An Isolated Stellar-mass Black Hole Detected through Astrometric Microlensing* Kailash C. Sahu1,70 , Jay Anderson1 , Stefano Casertano1, Howard E. Bond1,2 , Andrzej Udalski3,71 , Martin Dominik4,72 , Annalisa Calamida1 , Andrea Bellini1 , Thomas M. Brown1 , Marina Rejkuba5 , Varun Bajaj1, Noé Kains6,73 , Henry C. Ferguson1 , Chris L. Fryer7 , Philip Yock8 , Przemek Mróz3, Szymon Kozłowski3 , Paweł Pietrukowicz3 , Radek Poleski3 , Jan Skowron3 , Igor Soszyński3 , Michał K. Szymański3 , Krzysztof Ulaczyk3,9 , Łukasz Wyrzykowski3 (OGLE Collaboration), Richard K. Barry10 , David P. Bennett10,11 , Ian A. Bond12, Yuki Hirao13 , Stela Ishitani Silva10,14 , Iona Kondo13 , Naoki Koshimoto10 , Clément Ranc15 , Nicholas J. Rattenbury16 , Takahiro Sumi13 , Daisuke Suzuki13 , Paul J. Tristram17, Aikaterini Vandorou10,11 (MOA Collaboration), Jean-Philippe Beaulieu18,19, Jean-Baptiste Marquette20, Andrew Cole18 , Pascal Fouqué21 , Kym Hill18, Stefan Dieters18, Christian Coutures19, Dijana Dominis-Prester22 , Clara Bennett23, Etienne Bachelet24 , John Menzies25, Michael Albrow26 , Karen Pollard26 (PLANET Collaboration), Andrew Gould27,28, Jennifer C. Yee29 , William Allen30, Leonardo A. Almeida31,32 , Grant Christie33, John Drummond34,35, Avishay Gal-Yam36 , Evgeny Gorbikov37, Francisco Jablonski38, Chung-Uk Lee39 , Dan Maoz40, Ilan Manulis41, Jennie McCormick42, Tim Natusch33,43, Richard W. Pogge28 , Yossi Shvartzvald41 (μFUN Collaboration), Uffe G. Jørgensen44 , Khalid A. Alsubai45, Michael I. Andersen46, Valerio Bozza47,48 , Sebastiano Calchi Novati49, Martin Burgdorf50 , Tobias C. Hinse51,52 , Markus Hundertmark15,73 , Tim-Oliver Husser53, Eamonn Kerins54 , Penelope Longa-Peña55, Luigi Mancini27,56,57 , Matthew Penny58, Sohrab Rahvar59 , Davide Ricci60 , Sedighe Sajadian61 , Jesper Skottfelt62 , Colin Snodgrass63,73 , John Southworth64 , Jeremy Tregloan-Reed65, Joachim Wambsganss15 , Olivier Wertz66 (MiNDSTEp Consortium), and Yiannis Tsapras15 , Rachel A. Street24 , D. M. Bramich67,68, Keith Horne4,74 , and Iain A. Steele69 (RoboNet Collaboration) 1 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA; ksahu@stsci.edu 2 Department of Astronomy & Astrophysics, The Pennsylvania State University, University Park, PA 16802, USA 3 Astronomical Observatory, University of Warsaw, Al. Ujazdowskie 4, 00-478 Warszawa, Poland 4 University of St Andrews, Centre for Exoplanet Science, SUPA School of Physics & Astronomy, North Haugh, St Andrews, KY16 9SS, UK 5 European Southern Observatory, Karl-Schwarzschild-Straße 2, D-85748 Garching bei München, Germany 6 Department of Physics & Astronomy, Barnard College, Columbia University, 3009 Broadway, New York, NY 10027, USA 7 Center for Theoretical Astrophysics, Los Alamos National Laboratory, Los Alamos, NM 87545, USA 8 Department of Physics, University of Auckland, Auckland, New Zealand 9 Department of Physics, University of Warwick, Gibbet Hill Road, Coventry, CV4 7AL, UK 10 Code 667, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA 11 Department of Astronomy, University of Maryland, College Park, MD 20742, USA 12 Institute of Natural and Mathematical Sciences, Massey University, Auckland 0745, New Zealand 13 Department of Earth and Space Science, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan 14 Department of Physics, The Catholic University of America, Washington, DC 20064, USA 15 Zentrum für Astronomie der Universität Heidelberg, Astronomisches Rechen-Institut, Mönchhofstr. 12-14, D-69120 Heidelberg, Germany 16 Department of Physics, University of Auckland, Private Bag 92019, Auckland, New Zealand 17 University of Canterbury Mt. John Observatory, P.O. Box 56, Lake Tekapo 8770, New Zealand 18 School of Natural Sciences, University of Tasmania, Private Bag 37 Hobart, Tasmania 7001, Australia 19 Sorbonne Universite, UPMC Univ Paris 6 et CNRS, UMR 7095, Institut d’Astrophysique de Paris, 98 bis bd Arago, F-75014 Paris, France 20 Laboratoire d’astrophysique de Bordeaux, Univ. Bordeaux, CNRS, B18N, allée Geoffroy SaintHilaire, F-33615 Pessac, France 21 Université de Toulouse, UPS-OMP, IRAP, Toulouse, France 22 Faculty of Physics, University of Rijeka, HR-51000 Rijeka, Croatia 23 Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 24 Las Cumbres Observatory Global Telescope Network, 6740 Cortona Drive, Suite 102, Goleta, CA 93117, USA 25 South African Astronomical Observatory, PO Box 9, Observatory 7935, South Africa 26 University of Canterbury, Department of Physics & Astronomy, Private Bag 4800, Christchurch 8020, New Zealand 27 Max-Planck-Institute for Astronomy, Königstuhl 17, D-69117 Heidelberg, Germany 28 Department of Astronomy, Ohio State University, 140 W. 18th Avenue, Columbus, OH 43210, USA 29 Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA 30 Vintage Lane Observatory, Blenheim, New Zealand 31 Escola de Ciências e Tecnologia, Universidade Federal do Rio Grande do Norte, Natal—RN, 59072-970, Brazil The Astrophysical Journal, 933:83 (28pp), 2022 July 1 https://doi.org/10.3847/1538-4357/ac739e © 2022. The Author(s). Published by the American Astronomical Society. * This research is based in part on observations made with the NASA/ESA Hubble Space Telescope, obtained from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. 1 https://orcid.org/0000-0001-6008-1955 https://orcid.org/0000-0001-6008-1955 https://orcid.org/0000-0001-6008-1955 https://orcid.org/0000-0003-2861-3995 https://orcid.org/0000-0003-2861-3995 https://orcid.org/0000-0003-2861-3995 https://orcid.org/0000-0003-1377-7145 https://orcid.org/0000-0003-1377-7145 https://orcid.org/0000-0003-1377-7145 https://orcid.org/0000-0001-5207-5619 https://orcid.org/0000-0001-5207-5619 https://orcid.org/0000-0001-5207-5619 https://orcid.org/0000-0002-3202-0343 https://orcid.org/0000-0002-3202-0343 https://orcid.org/0000-0002-3202-0343 https://orcid.org/0000-0002-0882-7702 https://orcid.org/0000-0002-0882-7702 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https://orcid.org/0000-0001-8411-351X https://orcid.org/0000-0001-6279-0552 https://orcid.org/0000-0001-6279-0552 https://orcid.org/0000-0001-6279-0552 https://orcid.org/0000-0003-1728-0304 https://orcid.org/0000-0003-1728-0304 https://orcid.org/0000-0003-1728-0304 mailto:ksahu@stsci.edu https://doi.org/10.3847/1538-4357/ac739e https://crossmark.crossref.org/dialog/?doi=10.3847/1538-4357/ac739e&domain=pdf&date_stamp=2022-07-06 https://crossmark.crossref.org/dialog/?doi=10.3847/1538-4357/ac739e&domain=pdf&date_stamp=2022-07-06 32 Programa de Pós-Graduação em Física, Universidade do Estado do Rio Grande do Norte, Mossoró—RN, 59610-210, Brazil 33 Auckland Observatory, Auckland, New Zealand 34 Possum Observatory, Patutahi, New Zealand 35 Centre for Astrophysics, University of Southern Queensland, Toowoomba, Queensland 4350, Australia 36 Benoziyo Center for Astrophysics, Weizmann Institute of Science, 76100 Rehovot, Israel 37 School of Physics and Astronomy, Raymond and Beverley Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel Aviv 69978, Israel 38 Instituto Nacional de Pesquisas Espaciais, Astrophysics Division, Sao Jose dos Campos, Brazil 39 Korea Astronomy and Space Science Institute, Daejon 34055, Republic of Korea 40 School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv 6997801, Israel 41 Department of Particle Physics and Astrophysics, Weizmann Institute of Science, 76100 Rehovot, Israel 42 Farm Cove Observatory, Centre for Backyard Astrophysics, Pakuranga, Auckland, New Zealand 43 Institute for Radio Astronomy and Space Research (IRASR), AUT University, Auckland, New Zealand 44 Centre for ExoLife Sciences, Niels Bohr Institute, University of Copenhagen, Øster Voldgade 5, 1350 Copenhagen, Denmark 45 Qatar Environment and Energy Research Institute (QEERI), HBKU, Qatar Foundation, Doha, Qatar 46 Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen, Denmark 47 Dipartimento di Fisica “E.R. Caianiello,” Università di Salerno, Via Giovanni Paolo II 132, I-84084 Fisciano, Italy 48 Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, Napoli, Italy 49 IPAC, Mail Code 100-22, Caltech, 1200 E. California Blvd., Pasadena, CA 91125, USA 50 Universität Hamburg, Faculty of Mathematics, Informatics and Natural Sciences, Department of Earth Sciences, Meteorological Institute, Bundesstraße 55, D-20146 Hamburg, Germany 51 Institute of Astronomy, Faculty of Physics, Astronomy, and Informatics, Nicolaus Copernicus University in Toruń, ul. Grudziadzka 5, 87-100 Toruń, Poland 52 Chungnam National University, Department of Astronomy, Space Science and Geology, Daejeon, Republic of Korea 53 Institut fur Astrophysik, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, D-37077 Göttingen, Germany 54 Jodrell Bank Centre for Astrophysics, Alan Turing Building, University of Manchester, Manchester, M13 9PL, UK 55 Centro de Astronomía, Universidad de Antofagasta, Avenida Angamos 601, Antofagasta 1270300, Chile 56 Department of Physics, University of Rome “Tor Vergata,” Via della Ricerca Scientifica 1, I-00133 Roma, Italy 57 INAF—Astrophysical Observatory of Turin, Via Osservatorio 20, I-10025 Pino Torinese, Italy 58 Louisiana State University, 261-B Nicholson Hall, Tower Drive, Baton Rouge, LA 70803-4001, USA 59 Department of Physics, Sharif University of Technology, PO Box 11155-9161, Tehran, Iran 60 INAF—Padova Astronomical Observatory, Vicolo dell’Osservatorio 5, I-35122 Padova, Italy 61 Department of Physics, Isfahan University of Technology, Isfahan 84156-83111, Iran 62 Centre for Electronic Imaging, Department of Physical Sciences, The Open University, Milton Keynes, MK7 6AA, UK 63 Institute for Astronomy, University of Edinburgh, Royal Observatory, Edinburgh, EH9 3HJ, UK 64 Astrophysics Group, Keele University, Staffordshire, ST5 5BG, UK 65 Instituto de Investigación en Astronomía y Ciencias Planetarias, Universidad de Atacama, Copayapu 485, Copiapó, Atacama, Chile 66 Space Sciences, Technologies, and Astrophysics Research (STAR) Institute, University of Liège, Liège, Belgium 67 Center for Astro, Particle, and Planetary Physics, New York University Abu Dhabi, P.O. Box 129188, Saadiyat Island, Abu Dhabi, UAE 68 Division of Engineering, New York University Abu Dhabi, P.O. Box 129188, Saadiyat Island, Abu Dhabi, UAE 69 Astrophysics Research Institute, Liverpool John Moores University, Liverpool, L3 5RF, UK Received 2022 January 28; revised 2022 May 19; accepted 2022 May 24; published 2022 July 6 Abstract We report the first unambiguous detection and mass measurement of an isolated stellar-mass black hole (BH). We used the Hubble Space Telescope (HST) to carry out precise astrometry of the source star of the long-duration (tE; 270 days), high-magnification microlensing event MOA-2011-BLG-191/OGLE-2011-BLG-0462 (hereafter designated as MOA-11-191/OGLE-11-462), in the direction of the Galactic bulge. HST imaging, conducted at eight epochs over an interval of 6 yr, reveals a clear relativistic astrometric deflection of the background star’s apparent position. Ground-based photometry of MOA-11-191/OGLE-11-462 shows a parallactic signature of the effect of Earth’s motion on the microlensing light curve. Combining the HST astrometry with the ground-based light curve and the derived parallax, we obtain a lens mass of 7.1± 1.3Me and a distance of 1.58± 0.18 kpc. We show that the lens emits no detectable light, which, along with having a mass higher than is possible for a white dwarf or neutron star, confirms its BH nature. Our analysis also provides an absolute proper motion for the BH. The proper motion is offset from the mean motion of Galactic disk stars at similar distances by an amount corresponding to a transverse space velocity of∼45 km s−1, suggesting that the BH received a “natal kick” from its supernova explosion. Previous mass determinations for stellar-mass BHs have come from radial velocity measurements of Galactic X-ray binaries and from gravitational radiation emitted by merging BHs in binary systems in external galaxies. Our mass measurement is the first for an isolated stellar-mass BH using any technique. 70 PLANET Collaboration, MiNDSTEp Consortium. 71 OGLE Collaboration. 72 MiNDSTEp Consortium, RoboNet Collaboration. 73 RoboNet Collaboration. 74 PLANET Collaboration. Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 2 The Astrophysical Journal, 933:83 (28pp), 2022 July 1 Sahu et al. http://creativecommons.org/licenses/by/4.0/ Unified Astronomy Thesaurus concepts: Black holes (162); Gravitational microlensing (672) Supporting material: data behind figures 1. Measuring the Masses of Black Holes 1.1. Black Holes in Binary Systems Stars with initial masses greater than∼20Me are expected to end their lives as black holes (BHs; e.g., Fryer & Kalogera 2001; Woosley et al. 2002; Heger et al. 2003; Spera et al. 2015; Sukhbold et al. 2016). Objects of these masses constitute roughly 0.1% of all stars, leading to the expectation that the Galaxy should now contain of the order of≈108 BHs (Shapiro & Teukolsky 1983; van den Heuvel 1992; Brown & Bethe 1994; Samland 1998). However, the actual detection of stellar-mass BHs is observationally challenging, and determining their masses is even more so. BHs have been identified in the Galaxy and Local Group through X-ray emission due to accretion in short-period binary systems, most of them soft X-ray transients. In such cases, dynamical masses of the BHs can be measured or estimated through radial velocity measurements and light-curve modeling for the optical companion stars (the techniques are reviewed by Remillard & McClintock 2006; Casares & Jonker 2014). Masses of nearly two dozen BHs in X-ray binary systems have been determined using these methods, with varying degrees of precision. These “electromagnetically measured” BH masses show a distribution peaking near 7–8Me, with few if any below∼5Me (e.g., Özel et al. 2010; Farr et al. 2011; Kreidberg et al. 2012; Corral-Santana et al. 2016). This suggests that a “mass gap” exists between the lowest-mass BHs and the highest measured masses of neutron stars (NSs) in binary radio pulsars of∼2.1–2.3Me (Linares et al. 2018; Cromartie et al. 2020). Recently, however, a few nonaccreting or weakly accreting BHs have been discovered in longer-period spectroscopic binaries in the field (e.g., Thompson et al. 2019; Jayasinghe et al. 2021) and in globular clusters (Giesers et al. 2019), lying in the NS–BH gap with dynamical masses of ∼3–4.5Me. Precision astrometry of nearby stars by Gaia shows the promise of detecting additional wide-binary systems containing quiescent BHs (e.g., Chawla et al. 2021; Janssens et al. 2022, and references therein) and measuring their masses. At the high-mass end, the BH mass distribution falls off above∼10Me, and very few electromagnetic BH masses are known above∼15Me, the only exception in the Milky Way being an updated mass determination of 21.1± 2.2Me for the BH in Cygnus X-1 (Miller-Jones et al. 2021). Among extragalactic X-ray binaries, BH masses as high as 15.65± 1.45 Me and 17± 4Me have been reported for M33 X-7 (Orosz et al. 2007) and NGC 300 X-1 (Binder et al. 2021), respectively. An even higher mass of at least 23.1Me was reported for the compact object in IC 10 X-1 (Silverman & Filippenko 2008), but this has been questioned (Laycock et al. 2015). The first detections of gravitational waves (GWs) by the Laser Interferometry Gravitational Wave Observatory (LIGO) and Virgo Collaboration (Abbott et al. 2016) revealed a population of massive merging binary BHs, BH–NS pairs, and binary NSs at extragalactic distances. In the source catalogs from the third observing run of the Advanced LIGO and Advanced Virgo Collaboration (Abbott et al. 2021a, 2021b), the inferred masses of the BHs among the premerger systems range from≈6 to 95Me, with two low-mass outliers among the secondary components at ∼2.6 and 2.8Me, which could be either BHs or NSs. 1.2. Isolated Black Holes The electromagnetic and GW mass measurements described above are all for BHs in binary systems, including those undergoing mass accretion or mergers. However, there are reasons to believe that a substantial fraction of stellar-mass BHs are single, rather than belonging to binaries. First, about 30% of massive stars are born single (see Sana et al. 2012; de Mink et al. 2014). Second, in a close-binary system, the pair may enter into a common envelope and merge before the supernova (SN) explosion (e.g., Fryer et al. 1999; Zhang & Fryer 2001; Tutukov et al. 2011; Dominik et al. 2012). Lastly, in a wide binary, the “natal kick” imparted to the companion by the SN event may be large enough to detach the two components, producing an isolated BH (e.g., Tauris & van den Heuvel 2006; Belczynski et al. 2016). Being less altered by interactions with companions, single BHs potentially provide a more direct probe of BH formation than those in binaries. Isolated BHs are extremely difficult to detect directly. They emit no light of their own, and the accretion rate from the interstellar medium is generally likely to be too low to produce detectable X-ray or radio emission (see, however, Agol & Kamionkowski 2002; Fender et al. 2013; Tsuna & Kawanaka 2019; Scarcella et al. 2021, for the case of isolated BHs in dense environments). In fact, until now, no isolated stellar-mass BH has ever been unambiguously found within our Galaxy or elsewhere. Microlensing is the only available method for measuring the masses of isolated BHs. Astrometric microlensing—the relativistic deflection of the apparent position of a background star when a compact object passes in front of it—provides a direct method for measuring the masses of BH lenses. High spatial resolution interferometric observations of microlensing events (Dong et al. 2019; Zang et al. 2020) and observations of rare events where the lens passes over the surface of the source (Yoo et al. 2004) can also yield the masses of BH lenses. In this paper, we describe how the technique of astrometric microlen- sing is used to determine masses. We discuss our ongoing program of astrometric measurements of microlensing events with the Hubble Space Telescope (HST). Then, we report the first detection of an isolated BH and our measurement of its mass. 2. Measuring the Masses of Isolated Black Holes with Astrometric Microlensing 2.1. Microlensing Events and Black Hole Candidates A microlensing event occurs when a star or compact object (the lens) passes almost exactly in front of a background star (the source). As predicted by general relativity (Einstein 1936), the lens magnifies the image of the source, producing an apparent amplification of its brightness. The lens also slightly shifts the apparent position of the source (Miyamoto & Yoshii 1995; Høg et al. 1995; Walker 1995)—an analog of the deflection of stellar images during the 1919 solar eclipse 3 The Astrophysical Journal, 933:83 (28pp), 2022 July 1 Sahu et al. http://astrothesaurus.org/uat/162 http://astrothesaurus.org/uat/672 (Dyson et al. 1920), which provided support for the general theory of relativity. Microlensing survey programs, including OGLE (Udalski et al. 2015), MOA (Bond et al. 2001), and KMTNet (Kim et al. 2016), carry out photometric monitoring of rich stellar fields in the Galactic bulge. These surveys typically detect >2000 events toward the Galactic bulge annually. To date, more than 30,000 microlensing events have been discovered and monitored by these survey programs. The characteristic scale of gravitational microlensing is provided by the angular Einstein radius θE, given by ( )q p º GM c 4 1 au , 1E lens 2 LS where Mlens is the mass of the lens and ⎜ ⎟ ⎛ ⎝ ⎞ ⎠ ( ) ( )p p pº - = - D D 1 au 1 1 2LS L S L S is the relative lens–source parallax, with DL and DS being the distances from the observer to the lens and to the source, respectively. The Einstein radius θE, however, cannot be obtained directly from the magnification light curve, whose only characteristic that carries a physical dimension is the timescale. Given the relative proper motion μLS between lens and source, we can straightforwardly define tE , the time for the source to traverse an angular distance of θE in the barycentric reference frame, as  q m=tE E LS (more details in Section 2.4). The distribution of the timescale tE of the observed events peaks around 25 days, with tE ranging from a fraction of a day to several hundred days (e.g., Wyrzykowski et al. 2015). If BHs constitute a small but nonnegligible fraction of the total stellar mass of the Galaxy, as described above, then a few of the observed microlensing events are expected to be due to BHs. Equation (1) shows that the angular Einstein radius θE is proportional to the square root of the lens mass. Hence, all else being equal, events due to massive compact objects would preferentially tend to be characterized by longer event durations (tE 150 days), combined with an apparent lack of light contribution from the lens. However, a degeneracy between lens mass and proper motion remains. Thus, a long-duration event with no light contribution from the lens could arise from a high-mass, nonluminous BH lens with a large Einstein radius —but it could alternatively be due simply to an unusually slow- moving, faint, low-mass ordinary star. If the lens were a luminous massive star, this would generally be recognizable through the contribution of its light, particularly when observations are available in two different bandpasses. Indeed, several long-duration OGLE and MOA microlensing events have been suggested as being due to BHs (e.g., Bennett et al. 2002; Mao et al. 2002; Minniti et al. 2015; Wyrzykowski & Mandel 2020). However, these claims remain statistical in nature, being derived from assumptions about the transverse velocity distributions. The degeneracy between mass and relative velocity can be lifted if precise astrometry is added to the photometry of the microlensing event. The size of the expected astrometric shift is small—of the order of milliarcseconds—but it is proportional to the angular Einstein radius θE, and therefore if this small shift can be measured, then the mass of the lens can be determined unambiguously, as described in detail below. 2.2. Photometric Microlensing Photometric microlensing is the apparent transient bright- ening that results as a background source passes almost directly behind a foreground lens (see reviews by Paczyński 1996; Gaudi 2012, and Tsapras 2018). With θS and θL denoting the angular positions of the source and lens as seen by the observer, we can define a dimensionless source–lens separation ( )q q q º - u . 3S L E As the lens intervenes near the line of sight from the observer to the source, the gravitational bending of light leads to a time-varying magnification ( ) ( )= + + A u u u u 2 4 , 4 2 2 which depends solely on u= |u|. This expression holds as long as the finite angular size of the source star can be neglected, which we will assume in the following discussion. We will demonstrate in Section 9.3 that this is a valid approximation for the case analyzed in this paper. If FS is the intrinsic source flux and FB the background flux contributed by any other objects not resolved from the observed source star, the observed flux of the target for a specific telescope and filter is given by ( ) [ ( )] [ ( )] ( )= + = + + F t F A u t F F A u t g g1 , 5S B base where Fbase≡ FS+ FB is the baseline flux and g≡ FB/FS is the specific blend ratio. 2.3. Astrometric Microlensing Microlensing also produces an astrometric shift of the apparent position of the source. If we assume that we can observe the centroid of light formed by the images of the source without any contribution from other bodies such as the lens or other neighboring stars, its shift is described by the vector ( ) ( )d q= + u u u 2 6 2 E (Gould 1992; Paczyński 1998). In contrast to the photometric microlensing signature, the astrometric signature is explicitly proportional to the angular Einstein radius θE. Moreover, while the magnification diverges (for a pointlike source) as u→ 0, the astrometric shift becomes maximal for =u 2 . The light magnification falls rapidly with increasing u: for large separations, u? 1, the brightness enhancement, A(u)− 1, falls as 1/u4. On the other hand, the centroid shift, given by Equation (6), decreases more slowly with u, and for large separations it falls only as 1/u. The astrometric perturbation thus has a considerably longer duration than the photometric signal. For more details, see Dominik & Sahu (2000), Sahu et al. (2014), and Bramich (2018). The photometric and astrometric signatures of a microlen- sing event are connected because they arise from the same source–lens trajectory, u(t). Specifically, if a fit to the light curve of a microlensing event already yields u as a function of time, the astrometric data then provide a direct measurement of 4 The Astrophysical Journal, 933:83 (28pp), 2022 July 1 Sahu et al. the angular Einstein radius θE, as well as the orientation angle of the trajectory. 2.4. Parallax Effect and Proper Motion For constructing the source–lens trajectory u(t), we have to consider the proper motions μS and μL of the source and lens objects, as well as their parallaxes, πS and πL. Let γ(t) (1 au) denote the projection of Earth’s orbit onto a plane perpendicular to the direction toward the source star. The apparent geocentric positions of source and lens star are then given by (see An et al. 2002; Gould 2004) ( ) ( ) ( ) ( ) ( ) ( ) ( ) q q m g q q m g p p = + - - = + - - t t t t t t t t , , 7 S S,0 0 S S L L,0 0 L L so that for θ(t)≡ θS(t)− θL(t) one finds ( ) ( ) ( ) ( ) ( )q q q m gp= - - - +t t t t , 8S L 0 0 LS LS where μLS≡μL−μS and πLS≡ πL− πS are the relative proper motion and relative parallax between lens and source, while ( )q q q q- º -S L 0 S,0 L,0. Consequently, with the microlensing parallax parameter πE= πLS/θE, u(t)≡ θ(t)/θE takes the form (see Dominik et al. 2019) ( ) ( ) ( ) ( )gp d= + - +u u ut t t t , 90 0 0 E where ( ) ( ) ( ) ( )q q g q pº = - +u u t t , 100 0 S L 0 E E 0 ( ) ( ) ( ) m g q pº = - +  u u t t , 110 0 LS E E 0 as well as ( ) ( ) ( ) ( ) ( ) ( )g g g gd = - - - t t t t t t . 120 0 0 By construction, δγ(t0)= 0 and ( )gd = t 00 . By choosing t0 so that ^ u u0 0, we can write u(t) in its components toward northern and eastern directions as ( ) ( ) ( ) ( ) ( ) y y p dg y y p dg = - - + = - + + u t t t t u t u t t t t u t cos sin , sin cos , 13 n n e e 0 E 0 E 0 E 0 E where u0≡ |u0|, ∣ ∣= ut 1E 0 , and ψ denotes the direction angle of u0 measured from north toward east. Alternatively, the trajectory can be parameterized as ( ) ( ) ( ) ( ) ( )           y y p g y y p g = - - + = - + + u t t t t u t u t t t t u t cos sin , sin cos , 14 n n e e 0 E 0 E 0 E 0 E where ∣ ( ) ( )∣  q q qº -u t t0 S 0 L 0 E, ∣ ∣ mq=tE E LS , and ψå denotes the direction angle of−μLS=μS−μL measured from north toward east. Note that the starred quantities, ψå, tE , t0 , and u0 refer to parameters in the barycentric reference frame, whereas the corresponding unstarred quantities refer to parameters as seen by an observer on Earth. This applies to any orthonormal reference frame; the only difference is in the specific angle, e.g., ψeq, ψecl, and ψgal (and yeq, yecl, and ygal) for equatorial, ecliptic, and galactic coordinates, respectively, which are related by a rotation of the coordinate axes at the target position. While most discussions of photometric microlensing events choose an ecliptic coordinate frame, the observed astrometric data are more easily described in an equatorial coordinate frame, and therefore we adopt the latter in the following analysis. In this frame, ( )j y= + 180 15LS eq gives the position angle (PA) of the proper motion of the lens with respect to the source μLS, measured from equatorial north toward east. We illustrate the geometry of the source–lens trajectory u(t) in Figure 1. 2.5. Measuring the Lens Mass As stated above, the only useful physical parameter in a typical microlensing event is the timescale  q m=tE E LS, which is the time it takes the source to traverse the radius of the Einstein ring, which itself depends on the lens mass, Mlens, and the lens–source parallax, πLS. However, for long-duration events, the annual parallax tends to lead to prominent departures in the photometric signature (Gould 1992; Alcock et al. 1995), so that a microlensing parallax parameter πE≡ πLS/θE can be inferred. Coinciden- tally, the BH mass lenses tend to imply such long-duration events. On the other hand, the astrometric signature is proportional to the angular Einstein radius θE, so that by combining photometric and astrometric observations, Mlens, πLS, and μLS become fully decoupled. Specifically, with πE from the photometry and θE from the astrometry, the definition of θE, Equation (1), immediately gives us ( ) ( )q p q kp = =M c G 1 au 4 , 16lens E E 2 E E where [ ( )]k = - G c M4 1 au 8.144 mas2 1. In the case of microlensing toward the Galactic bulge, the source often lies at the distance of the bulge itself, which can be verified from its baseline position in a color–magnitude diagram (CMD). If spectroscopic observations are available in addition to baseline photometry—as is the case for the event discussed in this paper—a more accurate source distance can be determined. The lens–source relative parallax, πLS= πE θE, can then be used to estimate the distance to the lens, using Equation (2). As a bonus, the event timescale gives a direct Figure 1. Source–lens trajectory u(t) as seen by the observer, showing the effect of annual parallax. At epoch t0, the tangent to the source–lens trajectory u0 is not (anti)parallel to the direction of the lens–source proper motion m̂LS but differs by ( )gp  tE 0 (Equation (11)), related to the orbital velocity of Earth at t0. Consequently, we distinguish the direction angles ψ, ψå, and jLS, referring to u0 and m̂ LS, respectively. Furthermore, ∣ ∣= ut 1E 0 and ∣ ∣ mq=tE E LS . 5 The Astrophysical Journal, 933:83 (28pp), 2022 July 1 Sahu et al. measure of the relative transverse velocity of lens and source (which, for stellar remnants, might include “kicks” received in SN explosions). This method thus provides independent measurements of three separate physical parameters of the lens: its mass, distance, and transverse velocity. 2.6. Characteristics of Astrometric Deflections Some features of astrometric deflections under various scenarios are described by Dominik & Sahu (2000). To illustrate a typical case, we show in Figure 2 the calculated astrometric shifts and light magnification for a nominal event of a BH lens of mass 5Me, at a distance of 2 kpc from the Sun, passing in front of a background source situated in the Galactic bulge at a distance of 8 kpc. The closest angular approach is assumed to be at a separation of 0.05θE. In this case, the size of the angular Einstein ring is θE; 4 mas, so that the maximum astrometric shift is ∼1.4 mas (occurring at a separation of =u 2 ) and the maximum light magnification is a factor of ∼20 (at closest angular approach). As Figure 2 illustrates, and as discussed in Section 2.3, the duration of the astrometric deflection is considerably longer than that of the photometric magnification. This makes it necessary to carry out the astrometric measurements over a longer time interval than the photometry. Although the deflection measured at any given epoch provides in principle an estimate of θE, it is necessary to observe at multiple epochs in order to separate the shifts caused by microlensing from those caused by the proper motion of the source; observations at a late epoch are particularly useful for this purpose. The figure also shows that the astrometric shift is close to zero at the time of highest magnification; therefore, observations near the photometric peak are also very useful to constrain the source proper motion. In Figure 3, we plot the maximum astrometric shifts for a source in the Galactic bulge at 8 kpc, as functions of lens mass. The lenses are assumed to be located at distances of 2 and 4 kpc (“disk” lenses), and 6 kpc (“bulge” lenses). The dotted line at the bottom shows the nominal astrometric precision of 0.2 mas achievable with high signal-to-noise ratio (S/N) HST imaging, as discussed below. Therefore, the deflection is detectable at 1σ per epoch for lens masses down to∼0.5Me, except at lens distances larger than 6 kpc. The most favorable situation for a precise mass measurement, of course, would be for a nearby, high-mass lens. 2.7. High-precision Astrometry Although an unambiguous determination of lens mass is possible from a combination of photometry and astrometry, as we have just discussed, the expected astrometric shifts are extremely small, of the order of milliarcseconds or less. HST has demonstrated its capability to carry out submilliarcsecond astrometry through a variety of techniques. For example, high- S/N HST observations of isolated sources were used to achieve submilliarcsecond accuracy, leading to the measurement of proper motions for several distant hypervelocity stars (Brown et al. 2015). A collection of sources was used as probes to achieve an astrometric accuracy of ∼12 μas, in order to measure the transverse velocity of M31 (Sohn et al. 2012). Spatial-scan techniques have been used to achieve an astrometric accuracy of ∼30 μas (Casertano et al. 2016; Riess et al. 2018) in the trigonometric parallax of Cepheids used for accurate determination of H0 and to measure the distance to the globular cluster NGC 6397 (Brown et al. 2018). Recently, our group used the astrometric microlensing technique to measure the mass of the nearby white dwarf Stein 2051 B, achieving an Figure 2. Astrometric shift (top panel) and light magnification (bottom panel) for a microlensing event produced by a 5 Me BH at a distance of 2 kpc passing in front of a background star at 8 kpc. The assumed minimum impact parameter is u0 = 0.05. The maximum astrometric shift of the source is ∼1.4 mas, at =u 2 , and the maximum magnification is ∼20, at the time of closest angular approach. Note the much longer duration for the astrometric shift, compared to that of the light magnification. Figure 3. Maximum astrometric shift of a source in the Galactic bulge at 8 kpc as a function of lens mass, for lens distances of 2, 4, and 6 kpc. Since the maximum astrometric shift occurs at a lens–source separation of =u 2 , while the maximum light magnification occurs at the minimum value of u, all high-magnification microlensing events will pass through the point of maximum astrometric deflection. High-S/N imaging with HST allows measurements of astrometric shifts with a precision of ∼0.2 mas per observation epoch, shown by the dotted line at the bottom. 6 The Astrophysical Journal, 933:83 (28pp), 2022 July 1 Sahu et al. astrometric precision of ∼0.2 mas per epoch (Sahu et al. 2017). Kains et al. (2017) looked for astrometric deflections in HST observations of 10 microlensing events with timescales of <50 days. They achieved an astrometric precision of 0.2 mas per epoch (but did not detect any deflections). From the ground, Zurlo et al. (2018) used the Very Large Telescope (VLT) to measure the mass of Proxima Centauri through astrometric microlensing. Lu et al. (2016) employed the Keck telescope to look specifically for isolated BHs by monitoring three microlensing events, where they achieved a final positional error of 0.26–0.68 mas. The timescales of those events were 60–160 days, and there were no detections of astrometric deflections. 3. In Search of Isolated Black Holes with HST 3.1. Astrometry of Long-duration Microlensing Events In 2009, we began a multicycle HST program of astrometry of long-duration microlensing events in the direction of the Galactic bulge in order to detect isolated BHs and measure their masses. Our aim is to select events having timescales200 days, light curves showing no evidence for a light contribution by a luminous lens, and preferably a high magnification factor. We then obtain high-resolution HST imaging as the events proceed, in order to measure the astrometric deflections of the background sources. To date we have monitored eight long-duration events. For some of them, there is no clear detection of an astrometric signal, but our data analysis is still in progress, and the results will be discussed in separate publications. In the present paper we analyze and discuss our findings for an event that clearly shows a large astrometric deflection, consistent with a high- mass lens. 3.2. MOA-2011-BLG-191/OGLE-2011-BLG-0462 MOA-2011-BLG-191/OGLE-2011-BLG-0462 (hereafter designated MOA-11-191/OGLE-11-462) was a long-duration and high-magnification microlensing event in the direction of the Galactic bulge. It was discovered independently by both MOA and OGLE ground-based microlensing survey programs and announced by both teams nearly simultaneously on 2011 June 2, through their public-alert websites.75 The target was also covered by the Wise Microlensing Survey. Table 1 gives details of this remarkable event. MOA-11-191/OGLE-11-462 occurred in an extremely crowded Galactic bulge field, less than 2° from the Galactic center. The observed peak magnification factor of this event was only about 20 in the ground-based data, but this was strongly diluted by blending with neighboring stars. It soon became apparent, based on findings disseminated through internal communications in the microlensing groups, that the undiluted event actually had an extremely high magnification factor, approaching 400. Blending also made the apparent timescale of the event appear shorter than the actual value, which was inferred to be longer than 200 days. It was clear from the ground-based observations that there was blending, for two reasons. First, the light curve for a typical event has a characteristic shape that is completely determined by the timescale and the maximum magnification, except for distor- tions due, e.g., to the lens–source relative parallax. The shape of the observed light curve was inconsistent with the expected shape unless the light at baseline was highly diluted by a blend, thus implying that the real magnification was much larger than the observed value. Second, as the source brightened, its centroid position in the ground-based images was seen to change, again consistent with blending with a neighboring star. Table 1 Basic Data for MOA-11-191/OGLE-11-462 Microlensing Event Parameter Value Sources and Notesa Event designation (MOA) MOA-2011-BLG-191 (1) Event designation (OGLE) OGLE-2011- BLG-0462 (1) J2000 R.A., α 17:51:40.2082 (2) J2000 decl., δ −29:53:26.502 (2) Galactic coordinates, (l, b) 359°. 86, −1°. 62 (2) Baseline F606W magnitude 21.943 ± 0.014 (3) Baseline F814W magnitude 19.578 ± 0.012 (3) Baseline (F606W − F814W) color 2.365 ± 0.026 (3) Peak magnification, Amax 369 (4) Date of peak magnification, t0 2011 July 20.825 (4) Timescale, tE 270.7 ± 11.2 days (4) Note. a Sources and notes: (1) MOA and OGLE websites; the event was first alerted by MOA. (2) This paper, from astrometric analysis in Section 5.2 in Gaia EDR3 frame at average epoch 2013.5. (3) This paper, Vegamag scale, from photometric analysis in Section 5.3. (4) This paper, from Table 6. Figure 4. HST image in the F814W (I-band) filter of an ¢¢ ´ ¢¢8 8 region centered on MOA-11-191/OGLE-11-462, obtained at our final epoch in 2017 August. North is at the top, east on the left. Encircled in green is the source star, now returned to baseline luminosity. The site is resolved into the source, a much brighter neighboring star 0 4 to the W–NW, and several nearby fainter stars. The inner cyan circle has a diameter of ¢¢1 , corresponding to the typical best seeing in ground-based microlensing survey images; the outer cyan circle’s diameter is ¢¢2 , which is not unusual seeing. The source, bright neighbor, and several fainter stars are generally blended in ground-based frames, and the blending increases with seeing. 75 MOA alerts: https://www.massey.ac.nz/~iabond/moa/alerts. OGLE alerts: http://ogle.astrouw.edu.pl/ogle4/ews/2011/ews.html. 7 The Astrophysical Journal, 933:83 (28pp), 2022 July 1 Sahu et al. https://www.massey.ac.nz/~iabond/moa/alerts http://ogle.astrouw.edu.pl/ogle4/ews/2011/ews.html Note that this shift is due simply to blending and scales with the separation of the two stars; it is unrelated to the much smaller relativistic deflection of the source itself, which is discussed below. Figure 4 shows an ¢¢ ´ ¢¢8 8 region centered on the source, as imaged by us in the F814W (I-band) filter by HST with its Wide Field Camera 3 (WFC3). The source star is encircled in green. A conspicuous neighbor, nearly 20 times brighter than the unmagnified source, lies at a separation of only 0 4. The cyan circles in the figure have diameters of ¢¢1 and ¢¢2 , corresponding to the generally best seeing in the ground-based survey observations and more typical seeing, respectively. Thus, in ground-based images, the source is indeed blended with the bright neighbor and a number of fainter stars, depending on the seeing. High-magnification events are generally very sensitive to perturbations due to planets around the lensing objects (Mao & Paczyński 1991; Griest & Safizadeh 1998). Thus, considerable interest was aroused by MOA-11-191/OGLE-11-462 among groups engaged in searches for such planets. As a result, intensive photometric monitoring of this event was carried out by multiple groups, providing valuable data for our analysis. 4. HST Observations The MOA-11-191/OGLE-11-462 event satisfies all the selection criteria for our HST follow-up program described in Section 3.1, and thus we triggered our observing sequence. Our project had a “nondisruptive” target-of-opportunity status, requir- ing a lead time of about 2–3 weeks from activation to the first observations. The first-epoch HST data were obtained on 2011 August 8, some 19 days after the peak light magnification on 2011 July 20. The magnification was still reasonably high (∼12, corresponding to u; 0.08), so that the expected astrometric deflection was δ; 0.04 θE (see Equation (6)), i.e., close to zero at this epoch, but its correct value is taken into account in the model described in Section 8. Subsequent HST observations indicated departure from a linear proper motion for the source. Thus, we continued the imaging, ultimately over an interval of over 6 yr, long enough for robust separation of the relativistic deflection from proper motion. Table 2 gives the HST observing log. All our HST observations were obtained with the UVIS channel of WFC3, whose CCD detectors provide a plate scale of 39.6 mas pixel−1. To avoid buffer dumps during the orbital visibility period and thus maximize observing efficiency, we used the UVIS2-2K2C-SUB subarray, giving a field of view (FOV) of ¢¢ ´ ¢¢80 80 . This FOV is large enough to provide dozens of nearby astrometric reference stars surrounding the primary target. The WFC3 detectors are subject to an increasing amount of degradation of their charge transfer efficiency (CTE) as they are exposed to the space environment. The chosen subarray aperture places the target in the middle of the left half of the UVIS2 CCD, which lessens the impact of imperfect CTE relative to a placement closer to the center of the FOV. Nevertheless, a time-dependent correction for CTE must still be applied in the astrometric analysis of the images. Our HST observations were taken at a total of eight epochs, strategically scheduled for measurement and characterization of the astrometric deflections. At each epoch, we obtained images in two filters (to verify the achromatic nature of the event, and to test for blending by very close companions): “V” (F606W) and “I” (F814W). At the initial epoch, when the source was bright, we obtained nine exposures, four in F606W and five in F814W. At each subsequent epoch, using longer integration times because of the fading of the source, we obtained seven exposures, three in F606W and four in F814W. Individual exposure times were adjusted to take into account the brightness of the source and the orbital visibility of HST and ranged from a minimum of 60 s at the first epoch to a maximum of 285 s at the later epochs. The telescope pointing was dithered by ∼200 pixels (~ ¢¢8 ) between individual exposures; this allowed retention of a common set of reference stars in all the exposures, in order to mitigate errors in the distortion solution. To maximize the S/N for the most crucial astrometric measurements, we separated Epochs 3 and 4 by only 16 days in 2012 September, around the time when the deflection was expected to be near maximum. Figure 5 zooms in on the field around the source in Figure 4, showing a 2 1× 2 0 region as observed at all eight epochs. The bright source is marked with an arrow in the Epoch 1 (top left) image, and it can be seen to fade in the subsequent frames. The astrometric deflection was highest at Epochs 3 and 4, even though the photometric magnification was only about 10% at this epoch. There was very little photometric change in the subsequent epochs, but the astrometric deflections remained detectable until Epoch 7, demonstrating the need for astrometric monitoring over a much longer duration than the photometric variability period. 5. HST Data Analysis 5.1. Image Processing We used the flat-fielded and CTE-corrected (_flc) images produced by the Space Telescope Science Institute pipeline Table 2 Journal of HST Wide Field Camera 3 Observations Epoch Date MJD Year Proposal No. Frames No. Frames ID in F606Wa in F814Wa 1 2011 Aug 8 55,781.7 2011.600 GO-12322 4 5 2 2011 Oct 31 55,865.2 2011.829 GO-12670 3 4 3 2012 Sep 9 56,179.2 2012.689 GO-12670 3 4 4 2012 Sep 25 56,195.3 2012.733 GO-12986 3 4 5 2013 May 13 56,425.8 2013.364 GO-12986 3 4 6 2013 Oct 22 56,587.2 2013.806 GO-13458 3 4 7 2014 Oct 26 56,956.1 2014.816 GO-13458 3 4 8 2017 Aug 29 57,994.7 2017.660 GO-14783 3 4 Note. a Individual exposure times ranged from a minimum of 60 s at Epoch 1 to a maximum of 285 s at later epochs. 8 The Astrophysical Journal, 933:83 (28pp), 2022 July 1 Sahu et al. reductions (Sahu et al. 2021; Dressel 2021) for the analysis. As noted above, WFC3 suffered from increasingly poor CTE during this period, so it was essential to take it into account. The _flc products were produced using the v2.0 pixel-based CTE model described by Anderson (2021). 5.2. Astrometric Analysis To measure stellar positions in individual frames, we used an updated version of the star-measuring algorithm described in Anderson & King (2006). The routine goes through each exposure pixel by pixel and identifies as a potential star any local maximum that is sufficiently bright and isolated. The routine uses the spatially variable effective point-spread functions (PSFs) provided at the WFC3/UVIS website76 to fit the PSF to the star images in the individual _flc exposures, in order to determine a position and flux for each star in the raw pixel frame of that exposure. Finally, the positions are corrected for geometric distortion using the distortion solutions provided by Bellini et al. (2011). As the positions of individual stars are expected to change during the ∼6 yr course of our observations owing to their proper motions, we needed to determine their proper motions to properly specify the reference frame. For this, we began with the Gaia Early Data Release 3 (EDR3; Gaia Collaboration et al. 2021) positions and motions for the bright but unsaturated HST stars in the field. The reference frame was constructed to place the bright star close to MOA-11-191/OGLE-11-462 at the center of the reference frame at (x, y)= (1000, 1000) at the 2016.0 epoch, with a plate scale of 40 mas pixel−1 and north up. (Note that the Gaia catalog could be incomplete in this region because of the high source density.) Using the Gaia positions and motions, we determined the position for each Gaia star in this frame at each epoch in order to properly transform the distortion-corrected observations at that epoch into the reference frame. This ensures that the proper motions that we derive represent absolute proper motions. After this initial setup of the reference frame based on the brighter stars, we incorporated high-precision HST stars that were too faint to be found with high precision in the Gaia catalog and solved for their accurate positions and motions. We then used their time- dependent positions to improve the reference frame. Even after allowing individual solutions to improve on the basis of HST observations, there remains very good agreement between our proper motions and those of Gaia. Figure 6 plots the proper motions of our reference stars derived from our HST observations against the Gaia proper motions, where the red points are for brighter stars with G< 18 and blue points are for fainter stars with G� 18. The agreement is imperfect, of course, since the HST observations have a 6 yr baseline and have higher S/N in individual measurements, resulting in higher accuracy in proper-motion measurements, particularly at fainter magnitudes. The agreement is better for the brighter sample, for which Gaia proper-motion errors are typically Figure 5. 2 1 × 2 0 cutouts around MOA-11-191/OGLE-11-462 as observed by HST at all eight epochs. Exposures were taken in F606W and F814W from 2011 to 2017; see Table 2 for details. The source star is marked by an arrow in the first-epoch (top left) image. At this epoch, on 2011 August 8, the magnification was a factor of ∼12. The maximum astrometric deflection occurred at Epochs 3 and 4, when the photometric magnification was only about 10%. In 2017, the source had returned very close to its unmagnified brightness and undeflected position. Figure 6. Proper motions of the reference stars used in our analysis derived from our HST observations with a 6 yr baseline, versus the Gaia proper motions. Red points represent stars brighter than G = 18, for which the Gaia errors are typically <0.2 mas yr−1, and blue points represent fainter stars for which the errors are larger. Our analysis uses the Gaia reference frame, so it is natural that there is good agreement between them. But the individual HST measurements have higher precision, particularly at fainter magnitudes. 76 https://www.stsci.edu/hst/instrumentation/wfc3/data-analysis/psf 9 The Astrophysical Journal, 933:83 (28pp), 2022 July 1 Sahu et al. https://www.stsci.edu/hst/instrumentation/wfc3/data-analysis/psf smaller (<0.2 mas yr−1). Note that the MOA-11-191/OGLE- 11-462 source itself is too faint at baseline for inclusion in the Gaia catalog. In the next step of our analysis, we used only the HST observations because the Gaia measurements have much higher uncertainties for the fainter stars and the HST observations have a longer baseline of 6 yr compared to the 3 yr of Gaia. In this step of the transformation, we used stars (1)with brightness similar to the average brightness of the target, (2)with color similar to the source’s color (see Figure 7), and (3) lying within 350 pixels of the source. The first criterion minimizes any residual shift caused by CTE effects. We note that we already used the most recent CTE correction software for our analysis. Since the CTE effects on the position measurements are differential, using linear transformations based on stars of similar brightness should remove any residual CTE effects. (It Figure 7. Left panel: CMD (mF606W vs. (mF606W − mF814W)) for all stars in the HST/WFC3 field. The main-sequence turnoff occurs at mF606W ; 22, above which the disk and the bulge split into two sequences: the redder stars are mainly bulge objects, while the bluer ones are mainly disk objects. Stars marked by blue squares are selected as “disk-like” stars, and the stars marked by black crosses are selected as “bulge-like.” The position of the (unmagnified) microlensed source is shown as a green circle. The nearby bright star 0 4 away and about 20 times brighter than the unmagnified source is shown as a red circle. Magenta circles are the astrometric reference stars used in our analysis. Top right panel: proper motions of the disk-like stars (blue), the reference stars (magenta), and the source (green circle). Bottom right panel: proper motions of the bulge-like stars (black crosses), the reference stars (magenta), and the source (green circle). Only bulge-like stars (shown in magenta) with brightness similar to the observed brightness of the star were used in the final astrometric transformations. This ensures that the reference stars are very similar to the target source star, and uncorrected CTE and small parallax effects should cancel in the differential astrometric measurements. 10 The Astrophysical Journal, 933:83 (28pp), 2022 July 1 Sahu et al. is worth noting here that the images with the highest astrometric deflection were taken when WFC3/UIVS was young and when CTE losses were small.) The second criterion ensures that the stars used in the transformation belong to the bulge, which helps in minimizing errors due to parallax, as described in more detail below. The third criterion minimizes residuals in the distortion solution. We employed an iterative procedure to measure the positions and proper motions of the stars, starting from the revised values in each iteration. We rejected the highest-sigma point after each complete iteration. We repeated this procedure until the highest-sigma point was no more than a preset tolerance, for which we adopted 6σ. Only a small number of points were rejected by this procedure, mostly affected by cosmic-ray hits on the detector. Then, at each epoch the reference-star positions were corrected for proper motion, and the positions of the source were determined relative to this adjusted frame. The estimated uncertainty in the position of the source star relative to the adjusted frame is ∼0.4 mas in each individual exposure. As an illustration, Figure 8 shows the proper motions as measured for nine representative stars. Figure 9 shows the errors in the proper-motion measurements of the reference stars. We note that all the reference stars are within about 1.5 mag of each other (See Figure 7). Figure 10 shows the the histograms of the residuals of each measurement from that star’s proper-motion solution along the R.A. and decl. Figure 8. Motions of nine representative astrometric reference stars in R.A. and decl. The red lines are linear fits to the proper motions of the stars. 11 The Astrophysical Journal, 933:83 (28pp), 2022 July 1 Sahu et al. directions. Both distributions are consistent with a Gaussian distribution (the red curve). As shown by previous similar studies, the final reference-frame positions are expected to be internally accurate to better than 0.01 pixels (Anderson et al. 2008; Bellini et al. 2015). We specifically solved for the proper motions of the reference stars, but we ignored their parallaxes. The reason for adopting this approach is the following. Our choice of reference stars ensures that a large fraction of them belong to the Galactic bulge and hence have similar parallactic motion to the source star. We note that the source parallax is small to begin with (0.2 mas). Then, the source position is referenced to stars chosen to be at comparable distance, so any remaining impact of the source parallax on the astrometry or photometry of the event is expected to be negligible. As described earlier, there is a bright star ∼10 pixels away from our source. The target star is close to the brightness of this neighbor in the first two epochs and slowly fades to its nominal brightness, at which point it is about 3 mag fainter than the bright neighbor. For accurate astrometry of the source, we wanted to make sure that the position measurements of the source are not affected by the presence of this bright neighbor. So we wanted to subtract the PSF of the bright star before measuring the positions of the source at every epoch. However, subtracting the bright star is not just a matter of subtracting a standard PSF. The separation is ∼10 pixels, and the available library PSFs go out to 12 pixels and are tapered and not very accurate in the wings. Thus, we needed to make a more extended PSF model. To make an extended PSF, we carefully selected stars that (1) are within ∼350 pixels of the bright neighbor, (2) have brightness and color similar to the neighbor, and (3) are fairly isolated. We found 18 such stars (excluding the bright star itself), which provided a good sample to make the required extended PSFs. We used the images of these 18 stars to produce a separate well-sampled, extended PSF for each individual exposure. For illustration, Figure 11 shows the stacked PSFs in F606W and F814W. We have taken particular care to make sure that the PSF is well characterized in the wings since the source lies in the wings of the bright star, and subtracting the wings correctly is crucial for accurate astrometry. We then took this PSF model for each exposure and subtracted it from the neighbor star in each exposure. Figure 12 shows the original images (first and third rows) and the subtracted images (second and fourth rows) in the F606W and F814W filters. The residuals are very small, particularly in the wings of the PSF. We found that the astrometric position of the source changes by ∼0.03 pixels (1.2 mas) after this subtraction, which could have a significant effect on the mass determination of the lens, so this extra step of neighbor subtraction was Figure 9. Proper-motion errors of the reference stars. The y-axis shows the proper-motion error, defined as [ ]s s s= +m m mx y 2 2 0.5, where σμx and σμy are the proper-motion errors along the x- and y-axes, which are parallel to R.A. and decl., respectively. Reference stars cover a range of 1.5 mag around the (unamplified) magnitude of the source; their proper-motion uncertainties vary significantly from object to object, with a modest systematic increase with magnitude. Figure 10. Histogram of the position residuals for all the reference stars in R.A. and decl. from their proper-motion solution, as measured in each image separately, and scaled with the measured dispersion for that star. The scaled residuals closely follow a standard normal distribution (shown by the red curves). As noted in the caption to Figure 9, reference stars cover a range of magnitudes around the unamplified source magnitude. Figure 11. The stacked F606W (left) and F814W (right) PSFs, with the 10- real-pixel radius shown in green. Note that the F606W PSF has a diffraction spike bump very close to the location of the source star. The bump for the other diffraction spikes does not show up in the residual images shown in the second row of Figure 12, which implies that it is subtracted well beneath the source, making the position measurements of the source more robust. The F814W PSF has a very strong radial gradient (along with azimuthal structure) at the location of the source. Without subtracting a high-fidelity model PSF, there would be some impact on the measurement of the source positions. And as the source moves relative to the neighbor, then the source would move across the PSF halo, which could introduce artificial shifts if the PSF of the bright neighbor is not subtracted well. 12 The Astrophysical Journal, 933:83 (28pp), 2022 July 1 Sahu et al. crucial in improving the analysis/results. The resulting astrometric positions of the source were used for further analysis as described in the next section. 5.3. Photometric Analysis In addition to the measured positions, the analysis algorithm provides PSF-based photometry of all the stars in the field. To set a calibrated zero-point, we used standard aperture photometry to determine the fluxes of a few isolated stars in the field within an aperture with a 10-pixel radius. These fluxes were then corrected to an infinite aperture, using encircled- energy measurements from Calamida et al. (2021), and the photometric zero-point in the image headers (PHOTFLAM) was used to convert these fluxes to the Vegamag scale. The mean difference between these values and the values obtained by the PSF fitting was then applied to all of the PSF magnitudes to convert them to Vegamag. As described above, the source star lies on the wings of the PSF of the neighboring bright star. Hence, for accurate photometry of the source, it was critical to correctly subtract the light contribution from the bright neighbor. The photometry for the source was carried out after subtracting the superposed flux from the neighbor star using a high-fidelity PSF as described above. The resultant time-series HST photometry of the source is shown in Figure 13, along with the model light curve described below in Section 8. There is no detectable color change as the event progresses and the star fades: the color of the source has remained constant to within 0.01 mag during the 6 yr of observations with HST. There is also no detectable blending as described in more detail in Section 8.2. Figure 12. The source star lies in the PSF wings of a bright neighbor, marked in these frames with green circles with a radius of 10 pixels. Since the available WFC3 library PSFs do not extend to this large a radius, a special PSF extending to 20 pixels was constructed using the same HST images. This PSF was used to subtract the neighbor before measuring the position of the source in each exposure. The top row shows the original stacked F606W images of Epochs 1–8 (from left to right), and the second row shows the PSF-subtracted images. The third row shows the original stacked F814W images, and the bottom row shows the subtracted images. The stacks are overbinned by a factor of 2 to show details. Care was taken to assure that the PSF is well characterized in the wings of the neighbor star where the source lies. Figure 13. Top panel: photometry of MOA-11-191/OGLE-11-462, obtained with HST over 6 yr at eight epochs in the F606W (V ) and F814W (I) filters (black filled circles), along with our final model fits from Section 8.2 (red and blue curves). Errors on the photometry are smaller than the plotting points. The second panel plots the observed values of V − I, showing that the source color remained constant to within ∼0.01 mag during the entire 6 yr duration. The third and fourth panels show that the residuals, Vobs − Vmodel and Iobs − Imodel, at the epochs of the HST observations are consistent with zero within the measurement uncertainties (see Section 8.2 for more details and discussion). (The data used to create this figure are available.) 13 The Astrophysical Journal, 933:83 (28pp), 2022 July 1 Sahu et al. The source at baseline brightness has apparent magnitudes of mF606W= 21.943± 0.014 and mF814W= 19.578± 0.012. We made a stack of all the images for each filter for every epoch. The HST images allow us to detect and measure magnitudes of stars as faint as V; 25. Figure 7 shows the CMD based on this photometry, where we also show the position of the source and the bright neighbor 0 4 away. 6. Ground-based Light Curve MOA-11-191/OGLE-11-462 was monitored photometri- cally by several ground-based observatories. The coverage by MOA, OGLE, and Wise Microlensing Survey extended over several years. Moreover, as a high-magnification event, it attracted intensive monitoring by a number of additional ground-based telescopes—especially around the time of peak brightness, where the microlensing light curve is sensitive to planet detection. Table 3 gives a journal of the photometric observations and data used in our analysis. We use the data re- reduced by the surveys and other groups (Udalski et al. 1992; Bond et al. 2001; Sackett et al. 2004; Gould et al. 2006; Tsapras et al. 2009; Dominik et al. 2010). Figure 14 shows the light curve, both over a 300-day interval (top panel) and zooming in on the 7 days around peak magnification (bottom panel). Superposed is our model fit to the light curve, from the analysis described in the next two sections. 7. Blending, Relative Parallax, and Lens Trajectory The combination of ground-based photometric monitoring with long-term astrometric and photometric measurements from HST affords the ability to constrain all aspects of this event and obtain high-quality measurements of its parameters. In this section, we describe some characteristics of the data, illustrating the key information that can be obtained from photometry and astrometry separately through heuristic considerations. 7.1. Photometric Blending Ground-based photometry of MOA-11-191/OGLE-11-462 suffers from significant amounts of blending with neighboring stars, as shown in our HST images (Figure 4). Moreover, the amount of blending changes markedly depending on image quality. There is a bright neighbor star only 0 4 away, along with two fainter stars within 0 5 of the source. At larger separations, there are three more stars within ¢¢1 , whose combined brightness is greater than that of the baseline source, and there are several more stars within 1 5 that are also brighter than the source. Since in the available ground-based imaging data the measured image quality is seldom better than ¢¢1 , the ground-based photometry will always include the light from at least the three closest stars within 0 5. In a fraction of the data, taken under poorer seeing conditions, the source is blended with an increasing number of neighbors. Our HST images allow us to place constraints on the expected blending parameter, g, defined as the ratio of the flux from neighbors included in the photometry to the flux from the unmagnified source itself (Equation (5)), in the ground-based observations. The bright neighbor is 18.88 times brighter than the source at baseline in F814W, and the contribution from the two fainter stars is an additional 0.19 times that of the source. Thus, the expected blending factor due to these three stars is g= 19.07 in the F814W filter. Since the bright neighbor is similar in color to the source (see Figure 7), we adopt this value of g for both OGLE (I band) and MOA (R band) data as an initial estimate, but we keep it as a variable in our analysis. The final values (Section 5.2) differ significantly between OGLE and MOA, possibly due in part to differences in processing between the two data sets. The effect of blending can be reduced substantially by basing the photometry on difference images. It can be further reduced by restricting the analysis to images taken under good seeing. It is obvious, however, that variable blending will affect the noise characteristics of ground-based photometry; it is Table 3 Journal of Ground-based Photometry of MOA-11-191/OGLE-11-462 Data Set Telescope Aperture Filter No. of Date Range Location (m) Observations (HJD − 2,450,000) MOA New Zealand 1.8 R 46040 3824.093K9441.117 OGLE Chile 1.3 I 15546 5260.855K8787.509 Wise Survey Israel 1.0 I 953 5658.529K5722.543 Danish DFOSC Chile 1.54 I 921 5744.804K5782.554 Danish LuckyCam Chile 1.54 broad 10 5738.624K5742.706 MONET North Texas, USA 1.2 I 214 5762.722K5764.824 Faulkes North Hawaii, USA 2.0 SDSS i’ 99 5763.777K5768.955 Liverpool Canary Islands, Spain 2.0 SDSS i’ 254 5739.521K5768.438 SAAO 1.0 m South Africa 1.0 I 611 5751.264K5777.315 SAAO 1.0 m South Africa 1.0 V 44 5758.454K5765.385 U. Tasmania Australia 1.0 I 60 5761.055K5769.110 CTIO Chile 1.3 I 226 5757.511K5772.785 CTIO Chile 1.3 V 27 5757.515K5763.807 Auckland New Zealand 0.4 R 160 5759.873K5778.893 Farm Cove New Zealand 0.35 unfiltered 37 5741.856K5761.856 Kumeu Obs. New Zealand 0.35 R 63 5759.803K5762.134 Vintage Lane New Zealand 0.4 unfiltered 60 5762.792K5767.941 Weizmann Israel 0.4 I 167 5762.281K5764.423 Wise Israel 0.46 I 142 5762.266K5763.456 Note. The data were acquired by the MOA (Bond et al. 2001), OGLE (Udalski et al. 2015), Wise Microlensing Survey (Shvartzvald et al. 2016), MiNDSTEp (Dominik et al. 2010), RoboNet (Tsapras et al. 2009), PLANET (Sackett et al. 2004), and μFUN (Gould et al. 2006) teams. 14 The Astrophysical Journal, 933:83 (28pp), 2022 July 1 Sahu et al. difficult to include such effects deterministically because of the imperfect knowledge of the blending (at the subpercent level) for individual images. 7.2. Heuristic Considerations 7.2.1. Photometric Constraints on Parallax and Lens Trajectory As discussed in Section 2.3, the light curve of a long- duration microlensing event such as MOA-11-191/OGLE-11- 462 can show distortion by the relative parallactic motions of the source and lens (e.g., Gould 1992; Alcock et al. 1995). Specifically, the light curve is sensitive to πE and jLS because their combination modifies the relative path of source and lens and thus the shape of the light curve. We note that, in our formalism, jLS corresponds to the PA of the path of the lens relative to the source without parallax in equatorial coordinates (not to be confused with the instantaneous path of the lens at the time of closest angular approach in ecliptic coordinates; see Section 2.2). In principle, a sufficiently accurate light curve can provide good constraints on both πE and jLS. However, as discussed in the previous subsection, the photometry is significantly affected by blending. We attempted to model the light curve alone but found that it can be fitted with a range of parameter combinations, in which the values of πE and jLS are strongly correlated. In addition, the derived value of πE varies with the specific subset of photometric data chosen for analysis, as well as with the assumed blending factor for those data. The derived value of πE ranges from 0.07 to 0.12, with larger values corresponding to larger values of jLS, ranging from 330° to 358°. The reason is that increasing jLS makes the lens move in a more northerly direction as seen in the bottom panel of Figure 15. Since parallax is predominantly in the east–west direction, in order to produce a fixed change in u, the value of πE has to increase with jLS, so that the change in position due to parallax can compensate for a more northerly motion of the lens. Several different combinations of these quantities can reproduce the observed light curve, with differences between solutions of the order of 1 mmag at early and late times and ∼5 mmag near the peak. Systematic differences in the data at this level could be caused by small variations in blending associated with changes in the ground-based seeing, or other minor secular variations in the photometry. Therefore, we conclude that, when photometry alone is used to constrain the parameters of the event, only a reliable joint constraint on πE and jLS can be derived. Fortunately, astrometry provides a Figure 14. Ground-based photometric observations of MOA-11-191/OGLE-11-462, along with a best-fitting model light curve, shown over a 300-day interval in the top panel and over a zoomed-in region covering 7 days around peak magnification in the bottom panel. All data have been transformed to OGLE I magnitudes according to the inferred baseline magnitudes and blend ratio from the common model. 15 The Astrophysical Journal, 933:83 (28pp), 2022 July 1 Sahu et al. robust independent estimate of jLS, allowing us to break this degeneracy and determine the two quantities separately. 7.2.2. Astrometric Deflection and Orientation of the Relative Motion In order to understand how astrometry can constrain the direction of motion of the lens, it is useful to consider an illustrative plot of the motion of the lens relative to the source, as shown in Figure 15. North is at the top, east on the left; the (uRA, uDec) coordinates give the position of the lens relative to the source in units of θE, with uRA increasing to the east. We have used our actual final model described below in Section 8.2 for this illustration. The top panel shows the motion of the lens, with πE= 0.0894 and jLS= 342°.5, and an impact parameter of u0= 0.00271. The straight line represents the proper motion of the lens with respect to the source, while the wavy line adds the parallactic motion, computed using the JPL ephemeris of Earth.77 Red circles show the position of the lens at the eight epochs of our HST observations. The bottom panel in Figure 15 shows an enlarged view of the lens trajectory near the source position. The dotted black line represents the proper motion of the lens with respect to the source without parallax, while the solid black line includes the parallactic effect. (The red lines correspond to a less preferred u0,+ solution described in Section 8.3.) The plot shows that near the closest angular approach—and thus the peak magnification—the relative path is substantially affected by parallax; however, the astrometric deflection is very small at this time (see Section 2.3). Since the source deflection is always in the direction of the line joining the instantaneous position of the lens to the undeflected position of the source, the directions of the source deflections at late times will remain nearly constant with little parallax effect; thus, the late-time deflection directions robustly constrain the orientation, jLS, of the lens trajectory. 7.2.3. Constraining the Lens Trajectory Orientation Since the parallactic effect is unimportant for constraining jLS, we first fitted the photometry using a light-curve model that neglects parallax. The resultant model (with t0− 2,450,000= 5763.33, tE= 231.56 days, =A 372.62max , and g= 19.3) predicts the total deflection in units of θE as a function of time, through Equation (6). As described in Section 5.2, we have accurate measurements of the (x, y) positions of the source at the eight epochs of HST observations. These positions are affected by both the proper motion of the source and its deflections, which are a function of θE and jLS. We fitted a model to the positions, whose parameters are the x (R.A.) and y (decl.) components of the proper motion, θE, and jLS. This fit resulted in values of θE= 5.2± 0.5 mas and jLS= 337°.9± 5°.0. Components of the resultant deflections as a function of time, after subtracting the best-fitting proper motion, are shown in the top two panels of Figure 16. These panels plot the deflections in the R.A. and decl. directions. The bottom left panel shows the total amount of deflection, again as a function of time. Note that the total deflection reaches a maximum of ∼2 mas in late 2012. The bottom right panel of Figure 16 plots the R.A. versus decl. deflections. These deflections are always along the line Figure 15. Top panel: path of the lens with respect to the source, the position of the source being fixed at (0, 0). The direction of motion is shown by an arrow. The black straight line is the lens path without parallax, and the wavy line is the path including parallax. Red points mark the epochs of HST observations. uRA and uDec are the R.A. and decl. components of the lens–source separation, in units of θE. Bottom panel: enlarged view of the lens trajectory with respect to the source around closest angular approach. Black lines correspond to the u0,− solution, which is the preferred solution, and the red lines show the less preferred u0, + solution. In both cases, the barycentric trajectories (no parallax), shown by dotted lines, pass on the north side of the source. The geocentric trajectory (i.e., the trajectory as seen by an observer on Earth, which includes parallax) of the u0,+ solution (shown by the solid red line) passes on the north side of the source, but the trajectory of the u0,− solution (shown by the black solid line) passes to the south. The u0,− solution is used in our analysis presented here. It is worth noting, however, that the derived θE, and hence the mass of the lens, is nearly identical for both solutions. jLS is the PA of the barycentric proper motion of the lens with respect to the source (i.e., no parallax), as shown here. 77 https://ssd.jpl.nasa.gov/horizons/app.html#/ 16 The Astrophysical Journal, 933:83 (28pp), 2022 July 1 Sahu et al. https://ssd.jpl.nasa.gov/horizons/app.html#/ joining the lens to the source, and thus at large separations their direction is opposite to the direction of the relative motion of the lens. The solid black line passes through the origin, at the jLS angle derived above. The dotted blue lines indicate the allowed range based on the uncertainties. Although the best-fit value from photometry alone is jLS= 354°.8 (shown by the solid magenta line), the allowed range of jLS (shown by the dashed magenta lines) has a flat probability distribution. The combined constraint from photometry and astrometry is used in our subsequent analysis. Note that for clarity in Figure 16 we have shown the mean deflections at each epoch, and not the individual measurements. In particular, this avoids a confusing overlap of points in the bottom right panel, where the deflections are not a monotonic function of time. The individual measurements are shown in the next section. 8. Full Modeling of the Photometric and Astrometric Data In this section we give full details of our analysis, carried out independently by several coauthors using different parameter- izations, all leading to a consistent final model of the event. 8.1. First Approach: All Photometric Data Sets, and Robustness of Parallax Measurement In addition to OGLE and MOA, photometric time-series data were obtained at several more observatories (see Table 2). These data typically cover a relatively narrow range of ∼20 days around the peak, beíng primarily aimed at searching for planet-related distortions of the light curve. These photometric series, unlike those obtained by the survey programs, do not provide significant constraints on the lens–source model— especially since each set of observations can have a different baseline magnitude and blending parameter. Nevertheless, in our initial approach, we included all the data in our analysis but also carried out analyses separately for “OGLE-only,” “MOA- only,” and “OGLE+MOA-only” data sets. Ground-based time-series photometry is susceptible to systematic noise, and this must not be mistaken for real features of the light curve. To improve the robustness of our solution, we model the photometric uncertainties, and more- over force the model to follow the bulk of the data by explicitly down-weighting outliers. As implemented for the SIGNAL- MEN microlensing anomaly detector (Dominik et al. 2007, 2019), we specifically adopt a bi-square weight function Figure 16. The top two panels show the average values of the measured deflections in R.A. and decl. at each HST epoch. We use the values of t0, tE, and u0 as derived from a light-curve fit without parallax, and fit for the proper motion of the source, θE, and jLS. The solid black line is the best fit with θE = 5.2 mas and jLS = 337°. 9. The bottom left panel shows the total deflections at each epoch. The bottom right panel shows the R.A. vs. decl. deflections. These deflections are always along the line joining the lens to the source, and thus at large separations their direction is opposite to the direction of the relative motion of the lens with respect to the source. The solid black line passes through the origin, at the jLS angle derived above. The dotted blue lines indicate the allowed range from astrometry based on the uncertainties. The allowed range of jLS from photometry alone is shown by the dotted magenta lines, the best-fit value being jLS = 354°. 8 shown by the solid magenta line. The combined constraint from photometry and astrometry is used in our subsequent analysis. 17 The Astrophysical Journal, 933:83 (28pp), 2022 July 1 Sahu et al. with regard to the median residual and a Gaussian distribution for the uncertainties with revised standard deviation in magnitude of ˜ ( ) ( )s k s s= + , 172 0 2 where σ denotes the reported error bar, κ is a scaling factor, and σ0 corresponds to a systematic error added in quadrature. For the plot of the various data sets as shown in Figure 14, we give the respective estimated values of κ and σ0 in Table 4. If the size of the error bars does not vary substantially, there is a degeneracy between κ and σ0, and either of the parameters provides modified constant error bars (while it does not matter which). We find two viable models, significantly only distinguished by the sign of u0, and will refer in the following to the model with u0,−. The microlensing parallax parameter πE is constrained by the wing of the light curve and much less sensitive to the peak region, and therefore it is mostly constrained by the microlensing survey data. From various combinations of data sets, we consistently find πE= 0.10± 0.02 but some variation in the trajectory angle ψ, correlated with tE and the blend fraction, yielding visually indistinguishable model light curves. However, the angle of lens–source proper motion, jLS, follows robustly from the astrometric data (see Section 7.2.2), given that the centroid shift to first order (i.e., neglecting the small distortion caused by parallax) traces an ellipse (a highly flattened ellipse resembling a line in our case) whose semimajor axis is parallel to μLS. If we restrict this angle to the range 333°� jLS� 343°, as suggested by the astrometric data, the photometric light curve does not change substantially, and we find πE; 0.086. We emphasize here that it is incorrect to say that there is a discrepancy between the paths determined from photometry and astrometry, since there is a correlation between jLS and πE in the photometric solution. However, restricting the trajectory angle jLS as robustly derived in the last section from the orientation of the centroid shifts to the range 333°� jLS� 343° suggests πE; 0.086. 8.2. Second Approach: Simultaneous Fit of Photometric and Astrometric Data We now turn to a full analysis in which we fit the astrometric and photometric data simultaneously in order to obtain all of the parameters. Such a solution is important, since the crucial parameters of θE and πE are derived from two different types of data. A simultaneous solution is also essential for a correct estimate of the uncertainties in the model parameters. We follow the same plane-of-sky approach described in Section 7.2, which makes it easier to work with, and also show the actual paths of the lens and the source, as well as the deflections. We follow a different parameterization procedure where the model parameters we optimize contain all terms needed to characterize the positions of the lens and the source on the sky as a function of time; these include the reference positions and proper motions of both lens and source, their relative parallax, and the angular Einstein radius of the lens. In principle, the source parallax is also needed; however, its parallax in the reference system we use is close to zero, and it is not meaningfully constrained by the observations. As discussed below in Section 9.2, the best constraints on the source distance come instead from photometry and high-resolution spectroscopy. From these parameters, the undeflected paths of the source and lens can be determined. The deflection of the lensed image of the source is then computed, and the resulting deflected source positions are matched to the observed positions. The same calculation also yields the source magnification; in order to match the observed photometry, the model must include a baseline magnitude of the source and a blending parameter for each photometric data set. Consistent with the previous approach, we found that most of the photometric data sets cover too short a time interval to yield meaningful constraints on the event parameters in the presence of significant blending; therefore, we limit the model optimization to the MOA and OGLE photometric data sets and validate the resulting model for the other data sets separately (see Section 8.1). In addition, in order to avoid undue impact from any secular variations in photometric responses, we only include OGLE and MOA photometric measurements within ±2 yr from the peak of the event. We adopt the approximate values of jLS, tE, t0, and u0 from the analysis of the previous section as our initial estimates, but we leave all parameters free in the optimization; the baseline magnitude and blending parameters for MOA and OGLE are also separately optimized. As discussed in the previous subsection, the results of the optimization depend to some extent on the relative weighting of astrometry and photometry. Because the number of photometric measurements greatly exceeds that of astrometric measurements and the nominal photometric uncertainties are very small, an optimization using nominal errors disproportio- nately weights photometry, resulting in a poor match to the astrometry. In order to obtain a more balanced weighting of astrometry and photometry, we scaled the photometric errors by different amounts in different temporal bins making sure that the scaled errors are compatible with the statistical dispersions in the measurements, and validated the solution based on a reasonable match to the astrometric data. In our final Table 4 Revised Error Bars of Photometric Data Data Set κ σ0 OGLE 1.25 0.005 MOA 1.18 0.0025 Wise Survey 0.64 0.009 Danish 1.54 m DFOSC 2.75 10−5 (*) Danish 1.54 m LuckyCam 0.1 (*) 0.050 MONET North 1.2 m 0.57 0.006 Faulkes North 2.0 m 0.1 (*) 0.014 Liverpool 2.0 m 1.17 0.008 SAAO 1.0 m I 1.42 0.0014 SAAO 1.0 m V 0.59 0.009 Tasmania 1.0 m 0.28 0.017 CTIO 1.3 m I 0.62 10−5 (*) CTIO 1.3 m V 1.01 10−5 (*) Auckland 0.4 m 0.91 0.007 Farm Cove 0.35 m 1.00 0.003 Kumeu 0.35 m 1.87 10−5 (*) Vintage Lane 0.4 m 0.1 (*) 0.011 Weizmann 0.4 m 0.80 10−5 (*) Wise 0.46 m 0.22 0.009 Note. The adopted error bar (in magnitude) becomes ˜ ( )s k s s= +2 0 2 , where σ is the reported error bar of the photometric data. We have applied range constraints κ > 0.1 and σ0 � 10−5. An asterisk (*) indicates that the value is at the range boundary. 18 The Astrophysical Journal, 933:83 (28pp), 2022 July 1 Sahu et al. model, most photometric points are still at an uncertainty below 10 mmag. This solution has a total astrometric χ2 of 136 with 106 points. (The solution with nominal weights has a higher astrometric χ2 of 149.) The parameters of the final model are given in Table 5. Figure 17 shows the reconstructed motion of the lens (magenta) and of the source (black) in the plane of the sky, based on our adopted model. The predicted apparent source trajectory is shown by the green solid line. The astrometric measurements are shown individually by the small filled circles, and their epoch averages are shown as red triangles. Cyan squares show the model position at each epoch; gray lines connect the model lens position to the undeflected and deflected source positions at each epoch. Figure 18 presents the measured and predicted source positions separately for the x and y coordinates. To improve the legibility of the plot, the fitted proper motion of the source has been subtracted from both model and data; therefore, the points shown represent the deflection of the source. The black line is our final adopted model, which takes photometry and astrometry into account. We can now check the consistency of the final model with our measured HST photometry in the F606W and F814W filters and also constrain the amount of blending. We used the final model to calculate the magnifications at the HST observation epochs and varied the baseline magnitudes and the blending factor to Figure 17. Representation of the reconstructed motions of the lens and of the source on the plane of the sky. The reference point is the undeflected position of the source at time t0. The small circles are the individual measurements from HST images (green for F606W, orange for F814W); the larger red triangles represent the average positions for each epoch. The cyan squares are the fitted positions at each epoch. The gray lines connect the undeflected source and lens positions at each HST epoch. Table 5 Parameters of the Full Fit to Astrometry and Photometry Parameter Units Value Uncertainty (1σ) Notesa μS (R.A.) mas yr−1 −2.263 0.029 (1) μS (decl.) mas yr−1 −3.597 0.030 (1) θE mas 5.18 0.51 (2) tE days 270.7 11.2 (3) jLS deg 342.5 4.9 (4) t0 (HJD −2,450,000.0) days 5765.00 0.87 (5) πE 0.0894 0.0135 (6) u0 0.0422 0.0072 (7) MOA baseline R magnitude mag 16.5147 0.0016 MOA blending parameter 16.07 0.66 OGLE baseline I magnitude mag 16.4063 0.0015 OGLE blending parameter 18.80 0.79 Derived parameters: u0 0.00271 (8) t0 (HJD −2,450,000.0) days 5763.32 (9) Note. a Notes: (1) Undeflected proper motion of the source in Gaia EDR3 absolute frame. (2) Angular Einstein radius. (3) Angular Einstein radius θE divided by absolute value of lens– source proper motion μLS. (4) Orientation angle of the lens proper motion relative to the source (N through E). (5) Time of closest angular approach without parallax motion. (6) Relative parallax of lens and source in units of θE. (7) Impact parameter in units of θE without parallax motion. (8) Impact parameter derived using the model-fit parameters above and after including parallax motion, in units of θE. (9) Time of closest angular approach derived as above and after including parallax motion. 19 The Astrophysical Journal, 933:83 (28pp), 2022 July 1 Sahu et al. fit the HST photometry. The resulting fit is excellent (see Figure 13) and yields baseline source magnitudes of mF606W= 21.943± 0.012 and mF814W= 19.578± 0.012, with corresp- onding blending factors for the HST photometry of g= −0.012± 0.015 and −0.006± 0.012, respectively. The bottom two panels of Figure 13 show the residuals of the observations relative to the model in F606W and F814W at the epochs of HST observations, where we have assumed the minimum physically allowed value of g= 0. (It makes little difference if we instead use the slightly negative values of g from the formal fit.) The stringent constraints on blending and lack of color variation make it unlikely that our deflection measurements could be affected by blending with a binary companion or a field star lying within the HST PSF. 8.3. Lens Motion The path of the lens in the sky plane, as derived from the above analysis, is shown in Figure 15. We note that there are two solutions corresponding to u0,+ and u0,− (Gould 2004). However, at t0, the angular separation between the source and the lens is dominated by the parallactic motion of the lens (the separation caused by the parallactic motion is ∼0.04, compared to the actual value of u0; 0.00271). Thus, the path without parallax in both solutions lies on the u0,+ side. The paths of the lens for the u0,+ and u0,− solutions are shown in Figure 15, where the dashed lines show the path of the lens without parallax, and the solid lines show the path of the lens after the parallactic motion is taken into account. The respective paths are separated only by 0.02 mas at the time of maximum magnification, and they quickly merge. The u0,− solution is the preferred solution and is the one used in our analysis. However, we verified that both solutions provide practically identical deflections (since the deflection measure- ments are at much higher u, where the two paths nearly merge), and the results are the same for all practical purposes in both solutions. The first two rows of Table 5 give the proper motions of the source. We use the values of θE, tE , and jLS given in the next three rows to determine the proper motion of the lens with respect to the source as −2.10± 0.22 mas yr−1 and 6.66± 0.67mas yr−1 in R.A. and decl., respectively. The resulting absolute proper motions of the lens are given below in Table 6. 9. Properties of the Source As described in Sections 2.2–2.3, the mass determination for the lens does not depend on the individual distances to the lens and source, but only on the relative lens–source parallax, πLS, and the Einstein ring radius, θE—quantities that are directly determined from the light curve and the measured astrometric deflections. However, we still need an estimate of the distance to the source, DS, in order to determine the distance to the lens, DL, which is discussed below. Moreover, MOA-11-191/ OGLE-11-462 was a very high magnification event, with an impact parameter of only u0; 0.00271. Thus, it is desirable to estimate the angular diameter of the source, to verify that it is consistent with the point-source light-curve modeling adopted in the previous section. Figure 18. Predicted and measured positions of the source for our adopted joint astrometric and photometric fit. Small circles show the individual measurements from HST images (green for F606W, orange for F814W), while red triangles with error bars show the average and uncertainty at each HST epoch. The black line is our final adopted model. The fitted proper motion for the source has been subtracted from both model and measurements, in order to allow a better scaling of the plot. (The data used to create this figure are available.) 20 The As