A re-analysis of exomoon candidate MOA-2011-BLG-262lb using the Besançon Galactic Model

A re-analysis of exomoon candidate MOA-2011-BLG-262lb using the Besançon Galactic Model A dissertation submitted to the University of Manchester for the degree of Master of Science by Research in the Faculty of Science & Engineering 2017 By Parvin Mansour School of Physics and Astronomy

Contents Abstract 8 Declaration 9 Copyright 10 Acknowledgements 11 The Author 12 Dedication 13 Introduction 14 1 Microlensing 16 1.1 Lens equation............................. 16 1.2 Binary lenses............................. 20 1.3 Caustics................................ 22 1.4 Optical depth and Event rate.................... 23 1.5 Planetary microlensing light curve.................. 24 1.6 Microlensing degeneracy....................... 27 1.7 FFPs.................................. 31 1.8 Microlensing surveys......................... 32 Parvin Mansour 2

CONTENTS 2 MOA-2011-BLG-262lb 35 2.1 Observation.............................. 35 2.2 Light curve analysis.......................... 37 2.3 The relative proper motion...................... 38 2.4 Source and lens magnitude...................... 39 2.5 Bayesian priors............................ 41 2.6 Probability distribution........................ 43 2.7 MOA-2011-BLG-262lb conclusion.................. 44 3 Analysing MOA-2011-BLG-262lb using the Besançon Model 46 3.1 Besançon Galactic Model....................... 47 3.1.1 A brief description...................... 47 3.1.2 Interstellar extinction..................... 49 3.2 Initial catalogue parameters..................... 49 3.3 Accounting for the finite source effect................ 50 3.4 Optical depth............................. 51 3.5 Average Einstein radius crossing time and event rate....... 52 3.6 Comparison with MABµls microlensing maps........... 52 3.7 Preliminary probability density maps................ 57 3.7.1 Selection functions...................... 58 3.8 Application to MOA-2011-BLG-262lb................ 59 3.8.1 Refining Besançon catalogues................ 62 3.8.2 Final probability density maps................ 69 4 Discussion/ Results 77 5 Conclusion/ Future direction 81 BIBLIOGRAPHY 83 Parvin Mansour 3

List of Tables 1.1 Microlensing parameters....................... 32 2.1 Model parameters for planet-moon solution............. 37 2.2 Model parameters for star-planet solution.............. 37 3.1 The values for optical depth, event rate and average crossing time from our and MABµls simulations.................. 53 4.1 The χ 2 -weight for all four models (Bennett et al. 2014)..... 79 Parvin Mansour 4

List of Figures 1.1 Simple geometry of a lensing system................ 17 1.2 Geometry of a point mass lens.................... 18 1.3 Caustic configurations........................ 23 1.4 Planetary microlensing light curve.................. 25 1.5 Planetary microlensing light curve.................. 26 1.6 Parallax effect on the microlensing light curve........... 29 2.1 MOA-2011-BLG-262 light curve................... 36 2.2 K-band images from Kick-2 and VVV................ 39 2.3 The magnitude-distance relation for both models......... 40 2.4 Lens distance probability distribution................ 43 2.5 The planet-moon lens distance probability distribution...... 43 2.6 The star-planet lens distance probability distribution....... 44 3.1 Simulated map of the optical depth................. 54 3.2 Simulated map of the event rate................... 55 3.3 Simulated map of the average crossing time............ 56 3.4 Posterior probability map of Solar-mass lenses........... 57 3.5 Posterior probability map of Jupiter-mass lenses.......... 58 3.6 Posterior probability map of solar-mass lenses including the selection functions............................. 60 Parvin Mansour 5

LIST OF FIGURES 3.7 Posterior probability map of Jupiter-mass lenses including the selection functions............................ 61 3.8 Plot of source magnitude versus source motion........... 62 3.9 Plot of source distance versus lens distance for Solar-mass population 63 3.10 Plot of source distance versus lens proper motion for Solar-mass population............................... 64 3.11 Plot of source distance versus source proper motion for Solar-mass population............................... 64 3.12 Plot of lens distance versus lens proper motion for Solar-mass population................................. 65 3.13 Plot of lens distance versus source proper motion for Solar-mass lens population............................ 65 3.14 Plot of lens proper motion versus source proper motion for Solarmass population............................ 66 3.15 Plot of source distance versus lens distance for Jupiter-mass population................................. 66 3.16 Plot of source distance versus lens proper motion for Jupiter-mass population............................... 67 3.17 Plot of source distance versus source proper motion for Jupitermass population............................ 67 3.18 Plot of lens distance versus lens proper motion for Jupiter-mass population............................... 68 3.19 Plot of lens distance versus source proper motion for Jupiter-mass lens population............................ 68 3.20 Plot of lens proper motion versus source proper motion for Jupitermass population............................ 69 3.21 Final posterior probability map of the 0.12 solar-mass lenses... 70 3.22 Final posterior probability map of the 3.6 Jupiter-mass lenses.. 71 Parvin Mansour 6

LIST OF FIGURES 3.23 Contour plot of probability distribution of 0.12 Solar-mass lens population............................... 72 3.24 Contour plot of probability distribution of 3.6 Jupiter-mass lens population............................... 73 3.25 Probability density map of Solar-mass lenses with the relative proper motion line.......................... 74 3.26 Probability density map of Jupiter-mass lenses with the relative proper motion line.......................... 75 3.27 Probability density map of Jupiter-mass lenses with a more relaxed distance................................ 76 Parvin Mansour 7

The University of Manchester ABSTRACT OF DISSERTATION submitted by Parvin Mansour for the Degree of Master of Science by Research and entitled Towards Real Time Selection of Computing Exoplanet Microlensing Models September 2017 Gravitational microlensing is the bending of star light due to gravitational influence of a massive compact object, known as the lens, along the line of sight. The presence of any planet orbiting the lens can be detected via the microlensing method. Due to the fact that it does not rely on detection of photon from the star or the planet, this method provides a powerful tool for detecting free floating planets and cool exoplanets orbiting a wide range of stars with distances of order of several kpc. The physical characteristics of the lens system can be determined by constructing a model that matches with the observed data. Unfortunately, typical microlensing models suffer from a two fold degeneracy, which means that the mass and distance of the lens cannot be disentangled. Finding the best parameters set that provide a good description of the observed microlensing light curve is a challenging task. Different model fits can produce similar light curves with reasonable agreement with the observation, therefore it is essential to be able to compute the probability density of different model fits. We developed a software, using the Besançon population synthesis model of the Galaxy, that can predict the probability density of different microlensing event model fits. We used this software to compute the probability density of two models describing the microlensing event MOA-2011-BLG-262-lb, a free floating planet-moon system and a star-planet system with a super Earth orbiting a star. We calculated the relative posterior probability of both model fits by incorporating selection functions for the Einstein radius crossing time, relative proper motion, source apparent magnitude and χ 2 from the MOA-2011-BLG-262-lb event and found that the ratio of the planet-moon model posterior probability to that of the star-planet model is in order of 10 10. Parvin Mansour 8

Declaration No portion of the work referred to in this dissertation has been submitted in support of an application for another degree or qualification of this or any other university or other institution of learning. Parvin Mansour 9

Copyright The author of this dissertation (including any appendices and/or schedules to this dissertation) owns certain copyright or related rights in it (the Copyright ) and he has given The University of Manchester the right to use such Copyright, including for administrative, promotional, educational and/or teaching purposes. Copies of this dissertation, either in full or in extracts and whether in hard or electronic copy, may be made only in accordance with the regulations of the John Rylands University Library of Manchester. Details of these regulations may be obtained from the Librarian. This page must form part of any such copies made. The ownership of any patents, designs, trade marks and any and all other intellectual property rights except for the Copyright (the Intellectual Property Rights ) and any reproductions of copyright works, for example graphs and tables ( Reproductions ), which may be described in this dissertation, may not be owned by the author and may be owned by third parties. Such Intellectual Property Rights and Reproductions cannot and must not be made available for use without the prior written permission of the owner(s) of the relevant Intellectual Property Rights and/or Reproductions. Further information on the conditions under which disclosure, publication and exploitation of this dissertation, the Copyright and any Intellectual Property Rights and/or Reproductions described in it may take place is available from the Head of School of Physics and Astronomy. Parvin Mansour 10

Acknowledgements I would like to thank my supervisor, Dr Eamonn Kerins, for his support over the past year. This work could not have been completed without his guidance and encouragement. I enjoyed our weekly meetings and discussions of the research and life in general. I also would like to thank Safa Al-Hakeem and Charlie Walker for proof-reading this dissertation. Their comments have improved this work. My family means the world to me and I could not have completed my journey without them. Special thanks to my husband, Nabil, for his ongoing support over the years. Parvin Mansour 11

The Author The author was born in Tehran, Iran where she completed a high school diploma before moving to UK in 1995. She started her Bachelor of science, Astronomy and planetary path away at the Open University in 2012 and graduated in 2016. She started her MSc in September 2016 at the University of Manchester. The result of the research she undertook during her MSc are presented in this thesis. Parvin Mansour 12

The sky Is a suspended blue ocean. The stars are the fish That swim. The planets are the white whales I sometimes hitch a ride on, And the sun and all light Have forever fused themselves Into my heart and upon My skin. There is only one rule On this Wild Playground, For every sign Hafiz has ever seen Reads the same. They all say, Have fun, my dear; my dear, have fun, In the Beloved s Divine Game. Hafiz For my beloved husband, Nabil and our children, Reem, Hassan and Rasha who helped me find my playground. Parvin Mansour 13

Introduction By the 19th century all of the planets within the solar system were known. However it was not until the late 1900s when a new planet, an exoplanet (a planet orbiting a star other than the sun) was discovered. Exoplanetary science is a relatively new field in astronomy and in recent years has witnessed a huge increase in the number of detected exoplanets in the Galaxy. New discoveries are announced almost every week. To date we have more than 3600 confirmed exoplanets 1 and this number is expected to increase significantly when all of the data from the Kepler satellite is analysed. The overwhelming majority of these exoplanets are discovered by the transit and radial velocity techniques (Perryman 2014). By combining the information gathered from both methods, the planet s radius, mass, density, orbital inclination and the atmospheric composition can be measured. However these methods are biased towards massive planets in close orbit around the host. In fact, the first exoplanet discovered (by radial velocity) orbiting a sun like star was a Jupiter-mass planet orbiting the host at an orbit closer than that of Mercury around the sun (Mayor & Queloz 1995). Hence to be able to detect planets further away from the host we need to use different techniques. Unlike other detection methods, the microlensing technique does not rely on detection of photons from the host star or the planet and is capable of detecting planets beyond the snow line (a region around the star where volatiles can exist in ice form and where according to core accretion theory (Safronov 1972) (Gol- 1 The Extrasolar Planets Encyclopaedia: http://exoplanet.eu Parvin Mansour 14

dreich & Ward 1973) many planets are formed). This method is also capable of detecting planets over a wide range of Galactic distances and around a wide range of stars, making it complementary to other techniques. Together they can produce the full map of planetary distribution within the Galaxy, helping us gain a better understanding of planet formation and evolution. Much like other methods microlensing also suffers from limitations, it is rare and unpredictable and suffers from degeneracies. To find a correct set of parameters to describe a microlensing event is a challenging task. A typical binary microlensing event is described by a model with several parameters, hence in order to re-construct a model of the observed light curve, it is necessary to search in a multi-dimensional parameter space. However, even if one does find a set of parameters which seems to provide a good description of the event, many other parameter sets may also provide similar and sometimes indistinguishable light curves. These degenerate models require extra investigation, for instance using Bayesian analysis based on different Galactic models, when performing microlensing modelling. The motivation behind this research was to develop software that can calculate the probability density of measured lens properties within the context of an assumed Galactic model. The software was used on one case, MOA-2011-BLG- 262lb, where the observed microlensing light curve had two solutions (Bennett et al. 2014), a free floating 2 planet-moon system, where an Earth-mass moon orbits a Jupiter-mass planet and a star-planet system, where a super-earth planet orbits a brown dwarf. The structure of this dissertation is as follows. The theoretical bases of microlensing, microlensing observation, surveys and significant discoveries are discussed in Chapter 1. Chapter 2 covers the MOA-2011-BLG-262lb case. The Besançon Galactic Model, which was used for the analysis is described in Chapter 3. Whereas, Chapter 4 and 5 cover the discussion and conclusion respectively. 2 Free Floating Planets will be described further in section 1.7. Parvin Mansour 15

Chapter 1 Microlensing Microlensing is the temporary magnification of a background star, known as the source, due to gravitational field of a foreground star, known as the lens. The presence of the lens results in multiple images of the source with a very small separation. The image separation is so small that it cannot be resolved by optical telescopes. However, microlensing events can be detected due to the combined magnification of these images (Gaudi 2010). As the apparent brightness of the source changes, the relative proper motion of the lens with respect to the source, µ rel, will produce a unique light curve known as the Paczynski light curve (Paczynski 1996). If the lens has a companion, there will be further perturbation in the light curve. Such features can be used to measure or constrain the physical parameters of the lensing system such as the mass, distance and transverse velocity, υ, which is the relative transverse velocity between the source and the lens. 1.1 Lens equation The simplest case of microlensing involves a point-mass lens. The geometry of a point-mass lens can be seen in figure 1.1. Parvin Mansour 16

1.1: LENS EQUATION Figure 1.1: The simple geometry of a lensing system and the angles and distances used to derive the lens equation are shown here (Mao 2012). The light ray is deflected by the lens, D by an angle α. β and θ are the true and observed angles between the lens and the source, S, respectively. ξ and η are the distances between the lens and the light ray and the source position respectively. D d, D s and D ds are the distance between the observer and lens, the observer and source and source and lens respectively. In 1915 Einstein derived the formula for the deflection angle of a light ray due to the gravitational field of a point-mass M (Schneider, Ehlers & Falco 1992) as α = 4 G M ξ c 2. (1.1) where G and c are the gravitational constant and speed of light respectively. Assuming a point source and small angle (ξ D d ) we have β = θ D ds D s α, (1.2) and θ = ξ D d. (1.3) Parvin Mansour 17

1.1: LENS EQUATION Using equations (1.1) and (1.3), equation (1.2) can be written as β = θ 4 G M D ds θ c 2 D s D d. (1.4) When β=0, the images will form a ring known as an Einstein ring with an angular size θ E = Dividing equation (1.4) by θ E gives 4 G M Dds c 2 D s D d. (1.5) r s = r 1 r, (1.6) where r s β θ E and r θ θ E. This equation has two solutions r ± = 1 2 ( r s ± ) rs 2 + 4. (1.7) Figure 1.2: The two images created by the lens, M, of the source, S, are marked as I 1 corresponding to the minor image and I 2 corresponding to the major image. The Einstein radius is marked as a dashed circle (Paczynski 1996). Parvin Mansour 18

1.1: LENS EQUATION Figure 1.2 shows the position of the images produced during a microlensing event. The solution for r + corresponds to the major image outside the Einstein radius while the solution for r - gives the position of the minor image inside the Einstein radius on the opposite side of the lens (Gaudi 2012). The separation of these images is typically milliarcsecounds and cannot be resolved (Mao 2012). Nevertheless, the event can be observed through changing magnification of the images. The source brightness is conserved (Schneider, Ehlers & Falco 1992), therefore the magnification of each image (µ + and µ - ) is simply the ratio of image area to that of the source µ ± = θ dθ β dβ. (1.8) Using ( ) dθ dβ = 1 β 1 ±, (1.9) 2 β2 + 4 θe 2 the solution of equation (1.8) is The total magnification is given by µ ± = u2 + 2 2 u u 2 + 4 ± 1 2. (1.10) µ = µ + + µ - = u2 + 2 u u 2 + 4, (1.11) where u is the lens-source separation. For u 1 the magnification scales as µ 1 u, (1.12) and for u 0 the magnification scales as µ 1 + 2 u 4. (1.13) Parvin Mansour 19

1.2: BINARY LENSES Also as u, µ 1 and as u 1, µ diverges. Assuming a constant proper motion between the source and lens, the lens-source separation is given by u = ( ) 2 t u 2 t0 0 +, (1.14) t E where t 0 is the time of closest approach, u 0 is the minimum source-lens separation in units of θ E and t E is the Einstein radius crossing time (the time it takes the source to move by an angular distance equal to θ E ). Any Paczynski light curve can be described by these three parameters. t E is related to θ E and µ rel (the lens-source relative proper motion) as t E = θ E µ rel = R E υ, (1.15) where υ = µ rel D d, R E = θ E D d = 4 G M D d (D s D d ) c 2 D s. (1.16) 1.2 Binary lenses Derivation of the binary lens equation builds on that of the point-mass lens. The lens equation for an N point-mass lens can be written as β = ξ θ 2 E N i=1 m i ξ ξ i ξ ξ i 2, (1.17) where m i is the ratio of mass i to the total mass of the lens system (Mao 2012). Using the relation between the two-dimensional vectors and complex numbers, Witt (1990) shows that equation (1.17) can be written in a complex form by substituting the vectors with complex numbers as z s = z N i=1 m i z z i, (1.18) Parvin Mansour 20

1.2: BINARY LENSES where z and z s are the complex position of each point-like image and the source respectively and z i is the position of the i th mass component of the lens. For a binary system where the mass ratio is defined as equation (1.18) can be written as q = m 2 m 1, (1.19) Taking the complex conjugate of this equation gives z s = z m 1 z z 1 + m 2 z z 2. (1.20) z = z s + m 1 z z 1 + m 2 z z 2. (1.21) Substituting equation (1.21) into (1.20) will eliminate z and give a fifth order polynomial which can be solved numerically (Mao 2012). The solution to the binary lens equation and hence the number of images produced during the event depend on the position of the source relative to that of the lens (Gaudi 2012). Similarly to the point-mass lenses, the magnification of the images in binary lenses is the ratio of their area to that of the source. The magnification of the image is the inverse of the Jacobian determinant evaluated at the position of the image µ = 1 J, (1.22) where J is the determinant of Jacobian of the binary lens equation. J can be written as J = 1 m 1 (z z 1 ) 2 + m 2 (z z 2 ) 2 2. (1.23) Just as the single lens, the total magnification is the sum of the absolute magnification of all images. Parvin Mansour 21

1.3: CAUSTICS 1.3 Caustics From equation (1.22) the source will be infinitely magnified when J=0. In this case, equation (1.23) can be written as or more generally 1 = m 1 (z z 1 ) 2 + m 2 (z z 2 ) 2 e iφ = N i=1 2, (1.24) m i (z z i ) 2. (1.25) Solving equation (1.25) for 0 φ 2π will give a closed smooth curve in the lens plane known as a critical curve. The position of the corresponding curves in the source plane are known as caustics. The points where caustics meet are known as cusps (Gaudi 2012). Caustic number, size and shape is determined by the mass ratio, q, and the projected separation, s, of the binary lens (Griest & Safizadeh 1998). Gaudi (2008) puts caustics in three different configurations, close, resonant and wide (figure 1.3). The caustic closer to the massive component of the binary lens is known as the central caustic whilst the caustics further away from the massive component are known as the planetary caustics (Gaudi 2012). When a similar mass companion orbits the primary lens close to the Einstein ring, the central and planetary caustics merge and form a large caustic, close to the more massive component, known as the resonant caustic. During high magnification events (i.e. µ > 100), the probability of the source crossing the central caustic is very high (Griest & Safizadeh 1998). This makes high magnification events ideal for detecting planetary anomalies in the event light curve. Hearnshaw et al. (2006) state that as s increases, the size of the planetary caustic scales as s 2, however as s decreases the size of the caustic will scale as s 3. The central caustic size scale with s as (s + s 1 ) 2 (Chung et al. 2005). Parvin Mansour 22

1.4: OPTICAL DEPTH AND EVENT RATE Figure 1.3: The three caustic topologies, close (left), resonant (middle) and wide (right). The solid lines d c (q) and d w (q) separate these regions. The primary component of the binary is close to the diamond-shape/central caustic (Cassan 2008). 1.4 Optical depth and Event rate Mao (2012) defines the optical depth, τ, as the probability that at any given time any given source falls within the Einstein radius of any given lens. τ is therefore a cross section of θ E of all lenses between the observer and the source and is given by τ = Ds 0 N(D d ) D 2 d π θ E 2 dd d, (1.26) where N(D d ) is the number density of all lenses along the line of sight. In the simple limit of constant density along the line of sight equation (1.26) evaluates to τ = 2π G ρ 0 D s 2 3 c 2, (1.27) Parvin Mansour 23

1.5: PLANETARY MICROLENSING LIGHT CURVE whilst the measured optical depth towards the Galactic bulge is 10 6 (Sumi et al. 2013). The microlensing event rate, Γ, is the rate of microlensing events per unit time for a given star (Mao 2012). Γ is related to τ as Γ = 2 τ π < t E >, (1.28) where < t E > is the average microlensing event time scale. For sources with an apparent magnitude of I <20 at the Galactic positions b 3 o and 0 o l 2 o Sumi & Penny (2016) find Γ = 3.4 10 5 star 1 year 1, and increasing slightly at lower latitudes. The small event rate means that in order to be able to detect a few hundred events each year, microlensing surveys need to monitor hundreds of millions of stars continuously. 1.5 Planetary microlensing light curve Planetary microlensing events occur when a planet orbits the primary lens near one of the images produced during the event. The gravitational field of the planet will distort the shape of the images and produce a detectable anomaly in the event light curve (as seen in figure 1.4). Unlike point-mass lenses, binary lens light curves are usually asymmetrical with some deviations from the Paczynski form. A typical single-lens microlensing event could last anywhere between a few weeks to a few months (with a typical duration of 20 days towards the Galactic bulge). Planetary microlensing events are much shorter lasting a few days, for Jupiter-mass planets and a few hours, for Earth-mass planets. Parvin Mansour 24

1.5: PLANETARY MICROLENSING LIGHT CURVE Figure 1.4: The left panel shows two images (blue disks) of the source (red circle) which are produced due to presence of the lens (black dot). The right panel shows the magnification as a function of time. Further perturbation in the light curve is due to the presence of a planet orbiting the host near one of the images (Paczynski 1996). Figure 1.5 shows the light curve of the microlensing event OGLE-2003-BLG- 235/ MOA-2003-BLG-53, the first microlensing planet discovery (Bond et al. 2004). During the planetary microlensing event, as the source enters the caustic region, the total number of images increases to five and the light curve exhibits a discontinuous increase in magnification, followed by a smooth decrease inside the caustic region. When the source exits the caustic, the light curve display the opposite behaviour with a smooth increase in magnification followed by a discontinuous decrease as the source crosses the caustic curve. The light curve then has two large peaks with a U-shape in between the peaks. This is particularly useful for planetary caustic crossings. The star-planet mass ratio can be inferred from the size of the U-shape base (Griest & Safizadeh 1998). Parvin Mansour 25

1.5: PLANETARY MICROLENSING LIGHT CURVE Figure 1.5: The light curve of the first planetary microlensing event OGLE-2003-BLG- 235/ MOA-2003-BLG-53. The U-shape feature is due to the source entering and exiting the caustic (Bond et al. 2004). In order to model the light curve of a planetary microlensing event in addition to t E, t 0, u 0, s and q we also need α, the angle of source trajectory relative to the binary lens axis (Gaudi 2012). The magnification map of the event can be generated by mapping from the image to the source plane for a trial values of s and q, using the inverse ray shooting technique (Wambsganss 1997). The model light curve is then generated using a numerical grid search with a fixed set of values for s, q and α whilst minimising χ 2 over the other parameters. Due to many local minima, finding the best model is a difficult and time consuming task. Blend flux One additional parameter needed for modelling the event light curve is the source flux, F s. The source-lens alignment is very small and can not be resolved, there- Parvin Mansour 26

1.6: MICROLENSING DEGENERACY fore instead of collecting the magnified flux of the source alone, the telescope will collect all sources of light (the source+lens flux). Furthermore, microlensing surveys take place towards the Galactic bulge which is very crowded, so the neighbouring stars also contribute to the observed flux. The total flux then is F t = F s + F B, (1.29) where F B is the blend flux which is the sum of lens flux and the flux from neighboring stars. F s and F B may be separable a few years after the microlensing event when the angular separation between the lens and source increases to a few milliarcsec (Bennett, Anderson & Gaudi 2007). 1.6 Microlensing degeneracy Out of the three parameters that describes a point-mass microlensing event light curve, t E is the only one that contains information about the lens s mass, distance and velocity. Different combinations of M, D s, D d and υ can give the same t E, this is known as microlensing degeneracy. Fortunately this degeneracy can be broken by measuring parallax and the finite source effect. Finite source effect In order to derive the lens equation we made the assumption that the source is a point-mass object, which is incorrect. Real stars have a finite angular size, therefore as the source moves near the centre of the lens, one part of it can be magnified more significantly than other parts. Subsequently, in order to measure the apparent brightness of the source correctly, it is necessary to integrate the product of the source magnification and intensity over the face of the star (Gould 1994). Mao (2012) states that once we measure the finite source effect we can then find t (the source crossing time) and therefore ρ, the ratio of angular source Parvin Mansour 27

1.6: MICROLENSING DEGENERACY radius to the angular Einstein radius ρ = t t E. (1.30) The source s colour and magnitude can be used to measure θ and in turn θ E since This then gives the relative proper motion θ E = θ ρ. (1.31) µ rel = θ E t E. (1.32) Therefore, finite source effect can be used to partially break the microlensing degeneracy. However in order to break the microlensing degeneracy completely we need additional information such as parallax. Parallax The other assumption we made in order to derive the lens equation was that the observer, lens and source are in a rectilinear motion. While this assumption holds for most microlensing events, the Earth orbit and acceleration becomes important when the microlensing duration is equal to a large fraction of the Earth orbital period (Gaudi 2008). Also the Earth rotation can result in a measurable difference in the timing of data points collected at different locations, thus making the socalled terrestrial parallax measurement possible. Parallax can also be measured if the event is simultaneously observed by a ground-based telescope and a satellite, orbiting the Earth at large orbital separation ( 1 AU). The measurement of this so-called space-based parallax is made possible by the fact that the relative lenssource position as seen from the Earth differs to that seen by a satellite. Parallax, π E, is given by π E = AU, (1.33) R E Parvin Mansour 28

1.6: MICROLENSING DEGENERACY where R E = 4 G M D d c 2 R E, (1.34) where R E is the linear Einstein radius in the lens plane and R E Einstein radius in the observer plane. is the projected Figure 1.6: The light curve of a microlensing event OGLE-2014-BLG-0939 which was observed by OGLE (black dots) and Spitzer (red dots) (Yee et al. 2015). Figure 1.6 shows the difference in t 0 and u 0 of two light curves for the microlensing event OGLE-2014-BLG-0939 observed by space-based satellite, Spitzer and ground-based telescope, OGLE (the Optical Gravitational Lensing Experiment). Using the difference between these light curves π E can be measured. Combining π E measurements with θ E, we can find the lens mass, proper motion Parvin Mansour 29

1.6: MICROLENSING DEGENERACY and transverse velocity: where M = k θ E k π E, (1.35) 4 G c 2 AU, (1.36) µ l = µ E θ E + µ s, (1.37) and υ = µ E θ E + µ s π E θ E + π s, (1.38) where µ l, µ s, µ E and π s are the lens, source, Earth proper motions and the source parallax respectively (Gould 2000). For microlensing events where θ E measurement is not possible, a value for lens mass can still be obtained from π E measurement and source brightness (Bennett et al. 2006). Parallax measurements are the only way to find lens mass values for point-mass lenses. This fact becomes very important when we consider the distribution of isolated low-mass objects such as Free Floating Planets, FFPs, within the Galaxy. Xallarap and binary motion If the source or the lens are part of a binary system, the orbital motion of the companion can be detected in the event light curve. Unfortunately the effect of binary orbital motion due to the source companion, known as xallarap (Griest & Hu 1992) (Rahvar & Dominik 2009), and binary motion due to the lens companion (Dominik 1998) can mimic parallax which leads to difficulties pinpointing the true origin of the effect. On the positive note, however, when these effects are measured, they lead to direct constraints on physical parameters such as lens system s orbital parameters and separation between the lens component (Skowron et al. 2011). Parvin Mansour 30

1.7: FFPS 1.7 FFPs The microlensing method is most sensitive to planets orbiting the host at an orbital separation of a R E 2 AU (Griest & Safizadeh 1998). However, it is also sensitive to planets with orbital separation of a 0.5 AU- (Sumi et al. 2011), making this method ideal for detecting FFPs. FFPs are planets that do not orbit any star. Unfortunately, the processes of their formation are still unknown. Boss (2003) states that FFPs undergo a similar formation to stars (i.e. have formed due to gravitational collapse of gas clouds), whilst Lissauer, Levison & Duncan (1998) state that FFPs could form inside protoplanetary disks and be removed from the system via planet-planet interaction. Measurements such as the one made by Sumi et al. (2011) have put a strain on the latter theory. Sumi et al. (2011) claims that for each main-sequence star in the Galaxy there should be 2 FFPs. This means that each protoplanetary disk should produce 8 planets and nearly 40% of these planets should be ejected from the planetary system (Veras & Raymond 2012). Are all the protoplanetary disks in the Galaxy massive enough to produce this many planets? In order to be able to answer this question we need to gain a better understanding of the distribution of FFPs. Due to the fact that FFPs are not bound to any stars, other detection methods are not capable of detecting them (although young/ close-by FFPs can sometimes be detected by the direct imaging technique (Quanz et al. 2010)). FFPs can act as single-lenses and can be detected by the microlensing method. Sumi et al. (2011) show that the duration of an event scales with lens mass as t E M M J, (1.39) where M J is Jupiter mass. So while stellar microlensing events can last for weeks to months, the FFP event duration is much shorter (with a typical event duration of 1 day). Due to their short duration, space-based parallax measurement is the Parvin Mansour 31

1.8: MICROLENSING SURVEYS only way to obtain a mass value for FFPs. A summary of the main microlensing parameters are shown in table 1.1. Table 1.1: Microlensing parameters Parameters Description t E The Einstein radius crossing time t 0 The time of closest approach u 0 The minimum source-lens separation in units of θ E s Projected separation of the binary lens α The angle of the source trajectory relative to the binary lens axis q The binary lens mass ratio µ rel The relative proper motion of the lens with respect to the source θ E The Einstein radius The angular source radius θ 1.8 Microlensing surveys As mentioned in section 1.4, the microlensing event rate is 10 5. However the probability of detecting a planetary perturbation in the light curve is 10 2 (Mao & Paczynski 1991) 1. Therefore billions of stars need to be monitored in order to detect few planets. To increase the number of stars observed, all microlensing surveys take place towards the Galactic bulge, which has the highest surface density of stars. These surveys monitor hundreds of square degrees continuously in order to observe a few events. Ground-based surveys such as OGLE with a 1.4 m telescope and a 1.4 deg 2 field of view (Udalski 2003), MOA-II, Microlensing Observation in Astrophysics, with a 1.8 m telescope and a 2.2 deg 2 field of view (Hearnshaw et al. 2006) and KMT- Net, the Korean Microlensing Telescope Network, with a network of three 1.6 m telescopes each covering a field of view of 4 deg 2, located in Chile, South Africa and Australia (Poteet et al. 2012) monitor the sky every night. Once an event is 1 Mao & Paczynski (1991) states that the probability of planetary microlensing is roughly equal to half of the width of the caustics region. Assuming every star in the Galaxy is orbited by a planet with a mass of 10 3 M star and an orbital distance of 1-10 AU, they show that the probability of planetary microlensing is 0.03 Parvin Mansour 32

1.8: MICROLENSING SURVEYS detected and identified, a warning alert will go off and a network of smaller telescopes such as PLANET (Probing Lensing Anomalies NETwork) (Albrow et al. 1998), RoboNet (Tsapras et al. 2009) and µfun 2 start round the clock coverage of the event. These follow up observations are essential for planetary microlensing events. Surveys such as MOA have a cadence (observation frequency) of 50 minutes. However, planetary microlensing events are very short and in order to obtain the full shape of the light curve, much shorter cadence are needed. Followup observation can provide a cadence of 3-5 minutes. Space-based telescopes, Spitzer and Kepler, have recently been used for microlensing surveys (Udalski et al. 2015) (Han et al. 2016). In addition to measuring parallax and breaking microlensing degeneracy, space-based surveys can have a much better resolution and can detect significantly more events than groundbased surveys (Bennett 2004). Faint main-sequence stars are more easily observed from space and their small angular size results in longer and sharper planetary anomalies in the event light curve (Bennett and Rhie 2002), increasing the planet detection sensitivity. Space-based telescopes are capable of detecting planets with mass as low as 0.1 M (Barry et al. 2011). Future surveys Space-based microlensing observations will be carried out by WFIRST (Barry et al. 2011), the Wide Field Infrared Survey Telescope, and Euclid (Penny et al. 2013). WFIRST which is due to launch in 2025 3, will observe the Galactic bulge for 500 days, over a 6 year period, and is capable of detecting planets with mass of 0.1 M <M< few M J. WFIRST is predicted to detect around 5330 planets of which 2080 are FFPs. Euclid has a shorter observation period (around 180 days) but is capable of detecting Mars to Earth-mass planets. 2 http://www.astronomy.ohio-state.edu/microfun/ 3 WFIRST: https://wfirst.gsfc.nasa.gov Parvin Mansour 33

1.8: MICROLENSING SURVEYS Significant discoveries The first exoplanet discovered by the microlensing technique was observed simultaneously by OGLE and MOA in 2004 (Bond et al. 2004), with an event duration of 7 days. The event light curve was explained by a binary lens where a Jupiter-mass planet with a mass of 2.6 M J orbits a 0.63 M host at an orbital separation of around 4.3 AU. To date 59 exoplanets have been discovered by the microlensing method 4, including the first cool super-earth orbiting an M dwarf at an orbital separation of 2.6 AU (Beaulieu et al. 2006), a population of FFPs with t E < 2 days (Sumi et al. 2011), the first circumbinary planet (a planet orbiting both components of the binary system) (Bennett et al. 2016), a planetary microlensing event, where the source is part of a binary system (Bennett et al. 2017), a multiple-planet system (Gaudi et al. 2008) and even a planet possibly in the habitable zone of the host (Batista et al. 2014). 4 The Extrasolar Planets Encyclopaedia: http://exoplanet.eu Parvin Mansour 34

Chapter 2 MOA-2011-BLG-262lb MOA-2011-BLG-262lb is a very good example of microlensing degeneracy. The light curve of this event has two different solutions (Bennett et al. 2014). The first model fit is a free floating exoplanet-exomoon system with an Earth-mass moon orbiting a Jupiter-mass planet with a mass of around 4 M J. The second model fit is a star-planet system where a super-earth orbits a star. 2.1 Observation MOA-2011-BLG-262lb, located at l = -0.4 o and b = -3.9 o, was first observed by MOA-II 1.8 m telescope at Mt. John University Observatory (MJUO) in New Zealand on 26th June, 2011 (Hearnshaw et al. 2006). The data from MOA-II triggered the MOA alert system and follow up observations by the PLANET collaboration started immediately. In addition to MOA-II, the event was also observed by a 0.61 m telescope at MJUO, a 1.0 m telescope at Canopus observatory, the OGLE 1.3 m telescope, a 1.3 m telescope at CTIO (Cerro Tololo InterAmerican Observatory) and the FTS (Faulkes South Telescope) 2 m telescope, although CTIO and FTS did not cover the planetary anomaly. Due to the fact that the event was identified as a high magnification event with a short duration, the observing cadence on MOA-II was changed from every Parvin Mansour 35

2.1: OBSERVATION 50 minutes to every 2 minutes and later to every 7 minutes. MOA-2011-BLG- 262lb is a binary microlensing event with a mass ratio of q 4.7 10 4. As the source limb entered the caustic MOA-II, MJUO and Canopus telescopes had started observing the event and continued to do so until well after the source exited the caustic. The event had a peak magnification of A max 75 and the observation was continued until the magnification fell below 30. The event light curve consists of 4884 observations in MOA red-band, 562 observations in I-band and 143 observations in V-band. Figure 2.1 shows the event light curve with the planet-moon and star-planet models indicated by the magenta and black curves respectively. Figure 2.1: The data from MOA-II, MJUO, Canopus, CTIO, OGLE and Faulkes telescopes are shown in red, green & cyan, blue, magenta, black and gold respectively. The magenta curve indicates the exoplanet-exomoon model and the star-exoplanet model is indicated by the black curve (Bennett et al. 2014). Parvin Mansour 36

2.2: LIGHT CURVE ANALYSIS 2.2 Light curve analysis As previously mentioned, this event was identified as a high magnification event. The majority of planetary microlensing events with high magnification suffer from a 2-fold s 1/s degeneracy (Griest & Safizadeh 1998). This degeneracy results in two solutions for s > 1 and s < 1. The parameters of both solutions are shown in table 2.1. Table 2.1: Model parameters for planet-moon solution Parameters s < 1 s > 1 t E ( days) 3.827 3.846 t 0 ( HJD-2455700) 39.1312 39.1311 t ( days) 0.01316 0.01315 u 0 0.01465 0.01451 s 0.9578 1.0605 I s 19.929 19.935 V s 21.888 21.894 fit χ 2 5757.94 5758.58 This event also suffers from another degeneracy which results from the fact that the source radius crossing time, t, is similar to the caustic crossing time of the event. As a result of this degeneracy, two more model fits can also describe the observed light curve. The parameters of these solutions for s < 1 and s > 1 are given in table 2.2. Table 2.2: Model parameters for star-planet solution Parameters s < 1 s > 1 t E ( days) 3.858 3.855 t 0 ( HJD-2455700) 39.1309 39.1310 t ( days) 0.02217 0.02221 u 0 0.01470 0.01463 s 0.9263 1.0966 I s 19.937 19.937 V s 21.898 21.897 fit χ 2 5760.85 5763.82 Parvin Mansour 37

2.3: THE RELATIVE PROPER MOTION 2.3 The relative proper motion The fact that this event had a sharp caustic feature and mass ratio of less than 10 3 ensured that the lens relative proper motion with respect to the source, µ rel, could be measured. As mentioned in Section 1.6, θ can be measured from the source brightness and colour using the method from (Kervella & Fouqué 2008). Bennett et al. (2014) obtained a value for planet-moon and star-planet models of θ = 0.778 ± 0.059 µas and θ = 0.776 ± 0.059 µas respectively. Using θ E = θ t E t, (2.1) and values for t E and t from table 2.1 and 2.2, yields an angular Einstein radii of 0.205 ± 0.015 mas and 0.122 ± 0.009 mas for planet-moon and star-planet models respectively. Equation (1.32) and these θ E values gives µ rel = 19.6 ± 1.6 mas/yr for the planet-moon model and µ rel = 11.6 ± 0.9 mas/yr for the starplanet model. Both of these high µ rel values are an indication of a nearby lensing system. Source proper motion Due to the crowded nature of the Galactic bulge, the proper motion measurements are usually restricted to stars with I s < 18 (Sumi et al. 2004). Although the MOA-2011-BLG-262lb source is a faint star with I s 19.9, it was possible to obtain a value for its proper motion, using a dipole-fitting method developed by Skowron et al. (2014), which gave a source proper motion of (-2.3, -0.9) ± (2.8, 2.6) mas/yr in a (North, East) Galactic heliocentric coordinate frame. Assuming D s = 8.3 kpc, Bennett et al. (2014) obtain a geocentric source proper motion of (-2.3, -1.7) ± (2.8, 2.6) mas/yr. This result strongly disfavours a bulge-lens for the planet-moon system whilst only slightly disfavouring a bulge lens for the star-planet system. Parvin Mansour 38

2.4: SOURCE AND LENS MAGNITUDE 2.4 Source and lens magnitude Nearly one year after the event was identified by MOA-II, the Keck-2 telescope 1 took an AO (Adoptive Optics) image of the event in the J, H and K-bands. The source star was no longer magnified. Figure 2.2 shows a comparison between the image in K-band taken by Keck-2 and VVV 2, where S1 indicates the position of the source+lens. Figure 2.2: The K-band image from the VVV survey is shown in the left panel. The middle panel shows the K-band image from Keck-2 and a zoom-in of the field, where the position of the source+lens is shown by the arrow. The right panel shows the source star, S1, and the four closest stars to it in an AO image (Bennett et al. 2014). The right panel in figure 2.2 shows a relatively uncrowded field, which means negligible contribution to the event magnitude from the neighbouring stars. The AO image (the right panel in figure 2.2) which was taken less than a year after the microlensing event peak magnification, suggests a source- lens separation of less than 20 mas. Therefore the S1 flux is the combined flux of the source and the lens. PSF photometry yields a magnitude of 18.64 ± 0.10, 18.15 ± 0.07 and 18.15 ± 0.10 for S1 in J, H and K-band respectively. Using the H-band data from the CTIO telescope, taken during the event, gives the source magnitude in 1 Keck Observatory: http://keckobservatory.org/about/the observatory 2 The VVV survey: https://vvvsurvey.org/ Parvin Mansour 39

2.4: SOURCE AND LENS MAGNITUDE H-band for both models as H planet moon = 18.220 ± 0.040 and H star planet = 18.226 ± 0.040. Subtracting these values from S1 flux, gives a lens flux of 21.16 ± 1.10 and 21.07 ± 1.00 for the planet-moon and star-planet models respectively. Figure 2.3: The magnitude-distance relation for the planet-moon model (left) and the star-planet model (right). The black, dashed curves shows the magnitude-distance relation based on two different extinction laws and the shaded area is the allowed region. The range for brown dwarfs and super Jupiters are indicated by the red, dashed lines. The blue curves (top right corner) indicates some isochrones for main-sequence stars obtained using methods from An et al. (2007) (Bennett et al. 2014). Figure 2.3 shows the lens absolute magnitude-distance relations and limits for both models. The black, dashed curves shows the magnitude-distance relation based on two different extinction laws from Gonzalez et al. (2011) and Cardelli, Clayton & Mathis (1989) where the shaded area is the allowed region. The red dashed lines show the range for stars with M=0.07 M and planets with M=13 M J. The blue curve indicates some isochrones for main-sequence stars obtained using methods from An et al. (2007). The position of the isochrone curve requires a lens distance of 7 and 7.7 kpc for the planet-moon and star-planet models respectively. These values implies a host mass and transverse velocity of 0.36 M and 677 km/s for the planet-moon model and 0.41 M and 442 km/s for the Parvin Mansour 40

2.5: BAYESIAN PRIORS star-planet model respectively. At a distance of 0.5 kpc (the favoured planet-moon distance), stars would have ( J-K) 1 and would be very red, low-mass stars, which exclude any of the neighboring stars in the AO image as a possible host for the planet-moon system. Hence if the planet-moon model is the correct solution, it must be a free floating system. 2.5 Bayesian priors Apart from including a prior on source proper motion in the Bayesian analysis, Bennett et al. (2014) also made three prior assumptions, the relative proper motion, the host mass function and the fact that the mass of the primary lens has no effect on the probability of the host having a companion with the observed mass ratio, q 4.7 10 4 and at a host-companion separation of a R E. The relative proper motion The average proper motion of stars in the Galactic bulge is given by < µ star > = υ D, (2.2) where υ and D are the Galactic rotation speed and the distance to the centre of the Galaxy respectively. For a star in the bulge, equation (2.2) will give a mean proper motion of 6.4 mas/yr. Following the method given by Koz lowski et al. (2006), if both the source and the lens reside in the bulge, the relative proper motion is around 5.2 mas/yr. If the lens is a disk lens (D l < 2.7 kpc), the mean proper motion difference between the bulge and the disk will result in µ rel of around 8.2 mas/yr. Both of these values are smaller than the relative proper motion of the star-planet and planet-moon systems, which means a disk lens at D l < 2.7 kpc is preferred. However the planet-moon solution favours a disk Parvin Mansour 41

2.5: BAYESIAN PRIORS lens more than the star-planet solution. The proper motion value for the planetmoon model is about 3.5 times larger than the two bulge star s proper motion dispersion (which is around 5.6 mas/yr), whilst the measured proper motion for the star-planet model is only 2.1 times larger than the bulge-bulge proper motion dispersion. Mass function Bennett et al. (2014) use a mass function in their Bayesian analysis which follows the broken law with dn d logm = M 1 α, (2.3) α = 2.0 for 0.7 < M M 1 = 1.3 for 0.08 < M M 0.7 = 0.49 for 0.01 < M M 0.08 (2.4) = 1.3 for 10 5 < M M 0.01. This will give the ratio of stars to brown dwarfs to planets as 1 : 0.73 : 5.5. Assuming all planets have a mass of 1 M J, the Bayesian analysis gives similar densities for stars and planets, but this does not imply an equal lensing probability. From equation (1.16), we have R E M. (2.5) The fact that the lensing probability for stars is greater than that of planets, and the stellar density in the bulge is about 5 times larger than that of the disk means that the Bayesian analysis favours the star-planet solution over the planet-moon solution. Parvin Mansour 42

2.6: PROBABILITY DISTRIBUTION 2.6 Probability distribution The result of the Bennett et al. (2014) Bayesian analysis, including all solutions, is shown in Figure 2.4. Figure 2.5 and 2.6 show the lens distance probability for the planet-moon and star-planet models respectively. Figure 2.4: The probability distribution for planet-moon and star-planet models weighted by Galactic priors (Bennett et al. 2014). Figure 2.5: The lens distance probability distribution for the planet-moon model (Bennett et al. 2014). Parvin Mansour 43

2.7: MOA-2011-BLG-262LB CONCLUSION Figure 2.6: The probability distribution for star-planet model (Bennett et al. 2014). Figure 2.4 shows two distinct distributions, one for the planetary host at a distance of 0.64 kpc and one for the stellar host in the Galactic bulge at a distance of 7 kpc. The planetary mass distribution gives the host and companion masses and their separation as M h 3.6 M J, M c 0.54 M and a 0.13 AU respectively. For the stellar host distribution, M h, M c and a are approximately 0.12 M, 18 M and 0.84 AU respectively. 2.7 MOA-2011-BLG-262lb conclusion The data from this event is well fit by two different solutions with similar likelihoods. The best-fit model has a large relative proper motion, µ rel = 19.6 ± 1.6 mas/yr which implies a close lens distance of 0.56 kpc and a Jupiter-mass planet with M 3.6 M J as the primary lens. The other solution which is disfavoured by χ 2 = 2.91 has a lower relative proper motion, µ rel = 11.6 ± 0.9 mas/yr and a distance of D l = 7.2 ± 0.8 kpc, where a star with M= 0.12 M is the primary lens. Parvin Mansour 44

2.7: MOA-2011-BLG-262LB CONCLUSION Unfortunately even though the result from Skowron et al. (2014) moderately disfavours the star-planet model, the lack of parallax measurement has resulted in a great uncertainty in the Bayesian analysis, making it impossible to favour one model over the other. The 2000 km separation between the MJUO telescope in New Zealand and Canopus telescope in Tasmania would have been sufficient enough for measuring this event s terrestrial parallax. Unfortunately due to telescope hardware problems, the images obtained by the Canopus telescope were of poor quality and the photometry was not precise enough for parallax measurement. Therefore the only light curve parameters available for constraining the lensing system mass, distance and transverse velocity were t E and θ. Lensing systems with stellar-mass primaries would have a larger Einstein radius, therefore the mass function in the Bayesian analysis would prefer the star-planet over the planet-moon model. Hence Bennett et al. (2014) chose the star-planet model, stating that the probability of a free floating exoplanet-exomoon system is much less than the star-exoplanet solution of this event. Parvin Mansour 45

Chapter 3 Analysing MOA-2011-BLG-262lb using the Besançon Model We have re-analyzed MOA-2011-BLG-262lb using Galactic priors based on the Besançon Galaxy Population Synthesis Model (Robin et al. 2003, 2012). We used the latest version of the Besançon Galactic Model (Robin et al. 2014) to generate stellar catalogues and carry out a series of microlensing simulations. We calculated the posterior probability distribution of microlensing events with respect to the relative proper motion, µ rel, and lens distance, D l, for two different lens host populations with mass of 1M and 1M J. We also measured the probability density of both models which describe the MOA-2011-BLG-262lb microlensing event light curve. An overview of the Besançon model, the microlensing simulations, the preliminary probability density maps and the result of its application to the MOA-2011-BLG-262lb event are presented here. Parvin Mansour 46

3.1: BESANÇON GALACTIC MODEL 3.1 Besançon Galactic Model 3.1.1 A brief description The Besançon Galactic population synthesis model is a simulation tool which uses a theoretical model of the Galaxy, including stellar and Galactic evolution and dynamics, in order to constrain the observable parameters of the stellar population of the Galaxy. The Besançon model is used to produce catalogues of stars containing stellar information such as distance, mass, magnitude, proper motion and radius amongst other parameters. The Besançon model assumes four distinct stellar populations; a thin disk, a thick disk, a bulge and a stellar halo. Each population is modelled based on a set of evolutionary tracks, an initial mass function (IMF) and a star formation rate (Haywood, Robin & Creze 1997). The Besançon model is constantly refined as new physical constraints become available. We have used version 1307 of Besançon model (Robin et al. 2012, 2014) for our work. Thin disk The thin disk, which comprises seven components aged between 0-10 Gyr, is divided into two parts, the young disc, defined by a population of stars with age < 0.15 Gyr and the old disk which is defined by the other six components. The young and old disk density distributions are modelled based on the Einasto (1979) density law and the distribution of the components of both disks are defined by an ellipsoid. The functions describing the density law of the ellipsoid can be found in Robin et al. (2003). The disk has a scale length of 2.2 kpc and a large hole at its centre with a scale length of 1.3 kpc, therefore the maximum density of its population is located at 2.3 kpc from the centre of the Galaxy. The total mass of the thin disk is Parvin Mansour 47

3.1: BESANÇON GALACTIC MODEL 9.0 10 9 M with the IMF modeled by the following power law dn dm = M α, (3.1) where α = -1.6 for M < 1 M and α = -3 for M > 1 M. Thick disk Unlike the thin disk populations, which are formed during separate periods of star formation, the thick disk stars are assumed to be formed during a short period of time and age between 11-12 Gyr. The disk kinematics are described by observational constrains from Ojha et al. (1996). The thick disk has a much lower density compared to the thin disk and the bulge becomes significant only at Galactic latitude b < 8 o. The low density of the thick disk at lower latitudes means that it only lightly contributes to microlensing optical depth and event rate measurements. The bulge Covering a region of 20 o < l < 20 o, 10 o < b < 10 o and assuming a Sun- Galactocentric radius of 8 kpc, Robin et al. (2012) revisited the stellar density, luminosity function and bulge structure analysis by Picaud & Robin (2004) and showed that the bulge has two main components: the bar extending to latitudes of 5 o and the thick bulge which dominates the star counts at latitudes above 5 o. The exact nature of the thick bulge is not well understood, it is either a flattened spheroid or a counter part of the local thick disk. The bar is modelled by a boxy, S-shaped ellipsoid with scale length along the principle axis with ratio of 1.46:0.49:0.39 kpc and a position angle of 13 o with respect to the Sun-Galactic centre direction. The thick bulge is modelled by another ellipsoid with scale length of 4.44:1.31:0.80 kpc. The bulge IMF is also modelled by the equation (3.1) but with α= -2.35 for mass above 0.7 M. The total mass of the bar and Parvin Mansour 48

3.2: INITIAL CATALOGUE PARAMETERS the thick bulge is 6.1 10 9 M and 2.6 10 8 M respectively. The age of the bulge stellar population is 8 Gyr with the bar population having a higher metallicity compared to that of the thick bulge. Stellar halo The stellar halo and faint stellar remnants (i.e. white dwarfs) are believed to be an important part of the dark matter halo (Robin et al. 2004). The stellar halo is made up of a population of metal-poor stars with an age of 14 Gyr formed during a single, short period of star formation. Robin, Reylé & Crézé (2000) states that the stellar halo IMF also is modelled by the equation (3.1) with α = -1.9 ± 0.2 for mass of 0.1 M M 0.8 M. 95% of the stellar halo has a Galactocentric distance of less than 25 kpc (Robin et al. 2014). The stellar halo contribution to the microlensing optical depth and event rate is not significant due to its small density near the Galactic centre. 3.1.2 Interstellar extinction Interstellar extinction is the effect (absorbing and scattering) of dust on electromagnetic radiation, which results in reddening and dimming of stars. The extinction distribution along any line of sight can be computed, using a comparison between simulated, unreddened stars and observed, reddened stars. To achieve a realistic model Robin et al. (2012) adopts the method used by Marshall et al. (2006), where extinction is measured as a function of distance in the Galactic longitude l < 100 o and latitude b < 10 o. 3.2 Initial catalogue parameters Using the Besançon model, we produced a pair of source/lens catalogues for four different line of sights with l = 1 o and b = 1 o, 2 o, 3 o and 4 o. In order to Parvin Mansour 49

3.3: ACCOUNTING FOR THE FINITE SOURCE EFFECT ensure good statistics as well as a reasonable computational time, we choose the solid angle, Ω, in each catalogue so that every source catalogue contains around 2000 stars whilst every lens catalogue has around 10000 stars 1. All catalogues are truncated at a stellar distance of 15 kpc. We measured the average optical depth, average Einstein radius crossing time and average event rate. The parameters used for the microlensing calculations include, apparent magnitude in K-band along with I-K colour, proper motion in l and b, stellar mass, distance and radius (the latter was used in order to measure the finite source effect). The source catalogue was generated with a cut of 12-23 in I-band, ensuring that even stars four magnitudes fainter than ground-based detectability limits are included, whilst the lens catalogue was drawn with no cut since even the faintest stars and remnants can act as lenses. 3.3 Accounting for the finite source effect For source-lens pairs with u θ E 3 θ, the impact parameter, the finite source effect becomes important. Normally the rate and optical depth are computed for events with minimum impact parameter u 0 1, corresponding to magnification A 0 3/ 5. For finite source events we explicitly compute u 0 corresponding to A 0 3/ 5 for all source-lens pairs with 0.01 < ρ < 3.16, where ρ is the ratio of angular source radius to the angular Einstein radius, using a 2-D pre-computed look-up table of source magnification versus ρ and u. Otherwise we assume a point-mass source, point-mass lens model. The look-up table was produced for stars of uniform surface brightness by generating a grid of u and ρ, where each (u and ρ) bin is associated with a magnification, µ, obtained using the equation µ = 2 π ρ 2 ρ 0 π 0 A ρ dρ dφ, (3.2) 1 These numbers give around 10 6 lensing systems in each simulation and although a better statistic is always more desirable, however to achieve a reasonable computational time we had to limit ourselves to these numbers. Parvin Mansour 50

3.4: OPTICAL DEPTH where A = u 2 + ρ 2 2 ρ cosφ + 2 u2 + ρ 2 2 u ρ cosφ u 2 + ρ 2 2 u ρ cosφ + 4. (3.3) Here u is the distance between the lens and the point on the source face which is magnified, ρ is the distance between the source centre and the point and φ is the angle between u and ρ. The look-up table covered 1500 ρ and u values in ranges 0.01 3.16 and 0 1.59 respectively. To include the finite source correction in our computation we added u i to the rate weighting of each source-lens pair as w = µ rel D 2 l θ E u i, (3.4) where u i is the impact parameter corresponding to A > 3 5 for microlensing events with finite source effect and w is proportional to the overall rate contribution from lenses with parameters µ rel, D l, θ E and u i (Kerins, Robin & Marshall 2009). 3.4 Optical depth The source optical depth is calculated for all source-lens pairs with D l < D s, τ = N(D L <D s) i π θ 2 E,i u2 i Ω l, (3.5) where Ω l is the solid angle over which the lens catalogues are selected. The source-averaged microlensing optical depth of all sources along the line of sight was calculated as τ = Ns i N(DL <D s) i Ns i u 2 i where N s is the number of catalogue sources. π θ 2 E,i u2 i Ω l, (3.6) Parvin Mansour 51

3.5: AVERAGE EINSTEIN RADIUS CROSSING TIME AND EVENT RATE 3.5 Average Einstein radius crossing time and event rate In order to calculate the average Einstein radius crossing time, < t E measured t E for each lens-source pair as >, we t E = θ E µ rel, (3.7) where µ rel is calculated using stellar proper motions from the Besançon model µ rel = (µ l l µs l )2 + (µ l b µs b )2, (3.8) where µ l l and µs l are the lens and source proper motions in l respectively and µ l b and µ s b are lens and source proper motion in b respectively. The average crossing time is then calculated using the rate weighted t E < t E > = = N(DL <D s) i N(DL <D s) i w i N(DL <D s) i w i t E,i µ i Dl,i 2 θ E,i u i t E,i N(DL <D s) i µ i Dl,i 2 θ. E,i u i (3.9) The average event rate over all source stars is then simply Γ = 2 τ π < t E >. (3.10) 3.6 Comparison with MABµls microlensing maps To check our computation accuracy and before producing the probability density maps for different populations, we compared our values for average optical depth (τ s ), event rate (Γ s ) and Einstein radius crossing time (< t E > s ) along four differ- Parvin Mansour 52

3.6: COMPARISON WITH MABµLS MICROLENSING MAPS ent line of sights with the MABµls 2 simulated maps. MABµls, the Manchester- Besançon microlensing Simulator, is a multi-wavelength microlensing simulator (Penny et al. 2013) which calculates the optical depth, event rate, Einstein radius crossing time and generates microlensing maps of the inner Galaxy. Table 3.1 shows our results for optical depth, event rate and crossing time along four line of sights with l= 1 o, compared to that obtained from MABµls simulated maps 3. Table 3.1: The values for optical depth, event rate and average crossing time from our and MABµls simulations b τ s τ MAB Γ s Γ MAB < t E > s < t E > MAB b = 4 o 1 10 6 1.8 10 6 1.0 10 5 1.4 10 5 25 days 23 days b = 3 o 1.7 10 6 3.1 10 6 1.6 10 5 2.8 10 5 24 days 24 days b = 2 o 2.4 10 6 5.1 10 6 2.5 10 5 4.2 10 5 24 days 24 days b = 1 o 3.3 10 6 4.1 10 6 3.4 10 5 4.0 10 5 25 days 25 days Figures 3.1, 3.2 and 3.3 show the simulated microlensing maps of optical depth, event rate and average crossing time respectively. These maps cover the central regions of the Galaxy with 0 o < l < 2 o and 5 o < b < 0 o. Overall there is good agreement with MABµls but there are some differences due to different IMF assumptions and difference in finite source treatment and source counting. 2 the Manchester-Besançon microlensing Simulator:http://www.mabuls.net/ 3 All values in the table are approximate values. Parvin Mansour 53

3.6: COMPARISON WITH MABµLS MICROLENSING MAPS Figure 3.1: The simulated map of the optical depth towards the Galactic longitude 0 o < l < 2 o and latitude 5 o < b < 0 o from the MABµls simulation. Parvin Mansour 54

3.6: COMPARISON WITH MABµLS MICROLENSING MAPS Figure 3.2: The simulated map of the event rate towards the Galactic longitude 0 o < l < 2 o and latitude 5 o < b < 0 o from the MABµls simulation. Parvin Mansour 55

3.6: COMPARISON WITH MABµLS MICROLENSING MAPS Figure 3.3: The simulated map of the average crossing time in days towards the Galactic longitude 0 o < l < 2 o and latitude 5 o < b < 0 o from the MABµls simulation. Parvin Mansour 56

3.7: PRELIMINARY PROBABILITY DENSITY MAPS 3.7 Preliminary probability density maps In our computation of the microlensing optical depth, event rate and average duration, we used the masses generated by the Besançon model. To calculate the probability of the model fits from MOA-2011-BLG-262lb case, we replaced all lens masses with 3.6 M J and 0.12 M, the primary lens mass of the planetmoon and star-planet models respectively. We generated a new catalogue with l = 0.4 o, b = 3.9 o, the Galactic coordinates for the MOA-2011-BLG-262lb event, and calculated the rate-weight of each source-lens pair. Figures 3.4 and 3.5 show the posterior probability map of 0.12 M and 3.6 M J lenses respectively. Figure 3.4: The posterior probability map of Solar-mass lenses is shown here. Parvin Mansour 57

3.7: PRELIMINARY PROBABILITY DENSITY MAPS Figure 3.5: The posterior probability map of Jupiter-mass lenses is shown here. The highest posterior probability of 0.12 M lenses are about an order of magnitude larger than that of 3.6 M J lenses with significant weights corresponding to lens distance of around 4.5 kpc D l 9.5 kpc and relative proper motion of around 1 mas/yr µ rel < 10.5 mas/yr for both lens populations. 3.7.1 Selection functions To improve our calculation we constrain our model further by adding a selection function to the rate-weight of each source-lens pair. The selection function is Parvin Mansour 58

3.8: APPLICATION TO MOA-2011-BLG-262LB given by ψ = i = i P i exp ( 1 2 ( ) X i sim Xobs i 2 ), σ(x obs ) (3.11) where i is the product sum of all selection functions included in our computation and Xsim i and Xobs i are the simulated and observed parameters respectively. the σ(x obs ) is the uncertainty in the observed parameter. Since the impact parameter u 0 distribution is uniform and does not enter probability ratios, we remove it from rate-weight calculation before applying our model to the MOA-2011-BLG-262lb microlensing event. Therefore the rate-weight of each lensing system summed over all lenses j is simply Ψ = j ψ j θ E,j D 2 l,j µ j. (3.12) For a complete representation of the posterior probability of each model we consider all the available constraints, which for MOA-2011-BLG-262lb includes t E, I s and µ rel. 3.8 Application to MOA-2011-BLG-262lb The posterior probability maps in figures 3.4 and 3.5 were produced with ψ=1. However, for this event we need to constrain our model further by adding selection functions to the rate-weight, using the observed parameters given by Bennett et al. (2014). The MOA alert system gives the relative error of 1 10 3 t E for this event, so we set t E = 3.845 ± 0.015 days and I s = 19.93 ± 0.01. Figures 3.6 and 3.7 shows the new posterior probability maps for 0.12 solar-mass and 3.6 Jupiter-mass lens populations with constraints in event crossing time and source magnitude. Parvin Mansour 59

3.8: APPLICATION TO MOA-2011-BLG-262LB Figure 3.6: The posterior probability map of 0.12 M lenses with a constraint in Einstein radius crossing time and source magnitude. Parvin Mansour 60

3.8: APPLICATION TO MOA-2011-BLG-262LB Figure 3.7: The posterior probability map of 3.6 M J lenses with a constraint in Einstein radius crossing time and source magnitude. The mass, t E and I s constraints decrease the maximum posterior probability for 0.12 M and 3.6 M J populations by nearly 4 and 3 order of magnitude, respectively. Adding these constraints had no effect on the 0.12 M lens population distance range, but the lenses have a higher proper motion of 4 µ rel 14.5 mas/yr. For 3.6 M J lenses, these effects are more significant. The model now favours a population of lenses with a much lower relative proper motion of 1 µ rel 7 mas/yr and a D l region extending to a distance of 3 kpc D l 9 kpc. Parvin Mansour 61

3.8: APPLICATION TO MOA-2011-BLG-262LB 3.8.1 Refining Besançon catalogues In order to constrain our model further, we made more cuts in the source catalogue. Figure 3.8 shows the plot of source apparent magnitude versus source proper motion for all the lensing systems with significant contributions to the rate-weight, where the source magnitude is given by the catalogue K and I-K values I s = K + (I K). (3.13) The significant rate-weights correspond to a source magnitude of 19.5-20.32 and a proper motion of 0-14.5 mas/yr, so we produced a new larger source catalogue restricted to these cuts. Figure 3.8: The plot of source apparent magnitude versus source proper motion. The sources with significant contribution to the weight have µ s 14.5 mas/yr and 19.5 < I s < 20.32. We computed the rate-weight of all lensing systems for the new source catalogue and used plots of D s vs D l, µ l and µ s, D l vs µ l and µ s and µ l vs µ s to make further cuts in our source and lens catalogues. Figures 3.9-3.14 show these plots Parvin Mansour 62

3.8: APPLICATION TO MOA-2011-BLG-262LB for the 0.12 M population, whilst figures 3.15-3.20 show the plots for the 3.6 M J population. All rate-weighted systems that contribute significantly to the posterior probability fall within lens and source distance of 2-14 kpc for both populations and have a source proper motion of 0-14.5 mas/yr. Therefore the final lens catalogue used in our computation had a cut in D l = 2 14 kpc, whilst the source catalogue had the following cuts in D s = 2 14 kpc, I s = 19.5 20.32 and µ s = 0 14.5 mas/yr. With these restrictions we can re-run a larger simulation sampling only regions of significant microlens weighting. Figure 3.9: Plot of source distance versus lens distance for the 0.12 solar-mass population. All sources and lenses which contribute significantly to the weight have a distance of 2-14 kpc. Parvin Mansour 63

3.8: APPLICATION TO MOA-2011-BLG-262LB Figure 3.10: Plot of source distance versus lens proper motion for the 0.12 solar-mass lens population. The source-lens pairs with significant contribution to the weight have 2 D s 14 kpc and µ l < 19 mas/yr. Figure 3.11: Plot of source distance versus source proper motion for the 0.12 solar-mass lens population. Sources with significant contribution to the weight have a distance and proper motion of 2 D s 14 kpc and µ s < 14.5 mas/yr respectively. Parvin Mansour 64

3.8: APPLICATION TO MOA-2011-BLG-262LB Figure 3.12: Plot of lens distance versus lens proper motion for the 0.12 solar-mass lens population. Lenses with significant contribution to the weight have a distance and proper motion of 2 D l 14 kpc and µ l < 19 mas/yr respectively. Figure 3.13: Plot of lens distance versus source proper motion for the 0.12 solar-mass lens population. The source-lens pairs with significant contribution to the weight have 2 D l 14 kpc and µ s < 14.5 mas/yr. Parvin Mansour 65

3.8: APPLICATION TO MOA-2011-BLG-262LB Figure 3.14: Plot of source proper motion versus lens proper motion for the 0.12 solarmass lens population. The source-lens pairs with significant contribution to the weight have µ s < 14.5 and µ l < 19mas/yr. Figure 3.15: Plot of source distance versus lens distance for the 3.6 M J lens population. All sources and lenses which contribute significantly to the weight have a distance of 1.7-14 kpc. Parvin Mansour 66

3.8: APPLICATION TO MOA-2011-BLG-262LB Figure 3.16: Plot of source distance versus lens proper motion for the 3.6 M J lens population.the source-lens pairs with significant contribution to the weight have 1.7 D s 14 kpc and µ l < 18 mas/yr. Figure 3.17: Plot of source distance versus source proper motion for the 3.6 M J lens population. Sources with significant contribution to the weight have a distance and proper motion of 1.7 D s 14 kpc and µ s < 14.5 mas/yr respectively. Parvin Mansour 67

3.8: APPLICATION TO MOA-2011-BLG-262LB Figure 3.18: Plot of lens distance versus lens proper motion for the 3.6 M J lens population. Lenses with significant contribution to the weight have a distance and proper motion of 2 D l 14 kpc and µ l < 18 mas/yr respectively. Figure 3.19: Plot of lens distance versus source proper motion for the 3.6 M J lens population. The source-lens pairs with significant contribution to the weight have 2 D l 14 kpc and µ s < 14.5 mas/yr. Parvin Mansour 68

3.8: APPLICATION TO MOA-2011-BLG-262LB Figure 3.20: Plot of source proper motion versus lens proper motion for the 3.6 M J lens population. The source-lens pairs with significant contribution to the weight have µ s < 14.5 and µ l < 18 mas/yr. 3.8.2 Final probability density maps Figures 3.21 and 3.22 show the posterior probability maps for 0.12 M and 3.6 M J lens populations using source/lens catalogues with all the cuts. It can be seen from these figures that our model strongly favours a stellar bulge lens with µ rel > 5 mas/yr, however, at lower proper motion, a disk/bulge Jupiter-mass lens is favoured. Parvin Mansour 69

3.8: APPLICATION TO MOA-2011-BLG-262LB Figure 3.21: The final posterior probability map of the 0.12 solar-mass lenses is shown here. Parvin Mansour 70