The Formation and Evolution of Disk Galaxies

Dissertation

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

Jonathan C. Bird, M.S., B.S.

Graduate Program in Astronomy

The Ohio State University 2012

Dissertation Committee: Professor David H. Weinberg, Advisor Professor Jennifer A. Johnson Professor Marc H. Pinsonneault Copyright by

Jonathan C. Bird

2012 ABSTRACT

Current and next generation surveys of the Milky Way promise to revolutionize our observational perspective of the Galaxy. My dissertation uses a suite of N-body and SPH simulations of disk galaxies to make testable predictions of the assembly history of the Milky Way and identify observational probes that take advantage of the forthcoming data. APOGEE, an infrared survey of the Galaxy and a component of the SDSS-III, will measure the distance, radial velocity, and multi-element chemistry of ∼ 105 stars located throughout the Galaxy, making it particularly well suited for comparison with simulations. We first use a fuel-consumption argument to constrain the integrated luminosity of the TP-AGB phase; the energy release in this phase is a major uncertainty in stellar population synthesis models. Our initial numerical investigation explores how the minor mergers expected in ΛCDM and inherent properties of stellar disks affect the dynamics of stellar radial migration- an essential ingredient in understanding the evolution of the Milky Way and disk galaxies in general. We discover that the resonances and mechanisms responsible for migration are different in isolated and satellite-bombarded galaxies, resulting in distinct migration patterns and potential observational signatures of accretion

ii events. Continuing our development of tools to describe the chemo-dynamics of the disk, we construct statistics to measure overdensities and characterize outliers in the distance, radial velocity projection of phase space. I discuss mock APOGEE observations of our numerical simulations and demonstrate that our statistics can begin to discriminate between signiﬁcant galaxy formation mechanisms given the data available in the near term. Finally, we use a state of the art cosmological simulation to describe the evolution of mono-age stellar populations and their eventual assembly into a galaxy resembling the Milky Way. Galaxy formation theory faces the exciting challenge of an unprecedented level of statistical scrutiny: imminent and ongoing surveys such as SEGUE, RAVE, APOGEE, LAMOST, and

HERMES oﬀer an extraordinary opportunity to unravel the formation history of the

Milky Way.

iii To my number one fan, always.

iv ACKNOWLEDGMENTS

I would never consider a work representing the culmination of a signiﬁcant fraction of my life as an independent endeavor. None of this would have been possible if not for my advisor, David Weinberg. His thoughtfulness and vision have led me down an exciting path of scientiﬁc investigation; I am sure I will continue to learn from our interactions long after I leave Ohio State. I owe a special thanks to Stelios

Kazantzidis; I have enjoyed our deep collaboration. The Ohio State Astronomy faculty have had a strong, positive inﬂuence on my growth as a scientiﬁc researcher and as a person. In particular, I thank Professors Paul Martini, Kris Stanek, Marc

Pinsonneault, Jennifer Johnson, Scott Gaudi, and Todd Thompson.

I have not made it to this point without a great deal of help. A huge thank you to my Mother: I really appreciate that whole “giving birth to me” thing, the countless sacriﬁces you made to raise me, and your unwavering support of my passion. To my sister, Katrina, thank you for always telling me to “kick some astro-butt”; I am so thankful for deep connection and for how much I have learned through our relationship. Thank you for rooting me on Natasha, my sister, and

Narayan, her husband; and thank you for your help in obtaining this opportunity. A

v bittersweet, deep thank you to my now departed Father; thank you for showing me what a man could become. Thank you family, I love you all.

I am incredibly lucky to have so many wonderful friends who are constant sources of inspiration, joy, and support. To the other members of the ﬁve- Kevin,

Frankie, Ray, and Pat: thank you for what is turning into a lifetime of friendship; our brotherhood has carried me more than you know. Tony, thanks for the caped adventures; Kamilee, thank you for understanding the 20 year-old me. To the amazing set of friends I have accumulated throughout graduate school, especially

Jason, Rob, Linda, Kelly, Kate, and Roberto: you all rock and you have made this time so special. To the incomparable Katie “Tizzle B”: I think “I wanna dance with somebody” sums it up best. Finally, to the love of my life, Emily: Thank you for shining so brightly that I could ﬁnd my way even during the darkest days that I have ever known.

vi VITA

April 30, 1981 ...... Born – Santa Monica, Califonia, United States

2003 ...... B.S. Physics, California Institute of Technology

2006 – 2007 ...... Distinguished University Fellow, The Ohio State University 2007 – 2011 ...... Graduate Teaching and Research Associate, The Ohio State University 2011 – 2012 ...... Distinguished University Fellow, The Ohio State University

PUBLICATIONS

Research Publications

1. Nidever, David L., Zasowski, Gail, Majewski, Steven R., Bird, Jonathan C., and 32 coauthors, “The Apache Point Observatory Galactic Evolution Exper- iment: First Detection of High-velocity Milky Way Bar Stars”, ApJ, 755L, 25N, (2012).

2. J. C. Bird, S. Kazantzidis,D. H. Weinberg,“Radial Mixing in Galactic Disks: The Eﬀects of Disk Structure and Satellite Bombardment, MNRAS, 420, 913B, (2012).

3. C.J. Grier and 38 coauthors including Jonathan C. Bird, “A Reverbera- tion Lag for the High-Ionization Component of the Broad Line Region in the Narrow-Line Seyfert 1 Mrk 335”, ApJ, 744L, 4G2011, (2012).

4. Jonathan C. Bird and Marc H. Pinsonneault, “A Bound on the Light Emitted During the TP-AGB Phase”, ApJ, 733, 81B, (2011).

5. Y.S. Lee, T.C. Beers, D. An, Z. Ivezic, A. Just, C. M. Rockosi, H. L. Morrison, J. A. Johnson, R. Schonrich, J. Bird, B. Yanny, P. Harding, H. J.

vii Rocha-Pinto, “Formation and Evolution of the Disk System of the Milky Way: [alpha/Fe] Ratios and Kinematics of the SEGUE G-Dwarf Sample”, ApJ, 738, 187L, (2011).

6. K. D. Denney and 42 coauthors including J. C. Bird, “Reverberation Mapping Measurements of Black Hole Masses in Six Local Seyfert Galaxies”, ApJ, 721, 715D, (2010).

7. C. Villforth and 50 coauthors including J. C. Bird, “Variability and stability in blazar jets on time-scales of years: optical polarization monitoring of OJ 287 in 2005-2009”, MNRAS, 402, 2087, (2010).

8. Kelly D. Denney and 42 coauthors including Jonathan C. Bird, “Diverse Kinematic Signatures From Reverberation Mapping of the Broad-Line Regions in Active Galactic Nuclei”, ApJL, 704, L80, (2009).

9. J. C. Yee, A. Udalski, T. Sumi, S. Dong, S. KozÃlowski, J. C. Bird, and 80 coauthors, “Extreme Magniﬁcation Microlensing Event OGLE-2008-BLG-279: Strong Limits on Planetary Companions to the Lens Star”, ApJ, 703, 2082, (2009).

10. K. D. Denney and 32 coauthors including J. C. Bird, “A Revised Broad- Line Region Radius and Black Hole Mass for the Narrow-Line Seyfert 1 NGC 4051”, ApJ, 702, 1353, (2009).

11. M. J. Valtonen and 43 coauthors including J. C. Bird, “Tidally Induced Outbursts in OJ 287 during 2005-2008”, ApJ, 698, 781, (2009).

12. J. C. Bird, K. Z. Stanek, and J. L. Prieto, “Using Ultra Long Period Cepheids to Extend the Cosmic Distance Ladder to 100 Mpc and Beyond”, ApJ, 695, 874, (2009).

13. C. J. Grier and 16 coauthors including J. Bird, “The Mass of the Black Hole in the Quasar PG 2130+099”, ApJ, 688, 837, (2008).

14. J. C. Bird, P. Martini, and C. Kaiser, “The Lifetime of FR II Sources in Groups and Clusters: Implications for Radio-Mode Feedback”, ApJ, 676, 147, (2008).

FIELDS OF STUDY

Major Field: Astronomy

viii Table of Contents

Abstract...... ii

Dedication...... iv

Acknowledgments...... v

Vita ...... vii

ListofTables ...... xii

ListofFigures...... xiii

Chapter 1 Introduction ...... 1

1.1 ContextandScope ...... 4

Chapter 2 A Bound on the Light Emitted During the TP-AGB Phase 7

2.1 SampleandMethods ...... 9

2.1.1 White Dwarf Samples and Final Masses ...... 9

2.1.2 ClusterParameters ...... 11

2.1.3 InitialMasses ...... 12

2.1.4 ClusterAverages ...... 16

2.1.5 Fraction of Core Growth in the TP-AGB Phase ...... 17

2.1.6 Fuel Consumption: Mass-Light Coupling ...... 17

2.2 Results...... 19

2.2.1 Initial Final Mass Relation ...... 19

ix 2.2.2 CoreGrowthintheTP-AGBPhase...... 20

2.2.3 Fuel from Stellar Core Growth ...... 22

2.2.4 Fuel from Stellar Yields ...... 23

2.3 SummaryandDiscussion...... 24

Chapter 3 Radial Mixing in Galactic Disks: The Eﬀects of Disk Structure and Satellite Bombardment ...... 35

3.1 Methods...... 39

3.1.1 IsolatedDiskModels ...... 39

3.1.2 PerturbedDiskModels...... 41

3.1.3 NumericalParameters ...... 42

3.2 Results...... 44

3.2.1 RadialMigration ...... 45

3.2.2 OrbitalCharacteristics ...... 48

3.2.3 SolarAnnulus...... 54

3.3 SummaryandDiscussion...... 57

Chapter 4 Phase Space Substructure in the Milky Way’s Disk: Probe of Evolutionary History? ...... 73

4.1 GalaxyModelsandtheAPOGEESurvey...... 75

4.2 Phase-spaceSubstructureFinding ...... 76

4.2.1 Mock APOGEE Observations and Group Identiﬁcation . . . . 79

4.3 Results...... 81

4.3.1 FractionofDiskinSubstructure...... 81

4.3.2 RV as Accretion History Diagnostic ...... 82

4.4 Summary ...... 83

x Chapter 5 Inside Out and Upside Down: Tracing the Assembly of a Simulated Disk Galaxy Mono-Age Stellar Populations ...... 96

5.1 TheErisLTSimulation ...... 98

5.2 RedshiftZero ...... 99

5.3 EvolutionofAgeCohorts ...... 103

5.4 Migration ...... 106

5.5 Summary ...... 108

Bibliography ...... 126

xi List of Tables

2.1 Cluster Metallicity, Distance, and Age...... 32

2.2 Fractional Core Growth of each cluster...... 33

2.3 Corresponding Energy Output and Estimated Helium Yields. . . . . 34

5.1 Kinematic Decomposition ...... 125

xii List of Figures

2.1 TheIFMRforthe48WDsinoursample...... 29

2.2 The fraction of ﬁnal remnant mass built up in the TP-AGB phase . . 30

2.3 Fraction of ﬁnal remnant mass contributed by TP-AGB phase

according to our fMc,TP−AGB,⊙ calculations ...... 31

3.1 Surface density maps of the stellar distributions of galactic disks with diﬀerent initial scale-heights...... 61

3.2 The distribution of radius change for all disk particles in each of the sixisolateddisksimulations...... 62

3.3 The ∆r distribution for all disk particles in the four collisionless simulations...... 63

3.4 The fraction of particles that migrate more than various distances as afunctionofformationradius...... 64

3.5 Lindblad diagrams for all collisionless simulations...... 65

3.6 Possible values of circularity as a function of eccentricity in the solar annulus...... 66

3.7 The initial and ﬁnal E, Jz pairs for ten randomly selected particles in eachsimulation...... 67

3.8 Change in circularity as a function of change in radius...... 68

3.9 Change in maximum vertical displacement as a function of change in radius...... 69

3.10 The distribution of formation radius for all particles that reside in the solarannulus...... 70

3.11 The change in circularity as a function of change in radius for particles residinginthesolarannulus...... 71

xiii 3.12 The change in vertical displacement as a function of change in radius for particles residing in the solar annulus...... 72

4.1 Edge-on and face-on views of the three simulated galaxies analyzed in oursubstructureanalysis...... 85

4.2 The line of sight velocity (vlos) and distance d distributions for particles in the l = 60, b = 0 ﬁeld of all four simulations ...... 86

4.3 The line of sight velocity and distance distributions (no errors) for all particles along the l = 60, b = 4 field in the perturbed model, the phase-space density of all particles, and the field’s HOP group identifications ...... 87

4.4 The phase-space density of particles in three ﬁelds across four models usingourintuitivescheme ...... 88

4.5 Group identiﬁcations by HOP in three ﬁelds across four models using ourintuitivescheme ...... 89

4.6 Line of sight velocity and distance distributions of particles in three different fields and four models using our fiducial scheme ...... 90

4.7 The eﬀect of target selection and measurement error on phase-space density...... 91

4.8 The eﬀect of target selection and measurement error on group identiﬁcations ...... 92

4.9 The percentage of mock APOGEE surveys associated with substructure as a function of galaxy formation model ...... 93

4.10 The interquartile of mock APOGEE RV measurements as a function of galactocentric distance and formation model using unsampled, error- freedata...... 94

4.11 The interquartile of mock APOGEE RV measurements as a function of galactocentric distance and formation model using N = 1000 targets perﬁeldandassuming20%distanceerrors ...... 95

5.1 The spatial density of stellar particles in the simulated galaxy as a function of position and age at redshift zero ...... 111

5.2 The radial proﬁles of surface mass density and median height above thediskplaneatredshiftzero ...... 112

xiv 5.3 Vertical stellar mass density proﬁles for three annuli in the disk . . . 112

5.4 The radial proﬁles of vertical velocity dispersion, radial velocity dispersion, median circularity, and circularity dispersion at redshift zero ...... 113

5.5 The surface density of S0.0,0.5 asafunctionoftime...... 114

5.6 The radial proﬁles of the surface mass density, median height above the disk plane, and vertical velocity dispersion of S0.0,0.5 ...... 115

5.7 The surface density of S0.5,1.0 asafunctionoftime...... 116

5.8 The radial proﬁles of the surface mass density, median height above the disk plane, and vertical velocity dispersion of S0.5,1.0 ...... 117

5.9 The surface density of S1.0,2.0 asafunctionoftime...... 118

5.10 The radial proﬁles of the surface mass density, median height above the disk plane, and vertical velocity dispersion of S1.0,2.0 ...... 119

5.11 The surface density of S3.0,4.0 asafunctionoftime...... 120

5.12 The radial proﬁles of the surface mass density, median height above the disk plane, and vertical velocity dispersion of S3.0,4.0 ...... 121

5.13 The surface density of S7.0,8.0 asafunctionoftime...... 122

5.14 The radial proﬁles of the surface mass density, median height above the disk plane, and vertical velocity dispersion of S7.0,8.0 ...... 123

5.15 The formation radius of particles, binned into age cohorts,residing at threediﬀerentradialannuli ...... 123

5.16 The current guiding center radius of particles, binned into age cohorts,residing at three diﬀerent radial annuli ...... 123

5.17 The change in guiding center radius of particles, binned into age cohorts,residing at three diﬀerent radial annuli ...... 124

xv Chapter 1

Introduction

Galaxy formation and evolution is one of the most active areas of astrophysics research: from high redshift observations of primitive galaxy construction in the

HST UDF and COSMOS survey to detailed studies of chemical composition and kinematic structure of nearby galaxies and the multitudinous theoretical eﬀorts to understand these observations and the formation and evolution of galaxies over the age of the Universe (e.g., Madau et al. 1998). Galaxy construction begins with the hierarchical assembly of dark matter (Tinsley & Larson 1977) into halos and subsequent gas accretion onto the halo’s potential well (White & Rees 1978).

Gas shock heats during the accretion, converting its kinematic energy to thermal energy (Rees & Ostriker 1977, although some recent models suggest that galaxies can accrete cold gas along cosmic ﬁlaments, e.g., Kereˇset al. (2005)). The main diﬃculties remaining in galaxy formation theory center around the complicated gas physics that dictate how this accreted gas assembles into the galaxies we see today.

Before observations could inform theoretical models, the community had to interpret the empirical data. Baade (1944) was the ﬁrst to describe a galaxy as a

1 collection of stellar populations, each with shared characteristics. Later, O’Connell

(1958) associated separate stellar populations to morphological structures, claiming that stars forming the Milky Way’s halo and disk were distinct from one another.

Given this basic understanding, the seminal work of Eggen et al. (1962) showed that the disk and halo populations could naturally arise from the monolithic collapse of a gas cloud onto the dark matter halo. Searle & Zinn (1978) suggested that galaxies form the same way as dark matter halos but through the hierarchical assembly of stellar populations rather than dark matter. Interest in the formation of the

Milky Way (MW) grew as it was realized that insight into the growth of our Galaxy could be fundamental in our understanding of disk galaxy formation in the universe

(Sandage et al. 1970).

The MW, despite it being a singular outcome, aﬀords us unique insight into the myriad of processes responsible for the construction and evolution of galaxies because of the level of detail possible in observations of our own Galaxy. Current and upcoming spectroscopic surveys promise radial velocities and chemical abundances for 105–106 MW stars. These eﬀorts are revolutionizing our understanding of current MW structure and composition and present an unprecedented opportunity to uncover the major phases of the MW’s assembly history and place them in the broader context of generic galaxy formation. The current goal of many is to develop the theoretical framework needed to interpret these observations and to apply these tools to data to form links between the present, observable Galaxy and its past.

2 There are many approaches to galaxy modeling. Empirical models use star counts and population synthesis techniques to extrapolate a complete galaxy replica from observations (Bahcall 1986; Girardi et al. 2005), but they do not themselves connect this replica to galaxy formation theory. Classical, one-dimensional chemical evolution models (e.g., Matteucci and Francois 1989) use current observations to constrain, investigate, and balance their descriptions of gas inflow, star-formation, chemical enrichment, gas recycling, and gas outflow. These models can quickly explore a large parameter space with their flexible framework, however, they do not capture the dynamical processes, such as radial migration, now believed essential to explain many recent observations (though the state of the art models of Schönrich & Binney (2009a) include a parametrized form of radial migration).

Cosmological simulations of galaxy formation (e.g., Navarro et al. 1995, Governato et al. 2007) self-consistently track dynamics and chemical enrichment, but they are computationally expensive, and at most a handful of such simulations resolve the radial, azimuthal, and vertical structure of galaxy disks at z = 0 (Guedes et al. 2011) essential for detailed comparison with current observations. However, numerical experiments self-consistently capture the relevant dynamics of galaxy formation.

Much of my dissertation uses a suite of controlled N-body experiments to explore the observational consequences of diﬀerent formation histories.

Modern data sets have led to new insights and have unprecedented power to constrain galaxy formation models. Bell et al. (2008) were unable to ﬁt a smooth

3 model to the spatial distribution of ∼ 4 million turn-oﬀ stars observed in the Sloan

Digital Sky Survey (SDSS, York et al. 2000); they concluded that accreted satellite systems hosting independently-evolved stellar populations must contribute to a signiﬁcant fraction of the MW’s halo. Subsequent studies by e.g., Schlaufman et al.

(2009); Xue et al. (2011); Cooper et al. (2011) came to similar conclusions using a variety of stellar tracer populations to measure kinematic substructure in the halo.

However, data does not always lead to such clear conclusions. For example, the thick disk of the MW refers to a collection of kinematically hot (e.g. Chiba & Beers 2000;

Quillen & Garnett 2000), metal-poor (e.g. Bensby et al. 2003; Haywood 2008), and old (e.g. Bensby et al. 2005) stars populating a vertically extended (e.g. Gilmore &

Reid 1983; Juri´cet al. 2008) disk structure. Despite this rich knowledge, a plethora of formation scenarios can produce the thick disk structure: dynamic heating by satellite accretion events (e.g. Kazantzidis et al. 2008), stellar radial migration within the disk (Sch¨onrich & Binney 2009a; Loebman et al. 2011), and early time gas-rich mergers (Brook et al. 2004, 2005) are all consistent with the data.

The current chemodynamic properties of a star do not identify a unique origin; a complete galaxy formation theory must model the processes that blur the connection between a star’s present state and its history. We are still determining what mechanisms have a major impact on the evolution of galaxies. In the past decade, stellar radial migration, or stars moving away from their birth radii, has been established as a major dynamical process in disk galaxy evolution. In the MW,

4 observations of super metal rich ([Fe/H]> 0.3) stars in the solar vicinity (Grenon

1972, 1989) cannot be explained by chemical evolution models without invoking radial migration to some extent (Chiappini 2009). Furthermore, the large scatter in the stellar age-metallicity relationship (Casagrande et al. 2011) cannot be explained if stars cannot not move away from their formation radius. Following the analytical and numerical work by Sch¨onrich & Binney (2009a) and Roˇskar et al. (2008b), respectively, most MW formation models address the radial migration process. An exact description of the eﬀectiveness of radial migration in the MW remains elusive.

It is clear that the formation of disk galaxies is a complex process. We have now entered the regime in which the predictive power of galaxy formation models is overwhelmed by the possibilities presented by the data. A promising way forward are eﬀorts to describe the growth of disk galaxies from cosmological initial conditions

(Stinson et al. 2012; Brook et al. 2012); realistic models of MW-like galaxy formation have only now become technically feasible (Guedes et al. 2011). Despite knowing the cosmological parameters that serve as inputs to these simulations and govern the clustering of dark matter and the formation of galaxy halos to ∼ 5% (Weinberg et al.

2012), there are still many uncertainties regarding how these halos accrete gas and the subsequent formation and assembly of stellar populations. As both empirical stellar data and galaxy formation theory evolve, we will be able to understand the history of the MW and its satellites within the broader context of halo mass and luminosity functions classically reserved for cosmologists; this is truly an exciting

5 time for “Near-field Cosmology” as the community attempts to find a unifying theory describing the conception, evolution, and current configuration of the MW.

1.1. Context and Scope

Observations of similar galaxies over a range of redshifts can statistically constrain how the evolution of stellar populations in the average galaxy of the data set (e.g. Steidel et al. 1994). Many studies of this type rely on the analysis of integrated light from composite stellar populations to provide information regarding the galaxy’s average age and chemical composition. The dominant uncertainty in stellar population synthesis modeling, as identiﬁed by Conroy & Gunn (2010), is the luminosity of thermally-pulsating AGB (TP-AGB) stars. Stars in this penultimate stellar evolutionary phase account for up to 40% (Persson et al. 1983) of a stellar population’s energy output. The energy generation and spectral energy distribution of the TP-AGB stars must be properly modeled to maximize the informational content of integrated light studies at high redshift. In Chapter 2, we use a simple fuel consumption argument to constrain the integrated luminosity of thermally pulsating

AGB (TP-AGB) stars across a range of progenitor mass.

Sellwood & Binney (2002) showed that the radial migration of stars, via corotational resonant interactions with transient spiral waves, could impact the evolution of a stellar disk without disrupting it. Since Roˇskar et al. (2008b)

6 conﬁrmed the presence of radial migration in galaxy-scale numerical experiments, both numerical (e.g. Roˇskar et al. 2008a; Loebman et al. 2011; Minchev et al. 2012) and analytical (e.g. Sch¨onrich & Binney 2009b) models have employed the process to explain observational constraints in the solar neighborhood. Further investigation into the radial migration phenomenon revealed that bar resonances, in addition to spiral waves, could drive radial migration (Minchev & Famaey 2010; Minchev et al.

2011). Still, the radial mixing process is not well-quantified; we do not know how different physical processes and disk properties couple with the migration process and determine its impact on present day observations. We perform a comparative analysis of radial migration in a series of controlled galaxy formation experiments with different evolutionary histories; in particular, we are the first to investigate, using a full N-body analysis, how gravitational perturbations from satellite galaxy accretion events expected in ΛCDM affect the radial migration of stars (Chapter 3).

Several studies have used the clustering of various stellar populations in distance and radial velocity to determine that hierarchical assembly must play a major role in the formation of the MW’s halo (Bell et al. 2008; Schlaufman et al. 2009; Xue et al. 2011; Cooper et al. 2011); most reach their conclusion via comparison of the observationally measured signal with identically measured signals in N-body models.

In Chapter 4, we explore how kinematic substructure studies can determine the origin of our Galaxy’s disk. The required observational data for such a measurement will not exist until the mid-infrared survey APOGEE is close to completion (Eisenstein

7 et al. 2011). We perform mock APOGEE observations of N-body galaxy models with a range of formation mechanisms; measure a statistic capturing the distance, radial velocity substructure in each model; and predict the diﬀerent signals that

APOGEE should measure for a range of galaxy assembly histories.

Disk galaxy formation theories often quantify their success through their ability to describe the structural components of the MW. The bulge, thin disk, and thick disk of our Galaxy have been deﬁned observationally as distinct stellar populations on the basis of their kinematics (e.g., Veltz et al. 2008), chemistry (e.g.

Lee et al. 2011), or both (e.g. Bensby et al. 2005). These deﬁnitions inherently assume a distinct origin for each component. Recently, Brook et al. (2012) followed the assembly of a disk galaxy from cosmological initial conditions and found that their chemically selected galaxy components naturally arise from early gas-rich mergers and continued star formation. Bovy et al. (2012) dissect the observable disk

(extrapolated from survey data) into mono-abundance (small spread in both [Fe/H] and [α/H]) populations, argue for an isothermal description of these populations, and show that the thin and thick disk reveal themselves in a composite of all mono-abundance populations. The latter two studies suggest that the construction of the Galactic structural components may not be as independent as previously thought. In Chapter 5, we characterize the formation and assembly of stellar populations within a high resolution, state of the art, cosmological N-body + SPH model of the formation of a disk galaxy structurally similar to the MW. Instead of

8 chemistry or kinematics, we use stellar age to divide the galaxy into its fundamental constituents. We ﬁnd that mono-age populations have distinct kinematics that directly reﬂect their evolution; furthermore, many observational properties of the

MW, previously thought to discern amongst MW formation scenarios, may in fact be a generic result of gravity-driven gas accretion onto dark matter halos.

9 Chapter 2

A Bound on the Light Emitted During the TP-AGB Phase

Stars can experience a core-collapse supernova only if they are born with a mass much higher than the Sun, even though a chemically evolved core of order only one solar mass is required to ignite the advanced burning stages. The culprit is mass loss severe enough to strip off the stellar envelope before a sufficiently massive processed core develops. In intermediate mass stars, the vast majority of this mass loss occurs in the presence of thermal pulsations involving interactions between hydrogen and helium burning shells. We refer to this as the thermally pulsing AGB (TP-AGB) phase, and our understanding of even the basic properties of the TP-AGB phase, such as the lifetime or light emitted, is surprisingly limited. The inter-related problems of shell interactions, pulsations, and mass loss all make predictive models much more complex and difficult than in prior evolutionary phases.

Stellar interiors models can accurately predict evolutionary properties up to the onset of the TP-AGB phase, and such models have been extensively used to reconstruct star formation histories in both resolved and unresolved populations.

10 The uncertainties in TP-AGB evolution have profound consequences for stellar population studies, and the impact on integrated light may be signiﬁcant. There has been intriguing work suggesting that the fraction of light emitted by TP-AGB stars in LMC star clusters is of order 40 % (Persson et al. 1983), and such a substantial ﬂux, dominating the light in the near-IR for several gigayear old populations, would be especially important for interpreting intermediate redshift galaxy properties (Conroy et al. 2009). Correspondingly, there has been a strong emphasis on updating population synthesis codes to include the TP-AGB phase (e.g.

Bruzual & Charlot 2003; Maraston 2005). Maraston et al. (2006) demonstrated that modeling of the TP-AGB phase has become a defining characteristic of different stellar population synthesis (SPS) codes while Conroy et al. (2009) identified the substantial uncertainties in TP-AGB properties as a major component of the error budget for galaxy evolution models.

In this paper we employ a fuel consumption argument (cf. Renzini & Buzzoni

1986) to set a firm lower bound on the fraction of light emitted during the TP-AGB phase. The white dwarf initial-final mass relationship (IFMR) predicts the remnant mass as a function of initial mass. We update the IFMR to account for new data and cluster parameters (Section 2.2.1). We then use stellar interiors models to set the core mass at the onset of the TP-AGB phase (Section 2.2.2). During the TP-AGB phase, the nuclear processed core grows until the envelope is expelled, at which point the final white dwarf mass is set. The difference between the remnant mass and the

11 core mass at the onset of the TP-AGB phase therefore serves as a bound on the fuel consumed, and thus the emitted light (Section 2.2.3). We thus have two potential tests of interior models: do the cores grow enough, and do the models burn enough fuel?

Although it might seem as if these two cases are the same, processed fuel

(especially helium) can be ejected in winds, so the white dwarf data formally sets only a lower bound on the fuel consumed and the emitted light. This opens up a complimentary application of our result: given measurements of the integrated light of TP-AGB stars, our results would constrain helium and metal ejecta yields

(Section 2.2.4). This raises interesting links between the light emitted in this phase and stellar yields (quantified in Marigo & Girardi 2001). We discuss possible ramifications for chemical evolution studies in our conclusions. Although qualitative comparisons between the IFMR, TP-AGB onset core mass, and final core masses have been performed before, and rough agreement with theory reported (e.g. Marigo

& Girardi 2007; Salaris et al. 2009) we are the ﬁrst to quantify the mass deﬁcit.

We compare both the calculated core mass growth and the corresponding minimum luminosity with that predicted by the Padova (Bertelli et al. 2008, 2009) and BaSTI

(Pietrinferni et al. 2004) TP-AGB evolutionary models. The plan of our paper is straightforward. Our sample and methods are discussed in Section 2, our results are presented in Section 3, and we discuss their broader implications in Section 4.

12 2.1. Sample and Methods

Stellar evolution theory makes a robust prediction for the core mass at the onset of the TP-AGB phase as a function of composition and initial mass. Open clusters provide a laboratory in which we can constrain the progenitor masses of stellar remnants whose current mass is well measured. We therefore begin by discussing the open cluster white dwarf IFMR. We discuss the WD data in Section

2.1.1, update the S09 cluster parameter data in Section 2.1.2, summarize our cluster initial mass data in Section 2.1.3, and deﬁne our cluster averages in Section 2.1.4.

We then measure TP-AGB phase core growth and its uncertainty in Section 2.1.5, and close with how we relate fuel consumption to energy output (Section 2.1.6).

2.1.1. White Dwarf Samples and Final Masses

Our sample consists of 48 white dwarfs (WDs) in 9 open clusters. The primary reference for the majority of our initial mass (Mi), final mass (Mf ) data is the compilation of Salaris et al. (2009); hereafter S09. S09 provide an extensive investigation of the systematic errors involved in determining the IFMR and calculate initial and final masses self-consistently across their entire sample. We include initial and final mass pairs from nine of the ten open clusters in S09. The Pleiades is omitted because the white dwarf in question is very young, the progenitor was very

13 massive, and the theoretical errors associated with the mass of the core at the ﬁrst thermal pulse are large.

Final white dwarf masses are determined from spectra of their Balmer lines, which are particularly sensitive to changes in surface gravities (g) and effective temperature (Teff ). Theoretical WD cooling curves provide a mass-radius relationship which is used to infer the final white dwarf mass and the cooling age (τcool) of the remnant, with only a mild dependence on the WD composition. S09 use the cooling curves of Salaris et al. (2000) to determine the remnant mass (Mf ) and τcool. The average fractional errors in final mass and cooling age for the S09 sample are 8.4% and 34.9% respectively.

We adopt the WD masses and cooling ages from S09 for six of the nine clusters in our sample where S09 has compiled the most recent and high resolution data. S09 adopt the log g and Teﬀ measurements ﬁrst compiled in Ferrario et al. (2005) for the Hyades (Claver et al. 2001), NGC 2516 (Koester & Reimers 1996), M37 (Kalirai et al. 2005), NGC 6819 and NGC 7789 (Kalirai et al. 2008), and NGC1039 (Rubin et al. 2008).

There are more recent WD observations for three clusters in their sample. In

NGC3532, Dobbie et al. (2009) present high resolution spectroscopic and photometric observations of six WDs, including the three originally observed in Koester &

Reimers (1993) and reported in S09. With precise distance modulus measurements,

14 they determine that two of these six WDs are not associated with the cluster. The

ﬁnal masses and cooling times for the remaining four WDs are interpolated from the theoretical cooling curves of Fontaine et al. (2001). The average fractional errors in

ﬁnal mass and cooling age are 6.0% and 22.2% respectively.

Williams et al. (2009a) expand upon their previous study (Williams et al.

2004, reported in S09) of WDs in M35. They present high resolution spectroscopic observations and updated log g and Teﬀ measurements of 12 DA WDs. Three WDs in their sample were peculiar and we remove them from our sample. Williams et al.

(2009a) interpolate the WD cooling curves of Fontaine et al. (2001) to obtain their

WD masses and cooling times with a mean fractional error in ﬁnal mass for the nine normal WDs of 9.3% and 51%.

Casewell et al. (2009) present high-resolution spectroscopic observations of 9

WDs in Praesepe using the Very Large Telescope. They found that WD0836+201 was mislabeled by previous studies and has strong magnetic ﬁelds, making it unsuitable for IFMR studies. Using radial velocity measurements, they argue against the membership of another candidate, WD0837+218, as a cluster member. We use their remaining seven candidates as our Praesepe WD sample. Casewell et al.

(2009) measured line core velocity shifts of the Hα and Hβ lines and reﬁt model atmospheres to the spectra accounting for these shifts, resulting in lower χ2 ﬁts and used a grid of cooling curve models from Fontaine et al. (2001) to determine the WD

15 masses and cooling times. The average fractional uncertainty was 5.6% in WD mass and 11.6% in cooling age.

Our adopted WD masses for these three clusters were calculated using a diﬀerent set of theoretical cooling curves from the rest of our sample, which could potentially impact our results. However, the uncertainty in WD mass is dominated by observational errors rather than systematic ones stemming from the choice of

WD cooling curves (Salaris et al. 2009). To account for systematics, we assume a fractional error of 50% for the WD cooling ages obtained using the Fontaine et al. (2001) models. For nominal choices of WD composition, envelope thickness, energy loss rates, and opacities, the fractional uncertainty in WD cooling ages for the remainder of our sample is still < 20% in most cases. Improvements in the observational measurements of log g and Teﬀ and the removal of cluster non-members from the sample more than oﬀset the errors stemming from the inclusion of a second group of WD cooling curve models in our analysis.

2.1.2. Cluster Parameters

The accuracy of our initial-ﬁnal mass data set depends critically on constraining the cluster distance and age. Although a few clusters have measured parallaxes, the distances to most are inferred by main sequence ﬁtting. Photometric methods can be used to infer composition, reddening, and distance even for clusters with limited

16 membership and spectroscopic data (see e.g. Pinsonneault et al. 1998, 2004; An et al.

2007). Well-studied open clusters can have spectroscopic metallicity measurements, extinction inferred from polarization studies, and both radial velocity and proper motion membership data. The uncertainties in these basic cluster parameters are largely determined by the available information in each speciﬁc cluster, which we discuss below, and set the uncertainty in the inferred cluster distance. The measured distance to the cluster yields the turnoﬀ luminosity, which can be combined with isochrones to determine the age and the mass-main sequence lifetime relationship

(see S09 for a discussion). The errors in these two aspects are distinct in nature.

Core size deserves special comment. Convective core overshooting (Chiosi &

Maeder 1986) or rotational mixing (Maeder & Meynet 2000) are diﬃcult to model theoretically, and both mechanisms have the practical eﬀect of extending the main sequence lifetime by providing extra fuel. The main sequence is broader than that predicted by models without overshoot or mixing, which is evidence that this phenomenon is real to some degree (e.g. Andersen 1991; Torres et al. 2010). As a result, the main sequence lifetime-mass relationship has substantial theoretical uncertainties. However, as we will show in Section 2.2, the total fuel burned is more reliable than the cluster ages. For the purposes of this paper we adopt a conservative limiting case approach, independently inferring fuel consumption bounds from models with and without overshoot.

17 2.1.3. Initial Masses

The progenitor lifetime (τprog) is the diﬀerence between the cluster age (τclus) and the cooling age of the WD (τcool). Stellar evolutionary tracks spanning a range of stellar mass and metallicity are then interpolated to yield the best-ﬁtting progenitor mass (Mi) given τprog. WD spectra modeled by theoretical cooling curves constrain

τcool (Section 2.1.1). Set by the distance to the cluster and metallicity, the cluster turn-oﬀ luminosity is a direct indicator of τclus. The cluster color-magnitude diagram

(CMD), [Fe/H], and E(B − V ) are the three main data inputs in main sequence

ﬁtting algorithms used to constrain the cluster’s distance modulus. S09 ﬁrst assume, using measurements from the literature, a metallicity and reddening for each cluster

(see their Table 2 and 3 for their sources and values, respectively). To obtain the cluster distance, they empirically fit the main sequence of V -(B − V ) CMDs using a sample of of field dwarfs with known metallicities and Hipparcos parallaxes (Percival et al. 2003). Once the distance is known, the turn-off luminosity is measured.

S09 interpolate the isochrones (both with and without convective overshooting) of

Pietrinferni et al. (2004) in both turn-oﬀ luminosity and metallicity to constrain the best ﬁt cluster age.

We adopt the cluster parameters, namely cluster age and composition, presented in S09 for three of the nine clusters in our analysis. The composition measurements included in S09 of the Hyades, NGC 6819, and NGC 1039 are either taken from

18 the most recent, high resolution studies of these clusters or are in agreement with current measurements. Additionally, the computed distances to these clusters align with previous distance investigations (e.g. Perryman et al. 1998; Kalirai et al. 2001; and Jones & Prosser 1996 for the Hyades, NGC 6819, and NGC 1039 respectively).

We searched the literature for revised metallicity measurements of and distance determinations to all the clusters in our sample. For six of the clusters, we modify the cluster composition, distance, or both from the values found in S09. In two of these six clusters, S09 either had difficulty with their main sequence (MS) fitting technique due to poor data (in the case of NGC 3532) or chose a reddening value significantly different than that calculated in the recent literature (M37). In other clusters in our sample, several high resolution spectroscopic studies over the last decade have produced very precise metallicity measurements. We use these more recent values in our final calculations.

We put all of our cluster ages on the same relative scale. Percival et al. (2003) present the main sequence ﬁtting technique applied in S09, and derive a metallicity dependency of ∆(B − V ) = 0.154∆[Fe/H] for their procedure. Their algorithm produces distances relative to an assumed distance to the Hyades, principally by matching the shape of the Hyades MS to the cluster CMDs. Assuming a linear slope with a magnitude of ∼ 5 for the relevant portion of the Hyades MS (Percival et al.

2003), ∆(B − V ) = ∆Mv/5; therefore, ∆Mv = 0.77 × ∆[Fe/H]. Twarog et al. (2009) performed an investigation of MS ﬁtting techniques using nearby ﬁeld stars with

19 Hipparcos parallaxes and precise metallicities and find ∆Mv = 0.98 × ∆[Fe/H]. The metallicity dependency determined by Percival et al. (2003) and Twarog et al. (2009) are similar; implementing the Twarog et al. (2009) relation would not significantly impact our results. We modify cluster ages corresponding to changes in distance according to the errors presented in S09. For small changes in distance, the lifetime should scale linearly with turnoff luminosity (lifetime ∝ 1/L) and thus distance (the isochrones of Marigo et al. (2008) and others confirm this); we therefore assume that, for a given cluster, the fractional error in distance is equal to the fractional error in age . If ∆(m − M) is the S09 cluster distance subtracted from the new distance then

∆(m − M)/σ(m−M) × σtclus = ∆tclus, where σ(m−M) and στclus are from S09 and the new cluster age is tclus = tclus,S09 − ∆tclus. We adopt ages relative to those of S09 for consistency. Below we summarize our revisions to these six clusters.

Praesepe An et al. (2008) present a new distance measurement to this cluster using empirically calibrated isochrones. Additionally, they obtain high signal to noise spectroscopy of several Praesepe stars, reporting [Fe/H]= 0.11 ± 0.03.

An et al. (2008) ﬁnd (m − M)0 = 6.33 ± 0.04 assuming [Fe/H]= 0.14 ± 0.02 (a weighted mean of their result and literature values excluding non-members) and

E(B − V ) = 0.006 ± 0.002. We adopt the latter An et al. (2007) value for our metallicity. Comparing the metallicity used in this distance determination with that of S09 ([Fe/H]= 0.04 ± 0.06), we ﬁnd ∆[Fe/H] = 0.14 − 0.04 = 0.10 and thus the de-reddened S09 distance is (m − M)0 = 6.32 ± 0.04. The An et al. (2008) distance

20 is 0.01 mag larger than expected, implying a relatively younger cluster age (τclus) of 637 ± 50 Myr using isochrones incorporating convective overshoot (OS) and

440 ± 40 Myr assuming no overshoot (nOS). The errors in the cluster age remain unchanged.

NGC 2516 An et al. (2008) simultaneously best-ﬁt their photometry of NGC 2516 with [Fe/H]= −0.04 ± 0.05, E(B − V ) = 0.117 ± 0.002, and

(m − M)0 = 8.03 ± 0.04. This represents a substantial improvement in σ[Fe/H] over

S09 (σ[Fe/H] = 0.11). De-reddening and accounting for ∆[Fe/H] = 0.12, the S09 distance is (m − M)0 = 8.04. Hence, ∆(m − M)0 = −0.01 mag yielding lifetimes of

τOS = 137 ± 29 Myr and τnOS = 91 ± 26 Myr. The error in cluster age has been reduced from their S09 values by the factor σ(m−M),An08/σ(m−M),S09 = 0.57.

NGC 3532 S09 could not find sufficiently precise photometry of this cluster in the magnitude range necessary to match their WD templates; subsequently, they had to scale the derived distance and age from their results for Praesepe. Assuming the same reddening as S09, Kharchenko et al. (2005) find (m − M)V = 8.61 ± 0.2 using optical photometry and Hipparcos proper motions. Their distance measurement is

0.21 mag larger than S09, giving τOS = 316 ± 80 Myr and τnOS = 216 ± 80 Myr.

Note that the errors in cluster age are reduced by σ(m−M),K05/σ(m−M),S09 = 0.80. We could not ﬁnd a a recent, high resolution spectroscopic metallicity determination of this cluster. We use the S09 value: [Fe/H]= 0.02 ± 0.06 (Gratton 2000).

21 M37 S09 fit the main sequence of this cluster using two different sets of cluster parameters. The first, with super-solar metallicity and E(B − V ) = 0.30 produced a cluster age that was improbably young. The progenitor masses for this age were so large relative to the remnant masses that the theoretically predicted core mass at the onset of the TP-AGB phase was typically greater than the remnant mass. S09 tried a substantially lower metallicity, [Fe/H]= −0.20, and reddening, E(B − V ) = 0.23 and obtained a more reasonable cluster age in agreement with previous age determinations. We scale our results from this latter cluster characteristic set. Fortunately, Hartman et al. (2008) performed high resolution spectroscopic and photometric observations of M37 using the MMT, and these results clarify the cluster properties. They determine [Fe/H]=0.045 ± 0.044,

E(B − V ) = 0.227 ± 0.038, and (m − M)V = 11.57 ± 0.13. Comparing this distance with that of S09, ∆(m − M)0 = 0.17. However, adjusting the S09 distance to the same metallicity (∆[Fe/H] = 0.24), we ﬁnd (m − M)V = 11.58.

Thus, ∆(m − M)0 = 11.57 − 11.58 = −0.01 mag, and the cluster age for M37 is

τOS = 554 ± 54 Myr and τnOS = 354 ± 43 Myr. The errors in cluster age have increased by the factor σ(m−M),H08/σ(m−M),S09 = 1.083 over their S09 counterparts.

M35 We found one high resolution, high signal to noise measurement of

M35’s composition ([Fe/H]= −0.21 ± 0.10) by Barrado y Navascu´es et al. (2001).

∆[Fe/H] = −0.21 − (−0.19) = −0.02 and thus ∆(m − M) = −0.015. The

22 revised cluster ages are slightly older than those in S09: τOS = 124 ± 30 Myr and

τnOS = 88 ± 25 Myr. Cluster age errors remain unchanged from the S09 values.

NGC 7789 Tautvaiˇsien˙eet al. (2005) determine [Fe/H]= −0.04 ± 0.05 in

NGC 7789 using high resolution spectra of evolved cluster members while the measurement ([Fe/H]= −0.13 ± 0.08) quoted in SO9 is an average of photometric and low dispersion spectroscopic abundance measurements (Gratton 2000).

∆[Fe/H] = −0.04 − (−0.13) = 0.09; the accompanying change in distance is

∆(m − M) = 0.07. The cluster age is 58 Myr younger than stated in S09:

τOS = 1442 ± 100 Myr and τnOS = 1042 ± 100 Myr. Cluster age errors are repeated from S09.

The largest departure in cluster age from S09 is 21% (NGC 3532), which has the most limited data of the S09 sample. Typically, however, our cluster ages represent modiﬁcations to those of S09 on the order of a few percent. Although our global conclusions have not been strongly impacted by the cluster parameter updates here, the errors are reduced and some potentially interesting outlier cases are now consistent with the data.

For each WD, the lifetime of its progenitor is τprog = τclus − τcool. Errors are propagated from τcool and τclus to τprog. Using the stellar evolutionary models of

Pietrinferni et al. (2004), we construct a data cube of Mi as a function of τprog and [Fe/H]. For the six clusters with modiﬁed cluster ages or WD cooling ages,

23 6 we sampled a Gaussian distribution in τprog and [Fe/H] 2 × 10 times (the width

of each Gaussian is στprog and σ[Fe/H], respectively), interpolating the cube linearly in [Fe/H] and quadratically in τprog to obtain the progenitor mass. Our effective one sigma error bars encapsulate the central 68% of the values interpolated from the grid. S09 interpolate the Pietrinferni et al. (2004) models in the same fashion to obtain progenitor masses for WDs in the other three clusters. In their error budget, they also include the systematic offset in Mi from interpolating a different stellar evolutionary model set (specifically those of Girardi et al. (2000)). While our revised Mi measurements in Praesepe, NGC 2516, NGC 3532, M37, M35, and NGC

7789 do not include the systematic error associated with using the Padova models,

this potential contribution to the total σMi is small when compared to the cluster parameter uncertainties (see S09).

2.1.4. Cluster Averages

The ﬁnal and initial masses of all 48 WDs in our sample are shown in

Figure 2.1. In many of the clusters, there is considerable random scatter in these values, especially when Mi> 3M⊙. When τclus − τcool is small, errors in either quantity contribute more strongly to τprog, resulting in a spread of initial masses for a single cluster. We reduce this scatter by representing each cluster by its weighted mean Mi and Mf (ﬁrst proposed in Kalirai et al. 2008). To determine the inﬂuence of outlying measurements, we also compute the median Mi and Mf for each cluster and

24 adopt the standard error approximation σmed = π/2 × σmean. We ﬁnd no statistically signiﬁcant change in our results when adopting the median Mi, Mf for each cluster instead of the weighted mean. For the remainder of this paper we present our results using the weighted mean Mi and Mf of each cluster.

2.1.5. Fraction of Core Growth in the TP-AGB Phase

The core mass at the ﬁrst thermal pulse of the AGB stage (Mc,1TP) is ﬁxed by theory via the star’s initial mass and composition. We determine Mc,1TP in a manner identical to that of Mi, sampling gaussian distributions of Mi and [Fe/H]

2 × 106 times and interpolating the Pietrinferni et al. (2004) models linearly in

[Fe/H] and quadratically in Mi. Central values and errors are calculated similarly to that of Mi in Section 2.1.3. As the ﬁnal remnant mass is equal to the core mass at the tip of the AGB, the core grows by ∆Mc= Mf − Mc,1TP in the TP-AGB

phase. The uncertainties in Mc,1TP and Mf are propagated to determine σ∆Mc .

We deﬁne the fractional contribution of the TP-AGB phase to ﬁnal core mass as

fMc,TP−AGB= ∆Mc/Mf . Table 2.3 lists the Mi, Mf , ∆Mc, and fMc,TP−AGB of each cluster using both the OS and nOS models.

25 2.1.6. Fuel Consumption: Mass-Light Coupling

The total energy released during any evolutionary phase of a star’s life is directly proportional to the fuel consumed (Renzini & Buzzoni 1986). The core’s growth during the TP-AGB phase thus represents a direct lower bound on the fuel burned during this phase (Marigo & Girardi 2001). Contributions to the light budget from gravitational contraction of the core and neutrino loses only represent corrections on the percent level (Marigo & Girardi 2001) and we therefore neglect them.

Mathematically, we couple the consumed fuel to the energy released during the

TP-AGB phase via (cf. eq. 5 from Marigo & Girardi 2001):

1 F (Mf , Mi)= LMi (t)dt. (2.1) AH Z

The fuel, F (Mi), is expressed in solar masses and the integral is the sum of the energy released during the TP-AGB phase. The conversion from energy to mass is represented by AH , the eﬃciency of H burning reactions. Here, we adopt

10 −1 AH = 9.75 × 10 L⊙ yr M⊙ following Marigo & Girardi (2001). Above, we note that nucleosynthesis is the dominant stellar energy source. The fuel consumed in the

TP-AGB phase is therefore:

TP −AGB ′ P FTP −AGB ≃ (X1,2 + 0.1)(∆Mc)+ My (He)+(1.1 − Y )My (CO). (2.2)

26 X1,2 is the surface mass fraction of hydrogen after the second dredge-up event,

′ Y is deﬁned as Y1,2/(X1,2 + Y1,2) where Y1,2 is the mass fraction of helium in the

TP −AGB P envelope after the second dredge-up, and My (He) and My (CO) refer to the stellar yield of He and CO, respectively, during the TP-AGB phase. The term

(X1,2 + 0.1) takes into account both hydrogen and helium burning as helium burning releases roughly 10% of the energy generated via hydrogen burning. (1.1 − Y ′) is the mass fraction of envelope material that is originally hydrogen and subsequently burned to carbon and oxygen. We assume an initial H abundance (X0) of 0.71, an initial X typical for solar models (Bahcall et al. 2001), and map to X1,2 using the tabulated surface compositions in Bertelli et al. (2008, 2009). Irrespective of the yields of He and CO we can construct our strict lower limit using only the observed growth of the core during the TP-AGB phase. The minimum bound on the fuel consumed during this phase, and hence the energy released, is obtained by setting

F (Mi)= FTP −AGB =(X1,2 + 0.1)(∆Mc). Solving for LMi (t)dt in equation 2.1, we R obtain the minimum energy release (Lmin) necessary to increase the core mass by

∆Mc. The uncertainty in Lmin is propagated from σFTP −AGB . We list this value for each cluster in Table 2.3.

27 2.2. Results

Observations of white dwarfs in open clusters combined with established evolution models constrain the fuel consumption, and hence light output, during the TP-AGB phase. Our results demonstrate that the stellar core grows by a non-negligible amount during the TP-AGB phase, regardless of progenitor mass.

Below, we address the evolution of core mass growth in TP-AGB stars as a function of initial mass, couple this core mass increase to a lower bound on TP-AGB star light output, compare our results with recent theoretical models of the TP-AGB phase, and discuss how our results change with diﬀerent overshooting conditions.

2.2.1. Initial Final Mass Relation

We plot the IFMR for our sample in Figure 2.1. The individual and initial masses are represented by the gray points and the weighted mean Mf -Mi pair for each cluster is in black. Typically, the ﬁnal masses of a given cluster are consistent across diﬀerent IFMR studies despite coming from various theory groups (e.g.

Williams et al. 2009a). On the other hand, the cluster parameters that dictate the calculated initial masses are less well constrained; cluster age, in particular, can be notoriously diﬃcult to determine. However, progenitor masses calculated in our analysis rather than in S09 still show good agreement with those of other studies assuming the same cluster ages, e.g., in M37 our weighted mean Mi= 3.21 ± 0.11 M⊙

28 for τclus= 563 ± 31 Myr and Ferrario et al. (2005), who use τclus= 520 ± 80 Myr, ﬁnd

Mi= 3.27 ± 0.12 M⊙. However, some potential cluster anomalies (e.g. NGC3532 and

M37, see Section 2.1.3) vanish in the light of improved cluster studies. The average fractional uncertainty in Mf (Mi) in our new relation is 3%(7%).

The red line in Figure 2.1 connects the anticipated core mass at the ﬁrst thermal pulse (Mc,1TP) for each cluster’s weighted mean initial mass. We determine

Mc,1TP given Mi and [Fe/H] according to Pietrinferni et al. (2004) (see Section 2.1.5).

The relationship between the core mass at the ﬁrst thermal pulse and the progenitor mass, the basis for our analysis, is a robust theoretical prediction; e.g., Girardi et al. (2000) predict Mc,1TP within 2% of the models used here. In Figure 2.1, all the cluster mean ﬁnal masses are larger than the predicted core mass at the

ﬁrst thermal pulse. Given nominal assumptions regarding stellar evolution, current cluster observations demand that the core grows substantially during the TP-AGB phase; thus, TP-AGB stars must contribute signiﬁcantly to cluster luminosity.

2.2.2. Core Growth in the TP-AGB Phase

In Figure 2.2, we illustrate the fraction of the ﬁnal core mass grown during

the TP-AGB phase (fMc,TP−AGB, deﬁned in Section 2.1.5) as a function of initial mass. The data points are made using the weighted mean initial and ﬁnal masses for each cluster. Filled squares represent masses found with models and cluster ages

29 incorporating overshoot; open squares are masses that do not take overshooting into account. Dashed lines connect the results of these two cases for each cluster.

The error bars represent one sigma errors in fractional core mass gained during the

TP-AGB phase and initial mass. The red lines connect the moving weighted mean of each data point type (every OS or nOS data point is averaged with its two nearest neighbors) and clarify the impact of overshooting on our results. The solid red line follows the averages of the ﬁlled data points (OS); the dashed red line corresponds to the open square (nOS) averages.

In the case where convective overshooting is considered (OS), the weighted

mean fMc,TP−AGB= 0.20 ± 0.01 when 1.9 ≤ Mi ≤ 3.6 and drops by a factor of two

to 0.08 ± 0.02 elsewhere. This broad peak in fMc,TP−AGB above the remainder of the sample is signiﬁcant at the 3.5σ level and marginally coincides with the peak in the lifetime function predicted by recent TP-AGB models (e.g. Marigo & Girardi

2007, for a physical explanation). For progenitor masses calculated with models

and cluster ages that do not include convective overshooting (nOS), fMc,TP−AGB is marginally smaller than its OS counterpart and the fraction of core mass gained in the TP-AGB phase peaks at a slightly higher progenitor mass (3.5M⊙ versus

3M⊙). These minor shifts are a natural consequence of the inclusion or absence of convective overshooting in evolutionary models and isochrones. There is a relatively straightforward stellar physics explanations for this insensitivity: core overshoot impacts the fuel consumed on the main sequence, but there is a minimum core

30 mass required to ignite helium which is a weak function of the total mass. For stars at the lower end of our mass range, the core must grow prior to the ignition regardless of overshoot; for stars at the higher end the core must simply heat up and contract to ignite He. One would therefore expect the He core mass to be less sensitive to overshoot than the MS lifetime at the low mass end (M < 2 M⊙).

For higher mass stars the impact of overshooting on core mass is less than the impact on lifetime because the MS turnoﬀ He core mass is the product of the mean luminosity and lifetime; the latter is higher with overshoot but the former is lower, since stars with more overshoot evolve more strongly in luminosity. The nOS and

OS results are statistically consistent with one another: eight of the nine clusters have OS and nOS ∆Mc within 1σ of each other (the other is within 2σ). The precise extent of convective cores is still a matter of active debate, so it is encouraging that our results are surprisingly independent of the treatment of overshooting. Given this generality, and the fact that models with convective overshooting are favored by other observations, we adopt our OS results as the basis for discussion in the remainder of the paper.

Having established fMc,TP−AGB as a function of [Fe/H] and Mi using the latest

IFMR data, we now compare the core growth in the TP-AGB phase required by observations with that predicted by two sets of current evolutionary models from the BaSTI and Padova groups (Pietrinferni et al. (2004) and Bertelli et al. (2008,

2009), respectively). For this comparison and all others between the IFMR and

31 TP-AGB models, we assume all clusters have solar metallicity and scale Mc,1TP appropriately as TP-AGB phase core masses are readily available at solar metallicity for both models. Our sample’s open clusters span a narrow metallicity range surrounding the solar value, however, so these corrections are small: only the two largest metallicity outliers (NGC 2516 and M35) show changes in Mc,1TP> 0.003 M⊙

and the errors associated with their fMc,TP−AGB are already large. The fraction

of the core built up during the TP-AGB phase at solar metallicity (fMc,TP−AGB,⊙) is ∆Mc/Mf where ∆Mc= Mf − Mc,⊙,1TP; Mc,⊙,1TP is the core mass at the ﬁrst thermal pulse given Mi and solar composition. The analogous quantity predicted by

the BaSTI (fMc,TP−AGB,⊙,BAS) and Padova (fMc,TP−AGB,⊙,PAD) models are ∆Mc/Mf where ∆Mc= Mf,mod − Mc,⊙,1TP,mod; Mf,mod and Mc,⊙,1TP,mod are the ﬁnal core mass and Mc,1TP, respectively, predicted by each model for Mi and solar composition.

We use the same prescription to calculate the errors associated with the various

fMc,TP−AGB,⊙ as used for the original fMc,TP−AGB presented earlier in this section.

In Figure 2.3, we compare the predicted core growth in the TP-AGB phase by

both sets of models with our calculated fMc,TP−AGB. The fMc,TP−AGB,⊙ are shown as

open squares, fMc,TP−AGB,⊙,PAD as light gray regions, and fMc,TP−AGB,⊙,BAS as dark

gray regions. Earlier we established the rise of fMc,TP−AGB at intermediate masses

(2.0 < Mi < 4.0 M⊙) and fall at both low (Mi< 2.0 M⊙) and high (Mi> 4.0 M⊙)

Mi; in these three mass bins, we examine the weighted mean diﬀerence in core mass between the fc,TP−AGB,mod predicted by the models and that dictated by the IFMR

32 (Rf c,TP−AGB = fMc,TP−AGB,⊙ − fc,TP−AGB,⊙,mod). Using the Padova group models,

Rf c,TP−AGB is −0.037±0.027 M⊙, 0.041±0.016 M⊙, and 0.102±0.028 M⊙ in the low,

intermediate, and high mass bins, respectively. Rf c,TP−AGB for the BaSTI models, in order of increasing progenitor mass, is 0.024 ± 0.023 M⊙, 0.098 ± 0.015 M⊙, and 0.059 ± 0.028 M⊙. Both models systematically underpredict the core growth in the TP-AGB phase but are still consistent with the data in the majority of progenitor mass regimes. The BaSTI group’s models, however, fail to grow their cores suﬃciently at the > 6σ level for stars with 2.0 < Mi < 4.0 M⊙. To our knowledge we are the ﬁrst to make this detailed, quantitative comparison between the IFMR and TP-AGB models. In the past, this comparison between TP-AGB models and the IFMR has been more qualitative in nature (Figure 3 in Salaris et al.

(2009)). Future models can use our fMc,TP−AGB results as an important check on their characterization of core growth during the TP-AGB phase.

2.2.3. Fuel from Stellar Core Growth

Some processed fuel is ejected in winds, so TP-AGB models could burn enough fuel to be compatible with our constraints while not growing the core enough. This would indicate problems with core growth as opposed to the total fuel consumed in the phase. To test this, we compare the predicted integrated light in TP-AGB models with that required by core mass growth. We have already determined the mass added to the core during the TP-AGB phase (∆Mc) for each cluster. Using

33 the equations in Section 2.1.6, ∆Mc is converted to its equivalent energy output

(Lmin). We compute Lmin as a function of initial mass for each cluster assuming solar metallicity (listed in Table 2.3).

We can now determine whether TP-AGB evolutionary models predict enough

light to be consistent with Lmin. We deﬁne fE,∆Mc = Lmin/ LMi,X (t)dt where R

LMi,X (t)dt is the predicted total energy in the TP-AGB phase and is obtained by R integrating the Padova TP-AGB stellar evolutionary tracks. In Table 2.3, we list

fE,∆Mc and its error on a cluster by cluster basis for both model sets. The weighted

mean of fE,∆Mc in our sample using the Padova tracks is 0.54 ± 0.04. The individual

fE,∆Mc for each cluster is within 2σ of this mean while all but the cluster with the smallest progenitor mass, NGC 6819, and M37 are within 1σ of this mean. The

Padova set of models predict more than the required luminosity from the calculated core mass growth in the TP-AGB phase.

Integrating their predicted total energy to compute LMi,X (t)dt, the BaSTI R models produced a weighted mean fE,∆Mc of 1.28 ± 0.09 M⊙. While all but one individual cluster measurement is within 2σ of this mean, these models predict less light in the TP-AGB phase than is required by the core mass growth found in Section 2.2.2. As the BaSTI models systematically underpredict ∆Mc, these models could simply assume more core growth and remedy their discrepancy. Upon inspection, the BaSTI models predict shorter TP-AGB lifetimes by about a factor of three than the Padova models, caused by more severe mass loss in the former.

34 If both model sets had similar lifetimes, their fE,∆Mc would be consistent with one another.

2.2.4. Fuel from Stellar Yields

We are now in position to calculate the burned fuel predicted to be released to

the ISM for models of the TP-AGB phase with fE,∆Mc < 1. This serves as an example of how the yields of TP-AGB stars can be calculated given Lmin and a prediction or measurement of the total integrated light of TP-AGB stars. If we make the reasonable assumption (Section 2.1.6) that the available fuel in the star during the

TP-AGB phase comes primarily from nuclear burning, the quantity 1 − fE,∆Mc is the fraction of the total model predicted or observationally required energy accounted for by nucleosynthetic products that were expelled into the ISM rather than added to

the core. The product (1 − fE,∆Mc ) × LMi,X (t)dt is the amount of light unaccounted R for by core mass growth. Setting (1 − fE,∆Mc ) × LMi,X (t)dt = LMi (t)dt in R R equation 2.1, we obtain the available fuel, in solar masses (FTP−AGB,He), that is consumed and released to the ISM. In equation 2.2, we make a distinction between the yield of He and that of CO. In practice, the diﬀerence in binding energy per unit mass between H and He is 10 times that of He and C or O; thus, we make the approximation that He is the sole component of the stellar yield. The predicted

FTP−AGB,He of the Padova models for each cluster is shown in Table 2.3 (1 − fE,∆Mc is negative when using the BaSTI models). We ﬁnd our sample’s weighted mean

35 FTP−AGB,He= 0.08 ± 0.01 M⊙. Every individual measurement of the stellar yield, including the largest Mi cluster with the formally negative yield (NGC 2516), is within two σ of this mean.

2.3. Summary and Discussion

Using the WD IFMR and robust stellar evolutionary theory, we have shown that a signiﬁcant mass fraction of the ﬁnal, stellar core of intermediate mass stars

(∼ 1.7 < M⊙ < 6) is generated during the TP-AGB phase. There is measurable core growth during the TP-AGB phase in all nine clusters (spanning ∼ 0.30 dex near the solar [Fe/H]) of our sample. This result depends only upon the IFMR and the core mass at the ﬁrst thermal pulse (agreed upon by the vast majority of stellar evolutionary codes); it is thus independent of one’s evolutionary model choice and avoids the complex physics and uncertainties of the TP-AGB phase completely.

Further, we have demonstrated that our measurement is robust against the major uncertainty in stellar evolutionary models up to the onset of the TP-AGB phase: the depth of convective overshooting and whether or not it takes place. Through simple fuel consumption arguments, the TP-AGB phase must emit a substantial amount of light- proportional to the aforementioned core growth. Therefore, we have constructed a strict lower bound on the light output during the TP-AGB phase as a

36 function of initial mass that is predominantly dependent on observational constraints from white dwarfs and their host cluster properties.

The fractional contribution to the core mass during the TP-AGB phase peaks when Mi ∼ 2 − 3.5 M⊙. In general, ∼ 20% of the core is built during the TP-AGB phase in this mass range, double that of lower and higher mass stars. The maximum of the TP-AGB fractional energy output at ∼ 2 − 3.5 M⊙ has strong implications for interpreting the integrated light of elliptical galaxies. Taking into account the number of stars in each mass bin, one predicts that the most TP-AGB phase core mass growth occurs in ∼ 1.5 − 2 M⊙ stars. This mass range roughly corresponds to lifetimes of 2 to 3 Gyr. Consistent with our ﬁndings, Conroy et al. (2009) predict that the infrared light from TP-AGB stars will be one of the stronger principle components of the galaxy spectral energy distributions of galaxies harboring stellar populations of this age.

Constraining the core mass growth and thus the contribution of the TP-AGB phase to cluster light via the IFMR becomes increasingly diﬃcult for progenitor masses above ∼ 5M⊙. High mass stars have short lifetimes, implying that τclus −τcool is a small number. In that case, nominal errors in τclus and τcool have a fractionally larger impact on the uncertainty in progenitor age at high mass than at low mass.

Still, we find compelling evidence that ∼ 10% of the final core is built up during the TP-AGB stage at high mass regardless of convective overshoot efficiency. The upper limit in progenitor mass for stars experiencing the TP-AGB phase is still

37 controversial and empirical data is scare (Williams et al. 2009b). While this work suggests 5.5 M⊙ stars experience a TP-AGB phase, the IFMR will provide a strong means to determine the highest initial mass TP-AGB star if progenitor masses in this regime can be better constrained in the future.

Nuclear burning requires energy release and thus the measured core growth in the TP-AGB phase represents a lower limit to the integrated luminosity of the star over this time period. By quantifying the core growth in the TP-AGB phase and converting it to a minimum luminosity (Lmin, Sections 2.1.6, 2.2.2, 2.2.3), we assure that future observations of TP-AGB populations will ﬁnd that their luminosity (Lobs) is larger than or consistent with Lmin. Either result will provide the necessary boundary conditions to guide convincing models of TP-AGB evolution.

However, these two scenarios predict vastly diﬀerent TP-AGB stellar chemical yields to the ISM; speciﬁcally, the greater the observed integrated TP-AGB luminosity

(Lobs) relative to Lmin, the more fuel is burned and expelled to the ISM. This important distinction can be used to predict, or at least limit the range of, Lobs before observations become available. Using our procedure in Section 2.2.4, one could calculate the yields of TP-AGB stars that emit any given amount of integrated light

(in fact, Section 2.2.4 essentially calculates the yields of TP-AGB stars assuming the total integrated light of TP-AGB stars is correctly predicted by the Padova models). TP-AGB stars are numerous; their yields, if signiﬁcant, should impact the chemical composition of the ISM. With the establishment of Lmin, any model

38 that predicts the total integrated light of TP-AGB stars also predicts their yields.

These yields should be checked for agreement with the required ∆Y/∆Z relationship employed by chemical evolution models to recreate observed stellar lifetimes and luminosities. We can already place upper limits on Lobs by requiring it to not to be so large as to predict TP-AGB yields that would modify this empirically calibrated relationship. Models of the TP-AGB phase must predict more integrated light than

Lmin while chemical evolution considerations, and eventually direct measurement, provide constraints on the total integrated light in the TP-AGB phase.

Observations of TP-AGB populations are rare but are slowly becoming more common. Persson et al. (1983) determine that up to 40% of a cluster’s integrated

light comes from TP-AGB stars; this number exceeds our calculated fMc,TP−AGB by at least a factor of two for almost all Mi. If Lobs is in this range, at least as much integrated luminosity from TP-AGB stars can be accounted for by stellar yields than by core growth. More recently, Conroy & Gunn (2010) found that they must reduce the luminosity of the Padova group’s TP-AGB models by about half to match observations of Magellanic Cloud star clusters. While the relatively lower metallicity of the Magellanic Cloud cluster IFMR may have a slightly diﬀerent form than the one we produce here, theory predicts that Mf would change by less than 10% given the metallicity spread between the Galactic and Magellanic Cloud clusters (Meng et al. 2008). If conﬁrmed, the Conroy & Gunn (2010) result suggests that TP-AGB stars release little helium; further, evolutionary models would have to

39 modify the luminosity of the pulses in the TP-AGB phase along with their mass loss prescriptions in order to better match Lmin with Lobs.

The most prominent sources of error in this analysis are the WD and cluster observations. Measurements of WD surface gravities and temperatures provide constraints on remnant masses and cooling times while cluster observations yield the cluster age and composition. Uncertainties in theory linking these observations and the parameters of interest are typically small compared to the observational errors (S09). By assuming either perfect WD or cluster measurements and repeating our analysis, we ﬁnd that uncertainties in these two types of observables contribute similarly to the resulting error in core mass growth during the TP-AGB phase.

More precise WD or cluster measurements will provide a stricter lower bound on the integrated energy output of TP-AGB stars. Another potential method to improve the precision of the IFMR (and our results) would be a survey speciﬁcally designed to ﬁnd young white dwarfs in open clusters. In any given cluster, the youngest white dwarfs would correspond to the oldest progenitor lifetimes, reducing the impact of uncertainties in the WD and cluster observations on the implied initial mass and

Mc,1TP. More precise measurements of the IFMR would directly lead to more precise calculations in our procedure.

Determining the metallicity dependence of TP-AGB star total energy output is an intriguing next step. Ideally, we would construct our lower TP-AGB luminosity bound using IFMRs for a wide range of metallicity. However, there are not enough

40 open clusters at low metallicities and small distances to empirically calibrate such a dependence presently. An intermediate step would be to increase the number of clusters in our sample, even if the overall spread in metallicity does not change. With many more clusters, one could reproduce our results in small bins of metallicity.

After empirically characterizing how the energy output of the TP-AGB phase scales with composition (albeit over a limited range), theory may be able to provide a physical model for TP-AGB star integrated luminosity as a function of initial mass and stellar density. If said model could be extrapolated to low metallicity, it would provide a theoretical constraint on the contribution of TP-AGB stars to the spectral energy distributions of high redshift galaxies (where [Fe/H] is typically lower than our our sample). At high metallicity, the lowest mass stars may go directly to the white dwarf stage, missing the core He burning stages and beyond

(Kilic et al. 2007). This eﬀect is not currently included in population synthesis models and might be important for the low redshift properties of giant elliptical galaxies. Physically motivated priors on the correlation between TP-AGB star light output and composition would be a substantial advancement of semi-analytic galaxy evolution models as SPS codes still have widely varied characterizations of these stars.

We have shown that the WD IFMR places powerful constraints on the total energy released during the TP-AGB stage of stellar evolution over the metallicity range of our sample. These constraints are surprisingly independent of most

41 theoretical uncertainty in stellar interior models (notably the parametrization of convective overshoot). We ﬁnd that the calculated core mass growth and associated luminosity are important checks on modern TP-AGB evolutionary models. Additionally, we show that observations of TP-AGB populations will constrain the helium yields of these stars- an important factor for chemical evolution models to consider. Alternatively, nominal assumptions regarding the yield of these stars, in conjunction with this work, result in a narrower range of possible TP-AGB population luminosity than previously considered.

42 Fig. 2.1.— The IFMR for the 48 WDs in our sample. The progenitor masses are calculated using cluster ages and stellar evolutionary theory that incorporate a degree of convective overshooting (Pietrinferni et al. 2004). The Mf -Mi pairs for each WD are in light gray; error bars represent one σ deviations in each quantity. The open black squares show the weighted mean Mf and Mi for the nine clusters in our sample. Error bars are the one σ error of the means. Given an initial mass and cluster metallicity, theory predicts the mass of the core at the ﬁrst thermal pulse (Mc,1TP). The solid red line connects the nine Mc,1TP - Mi points created using the weighted mean Mi and the [Fe/H] of each cluster. The observed remnant masses in these clusters all lie above the red line. As Mf is the mass of the core at the tip of the AGB, stellar cores must grow during the TP-AGB phase.

43 Fig. 2.2.— The fraction of the observed final remnant mass built up in the TP-AGB phase as a function of initial mass. The squares are calculated using the weighted mean initial and final masses of each cluster. Filled squares use evolution theory and isochrones incorporating convective overshooting; open squares represent the limiting case of no overshooting. The dashed lines connect these two choices for each cluster. The solid (dashed) red line links the results of a 3 point moving weighted mean of the filled (open) squares. The fractional core mass growth in the TP-AGB phase has a broad peak between 2-3 M⊙.

44 Fig. 2.3.— The fraction of the ﬁnal remnant mass built up in the TP-AGB phase as a function of initial mass at solar metallicity according to our fMc,TP−AGB,⊙ calculations (open squares and error bars) and the Padova and BaSTI TP-AGB models (light and dark gray hatched regions, respectively). The squares and hatched regions are calculated using the weighted mean initial and ﬁnal masses of each cluster. Each hatched region is centered on the model value corresponding to the weighted mean Mi and Mf of each cluster. Its width is equal to the uncertainty in Mi and height set by the corresponding uncertainty in the fc,TP−AGB,⊙,mod. See text for details.

45 Cluster [Fe/H] (m − M)0 (mag) OS Age (Myr) nOS Age (Myr) References (1) (2) (3) (4) (5) (6)

NGC 6819 0.09 ± 0.03 12.15 ± 0.20 2000 ± 200 1500 ± 200 1,2 NGC 7789 −0.04 ± 0.05 12.19 ± 0.12 1442 ± 100 1042 ± 100 3,2 Hyades 0.13 ± 0.06 3.33 ± 0.01 640 ± 40 440 ± 40 4,2 Praesepe 0.14 ± 0.02 6.33 ± 0.04 637 ± 50 440 ± 40 5 M37 0.04 ± 0.04 10.83 ± 0.13 554 ± 54 354 ± 43 6 NGC 3532 0.02 ± 0.06 8.48 ± 0.20 316 ± 80 216 ± 80 4,7 NGC 1039 0.07 ± 0.04 8.58 ± 0.15 250 ± 25 150 ± 30 8,2 NGC 2516 −0.04 ± 0.05 8.03 ± 0.04 137 ± 29 91 ± 26 5 M35 −0.21 ± 0.10 10.48 ± 0.12 124 ± 30 88 ± 25 9,2 46

Note. — Cluster compositions, distances, and ages. (1): Cluster. (2): [Fe/H]. (3): Absolute distance modulus. (4): Cluster age using isochrones with overshooting. (5): Cluster age using isochrones without overshooting. (6): Reference for cluster metallicity and distance; if two references are listed, the first applies to metallicity and the second, distance. Note that the distances in column 3 may be modified from their references if different cluster compositions were adopted.

References. — 1. Bragaglia et al. (2001); 2. Salaris et al. (2009); 3. Tautvaiˇsien˙eet al. (2005); 4. Gratton (2000); 5. An et al. (2007); 6. Hartman et al. (2008); 7. Kharchenko et al. (2005); 8. Schuler et al. (2003); 9. Barrado y Navascu´es et al. (2001).

Table 2.1. Cluster Metallicity, Distance, and Age. Cluster Mf ( M⊙) Mi ( M⊙) fc,TP −AGB ∆Mc ( M⊙) (1) (2) (3:OS,nOS) (4:OS,nOS) (5:OS,nOS)

NGC 6819 0.55 ± 0.02 1.75 ± 0.08 , 1.87 ± 0.06 0.06 ± 0.03 , 0.04 ± 0.03 0.03 ± 0.02 , 0.02 ± 0.02 NGC 7789 0.61 ± 0.02 1.94 ± 0.05 , 2.02 ± 0.06 0.18 ± 0.04 , 0.13 ± 0.04 0.11 ± 0.02 , 0.08 ± 0.02 Hyades 0.71 ± 0.01 2.86 ± 0.03 , 3.28 ± 0.05 0.20 ± 0.02 , 0.19 ± 0.02 0.15 ± 0.01 , 0.14 ± 0.02 M37 0.80 ± 0.03 3.10 ± 0.07 , 3.79 ± 0.14 0.23 ± 0.04 , 0.14 ± 0.05 0.18 ± 0.03 , 0.11 ± 0.04 Praesepe 0.79 ± 0.02 3.21 ± 0.16 , 3.84 ± 0.38 0.20 ± 0.05 , 0.14 ± 0.09 0.15 ± 0.04 , 0.11 ± 0.07 NGC 3532 0.85 ± 0.02 3.57 ± 0.21 , 4.03 ± 0.37 0.16 ± 0.05 , 0.13 ± 0.08 0.13 ± 0.05 , 0.11 ± 0.06

47 NGC 1039 0.87 ± 0.04 4.00 ± 0.23 , 4.79 ± 0.40 0.09 ± 0.06 , 0.08 ± 0.06 0.08 ± 0.05 , 0.07 ± 0.05 M35 0.96 ± 0.03 5.14 ± 0.31 , 5.95 ± 0.54 0.10 ± 0.04 , 0.10 ± 0.05 0.09 ± 0.03 , 0.10 ± 0.04 NGC 2516 0.99 ± 0.03 5.47 ± 0.59 , 6.39 ± 1.64 0.13 ± 0.06 , 0.14 ± 0.11 0.12 ± 0.06 , 0.14 ± 0.11 Weighted mean 0.73 ± 0.01 2.65 ± 0.02 , 2.61 ± 0.03 0.16 ± 0.01 , 0.14 ± 0.01 0.12 ± 0.01 , 0.08 ± 0.01

Note. — Initial mass, ﬁnal mass, and fractional core growth of each cluster. (1): Cluster. (2): Remnant mass in M⊙. (3): Progenitor mass in M⊙ (OS, nOS). (4): Fraction of remnant mass added to the core in the TP-AGB phase (OS, nOS). (5): Core mass added in the TP-AGB phase in M⊙ (OS, nOS).

Table 2.2. Fractional Core Growth of each cluster. 9 Cluster Lmin (10 L⊙yr) fE,∆Mc fE,∆Mc FTP−AGB,He ( M⊙) (1) (2: [Fe/H]=⊙) (3: PAD) (4: BAS) (5: PAD)

NGC 6819 2.75 ± 1.23 0.29 ± 0.14 0.54 ± 0.25 0.07 ± 0.01 NGC 7789 7.96 ± 1.65 0.60 ± 0.14 1.13 ± 0.25 0.05 ± 0.02 Hyades 10.59 ± 1.00 0.52 ± 0.05 1.30 ± 0.12 0.10 ± 0.01 M37 13.87 ± 2.19 0.72 ± 0.11 1.87 ± 0.30 0.06 ± 0.02 Praesepe 11.59 ± 2.57 0.61 ± 0.14 1.65 ± 0.39 0.08 ± 0.03 NGC 3532 10.77 ± 3.41 0.60 ± 0.20 1.86 ± 0.63 0.07 ± 0.04 NGC 1039 6.01 ± 3.91 0.41 ± 0.27 1.36 ± 0.90 0.09 ± 0.04 M35 7.81 ± 2.59 0.85 ± 0.35 1.75 ± 0.58 0.01 ± 0.03

48 NGC 2516 8.10 ± 3.88 1.27 ± 0.95 1.86 ± 0.89 −0.02 ± 0.06 Weighted mean 8.36 ± 0.60 0.54 ± 0.04 1.28 ± 0.09 0.08 ± 0.01

Note. — Corresponding energy output and estimated helium yields of each cluster. Columns 2-4 are only calculated for models that include overshooting (OS). (1): Cluster. (2): Minimum total energy output required 9 by TP-AGB core growth in 10 L⊙yr assuming solar [Fe/H]. (3): Fraction of energy in TP-AGB phase predicted by Bertelli et al. (2008, 2009) (Padova) accounted for by core growth. (4): Fraction of energy in TP-AGB phase predicted by Pietrinferni et al. (2004) (BaSTI) accounted for by core growth.(9): Assuming the total energy output of TP-AGB stars in Bertelli et al. (2008, 2009), estimated stellar yields of helium in M⊙.

Table 2.3. Corresponding Energy Output and Estimated Helium Yields. Chapter 3

Radial Mixing in Galactic Disks: The Effects of Disk Structure and Satellite Bombardment

Under the inﬂuence of gravity, disk galaxies are expected to assemble in an

“inside-out” fashion: stars form first from high-density gas in the central region of the galaxy where the potential is deepest, and subsequently at increasing galacto-centric radii (e.g. Larson 1976). An immediate consequence of this formation scenario is that stars born at the same time and in the same region of a galaxy should have similar chemical compositions. However, observations in our Galaxy suggest that these initial conditions are not maintained. Wielen et al. (1996) argued that the Sun was substantially more metal rich than nearby solar age stars and the local interstellar medium (ISM). A recent recalibration of the Geneva Copenhagen Survey (GCS) using the infrared flux method finds no discrepancy with solar age stars (Casagrande et al. 2011), but even modern studies confirm that the age-metallicity relationships

(AMRs) of ﬁeld and solar neighborhood stars are characterized by higher dispersions than expected (Edvardsson et al. 1993; Nordstr¨om et al. 2004; Holmberg et al.

2007; Casagrande et al. 2011). In addition, simple chemical evolution models that divide the Galaxy into independently evolving concentric annuli predict many more

49 low metallicity G-dwarfs in our region of the disk compared to those observed, a discrepancy known as “the local G-dwarf problem” (van den Bergh 1962; Schmidt

1963). Evidence seemingly in contradiction to standard galaxy chemical evolution theory is not limited to our own Galaxy. Metallicity gradients in disk galaxies are shallower than predicted by classical models (e.g., Magrini et al. 2007). Ferguson &

Johnson (2001) and Ferguson et al. (2007) find unexpectedly old stellar populations on nearly circular orbits in the outskirts of M31 and M33, respectively. The outermost regions of NGC300 and NGC7739 show flattened or positive abundance gradients with radius (Vlajićet al. 2009, 2011). These perplexing observations cannot be readily explained within the confines of classic galaxy formation models.

A natural explanation for the observational challenges above arises if the present day radii of many stars could be signiﬁcantly diﬀerent from their birth radii.

One diﬃculty in establishing radial migration as a common phenomenon, from a dynamical standpoint, lies in ﬁnding a mechanism that can cause a substantial fraction of stars to migrate several kiloparsecs while retaining the observed approximately circular orbits. In a seminal paper, Sellwood & Binney (2002, hereafter SB02) investigated the relationship between changes in stellar angular momentum and disk heating. They found that radial migration is a ubiquitous process in spiral galaxies; stars naturally migrate (change angular momentum) as they resonantly interact with transient spiral waves. Stars in corotational resonance

(CR) with said waves are scattered without heating the disk and maintain their

50 nearly circular orbits (unlike Lindblad resonance (LR) scattering). In the present paper, we examine radial migration in simulations of disk galaxies. Our experiments include galactic disks evolved both in isolation and under the action of infalling satellites of the type expected in the currently favored cold dark matter (CDM) paradigm of hierarchical structure formation (e.g., Peebles 1982; Blumenthal et al.

1984). The latter set of experiments were presented in the studies of Kazantzidis et al. (2008, hereafter K08) and Kazantzidis et al. (2009) and were utilized to investigate the generic dynamical and morphological signatures of galactic disks subject to bombardment by CDM substructure.

Inspired by SB02, several groups have recently investigated the potential role of radial mixing in the chemical and dynamical evolution of disk galaxies. Sch¨onrich

& Binney (2009a) presented the first chemical evolution model to incorporate radial migration. The rate at which stars migrate via the SB02 mechanism is left as a free parameter constrained by the metallicity distribution function (MDF) of solar neighborhood stars in the GCS (Nordström et al. 2004). Their model successfully reproduced, within systematic uncertainties, the observed age-metallicity distribution of stars in the GCS (Holmberg et al. 2007) and the observed correlation between tangential velocity and abundance pattern described by Haywood (2008). However, there is partial degeneracy between the magnitude of radial migration and other parameters in the model such as star-formation rates and gas inflow characteristics.

51 Furthermore, it is unclear whether the level of migration required to ﬁt the data is consistent with theoretical expectations .

Numerical simulations have conﬁrmed the occurrence of radial migration under a variety of conditions. Roˇskar et al. (2008b,a) studied the migration of stars in a simulation of an isolated Milky Way (MW)-sized stellar disk formed from the

12 cooling of a pressure-supported gas cloud in a 10 M⊙ dark matter halo. In their simulations, some older stars radially migrated to the outskirts of the disk while maintaining nearly circular orbits, forming a population akin to that observed in

M31 and M33 (Ferguson & Johnson 2001; Ferguson et al. 2007). Roˇskar et al.

(2008a) found that ∼ 50% of all stars in the solar neighborhood were not born in situ; this is a natural explanation for the observed dispersion in the AMR and solar neighborhood metallicity distribution function (MDF).

More recently, Quillen et al. (2009) investigated radial migration in a stellar

9 disk perturbed by a low-mass (∼ 5 × 10 M⊙) orbiting satellite. Their numerical simulations integrated test particle orbits in a static galactic potential and highlighted the fact that mergers and perturbations from satellite galaxies and subhalos can induce stellar radial mixing. Although informative, test particle simulations in a static isothermal potential will not capture all the relevant physics of the process of stellar radial migration in disk galaxies; the interactions between the gravitational perturbations and the self-gravity of the disk are essential to a detailed analysis of the phenomenon. In our paper, we expand on the analysis of Quillen et al. (2009) by

52 investigating radial migration using fully self-consistent numerical simulations both with and without satellite bombardment.

There are now several established phenomena that can cause a star to populate a region of the disk diﬀerent from its birth radii. Stars on elliptical orbits maintain their guiding center and angular momentum (modulo asymmetries in the potential) but can be found over the range in galacto-centric radius deﬁned by their pericenter and apocenter. Changing a star’s angular momentum, and hence its guiding center, requires direct scattering or a resonant interaction with transient patterns in the disk. The local encounters of stars with molecular clouds (e.g. Spitzer &

Schwarzschild 1953) or Lindblad resonance (LR) scattering between stars and spiral waves (e.g. SB02) both change stellar guiding centers (albeit to a relatively small degree) and increase the random motions of stars over time, “blurring” the disk.

Stars scattered at CR with spiral waves can change their guiding centers by several kiloparsecs without increasing the amplitude of their radial motion. For any single spiral wave, SB02 predict that stars are scattered on each side of the CR, “churning” the contents of the disk1. Stars may undergo several encounters with transient spiral waves throughout their lifetimes. While SB02 investigate all resonant interactions between spiral waves and stars, we will refer to this special case of CR as the “SB02

1Blurring and churning are the terms proposed by Sch¨onrich & Binney (2009a) to describe these distinct aspects of radial migration.

53 mechanism”2. As SB02 note, migration due to spiral waves can be described by blurring and churning regardless of how the waves arise (satellites, e.g., could induce spiral structure that would lead to migration described by SB02). Simulations have shown that other transient wave patterns internal to a galaxy, such as those resulting from bar propagation, can produce resonance overlap with existing spiral patterns and induce radial migration (Minchev & Famaey 2010; Minchev et al. 2011).

Orbiting satellites, external to the galaxy and discussed above, will have a complex interaction with the disk as they provide a means of direct scattering over a large area and also induce spiral modes in the disk. In this work, we aim to characterize radial migration induced by satellite bombardment and compare its eﬀects on the stellar disk with those observed in secularly evolved galaxies.

Our investigation complements earlier and ongoing radial migration studies.

We perform a simulation campaign, including numerical experiments of isolated disk galaxies with different scale heights and gas fractions, which in turn lead to different levels of spiral structure. For the first time, we examine the effect of satellite bombardment on radial migration utilizing simulations where galactic disks are subjected to a cosmologically motivated satellite accretion history. Via a comparative approach, we determine how the magnitude and efficiency of radial migration depend on input physics, establish correlations between orbital parameters and migration,

2In the recent radial migration studies of secularly evolved simulations, signiﬁcant migration is almost always attributed to CR scattering.

54 and present evidence that each of the three migration mechanisms is distinct in the examined parameter space. These characteristics lead to possible observational signatures that may constrain the relative importance of each migration mechanism in the Milky Way.

3.1. Methods

3.1.1. Isolated Disk Models

We employ the method of Widrow & Dubinski (2005) to construct numerical realizations of self-consistent, multicomponent disk galaxies. These galaxy models consist of an exponential stellar disk, a Hernquist bulge (Hernquist 1990), and a

Navarro et al. (1996, hereafter NFW) dark matter halo. They are characterized by 15 free parameters that may be tuned to ﬁt a wide range of observational data for actual galaxies including the MW and M31. The Widrow & Dubinski (2005) models are derived from three-integral, composite distribution functions and thus represent self-consistent equilibrium solutions to the coupled Poisson and collisionless

Boltzmann equations. Owing to their self-consistency, these galaxy models are ideally suited for investigating the complex dynamics involved in the process of radial mixing. The Widrow & Dubinski (2005) method has been recently used in a variety of numerical studies associated with instabilities in disk galaxies, including the dynamics of warps and bars (Dubinski & Chakrabarty 2009; Dubinski et al.

55 2009), the gravitational interaction between galactic disks and infalling satellites

(Gauthier et al. 2006; K08; Purcell et al. 2009; Kazantzidis et al. 2009), and the transformation of disky dwarfs to dwarf spheroidal galaxies under the action of tidal forces from a massive host (Kazantzidis et al. 2011). We refer the reader to

Widrow & Dubinski (2005) for an overview of all relevant parameters and a detailed description of this technique.

For the majority of numerical experiments in the present study, we employ model “MWb” in Widrow & Dubinski (2005), which satisﬁes a broad range of observational constraints on the MW galaxy. Speciﬁcally, the stellar disk has a

10 2 mass of Mdisk = 3.53 × 10 M⊙, a radial scale length of Rd = 2.82 kpc, and a sech scale height of zd = 400 pc. We note that the adopted value for the scale height is consistent with that inferred for the old, thin stellar disk of the MW (e.g., Kent et al.

1991; Juri´cet al. 2008). The equivalent exponential scale height is approximately

200 pc, but the sech2 vertical distribution is more accurate. The bulge has a mass

10 and a scale radius of Mb = 1.18 × 10 M⊙ and ab = 0.88 kpc, respectively. The

NFW dark matter halo has a tidal radius of Rh = 244.5 kpc to keep the total mass

11 ﬁnite (K08), a mass of Mh = 7.35 × 10 M⊙, and a scale radius of rh = 8.82 kpc.

The total circular velocity of the galaxy model at the solar radius, R⊙ ≃ 8 kpc, is

−1 Vc(R⊙) = 234.1 km s , and the Toomre disk stability parameter is equal to Q = 2.2 at R = 2.5Rd. We note that direct numerical simulations of the evolution of model

MWb in isolation conﬁrm its stability against bar formation for 10 Gyr.

56 We wish to address the dependence of radial mixing in isolated disk galaxies upon initial disk thickness and the presence of gas in the galactic disk. Because of their smaller “eﬀective” Toomre Q stability parameter (e.g., Romeo & Wiegert

2011), thinner disks are more prone to gravitational instabilities and thus yield stronger and better deﬁned spiral structure. Therefore, we might expect them to cause enhanced radial mixing compared to their thicker counterparts. Similarly, because radiative cooling in the gas damps its random velocities, the presence of gas is associated with long-lived spiral structure in disks (e.g., Carlberg & Freedman

1985), which may act to increase the amount of radial mixing. Correspondingly, we initialize several additional disk galaxy models.

The first modified galaxy model was constructed with the same parameter set as MWb but with a scale height 2 times smaller (zd = 200 pc). Except for disk thickness and vertical velocity dispersion, all of the other gross properties of the three galactic components of this model are within a few percent of the corresponding ones for MWb. Not surprisingly, being more prone to gravitational instabilities, this model develops a bar inside ∼ 5 kpc at time t ∼ 1.2 Gyr. The second set of modified galaxy models are the same as MWb except for the fact that a fraction fg of the mass of the initial stellar disk is replaced by gas. Thus, the resulting gaseous component is constructed with the same initial density distribution as the stellar disk. In the present study, we employ two values for the gas fraction, fg = 0.2 and fg = 0.4, and the adopted methodology will be described in detail in a forthcoming

57 paper (Kazantzidis et al. 2011, in preparation). Brieﬂy, the construction of the gas disk is done by assuming that its vertical structure is governed by hydrostatic equilibrium and by computing the potential and the resulting force ﬁeld for the radially varying density structure of the gas component. By specifying the polytropic index and the mean molecular weight of the gas and using a tree structure for the potential calculation, the gas azimuthal streaming velocity is determined by the balance between gravity and centrifugal and pressure support.

3.1.2. Perturbed Disk Models

In the standard CDM paradigm of hierarchical structure formation, galaxies grow via continuous accretion of smaller satellite systems. To investigate the effect of accretion events on radial mixing in galactic disks, we have also analyzed N-body simulations of the gravitational interaction between a population of dark matter subhalos and disk galaxies. These simulations were first presented by K08, and we summarize them briefly here.

K08 performed high-resolution, collisionless N-body simulations to study the response of the galactic model MWb discussed above to a typical ΛCDM- motivated satellite accretion history. The speciﬁc merger history was derived from cosmological simulations of the formation of Galaxy-sized CDM halos and involved six dark matter satellites with masses, orbital pericenters, and tidal radii

58 9 10 of 7.4 × 10 ∼< Msat/M⊙ ∼< 2 × 10 , rperi ∼< 20 kpc, and rtid ∼> 20 kpc, respectively, crossing the disk in the past ∼ 8 Gyr. As a comparison, the Large Magellanic Cloud

10 has an estimated total present mass of ∼ 2 × 10 M⊙ (e.g., Schommer et al. 1992;

Mastropietro et al. 2005), so the masses of the perturbing dark matter satellites correspond to the upper limit of the mass function of observed satellites in the Local

Group. K08 modeled satellite impacts as a sequence of encounters. Speciﬁcally, starting with the ﬁrst subhalo, they included subsequent systems at the epoch when they were recorded in the cosmological simulation. We note that although K08 followed the accretion histories of host halos since z ∼ 1, when time intervals between subhalo passages were larger than the timescale needed for the disk to relax after the previous interaction, the next satellite was introduced immediately after the disk had settled from the previous encounter. Each satellite was removed from the simulation once it reached its maximum distance from the disk after crossing. This approach was dictated by the desire to minimize computational time and resulted in a total simulation time of ∼ 2.5 Gyr instead of ∼ 8 Gyr that would formally correspond to z = 1.

K08 and Kazantzidis et al. (2009) found that these accretion events severely perturbed the galactic disk of model MWb without destroying it and produced a wealth of distinctive morphological and dynamical signatures on its structure and kinematics. In this paper, we will investigate the magnitude of radial mixing induced in disk model MWb by these accretion events. As in the case of isolated disk models,

59 it is worthwhile to examine the dependence of radial mixing upon disk thickness.

For this purpose, we employed the thinner galaxy model with a scale height of zd = 200 pc described above and repeated the satellite-disk encounter simulations of

K08.

3.1.3. Numerical Parameters

All collisionless numerical simulations discussed in this paper were carried out with the multi-stepping, parallel, tree N-body code PKDGRAV (Stadel 2001). The hydrodynamical simulations were performed with the parallel TreeSPH N-body code

GASOLINE (Wadsley et al. 2004). In the gasdynamical experiments, we include atomic cooling for a primordial mixture of hydrogen and helium, star formation and (thermal) feedback from supernovae. Our star formation recipe follows that of

Stinson et al. (2006), which is based on that of Katz (1992). Gas particles in cold and dense regions which are simultaneously parts of converging ﬂows spawn star particles with a given eﬃciency c⋆ at a rate proportional to the local dynamical time.

Feedback from supernovae is treated using the blast-wave model described in Stinson et al. (2006), which is based on the analytic treatment of blastwaves described in

McKee & Ostriker (1977). In our particular applications, gas particles are eligible to form stars if their density exceeds 0.1 atoms/cm3 and their temperature drops

4 below Tmax = 1.5 × 10 K, and the energy deposited by each Type-II supernova into the surrounding gas is 4 × 1050 erg. We note that this choice of parameters

60 and numerical techniques is shown to produce realistic disk galaxies in cosmological simulations (Governato et al. 2007). Lastly, in an attempt to investigate the effect of star formation efficiency on radial mixing, for each of the gas fractions fg above, we adopted two different values for c⋆, namely c⋆ = 0.05 (which reproduces the slope and normalization of the observed Schmidt law in isolated disk galaxies) and a lower value of c⋆ = 0.01.

6 All N-body realizations of the disk galaxy models contain Nd = 10 particles

4 5 4 of md = 3.53 × 10 M⊙ in the disk; Nb = 5 × 10 , mb = 2.36 × 10 M⊙ in the bulge;

6 5 and Nh = 2 × 10 , md = 3.675 × 10 M⊙ in the dark matter halo. The gravitational softening lengths for the three components were set to ǫd = 50 pc, ǫb = 50 pc, and

ǫh = 100 pc, respectively. These are “equivalent Plummer” softenings; the force softening is a cubic spline. In the hydrodynamical simulations of isolated galaxies,

5 5 gaseous disks were represented with Ng = 2 × 10 and Ng = 4 × 10 for gas fractions of fg = 0.2 and fg = 0.4, respectively. In these cases, the gravitational softening length for the gas particles was set to ǫg = 50 pc. All numerical simulations of isolated and perturbed disks were analyzed at 2.5 Gyr. This time-scale corresponds to approximately 17.5 orbital times at the disk half-mass radius. For evaluating the impact of accretion events, it is fairest to compare isolated and perturbed disks after the same amount of integration, but we may underestimate the total amount of mixing in the isolated (and, to a lesser extent the perturbed) models. In the future, we plan to evolve selected simulations for the full 8 Gyr interval since z = 1,

61 but such simulations will require more than three times the computational resources used here.

Exchange of angular and linear momentum between the infalling satellites and the disk tilt the disk plane and cause the disk center of mass to drift from its initial position at the origin of the coordinate frame (K08; Kazantzidis et al. 2009).

Therefore, we calculate ∆r in the satellite-disk encounter experiments after removing the global displacement and tilt of the disk galaxy by determining the principal axes of the total disk inertia tensor and rotating our original coordinate system such that it is aligned with this tensor.

3.2. Results

Figure 3.1 presents surface density maps of the stellar distributions of our four collisionless galactic disks with different initial scale-heights zd. Each simulation exhibits a distinctive combination of morphological features that could affect radial migration. The initial smooth disks are unstable to spiral instabilities arising from the swing amplification of particle shot noise, an effect that leads to emerging spiral structure as seen in the final isolated disks (e.g. Julian & Toomre 1966). Owing to its smaller “effective” Toomre Q stability parameter (e.g., Romeo & Wiegert 2011), the zd = 200 pc disk (hereafter, disks with this initial scale height will be referred to as “thin”) develops prominent spiral structure and a strong bar. Conversely, the

62 zd = 400 pc disk (hereafter, disks with zd = 400 pc will be referred to as “thick”) has relatively little spiral structure and does not form a bar in isolation. We emphasize that the “thick”, zd = 400 pc disk is the one in best agreement with the old, thin disk of the MW, while zd = 200 pc is too thin (see Section 3.1.1).

The radial and vertical morphology of the perturbed disks is distinct from their isolated counterparts. Both perturbed disks develop bars and are characterized by prominent ﬂaring and much larger scale heights than those of the isolated disks

(K08; Kazantzidis et al. 2009). There is evidence that some spiral structure evident in the isolated disks has been washed out in the simulations with substructure bombardment: within 10 kpc of the galactic center, local enhancements of the stellar surface density evident in the isolated disks are substantially more diffuse after the action of the infalling satellites. Throughout this section, we will investigate how the growth and dissipation of spiral structure affect radial migration. While we have investigated similar maps in the four hydrodynamical simulations, we do not show them here as they are qualitatively similar to their collisionless counterparts. As expected, however, the magnitude of spiral structure increases as the gas fraction rises. We note that the differential effect of gas on the strength of spiral structure we find here is smaller than that reported in previous numerical investigations of isolated disk galaxies (e.g., Barnes & Hernquist 1996). This is mainly due to the effect of stellar feedback, which causes the ISM in our simulations to become turbulent and multi-phase (see also Stinson et al. 2006).

63 3.2.1. Radial Migration

We ﬁrst investigate where particles move throughout each simulation. Figure

3.2 shows the distribution of ∆r= rf − ri in the six isolated disk simulations, where rf and ri refer respectively to each particle’s ﬁnal and initial projected distance from the galactic center. Each panel contains the ∆r distribution, the median |∆r|

th (denoted ∆rmed), and 80 percentile |∆r| (denoted ∆r80) for two of the six isolated galaxies in our simulation suite.

The isolated, collisionless simulations (left panel) clearly demonstrate that disk scale height and gas content aﬀect the radial migration process. The thin disk’s

∆r distribution (black line) is considerably broader than the thick disk’s (gray hatch). Both ∆rmed= 0.91 kpc and ∆r80= 2.03 kpc of particles in the thin disk are approximately double those found in the thick disk.

The remaining panels of Figure 3.2 compare the ∆r distributions of the hydrodynamical simulations of our sample. We examine four isolated disks with two initial gas fractions: fg = 20% (middle panel) and fg = 40% (right panel). The two histograms in each panel represent star formation eﬃciencies (c⋆) of 1% (solid line) or 5% (gray, hatched histogram). Gas has a strong impact on particle radial migration: ∆rmed increases from 0.47 kpc in the collisionless case (all hydrodynamical simulations are of “thick” disks) to ∼ 0.70 kpc when fg = 20% and ∼ 0.80 kpc when fg = 40%. The extent of radial migration is less dependent on star formation

64 eﬃciency; there are only minor diﬀerences amongst each pair of histograms in the middle and right panels. However, in both panels, ∆rmed and ∆r80 are highest when c⋆= 1%. In these four hydrodynamical simulations, the time-averaged gas content of the disk is directly correlated with the percentage of particles that migrate and their typical displacement relative to their formation radius.

Decreasing the scale height of the stellar disk, increasing the fraction of the initial disk mass in gas, and lowering the star-formation eﬃciency all increase the midplane density of the disk. This enhanced density increases the coherence and self-gravity of spiral structure, supporting it against the dissipative nature of individual particles’ random motions (as described by, e.g., Toomre 1977). The correlation of spiral structure strength with the fraction of particles that migrate away from their birth radii and the median migratory distance traversed suggests that the SB02 mechanism, intimately linked with spiral waves, plays a major role in the migration of particles in the isolated systems. The symmetric shapes (to within

0.4% about 0 kpc) of the ∆r distributions are also consistent with migration via the SB02 mechanism. Resonances between the bar and spiral structure could also inﬂuence the timescale for migration in the thin disk models (Minchev & Famaey

2010), but the zd = 400 pc models do not develop bars, regardless of whether they include gas.

Figure 3.3 compares the ∆r distributions of the isolated collisionless disks to those of their perturbed counterparts. The ∆rmed and ∆r80 of each distribution

65 are labeled and color-coded to match the corresponding histogram. In each panel, the most sharply peaked histogram (red) shows the expected ∆r distribution from epicyclic motion alone, which we compute given the particle’s initial phase space coordinates. We model the epicycle as a simple harmonic oscillator about the particle’s guiding center radius (Rg) with an amplitude set by its initial radial energy

(Er) (Chapter 3, Binney & Tremaine 2008). The energy associated with the circular component of the particle’s orbit (Ecirc) is set by Rg, which is the radius of a circular orbit with angular momentum equal to the midplane component of the particle’s angular momentum. The radial energy of the orbit is Er = Etot − Ez − Ecirc where

Ez is the vertical energy component and Etot is the total energy . As Er sets the amplitude of the epicycle, we can solve for the position of the particle as function of time with respect to its guiding center radius. We choose two random phases of the epicycle oscillation, designating the ﬁrst as ri and the second as rf , and compute

∆r. The resulting “baseline” distribution is shown in red and is a result of observing the two random phases (akin to our “initial” and “ﬁnal” snapshots) of the initially elliptical orbits. It does not take into account the potential heating of these initial orbits and the subsequent increase in amplitude of their radial motion (blurring).

The shape of the distribution does not change if we change our random seed or average a larger number of phases to obtain rf and ri. For the isolated thick disk

(right panel), the ∆r distribution from full dynamical evolution is only marginally broader than the expected baseline distribution, suggesting that these stars are

66 not heated with time and continue to follow their initial orbits. However, satellite bombardment broadens the ∆r distribution dramatically, nearly doubling both

∆rmed and ∆r80, necessitating guiding center modiﬁcation of a substantial fraction of the orbits.

For the thin disk, isolated evolution produces a much broader ∆r distribution than the baseline distribution, demonstrating the impact of the bar3 and spiral structure in this more unstable disk. In this case, satellite bombardment only slightly increases the width of the ∆r distribution, despite the strong impact on disk structure that is visually evident in Figure 3.1. We suspect that the small net difference between these two histograms reflects a cancellation between two competing effects of satellite bombardment. Accretion events heat the stellar disk and thereby suppress the development of spiral structure, thus reducing the level of

SB02 migration. However, the accretion events also induce radial mixing directly via their dynamical perturbations. The ∆r histograms of both perturbed disks, while still approximately symmetric, are noticeably more asymmetric than those of the isolated disks, with 2% (3%) more particles moving outwards than inwards in the thin (thick) cases, compared to ≤ 0.4% for the isolated models.

3Bars can increase the eccentricity of particles, contributing to the “blurring” of the disk, and their potential resonance overlap with spiral structure can induce migration (see Minchev & Famaey

2010).

67 Figure 3.4 presents clear evidence that satellite-induced migration is distinct from that in the isolated experiments, acting in different environments from either epicyclic blurring or spiral-induced churning. In each panel, solid, dashed, and dotted lines show the fraction of particles at each birth radius Rform that migrate by more than |∆r| = 1.0, 2.0, or 3.0 kpc, respectively. Shaded regions show the scaled radial surface density profile of the initial disk. For both isolated disks, the migration probability decreases outwards and approximately traces the surface density profile.

This behavior is similar to that assumed in the chemical evolution models of

Schönrich & Binney (2009a), who parametrized the probability that a star migrates as proportional to the mass surrounding it. However, for both perturbed disks the probability of migration is flat or increasing outwards beyond Rform= 10 kpc, where the disk potential weakens, and it definitely does not trace the mass distribution. To better understand this difference between satellite- and spiral-induced radial mixing, we now investigate how changes in angular momentum and energy are correlated with change in radius.

3.2.2. Orbital Characteristics

Total energy and angular momentum are the classic two-dimensional integrals of motion (Binney & Tremaine 2008). Figure 3.5 shows Lindblad diagrams for the two initial and four final states of the collisionless simulations, plotting particle specific angular momentum projected along the axis of symmetry (Jz) versus specific

68 binding energy (E). Particles are grouped into 50 distinct linear bins of E. We calculate the median Jz in each bin. Red, yellow, and orange regions connect the

th th th 68 , 95 , and 99 percentile Jz intervals (centered on each median Jz), respectively, across all energy bins. The 1% most discrepant particles in Jz for a given E are plotted as individual points. Using each disk’s rotation curve, we plot Jz and E for circular orbits in the midplane of the disk (green line). Here, Jz = vc(r) × r , where

1 2 vc(r) is the circular velocity at radius r and E = 2 vc (r) + Φ(r), where Φ(r) is the potential in the disk midplane at radius r. Particles on a circular orbit have the maximum Jz allowed given their energy.

By construction, particles are initially (left column of Figure 3.5) on nearly circular orbits with a small radial velocity component. The red region, falling close

to the circular orbit curve in both initial disks, confirms that σvr is small. Particles significantly displaced from the green curve in the initial states either have extreme vr or are on highly inclined orbits (Jz is projected along the z axis). Due to this inclination effect, the initial thick disk has a lower median Jz as a function of E than the thin disk despite having the same vr distribution.

In the isolated thin disk (upper middle panel of Figure 3.5), the range of Jz grows relative to the initial values at every energy E. The change is largest at lower energies. Recall from Figure 3.1 that the isolated thin disk develops a bar, which is composed of particles on radial orbits with relatively low Jz, in the most bound region of the disk. Thus, this relatively large change towards less circular orbits at

69 low energies can be associated with bar formation. In contrast, the distribution of

Jz in the isolated thick disk is basically equivalent to its initial state. In the isolated thick disk there is no bar, relatively little spiral structure develops, and we ﬁnd the least radial migration among the four simulations. Ignoring the region of the

Lindblad diagram inﬂuenced by members of the bar, the isolated disk diagrams suggest that the extensive radial migration seen in the thin disk and associated with guiding center modiﬁcation decreases the angular momentum of particles.

The two perturbed disks (right column of Figure 3.5) show larger changes in their Lindblad diagrams. Bar formation in both simulations can explain the signiﬁcantly lower median Jz at lower energies. At higher energies, corresponding to less bound particles further out in the disk, both disks show a much larger dispersion in Jz than their isolated counterparts. Additionally, there is substructure in the

Lindblad diagrams (groups of relatively low Jz in a narrow range of energy) not seen in the isolated disks. Satellite bombardment in the perturbed disk simulations has a qualitatively discernible impact on the angular momentum distribution of the disk.

To quantify changes in Jz, we introduce the circularity (ǫ) quantity (e.g.,

Abadi et al. 2003). For a particle of energy Ei and angular momentum Ji in the z-direction, we deﬁne circularity as ǫ = Ji/Jcirc(Ei), the ratio of Ji to the speciﬁc angular momentum the particle would have if it were on a circular orbit with energy

Ei (obtained using the circular orbit curve). Circular orbits have ǫ = 1 and radial orbits have ǫ = 0. Negative circularities correspond to retrograde orbits. We note

70 that this method is formally different from that of Abadi et al. (2003). The circular orbit curve may not represent the highest Jz for a given E due to shot noise in the potential or gravitational potential asymmetries. Strictly defining the maximum Jz in each energy bin (as in Abadi et al. (2003)) results in the same qualitative trends discussed later. Our method benefits from the use of a mathematically constructed and reproducible rotation curve and systematically decreases ∆ǫ= ǫf − ǫi at the 0.01 level. The inset of each panel in Figure 3.5 shows the circularity distribution of each simulation.

Figure 3.6 plots the relation between circularity and eccentricity for orbits near the solar radius (8.0 kpc) in the zd = 400 pc disk. Circularity is related to eccentricity via the particle’s radial and vertical energies as well as its orbital inclination (since Jz is a projected quantity). Orbits in the midplane of the disk have the highest circularity for a given eccentricity. Shaded regions in Figure 3.6 show the 95th and 99th percentile circularity as a function of eccentricity at the solar radius. For reference in interpreting Figure 3.5 and subsequent ﬁgures, it is worth noting that changes of ∼ 0.02 in ǫ typically correspond to quite noticeable changes in eccentricity. The boundaries of these regions are not smooth due to binning and small number statistics for initially high eccentricity particles.

Returning to Figure 3.5, we see that the two isolated disks show markedly diﬀerent evolution of their circularity distributions. In the isolated thin disk, the fraction of stars with ǫ ∼ 1 (rightmost bin) drops from 0.38 to 0.26, and the median

71 ǫ drops from 0.980 to 0.955. Particles in the bar predominantly populate the newly formed low circularity tail. In the thick isolated disk, on the other hand, the ǫ distribution is nearly identical to the initial disk’s, with median ǫ dropping only

0.002. This lack of evolution in the circularity distribution is fully consistent with CR scattering. As described by SB02, individual stars may exchange angular momentum across CRs (churning) while the overall distribution would remain unchanged.

However, our results from Section 3.2.1 show that the Delta R distribution of this model, which reflects individual particles and their orbits, is nearly identical to that expected from observing the particles’ initially elliptical orbits. The circularity distribution of the isolated, thick disk combined with its Delta R distribution imply that, on average, individual particle guiding centers are not significantly modified.

The circularity distributions of the perturbed disks are demonstrably altered from their isolated counterparts but are similar to one another. The most common circularities are now in the range 0.96 ≤ ǫ ≤ 0.98 in both disks, with a decrease for

ǫ> 0.98. The thin disk starts with more ǫ ∼ 1 particles because of its smaller orbital inclinations, and its circularity distribution evolves more strongly, with median

ǫ dropping from 0.980 to 0.936 vs. 0.963 to 0.929 for the perturbed thick disk.

Notably, the perturbed thin and thick disks evolve to similar ∆r (Figure 3.3) and ǫ distributions despite starting with diﬀerent scale heights.

Figure 3.7 tracks the changes of selected individual particles in the (E, Jz) space of the Lindblad diagram. While E and Jz are not individually conserved in

72 the presence of a non-axisymmetric perturbation, the Jacobi invariant I = E − ΩbJz is, where Ωb is the pattern speed of the perturbation, assumed to be static and small

(SB02; Sellwood 2010). If ∆I = 0, then ∆Jz/∆E ≈ Ωb. The SB02 mechanism operates at the corotation resonance of the star/particle and the spiral wave, requiring that Ωb = Ωrot. Thus, in the galaxy, particles should move parallel to the line that is tangent to the circular orbit curve at their binding energy prior to scattering. In other words, the SB02 mechanism requires that changes in energy be accompanied by changes in angular momentum that preserve the orbital shape, modulo diﬀerences in the slope of the circular orbit curve over the range [Ei,Ef ] of a given particle.

Figure 3.7 zooms in on the area of the Lindblad diagram designated by the two small boxes drawn on the initial state diagrams in Figure 3.5. We randomly select ten particles within the same initial E and Jz range (ﬁlled, dark squares in

5 Figure 3.7) and plot their ﬁnal E and Jz. We require Ef − Ei = ∆E < −0.05 × 10 km2 s−2 to ensure that the individual tracks are visible. In the isolated thin disk

(top left), most particles move parallel to the circular orbit curve tangent (dashed line); this relationship between ∆Jz and ∆E is consistent with corotational resonant scattering (equation 2, SB02). When particles lose energy in the isolated thick disk

(bottom left), their Jz typically remains closer to the circular orbit curve than in the isolated thin disk. The changes in energy and angular momentum are relatively small, indicating that the guiding centers are modiﬁed to a modest extent (Binney &

73 Tremaine 2008). Note that these particles are in the top 1% of |∆Jz|; most particles in this particular experiment have ∆Jz ≈ 0. The randomly selected particles in both perturbed disks have slopes ∆Jz/∆E that are steeper than the slope of the tangent to the circular orbit curve at corotation. This distinct coupling of ∆E and

∆Jz, combined with our results concerning the migration probability as a function of Rform (Section 3.2.1), oﬀer compelling evidence that migration in the perturbed disks is driven, at least in part, by a mechanism that does not operate in the isolated systems. Figure 3.7 shows that the orbital characteristics of many particles in the perturbed disks are modiﬁed in a fashion inconsistent with either epicyclic motion or a single spiral wave scattering event.

We now examine the correlation between migration and changes in orbital properties. Since the metallicity of star-forming gas increases with time and decreases with radius, any such correlations also imply observable correlations between the present orbital parameters and metallicities of stars as a function of age and Galacto-centric radius. Figure 3.8 shows the median change in circularity,

∆ǫ = ǫf − ǫi, as a function of radial migration distance ∆r in the four collisionless simulations (open squares). In the isolated thick disk; changes in circularity are tiny

(median |∆ǫ| ≤ 0.005) at any ∆r; Figure 3.7 shows that particles in this simulation stay close to the circular velocity curve even if they change energy. In the other three simulations, particles with negative ∆r show substantial drops in circularity (typical median ∆ǫ< −0.06), which are almost certainly associated with bar formation. The

74 bars that form in these three simulations increase the orbital eccentricities of their members. To focus on behavior in the disk proper, the ﬁlled squares in each panel show the median ∆ǫ vs. ∆r for those particles that start and end the simulation at r ≥ 4 kpc, beyond the extent of the bar. Error bars mark the inter-quartile range

(25th to 75th percentile) at each ∆r.

In the isolated thin disk, particles that migrate inwards (∆r< 0) experience a modest decrease in circularity, stronger for more negative ∆r (and ∆E), consistent with the tracks shown in Figure 3.7. Particles that migrate outwards (∆r> 0) increase their energy and typically traverse areas of the Lindblad diagram with relatively little change in the slope of the circular orbit curve. Following the tangent to the circular orbit curve, particles will not signiﬁcantly change their circularity in such a scenario (note the relative lack of low circularity particles at

E > −1.5 km2 s−2 in Figure 3.5). The perturbed disks show a decrease in median circularity at every ∆r, and a much wider inter-quartile range indicating a greater range of orbital inclinations and eccentricities. Circularity drops more strongly for particles that have experienced strong radial migration, either inward or outward.

The minimum in these curves is slightly oﬀset to positive ∆r because dynamical heating slightly puﬀs up the disk radially, decreasing the potential at a given radius, thereby increasing its total energy, moving particles to the right on the Lindblad diagram , and lowering the circularity for particles with ∆r= 0. Figure 3.8 indicates that stars with anomalous chemistry for their age and current position should have

75 preferentially more eccentric orbits. The trend is smaller than the inter-quartile range, but similar in magnitude.

Sch¨onrich & Binney (2009a) and Loebman et al. (2010) discuss the possible role of radial migration in producing thick disks like the ones observed in the Milky

Way (Gilmore et al. 1989) and other edge-on galaxies (Dalcanton & Bernstein 2002).

Figure 3.9 is similar to Figure 3.8, but instead of ∆ǫ it plots the change in vertical energy, quantiﬁed by the maximum distance zmax that a particle can reach from the disk plane. We compute zmax approximately from each particle’s vertical velocity component assuming that the ﬁnal potential of each simulation is static, namely

2 vz zmax= |z| + 4πGΣ(r) where z is the vertical position of the particle at the ﬁnal output, vz is the velocity along the z axis, G is the gravitational constant, and Σ(r) is the surface density of the disk at radius r assuming all the mass of the disk is in the midplane. The change in zmax is ∆zmax= zmax,f − zmax,i where the subscripts i and f refer to the initial and ﬁnal simulation snapshots, respectively.

The two isolated simulations show a shallow linear trend between ∆zmax and

∆r (open squares for all particles; ﬁlled squares for those with ri,rf > 4 kpc). When particles move outwards (inwards) through the disk, they experience a weaker

(stronger) gravitational potential, thus increasing (decreasing) zmax. However, changes in zmax are small, less than 300 pc even when we consider the quartile range at the extremes of ∆r. The perturbed disks, by contrast, show larger changes in the median zmax and a dramatic increase in the inter-quartile range at all ∆r. K08

76 show that the perturbed 400 pc disk develops a two-component vertical structure in quantitative agreement with the observed thin/thick disk structure of the MW.

The perturbed 200 pc disk increases its scale height (to ≈ 500 pc at the solar annulus), but it can still be described by a single component model. This vertical heating by satellite perturbations is evident in Figure 3.9. Particles that have large positive ∆r have the largest increase in zmax, which is plausibly a consequence of moving outwards to regions of lower disk surface density and thus weaker vertical restoring force. For ∆r < 0, the trend of median ∆zmax with ∆r is approximately

ﬂat, suggestion a cancellation between the eﬀects of increased Σ(r) at smaller r and direct satellite-induced heating of those particles with the largest excursions.

Figure 3.9 implies that stars with high metallicity for their age and present location should have preferentially larger zmax, though the scatter is larger than the trend.

3.2.3. Solar Annulus

The solar neighborhood is easier to study than other regions of the Galaxy, since high-precision spectroscopy is easier for brighter stars and parallax and proper motion measurements are more accurate at smaller distances. Some of the most detailed chemo-dynamic surveys, such as the GCS (Nordstr¨om et al. 2004) and the Radial Velocity Experiment (RAVE, Steinmetz 2006) concentrate on the solar neighborhood. In this section, we repeat some of our earlier analysis speciﬁcally for stars that reside in the solar annulus (7 kpc ≤ rf ≤ 9 kpc) at the end of each

77 simulation. This focus on the solar annulus also removes much of the impact of the bars that dominate evolution of the inner disk (r < 3 kpc) in three of our simulations, though some particles from the bar region can migrate as far as the solar radius, and resonances between the bar and spiral structure may increase migration frequency (Minchev & Famaey 2010).

Figure 3.10 shows the radius of formation (Rform) distribution of particles residing in the solar annulus, marked by the vertical dashed lines. Only the isolated thick disk simulation predicts a ﬁnal solar annulus dominated by stars born in the solar annulus, with tails extending 1–2 kpc on either side. The broad ∆r distributions of the other three simulations show that their stars migrate to the solar annulus from a wide range of formation radii. The global ∆r (Figure 3.3) and solar annulus Rform distributions of the isolated and perturbed thin disks are remarkably similar, despite the diﬀerences in migration mechanisms discussed in Section 3.2.2.

In the isolated thin disk, 32% of solar annulus particles originated at Rform≤ 6 kpc and 4% at Rform≥ 10 kpc. Corresponding numbers for the perturbed thin disk are

36% and 3%. The Rform distribution of the perturbed thick disk is slightly narrower, but it still broad with respect to the isolated thick disk.

Figure 3.11 shows the correlations between ∆ǫ and ∆r for particles that end in the solar annulus. Consistent with results for the full disk (Figure 3.8), the solar annulus particles in the isolated thick disk show no signiﬁcant change in median ǫ regardless of ∆r. In the isolated thin disk, the range of ∆ǫ is much larger, with a

78 modest decrease in median ǫ. The median ∆ǫ drops at large positive ∆r because particles move to less inclined orbits as they migrate outwards and disk heating has slightly modified the galaxy’s circular velocity curve, allowing outward moving particles to potentially increase circularity. In the perturbed thick disk, there is a strong and nearly linear trend between ∆ǫ and ∆r. Particles that migrated to the solar annulus from the outer disk have experienced substantial drops in circularity, while particles migrating from the inner disk show only modest decreases. The range of ∆ǫ is large at all ∆r. Results for the perturbed thin disk are intermediate between those of the isolated thin and perturbed thick disks: quasi-linear trends at large |∆r| but a flat plateau at intermediate |∆r|. This behavior is plausibly a consequence of two different mechanisms contributing to migration, with spiral-induced mixing dominating at intermediate ∆r and satellite-induced mixing dominating at the extremes.

Figure 3.12 shows the solar annulus correlations of ∆zmax with ∆r, analogous to Figure 3.8 for the full disk. For the two isolated disks there is a clear linear trend of median ∆zmax with ∆r as expected from the arguments in Section 3.2.2: particles that migrate outward move to a region of lower disk surface density, so if their vertical velocities are not systematically changed by the radial migration they will attain higher zmax. Both perturbed disks show signs of the strong vertical heating induced by satellite accretion. The median ∆zmax is ≥ 0.2 kpc almost independent of ∆r, and the range of ∆zmax is much larger than in the isolated disks. Since

79 inwardly migrating particles experience a higher vertical potential at rf than ri, they must experience more vertical heating than outwardly migrating particles to keep the ∆zmax trend ﬂat.

Returning to Figure 3.10, gray histograms represent the Rform distributions of particles that end the simulations in the solar annulus at high z, 1.0 < |z| < 1.5 kpc.

In the isolated thick disk and both perturbed disks, the Rform distribution of high z particles resembles that of all solar annulus particles. In these three models, therefore, selecting high- z particles does not isolate a population with atypical radial migration. In the isolated thin disk, on the other hand, the Rform distribution of high-z particles is strongly skewed towards low formation radii, with a peak at

Rform= 5 kpc. Here, only initially hot particles that migrate signiﬁcantly outwards via churning or interactions with the bar and experience a weaker potential have enough vertical energy to overcome the local restoring force and reach high z. Only

0.1% of the solar annulus is at high z in the isolated thin disk (truly the tail of the initial vertical velocity dispersion); this number rises to 3.2% and 7.5% in the thin and thick perturbed disks, respectively. The radial migration mechanisms in the perturbed disks ensure that even the high z population of the solar annulus comes from a broad range of Rform. Measurements of the age-metallicity relation for high-z stars could be a valuable diagnostic for distinguishing models of radial migration and vertical heating.

80 3.3. Summary and Discussion

Observations suggest that radial migration plays an important role in the chemical evolution of the MW disk (Wielen et al. 1996; Sch¨onrich & Binney 2009a).

SB02 described a mechanism by which stars at corotational resonance with spiral waves can scatter, producing large changes in guiding center radius while keeping stars on nearly circular orbits. Previous simulations have shown that migration over several kiloparsecs can occur in isolated disks grown by smooth accretion (Roˇskar et al. 2008b) and that encounters with satellites can also induce migration (Quillen et al. 2009). Here we have carried out a systematic investigation of a variety of simulations to characterize the role of stellar disk properties, gas fractions, and satellite perturbations in producing radial migration. Most importantly, our suite of simulations includes experiments with a level of satellite bombardment expected in

ΛCDM models of galaxy formation, which earlier investigations (K08, Kazantzidis et al. 2009) have shown to produce vertical and in-plane structure resembling that seen in the MW.

For disks evolved in isolation, the degree of migration correlates with the degree of disk self-gravity, susceptibility to bar formation, and spiral structure. The collisionless stellar disk with zd = 400 pc, chosen to match that of the thin disk in the

MW, has no bar and minimal spiral structure. It exhibits limited radial migration; the median value of radial change |rf − ri| is ∆rmed= 0.47 kpc, consistent with the

81 level expected from epicyclic motion (Figure 3.3). The collisionless zd = 200 pc disk is much more unstable, develops a bar and spiral structure (Figure 3.1), and has a higher degree of radial migration, with ∆rmed= 0.91 kpc and ∆r80= 2.03 kpc.

The presence of gas is a catalyst for radial migration in the zd = 400 pc case.

Both ∆rmed and ∆r80 increase in models with larger gas fractions or lower star formation efficiency (which consumes gas more slowly). The galactic bar strongly influences, and perhaps dominates, the resultant individual particle dynamics in the inner galaxy. Substantial radial migration, consistent with the SB02 mechanism or being satellite-induced, occurs in the outer portion of the disk in all of our collisionless experiments except for the isolated zd = 400 pc disk. In the zd = 200 pc disk, stars that finish the simulation in the solar annulus (7

∆rmed= 1.36 kpc, ∆r80= 2.68 kpc.

Adding satellite perturbations dramatically increases radial migration in the zd = 400 pc disks, raising ∆rmed from 0.47 kpc to 0.89 kpc globally and from

0.60 kpc to 1.15 kpc in the solar annulus. Perturbations do not change the ∆r distribution so drastically in the zd = 200 pc disks, but there are other indications that the nature of radial migration is diﬀerent. In the isolated galaxies, migrating particles in the outer disk follow tracks in E,Jz space that parallel the tangent to the local circular velocity curve, as expected for the SB02 mechanism (see Section 3.2.2).

However, inwardly migrating particles in the perturbed disks lose more angular momentum for a given change in energy, causing migration associated with heating

82 and thus diﬀerent from the SB02 mechanism in this respect. The relationship between ∆E and ∆Jz found in the perturbed experiments is broadly more consistent with scattering at LR (see SB02), but a detailed decomposition of the modes in the disk and satellites (beyond the scope of this paper) is necessary to conﬁrm

LR scattering on a particle by particle basis. The radial distribution of migrating particles presents an even clearer distinction, one with important implications for chemical evolution models (Figure 3.4). In the isolated disks, the probability that a particle with a formation radius Rform undergoes signiﬁcant migration (|∆r| > 1 kpc) is approximately proportional to the disk surface density at Rform; i.e., migration follows mass. In the perturbed disks, the probability of migration is ﬂat or increasing with radius, so a much larger fraction of migration comes from the tenuous outer disk. Particles in the outer disk are overall more likely to migrate relative to the inner disk as 1) they feel a weaker potential and are thus more susceptible to direct heating by satellites and 2) their circular frequencies make them more likely to resonantly interact with satellites and migrate. Thus, satellite-induced migration patterns are distinct from those produced by purely secular evolution.

Satellite bombardment heats the disk both vertically and radially. Compared to the isolated disks, the perturbed disks experience a greater drop in median circularity

(hence growth of median eccentricity) during their evolution, and individual particles experience a wider range of circularity changes. The circularity change is moderately correlated with radial migration distance and direction, though the trend is smaller

83 than the inter-quartile range at a given ∆r (Figure 3.8). If we restrict our analysis to the solar annulus, the correlation between ∆r and ∆ǫ is more signiﬁcant. In particular, particles in the solar annulus of the zd = 400 pc disk that migrate

2–4 kpc inwards experience substantial drops in circularity, while those that migrate the same distance outwards nearly maintain their initial circularity (Figure 3.11).

The large inward excursions from the outer disk, driven by satellite perturbations, systematically remove angular momentum from particle orbits.

As shown by K08, vertical heating by satellite bombardment produces (in the case of the zd = 400 pc disk) a two-component vertical structure in quantitative agreement with the thin and thick disk profiles observed in the MW. In the isolated disks, we find the expected trend that as particles migrate outwards, they experience a weaker vertical potential and increase their vertical energy (characterized by zmax, the maximum distance from the plane that a particle can reach given its current location and velocity). However, in our simulations, this effect is not sufficient to produce a second, “thick-disk” component in the isolated systems even if they have substantial radial migration. The satellite perturbations increase the median and range of zmax considerably at all ∆r. The resulting overall trend of ∆zmax with ∆r is fairly flat (especially at a fixed rf , such as the solar annulus, see Figures 3.9 and 3.12), suggesting that systems with low vertical velocity dispersion and subjected to weaker potentials (as in the outer disk) are more susceptible to vertical heating (similar to the radial heating trends seen in Section 3.2.1). Particles with Rform interior or

84 exterior to a given rf can be found at signiﬁcant distances above the plane in the perturbed disks, while increases in zmax are smaller in the isolated systems and rely on particles moving outwards (Figure 3.10).

Our results conﬁrm earlier ﬁndings that spiral structure development in isolated disks (Roˇskar et al. 2008b; Loebman et al. 2010) and perturbations by satellites

(Quillen et al. 2009) can produce significant radial mixing of stellar populations while retaining reasonable orbital structure for disk stars. They strongly support the view (Wielen et al. 1996; SB02; Schönrich & Binney 2009a) that radial mixing is an essential ingredient in understanding the chemical evolution of the MW and disk galaxies in general. While a combination of metallicity and age can be used to estimate a star’s formation radius given a chemical evolution model, the uncertainties in this approach (including the difficulty of estimating ages for typical stars in a spectroscopic survey) will make it difficult to reconstruct formation radius distributions for observed stars, even in the solar neighborhood. However, our analysis provides theoretical guidance for chemical evolution models that incorporate radial mixing (e.g. Schönrich & Binney 2009a) and suggestions for the correlations one might search for between chemical abundances and orbital properties (though the predicted trends are fairly weak). We regard our perturbed zd = 400 pc disk as the most relevant simulation, as it includes the satellite accretion events expected in

ΛCDM and produces a ﬁnal vertical structure similar to that measured in the MW.

In this simulation 41% of particles in the 7 kpc < R < 9 kpc solar annulus were

85 “born” in that annulus, 20% migrated there from R< 6 kpc, and 7% migrated there from R> 10 kpc.

Radial mixing and orbital dynamics changes are sensitive to several different aspects of disk modeling, as shown by the variety of our results. Robust predictions should therefore come from calculations that self-consistently include star formation, chemical enrichment, gas accretion, and accretion events, and the interplay among these elements. We will move in this direction with our future simulations. Giant spectroscopic surveys such as SEGUE, RAVE, APOGEE, LAMOST, and HERMES offer an extraordinary opportunity to unravel the formation history of the MW, and they offer an exciting new challenge to theoretical models of galaxy formation.

86 Fig. 3.1.— Surface density maps of the stellar distributions of galactic disks with different initial scale-heights zd. Maps are presented only for the collisionless experiments and include the initial (left panels), final isolated (middle panels), and final perturbed disks (right panels). Each panel includes the face-on (bottom panels) and edge-on (upper panels) distributions of disk stars. Particles are color-coded on a logarithmic scale, with hues ranging from blue to white indicating increasing stellar density. Local density is calculated using an SPH smoothing kernel of 32 neighbors. 5 9 2 The density ranges from 7 × 10 to 2 × 10 M⊙/ kpc .

87 Fig. 3.2.— The distribution of radial shift (∆r= rf − ri) for all disk particles in each of the six isolated disk simulations. ∆rmed and ∆r80 specify the median and 80th percentile distance traveled, respectively. Gray values refer to the hatched histograms; black values are associated with the solid black line in each panel. We report the fraction of all particles in non-overlapping 500 pc bins of ∆r. The two collisionless simulations (left panel) illustrate the effect of disk structure on radial mixing, i.e., there is more radial migration in the smaller scale height (zd = 200 pc, black line) disk than in its thicker counterpart (zd = 400 pc, hatched histogram). The four hydrodynamical simulations are in the middle (fg = 20%) and right (fg = 40%) panels. In both of these panels, the gray, hatched histograms represent the simulations with higher star formation efficiency (c⋆= 0.05); those with c⋆= 0.01 are shown in black. Both increased gas fractions and smaller star formation efficiencies yield greater radial mixing.

88 Fig. 3.3.— The ∆r distribution for all disk particles in the four collisionless simulations. Each histogram show the fraction of particles in non-overlapping 500 pc bins of ∆r. The isolated (hatched, gray histogram) and perturbed (black line) zd = 200 pc (left panel) and zd = 400 pc (right panel) disks are shown. The perturbed disks show greater dispersion in ∆r compared to isolated galaxies with the same geometry. The red histograms denote the expected ∆r distribution from epicyclic motion alone, given the initial orbital conﬁgurations of the particles in each disk. ∆rmed and ∆r80 are labeled and color-coded for the three distributions in each panel.

89 Fig. 3.4.— The fraction of particles that migrate more than 1.0 kpc (solid line), 2.0 kpc (long dashed line), and 3.0 kpc (dotted line) as a function of formation radius. Results are binned such that the migration probability is calculated for particles in non-overlapping 1.0 kpc wide annuli. Lines connect the migration fraction in each bin (x-coordinates are bin centers, from 5.5 to 19.5 kpc). The gray regions are the radial mass proﬁles of the initial disks normalized such that the total mass contained in the ﬁrst annulus equals 2/3 on this scale. The migration probability follows the mass distribution in the isolated disks but is anti-correlated with mass in the perturbed disks.

90 Fig. 3.5.— Lindblad diagrams for all collisionless simulations. Each column shows the initial (left); final, isolated (middle); and final, perturbed (right) particle angular −1 momentum projected along the z axis (Jz, kpc km s ) vs. specific energy (E, km2 s−2). The red, yellow, and orange regions encompass 68%, 95%, and 99%, respectively, of particles centered on the median Jz as a function of E. Jz > 99% outliers are plotted as points. The green line denotes the theoretical angular momentum - energy curve for circular orbits. Each panel includes a histogram illustrating the distribution of circularity (ǫ) for all disk particles (see text for details). The isolated thin disk shows evolution towards slightly non-circular orbits. There is little change in ǫ in the isolated zd = 400 pc case. Simulations with satellite bombardment show a pronounced shift towards less circular orbits. For reference, the total energy of midplane orbits at 2, 5, 10, and 15 kpc from the galactic center is approximately −2.0, −1.6, −1.2, and −1.0 km2 s−2, respectively. 91 Fig. 3.6.— Possible values of circularity as a function of eccentricity in the solar annulus (7 ≥ r ≥ 9 kpc) of the zd = 400 pc disk. The top edge of the dark region represents orbits in the midplane of the disk. The dark region encompasses the 95th percentile of particle circularity in the solar annulus of each simulation; the bottom edge of the light region corresponds to the 99th percentile circularity.

92 Fig. 3.7.— The initial and final E, Jz pairs for ten randomly selected particles in 5 2 −2 each simulation with initial energy −1.405 ≤ Ei ≤ −1.395× 10 km s and angular 3 −1 momentum 1.645 ≤ Jz,i ≤ 1.655 × 10 kpc km s (large, filled square). For clarity we require that plotted particles lose at least 0.05 × 105 km2 s−2 in energy during the simulation. The final E,Jz pairs of the ten particles are plotted as open squares. Lines connect the initial and final E,Jz points of each particle. The E,Jz curve populated by circular orbits in the initial state of each simulation is indicated by the thick black curve. The dashed line is tangent to the circular orbit curve at the initial energy of all ten particles.

93 Fig. 3.8.— Median change in circularity ∆ǫ as a function of ∆r for the entire disk (open squares) and for particles with rf , ri > 4.0 kpc (ﬁlled squares). Particles are sorted into non-overlapping 0.5 kpc bins of ∆r. Error bars mark the 25th and 75th percentile ∆ǫ in each bin. Histograms at the bottom of each panel represent the relative fraction of particles in each bin for the case rf , ri > 4.0 kpc.

94 Fig. 3.9.— Like Figure 3.8, but the change in maximum vertical displacement, ∆zmax, is plotted in place of the change in circularity. Open squares show all disk particles, th th ﬁlled squares show those with rf , ri > 4 kpc, and error bars mark 25 and 75 percentile at a given ∆r.

95 Fig. 3.10.— The distribution of formation radius (Rform, in kpc) for all particles that are in the solar annulus (7 kpc ≤ r ≤ 9 kpc) at the end of each simulation. Histograms report the fraction of solar annulus particles emigrating from non-overlapping 500 pc annuli in Rform. Particles within the dashed lines remained in the solar annulus throughout the simulation. Both thin disks and the perturbed thick disk show a broad range of Rform at the solar annulus. The gray histogram in each panel is the Rform distribution for those particles that are in the solar annulus and at large heights above the plane (1.0 < |z| < 1.5 kpc) in the ﬁnal simulation output.

96 Fig. 3.11.— The change in circularity ∆ǫ as a function of ∆r for particles ending the four collisionless simulations in the solar annulus (7 kpc ≤ rf ≤ 9 kpc). Particles are sorted into non-overlapping 0.5 kpc bins of ∆r. We plot the median (squares) and the 25th and 75th percentile (error bars) ∆ǫ in each bin. Histograms at the bottom of each panel represent the relative fraction of particles in each bin.

97 Fig. 3.12.— Like Figure 3.11, but the change in vertical displacement ∆zmax is plotted in place of the change in circularity.

98 Chapter 4

Phase Space Substructure in the Milky Way’s Disk: Probe of Evolutionary History?

The formation mechanisms responsible for the Milky Way (MW) and other disk galaxies are still unknown to a large extent. Historically, stellar disks have been observed to be in approximate centrifugal equilibrium, suggesting that they formed out of gaseous disk. Eggen et al. (1962) explain the formation of this gas disk via the monolithic collapse of a gas cloud. Searle & Zinn (1978) built upon

Eggen et al. (1962) and added a secondary phase of galaxy formation in which other stellar systems evolve independently and then congregate in the Galaxy after the central halo collapsed. Some of these systems would have completely phase-mixed with the Galaxy, while others would still be satellite systems in the present day. We now known that galaxies must form in a hierarchical fashion to some extent; the

MW’s surviving globular clusters and disrupted dwarf galaxies oﬀer strong evidence of the idea. Still, the outstanding question remains: how much of an inﬂuence did hierarchical assembly have in the formation of the MW’s halo and disk?

99 There has been great progress in understanding the assembly of the MW’s halo via the identiﬁcation and characterization of kinematic substructure. Schlaufman et al. (2009) examined a SEGUE sample of metal poor main sequence turnoﬀ

(MPMSTO) stars for overdensities in their radial velocity distributions along individual sightlines; they conclude that roughly 34% of the halo is associated with

MPMSTO star substructure. Xue et al. (2011) measured the clustering statistics of blue horizontal branch (BHB) stars in position and radial velocity space and found that their measured clustering signal from SDSS is consistent with the signal measured in 11 diﬀerent ΛCDM N-body simulations that form halos purely from disrupted satellites. Cooper et al. (2011) construct a two-point correlation function using the distance and radial velocity of BHB stars and determine that the outer halo (rgc> 20 kpc) was formed from a few massive satellites but the inner halo rgc< 20 kpc is smoother than substructure-dominated models predict and suggest that this region of the galaxy has a smooth component with unknown origin. Recent studies all conclude that hierarchical assembly played a large role in the construction of the MW’s halo.

The assembly history of the thick disk remains diﬃcult to ascertain. Currently, four diﬀerent formation scenarios are discussed in the literature. Abadi et al. (2003) suggests that the thick disk could be formed via the accretion of stars from disrupted satellites on nearly co-planar orbits. In the disk heating scenario, merging satellites dynamically heat a pre-existing thin disk of stars into the thick disk (Kazantzidis

100 et al. 2008; Villalobos & Helmi 2008). Brook et al. (2004, 2005) show that a thick disk could be born in-situ from hot gas cloud formed during an early, gas-rich merger. The radial migration of stars from the inner galaxy outwards may also form a thick disk; simulations and analytical models show that hot, inner galaxy stars can migrate outwards, feel a weaker potential, and move away from the plane to form a thick disk (e.g. Sch¨onrich & Binney 2009a; Loebman et al. 2011). Parts of all of these scenarios are consistent with current data.

Inspired by the insight into the assembly of the halo via substructure studies, we perform a similar analysis for galactic disks. Using a suite of N-body galaxy formation models originally intended to explore the disk heating formation scenario discussed above (models described in Kazantzidis et al. 2008) and a fully cosmological, high-resolution N-body + SPH simulation (hereafter ErisLT, Guedes et al. 2011), we hope to predict the level of substructure in the disk as a function of the evolutionary history of each galaxy. While there is no data set capable of measuring substructure over a wide area of the disk yet, we tailor our calculations to forthcoming data sets. APOGEE, a part of SDSS-III and described in Eisenstein et al. (2011), is a mid/near IR survey of Milky Way giant stars. When complete, the survey will take spectra of over 105 stars across a large swath of the disk, bulge, and halo. APOGEE will uniformly observe an unprecedented Galactic footprint down to a limiting magnitude of H = 12.5. In this work, we apply the selection function of the APOGEE survey to the simulated galaxies so that any substructure statistic

101 measured in the simulations will take into account observational limitations and can be immediately compared with the same signal from the full APOGEE survey when it becomes available.

This work is organized as follows. In Section 4.1, we brieﬂy describe our models and the relevant aspects of the APOGEE selection function. We outline our steps in creating a phase space density statistic and our algorithm to identify over-densities in

Section 4.2. Section 4.2.1 examines the accuracy and robustness of our statistics in a variety of survey strategies. In Section 4.3, we discuss our calculation of the fraction of each galactic disk model associated with substructure. We use APOGEE’s radial velocity precision to determine the relative strength of heating processes in the disk

(Section 4.3.2). Finally, we summarize our results in Section 4.4.

4.1. Galaxy Models and the APOGEE Survey

In Figure 4.1, we show the surface density maps of the ﬁnal states of the three simulated galaxies used in our analysis. The isolated galaxy (left panel) and perturbed galaxy (middle panel) were evolved from identical initial conditions (we also include these smooth initial conditions as a control in our model sample).

These high-resolution simulations are zoom-ins of a cosmological dark matter-only simulation and are described in detail in Kazantzidis et al. (2008). The isolated

N-body model consists of a stellar Hernquist bulge and exponential disk embedded in a NFW dark matter halo and allowed to evolve in isolation for 3.5 Gyr. The

102 perturbed model is subjected to a cosmologically motivated accretion history; a

8 total of 5 satellites, the largest with a total mass of the 10 M⊙, perturb the initial stellar disk over the course of the ﬁrst 2.5 Gyr of the simulation. The disk is then allowed to evolve for another gigayear in isolation. The z = 0 snapshot of the ErisLT simulation is shown in the right panel of Figure 4.1. The Eris simulation suite contains the highest resolution simulations with full cosmological initial conditions to form realistic spiral galaxies like the MW (Guedes et al. 2011). The four galaxy models span a broad range of formation histories and show qualitative diﬀerences in their morphologies.

To make any measured signal as applicable as possible to APOGEE, we must construct a procedure by which to compare the eventual observations of the

APOGEE survey with these models, i.e., we measure our statistics using mock

APOGEE catalogs of the simulated galaxies. The first component to creating these mock catalogs is matching APOGEE’s observing geometry. APOGEE’s final field selection samples vertical columns of the disk (b = −8◦, −4◦, 0◦, 4◦, 8◦) at seven equally spaced galactic longitudes (l = 30◦, 60◦, 90◦, 120◦, 150◦, 180◦, 210◦). These

35 (l,b) combinations will be referred to as APOGEE’s “ﬁelds”, are 3◦ in diameter, and represent the fundamental unit of a mock survey. Next, each mock survey must have a vantage point, i.e., the solar position within the simulation. We assume the sun has a galactocentric distance of 8.0 kpc; this value is consistent with current constraints on the sun’s location in the MW (Nataf et al. 2012; Sch¨onrich 2012). We

103 do not assume a unique azimuthal position for the sun and uniformly sample the solar circle in 90 positions 4◦ apart. Each mock APOGEE survey (S) will consist of observations, from a single vantage point, of the 35 ﬁelds listed above.

4.2. Phase-space Substructure Finding

Our method is more analogous to that of Schlaufman et al. (2009) than Cooper et al. (2011), i.e., we look for phase space overdensities in individual fields (lines of sight). We examined fields in the four different galaxy models (three final states plus the initial state of the collisionless models) and found qualitative differences in the distribution of stellar particle distance, radial velocity pairs (Figure 4.2).

Figure 4.2 represents what APOGEE would measure in this field in each of the four simulations assuming APOGEE has no measurement error and observed every red giant star in a field. Before addressing how measurement error and incomplete target sampling would affect the data, we discuss how we assign phase-space densities and overdensities in this “error-free” data for clarity.

In our analysis, the dimensions of distance and radial velocity make up all of phase-space. To assign densities to positions on the phase space plane, we must choose a metric to establish the relative importance of clustering in each of the two dimensions. We attempt to assign equal importance to the information from each dimension and circularize the phase-space plane by choosing a metric for each ﬁeld

104 such that Xi = xi/σxi where x is the set of initial measurements, X are the data after the metric has been applied, and i represents either the distance (d) or radial velocity (rv) dimensions (see Maciejewski et al. (2009) for motivation). Once the metric is chosen, it follows that we can use the mass per geometric area on the modiﬁed plane to measure surface density. Following Monaghan & Lattanzio (1985), we choose a truncated spline kernel to measure the phase-space density at every particle.

2 3 2ri  1 − 6qi + 6qi if 0 ≥ h ≥ 1,  n  10  ρj = ×  1 3 2ri (4.1) 2  4 (2 − 2q) if 1 < h ≥ 2, 7πh Xi=1    0 otherwise   

where qi = ri/h, ri is the metric distance from particle i to the particle in question j, h is the truncation distance, and the sum is over all particles in particle j’s ﬁeld. Note that there is no contribution to the density of particle from sources outside the truncation radius. In our analysis, we choose h = 0.707 everywhere; only particles within a metric circle with r = 0.5 inﬂuence once another.

We use these particle phase-space densities (ρi) and the group ﬁnding algorithm

HOP (Eisenstein & Hut 1998) to identify particle associations in each ﬁeld. HOP begins by associating a particle with the densest of its closest Nhop neighboring particles; the process repeats itself as the algorithm “hops” to the densest neighbor of the previous densest neighbor. Each particle is then the origin of a hopping path

105 that goes in the direction of increasing density; the hopping terminates when it lands on a particle that has no denser neighbor. All path origins (particles) with a common destination are labeled a group. While this method distinguishes the dense regions of each field from the sparse regions, there are two problems left unaccounted for if we are to identify substructure in a physically realistic fashion: all particles are assigned to groups while, in reality, some particles likely belong to a smooth component; dense regions may have multiple local maxima, leading to unphysical group boundaries. HOP solves the former by adding a user-defined outer density (δouter) contour; particles below this density threshold cannot be assigned to groups. The issue of multiple local maxima is addressed by exploiting “boundary pairs” which exist whenever a particle and one of its Nmerge closest neighbors are in different groups. The two groups sharing the boundary pair are merged if the density of the pair, i.e., the average density of the two particles in the pair, is higher than another user-defined density contour δsaddle. In addition, if the density of a group’s most dense particle is lower than that of a third density contour, δpeak, it is disbanded unless said group has a boundary pair with density greater than δsaddle with another group whose peak density is greater than δpeak; in that case, the group is attached to the other group represented in its densest boundary pair.

We choose values for the ﬁve user-deﬁned parameters in HOP: Nhop, Nmerge,

δouter, δsaddle, and δpeak. Nominally, HOP requires a sixth parameter, Ndens, to compute densities but as we use our own density-measuring method outlined above,

106 we do not require Ndens. We choose Nhop= 8 and Ndens= 4 globally and δouter= 0.02,

δsaddle= 0.5, and δpeak= 0.5 in units of the mean particle phase-space density in the

field being processed. Most of these choices are optimal to find small, dense groups removed from the main locus of particles. The small δouter reflects the fact that almost all particles in these fields belong to an association of some kind. Because the lowest density particles represent a greater extreme than the higher density particles in each field, we must make the saddle density high to ensure that groups are not merged too easily. The high δpeak value demands that all the identified groups are truly significant.

At this point, the reader may take exception with our choice of HOP parameters.

However, we contend that no choice would appeal to all and the most important aspect in our analysis is consistency; we are only interested in diﬀerential signals from the various models. Once we account for observational errors and survey strategy, our choice of group ﬁnding parameters will become more obvious. We describe results using alternative parameters in Section 4.2.1.

4.2.1. Mock APOGEE Observations and Group

Identification

Figure 4.3 shows the complete process for an individual ﬁeld: mock APOGEE observation (with no errors or target sampling), density assignments calculated from

107 Equation 4.1, and group identifications using HOP. The raw radial velocity and distance measurements are shown, but, as described in Section 4.2, density and group calculations concern data after the metric has been applied. The middle panel shows several local maxima and the right panel illustrates how our fiducial choices of δouter, δsaddle, and δpeak influence the group identification. We place a premium on substructure off of the largest group of particles in each field (hereafter, the largest group will be referred to as the “locus”). Below, we further illustrate the group-finding process in a variety of mock observations and justify our choice of

HOP parameters.

Figure 4.3 warrants further attention. It seems more intuitive to pick our metric and group ﬁnding parameters speciﬁcally to identify the two large, apparent groups

−1 in the left and middle panels (with a saddle point at vlos∼ 140 km s ), yet our choice of δsaddle forced HOP to merge these two groups. In Figures 4.4 and 4.5, we illustrate our attempt to capture this larger substructure. Here, we emphasize voids in the d, vlos plane. This “intuitive” scheme differs from the fiducial scheme described above in three fundamental ways. First, the phase-space density of a particle is the surface density within a box on the d, vlos plane centered on the particle; this choice decreases the density measurement resolution when compared with using a spline kernel but it forces the density measurements into coarser bins. Second, the hopping process is not confined to the nearest Nhop neighbors, but rather a distance on the metric plane; the geometric hopping cutoff makes it difficult/impossible to

108 merge groups across gaps in vlos, d plane. Third, δsaddle is increased from 0.5 × µδ to 1.0 × µδ where µδ is the mean density of particles in the field; increasing δsaddle puts more emphasis on individual group detection in dense environments even at the cost of forming unphysical groups. Figures 4.4 and 4.5 show the phase-space density distributions and identified groups using our “intuitive” scheme for a number of fields across the four different galaxy models. The density peaks are well defined

(Figure 4.4), but HOP identiﬁes too many unphysical groups if we ensure the detection of obvious structures such as the larger groups in the perturbed model

(third row of both ﬁgures). The problem is only exacerbated when we make the

field data more realistic by sampling the fields and added errors to the distance measurements: groups identified in error-free data were rarely recovered and spurious groups began to dominate over intrinsic ones.

We abandon our “intuitive” scheme and return to our fiducial parameter choices that emphasize dense structure that is clearly distinct in d,vlos space. Figure 4.6 shows how the phase-space density distribution can dramatically change from model to model and field to field. Each row of the figure shows three fields with

(l, b) coordinates of (30,0), (60,0), and (60,4) observed in one of our four galaxy models (initial collisionless state, isolated collisionless, perturbed collisionless, and

ErisLT). The observations all assume the same vantage point. Note how the number, connectivity, and densities of particles varies with model and ﬁeld choice.

109 Thus far, we have shown mock observations with no measurement errors of all particles in a selected ﬁeld. We now account for the fact that APOGEE does not have inﬁnite exposure time and must choose a subset of its potential targets.

We randomly sample the particle targets along a line of sight to better match the expected number of APOGEE observations in any given ﬁeld; the middle column of

Figure 4.7 demonstrates how this approximation of the APOGEE selection function aﬀects the calculated phase-space densities (compare the left and middle columns).

The global features of the vlos, d, phase-space density distribution, including most local maxima, remain intact as one samples fewer particles along a sightline. Detailed statistics, such as the coordinates of the densest particle in each group, vary due to the stochastic nature of the sampling.

We also account for the expected error in APOGEE’s distance and radial velocity measurements. The most challenging aspect of our analysis is to choose a phase-space density and group-ﬁnding algorithm that can perform well on realistic data. APOGEE expects 20% errors in distance while the error in radial velocity will be ∼ 200 m s−1 (∼ 0.2%). We assume perfect radial velocity measurements as the distance errors dominate any calculation. Figure 4.7 shows the impact of realistic distance errors on our analysis. The left column shows the calculated phase-space densities using error-free data and no target sampling while the right column still shows phase-space density but now of a randomly selected N = 1000 particles subset

110 with 20% errors in distance. Comparing the right column with the left, some of the highest density peaks are retained, while many are completely washed out.

The effect on group identification is muted because of our chosen density thresholds (Figure 4.8). The fiducial scheme only picks out the most obvious substructure (see third row of Figure 4.8). Unfortunately, we cannot probe any hierarchy of substructure given our parameter choice. However, most of the identified substructure survives even after we apply the distance errors. Even with our conservative approach, sparser sampling can create spurious groups as the connectivity of the particles decreases (top row, middle panel of figure).

4.3. Results

4.3.1. Fraction of Disk in Substructure

We use our group finding procedure to determine the fraction of all stars associated with substructure (fsubs). One complete mock APOGEE survey S includes 35 fields; we define the locus of each field as the group with the largest number of members. Non-locus groups are labeled as substructure; we track the

number of particles in these substructures (nsubsi ). The process of identifying the locus, non-locus groups, and the number of members in each is repeated for all

35 ﬁelds. Finally, we calculate the fraction of the mock survey associated with

111 35 substructure fsubs(S)= i=1 nsubsi /N where N is total number of particles in S and P i refers to the ﬁelds of the survey numbered 1-35. Figure 4.9 shows fsubs as a function of galaxy model (initial, isolated, perturbed, and cosmological) and data quality

(error-free, error-free with random sampling of 1000 targets per field, and random sampling with distance errors). The 90 solar vantage points (Section 4.1) per mock observed simulation each yield a value of fsubs. Similar in concept to accounting for cosmic variance in cosmological studies, we must allow for the variance in fsubs due to different areas of the galaxy being observed from different vantage points. The perturbed and cosmological models generally have the highest fsubs; this result is robust to changes in data quality. In the most realistic case (sampling plus distance errors), the fsubs seen in the perturbed model drops steeply but remains higher than fsubs in either the initial or isolated models. Referring to Figure 4.8, most of the perturbed model’s substructure is at large line of sight distance and is easily disassociated with distance errors. Still, the correlation between the median fsubs measured and the galaxy model is preserved.

4.3.2. RV as Accretion History Diagnostic

APOGEE will measure radial velocities with an internal precision of σvlos = 200 ms−1 (Nidever et al. 2012) across a range of galactocentric radii; we explore how this data set can probe the accretion history of the MW’s disk. In the plane of the disk, the line of sight velocity at a galactic longitude is linearly proportional to the

112 diﬀerence of the angular speed at the radius of the star being measured and the angular speed of the sun. Following equation 9.3 of Binney & Merriﬁeld (1998)

v los = [Ω(R) − Ω(R )]R (4.2) sinl 0 0

where vlos is the line of sight velocity, l is galactic longitude, R0 is the galactocentric distance of the sun, Ω(R) is the angular velocity of material at the galactocentric radius of the star, and Ω(R0) is the angular velocity of the sun.

We use equation 4.2 to determine the degree of radial heating in each of the models as a function of galactocentric distance. Consider a galactocentric annulus of width δrgc. In the limit that all the stars/particles in the annulus were on perfectly circular orbits and δrgc was inﬁnitesimally small, the left side of equation 4.2, the

vlos measured sinl (hereafter, W ) of stars in that annulus would have a singular value.

As either the annulus widens or motions along the line of sight increase, the range of

W measured in the annulus increases. Any heating process aﬀecting stars/particles in the annulus will thus have a measurable impact on the W (R) distribution (where

R is the galactocentric radius of the annulus).

Figure 4.10 shows the result of measuring the inter-quartile (IQ) range of

W (R) in our four galaxy models. Note that we only use b = 0 ﬁelds in this part of the analysis. The points show the median values from the 90 Si per simulation.

While the normalization of the curves has limited value as stars in the MW may be

113 systematically heated to a larger or smaller extent than our models, the shape of the curves are telling. Speciﬁcally, the slope of IQ(W (R)) at rgc> 8 kpc is almost ﬂat in the perturbed and cosmological models and negative in the initial and isolated experiments. In Bird et al. (2012), we found that satellite accretion events increase the migration and heating of disk galaxy outskirts; this heating could cause the trends seen in Figure 4.10. The general result and trend holds if we only sample

1000 targets from each field and incorporate 20% distance errors for each particle measurement (Figure 4.11). The distance errors scatter stars in an out of any single annulus, but the effect is not strong enough to wash out the signal. APOGEE will be able to measure IQ(W (R)) at great precision and thus might be able to distinguish between different merger histories in the MW’s outer disk.

4.4. Summary

We have used mock APOGEE observations of a suite of galaxy models to construct and measure a substructure statistic that can be directly applied to

APOGEE data as it becomes available. The fraction of disk stars in an APOGEE-like survey that are associated with substructure (fsubs) is an excellent diagnostic of disk assembly history in the limit of perfect data, but may not be deﬁnitive given the

20% distance errors expected of APOGEE measurements. We are able to make use of our mock catalogs to construct another, more robust statistic: APOGEE will soon

114 measure the radial velocities of many stars across a range of galactocentric radius.

We use APOGEE’s exquisite vlos precision to determine the relative heating of the outer and inner disk of each galaxy model and, in turn, constrain the importance of gravitational perturbations (such as those from satellite accretions) in the recent history of the outer disk. In our experiments, galaxies with purely secularly evolution exhibit a negative IQ(W ) radial gradient while more active galaxy formation models have a ﬂat IQ(W ) radial gradient.

Our group finding parameter choices did not maximize our ability to detect substructure; furthermore, the choices were not made such that fsubs reflected the true amount of substructure in each disk model. Our goal was to present a statistic that differentiated between the four models and measured the same relationship between fsubs and galaxy formation model in both error-free and realistic data.

Figure 4.9 shows fsubs using our ﬁducial parameter choices: holding the data quality

ﬁxed, the models listed in increasing median fsubs are initial, isolated, cosmological, and perturbed. We speculate that interactions with merging satellites increase fsubs in the perturbed and cosmological models. Accretion events gravitationally perturb the stellar disk through resonant interactions and dynamical friction; these perturbing events have already been shown to increase the vertical and radial velocity dispersion of particles in N-body simulations (e.g. Kazantzidis et al. 2008; Minchev et al. 2012, Bird et al., in prep.). Satellite-induced disk resonances can also create substructure in the E, Lz plane (G´omez et al. 2012). A detailed investigation of the

115 resonances in the disk both before and after satellite bombardment is beyond the scope of this paper. However, both the cosmological and perturbed models feature stellar disks perturbed by minor satellite encounters; the gravitational perturbations induced by accretion events are a plausible explanation for the increased fsubs found in these simulations.

116 4 2 0 −2 z[kpc] −4

0 y[kpc] −10

−20

−30 −30 −20 −10 0 10 20 30 −30 −20 −10 0 10 20 30 −60 −40 −20 0 20 40 60 x[kpc] x[kpc] x[kpc]

Fig. 4.1.— Edge-on (upper panels) and face-on (bottom panels) views of the stellar surface density maps in the controlled experiments evolved in isolation (left panel), subjected to bombardment from a cosmologically motivated satellite population (middle panel), and a z = 0 snap shot of MW-like disk galaxy from the Eris cosmological simulation suite (ErisLT, Guedes et al. 2011). The dynamics and morphological structure of the stellar populations are as varied as each galaxy’s assembly history.

117 Initial Isolated 250

200

150

100 ] km/s [ Perturbed Cosmological 250 los v

200

150

100

5 10 15 20 25 5 10 15 20 25 Distance [kpc]

Fig. 4.2.— The mock APOGEE measurements of the line of sight velocity (vlos) and distance (d) distributions for particles in the l = 60, b = 0 field of the initial, isolated, perturbed, and cosmological models (see text for descriptions). Each panel assumes the same, randomly chosen vantage point on a circle with galactocentric radius of 8 kpc. The plots show error-free measurements of all particles along the line of sight. Models with different evolutionary histories show qualitative differences in this plane.

118 1.35

2.5 1.20 ] 1.05

2.0 0.90 km/s

2 0.75

10 1.5

× 0.60 [ 0.45 los 1.0 v 0.30 0.5 0.15 5 10 15 20 25 5 10 15 20 25 5 10 15 20 25 dLOS [kpc] dLOS [kpc] dLOS [kpc]

Fig. 4.3.— The line of sight velocity and distance distributions for all particles along the l = 60, b = 4 field in the perturbed model given a random vantage point on the solar circle and error-free measurements (left). The same particles are now color-coded by their phase-space density normalized to the mean density in this field (middle and colorbar at right). The group-finding algorithm HOP, using (δouter, δsaddle, δpeak) of (0.02 , 0.5, 0.5) in units of the mean phase-space density of the field, partitions the plane into connected groups (right, see text for details). Groups are color-coded to highlight their boundaries.

119 Fig. 4.4.— The error-free distances and radial velocities of particles in 3 ﬁelds (columns) and the four galaxy models (rows). The points are color-coded according to their phase-space density as deﬁned by our “intuitive” phase-space density algorithm (see text for details). Red points are the most dense; blue the most sparse. Note the resolution of structure within the main locus of points.

120 Fig. 4.5.— Group identiﬁcation by HOP of the error-free data in Figure 4.4 using our “intuitive” parameter choice (see text for detail). Particles of any particular color are labeled as a collective group; the dark blue particles are not in any group. Note the unphysical shape of many groups.

121 Fig. 4.6.— Line of sight velocity and distance distributions of particles in three different fields (columns) for the four models (rows). Particles are colored according to their density normalized to the mean density in each panel with red regions representing the most dense and blue the least dense. Note how the number, connectivity, and densities of particles changes with model and field choice.

122 Fig. 4.7.— The effect of target selection and measurement errors on density in the distance, radial velocity plane. Particles are colored according to their density normalized to the mean density in each panel with red representing the dense regions and blue coloring sparse regions. The left column shows the d,vlos pairs of all particles in the field using error-free measurements. The middle column also shows the same but for a subset of 1000 randomly selected particles. The right column illustrates how incorporation of the gaussian distributed 20% fractional distance error expected from APOGEE and N = 1000 randomly target sampling affect the phase-space density distribution. Comparing the middle column with the left, the global features of the vlos, d, phase-space density relationship remain intact as one samples fewer particles along the sightline; detailed statistics, such as the coordinates of the densest particles in each panel, vary due to stochastic nature of the sampling. In the right column, we find that distance errors can significantly alter the location of the maxima in the density distribution and wash out obvious substructure.

123 Fig. 4.8.— The effect of target selection and measurement errors on group identification in the d-vlos plane. The left, middle, and right columns show the vlos and distance distributions of all particles with error-free measurements, 1000 randomly selected particles with error-free measurements, and 1000 randomly selected particles with 20% distance errors, respectively, in one field across four models (rows). Particles are colored according to their group identification to highlight the results of the HOP algorithm. Comparing the second and third column with the first, sparser sampling can create spurious groups as the connectivity of the particles decreases (top middle panel). In this example, the observed group in the perturbed model is identifiable even with realistic data (right column, third row).

124 0.07 e r 0.06 u t c u r t .

s 0 05 ub S

n 0.04 i y e v r 0.03 u S f 0.02 on o ti

ac . r 0 01 F

0.00 Initial Isolated Perturbed Cosmological Simulation

Fig. 4.9.— The percentage of mock APOGEE surveys associated with substructure as a function of galaxy formation model. Median fsubs values are represented by the points, interquartile ranges by the inner brackets, and 10th,90th percentile ranges by the outer brackets. Our results using error-free data are shown with solid lines. Other lines reﬂect how our results change when we account for target sampling with 1000 randomly selected particles per ﬁeld (dashed) and convolving the target sampling with 20% distance errors (dotted).

125 180 Initial

xxxxxxx 160 Perturbed

] Cosmological s

/ Isolated

m 140 k xxxxxxx [ V

R 120 f o e

xxxxxxx

ng 100 a R e 80 til r a

xxxxxxx qu

r 60 e t n I 40

xxxxxxx

20 6 8 10 12 14 Galactocentric Radius [kpc]

xxxxxxx

Fig. 4.10.— The interquartile of mock APOGEE RV measurements as a function of galactocentric distance and formation model. Median values are represented by dots, interquartile ranges by the hashed regions. Distance measurements are error-free. Note how the isolated and initial models have a negative IQ gradient at large radii; the perturbed and cosmological models have a ﬂat gradient.

126 180 Initial

xxxxxxx 160 Perturbed

] Cosmological s

/ Isolated

m 140 k xxxxxxx [ V

R 120 f o e

xxxxxxx

ng 100 a R e 80 til r a

xxxxxxx qu

r 60 e t n I 40

xxxxxxx

20 6 8 10 12 14 Galactocentric Radius [kpc]

xxxxxxx

Fig. 4.11.— The same as Figure 4.10 but now using randomly selected N = 1000 subsets in each ﬁeld and 20% errors in distance measurements. Median values are represented by dots, interquartile ranges by the hashed regions. The trend found in error-free data is robust: the isolated and initial models still have a negative IQ gradient at large radii whereas the perturbed and cosmological models have a ﬂat gradient.

127 Chapter 5

Inside Out and Upside Down: Tracing the Assembly of a Simulated Disk Galaxy Mono-Age Stellar Populations

Many important phases of the construction of stellar populations and their assembly into disk galaxies is still uncertain. White & Rees (1978) provided a framework for galaxy formation theory: gas cools onto halos created by clustered dark matter and subsequently forms stars. Many have tried to model the details of baryonic cooling and assembly. The seminal work of Eggen et al. (1962) showed that a large gas cloud could monolithically collapse onto a halo and naturally form a disk via the conservation of angular momentum. Searle & Zinn (1978) added a second epoch of accretion to the monolithic collapse theory: independently evolved satellite halos would collapse onto the parent halo at later times. Modern data sets have ruled out a pure monolithic collapse scenario for the build up of at least the Milky

Way’s Halo, but a detailed blueprint for the origin of the bulge and disk components of spiral galaxies remains elusive.

128 The last decade has seen a dramatic increase in the number of Milky Way

(MW) stellar spectra and photometry and a corresponding rise in the number of theories proposed to explain why the MW (and other disk galaxies) has its current chemodynamic properties. The existence and explanation of the MW’s thick disk is a microcosm of the contention. Observation studies have identiﬁed a vertically extended stellar component in our disk via bimodal distributions in star counts (e.g.

Gilmore & Reid 1983), kinematics (e.g. Bensby et al. 2003), and chemistry (e.g. Lee et al. 2011) in the solar neighborhood. No fewer than four classes of theory have been developed to explain this structure and are consistent, to some extent, with the current data: the accretion of stars stripped from satellites (Abadi et al. 2003), a satellite dynamically heating a pre-existing stellar disk (e.g. Kazantzidis et al. 2008), a gas rich merger at early times (Brook et al. 2004), and stellar radial migration

(Sch¨onrich & Binney 2009a; Loebman et al. 2011). The lack of a unique solution probably has its basis in reality: the Galaxy undoubtedly has a complex history and there are a variety of physical processes that could, if they played a major role in the

Galaxy’s assembly, produce the observed signals in the solar neighborhood.

Amongst many ideas to push past tension between current observations and their explanation by a variety of Galactic history models, we focus on two: stellar observations in the disk and away from the solar neighborhood and the discussion of galaxy evolution in terms of fundamental units rather than the trend-based, holistic comparisons currently made between theory and observation. Galactic surveys like

129 the mid IR APOGEE survey (Eisenstein et al. 2011) will answer the call of the former. To the latter point, some studies have chosen to dissect Galactic observations or galaxy models by either observational or theoretical quantities and then examine how these binned systems assemble. For example, Bovy et al. (2012) divided the G dwarf SEGUE sample into bins of [Fe/H], [α/Fe] pairs; they were able to describe each mono-abundance population as isothermal and showed that the thick disk structure naturally arose from the superposition of these diﬀerent populations.

Using numerical simulations, we can dissect a disk galaxy into its constituents and not only determine how these components evolve into their current state and but also how their superposition produces the ﬁnal, observable galaxy. Cosmological simulations assume the least about the initial state of the galaxy and provide the ideal setting to follow self-consistent galaxy assembly (Stinson et al. 2012). Brook et al. (2012) independently track the evolution of the thin disk, thick disk, and halo

11 in a cosmological “zoom” simulation of the formation of a Mvir = 1.9 × 10 M⊙ disk galaxy. They assign particles to these galactic components based on lines drawn through the [Fe/H], [α/Fe] plane at z = 0 and ﬁnd that the thick disk arose primarily from stars formed in-situ from kinematically hot gas created by an early epoch of major mergers. Their model self-consistently explains how the major structural components can evolve from standard initial conditions into what is seen in the MW today.

130 In this work we follow the assembly of a realistic MW analogue in the cosmological, high-resolution, state of the art ErisLT simulation (ﬁrst introduced in

Guedes et al. 2011). We choose stellar age, rather than kinematics or chemistry, to dissect the galaxy into “age cohorts” and track each population’s evolution. We attempt to identify processes and features possibly generic to disk galaxy formation.

In essence, we follow how the disk galaxy is made as a function of time, rather than asking what material (i.e., stellar chemical composition) were added at what time. The former question is simple yet still uncertain, so this work provides a straightforward, self-consistent description of disk galaxy formation.

5.1. The ErisLT Simulation

The ErisLT simulation is identical to its twin Eris (Guedes et al. 2011) except for a lower star formation threshold density and lower star formation eﬃciency.

ErisLT was created using the standard “zoom-in” techniques. The initial conditions and merger history of ErisLT were taken from a low resolution, cosmological, dark matter simulation, speciﬁcally from a halo with no major mergers after z = 3 and with a mass similar to the MW at z = 0. The simulation initialized at z = 90 with 13

4 4 million mDM = 9.8 × 10 M⊙ dark matter particles and 13 million msph = 2 × 10 M⊙ gas particles. The force softening length is 120 pc for all particle species from z = 9 to present. The formation of stars is governed by the density threshold that gas

3 particles must reach to create a stellar particle (nSF = 0.1 atoms cm ) and the

131 eﬃciency at which gas above the density threshold is converted to stars (ǫSF = 0.05).

The star formation parameters were chosen to ensure a match to the normalization of star formation density observed in local galaxies (Governato et al. 2010). ErisLT forms a spiral galaxy with similar structural parameters to the MW. A detailed explanation of the ErisLT simulation will be in a forthcoming paper.

5.2. Redshift Zero

We denote stellar populations binned by age with a script S and subscripts corresponding to the age limits of the bin; e.g., stars born between 0.0 and 0.5 Gyr after the big bang will be collectively referred to as S0.0,0.5.

Figure 5.1 shows the spatial distribution of stars binned into age cohorts.

Speciﬁcally, we examine stars with formation times (tform) spanning 0.0 to 0.5 Gyr,

0.5 to 1.0 Gyr, 1.0 to 2.0 Gyr, 2.0 to 4.0 Gyr, 4.0 to 8.0 Gyr,8.0 to 12.8 Gyr, and the ﬁnal 500 Myr of star formation, 12.8 to 13.3 Gyr. Each panel denotes the range of tform considered and shows face-on and edge-on views of the galaxy.

Following the density distributions from left to right and top to bottom, younger stellar populations show longer, flatter structure than older populations. The oldest population (S0.0,0.5) currently resides in the central region of the galaxy. S0.5,1.0 show a spheroidal number density profile out to rgc∼ 10 kpc. Compared to the stars born in the first gigayear of cosmic time, the spatial distribution of stars in S1.0,2.0 extends

132 further in radius, and the edge-on view shows a distinctive but minor ellipsoidal component indicative of disk membership. S2.0,4.0 has a smooth, but signiﬁcant, departure from a spheroidal spatial distribution; the edge-on view reveals a vertically extended and relatively short disk. Edge-on views of S4.0,8.0 and S8.0,12.8 reveal that younger populations inhabit progressively longer and thinner disks.

Returning to the face-on views in the bottom row of Figure 5.1, we ﬁnd that the last half gigayear of star formation is entirely conﬁned to the spiral arms and has a very sharp break at rgc= 10 kpc. In time, the stars in these spiral arms will heat up and increase their random motions, eventually leading to the dissolution of the transient spiral wave. The azimuthal perturbations in the density distributions of

S4.0,8.0 and S8.0,12.8 show how the spiral overdensity fades with the increased random motions of the stars over time.

In the left panel of Figure 5.2, we show the mass surface density as a function of radius for the different age cohorts. The color scheme for this figure is repeated throughout the rest of the paper: red colors for the oldest cohorts, progressing through orange, yellow, green, and cyan for young cohorts and finally dark blue for the youngest stars. The mass surface density profile for all particles (black line) reveals a large concentration of mass in the central 2 kpc of the galaxy; starting at rgc∼ 4 kpc, the profile looks exponential and shows no break out to 20 kpc. As seen in Figure 5.1, S0.0,0.5 solely contributes to the central region of the galaxy. The mass density profiles of S0.5,1.0 and S1.0,2.0 are intermediate to both a power law or

133 exponential characterization. In addition to an underlying power law distribution,

S2.0,4.0 has an qualitatively exponential component at 3 < rgc < 10 kpc. Younger populations show dominant exponential components: S4.0,8.0’s mass density proﬁle is exponential at 3 < rgc < 20 kpc while the mass density of S8.0,12.8 is exponential over the same radial range but breaks at rgc ∼ 12.5 kpc. Stars born in the last

500 Myr have the mass profile of a thin disk with a break radius of 9 kpc. The profile of S12.8,13.4 deserves note: several authors have used integrated light studies to determine that the disk break radius scales inversely with age/metallicity in nearby spiral galaxies (e.g. Radburn-Smith et al. 2012; Yoachim et al. 2012). In the right panel of Figure 5.2, we find that older populations are systematically at greater heights above the plane than younger cohorts at all galacto-centric radii. Several groups have measured similar correlations at the solar annulus in the MW.

We show each age cohort’s vertical density profile at three different galactocentric radii in Figure 5.3. The profile of all the age cohorts in the solar annulus

(middle panel, black line) is consistent with the classic two exponential proﬁle

observed in the MW (Gilmore & Reid 1983); the scale heights are zh1 = 0.55 kpc

and zh2 = 1.47 kpc. The shape of the ensemble vertical profile is due to the sum of the steadily changing exponential descriptions of the individual cohorts; the younger cohorts have double exponential profiles of their own. Generally, the vertical profiles of each age cohort evolve away from this simple exponential parametrization in two respects: at any given radius, older stars have flatter vertical distributions

134 than younger stars; each individual cohort’s vertical proﬁle ﬂattens at larger radii.

The latter observation explains why the ratio zh1 /zh2 increases with galacto-centric radius.

The age cohorts’ radial proﬁles of velocity dispersion in the vertical and radial directions have several note-worthy features (top row of Figure 5.4). Older age cohorts are systematically hotter than their younger counterparts in the radial and vertical directions at nearly all radii; ErisLT qualitatively matches the stellar age-velocity relations established in the MW. The velocity dispersions of S0.5,1.0 and S1.0,2.0 are locally maximal at the edges of their bulge-like mass distributions

(rgc ∼ 3, 6 kpc, respectively). Stars with tform> 4 Gyr show remarkably ﬂat velocity dispersions in the disk out to rgc∼ 12.5 kpc; thereafter, there is a positive gradient

in σvr and σvz . Note that there are few stars in S12.8,13.4 with rgc> 12.5 kpc; this age cohort’s precipitous rise in velocity dispersion at large radii is not statistically signiﬁcant. Each cohort generally follows the age-dependent trends established in

each plot; however, compared to the other cohorts, S2.0,4.0 has a positive, large ∆σvr

gradient at rgc> 5 kpc (and to a lesser extent in the ∆σvz plane). Referring back to the surface mass density panel, S2.0,4.0 begins to show an exponential component at rgc= 5; it is plausible that the stars in this vertically puffed up disk experienced a significantly different evolutionary history than the younger populations.

The bottom row of Figure 5.4 examines the median circularity and the dispersion in circularity in each age cohort as a function of radius. Reﬂecting the

135 trends in velocity dispersion, older stars have lower, broader circularity distributions than younger populations. The median circularity of S2.0,4.0,S8.0,12.0, and S8.0,12.8 is remarkably constant over increasingly larger ranges in radius for younger populations.

The dispersion in circularity, however, rises with radius for this subset of cohorts; the radial onset of this increase is inversely correlated with age. Stars formed during the last 500 Myr are born on very circular orbits but show a greater dispersion in circularity than S8.0,12.8 at some radii.

To aid in our description of the current state of the galaxy, we kinematically separate the galaxy into components using the energy and angular momentum of the particles and following a modified version of the prescription outlined in Abadi et al. (2003). We describe these components as “thin disk”, “bulge”, etc., though this nomenclature implies, to some degree, sharp distinctions of origin that may not reflect true history. We first choose a coordinate system such that the origin is coincident with the center of mass of the stellar particles and the z-axis is aligned with the angular momentum vector of the stars. Each particle’s angular momentum in the plane of the disk is then Jz, and positive Jz denotes a corotating orbit.

Circular orbits have the maximal Jz (hereafter Jcirc) at all energies. A particle’s circularity (ǫ) is the ratio of Jz to Jcirc at its total energy.

The principal discriminants in our decomposition are circularity and total

1 2 speciﬁc energy (Etot), where an individual particle’s Etot = 2 v + Φ; v is the total velocity of the particle and Φ is the potential at the position of the particle. We

136 ﬁrst assign all particles with ǫ> 0.8 to the thin disk as it should contain mostly circular orbits (ǫ= 1 for a circular orbit). To the spheroid, we add all particles with circularity below a critical threshold (ǫcrit); ǫcrit is chosen such that the ensemble of all particles with ǫ< ǫcrit exhibit exhibit zero net rotation. Those members of the spheroid with more energy than the median stellar particle energy (E⋆) are further classiﬁed as the halo; the bulge contains the remaining spheroid members. The

ﬁnal, unclassiﬁed population has ǫcrit<ǫ< 0.8 and is divided into the thick disk

(Etotal ≥ E⋆) and pseudo bulge (Etotal

Our kinematic distinctions are arbitrary and may not perfectly describe the galaxy. The caveats to this decomposition are similar to those in Abadi et al. (2003): the spheroidal component could have a modest amount of rotation, the circularity cutoﬀ for the thin disk may not precisely match that in the MW, and the energy cuts do not ensure that our structural distinctions are neither complete nor free of contamination. However, we do not rely on the accuracy of this procedure in any of our conclusions. Furthermore, the process gives us a qualitative description of the galaxy’s structure and is a useful method to divide the galaxy that provides intuition and context to our analysis.

In Table 5.5, we show how each age cohort contributes to the galaxy structures identiﬁed in our kinematic decomposition. The results corroborate much of what our eyes see in Figure 5.1. Old stellar populations with tform< 2.0 Gyr predominantly contribute to the spheroid. After t = 4 Gyr, new stars overwhelmingly populate the

137 disk of the galaxy. S2.0,4.0 is an intermediate population: signiﬁcant fractions of its members have kinematics consistent with classical disk or spheroid orbits.

5.3. Evolution of Age Cohorts

We now examine the assembly history of individual age cohorts. The ﬁrst three tform bins (S0.0,0.5, S0.5,1.0, and S1.0,2.0) are illustrative of the relatively complex compilation of stellar populations at early times. S3.0,4.0 represents an “intermediate” population born during the major merger epoch. The ﬁnal age cohort described here, S7.0,8.0, has the longest history of any cohort that is overwhelmingly born in situ and evolves in a quiescent disk.

The First 500 Myr We consider the spatial distribution of S0.0,0.5 as a function of time, specifically at t=0.6, 1.1, 2.0, and 4.0 Gyr, in Figure 5.5. These stars populate a number of different subhalos at early times, but their dominant concentration is in the parent halo of the galaxy (note the changing scale on each panel in the figure). The final merger event involving a significant fraction of

S0.0,0.5 is well underway at t = 2 Gyr; by t = 4 Gyr, S0.0,0.5 has assembled into a conﬁguration much like what we see today (compare the bottom right of Figure 5.5 with the top left of Figure 5.1. We omitted snapshots at later times as they all showed smooth, low-level accretion.

138 The radial proﬁles of several physical parameters conﬁrm that the S0.0,0.5 population assembles fairly rapidly and evolves quiescently thereafter (Figure 5.6).

Both the satellite accretion seen in Figure 5.5 and the flattening of the mass profile within rgc= 5 kpc broaden the surface density distribution with time. The radial profiles of vertical velocity dispersion (right panel) and median height above the plane (middle panel) show that S0.0,0.5 kinematically heats up as it assembles but its velocity ellipsoid stays relatively constant after t = 4 Gyr. The radial profiles of several physical parameters confirm that S0.0,0.5 assembles fairly rapidly, remains centrally concentrated, and evolves quiescently after its last substantial merger event.

tform = [0.5, 1.0] Gyr Similar to S0.0,0.5, S0.5,1.0 is scattered amongst several subhalos at early times but is still always concentrated in the most massive halo

(Figure 5.7). At t = 2.0 Gyr, we find that the last significant merger for S0.0,0.5 has an even higher ratio of satellite to parent mass for S0.5,1.0. Still, after another 2 Gyr, the vast majority of these features is erased; the short dynamical time at the center of the potential well enables this quick phase-mixing. From t = 4 Gyr to t = 6 Gyr, the spatial distribution of stars outside rgc> 5 becomes more symmetric in the plane of the disk and the inner disk’s mass profile seems to flatten slightly. Snapshots at later times reveal slow enhancement of these last two phenomena.

The surface mass density radial proﬁle of S0.5,1.0 rapidly evolves to resemble its current state by t ∼ 4 Gyr (Figure 5.8, left panel) in the outer regions of the

139 disk. However, mass is continually lost in the innermost kiloparsec and gained at 2 < rgc < 7 kpc. There is similarly slow late-time evolution in the vertical velocity dispersion; over the last 10 Gyr, the most dramatic heating is in the

−1 innermost kiloparsec but σvz still only increases at a rate of ∼ 2 km s per gigayear

(Figure 5.8, middle panel). Only the inner few kpc host any appreciable change in the median height of the cohort after t = 4 Gyr; the median height change is on the order of 100 pc (Figure 5.8, right panel). S0.5,1.0, like S0.0,0.5, assembles quickly into the kinematically hot component seen in section 5.2.

tform = [1.0, 2.0] Gyr The projected density distribution of S1.0,2.0 is shown as a function of time in Figure 5.9. The merger event at t = 2.0 Gyr induces star formation in the parent galaxy and satellites; we also note an arc-like stellar structure associated with the satellite that is not seen in the older cohorts. By t = 4 Gyr, almost all of the S1.0,2.0 stars in merging satellites have been assimilated into the parent population. At t = 6.0 Gyr, S1.0,2.0’s conﬁguration is the ﬁrst to show an ellipsoidal disk component. The tilt and asymmetries to the ellipsoid are qualitatively symmetrized by t = 8.0 Gyr.

Despite the evident diﬀerences in their evolutionary history, S1.0,2.0’s radial proﬁle plots show similar behavior to that of the older cohorts. After the merger epoch, the cohort has very mild mass redistribution, typically emigration from the innermost kiloparsec. S1.0,2.0 is born hot and kinematically resembles its older counterparts over a large radial range; the inner 10 kpc. At later times, the

140 population at 1 ≤ rgc ≤ 5 kpc moderately increases its median height but the height of the outer disk remains unchanged.

tform = [3.0, 4.0] Gyr Many of the evolutionary trends established for stars with tform< 2 Gyr are modified when we examine S3.0,4.0 (Figure 5.11). Here, the vast majority of the cohort is born in situ in the galactic disk; the early disk shows a prominent m= 2 mode spiral (top left of figure). At t = 4 Gyr, we also find two satellites with significant S3.0,4.0 populations that will soon interact with the in situ population. Just 1.0 Gyr later (top right), one of the two satellites is fully disrupted and the strong spiral structure has dissipated. The snapshots at 7.0 and 9.0 show the continued disruption of spiral structure, a slight vertical “puffing up” of the disk, and the continued phase-mixing of outer disk asymmetries remaining from the merger events.

S3.0,4.0 is the oldest cohort to have a signiﬁcant exponential component in its surface density radial proﬁle. As seen in the density plots, this cohort’s disk was in place at early times but Figure 5.12 (left) reveals that some mass is redistributed from the inner disk to the outer disk over time. We investigate the potential

migration of this cohort and others in Section 5.4. The radial proﬁle of σvz (middle panel) shows that S3.0,4.0 is much colder kinematically than the older cohorts but the

population’s heating rate (∆σvz /∆t) is twice that of S1.0,2.0. The older populations

grew colder at larger radius but the S3.0,4.0 population has a positive σvz gradient; this feature is most likely due to disk-like kinematics at smaller radii giving way to

141 halo kinematics past the break radius of the S3.0,4.0 disk. During S3.0,4.0’s evolution, median height increases by several hundred parsecs at rgc< 1 kpc and perceptible increases continue out to rgc∼ 5 kpc.

tform = [7.0, 8.0] Gyr S7.0,8.0 is illustrative of stellar populations born after redshift one: it forms entirely in the disk, exhibits prominent spiral structure, and secularly evolves (Figure 5.13). The initial disk of S7.0,8.0 is thin and shows obvious warping in its outskirts at early times (top left). As time progresses, the random motions of the stars dissipate the spiral overdensities and a small ’bulge-like’ component grows in the central region of the galaxy. The edge-on view shows that the initial outer disk warps weaken and the cohort’s disk becomes more vertically extended with time.

This cohort has a steep central mass component that gives way to an exponential proﬁle in the range of 2.5 ≤ rgc ≤ 14 kpc (left panel, Figure 5.14).

Like S3.0,4.0, the inner disk loses mass while the outer disk accumulates it; here, however, the radius dividing the inner and outer disk is rgc= 11 kpc (7.0 kpc for

S3.0,4.0). The inner disk, born very cold, experiences relatively strong vertical heating

while the outer disk experiences negative ∆σvz /∆t until its σvz is only slightly higher than the inner disk’s. Edge-on density plots revealed a bulge-like structure in this cohort at late times; we ﬁnd that particles in the most central few hundred parsecs heat up signiﬁcantly with time and move further away from the plane (right panel, Figure 5.14). The median height of particles within rgc= 15 kpc increases

142 monotonically with time and the median height growth rate increases with radius in the range 1 < rgc < 15 kpc; i.e., S7.0,8.0’s disk is progressively more vertically extended as a function of both radius and time.

5.4. Migration

We now take a more microscopic view of the age cohorts’ evolution by examining how they move in and out of radial annuli over time. Each panel in Figure 5.15 isolates particles residing in diﬀerent areas of the disk at z = 0 and shows the distribution of formation radius (Rform) of these particles as a function of age; Rform is simply the two dimensional radius of the particle in the simulation snapshot most closely following the particle’s birth (∆t between snapshots never exceeded 0.5 Gyr).

In this section, we demand that particles reside within 4 kpc of the galactic plane at z = 0 to concentrate on the migratory patterns of particles that would be associated with the disk in an observational survey. The middle panel focuses on particles found in the present day solar annulus (7 ≤ rgc ≤ 9 kpc) at z = 0. Here, the eldest cohort, S0.0,0.2, comes from a broad distribution of Rform as many of its members were still accreting onto the parent halo after their formation; at the solar radius, the

S0.0,2.0 particles are predominantly associated with the halo (Section 5.2) and only contribute 7% of the mass at this annulus (see legend in panel). Older cohorts are born at smaller radii than younger cohorts when tform> 2 Gyr; the Rform distribution

143 peaks at approximately 4.0, 6.5, 7.5, and 8.0 kpc for S2.0,4.0, S4.0,8.0, S8.0,12.8, and

S12.8,13.4, respectively. The same correlation between age and formation radius exists both inside (left panel) and outside (right panel) the solar annulus. Older cohorts show less variation in their Rform distributions as a function of radius while the disparity in Rform amongst the age cohorts increases as one moves outwards in the disk; e.g., the overall shape of S2.0,4.0’s Rform distribution is similar in the three radial annuli while the peak in Rform for S12.8,13.4 almost matches the midpoint of the radial bins at rgc= 4.5, 8.0, and 11.5 kpc.

Figure 5.16 begins to investigate how particles moved from Rform to their current position. Particles can move away from their Rform via radial motion in their orbit or by changing their guiding center radius (Rg). If a particle has angular momentum Lz, its Rg is the radius of a purely circular orbit with Lz. There are several mechanisms that can modify a particle’s (or star’s) Rg including resonant interactions with corotational spiral waves (e.g. Sellwood & Binney 2002; Roˇskar et al. 2008b) and gravitational perturbations from satellite accretion events (Quillen et al. 2009; Bird et al. 2012). We plot each age cohort’s current Rg distribution as a function of final radius in Figure 5.16. Older cohorts still have smaller Rg than their younger counterparts in all three areas of the disk, but the collection of the cohorts’ Rg distributions differ less than their Rform distributions. We note an exception: the guiding centers of the S2.0,4.0 population at 11 ≤ rgc ≤ 12 kpc are significantly different and smaller than that of younger cohorts. The length of the

144 radial excursions from particles’ guiding centers correlates well with age across the entire disk.

We quantify the degree to which age cohorts change their Rg in Figure 5.17.

To compute ∆Rgc, we ﬁrst calculate the particle’s initial Rg at the snapshot corresponding to Rform; ∆Rgc is then Rg(z=0)− initial Rg. The inner galaxy is dominated by particles with no signiﬁcant change in their guiding center radii and the tails of the ∆Rgc distributions are symmetric about ∆Rgc= 0 (left panel). In the solar annulus, both the oldest and the youngest stars predominantly have ∆Rg= 0 while the intermediate-aged populations, 2.0 ≤ tform ≤ 8.0 Gyr, have a positively skewed ∆Rgc distribution; the median ∆Rgc is 0.74 kpc and 0.53 kpc for S2.0,4.0 and

S4.0,8.0, respectively. In the outer disk, almost every cohort has a positive ∆Rgc tail.

The ∆Rgc distributions are again most positive for the intermediate populations; the median ∆Rgc of S2.0,4.0 is 0.54 kpc, rises to 1.86 kpc for S4.0,8.0, and falls back to 0.60 kpc for S8.0,12.8. Still, the vast majority of particles in the outer disk annulus had initial Rg outside the solar annulus; only a subset of intermediate-aged particles make signiﬁcant Rg changes to travel to the outer disk. We defer a detailed investigation of the physical mechanisms responsible for this migration to a later work.

145 5.5. Summary

We examine the kinematic evolution of mono-age stellar particle populations in the state of the art, high resolution, cosmological ErisLT simulation of the formation of a realistic spiral galaxy. Dissecting the stellar component of the galaxy by age, rather than chemistry makes it easier to determine the dynamical processes relevant to stellar populations born at each epoch and form a connection between those dynamical processes and the current, observable conﬁguration of each age cohort.

At late times, younger age cohorts form structure with longer scale lengths and shorter scale heights than their older counterparts (Figure 5.1). We perform a kinematic decomposition of the galaxy at z = 0 and indeed ﬁnd that the oldest cohorts typically populate the bulge and halo, intermediately-aged cohorts reside in both the spheroid and disk structures, and younger cohorts form increasingly thinner disks. Recently, Bovy et al. (2012) found a smooth transition from short and thick to long and thin structures when looking at progressively more metal-rich, alpha-poor mono-abundance populations in the SEGUE survey. Here, we ﬁnd a similar transition from old spheroid to young disk that is remarkably smooth.

The stellar structure evident in the current epoch of ErisLT is quite similar to that our own MW; dissecting the galaxy by age at z = 0 breaks this complex structure into straightforward, well-described components. We recover many of the scaling relations observed in the Milky Way (MW) and other spiral galaxies.

146 Compared to young cohorts, older populations are kinematically hotter in the vertical and radial directions, have larger scale heights over a wide radial range, and are on less circular orbits. Despite the general trend of younger cohorts having longer scale lengths, the two youngest age cohorts show breaks in their radial mass proﬁles at a smaller galacto-centric radii than older cohorts. Integrated light and spectroscopic studies of nearby disk galaxy outskirts ﬁnd similar results (e.g.

Radburn-Smith et al. 2012). While we defer an investigation into the dynamical origin of the break radius to a future work, we note that the break radii of the older cohorts are larger than their younger counterparts even at early times; the migration of additional old stars outwards increases the diﬀerence in radial scale length for the younger and older populations.

The vertical mass profiles of individual age cohorts smoothly transition from older cohorts with flatter mass profiles to young cohorts with approximately exponential profiles. Remarkably, the superposition of all age cohorts in the solar annulus results in the familiar double-exponential profile observed in MW star counts by e.g., Gilmore & Reid (1983); Jurićet al. (2008). In ErisLT, the scale height of the older, thicker component at the solar annulus is 1.47 kpc and the younger, thinner

component is 0.55 kpc. While these results do not match the zh1 = 300 pc and

zh2 = 900 pc measured by Juri´cet al. (2008), the ratio of thick to thin component is the same in both ErisLT and the MW. The double-exponential shape seems to be a generic result from the mixture of old, kinematically hot and young, kinematically

147 cold populations. Moving outwards in the disk, the scale heights of all cohorts increase; as a result, the ratio of the thin to thick component’s scale height decreases with radius.

In Section 5.3, we ﬁnd that the late-time kinematic properties of older cohorts are in place shortly after their formation. Older stellar populations are born relatively hot and stay that way; furthermore, older cohorts have a lower late-time heating rate than younger populations. The age of the cohort relative to the major merger epoch (tMM ≈ 3-4 Gyr) is a relevant parameter. S0.0,0.2 has predominantly congregated in the parent halo by tMM but is relatively unaﬀected by the merger epoch because the high random motions of its stars decrease its phase-space density and lower the dynamical friction between the perturbing mergers and S0.0,0.2. The intermediate cohorts, S2.0,5.0, are born during the major merger epoch. Just prior to tMM, the existing stars of S2.0,5.0 represent the youngest, coldest stars in the galaxy and will have a larger dynamical cross-section during the imminent satellite accretion than the particles in S0.0,2.0. As the merger epoch proceeds, S2.0,5.0’s population grows via merger induced star formation in the disk (Figure 5.11) or star formation in major satellite galaxies that is subsequently stripped/accreted onto the parent

galaxy. The S2.0,5.0 population shows the largest σvz and σvr of any cohort harboring a signiﬁcant fraction of particles with disk-like kinematics. As time progresses , stars form in increasingly thinner, colder disks. These colder, younger populations are more susceptible to kinematic heating over time, i.e., ∆σvz /∆t is inversely correlated σvz

148 with age. Older cohorts have larger absolute velocity dispersions and kinematically evolve more quickly than the colder, younger cohorts that steadily heat up over time.

The spatial descriptions of the cohorts reﬂect these heating rates: the surface mass density and median height radial proﬁles of older age cohorts resemble their z = 0 state more quickly than younger cohorts.

We investigated each cohort’s migratory behavior in the inner disk, solar annulus, and outer disk (section 5.4). Older stars emigrated from their formation radii via strongly radial orbits, guiding center change, or some combination thereof.

The ratio of stars migrating and changing their guiding center radii to stars observed towards the apocenter of radial orbits increases with the ﬁnal galactocentric radius considered. S2.0,4.0 and S4.0,8.0 have the largest median ∆Rgc when migrating outwards. Still, the majority of stars in the outer disk (11 ≤ rgc ≤ 12 kpc) were born outside the solar annulus.

The spatial and kinematic descriptions of each stellar cohort vary smoothly.

This behavior is unexpected, especially for the older cohorts; some of these stars are born in completely independent halos and should therefore know little about one another. However, the old cohorts are subjected to a turbulent accretion process at early times, enabling the populations to fully phase-mix over time. At later times, the cohorts form singularly, similar to the monolithic collapse scenario presented in

Eggen et al. (1962). This early and late phase evolution, coupled with intermediate

149 populations bridging the merger-dominated past with the quiescent present, are most likely generic results for the formation of L⋆ galaxies.

150 ]

c 3

p 1 −1 [k −3 z

10 ] c p 0 [k y

−10 ]

c 3 −10 0 10

p 1 −1 [k −3 z 2.0 )

10 2 c p

1.2 / ] ⊙ c M p 0 ( [k og l

y 0.4

−10 −0.4

−10 0 10 −10 0 10 −10 0 10 x [kpc] x [kpc] x [kpc]

Fig. 5.1.— The z = 0 spatial density of stellar particles in the simulated galaxy as a function of position and age. Each panel shows the face-on and edge-on views of a single age cohort (age bins denoted in the bottom right corner of each panel); pixel color represents the logarithmic surface density at the pixel position and in the speciﬁed age bin (see colorbar). The coordinate system is such that the disk lies in the xy plane and the galaxy’s angular momentum vector is aligned with the z axis. Younger stellar populations have longer scale lengths and shorter scale heights than older populations.

151 τform=[0.0,0.5] Gyr 109 τ ] form=[0.5,1.0] Gyr ] c

2 0 8 p 10 τ c 10 form=[1.0,2.0] Gyr [k

kp τ z form=[2.0,4.0] Gyr / 7 n ⊙ 10 τ a form=[4.0,8.0] Gyr i M d [ τ 6 e form=[8.0,12.8] Gyr Σ 10 10−1 M τform=[12.8,13.4] Gyr 105 Total

0 5 10 15 20 0 5 10 15 20 rgc [kpc] rgc [kpc]

Fig. 5.2.— The radial proﬁles of surface mass density and median height above the disk plane as a function of age at redshift zero. The color of each radial proﬁle indicates the plotted age cohort; colors progress from red for old stars to blue for young stars (see legend for details). We also plot the surface mass density of all stars in the left panel (black line).

10−1 r = [4.0,5.0] kpc 10−2 r = [7.0,9.0] kpc r = [11.0,12.0] kpc gc gc gc τ z =0.34 z =0.55 z =0.87 form=[0.5,1.0] Gyr h1 h1 10−3 h1 zh2 =1.09 zh2 =1.47 zh2 =3.31 τ ] form=[1.0,2.0] Gyr 3 c −2 −3 τ 2 0 4 0 Gyr p 10 10 form=[ . , . ] / τ ⊙ form=[4.0,8.0] Gyr

M −4 [ 10 τform=[8.0,12.8] Gyr −4 ρ −3 10 10 τform=[12.8,13.4] Gyr Total

10−5 10−5 0.0 1.0 2.0 3.0 4.0 0.0 1.0 2.0 3.0 4.0 0.0 1.0 2.0 3.0 4.0 rgc [kpc] rgc [kpc] rgc [kpc]

Fig. 5.3.— Vertical stellar mass density profiles for the three labeled annuli in the disk (from left to right: rgc= [4.0, 5.0], [7.0, 9.0], and [11.0, 12.0] kpc). The y axis is the mass per unit volume in units of solar masses per cubic parsec. The x axis shows the same range in height above the plane (in kiloparsecs) for all panels. Within each panel, we show the vertical density profile for the individual age cohorts; the color-coding is the same as in Figure 5.2 (see legend for details). We plot the vertical mass density profile of all stars in each annuli with black lines. We fit two exponential profiles to the total mass density in each panel (gray, dashed lines); we note the scaleheight of each fit in the upper right of each panel. Note that there is no significant S0.0,0.5 population at these annuli.

152 160

140 140

120 120 ] ] s s / 100 / 100 m m [k 80 [k 80 r z v v σ 60 σ 60

40 40 τform=[0.0,0.5] Gyr 20 20 τform=[0.5,1.0] Gyr 0 5 10 15 20 0 5 10 15 20 τform=[1.0,2.0] Gyr rgc [kpc] rgc [kpc] τform=[2.0,4.0] Gyr τ =[4.0,8.0] Gyr 0.35 form 0.8 τform=[8.0,12.8] Gyr 0.30 τ ) form=[12.8,13.4] Gyr

ε ( 0.6 0.25

n ε a σ i

d 0.20

e 0.4

M 0.15

0.2 0.10

0.05 0 5 10 15 20 0 5 10 15 20 rgc [kpc] rgc [kpc]

Fig. 5.4.— The radial proﬁles of four kinematic parameters as a function of stellar age (at redshift zero). The parameters include vertical velocity dispersion, radial velocity dispersion, median circularity, and circularity dispersion from left to right and top to bottom, respectively. The color of each radial proﬁle indicates the age cohort; colors progress from red for old stars to blue for young stars (see legend for details).

153 ] 30 ] c c 20

p 10 p 0 −10 [k −30 [k

z z −40

20 ] ] c c

p p 0 [k [k 2.0 y y −20

−40 ) 2 c p

−20 20 −40 0 40 1.2 / ⊙

x [kpc] x [kpc] M ] ] (

c 3 c 3 og p 1 p 1 −1 −1 l [k −3 [k −3 z z 0.4

10 10 ] ] c c −0.4 p 0 p 0 [k [k y y

−10 −10

−10 0 10 −10 0 10 x [kpc] x [kpc]

Fig. 5.5.— The surface density of S0.0,0.5 as a function of time. Edge-on and face- on views are shown at the age of each snapshot (labeled in units of gigayears in the bottom right hand corner of each panel). Pixel color represents the logarithmic surface mass density at the pixel position (see colorbar). The coordinate system is such that the disk lies in the xy plane and the galaxy’s angular momentum vector is aligned with the z axis. Note the changing scale: for clarity, we choose axis limits that encompass 95% of the cohort’s mass in a given snapshot until reaching the limits used in Figure 5.1;i.e., (|x|, |y|, |z|) = (17.5, 17.5, 4) kpc.

154 108 160 140

7 ] ] 10 c 120 2 p ] c s

0 / [k 10 t=0.59 Gyr 100 kp m z

/ t=1.08 Gyr

6 n [k ⊙ 10

a 80 z i t=2.05 Gyr v M d [ σ e t=4.05 Gyr 60 Σ 105 M t=8.14 Gyr t=11.53 Gyr 40 t=13.33 Gyr 20 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 rgc [kpc] rgc [kpc] rgc [kpc]

Fig. 5.6.— The radial profiles of the surface mass density, median height above the disk plane, and vertical velocity dispersion for S0.0,0.5 as a function of time (left to right panels, respectively). The color of each line represents the age of the simulation when the profile was measured (see legend in middle panel). Note that the colors progress from red for early times to blue for late times; we adopt this color scheme throughout the rest of the figures.

155 ] ] 4 c 20 c 2 p 0 p 0

[k −20 [k −2

z z −4

40 10 ] ] c c

p 0 p 0 [k [k 2.0 y y

−10 −40 ) 2 c p

−40 0 40 −10 0 10 1.2 / ⊙

x [kpc] x [kpc] M ] ] 4 ( c c 3 og p p 1 0 −1 l [k −4 [k −3 z z 0.4

10 10 ] ] c c −0.4 p 0 p 0 [k [k y y

−10 −10

−10 0 10 −10 0 10 x [kpc] x [kpc]

Fig. 5.7.— The spatial conﬁguration of S0.5,1.0 as a function of time. All symbols and colors have the same meaning as in Figure 5.5.

156 109 140

8 10 ] 120 ] c 1 2 p 10 ] c s / 107 [k t=1.08 Gyr 100 kp m z

/ t=2.05 Gyr n [k ⊙ 80

6 a z 10 i t=4.05 Gyr v M d [ σ e 100 t=6.04 Gyr 60 Σ 105 M t=8.14 Gyr 40 t=11.53 Gyr 4 10 t=13.33 Gyr 20 0 10 20 0 10 20 0 10 20 rgc [kpc] rgc [kpc] rgc [kpc]

Fig. 5.8.— The radial proﬁles of the surface mass density, median height above the disk plane, and vertical velocity dispersion for S0.5,1.0 as a function of time (left to right panels, respectively). All symbols and colors have the same meaning as in Figure 5.6. The range in rgc is now [0, 20] kpc.

157 ] ]

c 10 c 5 p 0 p

[k [k − −10 5 z z

10 10 ] ] c c

p 0 p 0 [k [k 2.0 y y

−10 −10 ) 2 c p

−10 0 10 −10 0 10 1.2 / ⊙

x [kpc] x [kpc] M ] ] (

c 4 c 4 og p p

0 0 l [k −4 [k −4 z z 0.4

10 10 ] ] c c −0.4 p 0 p 0 [k [k y y

−10 −10

−10 0 10 −10 0 10 x [kpc] x [kpc]

Fig. 5.9.— Same as in Figure 5.5 but for S1.0,2.0.

158 101 109 140

] 120 ] c 2 p 8 ] c

10 s / [k

kp 100 m z

/ t=2.05 Gyr n [k ⊙

7 a z 10 i 0 t=4.05 Gyr v M 10 80 d [ σ e t=6.04 Gyr Σ M t=8.14 Gyr 60 106 t=11.53 Gyr t=13.33 Gyr 40 0 10 20 0 10 20 0 10 20 rgc [kpc] rgc [kpc] rgc [kpc]

Fig. 5.10.— Same as in Figure 5.8 but for S1.0,2.0.

159 ] ]

c 10 c 6

p p 2 0 −2 [k −10 [k −6 z z

10 10 ] ] c c

p p 0 [k [k 2.0 y y −10 −10 ) 2 c p

−10 10 −10 0 10 1.2 / ⊙

x [kpc] x [kpc] M ] ] 6 6 ( c c og p 2 p 2 −2 −2 l [k [k −6 − z z 6 0.4

10 10 ] ] c c −0.4 p p [k [k y y

−10 −10

−10 10 −10 10 x [kpc] x [kpc]

Fig. 5.11.— Same as in Figure 5.5 but for S3.0,4.0.

160 109 100

8 90

10 ] ] c 2 p ] c

s 80 /

[k 0 7 10 kp 10 m z / t=4.05 Gyr 70 n [k ⊙ a z i t=5.04 Gyr v M

6 d [ 10 σ 60 e t=7.04 Gyr Σ M t=9.04 Gyr 50 − 105 10 1 t=11.53 Gyr t=13.33 Gyr 40 0 10 20 0 10 20 0 10 20 rgc [kpc] rgc [kpc] rgc [kpc]

Fig. 5.12.— Same as in Figure 5.8 but for S3.0,4.0.

161 ] ]

c 3 c 3

p 1 p 1 −1 −1 [k −3 [k −3 z z

10 10 ] ] c c

p 0 p 0 [k [k 2.0 y y

−10 −10 ) 2 c

1 2 p −10 0 10 −10 0 10 . / ⊙

x [kpc] x [kpc] M ] ] (

c 3 c 3 og p 1 p 1 −1 −1 l [k −3 [k −3 z z 0.4

10 10 ] ] c c −0.4 p 0 p 0 [k [k y y

−10 −10

−10 0 10 −10 0 10 x [kpc] x [kpc]

Fig. 5.13.— Same as in Figure 5.5 but for S7.0,8.0.

162 108 70 ] ] c 0 60 2 p 7 10 ]

c 10 s / [k kp m z 50 / n [k

⊙ 6

10 a z i t=8.14 Gyr v M d

[ 40 σ e t=9.04 Gyr Σ

105 M −1 10 t=9.87 Gyr 30 t=11.53 Gyr 104 t=13.33 Gyr 20 0 10 20 0 10 20 0 10 20 rgc [kpc] rgc [kpc] rgc [kpc]

Fig. 5.14.— Same as in Figure 5.8 but for S7.0,8.0.

0.9 0.45 0.10 0.07 0.09 rgc=[4.0,5.0] rgc=[7.0,9.0] 0.7 rgc=[11.0,12.0]

y 0.31 0.15 0.11

it 0.7 0.35 s 0.37 0.34 0.27 τ =[0.0,2.0] Gyr n form

e 0.20 0.38 0.52 0.5 τ =[2.0,4.0] Gyr

D form 0.5 0.02 0.25 0.05 0.02 y τform=[4.0,8.0] Gyr

ilit τform=[8.0,12.8] Gyr b 0.3

a 0.3 0.15 τform=[12.8,13.4] Gyr ob r P 0.1 0.05 0.1

0 2 4 6 8 0 4 8 12 0 4 8 12 16 rgc [kpc] rgc [kpc] rgc [kpc]

Fig. 5.15.— The formation radius of particles, binned into age cohorts, residing at rgc= [4.0, 5.0], rgc= [7.0, 9.0], and rgc= [11.0, 12.0] at redshift zero (left to right, respectively). Each cohort’s percentage mass contribution to each ﬁnal annuli is listed in the legend of each panel.

0.9 0.5 0.7 0.10 0.07 0.09 rgc=[4.0,5.0] rgc=[7.0,9.0] rgc=[11.0,12.0]

y 0.31 0.15 0.11

it 0.7 s 0.37 0.34 0.27 τ =[0.0,2.0] Gyr n 0.5 form

e 0.20 0.38 0.52 τ =[2.0,4.0] Gyr D 0.3 form 0.5 0.02 0.05 0.02 y τform=[4.0,8.0] Gyr

ilit 0.3 τform=[8.0,12.8] Gyr b

a 0.3 τform=[12.8,13.4] Gyr ob

r 0.1 P 0.1 0.1

0 2 4 6 8 0 4 8 12 0 4 8 12 16 Rg [kpc] Rg [kpc] Rg [kpc]

Fig. 5.16.— The current guiding center radius of particles, binned into age cohorts, residing at rgc= [4.0, 5.0], rgc= [7.0, 9.0], and rgc= [11.0, 12.0] at redshift zero (left to right, respectively). The legends of each panel are the same as in Figure 5.15.

163 1.4 0.10 0.07 0.09 1.4 rgc=[4.0,5.0] 1.4 rgc=[7.0,9.0] rgc=[11.0,12.0] y 0.31 0.15 0.11 it s 0.37 0.34 0.27 τ =[0.0,2.0] Gyr n 1.0 form

e 0.20 0.38 0.52 1.0 1.0 τ =[2.0,4.0] Gyr D 0.02 0.05 0.02 form y τform=[4.0,8.0] Gyr

ilit 0.6 τform=[8.0,12.8] Gyr b 0.6 0.6 a τform=[12.8,13.4] Gyr ob r

P 0.2 0.2 0.2

−2 2 −2 2 −4 0 4 ∆Rg [kpc] ∆Rg [kpc] ∆Rg [kpc]

Fig. 5.17.— The change in guiding center radius of particles, binned into age cohorts, residing at rgc= [4.0, 5.0], rgc= [7.0, 9.0], and rgc= [11.0, 12.0] at redshift zero (left to right, respectively). Positive ∆Rgc indicates the particle has migrated outwards; particles migrating inwards have negative ∆Rgc. The legends of each panel are the same as in Figure 5.15.

164 Age Mass Fraction fthin fthick fpseudo fbulge fhalo

(0.0,0.5) 0.002 0.017 0.011 0.229 0.696 0.048 (0.5,1.0) 0.058 0.019 0.055 0.188 0.551 0.188 (1.0,2.0) 0.186 0.095 0.115 0.258 0.378 0.153 (2.0,4.0) 0.313 0.420 0.152 0.221 0.147 0.060 (4.0,8.0) 0.260 0.712 0.049 0.102 0.095 0.042 (8.0,12.8) 0.165 0.828 0.015 0.088 0.055 0.014 (12.8,13.4) 0.017 0.861 0.023 0.082 0.020 0.013 Total 1.000 0.486 0.088 0.171 0.184 0.072

Table 5.1. Kinematic Decomposition

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