11institutetext:  Department of Physics, Chungbuk National University,
Cheongju 28644, Republic of Korea 22institutetext: Astronomical Observatory,
University of Warsaw, Al. Ujazdowskie 4, 00-478 Warszawa, Poland
33institutetext: Institute of Natural and Mathematical Science, Massey
University, Auckland 0745, New Zealand 44institutetext: Korea Astronomy and
Space Science Institute, Daejon 34055, Republic of Korea 55institutetext: Max
Planck Institute for Astronomy, Königstuhl 17, D-69117 Heidelberg, Germany
66institutetext: Department of Astronomy, The Ohio State University, 140 W.
18th Ave., Columbus, OH 43210, USA 77institutetext: University of Canterbury,
Department of Physics and Astronomy, Private Bag 4800, Christchurch 8020, New
Zealand 88institutetext: Department of Particle Physics and Astrophysics,
Weizmann Institute of Science, Rehovot 76100, Israel 99institutetext: Center
for Astrophysics $|$ Harvard & Smithsonian 60 Garden St., Cambridge, MA 02138,
USA 1010institutetext: Department of Astronomy and Tsinghua Centre for
Astrophysics, Tsinghua University, Beijing 100084, China 1111institutetext:
School of Space Research, Kyung Hee University, Yongin, Kyeonggi 17104,
Republic of Korea 1212institutetext: Department of Physics, University of
Warwick, Gibbet Hill Road, Coventry, CV4 7AL, UK 1313institutetext: Institute
for Space-Earth Environmental Research, Nagoya University, Nagoya 464-8601,
Japan 1414institutetext: Code 667, NASA Goddard Space Flight Center,
Greenbelt, MD 20771, USA 1515institutetext: Department of Astronomy,
University of Maryland, College Park, MD 20742, USA 1616institutetext:
Department of Earth and Planetary Science, Graduate School of Science, The
University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
1717institutetext: Instituto de Astrofísica de Canarias, Vía Láctea s/n,
E-38205 La Laguna, Tenerife, Spain 1818institutetext: Department of Earth and
Space Science, Graduate School of Science, Osaka University, Toyonaka, Osaka
560-0043, Japan 1919institutetext: Department of Physics, The Catholic
University of America, Washington, DC 20064, USA 2020institutetext: Department
of Astronomy, Graduate School of Science, The University of Tokyo, 7-3-1
Hongo, Bunkyo-ku, Tokyo 113-0033, Japan 2121institutetext: Sorbonne
Université, CNRS, UMR 7095, Institut d’Astrophysique de Paris, 98 bis bd
Arago, 75014 Paris, France 2222institutetext: Department of Physics,
University of Auckland, Private Bag 92019, Auckland, New Zealand
2323institutetext: University of Canterbury Mt. John Observatory, P.O. Box 56,
Lake Tekapo 8770, New Zealand

# OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing
events with prominent orbital effects

Cheongho Han OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effectsOGLE-2018-BLG-0971,
MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing events with prominent
orbital effects Andrzej Udalski OGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital
effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effects Ian A. Bond
OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing
events with prominent orbital effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital effects Chung-
Uk Lee OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effectsOGLE-2018-BLG-0971,
MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing events with prominent
orbital effects Andrew Gould
(Leading authors)
Michael D. Albrow OGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital
effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effectsOGLE-2018-BLG-0971,
MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing events with prominent
orbital effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effectsOGLE-2018-BLG-0971,
MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing events with prominent
orbital effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effects Sun-Ju Chung
OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing
events with prominent orbital effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital effects Kyu-Ha
Hwang OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effectsOGLE-2018-BLG-0971,
MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing events with prominent
orbital effects Youn Kil Jung OGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital
effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effects Hyoun-Woo Kim
OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing
events with prominent orbital effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital effects Yoon-
Hyun Ryu OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effectsOGLE-2018-BLG-0971,
MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing events with prominent
orbital effects Yossi Shvartzvald OGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital
effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effects In-Gu Shin
OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing
events with prominent orbital effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital effects
Jennifer C. Yee OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effectsOGLE-2018-BLG-0971,
MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing events with prominent
orbital effects Hongjing Yang OGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital
effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effects Weicheng Zang
OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing
events with prominent orbital effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital
effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effectsOGLE-2018-BLG-0971,
MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing events with prominent
orbital effects Sang-Mok Cha OGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital
effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effectsOGLE-2018-BLG-0971,
MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing events with prominent
orbital effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effects Doeon Kim
OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing
events with prominent orbital effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital effects Dong-
Jin Kim OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effectsOGLE-2018-BLG-0971,
MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing events with prominent
orbital effects Seung-Lee Kim OGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital
effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effects Dong-Joo Lee
OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing
events with prominent orbital effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital effects
Yongseok Lee OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effectsOGLE-2018-BLG-0971,
MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing events with prominent
orbital effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effectsOGLE-2018-BLG-0971,
MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing events with prominent
orbital effects Byeong-Gon Park OGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital
effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effects Richard W. Pogge
(The KMTNet Collaboration)
Przemek Mróz OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effectsOGLE-2018-BLG-0971,
MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing events with prominent
orbital effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effectsOGLE-2018-BLG-0971,
MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing events with prominent
orbital effects Michał K. Szymański OGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital
effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effects Jan Skowron
OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing
events with prominent orbital effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital effects
Radosław Poleski OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effectsOGLE-2018-BLG-0971,
MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing events with prominent
orbital effects Igor Soszyński OGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital
effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effects Paweł Pietrukowicz
OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing
events with prominent orbital effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital effects Szymon
Kozłowski OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effectsOGLE-2018-BLG-0971,
MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing events with prominent
orbital effects Krzysztof A. Rybicki OGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital
effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effectsOGLE-2018-BLG-0971,
MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing events with prominent
orbital effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effects Patryk Iwanek
OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing
events with prominent orbital effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital effects
Krzysztof Ulaczyk OGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital
effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effects Marcin Wrona
OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing
events with prominent orbital effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital effects Mariusz
Gromadzki OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effectsOGLE-2018-BLG-0971,
MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing events with prominent
orbital effects Mateusz J. Mróz
(The OGLE Collaboration)
Fumio Abe OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effectsOGLE-2018-BLG-0971,
MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing events with prominent
orbital effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effectsOGLE-2018-BLG-0971,
MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing events with prominent
orbital effects Richard Barry OGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital
effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effects David P. Bennett
OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing
events with prominent orbital effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital
effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effectsOGLE-2018-BLG-0971,
MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing events with prominent
orbital effects Aparna Bhattacharya OGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital
effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effectsOGLE-2018-BLG-0971,
MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing events with prominent
orbital effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effects Hirosame Fujii
OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing
events with prominent orbital effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital effects Akihiko
Fukui OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effectsOGLE-2018-BLG-0971,
MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing events with prominent
orbital effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effectsOGLE-2018-BLG-0971,
MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing events with prominent
orbital effects Ryusei Hamada OGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital
effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effects Yuki Hirao
OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing
events with prominent orbital effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital effects Stela
Ishitani Silva OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effectsOGLE-2018-BLG-0971,
MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing events with prominent
orbital effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effectsOGLE-2018-BLG-0971,
MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing events with prominent
orbital effects Yoshitaka Itow OGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital
effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effects Rintaro Kirikawa
OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing
events with prominent orbital effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital effects Naoki
Koshimoto OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effectsOGLE-2018-BLG-0971,
MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing events with prominent
orbital effects Yutaka Matsubara OGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital
effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effects Shota Miyazaki
OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing
events with prominent orbital effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital effects Yasushi
Muraki OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effectsOGLE-2018-BLG-0971,
MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing events with prominent
orbital effects Greg Olmschenk OGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital
effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effects Clément Ranc
OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing
events with prominent orbital effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital effects
Nicholas J. Rattenbury OGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital
effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effects Yuki Satoh
OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing
events with prominent orbital effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital effects
Takahiro Sumi OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effectsOGLE-2018-BLG-0971,
MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing events with prominent
orbital effects Daisuke Suzuki OGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital
effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effects Mio Tomoyoshi
OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing
events with prominent orbital effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital effects Paul J.
Tristram OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effectsOGLE-2018-BLG-0971,
MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing events with prominent
orbital effects Aikaterini Vandorou OGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital
effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effectsOGLE-2018-BLG-0971,
MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing events with prominent
orbital effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effects Hibiki Yama
OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136: Microlensing
events with prominent orbital effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital effects Kansuke
Yamashita
(The MOA Collaboration) OGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136: Microlensing events with prominent orbital
effectsOGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136:
Microlensing events with prominent orbital effects

(Received ; accepted)

###### Abstract

Aims.  We undertake a project to reexamine microlensing data gathered from
high-cadence surveys. The aim of the project is to reinvestigate lensing
events with light curves exhibiting intricate anomaly features associated with
caustics, yet lacking prior proposed models to explain these features.

Methods.  Through detailed reanalyses considering higher-order effects, we
identify that accounting for orbital motions of lenses is vital in accurately
explaining the anomaly features observed in the light curves of the lensing
events OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136.

Results.  We estimate the masses and distances to the lenses by conducting
Bayesian analyses using the lensing parameters of the newly found lensing
solutions. From these analyses, we identify that the lenses of the events
OGLE-2018-BLG-0971 and MOA-2023-BLG-065 are binaries composed of M dwarfs,
while the lens of OGLE-2023-BLG-0136 is likely to be a binary composed of an
early K-dwarf primary and a late M-dwarf companion. For all lensing events,
the probability of the lens residing in the bulge is considerably higher than
that of it being located in the disk.

###### Key Words.:

Gravitational lensing: micro

Table 1: Coordinates of events. Event | (RA, Dec)J2000 | $(l,b)$ | Other ID references
---|---|---|---
OGLE-2018-BLG-0971 | (17:59:01.63, -28:13:42.20) | $(2^{\circ}\hskip-2.0pt.1082,-2^{\circ}\hskip-2.0pt.1727)$ | MOA-2018-BLG-173, KMT-2018-BLG-2336
MOA-2023-BLG-065 | (18:00:35.68, -29:13:23.95) | $(1^{\circ}\hskip-2.0pt.4140,-2^{\circ}\hskip-2.0pt.9645)$ | KMT-2023-BLG-2430
OGLE-2023-BLG-0136 | (18:09:16.97, -30:36:47.00) | $(1^{\circ}\hskip-2.0pt.1132,-5^{\circ}\hskip-2.0pt.2851)$ | KMT-2023-BLG-2849

##  1 Introduction

In general, light curves of microlensing events are modeled by assuming a
rectilinear relative motion between the lens and the source. However,
deviations from this assumption arise due to accelerations affecting the
motion of either the observer, lens, or source. For instance, an observer
experiences acceleration due to the Earth’s orbital motion around the Sun,
known as microlens-parallax effects (Gould, 1992, 2000). Additionally, if a
source of a lensing event is part of a binary system in which two stars orbit
a common barycenter, the source’s motion also undergoes acceleration (Han &
Gould, 1997; Rahvar & Dominik, 2009). Similarly, the lens’s orbital motion
induces acceleration, causing deviations from a rectilinear relative motion
between the lens and source, known as lens-orbital effects.

There have been instances of lensing events in which accounting for lens
orbital motions was crucial for accurately interpreting observed lensing light
curves. MACHO 97-BLG-41 (Alcock et al., 2000) notably marked the first binary-
lens single-source (2L1S) system displaying significant deviations from the
assumption of a static binary configuration. Initially, these deviations were
attributed to the presence of a third body of the lens, specifically a
circumbinary planet (Bennett et al., 1999). However, Albrow et al. (2000),
based on the analysis using an independent data set, later proposed a solution
involving an orbiting binary lens. The controversy was definitively resolved
by Jung et al. (2018), who, through a direct comparison of the two models
using a combined data set, found that the orbiting binary-lens interpretation
is preferred over the circumbinary planet model.

OGLE-2006-BLG-109 was the second instance in which lens orbital effects played
a crucial role in providing accurate explanations for the lensing light curve
(Gaudi et al., 2008; Bennett et al., 2010). The light curve of this event
exhibited a complex anomaly pattern, comprising multiple distinctive features.
A comprehensive understanding of these anomalies was achieved only after
accounting for combined higher-order effects resulting from the parallactic
motion of Earth and the orbital motion of the lens. From the analysis
considering these higher-order effects, the lens was proven to be the first
double planet system discovered with the gravitational microlensing method.

OGLE-2005-BLG-018 marked the third instance in which the significance of lens
orbital effects was established. The light curve of the event displayed
multiple anomaly features, comprising two neighboring strong anomalies and a
comparatively weak anomaly positioned apart from the stronger ones. Although a
model based on a static binary lens configuration could approximately explain
the two strong anomalies, accurately describing the separate weak anomaly was
challenging. As a result, this event had not been previously addressed until
Shin et al. (2011) revisited it and demonstrated that accounting for the lens
orbital motion was essential for accurately describing all the anomaly
features in the lensing light curve.

Prompted by the work of Shin et al. (2011), Park et al. (2013) revisited
microlensing data available until then and conducted thorough analyses of
binary-lens events. They focused on cases in which static binary models fell
short in accurately describing observed light curves. Through these analyses,
they revealed that the substantial residuals of the light curves of two
lensing events OGLE-2006-BLG-277 and OGLE-2012-BLG-0031 from static 2L1S
models were predominantly attributed to the influence of lens orbital motion.

OGLE-2009-BLG-020 was identified as a binary-lens event where orbital motion
was initially predicted through light curve analysis (Skowron et al., 2011).
Subsequent 3.5 years of radial velocity monitoring confirmed an orbit
consistent with the predictions derived from the microlensing light curve
analysis (Yee et al., 2016).

In the case OGLE-2013-BLG-0723, the lensing light curve was initially
interpreted by a triple-lens model, in which the lens consisted of a Venus-
mass planet and binary brown-dwarf host (Udalski et al., 2015a). Later, Han et
al. (2016a) reexamined the event, incorporating lens-orbital effects, and
proposed a revised interpretation involving two-body lens instead of the
previously suggested three-body lens solution. This updated model provided a
notably better fit to the observed light curve.

Gaia16aye, detected toward the northern Galactic disk field, was a binary
microlensing event and stood among the earliest instances detected from the
alerts issued by the Gaia space mission. The brightening of the source induced
by lensing endured for more than two years, and thus it was crucial to include
the orbital motion of the lens for the precise description of the lensing
light curve (Wyrzykowski et al., 2020).

The light curve of the lensing event KMT-2021-BLG-0322 exhibited multiple sets
of caustic-crossing features. The overall features of the light curve were
approximately described by a 2L1S model, but the model left substantial
residuals. From the reanalysis, Han et al. (2021) found that the residuals
could be explained either by considering a non-rectilinear lens-source motion
caused by the combination of microlens-parallax and lens-orbital effects or by
adding an additional low-mass companion to the binary lens, and hence three
lens components (3L1S system). The degeneracy between the higher-order 2L1S
model and the 3L1S model was very severe, making it difficult to single out a
correct solution based on the photometric data. This degeneracy was known
before for two previous events (MACHO-97-BLG-41 and OGLE-2013-BLG-0723), which
led to the false detections of planets in binary systems, and thus the
identification of the degeneracy for the event illustrated that this
degeneracy could be common.

In this study, we present comprehensive analyses on three 2L1S lensing events:
OGLE-2018-BLG-0971, MOA-2023-BLG-065, and OGLE-2023-BLG-0136. These events
bear resemblances, displaying anomalies in their light curves with intricate
and complex features that prove challenging to interpret using static binary-
lens models. We demonstrate the importance of considering orbital motions of
lenses in precisely describing the observed anomaly features in the lensing
light curves.

##  2 Event selections and data

We undertook a project involving the reexamination of microlensing data
obtained from three ongoing high-cadence microlensing surveys: the Korea
Microlensing Telescope Network (KMTNet: Kim et al., 2016), the Optical
Gravitational Lensing Experiment (OGLE: Udalski et al., 2015a), and the
Microlensing Observations in Astrophysics survey (MOA: Bond et al., 2001). In
this project, we directed our attention to lensing events displaying intricate
anomaly features related to caustics, yet lacking prior proposed models to
explain these features. We commenced our investigation by analyzing the KMTNet
data spanning from the 2016 season to the 2023 season, specifically focusing
on identifying anomalous lensing events characterized by conspicuous caustic-
crossing features. Subsequently, we examined the presence of lensing models
corresponding to these events, filtering out candidate events for which either
no models were proposed or presented models failed to accurately delineate the
anomalous features. Finally, in the concluding step, we verified the
availability of additional data obtained from the OGLE and MOA surveys for the
candidate events. Through this process, we identified three lensing events for
which the consideration of orbital motions of lenses played crucial role in
accurately describing the observed anomaly features in the lensing light
curves, including OGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136. In Table 1, we present the equatorial and Galactic
coordinates of the events. All these events were captured and observed by
multiple surveys. We provide ID references assigned by the respective surveys,
using the ID references from the initial discovery surveys in subsequent
discussions.

Observations of the events by the individual surveys were conducted using the
telescopes operated by the respective survey groups. The KMTNet group utilizes
three identical telescopes, each featuring a 1.6-meter aperture and equipped
with a camera capable of capturing a field spanning 4 square degrees. For
continuous coverage of lensing events, the KMTNet telescopes are strategically
distributed throughout three countries in the Southern Hemisphere: at the
Siding Spring Observatory in Australia (KMTA), the Cerro Tololo Interamerican
Observatory in Chile (KMTC), and the South African Astronomical Observatory in
South Africa (KMTS). The MOA survey employs a telescope with a 1.8 m aperture
located at the Mt. John Observatory in New Zealand. The camera mounted on the
MOA telescope has the capacity to capture a 2.2 square degree area of the sky
in a single shot. The OGLE survey operates the 1.3-meter Warsaw telescope
situated at the Las Campanas Observatory in Chile. The camera mounted on the
OGLE telescope provides a field of view that spans 1.4 square degrees. The
primary observations conducted by the KMTNet and OGLE surveys were done in the
$I$ band, whereas observations by the MOA survey were conducted in the custom
MOA-$R$ band. In all surveys, a portion of images was acquired in the $V$ band
for the color measurements of source stars.

The data of the events were processed using photometry pipelines that are
customized to the individual survey groups: KMTNet employed the Albrow et al.
(2009) pipeline, OGLE utilized the Udalski (2003) pipeline, and MOA employed
the Bond et al. (2001) pipeline. For the use of the optimal data, the KMTNet
data set was refined through a re-reduction process using the code developed
by Yang et al. (2023). For each data set, error bars estimated from the
photometry pipelines were recalibrated not only to ensure consistency of the
errorbars with the scatter of data but also to set the $\chi^{2}$ value per
degree of freedom (d.o.f.) for each data to unity. This normalization process
was done in accordance with the procedure outlined by Yee et al. (2012).

##  3 Light curve modeling

The light curves of all the analyzed events show anomalous features that are
characteristic of caustics. Caustics arise when a lens system comprises
multiple masses, and they represent the source positions at which the
magnification of a point source diverges to infinity. Consequently, the
presence of caustic-related features in the light curves implies the
involvement of lenses composed of multiple masses in producing these events.

Taking the caustic-related features into account, our analysis begins by
modeling the light curves within a static 2L1S framework. This framework
operates under the assumption of rectilinear relative motion between the lens
and the source. Within this static binary-lens model, a lensing light curve is
defined by seven basic parameters. The first three of these parameters
characterizes the source’s approach to the lens, denoted as
$(t_{0},u_{0},t_{\rm E})$. These parameters represent the time of the closest
lens-source approach, the separation at that instant (impact parameter), and
the event time scale, respectively. Here $u_{0}$ is scaled to the angular
Einstein radius $\theta_{\rm E}$, and $t_{\rm E}$ is defined as the duration
for the source to traverse the Einstein radius. The additional set of three
parameters $(s,q,\alpha)$ characterizes the binary lens configuration. These
parameters represent the projected separation (scaled to $\theta_{\rm E}$),
the mass ratio between the binary lens components ($M_{1}$ and $M_{2}$), and
the angle (source trajectory angle) formed between the lens-source proper
motion vector $\mu$ and the axis defined by $M_{1}$ and $M_{2}$. The last
parameter $\rho$, which is defined as the ratio of the angular source radius
$\theta_{*}$ to the Einstein radius, that is, $\rho=\theta_{*}/\theta_{\rm
E}$, quantifies the deformation of lensing light curves during caustic
crossings due to finite-source effects. In the 2L1S modeling, we begin by
exploring the binary parameters $s$ and $q$ using a grid approach, employing
multiple initial values of $\alpha$. Subsequently, we determine the remaining
parameters through a downhill method based on the Markov Chain Monte Carlo
(MCMC) technique. The lensing solutions are further refined by enabling
variation in all parameters.

As shown in the following section, interpreting the light curves of the events
examined in this study solely through static 2L1S models poses a significant
challenge. In such instances, we undertake additional modeling to account for
higher-order effects which induces deviations in the relative lens-source
motion from a rectilinear path. Our modeling approach explores the influences
of both lens orbital motion and microlens-parallax effects. To integrate these
higher-order effects into the modeling, additional parameters beyond the
fundamental set need to be included. The additional parameters introduced for
modeling with microlens-parallax effects encompass $(\pi_{{\rm E},N},\pi_{{\rm
E},E})$, representing the north and east components of the microlens-parallax
vector, $\mbox{\boldmath$\pi$}_{\rm E}$, respectively. The microlens-parallax
vector is defined as

$\mbox{\boldmath$\pi$}_{\rm E}=\left({\pi_{\rm rel}\over\theta_{\rm
E}}\right)\left({\mbox{\boldmath$\mu$}\over\mu}\right),$ (1)

where $\pi_{\rm rel}={\rm au}(1/D_{\rm L}-1/D_{\rm S})$ denotes the relative
lens-source parallax, and $(D_{\rm L},D_{\rm S})$ represent the distances to
the lens and source, respectively. Under the first-order approximation of
small changes in the positions of the lenses during lensing magnifications,
the lens-orbital effects can be characterized by two parameters:
$(ds/dt,d\alpha/dt)$. These parameters represent the rates of change in the
binary separation and the source trajectory angle, respectively. In the
parallax+orbit modeling, we enforced a condition where the projected kinetic-
to-potential energy ratio was required to remain below unity. This condition
ensured that the planet remained gravitationally bound to its host. The energy
ratio was computed from the higher-order lensing parameters by

$\left({{\rm KE}\over{\rm PE}}\right)_{\perp}={(a_{\perp}/{\rm au})^{3}\over
8\pi^{2}(M/M_{\odot})}\left[\left({1\over s}{ds/dt\over{\rm
yr}^{-1}}\right)^{2}+\left({d\alpha/dt\over{\rm yr}^{-1}}\right)^{2}\right].$
(2)

Here $a_{\perp}$ denotes the physical separation between the binary lens
components. Fully describing the Keplerian orbital motion necessitates the
inclusion of additional parameters.111To access a comprehensive description of
the orbital lensing parameters, refer to the summary provided in the Appendix
of Skowron et al. (2011). However, determining these extra parameters poses a
challenge due to the limitations of gravitational lensing, which is
insensitive to the motion of the lens along the line of sight together with
the partial light curve coverage spanning only a minor fraction of the
rotation period (Albrow et al., 2000).

Another higher-order effect that causes accelerations of the relative lens-
source motion is the orbital motion of the source, known as the ”xallarap
effect” (Griest & Hu, 1992; Han & Gould, 1997; Poindexter et al., 2005; Zhu et
al., 2017; Satoh et al., 2023). While the lens is confirmed to be binary,
there is no prior justification to assume a binary source, although this
possibility cannot be entirely dismissed. Therefore, we refrain from testing
the xallarap effect as long as the observed anomalies can be explained by
lens-orbital effects.

On occasion, light curves of 2L1S events affected by higher-order effects can
imitate the patterns seen in 3L1S event light curves. This confusion typically
arises when a distinct anomaly feature in a lensing light curve is isolated
from the primary ones, as seen in previous events such as MACHO-97-BLG-41,
OGLE-2013-BLG-0723, and KMT-2021-BLG-0322. For the analyzed events in this
work, we find that such degeneracies do not exist. In the following
subsections, we offer detailed analyses of each individual event.

Table 2: Lensing parameters of OGLE-2018-BLG-0971 Parameter | Static | Orbit | Orbit + Parallax
---|---|---|---
$\chi^{2}$ | $13921.7$ | $9973.2$ | $9972.3$
$t_{0}$ (${\rm HJD}^{\prime}$) | $58278.5148\pm 0.0085$ | $58279.0241\pm 0.0098$ | $58279.0245\pm 0.0097$
$u_{0}$ | $0.2771\pm 0.0006$ | $0.2956\pm 0.0005$ | $0.2955\pm 0.0008$
$t_{\rm E}$ (days) | $7.143\pm 0.012$ | $7.126\pm 0.009$ | $7.128\pm 0.011$
$s$ | $0.9552\pm 0.0006$ | $1.0070\pm 0.0009$ | $1.0074\pm 0.0010$
$q$ | $0.7161\pm 0.0062$ | $0.8669\pm 0.0064$ | $0.8653\pm 0.0066$
$\alpha$ (rad) | $2.1609\pm 0.0024$ | $2.2964\pm 0.0027$ | $2.2968\pm 0.0025$
$\rho$ ($10^{-2}$) | $1.078\pm 0.017$ | $1.218\pm 0.011$ | $1.221\pm 0.011$
$\pi_{{\rm E},N}$ | – | – | $0.07\pm 0.27$
$\pi_{{\rm E},E}$ | – | – | $-0.016\pm 0.091$
$ds/dt$ (yr-1) | – | $-1.722\pm 0.048$ | $-1.720\pm 0.094$
$d\alpha/dt$ (rad yr-1) | – | $0.931\pm 0.026$ | $0.939\pm 0.054$
Figure 1:  Light curve of the lensing event OGLE-2018-BLG-0971. The solid and
dotted curves drawn over data points represent the model curves obtained from
the 2L1S analyses with and without considering the lens orbital motion,
respectively. The arrows with labels $t_{1}$, $t_{2}$, and $t_{3}$ denote the
times of the major anomaly features.

###  3.1 OGLE-2018-BLG-0971

The OGLE group first identified the lensing event OGLE-2018-BLG-0971 on 4
June, 2018. Four days later, the MOA group verified the event, and the KMTNet
group later retrieved it from a post-season data examination. The MOA and
KMTNet groups designated the event as MOA-2018-BLG-173 and KMT-2018-BLG-2336,
respectively. Figure 1 shows the light curve of the event. The source of the
event was situated within the overlapping region of the three KMTNet fields –
BLG02, BLG03, and BLG43. To differentiate between the individual data sets, we
designate labels corresponding to the respective fields. The light curve
manifests multiple anomaly features centered at ${\rm HJD}\equiv{\rm
HJD}-2400000\sim 58277$ ($t_{1}$), $\sim 58280$ ($t_{2}$), and $\sim 58283$
($t_{3}$). The symmetric pattern with respect to $t_{1}$ suggests that the
anomaly around this epoch likely stems from the source’s approach to a caustic
cusp, while the U-shaped pattern spanning $t_{2}$ to $t_{3}$ indicates that
these epochs correspond to the times of caustic entrance and exit. Apart from
these anomaly features, there is an additional subtle anomaly feature
appearing just after the caustic exit. Figure 2 offers a detailed view of this
specific region.

After modeling the light curve using a static 2L1S framework, we identified a
solution that broadly captures the features of the anomaly. In Table 2, we
list the lensing parameters of the static solution. However, it is found that
the static model exhibits subtle residuals, particularly in the vicinity of
$t_{3}$, as illustrated in the magnified view presented in Figure 2.

Figure 2:  Enlarged view of the OGLE-2018-BLG-0971 light curve in the region
around $t_{2}$ and $t_{3}$ marked in Fig. 2.  Figure 3:  Scatter plots of
points in the MCMC chain on the $\pi_{{\rm E},E}$–$\pi_{{\rm E},N}$ planes for
the four events analyzed in this paper. Points are color-coded to represent
those with $<1\sigma$ (red), $<2\sigma$ (yellow), $<3\sigma$ (green), and
$<4\sigma$ (cyan).  Figure 4:  Lens system configuration for
OGLE-2018-BLG-0971. The arrowed line represents the source trajectory, and the
closed figures composed of concave curves represent caustics. The caustic
shape and location evolve over time due to the orbital motion of the lens. The
sets of caustics drawn in black, green, and red correspond to the three
anomaly epochs $t_{1}$, $t_{2}$, and $t_{3}$ marked in Fig. 1. The right
panels display still frames capturing the source’s approach to the caustic at
these three epochs. The orange circle on the source trajectory appearing in
each right panel is sized to indicate the angular dimension of the source
relative to the size of the caustic.

In light of deviations from the static model, we checked the feasibility of
explaining the deviation with higher-order effects. The event duration,
estimated at $t_{\rm E}\sim 7.1$ days from the static model, is short.
Consequently, we initially considered the lens-orbital effect in the modeling.
Through this approach, we derived a solution that accounts for all anomaly
features. The model curve of this solution is shown as a solid line in Figure
1, offering a comprehensive view, and in Figure 2, providing an enlarged view
around $t_{3}$. The fit significantly improved with the inclusion of lens
orbital motion, by $\Delta\chi^{2}=3949.4$ compared to the static model.
Subsequent examination of microlens-parallax effects through additional
modeling revealed a minimal fit improvement, by $\Delta\chi^{2}=0.9$,
indicating the predominant influence of lens-orbital effects. In Figure 3, we
plot the scatter plot of points in the MCMC chain on the $\pi_{{\rm
E},E}$–$\pi_{{\rm E},N}$ plane. In Table 2, we provide the lensing parameters
for both the orbit-only and orbit+parallax solutions. The parameters defining
the binary lens are $(s,q)\sim(1.01,0.87)$, indicating that the event was
generated a binary system composed of roughly equal masses with a separation
close to the Einstein radius of the lens system. Although the time scale of
the event, $t_{\rm E}\sim 7.1$ days, represents only a small fraction of the
Earth’s orbital period, we conducted separate modeling specifically to account
for the microlens-parallax effect. From this analysis, it was observed that
not only did the fit perform worse compared to the orbit-only model by
$\Delta\chi^{2}=42.6$, but also the derived parallax parameters $(\pi_{{\rm
E},E},\pi_{{\rm E},N})\sim(35.80,32.50)$ appeared absurdly large for a typical
Galactic lensing event.

The configuration of the lens system for the lensing event OGLE-2018-BLG-0971
is shown in Figure 4. This configuration reveals that the lens system forms a
single set of resonant caustics featuring six cusps – two along the binary
axis and four positioned away from the axis. The source initially approached
the left on-axis cusp around $t_{1}$, entered the caustic near $t_{2}$, and
exited the caustic at around $t_{3}$. These caustic approach and crossings
gave rise to anomalies at the corresponding epochs. The primary deviation of
the static 2L1S model, particularly around $t_{3}$, stems from its inability
to account for the variation in the caustic caused by the orbital motion of
the binary lens.

Table 3: Lensing parameters of MOA-2023-BLG-065 Parameter | Static | Parallax | Orbit | Orbit + Parallax
---|---|---|---|---
$\chi^{2}$ | $2399.0$ | $1876.6$ | $1882.7$ | $1872.6$
$t_{0}$ (${\rm HJD}^{\prime}$) | $60030.308\pm 0.024$ | $60030.140\pm 0.045$ | $60029.951\pm 0.079$ | $60030.356\pm 0.090$
$u_{0}$ | $-0.3893\pm 0.0053$ | $-0.426\pm 0.009$ | $-0.4539\pm 0.0079$ | $-0.4378\pm 0.0083$
$t_{\rm E}$ (days) | $30.74\pm 0.51$ | $37.27\pm 0.59$ | $39.03\pm 0.54$ | $37.81\pm 0.75$
$s$ | $1.327\pm 0.011$ | $1.452\pm 0.008$ | $1.471\pm 0.007$ | $1.467\pm 0.010$
$q$ | $1.089\pm 0.013$ | $0.877\pm 0.041$ | $0.939\pm 0.023$ | $0.887\pm 0.022$
$\alpha$ (rad) | $4.7757\pm 0.0001$ | $4.7668\pm 0.0015$ | $4.7705\pm 0.0050$ | $4.7477\pm 0.0062$
$\rho$ ($10^{-3}$) | $2.018\pm 0.039$ | $1.724\pm 0.041$ | $1.763\pm 0.033$ | $1.739\pm 0.037$
$\pi_{{\rm E},N}$ | – | $0.069\pm 0.119$ | – | $-0.20\pm 0.26$
$\pi_{{\rm E},E}$ | – | $-0.157\pm 0.045$ | – | $-0.214\pm 0.054$
$ds/dt$ (yr-1) | – | – | $0.015\pm 0.071$ | $-0.296\pm 0.082$
$d\alpha/dt$ (rad yr-1) | – | – | $-0.531\pm 0.159$ | $0.217\pm 0.506$
Figure 5:  Lensing light curve of MOA-2023-BLG-065. The left and right insets
show the zoom-in views of the regions around the caustic spikes at $t_{1}$ and
$t_{3}$.

###  3.2 MOA-2023-BLG-065

Figure 5 shows the lensing light curve of the MOA-2023-BLG-065 event. The
source flux magnification induced by lensing was initially identified on 17
March, 2023 (${\rm HJD}^{\prime}=60021$) through the survey conducted by the
MOA group, subsequently confirmed by the KMTNet group. The ID reference
designated by the KMTNet survey is KMT-2023-BLG-2430. The light curve displays
a complex pattern comprising multiple anomaly features. Notably, two spike
features at $t_{1}\sim 60020$ and $t_{3}\sim 60035$ appear to be a pair of
caustic-crossing spikes. Furthermore, the symmetry observed between the
ascending and descending segments of the anomaly feature centered at
$t_{4}\sim 60043$ implies that it originated from the source approach to a
cusp of the caustic. While the magnification between caustic spikes typically
follows a U-shape pattern, the region between the caustic spikes at $t_{1}$
and $t_{3}$ deviates significantly from the U-shape, displaying a distinctive
rise and fall in the region centered at $t_{2}\sim 60023$. This deviation is
indicative of a source asymptotically approaching a fold of the caustic.

From the 2L1S analysis of the lensing light curve under a static binary frame,
we found that the model falls short of precisely describing the data, even
though it approximately outlines the anomaly features. In Figure 5, the static
2L1S model is represented by a dotted curve. This static model exhibits a
notably inadequate fit especially in the region of the anomaly centered at
$t_{4}$, as highlighted in the enlarged view presented in Figure 6. The
complete lensing parameters for the static 2L1S solution are detailed in Table
3.

Figure 6:  Enlarged view of the MOA-2023-BLG-065 light curve in the region
around $t_{4}$ marked in Fig. 5.

Although the static solution cannot adequately describe the anomaly feature at
around $t_{4}$, it does exhibit an anomaly around the time of that anomaly.
This suggests the possibility that it can be described with a slight
deformation of the source trajectory caused by higher-order effects. In light
of this possibility, we conducted three additional sets of modeling: the first
two models separately incorporated microlens-parallax and lens-orbital
effects, while the third model encompassed both effects simultaneously. In
Table 3, we present the lensing parameters for the three models. Comparing the
static and higher-order solution demonstrates a notable enhancement in fit,
with a $\Delta\chi^{2}=526.4$ compared to the static solution. In Figure 3, we
present the scatter plot of points in the MCMC chain on the $\pi_{{\rm
E},E}$–$\pi_{{\rm E},N}$ plane. It shows that the east component of the
microlens-parallax vector, $\pi_{{\rm E},E}$, is constrained, although the
uncertainty of the north component, $\pi_{{\rm E},N}$, is large. Additionally,
the normalized source radius $\rho=(1.739\pm 0.037)\times 10^{-3}$ was
measured from the deformation of the light curve by finite source effects
during the epochs around $t_{2}$ and $t_{3}$. The magnified views of these
specific regions are shown in the two insets of Figure 5.

Figure 7:  Lens system configuration for MOA-2023-BLG-065.

Figure 7 illustrates the lens-system configuration of MOA-2023-BLG-065. It
shows that the binary lens, characterized by parameters $(s,q)\sim(1.5,0.9)$,
creates a resonant caustic, and the source traversed the left side of this
caustic. Initially, the source passed the upper fold, creating the first
caustic spike at $t_{1}$. Subsequently, it approached the upper left fold
asymptotically, resulting in rising and falling features around $t_{2}$. The
source then exited the caustic, generating the second caustic spike at
$t_{3}$, and proceeded to pass the tip of the lower left cusp, causing the
final anomaly feature at around $t_{4}$. The distortion of the caustic due to
the orbital motion of the lens led the light curve to deviate from the
anticipated behavior according to the static model. This discrepancy was
particularly noticeable in the part of the light curve during the final cusp
approach, enabling the detection of the lens orbital effect.

Table 4: Lensing parameters of OGLE-2023-BLG-0136 Parameter | Static | Parallax | Orbit | Orbit + Parallax
---|---|---|---|---
$\chi^{2}$ | $3180.0$ | $2773.6$ | $812.6$ | $809.4$
$t_{0}$ (${\rm HJD}^{\prime}$) | $60030.726\pm 0.093$ | $60029.065\pm 0.132$ | $60030.276\pm 0.107$ | $60030.08\pm 0.11$
$u_{0}$ | $0.2685\pm 0.0007$ | $0.2290\pm 0.0008$ | $0.2808\pm 0.0012$ | $0.2833\pm 0.0017$
$t_{\rm E}$ (days) | $66.70\pm 0.13$ | $86.00\pm 0.2999499$ | $61.25\pm 0.13$ | $59.35\pm 0.48$
$s$ | $0.66848\pm 0.00042$ | $0.68593\pm 0.00034$ | $0.71297\pm 0.00035$ | $0.71101\pm 0.00085$
$q$ | $0.3117\pm 0.0015$ | $0.2607\pm 0.0025$ | $0.2851\pm 0.0027$ | $0.2978\pm 0.0034$
$\alpha$ (rad) | $1.4503\pm 0.0009$ | $1.3623\pm 0.0019$ | $1.4268\pm 0.0016$ | $1.42832\pm 0.0027$
$\rho$ ($10^{-3}$) | $1.965\pm 0.029$ | $1.535\pm 0.03$ | $1.992\pm 0.029$ | $2.005\pm 0.029$
$\pi_{{\rm E},N}$ | – | $0.016\pm 0.003$ | – | $0.080\pm 0.021$
$\pi_{{\rm E},E}$ | – | $-0.220\pm 0.001$ | – | $0.030\pm 0.010$
$ds/dt$ (yr-1) | – | – | $-0.4403\pm 0.0037$ | $-0.4410\pm 0.0039$
$d\alpha/dt$ (rad yr-1) | – | – | $-0.237\pm 0.007$ | $-0.408\pm 0.050$
Figure 8:  Lensing light curve of OGLE-2023-BLG-0136. The insets provides
closer look at the regions surrounding $t_{2}$ and $t_{3}$.

###  3.3 OGLE-2023-BLG-0136

The lensing event OGLE-2023-BLG-0136 was first found during its early phase by
the OGLE group on 1 April, 2023 (${\rm HJD}^{\prime}=60036$). Subsequently,
the KMTNet survey validated the event, designating it as KMT-2023-BLG-2849.
Figure 8 illustrates the lensing light curve of the event. The curve exhibits
a complex pattern of anomalies, comprised of multiple distinct features: a
caustic-crossing feature around $t_{1}=59997$ and two additional features
centered at around $t_{2}=60089$ and $t_{3}=60103$. The first feature had
limited coverage because it was observed solely by the KMTC telescope, while
the other KMTNet telescopes and the OGLE telescope did not start observations
of the 2023 season at the time of the anomaly. However, the subsequent two
anomaly features were extensively observed by the combined data from both the
OGLE and KMTNet surveys.

We initially modeled the lensing light curve under a static 2L1S framework.
The model curve of the static solution is depicted by a dotted curve in Figure
8, and the lensing parameters of the solution are listed in Table 4. Upon
examination of the fit, it is observed that this model roughly captures the
anomaly features around $t_{2}$ and $t_{3}$, yet it lacks accuracy in
explaining the feature at $t_{1}$. Figure 9 offers a closer look at the model
fit around $t_{1}$. While the static model does not precisely capture the
first caustic-crossing anomaly, it does display a weak bump feature which
seems to result from a caustic approach. This suggests that the source might
pass over the caustic, potentially influenced by a slight shift in the caustic
position due to the orbital motion of the binary lens. Considering this, we
move forward with additional modeling, incorporating the effects of the lens
orbital motion.

Figure 9:  Enlarged view of the OGLE-2023-BLG-0136 light curve in the region
around $t_{1}$ marked in Fig. 8.

The solid curves in Figures 8 and 9 represent the model derived from the
orbital solution. The lensing parameters of the solution are listed in Table
4. From the inspection of the fit, it is found that the orbital solution
accurately accounts for the anomaly around $t_{1}$, improving the fit by
$\Delta\chi^{2}=2367.4$ compared to the static solution. The estimated event
time scale, $t_{\rm E}\sim 60$ days, is moderately long, and thus we further
examined whether incorporating the microlens-parallax effect could enhance the
fit. From the model derived with the consideration both higher-order effects,
we found a marginal enhancement in the fit, $\Delta\chi^{2}=4.5$, compared to
the orbital solution. This suggests that the dominant higher-order effect is
attributed to the lens orbital motion. Details of the lensing parameters for
the parallax and orbit+parallax solutions are outlined in Table 4 and the
scatter plot on the $\pi_{{\rm E},E}$–$\pi_{{\rm E},N}$ planet is shown in
Figure 3. The estimated binary lens parameters are $(s,q)\sim(0.71,0.30)$. The
normalized source radius, $\rho=(2.005\pm 0.029)\times 10^{-3}$, was precisely
measured from the deformed light curve during the anomaly at around $t_{2}$.
The enlarged view of this region is shown in the inset of Figure 8.

Figure 10:  Configuration of the lens system for OGLE-2023-BLG-0136.

In Figure 10, we illustrate the lens system configuration for
OGLE-2023-BLG-0136. At the time of the first anomaly, the caustic exhibited a
resonant form in which the central caustic and peripheral caustics were
interconnected by narrow bridges. The first anomaly feature at around $t_{1}$
was generated when the source traversed the lower bridge of this resonant
caustic. As the source proceeded, the binary separation decreased, leading to
the detachment of the peripheral caustics from the central caustic. The second
and third anomaly features arose from the successive passages of the source
over the lower and right cusps of the upper peripheral caustic. The static
model effectively captured the second and third anomaly features due to their
close temporal proximity, which minimized the deformation of the caustic
caused by the lens orbital motion. However, the time gap between these
features and the first anomaly feature exceeded 100 days, resulting in
substantial alterations to the caustic shape and position. As a result, the
static model fell short in accurately representing the observed light curve.

It is important to note that event time scales can significantly vary between
static and non-static models. For instance, in the case of MOA-2023-BLG-065,
the time scale shifts from $\sim 30.7$ days for the static solution to $\sim
37.8$ days for the higher-order solution. Similarly, for OGLE-2023-BLG-0136,
the time scale changes from $\sim 66.7$ days for the static solution to $\sim
59.4$ days for the higher-order solution. The time scale serves as a
fundamental observable for constraining the physical parameters of the lens.
Therefore, accounting for higher-order effects in modeling is essential for
accurately determining these parameters.

Table 5: Source parameters, Einstein radii, and relative proper motions  Parameter | OGLE-2018-BLG-0971 | MOA-2023-BLG-065 | OGLE-2023-BLG-0136
---|---|---|---
$(V-I)_{\rm S}$ | $2.446\pm 0.021$ | $1.730\pm 0.011$ | $1.488\pm 0.035$
$I_{\rm S}$ | $18.876\pm 0.002$ | $20.490\pm 0.002$ | $19.373\pm 0.005$
$(V-I,I)_{\rm RGC}$ | $(2.645,116.173)$ | $(2.131,15.788)$ | $(1.817,15.452)$
$(V-I,I)_{\rm RGC,0}$ | $(1.060,14.371)$ | $(1.060,14.386)$ | $(1.060,14.393)$
$(V-I)_{\rm S,0}$ | $0.861\pm 0.045,$ | $0.659\pm 0.041,$ | $0.731\pm 0.053,$
$I_{0}$ | $17.073\pm 0.020$ | $19.089\pm 0.020$ | $18.315\pm 0.021$
Type | K1IV | G0V | G4V
$\theta_{*}$ ($\mu$as) | $1.433\pm 0.119$ | $0.451\pm 0.037$ | $0.699\pm 0.061$
$\theta_{\rm E}$ (mas) | $0.117\pm 0.010$ | $0.256\pm 0.021$ | $0.351\pm 0.031$
$\mu$ (mas/yr) | $6.01\pm 0.50$ | $2.42\pm 0.20$ | $2.13\pm 0.19$

##  4 Source stars and Einstein radii

In this section, we specify the source stars and determine the angular
Einstein radii of the events. The source stars are specified based on their
colors and magnitudes, accounting for corrections due to reddening and
extinction. In this process, we first conducted photometry of the $I$ and
$V$-band data using the pyDIA code (Albrow, 2017), and then estimated the
instrumental source color and magnitude, $(V-I,I)_{\rm S}$, by regressing the
data of the individual passbands with respect to the model. Calibration of the
source color and magnitude follows the method outlined by Yoo et al. (2004),
leveraging the centroid of the red giant clump (RGC) in the color-magnitude
diagram (CMD) for this purpose. The RGC centroid can be used for a reference
because its de-reddened color and magnitude, represented as $(V-I,I)_{\rm
RGC,0}$, are established from studies conducted by Bensby et al. (2013) and
Nataf et al. (2013). In the calibration process, we first positioned the
source in the CMD constructed using the pyDIA code, measured the offsets of
the source in color and magnitude, $\Delta(V-I,I)$, from the RGC centroid, and
then estimated the de-reddened source color and magnitude as:

$(V-I,I)_{\rm S,0}=(V-I,I)_{\rm RGC,0}+\Delta(V-I,I).$ (3)

In Figure 11, we indicate the locations of source stars for the individual
events relative to the RGC centroids on the instrumental CMDs constructed
using the KMTC data sets. The values estimated for $(V-I,I)_{\rm S}$,
$(V-I,I)_{\rm RGC}$, $(V-I,I)_{\rm RGC,0}$, and $(V-I,I)_{\rm S,0}$ through
the described procedure are compiled in Table 5. Based on the derived colors
and magnitudes, we determine that the source star of OGLE-2018-BLG-0971 is a
K-type subgiant. Additionally, the sources of MOA-2023-BLG-065 and
OGLE-2023-BLG-0136 are G-type main-sequence stars.

Figure 11:  Positions of source stars with respect to the centroids of red
giant clump (RGC) in the instrumental color-magnitude diagrams. For
MOA-2023-BLG-065 and OGLE-2023-BLG-0136, the positions of the blends are
additionally marked.

The angular Einstein radius of each event was estimated from the relation

$\theta_{\rm E}={\theta_{*}\over\rho},$ (4)

where the angular source radius $\theta_{*}$ is deduced from the color and
magnitude, and the normalized source radius $\rho$ is measured from the light
curve analysis. To derive the source radius, we initially transformed the
measured $V-I$ color into $V-K$ color using the Bessell & Brett (1988)
relation. Subsequently, we determined $\theta_{*}$ using the relationship
provided by Kervella et al. (2004) between $(V-K,V)$ and $\theta_{*}$. The
estimated angular radii of source stars for the individual lensing events are
listed in Table 5, along with the corresponding angular Einstein radii
calculated using the relationship described in Eq. (3). Also listed in the
table are the relative lens-source proper motion estimated by the relation

$\mu={\theta_{\rm E}\over t_{\rm E}}.$ (5)

For MOA-2023-BLG-065 and OGLE-2023-BLG-0136, we were able to constrain the
blended light. In Figure 11, we mark the positions of blend in the CMDs. To
assess the likelihood of the lens being the primary source of the blended
flux, we measured the centroids of the source image during lensing
magnification and at the baseline. In the case of MOA-2023-BLG-065, the
measured astrometric offset between the source positions is
$\delta\theta=0.46\pm 0.08$ arcsec, and this excludes the possibility that the
blended light comes mainly from the lens. For OGLE-2023-BLG-0136, the
significant astrometric uncertainty prevents drawing a meaningful conclusion
regarding the origin of the blended light.

##  5 Physical lens parameters

The physical parameters of a lens are constrained by the lensing observables
of $t_{\rm E}$, $\theta_{\rm E}$, and $\pi_{\rm E}$. When all these parameters
are simultaneously measured, the mass and distance to the lens are uniquely
determined by

$M={\theta_{\rm E}\over\kappa\pi_{\rm E}};\qquad D_{\rm L}={{\rm
au}\over\pi_{\rm E}\theta_{\rm E}+\pi_{\rm S}}.$ (6)

Here $\kappa=4G/(c^{2}{\rm au})$ and $\pi_{\rm S}={\rm au}/D_{\rm S}$
represents the parallax of the source lying at a distance $D_{\rm S}$ (Gould,
2000). For all analyzed events, the observables of $t_{\rm E}$ and
$\theta_{\rm E}$ were precisely measured, but the constraint on the microlens
parallax was relatively weak because of its subtle effects. Due to these
incomplete measurements of the lensing observables, we determine the physical
lens parameters through Bayesian analyses of the individual events. This
approach integrates constraints from measured lensing observables with priors
derived from the physical and dynamic distributions, as well as the mass
function of lens objects within the Galaxy.

Table 6: Physical lens parameters  Parameter | OGLE-2018-BLG-0971 | MOA-2023-BLG-065 | OGLE-2023-BLG-0136
---|---|---|---
$M_{1}$ ($M_{\odot}$) | $0.215^{+0.231}_{-0.126}$ | $0.30^{+0.27}_{-0.15}$ | $0.88^{+0.23}_{-0.18}$
$M_{2}$ ($M_{\odot}$) | $0.186^{+0.200}_{-0.109}$ | $0.28^{+0.25}_{-0.14}$ | $0.259^{+0.068}_{-0.052}$
$D_{\rm L}$ (kpc) | $7.50^{+0.98}_{-1.01}$ | $7.62^{+0.89}_{-0.84}$ | $7.52^{+0.92}_{-1.13}$
$a_{\perp}$ (au) | $1.39^{+0.18}_{-0.19}$ | $3.10^{+0.36}_{-0.34}$ | $1.92^{+0.23}_{-0.29}$
$p_{\rm disk}$ | $26\%$ | $9\%$ | $21\%$
$p_{\rm bulge}$ | $74\%$ | $91\%$ | $79\%$

The Bayesian analysis began by generating a large number of synthetic events
via Monte Carlo simulation. Within this simulation, the physical parameters of
the lens mass were deduced from a model mass function, while the distances to
the lens and source, along with their relative proper motion, were derived
from a Galaxy model. Our approach incorporated the mass function model
suggested by Jung et al. (2018) and utilized the Galaxy model introduced by
Jung et al. (2021). For each synthetic event defined by physical parameters
$(M_{i},D_{{\rm L},i},D_{{\rm S},i},\mu_{i})$, we calculated the the values of
the corresponding lensing observables using the relations

$t_{{\rm E},i}={\theta_{{\rm E},i}\over\mu_{i}};\qquad\theta_{{\rm
E},i}=\sqrt{\kappa M_{i}\pi_{{\rm rel},i}}$ (7)

for the event time scale and Einstein radius, respectively, and using the
relation in Eq. (1) for the microlens parallax. Then, the posteriors of $M$
and $D_{\rm L}$ were obtained by assigning a weight to each event of
$w_{i}=\exp(-\chi_{i}^{2}/2)$, where $\chi_{i}^{2}$ value is computed by

$\chi_{i}^{2}={\Delta t_{{\rm E},i}^{2}\over\sigma^{2}(t_{\rm
E})}+{\Delta\theta_{{\rm E},i}^{2}\over\sigma^{2}(\theta_{\rm
E})}+\sum_{j=1}^{2}\sum_{k=1}^{2}b_{j,k}(\pi_{{\rm E},j,i}-\pi_{{\rm
E},i})(\pi_{{\rm E},k,i}-\pi_{{\rm E},i}).$ (8)

Here, $\Delta t_{{\rm E},i}=t_{{\rm E},i}-t_{\rm E}$, $\Delta\theta_{{\rm
E},i}=\theta_{{\rm E},i}-\theta_{\rm E}$, $(t_{\rm E},\theta_{\rm E})$ stand
for the measured values of the observables, $[\sigma(t_{\rm
E}),\sigma(\theta_{\rm E})]$ indicate their corresponding uncertainties, and
$b_{j,k}$ represents the inverse covariance matrix of the microlens-parallax
vector $\mbox{\boldmath$\pi$}_{\rm E}$, $(\pi_{{\rm E},1},\pi_{{\rm
E},2})_{i}=(\pi_{{\rm E},N},\pi_{{\rm E},E})_{i}$ are the parallax parameters
of each simulated event, and $(\pi_{{\rm E},N},\pi_{{\rm E},E})$ represent the
parallax parameters measured from modeling. Han et al. (2016b) showed that
parallax measurements can be important even when there is little $\chi^{2}$
improvement, because they can constrain $\pi_{\rm E}$ to be small. We,
therefore, consider the constraints given by the measured parallax parameters
in the Bayesian analyses.

Figure 12:  Bayesian posterior distributions of the primary lens mass
($M_{1}$) for the lensing events. In each panel, the event rate contributions
from the disk and bulge lens populations are depicted by the blue and red
curves, respectively, while black curve represents the sum of the
contributions from two lens populations.

In Figures 12 and 13, we present the Bayesian posteriors of the primary lens
masses and distances to the lens systems. In Table 6, we summarize the
estimated physical parameters for the individual lensing events. Among the
parameters, $M_{1}$ and $M_{2}$ denote the masses of the primary and companion
of the lens, and $a_{\perp}$ denotes the projected physical separation between
$M_{1}$ and $N_{2}$. We present the median value derived from the Bayesian
posterior distribution as a representative value for each physical parameter,
with uncertainties estimated within the 16% to 84% range of the distribution.
Also listed in the table are the relative probabilities for the lens being in
the Galactic disk, $p_{\rm disk}$, and in the bulge, $p_{\rm bulge}$.
According to the estimated masses, the lenses of the events OGLE-2018-BLG-0971
and MOA-2023-BLG-065 are binaries composed of M dwarfs. On the other hand, the
lens of OGLE-2023-BLG-0136 is likely to be a binary composed of an early
K-dwarf primary and a late M-dwarf companion. Across all the analyzed events,
$p_{\rm bulge}$ is substantially higher than $p_{\rm disk}$, suggesting a
higher likelihood of the all lenses being located in the bulge rather than the
disk. The probabilities $p_{\rm disk}$ and $p_{\rm bulge}$ were found by
analyzing the proportion of artificial lensing events where the lenses came
from either the disk or bulge distributions within the Galaxy model utilized
in the Monte Carlo simulation process.

Figure 13:  Bayesian posterior distributions of the distances to the lens
($D_{\rm L}$). The notations correspond to those used in Fig. 12.

##  6 Summary

We have analyzed microlensing data collected from high-cadence surveys to
reevaluate lensing events that lacked proposed interpretations for their
intricate anomaly features. Through detailed reanalyses considering higher-
order effects, we identified that accounting for the orbital motions of lenses
was vital in accurately explaining the anomaly features observed in the
lensing light curves of OGLE-2018-BLG-0971, MOA-2023-BLG-065, and
OGLE-2023-BLG-0136.

By conducting Bayesian analyses based on the lensing parameters from the newly
found solutions and together with the constraints derived from lensing
observables, we estimated masses and distances to the lenses. These analyses
revealed that the lenses for events OGLE-2018-BLG-0971 and MOA-2023-BLG-065
are binary systems consisting of M dwarfs. Additionally, for
OGLE-2023-BLG-0136, the lens is likely to be a binary system comprising an
early K-dwarf primary and a late M-dwarf companion. Notably, across all
observed lensing events, the likelihood of the lens being in the bulge
significantly outweighs its likelihood of being in the disk.

###### Acknowledgements.

Work by C.H. was supported by the grants of National Research Foundation of
Korea (2019R1A2C2085965). J.C.Y. and I.-G.S. acknowledge support from U.S. NSF
Grant No. AST-2108414. Y.S. acknowledges support from BSF Grant No. 2020740.
This research has made use of the KMTNet system operated by the Korea
Astronomy and Space Science Institute (KASI) at three host sites of CTIO in
Chile, SAAO in South Africa, and SSO in Australia. Data transfer from the host
site to KASI was supported by the Korea Research Environment Open NETwork
(KREONET). This research was supported by KASI under the R&D program (project
No. 2023-1-832-03) supervised by the Ministry of Science and ICT. W.Z. and
H.Y. acknowledge support by the National Natural Science Foundation of China
(Grant No. 12133005). W. Zang acknowledges the support from the Harvard-
Smithsonian Center for Astrophysics through the CfA Fellowship. The MOA
project is supported by JSPS KAKENHI Grant Number JP24253004, JP26247023,
JP16H06287 and JP22H00153.

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