Florida State University Libraries

Home , AB Doradus, Alpha Doradus, AS 314, Beta Doradus, Brightest Blue, Delta Doradus

Electronic Theses, Treatises and Dissertations The Graduate School

Constraining the Evolution of Massive StarsMojgan Aghakhanloo

Follow this and additional works at the DigiNole: FSU's Digital Repository. For more information, please contact [email protected] FLORIDA STATE UNIVERSITY

COLLEGE OF ARTS AND SCIENCES

CONSTRAINING THE EVOLUTION OF MASSIVE STARS

MOJGAN AGHAKHANLOO

A Dissertation submitted to the Department of Physics in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy

2020

Jeremiah Murphy Professor Directing Dissertation

Munir Humayun University Representative

Kevin Huﬀenberger Committee Member

Eric Hsiao Committee Member

Harrison Prosper Committee Member

The Graduate School has veriﬁed and approved the above-named committee members, and certiﬁes that the dissertation has been approved in accordance with university requirements.

ii I dedicate this thesis to my parents for their love and encouragement. I would not have made it this far without you.

iii ACKNOWLEDGMENTS

I would like to thank my advisor, Professor Jeremiah Murphy. I could not go through this journey without your endless support and guidance. I am very grateful for your scientiﬁc advice and knowledge and many insightful discussions that we had during these past six years. Thank you for making such a positive impact on my life. I would like to thank my PhD committee members, Professors Eric Hsiao, Kevin Huf- fenberger, Munir Humayun and Harrison Prosper. I will always cherish your guidance, encouragement and support. I would also like to thank all of my collaborators. In particular, many thanks to Professors Nathan Smith and Keivan Stassun for providing amazing opportunities throughout my graduate education. This dissertation would not be possible without your contribution. I would like to thank the staﬀ in the Physics Department, Jonathan Henry, Brian Wilcoxon, Dr. Felicia Youngblood, Shawana Elwood and Joe Ryan for all their help throughout these years. I have built up many friendships along the way and would not be where I am today without their love. In particular, a great thanks to Dr. Pegah Nasabian and Dr. Soheila Abrishami and Luz Jimenez Vela who have helped me with their valuable suggestions and motivated me during these years. Last but not least, many thanks to my lovely parents, Nosrat and Maloos and my partner Ramu for their constant support through every single challenge of my life. Thank you for encouraging me in all of my pursuits and inspiring me to follow my dreams. Words alone cannot express my gratitude.

iv TABLE OF CONTENTS

List of Tables ...... vi List of Figures ...... vii Abstract ...... ix

1 Background 1 1.1 Luminous Blue Variables ...... 1 1.2 Massive Stars and the Role of Binary Interaction ...... 6 1.3 Precise Distances to LBVs ...... 6 1.4 Westerlund 1 Cluster ...... 9

2 Modelling Luminous-Blue-Variable Isolation 12 2.1 Observations ...... 12 2.2 A Generic Model for the Spatial Distribution of the Stars in a Passive Dispersal Cluster ...... 16 2.3 Cluster Dissolution with Close Binary Interactions ...... 25

3 On the Gaia DR2 Distances for Galactic Luminous Blue Variables 38 3.1 Gaia Spacecraft ...... 38 3.2 Distances for LBVs in Gaia DR2 ...... 39 3.3 Notes on Individual LBVs ...... 57 3.4 Discussion ...... 68

4 Inferring the Parallax of Westerlund 1 from Gaia DR2 79 4.1 Method ...... 79 4.2 Parallax and Distance to Westerlund 1 ...... 92 4.3 Discussion ...... 93

5 Conclusions 104

Bibliography ...... 107 Biographical Sketch ...... 130

v LIST OF TABLES

2.1 List of LBVs and LBV candidates adapted from Smith and Tombleson [2015]. 14

2.2 The P -values for KS tests for the distributions of separation...... 16

3.1 Previous literature distances for Galactic LBVs and candidate LBVs (in parentheses)...... 42

3.2 Parameters from the Gaia DR2 and Bailer-Jones catalogs...... 45

3.3 LBVs and LBV candidate (in parentheses) Gaia DR2 distances...... 46

vi LIST OF FIGURES

2.1 Cumulative distributions for the projected separation to the nearest O star. . 17

2.2 Normality test...... 18

2.3 We propose a Monte Carlo model for the separations between O stars and LBVs by considering a random sample of dissolving clusters at random ages. . . . . 22

2.4 The mean (bottom panel) and std. deviation (top panel) distance to the nearest neighbour versus drift velocity...... 24

2.5 Two simple spatial-distribution models for the derivation of our analytic scalings. 27

2.6 LBV isolation is inconsistent with the single-star model and passive dissolution of the cluster...... 31

2.7 The relative isolation of LBVs is consistent with a binary merger in which the LBV is a rejuvenated star...... 33

2.8 Merger model outline...... 34

2.9 Kick model outline...... 34

2.10 LBV dispersion velocity as a function of LBV age ...... 36

3.1 Distances by Bayesian inference, dBayes (Table 3.3) compared to: (top) literature distances, dLit, (Table 3.3), (middle) distances given by 1/$ (Table 3.2), and (bottom) the Bailer-Jones Bayesian distances, dBJ [Bailer-Jones et al., 2018] (Table 3.2)...... 52

3.2 The HR diagram showing only Galactic LBVs (ﬁlled circles) and Galactic LBV candidates (unﬁlled circles) with their luminosities scaled by the revised Gaia DR2 distances (dBayes)...... 55 3.3 Same as Fig. 3.2, but including positions of LBVs based on both the old literature distances (red) and those inferred from Gaia DR2 distances (black). . . 56

3.4 An HR diagram similar to Fig. 3.5, but showing only W243 based on the old distance of around 5 kpc (red) and using a distance of 3.2 0.4 kpc (black) based on the distance inferred for the whole Wd1 cluster from DR2± data [Aghakhanloo et al., 2020]...... 66

vii 3.5 The HR diagram with LBVs (filled circles) and LBV candidates (unfilled circles), adapted from a similar figure in Smith and Tombleson [2015] and Smith and Stassun [2017]...... 78

4.1 Position of all Gaia stars within 10 arcmin of Westerlund 1...... 80

4.2 Histograms of parallaxes (top left panel), parallax uncertainties (top right panel), astrometric excess noise (, bottom left panel), and astrometric excess noise signiﬁcance (D, bottom right panel) of all stars in the inner circle and the outer annulus...... 82

4.3 Histogram of parallax over the uncertainty of all stars in the inner circle and the outer annulus...... 83

4.4 Estimates for the true mean parallax for each ring...... 84

4.5 Probabilistic graphical model for the Bayesian model...... 87

4.6 Posterior distribution for the six-parameter model...... 94

4.7 Bayesian inferred cluster parallax for each ring...... 95

4.8 Same as Fig. 4.6, but we assume that the oﬀset for the ﬁeld-star distribution, $os is equal to the instrumental zero-point, $zp...... 96 4.9 Histogram of scale factor x for all sources within 1 arcmin of centre of the cluster. 98

4.10 The HR diagram for evolved stars in Westerlund 1, including the LBV, W243. 102

viii ABSTRACT

Massive stars play a crucial role in the universe. Yet, our understanding of massive stars remains incomplete due to their rarity, short lifetimes, complexity of binary interactions, and imprecise Galactic distances. An important challenge is to understand the physics and relative importance of steady and eruptive mass loss in the most massive stars. For example, the luminous blue variable (LBV) is one such poorly constrained class of eruptive stars. LBVs are the brightest blue irregular variable stars in any large star-forming galaxy. They can achieve the highest mass-loss rates of any known types of stars, and they exhibit a wide diversity of irregular and eruptive variability. In the single-star scenario, the hypothesis is that most stars above 30 solar mass pass through an LBV phase. However, the relative ∼ isolation of LBVs from O stars challenges this interpretation, and another hypothesis is emerging that the LBV phenomenon is the product of binary evolution. To test these hypotheses, we modeled the dissolution of young clusters and the separation between O stars and LBVs. We find that the single-star scenario is inconsistent with the observed LBV environments. If LBVs are single stars, then the lifetimes inferred from their luminosity and mass are far too short to be consistent with their isolation from O stars. This implies that LBVs are likely products of binary evolution. To further constrain these hypotheses, we must first infer the fundamental properties of LBVs such as luminosity, mass, and age. Ultimately, these depend upon accurate Galactic distances. Using Gaia parallaxes, we find that nearly half of the Galactic LBVs are significantly closer than previous literature estimates; these new distances lower their luminosities and their initial masses. We also infer a closer distance to the massive cluster, Westerlund 1, which hosts an LBV, 24 Wolf- Rayet stars, 6 yellow hypergiants, and a magnetar. Together these Gaia-based distances are more accurate (at least a factor of ten) and have consequences for the late-stage evolution of massive stars.

ix CHAPTER 1

BACKGROUND

Massive stars dramatically impact the evolution of galaxies. The high photon flux of massive stars ionize galaxies and the Universe. Their winds and explosions influence the gas dynamics of entire galaxies, and their explosive deaths produce many of the elements in the universe heavier than hydrogen or helium. Therefore, it is important to understand their evolution. Stellar evolution theory predicts that massive stars, evolve rapidly through many dynamic phases, and how this evolution proceeds affects the massive star’s fate. In particular, this evolution can affect whether or not they explode. Stellar mass is one of the primary characteristics that determine a star’s evolution and fate [Woosley and Heger, 2015]. Yet, understanding the physics and relative importance of steady and eruptive mass-loss in the most massive stars remains a major challenge in the stellar evolution theory. There has been substantial progress in understanding mass-loss via steady line-driven winds of hot stars [Kudritzki and Puls, 2000, Puls et al., 2008], and this effect is included in the stellar evolution models [Vink et al., 2001, Woosley et al., 2002, Meynet and Maeder, 2005, Martins and Palacios, 2013]. However, the mass-loss rates of red supergiants (RSGs) and the role of eruptive mass-loss remain unclear, and the influence on stellar evolution remains uncertain [Smith and Owocki, 2006, Smith, 2014]. The luminous blue variable (LBV) is one such poorly constrained class of eruptive stars.

1.1 Luminous Blue Variables

Luminous blue variables (LBVs) are the brightest blue irregular variable stars in any large star-forming galaxy [Hubble and Sandage, 1953, Tammann and Sandage, 1968]. LBVs were originally known as the “Hubble-Sandage variables” but later [Conti, 1984a] related them to the famous Galactic objects like P Cygni and η Carinae and grouped them as “LBVs”. In the single-star scenario, stars above 30 solar mass pass through an LBV phase. Their ∼ 1 spectra were found to contain lines with P Cygni proﬁles and also show dramatic variations due to the photometric variations [Humphreys and Davidson, 1994]. LBVs can achieve the highest mass-loss rates of any known types of stars, and they exhibit a wide diversity of irregular and eruptive variability [Conti, 1984b, Humphreys and Davidson, 1994, van Genderen, 2001b, Clark et al., 2005, Smith et al., 2004, Smith and Owocki, 2006, Smith et al., 2011b, Smith, 2014]. Yet despite decades of study, the physical mechanism that causes their variability remains unknown. An important corollary is that there are stars that occupy the same parts of the Hertzsprung-Russell (HR) diagram that are not (currently) susceptible to the same instability. The degree to which a star in this region of the Hertzsprung-Russell (HR) diagram is unstable may depend on its initial mass, its age, its history of mass loss (and hence, its metallicity), and past binary interaction. Part of the reason that LBVs are poorly understood is that there are few examples. There are 25 in our Galaxy, 19 known in the nearest galaxies, the Large Magellanic Cloud (LMC) and Small Magellanic Cloud [SMC; Smith and Tombleson, 2015, Smith et al., 2019] and 34 in M31 and M33 [Humphreys et al., 2017]. Even this small sample includes ‘candidate’ LBVs (see below). Classifying various stars as LBVs or candidates can be somewhat controversial [Humphreys and Davidson, 1994, Weis, 2003, Vink, 2012]. Stars that resemble LBVs in their physical properties and spectra, but lack the tell-tale variability, are usually called ‘LBV candidates’. The reason they are sometimes grouped together is that it is suspected that the LBV instability may be intermittent, so that candidates are temporarily dormant LBVs [Smith et al., 2011a, Smith, 2014]. LBVs are relatively rare and still somewhat mysterious type of stars. But know matter what they are, they are collection of stars that exhibit wild variations in both their brightness and spectrum and our goal is to understand them as a group.

1.1.1 LBV Eruptions

Although the signature eruptive variability of LBVs was identiﬁed long ago, the physical theory of LBV eruptions is not yet clear. For the most part, LBVs seem to experience two

2 classes of eruptions: S Doradus (or S Dor) eruptions (1–2 magnitude increase in the visual brightness) and giant eruptions ( 2 magnitudes). ≥ S Doradus variables take their namesake from the prototypical LBV S Doradus [van Genderen, 2001a]. During S Dor outbursts, LBVs make transitions in the HR diagram (HRD) from their normal, hot quiescent state to lower temperatures (going from blue to red). In its quiescent state, an LBV has the spectrum of a B-type supergiant or a late Of-type/WN star [Walborn, 1977, Bohannan and Walborn, 1989]. In this state, LBVs are fainter (at visual wavelengths) and blue with temperatures in the range of 12000–30000 K [Humphreys and Davidson, 1994]. In their maximum visible state, their spectrum resembles an F-type supergiant with a relatively constant temperature of 8000 K. S Dor events were ∼ originally proposed to occur at constant bolometric luminosity [Humphreys and Davidson, 1994]. So a change in temperature implies a change in the photospheric radius, L = 4πσR2T 4. Humphreys and Davidson [1994] suggested that the eruption is so optically thick that a pseudo-photosphere forms in the wind or eruption. However, quantitative estimates of mass- loss rates show that they are too low to form a large enough pseudo-photosphere [de Koter et al., 1996, Groh et al., 2009b]. Similar studies also imply that the bolometric luminosity is not strictly constant [Groh et al., 2009b]. Instead, it has been suggested that the observed radius change of the photosphere can be a pulsation or envelope inflation driven by the Fe opacity bump [Gräfeneret al., 2012a]. The other distinguishing type of variability is in the form of giant eruptions like the 19th century eruption of η Car [Smith et al., 2011a]. The basic difference from S Dor events is that giant eruptions show a strong increase in the bolometric luminosity and are major eruptive mass-loss events, whereas S Dor eruptions occur at roughly constant luminosity and are not major mass-loss events. The mass-loss rate at S Dor maximum is of the order of

−4 −1 10 M yr or less [Wolf, 1989a, Groh et al., 2009b]. On the other hand, giant eruption

−1 −1 mass-loss rate is of the order of 10 –1 M yr [Owocki et al., 2004a, Smith and Owocki, 2006, Smith, 2014]. It is unlikely that a normal line-driven stellar wind is responsible for the giant eruptions because the material is highly dense and optically thick [Owocki et al., 2004a, Smith and

3 Owocki, 2006]. Instead, giant eruptions must be continuum-driven super-Eddington winds or hydrodynamic explosions [Smith and Owocki, 2006]. Both of these lack an explanation of the underlying trigger; the super-Eddington wind relies upon an unexplained increase in the star’s bolometric luminosity, whereas the explosive nature of giant eruptions would require signiﬁcant energy deposition. Signiﬁcant energy may be deposited in a star’s envelope by waves generated by convective motions deep in the core [Meakin and Arnett, 2007, Quataert and Shiode, 2012, Shiode et al., 2013], stellar collisions or mergers [Podsiadlowski et al., 2010, Smith, 2011b, Smith and Arnett, 2014], unsteady burning [Smith and Arnett, 2014], the pulsational pair instability [Woosley et al., 2002, 2007], or explosive shell burning instabilities [Dessart et al., 2010, Smith and Arnett, 2014].

1.1.2 Isolated Star Model of LBVs

The traditional view has been that most stars above 25-30 M pass through an LBV phase in transition from core H burning to He burning. In this brief phase, they experience eruptive mass-loss as a means to transition from a hydrogen-rich star to an H-poor Wolf– Rayet (WR) star [Humphreys and Davidson, 1994]. If LBVs mark a brief transitional phase at the end of the main sequence and before core-He burning WR stars, then they should be found near other massive O-type stars. However, Smith and Tombleson [2015] found that LBVs are quite isolated from O-type stars, and even farther away from O stars than the WR stars are. Given their isolation, Smith and Tombleson [2015] concluded that the LBV phenomenon is inconsistent with a single-star scenario and is most consistent with binary scenarios. The ﬁrst goal of this thesis is to develop models for the passive dispersal of clusters to see whether these scenarios are in good agreement with observations.

1.1.3 Luminous Blue Variables as Supernova (SN) Progenitors

If LBVs are single stars that are in the transition to the core He burning phase, then, can’t explode as core-collapse SNe. However, there was already earlier evidence that the simple LBV-to-WR-to-SN mapping is not entirely accurate [Smith et al., 2007a, 2008]. For

4 example, Kotak and Vink [2006] proposed an LBV and SN connection. Kotak and Vink [2006] suggested that modulations in the radio light curve of type IIb SN 2003bg and type Ic SN 1998bw reflected variations in the mass-loss rate similar to S Dor variations. In other cases, some Type IIn SNe may have LBV-like progenitors based on pre-SN mass-loss properties (mass, speed, H composition). For example, Ofek et al. [2013] reported a pre-supernova outburst 40 d before the Type IIn supernova SN 2010mc. Even though the progenitor of SN 2010mc was not directly identified as an LBV such an outburst is consistent with rare giant eruptions of LBVs. However, there has never been a direct connection between LBVs and Type IIn SNe. Instead, the connection is circumstantial in that narrow lines of Type IIn imply significant mass-loss from the progenitor, and even when the progenitor has been observed to vary, there are generally not enough observations to definitively classify a progenitor as an LBV. On the other hand, the isolation of directly identified LBVs provides a stronger constraint on their evolution [Smith and Tombleson, 2015].

1.1.4 Supernova Impostors

Extragalactic LBV giant eruptions are sometimes called “Supernova impostors”[Van Dyk et al., 2000]. These stars have been observed to undergo major eruptions like SN explosions. For example, SN 2009ip is classiﬁed as a SN impostor [Berger et al., 2009a, Miller et al., 2009, Li et al., 2009]. In 2009, SN 2009ip was discovered as an LBV-like outburst. But the ﬁrst 2009 outburst was very brief [a few days; Smith et al., 2010] compared to other LBV-like eruptions

[a few months or more; Smith, 2011a]. The progenitor was a very massive star 50-80 M ∼ [Smith et al., 2010, Foley et al., 2011] that experienced a series of LBV-like eruptions [Smith et al., 2010, Mauerhan et al., 2013, Pastorello et al., 2013]. But ﬁnally exploded as a SN IIn [Mauerhan et al., 2013, Smith et al., 2014]. SN 2009ip may be an example that LBV-like stars are linked to SNe IIn. But, give all of the tentative connections, it is still uncertain that SN impostors are associate with LBVs.

5 1.2 Massive Stars and the Role of Binary Interaction

A majority (50-70%) of massive stars are in interacting binary systems [Sana et al., 2012]. Therefore, the inﬂuence of binarity on massive stars should be considered in order to further understand their evolution. For example, binary evolution may explain the observed populations of WR stars and SNe Ibc. Smith et al. [2011b] found that the observed fraction of stripped envelope SNe is too high to be consistent with single-star evolution. They also found that if SN Ibc progenitors are assumed to be WR stars, then the observed fraction of

SNe Ibc would require progenitor masses around 22 M which is much lower than what is expected from single-star evolution models. The duration of the LBV phase may also be longer due to the binarity since it varies with initial mass. Therefore, binariy may explain the isolation of LBVs from O-type stars. One of the reasons that we are unable to constrain whether LBVs are product of binary evolution is due to their inaccurate initial masses and ages. To further constrain these hypotheses, we must infer the fundamental properties of LBVs such as luminosity, mass, and age which depends upon accurate distances.

1.3 Precise Distances to LBVs

For any class of stars, distances and true bolometric luminosities are important for understanding their physical nature. This is particularly true for LBVs, since their deﬁning instability, mass loss, and evolutionary state are thought to be a consequence of their high luminosity [Conti, 1984b, Humphreys and Davidson, 1994, Lamers and Fitzpatrick, 1988, Ul- mer and Fitzpatrick, 1998, Owocki et al., 2004b]. Precise distances are critical for inferring whether a star is in close proximity to the classical Eddington limit based on its position on the HR diagram compared to stellar evolution model tracks. Many LBVs seem to skirt the observed upper luminosity boundary on the HR diagram, the Humphreys-Davidson (HD) limit, oscillating between their hot quiescent states and cooler eruptive states when they cross that observational limit [Humphreys and Davidson, 1994]. Based on distances, luminosities, and temperatures estimated for a few LBVs in the Milky Way and nearby galaxies,

6 Wolf [1989b] proposed that LBVs in their hot states reside along the S Doradus instability strip on the HR Diagram. This zone in the HR Diagram is thought to be an important clue to their instability, perhaps related to the Eddington limit modified by opacity [Lamers and Fitzpatrick, 1988, Ulmer and Fitzpatrick, 1998] and rotation [Groh et al., 2009a, Langer et al., 1999]. Extragalactic LBVs in the Large or Small Magellanic Clouds (LMC/SMC) and in the nearby spirals M31/M33 have reliable distance estimates, and hence, fairly reliable estimates of their bolometric luminosity if detailed quantitative analysis has been used to estimate their bolometric corrections. In distant environments, however, we may be missing faint LBVs, if they exist, either due to selection effects (low amplitude variability, for example) or because they have not received as much observational attention as the most luminous stars. Noticing that a star is actually an LBV is often the result of detailed analysis and long-term monitoring; a typical S Doradus cycle of an LBV may last a decade. Moreover, difficulties associated with contaminating light from neighboring stars become more problematic for extragalactic LBVs. Distances to LBVs cannot be determined solely by detailed spectroscopic analysis of an individual star, because the relationship between the spectrum and absolute luminosity is ambiguous [Najarro et al., 1997, Hillier et al., 1998, Groh et al., 2009c]. The problem is that the emission-line spectra of LBVs can be dominated by wind emission, which depends on density and ionization of the outflow, not the absolute luminosity of the star or its surface gravity. Stars with dense winds can have very similar spectra across a wide range of luminosity. For instance, Groh et al. [2013] have shown that spectral models of an evolved

20-25 M star that is moving to the blue part of the HR diagram after experiencing high mass loss in the RSG phase can have a spectrum that closely resembles a vastly more luminous classical LBV like AG Car. As such, other types of evolved stars at lower mass that have eﬀective temperatures similar to LBVs and dense winds or disks like a B[e] star, post-AGB star, or various types of interacting binaries can have similar emission-line spectra that may masquerade as LBVs. These can be mistaken for more luminous LBVs if assumed to be at a distance that is too far, and vice versa. We will see below that this is indeed the case for a few

7 objects that have been considered LBVs or LBV candidates. In addition to the luminosity, other stellar parameters derived from spectroscopic analysis also depend on the assumed distance d. The stellar radius depends on d (relevant for e.g. binary interaction), and mass-loss rates scale as d1.5. This inﬂuences our interpretation of the mass-loss history, fate, circumstellar material properties in interacting supernovae (SNe), etc. Other properties like

the effective temperature Teff and the terminal wind speed v∞ have a negligible dependence on the distance (see Groh et al. 2009c). The second goal of this thesis is to infer distances to Galactic LBVs using Gaia DR2. Of course, star clusters have been a useful tool for estimating distances and ages for many classes of stars. A significant impediment to determining LBV distances by this method, however, is that many LBVs are not associated with clusters, counter to expectations for the origin of LBVs in single-star evolution (e.g., Lamers and Nugis 2002). Smith and Tombleson [2015] showed that LBVs are isolated from clusters of O-type stars in general, whereas the few that are in clusters seem to be too young (or overluminous) for their environment. These inferences about age and environments of LBVs were based on stars in the LMC, where the distance is reliable. Such environmental comparisons are harder in the Milky Way because of distance ambiguities and extinction. For this reason, Smith and Tombleson [2015] did not undertake a quantitative assessment of LBV isolation for Milky Way LBVs (although they did note anecdotal evidence that Galactic LBVs do appear remarkably isolated as well). A Milky Way star seen near other O-type stars on the sky might be at a different distance but seen nearby in projection, or alternatively, a lack of O stars in the vicinity might be a selection effect (LBVs are very bright at visual wavelengths, but hotter and visually fainter main- sequence O-type stars might be dim and possibly undetected because of extinction in the Galactic plane). These complications make it difficult to know if the statistical environments of LBVs in the LMC also apply in the Milky Way, where the metallicity sensitivity of the LBV instability might differ. Similarly, the lack of reliable distances for Milky Way LBVs has hampered our understanding of their true physical parameters, especially their bolometric luminosities. Since most LBVs are not associated with young clusters of O-type stars, many

8 of them have very uncertain distances in the literature (see Section 2.2), and similarly, highly uncertain ages and initial masses. Of the dozen or so LBVs in the Milky Way [Clark et al., 2005], only a few are seen to be associated with massive young clusters or associations. One is η Car, arguably the most massive and luminous star in the Milky Way. The others are FMM 362 and the Pistol Star, both apparently associated with the Quintuplet Cluster in the Galactic Center and visually obscured (and therefore not relevant to the statistical assessments of LBV isolation, since we do not have a meaningful sample of visually obscured LBVs in the ﬁeld). The other is W243 in the massive young cluster Westerlund 1 (Wd1). (Wra 751 is seen amid a cluster too, but not a massive young cluster.) The third goal of this thesis is to infer an independent and geometric distance to Wd1 using Gaia DR2.

1.4 Westerlund 1 Cluster

Westerlund 1 has an LBV [Clark and Negueruela, 2003], at least 24 WR stars [Clark et al., 2005, Crowther et al., 2006, Groh et al., 2006, Fenech et al., 2018], 6 yellow hypergiants [YHG; Clark et al., 2005], and a magnetar [Muno et al., 2005]; knowing the distance to this one cluster will help to constrain the luminosity, mass, and evolution of all of these late phases of stellar evolution. However, our sightline to the cluster suffers from substantial extinction and reddening, which has made its distance difficult to estimate using luminosity indicators [Piatti et al., 1998, Clark et al., 2005, Damineli et al., 2016]. Gaia parallaxes provide an independent distance indicator. The massive young star cluster, Westerlund 1 (Wd1), was detected by Westerlund [1961] during a survey of the Milky Way. Wd1 is located at RA(2000) = 16h47m04s.0, Dec.(2000) = 45°5100400.9, which corresponds to Galactic coordinates of ` = 339.55° and b = 00.40°. − − The first distance estimates mostly relied on reddening-distance relationships and ranged from 1.0 to 5 kpc [Westerlund, 1961, 1968, Piatti et al., 1998, Clark et al., 2005, Crowther et al., 2006]. Westerlund [1961] first suggested an extinction of AV = 12.0 mag, and reported a distance of 1.4 kpc. Later, Westerlund [1968] derived a significantly larger distance of 5 kpc by using VRI photographic photometry with near-infrared photometry of the brightest

9 stars. In contrast, Piatti et al. [1998] presented CCD imaging in the V and I bands, and using isochrone fitting, estimated a distance of 1.0 0.4 kpc. ± More recently, Clark et al. [2005] obtained spectra for the brightest members of Wd1 and improved our knowledge of Wd1. With more detailed spectra, many of the brightest members were identified as post-main-sequence stars. Since the isochrone fitting of Piatti et al. [1998] assumed that many of the stars are on the MS, the Piatti et al. [1998] distance estimate was incorrect. Six of the stars in Clark et al. [2005] are YHGs, and the most luminous YHGs are presumed to have relatively standard luminosity of around log(L/LM ) 5.7 [Smith et al., 2004]. Assuming that the YHGs in Wd1 were at the observed upper ∼ luminosity limit for cool hypergiants, and adopting an extinction of AV = 11.0, Clark et al. [2005] inferred a distance of 5.5 kpc. However, they noted that their reddening law is not ≤ entirely consistent with Wd1 data. To place a lower limit on the distance, they noted a lack of radio emission from the WR winds; this suggests a minimum distance of 2 kpc. Hence, ∼ Clark et al. [2005] reported a distance of 2 < R < 5.5 kpc. However, these constraints on the distance and reddening depend sensitively on assumptions of wind physics for evolved stars, which is still uncertain [Smith, 2014]. A variety of subsequent investigations produce similar results and accuracy. Crowther et al. [2006] inferred a similar distance using near-IR classification of WN (Wolf–Rayet stars with dominant lines of nitrogen and some carbon lines in their spectra) and WC (carbon dominant, no nitrogen lines) stars. More recently, Kothes and Dougherty [2007] derived a distance of 3.9 0.7 kpc based on the radial velocity of Hi features in the direction of Wd1. ± Brandner et al. [2007] studied the population of stars below 30 M and derived a distance ∼ of 3.55 0.17 kpc based on the apparent brightness and width of the colour magnitude ± diagram (CMD) using Pre-MS isochrones. Later, Gennaro et al. [2011] performed a similar analysis but also modelled the completeness, field star subtraction, and error propagation, and inferred a distance value of 4.0 0.2 kpc. Many of the previous techniques require ± some assumption about the luminosity of spectral classes. Recently, Koumpia and Bonanos [2012] used the dynamics and the geometry of an eclipsing binary (W13) to infer a distance of 3.7 0.6 kpc. Given all of difficulties explained above in estimating the distance, it is ± 10 necessary to use Gaia Data Release 2 (DR2) to find an independent distance estimate to Wd1. These distances have implications for the mass, age, and cluster members. For example, assuming a distance of 5 kpc, Clark and Negueruela [2003] inferred a total cluster mass of

5 10 M and an age of 3.5-5 Myr. With this age, the magnetar’s progenitor would have an ∼ initial mass of >40 M . The distance is critical, but estimates of the distance suffer from large uncertainties and systematics. It is therefore valuable to infer the distance using a more accurate and geometric technique. In the following chapters, I present the work of three papers that represent our initial efforts into constraining the luminosity, mass, and ages of LBVs. I am first author on two of the papers that are published [Aghakhanloo et al., 2017, 2020]. I am second author on the other paper [Smith et al., 2019] as I performed much of the data analysis.

11 CHAPTER 2

MODELLING LUMINOUS-BLUE-VARIABLE ISOLATION

In this chapter, we test whether the single star model can produce the observed isolation of LBVs from O stars [Aghakhanloo et al., 2017]. To do this, we model the dissolution of young clusters, and we find that the single star model can not reproduce the isolation of LBVs. Due to their short lifetime, LBVs do not have enough time to wander very far from other O stars. Instead, we find that LBVs must have evolved as lower mass stars and later gained mass via binary interactions. In fact, we find that LBVs are most likely the product of stellar mergers.

2.1 Observations

In the following sections, we explore which theoretical models are most consistent with the data, but before that, we clearly deﬁne, analyse and characterize the data in this section. First, we reproduce and verify the results of Smith and Tombleson [2015]. Secondly, we further characterize the data, noting that the distributions of nearest neighbours are lognormal. Since lognormal distributions have very few parameters, this restricts the complexity and parameters of our models in Section 2.2. Smith and Tombleson [2015] found that LBVs are much more isolated than O-type or WR stars, suggesting that LBVs are not an intermediary stage between these two evolutionary stages. In particular, they found that on average, the distance from LBVs to the nearest O star is quite large (0.05 deg). For comparison, the average distance from early O stars to the nearest O star is 0.002 deg, and from mid and late O stars are 0.008 and 0.010 deg respectively. If single early- and mid-type O stars are indeed the main-sequence progenitors of LBVs, then one would expect the spatial separations between LBVs and other O stars to

12 be not too different from the separation between early- and mid-type O stars. However, the LBV separations are an order of magnitude farther than the early- and mid-type separations. In fact, the LBV separations are five times larger than even the late-type O stars, which live longer and can in principle migrate farther. Smith and Tombleson [2015] quantified the difference in the distributions of separations by using the Kolmogorov–Smirnov (KS) test. Comparing the distributions of LBVs to early, mid and late types gives P -values of 5.5e–9, 1.4e–4 and 4.4e–6, respectively. These values imply that O-stars and LBV distributions are quite different. If true, then these results have profound consequences for our understanding of LBVs and their place in massive star evolution. In fact, Smith and Tombleson [2015] suggested that the most natural explanation is that LBVs are the result of extreme binary encounters. Later we will test this assertion, but for now we reproduce and verify their results. To verify the results of Smith and Tombleson [2015], we first define the data. The data consist of two main parts: LBVs and O stars. Their sample includes WR stars, sgB[e] stars and RSGs too, but we do not discuss them here because at the moment, we want to keep our models in Sections 2.2 and 2.3 simple and we will focus just on LBVs and O stars. Their LBV samples include 16 stars in the LMC, and three stars in the SMC. They did not consider Milky Way LBVs because the distances and intervening line-of-sight extinction in the plane of the Milky Way are uncertain [Smith and Stassun, 2017]. In their study, they included LBV candidates with a massive CSM (circumstellar medium) shell that likely indicates a previous LBV-like giant eruption. LBVs and their important parameters are summarized in Table 2.1. The masses of LBVs that are in Table 2.1 are uncertain; specifically, the uncertainty in masses due to distance uncertainties is at least 8%, but the systematic uncertainties due to the stellar evolution modelling are likely much larger. Currently, it is difficult to adequately quantify these uncertainties. None of them are kinematic mass measurements. Rather, they are based upon inferring the mass by comparing their colour and magnitude in the HRD with evolutionary tracks of various masses. In this modelling, the two main sources of uncertainties are modelling the uncertain physics of late-stage evolution and distance.

13 Table 2.1: List of LBVs and LBV candidates adapted from Smith and Tombleson [2015]. For the stars in the SMC, we rescale their angular separation by 1.2 as if they are located at the distance of the LMC. Parentheses in the name represent LBV candidates and parentheses in mass column specify the LBVs with relatively poorly constrained luminosity and mass.

LBV (name) Galaxy (name) S (deg) Meff (M ) R143 LMC 0.00519 60 R127 LMC 0.00475 90 S Dor LMC 0.0138 55 R81 LMC 0.1236 (40) R110 LMC 0.2805 30 R71 LMC 0.4448 29 MWC112 LMC 0.0892 (60) R85 LMC 0.0252 28 (R84) LMC 0.1575 30 (R99) LMC 0.0412 30 (R126) LMC 0.0358 (40) (S61) LMC 0.1432 90 (S119) LMC 0.3467 50 (Sk-69142a) LMC 0.0522 60 (Sk-69279) LMC 0.0685 52 (Sk-69271) LMC 0.040 50 HD5980 SMC 0.0191 150 R40 SMC 0.1112 32 (R4) SMC 0.0160 (30)

The distance uncertainty to the LMC is 3–4% [Marconi and Clementini, 2005, Walker, 2012, Klein et al., 2014], which would translate to a luminosity uncertainty of 6–8%. However, the systematic uncertainties in modelling LBVs and their luminosities and colour are unknown and could easily be much larger than the distance uncertainty. Therefore, like Smith and Tombleson [2015], we merely report rough estimates for the LBV masses in Table 2.1. Smith and Tombleson [2015] gathered the positions of O-type stars within 10◦ projected radius of 30 Dor from SIMBAD database. They also used the revised Galactic O-star Catalog [Ma´ızApell´anizet al., 2013] to check their O-star samples (not shown in their paper), but as they claimed this did not change their overall results. We collect the same O-star samples from SIMBAD database.

14 After gathering the data, we find the distance from one star to the nearest O star. The bottom panel of Fig. 2.1 shows the resulting cumulative distributions (the top panel shows results of our modelling, which we discuss in Section 2.2); note that the distributions for LBV and O-type separations are quite distinct. For example, the P -value for the comparison of the LBV and the mid-type distributions is 2.8 10−5. Our KS-test P -values are listed in Table × 2.2. We consider three KS tests. In one, we compare the separation for both confirmed and candidate LBVs with O-star distribution. In the second, the LBV distribution only includes ten confirmed LBVs, and in the third, the LBV distribution only contains nine LBV candidates. When we include both confirmed and candidate LBVs, the LBV and O- star distributions are clearly not drawn from the same parent distribution. However, omitting LBV candidates reduce the distinctions between the distributions. One might argue that since LBVs represent a later evolutionary stage, then the spatial separations should be larger, and therefore, the distributions of early-type O stars and LBVs should not represent the same distribution. However, we will show in section 2.3 that the lifetimes of massive stars are far too short to explain these large discrepancies. In our initial assessment, we agree with Smith and Tombleson [2015]; the large separations present a challenge to the single-star evolution scenario. In the next sections, we will present theoretical models to quantify this inconsistency. Before we constrain the models, note that the separation distributions are lognormal (see Fig. 2.2). In fact, this simple observation greatly restricts the complexity of the models that we may explore in the next sections. If a variable such as the separation between stars shows a lognormal distribution, then there are only two free parameters that describe the distribution, the mean and the variance. In addition, if the separation depends upon other variables such as a velocity distribution, then thanks to the central limit theorem, the separation distribution will only depend upon the mean of the variance of the secondary variables such as the velocity distribution. This means that we cannot propose overly complex models for the velocity distribution. We would only be able to infer the mean and the variance anyway. Fortunately, we may measure the separation for different types of O stars and other evolutionary stages. This means that we may infer the temporal evolution in addition to

15 Table 2.2: The P -values for KS tests for the distributions of separation. We are comparing the separations between LBVs and O stars and the separations between O stars of various types. Broadly, we reproduce the results of Smith and Tombleson [2015] who found that the distribution of separations between LBVs and the nearest O star is quite different from the distributions for the separations between O stars and the nearest O star. The second row shows the results of our KS tests between the LBV separations and the early-, mid- and late-type O stars. Our results are similar to those of [Smith and Tombleson, 2015, first row], first row. Like Smith and Tombleson [2015] we obtain the positions of O stars and their rayet spectral types from SIMBAD. Smith and Tombleson [2015] updated the spectral types with the Galactic O-star Catalog [Ma´ızApellánizet al., 2013]; however, we did not. This slight difference in spectral typing is what causes the modest difference in P -values. In either case, the LBV separations are inconsistent with any O-type separations. If we exclude the LBV candidates (third row), the conclusions remain the same, but the significance is greatly reduced. Data set Early O Mid O Late O LBV+LBVc [Smith and Tombleson, 2015] 5.5e–9 1.4e–4 6.4e–06 LBV+LBVc (this work) 8.2e–08 2.8e–05 8.4e–05 LBV (this work) 9.2e–04 2.3e–02 5.7e–02 LBVc (this work) 6.4e–06 5.1e–05 2e–04 the mean and variance. Whatever models we propose, they cannot be too elaborate; we will only be able to infer the mean and variance of one quantity as a function of time.

2.2 A Generic Model for the Spatial Distribution of the Stars in a Passive Dispersal Cluster

In order to model the relative isolation of LBVs, we need to model the dissolution of clusters and associations of massive stars. For several reasons, we model the dissolution of young stellar clusters with a minimum set of parameters. For one, the O-star distributions are lognormal. Therefore, there are only a few parameters that describe the data that one may ﬁt. The only data that we can reliably ﬁt are the mean, variance and time evolution of the separations. So whatever models we develop, they should not be overly complex. Also, as far as we know, there are no simple self-consistent and tested models for the dissolution of clusters. Therefore, we propose a simple model of cluster dissolution and adapt it to

16 1.0 O2-O5 (Early) Model O6+O7 (Mid) 0.8 O8+O9 (Late)

0.6

0.4

0.2

OBS

Fraction of0 total .8

0.6

0.4

0.2

LBV 0.0 3 2 1 0 10− 10− 10− 10 Projected separation to nearest O-type star [deg]

Figure 2.1: Cumulative distributions for the projected separation to the nearest O star. The top panel represents the modelled distribution for O stars and the bottom panel represents the data for both O stars and LBVs. Later, we will use the modelled O-star distributions to devise a general dispersion model, which we use to model the LBV separations (see Section 2.3). Broadly, the model reproduces the observations; both show a lognormal distribution, and the average separation increases with spectral-type because the later spectral type last longer.

17 5 p 0.5 O2-O5 (Early) 4 ∼

p 0.5 O6+O7 (Mid) 10 ∼ 8 6 4

Frequency 2

p 0.4 O8+O9 (Late) 20 ∼

0 4 3 2 1 0 10− 10− 10− 10− 10 S [deg]

Figure 2.2: Normality test. The distributions of separations for early, mid,and late O stars are consistent with a lognormal distribution. In each plot, we show the probability, p, that the parent distribution is a lognormal distribution.

18 consider two scenarios: cluster dissolution in the context of single-star evolution and cluster dissolution with close binary interactions. In this section, we present a cluster dispersal model considering only single-star evolution. While our dissolution models represent the spatial distributions reasonably well in certain respects, we note that our model fails to match the data in other ways. This implies that our model is missing something. In other words, we may be able to infer more physics about the dissolution of clusters from the simple spatial distribution of O stars. In the next section, we contrast the single-star model with a model that considers binarity. Our main goal is to introduce a model for young stellar clusters that predicts the spatial distribution of massive stars, especially O stars. We start by considering the simplest model. In the following, we model the average distance to the nearest O star by nothing more than the passive dispersal of a cluster. Before we dive into the details of the model, it is worth characterizing the scales of a typical cluster. We begin right after star formation ends and consider a system of gas and stars that is in virial equilibrium. In this case, we have 2T + U = 0, where T is the total thermal plus kinetic energy and U is the gravitational potential energy. Initially, the system with total mass MT and radius R is bound, and the stars have a velocity dispersion that 1 scales as the gravitational potential of the entire system σv (GMT /R) 2 . Then the system ∼ loses gas mass by some form of stellar feedback (UV radiation, stellar winds, etc.) and likely makes the stars unbound. If the system loses all of the gas quickly, then the stars will drift away with a speed roughly equal to the velocity dispersion when the cluster was

bound. Hence, vd σv. All that is left to do is estimate MT and R. A typical cluster has ∼ R 4 pc and about 40 O stars; if only 1% of the gas in giant molecular clouds form ∼ ∼ stars [Krumholz and Tan, 2007], the total mass of the molecular cloud, MT , is the order of

5 2 10 M . Given these approximations, we estimate that the drift velocity is the order of × M 4pc 1 T 2 vd 13.5( 5 ) . ∼ 2×10 M R Next, we present a more speciﬁc dissolution model to convert this dispersal velocity into a distribution of separations as a function of time. Rather than using this estimate for the dispersal velocity, we will use the data and our model to infer the dispersal velocities. We

19 propose a Monte Carlo model for the dissolution of the clusters. First, we randomly sample

Ncl clusters uniformly in time between 0 and 11 Myr. For each cluster, we draw a cluster mass from a distribution of cluster masses. Then, we estimate the total number of the stars

(N∗), and for each cluster, we draw a distribution of stellar masses (M∗) from the Salpeter distribution.

First, we randomly select a total number of O stars , N∗, for each cluster. The distribution from which we draw the size of each cluster is the Schechter function [Elmegreen and Efremov,

dNcl −2 1997], M , where Mcl is the mass of the cluster. However, we are most interested in dMcl ∝ cl the number of O stars for each cluster, so our ﬁrst order of business is to express the Schechter

R M∗2 −1.35 function in terms of the number of O stars. The mass of the cluster is Mcl = A M∗ , M∗1 where M∗1 and M∗2 are the minimum and maximum masses of O star that we consider. In terms of this, the total number of O stars becomes

Z M∗2 Mcl −2.35 N∗ = M∗ dM∗ . (2.1) R M∗2 −1.35 M∗ dM∗ M∗1 M∗1 Therefore, the total number of stars in the cluster is proportional to the mass of the cluster

(N∗ Mcl), and we can easily translate the distribution in mass to a distribution in the ∝ dNcl −2 number of stars for each cluster, N . If R∗ is drawn from the uniform distribution dN∗ ∝ ∗ between 0 and 1, then the total number of stars in the cluster is

1 N∗ = −1 −1 −1 , (2.2) R∗(N∗max N∗min ) + N∗min − where N∗max and N∗min are the maximum and minimum number of the stars in the cluster. For each star, we draw the mass from the Salpeter initial mass function (IMF),

1 0.74 M∗ = ( −1.35 −1.35 −1.35 ) , (2.3) [Rm(Mmax Mmin )] + Mmin − where Rm is a random number between 0 and 1. Having established the initial conditions, we now describe the evolution. The average separation between O stars depends upon how much the cluster has dispersed and how many O stars are left. So we need to model the dispersion of the O stars and their disappearance. Therefore, we need to model the spatial distribution (or spatial density) and time evolution

20 of massive stars in a cluster. Once we establish the spatial distribution, we then calculate the separation between stars. The distribution of separations in essence is a convolution of the density function with itself. Because this is a multiplicative process, the central limit theorem implies a lognormal distribution. The central limit theorem also dictates that any underlying spatial distribution with a well-deﬁned mean and variance results in a lognormal distribution. Therefore, we are free to choose a simple model for the spatial distribution, and we choose a Gaussian for the spatial distribution. For the time evolution we assume that each cluster is passively dispersing with a typical

velocity scale of vd.Therefore, the characteristic size scale of the Gaussian spatial distribution is σ = vdt. Given the assumption that stars are coasting then the individual velocities are r/t. With these assumptions, then the distribution of velocities is Gaussian too, p(v) =

2 2 √ t e−r /2σ . 2πσ2 Another important aspect of modelling these clusters is to model the age and disappearance of massive stars. For the lifetimes, we use the results of single-star evolutionary models from the binary population synthesis code, binary c [Izzard et al., 2004, 2006, 2009]. There- fore, the average separation between stars goes up both because the cluster is dispersing and O stars are disappearing. Fig. 2.3 shows the spatial distribution of an example model at several ages. With our model deﬁned, our ﬁrst task is to constrain whether the average distances between LBVs and O stars are consistent with the passive dissolution of a cluster with single- star evolution. To compare our models to the data, we calculate the angular separations, assuming that the clusters are at the distance of the LMC. Furthermore, to be consistent with Smith and Tombleson [2015], we subdivide the modelled O stars into early, mid and late types based upon their masses. To convert from mass to spectral type, we used Martins

et al. [2005] data. Early-type O stars have masses greater than 34.17 M , late-type O stars

have masses 24.15 M and mid-type O stars have masses in between. In the next section, ≤ we test whether our passive single-star dissolution model is consistent with the data.

21 80 60 1.8 Myrs 40 vdt 20 S 1/2 0 [NO(t)] 20 h i ∼ −40 −60 −80 − 80 60 40 20 0 20 40 60 80 80 − − − − 80

y [pc] 60 5.9 Myrs 60 9.9 Myrs 40 40 20 20 0 0 20 20 − −40 40 − −60 60 − −80 80 − − 80 60 40 20 0 20 40 60 80 80 60 40 20 0 20 40 60 80 − − − − − − − − x [pc]

Figure 2.3: We propose a Monte Carlo model for the separations between O stars and LBVs by considering a random sample of dissolving clusters at random ages. Here we show the O stars of three randomly generated clusters, each with its own age. Note that the average separation between the O stars increases with age for two reasons. First, the separations increase as the cluster disperses with a drift velocity vd over time t. Secondly, O stars disappear as they evolve.

2.2.1 Comparing the Passive Single-star Dissolution Model with the Data

Next, we compare the passive dissolution of single stars to the LMC and SMC nearest- neighbour distributions. Fig. 2.1 shows the cumulative distribution for the separations for our simple dissolution model (top panel) and for the observations (bottom panel). For illustration purposes, we set vd to 14.5 km/s, making the modelled distribution have about the same mean as the data. So far, our passive dissolution model is in good agreement with observations. Both the model and observations show a lognormal distribution in separations, and the average separation increases with spectral type, which is expected since later O stars live longer and have more time to disperse. Because the distributions are lognormal, there are only two parameters that describe the distribution, the mean and std. deviation. Therefore, we investigate how our model

22 reproduces these two distribution characteristics. The primary parameter in our model is

vd, so in Fig. 2.4 we plot the mean (bottom panel) and std. deviation (top panel) as a

function of vd. The dashed lines represent the modelled mean and std. deviation, and the solid bands indicate the observed values. The vertical axes in Fig. 2.4 are µS and σS.

µ(log S) First, we calculate the mean and std. deviation in log; then, we calculate µS = 10

σ(log S) and σS = 10 . The solid bands provide some estimate of uncertainty in our inferred drift velocity, we bootstrap (random sampling with replacement to estimate statistics on a population) the observations, giving a variance for both the mean and std. deviation. We draw three main conclusions from Fig. 2.4. For one, the drift velocities that we infer by comparing our simple model with the data are roughly what we would expect; see our order-of-magnitude estimate in Section 2.2. Secondly, we infer larger drift velocities for the late-type O stars (10–12 km/s) in comparison to early-type stars (6–8 km/s). However, this trend is not monotonic; the mid-type O stars have an inferred drift velocity (14–16 km/s) that is similar to but slightly higher than the late-type O stars. Thirdly, our simple model is not able to reproduce the variance in the distributions. This implies that something is missing from our model. In other words, there is more that we can learn about the evolution of massive stars in clusters from their spatial distributions. Despite the shortcomings, the model is able to reproduce the average separations with reasonable drift velocities. Therefore, we proceed with our analyses under these caveats. Since the early-type O stars are more massive and have lower velocities, it is natural to consider mass segregation as the reason for these lower velocities. However, the relaxation time is of the order of 100 Myr, which is more than the maximum age of late-type O stars (11 Myr). So, it is unlikely that these systems have enough time to reach equipartition and mass segregation. Despite this fact, we test this idea and we ﬁnd that the inferred velocities are not readily consistent with equipartition anyway. In equilibrium, the stars in a cluster are in equipartition in their kinetic energies. Therefore, the ratio of masses for two stars

2 should equal the inverse ratio squared of their velocities: mi/mj = (vj/vi) . Comparing late

to early, the ratio of masses is mlate/mearly 0.3 and the ratio of the squared velocities ∼ 2 is (vearly/vlate) 0.4. This seems consistent with mass segregation. However, the other ∼ 23 10

7 S σ 6

3 5 10 15 20 25

0.030 O2-O5 (Early) OBS O6 + O7 (Mid) Model 0.025 O8 + O9 (Late)

0.020 V 10-12 km/s Late ∼

S 0.015 µ

0.010

0.005 V 14-16 km/s Mid ∼

VEarly 6-8 km/s 0.000 ∼ 5 10 15 20 25 Vdrift [km/s]

Figure 2.4: The mean (bottom panel) and std. deviation (top panel) distance to the nearest neighbour versus drift velocity. We calculate the mean and std. µ (log S) σ (log S) deviation in log ﬁrst; then, we calculate the µS = 10 and σS = 10 . In both panels, dashed lines represent the passive dissolution model and solid lines represent the observational data [Smith and Tombleson, 2015]. We highlight three main conclusions. (1) The drift velocities that we infer by comparing our simple model with the data are roughly what we estimated in Section 2.2. (2) We infer larger drift velocities for the later type O stars, implying that binary evolution and kicks may be important. (3) The passive dissolution model is not able to reproduce the variance in the distributions, which implies missing physics from our model. In other words, there is room to improve our model and learn more about the interplay between O-star evolution and cluster dissolution.

24 2 comparisons do not. For mid and early, mmid/mearly 0.43 and (vearly/vmid) 0.21, which ∼ ∼ 2 is a factor of 2 off. The late-to-mid comparison gives mlate/mmid 0.7 and (vmid/vlate) ∼ ∼ 1.85, which is also a factor of 2 off. Furthermore, if equipartition in kinetic energy were valid, then all of these ratios should have similar values. We have yet to adequately assess the uncertainties in these ratios; that will take significant more modelling. Even so, the fairly large discrepancies seem to rule out kinetic energy equipartition in the cluster. We can use the results in Fig. 2.4 to also infer that LBV isolation puts interesting constraints on their evolution. The average separation for late-type O stars is 0.01 deg. For LBVs, the average separation is roughly five times bigger. Dimensionally, the average separation should be proportional to the dispersion velocity and the age, S vdt . If an LBV ∼ age comes from the most massive stars, then one would not expect them to have ages larger than the late-type O stars. Therefore, as a conservative estimate, let us assume that an LBV is an evolved massive star that has about the same age as a late O-type star. Under this assumption, since the separations for LBVs are five times bigger than late-type O stars, this implies that the dispersal velocity is five times bigger than the late-type O star, which is of the order of 100 km/s. To be more quantitative, in the next sections, we extend the passive model to infer the actual dispersal velocity for LBVs. Alternatively, we consider binary scenarios that may give an explanation for the relatively large isolation for LBVs.

2.3 Cluster Dissolution with Close Binary Interactions

In the previous section, we suggested that the single-star dispersal model is inconsistent with the isolation of LBVs. In this section, we put the passive single-star dispersal model to the test, and show that it is indeed inconsistent with observations. In addition, we consider models that involve binary interactions in a dispersing cluster. Our aim is to develop models to see whether binary scenarios are consistent with the LBV observed separations. At the moment, there is very little information other than the separations, so it is not worth developing an overly complex model for binary interaction. We would not be able to constrain the extra parameters of the model. Therefore, we develop the simplest binary

25 models to constrain the data. In particular, we consider two simple models that involve binary evolution in a dispersing cluster. In the first model, we consider that an LBV is the product of a merger and is a rejuvenated star; in the second model, we consider that an LBV is a mass gainer, which would also be a rejuvenated star, and receives a kick when its primary companion explodes. See Fig. 2.8 and 2.9. In Section 2.3.1, we first put together an analytic model for the average separation between two stars versus time. Then in Section 2.3.2 we use this model to show the inconsistency in the single-star model, and we show that LBVs are either overluminous given their mass or they are the product of a merger and are a rejuvenated star. Alternatively, in Section 2.3.3, we use the analytic model to develop a kick model, and in Section 2.3.4, we use this model to infer a potential kick velocity for LBVs. In summary, we illustrate that the isolation of LBVs is consistent with binary scenario and is inconsistent with the single-star model. To constrain the models, we first derive analytic scalings for the average separations, and then we explore whether these scalings are consistent with simple binary models. First, we consider simple models for the spatial distribution of two groups of stars, type O and type

L. Each has a Gaussian spatial distribution with its own velocity dispersion vd ,which we label as vO and vL. Later, we will consider two scenarios: one in which these average velocity dispersions are the same, and one in which they are diﬀerent. For a visual representation of these simple models, see Fig. 2.5. Given these distributions, we calculate the average separation between a star and the nearest star in the same group. Then we calculate the average separation between a star in group O and a star in group L. The average separation between stars in the same population is

Z ∞ S = 2π S p(r) rdr , (2.4) h i 0

1 −r2/2σ2 where S is the separation, and p(r) is the probability density function p(r) = 2πσ2 e where σ = vt. To calculate the mean value of the separation, we need to find the separation (S). One way to estimate the distance to the nearest neighbour is to use the spatial density of stars. In general, an estimate for the distance to the nearest neighbour is, Sˆ 1/n1/d, ≈ 26 S represent will is ‘L’ density later combined Obviously, the which case, first. stars, this the tracer In than of a LBVs. set distribution as rare density stars more different O a consider a represents later ‘L’, have other, will may The we suggests, stars. label tracer of the number As large stars. tracer of number largest the where is space two-dimensional distribution, in density separation simple a consider we If sky, the to on projected clusters viewing When 2014]. al., et where ˆ i qain 25 n hnw a aclt h vrg eaainfo qain (2.4). equation. from separation average the calculate can we then and (2.5) equation. via , ihti w-opnn xrsinfrtedniy ecneaut h oa separation, local the evaluate can we density, the for expression two-component this With represents which , ‘O’ with represent we One stars. of populations two consider we Now iue25 w ipesaildsrbto oesfrtedrvto foranalytic our of derivation the for models scalings. spatial-distribution simple Two 2.5: Figure N n stenme est and density number the is stettlnme fsasi h cluster. the in stars of number total the is

#stars/pc3 n OL ( r = ) σ O 2 h d N πσ S v = ˆ i stenme fdmnin htw osdr[Ivezic consider we that dimensions of number the is O O 2 S 2 = ˆ O e t ≈ − r 2 1 27 σ / /n 2 2 N r n L σ π ( [ O 2 v = 1 pc r / = ) + 2 1 ] . / L 2 2 t N πσ , σ 2 L πσ N L 2 2 e − exp r 2 d / 2 {− σ 2, = L 2 r . 2 / 2 σ 2 } hnteaverage the then , (2.5) (2.7) (2.6) Calculating the average separation is numerically straightforward. However, with a small but useful assumption, we can derive an analytic estimate for the average separation. To make it easier to calculate the integral analytically, we make two assumptions. First, we assume that the average separation is roughly given by the scale of one over the square root of the average density. Therefore,

1 SOL 1/2 . (2.8) h i ≈ nOL h i Secondly, because LBVs are extraordinarily rare compared to the O stars, we assume that

NO NL. By considering these two assumptions, the average density is

NO nOL 2 2 . (2.9) h i ≈ 2π(σO + σL)

Once we plug this into the equation for nOL , equation. (2.8) leads to the average distance h i from LBVs to the nearest O star:

2 2 1/2 2π(σO + σL) SOL (2.10) h i ≈ NO(t)

Soon we will use the separation between O stars to help constrain the models for LBVs, so we now derive an analytic model for SO . The average density for O stars is h i

NO(t) nO = 2 , (2.11) h i 4πσO and so the average separation between O stars is roughly

1 4π 1/2 SO 1/2 = σO . (2.12) h i ≈ nO NO(t) h i To make use of these expressions for the average separation, we need to compare the separations between two diﬀerent tracer populations. Because the masses of mid-type O stars correspond roughly to inferred minimum mass of LBVs, we use the mid-type O-star average separation as a reference:

2 2 SOL 1 σL NO(tO) h i = 1 + 2 , (2.13) SO 2 σ NO(tL) h i O

28 where we are careful to consider how the number of O stars changes with time and we evaluate this function at the age of the LBV population and the reference O-star population. This equation represents the general expression relating age, the average separations and the drift (or kick) velocity of each tracer population. In the expressions for the average separations, the separations grow due to two eﬀects: a drift velocity and the death of O stars. The drift part is simply proportional to t. Next, we

explicitly derive the number of O stars as a function of time, NO(t).Given a mass function

R M2 dN A −α+1 −α+1 dN/dM, the total number of O stars is NO = dM = (M M ), where M1 dM −α+1 2 − 1 M is the minimum mass for an O star ( 16 M ), M is the maximum mass for an O star, 1 ∼ 2 which is a function of the age of the cluster, α is the slope (we use Salpeter, 2.35) and A is a normalization constant. If we assume a power-law relationship between mass of an O star

and its lifetime as an O star, then we can relate the age of an M2 O star to the age of an M O star: M = M ( t )−1/β. From the binary population synthesis code, binary c [Izzard 1 2 1 t1 et al., 2004, 2006, 2009], we ﬁnd that the value of β 1.7. Combining these expressions, we ∼ get an equation for the number of O stars as a function of the age of the cluster, t, A t τ 1−α NO(t) = 1 ( ) M . (2.14) 1 α − t 1 − 1 With an explicit function for the number of O stars, we may now derive the equation relating separation, age and velocity, including explicitly all of the dependence on time.

Substituting the expression for NO(t), equation. (2.14) into the general analytic expression, equation. (2.13), we ﬁnally arrive at the general analytic formula, explicitly relating separation, age and velocity:

2 2! τ SOL 1 vLx (1 xO) h i = 1 + − τ , (2.15) SO 2 vOxO (1 x ) h i −

α−1 tO tL where τ = , xO = and x = . t (11 Myr) is the age of the minimum mass and tO β t1 t1 1 (3 Myr) is a reference age. We estimate these values from binary population synthesis code, binary c [Izzard et al., 2004, 2006, 2009]. In the next sections, we use our general analytic result, equation. (2.15), to explore what average separations one would expect when we consider the passive dissolution in three

29 scenarios, a single-star evolution scenario, a binary scenario that involves a merger and a binary scenario that involves a kick.

2.3.1 Passive Model

Using our analytic estimates for the average separation, we assume that the dispersal velocities for LBVs and O stars are the same and estimate the average separation for the passive single-star model. Comparing this model to the observations, we ﬁnd that the passive single-star model is inconsistent with the observations. If LBVs do passively disperse with the same velocity as the rest of the O stars, then we propose that LBVs are the product of a merger and are rejuvenated stars. In this case, we need to consider the average separation when the dispersal velocities for LBVs and O stars are the same. In this scenario, our general analytic expression, equation. (2.15), reduces to

1/2 " 2! τ # SL 1 x (1 xO ) = 1 + − τ . (2.16) SO 2 xO (1 x ) − This equation represents the passive model.

2.3.2 Inconsistency in the Passive Model Implies Merger and Rejuvenation

Next, we use the passively dissolving solution, equation. (2.16), to show that the isolation

of LBVs is inconsistent with the single-star scenario. If LBVs are massive stars above 21 M and evolve as isolated stars, then Fig. 2.7 and Fig. 2.8 demonstrate that the maximum ages of these LBVs are wholly inconsistent with the large separations observed for massive stars. The passive model predicts much lower separations than the observational data. See

Fig. 2.6 for an illustration. The orange curve represents the passive model, SLBV in equation. (2.16). The solid brown line illustrates the LBVs’ average separation obtained from the data compared to the reference average separation, SLBV . It is clear that most of the LBVs S0 have larger separations compared to what the passive model predicted.

30 Mdeath [M ] ⊙ 70 65 60 55 50 45 40 35 30 25 20 18

MTO [M ] ⊙ 100 55 50 45 40 35 30 25 20 17

1 10−

2 10−

SLBV log S [deg] LBVc 3 10− LBV passive model TO death 4 10− − 0 2 4 6 8 10 12 t [Myrs]

Figure 2.6: LBV isolation is inconsistent with the single-star model and passive dissolution of the cluster. The solid brown line shows the LBVs’ average separation obtained from the data. The orange curve shows our analytic description, equation. (2.16), for the average separation in the context of passive dissolution. This model requires a reference; we used the mid-type O observations as the reference. The line segments show the purported masses and allowable ages for the LBVs (solid segments) and LBV candidates (dashed segments). If LBV mass estimates are correct, then even when one considers the maximum age for LBVs, the separations are much larger than what the single-star passive model predicts.

Moreover, in the passive model, LBVs do not have enough time to get to the observed average separation. Fig. 2.7 shows the same passive model, the observed LBV separation, but this time we simplify the possible ages of LBVs by showing the ages for the average mass of our LBV sample. Clearly, if LBVs evolve as a normal single star, then they do not have enough time to reach the large separations. Instead, let us consider how old an LBV would have to be in order to passively disperse to the observed separations. Fig. 2.7 shows that the age would need to be about 9.2 Myr. Yet this age corresponds to the main-sequence

31 turnoff time for a 19 M star or the death of a 21 M star. Both of these values are below the average mass of the LBVs, 50 M (Section 2.1). It is clear that considering LBVs in the context of a standard single-star evolution is inconsistent with the isolation. In short, the luminosity-to-age mapping of single-star models is inconsistent with the extreme isolation of LBVs. One can consider this mapping in two steps: an age-to-mass mapping and a mass-to-luminosity mapping.Technically, the breakdown in the luminosity- to-age mapping could be a result of the breakdown in either one of these steps. In other words, LBVs could be far more luminous than their masses would suggest. At the moment, there is no known physics that would lead to this, so we instead consider how binary evolution may alter the mass-to-age mapping. Assuming that the drift velocities of the LBVs and the O stars are the same, then one possible solution is that LBVs are the result of mergers and are rejuvenated stars. See Fig. 2.8 to visualize the merger model. We are not the first to suggest that LBVs are linked to close binary interaction. For example, see Justham et al. [2014a], Vanbeveren et al. [2013] and Gallagher [1989a]. What is different here is that, following Smith and Tombleson [2015], we analyse how the spatial distribution of LBVs strongly suggests close binary interactions.

2.3.3 Kick Model

Another binary model that is consistent with the isolation of LBVs is the kick model. To visualize the kick model, consider the binary scenario in Fig. 2.9. In this model, the primary star, the more massive star, evolves ﬁrst and transfers mass to the secondary star. If the more massive star is massive enough to explode as a core-collapse SN, then the companion may receive a kick. This kick may be imparted by either an asymmetric explosion, the Blaauw mechanism [Blaauw, 1961] or a combination of both. In this thesis, we do not model the binary evolution and kick velocities. Rather we just assume that there are two populations, one more numerous and does not receive kicks (the O stars), and one that is less numerous and whose velocity distribution is dominated by kicks. Once again, we may use our general analytic expression, relating the separations, age and velocities, equation. (2.15), but this time we express vL in terms of the age, x = tL/t1,

32 Mdeath [M ] ⊙ 70 65 60 55 50 45 40 35 30 25 20 18

MTO [M ] ⊙ 0.25 55 50 45 40 35 30 25 20 17

0.20

MLBV(TO) 19 M 0.15 ∼ ⊙ MLBV(death) 21 M ∼ ⊙ t 9.2 Myrs LBV ∼

SS [deg] [deg] 0.10

MLBV 50 M OBS ∼ ⊙ 0.05

Passive Model

0.00 0 2 4 6 8 10 12 t [Myrs]

Figure 2.7: The relative isolation of LBVs is consistent with a binary merger in which the LBV is a rejuvenated star. The solid brown line shows the observed LBV average separation. If LBVs passively dissolve with the rest of the cluster (orange curve), then we infer an average age for LBVs of 9.2 Myr. This corresponds to the main-sequence turn-oﬀ time for a 19 M star and the death time for a 21 M star. However, the average mass for LBVs estimated from their luminosities is roughly 50 M . Stars this massive do not live long enough to passively disperse to large distances. On the other hand, if LBVs are the products of a merger, and the primary has a mass between about 19 and 21 M , then the rejuvenated star could have a high luminosity, high mass and old age allowing it to disperse to larger distances. and the measured values of the separations,

1/2 " 2 τ # vL xO SOL (1 x ) = 2 h i − τ 1 . (2.17) vO x SO (1 x ) − h i − O Smith and Tombleson [2015] showed that the average distance from LBVs to the nearest O star is 6.5 times larger than the average distance from O star to the nearest O star. If ∼

33 Figure 2.8: Merger model outline. Figure 2.9: Kick model outline. In In this binary scenario, LBVs are a this binary scenario, a pre-LBV star product of rejuvenation of two mas- gains mass from its more massive sive stars. For a given mass, a re- companion star. After mass trans- juvenated star has a larger maxi- fer, the mass gainer (LBV) receives a mum possible age than a single-star kick when its companion explodes in counterpart. These larger maximum an SN. ages allow a rejuvenated star enough time to drift farther from other O stars. This is one binary scenario that is consistent with the isolation of LBVs.

34 the age of LBVs are similar to the average mid-type O star, then in the assumption of the

kick model, this immediately implies that vL is roughly nine times larger than vO. In the next section, we estimate the LBVs’ drift velocity given this model.

2.3.4 Estimation and Interpretation of the Kick Velocity

Fig. 2.10 shows the inferred kick velocity as a function of the LBV age. If the mass gainer that eventually becomes the LBV gains little mass, then there is little discrepancy between the zero-age main-sequence mass and the final mass. In this case, there is little difference between its apparent age and its true main-sequence age. Then its true age is relatively short and the only way to get a large separation with a large kick velocity. In this scenario, we find that the kick can be as high as 105 km/s. If there is no mass gained and hence a larger kick (upper left in Fig. 2.10) then the star that was kicked has not necessarily had any anomalous evolution (no accretion and spin-up) and hence gives no special explanation for its observed LBV instability. On the other hand, if the mass gain is high, then the true main-sequence age would be much older than the current mass implies. With a much older age, the velocity required to get a large separation is much lower. It might even be zero, in which case, the LBV has gained so much mass that it is rejuvenated like a merger product. The horizontal black solid line represents the average observed separation for mid-type O stars. The solid blue line curve represents our model to infer the kick velocity, equation. (2.17). Though we predict that the kick velocities may be as high as 105 km/s, we note that ∼ the kick may be quite low, even near zero. Humphreys et al. [2016] argued that none of the LBVs in the LMC have high velocities. They suggest that most of the LBV velocities [listed in table 3 of Humphreys et al. [2016]] are consistent with the systemic velocities of the LMC, concluding that the observed velocities are inconsistent with the kick. In Fig. 2.10, we show that the kick velocity may be anywhere from 0 to 105 km/s depending on the orbital ∼ parameters at the time of the SN, and how much mass was transferred. To further constrain the mass-gainer and kick model, one will need to properly model binary evolution including explosions and kicks in the context of dispersing cluster. Current modelling efforts already indicate that the dispersal velocities from binary evolution could have a large range, even

35 Mdeath [M ] ⊙ 70 65 60 55 50 45 40 35 30 25 22

MTO [M ] ⊙ 120 50 45 40 35 30 25 20

100 Large Mass Gain Requires Low Kick 80

[km/s][km/s] 60 Merger No Kick

LBV LBV Low Mass Gain v v 40 Requires High Kick

20 vdrift of Mid O

0 3 4 5 6 7 8 9 Age of LBV [Myr]

Figure 2.10: LBV dispersion velocity as a function of LBV age. Another binary model that is consistent with observations is one in which the LBV is a mass gainer and receives a kick when its primary companion explodes. The solid blue curve represents our analytic model, equation. (2.17). For reference, the solid black line shows the drift velocity of mid-type O stars, see Fig. 2.4. As a mass gainer, the age of the LBV will be older than one would infer from its luminosity and mass. If LBVs gain little to no mass, then the kick required to match the observed separations is in the range 0–105 km/s. The lower end of this range corresponds to high mass transfer. Speciﬁcally, if LBVs have an age of the order of 9.2 Myr, then we suggest that LBVs are mergers and received no kick. low dispersal velocities [Eldridge et al., 2011, de Mink et al., 2014, Smith, 2016a]. However, putting these binary models in the context of cluster dispersion is yet to be done. In conclusion, we develop models for cluster dissolution and the spatial distribution of LBVs and O stars. These models suggest that single-star evolution in passively evolving clusters is inconsistent with the extreme isolation of LBVs. Instead, we ﬁnd that either

36 LBVs are less massive than their luminosities would imply or binary interaction is most consistent with LBV isolation. In particular, we crudely ﬁnd that two binary scenarios are consistent with the data. Either LBVs are mass gainers and received a kick when the primary exploded or they are rejuvenated stars, being the product of mergers.

37 CHAPTER 3

ON THE GAIA DR2 DISTANCES FOR GALACTIC LUMINOUS BLUE VARIABLES

In this chapter, we infer the distances to 25 Galactic LBVs from the Gaia Data Release 2 (DR2) [Smith et al., 2019]. Nearly half of the new Gaia inferred distances to Galactic LBVs are signiﬁcantly closer than previous estimates in the literature, lowering their luminosities. For the remaining half of the sample, the Gaia distances are formally consistent within the uncertainty with previously adopted values, either because there is little change or because the large uncertainty encompasses a wide range of values. These preliminary results already indicate signiﬁcant changes in the characterization of LBVs.

3.1 Gaia Spacecraft

Gaia is a space observatory of the European Space Agency (ESA), launched on 19 De- cember 2013 and expected to operate until c. 2022 . The main goal of Gaia is to measure the position, distance and the motion of stars and to determine their astrophysical properties to create a three-dimensional map of the Galaxy. Gaia operates at the second Lagrange (L2) point of the Sun- Earth-Moon system, ∼ 1.5 million km from Earth. The spacecraft moves around L2 in a Lissajous-type orbit to avoid blockage of the Sun by the Earth and for instance stable thermal conditions. Gaia is equipped with two identical, three-mirror anastigmatic telescopes, with apertures of 1.45 m 0.50 m with a ﬁxed, wide angle of 106°.5. This enables the spacecraft to make simultaneous × measurements of star positions at small and large angular scales. The ﬁrst data release, Gaia DR1, based on 14 months of observations made through September 2015, took place on 14 September 2016. The second data release occurred on 25

38 April 2018, which is based on 22 months of observations made between 25 July 2014 and 23 May 2016. The astrometric data in Gaia DR2 include the ﬁve astrometric parameters (position, parallax, and proper motions). 1332 million sources are treated as single stars and thus rep-

resentable by the five astrometric parameters (α,δ,$,µα∗,µδ). For an additional 361 million mostly faint sources, the sub-optimum quality of the fit resulted in only the mean positions being considered reliable enough for publication. Gaia DR2 parallax uncertainties are 0.04 ∼ milliarcsecond1 (mas) for sources with magnitude G < 15, around 0.1 mas for sources with G=17 and 0.7 mas at G = 20. The proper motion uncertainties are up to 0.06 mas/yr for G ∼ < 15 mag, 0.2 mas/yr for G = 17 mag and 1.2 mas/yr for G = 20 mag. The astrometric uncertainties are derived from the formal errors resulting from the astrometric data treatment. Gaia DR2 also contains significant spatial correlations, of up to 0.04 mas in parallax and 0.07 mas per year in proper motion on small (< 1 deg) and intermediate (20 deg) angular scales [Lindegren et al., 2018].

3.2 Distances for LBVs in Gaia DR2 3.2.1 Sample Selection

We searched the Gaia DR2 database [Brown et al., 2018]2 for all known Galactic LBVs and LBV candidates. As a convenient reference, we take the source list of Galactic LBVs and LBV candidates from the compilations by Clark et al. [2005] and Smith and Tombleson [2015]. To the list of Clark et al. [2005] we added SBW1 (candidate LBV), MWC 930, MN48, and WS1 (conﬁrmed LBVs). SBW1 was not listed in the compilation of LBVs and candidates by Clark et al. 2005 because it was discovered later in 2007 [Smith et al., 2007b]. It has a ring nebula similar to that of SN 1987A, and should be considered an LBV candidate for the same reason that Sher 25 and HD 168625 have been included in past lists of LBV candidates. MWC 930 was not included in the list by Clark et al. [2005] because its LBV-like

1 1 1 milliarcsecond= 3600000 degree 2http://gea.esac.esa.int/archive/

39 variability was discovered afterward in 2014 [Gvaramadze et al., 2012]. Our total sample of Galactic LBVs and LBV candidates consists of 25 objects. Of the LBV sources we checked, 4 conﬁrmed LBVs (η Car, GCIRS34W, FMM 362, and AFGL 2298) and 10 LBV candidates (GCIRS 16NW, 16C, 16SW, 33SE, 16NE, the Pistol Star, WR102ka, LBV 1806-20, G25.520+0.216, and G26.47+0.02) did not have parallax values in Gaia DR2. LBVs or LBV candidate stars in the vicinity of the Galactic Center are not listed in Gaia DR2 because they are visually obscured, including the Pistol Star, GCIRS 16NW, 16C, 16SW, 33SE, 16NE, 34W, etc. We note that FMM 362 has an almost coincident Gaia source where the DR2 parallax indicates a distance of only 1.6 kpc. This consumed our attention for some time, but detailed examination of images shows that this 18th magnitude Gaia source is oﬀset from FMM 362 by about 2.25 arcsec and is likely to be a foreground K or M-type star. FMM 362 itself is highly obscured. Some objects with IR-detected shells are also not found for similar reasons, including IRAS 18576+0341 (AFGL 2248), G25.520+0.216, and G26.47+0.02. MN44 [Gvaramadze et al., 2015] is a recent addition to the class of LBVs. However, in Gaia DR2 it has a negative parallax and an extreme astrometric noise parameter. Since there is no previous distance measurements (only example luminosities were given for assumed distances from 2 to 20 kpc), Gaia DR2 does not improve the situation and we do not discuss MN44 further here; we await DR3 for useful information on this source. Also not included in Gaia DR2 is the very massive star η Carinae. Its parallax is not available, but in this case the absence is presumably due to complications associated with its circumstellar Homunculus nebula. Fortunately, η Car already has a reliable distance of 2.3 kpc based on the expansion parallax of this nebula [Smith, 2006].

3.2.2 Previous Literature Distances for LBVs

Galactic distances are notoriously diﬃcult to measure. This is especially true for LBVs because (a) their luminosity cannot be uniquely determined from stellar features, (b) they often undergo signiﬁcant mass loss resulting in non-negligible circumstellar absorption, and (c) they sometimes exhibit peculiar velocities. As such, the previous literature estimates of

40 the distances to the LBVs and and candidates considered in this study arise from incredibly varied methods. For simplicity, we adopt the compilation of LBV and LBV candidate distances presented in the sample/review papers of Nazéet al. [2012] and van Genderen [2001b] as our “baseline” literature distances for each star. Due to their presentation in a unified location, their distances have been widely used when assessing the luminosity of the population of Galactic LBVs. It is these distances, supplemented by measurements from Gvaramadze et al. [2012], Kniazev et al. [2016], Smith [2007] for WS1, MN48, and SBW1, respectively, that are presented as dlit in the second column of Table 3.1 and the final column of Table 3.3. However, in attempting to present the “best” distance value for each star, these compilations have sometimes averaged together measurements from multiple literature studies, and in all cases the full allowable distance range, formal errors (if given), and measurement method from original works have been obscured. Therefore in Table 3.1 we additionally compile this information for publications whose measurements were utilized to compile our baseline distances. While not an exhaustive list of every published distance estimate, these values illustrate the typical range of distance measurements previously available for most galactic LBVs.

Table 3.1 serves as a reminder of just how unreliable and heterogeneous previous distance and luminosity estimates have been for Galactic LBVs. Once a distance was estimated in the literature, subsequent authors often engaged in more detailed study of the spectrum or variability of an LBV. In doing so, it was common to adopt a representative or average distance for the sake of discussion. These adopted values often propagated to subsequent papers as a ”standard” value, but often without emphasizing or retaining the uncertainty in the original distance measurement (indeed, such bookkeeping becomes cumbersome for a caveat-loaded discussion, and many HR diagrams of LBVs in the literature have no error bars on the luminosity). Nevertheless, the considerable uncertainty in distances has a profound impact on shaping our views of LBVs and their evolutionary scenarios. Even when uncertainties in luminosity or distance were given, the quoted error bars might not include the true uncertainty. For example, a distance estimate based on radial velocity and Galactic kinematics, or based on interstellar extinction, might yield a value with an error bar — but

41 Table 3.1: Previous literature distances for Galactic LBVs and candidate LBVs (in parentheses).

Baseline Distance Original Studies a Name dlit (kpc) Ref. dlit (kpc) Ref. Technique HR Car 5.20 N12 5.4±0.4 Hutsemekers and van Drom [1991] Kinematics 5±1 van Genderen et al. [1991] Reddening-Distance Relationship AG Car 6.00 N12 >5 Humphreys et al. [1989] Reddening-Distance Relationship 6.4−6.9 Humphreys et al. [1989] Kinematics 6±1 Hoekzema et al. [1992] Reddening-Distance Relationship Wra 751 6.00 N12 ∼6 Pasquali et al. [2006] Kinematics (private communication) >5 Hu et al. [1990] Reddening-Distance Relationship >4−5 van Genderen et al. [1992] Reddening-Distance Relationship Wd1 W243 5.00 N12 < 5 Clark et al. [2005] Cluster Membership - YHG Luminosity >2 Clark et al. [2005] Cluster Membership - WR Stars +0.5 5−1.0 Crowther et al. [2006] Cluster Membership - WR Stars 3.55±0.17 Brandner et al. [2008] Cluster Membership - MS Fitting HD 160529 2.50 N12 ∼2.5 Sterken et al. [1991] Comparison with R110 HD 168607 2.20 N12 DM=11.86 Humphreys [1978] Cluster Membership - MS fitting 2.2±0.2 Chini et al. [1980] Cluster Membership - MS Fitting P Cyg 1.70 N12 1.7±0.1 Najarro et al. [1997] Spectral modelling MWC 930 3.50 N12 3−4 Miroshnichenko et al. [2005] Associated with the Norma Spiral Arm Kinematics, Extinction arguments G24.73+0.69 5.20 N12 <5.2 Clark and Negueruela [2004] Reddening-Distance Relationship WS 1 11.0 G12 ∼11 Gvaramadze et al. [2012] Assume Luminosity MN48 5.00 K16 3−5 Kniazev et al. [2016] Assume association with spiral arm Luminosity, Kinematics arguments (HD80077) 3.00 N12 2.8±0.4 Steemers and van Genderen [1986] Cluster Membership - MS Fitting ∼3.2 Moffat and Fitzgerald [1977] Cluster Membership - MS Fitting (SBW1) 7.00 S7 ∼7 Smith et al. [2007b] Luminosity Class, Kinematics Arguments (Hen 3-519) 8.00 N12 >6 (∼8) Davidson et al. [1993] Reddening-Distance Relationship (Sher 25) 6.30 N12 6.3±0.6 Pandey et al. [2000] Cluster Membership - MS fitting 6.1±0.6 de Pree et al. [1999] Kinematics (ζ1Sco) 2.00 N12 2.0±0.2 Baume et al. [1999] Cluster Membership - MS fitting (HD 326823) 2.00 N12 >2 McGregor et al. [1988] Location in direction of galactic center 1.98 Kozok [1985] Luminosity Class, Color-Luminosity Relation (WRAY 17-96) 4.50 N12 <4.9 Egan et al. [2002] Reddening-Distance Relationship (HD 316285) 1.90 N12 ∼2 Hillier et al. [1998] Assumed, Reddening Arguments ∼3.4 van der Veen et al. [1994] Assumed, Reddening Arguments (HD 168625) 2.20 N12 DM=11.86 Humphreys [1978] Cluster Membership - MS fitting 2.2±0.2 Chini et al. [1980] Cluster Membership - MS Fitting (AS 314) 8.00 N12 ∼10 Miroshnichenko et al. [2000] Assume Luminosity (MWC 314) 3.00 N12 3.0±0.2 Miroshnichenko et al. [1998] Kinematics (W51 LS1) 6.00 N12 8.5 ±2.5 Schneps et al. [1981] maser proper motion - membership 6.1±1.3 Imai et al. [2002] maser proper motion - membership ∼5.5 Russeil [2003] kinematics <5.8 Barbosa et al. [2008] Radio Luminosity/Lyman Continuum Photons 2.0±0.3 Figuerêdoet al. [2008] spectroscopic parallax +2.9 5.1−1.4 Xu et al. [2009] trigonometric parallax + maser proper motion +0.31 5.41−0.28 Sato et al. [2010] trigonometric parallax + maser proper motion (G79.29+0.46) 2.00 N12 1−5, assume 2 Higgs et al. [1994] Reddening-Distance, Kinematics 1−4, assume 2 Voors et al. [2000] Reddening-Distance, Kinematics (CYG OB2 12) 1.70 N12 1.7±0.2 Torres-Dodgen et al. [1991] Cluster Membership - MS fitting DM=11.4±0.1 Massey and Thompson [1991] Cluster Membership - MS Fitting

aN12: Naz´eet al. [2012]; G12: Gvaramadze et al. [2012]; K16: Kniazev et al. [2016]; S7: Smith [2007].

42 without acknowledging that the method may not be valid for LBVs. As noted above, Galac- tic rotation might not work as a distance estimate if LBVs have peculiar velocities because they have received a kick, for example, and the interstellar extinction method might not work if LBVs have their own circumstellar extinction, or if their local region has interstellar dust grains with anomalous reddening properties because they have been processed by strong UV radiation. This serves to underscore the value of the new Gaia DR2 distances. Even though some of the objects have quite large DR2 error bars, they are most often comparable to or smaller than previous estimates. Moreover, they represent a single consistent method for all objects, and a direct (i.e., geometric) method that does not rely on sometimes dubious assumptions. The most commonly utilized methods in literature distance estimates were cluster association plus main sequence fitting (7 objects), kinematics plus inferences from the Galactic rotation curve (7 objects), and extinction measurements coupled with distance-reddening relations (7 objects). Other methods occasionally invoked include comparison to other LBVs/assumption of a luminosity on the S Dor instability strip [e.g. Gvaramadze et al., 2012, Miroshnichenko et al., 2000], spectroscopic modelling [Najarro et al., 1997], maser proper motions [Imai et al., 2002, Schneps et al., 1981, Xu et al., 2009], and spectroscopic parallax. A majority of LBVs and LBV candidates considered have distances quoted in the literature whose full ranges span >3 kpc. In particular, 12 of the 25 stars have only either approximate distances reported (no associated errors) or upper/lower limits to their distances quoted. The most precise literature measurements available come from cluster association and main sequence fitting, with typical quoted errors of .0.3 kpc. Even these, however, suffer from the sometimes questionable assumption of membership. Four stars in our sample have previous distance measurements that warrant specific discussion or mention:

• Wd1 W243: The distance to Wd1 W243 has been derived based on its presumed association with the cluster Westerlund 1. While the baseline distance to Wd1 given in Naz´eet al. [2012], Ritchie et al. [2009] is 5.0 kpc, a range of distance estimates to

43 Wd1 exist. Clark et al. [2005] initially quote a distance of >2 kpc and <5.5 kpc based largely on the luminosity of yellow hypergiants and the lack of identified Wolf Rayet +0.5 (WR) stars in their data. Crowther et al. [2006] subsequently quote a distance of 5−1.0 kpc based on modelling of WR stars, although they note an observed dispersion of σ 0.7 in the distance modulus for Wd1, and caution that WR stars do not represent ≈ ideal distance calibrators. Most recently, Brandner et al. [2008] fit main-sequence and pre-main sequence evolutionary tracks to near-infrared data, finding both a more precise, and significantly lower, distance of 3.55 0.17 kpc. (In the next chapter, we ± discuss the revised cluster distance to Wd1 based on Gaia DR2; Aghakhanloo et al. 2020.)

• W51 LS1: Similarly, the distance to LBV candidate W51 LS1 has been tied to the— surprisingly fraught—distance to the W51 complex (see Clark et al. 2009 and Figuerêdo et al. 2008 for recent summaries). Early maser proper motion measurements gave relatively large distances of 6.1 1.3 kpc and 8.5 2.5 kpc [Imai et al., 2002, Schneps ± ± et al., 1981]. These were broadly consistent with kinematic measurement of 5.5 kpc ∼ [Russeil, 2003], leading Clark et al. [2009] and Nazéet al. [2012] to adopt a baseline distance to the LBV candidate W51 LS1 of 6 kpc. However, using spectroscopic parallax measurements of 4 O-type stars, Figuerêdoet al. [2008] find a much smaller distance of 2.0 0.3 kpc, a distance they acknowledge is difficult to reconcile with ± previous measurements. Most recently, updated trigonometric parallax measurements coupled with maser proper motions have given distances slightly lower than our baseline +2.9 value, but still inconsistent with the spectroscopic parallax values: 5.1−1.4 kpc [Xu +0.31 et al., 2009] and 5.41−0.28[Sato et al., 2010].

• HD168607/HD168625: In the literature studies presented here, LBV HD168607 and LBV candidate HD168625 are both assumed to be members of the same stellar association, Ser OB1, and hence have the same distance.

3.2.3 New Gaia DR2 Data

Table 3.2 presents the Gaia DR2 data for the 25 LBVs and candidates (names for LBV candidates are given in parentheses) analyzed in this chapter, and Table 3.3 lists the distances that we infer from Gaia DR2 parallaxes. The last column of Table 3.3 gives a “baseline” previously adopted distance from the literature (see Section 3.2.2 for further elaboration on

44 Table 3.2: Parameters from the Gaia DR2 and Bailer-Jones catalogs.

a Name Gaia DR2 id $ (mas) σ$ (mas) N RUWE (mas) D ` (kpc) dBJ dBJ,low dBJ,high HR Car 5255045082580350080 0.171 0.033 18 1.04 0.000 0.0 1.66 4.89 4.20 5.82 AG Car 5338220285385672064 0.153 0.029 17 0.88 0.000 0.0 1.60 5.32 4.59 6.29 Wra 751 5337309477433273728 0.169 0.044 19 1.02 0.149 8.3 1.57 4.82 3.97 6.06 Wd1 W243 5940105830990286208 0.979 0.165 11 1.11 0.582 120.1 1.38 1.03 0.86 1.27 HD 160529 4053887521876855808 0.438 0.057 10 0.94 0.000 0.0 2.15 2.18 1.92 2.50 HD 168607 4097791502146559872 0.644 0.060 10 1.02 0.000 0.0 1.52 1.50 1.37 1.65 P Cyg 2061242908036996352 0.736 0.180 17 1.05 1.085 437.5 1.16 1.37 1.06 1.93 MWC 930 4159973866869462784 -0.162 0.094 11 0.74 0.217 10.0 1.36 7.81 5.78 10.64 G24.73+0.69 4255908794692238848 -0.329 0.223 10 1.03 1.302 233.6 1.44 5.44 3.58 8.25 WS 1 4661784273646151680 -0.051 0.031 18 1.25 0.222 8.0 0.48 8.68 7.50 10.10 MN48 5940216130049700480 0.323 0.135 12 1.02 0.469 77.9 1.38 2.82 2.00 4.40 (HD80077) 5325673208399774720 0.392 0.031 15 0.94 0.000 0.0 1.20 2.37 2.21 2.57 (SBW1) 5254478417451126016 0.128 0.030 17 0.87 0.000 0.0 1.58 6.01 5.10 7.26 (Hen 3-519) 5338229115839425664 0.042 0.030 17 1.03 0.000 0.0 1.61 9.57 7.72 12.17 (Sher 25) 5337418397799185536 0.072 0.033 19 0.91 0.000 0.0 1.59 7.88 6.39 10.03 (ζ1 Sco) 5964986649547042944 0.713 0.242 12 0.94 0.947 297.5 1.36 1.51 1.01 2.74 (HD 326823) 5965495757804852992 0.743 0.053 11 1.05 0.000 0.0 1.41 1.30 1.22 1.40 (WRAY 17-96) 4056941758956836224 0.827 0.181 10 0.85 0.467 54.4 2.34 1.26 0.97 1.78 (HD 316285) 4057682692354437888 0.169 0.092 10 0.94 0.000 0.0 2.28 4.90 3.35 7.90 (HD 168625) 4097796621733266432 0.621 0.064 9 1.02 0.000 0.0 1.52 1.55 1.41 1.73 (AS 314) 4103870014799982464 0.624 0.052 11 0.96 0.000 0.0 1.75 1.54 1.42 1.68 (MWC 314) 4319930096909297664 0.191 0.042 13 0.91 0.000 0.0 1.34 4.36 3.67 5.33 (W51 LS1) 4319942771347742976 0.508 0.116 14 1.08 0.741 133.4 1.32 1.91 1.53 2.54 (G79.29+0.46) 2067716793824240256 0.180 0.139 15 1.06 0.648 150.3 0.79 3.09 2.29 4.36 (CYG OB2 12) 2067782734461462912 1.175 0.128 16 1.52 0.588 122.4 0.80 0.84 0.75 0.95

a The following are from the Gaia DR2 catalog: $ is the parallax, σ$ is the expected uncertainty in the p parallax, N is the number of visibility periods, RUWE is the re-normalized goodness of fit ( χ2/(N 5)), is the excess astrometric noise, D is the significance of the excess astrometric noise. The following are− from the Bailer-Jones catalog: ` is the length scale of field stars in the direction of the LBV or candidate, dBJ is the most likely Bailer-Jones distance, dBJ,low and dBJ,high give the highest 68% density interval (HDI). this baseline value as well as a discussion of the methods, spread, and uncertainty in previous literature estimates).

Table 3.2 includes the Gaia DR2 data and distance estimates from Bailer-Jones et al. [2018]. The ﬁrst two columns present the parallax, $, and the theoretical uncertainty in

the parallax, σ$. One could simply report d = 1/$, and one could use σ$ to calculate an uncertainty for the distance. However, such a calculation would not fully account for all of the calibration issues or extra sources of uncertainty. Furthermore, the uncertainty for many objects in DR2 is quite large. In fact, the uncertainty can be so large that the parallax can be negative. In these circumstances, inferring the distance and uncertainty from d = 1/$ is either inaccurate or impossible. A preferred solution for all of these issues is to infer the distance using Bayesian inference [Luri et al., 2018].

45 Table 3.3: LBVs and LBV candidate (in parentheses) Gaia DR2 distances.

a Name RA(deg) DEC(deg) dBayes (kpc) dlow (kpc) dhigh (kpc) dlit (kpc) HR Car 155.72429 -59.62454 4.37 3.50 5.72 5.20 AG Car 164.04820 -60.45355 4.65 3.73 6.08 6.00 Wra 751 167.16688 -60.71436 3.81 2.49 6.26 6.00 Wd1 W243 251.78126 -45.87477 1.78 0.83 4.16 4.50 HD 160529 265.49594 -33.50381 2.10 1.80 2.51 2.50 HD 168607 275.31203 -16.37550 1.46 1.31 1.65 2.20 P Cyg 304.44665 38.03290 2.17 0.98 4.25 1.70 MWC 930 276.60514 -7.22165 4.46 2.89 6.93 3.50 G24.73+0.69 278.48031 -6.97742 3.04 1.52 5.47 5.20 WS 1 73.23917 -66.68708 2.47 1.83 3.38 11.00 MN48 252.40708 -45.59980 2.74 1.44 5.18 5.00 (HD80077) 138.97824 -49.97347 2.26 2.01 2.59 3.00 (SBW1) 160.08071 -59.81940 5.11 4.03 6.78 7.00 (Hen 3-519) 163.49819 -60.44564 7.12 5.45 9.65 8.00 (Sher 25) 168.78180 -61.25488 6.28 4.82 8.52 6.30 (ζ1 Sco) 253.49886 -42.36204 2.52 1.12 4.97 2.00 (HD 326823) 256.72461 -42.61104 1.27 1.17 1.40 2.00 (WRAY 17-96) 265.39765 -30.11078 3.02 1.16 7.16 4.50 (HD 316285) 267.05848 -28.01478 4.56 3.05 7.63 1.90 (HD 168625) 275.33145 -16.37392 1.51 1.34 1.73 2.20 (AS 314) 279.85874 -13.84646 1.50 1.36 1.68 8.00 (MWC 314) 290.39156 14.88245 3.93 3.15 5.11 3.00 (W51 LS1) 290.94849 14.61083 2.50 1.19 4.87 6.00 (G79.29+0.46) 307.92617 40.36639 1.87 1.07 3.27 2.00 (CYG OB2 12) 308.17065 41.24145 1.04 0.60 2.21 1.70

a dBayes is the most likely distance to the LBV or LBV candidate. dlow and dhigh give the highest 68% density interval (HDI). Details of the Gaia DR2 observations are in Table 3.2. dlit is the nominal distance typically adopted in the literature (see text section 3).

The fifth and sixth columns present the excess astrometric noise, , and the excess astrometric noise significance, D. See Lindegren et al. [2012] for a thorough discussion of the astrometric noise. In short, the excess astrometric noise represents variation in the astrometric data beyond the standard astrometric solution. The standard astrometric solution is composed of five parameters: two sky coordinates, two proper motion coordinates, and a parallax. Any unknown calibration issues or extra motions due to binaries, etc., lead to excess astrometric noise. D is a measure of the significance of the excess astrometric noise. Even in the ideal situation with no true excess noise, random noise produces excess noise in about half the measurements. In the ideal situation, 98.5% of measurements have a significance of D < 2. Column seven of Table 3.2 shows significant excess noise for several of the LBV and candidate sources. Therefore to properly infer the distribution for the distance, one must include the excess noise.

46 Columns 10 12 of Table 3.2 give the Bayesian inference for the distances by Bailer- − Jones et al. [2018]. Unfortunately, these distance estimates do not include the excess noise, (WELL, THEY DO - JUST NOT IN A GOOD WAY!) so we merely report them for comparison. To obtain the Bayesian-inferred distances, we searched the catalogue at http://gaia.ari.uni-heidelberg.de/tap.html which reports geometric distances inferred from Gaia DR2 parallaxes [Bailer-Jones et al., 2018]. Column nine gives an estimate for the length scale of field stars in the direction of the LBV or candidate. This length scale is a parameter in the prior of the Bailer-Jones et al. [2018] estimate. dBJ is the mode of their posterior, and dBJ,low and dBJ,high give the upper and lower bounds for the highest density 68% confidence interval. Bailer-Jones et al. [2018] used a parallax zero-point (an offset in the parallaxes due to various instrumental effects) of $ = 0.029 mas for their inference. zp − However, subsequent work indicates that the zero-point has significant variation [Lindegren et al., 2018, Riess et al., 2018, Stassun and Torres, 2018, Zinn et al., 2019].

3.2.4 New Bayesian-inferred Distances

The primary goal here is to use Bayesian inference to estimate the distance given by data in Gaia DR2. To infer the distance to any star, the posterior distribution for the distance, d, and the zero point, $zp is

P (d, $ $, σ$, , µ , σ , `) ($ d, σ$, , $ ) P ($ µ , σ ) P (d `) . (3.1) zp| zp zp ∝ L | zp × zp| zp zp × |

$ and σ$ are the parallax and expected uncertainty, respectively, is the excess astrometric noise, and µ and σ are the mean and variance of the zero point parallax. ($ d, σ$, , $ ) zp zp L | zp is the likelihood of observing parallax $ given the model parameters. P ($ µ , σ ) is the zp| zp zp distribution of zero points. P (d `) is the prior of observing d given the galactic length scale | of stars in the FOV.

The parameter σ$ is the expected theoretical uncertainty in the parallax, but it does not represent the full uncertainty. Lindegren et al. [2018] found that the empirical uncertainty is 1.081 times larger than the expected value. For many objects there is also an excess astrometric noise, . Formally, represents excess noise in the entire astrometric solution,

47 not just the parallax. The excess noise absorbs the residual in the astrometric solution. It is determined by adding in quadrature to the position uncertainty in the denominator of the χ2 minimization formula. Minimizing χ2 finds a simultaneous solution for the five astrometric parameters and the excess noise. The uncertainties of the astrometric solution are calculated using standard error propogation. Therefore, the uncertainty in the DR2 parallax includes the excess noise. However, the procedure for including the excess noise assumes that the excess noise is gaussian and uncorrelated among the observed positions. If the excess noise is a correlated systematic residual, then the excess noise model in the Gaia pipeline would not capture the true behavior of the residual. For example, when assuming that the excess noise is uncorrelated, the uncertainty in the parallax would go down by √N, where N is the number of observations. On the other hand, a systematic uncertainty may not reduce with added observations. Without knowing the source of the excess noise, it is difficult to determine whether the excess noise should be modeled as correlated or uncorrelated noise. Aghakhanloo et al. [2020] measured the empirical uncertainty distribution for the star cluster Westerlund 1 in the same way that Lindegren et al. [2018] measured the empirical uncertainty distribution for all quasars in Gaia DR2. Whereas the empirical uncertainty is 1.081 times larger for the quasars, Aghakhanloo et al. [2020] found that most of the stars have a large astrometric excess noise and the empirical uncertainty for the stars in Westerlund 1 is 1.6 times larger than the DR2 solution. This may suggest that the assumption of uncorrelated excess noise is inconsistent with the data. Instead, the excess noise may not be random noise but a correlated residual. Since in this chapter we analyze individual stars, it is impossible to empirically estimate the uncertainty for each. Yet, several LBVs have large astrometric excess noise, and it is likely that the excess noise contribution to the parallax uncertainty is underestimated. W243 illustrates this point. W243 is in Westerlund 1, which has a distance of 3.2 0.4 kpc ± [Aghakhanloo et al., 2020] as determined by a large statistical sample of DR2 parallaxes of +0.24 cluster members. Yet, the DR2 distance to the individual star W243 is 1.03−0.17 kpc. Using the DR2 uncertainty, 0.24 kpc, this would be 9 sigma from the statistically inferred distance

48 to Westerlund 1. Even using the statistical uncertainty of 0.4 kpc from Aghakhanloo et al. [2020], the Gaia DR2 estimate for W243 is still 5 sigma from the statistical cluster estimate. Clearly, the parallax uncertainty in DR2 is underestimated in some cases. Given the above problems with the astrometric excess noise, and our goal of estimating distances for individual objects, we choose to adopt a conservative uncertainty estimate. In our analysis, we found that making diﬀerent assumptions nevertheless led to robust results for sources with small or zero , but for sources with signiﬁcant astrometric noise, the resulting distance could jump by much more than the DR2 uncertainty. Hence, the excess residual that contributes to the parallax should be increased for such sources. In the absence of further information, we adopt the most conservative approach, and add the full excess noise,

in quadrature with the theoretical parallax uncertainty, σ$:

2 2 2 σ = (1.081σ$) + . (3.2)

While this yielded rather large uncertainty for some sources, we found that this uncertainty encompassed the full variation we had found in derived distances under different assumptions in the analysis. With the larger uncertainty included this way, we achieved consistent results. In many cases, these large uncertainties in the resulting distances for sources with large are still an improvement over estimates in the literature. Inferred distances are also affected by the adopted zero point offset in the astrometric solution for the parallax. µzp and σzp are the mean and variance for the zero point parallax. In the initial astrometric solution for DR2, Lindegren et al. [2018] used quasars to quantify the zero point. They found an average of 0.029 mas. In addition, they noted significant − variation in the zero point as a function of sky position and other possible parameters such as brightness and color. They did not quantify the variation in zero point; rather they encourage users of Gaia DR2 to model their problem-specific zero point. Riess et al. [2018] use Gaia DR2 parallaxes to calibrate Cepheid distances, and infer a zero point of 0.046 − ( 0.013) mas. Zinn et al. [2019] compare the distances inferred from astroseismology to infer ± a zero point of 0.0528 ( 0.0024) mas. Stassun and Torres [2018] use distances derived from − ± eclipsing binaries to infer a zero point of 0.082 ( 0.033) mas. These are all estimates for − ± 49 the mean zero point µzp; Lindegren et al. [2018] note a spatial variation of the offset of about 0.03 mas in the direction of the LMC. Therefore, we assume that σ = 0.03. Unfortunately, ± zp determining the zero point specifically for LBVs is difficult. They typically lie in the plane of the Galaxy, where there are no observable quasars, and there are no accurate distance measures for LBVs. Therefore, our best estimate for the zero point is the mean of the above four investigations, µzp = 0.05 mas, and we use the spatial variation from Lindegren et al.

[2018] for σzp = 0.03. These measures of the zero-point are far from ideal, but they are a good first attempt. The likelihood and distribution for the zero points are as follows: The likelihood for observing parallax $ is 2 1 ($ 1/d $zp) ($ d, σ$, , $zp) = exp − − − . (3.3) L | √2πσ 2σ2 Lindegren et al. [2018] show that the zero point has significant variation as a function of sky location. Therefore, we model the distribution of zero points as 2 1 ($zp µzp) P ($zp µzp, σzp) = exp − −2 . (3.4) | √2πσzp 2σzp P (d `) represents the prior for d based upon the Galactic distribution of stars and dust | extinction. Consider an image populated with Galactic stars. The total number of stars in the image is given by N = FOV R nr2dr, where FOV is the field of view in square radians, and n is the number density of stars. If n is constant, then any random star in the image is drawn from a probability distribution of P (r) r2. In the presence of dust extinction, ∝ this distribution will be attenuated by exp ( r/`) , where ` is an effective optical depth { − } for extinction. For these reasons, Bailer-Jones et al. [2018] use the following prior when calculating the geometric distance to stars in Gaia DR2: 1 P (d `) = d2 exp( d/`) , (3.5) | 2`3 − where ` is the attenuation length and depends upon the Galactic coordinates (l, b). This prior has a mode at 2`. To find the posterior distribution for only the distance, we marginalize eq. (3.1) over the nuisance parameter, the zero point, $zp. In practice, this marginalization involves a

50 convolution of the likelihood, eq. (3.3), with the distribution for zero points, eq. (3.4). The convolution of two Gaussian distributions is analytic and so is the ﬁnal posterior distribution.

2 1 ($ 1/d µzp) 1 2 P (d $, σ$, , µzp, σzp, `) q exp − − 2 −2 3 d exp( d/`) . | ∝ 2 2 2(σ + σzp) × 2` − 2π(σ + σzp) (3.6)

The geometric distances in columns 4-6 of Table 3.3 are the mode (column 4) and the highest density 68% interval (HDI, columns 5 & 6) for this posterior distribution. The 68% HDI speciﬁes an interval that spans 68% of the distribution such that every point inside the interval has higher probability density than any point outside the interval. The seventh column in Table 3.2 gives the attenuation scale, `, in the prior. These were calculated using the same technique as in Bailer-Jones et al. [2018].3 The posterior distribution seamlessly handles both accurate and inaccurate parallax measurements. In the limit of an accurate parallax measurement, the width of the posterior will be dominated by the Gaussian and consequently σ. In the limit of very inaccurate parallax measurements, the mode and uncertainty will be dominated by the prior. Therefore, in the inaccurate cases, the most likely distance will be 2`. Since LBVs may be atypical in their distance distribution, it is not clear that the Bailer- Jones et al. [2018] prior is the most appropriate prior to use. LBVs tend to be quite bright, so the distance scale, `, should probably be larger, and it is not clear that LBVs should trace the general population of stars. The prior mostly aﬀects the result of those with very uncertain parallaxes.

Fig. 3.1 compares all four distance estimates: our Bayesian-inferred distance (dBayes) is compared to (bottom panel) the Bailer-Jones Bayesian estimate, then (middle panel) the

simple 1/$ calculation (d$), and (top panel) literature distances (dlit). In all panels, the ﬁlled circles represent LBVs and the open circles represent the LBV candidates.

The bottom panel of Fig. 3.1 compares the mode and 68% HDI of eq. (3.6), dBayes with the mode and HDI of Bailer-Jones et al. [2018], dBJ. The primary diﬀerences are that

3The python code to calculate these distances is available at the following GitHub repository: https://github.com/curiousmiah/Gaia Distances.

51 101 [kpc] Lit d

100

101 [kpc] /̟ 1

100

101 [kpc] BJ d

100 100 101 dBayes [kpc]

Figure 3.1: Distances by Bayesian inference, dBayes (Table 3.3) compared to: (top) literature distances, dLit, (Table 3.3), (middle) distances given by 1/$ (Table 3.2), and (bottom) the Bailer-Jones Bayesian distances, dBJ [Bailer-Jones et al., 2018] (Table 3.2). Filled circles represent the LBVs and open circles represent the LBV candidates. For roughly half of the sample, dBayes is signiﬁcantly closer than the literature distances. This has consequences for the inferred luminosities and masses for many LBVs.

52 the posterior distribution for dBayes includes a more conservative prescription for the excess astrometric noise, a slightly more negative zero point, and a variation in the zero point. The zero point in the d distances is 0.029 mas, and the zero point for d is 0.05 BJ − Bayes − ( 0.03) mas. As a result, the uncertainties for d are larger when the excess noise is ± Bayes non-zero. Because the zero point for dBayes is slightly more negative, those distances tend to

be smaller than dBJ.

The middle panel of Fig. 3.1 compares dBayes with d$ = 1/$. Naturally, both methods

are roughly consistent; the derivation of dBayes uses $ after all. However, there are some

noteworthy diﬀerences. At distances larger than 4 kpc, the d$ distances are systematically ∼ larger than dBayes. This systematic discrepancy is mostly due to the parallax zero-point. For

some, the prior distribution dominates the posterior for dBayes; for these distance estimates,

dBayes tends to have much larger mode and uncertainty.

The top panel of Fig. 3.1 compares dBayes with the baseline previous literature estimates,

dlit, described in Section 3.2.2. On average, the literature distances are larger than the new inferred distances; the average literature distance is 4.1 0.6 kpc, and the average d ± Bayes distance is 3.1 0.3 kpc. For about half of the objects, the two estimates are consistent. For ± most of the other half, the literature distances significantly overestimate the distance. To be more quantitative, there are 25 LBVs and candidates. 11 of the literature distances are outside of the 68% confidence intervals (CIs). In particular, 2 are below the CIs, and 9 are above. The expectation is that roughly 16% should be below and 16% should be above; 16% of 23 is 4. The Poisson probability of 2 below when the expected value is 4 is 15%, which is roughly consistent. The Poisson probability of 9 above is 1.5%. As a population, there is a tension between the literature distances and the Gaia DR2 distances in that the literature distances tend to be too large. This has a significant impact on the inferred luminosities and masses of LBVs. Fig. 3.2 shows a revised HR diagram for LBVs using updated distances from Gaia DR2

from Table 3.3. To construct this, we adopted previously published values of Teﬀ and LBol compiled from the literature [Smith and Stassun, 2017, Smith and Tombleson, 2015, Clark et al., 2005, Naz´eet al., 2012] with their associated previous distances (see Section 3), and we

53 simply scaled the bolometric luminosities appropriate to the new DR2 distances4. Fig. 3.3 shows a similar HR diagram with the new DR2 values compared to previous literature estimates. Some LBVs changed little and others changed dramatically. LBV positions based on previous literature values are plotted in red, and those with their LBol scaled by the new Gaia DR2 distances are plotted in black. Fig. 3.5 then shows these same new values, but superposed with additional information for context, including extragalactic LBVs in nearby galaxies, the previously proposed S Doradus instability strip, locations of B supergiants and B[e] supergiants, a few SN progenitors, and representative stellar evolution model tracks. Extragalactic LBVs in the LMC, SMC, M31, and M33 are plotted in light purple for comparison. Representative single-star and binary evolution tracks are included for comparison, as in earlier versions of this ﬁgure by Smith and Stassun [2017] and Smith and Tombleson [2015]. These model tracks are from Brott et al. [2011] and Langer and Kudritzki [2014].

3.2.5 The Signiﬁcance of the Zero-point and the Lower Distances

The Gaia DR2 parallaxes provide the largest collection to date of LBV and LBV candidate distances that are measured in a uniform and direct way. As an ensemble, these distances are significantly lower than previous literature distances. In total, there are 25 LBVs and candidates, 11 have literature distances that are larger than the 68% HDI. The Gaia DR2 distances are closer than previous estimates for two reasons. First, on average, the measured parallaxes are larger. Second, there is a significant negative zero point offset for Gaia DR2. In other words, the true parallax is larger by 0.05 mas. HD168607 is an example for which the Gaia DR2 distance is closer even if one ignores the parallax zero-point. The previously reported distance to HD 168607 is 2.2 kpc. Including the parallax zero-point, the +0.15 Gaia DR2 distance is 1.46−0.14 kpc. Without applying the zero-point offset, the Gaia DR2 +0.19 distance is 1.57−0.15 kpc. AG Car is an example for which the offset changes the consistency between the Gaia DR2 distance and the previous estimate. The previous distance estimate was 6 kpc. With

4 Recall that determinations of Teﬀ from spectroscopic analysis have negligible dependence on the distance (Chapter 1).

54 Figure 3.2: The HR diagram showing only Galactic LBVs (ﬁlled circles) and Galac- tic LBV candidates (unﬁlled circles) with their luminosities scaled by the revised Gaia DR2 distances (dBayes). For this plot, we use the new DR2 distances from Table 3.3 (see text). Here we do not show the presumed location of the S Dor instability strip, stellar evolution model tracks, or any extragalactic LBVs.

+1.17 a zero point offset the Gaia DR2 distance is 4.73−0.81 kpc, and without the zero point, it is +1.5 6.0−1.1 kpc. With the offset, there is a mild tension between the Gaia DR2 distance and the previous distance esimate. Without the offset, both distances are consistent. Collectively, if one omits the offset, then the number of objects with previous distances that are outside the 68% HDI reduces from 11 to 10. The Poisson probability of 8 when 4 is expected is 3%, while for 9 it is 1.3%. Therefore, the tension only slightly depends upon the reliability of the zero-point estimate. It is difficult or impossible to measure the offset for each LBV directly. There are few accurate independent distance estimates for LBVs in the Galaxy, and because they reside in the Galactic plane, there are no bright background quasars to provide the zero-point calibration. Based upon the estimates of four independent groups, the global average offset is consistently measured to be about 0.05 mas. The − average offset for 556,869 quasars is 0.029 mas [Lindegren et al., 2018]; the average offset − for 50 Cepheid variables is 0.046 ( 0.013) mas; the average offset of 3500 giants with − ± ∼

55 Figure 3.3: Same as Fig. 3.2, but including positions of LBVs based on both the old literature distances (red) and those inferred from Gaia DR2 distances (black). The S Dor instability strip and constant temperature eruptive LBV strip are also indicated, as in Smith et al. [2004].

astroseismology is 0.0528 ( 0.0024) mas, and the average offset for 89 eclipsing binaries − ± is 0.082 ( 0.033) mas. The mean of these is 0.05 and the standard deviation is 0.02 − ± − mas. This represents the global average, but Lindegren et al. [2018] note a significant spatial variation in the offset. In the direction of the LMC the variation is about 0.03 mas. ±

56 3.3 Notes on Individual LBVs

Here we provide notes on each of the 25 LBVs and candidates discussed in this work and the impact of the new Gaia DR2 distances. In this sample of 25 LBVs and candidates, 11 have Gaia DR2 posterior distances for which the 68% HDI is closer than the literature distances. This represents almost half of the LBV and LBV candidate sample, even though the expectation is that only 4 should have closer distances. The Poisson probability of having 9 closer when 4 are expected is 1.3%. Some of these distance estimates changed by a large factor.

3.3.1 Conﬁrmed LBVs

HR Carinae: The Gaia DR2 Bayesian-inferred distance (Table 3.3) is 15% closer than its traditional value. Its luminosity therefore moves down by 28% on the HR Diagram. The Gaia DR1 parallax yielded a smaller distance around 2 kpc as well, although with a large uncertainty, so Smith and Stassun [2017] did not advocate a revision to the traditional distance. The new value is consistent within the uncertainty of DR1, although the distance from DR2 is more precise. Here, the new distance is somewhat lower than dlit and the star somewhat less luminous, but marginally consistent with the traditional value within the uncertainty. Therefore, the Gaia DR2 distance for HR Car does not signiﬁcantly alter our interpretation of this objects parameters [Groh et al., 2009a]. AG Carinae: As for HR Car, the Gaia DR2 distance derived in Table 3.3 is slightly closer than the traditional value. Adopting the most likely value in Table 3.3 (plotted in Fig. 3.5), AG Car is about 20% closer (moving from 6 kpc to 4.7 kpc) and is therefore about 40% less luminous than previously determined from detailed analysis by Groh et al. [2006, 2009c]. However,the large uncertainty means that there is only a weak tension between the Gaia DR2 and literature distance. In fact, accounting for the errors in the original kinematics and reddening-distances measurements [Humphreys et al., 1989, Hoekzema et al., 1992], there is overlap with the upper end of the Gaia DR2 68% HDI. This is in contrast to the much closer distance inferred from the Gaia DR1 parallax, which was reported to be about 2 kpc [Smith and Stassun, 2017]. That closer distance was evidently an underestimate, although

57 the range of distances indicated by the DR1 uncertainty of 1.3 to 3.7 kpc are marginally consistent with the DR2 68% HDI range found here of 3.7 tp 6.1 kpc. A much closer distance for AG Car would have been surprising and problematic, since it would have moved AG Car – a prototypical classical S Dor variable – to be far below the S Dor instability strip that it helped to deﬁne, and far from its near twin R 127 in the LMC. Gaia DR2 values suggest that more modest revision to its traditionally assumed distance is needed. The new DR2 value is consistent with the traditional S Dor strip, although its implied eﬀective initial mass

(compared to single-star models) would be around 60-70 M instead of 90 M . ∼ Wra 751: The Gaia DR2 distance for this LBV in Table 3.3 moves it from a baseline literature distance of 6 kpc to 3.8+2.4 kpc. While a significant nominal correction, this is ∼ −1.3 just barely consistent with the old value within the uncertainty, and has more overlap with the original distance measurements of van Genderen et al. [1992] who simply quote limits of 4 5 kpc. With the more reliable uncertainties of the Gaia DR2 distance, this provides a & − stronger case that Wra 751 is significantly below or hotter than the S Dor instability strip on the HR Diagram. If Wra 751 really is off the S Dor instability strip, then this is interesting because it is now considered a confirmed LBV, due to the development of apparent S Dor variability [Sterken et al., 2008, Clark et al., 2005]. This is also of interest because Wra 751 has been associated with a presumed host cluster [Pasquali et al., 2006]. It therefore bucks the trend that LBVs statistically avoid O-star clusters [Smith and Tombleson, 2015]. Although Wra 751 is in a cluster, the cluster’s age and turnoff mass do not agree with those of the LBV if it has evolved as a single star. From its position on the HR Diagram, we would expect Wra 751 to have an effective single-star initial mass around 30-40 M . As noted by Pasquali et al. [2006], however, the earliest spectral type main sequence star in the host cluster is O8 V, translating to a cluster turnoff mass around 20 M . This is consistent with results from LMC LBVs, where it has been noted that in the few cases when LBVs are seen in a cluster, they seem too young and massive for their environment [Smith and Tombleson, 2015]. This, in turn, reinforces ideas about the role of binary evolution and rejuvenation that make LBVs massive blue stragglers [Smith and Tombleson, 2015, Smith,

58 2016b, Aghakhanloo et al., 2017]. Wra 751 would be an excellent case study for investigating how its proper motion compares to those of the cluster members. Wd1 W243: This is a special case because it has a host cluster with a Gaia DR2 distance; see Section 3.3 below. +0.36 HD 160529: Gaia DR2 indicates that this object’s new distance is 2.09−0.27 kpc, which is smaller than its usually adopted value of 2.5 kpc. Its luminosity moves down by about 30%, although the old value is just outside the 68% confidence interval. This places HD 160529 at the very bottom of the traditional range of luminosities for LBVs. HD 168607 and HD 168625: Although HD 168625 is a candidate LBV (section 3.2 below), we discuss it here alongside HD 168607. These two are usually considered as a pair since they are only 1 arcmin apart on the sky. As described in Section 3.2.2, HD168607 and HD168625 have both been considered members of the same stellar association Ser OB1. HD168607 has long been considered as a confirmed LBV [Chentsov, 1980] in the group of low-luminosity LBVs [Smith et al., 2004], sometimes thought to arise from post-RSG evolution. HD168625 is a blue supergiant and considered an LBV candidate based on its dusty circumstellar nebula, which is observed to have a triple-ring structure that is very reminiscent of SN 1987A [Smith, 2007]. The new Gaia DR2 measurements for each star +0.17 +0.19 are both very consistent with one another (1.46−014 and 1.55−0.16; supporting their joint membership in Ser OB1) and significantly lower than the usual literature value of 2.2 kpc. The new Gaia DR2 distances (1.32-1.63 kpc) are consistent (within the uncertainty) with the Gaia DR1 distance of 0.75-1.89 kpc for HD168607 previously reported [Smith and Stassun, 2017]. With the new distances, the luminosity of each star drops by a factor of 2 from literature measurements. Importantly, both stars land well below the traditional range of luminosities for LBVs and well below the S Dor instability strip. For both stars, the lower luminosity would imply an effective single-star initial mass of around 20 M or below. In the case of HD168625, this lower luminosity is very close to that of the progenitor of SN 1987A, strengthening comparisons between these two objects that were previously based primarily on the triple-ring structure of the nebula around HD 168625 [Smith, 2007].

59 P Cyg: P Cyg has a large excess noise, = 1.1 mas, and the observed parallax is only $ = 0.736 mas. Therefore, the range of allowed distances is quite large; the 68% HDI goes from 0.98 to 4.25 kpc. This is consistent with the previous estimate of 1.7 kpc. Unfortunately, we cannot say anything new about the distance and luminosity of this LBV due to the large astrometric noise and large DR2 uncertainty. MWC 930 (V446 Sct): The distance for this conﬁrmed LBV has a negative parallax in Gaia DR2. A crude way to deal with the negative parallax is to treat the parallax uncertainty as a lower limit for the distance. Including both the statistical uncertainty and the excess noise, the total uncertainty is σ = 0.24 mas, and the corresponding lower limit for this LBV is 4.2 kpc. At face value it would seem that the lower limit is farther than previously adopted literature value. However, the more formal statistical Bayesian inference gives a distance of +2.4 4.5−1.5 kpc (Table 3.3). This value is dominated by the adpted prior. This is marginally consistent with the previous literature value of 3.5 kpc. G24.73+0.69: This LBV with a dusty shell has a negative value for the parallax listed in Gaia DR2, and is similarly dominated by the prior. The statistical uncertainty for

G24.73+0.69 is only σ$ = 0.223 mas, but the excess noise is quite large, = 1.302 making our formal uncertainty σ = 1.3 mas. In this case, the lower limit for G24.73+0.69 is 0.8 kpc. +2.6 Given this large uncertainty, the Bayesian distance in Table 3.3, 3.0−1.6 kpc, is dominated by the prior. This is lower than the literature value of 5.2 kpc, but consistent within the uncertainty, and so this star is still marginally consistent with being an LBV on the S Dor instability strip. The higher precision expected in future Gaia data releases is needed to say more. WS 1: WS 1 was classified as a bona-fide LBV by Kniazev et al. [2015] based on observations of significant photometric and spectroscopic variability. The Gaia DR2 distance of 2.48+0.73kpc is significantly lower than the baseline literature value of 11 kpc, although this −0.54 ∼ object has a negative parallax and like the previous two, is dominated by the prior. However, the literature luminosity is also highly uncertain because it is not based on a measurement — itwas derived by assuming that the star fell on the S Dor instability strip Gvaramadze et al. [2012], so perhaps a large revision is not so surprising. Gaia data suggests that the

60 star is significantly lower in luminosity, although again DR3 data are needed for a confident result. MN48: MN48 was classified as a bona fide LBV by Kniazev et al. [2016] who identified spectroscopic and photometric variability typical of LBVs. There is large excess noise in the Gaia DR2 data for this star, and the resulting 68% HDI spans 1.44 to 5.18 kpc. The upper end of this range is marginally consistent with the previous literature value of 5 kpc, ∼ although again, the literature value was highly uncertain as well.

3.3.2 Candidate LBVs

HD 80077: This blue hypergiant was included as an LBV candidate by van Genderen [2001b] based mainly on its high luminosity and spectrum. Its new Gaia DR2 distance is significantly closer than the baseline literature value, reducing its nominal luminosity by about 40%, although its luminosity is still above the S Dor strip. The Gaia DR2 distance and the original cluster-fitting distance of Steemers and van Genderen [1986] (2.8 0.4) agree ± within their quoted errors. SBW1: As in the cases of Sher 25 and HD 168625, this blue supergiant is an LBV candidate because of its circumstellar nebula that bears a remarkable resemblance to the ring nebula around SN 1987A. In fact, it has been argued that SBW1 is in some respects the best Galactic analog to the progenitor of SN 1987A, in terms of both its nebula and the properties of the weak-winded central star [Smith et al., 2013, 2017]. Based on various considerations, Smith et al. [2017] favored a distance of about 7 ( 1) kpc for SBW1. The ± +1.4 Gaia DR2 parallax indicates a distance of 5.2−1.0 kpc (Table 3.3). This distance is closer, but the range of 4.0-6.8 kpc overlaps with the uncertainty of 7 ( 1) kpc for the previous ± distance. This new distance corresponds to a somewhat lower effective single-star initial mass of 13-15 M , as compared to 18 M for the progenitor of SN 1987A [Arnett, 1989]. ∼ This confirms the notion that SBW1 is an analog of the progenitor of SN 1987A and its nebula [Smith et al., 2013], although at somewhat lower initial mass. Importantly, the new Gaia DR2 distance and uncertainty confirm that although SBW1 is seen to be projected

61 amid the young Carina Nebula at around 2 kpc, it is in fact a luminous background blue supergiant star and not a member of the Carina nebula population. Hen 3-519: The large Gaia DR2 distance for Hen 3-519 contradicts the closer distance from Gaia DR1 reported by Smith and Stassun [2017], which was around 2 kpc. The new +2.5 Gaia DR2 distance is 7.6−1.7 kpc, and is consistent with the traditionally assumed value of around 8 kpc. This indicates that Hen 3-519 is still a very luminous LBV candidate. Its new value barely overlaps with the S Dor instability strip within the uncertainty, although it may be below the S Dor strip (especially for some proposed locations of the S DOr strip, as shown in Fig. 3.5. The reason why the DR1 distance was too close likely has to do with the interpretation of the large uncertainty. The uncertainty in parallax for Hen 3-519 was large to begin with, having a DR1 value of $ = 0.796 0.575 mas. Adding a correction of 0.3 mas ± to this uncertainty, as noted by Smith and Stassun [2017], would give a negative parallax. In hindsight, taking the uncertainty as indicating a lower limit to the distance would have been a better choice. Smith and Stassun [2017] adopted the prior distribution for stars in the Milky Way from Astraatmadja and Bailer-Jones [2016], which was intended to account for systematic underestimate of the luminosity, but that prior distribution may have been inappropriate for a distant LBV. Sher 25: The blue supergiant Sher 25 is of interest because of its ring nebula that resembles the equatorial ring around SN 1987A, and because of its projected proximity to the massive young cluster NGC 3603. The large distance of 6.6 kpc (Table 3.3) is remarkably consistent with previously assumed values. This is important because Sher 25 was thought to be considerably more luminous than the progenitor of SN 1987A (also shown in Fig. 3.5), with a luminosity that corresponded to about twice the initial mass ( 40 M vs. 18 M ). ∼ It appears to be below or barely overlapping with the S Dor instability strip. These results are suggestive that Sher 25 is too massive to be considered as a good analog of SN 1987A’s progenitor in terms of initial mass. Importantly, at this luminosity, the ring around Sher 25 has probably not arisen from a fast blue supergiant wind that swept into a previous red supergiant (RSG) wind, since in that scenario the surrounding nebula would probably be more massive and younger. This is in agreement with its chemical abundances, which are

62 inconsistent with the level of enrichment expected if it had passed through a previous RSG phase [Smartt et al., 2002]. ζ1 Sco: This is a blue hypergiant star that was included as an “ex-dormant” LBV (i.e. an LBV candidate) by van Genderen [2001b] because of its hypergiant-like spectrum and microvariability. It was previously assumed to be at a distance of around 2 kpc, placing its luminosity just above the S Dor instability strip. The excess noise for this star is quite large, = 0.95 mas, making the distance uncertainty quite large and dominated by the prior. The old value for the distance and the Gaia DR2 most likely distance are above the S Dor instability strip. However, the new uncertainty overlaps the instability strip. HD 326823: This blue supergiant was considered as another “ex-dormant” LBV (candidate) by van Genderen [2001b], again because of its spectrum and microvariability. More recently, it has been suggested to be a close binary system with a period of 6.1 d [Richard- son et al., 2011]. Its new Gaia DR2 distance is reduced from the old value by about 37%, reducing its luminosity to less than half its previous value. The error bar on the Gaia DR2 distance (1.18-1.38 kpc) is small enough that this is a signiﬁcant revision. Its corresponding eﬀective single-star initial mass is reduced from &25 M to 17-18 M . This is yet another case of an LBV candidate with stellar properties very similar to those of SN 1987A’s progenitor or the putative surviving companion of SN 1993J. This similarity combined with its status as a close binary make this a potentially very interesting target for comparison with the progenitor systems of SN 1987A and SN 1993J. Wray 17-96 (B61): This classic B[e] supergiant is considered to be an LBV candidate based on its dusty circumstellar shell nebula discovered by IR surveys [Egan et al., 2002]. Its previously adopted distance of 4.5 kpc would imply an extreme luminosity for this object

6 above 10 L , placing it in the regime of classical high-luminosity LBVs, although it has not exhibited LBV-like variability. Unfortunately, the large excess astrometric noise limits the Gaia DR2 distance between 1.2 and 7.2 kpc. HDE 316285: This star is sometimes considered an LBV candidate due to is remarkable spectral similarity to η Car [Hillier et al., 2001] and its dusty nebula [Clark et al., 2005, Morris et al., 2008]. Although the uncertainty in distance is large, the new Gaia DR2

63 parallax suggests that HDE 316285 is signiﬁcantly farther away than previously assumed, moving it from about 1.9 kpc out to about 5 kpc. This raises its luminosity by more than a factor of 6. As such, it is pushed well above the upper luminosity limit for RSGs and into the regime of the classical LBVs, becoming potentially even more luminous than AG Car (although, again, the error bar is large). This larger distance and higher luminosity may help explain why the spectrum of HDE 316285 has such an uncanny resemblance to η Car [Hillier et al., 2001]. Interestingly, Morris et al. [2008] speculated that HDE 316285 may be coincident with Sgr D near the Galactic center at a distance of around 8 kpc. This is farther than the most likely Gaia DR2 distance, but permitted within the uncertainty, although it is unclear if its line of sight extinction is consistent with a distance this large. HD 168625: see above (Section 3.1). AS 314: This star was considered to be an LBV candidate based on its presumed high luminosity at a large 8-10 kpc distance, its hypergiant-like spectrum, and dust excess [Mirosh- nichenko et al., 2000, van Genderen, 2001b, Clark et al., 2005]. The new Gaia DR2 parallax +0.16 moves its distance from 8 kpc to only about 1.5−0.13 kpc, and with no excess astrometric

noise, lowering its luminosity by a factor of 25 to only log(L/L )=3.5. Its new luminosity is so low that it cannot be plotted in Fig. 3.5 because it is oﬀ the bottom of the plot (even its upper error bar is below the bottom of the plot). This is probably a post-AGB object from an intermediate-mass star, and it is most likely not related to LBVs. MWC 314: This is one of two sample stars with both a relatively small uncertainty on the distance, and where the distance has increased compared to values adopted in the literature. +1.03 This LBV candidate has moved from about 3 kpc to 4.0−0.7 kpc (Table 3.3), roughly doubling its luminosity. This moves it about one sigma oﬀ the S Dor instability strip and makes it similar to η Carinae on the HR diagram (Fig. 3.5). The distance calculated from the parallax makes it seem quite likely that MWC 314 may be one of the most luminous stars in the Milky Way. W51 LS1: This luminous blue supergiant was added to the list of LBV candidates by Clark et al. [2005], based on its supergiant spectrum and high luminosity. It has shown no variability or obvious shell nebula, but does have a near-IR excess. As described in Sec-

64 tion 3.2.2, the distance to the W51 complex has a complex history. The new Gaia DR2 par- +2.4 allax indicates a smaller distance for this source than most previous measurements: 2.5−1.3. The wide HDI is due to the large excess astrometric noise. With such a large excess noise, the posterior distribution is heavily influenced by the prior distribution, and while inconsistent with our baseline literature value 6 kpc, does just barely overlap with recent maser ∼ proper motion measurements of Xu et al. [2009], Sato et al. [2010]. Intriguingly, our Gaia measurements also show significant overlap with the spectroscopic parallax measurement of Figuerêdoet al. [2008]. W51 LS1 revised position lies within the region of normal blue supergiants and B[e] supergiants (green oval) in the HR diagram, but the large uncertainty still encompasses the lower end of the S Dor instability strip (Fig. 3.5). G79.29+0.46: This is an LBV candidate with a dust shell. The Gaia DR2 distance +1.4 1.9−0.8 kpc (Table 3.3) is consistent with the literature distance of 2 kpc. The distances are consistent, and so G79.29+0.46 is still likely a massive star, and its dust shell is not a planetary nebula. Cyg OB2 #12: Cygnus OB2 #12 is a B hypergiant that is usually considered as an LBV candidate because of its extremely high luminosity and cool temperature [Clark et al., 2005, Massey et al., 2001, Humphreys and Davidson, 1994]. It has rather mild variability, so van Genderen [2001b] classified it in the group of “weak-active” S Doradus variables (meaning low-amplitude <0.5 mag variability). Its status as one of the most luminous stars in the Milky Way (e.g., de Jager 1998) is based on its presumed association with Cyg OB2 at about 1.7 kpc [Clark et al., 2005, 2012]. Recently, however, Berlanas et al. [2019] have questioned this association. Berlanas et al. [2019] find two likely populations along the same line of sight from a recent analysis of Gaia DR2 data, with one at 1.76 kpc (close to the traditional distance for Cyg OB2) and a foreground group at 1.35 kpc. The excess astrometric noise for this star is quite large, 0.588 mas. The measured parallax is $=1.178 ( 0.128) mas, so ± while the measured parallax is larger than the excess noise, the excess noise does significantly impact the uncertainty. Moreover, the RUWE value of 1.52 makes this the only source in our sample that is bigger than the value of 1.4, above which Lindegren et al. [2018] caution against. The HDI for Cyg OB2 #12 is between 0.6 and 2.2 kpc. Hence, Gaia DR2 is

65 Figure 3.4: An HR diagram similar to Fig. 3.5, but showing only W243 based on the old distance of around 5 kpc (red) and using a distance of 3.2 0.4 kpc (black) based on the distance inferred for the whole Wd1 cluster from DR2± data [Aghakhanloo et al., 2020]. The reference single-star models are the same as in Fig. 3.5. consistent with the literature distance, or the two diuﬀerent distance of subgeroupos along the same line of sight of 1.76 or 1.35 kpc [Berlanas et al., 2019].

3.3.3 Westerlund 1 and W243

W243 is a confirmed LBV that is thought to be associated with the young massive star cluster Westerlund 1 [Clark and Negueruela, 2003, Clark et al., 2005]. It exhibits photometric variability and changes in spectral type that have been interpreted as LBV-like variability [Clark and Negueruela, 2003], but the variability that has been observed is not conclusively due to S Dor-like cycles. With a often-presumed distance of 5 kpc for that cluster, W243 would be a luminous LBV with an effective single-star initial mass around 50 M , similar to S Dor. With that presumed mass, it helps define the inferred turn-off mass, young age, and high total stellar mass of Wd1. In Gaia DR2, W243 has a large astrometric excess noise, 0.582 mas, which is almost as large as the measured parallax, $ = 0.979 0.165 mas. As ± 66 a result, the Gaia DR2 HDI for W243 alone is unfortunately large, between 0.83 and 4.15 kpc, the upper end of which is closer than the typical literature value (although to be fair, previous literature estimates had a wide dispersion from 2-5 kpc as well, a dispersion that is often not quoted). On the other hand, if W243 is truly a member of the Wd1 cluster, then the numerous other members of the cluster (many of which do not have such large astrometric noise) can be used to reduce the uncertainty in distance. A full analysis of the DR2 results for a sample of stars in Wd1 is discussed in the next chapter [Aghakhanloo et al., 2020]. In that study, we find a well-determined Gaia DR2 distance to Wd1 of 3.2 0.4 kpc. This cluster-based distance ± would be a less extreme revision to its distance and luminosity than for the Gaia DR2 data for the star W243 alone, but also clearly reduced from the usually adopted value around 5 kpc. At the new cluster distance, W243 would have a lower luminosity, appropriate to an

evolved single star of around 25-30 M (see Fig. 3.4) instead of 50 M (the implied cluster turn-off mass would be lower as well, and the cluster age would be older than previously assumed so that it is not such a young and massive cluster after all; see Aghakhanloo et al. 2020). W243 would be near the bottom of the traditional range of luminosities for LBVs, close to where the S Dor strip meets the constant-temperature outburst strip. However, at this low luminosity, W243’s presumed high temperature around 17-18 kK indicated by its B2 spectral type in quiescence [Clark and Negueruela, 2003, Clark et al., 2005] would move it off the S Dor instability strip at quiescence, and its observed variation in temperature would be too large to be consistent with traditional expectations for LBVs. For now, 3.2 ( 0.4) kpc ± is our preferred distance to W243 because it is more precise, although formally, it is fully consistent with the individual DR2 value we find between 0.83 and 4.15 kpc. AlthoughGaia DR2 data are not consistent with a large distance of 4.5 or 5 kpc that is often adopted in the literature, the new DR value is consistent with the original claimed distance of 2 kpc < d < 5.5 kpc [Clark et al., 2005]. The case of W243 illustrates the utility in having an independent estimate of the distance based on the parallax for many associated cluster members, especially in individual cases where high prohibits a reliable individual parallax. Unfortunately, however, it appears that

67 many LBVs simply do not reside in obvious clusters Smith and Tombleson [2015]. There are some LBVs with possible host clusters that have been noted, such as Wra 751. Even though Wra 751 appears to be overluminous for this cluster, it may help to more tightly constrain the distance and true age and initial mass of this object. Another possible case to investigate is ζ1Sco. HD 168625 and HD 168607 may be associated with Ser OB1, as noted earlier, but these already have quite good Gaia DR2 distances (with =0.0).

3.4 Discussion

About half of the LBVs and LBV candidates included in Gaia DR2 have literature distances that are within the 68% HDI. These are the LBVs HR Car, Wra 751, P Cygni, MWC930, MN48, and G24.73+0.69, and the LBV candidates Hen 3-519, Sher 25, SBW1, ζ1 Sco, WRAY 17-96, G79.29+0.46, and CYG OB2 #12. While the uncertainty overlaps with literature estimates, the value for the distance from Gaia DR2 is lower in most of these cases. Two LBV candidates have had their distances and luminosities increase significantly as a result of DR2. MWC314 has a significantly increased distance that makes its luminosity comparable to that of η Carinae. HD 316285 also has a significantly increased distance,

6 raising its luminosity above 10 L . For almost all the objects where the Gaia DR2 distance is significantly revised, however, we find that the objects have moved closer and their luminosity is lower than traditionally assumed. This reduction applies to 10 objects, including both confirmed and candidate LBVs. Of the five confirmed LBVs that have moved significantly closer (AG Car, W243, HD 160529, WS1, and HD 168607), only W243 and WS1 have large excess noise. As we noted, however, if we adopt the new Gaia DR2 parallax for the host cluster Wd1 as the distance to W243 [Aghakhanloo et al., 2020], then this object is also confidently closer and less luminous, but with a much smaller uncertainty. Five of the LBV candidates are moved to a significantly closer distance and lower luminosity, although one of them (AS314) has such a small distance and low luminosity that it is probably not related to LBVs). Overall, with a few getting brighter, many not changing significantly, and 10 shifting to significantly ∼ 68 closer distances and lower luminosities, the net effect is a widening and an overall downward shift of the observed luminosity range for Galactic LBVs. This broad conclusion is true whether or not we include sources with excess astrometric noise. Obviously the situation is expected to improve with DR3, but that is several years in the future, and it is worthwhile to consider the implications for LBVs now.

3.4.1 Luminosity Range and the S Dor Strip

With a larger spread in LBV luminosity range that now extends to lower luminosity than previously assumed, there are two divergent ways to interpret the result. One option is simply that these lower-luminosity stars were mistakenly classified as LBVs. The other view is that the original definition of the S Doradus strip, based on only a few objects, might have been too narrow; it might therefore fail to capture the diverse range of real physical variability and mass loss exhibited by luminous, blue, and irregularly variable stars. Which of these two options is chosen might have important implications for understanding the range of initial masses that yield LBVs and the physical mechanism(s) governing their instability and mass loss. Both options have some subjectivity. Following the first option, we might decide to strip these lower-luminosity stars of their LBV or LBV candidate status, demoting them to “normal” blue supergiants or B[e] supergiants and thus preserving the S Doradus instability strip. This demotion may be valid for some objects, where the main motivation for including them as LBV candidates in the first place was their high luminosity (like AS 314). However, it is less appealing to simply discount the lower-luminosity candidates with dusty shell nebulae, because these nebulae indicate substantial episodes of previous mass loss that are relatively rare among blue supergiants. It is also not so easy to discount lower luminosity stars that have strong emission-line spectra that resemble their very luminous cousins, since these emission-line spectra indicate strong current mass loss. Moreover, it is not so easy to “unconfirm” Wra 751 or the lower- luminosity objects that have been confirmed as LBVs based on their variability (HD 168607, W243, and WS1). If these are not lower-luminosity LBVs, then what type of blue irregular variable are they?

69 We note that previous efforts to classify a star as an “LBV”, “LBV candidate”, or “nei- ther” (not to mention various subtypes like classical S Doradus stars, SN impostors, ex- dormant, weak active, P Cygni stars, etc.) have been somewhat arbitrary and inconsistent among authors [Conti, 1984b, Humphreys and Davidson, 1994, van Genderen, 2001b, Smith et al., 2004, Clark et al., 2005]. This reflects the fact that LBVs are rare stars and that each one has some unique peculiarities. To classify them in a group or subgroups requires one to make choices about which observed properties to emphasize in a definition. The original definition of an LBV by Conti [1984b] was a hodgepodge of many different types of massive and variable hot stars – basically “not Wolf-Rayet stars” and “not red supergiants” (in fact, Conti used the term “other”) – including the Hubble-Sandage variables, S Doradus variables, η Car-like variables, P Cygni stars, etc. The motivation was that all these stars may play a similar transitional role in evolution once O-type stars leave the main sequence, and it is potentially useful to discuss them together. It is also a convenient oversimplification for the purpose of discussion. As study of these stars intensified, some observers, guided by stellar evolution models for single stars, favored a more precise definition of what is an LBV so that only very few objects were included [Wolf, 1989b, Humphreys and Davidson, 1994], whereas others chose to proliferate LBV subtypes to accommodate some of the diversity in observed characteristics [van Genderen, 2001b]. Some objects were included as LBVs or LBV candidates based on much more limited information than for the classical LBVs, as in the cases of the Galactic Center sources. In light of the fact that we still do not understand the physical mechanism that drives LBV variability or their place in evolution, it may be wise to lean toward being inclusive with respect to this diversity. Regardless of the name we choose to give them, revised distances and luminosities from Gaia DR2 seem to indicate that blue supergiants at lower luminosity than previously thought can also suffer episodes of mass ejection, variability, instability, and strong winds that could be similar to traditional expectations for LBVs. A few objects are also found to be off the S Dor instability strip, but above it. These include η Car (this has been known for a long time), MWC 314, HDE 316285, and HD 80077. The Pistol Star and FMM 362 were also known to be well above the S Dor instability strip,

70 similar to η Car. Demoting the lower-luminosity stars from the class of LBVs would not change the fact that these more luminous stars are also off the S Doradus instability strip, again arguing that its definition may have been too narrow in the past. Whether or not revised luminosities land a star on the S Dor instability strip depends, of course, on exactly where we choose to put that instability strip. Since the defining S Doradus variables, AG Car and HR Car, have slightly revised distances, perhaps the location of the S Dor strip needs to be adjusted as compared to the original position [Wolf, 1989b]. As noted earlier, Groh et al. [2009a] presented a revised S Dor strip defined by detailed modelling of physical parameters for AG Car and HR Car in their hot quiescent states (which we have slightly adjusted in Fig. 3.5 based on their new Gaia DR2 distances). If this placement of the S Dor instability strip is adopted, then η Car and MWC 314 fit nicely along an extension of its slope, as do P Cygni and HD 168625. This steeper slope also encompasses the general locations of B[e] supergiants and many LBV candidates. However, with that steeper slope, many other LBVs are then left far off the S Dor strip, including most of the known extragalactic LBVs, as well as W243, HD 160529, HD 168607, and G24.73+0.69. Wra 751 is far off the S Dor strip no matter what. It seems difficult to escape the conclusion that the S Dor instability strip must be broader and must extend over a wider luminosity range than previously appreciated. How shall we interpret this? One possible option is that the LBV instability does indeed occupy a wider

spread of luminosity and Teﬀ than the narrow strip originally deﬁned by Wolf [1989b] or the revised version proposed by Groh et al. [2009a]. When one examines Fig. 3.2, where no extragalactic LBVs are plotted and where we show no S Dor strip to guide the eye, it is not obvious that LBVs reside along any strip at all. Figures 3.2, 3.3, and 3.5 give the impression that the zone of instability for LBVs and related objects might include everything redward of the terminal age the main sequence, over to about 8000 K, spreading both above and below the S Dor strip (objects that are luminous and variable but cooler than 8000 K are not called LBVs because they are yellow or red). In other words, one might simply extend the locus of normal BSGs and B[e] supergiants upward to include the classical LBVs as well. This zone encompasses LBVs and LBV candidates, but

71 also includes many blue supergiants that are not highly variable and do not have signiﬁcant circumstellar material. In this view, perhaps the classical LBVs are just the most extreme end of a continuum of diverse variability and mass loss. Whether or not a star in this zone is an LBV may depend on its evolutionary history, as well as our choice for the threshold of variability needed to call that star an LBV. Some LBVs could conceivably be on a post-RSG blue loop, where previous strong mass loss as a RSG has increased their L/M ratio, making them more unstable. This may work for the stars in the

30-40 M initial mass range [Humphreys and Davidson, 1994, Smith et al., 2004], but at lower luminosities, stars remain far from the classical Eddington limit. The other viable option is that some stars arrive in this zone through single-star evolution whereas others arrive there as a product of binary interaction, or that they have diﬀerent rotation rates. Mass accretion and spin up through binary mass transfer or stellar mergers may provide a means for only some selected stars in this zone to experience peculiar and episodic mass loss, anomolous enrichment, rapid rotation, and instability [Kenyon and Gallagher, 1985, Gallagher, 1989b, Justham et al., 2014b, Smith and Tombleson, 2015, Smith, 2016b, Aghakhanloo et al., 2017]. There is no clear reason why such eﬀects would be limited to a narrow zone coincident with the S Dor instability strip, so in a binary context, the wider spread of luminosity would not be surprising.

3.4.2 Lower Luminosity LBVs

Gaia DR2 reveals a handful of LBVs and candidate LBVs that reside at lower luminosities than previously realized, well below log(L/L )=5.3 where the nominal S Doradus instability strip joins the constant temperature outburst temperature of LBVs. Why have analogs of these lower-luminosity LBVs not been found in the LMC/SMC or M31/M33 (the purple sources in Fig. 3.5)? This might easily be a selection effect since it is harder to detect subtle variability in fainter stars, especially if one is interested in the most luminous stars. Alternatively, “LBV-or-not” classifications in these galaxies may have been biased to high luminosities (the brightest objects are deemed LBVs, while fainter blue stars with variability may have been ignored or called something else). There may also be a real physical effect:

72 perhaps whatever mechanism is responsible for the LBV instability (such as Fe opacity; e.g., Gräfeneret al. 2012b, Jiang et al. 2018) can be triggered at different luminosities in a higher-metallicity environment or at different rotation rates. Deciding between these options is difficult, and a renewed and unbiased effort to characterize variable stars in these nearby galaxies may be warranted. This is an area where the Large Synoptic Survey Telescope may provide a significant advance. We note that recently, such a study has been undertaken for M51 by Conroy et al. [2018]. Using multi-epoch HST data, they found a continuum of variability over a wide luminosity range for luminous stars, where the observed diversity of variability among luminous blue stars was considerably broader than the narrow definition of traditional S Dor variables. The possible existence of LBV-like instability at lower initial mass and lower luminosity than previously recognized has at least three broader implications. 1. Physical cause of LBV instability: The traditional interpretation for the cause of normal S Dor instability has been that these stars are unstable because of their proximity to an opacity-modified Eddington limit [Lamers and Fitzpatrick, 1988, Ulmer and Fitzpatrick,

1998]. Single-star models suggest that a star of 60 M , for example, will develop a pro- ∼ gressively more LBV-like spectrum and instability as it approaches the Eddington limit in its mass-loss evolution [Groh et al., 2014, Jiang et al., 2018]. This may work for the most luminous LBVs, but it may not work so well for the lower-luminosity examples in the 30-40

M initial mass range. As noted above, these might plausibly reach a similar instability if one invokes rather severe previous RSG mass loss, so that they are now in a post-RSG phase [Humphreys and Davidson, 1994, Smith et al., 2004, Groh et al., 2013, 2014]. Gaia DR2 distances now suggest that there are LBV-like stars at even lower luminosities (in the 10-30

M initial mass range). At such low luminosities, this near-Eddington instability doesn’t work because their previous RSG mass loss is not strong enough, and their luminosities are not high enough [Beasor and Davies, 2018]. We must either conclude that they have a separate instability mechanism unrelated to the Eddington limit, or that perhaps some other instability governs all the LBVs. As noted above, post-merger or post-mass transfer

73 evolution may populate the whole relevant range of luminosities with massive blue stragglers. This may be an important clue. 2. Relation to low-luminosity SN impostor progenitors like SN 2008S: The possible existence of LBVs that push to lower luminosities and initial masses than previously thought may have interesting implications for a subset of SN impostors similar to the well-studied object SN 2008S. SN impostors were generally thought to be related to giant eruptions of LBVs [Humphreys et al., 1999, van Dyk, 2005, Smith et al., 2011b]. However, a few transients in the past decade, highlighted by the prototypes SN 2008S and NGC 300-OT [Prieto, 2008, Prieto et al., 2008, Bond et al., 2009], had dust-obscured progenitors. These had lower inferred luminosities and lower initial masses than traditional LBVs, so these have been sug-

gested to be transients that arise from super-AGB stars with initial masses around 8-10 M , including electron capture SNe [Thompson et al., 2009, Botticella et al., 2009]. However, if LBVs actually extend to lower luminosity, then they might also be dust-enshrouded LBV- like supergiants that reside at somewhat lower luminosity than previously recognized [Smith et al., 2009, Berger et al., 2009b, Bond et al., 2009]. For NGC 300-OT, the surrounding stellar population points to an age appropriate to an initial mass of 12-25 M [Gogarten et al.,

2009], inconsistent with an electron capture SN (8-10 M ) or a transient associated with an even lower-mass star. This would, however, be consistent with the implied initial masses for the lower-luminosity Galactic LBVs like W243, and HD 168607 found here. Perhaps these lower-luminosity Galactic LBVs are likely progenitors for some of these SN 2008S-like events, or products of them if they are merger events. The spectra for many of these objects look quite similar at various points in their evolution, including objects that have been suggested to be stellar mergers [Smith et al., 2011b, 2016b]. On the other hand, the SN impostors may be a mixed-bag across a wide mass range, since some, like SN 2008S itself and SN 2002bu, have surrounding star formation histories that translate to ages appropriate for initial masses less than 8 M [Williams et al., 2018]. 3. SN progenitors with pre-SN mass loss: The group of lower-luminosity Galactic LBVs and LBV candidates have interesting potential implications for some types of SN progenitors. First, we have noted that several LBV-like stars seem to be quite close to the location of

74 SN 1987A’s progenitor on the HR diagram, and few of these even have similar ring nebulae. This adds to speculation that some sort of LBV-like instability and mass loss could have played a role in forming the nebula around SN 1987A [Smith, 2007]. Previously, it was thought that the lower bound of LBV luminosities did not extend low enough to include SN 1987A, but Gaia DR2 shows that it reaches even lower. Second, there has been much discussion about LBVs as possible progenitors of Type IIn supernovae (SNe IIn), because their dense circumstellar material (CSM) seems to require some sort of eruptive pre-SN mass loss akin to LBV eruptions (see review by Smith 2014 and references therein). In seeming contradiction, host galaxy environments surrounding SNe IIn (and also SN impostors) do not favor very high mass stars in very young regions [Anderson and James, 2008, Anderson et al., 2012, Habergham et al., 2014] Even the special case of SN 2009ip, with a very luminous and eruptive LBV-like progenitor, is out in the middle of nowhere, with no sign of recent star formation [Smith et al., 2016a]. If the LBV phenomenon extends to much lower masses than previously thought, then perhaps SNe IIn can arise from LBV-like progenitors over a wide range of initial masses that even overlaps with progenitors of normal SNe II-P. The lower-luminosity LBVs in Fig. 3.5 overlap with single-star evolutionary tracks as low as 10-20 M . If they are the results of mass gainers or mergers in binary systems, then their true initial masses may extend even lower, and their lifetimes may potentially be quite long [Smith and Tombleson, 2015, Smith, 2016b, Aghakhanloo et al., 2017]. Such LBV-like progenitors of SNe IIn originating from this lower-mass range might vastly outlive any main-sequence O-type stars that could ionize surrounding gas, possibly explaining the lack of correlation between SN IIn locations and Hα emission [Anderson and James, 2008, Anderson et al., 2012, Habergham et al., 2014]. This last point seems to be in general agreement with the relative isolation of LBVs on the sky as compared to O-type stars [Smith and Tombleson, 2015]. Moreover, some of the ﬁrmer distance estimates for Galactic LBVs reported here have implications for the isolation of LBVs and implications for their evolutionary origin in binary systems [Smith and Tombleson, 2015, Smith, 2016b, Aghakhanloo et al., 2017].

75 3.4.3 LBVs and Clusters

Among conﬁrmed unobscured LBVs that are found in young massive clusters, now only η Car remains as a conﬁdent association, although its luminosity is so high that it is consistent with being a blue straggler as compared to the surrounding stars in Tr 16. The earliest-type

main sequence star in Tr16 is O3.5 V with an implied initial mass of around 60 M , whereas

η Car has an equivalent single-star initial mass of around 200 M or more. Indeed, there are a number of other clues, such as light echoes from the Great Eruption, that η Car’s present state is inconsistent with single-star evolution and might instead be the result of a stellar merger event [Smith et al., 2018b,a]. Although W243 is probably in the cluster Wd1, the revised nearer distance to Wd1 from Gaia DR2 [Aghakhanloo et al., 2020] means that the cluster is not as young and massive as previously thought. Wra 751’s distance makes it marginally consistent with its presumed host cluster [Pasquali et al., 2006], but this is not a young massive cluster either, and this is actually a problem for the single-star scenario. This is because Wra 751 has a luminosity indicating an eﬀective single-star initial mass that is more than 2 times higher than the turnoﬀ mass inferred from the late O-type stars still on the main sequence in that same cluster. AG Car and Hen 3-519 are at a large distance, but this means that they are not at the same distance as O-type stars that appear near them on the sky in the Car OB association, which is at around 2 kpc [Smith and Stassun, 2017]. This makes their apparent isolation even worse, and the discrepancy is exacerbated by the fact that the higher distance also gives them a higher luminosity and shorter lifetime. It is remarkable that AG Car has a luminosity consistent with an initial mass of around

80 M , but it is not known to be associated with any O-type stars at a similar distance. More detailed investigations of any possible birth populations associated with LBVs could be illuminating. Of course, there are a few luminous LBVs and candidate LBVs known in highly obscured massive clusters, such as the clusters in the Galactic Center. However, their signiﬁcance is diﬃcult to judge, and their implications for LBV evolution are unclear, since these were discovered as LBV-like stars based on studies of the clusters and are therefore a heavily biased sample. As noted by Smith and Tombleson [2015], there is no available census of visually-

76 obscured LBVs in the ﬁeld or a census of highly obscured O-star populations around them that can be used to infer what fraction obscured LBVs in the Galactic plane avoid clusters.

77 100 M8

80 M8

60 M8

50 M8

40 M8

SN1987A 30 M8 SN1993J companion 25 M8 mass gainer sgB[e] 20 M8 mass BSG donor

primary 16 M 8

15 M8

SN Ib secondary 14 M 8 d = Gaia DR2 d = literature (no DR2) d = extragalactic 10 M8

Figure 3.5: The HR diagram with LBVs (filled circles) and LBV candidates (unfilled circles), adapted from a similar figure in Smith and Tombleson [2015] and Smith and Stassun [2017]. Here, Galactic LBVs and LBV candidates are shown in black, with luminosities adjusted from old values as appropriate for the new Gaia DR2 distance. For this plot, we use the DR2 distances from Table 3.3 (see text). LBVs in nearby galaxies (LMC, SMC, M31, M33) are shown in light purple for comparison. Locations of blue supergiants and B[e] supergiants, the progenitors of SN 1987A and SN 1993J, and some example stellar evolutionary tracks are also shown for comparison. The gray boxes show the locations of the temperature dependent S Doradus instability strip [Wolf, 1989b] and the constant temperature strip of LBVs in outburst, as in Smith et al. [2004]. The thinner orange line shows the somewhat steeper S Doradus instability strip suggested by Groh et al. [2009a] based on AG Car and HR Car (although it has been shifted slightly here to accommodate their revised distances and luminosities from DR2, and we have extrapolated over a larger luminosity range with the same slope). The single-star model tracks (blue) are from Brott et al. [2011] and the pair of binary system model tracks (red and pumpkin colored) is from Langer and Kudritzki [2014].

78 CHAPTER 4

INFERRING THE PARALLAX OF WESTERLUND 1 FROM GAIA DR2

In this chapter, we infer the distance to Westerlund 1 (Wd1) cluster, which hosts an LBV. The Wd1 cluster is of significant interest, because at its typically adopted distance of 5 kpc, it would be one of the most massive young star clusters in the Local Group, and it contains a large population of evolved massive stars that serve as a test of stellar evolution models. It has one of the largest concentrations of evolved massive stars such as blue supergiants, yellow hypergiants, red hypergiants, WRS, one LBV and it even has a magnetar [Clark and Negueruela, 2003, Clark et al., 2005, Crowther et al., 2006, Muno et al., 2005, Groh et al., 2006, Fenech et al., 2018]. Using Gaia DR2 parallaxes, we infer a smaller distance to Wd1 [Aghakhanloo et al., 2020]. We use Bayesian inference to model both cluster members and +0.6 Galactic field stars, and we find that the distance to cluster is 2.6−0.4 kpc. Our new estimate reduces the uncertainty by more than an order of magnitude, from a factor of 3 to 18%. ∼ 4.1 Method

In this section, we describe the method to infer the distance to Wd1. The stars in the direction of Wd1 comprise cluster stars as well as Galactic ﬁeld stars. Therefore, to infer the distance to Wd1, our likelihood model must account for both cluster and ﬁeld stars. The following sections describe the data and methods required to model both components and infer the distance to Wd1.

4.1.1 Gaia Data Release 2 Data

The source of the data is the Gaia DR2 . We collect all Gaia DR2 sources within 10 arcmin of the position of Wd1; RA(2000) = 16h47m04s.0, Dec.(2000) = 45°5100400.9. −

−

− − − − −

Figure 4.1: Position of all Gaia stars within 10 arcmin of Westerlund 1. Axes are offsets from a given position in degrees on the sky; RA Offset=(RA − RAWd1) cos(Dec.Wd1) and Dec. Offset=Dec. Dec.Wd1, where RAWd1 = 251.77° − and Dec.Wd1 = 45.85°. Stars in the inner circle, with 1 arcmin radius, are mostly associated with− the cluster, while the stars in the outer ring are mostly field stars.

Fig. 4.1 presents the positions of all objects within 10 arcmin of Wd1. The inner circle marks a region that is 1 arcmin from the centre of the cluster. The outer annulus extends from 9 to 10 arcmin. The density of stars is mostly uniform throughout the field of view, it is slightly over dense towards the right; however, the density does noticeably increase towards the centre. This spatial separation between cluster and field stars suggest a strategy for constraining the parameters for each population. The inner circle contains both field and cluster stars, but is dominated by cluster stars. Therefore, the inner circle provides a good

80 constraint on the cluster population. The outer annulus are mostly field stars. Therefore, the outer region will constrain the field star distribution. Fig. 4.2 shows histograms of parallax, parallax uncertainty, astrometric excess noise, and astrometric excess noise significance of all stars in the inner circle and the outer annulus. The average parallax for the inner circle is 0.22 0.04 mas, and the average parallax for ± the outer annulus is 0.45 0.02 mas. These averages include all stars, no filtering, and ± does not include the parallax zero-point. Nevertheless, the difference in average parallax already indicates that the cluster is farther than the average field star. The distribution of expected parallax uncertainty (top right panel) is similar between the two regions. The minimum parallax uncertainty in the direction of Wd1 is 0.04 mas, but the vast majority of uncertainties are even larger (up to a few mas), including systematics such as parallax zero-point [Lindegren et al., 2018]. The distribution for astrometric excess noise (bottom left panel) shows a considerable difference between the two regions. The astrometric excess noise () represents extra variation in the data that is not included in the five-parameter astrometric model; the astrometric noise significance is D (bottom right panel). A higher fraction of objects in the inner circle have large excess noise. Therefore, we keep the sources with D 2 to avoid bad astrometric solutions. ≤ Although the astrometric excess noise and significance are useful for assessing the quality of the astrometric solutions, the renormalized unit error (RUWE) and visibility periods used are more useful in very crowded regions like Wd1. The RUWE is the re-normalized p goodness of fit ( χ2/(N 5)). N is the total number of along scan observations used in − the astrometric solution of the source. The high crowding is possibly responsible for the lower number of observations per star. A visibility period indicates the number of groups of observations separated from other groups by at least 4 days. Therefore, a higher number of visibility periods indicates that the solution is less vulnerable to errors. As recommended by Lindegren et al. [2018], we use astrometric solutions with at least eight visibility periods and RUWE < 1.40. The astrometric excess noise, visibility periods, and RUWE cuts reduce the number of sources in the inner circle from 435 to 42 and reduce the number of sources in the outer ring from 2127 to 1344.

ϖ[mas] σϖ[mas]

0 1 2 3 4 5 Astrometric_Excess_Noise

Figure 4.2: Histograms of parallaxes (top left panel), parallax uncertainties (top right panel), astrometric excess noise (, bottom left panel), and astrometric excess noise signiﬁcance (D, bottom right panel) of all stars in the inner circle and the outer annulus. The average parallaxes for the inner circle and the outer ring are clearly distinct; the average parallax $ = 0.22 0.04 mas for the inner circle, and $ = 0.45 0.02 mas for the outerh ring.i A small± fraction of the inner circle stars hhavei small± excess noise (D < 2), while the outer ring has a much larger fraction that satisfy this quality metric.

− ϖ/ϖ

Figure 4.3: Histogram of parallax over the uncertainty of all stars in the inner circle and the outer annulus. Many of the stars in the inner circle and the outer annulus have large uncertainties (ratios of order 1 or less). The large uncertainties and negative parallaxes indicate that inferring the distance to an individual star using parallax inversion d = 1/$ is either inaccurate or impossible.

If the uncertainties are relatively small, then direct inversion of the parallax is a reasonable inference of the distance, d = 1/$. However, Fig. 4.3 shows that most of the stars in the inner and the outer region have large uncertainties, and in some cases, the parallaxes are negative. Therefore, the straightforward approach of inverting the observed parallax is either inaccurate or impossible to an individual star. Averaging a collection of stars would mitigate the problem that a subset of the stars have negative parallaxes, but this average would be biased due to the large uncertainties. Therefore, a more sophisticated inference is required such as the Bayesian approach described in Section 4.1.3.

4.1.2 Average Statistics: a First Rough Statistical Inference

Before using Bayesian inference, we estimate the true mean parallax for several annuli to roughly infer the cluster parallax.

83 0.60

0.55

0.50 )[mas]

zp 0.45

ϖ Mostly Field Stars

− 0.40 > 0.35 Mostly Cluster Stars ϖ 0.30 < ( 0.25 0 2 4 6 8 10 rannulus[arcmin]

Figure 4.4: Estimates for the true mean parallax for each ring. The mean observed parallax is $ , and the parallax zero-point is $zp = 0.05 mas. The average parallax forh thei outer rings most likely represent the field− stars, and the inner rings represent the cluster. Below 3 arcmin, the average parallax transitions from being dominated by the field stars∼ to the cluster stars. The vertical error bars show the 68% confidence interval after bootstrapping the average 500 times for each annulus.

Fig. 4.4 shows the estimates for the true mean parallax as a function of radius from the cluster centre. After using the selection criteria (section 4.1.1), we calculate the mean parallax for each annulus, and adjust it by the parallax zero-point to get an estimate for the true parallax. These averages show a clear indication that the cluster has a smaller parallax (larger distance) than the average ﬁeld star. The blue line represents the observed mean parallax $ corrected by the parallax zero-point of $ = 0.05 mas for several annuli. h i zp − For the zero-point, we take the mean of recent estimates [Lindegren et al., 2018, Riess et al., 2018, Stassun and Torres, 2018, Zinn et al., 2019]. See section 4.1.3 for more details. Each annulus is 1 arcmin in width. To estimate the uncertainty on the average, the vertical error bars show the 68% conﬁdence interval after bootstrapping the average 500 times for each annulus. If the uncertainties are accurate, then one could just calculate the uncertainty using

84 standard error propagation. However, as we show below, the uncertainties are inconsistent with the scatter in the data. Therefore, we choose to use the actual data to report the uncertainties in the average. For radii above 3 arcmin, the average parallax seems to be ∼ dominated by the field stars. Interior to this radius, $ becomes more influenced by the h i cluster with decreasing radius. Fig. 4.4 also justifies our choice for the size of the inner circle and the outer annulus. To be clear, the cluster is not limited to a radius of 1 arcmin. Rather, this is the radius for which we restrict our inference of the cluster parallax. Based upon Fig. 4.4, the cluster clearly extends several arcminutes. However, our model assumes one density for the cluster. Therefore, we restrict our inference to the inner region where the density is roughly uniform and consistent with our model assumption. Conversely, since the cluster clearly extends several arcminutes, we choose the outer ring far enough to be dominated by field stars. Fig. 4.4 indicates that 10 arcmin is sufficiently far enough to satisfy this requirement. One obvious way to estimate the parallax is using the variance weighted mean. The variance-weighted mean parallax for the inner circle is 0.60 0.1 mas, which is inconsistent ± with the simple mean. 0.1 mas represents the uncertainty from bootstrapping, while the standard error propagation gives a smaller uncertainty of 0.01 mas. This suggests that the empirical uncertainty is quite a bit larger than the reported uncertainty (see section 4.3 for more details). Given that the variance-weighted mean is compromised by the inaccurate uncertainties, we should use a more complicated method like Bayesian inference to infer the parallax. In the next sections, we infer the cluster parallax through Bayesian inference.

4.1.3 Bayesian Analysis

To infer the Wd1 parallax, the posterior distribution for the model parameters, θ is the product of the likelihood (data θ) and a multidimensional prior probability P (θ): L |

P (θ data) (data θ)P (θ) . (4.1) | ∝ L |

Before we fully describe the likelihood model, we deﬁne the model parameters and data. The density of stars (see Fig. 4.1) suggests a model for two sets of stars. One set, the inner

85 circle, contains both cluster and field stars, and the other set, outer ring, includes only field stars. The outer ring constrains the parameters of the field-star distribution, one of which is the length scale (L). L gives an effective length scale for the distribution of field stars, equation (4.8). In practice, this length scale is set by many factors, and one of the main factors is an effective optical depth for extinction along the line of sight [Bailer-Jones et al., 2018]. The full set of observations, data, includes two data sets: the parallaxes of stars in the

inner circle, $j , and the parallaxes of stars in the outer annulus, $k . data also includes { } { } the number of stars in the inner circle, Ni, and the number of stars in the outer annulus, No.

σ$ and σ$ are the parallax uncertainties for the inner region and the outer annulus. { j } { k } We consider the parallax uncertainties to be ﬁxed parameters and the dependencies on the uncertainties are omitted in the following equations for brevity. The model parameters,

θ, are the cluster parallax, $cl (mas), density of the cluster stars in the inner circle, ncl

(number per square arcminute), density of the field stars in the outer ring, nf (number per square arcminute), which we assume to be similar in the inner ring, the parallax zero-point of the cluster, $zp (mas), the field-star length scale, L (kpc), and the field-star offset, $os

(mas). The field-star offset ($os) includes the parallax zero-point but it also includes other possible systematics that affects the field-star distribution but not the cluster parallax; we elaborate more on this later. Naturally, ncl is a function of radius from the centre of the cluster. Rather than assuming a radial profile, we consider annuli and infer an average number density for each annulus. Specifically, henceforth, ncl refers to the average density of the central arcminute of the cluster. Each set also has a nuisance parameter. The nuisance parameters, η, are the set of true parallaxes for the inner circle, $ˆ j (mas), and the set of { } true parallaxes for the outer annulus, $ˆ k (mas). { } The probabilistic graphical model (PGM), Fig. 4.5, shows the interdependence among the observations, the model parameters, and the nuisance parameters. The likelihood probability is:

(data θ) = (N n , n ) (N n ) ( $j θ) ( $k θ) . (4.2) L | L i| cl f L o| f L { }| L { }|

86 Figure 4.5: Probabilistic graphical model for the Bayesian model. Arrows show the dependence of variables, circles indicate continuous variables, and double circles show fixed values. The top row shows the model parameters; for the first six ( density of the cluster stars in the inner circle, ncl, density of the field stars, nf, the cluster parallax, $cl, the parallax zero-point of the cluster, $zp, the field-star length scale, L, and the field-star offset, $os) we infer a posterior distribution. The middle row shows the latent, or nuisance, parameters (the set of true parallaxes for the inner circle, $ˆ j , and the set of true parallaxes for the outer annulus, $ˆ k ). The bottom row{ shows} the observations (the number of stars in the inner{ circle,} Ni, the number of stars in the outer annulus, No, the parallaxes for the inner circle, $j , and the parallaxes for the outer annulus, $k ). This diagram maps out the {dependencies} when deriving the conditional probabilities.{ }

Each component of the likelihood on the RHS is a likelihood for a particular set of data. The parameters after represent the set of model parameters which determine the data | in the model. If the data depend upon the full set of model parameters, then θ appears, otherwise we include only the model parameters that matter for each likelihood component.

For example, the number of stars in the inner circle, Ni, depends only upon the cluster density, ncl, and ﬁeld density, nf . To describe the model for the observations, ﬁrst, consider the bottom row in Fig. 4.5; it shows all observed parameters. Near the centre is $j , which represents the data for { } 87 inner circle. Each observed parallax is drawn from a Gaussian distribution centred on the

true parallax; each star has its own true parallax,$ ˆ j; therefore, there is a set of many

true parallaxes for the inner region, $ˆ j . Since the central region has both field and cluster { } stars, each true parallax is either drawn from the cluster or the field-star parent distributions. A priori, it is not clear which star is associated with the cluster or field-star distributions.

However, the fraction of stars associated with the cluster is ncl/(ncl + nf ) and the fraction

associated with the field stars is nf /(ncl + nf ). Hence, the density of cluster and field stars are important parameters in the generation of the data. Unfortunately, modelling just the inner circle does not constrain the four model parameters along the top row. Modelling the outer ring provides constraints on the field-star parameters. This then, in combination with the inner circle data, provides a unique constraint on the cluster parame-

ters. Each observation of the outer annulus, $k , is also drawn from a Gaussian, and the { } true field-star parallax is drawn from the field-star distribution. Our assumption is that all stars in the outer ring are drawn from the field star distribution. The number of stars in the outer ring is simply given by the field-star density, while the inner circle is a weighted combination of the cluster and field-star densities. Hence, to constrain the cluster density and ultimately the fraction of cluster stars in the inner circle, the likelihood must model both the number of stars in the inner circle and the outer ring. The first two likelihoods in equation (4.2), (N n , n ) and (N n ), represent the L i| cl f L o| f number of stars in the inner circle and the outer ring:

Ni −λi λi e (Ni ncl, nf ) = , (4.3) L | Ni! and No −λo λo e (No nf ) = , (4.4) L | No! where the expected number of stars in the inner circle is λ = (n +n ) A and the expected i f cl × i number of stars in the outer ring is λ = n A .A is the area of the inner circle and A o f × o i o is the area of the outer ring. ( $ θ) in equation (4.2) is the likelihood for the outer set L { k}| of data: Y ( $k θ) = k($k θ) . (4.5) L { }| L | k 88 PGM provides a map of how to further deconstruct the likelihood using the conditional probability theorem: Z k($k θ) = Pk($k $ˆ k) Pk($ ˆ k θ)d$ˆ k . (4.6) L | | × |

The ﬁrst term in equation (4.6), Pk($k $ˆ k), is the probability of observing any parallax | for the kth star in the outer ring:

2 1 ($k $ˆ k) Pk($k $ˆ k) = exp − −2 , (4.7) | √2πσ$k 2σ$k where σ$k is the parallax uncertainties for the outer annulus. In this work, we consider the parallax uncertainties to be fixed parameters. The second term in equation 4.6, P ($ ˆ k θ), | represents the field star distribution. The PGM shows that this distribution only depends upon two model parameters, L and $ ; hence, P ($ ˆ k L, $ ). If one considers an image pop- os | os ulated with stars, then the total number of stars in the image is given by N = FOV R nr2dr, where FOV is the field of view in square radians, n is the number density of stars, and r is the distance from the sun. If n is constant, then any random star in the image is drawn from a probability distribution of P (r) r2. This distribution is assumed to fall off exponentially, ∝ exp( r/L), due to various effects including the Gaia selection function, attenuation due to − dust or a combination of both. Therefore, the distribution of the field stars is

1 P (r L) = r2 exp( r/L) . (4.8) | 2L3 −

After transforming from distance to parallax, the ﬁeld-star distribution becomes

1 exp[ 1/(($ ˆ k $os)L)] P ($ ˆ k L, $os) = 3 − 4 − . (4.9) | 2L $ˆ k

Together, equations (4.7) and (4.9) represent the likelihood for the outer ring, given in equation (4.6). The zero-point in equation (4.9), $os represents an offset for the field star distribution, and it may or may not be the same as the zero-point, $zp, for the cluster members. $zp represents a zero-point associated with instrumental and analysis biases [Lindegren et al., 2018]. The parallax distribution for field stars, equation (4.9), assumes that the distribution approaches zero at$ ˆ k = 0; however, it may not do so for several

89 reasons. Certainly, one part is due to the same instrumental and analysis biases that impact the cluster members. In addition, sightlines through the plane of the Galaxy likely have a distribution of field stars that is more complicated than equation (4.9). For example, while uniformly distributed dust may cause an exponential attenuation with one scale, L, a very dusty star-forming region in a spiral arm will likely present a wall, beyond which we cannot see any stars. This would manifest as an abrupt truncation of the field-star distribution at some finite, positive parallax. For this reason, the zero-points for the field stars and the cluster must remain separate variables. To find the likelihood distribution for the outer ring we marginalize over the nuisance parameters, η. In the PGM, Fig. 4.5, the nuisance parameters are the true parallaxes, $ˆ k . { } The convolution of the Gaussian and the true field distribution is not analytic and requires a numerical solution.

To derive the likelihood for the inner ring, ( $j θ) in equation (4.2), we use equa- L { }| tions (4.5) and (4.6), but with a change of index from k to j. The probability of observing any parallax for the jth star in the inner circle Pk($k $ˆ k) is also same as equation (4.7) | with a simple exchange of index from k to j. Next, we propose a distribution for the true parallaxes of the inner circle given the model parameters, Pj($ ˆ j θ). Once again, the inner circle is composed of cluster and ﬁeld stars. A | star in the inner circle is either drawn from the cluster or from the ﬁeld-star distributions, and the weighting for each draw is proportional to the density of the respective population. Hence, the distribution for the true parallaxes is ncl nf Pj($ ˆ j θ) = P ($ ˆ j $cl, $zp) + P ($ ˆ j L, $os) . (4.10) | ncl + nf | ncl + nf |

Since the size of the cluster is much smaller than the distance to the cluster, we assume a delta function for the cluster true-parallax distribution at $ ; P ($ ˆ j $ , $ ) = δ($ ˆ j $ $ ). cl | cl zp − cl − zp The second term represents the portion associated with the ﬁeld stars. To ﬁnd the likelihood distribution for the inner ring, we marginalize over the nuisance parameter for the inner ring,

$ˆ j . The convolution of a Gaussian and a delta function is analytical and a convolution of { } a Gaussian and the ﬁeld-star distribution requires a numerical integration.

90 To find the posterior distribution, equations (4.1), we choose uniform positive prior distributions for ncl, nf , $cl, $os, and L. Lindegren et al. [2018] found that the zero-point is a function of color, magnitude and position; hence, the zero-point has significant variance. However, there are no reference objects in our field to find the zero-point. Therefore, we must use prior information to estimate the zero-point of the cluster. For simplicity, we assume that the zero-point distribution for DR2 fields is Gaussian:

2 1 ($zp µzp) P ($zp µzp) = exp − −2 , (4.11) | √2πσzp 2σzp where µzp and σzp are the mean and variance for the parallax zero-point. The two most effective means to calculate zero-point are to either use background quasars or to use independent distance measurements. Wd1 is in the Galactic plane, so there are no background quasars, and the current distance estimates for Wd1 are too uncertain to use to constrain the zero-point. Therefore, we will use previous analyses to estimate the zero-point distribution. Lindegren et al. [2018] used quasars to infer the zero-point. They found an average of 0.029 − mas. Riess et al. [2018] inferred a zero-point of 0.046 0.013 mas for the Cepheid sample. − ± Zinn et al. [2019] compared the distances inferred from astroseismology to infer a zero-point of 0.0528 0.0024 mas. Stassun and Torres [2018] also reported the zero-point of 0.082 − ± − 0.033 mas from eclipsing binaries. The mean of the above four investigations is µ = ± zp 0.05 mas, and Lindegren et al. [2018] found that the variation for the zero-point across − many fields is σzp = 0.043 mas [Lindegren et al., 2018]. The data that we use in our inference cannot constrain the zero-point. Therefore, when we infer the posterior distribution for $zp, it will merely reflect this prior distribution.

4.1.4 Numerical Solution for the Posterior Distribution

To ﬁnd the posterior distribution, we use a six dimensional Monte Carlo Markov chain package (MCMC), emcee [Goodman and Weare, 2010, Foreman-Mackey et al., 2013] to infer six model parameters (ncl, nf , $cl, $zp, L, and $os). For each step in the chain, emcee evaluates the posterior by calculating the likelihood for the inner circle and the likelihood for the outer ring. Both likelihoods require the convolution of a Gaussian with the true ﬁeld

91 distribution. Evaluating these integrals is time intensive. Instead of calculating the integrals at every step in the chain, we create look-up tables for each integral. Each object has its own look-up table evaluated at a grid of points in L. For each trial of L in the MCMC, we find the convolution by first-order interpolation in the look-up table. To costruct each look-up table, we use trapezoid numerical integration, which requires bounds of integration. Formally, the bounds extend from$ ˆ = to$ ˆ = , but that is not −∞ ∞ practical for trapezoid numerical integration. Fortunately, the integrands in the likelihoods have a peak and fall off quickly on either side of this peak. To ensure that the numerical integration adequately samples this peak, we set the bounds of integration to be centred on the peak and have a width that extends just outside the peak. To roughly estimate the position of the peak and the extent of the bounds, we approximate the integrand as the convolution of two Gaussians. The mode and width of the first Gaussian is straightforward,

µ1 = $ and σ1 = σ$. The mode of the ﬁeld star distribution is µ2 = 0.25/L, and the width

is σ2 = 0.5/L. In the two Gaussian approximation, the mode of the integrand is roughly at

p 2 2 2 2 2 µ = pµ1 + qµ2 and the width is roughly σ = pσ1 + qσ2 + pµ1 + qµ2 µ where weighting 2 − factors are p = 4(Lσ) and q = 1 . Therefore, we integrate from$ ˆ = µ 2σ to 1+4(Lσ)2 1+4(Lσ)2 − $ˆ = µ + 2σ. For each MCMC run, we use 100 walkers, 2500 steps each, and we burn 500 of those. For the results presented in section 4.2, the acceptance fraction is in the α = 0.5 range.

4.2 Parallax and Distance to Westerlund 1

Fig. 4.6 shows the posterior distribution for $cl, ncl, nf, L, $os, and $zp. The two regions used to constrain these parameters are an inner circle centred on the position of Wd1 and with a radius of 1 arcmin, and an outer annulus from 9 to 10 arcmin. The values in the top right corner show the mode and the highest 68% density interval (HDI) for marginalized +0.07 distributions. The parallax of the cluster is $cl = 0.35−0.06 mas, which corresponds to a +0.6 +6.59 distance of 2.6−0.4 kpc, density of the cluster is ncl = 102.17−6.40 stars per square arcminute, +0.68 density of field stars is nf = 35.73−0.85 stars per square arcminute, the parallax zero-point of the cluster is $ = 0.05 0.04 mas, the field-star length scale is L = 1.21 0.02 zp − ± ± 92 kpc, and the field-star offset is $ = 0.12 0.02 mas. The posterior shows a single-peaked os ± marginalized probability distribution for all parameters. Presumably, the cluster density is a function of radius. Since we model the cluster density with one average density and not a radial profile, there is a potential for bias to affect the inference. As long as the width of each annulus is small compared to the change in density, then approximating the density in each ring with an average annulus should work well. To test this hypothesis, we infer the full posterior distribution for several inner annuli. For each inference, we use the outer annulus from 9 to 10 arcmin to constrain the field-star parameters. Fig. 4.7 shows the inferred parallaxes for each annulus. The rings extend from 0 to 0.5, 0.5 to 1.0, 1.0 to 1.5, and 1.5 to 2.0 arcmin with 8, 34, 105, and 141 total number of stars, respectively. In terms of parallax, the 68% highest density intervals are [0.25, 0.43], [0.28, 0.43], [0.34, 0.45], and [0.40, 0.54] mas. All rings below 1.5 arcmin are consistent with our +0.07 main result of 0.35−0.06 mas from using an inner circle with radius 1 arcmin (Fig. 4.6), and rings above 1.5 arcmin more likely represent the field stars parallax. Therefore, we conclude that modelling the average cluster density rather than a radial density profile is sufficient for the chosen inner region sizes.

4.3 Discussion

The results in Section 4.2 have significant implications for both the distance to Wd1 and +0.07 the distribution of field stars. The inferred L is 1.21−0.06 kpc, which is consistent with the model of Bailer-Jones et al. [2018], LBJ =1.38 kpc. The 2σ difference is not that surprising given that the Bailer-Jones et al. [2018] estimate for L varies as a function of Galactic longitude and latitude (l,b), and it does not include clusters. Also, given that the Bailer- Jones et al. [2018] length scale was derived from a model of the Galaxy before the era of accurate Gaia parallaxes, it is encouraging that the Bailer-Jones et al. [2018] model is only 2σ away from the inferred value. Using Bailer-Jones et al. [2018] length scale, L=1.38 kpc, and using the same parallax zero-point for the whole region, we find that the cluster parallax is 0.39 0.05 0.04(sys) mas, which differs by only 1σ from the value when we infer L ∼ ± ±

93 Figure 4.6: Posterior distribution for the six-parameter model. We report the mode and the highest density 68% confidence interval for the cluster parallax ($cl), the cluster density (ncl), the field-star density (nf), the field-star length scale (L), the field-star offset ($os), and the parallax zero-point of the cluster ($zp). The parallax +0.07 +0.6 of the cluster is $cl = 0.35−0.06 mas, which corresponds to a distance of R = 2.6−0.4 kpc. The posterior for the cluster zero-point, $zp, reflects the prior; there is no information in this data to constrain this parameter.

94 0.60

0.55

0.50

0.45 [mas]

cl 0.40 ϖ

0.35

0.30

0.25 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 rannulus[arcmin]

Figure 4.7: Bayesian inferred cluster parallax for each ring. Top and bottom dashed lines represent the highest density 68% conﬁdence interval. All rings bellow 1.5 arcmin are consistent with the inference from stars within 1 arcmin.

as well. While the results are formally consistent, using the Bailer-Jones et al. [2018] length scale, skews the cluster parallax towards the average ﬁeld-star distribution. +0.07 Our inferred parallax of the cluster is $cl = 0.35−0.06 mas, which corresponds to a distance +0.6 of 2.6−0.4 kpc. For this inference, we used Bayesian inference with a six-parameter model; there are other, simpler statistical inferences, but these typically ignore known contamination and systematics. These simpler inferences are useful in checking the results of our Bayesian inference. For example, we calculate the average distance for all stars within 1 arcmin from the cluster centre, where the individual distances comes from Bailer-Jones et al. [2018]. To estimate the uncertainty, we bootstrap the average. In this way, the average distance for these stars is 3.2 0.06(stat) 0.4(sys) kpc; 25% of our posterior distribution contains ± ± distances larger than this simple mean. The systematic uncertainty is due to the zero-point uncertainty [Lindegren et al., 2018] and it dominates the uncertainty. Or another approach is to calculate the true mean parallax for the inner circle with 1 arcmin radius (see the ﬁrst point in Fig. 4.4). This gives 0.36 0.09(stat) 0.04(sys) mas; this simple mean is within the ± ±

95 Figure 4.8: Same as Fig. 4.6, but we assume that the offset for the field-star distribution, $os is equal to the instrumental zero-point, $zp. With this assumption, the +0.04 +1.5 cluster parallax is $cl = 0.19−0.05 mas, which corresponds to distance of R = 4.9−1.0 kpc. However, the field-star distribution has systematics other than the instrumental systematic; therefore, Fig. 4.6 represents the most likely inference.

96 68% confidence interval of our inferred parallax. Once again, the systematic uncertainty is due to the uncertainty in the zero-point. While these simple approaches provide a good check on our result, these naive calculations are not sufficient. For one, many of the parallaxes are negative. Secondly, we infer that at least 1/4 of the stars in the inner arcminute are field stars. Therefore, to infer the distance to the cluster one must use Bayesian inference and model both the field stars and the cluster stars. Wd1 is located at a Galactic longitude of ` = 339.55°. Given this longitude and the new inferred distance of 2.6+0.6 kpc ( 8500 ly), Wd1 most likely lies in the Scutum-Centaurus −0.4 ∼ arm, which is one of the major spiral arms of the Milky Way [Bobylev and Bajkova, 2014, Urquhart et al., 2014, Vallée,2014]. This may be an independent way to confirm the new +0.6 inferred distance of 2.6−0.4 kpc to Wd1. Based upon the prior distribution, the parallax zero-point for the cluster is $ = 0.05 zp − ± 0.04 mas; in contrast, we infer a field-star offset of $ = 0.12 0.02 mas. In our model, os ± $os represents a combination of the instrumental zero-point, $zp, and a truncation of the field-star distribution. It could represent the zero-point for the entire region. However, 0.12 mas is much larger than the average of all other previous analyses (mean of 0.05 − mas). Therefore, the most likely conclusion is that $os is dominated by a truncation of the field star distribution. This would occur if the line of sight toward this region intersects an exceptionally dense region of dust at a true parallax around 0.17 mas. This scenario is consistent with the fact that this line of sight is in the plane of the Galaxy. Alternatively, +0.04 if we force $zp = $os, then we find that the parallax of the cluster is $cl = 0.19−0.05 mas, +1.5 which corresponds to a distance of R = 4.9−1.0 kpc (see Figure 4.8). Again, the most likely scenario is that $ = $ ; therefore, the most likely cluster parallax is $ = 0.35+0.07 mas. os 6 zp zp −0.06 σ$ is the expected parallax uncertainty, but the empirical uncertainty is different. For example, Lindegren et al. [2018] found that the empirical uncertainty is 1.081 times larger than the expected value for quasars in DR2. Similarly, we measure the empirical uncertainty distribution for all stars within 1 arcmin of Wd1. Fig. 4.9 shows the parallaxes centred on the cluster and weighted by the uncertainty: $ $ $ x = i − cl − zp . (4.12) σ$i 97

= −

− =

− − x = ϖ − ϖ − ϖ ϖ

Figure 4.9: Histogram of scale factor x for all sources within 1 arcmin of centre of the cluster. The empirical uncertainty for stars in Wd1 is 1.7 times larger than the DR2 solution. If uncertainties represents the full variation in the data, one would expect the distribution to have a 68% HDI that ranges from -1 to 1. However, the variances are signiﬁcantly smaller than expected variation for all of the stars within 1 arcmin of Wd1.

If the uncertainties are accurate, uncorrelated and random, then the distribution of x should

be Gaussian with σx = 1. In other words, 68% of x should be within the interval [ 1, 1]. − Fig. 4.9 reports the percent of the distribution inside of 1, 2, 3, and 4σ. If uncertainties represent the full variation and the distribution is Gaussian, then one would expect the distribution to have 68% of the data inside 1σ, 95% inside 2σ, etc. However, the probabilities are smaller than one would expect. In fact, we calculate that the 68% highest density interval

for x, HDIx is [ 2.4, 0.97]. The half width of this interval is 1.7. In other words, the empirical − uncertainties are 1.7 times larger than the expected uncertainties. This suggests that there are signiﬁcant problems with the uncertainty estimate in the region of Wd1. At the moment, it is not clear what is causing this inaccuracy. The high extinction (red colours), crowding, and binarity may be three important factors. Without having the raw data, it is diﬃcult to investigate this problem further, so we proceed with our inference, keeping in mind that the

98 DR2 uncertainties are likely too small by a factor of 1.7. The fact that the empirical uncertainties are 1.7 larger than the calculated uncertainties suggests that there is a problem with the ﬁve-parameter astrometric model for these stars. This could be due to any number of issues, including the excess noise model, binarity, or the degrees of freedom bug [Lindegren et al., 2018]. Formally, the astrometric excess noise is a way to incorporate extra empirical noise. The DR2 pipeline uses error propagation

to include the excess noise into σ$i of source i. However, even after the excess noise is included we find that there is still significant excess empirical noise. This discrepancy may be due to the specific model assumed for the excess noise in the DR2 pipeline. However, equation 120 of Lindegren et al. [2012] assumes that the excess noise for each observation is uncorrelated. Hence, more observations will reduce the uncertainty by the square root of the observations. However, if the true excess noise is correlated among the observations, more data will not reduce the uncertainty; the excess noise would represent a floor on the uncertainty. The assumption that the excess noise for each observation is uncorrelated may be why the empirical uncertainties for Wd1 are 1.7 times larger than the calculated uncertainties. +0.6 The inferred distance to Wd1 is 2.6−0.4 kpc, which represents the highest precision distance estimate for Wd1 that has been published. Historical estimates to Wd1 range from 1.0 to 5.5 kpc (see chapter 1). Recently, Clark et al. [2005] estimated that the distance to Wd1 ranged from 2 to 5.5 kpc. The bounds given by Clark et al. [2005] corresponds to a factor of 2.8 in distance; in contrast, the precision in the Gaia DR2 inferred distance is ∼ 18%. One can use the new distance to infer the fundamental parameters of the cluster such as luminosity, mass, and age via isochrone fitting. In this manuscript, we do not perform isochrone fitting. Instead, we estimate these fundamental parameters using two techniques. First, we scale previous estimates using the new distance, and we infer the luminosity, mass, and age of two bright stars in Wd1. The estimates of two bright stars provides a good proxy for the whole cluster.

The 18% precision in distance will lead to σL/L 36% precision in luminosity. For ≈ stars below about 20 M , the main-sequence luminosity for a given mass should scale as

99 3.5 L M . Therefore, the corresponding uncertainties in mass and age are σM /M 10.3% ∝ ≈ and σt/t 3.7%, respectively. For stars above about 55 M , L M, and in this case, the ≈ ∝ range of mass estimates has the same precision as the luminosity. For the highest masses, the age depends very weakly on mass or luminosity because these stars have very similar lifetimes around 3 Myr. The new Gaia DR2 distance also provides strong constraints on the luminosity, mass, and age of cluster members. Wd1 hosts a diverse population of evolved massive stars such as WR stars, red and blue supergiants, YHGs, an LBV, and a magnetar. Previous studies

have inferred a turn-oﬀ mass of around 40 M and cluster age of 3.5-5 Myr with a presumed distance of around 5 kpc [Clark et al., 2005, Crowther et al., 2006, Ritchie et al., 2009, Negueruela et al., 2010, Lim et al., 2013]. These estimates were based on modelling the luminosity and temperatures of YHGs, RSGs, and WR stars. By association, this would

imply that the magnetar progenitor had an initial mass of >40 M [Muno et al., 2005, Ritchie et al., 2010]. On the other hand, Koumpia and Bonanos [2012] found a progenitor

mass of 40 M and a distance value of 3.7 0.6 kpc by studying eclipsing binaries in the ∼ ± Wd1. Without re-evaluating the bolometric and extinction corrections for each star, the new +0.6 distance of 2.6−0.4 kpc reduces all luminosities by -0.58 dex as compared to a distance of 5 kpc. Using single-star stellar evolution models [Brott et al., 2011], we now infer the mass, age, and corresponding main-sequence turn-off mass for two of the brightest stars in Wd1, the LBV W243 and a YHG4. The inferred log(L/L ) for W243 is 5.2 0.1 and for YHG4 is M ± 5.4 0.1. The spectral type of W243 is B2I (to A2I) [Westerlund, 1987, Clark and Negueruela, ± 2003] and for YHG4 is F2Ia+. This corresponds to temperatures of 9.17 kK (to 17.58 kK) and 7.2 kK, respectively. For these temperatures, using single-star stellar evolution models +2.8 +4.1 [Brott et al., 2011], the masses are 23.9−3.2M for W243 and 28.6−4.8M for YHG4, the +1.0 +1.6 corresponding ages are 7.6−0.9 and 5.5−0.5 Myr, respectively. While we did not fit isochrones, +2.2 these ages would correspond to isochrones with main-sequence turn-off masses of 22.3−2.4M +2.8 (W243) and 25.9−3.2M (YHG4).

100 If we assume that LBV W243 is a representative of the cluster, then the age of the +1.0 +2.2 cluster is 7.6−0.9 Myr, the turn-off mass is 22.3−2.4M (down from 40M ), and the mass +4.1 of the most evolved stars is 28.6−4.8M . Fig. 4.10 shows that the new inferred luminosity brings the LBV W243 to the lower edge of the S Doradus instability strip [Smith et al., 2004]. However, Fig. 4.10 also clearly shows that there are RSGs in Wd1 with implied initial masses below 20 M , well below the presumed turn-off mass even at the nearer distance, and implied ages of around 10 Myr. This may suggest either uncertain bolometric corrections, a range of ages in Wd1, or may point to the influence of binary evolution on the evolved star population (see e.g. Beasor et al. 2019). Most of the prior distances for Wd1 relied on measuring an apparent magnitude, assigning an absolute magnitude based upon the stellar type, and calculating the distance modulus.

However, Wd1 suffers from high extinction, with an inferred AV of about 11 mag [Clark et al., 2005, Damineli et al., 2016]. The uncertainty in the reddening translates to highly uncertain true apparent magnitude estimates, and hence, highly uncertain luminosity-based distance estimates. With an independent geometric Gaia DR2 distance, the reddening and bolometric luminosities of cluster stars can be re-evaluated, although this is beyond the scope of this thesis. One final but important point concerns the total stellar mass of Wd1. This cluster has been discussed as potentially one of the most massive young star clusters in the Galaxy [Clark et al., 2005]. However, in addition to lowering the luminosities of the evolved stars, lowering the cluster turn-off mass, and raising the cluster’s age, the smaller distance from DR2 also lowers the total mass of the cluster. The inferred very high total stellar mass of

5 the cluster of 10 M was derived by integrating down a Kroupa IMF from the turn-off ∼ mass of around 40 M , and by scaling relative to the number of observed evolved supergiant stars of initially 30-40 M . If the revised DR2 distance lowers all luminosities by -0.58 dex, and hence the turn-off mass from 40 to 22 M as noted above, then the expected relative number of evolved stars increases by a factor of 4 at the turn-off mass. Normalizing to the ∼ observed number of evolved stars therefore lowers the extrapolated total stellar mass of the cluster by roughly the same amount, which would make Wd1’s initial mass comparable to

101 Figure 4.10: The HR diagram for evolved stars in Westerlund 1, including the LBV, W243. The open circles show the luminosities when d = 5 kpc, and the ﬁlled circles +0.6 show the luminosities with the new Gaia DR2 distance of 2.6−0.4 kpc. All ﬁlled circles have the same uncertainty as W243. The orange symbols represent YHGs, the red symbols represent RSGs, the purple symbols represent WNs, and the green circles represent WCs [Crowther et al., 2006, Fenech et al., 2018]. The gray boxes show the locations of the temperature-dependent S Doradus instability strip [Wolf, 1989a] and the constant temperature strip of LBVs in outburst, as in Smith et al. [2004]. The new Gaia DR2 distance brings the LBV W243 to the lower edge of the S Doradus instability strip. YHGs and RSGs [Clark et al., 2005, Mengel and Tacconi-Garman, 2007] have a wide range of zero-age-main-sequence masses which could be due to errors in bolometric correction [Davies and Beasor, 2017], variations in reddening, or could be due to binaries. The single-star model tracks (blue) are from Brott et al. [2011]. The evolutionary tracks do not reproduce the WR phases. +2.8 +1.0 The LBV W243 has an inferred mass of 23.9−3.2M , an age of 7.6−0.9 Myr. The +4.1 +1.6 brightest YHG has an inferred initial mass of around 28.6−4.8M , an age of 5.5−0.5 Myr.

102 or less than the mass of the Arches cluster in the Galactic centre, although Wd1 would be signiﬁcantly older [Kim et al., 2000, Stolte et al., 2002, Harfst et al., 2010].

103 CHAPTER 5

CONCLUSIONS

Smith and Tombleson [2015] found that LBVs are surprisingly isolated from other O stars. They suggested that the relative isolation is inconsistent with a single-star scenario in which the most massive stars undergo an LBV phase on their way to evolving into a WR star. Instead, they suggested that a binary scenario is likely more consistent with the relative isolation of LBVs. In this thesis, we test this hypothesis by developing crude models for single-star and binary scenarios in the context of cluster dissolution. Even with these crude models, we find that the LBVs’ isolation is mostly inconsistent with the standard passive single-star evolution model. In particular, if LBVs do evolve as single stars, then their isolation implies an age that is twice the maximum age of an average LBV. It may be the case that a small fraction of LBVs could evolve as single stars and still be consistent with the measured isolation. However, the fact that most LBVs are very isolated suggests that a large fraction is inconsistent with single-star evolution. For most LBVs, there is a clear problem in the single-star model’s mapping between luminosity and kinematic age, and this is either because there is a problem in the luminosity-to-mass mapping or there is a problem in the mass-to-age mapping. In this thesis, we consider how binary evolution might affect the latter, the mass-to-age mapping. We find that the LBV isolation is most consistent with two binary scenarios: either LBVs are mass gainers and receive a kick anywhere from 0 to 105 km/s or they are the product of mergers and are rejuvenated stars. Of course, LBVs ∼ may actually represent a combination of these two scenarios. Based on their environments, it is quite possible that some are mass gainers and some are the product of mergers. With current observations and theory, either binary model is consistent with the data. To further constrain which binary model is most consistent, we need to gather more data and develop better models. For example, detailed kinematic observations and theory would help to distinguish between these two models. Humphreys et al. [2016] suggest that the velocities

104 of LBVs are too low to be consistent with the kick scenario. However, there are binary scenarios that would produce low kick velocities. For example, if the secondary accretes so much mass that it becomes the more massive star in the binary, then this much more massive secondary will have a low orbital velocity in its binary orbit. When the low-mass primary explodes, the mass gainer drifts away at its low orbital speed. Hence, Smith [2016a] pointed out that large kick speeds are not necessarily expected, especially when there has been a significant amount of mass gained [Eldridge et al., 2011, de Mink et al., 2014]. We show that the mass-gainer scenario currently predicts a wide range of kick velocities. To truly test the consistency of the kick model, we must first model binary evolution and develop a model for the appearance of the kinematics, including randomness, projection, etc. The merger model would manifest as an inconsistency between the maximum age of the LBV and the surrounding stellar population. Therefore, to constrain the merger model, we need better mass estimates for the LBVs and age estimates for the surrounding stellar populations. Therefore, we must infer the fundamental properties of LBVs such as luminosity, mass, and age. Ultimately, these depend upon accurate Galactic distances. Using Gaia parallaxes, we infer the distances to 25 Galactic LBVs from the Gaia DR2[Smith et al., 2019]. The sample includes 11 LBVs and 14 LBV candidates. Nearly half of the new Gaia inferred distances are significantly closer than previous estimates in the literature, lowering their luminosities. For the remaining half of the sample, the DR2 distances are formally consistent within the uncertainty with previously adopted values, either because there is little change or because the large uncertainty encompasses a wide range of values. From the Gaia DR2 derived estimates, there are three main implications. 1) Most previous distance estimates (using many different methods with different systematic uncertainties) were so inaccurate that the authors did not calculate a formal uncertainty, so our Gaia DR2 distance estimates provide the first reliable uncertainties for most Galactic LBVs, measured in a consistent way. 2) Overall, there is a larger spread in luminosity that extends to lower luminosity than previously recognized for Galactic LBVs. The new luminosity estimates for the Galactic LBVs no longer define a tight S Dor instability strip, with several confirmed LBVs landing off the S Dor instability strip (often below it). 3) There are two divergent

105 ways to interpret the result. One option is simply that these lower-luminosity stars were mistakenly classified as LBVs. This is problematic because a few of the lower-luminosity sources are “confirmed” LBVs that have demonstrated LBV-like variability, and are not just candidate LBVs. The other way to view the spread is that whatever mechanism causes the LBV instability may apply to a larger luminosity spread, making it difficult to understand the LBV instability as arising when a star encounters the Eddington limit in single-star evolution. This would be a significant revision in our understanding of the post-main-sequence evolution of massive stars. The original demarkation of the S Dor strip might therefore fail to capture the diverse range of real physical variability exhibited by luminous, blue, and irregularly variable stars. Then, we use Gaia DR2 parallax measurements and Bayesian inference to estimate the distance to the Westerlund 1 (Wd1) massive star cluster, which also hosts an LBV. We model +0.07 both cluster stars and Galactic field stars, and we find that the cluster parallax is 0.35−0.06 +0.6 mas, which corresponds to a distance of 2.6−0.4 kpc. The new distance represents the highest precision, 18%, to Wd1 to date. Much of this precision is limited by the systematics such as parallax zero-point, which is included in the Bayesian model. However, the models are rough and conservative, and require improvement in the future. For example, rather than using one zero-point value, we consider a distribution of zero-points due to the observed variation of the parallax zero-point. We also consider different offsets for the field and cluster stars. This model is a rough estimate, and to further improve the parallax precision will either require better models for the systematics, or better calibration in subsequent data releases. Wd1 has been discussed as potentially one of the most massive young star clusters in the Galaxy, but revising the distance to this one cluster reduces its total mass and increases its age, and may have profound consequences for stellar evolution theory. An improved distance can significantly narrow the precision on luminosity, mass, and age of the cluster, which provides constraints on the post-main-sequence evolution of cluster members. Based

on the new Gaia distance, we infer turn-oﬀ mass of around 22 M , which implies that the progenitor mass of the magnetar CXO J164710.2–455216, and LBV W243 is a little bit above

22 M .

106 BIBLIOGRAPHY

Mojgan Aghakhanloo, Jeremiah W. Murphy, Nathan Smith, and Ren´eeHloˇzek.Modelling luminous-blue-variable isolation. MNRAS, 472(1):591–603, November 2017. doi: 10. 1093/mnras/stx2050.

Mojgan Aghakhanloo, Jeremiah W. Murphy, Nathan Smith, John Parejko, Mariangelly D´ıaz- Rodr´ıguez, Maria R. Drout, Jose H. Groh, Joseph Guzman, and Keivan G. Stassun. Inferring the parallax of Westerlund 1 from Gaia DR2. MNRAS, 492(2):2497–2509, Feb 2020. doi: 10.1093/mnras/stz3628.

J. P. Anderson and P. A. James. Constraints on core-collapse supernova progenitors from correlations with Hα emission. MNRAS, 390(4):1527–1538, Nov 2008. doi: 10.1111/j. 1365-2966.2008.13843.x.

J. P. Anderson, S. M. Habergham, P. A. James, and M. Hamuy. Progenitor mass constraints for core-collapse supernovae from correlations with host galaxy star formation. MNRAS, 424(2):1372–1391, Aug 2012. doi: 10.1111/j.1365-2966.2012.21324.x.

David Arnett. The Light Curve of Supernova 1987A in Context. ApJ, 343:834, Aug 1989. doi: 10.1086/167753.

Tri L. Astraatmadja and Coryn A. L. Bailer-Jones. Estimating Distances from Parallaxes. III. Distances of Two Million Stars in the Gaia DR1 Catalogue. ApJ, 833(1):119, Dec 2016. doi: 10.3847/1538-4357/833/1/119.

C. A. L. Bailer-Jones, J. Rybizki, M. Fouesneau, G. Mantelet, and R. Andrae. Estimating distance from parallaxes. iv. distances to 1.33 billion stars in gaia data release 2. AJ, 156(2):58, Jul 2018. ISSN 1538-3881. doi: 10.3847/1538-3881/aacb21. URL http: //dx.doi.org/10.3847/1538-3881/aacb21.

C. A. L. Bailer-Jones, J. Rybizki, M. Fouesneau, G. Mantelet, and R. Andrae. Estimating Distance from Parallaxes. IV. Distances to 1.33 Billion Stars in Gaia Data Release 2. AJ, 156(2):58, Aug 2018. doi: 10.3847/1538-3881/aacb21.

C´assioL. Barbosa, Robert D. Blum, Peter S. Conti, Augusto Damineli, and Elysandra Figuerˆedo. High Spatial Resolution Spectroscopy of W51 IRS 2E and IRS 2W: Two Very Massive Young Stars in Early Formation Stages. ApJL, 678(1):L55, May 2008. doi: 10.1086/588500.

107 G. Baume, R. A. V´azquez,and A. Feinstein. UBVI imaging photometry of NGC 6231. A&AS, 137:233–244, Jun 1999. doi: 10.1051/aas:1999245.

Emma R. Beasor and Ben Davies. The evolution of red supergiant mass-loss rates. MNRAS, 475(1):55–62, Mar 2018. doi: 10.1093/mnras/stx3174.

Emma R Beasor, Ben Davies, Nathan Smith, and Nate Bastian. Discrepancies in the ages of young star clusters; evidence for mergers? Monthly Notices of the Royal Astronomical Society, 486(1):266–273, Apr 2019. ISSN 1365-2966. doi: 10.1093/mnras/stz732. URL http://dx.doi.org/10.1093/mnras/stz732.

E. Berger, R. Foley, and I. Ivans. SN 2009ip is an LBV Outburst. The Astronomer’s Telegram, 2184:1, September 2009a.

E. Berger, A. M. Soderberg, R. A. Chevalier, C. Fransson, R. J. Foley, D. C. Leonard, J. H. Debes, A. M. Diamond-Stanic, A. K. Dupree, I. I. Ivans, J. Simmerer, I. B. Thompson, and C. A. Tremonti. An Intermediate Luminosity Transient in NGC 300: The Eruption of a Dust-Enshrouded Massive Star. ApJ, 699(2):1850–1865, Jul 2009b. doi: 10.1088/0004-637X/699/2/1850.

S. R. Berlanas, N. J. Wright, A. Herrero, J. E. Drew, and D. J. Lennon. Disentangling the spatial substructure of Cygnus OB2 from Gaia DR2. MNRAS, 484(2):1838–1842, Apr 2019. doi: 10.1093/mnras/stz117.

A. Blaauw. On the origin of the O- and B-type stars with high velocities (the ”run-away” stars), and some related problems. BAN, 15:265, May 1961.

V. V. Bobylev and A. T. Bajkova. The Milky Way spiral structure parameters from data on masers and selected open clusters. MNRAS, 437:1549–1553, January 2014. doi: 10.1093/mnras/stt1987.

B. Bohannan and N. R. Walborn. The Ofpe/WN9 class in the Large Magellanic Cloud. PASP, 101:520–527, May 1989. doi: 10.1086/132463.

Howard E. Bond, Luigi R. Bedin, Alceste Z. Bonanos, Roberta M. Humphreys, L. A. G. Berto Monard, Jos´eL. Prieto, and Frederick M. Walter. The 2008 Luminous Optical Tran- sient in the Nearby Galaxy NGC 300. ApJL, 695(2):L154–L158, Apr 2009. doi: 10.1088/0004-637X/695/2/L154.

108 M. T. Botticella, A. Pastorello, S. J. Smartt, W. P. S. Meikle, S. Benetti, R. Kotak, E. Cap- pellaro, R. M. Crockett, S. Mattila, M. Sereno, F. Patat, D. Tsvetkov, J. Th. van Loon, D. Abraham, I. Agnoletto, R. Arbour, C. Benn, G. di Rico, N. Elias-Rosa, D. L. Gor- shanov, A. Harutyunyan, D. Hunter, V. Lorenzi, F. P. Keenan, K. Maguire, J. Mendez, M. Mobberley, H. Navasardyan, C. Ries, V. Stanishev, S. Taubenberger, C. Trundle, M. Turatto, and I. M. Volkov. SN 2008S: an electron-capture SN from a super-AGB progenitor? MNRAS, 398(3):1041–1068, Sep 2009. doi: 10.1111/j.1365-2966.2009.15082.x.

W. Brandner, J. S. Clark, A. Stolte, R. Waters, I. Negueruela, and S. P. Goodwin. In- termediate to low-mass stellar content of westerlund 1. Astronomy & Astrophysics, 478(1):137–149, Nov 2007. ISSN 1432-0746. doi: 10.1051/0004-6361:20077579. URL http://dx.doi.org/10.1051/0004-6361:20077579.

W. Brandner, J. S. Clark, A. Stolte, R. Waters, I. Negueruela, and S. P. Goodwin. Inter- mediate to low-mass stellar content of Westerlund 1. A&A, 478(1):137–149, Jan 2008. doi: 10.1051/0004-6361:20077579.

I. Brott, S. E. de Mink, M. Cantiello, N. Langer, A. de Koter, C. J. Evans, I. Hunter, C. Trundle, and J. S. Vink. Rotating massive main-sequence stars. Astronomy & As- trophysics, 530:A115, May 2011. ISSN 1432-0746. doi: 10.1051/0004-6361/201016113. URL http://dx.doi.org/10.1051/0004-6361/201016113.

I. Brott, S. E. de Mink, M. Cantiello, N. Langer, A. de Koter, C. J. Evans, I. Hunter, C. Trun- dle, and J. S. Vink. Rotating massive main-sequence stars. I. Grids of evolutionary models and isochrones. A&A, 530:A115, Jun 2011. doi: 10.1051/0004-6361/201016113.

A. G. A. Brown, A. Vallenari, T. Prusti, J. H. J. de Bruijne, C. Babusiaux, C. A. L. Bailer- Jones, M. Biermann, D. W. Evans, L. Eyer, and et al. Gaiadata release 2. Astronomy & Astrophysics, 616:A1, Aug 2018. ISSN 1432-0746. doi: 10.1051/0004-6361/201833051. URL http://dx.doi.org/10.1051/0004-6361/201833051.

E. L. Chentsov. The star HD 168607 - an S Doradus object. Soviet Astronomy Letters, 6: 199–201, Jun 1980.

R. Chini, H. Elsaesser, and T. Neckel. Multicolour UBVRI photometry of stars in M17. A&A, 91:186, Nov 1980.

J. S. Clark and I. Negueruela. A newly identiﬁed luminous blue variable in the galactic starburst cluster westerlund 1. Astronomy & Astrophysics, 413(2):L15–L18, Dec 2003. ISSN 1432-0746. doi: 10.1051/0004-6361:20031700. URL http://dx.doi.org/10.1051/ 0004-6361:20031700.

109 J. S. Clark and I. Negueruela. A newly identiﬁed Luminous Blue Variable in the galactic starburst cluster ¡ASTROBJ¿Westerlund 1¡/ASTROBJ¿. A&A, 413:L15–L18, Jan 2004. doi: 10.1051/0004-6361:20031700.

J. S. Clark, V. M. Larionov, and A. Arkharov. On the population of galactic Luminous Blue Variables. A&A, 435(1):239–246, May 2005. doi: 10.1051/0004-6361:20042563.

J. S. Clark, I. Negueruela, P. A. Crowther, and S. P. Goodwin. On the massive stellar population of the super star cluster westerlund 1. Astronomy & Astrophysics, 434 (3):949–969, Apr 2005. ISSN 1432-0746. doi: 10.1051/0004-6361:20042413. URL http://dx.doi.org/10.1051/0004-6361:20042413.

J. S. Clark, I. Negueruela, P. A. Crowther, and S. P. Goodwin. On the massive stellar population of the super star cluster ¡ASTROBJ¿Westerlund 1¡/ASTROBJ¿. A&A, 434(3):949–969, May 2005. doi: 10.1051/0004-6361:20042413.

J. S. Clark, B. Davies, F. Najarro, J. MacKenty, P. A. Crowther, M. Messineo, and M. A. Thompson. The P Cygni supergiant [OMN2000] LS1 - implications for the star formation history of W51. A&A, 504(2):429–435, Sep 2009. doi: 10.1051/0004-6361/ 200911980.

J. S. Clark, F. Najarro, I. Negueruela, B. W. Ritchie, M. A. Urbaneja, and I. D. Howarth. On the nature of the galactic early-B hypergiants. A&A, 541:A145, May 2012. doi: 10.1051/0004-6361/201117472.

Charlie Conroy, Jay Strader, Pieter van Dokkum, Andrew E. Dolphin, Daniel R. Weisz, Jeremiah W. Murphy, Aaron Dotter, Benjamin D. Johnson, and Phillip Cargile. A Complete Census of Luminous Stellar Variability on Day to Decade Timescales. ApJ, 864(2):111, Sep 2018. doi: 10.3847/1538-4357/aad460.

P. S. Conti. Basic Observational Constraints on the Evolution of Massive Stars. In A. Maeder and A. Renzini, editors, Observational Tests of the Stellar Evolution Theory, volume 105 of IAU Symposium, page 233, January 1984a.

P. A. Crowther, L. J. Hadﬁeld, J. S. Clark, I. Negueruela, and W. D. Vacca. A census of the wolf-rayet content in westerlund 1 from near-infrared imaging and spectroscopy. Monthly Notices of the Royal Astronomical Society, 372(3):1407–1424, Nov 2006. ISSN 1365-2966. doi: 10.1111/j.1365-2966.2006.10952.x. URL http://dx.doi.org/10.1111/j. 1365-2966.2006.10952.x.

110 P. A. Crowther, D. J. Lennon, and N. R. Walborn. Physical parameters and wind properties of galactic early B supergiants. A&A, 446(1):279–293, Jan 2006. doi: 10.1051/0004-6361:20053685.

A. Damineli, L. A. Almeida, R. D. Blum, D. S. C. Damineli, F. Navarete, M. S. Rubinho, and M. Teodoro. Extinction law in the range 0.4–4.8 µm and the 8620 ˚adib towards the stellar cluster westerlund 1. Monthly Notices of the Royal Astronomical Society, 463(3):2653–2666, Aug 2016. ISSN 1365-2966. doi: 10.1093/mnras/stw2122. URL http://dx.doi.org/10.1093/mnras/stw2122.

Kris Davidson, Roberta M. Humphreys, Arsen Hajian, and Yervant Terzian. He 3-519: A Peculiar Post-LBV, Pre-WN Star? ApJ, 411:336, Jul 1993. doi: 10.1086/172833.

Ben Davies and Emma R Beasor. The initial masses of the red supergiant progenitors to type ii supernovae. Monthly Notices of the Royal Astronomical Society, 474(2): 2116–2128, Oct 2017. ISSN 1365-2966. doi: 10.1093/mnras/stx2734. URL http: //dx.doi.org/10.1093/mnras/stx2734.

Cornelis de Jager. The yellow hypergiants. A&A Rv, 8(3):145–180, Jan 1998. doi: 10.1007/ s001590050009.

A. de Koter, H. J. G. L. M. Lamers, and W. Schmutz. Variability of luminous blue variables. II. Parameter study of the typical LBV variations. A&A, 306:501, February 1996.

S. E. de Mink, H. Sana, N. Langer, R. G. Izzard, and F. R. N. Schneider. The Incidence of Stellar Mergers and Mass Gainers among Massive Stars. ApJ, 782:7, February 2014. doi: 10.1088/0004-637X/782/1/7.

C. G. de Pree, Melissa C. Nysewander, and W. M. Goss. NGC 3576 and NGC 3603: Two Luminous Southern H II Regions Observed at High Resolution with the Australia Telescope Compact Array. AJ, 117(6):2902–2918, Jun 1999. doi: 10.1086/300892.

Luc Dessart, Eli Livne, and Roni Waldman. Shock-heating of stellar envelopes: a possible common mechanism at the origin of explosions and eruptions in massive stars. MNRAS, 405(4):2113–2131, July 2010. doi: 10.1111/j.1365-2966.2010.16626.x.

Michael P. Egan, J. Simon Clark, Donald R. Mizuno, Sean J. Carey, Iain A. Steele, and Stephan D. Price. An Infrared Ring Nebula around MSX5C G358.5391+00.1305: The True Nature of Suspected Planetary Nebula Wray 17-96 Determined via Direct Imaging and Spectroscopy. ApJ, 572(1):288–299, Jun 2002. doi: 10.1086/340222.

111 J. J. Eldridge, N. Langer, and C. A. Tout. Runaway stars as progenitors of supernovae and gamma-ray bursts. MNRAS, 414:3501–3520, July 2011. doi: 10.1111/j.1365-2966. 2011.18650.x.

B. G. Elmegreen and Y. N. Efremov. A Universal Formation Mechanism for Open and Globular Clusters in Turbulent Gas. ApJ, 480:235–245, May 1997.

D. M. Fenech, J. S. Clark, R. K. Prinja, S. Dougherty, F. Najarro, I. Negueruela, A. Richards, B. W. Ritchie, and H. Andrews. An alma 3 mm continuum census of westerlund 1. Astronomy & Astrophysics, 617:A137, Sep 2018. ISSN 1432-0746. doi: 10.1051/ 0004-6361/201832754. URL http://dx.doi.org/10.1051/0004-6361/201832754.

E. Figuerˆedo,R. D. Blum, A. Damineli, P. S. Conti, and C. L. Barbosa. The Stellar Content of Obscured Galactic Giant H II Regions. Vi. W51A. AJ, 136(1):221–233, Jul 2008. doi: 10.1088/0004-6256/136/1/221.

Ryan J. Foley, Edo Berger, Ori Fox, Emily M. Levesque, Peter J. Challis, Inese I. Ivans, James E. Rhoads, and Alicia M. Soderberg. The Diversity of Massive Star Outbursts. I. Observations of SN2009ip, UGC 2773 OT2009-1, and Their Progenitors. ApJ, 732 (1):32, May 2011. doi: 10.1088/0004-637X/732/1/32.

Daniel Foreman-Mackey, David W. Hogg, Dustin Lang, and Jonathan Goodman. emcee: The mcmc hammer. Publications of the Astronomical Society of the Paciﬁc, 125(925): 306–312, Mar 2013. ISSN 1538-3873. doi: 10.1086/670067. URL http://dx.doi.org/ 10.1086/670067.

J. S. Gallagher. Close binary models for luminous blue variables stars. In K. Davidson, A. F. J. Moﬀat, and H. J. G. L. M. Lamers, editors, IAU Colloq. 113: Physics of Luminous Blue Variables, volume 157 of Astrophysics and Space Science Library, pages 185–192, 1989a. doi: 10.1007/978-94-009-1031-7 22.

J. S. Gallagher. Close Binary Models for Luminous Blue Variables Stars. In Kris Davidson, A. F. J. Moﬀat, and H. J. G. L. M. Lamers, editors, IAU Colloq. 113: Physics of Luminous Blue Variables, volume 157 of Astrophysics and Space Science Library, page 185. 1989b. doi: 10.1007/978-94-009-1031-7 22.

M. Gennaro, W. Brandner, A. Stolte, and Th. Henning. Mass segregation and elongation of the starburst cluster westerlund 1. Monthly Notices of the Royal Astronomical Society, 412(4):2469–2488, Feb 2011. ISSN 0035-8711. doi: 10.1111/j.1365-2966.2010.18068.x. URL http://dx.doi.org/10.1111/j.1365-2966.2010.18068.x.

112 Stephanie M. Gogarten, Julianne J. Dalcanton, Jeremiah W. Murphy, Benjamin F. Williams, Karoline Gilbert, and Andrew Dolphin. The NGC 300 Transient: An Alternative Method for Measuring Progenitor Masses. ApJ, 703(1):300–310, Sep 2009. doi: 10. 1088/0004-637X/703/1/300.

Jonathan Goodman and Jonathan Weare. Ensemble samplers with aﬃne invariance. Commun. Appl. Math. Comput. Sci., 5(1):65–80, Jan 2010. ISSN 1559-3940. doi: 10.2140/camcos.2010.5.65. URL http://dx.doi.org/10.2140/camcos.2010.5.65.

G. Gr¨afener,S. P. Owocki, and J. S. Vink. Stellar envelope inﬂation near the Eddington limit. Implications for the radii of Wolf-Rayet stars and luminous blue variables. A&A, 538:A40, February 2012a. doi: 10.1051/0004-6361/201117497.

J. H. Groh, A. Damineli, M. Teodoro, and C. L. Barbosa. Detection of additional wolf- rayet stars in the starburst cluster westerlund 1 with soar. Astronomy & Astrophysics, 457(2):591–594, Sep 2006. ISSN 1432-0746. doi: 10.1051/0004-6361:20064929. URL http://dx.doi.org/10.1051/0004-6361:20064929.

J. H. Groh, D. J. Hillier, and A. Damineli. AG Carinae: A Luminous Blue Variable with a High Rotational Velocity. ApJL, 638(1):L33–L36, Feb 2006. doi: 10.1086/500928.

J. H. Groh, A. Damineli, D. J. Hillier, R. Barbá,E. Fernández-Lajús,R. C. Gamen, A. P. Moisés,G. Solivella, and M. Teodoro. Bona Fide, Strong-Variable Galactic Luminous Blue Variable Stars are Fast Rotators: Detection of a High Rotational Velocity in HR Carinae. ApJL, 705(1):L25–L30, Nov 2009a. doi: 10.1088/0004-637X/705/1/L25.

J. H. Groh, D. J. Hillier, A. Damineli, P. A. Whitelock, F. Marang, and C. Rossi. On the Nature of the Prototype Luminous Blue Variable Ag Carinae. I. Fundamental Parameters During Visual Minimum Phases and Changes in the Bolometric Luminosity During the S-Dor Cycle. ApJ, 698:1698–1720, June 2009b. doi: 10.1088/0004-637X/ 698/2/1698.

J. H. Groh, D. J. Hillier, A. Damineli, P. A. Whitelock, F. Marang, and C. Rossi. On the Nature of the Prototype Luminous Blue Variable Ag Carinae. I. Fundamental Parameters During Visual Minimum Phases and Changes in the Bolometric Luminosity During the S-Dor Cycle. ApJ, 698(2):1698–1720, Jun 2009c. doi: 10.1088/0004-637X/ 698/2/1698.

113 Jose H. Groh, Georges Meynet, Cyril Georgy, and Sylvia Ekstr¨om.Fundamental properties of core-collapse supernova and GRB progenitors: predicting the look of massive stars before death. A&A, 558:A131, Oct 2013. doi: 10.1051/0004-6361/201321906.

Jose H. Groh, Georges Meynet, Sylvia Ekstr¨om,and Cyril Georgy. The evolution of massive stars and their spectra. I. A non-rotating 60 M star from the zero-age main sequence to the pre-supernova stage. A&A, 564:A30, Apr 2014. doi: 10.1051/0004-6361/201322573.

V. V. Gvaramadze, A. Y. Kniazev, A. S. Miroshnichenko, L. N. Berdnikov, N. Langer, G. S. Stringfellow, H. Todt, W. R. Hamann, E. K. Grebel, D. Buckley, L. Crause, S. Crawford, A. Gulbis, C. Hettlage, E. Hooper, T. O. Husser, P. Kotze, N. Loaring, K. H. Nordsieck, D. O’Donoghue, T. Pickering, S. Potter, E. Romero Colmenero, P. Vaisanen, T. Williams, M. Wolf, D. E. Reichart, K. M. Ivarsen, J. B. Haislip, M. C. Nysewander, and A. P. LaCluyze. Discovery of two new Galactic candidate luminous blue variables with Wide-ﬁeld Infrared Survey Explorer. MNRAS, 421(4):3325–3337, Apr 2012. doi: 10.1111/j.1365-2966.2012.20556.x.

V. V. Gvaramadze, A. Y. Kniazev, and L. N. Berdnikov. Discovery of a new bona ﬁde luminous blue variable in Norma. MNRAS, 454(4):3710–3721, Dec 2015. doi: 10.1093/ mnras/stv2278.

S. M. Habergham, J. P. Anderson, P. A. James, and J. D. Lyman. Environments of interacting transients: impostors and Type IIn supernovae. MNRAS, 441(3):2230–2252, Jul 2014. doi: 10.1093/mnras/stu684.

S. Harfst, S. Portegies Zwart, and A. Stolte. Reconstructing the arches cluster - i. constraining the initial conditions. Monthly Notices of the Royal Astronomical Society, 409(2): 628–638, Sep 2010. ISSN 0035-8711. doi: 10.1111/j.1365-2966.2010.17326.x. URL http://dx.doi.org/10.1111/j.1365-2966.2010.17326.x.

L. A. Higgs, H. J. Wendker, and T. L. Landecker. G 79.29+0.46: a young stellar-wind shell in Cygnus X. A&A, 291:295–309, Nov 1994.

D. J. Hillier, P. A. Crowther, F. Najarro, and A. W. Fullerton. An optical and near-IR spectroscopic study of the extreme P Cygni-type supergiant HDE 316285. A&A, 340: 483–496, Dec 1998.

D. John Hillier, K. Davidson, K. Ishibashi, and T. Gull. On the Nature of the Central Source in η Carinae. ApJ, 553(2):837–860, Jun 2001. doi: 10.1086/320948.

N. M. Hoekzema, H. J. G. L. M. Lamers, and A. M. van Genderen. The distance and reddening of stars near the luminous blue variable AG Carinae. A&A, 257:118–127, Apr 1992.

114 J. Y. Hu, D. de Winter, P. S. The, and M. R. Perez. WRA 751, a candidate for a new luminous blue variable. A&A, 227:L17–L20, Jan 1990.

E. Hubble and A. Sandage. The Brightest Variable Stars in Extragalactic Nebulae. I. M31 and M33. ApJ, 118:353, November 1953. doi: 10.1086/145764.

R. M. Humphreys. Studies of luminous stars in nearby galaxies. I. Supergiants and O stars in the Milky Way. ApJS, 38:309–350, Dec 1978. doi: 10.1086/190559.

R. M. Humphreys, H. J. G. L. M. Lamers, N. Hoekzema, and A. Cassatella. The distance and evolutionary phase of the luminous blue variable AG Car. A&A, 218:L17–L20, Jul 1989.

R. M. Humphreys, K. Davidson, and N. Smith. η Carinae’s Second Eruption and the Light Curves of the η Carinae Variables. PASP, 111:1124–1131, September 1999. doi: 10. 1086/316420.

R. M. Humphreys, K. Weis, K. Davidson, and M. S. Gordon. On the Social Traits of Luminous Blue Variables. ArXiv e-prints, March 2016.

Roberta M. Humphreys and Kris Davidson. The Luminous Blue Variables: Astrophysical Geysers. PASP, 106:1025, Oct 1994. doi: 10.1086/133478.

Roberta M. Humphreys, Kris Davidson, David Hahn, John C. Martin, and Kerstin Weis. Luminous and Variable Stars in M31 and M33. V. The Upper HR Diagram. ApJ, 844 (1):40, July 2017. doi: 10.3847/1538-4357/aa7cef.

D. Hutsemekers and E. van Drom. HR Carinae : a luminous blue variable surrounded by an arc-shaped nebula. A&A, 248:141, Aug 1991.

Hiroshi Imai, Teruhiko Watanabe, Toshihiro Omodaka, Masanori Nishio, Osamu Kameya, Takeshi Miyaji, and Junichi Nakajima. 3-D Kinematics of Water Masers in the W 51A Region. PASJ, 54:741–755, Oct 2002. doi: 10.1093/pasj/54.5.741.

Zeljko. Ivezic, Andrew . J. Connolly, Jacob . T VanderPlas, and Alexander Gra. Statis- tics, Data Mining, and Machine Learning in Astronomy. Princeton Series in Modern Observational Astronomy. Princeton University Press, 2014.

R. G. Izzard, C. A. Tout, A. I. Karakas, and O. R. Pols. A new synthetic model for asymptotic giant branch stars. MNRAS, 350:407–426, May 2004. doi: 10.1111/j.1365-2966.2004. 07446.x.

115 R. G. Izzard, L. M. Dray, A. I. Karakas, M. Lugaro, and C. A. Tout. Population nucleosyn- thesis in single and binary stars. I. Model. A&A, 460:565–572, December 2006. doi: 10.1051/0004-6361:20066129.

R. G. Izzard, E. Glebbeek, R. J. Stancliﬀe, and O. R. Pols. Population synthesis of binary carbon-enhanced metal-poor stars. A&A, 508:1359–1374, December 2009. doi: 10. 1051/0004-6361/200912827.

Yan-Fei Jiang, Matteo Cantiello, Lars Bildsten, Eliot Quataert, Omer Blaes, and James Stone. Outbursts of luminous blue variable stars from variations in the helium opacity. Nature, 561(7724):498–501, Sep 2018. doi: 10.1038/s41586-018-0525-0.

S. Justham, P. Podsiadlowski, and J. S. Vink. Luminous Blue Variables and Superluminous Supernovae from Binary Mergers. ApJ, 796:121, December 2014a. doi: 10.1088/ 0004-637X/796/2/121.

Stephen Justham, Philipp Podsiadlowski, and Jorick S. Vink. Luminous Blue Variables and Superluminous Supernovae from Binary Mergers. ApJ, 796(2):121, Dec 2014b. doi: 10.1088/0004-637X/796/2/121.

S. J. Kenyon and III Gallagher, J. S. Spectroscopy of the winds from Hubble-Sandage stars in M 31 and M 33. ApJ, 290:542–550, Mar 1985. doi: 10.1086/163010.

Sungsoo S. Kim, Donald F. Figer, Hyung Mok Lee, and Mark Morris. N-body simulations of compact young clusters near the galactic center. The Astrophysical Journal, 545(1): 301–308, Dec 2000. ISSN 1538-4357. doi: 10.1086/317807. URL http://dx.doi.org/10. 1086/317807.

C. R. Klein, S. B. Cenko, A. A. Miller, D. J. Norman, and J. S. Bloom. Probing the distance and morphology of the Large Magellanic Cloud with RR Lyrae stars. ArXiv e-prints, May 2014.

A. Y. Kniazev, V. V. Gvaramadze, and L. N. Berdnikov. WS1: one more new Galactic bona ﬁde luminous blue variable. MNRAS, 449:L60–L64, Apr 2015. doi: 10.1093/mnrasl/ slv023.

A. Y. Kniazev, V. V. Gvaramadze, and L. N. Berdnikov. MN48: a new Galactic bona ﬁde luminous blue variable revealed by Spitzer and SALT. MNRAS, 459(3):3068–3077, Jul 2016. doi: 10.1093/mnras/stw889.

R. Kotak and J. S. Vink. Luminous blue variables as the progenitors of supernovae with quasi- periodic radio modulations. A&A, 460:L5–L8, December 2006. doi: 10.1051/0004-6361: 20065800.

116 R. Kothes and S. M. Dougherty. The distance and neutral environment of the massive stellar cluster westerlund 1. Astronomy & Astrophysics, 468(3):993–1000, Apr 2007. ISSN 1432-0746. doi: 10.1051/0004-6361:20077309. URL http://dx.doi.org/10.1051/ 0004-6361:20077309.

E. Koumpia and A. Z. Bonanos. Fundamental parameters of four massive eclipsing binaries in westerlund 1. Astronomy & Astrophysics, 547:A30, Oct 2012. ISSN 1432-0746. doi: 10.1051/0004-6361/201219465. URL http://dx.doi.org/10.1051/0004-6361/201219465.

J. R. Kozok. Distances, reddenings and distribution of emission B-stars in the galactic centre region L <= 45. A&AS, 62:7–16, Oct 1985.

M. R. Krumholz and J. C. Tan. Slow Star Formation in Dense Gas: Evidence and Implica- tions. ApJ, 654:304–315, January 2007. doi: 10.1086/509101.

R.-P. Kudritzki and J. Puls. Winds from Hot Stars. ARA&A, 38:613–666, 2000. doi: 10.1146/annurev.astro.38.1.613.

H. J. G. L. M. Lamers and T. Nugis. An explanation for the curious mass loss history of massive stars: From OB stars, through Luminous Blue Variables to Wolf-Rayet stars. A&A, 395:L1–L4, Nov 2002. doi: 10.1051/0004-6361:20021381.

Henny J. G. L. M. Lamers and Edward L. Fitzpatrick. The Relationship between the Edding- ton Limit, the Observed Upper Luminosity Limit for Massive Stars, and the Luminous Blue Variables. ApJ, 324:279, Jan 1988. doi: 10.1086/165894.

N. Langer and R. P. Kudritzki. The spectroscopic Hertzsprung-Russell diagram. A&A, 564: A52, Apr 2014. doi: 10.1051/0004-6361/201423374.

Norbert Langer, Guillermo Garc´ıa-Segura,and Mordecai-Mark Mac Low. Giant Outbursts of Luminous Blue Variables and the Formation of the Homunculus Nebula around η Carinae. ApJL, 520(1):L49–L53, Jul 1999. doi: 10.1086/312131.

W. Li, N. Smith, A. A. Miller, and A. V. Filippenko. Rebrightening of SN 2009ip is Remi- niscent of eta Carinae Just Before 1843 Eruption. The Astronomer’s Telegram, 2212: 1, September 2009.

Beomdu Lim, Moo-Young Chun, Hwankyung Sung, Byeong-Gon Park, Jae-Joon Lee, Sangmo T. Sohn, Hyeonoh Hur, and Michael S. Bessell. The starburst cluster westerlund 1: The initial mass function and mass segregation. The Astronomical Jour- nal, 145(2):46, Jan 2013. ISSN 1538-3881. doi: 10.1088/0004-6256/145/2/46. URL http://dx.doi.org/10.1088/0004-6256/145/2/46.

117 L. Lindegren, U. Lammers, D. Hobbs, W. O’Mullane, U. Bastian, and J. Hern´andez. The astrometric core solution for the gaia mission. Astronomy & Astrophysics, 538:A78, Feb 2012. ISSN 1432-0746. doi: 10.1051/0004-6361/201117905. URL http://dx.doi. org/10.1051/0004-6361/201117905.

L. Lindegren, U. Lammers, D. Hobbs, W. O’Mullane, U. Bastian, and J. Hern´andez.The astrometric core solution for the Gaia mission. Overview of models, algorithms, and soft- ware implementation. A&A, 538:A78, Feb 2012. doi: 10.1051/0004-6361/201117905.

L. Lindegren, J. Hern´andez, A. Bombrun, S. Klioner, U. Bastian, M. Ramos-Lerate, A. de Torres, H. Steidelm¨uller,C. Stephenson, D. Hobbs, and et al. Gaia data release 2. Astronomy & Astrophysics, 616:A2, Aug 2018. ISSN 1432-0746. doi: 10.1051/0004-6361/201832727. URL http://dx.doi.org/10.1051/0004-6361/201832727.

X. Luri, A. G. A. Brown, L. M. Sarro, F. Arenou, C. A. L. Bailer-Jones, A. Castro-Ginard, J. de Bruijne, T. Prusti, C. Babusiaux, and H. E. Delgado. Gaia Data Release 2. Using Gaia parallaxes. A&A, 616:A9, Aug 2018. doi: 10.1051/0004-6361/201832964.

J. Ma´ızApell´aniz,A. Sota, N. I. Morrell, R. H. Barb´a,N. R. Walborn, E. J. Alfaro, R. C. Gamen, J. I. Arias, and A. T. Gallego Calvente. First whole-sky results from the Galactic O-Star Spectroscopic Survey. In Massive Stars: From alpha to Omega, page 198, June 2013. URL http://a2omega-conference.net.

M. Marconi and G. Clementini. A “Pulsational” Distance Determination for the Large Magellanic Cloud. AJ, 129:2257–2267, May 2005. doi: 10.1086/429525.

F. Martins and A. Palacios. A comparison of evolutionary tracks for single Galactic massive stars. A&A, 560:A16, December 2013. doi: 10.1051/0004-6361/201322480.

F. Martins, D. Schaerer, and D. J. Hillier. A new calibration of stellar parameters of Galactic O stars. A&A, 436:1049–1065, June 2005. doi: 10.1051/0004-6361:20042386.

Philip Massey and A. B. Thompson. Massive Stars in CYG OB2. AJ, 101:1408, Apr 1991. doi: 10.1086/115774.

Philip Massey, Kathleen DeGioia-Eastwood, and Elizabeth Waterhouse. The Progenitor Masses of Wolf-Rayet Stars and Luminous Blue Variables Determined from Cluster Turnoﬀs. II. Results from 12 Galactic Clusters and OB Associations. AJ, 121(2): 1050–1070, Feb 2001. doi: 10.1086/318769.

118 Jon C. Mauerhan, Nathan Smith, Alexei V. Filippenko, Kyle B. Blanchard, Peter K. Blan- chard, Chadwick F. E. Casper, S. Bradley Cenko, Kelsey I. Clubb, Daniel P. Cohen, Kiera L. Fuller, Gary Z. Li, and Jeﬀrey M. Silverman. The unprecedented 2012 outburst of SN 2009ip: a luminous blue variable star becomes a true supernova. MNRAS, 430(3):1801–1810, April 2013. doi: 10.1093/mnras/stt009.

P. J. McGregor, A. R. Hyland, and D. J. Hillier. Atomic and Molecular Line Emission from Early-Type High-Luminosity Stars. ApJ, 324:1071, Jan 1988. doi: 10.1086/165964.

Casey A. Meakin and David Arnett. Turbulent Convection in Stellar Interiors. I. Hydrody- namic Simulation. ApJ, 667(1):448–475, September 2007. doi: 10.1086/520318.

S. Mengel and L. E. Tacconi-Garman. Medium resolution 2.3 µm spectroscopy of the massive galactic open cluster westerlund 1. Astronomy & Astrophysics, 466(1):151–155, Feb 2007. ISSN 1432-0746. doi: 10.1051/0004-6361:20066717. URL http://dx.doi.org/10. 1051/0004-6361:20066717.

G. Meynet and A. Maeder. Stellar evolution with rotation. XI. Wolf-Rayet star populations at diﬀerent metallicities. A&A, 429:581–598, January 2005. doi: 10.1051/0004-6361: 20047106.

A. A. Miller, W. Li, P. E. Nugent, J. S. Bloom, A. V. Filippenko, and A. T. Merritt. A Previous Transient Consistent with the Location of SN 2009ip Suggests that SN 2009ip is Not a Supernova. The Astronomer’s Telegram, 2183:1, September 2009.

A. S. Miroshnichenko, Y. Fremat, L. Houziaux, Y. Andrillat, E. L. Chentsov, and V. G. Klochkova. High resolution spectroscopy of the galactic candidate LBV MWC 314. A&AS, 131:469–478, Sep 1998. doi: 10.1051/aas:1998283.

A. S. Miroshnichenko, E. L. Chentsov, and V. G. Klochkova. AS 314: A dusty A-type hypergiant. A&AS, 144:379–389, Jun 2000. doi: 10.1051/aas:2000216.

A. S. Miroshnichenko, K. S. Bjorkman, M. Grosso, H. Levato, K. N. Grankin, R. J. Rudy, D. K. Lynch, S. Mazuk, and R. C. Puetter. MWC 930 - a new luminous blue variable candidate. MNRAS, 364(1):335–343, Nov 2005. doi: 10.1111/j.1365-2966.2005.09567.x.

A. F. J. Moﬀat and M. P. Fitzgerald. Some very luminous supergiants associated with compact groups of luminous OB stars. A&A, 54:263–268, Jan 1977.

Patrick Morris, Morris, and Spitzer WRRINGS Team. Infrared Tracers of Mass-Loss Histo- ries and Wind-ISM Interactions in Hot Star Nebulae. In F. Bresolin, P. A. Crowther, and J. Puls, editors, Massive Stars as Cosmic Engines, volume 250 of IAU Symposium, pages 361–366, Jun 2008. doi: 10.1017/S174392130802070X.

119 Michael P. Muno, J. Simon Clark, Paul A. Crowther, Sean M. Dougherty, Richard de Grijs, Casey Law, Stephen L. W. McMillan, Mark R. Morris, Ignacio Negueruela, David Pooley, and et al. A neutron star with a massive progenitor in westerlund 1. The Astrophysical Journal, 636(1):L41–L44, Dec 2005. ISSN 1538-4357. doi: 10.1086/ 499776. URL http://dx.doi.org/10.1086/499776.

F. Najarro, D. J. Hillier, and O. Stahl. A spectroscopic investigation of P Cygni. I. H and HeI lines. A&A, 326:1117–1134, Oct 1997.

Y. Nazé,G. Rauw, and D. Hutsemékers. The first X-ray survey of Galactic luminous blue variables. A&A, 538:A47, Feb 2012. doi: 10.1051/0004-6361/201118040.

I. Negueruela, J. S. Clark, and B. W. Ritchie. The population of ob supergiants in the starburst cluster westerlund 1. Astronomy and Astrophysics, 516:A78, Jun 2010. ISSN 1432-0746. doi: 10.1051/0004-6361/201014032. URL http://dx.doi.org/10.1051/ 0004-6361/201014032.

E. O. Ofek, M. Sullivan, S. B. Cenko, M. M. Kasliwal, A. Gal-Yam, S. R. Kulkarni, I. Arcavi, L. Bildsten, J. S. Bloom, A. Horesh, D. A. Howell, A. V. Filippenko, R. Laher, D. Mur- ray, E. Nakar, P. E. Nugent, J. M. Silverman, N. J. Shaviv, J. Surace, and O. Yaron. An outburst from a massive star 40 days before a supernova explosion. Nature, 494: 65–67, February 2013. doi: 10.1038/nature11877.

S. P. Owocki, K. G. Gayley, and N. J. Shaviv. A Porosity-Length Formalism for Photon- Tiring-limited Mass Loss from Stars above the Eddington Limit. ApJ, 616:525–541, November 2004a. doi: 10.1086/424910.

Stanley P. Owocki, Kenneth G. Gayley, and Nir J. Shaviv. A Porosity-Length Formalism for Photon-Tiring-limited Mass Loss from Stars above the Eddington Limit. ApJ, 616 (1):525–541, Nov 2004b. doi: 10.1086/424910.

Anil K. Pandey, Katsuo Ogura, and Kaz Sekiguchi. Stellar Contents of the Galactic Giant H II Region NGC 3603. PASJ, 52:847–865, Oct 2000. doi: 10.1093/pasj/52.5.847. { } A. Pasquali, F. Comer´on,and A. Nota. The birth-cluster of the galactic luminous blue variable WRA 751. A&A, 448(2):589–596, Mar 2006. doi: 10.1051/0004-6361:20053977.

120 A. Pastorello, E. Cappellaro, C. Inserra, S. J. Smartt, G. Pignata, S. Benetti, S. Valenti, M. Fraser, K. Tak´ats, S. Benitez, M. T. Botticella, J. Brimacombe, F. Bufano, F. Cellier-Holzem, M. T. Costado, G. Cupani, I. Curtis, N. Elias-Rosa, M. Ergon, J. P. U. Fynbo, F. J. Hambsch, M. Hamuy, A. Harutyunyan, K. M. Ivarson, E. Kankare, J. C. Martin, R. Kotak, A. P. LaCluyze, K. Maguire, S. Mattila, J. Maza, M. McCrum, M. Miluzio, H. U. Norgaard-Nielsen, M. C. Nysewander, P. Ochner, Y. C. Pan, M. L. Pumo, D. E. Reichart, T. G. Tan, S. Taubenberger, L. Tomasella, M. Turatto, and D. Wright. Interacting Supernovae and Supernova Impostors: SN 2009ip, is this the End? ApJ, 767(1):1, April 2013. doi: 10.1088/0004-637X/767/1/1.

A. E. Piatti, E. Bica, and J. J. Clari´a. Fundamental parameters of the highly reddened young open clusters westerlund 1 and 2. Astronomy and Astrophysics Supplement Series, 127(3):423–432, Feb 1998. ISSN 1286-4846. doi: 10.1051/aas:1998111. URL http://dx.doi.org/10.1051/aas:1998111.

Philipp Podsiadlowski, Natasha Ivanova, Stephen Justham, and Saul Rappaport. Explosive common-envelope ejection: implications for gamma-ray bursts and low-mass black-hole binaries. MNRAS, 406(2):840–847, August 2010. doi: 10.1111/j.1365-2966.2010.16751. x.

Jose L. Prieto. Bright Infrared Progenitor of Luminous Transient in NGC300 implies Ex- plosion of a Massive Star. The Astronomer’s Telegram, 1550:1, May 2008.

Jos´eL. Prieto, Matthew D. Kistler, Todd A. Thompson, Hasan Y¨uksel,Christopher S. Kochanek, Krzysztof Z. Stanek, John F. Beacom, Paul Martini, Anna Pasquali, and Jill Bechtold. Discovery of the Dust-Enshrouded Progenitor of SN 2008S with Spitzer. ApJL, 681(1):L9, Jul 2008. doi: 10.1086/589922.

J. Puls, J. S. Vink, and F. Najarro. Mass loss from hot massive stars. A&A Rv, 16:209–325, December 2008. doi: 10.1007/s00159-008-0015-8.

E. Quataert and J. Shiode. Wave-driven mass loss in the last year of stellar evolution: setting the stage for the most luminous core-collapse supernovae. MNRAS, 423(1):L92–L96, June 2012. doi: 10.1111/j.1745-3933.2012.01264.x.

N. D. Richardson, D. R. Gies, and S. J. Williams. A Binary Orbit for the Massive, Evolved Star HDE 326823, a WR+O System Progenitor. AJ, 142(6):201, Dec 2011. doi: 10.1088/0004-6256/142/6/201.

121 Adam G. Riess, Stefano Casertano, Wenlong Yuan, Lucas Macri, Beatrice Bucciarelli, Mario G. Lattanzi, John W. MacKenty, J. Bradley Bowers, WeiKang Zheng, Alexei V. Filippenko, and et al. Milky way cepheid standards for measuring cosmic distances and application to gaia dr2: Implications for the hubble constant. The Astrophysical Journal, 861(2):126, Jul 2018. ISSN 1538-4357. doi: 10.3847/1538-4357/aac82e. URL http://dx.doi.org/10.3847/1538-4357/aac82e.

Adam G. Riess, Stefano Casertano, Wenlong Yuan, Lucas Macri, Beatrice Bucciarelli, Mario G. Lattanzi, John W. MacKenty, J. Bradley Bowers, WeiKang Zheng, Alexei V. Filippenko, Caroline Huang, and Richard I. Anderson. Milky Way Cepheid Standards for Measuring Cosmic Distances and Application to Gaia DR2: Implications for the Hubble Constant. ApJ, 861(2):126, Jul 2018. doi: 10.3847/1538-4357/aac82e.

B. W. Ritchie, J. S. Clark, I. Negueruela, and F. Najarro. Spectroscopic monitoring of the luminous blue variable westerlund1-243 from 2002 to 2009. Astronomy & Astrophysics, 507(3):1597–1611, Oct 2009. ISSN 1432-0746. doi: 10.1051/0004-6361/200912986. URL http://dx.doi.org/10.1051/0004-6361/200912986.

B. W. Ritchie, J. S. Clark, I. Negueruela, and F. Najarro. Spectroscopic monitoring of the luminous blue variable Westerlund1-243 from 2002 to 2009. A&A, 507(3):1597–1611, Dec 2009. doi: 10.1051/0004-6361/200912986.

B. W. Ritchie, J. S. Clark, I. Negueruela, and N. Langer. A vlt/ﬂames survey for massive binaries in westerlund 1. Astronomy and Astrophysics, 520:A48, Sep 2010. ISSN 1432- 0746. doi: 10.1051/0004-6361/201014834. URL http://dx.doi.org/10.1051/0004-6361/ 201014834.

D. Russeil. Star-forming complexes and the spiral structure of our Galaxy. A&A, 397: 133–146, Jan 2003. doi: 10.1051/0004-6361:20021504.

H. Sana, S. E. de Mink, A. de Koter, N. Langer, C. J. Evans, M. Gieles, E. Gosset, R. G. Izzard, J. B. Le Bouquin, and F. R. N. Schneider. Binary Interaction Dominates the Evolution of Massive Stars. Science, 337(6093):444, July 2012. doi: 10.1126/science. 1223344.

M. Sato, M. J. Reid, A. Brunthaler, and K. M. Menten. Trigonometric Parallax of W51 Main/South. ApJ, 720(2):1055–1065, Sep 2010. doi: 10.1088/0004-637X/720/2/1055.

M. H. Schneps, A. P. Lane, D. Downes, J. M. Moran, R. Genzel, and M. J. Reid. Proper motions and distances of H2O maser sources. III - W51NORTH. ApJ, 249:124–133, Oct 1981. doi: 10.1086/159267.

122 Joshua H. Shiode, Eliot Quataert, Matteo Cantiello, and Lars Bildsten. The observational signatures of convectively excited gravity modes in main-sequence stars. MNRAS, 430 (3):1736–1745, April 2013. doi: 10.1093/mnras/sts719.

S. J. Smartt, D. J. Lennon, R. P. Kudritzki, F. Rosales, R. S. I. Ryans, and N. Wright. The evolutionary status of Sher 25 - implications for blue supergiants and the progenitor of SN 1987A. A&A, 391:979–991, Sep 2002. doi: 10.1051/0004-6361:20020829.

N. Smith. Mass Loss: Its Eﬀect on the Evolution and Fate of High-Mass Stars. ARA&A, 52:487–528, August 2014. doi: 10.1146/annurev-astro-081913-040025.

N. Smith. The isolation of luminous blue variables: on subdividing the sample. MNRAS, 461:3353–3360, September 2016a. doi: 10.1093/mnras/stw1533.

N. Smith, W. Li, R. J. Foley, J. C. Wheeler, D. Pooley, R. Chornock, A. V. Filippenko, J. M. Silverman, R. Quimby, J. S. Bloom, and C. Hansen. SN 2006gy: Discovery of the Most Luminous Supernova Ever Recorded, Powered by the Death of an Extremely Massive Star like η Carinae. ApJ, 666:1116–1128, September 2007a. doi: 10.1086/519949.

N. Smith, R. J. Foley, J. S. Bloom, W. Li, A. V. Filippenko, R. Gavazzi, A. Ghez, Q. Konopacky, M. A. Malkan, P. J. Marshall, D. Pooley, T. Treu, and J.-H. Woo. Late-Time Observations of SN 2006gy: Still Going Strong. ApJ, 686:485-491, October 2008. doi: 10.1086/590141.

N. Smith, W. Li, J. M. Silverman, M. Ganeshalingam, and A. V. Filippenko. Luminous blue variable eruptions and related transients: diversity of progenitors and outburst properties. MNRAS, 415:773–810, July 2011a. doi: 10.1111/j.1365-2966.2011.18763.x.

Nathan Smith. The Structure of the Homunculus. I. Shape and Latitude Dependence from H2 and [Fe II] Velocity Maps of η Carinae. ApJ, 644(2):1151–1163, Jun 2006. doi: 10.1086/503766.

Nathan Smith. Discovery of a Nearby Twin of SN 1987A’s Nebula around the Luminous Blue Variable HD 168625: Was Sk -69 202 an LBV? AJ, 133(3):1034–1040, Mar 2007. doi: 10.1086/510838.

Nathan Smith. Explosions triggered by violent binary-star collisions: application to Eta Carinae and other eruptive transients. MNRAS, 415(3):2020–2024, August 2011a. doi: 10.1111/j.1365-2966.2011.18607.x.

Nathan Smith. Explosions triggered by violent binary-star collisions: application to Eta Carinae and other eruptive transients. MNRAS, 415(3):2020–2024, August 2011b. doi: 10.1111/j.1365-2966.2011.18607.x.

123 Nathan Smith. Mass loss: Its eﬀect on the evolution and fate of high-mass stars. ARA, 52 (1):487–528, Aug 2014. ISSN 1545-4282. doi: 10.1146/annurev-astro-081913-040025. URL http://dx.doi.org/10.1146/annurev-astro-081913-040025.

Nathan Smith. The isolation of luminous blue variables: on subdividing the sample. MNRAS, 461(3):3353–3360, Sep 2016b. doi: 10.1093/mnras/stw1533.

Nathan Smith and W. David Arnett. Preparing for an Explosion: Hydrodynamic In- stabilities and Turbulence in Presupernovae. ApJ, 785(2):82, April 2014. doi: 10.1088/0004-637X/785/2/82.

Nathan Smith and Stanley P. Owocki. On the Role of Continuum-driven Eruptions in the Evolution of Very Massive Stars and Population III Stars. ApJL, 645(1):L45–L48, Jul 2006. doi: 10.1086/506523.

Nathan Smith and Keivan G. Stassun. The Canonical Luminous Blue Variable AG Car and Its Neighbor Hen 3-519 are Much Closer than Previously Assumed. AJ, 153(3):125, Mar 2017. doi: 10.3847/1538-3881/aa5d0c.

Nathan Smith and Ryan Tombleson. Luminous blue variables are antisocial: their isolation implies that they are kicked mass gainers in binary evolution. MNRAS, 447(1):598–617, Feb 2015. doi: 10.1093/mnras/stu2430.

Nathan Smith, Jorick S. Vink, and Alex de Koter. The missing luminous blue variables and the bistability jump. The Astrophysical Journal, 615(1):475–484, Nov 2004. ISSN 1538-4357. doi: 10.1086/424030. URL http://dx.doi.org/10.1086/424030.

Nathan Smith, Jorick S. Vink, and Alex de Koter. The Missing Luminous Blue Variables and the Bistability Jump. ApJ, 615(1):475–484, Nov 2004. doi: 10.1086/424030.

Nathan Smith, John Bally, and Josh Walawender. And in the Darkness Bind Them: Equa- torial Rings, B[e] Supergiants, and the Waists of Bipolar Nebulae. AJ, 134(2):846–859, Aug 2007b. doi: 10.1086/518563.

Nathan Smith, Mohan Ganeshalingam, Ryan Chornock, Alexei V. Filippenko, Weidong Li, Jeﬀrey M. Silverman, Thea N. Steele, Christopher V. Griﬃth, Niels Joubert, Nicholas Y. Lee, Thomas B. Lowe, Martin P. Mobberley, and Dustin M. Winslow. SN 2008S: A Cool Super-Eddington Wind in a Supernova Impostor. ApJL, 697(1): L49–L53, May 2009. doi: 10.1088/0004-637X/697/1/L49.

124 Nathan Smith, Adam Miller, Weidong Li, Alexei V. Filippenko, Jeﬀrey M. Silverman, An- drew W. Howard, Peter Nugent, Geoﬀrey W. Marcy, Joshua S. Bloom, Andrea M. Ghez, Jessica Lu, Sylvana Yelda, Rebecca A. Bernstein, and Janet E. Colucci. Discov- ery of Precursor Luminous Blue Variable Outbursts in Two Recent Optical Transients: The Fitfully Variable Missing Links UGC 2773-OT and SN 2009ip. AJ, 139(4):1451– 1467, April 2010. doi: 10.1088/0004-6256/139/4/1451.

Nathan Smith, Weidong Li, Jeﬀrey M. Silverman, Mohan Ganeshalingam, and Alexei V. Filippenko. Luminous blue variable eruptions and related transients: diversity of progenitors and outburst properties. MNRAS, 415(1):773–810, Jul 2011b. doi: 10.1111/j.1365-2966.2011.18763.x.

Nathan Smith, W. David Arnett, John Bally, Adam Ginsburg, and Alexei V. Filippenko. The ring nebula around the blue supergiant SBW1: pre-explosion snapshot of an SN 1987A twin. MNRAS, 429(2):1324–1341, Feb 2013. doi: 10.1093/mnras/sts418.

Nathan Smith, Jon C. Mauerhan, and Jose L. Prieto. SN 2009ip and SN 2010mc: core- collapse Type IIn supernovae arising from blue supergiants. MNRAS, 438(2):1191– 1207, February 2014. doi: 10.1093/mnras/stt2269.

Nathan Smith, Jennifer E. Andrews, and Jon C. Mauerhan. Massive stars dying alone: the extremely remote environment of SN 2009ip. MNRAS, 463(3):2904–2911, Dec 2016a. doi: 10.1093/mnras/stw2190.

Nathan Smith, Jennifer E. Andrews, Schuyler D. Van Dyk, Jon C. Mauerhan, Mansi M. Kasliwal, Howard E. Bond, Alexei V. Filippenko, Kelsey I. Clubb, Melissa L. Gra- ham, Daniel A. Perley, Jacob Jencson, John Bally, Leonardo Ubeda, and Elena Sabbi. Massive star mergers and the recent transient in NGC 4490: a more massive cousin of V838 Mon and V1309 Sco. MNRAS, 458(1):950–962, May 2016b. doi: 10.1093/mnras/stw219.

Nathan Smith, Jose H. Groh, Kevin France, and Richard McCray. Ultraviolet spectroscopy of the blue supergiant SBW1: the remarkably weak wind of a SN 1987A analogue. MNRAS, 468(2):2333–2344, Jun 2017. doi: 10.1093/mnras/stx648.

Nathan Smith, Jennifer E. Andrews, Armin Rest, Federica B. Bianco, Jose L. Prieto, Tom Matheson, David J. James, R. Chris Smith, Giovanni Maria Strampelli, and A. Zenteno. Light echoes from the plateau in Eta Carinae’s Great Eruption re- veal a two-stage shock-powered event. MNRAS, 480(2):1466–1498, Oct 2018a. doi: 10.1093/mnras/sty1500.

125 Nathan Smith, Armin Rest, Jennifer E. Andrews, Tom Matheson, Federica B. Bianco, Jose L. Prieto, David J. James, R. Chris Smith, Giovanni Maria Strampelli, and A. Zenteno. Exceptionally fast ejecta seen in light echoes of Eta Carinae’s Great Eruption. MNRAS, 480(2):1457–1465, Oct 2018b. doi: 10.1093/mnras/sty1479.

Nathan Smith, Mojgan Aghakhanloo, Jeremiah W. Murphy, Maria R. Drout, Keivan G. Stassun, and Jose H. Groh. On the Gaia DR2 distances for Galactic luminous blue variables. MNRAS, 488(2):1760–1778, Sep 2019. doi: 10.1093/mnras/stz1712.

Keivan G. Stassun and Guillermo Torres. Evidence for a systematic oﬀset of -80 µas in the gaia dr2 parallaxes. The Astrophysical Journal, 862(1):61, Jul 2018. ISSN 1538-4357. doi: 10.3847/1538-4357/aacafc. URL http://dx.doi.org/10.3847/1538-4357/aacafc.

W. J. G. Steemers and A. M. van Genderen. VBLUW photometry of two hypergiants HD 80077 (B2Ia+) and HD 74180 (F2Ia+) and of the open cluster PISMIS 11. A&A, 154: 308–312, Jan 1986.

C. Sterken, E. Gosset, A. Juttner, O. Stahl, B. Wolf, and M. Axer. HD 160529 : a new galactic luminous blue variable. A&A, 247:383, Jul 1991.

C. Sterken, A. M. van Genderen, A. Plummer, and A. F. Jones. Wra 751, a luminous blue variable developing an S Doradus cycle. A&A, 484(2):463–467, Jun 2008. doi: 10.1051/0004-6361:200809763.

A. Stolte, E. K. Grebel, W. Brandner, and D. F. Figer. The mass function of the arches cluster from gemini adaptive optics data. Astronomy & Astrophysics, 394(2):459–478, Oct 2002. ISSN 1432-0746. doi: 10.1051/0004-6361:20021118. URL http://dx.doi.org/ 10.1051/0004-6361:20021118.

G. A. Tammann and Allan Sandage. The Stellar Content and Distance of the Galaxy NGC 2403 IN the M81 Group. ApJ, 151:825, March 1968. doi: 10.1086/149487.

Todd A. Thompson, Jos´eL. Prieto, K. Z. Stanek, Matthew D. Kistler, John F. Beacom, and Christopher S. Kochanek. A New Class of Luminous Transients and a First Census of their Massive Stellar Progenitors. ApJ, 705(2):1364–1384, Nov 2009. doi: 10.1088/ 0004-637X/705/2/1364.

A. V. Torres-Dodgen, M. Tapia, and M. Carroll. uvby and JHKL’ photometry of OB stars in the association Cygnus OB2. MNRAS, 249:1–12, Mar 1991. doi: 10.1093/mnras/ 249.1.1.

Andrew Ulmer and Edward L. Fitzpatrick. Revisiting the Modiﬁed Eddington Limit for Massive Stars. ApJ, 504(1):200–206, Sep 1998. doi: 10.1086/306048.

126 J. S. Urquhart, C. C. Figura, T. J. T. Moore, M. G. Hoare, S. L. Lumsden, J. C. Mottram, M. A. Thompson, and R. D. Oudmaijer. The RMS survey: galactic distribution of massive star formation. Monthly Notices of the Royal Astronomical Society, 437(2): 1791–1807, Jan 2014. doi: 10.1093/mnras/stt2006.

Jacques P. Vall´ee.THE SPIRAL ARMS OF THE MILKY WAY: THE RELATIVE LOCA- TION OF EACH DIFFERENT ARM TRACER WITHIN a TYPICAL SPIRAL ARM WIDTH. The Astronomical Journal, 148(1):5, may 2014. doi: 10.1088/0004-6256/148/ 1/5. URL https://doi.org/10.1088%2F0004-6256%2F148%2F1%2F5.

W. E. C. J. van der Veen, L. B. F. M. Waters, N. R. Trams, and H. E. Matthews. A study of dust shells around high latitude supergiants. A&A, 285:551–564, May 1994.

S. D. van Dyk. The η Carinae Analogs. In R. Humphreys and K. Stanek, editors, The Fate of the Most Massive Stars, volume 332 of Astronomical Society of the Paciﬁc Conference Series, page 49. 2005.

Schuyler D. Van Dyk, Chien Y. Peng, Jennifer Y. King, Alexei V. Filippenko, Richard R. Treﬀers, Weidong Li, and Michael W. Richmond. SN 1997bs in M66: Another Ex- tragalactic η Carinae Analog? PASP, 112(778):1532–1541, December 2000. doi: 10.1086/317727.

A. M. van Genderen. S Doradus variables in the Galaxy and the Magellanic Clouds. A&A, 366:508–531, February 2001a. doi: 10.1051/0004-6361:20000022.

A. M. van Genderen. S Doradus variables in the Galaxy and the Magellanic Clouds. A&A, 366:508–531, Feb 2001b. doi: 10.1051/0004-6361:20000022.

A. M. van Genderen, F. H. A. Robijn, B. P. M. van Esch, and H. J. G. Lamers. The distance to the luminous blue variable HR Carinae. A&A, 246:407, Jun 1991.

A. M. van Genderen, P. S. The, D. de Winter, A. Holland er, P. J. de Jong, F. C. van den Bosch, O. M. Kolkman, and M. A. W. Verheijen. On the distance to the new luminous blue variable WRA 751 andits variability. A&A, 258:316–322, May 1992.

D. Vanbeveren, N. Mennekens, W. Van Rensbergen, and C. De Loore. Blue supergiant progenitor models of type II supernovae. A&A, 552:A105, April 2013. doi: 10.1051/ 0004-6361/201321072.

J. S. Vink. Eta Carinae and the Luminous Blue Variables. In K. Davidson and R. M. Humphreys, editors, Eta Carinae and the Supernova Impostors, volume 384 of Astro- physics and Space Science Library, page 221, 2012. doi: 10.1007/978-1-4614-2275-4 10.

127 J. S. Vink, A. de Koter, and H. J. G. L. M. Lamers. Mass-loss predictions for O and B stars as a function of metallicity. A&A, 369:574–588, April 2001. doi: 10.1051/0004-6361: 20010127.

R. H. M. Voors, T. R. Geballe, L. B. F. M. Waters, F. Najarro, and H. J. G. L. M. Lamers. Spectroscopy of the candidate luminous blue variable at the center of the ring nebula G79.29+0.46. A&A, 362:236–244, Oct 2000.

N. R. Walborn. Spectral classiﬁcation of O and B0 supergiants in the Magellanic Clouds. ApJ, 215:53–57, July 1977. doi: 10.1086/155334.

A. R. Walker. The Large Magellanic Cloud and the distance scale. Ap&SS, 341:43–49, September 2012. doi: 10.1007/s10509-011-0961-x.

K. Weis. On the structure and kinematics of nebulae around LBVs and LBV candidates in the LMC. A&A, 408:205–229, September 2003. doi: 10.1051/0004-6361:20030921.

B. E. Westerlund. A heavily reddened cluster in ara. Publications of the Astronomical Society of the Paciﬁc, 73:51, Feb 1961. ISSN 1538-3873. doi: 10.1086/127618. URL http://dx.doi.org/10.1086/127618.

B. E. Westerlund. On the extended infrared source in ara. The Astrophysical Journal, 154: L67, Nov 1968. ISSN 1538-4357. doi: 10.1086/180270. URL http://dx.doi.org/10. 1086/180270.

B. E. Westerlund. Photometry and spectroscopy of stars in the region of a highly reddened cluster in Ara. A&AS, 70:311–324, Sep 1987.

Benjamin F. Williams, Tristan J. Hillis, Jeremiah W. Murphy, Karoline Gilbert, Julianne J. Dalcanton, and Andrew E. Dolphin. Constraints for the Progenitor Masses of Historic Core-collapse Supernovae. ApJ, 860(1):39, Jun 2018. doi: 10.3847/1538-4357/aaba7d.

B. Wolf. Empirical amplitude-luminosity relation of S Doradus variables and extragalactic distances. A&A, 217:87–91, Jun 1989a.

B. Wolf. Empirical amplitude-luminosity relation of S Doradus variables and extragalactic distances. A&A, 217:87–91, Jun 1989b.

S. E. Woosley and A. Heger. The Deaths of Very Massive Stars. In J. S. Vink, editor, Very Massive Stars in the Local Universe, volume 412 of Astrophysics and Space Science Library, page 199, 2015. doi: 10.1007/978-3-319-09596-7 7.

128 S. E. Woosley, A. Heger, and T. A. Weaver. The evolution and explosion of massive stars. Reviews of Modern Physics, 74:1015–1071, November 2002. doi: 10.1103/RevModPhys. 74.1015.

S. E. Woosley, S. Blinnikov, and Alexander Heger. Pulsational pair instability as an explanation for the most luminous supernovae. Nature, 450(7168):390–392, November 2007. doi: 10.1038/nature06333.

Y. Xu, M. J. Reid, K. M. Menten, A. Brunthaler, X. W. Zheng, and L. Moscadelli. Trigono- metric Parallaxes of Massive Star-Forming Regions: III. G59.7+0.1 and W 51 IRS2. ApJ, 693(1):413–418, Mar 2009. doi: 10.1088/0004-637X/693/1/413.

Joel C. Zinn, Marc H. Pinsonneault, Daniel Huber, and Dennis Stello. Confirmation of the gaia dr2 parallax zero-point offset using asteroseismology and spectroscopy in the kepler field. The Astrophysical Journal, 878(2):136, Jun 2019. ISSN 1538-4357. doi: 10.3847/1538-4357/ab1f66. URL http://dx.doi.org/10.3847/1538-4357/ab1f66.

Joel C. Zinn, Marc H. Pinsonneault, Daniel Huber, and Dennis Stello. Conﬁrmation of the Gaia DR2 Parallax Zero-point Oﬀset Using Asteroseismology and Spectroscopy in the Kepler Field. ApJ, 878(2):136, Jun 2019. doi: 10.3847/1538-4357/ab1f66.

129 BIOGRAPHICAL SKETCH

Mojgan Aghakhanloo was born and raised in Tehran, Iran. She earned a Bachelor of Science degree in Physics Engineering Plasma from Islamic Azad University, Science and Research branch in 2013. After graduation, she began her pursuit of a Doctoral degree in physics at Florida State University. She joined the FSU astrophysics group working under the guidance of Dr. Jeremiah Murphy. Her PhD research focused on massive stars.

130