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Theses and Dissertations--Physics and Astronomy Physics and Astronomy

2014

A New Set of Spectroscopic Metallicity Calibrations for RR Lyrae Variable Stars

Eckhart Spalding University of Kentucky, [email protected]

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Recommended Citation Spalding, Eckhart, "A New Set of Spectroscopic Metallicity Calibrations for RR Lyrae Variable Stars" (2014). Theses and Dissertations--Physics and Astronomy. 22. https://uknowledge.uky.edu/physastron_etds/22

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Eckhart Spalding, Student

Dr. Ron Wilhelm, Major Professor

Dr. Tim Gorringe, Director of Graduate Studies A NEW SET OF SPECTROSCOPIC METALLICITY CALIBRATIONS FOR RR LYRAE VARIABLE STARS

THESIS

A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in the College of Arts and Sciences at the University of Kentucky

By Eckhart Spalding Lexington, Kentucky

Director: Dr. Ron Wilhelm, Professor of Physics Lexington, Kentucky 2014

c Copyright￿ Eckhart Spalding 2014 ABSTRACT OF THESIS

A NEW SET OF SPECTROSCOPIC METALLICITY CALIBRATIONS FOR RR LYRAE VARIABLE STARS

RR Lyrae stars are old, iron-poor, Helium-burning variable stars. RR Lyraes are extremely useful for tracing phase-space structures and metallicities within the galaxy because they are easy to identify, have consistent luminosities, and are found in large numbers in the galactic disk, bulge, and halo. Here we present a new set of spectroscopic metallicity calibrations that use the equivalent widths of the Ca II K, Hγ,andHδ lines to calculate metallicity values. Applied to spectroscopic survey data, these calibrations will help shed light on the evolution of the Milky Way and other galaxies.

KEYWORDS: RR Lyraes, metallicity calibration, spectroscopy, galactic halo, Sloan Digital Sky Survey

Author’s signature: Eckhart Spalding

Date: June 21, 2014 A NEW SET OF SPECTROSCOPIC METALLICITY CALIBRATIONS FOR RR LYRAE VARIABLE STARS

By Eckhart Spalding

Director of Thesis: Ron Wilhelm

Director of Graduate Studies: Dr. Tim Gorringe

Date: June 21, 2014 To the villagers of Olkiramatian, who are wealthy in beautiful night skies, in which the developed world is increasingly impoverished ACKNOWLEDGMENTS

I owe a great deal of thanks to a number of key individuals for making this work possible. Those directly associated with the University of Kentucky include Robert Milton Huffaker (UK, ’57), who generously funded much of the travel costs through Huffaker Travel Scholarships; Dr. Keith MacAdam, who provided the MacAdam Student Observatory to the UK Physics and Astronomy Department; Tim Knauer and Kyle McCarthy, who provided assistance at the MSO; and department Chair Dr. Sumit Das, who released funds to enable me to present results at the 223rd AAS meeting. I am also grateful to Dr. John Kielkopf, for taking some of the photometry at the University of Louisville’s Moore Observatory (and for prodding his colleague Dr. Rhodes Hart to nab a star from Australia, making this project an international col- laboration); David Doss, the eminence grise of McDonald Observatory’s Mt. Locke, for his technical help (and his concern the morning after I hit my head on the spectro- graph); and, of course, the cherubic Dr. Nathan De Lee and my advisor, the pensively barbate Dr. Ron Wilhelm, for all their variegated forms of guidance. A special thanks goes to the viola virtuoso Dr. John Kuehne, the colorful and wild-haired resident elf of the 82-inch dome at McDonald Observatory, for his tireless technical assistance; his eclectic conversations; and for performing all the live music that went into this research. This work made use of the Expeditions and Discoveries: Sponsored Exploration and Scientific Discovery in the Modern Age collection of the Harvard University Library Open Collections Program, and the GCVS database of the Sternberg As- tronomical Institute of the Lomonosov Moscow State University. Here are further

iii official acknowledgements:

We acknowledge with thanks the variable star observations from the AAVSO International Database contributed by observers worldwide and used in this research.

This research has made use of the SIMBAD database, operated at CDS, Stras- bourg, France.

Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Coun- cil for England. The SDSS Web Site is http://www.sdss.org/.

The SDSS is managed by the Astrophysical Research Consortium for the Par- ticipating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel Uni- versity, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max- Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pitts- burgh, University of Portsmouth, Princeton University, the United States Naval Ob- servatory, and the University of Washington.

iv TABLE OF CONTENTS

Acknowledgments...... iii

TableofContents...... v

List of Tables ...... vii

ListofFigures ...... viii

Chapter1 :Introduction...... 1

Chapter2 : ABriefHistoryofRRLyraeResearch...... 2

Chapter 3 : The Pulsation Mechanism ...... 8

Chapter 4 : The Light Curve Shape ...... 11

Chapter5 : TheImportanceofMetallicityTracing ...... 14

Chapter6 : ABriefHistoryoftheCalibration ...... 17

Chapter 7 : The Saha and Boltzmann Equations and Line Formation . . . . 22

Chapter8 :DataAquisition ...... 25 8.1 Photometry ...... 25 8.2 Period Data ...... 27 8.3 Spectroscopy ...... 28

Chapter 9 : Data Reduction ...... 31 9.1 Photometry ...... 31 9.2 Metallicity Values ...... 31 9.3 Spectroscopy ...... 32 9.4 FunctionFitting...... 34

Chapter 10 : Results ...... 36

Chapter 11 : Conclusions ...... 42

Appendices ...... 43 Appendix A: Boyden Station During Revolution in Peru ...... 44 Appendix B: Heuristic Species Population Estimates ...... 46 Appendix C: Light Curves and Polynomial Overlays ...... 48 Appendix D: Author Metallicity Comparisons ...... 58 Appendix E: Stripe 82 [Fe/H] Error Calculation ...... 62

v Appendix F: H-K Plots Showing Phase ...... 63 Appendix G: Python Code for Synthetic Data Generation ...... 73 Appendix H: C++ Code for Function Fitting ...... 75

Bibliography ...... 79

Vita...... 89

vi LIST OF TABLES

7.1 Estimates of Species Proportions in an RR Lyrae Photosphere ...... 23

8.1 Epochs-of-Maximum ...... 27 8.2 Star Periods ...... 29 8.3 Metallicities from the Literature ...... 30

10.1 Calibration Coefficients and Standard Deviations ...... 36

vii LIST OF FIGURES

2.1 Mt.Harvard,nearChosica,Peru ...... 6 2.2 BoydenStation,outsidethetownofArequipa ...... 6 2.3 One of Shapley’s depictions of the distances to globular clusters . . . . . 7 2.4 Illustrative simulations of Type I Cepheid and RR Lyrae densities in the galaxy ...... 7

3.1 AnHRdiagramshowingthelocationsofvariablestars ...... 10

4.1 AnRRablightcurvetemplate...... 12 4.2 AnRRabempiricallightcurve ...... 12 4.3 AnRRclightcurvetemplate ...... 13 4.4 AnRRcempiricallightcurve ...... 13

6.1 VisualizationoftheLaydencalibraton ...... 20 6.2 Detail of a comparison of the Layden and SSPP metallicity values . . . . 20 6.3 EarlydivergenceoftheLaydencalibration ...... 21 6.4 AbsenceofanytrendinSSPPvalues ...... 21

7.1 Line strength changes as a function of temperature ...... 24

9.1 Trends of the Hβ and Hγ EWs with the Hδ EW ...... 35

10.1 SpectralchangesofXAriasafunctionofphase ...... 37 10.2 Metallicity values returned by an application of the ABC calibration on SDSSStripe82stars ...... 37 10.3 Distribution of metallicity errors (1) ...... 38 10.4 Distribution of metallicity errors (2) ...... 38 10.5 TheABCcalibration,withouterrorbars ...... 39 10.6 TheABCcalibration,witherrorbars ...... 40 10.7 TheseparateABandCcalibrations ...... 41

viii Chapter 1 : Introduction

This thesis is an effort to produce an improved set of spectroscopic metallicity cali- brations for RR Lyrae variable stars. The calibrations are three: one for RRab stars, one for RRcs, and one that makes no distinction between the two subtypes. These calibrations use the equivalent widths (EWs) of the Ca II K line and acombinationoftheHγ and Hδ lines to generate a value of [Fe/H] scaled to the values used in a previous publication (Layden, 1994). In turn, these values can be systematically rescaled as any researcher may see fit in order to provide the most accurate metallicities. The general procedure was, first, to gather photometry to enable determination of the science stars’ epochs-of-maximum. These were used to assign phase values to the stars’ spectra, which we took at McDonald Observatory in Texas. Then, after excluding spectra taken during the time of rising light (when NLTE effects are significant), a bilinear equation was fit to the stars’ metallicity values and the EWs of the Ca II K and Balmer lines. (The bilinear fit is appropriate in the temperature range of these stars, where the Ca II K and Balmer lines change linearly.) The resulting bilinear fits constitute the metallicity calibrations, which can be applied to single-epoch RR Lyrae spectra. If these calibrations are applied to the torrent of single-epoch spectra being produced by large-scale surveys such as the Sloan Digital Sky Survey, this will help provide better constraints on variations of evolutionary models of the Milky Way, and of cosmological ΛCDM models on sub- galactic scales.

c Copyright￿ Eckhart Spalding, 2014.

1 Chapter 2 : A Brief History of RR Lyrae Research

Ever since our prehistoric ancestors have had a sufficient degree of sentience, they have been aware of the variable nature of the night sky. In addition to the continual rolling of the star fields overhead, pre-modern humans were familiar with the sight of meteors, comets, and roving planets. Once in a great while, humans were witness to a cataclysmic variable—a one- off variation in the brightness of a star. The Chinese kept records of cataclysmic variables as early as 532 BCE (Xu, 2000, p. 129) and they, along with the Japanese, Koreans, Arabs, Europeans, and arguably pre-Columbian Americans, left records of true supernovae events (e.g., Stephenson and Green (2002, pp. 117-128, 162-167); Brandt et al. (1975)).1 Regular variables, however, appear to have left no unambiguous trace in the historical record. Their variations are just too small to notice without a telescope, except for the eclipsing binary star β Persei (Algol), which ranges between m =2.1 to 3.4 during a period of less than three days. The earliest recorded observation of a regular variable star may have been conducted by the German pastor David Fabricius, who observed the variable o Ceti, a star with a period of 11 months, in 1595 to 1596 (Ostlie and Carroll, 1996, p. 541). This star continued to be methodically observed in the 17th century, and pulsators with periods on the order of a week were observed in the 18th century. Jacobus Kapteyn’s 1890 discovery of the field variable star U Leporis may have been the first time a variable star was discovered which was later categorized as an RR Lyrae star (Smith, 1995, p. 2). By the end of the 19th century, astronomy had undergone enough development to require a greater level of standardization. E.C. Pickering, the director of Harvard College Observatory, set out to formalize photometric measurements by adopting a brightness scale and pegging the magnitude 2.1 to the ‘North Star,’ α Ursae Minoris (Jones and Boyd, 1971, p. 187).2 Pickering oversaw the measurement of the apparent magnitudes of 4,000 stars by comparing images of meridian-crossing stars with α UMi, and published∼ the results in 1884 (Pickering et al., 1884; Jones and Boyd, 1971, pp. 188-189). Soon, Pickering’s ambition redoubled to include precision photometry and all- sky surveys.3 He looked south, to the poorly-documented skies of the southern hemi- sphere, and sent Solon Bailey to South America in 1889 to scout out possible sites for 1Contemporaneous accounts of the seven or eight historical supernovae include descriptions of the supernova of 1006 CE in the southern constellation Lupus, which released 1050 ergs only 2kpc away. (1050 ergs is on the order of the total amount of the energy expended by the Sun in the∼ past 5 billion years.) This explosion, likely a Type Ia, is thought to have registered a peak magnitude of around m = 7.5, and remained in the sky for weeks (Winkler and Laird, 1976; Stephenson et al., 1977; Winkler− and Gupta, 2003). 2Hertzsprung later discovered that the star does exhibit some variability (Jones and Boyd, 1971, p. 189). 3Some of the following text in this section was submitted to the Griffith Observer magazine as part of the article “In Pursuit of RR Lyraes.”

2 an observatory. Bailey spent a year and a half on a Peruvian mountaintop christened Mt. Harvard, near the town of Chosica, living in a makeshift observatory cobbled together from wood, canvas, and cardboard (Fig. 2.1). Bailey shook scorpions from his clothes, and transported equipment on the backs of mules. He made photographic plates with an 8-inch telescope, and haphazardly posted them in crates back to Cam- bridge. In 1890, Bailey relocated to a more permanent site at an elevation of 8,000 feet at the foot of a dozing volcano near the town of Arequipa (Fig. 2.2).4 Pickering arrived at the new Boyden Station with a 13-inch telescope, and soon the facility saw the addition of a 24-inch, as well (Bailey, 1904, seq. 6 and 8). Bailey and others lived in relative comfort at Boyden Station compared to Mt. Harvard, though it was not without its challenges, including political upheaval.5 Starting in 1893, Bailey made concerted studies of globular clusters (e.g., Bai- ley (1902)). They were so crammed with stars that Bailey recommended using a microscope or scrutinizing blow-ups to make them out. By comparing plates, Bai- ley soon noticed three variable stars in the cluster 47 Tucanae. Back in Cambridge, Pickering and one of his female “computers,” Williamina Fleming, found two more in plates of ω Centauri (Jones and Boyd, 1971, pp. 321, 354-355). Then Fleming spotted another three in 47 Tucanae (Bailey, 1902). Detections were made by taking positive and negative plates on different nights, and covering one with the other to see if a star’s image underneath produced an edge around the one on top (Belkora, 2002, p. 255). Bailey and his assistants settled into a routine of marking variable star candidates, and letting Pickering and his computers do the checking (Jones and Boyd, 1971, pp. 320). Bailey’s interest in variables grew, and the initial trickle of variables became a torrent. Five hundred were found in plates taken at Boyden Station between 1895 and 1897 (Bailey, 1902), which effectively doubled the number of known variable stars in the Milky Way. In the meantime, Henrietta Leavitt—also a computer—used the Arequipa plates to find a plethora of variable stars in the more distant Magellanic Clouds. Some of them were bright enough and had plates taken at the right time to pin down the time periods of their light cycles (Leavitt, 1908). Leavitt proceeded to determine a relationship between her stars’ periods and their apparent magnitudes (Leavitt and Pickering, 1912), which, once ‘zeroed,’ would allow for distance measurements within the galaxy.6 (The implied assumption was that the variable stars in the Magellanic Clouds were the same distance away.) In the 1910s, with Bailey’s encouragement, a crime-reporter-turned-astronomer named Harlow Shapley did follow-up observations of known cluster variable stars from 4In the process of descending equipment from Mt. Harvard, four mules tumbled down the incline but by “some kind fate which seems to watch over observatories & mules they escaped without broken bones” (Bailey, 1890, seq. 10). 5See Appendix A: Boyden Station During Revolution in Peru. 6Hertzsprung made a valiant first attempt to zero this calibration by using the radial and proper motions of stars, and found that the Small Magellanic Cloud was 30 kly away (Bartusiak, 2009, pp. 121-122). The true value is around 200 kly.

3 Mt. Wilson near Los Angeles, and discovered some more of his own. He produced calculations suggesting that Cepheid variables would be unstable if they were eclipsing binaries, and surmised that they must be stars that are physically pulsating, an idea proposed as early as 1873 by the physicist Arthur Ritter (Leverington, 1995, p. 155) and later supported by the calculations of Arthur Eddington (Gingerich, 1973). Feeling cocky, Shapley decided to use Leavitt’s idea to trace out the extent of the Milky Way. Lumping together Bailey and Leavitt’s stars, Shapley made his own distance calibration based on stars’ proper motions (Ostlie and Carroll, 1996, p. 544). Shapley also made other assumptions concerning the luminosity of the brightest stars in the clusters, and the size of globular clusters as a whole (see Dewhirst and Hoskin (1997)). Finally he came to a stupefying conclusion: the distance from the Sun to the center of the galaxy was about 20 kpc, and the width of the galactic disk was around 100 kpc (Fig. 2.3) (Shapley, 1918b). Shapley wrote that these claims were “sketchy and arrogant, I know; and I haven’t much excuse for it” (quoted in Gingerich (1973)). Bailey and Leavitt, it eventually turned out, had been looking at two distinct types of variable stars. Bailey’s variables were more numerous but more faint (now known as RR Lyraes), so Bailey tended to detect them in the relatively nearby globu- lar clusters. Leavitt, on the other hand, tended to find brighter variables (now known as Cepheids) in the more distant Magellanic Clouds. Even though Shapley noticed some differences, he shunted them aside in making his calibration (Shapley, 1918a). (He also noticed that variable stars in a single cluster—and therefore roughly at the same distance—had the same magnitude, a hint that globular cluster variables might have a narrow range of luminosities.) Though Shapley’s calculations overestimated the galaxy’s scale by a factor of about two, his work drove home the newfound reality that the solar system did not lie at the center of the Milky Way. This had been mentioned as a possibility by the astronomer Karl Bohlin in 1909, who, without the benefit of distance measurements, noticed that globular clusters were concentrated in one area of the sky (Bohlin, 1909).7 Shapley wrote that “the solar system is off center and consequently man is too, which is a rather nice idea because it means that man is not such a big chicken” (Shapley, 1969, pp. 59-60). Since then, the distance from the Earth to the center of the Milky Way has been whittled down to about 8.0 kpc, using a variety of methods, including the use of RR Lyrae stars. The distance to the galactic center is not just an academic exercise, but has implications for determining the luminosity of variable stars (the reverse problem, as it were), stellar distances inferred from radial velocities, and calculations of the galaxy’s mass and gravitational potential (Reid, 1993; Eisenhauer et al., 2003). Shapley studied globular clusters at a time when the distinction between glob- ular clusters and the smudgy spiral nebulae was not clear (Belkora, 2002, p. 259), and this led to disagreements among astronomers as to whether the Milky Way repre- sented the entirety of the visible universe, or whether the Galaxy was just one ‘island’ among many. As far as Shapley was concerned, his globular cluster studies exploded the ‘island universe’ view, because the galaxy just seemed too large. Previously a be- 7John Herschel also noticed this as early as the 1830s (Ashman and Zepf, 1998).

4 liever of island universes, Shapley now took on the zeal of the converted, propounding the view that the universe consisted of the Milky Way only, and that spiral nebulae were just parts of the galaxy. He famously had a public debate in Washington, D.C. in 1920 with Heber Curtis, who claimed that the nebulae were other galaxies.8 In the succeeding years, there was a growing awareness in the astronomical community of the considerable (albeit imperfect) uniformity among the cluster-type variables, as well as the differences between them and Cepheids (see Fig. 2.4).9 For example, it was noticed that both cluster-type variables in the galactic field and in globular clusters have nearly the same luminosity, and in contrast to Cepheids, the luminosity did not seem to be a function of the period (Payne-Gaposchkin, 1954, p. 19). Eventually, as more of the cluster variables were discovered in the field, they became referred to as RR Lyrae stars, after their brightest member, located in the constellation Lyrae. In 1948 the International Astronomical Union made the designa- tion formal (Smith, 1995, p. 4). Soon after, RR Lyraes indirectly provided distance measurements even beyond the galaxy: Walter Baade’s failure to find RR Lyrae stars in the Andromeda galaxy in the early 1950s indicated that the Cepheids that had been found there were members of the brighter Type I class, and that therefore An- dromeda was larger and farther away than previously thought (Dewhirst and Hoskin, 1997, p. 337). The big chicken, no longer in the center of the coop, was finding itself in an increasingly obscure corral.

8Famous in retrospect only, that is; the ‘Great Debate’ is a widely-disseminated anachronism. At the time, few people (if any) saw it has an historic event, and Shapley himself described it only as “a sort of symposium” (Shapley, 1969, p. 79). At any rate, the veracity of the island universe hypothesis remained very uncertain until Hubble’s 1925 announcement of the discovery of apparent Cepheid variables in M13. After this, resistance to the idea of external galaxies ebbed, and vanished completely within the next decade (Smith, 1982, pp. 118-119, 136). 9Consider the lingering uncertainty in a text from 1925 which remarks that “it is really a question whether the cluster variables should be considered a separate class” (Moulton, 1925, p. 522).

5 Figure 2.1: Mt. Harvard, near Chosica, Peru. From 1889 to 1890, Solon Bailey took survey plates at this ramshackle facility with an 8-inch telescope, and lived a life “so isolated that man and animals were on terms of intimacy and equality” (Bailey, 1922). (Image courtesy of Harvard College Observatory.)

Figure 2.2: Boyden Station, outside the town of Arequipa. Activated in 1891, this installation was outfitted with 8-, 13-, and 24-inch telescopes. Plates of globular clusters taken at Boyden Station were the first to contain significant numbers of variable stars later identified as RR Lyraes. Though more comfortable than the Mt. Harvard station, Bailey experienced considerable anxiety when the observatory was nearly swept up in the Peruvian revolution of 1894-5. (Image courtesy of Harvard College Observatory.)

6 Figure 2.3: One of Shapley’s depictions of the distances to globular clusters. Shapley’s observations of the distribution of globular clusters in the Milky Way suggested that the galactic center lay some 20 kpc away from Earth. (Fig. 2 in Shapley (1918c). c AAS. Reproduced with permission.) ￿

Figure 2.4: Illustrative simulations of Type I Cepheid and RR Lyrae densities in the galaxy (left and right halves, respectively). These were made by Christopher Klein, “based on measurements of densities in various galactic components and statistical 3D density distributions of those components.” The colors correspond to galactic com- ponents including the disk and thick disk (yellow and goldenrod), bulge (olive), halo (grey), spiral arms (purple lines), and bar (red line). (Plots courtesy of Christopher Klein.)

7 Chapter 3 : The Pulsation Mechanism

RR Lyraes are low-mass stars that have evolved off the main sequence, undergone the Helium flash, and are in their core Helium-burning stage. Ultimately they will lose their variability, expand into a double-shell burning giant, and die quietly as planetary nebulae. RR Lyraes reside on the Hertzsprung-Russell diagram where the Horizontal Branch (HB) intersects the stellar instability strip, which runs nearly vertically on the HR diagram (Fig. 3.1). The instability strip also encompasses Cepheid and δ Scuti variables, the more- and less-massive counterparts to RR Lyraes. The period for which a star smolders in the strip is fleeting, perhaps 100 million years, and it is during this time that the envelope phenomena take place to make a star pulsate. Arthur Eddington initially suspected that stellar pulsation was actually con- fined to the core, where compression was acting to increase the rate of energy output. This is the so-called ‘ε-mechanism’. Later, in 1926, he proposed that it might instead be due to changes in the rate with which the star bleeds away its energy. It was not until the 1950s that anyone seriously proposed that the layer of second He ionization (He II He III) might be behind this, and computer simulations since the 1960s have provided→ strong evidence for its culpability (King and Cox (1968); for a full review, see the monograph by Cox (1980)). Eddington’s calculations also led to a theoretical justification for the relation

Q P = (3.1) ρ /ρ ∗ ⊙ ￿ where P is the star’s period, ρ is the mean density, and Q is a constant that depends on the properties of the star. Theoretical and empirical values of Q are on the order of an hour, which means that P is on the order of the free-fall time of the Sun, or alternatively, the time it takes for a pressure front to pass through the star (King and Cox, 1968). It can therefore be expected that stars in the instability strip will have periods on the order of hours to days. The primary engine of the pulsation is the ‘κ-mechanism.’ The Rosseland mean opacity is the wavelength-averaged sum of opacities due to absorption or scattering, or

κ ρn κ = 0 (3.2) T s

where T is temperature, and κ0 is a constant that depends on density, temperature, and stellar composition. In areas where the envelope is not experiencing ionization, n 1ands 3.5, so the material becomes more transparent as the temperature rises. This≈ changes≈ in ionization regions, however, where compression does not necessarily lead to a rise in temperature, and “s can become small or even negative[,] with the

8 result that the gas can become most opaque near maximum compression” (King and Cox (1968); Smith (1995, p. 21)). Upon expansion, recombination takes place, which increases the thermal energy of the material. Thus the κ-mechanism is often thought of as a piston that produces an interval of compression, heating, and rise in ionization- zone opacity, followed by expansion, cooling, and a drop in opacity that allows energy to radiate. Of secondary importance is the ‘γ-mechanism,’ where partial ionization leads to decreases in adiabatic exponents that prevent temperature rises during the com- pression phase. Energy is instead diverted towards ionization, which has the effect of delaying the pressure maximum until after the star reaches its minimum volume (King and Cox, 1968). The first and second ionization zones both move up and down as the star goes through its heating cycle. (The ε-mechanism is also active, but is insufficient for powering instability-strip pulsations (Ostlie and Carroll, 1996, p. 553).) Due to the temperatures necessary for these ionizations, only the outer 30 to 50 percent of the stellar radius is strongly pulsating (King and Cox, 1968), though a star pulsating in the fundamental mode still experiences some pulsation along its entire radius (Ostlie and Carroll, 1996, p. 551). Given the density gradients in these stars, this means that at least (and usually much more) than 99% of the star’s mass lies below the ionization zones (Ostlie and Carroll, 1996, p. 555). At temperatures of 7500 K, the partial ionization layers are high up in the envelope, in regions too rarefied∼ to cause pulsation. In stars with temperatures below 6000 K, convection disrupts the piston mechanism, particularly at minimum radius (Ostlie∼ and Carroll, 1996, p. 555). These temperature limits lead to the blue and relatively hazy red ends of the instability strip, respectively. RRab stars are likely pulsating in the fundamental mode: at any given time, all material is moving outward or inward from the center. The radius of such a star is around a quarter of the pressure wavelength (Majewski, 2012). RRcs appear to be pulsating in the first overtone mode. This may be visualized as consisting of a stationary, spherical node in the star at r 0.6, interior to which envelope material moves in while exterior material moves out,≈ and vice versa (Percy (2007, p. 136); Feast (1996, pp. 83-84)). The radius of the star is then 3/4ths of a pressure wavelength (Majewski, 2012). There are also rarer RRd stars, which are apparently pulsating in the second overtone mode; and even candidate RRe stars, which might be pulsating in the second overtone (OGLE; Kov´acs, 1998). One star, AC Andromeda, has even been found to pulsate in the fundamental, first, and second overtones (Berdnikov, 1993; Feast, 1996, p. 85). (There is also evidence for nonradial pulsations in RR Lyrae stars (Moskalik, 2013).) For a time, researchers were bedeviled by the existence of a ‘phase lag’ between the times of minimum radius and maximum luminosity. Schwarzschild found in 1938 that the luminosity phase lag was due to the fact that the maximum compression in the outer layers occurs after the maximum compression in the center (Schwarzschild, 1938a,b; Leverington, 1995, p. 162). The correct phase lag was reliably reproduced in numerical calculations in the 1960s, and appears to be due to movement of the H ionization region through the envelope: at the star’s minimum radius, radiant energy

9 is greatest on the underside of the H ionization layer, and pushes the layer outwards until the radiation can escape most easily (Castor, 1968; King and Cox, 1968; Ostlie and Carroll, 1996, p. 556).

Figure 3.1: An HR diagram showing the locations of variable stars. Isochrones cor- respond to different stellar masses. Note the almost vertical nature of the instability strip. (Fig. 14.6 in OSTLIE, DALE A.; CARROLL, BRADLEY W., AN INTRO- DUCTION TO MODERN STELLAR ASTROPHYSICS, 1st Edition, c 1996, p. 547. Reprinted by permission of Pearson Education, Inc., Upper Saddle River,￿ NJ.)

10 Chapter 4 : The Light Curve Shape

RR Lyraes have distinctive light curves that typically exhibit variations of 0.3 to 2 mag. Through shorter-wavelength filters such as U, G, or V, the characteristic light curves produced by RRab stars (Figs. 4.1, 4.2) exhibit a steady descent in apparent magnitude that shallows until becoming roughly flat around φ 0.7. (We define φ =0tobetheluminositymaximum.)Theremaybea‘bump’,orsmallbutsudden∼ drop, followed by a steep rise at φ 0.9. This final rise may involve a ‘hump’ along the way, and there may even be a∼ small ‘lump’ shortly before the bump (Chadid and Preston, 2013). RRc light curves (Figs. 4.3, 4.4) are more sinusoidal, more symmetric, and have smaller amplitude than those of RRabs. The steep luminosity rise is around φ 0.8, and may include a visible hump. In a sizeable∼ fraction of RR Lyrae stars, there are periodic changes in the light curve amplitude and morphology which are still not fully understood (e.g., (Kov´acs, 2009; Chadid et al., 2011; Chadid and Preston, 2013). This is broadly referred to as the ‘Blazhko effect,’ which has periods on the order of tens or hundreds of days (Percy, 2007, p. 177). One must keep in mind that it would be premature to claim that a star does or does not exhibit a hump based on a single light curve, since ground-based observations of RR Lyraes are never continuous over several periods.1 Even in some non-Blazhko stars, there are small variations from cycle to cycle, which may be due to the changing intensity of the κ-andγ-mechanisms (Chadid and Preston, 2013). It has also long been recognized that simulations must incorporate shock waves to fully explain light curve shapes (e.g., Hill (1972); Simon and Aikawa (1986)).2 As listed by Chadid (2011), the hump has been observed to exhibit hydrogen Balmer emission (also seen during the bump), helium emission, calcium emission, metallic absorption line broadening, and emission and absorption line doubling. In one star, neutral metal line disappearance and ionized line doubling during the hump indicated that a shock wave was ionizing Fe (Chadid et al., 2008). Emission lines are likely caused by de-exciting atoms as they cool behind the shock wave (Chadid, 2011). Fokin (1992) calculated that the bump may be due to a downward-moving shockwave that heats the envelope material and allows the photons to shine through. Supporting earlier calculations by Hill (1972), Gillet and co-authors suggested that the shockwave associated with the bump is due to the collision between infalling material and layers below, and the one associated with the hump is produced by the κ-mechanism itself (Gillet and Crowe, 1988; Gillet et al., 1989).

1This is put in perspective by the data haul from the planet-hunting, space-based Kepler mission, which stared at a waffle-iron-shaped area around the constellations Cygnus and Lyra for four years. At least 41 RR Lyraes were within the Kepler field (Nemec et al., 2013), and their light curves were observed with unprecedented precision and continuity, revealing considerable Blazhko variation. (See, for example, Figs. 3, 4, and 5 in Kinemuchi (2011). See also Fig. 5 in Chadid et al. (2011), for CoRoT observations.) 2This effect is in addition to the strong velocity gradients betrayed by the differential slosh of the Balmer lines (Preston, 2011), which are also largely velocity-decoupled from Fe lines (Preston, 1964).

11 RRab Sesar Template

0

0.2

0.4

0.6

Normalized Relative Magnitude 0.8

1 0 0.2 0.4 0.6 0.8 1 Phase

Figure 4.1: An RRab light curve template. (See Sesar et al. (2010).) Individual RRab stars display small variations in bump and hump size, and Blazkho-effect amplitude changes. (Data courtesy of Nathan De Lee.)

Figure 4.2: An RRab empirical light curve. This is the result of ‘folding’ observations from many different periods onto one. Note the bump and hump. (Image courtesy of OGLE [I. Soszy´nski].)

12 RRc Sesar Template

0

0.2

0.4

0.6

Normalized Relative Magnitude 0.8

1 0 0.2 0.4 0.6 0.8 1 Phase

Figure 4.3: An RRc light curve template. (See Sesar et al. (2010).) RRc light curves have smaller amplitude changes than RRab’s, and are more sinusoidal. (Data courtesy of Nathan De Lee.)

Figure 4.4: An RRc empirical light curve. Note the hump. (Image courtesy of OGLE [I. Soszy´nski].)

13 Chapter 5 : The Importance of Metallicity Tracing

As the stellar neighborhood becomes enriched with metals from the sputterings of planetary nebulae and the blasts of supernova explosions, new stars forming in col- lapsing gas clouds are stamped with a characteristic metallicity. Even though stellar populations can become highly mixed (especially in the disk of the galaxy), indi- vidual stars retain their photospherically-measured iron metallicity throughout their lifetimes, indicating their age and common origin. Metallicity tracing is therefore a crucial marker for disentangling the history of star populations. A half century ago, Eggen et al. (1962) examined the kinematics of a small sam- ple of suspected halo stars zipping through the solar neighborhood. They concluded that the Milky Way had undergone a rapid collapse that left behind a relatively homogenous halo. This view was challenged by Searle and Zinn (1978), who found that the heterogeneous metallicities and horizontal branch morphologies of globular clusters indicated that the history of the halo involved the gravitational convergence of discrete star clusters. Considerable subsequent efforts have been invested in reconstructing the his- tory of the halo, because it is a fossilized region of the galaxy where the shallow gravitational potential and weak tidal gradients inhibit mixing and allow structures to persist for a few billions of years. At least 20-40% of halo stars are bound up in this substructure (Bell et al., 2008; Sesar et al., 2010). Unraveling the mystery of the halo is a key to understanding the evolution of the Milky Way as a whole, which remains poorly mapped beyond 30 kpc, where “the largest discrepancy between simulated halos with different formation∼ histories occurs” (Sesar (2011), referring to Fig. 6 in Cooper et al. (2011)). In turn, precise characterization of the Milky Way’s history will help provide constraints on variations of the ΛCDM model on galactic scales. Estimates based on empirical measurements have suggested that RR Lyrae number densities range from 1000 kpc-3 at 1 kpc from the galactic center to 0.01 kpc-3 a few tens of kpc from the center (Wetterer and McGraw, 1996). This is in contrast to Type I Cepheids (which have lifetimes that are too short for them to be found in the halo) or Type II Cepheids (which are extremely rare in the field of the halo (Wallerstein, 2002)) (see Fig. 2.4). They are also very luminous and easy to detect, so RR Lyraes are a tool of choice for investigating the halo.1 Should a hierarchical model of galaxy assemblage be accurate, we should be able to use RR Lyraes to detect ‘lumpiness’ in the outer halo (Bullock et al., 2001; Bullock and Johnston, 2005; Font et al., 2006) as a result of different accretion times, satellite disruption times, and initial satellite masses. On a larger distance scale, a slow collapse of the thick disk would manifest itself in halo-wide metallicity gradients. (If the thick and thin disks have different origins, they, too, will exhibit differences in metallicity.) Failure to detect a halo metallicity gradient would provide support to the Eggen model of a monolithic collapse, because the collapse would have happened 1Shapley himself suspected RR Lyraes’ importance for halo studies (Preston, 1964).

14 too quickly for gradients to emerge (Majewski, 1993; Carney, 1993). Prior to the 1990s, however, most studies of the halo had the weakness of being limited to sample sizes of tens or hundreds of stars—whether they be presumptive RR Lyraes or not. Some of the stars were plucked from the solar neighborhood, which could introduce color degeneracies (leading to star type ambiguity), or metallic or kinematic selection bias (as was suffered by Eggen et al. (1962)).2 Still, it was possible to use RR Lyraes to quantify parameters such as the distances of globular clusters, and by extension the absolute magnitudes of stars at the clusters’ main-sequence turnoff points. This led to a calculation of the minimum age of the galaxy (Lee, 1992; Sandage, 1989, p. 122). Since RR Lyraes are plentiful outside the disk of the galaxy, they were also useful for measurements of things like interstellar reddening and absorption at high galactic latitudes, where luminous, early-type stars are rarefied (Preston, 1964).3 But in the past two decades, the importance of RR Lyraes has undergone a considerable renewal with the advent of large-scale variability surveys, which have paved the way for statistically significant in situ observations. Earlier attempts to find metallicity gradients in the halo with RR Lyraes (Suntzeff et al., 1991) were rapidly superseded by at least eight other large-scale variability surveys (Sesar et al. (2007); Table 1 of Mateu et al. (2012)), and more RR Lyraes continue to be turned up in surveys all the time (e.g., Abbas et al. (2014); Zinn et al. (2014)). Current surveys, which have a variety of science goals (see Paczy´nski (2000)), include the Catalina Sky Survey (CSS), La Silla QUEST survey, Lincoln Near-Earth Asteroid Research (LINEAR), and also the SDSS. The SDSS was not primarily de- signed with variability in mind—its main goals were to survey large-scale galactic structure, quasars, and nonpulsating stars (Castander, 1998)—but has multi-epoch data that can be used to unmask variable stars (e.g., Wilhelm et al. (2013)). In fact, RR Lyraes are so numerous in SDSS that around 1 in 4 variable stars in SDSS are RR Lyraes (Sesar et al., 2007).4 The largest single repository of RR Lyraes appears to be the University of Warsaw Astronomical Observatory’s Optical Gravitational Lensing Experiment (OGLE), which looks for exoplanetary microlensing events. Its current data haul includes more than 44,000 RR Lyrae stars in the Magellanic Clouds and the galactic bulge (OGLE; Soszy´nski et al., 2009, 2010, 2011). In the SDSS, RR Lyraes are detectable out to distances approaching 100 kpc, 2Observations of stars much further away would have required booking lots of time on large telescopes, which, even if practical, would also have kept sample sizes small. 3RR Lyraes have also been spotted in other Local Group galaxies. In the 1980s, a few RR Lyrae candidates were found in the halo of the nearby galaxy M31 (Andromeda) (Pritchet and van den Bergh, 1987). Confirmation of the presence of RR Lyrae stars in both M31 and M33 (Triangulum) followed in the 2000s. As of 2011, known RR Lyrae stars in sizeable nearby galaxies numbered 29 RRabs and 25 RRcs in M31, and at least 64 RRabs in M33 (Sarajedini, 2011). In the future, RR Lyraes may even be detected in numbers large enough in galaxies at distances sufficient for measuring the Hubble Constant (Klein, 2014). Future low-resolution spectroscopic studies of extragalactic RR Lyraes would be in need of a metallicity calibration like ours. 4Starting perhaps shortly after 2020, the successor to the SDSS, the Large Synoptic Survey Telescope (LSST), will survey 20,000 square degrees, one thousand times, in one decade. (By comparison, the SDSS has surveyed 14,555 unique square degrees since 1998.)

15 very deep into the halo (Newberg, 2003; Z.ˇ Ivezi´cet al., 2005; Sesar, 2011), and have been used since the early days of the SDSS to trace out halo overdensities irreconcilable with Poisson noise (e.g., Z.ˇ Ivezi´cet al. (2000); Sesar et al. (2007); Bell et al. (2008); Sesar et al. (2010)). They have confirmed the existence of suspected substructures (e.g., Simion et al. (2014)), and have uncovered new ones, such as the Pisces overdensity (Watkins et al., 2009). Other structures whose metallicity or phase-space characteristics have been traced out are the Sagittarius dwarf galaxy and its tidal debris (Newberg, 2003; Keller et al., 2008; Sesar et al., 2010), the Virgo overdensity (Keller et al., 2009), and the Monoceros stream (Sesar et al., 2010). RR Lyraes have also contributed to measurements of larger-scale parameters. Watkins et al. (2009) found that RR Lyrae number densities in SDSS indicate that the stellar density in the galaxy generally falls off as a piecewise asymptotic function, 2.4 proportional to r− from approximately 5 to 23 kpc from the galactic center, and 4.5 r− above 23 kpc out to at least around 100 kpc. Sesar et al. (2010) also came to the conclusion, based on an analysis of SDSS RR Lyraes, that the power-law dropoff becomes more steep at around 30 kpc. Zinn et al. (2014) used RR Lyraes from the QUEST survey to find the kink to be at 25 kpc. One of the underlying motivations∼ in all this is to investigate the hierarchical nature of structure formation on a sub-galactic scale. There is very strong observa- tional and theoretical evidence that cold dark matter (non-relativistic, non-luminous, gravitationally-interacting material) leads to the growth of large-scale cosmic struc- ture through a ‘bottom-up’ accrual of globs with masses of 105 M . This is in ⊙ contrast to an alternative top-down process of disintegration∼ of globs of 1017 M ⊙ of relativistic ‘hot’ dark matter (Elmegreen, 1998, p. 284). Observational∼ support for the CDM scenario includes galaxy rotation curves, the sinewy structure of galaxy groups, the existence of galaxies at high redshift, gravitational lensing, and CMB parameters (for the latter, see Hinshaw et al. (2013)). However, while the ΛCDM paradigm5 has been very successful at the cosmic scale, it remains poorly constrained at the galactic scale and below (e.g., Springel et al. (2006)). Outstanding problems include reconciling central density profile cusps in dark matter halo simulations with observations that suggest that the central density is actually flat (the ‘cusp-core problem’), and the fact that simulations make galaxies spin too fast, given their luminosity (the ‘Tully-Fisher relation zero point problem’) (see F´eron (2008)). Other problems that have seen some (or possible) theoretical resolution but require more refinement is the ‘missing satellite problem,’ whereby the 20 detected dwarf galaxies are at odds with the hundreds predicted by simulations (Klypin∼ et al., 1999); the differences in metallicity between the dwarf galaxies and the halo remnants of previous mergers (Robertson et al., 2005; Font et al., 2006); differences between observations and simulations of the spatial distribution of dwarf galaxies (Helmi, 2008); and of course, determination of whether collapse or accretion was the dominant mechanism in the formation of the Milky Way. 5The ‘Λ’ signifies the inclusion of dark energy. The ΛCDM paradigm has six fundamental parameters to describe the universe, including the energy densities of baryonic matter, dark matter, and dark energy, which are currently constrained to proportionately be Ωb = .046, ΩCDM =0.235, and ΩΛ = .719 (Hinshaw et al., 2013).

16 Chapter 6 : A Brief History of the Calibration

As early as 1959, George Preston took spectra of 159 RR Lyrae stars at a variety of phases, and settled on a parameter that he proposed could be used as a metallicity indicator. The parameter, ∆S,isthedifference in ‘tenths’ of a spectral type visually determined from the hydrogen lines and Ca II K, taken at minimum light, and can be written as the pseudoequation ∆S =10[Sp(H) Sp(K)]. For example, a star of apparent type F6 from the hydrogen lines and A7 from− the Ca II K line would have ∆S =9(Preston,1959). Later, Butler (1975) examined 47 RR Lyraes in globular clusters, and 67 in the field. He made the calibration

[Fe/H]= 0.23(9) 0.16(2)∆S (6.1) − − and used it to determine the metallicities of a collection of globular clusters. Butler’s sample made use of both RRabs and RRcs, and his calibration does not distinguish between the two types. Still later, Clementini et al. (1991) found that the Ca II K line alone could be used as a better metallicity indicator than ∆S. Layden (1994) used the Ca II K and Balmer lines in low-to-moderate resolution spectra (with “most” of the rising-light spectra excluded) to produce a set of three calibrations of the form

W (K)=a + bW (H)+c[Fe/H]+dW (H)[Fe/H](6.2) where W(K) is the Ca II K line EW and W(H) is the Balmer EW (Fig. 6.1). The one Layden calibration which included correction for interstellar Ca, and averaged Hδ and Hγ to find the hydrogen EW, had coefficients a =13.669(426), b = 1.125(77), c =4.147(291), and d = 0.300(53). In the same paper, Layden went on− to calculate the metallicities of 302 nearby− RRab stars. Following these studies, Clementini et al. (1995) took spectra of 10 field RRabs at minimum light, and revised Butler’s coefficients to find

[Fe/H]= 0.08(18) 0.194(11)∆S (6.3) − − They also found the V-band magnitude metallicity dependency

MV =0.20(3)[Fe/H]+1.06 (6.4)

17 which represents one among several V-band-metallicity relations published over the years, but whose applicability has been called into question by authors such as Catelan (2009). (Other relations find an additional dependence on log(P )(e.g.,Storm(2006).) Fernley and Barnes (1996) observed nine field RR Lyraes and used spectra taken at minimum light, revising Butler’s coefficients again to find

[Fe/H]= 0.16 0.195∆S (6.5) − − Also, Lambert et al. (1996) took high-resolution spectra of 18 field RR Lyraes at minimum light and found

[Fe/H]= 0.128(72) 0.195(12)∆S (6.6) − − But by this time, the ∆S method was becoming a thing of the past. The widespread use of CCD technology since the 1980s had made the direct measurement of EWs possible, and metallicity calibrations could be based on individual lines. In this respect, the Layden (1994) calibration was certainly a step in the right direction. However, the calibration quickly encounters problems at large Balmer EWs (i.e., high effective temperatures). If the linear regression lines in Fig. 6.1 were extended, the Ca II K EW would disappear at a Balmer EW of 9.5 A˚ (X Ari), 11.1 A(SUDra),11.6˚ A˚ (TU UMa), 12.1 A˚ (XZ Dra), and 12.4 A˚ (DX Del), all of which are well within the possible range of Balmer EWs. The Layden (1994) calibration was made shortly after the advent of large vari- able object surveys in the early 1990s. One large-scale survey which, as mentioned above, was not designed with variability in mind is the Sloan Digital Sky Survey (SDSS), which contains 760,000 stellar spectra as of Data Release 10 (released July 2013) (SDSS). SDSS calculates∼ stellar parameters through the SEGUE Stellar Pa- rameter Pipeline (SSPP), using both photometry and spectroscopy (Lee et al., 2008). The SDSS contains a certain stripe along one-third of the stretch of the celestial equator, Stripe 82, which has been imaged repeatedly for calibration purposes, and whose spectra have been averaged to provide high S/N. In 2007, as part of his study of halo RR Lyraes, Nathan De Lee observed close to 100 RR Lyrae stars within Stripe 82 with the Victor Blanco 4-meter telescope at the Cerro Tololo Inter-American Observatory (CTIO) in Chile. (Much of the follow- ing is from personal communications with De Lee; see also Lee (2008).) There were 14 RR Lyrae stars common to both Stripe 82 and the CTIO/Layden data set. In June 2008, when the SSPP was still under development, De Lee compared metallic- ity values between the CTIO spectra using the Layden (1994) calibration, and their (single-epoch) SDSS counterparts using the SSPP.1

1De Lee asked Ron Wilhelm to run the spectra through the Layden (1994) calibration, and Young Sun Lee, then a graduate student, to run the spectra through SSPP.

18 A linear comparison of phase-restricted spectra between Layden (1994) metal- licities (L) and those of SSPP (S) found

L =1.013(278)S +0.275(77) (6.7) with agreement as poor as 1dex.In2010,afterSSPPhadundergonemoredevel- opment, a second test yielded∼ a rather different

L =0.871(337)S 0.147(101) (6.8) − Finally, at the end of 2012, De Lee and Wilhelm ran 310 Stripe 82 spectra of 203 stars through both Layden (1994) and SSPP, resulting in a “pretty nasty cloud” (see Figs. 6.2, 6.3, 6.4). The SSPP is not necessarily more accurate than the Layden (1994) calibration, because it was designed for photospheres in an equilibrium state. De Lee also felt the geff values for RR Lyraes were questionable, since SSPP starts with 5-color photometry for finding Teff , and this photometry is likely taken out-of-phase with the spectra. So, both the Layden (1994) calibration and the application of SSPP to RR Lyrae spectra had weaknesses. There was a need for a fresh assault on this problem, one that would produce a spectroscopic RR Lyrae metallicity calibration that had a more realistic H-K trend, and was independent of the equilibrium considerations that went into the development of the SSPP.

19 Figure 6.1: Visualization of the Layden calibraton. (Fig. 5 in Layden (1994), c AAS. Reproduced with permission.) ￿

-0.5 0.8 Unity Fit 0.7 -1

0.6 -1.5

0.5 Obs. Phase

(Layden EW Method) -2

Ron 0.4 Num Points: 61

[Fe/H] -2.5 M: 0.1181 B: 0.1809 0.3 RMS: 0.9794 [Fe/H]Ron = 0.825525*[Fe/H]SSPP-0.130824 -3 0.2 -3 -2.5 -2 -1.5 -1 -0.5

[Fe/H]SSPP

Figure 6.2: Detail of a comparison of the Layden (1994) and SSPP metallicity val- ues. These data points correspond to a sampling of SDSS spectra which have been restricted to the phase region between 0.20 and 0.75, and Balmer EWs of 4.5 to 7.0 A.˚ (Plot courtesy of Nathan De Lee.)

20 200 RRab RRc

150

100

50 (Layden EW Method)

Ron 0 [Fe/H] -50

-100 2 4 6 8 10 12 14 16 18 Balmer EW

Figure 6.3: Early divergence of the Layden (1994) calibration. (Plot courtesy of Nathan De Lee.)

-0.8 RRab RRc -1

-1.2

-1.4

-1.6 SSPP

-1.8 [Fe/H]

-2

-2.2

-2.4

-2.6 2 4 6 8 10 12 14 16 18 Balmer EW

Figure 6.4: Absence of any trend in SSPP values. This is the counterpart to Fig. 6.3, with a different ordinate scale. The absence of a trend with increasing Balmer EWs should indeed be the case, but it is uncertain how reliable these metallicity values are. (Plot courtesy of Nathan De Lee.)

21 Chapter 7 : The Saha and Boltzmann Equations and Line Formation

In the 1810s, the Bavarian glassworker Joseph Fraunhofer used a prism to spread a shaft of sunlight into its component wavelengths. To his surprise, the spectrum was striated with dark, randomly-spaced lines. He labeled the most pronounced ones with letters, assigning ‘H’ to a close pair of lines at the violet end. Thirty years later, the chemist J.W. Draper named one of the pair ‘K,’ and assigned Greek letters to additional lines (Hearnshaw, 1986, pp. 27, 36). The ‘H’ and ‘K’ lines were later identified as singly-ionized calcium (Ca II) absorption lines, and the Greek-lettered lines were the hydrogen Balmer series. In our metallicity calibration, we followed Layden and made use of the Hγ (4340.5 A),˚ Hδ (4101.7 A),˚ and Ca II K (3933.7 A)˚ lines. (The Hβ line was found to contribute considerable scatter, apparently due to metal contamination (Layden, 1994), and the Ca II H line is unusable, as it is mixed with H￿.) These lines are convenient to measure, since they are of similar wavelengths, and usually have large EWs, unlike the iron lines. To get a sense of how these lines should trend with changing temperature, we can do back-of-the-envelope calculations of the comparative populations of H and Ca atoms capable of producing absorption lines. For a gas in thermal equilibrium and at low densities, the Saha equation gives the ratio between ions of an element:

3/2 Ni+1 2kTZi+1 2πmekT χ /kT = e− i (7.1) N P Z h2 i e i ￿ ￿ where Nn is the number of atoms in the nth ionization state, Zn is the corresponding partition function, Pe is the free electron pressure, and χi is the energy required for moving an electron from state i to i +1. The Boltzmann equation gives the population ratio between excitation states:

Nb gb (E Ea)/kT = e b− (7.2) Na ga where gn is the energy degeneracy, and En is the energy of the state. Let us look at the temperature range occupied by RR Lyraes. Following the arguments in Ostlie and Carroll (1996, pp. 231-40) as well as their heuristic electron dyne 1 pressure value Pe =15cm2 ,wehavetheapproximatepopulationratiosinTable1, where NI is the neutral atom, NII is the singly-ionized state, N1 is the ground state 2 (n =1),andN2 is the first excited state (n =2). From the table we see that most H atoms are in the neutral ground state. By contrast, far more Ca atoms are singly ionized (Ca II) than are neutral (Ca I), but

1 Ntot refers to the total number of atoms of that element. 2See Appendix B: Heuristic Species Population Estimates.

22 Table 7.1: Estimates of Species Proportions in an RR Lyrae Photosphere H (6100 K) Ca (6100 K) H (7400 K) Ca (7400 K) -4 NII/NI 3.77 10 2017 0.057 25,176 × -4 NII/Ntot 3.77 10 10.054 1 × ≈-4 ≈-5 NI /Ntot 14.9610 0.95 4.0 10 ≈-8 × -3 -7 × N2/N1 1.51 10 5.30 10 4.55 10 0.015 × -8 × -3 × -7 N2/Ntot 1.50 10 5.27 10 4.30 10 0.015 N /N × 1 × 10.950.99× 1 tot ≈ ≈ most are also in the ground state. The rarity of Ca compared to H atoms means that the small proportion of H atoms capable of producing Balmer absorption lines is counterbalanced by the large proportion of singly-ionized Ca atoms. If there are 5 105 H atoms per Ca atom, we see that ×

H atoms N 0.0075 5 105 2 0.22 (7.3) ￿ × Ca atoms N ￿ ￿ ￿￿ tot ￿H or equivalently, that there are about 5 to 130 line-producing Ca II atoms for every line-producing H I atom. In sum, we should expect the hydrogen Balmer and Ca II K lines to be of comparable intensity. A graph of the line intensities (Fig. 7.1) shows that within the temperature range of interest, they should vary in a fairly linear, inverse fashion. Outside the RR Lyrae temperature range, the peaks in the H and Ca II line strengths represent regions where certain excitation levels become predominant within the pop- ulation. The peak in the H line strength near 10,000 K is the area where the first excitation level is most common among the H atoms, and therefore the Balmer lines will be strongest. At yet higher temperatures, more H atoms will be ionized and unable to absorb photons. In the meantime, Ca II becomes increasingly ionized into species like Ca III, and the Ca II K line will vanish accordingly. Thus, within the temperature range of RR Lyrae stars, hotter temperatures mean stronger Balmer lines and weaker Ca II lines, and vice versa.

23 Figure 7.1: Line strengths as a function of temperature. RR Lyrae stars veer around within the temperature range 6100 to 7400 K, where H and Ca II trends are roughly linear, and the slopes are of opposite sign. (Fig. 8.9 in OSTLIE, DALE A.; CAR- ROLL, BRADLEY W., AN INTRODUCTION TO MODERN STELLAR ASTRO- PHYSICS, 1st Edition, c 1996, p. 240. Reprinted by permission of Pearson Educa- tion, Inc., Upper Saddle￿ River, NJ.)

24 Chapter 8 : Data Aquisition

We chose to observe bright RR Lyrae stars with a wide range of metallicities which had been determined, if possible, from high-resolution spectroscopy. The empirical data we had to gather included both spectroscopy and photometry: the spectroscopy was needed in order to find the EWs of the Ca II K and hydrogen Balmer lines, and the photometry (in the form of light curves) was needed in order to determine the pulsation phases, so as to isolate the phase region where the linear H-K relation breaks down.

8.1 Photometry

The first step in making independent determinations of pulsation phases was to pre- dict an epoch-of-maximum (EOM), based on previous observations. If they are avail- able, such data can save a considerable amount of time. Any attempt to determine an EOM of an RR Lyrae star ‘from scratch’ could take a total observation time ap- proaching the duration of the star’s period. Observations would likely be fragmentary and lie scattered over several nights, since it will take a while for a star’s maximum to ‘coast’ into visibility, if the periods are close to simple fractions of a day. Fortunately, there is a large online storehouse of variable star observations, maintained by the American Association of Variable Star Observers (AAVSO) and accessible (for free) through their website. Any interested individuals, including non-members of the AAVSO, can upload their own visually-determined or digitally- processed observations of the apparent or differential magnitudes of variable stars. As of 2012, there were more than 22 million observations in the AAVSO database, and the number of data-contributors totaled more than 700 (AAVSO, 2013, pp. 4, 43). Most of the RR Lyrae stars involved in this project already had earlier observa- tions in the AAVSO database, some of which included a magnitude maximum. When available, the past EOM (in JD) and the star’s period were used to calculate future EOMs (see Meeus (1991, p. 63)).1 Observations of the star were then carried out, typically starting before, and ending after, the predicted EOM by 45 minutes up to acouplehours.2 ∼ Helpful as they were, the predicted EOMs were not always reliable, and this highlighted the need for independent observations. The AAVSO carries out some automated and by-eye checks of data after submission (AAVSO, 2013, p. 5), but crucially, there is no independent check of the recorded times in uploaded data (Arne Henden, personal communication). Hobbyists who upload data may or may not be 1Thanks to Tim Knauer for preparing the spreadsheet program that makes these calculations. 2To be most exact, conversions of the EOM JDs to HJDs could have been made in order to account for the changing distance of the Earth from the star during its orbit around the Sun. These conversions were not made at this stage, but would only have meant differences in predicted EOMs of up to 17 minutes, well within the observation window. ∼

25 interested in recording the precise time, or may not be fastidious enough to realize their clocks were off.Itmayalsohappenonoccasionthatsomeoneinputsalocal time instead of UTC. Even if the time information is precise to the minute, the data may be suffi- ciently old that the predicted epoch has shifted significantly from the real one. Over the span of a single year, a star with a true period of exactly 0.5 day, and whose mea- sured period is only one minute off from this value, will see a shift in the predicted EOM from the true one by 1.02 periods. (A star with a true period of 0.25 day will see a shift of 4.05 periods.) A shift as small as a half hour would represent 3to 9% of the periods of the stars involved in this study. ∼ ∼ Unsurprisingly, it was not uncommon to find EOMs predicted from AAVSO data to be off by as much as an hour. One prediction (that of AV Peg) was off by 4 hours, 16 minutes. Nevertheless, weather permitting, it was possible for us to capture maxima in roughly one out of every two observing sessions of a particular star. Light curves were gathered from December 2012 to January 2014, from three lo- cations: the MacAdam Student Observatory (MSO), perched on a parking structure on the University of Kentucky campus in Lexington; the University of Louisville’s Moore Observatory (ULMO), in the woods 20 km northeast of downtown Louisville, KY; and the Mt. Kent Observatory, near the town of Toowoomba in eastern Aus- tralia.3 All three telescopes were 20-inch PlaneWave Corrected Dall-Kirkham f/6.8 models (CDK20). Filters used were V, G, and in one case, R. Software used in the photometry gathering at MSO included CCDOps ver. 5.47, TheSky ver. 6.0.0.65, and Guide ver. 9.0. The differential photometry data from the MSO CCD images were generated with AIP4WIN ver. 2.1.10 software, and the Moore data was reduced using AstroImageJ (K. Collins and J. Kielkopf, in preparation). The MSO is in a very light-polluted area, and some CCD images were ruined when cars drove in and out of the parking structure and caused the mount to shake. Nevertheless, since these were bright stars and the apparent magnitude variations were typically 0.2 mag, the differential photometry was relatively easy to acquire.4 The observing∼ conditions at ULMO were generally better, though winter snows could make the access road impassable (J. Kielkopf, personal communication). Another challenge was that many of the stars had periods close to simple fractions of a day, which meant that it took some time for the maxima of some stars to coast into visibility overhead. 3See Appendix C: Light Curves and Polynomial Overlays. 4Thanks to Kielkopf for introducing us to the defocusing technique. This involves slightly defocusing the telescope so as to allow longer integration times before the counts per pixel start to reach the CCD’s nonlinear regimes. For simple differential photometry, this greatly increases the precision. In our experience, we were able to reduce the precision from 1 mag down to 0.04 mag, which is on the edge of being able to see hot Jupiter exoplanet transits.∼ The one obstacle∼ to getting the precision down further appeared to be the telescope tracking.

26 Table 8.1: Epochs-of-Maximum Star Type Observatory Filter JD-2456000 HJD-2456000 RWAri c MSO V 313.551980 313.557503 XAri ab MSO V 600.614467 600.613225 UYCam c ULMO G 575.775563 575.771938 RR Cet ab MSO V 576.712696 576.711733 SV Eri ab Mt. Kent G 580.032443 580.030123 VX Her ab ULMO G 575.552860 575.556381 RRLeo ab ULMO G 604.946028 604.940543 TT Lyn ab or c MSO V 606.805239 606.800074 TV Lyn c MSO V 311.668063 311.668193 TW Lyn ab MSO V 606.914045 606.908996 RR Lyr ab MSO V 465.662852 465.661090 V535Mon c ULMO R 610.844004 610.839092 V445Oph ab MSO V 485.750582 485.753997 AVPeg ab ULMO G 593.585144 593.588878 BHPeg ab ULMO G 563.627096 563.627938 ARPer ab ULMO G 574.747057 574.742518 RUPsc c MSO V 602.762516 602.763589 TSex c MSO V 275.820087 275.814517 TUUMa ab MSO V 679.770138 679.766153

8.2 Period Data

The accuracy of the phase data relies on the accuracy of the known pulsation periods. The periods used in this work were found by averaging the values listed in Fernley et al. (1998); Feast et al. (2008); Hoffman et al. (2009) the International Variable Star Index (VSX, 2014); and the General Catalogue of Variable Stars (SAI, 2014). Not all the periods from these sources were independently obtained, how- ever, and when the periods were obtained second-hand, authors were not necessarily scrupulous in citing their sources. Indeed, the authors themselves may no longer remember where they came from, and “the original source has been lost in time” (Ken Janes, personal communication). However, the VSX and GCVS databases are fairly good at citing references, and, wherever it was possible to identify them, source repeats were removed. Differences between remaining periods were usually less than 10-4 day ( 10 seconds), and sometimes an order of magnitude less than that. Of course,∼ for the duration of an RR Lyrae’s pulsational lifetime, the period will not remain constant. This effect is negligible for this study, however, because RR Lyrae period changes typically represent an instantaneous rate on the order of 0.1 day per million years (Catelan, 2009). SV Eri has a relatively very long period and exhibits an extremely large rate of period change, possibly because it is passing out of the RR Lyrae region of the HB, but this is still 2.1 days per million years—less than one-fifth of a second per year (Borgne et al., 2007).

27 8.3 Spectroscopy

Spectroscopy was gathered at the 2.08 m Otto Struve Telescope at McDonald Ob- servatory, located in the hilly scrubland of western Texas. The spectrograph was the low-to-moderate resolution Electronic Spectrograph Number 2 (ES2), with a resolu- tion set at R 1, 000. The two∼ observing runs lasted from 20-27 December 2012, and 19-25 July 2013. During the first run, it was attempted to achieve a more or less general phase coverage of a collection of stars with a wide metallicity spread. The second observing run allowed for the observation of different stars (being at a different time of year), and we also had the goal of observing a single RRab and single RRc star over as much of the period as possible. All told, 13 stars were observed at McDonald in December, and 6 new ones in July. (One star, RU Psc, was observed on both runs.) A total of 168 useable spectra were reduced in the analysis. The stars whose periods were covered as much as possible were VX Her (an RRab, with 26 spectra) and RU Psc (an RRc, with 22 spectra). In terms of spectra taken every 0.03 of the phase, the phase coverages were 70-75% and 60-65%, respectively. ∼ Raw spectra from December ranged over wavelengths of 3900 to 5000 A,˚ and in July, 3610 to 4960 A.˚ (Spectra were later chopped to 3911 to 4950 Atolimit˚ distortions of the Ca II K and Hβ during the normalization process.) Integration times ranged from 40 to 1800 seconds, and always remained less than 10% of the phase in order to avoid ‘phase smearing.’ Of the 168 spectra, the integration times of six of them represented less than 1% of the period, 52 represented between 1% and 2%, 86 between 2% and 5%, and 24 between 5% and 9%. (In a few instances integration was paused due to a passing cloud, which may have increased the total elapsed time by a minute or two.)

28 Table 8.2: Star Periods Star Type AAVSO VSX F98 F08 GCVS addl. GCVS ref. H09 Average RWAri c 0.3543006 0.354301 0.3543 0.354300 XAri ab 0.6511796 0.651139 0.651154 0.651163 0.651159 UYCam c 0.26704234 0.267044 0.267042 0.267030 0.267040 RR Cet ab 0.5530305 0.553025 0.553030 0.553028 0.553410 0.553105 SV Eri ab 0.71387 0.713863 0.713865 0.713796 0.713849 VXHer ab 0.4553595 0.455362 0.45537282 0.45559 0.455421 RR Leo ab 0.4524021 0.452387 0.452392 0.452393 0.452290 0.452373 TT Lyn ab or c 0.597434355 0.597438 0.597429 0.597434 0.597434

29 TV Lyn c 0.24065119 0.2407 0.240676 TW Lyn ab 0.4818655 0.481853 0.481862 0.481860 0.481860 RR Lyr ab 0.566839 0.56686776 0.566853 V535Mon c 0.3328635 0.332864 0.3329 0.333 0.332907 V445Oph ab 0.3970242 0.397192 0.397023 0.3970227 0.39699 0.397050 AVPeg ab 0.3903814 0.389941 0.390378 0.3903747 0.39048 0.390311 BH Peg ab 0.640991 0.640993 0.64062 0.640868 ARPer ab 0.42554892 0.425549 0.425551 0.425549 0.425720 0.425584 RUPsc c 0.390317 0.390333 0.390385 0.39048 0.390379 TSex c 0.324706 0.324698 0.32478 0.324728 TUUMa ab 0.5576587 0.557658 0.557659 0.557730 0.557676

Abbreviations: F98, Fernley et al. (1998); F08, Feast et al. (2008); H09, Hoffman et al. (2009). Table 8.3: Metallicities from the Literature Star Type C95 F08 F96 F97 K82 L96 L94 S97 Avg, L94 RW Ari c -1.48 -1.52(23) XAri ab -2.50-2.43 -2.53-2.40 -2.49(11) UYCam c -1.33 -1.51 -1.06 -1.41(19) RR Cet ab -1.38 -1.45 -1.49 -1.52 -1.31 -1.62(15) SV Eri ab -1.70 -1.91 -2.04 -1.92(10) VX Her ab -1.58 -1.58 -1.68(13) RR Leo ab -1.60 -1.16 -1.57 -1.49(15) TT Lyn ab or c -1.56 -1.53 -1.53 -1.32 -1.34 -1.62(15) TV Lyn c -0.99 -0.77(23) TW Lyn ab -0.66 -0.09 -1.23 -0.66(19) 30 RRLyr ab -1.39 -1.39 -1.47 -1.37 -1.47(12) V535 Mon c -1.44 -1.45(23) V445Ophab 0.17-0.19 -0.23 -0.21(12) AV Peg ab -0.08 -0.03 -0.14 -0.12(15) BH Peg ab -1.22 -1.38 -1.33(18) ARPer ab -0.30 -0.29 -0.29 -0.22 -0.43 -0.23 -0.31(15) RU Psc c -1.75 -1.65 -1.81(19) TSex c -1.34 -1.37 -1.18 -1.49 -1.27 -1.46(17) TUUMa ab -1.51 -1.57 -1.57 -1.44 -1.44 -1.66 -1.69(15)

Abbreviations: C95, Clementini et al. (1995); F08, Feast et al. (2008); F96, Fernley and Barnes (1996); F97, Fernley and Barnes (1997); K82, Kemper (1982); L96, Lambert et al. (1996); L94, Layden (1994); S97, Solano et al. (1997). Note: The ‘Avg, L94’ column indicates the average of the values after having rescaled them to Layden (1994), with errors in brackets. These are the [Fe/H] values we used for our science stars. Chapter 9 : Data Reduction

9.1 Photometry

Once the light curves were taken, MS Excel was used to fit fourth-order polynomials to the relevant parts of the light curves in order to determine an epoch-of-maximum in JD. Time resolution of the light curves ranged from less than 30 seconds to 2.9 minutes. Comparison with light curve templates (De Lee, personal communication) that fit well to the data revealed a disagreement of 0.0009 to 0.004 day ( 80 secs to almost 6 minutes), which is equivalent to 0.1 to 1.5% of the phases of all∼ the stars involved in this project. This can be considered∼ ∼ the error in the epoch. The EOM in JD was converted to HJD using the NASA-distributed IDL script helio jd (ASD, 2014). These HJD values were used together with the HJD of the middle of every spectral integration to calculate the phases of the spectra.

9.2 Metallicity Values

It was necessary to find reliable, previously-published metallicities so as to base the calibration on them. It was attempted to use metallicity values generated with high- resolution spectroscopy, though use was made of a few other sources, as well. Pub- lished metallicities were gleaned from the following sources: From Fernley and Barnes (1996) and Fernley and Barnes (1997), who used spectra with R 60, 000; Feast et al. (2008), whose [Fe/H] values went into the HIPPARCOS magnitude∼ calibra- tion and included some values from Fernley et al. (1998); Solano et al. (1997), with R 20, 000; Clementini et al. (1995), who claim “moderately high resolution” spec- troscopy;∼ Kemper (1982), which was not intended to be a high-resolution study, but is a large collection of RRc stars; and Lambert et al. (1996), with R 23, 000. An averaging of these [Fe/H] values is not sufficient, because it can∼ be expected that there will be systematic differences between the authors’ values. Therefore, the calculated metallicities of all shared stars between any authors other than Layden (1994) were plotted against the values of Layden (1994). There was one exception to this: Kemper (1982) did not share any stars with Layden (1994), so the Kemper (1982) values were scaled to Feast et al. (2008), which was then scaled to Layden (1994). Most trends appeared fairly linear (albeit a couple had considerable scatter), which is to be expected from systematic effects. Linear fits were made between the authors, and these fits were used to convert authors’ values to the Layden (1994) scale.1 Layden (1994) was chosen not because his metallicity values are expected to be more accurate than those gleaned from high-resolution spectroscopy, but because his study involved such a large number of stars (302), that it provided a consistent basis. 1See Appendix D: Author Metallicity Comparisons.

31 9.3 Spectroscopy

The raw .fits files consisted of 2D CCD readouts, with the star’s spectrum stretching roughly along one axis of the CCD. Any CCD has physical warps and response defects, however, and these must be removed in the reduction process. The basic procedure in reducing the data is to remove response bias, divide out CCD defects, fit the correct trace of the spectrum along the dispersion axis, and then use the comparison lamp spectra (necessary to account for changes in the light travel path due to telescope flexure) to apply the correct wavelength solution to the object spectra. With small variations depending on the run, the data from December and July were reduced in IRAF in the manner below. These procedures are among those explained in Massey (1997).2 (More background can be found in Howell (2000).) First, it was necessary to remove any offset introduced by temperature or elec- tronic effects. After discarding CCD readouts that had been compromised for reasons such as dawn twilight, the ten ‘zero’ frames of each night were averaged together with IRAF ’s zerocombine task. The master zero was subtracted from the flats with the ccdproc task, an action which also trimmed the unresponsive edges of the CCD. In order to account for different pixel response levels, the ‘flat’ frames from the beginning of each observing night were combined into a master flat, using the flatcombine task. This was followed with an application of the ccdproc task to both the object and comparison lamp spectra, in order to subtract the master zeros and divide out the master flats. Next, it was necessary to extract the spectra from along the dispersion axis. For this, the task apall was used on both the object and comparison lamp readouts to find the correct apertures. (Object and corresponding comparison lamp spectra often had the same dispersion traces.) The lowest possible order polynomial fit was applied along the trace for extraction. The wavelength calibrations were constructed by running the task identify on the comparison lamp spectra with lists of known emission lines that were supposed to be present, and then adding or removing lines manually. Again, a lowest-possible order polynomial was fit to the residuals between the empirical and surmised line po- sitions. The wavelength solution for each object spectrum was applied after inserting the name of the reference spectrum in the corresponding object spectrum file header (tasks hedit and dispcor). The ‘blue end’ of the spectrum was particularly challeng- ing, due to the presence of no more than two reliable emission lines at wavelengths less than about 4150 A˚ (which includes the Hδ and Ca II K absorption lines in the object spectra), and a complete lack below 3950 A˚ (which includes Ca II K). Furthermore, the comparison spectra were produced by an aging argon lamp, and it may be that there was a presence of contaminants among the lines (and perhaps a leakage of argon, too). Both the December and July data reductions required recalibrated wavelength dispersions after early attempts produced absorption lines at wavelengths outside of what could be expected from radial velocities. While reducing the July data, the reduction was particularly challenging due to the difficulty of 2Thanks also to Kyle McCarthy for his condensed write-up of IRAF reduction procedures.

32 identifying the lamp emission lines in the first place. It was attempted to fit them backwards, by taking object spectra from close to zenith and matching known Hβ, Hγ,Hδ, H6, H7, and Ca II K lines with their corresponding pixel column, using the linear correspondence to find the wavelengths of observed lines in the lamp, and matching them with lists of lines that should be present in the lamp. This was still apoorwavelengthsolution,andwassummarilydiscarded.Thefinalsolutionwas constructed using some Ar I lines. Spectra were written out to ASCII files using the rspectext task. Cosmic rays were manually removed by deleting data lines in the ASCII file. Spectra were then normalized using the script bkgrnd.cc by Kenneth Carroll (personal communication) with a ‘smoothing parameter’ of 22 (see Fig. 10.1). They were then reconverted to .fits files with rspectext. The wavelength dispersion was pasted back onto the .fits file spectrum by altering the header. In order for the EW measurements to be as consistent and reproducible as pos- sible, attempts were made to measure the automated EWs with an automated fitting routine called Robospect (Waters and Hollek, 2013). However, neither Robospect’s Hjerting nor Humlicek fits (both approximations to the Voigt profile) handled the large wings of the Balmer lines well. Finally, a FORTRAN script by Ron Wilhelm (personal communication) was used, which interfaces with IRAF to measure EWs. This program takes the .fits files of normalized spectra, fits Voigt profiles, and returns the EWs, along with errors calculated using the flux values of an unnormalized spectrum in ASCII format. For each line, a combination of window widths from among 15, 25, 50, and 80 Awas˚ used that produced consistent EWs.3 These EWs for each± line were averaged together to produce a final EW. The EWs of the Hβ and Hγ lines were scaled to the EWs of the Hδ lines by plotting the Hβ and Hγ EWs against those of Hδ, and using the best linear fit as the scaling relation. The scaled EWs and those of Hδ were averaged to find a net Balmer line EW. As mentioned above, the Hβ line introduced considerable scatter, and was discarded 9.1. Thus the Balmer line EWs are an averaging of the Hδ EW and a rescaled Hγ EW. (Layden (1994) also excludes Hβ in two of his three calibrations. In the third, he uses a metal-corrected Hβ EW.) The total error of the metallicity values consisted of the addition, in quadrature, 2 2 4 of the EW measurement errors and the parameter fit errors, or σ[Fe/H] = σP + σM . A first method for finding σ averaged the errors returned by IRAF for each of the M ￿ window combinations of that line, and, in the case of Hγ, rescaled the errors to Hδ. The resulting, propagated error from Hγ was averaged with that of Hδ to find σH (Fig. 10.3). A second method for finding σM simply considered the measurement error to be 10% of the EWs (Fig. 10.4). 3They were: Hγ, 25 and 50; Hδ, 50, and 80; Ca II K, 15 only; and for completion’s sake, Hβ, 25, 50, and 80. ± ± ± 4±See Appendix E: Stripe 82 [Fe/H] Error Calculation.

33 9.4 Function Fitting

Plots of the H-K trend were made to determine the nonlinear phase region.5 After visual inspection, this region was found to be in 0.875 < φ < 0.980 for RRabs. However, none of the RRcs exhibited clear nonlinear behavior, though five spectra of RRc stars were in that phase region, and two in the case of the ambiguous star TT Lyn. It should be cautioned that the number of data points in the phase exclusion region for all stars were only 17, and the points displaying reasonably clear nonlinear behavior are only five (there were five other ambiguous ones), ranging from φ =0.880 to φ =0.976, except for one at φ =0.028. There were six other spectra between φ =0.976 and φ =0.028 that did not display nonlinear behavior, so the exclusion region was not extended beyond φ =0.980. In total, 13 spectra in the exclusion region were discarded even though they did not display nonlinear behavior. After excluding data from spectra taken within 0.875 < φ < 0.980, the Balmer EWs, Ca II K EWs, metallicities, and all associated errors were written to ASCII files. A Python6 was used to write out 1000 synthetic data sets, each one containing the same number of lines as the empirical data. Every line included a Balmer EW, Ca II K EW, and metallicity corresponding to the matching line in the empirical file. These values were randomly chosen from a Gaussian distribution centered around the empirical value, where the standard deviation of the distribution was the error listed in the empirical data file. A C++ script7 took each synthetic data file and interfaced with Eureqa (Nuto- nian) running on an Intel Xeon Processor E5530 to make a fit corresponding to Eqn. 6.2. Every fit was allowed 1200 secs to converge, after which the coefficients were written out to a line in another ASCII file. Further lines were appended until all 1000 fits had been made. Another script found the averages and standard deviations of each column (i.e., every coefficient) and wrote them out to yet another ASCII file.

5See Appendix F: H-K Plots Showing Phase. 6See Appendix G: Python Code for Synthetic Data Generation. 7See Appendix H: C++ Code for Function Fitting.

34 Balmer Line Width Comparison

20

H-" H-#

15

10 EW (Å)

5

0 5 10 15 H-! EW (Å)

Figure 9.1: Trends of the Hβ and Hγ EWs with the Hδ EW. Note the increased scatter in Hβ due to metallicity contamination, which made the line unsuitable for the making of this calibration.

35 Chapter 10 : Results

The fits yielded the coefficients listed in Table 10.1. A visualization of the ABC calibration (RRab and RRc stars together, with no distinction between the two) can be seen in Figs. 10.5 and 10.6, and the AB calibration (RRabs only) and C calibration (RRcs only) together in Fig. 10.7. By way of comparison with Layden (1994), the ABC calibration implies (using Layden’s metallicity values) a Ca II K EW of zero at a Balmer EW of 16.91 A˚ (X Ari), 18.22 A(SUDra),18.48˚ A˚ (TU UMa), 18.70 A˚ (XZ Dra), and 18.84 A˚ (DX Del). Thus our calibration is comparatively ‘shallower,’ and can handle a much larger range of Balmer EWs. The ABC calibration was tested on 249 SDSS spectra of 225 unique stars in Stripe 82. These spectra were processed exactly the same as our empirical data, except that: 1) there was no manual correction for cosmic rays, which can distort local regions of the spectra during normalization; and 2) this test included all spectra, and was blind to phase information. It should be noted that when fitting the Voigt functions, the fits for Ca II K were never allowed to pass into the region of wavelengths shorter than 3911 A,˚ just like our spectra, even though the SDSS spectra had data from 3900 to 6000 A.˚ The gain used was 1.10, which is averaged from the values in Table 4 of Smee et al. (2013). Fig. 10.2 shows that the Stripe 82 spectra have a clear peak in metallicity around 1.70 < [Fe/H] < 1.60, with a somewhat smooth descent toward solar metallicity,− and an additional,− older stellar component at 2.50 < [Fe/H] < 2.00. This compares favorably with an inner halo ( 10-15 kpc)− with a metallicity− peak thought to be at -1.6, and an outer halo (beyond∼ 15 kpc) with a peak at -2.2 (Carollo et al., 2007). Errors appear in Figs. 10.3 and∼ 10.4.

Table 10.1: Calibration Coefficients and Standard Deviations Calibration a b c d ABC 17.343(944) -0.9158(920) 5.690(511) -0.2909(522) AB 17.93(106) -0.950(108) 5.881(551) -0.3060(591) C8.68(303)-0.276(242)1.85(181)-0.012(147)

36 0.99

0.92

0.58

0.49

0.44

0.42

Normalized Counts 0.33

0.24

0.23

4000 4200 4400 4600 4800 Wavelength (Å)

Figure 10.1: Spectral changes of X Ari as a function of phase. Note the disruption to the lines during rising light.

Figure 10.2: Metallicity values returned by an application of the ABC calibration on SDSS Stripe 82 stars. These values are from 249 spectra of 225 unique stars. Spectra that lie outside the range of the plot include 11 with returned values of [Fe/H] < 5 and 14 with [Fe/H] > 1. −

37 Figure 10.3: Distribution of metallicity errors (1). The measured error σM was found using errors returned from each of the the IRAF windows. There are 50 spectra beyond the range of this plot, where returned values of σ[Fe/H] > 1.0.

Figure 10.4: Distribution of metallicity errors (2). This is the same as Fig. 10.3, except that measured error σM was set at 10% of the EWs. There are 40 spectra beyond the range of this plot, where σ[Fe/H] > 1.0.

38 EW Changes by Star (Hbeta excluded from Balmer average)

15 V445 Oph (-0.21) AR Per (-0.31) TW Lyn (-0.66) TV Lyn (-0.77) T Sex (-1.46) VX Her (-1.68) 10 RU Psc (-1.81) X Ari (-2.49) Ca II K EW (Å) 39 5

0 5 10 15 Balmer line EW (Å)

Figure 10.5: The ABC calibration, without error bars. The data points are empirical, and the lines correspond to the star’s metallicity. This sampling of stars is chosen to represent a wide range of metallicites, and stars with good phase coverage if possible. EW Changes by Star (Hbeta excluded from Balmer average)

15 V445 Oph (-0.21) AR Per (-0.31) TW Lyn (-0.66) TV Lyn (-0.77) T Sex (-1.46) VX Her (-1.68) 10 RU Psc (-1.81) X Ari (-2.49) Ca II K EW (Å) 40 5

0 5 10 15 Balmer line EW (Å)

Figure 10.6: The ABC calibration, with error bars. EW Changes by Star (Hbeta excluded from Balmer average)

15 V445 Oph (ab, -0.21) AR Per (ab, -0.21) TW Lyn (ab, -0.66) TV Lyn (c, -0.77) T Sex (c, -1.46) VX Her (ab, -1.68) 10 RU Psc (c, -1.81) X Ari (ab, -2.49) Ca II K EW (Å) 41 5

0 5 10 15 Balmer line EW (Å)

Figure 10.7: The separate AB and C calibrations. Each line corresponds to the calibration appropriate for that star. Chapter 11 : Conclusions

In this work, we have found that the calibrations for RRab and RRc stars are sig- nificantly different, and have the coefficients listed in Table 10.1. To our knowledge, no previous author has found that the two types of RR Lyrae stars require different spectroscopic metallicity calibrations. The phase region to avoid when calculating metallicities appears to be 0.875 < φ < 0.980. It should be noted, however, that we do not have evidence that a phase exclusion region exists in RRc stars, which may or may not be an artifact of having a smaller dataset of RRc’s than RRab’s. The AB, C, and ABC calibrations we have found are ‘shallower’ than the Layden (1994) calibration in the H-K plane, and the C calibration is the shallowest of the three. This can be expected to be the case for a calibration that is more realistic than Layden (1994), since it allows a greater range of Balmer EWs. It would be of interest to compare these calibrations’ metallicity values with those generated by other means, such as photometry (using phase terms in the Fourier components of a light curve (e.g., Sandage (2004)) by finding systematic differences and finding where the consistency fails. This might bring attention to areas where metallicities generated with either method are found to be questionable. This calibration might be improved by getting complete phase coverage at the same resolution of a small sampling of RRab and RRc stars. One might even consider going to the southern hemisphere, so as to overlap this data set with the 11 field RRab stars of For et al. (2011). Their stars have unmatched high-resolution spectroscopic phase coverage, and For et al.’s calculation of stellar parameters like log(g)would allow us to try to find correlations between them and the metallicities generated by our calibrations.

42 Appendices

43 Appendix A: Boyden Station During Revolution in Peru

In addition to the logistical difficulties of maintaining an observatory at Boyden Station, Solon Bailey had to contend with political turmoil in the shaky Peruvian state. His observations and anxieties during the revolution of 1894-5 are documented in his letters to Cambridge from this time, some of which are written in a narrative style that offers a substantial change from his habitually staid correspondence. Bailey’s mounting unease with the growing civil unrest in 1893 to 1894 came to a head in January 1895, when Bailey, along with his wife and son, were returning from an errand in a nearby town on a train. With cries of “Viva Pi´erola!” rebel forces sprang from the trackside, seized the train and forced it to reverse course, whereupon they captured the town and released the passengers. Bailey and his wife took refuge in a cellar in anticipation of a government counterattack, where, “[i]n spite of the general fright it was new and interesting and we enjoyed it after a fashion, at least I did” (seq. 7-8).1 After they returned to Boyden Station, the fighting reached the area around Arequipa and raged for weeks. On the observatory grounds, the audible crackle of gunshots went on for days, in “though irregular, pretty continuous” fashion (seq. 15, 22). Bailey fretted that stray bullets would shatter the telescope lenses, and that the seizure of trains would mean the loss or delay of crates of photographic plates sent to Cambridge (seq. 9, 17-18). Armed with a handful of pistols and “clubs,” Bailey mounted steel shutters on the observatory and frantically set about building defensive ramparts (seq. 17). After hoisting the American flag on the stockade, Bailey was deluged with requests from locals for “the privilege of seeking refuge” should the situation deteriorate further (seq. 20). The observatory continued taking meteorological data as normal, even to the sound of gunshots only a few yards away, when the telescope lenses were sequestered to prevent damage (seq. 22). During lulls in the fighting, Bailey made cautious forays into Arequipa, where he encountered rebel troops careering about on mules, “a trifle reckless” and “partic- ularly careless in the handling of their guns[,] pointing them in any direction while they are loading or examining them” (seq. 22). Rumors of an imminent government attack sent Bailey galloping on his horse back to the observatory, leaping over street barricades in his haste to find sanctuary, and, he guiltily hoped, a good view of the proceedings (seq. 21, 25). After Bailey’s descriptions of the situation reached distant Cambridge, Picker- ing sent a letter beginning with a mundane accounting of science stars and supplies. Then, with more than a touch of indignation that his observatory might be threat- ened by unruly mobs, Pickering recommended making building modifications so as to afford wider shooting angles, and the installation of slits from which the occupants could fire away at aggressors. He added, “Can you conveniently pour buckets of water (preferably hot) on persons attempting to force an entrance. The most dangerous 1Unless otherwise noted, ‘sequence’ page numbers in this section are those in Bailey (1895). See also Jones and Boyd (1971, pp. 346-349).

44 mobs are sometimes dispersed by a thundershower” (Pickering, 1895, seq. 329-330). After the fighting in the area ended in February and the government fell soon thereafter, Bailey continued work on the defensive wall “for the next revolution” (seq. 25, 59). One evening the observatory put on a lavish dinner for Se˜nor Nicol´as de Pi´erola, the leader of the revolt, and some of his officers. Bailey wrote to Cambridge that Pi´erola and his men ran up a bill of twenty American dollars—no doubt a small fortune by the standards of 1890s rural Peru—but that “if he lives, I think it was a wise act” (seq. 57).

45 Appendix B: Heuristic Species Population Estimates

Consider a pure hydrogen atmosphere at T = Teff =6100K, the lower limit of RR Lyrae stars.2 Then,

3/2 Ni+1 2kTZII 2πmekT χ /kT = e− i N P Z h2 ￿ i ￿H e I ￿ ￿

3/2 16 erg 28 16 erg 2 1.38 10− (6100K)(1) 2π (9.11 10− g) 1.38 10− (6100K) = × K × × K dyne 27 2 15 2 (2) ￿ (6.63 10− erg s) ￿ ￿ cm ￿ ×￿ − ￿ ￿ ￿ 13.6eV exp − × 8.62 10 5 eV (6100K) ￿ × − K ￿ ￿ ￿ 4 =3.77 10− × (1)

where Nn is the number of atoms in the nth ionization state, Zn is the corresponding partition function, Pe is the free electron pressure, and χi is the energy required for moving an electron from state i to i +1. To find the proportion between the first excited and ground states, we use the Boltzmann equation

N2 g2 (E2 E1)/kT = e − N1 g1

13.6eV 13.6eV 8 − 2 − 2 (2) = exp − 2 − 1 2 8.62 10 5 eV (6100K) ￿ ￿￿ × −￿ K ￿ ￿￿￿ ￿ ￿ 8 =1.51 10− × Relative to the total number of atoms, we make the approximation N1 + N2 NI and use the values found above to find ≈

N2 N2 NI N2/N1 1 8 = =1.50 10− N ≈ N + N N 1+N /N 1+N /N × ￿ tot ￿H ￿ 1 2 ￿￿ tot ￿ ￿ 2 1 ￿￿ II I ￿

N N N 1 = 2 1 1 N N N ≈ ￿ tot ￿H ￿ tot ￿￿ 2 ￿ 2Again, see Ostlie and Carroll (1996, pp. 231-240).

46 NII NII NI NII 1 4 = =3.77 10− N N N ≈ N 1+N /N × ￿ tot ￿H ￿ I ￿￿ tot ￿ ￿ I ￿￿ II I ￿ and finally,

N N N I = I II 1 N N N ≈ ￿ tot ￿H ￿ II ￿￿ tot ￿ The calculations are similar for the T=7400 K case. When making the calculations for calcium at T=6100 K and T=7400 K, the only changes are

ZI =1.32 ZII =2.30 g1 =2 g2 =4 χI =6.11eV E E =3.12eV 2 − 1

47 Appendix C: Light Curves and Polynomial Overlays

Below are the light curves of the 19 RR Lyrae stars used in this project, with the 4th-order polynomial fits (red lines) used to calculate the epoch-of-maximum. Error bars are from the counts alone, and in cases where the crowding is high, the riser bars have been removed. (Note that in the light curve for T Sex, the error bar information has been lost.)

RW Ari (MacAdam Observatory, 20-21 January 2013) 0.5

0.6

0.7

0.8 Relative Magnitude 0.9

1

0.5 0.55 0.6 0.65 JD - 2456313

X Ari (MacAdam Observatory, 3-4 November 2013)

-2.9

-2.85

-2.8

-2.75

Relative Magnitude -2.7

-2.65

-2.6

0.6 0.62 0.64 0.66 JD - 2456600

48 UY Cam (Moore Observatory, 9-10 October 2013)

0.305

0.3

0.295

Relative Flux 0.29

0.285

0.28

0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.8 JD - 2456575

RR Cet (MacAdam Observatory, 10-11 October 2013)

-6

-5.8

-5.6

-5.4 Relative Magnitude

-5.2

-5

0.69 0.7 0.71 0.72 0.73 JD - 2456576

49 SV Eri (Mt. Kent Observatory, 13-14 October 2013)

2.8

2.7

2.6

2.5 Relative Flux 2.4

2.3

2.2 1 1.05 1.1 1.15 JD - 2456579

VX Her (Moore Observatory, 9-10 October 2013) 1.5

1 Relative Flux

0.5

template from AAVSO data

0.5 0.55 0.6 JD - 2456575

50 RR Leo (Moore Observatory, 7-8 November 2013)

0.4

0.3 Relative Flux 0.2

0.1

0.89 0.9 0.91 0.92 0.93 0.94 0.95 0.96 JD - 2456604

TT Lyn (MacAdam Observatory, 9-10 November 2013)

-3.8

-3.6

-3.4

-3.2 Relative Magnitude

-3

-2.8 0.74 0.76 0.78 0.8 0.82 0.84 0.86 JD - 2456606

51 TV Lyn (Moore Observatory, 27-28 November 2013)

0.225

0.22

0.215

0.21 Relative Flux

0.205

0.2

0.76 0.78 0.8 0.82 0.84 JD - 2456624

TW Lyn (MacAdam Observatory, 9-10 November 2013)

-1

-0.5 Relative Magnitude

0

0.86 0.88 0.9 0.92 0.94 0.96 JD - 2456606

52 RR Lyr (MacAdam Observatory, 21-22 June 2013)

-4.9

-4.8

Relative Magnitude -4.7

-4.6

0.64 0.65 0.66 0.67 0.68 0.69 0.7 0.71 JD - 2456465

V535 Mons (Moore Observatory, 13-14 November 2013)

0.2

0.195

0.19

0.185 Relative Flux

0.18

0.175

0.81 0.82 0.83 0.84 0.85 0.86 0.87 JD - 2456610

53 V445 Oph (MacAdam Observatory, 11-12 July 2013)

-4.5

-4

-3.5 Relative Magnitude -3

-2.5

0.68 0.7 0.72 0.74 0.76 0.78 JD - 2456485

AV Peg (Moore Observatory, 27-28 October 2013)

0.3

0.25

Relative Flux 0.2

0.15

0.56 0.57 0.58 0.59 0.6 0.61 JD - 2456593

54 BH Peg (Moore Observatory, 28-29 September 2013)

0.25

0.24

0.23

0.22 Relative Flux

0.21

0.2

0.19 0.56 0.58 0.6 0.62 0.64 0.66 0.68 JD - 2456563

AR Per (Moore Observatory, 8-9 October 2013)

0.25

0.2 Relative Flux

0.15

0.1 0.72 0.74 0.76 0.78 JD - 2456574

55 RU Psc (MacAdam Observatory, 5-6 November 2013)

-3.2

-3

-2.8 Relative Magnitude -2.6

-2.4

0.65 0.7 0.75 0.8 JD - 2456602

T Sex (MacAdam Observatory, 13-14 December 2012)

-3.15

-3.1

-3.05

-3 Relative Magnitude -2.95

-2.9

-2.85 0.75 0.8 0.85 0.9 JD - 2456275

56 TU UMa (MacAdam Observatory, 21-22 January 2014)

-2.6

-2.4

-2.2

-2

-1.8 Relative Magnitude

-1.6

-1.4

0.72 0.74 0.76 0.78 0.8 JD - 2456679

57 Appendix D: Author Metallicity Comparisons

These plots compare literature metallicity values for stars that were shared between the authors. The best fit lines were used to scale values to (Layden, 1994).

0.5

0

-0.5

-1

-1.5 Clementini et al ’95

-2

-2.5

-2.5 -2 -1.5 -1 -0.5 0 Layden ’94

0

-0.5

-1

-1.5 Feast et al ’08

-2

-2.5

-3 -3 -2.5 -2 -1.5 -1 -0.5 0 Layden ’94

58 -0.5

-1

Fernley and Barnes ’96 -1.5

-2

-2 -1.5 -1 -0.5 0 Layden ’94

0

-0.5

-1 Fernley & Barnes ’97 -1.5

-2 -3 -2.5 -2 -1.5 -1 -0.5 0 Layden ’94

59 -1.5

-1.6

-1.7

-1.8 For et al ’11

-1.9

-2

-2.1 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 Layden ’94

0

-0.5

-1

-1.5 Lambert et al ’96

-2

-2.5

-2.5 -2 -1.5 -1 -0.5 0 Layden ’94

60 0

-0.5

-1 Solano et al ’97

-1.5

-2

-2 -1.5 -1 -0.5 0 Layden ’94

61 Appendix E: Stripe 82 [Fe/H] Error Calculation

Our calibration is in the form of Eqn. 6.2. We will simplify the notation by letting H be the Balmer EW, K the Ca II K EW, and F the metallicity. Solving for F ,

K (a + bH) F = − c + dH There are two sources of error: the parameter fit, and the EW measurement itself. The square of the error of the parameter fit is

4 ∂F ∂F σ2 = σ2 P ∂α ∂α αiαj i,j=1 i j ￿ ￿ ￿￿ ￿

where the αk are the coefficients, and

∂F 1 ∂F H = − , = − , ∂a c + dH ∂b c + dH ￿ ￿ ￿ ￿ ∂F [K (a + bH)] ∂F H[K (a + bH)] = − − , = − − ∂c (c + dH)2 ∂d (c + dH)2 ￿ ￿ ￿ ￿ and the covariance terms (where α p = α q)are i ≡ ￿ j ≡

1 N σ = (p p¯)(q q¯) pq N 1 l − l − ￿ − ￿ ￿l=1 where N =1000isthesizeofthedataset.Thesquareoftheerrorfromthemeasure- ment is

∂F 2 ∂F 2 σ2 = σ2 + σ2 M ∂H H ∂K K ￿ ￿ ￿ ￿ where

∂F b d[K (a + bH)] ∂F 1 = − − , = ∂H c + dH − (c + dH)2 ∂K c + dH ￿ ￿ ￿ ￿ Finally, the total error in the metallicity is

2 2 σ[Fe/H] = σP + σM ￿

62 Appendix F: H-K Plots Showing Phase

Note that in these and the following plots, if the data points are few in number, the axis ranges are the same so as to make easier comparisons.

RW Ari

X Ari

63 UY Cam

RR Cet

64 SV Eri

VX Her

65 RR Leo

TT Lyn

66 TV Lyn

TW Lyn

67 RR Lyr

V535 Mon

68 V445 Oph

AV Peg

69 BH Peg

AR Per

70 RU Psc

TSex

71 TU UMa

72 Appendix G: Python Code for Synthetic Data Generation import numpy as np import random import math import time import matplotlib.pyplot as plt

## Written by E. Spalding (spring 2014)

#Thepurposeofthisscriptistogenerateaseriesofdifferentsynthetic data tables, given a table of empirical data that includes errors. measDataFileName = raw input(‘Enter name of empirical data table (not including the number and .dat at the end; i.e., data table 1000.dat --> data table ): ’) synExpan = int(raw input(‘Enter number of synthetic data rows to generate per row of measured data (Enter 1 if synthetic table is to be same length as empirical table): ’)) synthWrite = int(raw input(‘Enter number of synthetic tables to generate: ’)) measData=np.loadtxt(measDataFileName, dtype=‘‘double’’,skiprows=0) measDataNrow=len(measData.T[0]) # number of rows measDataNcol=len(measData[0]) # number of columns measData = measData.T open(‘fits.dat’, ‘w’).close() # clears the fit coeffs g=open(‘fits.dat’,‘a’)#reopenforwritingnewcoeffs,appendingeach new row

#------Take rows of empirical data file, make synExpan synthetic rows for each row of measured data, write to synthetic.dat for synthIteration in range(synthWrite):

dataSetNumber = str(1000+synthIteration) f=open(‘synthetic%s.dat’%(dataSetNumber),’w’)#opentowrite synthetic data (previous data cleared)

#f.write(’%s %s %s’ %(H, F, K)) # synthetic x, y, z f.write(‘H F K n’) # synthetic x, y, z \ for rowNumber1 in range(measDataNrow): for iterate in range(synExpan):

random.seed((time.time()-abs(1388000000))+ (rowNumber1+synthIteration+iterate)) #makeseedindependent

#columnnamesinempiricaldatatableareasfollows: #0:x(W(H)) #1:dx #2:y([Fe/H]) #3:dy #4:z(W(K)) #5:dz

x=dx=xg=0 x=measData[0,rowNumber1]#x dx = measData[1,rowNumber1] # dx xg = random.gauss(x,dx)

y=dy=yg=0 y=measData[2,rowNumber1]#y dy = measData[3,rowNumber1] # dy

73 yg = random.gauss(y,dy)

z=dz=zg=0 z=measData[4,rowNumber1]#z dz = measData[5,rowNumber1] # dz zg = random.gauss(z,dz)

#writeoutnumberstofile f.write(’%f %f %f’ %(xg, yg, zg)) # synthetic x, y, z f.write(’ n’) \ f.close()

74 Appendix H: C++ Code for Function Fitting

This is my variation of the basic client open source code example by Nutonian (2010). The ‘xxx.x.x.x’ is a stand-in for the server name.

#include #include #include #include #include #include #include #include

#ifdef WIN32 inline void sleep(int sec) Sleep(sec*1000); #endif { }

//function declarations void pause exit(int code); void search(int argc, char *argv[]); int main(int argc, char *argv[])

{ eureqa::search progress progress; // receives the progress and new solutions //open file to write to std::ofstream myfile; myfile.open (‘‘solutions.dat’’);

// number of data sets to process int total number data sets; int duration seconds; std::cout << ‘‘Enter the total number of data sets to process: ’’; std::cin >> total number data sets; std::cout << ‘‘Enter the amount of time in seconds to spend on each solution: ’’; std::cin >> duration seconds;

for (int data set number=0; data set number

{ int start time = time(0); while (time(0)-start time < duration seconds)

std::cout{ << ‘‘The basic client attempts to connect to a Eureqa Server "; std::cout << ‘‘running on the local machine and perform a search. It "; std::cout << ‘‘is an extension of the minimal client, with informational "; std::cout << ‘‘messages and proper error checking. It is meant to be "; std::cout << ‘‘used as a source code example." << std::endl; std::cout << std::endl;

// initialize a data set std::cout << std::endl; std::cout << ‘‘> Importing data set from a text file" << std::endl;

eureqa::data set data; // holds the data std::string error msg; // error information during import

// import data from a text file std::string filename1 = ‘‘../synthetic data sets/synthetic"; std::string filename2 = boost::lexical cast(data set number+1000); std::string filename3 = ‘‘.dat"; if (!data.import ascii(filename1+filename2+filename3, error msg))

{ std::cout << error msg << std::endl; std::cout << ‘‘Unable to import this file" << std::endl; pause exit(-1);

75 } // print information about the data set std::cout << ‘‘Data set imported successfully" << std::endl; std::cout << data.summary() << std::endl;

// initialize search options eureqa::search options options(‘‘K = f0()+f1()*H+f2()*F+f3()*H*F"); // holds the search options std::cout << std::endl; std::cout << "> Setting the search options" << std::endl; std::cout << options.summary() << std::endl;

// connect to a eureqa server eureqa::connection conn; std::cout << std::endl; std::cout << ‘‘> Connecting to a eureqa server at xxx.x.x.x" << std::endl; if (!conn.connect(‘‘xxx.x.x.x"))

{ std::cout << ‘‘Unable to connect to server" << std::endl; std::cout << ‘‘Try running the eureqa server binary provided with "; std::cout << ‘‘the Eureqa API (¨server¨sub-directory)first." << std::endl; pause exit(-1); else} if (!conn.last result())

{ std::cout << ‘‘Connection made successfully, but "; std::cout << ‘‘the server sent back an error message:" << std::endl; std::cout << conn.last result() << std::endl; pause exit(-1); else}

{ std::cout << ‘‘Connected to server successfully, and "; std::cout << ‘‘the server sent back a success message:" << std::endl; std::cout << conn.last result() << std::endl; } // query the server’s information eureqa::server info serv; std::cout << std::endl; std::cout << ‘‘> Querying the server systems information" << std::endl; if (!conn.query server info(serv))

{ std::cout << ‘‘Unable to recieve the server information" << std::endl; pause exit(-1); else}

{ std::cout << ‘‘Recieved server information successfully:" << std::endl; std::cout << serv.summary() << std::endl; } // send data set std::cout << std::endl; std::cout << ‘‘> Sending the data set to the server" << std::endl; if (!conn.send data set(data))

{ std::cout << ‘‘Unable to transfer the data set" << std::endl; pause exit(-1); else} if (!conn.last result())

{ std::cout << ‘‘Data set transferred successfully, but "; std::cout << ‘‘the server sent back an error message:" << std::endl; std::cout << conn.last result() << std::endl;

76 pause exit(-1); else}

{ std::cout << ‘‘Data set transferred successfully, and "; std::cout << ‘‘the server sent back a success message:" << std::endl; std::cout << conn.last result() << std::endl; } // send options std::cout << std::endl; std::cout << ‘‘> Sending search options to the server" << std::endl; if (!conn.send options(options))

{ std::cout << ‘‘Unable to transfer the search options" << std::endl; pause exit(-1); else} if (!conn.last result())

{ std::cout << ‘‘Search options transferred successfully, but "; std::cout << ‘‘the server sent back an error message:" << std::endl; std::cout << conn.last result() << std::endl; pause exit(-1); else}

{ std::cout << ‘‘Search options transferred successfully, and "; std::cout << ‘‘the server sent back a success message:" << std::endl; std::cout << conn.last result() << std::endl; } // start searching std::cout << std::endl; std::cout << ‘‘> Telling server to start searching" << std::endl; if (!conn.start search())

{ std::cout << ‘‘Unable to send the start command" << std::endl; pause exit(-1); else} if (!conn.last result())

{ std::cout << ‘‘Start command sent successfully, but "; std::cout << ‘‘the server sent back an error message:" << std::endl; std::cout << conn.last result() << std::endl; pause exit(-1); else}

{ std::cout << ‘‘Start command sent successfully, and "; std::cout << ‘‘the server sent back a success message:" << std::endl; std::cout << conn.last result() << std::endl; } // monitor the search std::cout << std::endl; std::cout << ‘‘> Monitoring the search progress" << std::endl; eureqa::solution frontier best solutions; // filters out the best solutions

// continue searching (until user hits ctrl-c) while (conn.query progress(progress))

{ // print the progress (e.g. number of generations) std::cout << ‘‘> " << progress.summary() << std::endl; std::cout << std::endl;

77 // the eureqa server sends a stream of new solutions in the progress // here we filter out and store only the best solutions if (best solutions.add(progress.solution ))

{ std::cout << ‘‘New solution found:" << std::endl; std::cout << progress.solution << std::endl; } /* // print a display of the best solutions std::cout << std::endl; std::cout << best solutions.to string() << std::endl; std::cout << std::endl; */

// update every second sleep(1);

if (time(0)- start time > duration seconds)

{ std::cout << std::endl; std::cout << ‘‘Here’s the best fit:" << std::endl; std::cout << progress.solution << std::endl; myfile << progress.solution << std::endl; std::cout << std::endl; break; //pause exit(-1); } } // end while statement to constrain time } // end for loop setting number of data sets to process } myfile.close(); pause exit(0); return 0; // end main } void pause exit(int code)

{ std::cout << std::endl; #ifdef WIN32 system(‘‘pause"); #endif exit(code); }

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88 Vita

Place of Birth: Batavia, New York

Educational institutions attended: Illinois College, 2003-2007 (B.S. 2007, Physics and History) study abroad: Universit´eMarc Bloch and Universit´eRobert Schuman, 2004-2005, Strasbourg, France University of Illinois at Urbana-Champaign, 2008 (non-degree-seeking) University of Kentucky, 2011-2014 (M.S. degree-seeking in Physics)

Professional positions held: Physics/math tutor, Illinois College (2004-2007) Laboratory intern, SIU School of Medicine (2006, 2007) Sec. Math and Science Ed. volunteer, U.S. Peace Corps (2008-2010) Teaching Assistant, University of Kentucky (UK) (2011-2014) Research Assistant, UK (2011, 2012) Graduate Assistant, UK (2013-2014)

Scholastic and professional honors: Phi Beta Kappa Dean’s List (2003-2007)

Professional publications (e-prints): E. Spalding, R. Wilhelm, and N. De Lee. A new RR Lyrae metallicity calibra- tion including high-temperature phase regions. In K. Kinemuchi, C.A. Kuehn, N. De Lee, and H.A. Smith, editors, 40 Years of Variable Stars: A Celebration of Contributions by Horace A. Smith, pages 198-201, 2013. URL http://arxiv.org/pdf/1310.0590.pdf. R. Wilhelm, E. Spalding, and N. De Lee. Panning for RRL nuggets in the SDSS-DR9, single epoch spectra. In K. Kinemuchi, C.A. Kuehn, N. De Lee, and H.A. Smith, editors, 40 Years of Variable Stars: A Celebration of Contributions by Horace A. Smith, pages 113-121, 2013. URL http://arxiv.org/pdf/1310.0550.pdf.

Eckhart Spalding

89