Exploring the Extremes of Exoplanet Detection and Characterization in High-Magnification Microlensing Events

Dissertation

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

By

Jennifer C. Yee

Graduate Program in Astronomy

The Ohio State University 2013

Dissertation Committee: Professor Andrew P. Gould, Advisor Professor B. Scott Gaudi Professor Richard W. Pogge Copyright by

Jennifer C. Yee

2013 ABSTRACT

The field of microlensing planet searches is about to enter a new phase in which wide-field surveys will be the dominant mode of planet detection. In addition, there are now plans to execute microlensing surveys from space allowing the technique to reach smaller planets and resolve some of difficulties of ground-based microlensing where the resolution is poor. This new phase of observations also requires a new mode of analysis in which events are analyzed en masse rather than as individuals.

Until now, there has not been any investigation into the detection threshold for planets in real data. Some people have suggested that the threshold for detecting planets may be as small as ∆χ2 of 160, and that is frequently used in microlensing simulations of planet yields. However, no planets have been published with signals that small. I have done the first empirical investigation of the detection threshold for planets in high-magnification microlensing events. I found that MOA-2008-BLG-310

(∆χ2 = 880), MOA-2011-BLG-293 (∆χ2 = 500 without followup data), and

MOA-2010-BLG-311 (∆χ2 = 80) form a sequence that spans from detected with high confidence (mb310) to marginally detected (mb293) to something too small

ii to claim with confidence (mb311). This suggests that the detection threshold for planets in high-magnification events is 500 ∆χ2 < 880. ≤

I have also analyzed OGLE-2008-BLG-279 to determine the range of planets that are detectable for this event given the excellent data quality and the high- magnification. This event illustrates that high-magnification events will still be important in the era of surveys because each event is much more sensitive to planets than any individual low-magnification event. Because they probe the central caustics, high-magnification events are sensitive to planets at any angle, meaning that they place more stringent limits on the presence of planets. For this event,

Jupiter-mass companions can be ruled out from 0.5-20 AU.

As the field extends to new modes of observations, it is worth considering how we can maximize the information we can obtain for each microlensing event, particularly given the limitation that microlensing is primarily sensitive to mass ratio rather than planet mass. I propose a means to take advantage of the excellent light curves that will be available from space and combine them with ground-based observations to measure microlens parallax for a large fraction of the microlensing events that will be seen by a space-based microlensing survey. This measurement will enable the measurement of the planet masses for these events.

iii Dedicated to my parents and the memory of my grandfather

Thomas P. Vogl

iv ACKNOWLEDGMENTS

First and foremost, I would like to thank my advisor Andy Gould, who has carefully steered all of the projects presented in this dissertation. He has helped me develop from a very uncertain student to a scientist confidently pursuing a research career. He has challenged me to think and operate at a much higher level. His passion for science has inspired me to work harder and push my limits.

I would also like to thank the other members of my committee: Scott Gaudi and Rick Pogge. Scott Gaudi got me started as my first research advisor at Ohio

State. He also carefully read and commented on my papers and was an excellent

Graduate Studies Chair. Rick Pogge kept MicroFUN running smoothly for many years and always made time to help me with the latest crisis.

I would like to thank my microlensing collaborators. The OGLE, MOA,

PLANET, RoboNET, and MiNDSTEp have all provided data used in this dissertation as well as feedback on the analysis. I would also like to thank the

SMARTS team at Yale and CTIO who make our observations from the CTIO

SMARTS 1.3m possible. A special thanks to the members of MicroFUN, especially the amateur astronomers whose dedication to the project continues to inspire me.

v Thank you to the Department of Astronomy and its members past and present for maintaining a scientifically lively atmosphere dedicated to the success of its graduate students.

Thank you to my friends who have helped me maintain perspective and provided support and commiseration.

I would like to thank my parents for giving me the confidence to know I can succeed at anything: my mother for her strength and my father for sharing his enthusiasm for astronomy and science fiction. Thanks to my sister for sharing my morbid sense of humor and always providing moral support.

Finally, I would especially like to thank my husband Eric for his unwavering love and support and for sticking with me even when I was crabby and miserable.

This work was supported in part by an allocation of computing time from the

Ohio Supercomputing Center.

vi VITA

April 3, 1985 ...... Born – Poughkeepsie, NY

2007 ...... B.A. Astrophysics, Swarthmore College

2007 – 2008 ...... Distinguished University Fellow, The Ohio State University

2008 – 2006 ...... Graduate Teaching and Research Associate, The Ohio State University

2009 – 2012 ...... National Science Foundation Graduate Research Fellow, The Ohio State University

2010 ...... M.S. Astronomy, The Ohio State University

2012 – 2013 ...... Distinguished University Fellow, The Ohio State University

PUBLICATIONS

Research Publications

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

2. J.C. Yee and B. S. Gaudi, “Characterizing Long-Period Transiting Plan- ets Observed by Kepler”, ApJ, 688, 616 (2008).

3. J.A. Carter, J.C. Yee, J.D. Eastman, B.S. Gaudi, and J.N. Winn, “Ana- lytic Approximations for Transit Light-Curve Observables, Uncertainties, and Covariances”, ApJ, 689, 499 (2008).

4. M.J. Valtonen, et al. (40 coauthors including Yee, J.C.), “Tidally In- duced Outbursts in OJ 287 during 2005-2008”, ApJ, 698, 781 (2009).

vii 5. J.C. Yee, et al. (84 coauthors), “Extreme Magnification Microlensing Event OGLE-2008-BLG-279: Strong Limits on Planetary Companions to the Lens Star”, ApJ, 703, 2082 (2009).

6. C. Villforth, et al. (47 coauthors including J.C. Yee), “Variability and stability in blazar jets on time scales of years: Optical polarization monitoring of OJ287 in 2005-2009”, MNRAS, 402, 2087 (2010).

7. J.C. Yee and E.L.N. Jensen, “A Test of Pre-Main-Sequence Lithium De- pletion Models”, ApJ, 711, 303 (2010).

8. P. Fouque, et al. (112 coauthors including J.C. Yee), “OGLE 2008-BLG- 290: an accurate measurement of the limb darkening of a galactic bulge K Giant spatially resolved by microlensing”, A&A, 518, 51 (2010).

9. A. Gould, et al. (126 coauthors including J.C. Yee), “Frequency of Solar-like Systems and of Ice and Gas Giants Beyond the Snow Line from High-magnification Microlensing Events in 2005-2008”, ApJ, 720, 1073 (2010).

10. Y.-H. Ryu, et al. (107 coauthors including J.C. Yee), “OGLE-2009- BLG-092/MOA-2009-BLG-137: A Dramatic Repeating Event with the Second Perturbation Predicted by Real-time Analysis”, ApJ, 723, 81 (2010).

11. N. Miyake, T. Sumi, S. Dong, R. Street, L. Mancini, A. Gould, D.P. Bennett, Y. Tsapras, J.C. Yee, et al. (109 additional coauthors), “A Sub-Saturn Mass Planet, MOA-2009-BLG-319Lb”, ApJ, 728, 120 (2011).

12. V. Batista, et al. (125 coauthors including Yee, J. C.), “MOA-2009- BLG-387Lb: a massive planet orbiting an M dwarf”, A&A, 529, 102 (2011).

13. “Binary microlensing event OGLE-2009-BLG-020 gives a verifiable mass, distance and orbit predictions” Skowron, J., et al. (101 coauthors including Yee, J. C.) ApJ, 738, 87 (2011).

14. Y. Muraki, et al. (128 coauthors including Yee, J. C.), “Discovery and Mass Measurements of a Cold, 10 Earth Mass Planet and Its Host Star”, ApJ, 741, 22 (2011).

15. T. Bensby, et al. (23 coauthors including J.C. Yee), “Chemical evolution of the Galactic bulge as traced by microlensed dwarf and subgiant stars. IV. Two bulge populations”, A&A, 533, 134 (2011).

viii 16. I.-G. Shin, et al. (151 coauthors including J.C. Yee) “Microlensing Bina- ries Discovered through High-magnification Channel”, ApJ, 746, 127 (2012).

17. J.-Y. Choi, et al. (150 coauthors including J.C. Yee), “Characterizing Lenses and Lensed Stars of High-magnification Single-lens Gravitational Microlens- ing Events with Lenses Passing over Source Stars”, ApJ, 751, 41 (2012).

18. E. Bachelet, et al. (140 coauthors including J.C. Yee), “MOA 2010- BLG-477Lb: Constraining the Mass of a Microlensing Planet from Microlensing Parallax, Orbital Motion, and Detection of Blended Light”, ApJ, 754, 73 (2012).

19. A. Gould and J.C. Yee “Cheap Space-Based Microlens Parallaxes for High-Magnification Events”, ApJL, 755, 73 (2012).

20. I-G. Shin, et al. (120 coauthors including J.C. Yee), “Characterizing Low-mass Binaries from Observation of Long-timescale Caustic-crossing Gravita- tional Microlensing Events”, ApJ, 755, 91 (2012).

21. J.C. Yee, et al. (77 coauthors), “MOA-2011-BLG-293Lb: A test of pure survey microlensing planet detections”, ApJ, 755, 102 (2012).

22. J.-Y. Choi, et al. (120 coauthors including J.C. Yee), “A New Type of Ambiguity in the Planet and Binary Interpretations of Central Perturbations of High-Magnification Gravitational Microlensing Events”, ApJ, 756, 48 (2012).

23. I-G. Shin, et al. (155 coauthors including J.C. Yee), “Microlensing Bina- ries with Candidate Brown Dwarf Companions”, ApJ, 760, 116 (2012).

24. C. Han, A. Udalski, J.-Y. Choi, J.C. Yee, et al. (32 additional coau- thors), “The second multiple-planet system discovered by microlensing: OGLE-2012- BLG-0026Lb, c, a pair of jovian planets beyond the snow line”, ApJL, 763, 28 (2013).

25. A. Gould, J.C. Yee, I.A. Bond, A. Udalski, C. Han, U.G. Jorgensen, J. Greenhill, Y. Tsapras, M.H. Pinsonneault, T. Bensby, et al. (109 additional coau- thors), “MOA-2010-BLG-523: ‘Failed Planet’ = RS CVn Star ”, ApJ, 763, 141 (2013).

26. T. Bensby, J.C. Yee, et al. (21 additional coauthors), “Chemical evolu- tion of the Galactic bulge as traced by microlensed dwarf and subgiant stars V. Evidence for a wide age distribution and a complex MDF”, A&A, 549, 147 (2013).

27. R.A. Street, et al. (128 coauthors including J.C. Yee), “MOA-2010-

ix BLG-073L: An M-Dwarf with a Substellar Companion at the Planet/Brown Dwarf Boundary”, ApJ, 763, 67 (2013).

28. A. Gould and J.C. Yee, “Microlens Terrestrial Parallax Mass Measure- ments: A Rare Probe of Isolated Brown Dwarfs and Free-Floating Planets”, ApJ, 764, 107 (2013).

29. A. Gould and J.C. Yee, “Microlens Surveys are a Powerful Probe of As- teroids”, ApJ, 767, 42(2013).

30. A. Gould,I-G. Shin, C. Han, A. Udalski, J.C. Yee, “OGLE-2011-BLG- 0417: A Radial Velocity Testbed for Microlensing”, ApJ, submitted, (2013).

31. J.-Y. Choi, et al. (123 coauthors including J.C. Yee), “Microlensing Dis- covery of a Population of Very Tight, Very Low-mass Binary Brown Dwarfs”, ApJ, submitted (2013).

32. N. Kains, et al. (130 coauthors including J.C. Yee), “A Giant Planet beyond the Snow Line in Microlensing Event OGLE-2011-BLG-0251”, A&A, 552, 70 (2013).

33. J.C. Yee, et al. (147 additional coauthors), “MOA-2010-BLG-311: A planetary candidate below the threshold of reliable detection”, ApJ, 769, 77 (2013).

34. J.C. Yee, “WFIRST Planet Masses from Microlens Parallax”, ApJL, 770, 31 (2013).

FIELDS OF STUDY

Major Field: Astronomy

x Table of Contents

Abstract ...... ii

Dedication ...... iv

Acknowledgments ...... v

Vita ...... vii

List of Tables ...... xv

List of Figures ...... xvi

Chapter 1 Introduction ...... 1

1.1 HistoryofMicrolensing...... 1

1.2 Detecting Planets with Microlensing ...... 5

1.3 MicrolensingObservations ...... 7

1.4 ScopeofthisDissertation ...... 12

Chapter 2 Extreme Magnification Microlensing Event OGLE-2008- BLG-279: Strong Limits on Planetary Companions to the Lens Star 17

2.1 Introduction...... 17

2.2 DataCollection ...... 20

2.3 Point-LensAnalysis...... 22

xi 2.3.1 Angular Einstein Ring Radius ...... 22

2.3.2 Parallax ...... 25

2.4 TheBlendedLight ...... 27

2.4.1 AstrometricOffset ...... 29

2.4.2 SearchforShear...... 30

2.5 LimitsonPlanets...... 32

2.6 Summary ...... 37

Chapter 3 MOA-2011-BLG-293Lb: A test of pure survey microlensing planet detections ...... 46

3.1 Introduction...... 46

3.2 Data Collection and Reduction ...... 53

3.2.1 Data Binning and Error Normalization ...... 56

3.3 CMD...... 57

3.4 Effect of Faint Sources on Microlens Parameters ...... 58

3.4.1 Robustly Measured Parameters ...... 59

3.4.2 Parameters Vulnerable to Systematics ...... 63

3.5 Analysis ...... 64

3.5.1 Effect of Systematics in fbase ...... 68

3.5.2 Analysis with Survey-Only Data ...... 69

3.6 PhysicalPropertiesoftheEvent...... 71

3.7 PropertiesoftheLens ...... 72

3.7.1 LimitsontheLensBrightness ...... 72

3.7.2 Bayesian Analysis ...... 74

3.8 Possible Constraints from AO Observations ...... 77

xii 3.9 Discussion...... 80

3.9.1 Implications for Planet Formation Theory ...... 80

3.9.2 The Ongoing Importance of Followup ...... 81

Chapter 4 MOA-2010-BLG-311: A planetary candidate below the threshold of reliable detection ...... 94

4.1 Introduction...... 94

4.2 Data...... 96

4.2.1 Observations ...... 96

4.2.2 DataReduction...... 97

4.3 Color-MagnitudeDiagram ...... 98

4.4 Modeling...... 100

4.4.1 TheBasicModel ...... 100

4.4.2 Error Renormalization ...... 101

4.4.3 PointLensModels ...... 108

4.4.4 2-bodyModels ...... 109

4.4.5 Reliability of the Planetary Signal ...... 112

4.5 θE and µrel ...... 115

4.6 The Lens as a Possible Member of NGC 6553 ...... 116

4.7 Discussion...... 118

Chapter 5 Chapter 5: WFIRST Planet Masses from Microlens Parallax ...... 132

5.1 Introduction...... 132

5.2 Measuring πE,⊥ ...... 134

5.2.1 SimplifiedCase ...... 134

xiii 5.2.2 Exact Expression for πE,⊥ ...... 137

5.2.3 Constraints on πE,⊥ ...... 139

5.3 Discussion...... 141

Appendix A Uncertainty in θ⋆ µ, and θE ...... 149

Bibliography ...... 151

xiv List of Tables

2.1 LightCurveFits ...... 38

3.1 Data...... 85

3.2 Lightcurve“Invariants”...... 86

3.3 ModelParameters...... 87

4.1 Data Properties for Two Methods of Error Renormalization. The observatories are listed in order of longitude starting with the most Eastward. If data were taken in more than one filter at a given site, different filters are given on successive lines. The error renormalization coefficients and method for removing outliers are described in Section 4.4.2...... 122

4.2 Fits with 1-Parameter Errors. The ∆χ2 is given relative to the χ2 of the best-fit planetary model with s < 1 and without parallax (χ2 = 6637.96), i.e. model “C”; positive numbers indicate a worse fit and negative numbers indicate an improvement relative to this model. The point lens models are given first, followed by the planetary models; “A”, “B”, and “C” denote the three planetary models with distinct values of α corresponding to the three χ2 minima...... 123

4.3 Fits with 2-Parameter Errors. The ∆χ2 is given relative to the χ2 of the best-fit planetary model with s < 1 and without parallax (χ2 = 6751.93), i.e. model “C”; positive numbers indicate a worse fit and negative numbers indicate an improvement relative to this model. The point lens models are given first, followed by the planetary models; “A”, “B”, and “C” denote the three planetary models with distinct values of α corresponding to the three χ2 minima...... 124

xv List of Figures

1.1 GeometryofMicrolensing...... 16

2.1 LightcurveofOGLE-2008-BLG-279 ...... 39

2.2 Color-Magnitude Diagram of the OGLE-2008-BLG-279 field . . . . . 40

2 2.3 χ contours as a function of u0 and z0 ...... 41

2.4 Possible solutions for the blend plotted with Yale-Yonsei isochrones . 42

2.5 Shear as a function of α...... 43

2.6 Planet sensitivity as a function of distance from the lens...... 44

2.7 Detection efficiency map in the (d,q)plane...... 45

3.1 The light curve of MOA-2011-BLG-293 ...... 88

3.2 Color-Magnitude Diagram of MOA-2011-BLG-293 ...... 89

3.3 BaselineMOAandOGLEfluxes...... 90

3.4 Caustic structure and source trajectory of MOA-2011-BLG-293 . . . 91

3.5 Comparison of fits to all data and “survey-only” data ...... 92

3.6 Probability densities for the lens distance and mass ...... 93

4.1 Color-magnitude diagram and color-color diagram of MOA-2010-BLG- 311...... 125

4.2 Light curve of MOA-2010-BLG-311 ...... 126

4.3 Residuals to the best-fit models: 1-parameter errors ...... 127

4.4 Residuals to the best-fit models: 2-parameter errors ...... 128

4.5 Thecausticandthreesourcetrajectories ...... 129

xvi 4.6 The difference between the cumulative χ2 distribution and the expected N 2 distribution ( j χj = N) ...... 130 4.7 DSS image ofP NGC 6553 and Excess star density over the background 131

5.1 Projection of a Microlensing Event onto the Observer Plane . . . . . 147

5.2 Generalized Geometry of a Microlensing Event ...... 148

xvii Chapter 1

Introduction

1.1. History of Microlensing

“Microlensing” is a term used to describe the bending of light by the gravitational field of a star. The term was originally created by Paczynski (1986a) to distinguish between the lensing of a quasar into distinct images by an intervening galaxy (“macrolensing”) and the slight variations induced in the images by the lensing due to the individual stars in the galaxy, which occurs on microarcsecond scales. In this dissertation, the term will be used to describe the lensing of individuals stars, i.e., a background star (the source) being lensed by another star along the line of sight (the lens).

Einstein first derived the basics of microlensing in 1912 (Renn et al. 1997), but did not publish the result until 19361(Einstein 1936). In the meantime, other authors touched on some of the ideas contained in Einstein (1936) as a natural consequence of Einstein’s Theory of General Relativity. In the context of proposing tests of the 1Whether the 1936 result drew upon the Einstein’s 1912 work or whether it was a separate derivation is unknown (Renn et al. 1997).

1 theory, Eddington (1920) discussed the possibility that if a star is < 1′′ from another ∼ star at a different distance there will be two deflected rays from the more distant star leading to an increase in its magnification. Chwolson (1924) showed that if the two stars are perfectly superposed, the light from the background star would be deflected into a ring. In his 1936 paper, Einstein quantified these effects for the case that the foreground star (the lens) is much closer than the background star (the source). Extending this to the general case, the size of the ring is

4GM DL(DS DL) rE = 2 − (1.1) s c DS or written as an angle,

r 4GM D D θ = E = S − L . (1.2) E D c2 D D L r S L

Figure 1.1 illustrates the relationships between the various quantities. The definitions of the variables in the above equations are as follows:

G = gravitational constant,

c = speed of light,

M = the mass of the lensing star,

DL = the distance to the lensing star,

DS = the distance to the source star.

2 Einstein also calculated the combined magnification of the images of the source star, which in the general case for which DL is not much smaller than DS is given by

u2 + 2 A(u)= , (1.3) u√u2 + 4 where u is the projected separation between the source and the lens star as a fraction of the Einstein ring. Since the source and the lens stars will be moving, u changes as a function of time, creating a microlensing event whose maximum magnification is determined by the impact parameter, u0, between the source and the lens. Einstein’s brief paper was later expanded on by Tikhov (1938)2 who worked out in detail the positions and shapes of the images and also the effect of the finite size of the source.

None of these early calculations considered that the effect would be observable because the images would be too close together to resolve and the magnification would be so low as to be unmeasurable for a “typical” case for which the source and lens are separated by r . For a 1M lens and galactic scale distances of ∼ E ⊙ kiloparsecs (e.g., D = 4 kpc and D = 8 kpc), θ 1 mas. The magnification for a L S E ∼ source at 1 rE from the lens is only 1.3, and it is highly unlikely for two unrelated stars to pass that close to each other. Hence, the idea of observing the lensing of individual stars seemed highly implausible.

The idea was revisited by Liebes (1964) and Refsdal (1964) who developed the theoretical framework of lensing and calculated the probability of a lensing event

2Refsdal (1964) references an earlier paper Tikhov (1937), but this reference is difficult to locate.

3 given a population of stars. Liebes (1964) focused lensing by a globular cluster because of the high density of stars, considering both self-lensing of stars within the cluster and the case for which the cluster was seen superimposed on the bulge of the Milky Way. Refsdal (1964) considered the rate of lensing by nearby stars and proposed that events could be predicted by observing the proper motions of the stars. Still, while these authors found that it was possible for stellar lensing events to occur, their expected rates were very low, of order a few per decade. It was not until the late 1980s that surveys specifically for microlensing events were seriously considered. At this point, numerical techniques for analyzing photographic plates had finally developed to a point where it was feasible to measure small changes in the brightness of millions of stars (Paczynski 1986b). The later development of large

CCDs further simplified the problem.

Microlensing searches started out as a means to look for massive compact halo objects (MACHOs). These dark matter candidates were expected to be (1M ), O ⊙ but dark, e.g., stellar mass black holes or brown dwarfs. Paczynski (1986b) and

Griest (1991) both proposed that microlensing could be used for this purpose, and

Alcock et al. (1992) and Aubourg et al. (1993) were the first surveys to do so. These searches were primarily targeted towards the Magellanic clouds. However, Paczynski

(1991) and Griest et al. (1991) proposed searches toward the galactic bulge as a means to detect lensing by both MACHOs and disk stars. The lensing from the disk stars was guaranteed to be observed and would serve as proof-of-concept that

4 microlensing occurred even if no MACHOs were found. It was quickly realized that planets would also be detectable around the lens stars (Mao & Paczynski 1991), and this became a major scientific driver for microlensing surveys.

1.2. Detecting Planets with Microlensing

The lensing by planets orbiting other stars was first conceived of by Liebes

(1964) who calculated the cross-section for interaction and the magnification due to a hypothetical Earth located around another Sun3. He concluded, “It seems, therefore, that the primary effect of planetary deflectors bound to stars other than the sun will be to slightly perturb the lens action of these stars.” Mao & Paczynski

(1991) proposed that the microlensing surveys being undertaken toward the galactic bulge would detect many binary microlensing events that would include both binary star systems and planetary systems. By their rough calculation, 5–10% of all events could contain detectable signals from a planet. Gould & Loeb (1992) expanded on this work to give a more detailed calculation of the effect of hypothetical planets, and in particular, showed that there was a coincidence between the typical radius of the Einstein ring and the semi-major axis of Jupiter, such that if most exoplanet systems are similar to our own, the prior probability from the semi-major axis boosts

3Indeed, Liebes (1964) appears to have thought of many future applications of microlensing including searches for dark objects, lensing toward M31, and the detection of free-floating planets, although he did dismiss the last two as nearly impossible to observe.

5 the microlensing detectability of the planet. Hence, the likelihood of detecting a

Jupiter-analog was increased to 17% and in addition, there is a 3% probability of detecting a Saturn-analog. In total, Gould & Loeb (1992) showed that planets could be detected in 20% of all microlensing events if every star has a system of planets similar to the solar system.

Because the mass ratio, q, between a planet and a star is so small (10−3 between

Jupiter and the Sun and 10−5 between the Earth and the Sun) the dominant lensing effect comes from the star and the planet acts as a small perturbation on the primary, stellar lensing event. If the planet is near the Einstein ring of the lens star, that planet can interact with an image of the source, creating a detectable perturbation.

This can happen in one of two ways. First, as the source star moves past the lens, the images trace out a path that follows the Einstein ring and interacts with the planet. This is the case considered in Gould & Loeb (1992). Second, if the source has a small impact parameter and is highly magnified, then the images are very large, spanning most of the Einstein ring. Because of their large size, the images are unstable and hence, very sensitive to perturbations from a planet.

An alternative formulation is to consider the caustic structure. There are certain source positions for which the images will interact with the planet. In particular there are places for which a perfect point source will be infinitely magnified. These places form closed curves called the caustics. A given planet creates two sets of caustics: one large planetary caustic and a smaller caustic at the position of the

6 lens star called the central caustic. If the source passes near or over a caustic, the images interact with the planet creating a detectable perturbation. The planetary caustic corresponds to the Gould & Loeb (1992) case whereas the central caustic corresponds to the high-magnification case.

Griest & Safizadeh (1998) demonstrated that if an event has a high magnification

(u < 0.02) it will almost always probe the central caustic, leading to a 100% 0 ∼ detection rate for Jupiter-mass planets near the Einstein ring (0.618

This follows from Equation 1.3, since a high-magnification requires small u, and the central caustic is known to surround the position of the lens star. Griest & Safizadeh

(1998) showed that the size of this caustic is approximately

qs w , (1.4) ≈ (s 1)2 − which is 0.01 for a q = 0.03 planet at s 0.6. Hence, if u < 0.02, the source is ∼ ∼ likely to pass close enough to this caustic to produce a measurable perturbation to the normal point lens light curve.

1.3. Microlensing Observations

Paczynski (1991) and Griest et al. (1991) showed that the brightnesses of millions of stars must be monitored in order to detect just a handful of microlensing events toward the bulge each year. Given the resources available at the time, this meant that microlensing surveys to search for such events could have a cadence of

7 about one observation per night. This is sufficient to detect a microlensing event and measure its basic parameters, but too low to detect planets, whose signals last

1 day for Jupiter-mass ratios and get shorter as the mass ratio decreases. Hence, ∼ Gould & Loeb (1992) suggested a follow up program to observe known microlensing events with a cadence of several times or more per night in order to search for planetary signals in these events. The work of Griest & Safizadeh (1998) further showed that limiting the observations to high-magnification events would maximize the probability of detecting a planet and minimize the required resources, since only the peaks of the events (when u is small) would need to be monitored.

The Microlensing Follow-Up Network (MicroFUN) is a group dedicated to following high-magnification events as described in Griest & Safizadeh (1998).

MicroFUN observations rely on real-time announcements of ongoing microlensing events by the Microlensing Observations in Astrophysics (MOA; Bond et al. 2001) and Optical Gravitational Lens Experiment (OGLE; Udalski 2003) survey groups.

In practice, the threshold for an event to be of interest to MicroFUN is somewhat more strict than originally proposed: u0 < 0.01. This is driven by the increased ∼ sensitivity of the events with increasing peak magnification and also the fact that the telescopes are small (typically 0.25m in diameter), which gives an additional preference for higher magnification events since those events will be brighter. The recognition of the correspondence between the Griest & Safizadeh (1998) strategy and the nature of the MicroFUN resources (most of the telescopes are owned

8 and operated by amateur astronomers) has led to the intensive observation of many high-magnification events. The events analyzed in this dissertation are all high-magnification events observed by MicroFUN. The combination of the intrinsic sensitivity of these events with the intensive observations has allowed us to push the limits of the technique to search for smaller planets and planets farther from the

Einstein ring.

In the past, the driver for having two sets of groups, survey and followup, was set by the limitations in available resources. Detecting microlensing events requires monitoring millions or tens of millions of stars, but detecting planets requires an observing cadence of several times per hour in order to reach the smallest mass planets (e.g., Earths and Super-Earths). Previously, it was extremely difficult to both monitor enough stars and achieve a high enough cadence to detect microlensing planets using the same telescope, so the tasks were split between two groups. Only two of the 16 microlensing planet discoveries to date were made purely with data from survey telescopes (Bond et al. 2004; Bennett et al. 2008), and one of them involved a modification to the survey strategy to behave like a followup telescope and focus on the event in question.

However, the situation is now changing, and microlensing surveys will soon dominate the planet detections. Recently, OGLE and MOA have upgraded to larger

field of view cameras that allow them to cover more area with a faster cadence. The

MOA and OGLE groups have teamed up with Wise Observatory in Israel to begin

9 the first, second generation microlensing survey (Shvartzvald & Maoz 2011). Such a survey is defined as being able to monitor enough stars with a high enough cadence to both detect events and find and characterize planetary signals without any additional data from followup groups. The Korea Microlensing Telescope Network currently under construction will also conduct such a survey. Finally, space-based microlensing surveys are under consideration as part of the Euclid and WFIRST missions.

There are two major advantages to the pure survey approach over the previous survey+followup approach. First, many more microlensing events are being routinely monitored throughout the entire light curve than is possible from targeted followup observations. The planetary caustics are much larger than the central caustic, so given that a microlensing event occurs, i.e., u0 < 1.0, it is much more likely for a source to interact with the planetary caustic just based on its much larger cross-section. The difference is that unlike central caustic crossing events, the interaction cannot be predicted, so the events must be monitored over the entire light curve rather than just the peak. Hence, if a survey can monitor more events with a high enough cadence, more planets will be detected in a given sample of microlensing events than would be found by focusing solely on the high-magnification events because those events are rare.

The second advantage of a pure survey approach to microlensing planet detection is that the observing strategy is pre-determined and not influenced by

10 observed planet signals. In contrast, followup observations of low-magnification events are usually intensified if a planetary signal is suspected. The pre-determined observing strategy means that the sample forms a controlled experiment from which statistics of the discovered planets, such as their frequency and distribution in mass and semi-major axis, can be extracted. The combination of the larger numbers of planets expected to be found from these next generation surveys and the nature of these events (both with and without planets) as part of a controlled sample requires a new method of analysis in which all of these events are analyzed in a uniform manner.

A critical point in analyzing such surveys is choosing a detection threshold.

Planetary signals will have a continuum of magnitudes that extends into the noise. A detection threshold is necessary to distinguish between signals that have a high probability of being real and signals that are most likely due to noise.

Previous analyses have chosen a range of detection thresholds, most of which are described by their authors as “conservative.” The earliest analyses for detection efficiencies in microlensing events used ∆χ2 = 40 (Rhie et al. 2000) and ∆χ2 = 50

(Albrow et al. 2000). This is consistent with the threshold used in the first uniform analysis of a large sample of microlensing events to measure the frequency of Jovian planets (Gaudi et al. 2002). That work determined their threshold based on the observed ∆χ2 distribution for the full sample of events and also the distribution for non-microlensed stars fit with crude planetary models. They found that signals up

11 to ∆χ2 = 60 were likely to be produced by systematics in the photometry and so chose ∆χ2 = 60 as their threshold. Higher thresholds have been advocated as well.

The later analysis of Gould et al. (2010a), which focused only on high-magnification events, argued that for those events a threshold of ∆χ2 = 500 was more appropriate based on experience, e.g., the smallest planetary signal in a high-magnification event that had been published was ∆χ2 = 880 (Janczak et al. 2010). However, they also argued that the threshold could be anywhere in the range from ∆χ2 = 300 to

∆χ2 = 700. In the case of space-based microlensing surveys, Penny (2011) argued that ∆χ2 = 100 was appropriate based on simulations of a Euclid microlensing survey.

1.4. Scope of this Dissertation

This Dissertation takes on a variety of topics that will be of importance to future surveys, drawing on the analysis of high-magnification microlensing events observed by MicroFUN. While each of these events is of some interest in its own right, in context with other events they have the ability to help us understand what is truly detectable with microlensing. I will address the range of planets that can be detected with ground-based microlensing, in terms of both the minimum mass and the minimum and maximum separations that can be detected. I will also discuss the detectability of those planets, showing the limits under which they

12 are indeed detectable from the proposed and ongoing second-generation surveys.

Finally, I will address the problem of planet mass versus mass ratio, which is endemic to microlensing. I will propose a means for space-based microlensing surveys to routinely measure the microlens parallax, which will allow measurements of the lens star masses and the conversion of mass ratios, the fundamental microlensing observable, to true planet masses, which are of far more interest.

First, I present the analysis of OGLE-2008-BLG-279, which has the most sensitivity to planets of any microlensing event due to a combination of its extreme magnification and the coverage of the peak of the event. This event peaked at a maximum magnification of A 1600 on 30 May 2008. The peak of the event ∼ exhibits both finite-source effects and terrestrial parallax, from which I determine the mass of the lens, M = 0.64 0.10M , and its distance, D = 4.0 0.6kpc. I rule l ± ⊙ l ± out Jupiter-mass planetary companions to the lens star for projected separations in the range 0.5-20 AU. More generally, using a detection threshold of ∆χ2 = 160, this event was sensitive to planets with masses as small as 0.2 M 2 M with ⊕ ≃ Mars projected separations near the Einstein ring ( 3 AU). ∼

Second, I examine two events, MOA-2011-BLG-293 and MOA-2010-BLG-311, to confirm that the surveys are able to detect and characterize planets without additional data and to set empirical limits on the detection threshold. MOA-2011-BLG-293Lb is the first planet to be published from the new surveys, and it also has substantial followup observations. This planet is robustly detected in survey+followup data

13 (∆χ2 5400). The planet/host mass ratio is q = 5.3 0.2 10−3. The best fit ∼ ± × projected separation is s = 0.548 0.005 Einstein radii. However, due to the s s−1 ± ↔ degeneracy, projected separations of s−1 are only marginally disfavored at ∆χ2 = 3.

+0.35 A Bayesian estimate of the host mass gives ML = 0.59−0.29 M⊙, with a sharp upper limit of ML < 1.2 M⊙ from upper limits on the lens flux. Hence, the planet mass is m = 3.3+1.9M , and the physical projected separation is either r 1.0 AU p −1.6 Jup ⊥ ≃ or r 3.4 AU. I show that survey data alone predict this solution and are able ⊥ ≃ to characterize the planet, but the ∆χ2 is much smaller (∆χ2 500) than with ∼ the followup data. The ∆χ2 for the survey data alone is smaller than for any other securely detected planet.

MOA-2010-BLG-311 is a high magnification (Amax > 600) microlensing event with complete data coverage over the peak, making it very sensitive to planetary signals. I fit this event with both a point lens and a 2-body lens model and find that the 2-body lens model is a better fit but with only ∆χ2 80. The preferred mass ∼ ratio between the lens star and its companion is q = 10−3.7±0.1, placing the candidate companion in the planetary regime. Despite the formal significance of the planet, I show that because of systematics in the data the evidence for a planetary companion to the lens is too tenuous to claim a secure detection. When combined with analyses of other high-magnification events, this event helps empirically define the threshold for reliable planet detection in high-magnification events, which remains an open question.

14 Finally, I present a method using only a few ground-based observations of magnified microlensing events to routinely measure the parallaxes of WFIRST events if WFIRST is in an L2 orbit. This could be achieved for all events with

Amax > 30 using target-of-opportunity observations of select WFIRST events, or with a complementary, ground-based survey of the WFIRST field, which can push beyond this magnification limit. When combined with a measurement of the angular size of the Einstein ring, which is almost always measured in planetary events, these parallax measurements will routinely give measurements of the lens masses and hence, the absolute masses of the planets. They can also lead to mass measurements for dark, isolated objects such as brown dwarfs, free-floating planets, and stellar remnants if the size of the Einstein ring is measured.

15 ~ rE I

r θ E E OLS

DS

DL

Fig. 1.1.— Basic geometry of microlensing. Light from the source, S, at a distance DS is bent by a lens, L, at a distance DL from the observer, ‘O’. The observer sees an image of the source, I. In this case, the observer, lens, and source are perfectly aligned, so the source would appear as a perfect ring with angular radius, θE. This size may also be expressed as a physical size in the lens, rE, or observer,r ˜E, planes.

16 Chapter 2

Extreme Magnification Microlensing Event OGLE-2008-BLG-279: Strong Limits on Planetary Companions to the Lens Star

2.1. Introduction

A complete census of planets beyond the snow line will be crucial for testing the currently favored core-accretion theory of planet formation since that is the region where this model predicts that giant planets form. For example, Ida & Lin (2004)

find that gas giant planets around solar-type stars preferentially form in the region between the snow line at 2.7 AU and 10 AU. While radial velocity and transit ∼ searches account for most of the more than 300 planets known to date, microlensing has the ability to probe a different region of parameter space that reaches far beyond the snow line and down to Earth-mass planets. Microlensing is most sensitive to planets near the Einstein ring radius, which Gould & Loeb (1992) showed lies just outside the snow line:

M 1/2 r 4 l AU, (2.1) E ≃ M µ ⊙ ¶ 17 for reasonable assumptions. This sensitivity to planets beyond the snow line is demonstrated by the eight published planets found by microlensing, which range in mass from super-Earths to Jupiters and more massive objects (Bond et al. 2004;

Udalski et al. 2005; Beaulieu et al. 2006; Gould et al. 2006; Bennett et al. 2008;

Dong et al. 2009b; Gaudi et al. 2008).

In high magnification microlensing events (A > 100), the images finely probe ∼ the full angular extent of the Einstein ring, making these events particularly sensitive to planets over a wide range of separations (Griest & Safizadeh 1998). Additionally, because the time of maximum sensitivity to planets (the peak of the event) can be determined in advance, intensive observations can be planned resulting in improved coverage of the event, particularly given limited resources. Even when a planet is not detected, the extreme sensitivity of such an event can be used to put broad constraints on planetary companions.

High magnification events are also useful because it is more likely that secondary effects such as the finite-source effect and terrestrial parallax can be measured

(Gould 1997). These effects can be used to break several microlensing degeneracies and allow a measurement of the mass of the lens and its distance. This allows us to determine a true mass of a planet rather than the planet/star mass ratio and a true projected separation rather than a relative one. Thus, in addition to being more sensitive to planets, high magnification events allow us to make more specific inferences about the nature of the system.

18 Previous work has empirically demonstrated the sensitivity of high magnification events to giant planets by analyzing observed events without detected planets and explicitly computing the detection sensitivity of these events to planetary companions. The first high magnification event to be analyzed in such a way was

MACHO 1998-BLG-35 (Rhie et al. 2000). Rhie et al. (2000) found that planets with a Jupiter-mass ratio (q = 10−3) were excluded for projected separations in units of the Einstein ring radius of d =0.37–2.70. Since then, many other authors have analyzed the planet detection sensitivity of individual high-magnification events

(Bond et al. 2002; Gaudi et al. 2002; Yoo et al. 2004; Abe et al. 2004; Dong et al.

2006; Batista et al. 2009). In particular, prior to the work presented here, the most sensitive event with the broadest constraints on planetary companions was MOA

2003-BLG-32, which reached a magnification of 520 (Abe et al. 2004). Dong et al.

(2006) found that this event had sensitivity to giant planets out to d < 41. ∼

This chapter presents the analysis of OGLE-2008-BLG-279, which reached a magnification of A 1600 and was well-covered over the peak, making it extremely ∼ sensitive to planetary companions. In fact, as we will show, this event has the greatest sensitivity to planetary companions of any event yet analyzed, and we can exclude planets over a wide range of separations and masses. Furthermore, this

1Dong et al. (2006) also analyzed the event OGLE-2004-BLG-343, which reached a peak magnification of A 3000. Although this is the highest magnification event analyzed for planets, ∼ sparse observational coverage over the peak greatly reduced its sensitivity.

19 event exhibited finite-source effects and terrestrial parallax, allowing a measurement of the mass and distance to the lens. This allows us to place constraints on planets in terms of their mass and projected separation in physical units. We begin by describing the data collection and alert process in 2.2. In 2.3 we describe our fits § § to the light curve and the source parameters. We then go on to discuss the blended light and the shear contributed by a nearby star in 2.4. Finally, we place limits on § planetary companions in 2.5. We conclude in 2.6. § §

2.2. Data Collection

On 2008 May 13 (HJD′ HJD - 2,450,000 = 4600.3604), the OGLE ≡ collaboration announced the discovery of a new microlensing event candidate

OGLE-2008-BLG-279 at RA=17h58m36.s17 Dec=-30◦22′08.′′4 (J2000.0). This event was independently announced by the MOA collaboration on 2008 May 26 as MOA-2008-BLG-225. Based on the available OGLE and MOA data, µFUN began observations of this event on 2008 May 27 from the CTIO SMARTS 1.3m in Chile, acquiring observations in both the V and I bands, and the next day identified it as likely to reach very high magnification two days hence. This event was monitored intensively over the peak by MOA, the PLANET collaboration, and many µFUN observatories. Specifically, the µFUN observatories Bronberg, Hunters

Hill, Farm Cove, and Wise obtained data over the peak of this event (see Fig.

20 2.1). OGLE-2008-BLG-279 peaked on 2008 May 30 at HJD′ = 4617.3481 with a magnification A 1600. ∼

Because there were so many data sets, this analysis focuses on the µFUN data from observatories that covered the peak of the event (µFUN Bronberg (South

Africa), Hunters Hill (Australia), Farm Cove (New Zealand), and Wise (Israel)) and PLANET Canopus (Australia) combined with the data from OGLE and MOA which cover both peak and baseline. We used the data from CTIO to measure the colors of the event but not in other analyses. Early fits of the data indicated that the

Bronberg data from HJD′4617.0-4617.32 suffer from systematic residuals that are more severe than those seen in any of the other data, so these data were excluded from subsequent analysis.

The data were all reduced using difference imaging analysis (DIA; Wozniak

2000) with the exception of the CTIO data which were reduced using the DoPHOT package (Schechter et al. 1993). The uncertainties in all the data sets were normalized so that the χ2/degree-of-freedom 1, and we removed > 3σ outliers ∼ whose deviations were not confirmed by near simultaneous data from other observatories. The normalization factors for each observatory are as follows:

OGLE(1.8), MOA(1.0), µFUN Bronberg(1.4), µFUN Hunters Hill I(2.7) and U(1.5),

µFUN Farm Cove(2.1), µFUN Wise(3.8), PLANET Canopus(4.6), and µFUN CTIO

I(1.4) and V(2.0).

21 2.3. Point-Lens Analysis

The data for OGLE-2008-BLG-279 appear to be consistent with a very-high magnification, A = 1570 120, single lens microlensing event. We therefore begin by ± fitting the data with a point-lens model and then go on to place limits on planetary companions in 2.5. In this section, we describe our fits to the data and address the § second-order, finite-source and parallax effects on the light curve.

2.3.1. Angular Einstein Ring Radius

From the V - and I-band images taken with CTIO both during the peak and after the event, we construct a CMD of the event (Fig. 2.2). We calibrate this

CMD using stars that are also in the calibrated OGLE-III field. For the source, we measure [I, (V I)] = [21.39 0.09, 2.53 0.01]. If we assume that the source is in − ± ± the bulge and thus behind the same amount of dust as the clump, we can compute the dereddened color and magnitude. We measure the color and magnitude of the clump: [I, (V I)] = [16.48, 2.71]. The absolute color and magnitude of the clump − cl are [M , (V I) ] = [ 0.20, 1.05], which at a distance of 8.0 kpc would appear to I − 0 cl − be [I, (V I)] = [14.32, 1.05]. We find A = I I = 16.48 14.32 = 2.16 and − 0,cl I cl − 0,cl − E(V I)=(V I) (V I) = 1.66. We then calculate the dereddened color − − cl − − 0,cl and magnitude of the source to be [I, (V I)] = [19.23, 0.87]. − 0

22 The angular Einstein ring radius can be determined by combining information from the light curve and the color-magnitude diagram (CMD). Finite source effects in the light curve enable us to determine the ratio of the source size, θ⋆, to the

Einstein radius, θE:

ρ⋆ = θ⋆/θE. (2.2)

We can then estimate θ⋆ from the color and magnitude of the source measured from the CMD, and solve for θE.

Finite-Source Effects

If the source passes very close to the lens star, finite-source effects will smooth out the peak of the light curve and allow a measurement of the source size ρ⋆.

Although finite-source effects are not obvious from a visual inspection of the light curve, including them yields a dramatic improvement in χ2. In order to fit for

finite source effects, we first estimate the limb-darkening of the source from its color and magnitude. We combine the color and magnitude of the source with the Yale-Yonsei isochrones (Demarque et al. 2004), assuming a distance of Ds = 8 kpc and solar metallicity, to estimate Teff = 5250K and log g = 4.5. We use these values to calculate the limb-darkening coefficients, u, from Claret (2000), assuming a microturblent velocity of 2 km/s. We calculate the linear limb-darkening parameters

Γ and Γ using Γ = 2u/(3 u) to find Γ = 0.65 and Γ = 0.47. We use these V I − V I values in our finite-source fits to the data. We find that a point-lens fit including

23 finite-source effects is preferred by ∆χ2 of 2647.85 over a fit assuming a point source.

We search a grid of u0 and ρ⋆ near the minimum to confirm that this is a well constrained result. We use z0 = u0/ρ⋆ as a proxy for ρ⋆ following Yoo et al. (2004).

2 The resultant χ map in the u0-z0 plane is shown in Figure 2.3. Our best-fit value for ρ is 6.6 0.6 10−4. For this value of ρ , z is almost unity, indicating that ⋆ ± × ⋆ 0 the source just barely grazed the lens star. The other parameters for our best-fit including finite-source effects are given in Table 2.1.

Source Size

We convert the dereddened color and magnitude of the source to (V K) using − Bessell & Brett (1988), and combine them with the surface brightness relations in

Kervella et al. (2004) to derive a source size of θ = 0.54 0.4 µas. The uncertainty ⋆ ± in θ⋆ comes from two sources: the uncertainty in the flux and the uncertainty in the conversion from the observed (V I) color to surface brightness. The uncertainty − in the flux (i.e. the model fit parameter fs,I ) is 8.5%, and we adopt 7% as the uncertainty due to the surface brightness conversion. From equation (2.2), we find that θ = θ /ρ = 0.81 0.07 mas. We also calculate the (geocentric) proper motion E ⋆ ⋆ ± of the source: µ = θ /t = 2.7 0.2 mas/yr. Because the peak flux ( f /ρ ) geo E E ± ∝ s,I ⋆ and source crossing time (ρ⋆tE) are both essentially direct observables, and so are well constrained by the light curve, the fractional uncertainty in θE and µgeo are

24 comparable to the fractional uncertainty in θ⋆. This result is generally applicable to point-lens/finite-source events and is discussed in detail in the Appendix.

2.3.2. Parallax

Given that we have a measurement for θE, if we can also measure microlens parallax, πE, we can combine these quantities to derive the mass of the lens and its distance. The mass of the lens is given by

θE 4G mas Ml = , κ 2 8.14 . (2.3) κπE ≡ c AU ≃ M⊙

Its distance Dl is

1 AU = πl = πs + πrel, (2.4) Dl

where πl is the parallax of the lens, πs = 0.125 mas is the parallax of the source

(assuming a distance of Ds = 8 kpc), and πrel = θEπE.

Microlens parallax is the combination of two observable parallax effects in a microlensing event. Terrestrial parallax occurs because observatories located on different parts of the Earth have slightly different lines of sight toward the event and so observe slight differences in the peak magnification and in the timing of the peak, described by the parameters u0 and t0, respectively (Hardy & Walker

1995; Holz & Wald 1996). Orbital parallax occurs because the Earth moves in its orbit during the event, again, changing the apparent line of sight. Gould (1997)

25 argued that one might expect to measure both finite-source effects and terrestrial parallax in extreme high-magnification events. We fit the light curve for both of the sources of parallax, including finite-source effects. Fitting for both kinds of parallax simultaneously yields a ∆χ2 improvement of 165 (see Table 2.1). We

find π = (π , π )=( 0.15 0.02, 0.02 0.02), where π and π are the E E,E E,N − ± ± E,E E,N projections of πE in the East and North directions, respectively.

Smith et al. (2003) showed that for orbital parallax and a constant acceleration, u0 has a sign degeneracy. This degeneracy may be broken if terrestrial parallax is observed (see also Gould 2004). In the fits described above, we assumed u0 > 0. We repeat the parallax fit fixing u0 < 0. We find that the +u0 solution is preferred over the u case by ∆χ2 = 37 (see Table 2.1). − 0

We perform a series of fits in order to isolate the source of the parallax signal, i.e. whether it is primarily due to orbital parallax or terrestrial parallax. We first

fit the light curve for orbital parallax alone and then fit for terrestrial parallax alone. The results are given in Table 2.1. For +u0, the orbital parallax fit gives

(π , π ) = (1.5 0.4, 0.3 0.2) and a ∆χ2 improvement of 16 over the E,E E,N ± − ± ∼

finite-source fit without parallax. In contrast, the +u0 fit for terrestrial parallax alone yields ∆χ2 = 166 and (π , π )=( 0.16 0.02, 0.03 0.02). While the E,E E,N − ± ± orbital and terrestrial parallaxes are nominally inconsistent at more than 3σ, from previous experience (Poindexter et al. 2005) we know that low-level orbital parallax can be caused by small systematic errors or xallarap (the orbital motion of the

26 source due to a companion), so we ignore this discrepancy. From the ∆χ2 values, it is clear that terrestrial parallax dominates the microlens parallax signal in this event, so any spurious orbital parallax signal does not affect our final results.

We also confirm that the terrestrial parallax signal is seen in multiple observatories, and thus cannot be attributed to systematics in a single data set.

To test this, we repeat the fits for parallax excluding the data from an individual observatory. If a data set is removed and the parallax becomes consistent with zero, then that observatory contributed significantly to the detection of the signal. Using this process of elimination, we find that the signal comes primarily from the MOA and Bronberg data sets.

Given the results of these various fits, we conclude that the best fit to the data is for the +u0 solution, and we include both forms of parallax for internal consistency. Combining this parallax measurement with our measurement of θE from

2.3.1, we find M = 0.64 0.1M and D = 4.0 0.6kpc (π = 0.13 0.02 mas) § l ± ⊙ l ± rel ± using equations (2.3) and (2.4).

2.4. The Blended Light

The centroid of the light at baseline when the source is faint is different from the centroid at peak magnification, indicating that light from a third star is blended into the PSF. The measured color and magnitude of blended light are

27 [I, (V I)] = [17.21 0.01, 2.32 0.02]. Stars of this magnitude are relatively rare, − b ± ± and so the most plausible initial guess is that the third star is either a companion to the source or a companion to the lens. If the former, we can use the values of A and E(V I) we found above to derive the intrinsic color of the blend: I − [I, (V I)] = [15.05, 0.66]. This assumes that the blend is in the bulge at a − 0,b distance of 8 kpc, giving an absolute magnitude of MI,b = 0.53 and MV,b = 1.19.

Figure 2.4 shows this point (open square) compared to solar (Z=0.02) and sub-solar metallicity (Z=0.001) Yale-Yonsei isochrones at 1, 5, and 10 Gyrs (Demarque et al.

2004). These isochrones show that the values of [M , (V I) ] may be consistent V − 0 b with a sub-giant that is a couple Gyr old, but a more precise determination of age is not possible since the age is degenerate with the unknown metallicity of the blend.

If the blend is a companion to the lens, however, it lies in front of some fraction of the dust. In order to derive a dereddened color and absolute magnitude to this star, we need a model for the dust. We explore this scenario using a simple model for the extinction that is constant in the plane of the disk and decreases exponentially out of the plane with a scale height of H0 = 100 pc:

D sin b A (d)= K 1 exp − , (2.5) I 1 − H · µ 0 ¶¸ where D is the distance to a given point along the line of sight, b is the Galactic latitude, and K1 is a constant. We can solve for K1 by substituting in the value of

AI that we find for the source at 8 kpc. We then model the selective extinction in a

28 similar manner:

D sin b E(V I)= K 1 exp − , (2.6) − 2 − H · µ 0 ¶¸ and solve for K2 using the value of E(V-I) calculated for the source at 8 kpc. From equations 2.5 and 2.6, we can recover the intrinsic color and magnitude of the blend assuming it is at various distances. In Figure 2.4, we plot a point assuming the blend is at the distance of the lens, 4.0 kpc. By interpolating the isochrones and assuming a solar metallicity, we find that the blend is consistent with being a 1.4

M⊙ sub-giant companion to the lens with an age of 3.8 Gyr. For comparison, we also plot a line showing how the inferred color and magnitude of the blend vary with the assumed distance.

2.4.1. Astrometric Offset

From the measured blend flux, one can determine the astrometric offset of the source and blend by comparing the centroid of light during and after the event. At a given epoch, the centroid is determined by the ratio of the flux of the blend to the sum of the fluxes of the source and lens. That ratio depends on the magnification of the source. Thus, if we know the magnification of the source at two different epochs and the intrinsic magnitude of the source and the blend, we can solve for the separation of the lens and the blend. We find ∆θ = 153 18 mas. Given this ± offset, we will show below that based on the lack of shear observed in the light curve,

29 the blended light cannot lie far in the foreground and thus cannot be the sub-giant companion to the lens hypothesized above.

2.4.2. Search for Shear

Because all stars have gravity, if the blend described above lies between the observer and the source, it will induce a shear γ in the light curve. We can estimate the size of the shear using the observed astrometric offset and assuming that the blend is a 1.4M⊙ companion to the lens.

θ2 κπ M γ = E,b = rel,b b , ∆θ2 ∆θ2 π M ∆θ −2 = 6.2 10−5 rel b . (2.7) × 0.13 mas 1.4 M⊙ 153 mas ³ ´ µ ¶ µ ¶

Using the 1 σ upper limit on the separation (171 mas), we find a minimum shear of γ = 4.9 10−5 if the blend is a companion to the lens. To determine if this × value is consistent with the light curve, we perform a series of fits to the data using binary-lens models that cover a wide range of potential shears. The effect of the shear is to introduce two small bumps into the light curve as the small binary caustic crosses the limb of the source, and this is indeed what we see in the binary-lens models we calculate.

Because the separation between the lens and a companion is large

(B = ∆θ/θ 1), the shear can be approximated as γ Q/B2, where E ≫ ≃ 30 Q = Mb/Ml is the mass ratio of the companion and the lens. This reduces the number of parameters that need to be considered from three to two: γ and α, the angular position of the blend with respect to the motion of the source. We use a grid search of γ and α to place limits on the shear. For each combination of γ and

α, we generate a binary light curve in the limit B 1 that satisfies Q = γB2 and fit ≫ it to the data using a Markov Chain Monte Carlo with 1000 links. We bin the data over the peak to reduce computing time. We compute the difference in χ2 between the binary model and the best-fit finite-source point-lens model. Figure 2.5 shows the results of the grid search over-plotted with the upper and lower limits on the shear assuming the blend is a companion to the lens. From this figure, we infer that a shear of 6.2 10−5 is inconsistent with our data since it is in a region where the fit × is worse by ∆χ2 > 36.

The two minima in the χ2 map at γ 10−4, α = π/2, π are well-defined but ∼ appear to be due to a single, deviant data point. Fits to the data with these binary models show improvement in the fit to this data point, but the residuals from these

fits for the other data points are large and show increased structure. Thus, we believe these minima to be spurious and conclude that the maximum shear that is consistent with our data (∆χ2 9) is γ = 1.6 10−5. ≤ max ×

Since we have ruled out the scenario where the blend is a companion to the lens, we need to ask what possible explanations for the blend are consistent both with γmax and with the observed color and magnitude. Given γmax, we can place

31 constraints on the distance to the blend, Db, for a given mass. The distance is given by

1 Db = , (2.8) πb γ(∆θ)2 where πb = πs + πrel,b = πs + (2.9) κMb

If we assume Mb = 1M⊙, γ = γmax, and use previously stated values for the other parameters, we find Db > 5.8 kpc. A metal-poor sub-giant with this mass located at or beyond this distance would be consistent with the observed color and magnitude of the blend given the simple extinction model described above. However, other explanations are also possible. For example, if the mass of the blend were decreased,

πb would increase, and a slightly closer distance would be permitted. Thus, we cannot definitively identify the source of the blended light. However, given that γmax is very small, we can ignore any potential shear contribution in later analysis.

2.5. Limits on Planets

We use the method described by Rhie et al. (2000) to quantify the sensitivity of this event to planets. This approach is used for events such as this one for which the residuals are consistent with a point-lens. Rather than fitting binary models for planetary companions to our data as advocated by Gaudi & Sackett

(2000), we generate a binary model from the data and fit it with a point-lens

32 model. When the single-lens parameters are well constrained (as is the case with

OGLE-2008-BLG-279), these two approaches are essentially equivalent (see the discussion in Gaudi et al. 2002 and Dong et al. 2006). We create a magnification map assuming an impact parameter, d, and star/planet mass ratio, q, using a lens with the characteristics from our finite-source fit. The method for creating the magnification map is described in detail in Dong et al. (2006) and Dong et al.

(2009b). For each epoch of our data, we generate a magnification due to the binary lens assuming some position angle, α, of the source’s trajectory relative to the axis of the binary and assign it the uncertainty of the datum at that epoch. As in 2.4.2, § we use binned data for this analysis.

For q = 10−3, 10−4, 10−5, and 10−6 we search a grid of d, α and compute the

2 2 ∆χ . Based on the systematics in our data, we choose a threshold ∆χmin = 160

2 2 (Gaudi & Sackett 2000). For ∆χ > ∆χmin, the fit is excluded by our data, and we are sensitive to a planet of mass ratio q at that location. We repeat the analysis using unbinned data for a small subset of points and confirm that the ∆χ2 for

fits with the unbinned data is comparable to fits with binned data. Figure 2.6 shows the sensitivity maps for four values of q. These maps show good sensitivity to planets with mass ratios q = 10−3, 10−4, and 10−5 and some sensitivity to

−6 planets with q = 10 . For our measured value of Ml = 0.64M⊙, a mass ratio

−3 of q = 10 corresponds to a planet mass mp = 0.67MJup and a mass ratio of q = 10−6 corresponds to m 2M . The results bear a striking resemblance to p ≃ Mars

33 the hypothetical planet sensitivity of the A 3000 event OGLE-2004-BLG-343 max ∼ if it had been observed over the peak (Dong et al. 2006). In particular, this event shows nearly uniform sensitivity to planets at all angles α for large mass ratios. The hexagonal shape of the sensitivity map is the imprint of the difference between the magnification maps of planetary-lens models and their corresponding single-lens models (see upper panel of Fig. 3 in Dong et al. (2009b)).

Figure 2.7 shows a map of the planet detection efficiency for this event. The efficiency is the percentage of trajectories, α, at a given mass ratio and separation

2 2 that have ∆χ > ∆χmin (Gaudi & Sackett 2000). The efficiency contours are all quite close together because of the angular symmetry described above for the planet sensitivity maps. Because we measure the distance to the lens, we know the projected separation, r⊥, in physical units:

r⊥ = dθEDl. (2.10)

Since we know Ml, we also know the planet mass, mp = qMl. We can rule out

Neptune-mass planets with projected separations of 1.5–7.2 AU (d = 0.5–2.2) and Jupiter-mass planets with separations of 0.54–19.5 AU (d = 0.2–6.0). We are also able to detect Earth-mass planets near the Einstein ring, although the efficiency is low. The region where this event is sensitive to giant planets probes well beyond the snow line of this star, which we estimate to be at 1.1 AU assuming

2 asnow = 2.7AU(M⋆/M⊙) (Ida & Lin 2004). The observed absence of planets, especially Neptunes, immediately beyond the snow line of this star is interesting

34 given that core-accretion theory predicts that Neptune-mass planets should preferentially form around low-mass stars (Laughlin et al. 2004; Ida & Lin 2005).

It is also interesting to consider how the sensitivity of this event to planets compares to the sensitivity of other planet-search techniques. Obviously, because of the long timescales involved, most transit searches barely probe the region of sensitivity for this event. As a space-based mission, the Kepler satellite has the best opportunity to probe some of the microlensing parameter space using transits.

Using equation 21 from Gaudi & Winn (2007), we can estimate Kepler’s sensitivity to transits around this star:

S/N 3/2 a 3/4 m = 0.22 100.3(mV −12)M , (2.11) p 10 1 AU Earth µ ¶ ³ ´ where (S/N) is the signal-to-noise ratio, a is the semi-major axis of the planet, and mV is the apparent magnitude of the star. We have assumed that the density of the planet is the same as the density of the Earth and the stellar mass-radius relation

0.8 R⋆ = kM⋆ (Cox 2000). Kepler is also limited by its mission lifetime of 3.5 yrs. For periods longer than this, it becomes increasingly unlikely that Kepler will observe a transit (Yee & Gaudi 2008). This limits the sensitivity to planets within 2 AU ∼ where the period is less than the mission lifetime. These boundaries are plotted in

Figure 2.7.

For comparison, we can also estimate the sensitivity of the radial velocity technique to planets around a star of this mass assuming circular orbits and an

35 edge-on system. Radial velocity is sensitive to planets of mass

σ S/N N −1/2 a 1/2 m = 8.9 RV M , (2.12) p 1m/s 10 100 1 AU Earth µ ¶ µ ¶ µ ¶ ³ ´ where σRV is the precision, and N is the number of observations. The limit of radial velocity sensitivity is plotted in Figure 2.7 as a function of separation assuming a precision of 1 m/s. Additionally, we can consider how this microlensing event compares to the sensitivity of a space-based astrometry mission with microarcsecond precision (σa = 3 µas):

σ S/N N −1/2 a −1 d m = 6.4 a M . (2.13) p 3 µas 10 100 1 AU 10 pc Earth µ ¶ µ ¶ µ ¶ ³ ´ µ ¶

We assume circular face-on orbits. We show the limiting mass as a function of semi-major axis in Figure 2.7 for 3 µas precision. While these contours encompass a large region of the parameter space, they do not take into account the time it takes to make the observations, which increases with increasing semi-major axis.

Furthermore, we only expect this kind of astrometric precision from a future space mission, whereas this event shows that microlensing is currently capable of finding these planets from the ground. This discussion shows that microlensing is sensitive to planets in regions not probed by transits and radial velocity and will be particularly important for finding planets at wide separations where the periods are long. For example, for semi-major axis a = 4 AU (near the maximum sensitivity shown in Fig.

2.7), the period is P 10 yr. ≃ 36 2.6. Summary

The extreme magnification microlensing event OGLE-2008-BLG-279 allowed us to place broad constraints on planets around the lens star. Even with a more conservative detection threshold (∆χ2 > 160), this event is more sensitive than any previously analyzed event (the prior record holder was MOA-2003-BLG-32; Abe et al. 2004). Furthermore, because we observe both parallax and finite-source effects in this event, we are able to measure the mass and distance of an isolated star

(M = 0.64 0.10M ,D = 4.0 0.6kpc). Using these properties of the lens star, we l ± ⊙ l ± convert the mass ratio and projected separation to physical units. We can exclude giant planets around the lens star in the entire region where they are expected to form, out beyond the snow line. For example, Jupiter-mass planets are excluded from 0.54–19.5 AU. Events like this that can detect or exclude a broad range of planetary systems out beyond the snow line will be important for determining the planet frequency at large separations and constraining models of planet formation and migration.

37 Effects Fit Parameters Finite- Orbital Terrestrial t 4617.34 u 104 t ρ 104 π π 0 − 0 × E ⋆ × E,E E,N Source Parallax Parallax ∆χ2 (days) (θ ) (days) (θ ) − E E

Y 0.00 0.00783(7) 6.4(5) 111.(9) 6.6(6) ······ Y Y Y 164.50 0.00787(8) 6.6(5) 106.(9) 6.8(6) -0.15(2) 0.02(2) Y Y Y 127.97 0.0081(1) -6.4(6) 109.(9) 6.7(6) 0.11(2) 0.09(2) Y Y 15.52 0.00784(8) 8.(1) 84.(12) 9.(1) 1.5(4) -0.3(2) Y Y 15.51 0.00786(8) -8.(1) 84.(12) 9.(1) 1.5(4) -0.3(2) 38 Y Y 166.40 0.00787(8) 6.9(6) 101.(8) 7.2(6) -0.16(2) 0.03(2) Y Y 129.59 0.0081(1) -6.9(5) 102.(8) 7.1(6) 0.11(2) 0.11(3)

Note. — The first 3 columns indicate which effects were included in the point-lens fit. The ∆χ2 improvement for each fit (col. 4) is given relative to the best-fit including finite-source effects but without parallax. There are 5731 data points in the fit light curve. The numbers in parentheses indicate the uncertainty in the final digit or digits of the fit parameters.

Table 2.1. Light Curve Fits OGLE−2008−279 Peak 12.5 OGLE 13 MOA Bronberg HHO I 13.0 14 HHO U UTas

Wise OGLE I 13.5 15 Farm Cove OGLE I CTIO I CTIO V 16 14.0

0.04

17 0.00

−0.04 4610 4620 4630 Residuals 17.2 17.3 17.4 17.5 HJD−2,450,000 HJD−2,454,600

Fig. 2.1.— Light curve of OGLE-2008-BLG-279 near its peak. The left panel shows the entire event, while the right panel shows a close-up of the peak with residuals from the point-lens model including finite-source effects. The black solid line shows this best-fit model. For clarity, the data have been binned and rescaled to the OGLE flux.

39 Fig. 2.2.— Calibrated Color-Magnitude Diagram (CMD) constructed from the CTIO and OGLE data. The square indicates the centroid of the red clump, the open circle shows the blended light, and the solid circle indicates the source. The small black points are field stars. The error bars are shown but are smaller than the size of the points.

40 Fig. 2.3.— χ2 contours as a function of impact parameter, u , and z u /ρ where 0 0 ≡ 0 ⋆ ρ⋆ = θ⋆/θE is the normalized source size. The best fit is marked with a plus sign.

41 Fig. 2.4.— Possible absolute magnitudes and colors for the blend plotted with Yale-Yonsei isochrones (Demarque et al. 2004). The isochrones plotted are the Y2 isochrones for solar (thick) and sub-solar metallicities (thin) for populations 1 (dotted), 5 (dot-dashed), and 10 Gyr old (solid). The dashed line shows the color and magnitude of the blend for a continuous distribution of distances assuming a dust model that decreases exponentially with scale height. The square shows the absolute magnitude and color of the blend assuming it has the same distance (8 kpc) and reddening as the clump. The plus sign, diamond, and triangle show the absolute magnitude and color using our simple dust model and distances of 2, 4, and 6 kpc, respectively. If the blend is a companion to the lens, it would be at a distance of 4 kpc (diamond).

42 Fig. 2.5.— Shear as a function of α (angular position with respect to the motion of the source). Open symbols indicate an improved χ2 compared to the finite-source point-lens fit. Filled symbols indicate a worse fit. The magnitude of ∆χ2 is indicated by the color legend shown. The solid line indicates our calculated value for the shear assuming the blend is at the same distance as the lens. The shaded area shows the 1 σ limits on this value from the uncertainty in the centroid of the PSF (see text).

43 −3 −4 q=10 , mp=0.7MJup q=10 , mp=1.2MNep

5 2 α α 0 0 d sin d sin

−5 −2

−5 0 5 −2 0 2 d cos α d cos α −5 −6 q=10 , mp=2.1MEarth q=10 , mp=2.0MMars 1.0 1 0.5 ∆χ2

α α 60−100 0 0.0 100−160 160−250

d sin d sin 250−400 −0.5 > 400 −1 −1.0 −1 0 1 −1.0 −0.5 0.0 0.5 1.0 d cos α d cos α

Fig. 2.6.— Planet sensitivity as a function of distance from the lens in units of Einstein radii. The white/black circle indicates the Einstein ring (d = 1). The mass ratios and corresponding planet masses are indicated on each plot. The colors indicate the ∆χ2 that would be caused by a planet at that location.

44 Fig. 2.7.— Detection efficiency map in the (d,q) plane, i.e. projected separation in units of θE and planet-star mass ratio. The contours show detection efficiencies of 0.99, 0.90, 0.75, 0.50, 0.25, and 0.10 from inside to outside. The inner spike is due to resonant caustic effects at the Einstein ring. The upper and right axes translate (d,q) into physical units (r⊥,mp), i.e. physical projected separation and planet mass. The vertical solid line shows the position of the snow line for this star. The dotted line shows Kepler’s sensitivity to planets around the lens star assuming mV = 12. The cutoff in separation (d 0.6) occurs where a planet’s orbital period is equal to Kepler’s mission lifetime of≃ 3.5 yrs. The dashed line shows the sensitivity limit for radial velocity observations with 1 m/s precision. The dot-dashed line shows the sensitivity limit for a space-based astrometry mission with precision of 3 µas assuming the star is at 10 pc.

45 Chapter 3

MOA-2011-BLG-293Lb: A test of pure survey microlensing planet detections

3.1. Introduction

Large-format, wide-field cameras have placed microlensing on the cusp of joining RV and transits as a technique able to find dozens of planets at a time

(Shvartzvald & Maoz 2011), moving the field from the discovery of individual objects to the study of planet populations. Using these new cameras, “second generation” microlensing surveys will be able to effectively monitor an order of magnitude more events for anomalies due to planets. At the same time, they can maintain an observing strategy that makes no reference to whether or not a planetary signal is suspected, thus enabling a statistically robust sample of events whose detection efficiencies are well understood. One requirement for such a sample is that all events must be analyzed for planets, including signals at the limits of detectability. At present the detection threshold is poorly understood since the current practice is only to analyze the most obvious signals. However, it is known that microlensing data have systematics and correlated noise that make it difficult

46 to use standard statistical measures to set the detection thresholds. In this chapter, we analyze the microlensing event MOA-2011-BLG-293, which is covered by all three second-generation survey telescopes and also has substantial followup data, and we suggest a means to study the boundary of what is detectable in the second generation surveys.

Originally, the purpose of microlensing surveys was simply to identify ongoing microlensing events, which requires monitoring several million stars with a cadence of about once per night. Because a typical planetary signal lasts for only a few hours, it was nearly impossible to detect planets from the early survey data.

Thus, in order to detect planets, higher cadence followup data were needed1.

One followup strategy is to monitor one or more targets with increased cadence to provide additional coverage of the light curve and to search for anomalies. A second strategy is continuous or near-continuous monitoring of a single event of interest, usually because it is suspected to be high magnification (Amax > 100) or anomalous. These additional observations can be taken either by dedicated followup groups or the surveys themselves can go into followup mode (typically, continuous or near-continuous observations) if they deem an event to be of sufficient interest.

Therefore, in followup mode, both survey and followup groups may modify their target list and/or observing cadence in response to suspected planetary signals. This strategy has been effective at finding planets but makes understanding the detection

1Implicit in this is that the data quality is good enough for planet detection.

47 efficiencies complex, although this has been done successfully for high magnification events in Gould et al. (2010a). Additionally, Sumi et al. (2010) were able to derive a slope (but not the normalization) for the mass ratio function of planets from the planetary events known at the time. Of the 13 microlensing planets published to date2, only one was published from data taken in a pure survey mode (Bennett et al.

2008)3.

The new high-cadence, systematic surveys will have sufficient cadence and data quality to detect and characterize planets with masses as small as the Earth without additional followup data (Gaudi 2008). Such pure-survey detections require near-24-hour monitoring with a cadence of several observations per hour. Many of these future discoveries will be part of a rigorous experiment wherein the detection efficiencies are well understood because they will be found in blind or blinded (in which followup data are removed) searches. High cadence surveys, even without global coverage, also allow additional science such as the detection of very short timescale events (Sumi et al. 2011). Recent upgrades by the Optical Gravitational

2For completeness, we note that there are at least two other planets claimed in the literature, which are not considered secure detections. A possible circumbinary planet proposed to explain anomalies in MACHO 97-BLG-41 (Bennett et al. 1999) can also be explained by orbital motion of the binary lens (Albrow et al. 2000). There is also evidence for a planetary companion to the lens in

MACHO 98-BLG-35, but only with ∆χ2 = 20 (Gaudi et al. 2002 contains a discussion of why this is inadequate for detection). 3At least one other pure survey detection should be published soon (see Bennett et al. in prep).

48 Lensing Experiment (OGLE; Chile) and Microlensing Observations in Astrophysics

(MOA; New Zealand) collaborations augmented by the Wise Observatory (Israel) survey now allow near-continuous monitoring (observations every 15-30 minutes) of several fields in the Galactic Bulge, 22 hours/day (Shvartzvald & Maoz 2011).

The power of these new surveys comes from the combination of high-cadence, systematic observations, which were previously only achievable through followup for a small subset of events, and the ability to monitor millions of stars. At the same time, followup observations maintain some advantages over current surveys.

Because the followup networks have access to additional telescopes at various sites, followup observations often have redundancy. This makes them less vulnerable to bad weather, which can create gaps in the data. Additionally, multiple data sets at a given epoch provide a check on systematics or other astrophysical phenomena that may create false microlensing-like signals (see Gould et al. in prep). Simultaneous or near simultaneous observations from multiple sites are also required to measure terrestrial microlens parallax (e.g. Gould et al. 2009). Furthermore, since followup observations are targeted, they can achieve a much higher cadence, and are frequently continuous, although the current strategy for surveys is typically to switch to near continuous followup observations for events of interest. Finally, while followup groups routinely make an intensive effort to get observations in additional

49 filters4, survey groups are less aggressive about obtaining such observations. All of these additional bits of information can increase confidence in the microlensing interpretation and reduce ambiguity in the models. The trade off is that with somewhat sparser data coverage, surveys are able to systematically monitor more than an order of magnitude more events.

The additional planets detected by surveys, which are not currently being detected with followup, will fall into two categories related to the two kinds of caustics produced by a 2-body lens. First, there will be many more planetary caustic anomalies detected. These caustics are created along or near the planet-star axis at a distance from the lens star that depends on the projected separation, s, and the mass ratio, q, between the two bodies. Anomalies created by these caustics can be found with current followup but since the source trajectory is random with respect to the binary axis, these anomalies occur in a random place in the light curve. Hence, surveys will detect more of them because they can observe more stars.

Second, a planetary companion to the lens induces a caustic at the position of the lens star, the so-called “central” caustic. Source crossings of such caustics can be predicted in advance because they require that the source trajectory pass very close to the position of the lens star. Surveys will observe more central caustic events

4Microlensing observations are normally done in I-band (or similar filters) because that is the optical band that is most sensitive toward the Galactic Bulge. In order to derive source colors as in

Sec. 3.3, we need observations in a different filter, for which we typically use V -band.

50 that are too faint to observe with current followup or are not recognized to be high magnification quickly enough to organize followup observations. For both types of events, there is the question of whether the survey data alone are indeed sufficient to detect planets in individual microlensing light curves in spite of having sparser data. For central caustic events there is an additional question of whether or not the anomaly will be sufficiently characterized, since the models can be quite degenerate for these kinds of events, sometimes with little constraint on the mass ratio between the lens star and its companion (e.g. Choi et al. 2012). Given this much larger sample of events which will contain signals of all significance levels including ones that can be confused with systematics, the challenge is to create a subset of events for which the vast majority of planetary signals can be considered reliable, and secondarily for which the planets are well-characterized, in the case of central caustic-type events.

Gould et al. (2010a) estimate a threshold of ∆χ2 =350–700 would be appropriate, but the true value is unknown. In principle, such questions could be addressed with simulations. However, in simulations it is difficult to account for real effects such as data systematics and stellar variability. Hence, using actual microlens data provides

field testing that complements results from simulations.

MOA-2011-BLG-293 provides an opportunity for investigating survey-only detection thresholds. The planet is robustly detected in the survey+followup data

(∆χ2 5400), and the event was observed by all three current survey telescopes. ∼ Wise Observatory obtained data of the anomaly in their normal survey mode without

51 changing their observing cadence, and the rest of the light curve is reasonably well covered by OGLE and MOA survey data. For this event, we are able to determine whether the survey data alone can successfully “predict” the solution determined when all of the data are included.

This event also has the faintest source of any published planetary event. We show that for such faint sources small systematic errors in the flux measurements can radically affect the microlensing solution, even when all the anomalous features occur at high magnification when the source is bright. In particular, the source flux and the event timescale are determined primarily from data near baseline where small systematic errors may be of order the change in flux being measured. Because the systematic errors in the timescale propagate to many other quantities including the planet/star mass ratio, they must be investigated carefully. This is particularly important for future surveys where many of the events will be at or beyond the magnitude limit at baseline.

We begin by presenting the discovery and observations of MOA-2011-BLG-293 in Section 3.2. The color-magnitude diagram of the event is presented in Section

3.3 and used to derive the intrinsic source flux. In Section 3.4, we address the consequences of systematics in the measured flux when they are similar in magnitude to the source flux. Then, in Section 3.5 we present the analysis of the light curve of the event, and we compare the results with and without followup data in Section

3.5.2. Additional properties of the event are derived in Section 3.6, and the physical

52 properties of the lens star and planet are derived from a Galactic model in Section

3.7. We discuss the implications for future survey-only detections in Section 3.9.

Finally, the possibility of detecting the lens with adaptive optics (AO) observations is discussed in the Appendix.

3.2. Data Collection and Reduction

MOA issued an electronic alert for MOA-2011-BLG-293 [(RA,Dec) =

(17:55:39.35, 28:28:36.65), (l,b)=(1.52, 1.66)] at UT 10:27, 4 Jul 2011 − − (HJD′ = HJD-2450000 = 5746.94), based on survey observations from their 1.8m telescope with a broad R/I filter and 2.2 deg2 imager at Mt. John, New Zealand. At

UT 12:45, the Microlensing Follow-Up Network (µFUN) refit the data and announced that this was a possible high-magnification event, where “high-magnification” is

Amax > 100. At UT 17:28, µFUN upgraded to a full high-magnification alert ∼

(Amax > 270), emailing subscribers to their email alert service, which includes members of µFUN and other microlensing groups, to urge observations from Africa,

South America, and Israel. Additionally, a shortened version of the alert was posted to Twitter. This prompted µFUN Weizmann to initiate the first followup observations at UT 19:45, using their 0.4m telescope (I band) at the Martin S. Kraar

Observatory located on top of the accelerator tower at the Weizmann Institute of

Science Campus in Rehovot, Israel. At UT 23:25, µFUN Chile initiated continuous

53 observations using the SMARTS 1.3m telescope at CTIO. At UT 00:00 µFUN issued an anomaly alert based on the first four photometry points from CTIO, which were rapidly declining when the expected behavior was rapid brightening. The great majority of the CTIO observations were in I band, but seven observations were taken in V band to measure the source color. In addition, the SMARTS ANDICAM camera takes H band images simultaneously with each V and I image. These are not used in the light curve analysis but are important in the Appendix.

MOA-2011-BLG-293 lies within the survey footprint of the MOA, OGLE, and

Wise microlensing surveys and so was scheduled for “automatic observations” at high cadence at all three observatories. MOA observed this event at least 5 times per hour. Wise observed this field 10 times during the 4.6 hours that it was visible from their 1.0m telescope, equipped with 1 deg2 imager and I-band filter, at Mitzpe

Ramon, Israel. The event lies in OGLE field 504, one of three very high cadence

fields, which OGLE would normally observe about 3 times per hour. In fact, it was observed at a much higher rate, but with the same exposure time, in response to the high-magnification alert and anomaly alert. Unfortunately, high winds prevented opening of the telescope until UT 01:02. OGLE employs the 1.3m Warsaw telescope at Las Campanas Observatory in Chile, equipped with a 1.4 deg2 imager primarily using an I-band filter.

The data are shown in Figure 3.1. Several features should be noted. First, the pronounced part of the anomaly lasts just 4 hours beginning at HJD′ = 5747.40.

54 The main feature is quite striking, becoming about one magnitude brighter in about one hour. The coverage during the anomaly is temporally disjoint between the observatories in Israel and those in Chile, a point to which we return below.

Finally, the CTIO data show a discontinuous change of slope (“break”), which is the hallmark of a caustic exit, when the source passes from being partially or fully inside a caustic to being fully outside the caustic (see Fig. 3.4).

MOA and OGLE data were reduced using their standard pipelines (Bond et al.

2001; Udalski 2003) which are based on difference image analysis (DIA). In the case of the OGLE data, the source is undetected in the template image. Since the

OGLE pipeline reports photometry in magnitudes, an artificial blend star with a

flux of 800 units (IOGLE = 20.44) was added to the position of the event to prevent measurements of negative flux (and undefined magnitudes) at baseline when the source is unmagnified.

Data from the remaining three observatories were also reduced using DIA

(Wozniak 2000), with each reduction specifically adapted to that imager. Using comparison stars, the Wise and Weizmann photometry were aligned to the same

flux scale as the CTIO I band by inverting the technique of Gould et al. (2010b).

That is, the instrumental source color was determined from CTIO observations, and then the instrumental flux ratios (CTIO vs. Wise, or CTIO vs. Weizmann) were

55 measured for field stars of similar color. The uncertainties in these flux alignments are 0.016 mag for Wise and 0.061 mag for Weizmann.

3.2.1. Data Binning and Error Normalization

Since photometry packages typically underestimate the true errors, which have a contribution from systematics, we renormalize the error bars on the data, as is done for most microlensing events. After finding an initial model, we calculate the cumulative χ2 distribution for each set of data sorted by magnification. We renormalize the error bars using the formula

′ 2 2 σi = k σi + emin (3.1) q

2 2 and choosing values of k and emin such that the χ per degree of freedom χred = 1 and the cumulative sum of χ2 is approximately linear as a function of source magnification. Specifically, we sort the data points by magnification, calculate the

2 N 2 ∆χ contributed by each point, and plot i ∆χi as a function of N to create the cumulative sum of χ2, where N is the numberP of points with magnification less than or equal to the magnification of point N. Note that σi is the uncertainty in magnitudes (rather than flux). The values of k and emin for each data set are given in

Table 3.1. Except for OGLE, the values of emin are all zero. This term compensates for unrealistically small uncertainties in the measured magnitude, which can happen when the event is bright and the Poisson flux errors are small.

56 For the MOA data, we eliminate all observations with t outside the interval

5743.5 < t(HJD′) < 5749.5 (see Section 3.4.2). We also exclude all MOA points with seeing > 5′′ because these data show a strong nonlinear trend with seeing at baseline. After making these cuts, we renormalize the data as described above.

To speed computation, the OGLE and MOA data in the wings of the event were binned. In the process of the binning, 3σ outliers were removed. This binning does not account for correlations in the data, which if they exist can increase the

2 2 reduced χ above the nominal value of χred = 1.

3.3. CMD

We use the CTIO I and V band data to construct a color-magnitude diagram (CMD) of the event. We measure the instrumental (uncalibrated) source color by linear regression of the V and I fluxes (which is independent of the model) and the magnitude from the fS,CTIO of our best-fit model:

(V I,I) = (0.37, 22.27) (0.03, 0.05). The position of the source relative to the − S ± field stars within 60′′ of the source (small dots) is shown in Figure 3.2 as the solid black dot. We calibrate these magnitudes and account for the reddening toward the field by assuming the source is in (or at least suffers the same extinction as) the Bulge and using the centroid of the red clump as a standard candle. Because of strong differential extinction across the field, we use only stars within 60′′ of the source to measure the centroid of the red clump. Since the event is in a low

57 latitude field, there are more stars than is typical for bulge fields and the red-clump centroid can be reliably determined even with this restriction. In instrumental magnitudes, the centroid of the red clump is (V I,I) = (0.59, 16.90) compared − cl to its intrinsic value of (V I,I) = (1.06, 14.32) (Bensby et al. 2011; Nataf et al. − cl,0

2012), which assumes a Galactocentric distance R0 = 8kpc and that the mean clump distance toward l = 1.5 lies 0.1 mag closer than R0 (Rattenbury 2007). We can apply the offset between these two values to the source color and magnitude to obtain the calibrated, dereddened values (V I,I) = (0.84, 19.69) (0.05, 0.16). − S,0 ± The uncertainty in the color is derived from Bensby et al. (2011) by comparing the spectroscopic colors to the microlens colors of that sample. The uncertainty in the calibrated magnitude is the sum in quadrature of the uncertainty in fS,CTIO from the models (0.05 mag), the uncertainty in R (5% 0.1 mag), the uncertainty in the 0 → intrinsic clump magnitude (0.05 mag), and the uncertainty in centroiding the red clump (0.1 mag).

3.4. Effect of Faint Sources on Microlens Parameters

The source star in MOA-2011-BLG-293 is extremely faint with an apparent magnitude in the OGLE photometry of IS,OGLE = 21.7. Consequently, the measured

flux errors can be comparable to or larger than the source flux, particularly near baseline. Because of this, systematics in the baseline data must be carefully

58 accounted for so as not to bias the microlens results. Systematics in the measured

flux at the level of fS can lead to biases in the measured Einstein timescale, tE, of the same order. We begin by discussing robust parameters, which can be measured solely from the highly magnified portion of the light curve and so, are independent of uncertainties in the flux measured near baseline. Then in Section 3.4.2, we discuss in detail the effect of systematics in the measured baseline flux on the microlens parameters, particularly tE and the mass ratio between the components of the lens, q.

3.4.1. Robustly Measured Parameters

At high magnification, a microlensing light curve for a point source being lensed by a point lens can be described by the unmagnified, baseline flux of the event, fbase, and three parameters (“invariants”): the time of the peak, t0, the difference between fbase and the peak flux, flim, and the effective width of the light curve, teff

(Gould 1996, see Eq. (2.4) and (2.5) in). These parameters are nearly invariant under changes to the source flux and are robustly determined by the light curve.

The change in the observed flux due to the event can then be written as a function of these invariants:

f (t) f = G (t; t ,t , f ), (3.2) obs − base 3 0 eff lim 59 where fobs is the observed flux and the subscript on G refers to the number of parameters. Note that fbase is also an observable.

In the limit where the event is highly magnified, the exact form of G3(t) can be derived from the microlens variables. The three microlens variables of a point-source–point-lens microlensing model are t0, the impact parameter in units of the Einstein radius, u0, and the Einstein crossing time, tE. The observed flux is given by

f = f A(t)+ f = f [A(t) 1] + f , (3.3) obs S B S − base

where A(t) is the magnification of the source, fS is the flux of the source, and fB is the flux of all other stars blended into the PSF (including the flux from the lens).

By definition, fbase = fS + fB. For a point lens in the limit of high magnification

(A(t) 1), the magnification is given by ≫

1 A(t) (3.4) ≃ u0Q(t) where

t t 2 Q(t; t ,t )= 1+ − 0 , (3.5) 0 eff t s µ eff ¶ is a function of only time and the invariant:

t u t . (3.6) eff ≡ 0 E 60 In this limit, the evolution of the observed flux, G3(t; t0,teff , flim), is then given by

flim fS G3(t; t0,teff , flim)= , where flim . (3.7) Q(t) ≡ u0

If finite source effects are detected, the change in the observed flux is a more complicated function because of the additional microlens parameter ρ, which is the source size in units of the Einstein radius. However, there is also an additional invariant

t ρt , (3.8) ⋆ ≡ E the source crossing time, which determines the width of the peak of the light curve for a point lens. Hence, the change in the observed flux can be written as

f (t) f = G (t; t ,t , f ,t )= G (t)B (Qt /t ) , (3.9) obs − base 4 0 eff lim ⋆ 3 eff ⋆

where B(Qteff /t⋆)= B(u/ρ) is a function composed of elliptic integrals, whose exact form is derived in Gould (1994) and Yoo et al. (2004).

In the case of a two-body lens like MOA-2011-BLG-293, the invariants may not be obvious from the light curve, but they are still robustly measured as we show below. For example, the width of the peak is distorted by two-body perturbation, but based on the source trajectory (Fig. 3.4), it can be seen that the width of the

first bump at HJD′5747.46, which is caused by the cusp crossing, will be slightly larger than 2t⋆. For a two-body lens with a central caustic crossing, there is also

61 another invariant due to the mass ratio, q, between the two lensing bodies, so

f (t) f = G (t; t ,t , f ,t ,qt ). (3.10) obs − base 5 0 eff lim ⋆ E

5 This new invariant qtE can be understood as follows. For central caustics , like the one in MOA-2011-BLG-293, the caustic size is proportional to the mass ratio of the two lensing bodies, q, and the caustic shape is roughly constant for a given s. Therefore, the time between successive features in the light curve is set by qtE, i.e., the size of the caustic multiplied by the characteristic timescale, and since the observed times of the features can be well measured, the uncertainty in this quantity is extremely small. In this case, the main features are the two bumps in the light curve and the discontinuity in the slope that occurs between the bumps. [Note that a two-body lens introduces two parameters in addition to q: the separation between the two bodies projected onto the plane of the sky, s, and the angle of the source trajectory with respect to the binary axis, α. Because these parameters are of less interest, we do not discuss them in this context.]

Table 3.2 shows that for this event these quantities, teff , flim, t⋆, and qtE, do indeed have extremely small uncertainties and can approximately be considered invariants.

5Two-body lenses with unequal mass ratios will create one caustic at the position of the more massive body, the “central” caustic, and another set of caustics elsewhere, the “planetary” caustics.

62 3.4.2. Parameters Vulnerable to Systematics

The above invariants are determined by the data taken near the peak of the light curve where A 1. However, in order to extract the values of the ≫ microlens parameters, u0, ρ, and q, from the invariants, we must measure tE. The information on tE must necessarily come from the wings of the light curve where the magnification is small and A(t) = 1/u(t) (Dominik 2009). Since the magnification 6 is small, the change in the observed flux compared to the source flux is also small, so the measurement of tE may be considered to be the statistical sum of many measurements of a small change in flux. In order to get an accurate measurement of tE, the statistical and systematic errors in the flux must be smaller than the change we are trying to measure.

In the case of MOA-2011-BLG-293, because the source flux is extremely faint, it is difficult to measure accurately. When the magnification is a factor of a few or less (in the wings and at baseline), if the flux is not measured with an accuracy substantially smaller than the source flux, this can lead to bias in the measurement of fS or equivalently tE, since fStE = flimteff is robustly determined, and so to bias in quantities dependent on tE such as q. To check for this possible source of bias, we bin the OGLE and MOA data by 30 days to see if their measurements of the baseline

flux are stable (Fig. 3.3). We find that the OGLE measurement of the baseline

flux is stable at a level that is smaller than the observed source flux. Therefore,

63 we use all of the OGLE data in our models. We note that the flux after the event

(t> HJD′5790) appears to be at a lower level than the baseline before the event. In

Section 3.5.1, we discuss the effect of assuming the baseline decreases at a constant rate during the course of the event.

As seen in Figure 3.3, the MOA baseline flux exhibits scatter in excess of the measured photometric errors, and there is also variation in measured baseline flux from season to season. The magnitude of this scatter is similar to the magnitude of the source flux. Because of this variation, we conclude that the baseline flux is not sufficiently well measured in the MOA data to detect the small changes in

flux necessary to measure tE. As a result, to avoid biasing our results, we use only the MOA data from the peak of the light curve where the photometry is precise:

5743.5

3.5. Analysis

Without any modeling, we can make some basic inferences about the relevant microlens parameters from inspection of the light curve. MOA-2011-BLG-293 increases in brightness from I 19.7 to I 15.0, indicating a source magnification ∼ ∼ of at least 75. Additionally, except for the deviations at the peak, the event is symmetric about t0. From these two properties, we infer that only central or resonant caustics (both of which are centered on the position of the primary) are relevant to the search for microlens models.

64 We fit the light curve using a Markov Chain Monte Carlo (MCMC) procedure.

In addition to the parameters described in Section 3.4, a model with a two-body lens has two additional parameters: the angle of the source trajectory with respect to the binary axis defined to be positive in the clockwise direction6, α, and the projected separation between the two components of the lens scaled to the Einstein radius, s. Because they are approximately constants, we use the parameters teff and t⋆ in place of the microlens variables u0 and ρ. For a given model, Equation (3.3) must be evaluated for each observatory, i, so f , f f , f . We adopt the “natural” S B → S,i B,i linear limb-darkening coefficients Γ = 2u/(3 u) (Albrow et al. 1999). Based on the − measured position of the source in the CMD, we estimate that Teff = 5315K and log g = 4.5 cgs. We average the linear limb-darkening coefficients for Teff = 5250K

−1 and Teff = 5500K from Claret (2000) assuming vturb = 2 km s to find ΓV = 0.6368 and ΓI = 0.4602.

The magnifications are calculated on an (s,q) grid, using the “map-making” technique (Dong et al. 2006) in the strong finite-source regime and the “hexadecapole” approximation (Pejcha & Heyrovsk´y2009; Gould 2008) in the intermediate regime.

We began by searching a grid of s and q to obtain a basic solution for the light curve. For central caustic crossing events like this one, there is a well known degeneracy between models with close topologies (s< 1) and wide topologies (s> 1)

6The binary axis has its origin at the center of magnification and is positive in the direction of the planet.

65 (e.g. Griest & Safizadeh 1998). We initially searched a broad grid for close topologies and then used the results to inform our search for wide solutions, since to first order, s s−1. The basic model from this broad grid has s 0.55, q 0.005, → ∼ ∼ and α 220◦, such that the source passes over a cusp at the back end of a central ∼ caustic. This caustic is created by a two-body lens with a mass ratio similar to that of a massive Jovian planet orbiting a star. Figure 3.4 shows this basic geometry with the source trajectory relative to the caustic structure. The bump in the light curve at HJD′ 5747.45 is created when the source passes over the cusp of the caustic. ∼

Because the Wise and Weizmann data only overlap with other data sets where their errors are extremely large, there is some concern that the parameters of the models will be poorly constrained, since within the standard modeling approach the

flux levels of these data can be arbitrarily adjusted up or down relative to the other data. However, from the flux alignment described in Section 3.2, we have an estimate of fS,i for these data relative to fS,CTIO. This alignment gives us an independent means to test the validity of our model. If the model is correct, then the values of fS,Wise and fS,Weizmann should agree with fS,CTIO within the allowed uncertainties.

Alternatively, if we include the flux-alignment constraint in the MCMC fits, the solution should not change significantly.

66 We incorporate the flux-alignment constraint in a way that is parallel to the model constraints from the data, i.e., by introducing a χ2 penalty:

(f f )2 ln 10 f + f χ2 = S,CTIO − S,i ; σ = S,CTIO S,i σ , (3.11) b σ2 flux,i 2.5 2 i i flux,i X µ ¶ where i corresponds to the observatory with the constraint, and σi is the uncertainty in magnitudes of the flux alignment for that observatory. In the absence of any constraints, the flux parameters for each observatory, fS,i and fB,i, are linear and their values for a particular model can be found by inverting a block-diagonal covariance matrix, b. We include the flux constraints by adding half of the second

2 derivatives of χb to the b matrix:

2δik 1 ∆b(fS,i, fS,k)= 2 − , (3.12) σflux,i where k =CTIO and δik is a Kronecker-delta. This couples formerly independent

2 2 blocks. Strictly speaking, the equation for σ given in Equation (3.11) is a × flux,i numerical approximation. Therefore, we iterate the linear fit until the value of σflux,i is converged, which typically occurs in only a few iterations.

We refined the (s,q) grid around our initial close solution, fitting the data both with and without flux-alignment constraints. The mean and 1 σ confidence intervals for the parameters from these two fits are given in Table 3.3. There are only small quantitative differences between the two solutions, and nothing that changes the qualitative behavior of the model. The slight increase in χ2 is expected because of the additional term due to the flux constraints. After finding this close solution,

67 we repeated the grid with s s−1 to identify the wide solution. The parameters → of this solution are also given in Table 3.3 both with and without flux-alignment constraints. The close solution is mildly preferred over the wide solution by ∆χ2 3, ∼ so we quote the values for the flux-constrained close solution:

q = 5.3 0.2 10−3 s = 0.548 0.005, (3.13) ± × ± noting that the two topologies give very similar solutions (except s s−1). →

Additionally, we searched for a parallax signal in the event by adding two additional free parameters to the fit for the close solution: πE,N and πE,E, the North and East components of the parallax vector (e.g., Gould 2004). The parameters of this fit are given in Table 3.3. No parallax signal was detected, and we found no interesting constraints on these parameters. The χ2 improves for fits including parallax by only ∆χ2 = 7 for two additional degrees of freedom. In some cases, even when parallax is not detected, meaningful upper limits can be placed on the parallax, but in this case we have an uninteresting 3σ constraint of 0 π 7.8. ≤ | E| ≤

3.5.1. Effect of Systematics in fbase

As discussed in Section 3.4.2, it is possible that the underlying OGLE baseline

flux is changing during the course of the event. In order to test how that could introduce systematic effects in our results, we create a fake OGLE data set accounting for a constant decrease in baseline flux during the event. Specifically, we

68 assume that the baseline flux decreases at a constant rate between HJD′5710. and

′ HJD 5790. leading to an overall decrease in flux of 0.27fS,OGLE. We then repeat the

MCMC procedure for the close solution including flux-alignment constraints. We

find that the value of tE increases by 15%, and consequently, the values of q, u0, and ρ decrease by the same amount. In principle, this could represent a systematic error in our results. However, at the present time, the evidence for a change in the baseline flux is weak, so we only report these results for the sake of completeness.

3.5.2. Analysis with Survey-Only Data

From this analysis, we have a robustly detected planet (∆χ2 5400 compared ∼ to a point lens7) and a well-defined solution. Now we can ask whether the planet could have been detected from the survey data alone, whether the solution is well-constrained, and most importantly, whether it is the same solution. To begin, we construct a “survey only” subset of the data. We first eliminate the Weizmann and CTIO data. Second, we “thin out” the OGLE data to mimic OGLE survey data as they would have been if there had been no high-magnification or anomaly alerts.

OGLE data on several nights previous to (and following) the peak have a cadence of

7Note that the numbers quoted for the point lens models include constraints from the flux alignment in the fit. Removing the flux-alignment constraints improves the χ2, primarily because the Weizmann data can be scaled arbitrarily. However, compared to the planet fit, the point lens fit without flux-alignment constraints is still extremely poor, ∆χ2 4400. Flux-alignment constraints ∼ have very little effect on the point lens fit to survey only data.

69 1 observation per 0.015 days. We therefore adopt a subset of 18 (out of 44) OGLE points from the peak night with this sampling rate.

We repeat the analysis on this survey-only data set beginning with a broad grid search and then refining the solution following the same procedure used for analyzing the complete data set. We find that even without flux-alignment constraints, the global search isolates solutions in the general neighborhood of the solution found from the full data set. The fits to the survey-only data set are compared to fits with all data in Figure 3.5. Here, the ∆χ2 of the fit compared to a point lens fit for the survey-only data is 487, nearly all of which comes from data in the time-span shown in Figure 3.5 5747.1 < t(HJD′) < 5748.8. This is smaller than the ∆χ2 of any published microlensing planet. However, the parameters of the fit are well constrained with errors only a factor of 1.5-2 larger compared to fits with the full data set. Applying the flux-alignment constraint to this model confirms its validity, i.e., it does not appreciably change the solution (see Table 3.3). In this case, it is clear that the survey data are sufficient to robustly detect and characterize the planet.

In order to push farther into the limits of detectability, we also analyze this event without the Wise data, since those data contain most of the deviation from the underlying point lens. With only the MOA and thinned OGLE data, we find

∆χ2 70 between the 2-body lens and point lens models. We note that the ∼ point lens model has an unreasonably large value of parallax, π 20, making it E ∼ 70 somewhat suspicious. Without parallax, the ∆χ2 between the point lens and 2-body

fit increases to ∆χ2 170. Although this is a factor of 3 smaller than the ∆χ2 with ∼ the Wise data, the constraints on the planet-star mass ratio are still broadly confined to be planetary, assuming ML < 0.5M⊙, with a 3σ range of 0.001 < q < 0.025. ∼ ∼ ∼ However, because of the small ∆χ2, it appears unlikely to us that a planetary detection would have been claimed from solely the MOA and OGLE data even though the solution is formally well-constrained.

3.6. Physical Properties of the Event

Since finite source effects are measured in this event, we can determine the angular size of the Einstein ring, θE, and the lens-source relative proper motion, µ.

First, we estimate the angular size of the source, θ⋆, from the observed color and magnitude. We transform the (V I) color to (V K) using the dwarf relation − S,0 − from Bessell & Brett (1988). Then we use the (V K) surface brightness relations − from Kervella et al. (2004) to find θ = 0.42 0.03 µas. From this we derive the ⋆ ± lens-source relative proper motion and angular Einstein radius,

θ⋆ −1 µ = = 4.3 0.3 mas yr ; θE = µtE = 0.26 0.02 mas. (3.14) t⋆ ± ±

The uncertainties in these quantities come from a variety of factors. Specifically, the uncertainties in the Galactocentric distance, R0, and the measured intrinsic brightness of the red clump, the centroiding of the red clump from the CMD, and uncertainty in the surface brightness relations. The uncertainty contributed by the

71 surface brightness relations is 0.02 mag, and the uncertainties from the other factors are given in Section 3.3. The contribution of these factors can be understood from their relationship to θ⋆ (Yee et al. 2009):

√f θ = S , (3.15) ⋆ Z where fS is the source flux from the microlensing model and Z captures all other factors. Taking account of all factors mentioned above, we find σ(Z)/Z = 8%.

1/2 Since the statistical error in fS is only 2.3%, the error in Z completely dominates the uncertainty in θ∗. In general, the error in fS propagates in opposite directions for θE and µ (Yee et al. 2009). However, in the present case, since this error is small, the fractional error in these quantities is simply that of Z, as indicated in

Equation (3.14).

3.7. Properties of the Lens

3.7.1. Limits on the Lens Brightness

We can use the observed brightness of the event to place constraints on the lens mass. Since the source and lens are superposed, any light from the lens should be accounted for by the blend flux, fB,i, which sets an upper limit on the light from the lens. The unmagnified source is not seen in the OGLE data. From examination of an OGLE image at baseline with good seeing, we estimate the upper limit of the blend flux to be I 17.77 based on the diffuse background light and assuming B,0 ≥ 72 that the reddening is the same as the red clump. Assuming all of this light is due to the lens, the absolute magnitude of the lens is

DL R0 MI,L >IB,0 +(AI,S AI,L) 5 log = 3.25+(AI,S AI,L) + 5 log , (3.16) − − 10 pc − DL

where AI,S and AI,L are the reddening toward the source and lens, respectively, and

DL is the distance to the lens. Since the lens must be in front of the source, we have A A . Moreover, the lens should be closer than R (or at any rate, not I,S ≥ I,L 0 much farther). Hence, M 3.25 is a conservative lower limit. From the empirical I,L ≥ isochrones of An et al. (2007), this absolute magnitude corresponds to an upper limit in the lens mass of M 1.2M . We conclude from these flux-alignment constraints L ≤ ⊙ that either the lens is a main sequence star or, if it is more massive than our upper limit of 1.2M⊙, then it must be a stellar remnant such as a very massive white dwarf or a neutron star.

We can use our measurement of θE to estimate the distance to the lens based on its mass:

θ2 1 1 −1 4G D = E + with κ = 8.14 mas M −1, (3.17) L κM AU D ≡ c2AU ⊙ µ L S ¶ where DS is the distance to the source. If we assume the source is at 8 kpc

(i.e., about 0.1 mag behind the mean distance to the clump at this location) and

ML = 1.2M⊙, we find DL = 7.6 kpc. Hence, the lens could be an F/G dwarf or stellar remnant in the Bulge, or it could be a late-type star closer to the Sun.

73 3.7.2. Bayesian Analysis

Similar to Alcock et al. (1997) and Dominik (2006), we estimate the mass of the lens star and its distance using Bayesian analysis accounting for the measured microlensing parameters, the brightness constraints on the lens, and a model for the

Galaxy. The mathematics are similar to what is described in Section 5 of Batista et al. (2011), although the implementation is fundamentally different because we do not have meaningful parallax information. Specifically, we perform a numerical integral instead of applying the Bayesian analysis to the results of the MCMC procedure. We begin with the rate equation for lensing events:

d4Γ 2 = ν(x, y, z)(2RE)vrelf(µ)g(ML), (3.18) dDLdMLd µ where ν(x, y, z) is the density of lenses, RE is the physical Einstein radius, vrel is the lens-source relative velocity, f(µ) is the weighting for the lens-source relative proper motion, and g(ML) is the mass function. The vector form of the lens-source relative proper motion is µ, which can be described by a magnitude, µ, and an angle, φ, such that d2µ = µdµdφ. We transform variables (see Batista et al. 2011) to find

4 2 4 d Γ 2DLµ θE = ν(x, y, z)f(µ)g(ML). (3.19) dDLdθEdtEdφ κπrel

To find the probability density functions for the lens, we integrate this equation over the variables θE and φ, using a Gaussian prior for θE with the values given in Eq.

(3.14) and a flat prior for φ. We calculate µ from tE and θE using Equation (3.14).

We also integrate over DS, which appears implicitly in πrel and f(µ). For DS, we

74 include a prior for the density of sources based on our Galactic model (see below) assuming the source is in the Bulge.

8 Three functions remain to be defined : ν(x, y, z), f(µ), and g(ML). As in

Batista et al. (2011), we assume g(M) M −1. For the proper motion term, we ∝ follow Equation (19) of Batista et al. (2011):

2 2 1 (µN µexp,N ) (µE µexp,E ) f exp gal − gal gal − gal . (3.20) µ ∝ σ σ − 2σ2 − 2σ2 µ,Ngal µ,Egal " µ,Ngal µ,Egal #

Note that the variables in fµ are given in Galactic coordinates rather than Equatorial coordinates. The transformation between the two is simply a rotation by 60◦.

Still working in Galactic coordinates, the expected proper motion, µexp, takes into account the typical motion of a star in the Disk, v, and the motion of the Earth during the event, v =(v ,v )=( 0.80, 28.52) km s−1, ⊕ ⊕,Ngal ⊕,Egal −

vL (v⊙ + v⊕) vS (v⊙ + v⊕) µexp = − − , (3.21) DL − DS

−1 −1 where v⊙ = (7, 12) km s + (0,vrot) and vrot = 230 km s . For the Disk we use v = (0,v 10 km s−1) and σ =(σ , σ ) = (20, 30) km s−1, and for the Bulge rot − µ,Ngal µ,Egal −1 −1 v = (0, 0) km s and σ =(σµ,Ngal , σµ,Egal ) = (100, 100) km s .

For the stellar density ν(x, y, z), we use the model from Han & Gould (2003) including a bar in the Bulge. We assume the Disk has cylindrical symmetry with a hole of radius 1 kpc centered at R0 = 8 kpc. We limit the Bulge to 5

8We will neglect constants of proportionality as they are not relevant to a likelihood analysis.

75 For the Bayesian analysis, we use tE = 21.7 days measured from the microlensing

fit to the light curve. We also have the constraint from the lens brightness that

2 ML = θE/(κπrel) < 1.2M⊙. This analysis implicitly assumes that the lens is a main sequence star. The lens could be a stellar remnant, although this is much less likely because of their smaller relative space density. The possibility that the lens is a stellar remnant could be tested several years from now when the source and lens have moved sufficiently far apart so as to be separately resolved, i.e., in roughly

−1 10(λ/1.6µm)(Dtel/10m) years, where λ is the wavelength of the observations and

Dtel is the diameter of the telescope used, assuming the observations are diffraction limited (for a discussion of detecting light from a main sequence lens, see the

Appendix).

The results of the Bayesian analysis are shown in Figure 3.6. We find that

+0.35 if the lens is a main sequence star, its mass is ML = 0.59−0.29 M⊙ and its distance is D = 7.15 0.94 kpc (median and 68% confidence interval). Hence the planet L ± +1.9 mass is mp = 3.3−1.6MJup. In the close solution, the projected separation is sharply peaked at r = sD θ = 1.0 0.1 AU. However, the wide solution, which is not ⊥ L E ± strongly disfavored, gives an alternative r = 3.4 0.4 AU. If we assume a r , the ⊥ ± ∼ ⊥ planet would have a period of 1.3 or 8 years. ∼ ∼

76 3.8. Possible Constraints from AO Observations

Of the 13 previously published microlensing planets, two are very likely to be super-Jupiters orbiting M dwarfs9. In both cases, high-resolution imaging from space or the ground was needed to complete these determinations. OGLE-2005-BLG-071 has a mass ratio, q = 7.4 0.4 10−3 (Udalski et al. 2005; Dong et al. 2009a), similar ± × to that of MOA-2011-BLG-293 analyzed here. Dong et al. (2009a) subsequently combined Hubble Space Telescope (HST) and light curve data to determine the host mass M = 0.46 0.04 M , implying that this was an m 3.5 M planet L ± ⊙ p ∼ Jup orbiting an M dwarf. Batista et al. (2011) found an even higher mass ratio for

MOA-2009-BLG-387 of q = 0.0132 0.003. Their marginal detection or upper limit ± on the lens flux from 8m-class adaptive optics (AO) observations allowed them to place an upper limit M < 0.5 M⊙ (90% confidence) on the host, with a median estimate of ML = 0.19 M⊙, and so mp = 2.6 MJup.

Based on our Bayesian analysis (Section 3.7.2), it is possible that MOA-2011-

BLG-293L is another case of a super-Jupiter orbiting an M dwarf. As we now show,

8m-class adaptive optics (AO) observations could clarify the nature of the system.

The main uncertainty in an AO measurement of the lens flux is the uncertainty in the source flux, since the two objects are superposed unless many years have passed since the event. Because an alternate model for the baseline flux exists in which the

9Two more could be marginally included in this category (Bennett et al. 2006; Dong et al. 2009b).

77 flux is not constant (see Section 3.4.2), we cannot be certain it is possible to reliably measure the blended flux. Nevertheless, for the sake of argument, we are going to assume that the baseline flux is stable, so the uncertainty in fS is dominated by the statistical errors from the MCMC procedure. This assumption can be tested after OGLE-IV has collected several more years of baseline data on the event. If these data confirm that the baseline is stable, the calculations in this section are applicable. If not, the limits estimated here are overly optimistic.

As mentioned in Section 3.6, the statistical error in fS is 4.6%, which is due primarily to correlations of the source flux with other model parameters, rather than errors from fitting the light curve to an individual model. Thus, the H-band flux (in units of the instrumental scale of CTIO H-band images) is known to essentially the same precision. Using standard techniques (Janczak et al. 2010; Batista et al. 2011), this flux scale can be aligned to the AO flux scale to a few percent. Hence, the AO source flux fH,S can be known to about 7%. This means that excess light due to the lens can be securely detected at the 3 σ level, provided that ∆H H H < 1.7 ≡ L − S mag. This quantity can be related to the physical properties of the lens and source by

∆H H H = ∆M + ∆A + ∆Dmod (3.22) ≡ L − S H H where ∆M M M , ∆A A A , and ∆Dmod 5 log(D /D ). H ≡ H,L − H,S H ≡ H,L − H,S ≡ L S

Now, in the regime we will be considering, it is very likely that ∆AH = 0, but in any case ∆A 0, since the lens is in front of the source. Hence, we can conservatively H ≤ 78 ignore this term. The last term is

D θ2 D 0.45 M ∆Dmod = 5 log 1+ S E 0.3 S ⊙ (3.23) − AU κM → − 8 kpc M µ L ¶ L

where in the second step we have inserted Equation (3.14) and kept only the first term of the Taylor expansion of the logarithm10. Hence, the lens will be detectable provided

MH,L < MH,S + 2.0, (3.24) ∼

where 2.0 = (∆H ∆Dmod) assuming D = 8kpc and M = 0.45 M . From its − S L ⊙ color and magnitude, the source is a late G/early K Bulge dwarf, so M 4.2. H,S ∼

Hence, all dwarfs MH,L < 6.2 are detectable, which corresponds to ML > 0.43 M⊙. ∼ ∼

Such a detection or upper limit would not be absolutely secure. Detected light could in principle be due to a companion to the lens or source, or an unrelated star in this crowded, low-latitude Bulge field. Additionally, an upper limit on the lens

flux could be attributed to a remnant host rather than a late M dwarf. Nevertheless, the probabilities of these alternative interpretations can be quantified, and it is

10 This approximation assumes that (DS/DL 1) 1, where DL is implicitly embedded in θE. − ≪ 79 important to do so in order to estimate the frequency of very massive planets orbiting M dwarfs.

3.9. Discussion

3.9.1. Implications for Planet Formation Theory

The lens in MOA-2011-BLG-293 consists of a super-Jupiter orbiting a probable

M dwarf. The projected separation of the planet from the star is at most a few AU, making it difficult to form in situ if the host is indeed an M dwarf. Core accretion theory makes a general prediction that massive Jovian planets around M dwarfs should be rare (Laughlin et al. 2004; Ida & Lin 2005). While gravitational instability can form large planets around M dwarfs (Boss 2006), these typically form farther out, so if the planet formed by this mechanism, it would either be required to have migrated significantly or the projection effects must be severe. In the Appendix, we discuss how Adaptive Optics (AO) observations can confirm the microlensing measurement of the host mass or at least place upper limits on the host mass that would confine it to the M dwarf regime. Additionally, it should be noted that this planet joins a growing number of massive planets orbiting stars likely to be M dwarfs discovered by microlensing (see Appendix) and radial velocity (see Bonfils et al. 2011 for a summary and also Johnson et al. 2011).

80 3.9.2. The Ongoing Importance of Followup

Even with high-cadence surveys, followup data remain important for the interpretation of individual events. In this case, the event was high magnification with a faint source, undetected at baseline. This meant that the time period when the event was observable, to surveys and followup, was very brief. We have shown that followup data can vastly increase the signal and provide redundancy in the light curve coverage, which protects against weather. In Section 3.5.2, we demonstrated that with the survey data, even though the break in the light curve is missed because of bad weather, the same solution is recovered for MOA-2011-BLG-293, albeit at much lower significance than when all the data are included (∆χ2 500 compared ∼ to ∆χ2 5400). This is somewhat surprising since it is conceivable that because ∼ of the degeneracies possible for central caustic type events, the loss of this feature would leave the models relatively unconstrained or allow alternative solutions to the light curve. The followup data are beneficial in this case because in addition to adding to the signal-to-noise, they trace the sharp feature seen in the model light curves, increasing our confidence that the correct model has been found.

It is likely that without the real-time discovery of this event in the MOA data and the subsequent high magnification alert from µFUN, the planet would have remained undiscovered as of this writing. The planetary anomaly only becomes detectable when data from all three survey telescopes are combined, which requires

81 systematically reducing, combining, and searching all of the survey data, preferably using a difference imaging reduction in order to detect events with faint sources like this one. Routine reduction of all survey data is planned for the current

OGLE/MOA/Wise survey, but has not yet been fully implemented.

The multi-band data taken by followup groups can also be important for interpreting microlensing events. In order to determine the physical characteristics of the lens from the microlens variables, we used the (V I) color of the source to − estimate its angular size, thus providing a physical scale for the lens system. We also used the (V I) color to inform our choice of limb-darkening coefficients for our − model. Additionally, H-band data are important for comparison to AO observations that may be used to improve constraints on the lens mass as discussed for this event in the Appendix. H-band data are routinely taken as part of followup observations at CTIO but not part of the planned surveys. The OGLE survey regularly takes

V -band data every few days when the weather is good, and the Wise survey is planning similar observations for future seasons. Provided the event timescale is long, this will result in several points taken when the source is substantially magnified.

For the present case, the highly magnified part of the event is brief, and OGLE only obtained one V band point over peak, which was taken deliberately as part of followup observations. Because the V -band data were only taken as part of followup, we need to consider the effect of excluding these data in the context of a pure survey

82 detection. In principle, the (V I) color could be estimated following the method − in Gould et al. (2010b), which takes advantage of the difference in the OGLE and

MOA bandpasses to estimate the (V I) color from R I . For this event, − MOA − OGLE the uncertainty of this measurement from the fits to survey-only data is σR−I = 0.01 leading to a (V I) uncertainty of σ = 0.04. In this case, the precision is not − V −I much worse than what we found from the standard technique using CTIO V - and

I-band photometry, so the lack of survey V -band data would not have a major effect.

Finally, although this event clearly shows that a planet is detected at ∆χ2 500 ∼ without followup data, this is smaller than the ∆χ2 of any published microlensing planet, underlining the fact that low-significance signals have not been systematically explored. Events like MOA-2011-BLG-293 that are robustly characterized with followup data but with weaker signals in the survey data can be used to probe lower

∆χ2 signals and inform our understanding of the limits of what is detectable. For example, if we analyze a large number of events with a range of signal strengths in the survey data, we could determine a ∆χ2 threshold for which a known signal, seen in the complete data set including followup, can no longer be distinguised from the noise. Additionally, the results for central caustic events will help us understand a ∆χ2 threshold below which the model degeneracies mean that the “correct” solution, as determined from the complete data set, can no longer be recovered. We might also require that these events can be well-characterized, i.e., that degenerate central caustic models can be sufficiently disentangled so that the mass ratio is

83 well-constrained11. Understanding these thresholds will be important for analyzing large samples of events to study the planets as a population rather than individual discoveries. By analyzing a large sample of events similar to MOA-2011-BLG-293, we can empirically determine appropriate ∆χ2 thresholds or investigate other statistics for both detecting planets in all microlensing events and characterizing them in central caustic events.

11We will leave the exact definition of this phrase to future investigations but suggest that it might be along the lines of constraining the mass ratio to an order of magnitude at 2σ.

84 Error Renormalization Coefficients

Observatory Filter k emin Ndata

OGLE I 1.75 0.01 274a MOA MOA-Red 1.25 0.0 78b CTIO I 1.56 0.0 63 Wise I 1.57 0.0 49 Weizmann I 1.74 0.0 54 CTIOc V 9 ······

a Ndata after binning.

b ′ Ndata after binning. Restricted to 5743.5

Note. — The properties of each data set are given along with the error renormalization coefficients used to rescale the error bars (see Sec. 3.2.1).

Table 3.1. Data

85 Model teff flim t⋆ qtE (days) ()a (days) (days)

close 0.0756(5) 9.81(6) 0.0355(3) 0.115(2) close with 0.0754(5) 9.82(7) 0.0355(3) 0.115(2) flux constraints close with 0.0748(7) 9.95(11) 0.0355(7) 0.113(2) parallax wide 0.0754(5) 9.85(7) 0.0355(3) 0.116(2) wide with 0.0754(5) 9.85(7) 0.0355(3) 0.116(2) flux constraints survey only 0.075(2) 10.1(3) 0.039(3) 0.109(7) survey with 0.076(2) 9.9(3) 0.040(3) 0.110(8) flux constraints

af f /u , where f = 1 corresponds to a magnitude lim ≡ S,OGLE 0 S,OGLE I = 18, so flim has units of flux in this system.

Note. — Comparing the invariants of the lightcurve (Sec. 3.4.1) shows that they are robustly measured both in terms of their uncertainties and their variation among models.

Table 3.2. Lightcurve “Invariants”

86 2 fS,W ise fS,W eiz Model χ t0 5747. u0 tE ρ α s q πE,N πE,E − fS,CT IO fS,CT IO (HJD′) 103 (days) 103 (◦) 103 × × × close 658.9377 0.4935(7) 3.5(2) 21.67(96) 1.64(7) 221.3(5) 0.548(6) 5.3(2) 0.(.) 0.(.) 0.979(9) 1.09(2) close with 662.0860 0.4935(6) 3.5(2) 21.75(95) 1.63(7) 221.3(5) 0.548(5) 5.3(2) 0.(.) 0.(.) 0.990(4) 1.08(1) flux constraints close with 655.5644 0.4924(9) 3.5(2) 21.24(95) 1.68(8) 221.5(6) 0.552(6) 5.4(2) 1.7 -2.4 0.94(2) 1.04(3) parallax (1.1) (1.5) wide 662.8497 0.4931(7) 3.4(2) 22.49(98) 1.58(7) 221.1(5) 1.83(2) 5.2(2) 0.(.) 0.(.) 0.98(1) 1.08(2) wide with 665.9169 0.4931(6) 3.3(1) 22.64(98) 1.57(7) 221.1(5) 1.83(2) 5.1(2) 0.(.) 0.(.) 0.988(5) 1.07(1)

87 flux constraints survey only 497.3160 0.492(1) 3.8(2) 19.8(1.0) 2.0(2) 218(1) 0.55(1) 5.5(4) 0.(.) 0.(.) survey only with 498.8901 0.493(1) 3.8(2) 20.0(1.0) 2.0(2) 218(1) 0.55(2) 5.5(4) 0.(.) 0.(.) flux constraints

Note. — The mean and root mean square errors for the parameters of each model are given along with the χ2 for that model. The fits with “survey only” use only a subset of data representative of what would have been obtained without additional followup. Note that the parameters of these fits are very similar to the parameters of the other fits, but with slight increases in their uncertainties.

Table 3.3. Model Parameters MOA-2011-BLG-293 Peak

15 OGLE MOA 15.0 CTIO I 16 Wise Weizmann 15.5 17 MOA MOA I I 16.0 18

16.5 19

0.050 0.050 0.025 0.025 0.000 0.000 -0.025 -0.025 -0.050 -0.050 Residuals Residuals 44 45 46 47 48 49 50 0.3 0.4 0.5 0.6 0.7 HJD-2,455,700 HJD-2,455,747

Fig. 3.1.— The light curve of MOA-2011-BLG-293. The left-hand panel shows a broad view of the light curve, while the right-hand panel highlights the peak of the event where the planetary perturbation occurs. Data from different observatories are represented by different colors, see legend. The black curve is the best-fit model with a close topology (s< 1). The times are given in HJD′=HJD 2450000. −

88 16

18 CTIO I 20

22

-1.0 -0.5 0.0 0.5 1.0 1.5

(V-I)CTIO

Fig. 3.2.— Color-Magnitude Diagram of the event in instrumental (uncalibrated) magnitudes. The source is shown as the solid black point; the errors in the source color and magnitude are smaller than the size of the point. The centroid of the Red Clump is the open square with an X through it. The small points show the stars in the field, restricted to stars within 60′′ of the source because there is strong differential reddening on larger scales.

89 2 OGLE MOA 1 S 0 flux/f

-1

-2 4000 5000 5300 5400 5500 5600 5700 5800 5900 HJD' HJD'

Fig. 3.3.— Observed MOA (open squares) and OGLE (solid circles) fluxes at baseline. The fluxes have been scaled by the source flux and adjusted so that the baseline is approximately zero. The solid line shows the expected flux from the model. The data have been binned by 30 days (right panel) and semi-annually (left panel). Data taken when the event is significantly magnified (hashed region: 5710 < t(HJD′) < 5790) have been excluded. Note that the MOA data show significant variation at a level comparable to the source flux.

90 0.02

0.01

Y 0.00

-0.01

-0.02 -0.02 -0.01 0.00 0.01 0.02 X

Fig. 3.4.— Caustic structure and source trajectory of the best-fit model of MOA- 2011-BLG-293 in the source plane. The circle shows the physical size of the source, and its position at the time of the caustic exit (HJD′ 5747.5). The x-axis is the star-planet axis, and the origin is at the center of magnificatio∼ n. The scale of the axes is in units of the Einstein radius.

91 OGLE 15 MOA 15.0 15.0 CTIO I 15.5 15.5 Wise 16.0 16.0 Weizmann 16 16.5 16.5 17.0 17.0

I (mag) 0.2 0.3 0.4 0.5 0.6 0.2 0.3 0.4 0.5 0.6 17 HJD'-5747. HJD'-5747.

18 Point-Lens Fit Planet Fit Survey-Only Data Survey-Only Data

15 15.0 15.0

15.5 15.5

16.0 16.0 16 16.5 16.5 17.0 17.0

I (mag) 0.2 0.3 0.4 0.5 0.6 0.2 0.3 0.4 0.5 0.6 17 HJD'-5747. HJD'-5747.

18 Point-Lens Fit Planet Fit All Data All Data

5747.5 5748.0 5748.5 5747.5 5748.0 5748.5 HJD' HJD'

Fig. 3.5.— Comparison of point-lens fits (left) and planet fits (right) for “survey- only” data, (top) and all data (bottom). In both cases, the planet fit is clearly better than the point lens fit, but the difference is more significant when followup data are included. Note that for “survey-only” data the OGLE data have been thinned out to reflect the typical survey cadence.

92 0.5 1.2

1.0 0.4

0.8 0.3 0.6 0.2 0.4

0.1 0.2

0.0 0.0 0 2 4 6 8 10 0.01 0.10 1.00

DL (kpc) ML (MSun)

Fig. 3.6.— Probability densities for the lens (host) as function of its distance (left) and mass (right). The vertical scale is set so that the total area under each curve is equal to one. Masses ML > 1.2 M⊙ are excluded by the flux-alignment constraint on the lens brightness (bold vertical line). The 68% and 90% confidence intervals about the median are indicated by the shaded regions. The discontinuities in the slope of probability distribution for the lens distance arise from overlap between the Disk and Bulge stellar density distributions. From the Galactic model priors, there is a significant probability that the host is an M-dwarf. High-resolution imaging could confirm or contradict this by direct detection of the lens light (see Appendix).

93 Chapter 4

MOA-2010-BLG-311: A planetary candidate below the threshold of reliable detection

4.1. Introduction

High-magnification events, events in which the maximum magnification of the source, Amax, is greater than 100, have been a major focus of microlensing observations and analysis. Because the impact parameter between the source and the lens is very small in such cases, u 1/A , it is likely to probe a central caustic 0 ≃ max produced by a planetary companion to the lens star. Furthermore, such events can often be predicted in advance of the peak, allowing intensive observations of the event at the time when it is most sensitive to planets. Consequently, a substantial amount of effort has been put into identifying, observing, and analyzing such events.

Observed high-magnification events are classified into two groups for further analysis: events with signals obvious to the eye and events without. Only events in the first category are systematically fit with 2(or more)-body models. The other events are only analyzed to determine their detection efficiencies. As a result, no

94 planets have been found at or close to the detection threshold, and furthermore this detection threshold is not well understood1. Gould et al. (2010a) suggest a detection threshold in the range of ∆χ2 = 350–700 is required to both detect the signal and constrain it to be planetary, but they note that the exact value is unknown. With the advent of second-generation microlensing surveys, which will be able to detect planets as part of a controlled experiment with a fixed observing cadence, it is important to study the reliability of signals close to the detection threshold, since a systematic analysis of all events in such a survey will yield signals of all magnitudes, some of which will be real and some of which will be spurious.

In this chapter, we present the analysis of a high-magnification microlensing event, MOA-2010-BLG-311, which has a planetary signal slightly too small to claim as a detection. We summarize the data properties in Section 4.2 and present the color-magnitude diagram (CMD) in Section 4.3. In Section 4.4, we fit the light curve with both point lens and 2-body models. We then discuss why a planetary detection cannot be claimed in Section 4.4.5. We calculate the Einstein ring size and relative

1The need for a well-defined detection threshold is also discussed in Yee et al. 2012.

95 proper motion in Section 4.5 and discuss the possibility that the lens is a member of the cluster NGC 6553 in Section 4.6. We give our conclusions in Section 4.7.

4.2. Data

4.2.1. Observations

On 2010 June 15 (HJD′ 5362.967 HJD 2450000), the Microlensing ≡ − Observations in Astrophysics (MOA) collaboration (Bond et al. 2001; Sumi et al.

2011) detected a new microlensing event MOA-2010-BLG-310 at (R.A., decl.) =

(18h08m49.s98, -25◦57′04.′′27) (J2000.0), (l, b) = (5.17, -2.96), along our line of sight toward the Galactic Bulge. MOA announced the event through its email alert system and made the data available in real-time. Within a day, this event was identified as likely to reach high magnification. Because of MOA’s real-time alert system, the event was identified sufficiently far in advance to allow intensive follow up observations over the peak.

The observational data were acquired from multiple observatories, including members of the MOA, OGLE, µFUN, PLANET, RoboNet (Tsapras et al. 2009), and MiNDSTEp collaborations. In total, sixteen observatories monitored the event for more than one night, and thus their data were used in the following analysis.

Among these, there is the MOA survey telescope (1.8m, MOA-Red2, New Zealand)

2This custom filter has a similar spectral response to R-band

96 and the B&C telescope (60cm, V , I, New Zealand); eight of the observatories are from µFUN: Auckland (AO, 0.4m, Wratten #12, New Zealand), Bronberg (0.36m, unfiltered, South Africa), CTIO SMARTS (1.3m, V , I, H, Chile), Farm Cove (FCO,

0.36m, unfiltered, New Zealand), Kumeu (0.36m, unfiltered, New Zealand), Molehill

(MAO, 0.3m, unfiltered, New Zealand), Vintage Lane (VLO, 0.4m, unfiltered, New

Zealand), and Wise (0.46m, unfiltered, Israel); three are from PLANET: Kuiper telescope on Mt. Bigelow (1.55m, I, Arizona), Canopus (1.0m, I, Australia), and

SAAO (1.0m, I, South Africa); one is from RoboNet: Liverpool (2.0m, I, Canaries); and one is from MiNDSTEp: La Silla (1.5m, I, Chile). The event also fell in the footprint of the OGLE IV survey (1.3m, I, Chile), which was in the commissioning phase in 2010. The observatory and filter information is summarized in Table 4.1.

In particular, observations from the MOA survey telescope, MOA B&C,

PLANET Canopus, µFUN Bronberg, and µFUN VLO provided nearly complete coverage over the event peak between HJD′ = 5365.0 and HJD′ = 5365.4.

4.2.2. Data Reduction

The MOA and B&C data were reduced with the standard MOA pipeline

(Bond et al. 2001). The data from the µFUN observatories were reduced using the standard DoPhot reduction (Schechter et al. 1993), with the exception of Bronberg and VLO data, which were reduced using difference image analysis (DIA; Alard

97 2000; Wozniak 2000). Data from the PLANET and RoboNet collaborations were reduced using pySIS2 pipeline (Bramich 2008; Albrow et al. 2009). Data from

MiNDSTEp were also initially reduced using the pySIS2 pipeline. The OGLE data were reduced using the standard OGLE pipeline (Udalski 2003). Both the

MOA and MiNDSTEp/La Silla data were reduced in real-time, and as such the initial reductions were sub-optimal. In fact, the original MOA data over the peak were unusable because they were corrupted. After the initial analysis, both the

MiNDSTEp and MOA data were rereduced using optimized parameters.

4.3. Color-Magnitude Diagram

To determine the intrinsic source color, we construct a color-magnitude diagram

(CMD) of the field of view containing the lensing event (Figure 4.1) based on V - and I-band images from CTIO SMARTS ANDICAM camera. The field stars in the

CMD are determined from three V -band images and multiple I-band images. Four

V -band images were taken; however, only three of the images are of sufficient quality to contribute to the CMD. We visually checked each of the three images to make sure that there were no obvious defects such as cosmic-ray events in the images.

From the fit to the light curve, we find the instrumental magnitude of the source is I = 19.0 with an uncertainty of 0.05 mag due to differences between the planet and point lens models as well as the error parameterizations. In the

98 top right-hand corner of Figure 4.1, the red dot at (V I,I) = (0.40, 15.5) − cl marks the centroid of the red clump. The intrinsic color and magnitude of the clump are (V I,I) = (1.06, 14.3) (Bensby et al. 2011; Nataf et al. 2012). − cl,0 Using the offset between the intrinsic magnitude of the clump and the observed, instrumental magnitude, we can then calibrate the magnitude of the source to find

I = 17.8 0.1. S,0 ±

The color of the source is normally estimated from V - and I-band images using the standard technique in Yoo et al. (2004). However, with only one highly-magnified

V -band image, this method is unreliable, so we use an alternative technique to determine the instrumental (V I) color by converting from the instrumental − (I H) color. Using the simultaneous CTIO I- and H-band observations, we − measure the instrumental (I H) color of the source by linear regression of H − on I flux at various magnifications during the event. We then construct a VIH instrumental color-color diagram from stars in the field (bottom panel of Figure

4.1). The stars are chosen to be all stars seen in all three bands with instrumental magnitude brighter than HCTIO = 19.0 (note that the field of view for H-band observations is 2.4′ 2.4′ compared to 6′ 6′ for the optical bands). The field stars × × form a well defined track, which enables us to estimate the (V I) source color from − the observed (I H) source color. This yields (V I) = 0.75 0.05. Note that this − − 0 ± method would not work for red stars, (V I) > 1.3, because for these red stars, the − 0 VIH relation differs between giants and dwarfs (Bessell & Brett 1988). However, the

99 observed color is well blueward of this bifurcation. There is also a spectrum of the source taken at HJD′ = 5365.001 (Bensby et al. 2011). The “spectroscopic” (V I) − 0 reported in that work is 0.77, in good agreement with the value calculated here.

4.4. Modeling

4.4.1. The Basic Model

A casual inspection of the light curve does not show any deviations from a point lens, so we begin by fitting a point lens model to the data. A point lens model is characterized by three basic parameters: the time of the peak, t0, the impact parameter between the source and the lens stars, u0, and the Einstein timescale, tE. Since u0 is small, the finite size of the source can be important. To include this effect, we introduce the source size in Einstein radii, ρ, as a parameter in the model.

Additionally, we include limb-darkening of the source. The temperature of this slightly evolved source was determined from the spectrum to be T 5460 K by eff ∼ Bensby et al. (2011). Using Claret (2000) we found the limb-darkening coefficients to be uV = 0.7086, uI = 0.5470, and uH = 0.3624, assuming a microturbulent velocity

−1 = 1 km s , log g = 4.0, solar metallicity, and Teff = 5500K, which is the closest grid point given the Bensby et al. (2011) measurements. We then convert uV , uI , and uH

100 to the form introduced by Albrow et al. (1999)

2u Γ= (4.1) 3 u − to obtain ΓV = 0.62, ΓI = 0.45, and ΓH = 0.28. Because the various data sets are not on a common flux scale, there are also two flux parameters for each data set, fS,i and fB,i, such that

fmod,i = fS,iA(t)+ fB,i, (4.2) where A(t) is the predicted magnification of the model at time, t, and includes the appropriate limb-darkening for data set i. The source flux is given by fS,i, and fB,i is the flux of all other stars, including any light from the lens, blended into the PSF

(i.e., the “blend”).

4.4.2. Error Renormalization

As is frequently the case for microlensing data, the initial point lens fit reveals that the errors calculated for each data point by the photometry packages underestimate the true errors. Additionally, there are be outliers in the data that are clearly seen to be spurious by comparison to other data taken simultaneously from a different site. Simply taking the error bars at face value would lead to biases in the modeling. Because the level of systematics varies between different data sets, underestimated error bars can give undue weight to a particular set of data.

101 Additionally, if the errors are underestimated, the relative ∆χ2 between two models will be overestimated, making the constraints seem stronger than they actually are.

To resolve these issues, we rescale the error bars using an error renormalization factor (or factors) and eliminate outliers. We begin by fitting the data to a point lens to find the error renormalization factors. We remove the outliers according to the procedure described below based on the renormalized errors, refit, and recalculate the error renormalization factors. We repeat this process until no further outliers are found.

To first order, we can compensate for the underestimated error bars by rescaling them by a single factor, k. The rescaling factor is chosen for each data set, i, such

2 1/2 2 2 that ki =(χi /Ni) , producing a χ per degree of freedom of χdof = 1. We will refer to this simple scheme for renormalizing the errors as “1-parameter errors”.

Alternatively, we can use a more complex method to renormalize the errors, which we will call “2-parameter errors”. This method was also used in Miyake et al.

(2012) and Bachelet et al. (2012a). To renormalize the errors, we rank order the data by magnification and calculate two factors, k and emin, such that

2 2 σj = ki σorig,j + emin,i, (4.3) q where σorig is the original error bar and σ is the new error bar and the calculations are done in magnitudes. The index i refers to a particular data set, and j refers to a particular point within that data set. The additional term, emin, enforces a minimum

102 uncertainty in magnitudes, because at high magnification, the flux is large, so the formal errors on the measurement can be unrealistically small. The error factors, k

2 and emin, are chosen so that χdof for points sorted by magnification increases in a

2 uniform, linear fashion and χdof = 1 (Yee et al. 2012). Not all data sets will require

2 an emin term; it is only necessary in cases for which χdof has a break because the formal errors are too small when the event is bright. Note that the emin factor will be primarily affected by the points taken when the event is bright, whereas k is affected by all points, so if there are many more points at the baseline of the event, these will dominate the calculation of k.

To remove outliers in the data we begin by eliminating any points taken at airmass > 3 or during twilight. Additional outliers are defined as any point more than Xσ from the expected value, where X is determined by the number of data points, such that fewer than one point is expected to be more than Xσ from the expected value assuming a Gaussian error distribution. The normal procedure is to compare the data to the “expected value” from a point lens fit. We do this for data in the wings of the event, t(HJD′) < 5363 or t(HJD′) > 5367, when we do not expect to see any real signals. However, the peak of the event, 5365 t(HJD′) 5367, ≤ ≤ is when we would expect to see a signal from a planet if one exists. A planetary signal would necessarily deviate from the expectation for a point lens, so in this region instead of comparing to a point lens model, we use the following procedure to identify outliers:

103 1. For each point, we determine whether there are points from any other data set

within 0.01 days. If there are more than two points from a given comparison

dataset in this range, we keep only the point(s) immediately before and/or

after the time of the point in question (i.e., a maximum of 2 points). The point

is not compared to other points from the same data set.

2. If there are matches to at least two other data sets, we proceed to (3) to

determine whether or not the point is an outlier. Otherwise, we treat the point

as good.

3. We then determine the mean of the collected points, µ, including the point in

question, by maximizing a likelihood function for data with outliers (Sivia &

Skilling 2010):

N 2 1 e−Rj /2 L ln − (4.4) ∝ R2 j=1 Ã j ! X where R = (µ x )/σ and x is the datum and σ is its renormalized error j − j j j j bar. If the flux is changing too rapidly for the points to be described well by a

mean, we fit a line to the data.

4. We compare the point to the predicted value, using the likelihood function to

see if it is an outlier:

(a) If there is only one likelihood maximum, we calculate Rj. If Rj > X, the

point is rejected as an outlier.

104 (b) If there is more than one maximum, and the point in question falls in the

range spanned by the maxima, we assume the point is good. If it falls outside

the range, we calculate Rj using the nearest maximum to determine µ. If

Rj > X, we flag the point as an outlier.

Note that for this procedure, we use the point lens model values of FS,i and FB,i to place all of the data sets on a common flux scale.

This procedure is more complicated than usual, but because we compare the points only to other data, rather than some unknown model, it provides an objective means to determine whether a point is an outlier without destroying real signals corroborated by other data. We also visually inspect each set of points to confirm that the algorithm works as expected. Because finite source effects are significant in this event, one might expect slight differences among data sets due to the different

filters, so as part of this visual inspection we also checked that this did not play a significant role.

For both 1-parameter and 2-parameter errors, Table 4.1 lists the error normalization factors and number of points for each observatory that survived the rejection process. Which of these error parameterizations correctly describes the data depends on the nature of the underlying errors. In principle, the noise properties of the data are fully described by a covariance matrix of all data points, but we are unable to calculate such a matrix. Instead, we have two different error

105 parameterizations. The k factor is calculated primarily based on baseline data for which statistical errors dominate. In contrast, the emin factor is heavily influenced by points at the peak of the event when systematic errors are important. Thus,

2-parameter errors better reflect the systematic errors, whereas 1-parameter errors better reflect the statistical errors.

Correlated errors often have a major impact on our ability to determine whether or not the planetary signal in this event is real. We know that correlations in the microlensing data exist, but there has not been a systematic investigation of this in the microlensing literature. Correlated errors (red noise) are generally thought of as reducing the sensitivity to signals, because successive points are not independent, giving related information. But in fact, sharp, short timescale signals are not degraded by correlated noise and may still be robustly detected.

Consider the case of a short-timescale signal superimposed on a long-timescale correlation. Then a model may reproduce the short-timescale signal leading to an improvement in χ2 without actually passing through the data because of the overall offset caused by the correlations. Now suppose that the correlated, red noise has a larger-amplitude than the white noise (i.e., statistically uncorrelated noise). If we set the error bars by the large-amplitude deviations, which is correct for long timescales for which the data are uncorrelated, the significance of the short timescale jump will be diluted, possibly to the point of being considered statistically insignificant.

However, if the timescale of the signal is much shorter than the correlation length of

106 the red noise, the significance of the signal should actually be judged against the white noise, since on that timescale, the red noise will only contribute a constant offset.

In this case, we expect the planetary signal to be quite short, so if the systematic noise is dominated by correlated errors, the noise should be better described by the 1-parameter errors. Because the source in this event crosses the position of the lens and there are no obvious deviations due to a planet, we expect that any planetary signals will be due to very small caustics, which are detectable only at the limb-crossing times (t = t t ρ2 u2) when the caustic passes onto and limb 0 ± E − 0 p off-of the face of the source. Therefore, the timescale of such a perturbation will be very short, equal to tE times the caustic size w, which is < 15 minutes. In contrast, ∼ observed correlations in the microlensing data are typically on longer timescales,

(hour) (based on our experience with microlensing data which are usually sampled O with a frequency of 15 minutes). Hence, the timescale of the signal is likely to be ∼ less than the timescale of the correlated noise.

However, there are other sources of systematic errors that are unrelated to correlated noise such as flat-fielding errors. If such errors dominate over correlated noise, then the 2-parameter errors are a better description of the error bars over the peak.

107 Because the systematic errors, correlated and uncorrelated, have not been studied in detail, we are unable to determine which is the dominant effect. Hence, we are also unable to determine which error prescription better describes our data.

We will begin by analyzing the light curve using 1-parameter errors. We will then discuss how the situation changes for 2-parameter errors.

4.4.3. Point Lens Models

The best-fit point lens model and the uncertainties in the parameters are given in Table 4.2. This model is shown in Figure 4.2, and the residuals to this fit are shown in Figure 4.3. These exhibit no obvious deviations. These fits confirm that

finite source effects are important, since ρ is well measured and larger than the impact parameter, u0.

We also fit a point lens model that includes the microlens parallax effect, which arises either from the orbital motion of the Earth during the event or from the difference in sightlines from two or more observatories separated on the surface of the Earth. Microlens parallax enters as a vector quantity: πE = (πE,N, πE,E). The addition of parallax can break the degeneracy between solutions with u0 > 0 and u0 < 0, so we fit both cases. Parallax does improve the fit beyond what is expected simply from adding 2 more free parameters and shows a preference for u0 > 0.

However, we shall see in the next section that a planetary model without parallax

108 produces an even better fit and adding parallax in addition to the planet gives only a small additional improvement.

4.4.4. 2-body Models

We search for 2-body models over a broad range of mass ratios, from q = 10−6 to q = 10−1. For each value of q, we chose a range for the projected separation between the two bodies, s, for which the resulting caustic is smaller than ρ and s< 1.0. For each combination of s and q, we allow the angle of the source trajectory,

α, to vary, seeding each run with values of α from 0 to 360 degrees in steps of 5 degrees. For our models, we use the map-making method of Dong et al. (2006) when the source is within two source radii of the position of the center of magnification.

Outside this time range, we use the hexadecapole or quadrupole approximations for the magnification (Pejcha & Heyrovsk´y2009; Gould 2008). We used a Markov

Chain Monte Carlo to find the best-fit parameters and uncertainties for each s, q combination.

The grid search reveals an overall improvement in χ2 relative to the point lens model. We find three χ2 minima for different angles for the source trajectory. For central caustics with planetary mass ratios, the caustic is roughly triangular in shape with a fourth cusp where the short side of the triangle intersects the binary axis; the three trajectories roughly correspond to the three major cusps of the caustic. An example caustic is shown in Figure 4.5 along with the trajectories corresponding to

109 the three minima. The angles of the three trajectories are approximately α = 0, 115, and 235 degrees, and we will refer to them as trajectories “A”, “B”, and “C”, respectively.

We then refine our grid of s and q around each of these three minima. We repeat these fits accounting for various microlensing degeneracies. First, we fit without parallax and assuming s < 1. Then, we add parallax and fit both with u0 > 0 and u0 < 0 to see if this degeneracy is broken. Finally, we fit 2-body lens models with s > 1 and no parallax, since there is a well known microlensing degeneracy that takes s s−1. →

The best-fit solution has χ2 = 6637.96 and α = 236.4. This reflects an improvement in χ2 of ∆χ2 140 over the point lens solution. There is no preference ∼ for s < 1 over s > 1, but trajectory C is preferred by ∆χ2 > 35 over trajectories ∼ A and B. The parameters and their uncertainties for the planet fits are given in

Table 4.2. The mass ratio between the lens star and its companion is firmly in the planetary regime: q = 10−3.7±0.1. Furthermore, planetary mass ratios are clearly preferred over “stellar” mass ratios (q 0.1), which are disfavored by more than ∼ ∆χ2 = 60. Parallax further improves the fit by only ∆χ2 10 and has little effect ∼ on the other parameters.

To compare the point lens and planetary models, in Figure 4.6, we plot the “χ2 residuals”: the difference between the cumulative χ2 distribution and the expected

110 N 2 2 value j χj = N, i.e. each point is expected to contribute χj = 1. For both the pointP lens and the planet fits, the distribution rises gradually over the peak of the event. This is expected since 1-parameter errors do not account for correlated noise.

However, in the χ2 residuals for the point lens, there is a jump seen at the time of the first limb-crossing. This jump is even more pronounced when looking at the difference between the planet and point lens models. The jump is caused by MOA data at the time of the limb-crossing that do not fit the point lens well, thereby causing an excess increase in χ2. This is exactly the time when we expect to see planetary signals.

Finally, given the extreme finite source effects in this event, we might be concerned that uncertainties in the limb-darkening coefficients due to uncertainties in the source properties could influence our conclusions. The planetary signal has two components: a limb-crossing signal and an asymmetry. The limb-darkening could influence the first signal, but not the second. To check that the limb-darkening coefficients do not significantly influence our results, we repeat the point lens fits allowing the limb-darkening coefficients to be free parameters. In all cases (no parallax, parallax, 1-parameter or 2-parameter errors), the improvement to the

fit from free limb-darkeing is ∆χ2 < 10, much smaller than the planetary signal. ∼

Furthermore, the value of ΓV decreases by > 10%, which is excluded by the measured

111 source parameters. Thus, we conclude that our treatment of the limb-darkening is reasonable.

4.4.5. Reliability of the Planetary Signal

Although ∆χ2 140 appears to be significant, we are hesitant to claim a ∼ detection of a planet. The planetary signal is more or less equally divided between the jump at the first limb-crossing and a more gradual rise after the second limb-crossing (see the third panel of Fig. 4.6 showing the difference between the point lens and planet models). One could argue that the gradual rise, due to a slight asymmetry in the planet light curve, could be influenced by large-scale correlations in the data. Comparing Figures 4.3 and 4.6 shows that most of the signal at the first limb-crossing comes from only a few points. A careful examination of the residuals in Figure 4.3 shows that while the residuals to the planet model are smaller than for the point lens model, they are not zero, and the didactic residuals do not go neatly through the difference between the models as they do for MOA-2008-BLG-310

(Janczak et al. 2010). Hence, the evidence for the planet is not compelling.

If we repeat the analysis using 2-parameter errors, we find a similar planetary solution, although the exact values of the parameters are slightly different3. The

3For 2-parameter errors, model “A, wide” appears to be competitive with model “C”. However, this solution requires that the source trajectory pass over the planetary caustic at the exact time to compensate for a night for which the MOA baseline data are high by slightly more than 1σ

112 total signal from the planet is significantly degraded for 2-parameter errors, with only ∆χ2 80 between the best-fit planet and point lens models. Table 4.3 gives ∼ parameters for the complete set of point lens and planet fits for 2-parameter errors.

The residuals and error bars over peak are shown in Figure 4.4 and may be compared to 1-parameter errors in Figure 4.3.

We also show the χ2 residuals for 2-parameter errors in the bottom set of panels in Figure 4.6. They are more or less flat over the peak, showing that they track the data well in this region. The offset from zero is caused by systematics elsewhere in the light curve. The difference plot (bottom-most panel) shows that the planet fit is still an improvement over the point lens fit, but the signal from the planet at the

first limb-crossing is much weaker. This is a natural consequence of 2-parameter errors, since the data at the peak of the event, where the planetary signal is seen, have much larger renormalized error bars than for 1-parameter errors4.

Regardless of the error renormalization, this planetary signal is smaller than the ∆χ2 of any securely detected high-magnification microlensing planet.

Previously, the smallest ∆χ2 ever reported for a high-magnification event was for compared to other nights at baseline. If the data from this night are removed, the remaining data predict a different solution with the planetary caustic crossing 18 days earlier. Because this solution is pathological, we do not consider it further. 4Note that while the outliers are slightly different for 1-parameter and 2-parameter errors, no points were rejected in either case during the first limb-crossing, 5365.13

113 MOA-2008-BLG-310 with ∆χ2 = 880 (Janczak et al. 2010). Yee et al. (2012) discuss

MOA-2011-BLG-293, an event for which the authors argue the planet could have been detected from survey data alone with ∆χ2 = 500. However, although the planet is clearly detectable at this level, it is unclear with what confidence the authors would have claimed the detection of the planet in the absence of the additional followup data, which increases the significance of the detection to ∆χ2 = 5400.

At an even lower level, Rhie et al. (2000) find that a planetary companion to the lens improves the fit to MACHO-98-BLG-35 at ∆χ2 = 20, but they do not claim a detection. As previously mentioned, Gould et al. (2010a) suggest the minimum

“detectable” planet will have 350 < ∆χ2 < 700. However, this threshold has not been rigorously investigated; the minimum ∆χ2 could be smaller.

Because of the tenuous nature of the planetary signal, we do not claim to detect a planet in this event, but since including a planet in the fits significantly improves the χ2, we will refer to this as a “candidate” planet.

Finally, it is interesting to note that even though the ∆χ2 for the planetary model is too small to be considered detectable, for both 1-parameter and 2-parameter errors the parameters of the planet (s and q) are well defined (see Tables 4.2 and

4.3). Central caustics can be degenerate, especially when they are much smaller than the source size, so we might expect a wide range of possible mass ratios in this case since the limb-crossings are not well-resolved (Han 2009). However, perhaps we should not be surprised that the planet parameters are well-constrained: both

114 ∆χ2 = 80 and ∆χ2 = 140 are formally highly significant, which would plausibly lead to reasonable constraints on the parameters. In this case, because we believe that the signal could be caused by systematics, by the same token, the constraints on the parameters may be over-strong. We conjecture that the limb-crossing signal does not constrain q and that this constraint actually comes from the asymmetry of the light curve, since small, central caustics due to planets are asymmetric whereas those due to binaries are not.

4.5. θE and µrel

Because the source size, ρ, is well measured, we can determine the size of the

Einstein ring, θE, and the relative proper motion between the source and the lens,

µrel from the following relations:

θ⋆ θE θE = and µrel = . (4.5) ρ tE

Keeping the limb-darkening parameters fixed, we find the best fit for the normalized source size to be ρ = (2.70 0.06) 10−3 for the planetary fits; the value is ± × comparable for the point lens fits. We convert the (V I) color to (V K) using − − Bessell & Brett (1988) and obtain the surface brightness by adopting the relation derived by Kervella et al. (2004). Combining the dereddened I magnitude with this surface brightness yields the angular source size θ = 1.03 0.07 µas. The ⋆ ± error on θ⋆ combines the uncertainties from three sources: the uncertainty in flux

115 (f ), the uncertainty from converting (V I) color to the surface brightness, and s,I − the uncertainty from the Nataf et al. (2012) estimate of I0,cl. The uncertainty of fs,I is 3%, which is obtained directly from the modeling output. We estimate the uncertainty from the other factors (Z) to be 7%. The fractional error in θ⋆ is

2 2 1/2 given by [(1/4)(σfs,I ) +(σZ /Z) ] = 7%, which is also the fractional error of the proper motion µ and the Einstein ring radius θE (Yee et al. 2009). Thus, we find

θ = 0.38 0.03 mas and µ = 7.1 0.6 mas yr−1. E ± rel ±

4.6. The Lens as a Possible Member of NGC 6553

This microlensing system is close in projection to the globular cluster NGC

6553. The cluster is at (R.A., decl.) = (18h09m17.s60, -25◦54′31.′′3) (J2000.0), with a distance of 6.0 kpc from the Sun and 2.2 kpc from the Galactic Center (Harris

′ 1996). The half-light radius rh of NGC 6553 is 1.03 (Harris 1996), which puts the microlensing event 6.5 half-light radii (6.7′) away from the cluster center. By plotting the density of excess stars over the background, we find that about 6% of the stars at this distance are in the cluster (Figure 4.7).

Whether or not the lens star is a member of the cluster can be constrained by calculating the proper motion of the lens star. Zoccali et al. (2001) found a relative proper motion of NGC 6553 with respect to the Bulge of µ =(µl, µb)=(5.89, 0.42) mas yr−1. The typical motion of Bulge stars is about 100 km s−1 corresponding to about 3masyr−1 given an estimated distance of 7.7 kpc toward the source

116 along this line of sight (l = 5.1◦; Nataf et al. 2012). Therefore the expected amplitude of the lens-source relative proper motion if the lens is a cluster member is

µ = 7 3 mas yr−1, which is consistent with the measured value in Section (4.5). rel ±

Combining the measurement of the stellar density with the proper motion information, we find the probability that the lens is a cluster member is considerably higher than the nominal value based only on stellar density. First, of order half the stars in the field are behind the source, whereas the lens must be in front of the source. Second, the lens-source relative proper motion is consistent with what would be expected for a cluster lens at much better than 1 σ, which is true for only about

2/3 of events seen toward the Bulge. Combining these two effects, we estimate a roughly 18% probability that the lens is a cluster member.

One way to resolve this membership issue is by measuring the true proper motion of the lens as was done for a microlensing event in M22 for which the lens was confirmed to be a member of the globular cluster (Pietrukowicz et al. 2012). For this event, we have calculated the relative proper motion of the lens and the source to be 7.1 0.6 mas yr−1. This is consistent with the expected value if the lens were ± a member of the cluster. About 10 years after the event, the separation between the lens and the source star will be large enough to be measured with HST . Based on this followup observation, one will be able to clarify whether the lens is a member of the cluster by measuring the vector proper motion. If it is a member of the cluster,

117 its mass may be estimated from

2 θE Mlens = , (4.6) κπrel

2 where κ 4G/c AU = 8.14 mas/M and πrel is the source-lens relative parallax. ≡ ⊙ In addition, the metallicity of the lens could be inferred from the metallicity of the globular cluster.

4.7. Discussion

We have found a candidate planet signal in MOA-2010-BLG-311. The evidence in support of the planet is

1. The planet substantially improves the fit to the data,

2. In addition to a general improvement to the light curve, the planet produces a

signal when we most expect it, i.e. the time of the first limb-crossing,

3. The solution has a well-defined mass ratio and projected separation for the

planet (excepting the well known s s−1 degeneracy). →

The magnitude of the signal depends on whether the error bars are renormalized using 1-parameter (∆χ2 = 140) or 2-parameter error factors (∆χ2 = 80). We conservatively adopt ∆χ2 = 80 as the magnitude of the signal, but note that if correlated errors are the dominant source of systematic uncertainty, ∆χ2 = 140

118 should be adopted instead (see Section 4.4.2). Regardless, this signal is too small to claim as a secure detection.

Examining the residuals to the light curve and the χ2 residuals shows that the planetary signal is dispersed over many points at the peak of the light curve. It comes from an overall asymmetry near the peak plus a few points at the limb-crossing.

Because the signal is the sum of multiple cases of low-amplitude deviations, it is plausible that the microlensing model could be fitting systematics in the data, which is why the planet signal is not reliable.

Combined with other studies, this event suggests that central-caustic (high- magnification) events and planetary-caustic events require different detection thresholds. The detection threshold suggested in Gould et al. (2010a) of

350 < ∆χ2 < 700 was made in the context of high-magnification events, and our experience so far is consistent with this. A planet was clearly detectable in MOA-

2008-BLG-310 with ∆χ2 = 880 (Janczak et al. 2010). However, Yee et al. (2012) are uncertain if a planet would be claimed with ∆χ2 = 500 for MOA-2011-BLG-293.

Here, ∆χ2 = 80 is definitely insufficient to detect a planet. Hence, the detection threshold for planets in high-magnification events is around or just below ∆χ2 = 500.

In contrast, the planetary caustic crossing in OGLE-2005-BLG-390 produced a clear signal of ∆χ2 = 532 (Beaulieu et al. 2006), and the planet would most likely be detectable if the error bars were 50% larger (∆χ2 200) and might even be ∼ considered reliable if the error bars were twice as large.

119 We suspect the reason for the different detection thresholds is that the information about the microlens parameters and the planetary parameters comes from different parts of the light curve. For planetary-caustic events, the planet signal is a perturbation separated from the main peak. Thus, the microlens parameters can be determined from the peak data independently from the planetary parameters, which are measured from the separate, planetary perturbation. In contrast, for high-magnification events, the planetary perturbation occurs at the peak of the event, so the microlens and planetary parameters must be determined from the same data.

The detection threshold for planetary-caustic events will have to be investigated in more detail. If it is truly lower than for high-magnification events, this is good news for second generation microlensing surveys since that is how most planets will be found in such surveys. At the same time, it points to the continued need for followup data of high-magnification events since these seem to have a higher threshold for detection, requiring more data to confidently claim a planet. This is an important consideration because high-magnification events can yield much more detailed information about the planets.

Additionally, Han & Kim (2009) show that the magnitude of the planetary signal should decrease as the ratio between the caustic size and the source size (w/ρ)

120 decreases for the same photometric precision5. There are several cases of events for which w/ρ < 1: in this case, we have w/ρ = 0.12 and ∆χ2 = 140; the brown dwarf ∼ in MOA-2009-BLG-411L has w/ρ = 0.3 and ∆χ2 = 580 (Bachelet et al. 2012b);

MOA-2007-BLG-400 has w/ρ = 0.4 and ∆χ2 = 1070 (Dong et al. 2009b); and in the case of MOA-2008-BLG-310, the value is w/ρ = 1.1 with ∆χ2 = 880 (Janczak et al.

2010). This sequence is imperfect, but the photometry in the four cases is far from uniform, and it seems that in general the trend suggested by Han & Kim (2009) holds in practice.

Finally, we note that a large fraction of the planet signal comes from the MOA data, but in the original, real-time MOA data, this signal would not have been detectable since the data were corrupted over the peak of the event. It is only after the data quality was improved by rereductions, which in turn were undertaken only because the event became the subject of a paper, that we recovered the planet candidate. This points to the importance of rereductions of the data when searching for small signals. In the current system of analyzing only the events with the most obvious signals, this is not much of a concern. However, if current or future microlensing surveys are systematically analyzed to find signals of all sizes, this will become important.

5Chung et al. (2005) give equations for calculating the caustic size, w (see also Dong et al.

2009b).

121 1-parameter Errors 2-parameter Errors

Observatory Filter k Ndata k emin Ndata

Mt. Bigelow I 1.63 44 1.63 0.0 44 Molehill Unfiltered 0.72 69 0.72 0.0 69 Kumeu Wratten #12 1.19 188 1.18 0.0 188 Farm Cove Unfiltered 1.27 52 1.26 0.0 52 Auckland Wratten #12 0.98 84 1.00 0.0 84 Vintage Lane Unfiltered 4.87 112 3.05 0.004 112 B&C I 4.07 132 1.01 0.025 136 V 1.15 53 0.66 0.03 55 MOA MOA-Red 1.68 4452 1.55 0.003 4434 Canopus I 3.02 28 2.87 0.0 29 Wise Unfiltered 0.52 70 0.53 0.0 70 Bronberg Unfiltered 1.26 727 1.27 0.0 727 SAAO I 2.60 128 2.21 0.0015 127 Liverpool SDSS-i 1.85 120 1.04 0.007 119 La Silla I 10.02 169 3.79 0.004 174 CTIO I 1.33 22 1.34 0.0 22 V 0.50 3 0.50 0.0 3 Ha 74 74 ··· ······ OGLE I 1.36 429 1.32 0.008 429

aThese data were not used in the modeling. They were only used to determine the source color (see Sec. 4.3).

Table 4.1. Data Properties for Two Methods of Error Renormalization. The observatories are listed in order of longitude starting with the most Eastward. If data were taken in more than one filter at a given site, different filters are given on successive lines. The error renormalization coefficients and method for removing outliers are described in Section 4.4.2.

122 Model ∆χ2 t 5365. u t ρ α log s log q π π 0 − 0 E E,N E,E (HJD′) (days) (◦)

Point Lens 136.44 0.19615(4) 0.00152(3) 20.34(42) 0.00260(5) ··············· PL, parallax 69.61 0.1978(2) 0.00167(4) 19.37(41) 0.00275(6) 3.16(41) -1.34(20) ········· PL, parallax, u 123.89 0.19613(9) -0.00158(4) 20.23(42) 0.00262(5) -1.07(36) -0.98(27) − 0 ········· A 35.26 0.19615(4) 0.00167(3) 19.04(34) 0.00279(5) 347.7(6) -0.12(1) -4.46(8) ······ A, parallax -7.02 0.1976(2) 0.00172(4) 18.92(40) 0.00283(6) 347.4(3) -0.08(1) -4.9(1) 2.82(44) -1.29(22) A, parallax, u -3.98 0.19626(9) -0.00184(5) 18.51(38) 0.00289(6) -346.(1) -0.17(2) -4.17(8) -2.11(46) -2.43(40) − 0 A, wide 33.07 0.19615(4) 0.00167(4) 19.06(39) 0.00279(6) 348.1(6) 0.12(1) -4.44(8) ······ B 49.06 0.19614(4) 0.00159(4) 19.71(43) 0.00269(6) 118(1) -0.43(4) -3.5(1)

123 ······ B, parallax 21.45 0.1973(2) 0.00166(4) 19.40(41) 0.00275(6) 119(2) -0.40(5) -3.6(2) 2.21(45) -1.05(21) B, parallax, u 31.33 0.19617(9) -0.00169(4) 19.46(41) 0.00274(6) -115(1) -0.40(3) -3.5(1) -1.45(41) -1.50(35) − 0 B, wide 49.11 0.19615(4) 0.00159(3) 19.76(38) 0.00268(5) 118(1) 0.43(4) -3.5(1) ······ C 0.00 0.19613(4) 0.00159(3) 19.68(41) 0.00270(6) 236.4(7) -0.26(4) -3.7(1) ······ C, parallax -10.65 0.1968(3) 0.00164(4) 19.34(39) 0.00275(6) 235(1) -0.4(1) -3.4(3) 0.89(51) -0.78(24) C, parallax, u -12.63 0.19630(9) -0.00171(4) 19.20(38) 0.00278(6) -232(1) -0.51(8) -3.0(2) -1.11(46) -1.55(41) − 0 C, wide 0.00 0.19613(4) 0.00159(3) 19.73(39) 0.00269(5) 236.4(7) 0.26(4) -3.7(1) ······

Table 4.2. Fits with 1-Parameter Errors. The ∆χ2 is given relative to the χ2 of the best-fit planetary model with s< 1 and without parallax (χ2 = 6637.96), i.e. model “C”; positive numbers indicate a worse fit and negative numbers indicate an improvement relative to this model. The point lens models are given first, followed by the planetary models; “A”, “B”, and “C” denote the three planetary models with distinct values of α corresponding to the three χ2 minima. Model ∆χ2 t 5365. u t ρ α log s log q π π 0 − 0 E E,N E,E (HJD′) (days) (◦)

Point Lens 81.12 0.19618(5) 0.00152(3) 20.51(44) 0.00259(6) ··············· PL, parallax 25.43 0.1978(2) 0.00164(4) 19.80(43) 0.00270(6) 3.20(43) -1.14(21) ········· PL, parallax, u 72.23 0.1961(1) -0.00160(4) 20.23(44) 0.00263(6) -1.26(42) -0.96(32) − 0 ········· A 6.37 0.19611(5) 0.00160(4) 19.82(43) 0.00269(6) 354(2) -0.49(5) -3.1(1) ······ A, parallax -12.98 0.1972(3) 0.00168(5) 19.25(46) 0.00278(7) 349(3) -0.3(2) -3.7(6) 1.89(70) -1.16(25) A, parallax, u -8.07 0.1963(1) -0.00172(5) 19.27(42) 0.00278(6) -348(2) -0.46(7) -3.1(2) -0.98(52) -1.59(48) − 0 A, wide1 -1.59 0.19614(5) 0.00157(3) 20.05(37) 0.00266(5) 356(3) 0.539(4) -3.01(3) ······ B 13.25 0.19613(5) 0.00160(3) 19.79(41) 0.00270(6) 114(2) -0.51(6) -3.1(2) ······ B, parallax -6.42 0.1972(2) 0.00165(4) 19.54(41) 0.00274(6) 112(3) -0.5(1) -3.1(3) 1.97(48) -1.02(23)

124 B, parallax, u 4.22 0.1962(1) -0.00167(5) 19.54(41) 0.00274(6) -109(3) -0.50(6) -3.1(2) -0.99(47) -1.24(43) − 0 B, wide 13.10 0.19613(5) 0.00159(4) 19.82(42) 0.00269(6) 113(2) 0.51(6) -3.1(2) ······ C 0.00 0.19614(5) 0.00158(3) 19.94(40) 0.00268(5) 234(1) -0.40(7) -3.3(2) ······ C, parallax -14.90 0.1970(3) 0.00164(4) 19.51(41) 0.00274(6) 231(2) -0.56(8) -2.9(2) 1.26(54) -0.96(24) C, parallax, u -10.15 0.1963(1) -0.00166(5) 19.49(42) 0.00274(6) -230(2) -0.51(7) -2.9(2) -0.44(51) -1.04(46) − 0 C, wide -0.02 0.19613(5) 0.00158(3) 19.97(42) 0.00267(6) 234(1) 0.40(7) -3.3(2) ······

1This solution and its parameters should be treated with caution, since it corresponds to a pathological geometry. See footnote 3.

Table 4.3. Fits with 2-Parameter Errors. The ∆χ2 is given relative to the χ2 of the best-fit planetary model with s< 1 and without parallax (χ2 = 6751.93), i.e. model “C”; positive numbers indicate a worse fit and negative numbers indicate an improvement relative to this model. The point lens models are given first, followed by the planetary models; “A”, “B”, and “C” denote the three planetary models with distinct values of α corresponding to the three χ2 minima. Fig. 4.1.— Color-magnitude diagram (top) and color-color diagram (bottom) of the field of view of MOA-2010-BLG-311 constructed from the CTIO observations. The small black dots represent the field stars; stars used in constructing the color-color diagram are shown as the green squares in both panels. The black line in the bottom plot shows the measured instrumental (I H) color of the source. The (V I) color of the source is then derived from the intersection− of this line with the (V −I)—(I H) relation established by the field stars (green squares), producing the black− dot in− the bottom panel. The position of the source125 in the CMD (large black dot) is given by this (V I) measurement and the source flux from the fits to the light curve. The large red− dot shows the centroid of the red clump. MOA-2010-BLG-311 12 Mt. Bigelow Canopus Molehill Wise 13 Kumeu Bronberg Farm Cove SAAO Auckland Liverpool Vintage Lane La Silla 14 BC V CTIO V OGLE I BC I CTIO I MOA OGLE 15

16 5364.5 5365.0 5365.5 5366.0 HJD-2,450,000 Peak 12.0

12.5

13.0 OGLE I 13.5

14.0

14.5 5365.0 5365.1 5365.2 5365.3 5365.4 HJD-2,450,000

Fig. 4.2.— Light curve of MOA-2010-BLG-311. Data from different observatories are plotted in different colors. The data from Bronberg (medium pink) have been binned for clarity in the figures; only unbinned data were used in the fitting. The black line shows the best-fit point lens model; on this scale, the best-fit planetary model appears very similar. The error bars reflect 1-parameter errors (see Sec. 4.4.2).

126 Residuals: 1-parameter errors 0.04 0.02 0.00 Lens Point -0.02 -0.04 0.04 0.02 0.00 -0.02 Didactic -0.04 0.04 0.02 0.00 -0.02 Planet -0.04 5365.0 5365.1 5365.2 5365.3 5365.4 HJD-2,450,000

0.02 0.02 0.00 0.00 Lens Lens Point -0.02 -0.02 Point 0.02 0.02 0.00 0.00

Didactic -0.02 -0.02 Didactic 0.02 0.02 0.00 0.00

Planet -0.02 -0.02 Planet 5365.14 5365.15 5365.16 5365.17 5365.22 5365.23 5365.24 5365.25 5365.26 HJD-2,450,000 HJD-2,450,000

Fig. 4.3.— Residuals over the peak to the best-fit point lens and planetary microlensing models for 1-parameter error. Note that for the time range shown, which is the same as for the bottom panel for Fig. 4.2, the error bars tend to be larger for 2-parameter errors than for 1-parameter errors. Careful inspection of the residuals to the point lens and planet models shows improvement around the times of the limb-crossings of the source star and also on the falling side of the light curve. In the middle panel of each set, the red lines show the difference between the best planetary and best point lens models. The didactic residuals in the middle panel are the sum of the red line and the residuals to the best planet model (bottom panel). The dashed lines indicate the limb-crossing times. The dotted lines in the top panels of each set indicate the time ranges shown in the bottom panels. Note that the scales for the top and bottom residuals panels are not the same. The colors of the points are the same as in Fig. 4.2. The Bronberg data have been binned for clarity (as in Fig. 4.2)

.

127 Residuals: 2-parameter errors 0.04 0.02 0.00 Lens Point -0.02 -0.04 0.04 0.02 0.00 -0.02 Didactic -0.04 0.04 0.02 0.00 -0.02 Planet -0.04 5365.0 5365.1 5365.2 5365.3 5365.4 HJD-2,450,000

0.02 0.02 0.00 0.00 Lens Lens Point -0.02 -0.02 Point 0.02 0.02 0.00 0.00

Didactic -0.02 -0.02 Didactic 0.02 0.02 0.00 0.00

Planet -0.02 -0.02 Planet 5365.14 5365.15 5365.16 5365.17 5365.22 5365.23 5365.24 5365.25 5365.26 HJD-2,450,000 HJD-2,450,000

Fig. 4.4.— Same as Fig. 4.3 except for 2-parameter errors.

.

128 Fig. 4.5.— The caustic (red) and three source trajectories corresponding to the three minima discussed in Section 4.4.4. Trajectory C (black) is preferred by ∆χ2 > 35 over trajectories A (green) and B (blue) for models with 1-parameter errors and∼ without parallax. The source is shown to scale as the large circle; the lines with arrows indicate the trajectories of the center of the source. The abscissa in these plots is parallel to the binary axis with the lens star close to the origin and planet to the right. The scale is such that 1.0 equals the Einstein radius.

129 1−Parameter Errors 500 (a) 400 300 200

Residuals 100 Point Lens 2

χ 0 500 (b) 400 300 200 Planet

Residuals 100 2

χ 0 150 (c) 100

50 (a)−(b) 0 5365.0 5365.1 5365.2 5365.3 5365.4 HJD’

2−Parameter Errors 500 (a) 400 300 200

Residuals 100 Point Lens 2

χ 0 500 (b) 400 300 200 Planet

Residuals 100 2

χ 0 150 (c) 100

50 (a)−(b) 0 5365.0 5365.1 5365.2 5365.3 5365.4 HJD’

Fig. 4.6.— The difference between the cumulative χ2 distribution and the expected N 2 2 distribution ( j χj = N), i.e. the χ residuals, for both 1-parameter (top) and 2-parameter (bottom) errors. The top two panels of each set show the distributions for the point lensP model (a) and the planetary models (b). The bottom panels (c) of each set show the difference between the top and middle panels. The distributions for each data set are plotted separately and are shown over the same time range as the bottom panel of Fig. 4.2; data sets without points in this time range are not shown. The colors are as in Fig. 4.2. The thick black line shows the total distribution for all data. The limb-crossing times are indicated by the dashed lines. Note the jump in the MOA data (light green) at the time of the first limb-crossing (HJD′ 5361.5). The signal is much more pronounced for 1-parameter errors than for 2-parameter∼ errors. Note that the vertical scales in the two sets of panels are different.

130 103

102

101 excess counts / area with m < 18

100 100 101 r (arcmin)

Fig. 4.7.— DSS image of NGC 6553 (left). The position of the microlensing event is indicated by the circle. Excess star density over the background around the globular cluster NGC 6553 (right). The center of the cluster is placed at r = 0. The microlensing event is at r = 6.7′ (dashed line). The dip at 6′ is caused by a quasi-circular dust lane, which may be seen in the image. Even∼ though the density of stars drops quickly as a function of radius, cluster members still comprise about 6% of the stars at r = 6.7′.

131 Chapter 5

Chapter 5: WFIRST Planet Masses from Microlens Parallax

5.1. Introduction

The microlensing portion of the WFIRST mission will complete the census of planets by finding large populations of planets beyond the snow line with masses as small as Mars (Green et al. 2012). If the masses of the planets and their hosts are measured, this will permit a direct comparison to planet formation theories.

However, the primary observables in microlensing events are the mass ratio and projected separation (scaled to the Einstein ring) between the planet and its host star. A measurement of the lens mass is necessary to transform these to physical quantities.

If the lens is bright enough, WFIRST will be able to estimate its mass based on a measurement of the lens flux. However, there will be many cases for which the lens light will be too faint to be measured. Such cases will most likely be lenses at the bottom of the stellar mass function, but could also include brown dwarfs,

132 free-floating planets, or stellar remnants. For these events, with only an upper limit on the lens flux, the conclusions that can be drawn about the nature of the planet are limited. In addition, the WFIRST measurement of the lens flux will be a measurement of the total flux of the lens system, including any companions to the lens, which may or may not participate in the lensing event. Typically, companions within 10 AU will produce a measurable microlens perturbation and companions at more than 5 mas ( 40 AU) will be identifiable from a shift in the centroid relative ∼ to the lensing event. However, companions at intermediate separations are not easily identified. Hence, the WFIRST mass estimate will necessarily be an upper limit for any given system.

Fortunately, the lens masses can be measured if microlens parallax and finite source effects are observed. Microlens parallax is a vector quantity whose magnitude is the ratio of Earth’s orbit to the size of the Einstein ring projected onto the observer plane,r ˜E:

AU πE . (5.1) ≡ r˜E

If πE is measured, the mass of the lens, M, can be obtained with a measurement of the angular size of the Einstein ring, θE:

θ c2AU M = E . (5.2) π 4G µ E ¶ µ ¶ For any event in which the size of the source is resolved in time, i.e., it passes over a caustic or near a cusp, θE is measurable. Such finite source effects are almost always

133 measured in events with planets because detection of a planetary companion to the lens almost always requires a caustic interaction. Hence, if the microlens parallax can be measured, the planet masses are known. Finite source effects can also be measured in any event for which the source crosses the position of the lens.

In this chapter, I discuss a means to routinely measure the lens masses using microlens parallax if WFIRST is in an L2 orbit1. Because the WFIRST light curve will be measured so precisely, the orbital parallax effect will be routinely detected at high significance, effectively giving one extremely well-measured component of the parallax (Gould 2013). I show that only a few ground-based observations of each event are needed to complement the WFIRST observations and yield a complete parallax measurement for a large fraction of events.

5.2. Measuring πE, ⊥

5.2.1. Simplified Case

The microlens parallax vector can be written

πE =(πE,k, πE,⊥)=(πE cos θ, πE sin θ), (5.3) where θ is the angle between the lens trajectory and the projection of the Sun-Earth line on the plane of the sky, measured counter-clockwise. Because WFIRST will be

1See Gould (2013) for a discussion of WFIRST parallax measurements for a geocentric orbit.

134 in orbit about the Sun, there will be a measurable asymmetry in the light curve due to the orbital parallax effect (Gould 1992; Gould et al. 1994). This gives strong constraints on the component of the parallax parallel to the projected position of the Sun relative to the event, πE,k, but usually only very weak constraints on the other component, πE,⊥ (e.g., Gould 2013).

I will show that if WFIRST is at L2, as few as two observations of a microlensing event from Earth can be used to measure πE,⊥, leading to a measurement of πE. I begin with a simplified case to illustrate the problem. In the following section, I will present the full derivation and expression for πE,⊥ and show that it reduces to what is derived here.

Consider the projection of the microlensing event onto the observer plane

(Fig. 5.1). For the purposes of illustration, I assume that the size of Einstein ring projected onto this plane isr ˜E = 10 AU, u0 = 0.05, and that the observations are taken close to the equinox (the anticipated midpoint of WFIRST observations) so that the projection of the Earth-WFIRST line onto the sky, D, is equal to the true separation, i.e., D = 0.01 AU. The exact values for these quantities are irrelevant to the derivation; I will discuss their practical implications below. Finally, note that WFIRST at L2 puts it in line with Earth and Sun, so the projection of the

Earth-WFIRST line on the sky is parallel to the projection of the Sun’s position.

135 The WFIRST light curve will be extremely well measured, giving πE,k and the basic microlens parameters: the time of the peak, the source-lens impact parameter scaled to the Einstein ring, and the Einstein crossing time (t0, u0, and tE,

−1 respectively). Hence, the value of u0r˜E(cos θ) is also known. This fixes point ‘a’ on the lens trajectory projected onto the observer plane. Assume the event is observed from Earth when it is at the peak as seen from WFIRST (i.e., when the lens is at point ‘b’). Then, the fractional difference in the magnification is

∆A A A ∆u = − ⊕ (5.4) A A ≃ u0 where A is the magnification as seen from WFIRST, A⊕ is the magnification as seen from Earth, and I assume that the magnification is given by A u−1 (which applies ≃ 0 in the limit u 1) and that the difference between the impact parameter as seen 0 ≪ from Earth and from WFIRST is ∆u u . As illustrated in Figure 5.1, in the ≪ 0 regime where u r˜ D, (∆u)˜r D sin θ meaning that with some manipulation 0 E ≫ E ≃

πE,⊥ can be written:

∆A AU π = u . (5.5) E,⊥ 0 A D µ ¶ µ ¶ Note that all the variables in the right-hand side of the equation are known or measurable.

From the geometry in Figure 5.1, there is one degeneracy in Equation (5.5).

It is possible to change the sign of u0, i.e., reflect the figure over the x-axis, which changes the sign of πE,⊥. This leads to a degeneracy in the direction of πE, but not

136 in its magnitude, which is the relevant quantity for calculating masses. Out of the eight possible configurations one might consider as potentially degenerate with the geometry shown, only the u u degeneracy described here is permitted by the 0 → − 0 observables.

5.2.2. Exact Expression for πE, ⊥

I now derive a general expression for πE,⊥ that applies for an Earth-based measurement of the magnification at any time. In practice, this is the expression that will be used to calculate πE,⊥ from the observables.

Figure 5.2 shows the generalized geometry with all quantities scaled tor ˜E.

Consider an observation from Earth is taken at time t, i.e., when the lens is at point

‘c.’ The lens position is given by u and τ =(t t )t−1. The measured magnification 0 − 0 E is related to the separation of the lens, u, by

u2 + 2 A(u)= , (5.6) u√u2 + 4 where u is measured as a fraction of the Einstein ring. Thus, from the measured magnifications as seen from Earth and WFIRST, the lens separation from their projected positions is known, uE and uW, respectively. I can write

2 2 2 uE =(xW + D/r˜E) + yW, (5.7)

137 where xW and yW are the projections of uW onto the x- and y-axes. Equation (5.7) can be rewritten as

2 2 2 2D D uE uW = (u0 sin θ + τ cos θ)+ 2 . (5.8) − r˜E r˜E

−1 Recognizing that (πE,k, πE,⊥) = (cos θ, sin θ)(AU)˜rE (Eqn. 5.1 and 5.3), I evaluate

D2 AU τπ π = ∆u(2u + ∆u) E,k , (5.9) E,⊥ W − r˜2 2Du − u · E ¸ µ 0 ¶ 0 where ∆u u u . Sincer ˜−2 = (π2 + π2 )(AU)−2, this can be rewritten as a ≡ E − W E E,⊥ E,k quadratic equation for πE,⊥ with the solutions:

AU π = u E,⊥,± 0 D µ ¶ 1 D 2 D 1 1 π2 + 2 τπ ∆u(2u + ∆u) . v 2 E,k E,k W × − ± u − u0 " AU AU − # u µ ¶ µ ¶  t  (5.10)

Although there are formally two solutions for πE,⊥, these can readily be distinguished.

The solution πE,⊥,− corresponds to the case in which the lens passes between the projected positions of Earth and WFIRST. This scenario is expected to be very rare, but it can be definitively excluded with additional observations of the event from

Earth.

If πE is large, then the full expression must be evaluated. However, the present chapter is primarily focused on cases for which πE is small because those are the cases in which the WFIRST light curve will constrain only one component of the

138 parallax well. In that case, Equation (5.10) is well represented by the first term in the Taylor expansion:

1 D D 2 π = ∆u(2u + ∆u) 2 τπ π2 . (5.11) E,⊥,+ 2u2 W − AU E,k − AU E,k 0 " µ ¶ µ ¶ # Furthermore, the last term can generally be ignored because it is second order.

Finally, if the event is observed at peak (τ 0 and u u ) and we assume that → W → 0 ∆u u , | | ≪ | 0| AU π ∆u , (5.12) E,⊥ → D µ ¶ which is equivalent to Equation (5.5).

5.2.3. Constraints on πE, ⊥

The uncertainties from the WFIRST light curve are negligible compared to the uncertainties from the ground-based photometry, so the largest uncertainty in πE,⊥ comes from the measurement of (∆A)A−1. The actual observables from Earth are the magnified flux, fmag,⊕, and flux of the event at the baseline, fbase,⊕, such that

fmag,⊕ fbase,⊕ A⊕ = − + 1, (5.13) fS,⊕ where fS,⊕ is the source flux as seen from Earth. To solve for A⊕, the unknown source flux must be estimated by calibrating the ground-based photometry to the

WFIRST photometry using comparison stars. This situation is equivalent to the problem of measuring ∆u. Gould (1995), Boutreux & Gould (1996), and Gaudi &

139 Gould (1997) showed that ∆u is poorly constrained because u0 is correlated with t and f , where f = f f is the blended (non-varying) component of the E B B base − S

flux. However, most of the information about fB comes from the wings of the event

(Yee et al. 2012), which will not be well measured from the ground because the sky background is so high. Hence, flux calibration is necessary to find fS, constrain fB, and improve the precision of the measurement of u0, or equivalently A⊕.

If only one or two observations are taken of the magnified event, the flux calibration ultimately sets the limit on the precision of πE,⊥. Based on previous experience (e.g., Yee et al. 2012), the uncertainty in this calibration is limited by systematics to a precision of about 1%. This sets the fundamental noise floor on the measurement of (∆A)A−1. Given that by definition, (∆u)˜r D, the limit in the | E| ≤

flux precision means that a 3σ measurement of πE,⊥ is possible for

D r˜ −1 σ −1 u 0.03 E ∆A/A (5.14) 0 ≤ 0.01AU 10AU 0.01 µ ¶ µ ¶ ³ ´ where I again make the assumption that u 1, i.e., A u−1. 0 ≪ ≃ 0

In contrast, if there are many magnified points that can be seen above the baseline from the ground, the peak of the event will be resolved allowing a measurement of the effective timescale, teff = u0tE. Yee et al. (2012) showed that this quantity is invariant to uncertainties in fB, so flux calibration is unnecessary.

Because the velocity offset between WFIRST at L2 and Earth are quite small (0.3

−1 km s ), tE is approximately the same for the WFIRST and ground-based light

140 curves. Then, the uncertainty in ∆u is:

2 2 σt σ σ σ = u eff,⊕ + tE , (5.15) ∆u ≃ u0,⊕ 0,⊕ t t sµ eff,⊕ ¶ µ E ¶ where u0,⊕ is the impact parameter of the event as seen from Earth and I assume that u0,sat is known essentially perfectly. Hence, the method can be applied to events with u0 > 0.03 by reducing the uncertainty in teff,⊕ using additional observations.

Finally, I note that even if πE,⊥ is less well measured than πE,k, this does

not mean that the value of πE is not well measured. So long as πE,k > 3σπE,⊥ , ∼ the constraints on πE will be useful. This will be true for a large fraction of cases, depending on how well the lens trajectory aligns with the projection of the

Sun-Earth-WFIRST line, which is primarily a random effect.

5.3. Discussion

For pure satellite parallax measurements, the Earth-L2 baseline is not ideal because it is only a small fraction ofr ˜ (0.01 AU vs. 10 AU), limiting the precision E ∼ of the measurement that can be made. However, here I take advantage of several consequences of this special geometry that had not been previously considered for a microlensing satellite at L2 (Gould et al. 2003; Han et al. 2004) and the precision of the WFIRST light curve, which will allow the πE,k component of the parallax to be measured extremely well. The short baseline actually resolves the magnitude degeneracy described in Gould (1994) because it is extremely unlikely that the lens

141 will pass in between the two observatories, and if such a case occurred, the parallax would be so large as to be easily measured from the WFIRST light curve alone. In addition, L2 is in line with the projected position of the Sun, which when combined with the measurement of πE,k greatly simplifies the geometry (Figures 5.1 and 5.2).

Finally, because L2 is moving with Earth, the relative velocity offset between Earth and the satellite can be neglected relative to stellar motions (0.3 km s−1 versus

−1 300 km s ). This means that tE is essentially the same for the ground-based and space-based observations and the degeneracies discussed in Gould (1999); Dong et al.

(2007) are avoided.

I have shown that for an event discovered by WFIRST at L2, a measurement of the event magnification as seen from Earth yields a measurement of the component of the parallax perpendicular to the projection of Earth’s orbit, πE,⊥. Although the basic calculation was done assuming the event was observed from the ground at the peak as seen from WFIRST, I showed that in principle the Earth measurement can be made at any time. However, it is best to make the measurement as close to the peak of the event as possible, since the fractional difference in magnification will be largest at the peak, allowing for the best measurement of the parallax.

Measuring the magnification of the event as seen from Earth requires at least two observations: one when it is magnified and one at baseline. A third, magnified, observation would be beneficial in case the source is passing over a caustic at the time of the observations and for distinguishing between the two possible solutions for

142 πE,⊥ given in Equation (5.10). These ground-based observations can be made either with target-of-opportunity (ToO) observations of WFIRST events announced in real time or with a simultaneous, ground-based survey of the WFIRST microlensing

fields.

If there are only a few observations, as would likely be the case with ToO, πE,⊥ is measurable in all events with u0r˜E < 0.3AU, i.e., u0 < 0.03 or peak magnification

Amax < 30. This limit is set by systematics in the flux calibration between the ground-based data and the WFIRST data, which experience shows is limited to a precision of 1%. If only a few events can be observed from the ground, it is best to focus on the highest magnification events. In practice many of the WFIRST sources will be quite faint, so while 1% precision will be possible for the brighter sources, the limits on u0 are probably more stringent for the majority of events. This leads to a preference for higher magnification events. However, it is precisely these events for which parallax measurements are most desirable because these events are the most likely to have planetary signals (Griest & Safizadeh 1998). Furthermore, for point lens events, the higher the magnification, the more likely it is that finite source effects will be observed, allowing mass measurements for these lenses. These isolated objects could include stellar remnants, isolated stellar mass black holes, brown dwarfs, or the population of free-floating planets found by Sumi et al. (2011). Gould

& Yee (2013) also proposed a means to measure the mass of free-floating planets using terrestrial parallax. However, because the baseline for terrestrial parallax

143 measurements is R , parallaxes are only measurable for the closest objects. Here, ∼ ⊕ parallax measurements are possible for much more distant lenses (and hence, a larger volume and larger number of events) because of the larger Earth-WFIRST baseline.

Hence, targeted observations for measuring parallaxes should focus on the higher magnification events.

A NIR, ground-based survey simultaneous with WFIRST has the distinct advantage over ToO that it would not require real-time information. In half of all events with planets, the planetary signal will occur after peak, so the opportunity to measure the parallax with ToO will be missed. In addition, a survey would take multiple points throughout the light curve. At the very least, this will improve the precision of the ground-based flux measurement, allowing the limit of 1% precision to be reached for more events. However, the peaks of the events may also be resolved, leading to a measurement of teff and allowing πE,⊥ to be measured for events with

Amax < 30. Such a survey could be carried out with existing facilities such as the

UKIRT or VISTA telescopes or a purpose-built facility. Either way, it would cost a fraction of the WFIRST mission cost and yield substantial scientific benefit.

There is value in carrying out both a survey and a ToO program. The brighter events would be covered by the survey, but ToO with a larger telescope equipped with adaptive optics could reach fainter events. Combining the two approaches would maximize the number of events for which parallax observations can be made while minimizing the cost.

144 Although the method for obtaining parallaxes described here is observationally intensive, the potential scientific impact makes such observations invaluable. The core accretion theory of planet formation predicts that giant planets should be rare around M dwarfs (Laughlin et al. 2004; Ida & Lin 2005), which are typical microlensing hosts. Hence, measuring the masses of the lens stars and planets allows a direct comparison to the theory, which is otherwise very difficult if only mass ratios are known. Furthermore, when the lens mass is measured, solving:

AU 4GM D D r˜ = = L S , (5.16) E π c2 D D E r S − L

yields a measurement of its distance, DL (where the source distance, DS, is assumed to be in the bulge). Such measurements would allow a comparison of the planet populations in the bulge and disk. Given that the stars (and planets) in the bulge formed in a dense region of rapid star formation, one might expect a dearth of giant planets there (Thompson 2013). In addition, although WFIRST will measure the lens system fluxes for many events, whether the light comes from the lens itself or a stellar companion will be unknown. Systematic measurements of the microlens parallax can be used to measure the fraction of events for which the lens light is contaminated by the presence of a companion not involved in the microlensing event.

Finally, if microlens parallax is measured for many events, including ones with lens mass estimates from WFIRST, this will allow the first systematic test of the parallax effect.

145 Although this chapter has been written from the perspective of the WFIRST mission, it is broadly applicable to any microlensing satellite at L2. Thus, if a microlensing survey is included in the Euclid mission (cf. Penny et al. 2012; Beaulieu et al. 2013), it would also benefit from complementary ground-based observations.

In fact, for Euclid a ground-based parallax campaign is even more important for measuring the lens masses because its NIR resolution will make it more difficult to accurately measure the lens system fluxes.

146 Observer Plane (AU) 0.1 0.01 Sun -0.0 E W

-0.1 Sun E D W 0.00 θ ~ ~ -0.2 u rE (∆u)r ~ 0 E u0rE -0.3 cosθ −0.01 -0.4 ~θ b -0.5 a -0.6 −0.02 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 −0.02 −0.01 0.00 0.01

Fig. 5.1.— Left panel: basic geometry of a microlensing event projected onto the observer plane. Right panel: expanded view around the projected positions of Earth and WFIRST(‘E’ and ‘W’, respectively). The x-axis is parallel to the projection of the Sun-Earth-WFIRST line. The dotted line ab shows the lens trajectory. The value −1 of πE,⊥(AU) = sin θ/r˜E = ∆u/D can be derived from the observables ∆u and D, while πE,k = cos θ/r˜E can be measured by WFIRST alone.

147 Sun E W xW D ~ u rE θ W yW

uE u u0 cos0 θ τ c b

a

Fig. 5.2.— Generalized geometry of a microlensing event projected onto the Observer plane. This figure is analogous to Fig. 5.1 except that all quantities are scaled to the size of the Einstein ring,r ˜E. The line abc indicates the trajectory of the lens. The projected positions of Earth and WFIRST are labeled ‘E’ and ‘W’, respectively.

148 Appendix A

Uncertainty in θ⋆ µ, and θE

In the present case, the fractional errors in θ⋆, µ, and θE are all very nearly the same, although for somewhat different reasons. Since the same convergence of errors is likely to occur in many point-lens/finite-source events, we briefly summarize why this is the case. We first write (generally),

θ⋆ = fs/Z p where fs is the source flux as determined from the model, and Z is the remaining set of factors, which generally include the surface brightness of the source, uncertainties due to the calibration of the source flux, and numerical constants. Next, we write

θE θ⋆ √fs 1 θ⋆ 1 µ = = = θE = = fgrand tE t⋆ Z t⋆ ρ Z√fs where f f /ρ and t ρt . We note that for point-lens events with strongly grand ≡ s ⋆ ≡ E detected finite source effects, t⋆ and fgrand are quasi-observables, and so have extremely small errors. For example, if u0 = 0, then 2t⋆ is just the observed source crossing time while 2f [1 + (3π/8 1)Γ] is the observed peak flux. Even for grand − u = 0, these quantities are very strongly constrained, with errors σ = 0.4% and 0 6 fgrand 149 σt⋆ = 0.3% in the present case. Since the errors in fs and Z are independent, the

2 2 1/2 fractional errors in θ⋆, µ, and θE are each equal to [(1/4)(σfs /fs) +(σZ /Z) ] . In

the present case, σfs /fs is given by the fitting code to be 8.5%, while we estimate

σZ /Z to be 7%, and therefore find a net error in all three quantities (θ∗, θE, and µ) of 8%.

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