# The Variability of the Black Hole Image in M87 at the Dynamical Timescale

Kaushik Satapathy, Dimitrios Psaltis, Feryal Özel, Lia Medeiros, Sean T. Dougall, Chi-Kwan Chan, Maciek Wielgus, Ben S. Prather, George N. Wong, Charles F. Gammie, et al.

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Kaushik Satapathy, Dimitrios Psaltis, Feryal Özel, Lia Medeiros, Sean T. Dougall, et al. The Variability of the Black Hole Image in M87 at the Dynamical Timescale. The Astrophysical Journal, 2022, 925, 10.3847/1538-4357/ac332e. insu-03713346

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The Variability of the Black Hole Image in M87 at the Dynamical Timescale

Kaushik Satapathy<fnref n="1" />,<fnref n="2" /> ID, Dimitrios Psaltis<fnref n="1" />, Feryal Özel<fnref n="1" />, Lia Medeiros<fnref n="3" />,¹²⁷ ID, Sean T. Dougall<fnref n="1" /> ID, Chi-Kwan Chan<fnref n="1" /> ID,
Maciek Wielgus<fnref n="4" />,<fnref n="5" /> ID, Ben S. Prather<fnref n="6" />, George N. Wong<fnref n="6" /> ID, Charles F. Gammie<fnref n="6" />,<fnref n="7" /> ID, Kazunori Akiyama<fnref n="4" />,<fnref n="8" />,<fnref n="9" /> ID,
Antxon Alberdi<fnref n="10" /> ID, Walter Alef¹¹, Juan Carlos Algaba¹² ID, Richard Anantua<fnref n="4" />,<fnref n="5" /> ID, Keiichi Asada¹³, Rebecca Azulay¹¹,¹⁴,¹⁵ ID,
Anne-Kathrin Baczko¹¹ ID, David Ball¹⁶, Mislav Baloković<fnref n="4" />,<fnref n="5" /> ID, John Barrett<fnref n="8" /> ID, Bradford A. Benson¹⁷,¹⁸ ID, Dan Bintley¹⁹,
Lindy Blackburn<fnref n="4" />,<fnref n="5" /> ID, Raymond Blundell<fnref n="5" /> ID, Wilfred Boland²⁰, Katherine L. Bouman<fnref n="4" />,<fnref n="5" />,²¹ ID, Geoffrey C. Bower²² ID,
Hope Boyce²³,²⁴ ID, Michael Bremer²⁵, Christiaan D. Brinkerink²⁶ ID, Roger Brissenden<fnref n="4" />,<fnref n="5" /> ID, Silke Britzen¹¹ ID,
Avery E. Broderick²⁷,²⁸,²⁹ ID, Dominique Broguiere²⁵, Thomas Bronzwaer²⁶ ID, Sandra Bustamente³⁰, Do-Young Byun³¹,³² ID,
John E. Carlstrom¹⁸,³³,³⁴,³⁵, Andrew Chael³⁶,¹²⁸ ID, Koushik Chatterjee³⁷ ID, Shami Chatterjee³⁸ ID, Ming-Tang Chen²²,
Yongjun Chen (陈永军)³⁹,⁴⁰ ID, Ilje Cho³¹,³² ID, Pierre Christian⁴¹ ID, John E. Conway⁴² ID, James M. Cordes³⁸ ID,
Thomas M. Crawford¹⁸,³³ ID, Geoffrey B. Crew<fnref n="8" /> ID, Alejandro Cruz-Osorio⁴³ ID, Yuzhu Cui⁴⁴,⁴⁵ ID, Jordy Davelaar²⁶,⁴⁶,⁴⁷ ID,
Maria-feliciais De Lourentis⁴³,⁴⁸,⁴⁹ ID, Roger Deane⁵⁰,⁵¹,⁵² ID, Jessica Dempsey¹⁹ ID, Gregory Desvignes⁵³ ID, Jason Dexter⁵⁴ ID,
Sheperd S. Doeleman<fnref n="4" />,<fnref n="5" /> ID, Ralph P. Eatough¹¹,⁵⁵ ID, Heino Falcke²⁶ ID, Joseph Farah<fnref n="4" />,<fnref n="5" />,<fnref n="6" /> ID, Vincent L. Fish<fnref n="8" /> ID, Ed Fomalont⁵⁷ ID,
H. Alyson Ford¹⁶ ID, Raquel Fraga-Encinas²⁶ ID, Per Friberg¹⁹ ID, Christian M. Fromm<fnref n="4" />,<fnref n="5" />,⁴³ ID, Antonio Fuentes<fnref n="10" /> ID,
Peter Galison<fnref n="4" />,<fnref n="5" />,⁸⁵,⁹⁰ ID, Roberto García²⁵ ID, Olivier Gentaz²⁵, Boris Georgiev²⁸,²⁹ ID, Ciriaco Goddi²⁶,⁶⁰ ID, Roman Gold²⁷,⁶¹ ID,
Arturo I. Gómez-Ruiz⁶²,⁶³ ID, José L. Gómez⁶⁴ ID, Minfeng Gu (顾敏峰)³⁹,⁶⁵ ID, Mark Gurwell<fnref n="5" /> ID, Kazuhiro Hada⁴⁴,⁴⁵ ID,
Daryl Haggard²³,²⁴ ID, Michael H. Hecht<fnref n="8" />, Ronald Hesper⁶⁶ ID, Luis C. Ho (何子山)⁶⁷,⁶⁸ ID, Paul Ho¹³, Mareki Honma⁴⁴,⁴⁵,⁶⁹ ID,
Chih-Wei L. Huang¹³ ID, Lei Huang (黄磊)³⁹,⁶⁵ ID, David H. Hughes⁶², Shiro Ikeda<fnref n="9" />,⁷⁰,⁷¹,⁷² ID, Makoto Inoue¹³, Sara Issaoun²⁶ ID,
David J. James<fnref n="4" />,<fnref n="5" /> ID, Buell T. Jannuzi¹⁶, Michael Janssen¹¹ ID, Britton Jeter²⁸,²⁹ ID, Wu Jiang (江悟)³⁹ ID,
Alejandra Jimenez-Rosales²⁶, Michael D. Johnson<fnref n="4" />,<fnref n="5" /> ID, Svetlana Jorstad⁷³,⁷⁴ ID, Taehyun Jung³¹,³² ID, Mansour Karami²⁷,²⁸ ID,
Ramesh Karuppusamy¹¹, Tomohisa Kawashima⁷⁵ ID, Garrett K. Keating<fnref n="5" /> ID, Mark Kettenis⁷⁶ ID, Dong-Jin Kim¹¹ ID,
Jae-Young Kim¹¹,³¹ ID, Jongsoo Kim³¹ ID, Junhan Kim¹⁶,²¹ ID, Motoki Kino<fnref n="9" />,⁷⁷ ID, Jun Yi Koay¹³ ID, Yutaro Kofuji⁴⁴,⁶⁹,
Patrick M. Koch¹³ ID, Shoko Koyama¹³,⁷⁸ ID, Carsten Kramer²⁵ ID, Michael Kramer¹¹ ID, Thomas P. Krichbaum¹¹ ID,
Cheng-Yu Kuo¹³,⁷⁹ ID, Tod R. Lauer⁸⁰ ID, Sang-Sung Lee³¹ ID, Aviad Levis²¹ ID, Yan-Rong Li (李彦荣)⁸¹ ID, Zhiyuan Li
(李志远)⁸²,⁸³ ID, Michael Lindqvist⁴² ID, Rocco Lico<fnref n="10" />,¹¹ ID, Greg Lindahl<fnref n="5" /> ID, Jun Liu (刘俊)¹¹ ID, Kuo Liu¹¹ ID,
Elisabetta Liuzzo⁸⁴ ID, Wen-Ping Lo¹³,⁸⁵, Andrei P. Lobanov¹¹, Laurent Loinard⁸⁶,⁸⁷ ID, Colin Lonsdale<fnref n="8" />, Ru-Sen Lu
(路如森)¹¹,⁸⁸,⁸⁹ ID, Nicholas R. MacDonald¹¹ ID, Jirong Mao (毛基荣)⁹⁰,⁹¹,⁹² ID, Nicola Marchili¹¹,⁸⁴ ID, Sera Markoff³⁷,⁹³ ID,
Daniel P. Marrone¹⁶ ID, Alan P. Marscher⁷³ ID, Iván Martí-Vidal¹⁴,¹⁵ ID, Satoki Matsushita¹³ ID, Lynn D. Matthews<fnref n="8" />,
Karl M. Menten¹¹ ID, Izumi Mizuno¹⁹ ID, Yosuke Mizuno⁴³,⁹⁴ ID, James M. Moran<fnref n="4" />,<fnref n="5" /> ID, Kotaro Moriyama<fnref n="8" />,⁴⁴ ID,
Monika Moscibrodzka²⁶ ID, Cornelia Müller¹¹,²⁶ ID, Alejandro Mus Mejías¹⁴,¹⁵ ID, Gibwa Musoke²⁶,³⁷ ID, Hiroshi Nagai<fnref n="9" />,⁴⁵ ID,
Neil M. Nagar⁹⁵ ID, Masanori Nakamura¹³,⁹⁶ ID, Ramesh Narayan<fnref n="4" />,<fnref n="5" /> ID, Gopal Narayanan³⁰, Iniyan Natarajan⁵⁰,⁵²,⁹⁷ ID,
Antonios Nathanai¹⁴³, Joey Neilsen⁹⁸ ID, Roberto Neri²⁵ ID, Chunchong Ni²⁸,²⁹ ID, Aristeidis Noutsos¹¹ ID, Michael A. Nowak⁹⁹ ID,
Hiroki Okino⁴⁴,⁶⁹ ID, Héctor Olivares²⁶ ID, Gisela N. Ortiz-León¹¹ ID, Tomoaki Oyama⁴⁴, Daniel C. M. Palumbo<fnref n="4" />,<fnref n="5" /> ID,
Jongho Park¹³,¹²⁹ ID, Nimesh Patel<fnref n="5" />, Ue-Li Pen²⁷,¹⁰⁰,¹⁰¹,¹⁰² ID, Dominic W. Pesce<fnref n="4" />,<fnref n="5" /> ID, Vincent Piétu²⁵, Richard Plambeck¹⁰³ ID,
Aleksandar PopStefanija³⁰ ID, Oliver Porth³⁷,⁴³ ID, Felix M. Pötzl¹¹ ID, Jorge A. Preciado-López²⁷ ID, Hung-Yi Pu¹³,²⁷,¹⁰⁴ ID,
Venkatessh Ramakrishnan⁹⁵ ID, Ramprasad Rao²² ID, Mark G. Rawlings¹⁹ ID, Alexander W. Raymond<fnref n="4" />,<fnref n="5" />,
Luciano Rezzolla⁴³,¹⁰⁵,¹⁰⁶ ID, Bart Ripperda⁴⁷,¹⁰⁷ ID, Freek Roelofs<fnref n="5" />,²⁶ ID, Alan Rogers<fnref n="8" />, Eduardo Ros¹¹ ID, Mel Rose¹⁶ ID,
Arash Roshanineshat¹⁶, Helge Rottmann¹¹, Alan L. Roy¹¹ ID, Chet Ruszczyk<fnref n="8" /> ID, Kazi L. J. Rygl⁸⁴ ID, Salvador Sánchez¹⁰⁸,
David Sánchez-Arvelles⁶²,¹⁰⁹ ID, Mahito Sasada⁴⁴,¹¹⁰ ID, Tuomas Savolainen¹¹,¹¹¹,¹¹² ID, F. Peter Schloerb³⁰,
Karl-Friedrich Schuster²⁵ ID, Lijing Shao¹¹,⁶⁸ ID, Zhiqiang Shen (沈志强)³⁹,⁴⁰ ID, Des Small⁷⁶ ID, Bong Won Sohn³¹,³²,¹¹³ ID,
Jason SooHoo<fnref n="8" /> ID, He Sun (孙赫)²¹ ID, Fumie Tazaki⁴⁴ ID, Alexandra J. Tetarenko¹⁹ ID, Paul Tiede²⁸,²⁹ ID,
Remo P. J. Tilanus¹⁶,²⁶,¹¹⁴,¹¹⁵ ID, Michael Titus<fnref n="8" /> ID, Kenji Toma¹¹⁶,¹¹⁷ ID, Pablo Torne¹¹,¹⁰⁸ ID, Efthalia Traianou¹¹ ID,
Tyler Trent¹⁶, Sascha Trippe¹¹⁸ ID, Ilse van Bemmel⁷⁶ ID, Huib Jan van Langevelde⁷⁶,¹¹⁹ ID, Daniel R. van Rossum²⁶ ID,
Jan Wagner¹¹ ID, Derek Ward-Thompson¹²⁰ ID, John Wardle¹²¹ ID, Jonathan Weintroub<fnref n="4" />,<fnref n="5" /> ID, Norbert Wex¹¹ ID,
Robert Wharton¹¹ ID, Kaj Wiik¹²², Qingwen Wu (吴庆文)¹²³ ID, Doosoo Yoon³⁷ ID, André Young²⁶ ID, Ken Young<fnref n="5" /> ID,
Ziri Younsi⁴³,¹²⁴ ID, Feng Yuan (袁峰)³⁹,⁶⁵,¹²⁵ ID, Ye-Fei Yuan (袁业飞)¹²⁶ ID, J. Anton Zensus¹¹ ID, Guang-Yao Zhao<fnref n="10" /> ID, and
Shan-Shan Zhao³⁹ ID

<footnote n="1">¹ Astronomy Department, University of Arizona, 933 N. Cherry Ave, Tucson, AZ 85721, USA</footnote>

<footnote n="2">² Department of Physics, University of Arizona, 1118 E 4th St. Tucson, AZ 85721, USA</footnote>

<footnote n="3">³ School of Natural Sciences, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA</footnote>

<footnote n="4">⁴ Black Hole Initiative at Harvard University, 20 Garden Street, Cambridge, MA 02138, USA</footnote>

<footnote n="5">⁵ Center for Astrophysics, Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA</footnote>

<footnote n="6">⁶ Illinois Center for Advanced Studies of the Universe, Department of Physics, University of Illinois, 1110 West Green Street, Urbana, IL 61801, USA</footnote>

<footnote n="7">⁷ Department of Astronomy, University of Illinois, 1002 West Green Street, Urbana, IL 61801, USA</footnote>

<footnote n="8">⁸ Massachusetts Institute of Technology Haystack Observatory, 99 Millstone Road, Westford, MA 01886, USA</footnote>

<footnote n="9">⁹ National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan</footnote>

<footnote n="10">¹⁰ Instituto de Astrofísica de Andalucía-CSIC, Glorieta de la Astronomía s/n, E-18008 Granada, Spain</footnote>
¹¹Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

¹²Department of Physics, Faculty of Science, University of Malaya, 50603 Kuala Lumpur, Malaysia

¹³Institute of Astronomy and Astrophysics, Academia Sinica, 11F of Astronomy-Mathematics Building, AS/NTU No. 1, Sec. 4, Roosevelt Rd, Taipei 10617, Taiwan, R.O.C.

¹⁴Departament d'Astronomia i Astrofísica, Universitat de València, C. Dr. Moliner 50, E-46100 Burjassot, València, Spain

¹⁵Observatori Astronòmic, Universitat de València, C. Catedrático José Beltrán 2, E-46980 Paterna, València, Spain

¹⁶Steward Observatory and Department of Astronomy, University of Arizona, 933 N. Cherry Ave., Tucson, AZ 85721, USA

¹⁷Fermi National Accelerator Laboratory, MS209, P.O. Box 500, Batavia, IL 60510, USA

¹⁸Department of Astronomy and Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA

¹⁹East Asian Observatory, 660 N. A'ohoku Place, Hilo, HI 96720, USA

²⁰Nederlandse Onderzoekschool voor Astronomie (NOVA), PO Box 9513, 2300 RA Leiden, The Netherlands

²¹California Institute of Technology, 1200 East California Boulevard, Pasadena, CA 91125, USA

²²Institute of Astronomy and Astrophysics, Academia Sinica, 645 N. A'ohoku Place, Hilo, HI 96720, USA

²³Department of Physics, McGill University, 3600 rue University, Montréal, QC, H3A 2T8, Canada

²⁴McGill Space Institute, McGill University, 3550 rue University, Montréal, QC, H3A 2A7, Canada

²⁵Institut de Radioastronomie Millimétrique, 300 rue de la Piscine, F-38406 Saint Martin d'Hères, France

²⁶Department of Astrophysics, Institute for Mathematics, Astrophysics and Particle Physics (IMAPP), Radboud University, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands

²⁷Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON, N2L 2Y5, Canada

²⁸Department of Physics and Astronomy, University of Waterloo, 200 University Avenue West, Waterloo, ON, N2L 3G1, Canada

²⁹Waterloo Centre for Astrophysics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

³⁰Department of Astronomy, University of Massachusetts, Amherst, MA 01003, USA

³¹Korea Astronomy and Space Science Institute, Daedok-daero 776, Yuseong-gu, Daejeon 34055, Republic of Korea

³²University of Science and Technology, Gajeong-ro 217, Yuseong-gu, Daejeon 34113, Republic of Korea

³³Kavli Institute for Cosmological Physics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA

³⁴Department of Physics, University of Chicago, 5720 South Ellis Avenue, Chicago, IL 60637, USA

³⁵Enrico Fermi Institute, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA

³⁶Princeton Center for Theoretical Science, Jadwin Hall, Princeton University, Princeton, NJ 08544, USA

³⁷Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904, 1098 XH, Amsterdam, The Netherlands

³⁸Cornell Center for Astrophysics and Planetary Science, Cornell University, Ithaca, NY 14853, USA

³⁹Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, People's Republic of China

⁴⁰Key Laboratory of Radio Astronomy, Chinese Academy of Sciences, Nanjing 210008, People's Republic of China

⁴¹Physics Department, Fairfield University, 1073 North Benson Road, Fairfield, CT 06824, USA

⁴²Department of Space, Earth and Environment, Chalmers University of Technology, Onsala Space Observatory, SE-43992 Onsala, Sweden

⁴³Institut für Theoretische Physik, Goethe-Universität Frankfurt, Max-von-Laue-Straße 1, D-60438 Frankfurt am Main, Germany

⁴⁴Mizusawa VLBI Observatory, National Astronomical Observatory of Japan, 2-12 Hoshigaoka, Mizusawa, Oshu, Iwate 023-0861, Japan

⁴⁵Department of Astronomical Science, The Graduate University for Advanced Studies (SOKENDAI), 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan

⁴⁶Department of Astronomy and Columbia Astrophysics Laboratory, Columbia University, 550 W 120th Street, New York, NY 10027, USA

⁴⁷Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA

⁴⁸Dipartimento di Fisica "E. Pancini," Università di Napoli "Federico II", Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy

⁴⁹INFN Sez. di Napoli, Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy

⁵⁰Wits Centre for Astrophysics, University of the Witwatersrand, 1 Jan Smuts Avenue, Braamfontein, Johannesburg 2050, South Africa

⁵¹Department of Physics, University of Pretoria, Hatfield, Pretoria 0028, South Africa

⁵²Centre for Radio Astronomy Techniques and Technologies, Department of Physics and Electronics, Rhodes University, Makhanda 6140, South Africa

⁵³LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Université de Paris, 5 place Jules Janssen, F-92195 Meudon, France

⁵⁴JILA and Department of Astrophysical and Planetary Sciences, University of Colorado, Boulder, CO 80309, USA

⁵⁵National Astronomical Observatories, Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing 100101, People's Republic of China

⁵⁶University of Massachusetts Boston, 100 William T. Morrissey Boulevard, Boston, MA 02125, USA

⁵⁷National Radio Astronomy Observatory, 520 Edgemont Rd, Charlottesville, VA 22903, USA

⁵⁸Department of History of Science, Harvard University, Cambridge, MA 02138, USA

⁵⁹Department of Physics, Harvard University, Cambridge, MA 02138, USA

⁶⁰INAF—Osservatorio Astronomico di Cagliari, Via della Scienza 5, I-09047, Selargius, CA, Italy

⁶¹CP-3:Origins, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark

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

⁶³Consejo Nacional de Ciencia y Tecnología, Av. Insurgentes Sur 1582, 03940, Ciudad de México, México

⁶⁴Instituto de Astrofísica de Andalucía-CiSIC, Glorieta de la Astronomía s/n, E-18008 Granada, Spain

⁶⁵Key Laboratory for Research in Galaxies and Cosmology, Chinese Academy of Sciences, Shanghai 200030, People's Republic of China

⁶⁶NOVA Sub-mm Instrumentation Group, Kapteyn Astronomical Institute, University of Groningen, Landleven 12, 9747 AD Groningen, The Netherlands

⁶⁷Department of Astronomy, School of Physics, Peking University, Beijing 100871, People's Republic of China

⁶⁸Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, People's Republic of China

⁶⁹Department of Astronomy, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

⁷⁰The Institute of Statistical Mathematics, 10-3 Midori-cho, Tachikawa, Tokyo, 190-8562, Japan

⁷¹Department of Statistical Science, The Graduate University for Advanced Studies (SOKENDAI), 10-3 Midori-cho, Tachikawa, Tokyo 190-8562, Japan

⁷²Kavli Institute for the Physics and Mathematics of the Universe, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan

⁷³Institute for Astrophysical Research, Boston University, 725 Commonwealth Ave., Boston, MA 02215, USA

⁷⁴Astronomical Institute, St. Petersburg University, Universitetskij pr., 28, Petrodvorets, 198504 St. Petersburg, Russia

⁷⁵Institute for Cosmic Ray Research, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582, Japan

⁷⁶Joint Institute for VLBI ERIC (JIVE), Oude Hoogeveensdijk 4, 7991 PD Dwingeloo, The Netherlands

⁷⁷Kogakuin University of Technology & Engineering, Academic Support Center, 2665-1 Nakano, Hachioji, Tokyo 192-0015, Japan

⁷⁸Niigata University, 8050 Ikarashi-nino-cho, Nishi-ku, Niigata 950-2181, Japan

⁷⁹Physics Department, National Sun Yat-Sen University, No. 70, Lien-Hai Road, Kaosiung City 80424, Taiwan, R.O.C.

⁸⁰National Optical Astronomy Observatory, 950 N. Cherry Ave., Tucson, AZ 85719, USA

⁸¹Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Shijingshan District, Beijing, People's Republic of China

⁸²School of Astronomy and Space Science, Nanjing University, Nanjing 210023, People's Republic of China

⁸³Key Laboratory of Modern Astronomy and Astrophysics, Nanjing University, Nanjing 210023, People's Republic of China

⁸⁴Italian ALMA Regional Centre, INAF-Istituto di Radioastronomia, Via P. Gobetti 101, I-40129 Bologna, Italy
85 Department of Physics, National Taiwan University, No.1, Sect. 4, Roosevelt Rd., Taipei 10617, Taiwan, R.O.C

86 Instituto de Radioastronomía y Astrofísica, Universidad Nacional Autónoma de México, Morelia 58089, México
87 Instituto de Astronomía, Universidad Nacional Autónoma de México, CdMx 04510, México

88 Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, People's Republic of China

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

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

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

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

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

94 Tsung-Dao Lee Institute and School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, 200240, People's Republic of China

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

96 National Institute of Technology, Hachinohe College, 16-1 Uwanotai, Tamonoki, Hachinohe City, Aomori 039-1192, Japan

97 South African Radio Astronomy Observatory, Observatory 7925, Cape Town, South Africa

98 Villanova University, Mendel Science Center Rm. 263B, 800 E Lancaster Ave, Villanova, PA 19085, USA

99 Physics Department, Washington University CB 1105, St Louis, MO 63130, USA

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

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

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

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

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

105 Frankfurt Institute for Advanced Studies, Ruth-Moufang-Strasse 1, D-60438 Frankfurt, Germany

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

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

108 Instituto de Radioastronomía Milimétrica, IRAM, Avenida Divina Pastora 7, Local 20, E-18012, Granada, Spain

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

110 Hiroshima Astrophysical Science Center, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526, Japan

111 Alalto University Department of Electronics and Nanoengineering, PL 15500, FI-00076 Alalto, Finland

112 Alalto University Metsähovi Radio Observatory, Metsähovintie 114, FI-02540 Kylmälä, Finland

113 Department of Astronomy, Yonsei University, Yonsei-ro 50, SeoDaemun-gu, 03722 Seoul, Republic of Korea

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

115 Netherlands Organisation for Scientific Research (NWO), Postbus 93138, 2509 AC Den Haag, The Netherlands

116 Frontier Research Institute for Interdisciplinary Sciences, Tohoku University, Sendai 980-8578, Japan

117 Astronomical Institute, Tohoku University, Sendai 980-8578, Japan

118 Department of Physics and Astronomy, Seoul National University, Gwanak-gu, Seoul 08826, Republic of Korea

119 Leiden Observatory, Leiden University, Postbus 2300, 9513 RA Leiden, The Netherlands

120 Jeremiah Horrocks Institute, University of Central Lancashire, Preston, PR1 2HE, UK

121 Physics Department, Brandeis University, 415 South Street, Waltham, MA 02453, USA

122 Tuorla Observatory, Department of Physics and Astronomy, University of Turku, Finland

123 School of Physics, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, People's Republic of China

124 Mullard Space Science Laboratory, University College London, Holmbury St. Mary, Dorking, Surrey, RH5 6NT, UK

125 School of Astronomy and Space Sciences, University of Chinese Academy of Sciences, No. 19A Yuquan Road, Beijing 100049, People's Republic of China

126 Astronomy Department, University of Science and Technology of China, Hefei 230026, People's Republic of China

Received 2021 June 3; revised 2021 October 21; accepted 2021 October 24; published 2022 January 20

Abstract

The black hole images obtained with the Event Horizon Telescope (EHT) are expected to be variable at the dynamical timescale near their horizons. For the black hole at the center of the M87 galaxy, this timescale (5–61 days) is comparable to the 6 day extent of the 2017 EHT observations. Closure phases along baseline triangles are robust interferometric observables that are sensitive to the expected structural changes of the images but are free of station-based atmospheric and instrumental errors. We explored the day-to-day variability in closure-phase measurements on all six linearly independent nontrivial baseline triangles that can be formed from the 2017 observations. We showed that three triangles exhibit very low day-to-day variability, with a dispersion of ~3°–5°. The only triangles that exhibit substantially higher variability (~90°–180°) are the ones with baselines that cross the visibility amplitude minima on the u–v plane, as expected from theoretical modeling. We used two sets of general relativistic magnetohydrodynamic simulations to explore the dependence of the predicted variability on various black hole and accretion-flow parameters. We found that changing the magnetic field configuration, electron temperature model, or black hole spin has a marginal effect on the model consistency with the observed level of variability. On the other hand, the most discriminating image characteristic of models is the fractional width of the bright ring of emission. Models that best reproduce the observed small level of variability are characterized by thin ring-like images with structures dominated by gravitational lensing effects and thus least affected by turbulence in the accreting plasmas.

Unified Astronomy Thesaurus concepts: Black hole physics (159); High energy astrophysics (739)

1. Introduction

The Event Horizon Telescope (EHT) has produced horizon-scale images of the black hole in the center of the M87 galaxy using Very Long Baseline Interferometry (VLBI) at a wavelength of 1.3 mm (EHT Collaboration et al. 2019a, 2019b, 2019c, 2019d, 2019e, 2019f). Together with upcoming studies

<figure><img src="image_4.png" /></figure>

Original content from this work may be used under the terms of the [Creative Commons Attribution 4.0 licence](https://creativecommons.org/licenses/by/4.0/). Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

<footnote n="127">127 NSF Astronomy and Astrophysics Postdoctoral Fellow.</footnote>

<footnote n="128">128 NASA Hubble Fellowship Program, Einstein Fellow.</footnote>

<footnote n="129">129 EACOA fellow.</footnote>
of the black hole in the center of the Milky Way, Sgr A*, these images have opened up numerous avenues to studying physics at event-horizon scales related to both accretion processes and gravity. The structure of the accretion disks and jets observed around the black holes will improve our understanding of the behavior of plasma in these environments (EHT Collaboration et al. 2019e, 2019f). The measurements of the sizes and shapes of the black hole shadows give insights into deviations of the black hole spacetime from the Kerr metric (Psaltis 2019; Psaltis et al. 2020).

Accretion flows, which make up the horizon-scale environments of black holes, are expected to be highly variable by nature. The radiation observed from these regions at 1.3 mm is dominated by synchrotron emission from electrons that are heated and accelerated by magnetohydrodynamic turbulence (EHT Collaboration et al. 2019e). The accretion flow is also expected to evolve at the dynamical timescales near the innermost stable circular orbit (ISCO), which range from 4 to 56 minutes for Sgr A* (for $M_{BH}=4.148 \times 10^6 M_\odot$; see Gravity Collaboration et al. 2019) and 5 to 61 days for M87 (depending on the black hole spin; Bardeen et al. 1972). The timescales suggest that there can be substantial variability in the source structure over a single observing night for Sgr A*, which makes it difficult to obtain a static image. However, we expect substantial source variability in M87 only for observations that span about a week or longer. Evidence for long-term variability in the image structure of M87 has been reported recently, based on VLBI observations spanning nearly a decade (Wielgus & The EHT Collaboration 2020).

The EHT observed M87 in 2017 April for four nights with a maximum separation of six calendar days between observations. For a black hole of the inferred mass ($M_{BH} = 6.5 \times 10^9 M_\odot$; Gebhardt et al. 2011; EHT Collaboration et al. 2019f), the six days correspond to $16 GM/c^3$. This is approximately equal to ~3.5 dynamical timescales at the ISCO for a maximally spinning black hole. Over this observation span, only a limited amount of variability in source structure was inferred between days, reflected as a small change in the thickness or azimuthal brightness distribution of the observed bright emission ring (EHT Collaboration et al. 2019d).

Our goal in this paper is to use general relativistic magnetohydrodynamic (GRMHD) simulations of accretion flows, followed by general relativistic ray tracing and radiative transfer, with parameters that are relevant to the M87 black hole, in order to explore the short-term variability in the source structure as it is imprinted in the interferometric observables. The EHT, being an interferometer, measures complex visibilities, which denote the two-dimensional Fourier transform of the source image on the sky. The measurements are typically decomposed into visibility amplitudes and phases (Thompson et al. 2017). Image reconstruction techniques enable us to use these variables and create the map of the source brightness on the plane of the sky (see EHT Collaboration et al. 2019d).

The visibility measurements in VLBI are affected by time-dependent atmospheric and instrumental errors of the telescopes, particularly challenging for the short radio wavelengths (Thompson et al. 2017). The custom-built EHT data reduction pipeline addresses the specific requirements of the millimeter VLBI (EHT Collaboration et al. 2019c). The pipeline uses Atacama Large Millimeter/submillimeter Array (ALMA) as an anchor station, leveraging its extreme sensitivity for the calibration of the entire EHT array (Blackburn et al. 2019; Janssen et al. 2019). As a result, the measured visibilities

indicate sufficient phase-stability to enable long coherent averaging and building up the high signal-to-noise ratio. Remaining station-based errors can be modeled as complex gains multiplying the visibilities. In order to eliminate the effects of these errors in the data, one can construct closure quantities with amplitudes and phases (Jennison 1958; Kulkarni 1989; Thompson et al. 2017; Blackburn et al. 2020). For a set of three baselines forming a closure triangle, the closure phase is defined as the sum of the measured visibility phases in each of these baselines. This cancels out the station-based phase measurement errors for each station. As a result, the closure phases are optimal and robust probes of the intrinsic structure of the source.

Closure phases have indeed proven to be a powerful tool to study and quantify variability in synthetic data from GRMHD simulations. Medeiros et al. (2018) studied the dependence of variability in closure phases on the orientations and baseline lengths of the closure triangles (see also Roelofs et al. 2017). Triangles that involve large baselines (that probe small length scales in the source image) were shown to have a high degree of variability in closure phases. On the other hand, triangles that involve small baselines (that probe the overall size of the ring and source structure) exhibit less variability. The exception to this rule are triangles involving a baseline close to a deep visibility minimum. The visibility phases in regions of visibility minima are extremely sensitive to minor changes in source structure. This localization of variability in the Fourier space is vital in understanding variability in GRMHD simulations and in observations.

In this paper, we first present a data-driven model to quantify the variability of observed closure phases in the various linearly independent triangles of the M87 observations across the six days of observations in 2017. We apply this algorithm to the 2017 EHT data on M87 and identify three closure triangles that show a remarkably small degree of variability. We then explore the degree of variability produced in a large set of GRMHD simulations, with different magnetic field configurations and prescriptions of the plasma physics, and understand the effects of various model parameters on the variability in the models. Finally, we compare the predictions of the GRMHD simulations to the observations and discuss how the latter constrain the physical properties of the accretion flow in M87.

## 2. Closure-phase Observations of M87

The EHT measures complex visibilities, given by

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

where $I(x, y)$ represents the total intensity of radiation at a given spatial location $(x, y)$ on the image plane and $(u, v)$ denote the Fourier frequencies corresponding to the $(x, y)$ coordinates. However, due to atmospheric and instrumental effects, the measured visibilities do not represent the actual visibilities of the source. The measured visibilities are related to the source visibilities through complex gains. The measured visibility between the $i$th and the $j$th telescopes, $\gamma_{ij}$, can be written as

$$\gamma_{ij} = g_i g_j^* V_{ij}, \quad (2)$$

where $g_i$ and $g_j$ are the complex gains associated with the two telescopes, and the star superscript indicates complex conjugates. We define the closure phase as the argument of the
<figure><img src="image_5.png" /><figcaption>**Figure 1.** Left: the *u-v* coverage traced by the EHT for the M87 observations on 2017 April 11, with the dashed circles indicating locations of the two observed visibility minima. The colors indicate different orientations of the *u-v* coverage. Right: the visibility amplitude as a function of baseline length on the same day.</figcaption></figure>

complex bispectrum along a closed triangle of three telescopes (Jennison 1958). The closure phase $\phi_{ijk}^{CP}$ for a closure triangle formed by the ith, the jth, and the kth telescopes is hence given by

$$ 
\begin{aligned}
\phi_{ijk}^{CP} &= \operatorname{Arg}[\gamma_{ij} \gamma_{jk} \gamma_{ki}] \\
&= \operatorname{Arg}[g_i g_j^* V_{ij} g_k^* V_{jk} g_k V_{ki}^*] \\
&= \operatorname{Arg}[|g_i|^2 |g_j|^2 |g_k|^2 V_{ij} V_{jk} V_{ki}] \\
&= \operatorname{Arg}[V_{ij} V_{jk} V_{ki}] \\
&= \phi_{ij} + \phi_{jk} + \phi_{ki},
\end{aligned}
\quad (3) $$

where $\phi_{ij}$ denotes the visibility phase between the ith and the jth telescopes. Equation (3) shows that the measured closure phase in a triangle is equal to the actual closure phase represented by the source.

The EHT observed M87 on four nights in 2017 April 5, 6, 10, and 11. The observations involved seven stations spread across five geographical locations (EHT Collaboration et al. 2019b): the Atacama Large Millimeter Array (ALMA) and the Atacama Pathfinder Experiment (APEX) in Chile; the James Clerk Maxwell Telescope (JCMT) and the Submillimeter Array (SMA) in Hawaii; the Large Millimeter Telescope (LMT) in Mexico; the Submillimeter Telescope Observatory (SMT) in Arizona; and the IRAM 30 m telescope in Pico Veleta (PV) in Spain. On each night of observation, the rotation of the Earth implies that each baseline traces out an elliptical path in the *u-v* space with the progression of time.

Figure 1 shows the *u-v* coverage for M87 during one of the observing days (left panel) as well as the visibility amplitudes measured as a function of baseline length on the same day (right panel). Two deep minima in visibility amplitude can be seen at baseline lengths of ~3.4 and ~8.3 Gλ encountered in three baselines (LMT-SMA, SMT-SMA, and PV-SMA) along the east-west orientation (indicated in blue). In contrast, the minimum encountered by the ALMA-LMT baseline is only marginally deep (right panel of Figure 1).

We construct closure-phase quantities using all the baselines shown in Figure 1. However, closure triangles with two telescopes at the same geographical location yield trivial closure phases, with deviations from zero arising due to changes in the large scale source structure alongside

<table><caption>**Table 1**
Closure Triangles Used with Baseline Lengths and Inferred Variability in Closure Phases</caption><thead><tr><th rowspan="2">Triangle<sup>a</sup></th><th colspan="3">Baseline Lengths (Gλ)<sup>b</sup></th><th rowspan="2">Inferred Variability<sup>c</sup></th></tr><tr><th>1</th><th>2</th><th>3</th></tr></thead><tbody><tr><td>ALMA-SMA-LMT</td><td>6.7–7.2</td><td><strong>3.1–4.5</strong></td><td><strong>3.4–4.1</strong></td><td>~30°–60°</td></tr><tr><td>ALMA-LMT-SMT</td><td>3.4–4.2</td><td>1.3–1.5</td><td>4.8–5.5</td><td>3°.4</td></tr><tr><td>ALMA-PV-LMT</td><td>6.0–6.6</td><td>5.1–6.4</td><td>3.4–4.2</td><td>4°.6</td></tr><tr><td>ALMA-SMA-SMT</td><td>6.7–7.2</td><td><strong>2.4–3.5</strong></td><td>4.8–5.5</td><td>~180°</td></tr><tr><td>ALMA-PV-SMA</td><td>6.0–6.2</td><td><strong>8.2–8.3</strong></td><td>6.8–7.0</td><td>...</td></tr><tr><td>ALMA-PV-SMT</td><td>6.0–6.6</td><td>5.5–6.4</td><td>5.3–5.5</td><td>5°.4</td></tr></tbody></table>

**Notes.**

systematic and thermal errors (EHT Collaboration et al. 2019c). While these triangles are useful to quantify systematic errors in closure phases, they do not capture any information about the source structure at horizon scales. Hence, we only choose triangles with telescopes at different geographical locations for our study. Since Chile and Hawaii have two stations each, we pick the station with the higher signal-to-noise ratio. We, therefore, consider five stations—ALMA, SMA, LMT, SMT, and PV, out of which six linearly independent closure phases can be constructed. We list the six triangles in Table 1. We choose the triangles such that all of them involve ALMA, which has the highest signal-to-noise ratio, and hence smaller uncertainties in closure-phase measurements.

In Figure 2, we show the evolution of closure phases with time in Greenwich Mean Sidereal Time (GMST) coordinates over all four days of observations in three triangles that exhibit substantial day-to-day variability in closure phases over the span of the 2017 campaign (see also EHT Collaboration et al. 2019c, Section 7.3.2):

<footnote><sup>a</sup> The bold baselines indicate the triangles that cross one of the deep visibility minima in the E-W orientation at 3.4 and 8.3 Gλ.</footnote>

<footnote><sup>b</sup> Baselines labeled 1, 2, and 3 in a triangle labeled as A-B-C correspond to the baselines A-B, B-C, and C-A, respectively.</footnote>

<footnote><sup>c</sup> An estimated range of variation in the high-variability triangles, and the maximum likelihood value of the inferred Gaussian variability (Equation (7)) in the low-variability triangles.</footnote>
<figure><img src="image_6.png" /><figcaption>**Figure 2.** The evolution of closure phases plotted with time for all four days of observation for closure triangles in which baselines are known to encounter regions of deep visibility minima. All three triangles exhibit a high level of variability in closure phases across six days of observation.</figcaption></figure>

(i) The ALMA-SMA-LMT triangle shows a change of 30°–60° between the first two days and the later two days of observation.

(ii) The ALMA-SMA-SMT triangle shows a swing of 180° on each day of observation. In addition to this, the direction of this swing flips between the first two days and the later two days of observation.

(iii) In the ALMA-PV-SMA triangle, it is difficult to quantify the degree of variability because of the reduced

complex visibility amplitudes between PV and SMA baselines, and the low signal-to-noise ratio in some data points. However, a persistent structure in the evolution of closure phases is not observed.

We note that each of these closure triangles involves one of the baselines (i.e., LMT-SMA, SMT-SMA, and PV-SMA) that cross a deep visibility minimum in the E-W orientation. In addition, we note that, in the ALMA-SMA-SMT triangle, the closure phases show little day-to-day variability until ~19 GMST on each day. We attribute this behavior to the fact that the SMT-SMA baseline encounters the deep visibility minimum at ~19 GMST. We plan to analyze the dependence of variability in the observations on the depth of the visibility minima in a future study.

The closure phases in the remaining triangles (i.e., ALMA-LMT-SMT, ALMA-PV-LMT, and ALMA-PV-SMT) show a persistent and continuous evolution with time during each day of observation (see Figure 3). This behavior represents the evolution of closure phases as the various baselines trace their paths on the *u*-*v* plane following the rotation of the Earth. However, the same evolution with time is repeated in all days of observations; there is little scatter around the general trend set by the rotation of the baselines. The presence of substantial scatter from day to day would have been a signature of structure variability in the image.

In order to compare the observations to theoretical models, we quantify the degree of closure-phase day-to-day variability observed in this last set of triangles. Upon examination of the closure phases in the maximal set of closure triangles, we find that, outside the set of linearly independent triangles that we choose, only one triangle (LMT-SMT-PV) shows a low degree of day-to-day variability. We conclude that it is optimal and sufficient to quantify the day-to-day variability in the low-variability triangles that we choose in Table 1, since the LMT-SMT-PV triangle can be obtained by a linear combination of these three triangles.

We also note that the comparison between observations and models could, in principle, be carried out for both high-variability and low-variability triangles. However, as we will discuss in Section 4, images from theoretical models uniformly show a high degree of phase variability in baselines that cross a visibility minimum and, as a result, a comparison with the high-variability triangles in the data turns out not to have much discriminating power between various models. The degree of variability exhibited on the low-variability triangles identified in the data, on the other hand, is a more significant challenge to the GRMHD models and proves to be a useful tool to distinguish between them.

The quantification of day-to-day variability in the set of three low-variable triangles is not straightforward, because closure phases evolve with time on each observing day but the individual scans on each day are not aligned at the exact same times. As a result, a difference in closure phases measured in two scans separated by almost (but not exactly) one sidereal day incorporates both the change due to the slightly different locations on the *u*-*v* plane probed by the two scans and the change due to the structural changes in the image.

In order to disentangle the two sources of variability, we employ a data-driven analytic model for the evolution of the closure phase with time during a single day of observation. This is meant to capture the change in closure phases introduced by the changing location on the *u*-*v* plane of the
<figure><img src="image_7.png" /><figcaption>Figure 3. The parameters and functional forms of the data-driven closure-phase evolution model, applied to the closure triangles that exhibit low variability across the six days of 2017 EHT observations (see Equation (4)). Left: the posterior projected on the $\sigma_{c_0}-c_0$ parameter plane, where $\sigma_0$ is a measure of the variability across the six days. The red dots indicate the most likely values. The contours indicate the levels containing 68% and 95% of the posterior probability. The dashed horizontal lines indicate the systematic error of 2° in closure-phase observations. Right: closure-phase observations plotted as a function of time for the same triangles. The cyan band indicates the most likely value of the width of the model $\sigma_{c_0}$.</figcaption></figure>

closure triangles. We then compare the overall change of the parameters of this model from day to day, in order to quantify the variability due to structural changes of the image.

For our data-driven model, $\phi_{\text{model}}(t)$, we use a second-order polynomial in the observation time $t$ (in GMST coordinates), given by

$$\phi_{\text{model}}(t) = c_0 + c_1 t + c_2 t^2. \quad (4)$$

In order to account for potential structural changes from day to day, we allow for an intrinsic Gaussian spread in the zeroth-order coefficient of this polynomial ($c_0$), with a standard deviation of $\sigma_{c_0}$. In this prescription, we do not assign any physical significance to the polynomial coefficients. Since the

maximum separation between any two nights of observations is six days, $\sigma_{c_0}$ acts as a constraint on the overall change in closure phase at a given time in a given triangle across six days.

Using this model, we define the likelihood of making a particular closure-phase measurement using a Gaussian mixture model as

$$L_i(c_0, c_1, c_2, \sigma_{c_0}) \equiv \left\{ \frac{1}{\sqrt{2\pi(\sigma_i^2 + \sigma_{c_0}^2)}} \right\} \times \exp \left[ -\frac{1}{2} \frac{\{\phi_i - \phi_{\text{model}}(t_i)\}^2}{(\sigma_i^2 + \sigma_{c_0}^2)} \right], \quad (5)$$
where $\phi_i$ is the closure phase in a triangle at an observation time $t_i$, and $\sigma_i$ denotes the uncertainty in $\phi_i$. Note that we have assumed here Gaussian statistics for the measurements of closure phases. This assumption is not expected to introduce any significant changes in our quantitative results since the triangles we will be applying this method to are characterized by high signal-to-noise measurements. Applying this mixture model to data with large errors would require using an appropriate distribution for the closure-phase errors and their correlations (see Christian & Psaltis 2020).

The likelihood that n closure-phase observations in a given triangle across all four days of observation follow the model $\phi_{\text{model}}(t)$ is

$$L(c_0, c_1, c_2, \sigma_{c_0}) = \prod_{i=1}^{n} L_i(c_0, c_1, c_2, \sigma_{c_0}) \quad (6)$$

$$= \prod_{i=1}^{n} \left\{ \frac{1}{\sqrt{2\pi(\sigma_i^2 + \sigma_{c_0}^2)}} \right\} \exp \left[ -\sum_{i=1}^{n} \frac{1}{2} \frac{\{\phi_i - \phi_{\text{model}}(t_i)\}^2}{(\sigma_i^2 + \sigma_{c_0}^2)} \right]. \quad (7)$$

Assuming flat priors in the polynomial coefficients and in $\sigma_{c_0}$, we map the parameter space using the Markov-chain Monte Carlo sampling.

Figure 3 shows the posteriors over two of the model parameters, $c_0$ and $\sigma_{c_0}$, for each of the three triangles that exhibits low day-to-day variability. The left panels show the 68% and 95% contours of the posteriors; the right panels show the data-driven model with the most likely value of the parameters, overplotted on the closure-phase observations in these triangles. The most likely standard deviations of variability across the six days of observations are ~3°–5°. This is comparable to the systematic error in closure-phase observations of 2° as reported in EHT Collaboration et al. (2019c) and is remarkably small.

### 3. GRMHD Simulations and Library

A large suite of GRMHD simulations has been generated to model the accretion flow around M87. The simulations have been performed using the algorithms *harm* (see Gammie et al. 2003), *BHAC* (Olivares Sánchez et al. 2018), *KORAL* (Sądowski et al. 2014), etc. The simulations have been initialized with two magnetic field configurations that led to different field structures in the flows: Standard and Normal Evolution (SANE; see Igumenshchev et al. 2003) and Magnetically Arrested Disk (MAD; see Narayan et al. 2012 and Sądowski et al. 2013). A comprehensive comparison of the GRMHD algorithms is presented in Porth et al. (2019). The set of observables that results from these simulations has been obtained by general relativistic ray tracing and radiative transfer algorithms, such as *GRay* (Chan et al. 2013) and *ipole* (Mościbrodzka & Gammie 2018).

In this paper, we analyze the GRMHD simulations that were run using *harm* (see Gammie et al. 2003; Narayan et al. 2012). The simulations include SANE and MAD configurations of the magnetic fields and different black hole spins with corotating or counterrotating accretion disks. The flow properties, as calculated in the GRMHD simulations, scale with the mass of the black hole and, therefore, the latter does not enter the calculation. The dependence of the electron temperature on the

local value of the plasma $\beta$ parameter aims to capture the effects of sub-grid electron physics that are not resolved in the GRMHD simulations (see Chan et al. 2015). The accretion flows in the simulations are modeled as collisionless plasmas, where the ratio of ion temperature to electron temperature ($R \equiv T_i/T_e$) is given by

$$R = R_{\text{high}} \frac{\beta^2}{1 + \beta^2} + \frac{1}{1 + \beta^2}. \quad (8)$$

Here, $\beta \equiv P_{\text{gas}}/P_{\text{mag}}$ is the ratio of gas pressure ($P_{\text{gas}}$) to the magnetic pressure ($P_{\text{mag}}$). This prescription aims to simulate the effects of sub-grid electron heating, using this simple, physics-motivated parametric form (Mościbrodzka et al. 2016; and EHT Collaboration et al. 2019e).

General relativistic ray tracing and radiative transfer is performed on the simulated system at the wavelength of 1.3 mm in order to generate the images of the black hole using the different snapshots of the simulations. For radiative transfer, the black hole mass ($M_{\text{BH}}$) and the overall electron number-density scale in the accretion disk ($n_e$) set a scale for the mass and the optical depth in the accretion disk, which determines the mass accretion rate and the flux of emission in the source. It follows from the scaling properties of the emissivity of the accretion disk at 1.3 mm and from the natural scaling of the GRMHD simulations (see Chan et al. 2015) that $n_e$ and $M_{\text{BH}}$ are degenerate quantities with the resulting images and, therefore, the various interferometric observables, depending only on the product $M_{\text{BH}}n_e^2$ (see a derivation and discussion in the Appendix).

We use two sets of GRMHD models for studying closure-phase variability:

(i) The first set of models (henceforth Set A) is aimed at understanding the dependence of the closure-phase variability on the properties of the accretion flow, i.e., MAD versus SANE magnetic field configuration, the plasma prescription, and the electron number-density scale ($n_e$). In particular, they include SANE and MAD models for a single spin, three different values of $R_{\text{high}}$, and five different values of $n_e$ (defined at a fiducial mass of $M_{\text{BH}} = 6.5 \times 10^9 M_{\odot}$). For this value of the black hole mass, this range allows us to explore compact fluxes that are up to a factor of a few above and below the nominal value of 0.5 Jy. Each model is used to generate 1024 images with a time cadence of 10 GM/c³, amounting to a total of ~30,000 images in this set. Ray tracing and radiative transfer calculations for this simulation set are carried out using *GRay*.

(ii) The second set of models (henceforth Set B) explores a comprehensive set of black hole spins and $R_{\text{high}}$ of the accretion plasma, which are used in EHT Collaboration et al. (2019e) in the interpretation of the EHT observations. The electron number-density scale ($n_e$) in this simulation set is tuned a priori such that the total flux of radiation at 1.3 mm is equal to 0.5 Jy for a fiducial mass of $M_{\text{BH}} = 6.2 \times 10^9 M_{\odot}$. Each model in this simulation set is used to generate 600–1000 images at a time cadence of 5 GM/c³, amounting to ~50,000 images in this set. Ray tracing and radiative transfer calculations for this simulation set are carried out using *ipole*.

The ray tracing and radiative transfer tracing calculations on the GRMHD simulations performed using *ipole* and *GRay* ignore the effects of finite light travel time. Bronzwaer et al. (2018) showed that the effect of the approximation on the lightcurve of flux density in a simulation is restricted to only a few percent. Depending upon the nature of the spatial
<table><caption>**Table 2**
GRMHD Simulation Model Parameters</caption><thead><tr><th rowspan="2">Parameter<sup>a</sup></th><th colspan="2">Parameter Space</th></tr><tr><th>Set A</th><th>Set B</th></tr></thead><tbody><tr><td>Model type</td><td>MAD, SANE</td><td>MAD, SANE</td></tr><tr><td>Black hole spin<sup>b</sup></td><td>+0.9</td><td>0.0, ±0.5, ±0.94</td></tr><tr><td>Plasma R<sub>high</sub></td><td>1, 20, 80</td><td>1, 10, 20, 40, 80, 160</td></tr><tr><td>Electron number density<br>(n<sub>e</sub>)<sup>c</sup></td><td>1,2.5,5,7.5,10 (×10<sup>5</sup> cm<sup>-3</sup>)</td><td>Tuned to<br>constant flux</td></tr><tr><td>Time cadence</td><td>10 GM/c<sup>3</sup></td><td>5 GM/c<sup>3</sup></td></tr></tbody></table>

**Notes.**

<sup>a</sup> The u-v coordinates of the baselines are rotated to align the spin axis of the black hole at a position angle of 288° east of north.

<sup>b</sup> The positive spin models are ray traced at an inclination of 163° and negative spin models at 17° in Set A. Set B models are ray traced at an inclination of 17°.

<sup>c</sup> The electron number density is defined at a fiducial mass of 6.5 × 10⁹ M<sub>☉</sub> for Set A and 6.2 × 10⁹ M<sub>☉</sub> for Set B.

correlations in structural variability in the simulation, the computed images may carry either time-correlated or time-uncorrelated features. These features will have an effect on the amplitude of variability inferred. In this study, we ignore these effects in inferring the variability in the images.

The parameter spaces studied in sets A and B are listed in Table 2. Although the mass of the black hole $M_{BH}$ does not enter as an independent parameter in the calculation of the black hole images, it affects our interpretation of the observations in the following ways:

(i) The characteristic time unit ($\tau$) in the GRMHD simulations is set by the mass of the black hole as $\tau = GM_{BH}/c^3$. A higher mass of the black hole implies that an observation span of 6 days would translate to a shorter elapsed time in units of $\tau$.

(ii) The angular size of the image as seen by a distant observer depends on the black hole mass ($\Delta\theta \sim \lambda/M_{BH}$), which implies that the EHT probes different regions in the Fourier space for black holes of different masses. A higher mass of the black hole translates to the EHT probing structures at smaller length scales.

The mass of M87 has been measured using two methods prior to the EHT observations (EHT Collaboration et al. 2019e). Gebhardt et al. (2011) measured a mass of $M_{BH} = 6.6 \pm 0.4 \times 10^9 M_{☉}$ using stellar dynamics. Walsh et al. (2013) measured a mass of $M_{BH} = 3.5_{-0.7}^{+0.9} \times 10^9 M_{☉}$ by studying emission lines from ionized gas. The GRMHD models in simulation sets A and B give images for which the ring sizes and widths correspond to masses in the range inferred by the stellar dynamics measurement of Gebhardt et al. (2011; see EHT Collaboration et al. 2019f). Because of this, we allow the black hole mass to vary in the range $5 \times 10^9 M_{☉} \le M_{BH} \le 9.5 \times 10^9 M_{☉}$ in both sets of the simulations.

The orientation of the black hole spin axis has been constrained by long-wavelength observations of its jet at a position angle of ~288° north of east and at an inclination of 17° with respect to the line of sight (see discussion in EHT Collaboration et al. 2019f). We trace the Set A models at an inclination of 17°. In Set B, we trace the positive spin models at an inclination of 163° and the negative spin models at 17°.

The simulations typically run for lengths that correspond to ~$10^4 GM/c^3$. Here, we only use images obtained from the

relaxed turbulent state with a slowly varying mass accretion rate. This corresponds to simulation times in the range of $5 \times 10^3 \le t \le 10^4 GM/c^3$.

### 4. Closure-phase Variability in GRMHD Simulations

In order to analyze visibility phase and closure-phase variability in GRMHD models, we first transform the GRMHD snapshots into the visibility space by performing a two-dimensional Fourier transform on each of the images. We then compute the amplitude and phase from the complex visibilities at each location in the u-v plane.

Since the visibility phase is a directional quantity, we need to employ directional statistics to infer the standard deviation in a time-series. For a given time-series of $n$ phases $\theta_i$, we compute its directional dispersion as

$$D = \frac{1}{n} \sum_{i=1}^{n} \{1 - \cos(\theta_i - \bar{\theta})\}, \quad (9)$$

where $\bar{\theta}$ represents the circular mean, defined as

$$\bar{\theta} = \tan^{-1} \left\{ \frac{\sum_{i=1}^{n} \sin(\theta_i)}{\sum_{i=1}^{n} \cos(\theta_i)} \right\}. \quad (10)$$

In the limit of small deviations in the time-series from the circular mean $\bar{\theta}$, the directional dispersion can be approximated as

$$D \approx \frac{\sigma^2}{2}, \quad (11)$$

where $\sigma$ is inferred as a standard deviation of the time-series $\theta_i$ for small fluctuations about $\bar{\theta}$.

We use this quantity to construct heat maps of phase variability in order to understand the dependence of variability on the u-v coordinates (see also Medeiros et al. 2017). We present an example of such a heat map for one of the simulations in Set B in Figure 4. The left panel shows the mean visibility amplitude on the u-v plane computed across the simulation run and reveals deep visibility minima that are aligned with the spin axis of the black hole. The right panel shows a heat map of the visibility phase dispersion. There are *hot regions* of visibility phase variability, i.e., localized regions in u-v space that show large dispersions. These regions lie primarily along the spin axis of the black hole and at baseline lengths that correspond to the deep minima in visibility amplitudes. In fact, a natural anticorrelation between phase variability and mean visibility amplitude is observed in all the GRMHD models, which was also explored in Medeiros et al. (2017; see their Figures 3 and 4).

Based on this understanding, one expects the closure-phase variability computed along various triangles to depend on the locations of the three vertices in the u-v space. In particular, the localized nature of variability implies that closure triangles that have baselines crossing the hot regions are expected to exhibit high variability in closure phases, whereas triangles with vertices in quiet parts of the u-v space should show a low level of variability. In Figure 4, we show examples of two such triangles: the ALMA-SMA-LMT triangle has a baseline on a hot region and is expected to be highly variable, while a low-variability closure triangle (ALMA-LMT-SMT) avoids the hot regions.

The azimuthal extent of the hot regions of phase dispersion in the u-v plane depends on the degree of symmetry of the
<figure><img src="image_8.png" /><figcaption>**Figure 4.** Normalized mean visibility amplitude map on a logarithmic scale (left) and directional dispersion map (see Equation (9)) of visibility phases (right) for the SANE, $a = +0.5$, $R_{high} = 1$ model from simulation set B. The black hole spin points upwards as indicated by the white arrow labeled as $\Omega$. The EHT baseline coverage (on April 11) are shown by the white dots, and are rotated such that the spin points to a position angle of 288° east of north (with north indicated by the white arrow labeled as N and the $u_{rot}-v_{rot}$ plane representing the rotated u-v plane). The cyan and the red triangles indicate the ALMA-LMT-SMT (at UTC 3:32:5.0003 hr/ GMST 16:26:37.1531 hr) and ALMA-SMA-LMT (at UTC 5:00:5.0001 hr/GMST 17:54:51.6089 hr) triangles respectively. The second triangle includes the SMA- LMT baseline that crosses a visibility minimum, which translates to a high level of variability in the closure phase introduced by structural changes in the image. The mass of the black hole is set to $M_{BH} = 7.5 \times 10^9 M_{\odot}$.</figcaption></figure>

underlying image. This is shown in Figure 7, where each column corresponds to a simulation from Set A with a different electron-density scale $n_e$, and all other black hole and plasma parameters held fixed. Qualitatively, the two effects of increasing the electron-density scale are a higher level of symmetry in the bright emission ring and an increase in its fractional width. The more symmetric images lead to a larger azimuthal extent in the minima in the visibility amplitudes (middle panels) and in the hot regions of phase dispersion (lower panels). The increased fractional width of the images lead to the locations of both the visibility amplitude minima and the hot regions in phase dispersion to appear at smaller baseline lengths.

## 4.1. Closure-phase Variability and Compliance Fractions

The phase dispersions discussed in the previous section were calculated across the entire span of the simulations, which corresponds to many dynamical timescales. In order to compare the simulations to the observed variability in M87, however, we need to calculate the degree of variability for a time span of $\Delta t = 6$ days, which is the longest separation between the four observations in 2017. Furthermore, because observations only yield closure phases, hereafter we focus on this quantity to facilitate comparisons.

We define the variance in closure phase $\phi_{cp}(t)$ for a time span of $\Delta t = 6$ days using Equation (11) as

$$ \sigma^2(t_0) = \frac{1}{\Delta t} \int_{t_0}^{t_0+\Delta t} dt \, 2[1 - \cos\{\phi_{cp}(t) - \bar{\phi}_{cp}\}]. \quad (12) $$

In order to reduce the discretization noise when calculating this quantity, we perform a linear interpolation between simulation time-steps and construct a continuous function $\phi_{cp}(t)$ representing the evolution of closure phases. We use fixed u-v locations taken at a median time over the night of observations, in order to construct this function. This ensures that we separate the evolution of closure phases due to the changing locations of the baselines from the structural variability of the images that we are interested in inferring.

<figure><img src="image_9.png" /><figcaption>**Figure 5.** Closure phases for three triangles plotted as a function of time for a particular segment of the same simulation (Set B—MAD $a = +0.94$, $R_{high} = 20$). The black parallel lines indicate the timescale of 6 days and a standard deviation of $\sim 4^\circ$. The section within the green dashed vertical lines indicates a region of low variability, and hence counts toward the compliance fraction of the model.</figcaption></figure>

In Figure 5, we show the evolution of the closure phase with time during one segment of a simulation chosen from Set B, which has a MAD configuration with parameters $a = +0.94$ and $R_{high} = 20$. As is evident from this evolution, there are time spans where the closure phase shows little variability (e.g., the times between the green vertical lines) and others where there are substantial swings of order $\pi$ over a short period of time that are comparable to the length of the observations. Given the low degree of variability observed for some triangles in M87 data (see Section 2), this motivates us to define a metric that quantifies how common such periods are in a given model.

Figure 6 shows the standard deviation in closure-phase variability computed from all 6 day segments in the simulation shown in Figure 5, for the three triangles that have observationally shown a small degree of variability. The dashed lines correspond to thresholds $\sigma_1, \sigma_2, \sigma_3$ of this
<figure><img src="image_10.png" /><figcaption>**Figure 6.** Two-dimensional scatter plots of inferred standard deviation in closure phases in three triangles across one simulation (simulation set B–MAD *a* = +0.94, *R*<sub>high</sub> = 20). The horizontal and vertical dashed lines indicate the observational bounds for the standard deviation for each triangle, as described in Section 2.</figcaption></figure>

<table><caption>**Table 3**

Compliance Fraction for Set A</caption><thead><tr><th rowspan="2">Model</th><th rowspan="2">n<sub>e</sub> (×10<sup>5</sup>cm<sup>-3</sup>)</th><th colspan="3">R<sub>high</sub></th></tr><tr><th>1</th><th>20</th><th>80</th></tr></thead><tbody><tr><td rowspan="5">SANE a = +0.9</td><td>1</td><td>41%</td><td>30%</td><td>24%</td></tr><tr><td>2.5</td><td>37%</td><td>31%</td><td>24%</td></tr><tr><td>5</td><td>31%</td><td>28%</td><td>25%</td></tr><tr><td>7.5</td><td>24%</td><td>24%</td><td>22%</td></tr><tr><td>10</td><td>18%</td><td>18%</td><td>20%</td></tr><tr><td rowspan="5">MAD a = +0.9</td><td>1</td><td>52%</td><td>36%</td><td>22%</td></tr><tr><td>2.5</td><td>50%</td><td>32%</td><td>23%</td></tr><tr><td>5</td><td>40%</td><td>22%</td><td>20%</td></tr><tr><td>7.5</td><td>24%</td><td>15%</td><td>18%</td></tr><tr><td>10</td><td>12%</td><td>12%</td><td>18%</td></tr></tbody></table>

observed variability. In Figure 5, we choose as the most likely values of $\sigma_{c0}$ in the corresponding triangles as the thresholds (see Table 1). The green points indicate the occurrences of 6 day segments that are consistent with observations. We define the quantity $\mathcal{F}(\sigma_1, \sigma_2, \sigma_3)$ of a model as the fraction of 6 day segments that shows a level of variability consistent with the observed one; i.e., the ratio of green points to the total number of points in this figure.

Formally, $\mathcal{F}(\sigma_1, \sigma_2, \sigma_3)$ can be written as

$$ \mathcal{F}(\sigma_1, \sigma_2, \sigma_3) = \int_0^\infty d\sigma'_1 \int_0^\infty d\sigma'_2 \int_0^\infty d\sigma'_3 P(\sigma_{\text{sys}}) \times P_{\text{model}}(\sigma'_1, \sigma'_2, \sigma'_3|\sigma_{\text{sys}}), \quad (13) $$

where $P(\sigma_{\text{sys}})$ is the prior in the systematic uncertainty $\sigma_{\text{sys}}$, and $P_{\text{model}}(\sigma'_1, \sigma'_2, \sigma'_3|\sigma_{\text{sys}})$ is the three-dimensional distribution of circular standard deviations $\sigma'_1, \sigma'_2, \sigma'_3$ computed in the three low-variability triangles using Equation (12). The observed variability, as quantified in Section 2, is a combination of the true intrinsic variability of the source and the systematic errors in the measurement. Because of our limited knowledge of the systematic uncertainties and their priors, we assume a form

such that Equation (13) can be written as

$$ \mathcal{F}(\sigma_1, \sigma_2, \sigma_3) = \int_0^{\sigma_1} d\sigma'_1 \int_0^{\sigma_2} d\sigma'_2 \times \int_0^{\sigma_3} d\sigma'_3 P_{\text{model}}(\sigma'_1, \sigma'_2, \sigma'_3), \quad (14) $$

i.e., simply assuming that the combined intrinsic degree of variability and the systematic uncertainties cannot exceed the inferred values.

Because our inferred degree of variability from the observations themselves have associated formal errors as described by the marginalized posteriors $P(\sigma_1)$, $P(\sigma_2)$, and $P(\sigma_3)$ as shown in Figure 3; we compute the compliance fraction by integrating over those posteriors using

$$ \mathcal{C}(\text{model}) = \int d\sigma_1 d\sigma_2 d\sigma_3 P(\sigma_1) P(\sigma_2) P(\sigma_3) \mathcal{F}(\sigma_1, \sigma_2, \sigma_3). \quad (15) $$

The compliance fraction denotes the expectation value of $\mathcal{F}(\sigma_1, \sigma_2, \sigma_3)$ given the probability distributions of $\sigma_1, \sigma_2$, and $\sigma_3$.

We compute this compliance fraction for black hole masses in the range $5 \times 10^9 M_{\odot} \le M_{BH} \le 9.5 \times 10^9 M_{\odot}$ and use the maximum number as the compliance fraction for a given GRMHD model.

## 5. Comparison of GRMHD Simulations to M87 Observations

In this section, we use the two sets of simulations we discuss in Section 3 to explore the dependence of the compliance fraction on the magnetic field configuration, electron temperature prescription in the plasma, and electron-density scale (Set A), as well as on the black hole spin (Set B). These two sets are meant to serve as two slices in the large parameter space of simulations.

Table 3 shows the compliance fractions calculated for the 30 simulations in Set A. Overall, these compliance fractions are rather small, indicating that a lot of simulations show more frequent periods of high variability, even in the triangles that observationally do not exhibit large changes in the closure
<figure><img src="image_11.png" /><figcaption>**Figure 7.** The average images (top), normalized average visibility amplitudes (middle), and directional phase dispersion (bottom) of the MAD, $a = 0.9$, $R_{\text{high}} = 20$ simulation. Models with three electron number densities—$n_e = 1 \times 10^5 \text{ cm}^{-3}$ (left), $n_e = 5 \times 10^5 \text{ cm}^{-3}$ (middle), and $n_e = 1 \times 10^6 \text{ cm}^{-3}$ (right), are mapped in this figure. The white arrows labeled as $\Omega$ indicate the spin vector of the black hole, and the ones labeled as $N$ indicate the north direction. The coordinates ($u_{\text{rot}}$, $v_{\text{rot}}$) represent the $u-v$ coordinates rotated such that the spin points to a position angle of 288° east of north. It is evident that a model with lower $n_e$ exhibits an emission dominant from the photon ring, and has a lower variability in visibility phases outside the first visibility minimum. The three triangles plane indicate the low-variability triangles—ALMA-LMT-SMT (cyan), ALMA-PV-SMT (green), and ALMA-PV-LMT (white) at UTC 3:32:5.0003 hr/ GMST 16:26:37.1531 hr. The colormap in the visibility amplitude is in logarithmic scale with the maximum value normalized to unity.</figcaption></figure>

phase across the six days of the campaign. The dominant distinguishing characteristic of the simulations that show a higher compliance fraction is the small value for the electron-density scale. There is a smaller difference in compliance fractions between the MAD and SANE simulations. We observe that, for small values of $R_{\text{high}}$, the dependence of the compliance fraction on the fractional width is stronger.

We explored how the electron-density scale affects the image structure and variability. To this end, we calculated the characteristic properties of the average image in each simulation, using the image-domain techniques discussed in EHT Collaboration et al. (2019d, 2019f). We measured the fractional width of each average image $\delta w_f$ as the ratio of the width $\delta w$ to the ring diameter $d$, i.e., $\delta w_f \equiv \delta w/d$. We found the

fractional widths of the images correlate strongly with the electron-density scale, as shown in Figure 8. This is expected, given that increasing the electron-density scale also increases the emissivity and optical depth in the accretion flow, as shown in the top panels of Figure 7 (Chan et al. 2015).

The right panel of Figure 8 shows, indeed, that the compliance fraction is anticorrelated with the fractional width of the average images of the simulations. When the bright emission ring in an image is narrow, it traces the outline of the black hole shadow and, therefore, its shape and brightness distribution is determined primarily by gravitational lensing effects. On the other hand, when the bright emission ring is broad, variability due to the turbulence in the accretion flow introduces more apparent effects. The low level of variability
<figure><img src="image_12.png" /><figcaption>**Figure 8.** Left: the fractional width of the ring of emission in the average image of each simulation in Set A. Right: the anticorrelation between the fractional width and compliance fraction in the simulation set A.</figcaption></figure>

<table><caption>**Table 4**

Compliance Fraction for Set B</caption><thead><tr><th rowspan="2">Model</th><th rowspan="2">Spin</th><th colspan="6"><i>R</i><sub>high</sub></th></tr><tr><th>1</th><th>10</th><th>20</th><th>40</th><th>80</th><th>160</th></tr></thead><tbody><tr><td rowspan="5">SANE</td><td>-0.94</td><td>&lt;1% <sup>a,b,c</sup></td><td>2% <sup>a,b,c</sup></td><td>3%<sup>a,b,c</sup></td><td>2%<sup>a,b,c</sup></td><td>1%<sup>a,b,c</sup></td><td>&lt;1%<sup>a,b,c</sup></td></tr><tr><td>-0.5</td><td>&lt;1%<sup>a</sup></td><td>12%<sup>a</sup></td><td>12%<sup>a,b</sup></td><td>12%<sup>a,b</sup></td><td>8%<sup>a,b</sup></td><td>7%<sup>a,b</sup></td></tr><tr><td>0.0</td><td>&lt;1%<sup>a,b</sup></td><td>3%<sup>a,b</sup></td><td>8%<sup>a</sup></td><td>11%<sup>a,b</sup></td><td>11%<sup>a,b</sup></td><td>10%<sup>a,b</sup></td></tr><tr><td>+0.5</td><td>6%<sup>a,b</sup></td><td>2%<sup>a,b</sup></td><td>2%<sup>a,b</sup></td><td>6%<sup>a,b</sup></td><td>10%<sup>a,b</sup></td><td>10%<sup>a,b</sup></td></tr><tr><td>+0.94</td><td>21%<sup>b</sup></td><td>21%<sup>b</sup></td><td>15%<sup>a,b</sup></td><td>15%<sup>a,b</sup></td><td>19%<sup>a,b,c</sup></td><td>24%<sup>a,b,c</sup></td></tr><tr><td rowspan="5">MAD</td><td>-0.94</td><td>7%<sup>b,c</sup></td><td>6%<sup>a,b,c</sup></td><td>7%<sup>a,b,c</sup></td><td>6%<sup>a,b,c</sup></td><td>6%<sup>a,b,c</sup></td><td>9%<sup>a,b,c</sup></td></tr><tr><td>-0.5</td><td>10%<sup>b</sup></td><td>12%<sup>a,b</sup></td><td>13% <sup>a,b,c</sup></td><td>11% <sup>a,b,c</sup></td><td>11% <sup>a,b,c</sup></td><td>11% <sup>a,b,c</sup></td></tr><tr><td>0.0</td><td>12%<sup>b</sup></td><td>11% <sup>a,b</sup></td><td>15% <sup>a,b</sup></td><td>13% <sup>a,b</sup></td><td>10% <sup>a,b</sup></td><td>9% <sup>a,b</sup></td></tr><tr><td>+0.5</td><td>11%<sup>b</sup></td><td>16% <sup>a,b,c</sup></td><td>18% <sup>a,b,c</sup></td><td>19% <sup>a,b,c</sup></td><td>18% <sup>a,b,c</sup></td><td>17% <sup>a,b,c</sup></td></tr><tr><td>+0.94</td><td>20%<sup>c</sup></td><td>24% <sup>b,c</sup></td><td>24% <sup>a,b,c</sup></td><td>20% <sup>a,b,c</sup></td><td>16% <sup>a,b,c</sup></td><td>12% <sup>a,b,c</sup></td></tr></tbody></table>

**Notes.**

<sup>a</sup> Satisfies radiative efficiency constraint.

<sup>b</sup> Satisfies X-ray luminosity constraint.

<sup>c</sup> Satisfies jet power constraint.

across the six days observed in M87 argues in favor of images characterized by thin (~20%) rings of emission.

In the simulations of Set B, the electron number-density scale was tuned to generate a fixed source flux of 0.5 Jy. Different configurations were explored with both rotating and counterrotating black hole spins of various magnitudes. The compliance fractions are shown in Table 4 and are, overall, lower than those of the Set A simulations. As in the previous case, the MAD simulations have on average a higher compliance fraction than the SANE simulations, and there is only a marginal dependence on the electron temperature parameter *R*<sub>high</sub>. However, there is a significant dependence on the spin of the black hole, with corotating, high-spin simulations producing the highest compliance fractions.

The table also shows constraints imposed on the models of Set B by other considerations, not originating from the EHT observations (EHT Collaboration et al. 2019e). They include constraints related to (a) the simulations having achieved radiative equilibrium, (b) the predicted X-Ray flux to be *L*<sub>X</sub> = (4.4 ± 0.1) × 10<sup>40</sup> erg s<sup>-1</sup>, as measured during the nearly simultaneous observations with the Chandra X-Ray

Observatory and the Nuclear Spectroscopic Telescope Array (NuSTAR), and (c) the jet power being in range 10<sup>42</sup> to 10<sup>45</sup> erg s<sup>-1</sup>. Simulations with fast black hole spins, which are consistent with these constraints, are also characterized by larger compliance fractions, giving preference to these models.

## 6. Discussion

In this paper, we studied the variability of the black hole image structure of M87 at timescales comparable to the fastest dynamical timescale near the horizon, as it is imprinted on the closure phases measured during the EHT 2017 observations. We identified three linearly independent closure triangles that exhibit a persistent evolution pattern of closure phases over the course of each night of observation but show little variation around this pattern across the six days of observations, namely, ALMA-LMT-SMT, ALMA-PV-SMT, and ALMA-PV-LMT. In other words, the closure-phase evolution in these triangles follows set tracks determined by the rotation of the baselines on the *u*-v plane but shows very little scatter around these tracks from day to day. We quantified the degree of variability in these triangles to be ~3°–5°. This inferred level of variability is
comparable to the ~2° systematic error in measurement of closure phases (EHT Collaboration et al. 2019c).

We also found three triangles that exhibit a high level of day-to-day variability, namely, ALMA-SMA-LMT, ALMA-SMA-SMT, and ALMA-PV-SMA. The change in the closure phase across 6 days in a given location on the *u*-v plane for these triangles can be ~90°–180°. We identified them as triangles with at least one baseline that encounters a deep visibility minimum on the *u*-v plane. This is in agreement with expectations based on theoretical models that reveal the presence of highly variable but highly localized regions on the *u*-v plane associated with the locations of these minima.

We used GRMHD simulations to explore the dependence of closure-phase variability on different model parameters and at different locations in the *u*-v plane. We found that the most discriminating image characteristic of models in terms of the degree of closure-phase variability is the fractional width of the ring of high intensity on the image. Models that best reproduce the observed small level of variability are those with thin ring-like images with structures dominated by gravitational lensing effects and thus least affected by turbulence in the accreting plasmas.

Among the models we explored, there is marginal difference between SANE and MAD simulations, which explore different magnetic field configurations in the accretion flows, with some preference for the MAD models. There is also a small dependence on the black hole spin, with the high-spin corotating models showing the lowest level of day-to-day variability, in agreement with the observations.

These findings demonstrate that the method we introduced to quantify the day-to-day variability in the closure-phase data and compare it to the models is a useful tool in exploring the origin of variability in horizon-scale images of black holes and in discriminating between models.

This work was supported, in part, by the NSF PIRE award 1743747 and NASA ATP award 80NSSC20K0521. L.M. acknowledges support from an NSF Astronomy and Astrophysics Postdoctoral Fellowship under award No. AST-1903847. M.W. acknowledges the support of the Black Hole Initiative at Harvard University, which is funded by grants from the John Templeton Foundation and the Gordon and Betty Moore Foundation to Harvard University. All ray tracing calculations for Set A were performed with the El Gato GPU cluster at the University of Arizona that is funded by NSF award 1228509. All analyses for Set B were performed on CyVerse, supported by NSF grants DBI-0735191, DBI-1265383, and DBI-1743442.

The EHT Collaboration thanks the following organizations and programs: the Academy of Finland (projects 274477, 284495, 312496, and 315721); the Agencia Nacional de Investigación y Desarrollo, Chile via NCN19058 (TITANs) and Fondecyt 3190878; the Alexander von Humboldt Stiftung; an Alfred P. Sloan Research Fellowship; Allegro, the European ALMA Regional Centre node in the Netherlands, the NL astronomy research network NOVA, and the astronomy institutes of the University of Amsterdam, Leiden University, and Radboud University; the black hole Initiative at Harvard University, through a grant (60477) from the John Templeton Foundation; the China Scholarship Council; Consejo Nacional de Ciencia y Tecnología (Mexico, projects U0004-246083, U0004-259839, F0003-272050, M0037-279006, F0003-

281692, 104497, 275201, and 263356); the Delaney Family via the Delaney Family John A. Wheeler Chair at Perimeter Institute; Dirección General de Asuntos del Personal Académico-Universidad Nacional Autónoma de México (projects IN112417 and IN112820); 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); MICINN Research Project PID2019-108995GB-C22; the Gordon and Betty Moore Foundation (grant GBMF-3561); the Istituto Nazionale di Fisica Nucleare sezione di Napoli, iniziative specifiche TEONGRAV; the International Max Planck Research School for Astronomy and Astrophysics at the Universities of Bonn and Cologne; Joint Princeton/Flatiron and Joint Columbia/Flatiron Postdoctoral Fellowships; research at the Flatiron Institute is supported by the Simons Foundation; 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, QYZDJSSW-SYS008, and 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, and 25120007); the Malaysian Fundamental Research Grant Scheme (FRGS). FRGS/1/2019/STG02/UM/02/6; the MIT International Science and Technology Initiatives 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, 107-2119-M-110-005, 108-2112-M-001-048, and 109-2124-M-001-005); the National Aeronautics and Space Administration (NASA; Fermi Guest Investigator grant 80NSSC20K1567, NASA Astrophysics Theory Program grant 80NSSC20K0527, and NASA NuSTAR award 80NSSC20K0645); the National Institute of Natural Sciences of Japan; the National Key Research and Development Program of China (grants 2016YFA0400704 and 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-1555365, AST-1615796, AST-1715061, AST-1716327, AST-1903847, and AST-2034306); the Natural Science Foundation of China (grants 11573051, 11633006, 11650110427, 10625314, 11721303, 11725312, 11933007, 11991052, and 11991053); a fellowship of China Postdoctoral Science Foundation (2020M671266); the Natural Sciences and Engineering Research Council of Canada (NSERC; including a Discovery Grant and the NSERC Alexander Graham Bell Canada Graduate Scholarships-Doctoral Program); the National Youth Thousand Talents Program of China; the National Research Foundation of Korea (the Global PhD Fellowship Grant: grants NRF-2015H1A2A1033752 and 2015-R1D1A1A01056807; the Korea Research Fellowship Program: NRF-2015H1D3A1066561 and Basic Research Support Grant 2019R1F1A1059721); the Netherlands Organization for Scientific Research 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, which is a facility of the National Research Foundation, an agency of the Department of Science and Technology 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 Spanish Ministerio de Economía y Competitividad (grants PGC2018-098915-B-C21, AYA2016-80889-P, and PID2019-108995GB-C21); 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 Consejería de Economía, Conocimiento, Empresas y Universidad of the Junta de Andalucía (grant P18-FR-1769), the Consejo Superior de Investigaciones Científicas (grant 2019AEP112); 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 European Union's Horizon 2020 research and innovation program under grant agreement No. 730562 RadioNet; ALMA North America Development Fund; the Academia Sinica; Chandra DD7-18089X and 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 the National Institutes of Natural Sciences of Japan, together with National Research Council (Canada), MOST (Taiwan), the Academia Sinica Institute of Astronomy and Astrophysics (ASIAA; Taiwan), and the 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 OSO (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, CAS, 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 in 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 NSF 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. 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 the 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
### Approximate Degeneracy of the Radiative Transfer Solution

In this appendix, we present analytical arguments to show that the mm-wavelength images calculated from GRMHD simulations of accretion flows with parameters relevant to the M87 black hole are approximately invariant to the product $n_e^2 M_{BH}$, where $n_e$ is the electron-density scale and $M_{BH}$ is the mass of the black hole.

The radiative transfer equation at a frequency $\nu$ is given by

$$ \frac{dI_{\nu}}{ds} = \eta_{\nu} - \chi_{\nu} I_{\nu}, \qquad (A1) $$

where $I_\nu$ denotes the specific intensity at a frequency $\nu$, $s$ denotes the distance traveled along the path of a photon, $\eta_\nu$ denotes the emissivity of the medium, and $\chi_\nu$ denotes the opacity of the medium. Equation (A1) can be recast as

$$ \frac{dI_{\nu}}{ds} = \chi_{\nu} \left( \frac{\eta_{\nu}}{\chi_{\nu}} - I_{\nu} \right). \qquad (A2) $$
The fraction $\eta_{\nu}/\chi_{\nu}$ is the source function $S_{\nu}$ and, for thermal emission, is equal to the blackbody function, which depends solely on temperature $T$. We denote this as $\eta_{\nu}/\chi_{\nu} \equiv \mathcal{B}_{\nu}(T)$.

Throughout this work, we use the analytic approximation devised by Leung et al. (2011) for synchrotron emissivity from a thermal distribution of relativistic electrons

$$ \eta_{\nu} = n_e \frac{\sqrt{2} \pi e^2 \nu_s}{3K_2(1/\Theta_e)c} (X^{1/2} + 2^{11/12}X^{1/6})^2 \exp(-X^{1/3}), \quad (A3) $$

where $X \equiv \nu/\nu_s$, $\nu_s \equiv (2/9)\nu_c\Theta_e^2 \sin\theta$, $\nu_c \equiv eB/(2\pi m_e c)$, $\Theta_e \equiv k_b T_e/(m_e c^2)$ is the dimensionless electron temperature, and $K_2$ denotes the modified Bessel function of the second kind. In these expressions, $\theta$ denotes the angle between the magnetic field and the emitted photon, $B$ denotes the magnetic field strength, and $T_e$ denotes the electron temperature. We use the constants $m_e$, $e$, $k_b$, and $c$ to denote the electron mass, electron charge, the Boltzmann's constant, and the speed of light respectively. It is explicit in this expression that the synchrotron emissivity and opacity, for a given electron temperature and magnetic field, scale proportionally to the electron number density $n_e$. We show here that, for the parameters of the black hole at the center of the M87 galaxy, the synchrotron opacity and emissivity at a wavelength of $\lambda = 1.3$ mm also scale proportionally to a power of the magnetic field, i.e., $\propto B^{\alpha}$, with $\alpha \sim 2$.

We estimate the properties of the plasma in the inner accretion flow of M87 using the following analytic model. The electron density at a given radius $r$ is given by the continuity equation, assuming a spherical accretion rate $\dot{M}$ via the relation

$$ \dot{M} = 4\pi r^2 \left(\frac{h}{r}\right) m_p n_e u^r, \quad (A4) $$

where $h/r$ is the scale height of the accretion flow, $m_p$ is the proton mass (assuming fully ionized hydrogen), and $u^r$ is the radial component of the accretion flow. We take the latter to be a fraction $\xi$ of the freefall velocity, i.e.,

$$ u^r = \xi \left( \frac{GM_{BH}}{r} \right)^{1/2}. \quad (A5) $$

We also express the accretion rate as $\dot{M} \equiv \dot{m}\dot{M}_E$, in terms of the Eddington accretion rate defined by

$$ \dot{M}_E \equiv \frac{L_E}{\epsilon c^2} = \frac{4\pi GM_{BH}m_p}{\epsilon c \sigma_T}, \quad (A6) $$

where $L_E$ is the Eddington critical luminosity, $\epsilon$ is the radiative efficiency of the flow, $\sigma_T$ is the Thomson cross section, and $\dot{m}$ is the mass accretion rate in units of the Eddington accretion rate. Under these conditions, the electron density is

$$ \begin{aligned} n_e &= \left( \frac{c^2}{GM_{BH}\sigma_T} \right) \dot{m} \epsilon^{-1} \xi^{-1} \left( \frac{h}{r} \right)^{-1} \left( \frac{rc^2}{GM_{BH}} \right)^{-3/2} \\ &= 7 \times 10^5 \left( \frac{M_{BH}}{6.5 \times 10^9 M_{\odot}} \right)^{-1} \left( \frac{\dot{m}}{2 \times 10^{-5}} \right) \left( \frac{rc^2}{5GM_{BH}} \right)^{-3/2} \mathrm{cm}^{-3}, \end{aligned} \quad (A7) $$

where in the last expression we have used typical values for the M87 black hole and set $\xi = \epsilon = 0.1$ and $h/r = 0.4$.

Because of the radiatively inefficient character of the accretion flow around the M87 black hole, the ion temperature $T_i$ is a fraction of the virial temperature

$$ T_v = \frac{GM_{BH}m_p}{3k_B r}. \quad (A8) $$

It is also customary to write the electron temperature as a fraction $1/R$ of the ion temperature, i.e.,

$$ T_e = \frac{1}{R} \left( \frac{GM_{BH}m_p}{3k_B r} \right) \left( \frac{T_i}{T_v} \right). \quad (A9) $$

Finally, we write the magnetic field in terms of the plasma $\beta$ parameter as

$$ n_e k_B (T_i + T_e) = \beta \frac{B^2}{8\pi} \quad (A10) $$

such that

$$ \begin{aligned} B &= 2c^2 \left( \frac{2\pi m_p}{3GM_{BH}\sigma_T} \right)^{1/2} \left[ \frac{\dot{m}(1+R)}{\epsilon\xi\beta R} \right]^{1/2} \\ & \times \left( \frac{T_i}{T_v} \right)^{1/2} \left( \frac{h}{r} \right)^{-1/2} \left( \frac{rc^2}{GM_{BH}} \right)^{-5/4} \\ &= 9 \left( \frac{M_{BH}}{6.5 \times 10^9 M_{\odot}} \right)^{-1/2} \left( \frac{\dot{m}}{2 \times 10^{-5}} \right)^{1/2} \left( \frac{rc^2}{5GM_{BH}} \right)^{-5/4} \mathrm{G}, \end{aligned} \quad (A11) $$

and we have set $T_i/T_v = 1/3$ and $\beta = 10$.

Figure 9 shows the ratio $\eta_{\nu}/n_e$ evaluated at the observed frequency of the EHT ($\nu = 230$ GHz, $\lambda = 1.33$ mm) and at a radial distance $r = 5GM_{BH}/c^2$, as a function of the strength of the magnetic field $B$, for the values of the various parameters used in the previous equations, i.e., for the conditions of the inner accretion flow around the black hole at the center of M87. This figure demonstrates that, over a broad range of magnetic field strengths (spanning more than an order of magnitude), the synchrotron emissivity scales approximately as $\eta_{\nu} \sim n_e B^{\alpha}$, with $\alpha \approx 2$. This allows us to write the emissivity as

$$ \eta_{\nu} \approx n_e B^{\alpha} \mathcal{B}_{\nu}(T) f_{\nu}(T, \mathbf{r}), \quad (A12) $$

and the opacity as

$$ \chi_{\nu} \approx n_e B^{\alpha} f_{\nu}(T, \mathbf{r}), \quad (A13) $$

where $f_{\nu}(T, \mathbf{r})$ captures the scaling of the opacity with temperature $T$ and the location in the accretion flow (see similar arguments in Chan et al. 2015).

In order to calculate the various simulated images, we used the plasma properties from long GRMHD simulations. Nonradiative GRMHD simulations are invariant to a rescaling of the density by a factor $M$, as long as the magnetic field is also rescaled by a factor of $M^{1/2}$ and the internal energy by a factor of $M$. In other words, $B \sim n_e^{1/2}$ and the Alfvén speed $B/(m_p n_e)^{1/2}$ are the quantity that remains invariant under rescaling. Combined with the approximate expressions (A12)–(A13) derived above, this implies that the synchrotron emissivity and opacity evaluated using the simulation outputs are invariant to rescaling, as long as the product $n_e B^{\alpha} \sim n_e^{1+\alpha/2}$ remains constant.

Finally, the integration of the transfer equation is performed on a coordinate system in which the distances are expressed in
<figure><img src="image_13.png" /><figcaption>**Figure 9.** The ratio of the thermal synchrotron emissivity $\eta_v$ to the electron density $n_e$, as a function of the strength of the magnetic field, evaluated at an observing frequency of the EHT, $\nu = 230$ GHz, and for parameters appropriate to the inner accretion flow in the black hole at the center of the M87 galaxy (see text). The vertical dashed line shows the inferred strength of the magnetic field near the black hole horizon. The dashed line shows the approximate $\sim B^2$ scaling.</figcaption></figure>

terms of the length scale set by the mass of the black hole, i.e.,

$$ds \equiv ds' \left( \frac{GM_{BH}}{c^2} \right). \qquad (A14)$$

Rewriting Equation (A2) using the scaling (A12)–(A14), we find

$$\frac{dI}{ds'} = n_e^{1+\alpha/2} f(T, \mathbf{r}) \frac{GM_{BH}}{c^2} [\mathcal{B}(T) - I] \qquad (A15)$$

$$= (n_e^{1+\alpha/2} M_{BH}) \frac{G}{c^2} f(T, \mathbf{r}) [\mathcal{B}(T) - I]. \qquad (A16)$$

We drop the $\nu$ in subscript to indicate quantities calculated at $\nu = 230$ GHz. From this last expression, it is apparent that the electron number density $n_e$ and the black hole mass $M_{BH}$ are degenerate quantities in the solution of the radiative transfer problem at wavelengths where the dominant source of opacity is due to synchrotron processes, with a degeneracy in the product $n_e^{1+\alpha/2} M_{BH}$ and with $\alpha \approx 2$.

## ORCID iDs

Kaushik Satapathy https://orcid.org/0000-0003-0433-3585

Lia Medeiros https://orcid.org/0000-0003-2342-6728

Sean T. Dougall https://orcid.org/0000-0002-3769-1314

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

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

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

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

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

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

Juan Carlos Algaba https://orcid.org/0000-0001-6993-1696

Richard Anantua https://orcid.org/0000-0003-3457-7660

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

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

Bradford A. Benson https://orcid.org/0000-0002-5108-6823

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

Raymond Blundell https://orcid.org/0000-0002-5929-5857

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

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

Hope Boyce https://orcid.org/0000-0002-6530-5783

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

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

Thomas Bronzwaer https://orcid.org/0000-0003-1151-3971

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

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

Koushik Chatterjee https://orcid.org/0000-0002-2825-3590

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

Thomas M. Crawford https://orcid.org/0000-0001-9000-5013

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

Alejandro Cruz-Osorio https://orcid.org/0000-0002-3945-6342

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

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

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

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

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

Joseph Farah https://orcid.org/0000-0003-4914-5625

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

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

H. Alyson Ford https://orcid.org/0000-0002-9797-0972

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

Per Friberg https://orcid.org/0000-0002-8010-8454

Antonio Fuentes https://orcid.org/0000-0002-8773-4933

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

Roberto García https://orcid.org/0000-0002-6584-7443

Boris Georgiev https://orcid.org/0000-0002-3586-6424

Ciriaco Goddi https://orcid.org/0000-0002-2542-7743

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

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

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

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

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

Kazuhiro Hada https://orcid.org/0000-0001-6906-772X

Daryl Haggard https://orcid.org/0000-0001-6803-2138

Ronald Hesper https://orcid.org/0000-0003-1918-6098
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

Shiro Ikeda https://orcid.org/0000-0002-2462-1448

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

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

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

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

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

Mansour Karami https://orcid.org/0000-0001-7387-9333

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

Garrett K. Keating https://orcid.org/0000-0002-3490-146X

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

Dong-Jin Kim https://orcid.org/0000-0002-7038-2118

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

Jongsoo Kim https://orcid.org/0000-0002-1229-0426

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

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

Jun Yi Koay https://orcid.org/0000-0002-7029-6658

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

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

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

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

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

Cheng-Yu Kuo https://orcid.org/0000-0001-6211-5581

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

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

Aviad Levis https://orcid.org/0000-0001-7307-632X

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

Rocco Lico https://orcid.org/0000-0001-7361-2460

Greg Lindahl https://orcid.org/0000-0002-6100-4772

Jun Liu (刘俊) https://orcid.org/0000-0002-7615-7499

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

Elisabetta Liuzzo https://orcid.org/0000-0003-0995-5201

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

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

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

Jirong Mao (毛基荣) https://orcid.org/0000-0002-7077-7195

Nicola Marchili https://orcid.org/0000-0002-5523-7588

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

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

Alan P. Marscher https://orcid.org/0000-0001-7396-3332

Iván Martí-Vidal https://orcid.org/0000-0003-3708-9611

Satoki Matsushita https://orcid.org/0000-0002-2127-7880

Karl M. Menten https://orcid.org/0000-0001-6459-0669

Izumi Mizuno https://orcid.org/0000-0002-7210-6264

Yosuke Mizuno https://orcid.org/0000-0002-8131-6730

James M. Moran https://orcid.org/0000-0002-3882-4414

Kotaro Moriyama https://orcid.org/0000-0003-1364-3761

Monika Moscibrodzka https://orcid.org/0000-0002-4661-6332

Cornelia Müller https://orcid.org/0000-0002-2739-2994

Alejandro Mus Mejías https://orcid.org/0000-0003-0329-6874

Gibwa Musoke https://orcid.org/0000-0003-1984-189X

Hiroshi Nagai https://orcid.org/0000-0003-0292-3645

Neil M. Nagar https://orcid.org/0000-0001-6920-662X

Masanori Nakamura https://orcid.org/0000-0001-6081-2420

Ramesh Narayan https://orcid.org/0000-0002-1919-2730

Iniyan Natarajan https://orcid.org/0000-0001-8242-4373

Joey Neilsen https://orcid.org/0000-0002-8247-786X

Roberto Neri https://orcid.org/0000-0001-7176-4046

Chunchong Ni https://orcid.org/0000-0003-1361-5699

Aristeidis Noutsos https://orcid.org/0000-0002-4151-3860

Michael A. Nowak https://orcid.org/0000-0001-6923-1315

Héctor Olivares https://orcid.org/0000-0001-6833-7580

Gisela N. Ortiz-León https://orcid.org/0000-0002-2863-676X

Daniel C. M. Palumbo https://orcid.org/0000-0002-7179-3816

Jongho Park https://orcid.org/0000-0001-6558-9053

Ue-Li Pen https://orcid.org/0000-0003-2155-9578

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

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

Oliver Porth https://orcid.org/0000-0002-4584-2557

Felix M. Pötzl https://orcid.org/0000-0002-6579-8311

Jorge A. Preciado-López https://orcid.org/0000-0002-4146-0113

Hung-Yi Pu https://orcid.org/0000-0001-9270-8812

Venkatessh Ramakrishnan https://orcid.org/0000-0002-9248-086X

Ramprasad Rao https://orcid.org/0000-0002-1407-7944

Mark G. Rawlings https://orcid.org/0000-0002-6529-202X

Luciano Rezzolla https://orcid.org/0000-0002-1330-7103

Bart Ripperda https://orcid.org/0000-0002-7301-3908

Freek Roelofs https://orcid.org/0000-0001-5461-3687

Eduardo Ros https://orcid.org/0000-0001-9503-4892

Mel Rose https://orcid.org/0000-0002-2016-8746

Alan L. Roy https://orcid.org/0000-0002-1931-0135

Chet Ruszczyk https://orcid.org/0000-0001-7278-9707

Kazi L. J. Rygl https://orcid.org/0000-0003-4146-9043

David Sánchez-Arguelles https://orcid.org/0000-0002-7344-9920

Mahito Sasada https://orcid.org/0000-0001-5946-9960

Tuomas Savolainen https://orcid.org/0000-0001-6214-1085

Lijing Shao https://orcid.org/0000-0002-1334-8853

Zhiqiang Shen (沈志强) https://orcid.org/0000-0003-3540-8746

Des Small https://orcid.org/0000-0003-3723-5404

Bong Won Sohn https://orcid.org/0000-0002-4148-8378

Jason SooHoo https://orcid.org/0000-0003-1938-0720

He Sun (孙赫) https://orcid.org/0000-0003-1526-6787

Fumie Tazaki https://orcid.org/0000-0003-0236-0600

Alexandra J. Tetarenko https://orcid.org/0000-0003-3906-4354

Paul Tiede https://orcid.org/0000-0003-3826-5648

Remo P. J. Tilanus https://orcid.org/0000-0002-6514-553X

Michael Titus https://orcid.org/0000-0002-3423-4505

Kenji Toma https://orcid.org/0000-0002-7114-6010
Pablo Torne Ⓓ https://orcid.org/0000-0001-8700-6058

Efthalia Traianou Ⓓ https://orcid.org/0000-0002-1209-6500

Sascha Trippe Ⓓ https://orcid.org/0000-0003-0465-1559

Ilse van Bemmel Ⓓ https://orcid.org/0000-0001-5473-2950

Huib Jan van Langevelde Ⓓ https://orcid.org/0000-0002-0230-5946

Daniel R. van Rossum Ⓓ https://orcid.org/0000-0001-7772-6131

Jan Wagner Ⓓ https://orcid.org/0000-0003-1105-6109

Derek Ward-Thompson Ⓓ https://orcid.org/0000-0003-1140-2761

John Wardle Ⓓ https://orcid.org/0000-0002-8960-2942

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

Norbert Wex Ⓓ https://orcid.org/0000-0003-4058-2837

Robert Wharton Ⓓ https://orcid.org/0000-0002-7416-5209

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

Doosoo Yoon Ⓓ https://orcid.org/0000-0001-8694-8166

André Young Ⓓ https://orcid.org/0000-0003-0000-2682

Ken Young Ⓓ https://orcid.org/0000-0002-3666-4920

Ziri Younsi Ⓓ https://orcid.org/0000-0001-9283-1191

Feng Yuan (袁峰) Ⓓ https://orcid.org/0000-0003-3564-6437

Ye-Fei Yuan (袁业飞) Ⓓ https://orcid.org/0000-0002-7330-4756

J. Anton Zensus Ⓓ https://orcid.org/0000-0001-7470-3321

Guang-Yao Zhao Ⓓ https://orcid.org/0000-0002-4417-1659

Shan-Shan Zhao Ⓓ https://orcid.org/0000-0002-9774-3606

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