Stellar Multiplicity Analysis with Time-Resolved Spectroscopy and Markov Chain Monte Carlo Simulations

STELLAR MULTIPLICITY ANALYSIS WITH TIME-RESOLVED SPECTROSCOPY AND MARKOV CHAIN MONTE CARLO SIMULATIONS

Thomas Barrett Hettinger

A DISSERTATION

Submitted to Michigan State University in partial fulﬁllment of the requirements for the degree of

Astrophysics and Astronomy — Doctor of Philosophy

2015 ABSTRACT

STELLAR MULTIPLICITY ANALYSIS WITH TIME-RESOLVED SPECTROSCOPY AND MARKOV CHAIN MONTE CARLO SIMULATIONS

Thomas Barrett Hettinger

This dissertation examines the multiplicity properties of stars in the Milky Way and their relationship with metallicity. We present methods and techniques for data mining individual, raw, sub-exposure information from spectroscopic surveys as a statistical approach to performing scientiﬁc analyses in this era of Big Data. We also describe how Bayesian inference and Markov Chain Monte Carlo simulations work in conjunction with these sub-exposure spectroscopy techniques.

Binary interactions play a key role in many astrophysical processes, from altering surfaces abundances, to producing supernova. In Chapter 1, we give a brief introduction to stellar multiplicity, beginning with a description of the star formation process and possible scenarios for binary star formation. We discuss how binary stars interact through Roche-lobe overﬂow, and how binary systems lead to various astrophysical phenomena. We conclude the chapter with a look at our current understanding of multiplicity properties of stars in the Milky

Way as determined empirically from observations and surveys, and with a discussion for the future outlook of multiplicity studies.

In Chapter 2 we describe a methodology for measuring radial velocity variations in stellar sources using sub-exposure spectra from multi-ﬁber spectroscopic surveys. In particular, we describe a cross-correlation technique used on spectra that were observed as part of the SDSS survey. In Chapter 3 we give a brief introduction to Bayesian inference and the use of the MCMC python package emcee. We describe the methods used for detecting binarity in stellar sources from sparsely sampled radial velocity curves.

Chapter 4 contains the peer-reviewed article Hettinger et al. (2015) published in the

Astrophysical Journal Letters. In this Letter, we employ the sub-exposure radial velocity measurement techniques and the MCMC methods outlined in this dissertation to examine a population of F-type dwarf stars in the Milky Way. The sample was divided into three groups by metallicity, with the goal of investigating the metallicity dependence on multiplicity properties. We ﬁnd a higher fraction of short-period binaries for the metal-rich disk stars than the metal-poor halo stars.

Finally, in Chapter 5, we extend the work of Hettinger et al. (2015) to investigate possible constraints on the separation distribution of binaries in the F-dwarf population. For my mother, Rita Murray.

iv ACKNOWLEDGMENTS

Firstly, I would like to express my sincere gratitude to my advisors and collaborators, Carles

Badenes, Jay Strader, Timothy Beers, and Steven Bickerton, for their continuous support, motivation, and immense knowledge. I especially thank Carles for taking me on as his own student, and committing to work with me remotely across universities. His guidance was invaluable, and I am thankful to have had him as my advisor.

To my wife, Mengling Hettinger, I am deeply grateful. As a colleague, she provided me with direction, focus, and conﬁdence. Her tutoring in my course work and her support were crucial to my success. I could not have chosen a better partner to share the rest of my life experiences with. I love you Mengling.

I would like to thank my senior graduate students, Charles Kuehn, Chris Richardson,

Carolyn Peruta, and Aaron Hoﬀer for their advice; my oﬃce mates Alex Deibel, Brian

Crosby, Tom Connor, and Ryan Connolly for their help with brainstorming and for my addiction to coﬀee; and I would like to thank coﬀee.

For their work in outreach and community service, I thank John French, Shane Horvatin, and Dave Batch at the Abrams Planetarium, as well as Horace Smith and Laura Chomiuk for their work with public events at the telescope.

I am grateful to the NSF, NASA, and the DoE for funding the majority of all astrophysics research. Astronomy is the least applicable of the sciences, but it asks the biggest questions.

We are fortunate enough to have the capacity to ask these questions, therefore we must.

I thank all my family and friends for their continuing support and encouragement. Finally,

I want to thank Jason Linehan for introducing me to the night sky, without whom I would never have pursued astronomy.

v TABLE OF CONTENTS

LIST OF TABLES ...... viii

LIST OF FIGURES ...... ix

Chapter 1 A Brief Introduction to Stellar Multiplicity ...... 1 1.1 Introduction ...... 1 1.2 Star Formation and the Origin of Binary Stars ...... 2 1.2.1 Physical Principles ...... 2 1.2.2 Possible Binary Formation Mechanisms ...... 6 1.2.2.1 Capture ...... 6 1.2.2.2 Prompt Fragmentation ...... 6 1.2.2.3 Delayed Break-up ...... 7 1.2.3 Questions and Future Investigation ...... 8 1.3 Binary Evolution and Interactions ...... 10 1.3.1 Principles of the Evolution of Binary Systems ...... 10 1.3.2 Binary Interaction in Astrophysical Phenomena ...... 12 1.3.2.1 Hot Subdwarfs ...... 13 1.3.2.2 Chemically Peculiar Stars ...... 13 1.3.2.3 Symbiotic Binaries ...... 14 1.3.2.4 Blue Stragglers ...... 14 1.3.2.5 Thermonuclear Supernovae ...... 15 1.3.2.6 Core Collapse Supernovae ...... 15 1.4 Empirically Derived Multiplicity Properties ...... 16 1.4.1 Multiplicity Properties and Survey Methodology ...... 16 1.4.2 Trends and Characteristics of Multiple-Star Systems ...... 18 1.4.2.1 Multiplicity and Mass ...... 18 1.4.2.2 Multiplicity and Age ...... 22 1.4.3 Discussion ...... 23 1.5 Conclusion ...... 26

Chapter 2 Time-Resolved Spectroscopy ...... 28 2.1 Introduction ...... 28 2.2 The Sloan Digital Sky Survey ...... 30 2.2.1 Sub-Exposures ...... 31 2.2.2 SEGUE Stellar Parameter Pipeline And Sample Selection ...... 32 2.2.3 Plate Systematics ...... 34 2.3 Radial Velocities ...... 41 2.3.1 Continuum Normalization ...... 41 2.3.2 Spectral Template ...... 44

vi 2.3.3 Cross-Correlations ...... 45 2.4 Empirical Uncertainties ...... 48 2.5 e/i Variability ...... 53 2.6 Discussion ...... 53

Chapter 3 Markov Chain Monte Carlo ...... 55 3.1 Introduction ...... 55 3.2 Bayesian Inference and MCMC ...... 56 3.3 emcee: The MCMC Hammer ...... 57 3.3.1 An Aﬃne-Invariant Ensemble Sampler ...... 57 3.3.2 Using emcee ...... 58 3.4 Correcting Systematics in SDSS Spectra ...... 63 3.5 Modeling Multiplicity With Radial Velocity Curves ...... 64 3.5.1 Examples ...... 68

Chapter 4 Statistical Time-Resolved Spectroscopy: A Higher Fraction of Short-Period Binaries for Metal-Rich F-type Dwarfs in SDSS 77 4.1 Abstract ...... 77 4.2 Introduction ...... 78 4.3 Measurements ...... 80 4.3.1 SDSS Observations and Sample Selection ...... 80 4.3.2 Radial Velocities ...... 82 4.3.3 Uncertainties ...... 83 4.4 Multiplicity ...... 84 4.5 Discussion ...... 89

Chapter 5 Binary Fractions and Separation Distributions ...... 92 5.1 Introduction ...... 92 5.2 MCMC and Population-Wide Monte Carlo ...... 93 5.3 Discussion ...... 100

REFERENCES ...... 103

vii LIST OF TABLES

Table 2.1 Suspect Plates in SDSS ...... 39

Table 2.2 Absorption Features in F-dwarfs ...... 42

Table 3.1 Prior Limits for Hettinger et al. (2015) MCMC ...... 67

viii LIST OF FIGURES

Figure 1.1 Dependency of multiplicity fraction with primary mass for main sequence stars and VLM objects. Values used from the review by Duchˆene& Kraus (2013)...... 21

Figure 1.2 Left: Orbital period distribution in the solar neighborhood from Raghavan et al. (2010). The limit for RLOF in the MS is indicated by the black vertical dashed line, and the range corresponding to pre-CE systems is shaded in gray. The best-fit log-normal function is shown in black. The dashed red plot represents a modified function that also fits the data. Right: Comparing the number of systems in the best-fit log-normal distribution (solid black) and the modified model (dashed red). The period ranges corresponding to low-mass X- ray binary progenitors, stable habitable planets around binary stars, and SN Ia progenitors are shown with horizontal rulers. The yellow dash-dotted line marks the pre-outburst period of V1309 Sco. . . . . 25

Figure 2.1 Distribution of the number of sub-exposures (top) and the time lags (bottom) for the F-dwarf stars from the Hettinger et al. (2015) sample. Metallicity cutoﬀ values are [Fe/H] = −1.43 and [Fe/H] = −0.66. 33

Figure 2.2 Distribution of stellar parameters for the F-dwarf stars from the Het- tinger et al. (2015) sample, including metallicity (top), eﬀective temperature (middle), and surface gravity (bottom). Metallicity cutoﬀ values are [Fe/H] = −1.43 and [Fe/H] = −0.66...... 35

Figure 2.3 Scatter plot showing the distribution of metallicity and surface gravity for F-type stars in the SSPP DR9. The bimodal distribution of [Fe/H] traces the Halo and Disk components of the Milky Way. . . . 36

Figure 2.4 RVs for F-dwarf stars located on plate plugging 2085-53379 before and after correcting the plate for systematic sub-exposure offsets. Red points are sub-exposures that have low SNR. Fiber IDs are given for each fiber on the right axis. Sub-exposures in the plate are ordered chronologically. Corrections to systematic offsets are successful on this plate...... 38

Figure 2.5 Distribution of 10,264 systematic RV oﬀsets estimated for all plate sub-exposures in the F-dwarf sample...... 39

ix Figure 2.6 Same as Figure 2.4 for plate plugging 3002-54844. The left column of each subfigure shows RVs before correcting the plate for systematic offsets, and the right column shows RVs after correcting for offsets. Corrections are not successful on this plate due to anti-correlated subsets of similarly correlated fibers...... 40

Figure 2.7 Continuum normalization process applied to an F-type star showing: (a) the raw sub-exposure spectrum, (b) the spectrum with selected absorption features masked out, (c) a smoothed version of the spectrum, and (d) the ﬁnal continuum-normalized spectrum...... 43

Figure 2.8 Full continuum-normalized F-type dwarf template spectrum (top) with a detailed view of the blue end of the spectrum (bottom). Promi- nent spectral features are annotated...... 45

Figure 2.9 Example of the template creation process using 7 spectra. From top to bottom: blue end and full spectrum of the input co-add stellar spectra, blue end and full spectrum of the normalized, rest-frame input spectra, blue end and full spectrum of the input spectra resampled to a common wavelength solution, blue end and full spectrum of the template (averaged ﬂux of resampled input spectra)...... 46

Figure 2.10 Cross-correlation function (per pixel) for a single sub-exposure spectrum of an F-type star. Correlation values are calculated at integer pixel lags, and a spline interpolation of the function is ﬁt to these values. The function peaks at −3.275 pixels, or −229 km s−1. . . . . 47

Figure 2.11 Distribution of radial velocities (minus systemic velocity) for sub- exposures with [Fe/H] = −1.75 ± 0.25 and SNR = 30 ± 2.5. The distribution has a median absolute deviation of MAD = 3.04 km s−1, indicating measurement uncertainties of σ = 4.51 km s−1...... 49

Figure 2.12 Empirical uncertainties for cross-correlation measurements in F-type dwarf stars showing uncertainties (a) as a function of SNR independently determined for each [Fe/H], (b) as a function of SNR with a common constant oﬀset, and (c) as a function of SNR and [Fe/H] as in Equation 2.8...... 51

Figure 2.13 Values of coeﬃcient m from Equation 2.6 as a function of [Fe/H]. . . 52

Figure 2.14 Distribution of empirically assigned measurement uncertainties for the F-dwarf stars from the Hettinger et al. (2015) sample. Metallicity cutoﬀ values are [Fe/H] = −1.43 and [Fe/H] = −0.66...... 52

x Figure 2.15 Distribution of e/i, the ratio of the standard deviation of RVs to the typical measurement uncertainty values, for the F-dwarf stars from the Hettinger et al. (2015) sample. Metallicity cutoﬀ values are [Fe/H] = −1.43 and [Fe/H] = −0.66...... 54

Figure 3.1 Mock observations with X values drawn randomly from the range [0,1] and Y values drawn from Yi = 0.8Xi + 0.3. Simulated measurement uncertainties have been added with values drawn from a normal distribution with µ = 0.0, σ = 0.1...... 59

Figure 3.2 Top: values of m from the MCMC for the ﬁrst 60 steps taken by all 100 chain walkers. Convergence is reached by step 60. Bottom: same for the parameter b...... 60

Figure 3.3 Left: parameter-space location of the ﬁrst 60 steps taken for 7 of 100 chain walkers. Stars represent the inital position of each chain walker, and subsequent steps are displayed as circles of decreasing radius. Right: a closer view of the center of the chain step distribution. 61

Figure 3.4 Linear solutions for 300 randomly selected (m,b) pairs sampled from the MCMC posterior distribution...... 61

Figure 3.5 Posterior probability distribution for: (a) the join distribution of parameters m and b, (b) parameter m, marginalized over parameter b, and (c) parameter b, marginalized over parameter m. Blue lines indicate values of m and b used in the creation of the mock data points. 62

Figure 3.6 Two-parameter joint probability distributions for plate-plugging 2085- 53379. All values are in km s−1. Fully marginalized, single-parameter probability distributions occupy the subplots on the diagonal. Param- eters dyi (i in the text) represent the corrections to be applied to the ith exposure on the plate...... 65

Figure 3.7 Individual sub-exposure spectra (top) used in the production of the coadd spectrum (bottom) for ﬁber ID 2939-54515-194...... 69

Figure 3.8 Parameter value progression for all 200 chain walkers in the multiplicity MCMC for ﬁber ID 2939-54515-194. Chain samples have already been thinned. Samples earlier than the red dashed line were removed from analysis during the burn-in process...... 71

Figure 3.9 Orbits constructed from 200 random samples of the MCMC posterior distribution for ﬁber ID 2939-54515-194...... 72

xi Figure 3.10 Posterior probability distributions of parameters in the multiplicity MCMC for ﬁber ID 2939-54515-194...... 73

Figure 3.11 Changes in redshift with time for ﬁber ID 2960-54561-375. Absorp- tion lines from normalized sub-exposures are ordered chronologically from top to bottom. Velocities in each sub-ﬁgure are relative to the rest-frame wavelength of Calcium K (left), Calcium H (middle), and Hα (right). Dashed vertical lines represent the mean velocity of the star...... 74

Figure 3.12 Orbits constructed from 200 random samples of the MCMC posterior distribution for ﬁber ID 2960-54561-375...... 75

Figure 3.13 Posterior probability distributions of parameters in the multiplicity MCMC for ﬁber ID 2960-54561-375...... 76

Figure 4.1 Left: Metallicity distribution for 14,302 F-dwarfs. Right: Distribu- tion of maximum time lag between the ﬁrst and last exposure of a star...... 82

Figure 4.2 Mean (left) and standard deviation (right) of radial velocities within a star. Variations in the standard deviation of velocities are aﬀected, in part, by the larger measurement uncertainties for metal-poorer stars. 83

Figure 4.3 Distribution of η, the fraction of posterior samples using the binary model, for stars...... 86

Figure 4.4 Averaged probability distributions of log P for all binary detections (η > 0.80). These do not reﬂect actual distributions of periods, and should only be used as a guide to probe the region of MCMC sensitivity. The shaded region indicates where Roche lobe overﬂow and contact becomes relevant. The dashed line marks the circularization limit at a period of 12 days...... 88

Figure 4.5 Short-period binary fraction limits, relative to the metal-rich group. Binary companion detections are deﬁned by a cut in η, the fraction of posterior samples using the binary model. Group median values of [Fe/H] are used...... 89

xii Figure 5.1 Distribution of e/i values for a simulated population based on observations and uncertainties in the metal-rich population of Hettinger et al. (2015). Models illustrate variations in the e/i distribution from changes in separation distribution power law index α, while keeping ﬁxed the short-period binary fraction fb = 0.03 and the mass ratio distribution power law index β = 0.0...... 94

Figure 5.2 Distribution of e/i values as in Figure 5.1, with a varying fb, and ﬁxed α = −1.0 and β = 0.0...... 94

Figure 5.3 Distribution of e/i values as in Figure 5.1, with a varying β, and ﬁxed α = −1.0 and fb = 0.03...... 95

Figure 5.4 e/i distribution for the metal-poor group. Model e/i distributions are shown using values of α and fb randomly sampled from the MCMC posterior. The dashed line represents the cutoﬀ, below which bin heights were not used in the likelihood function...... 97

Figure 5.5 Same distribution as in Figure 5.4, for the metal-intermediate group. 97

Figure 5.6 Same distribution as in Figure 5.4, for the metal-rich group...... 98

Figure 5.7 Posterior distribution for the MCMC run of the metal-poor group, with parameters for the short-period binary fraction fb, and separation distribution α...... 98

Figure 5.8 Posterior distribution for the MCMC run of the metal-intermediate group, with parameters for the short-period binary fraction fb, and separation distribution α...... 99

Figure 5.9 Posterior distribution for the MCMC run of the metal-rich group, with parameters for the short-period binary fraction fb, and separation distribution α...... 99

Figure 5.10 Posterior distribution for the MCMC run of the metal-rich group, with an extended prior limit in α...... 100

Figure 5.11 Preliminary distribution of maximum RV variation for DR12 APOGEE targets...... 102

xiii Chapter 1

A Brief Introduction to Stellar

Multiplicity

1.1 Introduction

Stellar multiplicity plays a key role in star formation (Krumholz et al., 2012; Bate, 2014), stellar evolution (Paxton et al., 2015), the chemical evolution of galaxies (Kobayashi et al.,

2006), and the study of unresolved stellar populations (Conroy, 2013). Many interesting phenomena in astrophysics are related to interacting binary systems. These include the post-common envelope binaries (Schreiber & G¨ansicke, 2003): cataclysmic variables, classical novae, X-ray binaries, gamma-ray bursts, and SN Ia progenitors.

The current understanding of binary statistics is limited due to a diﬃculty in obtaining population samples that are complete and unbiased. Speciﬁcally, the short-period systems that will one day lead to common envelope episodes require expensive spectroscopic observing campaigns. Complete surveys have only been possible in the solar neighborhood (Duquennoy

& Mayor, 1991; Raghavan et al., 2010), and provide only a fraction of the global multiplicity properties of the Milky Way. This motivated Hettinger et al. (2015) to investigate a statistical approach for measuring multiplicity properties of stars through the raw, sub-exposures that comprise each individual spectrum in multi-ﬁber spectroscopic surveys.

1 This chapter acts as a brief introduction to binaries and multiple-star systems. We begin by discussing the basic theory of star formation, and by reviewing the contending mechanisms that aim to explain binary star formation. We also look at the important role that binaries and binary interactions play in our galaxy. We discuss the physical principles involved in mass transfer between companions. Additionally, we look at a few selected examples of interacting systems, and the phenomena resulting from binary interactions. Finally, we discuss what we know about the multiplicity properties of stellar systems in the Galaxy, as we have learned from observations using a variety of techniques. From these observations we outline the trends seen in multiplicity properties, and we discuss areas for improvement where the current understanding of multiple-star systems is limited.

1.2 Star Formation and the Origin of Binary Stars

In this section, we discuss the physical principles of star formation, which will allow us to consider the possible mechanisms for the formation of binary and multiple systems. For a more detailed discussion on the origin of binary stars, we recommend the review by Tohline

(2002), and the references therein.

1.2.1 Physical Principles

The formation of single stars from the collapse of a gravitationally bound molecular cloud can be broken up into a sequence of ﬁve stages, originally outlined by Shu et al. (1987). The molecular cloud is initially supported against collapse by a magnetic ﬁeld (Crutcher, 2012).

In the ﬁrst stage (Stage I), the magnetic ﬁeld leaks out of the over-dense regions of the cloud, allowing the dense regions to form denser cloud cores. In Stage II, a condensing cloud core

2 passes the criteria for Jeans instability (discussed below) and dynamically collapses towards

stellar densities, leading to the formation of a protostar. The surrounding envelope of gas

and dust continues to fall onto the protostar, initiating accretion and the formation of a

circumstellar disk. At Stage III, the infalling material weakens and the stellar winds are

able to break out, creating bipolar ﬂows. With Stage IV, we see a widening of the outﬂow

opening angle, and the protostar becomes a pre-main sequence (PMS) star. Finally, Stage

V is reached after the nebular disk disappears, and the PMS star continues to evolve toward

the main sequence.

It is useful to look at the physical parameters describing the collapse of a cloud core.

The protostellar cloud is characterized by its radius R, mean temperature T , total mass M, mean molecular weight µ, and rotational velocity ω. The mean density of the cloud is

3M ρ = . (1.1) 4πR3

The temperature varies with density upon compression by

T ∝ ρ γ−1 , (1.2)

where γ is the adiabatic exponent of the gas. This exponent is a function of the cloud’s

density, and plays an important role in the formation process. Key timescales involved in

the collapse include the free-fall time,

3π !1/2 t = , (1.3) ﬀ 32Gρ

3 the sound-crossing time, R ts = , (1.4) cs

and the rotation period of the cloud,

2π t = . (1.5) rot ω

Additionally, from Kepler’s Third Law, the binary orbital period is

1/2 4π2a3 ! P = , (1.6) GMtot

where Mtot is the total mass of the system and a is the semi-major axis. For a cloud in

equilibrium, the virial theorem states that

1 E + E = − E , (1.7) therm rot 2 grav

where the total thermal energy Etherm, total rotational energy Erot, and total gravitational potential energy Egrav, are deﬁned as

3RMT E ∼ , (1.8) therm 2µ

MR2ω2 E ∼ , (1.9) rot 5

and 3GM2 E ∼ − . (1.10) grav 5R

The constant R in Equation 1.8 is the gas constant. Finally, for a slowly rotating cloud, the

4 mass of a virialized cloud must be related to the density and temperature, from Equations

1.8 and 1.10, like RT !3/2 M ∼ 5.5 ρ −1/2 . (1.11) equil µG

If, for a given set of physical values, the left-hand side of Equation 1.7 is less than the

right-hand side, the cloud will collapse, on a free-fall timescale, due to a lack of kinetic

support. When this is true, M > Mequil (ignoring rotation), and the Jeans instability

criterion (Jeans, 1919) is met. In this case, Mequil is equivalent to the Jeans mass MJ. Stage

II of star formation begins when a cloud core acquires a mass that exceeds MJ.

The adiabatic exponent γ plays an important role in the collapse of a cloud. The condition for collapse, in the absence of rotation, requires

E 1 α ≡ therm < . (1.12) |Egrav| 2

Using Equations 1.2, 1.8, and 1.10, we reveal the relationship between α and γ,

α ∝ ρ γ−4/3 . (1.13)

According to this equation, if γ < 4/3, the energy ratio decreases during collapse, meaning

thermal pressure cannot stop the free-fall collapse of a cloud as long as the cloud evolves

while holding this relation. Indeed the values of γ during the collapse of a cloud core will

determine the conditions where fragmentation may occur (if any) to produce a binary system.

5 1.2.2 Possible Binary Formation Mechanisms

The possible mechanisms for producing binary stars fall into three broad categories: capture, prompt fragmentation, and delayed break-up. We will brieﬂy discuss these three mechanisms beginning with capture.

1.2.2.1 Capture

The capture mechanism states that most stars form into single-star systems, and that binaries are formed when a companion is captured through gravitational interactions after formation.

In order for two stars to bind, some fraction of energy must be dissipated (Clarke, 1992).

This can be accomplished through transfer of energy in a three-body system, but these interactions are rare outside of the dense environments of star clusters. The low frequency of these interactions indicates that the capture mechanism cannot be the main channel for binary formation. In fact, the PMS stellar population is rich in multiple systems, comparable to main-sequence stars, suggesting a primary mechanism that begins to work before Stage

IV of formation (Mathieu, 1994).

1.2.2.2 Prompt Fragmentation

In the prompt fragmentation mechanism, the initial angular momentum of the gas cloud causes it to spontaneously break into two pieces, during or just after the free-fall phase.

Collapse of the molecular cloud can be divided into two categories, homologous and non- homologous.

In a homologous collapse, the cloud is mostly spherical and uniform in density, with a cloud mass signiﬁcantly larger than MJ. Because α decreases during the isothermal phase of the collapse, the local MJ also decreases, and the cloud closely approximates a pressure-free

6 spheroid. Virtually all of the mass reaches the ﬁnal conﬁguration at the same time, and in

one free-fall time the mass accretion rate becomes very high.

On the other hand, a cloud can collapse in a non-homologous manner. This occurs if the

cloud is only marginally Jeans unstable and the density is centrally condensed. The regions

of higher density have shorter free-fall times, so the central regions can run away from the

less dense outer regions of the cloud. This will lead to an extended period of mass accretion

onto the core.

Three-dimensional hydrodynamical simulations strongly support the argument that, if

fragmentation occurs at all, it does so after the initial free-fall phase, after the core has

collapsed into a rotationally-ﬂattened, quasi-equilibrium state (Bate, 1998, 2012; Truelove

et al., 1998; Tsuribe & Inutsuka, 1999). Simulations have also shown that fragmentation

can occur easily after free-fall for homologous collapse, but non-homologous collapses have

generally not produced prompt fragmentation. Because of the limitations of current simula-

tions, further work is needed to determine if these fragmentations will lead to binary stars,

and at what frequency.

1.2.2.3 Delayed Break-up

After collapse in a non-homologous manner, the resulting core will often be stable against

fragmentation, and contain a small fraction of the cloud’s total mass. As mass is accreted

onto the core, the core’s angular momentum will increase, also increasing the ratio of Erot . |Egrav| Additionally, a disk will form if it hasn’t already. Instabilities in either the core or the disk may lead to a breakup of the protostar into a binary system. Various methods of break-up have been investigated.

7 Lebovitz (1974, 1984) revised the classical formulation of the ﬁssion theory of binary star formation (Lyttleton, 1954), where a bar-like structure is formed at the core which evolves into a pear-shaped or dumbbell-shaped structure. If conditions are right, the bar- like structure may break into separate components. Hydrodynamical codes have been used to study whether breakup occurs from an axisymmetric conﬁguration (Tohline et al., 1985;

Pickett et al., 1996; Brown, 2000). Such conﬁgurations have not lead to ﬁssion. More computationally-extensive studies of the originally proposed, non-axisymmetric case have yet to be carried out.

An alternative method of delayed break-up investigates the stability of the accretion disk.

Since most of the infalling gas will fall onto the accretion disk, the disk may become unstable when it acquires as much mass as the central core. Again, simulations of a rotating disk are computationally intensive, due in part to the dynamic range of size scales and the number of required time steps. These simulations suﬀer from limited scope, sometimes ending due to issues with boundary conditions, failing to fully evolve the tantalizing clumps in the disk

(Woodward et al., 1994; Laughlin & Rozyczka, 1996).

1.2.3 Questions and Future Investigation

Following the arguments of Boss (1988), there is a general agreement that the preferred mechanism is prompt fragmentation (Clarke & Pringle, 1993; Bodenheimer et al., 1993).

Infrared observations of protostars have been recorded for over a decade now (Dunham et al., 2014) and observed multiple systems at the earliest stages of formation (Pineda et al.,

2015) conﬁrm that fragmentation plays a crucial role in binary formation.

There is broad agreement on the following points (Tohline, 2002). Capture in not a strong candidate for the primary binary formation process. From observations, it is known

8 that stars are bound as binary systems before Stage IV of star formation, and capture is too inefficient. Numerical simulations have repeatedly shown that clouds do not fragment during the free-fall collapse phase. Instead, clouds collapse to a flattened configuration before fragmentation (if any) occurs. Prompt fragmentation works immediately after the free-fall phase if a significant fraction of the cloud’s mass falls onto the flattened configuration within a short time. Therefore, clouds may fragment if they begin as a uniform density cloud with more than a few Jeans masses. Non-homologous collapse doesn’t seem to produce prompt fragmentation. Axisymmetric cloud cores do not fragment, but will settle into a spinning bar-like structure, and a disk will form around the core. Protostellar disks become unstable toward long-wavelength structures when mass contained in the disk becomes comparable to the mass of the core.

Some questions on the formation processes of binary stars include the following. If prompt fragmentation is the primary mechanism, how do molecular clouds form such that they contain the required few Jeans masses without initially collapsing? Magnetic fields may play a significant role (Palau et al., 2013). Do the fragments of prompt formation lead to binary systems? Current simulations are unable to continue much further than the initial instant of fragmentation, and more work is needed to evolve these systems more. With the collapse of axisymmetric cores into ellipsoids with bar-mode instabilities, will further evolution of the instabilities produce the binaries predicted by the revised fission model hypothesized by Lebovitz (1974, 1984)? A method will have to be found to slowly evolve the configuration along a sequence of more and more distorted ellipsoids. Finally, how promising are disk instabilities as a binary formation mechanism? Recent numerical studies have suggested that disk fragmentation may lead to binary systems in Population III stars

(Stacy et al., 2010; Latif & Schleicher, 2015) and low-mass stars (Stamatellos et al., 2012).

9 Given that dynamical range is a problem for these simulations, we will need to wait for more sophisticated and eﬃcient simulations and hardware to ﬁnd out.

1.3 Binary Evolution and Interactions

In this section we discuss the basic principles of binary evolution and mass transfer. Although mass transfer can occur through the accretion of winds, we focus our attention on Roche-lobe overﬂow. We also discuss various systems and phenomena resulting from the interaction of binaries, highlighting the key role that binary systems play in the evolution and fates of stars, and in production of dramatic eruptions.

1.3.1 Principles of the Evolution of Binary Systems

In the co-rotating frame of a binary system, an effective potential can be derived. Within this effective potential, five points contain zero effective potential, the so-called Lagrangian points. The equipotential passing through the first Lagrangian point, a point between the two stars, defines the Roche Lobe. Matter can flow from one star to another through the first

Lagrangian point in a process called Roche-lobe overﬂow (RLOF). The Roche lobe radius only depends on the orbital separation a, and the mass ratio q = Mk/Mj. For a star with mass Mj, and a companion mass Mk, the Roche-lobe radius is approximated by

0.49q−2/3a RL = (1.14) 0.6q−2/3 + ln(1 + q−1/3)

(Eggleton, 1983). For larger companion masses and/or closer separations, RL for a given star decreases. Likewise, decreasing a star’s own mass will also decrease that star’s RL. If

10 a star’s radius is at least as large as the star’s RL, mass transfer can occur through RLOF.

In the case of one star ﬁlling its Roche lobe, the binary system is said to be semi-detached.

If neither star ﬁlls their Roche lobe, the system is a detached system. And, if both stars ﬁll their Roche lobes, the system is referred to as a contact binary, or a common-envelope (CE) binary.

As the radius of a star on the Main Sequence does not expand by a large amount, it is much more likely that RLOF begins after the donor star leaves the main sequence. Since stars spend the majority of their lives on the main sequence, most binaries observed in the sky have not yet had any strong interactions, but many will in the future.

RLOF can occur in one of two modes. In stable mass transfer, most of the mass transferred over is accreted by the companion star, ending when most of the hydrogen-rich envelope of the donor star has been removed. This process results in a hydrogen-depleted helium star. The recipient star will be rejuvenated if it were still on the main sequence.

Otherwise, the recipient may bypass the red giant phase and explode as a blue supergiant

(Podsiadlowski & Joss, 1989).

The second mode, unstable mass transfer, occurs when the recipient is unable to accrete all of the material transferred over. The transferred material builds up on the recipient, causing the recipient to overﬁll its own Roche lobe, resulting in a CE system. When the donor loses mass adiabatically, the convective envelope will expand rather than shrink. Meanwhile the mass transfer of the donor star causes its RL to shrink, leading to a runaway, or dynamical mass transfer. Friction within the CE leads to a spiral-in until the envelope can be ejected, resulting in orbital periods between about 0.1 d and 10 d. In some cases, the orbital energy may not be great enough to eject the envelope and a binary merger may occur, resulting in a single rapidly rotating star. The merger process and common-envelope evolution, in

11 general, are the least understood processes in multiplicity studies (Taam & Sandquist, 2000;

Podsiadlowski et al., 2001). Ivanova et al. (2013) present a review of the current standing of

CE evolution and a guide for moving forward.

Mass transfer can be driven by two mechanisms, either expansion of the donor star, or a loss of angular momentum from the system.

As a result of intrinsic stellar evolution, the donor star expands, and mass transfer occurs at a rate which depends on the mass ratio of the system. If the donor star is initially less massive, the mass transfer will occur on a nuclear timescale as the star evolves. On the other hand, if the donor star is more massive initially, the mass transfer will drive the separation down in order to conserve angular momentum. Since RL is more dependent on separation than mass ratio, mass transfer will occur at an accelerated rate until the minimum separation is reached at Mj = Mk, and further mass transfer begins to increase RL.

The second mechanism driving mass transfer does so by decreasing orbital separations through angular momentum loss, either by gravitational radiation, or magnetic braking.

Gravitational radiation becomes important for only the shortest periods (P < 12 hr). Mag- netic braking is understood to be important but the details are still uncertain.

1.3.2 Binary Interaction in Astrophysical Phenomena

Interaction among binaries play a key role in many areas of astrophysics, and understanding the global properties of binary systems can help elucidate the physical processes involved in the creation of various objects and eruptive events. Below we briefly discuss a few instances where binarity plays a significant role. Additional topics not discussed here include cataclysmic variable events resulting from mass transfer onto a white dwarf, binary star mergers and the production of gamma-ray bursts, the effects of mass transfer on the formation of

12 black holes vs. neutron stars, the various methods of mass transfer among x-ray binaries,

and more.

1.3.2.1 Hot Subdwarfs

These hot, compact objects are helium-core burning stars with masses 0.5 M , and are depleted of nearly all hydrogen. This single class of binaries illustrates a variety of diﬀerent types of binary interactions. Studies by Han et al. (2002, 2003) have concluded that there are three equally important formation channels, including RLOF, CE evolution, and binary mergers. As these hot subdwarfs are a dominant source of UV radiation, it has been proposed that they may be a contributing factor to the UV upturn seen in elliptical galaxies (Han et al., 2007) .

1.3.2.2 Chemically Peculiar Stars

Ba and CH stars contain an overabundance of carbon and other s-process elements in their atmospheres. The explanation for the origin of these overabundances was resolved by including mass transfer from a binary companion. In these systems, an AGB star produces carbon and s-process elements which are dredged up, through convection, to the surface.

Then, mass transfer from the AGB star onto the observed companion pollutes the observed star, transforming the companion into the Ba or CH star that we see today. Indeed, observations have shown that all Ba and CH systems contain binary companions (McClure &

Woodsworth, 1990). Similarly, there is a class of carbon-enhanced metal-poor (CEMP) stars residing in the Milky Way halo. CEMP radial velocity studies also return binary companion fractions (at least for the most common subclass, the s-process enhanced CEMP-s stars) consistent with a 100% binary fraction (Lucatello et al., 2005). Like the Ba and CH stars,

13 mass transfer from an AGB star is the favored explanation for their existence (Herwig, 2005;

Sneden et al., 2008). As CEMP stars trace some of the oldest stars in the Galaxy, they act as a fossil record allowing the exploration of the early stages of galaxy formation and galactic chemical evolution.

1.3.2.3 Symbiotic Binaries

These are interacting systems where a giant star (S-type) or a Mira variable (D-type) transfer mass onto a white dwarf. Both types of symbiotic binaries involve mass transfer in an atypical manner. The observed orbital-period distribution for S-type systems (∼ 10−1400 d;

Miko lajewska, 2007) is not easily explained by binary population synthesis models, which only involve stable or unstable transfer. Unstable transfer resulting in a CE phase would produce shorter periods than those that are observed, and stable mass transfer typically leads to a widening of the system. This issue, first realized by Webbink (1986), may have a solution with quasi-dynamical mass transfer. This may be achieved if the mass ratio of the binaries is close to 1, and the mass transfer rate is large enough to lead to a CE phase without significant spiral-in. D-type symbiotics, on the other hand, propose another issue for current binary evolution theories. Periods on the order of 1000 years seem too wide for binaries to be interacting. It seems that the Mira variables are able to fill their Roche lobes, not from the stellar radius itself, but rather through a slow wind, in what is referred to as wind Roche-lobe overflow.

1.3.2.4 Blue Stragglers

These exceptionally luminous and blue stars appear to be unevolved and live on the main sequence for longer times than their traditional main sequence counterparts. It appears that

14 blue stragglers arise from a variety of diﬀerent binary interactions. Indeed, blue stragglers are divided into two populations, based on colors, with one population likely resulting from binary collisions, and the other population from binary mass transfer (Piotto et al., 2004;

Lu et al., 2010).

1.3.2.5 Thermonuclear Supernovae

Type Ia supernovae are used as standardizable candles for distance measurements. They have provided the ﬁrst indication for an accelerating Universe, and they play an important role in constraining cosmological parameters (Riess et al., 1998). There is broad agreement that Type Ia explosions result from a thermonuclear explosion of a carbon oxygen white dwarf. Currently, however, there is continued debate on which progenitor system(s) lead to the explosion. It is clear though, that binary interactions of some kind are required. In the single-degenerate scenario, the white dwarf grows in mass due to the accretion of mass from a companion star such as a main sequence star or an evolved star (Whelan & Iben,

1973). Alternatively, Type Ia events may be explained through the merger of two white dwarfs, known as the double-degenerate scenario (Iben & Tutukov, 1984). One approach to determining the relative importance of each scenario is to compare the rates of Type

Ia explosions with progenitor rates based on binary population synthesis models. A better understanding of the multiplicity properties of the star formation process, and the subsequent evolution of binary systems would be invaluable in this regard.

1.3.2.6 Core Collapse Supernovae

As a massive star evolves, an iron core develops. When the iron core reaches a critical mass, the core collapses and bounces back driving a shock that ultimately results in a

15 supernova explosion. A supernova can be classiﬁed based on the observed characteristics of the explosion and the elements seen in the spectra. The diversity in the characteristics can be understood by a sequence of increased mass loss, with Type II-L having lost a fraction of their H-rich envelope before exploding, SN Ib with no H-rich envelope, and SN Ic with no H-rich or He-rich envelopes. While stellar winds may play an important role, binary interactions must play a bigger role in the removal of stellar envelopes through mass transfer preceding core collapse (Podsiadlowski et al., 1992).

1.4 Empirically Derived Multiplicity Properties

In this section, we explore empirically derived multiplicity properties and trends observed for stars in the Milky Way. We begin with a description of multiplicity survey techniques and methodologies, followed by what we know about the multiplicity properties of various populations, and how multiplicity varies with stellar mass and age. We end the section with a brief discussion on the implication of current observations as it pertains to formation mechanisms and the evolution of binary systems. Values for multiplicity fractions and companion fractions are taken from the review by Duchˆene& Kraus (2013), where the multiplicity properties of stellar groups are derived by collating works from across the discipline.

1.4.1 Multiplicity Properties and Survey Methodology

With regards to multiplicity, the most obvious properties of interest for stellar populations is the multiplicity fraction (MF), the fraction of stars with a companion, and the companion frequency (CF), the average number of companions per target. CF values can, in principle, be greater than 100%, for cases where a population has many systems with higher order multiple

16 systems (triple, quadruple, or higher). Either the orbital period P , or the separation a, for a binary system will be reported based on the method of observation. Period distributions within a population are described by either a power law form f(P ) ∝ P α, or a log-normal representation with parameters P and σlog P . The logarithmically-ﬂat α = −1 power law distribution, known as Opik’s¨ law (Opik,¨ 1924), is sometimes adopted. Opik’s¨ law suggests a

scale-free process, whereas a log-normal distribution has a scale preference, meaning that the

empirically derived choice in parameterization has an implication for the binary formation

process. Mass ratios, q = M2/M1 ≤ 1, can be estimated through ﬂux comparisons, and

are also described through a power law distribution f(q) ∝ qγ. Eccentricities e, can be

used to evaluate the dynamical evolution of a system, but can only be estimated for short

binaries where the orbit is fully mapped. At the shortest periods, tidal dissipation will

lead to a circularization of the orbit (Koch & Hrivnak, 1981). Three-body interactions can

lead to highly eccentric orbits through the Kozai mechanism (Kozai, 1962), and could be a

mechanism for mergers in the production of Type Ia supernovae (Thompson, 2011).

The observational techniques used to detect binaries are varied in implementation and in

range of sensitivity. A few of the most common techniques are described here. Visual binaries

(VB) are systems where both the primary and secondary are seen through photometry, and

their projected orbits can be measured through periodic motions on the sky. When only

the primary star is seen, and the secondary is inferred through the motion of the primary,

the system is referred to as an astrometric binary. Visual and astrometric binaries rely on

precise position measurements, and/or the ability to resolve the companion. Thus, these

observational techniques are limited to wide, long-period systems. Spectroscopic binaries

(SB) are systems where the Doppler shift of one (single-lined) or both (double-lined) of the

stars can be measured through line identiﬁcation or template correlations of stellar spectra.

17 Radial velocities (RV) are measured at multiple times, producing a RV curve that can be used to estimate orbital parameters. Strong changes in RV are easier to detect, making shorter-period systems more favorable to detection from spectroscopic methods. Eclipsing binaries require a nearly edge-on projection of the orbit, but allow the detection of a binary companion through variations in the primary’s flux as the secondary passes in front of and behind the primary. The eclipsing technique has the benefit of allowing a direct measurement of stellar radii and strong estimates of stellar masses, but unfortunately requires a specific inclination of the orbit for the observations to be made.

The optimal strategy for completing a binary survey is to gather a complete, uniform, volume-limited sample using a variety of the available observing techniques in order to probe the dynamic range of orbital periods. A sample size of 100 is required to measure MF values with a precision of 5%. To get a similar precision for orbital parameters for binaries, an even larger number of targets will be required to get a suﬃcient number of binary systems.

Thus far, complete and volume-limited surveys are rare, limited to the solar-neighborhood.

In general, surveys will attempt to correct for completeness with some assumptions about the properties of stellar systems for which they are not sensitive.

1.4.2 Trends and Characteristics of Multiple-Star Systems

1.4.2.1 Multiplicity and Mass

Solar-type stars (0.7 − 1.3 M ) are ideal for studying multiplicity on the main sequence due to their relatively high frequency and bright luminosities. It is not surprising that the most complete surveys of multiplicity are those of solar-type stars in the volume of space around our Sun. Duquennoy & Mayor (1991) coordinated the ﬁrst modern, volume-limited survey of

18 solar-type stars with analysis of 164 objects out to 22 pc, with new RV measurements taken

in addition to those from other works. However, incompleteness was still an issue, leading

Raghavan et al. (2010) to improve upon the work with additional objects for a total of 454

stars out to 25 pc, with very high completeness. Additional studies of solar-type stars were

conducted using individual detection techniques, such as surveys of spectroscopic binaries

(J. L. Halbwachs et al., 2003), visual binaries (Mason et al., 1998), and common proper

motion companions (Tokovinin, 2011). From Duquennoy & Mayor (1991) and Raghavan

et al. (2010), CF = 62 ± 3% and MF = 44 ± 2% for solar-type stars on the main sequence.

Furthermore, it appears stars that are slightly more massive than the Sun have a higher MF

than those just under 1 M , with values MF = 50±4% and MF = 41±3% respectively. This is a trend we will continue to see as we look at stars throughout the main sequence. These modern samples of solar-type stars exclude Opik’s¨ law, with periods showing a log-normal distribution with P = 250 yr and σlog P (d) = 2.3. The distribution in mass ratios follows

a ﬂat distribution (γ = 0.28 ± 0.05) in ratios all the way down to q = 0.1 with a marginal

peak at q > 0.95 for short-period systems.

At the lower-mass end of the main sequence (0.1−0.5 M ), stars are more abundant, but

less luminous. Observations from various direct imaging, speckle interferometry, and radial

velocity studies, totaling 166 stars, have been collected by Fischer & Marcy (1992) in oder to

perform a complete analysis of M dwarf multiplicity. Additionally, volume-limited surveys

of low-mass stars have been carried out (Reid & Gizis, 1997; Delfosse et al., 2004; Dieterich

et al., 2012), including a nearly complete sample by the RECONS consortium out to 10 pc

(Henry et al., 2006). These nearly complete surveys give multiplicity values of CF = 33±5%,

MF = 26 ± 3%, and separation distribution parameters a = 5.3 AU and σlog P (d) = 1.3 in a log-normal distribution for separations less than 500 AU. Like the solar-type binary

19 systems, short-period M-dwarf binaries are biased toward high-q systems. Very low mass

(VLM, < 0.1 M ) stars are even fainter, and have a lower MF, making complete studies of

a signiﬁcant number of binary systems all-the-more diﬃcult. Reviews of multiplicity in this

sub-stellar regime include Burgasser et al. (2007) and Luhman (2012). Estimates based on

incomplete surveys suggest MF ∼ 20−25%. The distribution in separations is much narrower than the higher-mass counterparts, and very few systems with separations a > 50 AU are known. The mass ratio distribution for VLM stars is heavily skewed towards equal-mass systems, with a power law index γ = 4.2 ± 1.0.

Issues with studies of intermediate-mass (1.5 − 5 M ) and high-mass (> 8 M ) stars

include their larger distances, lower numbers, and very high luminosity contrasts with their

companions. This makes multiplicity surveys diﬃcult, especially for mid-range periods,

where VB and SB techniques are limited. Additionally, some ∼ 30% of A-dwarf stars are

chemically peculiar, often because of binary interactions (Abt, 1965). SB searches among

intermediate- and high-mass stars include the works of Carquillat & Prieur (2007) and Chini

et al. (2012), and VB searches include the works such as Balega et al. (2011). In addition to

these ﬁeld-star studies, intermediate- and high-mass stars have been studied extensively in

the Scorpius-Centaurus OB association (Kouwenhoven et al., 2007) and other clusters and

associations (Sana et al., 2009; Kiminki et al., 2012). In fact, only about 20% of nearby O

stars are found in the ﬁeld, and multiplicity properties of the ﬁeld are assumed to be similar to

those in associations. Although incomplete, surveys conclude that MF > 50% (1.5 − 5 M ),

MF > 60% (8 − 16 M ), and MF > 80% (> 16 M ). The orbital period distribution for

intermediate-mass stars appears to be bimodal with peaks at P ∼ 10 d and a ∼ 350 AU, and

the mass ratio distribution has yet to be fully characterized. For high-mass stars, the period

distribution may have a complex functional form, possibly with a population of short-period

20 binaries (log P (d) < 1) for some 30% of all high-mass stars, and a power law distribution extending out to 104 AU. For high-mass stars we see, again, a relatively ﬂat mass-ratio distribution (γ = −0.1 ± 0.6 for M > 16 M ) with a peak around q = 0.8.

From studies spanning the entire main sequence mass range, it is clearly seen that there is a strong, monotonic dependence of multiplicity on mass, with MF increasing for higher mass primary stars. Values from Duchˆene& Kraus (2013) for multiplicity fraction at various stellar masses have been plotted in Figure 1.1. Higher order systems containing multiple stars also

1.0

0.8

0.6 MF 0.4

0.2

0.0 10-1 100 101

Stellar Mass (M ⊙)

Figure 1.1: Dependency of multiplicity fraction with primary mass for main sequence stars and VLM objects. Values used from the review by Duchˆene& Kraus (2013). favor higher mass primaries. Orbital period distributions for solar-type and lower-mass stars are unimodal, with the median separation and width both decreasing for lower-mass systems.

On the other hand, intermediate- and high-mass systems have complex distributions with strong peaks at log P (d) ∼ 0 − 1. A second peak for VBs is observed for intermediate-

21 mass stars, while a shallow power law is currently preferred for high-mass stars, although more work is needed in this area. The distribution in mass ratios is relatively ﬂat for all masses down to about 0.3 M , below which the mass ratio becomes skewed towards high-q systems. The hypothesis of random pairing of stars from an IMF to make up the binary population is excluded from the observations. Eccentricity shows very little dependence on mass, with a ﬂat distribution for all systems with periods greater than ∼ 100 d, and circularized eccentricities for orbits shorter than ∼ 10 d.

1.4.2.2 Multiplicity and Age

Population II (Pop II) stars are metal-poor stars that trace the Milky Way halo component, and generally act as a probe for investigating the formation processes at earlier chemical times. According to an analysis of 171 high proper motion, single-line systems, the properties of Pop II SBs appear to be very similar to those of the metal-rich population (Latham et al.,

2002). For VBs (a > 10 AU), however, the frequency of multiple systems is lower for metal- poorer stars (Zapatero Osorio & Mart´ın,2004; Lodieu et al., 2009). The overall frequency for

Pop II systems is CF = 39 ± 3% for primaries in the range 0.5 − 1.3 M and CF = 26 ± 6% for primaries in the range 0.1 − 0.6 M (Jao et al., 2009; Rastegaev, 2010). The period distribution for Pop II stars is characterized by a narrow peak around log P (d) ∼ 2 − 3 and a tail out to 104 AU. There is also a lack multiple systems with periods shorter than

P ∼ 10 d. The mass ratio distribution is not well constrained, but appears to be roughly uniform.

To investigate younger populations, studies have focused on open clusters with ages 50

Myr to 1 Gyr (Patience et al., 2002; Griﬃn, 2012), nearby associations with ages 77-100

Myr (Brandeker et al., 2003; Evans et al., 2012), PMS stars with ages 1-5 Myr (Luhman,

22 2012), and even protostars in the earliest stages of evolution (Duchˆene et al., 2007; Connelley et al., 2008). The total frequency for solar-type stars in open clusters is CF ∼ 65%, with a period distribution that is broad and unimodal and possibly indistinguishable from that of

field stars. We also see a flat distribution in mass ratios for open-cluster, solar-type stars. A uniform analysis of nearby stellar associations is still needed, but the characteristics of the multiplicity population seems to be consistent with that of the field. Surveys of PMS stars have measured CF values of solar-type stars (CF ∼ 65 − 80%) to be twice as high as those of the main sequence. Single-star systems still represent a quarter to a third of all PMS systems, suggesting an increase in the number of high-order multiple systems.

It is important to note that populations other than nearby field stars have not been studied well enough to draw strong conclusions on properties of multiplicity. For SBs, there are no significant differences among PMS stars, open cluster stars, Pop I stars and Pop II stars. For VBs, there appears to be a dichotomy of stellar multiplicity properties, with higher

CF values per decade of separation among lower density environments, such as associations, than among the denser environments of clusters. Within each environment though, there doesn’t seem to be any strong trends with age. For the older Pop II stars, a smaller frequency of wide binaries is observed, suggesting a metallicity dependence on binary formation or dynamic evolution on timescales of Gyr. It is clear that incompleteness and biases have not yet allowed a detailed picture of the global variation in multiplicity properties as a function of age, and there is much more work to be done in these areas of research.

1.4.3 Discussion

Here, we report the implications of current observations on binary formation and evolution, as well as the issues brought up for future research, as discussed by Duchˆene& Kraus

23 (2013). Multiplicity is a smooth function of primary mass for both the MS and earlier

phases, suggesting that prompt fragmentation appears to be the leading binary formation

mechanism. Fragmentation appears to be mild, as seen by the lack of higher-order multiple

systems, with a production of 1 to 3 cores per cloud. This is in good agreement with

numerical simulations (Bate, 2012).

The lower multiplicity observed in open clusters, compared to loose stellar associations,

hints at the possibility that multiplicity properties are not universal among birth environ-

ments. Unfortunately, current observations do not probe multiplicity at an early enough

evolutionary stage to be conclusive (dynamical evolution is over after only 1 Myr, Marks &

Kroupa, 2012). In fact, the diﬀerences in multiplicity could be explained through dynam-

ics, with associations representing ineﬀective binary disruption, and clusters representing

eﬀective disruption (Kroupa & Bouvier, 2003).

As stated above, post-CE binaries play a critical role in astrophysical phenomena. Solar-

type binaries undergo at least one CE episode if their initial period is below log P (d) ∼ 2.8

(Davis et al., 2010). In this regime, small deviations from the log-normal function (Raghavan et al., 2010) can result in large differences in the total number of pre-CE systems, by as much as a factor of three. This is illustrated in Figure 1.2 where the Raghavan et al. (2010) period distribution is compared with an alternate, yet data-consistent model. This poor handle on the multiplicity statistics for field stars with log P (d) ≤ 3 has profound implications. The disagreement between theoretical SN Ia rate calculations from binary population synthesis and observations is a factor of a few (Claeys et al., 2014), similar to the uncertainty in the number of CE systems in Figure 1.2. Models cannot be refined past a certain point without a better knowledge of the initial multiplicity statistics. Similar arguments can be made for cataclysmic variables and novae (Davis et al., 2012) and X-ray binaries (Podsiadlowski et al.,

24 Figure 1.2: Left: Orbital period distribution in the solar neighborhood from Raghavan et al. (2010). The limit for RLOF in the MS is indicated by the black vertical dashed line, and the range corresponding to pre-CE systems is shaded in gray. The best-fit log-normal function is shown in black. The dashed red plot represents a modified function that also fits the data. Right: Comparing the number of systems in the best-fit log-normal distribution (solid black) and the modified model (dashed red). The period ranges corresponding to low-mass X-ray binary progenitors, stable habitable planets around binary stars, and SN Ia progenitors are shown with horizontal rulers. The yellow dash-dotted line marks the pre-outburst period of V1309 Sco.

2003). In order to eﬃciently explore the statistics of stellar multiplicity over a wide range of stellar properties like metallicity, age, or disk/halo membership, it is necessary to use massively multiplexed spectrographs.

Works in the future will need to build more complete surveys of multiplicity in a variety of stellar populations, including very young PMS, protostellar, intermediate-mass, and high- mass stars. Big questions that are to be addressed include the following: Are the multiplicity properties resulting from the star formation process universal or do they depend on the native environment? Are there signiﬁcant diﬀerences in the multiplicity properties of highest-mass stars relative to all other stars, or do the properties follow smoothly across these masses?

The frequency and properties of visual binaries appear to be set by the PMS phase, but is this also true for spectroscopic binaries?

25 1.5 Conclusion

Over decades of theoretical work, and with the improvement in computational power, we have reached a broad agreement that prompt fragmentation is the likely scenario for binary star formation. The relative importance of the other mechanisms of star formation has yet been fully realized. The future will likely see improvement in the resolution and physical accuracy of simulations. We want to know how frequent, and it what number, binaries are formed, as well as their initial separations and mass ratios. These insights will help constrain population synthesis models and star formation models, ultimately allowing us to, for example, investigate how clusters and associations dissolve and interact, or how frequent the variety of potential progenitors of novae and supernovae may be.

It is clear that single-star systems produce only a limited number of the phenomena we observe in the skies. With the introduction of a binary companion, we begin to see a variety of interesting and varied interactions. We see instances of mass transfer between objects of all masses, from white dwarfs and black holes, to supergiants and everything in between.

Binary interactions produce spectacular explosions like novae, supernovae, and gamma-ray bursts. The binary interactions involved in chemically peculiar stars like Ba and CEMP stars illustrates the importance of understanding binary properties and the role that binarity plays in affecting observed surface abundances. Without a full understanding of the effects of mass transfer on observed abundances, the task of modeling the processes that occur to enrich the Galaxy with elements heavier than hydrogen and helium remains difficult. Thus, we will continue to see studies focused on understanding the frequency and properties of binaries, as well as the ways in which binaries can potentially interact.

From observations, we’ve found a smooth trend of increasing multiplicity with mass. It

26 is still too early to tell how older populations may diﬀer from younger populations in terms of multiplicity, and whether dynamical evolution and/or intrinsic formation processes play an important role, as modern studies have not yet built up a complete enough sample of these stars. Whether star formation is characterized by universal multiplicity distributions is an open question and has important bearing on star formation theory and the initial mass function (Oﬀner et al., 2014). The best approach for compiling complete samples is to look at multiplicity with a variety of techniques including photometry and spectroscopy.

Spectroscopy is time expensive however, and acquiring a large, complete dataset is an arduous task. In the meantime, we should be utilizing large multi-ﬁber spectrographs in order to do statistical analyses of large counts of stars, as a complement to the more rigorous target- selected surveys.

27 Chapter 2

Time-Resolved Spectroscopy

2.1 Introduction

With the increasing number of large survey telescopes, we are entering an era of large-data astronomy (National Research Council, 2010). With a greater throughput of celestial objects observed and a wider span of re-observation cadences, a new window of observation has opened. The accessibility of this time-domain window is growing with the recent introduction of programs such as CRTS (Djorgovski et al., 2011), PTF (Law et al., 2009), Pan-STARRS

(Kaiser et al., 2002), and DES (The Dark Energy Survey Collaboration, 2005), and with upcoming programs such as LSST (LSST Science Collaboration et al., 2009). The increased access to the time-domain dimension has provided astronomers an additional avenue to approach research in various areas including exoplanets, variable stars, microlensing events, novae and supernovae, and active galactic nuclei. We will also likely see the emergence of unknown phenomena that, until now, have been unobservable.

Before the adoption of multi-ﬁber spectrographs, obtaining spectra for targets in the sky was a time-consuming process. Due to the dispersion of the light source across the CCD, longer exposure times are needed to obtain high signal-to-noise ratios (SNR) in spectra.

Larger exposure times meant that gathering valuable spectra for a population of objects would be infeasible. Multi-ﬁber spectrographs introduced parallelization to the observing

28 process. Bundles of hundreds or thousands of optical ﬁbers can now be positioned in the

field-of-view to gather light from multiple sources simultaneously, greatly reducing the time to gather spectra for groups of objects, and increasing the efficiency of observing by orders of magnitude. We are seeing an increase in the development and use of multi-fiber spectroscopic instruments with LAMOST (Cui et al., 2012), BOSS (Smee et al., 2013), Hectospec (Fab- ricant et al., 2005), and others, and we will continue to see development with instruments such as the upcoming PFS (Sugai et al., 2012) and DESI (Levi et al., 2013).

With the increase in the number of astronomical sources and the amount of coverage of sources at each observation, it is important to consider the methods used for data extraction.

One goal of the work of Hettinger et al. (2015) was to investigate the usefulness of a method for identifying radial velocity (RV) variability in spectra obtained from the multi-fiber spectrograph used in the Sloan Digital Sky Survey (SDSS; York et al., 2000). In this chapter we present, in detail, the methods adopted in Hettinger et al. (2015) during the development of a pipeline designed for obtaining radial velocity measurements from sub-exposure spectra taken with the SDSS multi-fiber spectrograph. Section 2.2 gives a brief description of the SDSS survey, discussing the sub-exposure properties inherent to typical multi-fiber spectroscopic surveys, and also gives a cautionary warning with regards to the systematic uncertainties observed in the SDSS sub-exposures. Section 2.3 details the process for obtaining radial velocities from sub-exposure spectra using template cross-correlation. We discuss empirical estimates of the uncertainties in Section 2.4. And finally, one metric for measuring variability in a source is briefly discussed in Section 2.5, deferring the description of the

MCMC method of detecting variability to Chapter 3.

29 2.2 The Sloan Digital Sky Survey

SDSS is an imaging and multi-ﬁber spectroscopic survey program using the 2.5 m optical telescope at Apache Point Observatory in New Mexico. SDSS began data collection in 2000, and has operated through the present day in the optical and near-infrared with surveys focused on nearby and distant galaxies, supernovae, quasars, exoplanets, and stars in the

Milky Way.

The original Legacy survey from SDSS-I (2000-2005) imaged more than 8000 deg2 of the sky in five optical filters, and obtained spectra of galaxies and quasars. The original spectrograph was a 640-fiber optical spectrograph, operating in the 4000 A˚ – 9000 A˚ wavelength range, with a resolution R ∼ 2000 and a pixel scale of 70 km s−1 pixel−1. Upon comple- tion of the Legacy survey, SDSS had imaged over 2 million objects and obtained spectra for

800,000 galaxies and 100,000 quasars.

SDSS-II (2005-2008) saw the introduction of two new surveys using the original camera and spectrograph. The Supernova Survey (Frieman et al., 2008) scanned 300 deg2 of the sky, ﬁnding thousands of supernovae and variable objects. The second survey, the Sloan Ex- tension for Galactic Understanding and Exploration (SEGUE; Yanny et al., 2009), targeted

240,000 stars in the Milky Way, creating a map and providing a detailed picture of the age, composition, and distribution of stars in our Galaxy.

Additional surveys comprise SDSS-III (2011-2014). Using the original SDSS spectrograph, the SEGUE survey was expanded with SEGUE-2 (C. M. Rockosi et al., in preparation), focusing on the Milky Way Halo and adding 120,000 stars. The Multi-object APO

Radial Velocity Exoplanet Large-area Survey (MARVELS) monitored 11,000 bright stars with the MARVELS spectrograph, looking for the signatures of exoplanets, but had minimal

30 success. A high-resolution infrared spectrograph was added for the APO Galactic Evolution

Experiment (APOGEE; S. R. Majewski et al., in preparation) survey, aimed at 100,000 red giants across the Milky Way. SDSS-III also contained the Baryon Oscillation Spectroscopic

Survey (BOSS; Dawson et al., 2013), designed to measure the expansion rate of the universe through measurements of the spatial distribution of luminous red galaxies. With the BOSS survey came an upgrade to the SDSS spectrograph with new CCDs and an increase from

640 to 1000 simultaneous ﬁber pluggings.

SDSS-IV (2014-2020) continues today with the extension of two SDSS-III surveys; APOGEE-

2 and eBOSS continue to survey stars in the Milky Way and baryonic oscillations in the universe. Additionally, the new Mapping Nearby Galaxies at APO (MaNGA) survey will examine the detailed internal structure of 10,000 nearby galaxies with integral ﬁeld units.

2.2.1 Sub-Exposures

When a cosmic ray strikes a CCD detector, an artifact appears as many pixels become saturated. For spectra, these affected pixels often lead to incorrectly reported increases in flux at particular wavelengths. To facilitate the removal of these artifacts from spectra in SDSS, the original data processing pipeline uses the median flux values at each pixel averaged over multiple sub-exposures taken in succession. This requirement means that all single observations of an astronomical source actually contain several sub-exposures taken over some short period in time. Figure 3.7 of Section 3.5.1 provides an example, for an

F-type dwarf star, of a coadd spectrum with cosmic rays removed, along with the individual sub-exposure spectra used to construct the spectrum. Sub-exposure spectra, although lower in SNR and beset with cosmic rays, add a time dimension to the data that can be used for monitoring spectral variability.

31 In the Legacy and SEGUE surveys, typical spectra are composed of three sub-exposures, with typical exposure times of about 15 minutes. This presents the potential for detecting variability in spectra which is expected to occur at timescales on the order of hours. Addi- tionally, many areas of the sky were re-observed over the years for calibration and scientiﬁc purposes, yielding additional sub-exposures and increasing the range of timescales that can be probed. With these re-pointings included, sub-exposure counts per object in the sample of Hettinger et al. (2015) range from 3 to 47, with baselines ranging from 30 minutes to over

9 years (Figure 2.1).

Because cosmic-ray removal must be handled by all spectroscopic observations, sub- exposures are an expected product of all multi-fiber programs. As we enter an era of large spectroscopic datasets, there is an increasing potential for variability studies through data mining of sub-exposure spectra. Thus, we are motivated to develop the tools and techniques for extracting information from these sub-exposures. We encourage the designers of future multi-fiber programs to consider the choice in sub-exposure frequency and cadences, in addition to target selection and scheduling, in order to maximize scientific potential.

2.2.2 SEGUE Stellar Parameter Pipeline And Sample Selection

With the introduction of the SEGUE survey in SDSS-II, aimed at investigating the stellar properties and composition of the Milky Way components, there was a concerted eﬀort to build a pipeline for automatically determining stellar parameters of stars from the abundant spectra provided by the SDSS multi-ﬁber spectrograph. The SEGUE Stellar Parameter

Pipeline (SSPP; Lee et al., 2008) was designed to carry out these automated tasks. The

SSPP provides, using a variety of techniques, fundamental stellar atmospheric parameters such as metallicity [Fe/H], eﬀective temperature Teﬀ, and surface gravity log g, as well as

32 4 Metal-Poor Metal-Intermediate 3 Metal-Rich

2 log(N) 1

3 9 15 21 27 33 39 45 51 Exposure Count 4

1d 1m 1y

2 log(N) 1

3 4 5 6 7 8 9 ∆t (log s)

Figure 2.1: Distribution of the number of sub-exposures (top) and the time lags (bottom) for the F-dwarf stars from the Hettinger et al. (2015) sample. Metallicity cutoﬀ values are [Fe/H] = −1.43 and [Fe/H] = −0.66.

33 spectral classiﬁcation. A resource such as the SSPP is invaluable for increasing scientiﬁc gains

obtained from data mining techniques applied to large, multi-ﬁber spectroscopic surveys.

The SSPP was used for identifying the sample groups deﬁned in Hettinger et al. (2015).

Stars of spectral type F were selected using the (Hammer method) spectral classiﬁcations

provided by the SSPP. Unevolved, main sequence stars were used by selecting dwarfs with

surface gravities log g ≥ 3.75. The decision to use F-type stars was based on several factors.

The combination of high frequency, and suﬃciently bright luminosity, yields a statistically

robust sample with decent SNR values. The intrinsic variability in F-type RVs due to surface

activity is relatively low and will have minimal impact on measurement uncertainties. Also,

F-type stars have main sequence lifetimes greater than 5 Gyr, allowing us to select unevolved

stars from both the younger disk and the older halo. The Hettinger et al. (2015) sample was

further divided into three metallicity groups, aimed at tracing the Milky Way components,

using the stellar parameters provided by the SSPP (Figure 2.2). Figure 2.3 illustrates the

[Fe/H] and log g distribution of F-type stars in the SSPP DR9 dataset. The bimodal [Fe/H] distribution separates the Halo and Disk components, and the tails at lower values of log g identify the stars that have evolved away from the Main Sequence.

2.2.3 Plate Systematics

The spectra used in Hettinger et al. (2015) are from the SDSS Legacy, SEGUE-1, and

SEGUE-2 surveys, captured using the original SDSS spectrograph. The original spectrograph operated by plugging 640 optical fibers into holes (corresponding to each target’s position on the sky) on one of the many pre-drilled observing plates. Each spectrum was given an ID in the form of plate-mjd-fiber using the plate ID number, the Modified Julian Date of the observation, and the fiber ID number.

34 4.0

3.5

3.0

2.5

2.0

1.5 log(N)

1.0

0.5

0.0

−0.5 −4 −3 −2 −1 0 1 [Fe/H] 4.0

3.5

3.0

2.5

2.0

1.5 log(N)

1.0

0.5

0.0

−0.5 5000 5500 6000 6500 7000 7500 8000

Teff (K) 4.0 Metal-Poor 3.5 Metal-Intermediate 3.0 Metal-Rich

2.5

2.0

1.5 log(N)

1.0

0.5

0.0

−0.5 3.8 4.0 4.2 4.4 4.6 logg

Figure 2.2: Distribution of stellar parameters for the F-dwarf stars from the Hettinger et al. (2015) sample, including metallicity (top), eﬀective temperature (middle), and surface gravity (bottom). Metallicity cutoﬀ values are [Fe/H] = −1.43 and [Fe/H] = −0.66.

35 Figure 2.3: Scatter plot showing the distribution of metallicity and surface gravity for F- type stars in the SSPP DR9. The bimodal distribution of [Fe/H] traces the Halo and Disk components of the Milky Way.

36 After measuring RVs for F-type stars with the methods detailed in Section 2.3, correla-

tions in RV variations were found for ﬁbers residing on the same plate. Correlations were

typically seen among ﬁbers that were adjacent to each other on the CCD. Figure 2.4 illus-

trates the systematic issues with an example from plate plugging 2085-53379. This ﬁgure

shows the measured RVs of the four sub-exposures for each of the eight F-type dwarfs found

on this plate. It is clearly seen that the RVs of the stars decrease systematically from the

ﬁrst to second sub-exposures, decrease furthermore from the second to third sub-exposure,

and increase slightly on the last sub-exposure.

To correct the plate systematics, we adopted a Markov chain Monte Carlo (MCMC)

method to estimate the systematic shifts from sub-exposure to sub-exposure (Section 3.4).

10,264 ﬁber sub-exposures were examined across 2453 plates. The RV corrections applied

to these sub-exposures are distributed as seen in Figure 2.5, with a standard deviation of

2.2 km s−1 and corrections as large as 17 km s−1.

Unfortunately, not all plates could be corrected using the MCMC methods, because of multiple correlated subsets of ﬁbers on the same plate. Using plate 3002-54844 (Figure

2.6) as an example, we can see why simple techniques to correct systematic shifts are not successful. In this plate, the RVs from the F-dwarf stars that were exposed on one area of the

CCD (Panel a) are correlated with each other, but anti-correlated with the ﬁbers exposed on another area of the CCD (Panel d). Because of this, a systematic shift in RV for a single exposure on a plate does not increase the total value of the MCMC likelihood (Equation

3.7). It is shown from the correction column of Figure 2.6, that the systematics were not resolved. This lead to the inspection and removal of 25 plates (Table 2.1), requiring manual inspection of sub-exposure RV shifts.

It is unclear at this point where the source of the systematic shifts in radial velocity seen

37 Before Correction After Correction 120 120 110 110 100 100 90 90 RV (km/s) RV

80 80 2085-53379-194 70 70 60 60 50 50 40 40

RV (km/s) RV 30 30 2085-53379-215 20 20 100 100 90 90 80 80

RV (km/s) RV 70 70 60 60 2085-53379-219 190 190 180 180 170 170 160 160 RV (km/s) RV

150 150 2085-53379-246

80 80 70 70 60 60 50 50 RV (km/s) RV

40 40 2085-53379-438 40 40 30 30 20 20 10 10 RV (km/s) RV 0 0 2085-53379-533 30 30 20 20 10 10 0 0 RV (km/s) RV

−10 −10 2085-53379-572

60 60 50 50 40 40 30 30 RV (km/s) RV

20 20 2085-53379-597 Exposure Exposure

Figure 2.4: RVs for F-dwarf stars located on plate plugging 2085-53379 before and after correcting the plate for systematic sub-exposure offsets. Red points are sub-exposures that have low SNR. Fiber IDs are given for each fiber on the right axis. Sub-exposures in the plate are ordered chronologically. Corrections to systematic offsets are successful on this plate.

38 104

103

102 N

101

100

−20 −15 −10 −5 0 5 10 15 20 Systematic Exposure Offset (km/s)

Figure 2.5: Distribution of 10,264 systematic RV oﬀsets estimated for all plate sub-exposures in the F-dwarf sample.

Table 2.1: Suspect Plates in SDSS

Plate-MJD 0888-52339 2252-53565 2683-54153 2890-54495 2940-54508 1665-52976 2252-53613 2701-54154 2899-54568 3111-54800 2042-53378 2393-54156 2839-54461 2900-54569 3166-54830 2053-53446 2670-54115 2856-54463 2905-54580 3187-54821 2055-53729 2682-54401 2861-54583 2911-54631 3207-54850 Plates with systematic uncertainties removed from Hettinger et al. (2015), listed by plate number and Modiﬁed Julian Date. on plates in the SDSS sub-exposures can be traced to. We strongly encourage individuals who wish to use sub-exposure information from the SDSS spectrograph to be aware of these issues and take them into consideration when analyzing radial velocities.

39 (a) (b) (c) (d)

Figure 2.6: Same as Figure 2.4 for plate plugging 3002-54844. The left column of each subfigure shows RVs before correcting the plate for systematic offsets, and the right column shows RVs after correcting for offsets. Corrections are not successful on this plate due to anti-correlated subsets of similarly correlated fibers.

40 2.3 Radial Velocities

Radial velocities can be derived from spectra using the Doppler shift by comparing the wavelengths of the observed absorption features of the source λo, to the rest-frame expectations

λe. The velocity v, relative to the speed of light is

v λ − λ = o e . (2.1) c λe

Where appropriate, this is accomplished through a cross-correlation of the object’s spectrum with a theoretical or empirically derived template. In this section, we discuss the methods developed in this work for determining RVs through template cross-correlation, referencing the implementation of these methods in Hettinger et al. (2015). We address the processes for normalizing spectra, creating a spectral template, and performing the cross-correlation measurements.

2.3.1 Continuum Normalization

Cross-correlation requires that spectra be continuum normalized, such that signiﬁcant deviations from unity are attributed to absorption and emission features, rather than the blackbody continuum. The process of normalizing a spectrum is as follows.

A model of the continuum is estimated by generating a modified version of the spectrum, which is a greatly smoothed copy of the spectrum. The smoothed copy has all outlying pixels cleaned and set equal to the median flux value. Absorption lines must be masked out prior to modeling the continuum. The patched regions to be masked are specified explicitly with a set of wavelength limits where the absorption features are expected to appear for the

41 particular astronomical source. See Table 2.2, for example, for a list of the major regions that were masked out in the F-dwarf stars. The entirety of each region is replaced with a patch

Table 2.2: Absorption Features in F-dwarfs

Wavelength Range Feature 3600 A˚ – 4000 A˚ Ca H,K and others 4070 A˚ – 4130 AH˚ δ 4290 A˚ – 4360 AH˚ γ 4830 A˚ – 4890 AH˚ β 6540 A˚ – 6580 AH˚ α

ﬂux value, calculated by taking the median of all ﬂux values in the region where the values are greater than the median value. In other words, the region is patched with a continuous

ﬂux value equal to an average value, ignoring absorption features. See Figure 2.7 for an example of an F-type star with the absorption regions patched out. Masking absorption features this way, on a relatively simple spectrum such as that from an F-type star, can be done easily, however, spectra with more intense or more complicated features, such as those of an M-dwarf, would require an alternative method for continuum normalization (Ness et al., 2015). With the absorption regions masked out, a smoothed version of the spectrum is produced by removing high frequencies in the Fourier Transform using a FFT smoothing algorithm. Finally, the original spectrum is divided by the smooth continuum to obtain the continuum normalized spectrum. It is important to perform the normalization process correctly. Early attempts with poor continuum ﬁtting at the ends of the spectra resulted in asymmetric distributions in measured radial velocities.

42 800 700 (a) 600 500 400 Flux 300 200 100 0 800 700 (b) 600 500 400 Flux 300 200 100 0 800 700 (c) 600 500 400 Flux 300 200 100 0 1.4

1.2 (d)

1.0

0.8

0.6

0.4 Normalized Flux Normalized 0.2

0.0 4000 5000 6000 7000 8000 9000 Wavelength (Å)

Figure 2.7: Continuum normalization process applied to an F-type star showing: (a) the raw sub-exposure spectrum, (b) the spectrum with selected absorption features masked out, (c) a smoothed version of the spectrum, and (d) the ﬁnal continuum-normalized spectrum.

43 2.3.2 Spectral Template

Two common methods for measuring the RV of a source are, identiﬁcation of line indices,

and template cross-correlation. The former uses an algorithm to identify absorption and

emission features in spectra that are expected to be present, and compares the measured

centroids of the lines with the expected rest-frame wavelengths of the features to directly

measure a Doppler shift. Methods for locating centroids and correctly identifying lines

associated with particular features can be relatively complex. The latter method, using

template cross-correlation (Tonry & Davis, 1979), is more simple but requires an accurate

template spectrum for each source that you wish to measure RVs from. In Hettinger et al.

(2015), all sources are F-type dwarf stars, and can be ﬁt with a single template. One beneﬁt

of performing cross-correlation on a large dataset of similar sources, is that a template can

be constructed from the data itself. A master template (Figure 2.8) was constructed using

high quality spectra from 7207 F-type dwarf stars from SDSS DR10.

First attempts at making RV measurements with ﬁnely tuned templates yielded little

change in measured variations in RV. We created several templates composed of sources with

similar stellar parameters, allowing [Fe/H], log g, and Teﬀ to vary. The standard-deviation of

individual radial velocities within stars remained virtually the same across the entire [Fe/H] range, regardless of the choice in template. Similar behavior was observed for RV variations as a function of log g and Teﬀ. For simplicity, we used a single template composed of spectra from stars of varying stellar parameters, using all available science primary ﬁbers with a co-add SNR > 50.

Steps for creating a template spectrum are as follows. All input co-add spectra are continuum-normalized (Section 2.3.1). Next, the co-add spectra are de-shifted into the rest-

44 1.2

1.0

0.8

0.6

Hβ Hα Normalized Flux Normalized 0.4

0.2 4000 5000 6000 7000 8000 9000

1.2

1.0

0.8

0.6

Ca K Ca H Hδ Hγ Normalized Flux Normalized 0.4

0.2 3800 3900 4000 4100 4200 4300 4400 Wavelength (Å)

Figure 2.8: Full continuum-normalized F-type dwarf template spectrum (top) with a detailed view of the blue end of the spectrum (bottom). Prominent spectral features are annotated. frame using the “zBest” redshift values assigned to the co-adds in the SDSS pipeline. It should be noted here that while the SDSS pipeline provides RV estimates for the co-add spectrum of each ﬁber, these were obtained from an average of the individual sub-exposures, thereby averaging over the changes in RV between sub-exposures – the changes that we are seeking to measure. Once all of the spectra are shifted to rest-frame wavelengths, each spectrum is ﬁt by a third-order B-spline and resampled to a common wavelength solution.

Finally, all co-add spectra are averaged by taking the mean ﬂux at each wavelength in the resampling. Figure 2.9 illustrates the template creation process for a small set of input co-adds.

2.3.3 Cross-Correlations

Radial velocities obtained from the Doppler shift are measured by ﬁnding the wavelength shift, in pixels, of the source which maximizes the cross-correlation coeﬃcient of the source and template. This process is detailed here. For every star in the sample, each sub-exposure

45 Figure 2.9: Example of the template creation process using 7 spectra. From top to bottom: blue end and full spectrum of the input co-add stellar spectra, blue end and full spectrum of the normalized, rest-frame input spectra, blue end and full spectrum of the input spectra resampled to a common wavelength solution, blue end and full spectrum of the template (averaged ﬂux of resampled input spectra).

46 spectrum is normalized (Section 2.3.1) and cleaned of cosmic rays and other suspect pixels.

Next, the template spectrum is resampled so that the source and template spectra share the

same wavelength value at each pixel. The correlation coeﬃcient per pixel,

N 1 X C = y1,i × y2,i (2.2) N i=1

is calculated using the cleaned source spectrum and the template spectrum, repeated with

integer pixel shifts of the source spectrum from −20 to +20 pixels. This process produces

a cross-correlation function (CCF) that has a maximum value at the best estimate for the

Doppler shift of the source. An example CCF can be seen in Figure 2.10.

0.98510 Spline Interpolation Correlation Values 0.98505 Peak = -3.275

0.98500

0.98495 CCF / Pixel CCF/

0.98490

0.98485 −20 −15 −10 −5 0 5 10 15 20 Lag (Pixels)

Figure 2.10: Cross-correlation function (per pixel) for a single sub-exposure spectrum of an F-type star. Correlation values are calculated at integer pixel lags, and a spline interpolation of the function is ﬁt to these values. The function peaks at −3.275 pixels, or −229 km s−1.

The value obtained for the pixel lag can be converted to a Doppler shift in km s−1 with a conversion that is dependent on the resolution of the spectrum. For SDSS spectra, this conversion is 70 km s−1 pixel−1. To obtain sub-pixel precision, a smooth B-spline

47 interpolation is performed on the CCF, yielding a better estimate for the Doppler shift.

2.4 Empirical Uncertainties

Uncertainties in cross-correlation lag measurements must be estimated empirically or through

some Monte Carlo method. For an example of a Monte Carlo method, see Peterson et al.

(1998), where cross-correlations are performed over many iterations, taking random ﬂuctua-

tions in pixel ﬂux and selecting a random sub-set of pixels at each iteration. Hettinger et al.

(2015) employed an empirical method, looking at the spread in RV measurements for spectra

of similar quality. From these comparisons, we derived a function that assigns uncertainties

based on the SNR and [Fe/H] of each sub-exposure. Measurement uncertainties are expected

to increase for stars with lower metallicity, as these stars have less pronounced absorption

features, thereby reducing the signal of, and broadening the peak of, the CCF. The following

describes the steps taken to derive the empirical uncertainties.

To begin, the mean RV for a star is subtracted from all sub-exposure measurements of

the star, resulting in new RVs with a mean value of 0.0 km s−1. Similar measurements can

now be compared for all sub-exposures from all stars by identifying spectra with similar

[Fe/H] and SNR values; initial tests showed no signiﬁcant correlation between measurement uncertainty and other stellar parameters such as Teﬀ and log g. Once all measurements

are separated into sets of like [Fe/H] and SNR, an empirical estimate for the measurement

uncertainty for each [Fe/H]-SNR set is calculated using the median absolute deviation, or

MAD (Leys et al., 2013). The MAD value is the median deviation of all values from the

median value,

MAD = median(|RVi − median(RV)|), (2.3)

48 and is related to the empirical estimate of the measurement uncertainty by

σ = 1.4826MAD. (2.4)

One example of an empirical uncertainty estimate is shown in Figure 2.11. All sub-exposures used in this [Fe/H]-SNR set have [Fe/H] = −1.75 ± 0.25 and SNR = 30 ± 2.5. This process is repeated for all [Fe/H]-SNR sets that have at least 800 measurements.

0.09

0.08

0.07

0.06

0.05

0.04 N / Total / N

0.03

0.02

0.01

0.00 −40 −30 −20 −10 0 10 20 30 40

RV - Vo (km/s)

Figure 2.11: Distribution of radial velocities (minus systemic velocity) for sub-exposures with [Fe/H] = −1.75 ± 0.25 and SNR = 30 ± 2.5. The distribution has a median absolute deviation of MAD = 3.04 km s−1, indicating measurement uncertainties of σ = 4.51 km s−1.

Interpolation of the values obtained for the empirical uncertainty estimates of all [Fe/H]-

SNR sets yields a functional form describing the measurement uncertainty with respect to

[Fe/H] and SNR. First, a 1-dimensional solution is ﬁt to all [Fe/H]-SNR sets sharing the

49 same [Fe/H] value using the form

m[Fe/H] σ (SNR) = + b, (2.5) [Fe/H] SNR

where the uncertainty σ[Fe/H], is inversely proportional to SNR with constants m[Fe/H] and

b. This is performed independently for each of the nine [Fe/H] values used in the sets. The

median value b = 1.235 is adopted and used to reﬁt all solutions,

m[Fe/H] σ (SNR) = + 1.23. (2.6) [Fe/H] SNR

The results from this are shown in Figure 2.12(b). A linear relationship between [Fe/H] and

the constant m is described by

m ([Fe/H]) = −26.51[Fe/H] + 50.52, (2.7)

and is illustrated in Figure 2.13. Altogether, the relationship between the empirical estimate

of the measurement uncertainty and the sub-exposure [Fe/H] and SNR takes the form

(−26.51[Fe/H] + 50.52) σ ([Fe/H], SNR) = + 1.23. (2.8) SNR

Figure 2.12(c) illustrates the ﬁnal form of the empirical uncertainties with values plotted in lines of constant [Fe/H]. Hettinger et al. (2015) adopted Equation 2.8 to assign uncertainty values to every sub-exposure measurement in the sample. The distribution of the assigned uncertainties is shown in Figure 2.14.

50 6.0

5.5

5.0

4.5

(km/s) 4.0 RV σ 3.5

3.0

2.5 (a)

6.0

5.5

5.0

4.5

(km/s) 4.0 RV σ 3.5

3.0

2.5 (b)

[Fe/H] ≃ -2.25 6.0 [Fe/H] ≃ -2.00 [Fe/H] ≃ -1.75 5.5 [Fe/H] ≃ -1.50 [Fe/H] ≃ -1.25 5.0 [Fe/H] ≃ -1.00 [Fe/H] ≃ -0.75 4.5 [Fe/H] ≃ -0.50 [Fe/H] ≃ -0.25 (km/s) 4.0 RV σ 3.5

3.0

2.5 (c)

20 30 40 50 60 SNR

Figure 2.12: Empirical uncertainties for cross-correlation measurements in F-type dwarf stars showing uncertainties (a) as a function of SNR independently determined for each [Fe/H], (b) as a function of SNR with a common constant oﬀset, and (c) as a function of SNR and [Fe/H] as in Equation 2.8.

51 130

120

110

100

90 (km/s) m 80

50 −2.5 −2.0 −1.5 −1.0 −0.5 0.0 [Fe/H]

Figure 2.13: Values of coeﬃcient m from Equation 2.6 as a function of [Fe/H].

4 Metal-Poor Metal-Intermediate Metal-Rich 3

2 log(N)

0 2 4 6 8 10 σ (km/s)

Figure 2.14: Distribution of empirically assigned measurement uncertainties for the F-dwarf stars from the Hettinger et al. (2015) sample. Metallicity cutoﬀ values are [Fe/H] = −1.43 and [Fe/H] = −0.66.

52 2.5 e/i Variability

After sub-exposure RVs are determined and the uncertainties are well characterized, a metric

must be chosen to describe the variability of the source. One such metric, described by Geller

et al. (2008) and Milliman et al. (2014), detects variability in RV measurements by comparing

the RV deviations (external variation, e) to the typical measurement uncertainty (internal

variation, i), or e/i. This way, stars with diﬀering SNR, and therefore diﬀerent measurement

uncertainties, are weighted appropriately, preventing stars with large RV uncertainties from

being interpreted as true variables. The e/i metric is deﬁned as the ratio of the standard deviation of the star’s RVs to the mean of the measurement uncertainties. Values of e/i signiﬁcantly larger than 1.0 indicate an increasing likelihood that a star’s variability is not due to measurement uncertainties, but rather to the intrinsic changes in the star’s RV.

One disadvantage of the e/i method, however, is that it does not use a physical model to describe the system. For instance, RV variability of a star in the presence of a companion body is expected to show a periodic behavior. This lack of speciﬁcity is what led us to

ﬁnally adopt the MCMC methods detailed in Section 3.5. It is reassuring though, that examination of the e/i values of the stars in the Hettinger et al. (2015) sample (Figure

2.15) yields a conclusion consistent with that gathered from the MCMC method, with both methods indicating a higher fraction of short-period binaries in the metal-rich component of the Milky Way.

2.6 Discussion

Time resolved-spectroscopy is not limited to radial velocity measurements. The processes discussed in this chapter can be modiﬁed to look at changes, for example, in the emission and

53 4.0 Metal-Poor 3.5 Metal-Intermediate 3.0 Metal-Rich

2.5

2.0

1.5 log(N)

1.0

0.5

0.0

−0.5 0 5 10 15 20 e/i

Figure 2.15: Distribution of e/i, the ratio of the standard deviation of RVs to the typical measurement uncertainty values, for the F-dwarf stars from the Hettinger et al. (2015) sample. Metallicity cutoﬀ values are [Fe/H] = −1.43 and [Fe/H] = −0.66.

absorption features of stars such as those due to surface activity of cool stars or pulsation of

RR Lyrae stars. Future developments in telescopes and instruments may even lead to useful

studies of short-timescale variability of active galactic nuclei. We recommend that future

observing programs consider the sub-exposure properties of multi-ﬁber spectrographs, as

they add scientiﬁc value in regards to the time-domain dimension of the data. With careful

planning of the exposure times, counts, and frequencies adopted in target selection, the

scientiﬁc return obtained from survey programs can be maximized.

54 Chapter 3

Markov Chain Monte Carlo

3.1 Introduction

Many problems in astronomy and astrophysics are solved using complicated, costly models with large numbers of free parameters, often combined with low signal-to-noise observations.

Because of this, there has been an increased adoption of probabilistic analysis, such as

Bayesian inference. Additionally, with the increase in power and eﬃciency of modern day computers, numerical methods like Markov chain Monte Carlo (MCMC) methods are being used to solve problems that would have been previously unsolvable. Hogg et al. (2010) include more arguments for why one would want to adopt numerical data analysis techniques, and additionally provide instructions for using MCMC techniques for ﬁtting models to data.

In this chapter, we describe how MCMC methods can be used in an astrophysical context, referring to their use in Hettinger et al. (2015) as an example. We begin with a brief introduction to Bayesian inference in Section 3.2. In Section 3.3, we introduce the emcee

Python package and explain how an ensemble MCMC algorithm is used to ﬁnd expectation values of parameters of interest from a marginalized sampling of the posterior. In Section 3.4, we illustrate the use of the emcee package in identifying sub-exposure systematics in SDSS spectra. Section 3.5 oﬀers a description of the process for determining stellar variability from the RV curve of a star using a hierarchical MCMC method.

55 3.2 Bayesian Inference and MCMC

We begin with a brief introduction to Bayesian inference. Bayesian inference derives a

posterior probability that observations D, can be described by some model with a vector of

parameters Θ, as it relates to the likelihood function of the model and a prior probability

of the model parameters. Speciﬁcally, Bayesian inference computes the posterior probability

from Bayes’ theorem: 1 p(Θ|D) = p(D|Θ)p(Θ). (3.1) Z

Here, p(Θ|D) is the posterior probability, which specifies the probability that a set of model parameters can be inferred from the data, subject to the defined likelihood and prior. In other words, it describes the probability that a hypothesis is correct, after the observations are collected. The likelihood p(D|Θ), describes the compatibility of the observations with the model parameters. The prior probability p(Θ), specifies the previous estimate, if any, that the model parameters are correct, before any knowledge of the data is taken into account.

The model evidence Z, is a normalization factor which remains constant for all choices of parameter values, and can therefore be ignored for our purposes.

A posterior probability density function (PDF) gives the posterior probability for all of possible model parameters, having maximum values for sets of parameters which are most likely to yield the observations. Computing the PDF is often diﬃcult due to complex likelihood functions. MCMC methods can be adopted to approximate the PDF, numerically, by sampling from the posterior with a class of algorithms. With the sampling from the PDF,

MCMC methods easily allow marginalization over nuisance parameters (parameters that are required by the model, but are of less interest) to retrieve the probabilities and expectations for values of any parameter of interest to the problem.

56 Markov chain Monte Carlo methods are a class of algorithms designed to sample from a

probability distribution. They do so by constructing a Markov chain that allows a walker to

move between states in parameter space with some transition probability. At each step in

the chain, some set of parameters are compared with the data through a likelihood function.

The transition probabilities between steps are related to the relative likelihood values of each

state. MCMC chains have the property that in the limit that the chain takes an inﬁnite

number of steps, the density of states sampled represents the PDF for the model. When the

chain has reached the number of steps where this is approximately true, the chain is said to

have converged. Steps can ﬁnally be selected randomly, providing a sample of the PDF. In

the next section, we show how an MCMC algorithm can be used to ﬁnd the best choice of

model parameters from a set of observations.

3.3 emcee: The MCMC Hammer

3.3.1 An Aﬃne-Invariant Ensemble Sampler

A commonly used MCMC method is the Metropolis-Hastings (MH) method. The MH

method proposes chain steps based on some distribution (such as a multivariate Gaussian)

centered on the current chain position. This requires a number of tuning parameters which

scales as N(N + 1)/2, where N is the dimensionality of the model. Conﬁguring many tun-

ing parameters is costly and requires many burn-in steps, especially for highly anisotropic

densities. The Python package emcee (Foreman-Mackey et al., 2013) addresses this issue

by implementing the aﬃne-invariant ensemble sampling algorithm proposed by Goodman &

Weare (2010). emcee uses an ensemble of parallel chain walkers that take steps in series.

Before each walker takes a step, a step proposal is drawn using a stretch move. That is, a

57 random walker is selected from the remaining ensemble, and a step is proposed along the

vector connecting the two walkers. This process is aﬃne invariant, meaning the algorithm

performs equally well under all linear transformations. The beneﬁt of using the stretch move

algorithm is that only two tuning parameters are required, regardless of dimensionality of

the model. This allows an ensemble of chain walkers to explore anisotropic densities very

eﬃciently. Additionally, emcee implements multi-threading by running chain walkers as

separate threads, greatly increasing CPU eﬃciency when the code is run on a multi-core

machine. Because of the eﬃciency with anisotropic densities and the ease of parallelization,

Hettinger et al. (2015) uses emcee for testing binary models on the F-dwarf RV curves.

3.3.2 Using emcee

As an introduction to the emcee package and, more generally, the use of MCMC methods

for inference, we present a simple example problem. For more information on using emcee,

please consult Foreman-Mackey et al. (2013). In this example problem, we have simulated

30 observations (Xi, Yi) from the relationship

Yi = 0.8Xi + 0.3 , (3.2)

with simulated measurement uncertainties in Y folded in by drawing from a normal distribu-

tion with σY = 0.1 (Figure 3.1). We wish to use Bayesian inference and the emcee package

to determine the best values for the parameters of some hypothetical model.

We begin by suggesting a hypothetical model

y(x) = mx + b , (3.3)

58 1.2

1.0

0.8

Y 0.6

0.4

0.2

0.0

0.0 0.2 0.4 0.6 0.8 1.0 X

Figure 3.1: Mock observations with X values drawn randomly from the range [0,1] and Y values drawn from Yi = 0.8Xi + 0.3. Simulated measurement uncertainties have been added with values drawn from a normal distribution with µ = 0.0, σ = 0.1.

for which we are trying to determine the best values of m and b. The MCMC ensemble

sampler is created in emcee by specifying the number of chain walkers (say 100), the di-

mensionality of the model (in our case, 2), and a function used to calculate the posterior

probability at each step. For this problem we have deﬁned the posterior probability to be

p(Θ|D) = p(Θ)p(D|Θ) , (3.4)

with a likelihood   N 1 (Y − y(X ))2 Y √ i i p(D|Θ) = exp − 2  , (3.5) i=1 σY 2π 2σY

and a prior (1, (−10 < m < 10) and (−10 < b < 10) p(Θ) = . (3.6) 0, otherwise

We’ve adopted a uniform prior for both m and b. For most simple problems, an uninfor-

mative prior is a good choice, assuming most of the probability density lies within the limits

of the prior. Since the prior is multiplied by the likelihood, if no information about the prior

is given, the posterior will be equal to the likelihood, and therefore the peak in the posterior

59 will be equal to the maximum likelihood.

After the ensemble sampler is deﬁned, we initialize the starting positions for the 100

walkers. A simple choice for starting positions is a random distribution in parameter space

deﬁned by the prior limits. We can now run the ensemble sampler chain for some number of

steps S. The total number of steps needed for a robust solution depends on the complexity

of the problem and the starting positions of the walkers. We would like enough steps such

that the chains converge, and we want enough steps so as to produce a few independent

representative samplings of the posterior. We can see from the chain history (Figure 3.2

and Figure 3.3) that all chains converge in about 60 steps. Therefore, we want at least 60

steps per chain, while burning and removing the ﬁrst 60 steps from the ﬁnal distribution.

Additionally, we may want to thin the sample to reduce autocorrelation. Because chain

0 m

−5

−10 10

b 0

−5

−10 0 10 20 30 40 50 60 Step Number

Figure 3.2: Top: values of m from the MCMC for the ﬁrst 60 steps taken by all 100 chain walkers. Convergence is reached by step 60. Bottom: same for the parameter b. walkers often take steps proposed in their vicinity, there is some correlation among adjacent steps. To mitigate this eﬀect, we will keep every 3rd step from a walker, and discard the remainder. Figure 3.4 shows the solutions for 300 randomly selected (m,b) values sampled

60 8

6 0.5 4 0.4 2

b 0 b 0.3

−2 0.2 −4 0.1 −6

−8 0.0 −8 −6 −4 −2 0 2 4 6 8 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 m m

Figure 3.3: Left: parameter-space location of the ﬁrst 60 steps taken for 7 of 100 chain walkers. Stars represent the inital position of each chain walker, and subsequent steps are displayed as circles of decreasing radius. Right: a closer view of the center of the chain step distribution.

from the MCMC posterior. Figure 3.5 depicts the ﬁnal posterior distributions for our model

parameters. Sub-ﬁgure (a) is the joint PDF of the two variables m and b. If we ignore one

1.2

1.0

0.8

Y 0.6

0.4

0.2

0.0

0.0 0.2 0.4 0.6 0.8 1.0 X

Figure 3.4: Linear solutions for 300 randomly selected (m,b) pairs sampled from the MCMC posterior distribution. parameter and take the histogram for the other, we obtain the marginalized PDF for a single variable. Marginalizing over a nuisance parameter propagates the eﬀects of the uncertainty in the nuisance parameter into the PDF of the marginalized parameter. The marginalized

PDFs for m and b are also depicted Figure 3.5.

We ﬁnally arrive at our solution. Using the mean values of the marginalized PDFs, our

61 Figure 3.5: Posterior probability distribution for: (a) the join distribution of parameters m and b, (b) parameter m, marginalized over parameter b, and (c) parameter b, marginalized over parameter m. Blue lines indicate values of m and b used in the creation of the mock data points.

62 Bayesian inference yields the values m = 0.77 with a 68% credible interval 0.71 < m < 0.83,

and b = 0.30 with a 68% credible interval 0.27 < b < 0.34.

3.4 Correcting Systematics in SDSS Spectra

As discussed in Section 2.2.3, RV measurements of stars from the SDSS were observed to

have systematic uncertainties. Each plate-plugging contains 640 ﬁbers (M of which are

F-type stars) that were exposed simultaneously, N times, for a total of N × M F-type sub-

exposures per plugging. For some plate-pluggings, groups of ﬁbers had positively correlated

RV changes. To quantify the systematic shifts, we employed a MCMC chain with emcee.

Every plate-plugging was checked independently with a single ensemble sampler. The samplers consisted of N parameters i, one for each plate exposure. These parameters represent the optimal correction that should be applied to the ith RV of all M stars on the plate. The choice in the likelihood function seeks to reduce the scatter of RVs about the stars’ mean velocities, with the requirement that the oﬀset applied to exposure i is identical for all stars on the plate. The log-likelihood1 is

 2  M N 1 Yj,i − i − hY i X X  2 j  ln p(D|Θ) = − ln(σj,i) + 2  , (3.7) 2 j i σj,i

where Yj,i and σj,i are the RV and uncertainty for the ith exposure of the jth star, and hY ij is the weighted mean velocity of the jth star. We are assuming here that the contribution of scatter due to intrinsic variability in the stars is negligible. For the prior we use uniform

1To prevent issues with numerical precision, MCMC samplers, such as emcee, use the summation of log-likelihoods rather than the product of likelihoods.

63 distributions,

( 0, (−50 km s−1 < < 50 km s−1) for all i ln p(Θ) = i . (3.8) −∞, otherwise

The sampler for each plate uses 200 chain walkers, running 600 steps each, and thinned by a factor of 3. The ﬁrst 100 steps were removed as a burn-in. Figure 3.6 depicts the resulting PDF for an example plate-plugging, 2085-53379. This plate has 8 F-dwarf stars with 4 sub-exposures each. The corrections to the radial velocities are illustrated in Figure

2.4.

3.5 Modeling Multiplicity With Radial Velocity Curves

The work of Hettinger et al. (2015) examines the RV curves of all of the F-dwarf stars in the sample to determine the probability that a system has a binary companion detection. Once again we use emcee, creating a trans-dimensional, hierarchical MCMC ensemble sampler.

This sampler consists of two models: a single-star model with a sole parameter, and a binary- star model with four parameters. A hyperparameter λ, serves as an index for choosing one of the models at each step in the chain. After marginalizing over all model parameters for both models, the marginalized PDF of λ infers the posterior probability that a star’s RV curve supports a short-period binary detection.

The single-star model Ms, adopts a non-varying RV curve, parameterized by a systemic velocity V0,

Ms(t) = V0 . (3.9)

In the binary-star model Mb, the RV curve is ﬁt by a simple sinusoid deﬁned by four

64 Figure 3.6: Two-parameter joint probability distributions for plate-plugging 2085-53379. All values are in km s−1. Fully marginalized, single-parameter probability distributions occupy the subplots on the diagonal. Parameters dyi (i in the text) represent the corrections to be applied to the ith exposure on the plate.

65 parameters: the log of the semi-amplitude log A, the log of the period log P , a phase φ, and a systemic velocity V0, t M (t) = A sin 2π + φ + V . (3.10) b P 0

This model assumes circular orbits (eccentricity = 0). At short periods (P < 12 days), where our data is most sensitive, circular orbits are nearly ubiquitous (Raghavan et al., 2010), as orbits become circularized due to tidal forces.

In addition to the parameters describing the RV behavior of the primary star, we included a number of parameters to mitigate the effects of possible inter-plate systematics, discussed here. As mentioned in Section 2.2.3, intra-plate systematics were found to have been artificially shifting the RVs of large groups of fibers. These issues were addressed with

MCMC methods, but it remains diﬃcult to ﬁnd these systematics between plates, because

fiber plug positions were often changed between plate observations. It is reasonable to assume that systematics may be present from plate to plate, so additional parameters ωp are used to model possible shifts in RV between plate observations. This works by allowing all RVs of an individual star taken on plate p, to shift systematically relative to the first plate (p = 1). By introducing these free parameters, binary models that fit the RV curve based solely on plate-to-plate differences are penalized, while models that show strong RV variations within a plate are less affected. For most fibers, RV data is drawn from a single plate observation, and therefore do not contain any ωp parameters. Other fibers, which have

P > 1 plate observations, will have a total of P − 1 plate-shift parameters.

With the plate-shift parameters and the two RV models deﬁned, the log-likelihood used

66 in the emcee sampler becomes

 2  N 1 Yi − ωp − M(ti) X  2  ln p(D|Θ) = − ln(σi ) + 2  , (3.11) 2 i σi where M(t) is equivalent to Mb(t) whenever λ selects the binary model (λ ≥ 0.5), and

2 Ms(t) otherwise. Yi is the RV of the star at exposure i with a measurement uncertainty

σi.

The choice in prior distributions for the parameters (Table 3.1) were considered carefully.

In the regime where the period grows longer or the amplitude decreases in magnitude, the

Table 3.1: Prior Limits for Hettinger et al. (2015) MCMC

Parameter Lower Limit Upper Limit λ 0.0 1.0 φ 0 2π −1 ωi (km s ) -20 20 −1 V0 (km s ) -600 600 log A (km s−1) 0.48 2.40 log P (s) 4.0 7.0 Prior limits adopted in Hettinger et al. (2015) for the parameters in the MCMC ensemble sampler. Prior probabilities were uniformly distributed within these ranges. Parameters include the model selector index λ, orbital phase φ, plate shift parameters ωi, systemic velocities V0, the log of the semi-amplitude log A, and the log of the orbital period log P .

binary-star model gives the same likelihood values as the single-star model. Therefore, the limits on the prior probabilities should not be arbitrarily large, but should instead be set at a reasonable values that are representative of the sensitivity of the sparsely sample RVs.

For systems with periods log P (s) > 7.0, the expected RV amplitudes decrease to levels below the measurement uncertainty of our data. As expected, early MCMC trials found no

2Unfortunately, emcee does not handle binary data types as a model parameter, so we set λ as a parameter with uniform prior between 0 and 1, using 0.5 as the dividing point for determining which model to use at any step.

67 signiﬁcant probability for these large periods in the Mb model. A lower limit log P (s) = 4.0 is used; this is the orbital period at a = 1 R , for which stellar contact between two stars is certain. For the semi-amplitude, the lower limit on the prior is 3 km s−1, comparable to

the measurement uncertainties in the RVs. The upper limit on the prior is 250 km s−1, a

value greater than the maximum RV amplitude expected from an F-dwarf system. Systemic

velocities allowed are those less than the escape velocity of the Milky Way.

The MCMC samplers were executed independently for all stars in the sample using 200

chain walkers, taking a total of 2.4 million steps. The chains were burned to convergence,

and thinned down to a ﬁnal size of 600,000 samples per star. For a description of the ﬁndings

and a discussion of the results, please refer to Section 4.4 and Section 4.5. Instead, we will

show here the results for select stars, and discuss some features of the posteriors.

3.5.1 Examples

The ﬁrst example star is a binary candidate with the ﬁber ID 2939-54515-194. This star

is a F-type dwarf star from the Hettinger et al. (2015) sample, with stellar parameters

[Fe/H] = −1.50 and log g = 4.3. Twelve individual, raw sub-exposures (hSNRi = 40),

comprise the coadd spectrum (Figure 3.7).

68 Individual Raw Exposures Raw Individual Coadd

4000 5000 6000 7000 8000 9000 Wavelength in Å

Figure 3.7: Individual sub-exposure spectra (top) used in the production of the coadd spectrum (bottom) for ﬁber ID 2939- 54515-194.

69 Sub-exposures were taken on three separate nights within a week for a baseline of 143

hours. The measured sub-exposure RVs diﬀer by as much as 33.7 km s−1, indicating that

the star is a likely binary candidate.

Using the settings from Section 3.5, an ensemble of MCMC chain walkers compared the

RVs with the binary- and single-star models, ultimately ﬁnding the binary model to be the

most probable. Figure 3.8 reports the parameter values for all chain walkers as a function of

step number. All parameters converged by step 1000 (3000 before thinning), after which we

see virtually no samples with λ < 0.5, indicating a strong probability of the binary model over the single-star model. We’ve plotted the radial velocity curve for 2939-54515-194 in

Figure 3.9, along with potential orbits constructed from 200 randomly drawn samples from the posterior. We see the typical aliasing of the likely orbital periods that one expects to infer from a sparsely sampled RV curve. The gaps in coverage allow for models with orbits that have shorter periods, often at harmonic frequencies. Because of this, we may not be able to specify the exact period with conﬁdence, but the MCMC inference does allow us to rule out the single-star model. This can be seen, for example, when we marginalize over log P (and all other parameters) and look at the the marginalized PDF for the λ selection parameter (Figure 3.10, bottom-right). All values of λ are ≥ 0.5, indicating a signiﬁcant

preference for Mb. Examining the posterior PDFs in Figure 3.10, we see other features.

The period aliasing appears again in the marginalized posterior for log P as multiple peaks.

Additionally, a relationship among V0, φ, and log A exists. To ﬁt the data with models having extreme values of V0, the models must also have larger amplitudes with a phase shift. Also, these models can only work with longer periods, explaining the relationship seen between log P and log A. Larger amplitudes are supported only by longer periods.

With better RV coverage, the region of parameter space that can produce viable orbits to

70 Figure 3.8: Parameter value progression for all 200 chain walkers in the multiplicity MCMC for ﬁber ID 2939-54515-194. Chain samples have already been thinned. Samples earlier than the red dashed line were removed from analysis during the burn-in process.

71 Figure 3.9: Orbits constructed from 200 random samples of the MCMC posterior distribution for ﬁber ID 2939-54515-194.

fit the data diminishes, allowing for a more confident estimate of orbital parameters. Our next example has better coverage and demonstrates a case where the orbital parameters are specified more confidently.

The next star, fiber ID 2960-54561-375, is an F-type star like 2939-54515-194, but it has a slightly lower surface gravity with log g = 3.6. Because this star’s surface gravity is below the conservative definition for dwarf stars defined in Hettinger et al. (2015), it was not used in any of the analyses. However, this star is among the stars with the most constrained values, due to the large variations in RV, combined with adequate sampling.

Unlike most of the other F-type stars measured with our cross-correlation technique, star

2960-54561-375 has measured velocities that are greater than the dispersion velocity of the

SDSS spectra (70 km s−1 pixel−1), meaning changes in the absorption features have shifts that are greater than a pixel, and are visible by eye. We highlight this in Figure 3.11, where we have plotted a close-up view of three absorption features. The eight sub-exposures are ordered chronologically from top to bottom, and plotted as a function of redshift velocity.

72 Figure 3.10: Posterior probability distributions of parameters in the multiplicity MCMC for ﬁber ID 2939-54515-194.

73 Ca K Ca H Hα Normalized Flux Normalized

0 0 0 200 400 200 400 200 400 −800−600−400−200 −800−600−400−200 −800−600−400−200 Velocity (km/s) Velocity (km/s) Velocity (km/s)

Figure 3.11: Changes in redshift with time for ﬁber ID 2960-54561-375. Absorption lines from normalized sub-exposures are ordered chronologically from top to bottom. Velocities in each sub-ﬁgure are relative to the rest-frame wavelength of Calcium K (left), Calcium H (middle), and Hα (right). Dashed vertical lines represent the mean velocity of the star.

74 That is, each pixel’s wavelength in the spectrum corresponds to a measured redshift, had that pixel contained the centroid of the absorption feature for that particular line (e.g., Ca

K). To compute this, we convert each pixel wavelength to a velocity from the Doppler shift

(Equation 2.1) using the speciﬁc absorption feature as the rest-frame wavelength. Using the mean value of the star’s measured RVs (dashed vertical line) as a guide, shifts in the line centroids over time become apparent.

Looking at randomly selected orbits from the ensemble sampler (Figure 3.12), we see a strong preference for a model with a period of P = 7.94 days, followed by models with harmonic periods. In the posterior (Figure 3.13), we see the peaks in the marginalized prob-

Figure 3.12: Orbits constructed from 200 random samples of the MCMC posterior distribution for ﬁber ID 2960-54561-375. abilities for all parameters, corresponding to the best model. We also see strong probability concentrated in a second model with a shorter period at P = 1.14 days.

75 Figure 3.13: Posterior probability distributions of parameters in the multiplicity MCMC for ﬁber ID 2960-54561-375.

76 Chapter 4

Statistical Time-Resolved

Spectroscopy: A Higher Fraction of

Short-Period Binaries for Metal-Rich

F-type Dwarfs in SDSS

This Chapter contains an expanded version of the peer-reviewed article Hettinger et al.

(2015) published in the Astrophysical Journal Letters

4.1 Abstract

Stellar multiplicity lies at the heart of many problems in modern astrophysics, including the physics of star formation, the observational properties of unresolved stellar populations, and the rates of interacting binaries such as cataclysmic variables, X-ray binaries, and Type Ia supernovae. However, little is known about the stellar multiplicity of ﬁeld stars in the Milky

Way, in particular about the diﬀerences in the multiplicity characteristics between metal- rich disk stars and metal-poor halo stars. In this study we perform a statistical analysis of

∼14,000 F-type dwarf stars in the Milky Way through time-resolved spectroscopy with the

77 sub-exposures archived in the Sloan Digital Sky Survey. We obtain absolute radial velocity measurements through template cross-correlation of individual sub-exposures with temporal baselines varying from minutes to years. These sparsely sampled radial velocity curves are analyzed using Markov chain Monte Carlo techniques to constrain the very short-period binary fraction for ﬁeld F-type stars in the Milky Way. Metal-rich disk stars were found to be 30% more likely to have companions with periods shorter than 12 days than metal-poor halo stars.

4.2 Introduction

Stellar multiplicity plays a crucial role in many fields of astronomy. Star formation and evolution, Galactic chemical evolution, nuclear astrophysics, and cosmology are all influenced by our understanding of the multiplicity properties of an underlying stellar population. Binary interactions lead to phenomena as diverse as cataclysmic variables, classical novae, X-ray binaries, gamma-ray bursts, and Type Ia supernovae. Stellar interactions are also the cause of the anomalous surface abundances measured in Ba stars, CH stars, and the majority of carbon-enhanced metal-poor stars (Lucatello et al., 2005). The rates of these phenomena depend on the multiplicity properties such as the fraction of stars with companions and the distributions of separations and mass ratios. How these properties are in turn affected by variables such as stellar age, metallicity, and dynamical environment remains poorly understood. Moe & Di Stefano (2013) find no significant trends with metallicity for O- and

B-stars, but more work is needed for lower-mass stars.

The recent review by Duchˆene& Kraus (2013) summarizes the state of the art in multiplicity studies. The fraction of systems with companions is known to be a strong function

78 of stellar mass (Lada, 2006; Raghavan et al., 2010; Clark et al., 2012), and there are hints that lower mass systems have smaller separations (Duquennoy & Mayor, 1991; Allen, 2007;

Raghavan et al., 2010). Studies of the Solar neighborhood also indicate that lower metallicity stars are more likely to have stellar companions (Raghavan et al., 2010).

These results are based on heterogeneous samples of a few hundred stars at most, often dominated by wide systems which will never become interacting binaries. The spectroscopic surveys that reach small periods are labor intensive because large numbers of radial velocities

(RVs) are required to ﬁnd the orbital solution of each target. This leads to small sample sizes, which have only increased modestly in the past two decades, from 167 in Duquennoy & Mayor

(1991) to 454 in Raghavan et al. (2010). The drive to collect complete samples has limited previous spectroscopic studies to the Solar neighborhood or specific stellar clusters, but neither of these strategies can probe the full range of metallicities and ages spanning the field stars of the Milky Way (MW) disk and halo components. These limits bias the interpretation of data against the global properties of, and variation within, the MW field. Thus, we are motivated to take a statistical approach with a sample of stars located throughout the field in order to investigate their multiplicity properties with respect to age, [Fe/H], and component membership.

With the advent of multiplexed spectroscopic surveys like SDSS (York et al., 2000) and

LAMOST (Cui et al., 2012), we can use multiple RV measurements of thousands of stars to study the properties of stellar multiplicity that are more representative of the entire Galaxy.

SDSS Data Release 8 (Aihara et al., 2011) contains over 1.8 million optical spectra from the original SDSS spectrographs including over 600,000 stellar spectra. In this work we employ a lesser known SDSS feature, the time-resolved dimension. To facilitate cosmic ray removal, spectra were constructed through co-addition of several individual sub-exposures, typically

79 15 minutes in duration. Although under-utilized, the beneﬁt of the sub-exposure domain

is recognized in works such as Badenes et al. (2009) and Bickerton et al. (2012). Portions

of the sky were also re-observed for calibration and scientiﬁc purposes. These additional

pointings, combined with the sub-exposures, yield a time dimension where single stars have

exposure coverage ranging from 3 sub-exposures up to over 40 sub-exposures, and time gaps

from hours to nearly a decade. The techniques employed herein follow the time-resolved

work by Badenes & Maoz (2012) and Maoz et al. (2012).

4.3 Measurements

4.3.1 SDSS Observations and Sample Selection

F-type dwarfs are chosen for our sample because of the large number of stars targeted by

SDSS with repeat observations, and their relatively mild variability and activity. Addition-

ally, F-stars have main sequence (MS) lifetimes greater than 5 Gyr, allowing us to select

MS stars from both the younger disk and older halo. The Sloan Stellar Parameter Pipeline

(SSPP; Lee et al., 2008) was developed to determine parameters for stellar spectra in the

SDSS archive, including metallicity [Fe/H], eﬀective temperature Teﬀ, and surface gravity log g. Sample selection began with identifying science primary objects from SEGUE-1

(Yanny et al., 2009) and SEGUE-2 (Rockosi, C. M. et al., in prep.) in the SSPP that were classiﬁed as an F-type star by the “Hammer” classiﬁcation code (Covey et al., 2007).

To minimize the eﬀects of stellar evolution on multiplicity, we selected only dwarf stars

(log g ≥ 3.75). Stars with multiple fiber pluggings were identified astrometrically and joined with the appropriate science primary fibers.

After measuring stellar RVs (Section 4.3.2), systematics were revealed in the SDSS sub-

80 exposure spectra. These correlations appear as similar shifts in RVs for many fibers located on the same plate, typically affecting neighboring fibers on the CCD. After plate-wide comparisons of F-stars, RV correlations were corrected where possible. Corrections applied to the 104 RVs are as large as 17 km s−1 with a standard deviation of 2.2 km s−1. Not all correlations could be identified automatically because of multiple groups of correlated shifts, opposite in direction, on some plates. Visual inspection of plates containing numerous false binary detections lead to the removal of 25 plates including 1155 stars. We urge individuals using sub-exposure spectroscopy in SDSS to consider these systematic shifts in the wavelength solutions.

Quality control consisted of the removal of: stars without valid parameters in SSPP,

fibers located on ‘bad’ plates, sub-exposures with a median pixel signal-to-noise ratio (SNR) less than 20 or with fewer than 3000 unflagged pixels, stars with time lags ∆T < 1800 s, stars with less than three clean sub-exposures, and corrupt or misclassified spectra (from visual inspection of stars with the largest RV variation or non-characteristic Teff). The final sample consists of 14,302 stars (16,894 fibers) with as many as 47 sub-exposures, spanning up to nine years of observations (Figure 4.1).

Our cleaned sample is characterized by metallicities ranging from −3.41 ≤ [Fe/H] ≤

+0.52. To aid comparison in our analysis, the ﬁnal sample was sub-divided into three groups of equal size by cuts in metallicity at [Fe/H] = −1.43 and [Fe/H] = −0.66 (Figure

4.1). The majority of the stars have three or four sub-exposures (median = 4), typically taken about 15 minutes apart. The median time lag for a star is 2 hours, however more than three years between observations can be seen in more than 250 stars (Figure 4.1). SNRs for sub-exposures lie in the range 20 < SNR < 84 with a median value of 32.

81 4 4 Metal-Poor Metal-Intermediate 1d 1m 1y 3 Metal-Rich 3

2 2 log(N)

1 1

0 0

−4 −3 −2 −1 0 1 3 4 5 6 7 8 9 [Fe/H] ∆t (log s)

Figure 4.1: Left: Metallicity distribution for 14,302 F-dwarfs. Right: Distribution of maximum time lag between the ﬁrst and last exposure of a star.

4.3.2 Radial Velocities

RV measurements were attained through cross-correlation of sub-exposures with a master template constructed from 7207 sample-star, co-added spectra where the co-added SNR >

50. The spectra were de-shifted using the redshift value assigned to the co-adds by the SDSS pipeline, continuum-normalized, and averaged together.

Sub-exposures were independently prepared and cross-correlated with the template. Spec- tra were continuum-normalized by dividing the spectrum with a highly smoothed version of itself using a FFT smoothing algorithm, and then cross-correlated with the template at various integer pixel lags. Each spectrum had a cross-correlation function (CCF) that was ﬁt with a smooth spline interpolation. With spectral resolution of R ∼ 2000, the peak lag in pixels translates to the spectrum’s redshift at 70 km s−1 pixel−1. The mean and standard deviation of RVs for individual stars are shown in the Figure 4.2 distributions. The velocity dispersion of the mean RVs decreases with increasing [Fe/H], indicating that our [Fe/H]- groups sample both the disk and halo components of the MW. The standard deviation of

RVs within individual stars is larger for the metal-poor group; however, empirically estimated uncertainties also show larger measurement errors for metal-poor stars. This underscores the

82 importance of the use of proper error analysis in a method such as ours.

4 4 Metal-Poor Metal-Intermediate 3 3 Metal-Rich

2 2 log(N)

1 1

0 0

−600 −400 −200 0 200 400 600 0 10 20 30 40 50 60 RV (km/s) std(RV) (km/s)

Figure 4.2: Mean (left) and standard deviation (right) of radial velocities within a star. Vari- ations in the standard deviation of velocities are aﬀected, in part, by the larger measurement uncertainties for metal-poorer stars.

4.3.3 Uncertainties

It is well known that uncertainties in CCF peaks must be estimated empirically or through some Monte Carlo method (e.g., Peterson et al. 1998). For this work we determined RV uncertainties empirically by quantifying the spread in measurements for spectra of similar quality. The median absolute deviation (MAD) is a robust measure of the variability of a sample and is related to the standard deviation by σ = 1.4826MAD, where MAD = median(|RVi − median(RV)|) (Leys et al., 2013). All measurements were de-shifted into the rest frame using the SDSS estimates of the co-add redshift, and placed into bins of similar metallicity ([Fe/H] ± 0.25) and signal-to-noise (SNR ± 2.5). Initial tests showed no correlations between measurement spreads and either log g or Teﬀ. Estimates for the uncertainty of RV measurements within a bin were calculated using MAD values. Here, it is assumed that the majority of stars do not have detectable variability over the observed time baseline, and that eﬀects from intrinsic variations in RV are minimized by adopting

83 median values. After performing this process for all bins, a functional form for assigning RV measurement uncertainties σRV was ﬁt with an inverse proportionality to SNR, and with a linear correction in [Fe/H]. The measurement uncertainty as a function of [Fe/H] and SNR is, in km s−1, (−26.51[Fe/H] + 50.52) σ ([Fe/H], SNR) = + 1.23. (4.1) RV SNR

Uncertainties are sub-pixel, falling below the spectral resolution of 70 km s−1 pixel−1. For exposures with SNR < 25, uncertainties range from 3.0 to 8.0 km s−1, with a median value of 5.0 km s−1. Exposures with SNR > 40 have uncertainties in the range 1.9 to 4.4 km s−1, with a median value of 2.7 km s−1.

4.4 Multiplicity

The probability of a star having a companion was determined through model comparison using a trans-dimensional, hierarchical, Markov chain Monte Carlo (MCMC) method. Two models were compared: a single-star model Ms, and a binary-star model Mb. The hyperparameter λ, indexes the model choice at each step in the MCMC chain. We evaluated the hierarchical model using the Python package emcee, a MCMC ensemble sampler (Foreman-

Mackey et al., 2013).

The single-star model Ms, ﬁts a star with non-varying RVs, parameterized by a systemic velocity V0. Because intra-plate systematics are known to exist, it is reasonable to assume inter-plate systematics exist as well. In light of this, (P − 1) additional parameters psi, were included for each star, where P is the number of plate-MJD pluggings composing the star.

These plate-shift parameters allow all RVs from plate i to shift by some amount psi, relative to the ﬁrst plate P0. For the majority of stars P = 1, no plate-shift parameters are necessary,

84 and Ms is a 1-parameter model.

In the binary star model Mb, the sparsely sampled RVs are ﬁt by a sinusoid deﬁned by four-parameters: the log of the semi-amplitude log A, the log of the period log P , the phase

φ, and the systemic velocity V0. We assume circular orbits (eccentricity, e = 0), which is a

safe assumption for tidally circularized, short-period orbits (P < 12 days; Raghavan et al.,

2010), where we are most sensitive. A small number of the binaries found in this study may have longer periods and could have non-zero eccentricities, but this does not aﬀect our results. Plate-shift parameters were also adopted in Mb wherever P > 1.

Uninformative priors were used in the MCMC. The model index λ, has a ﬂat prior from

0 to 1, where λ < 0.5 denotes Ms and λ ≥ 0.5 denotes Mb. The semi-amplitude prior

−1 is log-uniform from 3 km s , comparable to the measurement uncertainties where Ms

−1 and Mb become degenerate, to 250 km s , greater than the largest RV diﬀerences in the

sample. The prior on the period is uniform in the range 4 ≤ log P (s) ≤ 7. The lower limit

log P (s) = 4.0 is equal to the orbital period at which stellar contact is certain for low-mass

companions. Above log P (s) = 7.0, RV amplitudes in binary systems are comparable to

the measurement uncertainties. Combined with the sparsity of the RV data, systems with

periods longer than log P (s) = 7.0 are outside our range of sensitivity. Priors are also

−1 uniform for the phase (0 ≤ φ ≤ 2π) and systemic velocity (−600 ≤ V0 (km s ) ≤ 600).

Markov chains were run independently on every star with an ensemble of 200 parallel chain

“walkers” for a total of 2.4 × 106 samples, then burned and thinned to 6 × 105 independent

samples of the posterior.

Evidence for detection of a companion star is reﬂected by the relative probabilities of λ.

We deﬁne the probability for the binary model, η as the fraction of samples in the marginal-

ized posterior having λ = Mb. We note that the value of η is dependent on the choice of

85 priors, and is sensitive to the treatment of the SDSS systematics. Moreover, a degeneracy

arises as the RV curve of a long-period, low-amplitude system becomes indistinguishable

from a single-star system. With this mind, we stress that values for η are not absolute

probabilities of a system having a companion, but reﬂect the ability of the data to rule out

models under the given prior. However, the [Fe/H]-groups can be compared, relatively, by considering the fraction of systems where η is large and Ms is strongly disfavored. The

results are shown in Figure 4.3

4.0 3.5 Metal-Poor Metal-Intermediate 3.0 Metal-Rich 2.5 2.0 1.5 log(N) 1.0 0.5 0.0

0.0 0.2 0.4 0.6 0.8 1.0 η

Figure 4.3: Distribution of η, the fraction of posterior samples using the binary model, for stars.

Various checks were implemented to ensure the robustness of our MCMC method. The

range of the limits on the priors were increased to search for signiﬁcant posterior probability

density at, for example, larger periods. With an increased range in semi-amplitudes and

periods, no signiﬁcant increase in probability was seen at values excluded in the current

prior limits, conﬁrming our sensitivity to short-period systems with log P < 12 (d). We

implemented a second method for comparing the single-star and binary-star models. A

posterior Bayes factor (POBF) was calculated on independent MCMC runs, using each of

the two models. Using the ratio of the POBF values as a metric for identifying likely

binary companions, we saw good agreement with stars predicted to have companions via

86 our hierarchical method. Additionally, we performed visual inspection of more than 500

stars with the highest values of η, as well as stars with the lowest values of η. As expected,

those stars with high values of η showed signiﬁcant RV variations expected from a binary

companion, and stars with the lowest values of η had little-to-no RV variations. For those stars with little RV variations, the distributions in the marginalized posteriors of the Mb parameters were ﬂat across the entire ranges speciﬁed by the priors.

We also investigated the e/i parameter proposed by Geller et al. (2008) as a metric for identifying the stars with large RV variations. We ﬁnd that the e/i parameter singles out many of the same stars as our more sophisticated MCMC-based inference. Our method not only takes into account deviations in RV from the mean, but also how well the data ﬁt the expected periodicity of a binary system.

Analysis of the posterior, and visual inspections of the binary model ﬁts, show that

681 stars with η > 0.65 are true spectroscopic binaries, though given the sparsity of the RV curve sampling, there are sometimes large uncertainties in the fitted values for specific model parameters. Another natural break point is η > 0.95; these are 209 stars for which the deter- mination and analysis of accurate individual model parameters should be possible (and will be characterized in future work). An intermediate cut at η > 0.80 is a compromise between these limits, yielding a larger sample of stars (406) with modest model constraints. The values of the binary fractions that we derive below are insensitive, within the uncertainties, to the exact choice of cut in η. This implies that the RV variations for our binary detections are sufficiently above the measurement uncertainties, and that the binary fractions reported are not biased due to differences in SNR or absorption features.

Figure 4.4 shows the log P posteriors for each [Fe/H]-group, marginalized over all binary systems (η > 0.80). The posterior distributions of log P for many of these stars are complex:

87 many are multimodal, aﬀected by aliasing or other issues related to the sparse, biased time sampling. One such eﬀect is the increase in probability at log P = 4. Here the metal- rich and -intermediate groups contain more stars than the metal-poor group with ∆t '

104 s. Systems with periods as short as this are extremely rare (Drake et al., 2014), and our increased probability in this area may be due to overﬁtting. Additionally, the gap at

∆t = 104.6 s = 12 hr (Figure 4.1) may aﬀect the estimate of a period. We defer a more sophisticated analysis to a future paper, but these eﬀects should not alter the ability to rule out a single-star model. For now, Figure 4.4 illustrates that we are mainly sensitive to periods in the range 4 < log P (s) < 6, or less than about 12 days. We emphasize that a more detailed analysis will be necessary to estimate the true underlying log P distribution in our sample.

Metal-Poor 0.20 Metal-Intermediate Metal-Rich 0.15

0.10

CombinedPDF 0.05

0.00 4.0 4.5 5.0 5.5 6.0 6.5 7.0 logP (s)

Figure 4.4: Averaged probability distributions of log P for all binary detections (η > 0.80). These do not reﬂect actual distributions of periods, and should only be used as a guide to probe the region of MCMC sensitivity. The shaded region indicates where Roche lobe overﬂow and contact becomes relevant. The dashed line marks the circularization limit at a period of 12 days.

88 4.5 Discussion

In Figure 4.5 we show fb, the measured lower bound for the fraction of stars with short-period companions (P <∼ 12 days) for each metallicity group, normalized to the metal-rich binary fraction. fb is a lower limit because of non-detections as a result of sparsely sampled RVs and high orbital inclinations, resulting in low amplitudes. We see agreement in fb measured for all three choices in η cutoff (0.65, 0.80, 0.95). With a cutoff of η = 0.80, values of fb for the metal-poor, -intermediate, and -rich groups respectively are: 2.5% ± 0.2%, 2.8% ± 0.2%, and 3.2% ± 0.3%. Since the observational biases that affect binary detection are mostly due to the sparsity of the RV coverage, which isn’t metallicity-dependent, we conclude that the

ﬁeld F-type MS stars in our metal-rich sample are, at a 2-sigma level, 30% more likely than those in our metal-poor sample to have close binary companions.

1.1 0.036

1.0 0.032

0.9 0.029 80) . rich b 0.8 0.026 0 / f η > ( b f b 0.7 0.023 f η > 0.65 0.6 0.019 η > 0.80 0.5 η > 0.95 0.016

−2.5 −2.0 −1.5 −1.0 −0.5 0.0 0.5 [Fe/H] Figure 4.5: Short-period binary fraction limits, relative to the metal-rich group. Binary companion detections are deﬁned by a cut in η, the fraction of posterior samples using the binary model. Group median values of [Fe/H] are used.

Our metal-rich and metal-poor samples mostly trace the MW disk and halo. Diﬀerences in

89 the fraction of short-period systems can stem from diﬀerences in the star-formation process,

dynamical interactions after star formation, or some combination of the two.

Three-dimensional hydrodynamic models from Machida et al. (2009) actually suggest a

higher frequency of binaries formed through cloud fragmentation for metal-poor clusters, due

to the decreased requirement of a cloud’s initial rotation energy to fragment. Moreover, their

models yield systems with shorter initial separations at lower metallicities. The increased

fb observed for metal-rich stars in this work can more likely be explained by dynamical

processes than by formation processes.

The observed diﬀerences in fb could be explained if the clusters that yielded halo ﬁeld

stars had larger stellar densities and/or gas densities than those of the disk. Korntreﬀ et al.

(2012) explore the eﬀects of gas-induced orbital decay on period distributions in clusters.

They note that an increased density of gas in a newly formed cluster will lead to a larger num-

ber of short-period system mergers shortly after formation. Parker et al. (2009) describe how

clusters with higher stellar densities destroy wide binaries through dynamical interactions.

An increase in the destruction of high-mass, wide-binary systems leads to the ejection of

former F-star secondaries into the ﬁeld. These orphaned, single-star systems would increase

the total number of F-star systems in the halo ﬁeld, eﬀectively decreasing the short-period

binary fraction measured. Observational evidence of these denser cluster environments is

needed to support these arguments for a lower fb in the halo.

Additionally, some close binaries may also transfer mass and convert themselves into blue

stragglers (Lu et al., 2010). Evidence for an abundance of blue stragglers in the halo has been

seen (Yanny et al., 2000), and may contribute to the lower fb observed in the metal-poor

group. Also, Duchˆene& Kraus (2013) show a decrease in fb with age for Solar-type stars, although this result is based on visual binaries with wider periods, and is poorly constrained

90 due to limited sample sizes.

We note that the recent results of Gao et al. (2014) and Yuan et al. (2015), using data from SDSS, show a larger binary fraction for metal-poor than metal-rich FGK stars in the

ﬁeld. In addition to probing longer periods, the former work does not make use of sub- exposure information (using only two RV epochs per star) and relies on the correctness of model values for the period distribution, mass ratio distribution, and initial mass function.

The latter work, which uses photometric color deviations to infer companions, shows a modest metallicity dependence on total binary fraction. Since their method is not sensitive to period, the binary fractions they report are strongly dominated by more common, wider- period systems near the peak of a log-normal period distribution (log P (s) = 10 for nearby,

Solar-like stars; Raghavan et al. 2010). It is clear that conclusions about binary fraction depend on a number of factors, especially the range of periods to which the search is sensitive and assumptions made about the overall period distribution.

Our MCMC analysis yields posterior probabilities in parameter space, allowing for a more detailed study of binary properties (e.g., period and separation distributions), which will be presented in future work. The techniques in this work have direct applications for current and future multiplexed spectroscopic surveys.

We thank Ewan Cameron, Dan Maoz, Jeﬀrey Newman, Chad Schafer, and the referee for useful discussions. T.H. and T.C.B. acknowledge partial support from grants PHY 08-

22648; Physics Frontier Center/JINA, and PHY 14-30152; Physics Frontier Center/JINA

Center for the Evolution of the Elements (JINA-CEE), awarded by the US National Science

Foundation. Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Oﬃce of Science.

91 Chapter 5

Binary Fractions and Separation

Distributions

5.1 Introduction

As mentioned in Section 1.4.3, the number of systems that will experience a CE phase depends on the details of the separation distribution and the fraction of stars that have short-period companions. Understanding binary fractions and separation distributions will impact the predicted merger rates and supernova rates (Badenes & Maoz, 2012). These properties also shed light onto the details of the star formation process and binary formation process, and speak to the dynamical history of stellar associations and clusters.

In Hettinger et al. (2015), we investigated the short-period binary fraction of F-type dwarf stars in the Milky Way halo and disk. In this chapter, we extend the work to include an investigation of the separation distribution. An MCMC sampler has been adopted in conjunction with population-wide Monte Carlo simulations, in order to constrain the separation distributions that are consistent with the data. We will look at the methodology in Section

5.2, and conclude with a discussion of the results and limitations in Section 5.3.

92 5.2 MCMC and Population-Wide Monte Carlo

In our approach to investigate the separation distribution of binaries in the F-dwarf sample,

we combined a MCMC sampler with population-wide Monte Carlo simulations. We assume

that the binary separations, in our range of sensitivity (0.01 AU < a < 0.16 AU), are distributed like a power law with an index parameter α,

f(a) ∝ aα . (5.1)

A power law distribution should be decent approximation to the often-used log-normal distribution, for the narrow range of separations to which we are sensitive.

In our MCMC, we have two free parameters describing the population of stars, the separation distribution index α, and the fraction of systems that have close-separation binary companions fb. We include fb, as there is a degeneracy in α and fb, aﬀecting the distribution of RV variations among the stars in the population. Figures 5.1 and 5.2 illustrate how changes in either α or fb aﬀect the distribution of e/i values for a population of stars.

The distribution of mass ratios in a population also has signiﬁcant implications on the star formation and binary formation processes, and the expected rates for various post-CE systems. Unfortunately, changes in mass ratios do not have a strong impact on the observed

RV variations (see Figure 5.3). The statistical approach for estimating multiplicity properties from sparsely sampled RVs is limited in this regard. Determining mass ratio distributions will likely be limited, for some time, to targeted campaigns consisting of detailed spectroscopic and eclipsing observations.

Since the parameters in our MCMC sampler deﬁne a population of stars, the entire population must be simulated at every step in the chain and compared with the data. This

93 Figure 5.1: Distribution of e/i values for a simulated population based on observations and uncertainties in the metal-rich population of Hettinger et al. (2015). Models illustrate variations in the e/i distribution from changes in separation distribution power law index α, while keeping ﬁxed the short-period binary fraction fb = 0.03 and the mass ratio distribution power law index β = 0.0.

Figure 5.2: Distribution of e/i values as in Figure 5.1, with a varying fb, and ﬁxed α = −1.0 and β = 0.0.

94 Figure 5.3: Distribution of e/i values as in Figure 5.1, with a varying β, and ﬁxed α = −1.0 and fb = 0.03. required us to adopt a single, easily calculated metric, to be used for each star in the population. The e/i metric (Section 2.5) provides a measurement of the variability of a star’s RV, and will change in magnitude depending on the system’s binary separation, orbital orientation, and exposure coverage.

To simulate an e/i distribution for the population, we run a population-wide Monte

Carlo simulation at each step in the MCMC chain. This is accomplished by simulating RV measurements for every stellar system in the population 60 times, calculating e/i for each system, and normalizing the e/i distribution so it can be compared with the data.

For individual stellar simulations, a binary companion is either absent or present, based on the value of the population-wide fb parameter at that step in the MCMC chain. Next a separation is chosen randomly from a power law distribution set by the α parameter at that step in the MCMC chain, with the separation conﬁned within the limits of our sensitivity (0.01 − 0.16 AU). Primary masses are drawn from an IMF (Kroupa, 2001), and secondary masses are drawn from a ﬂat distribution from 10% to 100% of the primary

95 mass. RV measurements are simulated using the same observing times that the star was

originally observed at. Finally, measurement uncertainties are folded in, using the empirically

estimated RV uncertainties previously derived for that particular star (Section 2.4).

To improve the statistics of the model at high e/i values, the population is simulated 60

times at every step in the MCMC chain. The result in a relatively smooth model, representing

a typical e/i distribution for the population of stars, given a speciﬁc pair of values for α and

fb. The likelihood function in the MCMC compares the histogram of e/i values for the population, calculated from the data, with the histogram of e/i values from the model using

Poisson probabilities at each bin. Thus, at each step in the MCMC, one possible population of stars is simulated robustly to produce a e/i curve, and that curve is compared to the histogram of true e/i values previously calculated from the dataset. This comparison allows the MCMC to accept and reject future proposals for α and fb.

The MCMC sampler was run on all three metallicity groups deﬁned in Hettinger et al.

(2015). In Figures 5.4, 5.5, and 5.6, we show the e/i distributions of the metal-poor, - intermediate, and -rich groups. Overplotted in each ﬁgure is a random selection of models, constructed from α and fb parameters drawn from the posterior. It should be noted that the

MCMC likelihood function only compared bin heights for bins with e/i > 3.0. The reasoning is that our empirically determined estimates for RV uncertainties had limited accuracy. This results in some deviations in the calculated e/i values. Since the majority of stars have low e/i values, small deviations in the assigned uncertainties lead to large deviations in the bin heights. Thus the likelihood function would be dominated by diﬀerences in the low-e/i values, and would have reduced sensitivity to the high-e/i binary systems. The posterior distributions for α and fb are depicted in Figures 5.7, 5.8, and 5.9. The results from the

MCMC run are discussed in the following section.

96 Figure 5.4: e/i distribution for the metal-poor group. Model e/i distributions are shown using values of α and fb randomly sampled from the MCMC posterior. The dashed line represents the cutoﬀ, below which bin heights were not used in the likelihood function.

Figure 5.5: Same distribution as in Figure 5.4, for the metal-intermediate group.

97 Figure 5.6: Same distribution as in Figure 5.4, for the metal-rich group.

Figure 5.7: Posterior distribution for the MCMC run of the metal-poor group, with parameters for the short-period binary fraction fb, and separation distribution α.

98 Figure 5.8: Posterior distribution for the MCMC run of the metal-intermediate group, with parameters for the short-period binary fraction fb, and separation distribution α.

Figure 5.9: Posterior distribution for the MCMC run of the metal-rich group, with parameters for the short-period binary fraction fb, and separation distribution α.

99 5.3 Discussion

Unfortunately, there is little additional information to be drawn from the posterior of these

MCMC runs. The short-separation binary fractions allowed by the data (2% – 8%) are

consistent with the values obtained in Hettinger et al. (2015) through the hierarchical MCMC

method.

These population-wide MCMC runs favored larger values of α, up to the prior limit imposed at α = 6. Previous MCMC runs with a much larger range in allowed α values

held that large values (α > 20) achieve the same posterior density as modest values (α ∼ 6;

Figure 5.10). This is likely due to the maximum separation (a = 0.16 AU) that was imposed

Figure 5.10: Posterior distribution for the MCMC run of the metal-rich group, with an extended prior limit in α.

from our deﬁnition of fb. For any fb, arbitrarily large values of α simply concentrate binaries

at a ' 0.16 AU. At larger α, all binary fractions within a reasonable range produce a similar e/i distribution.

As expected, at lower values of α, we do see a correlation with fb, where a smaller value

100 of fb is required for separation distributions favoring shorter periods. With higher quality data, this region of parameter space would be more tightly constrained.

Across the [Fe/H] groups, the values of α and fb are broadly consistent with one another.

An exception is, for the metal-intermediate group, there is increased density in the posterior at lower values of fb. This is likely due to the lack of objects with 8 < e/i < 12 in this

[Fe/H] group. Also, the MCMC seemed to ignore the two highest e/i objects.

The detection eﬃciency and period sensitivity of our method are determined by the survey parameters. The low spectral resolution of the SEGUE data requires stringent quality cuts to ensure that sub-pixel RV shifts can be detected at each epoch. The total number of stars in Hettinger et al. (2015) which survived quality cuts amounts to only ∼ 15, 000 among the

∼ 250, 000 SEGUE targets classiﬁed as F-type. Typical RV errors in SEGUE spectra are

∼ 4 km s−1, limiting the longest periods and separations that can probed.

The APOGEE survey has a higher resolution (R ∼ 20, 000) and smaller RV uncertainties

(∼ 0.5 km s−1). Applying our techniques to this dataset will result in a much higher detec-

tion eﬃciency, and will require less stringent quality cuts. Preliminary work with APOGEE

data yields a promising distribution of maximum RV variations, with thousands of likely

binaries with log P (d) as high as 3 (Figure 5.11).

We would like to see statistical time-resolved spectroscopy techniques continue to be em-

ployed in the future. Work on the APOGEE set of stars has begun already, and an extension

to other spectral classes within SEGUE can be accomplished with little modiﬁcation to the

original pipeline. Further down the road, we expect similar techniques described in this dis-

sertation will be applied to other current and upcoming multi-ﬁber spectroscopic missions

such as LAMOST, DESI, and PFS, which are expected to collect millions of stellar spectra

in the next few years.

101 Figure 5.11: Preliminary distribution of maximum RV variation for DR12 APOGEE targets.

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