Applications of High-Resolution Astrometry to Galactic Studies

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DISSERTATION

Presented in Partial Fulﬁllment of the Requirements for

the Degree Doctor of Philosophy in the

Graduate School of The Ohio State University

Samir Salim, M.S.

*****

The Ohio State University

2002

Dissertation Committee: Approved by

Professor Andrew Gould, Adviser

Professor Marc Pinsonneault Advisor Astronomy Graduate Program Professor David Weinberg

UMI Number: 3062651

______UMI Microform 3062651 Copyright 2002 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ______

ProQuest Information and Learning Company 300 North Zeeb Road PO Box 1346 Ann Arbor, MI 48106-1346 ABSTRACT

Astrometry is undergoing a revolution that started with Hipparcos satellite and is continuing with future missions like DIVA, FAME, GAIA or SIM;aswell as novel ground-based techniques. This dissertation investigates in detail several applications of high-resolution astrometry to studies of the Galaxy.

For many of the studies presented here a catalog of high proper motion stars is required. The largest such catalog, Luyten’s NLTT, constructed several decades ago, is of limited usefulness in its original form. We therefore construct its reﬁned version, containing improved astrometry and photometry for the vast majority of the ∼ 59, 000 stars in NLTT. The bright end is constructed by matching NLTT to Hipparcos, Tycho-2, and Starnet; the faint end by matching to USNO-A and

2MASS. We improve Luyten’s 6 positions to better than 0.1. Proper-motion errors in NLTT (∼ 25 mas yr−1) are improved to 5.5masyr−1. Further, with our improved optical/infrared photometry we construct a reduced proper motion (RPM) diagram that, in contrast to the original NLTT RPM diagram, allows for the ﬁrst time the classiﬁcation of NLTT stars into main-sequence (MS) stars, subdwarfs (SDs), and

ii white dwarfs (WDs). We use this diagram to analyze the properties of our revised, and of the original NLTT. We also produce a list of new candidate nearby WDs.

We use improved NLTT in selection of nearby microlensing candidates. SIM can measure a minute deﬂection in the source’s apparent position, and so provide a precise (1%) mass determination of the nearby lens star (in many cases a SD).

We search for lens-source encounters using Hipparcos, ACT and NLTT to select lenses, and USNO-A to select sources. Among 32 candidates from Hipparcos there are Proxima Centauri and Barnard’s star. For NLTT lenses, the distance and the impact parameter are more poorly known, leading to large uncertainties in the amount of SIM observing time. However, using single-epoch CCD observations of the candidates, and the information from the revised NLTT, we have considerably reduced the uncertainties and produced a reliable list of targets.

Three planned astrometry survey satellites, FAME, DIVA,andGAIA,all aim at observing magnitude-limited samples. We argue that substantial additional scientiﬁc opportunities are within a reach if a limited number of fainter targets is included. Thus, we can increase the number of late-M dwarfs, L dwarfs, and WDs with good parallaxes by an order of magnitude, and enable good determinations of local mass functions (MF). In most cases, the candidate dim dwarfs are not yet known, and we present various methods to identify them. The presented analysis applies to DIVA as well.

iii By combining SIM observations with ground-based photometry, one can completely solve microlensing events seen toward the Galactic bulge, allowing one to measure the mass, distance, and transverse velocity of ∼ 100 lenses to ∼ 5% precision. This would allow the MF of the bulge objects, both luminous and non-luminous (remnants), to be measured.

As an application of ground-based astrometry, we present a new geometrical method for measuring the distance to the Galactic center (R0) by solving for the orbit of individual stars bound to the Sgr A*. We identify stars to which the method may be applied, and show that two of them could produce 1-5% accuracy of

R0 after 15 years of positional and radial velocity measurements. Further, we show that combining the measurements of the three stars, while the common center and mass are constrained, produces considerable improvements in the R0 determination.

By modeling the probability distribution over allowed orbital parameter space, we

ﬁnd that by 2010 the achievement of 3% precision is almost certain, while there is a 30% chance of obtaining 1% precision. These estimates would improve if the astrometry errors are reduced from the current 2 mas.

For studying the stellar halo we propose observing proper motions of faint horizontal branch stars with FAME. Using them as standard candles, halo rotation can be precisely (2 km s−1) mapped out to 25 kpc, and the clumps in the kinematic space (halo substructure) detected. Finally, we propose using SIM astrometric

iv microlensing (in a fashion similar to that applied to get bulge MF) to shed light on the nature of MACHO objects and their relation to the dark halo. By simultaneous observations of microlensing events towards the LMC from SIM and the Earth, we expect to measure distances and masses of these objects.

v To my parents Mirjana and Yahya

vi ACKNOWLEDGMENTS

I wish to express my gratitude to Andrew Gould for being a truly great adviser. I am also thankful to everyone at the Department of Astronomy for always being helpful, and for creating an excellent research environment.

This work was supported by JPL contract 1226901, grant AST 97-27520 from the NSF, grant NAG5-3111 from NASA, and Program for the Enhancement of

Graduate Studies (PEGS) from The Ohio State University.

I thank Staˇsa for her love, which is the greatest support one can have.

vii VITA

May 16, 1971 ...... Born– Belgrade,Yugoslavia

1996 ...... B.S.Astrophysics,UniversityofBelgrade

2000 ...... M.S.Astronomy,TheOhioStateUniversity

1997 – 1998 ...... UniversityFellow,TheOhioStateUniversity

1998 – 2002 ...... GraduateTeachingandResearch Associate,

The Ohio State University

PUBLICATIONS

Research Publications

1. S. Salim and A. Gould, “Sagittarius A* “Visual Binaries”: A Direct Measurement of the Galactocentric Distance.” Astroph. J., 523, 633, (1999).

2. A. Gould and S. Salim, “Photometric Microlens Parallaxes with the Space Interferometry Mission.” Astroph. J., 524, 794, (1999).

3. S. Salim and A. Gould, “Nearby Microlensing Events: Identiﬁcation of the Candidates for the Space Interferometry Mission.” Astroph. J., 539, 241, (2000).

4. I. Ferrin et al.“Discovery of the Bright Trans-Neptunian Object 2000

EB173.” Astroph. J., 548, L243, (2001).

viii 5. C. Flynn, J. Sommer-Larsen, B. Fuchs, D. S. Graﬀ and S. Salim, “A Search for Nearby Counterparts to the Moving Objects in the Hubble Deep Field.” Monthly Notices of the Royal Astronomical Society, 322, 533, (2001).

6. Z. Zheng, C. Flynn, A. Gould, J. N. Bahcall and S. Salim, “M Dwarfs from Hubble Space Telescope Counts. IV.” Astroph. J., 555, 393, (2001).

7. A. Gould and S. Salim, “Searching for Failed Supernovae With Astrometric Binaries.” Astroph. J., 572, 944, (2002).

8. S. Salim, A. Gould and R. P. Olling, “Astrometry Survey Missions Beyond the Magnitude Limit.” Astroph. J., 573, 631, (2002).

9. S. Salim and A. Gould, “Classifying Luyten Stars Using an Optical- Infrared Reduced Proper-Motion Diagram” Astroph. J., 575, L86, (2002).

FIELDS OF STUDY

Major Field: Astronomy

ix Table of Contents

Abstract...... ii

Dedication...... vi

Acknowledgments...... vii

Vita...... viii

ListofTables...... xvi

ListofFigures...... xxi

1 Introduction 1

1.1AstrometryandAstronomy...... 1

1.2TheHipparcosRevolution...... 7

1.3FutureAstrometryMissions...... 10

x 1.4 Future of Ground-based Astrometry? ...... 13

2 Reﬁnement and Analysis of Luyten’s Catalog of High Proper

Motion Stars 16

2.1Introduction ...... 16

2.2BrightStarsinNLTT...... 23

2.2.1 Positional Errors of Bright NLTT Stars ...... 23

2.2.2 Strategy to Match Bright NLTT Stars ...... 28

2.2.3 Proper Motion Errors of Bright NLTT Stars ...... 33

2.3FaintStarsinNLTT...... 36

2.3.1 Strategy for Matching NLTT to USNO-A and 2MASS . . . . 36

2.3.2 AdditionalMatches ...... 47

2.3.3 Positional Errors of Faint NLTT Stars ...... 53

2.3.4 Proper Motion Errors of NLTT Stars ...... 55

2.3.5 CommonProperMotionBinaries...... 56

2.4RevisedNLTTCatalog ...... 58

xi 2.4.1 FormatoftheRevisedNLTT...... 58

2.4.2 Proper Motion Errors of the Revised NLTT ...... 63

2.5ClassiﬁcationofNLTTStars ...... 66

2.5.1 Classiﬁcation with the Original NLTT ...... 66

2.5.2 ClassiﬁcationwiththeRevisedNLTT ...... 68

2.5.3 CandidateNearbyWhiteDwarfs...... 69

2.6CompletenessofNLTT ...... 70

2.6.1 Completeness at Bright Magnitudes ...... 70

2.6.2 Bright-end Completeness as a Function of Galactic

Coordinates...... 73

2.6.3 CompletenessatFaintMagnitudes...... 75

2.6.4 Completeness of Diﬀerent Star Populations as a Function of

GalacticLatitude ...... 79

2.7DiscussionandConclusion...... 84

3 Precise Masses of Nearby Stars 111

xii 3.1Introduction ...... 111

3.2 Astrometric Microlensing with Space Interferometry Mission . . . . . 112

3.3SelectionofMicrolensingCandidates...... 114

3.3.1 PrinciplesofSelection...... 114

3.3.2 CatalogsUsedinSelection ...... 116

3.3.3 EstimatesofErrors ...... 123

3.3.4 SearchingfortheCandidateEvents ...... 134

3.3.5 CandidateEvents ...... 138

3.4ConﬁrmationofCandidateEvents ...... 143

3.4.1 ReﬁningEstimatesUsing2MASS...... 143

3.4.2 ObservingCampaign...... 144

3.5DiscussionandConclusion...... 146

4 Mass Function of Stellar-Mass Objects 183

4.1 Mass Function in the Solar Neighborhood ...... 183

4.1.1 Late-MandLdwarfs ...... 183

xiii 4.1.2 WhiteDwarfs ...... 187

4.2BulgeMassFunction...... 191

4.2.1 Bulge Mass Function and the Microlensing ...... 191

4.2.2 DegeneracyofMicrolensingEvents...... 192

4.2.3 Breaking the Degeneracy with Astrometric Microlensing . . . 196

4.2.4 Addition of Photometric Observations ...... 199

4.2.5 Simulations of SIM Observations ...... 203

4.2.6 Expected Precision of Mass Measurements ...... 213

5 Precise Distance to the Galactic Center 226

5.1Introduction ...... 226

5.1.1 Past Determinations of the Galactocentric Distance ...... 226

5.1.2 “VisualBinaries”AroundSgrA*...... 229

5.2PhysicalPrinciplesoftheMethod ...... 231

5.3MethodofErrorDetermination...... 234

xiv 5.4 Predictions of the Uncertainty of Galactocentric Distance

Determination ...... 238

5.4.1 IndividualStars ...... 238

5.4.2 CombinedSolution...... 242

5.4.3 Probability of Achieving a Given Precision ...... 243

5.5DiscussionandConclusion...... 247

6 Kinematics of Stellar and Nature of Dark Halo 258

6.1KinematicsoftheStellarHalo ...... 258

6.1.1 Introduction ...... 258

6.1.2 SelectingFaintA-typeStars ...... 259

6.1.3 HaloRotationMeasurement ...... 263

6.1.4 Substructure in the Galactic Halo ...... 267

6.2NatureoftheDarkMatterHalo ...... 269

6.2.1 Introduction ...... 269

xv 6.2.2 MACHO Masses and Distances from Astrometric

Microlensing ...... 270

6.2.3 Expected Precision of Distance and Mass Measurements . . . 272

Bibliography...... 284

xvi List of Tables

2.1BrightNLTTstarspositionprecisions ...... 86

2.2 Bright NLTT stars proper motion precisions ...... 87

2.3NumberofUSNO/2MASSmatches...... 88

2.4Newcandidatewhitedwarfscloserthan20pc...... 89

3.1 Hipparcos and ACT events - lens star properties (astrometry and

photometry) ...... 148

3.2 Hipparcos and ACT events - lens star properties (distance, kinematic

andphysicalproperties)...... 151

3.3 Hipparcos and ACT events - source star and event properties . . . . 154

3.4NLTTevents-lensstarproperties...... 157

3.5NLTT-sourcestarandeventproperties...... 167

xvii 4.1 Uncertainties of microlensing parameters, Bulge (I = 15, ψ = 225◦,

5hours)...... 219

6.1RotationofthehalofromBHBstars...... 276

6.2 Uncertainties of microlensing parameters, LMC (V = 20, arbitrary

ψ,20hours.)...... 277

xviii List of Figures

1.1 Astrometric accuracies of diﬀerent missions ...... 15

2.1 Positional errors of bright stars in NLTT – a close-up ...... 91

2.2PositionalerrorsofbrightstarsinNLTT...... 92

2.3 Diﬀerences between photographic photometry from NLTT and USNO 93

2.4PositionalerrorsoffaintstarsinNLTT...... 94

2.5 Fraction of NLTT stars found in a rectangle as a function of V .... 95

2.6 Fraction of NLTT stars found in a rectangle as a function of declination 96

2.7NLTTpropermotionerrors...... 97

2.8 Diﬀerence between separation vectors of CPM binaries in NLTT and

theRevisedcatalog...... 98

xix 2.9 Diﬀerence in relative proper motions for NLTT CPM binaries . . . . 99

2.10 Diﬀerences between proper motions as given by Tycho-2 and our

measurements based on identifying the stars in USNO and 2MASS . . 100

2.11 Diﬀerence in the relative proper motion of the components of

subdwarfbinaries...... 101

2.12 Original reduced proper motion (RPM) diagram for NLTT stars . . . 102

2.13 Optical-infrared RPM diagram for NLTT stars ...... 103

2.14 Original NLTT RPM diagrams with stars classiﬁed using optical-

infraredRPMdiagram...... 104

2.15 Completeness of the combined Hipparcos and Tycho-2 catalogs with

respect to NLTT, as a function of RNLTT ...... 105

2.16 Completeness of bright-end NLTT with respect to Hipparcos/Tycho-

2,asafunctionofpropermotion...... 106

2.17 Completeness of bright-end NLTT as a function of Galactic latitude . 107

2.18 Completeness of bright-end NLTT as a function of Galactic longitude 108

2.19 Completeness of faint-end NLTT as a function of magnitude . . . . . 109

xx 2.20 Surface density of three diﬀerent stellar populations as a function of

Galacticlatitude...... 110

3.1Mass-luminosityrelation...... 177

3.2 Reduced proper motion diagram for Hipparcos and NLTT stars . . . 178

3.3 Distance-modulus errors vs. distance modulus for Hipparcos stars . . 179

3.4 Hipparcos color-magnitude diagram with candidate events ...... 180

3.561CygA/Bmicrolensingevents...... 181

3.6 Apparent magnitude – proper motion diagram of candidate events . . 182

4.1 Histogram of late-type dwarfs with precise parallax estimates using

FAME ...... 224

4.2 Distribution over MV of white dwarfs with precise parallax estimates

using FAME ...... 225

5.1 Evolution of R0 fractional uncertainty from S0-1 orbits ...... 250

5.2 Evolution of σR0 from S0-1 orbit with diﬀerent values of σvrad .....251

5.3 Contour plot of σR0 from S0-1 orbit (tobs =15yr)...... 252

xxi 5.4 Contour plot of σR0 from S0-1 orbit (tobs =30yr)...... 253

5.5 Contour plot of σR0 from S0-2 orbit (tobs =15yr)...... 254

5.6 Contour plot of σR0 from S0-2 orbit (tobs =30yr)...... 255

5.7 Evolution of σR0 inacombinedsolution...... 256

5.8 Probability of obtaining speciﬁc σR0 ...... 257

6.1PropermotiondistributionofA-typestars...... 282

6.2 Estimate of the measurement error of halo rotation ...... 283

xxii Chapter 1

Introduction

1.1. Astrometry and Astronomy

The Merriam-Webster Dictionary defines “astrometry” as: “a branch of astronomy that deals with measurements (as of positions and movements) of celestial bodies.” It also lists that the term first appears around 1859. From this definition it is clear that astrometry is just one of the branches of astronomy, but its relation to astronomy and the role it plays in it, and also the way this relation and role have changed throughout the history requires further elaboration.

We can say that to a great extent, from its earliest beginnings until the late

19th century astronomy was astrometry. The ﬁrst time some ancient astronomer looked at the sky and recorded relative positions of stars in the form of a constellation, he or she was already making an astrometric measurement, albeit

1 a very crude one. And while there was not much to record in terms of stellar motions (except for the diurnal motion), following the apparently complex motions of planets prompted the ﬁrst sound ideas about the physical underpinnings of the universe. Actually it was the lack of any observable relative motion of stars that imposed constraints on these early models of the universe – namely the lack of parallax was one of the key arguments that the Earth was motionless, and as such occupying the central position in the universe.

Despite the fact that the change of planetary positions was more interesting to observe, it was soon realized that in order to determine these changes accurately enough, there must also exist a good catalog of stellar positions. Most likely using some very simple apparatus for measuring relative stellar angles and the altitudes of stars passing the meridian, Hipparchus, in the 2nd century BC, was able to compile a catalog of stellar positions and apparent magnitudes of 850 naked-eye stars, with a typical accuracy of half a degree. The original catalog of Hipparchus was included in Ptolemey’s “Alamgest” three centuries later, and possibly expanded with some of his own measurements. This catalog became a standard for centuries to come.

The next major advancement in astronomy (or astrometry) that had profound consequences for the science as a whole (and actually for laying the foundations of modern science), came with Tycho Brahe in the 16th century, and his precise measurements of stellar and planetary positions. Using large instruments for

2 measuring angles, he was able to achieve arcminute precision – the limit of naked-eye observations. His observations of some 800 stars were compiled by

Kepler. The level of accuracy allowed Kepler to challenge the assumption of circular planetary orbits, and ultimately to formulate his laws of planetary motion, which paved the way for one of the greatest achievements in the history of science

– Newton’s theory of universal gravity.

With the introduction of telescopes, more and more of astronomical observations had aspects diﬀerent from those of astrometry – telescopes allowed for the exploration of physical features of the celestial bodies, thus setting the stage for what we today call astrophysics, which practically became synonymous with astronomy. The introduction of telescopes meant further advances in astrometry as well, since more precise astrometric measurements become possible. Armed with a telescope and the Newton’s theory of gravity, a new ﬁeld emerged – celestial mechanics – determining orbits of Solar System bodies and predicting their future positions. The successes of the celestial mechanics were demonstrated early on by the discovery of the periodicity of the Halley’s comet and ultimately by the discovery of Neptune through gravitational perturbations on Uranus, by Le Verrier and Adams in mid 19th century.

The aspects of astrometry related to stars also came to ﬂourish. For the ﬁrst time it was possible to measure a star’s own (proper) motion – in 1718 Halley found

3 that Arcturus, Procyon and Sirius could no longer be considered “ﬁxed” stars. The

Earth’s motion was reflected in the abberation of light, discovered by Bradley, and more importantly, and what Bradley actually attempted to measure but it took a century before instruments became precise enough to do it, the parallax of another star was detected in 1838 by Bessell, thus obtaining the first distance to any star besides the Sun, and also ultimately proving that Earth orbits the Sun. This was also how astrometry touched upon the emerging field of astrophysics – measuring stellar distances enabled their physical brightness to be known and compared to the

Sun’s, leading to the concept of HR diagram, the main sequence, and giving clues to stellar evolution. Proper motions coupled with distances produce two components of the physical velocity, enabling studies of the kinematics and dynamics of the

Solar neighborhood, and later, of the Galaxy as a whole. Bessel is also credited with detecting wobbles in Sirius and Procyon, produced by companions, which at that time escaped optical detection .

Besides this aspect of high-precision (sub-arcsecond) astrometry, the charting of the skies that included more and more stars continued. Flamsteed produced a catalog of 3000 stars in the early 18th century (including Halley’s observations of stars in the southern hemisphere). This catalog reached a precision of 10. Around the mid-19th century, Argelander, after working for 25 years, produced a massive

4 catalog and charts containing 300,000 northern stars to 10th magnitude. This work was later expanded to the south.

At approximately the same time the photographic plates came to be used in astronomy. Now positions of many stars could be measured on a single plate, and with yet greater precision. The realization of this capability of astrophotography led to the creation of a huge international program of producing a photographic atlas of the sky at a scale good enough for astrometric measurements, known as Carte du

Ciel. Although the work was divided among many observatories (mostly in Europe and some in South Africa, South America and Australia), the project proved to be overwhelming, not only for its technical requirements where the quality and the number of plates was concerned (each was 2◦ square), but even more so when it came to measuring the positions of the stars from the plates to produce the accompanying catalog (Astrographic Catalog). The atlas aimed at reaching 14th magnitude, and the catalog which was never completed, was supposed to include stars to 12th magnitude. It is widely believed that the commitment of European astronomers to this burdensome project led to their falling behind their American colleagues, who at that time concentrated on the studies of extragalactic “nebulae” that resulted in one of the most profound discoveries, that of the expanding

Universe and the Big Bang. It was a century after its conception that the project of Carte du Ciel reached its conclusion.

5 Later in the 20th century more efficient methods of photographing large swaths of the sky became possible with the introduction of Schmidt cameras and their large 6◦ field of view. The Palomar Observatory Sky Survey released in 1958 was a treasure trove for discoveries of every kind, from local asteroids to distant galaxy clusters. It was this kind of plates that William Luyten used in his half a century long effort to catalog all stars with high proper motion. He achieved this by blinking plates taken at different epochs, either manually or using simple plate measuring machines. This endeavor resulted in the catalog of nearly 60,000 stars moving faster than 0.2yr−1, known as the New Luyten Two-Tenths (NLTT) catalog. In many aspects, such as depth, completeness and overall accuracy, this catalog is still unsurpassed. We found this catalog essential for many of the studies presented in this dissertation and that is why we pay special attention to it, and made considerable effort to bring this catalog up-to-date as to make it more useful for the variety of studies. Chapter 2 is thus completely devoted to this catalog.

Besides inability to produce deeper and more precise all-sky catalogs of stellar positions (and proper motions), diﬃculties also plagued further improvements in narrow-angle astrometry, which is responsible for yielding stellar parallaxes.

It is not only that the observations where technically diﬃcult, but the blurring of images because of the Earth’s atmosphere set the limit to the accuracy of individual astrometric measurement and also to the ultimate achievable accuracy,

6 thus limiting the volume of space that could be explored. As late as in the 1980s, the most sophisticated astrometric observations augmented by CCDs could at best reach a parallax precision of a few milli-arcseconds (mas), but even this for a very limited number of stars.

1.2. The Hipparcos Revolution

Close to the end of the 20th century it appeared that astrometry, and its role in astronomy was all but dead. However, this condition was radically changed with the release of results from the Hipparcos astrometric satellite in 1997. Launched in

1989, the satellite was not able to reach the correct orbit and it appeared that the entire mission would fail. However, useful data where obtained in this faulty orbit as well. Without blurring from the Earth’s atmosphere it was possible to achieve very high precision even with small aperture of Hipparcos telescopes by repeatedly making hundreds of positional measurements of the same star. The satellite carried two instruments. The main instrument performed observations of some 120,000 preselected stars, achieving the average positional, parallax and proper motion precision of 1 mas. This sample (approximately one half of the total number) was magnitude-complete to V ∼ 8. Of the fainter stars, to the limiting magnitude of the instrument of V ∼ 11.5, only some where observed – mainly high proper motion stars drawn from NLTT, but also many astrometric standards for which

7 improved positions were desirable, and even stars that are being occulted by the

Moon (presumably in order to use occultation timings to reﬁne the lunar orbit.)

This main catalog is known simply as the Hipparcos catalog (ESA 1997). The second instrument carried out measurements of all stars down to the magnitude limit, but with a 20 times lower precision. This is the Tycho catalog (ESA 1997), and it contains around 1 million stars.

Precise parallaxes and proper motions of tens of thousands of stars in the

Hipparcos catalog enabled many different studies. The HR diagram was calibrated with the unprecedented precision. The disputed distance to the Hyades was measured directly, leading to new estimates of the luminosity of stellar candles and redefining most of the cosmic distance ladder. Proper motions provided more secure estimates of Sun’s motion with respect to the local standard of rest, identified clumps of stars in the velocity space, and produced estimates of the local mass density. Many more important results stemming from Hipparcos catalog could be mentioned.

The larger, but less precise Tycho catalog also proved to be quite valuable.

Namely, it was realized that the density of Tycho stars is high enough to enable precise astrometric calibration of wide field photographic plates, like that of POSS, which are otherwise difficult to calibrate because of severe distortions. However, for first generation POSS plates, taken in the 1950s, the 20 mas Tycho positions

8 obtained in 1991 would degrade to several arcsec because of imprecise proper motions. This is where the material taken for Carte du Ciel Astrographic Catalog came to the rescue. With modern computer technology it was relatively easy to reduce these old plates. With the mean epoch of 1907, the Astrographic Catalog

(AC, Urban et al. 1998a) served as a perfect ﬁrst epoch to compare with Tycho and derive proper motions with accuracy of 4 mas yr−1. This combination of AC and Tycho catalogs came to be known as the ACT catalog (Urban, Corbin, &

Wycoﬀ 1998b). Later, the data from the Tycho experiment were rereduced to reach 0.5 mag fainter limit, doubling the number of stars, and again these positions were combined with AC and approximately 150 other smaller catalogs, to produce

Tycho-2 (Høg et al. 2000). With ACT stars serving as positional references, POSS-I and southern ESO/UKIRT plates were calibrated to produce the largest stellar catalog ever, USNO-A (Monet 1998), containing 5 × 108 stars. ACT also served as a basis for astrometric calibration of the two catalogs derived from truly digital sky surveys: optical SDSS (York et al. 2000), and infrared 2MASS (Skrutskie et al. 1997). We used USNO-A and 2MASS extensively to reﬁne astrometry and photometry of NLTT stars.

9 1.3. Future Astrometry Missions

Building on the success of Hipparcos, new astrometric missions having greater accuracy and sensitivity were proposed. In the ﬁrst group of these new missions we can include two missions aiming to achieve accuracies an order of magnitude better than that of Hipparcos – Full-Sky Astrometric Explorer (FAME) and DIVA.Both should be complete to R ∼ 15, totaling some 4 × 107 stars. The main expected scientiﬁc returns are that the primary distance calibrators such as Cepheids, RR

Lyrae and low-metallicity MS stars will be calibrated directly, and with a great precision. Astrometric monitoring over several years of mission lifetime will also enable detection of wobbles induced by planetary and substellar companions on the primaries. Unlike the radial velocity method, astrometric method produces masses without uncertainties due to orbital inclination. A wealth of information on the distribution of visible and dark mass will come from precise proper-motion measurements of tracer populations.

Neither FAME nor DIVA can transmit entire CCD frames they observe, therefore they require some pre-selection of stars that will be measured. FAME was supposed to have an input catalog with positions of stars brighter than R = 15, while DIVA has an on-board sensor that selects bright stars and transmits parts of the CCD array containing them. Our analysis has led us to conclude that

10 insisting on magnitude-limited samples, as it is planned, would lead to a loss of great scientific opportunities achievable by observing a limited number of fainter objects. These objects would have to be preselected based on scientific merit. In this dissertation (§4.1), I emphasize improvements achievable using this approach in obtaining the local mass function and luminosity calibration of late-M and L dwarfs, and of white dwarfs. Also, by selecting faint horizontal branch stars and measuring their proper motions, valuable information on the motion, and kinematic substructure of the Galactic halo is achievable (§6.1). Furthermore, observing proper motion of many faint quasars, whose motion is known a priori to be zero, enables multi-fold improvement in the definition of the reference frame. We have illustrated these principles using proposed characteristics of FAME. Recently, the funding for this satellite was terminated, however, the same principles can be applied to DIVA, scheduled for launch in 2004.

A more ambitious project, but using the same techniques as Hipparcos and

DIVA, is planned for early in the next decade. GAIA will have precision as high as

12 µas yr−1, and will measure 109 stars to V ∼ 20 (at which point precision falls to

160 µas yr−1). It will also measure radial velocities to V = 17. This will allow the full six-dimensional phase-space to be explored for a large part of the Galaxy. This will make feasible studies that give a full picture of the structure and evolution of the Galaxy. At this point it is too early to make any conclusions whether, as in the

11 case of FAME and DIVA, including still fainter samples of stars (beyond V = 20) is technically sound and scientiﬁcally justiﬁed.

Still greater accuracies will be achievable with Space Interferometry Mission

(SIM), scheduled for launch in 2011. It should be able to provide 4 µas yr−1 accuracy down to V = 20, regardless of magnitude. Unlike all the previous missions that are constantly scanning the sky and thus produce all-sky surveys, SIM will observe specific targets (some 40,000 during its 5 yr mission lifetime). Also, unlike the other missions, SIM uses optical interferometry to achieve this unprecedented accuracy. Scientific objectives, and therefore the targets to be observed will be defined by the winning proposals. Currently selected proposals (key projects) include a search for planets, obtaining direct distances to globular clusters, measuring proper motions of nearby galaxies, etc. One of the selected projects is astrometric measurement of microlensing events, headed by Andrew Gould.

Classical photometric microlensing is in itself a very powerful technique, especially in the statistical sense, but becomes even more powerful when astrometric eﬀects are observed as well so that distances and masses of lenses can be determined. In this respect, we investigate SIM’s performance and principles required to obtain the bulge mass function of luminous and non-luminous objects (§4.2), and the distances and masses of MACHOs (§6.2). For rare cases of lensing by nearby

12 stars the only observable eﬀect is astrometric and this allows their masses to be measured with high precision (Chapter 3).

Figure 1.1 compares the astrometric accuracies of diﬀerent missions as a function of magnitude.

1.4. Future of Ground-based Astrometry?

It might seem that with the advent os space-based astrometry there is no place for astrometric studies made from the ground. However, this is not entirely true.

The techniques such as speckle interferometry and adaptive optics in many cases eﬃciently counter the deleterious eﬀects of the atmospheric seeing. Even better results in achieving high-resolution imaging, and therefore enabling astrometry of high precision, are expected from ground-based optical and infrared interferometers that include telescopes of large apertures. Unlike space missions, these facilities will operate for longer periods of time, thus enabling studies that take more time

(for example, planets with longer periods). Also, they have greater collecting power than satellites and can operate in wavelengths not available to satellites.

Observing in the infrared is necessary in studying the Galactic center, which suﬀers too much extinction in the optical band. Astrometry of stars lying as close as 1000 AU from the radio source Sgr A* already give strong indications that these

13 stars move in the gravitational potential of a supermassive black hole associated with Sgr A*. In Chapter 5 we propose a method that would combine these astrometric observations with radial velocity measurements to achieve an accurate distance to the Galactic center, a fundamental parameter in Galactic astronomy.

In order for this method to be competitive, observations over a number of years are required, which is another reason why ground based observations are favored.

After decades of neglect, it is clear that once again astrometry, now of very high resolution, plays a crucial role in modern astronomy, and that no ﬁeld of astronomy or astrophysics can claim to be complete without results provided by and made possible by astrometry.

14 0.1“ FK5

10mas

Hipparcos 1mas

.1mas FAME

Mission Accuracy

10uas GAIA SIM SIM (exceptional targets) 1uas

0.1uas 0 5 10 15 20 V magnitude

Fig. 1.1.— Comparison of astrometric accuracies of diﬀerent missions as a function of magnitude. 15 Chapter 2

Refinement and Analysis of Luyten’s Catalog of High Proper Motion Stars

2.1. Introduction

More than two decades after its ﬁnal compilation, the New Luyten Two-Tenths

Catalog (NLTT) of high proper-motion stars (µ>0.18 yr) (Luyten 1979, 1980;

Luyten & Hughes 1980) and its better known subset, the Luyten Half-Second

Catalog (Luyten 1979, LHS) (µ>0.5 yr), continue to be a vital source of astrometric data. They are still mined for nearby stars (Reid & Cruz 2002;

Jahreiss, Scholz, Meusinger, & Lehmann 2001; Scholz, Meusinger, & Jahreiss 2001;

Gizis & Reid 1997; Henry, Ianna, Kirkpatrick, & Jahreiss 1997), subdwarfs (Gizis

& Reid 1997; Ryan 1992, 1989), and white dwarfs (Reid, Liebert, & Schmidt

2001; Schmidt et al. 1999; Liebert, Dahn, Gresham, & Strittmatter 1979; Jones

1972; Luyten 1970, 1977). NLTT is at the center of the controversy over whether

16 halo white dwarfs can contribute signiﬁcantly to the dark matter (Reid, Sahu, &

Hawley 2001; Oppenheimer et al. 2001; Flynn et al. 2001); and it is the primary source of candidates for astrometric microlensing events to be observed by the

Space Interferometry Mission (SIM) (Chapter 3). Despite the advent of many new proper-motion surveys, including Hipparcos (ESA 1997), Tycho-2 (Høg et al.

2000), Starnet (R¨oser 1996), UCAC1 (Zacharias et al. 2000), as well as deeper but more localized surveys: SuperCOSMOS Sky Survey in the south (Hambly et al. 2001; Hambly, Davenhall, Irwin, & MacGillivray 2001), Digital Sky Survey

(DSS)-based survey in the Galactic plane (L´epine, Shara, & Rich 2002), search for µ>0.4yr stars in ∼ 1400 deg2 (Monet et al. 2000), EROS 2 proper motion search in ∼ 400 deg2 of high latitude ﬁelds (EROS Collaboration et al. 1999), and

MACHO search in 50 deg2 towards the bulge and the LMC (Alcock et al. 2001),

NLTT remains unchallenged as a deep, all-sky, proper motion catalog.

NLTT is an all-sky catalog with position and proper-motions (PPM) for stars

−1 above a proper-motion threshold of µlim = 180 mas yr . It extends to V ∼ 19 over

◦ much of the sky, although it is less deep (V ∼< 15) within about 10 of the Galactic plane and also in the celestial south (δ<−30◦). In addition to PPM, NLTT lists somewhat crude photographic photometry in two bands (BNLTT, RNLTT), rough

“spectral types” (usually based on photographic colors), as well as important notes on some individual stars, primarily common proper motion (CPM) binaries.

17 To be sure, the newer catalogs have superseded NLTT in certain domains. By observing a large fraction of the brighter (V ∼< 11) NLTT stars, Hipparcos obtained vastly superior astrometry and photometry for about 13% of NLTT, although in its magnitude limited survey (V<7.3–9.0) it did not ﬁnd any signiﬁcant number of new high proper-motion stars not already catalogued by Luyten. Tycho-2, which combined re-reduced Tycho observations of 2.5 million stars with 144 ground-based catalogs (most notably the early 20-century Astrographic Catalog) to derive proper motions, includes PPM and photometry for several thousand additional NLTT stars, and also contains several hundred previously unknown bright (V ∼< 11) high proper-motion stars. However, neither Hipparcos nor Tycho-2 probes anywhere near the faint (V ∼ 19) limit of NLTT. Moreover, their overlap with NLTT has never been systematically studied. UCAC (US Naval Observatory CCD Astrograph

Catalog) is in the process of delivering a new all-sky PPM catalog down to R ∼ 16 based on CCD observations. The first release (UCAC1) covers 80% of the Southern hemisphere. While its photometry is only for a single band (close to R), UCAC1 can easily be matched to 2MASS (Skrutskie et al. 1997) with its JHK photometry, effectively creating a multi-wavelength PPM catalog. Unfortunately, the current release, UCAC1, excludes most of the NLTT high proper-motion stars, due to absence of these stars in Southern parts of USNO-A2 catalog (Monet 1998), which was used as a first epoch. (USNO-A2, and its earlier version USNO-A1,

Monet 1996, were derived by measuring ﬁrst generation Schmidt plates.) High

18 proper-motion stars pose an especially difficult problem for automated catalog construction: counterparts of slow-moving stars can be reliably identified at different epochs, but fast-moving ones are easily confused with pairs of unmatched, but unrelated stars, or even with spurious objects. These problems grow rapidly worse near the magnitude limit and towards the Galactic plane. Without either new and robust (but difficult to write) algorithms or a vast investment of manual labor, the only routes open in the face of these difficulties are to eliminate the potential high proper-motion stars from the new catalogs or to include them but to acknowledge that many may be spurious. These two approaches have been respectively adopted by UCAC1 and Starnet. (Starnet derives its proper motions for 4.3 million stars by combining the Nesterov, Kislyuk, & Potter 1990 reduction of Astrographic Catalog with GSC 1.0, Lasker et al. 1990.)

There are two other PPM catalogs whose release is promised in the near future, USNO-B (Monet 2000) and GSC-II (Lasker et al. 1998, also known as GSC

2.3 ). Both are based on photographic astrometry and photometry down to or beyond the NLTT limit of V ∼ 19. They use a combination of ﬁrst and second generation Schmidt plates to determine proper motions. Neither project has stated explicitly how they will handle high proper-motion stars.

Ultimately, space missions such as FAME, DIVA, and/or GAIA will produce reliable catalogs of high proper-motion stars, since each star is observed hundreds

19 of times with great astrometric precision, thus eliminating the possibility of false matching. However, NLTT stars that are fainter than the FAME or DIVA survey limit of R ∼ 15 cannot be observed by them unless they are securely located prior to the mission. Hence, this is yet another reason for obtaining improved astrometry for these stars.

While NLTT has proven incredibly valuable, it also has signiﬁcant shortcomings. As mentioned above, its two-band photometry is relatively crude, so that classiﬁcation of stars using the NLTT reduced proper motion (RPM) diagram

(one of the main motivations for constructing the catalog) is rather uncertain: white dwarfs are easily confused for subdwarfs, as are subdwarfs for main-sequence stars. Very few of the NLTT stars have their optical magnitudes in standard bands available in the literature. This problem is not easily resolved by matching with other catalogs that have better photometry, such as USNO-A (photographic BUSNO and RUSNO: σ ∼ 0.3) or 2MASS (JHK: σ ∼ 0.03). This is because although a large fraction of NLTT stars have fairly precise (∼ 10) positions, a significant minority have much larger errors, making automated identification of counterparts in other catalogs quite difficult.

Some applications require much higher precision proper motions than NLTT’s characteristic 20–30 mas yr−1. For example, even when faint NLTT stars can be identiﬁed in USNO-A (thus establishing their POSS I 1950s positions to ∼ 250 mas)

20 the NLTT proper-motion errors propagate to create ∼ 1.5 position errors in 2010, too large to reliably predict viable astrometric microlensing events to be observed by SIM (Chapter 3), and so measure precisely the mass of the lens (high proper motion star). These errors can be reduced to ∼ 100 mas, which is quite adequate for our purposes, by matching NLTT stars ﬁrst to USNO-A and then to 2MASS.

In §2.2 we present the first step in the construction of a refined NLTT catalog: matching bright NLTT entries with those in more recent and higher-precision catalogs, primarily Hipparcos and Tycho-2, but also Starnet. By doing so, we characterize both NLTT’s PPM errors and its completeness at the bright end for the first time. These results are of interest in their own right, but they also serve to guide our search procedure for USNO-A/2MASS counterparts at fainter magnitudes.

That faint-end search is discussed in §2.3. The general approach is be to match (circa 1950) USNO-A stars with their (circa 2000) 2MASS counterparts using NLTT (1950 epoch) positions as a rough guide as to where to ﬁnd USNO-A counterparts, and NLTT proper motions to predict the positions of their 2MASS counterparts. To correctly apply this approach and to understand its limitations, one must have a good grasp of the NLTT PPM error distributions, and for this it is best to compare NLTT with other PPM catalogs.

21 The bright-end and faint-end searches are complementary. On the one hand, Hipparcos/Tycho-2/Starnet become highly incomplete for V ∼> 12, and so cannot probe faint magnitudes. On the other hand, the POSS plates that were scanned to produce USNO-A saturate for V ∼< 11, leading to increasingly unreliable photometry, astrometry, and even identiﬁcations at bright magnitudes.

Nevertheless, these searches do have some overlap at intermediate magnitudes, and we will exploit this overlap to check each method against the other. Hence, we push the bright-end search as faint as we can, incorporating Starnet (with its spurious high-proper motion stars). And we push the faint-end search as bright as we can, eliminating only the Hipparcos/NLTT matches before beginning the search.

By combining the bright-end and faint-end searches we come to our revised catalog (Revised NLTT). The format and the properties of this catalog are described in §2.4.

The Revised NLTT has a wide range of potential applications. Most notably, the resulting V − J RPM diagram (where V is calibrated from BUSNO and RUSNO) permits a much more reliable separation of diﬀerent classes of stars than did the original BNLTT − RNLTT RPM diagram. The construction and the analysis of RPM diagram is discussed in §2.5. Study of this diagram is useful in its own right, but can also guide the interpretation of RPM diagrams constructed of NLTT stars present in SDSS (York et al. 2000), making use of excellent SDSS photometry and

22 SDSS/USNO-A derived proper motions. A reﬁned version of NLTT with improved astrometry and photometry could also be exploited to study binarity as a function of stellar population and to ﬁnd new wide-binary companions of NLTT stars.

Finally, in §2.6 we present a thorough study of NLTT’s completeness. NLTT’s completeness (or lack thereof) is central to the debate over the possibility that a significant fraction of the dark matter can be in halo white dwarfs. We first demonstrate that NLTT is in fact complete at the bright end, and explore this completeness as a function of the position in the Galactic coordinate system. Then we argue that if NLTT is complete at the bright end (V ∼< 13), then one could use a bootstrap-type procedure to show that it is also close to complete at the faint end. Finally, we investigate the relative completeness of different types of stars in the Revised NLTT as a function of Galactic latitude.

2.2. Bright Stars in NLTT

2.2.1. Positional Errors of Bright NLTT Stars

To match NLTT stars with counterparts found in other catalogs, we have to start with NLTT’s positions and proper motions. To devise a search strategy, it is therefore crucial that we ﬁrst evaluate the error distributions of these quantities.

23 Positions in NLTT are nominally given to two diﬀerent levels of precision, indicated by a ﬂag: in right ascension (RA), either 1 second or 6 seconds of time and in declination (DEC), either 0.1or1 of arc. The inferior precisions are the rule in the south (δ<−45◦), and the superior precisions are the rule in the north, although for δ>80◦, the RA is given at nominally inferior precision because 6 seconds of time still corresponds to a relatively small arc.

What is the actual PPM error distribution for the ∼ 54, 000 NLTT entries with nominally good positional precision? To investigate this, we ﬁrst select only

Tycho-2 stars with proper motions µ>180 mas yr−1, i.e., the same limit as of

NLTT. Using Tycho-2 proper motions (having typical errors of just 2 mas yr−1), we propagate their positions back to 1950, the epoch of NLTT. We then find all NLTT entries whose catalog coordinates lie within 3 of each Tycho-2 star – many times larger than the nominal positional accuracy. To ensure a high level of confidence that the counterpart is real, we restrict consideration to stars for which there is one and only one Tycho-2/NLTT match, and for which the unique NLTT match does not have a flag indicating an inferior-precision measurement. Finally, we eliminate

84 matches for which the Tycho-2 proper motion disagrees with the NLTT proper motion by more than 75 mas yr−1. Figures 2.1 and 2.2 show the position diﬀerences

(NLTT − Tycho-2) of the resulting 6660 unique matches, on small and large scales respectively. The central portion of this distribution (Fig. 2.1) exhibits an

24 obvious rectangle with dimensions (1 s × 6), i.e., exactly the precisions imposed by the rounding truncation of the catalog entries. Hence, one can deduce that for the majority of the entries within this rectangle, Luyten actually measured the positions to much better precision than they were recorded into NLTT. Also obvious in this ﬁgure is a halo of stars that have characteristic errors that are of order the discretization noise. We therefore model the distribution of Luyten’s position measurements as being composed of two populations, with intrinsic

Gaussian measurement errors (in arcsec) of σ1 and σ2. Given the discretization of the reported positions, the distribution of residuals in DEC (∆δ) should therefore be,

1 c + w/2 − ∆δ c − w/2 − ∆δ P (∆δ)= q erf −erf 2w σ σ 1 1 c + w/2 − ∆δ c − w/2 − ∆δ +(1 − q) erf −erf , (2.1) σ2 σ2 where c is the center and w is the width of the discretization box in DEC and q gives the relative normalizations of the two populations. We apply equation (2.1) to the N = 5495 stars lying in the strip 7.1cosδ +0.5 > ∆α>−7.9cosδ − 0.5, and fit only in the region shown in Figure 2.1, i.e., |∆δ| < 12. (The reason for offsetting the center of this box from 0 will be made clear in Table 2.1.) We find that when w is left as a free parameter, the best fit value is consistent with w =6, the value expected due to discretization. We then fix w and rederive the other

25 parameters. Next, we write a similar equation for the residuals in RA,

sec δ c + w/2 − ∆α c − w/2 − ∆α P (∆α)= q erf −erf 2w σ sec δ σ sec δ 1 1 c + w/2 − ∆α c − w/2 − ∆α +(1 − q) erf −erf . (2.2) σ2 sec δ σ2 sec δ

We apply equation (2.2) to the N = 5022 stars in the strip 3.3 > ∆δ>−3.7, and ﬁt only in the region |∆α| < 12 sec δ. The width is again consistent with the expected value, w =15, so we again hold w ﬁxed. Table 2.1 shows our results.

Here N is the total number of stars in the subsample and N1 = Nq is the number in the good-precision (σ1) population. For both the α and δ directions, we ﬁnd

that σ1 ∼ 1.1, σ2 ∼ 6 ,andN1 ∼ 4000. That is, for more than half the full sample

(4000/6660 = 60%), Luyten actually obtained the stellar positions to a precision of

1 even though he recorded his results much more coarsely. For most of the rest, his measurement errors were similar to the discretization noise.

The offsets c may result from small systematic errors in Luyten’s global astrometry, or from real offsets between his global frame and the ICRS that underlies Tycho-2 astrometry. In any case, these offsets are taken into account when we fix the intervals from which we draw stars to fit to equations (2.1) and

(2.2).

However, Figure 2.2 shows that there is an additional population, an “outer halo” beyond what would be predicted by extrapolating the behavior of the inner

26 two populations. The structure of the distribution shown in Figures 2.1 and 2.2 will lead us in § 2.2.2 to a “layered” approach to identifying NLTT stars.

Because some stars have much lower position errors than others, we suspect that they may also have much lower proper-motion errors. To test this, we divide the above sample into two subsamples, those lying with a slightly broadened

(16 × 8) rectangle and those lying outside it. For the stars in the rectangle, the RMS diﬀerence between the magnitudes of the proper motion as measured by NLTT and Tycho-2 is 18 mas yr−1, while for those outside the rectangle it is

24 mas yr−1. (Tycho-2 error are negligible in comparison.) Hence, the rectangle stars indeed have better proper motions. For the individual components of the proper motion vector, the corresponding RMS diﬀerences are 22 mas yr−1 and

27 mas yr−1 respectively. For random uncorrelated errors in RA and DEC, one would expect the magnitude RMS to equal the component RMS. The fact that the latter is larger is probably due to transcription errors in NLTT of the proper-motion direction (proper motion in NLTT is given as a magnitude and a position angle of direction).

While the dispersions of this cleaned sample probably realistically characterize the intrinsic errors in the NLTT proper motions, they are lower limits on the

RMS diﬀerences between NLTT and Tycho-2 for the catalog as a whole. This is mainly because binaries (which have been preferentially excluded from the

27 sample by demanding one-one matches) can cause proper-motions to diﬀer when measured over diﬀerent timescales. We use these estimates to guide our approach to matching, but will evaluate the dispersions on the matched sample again in §

2.2.3 after carrying out the match.

2.2.2. Strategy to Match Bright NLTT Stars

We match NLTT sequentially to three proper motion catalogs of bright stars:

Hipparcos, Tycho-2, and Starnet, each in succession containing more stars, but also generally poorer astrometry. That is, we ﬁrst match to Hipparcos. We then remove from consideration all matched NLTT stars and match the remainder to

Tycho-2 (or rather to the subset of Tycho-2 that is not associated with Hipparcos stars). We then repeat the procedure for Starnet.

Since NLTT has about 59,000 entries, of which almost 13,000 have counterparts in these three catalogs, we are especially interested in developing procedures that can match automatically as many of these as possible, thereby reducing to a minimum the number that require human intervention. For each catalog, we therefore begin by matching stars inside a (16 × 8) rectangle. The chance that two unrelated high proper-motion stars will fall so close together is miniscule.

We therefore place only very weak demands on matches: the magnitude of the vector proper-motion diﬀerence should be less than 100 mas yr−1,andthe“V ”

28 magnitudes should agree within 1.5 mag. For Hipparcos, we use the Johnson

V mag given in the catalog. For Tycho-2, we use the catalog’s VT . For NLTT we use the “red” magnitude RNLTT (which is actually quite close on average to

Johnson V , Flynn et al. 2001) except when it is not given, in which case we adopt (BNLTT − 1) for the “V ” mag. For Starnet, we use the red photographic magnitude which ultimately derives from the GSC 1.0. Of course, all these various magnitudes are not on exactly the same system, but because of the inaccuracy of photographic magnitudes, in particular those in NLTT, the systematic diﬀerences are not very important: we are interested only in a crude discrimination between stars of very diﬀerent brightnesses. We conduct our search only for NLTT stars with RNLTT ≤ 14.0.

Singular matches from this rectangle search are accepted without further review. There are, however, many multiple matches in both directions. For example, if one component, say A, of a CPM binary matches to a star in a given catalog, the other component, B, will almost always match as well. If the CPM binary is suﬃciently wide to have a separate entry in the catalog being searched

(Hipparcos/Tycho-2/Starnet), it may yield 4 “matches”: A–A, A–B, B–A, B–B, of which the second and third are false. All such multiple matches are investigated, but in many cases they cannot be fully resolved because the true match to the companion is outside the rectangle or outside the catalog altogether. We return

29 to this problem below. The matches are then removed from both catalogs, and a second search on the remaining NLTT stars is then conducted inside a radius

∆θ<120 but otherwise using the same criteria. These matches are also accepted without review: the main reason for separating the rectangle and circle searches at this stage is to reduce the number of multiple-matching candidates. After resolving double matches (where possible) we return to the rectangle, but loosen the criteria.

We now demand only that the magnitudes of the proper motions be consistent within 80 mas yr−1 but place no constraint on the direction. This is to allow for transcription errors in the angle recorded in NLTT. We also loosen the tolerance on the agreement in “V ” to 2.5 mag. The resulting new matches are then scanned manually. But again, since the chance that two unrelated high proper-motions stars will fall in the same rectangle is extremely small, virtually all of these matches are genuine. Next, we apply the same procedure to the 2 circle. All matches are again reviewed by hand, this time more critically. If, for example, the angle of the proper motion and the RA both disagree strongly, but the DEC agrees to within a few arcsec and the “V ” mags also agree well, we assume that the match is real and that there are multiple transcription errors. Of course, we are more liberal for the regions where NLTT has worse-precision positions. Finally, we extend the search to ∆θ<200 and with the weak constraints on proper-motion and magnitude agreement. We review the results with extreme caution. (We do not apply this last step to Starnet because it contains spurious high-proper motion stars).

30 After these automated, and semi-automated searches are completed on one catalog, we move on to the next. After they are all complete, we move on to one

final manual search. In it, we plot the unmatched NLTT, Hipparcos, and Tycho-2 stars, each using a different color, on a map of the sky using vectors whose length, orientation, and thickness represent respectively the magnitude and direction of the proper motion, and the “V ” mag of the star. This allows us to find counterparts of NLTT stars with major measurement and/or transcription errors. For example, we find three counterparts that disagree in DEC by exactly 1◦ but otherwise are in perfect agreement. We even find one that disagrees by exactly 11◦ in DEC and about 9 minutes of time in RA. That this entry is a transcription error is obvious from the name that Luyten assigned to the star “−65 : 2751”, which corresponds to its true declination, δ = −65◦.

We then return to the problem of binaries. We examine every pair of matches separated by less than 2 in NLTT. In a large fraction of cases, these are each single matches of well separated stars, but we still check that we do not have the matches reversed, using the relative separation and orientation reported in the NLTT notes on CPM binaries. However, there remain many multiple NLTT matches to single stars in other catalogs, especially Hipparcos. We resolve these whenever possible using the Tycho-2 Double Star Catalog (TDSC, Fabricius et al. 2002) which contains PPM and photometry for multiple-component objects in Tycho and

31 Hipparcos, both actually associated multiples and spurious optical doubles. This catalog also contains many binaries that are treated by NLTT as single stars. For some of these, NLTT contains a note that the entry is actually a binary and gives its separation and (usually) the magnitudes of its components. For others, NLTT regards the object as a single star. We make a note of all these cases for our future work on binaries, but for the present treat all single-entry NLTT stars as single stars. Some TDSC entries do not list a proper motion. For a large fraction of the cases we checked, the positions for these entries are also signiﬁcantly in error. We therefore do not make use of these entries unless there is corroborating information

(from NLTT or 2MASS) that the positions are correct. For cases where TDSC does not resolve an NLTT binary or where the TDSC entry does not contain a proper motion, we check to see if the star lies in the 47% of the sky covered by the 2MASS release. If it does, usually 2MASS resolves the binary and we substitute 2MASS coordinates for those of the other (e.g., Hipparcos) catalog. We also note whether we believe that the catalog’s photometry can really be applied to each component

(or whether this magnitude actually refers to a blend or to the other component).

In the latter case, we adopt “V ” from NLTT rather than the catalog. For binaries not covered by TDSC or 2MASS, or for which these catalogs do not resolve the binary, we record the two NLTT stars as an unresolved binary.

32 Finally, in the spirit of pushing our “bright” catalog as faint as possible for later comparison with the “faint” catalog (§2.3), we make a list of all NLTT CPM binaries for which one component is matched and whose fainter component has

RNLTT ≤ 14.0, but is not matched. We search for these directly in 2MASS, using the coordinates of the ﬁrst component and the separation vector given in the NLTT notes to predict the position of the second. We incorporate these by assuming that the proper motion of the ﬁrst component is also valid for the second.

2.2.3. Proper Motion Errors of Bright NLTT Stars

Altogether, we have matched 12,736 stars from NLTT to Hipparcos, Tycho-2, and Starnet (or in a few cases, to CPM companions of these stars that we found in 2MASS). We began this study by estimating the errors in positions and proper motions of the original NLTT catalog, but restricted to a relatively clean subset. Here we give the RMS differences between NLTT proper motions and those found in the three more modern catalogs. These differences can result from NLTT measurement errors or transcription errors, from real differences in the proper motion due to binarity, from misidentification of counterparts, or from proper-motion errors in the three modern catalogs.

The last of these four causes can be quite signiﬁcant for the relative handful of stars near the magnitude limits of these catalogs, particularly Hipparcos. Indeed,

33 we find that when Hipparcos reports errors larger than 10 mas yr−1, its true errors can be much larger than the tabulated errors. We also find that for the sample of stars common to NLTT, Hipparcos, and Tycho-2, the RMS tabulated Tycho-2 errors are substantially smaller than those of Hipparcos. Moreover, since these are established by a longer baseline of observation, they are more directly comparable to the NLTT proper motions in cases where the Hipparcos proper motion may be corrupted by a short-period (P ∼< 5 yr) binary. We therefore, first substitute

Tycho-2 for Hipparcos proper motions whenever the former are available. For purposes of comparing with NLTT we then eliminate all stars with tabulated proper-motion errors greater than 10 mas yr−1 in either direction. This removes 52

(0.6%) of Hipparcos stars, no Tycho-2 stars, and 23 (1.7%) of Starnet stars. After removal of these 75 stars, we expect that the proper-motion diﬀerences between

NLTT and the three modern catalogs are dominated by the ﬁrst three causes listed above.

Table 2.2 lists these RMS values for various subsets of the catalog. Here

“Hipparcos” refers to NLTT stars found in Hipparcos, “Tycho-2” to stars found in

Tycho-2 but not Hipparcos, and “Starnet” to stars found in Starnet, but not in either of the other two catalogs. “Better precision” stars are those with positions speciﬁed in NLTT to 1 second of time and 6 of arc (and generally lying in the range −45◦ <δ<80◦), and “worse precision” stars are the remainder. When two

34 NLTT stars, usually close components of a CPM binary, are matched to a single entry in the PPM catalogs, we compare the proper-motion measurements only once.

Table 2.2 shows that among the stars with nominally better positions, those inside the rectangle have consistently lower proper motion errors than those outside the rectangle. For Hipparcos and Tycho-2, the stars with nominally worse positions are intermediate in their proper motion errors between these two categories, indicating that they are also probably a mixture of intrinsically better and worse precision measurements. However, for Starnet, the proper motion errors of the stars outside the rectangle as well as those with nominally worse position errors both have signiﬁcantly worse proper motion errors. This may indicate that these stars suﬀer many more false matches. However these will be tracked down in our faint-end search (§2.3).

35 2.3. Faint Stars in NLTT

2.3.1. Strategy for Matching NLTT to USNO-A and

2MASS

As mentioned in §2.2, we wish to push our “faint” search as bright as possible, thereby maximizing overlap with the bright search. While we expect that the overwhelming majority of the bright-search matches between NLTT and the three

PPM catalogs (Hipparcos, Tycho-2, Starnet) are genuine, there is an important path to false matches. The true counterpart of the NLTT star may not be in any of the three PPM catalogs. Then, even though the real star may (most likely does) have a PPM that is very close to its NLTT values, our bright-star approach is to search farther and farther from the NLTT PPM in these catalogs. Eventually, we may ﬁnd a barely acceptable match in these catalogs. Such cases will be easily uncovered by cross-checking the bright and faint searches.

However, we cannot push our faint-star approach of matching counterparts from USNO-A and 2MASS too bright because bright stars are increasingly saturated in the POSS plates that were scanned to construct USNO-A, leading to increasingly unreliable PPMs and even identiﬁcations.

36 We therefore begin the “faint” search by removing from consideration all stars that were matched to Hipparcos in the ﬁrst two (“rectangle” and “circle”) searches carried out in §2.2.2. The Hipparcos completeness limit (V =7.3) assures us that essentially all the very bright stars in NLTT are accounted for in this way.

Moreover, the high-proper motion stars that entered the Hipparcos input catalog beyond its magnitude limit were drawn almost entirely from NLTT. (This is conﬁrmed by the fact that in §2.2 essentially all Hipparcos high proper-motion stars were successfully matched to NLTT. By contrast, for Tycho-2, whose magnitude range strongly overlaps that of Hipparcos but was not constructed from an input catalog, there were several hundred non-matches.) Hence, the probability that the

“best” match of a Hipparcos star to NLTT is a false match is very low. However, from the faint search we do not exclude Hipparcos stars that were matched using our more aggressive strategies, in order to have an additional check on the robustness of these identiﬁcations.

Next, we restrict our search to portions of the sky covered by the (circa 1950)

◦ POSS I survey (δ ∼> −33 ), since our entire approach fails farther south. USNO-A is a positional (and photometric) catalog, which is constructed by matching blue and red photographic plates. In order to minimize the number of artifacts masquerading as stars, USNO-A requires blue and red detections with position diﬀerences less than 2. Hence, in the non-POSS I areas of the sky, where the blue

37 and red plates were taken many years apart, all stars with sufficiently high proper motion will necessarily be absent. For example, if the plates were taken 8 years apart, then the catalog will not contain any stars with µ>250 mas yr−1.Thisdoes not pose any problem for regions covered by POSS I because its blue and red plates were taken sequentially. Farther south, however, the plate epochs differ by of order a decade. There is a further wrinkle here. The first version of the USNO-A catalog,

USNO-A1 (USNO-A1.0), is constructed by scanning POSS I plates all the way to their southern limit (δ ∼−33◦) and supplements POSS I with additional southern surveys only south of this limit. However, USNO-A2 (USNO-A2.0), which has somewhat better precision, exploits southern catalogs in place of POSS I for the

◦ region δ ∼< −20 . Hence, we conduct our search using USNO-A2 wherever possible, but supplement this with USNO-A1 where necessary.

We divide the search into two principle stages, which we dub the “rectangle” and the “circle”. In the rectangle stage, we ﬁrst identify all USNO-A stars (NLTT candidates) that lie within a 16 × 8 rectangle1 centered on the position predicted by using the reported NLTT proper motion to propagate the (epoch 1950) NLTT catalogued position to the epoch of the POSS plate underlying USNO-A. We then predict the position of the 2MASS counterpart of each such USNO-A star under the assumption that its proper motion is as given by NLTT. We use a map of

1We consider genuine USNO stars throughout: entries added from Tycho are ignored.

38 2MASS coverage to determine if the star should lie within the 47% of the sky covered by the second 2MASS incremental release and, if it should, what the exact

2MASS epoch is at that position. We then query the 2MASS data base for all stars lying within 5 of this position. The size of this error circle is inﬂuenced primarily by the errors in NLTT proper motions, which are expected from §2.2.3 to be about 25 mas yr−1 for stars in the rectangle. Since the diﬀerence in epochs is typically about 45 years, this translates to a 1.1 error in predicted position in each dimension, far larger than the ∼ 130 mas and ∼ 250 mas position errors in

2MASS and USNO-A, respectively. A 5 circle will therefore capture all but the extreme outliers. Ideally, only the USNO-A candidate actually corresponding to the NLTT star will have a 2MASS counterpart because static stars will not match to anything in 2MASS. In general, this query can have 3 possible outcomes, and we can therefore deﬁne 3 classes of stars: non-matches, unique matches, and multiple matches.

Unique matches are those NLTT stars for which our procedure generates one and only one 2MASS counterpart. Because the area covered by the rectangle is quite small, the overwhelming majority of unique matches should be genuine, even though we have not yet imposed any magnitude constraint on the selection procedure. We can therefore use the unique matches to study the color-color relations of counterparts in NLTT, USNO-A, and 2MASS. First, we ﬁnd from the

39 overlap of USNO-A1 and USNO-A2 stars that the BUSNO mags are not on the same system in the two versions of the catalog, which deviate by up to 1 mag. We obtain

− 2 − 3 4 BA2 = BA1 +49.056 9.5613BA1 +0.669BA1 0.0198BA1 +0.0002BA1,

(BA1 > 13.07) (2.3)

and BA2 = BA1 otherwise. Henceforth, when we write BUSNO it refers to magnitudes converted to the USNO-A2 system. However, in the actual catalog we report the original USNO-A1 magnitudes for stars identiﬁed in that catalog. Next we compare USNO and NLTT photometry. Figure 2.3 shows this comparison (for our full catalog, not just the rectangle matches) as a function of RNLTT. Clearly, there are strong nonlinearities. We model these by,

RUSNO =0.9333RNLTT +0.6932, (RNLTT > 10.39) (2.4)

and RUSNO = RNLTT for RNLTT < 10.39, and by

(B − R)NLTT − (B − R)USNO =0.2387 − 0.055RNLTT, (RNLTT < 16.5)

(B − R)NLTT − (B − R)USNO =7.3817 + 0.4067RNLTT, (RNLTT ≥ 16.5).(2.5)

Note that BNLTT is called “photographic magnitude” in NLTT.

40 Finally, we obtain very rough optical/IR color-color relations between USNO and 2MASS bands,

RUSNO − J = −0.62 + 3.908(J − Ks),

BUSNO − J =0.70 + 4.942(J − Ks), (J − Ks > 0.6),

RUSNO − J =0.37 + 1.536(J − Ks),

BUSNO − J =0.73 + 3.305(J − Ks), (J − Ks < 0.6), (2.6)

These last relations have a huge, 0.8 mag, scatter, and are not even continuous.

They nonetheless are extremely useful in assessing the quality of candidate matches.

Each multiple match is investigated and resolved by hand. The primary cause of multiple matches is common proper motion (CPM) binaries. Sometimes there are two USNO-A stars that each match to the same two 2MASS stars. In such cases, there is always a NLTT CPM binary. This is because USNO-A does

not resolve binaries with separations ∼< 5 , whereas Luyten was able to resolve these easily by eye. Hence these cases are always resolved unambiguously by making use of the NLTT supplementary notes on CPM binaries. More frequently, there are two 2MASS stars matched to one USNO-A “star”, which is actually an unresolved binary. In the great majority of these cases, NLTT shows a CPM binary, so these are also easily resolved. In such cases, however, we cannot use the USNO-A position to obtain a reliable new proper motion since it is a blend of

41 two stars. For such stars, our catalog has an improved 2000-epoch position and new J and Ks photometry, but not an improved proper motion nor better visual photometry (which is obtained from USNO). In some two dozen cases, we resolve the multiple match by ﬁnding a probable CPM binary that Luyten missed. Some of these binary candidates can be conﬁrmed by investigating POSS I and POSS

II images, but in most cases the companion is too faint in the optical band or is blended in the POSS images because the separation is of order 5 or less. We will present these new binary companions in future work on NLTT binaries. Finally, for each resolved multiple match, as well as for each unique match, we compare the BUSNO and RUSNO mags reported by USNO with the values that would be predicted on the basis of NLTT and of 2MASS photometry using the color-color relations established in using the unique matches. Even though these relations have large errors, they still help to identify wrong matches because random pairs of unrelated stars are likely to diﬀer by many magnitudes, so by this standard even measurements with 0.8 mag errors can be very eﬀective. The outliers are regarded as suspicious and put aside, while the remainder are accepted.

Operationally, there are two classes of non-matches: 1) there are no USNO-A stars in the rectangle or 2) there are such stars but they all fail to match with

2MASS stars. Class (1) can occur because A) the NLTT position is in error by more than the size of the rectangle (which is the case for a sizeable minority of

42 NLTT stars), or B) USNO failed to detect the star. Class (2) can occur because

A) the NLTT position is in error and there just happens to be an unrelated USNO star in the rectangle, B) the USNO star is the correct match but 2MASS has failed to detect the star, or C) the USNO star is correct but the NLTT proper motion is so far oﬀ that the 2MASS counterpart lies outside the 5 error circle. We ultimately attempt to distinguish among these ﬁve causes in § 2.3.2 but for the moment assume that that the correct USNO match lies outside the rectangle and continue our search for it in the circle.

We now remove from consideration all NLTT stars that were successfully matched to USNO-A stars in the rectangle. The great majority of the remainder are unmatched. However, a small number were matched in the rectangle search, but are regarded as questionable matches because they are photometric outliers as determined using the relations 2.4-2.6. We search for USNO-A/2MASS counterparts to both classes of stars by probing a circle of radius 2 in USNO-A centered on the NLTT position (updated using the NLTT proper motion to the epoch of the USNO-A POSS plate). Because the area being probed is now

350 times larger than in the rectangle search, the probability of false matches is proportionately larger. We therefore take several steps to reduce such false matches.

43 First, we demand that |RUSNO − RNLTT| < 2.2. These are approximately 5 σ limits. (Some NLTT stars do not contain an entry for RNLTT,inwhichcasewe us B mags.) Second, when we query the 2MASS data base at the 2MASS-epoch position, we ask whether the 2MASS star has any USNO-A stars within 3 and if so what the separation and mags of the closest one are. If the 2MASS star is associated with a USNO-A star at approximately the same position, we do not immediately eliminate the 2MASS star, but rather ﬂag it. If the association is real, the 2MASS star could not be the genuine counterpart of the NLTT star because its

− proper motion would be ∼< 70 mas yr 1, far below the NLTT threshold. However, the association might be due to chance proximity of a random field star. At high latitude, the probability of such a chance alignment is low, but in the difficult fields oftheGalacticplane,itisnot.Wealsodoareversecheck:weaskwhetherthere are 2MASS stars associated with USNO candidates (assuming no proper motion) within 2. The presence of a 2MASS star near the USNO 1950-epoch position will also indicate that the USNO candidate is not a NLTT star.

For a large fraction of the diﬃcult cases, we directly consult the digitized sky survey (DSS). This normally contains images from two well-separated epochs

(POSS I and II), and thus usually allows one to spot the high proper-motion star.

However, for the southern zones (approximately coincident with our USNO-A1 areas), the two DSS epochs are very close in time, making it extremely diﬃcult to

44 spot moving stars. In some cases, we therefore consult the USNO web site, which contains digitized POSS I, but the cycle time for such searches is about 20 minutes, so this cannot be done in every case. This mostly aﬀects regions very close to the

Galactic center.

The multiple matches are more complex than in the rectangle case. We again have large numbers of CPM binaries. These are usually easier to untangle than in the rectangle case because they are typically farther apart and so are resolved in USNO. It is then only necessary to check that the relative orientation of the

2MASS/USNO stars is the same as that given by the NLTT notes. However, there are now a large number of spurious matches. In a handful of cases, these turn out to be CPM companions of NLTT stars that were not found by Luyten. We will present these in the future. Unlike the CPM binaries found in the rectangle, these are mostly well enough separated that they are resolved in USNO. The great majority of the spurious matches are pairs of unrelated USNO-A/2MASS stars.

We track these down using a variety of photometric and astrometric indicators and by consulting DSS, as discussed above.

The nonmatches are of the same ﬁve classes that were described in the rectangle search. These are sorted out in § 2.3.2.

45 The entire procedure described above is carried out separately for USNO-A2

(δ>−37.◦5) and for USNO-A1 (−37.◦5 <δ<−15◦). In each case, we demand that the entry originate from POSS I. The two catalogs are then combined, giving precedence to USNO-A2 when a match is found in both. In practice, we ﬁnd a fairly sharp boundary at δ ∼−20◦. Table 2.3 shows the number of matches in the four categories (rectangles,circles)×(unique,resolved), for each of the two USNO catalogs, A1 and A2. Of the 23,681 USNO/2MASS matches, 18,442 (78%) lie inside the rectangle. The overwhelming majority of these are unique matches, and even the 3% that had multiple matches were relatively easy to resolve. Note that even for the majority of cases in which the star lay in the circle, there was only one USNO/2MASS pair whose separation was consistent with the NLTT proper motion. Thus, the high quality of the NLTT PPMs, combined with the paucity of stars over most of the sky, allowed us to match over 20,000 stars in two catalogs with very diﬀerent passbands, at widely separated epochs, with “only” about a

1000 problem cases.

However, these cases, which were mostly concentrated in Galactic plane, were often quite diﬃcult to resolve. As is well-known, optical and infrared images of the plane are each extremely crowded, but often not with the same objects, and this fact generally gave rise to dozens, sometimes more than 100, false matches for a single NLTT entry. The correct match had to be laboriously identiﬁed from among

46 these. Thus, our method, which works amazingly well away from the plane, tends to become extremely bogged down within it. An alternative approach to ﬁnding proper motion stars in the plane, albeit one that would miss many red stars, would be to compare blue plates in which the majority of background stars would be removed. In 2.6.4 we will see that that is exactly what Luyten did!

2.3.2. Additional Matches

Based on our analysis of the cases in which the PPM search found the NLTT star but the USNO/2MASS search did not, we conduct four additional types of searches, which we dub “annulus”, “2MASS-only”, “USNO-only”, and “CPM”.

The ﬁrst is aimed at ﬁnding 2MASS counterparts to USNO stars in cases for which the NLTT proper motion is seriously in error, including the possibility that there is a transcription error in the direction of proper motion. Hence the search is conducted in an annulus around the position of the unmatched USNO star.

In the bright-end search we found many examples of transcription errors of all types in NLTT, including in the position angle of the proper motion. We therefore search for 2MASS stars in an annulus around the position of all USNO counterparts lying within the NLTT rectangle but without previous 2MASS matches. To also allow for a substantial error in the NLTT value for the amplitude of the proper motion, µNLTT, we set the inner and outer and outer radii of the

47 −1 annulus at θ± =(T2MASS − TUSNO)(µNLTT ± 100 mas yr ). Here T2MASS and TUSNO are the respective epochs of the 2MASS and USNO observations. Queries of the

2MASS database allow one to ask for the nearest USNO star within 5. We accept

2MASS stars only if they do not have a plausible USNO counterpart as determined from the consistency of the optical/IR colors of the two catalog objects relative to the predictions of equation (2.6) and their relative position oﬀset. We also check that the USNO rectangle star does not have a plausible 2MASS counterpart at small separation, which would also indicate that it is not a proper motion star.

Occasionally we ﬁnd a 2MASS counterpart to the USNO star near the outer edge of the 5 search radius which, given the ∼ 45 year baseline, could only be a real

− counterpart if the proper motion were µ ∼< 100 mas yr 1.Inthiscasewepermitthe identiﬁcation provided that the position angle is reasonably consistent with NLTT.

In total, we ﬁnd 50 stars in the annulus, or 0.2%, substantially smaller than the 1%

(18/1800) that were identiﬁed in the overlap with Tycho-2+Starnet. This may be because the PPM search was more aggressive than the one we have conducted here

(since it could make use of additional proper-motion information).

When no USNO/2MASS match is found either in the circle or the rectangle, and when there is no USNO candidate within the rectangle, we hypothesize that the NLTT position is correct to within the precision of the rectangle, but that the star is absent from USNO. We therefore predict the position of the star at

48 the 2MASS epoch assuming that the NLTT 1950 position and proper motion are correct. We then search for 2MASS counterparts of the NLTT star in a radius of 12 around this position to allow for NLTT PPM errors. As in the annulus search, we reject 2MASS stars that have plausible USNO counterparts. Of course, the result of this search yields only a position and IR photometry, but not a new proper motion. In all, this search yields 862 2MASS-only identiﬁcations.

When a valid USNO counterpart (one without an associated 2MASS star having consistent optical/IR colors and lying within a few arcsec) of the NLTT star is found within the rectangle, but it is not matched with a 2MASS star either in the original search or the annulus search, we assume that the star is absent from 2MASS and accept the USNO/NLTT identiﬁcation, provided that the USNO/NLTT magnitudes are consistent. In these cases, we obtain a new position and new optical photometry, but no new proper motion. In all, this search yields 409 USNO-only identiﬁcations. Based on the USNO reduced proper motion diagram we conclude that most of these stars are WDs.

The notes to NLTT contain references to more than 2400 pairs that Luyten considered to be CPM binaries or possible binaries. Unfortunately, it is not straightforward to extract all of this information in a completely automated way.

The biggest single problem is that, for some reason, Luyten did not name a large numbers of the objects in his catalog. In these cases, he will have a note to one

49 star such as “comp. to prec. star”. Sometimes the companion will be the preceding star, but very often it will be several preceding, or occasionally following. Often, when there is a reference to a named star, the meaning of the reference is clear only from context. In brief, it seemed impossible to make sense of these references in an automated way. Fortunately, for stellar pairs that Luyten believed to be CPM binaries, he almost always recorded the same proper motion for both components, even when (as we will show in § 2.3.5) he was able to measure the diﬀerence. This permitted us to adopt a diﬀerent approach.

First, we create our own naming system: each NLTT star is named for its sequential number in the electronic version of the catalog, from 1 to 58,845. Next, we run a program to find all notes that might plausibly be interpreted as indicating that the star is a component of a binary or multiple system. Normally, these contain a separation and position angle relative to a primary, but sometimes they are more ambiguous. We then find all NLTT stars within one degree of the noted star that have identical NLTT proper motions (magnitude and position angle). If any of these stars had been identified in either the PPM or USNO/2MASS searches, we print out the catalog entries as well as all the notes that are associated with all of these stars. If two or more of these stars have been identified, we also print out the separations and position angles of all pairs. We then review this output, tracking down discrepancies when we have two or more identifications and our relative

50 astrometry is in conflict with NLTT’s, and we interactively query the 2MASS data base to search for additional matches when the companion is not already in our data base. After updating our catalog with these results, we run a second program that finds only those notes that it can parse well enough to determine a predicted separation and position angle. It assumes that this vector separation applies to all pairs of NLTT stars with the same proper motion and lying within 1 degree. If our catalog contains entry pairs that disagree with this separation by more than 20% or with the position angle by more than 20◦, or if we do not have an entry for one component but do have an entry with 2MASS identification for the other, it flags this system. All such flagged systems are reviewed and the 2MASS data base is searched for unmatched companions.

This search yields 137 CPM companion identiﬁcations, of which 59 are corrections to previously incorrect 2MASS identiﬁcations. Of these 59, 47 are components of binaries for which our USNO/2MASS search found the same star for both components. Our algorithm is prone to such errors because it treats each component separately, and we intended to sort out these discrepancies at the time we reviewed all binary matches. They therefore do not represent real mismatches.

However, the remaining 12 are genuine misidentiﬁcations: if these stars, which were treated by our algorithm as individuals, did not happen to belong to CPM binaries, we would never have recognized the error. Since, there are a total of

51 1235 stars in CPM binaries that were matched to USNO/2MASS, this represents a false-identification rate of about 1%. Alternatively, one might want to consider only those stars in CPM binaries with separations greater than 10 in order to probe a sample that is not severely affected by confusion with a nearby high-proper motion source. This would be a fairer proxy to the conditions that the algorithm faced for the majority of single stars. In this case, we get 6 misidentifications out of 784 stars. Or, for a more conservative 30, we get 4 misidentifications out of 442 stars.

That is, our estimated misidentiﬁcation rate is consistently 1%. This rate is about three times higher than the 0.3% rate we estimated from comparison with PPM stars. Most likely, this is because brighter stars are rarer, and hence the possibility that a random unrelated USNO/2MASS pair will mimic a given high-proper star is smaller. In any event, we consider a 1% misidentiﬁcation rate to be quite good.

Since the CPM search was conducted after the search for 2MASS-only stars, we can use it to check the reliability of the 2MASS-only identiﬁcations. There are

34 2MASS-only stars that lie in CPM binaries in addition to the 137 companion identifications discussed above. The relative positions of all 34 stars are consistent with the values in NLTT, and in this sense all 34 “pass the test”. However, of these 34, 26 are from 13 CPM pairs both components of which are identified only in 2MASS. Since we processed both stars from a pair simultaneously, we had knowledge of their CPM nature and offsets, and so these 26 do not constitute the a

52 fair test. The remaining 8 stars are companions of independently identiﬁed NLTT stars, and so are a good proxy for single stars. The statistical signiﬁcance of this test is modest because of small number statistics, but at least it has the right sign. One of the 8 is a companion to a USNO-only star, which does not have an independent proper motion. The remaining 7 are additional CPM companions each of whose proper motions can be inferred from that of its companion. This brings the total of such CPM companions to 144.

2.3.3. Positional Errors of Faint NLTT Stars

Here we use our improved measurements of NLTT stars to characterize the precision of its positions. Figure 2.4 shows the diﬀerences between the 1950 epoch positions of NLTT and those predicted based on the 2000 positions and proper motions in our catalog. It is restricted to faint (V>11) stars to maximize the potential contrast with Figures 2.2 and 2.1, which show the same quantities for bright NLTT stars found in Tycho-2. In fact, there is no qualitative diﬀerence between the bright and faint position errors. The majority of the residuals are roughly uniformly distributed in a rectangle whose dimensions are set by the decimal truncation of the NLTT entries. The “fuzziness” of the rectangle edges gives an estimate of the underlying measurement errors. As was true of the bright sample,thisisabout1. Figure 2.4b shows that the heavily populated rectangle

53 is surrounded by a diﬀuse halo extending out to at least 2, which was the limit of our faint search in most cases. Nevertheless, taken as a whole, the positions are extremely good. These good positions played a critical role in our ability to identify the vast majority of NLTT stars in USNO and 2MASS in an automatic fashion. Note that the claim by Bakos, Sahu, & N´emeth (2002) that the typical

LHS (subset of NLTT) position errors are larger than 10 is manifestly not true.

We further investigate the magnitude dependence of the NLTT position errors in Figure 2.5, which shows the fraction of NLTT stars whose actual position lies within a rectangle of dimensions (15 cos δ +2) × (6 +2) and centered on the

NLTT recorded position. The ﬁrst term in each dimension is the imprecision created by decimal truncation and the second allows for 1 measurement errors at each edge of the rectangle. The fraction remains mostly constant at about 80% over 19 magnitudes, except for a pronounced feature suddenly dipping to 45% at V = 10 and then gradually climbing back to 80% at V = 15. We speculate that the brighter stars V ≤ 9 were already well studied and that Luyten made use of literature data on these. For the remainder of the stars he made his own measurements from plates. Saturation badly aﬀects such measurements at V = 10, but gradually ameliorates at fainter mags.

Figure 2.6 shows the rectangle fraction as a function of declination. This

ﬁgure has a number of important features. Over most of the sky, −30◦ <δ<50◦,

54 the fraction is roughly 80%. As expected, it drops to essentially zero for δ<−45◦ where NLTT records positions only to 6 seconds of time and 1 of arc. Less expected, however, are the rapid dropoffs for δ<−30◦ and δ>50◦. The former dropoff has significant negative consequences for the prospects of extending our catalog to the south, δ<−33◦,.

2.3.4. Proper Motion Errors of NLTT Stars

Figure 2.7 shows the diﬀerences between proper motions in the α direction as given by NLTT and our values. The diagram for the δ direction looks extremely similar and is not shown. Binaries with separations closer than 10 are excluded from this plot because our measurements, at least, can be corrupted due to blending. The bracketing lines show the 1 σ scatter (with 3 σ outliers excluded).

At bright magnitudes, NLTT errors are extremely small, then rise typically to

20 mas yr−1. However, there is a curious “bump” of higher errors at V ∼ 10, which is reminiscent of the degradation in positions (see § 2.3.3) and which may have the same cause. At the very faintest mags, V ∼> 18, the errors deteriorate toward

30 mas yr−1.

The fraction of 3 σ outliers excluded from the ﬁt is fairly constant at ∼ 3%, which is much higher that would be characteristic Gaussian noise. These outliers may be due to transcription errors in NLTT, misidentiﬁcations of NLTT stars by

55 us, or possibly other causes. Whatever the cause, the reader should not apply

Gaussian norms to the interpretation of the tails of this error distribution.

2.3.5. Common Proper Motion Binaries

Figure 2.8 shows the diﬀerence in 339 separation vectors between binary components as given by NLTT versus our astrometry. The sample is restricted to separations ∆θ>10 to avoid problems in our astrometry due to confusion, and

∆θ<57 to avoid the regime in which NLTT errors induced by truncation of the position angle (given to integer degree) become larger than the errors induced by the truncation of the separation (given to integer arcsec). The inset shows that this relative astrometry is usually very good, better than 1.Anumberofthe large outliers are due to transcription errors in NLTT. This highly precise relative astrometry permitted us to reliably identify NLTT CPM components even when

USNO data were missing.

Our improved proper motions permit us to better determine the reality of the CPM binaries listed in the NLTT notes. Figure 2.9 shows the diﬀerences in component proper motions for 468 NLTT binaries with separations ∆θ>10

(again, to avoid problems with our astrometry due to confusion).

56 Of course, one does not expect the components of CPM binaries to have exactly the same proper motion. First they have orbital motion, which for face-on circular orbits induces a relative motion ∆µ =2π(M/D3∆θ)1/2,whereD is the distance, M is the total mass, ∆θ is the angular separation, with units of pc, M, arcsec, and years. Even if they had the same physical velocity, their proper motions would differ because the components are at different distances and because we see different components of motion projected on the sky. The order of these differences combined is ∆µ/µ ∼ ∆θ, in radians. We add these orbital and projection effects in quadrature and show them as error bars in Figure 2.9. We estimate distances using the brighter (but non-WD) component. We classify it as a SD or MS star according to position in the reduced proper motion diagram (see §2.5.2), and then assign it an absolute magnitude MV =2.7(V − J)+2.1orMV =2.09(V − J)+2.33 in the respective cases. In addition, there is a 6 mas yr−1 error (see § 2.4.2) in our relative proper-motion measurements that must be added in quadrature to the errors shown in the figure. We have not done this to avoid overwhelming the intrinsic scatter.

Also, we have not placed any error bars on the points lying below 12 mas yr−1 (i.e.,

2 σ) to avoid clutter and because in our view these CPM identiﬁcations can be accepted with good conﬁdence.

The ﬁgure shows that the overwhelming majority of NLTT CPM binaries with separations ∆θ<50 are real, but that close to half of those with ∆θ>100 are

57 unrelated optical pairs. In fact, a number of these with proper motion diﬀerences

∆µ>100 mas yr−1 could have been excluded by Luyten at the 2.5 σ level given his 20 mas yr−1 precision in each direction and for each component. However, he evidently decided to err on the side of not missing potential CPM binaries.

2.4. Revised NLTT Catalog

2.4.1. Format of the Revised NLTT

The Revised NLTT catalog contains information grouped in six sections, 1) summary, 2) NLTT, 3) source identiﬁcations, 4) USNO,

5) 2MASS, and 6) binaries. The catalog is available online from http://www.astronomy.ohio-state.edu/~gould/NLTT.

The summary section contains 11 entries: 1) the NLTT number (drawn consecutively from 1 to 58,845), 2) a letter code ‘A’, ‘B’, or ‘C’ if the NLTT “star” has been resolved into several sources, 3) α (2000, epoch and equinox), 4) δ (2000),

−1 5) µα,6)µδ,7)σ(µα), 8) σ(µδ) (all four in arcsec yr ), 9) V , 10) V − J, 11)

3-digit source code.

The three digits of the source code refer to the sources of the position, proper motion, and V photometry. 1 = Hipparcos, 2 = Tycho-2, 3 = Tycho Double Star

58 Catalog (TDSC, Fabricius et al. 2002), 4 = Starnet, 5 = USNO/2MASS, 6 =

NLTT, 7 = USNO (for position) or common proper motion companion (for proper motion). More speciﬁcally, “555” means 2MASS based position, USNO based V photometry, and USNO/2MASS based proper motion.

The (2000) position has been evolved forward from whatever epoch it was measured using the adopted proper motion. When the position source is Hipparcos,

Tycho-2, or TDSC, the position is given in degrees to 6 digits, otherwise to 5 digits.

For proper motions derived from a PPM catalog, the errors are adopted from that catalog. Proper motions from USNO/2MASS determinations have an estimated error of 5.5masyr−1 is accordance with the results from § 2.4.2. NLTT proper motions are 20 mas yr−1 as found in § 2.3.4. CPM binary companions (without other astrometry) are not assigned an error, and zeros are entered into the error

ﬁelds.

V refers to the Johnson V entry for Hipparcos, Tycho V for Tycho-2 and TDSC, and Guide-star catalog R for Starnet. The conversion from USNO photometry can be obtained from 3.6,

V = RUSNO +0.23 + 0.32(B − R)USNO. (2.7)

We remind the reader that for USNO-A1 photometry, we ﬁrst convert to USNO-A2 using equation (2.3) before applying equation (2.7). When NLTT photometry is

59 used, V is evaluated using equations (2.4) and (2.5) to convert to USNO mags, and then applying equation (2.7). In the rare cases for which NLTT photometry is employed and one of the two bands is not reported, a color of (B − R)NLTT =1 is assumed. No eﬀort has been made to “de-combine” photometry in the case of unresolved binaries. For example, if a binary is resolved into two stars in 2MASS, but is unresolved in USNO, then diﬀerent J band measurements will be reported for the two stars, but both with have the same, combined-light V photometry. The

V − J color reported in ﬁeld 10 will be the simple diﬀerence of these two numbers.

Similarly, if the NLTT “star” is resolved by TDSC but not 2MASS, then the V light will be partitioned between the two stars but not the J light. Finally, note that if no J band photometry is available (whether because of saturation, faintness, or the star being in an area outside the second incremental 2MASS release) V − J is given as −9.

When multiple sources of information are available, the priority for what is presented in the summary is as follows. Positions: 3,1,2,5,4,7,6; Proper Motions:

3,2,1,5,4,7,6; Photometry: 3,1,2,5,4,6.

Tycho-2 proper motions are given precedence over Hipparcos primarily because they better reﬂect the long-term motion when the stars are aﬀected by internal binary motions, but also because at faint magnitudes they are generally more precise. When Hipparcos proper motions are given, it is often because the

60 star is so faint that it does not show up in Tycho. In this case, the nominal

Hipparcos errors are often quite large and true errors can be even larger. We found a handful of cases by chance in which the Hipparcos proper motion was grossly in error and we removed the Hipparcos entry and substituted the USNO/2MASS value. However, we made no systematic eﬀort to identify bad Hipparcos proper motions.

Only stars for which we are providing additional information are recorded in the catalog. There are 36,020 entries for 35,662 NLTT stars including a total of

723 entries for 361 NLTT “stars” that have been resolved in TDSC.

The next 6 columns give information taken from NLTT, namely 12) α (2000),

13) δ (2000), 14) µα, 15) µδ, 16) BNLTT, 17) RNLTT. The coordinates and proper motions are precessed from the original 1950 equinox to 2000, and the position is updated to 2000 epoch using the NLTT proper motion.

The next 2 columns give source information. Column 18 is the Hipparcos number (0 if not in Hipparcos). Column 19 is the identiﬁer from TDSC, Tycho-2, or Starnet, whichever was used to determine the position in columns 3 and 4.

When the position comes from Hipparcos, 2MASS, or USNO, or NLTT, “null” is entered in this ﬁeld with one exception: when a Starnet measurement has been superseded by a 2MASS measurement, the Starnet identiﬁer has been retained

61 for ease of recovery of this source. It can easily be determined that the summary information comes from 2MASS because the ﬁrst digit in ﬁeld 11 will be a “5”.

The next six ﬁelds give USNO information: 20) Integer RA, 21) Integer DEC,

22) BA1 or A2, 23) RA1 or A2, 24) USNO Epoch, 25) 3-digit search-history code. The

Integer RA and DEC together serve as a unique USNO identiﬁer since that is the form RA and DEC are given in the original USNO-A1 and USNO-A2 releases.

They can also be converted into degree α and δ (at the USNO epoch) using the formulae: α = (Integer RA)/360000, δ = (Integer DEC)/360000 − 90. Regarding the 3-digit search history code, the ﬁrst digit tells which USNO catalog the entry is from: 1 = USNO-A1, 2 = USNO-A2. The second tells whether the USNO source was found in the rectangle (1) or the circle (2). The third tells whether it was a unique match (1), or had to be resolved by hand from among several possible matches (2). If there is no USNO information, all of these ﬁelds are set to zero.

The next six ﬁelds contain 2MASS information: 26) α, 27) δ (both at 2MASS

Epoch), 28) J, 29) H, 30) Ks, 31) 2MASS Epoch. If no 2MASS data are available, all ﬁelds are replaced by zeros. If there are 2MASS data, but not for a particular magnitude measurement, that value is replaced by −9.

The next six ﬁelds contain information about binarity: 31) binarity indicator,

32) NLTT number of binary companion, 33) NLTT estimated separation, 34)

62 NLTT estimated position angle, 35) our estimated separation, 36) our estimated position angle. Regarding the binarity indicator, 0 means NLTT does not regard this as a binary. Otherwise, it is a NLTT binary and the indicator is set according to whether the companion is (2) or is not (1) in our catalog. The NLTT estimates of the separation position angle come from the NLTT Notes. Our estimates come from the difference of the 2000 positions of the two stars. In cases for which the companion is not in our catalog, the fields with “our” separation and position angle are replaced by values found from the difference of the NLTT coordinates (i.e.,

fields 12 and 13). As discussed in § 2.3.5, the companion numbers are based on what we think is obviously what Luyten intended, rather than what was literally written down. However, no effort has been made to clean up any other transcription errors, even when these are equally obvious. No binary information is recorded in these fields about NLTT “stars” that were resolved by TDSC. Rather, the reader should recognize each of those binaries from the upper case letter appended to its

NLTT number (column 2).

2.4.2. Proper Motion Errors of the Revised NLTT

Once bright stars are identiﬁed with PPM catalog stars, they acquire the proper motions, and with them the proper motion errors, given in those catalogs.

These vary but are generally of order a few mas yr−1. Faint-star proper motions

63 are obtained by taking the diﬀerence of 2MASS and USNO positions. Again, these vary in quality but typically have errors of order 130 and 250 mas respectively.

Given the ∼ 45 year baseline, we expect proper motion errors of order 6 mas yr−1.

Nevertheless, one would like an independent experimental conﬁrmation of this estimate.

To obtain this, we compare in Figure 2.10 our USNO/2MASS proper motions with Tycho-2 proper motions for 1179 stars for which we have measurements from both. We exclude for this purpose binaries closer than 10 because, as we have emphasized several times, they can be corrupted by blending. The scatter, which is dominated by USNO/2MASS errors, is high at bright magnitudes and then plateaus at V ∼ 10 at about 6 mas yr−1 in each component. The poorer quality for bright stars is due to the problems that USNO astrometry has in dealing with saturated stars. Because we do not use USNO/2MASS proper motions for bright stars, our primary interest is in the asymptotic behavior of the error envelope toward faint magnitudes. Taking an average over the bins V>10 and taking account of the small contribution to the scatter due to Tycho-2 errors, we ﬁnd average errors of

−1 σµ =5.5masyr , (wide angle), (2.8)

64 in each direction. These error bars are calculated excluding 3 σ outliers, which constitute almost 5% of the points. Thus, as in the case of NLTT, these proper-motion errors have strong non-Gaussian tails.

USNO astrometry, which dominates the proper-motion error budget, is more accurate on small scales than large scales. On scales of several degrees it suﬀers from errors in the plate solutions while on smaller scales it is limited by centroiding errors. For some applications, notably studying the reality of binaries or their internal motions (see § 2.3.5), it is the small scale errors that are relevant.

To determine these narrow-angle errors, we plot in Figure 2.11 the diﬀerence in the proper motions of the two components of 52 SD binaries (based on the position in the reduced proper motion diagram, Figure 2.13) with separations

∆θ>10. We choose SDs because they are more distant, which in turn implies that these binaries will have wide physical separations (and so be relatively unaffected by internal motions) without being widely separated on the sky (and so prone to contamination by optical binaries – see Fig. 2.9). In a proper-motion selected sample, the mean distance is proportional to the mean transverse speed, which is of order five times larger for SDs than MS stars. The figure shows a tight clustering of points with two outliers at ∼> 10 σ, which are either not physical binaries or have extremely bad proper-motion measurements. After excluding these, we obtain a scatter in the two directions of 4.4masyr−1 and 4.2masyr−1.

65 Since each results from the combination of two proper motions measurements, we derive a narrow-angle proper-motion error of

−1 σµ =3.0masyr , (narrow angle). (2.9)

Hence, the RMS error in the magnitude of the proper motion diﬀerence of a binary is 6 mas yr−1, which was the value we adopted in analyzing Figure 2.9.

2.5. Classification of NLTT Stars

2.5.1. Classification with the Original NLTT

One of the principal motivations for undertaking a proper-motion survey is to construct a reduced proper motion (RPM) diagram, in which the RPM

(= m +5logµ), is plotted against color. Here m is the apparent magnitude and

µ is the proper motion measured in arcsec per year. The RPM serves as a rough proxy for the absolute magnitude M = m +5logπ +5,whereπ is the parallax measured in arcsec. Indeed, if all stars had the same transverse speed, the RPM diagram would be identical to a conventional color-magnitude diagram (CMD) up to a zero-point oﬀset.

66 Hence, one can hope to roughly classify stars using a RPM diagram, even when one has no parallax or spectroscopic information. In particular, subdwarfs

(SDs) should be especially easy to distinguish from main-sequence (MS) stars, since they are several magnitudes dimmer at the same color, and are generally moving several times faster. Each of these eﬀects tends to move SDs several magnitudes below the MS on a RPM diagram. White dwarfs (WDs) should also be easily distinguished from MS stars, since they are typically 10 magnitudes fainter at the same color, and are generally moving at similar velocities. Although WDs are closer to SDs on an RPM than they are to the MS, they still should be distinguishable.

The combination of NLTT photometry and astrometry permits the construction of a RPM diagram. See Figure 2.12. Unfortunately, this diagram is almost devoid of features that could be used to isolate individual populations.

There is a clump of stars oﬀ to the lower left that one might plausibly identify with WDs. However, there is no obvious separation between SDs and MS stars.

Moreover, it is common knowledge that reddish WDs are mixed in with the

SDs at intermediate colors, so that WDs must be spectroscopically culled from relatively large samples drawn from the lower reaches of this diagram (e.g., Liebert,

Dahn, Gresham, & Strittmatter 1979). There are three reasons for this mixing of populations. First, photographic B and R do not provide a very broad color baseline. Second, photographic photometry has intrinsically large errors. Third,

67 NLTT does not reach even the relatively limited precision that is in principle achievable with photography. Given the steepness of the color-magnitude relations of individual populations, the short color baseline and large errors combine to seriously smear out the diagram.

2.5.2. Classification with the Revised NLTT

In Figure 2.13 we show the RPM based on optical-infrared V − J color from the Revised NLTT. Note that the SDs and MS stars are clearly separated into two tracks, at least for V +5logµ ∼> 9. The WDs are also clearly separated from the other stars. Going towards the bright end, the SD track becomes vertical at

V +5logµ ∼ 10 and then turns to the right at V +5logµ ∼ 9. For a star having

−1 a transverse speed of v⊥ ∼ 250 km s , this latter value corresponds to MV ∼ 5, i.e., roughly a mag below the subdwarf turnoﬀ. In any event, the bright end is dominated by Hipparcos counterparts of NLTT stars (shown in yellow). Hipparcos stars generally have excellent parallaxes, so for them RPMs are superﬂuous. We therefore remove these stars from further consideration.

Using superior color-color plots of NLTT stars based on Sloan Digital Sky

Survey (SDSS, York et al. 2000) photometry, we have checked that the scheme shown in Figure 2.13 indeed properly classiﬁes NLTT stars that lie in the SDSS

Early Data Release (Stoughton et al. 2002) area.

68 Panels in Figure 2.14 show respectively where the stars classiﬁed as WDs,

SDs, and MS stars in Figure 2.13, lie in the original NLTT RPM. The points representing the WDs are enlarged for emphasis. Note that while the region to the lower left is indeed dominated by WDs, the red WDs are sprinkled among a much higher density of other faint red stars. From the distribution of SDs and MS stars it is clear that while the SDs do tend on average to be bluer and fainter than the MS stars, the two populations are severely mixed in the original NLTT RPM diagram.

2.5.3. Candidate Nearby White Dwarfs

Once stars are classified using the RPM diagram, it is possible to estimate their distances photometrically. Using this technique, in Table 2.4 we have tentatively identified 23 WD candidates that lie within 20 pc and are not listed in the online edition of the Gliese Catalog of Nearby Stars2. These WD candidates are also not listed in McCook & Sion (1999) catalog of spectroscopically confirmed

WDs. We determine photometric distances from J, V − J, and a CMR calibrated using Bergeron, Leggett, & Ruiz (2001) data. Stars with low values of ∆(V − J) lie close to the SD/WD discriminator line shown in Figure 2.13, and could therefore be extreme SDs–stars interesting in their own right. Spectroscopy will be ultimately required to conﬁrm the nature of these objects. If conﬁrmed, these WDs would

2http://www.ari.uni-heidelberg.de/aricns

69 represent a signiﬁcant addition to 109 WDs known to be closer than 20 pc (Holberg,

Oswalt, & Sion 2002). In fact, since for 10 of our candidate WDs we calculate d<13 pc, they could raise the local density derived by Holberg, Oswalt, & Sion

(2002) using a 13 pc sample (which they believe is complete) by as much as 20%

(but note that we do not cover the entire sky).

2.6. Completeness of NLTT

2.6.1. Completeness at Bright Magnitudes

We study the completeness of NLTT at the bright end by comparing it to

Hipparcos and Tycho-2. (Starnet cannot be used for this purpose because it contains spurious high proper-motion stars). That is, we count the fraction of

Hipparcos/Tycho-2 stars that were detected by NLTT as a function of various parameters. Before doing this, however, we ﬁrst ask the opposite question: what fraction of NLTT stars were detected by Hipparcos/Tycho-2 as a function of RNLTT

(i.e., NLTT’s proxy for “V ”)? The answer to this question, which is given by Figure

2.15, delineates the NLTT magnitude range to which our subsequent completeness tests apply.

70 The bold curve in Figure 2.15 shows the Hipparcos/Tycho-2 completeness as a function of RNLTT, i.e. it is the ratio of Hipparcos/Tycho-2 detections (shown by the upper of the two thin-line histograms) to NLTT detections (bold histogram).

This completeness falls to 50% at RNLTT =11.6, as a result of the Tycho-2 magnitude limit. Hence, our subsequent completeness tests apply approximately to

NLTT stars with V ∼< 11.5.

Figure 2.15 has several other features of note. First, there is a peak in

NLTT detections at RNLTT ∼ 8.7, which is then reproduced by the histogram of

Hipparcos/Tycho-2 matches, as well as that of the Hipparcos-only matches just below it. This turns out to be an artifact of systematic “bunching” of NLTT magnitudes: a histogram of Hipparcos/NLTT matches as a function of Hipparcos

V (not shown in the ﬁgure to avoid clutter) exhibits no such “premature” peak, but rather has a single, relatively broad peak at V ∼ 9.5.

Note that Hipparcos/Tycho-2 completeness is ∼ 100% only to about

RNLTT ∼ 8.5, falls to ∼ 95% at RNLTT ∼ 9.5, and then plummets rather sharply.

Given that Hipparcos/Tycho-2 is itself quite incomplete signiﬁcantly below

RNLTT =11.6, one might ask whether it can be used to reliably probe NLTT completeness all the way to this threshold. It can be so used if the reasons for NLTT non-detections are independent of the reasons for Tycho-2 non-detections. It is clear from the form of the completeness curve in Figure 2.15 that Hipparcos/Tycho-2

71 loses sensitivity with faintness. We will argue below that the NLTT non-detections are due to crowding, and not due to faintness, since NLTT goes much fainter than the limits of the modern catalogs. However, since crowding can exacerbate problems with detection of fainter objects, there could be some interplay between these two eﬀects. We will comment on the role of this interplay in § 2.6.2.

Figure 2.16 shows the fraction of Hipparcos/Tycho-2 recovered by NLTT as a function of proper motion µ (as measured by Hipparcos or Tycho-2). The solid

−1 vertical line is at µlim = 180 mas yr , the proper motion limit of NLTT. One expects completeness to fall by 50% at this point because half the stars that actually have this proper motion will scatter to lower values due to measurement error, and so will be excluded from the NLTT catalog. One therefore expects completeness to achieve its asymptotic value a few σ above this threshold. The two dashed vertical

−1 lines are at µlim ± 40 mas yr , which corresponds to ±2 σ for the better-precision

NLTT stars, and about ±1.5 σ for the others. The completeness curve does indeed reach an asymptotic value of ∼ 90% just beyond this point. Hence, at µ = µlim we expect the completeness to be 0.5 × 90% = 45%. The actual value is 42%. To avoid this threshold eﬀect, we will restrict future completeness tests to stars with

“true” (i.e., Hipparcos/Tycho-2) proper motions µ>250 mas yr−1.

72 2.6.2. Bright-end Completeness as a Function of

Galactic Coordinates

Figure 2.17 (bold curve) shows the fraction of Hipparcos/Tycho-2 proper- motion (µ>250 mas yr−1) stars recovered by NLTT as a function of sin b,whereb is

Galactic latitude. NLTT is virtually 100% complete away from the Galactic plane, but its completeness falls to about 75% close to the plane. This incompleteness is not symmetric: it is somewhat worse in the south. We will discuss the reasons for this in the following paragraph. The histogram shows the underlying distribution of proper-motion stars, from Hipparcos/Tycho-2, as a function of sin b. If these stars were distributed uniformly over the sky, then this histogram would be a horizontal line. One expects halo stars to be over-represented near the poles because the reflex of the Sun’s motion is most pronounced in those directions. One also expects disk stars to be over-represented near the plane because their density does not fall off with distance in these directions. Plausibly, one can see both effects in the

Hipparcos/Tycho-2 histogram. We will return to this conjecture in §2.6.4.

Figure 2.18 (bold curve) shows the fraction of Hipparcos/Tycho-2 proper- motion stars lying in the Galactic plane (|b| < 15◦) that are recovered by NLTT as a function of Galactic longitude. While the curve is somewhat noisy, there is

◦ ◦ a clear increase in incompleteness over the interval −80 ∼<∼< 20 . Thisisthe

73 brightest contiguous region of the Milky Way, which lends credence to the idea that NLTT incompleteness is traceable to crowding-induced confusion. The areas just south of the Galactic equator are on average brighter than the corresponding areas just to the north, so the asymmetric behavior seen in Figure 2.17 also lends credence to this hypothesis.

In § 2.6.1, we entertained the possibility that detection failures in NLTT and

Hipparcos/Tycho-2 might be correlated which, if it were the case, would undermine the completeness estimates obtained from the fraction of Hipparcos/Tycho-2 stars recovered in NLTT. Figure 2.18 shows that this eﬀect cannot be very strong, if it exists at all.

The expected number of high proper-motion stars as a function of Galactic longitude need not be uniform, and will in general depend on the model of the

Galaxy. However, in any plausible model, the number should be the same looking in directions separated by 180◦ because NLTT stars are not at suﬃciently large distances to probe the Galactic density gradients. Consequently, the distribution is a result of bulk kinematic eﬀects, which should be identical in antipodal directions.

Hence, if there were a correlation, one would expect that pairs of antipodal points with a positive diﬀerence in NLTT completeness would also have a positive diﬀerence in Hipparcos/Tycho-2 counts. No such pattern is seen in Figure 2.18.

74 In any event, the primary implication of Figure 2.17, namely that NLTT is essentially 100% complete away from the Galactic plane remains true, independent of these more subtle considerations.

2.6.3. Completeness at Faint Magnitudes

Here we perform a statistical test to investigate the completeness of the faint end of NLTT down to its nominal cutoﬀ of µ = 200 mas yr−1, and additionally to

µ = 500 mas yr−1, which is the cutoﬀ of LHS. It should be noted that the data in

LHS is a subset of data present in NLTT. In this test we assume that the local luminosity function is constant, and that the number density of stars does not change appreciably within the volume occupied by the majority of proper motion stars. We will discuss the validity of this assumption later on.

Consider two spheres centered around the Sun, the volumes of which stand in ratio 2:1. This is equivalent to radii being in relation r1/r2 =1.259, or distance modulus diﬀerence of 0.5 mag. If we deﬁne the outer edge of the bigger sphere as the distance at which a star of apparent magnitude RL,1 produces a proper motion

−1 µ1 = 200 mas yr , then this same star, if placed at distance r2,wouldhaveaproper

r1 −1 motion of µ2 = µ1 = 252 mas yr . Also, it would be 0.5 mag brighter. Therefore, r2

µ2 deﬁnes a proper motion limit at the distance r2 that is equivalent to proper motion limit µ1 at r1. These are the lower limits. For the upper proper-motion

75 lim −1 limit we adopt µ2 = 2500 mas yr , below which we know the sky was searched homogeneously. NLTT does contain stars with µ>2500 mas yr−1, possibly all that exist, but these were found by methods other than automated plate scanning.

Anyway, because of the small relative number of these stars, our statistical test is not very sensitive to the choice of upper proper-motion limit. So, if we take

lim −1 µ2 = 2500 mas yr as a limit below which we want to check for completeness, than this corresponds to some inner boundary of the smaller sphere (which we can

lim now call a shell). Everything closer than this inner boundary would have µ>µ2 and would not be included in NLTT. Now, in order to keep volumes of both shells in appropriate ratio, the outer shell has to have an inner edge corresponding to

r apropermotionofµlim = 2 µlim = 1986 mas yr−1. Thistestisperformedinthe 1 r1 2 same way when investigating completeness of LHS subset of NLTT, but with

−1 µ1 = 500 mas yr , and corresponding µ2.

Now that we have deﬁned the two shells in terms of the limiting proper motions, the statistical test consists of comparing the number of stars N1 of a given magnitude RNLTT (in a ∆RNLTT =0.5 mag bin) in the outer shell

−1 (µ1 <µ<1986 mas yr ), with the number of stars N2 of a magnitude

RNLTT − (RNLTT,1 − RNLTT,2)=RNLTT − ∆RNLTT = RNLTT − 0.5 in the inner shell

−1 (µ2 <µ<2500 mas yr ). The 0.5 mag shift (equal to one bin) brings the absolute

76 magnitudes of stars in the outer shell to that of the inner shell. The measure of completeness at magnitude RNLTT is given by the ratio

N1(RNLTT) f(RNLTT)= . (2.10) N2(RNLTT − 0.5)

If the sample of stars of apparent magnitude RNLTT is 100% complete with

3 respect to those of RNLTT − 0.5, then f(RNLTT) ≡ (r1/r2) = 2. Now we can deﬁne the completeness function F (RNLTT) for the stars of apparent magnitude RNLTT,in the following way

R R NLTT= NLTT f(RNLTT) F (RNLTT)= , (2.11) R R R 2 NLTT= NLTT,comp+∆ NLTT

where RNLTT,comp is some bright apparent magnitude at which we believe the catalogue is complete.

In Figure 2.19 we show the completeness function F (RNLTT) for the faint end of NLTT (dotted line) and LHS (solid line). More speciﬁcally, the test was performed on the subsample of NLTT that is believed to be spatially complete, that is, the part called the Completed Palomar Region (CPR) by Dawson (1986).

◦ This region covers northern declinations (δ ∼> −33 ), and avoids the galactic

◦ plane (| b |> 10 ). We take RNLTT,comp = 13. The choice is somewhat arbitrary, but we have reasons to believe that NLTT is complete at this magnitude. First,

77 when we plot f(RNLTT) against RNLTT, we get a ﬂat region around RL = 13.

Going to still brighter magnitudes might bring us into the part of NLTT that was not compiled from the photographic plates. Therefore, Figure 2.19 shows the completeness at RNLTT with respect to RL = 13. Dashed lines represent 100%, 75% and 50% completeness levels. The completeness of NLTT drops gradually from

90% at RNLTT =13.5 to 60% at RNLTT =18.5. Although one would not a priori expect the completeness to be the function of proper motion, for the stars with

µ>500 mas yr−1, i.e., those that are present in LHS, the completeness is much higher - in fact, it seems to be ∼ 100% complete to RNLTT = 18, and becomes incomplete at one magnitude fainter. (The solid line is much less smooth than the dotted one, because of the smaller number of stars that produced it.) The reason for diﬀering completeness of lower and higher proper motion stars might have to do with possibly better detection techniques used in LHS part of the catalog.

As mentioned, this test depends on number density being roughly constant in the volume investigated. Is this volume small enough for this condition to be valid, i.e. how far above the plane do we get? In a proper motion selected catalog the mean transverse velocity of stars is couple of times higher than the transverse

−1 velocity of the population itself, i.e. vt ∼ 90 km s . This means that the stars moving at µ>200 mas yr−1 will all be closer than 95 pc. Such stars will have disk scale height greater than that of normal population, so the distance of 95 pc is not

78 significant compared to that scale height. As for stars that move more slowly, they have to be placed even closer to make their way into NLTT, which means that they will be affected even less. We tested this by comparing completeness of the subsamples of NLTT and LHS from low (10◦ <| b |≤ 36◦) and high (| b |> 36◦) galactic latitudes. We see no significant difference.

2.6.4. Completeness of Different Star Populations as a

Function of Galactic Latitude

Our revised catalog recovers 95–97% of NLTT stars in areas that it covers.

This allows investigation of relative completeness of various types of stars (as determined using RPM diagram) as a function of Galactic latitude.

We showed in §2.6.1 by direct comparison with Hipparcos and Tycho-2, that

◦ at bright (V ∼< 11) magnitudes NLTT is close to 100% complete for |b| ∼> 15 , but that its completeness falls to 75% close to the plane, even for these bright stars.

For the faint stars we do not have an independent compilation of proper motions, so we cannot directly establish the absolute completeness of the catalog. (We will carry out an indirect absolute measurement in forthcoming work.) However, by making use of both the RPM classiﬁcation the much greater dynamic range of the

79 full catalog presented here, we can give a much more detailed picture of the relative completeness as a function of various variables.

Figure 2.20 shows the number of WDs, SDs, and MS stars per square degree as a function of Galactic latitude, for three diﬀerent magnitude ranges. To determine the surface density (stars deg−2), we calculate the fraction of each latitude bin that is covered by the 2MASS release and that lies to the north of our cutoﬀ in declination. (To allow for the somewhat ragged boundary of USNO-A1/POSS

I coverage near δ ∼ 33◦, we set this cutoﬀ at δ>−32.◦4.) However, we do not attempt to compensate for stars that fail to appear in 2MASS due either to saturation, faintness, or other causes. For 10 ∼

The results shown in Figure 2.20 are quite unexpected. One often hears of the severe incompleteness of NLTT close to the Galactic plane, but the real story

80 is more complex: NLTT is substantially less complete in the plane, but only for

MS stars. By contrast, NLTT coverage of SDs is near uniform over the sky and its coverage of WDs appears to be completely uniform. For MS stars, there is an evident dropoﬀ in counts over the interval −0.2 < sin b<0.3 in all three magnitude ranges. It is not very pronounced for bright stars, but one already knows that

NLTT is more than 85% complete averaged over this range. However, it is quite pronounced in the other two ranges, falling by a factor ∼ 10 over only ∼ 15◦ in each case. One also notices a more gradual decline in the MS density going from high to low latitudes. For example, in the faintest bin, it falls by a factor ∼ 2 between the poles and b ∼±15◦. A priori, one does not know if this is due to a genuine change in density or to an extension of the obvious incompleteness near the plane to higher latitudes. In our further analysis, we will adopt the former explanation for two reasons. First, one does expect a general trend of this sort because a large fraction of the fainter MS stars entering a proper-motion limited sample are from the old disk. These have on average substantially higher transverse speeds seen towards the poles than in the plane because of asymmetric drift, and are therefore selected over a larger volume. Second, as we now discuss, this trend is also seen in

SDs where it is expected on similar grounds, but this time without the problem of

“contamination” from a slower population.

81 There are too few SDs in the bright bin to draw any meaningful conclusion. In the middle bin 11

ﬂat, in very striking contrast to the MS distribution in the same magnitude range.

In particular, near the plane where the MS density suﬀers a factor 10 decline, the

SD distribution is virtually ﬂat and if it declines, does so by at most a few tens of percent. The fact that the MS counts gradually rise toward the poles while the

SD counts do not is easily explained. For a population of characteristic transverse speed v⊥, a survey will saturate its proper-motion limit µlim only for stars with MV brighter than,

v⊥ MV

−1 For SDs with v⊥ ∼ 300 km s surveyed at V = 15 to the NLTT limit

−1 µlim = 180 mas yr , this corresponds to MV < 7.3. There are very few subdwarfs at these bright magnitudes that could “take advantage” of the higher transverse velocities seen toward the poles. On the other hand, for typical MS speeds,

−1 v⊥ ∼ 50 km s , the limit is MV < 11.1. Since this is close to the peak of the

MS luminosity function (LF), a large fraction of MS stars are seen more readily toward the poles than the plane. The same eﬀect explains the patterns seen in the faintest bin. The limit imposed by equation (2.12) for subdwarfs at this magnitude is MV < 10.3, which includes a large fraction of the SD LF. Similarly, we expect

82 the eﬀect to be even stronger for MS stars in the faintest bin than the middle bin, because now the whole peak of the MS LF is included. On the other hand, in the brightest bin, the limit for MS stars is MV < 7.1 which includes a very small fraction of the MS LF. We therefore expect the slope to be small, and it is.

There are signiﬁcant numbers of WDs only in the faintest bin. Since WDs in

NLTT mostly have the same kinematics as the MS, the limit imposed by equation

(2.12) is MV < 14.1. Many WDs satisfy this constraint, so we expect to ﬁnd more

WDs near the poles, which is actually the case. Note that in the faintest bin, neither the WDs nor the SDs show any signiﬁcant tendency to drop oﬀ close to the plane.

Thus, a consistent picture emerges from Figure 2.20: MS completeness is very severely aﬀected by proximity to the plane but SD and WD completeness are barely aﬀected at all. The most plausible explanation for this is that while Galactic-plane

ﬁelds are extremely crowded and therefore in general subject to confusion, blue stars are no more common in the plane than anywhere else. Hence, by focusing on blue objects in the plane, Luyten was able to recover most high-proper motion SDs and WDs, even while he lost the overwhelming majority (∼ 90%) of the MS stars.

83 2.7. Discussion and Conclusion

Once the full 2MASS catalog is released, it will be relatively straightforward to extend our catalog to the other 33% of the sky that lies north of δ = −33◦.

However, the prospects for extending it to the south are less promising. First, as we discussed in § 2.3.1, the great majority of NLTT stars are missing from

USNO-A. In the north, even when a NLTT star was missing from USNO-A, we were frequently able to recover it by looking for the 2MASS counterpart at the position predicted by assuming that the star was inside the NLTT rectangle and that the NLTT proper motion was approximately correct. See § 2.3.2. However, the reason that this method was eﬀective is that over most of the northern sky, the star does actually lie in the rectangle about 80% of the time. From Figure

2.6, one sees that progressively fewer stars are in the rectangle in the south, and virtually none are for δ<−45◦. Moreover, a 2MASS-only identiﬁcation yields a position and infrared photometry, which are useful for many applications, but not a proper motion. At present only 13% of the area δ<−33◦ is covered by the

2MASS release, but the above factors lead us to believe that after full release, our technique will not be very eﬀective in this region. In any event, NLTT is generally limited to V ∼< 15 in the South and the brighter stars among these are already covered by our PPM search.

84 The catalog presented here gives improved astrometry and photometry for the great majority of stars in the NLTT that lie in the overlap of the areas covered by the second incremental 2MASS release and those covered by POSS I (basically

δ>−33◦). In addition, essentially all bright NLTT stars over the whole sky have been located in PPM catalogs and, whenever possible, the close binaries among them have been resolved using TDSC. We recover essentially 100% of NLTT stars

V<10, about 97% for 10

The new positions are accurate to 130 mas. The new proper motions are accurate to 5.5masyr−1, more than a 3-fold improvement over NLTT. Narrow angle proper motions are accurate to 3 mas yr−1. The catalog provides a powerful powerful means to investigate SDs, WDs, faint MS stars as well as CPM binaries.

85 Table 2.1. Bright NLTT stars position precisions

ﬁt σ1 σ2 cqNN1

RA 1.17.0 0.03s 0.774 5022 3887 DEC 1.15.80.15 0.735 5495 4040

86 Table 2.2. Bright NLTT stars proper motion precisions

Catalog Rectangle Position Number σ(|µ|) σ(µα) σ(µδ) quality mas yr−1 mas yr−1 mas yr−1

Hipparcos In Better 5261 20 22 25 Out Better 1623 31 33 32 All Worse 1284 24 28 34 Tycho-2 In Better 1470 25 24 24 Out Better 1005 32 35 35 All Worse 438 28 32 42 Starnet In Better 738 25 27 24 Out Better 435 44 55 65 All Worse 195 42 57 45

87 Table 2.3. Number of USNO/2MASS matches

rectangles circles total unique resolved unique resolved

USNO-A1 2426 78 517 174 3195 USNO-A2 15404 534 3574 974 20486 total 17830 612 4091 1148 23681

88 Table 2.4. New candidate white dwarfs closer than 20 pc

NLTT R.A. Decl. VB− VV− J ∆(V − J) d (pc) Notes

21351 09 16 35.85 +19 55 17.5 12.8 1.5 0.58 0.0 6.6 a 52890 22 05 33.03 +19 51 24.7 13.4 1.0 0.78 0.1 7.3 b 44850 17 29 17.38 −30 48 36.8 11.8 ··· 0.02 0.4 7.5 c 24703 10 34 02.98 +12 10 15.1 13.4 1.1 0.70 0.0 7.7 b 47807 19 27 21.31 −31 53 06.1 13.2 0.9 0.51 0.1 8.4 57038 23 29 43.45 −09 29 55.8 12.6 0.9 0.13 0.2 9.5 b 89 26398 11 08 14.50 −14 02 47.1 14.2 0.8 0.73 0.3 11.0 b 32147 12 52 10.43 +18 33 09.4 14.1 1.3 0.63 0.3 11.7 4828 01 26 48.89 −26 33 55.7 14.6 0.9 0.89 0.2 11.8 15314 05 34 20.18 −32 28 51.3 11.6 0.8 −0.41 1.0 11.8 b 11551 03 41 21.11 +42 52 33.9 15.8 1.2 1.40 0.1 13.9 b 56805 23 25 19.86 +14 03 39.4 15.8 1.2 1.37 0.1 14.2 8435 02 35 21.79 −24 00 47.0 15.8 0.6 1.32 0.5 14.3 d 23235 10 01 40.00 −24 04 41.2 15.8 1.5 1.18 0.3 15.7 e 19138 08 14 11.15 +48 45 29.8 15.1 0.8 0.76 0.5 16.1 47373 19 04 50.62 −03 43 07.0 14.6 1.9 0.49 0.8 16.8 46292 18 20 02.58 −27 45 50.5 14.6 0.4 0.46 0.8 17.3 (continued) Table 2.4—Continued

NLTT R.A. Decl. VB− VV− J ∆(V − J) d (pc) Notes

8581 02 39 19.68 +26 09 57.5 16.3 0.8 1.40 0.2 17.4 34826 13 40 00.80 +13 46 51.9 16.3 1.9 1.36 0.1 17.8 529 00 11 22.45 +42 40 40.8 15.2 0.5 0.69 0.5 18.4 f 53177 22 12 17.96 −14 29 46.1 15.0 0.5 0.51 0.8 19.8 g 31748 12 44 52.66 −10 51 08.9 14.4 0.5 0.24 1.1 19.9 h 40607 15 35 05.81 +12 47 45.2 15.9 0.8 0.95 0.4 20.0 i 90

Note. — NLTT numbers follow the record number in the electronic version of NLTT (ADC/CDS catalog I/98A). Positions are given for epoch and equinox 2000. V and B − V are calibrated from USNO photographic magnitudes, except for NLTT 44850 (GSC red magnitude). ∆(V − J) gives horizontal separation from the SD/WD discriminator line. Distances are photometric, based on Bergeron, Leggett, & Ruiz (2001) data. aUSNO photometry blended with NLTT 21352.

bNLTT magnitudes fainter.

c2MASS photometry possibly aﬀected by crowding.

dLHS 1421. eUSNO photometry blended with NLTT 23234.

f Giclas, Burnham, & Thomas (1965) : Suspected WD (GD 5).

gBeers et al. (1992) : V =15.09, B − V =0.23, ‘composite’ spectrum.

hGreen, Schmidt, & Liebert (1986) : Spectral type sdB (PG 1242–106).

iEggen (1968) : V =15.07, B − V =0.24 (WD 1532+12). 10

0 DEC (arcsec) ∆ −5

−10

−.5 0 .5 ∆RA (seconds)

Fig. 2.1.— Diﬀerences between stellar positions as reported in NLTT and the very accurate positions of the same stars from Tycho-2, both evaluated in the NLTT epoch of 1950. The (1 s × 6) rectangle that dominates this plot is caused by the fact that Luyten measured a majority of his stellar positions to a precision of 1, but reported them only to 1 second of time and 6 of arc.

91 100

0 DEC (arcsec) ∆

−100

−100 0 100 ∆RA (arcsec)

Fig. 2.2.— Same as Fig. 2.1 except that ﬁrst, RA is now plotted in arcsec rather than in seconds of time, and second, the scale is much larger. Although most NLTT positions are quite accurate (see Fig. 2.1), there is a substantial “halo” of outliers, some that go well beyond the dimensions of this plot.

92 2 USNO 0 − R NLTT R

−2

12 14 16 18

2 USNO

0 − (B−R) NLTT

(B−R) −2

12 14 16 18

RNLTT

Fig. 2.3.— Differences between photographic photometry from NLTT and USNO for stars in our catalog. Upper panel shows the difference in R magnitude and bottom panel shows the difference in (B − R) color. The bold curves give the mean difference and the solid lines give the 1 σ scatter (with 3 σ outliers removed from the fit) in 1 mag bins. This scatter ranges from 0.4 to 0.5 mag over most of the plot and is dominated by errors in NLTT. Note that errors in B and R are highly correlated in both NLTT and USNO.

93 10

0 DEC (arcsec) ∆ −5

−10

−.5 0 .5 ∆RA (time) 200

100

0 DEC (arcsec) ∆ −100

−200 −200 −100 0 100 200 ∆RA (arcsec)

Fig. 2.4.— Diﬀerences between NLTT listed position and true position (as determined by propagating the star back to its 1950 position using the 2000 position and proper motion from our catalog) for all faint, V>11, stars. The rectangle in the upper panel arises because, for most of his catalog, Luyten actually measured positions to about 1 but only recorded them to 1 s of time and 6 of arc. However, the lower panel shows that there is a substantial halo of outliers at least out to our search radius of 2. Compare to Figs. 2.2 and 2.1.

94 1

.4 Fraction in Rectangle .2

0 0 5 10 15 20 V

Fig. 2.5.— Fraction of NLTT stars found in a rectangle centered on the NLTT position and with dimensions (15 cos δ+2)×(6+2) as a function of V magnitude. The size is set to allow for 1 errors in addition to error caused by decimal truncation of the catalog entries. The feature at V ∼ 10 is probably caused by diﬃculty doing astrometry for stars saturated on plates.

95 −60 −45 −30 −15 0 15 30 45 60 1

.4 Fraction in Rectangle .2

0 −1 −.5 0 .5 1 sin δ

Fig. 2.6.— Fraction of NLTT stars found in a rectangle centered on the NLTT position and with dimensions (15 cos δ+2)×(6+2) as a function of of declination. The fraction is very high over most of our catalog area, δ>−33◦, but deteriorates drastically to the south.

96 100

50 (Catalog) α µ 0 (NLTT) − α

µ −50

−100 5 101520 V

Fig. 2.7.— Differences between proper motions in the α direction as given by NLTT and our catalog. The bold line shows the mean difference, which is consistent with 0. The solid lines indicate the 1 σ scatter (with 3σ outliers removed from the fit) in 0.5 mag bins. The NLTT errors are typically 20 mas yr−1 in the range 11

97 20

0 2

Tangential (arcsec) 0 ∆

−20 −1 −2 −2 −1 0 1 2

−20 0 20 ∆ Radial (arcsec)

Fig. 2.8.— Diﬀerence between separation vectors of 339 binaries with separations 10 < ∆θ<57 as given by NLTT and our catalog. The x-coordinateisalongthe direction of the separation and the y-coordinate is perpendicular to it. The great majority of NLTT separation vectors are accurate to 1.Seeinset.

98 200

150

100 (mas/yr) tot ∆µ 50

0 1 1.5 2 2.5 3 log(∆θ/")

Fig. 2.9.— Magnitude of the difference in the relative vector proper motion of binary members as given in our catalog. Error bars reflect the effects of internal binary motion at small separations, ∆θ, and projection effects at large separations. They do not include our measurement error of 6 mas yr−1.Pointswith∆µ<12 mas yr−1 are likely to be genuine pairs and do not have error bars to avoid clutter. At large separations, many NLTT “CPM” binaries are actually optical pairs.

99 ieec sdet rosi h SO2ASmaueet hc r thereby are the which measurement, of our USNO/2MASS Most 5 the and 2MASS. be in Tycho-2 and to errors evaluated USNO by to in given due stars as is the difference motions identifying proper on between based measurements Differences 2.10.— Fig. cnitn ih0 n h oi uvsso h 1 the show curves solid the and 0) with (consistent mte rmtefi)i . a is rosaemr eeea rgtmagnitudes bright at severe more astrometry. are USNO Errors affects bins. adversely mag saturation 0.5 because in fit) the from omitted . 5masyr µ µ µ µ δ,U2M − δ,Tyc2 (mas/yr) α,U2M − α,Tyc2 (mas/yr) −40 −20 −40 −20 20 40 20 40 0 0 − 1 for 1011129 1011129 V> 0 h odcresostema difference mean the shows curve bold The 10. 100 V σ cte wt 3 (with scatter σ outliers 100

0 (mas/yr) δ ∆µ

−50

−100 −100 −50 0 50 100 ∆µ α (mas/yr)

Fig. 2.11.— Diﬀerence in the vector relative proper motion of the components of 52 subdwarf binaries with separations ∆θ>10. For physical pairs the real relative proper motion is very close to 0, so these diﬀerences provide an estimate of our narrow-angle proper-motion errors, 3 mas yr−1 (with the two outliers omitted).

101 5 ) µ

10 + 5 log( L R

−101234

(B−R)L

Fig. 2.12.— Original reduced proper motion (RPM) diagram for NLTT stars. Both the proper motions, µ, and the photographic magnitudes, BNLTT and RNLTT,are taken from NLTT. However, since the magnitudes are originally given only to one decimal place, the color has been randomized by 0.1 mag to show the density of points. The apparent stripes result from discretization of color in the original NLTT data.

102 5 ) µ

10 V + 5 log(

0246 V−J

Fig. 2.13.— Optical-infrared RPM diagram for NLTT stars having 2MASS and Hipparcos counterparts (yellow), as identified in 2.2, or 2MASS and USNO-A counterparts (black), as identified in 2.3. For RPM V +5logµ ∼> 9, subdwarfs (SDs) and main-sequence (MS) stars clearly lie on different tracks. White dwarfs (WDs) are also clearly separated from SDs and MS stars. In the subsequent figure, the NLTT stars with 2MASS/USNO-A counterparts will be identified as MS (red), SD (green), or WD (cyan), depending on whether they lie to the right, between, or to the left of the two dashed lines shown in this figure.

103 5 5 5 ) µ

10 10 10 + 5 log( L R

15 15 15

−101234 −101234 −101234

(B−R)L (B−R)L (B−R)L

Fig. 2.14.— Left panel: Original NLTT RPM diagram (see Fig. 2.12) but with the stars identiﬁed in Fig. 2.13 as WDs shown as large cyan dots. While the region to the lower left is strongly dominated by WDs, many WDs lie further to the red where they are heavily contaminated by other stars. SDs are shown as small green dots in the middle panel, and MS stars are shown as small red dots in right panel. Comparison of SD and MS distribution shows that they strongly overlap in NLTT RPM. In all cases the other types of stars with 2MASS/USNO-A counterparts are shown as smaller black dots.

104 1.2

.8 Completeness .6

.4 NLTT Hip+T2 .2 Hipparcos Only 0 4 6 8 10 12

RNLTT

Fig. 2.15.— The completeness of the combined Hipparcos and Tycho-2 catalogs as a function of RNLTT (roughly Johnson V ) magnitude, measured from the fraction of NLTT stars (bold histogram) that are matched to one of these two catalogs (upper thin-line histogram). Also shown are the Hipparcos-only matches. The

“bump” in NLTT detections at RNLTT ∼ 8.7 is an artifact of NLTT mags. See text. Completeness falls to 50% at RNLTT =11.6. Hence, the subsequent tests on completeness of NLTT apply directly to its brighter stars, V ∼< 11.5. (All histograms are divided by 1000. The bin size is 0.1 mag.)

105 1.2

Completeness .4

0 .1 1 10 log µ (arcsec/yr)

Fig. 2.16.— Completeness of NLTT (i.e., the fraction of Hipparcos/Tycho-2 stars recovered by NLTT) as a function of Hipparcos/Tycho-2 proper motion, µ.The −1 solid vertical line shows the proper-motion limit of NLTT, µlim = 180 mas yr ,and −1 the two dashed lines show µlim ± 40 mas yr , i.e., roughly the 1.5 to 2 σ errors in NLTT. The eﬀect of this proper-motion threshold disappears by µ ∼ 250 mas yr−1. Hence, subsequent completeness tests will be restricted to stars moving faster than this value.

106 1.2

Completeness .8

.4 Hipparcos/Tycho−2 Counts

0 −1 −.5 0 .5 1 sin(b)

Fig. 2.17.— Completeness of NLTT (bold curve) as a function of sin b where b is Galactic latitude, i.e., the fraction of stars with µ>250 mas yr−1 in Hipparcos and Tycho-2 (whose distribution is shown by the histogram) that are recovered in NLTT. (The histogram has been divided by 200. The bin size is 0.04) Incompleteness is signiﬁcant only close to the plane, where it is somewhat skewed toward the south.

107 1.2

.8 Completeness

.6 Hipparcos/Tycho−2 Counts .4

0 0 100 200 300 Galactic Longitude

Fig. 2.18.— Completeness of NLTT (bold curve) as a function of Galactic longitude for the subset of stars lying close to the plane (|b| < 15◦). Also shown is a histogram (counts divided by 200, 20◦ bins) of Hipparcos and Tycho-2 stars with µ>250 mas yr−1 and |b| < 15◦.

108 Fig. 2.19.— Completeness relative to RNLTT = 13, in the Completed Palomar ◦ ◦ Region (δ ∼> −33 , | b |> 10 ). Solid line is the completeness of the LHS subset of NLTT, while the dotted one is the completeness of the entire NLTT. Dashed lines show 100%,75% and 50% completeness levels.

109 ftesbwrsi h i i n h ansqec tr ntebih i is bin bright the in stars sequence main the arguments. kinematic and simple bin from behavior all mid flat explained of the the easily counts to in compared the bin subdwarfs for faintest the tendency the in of The latitudes high not. at do rise to dwarfs factor populations white a and of subdwarfs drop Only but a plane, show bins. stars sequence magnitude Main different included. three latitude and Galactic of populations with function stars a stellar as different catalog our three in stars for of density Surface 2.20.— Fig. J htmty(eddfrsa lsicto ihteRMdarm are diagram) RPM the with classification star for (needed photometry log(number/deg2) log(number/deg2) log(number/deg2) −2.5 −1.5 −1.5 −1.5 −.5 −.5 −3 −2 −1 −2 −1 −2 −1 0 1−50. 1 .5 0 1 −.5 .5 −1 0 1 −.5 .5 −1 0 −.5 −1 White Dwarfs Subdwarfs Main Sequence sin b 110 11 15 ∼ 0i est ls othe to close density in 10 Chapter 3

Precise Masses of Nearby Stars

3.1. Introduction

The mass of a star is a single most important parameter determining many of its other physical parameters, its structure, and the course of its evolution. This is why the knowledge of mass is important by itself. However, since the advances in the theoretical understanding of stellar structure and evolution are today profound enough to be able to predict many of the stellar characteristics, the knowledge of a precise mass is also the best way to check the validity of the stellar theory, or to pose challenges for the reﬁnement of the stellar models.

We can illustrate this with a well known relation between mass and luminosity of stars on the main sequence. The bold line in Figure 3.1 shows the theoretical

M–L relation for solar-metallicity stars (Baraﬀe, Chabrier, Allard, & Hauschildt

111 1998), while the thin line shows the relation for metal-poor star with [Fe/H] = −1

(Baraffe, Chabrier, Allard, & Hauschildt 1997). The current mass measurements with their error bars are shown as circles (astrometric and spectroscopic binaries) and as triangles (eclipsing binaries). It is obvious that the observations offer very weak constraints on the theoretical models. This is especially true for low-metallicity stars for which there are few measurements to begin with. With precise mass measurements one would be able to map the mass-luminosity relation as a function of metallicity, or perhaps some other parameters that currently do not figure in the models (like the onset of full convection for low-mass stars). In addition to this, all currently used methods measure masses of stars in binary systems. Although we have no reason to believe that the masses of isolated stars would be different, this has never been actually tested.

3.2. Astrometric Microlensing with Space

Interferometry Mission

One is used to thinking of microlensing events as taking place towards the

Magellanic Clouds or the Galactic bulge. In both of these cases the lens is a faraway object, either a star belonging to the same system as the source star

(self-lensing), a distant star in the Milky Way’s disk, or a member of Milky Way’s

112 halo (whatever its nature might be). The effect that is routinely observed in such cases is the change in source’s brightness, but also present is an additional effect of the deflection of the source apparent position (Boden, Shao, & van Buren 1998).

The astrometric effect scales as the inverse of lens-source separation (∝ β−1), while the photometric effect decreases more severely (∝ β−4). This means that for the nearby lenses (d ∼< 100 pc), where lens-source separations are likely to be large, the only observable effect will be astrometric deflection. The amount of this deflection is directly proportional to the lens mass, allowing it to be measured in principle.

Initially proposed by Refsdal (1964), this idea was later examined by Paczy´nski

(1995, 1998) and Miralda-Escud´e (1996) in the context of rapid developments in space-based astrometry.

Obtaining masses using astrometric lensing has several advantages. First, it allows achievement of a greater precision than the one routinely achieved by measuring binary systems. Second, it is the only known method to obtain the masses of stars not residing in binary systems. Finally, the selection bias favors fast moving lenses, which are often low-metallicity halo subdwarfs, the masses of which are rarely measured.

The deﬂection is so small (∼ 100µas) as to be unobservable with present- day facilities. However the unprecedented astrometric precision of the Space

Interferometry Mission (SIM) of 4µas will enable such measurements.

113 We plan to measure some 20 stars using 200 hr of SIM time alloted to the

SIM Microlensing Key Project

3.3. Selection of Microlensing Candidates

3.3.1. Principles of Selection

Here we describe the original selection of candidate microlensing events. Later improvements for the part of the selection process came most notably from the use of 2MASS catalog, as described in Chapter 2. The impact of 2MASS on the selection process will be discussed in §3.4.1.

Our overall plan is to search for astrometric microlensing events (or ‘events’, for short) and rank these by the amount of SIM time required to measure the lens mass to a ﬁxed fractional error of 1%. To this end, we would like to consult a catalog containing the positions, parallaxes, proper motions, and magnitudes of all stellar sources in the sky. Unfortunately, there is no such catalog. To understand how to make use of existing catalogs, we review the basic requirements of the search.

First, while in principle the event depends on the relative proper motion of the source and lens, the lenses, being closer, almost always move much faster in

114 the sky than the sources. Hence, no proper-motion information is required for the sources in order to select candidate events.

The probability p that any individual lens will deﬂect light from a more distant star enough to measure the lens mass M to ﬁxed fractional accuracy is

p ∝ NsπµM, (3.1)

where Ns is the surface density of sources, and π and µ are the parallax and proper motion of the lens. One therefore expects events to be clustered near the Galactic plane, and for nearby, fast-moving stars to be over-represented as lenses. However, there are a greater number of distant than nearby stars and consequently more stars with low than high proper motions. The net of these two competing eﬀects is that for parallax-limited and proper-motion-limited catalogs, the total number of events scales as (Gould 2000a)

∝ −1 ∝ −1 Nevents πmin,Nevents µmin, (3.2)

where πmin and µmin are the limits of the respective types of catalogs of lenses.

Of course, the total number of potential lenses that one must examine scales as

−3 −3 πmin or µmin. Thus it is most eﬃcient to start with high π or high µ stars and move progressively to more distant or slower ones. In practice, one has available

115 magnitude-limited and not distance-limited catalogs, but for stars of ﬁxed absolute magnitude these are eﬀectively distance-limited.

3.3.2. Catalogs Used in Selection

The USNO-A2.0 all sky astrometric catalog , which is constructed from two photographic surveys [Palomar Observatory Sky Survey I (POSS I) for δ>−17.◦5

(‘north’ celestial hemisphere) and UK Science Research Council SRC-J survey plates and European Southern Observatory ESO-R survey plates (SERC/ESO) for δ<−17.◦5 (‘south’ celestial hemisphere)] is a nearly ideal catalog for sources, containing 526 million entries. To be included in the catalog, a star had to be detected on both the blue and red plates within a 2 coincidence radius aperture.

Hence the catalog begins to lose completeness at V ∼ 19 as stars fall below the detection threshold on one plate or the other. The catalog is also incomplete at bright magnitudes (V ∼< 11) because of poor astrometry of saturated stars, although for these stars USNO-A2.0 contains inserted entries from the ACT (Urban, Corbin,

& Wycoﬀ 1998b) or Tycho (ESA 1997) catalog. However, the epoch of these additional entries is 2000.0 and 1991.25 respectively, unlike the epoch of the other sources which is the mean epoch of the blue and the red plates (1950s for POSS I, and 1980s for SERC/ESO). In addition, USNO-A2.0 is by and large missing the

− stars with proper motions µ ∼> 250 mas yr 1 in the ‘south’, because the blue and

116 red plates of the SERC/ESO survey were on average taken 8 years apart, and so stars with µ>250 mas yr−1 moved outside the 2 error circle between the blue and red exposures. In reality, the time elapsed between the two plates varies from 0 to

15 years, leading to different proper-motion cutoffs for each plate. This problem does not affect POSS I because its blue and red plates were taken on the same night. Neither the incompleteness at bright magnitudes nor the incompleteness at high proper motions has any significant effect on USNO-A2.0 as a catalog for microlensing sources, since they are usually faint and move very slowly. However, both have substantial impact on our efforts to obtain critical information from this catalog about the lenses (see below).

The relative position errors, important for NLTT events, for USNO-A2.0 are about 150 mas. For Hipparcos and ACT events, it is the absolute errors

[USNO-A2.0 uses ICRS (International Celestial Reference System) as its reference frame] of about 250 mas that are relevant. There is, of course, an additional error in the position of the source in 2010 due to 60 years of proper motion in the case of POSS I and 30 years for the SERC/ESO plates. Since typical sources are on average 3 kpc distant and are moving at 25 km s−1 in each direction, this gives a proper motion of ∼ 2masyr−1. This proper motion adds about 100 mas in the

‘north’ and 50 mas in the ‘south’ to the total positional error. Hence, the total error on average is about 170 mas (260 mas in the absolute system). Note that this

117 will not be improved signiﬁcantly by the release of the USNO-B all-sky position and proper motion catalog (D. Monet 1998, private communication), since its proper-motion errors will be of the same order as the proper motions of typical source stars. Similar limitations will hold true in the case of GSC II (Guide Star

Catalog Two) (see Lasker et al. 1998, for example). USNO-B and GSC II will be compiled by comparing ﬁrst generation sky surveys with the second generation.

The absolute photometry errors in USNO-A2.0 are said to be about 0.25 mag for the stars that are not saturated. USNO-A2.0 lists photographic blue and red magnitudes. The equinox of the coordinates is ICRS J2000.

We search for lenses in three catalogs: Hipparcos, the ‘ACT Reference Catalog’

(ACT) and the ‘New Luyten Catalogue of Stars with Proper Motions Larger than

Two Tenths of an Arcsecond and First Supplement’ (NLTT). The three catalogs have substantially diﬀerent characteristics.

Hipparcos is a heterogeneous catalog with 118,000 entries. However, it has two approximate completeness characteristics that are very useful for understanding its role in the present study. First, it is approximately complete for V<8, with 41,000 stars to this limit. Second, it contains essentially all the NLTT stars brighter than its operational limit of V ∼ 12. As we mention below, NLTT is nominally complete for µ>180 mas yr−1. Based on statistical tests of the Hipparcos catalog, we ﬁnd that it (and thus presumably NLTT) is essentially complete for µ>220 mas yr−1

118 and V<11. In its last magnitude (11

Hipparcos. There are 6500 Hipparcos stars with µ>200 mas yr−1 and 15,000 with

µ>100 mas yr−1.

Hipparcos stars have trigonometric parallaxes with typical precisions of 1 mas.

As we discuss in §3.3.3, uncertainty in the distance to the lens is the main problem in estimating the amount of SIM time required for a lens mass measurement.

This uncertainty is virtually eliminated for Hipparcos stars. In addition, we use

Hipparcos parallaxes to calibrate our method for estimating distances of stars in the other two catalogs which lack trigonometric parallaxes. Hipparcos positions are accurate to 1 mas, while the proper motions have errors of order 1 mas yr−1, implying an error of about 20 mas in the star’s 2010 position. This is negligible compared to the error in the source position given in USNO-A2.0. Finally, most

Hipparcos stars have Tycho photometry which is accurate to of order 0.01 mag.

Even those stars lacking Tycho photometry usually have ground-based photometry of similar quality. Tycho photometry is far better than the minimum precision required for the present search.

The ACT catalog is constructed by matching stars common to both the

Astrographic Catalogue 2000 (AC 2000, Urban et al. 1998a) and Tycho, with

119 epochs circa 1910 and 1990 respectively. Such a long baseline combined with

Tycho’s precise positions, permits the proper motion accuracy of ACT to be

∼ 3masyr−1 (ten times better than Tycho itself). ACT is presently the largest

(nearly 1 million stars) all-sky catalog containing proper motions. It is limited at the faint end by incompleteness of the Tycho catalog which sets in over the range

11 ∼

− 50% for V ∼ 3. There is also a cutoﬀ at high proper motions (µ ∼> 1. 5yr 1), which results from the lack of proper-motion information about these stars in the

Tycho catalog. Typical errors of ACT proper motions imply an uncertainty in 2010 position of about 60 mas. This is still small compared to the uncertainty of the source position and so can be ignored. Tycho photometry is available for the great majority of ACT stars and, as stated above, this has much higher precision than is required for the present study. As we discuss in §3.3.3, we are able to estimate the distances to ACT stars with ∼ 30% accuracy which is quite adequate for our purposes.

The characteristics of NLTT are investigated in detail in Chapter 2, and here we present just the details relevant to lens selection. NLTT is nominally complete

− ◦ to µ>180 mas yr 1 and V<19 in the northern part of the sky (δ ∼> −33 ), and

120 away from the galactic plane (| b |> 10◦). In the south and near the plane, the incompleteness sets in at brighter magnitudes. NLTT α and δ are given only to 1 s and 0.1 respectively (in some cases to 0.1 min and 1 respectively) and so are not suﬃciently accurate to predict lens-source encounters which typically have impact parameters β ∼ 1. Hence to obtain improved positions of NLTT stars we search for the corresponding entries in USNO-A2.0. Recall that USNO-A2.0 entries have position errors of 250 mas. However, recall also that in the ‘south’ (δ<−17.◦5),

USNO-A2.0 is missing a large fraction of the NLTT stars. To recover this part of the NLTT catalog, it would be necessary to make new position measurements for the majority of NLTT stars in the ‘south’, or at least for all that pass within 6

(position error of NLTT) of some source star. This would be a major project which we do not attempt. The proper motion error present in NLTT (§2.3.4), propagated over the ∼ 50 year baseline implies errors of 1.2 in 2010 position. This is the dominant astrometric error for these stars and has important consequences as we discuss in §3.3.3.

Because NLTT stars must be found in USNO-A2.0 in order to be used, they automatically have available two sources of photometry, both photographic. As we discuss in §3.3.3, it is necessary to transform these photographic systems to the Johnson-like system used by Tycho in order to estimate distances. We ﬁnd that the transformation from USNO-A2.0 colors to Johnson B − V has somewhat

121 smaller scatter than the transformation from NLTT colors, and we therefore use the former. This scatter (0.25 mag) is still substantially larger than we would like.

As we discuss in §3.3.3, it leads to a factor 1.7 uncertainty in distance estimates for

NLTT stars.

There is one important additional source of incompleteness that aﬀects all searches based on USNO-A2.0. Suppose that a lens will pass close to a source in 2010 with a relative proper motion µ. The source must be identiﬁed from

USNO-A2.0 which is based on plates taken δt ∼ 60 yr earlier in the ‘north’ and

δt ∼ 20 yr earlier in (the later of the two plates) the ‘south’. At that time, the lens and source were separated by µδt. If the lens is suﬃciently bright, it will appear as a blob on the photographic plates and will therefore “blot out” the source at the epoch of the plate, and so the source will not appear in USNO-A2.0.

The exact blot-out radius depends on the magnitude of both the lens and the source (fainter stars will get blotted-out farther from the lens). However, the great majority of sources are relatively faint (V ∼ 17). For simplicity, we therefore identify this radius as a function of lens magnitude, θ(V ), the point where 50% of

V ∼ 17 stars are lost. We ﬁnd for V =2, 5, 8, 11, 15 that θ(V ) = 350, 80, 21, 11, 4 arcseconds respectively. Thus, for example, for a V = 8 lens (i.e., θ =21), the minimum proper motion it is required to have to allow an event to be detected is

−1 −1 µmin = θ(V )/δt = 350 mas yr in the ’north’ or 1000 mas yr in the ‘south’.

122 3.3.3. Estimates of Errors

The basic requirement for constructing a list of astrometric microlensing events is to rank order the events by the amount of telescope time (here speciﬁcally

SIM time) needed to make a mass measurement of a speciﬁed precision. At a later stage, one might decide to eliminate events with short observation times because of some diﬃculty in carrying out the observations, and one might choose to skip down the list to include an event with a long observation time because the lens in question is exceptionally interesting. However, for now we will be concerned primarily with the fundamental requirement of rank ordering the events.

The observation time needed for a 1% mass measurement is given by (Gould

2000a)

rβc2 2 µt µ∆t τ = T α2 100.4(Vs−17) γ 0 , , (3.3) 0 0 4GM β β

where r is the distance to the lens, β is the impact parameter of the event (the projected angular separation at the time t0 of closest approach), M is the mass of the lens, Vs is the apparent magnitude of the source, µ is the relative lens-source proper motion, ∆t(= 5 yr) is the duration of the experiment, γ is a known function which is discussed in detail by Gould (2000a), T0 =27hours,andα0 = 100 µas.

123 In order to estimate τ, one must first measure or estimate r, β, M, Vs,µ, and t0. Of course, there will be errors in all of these quantities, and these will in turn generate errors in τ. In most cases, these errors can be reduced by making additional observations (as will be done in §3.4.2) or carrying out additional investigations of various types. However, these refinements often require substantial legwork. Therefore, one should first decide what is an acceptable level of error in τ and what are the main contributors to it.

The list of events will be constructed in three stages. Stage 1 is an automated search of a pair of star catalogs (sources and lenses) for events with estimated observation times τ ≤ τmax,1. Stage 2 is a simple (but potentially very time consuming) check of this list to eliminate spurious candidates. In stage 3, additional observations are made of the remaining candidates. The estimate of

τ is reﬁned and the ﬁnal list is constructed with a more restrictive maximum observation time τ ≤ τmax,3,andτmax,3 <τmax,1.

What level of errors are acceptable at stage 1 and stage 3? At the outset it should be emphasized that errors in the estimate of τ do not cause errors in the

ﬁnal mass measurement by SIM. The cost of errors in stage 3 is that the SIM observations will be too short (causing larger than desired statistical errors in the mass measurement) or too long (wasting valuable SIM time pushing down the mass measurement errors below what is actually desired). Hence, a factor of two error is

124 acceptable. That is, if the SIM time were underestimated by a factor of 2, then the mass-measurement error would be 1.4% instead of 1%. This would be a bit worse than desired but on the other hand there would be a saving of SIM time that could be applied to other stars. If the SIM times were overestimated by a factor of 2, then one would waste some SIM time on the event, but one would reduce the error to 0.7% which is not completely without value. On the other hand, factor of 10 errors are not acceptable. Either one would waste a huge amount of SIM time, or one would obtain a mass measurement with an error much larger than desired. As a corollary, errors that are small compared to a factor of 2 can be ignored at any stage.

Much larger errors can be tolerated at stage 1 than stage 3. For example, if the stage-1 estimates could be in error by a factor of 10, then one must set

τmax,1 =10τmax,3 to avoid losing viable candidates. The cost is that the candidate

1/2 list is increased by a factor (τmax,1/τmax,3) ∼ 3 (Gould 2000a), and one must then sift through this larger list in stages 2 and 3. Clearly, however, this work load can become prohibitive for suﬃciently large errors.

We now show that of all the input parameters, only the distance r and the impact parameter β can induce suﬃciently large uncertainties in τ to warrant special attention. We examine the various parameters in turn.

125 If the lens is taken from the Hipparcos catalog, it will have a trigonometric parallax. In virtually all cases of interest, the lens will be close enough (r ∼< 200 pc) that the distance error will be less than 20%, which is quite adequate for present purposes. If the lens does not have a trigonometric parallax, its distance must be estimated from its measured ﬂux (in say V band) FV together with an estimate of its intrinsic luminosity, LV :

LV τ ∝ r2 = . (3.4) 4πFV

Equation (3.4) makes it appear as though the uncertainty in τ will be enormous.

For example, a star with a measured color V − I = 1 could plausibly be a clump giant with MV = 1, a main-sequence star with MV = 6, a subdwarf with MV =8,

5 or a white dwarf with MV = 14. This covers a range of 1.6 × 10 in luminosity and implies an uncertainty in τ of the same magnitude. Nevertheless, we will show below that with good two-band photometry, r can be determined with ∼ 30% accuracy which implies an error in τ of less than a factor of 2. Stars in the ACT catalog have good (Tycho) photometry. For stars in NLTT only photographic photometry is generally available. We will later show that for NLTT the 1 σ errors in LV (and so τ) are a factor of 3.

The ﬁrst step in estimating the distance to the lens is to determine its luminosity class (e.g., white dwarf, subdwarf, main-sequence, or giant star). If this

126 is properly determined, then the lens mass can be estimated quite accurately from the color. For the cases where the luminosity class is not correctly determined, the error induced in the distance is much greater than the error induced in the mass.

Thus, in either case, the error in the mass can be ignored.

The geometry of the event (µ, β,andt0) is determined from the astrometry.

These quantities aﬀect the estimate of τ through the β2 factor and the γ factor in equation (3.3). We focus ﬁrst on the β2 factor. As discussed in §3.3.2, the relative source-lens position error (and hence the error in β) is about 260 mas for lenses in Hipparcos and ACT and about 1.2 for NLTT. In §3.3.4, we discuss how these errors are incorporated into the search procedure.

According to equation (3.3) the 0.25 mag error in the source magnitude from

USNO-A2.0 induces a 25% error in τ. We ignore this.

Finally, since the launch date of SIM is not ﬁxed, we do not attempt to calculate γ based on the time of closest approach t0 relative to the midpoint of the mission, γ(µt0/β, µ∆t/β). Rather, we calculate γ for the optimal possible launch date for the given event when the midpoint of the mission coincides with the time of closest approach, i.e., t0 = 0. That is, we use γ(0,µ∆t/β). Some representative values are γ(0,x) = 10 for x ≥ 4, γ(0, 2) = 19, and γ(0, 1) = 99. When the launch date is ﬁxed and the time of minimum separation is better determined in the case

127 of NLTT events, one can substitute the correct first argument in place of 0. In some cases, γ may rise significantly but in others (particularly when µ∆t 4β)it will hardly be affected. In any event, because we are suppressing consideration of the first argument, any uncertainty in t0 does not enter our calculation.

Now we describe our method for estimating the distances to the lenses and evaluate the accuracy of these estimates. Our method has three distinct steps.

First, we assign a luminosity class (type) to each star based on its position in a reduced proper-motion diagram. Second, we assign a V band luminosity LV

(equivalently MV ) to each star based on its luminosity class and color. Third, we combine the LV with the measured ﬂux from the star FV (equivalently V )to obtain a distance. We apply this method to both the ACT and NLTT catalogs.

However, to calibrate and describe the method, we ﬁrst apply it to Hipparcos stars with parallax errors smaller than 20%. After the method is calibrated, we use it to

“predict” the distances to these stars and then compare the results to the measured

Hipparcos parallaxes.

Figure 3.2 is a reduced proper motion diagram of Hipparcos stars with parallax errors smaller than 20% (dots) and NLTT stars not present in Hipparcos

(crosses). (Please note that throughout this Chapter we will use V magnitudes in

Tycho system, and B − V colors in Johnson system. To get Johnson V magnitude, use the transformation (ESA 1997): VJ = V − 0.090(B − V ). Consequently

128 MV is in Tycho system as well.) If all stars had identical transverse speeds v∗, then this diagram would look exactly like a color-magnitude diagram (CMD),

−1 but with the vertical axis shifted by 5 log(v∗/47.4kms ). This means that disk stars (i.e., white dwarfs, main-sequence stars, and giants) which have typical

−1 v∗ ∼ 30 km s are shifted upward by 1 mag, while halo stars (i.e., subdwarfs)

−1 which have typical v∗ ∼ 240 km s are shifted downward by 3.5 mag. That is, the

∼ 2 mag separation between the main sequence and the subdwarfs in a “normal”

CMD is here augmented to ∼ 6.5 mag. (But note that in a proper motion selected sample, like that from NLTT, there is a bias which makes the mean transverse speed of disk stars several times greater than their true speed v∗.) We separate luminosity classes according to the bold lines shown in the diagram. Once these classes are chosen, we use color-magnitude relations for each class (white dwarfs, subdwarfs, main-sequence stars, and giants) to determine the absolute magnitude, and therefore the distance. Inevitably, stars of diﬀerent class have some overlap in the reduced proper motion diagram, especially in the red end of the diagram. In those cases our classiﬁcation is conservative, i.e. places a star in a class that will make it closer, and therefore producing smaller τ.

Figure 3.3 compares the distances of Hipparcos stars derived from these luminosity estimates (together with the measured V mags) to the true distances based on parallax. For the typical lens distance moduli of less than 2.7 (r =35pc),

129 the dispersion (excluding outliers) is 0.53 mag. This is equivalent to a distance uncertainty of 28%, and an error in the estimate of τ of 63%. For distance moduli greater than 2.7 , the dispersion is larger, but this is dominated by giants which are of little practical interest in the present search.

We directly apply this technique to the ACT catalog for which there is generally excellent photometry from Tycho.

For NLTT, generally only photographic photometry is available. Because of the large position errors in the NLTT catalog, we can search for astrometric lensing events only if we can identify the NLTT star with the corresponding object in

USNO-A2.0. Thus, in all cases we have photometry from USNO-A2.0. We convert from USNO-A2.0 mags to Johnson color and Tycho V using the relations:

B − V =0.38 + 0.55(B − R)USNO, (3.5)

V = RUSNO +0.23 + 0.32(B − R)USNO. (3.6)

The transformations were derived by comparing USNO-A2.0 photographic magnitudes to ground-based Johnson magnitudes of some sixty faint M dwarfs and white dwarfs. Here, BUSNO and RUSNO are blue and red photographic magnitudes, respectively. The scatter in the predicted versus actual B − V is ∼ 0.25 mag. Since the slope of the main sequence is ∆V/∆(B − V ) ∼ 5, the error in distance modulus

130 of NLTT stars is ∼ 1.2 mag or about a factor of 1.7 in distance. This corresponds to a factor 3 error (1 σ)inSIM time τ.

In the case of ACT and NLTT, the distance is also used to ﬁnd the luminosity, that in turn (using mass-luminosity relations) determines the masses of main sequence stars and subdwarfs. For giants and white dwarfs we adopt masses of

1M and 0.6M respectively.

With Hipparcos, the mass is found directly because their distances, and therefore luminosities are known from trigonometric parallax. Thus, we only need to correctly determine the luminosity type based on luminosity and color (CMD).

This is fairly straightforward for giants and white dwarfs, but can be ambiguous for main sequence stars vs. subdwarfs since they occupy not too different regions of CMD. We differentiate them by their transverse velocities, calling stars with v>85 km s−1 subdwarfs. Exact classification is only possible with additional information, such as a spectrum. In any case, as noted previously, the mass determination is not critical for the estimate of τ.

Next we explore the uncertainty in the predicted impact parameter. At ﬁrst sight these uncertainties appear so high (in the case of NLTT events 1.2) to make the estimate of τ impossible. For example, any source whose calculated impact parameter with respect to an NLTT lens is less than 2 might actually pass within

131 50 mas or even closer, thus reducing its SIM time by a factor of 1600 or more. In essence, one would seem to be forced to do follow-up observations of all encounters in this catalog having apparent impact parameters β<2 in order to ﬁnd the small subset with very close encounters. In fact, the situation is not quite so severe.

The size of the aperture stop for SIM has not yet been ﬁxed, but is likely to be at least 300 mas. This is about the size of the envelope of the SIM fringe pattern

(set by the 25 cm size of the mirrors). Hence, if the lens is as bright as the source then it would be diﬃcult to obtain reliable astrometry while the source is within

300 mas of the lens. Typically, the lens will be much brighter than the source so the problem will be even more severe. For events with β<300 mas, observations can be carried out during most of the event, but must be suspended during the period of closest approach. The precision of the mass measurement will then be approximately the same as for an event with β = 300 mas. That is, there is an eﬀective minimum impact parameter, βmin = 300 mas.

We account for the errors in distance and impact parameters as follows.

We aim for a ﬁnal catalog with τmax,3 = 100 hrs. For the Hipparcos and ACT lenses, we accept the lens distances and impact parameters at face value, but set τmax,1 = 300 hrs to allow for errors, primarily overestimation of the impact

132 parameter. For NLTT we set τmax,1 = 1000 hrs to allow for 1 σ photometry errors.

In addition, we calculate the τ ∗ (the best-case τ) by reducing β so that

β∗ → max(β − 1.8, 0). (3.7)

We always use the reduced β∗ to calculate the corresponding γ factorthatenters

τ ∗, but in cases when β∗ < 300 mas, we use β∗ = 300 mas for the value of the impact parameter, because of the discussed aperture stop. Finally, we allow all events where the lens is fainter than the source and the nominal impact parameter is β<1.8 on the oﬀ chance that the true impact parameter is very small. It is this best-case τ ∗ on which we impose the 1000 hr limit. These three adjustments to the NLTT-based catalog mean that it will contain a large number of spurious candidates. These must be eliminated by follow-up observations to obtain better photometry (which in some cases is available from the literature) and astrometry

(§3.4.2).

133 3.3.4. Searching for the Candidate Events

Although the basic strategy for searching for the candidate events is the same for all three catalogs (Hipparcos, ACT and NLTT) there are some speciﬁc details that apply to each of them. Also, there were certain problems associated with the raw lists of events produced by these catalogs. That is, each catalog’s initial list had its own set of ‘events’ that turned out not to be real.

The catalog of sources, USNO-A2.0, is written on 11 CD-ROMs, and the sky is divided into 24 zones each corresponding to 7.◦5 in declination. Each zone is written as one ﬁle. Our search program processes one zone at a time, checking every lens star that lies within that zone.

First, the initial position of the lens in J2000.0 coordinates is needed. In the case of Hipparcos and ACT this is straightforward as they both list coordinates in the ICRS J2000.0 system, the one used by USNO-A2.0. One only needs to apply proper motion in order to change the epoch of the coordinates from

1991.25 and 2000.0 (Hipparcos and ACT, respectively) to that of the search period

(2005-2015). In the case of NLTT, the procedure is much more involved, and is described in detail §2.3.1. We made a special eﬀort not to search NLTT stars that were included in Hipparcos or ACT: as discussed in §3.3.2, the NLTT data are of much lower quality and would generate many spurious events that are

134 eliminated by the better Hipparcos and ACT data. We screen for these duplicates by looking for Hipparcos stars around NLTT positions that have similar proper

−1 motions (∆µα, ∆µδ < 40mas yr ), and not too different magnitudes. We find 6233 matches, i.e., most of the Hipparcos stars with µ>200 mas yr−1. These matches are then flagged and skipped when identifying NLTT stars in USNO-A2.0. Also, if the match in USNO-A2.0 is associated with an ACT star, such NLTT star is also skipped. Occasionally, no match for an NLTT star is found because the input position was completely wrong [most likely a typo, since a machine-readable NLTT was produced by Optical Character Recognition (OCR)]. Identification efficiency is much worse in the ‘south’ (SERC/ESO) for reasons discussed in §3.3.2. Only 20% of NLTT stars are found within 10 of the expected position.

Next, the basic search strategy for events is to produce a box, the diagonal of which represents the lens’s proper motion from 2005 to 2015, the time span during which an event should take place. The size of the box is further increased by 5 years worth of proper motion (i.e, the largest possible impact parameter) to allow for events that take place near the starting and ﬁnal years. We then ﬁnd all the stars in USNO-A2.0 that are located within this box. A moving star, i.e. the lens, will pass by these stars, but not every encounter will produce a microlensing event. As discussed in §3.3.2 and §3.3.3 this depends on the physical parameters of the lens and on the brightness of the source star. Therefore, for each encounter we

135 calculate the required SIM time and keep only events with τ<τmax,1.Inthecase of NLTT, we use the reduced impact parameter β∗, as described in §3.3.3, to ﬁnd

τ ∗.

Additionally, when searching ACT we discard encounters with stars that were labeled in USNO-A2.0 as being associated with ACT, in order to avoid ﬁnding encounters of an ACT star with ‘itself’. It might not sound logical to ﬁnd an

ACT star approaching its USNO-A2.0 entry in the future, but this happens with some slowly moving ACT stars because the astrometry of bright USNO-A2.0 stars is poor. A similar problem is present with bright Hipparcos stars, for which

USNO-A2.0 sometimes contains multiple spurious entries. We discard these based on brightness and proximity of the Hipparcos star to the USNO-A2.0 entry at the epoch of the plate. Despite these automated rejection criteria, some ‘events’ that are nothing other than the lens and its entry in USN0-A2.0, make their way into a ﬁnal list. This most often happens because bright stars, having bad astrometry in USNO-A2.0, produce multiple entries if located in overlapping regions of the plates. These ‘events’ are characterized by very short SIM observing times (because the ‘source’ magnitude is bright). We check them by hand, by looking at the sky survey images themselves and making sure that there is only one star present.

Once an event satisfying all criteria is found, the output list containing all the information about the lens, the source, and the geometry of an event is produced.

136 We present these results in §3.3.5. However, the computer generated list is still far from containing only genuine events. One source of spurious entries aﬀecting searches with Hipparcos and ACT catalogs is discussed in the preceding paragraph.

Another problem is that since stars in these two catalogs are bright, their images in sky surveys have conspicuous diﬀraction spikes. These spikes in turn produce spurious entries in USNO-A2.0. Thus, sometimes an encounter will be reported in cases when the source is just an artifact from a diﬀraction spike. When we checked all of the Hipparcos and ACT events by comparing the sky survey images with

USNO-A2.0 generated star charts (http://ftp.nofs.navy.mil/data/), we were able to identify such occurrences. Also, since the diﬀraction spikes run along right ascension and declination, it was always the stars that had their proper motion along these directions that turned out to produce spurious events.

When it comes to NLTT, the most serious problem is with the encounters in the ‘south’, because the lens identification is often spurious. These are checked by calculating how much the lens has moved between the two plates. If that distance is less than the 2 error circle the chances are greater that the lens identification, and therefore the event, are real. Since there are not many of them, we check the ‘south’ NLTT events by hand. Finally, since the NLTT position is sometimes completely off, it could lead to the wrong USNO star be identified as a match for

NLTT star. Such a misidentiﬁed star might even produce an ‘event’. Since we do

137 not check entries in NLTT list by hand, a possibility exists that some entries might not be real.

As previously discussed, we try to eliminate doing NLTT stars that are present in either the Hipparcos or ACT catalogs. However, some survive our automated procedures. Therefore we check all NLTT events up to the Hipparcos/ACT detection limit and eliminate repetitions by hand. Thus, the NLTT list should contain only stars not present in the other two catalogs.

3.3.5. Candidate Events

The events produced by stars in the Hipparcos and ACT catalogs are presented in Tables 3.1–3.3. Tables 3.1 and 3.2 list the properties of the lens stars, while Table 3.3 lists those of the source stars and of the events themselves. Details about speciﬁc columns are given in the table notes. The events are ordered by the required SIM time. There are 32 events taking place between years 2005 and

2015. Eight of them are found using both the Hipparcos and ACT catalogs (in which case the results presented are from the Hipparcos catalog), as indicated by the last column in Table 3.3. There was only one event (associated with the star

AC368588) that was found in ACT and not in Hipparcos. However, inspection of

POSS I and POSS II plates lead us to conclude that its proper motion is much smaller than that reported in ACT, and that no event will be taking place. One

138 would expect all events detected by Hipparcos to be found in ACT, but this not the case. This is because in many cases of high-proper motion stars, the proper motion was not listed in Tycho, and therefore it is not listed in ACT either. In other cases, ACT was missing photometry because it was not available in Tycho.

These 32 events are produced by 25 diﬀerent stars. Therefore, seven entries in Tables 3.1 and 3.2 are repetitions, but we keep them in order to preserve compatibility with Table 3.3, i.e., the ‘Event #’.There are some notable stars among the lenses, such as Proxima Centauri (the closest star), Barnard Star (the highest proper motion), and the bright binary 61 Cyg A/B. They, together with the only white dwarf in the list (GJ 440), undergo multiple events that will both enable a more precise mass measurement and provide a check on systematics.

We classify 10 stars as subdwarfs, although some of them might be main sequence stars, and vice versa. A convenient way of presenting the types of

Hipparcos stars that will undergo microlensing is given in Figure 3.4. Plotted is the classical CMD of Hipparcos catalog stars with distances known to better than

10%. Superimposed as big dots are the Hipparcos/ACT stars that produce events listed in Tables 3.1– 3.3. As we see, except for a single white dwarf, the rest of the stars are uniformly distributed within the faint (MV > 5) portion of the main sequence, with subdwarfs located mostly below the densest concentration of stars.

139 The absence of stars with MV < 5 is the result of blotting out, as discussed in

§3.3.2.

Although the table includes events up to τmax = 300 hrs, they are concentrated towards shorter times. For example, 1/3ofeventshaveτ<20 hrs, and 1/2less than 70 hrs. In fact, when we investigate the number of events as a function of τ we see a behavior that is in line with the theoretical predictions of Gould (2000a).

As an example, in Figure 3.5 we show the 8 × 8 ﬁeld surrounding 61 Cyg

A/B as it appeared in 1951 (DSS 1/POSS I) and in 1991 (DSS 2/POSS II) (upper left and lower left panels, respectively). We can see that the pair has moved some

3.5 across the ﬁeld. In a 2 × 2 blow-up we show the region that the pair will transverse in the period 2005-2015 (from DSS 2/POSS II). The star chart (created from USNO-A2.0 data), corresponds to the 2 × 2 ﬁeld and has the lensed stars labeled with the number of the corresponding event from Tables 3.1– 3.3.

Additional features of the set of events found with Hipparcos and ACT will be discussed later in this section, together with the events from NLTT.

Tables 3.4 and 3.5 contain data about the 146 events found in NLTT, ordered by their nominal SIM observing time τ. Details about the columns are given in the table notes. These tables have many more entries than the Hipparcos/ACT

∗ tables partly because of the τmax = 1000 hr limit compared to τmax = 300 hrs

140 for Hipparcos/ACT. In fact, there are just 34 events with τ<300 hrs. That means that if we had perfect knowledge about the NLTT stars there would be approximately 34 events in such a ‘perfect’ list with τ<300 hrs, but those, of course, would not necessarily be the ﬁrst 34 from our present list. However, it should be noted that out of 146 events only 8 (5%) are detected in the SERC/ESO part of USNO-A2.0 which comprises 35% of the sky. Again, the nominal SIM observing times are concentrated toward the lower values, and the trend of the number of events vs. τ basically agrees with Gould (2000a) predictions.

NLTT events are produced evenly by stars that we classify as white dwarfs, subdwarfs, and late-type main-sequence stars. Such representation is not surprising having in mind that most intrinsically bright, fast-moving stars are also apparently bright and therefore already covered by Hipparcos and ACT, so the ones covered by NLTT represent a sample of relatively nearby, intrinsically faint stars. One should keep in mind that our classiﬁcation is conservative as not to miss a possible candidate, in the direction that some of our white dwarfs are actually subdwarfs or main-sequence stars, and some subdwarfs are main-sequence stars. This issue can be resolved in the stage 3 of list reﬁnement, when better photometry and astrometry is obtained, supplemented by what is known about these stars from previous studies.

141 Finally, both the Hipparcos/ACT and NLTT events can be investigated in the

V − µ plane. This allows us to see the characteristics of the catalogs and events combined. Figure 3.6 covers a wide range of visual magnitudes (2

µ =0.1yr−1 to that of Barnard’s star. The two long-dashed vertical lines show the nominal limit of the Hipparcos catalog of survey stars (V = 8), and the detection limit of Hipparcos non-survey stars, Tycho, and therefore of ACT (V = 12). The horizontal long-dashed line is the lower limit of µ =0.18 yr−1 for the NLTT. The lenses found only in Hipparcos are designated with ‘×’, and those found in both

Hipparcos and ACT look like asterisks. In order to present a more realistic relative number of NLTT lenses, we plot only those with nominal τ<300 hrs (circles).

As discussed in §3.3.2, the blotting out of images in USNO-A2.0 limits our ability to ﬁnd events moving slower than a speciﬁc value for the given lens magnitude.

We plot this function θ(V ) as a short-dashed line. Because of diﬀerent epochs of

POSS I and SERC/ESO, these cutoﬀs will be diﬀerent in the two parts of the sky.

The lower line corresponds to ‘north’ (POSS I). The region below these two lines is therefore excluded, and we can see that none of the lens stars is found there. The exclusion due to blot-out approximately follows the diagonal line corresponding

−1 toastarwithMV =6,v = 75km s . This shows that our survey cannot ﬁnd disk-star lenses with MV < 6, unless they are moving faster than average. Indeed, as shown in Figure 3.4, we ﬁnd no lenses with MV ∼< 5. However, halo stars with

142 −1 MV =6,v = 240km s (upper diagonal line), are comfortably away from this limit.

3.4. Confirmation of Candidate Events

3.4.1. Refining Estimates Using 2MASS

Currently 47% of the sky has been released in the 2MASS infrared survey.

These data, and more speciﬁcally, its Point Source Catalog, can be used in two ways to reﬁne the selection procedure outlined in §3.3. First, as shown in §2.5.2,

2MASS can be combined with USNO-A to produce optical-infrared reduced proper motion diagram of NLTT stars, allowing more secure classiﬁcation, and therefore the correct assignment of luminosity for a given color. Also, with a wider baseline for color, the distance estimates of NLTT stars becomes much more secure

(σd/d ≈ 20%), especially so for late dwarfs for which the optical color becomes degenerate with absolute magnitude. Second, the benefit of using 2MASS, where available, is its ∼ 2000 epoch positions, given with an adequate precision of 130 mas. This allows for a manifold refinement of the impact parameter of NLTT lensing candidates – by fixing the position at the current epoch, and by refining the proper motion vector (§2.4.2).

143 Taken together, there are two ways we can put 2MASS to use. First, conﬁrm the quality of events already selected in §3.3, and discard those that do not pass

2MASS criteria, thus reducing the observational burden needed to conﬁrm these events by 50–75%. Alternatively, use distances and proper motion vectors derived with 2MASS to predict additional events, but this time allowing for more than 1σ errors in astrometry. Both of these strategies where used to produce the list of events for the observing campaign.

3.4.2. Observing Campaign

The observation of microlensing candidates was performed on four observing runs – March, June and December 2001, and October 2001. Observations were carried out for a total of 8 photometric nights on a 2.4-m Hiltner telescope at the

MDM Observatory at Kitt Peak, Arizona. The detector used was 2k × 2k CCD

‘Echelle’, although due to the small lens-source separation, and in order to expedite the observing, only the central 512 × 512 pixels where read out. On the first two runs the observations where performed in BV I filters, but the B filter was excluded in the subsequent runs since it offered little additional information. Exposures where adjusted not to saturate the lens star (which is usually the brighter of the two stars), or not to exceed 30 s, whichever is shorter. A number of photometric standards was observed each night.

144 After the reduction of the CCD images and calibration of photometric standards, each image was checked against a finder chart in order to identify the candidate lens-source pair, and their photometry and relative astrometry where measured. Fields with good seeing and a sufficient number of stars where used to establish the orientation and scale of the CCD frame by fitting the chip coordinates to celestial coordinates of the stars (taken from USNO-A2). This was done for each night separately.

In a certain number of cases (∼ 10%) no event was taking place. This was either due to incorrect identiﬁcation of NLTT star, or in some more bizarre instances when the “lens” and the “source” actually belonged to a CPM binary, the separation vector of which happened to be parallel with the proper motion vector.

By measuring the separation vector today, and comparing it with the separation at POSS epoch, we obtained precise relative proper motion and therefore very accurate estimates of the projected impact parameter and the time of the event. The V − I color was used to determine accurate photometric distance after the star was classiﬁed as being WD, SD, or MS star. This ﬁnally allowed a new estimate of SIM observing time τ.

More than 400 candidate events where observed, including the Hipparcos events where we wanted to conﬁrm the assumed stationarity of the source. At the

145 moment 1/2 of the observations have been analyzed. Hipparcos events that were considered good (Tables 3.1– 3.3) were almost always conﬁrmed. As expected, the estimates of τ for NLTT events in some cases diﬀer substantially from those given in Table 3.5.

3.5. Discussion and Conclusion

If precise stellar masses are to be obtained using astrometric microlensing with

SIM, there is a necessity of ﬁnding microlensing candidates, as soon as possible, since the separation between the lens and the source is steadily getting closer, and it will become harder to produce a valid estimate of the likelihood of an event the longer we wait. With the currently available catalogs, we were able to produce a fairly reliable list of candidates from Hipparcos and ACT catalogs. Obtaining a list of similar quality of NLTT candidates required additional astrometric and photometric observations of the candidates in our list.

Another issue is getting more candidates. This can be assured with new catalogs of proper motions, having lower proper motion cutoﬀs and going to fainter magnitudes. The biggest such projects are USNO-B and GSC II which should list the proper motions of basically all the stars in POSS I/SERC/ESO.

Others are discussed in §2.1. Having a lower proper motion limit is particularly

146 important in V>12 range, where the blotting of stellar images no longer presents a limitation (at least not in the northern hemisphere), and where NLTT goes only to µ = 180 mas yr−1. USNO-B will also push the detection limit ∼ 1 mag fainter compared to NLTT. Since in USNO-B all the stars will have proper motions, the uncertainty of the source star’s position will also be reduced. There is also an issue of NLTT incompleteness at faint magnitudes, as discussed in §2.6.3. This is even more severe in the south and close to the galactic plane, where the incompleteness sets already at V ∼ 14. With USNO-B and GSC II this matter will most probably be resolved.

147 Table 3.1. Hipparcos and ACT events - lens star properties (astrometry and photometry)

Event HIP # RA DEC VB−V Other name #hms◦

1 104214 21 6 50.8350 +38 44 29.380 5.20 1.069 61 Cyg A 2 106122 21 29 46.4600 +45 53 37.083 7.986 0.759 HD 204814 3 57367 11 45 39.2635 −64 50 26.427 11.867 0.196 GJ 440

148 4 90959 18 33 17.8712 +22 18 55.449 9.016 1.181 V774 Her 5 86214 17 37 4.2404 −44 19 0.968 10.94 1.655 GJ 682 6 104217 21 6 52.1924 +38 44 3.890 6.208 1.309 61 Cyg B 7 28445 6 0 21.3792 +31 25 50.855 9.505 0.930 HD 250047 8 73734 15 4 19.2795 +60 23 2.956 11.00 1.500 Ross 1051 9 85523 17 28 39.4569 −46 53 34.986 9.38 1.553 GJ 674 10 104214 21 6 50.8350 +38 44 29.380 5.20 1.069 61 Cyg A 11 70890 14 29 47.7474 −62 40 52.867 11.01 1.807 Proxima Cen 12 64965 13 18 57.0885 −3 4 16.904 10.84 1.009 Ross 484 13 57367 11 45 39.2635 −64 50 26.427 11.867 0.196 GJ 440 (continued) Table 3.1—Continued

Event HIP # RA DEC VB−V Other name #hms◦

14 87937 17 57 48.9655 +4 40 5.837 9.54 1.570 Barnard’s Star 15 74234 15 10 13.5770 −16 27 15.521 9.44 0.850 HD 134440 16 76074 15 32 13.8455 −41 16 23.108 9.31 1.524 GJ 588 17 98906 20 5 3.3563 +54 26 11.144 11.98 1.524 V1513 Cyg

149 18 61629 12 37 53.1966 −52 0 5.580 10.767 1.470 GJ 479 19 33582 6 58 38.3423 −0 28 44.391 9.075 0.579 HD 51754 20 114622 23 13 14.7435 +57 10 3.498 5.57 1.000 HD 219134 21 27207 5 46 1.5287 +37 17 9.195 7.417 0.833 HD 38230 22 74926 15 18 39.2706 −18 37 32.607 10.643 1.214 BD−18 4031 23 70890 14 29 47.7474 −62 40 52.867 11.01 1.807 Proxima Cen 24 70890 14 29 47.7474 −62 40 52.867 11.01 1.807 Proxima Cen 25 104217 21 6 52.1924 +38 44 3.890 6.208 1.309 61 Cyg B 26 105090 21 17 17.7112 −38 51 52.468 6.69 1.397 AX Mic (continued) Table 3.1—Continued

Event HIP # RA DEC VB−V Other name #hms◦

27 102923 20 51 6.5386 +7 1 40.380 10.014 0.900 BD+06 4665 28 104214 21 6 50.8350 +38 44 29.380 5.20 1.069 61 Cyg A 29 87937 17 57 48.9655 +4 40 5.837 9.54 1.570 Barnard’s Star 150 30 79537 16 13 49.4874 −57 34 1.492 7.53 0.815 HD 145417 31 48336 9 51 8.9608 −12 19 34.728 10.093 1.446 SAO 155530 32 25878 5 31 26.9506 −3 40 19.712 8.144 1.474 HD 36395

Rows are ordered by increasing τ (see Table 3). HIP # is the Hipparcos catalog number. Right ascension and declination are taken from Hipparcos catalog. Equinox J2000, epoch 1991.25. Visual magnitude is in Tycho system if available, or Johnson (italics). Johnson colors are from Hipparcos catalog as well. Multiple entries arise from the fact that a single lens can produce multiple events. Table 3.2. Hipparcos and ACT events - lens star properties (distance, kinematic and physical properties)

Event µ p.a. vrad MV rMClass Sp −1 # / yr km s pc M

1 5.2807 52 −64.28 7.5 3.5 0.5 SD K5V 2 0.5531 50 −83.70 5.6 29.8 0.9 MS G8V 3 2.6876 97 13.5 4.6 0.6 WD DC:

151 4 0.5052 200 37.10 7.2 23.4 0.8 MS K4V 5 1.1765 217 −60.00 12.4 5.0 0.2 MS M5 6 5.1724 53 −63.48 8.5 3.5 0.4 SD K7V 7 0.3297 155 6.2 46.1 0.9 MS K2 8 0.6786 285 9.8 17.6 0.5 MS M: 9 1.0501 147 11.1 4.5 0.4 MS K5 10 5.2807 52 −64.28 7.5 3.5 0.5 SD K5V 11 3.8530 281 15.4 1.3 0.1 MS M5Ve 12 0.6517 258 126.00 8.1 35.9 0.5 SD K5 13 2.6876 97 13.5 4.6 0.6 WD DC: (continued) Table 3.2—Continued

Event µ p.a. vrad MV rMClass Sp −1 # / yr km s pc M

14 10.3577 356 −106.76 13.2 1.8 0.1 SD sdM4 15 3.6815 196 308.08 7.1 29.7 0.6 SD K0V: 16 1.5636 229 10.4 5.9 0.5 MS M0 17 1.4724 232 0.01 11.0 15.8 0.2 SD M3

152 18 1.0347 272 10.8 9.7 0.4 MS M3 19 0.6930 151 −80.59 4.9 68.4 0.8 SD G0 20 2.0952 82 −17.79 6.5 6.5 0.8 MS K3Vvar 21 0.7050 136 −30.90 5.9 20.6 0.9 MS K0V 22 0.5713 128 8.5 26.2 0.6 MS 23 3.8530 281 15.4 1.3 0.1 MS M5Ve 24 3.8530 281 15.4 1.3 0.1 MS M5Ve 25 5.1724 53 −63.48 8.5 3.5 0.4 SD K7V 26 3.4549 251 23.01 8.7 3.9 0.6 MS M1/M2V (continued) Table 3.2—Continued

Event µ p.a. vrad MV rMClass Sp −1 # / yr km s pc M

27 0.4333 147 6.6 48.3 0.6 SD K3 28 5.2807 52 −64.28 7.5 3.5 0.5 SD K5V 29 10.3577 356 −106.76 13.2 1.8 0.1 SD sdM4 30 1.6491 211 10.01 6.8 13.7 0.6 SD K0V 153 31 1.8487 142 61.01 9.4 13.7 0.4 SD M0 32 2.2277 160 10.61 9.4 5.7 0.6 MS M1V

Proper motions are given as intensity and position angle (from Hipparcos). Radial velocities are taken from SIMBAD. Absolute magnitude is in the same system as the corresponding visual magnitude. Distances come from Hipparcos trigonometric parallaxes. For mass estimate and class determination see §3.3.3 (MS - main sequence star, SD - subdwarf, WD - white dwarf). Spectral class is taken from Hipparcos catalog. Table 3.3. Hipparcos and ACT events - source star and event properties

Event RA DEC VB−Vτd2000 t0 β #hms ◦ hr yr mas

1 21 6 58.229 +38 45 41.14 10.7 0.1 66.2 2012.5 3064 H 2 21 29 47.366 +45 53 45.37 16.6 1.43 4.0 7.7 2013.9 366 H/A 3 11 45 48.968 −64 50 33.74 18.2 1.26 4.2 38.8 2014.4 738 H 4 18 33 17.603 +22 18 46.65 16.4 0.22 5.0 5.2 2010.2 456 H/A 154 5 17 37 2.520 −44 19 21.96 13.7 1.37 5.4 17.7 2014.9 2022 H 6 21 6 59.946 +38 45 14.23 18.8 0.93 5.5 69.6 2013.5 636 H 7 6 0 21.624 +31 25 44.73 17.0 1.76 10.7 4.0 2012.2 262 H/A 8 15 4 17.808 +60 23 5.38 17.1 0.77 12.6 5.2 2007.7 520 H 9 17 28 40.850 −46 53 50.64 13.8 1.92 12.9 12.2 2011.2 3401 H 10 21 6 56.938 +38 45 20.65 16.3 1.37 13.2 41.8 2007.9 3686 H 11 14 29 39.583 −62 40 42.81 17.2 1.70 13.3 23.4 2006.1 1360 H 12 13 18 56.199 −3 4 20.84 13.9 0.77 21.8 8.3 2012.6 1136 H 13 11 45 46.125 −64 50 29.29 17.8 1.26 41.6 20.4 2007.5 2805 H (continued) Table 3.3—Continued

Event RA DEC VB−Vτd2000 t0 β #hms ◦ hr yr mas

14 17 57 47.972 +4 43 0.77 18.8 0.49 58.0 83.3 2008.2 1286 H 15 15 10 11.993 −16 28 29.09 14.8 0.88 58.1 44.8 2012.2 1862 H 16 15 32 11.805 −41 16 47.52 16.5 1.37 67.2 20.0 2012.6 3261 H 17 20 5 1.087 +54 25 56.10 18.7 1.10 87.9 12.0 2008.2 243 H

155 18 12 37 50.912 −52 0 5.95 18.6 1.48 91.7 12.1 2011.6 974 H/A 19 6 58 38.763 −0 28 53.97 16.1 0.93 92.0 5.4 2007.8 863 H/A 20 23 13 18.727 +57 10 12.54 17.2 1.26 92.8 15.7 2007.2 4364 H 21 5 46 2.356 +37 17 0.80 17.8 1.21 113.5 6.9 2009.6 1349 H/A 22 15 18 40.012 −18 37 39.53 17.1 0.82 133.3 7.6 2013.2 1109 H/A 23 14 29 38.124 −62 40 35.97 17.9 0.99 147.4 34.7 2009.0 3341 H 24 14 29 36.081 −62 40 40.49 17.6 1.10 157.5 47.6 2012.3 3885 H 25 21 6 58.044 +38 44 44.13 16.4 0.66 187.4 34.9 2006.5 9655 H 26 21 17 13.381 −38 51 59.53 16.0 1.92 192.1 22.3 2005.7 10139 H (continued) Table 3.3—Continued

Event RA DEC VB−Vτd2000 t0 β #hms ◦ hr yr mas

27 20 51 6.792 +7 1 35.53 18.3 0.88 194.3 2.4 2005.4 464 H/A 28 21 6 58.833 +38 45 32.30 17.4 1.70 221.6 66.8 2012.6 8266 H 29 17 57 47.361 +4 44 5.21 16.7 1.10 231.6 150.0 2014.5 5503 H 30 16 13 47.751 −57 34 27.26 18.7 1.76 242.9 14.9 2009.0 1414 H 156 31 9 51 10.020 −12 19 57.92 17.0 0.49 261.4 11.8 2006.3 2040 H 32 5 31 27.623 −3 40 56.32 19.1 0.71 291.2 18.6 2008.2 3056 H

Numeration follows the numbers in tables 1 and 2. Source star’s right ascension and declination are from USNO-A2.0, at plate epoch, equinox J2000. Visual magnitude is in Tycho, and color in Johnson system, as calibrated from photographic magnitudes (see §3.3.3). Event is described by τ

(SIM observing time), d2000 lens-source separation in year 2000.0, t0 time of closest approach and β, the minimum impact parameter. If β<300 mas, τ is calculated using β = 300 mas. The last column designates whether the event was detected only using the Hipparcos catalog (H), or in both the Hipparcos and the ACT catalogs (H/A). Table 3.4. NLTT events - lens star properties

Name RA DEC epoch VB-Vµp.a. MV rM Cl. #hms◦ / yr pc M

1 0 35 49.472 +52 41 20.43 1954.8 12.3 1.75 0.789 102 14.9 3 0.1 MS 2 452- 1 19 21 41.879 +20 53 11.01 1953.6 13.6 1.97 1.751 213 18.8 1 0.1 MS 3 23 6 20.173 +65 3 40.05 1952.6 15.4 1.92 0.328 127 17.8 3 0.1 MS 4 755- 18 20 27 30.371 −13 17 36.29 1953.8 18.2 0.66 0.375 215 15.0 46 0.6 WD − 157 5 650-237 2 31 56.521 8 31 49.98 1953.9 16.0 1.54 0.302 164 11.0 104 0.2 SD 6 634- 1 19 56 30.645 −1 1 58.61 1951.6 13.6 0.49 0.790 211 14.5 7 0.6 WD 7 275- 67 16 35 14.667 +35 47 26.59 1954.5 13.8 1.98 0.221 236 18.8 1 0.1 MS 8 R 627 11 24 16.301 +21 21 36.50 1955.2 14.3 0.76 1.050 270 15.1 7 0.6 WD 9 543- 33 7 50 14.730 +7 12 55.88 1956.0 16.7 0.99 1.778 173 15.3 19 0.6 WD 10 23 18 6.829 +49 28 28.63 1954.6 13.2 1.15 0.320 177 8.0 107 0.5 SD 11 R 619 8 11 54.025 +8 50 31.39 1951.2 13.0 1.37 5.211 167 15.9 3 0.6 WD 12 388- 57* 17 36 11.388 +23 48 32.58 1951.5 19.4 1.21 0.184 196 15.6 59 0.6 WD 13 R 248 23 41 54.568 +44 11 54.44 1952.6 12.8 2.08 1.617 177 20.8 0 0.1 MS 14 707- 8 1 9 2.851 −10 42 12.41 1951.9 16.5 0.44 0.198 98 14.3 27 0.6 WD 15 816- 34 21 0 36.541 −18 16 49.96 1982.5 17.1 0.71 0.198 207 15.0 26 0.6 WD (cnt.) Table 3.4—Continued

Name RA DEC epoch VB-Vµp.a. MV rM Cl. #hms◦ / yr pc M

16 385- 32 16 6 36.066 +24 28 56.74 1950.4 18.4 1.43 0.316 191 16.1 28 0.6 WD 17 5 50 22.925 +17 19 40.47 1951.9 15.9 0.60 0.589 143 14.8 16 0.6 WD 18 197- 4 2 25 40.398 +42 27 9.20 1952.0 18.4 1.10 0.232 103 15.4 40 0.6 WD 19 795- 43 12 38 42.281 −19 21 39.48 1954.2 13.1 1.53 0.356 301 11.0 27 0.4 MS 20 19 38 48.802 +35 11 59.24 1952.5 15.0 1.48 0.786 359 10.0 101 0.3 SD

158 21 329- 21 16 1 47.561 +30 30 56.40 1950.4 19.1 1.21 0.217 151 15.6 51 0.6 WD 22 689- 11 17 55 49.486 −7 35 52.55 1954.5 12.4 1.31 0.253 234 8.2 70 0.7 MS 23 101- 15* 16 34 26.512 +57 8 51.70 1955.3 13.2 1.59 1.620 316 11.9 18 0.2 SD 24 * 1 47 57.166 +60 7 37.37 1954.8 13.4 1.59 0.238 228 11.9 19 0.3 MS 25 W 1471 17 42 13.355 −8 48 38.47 1954.5 13.5 1.64 0.965 240 12.9 13 0.2 MS 26 332- 17 17 19 4.649 +28 5 10.42 1950.5 16.8 1.76 0.237 239 14.9 24 0.1 MS 27 497- 4 13 8 26.657 +12 26 37.14 1955.4 14.3 1.70 0.288 268 13.9 12 0.1 MS 28 R 201 21 40 27.505 +54 0 27.20 1955.9 14.9 1.59 0.414 76 11.9 39 0.3 MS 29 R 28 4 13 0.374 +52 37 18.86 1954.8 13.7 1.65 0.910 203 12.9 14 0.2 MS 30 921- 25 18 6 31.135 −30 9 46.02 1977.5 16.1 0.16 0.252 158 12.7 48 0.6 WD (cnt.) Table 3.4—Continued

Name RA DEC epoch VB-Vµp.a. MV rM Cl. #hms◦ / yr pc M

31 627- 16 17 15 24.814 +1 19 17.37 1954.6 15.6 1.59 0.369 287 11.9 53 0.2 SD 32 29- 23 0 43 57.017 +75 12 26.58 1954.7 18.3 0.82 0.302 104 15.2 41 0.6 WD 33 785- 11 8 31 8.147 −20 41 59.95 1982.0 17.1 1.75 0.251 132 14.9 28 0.1 MS 34 575- 26 20 34 31.828 +7 57 32.55 1951.6 15.0 1.75 0.374 82 14.9 11 0.1 MS 35 722- 1 7 13 39.010 −13 27 8.92 1958.9 14.7 1.21 1.277 153 15.6 7 0.6 WD

159 36 5 10 28.686 +31 17 40.41 1955.8 17.0 1.21 0.690 104 15.6 19 0.6 WD 37 48-813 23 5 14.276 +71 23 4.05 1952.6 19.4 0.60 0.277 56 14.8 82 0.6 WD 38 W 1084 20 43 14.497 +55 19 31.92 1952.7 15.1 1.59 1.915 28 16.9 4 0.6 WD 39 447- 63 17 7 15.733 +19 25 51.92 1954.5 13.3 1.65 0.180 166 12.9 12 0.2 MS 40 206- 11 7 11 9.967 +43 30 24.24 1954.2 15.8 1.59 0.680 146 11.9 59 0.2 SD 41 572- 1 19 22 4.534 +7 2 51.55 1950.6 12.5 1.81 0.836 242 15.9 2 0.1 MS 42 22 36 37.955 +53 3 16.61 1953.8 17.2 0.05 0.260 226 11.7 127 0.6 WD 43 727- 3 9 9 54.157 −11 26 6.38 1954.2 14.7 1.65 0.483 118 12.9 23 0.2 MS 44 3 43 49.491 +63 40 30.53 1954.1 12.9 1.37 0.962 142 8.5 77 0.4 SD 45 St2051B 4 31 1.299 +59 0 32.22 1953.1 13.5 1.15 2.383 144 15.5 4 0.6 WD (cnt.) Table 3.4—Continued

Name RA DEC epoch VB-Vµp.a. MV rM Cl. #hms◦ / yr pc M

46 0 9 52.296 +53 1 12.88 1953.0 13.5 2.08 0.240 85 20.8 0 0.1 MS 47 106- 38 20 14 42.347 +61 46 2.31 1952.7 16.9 1.43 0.698 26 16.1 14 0.6 WD 48 20 34 2.957 +64 19 16.39 1952.6 13.5 1.43 0.436 254 9.0 78 0.4 SD 49 297- 12 2 11 57.902 +32 21 56.99 1954.8 15.9 1.87 0.567 114 16.8 6 0.1 MS 50 2 7 2.738 +49 39 3.27 1953.9 13.0 1.15 0.498 150 8.0 98 0.5 SD

160 51 399-299 22 1 5.921 +29 9 37.24 1951.7 16.2 0.82 0.598 94 15.2 16 0.6 WD 52 W 359* 10 56 41.064 +7 2 59.26 1953.3 13.9 2.14 4.696 234 21.7 0 0.1 MS 53 1 4 2.104 +59 38 5.31 1952.7 15.1 1.65 0.420 100 12.9 28 0.2 MS 54 T 9 17 18 46.499 −29 46 5.89 1981.4 13.1 1.97 0.242 109 18.8 1 0.1 MS 55 1 48 47.971 +55 2 7.76 1954.8 14.4 1.26 0.280 96 8.3 165 0.4 SD 56 15:4074B 20 11 14.683 +16 10 48.99 1951.7 13.8 1.42 0.572 313 9.0 90 0.4 SD 57 693- 14 19 38 31.486 −2 51 12.09 1953.8 11.1 0.83 0.286 111 7.4 55 0.5 SD 58 685- 55 16 34 41.539 −9 1 44.30 1954.3 12.2 0.60 0.185 176 6.1 167 0.7 SD 59 569- 98 18 2 32.457 +5 45 2.87 1950.5 18.8 0.76 0.472 205 15.1 55 0.6 WD 60 R 19 2 19 0.361 +35 21 39.42 1951.8 12.7 1.65 0.792 122 12.9 9 0.2 MS (cnt.) Table 3.4—Continued

Name RA DEC epoch VB-Vµp.a. MV rM Cl. #hms◦ / yr pc M

61 12 38 33.755 +35 13 19.73 1950.4 14.9 1.53 0.267 223 11.0 63 0.4 MS 62 0 28 51.956 +50 22 27.91 1954.7 13.3 1.98 0.440 74 18.8 1 0.1 MS 63 555- 5 12 21 49.935 +6 44 7.77 1956.2 14.5 1.53 0.732 174 11.0 52 0.2 SD 64 53:2911* 22 32 49.386 +53 47 35.92 1952.7 10.0 1.19 1.318 86 8.1 24 0.5 SD 65 44- 47 17 37 24.422 +71 4 16.12 1953.7 12.5 1.53 0.482 143 11.0 21 0.4 MS

161 66 751- 1 19 3 17.340 −13 33 51.30 1951.6 16.4 1.53 0.780 226 16.6 9 0.6 WD 67 5 44 0.906 +40 57 36.75 1953.0 15.8 0.33 1.229 147 13.8 25 0.6 WD 68 21 35 19.123 +46 33 41.32 1952.7 17.2 0.93 0.459 200 15.3 25 0.6 WD 69 187- 7* 21 35 19.123 +46 33 41.32 1952.7 17.2 0.93 0.459 200 15.3 25 0.6 WD 70 697- 45 21 31 21.313 −5 11 16.35 1954.5 15.0 1.54 0.374 96 11.0 66 0.2 SD 71 16- 36 5 37 59.137 +79 31 7.05 1955.0 18.9 0.99 1.192 143 15.3 51 0.6 WD 72 877- 22 22 52 25.884 −22 20 1.71 1982.8 13.5 1.70 0.291 192 13.9 8 0.1 MS 73 R 66 5 49 56.578 +36 50 46.56 1954.9 12.5 1.64 0.510 165 12.9 8 0.2 MS 74 5 48 23.867 +7 45 50.89 1955.9 14.5 1.53 0.276 165 11.0 52 0.4 MS 75 404- 7 23 57 50.184 +19 48 54.14 1954.7 17.0 1.04 0.308 33 15.4 21 0.6 WD (cnt.) Table 3.4—Continued

Name RA DEC epoch VB-Vµp.a. MV rM Cl. #hms◦ / yr pc M

76 747- 11 17 11 25.904 −14 47 40.78 1954.5 14.5 0.32 0.371 132 13.8 14 0.6 WD 77 382- 55 14 59 51.924 +21 24 57.91 1950.3 17.6 1.92 0.218 283 17.8 9 0.1 MS 78 192- 23 0 1 45.551 +41 36 2.69 1954.8 14.8 1.53 0.299 141 11.0 60 0.4 MS 79 787- 49 9 29 42.151 −17 32 36.06 1954.2 16.0 0.38 0.447 134 14.1 24 0.6 WD 80 15 27 44.993 −9 1 18.36 1955.4 15.5 1.70 0.318 172 13.9 20 0.1 MS

162 81 5 44 0.906 +40 57 36.75 1953.0 15.8 0.33 1.229 147 13.8 25 0.6 WD 82 18 39 28.561 +4 11 48.05 1950.5 15.6 1.26 0.506 240 8.3 287 0.4 SD 83 L 560-9 18 8 7.463 −30 55 37.12 1977.5 16.7 0.93 0.300 204 15.3 20 0.6 WD 84 813- 32 19 57 26.935 −17 30 16.64 1953.6 14.8 1.53 0.499 98 11.0 60 0.2 SD 85 544- 37 8 15 18.924 +4 55 46.72 1949.9 18.1 1.48 0.214 162 10.0 422 0.3 SD 86 * 21 10 59.850 +46 57 47.02 1952.5 14.6 1.26 0.395 218 8.3 181 0.4 SD 87 82- 44 3 0 58.173 +59 36 40.80 1954.1 15.2 1.59 0.223 92 11.9 44 0.3 MS 88 R 341 3 6 15.472 +51 3 45.96 1953.8 13.3 1.48 0.846 124 10.0 46 0.3 SD 89 * 13 14 9.166 +6 18 43.70 1956.2 15.8 1.37 0.334 216 8.5 294 0.4 SD 90 264- 49 11 24 9.491 +35 47 30.88 1953.2 18.5 1.54 0.277 269 16.6 25 0.6 WD (cnt.) Table 3.4—Continued

Name RA DEC epoch VB-Vµp.a. MV rM Cl. #hms◦ / yr pc M

91 336- 6* 19 7 38.155 +32 31 45.32 1950.5 12.2 1.43 1.635 49 9.0 43 0.4 SD 92 23 57 40.443 +23 19 8.49 1950.6 12.3 1.42 1.460 135 9.0 45 0.4 SD 93 18 47 15.693 −17 25 57.59 1954.5 14.5 1.32 0.480 201 8.4 167 0.4 SD 94 372- 4 10 20 42.745 +20 27 58.46 1955.3 19.0 0.71 0.265 202 15.0 61 0.6 WD 95 R 28 4 13 0.374 +52 37 18.86 1954.8 13.7 1.65 0.910 203 12.9 14 0.2 MS

163 96 4 48 8.475 +48 32 33.90 1953.8 16.5 1.32 0.503 123 15.8 14 0.6 WD 97 23 9 36.225 +33 12 40.10 1954.7 13.1 1.48 0.366 75 10.0 42 0.5 MS 98 642- 53* 23 21 16.024 +1 2 36.62 1953.8 19.1 0.82 0.267 207 15.2 60 0.6 WD 99 642- 52 23 21 16.024 +1 2 36.62 1953.8 19.1 0.82 0.267 207 15.2 60 0.6 WD 100 66- 58 13 32 49.849 +65 51 28.96 1955.4 15.6 1.59 0.254 334 11.9 53 0.3 MS 101 809- 20 18 15 13.220 −19 23 41.54 1954.5 12.8 0.93 0.382 162 7.6 112 0.5 SD 102 15 24 38.144 −6 49 7.71 1955.4 15.9 1.70 0.477 212 13.9 25 0.1 MS 103 152- 27 15 29 29.620 −61 46 28.80 1980.3 16.2 0.49 0.210 189 14.5 22 0.6 WD 104 358-663 4 18 24.239 +22 11 51.22 1950.9 18.0 1.54 0.404 136 16.6 20 0.6 WD 105 14 55 2.824 +43 1 45.61 1955.2 11.9 1.15 0.302 207 8.0 59 0.5 SD (cnt.) Table 3.4—Continued

Name RA DEC epoch VB-Vµp.a. MV rM Cl. #hms◦ / yr pc M

106 700- 35 22 32 47.803 −5 57 10.13 1954.5 11.8 1.10 0.266 242 7.9 61 0.5 SD 107 241- 23 0 31 3.915 +36 40 50.36 1951.8 15.8 1.48 0.257 174 10.0 146 0.3 SD 108 555- 21 12 27 54.387 +5 12 33.92 1956.2 14.1 1.70 0.576 244 13.9 11 0.1 MS 109 838- 25 6 14 16.133 −23 10 16.91 1982.6 15.1 0.98 0.388 156 7.7 298 0.5 SD 110 17 46 23.657 −18 6 57.04 1950.5 13.7 1.65 0.210 207 12.9 14 0.2 MS

164 111 29- 23 0 43 57.017 +75 12 26.58 1954.7 18.3 0.82 0.302 104 15.2 41 0.6 WD 112 782- 13 7 19 21.571 −19 4 53.58 1953.0 13.2 1.75 0.192 12 14.9 5 0.1 MS 113 152- 10 1 38 30.185 +47 32 24.66 1953.8 18.3 1.26 0.362 107 15.7 33 0.6 WD 114 458- 12 21 36 9.975 +19 5 7.41 1951.7 11.9 1.04 0.326 76 7.8 66 0.5 SD 115 23 10 3.421 +63 58 15.31 1952.6 14.7 0.38 0.400 175 14.1 13 0.6 WD 116 23 15 24.162 +9 44 42.71 1951.6 13.7 1.21 0.412 74 8.2 130 0.5 SD 117 -3:5711 23 49 22.223 −2 34 26.83 1954.6 10.8 0.67 0.260 93 6.6 72 0.6 SD 118 377- 12 12 33 50.545 +22 34 32.05 1955.4 18.6 1.59 0.307 260 16.9 22 0.6 WD 119 244- 7 1 48 42.049 +38 16 21.41 1954.7 13.9 1.75 0.276 127 14.9 6 0.1 MS 120 763- 7* 23 42 19.220 −13 56 29.82 1953.6 15.4 1.37 0.295 57 8.5 244 0.4 SD (cnt.) Table 3.4—Continued

Name RA DEC epoch VB-Vµp.a. MV rM Cl. #hms◦ / yr pc M

121 17 30 20.955 +19 12 37.12 1951.5 13.5 1.48 0.396 104 10.0 51 0.3 SD 122 R 600 4 41 20.420 +22 54 52.55 1950.9 13.0 1.32 0.650 145 8.4 84 0.4 SD 123 6 59 29.457 +19 30 43.79 1951.8 13.3 1.48 0.280 225 10.0 46 0.5 MS 124 629- 12 18 6 21.109 +2 3 21.23 1953.5 11.9 0.87 0.358 210 7.5 77 0.5 SD 125 552- 14 11 11 55.685 +3 37 32.06 1955.3 18.2 0.66 0.377 255 15.0 46 0.6 WD

165 126 173- 45 13 41 50.645 +47 0 7.27 1956.3 17.3 1.70 0.201 282 13.9 47 0.1 MS 127 285- 9 21 12 29.944 +35 55 58.31 1951.5 13.2 1.75 0.181 70 14.9 5 0.1 MS 128 757-135 21 13 8.753 −9 48 56.69 1953.7 17.3 1.75 0.181 216 14.9 30 0.1 MS 129 753- 7 19 36 8.097 −11 40 39.10 1951.6 17.9 1.31 0.248 186 15.8 26 0.6 WD 130 0 41 57.203 +57 48 4.82 1952.7 14.0 1.64 0.235 106 12.9 17 0.2 MS 131 80 -81 1 43 23.245 +62 39 32.99 1954.8 15.4 1.59 0.184 156 11.9 49 0.3 MS 132 2 8 43.243 +25 36 23.04 1953.8 14.1 1.48 0.332 79 10.0 67 0.3 SD 133 196- 61 2 25 29.837 +44 47 46.24 1952.0 12.4 0.99 0.204 238 7.7 86 0.5 SD 134 711- 11 2 50 15.703 −12 19 8.38 1955.9 16.2 1.70 0.226 84 13.9 28 0.1 MS 135 119- 44 5 28 3.987 +54 55 40.68 1955.0 15.7 1.59 0.206 142 11.9 56 0.3 MS (cnt.) Table 3.4—Continued

Name RA DEC epoch VB-Vµp.a. MV rMCl. #hms◦ / yr pc M

136 58-151 7 20 3.997 +68 27 48.50 1953.2 18.7 1.70 0.220 223 13.9 89 0.1 SD 137 256- 19 7 30 9.144 +32 48 32.43 1953.1 18.5 1.59 0.226 249 11.9 203 0.2 SD 138 678- 54 13 42 12.158 −5 59 1.36 1952.4 19.0 1.04 0.235 234 15.4 53 0.6 WD 139 679- 21 14 4 49.451 −5 31 20.97 1957.3 17.9 0.38 0.244 253 14.1 56 0.6 WD

166 140 329- 55 16 20 36.017 +29 15 18.04 1954.5 16.4 1.48 0.285 184 10.0 193 0.3 SD 141 16 50 22.992 −1 46 17.72 1950.5 13.8 0.93 0.257 198 7.6 178 0.5 SD 142 19 31 30.369 +32 20 51.08 1951.5 14.9 1.48 0.310 341 10.0 97 0.3 SD 143 338- 2 19 50 1.235 +32 34 51.31 1953.5 12.3 0.76 0.526 62 7.1 110 0.6 SD 144 105-523 20 3 9.374 +61 2 39.18 1952.6 12.7 1.26 0.187 13 7.9 88 0.7 MS 145 816- 34 21 0 36.809 −18 16 44.59 1954.6 17.5 1.37 0.198 207 8.5 643 0.4 SD 146 48-526 22 8 58.979 +70 41 41.01 1952.6 18.6 1.04 0.247 44 15.4 44 0.6 WD

Events are given in the order of increasing τ (see Table 5). Names are taken from NLTT. Right ascension and declination come from USNO-A2.0. Equinox J2000, epoch is that of the plate and is given in a separate column. Visual magnitude is in Tycho system, transformed from USNO-A2.0 photographic magnitudes. Color is in Johnson system, also transformed from photographic magnitudes. Proper motions are from NLTT, but in equinox J2000. For distance, physical parameters and class, see §3.3.3 (MS - main sequence star, SD - subdwarf, WD - white dwarf). Table 3.5. NLTT - source star and event properties

∗ Event RA DEC VB−Vτ τd2000 t0 β #hms ◦ hr hr yr mas

1 0 35 54.579 +52 41 10.80 15.8 0.76 0.8 0.8 11.7 2014.8 292 2 19 21 38.138 +20 51 49.73 18.4 1.81 2.3 2.3 15.5 2008.9 167 3 23 6 22.538 +65 3 28.89 16.2 2.14 3.2 3.2 3.1 2009.5 40 4 20 27 29.604 −13 17 52.59 17.6 0.44 4.0 0.0 2.4 2006.5 97 − 167 5 2 31 56.861 8 32 7.50 15.3 0.98 5.8 0.0 4.3 2014.2 62 6 19 56 29.051 −1 2 42.39 15.3 0.93 6.5 0.1 11.7 2014.7 1845 7 16 35 13.806 +35 47 19.73 19.4 1.21 6.9 6.9 2.5 2011.2 244 8 11 24 12.471 +21 21 39.64 14.1 0.99 9.4 0.6 7.2 2006.2 3094 9 7 50 15.485 +7 11 15.24 17.0 0.71 18.0 3.5 23.0 2012.9 677 10 23 18 6.898 +49 28 12.46 14.8 0.49 26.5 26.5 1.7 2005.2 201 11 8 11 58.445 +8 45 29.77 18.7 0.87 28.2 3.3 54.6 2010.5 2739 12 17 36 11.179 +23 48 22.91 18.4 0.93 28.7 0.0 1.2 2006.3 139 13 23 41 54.678 +44 10 17.02 17.8 1.21 43.3 7.0 21.1 2012.8 3970 14 1 9 3.653 −10 42 13.89 19.0 0.77 49.0 49.0 2.4 2012.1 196 15 21 0 36.351 −18 16 53.75 15.9 0.27 52.8 0.0 1.3 2005.7 699 (continued) Table 3.5—Continued

∗ Event RA DEC VB−Vτ τd2000 t0 β #hms ◦ hr hr yr mas

16 16 6 35.790 +24 28 36.69 19.4 0.54 73.2 73.2 4.7 2014.9 29 17 5 50 24.507 +17 19 11.32 18.6 1.04 73.6 12.3 8.6 2014.6 735 18 2 25 41.586 +42 27 5.97 18.6 1.21 74.0 74.0 2.4 2010.3 136 19 12 38 41.040 −19 21 28.98 19.1 1.59 117.7 117.7 4.1 2011.6 26 20 19 38 48.723 +35 12 44.22 15.7 1.48 120.4 120.4 7.7 2009.8 58

168 21 16 1 48.019 +30 30 44.81 16.9 0.44 121.0 0.0 2.3 2010.4 504 22 17 55 48.684 −7 36 1.16 18.2 1.32 128.7 128.7 3.2 2012.6 113 23 16 34 18.489 +57 9 59.07 17.0 0.66 150.1 35.0 21.4 2013.2 621 24 1 47 55.998 +60 7 28.82 17.3 0.60 161.5 24.9 1.5 2006.0 569 25 17 42 10.236 −8 49 4.78 17.9 1.48 162.7 41.9 9.3 2009.6 591 26 17 19 3.782 +28 5 3.45 17.1 0.66 163.7 163.7 1.7 2007.1 7 27 13 8 25.655 +12 26 36.56 19.2 0.55 171.9 171.9 1.8 2006.3 45 28 21 40 29.996 +54 0 33.33 16.7 0.93 178.8 58.1 4.6 2011.0 526 29 4 12 58.173 +52 36 35.13 16.6 0.93 180.3 15.7 7.0 2007.6 1018 30 18 6 31.383 −30 9 52.44 17.7 0.71 219.3 44.3 1.6 2006.0 537 (continued) Table 3.5—Continued

∗ Event RA DEC VB−Vτ τd2000 t0 β #hms ◦ hr hr yr mas

31 17 15 23.471 +1 19 23.59 16.7 0.76 236.5 236.5 4.3 2011.7 159 32 0 44 1.323 +75 12 21.45 16.3 1.32 252.8 0.0 3.7 2011.8 927 33 8 31 8.473 −20 42 4.44 17.4 1.32 270.0 270.0 1.9 2007.5 256 34 20 34 33.169 +7 57 34.37 15.9 1.21 290.9 10.0 2.1 2005.0 1047 35 7 13 41.229 −13 28 7.39 18.9 0.71 302.3 20.1 14.6 2011.2 2623

169 36 5 10 31.270 +31 17 30.97 19.3 0.82 333.5 31.4 4.0 2005.7 958 37 23 5 16.976 +71 23 13.35 17.8 0.77 335.4 0.0 2.8 2010.1 425 38 20 43 21.091 +55 21 11.31 19.2 0.76 340.5 65.8 23.9 2012.3 3761 39 17 7 15.943 +19 25 42.30 18.5 0.82 367.5 58.2 1.9 2010.4 496 40 7 11 12.028 +43 29 50.10 17.0 1.37 377.0 377.0 9.7 2014.2 241 41 19 22 1.631 +7 2 24.64 16.0 1.54 381.2 18.5 10.1 2011.4 3233 42 22 36 36.875 +53 3 6.69 17.7 0.66 405.2 334.9 1.9 2007.3 330 43 9 9 55.868 −11 26 19.66 19.2 0.49 432.9 432.9 6.3 2013.1 178 44 3 43 54.491 +63 39 48.85 18.4 1.37 451.3 451.3 9.1 2009.5 94 45 4 31 12.144 +58 58 47.85 17.6 0.76 568.5 197.6 23.1 2009.2 7577 (continued) Table 3.5—Continued

∗ Event RA DEC VB−Vτ τd2000 t0 β #hms ◦ hr hr yr mas

46 0 9 53.777 +53 1 15.84 18.7 0.82 587.7 0.5 2.9 2009.5 1790 47 20 14 44.986 +61 46 37.82 18.9 0.93 603.9 12.2 7.3 2010.2 1608 48 20 33 59.518 +64 19 8.75 15.4 1.32 613.7 34.9 3.1 2006.8 978 49 2 12 0.119 +32 21 43.15 17.6 0.82 628.7 31.4 5.7 2009.9 1126 50 2 7 4.257 +49 38 36.99 18.5 0.10 631.3 631.3 7.2 2014.4 245

170 51 22 1 8.543 +29 9 37.00 17.4 1.48 655.9 3.7 5.8 2009.1 2063 52 10 56 24.791 +7 0 27.92 15.8 0.76 658.7 421.7 68.5 2014.0 19571 53 1 4 5.115 +59 38 0.75 18.4 0.93 720.1 298.3 3.4 2008.1 466 54 17 18 47.071 −29 46 5.85 16.2 1.43 760.1 1.6 3.5 2010.5 2424 55 1 48 49.863 +55 2 6.08 17.5 1.26 771.6 771.6 3.7 2013.2 90 56 20 11 13.029 +16 11 11.76 17.3 0.82 825.3 256.9 5.4 2009.3 538 57 19 38 32.546 −2 51 17.35 17.7 0.93 846.7 80.8 3.6 2012.2 705 58 16 34 41.572 −9 1 54.26 18.4 1.15 859.0 845.8 1.5 2008.2 245 59 18 2 31.573 +5 44 38.70 14.9 1.43 898.2 0.2 4.5 2008.7 1866 60 2 19 3.789 +35 21 14.03 19.8 0.38 921.1 111.4 10.9 2013.7 863 (continued) Table 3.5—Continued

∗ Event RA DEC VB−Vτ τd2000 t0 β #hms ◦ hr hr yr mas

61 12 38 32.897 +35 13 6.02 12.7 1.48 947.8 0.0 4.3 2014.8 1646 62 0 28 54.282 +50 22 36.51 18.7 0.71 1080.0 3.1 4.4 2008.7 2155 63 12 21 50.167 +6 43 30.54 18.2 1.59 1180.4 504.0 5.3 2007.3 459 64 22 32 58.125 +53 47 43.99 17.7 1.32 1593.4 77.9 15.7 2011.7 2409 65 17 37 27.797 +71 3 52.41 19.2 1.09 1695.4 79.0 6.6 2013.5 1081

171 66 19 3 14.965 −13 34 17.86 15.7 0.82 1727.0 220.8 7.6 2007.2 5161 67 5 44 4.547 +40 56 36.51 18.5 0.82 1851.0 57.2 15.4 2012.4 2254 68 21 35 18.524 +46 33 17.42 16.3 0.93 1861.0 8.1 3.7 2006.3 2263 69 21 35 18.524 +46 33 17.42 16.3 0.93 1861.0 8.1 3.7 2006.3 2263 70 21 31 22.614 −5 11 19.26 16.0 0.98 1863.1 104.7 2.8 2007.0 920 71 5 38 15.352 +79 30 15.74 14.7 1.10 1872.0 206.6 14.6 2011.8 3887 72 22 52 25.820 −22 20 10.76 18.6 1.15 1963.9 47.6 4.2 2013.9 1000 73 5 49 57.019 +36 50 20.82 18.9 1.21 2080.0 42.5 3.5 2006.4 1393 74 5 48 24.167 +7 45 37.30 17.3 0.49 2098.8 86.2 2.3 2007.6 860 75 23 57 50.928 +19 49 8.44 19.7 0.76 2242.2 57.3 3.9 2012.2 1019 (continued) Table 3.5—Continued

∗ Event RA DEC VB−Vτ τd2000 t0 β #hms ◦ hr hr yr mas

76 17 11 26.777 −14 47 54.83 17.8 0.87 2586.4 4.1 2.8 2005.2 2057 77 14 59 51.067 +21 24 59.92 18.4 0.93 2959.1 180.1 1.5 2005.9 692 78 0 1 46.608 +41 35 51.44 13.1 0.76 3180.2 2.9 3.4 2008.9 2131 79 9 29 43.422 −17 32 55.88 18.9 0.77 3356.7 33.8 6.6 2014.3 1544 80 15 27 45.232 −9 1 36.89 18.1 0.60 3379.0 195.0 4.7 2014.6 848

172 81 5 44 4.731 +40 56 36.92 16.6 0.93 3418.2 429.3 16.6 2013.0 4224 82 18 39 26.907 +4 11 34.35 16.2 1.21 3428.8 745.7 3.3 2006.4 643 83 18 8 7.261 −30 55 47.57 17.4 2.36 3654.7 5.7 4.3 2012.9 1824 84 19 57 29.012 −17 30 19.74 17.4 0.88 4153.8 323.8 6.8 2013.5 931 85 8 15 19.197 +4 55 34.92 16.5 0.66 4355.4 0.1 1.8 2008.2 289 86 21 10 58.628 +46 57 29.33 16.1 0.76 4638.9 271.9 3.0 2007.4 931 87 3 0 59.817 +59 36 41.43 15.7 1.04 4766.7 29.8 2.5 2009.9 1150 88 3 6 19.919 +51 3 19.98 16.3 1.09 4919.1 73.7 10.4 2012.0 2182 89 13 14 8.421 +6 18 29.92 15.0 1.31 5696.5 0.0 3.2 2009.1 909 90 11 24 8.201 +35 47 31.74 19.7 0.66 6186.1 79.4 2.9 2009.8 1120 (continued) Table 3.5—Continued

∗ Event RA DEC VB−Vτ τd2000 t0 β #hms ◦ hr hr yr mas

91 19 7 44.041 +32 32 48.05 18.2 0.49 6590.8 144.9 16.4 2010.0 2023 92 23 57 45.331 +23 18 4.94 17.3 1.43 6680.6 539.1 20.6 2014.0 2665 93 18 47 14.940 −17 26 21.29 14.5 1.92 6935.0 54.1 4.5 2008.6 1701 94 10 20 42.195 +20 27 45.69 14.0 0.93 7497.0 12.9 3.7 2010.9 2351 95 4 12 58.125 +52 36 37.44 18.1 1.48 7537.2 198.2 5.5 2005.5 2343

173 96 4 48 11.065 +48 32 20.65 17.8 1.32 7898.8 158.6 6.3 2011.0 3102 97 23 9 37.763 +33 12 43.90 18.4 0.93 8163.2 91.1 3.3 2008.3 1351 98 23 21 15.663 +1 2 23.85 18.9 0.77 9066.2 0.0 1.8 2005.6 957 99 23 21 15.663 +1 2 23.85 18.9 0.77 9066.2 0.0 1.8 2005.6 957 100 13 32 48.813 +65 51 40.30 19.0 0.38 9804.3 863.4 1.8 2006.5 682 101 18 15 13.579 −19 24 0.98 16.9 1.65 10406.1 167.0 3.0 2007.0 1274 102 15 24 37.078 −6 49 28.27 14.6 1.15 11879.7 119.5 5.3 2009.6 2670 103 15 29 29.890 −61 46 35.06 15.4 1.21 12565.1 125.8 3.3 2008.4 2824 104 4 18 25.230 +22 11 33.41 18.4 1.15 13788.2 52.8 3.5 2006.4 2365 105 14 55 2.393 +43 1 29.84 13.7 1.15 14533.9 145.5 3.9 2008.9 2879 (continued) Table 3.5—Continued

∗ Event RA DEC VB−Vτ τd2000 t0 β #hms ◦ hr hr yr mas

106 22 32 46.989 −5 57 17.76 18.6 0.32 14750.1 269.9 2.4 2008.3 1014 107 0 31 4.126 +36 40 35.67 15.9 1.15 16817.1 296.5 2.7 2009.6 1002 108 12 27 52.465 +5 12 23.48 16.4 1.59 17022.7 456.5 6.1 2008.9 3215 109 6 14 16.562 −23 10 27.88 16.4 0.11 17980.3 752.3 5.7 2014.6 1014 110 17 46 23.181 −18 7 7.06 17.0 1.37 18248.0 22.5 2.2 2007.6 1558

174 111 0 44 1.377 +75 12 24.33 17.3 1.04 18700.1 3.4 3.6 2010.1 1915 112 7 19 21.877 −19 4 43.77 17.0 1.10 23243.6 9.9 2.6 2007.8 2154 113 1 38 32.012 +47 32 16.06 16.8 0.44 25131.9 199.0 4.4 2009.6 2753 114 21 36 11.207 +19 5 12.95 19.2 0.49 25135.2 534.1 2.8 2007.8 1097 115 23 10 4.171 +63 57 52.25 18.9 1.37 25253.3 272.5 5.3 2011.2 2858 116 23 15 25.657 +9 44 46.56 14.4 0.60 28041.6 127.1 3.4 2005.7 2415 117 23 49 23.228 −2 34 26.57 19.3 0.93 28674.9 436.4 3.4 2012.4 1041 118 12 33 49.375 +22 34 26.10 16.9 0.93 29929.6 436.2 4.5 2010.8 3033 119 1 48 43.205 +38 16 13.21 19.0 1.21 35659.6 63.4 3.7 2012.0 1691 120 23 42 20.200 −13 56 22.21 15.1 1.42 38018.7 0.0 2.8 2008.2 1397 (continued) Table 3.5—Continued

∗ Event RA DEC VB−Vτ τd2000 t0 β #hms ◦ hr hr yr mas

121 17 30 22.585 +19 12 29.43 16.9 0.71 41301.9 89.9 5.4 2012.8 2002 122 4 41 21.804 +22 54 21.73 18.3 1.21 41411.3 458.4 4.7 2006.7 1835 123 6 59 28.537 +19 30 33.30 18.1 1.10 46083.0 83.9 3.6 2011.2 1696 124 18 6 20.297 +2 3 3.82 17.3 1.21 58202.4 108.9 4.9 2012.6 1947 125 11 11 54.241 +3 37 28.13 19.4 1.21 59445.1 200.6 5.3 2013.4 1775

175 126 13 41 49.532 +47 0 8.19 15.6 1.48 62787.4 0.0 2.9 2012.7 1434 127 21 12 30.703 +35 56 4.50 17.3 1.09 76812.7 405.5 3.3 2011.1 2621 128 21 13 8.389 −9 49 6.11 17.2 1.04 78326.6 0.0 2.7 2013.3 1148 129 19 36 8.131 −11 40 53.07 19.4 0.82 79304.8 62.9 2.6 2007.4 1884 130 0 41 58.855 +57 48 3.19 19.3 0.77 >100000 245.6 2.9 2008.6 2092 131 1 43 23.601 +62 39 23.63 17.0 0.93 >100000 121.4 2.0 2006.7 1525 132 2 8 44.489 +25 36 28.48 17.8 0.66 >100000 594.0 3.1 2006.7 2175 133 2 25 29.051 +44 47 38.67 18.8 0.77 >100000 593.9 2.4 2006.4 2034 134 2 50 16.477 −12 19 8.52 18.2 0.38 >100000 389.9 1.8 2005.7 1288 135 5 28 4.644 +54 55 30.76 18.1 1.04 >100000 430.1 2.6 2009.9 1554 (continued) Table 3.5—Continued

∗ Event RA DEC VB−Vτ τd2000 t0 β #hms ◦ hr hr yr mas

136 7 20 2.327 +68 27 40.62 18.0 0.60 >100000 0.1 2.2 2007.9 1411 137 7 30 8.141 +32 48 28.61 17.3 1.43 >100000 0.1 2.7 2011.5 899 138 13 42 11.445 −5 59 11.55 18.6 0.27 >100000 35.6 3.9 2014.6 1959 139 14 4 48.737 −5 31 26.82 17.3 1.15 >100000 427.1 2.9 2006.2 2447 140 16 20 36.102 +29 15 0.76 14.1 0.82 >100000 206.5 4.8 2014.7 2221 176 141 16 50 22.520 −1 46 32.75 16.4 0.71 >100000 392.7 4.3 2014.6 2161 142 19 31 30.049 +32 21 8.21 16.8 0.99 >100000 293.5 3.1 2008.0 1844 143 19 50 3.063 +32 35 6.21 18.3 0.87 >100000 757.4 3.7 2005.6 2174 144 20 3 9.483 +61 2 49.76 19.0 0.87 >100000 403.4 2.2 2008.8 1521 145 21 0 36.351 −18 16 53.75 15.9 0.27 >100000 0.0 2.7 2010.7 1695 146 22 9 0.546 +70 41 52.27 17.9 1.09 >100000 66.8 2.8 2007.3 2144

Numeration follows the numbers in table 4. Source star’s right ascension and declination are at plate epoch, equinox J2000. Visual magnitude is in Tycho, and color in Johnson system (see §3.3.3). Event ∗ is described by τ (SIM observing time), τ (SIM observing time with reduced impact parameter), d2000

lens-source separation in year 2000.0, t0 time of closest approach and β, the impact parameter. 0

−.5 log M

−1

51015

Mbol

Fig. 3.1.— Mass-luminosity relation for main sequence stars. The bold line shows the theoretical M–L relation for solar-metallicity stars, while the thin line shows the relation for metal-poor star with [Fe/H] = −1. The current mass measurements with their error bars are shown as circles (astrometric and spectroscopic binaries) and as triangles (eclipsing binaries).

177 Fig. 3.2.— Reduced proper motion diagram for Hipparcos and NLTT stars. For

Hipparcos, only stars with σπ/π < 20% are plotted (dots). From NLTT, only stars that are not in Hipparcos are plotted (crosses). In both cases, every 10th star is plotted. For Hipparcos data, abscissa is B − V color as usually determined from Tycho photometry but sometimes from ground-based photometry, while for NLTT it is calibrated from photographic magnitudes, and then randomized to correctly show regions of diﬀerent density. If all stars had the same transverse speed, this ﬁgure would look like an ordinary CMD. Solid lines indicate the boundaries of our assignment of stars to classes: red giants are in the upper right corner, white dwarfs in the lower part, and subdwarfs between white dwarfs and the main sequence.

178 Fig. 3.3.— Distance-modulus errors versus distance modulus for the Hipparcos stars shown in Fig. 3.2. The distance-modulus of each star is estimated by ﬁrst classifying it according to the bold-line divisions in Fig. 3.2 and then assigning it an absolute magnitude using color-magnitude relations appropriate for each class. The distance- modulus error is then the diﬀerence between this estimate and the value based on the measured trigonometric parallax. For distance moduli less than 2.7 (35 pc), the typical errors are only ∼ 0.53 mag. Errors are larger for more distant stars, but these are dominated by giants which are not relevant in the present study.

179 Fig. 3.4.— A color-magnitude diagram of Hipparcos stars with distances measured to better than 10%. The event-producing stars (lenses) from the Hipparcos and ACT catalogs are superimposed as ﬁlled dots, some of which are labeled.

180 Fig. 3.5.— 8 × 8 fields around 61 Cyg A/B in 1951 (upper left panel) and 1991 (lower left panel). Shown magnified is a 2 × 2 region where events will take place during 2005-2015 period. The chart corresponds to the 2 ×2 field with source stars labeled with the numbers corresponding to ‘Event #’ in Tables 3.1–3.3. Events 6 and 25 are produced by 61 Cyg B, and the other three by 61 Cyg A.

181 Fig. 3.6.— Apparent magnitude – proper motion (V −µ) plane showing lenses found in Hipparcos (×), or both in Hipparcos and ACT catalogs (∗). NLTT events that have τ<300 hr are shown as circles. Vertical long-dashed lines are completeness limits for Hipparcos and ACT. Horizontal long-dashed line is NLTT proper-motion cutoﬀ. Short dashed lines delineate regions excluded due to blot-out (lower line -

POSS I, upper line - SERC/ESO). Solid lines represent an MV = 6 star at various distances if belonging to proper motion selected disk population (lower line), or halo population (upper line).

182 Chapter 4

Mass Function of Stellar-Mass Objects

4.1. Mass Function in the Solar Neighborhood

4.1.1. Late-M and L dwarfs

The usual method of obtaining the mass function (MF) is by first obtaining the luminosity function (LF), and then applying the mass-luminosity relation to convert it into MF. We discussed the possibilities of refining the mass-luminosity relation in Chapter 3. For obtaining the precise LF, the main obstacle is the relatively small and incomplete sample of stars with precise trigonometric parallaxes. Using the approach of going beyond the fixed magnitude limit when assembling the input catalogs for the astrometric satellites, described in Chapter 1, one can obtain volume-limited sample for some classes of objects. Here we present the benefits of this approach in the case of late-type M dwarfs, and L dwarfs. Construction

183 of a reliable MF in the Solar neighborhood will also allow comparisons with MF obtained from deep HSTstar counts (Zheng et al. 2001, and references therein).

The MF of substellar L-dwarfs is very poorly known.

In recent years there has been a major breakthrough in the discovery and study of late-type stars and substellar objects (brown dwarfs). Wide-area sky surveys, that either go much fainter than the previous ones (SDSS), or image the sky in the near-infrared where these cool stars emit most of their energy (2MASS,

DENIS), allow for the ﬁrst time a great number of these objects to be discovered.

These discoveries led to the introduction of two new spectral classes: L dwarfs that are cooler than the latest M dwarfs and some of which might have substellar mass, and T dwarfs that exhibit methane absorption and which are certainly substellar.

Currently, over 100 L dwarfs have been found, and some two dozen T dwarfs

(Burgasser et al. 2002; Leggett et al. 2002; Kirkpatrick et al. 2000).

Accurate distances are a key factor in understanding the structure and evolution of these objects. Besides establishing LF and luminosity calibration, distances would allow better constraints on other key parameters like the radius and the temperature. Distances are currently available for about 20 L dwarfs, and a few T dwarfs. In some cases a precise distance is known only because the object is a companion to a star with a measured trigonometric parallax.

184 Even for the more easily discovered late M dwarfs, the faint end of the main sequence (V − I>3) is deﬁned with the trigonometric parallaxes of only some 30 stars, mostly coming from the USNO CCD parallax program (Monet et al. 1992).

This is too few to accurately describe the diﬀerences in absolute magnitudes of diﬀerent populations of stars.

One of the already recognized mission goals of FAME is to reﬁne the absolute magnitude calibration in the entire HR diagram, including the faint main sequence stars. However, selecting only R<15 stars will leave unmeasured many stars for which good (< 10% accuracy) parallaxes are within the reach of FAME,withthe ratio of unobserved to observed stars getting worse as one goes further down the main sequence.

In Figure 4.1 we compare the number of stars for which FAME will be able to measure parallaxes with < 10% errors if no magnitude limit is imposed, compared to the number of such stars brighter than R = 15. Our ﬁrst bin is dM5.5 stars

(equivalent to MV = 14). There are intrinsically brighter stars (late K and early M dwarfs) for which including R>15 stars would also produce many additional good parallaxes. However, the number of such stars with R<15 is already high enough to allow calibration at the level of their intrinsic scatter. In calculating the number of stars in Figure 4.1 we use current estimates from the literature for the colors, magnitudes, and space densities for late-M, L and T dwarfs, coupled with FAME

185 astrometric precision. The numbers for L dwarfs are less certain (by a factor of two) because of their poorly known number densities. Since the stars fainter than

R = 15 must be deliberately selected to be included as nearby star candidates, and this selection might be difficult close to the Galactic plane, our estimate is based on the assumption that M dwarfs will be selected only if their Galactic latitude satisfies |b| > 20◦. On the other hand, we assumed that L dwarf candidates will be followed up and confirmed in all regions of the sky.

For dM5.5 stars, discarding the R = 15 magnitude limit augments the number of parallaxes from 600 to 4000 by extending the volume probed from

30 to 64 pc. For the very latest M-dwarfs (M7 to M9.5) one will obtain 500 measurements, compared to only 16 for R<15. Finally about 5 L dwarfs are present in the R>15, σπ/π < 0.1 sample, whereas the R<15 limit excludes all L dwarfs. It is obvious that, except for L dwarfs, which are very faint and whose parallaxes are possibly better determined by other means (ground-based campaigns, space telescopes), accepting this magnitude-limit-free selection leads to great improvements in calibrating low-mass stars. We also ﬁnd that T dwarfs are outside of reach of FAME, although a few lower precision measurements could possibly be made.

While most of the nearby L and T dwarfs will be found and conﬁrmed by dedicated searches, late M dwarfs need to be selected from some catalog.

186 Fortunately, 2MASS will contain all the late M dwarfs that FAME can observe.

We estimate the number of MV > 14 candidates by running a search through the

2nd Incremental Release of 2MASS PSC. We use the following search criteria:

H − Ks > 0.3, J − H>0.3andRUSNO − Ks > 3.5 (or no USNO detection) to select late M dwarfs based on color; J − Ks ≤ 4/3(H − Ks)+0.25 to eliminate

M giants (Gizis et al. 2000); J ≤ 14.5 to ensure selection of nearby objects, and

RUSNO > 15 to exclude stars already in the FAME main survey list. The search is further restricted to |b| > 20◦ to exclude contaminants in the Galactic plane.

We ﬁnd the average density of candidates to be 2.3deg−2, implying 64,000 total candidates, which is a modest addition to the FAME input catalog. In 1/5 of this area SDSS photometry will also be available which will permit reﬁnement of the selection The selection can be further augmented with GSC-2, which will contain photometry similar to I band. Any contaminants in this sample will at the end be obvious from the FAME parallax measurements themselves.

4.1.2. White Dwarfs

White dwarfs are one of the most numerous stellar populations in our neighborhood. Therefore they represent obvious targets for getting precise distances. A large sample of parallaxes would allow one to study the white dwarf mass function. The mass of a white dwarf is related to the mass of the progenitor

187 and can thus serve as an indicator of the IMF at different epochs. Current measurements of the mass function are still somewhat controversial both in terms of the peak mass (estimates range from 0.48M to 0.72M) and also concerning the width of the mass function (for recent results see Silvestri et el. 2001). At a given temperature (or color), WDs exhibit a range of absolute magnitudes reflecting a range of radii, since the absolute magnitude is then primarily a function of the white dwarf’s radius. Then the mass-radius relation, which is very well determined theoretically, allows one to find the masses. Other methods of getting masses either rely on a few binary systems or are of low precision (gravitational redshift).

Obtaining luminosities of the coolest white dwarfs is also useful for testing the predicted cooling scenarios.

Currently ∼ 200 precise (accuracy better than 10%) trigonometric parallaxes of white dwarfs are known (McCook & Sion 1999). Using the white dwarf luminosity function from Liebert, Dahn, & Monet (1988) and our estimates of

FAME parallax precision, we estimate that the FAME survey of R<15 stars will produce < 10% parallaxes of 400 white dwarfs. However, as in the case of late-M and L dwarfs, one can increase this number by including in the input catalog objects fainter than R = 15. Then the total number of white dwarfs with precise parallaxes measured by FAME rises to 2400, i.e., a six-fold increase. This increase is accentuated for fainter white dwarfs, as can be seen in Figure 4.2. For

188 example, for MV > 13 (T ∼< 8500 K) the increase with respect to R<15 sample is thirteen-fold. Such numbers would allow mass functions to be constructed as a function of age, and for diﬀerent atmospheric compositions.

The magnitude limit for 10% parallaxes will range from R =15.5 for the bluest white dwarfs (MV =9.5), to R =17.6 for the reddest (MV =16.5). These faint WDs must ﬁrst be selected and then added to the input catalog. Currently known white dwarfs account for ∼< 40% of the expected R>15 sample, based on white dwarfs with photometric information from McCook & Sion (1999). The most eﬃcient way of selecting white dwarfs is by using an all-sky proper motion catalog.

Then, the white dwarfs are selected from a reduced proper-motion diagram in which the reduced proper motion, Hm = m +5logµ + 5, is plotted against color.

For a stellar population with approximately the same transverse velocities, the proper motion becomes a proxy for the distance, and the reduced proper motion a proxy for the absolute magnitude. Because the white dwarfs are separated by

∼ 10 mag from the main sequence, they will stand out in a reduced proper-motion diagram by the same amount, since the two populations have on average the same transverse velocities. For redder white dwarfs, there will be some contamination from subdwarfs, since due to their greater velocity these stars will move down into the white dwarf region. However, even if some subdwarfs ‘contaminate’ the white dwarf observing list, these stars will be of considerable interest in their own right.

189 The only currently available all-sky proper motion catalog that contains faint stars, the NLTT catalog, has a proper motion cutoﬀ that is too high to include most of the white dwarfs with typical disk velocities that lie within the volume accessible to FAME. Thus, it will be necessary to draw white dwarf candidates from the yet to be released GSC-2 or USNO-B that will include all detectable proper motions with typical accuracy of 5 mas yr−1. This will be precise enough to select even the farthest WDs that produce 10% parallax measurement, since they typically move at 40 mas yr−1. In the part of the sky observed by SDSS, hot (B − V<0.3) white dwarf candidates can be selected solely based on their very blue color in all

SDSS bands (as distinct from other blue objects like the quasars and A-type stars, see Fan 1999), while SDSS photometry of red candidates from GSC-2/USNO-B will be good enough to distinguish subdwarfs from WDs. GSC-2/USNO-B white dwarf candidates can additionally be cross-correlated with GALEX UV sources, assuming that the GALEX survey is complete by the time required to deﬁne the input catalog. Some nearby white dwarfs might also end up in the input catalog as

‘contaminants’ from other projects suggested here (when selecting quasars and blue horizontal branch stars). Generally, we do not expect the white dwarf candidates to represent a significant increase in the input catalog size. It will be useful for spectroscopic follow-up to commence as soon as candidate nearby white dwarfs are identified. Besides offering spectroscopic confirmation, spectroscopy is necessary to

190 establish white dwarf atmosphere composition that in turn allows parallax data to be interpreted correctly.

4.2. Bulge Mass Function

4.2.1. Bulge Mass Function and the Microlensing

Here we present an alternative way of obtaining bulge MF. It relies on microlensing that is routinely observed towards the bulge. Since microlensing is equally sensitive to both dark and luminous massive objects, a MF obtained with microlensing will explore standard populations as well as remnant objects: white dwarfs, neutron stars and black holes (Gould 2000b). The MF of remnant objects in turn gives clues on MF of their massive progenitors. The white dwarf MF can be compared with one obtained locally (§4.1.2). Mass function of neutron stars is totally unknown. First, the vast majority of field neutron stars detected are radio-emitting pulsars with unknown masses. Neutron stars in binary systems have unusually narrow mass range (MNS =1.34 ± 0.04M), which however may be the result of evolutionary effects in binary systems. Black holes in binary systems have only rough mass estimates, and the masses and frequency of field black holes is entirely unknown.

191 The distribution of routinely Einstein timescales tE oﬀers no information on the details of the MF. Therefore, the masses of individual lenses are required. They can be obtained with the help of astrometric microlensing. As mentioned in §3.2, the photometric microlensing eﬀect is accompanied with astrometric deviation of the source position. As we will show here, the measurement of this deviation with

SIM, coupled with photometry from SIM and from the ground allows measuring of masses of bulge lenses. During 5 yr of SIM operation, we plan to obtain masses of

200 lenses.

4.2.2. Degeneracy of Microlensing Events

At present, the only quantity routinely measured for all events is the Einstein time scale, tE, which is a complicated combination of the physical parameters that one would like to know,

θE v tE = ,µ= (4.1) µ Dol

where v is the transverse speed of the lens relative to the observer-source line of sight, µ is the proper motion, and θE is the angular Einstein radius,

1/2 4GM ≡ DolDos θE = 2 ,D . (4.2) Dc Dls

192 Here, M is the mass of the lens, and Dol, Dos,andDls are the distances between the observer, lens, and source. There are numerous ideas on how to get additional information about individual events, but these often require special circumstances.

For example, if the source crosses a caustic in the lens geometry, then it is possible to measure the proper motion µ, and so from equation (4.1), the Einstein radius.

However, such caustic crossing events are rare and the great majority of them are binaries (and hence may not be representative of the lens population as a whole).

A second type of information can come from parallax measurements. If the event is suﬃciently long, then the normal light curve is distorted by the accelerated motion of the Earth about the Sun allowing one to measure the Einstein radius projected onto the observer plane,r ˜E,

r˜E = DθE, (4.3)

Gould (1992). Several parallaxes have been measured for bulge events, but all for events that are substantially longer than typical. It would be possible to routinely measure the parallaxes of microlensing events by launching a satellite into solar orbit (Refsdal 1966; Gould 1995b). The event would have a different time of maximum magnification, t0, and different impact parameter, β, as seen from the

Earth and the satellite. From the diﬀerences in these quantities, ∆t0 and ∆β,and using the known Earth-satellite separation, d⊕−s, and known angle γ between the

193 line of sight and the Earth-satellite vector, one could reconstruct both the size of the projected Einstein ring,r ˜E, and the direction of motion, φ,relativetothe satellite-Earth vector,

d⊕−s| sin γ| ∆β r˜E = 2 2 1/2 , tan φ = (4.4) [(∆β) +(∆t0/tE) ] ∆t0

A rather technical but in the present context very important point is that it is significantly easier to measure the difference in times of maximum, ∆t0,thanit is to measure the difference in impact parameters, ∆β. There are two interrelated reasons for this which are investigated in detail by Gould (1994, 1995b); Boutreux

& Gould (1996), and Gaudi & Gould (1997). First, the sign of the impact parameter, β, measured by a single observer is intrinsically ambiguous because the light curve contains no information about the side of the lens on which the source passes. Hence, from the two individual impact-parameter measurements, β⊕ and

βs, it is possible to reconstruct four diﬀerent values of ∆β = ±(β⊕ ± βs). Second, one must determine β and t0 from the light curve simultaneously with three other parameters, tE, F0,andFb, the latter two being the ﬂuxes from the source star and any unlensed background light that is blended into the photometric aperture of the source. While t0 is virtually uncorrelated with any of these three other parameters,

β is highly correlated with all of them, in particular with Fb. Hence β (and so

194 ∆β) is more poorly measured than t0 (and ∆t0). While it is possible to break the four-fold discrete degeneracy, this requires measurement of a higher-order eﬀect.

No dedicated parallax satellite is currently planned. However, the Space

Infrared Telescope Facility (SIRTF) could be used to measure parallaxes of at least some events. Because SIRTF makes its measurements in passbands that are inaccessible from the ground, the relative blending between the Earth and satellite is completely unconstrained, so measurement of ∆β is not simply diﬃcult, it is virtually impossible. Nevertheless, if ∆t0 is well constrained by Earth-satellite observations, then it is possible to determine ∆β from vigorous ground-based observations (Gould 1999).

In the best of all possible worlds, one would measure both θE andr ˜E.These

(together with the routinely measured tE and the approximately known source distance) would then yield a complete solution for M, Dol,andv. For example,

c2 M = θEr˜ . (4.5) 4G E

At present, this is possible by ground-based measurements only for certain rare classes of events. If there were a parallax satellite, then it would also be possible for those rare events which happened to be accessible to proper-motion measurement.

However, astrometric microlensing opens the possibility, at least in principle, that

195 such complete measurements might be made for a large unbiased sample of events in the future.

4.2.3. Breaking the Degeneracy with Astrometric

Microlensing

As Boden, Shao, & van Buren (1998) discuss, astrometric measurements are sensitive to two distinct eﬀects. First, the center of lensed light from the source is displaced from the actual position of the source by,

u δθθ = 2 θE (4.6) u +2

where u ≡ µ(t − t0)/θE is the projected position of the source relative to the lens in units of angular Einstein radius assuming rectilinear motion as would be observed from the Sun. That is,

− 2 1/2 (t t0) 2 u(t)= 2 + β (4.7) tE

This deviation traces out an ellipse with semi-major and semi-minor axes,

1 β θa = θ ,θb = θ . (4.8) 2(β2 +2)1/2 E 2(β2 +2) E

The major axis is aligned with the direction of motion of the lens relative to the source. Hence, by measuring this eﬀect, one can solve for both θE and the direction

196 of motion. A measurement of θE is often called a “proper motion” measurement because, from equation (4.1) it can be combined with the known Einstein time scale to yield the magnitude of the proper motion. However, in the case of astrometric measurements, it also yields the direction, φ, and so the full vector proper motion,

µ. Note that because the astrometric effect dies off very slowly (∝ u−1), stars not associated with the photometric microlensing event can cause significant shifts in the apparent position of the source. However, because this shift remains nearly constant during the event, it does not interfere with the measurement of µ.

The second eﬀect is a parallax deviation caused by motion of the Earth about

Sun. The exact formula for the combined parallax and proper motion eﬀect can be found by substituting

nˆ × nˆ × a u⊕ = u − (4.9) r˜E

into equation (4.6). Here nˆ is the unit vector in the direction of the source, and a is the position vector of the Earth relative to the Sun. Thus, the magnitude of the perturbative term is ∼ AU/r˜E,whichmightbe∼ 10%–30% for typical lensing events. This would seem to imply that one could determine the parallax (˜rE) about 10% to 30% as accurately as the proper motion (θE) for the same set of measurements.

197 Unfortunately, the situation is not so favorable. The perturbation in equation

(4.9) is not directly observable because there are no comparison observations from the Sun. Consider the limit tE yr/2π which is typical for events seen toward the

Galactic bulge. In this case, the Earth’s velocity would barely change during the event or even for the ﬁrst few tE after it. One would then see the same ellipse as described by equation (4.8), but with a diﬀerent β from the one that would have been seen from the Sun. Ninety days after the event, the direction of apparent source motion would have changed by an angle v⊕/v˜ wherev ˜ ≡ r˜E/tE is the speed

−1 of the lens projected onto the observer plane, and v⊕ =30kms is the speed of the

Earth. According to equation (4.6), this would introduce an astrometric deviation of order (v⊕/v˜)(tE/90 day)θE.SinceθE, is essentially determined from the major

−1/2 axis of the ellipse (eq. 4.8), which is approximately 2 θE, the relative size of the astrometric parallax and proper-motion measurement errors is roughly given by

σr˜E,ast/r˜E v˜ 60 day Γast/ast ≡ ∼ . (4.10) σθE,ast/θE v⊕ tE

In fact, equation (4.10) is too optimistic in that it implicitly assumes that the direction of the major axis of the ellipse can be determined with inﬁnite precision.

As we will show in §4.2.6, this is very far from the case. Hence, the true ratio of errors is generally larger than implied by equation (4.10). For typical bulge-bulge lensing events,v ˜ ∼ 800 km s−1.

198 The above analysis implies that astrometric microlens parallax measurements are several orders of magnitude less accurate than proper-motion measurements.

This would not present much of a problem if very accurate proper motion measurements could be made with a modest amount of observing time. However, as we will show in §4.2.6, even for bright (I ∼ 15) sources seen toward the bulge, proper-motion measurements accurate to 5% require about 5 hours of observations.

Hence, this analysis appears to imply that no more than a few accurate microlens parallaxes could be obtained in any reasonable observing program.

4.2.4. Addition of Photometric Observations

SIM is not designed to do photometry, and it would seem completely unwise to waste this precision astrometric instrument on measurements that could be done more eﬃciently from the ground using telescopes with collecting areas that are several orders of magnitude larger. Nevertheless, two unrelated factors combine to make SIM the ideal device to measure microlens parallaxes photometrically (rather than astrometrically).

First, SIM works by counting photons as a function of position in the interference pattern in order to ﬁnd the centroid of the central fringe. The photons

199 are distributed in this fringe as NF(θ)dθ,whereN is the total number of photons in the central fringe,

1 2 θ λ F (θ)= cos ,θf ≡ , (4.11) πθf 2θf 2πd d is the distance between the mirrors, and λ is the wavelength of the light. The astrometric precision is given by, (e.g. Gould 1995a),

2 −1/2 −1/2 d ln F −1/2 σθ = N dθF (θ) = N θf , (4.12) dθ

−1/2 and hence the fractional photometric precision (σph = N ) is related to the astrometric precision by

σθ σph = . (4.13) θf

Second, as discussed in §4.2.2, the real problem in obtaining microlens parallaxes photometrically is that the microlens parallaxes are inherently two- dimensional. In effect, by measuring ∆t0, one determines cos φ/r˜E,andby measuring ∆β, one determines sin φ/r˜E,whereφ is the angle of source-lens relative motion with respect to the direction of SIM-Earth axis at the moment when the event is a maximum as seen from the SIM-Earth midpoint. It is only by measuring both of these quantities that one can determiner ˜E.Since∆β is difficult to measure, obtaining a preciser ˜E is also difficult.

200 As discussed in §4.2.2, it is possible in principle to break the degeneracy in ∆β photometrically if the photometry is good enough. We will show in §4.2.6 that SIM photometry is sufficiently precise for this task provided that the observations are carefully planned. However, it is also the case that SIM astrometric measurements by themselves often determine φ (from the orientation of the ellipse) with sufficient precision to break the degeneracy in ∆β. In these cases,r ˜E can be determined from a measurement of ∆t0 alone. Later we will consider both methods of breaking the degeneracy, but in the remainder of this section we will assume that the degeneracy is broken astrometrically. This will allow us to estimate Γph/ast,the relative precision of SIM photometric parallax measurements to SIM astrometric proper-motion measurements. (Recall from the discussion following eq. 4.4 that blending does not significantly affect the measurement of ∆t0, and therefore will not be considered in this section. We give a thorough discussion of blending in

§4.2.5.)

As currently designed, SIM will ﬂy in a SIRTF-like orbit, drifting away from the Earth at about 0.1 AU per year. Let the distance at the time of the observations be d⊕−s.Then˜rE can be determined from the measured ∆t0 (and the known value of φ)by,

tE r˜E = d⊕−s | cos φ|. (4.14) |∆t0|

201 Gould (1999) analyzed how to optimize measurements of ∆t0 when he investigated microlens parallaxes with SIRTF. For photon-limited photometry, one should concentrate the measurements near times t± before and after the peak, where

1/2 t± = t0 ± (5/3) βtE. The error in t0,s is then approximately given by

∼ −1/2 β σt ,s n σph tE (β ∼< 0.5), (4.15) 0 0.5

where n is the total number of measurements in these two regions. We will assume that the ground-based measurements to determine t0,⊕ have similar precision as the SIM measurements. Equation (4.14) then implies that the fractional error in r˜E is given by,

σr σt vσ˜ t ˜E =21/2 0,s =21/2 0,s | sec φ|. (4.16) r˜E |∆t0| d⊕−s

Then, assuming that a similar number of measurements are used to determine the semi-major axis of the ellipse, θa, and to determine ∆t0 (they are somewhat the same measurements), the ratio of the photometric precision of the parallax to the astrometric precision of the proper motion is

| | σr˜E /r˜E r˜Eσph(β/0.3) sec φ /d⊕−s Γph/ast = ∼ , (4.17) σθa /θa σθ/(θE/3)

202 where we have approximated equation (4.8) as θa = θE/3. Using equations (4.5) and (4.13), this can be rewritten

− 4GM β M d⊕−s 1 Γph/ast ∼ β | sec φ|∼ | sec φ|, (4.18) c2d⊕−sθf 0.25 0.3M 0.2AU

wherewehaveadoptedθf =2.5 mas, which is appropriate if the ﬂux-weighted harmonic mean wavelength of the source is 0.8 µm, and the mirrors are separated by 10 m. Equation (4.18) implies that for typical lenses, the photometric parallax will be of comparable precision to the astrometric proper motion in sharp contrast to the large ratio for astrometric parallaxes found in equation (4.10).

In fact, the | sec φ| dependence in equation (4.17) is too pessimistic because we have ignored all photometric information about ∆β.Weshowin§4.2.6 that except for the case cos φ 0 (where the discrete degeneracy becomes astrometrically incorrigible) it is possible to essentially eliminate the | sec φ| term in equation (4.17) using a combination of astrometric and photometric data.

4.2.5. Simulations of SIM Observations

The estimates given in the previous two sections are useful because they elucidate the relation between the physics of the event and the measurement process on the one hand and the errors in the microlensing parameters on the other. By the same token, however, they cannot capture the full range of experimental conditions,

203 and so are necessarily approximate. The actual errors for any given event will depend both on the precise event parameters and on the observational strategy.

While a full investigation of the best observational strategy lies well beyond the scope of the present study, it is important to make a rigorous calculation of the statistical errors for some representative examples in order to obtain more precise estimates and to investigate more subtle eﬀects that are not captured by the rough analysis given above.

For this purpose, we consider a set of somewhat idealized observations.

First, we assume that the principal measurements are carried out at uniform time intervals that are short compared to tE, beginning when the magniﬁcation

ﬁrst reaches A =1.5 and ending at a time that will be determined below from signal-to-noise (S/N) considerations. However, for the bulge, we assume that the measurements are interrupted when the bulge is within 60◦ of the Sun. This measurement strategy can actually be very far from ideal, and we modify it somewhat in below. The assumption that events can be recognized at A =1.5is reasonable, but whether observations can begin as soon as the events are recognized requires additional discussion. SIM design characteristics are not yet ﬁxed but the current “requirement” for target-of-opportunity response is 4 days, with a “goal” of

2 days, where a “requirement” is what the project is using as a basis for planning and a “goal” is what the project will attempt to support if resources are available.

204 The mechanics of operating the spacecraft limit the response time absolutely to a minimum of about 16 hours, but to achieve this much faster response would be costly in dollars. Thus, the actual response time of SIM will be set by balancing the scientiﬁc returns against costs that ultimately limit other capabilities of the satellite. For simplicity, we here assume that observations can always begin at

A =1.5. Clearly, the ﬁnite response time will degrade this to some degree, particularly for the shortest events. Since the experiment being proposed here will probably stress the target-of-opportunity capability of SIM at least as much as any other, a more detailed study of the eﬀects of the delay should be made in the near future. However, this is beyond the scope of the present work.

Second, we assume that the exposure times are of equal durations, so that the S/N is better near the peak of the event. Third, we assume θf =2.5 mas and d⊕−s =0.2 AU, although the ﬁrst will clearly vary from star to star, and the second will change during the course of the mission. We will assume that the bulge ﬁeld lies at B = −6◦ from the ecliptic, close to the (northern) winter solstice. The

Earth-satellite separation projected onto the sky is therefore at a maximum at the

2 2 1/2 summer solstice and is d⊕−s|(1 − cos ψ cos B)] ∼ d⊕−s| sin ψ| at other times of the year, where ψ is the phase of the Earth’s orbit relative to the autumnal equinox. As discussed in detail by Gaudi & Gould (1997), the orbital phase ψ has two conﬂicting eﬀects. First, when cos ψ ∼ 0 (near the summer solstice), the

205 SIM-Earth projected separation is a maximum, and hence the measurement errors of ∆t0 and ∆β are reduced to a minimum. On the other hand, the relative velocity of SIM and the Earth projected onto the plane of the sky is at a minimum, and breaking the degeneracy in ∆β depends critically on this relative velocity. Hence, it is most diﬃcult to break the degeneracy at the summer solstice. As the phase moves away from the summer solstice, the projected separation slowly declines

(making the measurements of ∆t0 and ∆β less accurate) but the projected velocity diﬀerence rapidly increases (allowing more secure degeneracy breaking). If one relies on photometry to break the degeneracy, the optimal events are those that peak about 45 days from the summer solstice (see e.g. Fig. 6 from Gaudi & Gould

1997). On the other hand, if it is possible to break the degeneracy astrometrically, then events that peak at the solstice are optimal since the measurement errors are reduced by ∼ 2−1/2. In this section, we will primarily simulate events peaking at

ψ = 225◦, i.e. about May 7. However, we will also discuss events that peak at the summer solstice (ψ = 270◦).

Next, we will assume for deﬁniteness that the ground-based photometric observations have the same precision as the SIM observations. Finally, we will ignore blending in the SIM measurements, except blending by the lens. Blending will have an important impact on the overall precision and hence the strategy for

SIM measurements but as we show below, it will not aﬀect the main conclusions

206 which concern the relative precision of SIM astrometric and photometric microlens parallaxes. This is especially so toward the bulge which will be the main focus of analysis. A proper treatment of blending would therefore make the analysis substantially more complex without clarifying any of the central points. Hence, we defer consideration of this important eﬀect to a future work on observational strategy.

Why can blending be ignored to ﬁrst order in this analysis? First, Han &

Kim (1999) have shown that all potential blends lying more than 10 mas from the source can be eliminated by the SIM observations themselves. Since the density of ﬁeld stars having even a modest fraction of the source ﬂux is much less than

104 arcsec−2, this essentially eliminates all blends not directly associated with the event, namely the lens itself, binary companions to the lens, and binary companions to the source. For bulge events, 10 mas corresponds to 80 AU, so a substantial fraction of binary companions are also eliminated. Second, to minimize observation time, SIM observations must be almost entirely restricted to very bright stars

(relative to other stars in the field). This means observing clump giants. The chance that a bulge clump star has a companion within 80 AU and with more than a few percent of its own flux is small because their progenitors are about 50 times fainter than the stars themselves. The primary effect of a few percent blend would be to change the shape and orientation of the proper-motion ellipse and (assuming

207 the shape change went undetected) to therefore change the inferred direction of the lens-source relative proper motion, also by a few percent. This would in turn affect the parallax inferred from ∆t0 which depends on this direction through the angle φ. See equation (4.14). However, this effect on the parallax measurement will also be a few percent. As we will show in §4.2.6, it is quite possible to achieve accuracies of a few percent for bulge events, so a careful investigation of the effect of blending on parallax and proper-motion measurements should be undertaken as part of a more thorough analysis of the problem. Unfortunately, a proper analysis of blending from binary companions to the source requires simulated fits to the entire diffraction pattern, not just the centroid, and so is substantially more involved than the present study. By contrast, low-level blending by the lens can be treated within the framework of the centroid analysis given here and we therefore include it.

It is very unlikely that the lens itself will contribute more than a small fraction of the source light if the source is bright. We therefore allow for lens blending in our simulated ﬁts but assume that the actual blending is very small.

Note that we will not ignore blending in the ground-based photometric observations, since there is no way to eliminate ﬁeld star blends for the ground-based observations.

208 We will simulate simultaneous observations from SIM and from the ground.

There will be four measured quantities: 1) G1, the ﬂux observed from the ground,

2) G2, the ﬂux observed from SIM, 3) G3,thex astrometric position, and 4) G4, the y astrometric position. These give rise to four observational equations,

1 1 1 2 2 2 G (t)=Fs A(u⊕)+Fb ,G(t)=Fs A(uSIM)+Fb (4.19)

and

uSIM 3 4 − − [G (t),G (t)] = 2 θE αSIMπs + µst + θ0 θb (4.20) uSIM +2

2 2 Fb θE uSIM +3 b ≡ SIM SIM θ 2 2 2 u + καα (4.21) Fs A(uSIM)+Fb uSIM +2

2 2 1/2 where A(u)=(u +2)/[u(u +4) ] is the magniﬁcation, µs is the absolute proper motion of the source, θ0 is the true position of the source at time t =0,

nˆ × nˆ × a α = − , (4.22) AU

πs is the parallax of the source, and uSIM and u⊕ are deﬁned similarly to equation

(4.9), i.e.

AU uSIM = u + κααSIM, u⊕ = u + καα⊕,κ≡ , (4.23) r˜E

209 with

t − t0, u ≡ (τ cos φ − β sin φ, τ sin φ + β cos φ),τ ≡ . (4.24) tE,

The terms in equation (4.20) can be understood as follows. The ﬁrst term is the ellipse characterized by equation (4.8), modiﬁed by the motion of the Earth (i.e. u → uSIM). The second term is the parallactic motion of the source. The third and fourth terms represent the ordinary proper motion and position of the source.

The last term is the perturbation due to the luminosity of the lens which is written out explicitly in (4.21). Here the ﬁrst term is the diﬀerence between the “ellipse” in equation (4.20) and the relative lens-source position (−uSIMθE), while the second is the relative parallax of the lens and the source.

There are a total of 15 parameters: t0,, β, and tE, are the standard event parameters as it would be seen from the Sun, φ is the direction of source motion relative to the lens with respect to the SIM-Earth direction, θE is the Einstein radius, κ is the inverse projected Einstein radius (normalized in AU), πs is the parallax of the source, µs is the proper motion of the source, θ0 is its position

1 2 at t =0,Fs and Fs are the source ﬂuxes as received by the Earth and satellite

1 2 observatories, and Fb and Fb are the background ﬂuxes.

210 To determine the uncertainties in these parameters, we evaluate the covariance matrix cij

4 N l l −1 −2 ∂G (tk) ∂G (tk) c = b ,bij = σkl , (4.25) l=1 k=1 ∂ai ∂aj

where a1 ... a15 are the ﬁfteen parameters, tk are the times of the observations,

l and σkl is the error in the measurement of G at time tk. We enforce the condition

1 2 of weak blending by setting Fb = Fb =0after taking the derivatives in equation

(4.25).

The full problem of optimization of SIM microlensing observations lies outside the scope of this work, but it is straight forward to determine the optimum duration of observations once the (arbitrary) strategy of uniform observations has been adopted: one changes the interval over which the observations are carried while holding the total observing time ﬁxed and inspects the resulting errors.

We carry out this exercise and ﬁnd that the optimal duration to determiner ˜E is short, typically a few tens of days for various combinations of parameters, while the optimal duration to determine θE is well over 100 days. The reason for this is clear. The measurement ofr ˜E is determined primarily from photometry, and the photometric microlensing event is essentially over after 2tE. On the other hand

θE is determined from the astrometric event which lasts many tE. Only after the

211 astrometric event is essentially over is it possible to determine µs and so remove the correlation between this parameter and θE.

We address this inconsistency of time scales by modifying the observational strategy. We take observations uniformly over various intervals but with ﬁxed total observing time, T , and then take three additional observations, each with observing time T/20, at 3 months, 9 months, and 12 months after the peak of the event.

Thus, the total observing time is 1.15 T . We then ﬁnd that the precisions of both r˜E and θE are roughly constant when the continuous observations last anywhere from 30 to 120 days. Any choice in this range would lead to essentially the same result. We adopt 50 days since it is away from the edges of the interval but still on the shorter side (thus keeping the observations away from the time when the Sun comes close to bulge ﬁelds).

We assume that all bulge sources are I = 15 which is typical of the brighter microlensing events seen toward the bulge. Our ﬁducial I = 15 stars require 40 minutes to reach 4 µas. We allow a total of 5 hours of observation for each event.

It is then straight forward to scale our results to other assumed conditions.

212 4.2.6. Expected Precision of Mass Measurements

We consider two geometries. The ﬁrst is a bulge line of sight at 6◦ from the ecliptic. The source and lens are both in the bulge, with Dos =8kpcand

Dol = 6 kpc. Hence D = 24 kpc. (See eq. 4.2.) The speed of the lens relative to the observer-source line of sight is v = 200 km s−1.WevaryM, φ,andβ. Formally β is deﬁned as the impact parameter of the event as seen from an observer at the

Earth-SIM midpoint but in practice it is very similar to the β observed from the

Earth. All events are assumed to peak (as seen from the midpoint) on May 7, i.e.,

45 days before the (northern) summer solstice. This is the most favorable time to break the discrete degeneracy in ∆β photometrically (see Gaudi & Gould 1997),

1/2 but the intrinsic errors in ∆β and ∆t0 are larger by ∼ 2 than they would be at the summer solstice. We will therefore later investigate whether the discrete degeneracy in ∆β can be broken astrometrically so that observations could take place at the solstice. In order to better understand this and several other issues, we conduct two sets of simulations, one using SIM measurements alone (both astrometric and photometric) and the other combining astrometry with photometry from both SIM and the ground.

Table 4.1 shows the results. The ﬁrst three columns are the input parameters

M, β,andφ. Columns 4 and 5 show the errors in φ from SIM measurements only and with the addition of ground-based measurements. Columns 6 through 11 show

213 the fractional errors in various quantities both without and with ground-based photometry. Columns 6 and 7 show the errors inr ˜E, columns 8 and 9 show the

2 errors in θE, and columns 10 and 11 the errors in M =˜rEθEc /4G. Finally, columns

12 and 13 show the fractional errors in D =˜rE/θE,andinπs based on combined data from SIM and the ground.

These results are in rough qualitative agreement with the predictions of equations (4.10) and (4.18). In particular, they conﬁrm that the fractional error in

˜ r˜E is orders of magnitude larger than the fractional error in θE if one is restricted to SIM data, while the two errors are comparable if one combines astrometric with photometric data.

However, there are a number of additional important conclusions that can be drawn from Table 4.1. First, restricting consideration to impact parameters

β<0.5, the fractional errors in M, D,andπs are all typically about 5% although these errors do vary somewhat in particular cases. This means 5 hours of observation produce very precise individual solutions for the mass, distance, and velocity of the lens, and also for the distance and velocity of the source, implying that a few hundred hours of SIM time could yield a very detailed inventory of the material between the Sun and the Galactic center.

214 ◦ Second, while the errors inr ˜E do deteriorate toward φ =90, the trend is not as drastic as predicted by equation (4.18). For β<0.5, the errors are 50% to 100% worse at φ =90◦ compared to φ = 0, although equation (4.18) predicts they should be inﬁnite. The fundamental reason for this is that the continuous degeneracy in ∆β is not very severe, so that if one assumes that the discrete degeneracy is broken, then there is actually quite a lot of information about this component of r˜E in the photometric measurements. In fact, the values in Table 4.1 implicitly do assume that the discrete degeneracy is broken. This is because they are based on equation (4.25) which gives a purely local error analysis.

Recall that there are two discrete degeneracies. We focus initially on the one that aﬀects the magnitude of ∆β and defer consideration of the one that aﬀects only the sign of ∆β. To determine whether this discrete degeneracy is broken photometrically, we examine the work of Gaudi & Gould (1997), in particular their Figure 6. Under the observational conditions they considered, the discrete

◦ degeneracy is broken 90% of the time for M =0.3 M, ψ = 225 , and satellite separations d⊕−s =0.4 AU. This is twice the separation that we have assumed.

However, Gaudi & Gould (1997) have assumed photometry errors of 1% for the

Earth and 2% for the satellite for a total of about 70 observations. If our 5 hours of observing time were divided among 70 observations, the photometric precision would be 1.2%. That is, our assumed eﬀective errors are smaller by a factor

215 [(1.22 +1.22)/(12 +22)]1/2 =0.75. For small Earth-satellite separations, there is a direct trade oﬀ between measurement error and satellite separation, so our 0.2 AU separation corresponds to 0.25 AU in their simulations. Inspection of the Gaudi &

Gould (1997) Figure 6 shows that the degeneracies would be broken 70% of the time. To determine how this eﬀectiveness scales with lens mass, we turn to Figure

4 from Gaudi & Gould (1997). This shows that degeneracy breaking becomes more diﬃcult at lower masses. The ﬁgure is drawn for the case d⊕−s =1AU whereas the argument just given implies that with 5 hours of observation, our 0.2

AU separation is equivalent to 0.25 AU in the Gaudi & Gould (1997) simulations.

Hence, comparing Gaudi & Gould (1997) Figures 4 and 6, we estimate that 35% of the degeneracies would be broken for M =0.1 M. If the exposure times were multiplied by a factor of 4 to 20 hours, this fraction would rise to about 70%.

We now turn to the question of how well the degeneracies can be broken astrometrically. For the geometry considered here,r ˜E =7.5(M/0.3 M) AU. Hence

∆β ∼ d⊕−s/r˜E ∼ 0.025, which is quite small compared to typical values of β.This means that for most events the discrete degeneracy will be between solutions with

∆β ∼ 0.025 and ∆β 2β ∼ 0.5. Since tan φ =∆β/∆t0, the high ∆β solution will almost always correspond to angles 90◦ ± 2◦ or 270◦ ± 2◦, while the low ∆β solution (assuming that it is the real one) will be at some random angle. Thus to distinguish the two solutions, one must have independent information about φ

216 with errors that are a factor ∼ 3 smaller than |90◦ − φ| or |270◦ − φ|.Isφ this well constrained by the observations? Looking at column 4 of Table 4.1, φ seems to be very well constrained. However, this precision measurement is based primarily on the photometric measurements of ∆β and ∆t0 and so implicitly assumes that the discrete degeneracy has been broken. Hence we should use only SIM data (column

4).

We see from column 4 of Table 4.1 that the errors in φ are generally small for

φ ≤ 60◦ but deteriorate toward φ =90◦. This means that the discrete degeneracy is broken astrometrically at the 3 σ level for φ ≤ 60◦ but cannot be broken if the angle gets close enough to 90◦. Table 4.1 does not have suﬃcient resolution to determine the transition but we ﬁnd by more detailed calculations that for β =(0.2, 0.4)

◦ ◦ ◦ ◦ this occurs at (70 ,65 ), for M =0.1 M,at(75,70 ), for M =0.3 M,andat

◦ ◦ (80 ,75 ), for M =0.5 M. Hence, the degeneracy is usually broken astrometrically, but less frequently at low masses. Since the degeneracy is more diﬃcult to break at low masses both photometrically and astrometrically, it would be prudent to commit more observation time (say 20 hours rather than 4 hours) to the shortest events (which are likely to be low mass).

The fact that the degeneracy can be broken astrometrically for most events seems to argue against restricting observations to periods that are 45 days from the summer solstice. Recall that we adopted this restriction in order to permit

217 better photometric degeneracy breaking which now no longer seems so necessary.

However, we ﬁnd that for events peaking at the solstice, the errors in φ (when the Earth-based observations are ignored, column 4) are substantially higher than the values in column 5, implying that it is often not possible to break the discrete degeneracy astrometrically or photometrically at the solstice.

As we have noted, the above discussion actually applies to only one of two discrete degeneracies, the one involving two diﬀerent magnitudes of ∆β. This is the more important degeneracy because it aﬀects the estimate of the size ofr ˜E and so of the mass, distance, and speed of the lens. However, there is also another degeneracy involving the sign of ∆β but not its magnitude. Boutreux & Gould (1996) and

Gaudi & Gould (1997) refer to the ﬁrst of these as the “speed degeneracy”. We call the second the “direction degeneracy”. The direction degeneracy becomes

◦ ◦ difficult to break when |∆β| |∆t0|/tE, i.e., when φ ∼ 0 or φ ∼ 180 , so that the two degenerate solutions are close in φ. From column 4 in Table 4.1, we find that the error in φ from the astrometric data alone is generally quite small for φ =0, and hence is adequate to break the direction degeneracy unless φ lies within a few degrees of either 0◦ or 180◦. However, in this case the effect of the degeneracy is very small.

218 Table 4.1. Uncertainties of microlensing parameters, Bulge (I = 15, ψ = 225◦,5 hours)

Mβ φ σφ σr˜E /r˜E σθE /θE σM /M σD/D σπs /πs SS+⊕ SS+⊕ SS+⊕ SS+⊕ S+⊕ S+⊕

M deg deg deg

0.1 0.2 0 2.3 1.1 0.981 0.015 0.041 0.041 0.980 0.044 0.043 0.035 0.1 0.2 30 2.5 1.0 1.265 0.017 0.041 0.041 1.265 0.045 0.044 0.035

219 0.1 0.2 60 4.3 0.9 2.713 0.020 0.040 0.040 2.713 0.046 0.045 0.034 0.1 0.2 90 9.2 0.7 6.005 0.022 0.040 0.039 6.012 0.045 0.045 0.033

0.1 0.4 0 2.7 2.2 1.962 0.036 0.046 0.045 1.955 0.059 0.057 0.035 0.1 0.4 30 3.5 2.1 2.564 0.041 0.047 0.046 2.559 0.063 0.061 0.036 0.1 0.4 60 7.9 1.9 5.394 0.056 0.046 0.046 5.394 0.075 0.071 0.036 0.1 0.4 90 17.0 1.6 11.382 0.070 0.052 0.044 11.408 0.084 0.082 0.035

0.1 0.6 0 4.0 3.0 4.271 0.075 0.060 0.053 4.248 0.095 0.088 0.036 0.1 0.6 30 6.9 2.9 5.500 0.080 0.060 0.055 5.482 0.100 0.094 0.036 0.1 0.6 60 16.5 2.7 11.109 0.111 0.057 0.055 11.108 0.128 0.121 0.037 0.1 0.6 90 27.0 2.5 18.428 0.173 0.084 0.054 18.491 0.185 0.177 0.036 (continued) Table 4.1—Continued

Mβ φ σφ σr˜E /r˜E σθE /θE σM /M σD/D σπs /πs SS+⊕ SS+⊕ SS+⊕ SS+⊕ S+⊕ S+⊕

M deg deg deg

0.3 0.2 0 1.1 0.9 0.446 0.019 0.021 0.021 0.446 0.029 0.028 0.037 0.3 0.2 30 1.2 0.9 0.570 0.021 0.021 0.021 0.570 0.030 0.030 0.037 0.3 0.2 60 2.0 0.9 1.195 0.026 0.020 0.020 1.196 0.034 0.033 0.036 0.3 0.2 90 4.3 0.8 2.768 0.030 0.019 0.019 2.772 0.035 0.035 0.035 220 0.3 0.4 0 1.3 1.2 0.900 0.047 0.024 0.024 0.899 0.053 0.052 0.038 0.3 0.4 30 1.7 1.2 1.132 0.054 0.024 0.024 1.132 0.059 0.058 0.038 0.3 0.4 60 3.5 1.2 2.267 0.075 0.023 0.023 2.270 0.079 0.077 0.038 0.3 0.4 90 8.5 1.2 5.639 0.097 0.026 0.022 5.653 0.100 0.100 0.037

0.3 0.6 0 2.0 1.6 1.906 0.098 0.030 0.029 1.899 0.103 0.101 0.038 0.3 0.6 30 3.2 1.6 2.321 0.109 0.030 0.029 2.317 0.114 0.112 0.038 0.3 0.6 60 6.8 1.6 4.280 0.158 0.029 0.029 4.284 0.162 0.159 0.038 0.3 0.6 90 14.8 1.6 9.849 0.252 0.047 0.027 9.888 0.254 0.252 0.038 (continued) Table 4.1—Continued

Mβ φ σφ σr˜E /r˜E σθE /θE σM /M σD/D σπs /πs SS+⊕ SS+⊕ SS+⊕ SS+⊕ S+⊕ S+⊕

M deg deg deg

0.5 0.2 0 0.9 0.8 0.313 0.022 0.017 0.017 0.313 0.027 0.027 0.038 0.5 0.2 30 0.9 0.8 0.395 0.024 0.016 0.016 0.396 0.029 0.029 0.038 0.5 0.2 60 1.5 0.8 0.820 0.031 0.015 0.015 0.821 0.034 0.034 0.037 0.5 0.2 90 3.0 0.8 1.999 0.035 0.015 0.015 2.002 0.038 0.038 0.036 221 0.5 0.4 0 1.0 1.0 0.623 0.054 0.019 0.019 0.623 0.057 0.057 0.039 0.5 0.4 30 1.3 1.0 0.783 0.062 0.019 0.019 0.784 0.065 0.065 0.039 0.5 0.4 60 2.4 1.0 1.536 0.089 0.018 0.018 1.539 0.091 0.091 0.038 0.5 0.4 90 6.5 1.0 4.352 0.115 0.020 0.016 4.362 0.115 0.116 0.038

0.5 0.6 0 1.5 1.2 1.386 0.113 0.023 0.023 1.384 0.116 0.115 0.040 0.5 0.6 30 2.3 1.2 1.633 0.129 0.023 0.023 1.634 0.132 0.131 0.040 0.5 0.6 60 4.6 1.3 2.903 0.195 0.023 0.022 2.909 0.197 0.196 0.039 0.5 0.6 90 12.3 1.3 8.190 0.293 0.039 0.020 8.224 0.294 0.293 0.039 (continued) Table 4.1—Continued

Mβ φ σφ σr˜E /r˜E σθE /θE σM /M σD/D σπs /πs SS+⊕ SS+⊕ SS+⊕ SS+⊕ S+⊕ S+⊕

M deg deg deg

1.0 0.2 0 0.7 0.6 0.211 0.026 0.014 0.013 0.212 0.029 0.029 0.040 1.0 0.2 30 0.7 0.7 0.266 0.029 0.013 0.013 0.267 0.032 0.032 0.039 222 1.0 0.2 60 1.0 0.7 0.540 0.038 0.012 0.012 0.541 0.039 0.040 0.039 1.0 0.2 90 2.1 0.7 1.458 0.044 0.012 0.012 1.460 0.044 0.046 0.038

1.0 0.4 0 0.7 0.7 0.410 0.065 0.016 0.015 0.412 0.066 0.067 0.041 1.0 0.4 30 0.9 0.8 0.502 0.077 0.016 0.015 0.504 0.078 0.079 0.040 1.0 0.4 60 1.5 0.8 0.963 0.115 0.015 0.014 0.967 0.115 0.116 0.040 1.0 0.4 90 4.6 0.9 3.268 0.137 0.016 0.013 3.276 0.136 0.139 0.040 (continued) Table 4.1—Continued

Mβ φ σφ σr˜E /r˜E σθE /θE σM /M σD/D σπs /πs SS+⊕ SS+⊕ SS+⊕ SS+⊕ S+⊕ S+⊕

M deg deg deg

1.0 0.6 0 1.0 1.0 0.975 0.137 0.019 0.018 0.979 0.138 0.138 0.042

223 1.0 0.6 30 1.5 1.0 1.122 0.164 0.019 0.018 1.126 0.166 0.165 0.042 1.0 0.6 60 2.9 1.1 1.908 0.262 0.019 0.016 1.916 0.262 0.262 0.041 1.0 0.6 90 10.2 1.1 6.937 0.333 0.034 0.015 6.967 0.333 0.334 0.041

M is the mass of the lens, β is the impact parameter in Einstein radius units, φ is the angle of lens motion relative to satellite-Earth vector. S and S+⊕ designate two cases: ﬁrst, SIM astrometry and photometry measurements, and the second, SIM measurements plus ground-based photometry.

r˜E is the projected Einstein radius (See eq. 3.), θE is the angular Einstein radius (See eq. 4.2),

D =˜rE/θE is given in eq. 4.3, and πs is the parallax of the source. 10,000 noR limit 1,000 R<15

100

(stars with <10% plx error)

N 1 M5.5 M6 M7 M8.0 L0.0 L5 to to to M9.5 L4.5 L8

Fig. 4.1.— The histogram shows the number of late-type dwarfs of diﬀerent spectral classes for which one can measure parallaxes with fractional error < 10%, assuming FAME sensitivities. If objects are selected beyond the main survey magnitude limit of R = 15 (blank vs. grey bars), there is a signiﬁcant increase in the number of good parallaxes.

224 Fig. 4.2.— Distribution over MV of white dwarfs with precise parallax estimates using FAME. Observing WDs fainter than R = 15 greatly increases the number of good (< 10% fractional accuracy) parallaxes. The number (per magnitude) of good parallaxes (assuming FAME sensitivities) in the two cases is given as a function of WD absolute magnitude.

225 Chapter 5

Precise Distance to the Galactic Center

5.1. Introduction

5.1.1. Past Determinations of the Galactocentric

Distance

The distance to the center of the Galaxy (R0) is to Galactic astronomy what the Hubble constant (H0) is to extragalacatic astronomy and cosmology. Many of the parameters we determine for the Galaxy depend on R0. Fixing its value will directly give us a precise value of circular velocity Θ0 from the measured proper motion of Sgr A*. The circular velocity and distance in turn enter in estimates of the mass of the Galaxy. Having a precise value of R0 would allow us to recalibrate distance indicators such as RR Lyraes, and to compare them with local RR Lyraes.

226 This would give us an idea of applicability of the local RR Lyrae calibration to globular clusters and the LMC. This can to some extent also be done for the

Cepheids, which are an important step in the extragalactic distance ladder.

As with the Hubble constant, the distance to the center of the Galaxy is still debated (although with perhaps less controversy) 70 years after it was ﬁrst estimated by Shapley (1918). In Reid (1993) presented an overview of then current methods and suggested a weighted average of R0 =8.0 ± 0.5kpc.This is a convenient value which is still used by many researchers (although the IAU maintains its recommended value at R0 =8.5 kpc). In the closing sentence of his review, Reid (1993) expects R0 to be known with 3% uncertainty by the end of the decade. How close are we to this goal today? In the period after the Reid

(1993) article there were four new estimates of R0, one of which was based on a new method.

The ﬁrst new estimate uses RR Lyrae observations in the near-infrared (IR).

Apparent magnitudes suﬀer less interstellar extinction in the near-IR than in the optical. Smaller brightness amplitudes in the near-IR also permit an estimate of the mean magnitude with fewer measurements. Using 58 RR Lyrae stars in Baade’s

Window, Carney et al. (1995) report R0 =7.8 ± 0.4 kpc which is consistent with

Reid’s value and has 5% uncertainty. However, the real uncertainties turn out to be greater, as this result depends on the distance-scale model used. In fact,

227 the authors report two possible results that diﬀer by 14%. Metzger, Caldwell, &

Schechter (1998) measured R0 based on Cepheids and a model of the Galactic rotation. They find R0 =7.66 ± 0.32 kpc. The 4% uncertainty is a combination of many different systematic errors (Cepheid calibration, reddening, etc) that enter in the final result. The value of R0 itself is also model dependent, and an additional uncertainty comes from an unknown component of the ellipticity of the stellar orbits.

The Cepheid distance scale was recalibrated using the Hipparcos satellite.

This led to a new estimate of R0 by Feast & Whitelock (1997) . They report a

“long” scale of R0 =8.5 ± 0.5 kpc, i.e., a 6% uncertainty.

Hipparcos data also facilitated the introduction of a new method using red clump stars. Paczy´nski & Stanek (1998) Hipparcos parallaxes to calibrate nearby red clump stars and applied this calibration to ﬁnd the distances of the red clump stars near the Galactic center. Due to the great number of these stars in Baade’s

Window, the formal uncertainty of R0 is very small. However, various systematics produce a ﬁnal estimate of R0 =8.4 ± 0.4 kpc, i.e. 5% accuracy. Later, this result was corrected to R0 =8.2 ± 0.15 ± 0.15 kpc (Stanek & Garnavich 1998).

As we can see, there is still no agreement on a single value of R0, and all of the estimates have signiﬁcant uncertainties. Notably, none of these methods falls into

228 the category of primary methods – those that are not using a “standard candle” or a Galactic rotation model (Reid 1993). One would prefer to use a direct method in order to avoid the systematics associated with “standard candles” . Even the

H2O maser method (Genzel, Reid, Moran, & Downes 1981; Reid 1993) is model dependent. Also, the accuracy of this method seems to be limited to about 15%.

An attractive way to directly determine R0 is to measure the trigonometric parallax of Sgr A* using VLBI. However, signiﬁcant technical obstacles remain for this approach. The Space Interferometry Mission (SIM) will be sensitive enough to measure parallaxes with 3% precision at the Galactocentric distance, but since it operates in the optical part of the spectrum, SIM cannot look directly at the

Galactic center.

5.1.2. “Visual Binaries” Around SgrA*

Here we suggest a new, purely geometric method, which became feasible with the recent advances in speckle imaging of the stars surrounding the Galactic center

(Eckart & Genzel 1997; Genzel, Eckart, Ott, & Eisenhauer 1997; Ghez, Klein,

Morris, & Becklin 1998). In essence, it is the classic visual binary method which has been used for decades to measure the masses of and distances to binary stars.

We apply this method to stars orbiting around the massive object in the center of the Galaxy (presumably a black hole). This method requires observations of

229 the proper motions and radial velocities of several stars near Sgr A* during the next decade or so, in order to constrain the shape and the physical size of each of the orbits. These, together with the angular size of the orbit will in turn yield

R0. This is not a statistical method: each star individually leads to a value of

R0 whose uncertainties depend only on the precision of the proper-motion and radial-velocity measurements. Its basic advantages are that it is a direct method that overrides many of the systematic errors of the secondary methods and that it can possibly measure R0 accurate to 1-5% after 15 years of observing, and 0.5-2% after 30 years of observing. In the long term, it is important that its accuracy just keeps on improving, regardless of our ability to improve the quality of the position and velocity measurements themselves.

The orbital solution to a visual binary automatically gives the heliocentric radial velocity of the center of mass which in this case is nearly identical to that of Sgr A*. Since the velocity of the Local Standard of Rest (LSR) with respect to the Sun is known, the radial velocity of Sgr A* immediately gives the velocity of the LSR relative to Sgr A*. If this velocity is found to diﬀer signiﬁcantly from zero, it can only mean that either Sgr A* is not at rest with respect to the Galaxy or that the LSR is moving radially with respect to the Galaxy. Gould & Popowski

(1998) ﬁnd that the LSR is moving radially outward relative to local halo stars at

230 −4.0 ± 8.5kms−1. Measurement of the velocity of Sgr A* would therefore allow one to distinguish between these two possibilities.

5.2. Physical Principles of the Method

Genzel, Eckart, Ott, & Eisenhauer (1997) and Ghez, Klein, Morris, & Becklin

(1998) have measured the proper motions of many stars within 1 of Sgr A*. These proper motions are actually Keplerian orbital motion of stars around the central

6 mass of M =2.6(±0.2) × 10 M. (The assumption of Keplerian orbits will be discussed in more detail in §5.5). As yet, the time baseline of these observations

(∼ 2 years) is not suﬃcient to show these proper motions as making an arc around

Sgr A*. Let us consider star S0-1 from the Ghez, Klein, Morris, & Becklin (1998) catalogue (S1 in Genzel, Eckart, Ott, & Eisenhauer 1997 catalog), at a separation of 0.1 from Sgr A*, the closest so far detected. Its magnitude is K =14.9. Of all the stars in the vicinity of Sgr A*, S0-1 is the most likely to show signs of orbital motion in near-term future observations. How long do we have to wait until this happens? If we assume a circular orbit, and that the projected separation at which we now see S0-1 is close to the angular radius of its orbit, then Newton’s generalization of Kepler’s Third law gives us a period of 17 years. This is just a rough estimate, but it gives us an idea that a considerable part of the orbit of S0-1 can be traced in a reasonable amount of time.

231 What will learning about the orbit of S0-1 tell us about R0? For simplicity, assume that S0-1 is in a circular orbit of period P = 17 yr, with a major-axis a = R0α,whereα = 100 mas is the current separation. Let the angle of inclination be i, so that the projected semi-minor axis is b = a cos i. The observed dispersion in radial-velocity measurements is related to these quantities by

2πa sin i 2πα sin i [var(v )]1/2 =(v2 −v 2)1/2 = = R (5.1) rad rad rad P P 0

Let us for the moment assume that P and var(vrad) can be measured with perfect accuracy. Then the fractional error in the determination of R0 is equal to the fractional error in the measurements of α sin i. The fractional error in α is simply

− −1/2 δα σρ 2yr σρ α 1 P = =0.007 (5.2) α α P 2mas 100mas 17yr implying that the method is potentially very precise. However, for low-inclination orbits, the fractional error in α sin i is much greater than in α. Speciﬁcally

√ √ √ δ(α sin i) δ( α2 − β2) α2 + β2 1+cos2 i δα = √ = δα = , (5.3) α sin i α2 − β2 α2 − β2 sin2 i α

where β = b/R0. That is, the fractional error in R0 is larger than the fractional error in α by a factor of 2.5 for i =45◦, and by a factor of 5.3 for i =30◦.The fractional error in the period will be δP/P ∼ δα/α, so it is appropriate to ignore it in this simpliﬁed analysis.

232 How good do the radial-velocity measurements have to be so that they do not dominate the error in the determination of R0? The fractional error in the dispersion measurement must be smaller than the fractional error in α sin i.Thatis

√ −1/2 δ var(vrad) σv (P/2yr) δ(α sin i) 1+cos2 i δα rad = < = 2 , (5.4) var(vrad) v0 sin i α sin i sin i α

where v0 ≡ 2πa/P, and where we have assumed that there are annual radial-velocity

measurements with accuracy σvrad . Equation (4) can be expressed

√ σρ 1+cos2 i σvrad < v0 α sin i √ 2 σρ/α csc2 i +cot i v =56kms−1 0 . (5.5) 0.02 2 1400 km s−1

Hence, the velocity errors will not dominate unless they are greater than 50 km s−1.

This simple analysis for the case of a circular orbit demonstrates the potency of the method. In §5.3 we will do the exact (although intuitively less clear) treatment of this problem in the case of orbits with arbitrary eccentricity and orientation. We present the accuracies obtainable for R0 from the three stars closest to Sgr A*. Then we show how combining the measurements of the three stars considerably improves the accuracy, and simulate what we can expect to get.

In §5.5 we discuss the validity of our basic assumptions and some speciﬁc details.

233 5.3. Method of Error Determination

The distance to Sgr A*, R0, is one of the 13 parameters required to describe the Star-Sgr A* binary orbit, and the precision of its measurements depends not only on the observational errors, but on the actual values of the 13 parameters, i.e. on the specific orbit. We can only specify the precision of this distance measurement as a function of these 13 parameters. At first sight, this appears unwieldy. However, as we now show, 11 of the parameters are known sufficiently well that the uncertainty in R0 does not depend on them in a major way. They can therefore be considered fixed, and it is only necessary to specify the uncertainty in

R0 as a function of the two remaining free parameters.

The 13 parameters can be taken to be: (1) the six phase-space co-ordinates of

Sgr A*, (2) the six phase-space co-ordinates of the orbiting star relative to Sgr A*, and (3) the mass M of Sgr A*. The six phase-space co-ordinates of Sgr A* and the mass of Sgr A* can be considered known. For example, we take the distance to be 8 kpc. However, if the distance is 5% higher, 8.4 kpc, our estimate of the distance error will be oﬀ by a similar amount, which is quite small compared to the factor ∼ 5 range of values we will explore for the distance error. Similarly,

6 we take the mass of Sgr A* to be M =2.6 × 10 M. This value is currently uncertain by some 8%. But whether it is actually 8% higher or lower, our estimate

234 of the distance error is robust, since it would only have to be scaled by a similar amount. We performed trial runs with diﬀerent central masses and found this to be true. Nothing in the error analysis depends in any way on the values of the two-dimensional angular position, the two-dimensional angular velocity, and the radial velocity of Sgr A*, so these quantities can truly be ignored.

Four quantities are measured: the two-dimensional angular position and velocity of the star. Unfortunately, these measurements are relative to the frame of the near-IR speckle images, not Sgr A*. However, Menten, Reid, Eckart, & Genzel

(1997) have aligned this IR frame with the radio frame in which Sgr A*’s position and proper motion are measured. Using this alignment, Ghez, Klein, Morris, &

Becklin (1998) ﬁx the position of Sgr A* to ±10 mas. Ultimately, the visual-binary orbital solution will ﬁx this position with an order of magnitude better precision.

For the orbit to be completely speciﬁed, two additional parameters are required. For convenience, we take these two parameters to be the period P and inclination i. Thus, for each P and i we construct an orbit that is consistent with the observable quantities: the two-dimensional angular position and two-dimensional angular velocity. Not all values in the P − i parameter space produce an orbit.

When they do there are four possible solutions. There are two sets of two solutions that each comprise a pair of orbits that are symmetrical in the plane of the sky and

235 so have the same error structure. Thus, there are eﬀectively two diﬀerent solutions

(two diﬀerent orbits) for each P and i.

After we find the parameters for all the possible orbits of a given star we simulate a set of observations made once a year. Observations consist of measuring the position and the radial velocity. We introduce the radial-velocity measurements in the fifth year of observations (i.e. in the year 2000, since we set tobs =0at the epoch of the first Ghez, Klein, Morris, & Becklin (1998) observations, i.e., the year 1995), when we expect adaptive-optics spectroscopy to become good enough to produce these data. We assume the current positional measurement accuracy (2 mas) throughout, and an error in the radial-velocity measurements of

−1 § σvrad =20kms . The choice of σvrad will be discussed in 5.4.1.

We determine the evolution of the accuracy of R0 as the observations accumulate by calculating the 13 × 13 covariance matrix for the n = 13 parameters of the model. To do so, we ﬁrst write the three observables F m(t)(m =1, 2, 3 for

m F = ρx,ρy,vrad) as functions of the time t and of the 13 parameters which we

m m label ai.Thatis,F (t; a1,...,a13). We then linearize F in the neighborhood of

∗ the solution, ai .

n m − ∗ m m ∗ ∗ Flin(t)= (ai ai )fi (t)+F (t; a1,...,a13) (5.6) i=1

236 where

m m ∂F fi (t) ≡ . (5.7) ∂ai

We then specify the solution using seven classical parameters for the orbit of the star relative to Sgr A*, and six to specify the three-dimensional position and velocity of Sgr A* itself. Speciﬁcally, we have

m m F (t)=G (t)+ρm,0 + µmt, (m =1, 2) (5.8)

3 3 ∂G (t) F (t)=R + vr, (5.9) 0 ∂t 0

where (ρ1,ρ2)0 is the initial position of Sgr A* and (µ1,µ2) is its proper motion relative to the IR star frame, vr,0 is the heliocentric radial velocity of Sgr A*, and

Gm(t) is the three-dimensional orbit of the star relative to Sgr A*, in angular units.

That is,

m 1 2 3 r G (t)= Rmp(n)Rpq(i)Rqr(γ)H (t), (5.10) pqr

H1(t)=α(cos ψ − e),H2(t)=α(1 − e2)1/2 sin ψ, H3(t)=0, (5.11)

and

ψ − e sin ψ = ωt + φ. (5.12)

237 Here, α is the angular semi-major axis, e is the eccentricity, ω =2π/P, φ is the phase of the orbit at t =0,andR1, R2,andR3 are the rotations which we take to be around the 3, 1, and 3 axes respectively. The rotation angles are n, i,andγ.

We then create the matrices b and c with elements

m m ∂F (tk) ∂F (tk) 1 −1 ij b = m 2 ,c= b , (5.13) k,m ∂ai ∂aj (σk )

m m where σk is the error in the measurement of F (tk). The index k steps through the “data” – in our case the observations made each year. The square of fractional uncertainty of the parameter aj is just the covariance matrix element cjj. One of these parameters is R0. This allows us to predict the uncertainty in R0 after tobs years of observing. In §5.4 we present the results of these calculations for the three stars closest to Sgr A*.

5.4. Predictions of the Uncertainty of Galactocentric

Distance Determination

5.4.1. Individual Stars

How does the precision of the R0 determination evolve as the observations accumulate at the rate of one per year? Let us illustrate this using the star S0-1,

238 presently the closest to Sgr A*. For each P and i we will for now consider only one of the two possible orbits.

In Figure 5.1 we fix the inclination at i =45◦ and explore all the possible orbits with this inclination. Different lines correspond to different possible periods in integer increments. The short-dashed line corresponds to the shortest possible period of P = 20 yr, while the long-dashed line represents the orbit of P =40 yr, the upper limit that we explore in this section (but see §5.4.3). Each line represents the evolution of the fractional uncertainty of R0, as the observing time tobs progresses. The first thing to notice is that for the orbits of a fixed inclination,

σR0 /R0 will not depend strongly on the period. The plot shows that 5% precision is achieved after about 12 years of observing, 2% after some 20 years, and 1% precision after approximately 30 years. After this time, which corresponds to observing a star throughout its entire orbit, a plateau is reached, i.e. additional

−1/2 observations reduce the uncertainty of R0 only as tobs .

As we mentioned above, the radial-velocity errors are assumed to be

−1 σvrad =20kms . Since these measurements have not yet been made, we do not know what their precision is going to be. However, we can explore how sensitive

the determination of R0 is to a range of diﬀerent values of σvrad . In Figure 5.2 we

◦ show the evolution of σR0 /R0 for S0-1 for the P =30yr,i =45 orbit (one of the

−1 lines in Fig. 5.1) for various assumed values σvrad =5, 10, 20, 50 and 100 km s .

239 The basic result is that σR0 /R0 is not strongly aﬀected by σvrad over this range. At

late times (tobs > 30 yr) σR0 /R0 depends only very weakly on σvrad . In the region of

−1 tobs where this dependence is strongest we see that σvrad =20kms (bold line) is

the highest value that is still close to the lines of low σvrad . That is why we adopt

−1 σvrad =20kms throughout this paper. The σR0 /R0 dependence on σvrad is much less for the orbits with short periods (regardless of inclination), and only somewhat greater for longer periods.

Finally, we move on to exploring the entire P − i parameter space. We do

− so by making contour plots for S0-1 in the P i plane of σR0 /R0 after some

ﬁxed observing time. (We use contours showing 10, 5, 2, 1, and 0.5% distance error). We examine two cases: tobs =15yrandtobs = 30 yr. The outermost contour in all of the contour plots actually limits the parameter space of possible

orbits. Figure 5.3 displays contours of σR0 /R0 for orbits with diﬀerent P and i after tobs = 15 yr. It shows that orbits of higher inclination generally allow more a precise determination of R0 (although this is reversed for the highest inclinations at long periods). The situation after tobs = 30 yr is shown in Figure 5.4. The contours of high R0 precision (better than 1%) are again in the high inclination range and towards shorter periods. This corresponds to short orbits that are away from face-on orientation where the fractional error in α sin i is high. See §5.2. Since

240 for each P and i we have two possible orbits, there would be similar plots (that we do not show) for the second set of solutions.

Star S0-1 is the closest star from Sgr A* (0.114) found to date and has the highest measured two-dimensional velocity of ∼ 1400 km s−1 (Ghez, Klein, Morris,

& Becklin 1998). Therefore, it represents the ﬁrst logical choice for the analysis we have given. However, we ﬁnd two other stars in the Ghez, Klein, Morris, &

Becklin (1998) catalog that can also be used for ﬁnding R0 in tobs =30yr,andone

of them produces useful results in tobs = 15 yr. These are S0-2 (0. 151 from Sgr

A*, K =14.1) and S0-3 (0.218 from Sgr A*, K =14.7). They are also known as S2 and S4 in the Genzel, Eckart, Ott, & Eisenhauer (1997) catalog. (S3 from

Genzel, Eckart, Ott, & Eisenhauer 1997 is another candidate, but its detection is ambiguous in the Ghez, Klein, Morris, & Becklin 1998 catalog which we use throughout this paper).

We ﬁnd that the P − i parameter space for the star S0-2 looks substantially diﬀerent from S0-1. Highly inclined orbits (almost edge-on) are allowed for fairly short orbital periods. After tobs = 15 yr (Figure 5.5) some orbits already produce

R0 estimates with better that 1% errors. This is several times better than in the case of S0-1. After tobs = 30 yr (Figure 5.6), some orbits of S0-2 even give a better than 0.5% estimate of R0. Finally, for the star S0-3 (we do not show the results in

241 ﬁgures), after tobs = 30 yr some orbits yield a better than 1% estimate of R0.No useful estimate of R0 can be obtained from S0-3 alone after tobs =15yr.

5.4.2. Combined Solution

How much would the three stars together, compared to each of them individually, contribute to a more certain value of R0? One approach is to simply take a weighted average of the estimates of R0 from each single star. To illustrate this, in Figure 5.7 we choose some arbitrary values for the orbital parameters for each of the stars S0-1 (P =30yr,i =45◦), S0-2 (P =20yr,i =70◦) and S0-3

◦ (P =30yr,i =75). The evolution of σR0 /R0 from each of them is shown by thin solid lines. The uncertainty obtained by averaging is shown by a bold dashed line.

However, observing the three stars, even for a short time, can produce a much better estimate of R0 than the simple weighted average. This is because partial orbits of more than one star can together, in a rank 25 (= 3 × 6 + 7) covariance matrix, constrain the position, velocity and mass of Sgr A* much better than each orbit can individually: each star must orbit a common mass at a common center. We show the reduced uncertainty by a solid bold line. In the ﬁrst few years this combined solution produces a very signiﬁcant improvement compared to the naive statistical treatment. Afterwards, it remains better by some 40%.

Note that in this example, 2.5% precision can be obtained already in the eighth

242 year of observation, and 1% precision only two years after that, i.e. by 2003 and

2005, respectively. Ultimately, 0.2% accuracy of R0 could be achieved. We also performed the same analysis excluding the observations of S0-3. We ﬁnd that

S0-3 contributes somewhat (∼ 7%) (mostly by constraining the position and the velocity of Sgr A*) even at times when the observations of S0-3 alone do not tell us much about R0. This indicates that perhaps even the stars farther away, that alone cannot determine R0, might contribute by constraining the position and the velocity of Sgr A*. Therefore, it would be advisable to keep track of other stars near Sgr A* in addition to the three closest.

We note that in this example, the heliocentric radial velocity of Sgr A* is measured to 8 km s−1 precision after 15 years and 3 km s−1 after 30 years, roughly half the measurement errors of Gould & Popowski (1998). See §5.1.2.

5.4.3. Probability of Achieving a Given Precision

Figure 5.7 shows the evolution of distance errors for one particular combination of orbital parameters for the three stars S0-1, S0-2, and S0-3. However, Figures

5.3–5.6 show that the precision of the distance measurement from an individual star is a strong function of the actual value of the assumed parameter pair (P, i).

Thus, the precision obtained from combining measurements of the three stars will also depend strongly on the particular combination of orbital parameters. To assess

243 how likely it is that a given distance precision can be achieved after a given term of observations, we must calculate the relative probabilities of each pair of orbital parameters (P and i), for each solution (1 and 2), and for each star (S0-1, S0-2, and S0-3). To estimate these probabilities, we adopt a simple model of the stellar distribution around Sgr A* (assumed to be stationary): a spherically symmetric power-law density proﬁle dN/dr3 ∝ r−ν, and an eccentricity distribution that is uniform in e2. As we now show, these assumptions imply that the phase-space density is given by,

dN = CP1−(2/3)ν sin i, (5.14) dP di dη dγ de2 dφ where (i, η) speciﬁes the normal vector to the plane of the orbit, γ is the position oftheapsevector,φ is the phase of the orbit at t = 0 (see eqn. 5.12), and C is a constant.

By assumption, the distribution is uniform in e2. Since it is stationary, the distribution is uniform in phase φ. By spherical symmetry, the orientation of the orbital plane is uniform in d cos idη, and the position of the apse vector,γ, is uniformly distributed as well. Since a power-law has no scale, the distribution of semi-major axes will be proportional to the distribution of radial positions at a given moment, i.e., dN/da ∝ dN/dr or dN/d ln a ∝ a3−ν. Then by Kepler’s

Third Law, dN/d ln P ∝ dN/d ln a ∝ P 2−(2/3)ν, which completes the justiﬁcation of

244 equation 5.14. Following Bahcall & Wolf (1976), we adopt ν =7/4, so that the phase-space density scales as dN/dP ∝ P −1/6.

To determine the relative probabilities, we compute for each (P, i) pair, the partial derivatives of the “natural” phase-space coordinates used in equation 5.14 (η, γ, e2,φ) with respect to the observed phase-space coordinates

(r⊥, v⊥)=(rx,ry,vx,vy). The relative probability of a given orbit is then the product of equation 5.14 with the Jacobian of this transformation, i.e.,

2 −1/6 ∂(η, γ, e ,φ) P(P, i)=CP sin i . (5.15) ∂(rx,ry,vx,vy)

In order to normalize the probability densities, we extend our calculations up to periods of 200 yr, and extrapolate the trend of declining probabilities from that point (P = 200 yr) onward, with a simple power-law.

We ﬁnd that for S0-1 (both solutions) the probabilities that P<40 yr, 40 yr

200 yr are respectively 44%, 32%, and 24%. Recall that

Figures 5.3–5.6 show R0 uncertainties only for P<40 yr. One might expect that for P tobs, the orbits would be poorly determined and the uncertainties would grow dramatically. Since most of the probability is in the region P>40 yr, this would imply that the prospects for measuring R0 using this method would be poor.

245 In fact, we find that the distance errors do not depend significantly on period for P>40 yr. That is, the distance-error contours in Figures 5.3–5.6 simply extend horizontally to the right to arbitrarily large periods. The reason for this constancy is that these high-period orbits are approximately parabolic, with observations covering most of the “action” near periastron. We find that the semi-major axes and periods of these nearly parabolic orbits are poorly determined but the errors in all other orbital parameters (including the mass and distance of Sgr A*) are unaffected by the size of the orbit.

Now, with each orbit (P − i point) of each of S0-1, S0-2 and S0-3 (and their two diﬀerent solutions) having some absolute probability assigned to it, we run a Monte Carlo simulation that picks a random set of 3 orbits (one for each star) according to how probable they are. We run this simulation 20000 times in order to assure good sampling. For each simulation (set of three orbits) we calculate the

R0 uncertainty of the combined solution (which includes the constraints described above).

We present the results of this simulation in Figure 5.8. It shows the chances

(cumulative probability) of determining R0 with a fractional accuracy σR0 /R0 after tobs = 8yr (thin solid line), and tobs = 15yr (bold solid line). After 8 years (year

2003), there is a 40% chance that R0 will be determined with 5% accuracy. After

246 15 years, we can accomplish 2% accuracy with 70% chance, and 1% accuracy with

30% chance. Moreover, after 15 years, a 3% accuracy is almost guaranteed.

5.5. Discussion and Conclusion

Besides the possibility of giving us a precise measurement of R0, this method is very powerful because it does not rely on a calibration. The physical model has a single underlying assumption: stars in the vicinity of Sgr A* move in a Keplerian potential. We believe that this is a robust assumption which is consistent with the available observational data. Nevertheless, it is possible in principle that in addition to the point mass at the center of the potential there is a continuous mass distribution composed of many faint stars or perhaps some more exotic material.

This would cause the orbits to precess. However, in order for precession to be signiﬁcant (i.e. to amount to the positional measurement accuracy of 2 mas after one orbital period) we calculate using an “isochrone model” (Binney & Tremaine

1987) that the required stellar density at the position of S0-1 would have to be

10 −3 staggering 1.3 × 10 Mpc . If this material were in the form of a normal stellar population, it would correspond to a surface brightness of ∼ 5 mag arcsec−2 in K band, which is clearly not the case. The “isochrone model” is isotropic. Precession would also arise from a non-isotropic density distribution. In that case, accounting for any possible precession would be less straightforward.

247 Munyaneza, Tsiklauri, & Viollier (1998) have explored an exotic version of this scenario with an extended object at the Galactic center composed of degenerate, self-gravitating heavy neutrino matter. They show that such an object could alter the orbit of S0-1 so as to make it distinctly diﬀerent from a Keplerian orbit, and that positional measurements alone might resolve this eﬀect over the next 10 years.

Distinguishing between the black-hole and extended-object scenario will become even easier once the radial-velocity curve of S0-1 is measured. Although an unlikely prospect like this would complicate our method, it would in itself be a radical result with profound consequences.

In §5.2, we argued that in the case of circular orbits, only relatively crude

≤ −1 radial-velocity measurements (σvrad 50 km s ) are required. We now investigate the role of radial-velocity measurements more closely. In Figure 5.2 we show the

◦ fractional error in R0 as a function of time for S0-1 assuming i =45 and P =30

yr, and for various values of σvrad . Also shown in this plot (dashed curve) is the

fractional error in the quantity α sin i/P . This would be exactly equal to σR0 /R0, if there were perfect knowledge of var(vrad) (i.e. perfect velocity measurements) but none of the radial-velocity information were used to disentangle degeneracies among the orbital parameters.

At late times (tobs >P) all the curves converge, with the dashed curve lying slightly below the others. This shows that high-precision velocity measurements

248 are indeed of small value if the observations are carried out for at least one period.

However, at early times, the dashed curve lies well above the others. This shows that for observations shorter than the period, the radial-velocity measurements play a signiﬁcant role in disentangling the degeneracies among the orbital parameters, and hence can signiﬁcantly help determine the distance R0.

The degree of success of the method we present, in terms of precision, will eventually depend on the actual orbits of the stars in the vicinity of Sgr A*. Our

Monte Carlo simulations of possible orbits show that we will almost certainly soon obtain determinations of R0 that match those available today provided that observations are made on a yearly basis, and that they soon start to include radial velocities. Then, this method will produce a precise estimate of the distance to the

Galactic center which is free from the various systematic eﬀects that aﬀect other methods currently in use.

249 ◦ Fig. 5.1.— Evolution of R0 fractional uncertainty from S0-1 orbits with i =45

(tobs is in years). Different lines correspond to different integer orbital periods. The short-dashed line corresponds to the minimum possible period of P = 20 yr, while the long-dashed line represents the orbit of P = 40 yr, our upper cutoff in this example.

250 Fig. 5.2.— Evolution of R0 fractional uncertainty from S0-1 orbit (P =30yr, ◦ i =45, tobs is in years) with diﬀerent values of the radial-velocity measurement error −1 −1 σvrad =5, 10, 20, 50 and 100 km s . The upper solid line corresponds to 100 km s −1 and the bold line represents σvrad =20kms . The dashed line is the fractional error in α sin i/P.See§5.5.

251 Fig. 5.3.— Contour plot of R0 fractional uncertainty from S0-1 orbit (solution 1), after tobs = 15 yr. Contour in the upper right corner is of increasing uncertainty. The jagged edges of some contours are computational artifacts.

252 Fig. 5.4.— Contour plot of R0 fractional uncertainty from S0-1 orbit (solution 1), after tobs = 30 yr. The jagged edges of some contours are computational artifacts.

253 Fig. 5.5.— Contour plot of R0 fractional uncertainty from S0-2 orbit (solution 1), after tobs = 15 yr. Contours in the upper right corner are of increasing uncertainty.

254 Fig. 5.6.— Contour plot of R0 fractional uncertainty from S0-2 orbit (solution 1), after tobs =30yr.

255 Fig. 5.7.— An example of the evolution of R0 fractional uncertainty from each of ◦ ◦ the stars (S0-1: P1 =30yr,i1 =45, S0-2: P2 =20yr,i2 =70, S0-3: P3 =30yr, ◦ i3 =75) separately (thin solid lines), when combined in a straightforward statistical way (bold dashed line), and a combined solution with a common position, velocity and mass of Sgr A* (bold solid line).

256 Fig. 5.8.— Chances (cumulative probability) of obtaining a given R0 fractional uncertainty of a combined solution after tobs =8yrandtobs = 15 yr. Calculated using 20000 Monte Carlo simulations that pick orbits of S0-1, S0-2 and S0-3 randomly, according to their probability.

257 Chapter 6

Kinematics of Stellar and Nature of Dark Halo

6.1. Kinematics of the Stellar Halo

6.1.1. Introduction

Studies of the Galactic halo shed light on the formation of our galaxy.

The kinematics of the stellar halo are an important indicator of the formation mechanisms involved. Some of the more recent studies that derive the halo velocity ellipsoid in the inner halo include Layden et al. (1996) who used low-metallicity

RR Lyraes, and Gould & Popowski (1998) who used low-metallicity RR Lyraes as well as other low-metallicity stars. Gould & Popowski (1998) derive a halo rotation in prograde direction of 34.3 ± 8.7kms−1 with respect to the Galactic frame. This

258 estimate, which assumes that the Sun’s velocity in the Galactic plane is 232 km s−1 in the prograde direction, applies to halo stars within 3 kpc.

FAME will be able to greatly improve the range, the resolution and the accuracy of the halo velocity ellipsoid, thus allowing one to see any gradient in motion. FAME will be able to do this by measuring proper motions of many faint blue horizontal branch (BHB) stars in various directions of the sky. Horizontal branch stars are especially favorable for mapping purposes as their nearly constant luminosity permits a relatively precise estimate of their distances.

6.1.2. Selecting Faint A-type Stars

Since this faint stars are not within the magnitude limit of the mission, they have to be preselected and included in the observing list (the input catalog).

Candidate blue horizontal branch stars (BHB stars) can be eﬀectively selected from multi-band photometry. This was recently demonstrated by Yanny et al.

(2000) using SDSS ugr photometry. Their study showed that A-colored stars

(A-stars) trace huge substructures in the halo. Unfortunately, other types of blue stars, those with main sequence gravity (mostly ﬁeld blue stragglers (BSs)), have similar colors as BHB stars, and it is not clear whether distinguishing between these two populations can be done based on photometry alone. Mixing these two types

259 of stars that have very different absolute magnitudes (and with BS luminosities likely having a strong metallicity dependence as well) makes it difficult to use them as distance indicators. One certain way of distinguishing any individual star is by measuring the widths of Balmer lines that indicate surface gravity (BHB stars have lower surface gravity and narrower lines). However, spectroscopy will not be available in the majority of cases, and we must consider other methods. The stars of the two population with same apparent magnitudes lie at different distances, so they will on average exhibit different proper motions. It is this feature that we will use to distinguish between BS and BHB-star populations.

BHB stars with R<15 will already be included in the FAME input catalog.

In the 1/2 of the north polar cap for which SDSS data will be available one can employ color criteria similar to those used by Yanny et al. (2000). Selecting stars with 15

− 0.3

we derive a surface density of A-stars in the ∼ 500 deg2 of SDSS Early Release Data

(EDR) of ∼ 4.2deg−2. This implies that approximately 20,000 BHB star candidates will come from SDSS. For the rest of the northern cap and the entire southern cap, one can try to retrieve BHB candidates from some other all-sky catalog.

260 We perform this exercise using USNO-A2.0, although the Guide Star Catalog

2 (GSC-2), which will have better calibrated photometry, will be better suited.

Since in the case of USN0-A2.0 we have at our disposal only two-band photometry which is quite crude (color error of ∼ 0.4 mag), we need to estimate the efficiency of getting A-star candidates given some level of contamination from the many times more numerous turn-off stars. To do this we matched SDSS EDR A-stars to USNO-A2.0 objects and identified them on the USNO-A2.0 color-magnitude diagram (CMD) of all stars within the SDSS EDR sky coverage. SDSS selected

A-stars in this CMD lie on the blue side, as expected, but are not clearly separated from the turn-oﬀ star locus. Since USNO-A2.0 provides only single color on which to base the selection, the blue portion of the USNO-A2.0 CMD also contains white dwarf and quasar contaminants, as well as turn-oﬀ stars. In the case of SDSS selection these quasars and white dwarfs were eliminated using the u∗ − g∗ color.

Although in the present context WDs and QSOs are considered as contaminants, including them in the input catalog is useful for other aspects of this project. As stated previously, QSOs can be distinguished by their K-band excess, while WDs will stand out by their much higher proper motions.

The locus of both the turn-oﬀ stars and the SDSS selected A-stars in the

USNO-A2.0 CMD is tilted with a slope corresponding to the line,

BUSNO − RUSNO = a − 0.15RUSNO, (6.2)

261 where a is the zero point of the line. Now we can count the number of

A-stars are included in the region to the left of the line. As we move red-ward, both the number of real (SDSS selected) A-stars and the number of USNO-A2.0 A-star candidates will increase. We find that the ratio of USNO-A2.0 A-star candidates to SDSS A-stars remains constant for a<2.4, but that the turn-off contamination increases rapidly further red-ward of this point. For a<2.4 we find the ratio of

USNO candidates to SDSS A-stars to be 12. Such selection retrieves 54% of SDSS selected A-stars. To summarize, if this selection is applied to 15,000 deg2 of the northern and southern caps not covered by SDSS, one will end up with 340,000 candidates that will contain 28,500 actual A-stars. Note again that many of these

‘contaminants’ will be QSOs and WDs which one would want anyway.

In the case that the GALEX UV sky survey becomes available, we would be able to eliminate turn-oﬀ contaminants in USNO-A2.0 selected A-star candidates.

Alternatively, one can use GALEX to select A-star candidates and then match them in USNO-A2.0 (or better yet GSC-2).

262 6.1.3. Halo Rotation Measurement

We now come to the question of distinguishing A-stars as either BHB stars or BSs, using proper motions from FAME. For simplicity, we will assume that all

A-stars are located exactly in the direction of the Galactic pole, so that only two of the three components of the motion are expressed. In one direction, let us call it y, we will then see the rotational motion around the center of the Galaxy. The velocity corresponding to this motion is approximately 200 km s−1,andthevelocity dispersion in this direction according to Gould & Popowski (1998) is 109 km s−1.

In the perpendicular direction (x), parallel to Sun–Galactic center vector, we will assume no bulk motion, and the Gould & Popowski (1998) velocity dispersion of

160 km s−1.

At each apparent magnitude both BHB stars and BSs are sampled, but their ratio, even if constant in a given volume, is not independent of apparent magnitude.

If we assume, as for example implied by Yanny et al. (2000), that in a given volume the field BSs outnumber field BHB stars 2:1, and that both populations fall off with

Galactocentric distance as r−3.5, then from our vantage point 8 kpc from the center, at any given magnitude we will sample BSs at a different Galactocentric distance than BHB stars, and the magnitude bin will correspond to different volumes for the two types of stars because of different heliocentric distances. Thus at R =15

263 we ﬁnd NBS/NBHB =0.17, but at R = 19 the ratio is 2.3. Other values can be deduced from Table 6.1.

Next we use Monte Carlo techniques to simulate observations of all A-stars that one hopes to select using SDSS and USNO-A2.0, as previously described. We do this for R =15, 16, 17, 18, and 19. For each magnitude bin (R ± 0.5) we ﬁrst estimate the number of BHB stars and BSs that FAME will detect in that bin

(given in Table 6.1), then to BHB stars we assign an absolute magnitude drawn randomly from MR =0.8 ± 0.15, while to BSs we assign absolute magnitudes distributed as MR =3.2 ± 0.5, where the dispersion is meant to reﬂect the range of absolute magnitudes at a given color. This will determine the distance to that star. Then each star is assigned two components of physical velocity drawn from the halo velocity distribution described earlier in this section. Using the distance, we convert this velocity into a true proper motion. The observed proper motion is then obtained by adding in a measurement error based on the FAME accuracy.

Since at any given magnitude BSs lie three times closer than BHB stars, in each apparent magnitude bin there will be two peaks in proper motion, corresponding to BSs and BHB stars, shown in Figure 6.1. Because of proximity and because of a greater spread of absolute magnitudes, the distribution of proper motions corresponding to BSs will be wider at a given magnitude than that of

BHB stars. It is the position of the peak of BHB stars proper motion distribution

264 in y-direction that will yield the halo rotational velocity. To ﬁnd out how well this peak can be determined in the face of BS contamination, we calculate errors in

ﬁtting 2-dimensional Gaussians to each of the two peaks. Each Gaussian is deﬁned by six parameters – two for the x and y center of the peak, three for two diagonal

(σxx,σyy), and one oﬀ-diagonal (σxy) term in the covariance matrix describing peak widths, and one corresponding to the number of stars (amplitude). We calculate the errors of these parameters from our simulated measurements.

The results are summarized in Table 6.1. The second column lists the distance of BHB stars of magnitude R, which is given in the ﬁrst column. The third and fourth columns list the number of BHB stars and BSs that we expect FAME to measure in the two caps (|b| > 30◦). The next two columns list the errors that we derive for the position of the BHB proper motion peak, and columns 7 and

8 list estimates of the uncertainty of the BHB stars velocity dispersion along y and x directions. The ﬁnal column, derived from columns 2 and 5, shows the expected error in the stellar halo rotation velocity. In Figure 6.2 we present σrot as a function of R or d. One can see that within 20 kpc FAME achieves a precision

− of halo rotation measurement of ∼< 2kms 1 if the data are binned in 1-mag steps.

Better spatial sampling (resolution) can be achieved by choosing smaller bins, but with correspondingly larger errors. At distances to ∼ 30 kpc quite good estimates of stellar halo rotation can still be made. Only farther out do the measurement

265 errors and the preponderance of BSs of same magnitude as the BHB stars preclude obtaining a useful result. Note that here we assume that the BHB stars’ luminosity will be very well determined locally by FAME’s trigonometric parallaxes, and that this luminosity does not depend on distance from the plane. To get motions relative to the Galactic frame, it will of course be necessary to subtract the Sun’s circular velocity, which can also be determined precisely using FAME measurements of proper motions of a large number of bulge stars.

A more sophisticated analysis would show that the halo rotation can be mapped in two dimensions, instead of one as outlined here. Such probing of the halo potential would place more constraints on the dark halo models. Additional information on the still debated shape of the dark matter halo will be gleaned from the three-dimensional distribution of stars determined to be BHB stars from FAME proper motions. An eﬀect that might introduce systematic error in distances and thus the velocities is if BHB stars change in luminosity (due to age and metallicity eﬀects) as we move above the plane. This problem might be eliminated by imposing an axial symmetry–we would require vrot to be constant at a given Galactocentric radius, regardless of the direction. Such a treatment requires an analysis that is beyond the scope of this paper.

In the above analysis we have used BHB stars to probe halo kinematics and considered BSs as a contaminating factor. We note that with the halo motion

266 determined by BHB stars, the measured proper motions of BSs can be used to derive their luminosity calibration and distribution of metallicities.

6.1.4. Substructure in the Galactic Halo

Discoveries of clumps of RR Lyrae stars (Ivezi´c et al. 2000), and A-colored stars (Yanny et al. 2000) in the halo indicate that the Galaxy formation mechanism might be more complex than previously envisaged, and that accretion or merging of small galaxies might have played a crucial role. Some models even suggest that all halo stars come from disrupted satellite galaxies. The structures found by Yanny et al. (2000) lie at Galactocentric distances of 30 to 50 kpc, and extend over many kiloparsecs. The structures are seen as overdensities of BHB stars and

BSs, i.e., only positional information is used. The third coordinate, the distance, is compromised because of diﬃculties with BHB/BS distinction, leading to smearing of the features. Besides being clumped in space, the stars originating from the same disrupted satellite should cluster in velocity space as well. In fact, the velocity information is conserved much better than the spatial information, and it is possible to associate stars that are widely separated on the sky and that have mixed spatially with other streams. Helmi & White (1999) have shown that

10 Gyr after a merging event the spatial distribution of stars in the halo will be

267 −1 very smooth, while hundreds of halo streams, strongly (σv < 5kms ) clustered in velocity space will still be present.

FAME observations will yield two components of velocity that can be used to detect substructure in the halo. The old halo streams predicted by Helmi & White

(1999) will require local samples of subdwarfs. The best sensitivity for detecting such streams comes from analyzing the proper motions of G-subdwarfs (halo turn-oﬀ stars) which are bright and numerous. Helmi & White (1999) suggest that the velocity accuracy needed to resolve individual streams is < 5kms−1, but that they will be detectable at several times that accuracy. We believe that such studies will be possible with stars selected as part of the FAME main survey (R<15).

Including fainter stars might not be useful as the velocity accuracy would be limited by inaccurate distances. This is indicated by Helmi & de Zeeuw (2000) who

ﬁnd that FAME will be able to distinguish 15% of the nearby halo streams. Their sample of halo giants is limited by parallax errors to V<12.5, and the velocity measurements are augmented with an assumed ground-based radial velocity survey.

Bigger structures (tidal streams and remnants of recently disrupted satellites) might be detectable in the R>15 BHB star sample discussed previously. The speciﬁc structures found by Yanny et al. (2000) may be too distant to be detected by FAME, yet their survey covers only 1% of sky, so more nearby clumps are likely to be present in the rest of the sky. To ﬁnd them one should aim at ∼ 20 km s−1

268 accuracy per star. With the good photometric parallaxes achievable for BHB stars, substructure mapping might be possible to distances of 10 kpc (corresponding to

RBHB ≈ 16).

6.2. Nature of the Dark Matter Halo

6.2.1. Introduction

Massive compact halo objects (MACHOs) were suggested as possible constituents of the Milky Way’s dark halo. Paczynski (1986) suggested that this hypothesis can be tested by looking for microlensing of stars in the LMC/SMC by MACHOs. It was expected that the observed frequency of the events would unambiguously prove or disapprove MACHOs. After a decade of observations, the nature of the events currently being detected toward the Large Magellanic Cloud

(LMC) by the MACHO (Alcock et al. 1997) and EROS (Aubourg et al. 1993) collaborations are a complete mystery. On the one hand, the observed optical depth τ ∼ 2 × 10−7 is an order of magnitude higher than expected from known populations of stars. On the other hand, if the lenses lie in the Galactic halo and so comprise of order half the dark matter, then their masses (inferred from the event time scales and kinematic models of the halo) are of order half a solar mass. Thus, the objects could not be made of hydrogen or they would have easily

269 been discovered from star counts (Alcock et al. 1997 and references therein). Even the estimated contribution of MACHOs to the dark matter is not secure. The

MACHO group initial claim of 50% contribution (Alcock et al. 1997) was later revised to 20% (Alcock et al. 2000), which is more in accordance with the results of EROS (Lasserre et al. 2000). EROS group data, however, does not exclude 0% contribution, i.e. the complete absence of MACHOs. This then suggests LMC self-lensing, which is also problematic because the known populations in LMC are not capable of producing the required lensing rates. In the MACHO model, while some ranges of masses are ruled out, masses between 0.1M and 1M are still plausible. This all indicates that the direct measurements of the mass and distance of the lenses would unambiguously resolve this question.

6.2.2. MACHO Masses and Distances from Astrometric

Microlensing

The method we propose to measure masses and distances of LMC lenses by measuring astrometric deviation using SIM and observing the light curves from

SIM and Earth, is identical to the method proposed to measure the bulge mass function (§4.2.2–§4.2.4). However, there are some important diﬀerences. Very few events seen toward the LMC are brighter than V ∼ 20 and therefore ∼ 40 times more observing time is needed to achieve the same precision. Here, the source lies

270 at Dos = 50 kpc, while the lens is assumed to be in the halo at Dol = 15 kpc. Hence

D = 21 kpc, which is very similar to the bulge value. This means that at ﬁxed mass, the bulge events considered in §4.2.6 will have about the samer ˜E as the

LMC events considered here. The speed of the lens relative to the observer-source line of sight is v = 250 km s−1, slightly larger than for the bulge. Recall that we are assuming that the source is V = 20 and that the total observing time is 20 hours.

Further, the assumption of uniform observations is quite reasonable for LMC

(which is near the ecliptic pole). This also means that the time of year at which the event is discovered plays no role.

The blending situation is more complicated toward the LMC because the chance that an early main sequence star has a companion of comparable brightness is larger, probably a few tens of percent. Even clump stars have brighter companions toward the LMC than toward the bulge because they are younger and so have brighter progenitors. Also, the 10 mas limit on detecting blends directly

(Han & Kim 1999) corresponds to 500 AU toward the LMC compared to 80 AU toward the bulge. Nevertheless, even toward the LMC, the majority of sources will not have companions with more than 10% of the source ﬂux and hence even here it is appropriate to ignore blending by companions in a ﬁrst treatment.

271 Toward the LMC, we assume that the source is V = 20 which is near the bright end of the events detected in this direction. Because the LMC sources are fainter and the LMC events are rarer than those seen toward the bulge, we assume a total SIM integration time of 20 hours. Note that since about 50 times more photons are received from an I = 15 star than from a V = 20 star, and since we have assumed a four-fold increase in integration time toward the LMC, photon statistics alone will produce (50/4)1/2 ∼ 3.5 times larger errors toward the LMC compared to the bulge. There will be additional diﬀerences due to the diﬀerent geometries.

6.2.3. Expected Precision of Distance and Mass

Measurements

Table 6.2 shows the results. Apart from the factor ∼ 3.5 larger errors that results simply from photon statistics, they are qualitatively similar to those for the bulge. The largest diﬀerence is that the fractional error in πs is larger which simply reﬂects the fact that the LMC is more distant.

Of the eight microlensing events detected by Alcock et al. (1997) during their

ﬁrst two years of observations, three of the sources were brighter than V = 20 (after removing blended light) which is the nominal limit for SIM. Future microlensing

272 surveys could improve the rate of detection by an order of magnitude. However, only a factor of three of this improvement would be due to the coverage of a larger area. The rest would come from going deeper which would not yield any more bright sources. Hence, the total rate of events that are accessible to SIM will not be high. Most of the usable events that are detected are likely to be close to the magnitude limit. We ﬁnd that with our assumed source magnitudes and exposure times,the mass and distance estimates will be accurate to about 10% to 20%

(although they rise to ∼ 40% for the case of M = M, β =0.4). This would be an acceptable level of precision to resolve the question of the nature of the lenses assuming that more than a handful of events can be measured. The errors in the measurement of φ without making use of the Earth-based observations (column

4) are typically 8◦. Hence in many cases it will not be possible to break the ∆β degeneracy astrometrically.

To determine whether the photometry is suﬃciently precise to break the degeneracy, we compare our simulation with that of Boutreux & Gould (1996) who speciﬁcally considered an Earth-satellite separation of 0.26 AU which is close to our value of d⊕−s =0.2 AU. In their Monte Carlo simulation, the “speed degeneracy”

(between diﬀerent scalar values of ∆β) was broken 40% to 60% of the time in the mass range 0.1–1 M. We ﬁnd that their assumed photometric errors are about

273 twice the ones assumed here. Therefore, it seems likely that the SIM photometric observations would be adequate to break this degeneracy in the majority of cases.

Inspection of column 4 from Table 6.2 shows that the direction degeneracy will usually be broken astrometrically. The simulations of Boutreux & Gould (1996) show that it is about equally diﬃcult to break the direction degeneracy as the speed degeneracy. Thus, it should usually also be possible to break this degeneracy photometrically. In any event, as in the case of the bulge, the direction degeneracy is much less important than the speed degeneracy.

Of course, not all halo lenses can be expected to be at Dol =15kpc.We also considered Dol =10kpcandDol = 25 kpc. For the ﬁrst case, we ﬁnd that the fractional errors are smaller than those in Table 6.2 by a factor ∼ 0.8 for

M ≤ 0.5 M and ∼ 0.5 for M ∼ M (except for σπs which is unaﬀected). For the second case, we ﬁnd that these errors are larger than those in Table 6.2 by a factor

∼ 1.6 for M ≤ 0.5 M and ∼ 3 for M ∼ M. Thus, the results reported in Table

6.2 apply qualitatively to a broad range of halo distances for M ≤ 0.5 M, but not for M ∼ M.

If the lenses detected toward the LMC are in the LMC itself (rather than in the halo), then we ﬁnd that neitherr ˜E nor θE can be detected, let alone measured, in our ﬁducial 20 hours of observation. However, even non-detections of these two

274 quantities would be highly signiﬁcant as it would demonstrate that the lens was in the LMC.

In the end, we hope to be able to measure 5 events over 5 years of SIM mission, which would be suﬃcient to resolve the MACHO controversy.

275 Table 6.1. Rotation of the halo from BHB stars.

RdNBHB NBS σ(µy) σ(µx) σyy σxx σrot mag kpc 103 103 µas yr−1 µas yr−1 µas yr−1 µas yr−1 km/s

15 6.9 6.3 1.1 46 66 250 502 1.52 16 11.0 8.4 2.8 27 37 98 188 1.41 17 17.4 9.0 6.7 24 28 68 107 1.94 18 27.5 7.9 11.6 79 67 239 321 10.29 19 43.7 9.3 21.6 2632 1784 6601 ··· 545

276 Table 6.2. Uncertainties of microlensing parameters, LMC (V = 20, arbitrary ψ, 20 hours.)

Mβ φ σφ σr˜E /r˜E σθE /θE σM /M σD/D σπs /πs SS+⊕ SS+⊕ SS+⊕ SS+⊕ S+⊕ S+⊕

M deg deg deg

0.1 0.2 0 6.7 2.5 5.353 0.032 0.145 0.145 5.356 0.145 0.145 0.827 0.1 0.2 30 7.1 2.1 1.124 0.035 0.138 0.138 1.128 0.141 0.141 0.788

277 0.1 0.2 60 7.8 1.8 0.580 0.046 0.124 0.120 0.587 0.127 0.127 0.689 0.1 0.2 90 8.5 1.4 0.438 0.049 0.110 0.110 0.449 0.120 0.120 0.626

0.1 0.4 0 7.4 5.7 8.326 0.078 0.166 0.159 8.362 0.180 0.177 0.852 0.1 0.4 30 8.8 5.7 2.302 0.092 0.159 0.156 2.316 0.187 0.177 0.834 0.1 0.4 60 9.5 5.3 1.234 0.134 0.141 0.138 1.241 0.202 0.184 0.757 0.1 0.4 90 10.3 3.9 0.955 0.166 0.124 0.124 0.955 0.205 0.212 0.672

0.1 0.6 0 8.8 8.1 11.455 0.170 0.219 0.191 11.561 0.262 0.247 0.880 0.1 0.6 30 13.4 8.5 5.675 0.187 0.205 0.191 5.749 0.279 0.255 0.877 0.1 0.6 60 14.1 9.2 3.362 0.279 0.177 0.170 3.405 0.350 0.301 0.849 0.1 0.6 90 14.8 8.1 2.634 0.456 0.148 0.148 2.641 0.481 0.477 0.767 (continued) Table 6.2—Continued

Mβ φ σφ σr˜E /r˜E σθE /θE σM /M σD/D σπs /πs SS+⊕ SS+⊕ SS+⊕ SS+⊕ S+⊕ S+⊕

M deg deg deg

0.3 0.2 0 4.2 2.8 2.507 0.039 0.110 0.110 2.496 0.117 0.117 0.891 0.3 0.2 30 4.6 2.8 0.707 0.046 0.106 0.099 0.707 0.110 0.110 0.831 0.3 0.2 60 6.0 2.5 0.357 0.057 0.088 0.081 0.361 0.103 0.099 0.697 0.3 0.2 90 6.7 2.1 0.272 0.064 0.074 0.074 0.286 0.095 0.099 0.612 278 0.3 0.4 0 4.9 4.6 3.843 0.117 0.124 0.124 3.850 0.163 0.173 0.933 0.3 0.4 30 5.7 4.9 1.245 0.131 0.117 0.113 1.245 0.177 0.170 0.912 0.3 0.4 60 7.1 5.7 0.654 0.177 0.099 0.095 0.658 0.205 0.194 0.845 0.3 0.4 90 8.1 4.9 0.520 0.219 0.085 0.085 0.527 0.226 0.240 0.750

0.3 0.6 0 6.0 5.7 5.218 0.279 0.152 0.145 5.271 0.311 0.318 0.983 0.3 0.6 30 7.8 6.4 2.581 0.311 0.138 0.134 2.606 0.343 0.332 0.969 0.3 0.6 60 9.9 8.1 1.556 0.442 0.113 0.113 1.566 0.467 0.445 0.969 0.3 0.6 90 11.0 8.8 1.255 0.665 0.103 0.099 1.241 0.651 0.689 0.940 (continued) Table 6.2—Continued

Mβ φ σφ σr˜E /r˜E σθE /θE σM /M σD/D σπs /πs SS+⊕ SS+⊕ SS+⊕ SS+⊕ S+⊕ S+⊕

M deg deg deg

0.5 0.2 0 4.2 3.2 2.524 0.049 0.113 0.106 2.496 0.117 0.117 0.912 0.5 0.2 30 4.6 3.2 0.841 0.053 0.103 0.095 0.845 0.113 0.110 0.856 0.5 0.2 60 5.7 2.8 0.467 0.071 0.085 0.078 0.474 0.106 0.103 0.707 0.5 0.2 90 6.4 2.5 0.378 0.078 0.071 0.067 0.389 0.103 0.106 0.612 279 0.5 0.4 0 4.6 4.2 3.345 0.156 0.124 0.120 3.330 0.191 0.205 0.972 0.5 0.4 30 5.7 4.9 1.107 0.173 0.113 0.110 1.114 0.212 0.202 0.955 0.5 0.4 60 7.1 6.0 0.633 0.233 0.092 0.088 0.640 0.255 0.244 0.894 0.5 0.4 90 7.8 5.7 0.527 0.265 0.078 0.078 0.534 0.272 0.279 0.831

0.5 0.6 0 5.3 4.9 3.737 0.403 0.145 0.141 3.769 0.421 0.435 1.025 0.5 0.6 30 7.4 6.4 1.824 0.445 0.127 0.127 1.849 0.470 0.456 1.018 0.5 0.6 60 9.5 8.5 1.177 0.587 0.103 0.099 1.191 0.601 0.590 1.043 0.5 0.6 90 10.6 9.2 1.022 0.682 0.095 0.092 1.011 0.668 0.707 1.022 (continued) Table 6.2—Continued

Mβ φ σφ σr˜E /r˜E σθE /θE σM /M σD/D σπs /πs SS+⊕ SS+⊕ SS+⊕ SS+⊕ S+⊕ S+⊕

M deg deg deg

1.0 0.2 0 3.9 3.5 1.881 0.166 0.110 0.106 1.856 0.194 0.202 0.898 1.0 0.2 30 4.6 3.5 0.537 0.198 0.099 0.095 0.534 0.219 0.223 0.852

280 1.0 0.2 60 5.7 4.6 0.297 0.152 0.081 0.078 0.301 0.159 0.184 0.810 1.0 0.2 90 6.4 5.3 0.233 0.110 0.067 0.067 0.240 0.127 0.131 0.806

1.0 0.4 0 4.2 4.2 2.630 0.424 0.120 0.120 2.648 0.445 0.438 0.997 1.0 0.4 30 6.0 5.7 0.725 0.385 0.110 0.110 0.739 0.403 0.396 0.997 1.0 0.4 60 7.4 6.7 0.424 0.279 0.085 0.081 0.431 0.290 0.297 0.965 1.0 0.4 90 7.8 7.1 0.350 0.237 0.074 0.074 0.357 0.244 0.255 0.937 (continued) Table 6.2—Continued

Mβ φ σφ σr˜E /r˜E σθE /θE σM /M σD/D σπs /πs SS+⊕ SS+⊕ SS+⊕ SS+⊕ S+⊕ S+⊕

M deg deg deg

1.0 0.6 0 4.6 4.6 2.956 0.940 0.145 0.134 3.016 0.969 0.933 1.057

281 1.0 0.6 30 7.8 7.1 1.280 0.788 0.124 0.120 1.326 0.827 0.771 1.138 1.0 0.6 60 9.9 9.5 0.810 0.612 0.088 0.088 0.824 0.622 0.612 1.142 1.0 0.6 90 9.5 9.2 0.693 0.541 0.088 0.088 0.689 0.527 0.562 1.057

r˜E is the projected Einstein radius (See eq. 3.), θE is the angular Einstein radius (See eq. 4.2),

D =˜rE/θE is given in eq. 4.3, and πs is the parallax of the source. Fig. 6.1.— Proper motion distribution of A-type stars in four magnitude bins. Each of the panels show the simulation of proper motion measurements in the direction of Galactic rotation (in masyr) of BHB stars (solid line) and BS ‘contaminants’ (dashed line). The estimate of halo rotation measurement error is deﬁned by how well the value of the peak of BHB stars can be determined.

282 Fig. 6.2.— The kinematics of the halo, and especially its rotation, can be measured by selecting ﬁeld blue horizontal branch stars, which have good photometric distances, and measuring their mean proper motion. We show the precision of this estimate, assuming FAME sensitivities, as a function of BHB stars’ R magnitude, or equivalently, their heliocentric distance. The sample is binned in 1-mag bins. Deterioration beyond R ∼ 18 occurs mainly because the color-selected sample of A-stars becomes dominated by blue stragglers.

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