Astrophysical Applications of Gravitational Microlensing

Dissertation

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

Subo Dong, M.S.

Graduate Program in Astronomy

The Ohio State University 2009

Dissertation Committee: Professor Andrew Philip Gould, Advisor Professor Bernard Scott Gaudi Professor Krzysztof Zbigniew Stanek Copyright by

Subo Dong

2009 ABSTRACT

In this thesis, I present several astrophysical applications of Galactic and cosmological microlensing.

The first few topics are on searching and characterizing extrasolar planets by means of high-magnification microlensing events. The detection efficiency analysis of the A 3000 event OGLE-2004-BLG-343 is presented. Due to max ∼ human error, intensive monitoring did not begin until 43 minutes after peak, at which point the magnification had fallen to A 1200. It is shown that, had a ∼ similar event been well sampled over the peak, it would have been sensitive to almost all Neptune-mass planets over a factor of 5 in projected separation and even would have had some sensitivity to Earth-mass planets. New algorithms optimized for fast evaluation of binary-lens models with finite-sources effects have been developed. These algorithms have enabled efficient and thorough parameter-space searches in modeling planetary high-magnification events. The detection of the cool, Jovian-mass planet MOA-2007-BLG-400Lb, discovered from an Amax = 628 event with severe finite-source effects, is reported. Detailed analysis yields a fairly precise planet/star mass ratio of q = (2.5+0.5) 10−3, while the planet/star −0.3 × ii projected separation is subject to a strong close/wide degeneracy. Photometric and astrometric measurements from Hubble Space Telescope, as well as constraints from higher order effects extracted from the ground-based light curve (microlens parallax, planetary orbital motion and finite-source effects) are used to constrain the nature of planetary event OGLE-2005-BLG-071Lb. Our primary analysis leads to the conclusion that the host is an M = 0.46 0.04 M M dwarf and that the planet ± ⊙ has mass M = 3.8 0.4 M , which is likely to be the most massive planet yet p ± Jupiter discovered that is hosted by an M dwarf.

Next a spaced-based microlens parallax is determined for the ﬁrst time using

Spitzer and ground-based observations for binary-lens event OGLE-2005-SMC-001.

The parallax measurement yields a projected velocityv ˜ 230 km s−1, the typical ∼ value expected for halo lenses, but an order of magnitude smaller than would be expected for lenses lying in the Small Magellanic Cloud (SMC) itself.

Finally, I propose using quasar microlensing to probe Mg II and other absorption “cloudlets” with sizes 10−4.0 10−2.0pc in the intergalactic medium. ∼ − I show that signiﬁcant spectral variability on timescales of months to years can be induced by such small-scale absorption “cloudlets” toward a microlensed quasar.

With numerical simulations, I demonstrate the feasibility of applying this method to Q2237+0305, and I show that high-resolution spectra of this quasar in the near future would provide a clear test of the existence of such metal-line absorbing

“cloudlets”.

iii Dedicated to my mother and father

iv ACKNOWLEDGMENTS

In modern times, most of the necessities of people’s daily lives rely upon others, therefore, no matter how highly the lofty ideal of individual freedom is acclaimed, most people have to choose their jobs according to the needs of the society rather than following their inner voices (i.e., what they are interested in the most). I consider myself very lucky to have ﬁve years’ opportunity to pursue my childhood dream in becoming an astronomer with no ﬁnancial burden. I am thankful to everyone who has made this possible.

I thank Andy Gould for being an unparalleled advisor. He has taken immense care in guiding me along my path to becoming a scientist. His inﬂuence and help have permeated every aspect of my scientiﬁc activities as a graduate student.

Whenever I need discussions or advice (or whenever he perceives me as needing them), he is always ready to offer them in most timely fashion. His excitement by new findings, creativity in novel ideas, and self-driven “Gung-ho” working spirit exhibit no trace of his age. It is difficult to describe how exhilarating an experience it is to work with him.

v I thank Andy for being a truthful person – for being truthful to himself, to others, and to nature. I was initially intimidated by the way he spoke in morning coffee and colloquia. But I have come to realize that, if one is offended by sincere efforts to pursue the truth, or not offended by distortion or fabrication of facts and careless or superficial analyses, he/she is not a real scientist. I thank Andy for teaching me to be honest to data, to listen to what nature has to say rather than projecting one’s prejudices onto nature.

I came to the United States certainly at an interesting time, when “the best and brightest” in the White House and on the Wall Street have drastically transformed the world’s social and economic orders. As a “Stranger in a Strange Land”, I appreciate Andy for many illuminating discussions that shed light on the perplexing and intriguing events in the U.S. and on the world stage. He observes the society at a vantage point beyond his ethnical, national, and cultural background. I never have to worry about oﬀending him by expressing my honest views. I thank Andy for rallying a group of people (mostly astronomers) to have discussions on real-world events in a similar fashion as we discuss science. We attempt to distinguish between real and bogus information from various sources and analyze the events by confronting ideas against distilled evidence and data. Although I do not always agree with Andy, it has always been such a great time talking to him on almost anything.

Next I thank my thesis committee members, Scott Gaudi and Kris Stanek.

vi I ﬁrst met Scott back when he was a postdoc at CfA. He has been providing unreserved encouragement and support to me since then. I have beneﬁted greatly from his advice on science and professional development. I thank him for having led me to see the bright side when I was in depression or frustration. I admire his comprehensive knowledge in almost all branches of astronomy and his often amazingly deep physical intuition.

I thank Kris for being friendly and humorous and willing to talk to me on any topic. I thank him for inspiring discussions that led to my first original workable science idea. I appreciate his insistence that scientific pictures should fundamentally be simple. He always has great insights into complexity and sees through irrelevant artificial obscuration. I hope I will be able to acquire even a fraction of his ability in seizing the moments by recognizing fleeting opportunities in astronomy.

I am grateful to Rick Pogge, Chris Kochanek and many others in the department for providing a lot of great technical help and having interesting science discussions together. And I greatly appreciate many faculty in the department for creating an intellectually stimulating environment which is also very student-friendly. Daily coﬀee has become an indispensable part of my life. The department staﬀ are always helpful, and I especially thank David Will for solving many of my computer problems.

vii I have spent a great time with many graduate students and several postdocs at

OSU. I first thank Zheng for having provided me many great advice when I first got to OSU, and for him being a role-model student and astronomer. He shows me how high someone with a similar background as me can possibly achieve in astronomy. I thank Jose, Vimal, Dale, Himel and others for having a lot of interesting discussions and experience together. And I appreciate having many great office mates, Frank,

Molly, Ondrej, Kelly, Rob, Deokkeun, etc., during the last ﬁve years. Szymon and

Xinyu have been great friends and colleagues.

I could not possibly have ﬁnished my PhD work without many colleagues outside Ohio State. In particular, I thank Andrzej Udalski for being an exemplary astronomer. I never fail to be thrilled by the ﬁrst-rate data he and the OGLE team have produced. A lot of crucial data I have analyzed come from many “amateur” astronomers: Jennie McCormick, Grant Christie and Berto Monard, to name a few.

They are amateurs only in the sense that they are not paid for their observations, otherwise what they do are extraordinarily professional. Their fascination in the sky, enthusiasm and dedication have been a constant source of inspiration to me. I thank all my observer colleagues for oﬀering me the privilege in being the ﬁrst to see many beautiful light curves and realize the exotic extrasolar worlds that they reveal. The precious moments of discovery transcend anything else.

viii I wish to express my deep gratitude to Profs. Dawei Yang, Tianyi Huang,

Qiusheng Gu and Dr. Jin Zhu for their selﬂess help and great guidance at various important stages before I entered the graduate school.

At last, I thank my mother and father, who put my education, both in acquiring knowledge and in being a decent member of the society, as the ﬁrst priority in their life. Without their unequivocal support and sacriﬁce, I would have never been anywhere near the position I am at today to write this dissertation.

ix VITA

March 18, 1982 ...... Born – Chengde, Hebei, China

2004 ...... B.S. Astronomy, Nanjing University, China

2004 – 2005 ...... University Fellow, The Ohio State University

2005 – 2006 ...... Graduate Research Assistant,

The Ohio State University

2006 ...... M.S. Astronomy, The Ohio State University

2006 – 2008 ...... Graduate Research and Teaching Assistant,

The Ohio State University

2008 – 2009 ...... Presidential Fellow, The Ohio State University

PUBLICATIONS

Research Publications

1. A. Udalski, et al. “A Jovian-Mass Planet in Microlensing Event OGLE- 2005-BLG-071”, ApJL, 628, 109L, (2005).

2. Subo Dong, et al. “Planetary Detection Eﬃciency of the Magniﬁcation 3000 Microlensing Event OGLE-2004-BLG-343”, ApJ, 642, 842, (2006).

3. A. Gould, A. Udalski, D. An, D. P. Bennett, A.-Y. Zhou, S. Dong, et al. “Microlens OGLE-2005-BLG-169 Implies That Cool Neptune-like Planets Are Common”, ApJL, 644, L37, (2006).

x 4. Subo Dong, “Probing 100 AU Intergalactic Mg II Absorbing ‘Cloudlets’ ∼ with Quasar Microlensing”, ApJ, 660, 206, (2007).

5. Subo Dong, et al. “First Space-Based Microlens Parallax Measurement: Spitzer Observations of OGLE-2005-SMC-001”, ApJ, 664, 862, (2007).

6. B.S. Gaudi, D. P. Bennett, A. Udalski, A. Gould, G. W. Christie, D. Maoz, S. Dong, et al. “Discovery of a Jupiter/Saturn Analog with Gravitational Microlensing”, Science, 319, 927, (2008).

7. B. Scott Gaudi, Joseph Patterson, David S. Spiegel, Thomas Krajci, R. Koﬀ, G. Pojmanski, Subo Dong, et al. “Discovery of a Very Bright, Nearby Gravitational Microlensing Event”, ApJ, 677, 1268, (2008).

8. Subo Dong, et al. “OGLE-2005-BLG-071Lb, the Most Massive M Dwarf Planetary Companion?”, ApJ, 695, 970, (2009).

9. Subo Dong, et al. “Microlensing Event MOA-2007-BLG-400: Exhuming the Buried Signature of a Cool, Jovian-Mass Planet”, ApJ, 698, 1826, (2009).

10. A.Gould, A.Udalski, B.Monard, K.Horne, Subo Dong, et al. “The Extreme Microlensing Event OGLE-2007-BLG-224: Terrestrial Parallax Observation of a Thick-Disk Brown Dwarf”, ApJL, 698, L147, (2009).

FIELDS OF STUDY

Major Field: Astronomy

xi Table of Contents

Abstract...... ii

Dedication...... iv

Acknowledgments...... v

Vita ...... x

ListofTables ...... xvii

ListofFigures...... xix

Chapter 1 Introduction ...... 1

Chapter 2 Planetary Detection Eﬃciency of the Magniﬁcation 3000 Microlensing Event OGLE-2004-BLG-343 ...... 12

2.1 Introduction ...... 12

2.1.1 High-Magniﬁcation Events & Earth-Mass Planets ...... 13

2.1.2 Planet Detection Eﬃciencies: Philosophy and Methods . . . . 17

2.1.3 ”Seeing” the Lens in High-Magniﬁcation Events ...... 22

2.2 ObservationalData ...... 25

2.3 EventModeling ...... 27

2.3.1 Point-Source Point-Lens Model ...... 28

2.3.2 Source Properties from Color-Magnitude Diagram ...... 29

2.3.3 Finite-SourceEﬀects ...... 35

2.3.4 Monte-Carlo Simulation ...... 36

xii 2.4 DetectingPlanets...... 41

2.4.1 Detection Eﬃciency ...... 41

2.4.2 ConstraintsonPlanets ...... 46

2.4.3 NoPlanetDetected ...... 47

2.4.4 FakeData ...... 48

2.4.5 Detection Eﬃciency in Physical Parameter Space ...... 49

2.5 LuminousLens? ...... 50

2.6 SummaryandDiscussion ...... 55

Chapter 3 Microlensing Event MOA-2007-BLG-400: Exhuming the Buried Signature of a Cool, Jovian-Mass Planet ...... 72

3.1 Introduction ...... 72

3.2 Observations ...... 80

3.3 MicrolensModel ...... 83

3.3.1 Hybrid Pixel/Ray Map Algorithm ...... 86

3.3.2 The (w,q)GridofLensGeometries ...... 88

3.3.3 Best-FitModel ...... 88

3.4 Finite-SourceEﬀects ...... 91

3.5 LimbDarkening ...... 93

3.6 BlendedLight ...... 94

3.7 Discussion ...... 96

Chapter 4 OGLE-2005-BLG-071Lb, the Most Massive M-Dwarf Planetary Companion? ...... 111

4.1 Introduction ...... 111

4.2 Overview of Data and Types of Constraints ...... 115

xiii 4.3 Constraining the Physical Properties of the Lens and its Planetary Companion ...... 118

4.3.1 Microlens Parallax Eﬀects ...... 119

4.3.2 FittingPlanetaryOrbitalMotion ...... 121

4.3.3 Finite-source eﬀects and Other Constraints on θE ...... 122

4.3.4 HST Astrometry ...... 129

4.3.5 “Seeing” the Blend with HST ...... 133

4.3.6 Final Physical Constraints on the Lens and Planet ...... 139

4.3.7 Constraints on a Non-Luminous Lens ...... 146

4.3.8 Xallarap Eﬀects and Binary Source ...... 151

4.4 SummaryandFutureProspects ...... 154

4.5 Discussion ...... 158

Chapter 5 First Space-Based Microlens Parallax Measurement: Spitzer Observations of OGLE-2005-SMC-001 ...... 181

5.1 Introduction ...... 181

5.2 Observations ...... 186

5.2.1 ErrorRescaling ...... 189

5.2.2 Spitzer Data Reduction and Error Determination ...... 190

5.3 Complications Alter Strategy and Analysis ...... 192

5.3.1 Need for Additional Spitzer Observation ...... 193

5.3.2 Eight-foldWay ...... 196

5.4 BinaryOrbitalMotion ...... 197

5.4.1 Static Binary Lens Parameters ...... 199

5.4.2 Binary Orbital Parameters ...... 201

5.4.3 SummaryofParameters ...... 203

xiv 5.5 SearchforSolutions ...... 204

5.5.1 Convergence ...... 207

5.6 SolutionTriage ...... 207

5.6.1 Wide-Binary Solutions ...... 210

5.6.2 Close-Binary Solutions ...... 211

5.7 LensLocation ...... 211

5.7.1 HaloLenses ...... 213

5.7.2 SMCLenses ...... 215

5.7.3 SMCStructure ...... 216

5.7.4 Likelihood Ratios ...... 220

5.7.5 Kepler Constraints for Close Binaries ...... 221

5.7.6 Kepler Constraints for Wide Binaries ...... 223

5.7.7 Constraints From (Lack of) Finite-Source Eﬀects ...... 225

5.8 Discussion ...... 225

Chapter 6 Probing 100AU Intergalactic Mg IIAbsorbing “Cloudlets” with Quasar Microlensing∼ ...... 245

6.1 Introduction ...... 245

6.2 Varying Microlensed Quasar Image as A “Ruler” ...... 248

6.2.1 Basic Geometric Conﬁgurations and Motions ...... 249

6.2.2 Bulk Motion of the Un-microlensed “Ray Bundles” ...... 251

6.3 Application to Q2237+0305 ...... 252

6.4 Discussion and Conclusion ...... 260

Appendix A Two New Finite-Source Algorithms ...... 264

A.1 Map-Making ...... 264

xv A.2 Loop-Linking ...... 269

A.3 AlgorithmParameters ...... 272

Appendix B MOA-2003-BLG-32/OGLE-2003-BLG-219 ...... 275

Appendix C Extracting Orbital Parameters for Circular Planetary Orbit ...... 278

Appendix D Failure of Elliptical-Source Models for MOA-2007-BLG- 400 ...... 281

Bibliography ...... 283

xvi List of Tables

2.1 OGLE-2004-BLG-343 Best-Fit PSPL Model Parameters ...... 71

3.1 MOA-2007-BLG-400 Best-ﬁt Planetary Models ...... 110

4.1 OGLE-2005-BLG-071 Light Curve Parameter Estimations From Markov chain Monte Carlo Simulations A (Part 1). 174

4.2 OGLE-2005-BLG-071 Light Curve Parameter Estimations From Markov chain Monte Carlo Simulations A (Part 2). 175

4.3 OGLE-2005-BLG-071 Light Curve Parameter Estimations From Markov chain Monte Carlo Simulations B (Part 1). 176

4.4 OGLE-2005-BLG-071 Light Curve Parameter Estimations From Markov chain Monte Carlo Simulations B (Part 2). 177

4.5 OGLE-2005-BLG-071 Derived Physical Parameters . . . . . 178

4.6 Jovian-mass Companions to M Dwarfs (M∗ < 0.55 M⊙) . . . . 179

4.6 Jovian-mass Companions to M Dwarfs (M∗ < 0.55 M⊙) . . . . 180

5.1 OGLE-2005-SMC-001 Light Curve Models: Free Blending...... 229

5.1 OGLE-2005-SMC-001 Light Curve Models: Free Blending...... 230

5.1 OGLE-2005-SMC-001 Light Curve Models: Free Blending...... 231

5.1 OGLE-2005-SMC-001 Light Curve Models: Free Blending...... 232

5.1 OGLE-2005-SMC-001 Light Curve Models: Free Blending...... 233

5.2 OGLE-2005-SMC-001 Light Curve Models: Zero Blending...... 234

5.2 OGLE-2005-SMC-001 Light Curve Models: Zero Blending...... 235

5.2 OGLE-2005-SMC-001 Light Curve Models: Zero Blending...... 236

xvii 5.2 OGLE-2005-SMC-001 Light Curve Models: Zero Blending...... 237

5.2 OGLE-2005-SMC-001 Light Curve Models: Zero Blending...... 238

xviii List of Figures

2.1 Light curve of OGLE-2004-BLG-343 near its peak on 2004 June 19 (HJD 2,453,175.7467) ...... 59

2.2 CMDoftheOGLE-2004-BLG-343ﬁeld...... 60

2.3 Color-color diagram in a ﬁeld centered on OGLE-2004-BLG-343. . . . 61

2.4 Likelihood contours for ﬁnite-source points-lens models relative to the best-ﬁtPSPLmodelforOGLE-2004-BLG-343...... 62

2.5 Probability distributions of various parametrs for Monte Carlo events toward the line of sight of OGLE-2004-BLG-343...... 63

2.6 OGLE-2004-BLG-343 planetary detection eﬃciency for three mass ratios (q) as a function of the planet-star separation bx and by..... 64

2.7 Planet detection eﬃciency of OGLE-2004-BLG-343 as a function of b and q...... 65

2.8 OGLE-2004-BLG-343 planetary detection eﬃciency for four mass ratios with simulated peak data points as a function of bx and by. . . 66

2.9 OGLE-2004-BLG-343 planetary detection eﬃciency as a function of b and q with simulated peak data points...... 67

2.10 OGLE-2004-BLG-343 planetary detection eﬃciency as a function of the physical projected star-planet distance (r⊥) and the planetary mass (mp). 68

2.11 Planetary detection eﬃciency as a function of r⊥ and mp for OGLE- 2004-BLG-343 augmented by simulated data points covering the peak. 69

2.12 Planetary detection eﬃciency as a function of r⊥ andd mp for OGLE- 2004-BLG-343, by assuming that the blended light is due to the lens. 70

3.1 Lightcurve of MOA-2007-BLG-400 ...... 103

xix 3.2 Comparisons between two photometric reductions of µFUN H-band dataforMOA-2007-BLG-400...... 104

3.3 Magnification differences between of best-fit planetary model and single-lens models for MOA-2007-BLG-400 ...... 105

3.4 ∆χ2 Contours as a function of planet-star mass ratio q and projected planet-star separation d, as well as “short caustic diameter” w. . . . . 106

3.5 Instrumental color-magnitude diagram of ﬁeld containing MOA-2007- BLG-400...... 107

3.6 Bayesian relative probability densities for the physical properties of the planet MOA-2007-BLG-400Lb and its host star...... 108

3.7 Constraining the blended ﬂux from CTIO I-band images for MOA- 2007-BLG-400...... 109

4.1 Light curve of planetary microlensing event OGLE-2005-BLG-071 . . 162

4.2 Probability contours of OGLE-2005-BLG-071 microlens parallax parameters derived from MCMC simulations...... 163

4.3 CMDfortheOGLE-2005-BLG-071ﬁeld ...... 164

4.4 HST ACS astrometric measurements of OGLE-2005-BLG-071 . . . . 165

4.5 Posterior probability contours for OGLE-2005-BLG-071 relative lens- sourcepropermotion...... 166

4.6 Diﬀerences between OGLE V and HST F555W magnitudes for the matched stars in the ﬁeld of OGLE-2005-BLG-071...... 167

4.7 Probability contour of OGLE-2005-BLG-071 lens mass M and relative lens-source parallax πrel without assuming the HST blend is the lens. 168

4.8 Probability contour of OGLE-2005-BLG-071 lens mass M and relative lens-source parallax πrel byassumingtheblendisthelens...... 169

4.9 Probability contours of OGLE-2005-BLG-071Lb projected velocity r⊥γ in the units of vc,⊥ ...... 170

4.10 Probability distributions of OGLE-2005-BLG-071Lb planetary parameters from MCMC realizations assuming circular orbital motion. 171

xx 4.11 χ2 distributions for OGLE-2005-BLG-071 best-ﬁt xallarap solutions at ﬁxed binary-source orbital periods P ...... 172

4.12 OGLE-2005-BLG-071 xallarap ﬁts results by ﬁxing binary orbital phase λ and complement of inclination β at 1 yr period and u0 > 0. . 173

5.1 Standard (Paczy´nski 1986) microlensing ﬁt to the light curve of OGLE- 2005-SMC-001...... 239

5.2 Why 4 (not 3) Spitzer observations are needed to measure πE = (πE,k,πE,⊥)forOGLE-2005-SMC-001...... 240

5.3 Parallax πE = (πE,N ,πE,E) 1 σ error ellipses for all discrete solutions forOGLE-2005-SMC-001...... 241

5.4 Best-ﬁt binary microlensing model for OGLE-2005-SMC-001...... 242

5.5 Likelihood contours of the inverse projected velocity Λ v˜/v˜2 for SMC lenses together with Λ values for OGLE-2005-SMC-001.≡ . . . . 243

5.6 Likelihood contours of the inverse projected velocity Λ v˜/v˜2 for halo lenses together with Λ values for OGLE-2005-SMC-001.≡ ...... 244

6.1 Quasar microlensing caustics network, light curve and images. . . . . 262

6.2 Physical positions of “ray bundles” from a microlensed quasar in the frame of cloud at redshifts 0.1, 0.57, 0.83 and 1.69...... 263

B.1 Planetary exclusion regions for microlensing event MOA-2003- 32/OGLE-2003-BLG-219...... 277

xxi Chapter 1

Introduction

The spatial and temporal scales tangible to mankind are insigniﬁcant compared to those of the observable universe. Nevertheless, we aspire to apprehend all wonders of the nature and understand her secrets, ranging from the sub-atomic fabric of matter to the origin and evolution of the entire universe. To break human beings’ in-born limits, scientists utilize new tools designed with the knowledge we have learned from nature to probe nature. Modern astronomy was born four centuries ago when Galileo built his telescope according to newly discovered laws of optics and pointed it to the sky. Moreover, earthbound astronomers also attempt to use special celestial circumstances bestowed by the universe to study what is often unreachable from available technologies. Astronomers have long mastered the parallax technique by using the Earth’s motion to measure distances to faraway stars. Until recently the cosmic alignment among the Earth, Moon and Sun had been the only chance for us to glimpse the part of solar atmosphere normally overwhelmed by the glaring photosphere.

1 Gravitational microlensing is an elegant method that follows the great astronomy tradition by making use of cutting-edge technologies as well as capturing rare celestial opportunities. In the last two decades, the technique has opened up new windows to many aspects of the universe which are often times inaccessible via other means.

This dissertation is composed of several astrophysical applications of gravitational microlensing. Chapters 2, 3 and 4 are on discovering and characterizing extrasolar planets with microlensing toward the Galactic bulge. Chapter 5 presents the ﬁrst space-based microlens parallax measurements. The parallax determinations constrain the distance of a MACHO (MAssive Compact Halo Object) candidate in the direction of the Small Magellanic Cloud. In Chapter 6, I propose a method to probe small-scale structures of the intergalactic medium with quasar microlensing.

Microlensing as a ﬁeld has been rapidly developing during the course of completion of this dissertation. An overview of microlensing is given in the following along with brief discussions on how the dissertation topics ﬁt into ongoing microlensing research and broader context of astrophysics.

Gravitational microlensing occurs when a background “source” (a star for microlensing in the Local Group or a quasar for cosmological microlensing) and a foreground object (“lens”) get very closely aligned (within the “Einstein radius” rE) in the sight line of the observer. The light beams from the source are bent during

2 the passage near the lens according to the General Relativity, and the ﬂux from the source seen by the observer gets magniﬁed. Typically the angular size of the Einstein radius θE for a Galactic microlensing event is on the order of a milli-arcsecond, which makes the “optical depth”, the probability of microlensing events, extremely small in our Galaxy. Therefore, although the theoretical possibility of Galactic microlensing has been discussed by numerous people since early last century (e.g., Einstein 1936;

Liebes 1964), it was not until the visionary paper by Paczy´nski (1986) that the observational plausibility of stellar microlensing was brought into the sight of most astronomers.

Overwhelming astronomical evidences suggest that most of the matter in the universe is in the form of “dark matter”, whose existence has so far only exhibited from its gravitational effects. Paczyński (1986) demonstrated that, if most of the dark matter in the Galactic halo were made of dim massive compact objects, the optical depth of microlensing would be significantly boosted to 10−6. Paczyński ∼ (1986) proposed that one could search for MACHOs by identifying microlensing variability, which follows a distinct analytical form, from simultaneously monitoring brightness changes of millions of stars on a daily basis. It has also been shown that the optical depth solely due to the contributions from the known stellar populations toward the Galactic bulge should be of the same order (Kiraga & Paczynski 1994).

The massive data collection and processing required to search for microlensing events was unprecedented in astronomy. Fortunately, the silicon revolution brought

3 the relevant technology to astronomers in the last decade of 20th century. In particular, the advent of large format CCD, powerful computer hardware as well as sophisticated data analysis software that could obtain fast and precise relative photometry in wide and crowded stellar ﬁelds made microlensing surveys feasible.

In 1993, three collaborations announced their independent discoveries of the ﬁrst microlensing events (Udalski et al. 1993; Alcock et al. 1993; Aubourg et al. 1993).

Nowadays, OGLE-III and MOA-II surveys discover a total of about one thousand microlensing events toward the Galactic bulge every year.

Mostly driven by observations, rich astrophysical applications of microlensing have been revealed theoretically. An important and exciting observational prospect is to search for extrasolar planets with microlensing, which was recognized by Mao

& Paczynski (1991) (see also earlier work by Liebes 1964). At that time, the Sun was the only known star that harbored planets. Mao & Paczynski (1991) found that lenses orbited by planetary companions near rE could give rise to microlensing light curves easily distinguishable from those by single stellar lenses. In fact, the

ﬁrst microlensing planet discovered by OGLE and MOA more than a decade later

(Bond et al. 2004) was via an event perturbed by a resonant caustic, which bore close similarity to what was illustrated in Fig. 1 of Mao & Paczynski (1991).

Gould & Loeb (1992) did a comprehensive study on generic planetary perturbations in microlensing. They gave the characteristics of the most common type of planetary signature, whose duration is proportional to the square root of

4 planet-to-star mass ratio. The signal of a Jupiter lasts for a couple of days, so daily observing cadence by a microlensing survey group is generally not suﬃcient to capture a planet. They proposed to form follow-up groups who would do round-of-clock, intensive monitoring on a selective number of events found by the survey group to speciﬁcally search for planets.

Gould & Loeb (1992) found that microlensing was most sensitive to planets in the so-called “lensing zone” ([0.6 1.6]r ), which corresponds to a few Astronimical − E Unit (AU) for a typical Galactic microlensing event. Intriguingly, this is coincident with the orbital radius of the Jupiter, the dominant planet of our Solar System.

Therefore, solar system analogs should be the easiest targets to ﬁnd via microlensing.

Furthermore, Bennett & Rhie (1996) found that, an Earth-mass planet could produce discernible hour-long signals in a Gould & Loeb type event. By contrast, the most productive planet ﬁnding technique, Radial Velocity (RV), which was still then in its cradle, needs about 10 years of observations to ﬁnd a Jupiter. And the RV precision that could enable Earth-mass planet detections was beyond a spectroscopist’s wildest dreams. Even today, microlensing remains the most powerful technique to search for

Solar System analogs and Earth-mass planets from the ground.

Inspired by these theoretical considerations, the ﬁrst microlensing planet follow-up groups were formed according to the Gould & Loeb (1992) strategy in the mid 1990s. In their early years, these groups made many great discoveries in areas such as stellar-binary lensing (Albrow et al. 2000a), stellar atmosphere (Albrow et al.

5 1999b), etc, thanks to the high-cadence, high-precision monitoring. However, several early claims on discoveries of microlensing planets (e.g., Bennett et al. 1999; Sahu et al. 2001; Rhie et al. 2000) fail to gain acceptance due to inadequate modeling

(Albrow et al. 2000a), erroneous data analysis (Sahu et al. 2002), contradictory observational data, or simply insuﬃcient statistical signiﬁcance.

Gaudi et al. (2002) analyzed ﬁve-years (1995-2000) of well-covered microlensing events by the PLANET collaboration. They found no convincing detections of planets, and they concluded that Jupiter-mass planets at the lensing zone around stellar lenses are not very common (less than 33%). The ﬁrst Gould & Loeb ∼ type planet discovery, OGLE-2005-BLG-390, was led by PLANET and OGLE in

+5.5 2005 (Beaulieu et al. 2006), which was a Neptune-mass planet (5.5−2.7M ). Since ⊕ microlensing is much more sensitive to Jupiters than Neptunes, this appears to indicate Neptune-mass planets are much more common at a few AU than Jupiters around M dwarfs – the most common stellar lenses. This conclusion was further supported by additional microlensing planet discoveries (Gould et al. 2006).

The seminal work by Griest & Safizadeh (1998) proposed a different approach microlensing planet searches from that of Gould & Loeb (1992). Their idea was to find planets in high-magnification events (HMEs) via central caustics – then an overlooked type of planetary perturbation. Central caustics have smaller cross sections than those of planetary caustics, which are responsible for Gould & Loeb events. Thus an uniform sample of planetary microlensing events will mostly consist

6 of Gould & Loeb type events. However, Griest & Saﬁzadeh (1998) pointed out that central caustics always perturb the peak of events that reach high magniﬁcation

(Amax > 100), and an individual HME has nearly 100% sensitivity to any Jupiters ∼ ∼ within the lensing zone. An additional practical advantage of this channel is that the time that a HME reaches maximum magniﬁcation can be predicted in advance so that one can focus limited resource to observe over the peak only, whereas one must monitor the entire microlensing event to eﬀectively track down Gould & Loeb-type perturbations.

The signiﬁcance of Griest & Saﬁzadeh (1998) was immediately grasped by the microlensing community. Several follow-up studies on HMEs (e.g., Gaudi et al. 1998;

Rattenbury et al. 2002; Bond et al. 2002; Abe et al. 2004) have further enriched the scope of this planetary detection channel and further demonstrated its eﬀectiveness.

For example, it is found that HMEs are especially sensitive to multiple-planet systems (Gaudi et al. 1998). However, the potential of HMEs was not fully developed immediately after Griest & Saﬁzadeh (1998). One major practical diﬃculty was that, HMEs are very rare, and the capacity of the survey groups only permitted detections of a handful of them per year. Furthermore, the perturbations by central caustics is more complex and much less intuitive than those by planetary caustics.

Central-caustic perturbations were also much less understood than the planetary caustics ones, so many practitioners of microlensing were concerned with uniqueness of models from planetary HMEs. Finally, modeling planetary high-magniﬁcation

7 events was non-trivial as the two-point-mass-lens calculations with finite-source effects at high magnification were very time-consuming. The less intuitive light-curve features and complicated parameter-space structure only exacerbated the modeling difficulties.

The work presented in Chapter 2 was started in 2004, and it was on analyzing

OGLE-2004-BLG-343 – the single-lens microlensing event with highest peak magniﬁcation ( 3000) that had been ever analyzed. Due to observer error, most of ∼ the peak region was not well covered, which signiﬁcantly reduced its sensitivity to planets. However, it is shown that had the event been well sampled, it would have been sensitive to almost all Neptune-mass planets over a factor of 5 of projected separation and would even have had some sensitivity to Earth-mass planets. This event highlighted the importance of aggressively covering the peak of HMEs and showed the great potential of these events achievable with the ongoing projects.

To derive the planetary detection efficiency, I performed the most intensive yet binary-lens finite-source calculations at high magnification. As a “by-product”, I developed new algorithms (see APPENDIX A) that improved the speed of such calculations by a few orders of magnitudes for high-magnification events compared to previous codes. The algorithms and improved version (see Chapter 3) have played crucial roles in analyzing the majority of planets discovered from HMEs.

The OSU-based Microlensing Follow-Up Network (µFUN) has started to primarily focus on HMEs since 2004-2005 season. We have gradually enlarged a

8 global network of 1m-class and smaller ( 30 50 cm) telescopes that oﬀers 24-hour ∼ − coverage with some geographical redundancy in longitude. Such redundancy helps to alleviate weather problems and sometimes provides additional constraints on microlensing parameters (e.g. terrestrial parallax measurements by Gould et al.

2009). Due to their high peak magnifications, one can more easily obtain precise photometry over the peak from telescopes of all sizes. And thanks to the increasing number of HMEs due to upgraded survey telescopes of OGLE and MOA, we are currently able to intensively follow up a dozen or so HMEs per year. This strategy has proven quite fruitful. µFUN has led the discoveries of 10 microlensing ∼ planets in the last 4 years, including the first multiple planetary system through microlensing, which turned out to be the first extrasolar Jupiter-Saturn analog

(Gaudi et al. 2008).

In Chapter 3, a planetary microlensing event discovered by µFUN is analyzed.

The planet, MOA-2007-BLG-400Lb, was detected in a magnification Amax = 628 event, which exhibited severe finite-source effects. This was the first planetary high-magnification event in which the central caustics was smaller than the size of the source. A new parametrization based on central-caustic size was developed that enabled efficient and thorough parameter space search. It was demonstrated that the planetary nature of the deviation can be unambiguously ascertained from the gross features of the residuals. Detailed analysis yields a fairly precise planet/star

9 mass ratio of q = (2.5+0.5) 10−3, in accord with the large signiﬁcance (∆χ2 = 1070) −0.3 × of the detection.

Chapter 4 is on characterizing the physical properties of Jupiter-mass planet

OGLE-2005-BLG-071Lb and its host. One can directly infer planet/star mass ratio q and the planet-star projected separation d (in units of the angular Einstein radius) from a planetary microlensing light curve. By combining the parallax

(Gould 1992) and angular Einstein radius measurements derived from finite-source effects, one can directly determine the lens mass and distance and subsequently the planet mass and star-planet physical separation. In this chapter, we obtain such constraints from microlens parallax, finite-source effects and also planetary orbital motion from the light curve. In addition, we obtained photometric and astrometric measurements from Hubble Space Telescope. Our primary analysis leads to the conclusion that the host of Jovian planet OGLE-2005-BLG-071Lb is an M dwarf in the foreground disk with mass M = 0.46 0.04 M , distance D = 3.2 0.4 kpc, and ± ⊙ l ± thick-disk kinematics v 103 km s−1. From the best-fit model, the planet has LSR ∼ mass M = 3.8 0.4 M . These results from the primary analysis suggest that p ± Jupiter OGLE-2005-BLG-071Lb was likely to be the most massive planet yet discovered that is hosted by an M dwarf.

Chapter 5 presents the ﬁrst space-based microlens parallax measurements of OGLE-2005-SMC-001. The parallax measurement yields a projected velocity v˜ 230 km s−1, the typical value expected for halo lenses, but an order of magnitude ∼ 10 smaller than would be expected for lenses lying in the Small Magellanic Cloud (SMC) itself. The lens is a weak (i.e., non-caustic-crossing) binary, which complicates the analysis considerably but ultimately contributes additional constraints. It is found that the likelihood ratio is / = 20. Hence, halo lenses are strongly favored Lhalo LSMC but SMC lenses are not deﬁnitively ruled out.

Intergalactic Mg II absorbers are known to have structures down to scales

102.5pc, and there are now indications that they may be fragmented on scales ∼ < 10−2.5pc (Hao et al. 2006). Chapter 6 presents a method to probe Mg II ∼ and other absorption “cloudlets” with sizes 10−4.0 10−2.0pc using quasar ∼ − microlensing. When a lensed quasar is microlensed, the micro-images of the quasar experience creation, destruction, distortion, and drastic astrometric changes during caustic-crossings. It is shown that these small-scale structures in the intergalactic medium could induce signiﬁcant spectral variability on timescales of months to years. With numerical simulations, I demonstrate the feasibility of applying this method to Q2237+0305, and I show that high-resolution spectra of this quasar in the near future would provide a clear test of the existence of such metal-line absorbing

“cloudlets”.

11 Chapter 2

Planetary Detection Efficiency of the Magnification 3000 Microlensing Event OGLE-2004-BLG-343

2.1. Introduction

Current microlensing planet searches focus significant effort on high- magnification events, which have great promise for detecting low-mass extrasolar planets. It is therefore crucial to understand the potential for discovering planets and to optimize the early identifications and observational strategy of such events.

In previous planetary detection eﬃciency analyses of high-magniﬁcation events,

ﬁnite-source eﬀects have often been ignored mainly due to computational limitations.

However, such effects are intrinsically important in these events because the sources are more likely to be resolved at very low impact parameters. In this study, we improve the method of Yoo et al. (2004b) by incorporating finite-source effects to characterize the planetary detection efficiency of the extremely high-magnification event OGLE-2004-BLG-343, and we develop new efficient algorithms to make the calculations possible. Moreover, we attempt to find useful observational signatures

12 of high-magniﬁcation events so as to help alleviate the diﬃculties in their early recognition.

2.1.1. High-Magnification Events & Earth-Mass Planets

Apart from pulsar timing (Wolszczan & Frail 1992), microlensing is at present one of the few planet-ﬁnding techniques that is sensitive to Earth-mass planets.

A planetary companion of an otherwise isolated lens star introduces two kinds of caustics into the magniﬁcation pattern: “planetary caustics” associated with the planet itself and a “central caustic” associated with the primary lens. 1 When the source passes over or close to one of these caustics, the light curve deviates from its standard Paczy´nski (1986) form, thus revealing the presence of the planet (Mao &

Paczy´nski 1991; Liebes 1964).

Since planetary caustics are generally far larger than central caustics, a

“fair sample” of planetary microlensing events would be completely dominated by planetary-caustic events. Nevertheless, central caustics play a crucial role in current microlensing planet searches, particularly for Earth-mass planets (Griest &

Saﬁzadeh 1998), for the simple reason that it is possible to predict in advance that the source of a given event will arrive close to the center of the magniﬁcation pattern where it will probe for the presence of these caustics. Hence, one can organize the

1When the planetary companion is close to the Einstein ring, the planetary and central caustics merge into a single “resonant caustic”.

13 intensive observations required to characterize the resulting anomalies. By contrast, the perturbations due to planetary caustics occur without any warning. The lower the mass of the planet, the shorter the duration of the anomaly, and so the more crucial is the warning to intensify the observations. This is the primary reason that planet-searching groups give high priority to high-magniﬁcation events, i.e., those that probe the central caustics 2. As a bonus, high-magniﬁcation events are also more sensitive to planetary-caustic perturbations than are typical events (Gould

& Loeb 1992) since their larger images increase the chances that they will be perturbed by planets. However, this enhancement is relatively modest compared to the rich potential of central-caustic crossings.

In principle, it is also possible to search for Earth-mass planets from perturbations due to their larger (and so more common) planetary and resonant caustics, but this would require a very diﬀerent strategy from those currently being carried out. The problem is that these perturbations occur without warning during otherwise normal microlensing events, and typically last only 1 or 2 hr. Hence, one

2 Note that although high-magnification events are guaranteed to have low impact parameters, peak magnification for events with low impact parameters are not necessarily high if they have relatively large source sizes. And large sources will tend to smear out the perturbations induced by the central caustics, thereby decreasing the planetary sensitivity (Griest & Safizadeh 1998; Chung et al. 2005). So when central caustics are important in producing planetary signals, the maximum magnification of microlensing events serves as a better indicator of planetary detection efficiency than the impact parameter.

14 would have to intensively monitor the entire duration of many events. The only way to do this practically is to intensively monitor an entire ﬁeld containing many ongoing microlensing events roughly once every 10 minutes in order to detect and properly characterize the planetary deviations. Proposals to make such searches have been advanced for both space-based (Bennett & Rhie 2002) and ground-based

(Sackett 1997) platforms.

At present, two other microlensing planet-search strategies are being pursued.

Both strategies make use of wide-area (> 10 deg2) searches for microlensing events ∼ toward the Galactic bulge. Observations are made once or a few times per night by the OGLE-III3 (Udalski 2003) and MOA4 (Bond et al. 2001) surveys. When events are identified, they are posted as “alerts” on their respective Web sites. In the first approach, these groups check each ongoing event after each observation for signs of anomalous behavior, and if their instantaneous analysis indicates that it is worth doing so, they switch from survey mode to follow-up mode. This approach led to the first reliable detection of a planetary microlensing event, OGLE

2003-BLG-235/MOA-2003-BLG-53 (Bond et al. 2004).

In the second approach, follow-up groups such as the Probing Lensing Anomalies

NETwork (PLANET; Albrow et al. 1998) and the Microlensing Follow-Up Network

(µFUN; Yoo et al. 2004b) monitor a subset of alerted events many times per day

3 OGLE Early Warning System: http://www.astrouw.edu.pl/˜ogle/ogle3/ews/ews.html 4MOA Transient Alert Page: http://www.massey.ac.nz/˜iabond/alert/alert.html

15 and from locations around the globe. Generally these groups focus to the extent possible on high-magniﬁcation events for the reasons stated above. The survey groups can also switch from “survey mode” to “follow-up” mode to probe newly emerging high-magniﬁcation events.

Over the past decade several high-magnification events have been analyzed for planets. Gaudi et al. (2002) and Albrow et al. (2001) placed upper limits on the frequency of planets from the analysis of 43 microlensing events, three of which reached magnifications A 100, including OGLE-1998-BUL-15 max ≥ (A = 170 30), MACHO-1998-BLG-35 (A = 100 5), and OGLE-1999-BUL- max ± max ± 35 (A = 125 15). However, the first of these was not monitored over its peak. max ± MACHO-1998-BLG-35 was also analyzed by Rhie et al. (2000) and Bond et al.

(2002), who incorporated all available data and found modest (∆χ2 = 63) evidence for one, or perhaps two, Earth-mass planets.

Yoo et al. (2004b) analyzed OGLE-2003-BLG-423 (A = 256 43), which max ± at the time was the highest magniﬁcation event yet recorded. However, because this event was covered only intermittently over the peak, it proved less sensitive to planets than either MACHO-1998-BLG-35 or OGLE-1999-BUL-35.

Abe et al. (2004) analyzed MOA-2003-BLG-32/OGLE-2003-BLG-219, which at

A = 525 75, is the current record-holder for maximum magniﬁcation. Unlike max ± OGLE-2003-BLG-423, this event was monitored intensively over the peak: the Wise

16 Observatory in Israel was able to cover the entire 2.5 hr FWHM during the very brief interval that the bulge is visible from this northern site. The result is that this event has the best sensitivity to low-mass planets to date.

Recently, Udalski et al. (2005) detected a 3-Jupiter mass planet by intensively ∼ monitoring the peak of the high-magniﬁcation event OGLE-2005-BLG-071. This was the second robust detection of a planet by microlensing and the ﬁrst from perturbations due to a central caustic.

2.1.2. Planet Detection Efficiencies: Philosophy and

Methods

The fundamental aim of microlensing planet searches is to derive meaningful conclusions about the presence of planets (or lack thereof) from these searches.

Therefore, it is essential to quantitatively assess what planets could have been detected from the observations of individual non-planetary events if such planets had been present. Actually, this problem is not as easy to properly formulate as it might first appear. For example, the event parameters are measured with only finite precision. Among these, the impact parameter u0 (in units of the angular Einstein radius θE) is particularly important: if the event really did have a u0 equal to its best-fit value, then one could calculate whether a planet at a certain separation and position angle would have given rise to a detectable signal in the observed light

17 curve. But the true value of u0 may differ from the best-fit value by, say, 1 σ, and the same planet may not give rise to a detectable signal for this other, quite plausible geometry. (In principle, an error in the time of maximum, t0, would cause a similar problem, displacing the assumed path through the Einstein ring by δt0/tE, where tE is the Einstein crossing time. However, because u0 is strongly correlated with several other parameters while t0 is not, the error in u0 is substantially larger than the error in t0 divided by tE.) Or, as another example, consider finite-source effects. Planetary perturbations have a fairly high probability of exhibiting finite-source effects, which then have a substantial impact on whether the deviation can be detected in a given data stream. If there is such a planetary perturbation, one can measure

ρ∗ = θ∗/θE, the size of the source relative to the Einstein radius. But if there is no planet detected, no finite-source effects are typically detected, and hence there is no direct information on ρ∗. Therefore, one cannot reliably determine whether a given planetary perturbation would have been affected by finite-source effects and so whether it would have been detected. Finally, there are technical questions as to what exactly it means that a planet “would have been” detected.

The past decade of microlensing searches has been accompanied by a steady improvement in our understanding of these questions. Gaudi & Sackett (2000) developed the first method to evaluate detection efficiencies, which was later implemented by Albrow et al. (2000b) and Gaudi et al. (2002). In this approach, binary models are fitted to the observed data with the three “binary parameters”

18 (b, q, α) held ﬁxed and the three “point-lens parameters” (t0, u0,tE) allowed to vary.

Here b is the planet-lens separation in units of θE, q is the planet-star mass ratio, α is the angle of the source trajectory relative to the binary axis, t0 is the time of the source’s closest approach to the center of the binary system, u0 = u(t0) is the impact parameter, tE = θE/µ is the Einstein timescale, and µ is the source-lens relative proper motion. If a particular (b, q, α) yielded a χ2 improvement ∆χ2 <χ2 = 60, min − a planet could be said to be detected. If not, then the ensemble of (b, q, α) for which

2 2 ∆χ >χmin = 60 was said to be excluded for that event. For each (b, q), the fraction of angles 0 α 2π that was excluded was designated the “sensitivity” for that ≤ ≤ geometry.

Gaudi et al. (2002) argued that this method underestimated the sensitivity because it allowed the ﬁt to move u0 to values for which the source trajectory would

“avoid” the planetary perturbation but still be consistent with the light curve. That is, u0 has some deﬁnite value, even if it were not known to the modelers exactly what that value should be. Yoo et al. (2004b) followed up on this by holding u0

fixed at a series of values and estimated planetary detection efficiency at each. The total efficiency would then be the average of these weighted by the probability of each value of u0. In principle, one should also integrate over t0 and tE. In practice,

Yoo et al. (2004b) found that, at least for the event they analyzed, t and t u t 0 eﬀ ≡ 0 E were determined very well by the data, so that once u0 was ﬁxed, so were t0 and tE.

19 Yoo et al. (2004b) departed from all previous planet-sensitivity estimates by incorporating a Bayesian analysis that accounts for priors derived from a Galactic model of the mass, distance, and velocity properties of source and lens population into the analysis. They simulated an ensemble of events and weighted each by both the prior probability of the various Galactic model parameters (lens mass, lens and source distances, lens and source velocities) and the goodness of fit of the resulting magnification profile to the observed light curve. This approach was essential to enable a proper weighting of different permitted values of u0. As a bonus, it allowed one, for the first time, to determine the sensitivities as a function of the physical planetary parameters (such as planet mass mp and planet-star separation r⊥) as opposed to the microlensing parameters, the planet-star mass ratio q and the planet-star projected separation (in the units of θE) b.

Rhie et al. (2000) introduced a procedure for evaluating planet sensitivities that diﬀers qualitatively from that of Gaudi & Sackett (2000). For each trial (b, q, α) and observed point-lens parameters (t0, u0,tE), they created a simulated light curve with epochs and errors similar to those of the real light curve. They then ﬁtted this light curve to a point-lens model with (t0, u0,tE) left as free parameters. If the point-lens

2 2 model had ∆χ > χmin, then this (b, q, α) combination was regarded as excluded.

That is, they mimicked their planet-detection procedure on simulated planetary events.

20 Abe et al. (2004) carried out a similar procedure except that they did not ﬁt for

(t0, u0,tE), but rather just held these three parameters fixed at their point-lens-fit values. Of course, this procedure necessarily yields a higher ∆χ2 than that of Rhie et al. (2000), but Abe et al. (2004) expected that the difference would be small.

While all workers in this field have recognized that finite-source effects are important in principle, they have generally concluded that these did not play a major role in the particular events that they analyzed. This has proved fortunate because the source size is generally unknown, and even a single trial value for the source size typically requires several orders of magnitude more computing time than does a point-source model. Gaudi et al. (2002) estimated angular sizes θ∗ of each of their 43 source stars from their positions on an instrumental color-magnitude diagram (CMD) by adopting µ = 12.5 km s−1 kpc−1 for all events and evaluating

ρ∗ = θ∗/(µtE). They made their sensitivity estimates for both this value of ρ∗ and for a point source (ρ∗ = 0) and found that generally the differences were small. They concluded that a more detailed finite-source evaluation was unwarranted (and also computationally prohibitive). Using their Monte Carlo technique, Yoo et al. (2004b) were able to evaluate the probability distribution of the parameter combination z0, which is equal to impact parameter over source size. This analysis showed that z 1 (no finite-source effects) with high confidence for their event. This implied 0 ≫ that the source did not pass close to the central caustic and hence that finite-source

21 eﬀects were not important. Again, computation for additional values of ρ∗ would have been computationally prohibitive.

2.1.3. ”Seeing” the Lens in High-Magnification Events

In the very first paper on microlensing, Einstein (1936) already realized that it might be difficult to observe the magnified source due to “dazzling by the light of the much nearer [lens] star.”

Seventy years later, more than 2000 microlensing events have been discovered, but only for two of these has the “dazzling” light of the lens star been deﬁnitively observed. The best case is MACHO-LMC-5, for which the lens was directly imaged by the Hubble Space Telescope (HST) (Alcock et al. 2001a; Drake et al. 2004), which yielded mass and distance estimates of the lens that agreed to good precision with those derived from the microlensing event itself (Gould et al. 2004).

The next best case is OGLE-2003-BLG-175/MOA-2003-BLG-45, for which

Ghosh et al. (2004) showed that the blended light was essentially perfectly aligned with the source. This would be expected if the blend actually was the lens, but it would be most improbable if it were just an ambient ﬁeld star. In this case, the blend was about 1 mag brighter than the baseline source in I and 2 mag brighter in

V , perhaps ﬁtting Einstein’s criterion of a “dazzling” presence.

22 Intriguingly, the above two events positively identified to harbor luminous lenses are both high-magnification. It is quite plausible that events with luminous lenses are biased towards high magnification since they will more likely be missed if the magnifications are too low. This raises the question of whether OGLE-2003-BLG-423 has a luminous lens. In addition, identifying the lens star would allow us to directly determine the physical properties of the lens, which in turn would help better constrain parameters in the planetary detection analysis.

Here we analyze OGLE-2004-BLG-343, whose maximum magniﬁcation

A 3000 is by far the highest of any observed event and the ﬁrst to exceed the max ∼ A = 1000 benchmark initially discussed by Liebes (1964) as roughly the maximum possible magniﬁcation for typical Galactic sources and lenses.5

As we describe below, the event was alerted as a possibly anomalous, very- high-magniﬁcation event in time to trigger intensive observations over the peak, but due to human error, the actual observations caught only the falling side of the peak.

We analyze both the actual observations made of this event (in order to evaluate its actual sensitivity to planets) and the sequence of observations that should have been initiated by the trigger. The latter calculation illustrates the potential of

5 Liebes (1964) derived that for perfect lens-source alignment, Au=0 = 2θE/θ∗ by approximating the source star as having uniform surface brightness, and he evaluated this expression for several typical examples.

23 state-of-the-art microlensing observations, although unfortunately this potential was not realized in this case.

We analyze the event within the context of the Yoo et al. (2004b) formalism with several major modifications. First, we adopt the Rhie et al. (2000) criterion of planet-sensitivity in place of that of Gaudi & Sackett (2000). That is, we say a planet configuration is ruled out if simulated data generated by this configuration

2 2 are inconsistent with a point-lens light curve at ∆χ > ∆χmin . Second, using a

Monte Carlo simulation, we show that for this event, z 1, and hence finite-source 0 ∼ effects are quite important. This requires us to generalize the Yoo et al. (2004b) procedure to include a two-dimensional grid of trial parameters (u0,ρ∗) in place of the one-dimensional u0 sequence used by Yoo et al. (2004b). From what was said above, it should be clear that the required computations would be completely prohibitive if they were carried out using previous numerical techniques. Therefore, third, we develop new techniques for finite-source calculations that are substantially more efficient than those used previously.

In 2.2, we describe the data. Next, we discuss modeling of the event in 2.3. § § Then in 2.4, we present our procedures and results related to planet detection. § We explore the possibility that the blended light is due to the lens in 2.5. In § 2.6 we summarize our results and make suggestions on monitoring extremely § high-magniﬁcation events in the future. Finally, the two new binary-lens ﬁnite-source algorithms that we have developed are described in Appendix A.

24 2.2. Observational Data

The ﬁrst alert on OGLE-2004-BLG-343 was triggered by the OGLE-III Early

Warning System (EWS; Udalski 2003) on 2004 June 16, about 3 days before its peak on HJD′ HJD 2450000 = 3175.7467. On June 18, after the ﬁrst observation ≡ − of the lens was taken, the OGLE real time lens monitoring system (Early Early

Warning System [EEWS]; Udalski 2003) triggered an internal alert, indicating a deviation from the single lens light curve based on previous data. Two additional observations were made after that, but the new ﬁts to all of the data were still fully compatible with single-mass lens albeit suggesting high magniﬁcation at maximum.

Therefore, an alert to the microlensing community was distributed by OGLE on

HJD′ = 3175.1 suggesting OGLE-2004-BLG-343 as a possible high-magnification event. The observation at UT 0:57 (HJD′ 3175.54508) the next night showed a large deviation from the light-curve prediction based on previous observations, and an internal EEWS alert was triggered again. Usually further observations would have been made soon after such an alert, but unfortunately no observations were performed until UT 6:29 (HJD′ 3175.77626), about 0.71 hr after the peak. At that time, the event had brightened by almost 3 mag in I relative to the previous night’s observation, and therefore it was regarded very likely to be the first caustic crossing of a binary-lens event. As a consequence, no V -band photometry was undertaken to save observation time and in the hope that observations in V could be done when it brightened again. After two post-peak observations confirmed the event’s

25 extremely high magniﬁcation, OGLE began maximally intensive observations with a cadence of 4.3 minutes. However, after it was clear that the event was falling in a regular fashion, it was then observed less intensively. A total of 14 observations were performed during 3.39 hr, and a new alert to the microlensing community was immediately released by OGLE as well. However, the next day the event faded drastically, by about 3 mag from the maximum point of the previous night, implying that if the event were a binary, the peak had probably been a cusp crossing rather than a caustic crossing. After being monitored for a few more days, it became clearer that OGLE-2004-BLG-343 was most probably a point-lens event of very high magniﬁcation and therefore very sensitive to planets. This recognition prompted

OGLE to obtain a V -band point, but by this time (HJD′ 3179.51) the source had fallen 6 mag from its peak, so that only a weak detection of the V ﬂux was possible.

Hence, this yielded only a lower limit on the V I color. −

By chance, µFUN made one dual-band observation in I and H 1 day before peak

(HJD′ 3174.74256) solely as a reference point to check on the future progress of the event. After the event peaked, µFUN also concluded that it was uninteresting until

OGLE/µFUN email exchanges led to the conclusion that the event was important.

Since the source was magniﬁed by A 40 at this pre-peak µFUN observation, it ∼ enabled a clear H-band detection and so yielded an (I H) color measurement, − which can be translated to (V I). −

26 The OGLE data are available at the OGLE EWS Web site mentioned above and the µFUN data are available at the µFUN Web site 6.

There were 195 images in I and eight images in V , both from OGLE, as well as three images in H from µFUN. Since only OGLE I-band observational data are available near the peak, the following analysis is entirely based on the OGLE I-band data except that the OGLE V -band data and µFUN H-band data are used to constrain the color of the source star. The OGLE errors are renormalized by a factor of 1.42 so that the χ2 per degree of freedom for the best-ﬁt point-source/point-lens

(PSPL) model is close to unity. We also eliminate the two OGLE points that are 3 σ outliers. These are both well away from the peak and therefore their elimination has no practical impact on our analysis.

2.3. Event Modeling

Yoo et al. (2004b) introduced a new approach to model microlensing events for which u0 is not perfectly measured. As distinguished from previous analyses, this method establishes the prior probability of the event parameters by performing a Monte-Carlo simulation of the event using a Galactic model rather than simply assuming uniform distributions. This approach is not only more realistic but also makes possible the estimation of physical parameters, which are otherwise completely

6http://www.astronomy.ohio-state.edu/˜microfun

27 degenerate. Following the procedures of Yoo et al. (2004b), we begin our modeling by fitting the event to a PSPL model, evaluating the finite-source effects and performing a Monte-Carlo simulation. We then improve that method by considering

finite-source effects when combining the simulation with the light-curve fits.

2.3.1. Point-Source Point-Lens Model

The PSPL magniﬁcation is given by (Paczy´nski 1986)

u2 + 2 (t t )2 A(u)= , u(t)= u2 + − 0 , (2.1) 2 v 0 2 u√u + 4 u tE u t where u is the projected lens-source separation in units of the angular Einstein radius

θE, t0 is the time of maximum magniﬁcation, u0 = u(t0) is the impact parameter, and tE is the Einstein timescale.

The predicted ﬂux is then

F (t)= FsA[u(t)] + Fb, (2.2)

where Fs is the source ﬂux and Fb is the blended-light ﬂux.

The observational data are ﬁtted in the above model with ﬁve free parameters

(t0, u0, tE, Fs, and Fb). The results of the ﬁt are shown in Table 2.6 (also see

Fig. 2.1). The best-fit u is remarkably small, u = 0.000333 0.000121, which 0 0 ± indicates that the maximum magnification is A = 3000 1100. As discussed max ± 28 below in 2.3.3, the 3σ lower limit is Amax > 1450. This is the first microlensing § ∼ event ever analyzed in the literature with peak magnification higher than 1000. The uncertainties in u0, tE and Fs are fairly large, roughly 35%. As pointed out in Yoo et al. (2004b), these errors are correlated, while combinations of these parameters, t u t and F F /u , have much smaller errors: eff ≡ 0 E max ≡ s 0

t = 0.0141 0.0008 days, I = 13.805 0.065. (2.3) eﬀ ± min ±

Here Imin is the calibrated I-band magnitude corresponding to Fmax.

2.3.2. Source Properties from Color-Magnitude Diagram

It is by now standard practice to determine the dereddened color and magnitude of a microlensed source by putting the best-fit instrumental color and magnitude of the source on an instrumental (I, V I) CMD. The dereddened color and magnitude − can then be determined from the offset of the source position from the center of the red clump, which is locally measured to be [M , (V I) ]=( 0.20, 1.00). We adopt I − 0 − a Galactocentric distance R0 = 8 kpc. However, at Galactic longitude l = +4.21, the red clump stars in the OGLE-2004-BLG-343 field are closer to us than the Galactic center by 0.15 mag (Stanek et al. 1997). We derive (I, V I) = (14.17, 1.00). − 0,clump Although the source instrumental color and magnitude are both fit parameters, only the magnitude is generally strongly correlated with other fit parameters. By contrast, the source instrumental color can usually be determined directly by a

29 regression of V on I flux as the magnification changes. No model of the event is actually required to make this color determination. In the present case, we exploit both (V I) and (I H) data. Hence, in order to make use of this technique, we − − must convert the (I H) to (V I). This will engender some difficulties. − −

As discussed in 2.2, however, V -band measurements were begun only § when the source had fallen nearly to baseline. Hence, the measurement of the (V I) color obtained by this standard procedure has very large errors − and indeed is consistent with inﬁnitely red (Fs,V = 0) at the 2 σ level (see

Fig. 2.2). The CMD itself is based on OGLE-II photometry, and we have therefore shifted the OGLE-III-derived ﬂuxes by ∆I = I I = 0.26 mag. OGLE−II − OGLE−III On this now calibrated CMD, the clump is at (I, V I) = (15.51, 2.04). − clump Hence, the dereddened source color and magnitude are given by

(I, V I) =(I, V I)+(I, V I) (I, V I) =(I, V I) (1.34, 1.04), − 0 − − 0,clump − − clump − − the ﬁnal oﬀset being the reddening vector. This vector corresponds to

RVI = 1+1.34/1.04 = 2.29, which is somewhat high compared to values obtained by Sumi (2004) for typical bulge ﬁelds. However, we will present below independent evidence for this or a slightly higher value of RVI . Figure 2.2 also shows the position of the blended light, which lies in the so-called reddening sequence of foreground disk main sequence stars. This raises the question as to whether this blended star is actually the lens. We return to this question in 2.5. §

30 The source star is substantially fainter than any of the other stars in the

OGLE-II CMD. In order to give a sense of the relation between this source CMD position and those of main-sequence bulge stars, we also display the Hipparcos main

−0.15/5 sequence (ESA 1997), placed at 10 R0 = 7.5 kpc and reddened by the reddening vector derived from the clump. At the best-ﬁt value, V I = 3.09, the source lies − well in front of (or to the red of) the bulge main-sequence. However, given the large color error, it is consistent with lying on the bulge main sequence at the 1 σ level.

To obtain additional constraints on the color, we consider the µFUN instrumental H-band data. The single highly-magniﬁed (A 40) H-band point ∼ (together with a few baseline points) yields I H = 0.59 0.11 source OGLE−II − µFUN ± color. To be of use, this must be translated to a (V I) color using a − OGLE−II (V I)/(I H) color-color diagram of the stars in the ﬁeld. − −

Unfortunately, there are actually very few ﬁeld stars in the appropriate color range. This partly results from the small size ( 2 arcmin2) of the H-band image ∼ and partly from the fact that a large fraction of stars are either too faint to measure in V -band or saturated in H-band. We therefore calibrate the µFUN

H-band data by aligning them to Two Micron All Sky Survery (2MASS) data and generate a (V I)/(I H ) color-color diagram by matching stars from the − − 2MASS 2MASS H-band data with OGLE-II V,I photometry in a larger ﬁeld centered on

OGLE-2004-BLG-343. We ﬁnd that (H H ) = 1.99 0.01 from 48 2MASS − µFUN − ± stars in common in the ﬁeld, with a scatter of 0.08 mag. We transform the above

31 I H color to I H and plot it as a vertical line ona(V I)/(I H ) − µFUN − 2MASS − − 2MASS color-color diagram (see Fig. 2.3). From the intersection of the vertical line with the diagonal track of stars in the ﬁeld, we infer V I = 2.40 0.15. − ±

Since the ﬁeld stars used to make the alignment are giants, this transformation would be strictly valid only if the source were a giant as well. However, the source star is certainly a dwarf (see Fig. 2.2). After transforming 2MASS to standard infrared bands (Carpenter 2001), we use the data from Tables II and III of Bessell &

Brett (1988) to construct dwarf and giant tracks on a (V I) /(I H) color-color − 0 − 0 diagram. These are approximately coincident for blue stars (I H) < 1.6 but − 0 rapidly separate by 0.28 mag in (V I) by (I H) = 1.7. In principle, we should − 0 − 0 just adjust our estimate (V I) by the diﬀerence between these two tracks at the − dereddened (I H) color of the source. Unfortunately, there are two problems − 0 with this seemingly straightforward procedure. First, the Bessell & Brett (1988) giant track displays a modest deviation from its generally smooth behavior close to the color of our source, a deviation that is not duplicated by either the giants in our ﬁeld or the color-color diagram formed by combining Hipparcos and 2MASS data, which both show the same smooth behavior at this location. Second, if the

Hipparcos/2MASS diagram or the Bessell & Brett (1988) diagram is reddened using the selective and total extinctions determined above from the position of the clump, then the giant tracks do not align with our ﬁeld giants. To obtain alignment, one

32 7 must use RVI = 2.4. The conﬂict among these three determinations of RVI (1.9–2.1

[Sumi 2004], 2.29 [clump], 2.4 [Gould et al. 2001]) is quite a puzzle, but not one ∼ that we can explore here.

The bottom line is that there is considerable uncertainty in the dwarf-minus- giant adjustment but only in the upward direction. To take account of this, we add 0.2 mag error in quadrature to the upward error bar and ﬁnally adopt

V I = 2.4+0.25 for the indirect color determination via the (I H) measurement. − −0.15 − Finally, we combine this with the direct measurement of V I = 3.09 based on − the combined V and I light curve. Because the errors on the latter measurement are extremely large (and are Gaussian in flux rather than magnitudes), we determine the probability distribution for the combined determination numerically in a flux-based calculation and then convert to magnitudes. We finally find

V I =(V I) σ(V I) = 2.60 0.20, which we show as a magenta point in − − best ± − ± Figure 2.2. Hence,

(V I) = 1.56 0.20. (2.4) − 0 ± 7 Gould et al. (2001) found a similar value using the same method but a diﬀerent data set.

However, the RV I we obtained at the beginning of this section is based on the dereddened magnitude of the red clump, which depends on the distance to the Galactic center R0. If we were to adopt the new geometric measurement of R0 = 7.62 kpc (Eisenhauer et al. 2005), rather than the standard value of R0 = 8.0 kpc, we would then have I0,clump = 14.07, which would give RV I = 2.39. However this value conﬂicts still more severely with the typical values of RV I = 1.9–2.1 in bulge ﬁelds found by Sumi (2004).

33 In contrast to most microlensing events that have been analyzed for planets, the color of OGLE-2004-BLG-343 is fairly uncertain. The color enters the analysis in two ways. First, it indicates the surface brightness and so determines the relation between dereddened source ﬂux and angular size. Second, it determines the limb-darkening coeﬃcient.

Given the color error, we consider a range of colors in our analysis and integrate over this range, just as we integrate over a range of impact parameters u0 and source sizes (normalized to θ ) ρ . We allow colors over the range 2.2 < (V I) < 3.0 E ∗ − corresponding to 1.16 < (V I) < 1.96. We integrate in steps of 0.1 mag. For each − 0 color, we adopt a surface brightness such that the source size θ∗ is given by

−0.2(I−Ibest) θ∗ = θ(V −I) 10 , (2.5)

where I is the (reddened) apparent magnitude in the model, Ibest = 22.24, and

θ2.2 ...θ3.0 = (0.350, 0.371, 0.391, 0.421, 0.466, 0.515, 0.546, 0.580, 0.615)µas. These values are derived from the color/surface-brightness relations for dwarf stars given in Kervella et al. (2004) using the method as described in Yoo et al. (2004a). In our actual calculations, we use the full distribution of source radii, but for reference we note that the 1σ range of this quantity is

θ = 0.47 0.13µas. (2.6) ∗ ± 34 We ﬁnd from the models of Claret (2000) and Hauschildt et al. (1999) that the linear limb-darkening coeﬃcients for dwarfs in our adopted color range vary by only a few hundredths. Therefore, for simplicity, we adopt the mean of these values

ΓI = 0.50 (2.7)

for all colors. This corresponds to c = 3Γ/(2+Γ) = 0.60 in the standard limb-darkening parameterization (Afonso et al. 2000).

Finally, each model speciﬁes not only a color and magnitude for the source, but also a source distance. Evaluation of the likelihood of each speciﬁc combination of these requires a color-magnitude relation. We adopt (Reid 1991)

M = 2.37(V I) + 2.89 (2.8) I − 0 with a scatter of 0.6 mag. The ridge of this relation is shown as a red line segment in

−0.15/5 Figure 2.2, with the sources placed at 10 R0 = 7.5 kpc and reddened according to the red-clump determination, just as was done for the Hipparcos stars. This track is in reasonable agreement with the Hipparcos stars.

2.3.3. Finite-Source Effects

Yoo et al. (2004b) deﬁne z u /ρ (where ρ = θ /θ is the angular size 0 ≡ 0 ∗ ∗ ∗ E of the source θ∗ in units of θE), which is a useful parameter to characterize the

35 finite-source effects in single-lens microlensing events. We fit the observational data

2 to a set of point-lens models on a grid of (u0, z0) and then compare the resulting χ with the best-ﬁt PSPL model. As mentioned in 2.3.2, we adopt the limb-darkening § formalism of Yoo et al. (2004b) and for simplicity choose ΓI = 0.50.

Figure 2.4 displays the resulting ∆χ2 contours. It shows that the 1 σ contour extends from z 0.2 to arbitrarily large z . This is qualitatively similar to 0 ≃ 0 OGLE-2003-BLG-423 as analyzed in Yoo et al. (2004b). However, as we demonstrate in 2.3.4, the range of z that is consistent with the Galactic model is quite diﬀerent § 0 for these two events. This is to be expected since z0 = u0/ρ∗ =(u0/θ∗)θE, and u0/θ∗ is roughly 8 times smaller for this event, while θE is generally of the same order.

Figure 2.4 shows contours for both A and u−1. For z u /ρ > 2, these max 0 0 ≡ 0 ∗ are very similar, which is expected because in the absence of ﬁnite-source eﬀects

(and for u 1), A = u−1. Note that the A contours are roughly rectangular, 0 ≪ max 0 max so that while z is not well constrained, the 3σ lower limit A > 103.16 1450 is 0 max ∼ quite well deﬁned. This shows that although the blending is very severe, it is also very well constrained, implying that the event’s high magniﬁcation is secure.

2.3.4. Monte-Carlo Simulation

We perform a similar Monte-Carlo simulation using a Han & Gould (1996, 2003) model as described in Yoo et al. (2004b) by taking into account all combinations of

36 source and lens distances, Dl < Ds, uniformly sampled along the line of sight toward the source (l, b) = (4.21, 3.47). The simulation adopts the Gould (2000a) mass − function taking into account the bulge main sequence stars, white dwarfs (distributed around 0.6 M⊙), neutron stars (narrowly peaked at 1.35 M⊙), and stellar-mass black holes. This mass function is adequate to describe mass distributions of disk lenses except that the disk contains stars with masses greater than 1 M⊙ while the bulge does not. However, a disk main sequence star more massive than the Sun will be too bright to be the lens star for this event (see Fig. 2.2); so for simplicity, we use this mass function for both disk and bulge in our simulation. In Yoo et al. (2004b), the source ﬂux is determined from the tE for each Monte-Carlo event since only the

PSPL model is considered at this step. However, when ﬁnite-source eﬀects are taken into account, each tE corresponds to a series of Fs depending on the source size ρ∗, so there is no longer a 1-1 correspondence between Fs and tE. As discussed in Yoo et al. (2004a), θ∗ can be deduced from the source’s dereddened color and magnitude.

Since θE is known for each simulated event, ρ∗ is a direct function of Fs and the

(V I) color of the source, −

θ(V −I) Fs ρ∗ = , (2.9) θE sFbest

where Fbest corresponds to Ibest in equation (2.5). Using this constraint, we fit the k-th Monte-Carlo event to a point-lens model with finite-source effects, holding tE,k

ﬁxed at the value given by the simulation, for a variety of (V I) color values − 37 inferred from 2.3.2. Hence, for the j-th (V I) color and k-th Monte-Carlo § − event, we have best-ﬁt single-lens light-curve parameters t0,j,k, u0,j,k, ρ∗,j,k, Fs,j,k,

F , as well as ∆χ2 χ2 χ2 . We construct a three-dimensional table that b,j,k j,k ≡ j,k − PSPL includes these six quantities as well as the other parameters from the Monte-Carlo simulation (tE,k, θE,k, Ds,k, Dl,k, Ml,k, Γk), the Einstein timescale and radius, the source and lens distances, the lens mass, and the event rate. From these can also be derived two other important quantities, the source absolute magnitude MI,j,k and the physical Einstein radius r θ D . This three-dimensional table E,k ≡ E,k × l,k is composed of nine two-dimensional tables, one for each (V I) color. Each − j table contains approximately 200,000 rows, one for each simulated event. To make the notation more compact, we refer to the parameters a, b, c, ... lying in the bin a [a ,a ]; b [b , b ]; c [c ,c ]... as bin( a, b, c, ... ). ∈ min max ∈ min max ∈ min max { }

Similarly to Yoo et al. (2004b), the posterior probability of ai lying in bin(ai) is given by

P [bin(a )] (P ) (P ) exp[ ∆χ2 /2] (2.10) i ∝ V −I j × Reid j,k × − j,k Xj,k BC[bin( a )] Γ , × { i}j,k × k

where (P ) = exp( (M ) M [(V I) ] 2/2[σ2 +(σ )2 ]) accounts Reid j,k −{ I j,k − I,Reid − 0,j } Reid MI j,k for the scatter (σReid = 0.6) in MI about the Reid relation plus the dispersion

(σ ) from light-curve fitting, (P ) = exp [(V I) (V I) ]2/2σ2 MI j,k V −I j {− − j − − best (V −I)} 38 reflects the uncertainty in V I color, and BC is a boxcar function defined by − BC[bin(a)] Θ(a a ) Θ(a a). ≡ − min × max −

Figure 2.5 shows the posterior probability distributions of various parameters, including u0, dereddened apparent I-band magnitude of the source I0, proper motion

µ, z0, source distance modulus, lens distance modulus, absolute I-band magnitude of the source MI , and lens mass. The blue and red histograms represent bulge-disk events and bulge-bulge events, respectively. The relative event rate is normalized in the same way for both bulge-disk and bulge-bulge events. The total rate for bulge-disk events is about 6 times larger than that for bulge-bulge events, which means that the Monte-Carlo simulation tends strongly to favor bulge-disk events.

The Einstein radii are on average smaller for bulge-bulge events than for bulge-disk events, and as a result, the bulge-bulge events tend to have bigger ρ∗ and hence smaller z0. However, the top right panel of Figure 2.5 shows that the z0 probability distributions have similar shapes for both bulge-bulge and bulge-disk events. This is because the (lack of) finite-source effects constrain z0 > 0.7 at ∼ the 3σ level (see Fig. 2.4), which cuts off the lower end of the z0 distributions for both categories of events. Since bulge-disk events have smaller ρ∗ than bulge-bulge events, the u0 posterior probability distribution peaks at a lower value for the former. Furthermore, since z0 > 0.7, the proper motion is constrained to be ∼

µ = θ∗z0/teﬀ > 7mas/yr, which is typical of bulge-disk events but > 2 times the ∼ ∼ proper motion of typical bulge-bulge events.

39 Figure 2.5 also shows the distributions of u0 and I0 from the light-curve data alone by a black solid line. In strong contrast to the corresponding diagrams for OGLE-2003-BLG-423 presented by Yoo et al. (2004b), the light-curve based parameters agree quite well with the Monte-Carlo predictions. In the source distance-modulus panel, the prior distributions for bulge-disk and bulge-bulge events are shown in purple and green histograms, respectively. Again, distinct from OGLE-2003-BLG-423, the most likely source distances of this event agree reasonably well with typical values from the prior distributions. Moreover, from

Figure 2.5, the peak values of source MI distributions are also in good agreement with those derived from the Reid relation (M = 6.59 for [V I] = 1.56, see I − 0 eq. [2.8]). Therefore, the source of this event shows very typical characteristics as represented by the Monte-Carlo simulation. Also unlike OGLE-2003-BLG-423, the probability that z0 < 1 is very high for both bulge-disk and bulge-bulge events. ∼ Therefore, ﬁnite-source eﬀects must be taken into account in the analysis of this event. In addition to the posterior probability distribution (orange) of the lens mass, the prior distribution (dark green) is displayed in Figure 2.5 as well. In microlensing analyses, the lens mass is commonly assigned a “typical” value (for example, 0.3 M⊙). However, Figure 2.5 shows that, lenses with relatively high mass are strongly favored for this event as compared to the prior distribution. Detailed discussions on the lens properties are presented in 2.5. §

40 2.4. Detecting Planets

While there are no obvious deviations from point-lens behavior in the light

2 curve of OGLE-2004-BLG-343 at our adopted threshold of ∆χmin = 60, planetary deviations might be difficult to recognize by eye. We must therefore conduct a systematic search for such deviations. Logically, this search should precede the second step of testing to determine what planets we could have detected had they been there. However, as a practical matter it makes more sense to first determine the range of parameter space for which we are sensitive to planets because it is only this range that needs to be searched for planets. We therefore begin with this detection efficiency calculation.

2.4.1. Detection Efficiency

As reviewed in 2.1.2, a variety of methods have been proposed to calculate § the planetary sensitivities of microlensing events, either in predicting planetary detection efficiencies theoretically or in analyzing real observational data sets. In those methods, ∆χ2 is often calculated by subtracting the χ2 of single-lens models from that of the binary-lens models to evaluate detection sensitivities. However, the ways in which single-lens and binary-lens models are compared differ from study to study. As noted by Griest & Safizadeh (1998) and Gaudi & Sackett

(2000), for real planetary light curves, the lens parameters are not known a priori.

41 Therefore, ∆χ2 will generally be exaggerated if one subtracts from the binary lens model the single-lens model that has the same t0, u0, and tE instead of the best-ﬁt single-lens model to the binary light curve. One important factor contributing to this exaggeration is that the center of the magniﬁcation pattern (referred to as the center of the caustic in Yoo et al. 2004b) in the binary-lens case is no longer the position of the primary star as it is in the single-lens model (Dominik 1999b; An & Han 2002).

Therefore light-curve parameters such as u0 and t0 will shift correspondingly. We

find that by not taking into account this effect and directly comparing the simulated binary (i.e., planetary) light curve with the best-fit single-lens model to the data,

Abe et al. (2004) exaggerate the planetary sensitivity of MOA 2003-BLG-32/OGLE

2003-BLG-219, although it remains the most sensitive event analyzed to date (see

Appendix B).

Following Gaudi & Sackett (2000), planetary systems are characterized by a planet-star mass ratio q, planet-star separation in Einstein radius b, and the angle α of the source trajectory relative to the planet-star axis. In our binary-lens calculations, u0 and t0 are defined with respect to the center of magnification discussed above. According to Gaudi & Sackett (2000), the next step is to fit the data to both PSPL models and binary-lens models with a variety of (b, q, α) and calculate ∆χ2(b, q, α) = χ2(b, q, α) χ2 . Then ∆χ2(b, q, α) is compared with a − PSPL 2 2 2 threshold value χthres: if∆χ > χthres then a planet with parameters b, q, and α is claimed to be excluded while it is detected if ∆χ2 < χ2 . The (b, q) detection − thres

42 efficiency is then obtained by integrating Θ(∆χ2 ∆χ2 ) over α in the exclusion − thres region at fixed (b, q), where Θ is a step function. However, Gaudi et al. (2002) point out that for events with poorly constrained light-curve parameters, which is the case for OGLE-2004-BLG-343, this method will significantly underestimate the sensitivities since the binary-lens models will minimize the χ2 over the relatively large available parameter space. As discussed in Yoo et al. (2004b), the detection efficiency should be evaluated at a series of allowed u0 values. To take finite-source effects into account, we generate a grid of permitted (u0,ρ∗), and each (u0,m,ρ∗,m) bin is associated with the probability P [bin( u , ρ )] obtained using the following { 0,m ∗,m} equation:

P [bin( u , ρ )] P (2.11) { 0,m ∗,m} ∝ m,j,k Xj,k where

P = (P ) (P ) exp[ ∆χ2 /2] BC[bin( u )] m,j,k V −I j × Reid j,k × − j,k × { 0,m}j,k BC[bin( ρ )] Γ (2.12) × { ∗,m}j,k × k

If the light-curve parameters were well-constrained, the approaches of Gaudi

& Sackett (2000) and Rhie et al. (2000) would be very nearly equivalent, with the former retaining a modest philosophical advantage, since it uses only the observed light curve and does not require construction of light curves for hypothetical events.

43 However, because in our case these parameters are not well constrained, the Gaudi

& Sackett (2000) approach would require integration over all binary-lens parameters

(except Fs and Fb). Regardless of its possible philosophical advantages, this approach is therefore computationally prohibitive in the present case. We therefore do not follow Gaudi & Sackett (2000), but instead construct a binary light curve with the same observational time sequence and photometric errors as the OGLE observations of OGLE-2004-BLG-343, for each (b, q, α ; u0,ρ∗) combination and the associated probability-weighted parameters alc: t0, tE, Fs and Fb in the m-th (u0,ρ∗) bin,

Pm,j,kalc,j,k j,k alc(weighted),m = P . (2.13) Pm,j,k j,k P

Then each simulated binary light curve (b, q, α ; u0,m,ρ∗,m) is ﬁtted to a

2 single-lens model with finite-source effects whose best fit yields χ (b, q, α ; u0,m,ρ∗,m).

Another set of artiﬁcial binary light curves is generated under the assumption that OGLE had triggered a dense series of observations following the internal alert at HJD′ 3175.54508. These cover the peak of the event with the normal

OGLE frequency and are used to compare results with those obtained from the real observations.

Magnification calculations for a binary lens with finite-source effects are very time-consuming. Besides (b, q, α), our calculations are also performed on (u0,ρ∗) grids, two more dimensions than in any previous search of a grid of models with

ﬁnite-source eﬀects included. This makes our computations extremely expensive,

44 comparable to those of Gaudi et al. (2002), which equaled several years of processor time. Therefore, we have developed two new binary-lens ﬁnite-source algorithms to perform the calculations, as discussed in detail in Appendix A.

In principle, we should consider the full range of b, i.e., 0 1 results onto b < 1 except for the isolated sensitive zones along the x axis caused by planetary caustics − perturbations.

We define the planetary detection efficiency ǫ(b, q) as the probability that an event with the same characteristics as OGLE-2004-BLG-343, except that the lens is a planetary system with configuration of (b, q), is inconsistent with the single-lens model (and hence would have been detected),

2 2 ǫ(b, q, α) = Θ χ (b, q, α ; u0,m,ρ∗,m) ∆χthres P [bin( u0,m, ρ∗,m )] { m − × { } } X h i −1 P [bin( u0,m, ρ∗,m )] (2.14) ×{ m { } } X

and

1 2π ǫ(b, q)= ǫ(b, q, α)dα. (2.15) 2π Z0

45 2.4.2. Constraints on Planets

Figure 2.6 shows the planetary detection efficiency of OGLE-2004-BLG-343 for planets with mass ratios q = 10−3, 10−4, and 10−5, as a function of b, the planet-star separation (normalized to θE), and α, the angle that the moving source makes with the binary axis passing the primary lens star on its left. Different colors indicate 10%, 25%, 50%, 75%, 90% and 100% efficiency. Note that the contours are elongated along an axis that is roughly 60◦ from the vertical (i.e., the direction of the impact parameter for α = 0). This reflects the fact that the point closest to the peak occurs at t = 2453175.77626 when (t t )/t = 2.16u , and so when the − 0 E 0 source-lens separation is at an angle tan−1 2.16 = 65◦. For q = 10−3, the region of 100% efficiency extends through 360◦ within about one octave on either side of the Einstein ring. However, at lower mass ratios there is 100% efficiency only in restricted areas close to the Einstein ring and along the above-mentioned principal axis.

Figure 2.7 summarizes an ensemble of all ﬁgures similar to Figure 2.6, but with q ranging from 10−2.5 to 10−5.0 in 0.1 increments. To place this summary in a single

figure, we integrate over all angles α at fixed b. Comparison of this figure to Figure 8 from Gaudi et al. (2002) shows that the detection efficiency of OGLE-2004-BLG-343 is similar to that of MACHO-1998-BLG-35 and OGLE-1999-BUL-35 despite the fact that their maximum magnifications are A 100, roughly 30 times lower than max ∼

46 OGLE-2004-BLG-343. Of course, part of the reason is that OGLE-2004-BLG-343 did not actually probe as close as u = u 1/3000 because no observations were 0 ∼ taken near the peak. However, observations were made at u 1/1200, about 12 ∼ times closer than in either of the two events analyzed by Gaudi et al. (2002). One problem is that because the peak was not well covered, there are planet locations that do not give rise to observed perturbations at all. But this fact only accounts for the anisotropies seen in Figure 2.6. More fundamentally, even perturbations that do occur in the regions that are sampled by the data can often be ﬁtted to a point-lens light curve by “adjusting” the portions of the light curve that are not sampled.

Note the central “spike” of reduced detection eﬃciency plots near b = 1. As

ﬁrst pointed out by Bennett & Rhie (1996), this is due to the extreme weakness of the caustic for nearly resonant (b 1) small mass-ratio (q 1) binary lenses. ∼ ≪

2.4.3. No Planet Detected

Based on the detection efficiency levels we obtained in 2.4.2, we fit the § observational data to binary-lens models to search for a planetary signal in the regions with efficiency greater than zero from q = 10−5 to q = 10−2.5. We find no binary-lens models satisfying our detection criteria. In fact, the total χ2 contributions to the best-fit single-lens model of the observational points over the peak ( HJD = 2453175.5 2453176.0) are no more than 30, so even if all of these −

47 deviations were due to a planetary perturbation, such a binary-lens solution would not easily satisfy our ∆χ2 = 60 detection criteria. Therefore there are no planet detections in OGLE-2004-BLG-343 data.

2.4.4. Fake Data

Partly to explore further the issue of imperfect coverage of the peak, and partly to understand how well present microlensing experiments can probe for planets, we now ask what would have been the detection eﬃciency of OGLE-2004-BLG-343 if the internal alert issued on HJD′ 3175.54508 had been acted upon.

Of course, since the peak was not covered, we do not know exactly what u0 and ρ for this event are. However, for purposes of this exercise, we assume that they are near the best fit as determined from a combination of the light-curve fitting and the Galactic Monte Carlo, and for simplicity, we choose u0 = 0.00040,ρ∗ = 0.00040 which is very close to the best-fit combination. We then form a fake light curve sampled at intervals of 4.3 minutes, starting from the alert and continuing to the end of the actual observations that night. This sampling reflects the intense rate of OGLE follow-up observations actually achieved during this event (see 2.2). We § assume errors similar to those of the actual OGLE data at similar magnifications.

For those points that are brighter than the brightest OGLE point, the minimum actual photometric errors are assigned. We also assume that the color information is

48 known exactly in this case to be V I = 2.6. We then analyze these fake data in − exactly the same way that we analyze the real data. In contrast to the real data, however, we do not find a finite range of z u /ρ that are consistent with the fake 0 ≡ 0 ∗ data. Rather, we find that all consistent parameter combinations have z0 = 1 almost identically. We therefore consider only a one-dimensional set of (u0,ρ∗) combinations subject to this constraint.

Figure 2.8 is analogous to Figure 2.6 except that the panels show planet sensitivities for q = 10−3, 10−4, 10−5, and 10−6, that is, an extra decade. In sharp contrast to the real data, these sensitivities are basically symmetric in α, except for the lowest value of q. Sensitivities at all mass ratios are dramatically improved.

For example, at q = 10−3, there is 100% detection eﬃciency over 1.7 dex in b

(1/7 < b < 7). Even at q = 10−5 (corresponding to an Earth-mass planet around an ∼ ∼ M star), there is 100% eﬃciency over an octave about the Einstein radius.

Figure 2.9 is the fake-data analog of Figure 2.7. It shows that this event would have been sensitive to extremely low mass-ratios, lower than those accessible to any other technique other than pulsar timing.

2.4.5. Detection Efficiency in Physical Parameter Space

One of the advantages of the Monte Carlo approach of Yoo et al. (2004b) is that it permits one to evaluate the planetary detection eﬃciency in the space of

49 the physical parameters, planet mass and projected physical separation (mp, r⊥), rather than just the microlensing parameters (b, q). Figures 2.10 and 2.11 show this detection eﬃciency for the real and fake data, respectively. The fraction of Jupiter-mass planets that could have been detected from the actual data stream is greater than 25% for 0.8AU < r⊥ < 10AU and is greater than 90% for ∼ ∼

2AU < r⊥ < 6AU. There is also marginal sensitivity to Neptune-mass planets. ∼ ∼ However, the detection eﬃciencies would have been signiﬁcantly enhanced had the

FWHM around the peak been observed, as previously discussed by Rattenbury et al. (2002). For the fake data, more than 90% of Jupiter-mass planets in the range

0.7AU < r⊥ < 20AU and more than 25% with 0.3AU < r⊥ < 30AU would have ∼ ∼ ∼ ∼ been detected. Indeed, some sensitivity would have extended all the way down to

Earth-mass planets.

2.5. Luminous Lens?

Understanding the physical properties of their host stars is a major component of the study of extra-solar planets. It is especially important to know the mass and distance of the lens star for planets detected by microlensing because only then can we accurately determine the planet’s mass and physical separation from the star. Obtaining similar information for microlensing events that are unsuccessfully searched for planets enables more precise estimates of the detection eﬃciency. There

50 are only two known ways to determine the mass and distance of the lens: either measure both the microlensing parallax and the angular Einstein radius (which are today possible for only a small subset of events) or directly image the lens. In most cases the lens is either entirely invisible or is lost in the much brighter light of the source.

A simple argument suggests, however, that in extremely high-magniﬁcation events like OGLE-2004-BLG-343, the lens will often be easily visible and, indeed, it is the lens that is unknowingly being monitored, with the source revealing itself only in the course of the event. Events of magniﬁcation Amax require that the source be much smaller than the Einstein radius, θ∗ < 2θE/Amax. Since θE = √κMπrel, large ∼

θE requires a lens that is either massive or nearby, both of which suggest that it is bright. On the other hand, a small θ∗ implies that the source is faint. Generally, if a faint source and a bright potential lens are close on the sky, only the lens will be seen, until it starts to strongly magnify the source. This has important implications for the real time recognition of extreme magniﬁcation events, as we discuss in 2.6. § Here we review the evidence as to whether the blended light in OGLE-2005-BLG-343 is in fact the lens.

As was true for OGLE-2003-BLG-175/MOA-2003-BLG-45 mentioned in 2.1.3, § the blended light in OGLE-2004-BLG-343 lies in the “reddening sequence” of foreground disk stars. It is certainly “dazzling” by any criterion, being about 50

51 times brighter than the source in I and 150 times brighter in V (see Fig. 2.2). Is the blended light also due to the lens in this case?

There is one argument for this hypothesis and another against. We initiate the ﬁrst by estimating the mass and distance to the blend as follows. We model the extinction due to dust at a distance x along the line of sight by

−1 −qx −1 dAI /dx = 0.4 kpc e and set q = 0.26 kpc in order to reproduce the measured extinction to the bulge AI (8kpc) = 1.34. Using the Reid (1991) color-magnitude relation, we then adjust the distance to the blend until it reproduces the observed color and magnitude of the blend. We find a distance modulus of 12.6 ( 3.3 kpc), ∼ and with the aid of the Cox (2000) mass-luminosity relation, we estimate a corresponding mass Ml = 0.9 M⊙. Inspecting Figure 2.5, we see that this is almost exactly the peak of the lens-distance distribution function predicted by combining light-curve information and the Galactic model. This is quite striking because, in the absence of light-curve information, the lens would be expected to be relatively close to the source. From our Monte-Carlo simulation toward the line of sight of this event, the total prior probability of the bulge-bulge events is about 1.5 times higher than the prior probability of the bulge-disk events, and furthermore, only about 7% of all events have lenses less than 3.3 kpc away (see green and purple histograms in source and lens distance modulus panels of Fig. 2.5). It is only because the light curve lacks obvious finite-source effects (despite its very high-magnification) that one is forced to consider lenses with large θE, which generally drives one toward nearby

52 lenses in the foreground disk. Based on our experience analyzing many blended microlensing events, the blended light is most often from a bulge star rather than a disk star, which simply reﬂects the higher density of bulge stars. In brief, it is quite unusual for lenses to be constrained to lie in the disk, and it is quite unusual for events to be blended with foreground disk stars. This doubly unusual set of circumstances would be more easily explained if the blend were the lens.

However, if the blend were the lens, then the source and lens would be aligned to better than 1 mas during the event, and one would therefore expect that the apparent position of the source would not change as the source ﬁrst brightened and then faded. In fact, we ﬁnd that the apparent position does change by about

73 9 mas. However, since the apparent source (i.e., combined source and blended ± light at baseline) has a near neighbor at 830 mas, which is almost as bright as the source/blend, it is quite possible that the lens actually is the blend, but that this neighbor is corrupting the astrometry.

Thus, the issue cannot be definitively settled at present. However, it could be resolved in principle by, for example, obtaining high-resolution images of the field a decade after the event when the source and lens have separated sufficiently to both be seen. If the blend is the lens, then they will be seen moving directly apart with a proper motion given µ = θE/tE, where θE is derived from the estimated mass and distance to the lens and tE is the event timescale.

53 Since the blend cannot be positively identiﬁed as the lens, we report our main results using a purely probabilistic estimate of the lens parameters. However, for completeness, we also report results here based on the assumption that the lens is the blend. Compared to the previous simulation, in which we considered the full mass function and full range of distances, we sample only the narrow intervals of mass and distance that are consistent with the observed color and magnitude of the lens/blend. To implement these restrictions, we repeat the Monte Carlo, but with the additional constraint that the predicted apparent magnitudes agree with the observed blend magnitude (with an error of 0.5 mag) and that the predicted colors

(using the above extinction law and the Reid 1991 color-magnitude relation) also show good agreement with the observed color (with 0.2 mag error). These errors are, of course, much larger than the observational errors. They are included to reﬂect the fact that the theoretical predictions for color and magnitude at a given mass are not absolutely accurate.

Figure 2.12 is the resulting version of Figure 2.10 when the Monte Carlo is constrained to reproduce the blend color and magnitude. The sensitivity contours are narrower and deeper, reﬂecting the fact that the diagram no longer averages over a broad range of lens masses but rather is restricted eﬀectively to a single mass (and single distance).

54 2.6. Summary and Discussion

In this paper we present our analysis of microlensing event OGLE-2004-BLG-

343, with the highest peak magnification (A = 3000 1100) ever analyzed to date. max ± The light curve is consistent with the single-lens microlensing model, and no planet has been detected in this event. We demonstrate that if the peak had been well covered by the observations, the event would have had the best sensitivity to planets to date, and it would even have had some sensitivity to Earth-mass planets ( 2.4.4, § 2.4.5). However, this potential has not been fully realized due to human error § ( 2.2), and OGLE-2004-BLG-343 turns out to be no more sensitive to planets than § a few other high-magnification events analyzed before ( 2.4.2, 2.4.5). Thus, while § § ground-based microlensing surveys are technically sufficient to detect very low-mass planets, the relatively short timescale of the sensitive regime of high-magnification microlensing events demands a rapidity of response that is not consistently being achieved. In the final paragraph below, we develop several suggestions to rectify this situation.

In 2.3 we show that finite-source effects are important in analyzing this § event, so we extend the method of Yoo et al. (2004b) to incorporate such effects in planetary detection efficiency analysis. Moreover, since magnification calculations of binary-lens models with finite-source effects are computationally remarkably expensive, and applying previous finite-source algorithms, it would have taken

55 of order a year of CPU time to do the detection eﬃciency calculation required by this event. We therefore develop two new binary-lens ﬁnite-source algorithms

(Appendix A) that are considerably more eﬃcient than previous ones. The

“map-making” method (Appendix A.1) is an improvement on the conventional inverse ray-shooting method, which proves to be especially efficient for use in detection efficiency calculations, while the “loop-linking” method (Appendix A.2) is more versatile and could be easily implemented in programs aimed at finding best-fit finite-source binary-lens solutions. Using these algorithms, we were able to complete the computations for this paper in about 4 processor-weeks, roughly an order of magnitude faster than would have been required using previous algorithms.

Finally, we show in 2.5 that the blend, which is a Galactic disk star, might § very possibly be the lens, and that this case also proves to be highly probable from the Monte-Carlo simulation. However, it seems to contradict the astrometric evidence, and we point out that this issue could in principle be solved by future high-resolution images. Among the high-magnification events discovered by current microlensing survey groups, it is very likely that the lens star, which is also the apparent source, of those events is in the Galactic disk. Thus the blended light is usually far brighter than the source, thereby increasing the difficulty in early identification of such events. This fact motivates the first of several suggestions aimed at improving the recognition of very high-magnification events:

56 1) When events are initially alerted they should be accompanied by instrumental

CMD of the surrounding ﬁeld, with the location of the apparent “source”

highlighted. Events whose apparent sources lie on the “reddening sequence”

of foreground disk stars (see Fig. 2.2) have a high probability to actually be

lenses of more distant (and fainter) bulge sources. These events deserve special

attention even if their initial light curves appear prosaic.

2) For each such event it is possible to measure the color (but not immediately

the magnitude) of the source by the standard technique of obtaining two-band

photometry and measuring the slope of the relative ﬂuxes in the two bands.

If the color is diﬀerent from that of the apparent “source” at baseline, that

will prove that this baseline light is not primarily due to the source, and it will

increase the probability that this baseline object is the lens. Moreover, if the

source color is relatively red, it will show that the source is probably faint and

so is (1) most likely already fairly highly magniﬁed (thereby making it possible

to detect above the foreground blended light) and (2) capable, potentially at

least, of being magniﬁed to very high magniﬁcation (see 2.5). This would § motivate obtaining more data while the event was still faint to help predict

its future behavior and would enable a guess as to how to “renormalize” the

event’s apparent magniﬁcation to its true magniﬁcation. This is important

because generally one cannot accurately determine this renormalization until

57 the event is within 0.4 mag (when the event is teﬀ before its peak), at which

point it may well be too late to act on this knowledge.

3) Both survey groups and follow-up groups should issue alerts on suspected

high-magniﬁcation events guided by a relatively low threshold of conﬁdence,

recognizing that this will lead to more “false alerts” than at present. If such

alerts are accompanied by a cautionary note, they will promote intergroup

discussions that could lead to more rapid identiﬁcation of high-magniﬁcation

events without compromising the credibility of the group.

58 Fig. 2.1.— Light curve of OGLE-2004-BLG-343 near its peak on 2004 June 19 (HJD 2,453,175.7467). Only OGLE I-band data (open circles) are u sed in most of the analysis, except OGLE V -band data (open triangles) and µFUN H-band data (crosses) are used to constrain the color of the source star. All bands are linearly rescaled so that Fs and Fb are the same as the OGLE I-band observations. The solid line shows the bes t-ﬁt PSPL model. The upper right inset shows the peak of the light curve, wit h the range of the simulated data points plotted by the thick line.

59 Fig. 2.2.— CMD of the OGLE-2004-BLG-343 field. Hipparcos main-sequence stars −0.15/5 (blue dots), placed at 10 R0 = 7.5 kpc and reddened by the reddening vector derived from the clump, are displayed with the OGLE-II stars (black dots). The Reid (1991) relation is plotted by the red solid line over the Hipparcos stars. On the CMD, the magenta filled circle is the red clump and the green filled circle is the blended star. The large black filled circle is the OGLE V measurement of the source with of 1σ error bars, which sets a lower limit for the source V I color. The magenta filled circle with error bars is the result of combining the (I −H)/(V I) information (see Fig. 2.3) with the OGLE measurement. − −

60 Fig. 2.3.— (VOGLE IOGLE)/(IOGLE H2MASS) color-color diagram. All points are from matching 2MASS− H-band data with− OGLE-II V,I photometry in a ﬁeld centered on OGLE-2004-BLG-343. Stars on the giant branch are shown by open circles. The solid and dashed vertical lines represent the source IOGLE H2MASS color transformed from its I H value and its 1σ ranges. Their intersections− with the diagonal OGLE − µFUN track of stars give corresponding VOGLE IOGLE colors, which are represented by the horizontal lines. −

61 Fig. 2.4.— Likelihood contours (1σ, 2σ, 3σ) for ﬁnite-source points-lens models −1 relative to the best-ﬁt PSPL model. Contours with x-axis as log u0 and log Amax are displayed in solid and dashed lines, respectively.

62 b-b(posterior) b-d(posterior)

bigger finite-source effects

b-b(prior) b-d(prior)

prior posterior

Fig. 2.5.— Probability distributions of u0, dereddened apparent I-band magnitude of the source I0, proper motion µ, log(z0), source distance modulus, lens distance modulus, absolute I-band magnitude of the source MI , and lens mass for Monte Carlo events toward the line of sight of OGLE-2004-BLG-343. Blue histograms represent the posterior probability distributions for bulge-disk microlensing events while red ones represent the posterior probability distributions for bulge-bulge events. In the source and lens distance-modulus panels, histograms in purple and green represent the prior probability distributions for bulge-disk and bulge-bulge events, respectively. The black Gaussian curves in the u0 and I0 panels show probability distributions from PSPL light-curve ﬁtting alone. In the lens mass panel, the dark green histogram shows the prior probability distribution, while the orange histogram represents the posterior distribution.

63 Fig. 2.6.— (For real data) Planetary detection efficiency for mass ratios q = 10−3, 10−4, and 10−5 for OGLE-2004-BLG-343 as a function of the planet-star separation bx = b cos α and by = b sin α in the units of θE where α is the angle of planet-star axis relative× to the source-lens× direction of motion. Different colors indicate 10% (red), 25% (yellow), 50% (green), 75% (cyan), 90% (blue) and 100% (magenta) efficiency. The black circle is the Einstein ring, i.e., b = 1.

64 Fig. 2.7.— (For real data) Planet detection eﬃciency of OGLE-2004-BLG-343 as a function of the planet-star separation b (in the units of rE) and planet-star mass ratio q. The contours indicate 25%, 50%, 75%, and 90% eﬃciency.

65 Fig. 2.8.— (For fake data) Planetary detection efficiency for mass ratios q = 10−3, 10−4, 10−5 and 10−6 of OGLE-2004-BLG-343 augmented by simulated data points covering the peak as a function of the planet-star separation bx and by in the units of θE. Different colors indicate 10% (red), 25% (yellow), 50% (green), 75% (cyan), 90% (blue) and 100% (magenta) efficiency. The black circle is the Einstein ring, i.e., b = 1.

66 Fig. 2.9.— (For fake data) Planetary detection eﬃciency of OGLE-2004-BLG-343 augmented by simulated data points covering the peak. as a function of the planet- star separation b (in the units of θE) and planet-star mass ratio q. The contours represent 25%, 50%, 75%, and 90% eﬃciency.

67 Fig. 2.10.— (For real data) Planetary detection eﬃciency as a function of r⊥, the physical projected star-planet distance and mp, the planetary mass for OGLE-2004- BLG-343. The contours represent 25%, 50%, 75%, and 90% eﬃciency.

68 Fig. 2.11.— (For fake data) Planetary detection eﬃciency as a function of r⊥, the physical projected star-planet distance and mp, the planetary mass for OGLE- 2004-BLG-343 augmented by simulated data points covering the peak. The contours represent 25%, 50%, 75%, and 90% eﬃciency.

69 Fig. 2.12.— Planetary detection eﬃciency as a function of r⊥, the physical projected star-planet distance, and mp, the planetary mass for OGLE-2004-BLG-343, by assuming that the blended light is due to the lens. The contours represent 25%, 50%, 75%, and 90% eﬃciency.

70 2 t0(HJD’) u0 tE(days) Is Ib χ Degree of Freedom 71

3175.7467 0.0005 0.000333 0.000121 42.5 15.6 22.24 0.40 18.08 0.01 200.1 188 ± ± ± ± ±

Table 2.1. OGLE-2004-BLG-343 Best-Fit PSPL Model Parameters Chapter 3

Microlensing Event MOA-2007-BLG-400: Exhuming the Buried Signature of a Cool, Jovian-Mass Planet

3.1. Introduction

In the currently favored paradigm of planet formation, the location of the snow line in the protoplanetary disk plays a pivotal role. Beyond the snow line, ices can condense, and the surface density of solids is expected to be higher by a factor of several relative to its value just inside this line. As a result of this increased surface density, planet formation is expected to be most eﬃcient just beyond the snow line, whereas for increasing distances from the central star the planet formation eﬃciency drops, as the surface density decreases and the dynamical time increases

(Lissauer 1987). In this scenario, gas-giant planets must form in the region of the protoplanetary disk immediately beyond the snow-line, as the higher surface density is required to build icy protoplanetary cores that are suﬃciently massive to accrete a substantial gaseous envelope while there is remaining nebular gas (Pollack et al.

1996). Low-mass primaries are expected to be much less eﬃcient at forming gas

72 giants because of the longer dynamical times and lower surface densities at the snow lines of these stars (Laughlin et al. 2004; Ida & Lin 2005; Kennedy & Kenyon 2008).

Migration due to nebular tides and other dynamical processes can then bring the icy cores or gas giants from their formation sites to orbits substantially interior to the snow line (Lin et al. 1996; Ward 1997; Rasio & Ford 1996).

The precise location of the snow line in protoplanetary disks is a matter of some debate (e.g., Lecar et al. 2006), and is even likely to evolve during the epoch of planet formation, particularly for low-mass stars (Kennedy et al. 2006; Kennedy

& Kenyon 2008). The condensation temperature of water is 170 K, and a ﬁducial ∼ value for the location of the snow line in solar-mass stars motivated by observations in our solar system is 2.7 AU. This may scale linearly with the stellar mass M, ∼ since the stellar luminosity during the epoch of planet formation scales as M 2 ∼ for stars with M < M⊙ (Burrows et al. 1993, 1997). Whereas the radial velocity ∼ and especially transit methods are most sensitive to planets that are close to their parent star at distances well inside the snow line, the sensitivity of the microlensing method peaks at planetary separations near the Einstein ring radius of the primary lens (Mao & Paczy´nski 1991; Gould & Loeb 1992), which is 3.5 AU(M/M )1/2 for ∼ ⊙ typical lens and source distances of 6 kpc and 8 kpc, respectively. This corresponds to a peak sensitivity at equilibrium temperatures of T 150 K(M/M ) for a eq ∼ ⊙ mass-luminosity relation of the form L M 5, and distances relative to the snow line ∝ of 1.3(M/M )−1/2 if the location of the snow line at the epoch of planet formation ∼ ⊙

73 scales as M. Thus microlensing is currently the best method of probing planetary systems in the critical region just beyond the snow line (Gould & Loeb 1992).

Planetary perturbations in microlensing events come in two general classes. The majority of planetary perturbations are expected to occur when a planet directly perturbs one of the two images created by the primary lens, as the image sweeps by the planet during the microlensing event (Gould & Loeb 1992). Although these perturbations are more common, they are also unpredictable and can occur at any time during the event. Early microlensing planet searches focused on this class of perturbations, as it was the ﬁrst to be identiﬁed and explored theoretically (Gould

& Loeb 1992; Bennett & Rhie 1996; Gaudi & Gould 1997). The second class of planetary perturbations occurs in high-magniﬁcation events, in which the source becomes very closely aligned with the primary lens (Griest & Saﬁzadeh 1998).

In such events, the two primary-lens images become highly distorted and sweep along nearly the entirety of the Einstein ring (Liebes 1964). These sweeping images probe subtle distortions of the Einstein ring caused by nearby planets, which will give rise to perturbations within the full-width half-maximum of the event (Bond et al. 2002; Rattenbury et al. 2002). Although high-magniﬁcation events are rare and so contribute a minority of the planetary perturbations, they are individually more sensitive to planets because the images probe nearly the entire Einstein ring.

Furthermore, since the perturbations are localized to the peak of the event which

74 can be predicted beforehand, they can be monitored more eﬃciently with limited resources than the more common low-magniﬁcation events.

For these reasons, current microlensing planet searches tend to deliberately focus on high-magniﬁcation events. Thus, of the seven prior microlensing planets discovered to date (Bond et al. 2004; Udalski et al. 2005; Beaulieu et al. 2006;

Gould et al. 2006; Gaudi et al. 2008; Bennett et al. 2008), five have been found in high-magnification events, with peak magnifications ranging from A = 40 to

A = 800. However, despite the fact that this planet-search strategy has proven to be so successful, the properties of the planetary perturbations generated in high-magniﬁcation events are less well-understood than those in low-magniﬁcation events.

Most of the studies of the properties of planetary perturbations in high- magnification events have focused on the properties of the caustics, the locus of points defining one or more closed curves, upon which the magnification of a point source is formally infinite. The morphology and extent of the region of significant perturbation by the planetary companion can be largely understood by the shape and size of these caustic curves. Planetary perturbations in high-magnification events are caused by a central caustic located near the position of the primary, and thus several authors have considered the size and shape of these central caustics as a function of the parameters of the planet (Griest & Safizadeh 1998; Dominik 1999b; Chung et al. 2005). These and other authors have identified several potential degeneracies

75 that complicate the unique interpretation of central caustic perturbations. The

first to be identified is a degeneracy such that the caustic structure (and so light curve morphology) of a planet with mass ratio q 1 and projected separation in ≪ units of the Einstein ring d not too close to unity is essentially identical under the transformation d d−1 (Griest & Safizadeh 1998). A second degeneracy arises ↔ from the fact that very close or very wide roughly equal-mass binaries also produce perturbations near the peak of the light curves. These give rise to perturbations that have the same gross observables as planetary perturbations.

The severity of these degeneracies depends on both the speciﬁc parameters of the planetary/binary companion, as well as on the data quality and coverage.

Griest & Saﬁzadeh (1998) and Chung et al. (2005) demonstrated that the d d−1 ↔ degeneracy is less severe for more massive planets with separations closer to the

Einstein ring (d 1). Empirically, this degeneracy was broken at the ∆χ2 4 level ∼ ∼ for the relatively large mass-ratio planetary companion OGLE-2005-BLG-071Lb

(Dong et al. 2009), for which the light curve was well-sampled, but was essentially unresolved for the low mass-ratio planetary companion MOA-2007-BLG-192Lb

(Bennett et al. 2008), for which the planetary perturbation was poorly sampled.

For the planetary/equal-mass binary degeneracy, Han & Gaudi (2008) argued that, although the gross features of central caustic planetary perturbations can be reproduced by very close or very wide binary lenses, the morphologies diﬀer in detail, and thus this degeneracy can be resolved with reasonable light curve coverage and

76 moderate photometric precision. Indeed for every well-sampled high-magniﬁcation event containing a perturbation near the peak (and that is not in the Chang-Refsdal

(1979) limits), this degeneracy has been resolved (Albrow et al. 2002; Gould et al. 2006; Dong et al. 2009). Even for the relatively poorly-sampled light curve of

MOA-2007-BLG-192Lb, an equal-mass binary model is ruled out at ∆χ2 120 ∼ (Bennett et al. 2008), although in this case this is partially attributable to the exquisite photometric precision (< 1%).

One complication with searching for planets in high-magnification events is that, the higher the magnification, the more likely it is that the primary lens will transit the source. When this happens, the peak of the event is suppressed and smoothed out, as the lens strongly magnifies only a small portion of the source.

If the source is also larger than the region of signiﬁcant perturbations due to a planetary companion (roughly the size of the central caustic), then the planetary deviations will also be smoothed out and suppressed (Griest & Saﬁzadeh 1998;

Han 2007). These finite source effects have potential implications for both the detectability of central caustic perturbations, as well as the ability to uniquely determine the planetary parameters, and in particular resolve the two degeneracies discussed above. In practice, the caustic structures of all four high-magnification planetary events (containing five planets) were larger than the source. Hence, while there were detectable finite-source effects in all cases (which helped constrain the angular Einstein radius and so the physical lens parameters), the planetary

77 perturbations were in all four cases quite noticeable. Thus the eﬀect of large sources on the the detectability and interpretation of central-caustic perturbations has not been explored in practice.

Theoretical studies of detectability of central caustic perturbations when considering finite source effects have been performed by Griest & Safizadeh (1998),

Chung et al. (2005), and Han (2007). These authors demonstrated that the qualitative nature of planetary perturbations from central caustics is dramatically different for sources that are larger than the caustic. In particular, the detailed structure of the point-source magnification pattern, which generally follows the shape of the caustic, is essentially erased or washed out. Rather, the perturbation structure is characterized by a roughly circular region of very low-level, almost imperceptible deviation from the single-lens form that is roughly the size of the source and centered on the primary lens. This region is surrounded by an annular rim of larger deviations that has a width roughly equal to the width of the caustic (Griest & Safizadeh 1998;

Chung et al. 2005). Finally, there are less pronounced deviations that extend to a few source radii. Planets are detectable even if their central caustics are quite a bit smaller than the source, provided that the χ2 deviation is suﬃciently high. (Often

∆χ2 > 60 is adopted, although ∆χ2 > 150 may be more realistic.) The magnitude of these perturbations decrease as the ratio between the source size and caustic size increases, making it diﬃcult to detect very small planets for large sources (Chung et al. 2005; Han 2007).

78 Although central caustics may formally be detectable when the source is substantially larger than the caustic, it remains a significant question whether these very washed out caustics can be recognized in practice, and even if they can, whether they can be uniquely interpreted in terms of planetary parameters. Indeed, it is unknown whether a washed out central caustic due to a planet can actually be distinguished from one due to a binary companion. This question is especially important with regard to low-mass planets. The size of the central caustic scales as the product of the planet/star mass ratio and a definite function of planet-star separation. Hence, taken as a whole, smaller planets produce smaller caustics, meaning that events of higher magnification are required to detect them. These are just the events that are most likely to have their peaks washed out by finite-source effects.

Here we analyze the ﬁrst high-magniﬁcation event with a buried signature of a planet, in which the source size is larger than the central caustic of the planet.

The caustic is indeed so washed out that the event appears unperturbed upon casual inspection. However, the residuals to a point-lens ﬁt are clear and highly signiﬁcant. We show that one can infer the planetary (as opposed to binary) nature of the perturbation from the general pattern of these residuals, and that a detailed analysis constrains the mass ratio of the planet quite well, but leaves the close/wide

(d d−1) degeneracy intact. Hence, at least in this case, the fact that the caustic is ↔

79 buried in the source does not signiﬁcantly hinder one’s ability to uncover the planet and measure its mass ratio.

3.2. Observations

MOA-2007-BLG-400 [(α, δ) = (18h09m41s.98, 29◦13′26.95′′), J2000.0 − (l,b)=(2.38◦, 4.70◦)] was announced as a probable microlensing event by the − Microlensing Observations in Astrophysics (MOA) collaboration on 5 Sept 2007

(HJD′ HJD - 2450000 = 4349.1), about 5 days before peak. The source star proves ≡ to be a bulge subgiant and so is somewhat brighter than average, but the event timescale was relatively short (t 15 days) and observations had been interrupted E ∼ for 6 days by bad weather. Taken together, the two latter facts account for the relatively late alert. By coincidence, the triggering observations took place on the same night that another event, OGLE-2007-BLG-349 (aka MOA-2007-BLG-379), was peaking at extremely high magnification with an already-obvious planetary anomaly. After focusing exclusively on the latter event for the first 5 hours of the night, MOA resumed its normal field rotation for the last 1.5 hours, which led to the discovery of MOA-2007-BLG-400.

The Microlensing Follow Up Network (µFUN) began observing this as a possible high-magniﬁcation event on 7 Sept, but did not mobilize intensive observations until

UT 08:55 10 Sept, just 15 hours before peak, following a high-mag alert issued

80 by MOA a few minutes earlier. Even at that point, the predicted minimum peak magniﬁcation was only Amax > 90, which would have enabled only modest sensitivity to planets. Nevertheless, all stops were pulled and it was observed as intensively as possible from 7 observatories, µFUN CBA Perth (Australia) 0.25m unﬁltered, µFUN

Bronberg (South Africa) 0.35m unﬁltered, µFUN SMARTS (CTIO, Chile) 1.3m V ,

I, H, µFUN Campo Catino Austral (CAO, Chile) 0.50m unﬁltered, µFUN Farm

Cove (New Zealand) 0.35m unﬁltered, µFUN Auckland (New Zealand) 0.4m R, and

µFUN Southern Stars (Tahiti) 0.28m unﬁltered.

The source star lies just outside one of the OGLE ﬁelds (as deﬁned by their

field templates) and for this reason was not recognized as a microlensing event by the OGLE Early Warning System. However, due to small variations in pointing, there are a total of 452 OGLE images containing this source. Only two of these are significantly magnified, ten days and nine days before peak. Hence, the OGLE data do not help constrain the light curve parameters. However, they are useful to study of the baseline behavior of the source (see Appendix D).

Essentially all of the “action”, both the peak of the event and the planetary anomalies, occurred during the µFUN SMARTS (Chile) observations at CTIO,

4354.47 < HJD′ < 4354.69, using the ANDICAM optical/IR dual-channel camera, and µFUN CAO (Chile) observations 4354.50 < HJD′ < 4354.70. Most (45) of the optical CTIO observations over the peak were carried out in I band, with a few (8) taken in V in order to measure the (V I) color. Each of these was a 5 minute − 81 exposure, with approximately 1 minute read-out time between exposures. During each optical exposure, there were 5 dithered H-band exposures, each of 50 seconds, almost equally spaced over the 6 minute cycle time. That is, 53 5 = 265 H-band × observations in all. Unfortunately, the source became so bright as it transited the lens (i.e., when the planetary anomalies were the strongest), that 14 I-band images were aﬀected by non-linearities and saturation in the detector response. We exclude these 14 I-band data points from analysis. The H-band photometry are not aﬀected by this problem, therefore, with higher time resolution and more continuous coverage than the I-band data, the H-band data provide most of the constraining power to the microlens model.

There were 84 CAO un-ﬁltered observations taken during the peak night.

Unfortunately, the peak of the event was severely saturated and the clock zero point is not securely known, therefore, these data are not used in the analysis. However, the exposures during the times of maximum deviation from a point lens (i.e., when the caustic was crossing the stellar limb) are not saturated, and these qualitatively conﬁrm the interpretation from the more detailed CTIO data.

The µFUN CTIO and MOA data are used in the analysis since they provide essentially all the constraints to the microlens model. Data from other µFUN sites are checked for consistency with the final models, and they are found to be well fitted by the best-fit model. The MOA data were reduced using the standard

MOA diﬀerence imaging analysis (DIA) pipeline. The µFUN CTIO data were

82 reduced using DIA developed by Wo´zniak (2000). The H-band data are aﬀected by intrapixel sensitivity variations at the 1% level. Fortunately, the dither pattern was repeated almost exactly over the night of the peak, so that these variations follow the 5-element dither pattern quite well. We therefore treat the H-band data as 5 independent data sets, which reduces χ2 by 180 for 8 degrees of freedom. The

H-band images show a triangular PSF, which is likely to introduce systematic errors into the photometry. As a crosscheck, we also use the DIA package developed by

Bond et al. (2001) to independently reduce these images.

3.3. Microlens Model

Despite the fact that the peak of MOA-2007-BLG-400 was “flattened” by finite- source effects, it nevertheless reached a very high peak magnification, Amax = 628.

However, even to the experienced eye, it looks like an ordinary point-lens light curve with pronounced ﬁnite-source eﬀects. More detailed modeling is required to infer that it actually contains a Jovian mass-ratio planet.

Figure 3.1 shows the light curve together with the best-ﬁt point-lens model

(blue) and planetary model (red). Both include ﬁnite-source eﬀects. The most pronounced features of the point-lens model residuals are a short positive spike on the rising side and a short negative spike on the falling side, each lasting about 30 minutes, which leave very similar traces in I and H. As displayed in Figure 3.2, these

83 features clearly stand out in the reductions using both the Wozniak (top panel) and

Bond (bottom panel) DIA packages. Each package introduces its own systematic deviations, but there are no obvious trends besides the above features that are supported by both reductions. These occur very close to the times that the point lens begins and ends its transit of the source (within the framework of this model).

The timing of these deviations strongly suggests that they are due to microlensing rather than stellar variability. There are then two possible explanations: either the source is actually being transited by a more complicated caustic than a point lens

(due to a binary or planetary companion) or the limb of the source is not being properly modeled. However, if one assumes a circular source, the latter explanation would imply symmetric residuals, whereas the actual residuals are closer to being antisymmetric. (We address the possibility of an elliptical source in Appendix D.)

Indeed, this approximate antisymmetry extends to the less pronounced residual features, including the sustained deﬁcit prior to the ﬁrst spike (and sustained excess following the second one) as well as the declining residuals between the two spikes.

The similarity of the I and H residuals in itself argues that the deviations are due to microlensing rather than some sort of stellar variability, which would not generally be expected to be achromatic.

The short durations of the spikes tell us that the central caustic is quite small, with “caustic width” w 30 min/15 day = 1.4 10−3. This implies that ∼ × the companion either has low mass ratio, or is a very wide or very close binary

84 companion. Formally, w is given by equation (12) of Chung et al. (2005) as a very good approximation to the “short diameter” or “width” of a central caustic (see

Fig. 3.3). However, here the estimate is quite inexact not only because the width of the spikes is not precisely deﬁned, but also because we do not know, at this point, the exact orientation of the caustic.

After some algebra, one ﬁnds that in two limiting regimes, the Chung et al.

(2005) formula takes the forms,

4q w(d,q) (d 1), w(d,q) 4qd2 (d 1), (3.1) → d2 ≫ → ≪ and

27 q w(d,q) (d 1), (3.2) → s16 d 1 ∼ | − | where q is the companion/primary mass ratio and d is the separation in units of the Einstein ring. Note that in the ﬁrst limit (eq. [3.1]), w 4γ, where γ is the → shear. The crossover point for these approximations is d = 2.3 (or d = 0.43), at which point each is in error by about 15%. (For simplicity, we restrict the discussion here to the case d > 1. There is a well-known d d−1 degeneracy between the ↔ d 1 and d 1 limits, as can be guessed from the forms for w in eq. [3.1]. This ≪ ≫ degeneracy will prove to be almost perfect in this case, see 3.3.3). Hence, in the § ﬁrst limit, d 50q1/2, implying that if q were in the “binary range” ( log q < 1), ∼ | | then d would be quite large. That is, the central caustic would be generated by a

85 nearly pure shear and therefore would have a nearly symmetric, diamond-shaped,

Chang-Refsdal (1979) form. In the point-lens model, the lens passes almost directly over the center of the source. For this trajectory, a symmetric caustic would yield symmetric residuals, in sharp contrast to Figure 3.1. On the other hand, for the opposite limit, q 1 10−3(d 1), which lies squarely in the planetary regime. ∼ × − Thus, simple arguments already argue strongly in favor of a planetary companion that is fairly near the Einstein ring.

3.3.1. Hybrid Pixel/Ray Map Algorithm

Notwithstanding these arguments, we conduct a massive blind search for companions over a very broad range of masses using a modiﬁed version of the

“magnification map” technique of Dong et al. (2006), which was specifically designed for high-magnification events. The original approach was, for each given (d,q) pair, to shoot rays over a fairly narrow annulus (say, 0.01 Einstein radii) around the Einstein ring in the image plane and to sort these rays in pixels on the source plane. Then for each source position being modelled (i.e., each data point), one would identify the pixels that intersected the source and would check each ray contained in these pixels to determine whether it landed on the source and, if so, evaluate the source surface brightness at that position. In the initial broad search, three parameters (d,q,α) are held fixed on a grid of values, while the remaining

2 parameters (t0, u0,tE,ρ, and possibly others) are varied to minimize χ at each grid

86 point. Here, α is the angle of the source trajectory relative to the binary axis, t0 is the time of closest approach to the adopted center of the lens geometry (usually the center of mass), u0 is the source-lens separation at this time in units of the Einstein radius, ρ is the source radius in the same units, and tE is the Einstein crossing time.

This division is efficient because 1) (t0, u0,tE,ρ) are usually approximately known from the general structure of the light curve, so minimization over these parameters is straightforward once (d,q,α) are fixed; 2) (d,q) define the map, which naturally facilitates minimization of other parameters except for α, whose value is not usually even approximately obvious from the light curve.

The new approach diﬀers principally in that the pixels that are contained entirely within the source are now evaluated as a whole, i.e., by the total number of rays in that pixel. Pixels that cross the source boundary are still evaluated ray-by-ray, as previously. This primary change then leads to several other changes.

First, for each pixel, we record not only the geometric center but also the centroid of the rays. The surface brightness is then evaluated at the latter position. Second, the pixels are made much smaller, to minimize both the number of rays that must be evaluated individually and the surface-brightness variations across the pixel

(which are corrected only to ﬁrst order by the ray-centroid scheme just mentioned).

Typically, there are a few hundreds of pixels per source. Third, the pixels are hexagonal, since this is the most compact tiling possible, i.e., the closest tile shape to a circle. Fourth, the source positions outside the map region are evaluated using

87 the hexadecapole and quadrupole approximation of Gould (2008) (see also Pejcha &

Heyrovsky 2008). Finally, we use Markov chain Monte Carlo (MCMC) for the χ2 minimization.

3.3.2. The (w, q) Grid of Lens Geometries

The initial search for solutions is conducted over a rectilinear grid in (w,q) rather (d,q). Since the short diameter, or “caustic width”, w, is a monotonic function of the star-planet separation d (at fixed planet/star mass ratio q), these formulations are in some sense equivalent. However, for many events (including the present one) the short diameter w can be estimated by simple examination of the data. In these cases, the search space is both more regular and easier to define in terms of the (w,q) grid. In particular, equation (3.2) shows that at fixed w, d moves very close to 1 for very low q.

3.3.3. Best-Fit Model

We consider short diameters w over the range 3.5 log w 2 and − ≤ ≤ − companion mass ratios 4 log q 0, focusing on the regime d 1. We ﬁnd that − ≤ ≤ ≥ there is only one local minimum in this range. The range of allowed solutions is well localized around this minimum, with

q = (2.5+0.5) 10−3, d = 2.9 0.2, w = (1.30 0.06) 10−3, (3.3) −0.3 × ± ± × 88 with the last quantity being, of course, dependent on the ﬁrst two. Figure 3.4 shows the ∆χ2 = 1, 4, 9 contours with respect to the mass ratio q and the projected planet-star separation d along with the short diameter w. The main point to note is that these parameters are quite well constrained. Note that, as expected, d and q are strongly correlated, while w and q are basically uncorrelated. We also perform similar searches using the alternative H-band reduction by Bond’s DIA package. The solutions agree with the above to well within one sigma, but the parameters have larger uncertainties: q = (2.6 0.7) 10−3,d = 2.9 0.3,w = (1.26 0.09) 10−3. ± × ± ± × We therefore adopt results from the Wozniak-based reductions, noting that they may subject to systematic errors < 1σ. ∼

Figure 3.3 shows the source trajectory and the central caustics as well as the differences in magnification between the best-fit planetary model and its corresponding single-lens model. This geometry nicely accounts for the main features of the point-lens residuals seen in Figure 3.1. The regions beyond the “back walls”

(long segments) of the caustic are somewhat de-magnified, which accounts for the initial depression of the light curve. As the source crosses the “back wall” of the caustic, it spikes. After the source has exited the caustic, it continues to suffer additional magnification due to the “ridge” of magnification that extends from the trailing cusp.

We also conducted a similar blind search as above, but concentrating on the regime d < 1. As expected, we recover the well-known d d−1 degeneracy, and ↔ 89 ﬁnd a solution with essentially the same mass ratio q = (2.6 0.4) 10−3, but with ± × −1 +0.03 2 d = 2.9 = 0.34−0.02, and the wide solution is slightly preferred by ∆χ = 0.2.

Thus, although each solution is well-localized to its respective minimum, this discrete degeneracy implies that the projected separation can take on two values that diﬀer by a factor of 8.5. The severity of the degeneracy can be traced to ∼ the planetary parameters. Although the planet/star mass ratio is quite large, which tends to reduce the severity of the degeneracy, the planet lies quite far from the

Einstein ring, which tends to make it more severe. Actually, a better measure of the overall expected asymmetry between the d and d−1 solutions is the short diameter w, which in this case is small, implying a severe degeneracy. Indeed, the caustic structure and magnification pattern of the two solutions are nearly identical. In this case, the large size of the source has competing influences on the ability to resolve the degeneracy. On one hand, the large size of the source serves to suppress the planetary deviations, thus making subtle differences more difficult to distinguish.

On the other hand, the large source implies that a large fraction of the planetary perturbation region is probed. In this case, the source probes essentially the entire region of significant planetary perturbation, as can be seen in Figure 3.3. This is important for distinguishing between the solutions, as the largest difference between the magnification patterns of the two degenerate solutions occurs in the region near the tip of the arrow-shaped caustic (Griest & Safizadeh 1998). From Figure 3.3

90 it is clear that this region would have been entirely missed if the source had been substantially smaller than the caustic.

3.4. Finite-Source Effects

In addition to (d,q), the model also yields the source radius relative to the

Einstein radius,

ρ = θ /θ = (3.29 0.08) 10−3, (3.4) ∗ E ± ×

We then follow the standard (Yoo et al. 2004a) technique to determine the angular source radius,

θ = 1.05 0.05 µas. (3.5) ∗ ±

That is, we ﬁrst adopt [(V I) ,I ] = (1.00, 14.32) for the dereddened position − 0 0 clump of the clump. We then measure the oﬀset of the source relative to the clump centroid

∆[(V I),I]=( 0.19, 3.25), to obtain [(V I) ,I ] = (0.81, 17.57). See Figure − − − 0 0 s 3.5. The instrumental source color is derived from model-independent regression of the V and I ﬂux, while the instrumental magnitude is obtained from the light-curve model. We convert (V I) to (V K) using the color-color relations of Bessell & − − Brett (1988), yielding (V K) = 1.75, and then obtain equation (3.5) using the − 0 color/surface-brightness relations of Kervella et al. (2004). Combining equations

91 (3.4) and (3.5) gives θE = θ∗/ρ = 0.32 mas. And combining this with the deﬁnition

2 θE = κMπrel, where M is the lens mass, πrel is the source-lens relative parallax, and κ = 4G/c2AU 8.1 mas M −1, together with the measured Einstein timescale, ∼ ⊙ t = 14.3 0.3days, we obtain E ±

−1 πrel M = 0.10 M⊙ (3.6) 125 µas

and

θE −1 µrel = = 8masyr . (3.7) tE

The relatively high lens-source relative proper motion µrel is mildly suggestive of a foreground disk lens, but still quite consistent with a bulge lens. Since πrel = 125 µas corresponds to a lens distance DL = 4 kpc (assuming source distance DS = 8kpc), equation (3.6) implies that if the lens did lie in the foreground, then it would be a very low-mass star or a brown dwarf.

Assuming that the source lies at a Galactocentric distance modulus 14.52, its dereddened color and magnitude imply that [(V I) , M ]=(0.81, 3.07), making it − 0 I a subgiant.

92 3.5. Limb Darkening

As illustrated in Figure 3.1, the principal deviations from a point-lens light curve occur at the limb of the star. This prompts us to investigate the degree to which the planetary solution is influenced by our treatment of limb darkening. The results that we report are based on a fit to the H-band surface brightness profile of the form

S(ϑ) 3 5 1/2 ¯ = 1 Γ 1 cos ϑ Λ 1 cos ϑ , (3.8) Sλ − − 2 − − 4 where Γ and Λ are the linear and square-root parameters, respectively, and where

ϑ is the angle between the normal to the stellar surface and the line of sight, i.e., sin ϑ = θ/θ∗. See An & Han (2002) for the relation between (Γ, Λ) and the usual

(c,d) formalism.

In deriving the reported results, we ﬁx the H-band limb-darkening parameters

(Γ, Λ) = ( 0.15, 0.69), corresponding to (c,d)=( 0.21, 0.79) from Claret (2000) − − for a star with effective temperature Teff = 5325 K and log g = 4.0. These stellar parameters are suggested by comparison to Yale-Yonsei isochrones (Demarque et al. 2004) for the dereddened color and absolute magnitude reported in 3.4. We § also perform fits in which Γ and Λ are allowed to be completely free. From these

fits, we find that our best-fit model has (Γ, Λ) = ( 0.64, 1.47). Γ and Λ are highly − correlated, so their individual values are not of interest, and the surface-brightness

93 profiles generated by these two sets of (Γ, Λ) are qualitatively similar. In the present context, however, the key point is that when we fix the limb-darkening parameters at the Claret (2000) values, the contours in Figure 3.4 remain essentially identical and the best fit values change by much less than 1 σ.

Because of lower point-density and the aforementioned problems with the

I data over the peak, we only attempt a linear limb-darkening ﬁt, i.e., we use equation (3.8) with Λ 0, and we adopt Γ = 0.47 from Claret (2000). ≡

3.6. Blended Light

In the crowded ﬁelds of the Galactic bulge, the photometered light of a microlensing event rarely comes solely from the lensed source. Rather there is typically additional light that is blended with the source but is not being lensed.

This light can arise from unrelated stars that happen to be projected close enough to the line of sight to be blended with the source, or it can come from companions to the source, companions to the lens, or the lens itself. This last possibility is most interesting because, if the lens ﬂux can be isolated and measured, it provides strong constraints on the lens properties, and in this case would enable a complete solution of the lens mass and distance, when combined with the measurement of θE (e.g.,

Bennett et al. 2007).

94 To investigate the blended light, we begin by using the method of Gould &

An (2002) to construct an image of the ﬁeld with the source (but not the blended light) removed, and compare this to a baseline image, which of course contains both the source and the blended light (see Fig. 3.7). In these images, the source/blend is immersed in the wings of a bright star (roughly 3.7 mag brighter than the source), which lies about 2′′ away. On the baseline image, the source/blend is noticeable against this background, but hardly distinct. On the source-subtracted image, the blend is not directly discernible.

To make a quantitative estimate of the blend flux, we fit the region in the immediate vicinity of the bright star to the form F = a + a PSF, where “PSF” 1 2 × is the point-spread-function determined from the DIA analysis. We then subtract the best fit flux profile from the image. This leaves a clear residual at the position of the source/blend in the baseline image, but just noise in the source-subtracted image. We add all the flux in a 1.8′′ square centered on the lens, finding 564 ADU and 28 ADU, respectively. We conclude that the I-band flux blend/source ratio is − fb/fs < 0.05. Of course, even if we had detected blended light, it would be impossible to tell whether it was directly coincident with the source. If it were, this would imply that this light would be directly associated with the event, i.e., being either the lens itself or a companion to the lens or the source. Hence, this measurement is an upper limit on the light from the lens in two senses: no light is definitively measured, and if it were we do not know that it came from the lens. Combining this limit with

95 equation (3.6), and assuming the lens is a main-sequence star, it must then be less massive than M < 0.75 M⊙, and so must have relative parallax πrel > 15 mas. This implies a lens-source separation D D > 1 kpc, which certainly does not exclude S − L bulge lenses. Indeed, if the lens were a K dwarf in the Galactic bulge, it would saturate this limit.

3.7. Discussion

MOA-2007-BLG-400 is the ﬁrst high-magniﬁcation microlensing event for which the central caustic generated by a planetary companion to the lens is completely enveloped by the source. As a comparison, the planetary caustic of OGLE-2005-

BLG-390 (Beaulieu et al. 2006) is smaller than its clump-giant source star in angular size. When the planetary caustics is covered by the source, the ﬁnite-source eﬀects broaden the “classic” Gould & Loeb (1992) planetary perturbation features (Gaudi

& Gould 1997). By contrast, planet-induced deviations in MOA-2007-BLG-400 are mostly obliterated, rather than being broadened, because the source crosses the central caustic rather than the planetary caustic. We showed, nevertheless, that the planetary character of the event can be inferred directly from the light-curve features and that the standard microlensing planetary parameters (d,q) = (2.9, 2.5 10−3) × can be measured with good precision, up to the standard close/wide d d−1 ↔ degeneracy. We demonstrated that, in this case, the close/wide degeneracy is quite

96 severe, and the wide solution is only preferred by ∆χ2 = 0.2. This is unfortunate, since the separations of the two solutions diﬀer by a factor of 8.5. We argued that ∼ the severity of this degeneracy was primarily related to the intrinsic parameters of the planet, rather than being primarily a result of the large source size.

Although the mass ratio alone is of considerable interest for planet formation theories, one would also like to be able to translate the standard microlensing parameters to physical parameters, i.e., the planet mass mp = qM, and planet-star projected separation r⊥ = dθEDL. Clearly this requires measuring the lens mass

M and distance DL. In this case, the pronounced ﬁnite source eﬀects have already permitted a measurement of the Einstein radius θE = 0.32 mas, which gives a relation between the mass and lens-source relative parallax (eq. [3.6]). This essentially yields a relation between the lens mass and distance, since the source distance is close enough to the Galactic center that knowing DL is equivalent to knowing πrel.

Therefore, a complete solution could be determined by measuring either M or DL, or some combination of the two.

One way to obtain an independent relation between the lens mass and distance is to measure the microlens parallax, πE. There are two potential ways of measuring

πE. First, one can measure distortions in the light curve arising from the acceleration of the Earth as it moves along its orbit. Unfortunately, this is out of the question in this case because the event is so short that these distortions are immeasurably small. Second, one can measure the eﬀects of terrestrial parallax, which gives

97 rise to diﬀerences between the light curves simultaneously observed from two or more observatories separated by a signiﬁcant fraction of the diameter of the Earth.

Practically, measuring these differences requires a high-magnification event, which would appear to make this event quite promising. Unfortunately, although we obtained simultaneous observations from two observatories separated by several hundred kilometers during the peak of the event, one of these datasets suffers from large systematic errors and an unknown time zero point, rendering it unusable for this purpose.

The only available alternative for breaking the degeneracy between the lens mass and distance would be to measure the lens flux, either under the glare of the source or, at a later date, to separately resolve it after it has moved away from the line of sight to the source (Alcock et al. 2001a; Kozlowski et al. 2007). Panels (e) and (f) of Figure 3.6 show the Bayesian estimates of the lens brightness in I-band and H-band, respectively. If the lens flux is at least 2% of the source flux, then the former kind of measurement could be obtained from a single epoch Hubble Space

Telescope observation, provided it were carried out in the reasonably near future. At roughly 99% probability, the blended light would be either perfectly aligned with the source (and so associated with the event) or well separated from it. HST images can be photometrically aligned to the ground-based images using comparison stars with an accuracy of better than 1%. Hence, photometry of the source+blend would detect the blend, unless it were at least 4 mag fainter than the source. In principle,

98 the blend could be a companion to either the source or lens. Various arguments can be used to constrain either of those scenarios. We do not explore those here, but see

Dong et al. (2009). If the lens is not detectable by current epoch HST observations

(or no HST observations are taken), then it will be detectable by ground-based AO

H-band observations in about 5 years. This is because the lens-source relative proper

−1 motion is measured to be µrel = 8masyr , and the diﬀraction limit at H band on a 10m telescope is roughly 35 mas. If the lens proves to be extremely faint, then a wider separation (and hence a few years more time baseline) would be required.

In the absence of additional observational constraints, we must rely on a

Bayesian analysis to estimate the properties of the host star and planet, which incorporates priors on the distribution of lens masses, distances, and velocities

(Dominik 2006; Dong et al. 2006). This is a standard procedure, which we only brieﬂy summarize here. We adopt a Han & Gould (1995) model for the Galactic bar, a double-exponential disk with a scale height of 325 pc, and a scale length of 3.5 kpc, as well as other Galactic model parameters as described in Bennett &

Rhie (2002). We incorporate constraints from our measurement of the lens angular

Einstein radius θE and the event timescale, as well as limits on the microlens parallax and I-band magnitude of the lens. In practice, only the measurements of θE and tE provide interesting constraints on these distributions. In addition, we include the small penalty on the close solution, exp( ∆χ2/2), where the wide solution is favored −

99 by ∆χ2 = 0.2. For the estimates of the planet semimajor axis, we assume circular orbits and that the orbital phases and cos(inclinations) are randomly distributed.

The resulting probability densities for the physical properties of the host star, as well as selected properties of the planet, are shown in Figure 3.6. The

+0.19 Bayesian analysis suggests a host star of mass M = 0.30−0.12M⊙ at distance of

+0.6 DL = 5.8−0.8 kpc. In other words, given the available constraints, the host is most likely an M-dwarf, probably in the foreground Galactic bulge. Given that the planet/star mass ratio is measured quite precisely, the probability distribution for the planet mass is essentially just a rescaled version of the probability distribution

+0.49 for the host star mass. We ﬁnd mp = 0.83−0.31 MJup. The close/wide degeneracy is apparent in the probability distribution for the semimajor axis a. We estimate

+0.38 +3.2 aclose = 0.72−0.16 AU for the close solution, and awide = 6.5−1.2 AU for the wide

+28 solution. The equilibrium temperatures for these orbits are Teq.,close = 103−26 K and

T = 34 9 K for the close and wide solutions, respectively. eq.,wide ±

Thus our Bayesian analysis suggests that this system is mostly likely a bulge mid-M-dwarf, with a Jovian-mass planetary companion. The semimajor axis of the planetary companion is poorly constrained primarily because of the close/wide degeneracy, but the implied equilibrium temperatures are cooler than the condensation temperature of water. Specifically we find that Teq < 173 K at 2 σ ∼ level. Alternatively, if we assume the snow line is given by asnow = 2.7 AU(M/M⊙), we find for this system a snow line distance of 0.84 AU, essentially the same as ∼ 100 the inferred semimajor axis of the close solution. Thus this planet is quite likely to be located close to or beyond the snow line of the system.

Although we cannot distinguish between the close and wide solutions for the planet separation, theoretical prejudice in the context of the core-accretion scenario would suggest that a gas-giant planet would be more likely to form just outside the snow line, thus preferring the close solution. However, we have essentially no observational constraints on the frequency and distribution of Jupiter-mass planets at the separations implied by the wide solution ( 5.3 9.7 AU), for such low-mass ∼ − primaries. Unfortunately, the prospects for empirically resolving the close/wide degeneracy in the future are poor. The only possible method of doing this would be to measure the radial velocity signature of the planet. Given the faintness of the host star (see 3.6 and Fig. 3.6), this will likely be impossible with current or § near-future technology.

The mere existence of a gas-giant planet orbiting a mid-M-dwarf is largely unexpected in the core-accretion scenario, as formation of such planets is thought to be inhibited in such low-mass primaries (Laughlin et al. 2004). Observationally, however, although the frequency of Jovian companions to M-dwarfs with a < 3 AU ∼ does appear to be smaller than the corresponding frequency of such companions to FGK dwarfs (Endl et al. 2006; Johnson et al. 2007b; Cumming et al. 2008), several Jovian-mass companions to M dwarfs are known (see Dong et al. 2009 for a discussion), so this system would not be unprecedented. Furthermore, it must

101 be kept in mind that the estimates of stellar (and so planet) mass depend on the validity of the priors, and even in this context have considerable uncertainties.

Most of the ambiguities in the interpretation of this event would be removed with a measurement of the host star mass and distance, which could be obtained by combining our measurement of θE with a measurement of the lens light as outlined above. The Bayesian analysis informs the likelihood of success of such an endeavor.

This analysis suggests that, if the host is a main-sequence star, its magnitude will

+0.8 +0.7 be IL = 23.9−1.0 and HL = 21.4−1.0, which corresponds to 0.6% and 1.7% of the source ﬂux, respectively. If initial eﬀorts to detect the lens fail, more aggressive observations would certainly be warranted: microlensing is the most sensitive method for detecting planets around very low-mass stars simply because it is the only method that does not rely on light from the host (or the planet itself) to detect the planet. And given equation (3.6), even an M dwarf at the very bottom of the main sequence M = 0.08 M , would lie at D = 3.5 kpc and so would be H 24. ⊙ L ∼

102 HJD - 2454354.0

Fig. 3.1.— Top: Lightcurve of MOA-2007-BLG-400 with data from µFUN CTIO (Chile) simultaneously taken in H (cyan), I (DIA, black). Models are shown for a point lens (blue) and planet-star system (red). There are 5 50-second H exposures for each 300-second I (or V – not shown) exposure in 6 minutes cycles. Some I-band data at the peak suffer from saturation, and those points are therefore removed from the analysis (see text). Middle: Residuals for best-fit point-lens model and its difference with the planetary model. Note that in the top panel, the H data are shown as observed, while the I data are aligned. Normally, such alignment is straightforward because microlensing of point sources is achromatic. However, here there is significant chromaticity due to different limb-darkening. The I-band points in the top panel are actually the residuals to the I-band limb-darkened model (middle panel), added to the H-band model curve (top panel). Bottom: Residuals from a point-lens model with the same parameters as the planetary model, which can be directly compared to the “magnification map” in Fig. 3.3. These “didactic residuals” are naturally more pronounced than those from the best-fit point lens.

103 Fig. 3.2.— Comparisons of residuals to the best-fit point-lens models between two photometric reductions of µFUN H-band data, using the DIA packages developed by Wo´zniak (2000) (top) and Bond et al. (2001) (bottom). The red curve in each panel represents the difference of the best-fit planetary and point-lens models for that panel’s reduction. Both reductions agree on the main planetary features, but each package introduces its own systematics. For example, the systematic deviations from the planetary model at HJD 2454354.57 shown in the top panel are not supported by the reductions of the Bond’s∼ package. During 2454354.46 < HJD < 2454354.51, most images have low transparency (< 50%), which causes relatively large scatter in Bond’s DIA reductions. In comparison,∼ the reduction by Wozniak’s DIA has smaller scatter during this period. However, the data exhibit some low-level systematics, which are not supported by the other reduction.

104 Fig. 3.3.— Magnification differences between of best-fit planetary model [(q,d) = (0.0025, 2.9) and (q,d)=(0.0026, 0.34) being nearly identical] and single-lens models, in units of the measured source size, ρ = 0.0033 Einstein radii. Contours show 1%, 2%, 3%, and 4%, deviations in the positive (brown) and negative (blue) directions. Top panel: Single-lens geometry (t0, u0,tE) is taken to be the same as in the planetary model, with no finite source effects. Caustic (contour of infinite magnification) is shown in white. The deviations are very pronounced. Bottom Panel: Same as top panel, but including finite-source effects, which now explain the main features of the light curve. The trajectory begins with a negative deviation, then hits a narrow “brown ridge” causing the spike seen in the bottom panel of Fig. 3.1, as the edge of the source first hits the caustic. Then there are essentially no deviations (white) while the source covers the caustic. The caustic exit induces a narrow “blue ridge” corresponding to the negative-deviation spike seen in Fig 3.1. Finally, the source runs along the long “brown ridge” corresponding to the prolonged post-peak mild excess seen in Fig 3.1.

105 Fig. 3.4.— Contours of ∆χ2 = 1, 4, 9 relative to the minimum as a function of planet- star mass ratio q and projected planet-star separation d (top), as well as “short caustic diameter” (see Fig. 3.3) w (bottom). w (in units of θE) is a function of q and d (see +0.5 −3 text). The solution shown here corresponds to q = (2.5−0.3) 10 and d = 2.9 0.2 (or +0.03 × ± d = 0.34−0.02). These values of d correspond to physical separations and equilibrium temperatures of 5.3 9.7 AU, 34 K and 0.6 1.1 AU, 103 K for the close ∼ − ∼ ∼ − ∼ and wide solutions, respectively.

106 16

I [instrumental] 20

−.6 −.4 −.2 0 .2 (V−I) [instrumental]

Fig. 3.5.— Instrumental color-magnitude diagram of field containing MOA-2007- BLG-400. The color and magnitude of the source (black filled circle) are derived from the fit to the light curve, which also yields an upper limit for the I-band blended flux (green triangle). The large error bar on the latter point indicates a complete lack of information about its V -band flux. The clump centroid is indicated in red square. From the source-clump offset, we estimate [I, (V I)] = (17.57, 0.81), − 0,s implying it has angular radius θ∗ = 1.05 µas. Assuming the source lies at 8 kpc, it has [MI , (V I)]0,s = (3.07, 0.81), making it a subgiant. The lack of blended light allows us to− place an upper limit on the lens flux, which implies that it has mass M < 0.75 M⊙. See text.

107 Fig. 3.6.— Bayesian relative probability densities for the physical properties of the planet MOA-2007-BLG-400Lb and its host star. (a) Mass of the host star. (b) Planet semimajor axis. (c) Distance to the planet/star system. (d) Equilibrium temperature of the planet. (e) I-band magnitude of the host star. (f) H-band magnitude of the host star. In panel (a), we also show the probability density for the planet mass, which is essentially a rescaling of that of the host star, because the mass ratio is measured so precisely q = (2.5+0.5) 10−3. In all panels, the solid vertical lines show the −0.3 × medians, and the 68.3% and 95.4% conﬁdence intervals are enclosed in the dark and light shaded regions, respectively. In panel (b) and (d), the probability distributions for wide and close degenerate solutions are computed separately, and then the close solution is weighted by exp( ∆χ2/2), where ∆χ2 = 0.2 is the diﬀerence between them. These distributions are− derived assuming priors obtained from standard models of the mass, velocity, and density distributions of stars in the Galactic bulge and disk, and include constraints from the measurements of lens angular Einstein radius θE and the timescale of the event tE, as well as limits on the I-band magnitude of the lens. In practice, only the measurements of θE and tE provide interesting constraints on these distributions. 108 A) PEAK B) BASELINE PEAK

C) PEAK - BASELINE D) BASELINE - SOURCE

Fig. 3.7.— Constraining the blended flux from CTIO I-band images. A good-seeing baseline image B (upper right) is subtracted from an image taken at magnification 245 (A, upper left), and the resulting image C is shown in the bottom-left panel. The white circles on images A and C indicate the source positions. Then C is scaled by 1/244 in flux and subtracted from image B to generate image D, which has the source contribution removed from the baseline image. As described in the text, we fit PSF to the stars close to the source, and subtract them from images B and D. After subtraction, the flux sums in a 1.8” square (shown as black boxes on right two panels) are 564 ADU and -28 ADU, respectively. Non-detection of the blend constrains the blend-source flux ratio to be less than 5%.

109 Model1 t t 2 u t d q α3 ρ 0 − ref 0 E day day deg

Close 0.08107 0.00025 14.41 0.34 0.0026 227.06 0.00326

Wide 0.08106 0.00027 14.33 2.87 0.0025 226.99 0.00329

1The wide solution is preferred over best-ﬁt single-lens model by∆χ2 =

1070.04 and the close solution by ∆χ2 = 1069.84

2 tref = HJD 2454354.5 (Sep 11, 2007, 00:00 UT). 3The geometry of the source trajectory is visualized in Figure 3.3, in

which the planet is to the right of the lens star. t0, u0, and α are deﬁned

with respect to the “center of magniﬁcation”, which is the center of mass

of the star/planet system for the close model and q/(1 + q)/d away from

the position of the lens star toward the direction of the planet for the wide

model.

Table 3.1. MOA-2007-BLG-400 Best-ﬁt Planetary Models

110 Chapter 4

OGLE-2005-BLG-071Lb, the Most Massive M-Dwarf Planetary Companion?

4.1. Introduction

Microlensing provides a powerful method to detect extrasolar planets. Although only six microlens planets have been found to date (Bond et al. 2004; Udalski et al.

2005; Beaulieu et al. 2006; Gould et al. 2006; Gaudi et al. 2008; Dong et al. 2009), these include two major discoveries. First, two of the planets are “cold Neptunes”, a high discovery rate in this previously inaccessible region of parameter space, suggesting this new class of extrasolar planets is common (Gould et al. 2006; Kubas et al. 2008). Second, the discovery of the ﬁrst Jupiter/Saturn analog via a very high-magniﬁcation event with substantial sensitivity to multiple planets indicates that solar system analogs may be prevalent among planetary systems (Gaudi et al.

2008). Recent improvements in search techniques and future major upgrades should increase the discovery rate of microlensing planets substantially (Gaudi 2008).

111 Routine analysis of planetary microlensing light curves yields the planet/star mass ratio q and the planet-star projected separation d (in units of the angular

Einstein radius). However, because the lens-star mass M cannot be simply extracted from the light curve, the planet mass Mp = qM remains, in general, equally uncertain.

The problem of constraining the lens mass M is an old one. When microlensing experiments were initiated in the early 1990s, it was generally assumed that individual mass measurements would be impossible and that only statistical estimates of the lens mass scale could be recovered. However, Gould (1992) pointed out that the mass and lens-source relative parallax, π π π , are simply related rel ≡ l − s to two observable parameters, the angular Einstein radius, θE, and the Einstein radius projected onto the plane of the observer,r ˜E,

θE M = , πrel = θEπE. (4.1) κπE

Here, π = AU/r˜ is the “microlens parallax” and κ 4G/(c2 AU) 8.1 mas/M . E E ≡ ∼ ⊙ See Gould (2000b) for an illustrated derivation of these relations.

In principle, θE can be measured by comparing some structure in the light curve to a “standard angular ruler” on the sky. The best example is light-curve distortions due to the ﬁnite angular radius of the source θ∗ (Gould 1994a), which usually can be estimated very well from its color and apparent magnitude (Yoo et al. 2004a). While

112 such finite-source effects are rare for microlensing events considered as a whole, they are quite common for planetary events. The reason is simply that the planetary distortions of the light curve are typically of similar or smaller scale than θ∗. In fact, all six planetary events discovered to date show such effects. Combining θE with the

(routinely measurable) Einstein radius crossing time tE yields the relative proper motion µ in the geocentric frame,

θE µgeo = , (4.2) tE

2 From equation (4.1), measurement of θE by itself ﬁxes the product Mπrel = θE/κ.

Using priors on the distribution of lens-source relative parallaxes, one can then make a statistical estimate of the lens mass M and so the planet mass Mp.

To do better, one must develop an additional constraint. This could be measurement of the microlens parallax πE, but this is typically possible only for long events. Another possibility is direct detection of the lens, either under the “glare” of the source during and immediately after the event, or displaced from the source well after the event is over. Bennett et al. (2006) used the latter technique to constrain the mass of the ﬁrst microlensing planet, OGLE-2003-BLG-235/MOA-2003-BLG-53Lb.

They obtained Hubble Space Telescope (HST) Advanced Camera for Surveys (ACS) images in B, V , and I at an epoch ∆t = 1.78 years after the event. They found astrometric oﬀsets of the (still overlapping) lens and source light among these images of up to 0.7 mas. Knowing the lens-source angular separation ∆θ = µ∆t from the

113 already determined values of θE and tE, they were able to use these centroid oﬀsets to

ﬁx the color and magnitude of the lens and so (assuming that it was a main-sequence star) its mass.

While the planet mass is usually considered to be the most important parameter that is not routinely derivable from the light curve, the same degeneracy impacts two other quantities as well, the distance and the transverse velocity of the lens. Knowledge of these quantities could help constrain the nature of the lens, that is, whether it belongs to bulge, the foreground disk, or possibly the thick disk or even the stellar halo. Since microlensing is the only method currently capable of detecting planets in populations well beyond the solar neighborhood, extracting such information would be quite useful. Because the mass, distance, and transverse velocity are all aﬀected by a common degeneracy, constraints on one quantity are simultaneously constraints on the others. As mentioned above, simultaneous measurements of θE and πE directly yield the mass. However, clearly from equation (4.1) they also yield the distance, and hence (from eq. [4.2]) also the transverse velocity. Here, we assemble all available data to constrain the mass, distance and transverse velocity of the second microlensing planet,

OGLE-2005-BLG-071Lb, whose discovery we previously reported (Udalski et al.

2005, hereafter Paper I).

114 4.2. Overview of Data and Types of Constraints

The light curve consists of 1398 data points from 9 ground-based observatories

(see Fig. 2.1), plus two epochs of HST ACS data in the F814W (I) and F555W

(V ) ﬁlters. The primary ground-based addition relative to Paper I is late-time data from OGLE, which continued to monitor the event down to baseline until HJD

= 2453790.9.

These data potentially provide constraints on several parameters in addition to those reported in Paper I. First, the light curve shows a clear asymmetry between the rising and falling parts of the light curve, which is a natural result of microlens parallax due to the Earth’s accelerated motion around the Sun (see the best-ﬁt model without parallax eﬀects plotted in dotted line in Fig. 2.1). However, it is important to keep in mind that such distortions are equally well produced by

“xallarap” due to accelerated motion of the source around a companion. Poindexter et al. (2005) showed that it can be diﬃcult to distinguish between the two when, as in the present case, the eﬀect is detected at ∆χ2 < 100. ∼

Second, the two pronounced peaks of the light curve, which are due to “cusp approaches” (see the bottom inset of Fig. 2.1), are relatively sharp and have good coverage. These peaks would tend to be “rounded out” by ﬁnite-source eﬀects, so in principle it may be possible to measure ρ (i.e., θ∗ in the units of θE) from these distortions.

115 Third, the orbital motion of the planet can give rise to two effects: rotation of the caustic about the center of the mass and distortion of the caustic due to expansion/contraction of the planet-star axis. The first changes the orientation of the caustic structure as the event evolves while the second changes its shape. These effects are expected to be quite subtle because the orbital period is expected to be of an order of 10 years while the source probes the caustic structure for only about 4 days. Nevertheless, they can be very important for the interpretation of the event.

Finally, the HST data cover two epochs, one at 23 May 2005 (indicated by the arrow in Fig. 2.1) when the magniﬁcation was about A = 2 and the other at 21 Feb

2006 when the event was very nearly at baseline, A 1. These data could potentially ∼ yield four types of information. First, they can effectively determine whether the blended light is “associated” with the event or not. The blended light is composed of sources in the same photometric aperture as the magnified source, but that do not become magnified during the event. If this light is due to the lens, a companion to the lens, or a companion to the source, it should fall well within the ACS point spread function (PSF) of the source. On the other hand, if it is due to a random interloper along the line of sight, then it should be separately resolved by the ACS or at least give rise to a distorted PSF. Second, the HST data can greatly improve the estimate of the color of the blended light. The original model determined the source fluxes in both OGLE V and I very well, and of course the baseline fluxes are also quite well determined. So it would seem that the blended fluxes, which are

116 the differences between these two, would also be well determined. This proves to be the case in the I band. However, while the source flux is derivable solely from flux differences over the light curve (and so is well determined from OGLE difference image analysis – DIA – Wo´zniak 2000), the baseline flux depends critically on the zero point of PSF-fitting photometry, whose accuracy is fundamentally limited in very crowded bulge fields. The small zero-point errors turn out to have no practical impact for the relatively bright I background light, but are important for the V band. Third, one might hope to measure a centroid shift between the two colors in the manner of Bennett et al. (2006). Last, one can derive the source proper motion

µs from HST data (at least relative to the mean motion of bulge stars). This is important, because the event itself yields the source-lens relative proper motion,

µgeo. Hence, precise determination of µl requires knowledge of two proper-motion diﬀerences, ﬁrst the heliocentric proper motion

µ = µ µ , (4.3) hel l − s and second, the oﬀset between the heliocentric and geocentric proper motions

v π µ µ = rel . (4.4) hel − geo ⊕AU

Here, v is the velocity of the Earth relative to the Sun at the time of peak ⊕ magniﬁcation t0. Note that, if the lens-source relative parallax πrel is known, even

117 approximately, then the latter diﬀerence can be determined quite well, since its total

−1 magnitude is just 0.6 mas yr (πrel/0.17 mas).

4.3. Constraining the Physical Properties of the Lens

and its Planetary Companion

In principle, all the effects summarized in 4.2 could interact with each § other and with the parameters previously determined, leading potentially to a very complex analysis. In fact, we will show that most effects can be treated as isolated from one another, which greatly facilitates the exposition. In the following sections, we will discuss the higher order microlens effects in the order of their impact on the ground-based light curve, starting with the strongest, that is, parallax effects

( 4.3.1), followed by planetary orbital motion ( 4.3.2) and finally the weakest, § § finite-source effects ( 4.3.3). To study these effects, we implement Markov chain § Monte Carlo (MCMC) with an adaptive step-size Gaussian sampler (Doran &

Mueller 2003) to perform the model ﬁtting and obtain the uncertainties of the parameters. The HST astrometry is consistent with no (V I) color-dependent − centroid shift in the ﬁrst epoch, while such a shift is seen in the second epoch observations ( 4.3.4). In addition, the PSF of the source shows no sign of broadening § due to the blend, suggesting that the blend is associated with the event ( 4.3.5). § Therefore, the HST observations provide good evidence that the blend is likely due

118 to the lens. In 4.3.4, it is shown, under such an assumption, how the astrometry § can be used in conjunction with ﬁnite-source and microlens parallax measurements to constrain the angular Einstein radius and proper motion ( 4.3.4). In 4.3.5, § § we discuss using HST photometric constraints in the form of χ2 penalties to the

MCMC runs to extract the color and brightness of the blend. The results of these runs, which include all higher order eﬀects of the ground-based light curve and the

HST photometric constraints, are summarized as “MCMC A” in Table 4.1 and

4.2. Subsequently in 4.3.6, by making the assumption that the blended light seen § by HST is due to the lens, we combine all constraints discussed above to obtain physical parameters of the lens star and its planet. The corresponding best-ﬁt model parameters are reported as “MCMC B” in Table 4.3 and 4.4. The results for the physical parameters from these runs are given in Table 4.5. Finally we discuss some caveats in the analysis in 4.3.7 and 4.3.8. § §

4.3.1. Microlens Parallax Effects

A point-source static binary-lens model has 6 “geometric-model” parameters: three “single-lens” parameters (t0, u0, tE), where we define the time of “peak” magnification (actually lens-source closest approach) t0 and the impact parameter u0 with respect to the center of mass of the planet-star systems; and three “binary-lens” parameters (q, d, α), where α is the angle between the star-planet axis and the trajectory of the source relative to the lens. In addition, flux parameters are included

119 to account for light coming from the source star (Fs) and the blend (Fb) for each dataset. In this paper, we extend the fitting by including microlens parallax, orbital motion and finite-source effects. Paper I reported that, within the context of the point-source static binary-lens models, the best-fit wide-binary (d > 1) solution is preferred by ∆χ2 = 22 over the close-binary (d< 1) solution. Remarkably, when we take account of parallax, finite-source and orbital effects, this advantage is no longer as significant. We discuss the wide/close degeneracy with more detail in 4.3.6. §

The microlens parallax eﬀects are parametrized by πE,E and πE,N, following the geocentric parallax formalism by An et al. (2002) and Gould (2004). To properly model the parallax eﬀects, we characterize the “constant acceleration degeneracy”

(Smith et al. 2003) by probing models with u u and α α. We ﬁnd that 0 → − 0 → − all other parameters remain essentially unchanged under this form of degeneracy.

In the following sections, if not otherwise speciﬁed, parameters from models with positive u0 are adopted.

As shown in Figure 4.2, microlens parallax is ﬁrmly detected in this event at

> 8σ level. Not surprisingly, the error ellipse of πE is elongated toward πE,⊥, i.e., the direction perpendicular to the position of the Sun at the peak of event, projected onto the plane of the sky (Gould et al. 1994; Poindexter et al. 2005). As a result,

πE,E is much better determined than πE,N,

π = 0.26 0.05, π = 0.30+0.24 (wide), (4.5) E,E − ± E,N − −0.28 120 π = 0.27 0.05, π = 0.36+0.24 (close). (4.6) E,E − ± E,N − −0.27

Xallarap (light-curve distortion from reﬂex motion of the source due to a binary companion) could provide an alternate explanation of the detected parallax signals.

In 4.3.8, we ﬁnd that the best-ﬁt xallarap parameters are consistent with those § derived from the Earth’s orbit, a result that favors the parallax interpretation.

4.3.2. Fitting Planetary Orbital Motion

To model orbital motion, we adopt the simplest possible model, with uniform expansion rate b˙ in binary separation b and uniform binary rotation rate ω. Because orbital eﬀects are operative only for about 4 days, while the orbital period is of order 10 years, this is certainly adequate. Interestingly, the orbital motion is more strongly detected for the close solutions (at > 5.5σ level) than the wide solutions (at ∼ 3σ level), and as a result, it signiﬁcantly lessens the previous preference of the ∼ wide solution that was found before orbital motion was taken into account. Further discussions on planetary orbital motion are given in 4.3.6. §

121 4.3.3. Finite-source effects and Other Constraints on

θE

Color-Magnitude Diagram

We follow the standard procedure to derive dereddened source color and magnitude from the color-magnitude diagram (CMD) of the observed ﬁeld.

Figure 4.3 shows the calibrated OGLE CMD (black), with the baseline source being displayed as a green point. The V I color of the source can be determined − in a model-independent way from linear regression of the I-band and V -band observations. The I-band magnitude of the source is also precisely determined from the microlens model, and it is hardly aﬀected by any higher order eﬀects. The center of red clump (red) is at (V I,I) = (1.89, 15.67). The Galactic coordinates of − clump the source are at (l, b) = (355.58, 3.79). Because the Galactic bulge is a bar-like − structure that is inclined relative to the plane of the sky, the red clump density at this sky position peaks behind the Galactic center by 0.15 mag (Nishiyama

2005). Hence, we derive (V I,I) = (1.00, 14.47), by adopting a Galactic − 0,clump distance R0 = 8 kpc. We thereby obtain the selective and total extinction toward the source [E(V I),A ]=(0.89, 1.20) and thus R = A /E(V I) = 2.35. The − I VI V − dereddened color and magnitude of the source is ((V I),I) = (0.45, 18.31). From − s,0 its dereddened color (V I) = 0.45, as well as its absolute magnitude (assuming it − 0 is in the bulge) M 3.65, we conclude that the source is a main-sequence turnoﬀ I ∼

122 star. Following the method of Yoo et al. (2004a), we transform (V I) = 0.45 to − 0 (V K) = 0.93 (Bessell & Brett 1988), and based on the empirical relation between − 0 the color and surface brightness for subgiant and main-sequence stars (Kervella et al. 2004), we obtain the angular size of the source

θ = 0.52 100.2(19.51−Is) 0.05 µas, (4.7) ∗ × ±

where Is is the apparent magnitude of the source in the I band. Other features on the CMD shown in Figure 4.3 are further discussed in 4.3.5. §

Photometric Systematics of the Auckland Data set

The Auckland data set’s excellent coverage over the two peaks makes it particularly useful for probing the ﬁnite-source eﬀects. Unlike the more drastic

“caustics crossings” that occur in some events, the finite-source effects during “cusp approaches” are relatively subtle. Hence one must ensure that the photometry is not affected by systematics at the few percent level when determining ρ = θ∗/θE. The

Auckland photometry potentially suﬀers from two major systematic eﬀects.

First, the photometry of constant stars reduced by µFUN’s DoPHOT pipeline are found to show sudden “jumps” of up to 10% when the ﬁeld crossed the ∼ meridian each night. The signs and amplitudes of the “jumps” depend on the stars’ positions on the CCD. The Auckland telescope was on a German equatorial mount, and hence the camera underwent a meridian ﬂip. Due to scattered light,

123 the flat-fielded images were not uniform in illumination for point sources, an effect that can be corrected by making “superflats” with photometry of constant stars

(Manfroid 1995). We have constructed such “superﬂats” for each night of Auckland observations using 71 bright isolated comparison stars across the frame. The

DoPHOT instrumental magnitude mi,j for star i on frame j is modeled by the following equation:

m = m f(x , y ) Z fwhm s (4.8) i,j 0,i − i,j i,j − j − i,j × i

where m0,i is the corrected magnitude for star i, f(x, y) is a biquartic illumination correction as a function of the (x, y) position on the CCD frame with 14 parameters,

Zj is a zero-point parameter associated with each frame (but with Z1 set to be zero), and si is a linear correlation coefficient for the seeing fwhmi,j. A least-squares fit that recursively rejects 4-σ outliers is performed to minimize χ2. The best-fit f(x, y) is dominated by the linear terms and has small quadratic terms, while its cubic and quartic terms are negligible. The resulting reduced χ2 is close to unity, and the

“jumps” for all stars are eﬀectively eliminated. We apply the biquartic corrections to the images and then reduce the corrected images using the DIA pipeline. The resulting DIA photometry of the microlens target is essentially identical (at the

1% level) to that from the DIA reductions of the original Auckland images. ∼

As we now show, this is because DIA photometry automatically removes any artifacts produced by the ﬁrst- and second-order illumination distortions if the

124 sources are basically uniformly distributed across the frame. For the first-order effect, a meridian flip about the target (which is very close to center of the frame) will induce a change in the flux from the source, but it will also induce a change in the mean flux from all other stars in the frame, which for a linear correction will be the same as the change in position of the “center of light” of the frame light.

If the frame sources are uniformly distributed over the frame, the “center of light” will be the center of the frame, which is the same position as the source, therefore introducing no eﬀects. The second-order transformation is even under a rotation of

180 degrees, whereas a meridian flip is odd under this transformation. Hence the flip has no effects at second order.

Second, the Auckland observations were unfiltered. The amount of atmospheric extinction differs for stars with different colors. As shown in Figure 4.3, the source is much bluer than most of the bright stars in the field, which dominate the reference image. So the amount of extinction for the source is different from the average extinction over the whole frame. This difference varies as the airmass changes over time during the observations. Coincidentally, the times of the two peaks were both near maximum airmass when the “differential extinction” effect is expected to be the most severe. To investigate this effect, we match the isolated stars in the Auckland frame with CTIO I and V photometry. We identify 33 bright, reasonably isolated stars with (V I) (V I) < 0.25. We obtain a “light curve” for each of these | − − − s| stars, using exactly the same DIA procedure as for the source. We measure the

125 mean magnitude of each of the 33 light curves and subtract this value from each of the 508 points on each light curve, thereby obtaining residuals that are presumably due primarily to airmass variation. For each of the 508 epochs, we then take the mean of all of these residuals. We recursively remove outliers until all the remaining points are within 3 σ of the mean, as deﬁned by the scatter of the remaining points.

Typically, 1 or 2 of the 33 points are removed as outliers. The deviations are well

ﬁtted by a straight line,

dMag = 0.0347 0.0016 (4.9) dZ ± where Z is airmass. The sense of the eﬀect is that stars with the color of the microlensed source are systematically fainter at high airmass, as expected. (We also tried ﬁtting the data to a parabola rather than a line, but the additional [quadratic] parameter was detected at substantially below 1 σ.) Finally, we apply these

“diﬀerential extinction” corrections to the “superﬂat”-adjusted DIA photometry to remove both photometric systematics.

In general, the finite-source effects depend on the limb-darkening profile of the source star in the observed passbands. We find below that in this case, the impact proves to be extremely weak. Nevertheless, using the matched Auckland and CTIO stars, we study the difference between Auckland magnitudes and I-band magnitude as a function of the V I color. We find the Auckland clear filter is close to the R − band.

126 Blending in Palomar and MDM Data

Palomar data cover only about 80 minutes, but these include the cresting of the second peak, from which we derive essentially all information about finite-source effects. The Palomar data are sensitive to these effects through their curvature.

The curvature derived from the raw data can be arbitrarily augmented in the ﬁt

(and therefore the finite-source effects arbitrarily suppressed) by increasing the blending. In general, the blending at any observatory is constrained by observations at substantially different magnifications, typically on different nights. However, no such constraints are available for Palomar, since observations were carried out on only one night.

We therefore set the Palomar blending fb = 0.2 fs, that is, similar to the OGLE blending. That is, we assume that the observed flux variation of 9%, over the Palomar night, actually reflects a magnification variation of 9%/[1 f /Af ] = 9% + 0.026%, − b s where A 70 is the approximate magnification on that night. If our estimate of the ∼ blending were in error by of order unity (i.e. either fb =0 or fb = 0.4 fs), then the implied error in the magnification difference would be 0.026%, which is more than an order of magnitude below the measurement errors. Hence, the assumption of fixed blending does not introduce “spurious information” into the fit even at the 1 σ level.

MDM data cover the second peak for only 18 minutes. For consistency, we treat ∼

127 its blending in the same way as Palomar, although the practical impact of this data set is an order of magnitude smaller.

Modeling the Finite-Source Effects

After careful tests that are described immediately below, we determined that all ﬁnite-source calculations can be carried out to an accuracy of 10−4 using the hexadecapole approximation of Gould (2008) (see also Pejcha & Heyrovsky 2008).

This sped up calculations by several orders of magnitude. We began by conducting

MCMC simulations using the “loop linking” ﬁnite-source code described in Appendix

A of Dong et al. (2006). From these simulations, we found the 4.5 σ upper bound on the finite-source parameter ρ(4.5 σ) = 0.001. We then examined the differences between loop-linking (set at ultra-high precision) and hexadecapole for light curves at this extreme limit and found a maximum difference of 10−4. Based on Claret (2000), we adopt linear limb-darkening coefficients ΓI = 0.35 for the I-band observations and

ΓR = 0.43 for the observations performed in the R-band and the clear ﬁlters, where the local surface brightness is given by S(θ) 1 Γ[1 1.5(1 θ2/θ2)1/2]. Ten ∝ − − − ∗ additional MCMC runs are performed with ΓI and ΓR that diﬀer from the above values by 0.1 or 0.2. They result in essentially the same probability distributions of

ρ. Therefore, the choice of limb-darkening parameters has no eﬀect on the results.

+1.8 +1.7 The source size is found to be ρ = 3.9−2.7 for the wide solution and ρ = 3.1−2.5 for the close solution. Solutions with ρ> 0.0009 are ruled out at more than 3σ. The angular

128 Einstein radius is given by θE = θ∗/ρ. Hence, the lack of pronounced ﬁnite-source eﬀects yields a 3σ lower limit: θE > 0.6 mas. The lens-source relative proper motion in the geocentric frame is simply µgeo = θE/tE. The posterior probability distributions of µgeo derived from these MCMC simulations are compared with those derived from astrometry in 4.3.4. §

4.3.4. HST Astrometry

HST observations were taken at two epochs (HJD = 2453513.6 and

HJD = 2453788.2) with the ACS High Resolution Camera (HRC). For each epoch,

4 dithered images were acquired in each of F814W and F555W with individual exposure times of 225s and 315s respectively. The position of the microlens on the

HST frame is in excellent agreement with its centroid on the OGLE diﬀerence image

(within 0′′.01). The closest star to the source is about 0′′.6 away. This implies ∼ that the OGLE photometry of the target star does not contain additional blended light that would be identiﬁable from the HST images. Data analysis was carried out using the software program img2xym HRC (Anderson & King 2004) in a manner similar to that described in Bennett et al. (2006). Stars are ﬁtted with an empirical

“library” PSF that was derived from well-populated globular cluster ﬁelds. These positions are then corrected with precise distortion-correction models (accurate to

0.01 pixel). We adopted the ﬁrst F555W frame of the ﬁrst epoch as the reference ∼ frame, and used the measured positions of stars in this frame and the frame of each

129 exposure to define a linear transformation between the exposure frame and the reference frame. This allowed us to transform the position of the target star in each exposure into the reference frame, so that we see how the target star had moved relative to the other stars. The centroid positions of the target star in each filter and epoch are shown in Figure 4.4. For convenience, in this figure, the positions are displayed relative to the average of the centroid positions. The error bars are derived from the internal scatter of the four dithered images. The probability is P = 38% of measuring the observed separation (or larger) between F814W and F555W under the assumption that the true offset is zero. The fact that the blended light is aligned with the source argues that it is associated with the event (either it is the lens itself or a companion to the lens or the source). We give a more quantitative statement of this constraint in 4.3.5. For the present we simply note that the P = 38% § probability is compatible with the picture that the blend is due to the lens since the

first epoch was only about half of the Einstein-radius crossing time after t0, implying that the lens-source separation induces only a very small centroid offset, well below the HST detection limit. For the second epoch, the centroid offset is,

∆r = 0.52 0.20 mas, F814W−F555W,East − ±

∆r = 0.22 0.20 mas. (4.10) F814W−F555W,North ±

130 We also calculate the error in the centroid offsets from the scatter in such offsets among all comparison stars with F555W magnitudes within 0.5 mag of the target and find that it is consistent with the internally-based error quoted above.

At the peak of the event, the angular separation between the lens and the source was negligible, since u 1. We therefore ﬁx the angular positions of the 0 ≪ lens and source at a common θ0. From the CMD (Fig. 4.3), most of the stars in the

HST field are from the bulge. So we set a reference frame that is fixed with respect to the bulge field at distance Ds. The source and lens positions at time t are then,

θs(t) = θ0 + µs(t t ), − 0

θl(t) = θ0 + µl(t t )+ π [s(t) s(t )], (4.11) − 0 rel − 0

where s(t) is the Earth-to-Sun vector deﬁned by Gould (2004). Then by applying equations (4.3) and (4.4), the angular separation between the lens and source is,

θrel(t)= θl(t) θs(t)= µ (t t )+ π ∆s(t) (4.12) − geo − 0 rel

where ∆s(t) is given by eq. (5) in Gould (2004).

′ The centroid of the source images θs is displaced from the source position by

(Walker 1995),

131 ′ θrel(t) θ θ θs ∆ s(t)= s(t) (t)= − 2 (4.13) − [θrel(t)/θE] + 2

Therefore, one can obtain the centroid position of the lens and the source at time t,

θc(t) = [1 f (t)][θs(t)+ ∆θ (t)] + f (t)θl(t) − l s l

1 fl(t) θ0 µs θ = + (t t0)+ rel(t)[fl(t)+ − 2 ] (4.14) − [θrel(t)/θE] + 2

where fl(t) is the fraction of the total ﬂux due to the lens.

The centroid oﬀset between the two passbands, F814W and F555W, is related to the properties of the system by,

1 θ θ ∆ c(t)F814W−F555W = [f(t)l,F814W f(t)l,F555W] [1 2 ] rel(t).(4.15) − × −[θrel(t)/θE] + 2

The difference of the blend’s fractional flux between F814W and F555W is obtained from “MCMC A” described in 4.3.5. Consequently, under the assumption that the § blend is the lens, we can use the measurement of the second-epoch HST centroid offset to estimate the relative proper motion from equation (4.15) for a given πrel.

For purposes of illustration, we temporarily adopt πrel = 0.2 when calculating the probability distribution of µgeo (black contours in the upper panels of Fig. 4.5). The

132 centroid shift generally favors faster relative proper motion than that derived from the source size measurement (green contours in Fig. 4.5), but the difference is only at the 1σ level. We then get a joint probability distribution of µ from both ∼ geo finite-source effects and astrometry, which is shown as the red contours in the upper panels of Figure 4.5.

µ We then derive the distribution of the geo position angle φµgeo (North through

East), which is shown by the red histograms in the lower panels of Figure 4.5. Since the direction of the lens-source relative proper motion µgeo is the same as that of the microlens parallax πE in the geocentric frame, we have an independent check

on the φµgeo from our parallax measurements, whose distribution is plotted as blue histograms in Figure 4.5. Both constraints favor the lens-source proper motion to be generally West, but they disagree in the North-South component for which both constraints are weaker. The disagreements between two histograms is at about 2.5σ level.

4.3.5. “Seeing” the Blend with HST

If the blend were not the lens (or otherwise associated with the event), the

PSF of the source would likely be broadened by the blended star. We examine the

HST F814W images of the target and 45 nearby stars with similar brightness for each available exposure. We ﬁt them with the library PSF produced by Anderson

133 & King (2004). In order to account for breathing-related changes of focus, we fit each of these 45 nearby stars with the library PSF, and construct a residual PSF that can be added to the library PSF to produce a PSF that is tailor-made for each exposure. For both epochs, the source-blend combination shows no detectable broadening relative to the PSFs of other isolated stars in the field. From the ground-based light curve, it is already known that 16% of this light comes from ∼ the blend. We add simulated stars with the same flux as the blended light from 0 to 2.0 pixels away from the center of the source. We find that the blend would have produced detectable broadening of the PSF if it were more than 15 mas apart from the source at the second epoch. Hence, the source-blend separation must then be less than about 15 mas. From the HST image itself, the density of ambient stars at similar magnitudes is < 1 arcsec−2. The probability of a chance interloper is ∼ therefore < 0.07%, implying that the blended light is almost certainly associated with the event, i.e., either the lens itself, a companion to the lens, or a companion to the source. Both of the latter options are further constrained in 4.3.7 where, in § particular, we essentially rule out the lens-companion scenario.

As discussed in 4.2, the blended flux in I is relatively well determined from § the ground-based OGLE data alone, but the blended V flux is poorly determined, primarily because the systematic uncertainty in the zero point of the baseline flux

(determined from PSF ﬁtting) is of the same order as the blended ﬂux. Because the

HST image is very sparse, there is essentially no zero-point error in the HST V -band

134 flux. The problem is how to divide the baseline V flux into source and blend fluxes,

Fbase = Fs + Fb.

The standard method of doing this decomposition would be to incorporate the

HST V light curve into the overall ﬁt, which would automatically yield the required decomposition. Since this “light curve” consists of two points, the “ﬁt” can be expressed analytically

F (t ) F (t ) F = 1 − 2 ,F = F (t ) F . (4.16) s A 1 b 2 − s 1 − where we have made the approximation that the second observation is at baseline. Let us then estimate the resulting errors in Fs and Fb, ignoring for the moment that there is some uncertainty in A1 due to uncertainties in the general model. Each of the individual ﬂux measurement is determined from 4 separate subexposures, and this permits estimates of the errors from the respective scatters.

These are σ1 = 0.01 and σ2 = 0.03 mag. Hence, the fractional error in Fs is

(2.5/ ln 10)σ(F )/F = [σ2(A + r)2 + σ2(1 + r)2]1/2/(A 1), where r F /F . s s 1 1 2 1 − ≡ b s

Adopting, for purposes of illustration, A1 = 2 and r = 0.1, this implies an error

σ(Vs,HST ) of 0.04 mag. This may not seem very large, but after the subtraction in equation (4.16), it implies an error σ(V ) σ(V )/r 0.4 mag. And b,HST ∼ s,HST ∼ taking into account of the uncertainties introduced by model ﬁtting in determining the magniﬁcations, the error is expected to be even larger. Hence, we undertake an alternate approach.

135 Because the HST and OGLE V filters have very nearly the same wavelength center, Vs,HST should be nearly identical to Vs,OGLE up to a possible zero-point offset on their respective magnitude scales. Because the OGLE data contain many more points during the event, some at much higher magnification than the single

HST event point, Vs,OGLE is determined extremely well (for fixed microlensing model), much better than the 0.04 mag error for Vs,HST . Thus, if the zero-point offset between the two systems can be determined to better than 0.04 mag, this method will be superior. Although the I-band blend is much better measured than the V -band blend from the ground-based data, for consistency we determine the zero-point offset in I by the same procedure.

Figure 4.6 shows diﬀerences between OGLE and HST V magnitudes for matched stars in the HST image. The error for each star and observatory is determined from the scatter among measurements of that star. We consider only points with V < 19.5 because at fainter magnitudes the scatter grows considerably.

Each star was inspected on the HST images, and those that would be significantly blended on the OGLE image were eliminated. The remaining points are fit to an average offset by adding a “cosmic error” in quadrature to the errors shown. We carry out this calculation twice, once including the “outlier” (shown as a filled circle) and once with this object excluded. For the V band, we find offsets of

136 V V = 0.17 0.01 and 0.18 0.01, respectively. We adopt the following HST − OGLE ± ± the V -band oﬀset

∆V = V V = 0.18 0.01. (4.17) HST − OGLE ±

A similar analysis of the I band leads to

∆I = I I = 0.08 0.01. (4.18) HST − OGLE ±

We find no obvious color terms for either the V -band or I-band transformations. As a check, we perform linear regression to compare the OGLE and HST (V I) colors, − and we find they agree within 0.01 mag, which further confirms the color terms are unlikely to be significant in the above transformations.

We proceed as follows to make HST-based MCMC (“MCMC A”) estimates of

Vb,OGLE and Ib,OGLE that place the blending star on the OGLE-based CMD. Since

flux parameters are linear, they are often left free and fitted by linear least-squares minimization, which significantly accelerates the computations. However, for

“MCMC A”, the source ﬂuxes from OGLE and HST are treated as independent

MCMC parameters so that they can help align the two photometric systems as described below. Since HST blended light is not aﬀected by light from ambient stars (as OGLE is), we also leave HST blended ﬂuxes as independent. Therefore, in

“MCMC A”, we include the following independent MCMC ﬂux parameters, FI,s,OGLE,

FV,s,OGLE, FI,s,HST , FV,s,HST , FI,b,HST , and FV,b,HST , which for convenience we express

137 here as magnitudes. For each model on the chain, we add to the light-curve based χ2 two additional terms ∆χ2 = (V V ∆V )2/[σ(∆V )]2 and V s,HST − s,OGLE − 2 2 2 ∆χ =(I HST I ∆I) /[σ(∆I)] to enforce the measured oﬀset between the I s, − s,OGLE − two systems. Finally, we evaluate the V -band blended ﬂux from HST and convert it to OGLE system, V HST = V V HST + V HST (and similarly for I b,OGLE/ s,OGLE − s, b, band), where all three terms on the rhs are the individual Monte Carlo realizations of the respective parameters.

The result is shown in Figure 4.3, in which the blend (magenta) is placed on the OGLE CMD. Also shown, in cyan points, is HST photometry (aligned to the

OGLE system) of the stars in the ACS subﬁeld of the OGLE ﬁeld. Although this

ﬁeld is much smaller, its stars trace the main sequence to much fainter magnitudes.

The blend falls well within the bulge main sequence revealed by the HST stars on the CMD, so naively the blend can be interpreted as being in the bulge. Hence, this diagram is, in itself, most simply explained by the blend being a bulge lens or a binary companion of the source. However, the measurement of V I color has − relatively large uncertainty, and it is also consistent with the blend being the lens (or a companion to it) several kpc in front of the bulge, provided the blend is somewhat redder than indicated by the best-ﬁt value of its color. In the following section, we assume the blended light seen by HST is the foreground lens star, and the HST photometry is combined with other information to put constraints on the lens star under this assumption.

138 4.3.6. Final Physical Constraints on the Lens and

Planet

Constraints on a Luminous Lens

In the foregoing, we have discussed two types of constraints on the host star properties: the ﬁrst class of constraints, consisting of independent measurements of

πE, θE and µ, relate the microlens parameters to the physical parameters of the lens; the second class are HST and ground-based observations that determine the photometric properties of the blend.

In this section, we ﬁrst describe a new set of MCMC simulations taking all these constraints into account. Similarly to what is done to include HST photometry in the “MCMC A” (see 4.3.5), we incorporate HST astrometry constraints by § adding χ2 penalties to the ﬁttings. For a given set of microlens parameters, we can derive the physical parameters, namely, M, πrel, µgeo, and so calculate ρ = θ∗/θE

(from eq. [4.1]) and the F814W F555W centroid oﬀset (from eq. [4.15]). Then we − assign the χ2 penalties based on the observed centroid oﬀset from 4.3.4. In this § way, the MCMC simulations simultaneously include all microlens constraints on the lens properties. The posterior probability distribution of M and πrel are plotted in

Figure 4.7. The πrel determination very strongly excludes a bulge (πrel < 0.05) lens. ∼

139 Note that by incorporating HST astrometry, we implicitly assume that the blend is the lens.

If the blend is indeed the lens itself, we can also estimate its mass and distance from the measured color and magnitude of the blend. In doing so, we use theoretical stellar isochrones (M. Pinsonneault 2007, private communication) incorporating the color-temperature relation by Lejeune et al. (1997, 1998). We ﬁrst use an isochrone that has solar metal abundance, with stellar masses ranging from

0.25M 1.0M , and an age of 4 Gyrs. The variation in stellar brightness due to ⊙ − ⊙ stellar age is negligible for our purpose. Extinction is modeled as a function of Dl by dA /dD = (0.4 kpc−1) exp( wD ), where w is set to be 0.31 kpc−1 so that the I l − l observed value A (8.6kpc) = 1.20 (as derived from CMD discussed in 4.3.3) is I § reproduced. Again, the distance to the source is assumed to be 8.6 kpc, implying

πs = 0.116 mas, and hence that the lens distance is Dl/kpc = mas/(πrel + πs). In

Figure 4.7, we show the lens mass M and relative parallax πrel derived from the isochrone that correspond to the observed I-band magnitude I = 21.3 in black line and a series of V I values V I = 1.8 (best estimate), 2.0 (0.5 σ), 2.1 (1 σ), − − 2.3 (1.5 σ) and 2.6 (2 σ) as black points. The observed color is in modest disagreement

< 2σ with the mass and distance of the lens at solar metallicity. We also show analogous trajectories for [M/H] = 0.5 (red) and [M/H] = 1.0 (green). The level − − of agreement changes only very weakly with metallicity.

140 We then include the isochrone information in a new set of MCMC runs

(“MCMC B”). To do so, the HST blended fluxes in I and V bands can no longer be treated as independent MCMC parameters. Instead, based on the isochrone with solar metallicity, the lens V I color and I magnitude are predicted at the − lens mass and distance determined from MCMC parameters. Then the HST I-band and V -band fluxes are fixed at the predicted values in the fitting for each MCMC realization.

Figure 4.8 illustrates the constraints on M and πrel from the MCMC, which are essentially the same for both wide-binary (solid contours) and close-binary (dashed contours) solutions:

M = 0.46 0.04 M , π = 0.19 0.03 mas. (4.19) ± ⊙ rel ±

Assuming the source distance at 8.6kpc, the πrel estimates translate to the following lens distance measurement:

D = 3.2 0.4 kpc. (4.20) l ±

Furthermore, we can derive constraints on the planet mass Mp and the projected separation between the planet and the lens star r⊥,

M = 3.8 0.4M , r = 3.6 0.2AU (wide), (4.21) p ± Jupiter ⊥ ± 141 and

M = 3.4 0.4M , r = 2.1 0.1AU (close). (4.22) p ± Jupiter ⊥ ±

The wide solution is slightly preferred over close solution by ∆χ2 = 2.1.

To examine possible uncertainties in extinction estimates, we reran our MCMC with AI and AV that are 10% higher and lower than the ﬁducial values. These runs result in very similar estimates as when adopting the ﬁducial values.

From equations (4.14) and (4.13), one can easily obtain the centroid shift

1 between two epochs in a given passband by ignoring ∆θs(t) ,

θc(t ) θc(t ) = µs(t t )+ µ [f (t )(t t ) f (t )(t t )] 2 − 1 2 − 1 geo l 2 2 − 0 − l 1 1 − 0 +π [f (t )∆s(t ) f (t )∆s(t )] (4.23) rel l 2 2 − l 1 1

Because µgeo, πrel and fl in a given passband can be extracted from the MCMC realizations (“MCMC B”), we can use the above equation to measure the source proper motion by making use of the centroid shift in F814W between two epochs.

1 The angular separations between the source and the lens are 0.47θE and 4.4θE for the ∼ ∼ two HST epochs, respectively. Thus the angular position offsets between the centroids of the source images and the source are both 0.21θE and the directions of the offset relative to the source are ∼ almost the same due to the small impact parameter u0. The difference between lens flux fractions of the two epochs are about 7% in I band, so the offsets can be confidently ignored in deriving the source proper motion using the relative astrometry in F814W at two different epochs.

142 The source proper motion with respect to the mean motion of stars in the HST ﬁeld is measured to be

+0.2 −1 µs =(µ , µ )=(2.0 0.2, 0.5 ) mas yr . (4.24) s,E s,N ± − −0.7

We obtain similar results with F555W, but with understandably larger errorbars since the astrometry is more precise for the microlens in F814W.

Combining equations (4.3) and (4.4), the lens proper motion in the heliocentric frame is therefore

v π µ = µ + µ + rel . (4.25) l geo s ⊕AU

For each MCMC realization, πrel is known, so we can convert the lens proper motion to the velocity of the lens in the heliocentric frame vl,hel and also in the frame of local standard of rest vl,LSR (we ignore the rotation of the galactic bulge). The lens velocity in the LSR is estimated to be v = 103 15 km s−1. This raises l,LSR ± the possibility of the lens being in the thick disk, in which the stars are typically metal-poor. As shown in Figure 4.7, the constraints we have cannot resolve the metallicity of the lens star.

143 Planetary Orbital Motion

Wide/Close Degeneracy Binary-lens light curves in general exhibit a well-known “close-wide” symmetry (Dominik 1999b; An 2005). Even for some well-covered caustics-crossing events (e.g., Albrow et al. 1999a), there are quite degenerate sets of solutions between wide and close binaries. In Paper I, we found that the best-fit point-source wide-binary solution was preferred over close-binary solutions by ∆χ2 22. But this did not necessarily mean that the wide-close binary ∼ degeneracy was broken, since the two classes of binaries may be influenced differently by higher order effects. We find that the χ2 difference between best-fit wide and close solutions is within 1 from “MCMC A” and 2.1 (positive u0) or 2.2 (negative u0) from “MCMC B”.

However, orbital motion of the planet is subject to additional dynamical constraints: the projected velocity of the planet should be no greater than the escape velocity of the system: v v , where, ⊥ ≤ esc

2 2 AU v⊥ = d˙ +(ωd) θE, (4.26) q πl

2GM 2GM πl vesc = vesc,⊥ = c, (4.27) s r ≤ ≡ sdθEDl s2dπE and where r is the instantaneous 3-dimensional planet-star physical separation. Note that in the last step, we have used equation (4.1).

144 We then calculate the probability distribution of the ratio

2 2 3 ˙ 2 2 v⊥ AU d [(d/d) + ω ] πE 2 = 2 2 3 (4.28) vesc,⊥ c [πE +(πs/θE)] θE for an ensemble of MCMC realizations for both wide and close solutions. Figure 4.9 shows probability distributions of the projected velocity r⊥γ in the units of critical velocity vc,⊥, where r⊥γ is the instantaneous velocity of the planet on the sky, which is further discussed in Appendix C and vc,⊥ = vesc,⊥/√2. The dotted circle encloses the solutions that are allowed by the escape velocity criteria, and the solutions that are inside the solid line are consistent with circular orbital motion. We find that the best-fit close-binary solutions are physically allowed while the best-fit wide-binary solutions are excluded by these physical constraints at 1.6 σ. The physically excluded best-fit wide solutions are favored by ∆χ2 = 2.1 (or 2.2) over the close solutions, so by putting physical constraints, the degenerate solutions are statistically not distinguishable at 1 σ.

Circular Planetary Orbits and Planetary Parameters Planetary deviations in microlensing light curves are intrinsically short, so in most cases, only the instantaneous projected distance between the planet and the host star can be extracted. As shown in 4.3.6, for this event, we tentatively measure the § instantaneous projected velocity of the planet thanks to the relatively long

( 4 days) duration of the planetary signal. One cannot solve for the full set of ∼ 145 orbital parameters just from the instantaneous projected position and velocity.

However, as we show in Appendix C, we can tentatively derive orbital parameters by assuming that the planet follows a circular orbit around the host star. In Figure 4.10, we show the probability distributions of the semimajor axis, inclination, amplitude of radial velocity, and equilibrium temperature of the planet derived from “MCMC

B” for both wide and close solutions. The equilibrium temperature is defined to be T (L /L )1/4(2a/R )−1/2T , where L is the bolometric luminosity of eq ≡ bol bol,⊙ ⊙ ⊙ bol the host, a is the planet semimajor axis, and Lbol,⊙, R⊙, and T⊙ are the luminosity, radius, and effective temperature of the Sun, respectively. This would give the Earth an equilibrium temperature of Teq = 285 K. In calculating these probabilities, we assign a flat (Opik’s¨ Law) prior for the semimajor axis and assume that the orbits are randomly oriented, that is, with a uniform prior on cos i.

4.3.7. Constraints on a Non-Luminous Lens

In 4.3.5, we noted that the blended light must lie within 15 mas of the source: § otherwise the HST images would appear extended. We argued that the blended light must be associated with the event (either the lens itself or a companion to either the source or lens), since the chance of such an alignment by a random

ﬁeld star is < 0.07%. In fact, even stronger constraints can be placed on the blend-source separation using the arguments of 4.3.4. These are somewhat more § complicated and depend on the blend-source relative parallax, so we do not consider

146 the general case (which would only be of interest to further reduce the already very low probability of a random interloper) but restrict attention to companions of the source and lens. We begin with the simpler source-companion case.

Blend As Source Companion

As we reported in 4.3.4, there were two HST measurements of the astrometric § oﬀset between the V and I light centroids, dating from 0.09 and 0.84 years after peak, respectively. In that section, we examined the implications of these measurements under the hypothesis that the blend is the lens. We therefore ignored the ﬁrst measurement because the lens source separation at that epoch is much better constrained by the microlensing event itself than by the astrometric measurement.

However, as we now examine the hypothesis that the blend is a companion to the source, both epochs must be considered equally. Most of the weight (86%) comes from the second observation, partly because the astrometric errors are slightly smaller, but mainly because the blend contributes about twice the fractional light, which itself reduces the error on the inferred separation by a factor of 2. Under this hypothesis, we ﬁnd a best-ﬁt source-companion separation of 5 mas, with a companion position angle (north through east) of 280◦. The (isotropic) error is

3 mas. Approximating the companion-source relative motion as rectilinear, this measurement strictly applies to an epoch 0.73 years after the event, but of course

147 the intrinsic source-companion relative motion must be very small compared to the errors in this measurement.

There would be nothing unusual about such a source-companion projected separation, roughly 40 25 AU in physical units. Indeed, the local G-star binary ± distribution function peaks close to this value (Duquennoy & Mayor 1991).

The derived separation is also marginally consistent with the companion generating a xallarap signal that mimics the parallax signal in our dominant interpretation. The semi-major axis of the orbit would have to be about 0.8 AU to mimic the 1-year period of the Earth, which corresponds to a maximum angular separation of about 100 µas, which is compatible with the astrometric measurements at the 1.6 σ level.

Another potential constraint comes from comparing the color difference with the magnitude difference of the source and blend. We find that the source is about

0.5 0.5 mag too bright to be on the same main sequence. However, first, this is ± only a 1 σ difference, which is not significant. Second, both the sign and magnitude of the difference are compatible with the source being a slightly evolved turnoff star, which is consistent with its color.

The only present evidence against the source-companion hypothesis is that the astrometric oﬀset between V and I HST images changes between the two epochs, and that the direction and amplitude of this change is consistent with other evidence

148 of the proper motion of the lens. Since this is only a P = 1.7% effect, it cannot be regarded as conclusive. However, additional HST observations at a later epoch could definitively confirm or rule out this hypothesis.

Blend As A Lens Companion

A similar, but somewhat more complicated line of reasoning essentially rules out the hypothesis that the blend is a companion to the lens, at least if the lens is luminous. The primary diﬀerence is that the event itself places very strong lower limits on how close a companion can be to the lens.

A companion with separation (in units of θ ) d 1 induces a Chang-Refsdal E ≫ (1979) caustic, which is fully characterized by the gravitational shear γ = q/d2. We

ﬁnd that the light-curve distortions induced by this shear would be easily noticed unless γ < 0.0035, that is,

2 qc qcθE γ = 2 = 2 < 0.0035, (4.29) dc θc

where qc = Mc/M is the ratio of the companion mass to the lens mass and dc = θc/θE is the ratio of the lens-companion separation to the Einstein radius. Equivalently,

1/2 qc θc > 19 θE. (4.30) 1.3 149 Here, we have normalized qc to the minimum mass ratio required for the companion to dominate the light assuming that both are main-sequence stars. (We will also consider completely dark lenses below).

We now show that equation (4.30) is inconsistent with the astrometric data. If a lens companion is assumed to generate the blend light, then essentially the same line of reasoning given in 4.3.7 implies that 0.73 years after the event, this companion § lies 5 mas from the source, at position angle 280◦ and with an isotropic error of 3 mas. The one wrinkle is that we should now take account of the relative-parallax term in equation (4.15), whereas this was identically zero (and so was ignored) for the source-companion case. However, this term is only about 1.8πrel and hence is quite small compared to the measurement errors for typical πrel < 0.2mas. We will ∼ therefore ignore this term in the interest of simplicity, except when we explicitly consider the case of large πrel further below.

Of course, the lens itself moves during this interval. From the parallax measurement alone (i.e. without attributing the V/I astrometric displacement to lens motion), it is known that the lens is moving in the same general direction, i.e., with position angle roughly 210◦. In assessing the amplitude of this motion we consider only the constraints from finite-source effects (and ignore the astrometric displacement). These constraints yield a hard lower limit on θE (from lack of pronounced finite-source effects) of θE > 0.6 mas, which corresponds to a proper motion µ = 3.1 mas yr−1. At this extreme value (and allowing for 2 σ uncertainty

150 in the direction of lens motion as well in the measurement of the companion position), the maximum lens-companion separation is 11.4 mas (i.e., 19 θE), which is just ruled out by equation (4.30). At larger θE, the lens-companion scenario is excluded more robustly. For example, in the limit of large θE, we have

θ = µ 0.73 yr = θ (0.73 yr/t ) = 3.9θ , which is clearly ruled out by equation c × E E E (4.30).

Then we note that any scenario involving values of πrel that are large enough that they cannot be ignored in this analysis (πrel > 0.5 mas), must also have very ∼ large θE = πrel/πE > 1 mas, a regime in which the lens-companion is easily excluded. ∼

The one major loophole to this argument is that the lens may be a stellar remnant (white dwarf, neutron star, or black hole), in which case it could be more massive than the companion despite the latter’s greater luminosity.

4.3.8. Xallarap Effects and Binary Source

Binary source motion can give rise to distortions of the light curve, called

“xallarap” eﬀects. One can always ﬁnd a set of xallarap parameters to perfectly mimic parallax distortions caused by the Earth’s motion (Smith et al. 2003).

However, it is a priori unlikely for the binary source to have such parameters, so if the parallax signal is real, one would expect the xallarap ﬁts to converge to the

Earth parameters. For simplicity, we assume that the binary source is in circular

151 orbit. We extensively search the parameter space on a grid of 5 xallarap parameters, namely, the period of binary motion P , the phase λ and complement of inclination

β of the binary orbit, which corresponds to the ecliptic longitude and latitude in the parallax interpretation of the light curve, as well as (ξE,E, ξE,N), which are the counterparts of (πE,E,πE,N) of the microlens parallax. We take advantage of the two exact degeneracies found by Poindexter et al. (2005) to reduce the range of the parameter search. One exact degeneracy takes λ′ = λ + π and χ ′ = χ , while E − E all other parameters remain the same. The other takes β′ = β, u ′ = u and − 0 − 0 ξ′ = ξ (the sign of α should be changed accordingly as well). Therefore we E,N − E,N restrict our search to solutions with positive u and with π λ 2π. In modeling 0 ≤ ≤ xallarap, planetary orbital motion is neglected. In Figure 4.11, the χ2 distribution for best-ﬁt xallarap solutions as a function of period is displayed in a dotted line, and the xallarap solution with a period of 1 year has a ∆χ2 = 0.5 larger than the best ﬁt at 0.9 year. Figure 4.12 shows that, for the xallarap solutions with period of

1 year, the best fit has a ∆χ2 = 3.2 less than the best-fit parallax solution (displayed as a black circle point) and its orbital parameters are close to the ecliptic coordinates of event (λ = 268◦, β = 11◦). Therefore, the overall best-fit xallarap solution has − ∆χ2 = 3.7 smaller than that of the parallax solution (whose χ2 value is displayed as a filled dot in Fig. 4.11) for 3 extra degrees of freedom, which gives a probability of 30%. The close proximity between the best-fit xallarap parameters and those of the Earth can be regarded as good evidence of the parallax interpretation. The

152 slight preference of xallarap could simply be statistical ﬂuctuation or reﬂect low-level systematics in the light curve (commonly found in the analysis by Poindexter et al.

2005).

We also devise another test on the plausibility of xallarap. In 4.3.5, we argued § that the blend is unlikely to be a random interloper unrelated to either the source or the lens. If the source were in a binary, then the blend would naturally be explained as the companion of the source star. Then from the blend’s position on the CMD, its mass would be m 0.9 M . By deﬁnition, ξ is the size of the source’s orbit a c ∼ ⊙ E s in the units ofr ˆE (the Einstein radius projected on the source plane),

as amc ξE = = , (4.31) rˆE (mc + ms)ˆrE

where a is the semimajor axis of the binary orbit, and ms and mc are the masses of the source and its companion, respectively. Then we apply Kepler’s Third Law:

2 3 3 P mc ξErˆE 2 = . (4.32) yr M⊙(mc + ms) AU

Once the masses of the source and companion are known, the product of ξE and rÊ are determined for a given binary orbital period P . And in the present case, rˆ /AU = θ D = θ /ρD = 4.5 10−3/ρ. By adopting m = 1M , m = 0.9M , for E E s ∗ s × s ⊙ c ⊙ each set of P and ρ, there is a uniquely determined ξE from equation (4.32). We then apply this constraint in the xallarap fitting for a series of periods. The minimum χ2s for each period from the fittings are shown in solid line in Figure 4.11. The best-fit

153 solution has ∆χ2 1.0 less than the best-ﬁt parallax solution for two extra degrees ∼ of freedom. Although as compared to the test described in the previous paragraph, the current test implies a higher probability that the data are explained by parallax

(rather than xallarap) eﬀects, it still does not rule out xallarap.

4.4. Summary and Future Prospects

Our primary interpretation of the OGLE-2005-BLG-071 data assumes that the light-curve distortions are due to parallax rather than xallarap and that the blended light is due to the lens itself rather than a companion to the source. Under these assumptions, the lens is fairly tightly constrained to be a foreground M dwarf, with mass M = 0.46 0.04 M and distance D = 3.2 0.4 kpc, which has thick-disk ± ⊙ l ± kinematics (v 103 km s−1). As we discuss below, future observations might LSR ∼ help to constrain its metallicity. The microlens modeling suﬀers from a well-known wide-close binary degeneracy. The best-ﬁt wide-binary solutions are slightly favored over the close-binary solutions, however, from dynamical constraints on planetary orbital motion, the physically allowed solutions are not distinguishable within 1 σ.

For the wide-binary model, we obtain a planet of mass M = 3.8 0.4 M p ± Jupiter at projected separation r = 3.6 0.2 AU. The planet then has an equilibrium ⊥ ± temperature of about T = 55 K, i.e. similar to Neptune. In the degenerate

154 close-binary solutions, the planet is closer to the star and so hotter, and the estimates are: M = 3.4 0.4 M , r = 2.1 0.1 AU and T 71 K. p ± Jupiter ⊥ ± ∼

As we have explored in considerable detail, it is possible that one or both of these assumptions is incorrect. However, future astrometric measurements that are made after the lens and source have had a chance to separate, will largely resolve both ambiguities. Moreover, such measurements will put much tighter constraints on the metallicity of the lens (assuming that it proves to be the blended light).

First, the astrometric measurements made 0.84 yr after the event detected motion suggests that there was still 1.7% chance that the blend did not move relative to the source. A later measurement that detected this motion at higher conﬁdence would rule out the hypothesis that the blend is a companion to the source. We argued in 4.3.7 that the blend could not be a companion to a main-sequence § lens. Therefore, the only possibilities that would remain are that the lens is the blend, that the lens is a remnant (e.g., white dwarf), or that the blend is a random interloper (probability < 10−3). As we brieﬂy summarize below, a future astrometric measurement could strongly constrain the remnant-lens hypothesis as well.

Of course, it is also possible that future astrometry will reveal that the blend does not moving with respect to the source, in which case the blend would be a companion to the source. Thus, either way, these measurements would largely resolve the nature of the blended light.

155 Second, by identifying the nature of blend, these measurements will largely, but not entirely, resolve the issue of parallax vs. xallarap. If the blend proves not to be associated with the source, then any xallarap-inducing companion would have to be considerably less luminous, and so (unless it were a neutron star) less massive than the mc = 0.9 M⊙ that we assumed in evaluating equation (4.32). Moreover, stronger constraints onr ˆE (rhs of eq. [4.32]) would be available from the astrometric measurements. Hence, the xallarap option would be either excluded or very strongly constrained by this test.

On the other hand, if the blend were conﬁrmed to be a source companion, then essentially all higher order constraints on the nature of the lens would disappear.

The parallax “measurement” would then very plausibly be explained by xallarap, while the “extra information” about θE that is presently assumed to come from the blend proper-motion measurement would likewise evaporate.

These considerations strongly argue for making a future high-precision astrometric measurement. Recall that in the HST measurements reported in 4.3.5, § the source and blend were not separately resolved: the relative motion was inferred from the offset between the V and I centroids, which are displaced because the source and blend have different colors. Due to its well-controlled PSF, HST is capable of detecting the broadening of the PSF even if the separation of the lens and source is a fraction of the FWHM. Assuming that the proper motion is µ 4.4 mas yr−1, geo ∼ and based on our simulations in 4.3.5, such broadening would be confidently § 156 detectable about 5 years after the event (see also Bennett et al. 2007 for analytic

PSF broadening estimates). Ten years after the event, the net displacement would be 40 mas. This compares to a diﬀraction limited FWHM of 40 mas for H band on ∼ a ground-based 10m telescope and would therefore enable full resolution. The I H − color of the source is extremely well determined (0.01 mag) from simultaneous I and

H data taken during the event from the CTIO/SMARTS 1.3m in Chile. Hence, the ﬂux allocation of the partially or fully resolved blend and source stars would be known. The direct detection of a partially or fully resolved lens will provide precise photometric and astrometry measurements (see Kozlowski et al. 2007 for one such example), which will enable much tighter constraints on the mass, distance and projected velocity of the lens. It also opens up the possibility of determining the metallicity of the host star by taking into account both non-photometric and photometric constraints. If, as indicated by the projected velocity measurement, it is a thick-disk star, then it will be one of the few such stars found to harbor a planet

(Haywood 2008).

As remarked above, a deﬁnitive detection of the blend’s proper motion would still leave open the possibility that it was a companion to the lens, and not the lens itself. In this case, the lens would have to be a remnant. Without going into detail, the astrometric measurement would simultaneously improve the blend color measurement as well as giving a proper-motion estimate (albeit with large errors because the blend-source oﬀset at the peak of the event would then not be known).

157 It could then be asked whether the parallax, proper-motion, and photometric data could be consistently explained by any combination of remnant lens and main-sequence companion. This analysis would depend critically on the values of the measurements, so we do not explore it further here. We simply note that this scenario could also be strongly constrained by future astrometry.

4.5. Discussion

With the measurements presented here, and the precision with which these measurements allow us to determine the properties of the planet OGLE-2005-BLG-

071Lb and its host, it is now possible to place this system in the context of similar planetary systems discovered by radial velocity (RV) surveys. Of course, the kind of information that can be inferred about the planetary systems discovered via RV diﬀers somewhat from that presented here. For example, for planets discovered via

RV, it is generally only possible to infer a lower limit to the planet mass, unless the planets happen to transit or produce a detectable astrometric signal. Mutatis mutandis, for planets discovered via microlensing, it is generally only possible to measure the projected separation at the time of the event, even in the case for which the microlensing mass degeneracy is broken as it is here (although see Gaudi et al.

2008).

158 With these caveats in mind, we can compare the properties of OGLE-2005-

BLG-071Lb and its host star with similar RV systems. It is interesting to note that the fractional uncertainties in the host mass and distance of OGLE-2005-BLG-071Lb are comparable to those of some of the systems listed in Table 4.6.

OGLE-2005-BLG-071Lb is one of only eight Jovian-mass (0.2MJupiter <

Mp < 13MJupiter) planets that have been detected orbiting M dwarf hosts (i.e.,

M∗ < 0.55 M⊙) (Marcy et al. 1998, 2001; Delfosse et al. 1998; Butler et al. 2006;

Johnson et al. 2007b; Bailey et al. 2008). Table 4.6 summarizes the planetary and host-star properties of the known M dwarf/Jovian-mass planetary systems.

OGLE-2005-BLG-071Lb is likely the most massive known planet orbiting an M dwarf.

As suggested by the small number of systems listed in Table 4.6, and shown quantitatively by several recent studies, the frequency of relatively short-period

P < 2000 days, Jupiter-mass companions to M dwarfs appears to be 3 5 times ∼ ∼ − lower than such companions to FGK dwarfs (Butler et al. 2006; Endl et al. 2006;

Johnson et al. 2007b; Cumming et al. 2008). This paucity, which has been shown to be statistically signiﬁcant, is expected in the core-accretion model of planet formation, which generally predicts that Jovian companions to M dwarfs should be rare, since for lower mass stars, the dynamical time at the sites of planet formation is longer, whereas the amount of raw material available for planet formation is smaller

(Laughlin et al. 2004; Ida & Lin 2005; Kennedy & Kenyon 2008, but see Kornet

159 et al. 2006). Thus, these planets typically do not reach suﬃcient mass to accrete a massive gaseous envelope over the lifetime of the disk. Consequently, such models also predict that in the outer regions of their planetary systems, lower mass stars should host a much larger population of ‘failed Jupiters,’ cores of mass < 10 M⊕ ∼ (Laughlin et al. 2004; Ida & Lin 2005). Such a population was indeed identiﬁed based on two microlensing planet discoveries (Beaulieu et al. 2006; Gould et al.

2006).

Our detection of a 4 M companion to an M dwarf may therefore present ∼ Jupiter a diﬃculty for the core-accretion scenario. While we do not have a constraint on the metallicity of the host, the fact that it is likely a member of the thick disk suggests that its metallicity may be subsolar. If so, this would pose an additional diﬃculty for the core-accretion scenario, which also predicts that massive planets should be rarer around metal-poor stars (Ida & Lin 2004), as has been demonstrated observationally

(Santos et al. 2004; Fischer & Valenti 2005). This might imply that a diﬀerent mechanism is responsible for planet formation in the OGLE-2005-BLG-071L system, such as the gravitational instability mechanism (Boss 2002, 2006).

One way to escape these potential diﬃculties is if the host lens is actually a stellar remnant, such as white dwarf. The progenitors of remnants are generally more massive stars, which are both predicted (Ida & Lin 2005; Kennedy & Kenyon

2008) and observed (Johnson et al. 2007a,b) to have a higher incidence of massive planets. As we discussed above, future astrometric measurements could constrain

160 both the low-metallicity and remnant-lens hypotheses. These measurements are therefore critical.

Although it is diﬃcult to draw robust conclusions from a single system, there are now four published detections of Jovian-mass planetary companions with microlensing (Bond et al. 2004; Gaudi et al. 2008), and several additional such planets have been detected that are currently being analyzed. It is therefore reasonable to expect several detections per year (Gould 2009), and thus that it will soon be possible to use microlensing to constrain the frequency of massive planetary companions. These constraints are complementary to those from RV, since the microlensing detection method is less biased with respect to host star mass (Gould

2000a), and furthermore probes a diﬀerent region of parameter space, namely cool planets beyond the snow line with equilibrium temperatures similar to the giant planets in our solar system (see, e.g. Gould et al. 2007 and Gould 2009).

161 OGLE I OGLE V Clear I V Clear I I MOA I PLANET Canopus I RoboNet FTN R

15 15.2 15.4 15.6 15.8 16 3479 3480 3481 3482 3483

Fig. 4.1.— Main panel: all available ground-based data of the microlensing event OGLE-2005-BLG-071. HST ACS HRC observations in F814W and F555W were taken at two epochs, once when the source was magnified by A 2 (arrow), and again at HJD = 2453788.2 (at baseline). Planetary models that include∼ (solid) and excludes (dotted) microlens parallax are shown. Zoom at bottom: triple-peak feature that reveals the presence of the planet. Each of the three peaks corresponds to the source passing by a cusp of the central caustic induced by the planet. Upper inset: trajectory of the source relative to the lens system in the units of angular Einstein radius θE. The lens star is at (0, 0), and the star-planet axis is parallel to the x-axis. The best-fit angular size of the source star in units of θE is ρ 0.0006, too small to be resolved in this figure. ∼

162 Fig. 4.2.— Probability contours (∆χ2 = 1, 4) of microlens parallax parameters derived from MCMC simulations for wide-binary (in solid line) and close-binary (in a dashed line) solutions. Fig. 2 and eq. (12) in Gould (2004) imply that πE,⊥ is deﬁned so that πE,k and πE,⊥ form a right-handed coordinate system.

163 Fig. 4.3.— CMD for the OGLE-2005-BLG-071 field. Black dots are the stars with the OGLE I-band and V -band observations. The red point and green points show the center of red clump and the source, respectively. The errors in their fluxes and colors are too small to be visible on the graph. Cyan points are the stars in the ACS field, which are photometrically aligned with OGLE stars using 10 common stars. The magenta point with error bars show the color and magnitude of the blended light.

164 Fig. 4.4.— HST ACS astrometric measurements of the target star in F814W (red) and F555W (blue) filters in 2005 (filled dots) and 2006 (open dots). The center positions of the big circles show mean values of the 4 dithered observations in each filter at each epoch while radii of the circle represent the 1 σ errors.

165 finite-source wide finite-source close astrometry astrometry joint joint

parallax wide parallax close joint joint

Fig. 4.5.— Upper two panels show posterior probability contours at ∆χ2 = 1 (solid line) and 4 (dotted line) for relative lens-source proper motion µgeo. The left panel is for wide-binary solutions and the right one is for close-binary. The green contours show the probability distributions constrained by the finite-source effects. The black contours are derived from HST astrometry measurements assuming πrel = 0.2 mas. The red contours show the joint probability distributions from both constraints. The lower two panels show the posterior probability distribution of the position angle φµgeo of the relative lens-source proper motion for wide-binary and close-binary solutions, respectively. The histogram in red is derived from the red contours of joint probability for finite source and astrometry constraints in the upper panel. The blue histogram represents that of the microlens parallax. They mildly disagree at 2.5σ.

166 Fig. 4.6.— Differences between OGLE V and HST F555W magnitudes for the matched stars are plotted against their V magnitudes measured by OGLE. To calculate the offset, we add a 0.017 mag “cosmic error” in quadrature to each point in order to reduce χ2/dof to unity. The open circles represent the stars used to establish the final transformation, and the filled point shows an “outlier”.

167 V-I

1.8 (best-estimate) 2.0

2.1 2.3 2.6

wide close

Fig. 4.7.— Posterior probability distribution of lens mass M and relative lens- source parallax πrel from MCMC simulations discussed in 4.3.6. The constraints include those from parallax effects, finite-source effects and§ relative proper motion measurements from HST astrometry. The ∆χ2 = 1, 4, 9 contours are displayed in solid, dotted and dashed lines, respectively. Both wide-binary (magenta) and close- binary (blue) solutions are shown. The lines in black, red and green represent the predicted M and πrel from the isochrones for different metal abundances: [M/H] = 0 (black), 0.5 (red), 1.0 (green). The points on these lines correspond to the observed I-band magnitude− I−= 21.3 and various V I values V I = 1.8 (best estimate, filled dots), 2.0 (0.5 σ, filled triangle), 2.1 (1.0 σ,− filled squares),− 2.3 (1.5 σ, filled pentagons), and 2.6 (2.0 σ, filled hexagons)

168 Fig. 4.8.— Posterior probability distribution of lens mass M and relative lens-source parallax πrel from MCMC simulations assuming that the blended light comes from the lens star. The ∆χ2 = 1, 4 contours are displayed in a solid line for wide solutions, and in a dotted line for close solutions.

169 Fig. 4.9.— Probability contours of projected velocity r⊥γ (deﬁned in Appendix C) in the units of vc,⊥ for both close-binary (upper panel) and wide-binary (lower panel) solutions. All the solutions that are outside the dotted circle are physically rejected as the velocities exceed the escape velocity of the system. The boundary in a solid line inside the dotted circle encloses the solutions for which circular orbits are allowed.

170 Fig. 4.10.— Probability distributions of planetary parameters (semimajor axis a, equilibrium temperature, cosine of the inclination, and amplitude of radial velocity of the lens star) from MCMC realizations assuming circular orbital motion. Histograms in black and red represent the close-binary and wide-binary solutions, respectively. Dotted and dashed histograms represent the two degenerate solutions for each MCMC realization discussed in Appendix C.

171 Fig. 4.11.— χ2 distributions for best-fit xallarap solutions at fixed binary-source orbital periods P . The solid and dotted lines represent xallarap fits with and without dynamical constraints described in 4.3.8. The best-fit parallax solution is shown as a filled dot at period of 1 year. All of§ the fits shown in this figure assume no planetary orbital motion.

172 Fig. 4.12.— Results of xallarap fits by fixing binary orbital phase λ and complement of inclination β at period P = 1yr and u0 > 0. The plot is color-coded for solutions with ∆χ2 within 1 (black), 4 (red), 9 (green), 16 (blue), 25 (magenta), 49 (yellow) of the best fit. The Earth parameters are indicated by black circles. Because of a perfect symmetry (u0 u0 and α α), the upper black circle represents Earth parameter (λ = 268◦,→ β = − 11◦) for→ the − case u < 0. Comparison of parallax with − 0 xallarap must be made with the better of the two, that is, the lower one.

173 Model t0 u0 tE d q α ρπE,N πE,E ω d/d˙

χ2 (HJD’) (day) 103 (deg) 104 (yr−1) (yr−1) × ×

Wide+ 3480.7024 0.0282 71.1 1.306 7.5 273.63 3.9 -0.30 -0.26 -1.328 -0.256

1345.0 +0.0058 +0.0008 +2.3 +0.002 0.2 +0.16 +1.8 +0.24 0.05 +0.274 +0.134 −0.0054 −0.0009 −2.4 −0.004 ± −0.15 −2.7 −0.28 ± −0.165 −0.129 Wide 3480.7028 -0.0283 70.6 1.307 7.5 86.21 3.9 -0.34 -0.26 1.117 -0.277 − 174 1345.3 +0.0054 +0.0010 2.2 +0.003 0.3 +0.13 +1.8 +0.30 0.05 +0.130 +0.152 −0.0056 −0.0008 ± −0.004 ± −0.16 −2.6 −0.23 ± −0.293 −0.113 Close+ 3480.6789 0.0239 70.1 0.763 6.9 274.27 3.1 -0.36 -0.27 0.301 0.502

1345.8 +0.0055 +0.0009 +2.1 +0.004 0.3 +0.25 +1.7 +0.24 0.05 +0.486 +0.148 −0.0043 −0.0007 −2.4 −0.006 ± −0.36 −2.5 −0.27 ± −0.788 −0.101 Close 3480.6799 -0.0241 69.2 0.762 6.9 85.53 2.7 -0.33 -0.26 -0.405 0.528 − 1345.2 +0.0042 +0.0009 +2.3 +0.004 0.3 +0.39 2.2 +0.28 0.05 +0.696 +0.127 −0.0051 −0.0007 −1.9 −0.006 ± −0.26 ± −0.26 ± −0.622 −0.118

Table 4.1. OGLE-2005-BLG-071 Light Curve Parameter Estimations From Markov chain Monte

Carlo Simulations A (Part 1). Model Is Ib Vs Vb

χ2 (mag) (mag) (mag) (mag)

Wide+ 19.51 21.29 20.85 23.11

1345.0 +0.04 +0.22 0.04 +0.43 −0.03 −0.17 ± −0.26 Wide 19.51 21.30 20.85 23.11 − 1345.3 0.04 0.19 0.03 +0.43 ± ± ± −0.26 Close+ 19.52 21.28 20.85 23.13

1345.8 0.03 +0.21 +0.04 +0.40 ± −0.16 −0.03 −0.28 Close 19.52 21.30 20.85 23.17 − 1345.2 0.03 0.18 0.04 +0.35 ± ± ± −0.32

Table 4.2. OGLE-2005-BLG-071 Light Curve Parameter Estimations

From Markov chain Monte Carlo Simulations A (Part 2).

175 Model t0 u0 tE d q α ρπE,N πE,E ω d/d˙

χ2 (HJD’) (day) 103 (deg) 104 (yr−1) (yr−1) × ×

Wide+ 3480.7015 0.0287 69.3 1.305 7.7 273.67 6.1 -0.02 -0.22 -1.242 -0.283

1353.4 +0.0050 0.0007 +1.6 +0.003 0.2 +0.17 0.4 0.12 0.03 +0.321 0.129 −0.0059 ± −1.7 −0.005 ± −0.11 ± ± ± −0.125 ± Wide 3480.7012 -0.0287 69.2 1.305 7.7 86.29 6.0 0.02 -0.21 1.193 -0.293 − 176 1353.3 +0.0052 0.0007 +1.7 +0.002 0.2 +0.12 +0.5 +0.10 0.03 +0.127 +0.131 −0.0060 ± −1.8 −0.005 ± −0.15 −0.3 −0.13 ± −0.342 −0.121 Close+ 3480.6792 0.0245 68.3 0.763 7.0 274.38 6.0 -0.01 -0.22 0.415 0.569

1355.5 +0.0041 +0.0005 1.6 +0.003 +0.3 +0.23 0.4 +0.12 0.02 +0.503 +0.112 −0.0051 −0.0006 ± −0.006 −0.2 −0.39 ± −0.15 ± −0.744 −0.130 Close 3480.6793 -0.0245 68.2 0.762 7.1 85.63 6.0 0.04 -0.22 -0.179 0.561 − 1355.5 +0.0042 0.0006 +1.8 +0.004 0.3 +0.46 0.4 +0.09 0.03 +0.703 +0.126 −0.0051 ± −1.5 −0.006 ± −0.24 ± −0.16 ± −0.722 −0.112

Table 4.3. OGLE-2005-BLG-071 Light Curve Parameter Estimations From Markov chain Monte

Carlo Simulations B (Part 1). Model Is Ib Vs Vb

χ2 (mag) (mag) (mag) (mag)

Wide+ 19.49 21.40 20.82 23.91

1353.4 0.03 0.19 0.03 +0.24 ± ± ± −0.20 Wide 19.49 21.40 20.82 23.97 − 1353.3 0.02 0.19 0.03 +0.19 ± ± ± −0.24 Close+ 19.49 21.35 20.83 23.88

1355.5 0.03 +0.20 0.02 +0.24 ± −0.16 ± −0.18 Close 19.50 21.36 20.83 23.90 − 1355.5 0.02 0.18 0.02 0.21 ± ± ± ±

Table 4.4. OGLE-2005-BLG-071 Light Curve Parameter Estimations

From Markov chain Monte Carlo Simulations B (Part 2).

177 Model M πrel Dl µN µE θE Mp r⊥

2 −1 −1 χ M⊙ mas kpc mas yr mas yr mas MJupiter AU

Wide+ 0.46 0.19 3.2 -0.4 -4.3 0.84 3.8 3.6

1353.4 0.04 0.04 0.4 +2.7 0.3 +0.06 +0.3 0.2 ± ± ± −3.1 ± −0.04 −0.4 ± Wide 0.46 0.19 3.2 0.3 -4.3 0.85 3.8 3.6 − 1353.3 0.04 +0.04 0.4 +2.3 +0.3 0.05 +0.3 0.2 ± −0.03 ± −3.6 −0.2 ± −0.4 ± Close+ 0.46 0.19 3.1 -2.6 -4.4 0.86 3.4 2.1

1355.5 0.04 +0.04 0.4 +4.8 0.3 0.05 +0.3 0.1 ± −0.03 ± −1.1 ± ± −0.4 ± Close 0.46 0.20 3.1 -0.2 -4.4 0.87 3.4 2.1 − 1355.5 0.04 0.04 0.3 +3.6 0.3 0.04 0.3 0.1 ± ± ± −3.4 ± ± ± ±

Table 4.5. OGLE-2005-BLG-071 Derived Physical Parameters

178 Name M∗ Metallicity Dist. Mp P a Ref.

(M⊙) (pc) (MJup) (days) (AU)

GJ 876c 0.32 0.12 4.660 0.6 30.340 0.13030 1,2,3 − − 0.03 0.12 0.004 0.8 0.013 ± ± ± ± GJ 876b – – – 1.9 60.940 0.20783 – − 2.5 0.013 ± GJ 849b 0.49 0.16 8.8 0.82/ sin i 1890 2.35 4

0.05 0.2 0.2 130 ± ± ± ± GJ 317b 0.24 0.23 9.2 1.2/ sin i 692.9 0.95 5 − 0.04 0.2 1.7 4 ± ± ± ± GJ 832b 0.45 0.7 4.93 0.64/ sin i 3416 3.4 6 ∼ − ∼ 0.05 /-0.3 131 0.4 ± ± ± OGLE-2006 0.50 ? 1490 0.71 1830 2.3 7

-BLG-109Lb 0.05 ? 130 0.08 370 0.2 ± ± ± ± ± OGLE-2006 – – – 0.27 5100 4.6 –

-BLG-109Lc 0.03 730 0.5 ± ± ±

(cont’d)

Table 4.6. Jovian-mass Companions to M Dwarfs (M∗ < 0.55 M⊙)

179 Table 4.6—Continued

Name M∗ Metallicity Dist. Mp P a Ref.

(M⊙) (pc)(MJup) (days) (AU)

OGLE-2005 0.46 Subsolar?a 3300 3.8b – 3.6b,c This

-BLG-071Lb 0.04 300 0.4 0.2 Paper ± ± ± ±

aWhile the metallicity of the OGLE-2005-BLG-071Lb host star is not directly

constrained by our data, its kinematics indicate it is likely a member of the metal-

poor thick disk. bWe give the planet mass and projected separation for the wide solution, which

is favored by ∆χ2 = 2.1. The second, close solution has M = 3.4 0.3 M p ± Jupiter and r = 2.1 0.1 AU. ⊥ ± cWe give the the projected separation between the host and planet at the

time of event, which is the orbital parameter most directly constrained by our

observations. However, assuming a circular orbit, we infer that the semi-major

axis is likely only 10 20% larger [a(wide) 4.1AU,a(close) 2.5AU]. ∼ − ∼ ∼

References. — (1)Rivera et al. 2005; (2)Bean et al. 2006; (3)Benedict et al.

2002; (4)Butler et al. 2006; (5) Johnson et al. 2007b (6) Bailey et al. 2008 (7)

Gaudi et al. 2008

180 Chapter 5

First Space-Based Microlens Parallax Measurement: Spitzer Observations of OGLE-2005-SMC-001

5.1. Introduction

In a visionary paper written more than 40 years ago, Refsdal (1966) argued that two important but otherwise unmeasurable parameters of microlensing events could be determined by simultaneously observing the event from the Earth and a satellite in solar orbit. In modern language, these are the Einstein radius projected onto the observer plane,r ˜E, and the direction of lens-source relative proper motion.

Since the Einstein timescale, tE, is routinely measured for all events, these parameter determinations are equivalent to knowing the projected relative velocity, v˜, whose magnitude is simplyv ˜ r˜ /t . Here,r ˜ AU/π , t = θ /µ, and ≡ E E E ≡ E E E

π π = rel , θ = κMπ , (5.1) E κM E rel r q 181 where πE is the microlens parallax, M is the mass of the lens, θE is the angular

Einstein radius, πrel and µ are the lens-source relative parallax and proper motion, respectively, and κ 4G/(c2AU). ≡

The practical importance of this suggestion became clear when the MACHO

(Alcock et al. 1993) and EROS (Aubourg et al. 1993) collaborations reported the detection of microlensing events toward the Large Magellanic Cloud (LMC). Over the course of time, MACHO (Alcock et al. 1997, 2000) has found about 15 such events and argued that these imply that about 20% of the Milky Way dark halo is composed of compact objects (“MACHOs”), while EROS (Afonso et al. 2003;

Tisserand et al. 2006) has argued that their relative lack of such detections was consistent with all the events being due to stars in the Milky Way disk or the

Magellanic Clouds (MCs) themselves. For any given individual event, it is generally impossible to tell (with only a measurement of tE) where along the line of sight the lens lies, so one cannot distinguish among the three possibilities: Milky Way disk,

Milky Way halo, or “self-lensing” in which the source and lens both lie in the same external galaxy.

However, as Boutreux & Gould (1996) argued, measurement of v˜ might allow one to distinguish among these populations with good conﬁdence: disk, halo, and

MC lenses typically havev ˜ values of 50, 300, and 2000 km s−1, respectively. The high projected speed of MC lenses derives from the long “lever arm” that multiplies their

182 small local transverse speed by the ratio of the distances from the observer and the lens to the source.

There are serious obstacles, both practical and theoretical to measuring v˜. One obvious practical problem is simply launching a spacecraft with a suitable camera into solar orbit. But the theoretical diﬃculties also place signiﬁcant constraints on the characteristics of that spacecraft. To understand these properly, one should think in terms of the “microlens parallax” π , whose magnitude is π AU/r˜ E E ≡ E and whose direction is the same as v˜. Choosing a coordinate system whose x-axis is aligned with the Earth-satellite separation at the peak of the event, we can write

πE =(πE,τ ,πE,β). Then to good approximation,

AU ∆t0 πE =(πE,τ ,πE,β)= , ∆u0 , (5.2) d⊥ tE

where d⊥ is the Earth-satellite separation (projected onto the plane of the sky), ∆t0 is the difference in time of event maximum as seen from the Earth and satellite, and ∆u0 is the difference in dimensionless impact parameter (determined from the maximum observed magnification).

Refsdal (1966) already realized that equation (5.2) implicitly contains a four-fold degeneracy: while ∆t0/tE is unambiguously determined, there are four diﬀerent values of ∆u0 that depend on whether the individual impact parameters are positive or negative (on one side of the lens or the other; see Fig. 2 of Gould 1994b).

183 In fact, the situation is considerably worse than this. While t0 is usually measured very precisely in individual microlensing events, u0 typically has much larger errors because it is strongly correlated with three other parameters, the timescale, tE, the source ﬂux f , and the blended ﬂux, f . For a satellite separated by d 0.2AU, and s b ⊥ ∼ a projected Einstein radiusr ˜ 5 AU, errors in the impact-parameter determinations E ∼ of only σ(u ) 2% would lead to fractional errors σ(π )/π √2σ(u )˜r /d 70%. 0 ∼ E E ∼ 0 E ⊥ ∼ However, Gould (1995) showed that if the two cameras had essentially identical spectral responses and similar point-spread-functions, so that one knew a priori that the blended light was virtually identical for the Earth and satellite measurements, then the error in ∆u0 would be reduced far below the individual errors in u0, making the parallax determination once again feasible.

Unfortunately, this trick cannot be used on Spitzer, the ﬁrst general purpose camera to be placed in solar orbit. The shortest wavelength at which Spitzer operates is the L band (3.6 µm), implying that the camera’s sensitivity cannot be duplicated from the ground, because of both higher background and diﬀerent throughput as a function of wavelength.

In principle, microlens parallaxes can also be measured from the ground.

As with space-based parallaxes, one component of πE can generally be measured much more precisely than the other. For most events, t yr, and for these the E ≪ Earth’s acceleration can be approximated as constant during the event. To the degree that this acceleration is aligned (anti-aligned) with the lens-source relative

184 motion, it induces an asymmetry in the light curve, since the event proceeds faster

(slower) before peak than afterward (Gould et al. 1994). This is characterized by the “asymmetry parameter” Γ π α = π α, where α is the apparent ≡ E · E,k acceleration of the Sun projected onto the sky and normalized to an AU, and πE,k is the component of πE parallel to α. Since Γ is directly measurable from the light curve, one can directly obtain 1-D parallax information πE,k =Γ/α for these events, while the orthogonal component πE,⊥ is measured extremely poorly (e.g., Ghosh et al. 2004; Jiang et al. 2005). While there are a few exceptions (Alcock et al. 2001a;

Gould et al. 2004; Park et al. 2004), 2-D parallaxes can generally only be obtained for relatively long events tE > 90 days, and even for these, the πE error ellipse is ∼ generally elongated in the πE,⊥ direction (Poindexter et al. 2005).

Since Spitzer is in an Earth-trailing orbit and the SMC is close to the ecliptic pole, the πE,τ direction (deﬁned by the Earth-satellite separation vector) is very nearly orthogonal to the πE,k direction (deﬁned by the direction of the Sun).

Recognizing this, Gould (1999) advocated combining the two essentially 1-D parallaxes from the Earth-Spitzer comparison and the accelerating Earth alone to produce a single 2-D measurement of πE. He noted that once the diﬃcult problem of measuring πE,β was jettisoned, the satellite observations could be streamlined to a remarkable degree: essentially only 3 observations were needed, 2 at times placed symmetrically around the peak, which are sensitive to the oﬀset in t0 between the

Earth and satellite, and a third at late times to set the ﬂux scale. This streamlining is

185 important from a practical point of view because Target-of-Opportunity (ToO) time on Spitzer incurs a large penalty. Gould (1999) noted that the components of πE measured by the two techniques were not exactly orthogonal but argued (incorrectly as it turns out) that this had no signiﬁcant consequences for the experiment. We return to this point below.

Here we analyze Spitzer and ground-based observations of the microlensing event OGLE-2005-SMC-001 to derive the ﬁrst microlens parallax measurement using this technique.

5.2. Observations

On 2005 July 9 (HJD′ HJD-2450000 = 3561.37), the OGLE-III Early ≡ Warning System (EWS, Udalski 2003) alerted the astronomical community that

OGLE-2005-SMC-001 (α = 0h40m28s.5, δ = 73◦44′46.1′′) was a probable J2000.0 J2000.0 − microlensing event, approximately 23 days (and seven observations) into the 2005

OGLE-III observing season for the SMC. In fact, the EWS issued an internal alert

ﬁve days earlier, when there were only three 2005 points, but the OGLE team reacted cautiously because of the high rate of questionable alerts toward the SMC and because the source lies projected against a background galaxy, making it a potential supernova candidate. However, the event shows a modest, but unambiguous rise 140 days earlier, at the end of the 2004 season, which is inconsistent with a supernova,

186 and the unmagniﬁed source sits right in the middle of the red giant clump on the color-magnitude diagram (CMD), with (V I,I) = (0.92, 18.4). Moreover, − the light curve is achromatic. These factors convinced us that this was genuine microlensing, leading us to exercise our Spitzer ToO option, which consisted of three 2-hr observations: two placed symmetrically around the peak and one at the baseline. Once this decision was made, OGLE increased its density of coverage to

3–5 I-band observations per clear night. OGLE observations were obtained using the 1.3 m Warsaw telescope at Las Campanas Observatory in Chile, operated by the Carnegie Institution of Washington. The photometry was reduced using the standard OGLE-III data pipeline (Udalski 2003) based on the image subtraction technique, DIA (Wo´zniak 2000). Also many V -band observations were obtained during the event for monitoring achromaticity. There are Spitzer observations at four epochs (not three, as originally envisaged). These were centered at 2005 July

15 UT 20:02:40, 2005 August 25 UT 12:44:25, 2005 September 15 UT 20:13:53, and 2005 November 29 UT 10:24:40. The ﬁrst, third, and fourth observations each lasted 2 hr and consisted of two sets of about 100 dithered exposures, each of 26.8 s. The second observation (in August) was 1 hr, consisting of one set of 99 dithered exposures, each of 26.8 s. It was obtained with director’s discretionary time (DDT).

All four were carried out simultaneously at 3.6 and 5.8 µm. However, the third observation (which took place at relatively high magniﬁcation) was supplemented by

30 minutes of very short exposures in all four Infrared Array Camera (IRAC) ﬁlters

187 to probe the detailed spectral energy distribution of the source. The reason for the additional DDT observation is discussed in detail in 5.3. §

As originally conceived, the experiment was to consist only of OGLE and

Spitzer observations. However, unexpected complications led us to take additional data from other ground-based observatories as well as the Hubble Space Telescope

(HST).

Initially, we obtained some data using the 1.3 m SMARTS (former Two Micron

All Sky Survey [2MASS]) telescope at the Cerro Tololo Inter-American Observatory

(CTIO) in Chile simply as a precaution against possible future problems with the

OGLE telescope. (In order to align different light curves, it is generally necessary that they have some overlap; one cannot wait for the problems to arise before beginning to take data.) However, as the event approached peak, we found that it could not be fit with a classical Paczyński (1986) model, even when augmented by parallax. We therefore began to intensively observe the event from both the OGLE and SMARTS telescopes in the hopes of obtaining enough data to determine the nature of the light curve anomaly. Similar considerations led us to begin observations using the 0.35m Nustrini telescope at the Auckland Observatory in New Zealand, which lies at a substantially different longitude and suffers from substantially different weather patterns from those experienced by the two Chile telescopes.

188 Additionally, several high-dispersion spectra were obtained at Las Campanas

Observatory using 6.5 m Magellan and 2.5 m du Pont telescopes with echelle spectrographs at different magnifications in order to check for potential radial- velocity variations in the spectra of the magnified source.

Finally, the anomalous behavior made it prudent to get high-resolution images using HST, both to improve the modeling of blended light in the ground-based and

Spitzer images, and to determine whether the apparent ground-based source could actually be resolved into multiple sources. We had HST ToO time to complement low-resolution space-based microlensing parallax observations, which was originally to be applied to observations by Deep Impact. With the probability low that these would be triggered as originally planned, we applied this time to obtain two orbits of observations of OGLE-2005-SMC-001. There were two epochs of (V,I,J,H,K) exposures, on 2005 October 1 and 2006 May 17/18, with exposure times of (300,

200, 351, 351, and 639) seconds. The infrared observations were then repeated on

2006 June 25.

5.2.1. Error Rescaling

Errors from each ground-based observatory are rescaled to force χ2 per degree of freedom close to unity. For the OGLE data, we ﬁnd by inspecting the cumulative distribution of the normalized residuals (δ/σ)2, that the rescaling factor is not

189 uniform over the data set. Here σ is the error reported by OGLE and δ is the deviation of the data from the model. We therefore rescale in 4 segments, which are separated at HJD = 2453100, 2453576, and 2453609, with rescaling factors 1.4, 1.9,

3.6, and 2.0. We tested two other error-rescaling schemes, one with no rescaling and the other with uniform rescaling of the OGLE data. We found that the solutions do not diﬀer qualitatively when these alternate schemes are used.

5.2.2. Spitzer Data Reduction and Error Determination

Our scientiﬁc goals critically depend on obtaining high-precision IRAC photometry for each of the 4 epochs. See 5.3.1. These 4 epochs are divided into § 7 1-hour sub-epochs, each consisting of about 100 dithers, one sub-epoch for the second epoch and two for each of the other three epochs. Based on photon statistics alone, the best possible precision would be about 0.2% for the ﬁve sub-epochs near peak, and 0.4% for the last two sub-epochs. However, there are three interrelated ∼ problems that must be overcome to even approach this potential. First, the images contain “stripes” produced by nearby bright stars, perhaps AGB stars, which

(because the 3 near-peak observations took place over 60 days) appear at several diﬀerent rotation angles. Indeed, we expended considerable eﬀort repositioning each successive image to avoid having these stripes come too close to, or actually overlap, the microlensed target, but they inevitably did overlap some reference stars. Second, the microlensed source is blended with a neighboring star within 1.′′3, which is

190 easily resolved in HST images and clearly resolved in OGLE images as well. Third, this problem is signiﬁcantly complicated by the well-known fact that IRAC 3.6 µm images are undersampled.

We apply the procedures of Reach et al. (2005) to perform aperture photometry on the Basic Calibrated Data (BCD), which includes array-location-dependent and

“pixel-phase” photometric corrections at the few percent level. We choose 7 bright and isolated stars, which we select from the OGLE images. The HST frames are of course even better resolved, but they are too small to contain a big enough sample of reference stars. The centroid position of the target-star aperture on each of the (roughly 100) BCD dithers is determined by aligning the comparison stars with the OGLE coordinates. We determine the “internal error” for the target star and the comparison stars at each sub-epoch from the internal scatters in their measurements. This is typically very close to the photon limit. However, we find that the epoch-to-epoch scatter in the comparison stars is about 0.7%. While in principle this could be due to intrinsic variability, such variability is unlikely to be so pervasive at this level, particularly since any star showing variability in the I band over several years was excluded as a reference star. Hence, we attribute this variation to unknown epoch-to-epoch systematics, and we assume that these affect the target in the same way that they affect the reference stars. Hence, we adopt

0.7% as our photometric error for each of the 7 Spitzer sub-epochs.

191 We also attempted to do point-spread-function (rather than aperture) photometry, making use of “Point Response Functions” available at the Spitzer

Science Center Web site. However, we found that the reference stars showed greater scatter between sub-epochs with this approach and so adopted the results from aperture photometry.

5.3. Complications Alter Strategy and Analysis

Figure 5.1 shows the ground-based light curve with a ﬁt to a standard

(Paczy´nski 1986) model. The residuals are severe. The model does not include parallax. However, models that include parallax are quite similar. In the period before the peak, we were constantly reﬁtting the light curve with every new night’s data in order to be able to predict the time of peak and thus the time of the second

ToO observation (which was supposed to be symmetric around the peak with the

ﬁrst). It became increasingly clear that the event was not standard microlensing, and we began to consider alternate possibilities, including binary source (also called

“xallarap”), binary lens, and variable source. The last was especially alarming because if the variability were irregular, it would be almost impossible to model at the high precision required to carry out this experiment, particularly because the source might vary diﬀerently at I and L. Our concern about xallarap led us to obtain radial-velocity measurements at several epochs near the peak. These turned

192 out to be the same within less than 1kms−1, which ruled out xallarap for all but the most pathologically face-on orbits. But regardless of the nature of the anomaly, it could potentially cause serious problems because much more and better data are required to accurately model complicated light curves compared to simple ones.

Hence, as described in 5.2, recognition of the anomaly caused us to significantly § intensify our ground-based observations. Moreover, it also caused us to think more carefully about how we would extract parallax information from a more complicated light curve, and this led us to recognize a complication that affects even light curves that do not suffer from additional anomalies.

5.3.1. Need for Additional Spitzer Observation

Recall that πE,τ is derived from the diﬀerent peak time t0 as seen from Earth and Spitzer: if the two Spitzer observations are timed so that the ﬂuxes seen at

Earth are equal to each other (one on the rising and one on the falling wing of the light curve), then the Spitzer fluxes will nevertheless be different, the first one being higher if the event peaks at Spitzer before the Earth. However, the two Spitzer fluxes may differ not only because Spitzer is displaced from the Earth along the direction of lens-source relative motion, but also if it is displaced in the orthogonal direction by different amounts at the two epochs. Gould (1999) recognized this possibility, but argued that the amplitude of this displacement could be determined from the measurement of πE,k, which is derived from the ground-based parallax measurement

193 (i.e., from the asymmetry of the light curve). Hence, he argued that it would be possible to correct for this additional oﬀset and still obtain a good measurement of

πE,τ . Unfortunately, while it is true that the amplitude can be so derived, the sign of this correction is more diﬃcult to determine.

The problem can be understood by considering the work of Smith et al. (2003), who showed that ground-based microlensing parallaxes are subject to a two-fold degeneracy, essentially whether the source passes the lens on the same or opposite side of the Earth compared to the Sun. Within the geocentric formalism of Gould

(2004), this amounts to switching the sign of the impact parameter u0 (which by convention is normally positive) and leaving all other parameters essentially unchanged. Smith et al. (2003) derived this degeneracy under the assumption that the Earth accelerates uniformly during the course of the event. This is a reasonable approximation for short events, but is grossly incorrect for OGLE-2005-SMC-001, with its timescale of t 0.5 yr. Nevertheless, this degeneracy can hold remarkably E ∼ well even for relatively long events, particularly for u 1. The sign of the | 0| ≪ correction to πE,τ depends essentially on whether the absolute value of the impact parameter as seen from Spitzer is higher or lower than that seen from Earth. While the algebraic displacement of Spitzer along this direction can be predicted from the ground-based measurement of the parallax asymmetry (just as Gould 1999 argued), its eﬀect on the absolute value of u0 depends on whether u0 is positive or negative.

194 If the parallax is sufficiently large, then the ground-based light curve alone can determine the sign of u0, and if it is sufficiently small, the difference between the two solutions is also very small and may not be significant. However, for intermediate values of the parallax, this degeneracy can be important. To understand how an additional Spitzer observation can help, consider an idealized set of four observations, one at peak, one at baseline, and two symmetrically timed around peak (as seen from Earth). Call these fluxes fP , fb, f−, and f+. Consider now the ratio [f f ]/[(f + f )/2 f ]. If the impact parameter seen from Spitzer P − b − + − b is higher than that from Earth, this ratio will be lower for the Spitzer data than for the ground-based data. (Note that blended light, which may be different for the two sets of observations, cancels out of this expression.) In practice, we found from simulations that it was not necessary to have the three observations timed so perfectly. Hence, it was possible to plan both the additional DDT observation, as well as the second ToO observation to occur during regularly scheduled IRAC campaigns, so there was no 6.5 hour penalty for either observation. Hence, the net cost to Spitzer time was less than would have been the case for a single precisely timed ToO observation.

In brief, the above considerations demonstrate that the Gould (1999) technique requires a total of four observations, not three as originally proposed. Moreover, these observations do not have to be so precisely timed as Gould (1999) originally imagined. See Figure 5.2 for a visual explanation of these arguments.

195 5.3.2. Eight-fold Way

Ultimately, we found that the anomaly was caused by a binary lens. Binary lenses are subject to their own discrete degeneracies. This means that analysis of the event is impacted by two distinct classes of discrete degeneracies: those due to parallax and thoese due to binarity. The discrete parallax degeneracy, as summarized in 5.3.1, takes the impact parameter u u and (because u 1) leaves other § 0 → − 0 0 ≪ parameters changed by very little (see Gould 2004).

The discrete binary degeneracy is between wide and close binaries. Here “wide” means b 1 and “close” means b 1, where b is the angular separation between the ≫ ≪ two binary components in units of θE, (for which we give and justify our convention in 5.4.1). It gives (b ,q ) (b ,q ), § c c → w w

1+ q q b = c b−1, q = c (5.3) w 1 q c w (1 q )2 − c − c and leaves other parameters roughly unchanged. Here q is the mass ratio of the lens, with the convention that for qw, the component closer to the source trajectory goes in the denominator of the ratio. In both cases, the central magniﬁcation pattern is dominated by a 4-cusp caustic.

This degeneracy was ﬁrst discovered empirically by Albrow et al. (1999b) and theoretically by Dominik (1999b) and can be incredibly severe despite the fact that

196 the two caustics are far from identical: the solutions can remain indistinguishable even when there are two well-observed caustic crossings (An 2005).

In the present case, the deviations from a simple lens are not caused by caustic crossings, but rather by a close approach to a cusp, which makes this degeneracy even more severe. In fact, the caustic is symmetric enough that the approach may almost equally well be to either of two adjacent cusps. That is, the cusp degeneracy would be “perfect” if the cusp were four-fold symmetric, and it is only the deviation from this symmetry that leads to distinct solutions for diﬀerent cusp approaches.

In brief, the lens geometry is subject to an 8-fold discrete degeneracy, 2-fold for parallax, 2-fold for wide/close binary, and 2-fold for diﬀerent cusp approaches.

5.4. Binary Orbital Motion

Of course all binaries are in Kepler orbits, but it is usually possible to ignore this motion in binary-lens analyses. Stated less positively, it is rarely possible to constrain any binary orbital parameters from microlensing light curves. In the few known exceptions, (Albrow et al. 2000a; An et al. 2002), the light curve contained several well-measured caustic crossings that pinned down key times in the trajectory to O(10−5) of an Einstein crossing time. Hence, we did not expect to measure binary rotation in the present case in which there are no such crossings.

197 We were nevertheless led to investigate rotation by the following circumstance.

When we initially analyzed the event using only ground-based data, we found that the best ﬁt (for all 8 discrete solutions) had negative blended light, roughly 10% − of the source light, but with large errors and thus consistent with zero at the 1.5 σ level. This was not unexpected. As mentioned in 5.1, the component of the § microlens parallax perpendicular to the Sun, πE,⊥, is generally poorly constrained by ground-based data alone. The reason for this is that small changes in πE,⊥, the

Einstein timescale tE, the impact parameter u0, the source ﬂux fs, and the blended

ﬂux, fb, all induce distortions in the light curve that are symmetric about the peak, and hence all these parameters are correlated. Thus, the large errors (and consequent possible negative values) of fb are just the obverse of the large errors in

πE,⊥. Indeed, this is the reason for adding in Spitzer observations.

However, we found that when the Spitzer observations were added, the blending errors were indeed reduced, but the actual value of the blending remained highly negative, near 10%. This prompted us to look for other physical eﬀects that could − induce distortions in the light curve that might masquerade as negative blending.

First among these was binary orbital motion. Before discussing this motion, we ﬁrst review microlensing by static binaries.

198 5.4.1. Static Binary Lens Parameters

Point-lens microlensing is described by three geometric parameters, the impact parameter u0 (smallest lens-source angular separation in units of θE), the time at which the separation reaches this minimum t0, and the Einstein crossing time tE = θE/µ, where µ is the lens-source relative proper motion.

In binary lensing, there are three additional parameters, the projected separation of the components (in units of θE) b, the mass ratio of the components q, and the angle of the source-lens trajectory relative to the binary axis, φ. Moreover the first three parameters now require more precise definition because there is no longer a natural center to the system. One must therefore specify where the center of the system is. Then u0 becomes the closest approach to this center and t0 the time of this closest approach. Finally, tE is usually taken to be the time required to cross the Einstein radius defined by the combined mass of the two components.

In fact, while computer programs generally adopt some fairly arbitrary point

(such as the midpoint between the binaries or the binary center of mass), the symmetries of individual events can make other choices much more convenient. That is certainly the case here. Moreover, symmetry considerations that are outlined below will also lead us to adopt a somewhat non-standard tE for the wide-binary case, namely the timescale associated with the mass that is closer to the source trajectory, rather than the total mass. To be consistent with this choice, we also

199 express b as the separation between wide components in units of the Einstein radius associated with the nearest mass (rather than the total mass).

As is clear from Figure 5.1, the light curve is only a slightly perturbed version of standard (point-lens) microlensing, which means that it is generated either by the source passing just outside the central caustic of a close binary (that surrounds both components) or just outside one of the two caustics of a wide binary (each associated with one component). In either case, the standard point-lens parameters u0 and t0 will be most closely reproduced if the lens center is placed at the so-called “center of magniﬁcation”. For close binaries this lies at the binary center of mass. For wide binaries, it lies q(1 + q)−1/2b−1 from the component that is closer to the trajectory.

Hence, it is separated by approximately bq/(1+ q) from the center of mass. This will be important in deriving equation (5.4), below.

For light curves passing close to the diamond caustic of a close binary, the standard Einstein timescale (corresponding to the total mass) will be very close to the timescale derived from the best-ﬁt point lens of the same total mass. However, for wide binaries, the standard Einstein timescale is longer by a factor (1 + q)1/2, where q is the ratio of companion (whether heavier or lighter) to the component that is approached most closely. This is because the magniﬁcation is basically due just to this latter component (with the companion contributing only minor deviations via its shear), while the usual Einstein radius is based on the total mass. For wide

200 binaries, we therefore adopt an Einstein radius and Einstein timescale reduced by this same factor.

The advantage of adopting these parameter deﬁnitions is that, being fairly well

ﬁxed by the empirical light curve, they are only weakly correlated with various other parameters, some of which are relatively poorly determined.

5.4.2. Binary Orbital Parameters

While close and wide binaries can be (and in the present case are) almost perfectly degenerate in the static case, binary orbital motion has a radically different effect on their respective light curves. Note first that while 7 parameters would be required to fully describe the binary orbital motion, even in the best of cases it has not proven possible to constrain more than 4 of them (Albrow et al. 2000a; An et al.

2002). Two of these have already been mentioned, i.e., b and q, from the static case.

For the two binaries for which additional parameters have been measured, these have been taken to be a uniform rotation rate ω and a uniform binary-expansion rate b˙.

This choice is appropriate for close binaries because for these, the center of mass is the same as the center of magnification. Hence the primary effects of binary motion are rotation of the magnification pattern around the center of magnification and the change of the magnification pattern due to changing separation. Both

201 of these changes may be (probably are) nonuniform, but as the light curves are insensitive to such subtleties, the simplest approximation is uniform motion.

However, the situation is substantially more complicated for wide binaries, consideration of which leads to a different parameterization. Recall that the wide-binary center of magnification is not at the center of mass, and indeed is close to one of the components. Hence, as the binary rotates, the center of magnification rotates basically with that component. Nominally, the biggest effect of this rotation is the resulting roughly linear motion of the lens center of magnification relative to the source. However, the linear component of this motion, i.e., the first derivative of the motion at the peak of the event, is already subsumed in the source-lens relative motion in the static-binary fit. The first new piece of information about the binary orbital motion is the second derivative of this motion, i.e., the acceleration. Note that the direction of this acceleration is known: it is along the binary axis. Moreover, for wide binaries, Kepler’s Third Law predicts that the periods will typically be much longer than the Einstein timescale, so to a reasonable approximation, this direction remains constant during the event. We designate the acceleration (in

Einstein radii per unit time squared) as αb.

The parameter αb is related to the distance to the lens in a relatively straightforward way. For simplicity, assume for the moment that the center of magniﬁcation is right at the position of the component that is closer to the source trajectory (instead of just near it). The 3-dimensional acceleration of the component

202 2 is a = GM2/(brE csc i) , where M2 is the mass of the companion to the closer component, rE is the physical Einstein radius, and i is the angle between the binary

2 3 3 axis and the line of sight. Hence, αb = a sin i/rE = (q/b )GM1 sin i/rE, where M1 is the mass of the closer component. This can be simpliﬁed with the aid of the

2 following three identities: 1) 4GM1/c =r ˜EθE, 2) rE = DLθE, 3) rE/r˜E = DLS/DS.

Here D and D are the distances to the lens and source, and D = D D . We L S LS S − L then ﬁnd,

2 3 2 αb γc sin i DS = 2 , (5.4) πE 4DLAU DLS where γ q/b2 is the shear. Note that the shear determines the size of the caustic ≡ and so is one of the parameters that is most robustly determined from the light curve.

If we were to take account of the oﬀset between M1 and the center of magniﬁcation, the r.h.s. of equation (5.4) would change fractionally by the order of b−2.

5.4.3. Summary of Parameters

Thus, the model requires a total of 10 geometrical parameters in addition to the 8 ﬂux parameters (fs and fb for each of the 3 ground-based observatories plus Spitzer). These are the three standard microlensing parameters, t0, u0,tE

(the time of closest approach, separation at closest approach in units of θE, and

Einstein timescale), the three additional static-binary parameters b, q, φ (the binary separation in units of θE, the binary mass ratio, and the angle of the source trajectory

203 relative to the binary axis), the two binary-orbit parameters, b˙ and either ω (close) or αb (wide), and the two parallax parameters, πE = (πE,N ,πE,E), where N and E represent the North and East directions. These must be speciﬁed for 8 diﬀerent classes of solutions.

5.5. Search for Solutions

We combine two techniques to identify all viable models of the observed microlensing light curve: stepping through parameter space on a grid (grid-search) and Markov Chain Monte Carlo (MCMC) (Doran & Mueller 2003).

We begin with the simplest class of binary models, i.e., without parallax or rotation. Hence, there are six geometric parameters, t0, u0, tE, b, q, and φ. We consider classes of models with (b, q, φ) held ﬁxed, and vary (t0, u0,tE) to minimize

2 χ . (Note that for each trial model, fs and fb can be determined algebraically from a linear fit, so their evaluations are trivial.) This approach identifies four solutions, i.e. (2 cusp-approaches) (wide/close degeneracy). We then introduce parallax, and × so step over models with (b, q, φ, πE,N ,πE,E) held fixed, working in the neighborhood of the (b, q, φ) minima found previously. The introduction of parallax brings with it the u degeneracy, and so there are now 8 classes of solutions. ± 0

Next we introduce rotation. We begin by employing grid search and ﬁnd, somewhat surprisingly, that several of the eight (close/wide, u , on/oﬀ-axis cusp) ± 0

204 classes of solutions have more than one minimum in (ω, b˙) for close binaries or (αb, b˙) for wide binaries. We then use each of these solutions as seeds for MCMC and ﬁnd several additional minima that were too close to other minima to show up in the grid search. Altogether there are 20 separate minima, 12 for close binaries and 8 for wide binaries.

We use MCMC to localize our solutions accurate to about 1 σ and to determine the covariance matrix of the parameters. In MCMC, one moves randomly from one point in parameter space to another. If the χ2 is lower, the new point is added to the “chain”. If not, one draws a random number and adds the new point only if this number is lower than relative probability (exp( ∆χ2/2)). If the parameters − are highly correlated (as they are in microlensing) and the random trial points are chosen without reference (or without proper reference) to these covariances, then the overwhelming majority of trial points are rejected. We therefore sample parameter space based on the covariance matrix drawn from the previous “links” in the chain.

During the initial ”burning in” stage of the MCMC, we frequently evaluate the covariance matrix (every 100 “links”) until it stabilizes. Then we hold the covariance matrix ﬁxed in the simulation (Doran & Mueller 2003). From the standpoint of

2 finding the best χ , one can combine linear fits for the flux parameters fs and fb with

MCMC for the remaining parameters. However, since part of our MCMC objective is to find the covariances, we treat (fs, fb)OGLE as MCMC parameters while fitting for the remaining four flux parameters analytically.

205 In order to reduce the correlations among search parameters, we introduce the following parameter combinations into the search: t u t , f = f + f , eff ≡ 0 E base s b f f /u , and γ q/b2 (wide) or Q b2q/(1 + q)2 (close). Because these are max ≡ s 0 ≡ ≡ directly related to features in the light curve, they are less prone to variation than the naive model parameters. The effective timescale teff is 1/√12 of the full-width at half-maximum, fbase is just the flux at baseline, and fmax is the flux at maximum.

The scale of the Chang-Refsdal (1979, 1984) distortion (which governs the binary perturbation) is given by the shear γ for wide binaries and by the quadrupole Q for close binaries.

The MCMC “chain” automatically samples points in the neighborhood of the minimum with probability density proportional to their likelihoods, exp( χ2/2). − Somewhat paradoxically, this means that for higher-dimensional problems, it does not actually get very close to the minimum. Speciﬁcally, for a chain of length

N sampling an m-dimensional space, there will be only one point for which ∆χ2

(relative to the minimum) obeys ∆χ2

MCMC, we use the exp( χ2/2) chain. −

206 5.5.1. Convergence

A general problem in MCMC ﬁtting is to determine how well the solution

“converges”, that is, how precisely the “best fit” solution is reproduced when the initial seed solution is changed. In our case, the problem is the opposite: MCMC clearly does not converge to a single minimum, and the challenge is to find all the local minima. We described our procedure for meeting this challenge above, first by identifying 8 distinct regions of parameter space semi-analytically, and then exploring these with different MCMC seeds. This procedure led to well-defined minima (albeit a plethora of them), whose individual structures were examined by putting boundaries into the MCMC code that prevented the chain from “drifting” into other minima. We halted our subdivision of parameter space when the structure on the χ2 surface fell to of order ∆χ2 1, regarding the ∆χ2 < 1 region as the zone ∼ ∼ of convergence. As mentioned above, we located the final minimum by artificially decreasing the errors by a factor of 5, again making certain that the resulting

(exaggerated) χ2 surface was well behaved.

5.6. Solution Triage

Table 1 gives parameter values and errors for a total of 20 diﬀerent discrete solutions, which are labeled by (C/W) for close/wide binary, (+/ ) for the sign − of u , ( / ) for solutions that are approximately parallel or perpendicular to the 0 k ⊥

207 binary axis, and then by alphabetical sequential for diﬀerent viable combinations of rotation parameters. In Table 1, we allow a free ﬁt to blending.

Note that some solutions have severe negative blending. While it is possible in principle that these are due to systematic errors, the fact that other solutions have near-zero blending and low χ2 implies that the negative-blending solutions probably have lens geometries that do not correspond to the actual lens. Other solutions have relatively high χ2 and so are also unlikely. Solutions with χ2 that is less than 9 above the minimum have their χ2 displayed in boldface, while the remaining solutions are shown in normal type.

There are several reasons to believe that the blending is close to zero. First, the source appears isolated on our K-band NICMOS HST images, implying that if it is blended, this blended light must be within of order 100 mas of the source. As the density of sources in the HST images is low, this is a priori very unlikely unless the blended light comes from a companion to the source or the lens. Moreover, all the near neighbors of the source on the HST image are separately resolved by the OGLE photometry, so if there is blended light in the OGLE photometry then it must also be blended in the HST images. Second, the V I color of the source, which can be − derived by a model-independent regression of V ﬂux on I ﬂux, is identical within measurement error to the color of the baseline light from the combined source and

(possible) blend. This implies that either 1) the source is unblended, 2) it is blended by another star of nearly the same color as the source, or 3) it is blended by a star

208 that is so faint that it hardly contributes to the color of the blend. The second possibility is strongly circumscribed by the following argument. The source is a clump star. On the SMC CMD, there are first ascent giants of color similar to the source from the clump itself down to the subgiant branch about 2.5 mag below the clump. Thus, in principle, the blended light could lie in the range 0.1 < fb/fs < 1 ∼ ∼ without causing the baseline color to deviate from the source color. However, there are no solutions in Table 1 with blending this high. There is one low-χ2 solution with f /f 0.06, but this would be 3 mag below the clump and so below the b s ∼ subgiant branch. If we restrict the blend to turnoff colors, i.e., V I = 0.6, then − the color constraint implies f /f = 0.01 0.01, which is negligibly small from our b s ± perspective. Thus, while it is possible in principle that the source is blended with a reddish subgiant, the low stellar density in the HST image, the low frequency of such subgiants on the CMD, and the difficulty of matching color constraints even with such a star, combine to make this a very unlikely possibility. We therefore conduct the primary analysis assuming zero blending, as listed in Table 2. Again, the ∆χ2 < 9 solutions are marked in boldface. Note that most solutions in Table

1 are reasonably consistent with zero blending, which is the expected behavior for the true solution provided it is not corrupted by systematic errors. For these, the χ2 changes only modestly from Table 1 to Table 2. However, several solutions simply disappear from Table 2. This is because, in some cases, forcing the blending to zero has the eﬀect of merging two previously distinct binary-rotation minima.

209 The main parameters of interests are πE and the closely related quantity v˜.

However, for reasons that will be explained below, v˜ can be reliably calculated only for the close solutions, but not the wide solutions. Hence, we focus ﬁrst on πE.

Figure 5.3 shows error ellipses for all solutions, color-coded by according to ∆χ2 relative to the global minimum. The right-hand panels show the solutions presented in Tables 1 and 2, which include the Spitzer data. The left-hand panels exclude these data. The upper panels are based on a free ﬁt for blending whereas the lower panels are constrained to zero blending for the OGLE dataset. Comparing the two upper panels, it is clear that when blending is a ﬁtted parameter, the Spitzer data reduce the errors in the πE,⊥ direction by about a factor of 3. However, once the blending is

fixed (lower panels), the Spitzer data have only a modest additional effect. This is expected since πE,⊥ is correlated with blending and one can simultaneously constrain both parameters either by constraining πE,⊥ with Spitzer data or just by fixing the blending by hand. Figure 5.4 shows the best overall zero-blending fit to the data.

5.6.1. Wide-Binary Solutions

All eight wide solutions are effectively excluded. When a free fit to blending is allowed, their χ2 values are already significantly above the minimum. When zero blending is imposed, only five independent solutions survive and those that have negative blending are driven still higher. In all five cases to ∆χ2 > 16.

210 5.6.2. Close-Binary Solutions

Eight Of the 11 close solutions survive the imposition of zero blending. Of these, only one has ∆χ2 < 4, and only another two have ∆χ2 < 8.7 relative to the best zero-blending solution. We focus primarily on these three, which all have best-ﬁt parallaxes in the range 0.030 < πE < 0.047 and projected velocities in the range 210 km s−1 < v˜ < 330 km s−1. These projected velocities are of the order expected for halo lenses but are about 1 order of magnitude smaller than those expected for SMC self lensing. Note that because there are multiple solutions, the errors in v˜ are highly non-Gaussian and are best judged directly from Figures 5.5 and 5.6 (below) rather than quoting a formal error bar.

5.7. Lens Location

When Alcock et al. (1995) made the ﬁrst measurement of microlensing parallax, they developed a purely kinematic method of estimating the lens distance (and thus mass) based on comparison of the measured value of v˜ with the expected kinematic properties of the underlying lens population. Starting from this same approach,

Assef et al. (2006) devised a test that uses the microlens parallax measurement to assign relative probabilities to diﬀerent lens populations (e.g., SMC, Galactic halo,

Galactic disk) based solely on the kinematic characteristics of these populations, and without making prior assumptions about either the mass function or the density

211 normalization of any population. This is especially useful because, while a plausible guess can be given for the mass function and normalization of SMC lenses, nothing is securely known about a putative Galactic halo population. In the present case, the high projected velocityv ˜ immediately rules out Galactic-disk lenses, so we restrict consideration to the other two possibilities.

We begin by recapitulating the Assef et al. (2006) test in somewhat more general form. The diﬀerential rate of microlensing events of ﬁxed mass M (per steradian) is

6 2 (6) d Γ(M) 2 DLS d Γ(M) 2 2 = fL(vL)fS(vS)DSνS(DS)νL(DL)2˜vr˜E 2 , (5.5) ≡ d vL d vSdDLdDS DS

where fL(vL) and fS(vS) are the two-dimensional normalized velocity distributions of the lenses and sources, νL and νS are the density distributions of the lenses and sources,v ˜ is an implicit function of (DL,DS, vL, vS), andr ˜E is an implicit function of (DL,DS, M). The method is simply to evaluate the likelihood

2 2 (6) 2 d vL d vSdDLdDS d Γ(M) exp[ ∆χ (v˜)/2] = 2 2 (6) − (5.6) L R d vL d vSdDLdDS d Γ(M) R for each population separately, and then take the ratio of likelihoods for the two populations: / . Here, ∆χ2(v˜) is the diﬀerence of χ2 relative to Lratio ≡ Lhalo LSMC the global minimum that is derived from the microlensing light curve. Note that all

212 dependence on M disappears from . Equation (5.6) can be simpliﬁed in diﬀerent L ways for each population. In both cases, we express the result in terms of Λ,

v˜ π t Λ = E E , (5.7) ≡ v˜2 AU rather than v˜, because it is better behaved in the neighborhood of Λ = 0, just as trigonometric parallax is better behaved near zero than its inverse, distance.

5.7.1. Halo Lenses

For halo lenses, the depth of the SMC is small compared to DLS and the internal dispersion of SMC sources is small compared to the bulk motion of the SMC. Hence, one can essentially drop the three integrations over SMC sources, implying

DS DL v˜ = vL v⊕ vSMC, (5.8) DLS − − DLS

where vL, v⊕, and vSMC are the velocities of the lens, the “geocentric frame”, and the

SMC (all in the Galactic frame) projected on the plane of the sky. The “geocentric frame” is the frame of the Earth at the time of the peak of the event. It is the most convenient frame for analyzing microlensing parallax (Gould 2004) and for this event is oﬀset from the heliocentric frame by

(v ,v ) (v ,v )=( 24.9, 15.5) km s−1. (5.9) ⊙,N ⊙,E − ⊕,N ⊕,E − − 213 We assume an isotropic Gaussian velocity dispersion for the lenses with

−1 σhalo = vrot/√2, where vrot = 220 km s . After some manipulations (and dropping constants that would cancel out between the numerator and denominator) we obtain

2 exp[ ∆χ (Λ)/2]ghalo(Λ,DL)dDLdΛNorthdΛEast halo = − , (5.10) L R ghalo(Λ,DL)dDLdΛNorthdΛEast R where

g (Λ,D ) = exp( v2 /2σ2 )ν (D )Λ−5D7/2D1/2, (5.11) halo L − L halo halo L LS L

and where vL is an implicit function of Λ through equations (5.7) and (5.8). We adopt

2 2 νhalo(r)= const/(ahalo + r ), where ahalo = 5 kpc and the Galactocentric distance is

−1 R0 = 7.6 kpc. We adopt v⊙ = (10.1, 224, 6.7) km s in Galactic coordinates, which leads to a 2-dimensional projected velocity of (v ,v ) = (126, 126) km s−1 ⊙,N ⊙,E − toward the SMC source. From our assumed distances DSMC = 60kpc and the SMC’s measured proper motion of ( 1.17 0.18, +1.16 0.18) (Kallivayalil et al. 2006), we − ± ± obtain,

(v ,v ) (v ,v )=( 333, 330) km s−1. (5.12) SMC,N SMC,E − ⊙,N ⊙,E − 214 5.7.2. SMC Lenses

For the SMC, we begin by writing

vL/DL vS/DS DS DSMC v˜ = −1 − −1 = vS + ∆v vSMC v⊕ + ∆v, (5.13) D D DLS → − DLS L − S where v and v are now measured in the geocentric frame and ∆v v v . L S ≡ L − S The last step in equation (5.13) is an appropriate approximation because the SMC velocity dispersion is small compared to its bulk velocity and D D . LS ≪ S

We now assume that the sources and lenses are drawn from the same population,

1/2 which implies that the dispersion of ∆v is larger than those of vL and vS by 2 .

We assume that this is isotropic with Gaussian dispersion σSMC. Again making the approximation D D , we can factor the integrals in equation (5.6) by S → SMC evaluating the density integral

η(DLS)= dDLν(DL)ν(DL + DLS), (5.14) Z

where ν = νL = νS. We then obtain

2 exp[ ∆χ (Λ)/2]gSMC(Λ,DLS)dDLSdΛNorthdΛEast SMC = − , (5.15) L R gSMC(Λ,DLS)dDLSdΛNorthdΛEast R where

g (Λ,D ) = exp[ (∆v)2/4σ2 ]η(D )D7/2Λ−5, (5.16) SMC L − SMC LS LS 215 and where ∆v is an implicit function of Λ through equations (5.7) and (5.13).

5.7.3. SMC Structure

In order to evaluate equation (5.16) one must estimate σSMC as well as the

SMC density ν along the line of sight, which is required to compute η(DLS). This requires an investigation of the structure of the SMC.

In sharp contrast to its classic “Magellanic irregular” appearance in blue light, the SMC is essentially a dwarf elliptical galaxy whose old population is quite regular in both its density (Zaritsky et al. 2000) and velocity (Harris & Zaritsky 2006) distributions. Harris & Zaritsky (2006) ﬁnd that after removing an overall gradient

(more below), the observed radial velocity distribution is well ﬁt by a Gaussian with

σ = 27.5 km s−1, which does not vary signiﬁcantly over their 4◦ 2◦ (RA,Dec) ﬁeld. ×

We adopt the following SMC parameters for the 1-dimensional dispersion

σSMC, the tidal radius rt, and the density proﬁle (along the line of sight),

2 2 n/2 ν(r)= const/(aSMC + r ) ,

−1 σSMC = 25.5 km s , rt = 6.8 kpc, aSMC = 1 kpc, n = 3.3, (5.17) as we now justify.

The σ = 27.5 km s−1 dispersion reported by Harris & Zaritsky (2006) includes measurement errors. When the reported errors (typically 10 km s−1 per star) are

216 −1 included in the ﬁt, this is reduced to σSMC = 24.5 km s . However, these reported errors may well be too generous: the statistical errors (provided by D. Zaritsky 2006, private communication) are typically only 2–3 km s−1, the reported errors being augmented to account for systematic errors. If the statistical errors are used, we

−1 ﬁnd σSMC = 26.5 km s . D. Zaritsky (2006, private communication) advocates an

−1 intermediate value for this purpose, which leads to σSMC = 25.5 km s .

The old stellar population in the SMC is rotating at most very slowly. Harris

& Zaritsky (2006) report a gradient across the SMC of 8.3 km s−1 deg−1, which they note is a combination of the traverse velocity of the SMC and the solid-body component of internal bulk motion. As the SMC proper motion was poorly determined at the time, Harris & Zaritsky (2006) did not attempt to disentangle these two. However, Kallivayalil et al. (2006) have now measured the SMC proper motion to be (µ ,µ )=( 1.16 0.18, 1.17 0.18) mas yr−1. We reﬁt the Harris N E − ± ± & Zaritsky (2006) data and ﬁnd v = ( 10.5 1.4, 5.0 0.7) km s−1 deg−1. ∇ r − ± ± Subtracting these two measurements (including errors and covariances) we obtain a net internal rotation of 5.2 1.6 km s−1 deg−1 with a position angle of 183◦ 30◦. ± ± Since this rotation is due north-south (within errors), a direction for which the data have a baseline of only 1◦, and since solid-body rotation is unlikely to extend ∼ ± much beyond the core, it appears that the amplitude of rotational motion is only

−1 about 5 km s , which is very small compared to the dispersion, σSMC. Hence, we ignore it. In addition, we note that this rotation is misaligned with the HI rotation

217 axis (Stanimirovi´cet al. 2004) by about 120◦, so its modest statistical signiﬁcance may indicate that it is not real.

As we describe below, our likelihood estimates are fairly sensitive to the tidal radius rt of the SMC. Proper determination of the tidal radius is a complex problem.

Early studies, made before dark matter was commonly accepted, were carried out for

Kepler potentials and in analogy with stellar and solar-system problems (e.g. King

1962). Read et al. (2006) have calculated tidal radii for a range of potentials and also for an orbital parameter α that ranges from 1 for retrograde to +1 for prograde. − We choose α = 0 as representative and evaluate their expression for an isothermal potential and for the satellite being close to pericenter (as is appropriate for the

SMC):

r σ 2 ln ξ −1/2 t = sat 1+ . (5.18) D σ 1 ξ−2 host −

Here, ξ is the ratio of the apocenter to the pericenter of the satellite orbit, D is the pericenter distance, and σsat and σhost are the respective halo velocity dispersions.

Because we adopt an n = 3.3 proﬁle, the SMC halo velocity dispersion is larger than

1/2 1/2 its stellar dispersion by (3.3/2) , implying that σsat/σhost = 3.3 σSMC/vrot = 0.21.

We adopt ξ = 3 based on typical orbits found by Kallivayalil et al. (2006), which yields r = 0.107 D = 6.8 kpc. Note, moreover, that at ξ = 3, d ln r /d ln ξ 0.24, t SMC t ∼ so the tidal radius is not very sensitive to the assumed properties of the orbit.

218 The most critical input to the likelihood calculation is the stellar density along the line of sight. Of course, images of the SMC give direct information only about its surface density as a function of position. One important clue to how the two are related comes from the HI velocity map of Stanimirovićet al. (2004), which shows an inclined rotating disk with the receding side at a position angle of 60◦ ∼ (north through east). This is similar to the 48◦ position angle of the old-star ∼ optical profile found by Harris & Zaritsky (2006) based on data from Zaritsky et al. (2000). Hence, the spheroidal old-stellar population is closely aligned to the HI disk, although (as argued above) the stars in the SMC are pressure- rather than rotationally-supported. Our best clue to the line-of-sight profile is the major-axis profile exterior to the position of the source, which lies about 1 kpc to the southwest of the Galaxy center, roughly along the apparent major axis. The projected surface density is falling roughly as r−2.3 over the 1 kpc beyond the source position, from ∼ which we derive a deprojected exponent of n = 3.3. This is similar to the exponent for Milky Way halo stars. While there is a clear core in the star counts, this may be affected by crowding, and the core seems to have little impact on the counts beyond the source position (which is what is relevant to the density profile along the line of sight). Hence, we adopt aSMC = 1 kpc.

The likelihood ratio is most sensitive to the assumptions made about the stellar density in the outskirts of the SMC, hence to the power law and tidal radius adopted. This seems strange at ﬁrst sight because the densities in these outlying

219 regions are certainly extremely small, whatever their exact values. This apparent paradox can be understood as follows. For DLS < rt, the leading term in η is ∼ η(D ) [D2 + 4(b2 + a2 )]−n/2, where b = 1.0 kpc is the impact parameter. LS ∼ LS SMC In the outskirts, this implies η(D ) D−n. The integrand in the numerator of LS ∼ LS equation (5.15) then scales as exp [∆v(Λ,D )]2/4σ2 D7/2−n. For fixed Λ, ∆v {− LS SMC} LS is a rising function of DLS, and so for sufficiently large DLS, the exponential will eventually cut off the integral. However, for the measured value of Λ (corresponding to ∆˜v Λ/Λ2 (v v ) 300 km s−1, the cutoff does not occur until ≡ | − SMC − ⊙ | ∼ D (2σ /v˜)D 10 kpc. Thus, as long as the density exponent remains LS ∼ SMC SMC ∼ n < 3.5 and as long as the density is not actually cut off by rt, the integral keeps ∼ growing despite the very low density. On the other hand, a parallel analysis shows that the denominator in equation (5.15) is quite insensitive to assumptions about the outer parts of the SMC.

5.7.4. Likelihood Ratios

Using our adopted SMC parameters (eq. [5.17]), we find / = Lratio ≡ Lhalo LSMC 27.4. As discussed in 5.7.3, this result is most sensitive to the outer SMC density § profile, set by the exponent n and the cutoff rt. If the tidal radius is increased from r = 6.8 kpc to r = 10 kpc and other parameters are held fixed, = 30.3. If the t t Lratio exponent is reduced from n = 3.3 to n = 2.7, then = 29.1. Hence, halo lensing Lratio

220 is strongly favored in any case, but by an amount that would vary noticeably if any of our key model parameters were markedly oﬀ.

In carrying out these evaluations, we integrated equations (5.10) and (5.15) over all solutions with ∆χ2 < 9. These solutions are grouped around three close-binary minima shown in Table 2. We carried out the integration in two different ways: over the discrete ensemble of solutions found by the Markov chains and uniformly over the three error ellipses that were fit to these chains. The results do not differ significantly. Figures 5.5 and 5.6 show Λ for the Markov-chain solutions superposed on likelihood contours for SMC and halo lenses respectively.

5.7.5. Kepler Constraints for Close Binaries

Binaries move in Kepler orbits. In principle, if we could measure all the Kepler parameters then these, together with the measured microlens parallax πE (and the approximately known source distance DS), would ﬁx the mass and distance to the lens. While the two orbital parameters that we measure are not suﬃcient to determine the lens mass and distance, they do permit us to put constraints on these quantities.

Consider ﬁrst the special case of a face-on binary in a circular orbit. Kepler’s

3 2 Third Law implies that GM/(brE) = ω where M is the mass of the lens, b is the

221 binary separation in units of the local Einstein radius rE, and ω is the measured rotation parameter. Since rE =(DLS/DS)˜rE, this implies,

2 2 c DS 2 3 2 = ω (face on circular) (5.19) 4˜rEb DLDLS −

If one considers other orientations but remains restricted to face-on orbits, then this equation becomes a (“greater than”) inequality because at ﬁxed projected separation, the apparent angular speed can only decrease.

Further relaxing to the case of non-circular orbits but with b/b˙ = 0 (i.e., the event takes place at pericenter), the rhs of equation (5.19) is halved, ω2 ω2/2, → because escape speed (appropriate for near-parabolic orbits) is √2 larger than circular speed. Finally, after some algebra, and again working in the parabolic limit, one ﬁnds that including non-zero radial motion leads to ω2/2 [ω2 +(b/b˙ )2/4]/2, → and thus to

2 2 ˙ 2 −1 DLDLS c 2 (b/b) 2 3 ω + . (5.20) DS ≤ 2˜rEb 4

If the rhs of this equation is suﬃciently small, then only lenses that are near the Sun (D D ) or near the SMC (D D ) will satisfy it. In practice, we L ≪ S LS ≪ S ﬁnd that this places no constraint on SMC lenses, but does restrict halo lenses to be relatively close to the Sun, with the limit varying from 2 kpc to 10 kpc ∼ ∼ depending on the particular solution being probed and the MCMC realization of

222 that solution. Integrating over the entire Markov chain, we ﬁnd that is reduced Lratio by a factor three, from 27.4 to 11.2.

5.7.6. Kepler Constraints for Wide Binaries

As discussed in 5.6.1, all wide solutions are ruled out by their high χ2. § Nevertheless, for completeness it is instructive to ask what sort of constraints could be put on the lens from measurement of the acceleration parameter αb if these solutions had been accepted as viable. We rewrite equation (5.4) in terms of the parameter combination T (π γ/α )1/2, which has units of time, ≡ E b

3 DL DLS cT sin i T = v < 0.28 . (5.21) s60 kpc DS 2 u60 kpc AU yr u t

The ﬁve wide-binary zero-blending solutions listed in Table 2 have, respectively,

T = (0.039, 0.032, 0.035, 0.017, 0.015) yr. Thus, if these solutions had been viable, the lens would have been firmly located in the SMC (or else, improbably, within 6 pc of the Sun). This demonstrates the power of this constraint for wide binaries, which derives from light-curve features that arise from the motion of the center of magnification relative to the center of mass. These obviously do not apply to close binaries for which the center of magnification is identical to the center of mass.

However, this same relative motion makes it more diﬃcult to constrain the nature of the lens from measurements of the projected velocity, v˜. This is because

223 the parallax measurement directly yields only the projected velocity of the center of magniﬁcation, whereas models of the lens populations constrain the motion of the center of mass. If b˙ is measured, it is straight forward to determine the component of the diﬀerence between these two that is parallel to the binary axis, namely

b˙ AU ∆˜vk = . (5.22) 1+ q πE

On the other hand, the measurement of αb indicates that there is motion in the transverse direction, but speciﬁes neither the amplitude nor sign. For example, for circular motion with the line of nodes perpendicular to the binary axis,

αbb AU ∆˜v⊥ = csc i, (5.23) ±s1+ q πE and hence,

∆˜v⊥ αbb = tE csc i, (5.24) v˜ ±s1+ q

If this ratio (modulo the csc i term) is small, then the internal motion can be ignored

(assuming the inclination is not unluckily low). In the present case, the quantity

1/2 [αbb/(1 + q)] tE for the ﬁve respective wide-binary solutions shown in Table 2 is

(0.23,0.36,0.25,1.04,0.77), implying that constraints arising from measurement ofv ˜ would be signiﬁcantly weakened.

Again, however, since the wide solutions are in fact ruled out, all of these results are of interest only for purposes of illustration.

224 5.7.7. Constraints From (Lack of) Finite-Source Effects

Finite source eﬀects are parameterized by ρ θ /θ , where θ is the angular ≡ ∗ E ∗ size of the source. Equivalently, ρ = (DL/DLS)(r∗/r˜E), where r∗ = DSθ∗ is the physical source radius. Since the source is a clump giant, with r 0.05 AU, this ∗ ∼ implies

D 5 kpc π ρ = 0.021 L E (5.25) 60 kpc DLS 0.035

All models described above assume ρ = 0. Since the impact parameter is u 0.08 0 ∼ and the semi-diameter of the caustic is 2Q 0.02, finite source effects should be ∼ pronounced for ρ > u0 2Q 0.06. In fact, we find that zero-blending models with ∼ − ∼ ρ < 0.05 are contraindicated by ∆χ2 (ρ/0.0196)4. This further militates against ∼ ∼ SMC lenses, which predict acceptable values of ρ only for relatively large source-lens separations. If we penalize solutions with finite source effects by ∆χ2 =(ρ/0.0196)2, we find that rises from 11.2 to 20.3. Lratio

5.8. Discussion

By combining ground-based and Spitzer data, we have measured the microlensing parallax accurate to 0.003 units, by far the best parallax measurement yet for an event seen toward the Magellanic Clouds. Our analysis signiﬁcantly favors

225 a halo location for the lens over SMC self lensing. It excludes altogether lensing by

Galactic disk stars. Of course, with only one event analyzed using this technique, an

SMC location cannot be absolutely excluded based on the 5% probability that we ∼ have derived. The technique must be applied to more events before ﬁrm conclusions can be drawn. Spitzer itself could be applied to this task. Even better would be observations by the Space Interferometry Mission (SIM) (Gould & Salim 1999), for which time is already allocated for 5 Magellanic Cloud events. SIM would measure both πE and θE and so determine (rather than statistically constrain) the position of the lens.

Assuming that the lens is in the halo, what are its likely properties? The mass is

−2 πE πrel M = 10.0 M⊙ , (5.26) 0.047 180 µas

where the fiducial πE is the best-fit value and the fiducial πrel is for a “typical” halo lens, which (after taking account of the constraint developed in 5.7.5) would § lie at about 5 kpc. In the best-fit model, the mass ratio is q = 2.77, which would imply primary and secondary masses of 7.3 and 2.7 M⊙ respectively. The projected separation would be b = 0.22 Einstein radii, i.e., 4.7AU(DLS/DS). Note that at these relatively close distances, main-sequence stars in this mass range would shine far too brightly to be compatible with the strict constraints on blended light. Hence, the lenses would have to be black holes.

226 We must emphasize that the test carried out here uses only kinematic attributes of the lens populations and assumes no prior information about their mass functions.

Note, in particular, that if we were to adopt as a priori the lens mass distribution inferred by the MACHO experiment (Alcock et al. 2000), then the MACHO hypothesis would be strongly excluded. Recall from 5.7.5 (eq. [5.20]), that the § lens is either very close to the SMC or within 10 kpc of the Sun. However, from equation (5.26), for DL < 10 kpc, we have M > 4.6 M⊙, and hence a primary mass ∼

Mq/(1 + q) > 3.4 M⊙. From Figures 12–14 of Alcock et al. (2000), such masses are ∼ strongly excluded as the generators of the microlensing events they observed toward the LMC.

Hence, if the MACHO hypothesis favored by this single event is correct, the

MACHO population must have substantially diﬀerent characteristics from those inferred by Alcock et al. (2000), in particular, as mentioned above, a mass scale of order 10 M⊙ or perhaps more. Alcock et al. (2001b) limit the halo fraction of such objects to < 30% at M = 10 M⊙ and to < 100% at M = 30 M⊙. At higher masses, the sensitivity of microlensing surveys deteriorates drastically. However, Yoo et al.

(2004c) derived important limits in this mass range from the distribution of wide binaries in the stellar halo, putting an upper limit at 100% for M = 40 M⊙ and at

20% for M > 200 M⊙. Thus, if the MACHO hypothesis is ultimately conﬁrmed, this would be a new population in the mass “window” identiﬁed in Figure 7 of Yoo et al. (2004c) between the limits set by microlensing and wide-binary surveys. This

227 again argues for the importance of obtaining space-based parallaxes on additional microlensing events.

As we discussed in some detail in 5.7.3 (particularly the last paragraph) our § conclusion regarding the relatively low probability of the lens being in the SMC rests critically on the assumption that the SMC lens population falls oﬀ relatively rapidly along the line of sight. (We adopted an n = 3.3 power law.) This is because equation (5.16) scales D7/2−n. If the SMC had a halo lens population with a ∝ LS much shallower falloﬀ than we have assumed, this term could dominate the integral in equation (5.15) even if the overall normalization of the halo were relatively low.

Hence, Magellanic Cloud halo lensing could provide an alternate explanation both for this SMC event and the events seen by MACHO toward the LMC (Calchi

Novati et al. 2006). As mentioned above, SIM could easily distinguish between this conjecture and the halo-lens hypothesis.

Finally, we remark that analysis of this event was extraordinarily diﬃcult because it was a “weak” (i.e., non-caustic-crossing) binary. If it had been either a single-mass lens or a caustic-crossing binary, it would have been much easier to analyze and the inferences regarding its location (in the halo or the SMC) would have been much more transparent. We therefore look forward to applying this same technique to more typical events.

228 1/2 ˙ Model t0 u0 tE q Q : γ φ πE,N πE,E ω : αb b/b Fb/Fbase

χ2 (day) ( 100) (day) ( 100) (deg) (yr−1) (yr−1) × ×

C- a 3593.751 8.729 174.17 2.77 0.980 21.70 0.0342 0.0319 0.073 5.60 0.003 k − − 1455.40 0.040 0.532 8.13 0.33 0.066 1.10 0.0076 0.0047 0.047 0.46 0.061

C+ a 3593.687 7.763 190.03 2.15 0.850 19.34 0.0120 0.0236 0.141 5.83 0.115 k − −

229 1475.74 0.033 0.351 6.58 0.26 0.046 1.26 0.0039 0.0035 0.027 0.48 0.040

C- b 3593.560 9.567 160.61 0.98 0.999 9.23 0.0069 0.0298 0.331 0.40 0.101 k − − − 1462.89 0.036 0.509 6.69 0.25 0.063 1.89 0.0068 0.0049 0.032 0.61 0.060

C+ b 3593.643 9.321 163.41 1.11 1.002 11.24 0.0068 0.0199 0.167 1.33 0.071 k − − − 1466.84 0.040 0.346 4.52 0.15 0.046 1.12 0.0036 0.0032 0.029 0.26 0.040

(cont’d)

Table 5.1. OGLE-2005-SMC-001 Light Curve Models: Free Blending. Table 5.1—Continued

1/2 ˙ Model t0 u0 tE q Q : γ φ πE,N πE,E ω : αb b/b Fb/Fbase

χ2 (day) ( 100) (day) ( 100) (deg) (yr−1) (yr−1) × ×

C- c 3593.473 9.020 171.06 0.29 0.837 3.34 0.0070 0.0069 0.746 3.87 0.030 k − − − − 1463.52 0.042 0.352 5.89 0.04 0.041 1.31 0.0051 0.0039 0.069 0.44 0.043

230 C+ c 3593.460 9.207 167.46 0.29 0.853 3.78 0.0038 0.0029 0.743 3.92 0.053 k − − − − 1465.15 0.042 0.276 4.15 0.03 0.035 1.24 0.0032 0.0031 0.070 0.41 0.033

C- d 3593.639 9.831 157.29 0.90 1.052 8.53 0.0121 0.0166 0.146 0.04 0.131 k − − − − 1464.13 0.058 0.480 6.18 0.15 0.062 1.18 0.0067 0.0062 0.096 0.27 0.057

C+ d 3593.676 9.406 162.74 0.79 1.001 7.86 0.0042 0.0135 0.144 0.12 0.080 k − − − − 1467.70 0.056 0.333 4.42 0.14 0.043 1.47 0.0036 0.0043 0.048 0.50 0.040

(cont’d) Table 5.1—Continued

1/2 ˙ Model t0 u0 tE q Q : γ φ πE,N πE,E ω : αb b/b Fb/Fbase

χ2 (day) ( 100) (day) ( 100) (deg) (yr−1) (yr−1) × ×

C+ e 3593.655 9.161 165.60 1.10 0.983 11.36 0.0055 0.0197 0.162 1.50 0.051 k − − − 1474.32 0.047 0.498 7.24 0.20 0.060 1.42 0.0038 0.0045 0.038 0.74 0.058

231 C+ f 3593.769 7.516 194.33 1.86 0.826 19.21 0.0078 0.0170 0.018 5.94 0.140 k − − 1474.40 0.038 0.372 7.10 0.33 0.048 1.46 0.0042 0.0038 0.041 0.72 0.042

C- a 3593.251 8.026 176.25 1.74 0.719 280.21 0.0101 0.0309 0.758 1.10 0.061 ⊥ − − 1457.55 0.030 0.370 6.94 0.20 0.041 0.53 0.0054 0.0043 0.049 0.34 0.046

C+ a 3593.272 7.900 177.32 1.40 0.752 280.03 0.0079 0.0198 0.599 1.45 0.073 ⊥ − − 1470.74 0.035 0.298 5.33 0.25 0.052 0.58 0.0035 0.0038 0.084 0.32 0.037

(cont’d) Table 5.1—Continued

1/2 ˙ Model t0 u0 tE q Q : γ φ πE,N πE,E ω : αb b/b Fb/Fbase

χ2 (day) ( 100) (day) ( 100) (deg) (yr−1) (yr−1) × ×

W- a 3593.708 8.470 170.22 2.49 0.936 6.38 0.0073 0.0041 0.241 0.49 0.004 k − − 1471.32 0.037 0.306 5.70 0.40 0.036 0.51 0.0056 0.0033 0.175 0.02 0.038

232 W+ a 3593.703 8.568 167.72 2.48 0.950 6.37 0.0050 0.0017 0.273 0.49 0.016 k − 1471.29 0.035 0.273 4.70 0.40 0.034 0.49 0.0033 0.0029 0.174 0.02 0.033

W- a 3593.562 8.666 174.41 2.28 1.154 279.74 0.0039 0.0180 0.841 0.75 0.009 ⊥ − − − 1474.79 0.067 0.361 6.31 1.03 0.048 0.61 0.0057 0.0054 0.385 0.06 0.043

W+ a 3593.641 9.281 165.12 7.39 1.158 279.74 0.0036 0.0148 0.804 0.23 0.060 ⊥ − − − 1480.79 0.062 0.293 4.04 5.71 0.048 0.60 0.0036 0.0042 0.339 0.18 0.034

(cont’d) Table 5.1—Continued

1/2 ˙ Model t0 u0 tE q Q : γ φ πE,N πE,E ω : αb b/b Fb/Fbase

χ2 (day) ( 100) (day) ( 100) (deg) (yr−1) (yr−1) × ×

W- b 3593.570 9.209 166.04 3.21 1.207 279.24 0.0074 0.0204 0.909 0.16 0.054 ⊥ − − − −

233 1475.50 0.062 0.385 5.83 1.92 0.053 0.58 0.0056 0.0051 0.360 0.18 0.046

W+ b 3593.659 8.897 171.08 5.28 1.125 280.32 0.0023 0.0112 0.826 0.77 0.016 ⊥ − − − 1480.16 0.068 0.255 3.89 3.53 0.042 0.61 0.0035 0.0046 0.305 0.06 0.030

W- c 3593.492 9.443 163.30 2.20 1.291 278.45 0.0108 0.0195 1.094 0.49 0.086 ⊥ − − − 1474.90 0.046 0.329 5.24 0.51 0.052 0.48 0.0053 0.0050 0.244 0.05 0.040

W+ c 3593.534 9.492 162.68 2.96 1.267 278.50 0.0050 0.0155 1.054 0.44 0.091 ⊥ − − 1486.02 0.049 0.249 3.70 0.69 0.043 0.49 0.0035 0.0051 0.252 0.05 0.030 1/2 ˙ Model t0 u0 tE q Q : γ φ πE,N πE,E ω : αb b/b

χ2 (day) ( 100) (day) ( 100) (deg) (yr−1) (yr−1) × ×

C- a 3593.751 8.755 173.72 2.77 0.984 21.69 0.0347 0.0316 0.075 5.59 k − − 1455.38 0.039 0.028 0.88 0.32 0.018 1.15 0.0027 0.0048 0.047 0.29

234 C+ a 3593.648 8.728 171.75 1.42 0.940 14.45 0.0063 0.0245 0.191 3.21 k − − 1474.91 0.041 0.025 0.58 0.24 0.019 1.51 0.0037 0.0042 0.039 0.59

C- b 3593.612 8.708 173.51 0.85 0.898 9.17 0.0026 0.0300 0.368 1.04 k − 1463.48 0.028 0.020 0.48 0.18 0.017 1.68 0.0021 0.0046 0.030 0.59

(cont’d)

Table 5.2. OGLE-2005-SMC-001 Light Curve Models: Zero Blending. Table 5.2—Continued

1/2 ˙ Model t0 u0 tE q Q : γ φ πE,N πE,E ω : αb b/b

χ2 (day) ( 100) (day) ( 100) (deg) (yr−1) (yr−1) × ×

C+ b 3593.687 8.711 172.11 0.93 0.924 10.57 0.0008 0.0180 0.172 1.39 k − − 1469.51 0.034 0.021 0.45 0.11 0.016 1.08 0.0016 0.0030 0.028 0.29 235 C- c 3593.488 8.769 175.52 0.28 0.813 3.31 0.0103 0.0074 0.753 3.89 k − − − 1464.10 0.038 0.023 0.63 0.03 0.017 1.34 0.0019 0.0039 0.062 0.47

C+ c 3593.493 8.770 174.27 0.27 0.809 3.67 0.0084 0.0040 0.735 3.88 k − − − 1467.54 0.038 0.023 0.67 0.03 0.017 1.22 0.0015 0.0030 0.065 0.43

(cont’d) Table 5.2—Continued

1/2 ˙ Model t0 u0 tE q Q : γ φ πE,N πE,E ω : αb b/b

χ2 (day) ( 100) (day) ( 100) (deg) (yr−1) (yr−1) × ×

C+ b 3593.687 8.711 172.11 0.93 0.924 10.57 0.0008 0.0180 0.172 1.39 k − − 1469.51 0.034 0.021 0.45 0.11 0.016 1.08 0.0016 0.0030 0.028 0.29 236 C+ d 3593.751 8.725 172.80 0.65 0.919 7.20 0.0025 0.0115 0.137 0.11 k − − 1470.98 0.051 0.027 0.50 0.10 0.015 1.47 0.0015 0.0041 0.047 0.55

C- a 3593.233 8.524 166.86 1.77 0.762 279.76 0.0168 0.0310 0.746 0.73 ⊥ − − 1460.01 0.031 0.019 0.49 0.23 0.030 0.48 0.0020 0.0049 0.055 0.25

(cont’d) Table 5.2—Continued

1/2 ˙ Model t0 u0 tE q Q : γ φ πE,N πE,E ω : αb b/b

χ2 (day) ( 100) (day) ( 100) (deg) (yr−1) (yr−1) × ×

C+ a 3593.269 8.524 166.68 1.56 0.799 279.87 0.0119 0.0211 0.597 0.73 ⊥ − − 1470.98 0.032 0.019 0.50 0.25 0.034 0.49 0.0017 0.0039 0.058 0.22 237 W+ a 3593.590 8.522 167.60 8.33 0.953 8.38 0.0062 0.0092 0.267 0.21 k − 1479.54 0.059 0.020 0.46 6.18 0.016 0.78 0.0021 0.0035 0.161 0.20

W- a 3593.718 8.449 170.56 2.40 0.937 6.51 0.0088 0.0033 0.296 0.49 k − 1471.40 0.033 0.040 0.71 0.38 0.014 0.44 0.0042 0.0032 0.175 0.02

(cont’d) Table 5.2—Continued

1/2 ˙ Model t0 u0 tE q Q : γ φ πE,N πE,E ω : αb b/b

χ2 (day) ( 100) (day) ( 100) (deg) (yr−1) (yr−1) × ×

238 W+ b 3593.709 8.427 170.01 2.41 0.931 6.54 0.0055 0.0021 0.210 0.50 k 1471.49 0.034 0.037 0.90 0.37 0.014 0.44 0.0030 0.0028 0.163 0.02

W- a 3593.552 8.742 173.11 2.20 1.169 279.61 0.0051 0.0179 0.880 0.73 ⊥ − − − 1474.83 0.067 0.028 1.11 0.97 0.038 0.59 0.0030 0.0055 0.398 0.25

W+ a 3593.642 8.759 172.88 4.51 1.115 280.27 0.0010 0.0126 0.769 0.76 ⊥ − − 1480.41 0.069 0.025 1.15 2.76 0.036 0.62 0.0025 0.0051 0.359 0.21 OGLE-2005-SMC-001 15 OGLE I OGLE V CTIO I 15.5 AUCKLAND CTIO V SPITZER 16

16.5

17 3560 3580 3600 3620 3640 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 3560 3580 3600 3620 3640

Fig. 5.1.— Standard (Paczyński 1986) microlensing fit to the light curve of OGLE- 2005-SMC-001, with data from OGLE I and V in Chile, µFUN I and V in Chile, Auckland clear-filter in New Zealand, and the Spitzer satellite 3.6 µm at 0.2 AU from Earth. The data are binned by the day. All data are photometrically∼ aligned with the (approximately calibrated) OGLE data. The residuals are severe indicating that substantial physical effects are not being modeled. The models do not include parallax, but when parallax is included, the resulting figure is essentially identical.

239 2 1 (B) Earth 1.5 (A) .5

1 0 2.5 log(A) .5 −.5

0 −1 −40 −20 0 20 40 −1 −.5 0 .5 1 Time (days) 2 2

Spitzer Spitzer ) b 1.5 u0 < 0 (C) 1.5 u0 > 0 (D) *A+F s 1 1

2.5 log(F .5 .5

0 0 −40 −20 0 20 40 −40 −20 0 20 40 Time (days) Time (days)

Fig. 5.2.— Why 4 (not 3) Spitzer observations are needed to measure πE = (πE,k,πE,⊥). Panel A shows Earth-based light curve of hypothetical event (black curve) with πE = (0.4, 0.2), u0 = 0.2, and tE = 40 days, together with the corresponding (red) lightcurve− with zero parallax.− From the asymmetry of the lightcurve, one can measure π = 0.4 and u = 0.2, but no information can be extracted about E,k | 0| πE,⊥ or the sign of u0. Indeed, 9 other curves are shown with various values of these parameters, and all are degenerate with the black curve. Panel (B) shows the trajectories of all ten models in the geocentric frame (Gould 2004) that generate these degenerate curves. Solid and dashed curves indicate positive and negative u0, respectively, with πE,⊥ = 0.4, 0.2, 0, +0.2, +0.4(green, black, magenta, cyan, blue). Motion is toward positive− x, while− the Sun lies directly toward negative x. Dots indicate 5 day intervals. Panel C shows full light curves as would be seen by Spitzer, ◦ located 0.2 AU from the Earth at a projected angle 60 from the Sun, for the 5 u0 < 0 trajectories in Panel B. The source flux Fs and blended flux Fb are fit from the two filled circles and a third point at baseline (not shown) as advocated by Gould (1999). Note that these two points (plus baseline) pick out the “true” (black) trajectory, from among other solutions that are consistent with the ground-based data with u0 < 0, but Panel D shows that these points alone would pick out the magenta trajectory among u0 > 0 solutions, which has a different πE,⊥ from the “true” solution. However, a fourth measurement open circle would rule out this magenta u0 > 0 curve and so confirm the black u0 < 0 curve.

240 Fig. 5.3.— Parallax πE = (πE,N ,πE,E) 1 σ error ellipses for all discrete solutions for OGLE-2005-SMC-001. The left-hand panels show fits excluding the Spitzer data, while the right-hand panels include these data. The upper panels show fits with blending as a free parameter whereas the lower panels fix the OGLE blending at zero. The ellipses are coded by ∆χ2 (relative to each global minimum), with ∆χ2 < 1 (red), 1 < ∆χ2 < 4 (green), 4 < ∆χ2 < 9 (cyan), 9 < ∆χ2 < 16 (magenta), ∆χ2 > 16 (blue). Close- and wide-binary solutions are represented by bold and dashed curves, respectively. Most of the “free-blend, no-Spitzer” solutions are highly degenerate ◦ along the πE,⊥ direction (33 north through east), as predicted from theory, because only the orthogonal (πE,k) direction is well constrained from ground-based data. At seen from the two upper panels, the Spitzer observations reduce the errors in the πE,⊥ direction by a factor 3 when the blending is a free parameter. However, fixing the blending (lower panels)∼ already removes this freedom, so Spitzer observations then have only a modest additional effect.

241 OGLE-2005-SMC-001 15

OGLE I 3 OGLE V CTIO I 15.5 AUCKLAND 2.5 CTIO V SPITZER 16

16.5

1.5

17 3560 3580 3600 3620 3640 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 3560 3580 3600 3620 3640

Fig. 5.4.— Best-ﬁt binary microlensing model for OGLE-2005-SMC-001 together with the same data shown in Fig. 5.1. The model includes microlens parallax (two parameters) and binary rotation (two parameters). The models for ground-based and Spitzer observations are plotted in blue and red, respectively. All data are in the units of 2.5 log(A), where A is the magniﬁcation. Ground-based data are also photometrically aligned with the (approximately calibrated) OGLE data. The residuals show no major systematic trends.

242 6

-2

-4

-6 6 4 2 0 -2 -4 -6

Fig. 5.5.— Likelihood contours of the inverse projected velocity Λ v˜/v˜2 for SMC lenses together with Λ values for light-curve solutions found by MCMC.≡ The latter are color-coded for solutions with ∆χ2 within 1, 4, and 9 of the global minimum. The likelihood contours are spaced by factors of 5.

243 6

-2

-4

-6 6 4 2 0 -2 -4 -6

Fig. 5.6.— Likelihood contours of the inverse projected velocity Λ v˜/v˜2 for halo lenses together with Λ values for light-curve solutions found by MCMC.≡ Similar to Figure 5.5 except in this case the contours are color coded, with black, red, yellow, green, cyan, blue, magenta, going from highest to lowest.

244 Chapter 6

Probing 100AU Intergalactic Mg II Absorbing ∼ “Cloudlets” with Quasar Microlensing

6.1. Introduction

Mg II absorbers toward quasar sight lines have been systematically studied since

Lanzetta et al. (1987) (for more recent studies, see Zibetti et al. 2005 and references therein). Similar Mg II absorbers were subsequently seen in gamma-ray burst (GRB) spectra. Prochter et al. (2006) compared strong Mg II absorbers toward GRB and quasar sight lines, and found a signiﬁcantly higher incidence toward the former.

They proposed three possible eﬀects to explain this discrepancy: (1) obscuration of faint quasars by dust associated with the absorbers; (2) Mg II absorbers being intrinsic to GRBs; (3) gravitational lensing of the GRB by the absorbers. However, they concluded that none of these eﬀects provide a satisfactory explanation.

Frank et al. (2006) proposed a simple geometric solution to this puzzle. They argued that if Mg II absorption systems are fragmented on scales < 1016cm, similar ∼ to the beam sizes of GRBs, then the observed diﬀerence in the frequency of Mg II

245 absorbers would simply reﬂect the diﬀerence in the average beam sizes between

GRBs and quasars, with quasars being several times larger. This explanation predicts that absorption features due to intervening Mg II cloud fragments should evolve as the size of GRB afterglow changes, an eﬀect that has now been observed by Hao et al. (2006). However, the structures of Mg II absorbers down to scales of

1016cm cannot be directly inferred from their spectral features. Rauch et al. (2002) ∼ put the strongest upper limits on Mg II absorber sizes to date. They observed the spectra of three images of Q2237+0305 (Huchra et al. 1985) and found that each line of sight contained individual Mg II absorbers at approximately the same redshift, but with distinct spectral features. Thus these absorbers are part of a complex that extends at least 500pc, but the sizes of the individual “cloudlets” must be smaller ∼ than 200 300pc, corresponding to the separations of the macro-images. −

Some, if not all, strongly lensed quasars are also gravitationally microlensed by compact stellar-mass objects in the lensing galaxy (Wambsganss 2006), and

Q2237+0305 was the ﬁrst lensed quasar found to exhibit signiﬁcant microlensing variability (Corrigan et al. 1991; Wo´zniak et al. 2000). The macro-image of a microlensed quasar is split into many micro-images, and when the source moves over the caustic networks induced by the microlenses, those micro-images expand, shrink, appear, disappear and experience drastic astrometric shifts over timescales of months or years (Treyer & Wambsganss 2004). The angular sizes of major micro-images are usually of the same order as those of the quasars, and during the shape and position

246 changes of these images, absorption structures of similar scale along their sight lines will likely imprint signiﬁcant variations on the spectrum.

Brewer & Lewis (2005) pioneered the theoretical investigation of quasar microlensing as a probe of the sub-parsec structure of intergalactic absorption systems. They concluded that variation in the strength of the absorption lines over timescales of years or decades caused by microlensing can be used to probe the structures of Lyman α clouds and associated metal-line absorption systems on scales

< 0.1pc. However, as I will show, they signiﬁcantly underestimated the relevant ∼ timescales for spectral variability given the sizes of the systems they considered.

Thus, they substantially overestimated the scales of absorption structures that microlensing can eﬀectively probe.

In 6.2, I lay out the basic theoretical framework of the method. Then in § 6.3, I present a numerical simulation of the microlensing of Q2237+0305. I show § that micro-images of this quasar can be used to probe structures of Mg II and other metal-line absorbing clouds on scales of 1014 1016cm by monitoring the spectral ∼ − variations of absorption lines over months or years. Finally in 6.4, I summarize the § results and discuss their implications.

247 6.2. Varying Microlensed Quasar Image as A “Ruler”

I begin with a brief summary of notation. Subscripts “l”, “s”, “o” and “c” refer to the lens, source, observer and absorption cloud planes, respectively. The superscript “ray” is used to refer to the light ray on the cloud plane to distinguish it from the cloud. The angular diameter distance between object x and y is denoted

Dxy and is always positive regardless of which is closer; in particular, Dx refers to the angular diameter distance between the observer and object x. The vector angular position of object x is denoted θx, while its redshift is denoted zx.

Consider an absorbing cloud that is confronted with a “bundle of light rays”

ray making their way from the source to the lens to the observer. Let rc be the position vector of a point on the plane of the cloud. The line depth A of an absorption h λi line centered at wavelength λ is given by:

ray ray 2 ray ray 2 ray Aλ = σλ(rc )Aλ(rc )d rc σλ(rc )d rc (6.1) h i Z Z

ray where σλ(rc ) is the surface density of the “ray bundles” on the plane of the absorption cloud with the rays weighted by the surface brightness proﬁle of the

ray ray quasar and Aλ(rc ) is the absorption fraction at rc for light of wavelength λ

(Brewer & Lewis 2005).

248 6.2.1. Basic Geometric Configurations and Motions

The absorption cloud can be located either between the lens and observer or between the lens and source. In the former case, the angular position of the light ray

ray ray at the cloud plane is θc = θi, so rc = θiDc. The projected light rays on the cloud plane maintain the exact shapes of the quasar images, and their physical extents are proportional to the distance to the observer.

ray If the cloud is between the lens and the source, it can be easily shown that θc is given by (Brewer & Lewis 2005):

ray DlcDs DlcDs θc = 1 θi + θs, (6.2) − DlsDc DlsDc so

ray Dlc Dlc rc = Dc Ds θi + rs. (6.3) − Dls Dls

The lens-source relative proper motion, with time as measured by the observer, is given (in other notation) by Kayser et al. (1986),

1 vs 1 vl 1 voDls µls = + (6.4) 1+ zs Ds − 1+ zl Dl 1+ zl DlDs

where vs, vl, vo are the transverse velocities of the source, lens and observer, relative to the cosmic microwave background (CMB). In particular,

v = v (v zˆ)zˆ, (6.5) o CMB − CMB · 249 where zˆ is the unit vector in the direction of the lens and vCMB is the heliocentric

CMB dipole velocity (Kochanek 2004).

The formulae in this subject can be greatly simpliﬁed by proper choice of notation. To this end, I deﬁned the “absolute” proper motion of an object x moving at transverse velocity vx to be:

1 vx µabs,x = , (6.6) 1+ zx Dx

I also deﬁne the “reﬂex proper motion” of an object x relative to the observer-lens axis to be:

1 voDlx µo,l,x = sgn(zx zl) , (6.7) − 1+ zl DlDx

Then equation (6.4) can then be simpliﬁed,

µ = µ µ + µ . (6.8) ls abs,s − abs,l o,l,s

Similarly, the lens-cloud relative proper motion is given by:

µ = µ µ + µ (6.9) lc abs,c − abs,l o,l,c

Note that the last term has a diﬀerent sign depending on whether the cloud is farther or closer than the lens (see eq. [6.7]).

250 6.2.2. Bulk Motion of the Un-microlensed “Ray

Bundles”

Equations (6.2), (6.8) and (6.9) are the key formulae needed to carry out the simulation in 6.3. In this section, I discuss the bulk motion and relevant timescales § of the macro-image and its associated “ray bundles” on the cloud plane for the underlying case that the image is not perturbed by microlenses.

When the source moves at µls, the angular positions of the rays that compose the images are also changing with respect to the observer-lens axis. If the macro-image is unperturbed by the microlenses, the relative proper motion µli is simply given by (Kochanek et al. 1996),

µ = µ , (6.10) li M· ls

where is the magniﬁcation tensor. At the same time, the intersection of “ray M bundles” with the cloud plane also change its angular position relative to the lens.

If the cloud plane is between the lens and the source, then from equation (6.2), the

ray “ray bundle”-lens relative proper motion µlc is a linear combination of µls and µli weighted by distances,

ray DlcDs DlcDs µlc = 1 µli + µls. (6.11) − DlsDc DlsDc 251 Substituting equation (6.10) into equation (6.11) yields the bulk proper motion of the un-microlensed “ray bundle” relative to the lens,

ray DlcDs DlcDs µlc,bulk = 1 + µls. (6.12) − DlsDc M DlsDc I · where is the unit tensor. I

When the “ray bundles” are between the source and the lens, their relative bulk proper motion is simply,

µray = µ = µ . (6.13) lc,bulk li M· ls

There are two major eﬀects that may induce variability of absorption lines toward a lensed quasar. One is the creation, destruction, distortion and astrometric shifts of micro-images, which probe structures similar to the size of the micro-images.

The other is the bulk motion of the “ray bundles” relative to the cloud. On the cloud plane, this motion has an angular speed of ∆µ = µray µ , and the lc,bulk lc,bulk − lc time tcc required for the “ray bundles” to cross a cloud of transverse size Rc is,

R R t = c = c . (6.14) cc D ∆µ D µray µ c| bulk,lc| c| bulk,lc − lc|

6.3. Application to Q2237+0305

In October 1998, Rauch et al. (2002) obtained high-resolution Keck spectra of images A, B and C of Q2237+0305. They found Mg II absorption lines at redshifts

252 of z = 0.5656 and z = 0.827 in the spectra of all three images, but absorption proﬁles of the individual sight lines diﬀered (e.g., Figs. 6 and 10 of their paper). Therefore, they concluded that the Mg II complexes giving rise to these absorption features must be larger than 0.5kpc, while the individual Mg II components must be ∼ smaller than 200 300h−1pc. Q2237+0305 is also one of the most observed and ∼ − 50 best studied lensed quasars with obvious microlensing features, and the properties of the system are well known. These factors make it an ideal object to investigate.

The comprehensive statistical study of this lens by Kochanek (2004) showed that the size of the quasar is 1015h−1cm 1016h−1cm. Mortonson et al. (2005) ∼ − demonstrated that the source size has a much more significant effect on microlensing models than the source brightness profile. For simplicity, in the simulation, I model the source as uniform disks with four different sizes: 1015h−1cm, 3 1015h−1cm, × 5 1015h−1cm and 1016h−1cm. Uniform grids of rays are traced from the image × plane to the source plane (Wambsganss 1990). Because structures of interest have similar sizes as the source, finite-source effects must be taken into account. The grid size used has an angular scale 1/10 of the smallest source. Kochanek (2004) demonstrated that a Salpeter mass function cannot be distinguished from a uniform mass distribution and found the mean stellar mass to be M 0.037h−1M . For h i ∼ ⊙ −1 simplicity, I assign all stars in the simulation the same mass of 0.04h M⊙. Rays are shot from a region extending 47 θ on each side, where θ is the Einstein h Ei h Ei −1 radius of a 0.04h M⊙ star. I adopt a convergence and shear for image A of

253 (κ,γ) = (0.394, 0.395) from Kochanek (2004), and set the stellar surface density

κ∗ = κ. Four diﬀerent trajectories, oriented at 0, 30, 60 and 90 degrees with respect to the direction of the shear are studied. Positions on both the image plane and the source plane are recorded once a ray falls within a distance of 2 times the largest source size from any trajectory on the source plane. A total length of 5 θ along h Ei each trajectory is considered.

Throughout the paper, I adopt a ΛCDM cosmology with Ωm = 0.3, ΩΛ = 0.7

−1 −1 and H0 = 100 h km s Mpc . The lens and source are at redshifts zl = 0.0394 and zs = 1.695 (Huchra et al. 1985). These imply (Ds,Dl,Dls) =

(1223, 113, 1180) h−1 Mpc. Based on Kochanek (2004), I adopt transverse velocities

−1 of the lens, source and observer of (vl, vs, vo) = (300, 140, 62)km s . The lens and source absolute proper motions and the source reﬂex proper motion are

−1 (µabs,l,µabs,s,µo,l,s) = (0.54, 0.009, 0.11) hµas yr . So the lens absolute proper motion dominates the lens-source relative proper motion. It can easily be shown that the absolute and reﬂex proper motions of the cloud are much smaller than the absolute proper motion of the lens unless the cloud redshift is close to or smaller than the lens redshift. For Q2237+0305, the lens redshift is very small compared to the source, so most likely z z . Thus, in following analysis, I focus on clouds c ≫ l with redshift zc > zl, and ignore the absolute and reﬂex proper motions of the source

254 and the cloud. Under these assumptions, the source and the cloud share the same relative proper motion,

µ = µ = µ (6.15) lc ls − abs,l

In the simulation, as a practical matter, I hold the positions of the observer, lens galaxy (as well as its microlensing star ﬁeld) ﬁxed, and allow the source to move through the source plane at µls. By equation (6.15), the cloud then has the same relative proper motion as the source. At any given time, the angular positions of the “ray bundles” are calculated using equation (6.2). Then by simply subtracting the angular position of the source at that time, the “ray bundles” positions are transformed to the reference frame of the cloud.

During short timescales, microlensing causes centroid shifts of the macro-image

(Lewis & Ibata 1998; Treyer & Wambsganss 2004) with respect to the steady bulk motion of “ray bundles” relative to the cloud, which is described in 6.2.2. If the § direction of relative lens-source proper motion is the same as the lens shear, then by subtracting µls from equation (6.12), one ﬁnds that the “ray bundles” have their maximum bulk proper motion relative to the cloud,

DlcDs 1 ∆µbulk,lc,max = 1 1 µls − D D 1 κ γ − ! ls c − − D 2 1 1.037 lc hµas yr−1. (6.16) ∼ − Dc 255 If the source moves perpendicular relative to the lens shear, ∆µlc =

[1 D D /(D D )] 1/(1 κ + γ) 1 µ , which is approximately 0 for − LC S LS C | − − | ls the (κ,γ) of image A.

Figures 6.1 and 6.2 show the results for a source trajectory that is parallel to the shear direction. The bottom panel of Figure 6.1 shows the magnification pattern on the source plane, with a series of 4 concentric circles centered at 5 source positions; and the top panels show the images at these positions relative to the source (which has the same proper motion as the cloud). Different colors represent different source sizes. The middle panel of Figure 6.1 shows the light curves for the

4 source sizes with the blue dashed lines used to mark the times for the ﬁve source positions. Figure 6.2 shows the “ray bundle” positions in the cloud frame at redshifts

1.69, 0.83, 0.57, 0.1. The 5 diﬀerent columns show the 5 positions corresponding to those in Figure 6.1.

In the top row of Figure 6.2, one can see that the “ray bundles” at zc = 1.69, which is very close to the quasar, have almost exactly the same size and shape as the source and that the bundles show almost no bulk motion. The density of

“ray bundles” clearly have the imprints from the magniﬁcation pattern shown in the bottom panel of Figure 6.1. So if an absorption cloudlet has a size similar to or somewhat smaller than a source that is sitting directly behind it, the depth of its corresponding absorption line will change dramatically as the source crosses the caustics. The magniﬁcation close to a fold caustic is proportional to the inverse

256 square root of the distance from it, so for a cloud with angular size θc < θs, the

3/4 fractional change in absorption line depth caused by the caustics scales as (θc/θs) .

Hence, for a cloud close to the quasar redshift, structures on scale 1014 1016h−1cm ∼ − (depending on the source size) will be probed over few-month to few-year timescales

(i.e., the timescale of typical caustics crossings).

The fourth row of panels of Figure 6.2 shows “ray bundles” for zc = 0.1, which is close to the lens redshift. A distinct difference between these “ray bundles” and the ones at zc = 1.69 is that they are split into many groups of bundles, which correspond to the micro-images in the top panel of Figure 6.1. I dub these groups of “ray bundles” as micro-images in the following discussions. Most of these micro-images are stretched one-dimensionally, and most rays are concentrated in a few major micro-images. The micro-images in different columns have drastically different morphologies and positions. If there are cloudlets of similar size as these micro-images distributed on the cloud plane, then the absorption spectra will show multi-component absorption features at any given time. These components will experience drastic changes in line depth, with some components disappearing and other new components appearing during the course of months or years, as the source crosses the microlensing caustics. Another important characteristic is that the bulk of these micro-images are moving in the same direction as the source. This motion is described by equation (6.16), which yields 0.8 hµas yr. So from equation (6.14), ∼ structures as large as 3.0n 1016 h−1 cm will be crossed in n 10 h−1 yr by the ∼ × ×

257 bulk motion of the “ray bundles”. Hence, the eﬀects caused by the micro-images and the bulk motion of the bundles together probe scales of 1014 1016h−1cm on ∼ − timescales of months to years.

The second and third row of panels in Figure 6.2 refer to clouds at intermediate redshifts between the lens and the source. Their redshifts, zc = 0.83 and zc = 0.57, are close to the Mg II absorption systems observed by Rauch et al. (2002). As expected, the characteristics of these “ray bundles” are intermediate between those shown in rows 1 and 4. The overall shapes of the micro-images are close to that of the source, with magniﬁcation patterns imprinted on them. And they also clearly show multiple components, which appear and disappear as the source crosses caustics. The angular sizes of the images are close to that of the (unmagniﬁed) quasar. These micro-images could probe clouds with angular size from a factor of few smaller to a factor of few larger than the source size, which corresponds to scales of 1014 1016h−1cm. From equation (6.14) and (6.16), the bulk motion of the ∼ − bundles will probe cloud structures 0.8n 1016 h−1 cm and 1.3n 1016 h−1 cm ∼ × ∼ × during n 10 h−1 yr for z = 0.83 and z = 0.57, respectively. × c c

For a source trajectory that is perpendicular to the shear, the “ray bundles” will exhibit almost no bulk motion relative to the cloud, while the magnitude of bulk motion for intermediate trajectories is a fraction that is of the parallel case, depending on the angles of motion relative to the shear direction. And the micro-images of these trajectories share similar properties with those of the

258 trajectory that is parallel to the shear. Figure 6.2 shows that micro-image motions are on the same order of magnitude as the bulk motion of the macro-image over the timescales considered. Therefore, trajectory direction only has a modest impact on the scales at which cloud can be probed.

These results are contrast signiﬁcantly with those of Brewer & Lewis (2005), who claimed that quasar microlensing for image A of Q2237+0305 can induce considerable variability of absorption lines associated with structures as large as

0.1pc during the course of years to decades. According to their analysis, the effect is largest when the cloud is very close to the source, and for example, the timescale of line strength variation for a 0.1pc cloud very close to the source is given as 16.2yr. ∼ However, in their analysis, they effectively assumed the relative lens-cloud proper motion µ = 0. This would lead to a timescale t R D /(v D ) 16(R /0.1pc)yr lc cc ∼ c l l s ∼ c −1 for an absorption cloud near the source redshift (they adopted vl = 600km s ), which is in agreement with column 4 of their Table 1. In fact, I showed that, when peculiar velocities of the source and cloud are ignored, µlc = µls (eq. [6.15]), which is not negligible. Even considering realistic peculiar motions of the source and the cloud, it still leads to time scales that are more than one order of magnitude slower than those predicted for µlc = 0. In addition, when the cloud is close to the source, the angular sizes of the clouds they considered are orders of magnitudes larger than their source size, so effects of changes in magnification pattern on the source plane

259 alone have very little impact as well. Therefore, Brewer & Lewis (2005) signiﬁcantly overestimated the cloud size to which microlensing is eﬀectively sensitive.

6.4. Discussion and Conclusion

I have shown that there are two effects that might induce variation of absorption lines along the sight lines to lensed quasars. One effect is caused by the drastic morphological and positional changes of micro-images when the source crosses the caustic network. The other effect is due to the bulk motion of the

“ray bundles” relative to the absorption clouds. I have laid out a basic framework in studying these eﬀects for microlensed quasars in general. And in particular, I performed numerical simulations to apply the method to image A of Q2237+0305.

I demonstrated that the combinations of these two effects probe 1014 1016 h−1 cm − absorption cloudlets between the lens and the source over timescales of months to years. The existence of these cloudlets will be revealed by either changes in line depths or appearances/disappearances of multi-absorption components. Spectra should preferably be taken during the course of caustic crossings, which can be inferred from photometric monitoring programs of lensed quasars. In fact, the Mg II lines observed by Rauch et al. (2002) about 8 years ago already show different multi-components along sight lines to three different macro-images, implying they might be caused by fragmented cloudlets with similar sizes as the micro-images. A

260 similar high-resolution spectrum taken in the near future would provide a deﬁnitive test of the existence for structures of Mg II or other metal-line absorbers at the scales of 1014 1016 h−1 cm. If the spectral variations are indeed observed, a statistical − study similar to Kochanek (2004) will be required to infer the properties of the cloudlets. Moreover, a time series of spectra may provide additional constraints to quasar models that are currently based only on photometric-monitoring data.

261 Fig. 6.1.— Caustics network (bottom panel), light curve (middle panel) and images relative to the source position (top panel) for a source trajectory that is parallel to the lens shear (the direction of x-axis). Diﬀerent colors represent the 4 diﬀerent source sizes.

262 Fig. 6.2.— Physical positions of “ray bundles” in the frame of cloud at redshifts 0.1, 0.57, 0.83 and 1.69. The 5 diﬀerent columns correspond to the source positions shown in Figure 6.1. x-axis is the direction of the lens shear.

263 Appendix A

Two New Finite-Source Algorithms

To model planetary light curves, we develop two new binary-lens finite-source algorithms. The first algorithm, called “map-making”, is the main work horse. For a fixed (b, q) geometry, it can successfully evaluate the finite-source magnification of almost all data points on the light curve and can robustly identify those points for which it fails. The second algorithm, called “loop-linking”, is much less efficient than map-making but is entirely robust. We use loop-linking whenever the map-making routine decides it cannot robustly evaluate the magnification of a point. In addition, at the present time, map-making does not work for resonant lensing geometries, i.e., geometries for which the caustic has six cusps. For planetary mass ratios, resonant lensing occurs when the planet is very close to the Einstein radius, b 1. We use ∼ loop-linking in these cases also.

A.1. Map-Making

Map-making has two components: a core function that evaluates the magniﬁcation and a set of auxiliary functions that test whether the measurement

264 is being made accurately. If a light-curve point fails these tests, it is sent to loop-linking.

Finite-source effects are important when the source passes over or close to a caustic. Otherwise, the magnification can be evaluated using the point-source approximation, which is many orders of magnitude faster than finite-source evaluations. Hence, the main control issue is to ensure that any point that falls close to a caustic is evaluated using a finite-source algorithm or at least is tested to determine whether this is necessary. For very high magnification events, the peak points will always pass close to the central caustic. Hence, the core function of map-making is to “map” an Einstein-ring annulus in the image plane that covers essentially all of the possible images of sources that come close to the central caustic.

The method must also take account of the planetary caustic(s), but we address that problem further below.

We begin by inverse ray-shooting an annulus deﬁned by APSPL > Amin, where

APSPL is the Paczyński (1986) magnification due to a point source by a point lens and Amin is a suitably chosen threshold. For OGLE-2000-BLG-343, we find that

Amin = 75 covers the caustic-approaching points in essentially all cases. The choice of the density of the ray-shooting map is described below. Each such “shot” results in a four-element vector (xi, yi, xs, ys). We divide the portion of the source plane covered by this map into a rectangular grid with k = 1,...,Ng elements. We choose the size of each element to be equal to the smallest source radius being evaluated by

265 the map. Hence, each “shot” is assigned to some definite grid element k(xs, ys). We then sort the “shots” by k. For each light-curve point to be evaluated, we first find the grid elements that overlap the source. We then read sequentially through the sorted file1 from the beginning of the element’s “shots” to the end. For each “shot”, we ask whether its (xs, ys) lies within the source. If so, we weight that point by the limb-darkened profile of the source at that radius. Note that a source with arbitrary shape and surface brightness profile would be done just as easily.

For each light-curve point, we ﬁrst determine whether at least one of the images of the center of the source lies in the annulus. In practice, for our case, the points satisfying this condition are just those on the night of the peak, but for other events this would have to be determined on a point-by-point basis. We divide those points with images outside the annulus into two classes, depending on whether they lie inside or outside two or three rectangles that we “draw”, one around each caustic.

Each rectangle is larger than the maximum extent of the caustics by a factor of 1.5 in each direction. If the source center lies outside all of these rectangles, we assume that the point-source approximation applies and evaluate the magniﬁcation accordingly.

If it lies inside one of the rectangles (and so either near or inside one of the caustics), we perform the following test to see whether the point-source approximation holds.

1Whether this “ﬁle” should actually be an external disk ﬁle or an array in internal memory depends on both the size of the available internal memory and the total number of points evaluated in each lens geometry. In our case, we used internal arrays.

266 We evaluate the point-source magniﬁcations at ﬁve positions, namely, the source center A(0, 0), two positions along the source x-axis A( λρ , 0), and two positions ± ∗ along the source y-axis A(0, λρ ), where λ 1 is a parameter. We demand that ± ∗ ≤

A(λρ , 0) + A( λρ , 0) A(0,λρ )+ A(0, λρ ) ∗ − ∗ 1 + ∗ − ∗ 1 < 4σ, (A.1) 2A(0, 0) − 2A(0, 0) −

where σ is the maximum permitted error (defined in A.3). We use a minimum of § five values in order to ensure that the magnification pattern interior to the source is reasonably well sampled; the precise value is chosen empirically and is a compromise between computing speed and accuracy. We require the number of values to be at least 2[ρ∗/√q] in order to ensure that small, well-localized perturbations interior to the source caused by low mass ratio companions are not missed.

If a point passes this test, the magniﬁcation pattern in the neighborhood of the point is adequately represented by a gradient and so the point-source approximation holds. Points failing this test are sent to loop linking.

The remaining points, those with at least one image center lying in the annulus, are almost all evaluated using the sorted grid as described above. However, we must ensure that the annulus really covers all of the images. We conduct several tests to this end.

First, we demand that no more than one of the three or ﬁve images of the source center lie outside the annulus. In a binary lens, there is usually one image

267 that is associated with the companion and that is highly demagniﬁed. Hence, it can generally be ignored, so the fact that it falls outside the annulus does not present a problem. If more than one image center lies outside the annulus, the point is sent to loop-linking. Second, it is possible that an image of the center of the source could lie inside the annulus, but the corresponding image of another point on the source lies outside. In this case, there would be some intermediate point that lay directly on the boundary. To guard against this possibility, we mark the “shots” lying within one grid step of the boundaries of the annulus, and if any of these boundary “shots” fall in the source, we send the point to loop-linking. Finally, it is possible that even though the center of the source lies outside the caustic (and so has only three images), there are other parts of the source that lie inside the caustic and so have two additional images. If these images lay entirely outside the annulus, the previous checks would fail. However, of necessity, some of these source points lie directly on the caustic, and so their images lie directly on the critical curve. Hence, as long as the critical curve is entirely covered by the annulus, at least some of each of these two new images will lie inside the annulus and the “boundary test” just mentioned can robustly determine whether any of these images extend outside the annulus. For each (b, q) geometry, we directly check whether the annulus covers the critical curve associated with the central caustic by evaluating the critical curve locus using the algorithm of Witt (1990).

268 A.2. Loop-Linking

Loop-linking is a hybrid of two methods: inverse ray-shooting and Stokes’s theorem. In the ﬁrst method (which was also used above in “map-making”), one

ﬁnds the source location corresponding to each point in the image plane. Those that fall inside the source are counted (and weighted according to the local surface brightness), while those that land outside the source are not. The main shortcoming of inverse ray-shooting is that one must ensure that the ensemble of “shots” actually covers the entire image of the source without covering so much additional “blank space” that the method becomes computationally unwieldy.

In the Stokes’s theorem approach, one maps the boundary of (a polygon- approximation of) the source into the image plane, which for a binary lens yields either three or ﬁve closed polygons. These image polygons form the (interior or exterior) boundaries of one to ﬁve images. If one assumes uniform surface brightness, the ratio of the combined areas of these images (which can be evaluated using

Stokes’s theorem) to the area of the source polygon is the magniﬁcation.

There are two principal problems with the Stokes’s theorem approach. First, sources generally cannot be approximated as having uniform surface brightness.

This problem can be resolved simply by breaking the source into a set of annuli, each of which is reasonably approximated as having uniform surface brightness. However, this multiplies the computation time by the number of annuli. Second, there can

269 be numerical problems of several types if the source boundary passes over or close to a cusp. First, the lens solver, which returns the image positions given the source position, can simply fail in these regions. This at least has the advantage that one can recognize that there is a problem and perhaps try some neighboring points. The second problem is that even though the boundary of the source passes directly over a cusp, it is possible that none of the vertices of its polygon approximation lie within the caustic. The polygonal image boundary will then fail to surround the two new images of the source that arise inside the caustic, so these will not be included in the area of the image. Various steps can be taken to mitigate this problem, but the problem is most severe for very low-mass planets (which are of the greatest interest in the present context), so complete elimination of this problem is really an uphill battle.

The basic idea of loop-linking is to map a polygon that is slightly larger than the source onto the image plane, and then to inverse ray shoot the interior regions of the resulting image-plane polygons. This minimizes the image-plane region to be shot compared to other inverse-ray shooting techniques. It is, of course, more time-consuming than the standard Stokes’s theorem technique, but it can accommodate arbitrary surface-brightness proﬁles and is more robust. As we detail below, loop-linking can fail at any of several steps. However, these failures are always recognizable, and recovery from them is always possible simply by repeating the procedure with a slightly larger source polygon.

270 Following Gould & Gaucherel (1997), the vertices of the source polygon are each mapped to an array of three or five image positions, each with an associated parity. If the lens solver fails to return three or five image positions, the evaluation is repeated beginning with a larger source polygon. For each successive pair of arrays, we “link” the closest pair of images that has the same parity and repeat this process until all images in these two arrays are linked. An exception occurs when one array has three images and the other has five images, in which case two images are left unmatched. Repeating this procedure for all successive pairs of arrays produces a set of linked “strands”, each with either positive or negative parity. The first element of positive-parity strands and the last element of negative-element strands are labeled

“beginnings” and the others are labeled “ends”. Then the closest “beginning” and

“end” images are linked and this process is repeated until all “beginnings” and

“ends” are exhausted. The result is a set of two to ﬁve linked loops. As with the standard Stokes’s theorem approach, it is possible that a source-polygon edge crosses a cusp without either vertex being inside the caustic. Then the corresponding image-polygon edge would pass inside the image of the source, which would cause us to underestimate the magniﬁcation. We check for this possibility by inverse ray shooting the image-polygon boundary (sampled with the same linear density as we later sample the images) back into the source plane. If any of these points land in the source, we restart the calculation with a larger source polygon.

271 We then use these looped links to efficiently locate the region in the image plane to do inverse ray shooting. We first examine all of the links to find the largest difference, ∆ymax, between the y-coordinates of the two vertices of any link. Next, we

− sort the m = 1,...,n links by the lower y-coordinate of their two vertices, ym. One then knows that the upper vertex obeys y+ y− +∆y . Hence, for each y-value m ≤ m max of the inverse ray shooting grid, we know that only links with y− y y− +∆y m ≤ ≤ m max can intersect this value. These links can quickly be identiﬁed by reading through the sorted list from y− = y ∆y to y− = y. The x value of each of these m − max m crossing links is easily evaluated. Successive pairs of x’s then bracket the regions (at this value of y) where inverse rays must be shot. As a check, we demand that the

ﬁrst of each of these bracketing links is an upward-going link and the second is a downward-going link.

A.3. Algorithm Parameters

Before implementing the two algorithms described above, one must first specify values for certain parameters. Both algorithms involve inverse ray shooting and hence require that a sampling density by specified. Let g be the grid size in units of the Einstein radius. For magnification A 1, the image can be crudely ≫ approximated as two long strands whose total length is ℓ = 4Aρ∗ and hence of mean

2 width (πρ∗A)/ℓ = (π/4)ρ∗. If, for simplicity, we assume that the strand is aligned

272 with the grid, then there will be a total of ℓ/g grid tracks running across the strand.

Each will have two edges, and on each edge there will be an “error” of 12−1/2 in the

“proper” number of grid points due to the fact that this number must be an integer, whereas the actual distance across the strand is a real number. Hence, the total number of grid points will be in error by [(2ℓ/g)/12]1/2, while the total number itself

2 is π(ρ∗/g) A. This implies a fractional error σ,

[π(ρ /g)2A]2 3π2 σ−2 = ∗ = A(ρ /g)3. (A.2) (2ℓ/g)/12 2 ∗

In fact, the error will be slightly smaller than given by equation (A.2) in part because the strand is not aligned with the grid, so the total number of tracks across the strands will be lower than 2ℓ/g, and in part because the “discretization errors” at the boundary take place on limb-darkened parts of the star, which have lower surface brightness, so ﬂuctuations here have lower impact. Hence, an upper limit to the grid size required to achieve a fractional error σ is

g 3π2 1/3 = σ2/3A1/3. (A.3) ρ∗ 2

For each loop-linking point, we know the approximate magniﬁcation A because we know the weighted parameters of the single-lens model. We generally set σ at

1/3 of the measurement error, so that the (squared) numerical noise is an order of magnitude smaller than that due to observational error. Using equation (A.3) we can then determine the grid size.

273 For the map-making method, the situation is slightly more complicated. Instead of evaluating one particular point as in the loop-linking method, all of the points on the source plane with the same (b, q, u0,ρ) are evaluated within one map. Therefore, the grid size for this map is the minimum value from equation (A.3) to achieve the required accuracy for all of these points. We derive the following equation from equation (A.3) to determine the grid size in the map-making method:

2 1/3 3π Q(u0) g = ρ∗ (A.4) 2 u0 where

F (ti) Fb(u0) 2 Q(u0)= minAi>75 − σi (A.5) (F (t ) F (u ) ) max i − b 0

We ﬁnd that Q(u ) = 2.65 10−7 is independent of u for both the observational 0 × 0 and simulated data of OGLE 2004-BLG-343. In principle, one could determine a minimum g for all (u0,ρ) combinations and generate only one map for a given (b, q) geometry, but this would render the calculation unnecessarily long for most (u0,ρ) combinations. Instead we evaluate g for each (u0,ρ) pair and create several maps, one for each ensemble of (u0,ρ) pairs with similar g’s. The sizes of the ensembles should be set to minimize the total time spent generating, loading and employing maps. Hence, they will vary depending on the application.

274 Appendix B

MOA-2003-BLG-32/OGLE-2003-BLG-219

MOA-2003-BLG-32/OGLE-2003-BLG-219, with a peak magniﬁcation

A = 525 75 is most sensitive to low-mass planets to date (Abe et al. 2004). max ± However, instead of ﬁtting the simulated binary-lens light curves to single-lens models, Abe et al. (2004) obtain their ∆χ2 by directly subtracting the χ2 of a simulated binary-lens light curve from that of the light curve that is the best ﬁt to the data. Since the source star of this event could reside in the Sagittarius dwarf galaxy, which makes the Galactic modeling rather complicated, we do not attempt to apply our entire method to this event. We calculate planet exclusion regions with the same ∆χ2 thresholds (60 for q = 10−3 and 40 for the q < 10−3) as

Abe et al. (2004) but using our method of obtaining ∆χ2 by ﬁtting the simulated binary-lens light curves to single-lens models. Figure B.1 shows our results for the exclusion regions at planet-star mass ratios q = 10−5, 10−4, and 10−3 for

MOA-2003-BLG-32/OGLE-2003-BLG-219. The exclusion region we have obtained at q = 10−3 is about 1/4 in vertical direction and 1/9 in horizontal direction relative to the corresponding region in Abe et al. (2004), and the size of our exclusion region at 10−4 is about 60% in each dimension relative to that in Abe et al. (2004).

275 Although according to our analysis, Abe et al. (2004) overestimate the sensitivity of

MOA-2003-BLG-32/OGLE-2003-BLG-219 to both Jupiter-mass and Neptune-mass planets, their estimates of sensitivity to Earth-mass planets are basically consistent with our results and MOA-2003-BLG-32/OGLE-2003-BLG-219 nevertheless retains the best sensitivity to planets to date.

276 Fig. B.1.— Planetary exclusion regions for microlensing event MOA-2003-32/OGLE- 2003-BLG-219 as a function of projected coordinate bx and by at planet-star mass −5 −4 −3 ratio q = 10 (green), 10 (red) and 10 (blue). The source size (normalized to θE) ρ∗ is equal to 0.0007. The black circle is the Einstein ring, i.e., b = 1.

277 Appendix C

Extracting Orbital Parameters for Circular Planetary Orbit

OGLE-2005-BLG-071 is the ﬁrst planetary microlensing event for which the eﬀects of planetary orbital motion in the light curve have been fully analyzed. The distortions of the light curve due to the orbital motion are modeled by ω and b˙ as discussed in 4.3.2. In addition, the lens mass M and distance D are determined, § l so we can directly convert the microlens light-curve parameters that are normalized to the Einstein radius to physical parameters. In this section, we show that under the assumption of a circular planetary orbit, the planetary orbital parameters can be deduced from the light-curve parameters. Let r⊥ = DlθEd be the projected star-planet separation and let r⊥γ be the instantaneous planet velocity in the plane of the sky, i.e. r⊥γ⊥ = r⊥ω is the velocity perpendicular to this axis and

˙ r⊥γk = r⊥d/d is the velocity parallel to this axis. Let a be the semi-major axis and deﬁne the ˆı, ˆ, kˆ directions as the instantaneous star-planet-axis on the sky plane,

278 the direction into the sky, and kˆ = ˆı ˆ. Then the instantaneous velocity of the × planet is

GM v = [cos θkˆ + sin θ(cos φˆı sin φˆ)], (C.1) s a −

where φ is the angle between star-planet-observer (i.e., r⊥ = a sin φ) and θ is the angle of the velocity relative to the kˆ direction on the plane that is perpendicular to the planet-star-axis. We thus obtain

GM cos θ GM γ⊥ = , γk = sin θ cot φ. (C.2) s a3 sin φ s a3

To facilitate the derivation, we deﬁne

γ r3 γ2 A k = tan θ cos φ, B ⊥ ⊥ = cos2 θ sin φ, (C.3) ≡ γ⊥ − ≡ GM which yield as an equation for sin φ:

x(1 x2) B = F (sin φ); F (x)= − . (C.4) A2 + 1 x2 −

Note that F ′(sin φ) = 0 when sin2 φ = (3/2)A2 + 1 A (9/4)A2 +2. So ∗ − | | q equation (C.4) has two degenerate solutions when B < F (sin φ∗) and has no solutions when B > F (sin φ∗). Subsequently, one obtains,

r GM a = ⊥ , cos i = sin φ cos θ, K = q sin i, (C.5) sin φ − s a

279 where i is the inclination and K is the amplitude of radial velocity.

The Jacobian matrix used to transform from P (r⊥,γ⊥,γk) to P (a,φ,θ) is given below,

sin φ a cos φ 0

∂(r⊥,γ⊥,γk) GM 3 cos θ cos θ cos φ sin θ = 2 ∂(a,φ,θ) a3 − 2a sin φ − sin φ − sin φ

3 sin θ sin θ cot φ 2 cos θ cot φ − 2a − sin φ GM 2 1 2 2 = 3 cot φ sin θ tan φ . (C.6) a 2 −

Then for an arbitrary function H(a),

∂(r ,γ ,γ ) ∂(r ,γ ,γ ) 1 ⊥ ⊥ k = ⊥ ⊥ k , (C.7) ∂(H(a), cos φ, θ) ∂(a,φ,θ) × sin φH′(a) which, for the special case of a ﬂat distribution, H(a)=ln a, yields,

2 ∂(r⊥,γ⊥,γk) GM cos φ 1 2 2 = 2 . sin θ tan φ (C.8) ∂(ln(a), cos φ, θ) r⊥ sin φ 2 −

280 Appendix D

Failure of Elliptical-Source Models for MOA-2007-BLG-400

Because MOA-2007-BLG-400 is the ﬁrst microlensing event with a completely buried caustic, it is important to rule out other potential causes of the deviations seen in the light curve (apart from a planetary companion to the lens). The principal features of these deviations are the twin “spikes” in the residuals, which are approximately centered on the times when the lens enters and exits the source. In the model, these crossings occur at about HJD’ = 4354.53 and HJD’ = 4354.63, i.e., very close to the spikes in Figure 2.1. In principle, one might be able to induce such spikes by displacing the model source crossing times from the true times. The only real way to achieve this (while still optimizing the overall ﬁt parameters) would be if the source were actually elliptical, but were modeled as a circle (which, of course, is the norm).

One argument against this hypothesis is the similarity of the I and H residuals

( 2.3). If the source were an ellipsoidal variable, then one would expect color § gradients due to “gravity darkening”.

281 Nevertheless, we carried out two types of investigation of this possibility. First, we modeled the light curve as an elliptical source magnified by a point (non-binary) lens. In addition to the linear flux parameters (source flux plus blended flux for each observatory) there are 6 model parameters, the three standard point-lens parameters

(t0, u0,tE), plus the source semimajor and semiminor axes (ρa, ρb) and the angle of the source trajectory relative to the source major axis, α. We ﬁnd that the elliptical source reduces χ2 by about 200, but it does not remove the “spikes” from the residuals, which was the primary motivation for introducing it. Instead, essentially all of the χ2 improvement comes from eliminating the asymmetries from the rest of the light curve. Recall, however, that the planetary model removes both these asymmetries and the “spikes”. Moreover, the best-ﬁt axis ratio is quite extreme,

ρb/ρa = 0.7, which would produce very noticeable ellipsoidal variations unless the binary were being viewed pole on.

Next we looked for sinusoidal variations in the baseline light curve. The individual OGLE errorbars at baseline are smaller than for MOA, and since ellipsoidal variations are strictly periodic, the longer OGLE baseline (about

T = 2000 days versus T = 800 days for MOA) does a better job of isolating this signal from various possible systematics. Therefore, for this purpose, the OGLE data are more suitable than MOA. The OGLE data are essentially all baseline (only two magniﬁed points out of 452). Their periodogram shows several spikes at the

0.01 mag level, and a maximum ∆χ2 = 20, which are consistent with noise. The

282 width of the spikes is extremely narrow, consistent with the theoretical expectation for uniformly sampled data of σ(P )/P 2 24/∆χ2/(2πT ) 10−4 day−1, indicating ∼ ∼ q that the data set is behaving normally.

In brief, our investigation ﬁnds no convincing evidence for ellipticity of the source, certainly not for the several tens of percent deviation from circular that would be needed to signiﬁcantly ameliorate the deviations seen near peak in the light curve. Moreover, even arbitrary source ellipticities cannot reproduce the light curve’s most striking features: the two “spikes” in the residuals that occur when the lens crosses the source boundary.

283 BIBLIOGRAPHY

Abe, F., et al. 2004, Science, 305, 1264

Afonso, C., et al. 2000, ApJ, 532, 340

Afonso, C. et al. 2003, A&A, 400, 951

Albrow, M. D., et al. 1998, ApJ, 509, 687

Albrow, M. D., et al. 1999, ApJ, 512, 672

Albrow, M.D. et al. 1999, ApJ, 522, 1022

Albrow, M.D. et al. 2000, ApJ, 534, 894

Albrow, M. D., et al. 2000, ApJ, 535, 176

Albrow, M. D., et al. 2001, ApJ, 556, L113

Albrow, M. D., et al. 2002, ApJ, 572, 1031

Alcock, C. et al. 1993, Nature, 365, 621

Alcock, C. et al. 1995, ApJ, 454, L125

Alcock, C. et al. 1997, ApJ, 491, 436

Alcock, C. et al. 2000, ApJ, 542, 281

Alcock, C., et al. 2001, Nature, 414, 617

Alcock, C. et al. 2001, ApJ, 550, L169

An, J. H. 2005, MNRAS, 356, 1409

An, J. H. & Han, C. 2002, ApJ, 573, 351

An, J. H., et al. 2002, ApJ, 572, 521

284 Anderson, J. & King, I. R. 2004, Hubble Space Telescope Advanced Camera for Surveys Instrument Science Report 04-15

Assef, R. J. et al. 2006, ApJ, 649, 954

Aubourg, E. et al. 1993, Nature, 365, 623

Bailey, J., et al. 2009, ApJ, 690, 743

Bean, J. L., Benedict, G. F., & Endl, M. 2006, ApJ, 653, L65

Beaulieu, J.-P., et al. 2006, Nature, 439, 437

Benedict, G. F., et al. 2002, ApJ, 581, L115

Bennett, D. P., & Rhie, S. H. 1996, ApJ, 472, 660

Bennett, D. P., & Rhie, S. H. 2002, ApJ, 574, 985

Bennett, D. P., et al. 2006, ApJ, 647, L171

Bennett, D. P., Anderson, J., & Gaudi, B. S. 2007, ApJ, 660, 781

Bennett, D. P., et al. 1999, Nature, 402, 57

Bennett, D.P. et al., 2008, ApJ, 684, 663

Bessell, M. S., & Brett, J. M. 1988, PASP, 100, 1134

Bond, I. A., et al. 2001, MNRAS, 327, 868

Bond, I. A., et al. 2002, MNRAS, 333, 71

Bond, I. A., et al. 2004, ApJ, 606, L155

Boss, A. P. 2002, ApJ, 567, L149

Boss, A. P. 2006, ApJ, 643, 501

Boutreux, T. & Gould, A. 1996, ApJ, 462, 705

Brewer, B. J., & Lewis, G. F. 2005, MNRAS, 356, 703

Burrows, A., Hubbard, W. B., Saumon, D., & Lunine, J. I. 1993, ApJ, 406, 158

Burrows, A., et al. 1997, ApJ, 491, 856

Butler, R. P., et al. 2006, PASP, 118, 1685

Carpenter, J. 2001, AJ, 121, 2851

285 Calchi Novati, S., de Luca, F., Jetzer, P., & Scarpetta, G. 2006, A&A, 459, 407‘

Chang, K., & Refsdal, S. 1979, Nature, 282, 561

Chang, K. & Refsdal, S. 1984, A&A, 130, 157

Chung, S.-J., et al. 2005, ApJ, 630, 535

Claret, A. 2000, A&A, 363, 1081

Corrigan, R. T., et al. 1991, AJ, 102, 34

Cox, A. N. 2000, Allen’s Astrophysical Quantities (4th Ed.; New York; Springer)

Cumming, A., et al. 2008, PASP, 120, 531

Delfosse, X., et al. 1998, A&A, 338, L67

Demarque, P., Woo, J.-H., Kim, Y.-C., & Yi, S. K. 2004, ApJS, 155, 667

Dominik, M. 1999a, A&A, 341, 943

Dominik, M. 1999b, A&A, 349, 108

Dominik, M. 2006, MNRAS, 367, 669

Dong, S., et al. 2006, ApJ, 642, 842

Dong, S., et al. 2009, ApJ, 695, 970

Dong, S., et al. 2009, ApJ, 698, 1826

Doran, M., & Mueller, C. M. 2004, Journal of Cosmology and Particle Astrophysics, 9, 3

Drake, A. J., Cook, K. H., & Keller, S. C. 2004, ApJ, 607, L29

Duquennoy, A., & Mayor, M. 1991, A&A, 248, 485

Einstein, A. 1936, Science, 84, 506

Eisenhauer, F., et al. 2005, ApJ, submitted (astro-ph/0502129)

Endl, M., et al. 2006, ApJ, 649, 436

European Space Agency, 1997, The Hipparcos and Tycho Catalogues (SP-1200; Noordwijk: ESA)

Fischer, D. A., & Valenti, J. 2005, ApJ, 622, 1102

286 Frank, S. et al. 2007, Astrophysics and Space Science, 312, 3

Gaudi, B. S., & Gould, A. 1997, ApJ, 486, 85

Gaudi, B. S., & Sackett, P. D. 2000, ApJ, 528, 56

Gaudi, B. S., et al. 2002, ApJ, 566, 463

Gaudi, B. S. 2008, Astronomical Society of the Paciﬁc Conference Series, 398, 479, ed. D. Fischer, F. Rasio, S. Thorsett, & A. Wolszczan, Extreme Solar Systems (arXiv:0711.1614)

Gaudi, B. S., Naber, R. M., & Sackett, P. D. 1998, ApJ, 502, L33

Gaudi, B. S., et al. 2008, Science, 319, 927

Ghosh, H., et al. 2004, ApJ, 615, 450

Gould, A. 1992, ApJ, 392, 442

Gould, A. 1994a, ApJ, 421, L71

Gould, A. 1994b, ApJ, 421, L75

Gould, A. 1995, ApJ, 441, L21

Gould, A. 1999, ApJ, 514, 869

Gould, A. 2000a, ApJ, 535, 928

Gould, A. 2000b, ApJ, 542, 785

Gould, A. 2004, ApJ, 606, 319

Gould, A. 2008, ApJ, 681, 1593

Gould, A. 2009, Astronomical Society of the Paciﬁc Conference Series, 403, 86, ed. K.Z. Stanek, The Variable Universe: A Celebration of Bohdan Paczynski (arXiv:0803.4324)

Gould, A. & An, J.H. 2002, ApJ, 565, 1381

Gould, A. & Gaucherel, C. 1997, ApJ, 477, 580

Gould, A., & Loeb, A. 1992, ApJ, 396, 104

Gould, A. & Salim, S. 1999 ApJ, 524, 794

Gould, A., Miralda-Escude, J., & Bahcall, J. N. 1994, ApJ, 423, L105

287 Gould, A., Stutz, A., & Frogel, J. A. 2001, ApJ, 547, 590

Gould, A., Bennett, D. P., & Alves, D. R., 2004, ApJ, 614, 404

Gould, A. et al. 2006, ApJ, 644, L37

Gould, A., Gaudi, B. S., & Bennett, D. P. 2007, NASA/NSF Exoplanet Task Force White Paper (arXiv:0704.0767)

Gould, A., et al. 2009, ApJ, 698, L147

Griest, K. & Saﬁzadeh, N. 1998, ApJ, 500, 37

Han, C. 2007, ApJ, 661, 1202

Han, C., & Gould, A. 1995, ApJ, 447, 53

Han, C. & Gould, A. 1996, ApJ, 467, 540

Han, C. & Gould, A. 2003, ApJ, 592, 172

Han, C., & Gaudi, B. S. 2008, ApJ, 689, 53

Hao, H., et al. 2007, ApJ, 659, 99

Harris, J. & Zaritsky, D. 2006, AJ, 131, 2514

Hauschildt, P. H., Allard, F., & Baron, E. 1999, ApJ, 512, 377

Haywood, M. 2008, A&A, 482, 673

Huchra, J. et al. 1985, AJ, 90, 691

Ida, S., & Lin, D. N. C. 2004, ApJ, 616, 567

Ida, S., & Lin, D. N. C. 2005, ApJ, 626, 1045

Jiang, G. et al. 2005, ApJ, 617, 1307

Johnson, J. A., et al. 2007a, ApJ, 665, 785

Johnson, J. A., et al. 2007b, ApJ, 670, 833

Kallivayalil, N., van der Marel, R. P., & Alcock, C. 2006, ApJ, 652, 1213

Kayser, R. et al. 1986, A&A, 166, 36

Kennedy, G. M., Kenyon, S. J., & Bromley, B. C. 2006, ApJ, 650, L139

Kennedy, G. M., & Kenyon, S. J. 2008, ApJ, 673, 502

288 Kervella, P., et al. 2004, A&A, 426, 297

King, I. 1962, AJ, 67, 471

Kiraga, M., & Paczynski, B. 1994, ApJ, 430, L101

Kochanek, C. S. 1996, ApJ, 473, 610

Kochanek, C. S. 2004, ApJ, 605, 58

Kornet, K., Wolf, S., & R´o˙zyczka, M. 2006, A&A, 458, 661

Kozlowski, S., Wo´zniak, P. R., Mao, S., & Wood, A. 2007, ApJ, 671, 420

Kubas, D., et al. 2008, A&A, 483, 317

Lanzetta, K. M. et al. 1987, ApJ, 322, 739

Laughlin, G., Bodenheimer, P., & Adams, F. C. 2004, ApJ, 612, L73

Lecar, M., Podolak, M., Sasselov, D., & Chiang, E. 2006, ApJ, 640, 1115

Lejeune, T., Cuisinier, F., & Buser, R. 1997, A&AS, 125, 229

Lejeune, T., Cuisinier, F., & Buser, R. 1998, A&AS, 130, 65

Lewis, G. F., & Ibata, R. A. 1998, ApJ, 501, 478

Liebes, S., Jr. 1964, Phys.Rev., 133, 835

Lin, D. N. C., Bodenheimer, P., & Richardson, D. C. 1996, Nature, 380, 606

Lissauer, J. J. 1987, Icarus, 69, 249

Manfroid, J. 1995, A&AS, 113, 587

Mao, S., & Paczynski, B. 1991, ApJ, 374, L37

Marcy, G. W., et al 1998, ApJ, 505, L147

Marcy, G. W., et al 2001, ApJ, 556, 296

Mao, S. & Paczy´nski, B. 1991, ApJ, 374, L37

Mortonson, M. J. et al. 2005, ApJ, 628, 594

Nishiyama, S., et al. 2005, ApJ, 621, L105

Paczy´nski, B. 1986, ApJ, 304, 1

289 Park, B.-G. et al. 2004, ApJ, 609, 166

Park, B.-G., Jeon, Y.-B., Lee, C.-U., & Han, C. 2006, ApJ, 643, 1233

Pejcha, O., & Heyrovsk´y, D. 2009, ApJ, 690, 1772

Pollack, J. B., Hubickyj, O., Bodenheimer, P., Lissauer, J. J., Podolak, M., & Greenzweig, Y. 1996, Icarus, 124, 62

Poindexter, S. et al. 2005, ApJ, 633, 914

Prochter, G. E., et al. 2006, ApJ, 648, L93

Rasio, F. A., & Ford, E. B. 1996, Science, 274, 954

Rattenbury, N. J., Bond, I. A., Skuljan, J., & Yock, P. C. M. 2002, MNRAS, 335, 159

Rauch, M. et al. 2002, ApJ, 576, 45

Reach, W. T., et al. 2005, PASP, 117, 978

Read, J.I., Wilkinson, M.I., Evans, N.W., Gilmore, G., Kleyna, J.T. 2006, MNRAS, 366, 429

Refsdal, S. 1966, MNRAS, 134, 315

Reid, N. 1991, AJ, 102, 1428

Rhie, S. H., et al. 2000, ApJ, 533, 378

Rivera, E. J., et al. 2005, ApJ, 634, 625

Sackett, P. D. 1997, ESO Document: SPG-VLTI-97/002 (astro-ph/9709269)

Sahu, K. C., et al. 2001, Nature, 411, 1022

Sahu, K. C., Anderson, J., & King, I. R. 2002, ApJ, 565, L21

Santos, N. C., Israelian, G., & Mayor, M. 2004, A&A, 415, 1153

Schechter, P.L., Mateo, M. & Saha, A. 1993, PASP, 105 1342

Smith, M. C., Mao, S., & Paczy´nski, B. 2003, MNRAS, 339, 925

Stanek, K. Z., et al. 1997, ApJ, 477, 163

Stanimirovi´c, S. Staveley-Smith, L., & Jones, P.A. ApJ, 604, 176

Sumi, T. 2004, MNRAS, 349, 193

290 Tisserand, P. et al. 2007, A&A, 469, 387

Treyer, M., & Wambsganss, J. 2004, A&A, 416, 19

Udalski, A. 2003, Acta Astron., 53, 291

Udalski, A., et al. 1993, Acta Astronomica, 43, 289

Udalski, A., et al. 2005, ApJ, 628, L109

Walker, M. A. 1995, ApJ, 453, 37

Wambsganss, J. 1990, Ph.D. Thesis

Wambsganss, J. 2006, Saas-Fee Advanced Course 33: Gravitational Lensing: Strong, Weak and Micro, 453

Ward, W. R. 1997, Icarus, 126, 261

Witt, H. J. 1990, A&A, 236, 311

Wolszczan, A., & Frail, D. A. 1992, Nature, 355, 145

Wo´zniak, P. R. 2000, Acta Astron., 50, 421

Wo´zniak, et al. 2000, ApJ, 529, 88

Yoo, J., et al. 2004a, ApJ, 603, 139

Yoo, J., et al. 2004b, ApJ, 616, 1204

Yoo, J., Chanam´e, J., & Gould, A. 2004, ApJ, 601, 311

Zaritsky, D., Harris, J., Grebel, E.K., & Thompson, I.B. 2000, ApJ, 534, L53

Zibetti, S., et al. 2005, ApJ, 631, L105

291