Astrophysical applications of gravitational microlensing in the Milky Way

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A STROPHYSICAL APPLICATIONS

OF GRAVITATIONAL MICROLENSING IN THE M ILKY W AY

Przemysław Mróz

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

Warsaw University Observatory

Warsaw, April 2019

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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 Szyma´nski, Prof. Igor Soszy´nski, 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.

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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.

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Streszczenie

Współczesne teorie opisuj ˛ace powstawanie pozasłonecznych układów planetarnych przewiduj ˛a istnienie planet swobodnych, wyrzuconych z macierzystych układów i niezwi ˛azanych grawitacyjnie z ˙zadn ˛a gwiazd ˛a. Poniewa˙z te obiekty nie emituj ˛a praktycznie

´swiatła, jedyn ˛a metod ˛a pozwalaj ˛ac ˛a na ich detekcj˛e jest mikrosoczewkowanie grawitacyjne.

W pierwszej cz˛e´sci rozprawy doktorskiej przedstawiłem wyniki moich bada´n dotycz ˛acych poszukiwania i mierzenia cz˛esto´sci wyst˛epowania planet swobodnych w Drodze Mlecznej na podstawie analizy zjawisk mikrosoczewkowania zaobserwowanych przez przegl ˛ad nieba OGLE w latach 2010–2015.

Moja analiza pokazała, ˙ze planety swobodne o masach Jowisza s ˛a znacznie rzadsze ni˙z wcze´sniej szacowano (na ka˙zd ˛a gwiazd˛e w Galaktyce przypada co najwy˙zej 0,25 masywnych planet swobodnych). Dzi˛eki danym fotometrycznym zebranym przez przegl ˛ad OGLE mogłem równie˙z zbada´c zjawiska o najkrótszych skalach czasowych. Udało mi si˛e wykry´c kilka zjawisk wywołanych prawdopodobnie przez planety swobodne (lub znajduj ˛ace si˛e na szerokich orbitach) o masach Ziemi, zgodnie z przewidywaniami teorii formowania si˛e planet.

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 pier´scienia Einsteina. Te pomiary daj ˛a lepsze ograniczenia na masy soczewkuj ˛acych obiektów, poniewa˙z umo˙zliwiaj ˛a oszacowanie ich pr˛edko´sci.

W drugiej cz˛e´sci rozprawy wykorzystałem zjawiska mikrosoczewkowania zaobserwowane przez OGLE do badania struktury Drogi Mlecznej. Przygotowałem najwi˛eksze i najdokładniejsze mapy gł˛eboko´sci optycznej i cz˛esto´sci zjawisk mikrosoczewkowania w kierunku centrum Galaktyki. Te mapy znajd ˛a liczne zastosowania: ograniczenia na modele Drogi Mlecznej, ograniczenia na zawarto´sci ciemnej materii, pomiary funkcji mas gwiazd w Drodze Mlecznej, czy planowanie przyszłych satelitarnych przegl ˛adów mikrosoczewkowych.

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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).

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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

δθ = 4GM

Rc² = 1.75⁰⁰, (1.1)

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 Paczy´nski (1986b), who proposed monitoring the brightness of millions of stars in the Magellanic Clouds

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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 Paczy´nski (1986a) to describe the effect of lensing by individual stars rather than by the entire galaxies.

Paczy´nski’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 10⁻⁵yr⁻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 Paczy´nski (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 radiusθE, 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∼ θE.

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O L S r

_E

˜ r

_E

θ

_E

ˆ α

_d

D

_l

D

_s

Figure 1.1. Natural formalism for gravitational microlensing. A lens (L) at a distanceDlfrom an observer (O) deflects light from a source (S) (at a distanceDs) by an angleαˆd. 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 massM and the relative lens-source parallaxπrel:

θE =pκMπrel, (1.2)

where κ = 4G/c²au = 8.144 mas M⁻¹ and π_rel = π_l − πs, π_l and π_s are parallaxes to the lens and source, respectively. For typical microlensing events in the Milky Wayπs ∼ 0.1 mas, π_l∼ 0.2 mas, and so:

θ_E= 0.49 mas

M

0.3 M

1/2

πrel

0.1 mas

1/2

. (1.3)

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 asπE= πrel/θEand can be interpreted as the inverse of the Einstein radius projected onto observer’s plane ˜r_E: π_E = au/˜r_E(Gould 2000a), see Figure 1.1. In typical events:

˜

rE= 4.9 au

M

0.3 M

1/2

πrel

0.1 mas

−1/2

. (1.4)

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The microlens parallax is a vector quantity:

π_E= π_rel θE

µ_rel µrel

, (1.5)

and has the direction of µ_rel– 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:

M = θ_E²

κπ_rel = θE

κπ_E, (1.6)

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 timescaletE, defined as the time it takes the source to move with respect to the lens by one Einstein ring radius:

t_E= θE

µrel

. (1.7)

As the typical lens-source proper motion is on the order ofµ_rel ∼ 5 mas yr⁻¹: t_E = 36 d

M

0.3 M

1/2

πrel

0.1 mas

1/2

µrel

5 mas yr⁻¹

−1

, (1.8)

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 π_rel and µ_rel 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 ˆαdwas derived by Einstein (1916) based on his theory of general relativity:

ˆ

αd = 4GM

|b|²c²b. (1.9)

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− α^d(θ), (1.10)

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O L

S I

θ β

α

d

D

_l

D

s

b

ˆ α

d

Figure 1.2. Geometry of gravitational microlensing events. A lens (L) at a distanceDlfrom an observer (O) deflects light from a source (S) at a distanceDsby an angleαˆd. For a point lens with a massM , ˆαd = ^4GM_c2b as derived by Einstein (1916). An image of the source (I) is located at an angular separationθ from the lens.

which is known as the lens equation (e.g. Gaudi 2012). Because all angles are very small, we can write αd = ˆα_d(1− D^l/Ds). If the lens consists of Nl point masses, each with massMi

and angular position θ_M,i, the lens equation can be rewritten as:

β= θ− 4G c²

1 Dl − 1

Ds

^Nl

X

i

Mi

θ− θ^M,i

|θ − θ^M,i|², (1.11) because bi/Dl = θ− θ^M,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 N^l > 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:

Aj = 1 det J

θ=θj

, where det J =

∂β

∂θ

. (1.12)

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major image

minor image lens

−1 0 1

(t− t0)/t_E 1

2 5 10

Magnification

u0= 0.1 u0= 0.2 u0= 0.3 u0= 0.4 u0= 0.5 u0= 0.8 u0= 1.0

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 wheredet 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:

β = θ−4GM c²

1 Dl − 1

Ds

1

θ. (1.13)

If the lens and source are perfectly aligned (β = 0), the images will form a circle with a radius of θ =

r

4GM c²

1

Dl − _D¹_s

= θ_E, which is, of course, the angular Einstein radius that was defined earlier. If we normalize all angles byθE,u = β/θEandy = θ/θE, the lens equation has a simple form:

u = y− 1

y. (1.14)

Two images are formed at separations y±= 1

2

u±√

u²+ 4

. (1.15)

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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 hencey−< 0), see Figure 1.3. The angular separation of the images is on the order ofθE, 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):

A± = 1 2

u²+ 2 u√

u²+ 4 ± 1

. (1.16)

The total magnification is the sum of the magnification of both images:

APSPL(u) = u²+ 2 u√

u²+ 4, (1.17)

where the subscript PSPL stands for “point-source point-lens.” Ifu → ∞, then A⁺ → 1 and y₊ → u, so the major image becomes coincident with the unlensed position of the source.

The minor image (A− → 0) vanishes. If u → 0, the total magnification A → u⁻¹. 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 magnificationAPSPL(u = 1) ≈ 1.34, 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:

u(t) = s

t− t0

tE

2

+ u²₀, (1.18)

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 Paczy´nski’s curves (Figure 1.3).

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−2 −1 0 1 2 Time ((t− t⁰)/tE)

0 5 10 15 20 25

Magnification

ρ = 0.0 ρ = 0.1 ρ = 0.3 ρ = 0.5

Figure 1.4. Finite source effects. The blue curve is the standard Paczy´nski 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 tou0 = 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 radiusθE. The ratio of these quantities is known as a normalized source radiusρ = θ∗/θE. Finite source effects can be only detected whenρ≈ u0 (i.e., when the limb of the source passes over/near the 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 ofρ. In typical microlensing events, θ∗ ∼ O(1 µas) and θ^E ∼ O(1 mas). Therefore, ρ ∼ 10⁻³ and the chances of observing finite source effects are slim. However, the angular Einstein radii of planetary-mass objects are much smaller (O(1 µas)) and they are comparable to the angular radii of giant source stars in the 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 integratingAPSPLover its area:

A(u, ρ) = RR

source

A_PSPLdS RR

source

dS . (1.19)

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Gould (1994) derived the formula for A(u, ρ) in the high-magnification regime (when u ≈ 0 andAPSPL(u)≈ 1/u). The exact, although cumbersome, expression for A(u, ρ) for a uniform 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:

S (ϕ) / ¯S = 1− Γ

1− 3

2cos ϕ

− Λ

1−5

4

√cos ϕ

, (1.20)

whereΓ 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:

Apeak ≈ 1 + 2

ρ² (1.21)

(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.

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The microlens parallax is a vector quantity π_E. It is customary to express its components in the north and east directions: π_E = (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 timet0,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(u0, πE,⊥) ↔ −(u⁰, πE,⊥) (Jiang et al. 2004; Poindexter et al. 2005), whereπE,⊥is the parallax component perpendicular to the apparent acceleration of the Sun (projected on the sky) in the Earth frame att0,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 the observer planer˜E = au π_E⁻¹. 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 typicallyr˜E ∼ 1 − 10 au. The first such measurement was carried out by Dong et al. (2007), 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).

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6700 6750 6800 6850 6900 HJD-2450000

−0.2

−0.1 0.0 0.1 0.2

Residual (mag)

16.0 16.2 16.4 16.6 16.8 17.0 17.2 17.4

I (OGLE) [mag]

OGLE

^I

Spitzer 3.6

^µ

m

6820 6830 6840 6850 15.6

16.0 16.4 16.8

A B C D 16.1

16.2 16.3

C

+6837

0 1 2

16.2 16.3

D

+6844

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).

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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/θEand images y = θ/θE, whereθEis the angular Einstein radius corresponding to the total mass of the lensM =PNl

i Mi: u= y−

N_l

X

i

mi

y− ym,i

|y − ym,i|², (1.22)

where y_m,i is the position of lens massi 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 andz = y1+ iy2). Thus:

ζ = z−

Nl

X

i

mi

z− zi

(z− zⁱ)(¯z− ¯zⁱ) = z−

Nl

X

i

mi

¯

z− ¯zⁱ, (1.23)

whereziis the complex position of the massi. 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¯ complex polynomial of degree N_l² + 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 N^l > 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):

Aj = 1 det J

z=zj

, where det J = 1−∂ζ

∂ ¯z

∂ζ¯

∂ ¯ζ = 1−

Nl

X

i

mi

(¯z− ¯zi)²

2

. (1.24)

The total magnification is the sum of magnifications of individual images. There are some points satisfyingdet 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

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−2 −1 0 1 2 βx(θE)

−2

−1 0 1 2

βy(θE)

q = 0.5 s = 2.0

u0= 0.10 α = 0.94

q = 0.5 s = 2.0

u0= 1.00 α = 2.04

q = 0.5 s = 2.0

u0=−0.05 α = −0.13

−4 −3 −2 −1 0 1 2 3 4

(t− t0)/tE

0 1 2 3 4

2.5logA

−2 −1 0 1 2

βx(θE)

−2

−1 0 1 2

βy(θE)

q = 0.2 s = 0.7

u0= 0.10 α = 0.94

q = 0.2 s = 0.7

u0=−0.50 α = 0.31

q = 0.2 s = 0.7

u0=−0.09 α = −0.13

−4 −3 −2 −1 0 1 2 3 4

(t− t0)/tE

0 1 2 3 4

2.5logA

−2 −1 0 1 2

βx(θE)

−2

−1 0 1 2

βy(θE)

q = 0.8 s = 1.1

u0= 0.10 α = 0.94

q = 0.8 s = 1.1

u0=−0.50 α = 0.31

−4 −3 −2 −1 0 1 2 3

(t− t0)/tE

0 1 2 3

2.5logA

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 hasβx < 0).

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11.6 12.0 12.4 12.8 13.2 13.6 14.0 14.4 14.8

Magnitude

GaiaBialkow I APT2 I LT iDEMONEXT I Swarthmore I UBT60 I ASAS-SN V

7200 7300 7400 7500 7600 7700 7800 7900 8000

HJD - 2450000

−0.20

−0.15

−0.10

−0.05 0.000.05 0.100.15

Residual

11.6 12.0 12.4 12.8 13.2 13.6 14.0 14.4 14.8

Magnitude

Gaia

ground-based follow-up

7600 7605 7610 7615 7620

HJD - 2450000

−0.20

−0.15

−0.10

−0.05 0.000.05 0.100.15

Residual

7649 7651

HJD - 2450000

7713 7715

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 masses0.57 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).

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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 and light curve shapes (see Figure 1.6). If s > sw = (1 + q^1/3)^3/2/(1 + q)^1/2 (“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 point-mass objects. Ifs < sc = s^−1/2w (“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 separations(t) = s0 + ˙s(t− t0,kep), and the angle α(t) = α0+ ˙α(t− t0,kep), t0,kepcan be any arbitrary moment of time and 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.

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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 & Paczy´nski (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_E ∼ 20 d), anomalies caused by Jupiter-mass planets (q ∼ 10⁻³) 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

r_E= θ_EDl = 2.6 au

M

0.5 M

1/2

π_rel 0.1 mas

1/2 Dl

4 kpc

, (1.25)

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

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16.5

17.0

17.5

18.0

18.5

19.0

Magnitude

OGLE I OGLE V MOA

6320 6340 6360 6380 6400 6420 6440 6460

HJD - 2450000

−0.4−0.3

−0.2−0.10.0

0.10.2

0.30.4

Residual

6370 6372 6374 6376 6378 6380

16.6 16.8 17.0 17.2 17.4 17.6 17.8 18.0 18.2

17.0

17.5

18.0

18.5

19.0

19.5

20.0

20.5

21.0

Magnitude

OGLE I MOA

6525 6530 6535 6540 6545 6550 6555

HJD - 2450000

−0.4

−0.2 0.0 0.2 Residual 0.4

6530 6531 6532 6533 6534 6535 6536 17.0

17.5 18.0 18.5 19.0 19.5

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).

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10

²

10

¹

10

⁰

10

¹

Semimajor axis (au)

10

¹

10

⁰

10

¹

10

²

10

³

10

⁴

Mass (Earth masses)

V E M

J S

U N

Transits Radial velocity Microlensing

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).

10 ⁶ 10 ⁵ 10 ⁴ 10 ³ 10² 10 ¹ 10⁰ Mass ratio

10⁰ 10¹ 10² 10³

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

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 aboutq ≈ 2 × 10⁻⁴. The black histogram shows the observed planets, purple – corrected for detection efficiency. Adapted from Suzuki et al. (2016).

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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 Spitzermicrolensing 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.

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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 dNpl/d log q ∝ qⁿ with n =

−0.68 ± 0.2 (q is the planet-to-host mass ratio). Cassan et al. (2012) found that, on average, every star has1.6^+0.72_−0.89 planets in a range of0.5− 10 au and 5 M^⊕to10 MJup. Shvartzvald et al.

(2016) analyzed 224 microlensing events (of which eight had planetary anomalies), which were observed by three microlensing surveys (OGLE, MOA, and Wise). They found that55⁺³⁴₋₂₂% of microlensed stars host a snowline planet and that the mass-ratio distribution can be described by a single power-law withn =−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 (dNpl/d log q ∝ q^p) with p = 0.6^+0.5_−0.4 below the break. A similar conclusion was reached by Udalski et al. (2018), who found p = 0.73^+0.42_−0.34 at q < 2× 10⁻⁴ 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^Jupand∼ 0.27 M^Juporbiting a0.5 Mhost. This system is thought to 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 MJup and

∼ 0.86 M^Jup orbiting a G-type star (Han et al. 2013b; Beaulieu et al. 2016). Two possible planets (∼ 0.18 MJup and ∼ 0.27 MJup) were found also around OGLE-2014-BLG-1722 (Suzuki et al. 2018a), but these detections are less certain. Suzuki et al. (2018a) estimated that6± 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).

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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.

Thus, πr_E² is a natural cross-section for microlensing (Vietri & Ostriker 1983; Nityananda &

Ostriker 1984; Paczy´nski 1991). The optical depth can be expressed as:

τ (Ds) = Z Ds

0

n(Dl)(πr²_E)dDl = 4πG c²

Z Ds

0

ρ(Dl)Dl(Ds− D^l) Ds

dDl, (1.26) wheren(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:

τ = 1 N_s

Z ∞ 0

τ (Ds)dn(Ds), (1.27)

where dn(Ds) is the number of detectable sources in the range [Ds, Ds + dDs] (Kiraga &

Paczy´nski 1994) andNs=R∞

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 (Paczy´nski 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 2θE× µrel in a unit time, integrated over all lenses along the line of sight. The differential event rate toward a given source is:

d⁴Γ

dDldM d²µ_rel = 2rEvreln(Dl)f (µrel)g(M ), (1.28) whereM is the lens mass, rE = DlθE is its Einstein radius,n(D) is the local number density of lenses, v_rel = Dl|µ^rel| is the lens-source relative velocity, f(µ^rel) is the two-dimensional probability density for a given lens-source relative proper motion (µ_rel), andg(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

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mean Einstein timescale of microlensing events is:

htEi = 2 π

τ

Γ. (1.29)

Because typical Einstein timescales toward the Galactic bulge aretE ∼ 20 days, the event rate 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:

δu = u

u²+ 2θE (1.30)

(Hog et al. 1995; Miyamoto & Yoshii 1995; Walker 1995; Dominik & Sahu 2000). Ifu√ 2, the centroid shift scales as δ(u) ≈ θ^E/u and it falls much more slowly than the photometric magnification (A(u)→ 1 + 2/u⁴foru 1) for large impact parameters.

A measurement of the astrometric shift due to microlensing allows measuring the angular Einstein radiusθEand, in turn, the mass of the lens:

M = θ²_E κπrel

(1.31) 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

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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).

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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 Paczy´nski 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 (Paczy´nski 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 one2048× 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).

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Phase Duration Telescope Detector Area OGLE-I 1992–1995 Swope (1 m) 2048× 2048, 0.44⁰⁰per pixel 4 OGLE-II 1997–2000 Warsaw (1.3 m) 2048× 2048, 0.417⁰⁰per pixel 10 OGLE-III 2001–2009 Warsaw (1.3 m) 8× 2048 × 4096, 0.26⁰⁰per pixel 31 OGLE-IV 2010–present Warsaw (1.3 m) 32× 2048 × 4102, 0.26⁰⁰per 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 of0.26⁰⁰per 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 322048× 4102 pixel CCD detectors and covers a field of view of 1.4 square degrees with a scale of0.26⁰⁰per 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.

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Astrophysical applications of gravitational microlensing in the Milky Way