = 1 ! secular shear instability

145 APPENDIX B. MESA V8845 INLIST

D_ES_factor = 1 ! Eddington-Sweet circulation

D_GSF_factor = 1 ! Goldreich-Schubert-Fricke

D_ST_factor = 0 ! Spruit-Tayler dynamo

! this is for ang.mom. transport

am_nu_DSI_factor = 1

am_nu_SH_factor = 0

am_nu_SSI_factor = 1

am_nu_ES_factor = 1

am_nu_GSF_factor = 1

am_nu_ST_factor = 0

am_gradmu_factor = 0.1d0 ! f_mu from Brott et al

am_nu_factor = 1d0 ! this factor accounts for angular momentum transfer

am_D_mix_factor = 0.0333333333333333d00

/ ! End of controls namelist

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