Astrophysical Applications of Gravitational Microlensing in the Milky

ASTROPHYSICALAPPLICATIONS

OF GRAVITATIONAL MICROLENSING

INTHE MILKY WAY

Przemysław Mróz

Ph.D. thesis written under the supervision of prof. dr hab. Andrzej Udalski Warsaw University Observatory

Warsaw, April 2019

Acknowledgements

First and foremost, I would like to thank my supervisor, Prof. Andrzej Udalski, for the encouragement and advice he has provided throughout my time as his student. I have been extraordinarily lucky to have the supervisor who gave me immeasurable amount of his time, as a researcher and a mentor. This dissertation would not be possible without the sheer amount of work from all members of the OGLE team and their time spent at Cerro Las Campanas. In particular, I would like to thank Prof. Michał Szymanski,´ Prof. Igor Soszynski,´ Łukasz Wyrzykowski, Paweł Pietrukowicz, Szymon Kozłowski, Radek Poleski, and Jan Skowron, who have helped me since my very first steps at the Warsaw University Observatory. I thank all my collegues from the Warsaw Observatory for many helpful discussions and support. I am also grateful to Andrew Gould, Takahiro Sumi, and Yossi Shvartzvald, who shared the photometric data that are a part of this thesis. I thank Calen Henderson and all Pasadena-based microlensers for their hospitality during my stay at Caltech. I also thank my family for their support in my effort to pursue my chosen field of astronomy. I acknowledge financial support from the Polish Ministry of Science and Higher Education (“Diamond Grant” number DI2013/014743), the Foundation for Polish Science (Program START), and the National Science Center, Poland (grant ETIUDA 2018/28/T/ST9/00096). I also received support from the European Research Council grant No. 246678 and the National Science Center, Poland, grant MAESTRO 2014/14/A/ST9/00121 that were awarded to Prof. Andrzej Udalski.

iii

Abstract

The first part of my thesis focuses on searching for and constraining the frequency of rogue planets in the Milky Way. The existence of free-floating planets, which are not gravitationally tethered to any star, is predicted by current planet formation theories. Although rogue planets emit little or no light, they can be detected during gravitational microlensing events. I led the analysis of a large sample of microlensing events that were detected by the OGLE survey during the years 2010-2015. My statistical analysis showed that Jupiter-mass free-floating planets are much less common than previously thought (less than 0.25 objects per star). For the first time, I was able to study the population of the shortest microlensing events and I have found a few events that were likely caused by free-floating (or wide-orbit) Earth- and super-Earth-mass objects, as predicted by planet-formation theories. Recognizing the potential importance for planet formation and evolution of such a huge population of ejected (or very distant) low-mass planets, I developed a new technique to characterize them. My subsequent studies, in collaboration with other microlensing surveys (KMTNet, MOA, Wise), led to the first measurements of the angular Einstein radius of free-floating planet candidates. These measurements enabled me to constrain masses of free-floating planet candidates as they remove a degeneracy between the mass and velocity of the lens. In the second part of my thesis, I used microlensing events detected by OGLE to study the structure of the Milky Way. I created the largest and the most accurate microlensing optical depth and event rate maps of the Galactic bulge. These maps will have numerous applications: constraints on Galaxy models, constraints on the dark matter content in the Milky Way center, measurement of the initial mass function in the Galactic bulge, or planning the future space-based microlensing experiments.

Streszczenie

Współczesne teorie opisuj ˛ace powstawanie pozasłonecznych układów planetarnych przewiduj ˛a istnienie planet swobodnych, wyrzuconych z macierzystych układów i niezwi ˛azanych grawitacyjnie z zadn˙ ˛agwiazd ˛a.Poniewaz˙ te obiekty nie emituj ˛apraktycznie swiatła,´ jedyn ˛ametod ˛apozwalaj ˛ac˛ana ich detekcj˛ejest mikrosoczewkowanie grawitacyjne. W pierwszej cz˛esci´ rozprawy doktorskiej przedstawiłem wyniki moich badan´ dotycz ˛acych poszukiwania i mierzenia cz˛estosci´ wyst˛epowania planet swobodnych w Drodze Mlecznej na podstawie analizy zjawisk mikrosoczewkowania zaobserwowanych przez przegl ˛adnieba OGLE w latach 2010–2015. Moja analiza pokazała, ze˙ planety swobodne o masach Jowisza s ˛aznacznie rzadsze niz˙ wczesniej´ szacowano (na kazd˙ ˛agwiazd˛ew Galaktyce przypada co najwyzej˙ 0,25 masywnych planet swobodnych). Dzi˛ekidanym fotometrycznym zebranym przez przegl ˛adOGLE mogłem równiez˙ zbadac´ zjawiska o najkrótszych skalach czasowych. Udało mi si˛e wykryc´ kilka zjawisk wywołanych prawdopodobnie przez planety swobodne (lub znajduj ˛acesi˛ena szerokich orbitach) o masach Ziemi, zgodnie z przewidywaniami teorii formowania si˛eplanet. W celu lepszego zbadania populacji tych małomasywnych obiektów, zaproponowałem now ˛a metod˛e poszukiwania bardzo krótkich zjawisk mikrosoczewkowania. Dzi˛eki współpracy z innymi przegl ˛adami(KMTNet, MOA, Wise) odkryłem trzy zjawiska wywołane prawdopodobnie przez planety swobodne i po raz pierwszy zmierzyłem ich rozmiar k ˛atowy pierscienia´ Einsteina. Te pomiary daj ˛alepsze ograniczenia na masy soczewkuj ˛acych obiektów, poniewaz˙ umozliwiaj˙ ˛aoszacowanie ich pr˛edkosci.´ W drugiej cz˛esci´ rozprawy wykorzystałem zjawiska mikrosoczewkowania zaobserwowane przez OGLE do badania struktury Drogi Mlecznej. Przygotowałem najwi˛eksze i najdokładniejsze mapy gł˛ebokosci´ optycznej i cz˛estosci´ zjawisk mikrosoczewkowania w kierunku centrum Galaktyki. Te mapy znajd ˛aliczne zastosowania: ograniczenia na modele Drogi Mlecznej, ograniczenia na zawartosci´ ciemnej materii, pomiary funkcji mas gwiazd w Drodze Mlecznej, czy planowanie przyszłych satelitarnych przegl ˛adówmikrosoczewkowych.

vii

Supporting publications

Much of the work in this thesis has been previously presented in following papers: 1. Mróz, P., Udalski, A., Skowron, J., et al. 2017. No large population of free-floating or wide-orbit Jupiter-mass planets, Nature 548, 183. 2. Mróz, P., Ryu, Y.-H., Skowron, J., et al. 2018. A Neptune-mass free-floating planet candidate discovered by microlensing surveys, AJ 155, 121. 3. Mróz, P., Udalski, A., Bennett, D. P., et al. 2019. Two new free-floating or wide-orbit planets from microlensing, A&A 622, 201. Paper 1 contains the work detailed in Chapter 3 of this thesis. Chapter 4 presents the work published in papers 2 and 3. The publication based on findings reported in Chapter 5 is under preparation. The vast majority of the work presented in this thesis was performed by the author, except where explicitly mentioned in the text. In addition, I was the lead author of the following papers on gravitational microlensing that are not a part of this thesis: 1. Mróz, P., Han, C., Udalski, A., et al. 2017. OGLE-2016-BLG-0596Lb: A high-mass planet from a high-magnification pure-survey microlensing event, AJ 153, 143. 2. Mróz, P., Udalski, A., Bond, I. A. et al. 2017. OGLE-2013-BLG-0132Lb and OGLE-2013- BLG-1721Lb: Two Saturn-mass planets discovered around M-dwarfs, AJ 154, 205. 3. Mróz, P. & Poleski, R. 2018. New self-lensing models of the Small Magellanic Cloud: Can gravitational microlensing detect extragalactic exoplanets?, AJ 155, 154. 4. Wyrzykowski, Ł., Mróz, P., Rybicki, K. A., et al. 2019. Full orbital solution of the binary system in the Northern Galactic disk microlensing event Gaia16aye, A&A, submitted (arXiv:1901.07281).

Contents

Acknowledgements ...... iii

Abstract ...... v

Streszczenie ...... vii

Supporting publications ...... ix

1. Introduction ...... 1

1.1. Gravitational microlensing ...... 1 1.2. Lens equation ...... 4 1.3. Point-source points-lens microlensing ...... 6 1.4. Microlensing in the extended-source point-lens regime ...... 8 1.5. Microlens parallax ...... 9 1.6. Binary lens microlensing ...... 12 1.7. Planetary microlensing ...... 16 1.8. Microlensing optical depth and event rate ...... 21 1.9. Astrometric microlensing ...... 22

2. Optical Gravitational Lensing Experiment (OGLE) survey ...... 25

3. Measuring the frequency of free-ﬂoating planets in the Milky Way ...... 31

3.1. Motivation ...... 31 3.2. Data ...... 33 3.3. Selection of events ...... 34 3.4. Detection efﬁciency ...... 46 3.5. Parameter recovery ...... 49 3.6. Modeling event timescale distribution ...... 49 3.7. Mass function ...... 54 3.8. Results and conclusions ...... 57

xi 4. Measuring the angular Einstein radii of free-ﬂoating planet candidates ...... 61

4.1. Motivation ...... 61 4.2. Search for ultra-short timescale events ...... 62 4.3. Data ...... 66 4.4. Modeling ...... 69 4.5. Properties of source stars ...... 72 4.6. Proper motion of source stars ...... 75 4.7. Limits on stellar companions ...... 78 4.8. Discussion and conclusions ...... 82 4.8.1. OGLE-2016-BLG-1540 ...... 82 4.8.2. OGLE-2012-BLG-1323 and OGLE-2017-BLG-0560 ...... 84

5. Microlensing optical depth and event rate from OGLE-IV ...... 87

5.1. Motivation ...... 87 5.2. Data ...... 91 5.3. Selection of events ...... 92 5.4. Star counts ...... 98 5.5. Distribution of the blending parameter ...... 104 5.6. Catalog-level simulations ...... 107 5.7. Image-level simulations ...... 109 5.8. Results and conclusions ...... 111 5.8.1. Timescale distribution ...... 111 5.8.2. Microlensing optical depth and event rate ...... 116 5.8.3. Microlensing events in the direction of the Sagittarius Dwarf Spheroidal Galaxy 127

Summary ...... 129

Appendix A. OGLE-IV ﬁelds ...... 131

Appendix B. Microlensing optical depths and event rates in the OGLE-IV ﬁelds ...... 135

Bibliography ...... 139

xii 1. Introduction

1.1. Gravitational microlensing

The deflection of light by the gravity of massive objects was one of the key predictions of Einstein’s theory of general relativity (Einstein 1916). Einstein found that a light ray from a background source that is passing near the surface of the Sun should be deflected by an angle 4GM 00 (1.1) δθ = 2 = 1.75 , R c which is two times larger than that predicted by the “classical” corpuscular theory of light. The first observation of deflection of light of distant stars near the Sun, and confirmation of Einstein’s predictions, was carried out during the famous expedition by Arthur Eddington and his collaborators during the total solar eclipse of May 29, 1919 (Dyson et al. 1920). Although Eddington’s observations were affected by large error bars and, as some have suggested, confirmation bias, they served as one of the first proofs of general relativity (Will 2006). Einstein (1936) also studied the gravitational deflection of light from a background star (source) in the gravitational field of a foreground star (lens). He found that if the source, lens, and observer are perfectly aligned, the source will appear as a small ring, currently known as the Einstein ring. If the alignment is not perfect, two images of the source will form. Einstein calculated positions and magnifications of images by assuming that the distance to the source is much larger than the lens distance. He found that the source star will appear brighter during lensing events, but “there is no great chance of observing this phenomenon.” Fortunately for us, he was wrong. Gravitational lensing of stars was also studied by Refsdal (1964) and Liebes (1964), who derived general formulae for the magnifications and positions of the images caused by point-mass lenses. They also discussed the potential astrophysical applications of gravitational lensing (e.g., searching for dark objects in the Milky Way). This idea was revived by Paczynski´ (1986b), who proposed monitoring the brightness of millions of stars in the Magellanic Clouds

1 to search for gravitational microlensing caused by hypothetical dark, compact objects in the Milky Way halo, which – as suspected at that time – may have constituted dark matter. The term “gravitational microlensing” was introduced earlier by Paczynski´ (1986a) to describe the effect of lensing by individual stars rather than by the entire galaxies. Paczynski’s´ idea was put into practice by three groups – OGLE (Udalski et al. 1993), MACHO (Alcock et al. 1993), and EROS (Aubourg et al. 1993) – that reported the first detections of microlensing events in 1993, over 25 years ago, beginning the era of modern microlensing surveys. Searches for microlensing events in the direction of the Magellanic Clouds led to the detection of only a few genuine events and ultimately demonstrated that dark matter is unlikely to be composed of compact low-mass objects (Tisserand et al. 2007; Wyrzykowski et al. 2009, 2010, 2011a,b). However, over the years, new applications of microlensing have emerged. Microlensing events are rare and do not repeat. The typical microlensing event rate in the Galactic bulge direction is on the order of 10−5 yr−1, meaning that one would have to wait 105 years to see a given source star being microlensed. Current microlensing ∼ surveys are mostly observing stars located in our Galaxy – namely, the hundreds of millions of stars in the direction of the Galactic center, where the chances of lensing are the largest. This observing strategy enables the detection of about 2,000 microlensing events annually. Moreover, long-term photometric observations of millions of stars by microlensing surveys have revolutionized many other fields of astronomy and enabled, for example, the detection of hundreds of thousands of variable stars, studies of the structure and formation history of the Milky Way and Magellanic Clouds, and the search for extrasolar planets. In this Chapter, we provide a general overview of the basic equations and phenomenology of microlensing and briefly discuss the potential astrophysical applications of microlensing in the Milky Way. Comprehensive reviews of gravitational microlensing have been published by Paczynski´ (1996), Wambsganss (2006), Mao (2008), Gaudi (2012), and Gould (2016b). Throughout this dissertation, we use a modern natural formalism for gravitational microlensing that was developed by Gould (2000a), see Figure 1.1. The central quantity is the angular Einstein radius θE, which is the radius of the image of the source star created when the source and the lens are perfectly aligned. A measurable lensing signal can be observed if the angular distance between the lens and source is smaller than θ . ∼ E

2 r˜E

αˆd

θE

O Dl L S

Figure 1.1. Natural formalism for gravitational microlensing. A lens (L) at a distance Dl from an observer (O) deﬂects light from a source (S) (at a distance Ds) by an angle αˆd. rE = θEDl is the Einstein radius, θE – angular Einstein radius, r˜E – Einstein radius projected onto the observer’s plane.

The angular Einstein radius of the lens depends on its mass M and the relative lens-source

parallax πrel: p θE = κMπrel, (1.2) where κ = 4G/c2au = 8.144 mas M −1 and π = π π , π and π are parallaxes to the rel l − s l s lens and source, respectively. For typical microlensing events in the Milky Way π 0.1 mas, s ∼ π 0.2 mas, and so: l ∼ / 1 2 1/2 M πrel θE = 0.49 mas . (1.3) 0.3 M 0.1 mas The angular Einstein radius is usually measured from light curves of microlensing events thanks to ﬁnite-source effects (Section 1.4), but it can be also measured using astrometric microlensing (Section 1.9) or by resolving lens and source in high-resolution images taken after the event.

The microlens parallax (Section 1.5) is deﬁned as πE = πrel/θE and can be interpreted as the

inverse of the Einstein radius projected onto observer’s plane r˜E: πE = au/r˜E (Gould 2000a), see Figure 1.1. In typical events:

/ 1 2 −1/2 M πrel r˜E = 4.9 au . (1.4) 0.3 M 0.1 mas

3 The microlens parallax is a vector quantity:

πrel µrel πE = , (1.5) θE µrel

and has the direction of µrel – the relative lens-source proper motion. The methods of measuring microlens parallaxes are discussed in Section 1.5. If the angular Einstein radius and microlens parallax are measured from the light curve, it is possible to calculate the mass of the lens: θ2 θ M = E = E , (1.6) κπrel κπE but these quantities are rarely measured together, at least in events caused by single lenses. Usually, the only physical parameter that can be measured for the majority of microlensing events is its Einstein timescale tE, deﬁned as the time it takes the source to move with respect to the lens by one Einstein ring radius:

θE tE = . (1.7) µrel As the typical lens-source proper motion is on the order of µ 5 mas yr−1: rel ∼ / − M 1 2 π 1/2 µ 1 rel rel (1.8) tE = 36 d −1 , 0.3 M 0.1 mas 5 mas yr typical timescales of microlensing events toward the Galactic center are on the order of one month. The distribution of Einstein timescales of a large sample of microlensing events carries

information about the mass function of lenses, provided that distributions of πrel and µrel are known.

1.2. Lens equation

The geometry of microlensing events is illustrated in Figure 1.2. The lens is located at

a distance Dl from the observer, and it deﬂects light from the source at a distance Ds. The

deﬂection angle αˆd was derived by Einstein (1916) based on his theory of general relativity: 4GM αˆ = b. (1.9) d b 2c2 | | The angular positions θ of the images of the source and the angular separation β between the lens and source in the absence of lensing are related by:

β = θ αd(θ), (1.10) −

4 I

αˆd

b S αd θ β O L

Figure 1.2. Geometry of gravitational microlensing events. A lens (L) at a distance Dl from an observer (O) deﬂects light from a source (S) at a distance Ds by an angle αˆd. For a point lens 4GM with a mass M, αˆd = c2b as derived by Einstein (1916). An image of the source (I) is located at an angular separation θ from the lens.

which is known as the lens equation (e.g. Gaudi 2012). Because all angles are very small, we

can write αd = αˆd(1 Dl/Ds). If the lens consists of Nl point masses, each with mass Mi − and angular position θM,i, the lens equation can be rewritten as:

Nl 4G 1 1 X θ θM,i β θ (1.11) = 2 Mi − 2 , − c Dl − Ds θ θM,i i | − | because bi/Dl = θ θM,i. From the mathematical point of view, the lens equation describes − the mapping β θ between the source plane and the lens plane. Multiple images of the source → are usually created. The number of images cannot exceed 5(Nl 1) if Nl > 1 (Rhie 2001, − 2003). If the lens is composed of one object, two images are usually formed; if the lens is a binary, three or five images are formed. Gravitational lensing conserves the surface brightness of the source. The total area of images is larger than the source area, so the combined flux from the images is larger than the flux of

the unlensed source. The magniﬁcation Aj of each image j is given by the inverse of the

determinant of the Jacobian of the lens equation, evaluated at the image position θj:

1 ∂β Aj = , where det J = . (1.12) det J ∂θ θ=θj

5 10 u0 = 0.1 major u0 = 0.2 image u0 = 0.3 u0 = 0.4 u0 = 0.5 5 u0 = 0.8 u0 = 1.0

lens

minor Magniﬁcation image 2

1 1 0 1 − (t t )/t − 0 E Figure 1.3. Microlensing by a point-mass lens. Left: The lens (black dot) is located in the center of the image. Positions of the source are marked with gray circles and images of the source are black closed arcs. Two images of the source are formed – one outside (major image) and one inside (minor image) the Einstein ring (dashed circle). Right: Microlensing magniﬁcation as a function of time for eight selected impact parameters.

There are certain positions of the source, where the Jacobian is zero and the magnification is formally infinite. The set of all source positions where det J = 0 defines closed curves known as caustics. In the case of microlensing by a single lens, the caustic curve degenerates to a point.

1.3. Point-source points-lens microlensing

In the simplest case, when the lens is composed of a single point mass object, the lens equation can be rewritten as: 4GM 1 1 1 (1.13) β = θ 2 . − c Dl − Ds θ If the lens and source are perfectly aligned (β = 0), the images will form a circle with a radius r 4GM 1 1 of θ = 2 = θE, which is, of course, the angular Einstein radius that was c Dl − Ds deﬁned earlier. If we normalize all angles by θE, u = β/θE and y = θ/θE, the lens equation has a simple form: 1 u = y . (1.14) − y Two images are formed at separations 1 2 y± = u √u + 4 . (1.15) 2 ±

6 The positive solution (also known as the major image) is always located outside the Einstein ring ( y > 1) on the same side of the lens as the source. The second image (minor image) is | +| located within the Einstein radius ( y− < 1) and is located on the opposite side of the lens as the | | source (and hence y− < 0), see Figure 1.3. The angular separation of the images is on the order

of θE, which renders their detection challenging with current techniques. Microlensing images were resolved for only one event, TCP J05074264+2447555, using the Very Large Telescope Interferometer GRAVITY (Dong et al. 2019). The magnifications of both images can be calculated using Equation (1.12): 1 u2 + 2 A± = 1 . (1.16) 2 u√u2 + 4 ± The total magnification is the sum of the magnification of both images: u2 + 2 APSPL(u) = , (1.17) u√u2 + 4 where the subscript PSPL stands for “point-source point-lens.” If u , then A 1 and → ∞ + → y u, so the major image becomes coincident with the unlensed position of the source. + → −1 The minor image (A− 0) vanishes. If u 0, the total magnification A u . The → → → magnification is formally infinite if u = 0 and the source is a point. However, in reality, sources have finite size and the point-source approximation is no longer valid (Section 1.4). When the source is located at the Einstein ring, the total magnification A (u = 1) 1.34, PSPL ≈ corresponding to the brightening by 0.32 mag (in the absence of unmagnified blended light). During high-magnification events, the source can be amplified by a factor of 100 or larger. Because microlensing events are typically relatively short, the motion of the source with respect to the lens can be approximated as rectilinear. The lens-source separation varies with time: s 2 t t0 2 u(t) = − + u0, (1.18) tE where u0 is the impact parameter, t0 is the moment of the closest approach, and tE is the Einstein timescale. These three parameters describe the light curve of a microlensing event in the point-source point-lens approximation. The characteristic light curves have symmetric bell shapes and are also known as Paczynski’s´ curves (Figure 1.3).

7 25 ρ = 0.0 ρ = 0.1 ρ = 0.3 20 ρ = 0.5

10 Magniﬁcation

0 2 1 0 1 2 − − Time ((t t )/t ) − 0 E Figure 1.4. Finite source effects. The blue curve is the standard Paczynski´ light curve for a point source. ρ is the ratio of the angular radius of the source to the angular Einstein radius of the lens. Light curves correspond to u0 = 0.04, Γ = 0.6, and Λ = 0.0.

1.4. Microlensing in the extended-source point-lens regime

The point-source point-lens approximation breaks down in a high-magniﬁcation regime

(u 0) or when the angular radius of the source, θ∗, is similar to the angular Einstein radius θ . ≈ E The ratio of these quantities is known as a normalized source radius ρ = θ∗/θE. Finite source effects can be only detected when ρ u (i.e., when the limb of the source passes over/near the ≈ 0 position of the lens and each point on the source surface is magniﬁed by a different amount). A probability of such a chance alignment is on the order of ρ. In typical microlensing events,

−3 θ∗ (1 µas) and θ (1 mas). Therefore, ρ 10 and the chances of observing ﬁnite ∼ O E ∼ O ∼ source effects are slim. However, the angular Einstein radii of planetary-mass objects are much smaller ( (1 µas)) and they are comparable to the angular radii of giant source stars in the O Galactic bulge (θ∗ = 6 µas (R∗/10 R ) for a source located 8 kpc from the Sun), meaning that in such cases the ﬁnite source effects cannot be neglected.

When the source is extended, the magniﬁcation can be calculated by integrating APSPL over its area: RR APSPLdS A(u, ρ) =source . (1.19) RR dS source

8 Gould (1994) derived the formula for A(u, ρ) in the high-magnification regime (when u 0 ≈ and A (u) 1/u). The exact, although cumbersome, expression for A(u, ρ) for a uniform PSPL ≈ source was derived by Witt & Mao (1994) as a sum of elliptic integrals. For the purpose of this work, the finite-source magnifications are calculated by the direct integration of formulae derived by Lee et al. (2009). We assume that the limb-darkening profile of the source can be described using the formula: 3 5 S (ϕ) /S¯ = 1 Γ 1 cos ϕ Λ 1 √cos ϕ , (1.20) − − 2 − − 4 where Γ and Λ are (wavelength-dependent) parameters and ϕ is the angle between the normal to the stellar surface and the line of sight. We use the QUADPACK library (Piessens et al. 1983) for the numerical integration of Equation (1.19). Finite source effects distort the shape of the light curve (Figure 1.4); the peak magnification can be smaller or larger than for a point source, depending on the impact parameter u0 and radius of the source ρ. In particular, when the source is much larger than the Einstein ring (ρ 1), only a small fraction of its area is magnified and the peak magnification is suppressed: 2 A 1 + (1.21) peak ≈ ρ2 (Witt & Mao 1994; Gould & Gaucherel 1997).

1.5. Microlens parallax

The microlens parallax can be directly measured through three effects: the annual, terrestrial, and space-based microlens parallax effect. The subtle deviations from the standard microlensing light curve due to parallax can be detected in long-timescale events (Figure 1.5), as an Earth-based observer moves along the orbit (Gould 1992) and the orbital acceleration of Earth displaces the position of the observer relative to rectilinear lens-source motion. The annual parallax measurements are difﬁcult to make because the effect is usually small and can be mistaken with other high-order effects (e.g., the orbital motion of the lens) and/or systematics in the data. Moreover, the effect can be measured only if the duration of the event is long enough so that Earth subtends a large angle on its orbit. Thus, the annual microlens parallax measurements are biased toward long-timescale events and events caused by nearby lenses.

9 The microlens parallax is a vector quantity πE. It is customary to express its components in the north and east directions: πE = (πE,N , πE,E) (Gould 2004). The parallax is usually measured in the “geocentric frame” in which all parameters are measured in the instantaneous frame that is at rest with respect to the Earth at a specifically adopted time t0,par (t0,par is not a fit parameter) (An et al. 2002; Gould 2004). It is usually difficult to uniquely measure the annual microlens parallax from the light curve alone. Microlens parallaxes are subject to four degeneracies, which are described in detail by

Skowron et al. (2011). The most common is the ecliptic degeneracy (u , π ,⊥) (u , π ,⊥) 0 E ↔ − 0 E (Jiang et al. 2004; Poindexter et al. 2005), where πE,⊥ is the parallax component perpendicular to the apparent acceleration of the Sun (projected on the sky) in the Earth frame at t0,par. Events located near the ecliptic (i.e., all events in the Galactic bulge) may suffer from this degeneracy. The microlens parallax can also be measured whenever observations are carried out from two or more locations roughly separated by a large fraction of the Einstein ring projected onto

−1 the observer plane r˜E = au πE . In this case, the lensing light curve simultaneously observed from two locations can appear to be different (Figure 1.5) because the lens-source configurations seen from two observatories are different (Refsdal 1966). Refsdal (1966) proposed measuring microlens parallaxes using a satellite in solar orbit, as typically r˜ 1 10 au. The first such measurement was carried out by Dong et al. (2007), E ∼ − who measured the microlens parallax of the event OGLE-2005-SMC-001 using simultaneous ground-based and Spitzer satellite observations. A large program to carry out microlensing parallax measurements toward the Galactic bulge using the Spitzer satellite was started in 2014 (Yee et al. 2015), with the final observations scheduled for the summer of 2019. The primary goal of the campaign is the measurement of the frequency of extrasolar planets in different environments of the Galactic disk and bulge (Calchi Novati et al. 2015). At the moment of writing, nearly a thousand microlensing events have been monitored using simultaneous space- and ground-based observations, and about a dozen of microlensing planets were characterized (e.g., Udalski et al. 2015b; Street et al. 2016). The space-based parallax has also been measured using K2 (Zhu et al. 2017a,b) and Gaia (Wyrzykowski et al. 2019) satellite data. If the lens is located close enough so that the projected Einstein radius is small, the microlens parallax signal can be detected from two ground-based observatories (Gould 1997a; Gould & Yee 2013). This terrestrial parallax effect was detected in two events: OGLE-2007-BLG-224 (Gould et al. 2009) and OGLE-2008-BLG-279 (Yee et al. 2009).

10 16.0 16.1 15.6 A B C D C 16.2 16.2 16.0 16.3 +6837 16.4 16.2 D 16.4 16.3 16.6 16.8 +6844 6820 6830 6840 6850 0 1 2 16.8 I (OGLE)[mag] I 17.0

17.2 OGLE I Spitzer 3.6 µm 17.4

(mag) 6700 6750 6800 6850 6900 HJD-2450000

Figure 1.5. Microlens parallax. Upper panel: Light curve of the event OGLE3-ULENS-PAR-02 exhibiting strong annual parallax effect. Green curve shows the standard point-lens point-source model, red – model with parallax. From Wyrzykowski et al. (2016). Lower panel: Light curve of the planetary event OGLE-2014-BLG-0124 as observed from Earth by OGLE (black) and by Spitzer (red), which was located 1 au from Earth in projection at the time of the observations. From Udalski et al. (2015b). ∼

11 1.6. Binary lens microlensing

The lens equation (Section 1.2) takes a simple form if the lens is composed of one point-mass object. Things get complicated when the lens consists of a binary or multiple system. Equation (1.11) can be simpliﬁed if we deﬁne dimensionless vectors describing the

position of the source u = β/θE and images y = θ/θE, where θE is the angular Einstein radius

PNl corresponding to the total mass of the lens M = i Mi:

Nl X y ym,i u = y mi − , (1.22) − y y 2 i | − m,i|

where ym,i is the position of lens mass i and mi = Mi/M. Witt (1990) demonstrated that this equation can be expressed in the complex notation. The two-dimensional position of the source

u and images y can be written with complex quantities (ζ = u1 + iu2 and z = y1 + iy2). Thus:

Nl Nl X z zi X mi ζ = z mi − = z , (1.23) − (z zi)(¯z z¯i) − z¯ z¯i i − − i − where zi is the complex position of the mass i. By taking the complex conjugate of this equation, one can obtain an expression for z¯. Substituting this back into Equation (1.23) leads to a

2 complex polynomial of degree Nl + 1. Not all roots of this polynomial are the roots of the lens equation. As demonstrated by Rhie (2001, 2003), the maximum number of images is

5(Nl 1) if Nl > 1. Finding positions of images caused by a binary lens requires solving − a fifth-order complex polynomial. Special numerical techniques were developed to efficiently handle this task (Skowron & Gould 2012). The magnification of each image can be evaluated using Equation (1.12):

2 Nl 1 ∂ζ ∂ζ¯ X mi J (1.24) Aj = , where det = 1 = 1 2 . det J − ∂z¯ ∂ζ¯ − (¯z z¯i) z=zj i − The total magnification is the sum of magnifications of individual images. There are some points satisfying det J = 0, which indicates infinite magnification. These points form closed curves in the source plane known as caustics. The calculation of magnification for extended sources is a time-consuming task. Therefore, dedicated numerical techniques were developed. The point-source approximation may be valid in regions of low magnification, far from the caustics. In regions of high magnification, the semi-analytic quadrupole (Pejcha & Heyrovský 2009) and hexadecapole (Gould 2008; Cassan 2017) approximations can be used. Lensing magnification over the caustic may be computed

12 2 4 q = 0.5 s = 2.0 1 3 ) A E

θ 2 ( 0 y 5 log . β 2

1 1 − u0 = 0.10 α = 0.94 u0 = 1.00 α = 2.04 u0 = 0.05 α = 0.13 0 2 − − − 2 1 0 1 2 4 3 2 1 0 1 2 3 4 − − β (θ ) − − − −(t t )/t x E − 0 E 2 4 q = 0.2 s = 0.7 1 3 ) A E

θ 2 ( 0 y 5 log . β 2

1 1 − u0 = 0.10 α = 0.94 u = 0.50 α = 0.31 0 − u0 = 0.09 α = 0.13 0 2 − − − 2 1 0 1 2 4 3 2 1 0 1 2 3 4 − − β (θ ) − − − −(t t )/t x E − 0 E 2 3 q = 0.8 s = 1.1 1 2 ) A E θ ( 0 y 5 log . β 2 1

1 − u0 = 0.10 α = 0.94 u = 0.50 α = 0.31 0 − 0 2 − 2 1 0 1 2 4 3 2 1 0 1 2 3 − − β (θ ) − − − −(t t )/t x E − 0 E Figure 1.6. Examples of light curves of binary microlensing events in three topologies: wide (upper panels), close (middle), and intermediate (bottom). Corresponding source trajectories are marked with color lines and caustic curves are black. Green dots mark the positions of lens components (the more massive component has βx < 0).

13 11.6 Gaia 12.0 Bialkow I APT2 I 12.4 LT i DEMONEXT I 12.8 Swarthmore I UBT60 I 13.2 ASAS-SN V

Magnitude 13.6

14.0

14.4

14.8 0.20 −0.15 −0.10 −0.05 −0.00 0.05

Residual 0.10 0.15 7200 7300 7400 7500 7600 7700 7800 7900 8000 HJD - 2450000

11.6 Gaia 12.0 ground-based follow-up 12.4

12.8

13.2

13.6 Magnitude 14.0

14.4

14.8 0.20 −0.15 −0.10 −0.05 −0.00 0.05 0.10 Residual 0.15 7600 7605 7610 7615 7620 7649 7651 7713 7715 HJD - 2450000 HJD - 2450000 HJD - 2450000

Figure 1.7. Top: Light curve of the binary microlensing event Gaia16aye. The complex light curve is caused by the orbital motion of the lens. I found that Gaia16aye was caused by two M-dwarfs with masses 0.57 M and 0.36 M orbiting with a period of about 2.88 years, and I was able to precisely measure all orbital elements of the binary (Wyrzykowski et al. 2019). Bottom: As the Gaia satellite is 0.01 au from Earth, the Gaia light curve (black) differs from that seen from Earth (gray).

14 using the inverse ray-shooting method (Schneider & Weiss 1986; Kayser et al. 1986) or contour integration (Schramm & Kayser 1987; Dominik 1995, 1998; Bozza 2010). Description of binary lens events requires three additional parameters: q (the mass ratio of the two components), s (their projected separation in units of the Einstein radius), and α (the angle between the direction of lens-source relative motion with respect to the binary axis). Depending on the mass ratio, q, and projected separation, s, one to three caustic curves may form (Erdl & Schneider 1993; Dominik 1999). This leads to a variety of possible conﬁgurations

1/3 3/2 1/2 and light curve shapes (see Figure 1.6). If s > sw = (1 + q ) /(1 + q) (“wide topology”), two four-cusp caustics are formed near the positions of lens components. When s , these caustics degenerate into points, which correspond to lensing by two independent → ∞ −1/2 point-mass objects. If s < sc = sw (“close topology”), three caustics are formed: a central diamond-shaped caustic with four cusps and two triangular caustics with three cusps. The latter two are located symmetrically to the binary (βx) axis. When sc < s < sw (“intermediate topology”), only one large six-cusp caustic (known as a resonant caustic) is created. Additional parameters are required to describe the orbital motion of the lens, which, in the simplest scenario, can be approximated as linear changes of the separation s(t) = s +s ˙(t 0 − t , ), and the angle α(t) = α +α ˙ (t t , ), t , can be any arbitrary moment of time and 0 kep 0 − 0 kep 0 kep is not a ﬁt parameter (Albrow et al. 2000). This approximation works well for the majority of binary microlensing events because the orbital period of the lens is usually much longer than the duration of a typical event. There are a few known cases (Skowron et al. 2011; Shin et al. 2012; Wyrzykowski et al. 2019) for which the full orbital solution of the lens was found (Figure 1.7). An in-depth discussion of the parameterization of binary microlensing events is discussed by Skowron et al. (2011). The most important astrophysical application of binary lens microlensing is the search for extrasolar planets (Section 1.7). Microlensing has also been used to study brown dwarf binaries (e.g., Choi et al. 2013; Han et al. 2013a; Jung et al. 2018). This is important because knowledge of the binary fraction can provide important constraints on rival theories of brown dwarf formation (Chabrier 2001; Kroupa et al. 2013). On the high-mass end, microlensing can be used to detect neutron stars and black holes in binaries. For example, Shvartzvald et al.

(2015) found a massive stellar remnant (> 1.35 M ) orbiting a main-sequence star.

15 1.7. Planetary microlensing

One of the most important astrophysical applications of gravitational microlensing is the search for (bound) extrasolar planets. We discuss this topic relatively brieﬂy, as in this thesis we focus mostly on other applications of microlensing. Excellent reviews of microlensing searches for exoplanets were written by Gaudi (2012), Gould (2016b), and Udalski (2018). The idea to use gravitational microlensing to search for extrasolar planets was ﬁrst proposed by Mao & Paczynski´ (1991) and Gould & Loeb (1992). The presence of a planet can be revealed by a short-duration anomaly on top of a smooth single-lens light curve (Figure 1.8).

Timescales of planetary anomalies scale with the mass ratio q as ∆tanomaly/tE √q. For ∼ typical microlensing events (t 20 d), anomalies caused by Jupiter-mass planets (q 10−3) E ∼ ∼ should last about a day and those for Earth-mass planets (q 3 10−6) should last about ∼ × an hour. Because microlensing events are unpredictable (but see Section 1.9) and planetary anomalies are also unpredictable, putting this idea into practice seemed difﬁcult. Gravitational microlensing is most sensitive to planets located near the Einstein ring of a lens because the presence of a planet can be revealed by its perturbation on the images of the source star. Fortunately, typical Einstein radii of stars in the Milky Way are on the order of

/ 1 2 1/2 M πrel Dl rE = θEDl = 2.6 au , (1.25) 0.5 M 0.1 mas 4 kpc which happens to be near or beyond the snow line of the majority of planetary systems. This is a location in the proto-planetary disk where water ice may condense and where gas giant planets are believed to be formed (Mizuno 1980; Pollack et al. 1996). If light curves of microlensing events are sufficiently well sampled, about 4% of them exhibit signatures of planets (Shvartzvald et al. 2016). The first generation microlensing searches for planets involved a complex process. Survey telescopes, which used large-field-of-view detectors, were regularly monitoring the Galactic bulge with cadence of at most a few observations per night. They analyzed their data in real-time and searched for and alerted ongoing microlensing events. Subsequently, follow-up networks used many telescopes in different locations to fully cover light curves of the most promising events and to search for planetary anomalies. Nearly all microlensing alerts were issued by the Optical Gravitational Lensing Experiment (OGLE) (Chapter 2) and Microlensing Observations in Astrophysics (MOA) collaborations (Section 4.3). Follow-up groups include the Probing Lensing Anomalies NETwork (PLANET; Albrow et al. 1998), RoboNet (Tsapras

16 16.5

OGLE I 16.6 OGLE V 16.8 17.0 MOA 17.0 17.2 17.4 17.6 17.5 17.8 18.0 18.2 6370 6372 6374 6376 6378 6380 18.0 Magnitude

18.5

19.0 0.4 −0.3 −0.2 −0.1 −0.0 0.1 0.2 Residual 0.3 0.4 6320 6340 6360 6380 6400 6420 6440 6460 HJD - 2450000

17.0

OGLE I 17.0 17.5 MOA 17.5

18.0 18.0 18.5

18.5 19.0

19.5 19.0 6530 6531 6532 6533 6534 6535 6536

Magnitude 19.5

20.0

20.5

21.0 0.4 − 0.2 − 0.0 0.2 Residual 0.4 6525 6530 6535 6540 6545 6550 6555 HJD - 2450000

Figure 1.8. Light curves of two planetary microlensing events: OGLE-2013-BLG-0132 (upper panel) and OGLE-2013-BLG-1721 (lower panel). Anomalies in both events are caused by Saturn-mass planets (Mróz et al. 2017b).

17 104

103

102 S

U N 101 Mass (Earth masses)

0 E 10 V Transits Radial velocity Microlensing 10 1 M 10 2 10 1 100 101 Semimajor axis (au)

Figure 1.9. Known exoplanets (data were taken from the NASA Exoplanet Archive). Solar System planets are marked with black letters. Gravitational microlensing is sensitive to low-mass planets at large separations (i.e., a region of the parameter space that is inaccessible for other planet-detection techniques).

103

102

101 Number of planets / Planet frequency (arb. u.)

100 10 6 10 5 10 4 10 3 10 2 10 1 100 Mass ratio

Figure 1.10. The mass-ratio distribution of planets detected by Suzuki et al. (2016), who found a break in the power-law mass-ratio distribution at about q 2 10−4. The black histogram shows the observed planets, purple – corrected for detection≈ efﬁciency.× Adapted from Suzuki et al. (2016).

18 et al. 2009), Microlensing Network for the Detection of Small Terrestrial Planets (MiNDSTEp; Dominik et al. 2010), and the Microlensing Follow Up Network (µFUN; Gould et al. 2010). The next (“second”) generation of microlensing surveys for exoplanets was made possible thanks to technical upgrades and the installation of new, larger detectors. This enabled nearly continuous monitoring of the Galactic bulge with cadences of 15–30 minutes, which is sufﬁcient to detect and characterize planetary anomalies without the need for follow-up observations. The four survey groups include OGLE, MOA, Wise, and Korean Microlensing Telescope Network (KMTNet). All surveys are described in Chapter 2 and Section 4.3. Gravitational microlensing is sensitive to low-mass planets at large separations (a few au) – a region of the parameter space that is inaccessible to other planet-detection techniques (Figure 1.9). Most exoplanets have been detected using radial velocity measurements or transits, and these techniques are sensitive mostly to planets located near their host stars. Moreover, the microlensing signal does not depend on the host brightness, and the majority of microlensing planets have been discovered around low-mass faint M-dwarfs, which are the most common type of stars. The other advantage of microlensing is that it can probe planets across the entire Galaxy, whereas other planet-detection techniques are sensitive only to planets in the solar “neighborhood” (i.e., within 1 kpc of the Sun). Thus, using gravitational microlensing, one ∼ can study the planet populations both in the Galactic disk and bulge, which are environments of different metallicity, star formation history, etc. This is the primary scientiﬁc driver of the Spitzer microlensing campaign. It was also proposed to use gravitational microlensing to search for extragalactic planets in the Andromeda Galaxy (Covone et al. 2000; Baltz & Gondolo 2001; Ingrosso et al. 2009) and the Small Magellanic Cloud (Mróz & Poleski 2018). The fact that gravitational microlensing events do not repeat is the primary drawback of the method because it prevents a detailed characterization of many planets. However, microlensing is the best-suited method to conduct an unbiased systematic survey of planets and to analyze their population as a whole. The anticipated Wide-Field Infrared Survey Telescope (WFIRST), which will conduct a space-based third generation microlensing survey, will complete the statistical census of exoplanets that was started with radial velocity and transit surveys (National Academies of Sciences, Engineering, and Medicine 2018). WFIRST is designed to measure the frequency of planets with masses as low as that of Mars, but the current experiments have already provided some constraints on the frequency and mass function of exoplanets beyond the snow line.

19 Sumi et al. (2010) analyzed ten microlensing planets (the entire sample that was available at that time) and derived a power-law mass-ratio distribution dN /d log q qn with n = pl ∝ 0.68 0.2 (q is the planet-to-host mass ratio). Cassan et al. (2012) found that, on average, − ± +0.72 every star has 1.6 planets in a range of 0.5 10 au and 5 M⊕ to 10 M . Shvartzvald et al. −0.89 − Jup (2016) analyzed 224 microlensing events (of which eight had planetary anomalies), which were

+34 observed by three microlensing surveys (OGLE, MOA, and Wise). They found that 55−22% of microlensed stars host a snowline planet and that the mass-ratio distribution can be described by a single power-law with n = 0.50 0.17. − ± In recent work, Suzuki et al. (2016) analyzed 23 planetary events (out of 1474 microlensing events) that were found and characterized by the MOA survey data from 2007 through 2012. They argued for a break in the power-law mass-ratio distribution at about q 2 10−4 ≈ × (Figure 1.10). They measured n = 0.93 0.13 above the break and found a sign reversal − ± in the power-law index (dN /d log q qp) with p = 0.6+0.5 below the break. A similar pl ∝ −0.4 conclusion was reached by Udalski et al. (2018), who found p = 0.73+0.42 at q < 2 10−4 −0.34 × based on analysis of eight low-mass-ratio planets. The planetary mass function of Suzuki et al. (2016) cannot be explained by theoretical population synthesis models (Suzuki et al. 2018b, and references therein). However, the location of the break and the shape of the planetary mass function below it are still poorly constrained by the current data. Microlensing can also be used to detect multi-planet systems. At the moment of writing, three such systems have been published. The ﬁrst double planet system discovered with gravitational microlensing is OGLE-2006-BLG-109. The system consists of two giant planets with masses of 0.73 M and 0.27 M orbiting a 0.5 M host. This system is thought to ∼ Jup ∼ Jup resemble our Solar System in that the mass ratio, separation ratio, and equilibrium temperatures of the planets are similar to those of Jupiter and Saturn (Gaudi et al. 2008; Bennett et al. 2010). OGLE-2012-BLG-0026 consists of two planets with masses of 0.15 M and ∼ Jup 0.86 M orbiting a G-type star (Han et al. 2013b; Beaulieu et al. 2016). Two possible ∼ Jup planets ( 0.18 M and 0.27 M ) were found also around OGLE-2014-BLG-1722 ∼ Jup ∼ Jup (Suzuki et al. 2018a), but these detections are less certain. Suzuki et al. (2018a) estimated that 6 2% of microlensed stars host multiple cold gas giant planets. Moreover, gravitational ± microlensing has been used to detect planets in binary star systems, either orbiting one of the components (e.g., OGLE-2013-BLG-0341, Gould et al. 2014; OGLE-2008-BLG-092, Poleski et al. 2014b) or circumbinary (OGLE-2007-BLG-349, Bennett et al. 2016).

20 1.8. Microlensing optical depth and event rate

The microlensing optical depth toward a given source describes the probability that the

source falls into the Einstein radius (rE = θEDl) of some lensing foreground object. If the source is located exactly at the Einstein ring of a point lens, the source magniﬁcation of 1.34 can be easily measured. When the separation is larger, the ampliﬁcation decreases rapidly.

2 Thus, πrE is a natural cross-section for microlensing (Vietri & Ostriker 1983; Nityananda & Ostriker 1984; Paczynski´ 1991). The optical depth can be expressed as: Z Ds 4πG Z Ds D (D D ) 2 l s l (1.26) τ(Ds) = n(Dl)(πrE)dDl = 2 ρ(Dl) − dDl, 0 c 0 Ds where n(D) and ρ(D) are the number density and density of lenses along the line of sight. The microlensing optical depth can be also interpreted as the fraction of sky covered by the angular Einstein rings of all lenses. In theory, the optical depth depends only on the mass distribution and is independent of other parameters (mass function, kinematics). However, the observed optical depth is averaged over all detectable stars: 1 Z ∞ τ = τ(Ds)dn(Ds), (1.27) Ns 0 where dn(Ds) is the number of detectable sources in the range [Ds,Ds + dDs] (Kiraga & R ∞ Paczynski´ 1994) and Ns = 0 dn(Ds). The ﬁrst theoretical estimates predicted that the microlensing optical depth in the Galactic bulge direction should be on the order of τ 4 10−7 (Paczynski´ 1991), but the ﬁrst ∼ × observations (e.g., Udalski et al. 1994c) showed that the real value is an order of magnitude larger. We defer a detailed discussion of previous optical depth measurements to Chapter 5. The microlensing event rate Γ (i.e., the number of microlensing events per unit time for a given source) is the fraction of the sky that is covered by the solid angle 2θ µ in a unit E × rel time, integrated over all lenses along the line of sight. The differential event rate toward a given source is: d4Γ µ (1.28) 2 = 2rEvreln(Dl)f( rel)g(M), dDldMd µrel

where M is the lens mass, rE = DlθE is its Einstein radius, n(D) is the local number density

of lenses, v = Dl µrel is the lens-source relative velocity, f(µrel) is the two-dimensional rel | | probability density for a given lens-source relative proper motion (µrel), and g(M) is the mass function of lenses (Batista et al. 2011). Contrary to the optical depth, the event rate explicitly depends on the mass function of lenses and their kinematics. It can be demonstrated that the

21 mean Einstein timescale of microlensing events is: 2 τ t = . (1.29) h Ei π Γ Because typical Einstein timescales toward the Galactic bulge are t 20 days, the event rate E ∼ Γ 10−5 yr−1(τ/10−6). ∼

1.9. Astrometric microlensing

Images of the source star that are created during gravitational microlensing events are usually unresolved (but Dong et al. 2019) and we can only measure their combined magnification. However, thanks to precise astrometric measurements, it is possible to observe the shift of the light centroid of created images. In the simplest case of microlensing by a point-mass lens, positions and magnifications of images are given by Equations (1.15) and (1.16) and centroid shift of the source for a dark lens is given by: u δu = θ (1.30) u2 + 2 E (Hog et al. 1995; Miyamoto & Yoshii 1995; Walker 1995; Dominik & Sahu 2000). If u √2, the centroid shift scales as δ(u) θ /u and it falls much more slowly than the photometric ≈ E magnification (A(u) 1 + 2/u4 for u 1) for large impact parameters. → A measurement of the astrometric shift due to microlensing allows measuring the angular

Einstein radius θE and, in turn, the mass of the lens: θ2 M = E (1.31) κπrel provided that parallaxes of the lens and source are known. Currently, two mass measurements employing astrometric microlensing have been published. Sahu et al. (2017) used the Hubble Space Telescope (HST) observations to measure the mass of a nearby single white dwarf Stein 2051 B with accuracy of 8%. Zurlo et al. (2018) (using the combination of HST and Very Large Telescope data) detected astrometric shift caused by microlensing by Proxima Centauri and estimated its mass with an error of about 40%. More measurements are expected in the future, as several groups try ﬁnding stellar remnants using astrometric microlensing using either HST or Keck adaptive optics observations. For example, Lu et al. (2016) demonstrated that ground-based adaptive optics observations can achieve astrometric precision of 0.15 mas, which is sufﬁcient to detect the effect. It is also expected that precise astrometric measurements from

22 the Gaia satellite will yield a few detections of astrometric centroid shift (e.g., Rybicki et al. 2018). Precise measurements of proper motions and parallaxes by the Gaia satellite also allow one to predict future microlensing events, both photometric and astrometric. For example, Bramich & Nielsen (2018) published an almanac of 2509 predicted microlensing events until the end of the 21st century. Other works on predicting microlensing events include McGill et al. (2018), Bramich (2018), Klüter et al. (2018), Mustill et al. (2018), and Nielsen & Bramich (2018).

2. Optical Gravitational Lensing Experiment (OGLE) survey

The majority of photometric data analyzed in this dissertation were collected as part of the Optical Gravitational Lensing Experiment (OGLE) sky survey, which is one of the largest long-term photometric sky surveys worldwide. The OGLE survey was founded in the early 1990s by astronomers from the Warsaw University Observatory, who put into practice the early idea of Prof. Bohdan Paczynski´ to regularly monitor brightness of millions of stars to search for sudden brightenings caused by gravitational microlensing by hypothetical dark massive objects in the Milky Way halo (Paczynski´ 1986b). The history of OGLE operations is divided into four distinct phases, which are characterized by major instrumental upgrades (see Table 2.1). The first phase (OGLE-I) was conducted during the years 1992–1995 with the 1.0-m Swope telescope located at the Las Campanas Observatory, Chile (Udalski et al. 1992). These pioneering observations brought about discoveries of the first microlensing event toward the Galactic bulge (Udalski et al. 1993), the first binary microlensing event (Udalski et al. 1994b), the first measurement of the microlensing optical depth toward the Galactic center (Udalski et al. 1994c), and the first implementation of the alert system (Udalski et al. 1994a), among others. Thanks to these encouraging first results, it was possible to build a new 1.3-m Warsaw Telescope dedicated to the OGLE project (Figure 2.1). The new telescope was erected in the Las Campanas Observatory in Chile, in one of the best astronomical observing sites in the world. The first observations were taken in 1996 and the second phase of the project (OGLE-II) was conducted during the years 1997–2000 (Udalski et al. 1997). The OGLE-II camera consisted of only one 2048 2048 CCD detector, but thanks to using the drift-scan technique, the survey × monitored over 10 square degrees toward the Galactic bulge. From 40 to 70 microlensing events per year were discovered, which allowed the construction of the first microlensing optical depth maps of the Galactic bulge (Sumi et al. 2006).

25 Phase Duration Telescope Detector Area OGLE-I 1992–1995 Swope (1 m) 2048 2048, 0.4400 per pixel 4 OGLE-II 1997–2000 Warsaw (1.3 m) 2048 × 2048, 0.41700 per pixel 10 OGLE-III 2001–2009 Warsaw (1.3 m) 8 2048× 4096, 0.2600 per pixel 31 OGLE-IV 2010–present Warsaw (1.3 m) 32× 2048× 4102, 0.2600 per pixel 160 × × Table 2.1. Summary of OGLE phases (duration of each phase, telescope, detector, monitored area (in square degrees) in the direction of the Galactic bulge).

The next major technical upgrade occurred in 2001, starting the OGLE-III phase. A new mosaic, eight detector camera was built and installed at the Warsaw Telescope. The new detector covered the field of view of 0.34 deg2 with a pixel scale of 0.2600 per pixel (Udalski 2003). The new instrumental setup allowed OGLE to observe an area of about 31 square degrees around the Galactic center and to discover a few hundred microlensing events annually. The boosted stream of microlensing alerts (Udalski 2003) led to the first detections of extrasolar planets using gravitational microlensing (e.g., Bond et al. 2004; Udalski et al. 2005; Beaulieu et al. 2006; Gould et al. 2006; Gaudi et al. 2008; Dong et al. 2009; Sumi et al. 2010). A statistical analysis of microlensing data showed that extrasolar planets are very common around stars (e.g., Cassan et al. 2012; Tsapras et al. 2016). Besides the characterization of individual events, the large number of detections allowed statistical studies of the distribution of timescales of microlensing events across the monitored area (Wyrzykowski et al. 2015) and the search for stellar remnants (Wyrzykowski et al. 2016). The OGLE-III operations ended in 2009. The observing capabilities of the OGLE-III camera, although an order of magnitude better than those of its predecessor, did not fully utilize the technical capabilities of the Warsaw Telescope. A new mosaic camera, which filled the entire field of view of the telescope, was installed in 2009 and the OGLE-IV phase commenced soon after, with the first science operations starting in March 2010 (Udalski et al. 2015a). The OGLE-IV camera consists of 32 2048 4102 pixel CCD detectors and covers a field of view of × 1.4 square degrees with a scale of 0.2600 per pixel. About 2000 microlensing events are announced annually by the OGLE Early Warning System (Udalski 2003) during the OGLE-IV phase. Thanks to large scale observations, a dozen of new microlensing planets have been discovered every year (e.g., Mróz et al. 2017c; Udalski 2018). While it is not possible to mention all discoveries, the OGLE-IV survey led to the detection of an Earth-mass planet around an M-dwarf (Bond et al. 2017; Shvartzvald et al. 2017), massive planets around low-mass stars (e.g., Poleski et al. 2014a; Skowron et al.

26 Figure 2.1. Dome and Control Room of the Warsaw Telescope, Las Campanas Observatory, Chile (July 2013). The Swope Telescope, which hosted the ﬁrst phase of the project (OGLE-I), is located in the upper right corner of the image.

2015; Mróz et al. 2017b), a two planet system orbiting a low-mass star (Han et al. 2013b), and planets in binary systems (Gould et al. 2014; Poleski et al. 2014b). A large sample of planetary events enables more detailed studies of planet frequency and planetary mass function (e.g., Shvartzvald et al. 2016; Udalski et al. 2018). Another important contribution of OGLE-IV is the Spitzer microlensing campaign; the majority of targets observed during six Spitzer campaigns (2014–2019) were discovered by OGLE. Simultaneous space-based observations enable the measurement of the microlensing parallax, and hence the mass and distance of a lens. The ultimate goal of the campaign is to measure the frequency of planets in different environments of the Galactic disk and bulge. At the moment of writing, about a dozen of microlensing planets were characterized using simultaneous space- and ground-based observations (e.g., Udalski et al. 2015b; Street et al. 2016). Recent scientiﬁc highlights of the OGLE-IV microlensing survey also include the detection of ultra-short microlensing events that can be attributed to free-ﬂoating or wide-orbit planets (Mróz et al. 2017a, 2018, 2019, see also Chapters 3 and 4). All OGLE observations analyzed in this dissertation were collected during the fourth phase of the project during the years 2010–2017. I have been a member of the OGLE team since 2013 and I have taken part in the collection of photometric data analyzed hereinafter. In total, I spent over 250 nights at the Warsaw Telescope.

27 20◦ 10◦ 0◦ 10◦ 20◦ − − 477 478 483 479 488 484 480 791 489 485 481 857 798 792 493 490 486 482 859 854 793 494 868 858 805 799 491 487 860 855 811 806 800 794 495 492 879 869 861 10◦ 890 880 852 856 818 812 807 801 795 496 870 862 853 808 802 796 901 891 881 871 819 813 911 902 863 820 814 809 803 797 892 882 872 825 912 903 893 864 826 821 815 810 804 969 883 873 865 829 822 816 964 968 913 904 894 884 830 827 970 914 874 866 828 823 817 965 922 905 895 885 875 834 831 971 915 618 867 832 627 824 620 966 923 972 896 886 876 840 835 623 621 967 616 906 836 833 628 619 497 897 887 877 841 634 629 624 622 665 921 617 907 898 888 842 837 715 498 916 878 847 838 635 630 625 666 614 499 908 899 889 848 843 631 626 667 917 844 839 636 612 615 909 900 997 849 640 637 632 714 611 613 920 918 910 770 766 850 845 652 993 998 776 767 846 641 638 633 653 662 977 919 777 771 851 683 675 994 785 772 768 642 639 654 668 978 976 988 786 778 769 999 643 712 648 676 979 995 0◦ 779 773 760 677 655 973 989 787 774 762 713 647 649 974 980 996 788 780 763 761 533 650 678 669 924 990 790 781 775 716 644 679 670 975 789 765 764 645 500 651 928 925 991 581 574 782 717 664 659 661 671 575 569 783 542 646 504 501 672 929 926 992 582 576 570 543 663 534 660 609 930 927 583 571 784 580 511 505 603 610 673 589 577 656 544 512 506 535 674 931 986 590 584 578 572 545 518 502 604 680 718 585 549 527 519 513 507 681 932 981 987 719 591 579 528 522 508 503 605 933 982 592 586 557 550 523 520 514 606 682 726 720 587 551 529 515 509 599 934 984 983 721 593 563 558 530 524 521 510 600 607 727 594 559 552 525 573 516 608 935 985 728 722 595 564 553 531 517 536 601 735 723 565 560 532 526 588 602 936 736 729 596 561 554 597 539 537 937 730 724 750 566 555 657 546 538 938 28 737 725 567 562 547 598 540 939 738 731 751 691 556 658 948 541 941 742 732 756 752 568 684 548 942 940 743 739 757 698 692 687 956 949 944 740 733 758 753 693 688 685 945 943 10◦ 746 744 734 754 699 686 957 950 − 745 741 406 759 700 694 689 951 946 747 417 399 755 690 958 947 748 428 407 394 701 695 959 952 429 418 408 400 696 953 749 419 401 705 702 960 442 430 420 409 703 697 961 954 431 410 706 395 955 443 432 421 707 396 704 962 444 422 411 963 445 433 708 402 397 434 423 403 398 446 709 412 447 435 413 404 456 436 710 457 448 424 414 405 458 449 711 425 415 467 450 437 468 459 438 426 416 469 460 451 439 427 461 452 470 453 440 471 462 463 454 441 20◦ 472 455 − 473 464 474 465 475 466 476

Figure 2.2. OGLE-IV fields toward the Galactic center. Colors mark the typical cadence of observations: red – one observation every 20 min, yellow – one observation every 60 min, green – 2–3 observations per night, blue – one observation per night, cyan – one observation per two nights. Silver fields were regularly observed during the years 2010–2013, usually once every 2–3 days. Transparent fields are a part of the OGLE Galaxy Variability Survey and are not analyzed in this dissertation. OGLE is monitoring over 600 fields located toward the Galactic center (Figure 2.2), 121 of which have been observed for a sufficiently long time to allow a search for microlensing events. Fields are observed with a different cadence, ranging from one observation per 20 minutes to one observation every few days, see Figure 2.2 for details. In particular, nine fields (covering in total 12.6 square degrees) are observed with a cadence of either 20 min (BLG501, BLG505, and BLG512) or 60 min (BLG500, BLG504, BLG506, BLG511, BLG534, and BLG611), which enables us to detect short-lived planetary anomalies and short-duration microlensing events that can be caused by free-floating planets. Typical exposure times are 100–120 s and the vast majority of observations is taken through the I-band filter, closely resembling that of a standard Cousins system. The magnitude range of the survey is 12 < I < 21, but the limiting magnitude depends on the crowding of a given field (see Section 5.4). A small fraction of images (about 10%) is taken in the V -band to enable the characterization of sources. The remaining 485 fields in an extended area around the Galactic bulge are monitored as a part of the OGLE Galaxy Variability Survey (GVS) (Udalski et al. 2015a) with shorter exposure times (25 35 s) and lower depth (10 < I < 18.5). The analysis of these fields is not included − in this dissertation, because the GVS survey has not been completed yet. Basic information about all analyzed fields (equatorial and Galactic coordinates, number of monitored sources, number of epochs) is presented in Table A.1 in Appendix A. OGLE photometric pipeline is based on the Difference Image Analysis (DIA) method (Alard & Lupton 1998; Wo´zniak2000), which allows obtaining very accurate photometry in the dense stellar fields. For each field, a reference image is constructed by stacking several highest quality and seeing frames. This reference image is then subtracted from incoming frames and the photometry is performed on subtracted images. Variable and transient objects that are detected on subtracted images are stored in two databases. The “standard” database consists of all stellar-like objects detected on the reference frame, while “new” objects (those that do not correlate with any identified stars) are stored separately. The detailed description of image reductions and OGLE photometric pipeline is included in Wo´zniak(2000), Udalski (2003) and Udalski et al. (2015a).

3. Measuring the frequency of free-ﬂoating planets in the Milky Way

3.1. Motivation

Theories of planet formation predict the existence of free-floating (rogue) planets that are not gravitationally tethered to any host star. These objects could have formed in protoplanetary disks around stars, as “ordinary” planets, and could have been ejected as a result of various mechanisms, including planet-planet dynamical interactions (e.g., Rasio & Ford 1996; Weidenschilling & Marzari 1996; Marzari & Weidenschilling 2002; Chatterjee et al. 2008; Scharf & Menou 2009; Veras et al. 2009), ejections from multiple-star systems (e.g., Kaib et al. 2013; Sutherland & Fabrycky 2016), stellar flybys (e.g., Malmberg et al. 2011; Boley et al. 2012; Veras & Moeckel 2012), dynamical interactions in stellar clusters (e.g., Hurley & Shara 2002; Spurzem et al. 2009; Parker & Quanz 2012; Hao et al. 2013; Liu et al. 2013; Fujii & Hori 2018; Li et al. 2019), or the post-main-sequence evolution of the host star(s) (e.g., Veras et al. 2011, 2016; Kratter & Perets 2012; Voyatzis et al. 2013). It is believed that low-mass planets are more likely to be scattered to wide orbits or ejected than giant, Jupiter-mass planets. Calculations of Ma et al. (2016), which are based on the core accretion theory of planet formation, predict that most free-floating planets should be of Earth mass. Rogue planets are more likely to form around FGK-type stars, because they are scattered into wide orbits following close encounters with gas giant planets, which are more likely to form around massive stars. The typical total ejected mass is about 5 20 M⊕ and about 10-20% − of all planetary systems should give rise to rogue planets. Similarly, Barclay et al. (2017), using N-body simulations of terrestrial planet formation around solar-type stars, estimated that about 2.5 terrestrial-mass planets are ejected per star in the Galaxy during late-stage planet formation, but these numbers strongly depend on the adopted initial conditions.

31 Free-floating planetary-mass objects can also be formed by the fragmentation of gas clouds, in a way similar to that in which stars form. Star formation processes are believed to extend down to 1 4 M (Boyd & Whitworth 2005; Whitworth & Stamatellos 2006). This parameter − Jup space cannot be probed with the current surveys of young stellar clusters and star-forming regions, which are unable to detect objects less massive than 5 6 M (Peña Ramírez − Jup et al. 2012; Lodieu et al. 2013; Mužic´ et al. 2015; Kirkpatrick et al. 2019). Free-floating planetary-mass objects may also form from small molecular cloudlets that have been found in H II regions, although it is unclear whether these clouds may contract (Gahm et al. 2007; Grenman & Gahm 2014). Gravitational microlensing is the only method that enables us to find Earth-mass free-floating planets. A typical Einstein radius of the super-Earth-mass lens is θE = p p 5 µas M/10 M⊕ π /0.1 mas (here, M is the lens mass, π = π π is the relative rel rel l − s lens-source parallax, and πl and πs are parallaxes to the lens and source, respectively). As −1 typical lens-source proper motion in the direction of the Galactic center is µrel = 5 mas yr , timescales of microlensing events due to Earth-mass lenses are very short t = θ /µ E E rel ≈ 10−3 yr 0.4 d. Microlensing events with Einstein timescales shorter than 2 d have been ≈ traditionally attributed to unbound planets. Information about the mass function of lenses, including free-floating planets, can be inferred from a statistical analysis of the distribution of timescales of a large sample of microlensing events. The possibility of detecting free-floating planets has been discussed in the microlensing community for several years, but it required high-cadence observations. The first such measurement was attempted by Sumi et al. (2011), who analyzed a sample of 474 microlensing events detected by the Microlensing Observations in Astrophysics (MOA) group. They found an excess of nine short1 events, relative to what was expected from brown dwarfs and stars, and they attributed this excess to a large population of Jupiter-mass rogue planets, which should be nearly twice as common as main-sequence stars. Clanton & Gaudi (2017) modeled the microlensing signal expected from exoplanets on wide orbits using constraints from microlensing, radial velocity, and direct imaging surveys and concluded that at most 40% of short-timescale events detected by Sumi et al. (2011) can be ∼ interpreted as due to wide-orbit planets. However, the statistical significance of Sumi et al.’s results is largely based on three shortest-timescale events (tE < 1 days). As we mentioned

1 They reported ten events with timescales shorter than 2 days, but one measurement, for MOA-ip-1, is incorrect (see Section 3.5).

32 above, one measurement is incorrect and the model by Clanton & Gaudi (2017) shows that the remaining two are statistically consistent with being wide-orbit planets. That model still cannot account for a small overabundance of events with timescales between 1–2 days (see Figures 4 and 5 from Clanton & Gaudi 2017), but the statistical significance of the remaining excess relative to the short-timescale events expected from stars and brown dwarfs is small. A large population of Jupiter-mass free-floating planets suggested by Sumi et al. (2011) was difficult to reconcile with censuses of substellar objects in young clusters and star-forming regions and with predictions of planet-formation theories. For example, Peña Ramírez et al. (2012) and Scholz et al. (2012) analyzed substellar mass functions of the young clusters σ Orionis and NGC 1333, finding that free-floating planetary-mass objects are at least an order of magnitude less common than main-sequence stars. These observations are incomplete for masses below 6 M , so direct comparisons with microlensing surveys are difficult. Several ∼ Jup mechanisms of free-floating planet production have been proposed (e.g., Veras & Raymond 2012), but none of them is capable of explaining the large number of Jupiter-mass free-floaters suggested by Sumi et al. On the other hand, Earth- and super-Earth-mass planets can be scattered and ejected much more efficiently (Pfyffer et al. 2015; Ma et al. 2016; Barclay et al. 2017). These problems prompted us to carry out an independent analysis of a much larger sample of microlensing events that were detected by the OGLE-IV survey during the years 2010–2015. Thanks to technical upgrades, which were described in Chapter 2, the observing capabilities of the OGLE telescope increased considerably in 2010. That allowed us to monitor the sky with a very high cadence and to check the previous early results. In this Chapter, we describe the analysis of microlensing events located in high-cadence OGLE fields and present our conclusions on the frequency of free-floating planets in the Milky Way.

3.2. Data

All data analyzed in this Chapter were collected as part of the OGLE-IV sky survey (Udalski et al. 2015a) during the years 2010–2015. We analyzed light curves of objects located in nine OGLE ﬁelds, which are observed with the highest cadence of either 20 min (BLG501, BLG505, and BLG512) or 60 min (BLG500, BLG504, BLG506, BLG511, BLG534, and BLG611) and cover in total 12.6 square degrees (see Figure 2.2). We analyzed data collected between 2010 June 29 and 2015 November 8, that is, ﬁve and a half Galactic bulge observing seasons. Light

33 curves consist of 4,500 – 12,000 data points, depending on the field, which gives a total of 380 billion photometric measurements. All analyzed data were taken through the I-band filter. Basic information about the fields (equatorial and Galactic coordinates, number of objects in databases, number of epochs) is presented in Table A.1 in Appendix A. As explained in Chapter 2, OGLE photometric pipeline is based on the Difference Image Analysis (DIA) method (Alard & Lupton 1998; Wo´zniak2000). For each field, a reference image is constructed by stacking several highest-quality and seeing frames. This reference image is then subtracted from incoming frames and the photometry is performed on subtracted images. Variable and transient objects that are detected on subtracted images are stored in two databases. The “standard” database consists of all stellar-like objects detected on the reference frame, whereas “new” objects (those that do not correlate with any identified stars) are stored separately.

3.3. Selection of events

We analyzed 50 million light curves, from all the objects from the “standard” database. We began our analysis by correcting photometric uncertainties and transforming magnitudes into flux. It is known that uncertainties returned by the DIA are underestimated and do not reflect the actual observed scatter in the data (Yee et al. 2012). Skowron et al. (2016) provide an algorithm for their correction, so that these uncertainties now reflect the real scatter in the light curves. For stars fainter than approximately I = 15, the error bars are corrected using

p 2 2 formula δmi,new = (γδmi) + ε , where γ and are parameters determined for each ﬁeld separately. They are measured based on the scatter of constant stars (typically, γ = 1.2 1.6 − and ε = 0.002 0.004). For the brightest stars, there is an additional correction resulting from − non-linear response of the detector. The selection of events was conducted in three steps, described in detail below. The selection criteria for high-quality microlensing events are summarized in Table 3.1.

34 Criteria Remarks Number 2 χout/dof 2.0 No variability outside the 360-day window centered ≤ on the event nDIA 3 Centroid of the additional flux coincides with the source ≥ star centroid P χ3+ = (Fi Fbase)/σi 32 Significance of the bump 43,158 i − ≥ s < 0.4 Rejecting photometry artifacts A > 0.1 mag Rejecting low-amplitude variables nbump = 1 Rejecting objects with multiple bumps 11,989 Fit quality: 2 2 χfit/dof 2.0 χ for all data 2 ≤ 2 χ /dof 2.0 χ for t t0 < tE fit,tE ≤ | − | χ2 / . χ2 for t t < t fit,2tE dof 2 0 0 2 E 2 ≤ 2 | − | χfit,1/dof 2.0 χ for t t0 < 1 day 2 ≤ 2 | − | χ /dof 2.0 χ for t t0 < 5 days fit,5 ≤ | − | 2455377 t0 2457388 Event peaked between 2010 June 29 and 2015 December ≤ ≤ 31 u0 1 The minimum impact parameter ≤ Is 22.0 The minimum I-band source magnitude ≤ nr 2 if nd 2 Rising and descending parts of the light curve should be ≥ ≥ sufficiently sampled nr 4 if nd < 2 ≥ Fb > 0.251 The maximum negative blend flux, corresponding − to I = 19.5 mag star fs > 0.1 Rejecting highly-blended events 2617 Table 3.1. Selection criteria for high-quality microlensing events.

Figure 3.1. Selection of microlensing events. We placed a 360-day moving window on each light curve and measured the baseline ﬂux Fbase and its dispersion σbase using data points outside the window. We then searched for at least three consecutive points at least 3σbase above the baseline ﬂux.

35 Figure 3.2. To minimize the contamination from moving objects (like asteroids) and photometry artifacts, we required that the center of the additional ﬂux coincided with the center of the source (i.e., nDIA 3). The image shows the asteroid (7294) 1992LM passing in front of a bulge star. The≥ brightening of the source could be mistaken with a short-timescale microlensing event.

Figure 3.3. Reﬂections within the telescope (left panel) might cause spurious, short brightenings of neighboring stars (right panel).

36 Cut 1. We placed a 360-day moving window on each light curve and measured the baseline

flux Fbase and its dispersion σbase using data points outside the window (after rejecting 5σ outliers such as cosmic ray hits, see Figure 3.1). We required χ2 /d.o.f. 2.0, where d.o.f. out ≤ are degrees of freedom, outside the window, so we could reject most of the variable stars. Some genuine microlensing events with variable baseline or those longer than one year may have not passed this criterion. We defined a bump as a brightening with at least three consecutive points at P least 3σ above the baseline flux. For each bump we calculated χ = (Fi F )/σi (i is base 3+ i − base the index within a bump) and nDIA, the number of detections on subtracted images. We required χ 32 and n 3 to pass this cut. (We note that with the current data we were able to set 3+ ≥ DIA ≥ a lower threshold than in Sumi et al. (2011), who used χ 80). The introduction of the cut on 3+ ≥ nDIA allowed us to eliminate contamination from asteroids (Figure 3.2), photometry artifacts, and “ghost” microlensing events, which are stars affected by real variability of neighboring stars (Wyrzykowski et al. 2015). Cut 2. Cut 1 criteria were insufficient to remove all artifacts. For example, reflections within the telescope might cause spurious, short brightenings of neighboring stars correlated in time (Figures 3.3 and 3.4). Reflections were especially troublesome near the edges of CCD detectors #1, #7, #8, #16, #17, #25, #26, and #32 of the OGLE-IV mosaic camera, located at the edges of the telescope field of view (Udalski et al. 2015a). To quantify the concurrence of bumps, we

deﬁned the similarity of two bumps as s = N1/N2, where N1 is the number of individual frames

when both bumps were detected on subtracted images and N2 is the number of frames when at least one bump was detected. We calculated similarities for all possible pairs of bumps shorter than ﬁve days and then rejected objects with s 0.4. This threshold value was chosen after ≥ visual inspection of light curves and images of possible short events. It allowed us to reject over 95% of artifacts, while removing none of the genuine microlensing events from the sample. A number of stars that passed cut 1 criteria were OGLE small amplitude red giants (OSARGs) (Wray et al. 2004) which are red giant variable stars showing low-amplitude pulsations (< 0.13 mag in the I band) with (frequently multiple) periods in the range of 10 < P < 100 d. Some pulsation cycles in OSARGs might have slightly higher amplitudes so they were detected by our algorithm as potential microlensing events (Figure 3.5). We therefore rejected all objects with a bump amplitude A 0.1 mag, so only a few genuine microlensing ≤ events were discarded in this step, most of which suffer from large blending or have impact

37 parameters u0 > 1. The remaining OSARGs were easily rejected in the next step, because the microlensing light curve fit yielded nonphysical parameters. Finally, we rejected all objects with more than one bump in the light curve. These were mostly dwarf novae (Figure 3.5) and some remaining photometry artifacts. Twenty-nine genuine microlensing events were also rejected, most of them binary source or binary lens events, and some microlensing events with variable baseline. Cut 3. For the remaining 11,989 event candidates, we fitted the microlensing point-source point-lens model. The lensing model has three parameters: the time t0 and projected separation u0 (in Einstein radius units) between the lens and the source during the closest approach, and the Einstein radius crossing time tE. Two additional parameters describe the source flux Fs and blended unmagnified flux Fb from possible unresolved neighbors and/or the lens itself. The observed flux is Fi = FsAi + Fb, where Ai = A(ti; t0, tE, u0) is the model amplification. The

ﬂux parameters Fs and Fb were found analytically using the least-squares method. For a given set of (t0, tE, u0), ﬂux parameters were calculated using:

P FiAi P 1 P Ai P Fi σ2 σ2 σ2 σ2 i i i i − i i i i Fs = 2 , (3.1) 2 P Ai P 1 P Ai σ2 σ2 σ2 i i i i − i i 2 P Ai P Fi P Ai P AiFi σ2 σ2 σ2 σ2 i i i i − i i i i Fb = 2 , (3.2) 2 P Ai P 1 P Ai σ2 σ2 σ2 i i i i − i i

where (Fi σi) is the i-th data point. We also calculated the four-parameter fits, where the ± blend flux was set to zero, Fb = 0. Then the source flux is given by:

P FiAi σ2 i i Fs = 2 . (3.3) P Ai σ2 i i The best-ﬁtting parameters were found by minimizing the function

2 X (Fi Fmodel(ti)) χ2 = − (3.4) σ2 i i using the Nelder-Mead algorithm2 (Nelder & Mead 1965; O’Neill 1971). We performed the initial ﬁt using the data from a 360-day window centered on the event and later reﬁned the parameters using all available data.

2 We use the C implementation of the algorithm by John Burkardt, which is distributed under the GNU LGPL license (https://people.sc.fsu.edu/ jburkardt/c_src/asa047/asa047.html). ∼

38 39

Figure 3.4. Candidate microlensing events selected in field BLG500. Gray points show locations of artifacts that were rejected by s < 0.4 condition. Artifacts cluster mostly near the edge of the telescope field of view. Figure 3.5. Examples of light curves of variable stars that may mimic microlensing events: flaring stars, dwarf novae, OGLE small amplitude red giants (OSARGs).

40 log tE 1 0 1 2 3 − Best-ﬁtting model 95% conf. limit Sumi et al. (2011) 10 Earths per star 100 5 Earths per star

10 Number of events per bin

0.1 1 10 100 1,000 tE (days)

Figure 3.6. Observed distribution of timescales of 2,617 high-quality microlensing events discovered by OGLE in 2010–2015. The purple line is the best-fitting model. The dotted line constrains the 95% confidence limit on the number of wide-orbit or unbound Jupiter-mass planets of 0.25 planets per star. The dashed red line is the best-fitting model from Sumi et al. (2011) predicting almost two Jupiter-mass free-floating planets per star. According to that model we should find 64 events with 0.3 < tE < 1.8 d, but only 21 were observed (the discrepancy is even larger for events with 0.3 < tE < 1.3 d, where 6 events were found out of 42 expected). We detected six possible ultrashort-timescale events (tE < 0.5 d), which may be due to Earth-mass free-floating planets (gray histogram). Solid (dotted) green lines mark the expected microlensing signal assuming 5 M⊕ planets five (ten) times more frequent than stars. Error bars are the 1σ Poisson uncertainties on the counts of the number of events observed in a given tE bin.

41 Bin log tE BLG500 BLG501 BLG504 BLG505 BLG506 BLG511 BLG512 BLG534 BLG611 1 –0.93 0 1 0 0 0 0 0 0 0 2 –0.79 0 0 0 1 0 0 1 0 0 3 –0.65 1 1 0 0 0 0 0 0 0 4 –0.51 0 1 0 0 0 0 0 0 0 5 –0.37 0 0 0 0 0 0 0 0 0 6 –0.23 0 1 0 0 0 0 0 0 0 7 –0.09 1 0 0 0 1 0 0 0 0 8 0.05 1 1 1 0 0 0 0 0 0 9 0.19 0 4 0 2 0 4 1 1 3 10 0.33 3 5 4 4 1 0 3 5 2 11 0.47 4 9 7 8 5 8 3 8 5 12 0.61 10 19 13 28 10 10 13 6 3 13 0.75 17 40 17 39 19 11 13 17 9

14 0.89 22 32 24 55 25 19 28 20 20 42 15 1.03 25 37 30 78 34 22 40 22 25 16 1.17 26 35 46 57 44 33 46 24 23 17 1.31 28 62 38 62 39 30 40 24 38 18 1.45 23 42 39 53 32 32 40 33 36 19 1.59 15 39 27 40 32 24 25 20 21 20 1.73 12 25 20 39 19 21 31 18 11 21 1.87 7 13 11 20 10 10 12 6 8 22 2.01 3 9 6 11 6 7 3 2 4 23 2.15 5 2 3 2 7 1 4 2 2 24 2.29 0 0 1 1 3 2 3 0 1 25 2.43 0 0 0 1 2 0 0 0 0 Table 3.2. Number of events detected in individual timescale bins. There are 25 bins equally spaced in log t between 1.0 and 2.5. E − During modeling, we iteratively removed 4σ outliers (provided that the adjacent data points are within 3σ of the model, which conserves any systematic anomalies in the light curve). We calculated a number of goodness-of-fit statistics. 2 for the entire data set, 2 for χfit χfit,tE t t < t , χ2 for t t < 2t , and χ2 for t t < k (where k = 1 or k = 5 | − 0| E fit,2tE | − 0| E fit,k | − 0| days). We removed 4σ outliers provided that adjacent data points are within 1σ from the

2 best-fitting model and ti± ti < 1 day. We required χ /d.o.f. 2.0, which removes | 1 − | ≤ the majority of non-standard microlensing events (finite source, parallax, binary) in addition to non-microlensing events. We allowed for some amount of negative blending, that is, the blend flux F > F was allowed, where F = 0.251 is the flux corresponding to an 19.5-mag star b − 0 0 (here F = 1 corresponds to an 18-mag star). If F < F and the four-parameter model b − 0 was marginally worse (∆χ2 < 4) than the five-parameter model, we chose the four-parameter model. Usually, a high negative blending indicates that the single lensing model has been fitted to a non-microlensing event (like a dwarf nova outburst, OSARG, or stellar flare). However, a small amount of negative blending does not necessarily mean that the model is unphysical. The background (mainly unresolved main-sequence stars) in crowded fields of the Galactic bulge is not uniform and if the source happens to be located in a lower-density region, the blend flux might be negative. The issue of negative blending is discussed by Park et al. (2004), Jiang et al. (2004), and Smith et al. (2007). We checked that our prior on the negative blending has no impact on the final event timescale distribution (which remains the same after choosing

F0 = 0.1, that is, the flux corresponding to a 20.5-mag star). We also required at least n 2 data points on the rising part of the light curve (t t < r ≥ 0 − E t < t ) and at least n 2 data points on the descending branch (t < t < t + t ). If n < 2, 0 d ≥ 0 0 E d we required n 4. These cuts allowed us to eliminate contamination from flaring stars, which r ≥ can rise very steeply (Hawley et al. 2014) (within minutes), but fade slowly (on a timescale of hours), see also Figure 3.5. If the rising part of the light curve is not sufficiently sampled, a flare might be mistaken for a very short microlensing event. Our image-level simulations (see below) showed that we were unable to robustly measure the true timescale of an event if the event is faint and the blending is high (fs < 0.1, that is, less than 10% of baseline flux comes from the source). Therefore, to ensure that the final results are sound we did not include events with blending parameter fs < 0.1. The inclusion of highly-blended events had little effect on the final results, although we found an increased number of long-timescale events (tE > 100 d).

43 Star RA Decl. t0 (HJD) tE (d) u0 Is fs EWS ID h m s ◦ 0 00 +0.12 +6.86 +0.012 +0.10 +0.06 BLG611.25.88001 17 32 30.27 −27 06 41.8 2456796.27−0.12 99.59−5.97 0.135−0.011 20.78−0.10 0.61−0.06 OB140426 h m s ◦ 0 00 +0.43 +4.68 +0.139 +0.41 +0.77 BLG611.16.129767 17 32 34.87 −27 15 50.1 2456075.84−0.44 28.34−3.62 0.603−0.131 20.28−0.38 1.84−0.58 OB120671 h m s ◦ 0 00 +0.08 +1.33 +0.075 +0.30 +0.61 BLG611.16.102416 17 32 36.04 −27 10 37.0 2457115.89−0.08 7.98−1.10 0.303−0.061 20.25−0.31 1.82−0.44 OB150540 h m s ◦ 0 00 +0.04 +2.64 +0.009 +0.13 +0.02 BLG611.16.101082 17 32 37.65 −27 10 51.3 2456771.36−0.04 27.15−2.41 0.075−0.008 21.13−0.13 0.15−0.02 OB140609 h m s ◦ 0 00 +0.20 +0.76 +0.126 +0.49 +0.42 BLG611.25.78304 17 32 40.85 −26 53 31.6 2456352.66−0.19 3.44−0.47 0.592−0.153 19.74−0.35 1.12−0.41 h m s ◦ 0 00 +0.19 +2.58 +0.193 +0.68 +0.78 BLG611.16.49958 17 32 47.62 −27 20 43.0 2456062.57−0.19 7.01−1.49 0.448−0.169 20.86−0.61 1.05−0.49 h m s ◦ 0 00 +0.10 +1.87 +0.027 +0.13 +0.12 BLG611.16.46353 17 32 54.75 −27 20 41.5 2456800.39−0.10 28.10−1.79 0.261−0.025 20.05−0.13 0.92−0.10 OB140770 h m s ◦ 0 00 +0.01 +0.63 +0.004 +0.06 +0.06 BLG611.25.46594 17 32 56.39 −26 56 02.2 2456017.85−0.01 13.88−0.56 0.073−0.004 20.15−0.06 1.07−0.06 OB120268 h m s ◦ 0 00 +2.30 +4.52 +0.186 +0.90 +0.51 BLG611.25.2758 17 33 04.26 −27 07 43.5 2457055.78−2.45 14.38−2.64 0.291−0.151 20.07−0.51 0.85−0.48 h m s ◦ 0 00 +0.46 +1.78 +0.158 +0.56 +0.51 BLG611.07.145346 17 33 14.12 −27 32 00.3 2455802.11−0.45 8.03−1.17 0.663−0.186 20.09−0.41 1.11−0.45 h m s ◦ 0 00 +0.07 +2.09 +0.013 +0.10 +0.02 BLG611.07.147891 17 33 15.43 −27 30 12.2 2457083.37−0.07 32.12−2.09 0.137−0.012 20.12−0.11 0.21−0.02 OB150069 h m s ◦ 0 00 +0.19 +3.46 +0.070 +0.28 +0.44 BLG611.15.118726 17 33 15.51 −27 14 21.6 2455417.88−0.18 23.32−2.87 0.313−0.060 20.16−0.29 1.44−0.33 h m s ◦ 0 00 +0.28 +1.87 +0.172 +0.57 +0.47 BLG611.15.77644 17 33 19.36 −27 18 16.7 2456119.92−0.29 9.12−1.13 0.810−0.211 19.92−0.41 1.03−0.42 h m s ◦ 0 00 +0.16 +4.18 +0.076 +0.33 +0.09 BLG611.07.99111 17 33 22.74 −27 34 39.6 2455391.32−0.16 22.18−3.49 0.264−0.057 20.37−0.36 0.24−0.06 h m s ◦ 0 00 +0.49 +5.07 +0.198 +0.53 +0.31 BLG611.07.73806 17 33 23.53 −27 45 30.1 2457300.97−0.54 22.89−3.93 0.531−0.154 18.84−0.56 0.45−0.17 h m s ◦ 0 00 +0.47 +2.25 +0.033 +0.15 +0.12 BLG611.07.77424 17 33 23.65 −27 45 03.9 2455488.89−0.42 38.57−1.92 0.318−0.033 18.49−0.14 0.87−0.12 h m s ◦ 0 00 +0.17 +1.31 +0.119 +0.43 +0.37 BLG611.15.82655 17 33 24.18 −27 15 36.1 2456493.50−0.16 7.41−0.79 0.612−0.138 19.69−0.33 1.04−0.34 OB131274 h m s ◦ 0 00 +0.02 +1.47 +0.047 +0.38 +0.24 BLG611.24.83027 17 33 25.82 −27 03 47.3 2455405.10−0.02 5.13−1.14 0.095−0.030 20.88−0.42 0.52−0.15 h m s ◦ 0 00 +0.10 +3.74 +0.025 +0.18 +0.25 BLG611.24.72010 17 33 29.52 −26 52 43.6 2456179.10−0.11 30.06−3.24 0.143−0.021 20.76−0.19 1.33−0.20 OB121381 h m s ◦ 0 00 +0.51 +6.45 +0.261 +0.53 +0.30 BLG611.24.52441 17 33 31.32 −26 59 15.3 2456889.19−0.51 34.18−5.76 0.803−0.195 18.84−0.61 0.40−0.15 OB141491 h m s ◦ 0 00 +0.09 +2.13 +0.101 +0.42 +0.30 BLG611.07.52241 17 33 32.63 −27 39 10.4 2456128.18−0.09 8.89−1.61 0.286−0.079 20.56−0.43 0.60−0.19 OB121164 h m s ◦ 0 00 +0.12 +1.95 +0.040 +0.20 +0.18 BLG611.24.71071 17 33 34.89 −26 52 23.0 2457120.75−0.12 17.51−1.70 0.240−0.035 20.36−0.20 0.88−0.15 OB150541 h m s ◦ 0 00 +0.05 +1.61 +0.185 +0.87 +0.60 BLG611.07.63071 17 33 36.45 −27 34 06.3 2456436.14−0.05 2.75−0.76 0.285−0.144 20.80−0.72 0.63−0.35 OB130813 h m s ◦ 0 00 +0.07 +0.92 +0.018 +0.06 +0.06 BLG611.24.63086 17 33 36.86 −26 54 03.4 2456043.82−0.07 39.61−0.87 0.515−0.018 18.20−0.06 1.13−0.06 OB120120 h m s ◦ 0 00 +1.22 +9.26 +0.133 +0.47 +0.35 BLG611.15.33185 17 33 37.16 −27 24 43.6 2455438.30−1.15 51.86−5.63 0.725−0.166 19.64−0.34 0.97−0.34 Table 3.3. Best-fitting parameters of the analyzed microlensing events in high-cadence OGLE fields (first 25 objects). For each parameter we provide the median and 1σ confidence interval derived from the marginalized posterior distribution from the Monte Carlo chain. Is is the source brightness and fs = Fs/(Fs + Fb) is the blending parameter. Equatorial coordinates are given for the epoch J2000. OBNNMMMM stands for OGLE-20NN-BLG-MMMM. The full table is available in electronic form (www.astrouw.edu.pl/~pmroz/tables.tar.gz).

The purity of our sample is almost 100%. Over 90% of microlensing events detected in the real-time by the OGLE Early Warning System (Udalski 2003) passed our “cut 2” criteria. We detected additional 20–30% events (depending on the field) compared to Early Warning System detections. The final distribution of timescales of detected microlensing events is shown in Figure 3.6. Table 3.2 presents the number of events detected in individual fields and timescale bins. Finally, for all events passing our selection criteria, we also estimated the uncertainties of all parameters using the Markov chain Monte Carlo (MCMC) technique (Foreman-Mackey et al. 2013). We added the following prior on the negative blending:  1 if Fb 0 prior = ≥ (3.5) F 2 L  b exp σ2 if Fb < 0 − 2 0 where σ0 = 0.251/3 and 0.251 is a flux corresponding to a 19.5-mag star. The uncertainties are reported in Table 3.3.

44 a 104

HST LF (Holtzman+ 1998)

OGLE-IV LF

103 g a 102 m

n i m c r a

r 1

a 10 t s

100

10 1 12 14 16 18 20 22 24 magnitude

b 104

OGLE-IV LF for field BLG512.32

Simulations (LF shifted by 0.50 mag)

103 g a 102 m

n i m c r a

r 1

a 10 t s

100

10 1 12 14 16 18 20 22 magnitude

Figure 3.7. Galactic bulge luminosity function used for simulations. a, Deep luminosity function (LF) for the subfield BLG513.12, which was observed both by the OGLE-IV survey and the Hubble Space Telescope (HST) (Holtzman et al. 1998). Both LFs overlap in the range 16 < I < 18 mag. This deep LF was used as a template to generate artificial microlensing events in the analyzed fields, after shifting to match the red clump’s centroid in a given field. b, Comparison between the observed LF for subfield BLG512.32 and the simulated LF.

45 3.4. Detection efﬁciency

The observed distribution of timescales of microlensing events has to be corrected for the detection bias. Long timescale events are more likely to pass our selection criteria than short timescale events. To calculate the event detection efficiency, we carried out extensive image-level simulations in which we injected artificial microlensing events into real OGLE frames using the point spread function derived from neighboring stars. Subsequently, we created new reference images and used the OGLE implementation of the Difference Image Analysis algorithm to derive the photometry of simulated events. Image level simulations (event injection, image reductions, creation of photometric databases) were carried out by Prof. Andrzej Udalski. In each iteration we simulated 5,000 events per CCD detector, so the star density did not increase much (by 5–10%). We carried out six iterations for each field, so in total 8.6 million of events were simulated in all fields. Parameters t and u were drawn from uniform distributions: 0.0 u < 1.5 and 0 0 ≤ 0 2455377 t < 2457388. Einstein timescales were drawn from a log-uniform distribution ≤ 0 1.0 log t < 2.5. Sources were taken from the range 14 I < 22 mag from the − ≤ E ≤ s luminosity function of each subfield, which was created as follows. We constructed a very deep luminosity function for the subfield BLG513.12, which was observed both by the OGLE-IV survey and the Hubble Space Telescope (Holtzman et al. 1998). The OGLE-IV luminosity function and the Hubble Space Telescope luminosity function overlap in the range 16 < I < 18 mag (Figure 3.7). This deep luminosity function was used as a template to generate artificial microlensing events in other fields, after shifting it so that the centroid of the red clump matched the observed centroid. We therefore took into account variable bulge geometry and reddening. If there was an evidence for differential reddening, we divided subfields into smaller parts. There were a few subfields (7% of the total analyzed area) where we were not able to detect the red clump owing to extremely high extinction; these were omitted from the final calculations (we detected only a negligible number of 48 microlensing events in these fields). To measure the centroid of red clump stars in each subfield, we used the algorithm of Nataf et al. (2013). For the simulated events we applied exactly the same selection criteria as for the real events (Table 3.1). The detection efficiency curves for all analyzed fields are shown in Figure 3.8 and listed in Table 3.4. We note that the detection efficiency for events with tE = 2 d is very high, up to 53% of the maximum efficiency for field BLG512. Efficiencies for fields observed with

20-min and 60-min cadence are very similar, except for the shortest events with tE < 0.5 day. In

46 10 1

BLG500 BLG501 10 2 BLG504 BLG505 Detection efficiency BLG506 BLG511 BLG512 BLG534 BLG611

10 3 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5 ( / )

Figure 3.8. Detection efficiency curves. Detection efficiencies as a function of the Einstein timescale tE for all analyzed fields (averages for all subfields in the given field). Fields BLG501, BLG505, and BLG512 were observed with a 20-min cadence, and the remaining fields with a 60-min cadence. Error bars are the 1σ Poisson uncertainties on the counts of the number of simulated events in a given tE bin. general, we found that the detection efficiencies are most sensitive to crowding and interstellar reddening toward the given field (fields with higher reddening and higher crowding have lower efficiencies). We note that events were simulated using a standard point-lens point-source model. Higher-order effects, like the parallax (causing deviations in the light curve induced by the Earth’s motion), were not included and so detection efficiencies for long events (t 100 d) E ≥ may be slightly overestimated. Similarly, we did not include the finite source effects, which may reduce our detection efficiency for the shortest events (t 0.1 d), when the Einstein ring size E ∼ becomes similar to the source star radius (Chapter 4).

47 Bin log tE BLG500 BLG501 BLG504 BLG505 BLG506 BLG511 BLG512 BLG534 BLG611 1 –0.93 0.0016 0.0033 0.0021 0.0045 0.0015 0.0016 0.0039 0.0013 0.0013 2 –0.79 0.0030 0.0071 0.0046 0.0078 0.0043 0.0038 0.0085 0.0033 0.0041 3 –0.65 0.0041 0.0086 0.0061 0.0110 0.0057 0.0057 0.0126 0.0047 0.0053 4 –0.51 0.0061 0.0118 0.0089 0.0139 0.0086 0.0077 0.0144 0.0068 0.0084 5 –0.37 0.0096 0.0144 0.0126 0.0180 0.0120 0.0119 0.0186 0.0095 0.0121 6 –0.23 0.0130 0.0209 0.0176 0.0248 0.0189 0.0181 0.0297 0.0160 0.0180 7 –0.09 0.0194 0.0279 0.0255 0.0343 0.0299 0.0278 0.0381 0.0226 0.0290 8 0.05 0.0278 0.0365 0.0368 0.0423 0.0396 0.0388 0.0503 0.0335 0.0390 9 0.19 0.0371 0.0423 0.0461 0.0506 0.0495 0.0486 0.0603 0.0395 0.0525 10 0.33 0.0447 0.0506 0.0559 0.0571 0.0593 0.0596 0.0705 0.0484 0.0631 11 0.47 0.0508 0.0557 0.0630 0.0692 0.0675 0.0680 0.0790 0.0592 0.0755 12 0.61 0.0608 0.0630 0.0701 0.0753 0.0784 0.0758 0.0876 0.0641 0.0863 13 0.75 0.0658 0.0669 0.0750 0.0816 0.0866 0.0832 0.0940 0.0746 0.0874

14 0.89 0.0737 0.0746 0.0855 0.0876 0.0937 0.0907 0.0990 0.0772 0.1025 48 15 1.03 0.0760 0.0769 0.0910 0.0940 0.1011 0.0949 0.1056 0.0838 0.1107 16 1.17 0.0858 0.0826 0.0939 0.0950 0.1035 0.1035 0.1113 0.0899 0.1204 17 1.31 0.0872 0.0831 0.1026 0.1014 0.1079 0.1067 0.1131 0.0913 0.1252 18 1.45 0.0949 0.0898 0.1099 0.1055 0.1184 0.1151 0.1206 0.1012 0.1361 19 1.59 0.0964 0.0940 0.1145 0.1108 0.1191 0.1212 0.1286 0.1048 0.1389 20 1.73 0.1024 0.0973 0.1192 0.1134 0.1264 0.1249 0.1302 0.1105 0.1470 21 1.87 0.1000 0.1004 0.1207 0.1174 0.1288 0.1254 0.1336 0.1111 0.1525 22 2.01 0.1029 0.0965 0.1182 0.1124 0.1253 0.1218 0.1331 0.1085 0.1500 23 2.15 0.0989 0.0928 0.1122 0.1072 0.1148 0.1146 0.1160 0.1029 0.1458 24 2.29 0.0853 0.0788 0.0979 0.0890 0.0998 0.0914 0.0906 0.0888 0.1295 25 2.43 0.0618 0.0539 0.0638 0.0538 0.0596 0.0560 0.0548 0.0578 0.0891 Table 3.4. Detection efﬁciencies for the analyzed ﬁelds. There are 25 bins equally spaced in log t between 1.0 and 2.5. E − 3.5. Parameter recovery

We also used our simulations to ensure that there is no systematic difference between measured and real timescales. In Figure 3.9 we plot timescales for simulated events passing all criteria from Table 3.1. We found there is no systematic bias in measured timescales, unless events were faint and highly blended. This effect was predicted by Wo´zniak& Paczynski´ (1997), where it was found theoretically that in such cases the event timescale, impact parameter and blending parameter may be severely correlated, because information on the event timescale comes mostly from wings of the light curve that can be more easily affected by the photometric noise. In Figure 3.10a we show the ratio between measured and “real” (simulated) timescale

tE,out/tE,in versus the blending parameter fs = Fs/(Fs +Fb). It is clear that timescales of highly blended and faint events are not well measured and systematically overestimated. A similar

effect was also noticed in the earlier work (Sumi et al. 2011), where it was found that tE,in was systematically about 5% smaller than tE,out regardless of tE. Strong correlations between blending, impact parameter, and event timescale may also lead to the incorrect determination of parameters. For example, we found that one of short events reported by Sumi et al. (2011),

+8.1 MOA-ip-1, has an incorrectly measured timescale. The best-ﬁtting model with tE = 8.2−3.6 d is better by ∆χ2 = 9 than the model presented in the original paper (t = 0.73 0.08 d). E ± To be conservative, we decided not to include highly-blended events (fs < 0.1) in our ﬁnal sample of high-quality events. Thanks to this selection cut, there is almost no bias in the measured timescales (see Figures 3.9 and 3.10b).

3.6. Modeling event timescale distribution

The observed distribution of timescales of microlensing events carries information about the mass function of lenses. As we discussed in Section 1.8, the actual timescale distribution depends on the distribution and kinematics of lenses and sources as well as the underlying mass function (Kiraga & Paczynski´ 1994; Han & Gould 1996; Mao & Paczynski´ 1996). To compare the observed distribution of timescales with models we need to convolve the Galactic model, mass function of lenses and detection efﬁciencies. The timescale distribution can be computed

49 100

100 )

s 10 y a d (

10 Number density 1

0.1 1 0.1 1 10 100

, (days)

Figure 3.9. Comparison between measured Einstein timescales tE,out and “real” (simulated) timescales tE,in for simulated events. Only events passing selection criteria from Table 3.1 (including the cut on the blending parameter fs > 0.1) are shown. Note that the color scale is logarithmic. There is no systematic offset between measured and real timescales. from a multi-dimensional integral (Han & Gould 1996; Bissantz et al. 2004): Z f(tE) ρ(Ds)ρ(Dl)rE(Dl,Ds,M)Φ(M) ∝ (3.6) rE vrelf(vrel)δ(tE )dDldDsdvreldM, × − vrel

where ρ(D) is the distribution of lenses and sources along the line-of-sight, rE = θEDl the

Einstein radius, vrel is the lens-source relative velocity projected onto the plane of the sky, and

Φ(M) is the mass function. Ds and Dl are distances to the source and lens, respectively. We expect the timescale distribution to have power-law tails with slopes of +3 and 3 at short and − long timescales, respectively (Mao & Paczynski´ 1996; Wood & Mao 2005).

50 A stemda bouedvainfo h aasmedian. data’s the events from simulated deviation of absolute 90% median the For is 1%. MAD within blending timescales the real on cut and the (including 3.1 Table from parameter criteria selection events. passing all events of simulated fraction negligible for a a comprise events showing such events and of small Timescales ( parameters. blending negative impact high and blending timescale, Einstein between parameter ( parameters. blending highly-blended (simulated) the timescale versus “real” events Einstein simulated and measured measured the between between Comparison 3.10. Figure b Fraction of events a 0.4 0.6 0.0 0.8 0.2 1.0 0.0 2.0 0.5 1.0 1.5 0.0 0.0 0.5 0.5

f 1.0 s >

, 1.5 , 1.0 All All / / 0 2.0 , f , . 1 s median=1.00 MAD=0.12 2.5 .Rgrls ftetmsae hr sn ytmtcba ewe measured between bias systematic no is there timescale, the of Regardless ). 1.5 < 3.0 0

. 3.5 2.0 1 f vnsaentwl esrdadaebae yasrn correlation strong a by biased are and measured well not are events )

s 4.0 0.0 > 0.0

1 0.5 . 0.5 5

r ytmtclyudrsiae,btteba srelatively is bias the but underestimated, systematically are ) 1.0 , 1.5 , , , 1.0 < < / / 2.0 , ,

MAD=0.15 median=1.01 2.5 1.5 3.0 t

E 3.5 51 2.0 , out 4.0 f 0.0 s

n ra”(iuae)timescale (simulated) “real” and 0.0 = 0.5 0.5 F 1.0 s < < / 1.5 ( , , , 1.0 , F / / 2.0 s < < , , 0 + MAD=0.13 median=1.01

. 2.5 63 1.5

F b, 3.0 b t < ) itiuin of Distributions 3.5 2.0 ieclso an and faint of Timescales . 4.0 E 0.0 , out 0.0

/t 0.5 0.5 E 1.0 , in , 1.5 , , , 1.0 < > > / / 2.0 , t , 1 E MAD=0.12 median=1.00

. 2.5 a, , t 65 1.5 out E 3.0 , in The . Ratio /t 3.5 2.0 E for ,

in 4.0 In practice, Equation (3.6) can be used to generate a random ensemble of microlensing events from the adopted model (Figure 3.11). This procedure can be divided into ﬁve steps: 1. we draw a random source from the distribution ρ(D) = ρ(D; l, b) from the range [0, 8 kpc];

2. we draw a random lens from the distribution ρ(D) = ρ(D; l, b) from the range [0,Ds]; 3. we assign random velocities of the lens and the source from the appropriate distributions

and calculate the relative velocity vrel; 4. we draw a random lens mass M from the mass distribution;

5. we evaluate the Einstein radius rE and the event timescale tE = rE/vrel, for each event we

assign a weight w = rEvrel. For each model we generate 107 events. To generate random distances from the one dimensional distribution ρ(D), we calculate the cumulative distribution function CDF(D) and then invert it numerically. We adopted a standard Galactic model (Han & Gould 1995b, 2003), which incorporates the boxy-shaped bulge model (Dwek et al. 1995) and the double exponential model of the Galactic disk (Zheng et al. 2001). The velocity is expressed in the (x, y, z) coordinate system, centered on the Galactic center, where x and z axes point to the East and North Galactic pole, respectively. We assume that the velocity distributions of the lenses and sources are Gaussian: (v v˜ )2 (v v˜ )2 y y z z (3.7) f(vy, vz) = f(vy)f(vz) exp − 2 exp − 2 . ∝ − 2σy − 2σz

−1 As given in Han & Gould (1995b), we adopted v˜z,disk =v ˜z,bulge = 0 km s and σz,disk = −1 −1 20 km s and σz,bulge = 100 km s for the z component of the velocity. For the y direction, −1 −1 −1 −1 v˜y,disk = 220 km s , v˜y,bulge = 0 km s and σy,disk = 30 km s , σy,bulge = 100 km s , depending on whether the object is situated in the disk or in the bulge. Following Han & Gould (2003), the density distribution ρ(D) = ρ(x, y, z) is given by:

−3 2 ρbulge(rs) = 1.23M pc exp( rs /2), − (3.8) −3 ρ (R, z) = 1.07M pc exp( R/H)((1 β) exp( z /h ) + β exp( z /h )). disk − − −| | 1 −| | 2 p R = x2 + y2 is the Galactocentric distance projected onto the Galactic plane. We assume

H = 2.75 kpc, h1 = 156 pc (thin disk), h2 = 439 pc (thick disk), and β = 0.381. For the 0 2 0 2 2 0 4 1/4 0 0 0 bulge model, rs = (((x /x0) + (y /y0) ) + (z /z0) ) . The coordinates (x , y , z ) have their center at the Galactic center, the longest axis x0 is rotated 20◦ from the Sun–Galactic center axis

52 log tE 1 0 1 2 3 −

102 1 M 0.3 M

t 0.1 M

log

/d 1

Γ 10 d

100 0.1 1 10 100 1000 tE (days)

Figure 3.11. Theoretical distribution of timescales of microlensing events in the Galactic model of Han & Gould (1995b, 2003) for a delta-like mass function. If all lenses had a mass of 1 M , the timescale distribution would peak at 25 d (blue curve). If all lenses had a lower mass (0.3 M – green, 0.1 M – red), the maximum of the distribution would shift toward shorter timescales and the probability of lensing would decrease. The actual distribution of timescales is the convolution of the Galactic model and the mass function of lenses.

toward positive longitude, and the shortest axis is the z0 axis. Following Han & Gould (2003), we adopted scale lengths of x0 = 1.58 kpc, y0 = 0.62 kpc, and z0 = 0.43 kpc. The adopted mass function of lenses is discussed in Section 3.7. To compare the measured distribution of Einstein timescales with models, we maximized the following log-likelihood function: X ln = ln p(tE,i), (3.9) L i where p(tE) = pmodel(tE)ε(tE) is the normalized predicted timescale distribution, which serves as our likelihood function. Here pmodel(tE) is the timescale distribution from the Galactic model and ε(tE) is the detection efﬁciency. The summation was performed over all events.

53 a 2.0 100 b 2.0 100

1.8 1.8 80 80 1.6 1.6 4σ 1.4 60 1.4 60 3

2 σ 2 χ χ ms ms 2σ α α 1.2 ∆ 1.2 ∆ 40 40 1.0 2σ 1.0 3σ 0.8 20 0.8 20 4σ 0.6 0.6 0 0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 αbd αbd

Figure 3.12. Constraints on IMF slopes: a, Assuming that all lenses are single; b, assuming binary fraction fbin = 0.4.

3.7. Mass function

A detailed modeling of the initial mass function (IMF) would require population synthesis calculations, in addition to more sophisticated Galactic models, which is beyond the scope of this work. However, we can obtain useful constrains on slopes of the IMF using a simple model. We fitted the following initial mass function:  a M −αbd 0.01M M < 0.08M  1  ≤ −αms Φ(M) = a2M 0.08M M < Mbreak . (3.10)  ≤  −2.0 a3M M Mbreak ≥ We allowed the parameters α and α to vary, but we assumed a fixed IMF slope of 2.0 bd ms − above M > Mbreak = 0.5M (Zoccali et al. 2000), because our experiment was designed to analyze the low-mass end of the IMF. We also considered models with Mbreak = 0.7M and models with binary fraction f = 0, where we assumed a flat mass ratio distribution bin 6 (Belczynski´ et al. 2008) f(q) = 1 in a range 0 q 1. ≤ ≤ Moreover, we followed the approach of Gould (2000b) and we assumed that all stars with initial masses 1 < M/M 8 evolved into white dwarfs following the empirical initial-final ≤ mass relation for white dwarfs (Williams et al. 2009) Mfinal = 0.339 + 0.129 Minit. Masses of neutron stars (with initial masses in the range 8 < M/M 20) peak around 1.33 M with ≤

54 a 68% conﬁdence interval of (1.21, 1.43) M (Kiziltan et al. 2013), while for black holes we

assumed a Gaussian distribution at 7.8 1.2 M (Özel et al. 2010). ± We conducted modeling using events with tE > 0.5 and tE > 2.0 days and in both cases we obtained virtually identical results. Constraints on slopes of the IMF are shown in Figure 3.12. In general, we found that models with non-zero binary fraction describe the event timescale distribution better than models with fbin = 0. The standard IMF (Kroupa

2001) with fbin = 0 does not describe the entire timescale distribution well, especially at long timescales tE > 50 d, which has already been noted by Wegg et al. (2016, 2017). This may indicate that the current Galactic model underpredicts the number of long-timescale events, or the mass function of remnants (especially black holes) is underestimated, or remnants have distinct kinematics from brown dwarf and stellar lenses. The discrepancy can be also explained, if we assume that some fraction of lenses (fbin) are binary systems. Our models with fbin = 0.4 are substantially better than with fbin = 0.0 (with improvement in log-likelihood 2 ∆χ = 2.0(ln , ln , ) = 18.6). For the best-fitting models α 0.8 and α 1.3 Lmax 1 − Lmax 2 bd ≈ ms ≈ with 3σ confidence intervals: 0.2 < αbd < 1.3 and 1.1 < αms < 1.5. This corresponds to 0.90 0.05 (1σ) brown dwarfs per main-sequence star. Sumi et al. (2011) obtained a slightly ± +0.24 lower IMF slope in the brown dwarf regime of αbd = 0.49−0.27, but they used fixed αms = 1.3 and fbin = 0 (their slope αbd is in fact consistent with our models from Figure 3.12a for fixed

αms). Moreover, the slope αbd is correlated with the lower limit of the brown dwarf mass function. For example, for the low-mass cutoff of 0.005 M = 5 MJup, the best-fit slope is lower α 0.6. bd ≈ The IMF slope derived in the stellar regime is consistent with the “canonical” Kroupa (2001) value of 1.3. Observations of brown dwarfs in open clusters and star-forming regions indicate − α 0.6 0.7 (Alves de Oliveira 2013, and references therein) and our models are consistent bd ≈ − with those values. On the other hand, censuses of nearby field brown dwarfs tend to prefer lower slopes. Allen et al. (2005) found a 60% confidence interval of α 0.3 0.6 and other bd ≈ ± studies support α 0 (Alves de Oliveira 2013). However, mass function measurements for bd ∼ isolated field brown dwarfs are affected by difficulties in measuring their ages, distances, and masses. Recently, Kirkpatrick et al. (2019) measured the slope of α 0.6 based on a census bd ≈ of nearby ( 20 pc) brown dwarfs. ≤

55 log tE 1 0 1 2 3 − Best-ﬁtting model 95% conf. limit Sumi et al. (2011) 10 Earths per star 5 Earths per star 103

102 Number of events per bin

101 0.1 1 10 100 1,000 tE (days)

Figure 3.13. Distribution of event timescales corrected for the detection efficiency. This distribution, at short timescales, can be well approximated as a power-law with a slope of +3, consistent with theoretical expectations (Mao & Paczynski´ 1996). There remains a small possible excess of events with timescales 0.5 < tE < 1 d. If they were caused by the Jupiter-mass lenses, the best-fitting models predict their frequency of 0.05 Jupiter-mass planets per star with a 95% confidence limit of 0.25 planets per star (dotted purple line). All symbols are the same as in Figure 3.6. Error bars are the 1σ Poisson uncertainties on the counts of the number of events observed in a given tE bin.

56 3.8. Results and conclusions

The detection-efficiency-corrected histogram of event timescales is presented in Figure 3.13 and, clearly, does not show the excess of events with timescales t 1 2 d, claimed by Sumi E ≈ − et al. (2011). The difference (at a confidence level of 2.5 3σ) can be explained in part by the − relatively small number of events found in the earlier analysis Sumi et al. (2011). In addition to the 2,617 events analyzed in this work, we detected over twenty short-duration events that showed clear signatures of binarity (Bennett et al. 2012) and did not pass our strict selection criteria for the fit quality. Owing to lower photometric precision, such events may have been mistaken for single short-timescale events. It is also possible that event timescales measured in the previous work suffer from systematic effects (differential refraction, unphysical treatment of negative blending). Thanks to better image quality (smaller pixel scale, better seeing) and a narrower filter, our photometry is less prone to such systematic effects. To explain the excess of short events, Sumi et al. (2011) modeled their event timescale distribution using a stellar IMF with αbd = 0.5, αms = 1.3, and Mbreak = 0.7M with −3 additional planetary component, approximated as a delta function at M = 10 M . That model is shown in Figures 3.6 and 3.13 as dashed red line. According to that model we should

ﬁnd 64 events with 0.3 < tE < 1.8 d, but only 21 were observed (the discrepancy is even larger

for events with 0.3 < tE < 1.3 d, where 6 events were found out of 42 expected). Moreover, model of Sumi et al. (2011) systematically underpredicts the number of long-timescale events

(because of its very low sensitivity to long events, tE > 100 d, they found only ﬁve events in this range). Our best-ﬁtting model describes the observed timescale distribution well, but there remains

a small possible excess of events with timescales 0.5 < tE < 1 d (Figures 3.6 and 3.13). If we −3 assume, following Sumi et al. (2011), they are due to Jupiter-mass lenses (Mlens = 10 M ), the best-fitting models predict their frequency of 0.05 Jupiter-mass planet per star with 68% confidence interval of [0, 0.12] planets per star. The 95% confidence limit is 0.25 Jupiter-mass planet per star (dotted purple line in Figures 3.6 and 3.13). Our results agree with upper limits on the frequency of Jovian-mass planets inferred from direct imaging surveys (Lafrenière et al. 2007; Quanz et al. 2012). For example, a high-contrast adaptive imaging search (Bowler et al. 2015) for giant planets around nearby M-dwarf stars did not find any planets, providing very strong upper limits (at the 95% confidence limit) of 10-16% (depending on the model) for planets of between 1 and 13 Jupiter masses, at a distance of approximately 10 100 au. This −

57 suggests that almost the entire possible excess of events with timescales 0.5 < tE < 1 d can be attributed to planets on wide orbits (Clanton & Gaudi 2017). The timescales of six events passing our criteria for high-quality events are shorter than 0.5 day and these events last less that one night (Figure 3.14). We carefully checked CCD images by eye to ensure that these brightenings are real, which rules out problems such as photometry artifacts or asteroids. We also analyzed historical light curves for these events; four of the six have been observed by the OGLE survey for 20 years and we did not find any evidence for other outbursts in archival data. Nevertheless, because these events were so short and the light curves were not fully covered, we cannot rule out the possibility that some of them might be flaring stars (especially BLG512.18.22725 and BLG500.10.140417). The best-fitting microlensing models of six short events constrain their Einstein timescales in the range 0.1 < tE < 0.4 d (Table 3.5). Such short events should be caused by Earth- and super-Earth-mass objects, provided that they have kinematics that are similar to the brown dwarf, stellar and remnant lenses. They might be gravitationally unbound to any star or located at wide orbits (at least several astronomical units from the host star), given no signs of binarity in their light curves. Because the number of ultrashort events is very small and their nature is uncertain, we do not attempt to model their mass function. However, a mere detection of such ultra-short events means that Earth-mass lenses must be very common. If we assume that

5 M⊕-mass planets are five times more common than main-sequence stars, the expected number of ultrashort microlensing events is 2.2. For a more realistic mass function in which Earth-mass planets (Ma et al. 2016) are five times more common than main-sequence stars, the expected number of detections is 25% smaller. According to planet formation theories, most Earth- and super-Earth-mass planets should form at relatively small orbital separations (< 10 au) (Ida & Lin 2004). The most likely sources of wide-orbit and free-floating Earth-mass planets are dynamical interactions in young

Star RA Decl. t0 (HJD) tE (d) tE 1σ conf.int. u0 Is fs BLG501.31.5900 17:50:42.45 -29:24:49.7 2456175.648 0.241 [0.21,0.78] 0.772 18.20 0.97 BLG501.02.127000 17:53:13.44 -30:18:59.6 2457172.692 0.146 [0.12,0.26] 0.517 19.13 0.77 BLG500.10.140417 17:53:16.89 -28:40:51.4 2456116.554 0.246 [0.23,0.37] 0.377 19.08 1.24 BLG501.26.33361 17:54:17.54 -29:18:17.0 2455671.124 0.320 [0.29,0.79] 0.471 18.04 1.11 BLG505.27.114211 17:59:04.18 -28:36:51.7 2457157.780 0.158 [0.15,0.21] 0.597 19.14 1.38 BLG512.18.22725 18:05:25.00 -28:28:23.9 2456064.921 0.128 [0.08,0.19] 0.138 20.95 0.16

Table 3.5. Best-fitting parameters for ultrashort microlensing event candidates. Is is the source brightness and fs = Fs/(Fs + Fb) is the blending parameter. The inclusion of the finite source effects does not improve χ2 much (typically ∆χ2 = 0.0 3.3). Equatorial coordinates are given − for the epoch J2000. We also show 1σ confidence intervals for tE.

58 Figure 3.14. Light curves of ultrashort microlensing event candidates. The left panels show a close-up of the light curve at the event and the right panels show 5.5-year long light curves from OGLE-IV. Some of those events have been observed by OGLE for 20 years with no trace of other variability, but we nevertheless cannot exclude the possibility that some of them may be flaring stars. The shortest-timescale events are not well covered by observations and it is difficult, if not impossible, to either prove or disprove their nature as free-floating planets. The detection efficiency at these timescales is very low, meaning that a very few detections imply the existence of a large population of Earth-mass free-floating or wide-orbit planets. Future space-based missions, like WFIRST and Euclid, will enable the exploration of these short events in more detail. Error bars represent 1σ uncertainties. HJD, Heliocentric Julian date.

59 multi-planet systems (Chatterjee et al. 2008; Pfyffer et al. 2015; Ma et al. 2016; Barclay et al. 2017). Other mechanisms (including ejections from multiple-star systems, stellar fly-bys, interactions in stellar clusters, and post-main-sequence evolution of the host star(s)) have also been proposed (Veras & Raymond 2012). Although these processes are unlikely to produce a sizable population of Jupiter-mass free-floating planets, Earth-mass planets can be scattered and ejected much more efficiently. Thanks to the superb photometry quality and the possibility of continuous observations during approximately 100-day-long windows, future space-based missions, such as WFIRST (Spergel et al. 2015) and Euclid (Penny et al. 2013), will have the potential to explore the population of free-floating Earth-mass planets in more detail.

60 4. Measuring the angular Einstein radii of free-ﬂoating planet candidates

4.1. Motivation

The Einstein timescale is the only physical parameter that can be measured for the majority of microlensing events. As the timescale is proportional to the square root of mass, it is

expected that events caused by free-ﬂoating planets are very short (tE . 2 days). However, the mass measurement requires the knowledge of two additional physical parameters: the angular

Einstein radius θE and the microlens parallax πE: θ M = E (4.1) κπE −1 where κ = 8.144 mas M . Although the angular Einstein radius is routinely measured in binary microlensing events via the finite-source effects (Mao et al. 1994; Nemiroff & Wickramasinghe 1994), such a measurement is much harder for single lensing events because it requires that the source passes almost exactly over the lens to produce a detectable finite-source signal (Alcock et al. 1997a; Yoo et al. 2004), see Section 1.4. A typical Einstein radius of a super-Earth-mass lens is p p θ = 5 µas M/10 M⊕ π /0.1 mas (here, π = θ π = π π is the relative lens-source E rel rel E E l − s parallax, and πl and πs are parallaxes to the lens and source, respectively). Because angular radii of giant source stars in the Galactic bulge ρ∗ = 6 µas(R/10 R )(πs/0.125 mas) are comparable to angular Einstein radii of planetary-mass lenses, light curves of giant-source events attributed to free-floating planets should exhibit strong finite source effects (Bennett & Rhie 1996; Ma et al. 2016). Until now, however, no such measurements have been reported. The microlens parallax measurements are even harder for free-floating planets. The subtle deviations from the standard microlensing light curve due to parallax can be detected in long-timescale events, as the Earth-based observer moves along the orbit (Gould 1992). Parallax

61 can be also measured using simultaneous ground- and space-based observations (Refsdal 1966), for example, with the Spitzer satellite (Dong et al. 2007; Udalski et al. 2015b). However, Spitzer operations require the targets to be uploaded to the spacecraft at least three days in advance, making observations of short events nearly impossible. The problem can be overcome with continuous, survey-mode observations (e.g., Henderson & Shvartzvald 2016; Gould 2016a). Such an experiment was conducted during the K2 Campaign 9 (Henderson et al. 2016; Penny et al. 2017), but owing to the difficulties in extracting the photometry from crowded regions of the Galactic bulge (Zhu et al. 2017b; Poleski et al. 2018), no observations of short-timescale microlensing events from K2C9 were reported so far. In this Chapter, we describe the characterization and properties of three short-timescale microlensing events that exhibit finite source effects: OGLE-2012-BLG-1323, OGLE-2016-BLG-1540, and OGLE-2017-BLG-0560. For the first time, we were able to measure the angular Einstein radii of free-floating planet candidates.

4.2. Search for ultra-short timescale events

After the discovery of a few ultra-short timescale microlensing events in the OGLE data from 2010–2015 (Chapter 3), we decided to carry out an additional dedicated search for very-short-duration events in data from the 2016 season. We aimed to detect finite source effects, which would allow us to measure the angular Einstein radius of the lens. Such a measurement would enable us to constrain masses of free-floating planet candidates, as it would remove a degeneracy between the mass and velocity of the lens. We decided to supplement OGLE observations with photometric data from the Korea Microlensing Telescope Network (KMTNet) survey, a network of three telescopes located in Chile, Australia, and South Africa (see below). Additional data provided us with a better coverage of short-timescale microlensing events and allowed us to precisely measure parameters of the most interesting events. We used the event finder algorithm that was described in Section 3.3 to identify all possible candidate microlensing events in the OGLE data from the 2016 season using Cut 1 criteria (no variability outside the 360-day window centered on the event, at least three detections on subtracted images, and χ 32, see Table 3.1 for details). Subsequently, we chose 3+ ≥ 15 candidate events that lasted at most three days (Table 4.1) and we obtained KMTNet light curves of all candidates. Additional photometric data from KMTNet allowed us to identify genuine microlensing events (one candidate, BLG505.14.65844, turned out to be a flaring star)

62 Event RA Decl. Remarks BLG500.02.198445 17:53:10.80 –29:04:11.9 tE = 0.87 0.16 d, KMT SAAO light curve shows asymmetry,± likely binary BLG504.05.160580 17:56:46.44 –28:22:52.5 tE 3.3 1.0 d BLG505.14.65844 17:56:12.58 –29:26:55.1 flaring≈ star± BLG505.28.206873 17:57:58.20 –28:43:46.9 genuine microlensing event, but tE is poorly constrained because of high blending (low amplitude) BLG505.28.87130 17:58:22.67 –28:44:23.8 tE = 11.0 4.5 d +0±.72 BLG505.30.22841 17:57:05.57 –28:46:14.1 tE = 0.54 0.31 d BLG512.10.215221 18:04:16.98 –28:39:39.5 the best-fit− solution (with a prior on negative blending Fb > 0) tE 0.075 d, but there are no data on the rising branch≈ of the light curve. BLG512.24.154394 18:00:47.00 –28:21:35.2 tE 0.33 d, very strong finite source effects ≈ BLG534.18.184244 17:53:44.64 –30:47:58.6 tE = 1.44 0.42 d BLG501.12.117610 17:51:47.10 –29:58:56.4 binary ± BLG504.03.214906 17:58:00.12 –28:20:29.5 binary BLG511.20.32339 18:03:57.24 –27:10:01.3 binary BLG511.26.45384 18:05:07.84 –26:59:57.6 binary BLG534.09.125022 17:53:52.61 –31:07:15.7 binary BLG500.17.22074 17:55:08.51 –28:26:25.0 artifact Table 4.1. Candidates for short-timescale events from the 2016 season. Only one event (BLG512.24.154394 = OGLE-2016-BLG-1540) exhibited finite source effects and was analyzed in detail.

and to reject events with asymmetric light curves, which are likely caused by binary lenses (Figure 4.1). All candidate events are listed in Table 4.1. We fitted point-source point-lens and extended-source point-lens models to the light curves and estimated the uncertainties of model parameters using the Markov chain Monte Carlo technique. This procedure led to the identification of one microlensing event, BLG512.24.154394 (OGLE-2016-BLG-1540) that had a short timescale (t 0.5 d) and showed very strong finite source effects. This event E ≤ was probably caused by a Neptune-mass free-floating or wide-orbit planet, as inferred from the measurement of the angular Einstein radius of the lens. The detailed analysis of this event is presented below. Our search revealed two additional short-timescale events (BLG505.30.22841 with a timescale of 0.54+0.72 d and BLG512.10.215221 with a timescale of 0.075 d), but their −0.31 ≈ light curves did not exhibit finite source effects. Encouraged by the discovery of OGLE-2016-BLG-1540, we searched for short timescale events in the OGLE data from the 2017 and 2018 observing seasons and again we complemented them with photometric observations from the KMTNet survey (see Tables 4.2 and 4.3 for all candidates). Additional observations were crucial for robust characterization of candidate events, because the cadence of OGLE observations in fields BLG501, BLG505, and BLG512 (Figure 2.2) has been decreased to 60 minutes since 2017. In turn, our sensitivity

63 BLG505.14.65844 BLG511.26.45384 18.0 OGLE OGLE 18.2 KMT SSO KMT SSO KMT CTIO 15.70 KMT CTIO KMT SAAO KMT SAAO 18.4

18.6 15.75 18.8 magnitude magnitude

I 19.0 I 15.80

19.2

19.4 15.85 7490 7492 7494 7546 7548 7550 HJD-2450000 HJD-2450000

Figure 4.1. We selected candidate short-timescale microlensing events using OGLE data, but additional photometric observations from the KMTNet survey were used to vet all light curves. For example, event BLG505.14.65844 is likely a flaring star (the rise to the maximum is very fast) and BLG511.26.45384 is a binary microlensing event. KMTNet has three telescopes: at Siding Spring Observatory, Australia (KMT SSO), Cerro Tololo Inter-American Observatory, Chile (KMT CTIO), and the South African Astronomical Observatory, South Africa (KMT SAAO). to short-timescale events also decreased. Nonetheless, we found another microlensing event (BLG534.29.86668 = OGLE-2017-BLG-0560) that exhibited strong finite source effects and we were able to measure its angular Einstein radius, as described below. The only short-duration event in the 2018 data set was BLG534.19.69242 (OGLE-2018-BLG-0850) with a timescale t = 0.25 0.10 d, but the finite source effects were not detected. E ± Microlensing events OGLE-2016-BLG-1540 and OGLE-2017-BLG-0560 share a number of similarities. Both events occurred on bright giants (with estimated angular radii of 15.1 and 34.9 µas, respectively, see Section 4.5) and both events show prominent finite source effects. We realized that when the angular size of the source star is much larger than the Einstein ring (as is the case for giants), the maximum magnification may be relatively small (see Figure 1.4) and the light curve of the event may look like a “reversed” planetary transit. Such low-amplitude events could have been missed during our analysis of 2010–2015 data. We therefore searched for short-duration, low-amplitude microlensing events with bright giant sources in the archival OGLE data collected during the 2010–2015 period. Our efforts proved to be successful, as we found that another event, OGLE-2012-BLG-1323, showed very strong finite source effects.

64 Event RA Decl. Remarks BLG500.18.168028 17:53:49.60 –28:29:19.1 tE = 2.0 0.5 d, KMTA data are noisy BLG501.02.62615 17:53:27.17 –30:22:31.8 binary ± BLG505.06.142911 17:55:59.81 –29:42:31.5 binary BLG505.15.89715 17:55:33.12 –29:19:20.6 tE = 61 25 d, no signal in KMT SSO ±+4.3 BLG505.30.176736 17:56:34.03 –28:45:14.0 tE = 8.3 2.0 d, no signal in KMT SAAO, high blending − +1.0 BLG506.07.88060 17:55:24.90 –30:49:19.8 tE = 6.4 0.8 d BLG506.28.175643 17:58:00.41 –30:00:15.1 asymmetric− light curve, no signal in KMT SAAO, KMT SSO BLG506.28.194135 17:58:02.89 –29:52:49.8 asymmetric light curve, no signal in KMT SSO BLG511.09.171553 18:04:52.43 –27:29:23.9 4.7 0.3 d BLG512.01.129014 18:05:02.56 –28:58:29.6 1.41± 0.38 d, no finite source effects BLG534.26.9868 17:54:18.23 –30:41:31.3 binary± BLG534.29.86668 17:51:51.33 –30:27:31.4 tE = 0.92 0.01 d, log ρ = 0.044 0.004, strong finite± source effects, variable− baseline± +10.8 BLG611.20.81000 17:36:19.13 –26:54:41.5 tE = 21.0 5.2 d − Table 4.2. Candidates for short-timescale microlensing events from the 2017 season. Only one event (BLG534.29.86668 = OGLE-2017-BLG-0560) exhibited finite source effects and was analyzed in detail.

Event RA Decl. Remarks BLG611.28.128718 17:36:00.70 –26:48:37.0 not in the KMTNet footprint BLG500.26.9208 17:54:23.15 –28:11:28.3 tE = 12.9 3.0 d ± BLG534.19.69242 17:53:21.83 –30:56:15.7 tE = 0.25 0.10 d, no ﬁnite source effects, anomaly near± the peak (could be a wide orbit planet) BLG654.29.76828 17:35:17.30 –29:27:37.0 tE = 11.2 5.0 d +9±.5 BLG504.17.123564 18:00:13.70 –27:50:56.3 tE = 17.4 5.5 d +11− .7 BLG513.02.75022 18:04:29.97 –30:19:57.8 tE = 17.6 6.2 d − BLG500.12.57147 17:52:01.70 –28:46:02.0 tE = 2.91 0.18 d +0±.28 BLG501.14.29528 17:50:41.64 –29:51:42.7 tE = 1.95 0.15 d BLG506.22.9351 17:56:59.23 –30:21:00.6 ﬂaring star− +0.79 BLG506.23.56824 17:56:13.48 –30:19:48.6 tE = 8.53 0.65 d +10− .0 BLG511.04.70951 18:03:09.50 –27:51:00.9 tE = 19.7 7.4 d, asymmetric light curve − Table 4.3. Candidates for short-timescale events from the 2018 season.

Survey Latitude Longitude Telescope Events OGLE 29.0097 70.7017 Warsaw (1.3 m) OB121323,OB161540,OB170560 KMT CTIO −30.1697 −70.8065 KMT CTIO (1.8 m) OB161540,OB170560 KMT SSO −31.2733− 149.0644 KMT SSO (1.8 m) OB170560 KMT SAAO −33.9347 18.4776 KMT SAAO (1.8 m) OB161540,OB170560 MOA −43.9867 170.4650 MOA-II (1.8 m) OB121323 Wise −30.5973 34.7623 Wise (1.0 m) OB121323 Table 4.4. List of survey telescopes used for observations of free-ﬂoating planet candidates. OBNNMMMM stands for event OGLE-20NN-BLG-MMMM.

65 Wise

OGLE KMT SSO KMT CTIO KMT SAAO MOA

Figure 4.2. Survey telescopes used for observations of free-ﬂoating planet candidates.

Because the angular size of the source was about ﬁve times larger than the Einstein ring, the amplitude of the event was smaller than 0.1 mag, and so it could not have been identiﬁed previously. The KMTNet telescopes were not operational in 2012, but we used additional photometric observations from MOA and Wise surveys to characterize the event. The list of all telescopes used in the analysis is presented in Table 4.4. Locations of observatories are also shown in Figure 4.2.

4.3. Data

Microlensing event OGLE-2012-BLG-1323 was discovered by the OGLE Early Warning System (Udalski 2003) on 2012 August 21. This event is located at equatorial coordinates of R.A. = 18h00m18.s51, Dec. = 28◦3500100.7 (J2000.0) (Galactic coordinates l = 1.939◦, − b = 2.594◦) in the OGLE field BLG512, which was monitored with a cadence of 20 minutes. − This event was not previously identified as a free-floating planet candidate (Mróz et al. 2017a), owing to its extremely low amplitude (below 0.1 mag). OGLE observations (Figure 4.3) showed that the event was magnified during two nights 2012 August 20/21 and 21/22, albeit with a very small amplitude (about 0.08 mag). Unfortunately, the tails of the light curve were not observed and there were no observations collected between August 22 and 25, due to bad weather in Chile. Our preliminary model,

66 based on OGLE data only, gave a timescale of t 0.16 d and angular Einstein radius of about E ≈ θ 2.7 µas, which could have corresponded to an Earth-mass lens and warranted thorough E ≈ investigation. To characterize the event we supplemented OGLE observations with the data from the Microlensing Observations in Astrophysics (MOA) (Bond et al. 2001; Sumi et al. 2013) and Wise groups (Shvartzvald et al. 2016), which happened to observe this microlensing event. The MOA observations were carried out using the 1.8 m MOA-II telescope located at the Mount John University Observatory in New Zealand. The telescope is equipped with the wide-ﬁeld camera, which consists of ten 2000 4000 CCD detectors with the pixel scale of 0.5800 per pixel × (Sako et al. 2008). The median seeing for this data set was 2.000. The MOA observations were reduced by Takahiro Sumi and Ian A. Bond using the custom implementation of the difference image analysis technique (Bond et al. 2001). For the modeling, we used observations collected between July 1 and October 20, 2012. The typical cadence of MOA observations is 15 minutes. Wise observations were taken with the 1 m telescope at Wise Observatory in Israel equipped with the Large Area Imager for the Wise Observatory (LAIWO) camera, with a 1 square degree ﬁeld of view. Although the telescope is located in the northern hemisphere, the Wise group secured 5–6 data points per night during the event. All Wise data were reduced by Yossi

Shvartzvald using the PYSIS software (Albrow et al. 2009; Albrow 2017). The entire data set (Figure 4.3) shows the power of high-cadence continuous survey observations, with Wise covering the rise from the baseline, OGLE – the finite source effects over the peak, and MOA – the fall to the baseline. Microlensing events OGLE-2016-BLG-1540 and OGLE-2017-BLG-0560 were also discovered by the OGLE Early Warning System (Udalski 2003) on 2016 August 6 and 2017 April 16, respectively. The former event was located at equatorial coordinates of R.A. = 18h00m47.s00 and Decl. = 28◦2103500.2 (J2000.0) (Galactic coordinates l = 2.186◦, b = − 2.574◦) in the field BLG512, which was observed with a cadence of 20 minutes. The OGLE − light curve (Figure 4.4) shows that the event peaked in the night 2016 August 5/6. OGLE has also captured the rise to the peak and fall to the baseline during the nights August 4/5 and 6/7. The second event (OGLE-2017-BLG-0560) was located at equatorial coordinates of R.A. = 17h51m51.s33, Dec. = 30◦2703100.4 (J2000.0) (Galactic coordinates l = 0.607◦, b = 1.945◦) − − − in the field BLG534, which was observed with a cadence of 60 minutes. This event was magnified during two nights (2017 April 13/14 and 14/15, see Figure 4.5). There are no OGLE

67 15.30 OGLE OGLE-2012-BLG-1323 15.32 MOA Wise tE = 0.155 0.005 d 15.34 ± ρ = 5.03 0.07 15.36 ±

15.38

Magnitude 15.40

15.42

15.44

15.46 0.04 − 0.02 − 0.00 0.02 Residual 0.04 6157 6158 6159 6160 6161 6162 6163 6164 6165 HJD - 2450000

Figure 4.3. Light curve of microlensing event OGLE-2012-BLG-1323. The light curve can be accurately described using the extended-source point-lens model (black solid line). This event was observed by OGLE (blue points), MOA (red), and Wise (green). All measurements were transformed to the OGLE magnitude scale.

observations collected between April 15/16 and 17/18 due to the passage of the Moon through the Galactic bulge fields. Both events were also observed by three identical 1.6 m telescopes from the KMT Network (Kim et al. 2016), which are located at the Cerro Tololo Inter-American Observatory (CTIO; Chile), the South African Astronomical Observatory (SAAO; South Africa), and the Siding Spring Observatory (SSO; Australia). Each telescope is equipped with a mosaic CCD camera, consisting of four 9k by 9k detectors with a pixel scale of 0.4000 per pixel and the total field of view of about 4 square degrees. The network started operations in 2015. The KMT data were reduced using their custom pipeline (Albrow 2017) and provided to us by Andrew Gould. OGLE-2016-BLG-1540 was located in two overlapping KMT fields BLG03 and BLG43, monitored with a cadence of 14 minutes. We omitted KMT SSO observations, because they did not cover the peak and did not contribute to constraining the model. We also excluded KMT SAAO data taken before August 3 or after August 16, because the baseline light curve showed systematic variability connected with passages of the Moon near the bulge fields. OGLE-2017-BLG-0560 was located in the two overlapping fields BLG01 and BLG41, each

68 14.2 OGLE I OGLE-2016-BLG-1540 KMT CTIO I 14.3 KMT CTIO V tE = 0.320 0.003 d ± KMT SAAO I ρ = 1.65 0.01 14.4 ±

14.5

Magnitude 14.6

14.7

14.8

0.04 − 0.02 − 0.00 0.02 Residual 0.04 7603 7604 7605 7606 7607 7608 7609 7610 HJD - 2450000

Figure 4.4. Light curve of microlensing event OGLE-2016-BLG-1540. The light curve can be accurately described using the extended-source point-lens model (black solid line in the I-band, gray dashed line in the V -band). I- and V -band models differ because of different limb-darkening profiles of the source star in two filters. This event was observed by OGLE (red points), KMT CTIO (green and orange), and KMT SAAO (blue). All measurements were transformed to the OGLE magnitude scale. observed with a cadence of 30 minutes. For the modeling, we used observations collected between March 7 and May 26, 2017. All data were taken in the I band except for MOA data; the MOA group uses a custom wide filter, which is effectively the sum of the standard R and I filters. Additionally, OGLE-2016-BLG-1540 was observed in the V -band by one of KMTNet telescopes (KMT CTIO). The KMT CTIO V - and I-band images were additionally reduced using DoPhot (Schechter et al. 1993), which allowed us to determine the source color. No magnified V -band observations were collected for OGLE-2012-BLG-1323 and OGLE-2017-BLG-0560.

4.4. Modeling

The light curves of all events (Figures 4.3, 4.4 and 4.5) can be accurately described using the extended-source point-lens model. We will describe the modeling procedures using OGLE-2016-BLG-1540 as an example. The model has four parameters: the time and projected

69 OGLE OGLE-2017-BLG-0560 13.95 KMT CTIO 14.10 KMT SAAO tE = 0.905 0.005 d ± KMT SSO ρ = 0.901 0.005 14.25 ±

14.40

Magnitude 14.55

14.70

14.85

15.00 0.04 − 0.02 − 0.00 0.02 Residual 0.04 7856 7857 7858 7859 7860 7861 7862 7863 HJD - 2450000

Figure 4.5. Light curve of microlensing event OGLE-2017-BLG-0560. The light curve can be accurately described using the extended-source point-lens model (black solid line). This event was observed by OGLE (blue points), KMT CTIO (red), KMT SAAO (green), and KMT SSO (orange). All measurements were transformed to the OGLE magnitude scale.

separation of closest approach of the source to the lens t0 and u0, the Einstein timescale tE, and the normalized angular radius of the source ρ = θ∗/θE (θ∗ is the angular radius of the source).

Two additional parameters (for each observatory), Fs and Fb, describe the source and

(unmagnified) blend fluxes, respectively. When we allowed Fb to vary, we found that in the best-fit solution the blend flux is negative (with the absolute value corresponding to a 16–17-mag star). Although such solutions are mathematically possible, this negative blending is too big to be due to normal fluctuations in the background. The best-fit solution is only

2 ∆χ = 5 better than the solution with fixed Fb = 0, which can easily be due to statistical noise, or possibly low-level systematics in the data. Given the absence of evidence for blending and the low prior probability for ambient superposed bright source, our best estimate for the blended light is zero, i.e., Fb = 0. The only way that the source flux enters the characterization of the lens is via θ∗ (see Section 4.5). To account for this, while we fix Fb = 0 in the fits, we also add in quadrature 0.05 mag to the uncertainty in centroiding the clump, when we compute our errors of these quantities.

70 Parameter OGLE-2012-BLG-1323 OGLE-2017-BLG-0560 OGLE-2016-BLG-1540 Microlensing model: t0 (HJD0) 6161.107 0.008 7859.523 0.003 7606.726 0.002 ± ± ± tE (days) 0.155 0.005 0.905 0.005 0.320 0.003 ± +0.64 ±+0.031 ± u0 0.63 0.44 0.105 0.045 0.53 0.04 ρ 5.03 −0.07 0.901 −0.005 1.65 ± 0.01 ± ± ± Is 15.43 0.05 14.91 0.05 14.76 0.05 ± ± ± fs 1.00 (fixed) 1.00 (fixed) 1.00 (fixed) Source star: IS,0 14.09 0.06 12.47 0.05 13.51 0.09 ± ± ± (V I)S,0 1.73 0.02 2.31 0.02 1.67 0.02 − ± ± ± (V K)S,0 3.77 0.03 4.73 0.06 3.67 0.03 − ± ± ± Teff (K) 3800 200 3600 200 3900 200 Γ (limb darkening, I band)± 0.40± 0.41± 0.36 Λ (limb darkening, I band) 0.30 0.28 0.34 θ (µas) 11.9 0.5 34.9 1.5 15.1 0.8 Physical∗ parameters: ± ± ± θE (µas) 2.37 0.10 38.7 1.6 9.2 0.5 1 ± ± ± µrel,geo (mas yr− ) 5.6 0.3 15.6 0.7 10.5 0.6 ± ± ± Table 4.5. Short-timescale microlensing events exhibiting finite source effects. 0 HJD =HJD–2450000. fs = Fs/(Fs + Fb) is the blending parameter.

Two additional (wavelength-dependent) parameters Γ and Λ may be used to describe the limb-darkening profile (Equation (1.20)). The two-parameter limb darkening law provides a more accurate description of a brightness profile than a simple linear law (e.g., Albrow et al. 1999; Fields et al. 2003; Abe et al. 2003). For OGLE-2016-BLG-1540 we used a fixed

ΓI = 0.36 and ΛI = 0.34 which correspond to the physical parameters of the source star (c.f., Section 4.5). When we allowed Γ and Λ to vary, we found Γ = 0.25 0.20 and Λ = 0.36 0.40, ± ± consistent at 1.5 2σ level with the adopted values. − The finite-source magnifications were calculated by the direct integration of formulae derived by Lee et al. (2009), which remain valid in the low-magnification regime (Section 1.4). The uncertainties were estimated using the Markov Chain Monte Carlo method using the

EMCEE code (Foreman-Mackey et al. 2013) and represent 68% conﬁdence intervals of marginalized posterior distributions. We used the same procedure for modeling OGLE-2012-BLG-1323 and OGLE-2017-BLG-0560. The best-ﬁtting parameters and their 1σ error bars are shown in Table 4.5. The archival light curve of OGLE-2017-BLG-0560 shows low-amplitude (0.02 mag), semi-regular variability that is typical of OGLE small amplitude red giants (Wray et al. 2004). The strongest pulsation period in the 2017 data is 18.9 d. As the effective duration of the event (3 days) is much shorter than the pulsation period, we expected that the inferred model

71 parameters should not be strongly influenced by the variability of the source. Additional modeling, in which we assumed that the flux of the source varies sinusoidally with a period of 18.9 d, results in almost identical microlensing parameters (within the error bars) to those of the model with the constant source. We also considered models with terrestrial parallax (Gould et al. 2009; Yee et al. 2009). For OGLE-2016-BLG-1540 the microlens parallax in the best-fitting solution was π = 3200 700, E ± but the χ2 improvement was modest (∆χ2 = 18). The parallax signal came mostly from one observatory (KMT CTIO) from one night and the OGLE data from that night did not provide strong evidence for parallax. Thus, the terrestrial parallax signal may be mimicked by some low-level systematics in the data and cannot be trusted. For OGLE-2012-BLG-1323, the χ2 improvement was insignificant (∆χ2 = 1) and the limits on the microlens parallax were very poor. We did not fit the parallax model to the light curve of OGLE-2017-BLG-0560 because of the low-level variability of the source. Finally, we also searched for possible binary lens models. Short-duration events may be caused by close binary lenses (when the projected separation s, in Einstein radius units, is much smaller than 1), when the source crosses a small triangle-shaped caustic that is far ( 1/s) from the center of mass (Figure 1.6). The expected light curves are asymmetric, ∼ unless the source is larger than the caustic. In that case the light curve may superficially look like an extended-source point-lens event, except that it has a more extended tail. We found that the best-fitting close binary models are disfavored by ∆χ2 of several hundred for OGLE-2012-BLG-1323 and even more for OGLE-2017-BLG-0560. The latter event has a large amplitude ( 1 mag), but the peak magnification in close binary models is usually much ≈ lower than that, unless the source is small (ρ < 0.001) and the light curve is asymmetric. We cannot rule out that the lens is a wide-orbit planet; we discuss these cases in Section 4.7.

4.5. Properties of source stars

Model parameters can be translated into physical parameters of the lens provided that the

angular radius θ∗ of the source star is known. Here we use a standard technique of estimating

θ∗ by measuring the offset of the source star from the centroid of red clump giants in the color–magnitude diagram (CMD) (Yoo et al. 2004). We will describe the method in detail for OGLE-2016-BLG-1540.

72 This event was observed in the V -band by the KMT CTIO on the peak night (Figure 4.4), which allowed us to measure the color of the source. Because the finite-source effects are prominent and the event may no longer be achromatic, we have not used the model-independent regression to estimate the source color. Instead, we calculated the color for each link of the MCMC chain. The procedure of model fitting, calculating the source color and the limb-darkening coefficients was iterated, until the color measurement converged. We created an instrumental CMD based on KMT CTIO data for stars in a 20 20 region × around the event (about 5 pc 5 pc at the Galactic center distance; Figure 4.6). We measured × the centroid of red clump stars and we calculated the offset of the source from the red clump centroid. The de-reddened brightness and color of the source are calculated under the assumption that the source suffers the same amount of extinction as the red clump stars. Indeed, the position of the source in the CMD suggests that it is a giant located in (or behind) the Galactic bulge (see Section 4.6 for a detailed discussion). If the source were a nearby M-dwarf, it would have absolute I-band magnitude of 9.66 (Pecaut & Mamajek 2013) and would be located at a distance of 106 pc, which contradicts the Gaia DR2 parallax ( 0.76 0.14 mas). − ± We found that the source is ∆(V I) = 0.61 0.02 redder and ∆I = 0.85 0.09 − ± − ± brighter than the red clump (Figure 4.6). Assuming the intrinsic color of (V I) , = 1.06 − RC 0 of red clump stars (Bensby et al. 2011) and their mean de-reddened brightness in this direction of I , = 14.36 (Nataf et al. 2013), we calculated the intrinsic brightness I , = 13.51 0.09 RC 0 S 0 ± and color (V I) , = 1.67 0.02 of the source. − S 0 ± We then found (V K) , = 3.67 0.03 from color–color relations from Bessell & Brett − S 0 ± (1988) and estimated the angular radius of the source star θ∗ = 15.1 0.8 µas from color–surface ± brightness (CSB) relation for giants (Kervella et al. 2004). The latter estimate allowed us to measure the angular Einstein radius

θ = θ∗/ρ = 9.2 0.5 µas (4.2) E ± and the relative lens-source proper motion (in the geocentric frame)

−1 µ , = θ /t = 10.5 0.6 mas yr . (4.3) rel geo E E ±

73 12

red clump 13 source

15 -0.85 mag

16 0.61 mag

17 (instrumental) I 18

21 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 − − − − V I (instrumental) −

Figure 4.6. The KMT CTIO color–magnitude diagram for stars in a 20 20 region around OGLE-2016-BLG-1540. The source (blue square) is ∆(V I) = 0.61× 0.02 redder and ∆I = 0.85 0.09 brighter than the red clump centroid− (red disk). As± the de-reddened brightness− and± color of red clump stars are known (Bensby et al. 2011; Nataf et al. 2013), we can calculate the de-reddened brightness and color of the source, assuming the extinction is the same.

The heliocentric correction (v⊕,⊥πrel/au, where v⊕,⊥ is the Earth’s velocity projected on the −1 sky), which should be added vectorially, is of the order of 3πrel yr and is negligible unless the lens is nearby (closer than 1 kpc from the Sun). We can also estimate the effective temperature of the source of T = 3900 200 K using eﬀ ± the color–temperature relations of Houdashelt et al. (2000a,b) and Ramírez & Meléndez (2005). The corresponding limb-darkening coefﬁcients (Claret & Bloemen 2011) are:

ΓI =0.36 ΛI =0.34

ΓV =0.94 ΛV = 0.21 −

We used ATLAS models and assumed a solar metallicity, microturbulent velocity of 2 km/s and surface gravity of log g = 2.0. Limb-darkening coefﬁcients (c, d) from Claret & Bloemen (2011) were transformed to (Γ, Λ) using formulae derived by Fields et al. (2003).

74 We used the same procedure to estimate the angular Einstein radii of two remaining microlensing events OGLE-2012-BLG-1323 and OGLE-2017-BLG-0560, although with one important difference. Because we lacked color observations collected during these two events, our best estimate for the color of the source is the color of the baseline star. This assumption is true for OGLE-2016-BLG-1560. Furthermore, it is supported by the lack of evidence for blending in the I-band light curves and the low probability of bright unmagniﬁed blends. Both sources are giants likely located in the Galactic bulge (Figure 4.7). If sources were M dwarfs, they would have absolute I-band magnitudes of 9.75 (OGLE-2012-BLG-1323) and 13.90 (OGLE-2017-BLG-0560) (Pecaut & Mamajek 2013) and would be located at a distance of 140 pc and 16 pc, respectively, which contradicts the Gaia DR2 parallaxes (0.15 0.14 mas ± and 0.23 0.19 mas, respectively). ± As both sources are very red, it is important to determine how well the empirical CSB relations are calibrated in this range. The relation of Kervella et al. (2004) was derived for giants with colors 0.9 < (V K) < 2.5, but it agrees well with the earlier relation by − 0 Fouque & Gieren (1997), which is valid in a wider color range. Groenewegen (2004) published a CSB relation for M giants (3.2 < (V K) < 6.1), which gives angular radii that are − 0 systematically 10% lower than those based on Kervella et al. (2004): θ∗ = 10.9 0.7 µas ± (OGLE-2012-BLG-1323) and θ∗ = 29.8 1.9 µas (OGLE-2017-BLG-0560). Adams et al. ± (2018) recently published a new CSB relation for giants ( 0.01 < (V I) < 1.74), − − 0 from which we ﬁnd θ∗ = 11.5 0.9 µas (OGLE-2012-BLG-1323) and θ∗ = 32.3 2.3 µas ± ± (OGLE-2017-BLG-0560), in good agreement with our determination. The physical parameters of the source star and lens are given in Table 4.5.

4.6. Proper motion of source stars

As source stars are relatively bright and in all cases the contribution of the lens to the total light is negligible, it is possible to measure the absolute proper motion of the source. The proper motion of OGLE-2016-BLG-1540 was measured using the OGLE-IV data by Jan Skowron. For the measurement, we chose a subset of 363 best-seeing (0.7 100) and low-background images − (out of 11,276 total epochs of this ﬁeld) spanning 2010–2017 taken with the 24th CCD detector of the OGLE-IV camera. We used the CMD to identify 3818 candidate red clump stars, which served as anchors for the coordinates transformations between the CCD frames. This allowed us to measure proper motions with respect to the mean motion of the Galactic bulge.

75 12 red clump OGLE-2012-BLG-1323 13 source

16 (OGLE-IV) I 17

20 0.5 1.0 1.5 2.0 2.5 3.0 3.5 V I (OGLE-IV) −

13 red clump OGLE-2017-BLG-0560 14 source

17 (OGLE-IV) I 18

21 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 V I (OGLE-IV) −

Figure 4.7. OGLE-IV color–magnitude diagrams for stars in 20 20 regions around OGLE-2012-BLG-1323 and OGLE-2017-BLG-0560. Sources are marked× with blue squares and are likely located in the Galactic bulge.

76 10 12

red clump 13 source

5 14 ]

1 15 − r y

s 0 a 16 m [ (OGLE-IV) N I

µ 17

5 18

Red clump stars (3722) Main Sequence stars (1251) 19 10 source 20 10 5 0 5 10 0.5 1.0 1.5 2.0 2.5 3.0 3.5 µ [mas yr 1 ] V I (OGLE-IV) E − −

Figure 4.8. Left panel: Proper motions of stars in the OGLE-2016-BLG-1540 ﬁeld (90 180). Orange contours mark proper motions of red clump stars (bulge population), black contours× mark main-sequence stars (which represent the Galactic disk population). The dashed red line shows the direction of increasing Galactic longitude and the proper motion of the source star is marked with a blue dot. The gray dashed circle corresponds to the relative proper motion of 10.5 mas yr−1 with respect to the source. As the lens should be located on this circle, it likely belongs to the Galactic disk population. Gray crosses mark stars located near the source in the CMD; they follow the bulge distribution. Right panel: OGLE-IV color–magnitude diagram for stars in a 40 40 region around OGLE-2016-BLG-1540. The source (blue square) is located in a relatively lowly× populated region of the diagram. Blue and red areas mark stars used for the proper motion measurements (see Section 4.6).

We used the DoPhot PSF photometry package (Schechter et al. 1993) to measure positions of all stars in all 363 images. Then, we calculated the third-order polynomial coordinate transformations between each frame and the first frame by minimizing the scatter for the anchor red clump stars. The proper motions were fitted with the least-squares method (with outlier rejection). The formal uncertainties of the fit were typically 0.2 0.3 mas yr−1. However, − the comparison with proper motion measurements based on the OGLE-III data (2001–2009) showed discrepancies larger than the pure statistical error. We decided to employ 0.5 and 0.7 mas yr−1 for N and E directions as our measure of uncertainty; hence, the proper motion of the source is µ = (µ , µ ) = ( 5.6 0.5, 3.0 0.7) mas yr−1 with respect to the Galactic S N E − ± − ± bulge (see Figure 4.8). As both the position of the source star on the CMD is uncommon (the star is located below the red giant branch and redwards of red clump giants) and its proper motion is counter to the Galactic rotation, we consider whether this evidence indicates that the source belongs to the far

77 disk population. First, we investigated the CMD position of the source. We identified about 45 stars with similar CMD positions (the analyzed region is marked with a blue rectangle in left panel of Figure 4.8) and measured their kinematics (their proper motions are marked with gray crosses in Figure 4.8), finding that these are consistent with all other red giant (i.e., bulge) stars in the field1. Thus, the unusual position of the source star on the CMD cannot be taken as evidence for belonging to some other population. In 2018, the Gaia consortium published astrometric solutions (parallaxes and proper motions) of over 1.3 billion sources as part of the second Gaia data release (DR2) (Gaia Collaboration et al. 2016, 2018b). Unfortunately, the Gaia performance in the crowded regions of the Galactic center is poor, especially for faint sources. Objects located in the Galactic center may suffer from crowding effects, because their observations are contaminated by near neighbors, which may lead to spurious astrometric parameters. Nonetheless, we found that OGLE-2016-BLG-1540 is included in the Gaia DR2 as DR2 4062678804506277376. It has

−1 −1 the absolute proper motion of µα = 5.45 0.23 mas yr and µδ = 10.44 0.20 mas yr , − ± − ± which corresponds to µ = 5.0 0.3 mas yr−1 and µ = 2.7 0.3 mas yr−1 relative to the N − ± E − ± Galactic bulge stars. This is consistent with our ﬁndings based on OGLE-IV observations. The proper motions of OGLE-2012-BLG-1323 and OGLE-2017-BLG-0560 were taken from the Gaia DR2. Figure 4.9 shows proper motions of stars located within 40 of the sources. In both cases source proper motions are consistent with those of Galactic bulge stars (represented by red clump and red giant branch stars), although proper motions measured relative to the mean velocity of bulge stars are high: (µ , µ ) = ( 4.6 0.2, 0.5 N E − ± ± 0.2) mas yr−1 for OGLE-2012-BLG-1323 and (µ , µ ) = ( 6.2 0.3, 0.8 0.3) mas yr−1 N E − ± ± for OGLE-2017-BLG-0560. This contributes to the high relative lens-source proper motion of OGLE-2017-BLG-0560 and its very short timescale.

4.7. Limits on stellar companions

Additional features in the event light curves would have been detected if the trajectory of the source were fortunate enough to pass near a putative host star (see Figure 4.10). Because the light curves do not exhibit any signatures of the host star, we can only provide a lower limit

1 We compared both distributions of proper motions using the two-sample Anderson–Darling test and found p-values of 0.21 (for µl component) and 0.65 (µb). Similarly, the two-sample Kolmogorov–Smirnov test yields p-values of 0.29 and 0.34, respectively. Therefore, there is no evidence that these distributions are different.

78 20 OGLE-2012-BLG-1323 main sequence giants 15 source

0 [mas/yr] N µ 5 −

10 −

15 −

20 − 20 15 10 5 0 5 10 15 20 − − − − µE [mas/yr]

20 OGLE-2017-BLG-0560 main sequence giants 15 source

0 [mas/yr] N µ 5 −

10 −

15 −

20 − 20 15 10 5 0 5 10 15 20 − − − − µE [mas/yr]

Figure 4.9. Gaia DR2 proper motions of stars within 40 of OGLE-2012-BLG-1323 (top) and OGLE-2017-BLG-0560 (bottom). Blue contours correspond to the main-sequence stars (Galactic disk population) and red contours to giants (bulge population). Solid contours enclose 68% and 95% of all objects. The source is marked with a black dot. The black dashed circle corresponds to the relative source-lens proper motion of 5.6 mas yr−1 (top) and 15.6 mas yr−1 (bottom). 79 on the planet-host separation, using a variation of the method proposed by Gaudi & Sackett (2000). We will describe our procedure in detail for OGLE-2016-BLG-1540. We simulated artificial OGLE light curves (spanning from 2010 March 4, through 2017 October 10) for a given binary model, defined by three additional parameters as compared to the single-lens case: mass ratio q, star-planet separation s (expressed in Einstein radii of the system), and angle α between the source trajectory and binary axis. The remaining parameters were calculated based on the best-fitting single-lens model from Table 4.5. To make sure that the source always passes near the planet at a right distance and moment, we transformed parameters from the point lens model (i.e., relative to the planet; t0,pl, u0,pl, tE,pl, ρpl) to (t0, u0, tE, ρ) in the center-of-mass system of the binary. If q 1, the following transformation is valid: tE,pl tE = , √q ρ = ρ √q, pl (4.4)

t = t , t (s 1/s) cos α, 0 0 pl − E −

u = u , + (s 1/s) sin α. 0 0 pl − We ﬁtted binary and single lensing models to the artiﬁcial data and calculated the difference ∆χ2 = χ2 χ2 . We used q = 2 10−4, which corresponds to the Einstein radius of the single − binary × host of θE,host = θE/√q = 0.65 mas. If the lens is located in the Galactic disk (πrel = 0.1 mas),

the corresponding host mass is Mhost = 0.5 M . For each value of s, we simulated 180 light curves with uniformly distributed α [0, 2π], and calculated the probability of detecting the ∈ 2 2 host star as the fraction of light curves which fulﬁll ∆χ > ∆χthresh = 225. This probability drops below 90% when s > 5.1, which corresponds to projected separation of 15 au. The lower limit on the host separation is slightly weaker for larger mass ratios (because the host event is

−3 shorter). For q = 10 , corresponding to the 0.5 M host in the bulge (πrel = 0.01 mas), we found s > 4.8 (10 au). We note that the presence of a putative host may also be revealed by perturbations to the point-lens light curve due to the planetary caustic caused by the central star (Han & Kang 2003; Han et al. 2005). The angular size of the planetary caustic (relative to the Einstein radius of the planet) is 4/s2 0.16 for s 5. Because the source star is 10 times larger, the signatures of ≤ ≥ ∼ the caustic are washed out by the ﬁnite-source effects. We used a similar procedure to calculate constraints on the projected separation of the

host star for OGLE-2012-BLG-1323 and OGLE-2017-BLG-0560. We considered a 0.3 M

80 4 q = 10− α = 0.2π 4 0.5 q = 10− α = 0.1π 3 q = 10− α = 0.2π 0.4

A 0.3 5 log . 2 0.2

0.1

0.0 7200 7400 7600 7800 7605 7610 HJD-2450000 HJD-2450000

Figure 4.10. Additional features in the light curve of OGLE-2016-BLG-1540 would have been detected if the trajectory of the source were fortunate enough to pass near a putative host star. The light curves correspond to star-planet separation s = 5 and different mass ratios and trajectories of the source relative to the binary axis. The source always passes near the planet at a right distance and moment, so the three light curves overlap in the right panel.

1.0

0.8

0.6

OGLE-2012-BLG-1323 0.4 OGLE-2017-BLG-0560

Probability of0 detecting the. host2 star

0.0 1 2 3 4 5 6 7 8 9 10 Separation (Einstein radii of the host)

Figure 4.11. Probability of detecting the putative host star as a function of star-planet separation for OGLE-2012-BLG-1323 and OGLE-2017-BLG-0560. Solid curves correspond to the lens located in the Galactic bulge (πrel = 0.01 mas) and dashed curves to the lens in the Galactic disk (πrel = 0.1 mas).

81 host located either in the Galactic disk (πrel = 0.1 mas) or in the bulge (πrel = 0.01 mas), which corresponds to θE,host = 0.49 mas or 0.16 mas, respectively. Then, for each pair of p mass ratio q = θE/θE,host and separation s, we simulated 180 OGLE light curves (spanning from 2010 March 4 through 2017 October 30) with uniformly distributed α, and calculated the fraction of light curves that show signatures of the putative host star (see Figure 4.11). For OGLE-2012-BLG-1323 we ﬁnd a 90% lower limits of 11.8 au for the disk case (4.9 Einstein radii of the host) and 6.0 au for the bulge host (4.5 Einstein radii of the host). The formal 90% limits for OGLE-2017-BLG-0560 are 9.3 au and 3.9 au, respectively, but the sensitivity to additional anomalies in the light curve is reduced, owing to low-level variability of the source.

4.8. Discussion and conclusions

4.8.1. OGLE-2016-BLG-1540

Current microlensing surveys are capable of detecting free-ﬂoating planets down to Earth-mass objects. To this day, however, all reported free-ﬂoating planet candidates were based

on the very short timescale of an event (tE . 2 days) and lacked direct measurements of the angular Einstein ring size (Sumi et al. 2011; Mróz et al. 2017a). OGLE-2016-BLG-1540 was the first case for which we procured such a measurement, owing to the fortuitous fact that the source was a giant. If the source were a dwarf (with at least ten times smaller angular radius), as in the case of ultrashort candidate events detected by Mróz et al. (2017a), the finite-source effects would be significantly weaker. We simulated the OGLE light curve and found that the finite-source model would be preferred only by ∆χ2 = 1.6 over the point-lens model. The short timescale of the event can be explained in part by the unusual kinematics of

−1 the system (see Figure 4.8). The source is moving at µS = 6.4 mas yr in the direction

opposite to the Galactic rotation and the relative lens-source proper motion is large (µrel = 10.5 0.6 mas yr−1). One possible explanation is that the source is located behind the Galactic ± center in the far disk, in which case we expect the proper motion direction to be opposite compared to closer stars. To test the “far disk” hypothesis, we have studied proper motions of stars located near the source in the CMD (Section 4.6). These stars follow exactly the same distribution of proper motions as the bulge stars. It appears that, although the source proper motion has an unusual direction, the source belongs to the bulge population.

82 The large lens-source proper motion indicates that the lens is moving in the opposite

−1 direction than the source (along the Galactic rotation) at µL & 5 mas yr relative to red clump

stars. The gray dashed circle in Figure 4.8 marks the relative proper motion of µrel = 10.5 mas yr−1 with respect to the source. As the lens should be located on this circle, it likely belongs to the Galactic disk population. Only 15% of bulge stars (58% of the disk stars) are located outside the dashed circle in Figure 4.8, i.e., their proper motions with respect to the source star are higher than µrel.

Because the distance to the lens, and so the relative parallax πrel, is unknown, we cannot uniquely measure the lens mass:

2 θE 0.1 mas M = = 35 M⊕ (4.5) κπrel πrel If the lens is located in the disk (π 0.1 mas), it should be a Neptune-mass planet. However, rel ≈ we cannot rule out that the lens belongs to the bulge population (π 0.02 mas), in which rel ≈ case it should be a Saturn-mass object. The geometry required for the lens to be a brown dwarf (π 10−3 mas; i.e., D D 60 pc) would require significant fine-tuning. We note rel . S − L . that the event occurred inside the K2C9 superstamp (Henderson et al. 2016), but the parallax measurement was impossible as the K2C9 campaign finished a few weeks before the event. Microlensing alone cannot distinguish between wide-orbit and unbound planets and, in principle, the lens may be located at a wide orbit, like Uranus or Neptune. Our lower limit for the planet-host separation is 5.1 Einstein radii, which corresponds to the projected separation of 15 au at πrel = 0.1 mas. Owing to the relatively large lens-source proper motion, any stellar companions to the lens can be detected in the future, when the lens and source separate. However, the brightness of the source will make detection of the putative host light difficult. The characterization of this event would have been impossible without nearly continuous observations from the OGLE and KMTNet surveys. This event also shows the importance of securing the color information for short-timescale events and anomalies (see discussion in Hwang et al. 2018). Although we could not have measured the precise mass of the lens, such measurements will be possible in the future with the Euclid (Penny et al. 2013) and WFIRST (Spergel et al. 2015) satellites, but will require simultaneous ground- and space-based observations (Gould & Yee 2013; Yee 2013; Zhu et al. 2016). Current ground-based experiments are already sensitive to ultrashort microlensing events, but a bigger sample is needed to fully understand their origin.

83 4.8.2. OGLE-2012-BLG-1323 and OGLE-2017-BLG-0560

These two microlensing events and OGLE-2016-BLG-1540 share a number of similarities (Table 4.5). All events occurred on bright giant stars (with estimated angular radii of 9.2 − 34.9 µas) and their relative lens-source proper motions are high (5.6 15.6 mas yr−1). All − three events show prominent finite source effects, which led to the measurement of the angular Einstein radius. The fact that all three events occurred on bright, large sources is surprising as less than 3% of all known events are found with sources brighter than I = 16 mag. Moreover, the microlensing event rate Γ is proportional to the area of the sky swept by the Einstein ring: Γ θ µ . The high lens-source relative proper motion makes an event more likely to be ∝ E rel −1 found, but events with µrel > 10 mas yr are very rare (Han et al. 2017). Strong finite source effects make the duration of an event longer, especially if ρ 1, which makes giant source events easier to detect. The typical timescale t∗ of such an event is comparable to the time needed for the lens to cross the chord of the source: s 2 u0 t∗ = 2t ρ 1 . (4.6) E − ρ Monitoring of giant-star microlensing events, as advocated by the Hollywood strategy of Gould (1997b), is therefore a promising way of studying free-floating planets.

On the other hand, the peak magniﬁcation Apeak in the absence of blending declines with the source size (Witt & Mao 1994; Gould & Gaucherel 1997): 2 A 1 + (forρ 1). (4.7) peak ≈ ρ2 The OGLE Early Warning System alerts events that brighten by at least 0.06 mag (Udalski

2003), which corresponds to ρ . 6, but my search algorithm (Mróz et al. 2017a) is sensitive to lower magniﬁcations. Equation (4.6) also explains why the impact parameter of OGLE-2012-BLG-1323 is poorly measured (cf. Table 4.5): for large sources (ρ = 5) changing

the impact parameter from u0 = 0 to u0 = 1 leads to an increase in tE of only 2%, which is already included in the reported uncertainties. WFIRST is expected to detect dozens of microlensing events caused by free-ﬂoating planets (Johnson et al. 2019, in preparation), but the inability to precisely measure the impact parameter may impact the space-parallax, and hence mass, measurements (Gould & Yee 2013; Yee 2013).

84 The mass of the lens depends on the angular Einstein radius θE and the relative parallax πrel: θ2 M = E . (4.8) κπrel For both events, the masses cannot be unambiguously determined because the microlens parallaxes cannot be measured. For lenses located in the Galactic disk (π 0.1 mas) rel ≈ the masses are 2.3 M⊕ and 1.9 MJup for OGLE-2012-BLG-1323 and OGLE-2017-BLG-0560, respectively. If the lenses are located in the Galactic bulge (π 0.01 mas), they have higher rel ≈ masses of about 23 M⊕ or 20 MJup, respectively. Regardless of the lens location, the ultrashort timescale event OGLE-2012-BLG-1323 (t = 0.155 0.005 d) is almost certainly caused by a E ± planetary-mass object (Earth- to super-Earth-mass). It is not possible to determine, without further high-resolution follow-up observations, whether these planets are free-ﬂoating or are located at very wide orbits. Owing to their high relative lens-source proper motions, such searches will be possible in the near future with current instruments or next-generation telescopes (Gould 2016a). As the sources are bright, separations of 100 mas are required to resolve the putative host stars; such separations will ∼ be reached in the late 2020s. Presently, there are no observational constraints on the frequency of bound Earth- and super-Earth-mass wide-orbit planets as their detection is challenging with the current techniques. For example, Poleski et al. (2014b) found a 4 MUranus planet at projected separation of 5.3 Einstein radii and Sumi et al. (2016) discovered a Neptune analog at projected separation of 2.4 Einstein radii. Recently, the DSHARP project (Andrews et al. 2018) published deep, high resolution images of 20 nearby protoplanetary disks, which show annular substructures that are believed to result from planet-disk interactions (Huang et al. 2018; Zhang et al. 2018). Using these observations, Zhang et al. (2018) found that about 50% of analyzed objects host a Neptune to Jupiter mass planet beyond 10 au. Planet-formation theories, such as the core accretion model (Ida & Lin 2004), predict very few low-mass planets at wide orbits because the density of solids and gas in a protoplanetary disk is very low at such large separations. It is believed that Uranus and Neptune formed closer to the Sun, near Jupiter and Saturn, and subsequently migrated into wide orbits (Thommes et al. 1999, 2002). Likewise, multiple protoplanets of up to a few Earth masses can be scattered to wide orbits and eventually ejected by growing gas giants (e.g., Chatterjee et al. 2008; Izidoro et al. 2015; Bromley & Kenyon 2016; Silsbee &

85 Tremaine 2018). From the point of view of microlensing observations, these objects, whether bound or free-floating, are in practice indistinguishable. While making statistical inferences out of such a small sample of events is risky, we show that these detections are consistent with low-mass lenses being common in the Milky Way, unless it is just a coincidence that the events occurred on bright giant stars. According to models presented by Mróz et al. (2017a), about 2.8 10−3 of all events should be caused by Earth- and × super-Earth-mass lenses (on timescales tE < 0.5 d) if there were one such object per each star. About 50 events with giant sources brighter than I = 16 are found in OGLE high-cadence fields annually, thus we expect to find 2.8 10−3 50 = 0.1 very short microlensing events with giant × × sources annually (about one event during the entire OGLE-IV time span) if the probability of detection is the same for events due to free-floating planets and stars. In reality, the detection efficiency for bright events on timescales of O(1 d) is a factor of 2 4 lower than for stellar − events (Mróz et al. 2017a). Thus, our findings support the conclusions of Mróz et al. (2017a) that such Earth-mass free-floating (or wide-orbit) planets are more common than stars in the Milky Way.

86 5. Microlensing optical depth and event rate from OGLE-IV

5.1. Motivation

The microlensing optical depth toward a given source star depends only on the distribution of matter along the line of sight: 4πG Z Ds D (D D ) l s l (5.1) τ(Ds) = 2 ρ(Dl) − dDl, c 0 Ds

where ρ(D) is the density of lenses and Dl and Ds are distances to the lens and source, respectively. As the optical depth is independent of the mass function and kinematics of lenses, its measurements allow us to study the distribution of stars and other compact objects toward the Galactic center. In practice, however, the observed optical depth is averaged over all detectable stars and so it may weakly depend on their mass function and the star formation history, as well as interstellar extinction: 1 Z ∞ τ = τ(Ds)dn(Ds), (5.2) Ns 0

where dn(Ds) is the number of detectable sources in the range [Ds,Ds + dDs] and Ns = R ∞ 0 dn(Ds) (Kiraga & Paczynski´ 1994). The differential microlensing event rate toward a given source is: d4Γ µ (5.3) 2 = 2rEvreln(Dl)f( rel)g(M), dDldMd µrel

where M is the lens mass, rE = DlθE is its Einstein radius, n(D) is the local number density

of lenses, v = Dl µrel is the lens-source relative velocity, f(µrel) is the two-dimensional rel | | probability density for a given lens-source relative proper motion µrel, and g(M) is the mass function of lenses (Batista et al. 2011). Contrary to the optical depth, the event rate explicitly depends on the mass function of lenses and their kinematics.

87 Collaboration Location Optical Depth Nstars Nevents ∆T Source 6 6 (l, b)( 10− )( 10 ) (yr) × × OGLE-I (1◦, 4◦) 3.3 1.2 0.95 9 2 Udalski et al. (1994c) − ± MACHO (2.30◦, 2.65◦) > 1.3 0.43 4 1 Alcock et al. (1995) − +1.8 MACHO (2.55◦, 3.64◦) 3.9 1.2 1.3 45 1 Alcock et al. (1997b) − −+0.39 MACHO (2.68◦, 3.35◦) 2.43 0.38 17 99 3 Alcock et al. (2000) − − MACHO (3.9◦, 3.8◦) 2.0 0.4 2.1 52 5 Popowski et al. (2001) − ±+0.34 MACHO (2.22◦, 3.18◦) 2.01 0.32 17 99 2 Popowski (2002) − +0− .84 MOA (3.0◦, 3.8◦) 2.59 0.64 230 28 2 Sumi et al. (2003) − − EROS-2 (2.5◦, 4.0◦) 0.94 0.29 1.42 16 3 Afonso et al. (2003) − ±+0.47 MACHO (1.50◦, 2.68◦) 2.17 0.38 6 62 7 Popowski et al. (2005) − +0− .57 OGLE-II (1.16◦, 2.75◦) 2.55 0.46 1.5 32 4 Sumi et al. (2006) EROS-2 EROS-2− fields 1.68 − 0.22 5.6 120 6 Hamadache et al. (2006) ±+0.15 MOA-II MOA-II fields 1.87 0.13 90.4 474 2 Sumi et al. (2013) +0− .12 MOA-II MOA-II fields 1.53 0.11 110.3 474 2 Sumi & Penny (2016) − Table 5.1. Previous measurements of the microlensing optical depth toward the Galactic bulge.

Direct studies of the central regions of the Milky Way are difﬁcult because of high interstellar extinction and crowding. Precise measurements of the microlensing optical depth and event rate toward the Galactic center, although difﬁcult, provide strong constraints on theoretical models of the Galactic structure and kinematics (e.g., Han & Gould 2003; Wood & Mao 2005; Kerins et al. 2009; Awiphan et al. 2016; Wegg et al. 2016; Binney 2018). From the observational point of view, the optical depth can be estimated using the following formula that was derived by Udalski et al. (1994c):

π X tE,i τ = , (5.4) 2N ∆T ε(t ) s i E,i where Ns is the total number of monitored source stars, ∆T is the duration of the survey, tE,i is the Einstein timescale of the i-th event, and ε(tE,i) is the detection efficiency (probability of finding an event) at that timescale. The event rate is given by: 1 X 1 Γ = . (5.5) N ∆T ε(t ) s i E,i The first measurement of the microlensing optical depth toward the Galactic bulge was carried out by Udalski et al. (1994c) and was based on OGLE-I data from 1992–1993 (see Table 5.1 for a compilation of previous measurements). They found nine microlensing events in a systematic search of 106 light curves, and they calculated τ = (3.3 1.2) 10−6, which ∼ ± × was greater than contemporary theoretical estimates ((0.4 1) 10−6; Paczynski´ 1991; Griest − × et al. 1991; Kiraga & Paczynski´ 1994). A similar conclusion was reached by Alcock et al. (1995, 1997b) based on MACHO project observations of the Galactic bulge. These seminal papers boosted the development of the field, but as we now know, the calculated optical depths

88 should be treated with caution. The photometry was calculated using the point-spread function fitting method, which is more prone to systematic errors than the difference image analysis that is normally used in modern microlensing surveys. These first measurements of the optical depth led to the realization that most of the observed microlensing events are caused by lenses located in the Galactic bulge and that the inner regions of the Milky Way have a bar-like structure elongated along the line of sight (Paczynski´ et al. 1994; Zhao et al. 1995). The first measurements stimulated the development of improved models of the Galactic center (e.g., Zhao & Mao 1996; Fux 1997; Nikolaev & Weinberg 1997; Peale 1998; Gyuk 1999; Sevenster et al. 1999; Grenacher et al. 1999). Nonetheless, all of these models predicted the optical depth in the direction of MACHO fields in the range (1.1 2.2) 10−6, a factor of two–four lower than the reported values. − × The implementation of the difference image analysis technique (Alard & Lupton 1998) led to the improvement of the quality of the photometry in very dense stellar fields toward the Galactic center. This enabled the surveys to detect more microlensing events and to precisely measure their parameters. The optical depth measurements based on MACHO (2.43+0.39 10−6; −0.38× Alcock et al. 2000) and MOA-I (2.59+0.84 10−6; Sumi et al. 2003) data were still higher −0.64 × than the theoretical predictions. Binney et al. (2000) and Bissantz & Gerhard (2002) argued that such high optical depths cannot be easily reconciled with other constraints, such as the Galactic rotation curve and the mass density near the Sun. Nearly two decades later, Sumi & Penny (2016) suggested these measurements suffer from biased source star counts and are overestimated. Popowski et al. (2001) and Popowski (2002) noticed that previous microlensing optical depth measurements underestimated (or completely ignored) the influence of blending. The Galactic bulge fields are extremely crowded and there should be many faint unresolved stars within the seeing disk of any bright star. The omission of blending results in underestimated Einstein timescales. In highly blended events, as demonstrated by Wo´zniak& Paczynski´ (1997), the event timescale, impact parameter, and blending parameter may be severely correlated, which renders robust timescale measurements difficult (see also Figure 3.10). Popowski et al. (2001) proposed determining the microlensing optical depth using exclusively red clump giants as sources, because they are subject to little blending and it is easy to estimate their total number. Several measurements of the microlensing optical depth toward the Galactic bulge based on red clump giants were published by EROS (0.94 0.29 10−6, ± ×

89 Afonso et al. 2003 ; 1.68 0.22 10−6, Hamadache et al. 2006), MACHO (2.17+0.47 10−6; ± × −0.38 × Popowski et al. 2005) and OGLE-II (2.55+0.57 10−6; Sumi et al. 2006) groups. These estimates −0.46× were lower than those based on all-star samples of events (Alcock et al. 2000; Sumi et al. 2003). The current largest microlensing optical depth and event rate maps are based on two years (2006–2007) of observations of the Galactic center by the MOA-II survey (Sumi et al. 2013). Sumi et al. (2011) and Sumi et al. (2013) found over 1000 microlensing events in that data set, but only 474 events were used for the construction of event rate maps. All events are located in 22 bulge fields covering about 42 square degrees between 5◦ < l < +10◦ and − 7◦ < b < 1◦. Three years after the MOA-II publication, Sumi & Penny (2016) realized − − that the sample of red clump giants, which was used to scale the number of observed sources and thus optical depths and event rates, was incomplete, most likely due to crowding and high interstellar extinction. The completeness increased with the Galactic latitude – from 70% at b = 1.5◦ to 100% in fields located far from the Galactic plane (b = 6◦). This affected − − the measured optical depth and event rates, which were systematically overestimated at low Galactic latitudes. The revised all-source optical depth measurements were much lower than those published by Sumi et al. (2013), which alleviated (but did not completely remove) the tension with the previous measurements based on red clump giant stars (Popowski et al. 2005; Hamadache et al. 2006; Sumi et al. 2006). A similar bias may have affected the early MACHO and MOA measurements (Alcock et al. 2000; Sumi et al. 2003). Large samples of microlensing events were also recently reported by Wyrzykowski et al. (2015, 2016, OGLE-III), Navarro et al. (2017, 2018, VVV), and Kim et al. (2018a,b, KMTNet), but these authors did not attempt to calculate optical depths and event rates. The original MOA-II optical depth maps (Sumi et al. 2013) were used by Awiphan et al. (2016) to modify the Besançon Galactic model (Robin et al. 2014). For example, they needed to include M dwarfs and brown dwarfs in the mass function to match the timescale distribution of microlensing events. Awiphan et al. (2016) noticed that the predicted optical depths at low Galactic latitudes were about 50% lower than those reported by Sumi et al. (2013). This discrepancy can only be partially explained by Sumi & Penny (2016) findings; the theoretical optical depth is a factor 1.6 lower than the revised MOA-II measurements. The revised ∼ MOA-II data (Sumi & Penny 2016) were also used by Wegg et al. (2016) to constrain the dark matter fraction in the inner Galaxy.

90 The accurate microlensing event rates are also of interest for the astronomical community, for example, for the preparation of the future space-based microlensing surveys like the Wide Field Infrared Survey Telescope (WFIRST; Spergel et al. 2015) or Euclid (Penny et al. 2013). The current Galactic models are not precise enough to predict reliable event rates, and they have to be scaled to match the observations (Penny et al. 2013, 2019a). For example, Penny et al. (2019a) had to multiply the predicted rates by a factor of 2.11 to match Sumi & Penny’s (2016) results. All these model constraints and predictions are still based on a relatively small sample of microlensing events and many authors have raised the need for optical depths from the larger OGLE sample (e.g., Wegg et al. 2016; Penny et al. 2019a). In this Chapter, we address these needs.

5.2. Data

All photometric data analyzed in this Chapter were collected as part of the OGLE-IV sky survey (Udalski et al. 2015a) from 2010 to 2017. As explained in Chapter 2, we search for microlensing events in only 121 fields located toward the Galactic center that have been observed for at least two observing seasons (Figure 2.2). These fields cover an area of over 160 square degrees and contain over 400 million sources in OGLE databases. Fields are observed with different cadence and for different amount of time. Basic information about the fields (equatorial and Galactic coordinates, number of objects in databases, number of epochs) is presented in Table A.1 in Appendix A. Nine fields that are observed with the highest cadence (BLG500, BLG501, BLG504, BLG505, BLG506, BLG511, BLG512, BLG534, BLG611) have been already analyzed in Chapter 3 with the aim of measuring the frequency of free-floating planets. We use the sample of microlensing events presented in that Chapter to calculate optical depths and event rates. We also make use of image-level simulations that have been carried out to measure the detection efficiency of microlensing events. These data were collected between 2010 June 29 and 2015 November 8. Each light curve consists of 4,500–12,000 single photometric measurements, depending on the field. For the remaining 112 fields, we use data collected during a longer period (between 2010 June 29 and 2017 November 1) whenever available. Because of the changes in the observing strategy of the survey, some of these fields were observed for shorter periods of time (from two

91 to ﬁve Galactic bulge seasons). Seventy-six of these ﬁelds (or 68%), however, were monitored for nearly eight years. The majority of light curves consist of a hundred to two thousand data points.

5.3. Selection of events

The selection procedure and final selection cuts were similar to those used in Section 3.3 although with some small differences. Because the contamination from instrumental artifacts (such as reflections within the telescope optics) in the analyzed fields is much less severe than in high-cadence fields, we were able to relax the selection criteria compared to earlier work (Mróz et al. 2017a, Chapter 3). All criteria are summarized in Table 5.2. We began the analysis by correcting the reported photometric uncertainties using the procedure proposed by Skowron et al. (2016). The error bar correction coefficients were not available for 11 fields, and we closely followed Skowron et al. (2016) to calculate the missing values. Subsequently, we transformed magnitudes into flux. The search procedure consisted of three steps: 1. First, we searched for any kind of brightening in the light curves. We searched for at least

three consecutive data points that are at least 3σbase above the baseline flux Fbase. The baseline flux and its dispersion were calculated using data points outside a 720-day window centered on the event, after removing 5σ outliers (if the light curve was shorter than six years, we used a 360-day window instead). We required the light curve outside the window to be flat (χ2 /d.o.f. 2.0), which allowed us to remove the majority of variable stars out ≤ and image artifacts. We also required at least three magnified data points to be detected on subtracted images during the candidate event (n 3), meaning that the centroid of DIA ≥ the additional flux coincided with the source star centroid. That selection cut enabled us to remove any contamination from asteroids in some fields that have been observed a few times per night, as well as the contamination from spurious events and photometric artifacts. For P each candidate event, we calculated χ = (Fi F )/σi (the summation is performed 3+ i − base over all consecutive data points 3σ above the baseline). We required χ 32. These base 3+ ≥ simple selection criteria allowed us to reduce the number of candidate microlensing events from 350 million to 23,618. 2. Subsequent cuts were devised to remove any additional obvious non-microlensing light curves. We removed all objects with two or more brightenings in the light curve – mostly

92 Criteria Remarks Number 2 χout/dof 2.0 No variability outside the 720-day (or 360-day) window ≤ centered on the event nDIA 3 Centroid of the additional flux coincides with the source star ≥ centroid P χ3+ = (Fi Fbase)/σi 32 Significance of the bump 23,618 i − ≥ s < 0.4 Rejecting photometry artifacts A 0.1 mag Rejecting low-amplitude variables ≥ nbump = 1 Rejecting objects with multiple bumps 18,397 Fit quality: 2 2 χfit/dof 2.0 χ for all data 2 ≤ 2 χ /dof 2.0 χ for t t0 < tE fit,tE ≤ | − | 2455377 t0 2458118 Event peaked between 2010 June 29 and 2017 December 31 ≤ ≤ u0 1 The minimum impact parameter ≤ Is 21.0 The minimum I-band source magnitude ≤ Fb > 0.1 The maximum negative blend flux, corresponding − to I = 20.5 mag star fs > 0.01 Rejecting highly-blended events 5790 Table 5.2. Selection criteria for high-quality microlensing events for optical depth and event rate measurements.

dwarf novae and other erupting variable stars. We discarded all candidate events with amplitudes smaller than 0.1 mag to minimize the contamination from pulsating red giants. As in Section 3.3, we also removed all candidates that were located close to each other and were magnified in the same images – these are spurious detections caused by reflections within the telescope or non-uniform background. Examples of the light curves of rejected objects are provided in Figures 3.3 and 3.5. In this step, we removed 5221 objects from the sample. 3. Finally, we fitted the microlensing point-source point-lens model to the light curves of the remaining 18,397 candidates. The procedure is described in detail in Section 3.3. In short,

2 P 2 2 we found the model parameters that minimize the function χ = (Fi F (ti)) /σ , i − model i the summation is performed over all data points Fi σi. During the modeling, we iteratively ± removed any 4σ outliers provided that the adjacent data points were within 3σ of the model. To quantify the quality of the fit, we calculated 2 for the entire data set and 2 for data χfit χfit,tE points within t of the maximum of the event (i.e., t t < t ). We required χ2/d.o.f. 2. E | − 0| E ≤ We calculated five- and four-parameter models (with the blend flux set to zero). We allowed for some amount of the negative blending in five-parameter fits (F F , where F = 0.1 b ≥ − 0 0 is the flux corresponding to a 20.5-mag star). If F < F and the four-parameter model b − 0 was marginally worse (∆χ2 < 9) than the five-parameter model, we chose the former. We were left with 5790 objects, which will constitute our final sample of microlensing events used for the construction of optical depth and event rate maps in low-cadence fields.

93 The uncertainties of model parameters were estimated using the Markov chain Monte Carlo (MCMC) technique (Foreman-Mackey et al. 2013). To take into account our limits on negative blending, we added the following prior on the blend flux:  1 if Fb 0 prior = ≥ (5.6) F 2 L  b exp σ2 if Fb < 0 − 2 0 where σ0 = F0/3 (F0 = 0.1, which corresponds to I = 20.5). The best-fit parameters and their uncertainties are reported in Table 5.3. The uncertainties represent the 68% confidence range of marginalized posterior distributions. The purity of our sample of microlensing events is very high ( 99.5%). Figure 5.1 shows ∼ the distribution of fractional uncertainties of Einstein timescales. The median uncertainty is 16% and for 98% of events in the analyzed sample σ(t )/t 0.5. The error bars on t depend E E ≤ E mostly on the source brightness and impact parameter (amplitude of the event); the fainter the source is and the larger is the impact parameter, the larger are the uncertainties (Figure 5.2). Of the events from our sample, 3958 out of 5790 (68%) have been announced in a real-time by the OGLE Early Warning System (EWS) (Udalski 2003); the remaining 1832 events (32%) are new discoveries. For comparison, from 2011–2017, 6959 microlensing alerts in low cadence fields were announced by the EWS, and about 10–15% of these are anomalous or binary. We calculated more detailed statistics for the field BLG660 as an example. 180 candidate microlensing events were selected by our cut 1 criteria; the visual inspection of light curves showed that 138 were indeed microlensing events, while 125 were found by the EWS. Two objects reported in the EWS are not microlensing events (variable stars), and three were detected in “new” databases. Nine genuine EWS events were not identified by our search algorithm (mostly because of the variability in the baseline, the low significance of the event, or the small number of magnified data points), and 27 events were detected only by our search algorithm. Furthermore, 111 objects are common. Thus, our search algorithm was able to find (111 + 27)/(111 + 27 + 9) = 94% of events in that field. However, only 94 events from the field BLG660 (i.e., 64%) satisfied all our selection criteria and were included in the final sample of events. Half of the rejected events have very faint sources (I 21) and, as a consequence, s ≥ their parameters are not well measured. The remaining events are anomalous, do not fulfill the

2 constraints on t0 or u0, or their light curves are noisy and thus they do not meet the χ ﬁt quality criterion.

94 200 Median error: 16%

150

100

50 Number of events per bin

0 0.0 0.2 0.4 0.6 0.8 1.0 Fractional error σ(tE)/tE

Figure 5.1. Distribution of fractional uncertainties of Einstein timescales of microlensing events in low-cadence OGLE-IV ﬁelds. The median uncertainty is 16%, while σ(tE)/tE 0.5 for 98% of events in the analyzed sample. ≤

14 0.5

15 0.4 E s 16 /t I ) E t (

0.3 σ 17

18 0.2

Source magnitude 19 Fractional error 0.1 20

21 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Impact parameter u0

Figure 5.2. Fractional uncertainties of Einstein timescales as a function of the impact parameter and the brightness of the source. For a given source brightness, the larger the impact parameter, the larger the error bars. Similarly, for a given impact parameter, the fainter the source, the larger the error bars.

95 Star RA Decl. t0 (HJD) tE (d) u0 Is fs EWS ID h m s ◦ 0 00 +0.23 +4.55 +0.041 +0.31 +0.22 BLG617.16.73378 17 13 08.00 −29 48 13.0 2455434.91−0.22 24.41−2.85 0.206−0.045 20.11−0.23 0.91−0.22 h m s ◦ 0 00 +0.57 +18.49 +0.019 +0.14 +0.04 BLG617.24.41328 17 13 54.30 −29 36 35.5 2457462.79−0.66 199.66−17.42 0.139−0.016 20.29−0.16 0.25−0.03 OB160231 h m s ◦ 0 00 +0.28 +15.65 +0.010 +0.44 +0.39 BLG616.32.9872 17 13 56.45 −28 08 40.8 2455840.42−0.30 38.60−9.95 0.021−0.007 20.86−0.40 0.87−0.29 h m s ◦ 0 00 +0.24 +4.75 +0.048 +0.34 +0.28 BLG617.24.47717 17 13 56.92 −29 32 34.2 2457594.85−0.24 21.55−2.77 0.211−0.052 20.34−0.25 1.07−0.29 OB161459 h m s ◦ 0 00 +1.62 +18.74 +0.336 +1.01 +0.41 BLG617.07.43804 17 14 00.26 −30 08 42.2 2457842.43−1.62 38.77−8.75 0.691−0.317 19.74−0.83 0.36−0.22 OB170395 h m s ◦ 0 00 +0.52 +6.76 +0.068 +0.33 +0.26 BLG617.07.22676 17 14 06.09 −30 04 59.8 2456434.71−0.51 39.08−4.41 0.349−0.074 20.16−0.26 0.95−0.25 OB130673 h m s ◦ 0 00 +0.44 +5.42 +0.099 +0.43 +0.26 BLG617.15.7193 17 14 06.52 −29 56 21.5 2456038.68−0.43 26.22−3.22 0.438−0.111 19.60−0.33 0.72−0.24 OB120420 h m s ◦ 0 00 +0.03 +1.07 +0.008 +0.10 +0.04 BLG617.14.111445 17 14 18.39 −29 48 49.5 2455652.64−0.03 15.53−1.00 0.087−0.007 19.14−0.10 0.40−0.04 h m s ◦ 0 00 +0.01 +0.72 +0.001 +0.03 +0.03 BLG617.31.97087 17 14 18.45 −29 20 03.2 2456479.74−0.01 28.27−0.71 0.034−0.001 18.75−0.03 0.98−0.03 OB130992 h m s ◦ 0 00 +0.15 +2.18 +0.031 +0.43 +0.26 BLG617.14.111841 17 14 21.39 −29 50 01.3 2457834.40−0.13 7.97−1.46 0.056−0.024 19.76−0.38 0.61−0.20 OB170396 h m s ◦ 0 00 +0.52 +8.89 +0.060 +0.32 +0.20 BLG616.14.35260 17 14 40.27 −28 37 58.6 2456059.85−0.54 54.45−5.39 0.356−0.072 19.53−0.23 0.85−0.21 h m s ◦ 0 00 +0.62 +3.23 +0.144 +0.69 +0.30 BLG617.31.40734 17 14 41.78 −29 17 31.7 2457174.55−0.61 10.26−1.46 0.616−0.211 19.59−0.39 0.69−0.33 h m s ◦ 0 00 +0.81 +9.78 +0.209 +0.72 +0.33 BLG617.14.6899 17 14 47.57 −29 56 44.3 2456829.48−0.80 29.53−5.38 0.590−0.211 19.18−0.57 0.48−0.23 OB141193 h m s ◦ 0 00 +0.06 +6.88 +0.013 +0.27 +0.08 BLG617.31.20057 17 14 49.02 −29 12 40.2 2456785.64−0.07 34.09−5.22 0.052−0.011 20.70−0.25 0.29−0.06 OB140835 h m s ◦ 0 00 +0.05 +0.75 +0.051 +0.39 +0.20 BLG617.05.69384 17 15 09.42 −30 12 28.7 2456104.16−0.06 3.34−0.38 0.267−0.071 19.22−0.23 0.82−0.24 OB121027 h m s ◦ 0 00 +1.51 +50.51 +0.048 +0.30 +0.02 BLG617.05.57777 17 15 10.18 −30 20 07.5 2457830.83−1.63 270.00−40.90 0.186−0.040 20.37−0.30 0.07−0.02 OB170165 h m s ◦ 0 00 +0.38 +0.63 +0.017 +0.11 +0.05 BLG616.05.90565 17 15 10.89 −28 54 27.7 2455598.78−0.30 17.17−0.43 0.095−0.025 17.00−0.06 0.94−0.09 h m s ◦ 0 00 +0.13 +2.52 +0.018 +0.17 +0.20 BLG617.22.46738 17 15 19.23 −29 36 58.2 2456574.28−0.13 21.19−2.00 0.128−0.017 19.89−0.15 1.38−0.20 h m s ◦ 0 00 +0.02 +0.66 +0.004 +0.04 +0.03 BLG617.22.38934 17 15 19.76 −29 38 32.3 2457132.60−0.02 25.79−0.60 0.118−0.004 17.88−0.04 0.93−0.03 OB150455 h m s ◦ 0 00 +0.07 +2.30 +0.007 +0.12 +0.11 BLG617.05.8156 17 15 31.67 −30 16 59.5 2456436.94−0.08 27.86−1.77 0.068−0.007 19.85−0.10 1.15−0.12 OB130699 h m s ◦ 0 00 +0.54 +2.88 +0.018 +0.08 +0.06 BLG617.12.109866 17 15 40.52 −29 49 51.4 2457845.65−0.52 81.52−2.57 0.414−0.020 18.65−0.07 1.04−0.07 OB170208 h m s ◦ 0 00 +0.70 +5.11 +0.034 +0.16 +0.11 BLG617.29.115121 17 15 42.04 −29 12 06.9 2456143.38−0.75 75.17−3.59 0.494−0.047 19.08−0.11 1.02−0.14 OB120851 h m s ◦ 0 00 +0.63 +5.36 +0.122 +0.56 +0.28 BLG617.29.102385 17 15 43.18 −29 18 44.1 2457484.80−0.58 21.29−2.67 0.506−0.158 19.35−0.36 0.70−0.28 OB160624 h m s ◦ 0 00 +1.69 +5.47 +0.107 +0.51 +0.24 BLG617.04.62618 17 15 50.74 −30 18 46.9 2457275.58−1.84 22.81−3.27 0.566−0.151 19.55−0.29 0.78−0.29 h m s ◦ 0 00 +0.35 +4.20 +0.063 +0.33 +0.18 BLG617.29.88791 17 15 55.68 −29 08 20.3 2457941.92−0.36 28.37−2.23 0.451−0.089 19.29−0.20 0.85−0.22 OB171155 h m s ◦ 0 00 +0.71 +19.29 +0.235 +0.90 +0.41 BLG617.21.47343 17 16 00.70 −29 32 38.2 2457107.33−0.67 33.90−9.26 0.344−0.172 20.11−0.81 0.37−0.21 OB150454 h m s ◦ 0 00 +0.42 +1.23 +0.016 +0.10 +0.05 BLG617.29.41603 17 16 01.07 −29 17 42.0 2455857.41−0.40 35.73−0.92 0.492−0.029 16.93−0.05 0.95−0.08 OB111302 h m s ◦ 0 00 +0.42 +2.83 +0.085 +0.46 +0.31 BLG617.04.54041 17 16 09.35 −30 06 06.4 2457963.81−0.43 10.90−1.74 0.357−0.102 20.35−0.30 0.98−0.34 OB171467 h m s ◦ 0 00 +0.37 +3.27 +0.068 +0.37 +0.17 BLG617.28.101826 17 16 28.47 −29 20 36.4 2456462.98−0.37 20.63−1.65 0.563−0.114 19.21−0.20 0.86−0.25 OB130950 h m s ◦ 0 00 +1.70 +19.59 +0.227 +0.98 +0.44 BLG617.11.107345 17 16 31.47 −29 46 55.6 2457155.46−1.68 38.30−7.74 0.557−0.266 20.37−0.63 0.55−0.33 OB151021 h m s ◦ 0 00 +0.04 +3.17 +0.001 +0.08 +0.07 BLG617.03.87926 17 16 31.48 −30 18 45.1 2455984.69−0.04 51.71−2.68 0.006−0.001 20.04−0.07 1.03−0.07 OB120030 h m s ◦ 0 00 +0.04 +0.87 +0.003 +0.07 +0.06 BLG617.20.63992 17 16 39.97 −29 39 56.8 2457835.07−0.03 18.48−0.73 0.052−0.003 18.78−0.06 0.98−0.06 OB170284 h m s ◦ 0 00 +0.28 +3.29 +0.057 +0.52 +0.24 BLG617.03.55959 17 16 40.95 −30 20 47.5 2457988.33−0.28 11.19−1.58 0.224−0.081 19.59−0.29 0.80−0.30 h m s ◦ 0 00 +0.04 +2.95 +0.003 +0.07 +0.03 BLG617.03.26929 17 16 44.41 −30 19 46.3 2456135.78−0.05 54.31−2.70 0.049−0.003 19.78−0.07 0.38−0.02 OB121056 h m s ◦ 0 00 +3.97 +14.56 +0.070 +0.30 +0.15 BLG617.28.43862 17 16 44.54 −29 17 37.6 2456230.88−4.26 141.95−8.21 0.838−0.116 19.22−0.16 0.93−0.22 OB121403 h m s ◦ 0 00 +0.02 +1.44 +0.022 +0.19 +0.13 BLG616.20.2812 17 16 46.34 −28 25 36.4 2456034.77−0.02 13.76−0.97 0.073−0.029 19.05−0.14 0.93−0.15 h m s ◦ 0 00 +0.04 +0.53 +0.002 +0.02 +0.01 BLG616.20.31331 17 16 52.76 −28 20 29.3 2456417.81−0.04 52.37−0.37 0.218−0.003 16.48−0.01 0.99−0.02 OB131368 h m s ◦ 0 00 +0.46 +15.58 +0.054 +0.26 +0.12 BLG617.28.21305 17 16 55.48 −29 11 04.1 2457167.08−0.47 98.73−13.30 0.238−0.043 19.89−0.28 0.42−0.09 h m s ◦ 0 00 +0.05 +2.05 +0.006 +0.15 +0.12 BLG617.28.17554 17 16 56.59 −29 15 02.8 2455992.87−0.05 18.17−1.77 0.045−0.006 20.09−0.14 0.84−0.11 OB120156 h m s ◦ 0 00 +0.15 +5.35 +0.175 +0.67 +0.22 BLG616.19.79418 17 17 24.57 −28 23 07.7 2456102.24−0.16 10.87−3.63 0.223−0.094 20.03−0.80 0.20−0.09 Table 5.3. Best-fitting parameters of the analyzed microlensing events in low-cadence OGLE fields (first 40 objects). For each parameter, we provide the median and 1σ confidence interval derived from the marginalized posterior distribution from the Monte Carlo chain. Is is the source brightness and fs = Fs/(Fs + Fb) is the blending parameter. Equatorial coordinates are given for the epoch J2000. OBNNMMMM stands for OGLE-20NN-BLG-MMMM. The full table is available in electronic form (www.astrouw.edu.pl/~pmroz/tables.tar.gz).

96 Figure 5.3. Reference image of the ﬁeld BLG506. The observed surface density of sources may vary on small angular scales due to variable and patchy extinction. Each CCD detector covers an area of 8.80 17.70. Credit: Szymon Kozłowski. ×

HST HST LF LF simul HST ﬁeld OGLE ﬁeld Σ20 Σ21 Σ20 Σ21 Σ21 mw_bulge_u2c901 BLG505.19 2429.2 4592.9 2395.9 4390.0 3860.0 mw_bulge_u2cl02 BLG513.12 1603.8 2825.3 1476.1 2636.5 2769.4 mw_bulge_u2tw01 BLG513.12 1610.8 2841.2 1476.1 2636.5 2769.4 mw_bulge_u66h01 BLG513.12 1636.6 2848.4 1476.1 2636.5 2769.4 mw_bulge_u6ls02 BLG511.01 1521.3 2760.3 1610.2 2912.0 2950.3 mw_bulge_u2oq03 BLG532.08 – 524.0∗ – – 533.4∗

∗ Surface density in the range 19.5 < I < 21. Table 5.4. Star surface density (per arcmin2) in selected Hubble Space Telescope (HST) ﬁelds toward the Galactic bulge (Holtzman et al. 2006). Σ20 and Σ21 are the surface densities in the ranges 14 < I < 20 and 14 < I < 21, respectively. The star surface density was calculated using the HST images, matching the template luminosity function (LF), and image-level simulations.

97 5.4. Star counts

The number of monitored sources is an essential quantity in microlensing optical depth calculations. While it is usually presumed that star catalogs are nearly complete at the bright end of the luminosity function, this assumption no longer holds for faint sources. (In fact, the incompleteness in red clump giant counts led to the discrepancy between optical depths based on bright and faint sources; Sumi et al. 2016). The density of stars brighter than I = 21 in the most crowded regions of the Galactic bulge exceeds 4000 stars per arcmin2, which corresponds to about 0.7 unresolved blends in a typical seeing disk (FWHM = 0.8 0.900) of a star. A − faint star can be hidden in a glow of a bright neighbor or two faint stars cannot be resolved and the total brightness of the blend is higher than the brightness of both unresolved sources. Star catalogs might be therefore highly incomplete, especially in crowded fields. The most robust approach to counting the source stars is to use deep, high-resolution images of a given field taken with the Hubble Space Telescope (HST). This method is, however, impractical in our study, because sufficiently deep HST pointings are available for only a few sightlines toward the Galactic center (e.g., Holtzman et al. 1998, 2006). Moreover, the observed number density of stars may vary on small angular scales (Figure 5.3) due to variable and patchy interstellar extinction. We used several HST pointings as a “ground-truth” to test the accuracy of other methods of counting source stars. We used the database of stellar photometry of several Galactic bulge fields obtained using the Wide Field Planetary Camera 2 (WFPC2) onboard HST (Holtzman et al. 2006). The WFPC2 camera has a field of view of 4.97 arcmin2 and a pixel scale of 0.045500 or 0.100 per pixel, depending on the detector (Holtzman et al. 1995). The ∼ ∼ observations were taken through the F814W filter and transformed to the Cousins I magnitudes. Holtzman et al. (2006) also provided information on the completeness of the photometry as a function of brightness based on image-level simulations, which allowed us to correct the observed luminosity functions. Our results for six HST fields are reported in Table 5.4. The most common approach to assess the number of monitored sources is to use one deep luminosity function of a single field as a template (e.g., Alcock et al. 2000; Sumi et al. 2003). The template luminosity function is shifted in brightness and rescaled so that the brightness and number of red clump stars match the observed bright end of the luminosity function in a given direction (see Figure 5.4). We used this method to calculate the number of monitored sources in fifteen selected fields. The template luminosity function was constructed using deep HST

98 OGLE+HST LF OGLE+HST LF OGLE+HST LF 3 3 3 10 field BLG505 10 field BLG505 10 field BLG505 / mag / mag / mag 2 2 2

102 102 102 stars / arcmin stars / arcmin stars / arcmin 101 101 101

14 16 18 20 22 14 16 18 20 22 14 16 18 20 22 I magnitude I magnitude I magnitude

Figure 5.4. Estimating the number of monitored sources in the ﬁeld BLG505.09 using the deep template luminosity function (black). The OGLE-IV luminosity function of the ﬁeld BLG505.09 is blue. The template is shifted by 0.57 mag along the x-axis and multiplied by 1.59 so that the position and number of red clump stars match the observations (middle panel). The rescaled luminosity function is then integrated (right).

observations (Holtzman et al. 1998) of the field BLG513.12 for faint stars (I 17) and the ≥ OGLE-IV luminosity function for bright stars (I < 17). While the presented method can work well for neighboring regions, it may fail for fields located at or above the Galactic equator (as well as in the Galactic plane, far from the bulge), where the shape of the luminosity function may be different. We therefore tried a novel approach. The pixel size ( 0.2600) of the OGLE-IV camera ∼ and typical PSF size of stars on reference images (0.8 0.900) are much better than in previous − experiments (although still inferior to the HST images). We carried out a series of image-level simulations to estimate the completeness of our star catalogs. We injected artificial stars (14 < I < 21) into random locations on real OGLE images, stacked the images into the deep reference image, and ran our star-detection pipeline (Wo´zniak 2000) exactly in the same way as real star catalogs were created. Simulations were conducted by Prof. Andrzej Udalski. We injected 5000 stars per frame so that the density of the stars tended to increase by less than 5%. We consider the artificial star as detected if 1) the measured centroid is consistent (within 1.5 pix) with the location where the star was placed and the closest star from the original catalog is at least 2.1 pix away (the artificial star is in an “empty” field), or if 2) the measured brightness of the artificial star Isim is closer to the input brightness Iin than to the brightness of the real neighboring star I (i.e., I I < I I ) if such neighbor was detected within nei | sim − in| | sim − nei| 2.1 pix on the original frame (in other words, the real star from the original catalog is a blend). The star-detection algorithm can separate neighboring objects as close as less than 2 pix away, but its effectiveness depends on the flux and flux ratio.

99 HST LF (Holtzman+ 1998) 103 OGLE-IV LF (corrected) OGLE-IV LF (observed) / mag 2 102

101 stars / arcmin

100 12 14 16 18 20 22 I magnitude

Figure 5.5. Comparison between the OGLE-IV luminosity function (LF) of the ﬁeld BLG513.12 and the Hubble Space Telescope (HST) LF of the same region (Holtzman et al. 1998). The detection-efﬁciency-corrected LF (blue histogram) is consistent with deep HST observations.

BLG513.12 BLG503.09 BLG672.20

1.0 1.0 1.0

0.8 0.8 0.8

0.6 0.6 0.6

0.4 0.4 0.4 Completeness Completeness Completeness

0.2 0.2 0.2

0.0 0.0 0.0 14 15 16 17 18 19 20 21 14 15 16 17 18 19 20 21 14 15 16 17 18 19 20 21 I magnitude I magnitude I magnitude

Figure 5.6. Completeness of star counts toward three OGLE ﬁelds as a function of brightness. The selected three ﬁelds represent regions of high (BLG513.12: 2800 arcmin−2), medium (BLG503.09: 1800 arcmin−2), and low (BLG672.20: 500 arcmin−2) star density. OGLE star counts are nearly complete down to I 18 (BLG513.12), I 19 (BLG503.09), and I 20 (BLG672.20), respectively. ≈ ≈ ≈

100 We calculated completeness of our star catalogs in fourteen 0.5-mag-wide bins and corrected the observed luminosity functions for each subfield. This approach works well for I < 20.5. In a few cases of the most crowded fields, we needed to extrapolate the luminosity function for the faintest sources (I > 20.5) based on two or three earlier bins. Our luminosity function of the field BLG513.12 is consistent with the HST results (Holtzman et al. 1998) (Figure 5.5). Star catalogs are nearly complete down to I = 18 in the most crowded fields and even to I = 20 in relatively empty fields (Figure 5.6). The overall completeness (down to I = 21) typically varies from 30% to 80%, depending on the field. Figure 5.7 shows the comparison between the star surface density (down to I = 21)

simul measured using image-level simulations (Σ21 ) and that estimated from matching the template luminosity function (ΣLF). On average, ΣLF/Σsimul = 1.01 0.12, although we noticed a small 21 21 21 ± LF simul bias. In ﬁelds located close to the Galactic plane, Σ21 > Σ21 , while far from the plane, LF simul Σ21 < Σ21 . Similarly, the comparison of measured star densities to those inferred directly from HST images (Table 5.4) indicates that both proposed methods (template matching and image-level simulations) are accurate to about 10–15%. These tests demonstrate that we are presently unable to measure the number of monitored sources with accuracy better than 10%. In turn, optical depths and event rates may suffer from systematic errors at the 10–15% level. Because our sample of microlensing events is large, the accuracy of inferred optical depths and event rates will be mostly limited by the accuracy of the determination of the number of sources, not by small numbers of events as in the previous studies. The number and surface density of stars in the analyzed subﬁelds calculated using image-level simulations are presented in Table 5.5 and Figure 5.8.

101 Subﬁeld R.A. Decl. l b Σ18 Σ21 N18 N21 2 2 (J2000.0) (J2000.0) (deg) (deg) (arcmin− ) (arcmin− ) BLG500.01 268.526 –29.081 0.829 –1.666 356.6 4333.5 59437 722294 BLG500.02 268.351 –29.081 0.751 –1.534 276.5 3174.1 46075 528914 BLG500.03 268.175 –29.081 0.673 –1.401 225.8 2704.4 37609 450453 BLG500.04 267.999 –29.081 0.595 –1.269 166.7 1588.2 27758 264465 BLG500.05 267.823 –29.081 0.516 –1.137 85.2 829.9 14184 138160 BLG500.08 268.694 –28.759 1.180 –1.630 270.6 4225.4 45092 704120 BLG500.09 268.518 –28.759 1.102 –1.497 277.4 3963.2 46193 659959 BLG500.10 268.342 –28.759 1.024 –1.365 231.3 2724.2 38507 453530 BLG500.11 268.167 –28.759 0.946 –1.232 185.8 1757.4 30932 292575 Table 5.5. Surface density of stars in OGLE-IV subﬁelds calculated using image-level simulations. Σ18 and Σ21 are the surface densities of stars brighter than I = 18 and I = 21, respectively. N18 and N21 are the numbers of stars brighter than I = 18 and I = 21, respectively. The full table is available in electronic form (www.astrouw.edu. pl/~pmroz/tables.tar.gz).

5000 ) 2 − 4000

3000

2000

1000 star density - LF matching (arcmin

0 0 1000 2000 3000 4000 5000 2 star density - simulations (arcmin− )

Figure 5.7. Comparison between the star surface density calculated using image-level simulations and that measured by matching the template luminosity function.

102 5000 10

2000 5 2 1000 0

103 500 stars/arcmin 5

Galactic latitude (deg) − 200 10 − 100 10 5 0 5 10 − − Galactic longitude (deg)

Figure 5.8. Density of stars down to I = 21 in OGLE-IV ﬁelds toward the Galactic bulge. 5.5. Distribution of the blending parameter

Due to the high density of stars toward the Galactic bulge and the point spread function size of objects on reference images, some sources cannot be resolved on OGLE template images (this phenomenon is called “blending”). A faint star can be hidden in a glow of a bright neighbor or two faint stars cannot be resolved and the total brightness of the blend is higher than the brightness of both unresolved sources. We used image-level simulations that were described

in the previous Section to derive the distribution of the blending parameter fs as a function of the brightness of the baseline star. These distributions will be necessary for catalog-level simulations of detection efficiency of microlensing events in our experiment. The blending parameter is defined as the ratio between the source flux and the total flux of the detected object (i.e., the sum of fluxes of the source and unrelated blends). Previously, Wyrzykowski et al. (2015) used archival HST observations of the OGLE-III field BLG206 to obtain the distribution of the blending parameter in that field. They matched OGLE stars to individual stars present on the HST image and calculated the ratio of their flux to the total brightness of the object detected on the OGLE template image. Then, they assumed that the distribution of blending is the same across all analyzed OGLE-III fields. We used image-level simulations to construct distributions of the blending parameter in all analyzed fields. We matched stars injected into images with those detected on reference images

(we used a matching radius of 1.5 pix). The blending parameter is simply fs = Fin/Fout, where

Fin is the input flux and Fout is the flux measured on the template image. Sources were drawn from luminosity functions of corresponding fields. Figure 5.9 presents the comparison between the distribution of the blending parameter obtained from our image-level simulations and that from the empirical study of Wyrzykowski et al. (2015) based on HST images. Both distributions are very similar. The distribution of f of bright stars is bimodal – either the entire flux comes from the source (f 1) or the s s ≈ source is much fainter than the blend (f 0). For fainter stars, the blending parameter is s ≈ distributed more uniformly. There are small differences between the results of our simulations and distributions of Wyrzykowski et al. (2015), which are likely caused by different template images (OGLE-III reference images are deeper and have better resolution than OGLE-IV templates). Figure 5.10 shows the distributions of fs in three fields with different star densities.

104 Simulations HST observations 14 14

15 15 ) ) s s s s e

e 16 16 n n t t h h g i g i r

r 17 17 b

e e n i n l i l

18 e 18 e s s a a b b (

(

19 E 19 t L u G o I O I 20 20

21 21 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 fs = Fin/Fout (blending parameter) fs = FHST/FOGLE (blending parameter)

Simulations HST observations 0.40 0.40

14 < Iout < 17 14 < IOGLE < 17 0.35 0.35 17 < Iout < 19 17 < IOGLE < 19 19 < I < 20.5 19 < I < 20.5 0.30 out 0.30 OGLE

0.25 0.25

0.20 0.20

Probability 0.15 Probability 0.15

0.10 0.10

0.05 0.05

0.00 0.00 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 fs = Fin/Fout (blending parameter) fs = FHST/FOGLE (blending parameter)

Figure 5.9. Comparison between distributions of blending parameter found using image-level simulations (left panels) and the Hubble Space Telescope (HST) observations (right panels). The HST observations are taken from Wyrzykowski et al. (2015). Note that Wyrzykowski et al. (2015) analyzed images obtained with the OGLE-III camera, which has the same pixel size as the OGLE-IV camera, but the reference images are different.

105 BLG513.12 BLG513.12 14 1.0

14 < Iout < 17

15 17 < Iout < 19

) 0.8

s 19 < Iout < 20.5 s

e 16 n t h g

i 0.6

r 17 b

e n i l 18 e 0.4 Probability s a b (

19 t u o I 0.2 20

21 0.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 fs = Fin/Fout (blending parameter) fs = Fin/Fout (blending parameter)

BLG503.09 BLG503.09 14 1.0

14 < Iout < 17

15 17 < Iout < 19

) 0.8

s 19 < Iout < 20.5 s

e 16 n t h g

i 0.6

r 17 b

e n i l 18 e 0.4 Probability s a b (

19 t u o I 0.2 20

21 0.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 fs = Fin/Fout (blending parameter) fs = Fin/Fout (blending parameter)

BLG672.20 BLG672.20 14 1.0

14 < Iout < 17

15 17 < Iout < 19

) 0.8

s 19 < Iout < 20.5 s

e 16 n t h g

i 0.6

r 17 b

e n i l 18 e 0.4 Probability s a b (

19 t u o I 0.2 20

21 0.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 fs = Fin/Fout (blending parameter) fs = Fin/Fout (blending parameter)

Figure 5.10. The distribution of blending parameter as a function of the baseline brightness for three selected OGLE ﬁelds, which represent regions of high, medium, and low star density.

106 5.6. Catalog-level simulations

Image-level simulations provided us with robust measurements of the detection efficiency of microlensing events (Section 3.4). These calculations (i.e., injecting microlensing events into real images, performing image-subtraction photometry, and creating photometric databases) require significant amount of computational resources. In fact, simulations of event detection efficiency in nine high-cadence fields (Section 3.4) lasted nearly four months. As we aimed to measure detection efficiencies in the remaining 112 fields in a finite amount of time, we decided to carry out catalog-level simulations. We injected microlensing events on top of light curves of objects from the OGLE-IV photometric databases. Each data point and its error bar were rescaled by the expected magnification, which depends on microlensing model and blending. Our method conserves the variability and noise in original light curves, as well as information on the quality of individual

measurements. Let Fs be the flux of the source and Fb – unmagnified flux from possible blended

stars and/or the lens itself. The ﬂux of the baseline object (F0 = Fs + Fb) is magniﬁed during a microlensing event by a factor:

0 FsA(t) + Fb Fs(A(t) 1) + Fs + Fb A (t) = = − = 1 + fs(A(t) 1), (5.7) Fs + Fb Fs + Fb − where A(t) is the model magniﬁcation and fs = Fs/(Fs + Fb) is the blending parameter. If there is no blending (f = 1) then A0(t) = A(t); if the blending is very strong (f 0), the s s → observed magniﬁcation A0(t) 1. → To inject a microlensing event into the database light curve, we needed to transform the

0 0 0 observed ﬂux and its uncertainty (Fi, σi) to (F , σ ). The naive transformation Fi FiA (t) is i i → incorrect because it preserves the original photon noise during the magniﬁed part of the event. Let us consider a constant I = 19.5 star with a typical root mean square light curve scatter of 0.1 mag (left panel of Figure 5.11). The transformation A0(t) = 100 would shift the mean magnitude to I0 = 14.5, but it would preserve the original scatter, whereas the observed scatter of constant stars of that magnitude is much smaller (Figure 5.11). The following transformation ensures that the photon noise is properly scaled:

0 0 0 σ(F0) Fi = F0A (t) + (Fi F0) , (5.8) − σ(F0) 0 0 where F0 and F0 = F0A (t) are the mean flux in the baseline and the mean magnified flux, 0 0 respectively (the corresponding magnitudes are m0 and m0). The ratio σ(F0)/σ(F0) can be

107 calculated by assuming the photometric noise model of Skowron et al. (2016):

0 . m −m0 σ(F ) 1 + 100 4( B 0) 0 (5.9) = 0.4(m −m ) . σ(F0) 1 + 10 B 0 In this simple model, the observed scatter is the sum of Poisson noise contributions from the object and from the background (parameterized by mB). To verify the proposed model, we injected “constant” stars on top of real light curves from the database (which correspond to the transformation A0(t) = const). The right panel of Figure 5.11 shows the root mean square scatter of simulated light curves, which is consistent with that of real data. Our simple model underpredicts the scatter of the brightest stars (I . 15), likely because the accuracy of their photometry is limited by the accuracy of modeling the point spread function and so the quality of image subtractions, not by the photon noise. This effect does not inﬂuence our detection efﬁciency simulations, because the vast majority of events are fainter than I = 15.

Figure 5.11. Root mean square light curve scatter as a function of the mean magnitude of a star in the I-band in the OGLE-IV ﬁeld BLG521.12. Left panel: Real data. Right panel: Constant stars injected on top of real light curves (m m + const). Orange line is the best-ﬁt model from Skowron et al. (2016). →

Similarly, we used the noise model of Skowron et al. (2016) to transform the uncertainties:

0 0.4(mB −m ) 0 1 + 10 σi = σi. (5.10) 1 + 100.4(mB −m)

0 When the predicted uncertainties were equal to or below 0.003 mag, we assumed σi = 0.003 mag, as in the photometric databases. The proposed transformations preserve information on seeing and sky transparency (included in the reported error bars), as well as the variability in the original data.

108 The catalog-level simulations were carried out with the following steps: 1. We drew random parameters of a microlensing event from uniform distributions: t 0 ∼ U(2455377, 2458118) (i.e., between June 29, 2010 and December 31, 2017), u 0 ∼ U(0.0, 1.0), and log t U(0.0, 2.5). E ∼ 2. We drew a random star from the database and calculated its mean magnitude.

3. We drew a random blending parameter fs corresponding to the mean magnitude of the baseline object (Section 5.5). This parameter describes what fraction of light comes from the source. 4. We simulated a microlensing event on top of the light curve of the selected object using the procedure that was described above. Subsequently, we checked if the event passes our selection criteria (Table 5.2).

We properly weighted simulated events so that the simulated distribution of Is is consistent with the luminosity function of the given field. We took into account sources brighter than Is = 21. We simulated 25,000 events per each CCD detector, which yielded a total of 800,000 events per field. Examples of detection efficiency curves are shown in the left panel of Figure 5.12.

5.7. Image-level simulations

We carried out additional image-level simulations to check the accuracy of the catalog-level detection efficiencies. The image-level simulations were conducted using the pipeline described in Section 3.4, which was prepared for high-cadence fields. We injected artificial microlensing events into real images; sources were drawn from the luminosity function of a given subfield and were placed in random locations within the field. We then constructed reference images and calculated image-subtraction photometry for all injected events. (Image-level simulations – event injection, image reductions, creation of photometric databases – were carried out by Prof. Andrzej Udalski). Finally, we measured the fraction of events that pass our selection criteria (Table 5.2) as a function of an event timescale. Simulations span the period of 2010–2015. Image-level simulations were carried out for six low-cadence fields (BLG513, BLG518, BLG521, BLG535, BLG612, and BLG660). The right panel of Figure 5.12 presents the measured detection efficiencies in two representative fields. For comparison, we also present detection efficiencies calculated using catalog-level simulations for the same range of event parameters. Both curves agree reasonably well. We estimate that microlensing optical depths

109 0.1 0.1 BLG502 BLG535 BLG603 BLG535 BLG609 BLG612 BLG612 catalog-level simulations image-level simulations Detection eﬃciency BLG617 Detection eﬃciency BLG662 BLG668 BLG670 0.01 0.01 1 10 100 1 10 100 tE (d) tE (d)

Figure 5.12. Left: Image-level detection efficiencies for selected OGLE-IV fields for sources brighter than I = 21 for the period of 2010–2017. Right: Comparison between detection efficiencies calculated based on image-level and catalog-level simulations. These simulations span the period of 2010–2015. and event rates measured using image- and catalog-level simulations agree within 3%, a difference much smaller than other sources of statistical and systematic errors. We used catalog-level simulations to measure the detection efficiencies for 5790 events in low-cadence fields. For nine high-cadence fields (BLG500, BLG501, BLG504, BLG505, BLG506, BLG511, BLG512, BLG534, BLG611), we used image-level simulations that were discussed in Section 3.4. To facilitate the analysis, the sample of events from high-cadence

ﬁelds was restricted to sources brighter than Is = 21 and events longer than tE = 0.5 d; the detection efﬁciencies were accordingly recalculated. The restricted sample is comprised of 2212 events (i.e., 85% of the original data set). All 5790 + 2212 = 8002 events were thus used for the construction of event rate and optical depth maps.

110 5.8. Results and conclusions

5.8.1. Timescale distribution

The distribution of timescales of 5790 microlensing events detected in low-cadence ﬁelds is shown in the upper panel of Figure 5.13. The majority of events have timescales between 10 and 40 days and the number of events falls smoothly at shorter and longer timescales. The

short-tE end of the distribution appears to be steeper than the long-tE tail, which reflects the fact that the detection efficiency quickly declines as the timescale decreases (Figure 5.12). Indeed, the distribution of event timescales corrected for detection bias (which is constructed by assigning each event a weight 1/ε(tE,i), where ε(tE,i) is the detection efficiency) is more symmetric (see the lower panel of Figure 5.13). The short- and long-timescale distribution tails can be well described by power-law distributions with slopes of +3 and 3, as expected from − theory (Mao & Paczynski´ 1996). The timescale distribution appears broader than that presented in Figure 3.13 because the current sample contains events from a much larger region. As we will demonstrate, the mean timescales of microlensing events grow with increasing angular distance from the Galactic center. Moreover, the sensitivity to long-timescale events is larger than in Chapter 3 because we were able to use longer light curves (and so we searched for microlensing events using longer 2-year windows). The short-t end of the timescale distribution reveals no excess of short-duration (t E E ≤ 2 d) events. Because some analyzed fields are observed 2–3 times per night (these fields are marked green in Figure 2.2), there is still some sensitivity to short-timescale events, as shown in Figure 5.12. The non-detection of any excess of short-timescale events strengthens our conclusions from Chapter 3 that there is no large population of free-floating or wide-orbit Jupiter-mass planets (Sumi et al. 2011).

The shape of the long-tE end of the timescale distribution is more uncertain, even though it can be well described by a power law. A number of long-timescale events exhibit a strong annual parallax effect due to the orbital motion of Earth (Section 1.5). Therefore, they do not pass our strict selection criteria on fit quality. However, our detection efficiency simulations did not include the parallax effect (we simulated only point-source point-lens events without any second-order effects), and thus the detection efficiencies of long-timescale events are systematically overestimated.

111 102

101 Number of events per bin

100 1 10 100 tE (days)

103

(+3) (-3)

102 Number of events per bin

1 10 100 tE (days)

Figure 5.13. Distribution of timescales of 5790 microlensing events detected in low-cadence OGLE-IV ﬁelds. Upper panel: observed timescales. Lower panel: timescales corrected for the detection efﬁciency. The short and long timescale distribution tails can be well described by power-law distributions with slopes of +3 and 3 as expected from theory (Mao & Paczynski´ 1996). −

112 10 35 10 35

30 30 5 5

25 25 (d) (d)

0 i 0 i E E t t 20 h 20 h

5 5 Galactic latitude (deg) − 15 Galactic latitude (deg) − 15

10 10 10 10 − 10 5 0 5 10 − 10 5 0 5 10 − − − − Galactic longitude (deg) Galactic longitude (deg)

113 10 35 10 35

30 30 5 5

25 25 0 0 20 20 (d) - Gaussian ﬁt (d) - Gaussian ﬁt i i E E t t

5 h 5 h Galactic latitude (deg) − 15 Galactic latitude (deg) − 15

10 10 10 10 − 10 5 0 5 10 − 10 5 0 5 10 − − − − Galactic longitude (deg) Galactic longitude (deg)

Figure 5.14. Mean timescales of microlensing events in 600 600 (left column) or 300 300 (right) bins. The upper panels show the detection efficiency-corrected mean timescales and the lower panels report× the mean timescales× from the Gaussian fit. All maps were smoothed with a Gaussian with σ = 0.5◦. Galactic models predict that the mean timescales of microlensing events should depend on the location (because t = 2τ/πΓ). We divided the analyzed area into 300 300 and 600 600 h Ei × × bins and calculated the mean Einstein timescale provided at least five events were located in the bin: P tE,i ε(tE,i) t = i . (5.11) h Ei P 1 ε(tE,i) i Following Wyrzykowski et al. (2015), we also calculated the timescale corresponding to the mean log t , which they call “the mean timescale based on the Gaussian model” (i.e., h Ei the maximum of the event timescale histograms corrected for the detection efficiency – see Figures 3.13 and 5.13). The latter value is smaller than t and is less prone to large statistical h Ei fluctuations due to rare very long (or very short) timescale events. Figure 5.14 shows how mean timescales of microlensing events vary with the location in the sky. Low resolution bins (600 600) contain 5 to 324 events (median 33), high resolution (300 300) – from 5 to 106 × × (median 13.5). All maps were smoothed with a Gaussian with σ = 300. The distribution of average timescales agrees well with that found by Wyrzykowski et al. (2015) based on a smaller OGLE-III data set. Their sample of standard microlensing events comprised 3718 events from 2001 to 2009, most of which were located primarily in the Southern Galactic hemisphere in the region 5◦ < l < 5◦, 5◦ < b < 1◦. Similarly, Sumi & Penny − − − (2016) analyzed the distribution of mean timescales of 474 microlensing events in the MOA-II sample, all of which are located below the Galactic plane. The average timescale map of Sumi & Penny (2016) does not show any systematic trends, however, because their sample of events is too small. The mean timescales of microlensing events are the shortest in the central bins (located within 3◦ of the Galactic center) and they grow with the increasing angular distance from the ∼ Galactic center from 22 to 32 d (Figures 5.14 and 5.15). A similar trend was noticed earlier by Wyrzykowski et al. (2015) based on OGLE-III data, although they did not analyze events at l 5◦ and b > 0◦. One may argue this is a systematic effect arising from the fact that we | | ≥ were unable to detect the shortest-timescale events in low-cadence fields, but this is not the case. The average event timescales in fields BLG580, BLG518, and BLG522 (l 5◦, b 3◦) are ≈ ≈ − 28.5–30.0 d, and the shortest detected events have t 3 d. Each of these three fields contains E ≈ over a hundred events, so the low-number statistics cannot also be blamed.

114 36 36

34 34

(d) 32 (d) 32 i i E E t t h 30 h 30

28 28

26 26 Mean timescale Mean timescale

24 24

22 22 10 8 6 4 2 0 2 4 6 8 10 10 8 6 4 2 0 2 4 6 8 10 Galactic longitude− (deg)− − − − − − − Galactic− − latitude (deg)

Figure 5.15. Mean timescales of microlensing events as a function of Galactic longitude ( 6◦ b 1◦; left panel) and Galactic latitude ( 6◦ l +10◦; right). − ≤ ≤ − − ≤ ≤

The average timescales increase with increasing Galactic latitude (left panel of Figure 5.15) with t 32 d at l 8◦ and l 6◦, near the edge of the analyzed fields. Currently, OGLE h Ei ≈ ≈ ≈ − is observing a larger area around the Galactic center as part of the OGLE Galaxy Variability Survey (Chapter 2). These observations will tell us whether the average event timescales outside the analyzed area reach a plateau or increase. The distribution of timescales is asymmetric in Galactic longitude (Figure 5.14) – events located at positive longitudes appear to be on average slightly shorter than those at negative longitudes, which is qualitatively consistent with the theoretical mean timescale maps of Wegg et al. (2016, 2017) and Awiphan et al. (2016). This asymmetry stems from the fact that the Galactic bar is slightly inclined, resulting in typically larger Einstein radii at negative longitudes (Awiphan et al. 2016). The mean event timescales also vary with Galactic latitude (see Figure 5.14 and the right panel of Figure 5.15) with shorter average values closer to the Galactic plane, which is in agreement with theoretical expectations. Events located very close to the Galactic plane ( b | | ≤ 1◦) are on average much longer ( t 34 d) than those in neighboring fields ( t 25 d). h Ei ≈ h Ei ≈ They are probably caused by sources located in the foreground disk (not the Galactic bulge, which are invisible due to large extinction). The previous MOA-II (Sumi & Penny 2016) and OGLE-III (Wyrzykowski et al. 2016) average timescale maps covered the area below the Galactic plane. Although the extent of OGLE-IV fields in the Northern Galactic hemisphere is smaller than in the South, Figure 5.14

115 suggests the average timescale distribution may be slightly asymmetric about the Galactic plane. Events located above the Galactic equator appear to be slightly longer than those below it. We compared the timescale distributions of events below and above the Galactic plane using the Kolmogorov-Smirnov test (the test implementation for weighted data is discussed by Monahan 2011) and found p-values of 0.01 and 0.27 for 0◦ l 5◦ and 5◦ l 0◦, respectively. ≤ ≤ − ≤ ≤ This conﬁrms a small asymmetry for positive longitudes. The difference may be partly caused by the non-uniform interstellar extinction.

5.8.2. Microlensing optical depth and event rate

The microlensing optical depths and event rates were calculated using Equations (5.4) and (5.5) for each OGLE ﬁeld (see Table B.1 in Appendix B). The uncertainties of these quantities can be calculated as follows. Han & Gould (1995a) derived the formula for the statistical error in estimating the optical depth and they demonstrated that it is substantially higher than the naive Poisson estimate. We may derive a similar expression for the statistical error of the event rate. Recall that the event rate can be written as:

1 X nj Γ = , (5.12) N ∆T ε s j j

where Ns is the number of monitored sources, ∆T is the duration of the survey, nj is the

number of events in a j-th timescale bin, and εj is the event detection efﬁciency in that bin. The

summation is performed over all timescale bins. Since the εj-s are constants and the number

of events obeys Poisson statistics (σnj = √nj), the uncertainty of Γ can be evaluated using the standard error propagation: 2 2 X ∂Γ 1 X nj (σΓ) = √nj = , (5.13) ∂n N 2∆T 2 ε2 j j s j j and hence: rP nj ε2 σ j j Γ = . (5.14) Γ P nj εj j

116 10 30

20 )

5 1 −

10 yr 6 −

0 5

5 Event rate (10 Galactic latitude (deg) − 2

10 1 − 10 5 0 5 10 − − Galactic longitude (deg)

10 30

20 )

5 1 −

10 yr 6 −

0 5

5 Event rate (10 Galactic latitude (deg) − 2

10 1 − 10 5 0 5 10 − − Galactic longitude (deg)

Figure 5.16. Microlensing event rate per star in 100 100 bins. The lower map was smoothed with a Gaussian with σ = 100. ×

117 10 4.0 3.0

2.0 5 ) 6 −

1.0 0

0.5

5 Optical depth (10 Galactic latitude (deg) −

10 0.2 − 10 5 0 5 10 − − Galactic longitude (deg)

10 4.0 3.0

2.0 5 ) 6 −

1.0 0

0.5

5 Optical depth (10 Galactic latitude (deg) −

10 0.2 − 10 5 0 5 10 − − Galactic longitude (deg)

Figure 5.17. Microlensing optical depth per star in 100 100 bins. The lower map was smoothed with a Gaussian with σ = 100. ×

118 6 OGLE ( l < 3◦) | | 5 MOA-II All

) MOA-II RCG 6 − 4

2 Optical depth (10 1

0 8 6 4 2 0 2 4 6 8 − − − Galactic− latitude (deg) 60 OGLE ( l < 3◦) | | 50 MOA-II All )

1 MOA-II RCG −

yr 40 6 − 30

20 Event rate (10 10

0 8 6 4 2 0 2 4 6 8 − − − Galactic− latitude (deg)

Figure 5.18. Comparison between microlensing optical depth and event rates measured using OGLE data (black data points) and previous measurements based on MOA-II observations (Sumi & Penny 2016). Filled gray circles and a solid line are MOA measurements based on red clump giant (RCG) stars; open points and a dashed line are based on all-source sample of events.

119 For the construction of microlensing maps, we used the source counts estimated from our image-level simulations (Table 5.5). As discussed in Section 5.4, these counts may suffer from systematic errors at the 10% level. We assumed ∆T = 2011 d or ∆T = 2741 d when using 2010–2015 or 2010–2017 light curves, respectively. Our microlensing maps were constructed using the sample of point-source point-lens events with timescales shorter than t 300 d. About 10–15% of all events are anomalous (binary E ≈ lenses, events with parallax); although they were initially detected by our search algorithm, they were rejected by cuts imposing conditions on the point-source point-lens model fit quality. The measured optical depths and event rates may be thus slightly ( 10%) underestimated. We have ∼ not explicitly corrected the measured quantities for anomalous events. The optical depths and event rates were calculated in individual OGLE fields, and the most robust comparisons with the Galactic models can be performed on a field-to-field basis. For illustration purposes, we constructed high-resolution maps (100 100) showing the distribution × of Γ and τ in the sky in the Galactic coordinates (Figures 5.16 and 5.17). There are two versions of each map – unbinned and smoothed with a Gaussian with σ = 100. The first important conclusion that can be drawn from these images is that both maps are continuous. Recall that maps were constructed from two independent samples of microlensing events that were selected using different criteria. Moreover, for events from high cadence fields, we used image-level simulations to assess the detection efficiencies, while for low cadence fields, catalog-level simulations were used. Figures 5.16 and 5.17 revealed no discontinuities, which could indicate systematic errors in the analysis. The optical depth map is more granular (especially the Gaussian smoothed version) because optical depths are prone to large statistical fluctuations due to rare very long-timescale events. Event rates do not directly depend on timescales, so Figure 5.16 is smoother. Figures 5.16, 5.17, and 5.18 show that both the optical depth and event rate decrease with increasing angular distance from the Galactic center. However, the number of observed events sharply decreases at low Galactic latitudes ( b 1◦) owing to extremely large interstellar | | ≤ extinction. In these regions, we detected only a few microlensing events with nearby sources. Figure 5.18 shows that both τ and Γ turn over at b 1.5◦ because extinction limits the number | | ≈ of observable sources (in the optical band). Source stars of detected events are located closer than those at larger Galactic latitudes, and hence the number of potential lenses (and so the optical depth) is smaller. The situation should be different in the infrared bands. We expect that

120 10

621 619 616 5 629 624 622 665 617 630 625 715 666 614 636 631 626 667 612 615 637 632 714 611 613 652 641 638 633 653 662 675 642 639 683 654 668 648 676 643 655 677 647 649 0 533 650 644 670 651 645 500 717 664 gb6 659 661 504 gb5501 672 663 646 gb10 660 609 543 505 534 580gb15 511 gb4 610 gb21 544 gb9 535 603 Galactic latitude (deg) gb14 512 506 gb1 680 545 gb18 518 gb3 502 604 527gb20 513 gb8 507 522 519 gb13 503 605 528 gb17 508 gb19 520 514 gb7 gb2 606 529 523 gb12 509 599 5 gb16 521 515 530 524 510 600 − 573 gb11 516 531 525 536 588 517 532 526 597 539 566 657 546 547

10 − 10 5 0 5 10 Galactic longitude (deg) − −

Figure 5.19. MOA-II ﬁelds (color polygons) overlaid on the analyzed OGLE-IV ﬁelds.

τ and Γ should follow the rising trend that is observed at larger latitudes. The ongoing infrared surveys, such as VVV or UKIRT, will be able to probe these regions.

◦ ◦ We fitted a simple exponential models τ = τ exp(cτ (3 b ) and Γ = Γ exp(c (3 b ) 0 − | | 0 Γ − | | for fields located within l < 3◦ and b > 2◦, where variations of τ and Γ with Galactic | | | | −6 longitude are small. We found τ = (1.25 0.04) 10 and cτ = 0.39 0.03, the optical depth 0 ± × ± is symmetric (within the error bars) with respect to the Galactic plane: τ = (1.22 0.06) 10−6, 0 ± × ◦ −6 ◦ cτ = 0.35 0.08 for b > 0 and τ = (1.28 0.05) 10 , cτ = 0.41 0.03 for b < 0 . The ± 0 ± × ± event rate can also be described using the exponential model with Γ = (12.3 0.3) 10−6 yr−1 0 ± × −6 −1 and cτ = 0.49 0.02. The best-fitting models for the northern (Γ = (10.8 0.5) 10 yr , ± 0 ± × −6 −1 cτ = 0.49 0.07) and southern (Γ = (13.1 0.4) 10 yr and cτ = 0.52 0.02) hemispheres ± 0 ± × ± are marginally consistent. Our optical depths and event rates are smaller than previous determinations based on all-star samples of events (Table 5.1), but they are consistent (within 1.5σ) with EROS-2 measurements

121 Stars down to I = 20 Stars down to I = 20

) nOGLE/nMOA = 1.06 ) nOGLE/nMOA = 1.50 2 2 − −

1500 1500

1000 1000

500 500 Star density (OGLE-IV) (stars arcmin Star density (OGLE-IV) (stars arcmin

0 0 0 500 1000 1500 0 500 1000 1500 2 2 Star density (Sumi & Penny 2016) (stars arcmin− ) Star density (Sumi & Penny 2016) (stars arcmin− )

Figure 5.20. Comparison between the source star surface density in MOA-II fields from Sumi & Penny (2016) and that calculated using OGLE star catalogs – uncorrected (left panel) or corrected for incompleteness (right). based on bright (red clump) stars (Afonso et al. 2003; Hamadache et al. 2006). Figure 5.18 shows the comparison between the measured τ and Γ in the central Galactic bulge fields ( l < 3◦) and the recent measurements of Sumi & Penny (2016), which are based on a | | sample of 474 events from the MOA-II survey. Sumi & Penny (2016) carried out two types of measurements – one based on red clump giant stars and another using all stars brighter than I = 20. Their microlensing optical depths based on red clump stars were systematically lower than those based on the all-source sample (see Figure 5.18). Our measurements are consistent with the MOA-II red clump sample and are a factor of 1.5 lower than those based on all ∼ MOA-II events. Similarly, our event rates are systematically lower (by a factor of 1.6) than ∼ those measured by Sumi & Penny (2016). We tried to determine the cause of this difference. We suspected the cause was the number of sources used for optical depth and event rate calculations. MOA-II fields toward the Galactic bulge (with the exception of gb21) overlap with the currently analyzed OGLE-IV fields, as shown in Figure 5.19. First, we used the data (number of stars down to I = 20 and number of subfields) reported in Table 1 of Sumi & Penny (2016) to estimate the surface density of stars in their fields. Each MOA-II subfield has an area of 98.1 arcmin2, so the reported surface density varies from 338 (gb21) to 1275 (gb9) stars per arcmin2. Then, we measured the number of sources in MOA-II fields that are detected on OGLE reference images and are brighter than I = 20. Note that we did not correct our star counts for incompleteness. As shown in the

122 30 3.0 OGLE ( l < 3◦) OGLE ( l < 3◦)

) | | | |

1 25 Besan¸conmodel ) 2.5 Besan¸conmodel 6 − − yr

6 20 2.0 − 15 1.5

10 1.0 Optical depth (10 Event rate (10 5 0.5

0 0.0 5 0 5 5 0 5 − Galactic latitude (deg) − Galactic latitude (deg)

Figure 5.21. Comparison between the observed microlensing event rate and optical depth and predictions of the Besançon model from Awiphan et al. (2016) (for sources brighter than I = 21 ◦ in the region l < 3 and events shorter than tE = 300 d). Dashed lines are the best-fit exponential models| | to the OGLE data: τ = (1.25 0.04) 10−6 exp((0.39 0.03) (3◦ b )) and Γ = (12.3 0.3) 10−6 yr−1 exp((0.49 0±.02) (3×◦ b )). ± × −| | ± × ± × − | | left panel of Figure 5.20, the surface density of OGLE sources is 6% larger than that used by Sumi & Penny (2016). Some OGLE sources may be blended with fainter stars. However, as we showed in Section 5.4, OGLE star catalogs are not complete down to I = 20, especially in the most crowded fields. Thus, the discrepancy between the numbers of sources should be even larger. Indeed, we found that star counts (down to I = 20) estimated using our image-level simulations were a factor of 1.5 larger than those reported by Sumi & Penny (2016) (see the right panel of Figure 5.20), which explains the constant systematic difference between the optical depths and event rates. For example, for the MOA field gb9 (with 1275 stars per arcmin2 according to Sumi & Penny 2016), we measured the star density of 1896 arcmin−2 using our image-level simulations and 1747 arcmin−2 by matching the luminosity function template. As shown in Table 5.4, star counts calculated using our two independent approaches are consistent within 10% with the “ground truth” based on very deep images taken with HST. We are therefore confident that the larger source star counts (and so smaller optical depths and event rates) are correct. Recall that theoretical models of the Galactic bulge were not able to explain the large optical depths calculated using MOA-II observations. For example, the revised optical depths (Figure 14 in Sumi & Penny 2016) were a factor of 1.5 larger than predictions of the Besançon model by Awiphan et al. (2016). The Galactic model of Penny et al. (2019a) underpredicted the microlensing event rate (Sumi & Penny 2016) by a factor of 2.11. Our new measurements

123 of the microlensing optical depth and event rate based on a large sample of 8002 events from OGLE-IV will allow strict tests of the current models. As an example, we used the Manchester-Besançon Microlensing Simulator1 (Awiphan et al. 2016) to confront the predictions of the recent version of the Besançon Galactic model (Robin et al. 2014) with our observations. We simulated events with sources brighter than I = 21 and timescales shorter than tE = 300 d using the version 1307 of the Besançon model. The model is described in detail by Robin et al. (2014) and Awiphan et al. (2016). It consists of a thin disk, a thick disk, a boxy bulge (bar), and a stellar halo. The model also includes a 3D extinction map (Marshall et al. 2006) based on star counts from 2MASS (the Two Micron All Sky Survey). Our observations agree reasonably well with the Besançon predictions (see Figures 5.21, 5.22 and 5.23), especially at positive Galactic latitudes. We did not adjust the model parameters; instead, we overplotted its predictions on our data. Our optical depths and event rates are symmetric about the Galactic plane, whereas the model predicts that there should be more events at negative latitudes. However, the differences are small and can be partially explained by binary and anomalous events, which are not included in our measurements. We also estimate that systematic errors on stars counts may be on the order of 10 15%. It is also noteworthy − that the model predicts some detailed features of the map, such as the increased event rate at (l, b) (1◦, 2◦) (see Figure 5.22). According to model predictions, Γ and τ should turn over ≈ − at b < 1.5◦ owing to the increasing impact of extinction, which agrees well with our data. | | However, the minuscule details of maps at low Galactic latitudes are different and can be ﬁxed by incorporating recent extinction maps (e.g., Gonzalez et al. 2012; Nataf et al. 2013) into the model.

1 http://www.mabuls.net/

124 10 30

20 )

5 1 −

10 yr 6 −

0 5

5 Event rate (10 Galactic latitude (deg) − 2

10 1 − 10 5 0 5 10 − − Galactic longitude (deg)

10 30

20 )

5 1 −

10 yr 6 −

0 5

3 Event rate (10 Galactic latitude (deg) 5 − 2

10 1 − 10 5 0 5 10 Galactic longitude (deg)− −

Figure 5.22. Comparison between the observed microlensing event rate (upper panel) and predictions of the Besançon model from Awiphan et al. (2016) for sources brighter than I = 21 and events shorter than tE = 300 d (lower panel). Both maps have the same color scale.

125 10 4.0 3.0

2.0 5 ) 6 −

1.0 0

0.5

5 Optical depth (10 Galactic latitude (deg) −

10 0.2 − 10 5 0 5 10 − − Galactic longitude (deg)

10 4.0 3.0

2.0 5 ) 6 −

1.0 0

0.5 Optical depth (10

Galactic latitude (deg) 5 −

10 0.2 − 10 5 0 5 10 Galactic longitude (deg)− −

Figure 5.23. Comparison between the observed microlensing optical depth (upper panel) and predictions of the Besançon model from Awiphan et al. (2016) for sources brighter than I = 21 and events shorter than tE = 300 d (lower panel). Both maps have the same color scale.

126 5.8.3. Microlensing events in the direction of the Sagittarius Dwarf Spheroidal Galaxy

The main focus of this thesis is the study of microlensing events toward the Galactic bulge. In this section, however, we will discuss microlensing events in the direction of the Sagittarius Dwarf Spheroidal (Sgr dSph) galaxy (Ibata et al. 1994), for reasons that will become clear later. Sgr dSph is one of the closest neighbors of the Milky Way, it is located at a distance of 26.7 1.3 kpc (Hamanowicz et al. 2016) on the opposite site of the Galactic center from Earth, ± near the Galactic bulge. The galaxy is extended, but its core (corresponding to the globular cluster M54) is located at the Galactic coordinates of (l, b) = (+5.6◦, 14.1◦). − During 2011–2014, OGLE carried out a dedicated survey of the central regions of Sgr dSph with the aim of detecting variable stars and constructing the 3D picture of the galaxy and its stream. Seven fields (BLG705–BLG711) covering the area of about 10 square degrees were observed. The survey’s results were published by Hamanowicz et al. (2016). We used these observations to search for microlensing events. Figure 5.24 shows the surface density of stars in Sgr dSph fields, which decreases from 250 to 80 stars per arcmin2 with the increasing Galactic latitude. This suggests that the majority of observed sources are in fact located in the Milky Way, which will allow us to estimate the optical depth and event rate in the “field”, far from the Galactic plane. It has to be stressed, however, that the center of Sgr dSph (globular cluster M54) is clearly visible in the star counts map. We detected two microlensing events. Event 1 (Galactic coordinates l = 4.86◦, b = 14.90◦) is located close to the core of Sgr dSph and so we checked if the source can − belong to the dwarf galaxy. According to the microlensing model the majority of light came

from the source, which is included in the Gaia DR2. Its proper motion (µα = 1.00 − ± −1 1.51, µδ = 9.77 1.71 mas yr ) is inconsistent with that of Sgr dSph (µα 2.69, − ± ≈ − −1 µδ 1.36 mas yr ) (Gaia Collaboration et al. 2018a). Moreover, because the event was ≈ − simultaneously observed in V - and I- bands, we were able to measure its color (V I) = − s +0.23 +0.64 0.92−0.27 (from the model-independent regression) and brightness Is = 20.18−0.76. The source position on the color–magnitude diagram is also consistent with that of Milky Way stars. Event 2 (l = 5.51◦, b = 11.61◦) is located 13.700 from a bright V = 11.1 star that is − saturated in OGLE images. We nonetheless checked the individual CCD images of the field and verified this a genuine transient event, not a diffraction spike from the neighboring star. The event has a timescale of t = 10.0+2.4 d and source brightness I = 20.73 0.29. There E −1.6 s ± are no magnified V -band observations, nor the source is included in the Gaia DR2 catalog. We

127 8 300 −

10 − 200 event 2 2 12 −

M54 14 100 − event 1 90 stars/arcmin 80 Galactic latitude (deg) 16 70 − 60 18 50 − 10 8 6 4 2 0 Galactic longitude (deg)

Figure 5.24. Star surface density (down to I = 21) in OGLE ﬁelds toward the Sagittarius Dwarf Spheroidal (Sgr dSph) galaxy. Stars mark the location of detected microlensing events. The core of the galaxy (globular cluster M54) is marked with a black circle. cannot rule out the source is located in Sgr dSph, but the event is located far from its center in the region of high density of Galactic stars. We will assume it also belongs to the Milky Way. Both detected events are likely caused by stars located in the thick disk of the Milky Way. Taking into account the number of observed sources down to I = 21 and the duration of the survey, we estimate that the microlensing optical depth in this direction (l 5◦, b 14◦) is ≈ ≈ − τ = (0.09 0.07) 10−6, while the event rate Γ = (0.8 0.6) 10−6 yr−1. These estimates ± × ± × are consistent with the predictions of the Besançon model (Awiphan et al. 2016). Awiphan et al. (2016) calculated their model in the latitude range b < 10◦, but from extrapolation to (l 5◦, | | ≈ b 14◦) we found τ = (0.044 0.003) 10−6 and Γ = (0.29 0.03) 10−6 yr−1. ≈ − model ± × model ± × Cseresnjes & Alard (2001), who studied microlensing toward Sgr dSph, argued that at high Galactic latitudes (b 9◦) microlensing events with Sgr dSph sources may outnumber Milky ≈ − Way events by a factor of 5 or larger. Our observations suggest that the optical depth in these regions is smaller and is consistent with Milky Way events.

128 Summary

Gravitational microlensing is a powerful technique that allows us to study dark objects in a large mass range – from Earth-sized planets to brown dwarfs and black holes. It is also one of the most recent astrophysical techniques; although its theoretical foundations were developed in the first half of the 20th century, it was put into practice in the 1990s. However, only the last decade has brought about developments that have allowed us to fully appreciate the power of microlensing. Thanks to the installation of new, large field-of-view detectors, microlensing surveys have been able to monitor hundreds of millions of stars toward the Galactic center with a cadence as short as several minutes. Thousands of detected microlensing events allow us to make robust statistical inferences and to detect rare phenomena. In this thesis, I used long-term photometric observations collected as part of the OGLE-IV survey to search for gravitational microlensing events toward the Galactic center. These events are then used to solve important astrophysical problems. The first part of my thesis (Chapters 3 and 4) focuses on searching for and constraining the frequency of rogue planets in the Milky Way. The possibility of detecting the free-floating planets has been discussed in the microlensing community for several years, but it required high-cadence observations. The first results, published in 2011, claimed that free-floating Jupiters should be nearly twice as common as main-sequence stars (Sumi et al. 2011). That paper attracted considerable attention and triggered dozens of follow-up studies. However, over the years, serious doubts were cast over claims of a large population of Jovian-mass free-floaters; infrared surveys of young stellar clusters discovered significantly fewer substellar-mass objects, while theorists predicted relatively few Jupiter-mass free-floating planets (most of the ejected planets should be of Earth mass). I led the analysis of a much larger sample of microlensing events that were detected by OGLE between 2010 and 2015. My statistical analysis demonstrated that Jovian-mass free-floating planets are much less common than previously thought (less than 0.25 objects per star). For the first time, I was able to study the population of the shortest microlensing events

129 (with timescales between 0.1 and 0.5 days) and I have found a few ultra-short microlensing events that are likely caused by Earth- and super-Earth-mass objects (either free-floating or on wide orbits). My subsequent studies, in collaboration with other microlensing surveys (KMTNet, MOA, Wise), led to the first measurements of the angular Einstein radii of free-floating planet candidates. These measurements enabled me to constrain masses of free-floating planet candidates, as they remove a degeneracy between the mass and velocity of the lens. These findings were published in three papers in peer-reviewed journals (Mróz et al. 2017a, 2018, 2019) and made a significant impact on the microlensing community; they have been cited nearly 80 times at the moment of writing. The third paper (Mróz et al. 2019) has been selected by Astronomy & Astrophysics Editors as an A&A Highlight. Moreover, the future prospects for exploring the population of the low-mass free-floating (or wide-orbit) planets have been discussed in all five microlensing white papers that were prepared for the Astronomy and Astrophysics Decadal Survey (Astro2020) (Bennett et al. 2019; Bhattacharya et al. 2019; Lee et al. 2019; Penny et al. 2019b; Yee et al. 2019). In the second part of the thesis (Chapter 5), I used microlensing events detected by OGLE to study the structure of the Milky Way. I created the largest and the most accurate microlensing optical depth and event rate maps of the Galactic bulge. The new maps ease the tension between the previous measurements and Galactic models – the previous measurements of the optical depth were too large and difficult to reconcile with other constraints on the Galactic structure. The new measurements are consistent with some earlier calculations based on bright stars (from EROS and MOA) and systematically 30% smaller than the other estimates based ∼ on “all-source” samples. The difference is caused by the wrong number of source stars used for optical depth and event rate calculations. The new maps agree well with predictions of the Besançon model of the Galaxy. Our new measurements of the microlensing optical depth and event rate based on a large sample of over 8,000 events will enable strict tests of competing Galactic models. They will also have numerous other applications, such as the measurement of the initial mass function in the Galactic bulge or constraining the dark matter content in the Milky Way center. The new maps will also allow planning of the future space-based microlensing experiments by revising the expected number of events. The publication based on findings reported in Chapter 5 is under preparation. A. OGLE-IV fields

Table A.1: Basic information about analyzed ﬁelds.

Field RA Decl. l b Nstars Nepochs h m s BLG500 17 51 60 28◦3603500 0.9999 1.0293 4.1 4701 − − h m s BLG501 17 51 56 29◦5000000 0.0608 1.6400 5.3 12099 − − − h m s BLG502 17 51 39 33◦3201500 3.2832 3.4735 5.0 1700 − − − h m s BLG503 17 51 34 34◦4600500 4.3547 4.0831 4.9 745 − − − h m s BLG504 17 57 33 27◦5904000 2.1491 1.7747 5.9 6429 − − h m s BLG505 17 57 34 29◦1301500 1.0870 2.3890 7.0 12066 − − h m s BLG506 17 57 31 30◦2702300 0.0103 2.9974 5.4 4706 − − h m s BLG507 17 57 30 31◦4103000 1.0641 3.6101 5.5 1882 − − − h m s BLG508 17 57 30 32◦5502000 2.1341 4.2222 4.3 1490 − − − h m s BLG509 17 57 30 34◦0901000 3.2058 4.8329 4.6 807 − − − h m s BLG510 17 57 30 35◦2300000 4.2794 5.4419 3.9 559 − − − h m s BLG511 18 03 02 27◦2204900 3.2835 2.5219 5.6 4588 − − h m s BLG512 18 03 04 28◦3603900 2.2154 3.1355 7.1 10253 − − h m s BLG513 18 03 06 29◦5004000 1.1399 3.7432 4.6 1909 − − h m s BLG514 18 03 11 31◦0402700 0.0747 4.3626 4.6 1532 − − h m s BLG515 18 03 15 32◦1802500 0.9993 4.9741 4.4 1350 − − − h m s BLG516 18 03 20 33◦3201500 2.0711 5.5870 4.2 634 − − − h m s BLG517 18 03 25 34◦4600500 3.1453 6.1976 3.6 197 − − − h m s BLG518 18 08 26 26◦4601000 4.4046 3.2761 4.9 1937 − − h m s BLG519 18 08 30 28◦0000000 3.3316 3.8823 5.8 1988 − − h m s BLG520 18 08 36 29◦1305000 2.2603 4.4933 5.6 1625 − − h m s BLG521 18 08 44 30◦2704000 1.1905 5.1086 5.1 738 − − h m s BLG522 18 13 47 26◦0901500 5.5202 4.0329 4.7 999 − − h m s BLG523 18 13 55 27◦2300500 4.4477 4.6423 6.3 779 − − h m s BLG524 18 14 02 28◦3605500 3.3712 5.2462 6.1 644 − − h m s BLG525 18 14 12 29◦5004500 2.2975 5.8574 4.6 558 − − Continued on next page

131 Field RA Decl. l b Nstars Nepochs h m s BLG526 18 14 25 31◦0403500 1.2260 6.4752 3.1 516 − − h m s BLG527 18 19 05 23◦0404000 8.8082 3.6426 3.2 385 − − h m s BLG528 18 19 08 24◦1803000 7.7241 4.2297 4.4 439 − − h m s BLG529 18 19 11 25◦3202000 6.6383 4.8152 5.6 449 − − h m s BLG530 18 19 14 26◦4601000 5.5505 5.3987 5.6 441 − − h m s BLG531 18 19 26 28◦0000000 4.4760 6.0094 5.3 396 − − h m s BLG532 18 19 36 29◦1305000 3.3951 6.6107 3.8 358 − − h m s BLG533 17 52 00 27◦2300500 2.0542 0.4054 0.7 46 − − h m s BLG534 17 51 51 31◦0401500 1.1356 2.2547 4.3 4646 − − − h m s BLG535 17 51 44 32◦1802500 2.2129 2.8632 3.7 1947 − − − h m s BLG536 17 57 30 36◦3605000 5.3552 6.0490 3.2 197 − − − h m s BLG539 18 03 30 35◦5905500 4.2223 6.8055 2.8 191 − − − h m s BLG543 18 13 47 22◦2704500 8.7716 2.2752 3.4 395 − − h m s BLG544 18 13 47 23◦4103500 7.6890 2.8622 3.0 384 − − h m s BLG545 18 13 47 24◦5502500 6.6053 3.4482 3.9 731 − − h m s BLG546 18 14 39 32◦1802500 0.1530 7.0928 2.8 244 − − h m s BLG547 18 14 52 33◦3201500 0.9252 7.7037 3.3 195 − − − h m s BLG566 18 34 55 23◦4103500 9.9310 7.1538 2.4 198 − − h m s BLG573 18 08 54 31◦4103000 0.1216 5.7276 4.5 584 − − h m s BLG580 18 08 22 25◦3202000 5.4762 2.6684 4.7 1803 − − h m s BLG588 18 09 02 32◦5502000 0.9534 6.3377 3.7 492 − − − h m s BLG597 18 09 12 34◦0901000 2.0280 6.9510 2.6 359 − − − h m s BLG599 17 51 29 35◦5905500 5.4275 4.6916 4.1 374 − − − h m s BLG600 17 51 23 37◦1304500 6.5036 5.2961 3.6 370 − − − h m s BLG603 17 45 46 34◦0901000 4.4401 2.7412 4.7 1631 − − − h m s BLG604 17 45 36 35◦2300000 5.5113 3.3505 4.4 817 − − − h m s BLG605 17 45 25 36◦3605000 6.5850 3.9567 3.8 371 − − − h m s BLG606 17 45 13 37◦5004000 7.6615 4.5597 2.9 195 − − − h m s BLG609 17 39 50 34◦4600500 5.6069 2.0233 3.6 725 − − − h m s BLG610 17 39 35 35◦5905500 6.6777 2.6334 3.4 524 − − − h m s BLG611 17 35 33 27◦0904100 0.3282 2.8242 5.1 4526 − h m s BLG612 17 30 00 28◦0000000 1.0464 3.4008 3.1 768 − − h m s BLG613 17 30 00 29◦1305000 2.0762 2.7248 4.1 811 − − h m s BLG614 17 24 27 28◦3605500 2.2382 4.0765 3.6 620 − − h m s BLG615 17 24 23 29◦5004500 3.2679 3.3992 4.0 800 − − h m s BLG616 17 16 00 28◦3000000 3.1997 5.6697 3.6 225 − − h m s BLG617 17 16 00 29◦4305000 4.2100 4.9609 3.8 726 − − Continued on next page

132 Field RA Decl. l b Nstars Nepochs h m s BLG619 17 25 00 26◦0000000 0.0082 5.4341 2.8 59 − h m s BLG621 17 35 00 22◦1305000 4.4323 5.5773 3.6 477 − h m s BLG622 17 35 00 23◦2704000 3.3887 4.9182 3.3 449 − h m s BLG624 17 40 14 21◦3605500 5.6025 4.8747 3.8 502 − h m s BLG625 17 40 17 22◦5004500 4.5592 4.2168 3.9 537 − h m s BLG626 17 40 20 24◦0403500 3.5176 3.5577 4.6 755 − h m s BLG629 17 45 23 19◦4601000 7.8134 4.8089 3.2 185 − h m s BLG630 17 45 25 21◦0000000 6.7610 4.1659 3.5 564 − h m s BLG631 17 45 30 22◦1305000 5.7162 3.5117 4.1 599 − h m s BLG632 17 45 36 23◦2704000 4.6747 2.8534 5.2 877 − h m s BLG633 17 45 42 24◦4103000 3.6341 2.1945 3.7 672 − h m s BLG636 17 50 35 20◦2300500 7.9121 3.4452 2.8 200 − h m s BLG637 17 50 39 21◦3605500 6.8600 2.8041 3.9 522 − h m s BLG638 17 50 47 22◦5004500 5.8168 2.1491 3.7 575 − h m s BLG639 17 50 56 24◦0403500 4.7762 1.4906 2.9 543 − h m s BLG641 17 55 48 21◦0000000 7.9992 2.0816 3.0 358 − h m s BLG642 17 55 55 22◦1305000 6.9487 1.4394 1.9 362 − h m s BLG643 17 56 06 23◦2704000 5.9062 0.7838 1.7 358 − h m s BLG644 17 57 30 25◦3202000 4.2691 0.5348 0.8 36 − − h m s BLG645 17 57 30 26◦4601000 3.2039 1.1511 2.0 550 − − h m s BLG646 18 02 55 26◦0901500 4.3401 1.8981 2.6 690 − − h m s BLG647 17 52 04 26◦0901500 3.1204 0.2092 0.5 198 − h m s BLG648 17 46 36 26◦4601000 1.9634 0.9419 1.6 644 − h m s BLG649 17 46 28 28◦0000000 0.8963 0.3281 0.8 37 − h m s BLG650 17 46 22 29◦1305000 0.1665 0.2925 0.7 45 − − − h m s BLG651 17 46 14 30◦2704000 1.2329 0.9073 1.1 41 − − − h m s BLG652 17 41 10 26◦0901500 1.8508 2.3000 3.0 646 − h m s BLG653 17 35 32 28◦3605500 0.8998 2.0442 3.2 778 − − h m s BLG654 17 35 36 29◦5004500 1.9286 1.3682 2.7 920 − − h m s BLG655 17 35 41 31◦0403500 2.9551 0.6890 1.3 56 − − h m s BLG657 18 19 50 30◦2704000 2.3179 7.2216 4.3 208 − − h m s BLG659 17 46 04 31◦4103000 2.3030 1.5164 1.0 37 − − − h m s BLG660 17 45 56 32◦5502000 3.3695 2.1318 3.2 687 − − − h m s BLG661 17 40 05 33◦3201500 4.5361 1.4138 2.2 531 − − − h m s BLG662 17 30 00 30◦2704000 3.1047 2.0479 3.1 573 − − h m s BLG663 18 08 18 24◦1803000 6.5466 2.0595 1.6 230 − − h m s BLG664 18 02 55 24◦5502500 5.4117 1.2924 1.4 230 − − Continued on next page

133 Field RA Decl. l b Nstars Nepochs h m s BLG665 17 30 04 25◦1901500 1.2091 4.8559 1.9 114 − h m s BLG666 17 30 03 26◦3300500 0.1737 4.1858 1.6 89 − h m s BLG667 17 35 28 25◦5601000 1.3523 3.4996 1.8 456 − h m s BLG668 17 29 57 31◦4103000 4.1383 1.3792 1.4 323 − − h m s BLG670 17 33 57 34◦0901000 5.7392 0.6699 1.3 338 − − − h m s BLG672 17 33 02 36◦3700000 7.9091 1.8529 1.8 343 − − − h m s BLG675 17 40 58 27◦2300500 0.7819 1.6875 3.3 900 − h m s BLG676 17 40 49 28◦3605500 0.2799 1.0642 2.4 69 − − h m s BLG677 17 40 37 29◦5004500 1.3467 0.4490 0.6 66 − − h m s BLG680 17 39 06 37◦1304500 7.7734 3.2045 2.0 175 − − − h m s BLG683 17 46 39 25◦3202000 3.0214 1.5708 2.4 105 − h m s BLG705 18 41 50 30◦2704000 4.4295 11.4798 1.2 167 − − h m s BLG706 18 47 06 29◦5004500 5.4859 12.2582 1.0 164 − − h m s BLG707 18 47 31 31◦0403500 4.3744 12.8458 1.0 160 − − h m s BLG708 18 52 50 30◦2704000 5.4276 13.6408 1.0 157 − − h m s BLG709 18 58 02 29◦5004500 6.4709 14.4267 0.7 160 − − h m s BLG710 18 58 31 31◦0403500 5.3429 15.0054 0.7 153 − − h m s BLG711 19 03 50 30◦2704000 6.3893 15.8210 0.6 156 − − h m s BLG714 17 40 24 25◦1802500 2.4794 2.8944 2.8 345 − h m s BLG715 17 35 00 24◦4103000 2.3472 4.2575 2.4 444 − h m s BLG717 18 08 30 23◦0404000 7.6458 1.5033 1.8 156 − −

Equatorial coordinates are given for the epoch J2000. Nstars is the number of stars in the database in millions and Nepochs is the number of collected frames used in the analysis. l and b are Galactic longitude and latitude, respectively. B. Microlensing optical depths and event rates in the OGLE-IV ﬁelds

Table B.1: Microlensing optical depths and event rates in the OGLE-IV ﬁelds.

Field l b τ Γ tE Nev Ns h i 6 6 1 (deg) (deg) (10− ) (10− yr− ) (d) BLG500 0.9999 1.0293 1.77 0.19 21.9 1.8 18.8 164 6.78 − ± ± BLG501 0.0608 1.6400 1.95 0.14 22.1 1.3 20.5 317 13.31 − − ± ± BLG502 3.2832 3.4735 1.12 0.11 10.1 0.8 25.7 171 10.02 − − ± ± BLG503 4.3547 4.0831 0.68 0.09 5.0 0.6 31.4 91 10.70 − − ± ± BLG504 2.1491 1.7747 1.33 0.11 15.5 1.1 20.0 225 11.86 − ± ± BLG505 1.0870 2.3890 1.92 0.16 20.4 1.0 21.8 441 19.02 − ± ± BLG506 0.0103 2.9974 1.81 0.23 15.1 1.0 28.0 247 12.85 − ± ± BLG507 1.0641 3.6101 1.11 0.09 11.3 0.8 22.9 216 11.22 − − ± ± BLG508 2.1341 4.2222 0.76 0.08 6.2 0.6 28.5 119 9.53 − − ± ± BLG509 3.2058 4.8329 0.67 0.09 4.2 0.5 36.6 79 9.82 − − ± ± BLG510 4.2794 5.4419 0.44 0.08 3.3 0.6 31.2 38 7.67 − − ± ± BLG511 3.2835 2.5219 1.31 0.14 12.4 0.9 24.5 204 13.45 − ± ± BLG512 2.2154 3.1355 1.32 0.12 12.8 0.8 24.0 280 17.48 − ± ± BLG513 1.1399 3.7432 0.82 0.07 7.8 0.6 24.4 213 13.71 − ± ± BLG514 0.0747 4.3626 0.62 0.07 5.7 0.6 25.2 108 10.27 − ± ± BLG515 0.9993 4.9741 0.68 0.10 5.2 0.7 30.5 80 8.10 − − ± ± BLG516 2.0711 5.5870 0.46 0.07 4.2 0.6 25.3 48 7.61 − − ± ± BLG517 3.1453 6.1976 0.33 0.10 2.4 0.8 32.6 12 5.96 − − ± ± BLG518 4.4046 3.2761 0.86 0.09 6.7 0.5 30.0 160 12.34 − ± ± BLG519 3.3316 3.8823 0.95 0.09 8.5 0.6 25.9 223 12.93 − ± ± BLG520 2.2603 4.4933 0.65 0.08 5.1 0.5 29.5 124 11.63 − ± ± BLG521 1.1905 5.1086 0.54 0.07 4.1 0.5 30.9 69 9.18 − ± ± BLG522 5.5202 4.0329 0.83 0.10 6.8 0.7 28.5 108 8.62 − ± ± Continued on next page

135 Field l b τ Γ tE Nev Ns h i 6 6 1 (deg) (deg) (10− ) (10− yr− ) (d) BLG523 4.4477 4.6423 0.80 0.10 5.5 0.6 34.1 99 10.40 − ± ± BLG524 3.3712 5.2462 0.66 0.10 5.5 0.7 28.2 75 9.66 − ± ± BLG525 2.2975 5.8574 0.37 0.07 2.9 0.5 29.9 36 7.84 − ± ± BLG526 1.2260 6.4752 0.28 0.07 2.1 0.5 31.3 20 5.25 − ± ± BLG527 8.8082 3.6426 0.84 0.14 5.0 0.8 39.5 48 6.36 − ± ± BLG528 7.7241 4.2297 0.58 0.10 5.0 0.8 26.9 50 7.39 − ± ± BLG529 6.6383 4.8152 0.60 0.11 5.1 0.9 27.2 39 7.79 − ± ± BLG530 5.5505 5.3987 0.92 0.15 7.8 1.2 27.6 54 7.37 − ± ± BLG531 4.4760 6.0094 0.62 0.12 4.7 0.9 30.7 34 6.85 − ± ± BLG532 3.3951 6.6107 0.38 0.09 2.5 0.6 34.8 23 6.02 − ± ± BLG533 2.0542 0.4054 1.13 1.13 6.5 6.5 40.5 1 1.02 − ± ± BLG534 1.1356 2.2547 1.42 0.14 15.8 1.2 20.8 176 9.06 − − ± ± BLG535 2.2129 2.8632 1.53 0.14 13.9 1.0 25.7 189 7.57 − − ± ± BLG536 5.3552 6.0490 0.47 0.13 3.5 1.0 31.3 15 5.21 − − ± ± BLG539 4.2223 6.8055 0.56 0.19 3.7 1.3 34.9 13 4.52 − − ± ± BLG543 8.7716 2.2752 0.98 0.18 5.4 1.0 41.8 36 4.76 − ± ± BLG544 7.6890 2.8622 0.71 0.12 5.6 0.9 29.2 40 5.66 − ± ± BLG545 6.6053 3.4482 0.78 0.12 5.6 0.8 32.3 57 6.47 − ± ± BLG546 0.1530 7.0928 0.41 0.12 3.1 0.9 30.7 12 4.25 − ± ± BLG547 0.9252 7.7037 0.17 0.10 1.2 0.7 32.6 3 3.67 − − ± ± BLG566 9.9310 7.1538 0.13 0.09 0.8 0.6 38.5 2 3.03 − ± ± BLG573 0.1216 5.7276 0.37 0.07 3.4 0.6 25.0 33 7.16 − ± ± BLG580 5.4762 2.6684 0.92 0.10 7.3 0.7 29.1 116 8.47 − ± ± BLG588 0.9534 6.3377 0.37 0.08 2.5 0.5 33.9 23 5.39 − − ± ± BLG597 2.0280 6.9510 0.51 0.10 3.7 0.7 31.6 27 4.28 − − ± ± BLG599 5.4275 4.6916 0.48 0.09 3.6 0.6 31.4 41 7.80 − − ± ± BLG600 6.5036 5.2961 0.40 0.11 2.7 0.7 35.1 18 5.29 − − ± ± BLG603 4.4401 2.7412 1.11 0.11 9.7 0.8 26.7 167 10.02 − − ± ± BLG604 5.5113 3.3505 0.88 0.10 6.4 0.7 32.0 100 9.80 − − ± ± BLG605 6.5850 3.9567 0.71 0.16 3.6 0.7 45.9 29 6.13 − − ± ± BLG606 7.6615 4.5597 0.90 0.46 3.5 1.1 60.3 11 4.48 − − ± ± BLG609 5.6069 2.0233 1.14 0.14 7.7 0.9 34.4 81 6.77 − − ± ± BLG610 6.6777 2.6334 0.81 0.15 5.7 1.2 32.8 38 5.78 − − ± ± BLG611 0.3282 2.8242 1.39 0.14 14.9 1.2 21.8 158 6.93 ± ± BLG612 1.0464 3.4008 1.09 0.14 9.3 1.1 27.0 87 7.34 − ± ± BLG613 2.0762 2.7248 1.20 0.13 11.4 1.1 24.4 119 7.99 − ± ± Continued on next page

136 Field l b τ Γ tE Nev Ns h i 6 6 1 (deg) (deg) (10− ) (10− yr− ) (d) BLG614 2.2382 4.0765 0.72 0.13 5.0 0.8 33.2 43 6.32 − ± ± BLG615 3.2679 3.3992 1.00 0.15 6.2 0.8 37.5 74 8.09 − ± ± BLG616 3.1997 5.6697 0.38 0.11 3.1 0.9 28.2 12 5.08 − ± ± BLG617 4.2100 4.9609 0.64 0.13 4.8 1.0 30.8 43 6.32 − ± ± BLG619 0.0082 5.4341 0.63 0.37 3.0 1.8 49.0 3 4.47 ± ± BLG621 4.4323 5.5773 0.65 0.11 6.2 1.1 24.5 44 5.15 ± ± BLG622 3.3887 4.9182 0.69 0.16 3.3 0.7 49.4 25 4.76 ± ± BLG624 5.6025 4.8747 0.56 0.12 3.9 0.8 33.6 31 6.02 ± ± BLG625 4.5592 4.2168 0.72 0.14 5.0 0.8 33.4 46 6.47 ± ± BLG626 3.5176 3.5577 0.87 0.10 7.6 0.9 26.8 92 8.32 ± ± BLG629 7.8134 4.8089 0.49 0.16 3.1 1.0 36.7 11 4.56 ± ± BLG630 6.7610 4.1659 0.67 0.16 4.0 0.7 39.1 30 5.11 ± ± BLG631 5.7162 3.5117 0.80 0.12 6.5 0.9 28.7 61 7.09 ± ± BLG632 4.6747 2.8534 1.02 0.11 8.3 0.8 28.5 128 9.59 ± ± BLG633 3.6341 2.1945 1.58 0.18 12.0 1.3 30.4 106 6.30 ± ± BLG636 7.9121 3.4452 0.59 0.16 4.7 1.3 29.0 14 4.26 ± ± BLG637 6.8600 2.8041 0.89 0.15 4.8 0.8 43.0 40 5.84 ± ± BLG638 5.8168 2.1491 0.80 0.12 6.3 0.9 29.4 64 7.96 ± ± BLG639 4.7762 1.4906 1.27 0.17 9.5 1.2 30.9 70 5.08 ± ± BLG641 7.9992 2.0816 0.75 0.15 4.5 0.9 38.8 31 5.33 ± ± BLG642 6.9487 1.4394 1.15 0.21 8.0 1.4 33.5 36 3.22 ± ± BLG643 5.9062 0.7838 1.07 0.30 5.2 1.3 47.6 17 2.11 ± ± BLG644 4.2691 0.5348 – – – 0 1.02 − BLG645 3.2039 1.1511 1.85 0.29 12.5 1.7 34.3 63 3.33 − ± ± BLG646 4.3401 1.8981 1.03 0.16 7.6 1.1 31.7 57 5.09 − ± ± BLG647 3.1204 0.2092 – – – 0 0.68 BLG648 1.9634 0.9419 1.74 0.26 16.8 2.2 24.0 67 2.86 ± ± BLG649 0.8963 0.3281 – – – 0 0.93 BLG650 0.1665 0.2925 – – – 0 0.91 − − BLG651 1.2329 0.9073 – – – 0 1.56 − − BLG652 1.8508 2.3000 1.39 0.16 13.0 1.5 24.9 100 6.55 ± ± BLG653 0.8998 2.0442 2.10 0.22 18.4 1.8 26.6 127 5.37 − ± ± BLG654 1.9286 1.3682 1.69 0.18 15.3 1.5 25.6 110 4.92 − ± ± BLG655 2.9551 0.6890 0.98 0.57 8.7 5.6 26.3 3 2.13 − ± ± BLG657 2.3179 7.2216 0.29 0.12 2.4 1.0 27.2 6 4.43 − ± ± BLG659 2.3030 1.5164 – – – 0 1.05 − − Continued on next page

137 Field l b τ Γ tE Nev Ns h i 6 6 1 (deg) (deg) (10− ) (10− yr− ) (d) BLG660 3.3695 2.1318 2.38 0.31 21.7 2.4 25.5 94 4.63 − − ± ± BLG661 4.5361 1.4138 2.04 0.29 13.9 2.0 34.0 80 4.22 − − ± ± BLG662 3.1047 2.0479 1.38 0.17 11.2 1.3 28.6 87 6.16 − ± ± BLG663 6.5466 2.0595 0.37 0.21 3.4 2.1 25.2 4 2.15 − ± ± BLG664 5.4117 1.2924 0.47 0.20 5.2 2.5 21.0 6 1.85 − ± ± BLG665 1.2091 4.8559 0.38 0.38 1.1 1.1 79.6 1 2.72 ± ± BLG666 0.1737 4.1858 1.44 0.86 9.0 5.3 37.3 3 2.04 ± ± BLG667 1.3523 3.4996 1.13 0.24 10.4 2.3 25.3 27 2.41 ± ± BLG668 4.1383 1.3792 0.74 0.20 8.1 2.7 21.3 15 1.93 − ± ± BLG670 5.7392 0.6699 1.56 0.50 9.8 2.4 37.0 19 1.65 − − ± ± BLG672 7.9091 1.8529 1.14 0.32 5.1 1.3 51.8 17 2.45 − − ± ± BLG675 0.7819 1.6875 2.20 0.21 24.3 2.1 21.0 160 5.64 ± ± BLG676 0.2799 1.0642 0.17 0.17 3.6 3.6 10.8 1 4.58 − ± ± BLG677 1.3467 0.4490 – – – 0 0.96 − BLG680 7.7734 3.2045 0.46 0.17 4.6 1.8 23.4 8 3.57 − − ± ± BLG683 3.0214 1.5708 3.33 0.81 24.0 5.4 32.3 21 4.28 ± ± BLG705 4.4295 11.4798 – – – 0 1.10 − BLG706 5.4859 12.2582 0.13 0.13 3.1 3.1 9.9 1 0.94 − ± ± BLG707 4.3744 12.8458 – – – 0 0.81 − BLG708 5.4276 13.6408 – – – 0 0.84 − BLG709 6.4709 14.4267 – – – 0 0.61 − BLG710 5.3429 15.0054 0.58 0.58 2.4 2.4 56.0 1 0.59 − ± ± BLG711 6.3893 15.8210 – – – 0 0.48 − BLG714 2.4794 2.8944 1.14 0.18 10.1 1.6 26.4 54 5.84 ± ± BLG715 2.3472 4.2575 1.14 0.20 9.4 1.7 28.0 36 3.42 ± ± BLG717 7.6458 1.5033 0.74 0.31 3.6 1.5 48.1 6 2.06 − ± ±

OGLE-IV Galactic bulge ﬁelds with Galactic coordinates of the ﬁeld center (l,b), microlensing evenr rate Γ, optical depth τ, average Einstein timescale tE , number of detected events Nev, and number of sources is millions Ns. h i

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