ASTROPHYSICAL APPLICATIONS OF GRAVITATIONAL MICROLENSING IN THE 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. v 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). ix 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