Interferometric observations of gravitational microlensing events offer an opportunity for precise efficient and direct mass and distance measurements of lensing objects especially those of isolated neutron stars and black holes. However such observations were previously possible for only a handful of extremely bright events. The recent development of a dualfield interferometer GRAVITY Wide has made it possible to reach out to significantly fainter objects and increase the pool of microlensing events amenable to interferometric observations by two orders of magnitude. Here we present the first successful observation of a microlensing event with GRAVITY Wide and the resolution of microlensed images in the event OGLE2023BLG0061KMT2023BLG0496. We measure the angular Einstein radius of the lens with a subpercent precision thetarm E  1.280 pm 0.009 mas. Combined with the microlensing parallax detected from the event light curve the mass and distance to the lens are found to be 0.472 pm 0.012 Modot and 1.81 pm 0.05 kpc respectively. We present the procedure for the selection of targets for interferometric observations and discuss possible systematic effects affecting GRAVITY Wide data. This detection demonstrates the capabilities of the new instrument and it opens up completely new possibilities for the followup of microlensing events and future routine discoveries of isolated neutron stars and black holes.
['Astrophysics - Instrumentation and Methods for Astrophysics', 'Astrophysics - Solar and Stellar Astrophysics']
Observations of microlensed images with dualfield interferometry onsky demonstration and prospects
{'No Header': 'Draft version September 20, 2024', 'Observations of microlensed images with dual-field interferometry: on-sky demonstration and prospects': "Przemek Mróz, 1 Subo Dong, 2, 3 Antoine Mérand, 4 Jinyi Shangguan, 5 Julien Woillez, 4 Andrew Gould, 6, 7 Andrzej Udalski, 1 Frank Eisenhauer, 5 Yoon-Hyun Ryu, 8 Zexuan Wu, 2, 3 Zhuokai Liu 2, 3 And Hongjing Yang 9  \n-  \nGuillaume Bourdarot, 5 Denis Defrère, 10 Antonia Drescher, 5 Maximilian Fabricius, 5 Paulo Garcia, 11, 12 Reinhard Genzel, 5 Stefan Gillessen, 5 Sebastian F. Hönig, 13 Laura Kreidberg, 7 Jean-Baptiste Le Bouquin, 14 Dieter Lutz, 5 Florentin Millour, 15 Thomas Ott, 5 Thibaut Paumard, 16 Jonas Sauter, 5, 7 T. Taro Shimizu, 5 Christian Straubmeier, 17 Matthias Subroweit 17 And Felix Widmann 5  \n(The GRAVITY+ Collaboration)  \nMichał K. Szymański, 1 Igor Soszyński, 1 Paweł Pietrukowicz, 1 Szymon Kozłowski, 1 Radosław Poleski, 1 Jan Skowron, 1 Krzysztof Ulaczyk, 18, 1 Mariusz Gromadzki, 1 Krzysztof Rybicki, 19, 1 Patryk Iwanek, 1 Marcin Wrona 20, 1 And Mateusz J. Mróz 1  \n(The OGLE Collaboration)  \nMichael D. Albrow, 21 Sun-Ju Chung, 8 Cheongho Han, 22 Kyu-Ha Hwang, 8 Youn Kil Jung, 8, 23 In-Gu Shin, 24 Yossi Shvartzvald, 25 Jennifer C. Yee, 24 Weicheng Zang, 24 Sang-Mok Cha, 8, 26 Dong-Jin Kim, 8 Seung-Lee Kim, 8 Chung-Uk Lee, 8 Dong-Joo Lee, 8 Yongseok Lee, 8, 26 Byeong-Gon Park, 8 Richard W. Pogge 27, 28  \n(The KMTNet Collaboration)  \n1 Astronomical Observatory, University of Warsaw, Al. Ujazdowskie 4, 00-478 Warszawa, Poland  \n2 Department of Astronomy, School of Physics, Peking University, Yiheyuan Rd. 5, Haidian District, Beijing, China, 100871 3 Kavli Institute for Astronomy and Astrophysics, Peking University, Yiheyuan Rd. 5, Haidian District, Beijing, China, 100871 4 European Southern Observatory, Karl-Schwarzschild-Straße 2, D-85748 Garching, Germany 5 Max Planck Institute for Extraterrestrial Physics, Giessenbachstraße 1, D-85748 Garching, Germany 6 Department of Astronomy, Ohio State University, 140 W. 18th Ave., Columbus, OH 43210, USA 7 Max Planck Institute for Astronomy, Königstuhl 17, 69117 Heidelberg, Germany 8 Korea Astronomy and Space Science Institute, Daejeon 34055, Republic of Korea 9 Department of Astronomy, Tsinghua University, Beijing 100084, China 10 Institute of Astronomy, KU Leuven, Celestijnenlaan 200D, 3001, Leuven, Belgium 11 CENTRA - Centro de Astrofísica e Gravitação, IST, Universidade de Lisboa, 1049-001 Lisboa, Portugal 12 Faculdade de Engenharia, Universidade do Porto, Rua Dr Roberto Frias, 4200-465 Porto, Portugal 13 School of Physics & Astronomy, University of Southampton, Southampton, SO17 1BJ, UK 14 Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France 15 Université Côte d'Azur, Observatoire de la Côte d'Azur, CNRS, Laboratoire Lagrange, Nice, France 16 LESIA, Observatoire de Paris, Université PSL, Sorbonne Université, Université Paris Cité, CNRS, 5 place Jules Janssen, 92195 Meudon, France 17 1st Institute of Physics, University of Cologne, Zülpicher Straße 77, 50937 Cologne, Germany 18 Department of Physics, University of Warwick, Coventry CV4 7 AL, UK 19 Department of Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot 76100, Israel 20 Department of Astrophysics and Planetary Sciences, Villanova University, 800 Lancaster Ave., Villanova, PA 19085, USA 21 University of Canterbury, School of Physical and Chemical Sciences, Private Bag 4800, Christchurch 8020, New Zealand 22 Department of Physics, Chungbuk National University, Cheongju 28644, Republic of Korea 23 National University of Science and Technology (UST), Daejeon 34113, Republic of Korea  \n24 Center for Astrophysics | Harvard & Smithsonian, 60 Garden St.,Cambridge, MA 02138, USA  \nCorresponding author: Przemek Mróz  \npmroz@astrouw.edu.pl  \n28  \n25 Department of Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot 7610001, Israel 26 School of Space Research, Kyung Hee University, Yongin, Kyeonggi 17104, Republic of Korea 27 Department of Astronomy, Ohio State University, 140 West 18th Ave., Columbus, OH 43210, USA Center for Cosmology and AstroParticle Physics, Ohio State University, 191 West Woodruff Ave., Columbus, OH 43210, USA  \n(Received; Revised; Accepted)", 'ABSTRACT': 'Interferometric observations of gravitational microlensing events offer an opportunity for precise, efficient, and direct mass and distance measurements of lensing objects, especially those of isolated neutron stars and black holes. However, such observations were previously possible for only a handful of extremely bright events. The recent development of a dual-field interferometer, GRAVITY Wide, has made it possible to reach out to significantly fainter objects, and increase the pool of microlensing events amenable to interferometric observations by two orders of magnitude. Here, we present the first successful observation of a microlensing event with GRAVITY Wide and the resolution of microlensed images in the event OGLE-2023-BLG-0061/KMT-2023-BLG-0496. We measure the angular Einstein radius of the lens with a sub-percent precision, θ E = 1 . 280 ± 0 . 009 mas. Combined with the microlensing parallax detected from the event light curve, the mass and distance to the lens are found to be 0 . 472 ± 0 . 012 M ⊙ and 1 . 81 ± 0 . 05 kpc, respectively. We present the procedure for the selection of targets for interferometric observations, and discuss possible systematic effects affecting GRAVITY Wide data. This detection demonstrates the capabilities of the new instrument and it opens up completely new possibilities for the follow-up of microlensing events, and future routine discoveries of isolated neutron stars and black holes.  \nKeywords: Gravitational microlensing (672), Optical interferometry (1168)', '1. INTRODUCTION': "In his seminal paper on gravitational microlensing, Einstein (1936) realized that the angular separation between the two images of the source star created by a Galactic gravitational lens would be well below the resolution limits of contemporary telescopes. That led him to believe that observing the microlensed images would be, in practice, impossible. This notion persisted for many decades until the first optical interferometry facilities were developed.  \nIn point-source point-lens microlensing events, the gravitational lens creates two images of the source star (called the major and minor image). They are separated by approximately 2 θ E , where θ E = √ κMπ rel is the angular Einstein radius. Here, M is the lens mass, π rel is the lens-source relative parallax, and κ = 8 . 144 mas M -1 ⊙ is a constant. For typical configurations of stellar-mass microlensing events in the Milky Way, θ E ∼ 1 mas , and so the expected separation between the minor and major images is usually smaller than a few milliarcseconds. Interferometric observations of microlensing events, therefore, provide a direct way to precisely measure angular Einstein radii by resolving the images (and, as a consequence, measure the masses of lensing objects).  \nAlthough a few tens of thousands of single-lens microlensing events have been discovered so far, only a  \nfew had the angular Einstein radius measured. That was possible for only a small number of events exhibiting finite-source effects (Gould 1994; Nemiroff & Wickramasinghe 1994; Witt & Mao 1994), and the probability of such measurements is biased towards low-mass (planetary-mass) lenses. Another route for θ E measurements involved astrometric observations (Hog et al. 1995; Miyamoto & Yoshii 1995; Walker 1995), which led to the first detection of an isolated black hole (Sahu et al. 2022; Lam et al. 2022; Mróz et al. 2022; Lam & Lu 2023). These observations, however, required longterm astrometric monitoring, which is not feasible for a large sample of events. Conversely, interferometric observations are much more efficient as only one exposure is sufficient to resolve the microlensed images and measure the angular Einstein radius. When combined with the microlensing parallax ( π E ≡ π rel /θ E ) measurements from the light curve, they allow us to precisely determine the masses, distances, and transverse velocities of isolated objects, including neutron stars and black holes.  \nDelplancke et al. (2001) were the first to discuss the prospects of using the European Southern Observatory (ESO) Very Large Telescope Interferometer (VLTI) to study gravitational microlensing events. They estimated that the first-generation VLTI instruments would be able to observe dozens of events every year in the mid-  \n2000s. Yet, the first successful resolution of the microlensed images did not materialize until late 2017, when Dong et al. (2019) observed the microlensing event TCP J05074264+2447555 (a.k.a., Kojima-1) with the second-generation VLTI instrument GRAVITY (GRAVITY Collaboration et al. 2017).  \nWhy did it take almost two decades to achieve this milestone? The early predictions on the VLTI performance underestimated the role and impact of many practical problems encountered during system operations, such as beam stability, vibrations, or air dispersion (e.g., Delplancke 2008). Moreover, even when these technical problems were solved or mitigated, the performance of a ground-based interferometer is ultimately limited by atmospheric turbulence, which breaks the coherence of wavefronts arriving at each telescope. Therefore, the observations must be carried out with exposure times ( ∼ 1 ms) shorter than the typical wavefront coherence timescale (on the order of 20-30 ms) and shorter than the mechanical vibrations timescale, which dominate in the 10-1000 Hz regime. Even for the largest modern 10-m-class telescopes, this requirement sets a limiting magnitude of K ≈ 10 for interferometric observations. Gravitational microlensing events brighter than this limit are exceedingly rare (Figure 1). They occur at most a few times a year, and they exceed the GRAVITY limiting magnitude for a short period of time. The three events with the published interferometric observations were unusually bright: TCP J05074264+2447555 (Dong et al. 2019; Zang et al. 2020), ASASSN-22av (Wu et al. 2024, submitted), and Gaia19bld (Cassan et al. 2022; Rybicki et al. 2022; Bachelet et al. 2022) reached V ≈ 11 . 5 , g ' ≈ 12 . 5 , and I ≈ 9 . 0 , respectively. The former two were observed using the standard on-axis mode of GRAVITY, whose fringe tracker allow minute-long science exposures (Lacour et al. 2019) that significantly enhance the VLTI sensitivity (Eisenhauer et al. 2023).  \nThe dual-field interferometric observations were designed to alleviate this problem and reach fainter sources (e.g., Shao & Colavita 1992; Colavita et al. 1999; Delplancke 2008; Woillez et al. 2014). Still, a nearby bright ( K ≲ 10 ) reference (fringe-tracking) star is needed to stabilize the optical path difference between the telescopes and phase-reference the interferogram of the science target. The angular separation between the science object and the fringe-tracking star must be smaller than 20 -30 '' , which is set by the atmospheric conditions. In 2019-2022, the GRAVITY instrument was upgraded to enable such dual-field interferometric observations (GRAVITY+ Collaboration et al. 2022), hence the name of this observing mode, GRAVITY Wide.  \nFigure 1. Cumulative distribution of the expected K -band peak magnitudes of events detected by the OGLE EWS in 2023 (blue line). For comparison, the red line shows the cumulative distribution of the K -band magnitudes of the brightest star within 30 '' of the event (fringe-tracking star). Only events with t E ≥ 50 day and an adaptive-optics guide star brighter than G = 14 are plotted.  \n<!-- image -->  \nThe upgraded instrument opened up an entirely new pathway for characterizing and studying microlensing events because, in the dense fields of the Galactic bulge, the probability of finding a suitable ( K ≲ 10 , within 30 '' ) fringe-tracking star is as large as 50-70%. According to the GRAVITY manual (Issue 110), 1 the GRAVITY Wide observations are possible for science targets as faint as K SC ≈ 16 -17 . However, the limiting magnitude strongly depends on the atmospheric conditions and the separation between the science object and the fringe-tracking star. In addition, closure phase observations lose the signal-to-noise ratio faster than the normal visibility data as the targets become fainter. Thus, in practice, we rarely considered observing targets fainter than K ≈ 13 . 5 -14 , which still left us with plenty of candidate events to scrutinize (Figure 1).  \nAs soon as ESO offered regular GRAVITY Wide observations since October 2022 ('Period 110'), we initiated a program (PI: Mérand) of interferometric observations of microlensing events. Our primary science goal was to detect and measure precise masses, distances, and  \ntransverse velocities of isolated stellar remnants - neutron stars and black holes. Our observations also served as a testbed for verifying the capabilities of the new instrument, and planning, executing, and analyzing the interferometric follow-up observations of a large number of microlensing events.  \nThis paper presents the first resolution of microlensed images with GRAVITY Wide in the microlensing event OGLE-2023-BLG-0061/KMT-2023-BLG-0496.", '2. SELECTION OF THE TARGETS FOR THE INTERFEROMETRIC FOLLOW-UP': "The VLTI observations can be carried out with either four 8.2-m Unit Telescopes (UTs) or four 1.8-m Auxiliary Telescopes (ATs). The VLTI UTs observing runs are organized every month (around the full Moon) and typically last one week. During the rest of the month, observations are possible with ATs, which may be relocated to more than ten observing stations. GRAVITY Wide observations may be conducted with ATs, provided that the fringe-tracking star is brighter than K FT ≈ 9 -9 . 5 and the science target is brighter than K SC ≈ 13 -14 . For UTs, the limiting magnitudes are larger, K FT ≈ 10 -10 . 5 and K SC ≈ 16 -17 , respectively. As discussed above, the limiting magnitudes of science targets strongly depend on the atmospheric conditions (isoplanatic angle) and the separation of the fringetracking star.  \nWe began the selection of microlensing targets about 7-10 days before the planned start of each VLTI UT run. The candidates were chosen from publicly available lists of microlensing alerts published by the Optical Gravitational Lensing Experiment (OGLE) Early Warning System 2 (EWS; Udalski 2003; Udalski et al. 2015), Korean Microlensing Telescope Network (KMTNet) Alert System 3 (Kim et al. 2018), and Microlensing Observations in Astrophysics (MOA) Transient Alerts 4 . In addition, we also checked transient alerts published by allsky surveys, such as Gaia (Hodgkin et al. 2013, 2021) or All Sky Automated Survey for SuperNovae (ASAS-SN) (Shappee et al. 2014).  \nThe selection of targets was based on several scientific and technical criteria. First, we selected events near or past their maximum brightness so that parameters describing their light curves were reasonably well measured. However, the observations had to be secured before the event faded, when the contrast ratio between the microlensed images became too large. We required  \nthe contrast ratio to be smaller than 10:1, which corresponds to the maximum lens-source separation (in Einstein radius units) of u max = 1 . 22 or, equivalently, the minimum amplitude of ∆ I min = 0 . 13 .  \nBecause the primary scientific motivation of our project was searching for stellar remnants, which are expected to give rise to long-duration events, we selected events with Einstein timescales longer than t E = 50 day. However, the nature of the lens cannot be known at the time of selecting the targets. The mass of the lens can be determined only after the interferometric data are combined with the full light curve. That usually means waiting several weeks after the interferometric observations are taken because they are collected close to the maximum magnification.  \nIn the next step, we checked if suitable fringe-tracking and adaptive-optics reference stars are located within 30 '' of the event. For possible fringe-tracking stars, we queried the Two Micron All Sky Survey (2MASS) Point Source Catalog (Skrutskie et al. 2006), while for adaptive-optics guide stars - the Gaia Data Release 3 (Gaia Collaboration et al. 2016, 2023). The guide star had to be brighter than G ≈ 14 for the Multi Application Curvature Adaptive Optics (Arsenault et al. 2003) system, or brighter than K = 8 for the Coudé Infrared Adaptive Optics (Kendrew et al. 2012) system. Finally, we estimated the expected K -band brightness of the event during the planned observations by assuming that the blending parameter in the K -band is identical to that in the I -band. The baseline K -band brightness of the event was taken from the VISTA Variables in the Via Lactea survey (VVV; Minniti et al. 2010), United Kingdom Infra-Red Telescope (UKIRT) Galactic Plane Survey (Lawrence et al. 2007; Lucas et al. 2008), or 2MASS (Skrutskie et al. 2006). We also examined the estimated K -band brightness of the source by using the best-fit I -band source fluxes as cross checks. For such estimates, we assumed de-reddened color ( I -K ) 0 , giant = 1 . 4 and ( I -K ) 0 , dwarf = 1 . 0 for sources roughly classified using the extinction-corrected source magnitudes as giants ( I 0 < 16 . 5 ) and dwarfs ( I 0 > 16 . 5 ), respectively.  \nCandidate targets were selected independently by two teams (P.M. and S.D.) and subsequently investigated in more detail. In particular, we paid special attention to the microlensing parallax measurements, which are necessary for the lens mass determination. Since the value of the microlensing parallax is inversely proportional to the square root of the lens mass, we required it to be consistent with zero (or close to zero) during the trigger. We also run light curve simulations to ensure that the parallax will be precisely measured (or constrained) by the end of the observing season. We did not consider  \nevents with variable source stars because the variability may affect the microlensing parallax measurements.  \nThe blue solid line in Figure 1 shows the cumulative distribution of the expected K -band peak magnitudes of events detected by the OGLE's EWS in 2023. Only events with relatively long timescales ( t E ≥ 50 day) and suitable adaptive-optics guide stars ( G ≤ 14 ) within 30 '' are presented. Only a few events were bright enough ( K < 10 . 5 ) for standard GRAVITY on-axis observations. In contrast, the solid red line in Figure 1 shows the distribution of magnitudes of the brightest star within 30 '' of the event (which may serve as a fringe-tracking star). Nearly 100 events could have been considered for GRAVITY Wide observations.", '3. DATA': "The detection of the microlensing event OGLE-2023BLG-0061 was announced by the OGLE Early Warning System (Udalski 2003; Udalski et al. 2015) on 2023 Mar 13.60 UT ( HJD ' ≡ HJD -2460000 = 17 . 10 ). It was independently identified by the KMTNet Alert System (Kim et al. 2016, 2018) on 2023 Apr 20, and it was designated KMT-2023-BLG-0496. The event occurred on a bright red clump star ( I = 16 . 389 ± 0 . 001 , V -I = 2 . 26 ± 0 . 02 ) with equatorial coordinates (R.A., Decl.) J2000 = ( 17 h 43 m 04 . s 01 , -35 · 15 ' 32 . '' 3 ). According to data from the VVV survey, the event had K = 13 . 515 ± 0 . 011 in the baseline (Minniti et al. 2010).", '3.1. Photometric Data': "OGLE operates the 1.3-m Warsaw Telescope located at Las Campanas Observatory, Chile. The telescope is equipped with a mosaic camera covering a field of view of 1.4 deg 2 with the pixel scale of 0 . 26 arcsec pixel -1 . The event has been observed by the OGLE-IV survey since 2010. However, in this paper, we analyze the OGLE data collected from 2016 through 2023 because earlier observations do not contribute to constraining the parameters of the model. Observations were reduced using the OGLE-IV data reduction pipeline (Udalski et al. 2015), which employs a custom implementation of the Difference Image Analysis (DIA) method (Woźniak 2000).  \nKMTNet uses three 1.6-m telescopes located at Cerro Tololo Inter-American Observatory (KMTC, Chile), South African Astronomical Observatory (KMTS, South Africa), and Siding Spring Observatory (KMTA, Australia). Each of the KMTNet telescopes is equipped with a camera with 4 deg 2 field of view and pixel scale of 0 . 40 arcsec pixel -1 . The analyzed KMTNet observations cover the years 2021-2023. However, because of saturation, we deleted data points near the peak of the  \nevent. The KMTNet photometric data were reduced with the tender-loving-care (TLC) DIA-based pipeline (Yang et al. 2024), which was developed from pySIS (Albrow et al. 2009). The vast majority of OGLE and KMTNet images were taken in the I -band filter, with additional V -band observations to characterize the color of the source star.  \nSome additional observations were taken in the R ' band with the 0.18-m Newtonian telescope (CHI-18; Wu et al. 2024) located at the El Sauce Observatory in Chile to cover the peak of the event. The CHI-18 images were reduced with the TLC pipeline. The original observations were taken at a 2-minute cadence. Because the event did not exhibit variability on such short timescales, we binned the CHI-18 data into 1-hour-long bins.", '3.2. VLTI Data': "OGLE-2023-BLG-0061/KMT-2023-BLG-0496 was considered a promising candidate for VLTI observations on 2023 July 20. Although the event was still before the peak, the available data predicted a relatively long timescale ( t E ≈ 100 d). At the same time, the optical brightness indicated that the microlens parallax should be robustly constrained. A nearby ( 13 . 2 '' ) bright ( G = 13 . 7 , K = 9 . 8 ) star 2MASS 17430426-3515195 could serve as a fringe-tracking and adaptive-optics guide star. The first GRAVITY Wide observations were secured with UTs on 2023 July 29 ('epoch 1'). We obtained four sets of medium-resolution observations ( R = λ/ ∆ λ ≈ 500 ) and one set of low-resolution data ( R ≈ 22 ), a new mode never tried before with GRAVITY Wide. The first three medium-resolution exposures were of low quality, and the resulting closure phases were very noisy. We, therefore, decided not to use them in the subsequent analysis. The preliminary modeling carried out at that time indicated that the medium- and low-resolution data were inconsistent with each other.  \nWe thus attempted to collect additional VLTI data to study the magnitude of possible systematics affecting the low-resolution data. We triggered on-axis GRAVITY observations with ATs on 2023 August 23, when the event was approaching the peak (at magnification ∼ 90 ), but the data could not be collected. The next (successful) attempt to gather additional VLTI UTs observations took place on 2023 September 29 ('epoch 2'), when we secured six low-resolution exposures with GRAVITY Wide. Table 1 presents the log of VLTI observations. Each low-resolution observation consisted of twelve 30 s exposures; the medium-resolution observation consisted of four 100 s exposures. For each night, a bright star pair  \nTable 1. Log of VLTI Observations  \nwas observed to center the science fringe with GRAVITY differential delay line before the microlens observation. We did not adjust the science fringe when the telescope was moved to the science target.  \nThe data were reduced with the standard GRAVITY pipeline (version 1.4.2). We first used the Python script run\\_gravi\\_reduced.py to reduce the raw data up to applying the pixel-to-visibility matrix (P2VM). The default options were used except that we adopted -gravity\\_vis.output-phase-sc=SELF\\_VISPHI to calculate the internal differential phase between each spectral channel and -gravity\\_vis.opd-pupil-stddev-maxsc=9999 to ignore the poor pupil measurements in the acquisition camera which do not affect our closure phase measurements. The pipeline performed the bias and sky subtraction, flat fielding, wavelength calibration, and spectral extraction. Application of the P2VM converts the pixel detector counts into complex visibilities, taking into account all instrumental effects, including relative throughput, coherence, phase shift, and cross-talk. The dark, bad pixel, flat field, wavelength calibration, and P2VM matrix data were reduced from the daily calibration data obtained close in time to our observations. We then used run\\_gravi\\_trend.py to calibrate the closure phase data. For epoch 1, we used the star pair to center the science fringe and calibrate the medium-resolution data. Unfortunately, we did not have calibrator data observed in the low-resolution data on the same night. For epoch 2, we used a bright star pair observed after the microlens observation for the calibration. The following analyses were based on the closure phase data from the calibrated medium-resolution and uncalibrated low-resolution data from epoch 1, and calibrated lowresolution data from epoch 2. In this way, we used all the data with good quality. Meanwhile, when modeling the low-resolution data from epoch 2, we found that the calibration makes little difference to the closure phase.", '4. LIGHT CURVE MODEL': "The light curve of the event can be well fitted by a standard point-source point-lens model with the annual parallax effect (which is caused by the orbital motion of the Earth). This model has five free parameters: time of the closest approach between the lens and the source t 0 , their minimum separation (in Einstein radius units) u 0 , the Einstein radius crossing timescale t E , and North and East components of the microlensing parallax vector π E = ( π E ,N , π E ,E ) . The latter is a vector quantity whose direction is parallel to the direction of the relative lens-source proper motion µ rel . The magnification is calculated using the formula:  \nA ( t ) = u ( t ) 2 +2 u ( t ) √ u ( t ) 2 +4 , (1)  \nwhere u ( t ) = √ τ ( t ) 2 + β ( t ) 2 . The latter quantity is evaluated in the geocentric frame that is moving with a velocity equal to the Earth's velocity at t 0 , par = 2460179 (Gould 2004) in which  \nτ ( t ) = t -t 0 t E + δτ ( t ) , β ( t ) = u 0 + δβ ( t ) (2)  \nand  \n( δτ, δβ ) = ( π E · ∆ s, π E × ∆ s ) , (3)  \nwhere ∆ s is the projected position of the Sun. Two models with different signs of u 0 are possible; both models are almost perfectly degenerate, with the u 0 < 0 being preferred by only ∆ χ 2 = 3 . 1 . The best-fit model (with u 0 > 0 ) is presented in Figure 2. The best-fit parameters and their uncertainties are reported in Table 2. In Table 2, we also report the best-fit source magnitude I s , baseline magnitude I 0 , and dimensionless blending parameter f s .  \nFigure 3 shows the constraints on the microlensing parallax vector derived using data from different observatories. Blue, red, green, and orange contours mark  \nFigure 2. Light curve of OGLE-2023-BLG-0061/KMT2023-BLG-0496. The black line marks the best-fit model with u 0 > 0 . Arrows mark the two epochs of VLTI observations.  \n<!-- image -->  \nTable 2. Best-fit Parameters of the Light Curve Model  \nconstraints from OGLE, KMTC, KMTA, and KMTS data, respectively, whereas the solid black contours mark  \nFigure 3. Constraints on the microlensing parallax derived from different photometric data sets (blue - OGLE, red KMTC, orange - KMTS, green - KMTA) and all light curve data (black contours). The dotted and dashed lines mark constraints from the two epochs of VLTI data, respectively.  \n<!-- image -->  \nthe best-fit model to all data. The East component of π E , which is parallel to the projected acceleration of the Sun, is relatively well measured in all data sets. However, the North component π E ,N (perpendicular to the projected acceleration of the Sun) has a considerably larger uncertainty. The North component is also more susceptible to noise in the data (Gould et al. 1994; Smith et al. 2003; Gould 2004); hence, slightly different values of π E ,N are determined using different data sets. The best-fit parameters for individual data sets are reported in Tables 3 and 4.  \nThe light curve model of a microlensing event makes it possible to predict the brightness ratio η between the minor and major image at a given epoch of interferometric observations. Moreover, the detection of the microlensing parallax in the light curve allows us to predict the orientation of microlensed images in the sky. The position angle (north through east) of the microlensing parallax vector can be calculated using formula  \nΦ π = arctan π E ,E π E ,N . (4)  \nThe angle ϕ between the source-lens relative proper motion and source-lens relative position at a given time can also be calculated directly from the parameters of the light curve model:  \nϕ ( t ) = arctan β ( t ) τ ( t ) . (5)  \nTable 3. Best-fit Parameters of the Light Curve Model ( u 0 > 0) for Individual Data SetsTable 4. Best-fit Parameters of the Light Curve Model ( u 0 < 0) for Individual Data Sets  \nNote that in the limit of no parallax, this equation simplifies to ϕ ( t ) = arctan( u 0 t E / ( t -t 0 )) , which is equivalent to Equation (8) derived by Dong et al. (2019). Then, following Dong et al. (2019), the position angle of the minor image relative to the major image (north through east) is simply PA = Φ π + ϕ if u 0 > 0 or PA = Φ π -ϕ if u 0 < 0 . Note that Dong et al. (2019) introduced a different definition of the position angle of the major image relative to the minor image ψ = PA+ π .  \nTable 2 presents the predicted lens-source separation u (in Einstein radius units), the flux ratio between the minor and major image η , and the position angle of the images PA for both epochs of VLTI observations. Note that the expected lens-source separation and flux ratio of the images are virtually identical for u 0 > 0 and u 0 < 0 models. However, the expected position angles are different. For the positive u 0 solution, we can predict PA = -164 . 83 ± 0 . 95 deg and PA = 31 . 52 ± 0 . 58 deg for both VLTI epochs. For the negative u 0 solution, we predict PA = -156 . 60 ± 0 . 68 deg and PA = 14 . 29 ± 1 . 06 deg, respectively.  \nWe also note that the flux ratio of the images is tightly constrained by the light curve model, with a precision better than 0.3%. Conversely, the uncertainty of the position angle is dominated by the uncertainty in the determination of the angle Φ π , which itself is dominated by the uncertainty of π E ,N . As we observed small systematic differences between π E ,N determined using different data sets (Figure 3), these differences propagate to systematic variations in Φ π and therefore PA . In particular, the angle Φ π can vary from 18 . 56 to 25 . 90 deg (Tables 3 and 4).", '5. CLOSURE PHASE MODELS': 'We first separately analyze the VLTI data collected during epochs 1a, 1b, and 2, because they were taken at different times and using different instrument configurations. That will allow us to study the consistency between the model parameters derived using different data sets and the consistency with the light curve model (Sections 5.1, 5.2, and 5.3). In Section 5.4, we combine all VLTI data sets to derive the final parameters of the system. The results presented in this section were independently checked using the PMOIRED software 5 (Mérand 2022) and we found virtually identical results.', '5.1. Binary-star Model': 'We start by fitting a simple binary-star model to the closure phase data. We assume that both images of the source star can be considered as pointlike. We place  \nTable 5. Best-fit Parameters of the Binary-star Closure Phase Model  \nthe major image in the origin of the coordinate system, whereas the position of the minor image is parameterized by a vector (∆ α, ∆ δ ) . If we denote the flux ratio between the minor and major image as η , then the complex visibility is  \nVIS = 1 + η exp ( -2 πi λ ( U ∆ α + V ∆ δ ) ) 1 + η , (6)  \nwhere λ is the wavelength of observations, and ( U, V ) is the separation of the telescopes in the UV plane. The visibility is calculated for each pair of telescopes, and then the closure phase T3 is calculated for each triangle. Four possible closure phases can be calculated for VLTI; in theory, one of them is not independent. However, since the measured closure phases may be affected by noise in the data, we employ all four closure phase sets in the fits.  \nWe find the best-fit parameters by maximizing the log-likelihood function defined as:  \nln L = -1 2 4 n exp ∑ i =1 Λ ∑ j =1 (T3 ij -T3 model ij ) 2 σ (T3) 2 ij + σ 2 0 + -1 2 4 n exp ∑ i =1 Λ ∑ j =1 ln ( σ (T3) 2 ij + σ 2 0 ) , (7)  \nwhere n exp is the number of exposures in the epoch and Λ is the number of spectral channels. Since the error bars calculated from the pipeline may be underestimated, we add a constant error term σ 0 in quadrature. The best-fit parameters and their uncertainties are calculated using the Markov chain Monte Carlo (MCMC) algorithm coded by Foreman-Mackey et al. (2013). We assume flat (non-informative) priors on all parameters of the model.  \nFigure 4. Closure phase data for the VLTI epoch 1. Color solid lines mark the best-fit binary-star model.  \n<!-- image -->  \nResults of the fits, separately for VLTI epochs 1a, 1b, and 2, are reported in Table 5. The best-fitting closure phase models are shown in Figures 4 and 5. In addition to four model parameters, we report four derived quantities: separation between the images s = √ ∆ α 2 +∆ δ 2 , their position angle (north through east) PA = arctan∆ α/ ∆ δ , source-lens separation u (in Einstein radius units), and the angular Einstein radius θ E . The latter quantity is calculated using the formula  \nθ E = sη 1 / 4 √ η +1 (8)  \nderived under the assumption that the flux ratio between the microlensed images determines their separation in Einstein radius units. Similarly, the source-lens separation is √  \nu = 1 -η η 1 / 4 . (9)  \nResults of the fits to individual epoch 2 exposures are presented in Table 6.  \nThere is some tension between the medium- and lowresolution interferograms obtained during epoch 1: the measured ∆ α position differs by 3 . 3 σ , ∆ δ position2 . 3 σ , position angle3 . 2 σ . On the other hand, the inferred angular Einstein radii are formally consistent between the three different data sets, although the differences between individual measurements can amount up to 1 . 5 -2 . 9 σ . These tensions indicate that at least one of the analyzed data sets may suffer from unaccounted, low-level systematic errors. Thus, error bars reported in Tables 5 and 6 may be underestimated.  \nFurther checks are possible because the flux ratio η and position angle PA of the images can be independently measured using the light curve data (Section 4). In particular, there are two possible light curve models differing by the sign of u 0 , which predict the position angles of the microlensed images of ( -164 . 8 · , 31 . 5 · ) (positive u 0 ) or ( -156 . 6 · , 14 . 3 · ) (negative u 0 ), during the VLTI epochs 1 and 2, respectively. The measured angles ( -162 . 9 · , 26 . 7 · ) (Table 5) seem to favor the u 0 > 0 model.  \nHowever, while the light curve and closure phase models agree well for epoch 1, the expected position angles differ by almost 4 . 8 · (that is, 7 . 5 σ ) during epoch 2. That is illustrated in Figure 6, which presents 1 , 2 , 3 σ confidence ellipses in the (PA , η ) plane. Gray contours mark the constraints from three different VLTI data sets, whereas black contours were calculated based on the light curve model ( u 0 > 0 ). Blue, red, green, and orange contours mark constraints on (PA , η ) from OGLE, KMTC, KMTA, and KMTS data, respectively.  \nThe position angle of the images measured from interferometric observations can be projected onto the ( π E ,N , π E ,E ) plane (Figure 3). The dotted and dashed lines in this figure correspond to the best-fitting values of PA during epochs 1a and 2, respectively. The dashed line does not intersect color contours from the light curve model, which further exemplifies the tension discussed above.  \nThe light curve model also predicts the flux ratios between the microlensed images: η = 0 . 5907 ± 0009 for epoch 1 and η = 0 . 4264 ± 0 . 0013 for epoch 2. The for-  \nFigure 5. Closure phase data for the VLTI epoch 2. Color solid lines mark the best-fit binary-star model.  \n<!-- image -->  \nmer value is in 2 . 7 -3 . 5 σ tension with closure phase fits to epoch 1a and 1b data, respectively.', '5.2. Luminous-lens Model': 'Despite the slight tensions discussed above, the binary-star model overall describes the closure phase data well (Figures 4 and 5). We now modify the model to include a possibility of blended light coming from the lens itself or a luminous companion (either to the lens or source). We changed the primary parameters of the model to the lens-source separation u , the position angle of the lens relative to the source PA (which is equal to the position angle of the minor image relative to the major image), and the angular Einstein radius θ E , which provide a more natural description of a microlensing event than the binary-star model parameters. We keep σ 0 fixed at 6.990, 2.593, and 2.509 deg for epoch 1a, 1b, and 2, respectively.  \nWe first consider models with a luminous lens. We place the lens in the origin of the coordinate system and calculate the positions of the minor and major image relative to it (Dong et al. 2019). We parameterize the ratio of the lens flux to the (unmagnified) source flux as η b .  \nWe found that the closure phase data do not provide strong evidence for the light from the lens. Including the lens light in the fits improves the χ 2 by 1.3, 0.1, and 0.9 for epochs 1a, 1b, and 2, respectively. The corresponding 95% upper limits on η b are 0.74, 0.81, and 0.44, respectively. Moreover, we noticed that the lens flux is correlated with the projected lens-source separation (and so, the flux ratio between the microlensed images) and the angular Einstein radius.  \nThus, we explored the possibility that the light from the lens may be a source of tensions between the photometric and interferometric data discussed above. We repeated modeling, taking into account priors on u from the light curve model. The best-fit parameters are reported in Table 7. While the tension between u determined from the light curve model and that from the closure phase model is removed, the position angles are still slightly different.', '5.3. Luminous-blend Model': 'We also consider the model with a luminous blend, which has two additional parameters compared to the luminous-lens model, the offset of the blend (∆ α b , ∆ δ b ) relative to the lens in the sky. We searched for possible blends on a grid of 201 × 201 positions uniformly spread over the range -20 ≤ ∆ α b , ∆ δ b ≤ 20 mas . We kept the position of the blend fixed but allowed the other parameters ( u , PA, θ E , and η b ) to vary. The best-fit  \nparameters were found using a downhill approach using the Nelder-Nead algorithm. We adopted a prior on the lens-source separation u from the light curve model, and we required η b ≥ 0 .  \nThe grid search results for the epoch 2 data are presented in Figure 7. There is only one local minimum around (∆ α b , ∆ δ b ) = ( -6 . 0 , 15 . 8) mas that may be statistically significant (the χ 2 statistics is improved by ∆ χ 2 = 26 . 6 ). We explored this local minimum using the MCMC approach and found u = 0 . 4295 ± 0 . 0016 , PA = 26 . 00 ± 0 . 24 deg, θ E = 1 . 2672 ± 0 . 0054 mas, ∆ α b = -5 . 91 ± 0 . 18 mas, ∆ δ b = 15 . 73 ± 0 . 32 mas, and η b = 0 . 0426 ± 0 . 0094 . Thus, even if we consider this solution statistically significant, the blend does not explain the tension between the position angle measured using the light curve and interferometric data.', '5.4. Final Closure Phase Models': 'Finally, we measure the angular Einstein radius of the event θ E using all VLTI data combined. The baseline model has five parameters: the source-lens separation u and position angle of the images PA during epochs 1 and 2, and θ E (the same during both epochs). To estimate the impact of systematic errors on the parameter uncertainties, we employ the bootstrapping method (e.g. Efron 1979; Kervella et al. 2004; Lachaume et al. 2019). It involves randomly selecting eight VLTI interferograms (with replacement), feeding them to our fitting routine and then repeating the procedure multiple times (in this case, 5000 times) to obtain the multivariate probability density function for the parameters of the model. Such a procedure enables us to retrieve more realistic uncertainties on the model parameters. The results of the bootstrapping method are reported in the second column of Table 8.  \nWe then repeat the modeling, allowing an additional flux from the lens. We consider two models: with and without the prior on u from the light curve model (third and fourth columns of Table 8.) As in Section 5.2, we find that including the lens light does not significantly improve the fits, and we can only measure upper limits on η b (95% confidence).', '6. PHYSICAL PARAMETERS OF THE LENS': 'The mass and distance to the lens can be obtained from  \nM = θ E κπ E , D l = au π E θ E +au /D s , (10)  \nwhere D s ≈ 8 kpc is the source distance. As the interferometric data do not provide strong evidence for the light from the lens, we adopt θ E = 1 . 2802 ± 0 . 0085 mas (Table 8) as our final measurement. Using π E =  \nFigure 6. Constraints on the flux ratio and position angle of the microlensed images measured from the VLTI data (gray contours). The color contours mark the constrains from the individual photometric data sets (blue - OGLE, red - KMTC, orange - KMTS, green - KMTA), the black contours are measured using all photometric data combined.  \n<!-- image -->  \nTable 6. Best-fit Parameters of the Binary-star Closure Phase Model for Individual Epoch 2 DataTable 7. Best-fit Parameters of the Luminous-Lens Closure Phase Model  \nTable 8. Final Closure Phase Models  \n0 . 3330 ± 0 . 0084 , we find M = 0 . 472 ± 0 . 012 M ⊙ , π rel = 0 . 426 ± 0 . 011 mas, and D l = 1 . 81 ± 0 . 05 kpc. The lens is,  \ntherefore, most likely a main-sequence star located in the  \nFigure 7. Detection map for possible luminous blends in the VLTI epoch 2 data. The color codes the χ 2 improvement over the model without the blend. The lens is located in the origin of the coordinate system.  \n<!-- image -->  \nEpoch 2  \nnearby Galactic disk. Alternatively, it may be a white dwarf, but because white dwarfs are much less frequent than main-sequence stars, this possibility is unlikely.  \nIf the lens is a main-sequence star, its absolute magnitude is M K = 5 . 76 ± 0 . 04 (Pecaut & Mamajek 2013), and its apparent K -band magnitude is 17 . 20 ± 0 . 07 , assuming the K -band extinction of 0.15 mag toward the lens (Nataf et al. 2013). That corresponds to η b ≈ 0 . 03 in the K -band, consistent with the limits on the lens flux from the VLTI data.  \nThe trajectory of the source in the sky is presented in Figure 8 by a gray solid line. Open symbols (red square and blue circle) mark the position of the source during the two epochs of VLTI observations, whereas the filled symbols mark the positions of the images of the source.', '7. DISCUSSION AND CONCLUSIONS': "The start of the scientific operations of GRAVITY Wide opens up completely new possibilities for the follow-up of microlensing events. Only very few of the brightest events could have been observed with the standard on-axis GRAVITY mode. In contrast, GRAVITY Wide observations provide the opportunity for studies of a large sample of events and measurements of masses, distances, and transverse velocities of isolated objects, including neutron stars and black holes. Interferometric observations are very efficient at the same time-only one-hour exposure is sufficient for determining the angular Einstein radius. Thus, interferometric observations  \nFigure 8. Geometry of the event. The gray solid line marks the trajectory of the source in the sky. Open symbols (red square and blue circle) mark the position of the source during the two epochs of VLTI observations, whereas the filled symbols mark the positions of the images of the source. The dashed circle shows the Einstein radius.  \n<!-- image -->  \nwith GRAVITY Wide (and, its successor, GRAVITY+) hold promise for routine detections of isolated neutron stars and stellar remnants.  \nIn this paper, we report the first successful observation of a microlensing event with GRAVITY Wide and the resolution of microlensed images in the event OGLE2023-BLG-0061/KMT-2023-BLG-0496. These observations serve as a testbed for verifying the capabilities of the new instrument. In particular, some observations were carried out using the low-resolution mode, which had never been tried before with GRAVITY Wide. The comparison of medium- and low-resolution interferograms (epochs 1a and 1b) reveals low-level systematic differences between the inferred parameters of the lensing system (Table 5). Similar systematic differences can be seen in models fitted to individual exposures taken during epoch 2 (Table 6). Judging from Figures 4 and 5, the systematic and correlated errors in the closure phase data can reach up to 5-10 deg. We defer a detailed investigation of systematic errors in the closure phase data to a separate study.  \nWe use the bootstrapping method to deal with these systematic and correlated errors. Thanks to a large number of observations, we are still able to measure the angular Einstein radius with a sub percent precision, θ E = 1 . 2802 ± 0 . 0085 mas. By combining the informa-  \non from the light curve and the closure phase data, we measure the mass of the lens with a precision of 2.6%, M = 0 . 472 ± 0 . 012 M ⊙ .  \nIndependent tests of the accuracy of the closure phase model are possible with the light curve model. The model of the microlensing event light curve predicts the brightness ratio of the microlensed images and their position angle in the sky. While the predicted flux ratio matches that inferred from the closure phase data reasonably well (Figure 6), there is a tension in the position angles measured using the epoch 2 data. This tension cannot be explained by additional blended light in the closure phase data (Section 5.3).  \nOne possible explanation for this systematic difference is the unaccounted systematic errors in the closure phase data. Alternatively, the problem may lie in the light curve model. However, we find this unlikely, because the residuals from the best-fit light curve model are smaller than ≈ 0 . 01 mag, and there is no strong evidence that the light curve model is inadequate. A binary-lens model can be ruled out because the binary lens would produce additional images of the source, which should have been detected in our grid search. Still, the unaccounted orbital motion of the source ('xallarap effect') could, in principle, partly explain the discrepancy. The xallarap effect would not create additional images but would deflect the path of the source in the sky. Our modeling, however, do not provide strong evidence for xallarap in the light curve data (Appendix A).", 'ACKNOWLEDGEMENTS': "Based on observations collected at the European Southern Observatory under ESO program 108.220D. This research was funded in part by National Science Centre, Poland, grants OPUS 2021/41/B/ST9/00252 and SONATA 2023/51/D/ST9/00187 awarded to P.M. This research is partly supported by the National Natu-  \nral Science Foundation of China (Grant No. 12133005) and the science research grants from the China Manned Space Project with No. CMS-CSST-2021-B12. S.D. acknowledges the New Cornerstone Science Foundation through the XPLORER PRIZE.  \nGRAVITY+ is developed by the Max Planck Institute for extraterrestrial Physics, the Institute National des Sciences de l'Univers du CNRS (INSU) with its institutes LESIA / Paris Observatory-PSL, IPAG / Grenoble Observatory, Lagrange / Côte d'Azur Observatory and CRAL / Lyon Observatory, the Max Planck Institute for Astronomy, the University of Cologne, the CENTRA - Centro de Astrofisica e Gravitação, the University of Southampton, the Katholieke Universiteit Leuven and the European Southern Observatory. D.D. has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement CoG - 866070).  \nThis research has made use of the KMTNet system operated by the Korea Astronomy and Space Science Institute (KASI) at three host sites of CTIO in Chile, SAAO in South Africa, and SSO in Australia. Data transfer from the host site to KASI was supported by the Korea Research Environment Open NETwork (KREONET). This research was supported by KASI under the R&D program (project No. 2024-1-832-01) supervised by the Ministry of Science and ICT. W.Zang, H.Y., S.M., R.K., J.Z., and W.Zhu acknowledge support by the National Natural Science Foundation of China (Grant No. 12133005). W.Zang acknowledges the support from the Harvard-Smithsonian Center for Astrophysics through the CfA Fellowship. J.C.Y. and I.-G.S. acknowledge support from U.S. NSF Grant No. AST-2108414. Y.S. acknowledges support from BSF Grant No. 2020740. Work by C.H. was supported by the grants of National Research Foundation of Korea (2019R1A2C2085965 and 2020R1A4A2002885). J.C.Y. acknowledges support from a Scholarly Studies grant from the Smithsonian Institution.", 'REFERENCES': 'Albrow, M. D., Horne, K., Bramich, D. M., et al. 2009,  \nMNRAS, 397, 2099  \nArsenault, R., Alonso, J., Bonnet, H., et al. 2003, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 4839, Adaptive Optical System Technologies II, ed. P. L. Wizinowich & D. Bonaccini, 174-185  \nBachelet, E., Zieliński, P., Gromadzki, M., et al. 2022,  \nA&A, 657, A17  \nCassan, A., Ranc, C., Absil, O., et al. 2022, Nature  \nAstronomy, 6, 121 Colavita, M. M., Wallace, J. K., Hines, B. E., et al. 1999, ApJ, 510, 505 Delplancke, F. 2008, NewAR, 52, 199 Delplancke, F., Górski, K. M., & Richichi, A. 2001, A&A, 375, 701 Dong, S., Mérand, A., Delplancke-Ströbele, F., et al. 2019, ApJ, 871, 70 Efron, B. 1979, The Annals of Statistics, 7, 1', 'A. XALLARAP MODELS': "In this section, we explore whether the orbital motion of the source (xallarap effect) can explain the discrepancy between the position angle of the microlensed images measured using VLTI data and calculated from the light curve model. Compared to the standard point-source point-lens model, models including the xallarap effect have seven additional parameters describing the shape of the source's orbit. These are P ξ - the orbital period, a ξ - the semimajor axis (expressed in Einstein radius units), i ξ - the inclination of the orbit, Ω ξ - the longitude of the ascending node of the orbit, u ξ - the argument of latitude at the reference epoch (which is taken to be t 0 ,ξ = t 0 , par ), e ξ - the eccentricity of the orbit, and ω ξ - the argument of periapsis of the orbit.  \nFor simplicity, we consider only circular orbits, keeping e ξ = 0 and ω ξ = 0 . We also assume that the companion to the source is dark, given there is no strong evidence for the additional light in the VLTI data (Section 5.3). We employ the MulensModel package by Poleski & Yee (2019) to calculate the source trajectory and magnification.  \nWe search for best-fit xallarap models on a grid of orbital periods P ξ spanning from 10 to 251 days. We keep the orbital period fixed, but the remaining parameters ( a ξ , i ξ , Ω ξ , u ξ , and the standard microlensing parameters) are allowed to vary. The best-fit parameters are found using the MCMC approach (Foreman-Mackey et al. 2013). Simultaneously, we calculate the position angle of the minor image relative to the major image during the two epochs of VLTI observations (Section 4). We assume uniform priors on all parameters.  \nThe results of our calculations for the OGLE data are presented in Table 9, where we report the χ 2 statistic of the best-fit model and the predicted position angles. There is no strong evidence for xallarap in the data. The best-fit model, for the orbital period of P ξ = 79 . 4 days, is favored over the model without xallarap by only ∆ χ 2 = 11 . 6 , which is not statistically significant given the increased number of free parameters. The predicted position angles are all similar to those calculated using the standard point-source point-lens model. However, their uncertainties are typically larger, which quantifies the additional degrees of freedom on the position of the source due to orbital motion.  \nTable 9. Results of Xallarap Fits to the OGLE Data"}