<figure><img src="image_1.png" /></figure>

Monitoring the Morphology of M87* in 2009–2017 with the Event Horizon Telescope

Maciek Wielgus<fnref n="1" />,<fnref n="2" /> ID, Kazunori Akiyama<fnref n="1" />,<fnref n="3" />,<fnref n="4" />,<fnref n="5" /> ID, Lindy Blackburn<fnref n="1" />,<fnref n="2" /> ID, Chi-kwan Chan<fnref n="6" />,<fnref n="7" /> ID, Jason Dexter<fnref n="8" /> ID,
Sheperd S. Doeleman<fnref n="1" />,<fnref n="2" /> ID, Vincent L. Fish<fnref n="4" /> ID, Sara Issaoua<fnref n="9" /> ID, Michael D. Johnson<fnref n="1" />,<fnref n="2" /> ID, Thomas P. Krichbaum<fnref n="10" /> ID, Ru-Sen Lu
(路如森)<fnref n="11" />,<fnref n="10" /> ID, Dominic W. Pesce<fnref n="1" />,<fnref n="2" /> ID, George N. Wong¹²,<fnref n="13" /> ID, Geoffrey C. Bower<fnref n="14" /> ID, Avery E. Broderick<fnref n="15" />,<fnref n="16" />,¹⁷ ID,
Andrew Chael¹⁸,<fnref n="10" />,¹⁸ ID, Koushik Chatterjee¹⁹ ID, Charles F. Gammie¹²,²⁰ ID, Boris Georgiev<fnref n="16" />,¹⁷ ID, Kazuhiro Hada²¹,²² ID,
Laurent Loinard²³,²⁴ ID, Sera Markoff¹⁹,²⁵ ID, Daniel P. Marrone<fnref n="6" /> ID, Richard Plambeck²⁶ ID, Jonathan Weintroub<fnref n="1" />,<fnref n="2" /> ID,
Matthew Dexter²⁶ ID, David H. E. MacMahon²⁶ ID, Melvyn Wright²⁶ ID, Antxon Alberdi²⁷ ID, Walter Alef<fnref n="10" /> ID, Keiichi Asada²⁸ ID,
Rebecca Azulay<fnref n="10" />,²⁹,³⁰ ID, Anne-Kathrin Baczko<fnref n="10" /> ID, David Ball<fnref n="6" /> ID, Mislav Baloković<fnref n="1" />,<fnref n="2" /> ID, Enrico Barausse³¹,³² ID, John Barrett<fnref n="4" /> ID,
Dan Bintley³³ ID, Wilfred Boland³⁴ ID, Katherine L. Bouman<fnref n="1" />,<fnref n="2" />,³⁵ ID, Michael Bremer³⁶ ID, Christiaan D. Brinkerink<fnref n="9" /> ID,
Roger Brissenden<fnref n="1" />,<fnref n="2" /> ID, Silke Britzen<fnref n="10" /> ID, Dominique Broguiere³⁶ ID, Thomas Bronzwaer<fnref n="9" /> ID, Do-Young Byun³⁷,³⁸ ID,
John E. Carlstrom³⁹,⁴⁰,⁴¹,⁴² ID, Shami Chatterjee⁴³ ID, Ming-Tang Chen<fnref n="14" /> ID, Yongjun Chen (陈永军)<fnref n="11" />,⁴⁴ ID, Ilje Cho³⁷,³⁸ ID,
Pierre Christian<fnref n="2" />,<fnref n="6" /> ID, John E. Conway⁴⁵ ID, James M. Cordes⁴³ ID, Geoffrey B. Crew<fnref n="4" /> ID, Yuzhu Cui²¹,²² ID, Jordy Davelaar<fnref n="9" /> ID,
Mariafelicia De Laurentis⁴⁶,⁴⁷,⁴⁸ ID, Roger Deane⁴⁹,⁵⁰ ID, Jessica Dempsey³³ ID, Gregory Desvignes⁵¹ ID, Sergio A. Dzib<fnref n="10" /> ID,
Ralph P. Eatough<fnref n="10" /> ID, Heino Falcke<fnref n="9" /> ID, Ed Fomalont<fnref n="3" /> ID, Raquel Fraga-Encinas<fnref n="9" /> ID, Per Friberg³³ ID, Christian M. Fromm⁴⁷ ID,
Peter Galison<fnref n="1" />,<fnref n="5" />,²⁵ ID, Roberto García³⁶ ID, Olivier Gentaz³⁶ ID, Ciriaco Goddi<fnref n="9" />,⁵⁴ ID, Roman Gold<fnref n="15" />,⁵⁵ ID, José L. Gómez²⁷ ID,
Arturo I. Gómez-Ruiz⁵⁶,⁵⁷ ID, Minfeng Gu (顾敏峰)<fnref n="11" />,⁵⁸ ID, Mark Gurwell<fnref n="2" /> ID, Michael H. Hecht<fnref n="4" /> ID, Ronald Hesper⁵⁹ ID, Luis C. Ho
(何子山)⁶⁰,⁶¹ ID, Paul Ho²⁸ ID, Mareki Honma²¹,²² ID, Chih-Wei L. Huang²⁸ ID, Lei Huang (黄磊)<fnref n="11" />,⁵⁸ ID, David H. Hughes⁶² ID,
Makoto Inoue²⁸ ID, David J. James⁶³ ID, Buell T. Jannuzi<fnref n="6" /> ID, Michael Janssen<fnref n="9" /> ID, Britton Jeter<fnref n="16" />,¹⁷ ID, Wu Jiang (江悟)<fnref n="11" /> ID,
Alejandra Jimenez-Rosales⁶⁴ ID, Svetlana Jorstad⁶⁵,⁶⁶ ID, Taehyun Jung³⁷,³⁸ ID, Mansour Karami<fnref n="15" />,<fnref n="16" /> ID, Ramesh Karuppusamy<fnref n="10" /> ID,
Tomohisa Kawashima<fnref n="5" /> ID, Garrett K. Keating<fnref n="2" /> ID, Mark Kettenis⁶⁷ ID, Jae-Young Kim<fnref n="10" /> ID, Junhan Kim<fnref n="6" />,³⁵ ID, Jongsoo Kim³⁷ ID,
Motoki Kino<fnref n="5" />,⁶⁸ ID, Jun Yi Koay²⁸ ID, Patrick M. Koch²⁸ ID, Shoko Koyama²⁸ ID, Michael Kramer<fnref n="10" /> ID, Carsten Kramer³⁶ ID,
Cheng-Yu Kuo⁶⁹ ID, Tod R. Lauer⁷⁰ ID, Sang-Sung Lee³⁷ ID, Yan-Rong Li (李彦荣)⁷¹ ID, Zhiyuan Li (李志远)⁷²,⁷³ ID,
Michael Lindqvist⁴⁵ ID, Rocco Lico<fnref n="10" /> ID, Kuo Liu<fnref n="10" /> ID, Elisabetta Liuzzo⁷⁴ ID, Wen-Ping Lo²⁸,⁷⁵ ID, Andrei P. Lobanov<fnref n="10" /> ID,
Colin Lonsdale<fnref n="4" /> ID, Nicholas R. MacDonald<fnref n="10" /> ID, Jirong Mao (毛基荣)⁷⁶,⁷⁷,⁷⁸ ID, Nicola Marchili<fnref n="10" />,⁷⁴ ID, Alan P. Marscher⁶⁵ ID,
Iván Martí-Vidal²⁹,³⁰ ID, Satoki Matsushita²⁸ ID, Lynn D. Matthews<fnref n="4" /> ID, Lia Medeiros<fnref n="6" />,⁷⁹ ID, Karl M. Menten<fnref n="10" /> ID,
Yosuke Mizuno⁴⁷,⁸⁰ ID, Izumi Mizuno³³ ID, James M. Moran<fnref n="1" />,<fnref n="2" /> ID, Kotaro Moriyama<fnref n="4" />,²¹ ID, Monika Moscibrodzka<fnref n="9" /> ID,
Cornelia Müller<fnref n="9" />,<fnref n="10" /> ID, Gibwa Musoke<fnref n="9" />,¹⁹ ID, Hiroshi Nagai<fnref n="5" />,²² ID, Neil M. Nagar⁸¹ ID, Masanori Nakamura²⁸ ID,
Ramesh Narayan<fnref n="1" />,<fnref n="2" /> ID, Gopal Narayanan⁸², Iniyan Natarajan⁵⁰ ID, Antonios Nathanail⁴⁷ ID, Roberto Neri³⁶ ID, Chunchong Ni<fnref n="16" />,¹⁷ ID,
Aristeidis Noutsos<fnref n="10" /> ID, Hiroki Okino²¹,⁸³ ID, Héctor Olivares⁴⁷ ID, Gisela N. Ortiz-León<fnref n="10" /> ID, Tomoaki Oyama²¹ ID, Feryal Özel<fnref n="6" /> ID,
Daniel C. M. Palumbo<fnref n="1" />,<fnref n="2" /> ID, Jongho Park²⁸ ID, Nimesh Patel<fnref n="2" /> ID, Ue-Li Pen<fnref n="15" />,⁸⁴,⁸⁵,⁸⁶ ID, Vincent Piétu³⁶ ID, Aleksandar PopStefanija⁸²,
Oliver Porth¹⁹,⁴⁷ ID, Ben Prather¹² ID, Jorge A. Preciado-López<fnref n="15" /> ID, Dimitrios Psaltis<fnref n="6" /> ID, Hung-Yi Pu<fnref n="15" />,²⁸,⁸⁷ ID,
Venkatessh Ramakrishnan⁸¹ ID, Ramprasad Rao<fnref n="14" /> ID, Mark G. Rawlings³³ ID, Alexander W. Raymond<fnref n="1" />,<fnref n="2" /> ID, Luciano Rezzolla⁴⁷,⁸⁸ ID,
Bart Ripperda⁸⁹,⁹⁰ ID, Freek Roelofs<fnref n="9" /> ID, Alan Rogers<fnref n="4" /> ID, Eduardo Ros<fnref n="10" /> ID, Mel Rose<fnref n="6" /> ID, Arash Roshanineshat<fnref n="6" /> ID, Helge Rottmann<fnref n="10" /> ID,
Alan L. Roy<fnref n="10" /> ID, Chet Ruszczyk<fnref n="4" /> ID, Benjamin R. Ryan<fnref n="13" />,⁹¹ ID, Kazi L. J. Rygl⁷⁴ ID, Salvador Sánchez⁹²,
David Sánchez-Arguelles⁵⁷,⁶² ID, Mahito Sasada²¹,⁹³ ID, Tuomas Savolainen<fnref n="10" />,⁹⁴,⁹⁵ ID, F. Peter Schloerb⁸²,
Karl-Friedrich Schuster³⁶ ID, Lijing Shao<fnref n="10" />,⁶¹ ID, Zhiqiang Shen (沈志强)<fnref n="11" />,⁴⁴ ID, Des Small⁶⁷ ID, Bong Won Sohn³⁷,³⁸,⁹⁶ ID,
Jason SooHoo<fnref n="4" /> ID, Fumie Tazaki²¹ ID, Paul Tiede<fnref n="16" />,¹⁷ ID, Remo P. J. Tilanus<fnref n="6" />,<fnref n="9" />,⁵⁴,⁹⁷ ID, Michael Titus<fnref n="4" /> ID, Kenji Toma⁹⁸,⁹⁹ ID,
Pablo Torne<fnref n="10" />,⁹² ID, Tyler Trent<fnref n="6" /> ID, Efthalia Traianou<fnref n="10" /> ID, Sascha Trippe¹⁰⁰ ID, Shuichiro Tsuda²¹ ID, Ilse van Bemmel⁶⁷ ID,
Huib Jan van Langevelde⁶⁷,¹⁰¹ ID, Daniel R. van Rossum<fnref n="9" /> ID, Jan Wagner<fnref n="10" /> ID, John Wardle¹⁰² ID, Derek Ward-Thompson¹⁰³ ID,
Norbert Wex<fnref n="10" /> ID, Robert Wharton<fnref n="10" /> ID, Qingwen Wu (吴庆文)¹⁰⁴ ID, Doosoo Yoon¹⁹ ID, André Young<fnref n="9" /> ID, Ken Young<fnref n="2" /> ID,
Ziri Younsi⁴⁷,¹⁰⁵ ID, Feng Yuan (袁峰)<fnref n="11" />,⁵⁸,¹⁰⁶ ID, Ye-Fei Yuan (袁业飞)¹⁰⁷ ID, J. Anton Zensus<fnref n="10" /> ID, Guangyao Zhao³⁷ ID,
Shan-Shan Zhao<fnref n="9" />,⁷² ID, and Ziyan Zhu⁵³

<footnote n="1">¹ Black Hole Initiative at Harvard University, 20 Garden Street, Cambridge, MA 02138, USA; maciek.wielgus@gmail.com</footnote>

<footnote n="2">²Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA</footnote>

<footnote n="3">³National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, VA 22903, USA</footnote>

<footnote n="4">⁴Massachusetts Institute of Technology Haystack Observatory, 99 Millstone Road, Westford, MA 01886, USA</footnote>

<footnote n="5">⁵National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan</footnote>

<footnote n="6">⁶Steward Observatory and Department of Astronomy, University of Arizona, 933 N. Cherry Avenue, Tucson, AZ 85721, USA</footnote>

<footnote n="7">⁷Data Science Institute, University of Arizona, 1230 N. Cherry Avenue, Tucson, AZ 85721, USA</footnote>

<footnote n="8">⁸JILA and Department of Astrophysical and Planetary Sciences, University of Colorado, Boulder, CO 80309, USA</footnote>

<footnote n="9">⁹Department of Astrophysics, Institute for Mathematics, Astrophysics and Particle Physics (IMAPP), Radboud University, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands</footnote>

<footnote n="10">¹⁰Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany</footnote>

<footnote n="11">¹¹Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, People's Republic of China
¹²Department of Physics, University of Illinois, 1110 West Green Street, Urbana, IL 61801, USA</footnote>

<footnote n="13">¹³CCS-2, Los Alamos National Laboratory, P.O. Box 1663, Los Alamos, NM 87545, USA</footnote>

<footnote n="14">¹⁴Institute of Astronomy and Astrophysics, Academia Sinica, 645 N. A'ohoku Place, Hilo, HI 96720, USA</footnote>

<footnote n="15">¹⁵Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON, N2L 2Y5, Canada</footnote>

<footnote n="16">¹⁶Department of Physics and Astronomy, University of Waterloo, 200 University Avenue West, Waterloo, ON, N2L 3G1, Canada</footnote>
¹⁷ Waterloo Centre for Astrophysics, University of Waterloo, Waterloo, ON N2L 3G1 Canada

¹⁸ Princeton Center for Theoretical Science, Jadwin Hall, Princeton University, Princeton, NJ 08544, USA

¹⁹ Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904, 1098 XH, Amsterdam, The Netherlands

²⁰ Department of Astronomy, University of Illinois at Urbana-Champaign, 1002 West Green Street, Urbana, IL 61801, USA

²¹ Mizusawa VLBI Observatory, National Astronomical Observatory of Japan, 2-12 Hoshigaoka, Mizusawa, Oshu, Iwate 023-0861, Japan

²² Department of Astronomical Science, The Graduate University for Advanced Studies (SOKENDAI), 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan

²³ Instituto de Radioastronomía y Astrofísica, Universidad Nacional Autónoma de México, Morelia 58089, México

²⁴ Instituto de Astronomía, Universidad Nacional Autónoma de México, CdMx 04510, México

²⁵ Gravitation Astroparticle Physics Amsterdam (GRAPPA) Institute, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands

²⁶ Radio Astronomy Laboratory, University of California, Berkeley, CA 94720, USA

²⁷ Instituto de Astrofísica de Andalucía-CSIC, Glorieta de la Astronomía s/n, E-18008 Granada, Spain

²⁸ Institute of Astronomy and Astrophysics, Academia Sinica, 11F of Astronomy-Mathematics Building, AS/NTU No. 1, Section 4, Roosevelt Road, Taipei 10617, Taiwan, R.O.C.

²⁹ Departament d'Astronomia i Astrofísica, Universitat de València, C. Dr. Moliner 50, E-46100 Burjassot, València, Spain

³⁰ Observatori Astronòmic, Universitat de València, C. Catedrático José Beltrán 2, E-46980 Paterna, València, Spain

³¹ SISSA, Via Bonomea 265, 34136 Trieste, Italy and INFN Sezione di Trieste, Italy

³² IFPU—Institute for Fundamental Physics of the Universe, Via Beirut 2, 34014 Trieste, Italy

³³ East Asian Observatory, 660 N. A'ohoku Place, Hilo, HI 96720, USA

³⁴ Nederlandse Onderzoekschool voor Astronomie (NOVA), P.O. Box 9513, 2300 RA Leiden, The Netherlands

³⁵ California Institute of Technology, 1200 East California Boulevard, Pasadena, CA 91125, USA

³⁶ Institut de Radioastronomie Millimétrique, 300 rue de la Piscine, F-38406 Saint Martin d'Hères, France

³⁷ Korea Astronomy and Space Science Institute, Daedok-daero 776, Yuseong-gu, Daejeon 34055, Republic of Korea

³⁸ University of Science and Technology, Gajeong-ro 217, Yuseong-gu, Daejeon 34113, Republic of Korea

³⁹ Kavli Institute for Cosmological Physics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA

⁴⁰ Department of Astronomy and Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA

⁴¹ Department of Physics, University of Chicago, 5720 South Ellis Avenue, Chicago, IL 60637, USA

⁴² Enrico Fermi Institute, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA

⁴³ Cornell Center for Astrophysics and Planetary Science, Cornell University, Ithaca, NY 14853, USA

⁴⁴ Key Laboratory of Radio Astronomy, Chinese Academy of Sciences, Nanjing 210008, People's Republic of China

⁴⁵ Department of Space, Earth and Environment, Chalmers University of Technology, Onsala Space Observatory, SE-43992 Onsala, Sweden

⁴⁶ Dipartimento di Fisica “E. Pancini”, Università di Napoli “Federico II”, Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy

⁴⁷ Institut für Theoretische Physik, Goethe-Universität Frankfurt, Max-von-Laue-Straße 1, D-60438 Frankfurt am Main, Germany

⁴⁸ INFN Sez. di Napoli, Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy

⁴⁹ Department of Physics, University of Pretoria, Lynnwood Road, Hatfield, Pretoria 0083, South Africa

⁵⁰ Centre for Radio Astronomy Techniques and Technologies, Department of Physics and Electronics, Rhodes University, Grahamstown 6140, South Africa

⁵¹ LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Université de Paris, 5 place Jules Janssen, 92195 Meudon, France

⁵² Department of History of Science, Harvard University, Cambridge, MA 02138, USA

⁵³ Department of Physics, Harvard University, Cambridge, MA 02138, USA

⁵⁴ Leiden Observatory—Allegro, Leiden University, P.O. Box 9513, 2300 RA Leiden, The Netherlands

⁵⁵ CP3-Origins, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark

⁵⁶ Instituto Nacional de Astrofísica, Óptica y Electrónica, Luis Enrique Erro I, Tonantzintla, Puebla, C.P. 72840, México

⁵⁷ Consejo Nacional de Ciencia y Tecnología, Av. Insurgentes Sur 1582, 03940, Ciudad de México, México

⁵⁸ Key Laboratory for Research in Galaxies and Cosmology, Chinese Academy of Sciences, Shanghai 200030, People's Republic of China

⁵⁹ NOVA Sub-mm Instrumentation Group, Kapteyn Astronomical Institute, University of Groningen, Landleven 12, 9747 AD Groningen, The Netherlands

⁶⁰ Department of Astronomy, School of Physics, Peking University, Beijing 100871, People's Republic of China

⁶¹ Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, People's Republic of China

⁶² Instituto Nacional de Astrofísica, Óptica y Electrónica. Apartado Postal 51 y 216, 72000. Puebla Pue., México

⁶³ ASTRAVEO LLC, P.O. Box 1668, MA 01931, USA

⁶⁴ Max-Planck-Institut für Extraterrestrische Physik, Giessenbachstr. 1, D-85748 Garching, Germany

⁶⁵ Institute for Astrophysical Research, Boston University, 725 Commonwealth Avenue, Boston, MA 02215, USA

⁶⁶ Astronomical Institute, St. Petersburg University, Universitetskij pr., 28, Petrodvorets, 198504 St. Petersburg, Russia

⁶⁷ Joint Institute for VLBI ERIC (JIVE), Oude Hoogeveensedijk 4, 7991 PD Dwingeloo, The Netherlands

⁶⁸ Kogakuin University of Technology & Engineering, Academic Support Center, 2665-1 Nakano, Hachioji, Tokyo 192-0015, Japan

⁶⁹ Physics Department, National Sun Yat-Sen University, No. 70, Lien-Hai Road, Kaosiung City 80424, Taiwan, R.O.C

⁷⁰ National Optical Astronomy Observatory, 950 North Cherry Avenue, Tucson, AZ 85719, USA

⁷¹ Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Shijingshan District, Beijing, People's Republic of China

⁷² School of Astronomy and Space Science, Nanjing University, Nanjing 210023, People's Republic of China

⁷³ Key Laboratory of Modern Astronomy and Astrophysics, Nanjing University, Nanjing 210023, People's Republic of China

⁷⁴ Italian ALMA Regional Centre, INAF-Istituto di Radioastronomia, Via P. Gobetti 101, I-40129 Bologna, Italy

⁷⁵ Department of Physics, National Taiwan University, No.1, Sect.4, Roosevelt Road, Taipei 10617, Taiwan, R.O.C.

⁷⁶ Yunnan Observatories, Chinese Academy of Sciences, 650011 Kunming, Yunnan Province, People's Republic of China

⁷⁷ Center for Astronomical Mega-Science, Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing, 100012, People's Republic of China

⁷⁸ Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese Academy of Sciences, 650011 Kunming, People's Republic of China

⁷⁹ School of Natural Sciences, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA

⁸⁰ Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai, 200240, People's Republic of China

⁸¹ Astronomy Department, Universidad de Concepción, Casilla 160-C, Concepción, Chile

⁸² Department of Astronomy, University of Massachusetts, 01003, Amherst, MA, USA

⁸³ Department of Astronomy, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

⁸⁴ Canadian Institute for Theoretical Astrophysics, University of Toronto, 60 St. George Street, Toronto, ON M5S 3H8, Canada

⁸⁵ Dunlap Institute for Astronomy and Astrophysics, University of Toronto, 50 St. George Street, Toronto, ON M5S 3H4, Canada

⁸⁶ Canadian Institute for Advanced Research, 180 Dundas St West, Toronto, ON M5G 1Z8, Canada

⁸⁷ Department of Physics, National Taiwan Normal University, No. 88, Sec. 4, Tingzhou Road, Taipei 116, Taiwan, R.O.C.

⁸⁸ School of Mathematics, Trinity College, Dublin 2, Ireland

⁸⁹ Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ 08544, USA

⁹⁰ Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA
Received 2020 June 4; revised 2020 July 28; accepted 2020 August 2; published 2020 September 23

## Abstract

The Event Horizon Telescope (EHT) has recently delivered the first resolved images of M87*, the supermassive black hole in the center of the M87 galaxy. These images were produced using 230 GHz observations performed in 2017 April. Additional observations are required to investigate the persistence of the primary image feature—a ring with azimuthal brightness asymmetry—and to quantify the image variability on event horizon scales. To address this need, we analyze M87* data collected with prototype EHT arrays in 2009, 2011, 2012, and 2013. While these observations do not contain enough information to produce images, they are sufficient to constrain simple geometric models. We develop a modeling approach based on the framework utilized for the 2017 EHT data analysis and validate our procedures using synthetic data. Applying the same approach to the observational data sets, we find the M87* morphology in 2009–2017 to be consistent with a persistent asymmetric ring of ~40 µas diameter. The position angle of the peak intensity varies in time. In particular, we find a significant difference between the position angle measured in 2013 and 2017. These variations are in broad agreement with predictions of a subset of general relativistic magnetohydrodynamic simulations. We show that quantifying the variability across multiple observational epochs has the potential to constrain the physical properties of the source, such as the accretion state or the black hole spin.

*Unified Astronomy Thesaurus concepts: Black holes (162); Galaxy accretion disks (562); Galaxy accretion (575); Supermassive black holes (1663); Active galactic nuclei (16); Low-luminosity active galactic nuclei (2033); Very long baseline interferometry (1769); Astronomy data modeling (1859); Radio interferometry (1346)*

## 1. Introduction

The compact radio source in the center of the M87 galaxy, hereafter M87*, has been observed at 1.3 mm wavelength (230 GHz frequency) using very long baseline interferometry (VLBI) since 2009. These observations, performed by early configurations of the Event Horizon Telescope (EHT; Döeleman et al. 2009) array, measured the size of the compact emission to be ~40 µas, with large systematic uncertainties related to the limited baseline coverage (Döeleman et al. 2012; Akiyama et al. 2015). The addition of new sites and sensitivity improvements leading up to the April 2017 observations yielded the first resolved images of the source (Event Horizon Telescope Collaboration et al. 2019a, 2019b, 2019c, 2019d, 2019e, 2019f, hereafter EHTCI-VI). These images revealed an asymmetric ring (a crescent) with a diameter $d = 42 \pm 3$ µas and a position angle of the bright side $\phi_B$ between 150° and 200° east of north (counterclockwise from north/up as seen on the sky; EHTC VI); see the left panel of Figure 1. The apparent size and appearance of the observed ring agree with theoretical expectations for a $6.5 \times 10^9 M_\odot$ black hole driving a magnetized accretion inflow/outflow system, inefficiently

radiating via synchrotron emission (Yuan & Narayan 2014, EHTC V). Trajectories of the emitted photons are subject to strong deflection in the vicinity of the event horizon, resulting in a lensed ring-like feature seen by a distant observer—the anticipated shadow of a black hole (Bardeen 1973; Luminet 1979; Falcke et al. 2000; Broderick & Loeb 2009).

General relativistic magnetohydrodynamic (GRMHD) simulations of relativistic plasma in the accretion flow and jet-launching region close to the black hole (EHTC V; Porth et al. 2019) predict that the M87* source structure will exhibit a prominent asymmetric ring throughout multiple years of observations, with a mean diameter $d$ primarily determined by the black hole mass-to-distance ratio and a position angle $\phi_B$ primarily determined by the orientation of the black hole spin axis. The detailed appearance of M87* may also be influenced by many poorly constrained effects, such as the black hole spin magnitude, magnetic field structure in the accretion flow (Narayan et al. 2012, EHTC V), the electron heating mechanism (e.g., Mościbrodzka et al. 2016; Chael et al. 2018a), nonthermal electrons (e.g., Davelaar et al. 2019), and misalignment between the jet and the black hole spin (White et al. 2020; Chatterjee et al. 2020). Moreover, turbulence in the accretion flow, perhaps driven by magnetorotational instability (Balbus & Hawley 1991), is expected to cause stochastic variability in the image with correlation timescales of up to a few weeks (~dynamical time for M87*). The model uncertainties and expected time-dependent variability of these

<figure><img src="image_2.png" /></figure>

<footnote n="91">⁹¹ Center for Theoretical Astrophysics, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA</footnote>

<footnote n="92">⁹² Instituto de Radioastronomía Milimétrica, IRAM, Avenida Divina Pastora 7, Local 20, E-18012, Granada, Spain</footnote>

<footnote n="93">⁹³ Hiroshima Astrophysical Science Center, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526, Japan</footnote>

<footnote n="94">⁹⁴ Aalto University Department of Electronics and Nanoengineering, PL 15500, FI-00076 Aalto, Finland</footnote>

<footnote n="95">⁹⁵ Aalto University Metsähovi Radio Observatory, Metsähovintie 114, FI-02540 Kylmälä, Finland</footnote>

<footnote n="96">⁹⁶ Department of Astronomy, Yonsei University, Yonsei-ro 50, Seodaemun-gu, 03722 Seoul, Republic of Korea</footnote>

<footnote n="97">⁹⁷ Netherlands Organisation for Scientific Research (NWO), Postbus 93138, 2509 AC Den Haag, The Netherlands</footnote>

<footnote n="98">⁹⁸ Frontier Research Institute for Interdisciplinary Sciences, Tohoku University, Sendai 980-8578, Japan</footnote>

<footnote n="99">⁹⁹ Astronomical Institute, Tohoku University, Sendai 980-8578, Japan</footnote>

<footnote n="100">¹⁰⁰ Department of Physics and Astronomy, Seoul National University, Gwanak-gu, Seoul 08826, Republic of Korea</footnote>

<footnote n="101">¹⁰¹ Leiden Observatory, Leiden University, Postbus 2300, 9513 RA Leiden, The Netherlands</footnote>

<footnote n="102">¹⁰² Physics Department, Brandeis University, 415 South Street, Waltham, MA 02453, USA</footnote>

<footnote n="103">¹⁰³ Jeremiah Horrocks Institute, University of Central Lancashire, Preston PR1 2HE, UK</footnote>

<footnote n="104">¹⁰⁴ School of Physics, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, People's Republic of China</footnote>

<footnote n="105">¹⁰⁵ Mullard Space Science Laboratory, University College London, Holmbury St. Mary, Dorking, Surrey, RH5 6NT, UK</footnote>

<footnote n="106">¹⁰⁶ School of Astronomy and Space Sciences, University of the Chinese Academy of Sciences, No. 19A Yuquan Road, Beijing 100049, People's Republic of China</footnote>

<footnote n="107">¹⁰⁷ Astronomy Department, University of Science and Technology of China, Hefei 230026, People's Republic of China</footnote>

<footnote n="108">¹⁰⁸ NASA Hubble Fellowship Program, Einstein Fellow.</footnote>
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Left panel: one of the images of M87* obtained in EHT IV (see Section 4.2 for details). A 42 µas circle is plotted with a dashed line for reference. The observed position angle of the approaching jet $\phi_{jet}$ is 288° east of north (Walker et al. 2018). Under the assumed physical interpretation of the ring, we expect to find the bright side of the crescent on average approximately 90° clockwise from $\phi_{jet}$ (EHT V). We assume a convention $\phi_{B,exp} = 198^\circ$, indicated with a blue dashed line. Right panel: a random snapshot (note that this is not a fit to the EHT image) from a GRMHD simulation adopting the expected properties of M87* (Section 4.1). The spin vector of the black hole is partially directed into the page, counteraligned with the approaching jet (and aligned with the deboosted receding jet); its projection onto the observer's screen is located at the position angle of $\phi_{spin} = \phi_{jet} - 180^\circ$.</figcaption></figure>

theoretical predictions strongly motivate the need for additional observations of M87*, especially on timescales long enough to yield uncorrelated snapshots of the turbulent flow.

To this end, we analyze archival EHT observations of M87* from observing campaigns in 2009, 2011, 2012, and 2013. While these observations do not have enough baseline coverage to form images (EHT IV), they are sufficient to constrain simple geometrical models, following procedures similar to those presented in EHT VI. We employ asymmetric ring models that are motivated by both results obtained with the mature 2017 array and the expectation from GRMHD simulations that the ring feature is persistent.

We begin, in Section 2, by summarizing the details of these archival observations with the “proto-EHT” arrays. In Section 3, we describe our procedure for fitting simple geometrical models to these observations. In Section 4, we test this procedure using synthetic proto-EHT observations of GRMHD snapshots and of the EHT images of M87*. We then use the same procedure to fit models to the archival observations of M87* in Section 5. We discuss the implications of these results for our theoretical understanding of M87* in Section 6 and briefly summarize our findings in Section 7.

## 2. Observations and Data

Our analysis covers five separate 1.3 mm VLBI observing campaigns conducted in 2009, 2011, 2012, 2013, and 2017. The M87* data from 2011 and 2013 have not been published previously. For all campaigns except 2012, M87* was observed on multiple nights. For the proto-EHT data sets (2009–2013), we simultaneously utilize the entire data set from each year, fitting to data from multiple days with a single source model, when available. This is motivated by the M87* dynamical timescale argument, little visibility amplitude variation reported by EHT III on a one-week timescale, as well as by the limited amount of available data and lack of evidence for interday variability in the proto-EHT data sets. We use incoherent averaging to estimate visibility amplitudes on each scan (~few minutes of continuous observation) and bispectral averaging to

estimate closure phases (CPs; Rogers et al. 1995; Johnson et al. 2015; Fish et al. 2016). The frequency setup in 2009–2013 consisted of two 480 MHz bands, centered at 229.089 and 229.601 GHz. Whenever both bands or both parallel-hand polarization components were available, we incoherently averaged all simultaneous visibility amplitudes. The data sets are summarized in Table 1, where the number of detections for nonredundant baselines of different projected baseline lengths is given, with the corresponding (u, v) coverage shown in Figure 2. Redundant baselines yield independent observations of the same visibility. In Table 1 we also indicate the number of available nonredundant CPs (not counting redundant and intrasite baselines, minimal set; see Blackburn et al. 2020). As is the case for non-phase-referenced VLBI observations (Thompson et al. 2017), we do not have access to absolute visibility phases. All visibility amplitudes observed in 2009–2013 are presented in Figure 3.

A more detailed summary of the observational setup of the proto-EHT array in 2009–2013 and the associated data reduction procedures can be found in Fish et al. (2016). All data sets discussed in this paper are publicly available.<fnref n="109" />

### 2.1. 2009–2012

Prior to 2013, the proto-EHT array included telescopes at three geographical locations: (1) the Combined Array for Research in Millimeter-wave Astronomy (CARMA, CA) in Cedar Flat, California, (2) the Submillimeter Telescope (SMT, AZ) on Mt. Graham in Arizona, and (3) the Submillimeter Array (SMA, SM), the James Clerk Maxwell Telescope (JCMT, JC), and the Caltech Submillimeter Observatory (CSO, CS) on Maunakea in Hawai'i. These arrays were strongly east-west oriented, and the longest projected baselines, between SMT and Hawai'i, reached about 3.5 Gλ, corresponding to the instrument resolution (maximum fringe spacing) of ~60 µas.

The 2011 observations of M87* have not been published but follow the data reduction procedures described in Lu et al. (2013). The 2009 and 2012 observations and data processing of M87* have been published in Doeleman et al. (2012) and Akiyama et al. (2015), respectively. However, our analysis uses modified processing of the 2012 data because the original processing erroneously applied the same correction for atmospheric opacity at the SMT twice.<fnref n="110" /> The SMT calibration procedures have been updated since then (Issaoun et al. 2017).

Each observation included multiple subarrays of CARMA as well as simultaneous measurements of the total source flux density with CARMA acting as a connected-element interferometer; these properties then allow the CARMA amplitude gains to be “network-calibrated” (Fish et al. 2011; Johnson et al. 2015, EHT III). Of these three observing campaigns, only 2012 provides CP information for M87*, and all CPs measured on the single, narrow triangle SMT–SMA–CARMA were consistent with zero to within $2\sigma$ (Akiyama et al. 2015); see Figure 4.

### 2.2. 2013

The 2013 observing epoch did not include the CSO, but added the Atacama Pathfinder Experiment facility (APEX, AP) in the Atacama Desert in Chile. This additional site brought for

<footnote n="109">¹⁰⁹ https://eventhorizontelescope.org/for-astronomers/data</footnote>

<footnote n="110">¹¹⁰ An opacity correction raises visibility amplitudes on SMT baselines by ~10% in nominal conditions; our visibility amplitudes on SMT baselines are, thus, slightly lower than those reported by Akiyama et al. (2015). However, the calibration error does not change the primary conclusions of Akiyama et al. (2015).</footnote>
<figure><img src="image_4.png" /><figcaption>**Figure 2.** (u, v) coverage of the M87* observations performed in 2009–2013 with various proto-EHT arrays. Gray lines indicate detections obtained during the 2017 observations with a mature EHT array, including several new sites, but without the baselines to CARMA. Dashed circles correspond to angular scales of 50 µas (inner) and 25 µas (outer).</figcaption></figure>

<table><caption>**Table 1**
M87* Data Sets Analyzed in This Paper</caption><thead><tr><th rowspan="2">Year</th><th rowspan="2">Telescopes</th><th rowspan="2">Dates</th><th rowspan="2">Baselines<sup>a</sup></th><th colspan="5">Detections on Nonredundant Baselines</th><th rowspan="2">CPs</th></tr><tr><th>Zero<br>&lt;0.1Gλ</th><th>Short<br>&lt;1Gλ</th><th>Medium<sup>b</sup><br>&lt;3.6Gλ</th><th>Long<sup>c</sup><br>&gt;3.6Gλ</th><th>Total<br>&gt;0.1Gλ</th></tr></thead><tbody><tr><td>2009</td><td>CA, AZ, JC</td><td>Apr 5, 6</td><td>3/3/3</td><td>...</td><td>12</td><td>16/5</td><td>...</td><td>28</td><td>...</td></tr><tr><td>2011</td><td>CA, AZ, JC, SM, CS</td><td>Mar 29, 31; Apr 1, 2, 4</td><td>10/6/3</td><td>52</td><td>33</td><td>21/6</td><td>...</td><td>54</td><td>...</td></tr><tr><td>2012</td><td>CA, AZ, SM</td><td>Mar 21</td><td>3/3/3</td><td>14</td><td>11</td><td>19/6</td><td>...</td><td>44</td><td>7</td></tr><tr><td>2013</td><td>CA, AZ, SM, JC, AP</td><td>Mar 21–23, Mar 26</td><td>10/7/5</td><td>39</td><td>41</td><td>23/4</td><td>19/1</td><td>83</td><td>...</td></tr><tr><td>2017</td><td>AZ, SM, JC, AP, LM, PV, AA</td><td>Apr 6<sup>d</sup></td><td>21/21/10</td><td>24</td><td>...</td><td>33/13</td><td>92/16</td><td>125</td><td>67</td></tr><tr><td>2017</td><td>AZ, SM, JC, AP, LM, PV, AA</td><td>Apr 11<sup>d</sup></td><td>21/21/10</td><td>22</td><td>...</td><td>28/9</td><td>72/16</td><td>100</td><td>54</td></tr></tbody></table>

**Notes.**

<sup>a</sup> Theoretically available/with detections/nonredundant, nonzero with detections.

<sup>b</sup> All/SMT-Hawai’i.

<sup>c</sup> All/SMT-Chile.

<sup>d</sup> Single-day data set.

the first time the long-baselines (≈5–6 Gλ) CARMA–APEX and SMT–APEX, which are roughly orthogonal to the CARMA–Hawai’i and SMT–Hawai’i baselines; see Figure 2. The addition of APEX increased the instrument resolution (maximum fringe spacing) to ~35 µas. While the 2013 observations of Sgr A* were presented in several publications (Johnson et al. 2015; Fish et al. 2016; Lu et al. 2018), the M87* observations obtained during the 2013 campaign have not been published previously.

The proto-EHT array observed M87* on March 21, 22, 23, and 26 2013. CARMA–APEX detections were found on March 22 (11 detections) and 23 (7 detections) with a single SMT–APEX detection on March 23. March 23 (MJD 36374) was the only day with detections on baselines to each of the four geographical sites. No detections between Hawai’i and APEX were found, and there were no simultaneous detections over a closed triangle that would allow for the measurement of CP.

## 2.3. 2017

In 2017, the EHT observed M87* with five geographical sites (EHTC I; EHTC II), without CSO and CARMA, but with the addition of the Large Millimeter Telescope Alfonso Serrano (LMT, LM) on the Volcán Sierra Negra in Mexico, the IRAM

30 m telescope (PV) on Pico Veleta in Spain, and the phased-up Atacama Large Millimeter/submillimeter Array (ALMA, AA; Matthews et al. 2018; Goddi et al. 2019). The expansion of the array resulted in significant improvements in (u, v) coverage, shown with gray lines in Figure 2, and instrument resolution raised to ~25 µas. In addition to hardware setup developments (EHTC II), the recorded bandwidth was increased from 2 × 0.5 GHz to 2 × 2 GHz (226–230 GHz). The 2017 data processing pipeline used ALMA as an anchor station (EHTC III). Its high sensitivity greatly improved the signal phase stability (Blackburn et al. 2019; Janssen et al. 2019; EHTC III) and enabled data analysis based on robustly detected closure quantities obtained from coherently averaged visibilities (EHTC IV; Blackburn et al. 2020) rather than on visibility amplitudes alone. These improvements allowed for an unambiguous analysis of the M87* image by constraining the set of physical (EHTC V) and geometric (EHTC VI) models representing the source morphology.

## 2.4. M87* Data Properties

VLBI observations sample the Fourier transform of the intensity distribution on the sky I(x, y) via the van Cittert–
<figure><img src="image_5.png" /><figcaption>**Figure 3.** Top: visibility amplitudes of M87* detections in 2009–2013 as a function of projected baseline length $\sqrt{u^2 + v^2}$. The source model derived from the EHT 2017 observations is shown with gray dots. Gray dots with black borders show the predicted visibility amplitudes of the source model at the baselines of the prior observations in 2009–2013. Dashed black lines correspond to the family of Fourier transforms of a symmetric, infinitely thin ring of diameter $d_0 = 45.0~\mu\text{as}$. Bottom: total arcsecond-scale flux density (on intrasite baselines, network-calibrated) and compact emission flux density from the short CARMA–SMT baseline. In the case of the short baselines in 2017, predictions of the source model are given.</figcaption></figure>

Zernike theorem (van Cittert 1934; Zernike 1938):

$$V(u, v) = \iint I(x, y)e^{-2\pi i(xu + vy)} dxdy, \quad (1)$$

where the measured Fourier coefficients $V(u, v)$ are referred to as “visibilities” (Thompson et al. 2017). When an array of N telescopes observes a source, $N(N-1)/2$ independent visibility measurements are obtained, provided detections on all baselines are found. Certain properties of the geometry described by $I(x, y)$ can be inferred directly from inspecting the visibility data.

In the top panel of Figure 3, we summarize all the M87* detections obtained during 2009–2013 observations as

a function of projected baseline length $\sqrt{u^2 + v^2}$. Dashed lines represent $\hat{R}$, the analytic Fourier transform of an infinitely thin ring with a total intensity $I_0$ and a diameter $d_0$,

$$\hat{R}(\sqrt{u^2 + v^2}) = I_0 J_0(\pi d_0 \sqrt{u^2 + v^2}), \quad (2)$$

where $J_0$ is a zeroth-order Bessel function of the first kind. This simple analytic model predicts the visibility null located at

$$b_0 \approx 3.51 \left( \frac{d_0}{45~\mu\text{as}} \right)^{-1} \text{G}\lambda, \quad (3)$$
<figure><img src="image_6.png" /><figcaption>**Figure 4.** Consistency of the closure phases on the SMT-SMA-CARMA triangle between the values observed in 2012 (Akiyama et al. 2015) and numerically resampled source models constructed based on the 2017 observations. The predictions of the asymmetric ring models RT and RG fitted to 2012 observations are also given; see Sections 3 and 5. Data corresponding to two CARMA subarrays, C1 and C2, are shown separately.</figcaption></figure>

and a wide plateau around the first maximum, located at

$$b_1 \approx 5.59 \left( \frac{d_0}{45 \mu\text{as}} \right)^{-1} \text{G}\lambda, \quad (4)$$

recovering about 40% of the flux density seen on short baselines. In Figure 3, we use $d_0 = 45.0 \mu\text{as}$ and show $\hat{R}$ curves corresponding to $I_0 = 0.25, 0.5, 1.0$, and $1.5$ to guide the eye. The behaviors of the visibility amplitudes, particularly the fall-off rate seen on medium-length baselines (1.0–3.6 G$\lambda$) in all data sets and the flux density recovery on long baselines to APEX in 2013, are roughly consistent with that of a simple ring model. Moreover, all detections on baselines to APEX have a similar flux density of ~0.2 Jy, while the projected baseline length varies between 5.2 and 6.1 G$\lambda$. In the analytic thin ring model framework, this can be readily understood, because baselines to APEX sample the wide plateau around the maximum located at $b_1$, Equation (4).

The gray dots in Figure 3 correspond to the source model constructed based on the 2017 EHT observations—the mean of the four images reconstructed for April 5, 6, 10, and 11 2017 with the eht-imaging pipeline (Chael et al. 2016; EHT-C IV). In the 2017 model, east-west baselines, such as SMT-Hawai'i, probe a deep visibility null located around $b_0$ (Equation (3)), where sampled amplitudes drop below 0.01 Jy. North-south baselines do not show a similar feature, which indicates source asymmetry. Irrespective of the orientation, visibility amplitudes flatten out around $b_1$. Gray dots with black envelopes represent the 2017 source model sampled at the (u, v) coordinates of the past observations, for which all medium-length baselines were oriented in the east-west direction.

One can immediately notice interesting discrepancies. The visibility amplitudes measured on long baselines to APEX

(projected baseline length ~$b_1$) in 2013 were about a factor of 2 larger than the corresponding 2017 source model predictions. At the same time, the flux density on the short CARMA-SMT baseline is consistent between the 2013 measurements and the 2017 model predictions. This shows that the image on the sky has changed between 2013 and 2017 in a structural way, which cannot be explained with a simple total intensity scaling. We also notice that several detections obtained in 2009–2011, corresponding to projected (u, v) -distances of 3.2–3.5 G$\lambda$ on Hawai'i-USA baselines, record flux density above 0.1 Jy. At the same time, the 2017 model predicts that these baselines sample a visibility null region around $b_0$, with a flux density lower by an order of magnitude. However, the compact flux (on short baselines) did not change by more than a factor of 2, remaining between 0.5 and 1.0 Jy throughout the 2009–2017 observations; see the bottom panel of Figure 3. This suggests that the null location in the past (if present) was different than that observed in 2017, which may correspond to a fluctuation of the crescent position angle or a changing degree of source symmetry.

Apart from the visibility amplitude data, a limited number of CPs from the narrow triangle SMT-SMA-CARMA has been obtained from the 2012 data set (Akiyama et al. 2015). All of these CPs are measured to be consistent with zero, which suggests a high degree of east-west symmetry in the geometry of the source observed in 2012. While the CPs on this triangle were not observed in 2017 (CARMA was not part of the 2017 array), we can numerically resample the 2017 images (eht-imaging reconstructions; EHTC IV) to verify the consistency. In Figure 4, we show the CPs obtained in 2012, averaged between bands, with the two CARMA subarrays shown separately. Near-zero CPs observed in 2012 are roughly consistent with at least some models from 2017. Unfortunately, technical difficulties that occurred during the 2012 campaign precluded obtaining measurements between UTC 7.5 and 10.5, where nonzero CPs are predicted by all 2017 models.

Altogether, we see strong suggestions that the 2009–2013 data sets describe a similar geometry to the 2017 results, but there are also substantial hints that the detailed properties of the source structure evolved between observations. These differences can be quantified with geometric modeling of the source morphology.

### 3. Modeling Approach

The sparse nature of the pre-2017 data sets precludes reconstructing images in the manner employed for the 2017 data (EHTC IV). However, the earlier data are still capable of providing interesting constraints on more strictly parameterized classes of models. Figure 5 shows the 2013 data set overplotted with a best-fit ring model<fnref n="111" /> (in blue; five degrees of freedom) and asymmetric Gaussian model (in red; four degrees of freedom). Both models attain similar fit qualities, as determined by Bayesian and Akaike information criteria (see, e.g., Liddle 2007). In the absence of prior information, we would be unable to confidently select a preferred model. However, the robust image morphology reconstructed from the 2017 data provides a natural and strong prior for selecting an appropriate parameterization. The “generalized crescent” (GC) geometric models considered in EHTC VI yielded fit qualities comparable

<footnote n="111">¹¹¹ This is the maximum likelihood estimator for the slashed thick ring model (RT), as discussed in Section 3.1.</footnote>
<figure><img src="image_7.png" /><figcaption>**Figure 5.** Comparison of maximum likelihood (ML) asymmetric Gaussian and asymmetric ring models fitted to the 2013 data. Data are shown as points with error bars corresponding to the thermal errors. The shaded regions cover all amplitudes for a given model. The red and blue lines represent models evaluated at the (u, v) coordinates of the observations. Both models offer a very similar fit quality. ML estimators are shown as inset figures. The model of a ring with roughly double the diameter (dashed curve) fits the intermediate baselines but is excluded by long-baseline amplitudes.</figcaption></figure>

to those of image reconstructions for the 2017 data, and in this paper, we apply two variants of the GC model to the pre-2017 data sets. Owing to data sparsity, we restrict the parameter space of the models to a subset of that considered in EHTC VI containing only a handful of parameters of interest.

Throughout this paper, we use perceptually uniform color maps from the ehtplot library<fnref n="112" /> to display the images. In some of the figures, we present models blurred to a resolution of 15 µas, adopted in this paper as the effective resolution of the EHT. The EHT instrument resolution measured as the maximum fringe spacing in 2017 is about 25 µas; however, for the image reconstruction methods employed in EHTC IV, a moderate effect of superresolution can be expected (Honma et al. 2014; Chael et al. 2016). The effect may be much more prominent for the geometric models, which are not fundamentally limited by the resolution.

### 3.1. Model Specification

Given the image morphology inferred from the 2017 data, the primary parameters of interest we would like to constrain using the earlier data sets are the size of the source, the orientation of any asymmetry, and the presence or absence of a central flux depression. The analyses presented in this paper use two simple ring-like models—similar to those presented in Kamruddin & Dexter (2013) and Benkevitch et al. (2016)—both of which are subsets of the GC models from EHTC VI.

The first model we consider is a concentric “slashed” ring, where the ring intensity is modulated by a linear gradient, hereafter denoted as RT. In this model, the flux is contained within a circular annulus with the inner and outer radii $R_{in}$ and $R_{out}$, respectively. The model is described by five parameters:

1. the mean diameter of the ring $d = R_{in} + R_{out}$,

2. the position angle of the bright side of the ring $0 \le \phi_B < 2\pi$,

3. the fractional thickness of the ring $0 < \psi = 1 - R_{in}/R_{out} < 1$,

4. the total intensity $0 < I_0 < 2$ Jy, and

5. $\beta$, an intensity gradient (“slash”) across the ring in the direction given by $\phi_B$, corresponding to the ratio between the dimmest and brightest points on the ring, $0 < \beta < 1$. A ring of uniform brightness has $\beta = 1$, while a ring with vanishing flux at the dimmest part has $\beta = 0$.

This model reduces to a slashed disk for $\psi \to 1$. The assumed definition of mean diameter is consistent with the one used in EHTC VI, allowing for direct comparisons. Except where otherwise specified, we use this first model for the analyses discussed in this paper.

As a check against model-specific biases, we consider a second model consisting of an infinitesimally thin slashed ring, blurred with a Gaussian kernel (EHTC IV). The equivalent five parameters for this model are:

1. the mean diameter of the ring $d = 2R_{in} = 2R_{out}$,

2. the position angle of the bright side of the ring $0 \le \phi_B < 2\pi$,

3. the width of the Gaussian blurring kernel $0 < \sigma < 40$ µas,

4. the total intensity $0 < I_0 < 2$ Jy, and

5. the slash $0 < \beta < 1$.

This second model, hereafter referred to as RG, reduces to a circular Gaussian for $d \ll \sigma$.

Both the RT and RG models provide a measure of the source diameter ($d$), the orientation of the brightness asymmetry ($\phi_B$), and the presence of a central flux depression. We quantify the latter property using the following general measure of relative ring thickness (from EHTC VI):

$$f_w = \frac{R_{out} - R_{in} + 2\sigma\sqrt{2\ln 2}}{d}, \quad (5)$$

where $R_{out} = R_{in}$ for the RG model and $\sigma = 0$ for the RT model.

All data sets except 2009 contain observations from intrasite baselines (“zero baselines”); see Table 1. For M87*, these baselines are sensitive to the flux from the extended jet emission on ~arcsecond scales (EHTC IV; see also the bottom panel of Figure 3) and do not directly inform us about the compact source structure on scales of ~tens of microarcseconds. However, the intrasite baselines still provide useful constraints on station gain parameters (see Section 3.2), and hence, we do not flag them. Rather, we parameterize this large-scale flux using a large symmetric Gaussian component consisting of two parameters, flux and size. This component is entirely resolved out on intersite baselines and thus has no direct impact on the compact source geometry. Ultimately, the models that we use have 5 geometric parameters for the 2009 data set and $5 + 2 = 7$ geometric parameters in all other cases.

### 3.2. Fitting Procedure and Priors

We perform the parameter estimation for this paper using THEMIS, an analysis framework developed by Broderick et al. (2020) for the specific requirements of EHT data analysis. THEMIS operates within a Bayesian formalism, employing a differential evolution Markov Chain Monte Carlo (MCMC) sampler to explore the posterior space. Prior to model fitting, the data products are prepared in a manner similar to that

<footnote n="112">¹¹² https://github.com/liamedeiros/ehtplot</footnote>
described in EHTC VI. Descriptions of the likelihood con-
structions for different classes of data products are given in
Broderick et al. (2020).

One important difference between the 2017 and pre-2017
data sets is that the latter contain almost exclusively visibility
amplitude information, rather than having access to the robust
closure quantities in both phase and amplitude (Thompson
et al. 2017; Blackburn et al. 2020) that aided interpretation of
the 2017 data. In addition to thermal noise, visibility
amplitudes suffer from uncertainties in the absolute flux
calibration, including potential systematic effects such as
losses related to telescope pointing imperfections (EHTC III).
These uncertainties are parameterized within THEMIS using
station-based amplitude gain factors gᵢ, representing the scaling
between the geometric model amplitudes |V̂ᵢⱼ| and the gains-
corrected model amplitudes |V̂ᵢⱼ|,

$$
|\hat{V}_{ij}| = (1 + g_i)(1 + g_j)|\tilde{V}_{ij}|. \qquad (6)
$$

Model amplitudes |V̂ᵢⱼ| are then compared with the measured
visibility amplitudes |Vᵢⱼ|. Within THEMIS, the number of
amplitude gain parameters N_g is equal to the number of
(station, scan) pairs, i.e., the gains are assumed to be constant
across a single scan but uncorrelated from one scan to another.
By explicitly modeling station-based gains, we correctly
account for the otherwise covariant algebraic structure of the
visibility calibration errors (Blackburn et al. 2020). At each
MCMC step, THEMIS marginalizes over the gain amplitude
parameters (subject to Gaussian priors) using a quadratic
expansion of the log-likelihood around its maximum given the
current parameter vector; see Broderick et al. (2020) for details.
For the analysis of the 2009–2013 data sets presented in this
paper, we have adopted rather conservative 15% amplitude
gain uncertainties for each station, represented by symmetric
Gaussian priors with a mean value of 0.0 and standard
deviation of 0.15. The width of these priors reflects our
confidence in the flux density calibration rather than the
statistical variation in the visibility data.

The RT model is parameterized within THEMIS in terms of R<sub>out</sub>, φ<sub>B</sub>, ψ, I<sub>0</sub>, and β. Uniform priors are used for each of these parameters, with ranges of [0, 200] µas for R<sub>out</sub>, [0, 2π] for φ<sub>B</sub>, [0, 1] for ψ, [0, 2] Jy for I<sub>0</sub>, and [0, 1] for β. We achieve the “infinitesimally thin” ring of the RG model within THEMIS by imposing a strict prior on ψ of [10<sup>-7</sup>, 10<sup>-6</sup>], and the prior on σ is uniform in the range [0, 40] µas. Because d and f<sub>w</sub> are derived parameters, we do not impose their priors directly but rather infer them from appropriate transformations of the priors on R<sub>out</sub>, ψ, and σ. The effective prior on d = R<sub>out</sub> (2 − ψ) is given by

$$
\pi(d) = \begin{cases} \frac{\ln(2)}{200}, & 0 \le \mu\text{as} \le d < 200 \mu\text{as} \\ \frac{1}{200} \ln\left(\frac{400 \ \mu\text{as}}{d}\right), & 200 \ \mu\text{as} \le d < 400 \ \mu\text{as}' \\ 0, & \text{otherwise} \end{cases} \tag{7}
$$

which is uniform within the range [0, 200] µas. For the RT
model, the effective prior on f<sub>w</sub> = ψ/(2 − ψ) is given by

$$
\pi_{\text{RT}}(f_w) = \begin{cases} \frac{2}{(1 + f_w)^2}, & 0 \le f_w \le 1 \\ 0, & \text{otherwise} \end{cases}, \qquad (8)
$$

which is not uniform but rather increases toward smaller
values. For the RG model, the effective prior on
f<sub>w</sub> = σ√(2 ln(2)/R<sub>out</sub> is given by

$$
\pi_{\text{RG}}(f_w) = \begin{cases} \frac{1}{2\alpha}, & 0 \le f_w \le \alpha \\ \frac{\alpha}{2f_w^2}, & f_w > \alpha \\ 0, & \text{otherwise} \end{cases}, \qquad (9)
$$

where α = √(2 ln(2))/5 ≈ 0.235 for our specified priors on σ and R<sub>out</sub>; this prior is uniform within the range [0, α].

3.3. Degeneracies and Limitations

Modeling tests revealed the presence of a large-diameter
secondary ring mode in the posterior distributions for the
2009–2012 data sets, corresponding to the dashed green line in
Figure 5. This mode is excluded by the detections on long
baselines (APEX baselines in 2013, multiple baselines in 2017)
and detections on medium-length (~1.5 Gλ) baselines (LMT–
SMT in 2017). Excising this secondary mode, as we do for the
posteriors presented in Figure 6, effectively limits the diameter
d to be less than ~80 µas. In all cases, the prior range is
sufficient to capture the entire volume of the primary posterior
mode, corresponding to an emission region of radius ~20 µas.
We have verified numerically that this procedure produces the
same results as restricting the priors on R<sub>out</sub> to [0, 45] µas for
the analysis of the 2009–2012 data sets.

As a consequence of the Fourier symmetry of a real domain input signal, we have V*(u, v) = V(-u, -v). Hence, visibility amplitude data alone cannot break the degeneracy between the orientation of φ<sub>B</sub> and φ'<sub>B</sub> = φ<sub>B</sub> + 180°, and effectively, we only constrain the axis of the crescent asymmetry. This is how the reported uncertainties should be interpreted. Having that in mind, for 2009, 2011, and 2013, consisting exclusively of the visibility amplitude data, we choose the reported φ<sub>B</sub> using the prior information about the position angle of the jet φ<sub>jet</sub> to select the φ<sub>B</sub> such that φ'<sub>jet</sub> − 180° < φ<sub>B</sub> < φ'<sub>jet</sub>, where φ<sub>jet</sub> = 288° (Walker et al. 2018). In other words, between the orientations φ<sub>B</sub> and φ'<sub>B</sub>, we choose the one that is closer to the expected bright side position φ<sub>B,exp</sub> = 198°. This is motivated by the theoretical interpretation of the asymmetric ring feature (EHTC V). In the case of the 2012 data set, for which a very limited number of CPs is available, we report the orientation φ<sub>B</sub> of the maximum likelihood (ML) estimators, noting the bimodal character of the posterior distributions and the aforementioned 180° degeneracy. These caveats do not apply to the 2017 data set, for which substantial CP information is available and breaks the degeneracy.

It is important to recognize that the parameters of
a geometric model have no direct relation to the physical
parameters of the source, unlike direct fitting using GRMHD
simulation snapshots (Dexter et al. 2010; Kim et al. 2016;
<figure><img src="image_8.png" /><figcaption>**Figure 6.** Top two rows: marginalized distributions of the mean diameter *d* and brightness maximum position angle *φ*<sub>B</sub> for the RT model fits to GRMHD simulation snapshots GRMHD1 (first row) and GRMHD2 (second row). The 2017 posteriors are contained within the dark gray bands. The dashed vertical line in the left panels denotes the diameter of $2\sqrt{27} M/D$. The vertical dashed lines in the right panels denote the convention angle $\phi_{B,exp}$, $\phi_{B,exp} - 90^{\circ}$, and the approaching jet position angle $\phi_{jet} = \phi_{B,exp} + 90^{\circ}$. The range of ($\phi_{jet} - 180^{\circ}$, $\phi_{jet}$) is highlighted. Two bottom rows: similar to above, but for the RT model fitted to MODEL1 and MODEL2. Lightly shaded areas correspond to values reported in EHTC I, diameter *d* = 42 ± 3 µas and position angle $150^{\circ} < \phi_{B} < 200^{\circ}$.</figcaption></figure>

Fromm et al. 2019, EHTC V) or ray-traced geometric source
models (Broderick & Loeb 2009; Broderick et al. 2016;
Vincent et al. 2020) to the data. The crescent model is
a phenomenological description of the source morphology in

the observer's plane. If physical parameters (such as black hole
mass) are to be extracted, additional calibration, in general
affected by the details of the assumed theoretical model and the
(u, v) coverage, needs to be performed (EHTC VI).
## 4. Modeling Synthetic Data

In order to verify whether the (u, v) coverage and signal-to-noise ratio of the 2009–2013 observations are sufficient to constrain source geometric properties with simple asymmetric ring models, we have designed tests using synthetic VLBI observations. The synthetic observations are generated with the eht-imaging software (Chael et al. 2016, 2018b) by sampling four emission models (GRMHD1, GRMHD2, MODEL1, MODEL2) with the (u, v) coverage and thermal error budget reported for past observations. Additionally, corruption from time-dependent station-based gain errors has been folded into the synthetic observations. The ground-truth images that we use correspond to ray-traced snapshots of a GRMHD simulation and published images of M87* (EHTC IV), reconstructed based on the 2017 observations.

### 4.1. GRMHD Snapshots

For the first two synthetic data tests, we use a random snapshot from a GRMHD simulation of a low-magnetic-flux standard and normal evolution (SANE) accretion disk (Narayan et al. 2012; Sądowski et al. 2013) around a black hole with spin $a_* \equiv Jc/GM^2 = 0.5$, shown in Figure 1 (second panel) and in Figure 7 (first panel). The GRMHD simulation was performed with the iharm code (Gammie et al. 2003), and the ray tracing was done with ipole (Mościbrodzka & Gammie 2018). Following Mościbrodzka et al. (2016) and EHTC V, we assume a thermal electron energy distribution function and relate the local ratio of ion ($T_i$) and electron temperature ($T_e$) to the plasma parameter $\beta_p$ representing the ratio of gas to magnetic pressure:

$$ \frac{T_i}{T_e} = R_{\text{high}} \frac{\beta_p^2}{1 + \beta_p^2} + R_{\text{low}} \frac{1}{1 + \beta_p^2}, \quad (10) $$

with $R_{\text{high}} = 40$ and $R_{\text{low}} = 1$ for the considered snapshot. The prescription given by Equation (10) parameterizes complex plasma microphysics, allowing us to efficiently survey different models of electron heating, resulting in a different geometry of the radiating region. As an example, for the SANE models with large $R_{\text{high}}$, the emission originates predominantly in the strongly magnetized jet base region, while for a small $R_{\text{high}}$, disk emission dominates (EHTC V).

The image considered here is a higher-resolution version (1280 × 1280 pixels) of one of the images generated for the Image Library of EHTC V and corresponds to a 6.5 × 10⁹ M☉ black hole at a distance $D = 16.9$ Mpc. This choice results in an $M/D$ ratio<fnref n="113" /> of 3.80 µas and an observed black hole shadow that is nearly circular with an angular diameter not substantially different from the Schwarzschild case, which is $2\sqrt{27} M/D = 39.45$ µas (Bardeen 1973). For reference, the dashed circles plotted in Figures 1 and 7 have a diameter of 42.0 µas. These parameters were chosen to be consistent with the ones inferred from the EHT 2017 observations (EHTC I). The camera is oriented with an inclination angle of 22°. The viewing angle was chosen to agree with the expected inclination of the M87* jet (Walker et al. 2018). The choices of spin $a_*$, electron temperature parameter $R_{\text{high}}$, and the

SANE accretion state are arbitrary. The choice of $R_{\text{low}}$ follows the assumptions made in EHTC V. We also assume that the accretion disk plane is perpendicular to the black hole jet (the disk is not tilted). The first image, GRMHD1, has been rotated in such a way that the projection of the simulated black hole spin axis counteraligns with the observed position angle of the approaching M87* jet, $\phi_{\text{jet}} = 288^\circ$ (Walker et al. 2018; Kim et al. 2018). The GRMHD2 test corresponds to the same snapshot but rotated counterclockwise by 90° to $\phi_{\text{jet}} = 18^\circ$, hence displaying a brightness asymmetry in the east-west rather than in the north-south direction; see the second panel of Figure 7. Because the (u, v) coverage in 2009–2013 was highly anisotropic, a dependence of the fidelity of the results on the image orientation may be expected.

### 4.2. M87* Images

For additional synthetic data tests, we consider images of M87* generated based on the 2017 EHT observations, published in EHTC IV. We consider two days of the 2017 observations with good coverage and reported structural source differences (EHTC III; EHTC IV; Arras et al. 2020): 2017 April 6 (MODEL1) and 2017 April 11 (MODEL2); see the first row of Figure 7. MODEL2 was also shown in the first panel of Figure 1. While these models are constructed based on the observational data, we resample them numerically to obtain synthetic data sets considered in this section. Synthetic CPs on the SMT-SMA-CARMA triangle computed from these models were shown in Figure 4. Note that there is a subtle difference between resampling a model constructed based on the 2017 data with a numerical model of the 2017 array and direct modeling of the actual 2017 data, considered in Section 5. The sampled images were generated utilizing the eht-imaging pipeline through the procedure outlined in EHTC IV, with a resolution of 64 × 64 pixels. This test can be viewed as an attempt to evaluate what the outcome of the modeling efforts would have been had the 2017 EHT observations been carried out with one of the proto-EHT 2009–2013 arrays rather than with the mature 2017 array.

### 4.3. Results for the Synthetic Data Sets

Figure 7 shows a summary of the maximum likelihood (ML) RT model fits to the synthetic data sets; each column shows the fits for a single ground-truth image, and each row shows the fits for a single array configuration. Though our simple ring models cannot fully reproduce the properties of the abundant and high signal-to-noise data sampled with the 2017 array (i.e., fits to these data sets are characterized by poor reduced-$\chi^2$ values of $\chi_n^2 \sim 5$), they nevertheless recover diameter and orientation values that are reasonably consistent with those reported in EHTC VI for the 2017 observations, including the counterclockwise shift of the brightness position angle between 2017 April 6 (MODEL1) and 11 (MODEL2). We note that the underfitting of the data set sampled with the 2017 array results in an artificial narrowing of the parameter posteriors (the full posterior is captured in EHTC VI by considering a more complicated GC model). ML estimators for 2009–2013 data sets, on the other hand, typically fit the data much closer than the thermal error budget. Because we model time-dependent station gains in a

<footnote n="113">¹¹³ Hereafter, we use natural units in which $G = c = 1$.</footnote>
<figure><img src="image_9.png" /><figcaption>**Figure 7.** ML estimators corresponding to the fits to four synthetic images, shown in the first row (no blurring). Estimators were obtained through synthetic VLBI observations with (u, v) coverage and uncertainties identical to those of the real observations performed in 2009–2017. The thick ring model (RT) was used, and the presented images of ML estimators were blurred to 15 µas resolution. Blue dashed lines indicate the convention for the expected position angle of the bright component $\phi_{B,\text{exp}}$. The gray bar represents the ML estimate of $\phi_B$. For the 2009, 2011, and 2013 data sets, the orientation is determined assuming that $|\phi_{B,\text{exp}} - \phi_B| < 90^\circ$. The dashed circles correspond to a diameter of 42 µas.</figcaption></figure>

small array (often only two to three telescopes observing at the same time in 2009–2013 with missing detections on some baselines), the number of model parameters may be formally larger than the number of data points, and this complicates our estimation of the number of effective degrees of freedom (see also Section 5.3). As a consequence, we cannot

generally utilize a $\chi_n^2$ goodness-of-fit statistic as was done in EHTC IV and EHTC VI.<fnref n="114" />

<footnote n="114">¹¹⁴ See, e.g., Andrae et al. (2010) for further comments about the problems with the $\chi_n^2$ metric and counting the degrees of freedom.</footnote>
<table><caption>**Table 2**
Parameter Estimates from Fitting the RT Model to the Synthetic Data Sets</caption><thead><tr><th rowspan="2">Estimator<br>Confidence</th><th rowspan="2">Coverage</th><th colspan="2">d(µas)</th><th rowspan="2">φ<sub>B</sub> (deg)<br>ML<br>68%</th><th rowspan="2">f<sub>w</sub><br>At Most<br>95%</th></tr><tr><th>Median<br>68%</th><th>ML<br>95%</th></tr></thead><tbody><tr><td rowspan="5">GRMHD1</td><td>2009</td><td>46.4<sup>+5.3</sup><sub>-7.9</sub></td><td>51.1<sup>+4.9</sup><sub>-18.5</sub></td><td>194<sup>+56</sup><sub>-5</sub></td><td>0.88</td></tr><tr><td>2011</td><td>40.9<sup>+5.6</sup><sub>-5.5</sub></td><td>46.8<sup>+5.1</sup><sub>-13.9</sub></td><td>217<sup>+55</sup><sub>-35</sub></td><td>0.93</td></tr><tr><td>2012</td><td>44.0<sup>+4.9</sup><sub>-5.2</sub></td><td>49.3<sup>+4.3</sup><sub>-15.1</sub></td><td>244<sup>+35</sup><sub>-24</sub></td><td>0.90</td></tr><tr><td>2013</td><td>39.2<sup>+2.0</sup><sub>-0.8</sub></td><td>41.3<sup>+2.1</sup><sub>-4.0</sub></td><td>(246)</td><td>0.56</td></tr><tr><td>2017<sup>b</sup></td><td>43.8<sup>+0.1</sup><sub>-0.1</sub></td><td>43.8<sup>+0.1</sup><sub>-0.1</sub></td><td>225<sup>+1</sup><sub>-1</sub></td><td>0.33</td></tr><tr><td rowspan="5">GRMHD2</td><td>2009</td><td>43.1<sup>+3.4</sup><sub>-3.0</sub></td><td>37.2<sup>+11.8</sup><sub>-2.7</sub></td><td>(302)</td><td>0.95</td></tr><tr><td>2011</td><td>44.3<sup>+2.9</sup><sub>-5.4</sub></td><td>47.4<sup>+2.2</sup><sub>-12.4</sub></td><td>(265)</td><td>0.94</td></tr><tr><td>2012</td><td>42.2<sup>+3.3</sup><sub>-3.8</sub></td><td>36.3<sup>+10.0</sup><sub>-0.5</sub></td><td>310<sup>+6</sup><sub>-55</sub></td><td>0.96</td></tr><tr><td>2013</td><td>43.6<sup>+1.1</sup><sub>-0.6</sub></td><td>43.5<sup>+2.4</sup><sub>-1.2</sub></td><td>322<sup>+38</sup><sub>-55</sub></td><td>0.50</td></tr><tr><td>2017<sup>b</sup></td><td>42.4<sup>+0.1</sup><sub>-0.1</sub></td><td>42.4<sup>+0.1</sup><sub>-0.1</sub></td><td>295<sup>+1</sup><sub>-1</sub></td><td>0.33</td></tr><tr><td rowspan="5">MODEL1</td><td>2009</td><td>45.7<sup>+3.0</sup><sub>-4.7</sub></td><td>46.5<sup>+4.6</sup><sub>-9.1</sub></td><td>169<sup>+10</sup><sub>-80</sub></td><td>0.95</td></tr><tr><td>2011</td><td>47.1<sup>+2.7</sup><sub>-4.6</sub></td><td>49.6<sup>+2.2</sup><sub>-11.4</sub></td><td>190<sup>+65</sup><sub>-5</sub></td><td>0.90</td></tr><tr><td>2012</td><td>41.3<sup>+3.6</sup><sub>-4.0</sub></td><td>36.4<sup>+9.9</sup><sub>-1.2</sub></td><td>352<sup>+38</sup><sub>-47</sub></td><td>0.96</td></tr><tr><td>2013</td><td>43.1<sup>+2.1</sup><sub>-1.0</sub></td><td>45.1<sup>+1.8</sup><sub>-3.9</sub></td><td>177<sup>+36</sup><sub>-27</sub></td><td>0.63</td></tr><tr><td>2017<sup>b</sup></td><td>41.0<sup>+0.1</sup><sub>-0.1</sub></td><td>41.0<sup>+0.1</sup><sub>-0.1</sub></td><td>156<sup>+1</sup><sub>-1</sub></td><td>0.47</td></tr><tr><td rowspan="5">MODEL2</td><td>2009</td><td>42.7<sup>+3.8</sup><sub>-3.7</sub></td><td>36.0<sup>+11.9</sup><sub>-1.0</sub></td><td>169<sup>+30</sup><sub>-69</sub></td><td>0.97</td></tr><tr><td>2011</td><td>48.3<sup>+2.1</sup><sub>-4.7</sub></td><td>50.4<sup>+2.0</sup><sub>-10.7</sub></td><td>180<sup>+72</sup><sub>-5</sub></td><td>0.91</td></tr><tr><td>2012</td><td>41.0<sup>+3.3</sup><sub>-2.7</sub></td><td>37.0<sup>+9.7</sup><sub>-0.3</sub></td><td>(179)</td><td>0.98</td></tr><tr><td>2013</td><td>47.0<sup>+1.1</sup><sub>-1.3</sub></td><td>47.3<sup>+1.8</sup><sub>-2.4</sub></td><td>174<sup>+2</sup><sub>-10</sub></td><td>0.53</td></tr><tr><td>2017<sup>b</sup></td><td>41.2<sup>+0.1</sup><sub>-0.1</sub></td><td>41.2<sup>+0.1</sup><sub>-0.1</sub></td><td>180<sup>+1</sup><sub>-1</sub></td><td>0.50</td></tr></tbody></table>

**Notes.**

a $\eta$ calculated with Equation (11).

b Using the (u, v) coverage of 2017 April 6.

The relevant parameter estimates and uncertainties from the RT model fits are listed in Table 2. In Figure 6, we show the marginalized posteriors for the diameter and position angle parameters. For each synthetic data set, we also indicate the image domain position angle $\eta$, defined as

$$ \eta = \operatorname{Arg} \left( \frac{\sum_k I_k e^{i\phi_k}}{\sum_k I_k} \right), \qquad (11) $$

where $I_k$ is the intensity and $\phi_k$ is the position angle of the kth pixel in the image. A similar image domain position angle estimator was considered in EHTC IV. We notice that the image- and model-based estimators may occasionally display significant differences (e.g., GRMHD1). However, they are both sensitive to global properties of the brightness distribution, unlike some other estimators that could be considered, such as, e.g., the location of the brightest pixel. For the diameter $d$ estimates reported in Table 2, we list both the median and ML values, with 68% and 95% confidence intervals, respectively. For the orientation angle $\phi_B$, we list the ML values with 68% confidence intervals, and for the fractional thickness $f_w$, we list the 95th distribution percentile. Values of $\phi_B$ contained in parentheses indicate that the 68% confidence interval exceeds 100°, in which case we have concluded that the orientation is effectively unconstrained. We find that the diameter is well constrained in general, with the GRMHD data sets recovering a typical value of ~44 µas and

95% confidence intervals that never exceed ±12 µas from this value; the analogous measurement for the MODEL data sets is 44 ± 9 µas. Biases related to the array orientation can be seen —particularly with the 2013 coverage, the GRMHD2 test estimates an appreciably larger diameter than GRMHD1, inconsistent within the 68% confidence interval.

The orientation $\phi_B$ is poorly constrained, with posterior distributions that depend strongly on the details of the (u, v) coverage. Nevertheless, the 2009–2013 ML estimates provide orientations of the axis of asymmetry that are consistent within ±35° with the results obtained using the 2017 synthetic coverage. We note that in three out of four synthetic data sets, the limited number of CPs provided by the simulated data sets with the 2012 coverage is enough to correctly break the degeneracy in the position angle $\phi_B$, discussed in Section 3.3. For the synthetic GRMHD data sets, the preference for the correct brightness position angle is very strong (see Figure 6). For the MODEL data sets, the effect of CPs is much less prominent, the distributions remain bimodal, and in the case of the MODEL1 data set, the ML estimator points at the wrong orientation, suggesting brightness located in the north.

We also consider the fractional thickness $f_w$ of the ring, as defined in Equation (5). The fractional thickness provides a measure of whether the data support the presence of a central flux depression, a signature feature of the black hole shadow, or if it is consistent with a disk-like morphology (i.e., $f_w \approx 1$ for the RT model). We find that $f_w$ is less well constrained than the diameter $d$, consistent with the conclusions of EHTC VI. In some cases, the ML estimator corresponds to a limit of a disk-like source morphology without a central depression (see Figure 7). Only the 2013 and 2017 synthetic data sets allow us to confidently establish the presence of a central flux depression, with posterior distributions excluding $f_w > 0.7$ for all synthetic data sets (see Table 2). For the 2009–2012 coverage synthetic data sets, $f_w$ is not constrained sufficiently well to permit similar statements. We find that the RG model produces results that are typically consistent with those of the RT model (see Appendix A, Figure A1).

## 5. Modeling Real Data

Encouraged by the results of the tests on synthetic data sets, we performed the same analysis on the 2009–2013 proto-EHT M87* observations. We also present the analysis of the 2017 observations with the RT and RG models. In the latter case, only lower band data (2 GHz bandwidth centered at 227 GHz) were used.

### 5.1. Source Geometry Estimators

In the first row of Figure 8, we show the ML estimators obtained by fitting the RT model to each observational data set; in Figure 9, we show the same for the RG model. For the 2009, 2011, and 2013 data sets, which only constrain the axis of the crescent asymmetry, the orientation of the brightness peak was selected with a prior derived from the approaching jet orientation on the sky, $\phi_B \in (\phi_{jet} - 180^\circ, \phi_{jet})$, Section 3.3. The 2012 data set, for which some CPs are available, indicates a weak preference toward the brightness position located in the north rather than in the south (Figure 10 and Appendix B, Figure B1). However, the posterior distribution remains bimodal and 47% of its volume remains consistent with the jet-orientation-based prior. Hence, the distinction is not very
<figure><img src="image_10.png" /><figcaption>**Figure 8.** First row: ML estimators obtained from fitting the RT model to the 2009–2017 observations. The position angle $\phi_B$ is indicated with a bar. For the 2009, 2011, and 2013 data sets, the orientation is determined assuming that $\phi_{jet} - 180^{\circ} < \phi_B < \phi_{jet}$, where $\phi_{jet} = 288^{\circ}$. Position angle 68% confidence intervals are shown for the 2009–2013 data sets. Second row: RT models from the first row blurred to a 15 $\mu$as resolution, indicated with a beam circle in the bottom-right corner of the first panel. The dashed circle of 42 $\mu$as diameter is plotted for reference. Third row: mean of the $2 \times 10^4$ images drawn from the posterior of the RT model fits.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Figure 9.** Same as Figure 8, but for the RG model. The 2009–2012 posterior distributions contain a Gaussian mode, manifesting as a bright ring interior in the mean images. This is related to the lower spatial resolution of the 2009–2012 observations.</figcaption></figure>

significant—it is entirely dependent on the sign of CPs shown in Figure 4, which are all consistent with zero to within $2\sigma$. CPs predicted by the ML estimators for the RT and RG models are indicated in Figure 4. The 2017 data sets are consistent with the orientation imposed by the jet position prior. The second rows of Figures 8 and 9 show the ML estimators blurred to

a resolution of 15 $\mu$as. In the third rows of Figures 8 and 9, we present “mean images” for each data set, obtained by sampling $2 \times 10^4$ sets of model parameters from the MCMC chains and averaging the corresponding images. The mean images highlight structure that is “typical” of a random draw from the posterior distribution, though we note that a mean image itself
<figure><img src="image_12.png" /><figcaption>**Figure 10.** Top: marginalized distributions of mean diameter *d* and brightness maximum position angle φ<sub>B</sub> for the RT model fits to 2009–2017 observational data sets. Lightly shaded areas correspond to values reported in EHTC I, diameter *d* = 42 ± 3 µas and position angle 150° < φ<sub>B</sub> < 200°. The 2017 posteriors are contained within the dark gray bands. The dashed vertical line in the left panels denotes the diameter of 2√27 M/D. The vertical dashed lines in the right panels denote the convention angle φ<sub>B,exp</sub> = 198°, φ<sub>B,exp</sub> − 90°, and the approaching jet position angle φ<sub>jet</sub> = 288°. The range of (φ<sub>jet</sub> − 180°, φ<sub>jet</sub>) is highlighted. Bottom: same, but for the RG model.</figcaption></figure>

does not necessarily provide a good fit to the data. Because of
the rotational degeneracy, the orientation is always assumed to
be the one closer to the orientation given by the ML estimate of
φ<sub>B</sub> for the construction of these images.

5.2. *Estimated Parameters*

The marginalized posteriors for the mean diameter *d* and
position angle φ<sub>B</sub> for the observational data sets are shown in
Figure 10 for both the RT and RG models, and tabulated values
of the relevant estimates for the RT model are given in Table 3.
The posterior distributions for the 2009–2012 data sets have
complex shapes, not all parameters are well constrained, and
ML estimators do not necessarily coincide with the margin-
alized posteriors maxima of the individual model parameters,
which can be seen in the corner plots (Appendix B, Figure B2).
The behavior of the posterior distributions is much improved
for the 2013 data set and becomes exemplary in the case of the
2017 data sets (Figure B3).

Similar to the case of the synthetic data sets, we find that the
diameter *d* is well constrained; the RT model 95% confidence
intervals across all observational data sets always fall within
±12 µas from *d* = 40 µas. The 2013 proto-EHT observations
provide meaningful constraints on *φ*<sub>B</sub>, indicating that the source
asymmetry in 2013 was in the east–west direction rather than in
the north–south direction, as in the case of the 2017 data set. The
2009 and 2011 data sets do not constrain the orientation well.

All ML estimators and mean images from the RT model fits
show a clear shadow feature, indicating that a disk-like, filled-
in structure is disfavored by all observations (however, for
2009–2012, it cannot be excluded with high confidence based
on the relative thickness parameter *f*<sub>w</sub> distribution; Table 3 and

Appendix B). This is contrary to the synthetic data results
shown in Figure 7, where some of the ML estimators
correspond to a disk-like morphology. On the other hand, the
mean images for the 2009–2012 RG model fits show a
significant flux density interior to the ring, indicating that these
data sets are consistent with a symmetric Gaussian source
model, having no central flux depression. This is a conse-
quence of the resolution being limited by the lack of long
baselines prior to 2013. Short and medium-length baselines
alone provide insufficient information to fully exclude the
Gaussian mode allowed by the RG model or the disk-like mode
allowed by the RT model. For the same reason we see flattened
posterior distributions of the RG diameter for 2009–2012 in
Figure 10—these indicate consistency with a small, strongly
blurred ring with *f*<sub>w</sub> > 1, becoming a Gaussian in the limit
of *σ* ≫ *d*.

The slash parameter $\beta$ can be measured to be 0.3 $\pm$ 0.1 for the 2013 data set, which is consistent with the fits to the 2017 data sets that give $\beta \sim 0.20$ (RT) or $\beta \sim 0.35$ (RG). Fits to the 2012 data set indicate preference toward more symmetric brightness distribution; 2009 and 2011 do not provide meaningful constraints on $\beta$.

5.3. *Quality of the Visibility Amplitude Fits*

The quality of fits and their behavior in terms of $\chi_n^2$ are
similar to the synthetic data sets (see the comments in
Section 4.3 and Table 4). In Figures 11–12, we explicitly give
the number of independent visibility amplitude observations for
each data set $N_{\text{ob}}$ and the number of independent visibilities on
nonzero (intersite) baselines $N_{\text{nz}}$. Note that the latter is larger
than the number of detections on nonzero, nonredundant
<table><caption>**Table 3**
Parameter Estimates from Fitting the RT Model to the Observational Data Sets</caption><thead><tr><th rowspan="2">Estimator<br>Confidence</th><th colspan="2">d(μas)</th><th rowspan="2">φ<sub>B</sub>(deg)<br>ML<br>68%</th><th rowspan="2">f<sub>w</sub><br>At Most<br>95%</th></tr><tr><th>Median<br>68%</th><th>ML<br>95%</th></tr></thead><tbody><tr><td>2009</td><td>39.8<sup>+5.5</sup><sub>-5.2</sub></td><td>47.3<sup>+3.1</sup><sub>-16.0</sub></td><td>202<sup>+68</sup><sub>-23</sub></td><td>0.93</td></tr><tr><td>2011</td><td>36.5<sup>+2.0</sup><sub>-4.3</sub></td><td>38.6<sup>+2.8</sup><sub>-9.9</sub></td><td>(130)</td><td>0.91</td></tr><tr><td>2012<sup>a</sup></td><td>40.1<sup>+2.1</sup><sub>-4.8</sub></td><td>40.6<sup>+2.5</sup><sub>-8.3</sub></td><td>342<sup>+42</sup><sub>-40</sub></td><td>0.92</td></tr><tr><td>2013</td><td>46.8<sup>+2.3</sup><sub>-2.9</sub></td><td>47.6<sup>+3.0</sup><sub>-5.5</sub></td><td>281<sup>+17</sup><sub>-34</sub></td><td>0.28</td></tr><tr><td>2017<sup>b</sup></td><td>41.1<sup>+0.1</sup><sub>-0.1</sub></td><td>41.1<sup>+0.2</sup><sub>-0.2</sub></td><td>155<sup>+1</sup><sub>-1</sub></td><td>0.37</td></tr><tr><td>2017<sup>c</sup></td><td>40.7<sup>+0.1</sup><sub>-0.1</sub></td><td>40.7<sup>+0.1</sup><sub>-0.1</sub></td><td>184<sup>+1</sup><sub>-1</sub></td><td>0.44</td></tr></tbody></table>

**Notes.**

a Secondary mode present at $\phi_B$ – 180° (see the text).

b 2017 April 6.

c 2017 April 11.

<table><caption>**Table 4**
Comparison between Parameters Extraction Results Reported in This Paper, EHTC IV, and EHTC VI</caption><thead><tr><th>Source</th><th>Method</th><th>d(μas)</th><th>φ<sub>B</sub> or η (deg)<sup>a</sup></th><th>χ<sup>2</sup><sub>n</sub></th></tr></thead><tbody><tr><td colspan="5" style="text-align: center;">2017 Apr 6</td></tr><tr><td rowspan="2">This work</td><td>RT</td><td>41.1<sup>+0.09</sup><sub>-0.08</sub></td><td>154.8 ± 1.1</td><td>2.99</td></tr><tr><td>RG</td><td>41.6 ± 0.07</td><td>154.3<sup>+1.2</sup><sub>-1.1</sub></td><td>3.04</td></tr><tr><td>EHTC VI<br>(GC)</td><td>THEMIS</td><td>43.5 ± 0.14</td><td>153.0<sup>+2.0</sup><sub>-2.4</sub></td><td>1.32</td></tr><tr><td></td><td>dynesty</td><td>43.4<sup>+0.27</sup><sub>-0.26</sub></td><td>148.5<sup>+1.4</sup><sub>-1.2</sub></td><td>1.29</td></tr><tr><td>EHTC IV<br>(image<br>domain)</td><td>DIFMAP</td><td>40.1 ± 7.4</td><td>162.1 ± 9.7</td><td>2.10</td></tr><tr><td></td><td>eht-imaging</td><td>39.6 ± 1.8</td><td>151.1 ± 8.6</td><td>1.28</td></tr><tr><td></td><td>SMILI</td><td>40.9 ± 2.4</td><td>151.7 ± 8.2</td><td>1.34</td></tr><tr><td colspan="5" style="text-align: center;">2017 Apr 11</td></tr><tr><td rowspan="2">this work</td><td>RT</td><td>40.7 ± 0.05</td><td>184.2<sup>+0.6</sup><sub>-0.7</sub></td><td>7.26</td></tr><tr><td>RG</td><td>41.6<sup>+0.05</sup><sub>-0.04</sub></td><td>185.0 ± 0.6</td><td>7.49</td></tr><tr><td>EHTC VI<br>(GC)</td><td>THEMIS</td><td>42.2<sup>+0.43</sup><sub>-0.41</sub></td><td>201.1<sup>+2.5</sup><sub>-2.3</sub></td><td>1.07</td></tr><tr><td></td><td>dynesty</td><td>41.6<sup>+0.51</sup><sub>-0.46</sub></td><td>175.9<sup>+2.1</sup><sub>-2.0</sub></td><td>0.89</td></tr><tr><td>EHTC IV<br>(image<br>domain)</td><td>DIFMAP</td><td>40.7 ± 2.6</td><td>173.3 ± 4.8</td><td>2.19</td></tr><tr><td></td><td>eht-imaging</td><td>41.0 ± 1.4</td><td>168.0 ± 6.9</td><td>0.97</td></tr><tr><td></td><td>SMILI</td><td>42.3 ± 1.6</td><td>167.6 ± 2.8</td><td>1.08</td></tr></tbody></table>

**Notes.**

a Results from this work and EHTC VI correspond to visibility domain-based estimator $\phi_B$, while results from EHTC IV correspond to the image domain estimator $\eta$, similar to the one given by Equation (11).

baselines given in Table 1, as some detections are independent but redundant. We also provide the number of explicitly modeled amplitude gains $N_g$ for each data set (see Sections 3.2 and 4.3). Given the pathologies in the $\chi_n^2$ metric described in Section 4.3, we characterize the quality of the ML estimator fits to data using the two following metrics:

$$ \bar{e} = \frac{1}{N_{nz}} \sum_{i=1}^{N_{nz}} \left| \frac{|V_i| - |\bar{V}_i|}{\sigma_i} \right|, \qquad (12) $$

$$ \hat{e} = \frac{1}{N_{nz}} \sum_{i=1}^{N_{nz}} \left| \frac{|V_i| - |\hat{V}_i|}{\sigma_i} \right|. \qquad (13) $$

In Equations (12)–(13) we follow the notation of Equation (6), that is, $\bar{V}_i$ represents the visibilities of the geometric model while

$\hat{V}_i$ corresponds to the model modified by applying the estimated gains, representing the final fit to observations $V_i$. Uncertainties $\sigma_i$ correspond to the thermal error budget. We only account for nonzero baselines, which describe the compact source properties. In the bottom rows of Figures 11–12, we indicate two error bars. Black error bars correspond to the thermal uncertainties $\sigma_i$, while the red ones correspond to inflated uncertainties

$$ s_i = \sqrt{\sigma_i^2 + 2 \cdot (0.15 V_i)^2}, \qquad (14) $$

approximately capturing the uncertainty related to the amplitude gains. For all 2009–2013 data sets, the flexibility of the full model is sufficient to fit the sparse data to within the thermal uncertainty level with a best-fit ML estimator.

## 5.4. Consistency with the Prior Analysis

In order to assess model-related biases and verify to what extent our simplified models recover geometric parameters consistent with the ones reported by EHT, i.e., image domain results given in EHTC IV Tables 5 and 7, and geometric modeling results given in EHTC VI Tables 2 and 3, we gather these results in Table 4. For the details of the methods and algorithms, see explanations and references in EHTC IV and EHTC VI. We notice that (1) differences between methods may be as large as 30° for the same data set, (2) models considered in this work measure diameter and position angle consistently with more complex crescent models (GC) and with the image domain methods within the expected intermodel variation, (3) RT and RG models are too simplistic to fully capture the properties of the 2017 data sets, resulting in underfitting, indicated by higher values of $\chi_n^2$, and (4) posteriors of the RT and RG models are narrower for the 2017 data sets than the GC posteriors as an effect of the underfitting.

# 6. Discussion and Conclusions

Processing five independent observations of M87* in a unified framework offers important insights into the source morphology and stability. While the constraining power of the 2009–2013 data sets is rather weak in comparison to the 2017 observations, we find evidence for a persistent ring structure that shows modest structural variability. Both the persistence and variability offer important constraints for models of M87*. We will now discuss our evidence and theoretical implications for the presence of the shadow feature in 2009–2017 (Section 6.1), the persistence of the source geometry as an argument for its theoretical interpretation (Section 6.2), and the variability of the source geometry (Section 6.3). We also summarize the limitations and caveats of the theoretical interpretation within the GRMHD framework (Section 6.4).

## 6.1. Presence of the Shadow Feature

In Section 5 we have discussed the fits of the asymmetric ring models to the M87* observations, indicating that all data sets are consistent with such a geometry. Within the framework of a ring model, a key question for the archival observations is whether we unambiguously detect the inner flux depression seen in the 2017 results, the expected feature of a black hole shadow. While maximum likelihood estimators clearly indicate this feature in all cases (Figure 8), a detailed inspection of the posterior distributions shown in Appendix B allows us to
<figure><img src="image_13.png" /><figcaption>**Figure 11.** Top row: visibility amplitudes measured in 2009–2012 are shown as black diamonds with error bars corresponding to thermal uncertainties. Blue shaded regions correspond to the range of visibility amplitudes of the asymmetric ring RT model ML estimators, shown in the first row of Figure 9. The total number of observed visibility amplitudes $N_{\text{ob}}$ is given, along with the number of nonzero baseline visibility amplitudes $N_{\text{nz}}$ and the number of modeled gains $N_g$. Bottom row: differences between measured amplitudes $|V_i|$ and the geometric model amplitudes $|\tilde{V}_i|$. Black error bars correspond to thermal uncertainties, while red ones correspond to error budget inflated by adding systematics approximately capturing the gains uncertainties (Equation (14)). Two fit quality metrics, defined with Equations (12)–(13), are provided for each data set.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Figure 12.** Same as Figure 11, but for the 2013–2017 data sets.</figcaption></figure>

conclude that only the 2013 archival data set provides a robust detection of the central flux depression, constraining the relative thickness of the ring $f_w$ to less than 0.5 (see Table 3). For the 2009–2012 data sets, the RT model posteriors indicate some preference toward the presence of a flux depression;

however, a disk-like filled-in geometry cannot be excluded with a high degree of confidence (Appendix B). This is not entirely surprising, as the 2009–2012 observations lack baselines with projected lengths >3.6 Gλ, probing spatial frequencies higher than the visibility null $b_0$ (Section 2.4),
<figure><img src="image_15.png" /><figcaption>**Figure 13.** Top row: a single GRMHD simulation snapshot, ray-traced with total compact flux density adjusted to values between 0.1 Jy and 1.1 Jy. The brightness distribution in each panel has been normalized by the brightness maximum. EHT observations of M87\* found a total compact flux density of 0.8–0.9 Jy in 2009–2012 and 0.5–0.6 Jy in 2013 and 2017; see the bottom panel of Figure 3. Bottom row: same snapshots blurred to the resolution of 15 µas. The 42 µas diameter ring is indicated with a dashed circle. The position angle convention $\phi_{B,exp}$ is shown with a dashed blue bar. The gray bar indicates the image domain position angle $\eta$ calculated using Equation (11), and values of $\eta$ obtained are given in the bottom-right corner of each panel.</figcaption></figure>

which are more sensitive to differences between an empty ring, a filled-in disk, and a Gaussian.

However, the presence of the shadow feature is sensitive to changes in the optical depth. The total compact flux density in 2009–2012 was measured to be 0.8–0.9 Jy, significantly higher than the 0.5–0.6 Jy observed in 2013 and 2017 (Figure 3, bottom panel). Mildly elevated levels of X-ray emission from the nucleus of M87 before 2016 were also reported (Sun et al. 2018). These measurements suggest a higher mass accretion rate in 2009–2012 and hence a larger density scale, in turn increasing the opacities and the optical depth, possibly changing the appearance of the black hole shadow (Mościbrodzka et al. 2012; Chan et al. 2015). Is it possible that the shadow feature had been obscured by a more optically thick medium in 2009–2012 than in the case of the more recent 2013 and 2017 observations? While the fully general answer to that question would require extensive testing of a variety of GRMHD models, we address this concern by analyzing a representative example from the library of simulated M87\* images. We consider a random snapshot from an SANE simulation with spin $a^* = 0.5$ and electron temperature parameter $R_{\text{high}} = 20$. The $R_{\text{low}}$ parameter is equal to 1, as it is throughout this paper. The snapshot is then repeatedly ray-traced with its density scale (and hence opacities and emissivities) adjusted to give a total compact flux density equal to 0.1, 0.3, 0.5, 0.7, 0.9, and 1.1 Jy. The resulting images, normalized by the brightness maximum, are shown in Figure 13. These findings indicate that variation in the total compact flux density between 0.1 and 1.1 Jy does not eliminate the central brightness depression. Judging from the similarity between blurred images, the flux density scaling is also not expected to influence the estimated diameter or the position angle appreciably. However, it is likely to influence measures of asymmetry, such as the slash parameter $\beta$ in the RT/RG models, discussed in Section 3.1. Interestingly, we see a preference toward more symmetric source geometry (larger $\beta$) in the 2012 posteriors (Appendix B). We conclude that the central flux depression was most likely present throughout the 2009–2017 observations and that a lack of a high-confidence

<figure><img src="image_16.png" /><figcaption>**Figure 14.** Year-to-year consistency of the best-fitting (ML) asymmetric Gaussian model to the M87\* data sets.</figcaption></figure>

detection of this feature in 2009–2012 is presumably a consequence of the very limited $(u, v)$ coverage.

None of the EHT observations so far took place during an unambiguous flaring activity period of the M87 nucleus, such as the events discussed in Abramowski et al. (2012). We note that such an event could potentially influence the image morphology more strongly than the moderate increase of total brightness considered in Figure 13. Future simultaneous multiwavelength observational campaigns will shed light on the structural changes in the M87\* compact radio image in relation to the enhanced activity in different parts of the spectrum, allowing the site of particle acceleration to be localized.

## 6.2. Black Hole Shadow or a Transient Feature

As discussed in Section 3, the 2009–2013 data sets can be successfully modeled with both an asymmetric Gaussian and a ring model, with each giving a similar fit quality. However, the ML Gaussian models are very inconsistent in size, shape, and orientation across different years; see Figure 14. In contrast, the best-fitting ring models, as seen in Figure 8, are similar in size over all epochs under the priors described in Sections 3.2 and 3.3. Even when considered separately from the 2017 results, this consistency supports our choice to interpret the 2009–2013 morphology as a ring and to draw conclusions from the results of fitting asymmetric ring models to the data. The 95% confidence intervals of all 2009–2017 diameter posteriors lie within $40 \pm 12 \mu\text{as}$, while ML estimators of the diameter from all years lie within $43 \pm 5 \mu\text{as}$; see Table 3. These values are also consistent with the M87\* diameter measurement of $42 \pm 3 \mu\text{as}$ reported by EHTC VI
and with the expected size of an observed black hole of mass/
distance corresponding to the stellar dynamics measurement
(Blakeslee et al. 2009; Gebhardt et al. 2011). All 2009–2017
diameter measurements are inconsistent with the gas dynamics
mass measurement (Walsh et al. 2013), which predicts
a diameter roughly half as large. The consistency in the
diameter across multiple observational epochs supports the
interpretation of the ring-like feature as emission from the
immediate surroundings of the supermassive black hole.

The four EHT observations of M87* in 2017 spanned about a
week, corresponding to a timescale of ~15 M. With such a short
time span, we cannot exclude a transient origin for the source
morphology using 2017 data alone. However, such a feature
would need to attain the geometry and apparent size expected of
a shadow of a black hole (of independently measured mass-to-
distance ratio) through an unusual coincidence. Moreover,
transient features are unlikely to persist with a similar geometry
throughout multiple years of observations, corresponding to
~10³⁻¹⁰⁴ M timescales (the total span of our analyzed
observations is 7900 M). For example, a lensed background
source would need a low transverse velocity of v ~ 40 km s⁻¹ to
travel ~1 M between 2009 and 2017. This is much smaller than
the gas velocities seen in the nucleus of M87 (e.g., Macchetto
et al. 1997). A bright feature moving with the jet (v ~ c) should
travel ~1 pc on that timescale, a factor of roughly 10³ larger than
the physical size of the ring itself. A bright knot or other
stationary jet feature would need to persist with a similar
location, flux density, and ring morphology to remain consistent
with these results. The 8 yr span of the 2009–2017 monitoring is
also much longer than the typical variability timescale of the
M87 nucleus observed at 230 GHz, which is ~50 days (Bower
et al. 2015). While features remaining stationary for many years
in otherwise rapidly flowing jets have been reported and
interpreted as standing recollimation shocks (Lister et al.
2009), such a configuration would constitute one more unusual
coincidence. Thus, we conclude that with multiple years of
observations remaining consistent with a ~40 µas ring model, it
is highly unlikely that the origin of the observed geometry could
be a transient feature.

6.3. Time Variability of the Source Geometry

In addition to conclusions from the persistence of the ring
structure, we can also draw inferences from the variability
observed in the ring structure across the 2009–2017 data sets.
In particular, the spread of the diameter and brightness position
angle estimates (Table 3) are significantly larger than the spread
for corresponding static synthetic data sets (Table 2). As a
specific example, the circular standard deviation of the ML
position angle estimators given in Table 2 is equal to 19°, 19°,
11°, and 4° for GRMHD1, GRMHD2, MODEL1, and
MODEL2, respectively. For the observational data (Table 3),
the circular standard deviation is equal to 48°. This larger
spread suggests that we are detecting intrinsic structural
variability despite the large uncertainties in the parameters
estimated with pre-2017 observations. Moreover, unambiguous
signatures of intrinsic variability on a timescale of years can be
seen directly in the visibility data (Section 2.4).

Because GRMHD simulations naturally model both the source
structure and its variability, they provide an important pathway for
drawing conclusions from the observed variability. As a
preliminary demonstration for a comprehensive study that will
be published separately, we consider a small subset of the EHT

<figure><img src="image_17.png" /><figcaption>**Figure 15.** Histograms of the brightness position angle $\eta$, measured in the image domain for 500 ray-traced snapshots of GRMHD simulations from the EHT M87\* Simulation Library (EHTC V). In each panel, the blue histogram represents the same fixed model S1: MAD, $a_*=0.5$, $R_{\text{high}}=10$, and the red histogram represents a model in which a single parameter has been altered with respect to the fixed model (S2–S4). Orientations measured in the 2009–2017 observational data sets with an ML estimator are indicated, with 68% confidence intervals (Table 3, two results from 2017 are shown without their very narrow error bars). The gray area around $\phi_{\text{jet}}$ corresponds to the observed jet position angle variation (Walker et al. 2018).</figcaption></figure>

Image Library (EHTC V). The simulations are parameterized with
the black hole spin $a_*$, the electron temperature parameter $R_{\text{high}}$
(see Section 4.1), and the accretion state—strongly magnetized
magnetically arrested disk (MAD; Narayan et al. 2003) or SANE
flow, such as those considered in Sections 4.1 and 6.1. Other
parameters (e.g., the total compact flux density of 0.5 Jy,
inclination, jet position angle) are adjusted to match the observed
properties of M87*. For our exploratory study, we utilize the
following four simulations: (S1) MAD, $a_*=0.5$, $R_{\text{high}}=10$;
(S2) SANE, $a_*=0.5$, $R_{\text{high}}=10$; (S3) MAD, $a_*= -0.5$,
$R_{\text{high}}=10$; (S4) MAD, $a_*=0.5$, $R_{\text{high}}=160$. For each simula-
tion, we take 500 ray-traced snapshots with 5 M separation in time.
For each snapshot, we calculate the position angle of the bright
component $\eta$ using Equation (11). We then construct histograms
of $\eta$ for each of the four simulations, shown in Figure 15.

Each of the simulation parameters influences the distribution of
η, both in terms of its mean and spread. Some of these differences
can be readily understood, for instance, in the case of prograde
accretion onto a spinning black hole, the radiation is boosted both
with Doppler and with frame-dragging effects. The position angle
of the bright component is thus expected to be relatively more
influenced by the geometry (assumed to be fixed) and not by the
stochastic component than the retrograde case, in which Doppler
effect and frame-dragging counteract, and the geometry becomes
relatively less important. Irrespective of the mechanism, some
variants of simulations have great difficulties explaining either
source orientation on 2017 April 6 (bright component too far
clockwise), or in 2013 (too far counterclockwise), as indicated in
Figure 15. Thus, continued EHT observations, with tight
constraints on $\eta$ spaced over multiple years, will constrain these
types of models on the basis of variability in $\eta$.

6.4. Limitations of the Current Approach

There are caveats to the simple analysis outlined above that
should be addressed in more focused future studies. The
simulations that we consider do not capture the full extent of
the physics relevant for M87*. In particular, the electron
temperature could be calculated in a more self-consistent fashion
than via a temperature ratio prescription of Equation (10) by
separately evolving the energy of ions and electrons (Sądowski
et al. 2017). Then, one could evolve a population of nonthermal
electrons (Chael et al. 2017), determine the heating of the
electrons with a well-motivated subgrid prescription (Chael et al.
2018a; Davelaar et al. 2019), or even employ nonideal MHD to
model nonthermal emission caused by the particles accelerated
through magnetic reconnection in a more self-consistent manner
(Ripperda et al. 2019). In the current analysis, we also make an
assumption of no tilt between the plane of accretion and the black
hole spin. If the tilt is present, an additional degree of freedom
("camera longitude") corresponding to a position angle of the
black hole spin (misaligned with the jet position angle $\phi_{jet}$) in the
image plane will influence the observed crescent orientation
(Chatterjee et al. 2020). In that case, analysis of the position angle
distributions could place joint constraints on the tilt magnitude,
the longitude, and other parameters of the simulation. Large-scale
parameter surveys with these extensions to the GRMHD setup are
currently precluded by the immense computational costs.

A separate concern is whether the orientation of crescent
models fitted to the VLBI data is consistent with the image
domain $\eta$ (Equation (11)). In the case of the synthetic data sets
considered in Section 4, the two GRMHD data sets exhibited
quite large biases while the two MODEL data sets showed a
high level of consistency, so this issue requires further study.
Characterizing GRMHD simulations in terms of VLBI
observables is the subject of continued work.

7. Summary

We have performed geometric modeling of the 2009–2017
EHT observations of M87*. Motivated by EHT imaging and
modeling results using the 2017 observations and the stability
of fits across the archival observations, we have used a simple
asymmetric ring model. We found that the fitted ring diameter
is stable throughout these observations, which strongly argues
in favor of its association with the shadow of a supermassive
black hole. We observe indications of modest intrinsic
variability in the total flux density of the ring and in its
position angle.

Specifically, we find the brightness asymmetry along the
east–west direction in the previously unpublished 2013
observations, while all other data sets are consistent with the
north–south asymmetry direction seen in the 2017 EHT
images. This degree of position angle variation is seen in
some GRMHD simulations of M87*, while others do not show

position angle variations as broad as those observed between
2013 and 2017. Thus, the source variability over these
observations provides new constraints on the simulation
parameters, including the black hole spin, accretion flow
magnetization, and electron heating model. As an example, the
GRMHD MAD model with spin $a_*=0.5$ and $R_{high}=160$
(last panel of Figure 15), which was determined to be viable
by EHTC V, is inconsistent with the presented position angle
measurements. We expect that unmodeled physical effects such
as black hole and accretion flow spin misalignment may also be
important in interpreting this variability.

Our results extend the temporal span of EHT constraints on
the ring morphology by nearly three orders of magnitude, from
~15*M* over the 2017 observations to ~7900*M* between the
2009 and 2017 campaigns. Because the correlation timescale
for M87* is expected to be at least a few tens of *M*, the longer
span is critical for decoupling stable image features such as the
black hole shadow from transient features associated with the
turbulent accretion flow. As continued EHT observations
become available, the variation of the estimated position angle
should allow us to discriminate between viable GRMHD
models, providing constraints on the physical parameters of
M87* and opening an exciting new avenue for quantitative
time-domain studies of structural variability in M87*.

The authors of the present paper thank the following
organizations and programs: the Academy of Finland (projects
274477, 284495, 312496); the Advanced European Network of
E-infrastructures for Astronomy with the SKA (AENEAS) project,
supported by the European Commission Framework Programme
Horizon 2020 Research and Innovation action under grant
agreement 731016; the Alexander von Humboldt Stiftung; the
Black Hole Initiative at Harvard University, through a grant
(60477) from the John Templeton Foundation; the China
Scholarship Council; Comisión Nacional de Investigación Cientí-
fica y Tecnológica (CONICYT, Chile, via PIA ACT172033,
Fondecyt projects 1171506 and 3190878, BASAL AFB-170002,
ALMA-conicyt 31140007); Consejo Nacional de Ciencia y
Tecnología (CONACYT, Mexico, projects 104497, 275201,
279006, 281692); the Delaney Family via the Delaney Family
John A. Wheeler Chair at Perimeter Institute; Dirección General de
Asuntos del Personal Académico—Universidad Nacional Autón-
oma de México (DGAPA—UNAM, project IN112417); the
European Research Council Synergy Grant “BlackHoleCam:
Imaging the Event Horizon of Black Holes” (grant 610058); the
Generalitat Valenciana postdoctoral grant APOSTD/2018/177
and GenT Program (project CIDEGENT/2018/021); the Gordon
and Betty Moore Foundation (grants GBMF-3561, GBMF-5278);
the Istituto Nazionale di Fisica Nucleare (INFN) sezione di Napoli,
iniziative specifiche TEONGRAV; the International Max Planck
Research School for Astronomy and Astrophysics at the
Universities of Bonn and Cologne; the Jansky Fellowship program
of the National Radio Astronomy Observatory (NRAO); the
Japanese Government (Monbukagakusho: MEXT) Scholarship;
the Japan Society for the Promotion of Science (JSPS) Grant-in-
Aid for JSPS Research Fellowship (JP17J08829); the Key
Research Program of Frontier Sciences, Chinese Academy of
Sciences (CAS, grants QYZDJ-SSW-SLH057, QYZDJ-SSW-
SYS008, ZDBS-LY-SLH011); the Leverhulme Trust Early Career
Research Fellowship; the Max-Planck-Gesellschaft (MPG); the
Max Planck Partner Group of the MPG and the CAS; the MEXT/
JSPS KAKENHI (grants 18KK0090, JP18K13594, JP18K03656,
JP18H03721, 18K03709, 18H01245, 25120007); the MIT Inter-
national Science and Technology Initiatives (MISTI) Funds; the
Ministry of Science and Technology (MOST) of Taiwan (105-
2112-M-001-025-MY3, 106-2112-M-001-011, 106-2119-M-001-
027, 107-2119-M-001-017, 107-2119-M-001-020, and 107-2119-
M-110-005); the National Aeronautics and Space Administration
(NASA, Fermi Guest Investigator grant 80NSSC17K0649 and
Hubble Fellowship grant HST-HF2-51431.001-A awarded by the
Space Telescope Science Institute, which is operated by the
Association of Universities for Research in Astronomy, Inc., for
NASA, under contract NAS5-26555); the National Institute of
Natural Sciences (NINS) of Japan; the National Key Research and
Development Program of China (grant 2016YFA0400704,
2016YFA0400702); the National Science Foundation (NSF, grants
AST-0096454, AST-0352953, AST-0521233, AST-0705062,
AST-0905844, AST-0922984, AST-1126433, AST-1140030,
DGE-1144085, AST-1207704, AST-1207730, AST-1207752,
MRI-1228509, OPP-1248097, AST-1310896, AST-1312651,
AST-1337663, AST-1440254, AST-1555365, AST-1715061,
AST-1615796, AST-1716327, OISE-1743747, AST-1816420);
the Natural Science Foundation of China (grants 115730511,
11633006, 11650110427, 10625314, 11721303, 11725312,
11933007); the Natural Sciences and Engineering Research
Council of Canada (NSERC, including a Discovery Grant and
the NSERC Alexander Graham Bell Canada Graduate Scholar-
ships-Doctoral Program); the National Youth Thousand Talents
Program of China; the National Research Foundation of Korea (the
Global PhD Fellowship Grant: grants NRF-2015H1A2A1033752,
2015-R1D1A1A01056807, the Korea Research Fellowship Pro-
gram: NRF-2015H1D3A1066561); the Netherlands Organization
for Scientific Research (NWO) VICI award (grant 639.043.513)
and Spinoza Prize SPI 78-409; the New Scientific Frontiers with
Precision Radio Interferometry Fellowship awarded by the South
African Radio Astronomy Observatory (SARAO), which is a
facility of the National Research Foundation (NRF), an agency of
the Department of Science and Technology (DST) of South Africa;
the Onsala Space Observatory (OSO) national infrastructure, for
the provisioning of its facilities/observational support (OSO
receives funding through the Swedish Research Council under
grant 2017-00648) the Perimeter Institute for Theoretical Physics
(research at Perimeter Institute is supported by the Government of
Canada through the Department of Innovation, Science and
Economic Development and by the Province of Ontario through
the Ministry of Research, Innovation and Science); the Russian
Science Foundation (grant 17-12-01029); the Spanish Ministerio
de Economía y Competitividad (grants PGC2018-098915-B-C21,
AYA2016-80889-P); the State Agency for Research of the
Spanish MCIU through the "Center of Excellence Severo Ochoa"
award for the Instituto de Astrofísica de Andalucía (SEV-2017-
0709); the Toray Science Foundation; the US Department of
Energy (USDOE) through the Los Alamos National Laboratory
 operated by Triad National Security, LLC, for the National
 Nuclear Security Administration of the USDOE (contract
 89233218CNA000001)); the Italian Ministero dell'Istruzione
 Università e Ricerca through the grant Progetti Premiali 2012-
iALMA (CUP C52113000140001); the European Union's Horizon
 2020 research and innovation program under grant agreement No.
 730562 RadioNet; ALMA North America Development Fund; the
 Academia Sinica; Chandra TM6-17006X; the GenT Program
 (Generalitat Valenciana) Project CIDEGENT/2018/021. This
 work used the Extreme Science and Engineering Discovery
 Environment (XSEDE), supported by NSF grant ACI-1548562,

and CyVerse, supported by NSF grants DBI-0735191, DBI-
1265383, and DBI-1743442. XSEDE Stampede2 resource at
TACC was allocated through TG-AST170024 and TG-
AST080026N. XSEDE JetStream resource at PTI and TACC
was allocated through AST170028. The simulations were
performed in part on the SuperMUC cluster at the LRZ in
Garching, on the LOEWE cluster in CSC in Frankfurt, and on the
HazelHen cluster at the HLRS in Stuttgart. This research was
enabled in part by support provided by Compute Ontario (http://
computeontario.ca), Calcul Quebec (http://www.calculquebec.ca),
and Compute Canada (http://www.computecanada.ca). We thank
the staff at the participating observatories, correlation centers, and
institutions for their enthusiastic support. This paper makes use of
the following ALMA data: ADS/JAO.ALMA#2016.1.01154.V.
ALMA is a partnership of the European Southern Observatory
(ESO; Europe, representing its member states), NSF, and National
Institutes of Natural Sciences of Japan, together with the National
Research Council (Canada), Ministry of Science and Technology
(MOST; Taiwan), Academia Sinica Institute of Astronomy and
Astrophysics (ASIAA; Taiwan), and Korea Astronomy and Space
Science Institute (KASI; Republic of Korea), in cooperation with
the Republic of Chile. The Joint ALMA Observatory is operated
by ESO, Associated Universities, Inc. (AUI)/NRAO, and the
National Astronomical Observatory of Japan (NAOJ). The NRAO
is a facility of the NSF operated under cooperative agreement by
AUI. APEX is a collaboration between the Max-Planck-Institut für
Radioastronomie (Germany), ESO, and the Onsala Space
Observatory (Sweden). The SMA is a joint project between the
SAO and ASIAA and is funded by the Smithsonian Institution and
the Academia Sinica. The JCMT is operated by the East Asian
Observatory on behalf of the NAOJ, ASIAA, and KASI, as well as
the Ministry of Finance of China, Chinese Academy of Sciences,
and the National Key R&D Program (No. 2017YFA0402700) of
China. Additional funding support for the JCMT is provided by
the Science and Technologies Facility Council (UK) and
participating universities in the UK and Canada. The LMT is a
project operated by the Instituto Nacional de Astrofísica, Óptica, y
Electrónica (Mexico), and the University of Massachusetts at
Amherst (USA). The IRAM 30 m telescope on Pico Veleta, Spain
is operated by IRAM and supported by CNRS (Centre National de
la Recherche Scientifique, France), MPG (Max-Planck-
Gesellschaft, Germany), and IGN (Instituto Geográfico Nacional,
Spain). The SMT is operated by the Arizona Radio Observatory, a
part of the Steward Observatory of the University of Arizona, with
financial support of operations from the State of Arizona and
financial support for instrumentation development from the NSF.
The SPT is supported by the National Science Foundation through
grant PLR-1248097. Partial support is also provided by the NSF
Physics Frontier Center grant PHY-1125897 to the Kavli Institute
of Cosmological Physics at the University of Chicago, the Kavli
Foundation, and the Gordon and Betty Moore Foundation grant
GBMF 947. The SPT hydrogen maser was provided on loan from
the GLT, courtesy of ASIAA. The EHTC has received generous
donations of FPGA chips from Xilinx Inc., under the Xilinx
University Program. The EHTC has benefited from technology
shared under an open-source license by the Collaboration for
Astronomy Signal Processing and Electronics Research (CAS-
PER). The EHT project is grateful to T4Science and Microsemi for
their assistance with hydrogen masers. This research has made use
of NASA's Astrophysics Data System. We gratefully acknowledge
the support provided by the extended staff of the ALMA, both
from the inception of the ALMA Phasing Project through the
observational campaigns of 2017 and 2018. We would like to thank A. Deller and W. Brisken for EHT-specific support with the use of DiFX. We acknowledge the significance that Maunakea, where the SMA and JCMT EHT stations are located, has for the indigenous Hawaiian people.

## Appendix A
### ML Estimators for the RG Model

Figure A1 presents RG models best fitting the synthetic data sets, following the procedures described in Section 4.

<figure><img src="image_18.png" /><figcaption>**Figure A1.** ML estimators corresponding to the fits to four synthetic images, shown in the first row (no blurring). Estimators were obtained through synthetic VLBI observations with (u, v) coverage and uncertainties identical to those of the real observations performed in 2009–2017. The blurred ring model (RG) was used, and the presented images of ML estimators were blurred to a 15 µas resolution. Blue dashed lines indicate the convention for the expected position angle of the bright component $\phi_{B,exp}$. The gray bar represents the ML estimate of $\phi_B$. The dashed circles correspond to a diameter of 42 µas.</figcaption></figure>
## Appendix B
### Corner Plots for the RT and the RG Models

We present the posterior probability distributions corresponding to fitting the 2009–2013 data sets with the RT model (Figure B1) and with the RG model (Figure B2). Similarly, for the 2017 data sets, we show the posterior distributions obtained for the RT model (Figure B3) and for the RG model (Figure B4).

<figure><img src="image_19.png" /><figcaption>**Figure B1.** Posterior distributions of RT model parameters resulting from fitting to M87* 2009–2013 data, obtained using THEMIS. The maximum likelihood estimate is indicated with red lines. Contours indicate 0.2, 0.5, 0.8, and 0.95 of the posterior volume.</figcaption></figure>
<figure><img src="image_20.png" /></figure>

2009

<figure><img src="image_21.png" /></figure>

2011

<figure><img src="image_22.png" /></figure>

2012

<figure><img src="image_23.png" /></figure>

2013

<caption>Figure B2. Same as Figure B1, but for the RG model.</caption>
<figure><img src="image_24.png" /><figcaption>Figure B3. Same as Figure B1, but for the M87* observations on 2017 April 6 and April 11 fitted with the RT model.</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>Figure B4. Same as Figure B1, but for the M87* observations on 2017 April 6 and April 11 fitted with the RG model.</figcaption></figure>

ORCID iDs

Maciek Wielgus https://orcid.org/0000-0002-8635-4242

Kazunori Akiyama https://orcid.org/0000-0002-9475-4254

Lindy Blackburn https://orcid.org/0000-0002-9030-642X

Chi-kwan Chan https://orcid.org/0000-0001-6337-6126

Jason Dexter https://orcid.org/0000-0003-3903-0373

Sheperd S. Doeleman https://orcid.org/0000-0002-9031-0904

Vincent L. Fish https://orcid.org/0000-0002-7128-9345

Sara Issaoun https://orcid.org/0000-0002-5297-921X

Michael D. Johnson https://orcid.org/0000-0002-4120-3029

Thomas P. Krichbaum https://orcid.org/0000-0002-4892-9586

Ru-Sen Lu (路如森) https://orcid.org/0000-0002-7692-7967

Dominic W. Pesce https://orcid.org/0000-0002-5278-9221

George N. Wong https://orcid.org/0000-0001-6952-2147

Geoffrey C. Bower https://orcid.org/0000-0003-4056-9982

Avery E. Broderick https://orcid.org/0000-0002-3351-760X

Andrew Chael https://orcid.org/0000-0003-2966-6220

Charles F. Gammie https://orcid.org/0000-0001-7451-8935

Boris Georgiev https://orcid.org/0000-0002-3586-6424
Kazuhiro Hada https://orcid.org/0000-0001-6906-772X

Laurent Loinard https://orcid.org/0000-0002-5635-3345

Sera Markoff https://orcid.org/0000-0001-9564-0876

Daniel P. Marrone https://orcid.org/0000-0002-2367-1080

Richard Plambeck https://orcid.org/0000-0001-6765-9609

Jonathan Weintroub https://orcid.org/0000-0002-4603-5204

Melvyn Wright https://orcid.org/0000-0002-9154-2440

Antxon Alberdi https://orcid.org/0000-0002-9371-1033

Rebecca Azulay https://orcid.org/0000-0002-2200-5393

Anne-Kathrin Baczko https://orcid.org/0000-0003-3090-3975

Mislav Baloković https://orcid.org/0000-0003-0476-6647

Enrico Barausse https://orcid.org/0000-0001-6499-6263

John Barrett https://orcid.org/0000-0002-9290-0764

Katherine L. Bouman https://orcid.org/0000-0003-0077-4367

Christiaan D. Brinkerink https://orcid.org/0000-0002-2322-0749

Roger Brissenden https://orcid.org/0000-0002-2556-0894

Silke Britzen https://orcid.org/0000-0001-9240-6734

Do-Young Byun https://orcid.org/0000-0003-1157-4109

Shami Chatterjee https://orcid.org/0000-0002-2878-1502

Ilje Cho https://orcid.org/0000-0001-6083-7521

Pierre Christian https://orcid.org/0000-0001-6820-9941

John E. Conway https://orcid.org/0000-0003-2448-9181

James M. Cordes https://orcid.org/0000-0002-4049-1882

Geoffrey B. Crew https://orcid.org/0000-0002-2079-3189

Yuzhu Cui https://orcid.org/0000-0001-6311-4345

Jordy Davelaar https://orcid.org/0000-0002-2685-2434

Mariafelicia De Laurentis https://orcid.org/0000-0002-9945-682X

Roger Deane https://orcid.org/0000-0003-1027-5043

Jessica Dempsey https://orcid.org/0000-0003-1269-9667

Gregory Desvignes https://orcid.org/0000-0003-3922-4055

Sergio A. Dzib https://orcid.org/0000-0001-6010-6200

Ralph P. Eatough https://orcid.org/0000-0001-6196-4135

Heino Falcke https://orcid.org/0000-0002-2526-6724

Ed Fomalont https://orcid.org/0000-0002-9036-2747

Raquel Fraga-Encinas https://orcid.org/0000-0002-5222-1361

Peter Galison https://orcid.org/0000-0002-6429-3872

Roman Gold https://orcid.org/0000-0003-2492-1966

José L. Gómez https://orcid.org/0000-0003-4190-7613

Arturo I. Gómez-Ruiz https://orcid.org/0000-0001-9395-1670

Minfeng Gu (顾敏峰) https://orcid.org/0000-0002-4455-6946

Mark Gurwell https://orcid.org/0000-0003-0685-3621

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Luis C. Ho (何子山) https://orcid.org/0000-0001-6947-5846

Mareki Honma https://orcid.org/0000-0003-4058-9000

Chih-Wei L. Huang https://orcid.org/0000-0001-5641-3953

Lei Huang (黄磊) https://orcid.org/0000-0002-1923-227X

David J. James https://orcid.org/0000-0001-5160-4486

Michael Janssen https://orcid.org/0000-0001-8685-6544

Britton Jeter https://orcid.org/0000-0003-2847-1712

Wu Jiang (江悟) https://orcid.org/0000-0001-7369-3539

Svetlana Jorstad https://orcid.org/0000-0001-6158-1708

Taehyun Jung https://orcid.org/0000-0001-7003-8643

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Tomohisa Kawashima https://orcid.org/0000-0001-8527-0496

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Mark Kettenis https://orcid.org/0000-0002-6156-5617

Jae-Young Kim https://orcid.org/0000-0001-8229-7183

Junhan Kim https://orcid.org/0000-0002-4274-9373

Motoki Kino https://orcid.org/0000-0002-2709-7338

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Patrick M. Koch https://orcid.org/0000-0003-2777-5861

Shoko Koyama https://orcid.org/0000-0002-3723-3372

Michael Kramer https://orcid.org/0000-0002-4175-2271

Carsten Kramer https://orcid.org/0000-0002-4908-4925

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Tod R. Lauer https://orcid.org/0000-0003-3234-7247

Sang-Sung Lee https://orcid.org/0000-0002-6269-594X

Yan-Rong Li (李彦荣) https://orcid.org/0000-0001-5841-9179

Zhiyuan Li (李志远) https://orcid.org/0000-0003-0355-6437

Michael Lindqvist https://orcid.org/0000-0002-3669-0715

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Kuo Liu https://orcid.org/0000-0002-2953-7376

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Nicholas R. MacDonald https://orcid.org/0000-0002-6684-8691

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Qingwen Wu (吴庆文) https://orcid.org/0000-0003-4773-4987

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