First Sagittarius A^* Event Horizon Telescope Results. IV. Variability, morphology, and black hole mass

Variability, morphology, and black hole mass

0000-0002-9475-4254]Kazunori Akiyama
Massachusetts Institute of Technology Haystack Observatory, 99 Millstone Road, Westford, MA 01886, USA
National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
Black Hole Initiative at Harvard University, 20 Garden Street, Cambridge, MA 02138, USA

0000-0002-9371-1033]Antxon Alberdi
Instituto de Astrofísica de Andalucía-CSIC, Glorieta de la Astronomía s/n, E-18008 Granada, Spain

Walter Alef
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

0000-0001-6993-1696]Juan Carlos Algaba
Department of Physics, Faculty of Science, Universiti Malaya, 50603 Kuala Lumpur, Malaysia

0000-0003-3457-7660]Richard Anantua
Black Hole Initiative at Harvard University, 20 Garden Street, Cambridge, MA 02138, USA
Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA
Department of Physics & Astronomy, The University of Texas at San Antonio, One UTSA Circle, San Antonio, TX 78249, USA

0000-0001-6988-8763]Keiichi Asada
Institute 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.

0000-0002-2200-5393]Rebecca Azulay
Departament d'Astronomia i Astrofísica, Universitat de València, C. Dr. Moliner 50, E-46100 Burjassot, València, Spain
Observatori Astronòmic, Universitat de València, C. Catedrático José Beltrán 2, E-46980 Paterna, València, Spain
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

0000-0002-7722-8412]Uwe Bach
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

0000-0003-3090-3975]Anne-Kathrin Baczko
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

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

0000-0003-0476-6647]Mislav Baloković

Yale Center for Astronomy & Astrophysics, Yale University, 52 Hillhouse Avenue, New Haven, CT 06511, USA

0000-0002-9290-0764]John Barrett
Massachusetts Institute of Technology Haystack Observatory, 99 Millstone Road, Westford, MA 01886, USA

0000-0002-5518-2812]Michi Bauböck
Department of Physics, University of Illinois, 1110 West Green Street, Urbana, IL 61801, USA

0000-0002-5108-6823]Bradford A. Benson
Fermi National Accelerator Laboratory, MS209, P.O. Box 500, Batavia, IL 60510, USA
Department of Astronomy and Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA

Dan Bintley
East Asian Observatory, 660 N. A'ohoku Place, Hilo, HI 96720, USA
James Clerk Maxwell Telescope (JCMT), 660 N. A'ohoku Place, Hilo, HI 96720, USA

0000-0002-9030-642X]Lindy Blackburn
Black Hole Initiative at Harvard University, 20 Garden Street, Cambridge, MA 02138, USA
Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA

0000-0002-5929-5857]Raymond Blundell
Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA

0000-0003-0077-4367]Katherine L. Bouman

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

0000-0003-4056-9982]Geoffrey C. Bower
Institute of Astronomy and Astrophysics, Academia Sinica,
645 N. A'ohoku Place, Hilo, HI 96720, USA
Department of Physics and Astronomy, University of Hawaii at Manoa, 2505 Correa Road, Honolulu, HI 96822, USA

0000-0002-6530-5783]Hope Boyce
Department of Physics, McGill University, 3600 rue University, Montréal, QC H3A 2T8, Canada
McGill Space Institute, McGill University, 3550 rue University, Montréal, QC H3A 2A7, Canada

Michael Bremer
Institut de Radioastronomie Millimétrique (IRAM), 300 rue de la Piscine, F-38406 Saint Martin d'Hères, France

0000-0002-2322-0749]Christiaan D. Brinkerink
Department of Astrophysics, Institute for Mathematics, Astrophysics and Particle Physics (IMAPP), Radboud University, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands

0000-0002-2556-0894]Roger Brissenden
Black Hole Initiative at Harvard University, 20 Garden Street, Cambridge, MA 02138, USA
Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA

0000-0001-9240-6734]Silke Britzen
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

0000-0002-3351-760X]Avery E. Broderick
Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON, N2L 2Y5, Canada
Department of Physics and Astronomy, University of Waterloo, 200 University Avenue West, Waterloo, ON, N2L 3G1, Canada
Waterloo Centre for Astrophysics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

0000-0001-9151-6683]Dominique Broguiere
Institut de Radioastronomie Millimétrique (IRAM), 300 rue de la Piscine, F-38406 Saint Martin d'Hères, France

0000-0003-1151-3971]Thomas Bronzwaer
Department of Astrophysics, Institute for Mathematics, Astrophysics and Particle Physics (IMAPP), Radboud University, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands

0000-0001-6169-1894]Sandra Bustamante
Department of Astronomy, University of Massachusetts, 01003, Amherst, MA, USA

0000-0003-1157-4109]Do-Young Byun
Korea Astronomy and Space Science Institute, Daedeok-daero 776, Yuseong-gu, Daejeon 34055, Republic of Korea
University of Science and Technology, Gajeong-ro 217, Yuseong-gu, Daejeon 34113, Republic of Korea

0000-0002-2044-7665]John E. Carlstrom
Kavli Institute for Cosmological Physics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA
Department of Astronomy and Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA
Department of Physics, University of Chicago, 5720 South Ellis Avenue, Chicago, IL 60637, USA
Enrico Fermi Institute, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA

0000-0002-4767-9925]Chiara Ceccobello
Department of Space, Earth and Environment, Chalmers University of Technology, Onsala Space Observatory, SE-43992 Onsala, Sweden

0000-0003-2966-6220]Andrew Chael
Princeton Center for Theoretical Science, Jadwin Hall, Princeton University, Princeton, NJ 08544, USA
NASA Hubble Fellowship Program, Einstein Fellow

0000-0001-6337-6126]Chi-kwan Chan
Steward Observatory and Department of Astronomy, University of Arizona,
933 N. Cherry Ave., Tucson, AZ 85721, USA
Data Science Institute, University of Arizona, 1230 N. Cherry Ave., Tucson,
AZ 85721, USA
Program in Applied Mathematics, University of Arizona, 617 N. Santa Rita,
Tucson, AZ 85721

0000-0002-2825-3590]Koushik Chatterjee

Black Hole Initiative at Harvard University, 20 Garden Street, Cambridge, MA 02138, USA
Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA

0000-0002-2878-1502]Shami Chatterjee
Cornell Center for Astrophysics and Planetary Science, Cornell University, Ithaca, NY 14853, USA

0000-0001-6573-3318]Ming-Tang Chen
Institute of Astronomy and Astrophysics, Academia Sinica, 645 N. A'ohoku Place, Hilo, HI 96720, USA

0000-0001-5650-6770]Yongjun Chen (陈永军)
Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, People's Republic of China
Key Laboratory of Radio Astronomy, Chinese Academy of Sciences, Nanjing 210008, People's Republic of China

0000-0003-4407-9868]Xiaopeng Cheng
Korea Astronomy and Space Science Institute, Daedeok-daero 776, Yuseong-gu, Daejeon 34055, Republic of Korea

0000-0001-6083-7521]Ilje Cho
Instituto de Astrofísica de Andalucía-CSIC, Glorieta de la Astronomía s/n, E-18008 Granada, Spain

0000-0001-6820-9941]Pierre Christian
Physics Department, Fairfield University, 1073 North Benson Road, Fairfield, CT 06824, USA

0000-0003-2886-2377]Nicholas S. Conroy
Department of Astronomy, University of Illinois at Urbana-Champaign, 1002 West Green Street, Urbana, IL 61801, USA
Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA

0000-0003-2448-9181]John E. Conway
Department of Space, Earth and Environment, Chalmers University of Technology, Onsala Space Observatory, SE-43992 Onsala, Sweden

0000-0002-4049-1882]James M. Cordes
Cornell Center for Astrophysics and Planetary Science, Cornell University, Ithaca, NY 14853, USA

0000-0001-9000-5013]Thomas M. Crawford
Department of Astronomy and Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA
Kavli Institute for Cosmological Physics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA

0000-0002-2079-3189]Geoffrey B. Crew
Massachusetts Institute of Technology Haystack Observatory, 99 Millstone Road, Westford, MA 01886, USA

0000-0002-3945-6342]Alejandro Cruz-Osorio
Institut für Theoretische Physik, Goethe-Universität Frankfurt, Max-von-Laue-Straße 1, D-60438 Frankfurt am Main, Germany

0000-0001-6311-4345]Yuzhu Cui (崔玉竹)
Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shengrong Road 520, Shanghai, 201210, People’s Republic of China
Mizusawa VLBI Observatory, National Astronomical Observatory of Japan, 2-12 Hoshigaoka, Mizusawa, Oshu, Iwate 023-0861, Japan
Department of Astronomical Science, The Graduate University for Advanced Studies (SOKENDAI), 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan

0000-0002-2685-2434]Jordy Davelaar
Department of Astronomy and Columbia Astrophysics Laboratory, Columbia University, 550 W 120th Street, New York, NY 10027, USA
Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA
Department of Astrophysics, Institute for Mathematics, Astrophysics and Particle Physics (IMAPP), Radboud University, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands

0000-0002-9945-682X]Mariafelicia De Laurentis
Dipartimento di Fisica “E. Pancini”, Universitá di Napoli “Federico II”, Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy
Institut für Theoretische Physik, Goethe-Universität Frankfurt, Max-von-Laue-Straße 1, D-60438 Frankfurt am Main, Germany
INFN Sez. di Napoli, Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy

0000-0003-1027-5043]Roger Deane
Wits Centre for Astrophysics, University of the Witwatersrand, 1 Jan Smuts Avenue, Braamfontein, Johannesburg 2050, South Africa
Department of Physics, University of Pretoria, Hatfield, Pretoria 0028, South Africa
Centre for Radio Astronomy Techniques and Technologies, Department of Physics and Electronics, Rhodes University, Makhanda 6140, South Africa

0000-0003-1269-9667]Jessica Dempsey
East Asian Observatory, 660 N. A'ohoku Place, Hilo, HI 96720, USA
James Clerk Maxwell Telescope (JCMT), 660 N. A'ohoku Place, Hilo, HI 96720, USA
ASTRON, Oude Hoogeveensedijk 4, 7991 PD Dwingeloo, The Netherlands

0000-0003-3922-4055]Gregory Desvignes
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany
LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Université de Paris, 5 place Jules Janssen, 92195 Meudon, France

0000-0003-3903-0373]Jason Dexter
JILA and Department of Astrophysical and Planetary Sciences, University of Colorado, Boulder, CO 80309, USA

0000-0001-6765-877X]Vedant Dhruv
Department of Physics, University of Illinois, 1110 West Green Street, Urbana, IL 61801, USA

0000-0002-9031-0904]Sheperd S. Doeleman
Black Hole Initiative at Harvard University, 20 Garden Street, Cambridge, MA 02138, USA
Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA

0000-0002-3769-1314]Sean Dougal
Steward Observatory and Department of Astronomy, University of Arizona, 933 N. Cherry Ave., Tucson, AZ 85721, USA

0000-0001-6010-6200]Sergio A. Dzib
Institut de Radioastronomie Millimétrique (IRAM), 300 rue de la Piscine, F-38406 Saint Martin d'Hères, France
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

0000-0001-6196-4135]Ralph P. Eatough
National Astronomical Observatories, Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing 100101, PR China
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

0000-0002-2791-5011]Razieh Emami
Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA

0000-0002-2526-6724]Heino Falcke
Department of Astrophysics, Institute for Mathematics, Astrophysics and Particle Physics (IMAPP), Radboud University, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands

0000-0003-4914-5625]Joseph Farah
Las Cumbres Observatory, 6740 Cortona Drive, Suite 102, Goleta, CA 93117-5575, USA
Department of Physics, University of California, Santa Barbara, CA 93106-9530, USA

0000-0002-7128-9345]Vincent L. Fish
Massachusetts Institute of Technology Haystack Observatory, 99 Millstone Road, Westford, MA 01886, USA

0000-0002-9036-2747]Ed Fomalont
National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville,
VA 22903, USA

0000-0002-9797-0972]H. Alyson Ford
Steward Observatory and Department of Astronomy, University of Arizona, 933 N. Cherry Ave., Tucson, AZ 85721, USA

0000-0002-5222-1361]Raquel Fraga-Encinas
Department of Astrophysics, Institute for Mathematics, Astrophysics and Particle Physics (IMAPP), Radboud University, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands

William T. Freeman
Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, 32-D476, 77 Massachusetts Ave., Cambridge, MA 02142, USA
Google Research, 355 Main St., Cambridge, MA 02142, USA

0000-0002-8010-8454]Per Friberg
East Asian Observatory, 660 N. A'ohoku Place, Hilo, HI 96720, USA
James Clerk Maxwell Telescope (JCMT), 660 N. A'ohoku Place, Hilo, HI 96720, USA

0000-0002-1827-1656]Christian M. Fromm
Institut für Theoretische Physik und Astrophysik, Universität Würzburg, Emil-Fischer-Str. 31,
97074 Würzburg, Germany
Institut für Theoretische Physik, Goethe-Universität Frankfurt, Max-von-Laue-Straße 1, D-60438 Frankfurt am Main, Germany
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

0000-0002-8773-4933]Antonio Fuentes
Instituto de Astrofísica de Andalucía-CSIC, Glorieta de la Astronomía s/n, E-18008 Granada, Spain

0000-0002-6429-3872]Peter Galison
Black Hole Initiative at Harvard University, 20 Garden Street, Cambridge, MA 02138, USA
Department of History of Science, Harvard University, Cambridge, MA 02138, USA
Department of Physics, Harvard University, Cambridge, MA 02138, USA

0000-0001-7451-8935]Charles F. Gammie
Department of Physics, University of Illinois, 1110 West Green Street, Urbana, IL 61801, USA
Department of Astronomy, University of Illinois at Urbana-Champaign, 1002 West Green Street, Urbana, IL 61801, USA
NCSA, University of Illinois, 1205 W Clark St, Urbana, IL 61801, USA

0000-0002-6584-7443]Roberto García
Institut de Radioastronomie Millimétrique (IRAM), 300 rue de la Piscine, F-38406 Saint Martin d'Hères, France

0000-0002-0115-4605]Olivier Gentaz
Institut de Radioastronomie Millimétrique (IRAM), 300 rue de la Piscine, F-38406 Saint Martin d'Hères, France

0000-0002-3586-6424]Boris Georgiev
Department of Physics and Astronomy, University of Waterloo, 200 University Avenue West, Waterloo, ON, N2L 3G1, Canada
Waterloo Centre for Astrophysics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada
Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON, N2L 2Y5, Canada

0000-0002-2542-7743]Ciriaco Goddi
Dipartimento di Fisica, Università degli Studi di Cagliari, SP Monserrato-Sestu km 0.7, I-09042 Monserrato, Italy

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

0000-0003-2492-1966]Roman Gold
CP3-Origins, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark
Institut für Theoretische Physik, Goethe-Universität Frankfurt, Max-von-Laue-Straße 1, D-60438 Frankfurt am Main, Germany

0000-0001-9395-1670]Arturo I. Gómez-Ruiz
Instituto Nacional de Astrofísica, Óptica y Electrónica. Apartado Postal 51 y 216, 72000. Puebla Pue., México
Consejo Nacional de Ciencia y Tecnologìa, Av. Insurgentes Sur 1582, 03940, Ciudad de México, México

0000-0003-4190-7613]José L. Gómez
Instituto de Astrofísica de Andalucía-CSIC, Glorieta de la Astronomía s/n, E-18008 Granada, Spain

0000-0002-4455-6946]Minfeng Gu (顾敏峰)
Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, People's Republic of China
Key Laboratory for Research in Galaxies and Cosmology, Chinese Academy of Sciences, Shanghai 200030, People's Republic of China

0000-0003-0685-3621]Mark Gurwell
Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA

0000-0001-6906-772X]Kazuhiro Hada
Mizusawa VLBI Observatory, National Astronomical Observatory of Japan, 2-12 Hoshigaoka, Mizusawa, Oshu, Iwate 023-0861, Japan
Department of Astronomical Science, The Graduate University for Advanced Studies (SOKENDAI), 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan

0000-0001-6803-2138]Daryl Haggard
Department of Physics, McGill University, 3600 rue University, Montréal, QC H3A 2T8, Canada
McGill Space Institute, McGill University, 3550 rue University, Montréal, QC H3A 2A7, Canada

Kari Haworth
Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA

0000-0002-4114-4583]Michael H. Hecht
Massachusetts Institute of Technology Haystack Observatory, 99 Millstone Road, Westford, MA 01886, USA

0000-0003-1918-6098]Ronald Hesper
NOVA Sub-mm Instrumentation Group, Kapteyn Astronomical Institute, University of Groningen, Landleven 12, 9747 AD Groningen, The Netherlands

0000-0002-7671-0047]Dirk Heumann
Steward Observatory and Department of Astronomy, University of Arizona, 933 N. Cherry Ave., Tucson, AZ 85721, USA

0000-0001-6947-5846]Luis C. Ho (何子山)
Department of Astronomy, School of Physics, Peking University, Beijing 100871, People's Republic of China
Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, People's Republic of China

0000-0002-3412-4306]Paul Ho
Institute 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.
James Clerk Maxwell Telescope (JCMT), 660 N. A'ohoku Place, Hilo, HI 96720, USA
East Asian Observatory, 660 N. A'ohoku Place, Hilo, HI 96720, USA

0000-0003-4058-9000]Mareki Honma
Mizusawa VLBI Observatory, National Astronomical Observatory of Japan, 2-12 Hoshigaoka, Mizusawa, Oshu, Iwate 023-0861, Japan
Department of Astronomical Science, The Graduate University for Advanced Studies (SOKENDAI), 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
Department of Astronomy, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

0000-0001-5641-3953]Chih-Wei L. Huang
Institute 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.

0000-0002-1923-227X]Lei Huang (黄磊)
Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, People's Republic of China
Key Laboratory for Research in Galaxies and Cosmology, Chinese Academy of Sciences, Shanghai 200030, People's Republic of China

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

0000-0002-2462-1448]Shiro Ikeda
National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
The Institute of Statistical Mathematics, 10-3 Midori-cho, Tachikawa, Tokyo, 190-8562, Japan
Department of Statistical Science, The Graduate University for Advanced Studies (SOKENDAI), 10-3 Midori-cho, Tachikawa, Tokyo 190-8562, Japan
Kavli Institute for the Physics and Mathematics of the Universe, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan

0000-0002-3443-2472]C. M. Violette Impellizzeri
Leiden Observatory, Leiden University, Postbus 2300, 9513 RA Leiden, The Netherlands
National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville,
VA 22903, USA

0000-0001-5037-3989]Makoto Inoue
Institute 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.

0000-0002-5297-921X]Sara Issaoun
Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA
NASA Hubble Fellowship Program, Einstein Fellow

0000-0001-5160-4486]David J. James
ASTRAVEO LLC, PO Box 1668, Gloucester, MA 01931

0000-0002-1578-6582]Buell T. Jannuzi
Steward Observatory and Department of Astronomy, University of Arizona, 933 N. Cherry Ave., Tucson, AZ 85721, USA

0000-0001-8685-6544]Michael Janssen
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

0000-0003-2847-1712]Britton Jeter
Institute 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.

0000-0001-7369-3539]Wu Jiang (江悟)
Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, People's Republic of China

0000-0002-2662-3754]Alejandra Jiménez-Rosales
Department of Astrophysics, Institute for Mathematics, Astrophysics and Particle Physics (IMAPP), Radboud University, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands

0000-0002-4120-3029]Michael D. Johnson
Black Hole Initiative at Harvard University, 20 Garden Street, Cambridge, MA 02138, USA
Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA

0000-0001-6158-1708]Svetlana Jorstad
Institute for Astrophysical Research, Boston University, 725 Commonwealth Ave., Boston, MA 02215, USA

0000-0002-2514-5965]Abhishek V. Joshi
Department of Physics, University of Illinois, 1110 West Green Street, Urbana, IL 61801, USA

0000-0001-7003-8643]Taehyun Jung
Korea Astronomy and Space Science Institute, Daedeok-daero 776, Yuseong-gu, Daejeon 34055, Republic of Korea
University of Science and Technology, Gajeong-ro 217, Yuseong-gu, Daejeon 34113, Republic of Korea

0000-0001-7387-9333]Mansour Karami
Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON, N2L 2Y5, Canada
Department of Physics and Astronomy, University of Waterloo, 200 University Avenue West, Waterloo, ON, N2L 3G1, Canada

0000-0002-5307-2919]Ramesh Karuppusamy
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

0000-0001-8527-0496]Tomohisa Kawashima
Institute for Cosmic Ray Research, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582, Japan

0000-0002-3490-146X]Garrett K. Keating
Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA

0000-0002-6156-5617]Mark Kettenis
Joint Institute for VLBI ERIC (JIVE), Oude Hoogeveensedijk 4, 7991 PD Dwingeloo, The Netherlands

0000-0002-7038-2118]Dong-Jin Kim
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

0000-0001-8229-7183]Jae-Young Kim
Department of Astronomy and Atmospheric Sciences, Kyungpook National University,
Daegu 702-701, Republic of Korea
Korea Astronomy and Space Science Institute, Daedeok-daero 776, Yuseong-gu, Daejeon 34055, Republic of Korea
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

0000-0002-1229-0426]Jongsoo Kim
Korea Astronomy and Space Science Institute, Daedeok-daero 776, Yuseong-gu, Daejeon 34055, Republic of Korea

0000-0002-4274-9373]Junhan Kim
Steward Observatory and Department of Astronomy, University of Arizona, 933 N. Cherry Ave., Tucson, AZ 85721, USA
California Institute of Technology, 1200 East California Boulevard, Pasadena, CA 91125, USA

0000-0002-2709-7338]Motoki Kino
National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
Kogakuin University of Technology & Engineering, Academic Support Center, 2665-1 Nakano, Hachioji, Tokyo 192-0015, Japan

0000-0002-7029-6658]Jun Yi Koay
Institute 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.

0000-0001-7386-7439]Prashant Kocherlakota
Institut für Theoretische Physik, Goethe-Universität Frankfurt, Max-von-Laue-Straße 1, D-60438 Frankfurt am Main, Germany

Yutaro Kofuji
Mizusawa VLBI Observatory, National Astronomical Observatory of Japan, 2-12 Hoshigaoka, Mizusawa, Oshu, Iwate 023-0861, Japan
Department of Astronomy, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

0000-0003-2777-5861]Patrick M. Koch
Institute 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.

0000-0002-3723-3372]Shoko Koyama
Niigata University, 8050 Ikarashi-nino-cho, Nishi-ku, Niigata 950-2181, Japan
Institute 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.

0000-0002-4908-4925]Carsten Kramer
Institut de Radioastronomie Millimétrique (IRAM), 300 rue de la Piscine, F-38406 Saint Martin d'Hères, France

0000-0002-4175-2271]Michael Kramer
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

0000-0002-4892-9586]Thomas P. Krichbaum
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

0000-0001-6211-5581]Cheng-Yu Kuo
Physics Department, National Sun Yat-Sen University, No. 70, Lien-Hai Road, Kaosiung City 80424, Taiwan, R.O.C.
Institute 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.

0000-0002-8116-9427]Noemi La Bella
Department of Astrophysics, Institute for Mathematics, Astrophysics and Particle Physics (IMAPP), Radboud University, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands

0000-0003-3234-7247]Tod R. Lauer
National Optical Astronomy Observatory, 950 N. Cherry Ave., Tucson, AZ 85719, USA

0000-0002-3350-5588]Daeyoung Lee
Department of Physics, University of Illinois, 1110 West Green Street, Urbana, IL 61801, USA

0000-0002-6269-594X]Sang-Sung Lee
Korea Astronomy and Space Science Institute, Daedeok-daero 776, Yuseong-gu, Daejeon 34055, Republic of Korea

0000-0002-8802-8256]Po Kin Leung
Department of Physics, The Chinese University of Hong Kong, Shatin, N. T., Hong Kong

0000-0001-7307-632X]Aviad Levis
California Institute of Technology, 1200 East California Boulevard, Pasadena, CA 91125, USA

0000-0003-0355-6437]Zhiyuan Li (李志远)
School of Astronomy and Space Science, Nanjing University, Nanjing 210023, People's Republic of China
Key Laboratory of Modern Astronomy and Astrophysics, Nanjing University, Nanjing 210023, People's Republic of China

0000-0001-7361-2460]Rocco Lico
Instituto de Astrofísica de Andalucía-CSIC, Glorieta de la Astronomía s/n, E-18008 Granada, Spain

INAF-Istituto di Radioastronomia, Via P. Gobetti 101, I-40129 Bologna, Italy

0000-0002-6100-4772]Greg Lindahl
Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA

0000-0002-3669-0715]Michael Lindqvist
Department of Space, Earth and Environment, Chalmers University of Technology, Onsala Space Observatory, SE-43992 Onsala, Sweden

0000-0001-6088-3819]Mikhail Lisakov
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

0000-0002-7615-7499]Jun Liu (刘俊)
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

0000-0002-2953-7376]Kuo Liu
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

0000-0003-0995-5201]Elisabetta Liuzzo
INAF-Istituto di Radioastronomia & Italian ALMA Regional Centre, Via P. Gobetti 101, I-40129 Bologna, Italy

0000-0003-1869-2503]Wen-Ping Lo
Institute 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.
Department of Physics, National Taiwan University, No.1, Sect.4, Roosevelt Rd., Taipei 10617, Taiwan, R.O.C

0000-0003-1622-1484]Andrei P. Lobanov
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

0000-0002-5635-3345]Laurent Loinard
Instituto de Radioastronomía y Astrofísica, Universidad Nacional Autónoma de México, Morelia 58089, México
Instituto de Astronomía, Universidad Nacional Autónoma de México (UNAM), Apdo Postal 70-264, Ciudad de México, México

0000-0003-4062-4654]Colin J. Lonsdale
Massachusetts Institute of Technology Haystack Observatory, 99 Millstone Road, Westford, MA 01886, USA

0000-0002-7692-7967]Ru-Sen Lu (路如森)
Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, People's Republic of China
Key Laboratory of Radio Astronomy, Chinese Academy of Sciences, Nanjing 210008, People's Republic of China
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

0000-0002-7077-7195]Jirong Mao (毛基荣)

Yunnan Observatories, Chinese Academy of Sciences, 650011 Kunming, Yunnan Province, People's Republic of China
Center for Astronomical Mega-Science, Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing, 100012, People's Republic of China
Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese Academy of Sciences, 650011 Kunming, People's Republic of China

0000-0002-5523-7588]Nicola Marchili
INAF-Istituto di Radioastronomia & Italian ALMA Regional Centre, Via P. Gobetti 101, I-40129 Bologna, Italy
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

0000-0001-9564-0876]Sera Markoff
Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904, 1098 XH, Amsterdam, The Netherlands
Gravitation and Astroparticle Physics Amsterdam (GRAPPA) Institute, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands

0000-0002-2367-1080]Daniel P. Marrone
Steward Observatory and Department of Astronomy, University of Arizona, 933 N. Cherry Ave., Tucson, AZ 85721, USA

0000-0001-7396-3332]Alan P. Marscher
Institute for Astrophysical Research, Boston University, 725 Commonwealth Ave., Boston, MA 02215, USA

0000-0003-3708-9611]Iván Martí-Vidal
Departament d'Astronomia i Astrofísica, Universitat de València, C. Dr. Moliner 50, E-46100 Burjassot, València, Spain
Observatori Astronòmic, Universitat de València, C. Catedrático José Beltrán 2, E-46980 Paterna, València, Spain

0000-0002-2127-7880]Satoki Matsushita
Institute 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.

0000-0002-3728-8082]Lynn D. Matthews
Massachusetts Institute of Technology Haystack Observatory, 99 Millstone Road, Westford, MA 01886, USA

0000-0003-2342-6728]Lia Medeiros
NSF Astronomy and Astrophysics Postdoctoral Fellow
School of Natural Sciences, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA
Steward Observatory and Department of Astronomy, University of Arizona, 933 N. Cherry Ave., Tucson, AZ 85721, USA

0000-0001-6459-0669]Karl M. Menten
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

0000-0002-7618-6556]Daniel Michalik
Science Support Office, Directorate of Science, European Space Research and Technology Centre (ESA/ESTEC), Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands
Department of Astronomy and Astrophysics, University of Chicago,
5640 South Ellis Avenue, Chicago, IL 60637, USA

0000-0002-7210-6264]Izumi Mizuno
East Asian Observatory, 660 N. A'ohoku Place, Hilo, HI 96720, USA
James Clerk Maxwell Telescope (JCMT), 660 N. A'ohoku Place, Hilo, HI 96720, USA

0000-0002-8131-6730]Yosuke Mizuno
Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shengrong Road 520, Shanghai, 201210, People’s Republic of China
School of Physics and Astronomy, Shanghai Jiao Tong University,
800 Dongchuan Road, Shanghai, 200240, People’s Republic of China

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

0000-0002-3882-4414]James M. Moran
Black Hole Initiative at Harvard University, 20 Garden Street, Cambridge, MA 02138, USA
Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA

0000-0003-1364-3761]Kotaro Moriyama
Institut für Theoretische Physik, Goethe-Universität Frankfurt, Max-von-Laue-Straße 1, D-60438 Frankfurt am Main, Germany
Massachusetts Institute of Technology Haystack Observatory, 99 Millstone Road, Westford, MA 01886, USA
Mizusawa VLBI Observatory, National Astronomical Observatory of Japan, 2-12 Hoshigaoka, Mizusawa, Oshu, Iwate 023-0861, Japan

0000-0002-4661-6332]Monika Moscibrodzka
Department of Astrophysics, Institute for Mathematics, Astrophysics and Particle Physics (IMAPP), Radboud University, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands

0000-0002-2739-2994]Cornelia Müller
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany
Department of Astrophysics, Institute for Mathematics, Astrophysics and Particle Physics (IMAPP), Radboud University, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands

0000-0003-0329-6874]Alejandro Mus
Departament d'Astronomia i Astrofísica, Universitat de València, C. Dr. Moliner 50, E-46100 Burjassot, València, Spain
Observatori Astronòmic, Universitat de València, C. Catedrático José Beltrán 2, E-46980 Paterna, València, Spain

0000-0003-1984-189X]Gibwa Musoke
Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904, 1098 XH, Amsterdam, The Netherlands
Department of Astrophysics, Institute for Mathematics, Astrophysics and Particle Physics (IMAPP), Radboud University, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands

0000-0003-3025-9497]Ioannis Myserlis
Institut de Radioastronomie Millimétrique (IRAM), Avenida Divina Pastora 7, Local 20, E-18012, Granada, Spain

0000-0001-9479-9957]Andrew Nadolski
Department of Astronomy, University of Illinois at Urbana-Champaign, 1002 West Green Street, Urbana, IL 61801, USA

0000-0003-0292-3645]Hiroshi Nagai
National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
Department of Astronomical Science, The Graduate University for Advanced Studies (SOKENDAI), 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan

0000-0001-6920-662X]Neil M. Nagar
Astronomy Department, Universidad de Concepción, Casilla 160-C, Concepción, Chile

0000-0001-6081-2420]Masanori Nakamura
National Institute of Technology, Hachinohe College, 16-1 Uwanotai, Tamonoki, Hachinohe City, Aomori 039-1192, Japan
Institute 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.

0000-0002-1919-2730]Ramesh Narayan
Black Hole Initiative at Harvard University, 20 Garden Street, Cambridge, MA 02138, USA
Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA

0000-0002-4723-6569]Gopal Narayanan
Department of Astronomy, University of Massachusetts, 01003, Amherst, MA, USA

0000-0001-8242-4373]Iniyan Natarajan
Wits Centre for Astrophysics, University of the Witwatersrand,
1 Jan Smuts Avenue, Braamfontein, Johannesburg 2050, South Africa
South African Radio Astronomy Observatory, Observatory 7925, Cape Town, South Africa

Antonios Nathanail
Institut für Theoretische Physik, Goethe-Universität Frankfurt, Max-von-Laue-Straße 1, D-60438 Frankfurt am Main, Germany
Department of Physics, National and Kapodistrian University of Athens, Panepistimiopolis, GR 15783 Zografos, Greece

Santiago Navarro Fuentes
Institut de Radioastronomie Millimétrique (IRAM), Avenida Divina Pastora 7, Local 20, E-18012, Granada, Spain

0000-0002-8247-786X]Joey Neilsen
Department of Physics, Villanova University, 800 Lancaster Avenue, Villanova, PA 19085, USA

0000-0002-7176-4046]Roberto Neri
Institut de Radioastronomie Millimétrique (IRAM), 300 rue de la Piscine, F-38406 Saint Martin d'Hères, France

0000-0003-1361-5699]Chunchong Ni
Department of Physics and Astronomy, University of Waterloo, 200 University Avenue West, Waterloo, ON, N2L 3G1, Canada
Waterloo Centre for Astrophysics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada
Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON, N2L 2Y5, Canada

0000-0002-4151-3860]Aristeidis Noutsos
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

0000-0001-6923-1315]Michael A. Nowak
Physics Department, Washington University CB 1105, St Louis, MO 63130, USA

0000-0002-4991-9638]Junghwan Oh
Sejong University, 209 Neungdong-ro, Gwangjin-gu, Seoul, Republic of Korea

0000-0003-3779-2016]Hiroki Okino
Mizusawa VLBI Observatory, National Astronomical Observatory of Japan, 2-12 Hoshigaoka, Mizusawa, Oshu, Iwate 023-0861, Japan
Department of Astronomy, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

0000-0001-6833-7580]Héctor Olivares
Department of Astrophysics, Institute for Mathematics, Astrophysics and Particle Physics (IMAPP), Radboud University, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands

0000-0002-2863-676X]Gisela N. Ortiz-León

Instituto de Astronomía, Universidad Nacional Autónoma de México (UNAM), Apdo Postal 70-264, Ciudad de México, México
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

0000-0003-4046-2923]Tomoaki Oyama
Mizusawa VLBI Observatory, National Astronomical Observatory of Japan, 2-12 Hoshigaoka, Mizusawa, Oshu, Iwate 023-0861, Japan

0000-0002-7179-3816]Daniel C. M. Palumbo
Black Hole Initiative at Harvard University, 20 Garden Street, Cambridge, MA 02138, USA
Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA

0000-0001-6757-3098]Georgios Filippos Paraschos
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

0000-0001-6558-9053]Jongho Park
Institute 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.
EACOA Fellow

0000-0002-6327-3423]Harriet Parsons
East Asian Observatory, 660 N. A'ohoku Place, Hilo, HI 96720, USA
James Clerk Maxwell Telescope (JCMT), 660 N. A'ohoku Place, Hilo, HI 96720, USA

0000-0002-6021-9421]Nimesh Patel
Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA

0000-0003-2155-9578]Ue-Li Pen
Institute 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.
Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON, N2L 2Y5, Canada
Canadian Institute for Theoretical Astrophysics, University of Toronto, 60 St. George Street, Toronto, ON, M5S 3H8, Canada
Dunlap Institute for Astronomy and Astrophysics, University of Toronto, 50 St. George Street, Toronto, ON, M5S 3H4, Canada
Canadian Institute for Advanced Research, 180 Dundas St West, Toronto, ON, M5G 1Z8, Canada

0000-0002-5278-9221]Dominic W. Pesce
Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA
Black Hole Initiative at Harvard University, 20 Garden Street, Cambridge, MA 02138, USA

Vincent Piétu
Institut de Radioastronomie Millimétrique (IRAM), 300 rue de la Piscine, F-38406 Saint Martin d'Hères, France

0000-0001-6765-9609]Richard Plambeck
Radio Astronomy Laboratory, University of California, Berkeley, CA 94720, USA

Aleksandar PopStefanija
Department of Astronomy, University of Massachusetts, 01003, Amherst, MA, USA

0000-0002-4584-2557]Oliver Porth
Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904, 1098 XH, Amsterdam, The Netherlands
Institut für Theoretische Physik, Goethe-Universität Frankfurt, Max-von-Laue-Straße 1, D-60438 Frankfurt am Main, Germany

0000-0002-6579-8311]Felix M. Pötzl
Department of Physics, University College Cork, Kane Building, College Road, Cork T12 K8AF, Ireland
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

0000-0002-0393-7734]Ben Prather
Department of Physics, University of Illinois, 1110 West Green Street, Urbana, IL 61801, USA

0000-0002-4146-0113]Jorge A. Preciado-López
Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON, N2L 2Y5, Canada

0000-0001-9270-8812]Hung-Yi Pu
Department of Physics, National Taiwan Normal University, No. 88, Sec.4, Tingzhou Rd., Taipei 116, Taiwan, R.O.C.
Center of Astronomy and Gravitation, National Taiwan Normal University, No. 88, Sec. 4, Tingzhou Road, Taipei 116, Taiwan, R.O.C.
Institute 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.

0000-0002-9248-086X]Venkatessh Ramakrishnan
Astronomy Department, Universidad de Concepción, Casilla 160-C, Concepción, Chile
Finnish Centre for Astronomy with ESO, FI-20014 University of Turku, Finland
Aalto University Metsähovi Radio Observatory, Metsähovintie 114, FI-02540 Kylmälä, Finland

0000-0002-1407-7944]Ramprasad Rao
Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA

0000-0002-6529-202X]Mark G. Rawlings
Gemini Observatory/NSF NOIRLab, 670 N. A’ohōkū Place, Hilo, HI 96720, USA
East Asian Observatory, 660 N. A'ohoku Place, Hilo, HI 96720, USA
James Clerk Maxwell Telescope (JCMT), 660 N. A'ohoku Place, Hilo, HI 96720, USA

0000-0002-5779-4767]Alexander W. Raymond
Black Hole Initiative at Harvard University, 20 Garden Street, Cambridge, MA 02138, USA
Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA

0000-0002-1330-7103]Luciano Rezzolla
Institut für Theoretische Physik, Goethe-Universität Frankfurt, Max-von-Laue-Straße 1, D-60438 Frankfurt am Main, Germany
Frankfurt Institute for Advanced Studies, Ruth-Moufang-Strasse 1, 60438 Frankfurt, Germany
School of Mathematics, Trinity College, Dublin 2, Ireland

0000-0001-5287-0452]Angelo Ricarte
Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA
Black Hole Initiative at Harvard University, 20 Garden Street, Cambridge, MA 02138, USA

0000-0002-7301-3908]Bart Ripperda
Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ 08544, USA
Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA

0000-0001-5461-3687]Freek Roelofs
Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA
Black Hole Initiative at Harvard University, 20 Garden Street, Cambridge, MA 02138, USA
Department of Astrophysics, Institute for Mathematics, Astrophysics and Particle Physics (IMAPP), Radboud University, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands

0000-0003-1941-7458]Alan Rogers
Massachusetts Institute of Technology Haystack Observatory, 99 Millstone Road, Westford, MA 01886, USA

0000-0001-9503-4892]Eduardo Ros
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

0000-0001-6301-9073]Cristina Romero-Cañizales
Institute 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.

0000-0002-8280-9238]Arash Roshanineshat
Steward Observatory and Department of Astronomy, University of Arizona, 933 N. Cherry Ave., Tucson, AZ 85721, USA

Helge Rottmann
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

0000-0002-1931-0135]Alan L. Roy
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

0000-0002-0965-5463]Ignacio Ruiz
Institut de Radioastronomie Millimétrique (IRAM), Avenida Divina Pastora 7, Local 20, E-18012, Granada, Spain

0000-0001-7278-9707]Chet Ruszczyk
Massachusetts Institute of Technology Haystack Observatory, 99 Millstone Road, Westford, MA 01886, USA

0000-0003-4146-9043]Kazi L. J. Rygl
INAF-Istituto di Radioastronomia & Italian ALMA Regional Centre, Via P. Gobetti 101, I-40129 Bologna, Italy

0000-0002-8042-5951]Salvador Sánchez
Institut de Radioastronomie Millimétrique (IRAM), Avenida Divina Pastora 7, Local 20, E-18012, Granada, Spain

0000-0002-7344-9920]David Sánchez-Argüelles
Instituto Nacional de Astrofísica, Óptica y Electrónica. Apartado Postal 51 y 216, 72000. Puebla Pue., México
Consejo Nacional de Ciencia y Tecnologìa, Av. Insurgentes Sur 1582, 03940, Ciudad de México, México

0000-0003-0981-9664]Miguel Sánchez-Portal
Institut de Radioastronomie Millimétrique (IRAM), Avenida Divina Pastora 7, Local 20, E-18012, Granada, Spain

0000-0001-5946-9960]Mahito Sasada
Department of Physics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan
Mizusawa VLBI Observatory, National Astronomical Observatory of Japan, 2-12 Hoshigaoka, Mizusawa, Oshu, Iwate 023-0861, Japan
Hiroshima Astrophysical Science Center, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526, Japan

0000-0003-0433-3585]Kaushik Satapathy
Steward Observatory and Department of Astronomy, University of Arizona, 933 N. Cherry Ave., Tucson, AZ 85721, USA

0000-0001-6214-1085]Tuomas Savolainen
Aalto University Department of Electronics and Nanoengineering, PL 15500, FI-00076 Aalto, Finland
Aalto University Metsähovi Radio Observatory, Metsähovintie 114, FI-02540 Kylmälä, Finland
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

F. Peter Schloerb
Department of Astronomy, University of Massachusetts, 01003, Amherst, MA, USA

0000-0002-8909-2401]Jonathan Schonfeld
Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA

0000-0003-2890-9454]Karl-Friedrich Schuster
Institut de Radioastronomie Millimétrique (IRAM), 300 rue de la Piscine,
F-38406 Saint Martin d'Hères, France

0000-0002-1334-8853]Lijing Shao
Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, People's Republic of China
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

0000-0003-3540-8746]Zhiqiang Shen (沈志强)
Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, People's Republic of China
Key Laboratory of Radio Astronomy, Chinese Academy of Sciences, Nanjing 210008, People's Republic of China

0000-0003-3723-5404]Des Small
Joint Institute for VLBI ERIC (JIVE), Oude Hoogeveensedijk 4, 7991 PD Dwingeloo, The Netherlands

0000-0002-4148-8378]Bong Won Sohn
Korea Astronomy and Space Science Institute, Daedeok-daero 776, Yuseong-gu, Daejeon 34055, Republic of Korea
University of Science and Technology, Gajeong-ro 217, Yuseong-gu, Daejeon 34113, Republic of Korea
Department of Astronomy, Yonsei University, Yonsei-ro 50, Seodaemun-gu, 03722 Seoul, Republic of Korea

0000-0003-1938-0720]Jason SooHoo
Massachusetts Institute of Technology Haystack Observatory, 99 Millstone Road, Westford, MA 01886, USA

0000-0001-7915-5272]Kamal Souccar
Department of Astronomy, University of Massachusetts, 01003, Amherst, MA, USA

0000-0003-1526-6787]He Sun (孙赫)
California Institute of Technology, 1200 East California Boulevard, Pasadena, CA 91125, USA

0000-0003-0236-0600]Fumie Tazaki
Mizusawa VLBI Observatory, National Astronomical Observatory of Japan, 2-12 Hoshigaoka, Mizusawa, Oshu, Iwate 023-0861, Japan

0000-0003-3906-4354]Alexandra J. Tetarenko
Department of Physics and Astronomy, Texas Tech University, Lubbock, Texas 79409-1051, USA
NASA Hubble Fellowship Program, Einstein Fellow

0000-0003-3826-5648]Paul Tiede
Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA
Black Hole Initiative at Harvard University, 20 Garden Street, Cambridge, MA 02138, USA

0000-0002-6514-553X]Remo P. J. Tilanus
Steward Observatory and Department of Astronomy, University of Arizona, 933 N. Cherry Ave., Tucson, AZ 85721, USA
Department of Astrophysics, Institute for Mathematics, Astrophysics and Particle Physics (IMAPP), Radboud University, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands
Leiden Observatory, Leiden University, Postbus 2300, 9513 RA Leiden, The Netherlands
Netherlands Organisation for Scientific Research (NWO), Postbus 93138, 2509 AC Den Haag, The Netherlands

0000-0001-9001-3275]Michael Titus
Massachusetts Institute of Technology Haystack Observatory, 99 Millstone Road, Westford, MA 01886, USA

0000-0001-8700-6058]Pablo Torne
Institut de Radioastronomie Millimétrique (IRAM), Avenida Divina Pastora 7, Local 20, E-18012, Granada, Spain
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

0000-0002-1209-6500]Efthalia Traianou
Instituto de Astrofísica de Andalucía-CSIC, Glorieta de la Astronomía s/n, E-18008 Granada, Spain
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

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

0000-0003-0465-1559]Sascha Trippe
Department of Physics and Astronomy, Seoul National University, Gwanak-gu, Seoul 08826, Republic of Korea

0000-0002-5294-0198]Matthew Turk
Department of Astronomy, University of Illinois at Urbana-Champaign, 1002 West Green Street, Urbana, IL 61801, USA

0000-0001-5473-2950]Ilse van Bemmel
Joint Institute for VLBI ERIC (JIVE), Oude Hoogeveensedijk 4, 7991 PD Dwingeloo, The Netherlands

0000-0002-0230-5946]Huib Jan van Langevelde
Joint Institute for VLBI ERIC (JIVE), Oude Hoogeveensedijk 4, 7991 PD Dwingeloo, The Netherlands
Leiden Observatory, Leiden University, Postbus 2300, 9513 RA Leiden, The Netherlands
University of New Mexico, Department of Physics and Astronomy, Albuquerque, NM 87131, USA

0000-0001-7772-6131]Daniel R. van Rossum
Department of Astrophysics, Institute for Mathematics, Astrophysics and Particle Physics (IMAPP), Radboud University, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands

0000-0003-3349-7394]Jesse Vos
Department of Astrophysics, Institute for Mathematics, Astrophysics and Particle Physics (IMAPP), Radboud University, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands

0000-0003-1105-6109]Jan Wagner
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

0000-0003-1140-2761]Derek Ward-Thompson
Jeremiah Horrocks Institute, University of Central Lancashire, Preston PR1 2HE, UK

0000-0002-8960-2942]John Wardle
Physics Department, Brandeis University, 415 South Street, Waltham, MA 02453, USA

0000-0002-4603-5204]Jonathan Weintroub
Black Hole Initiative at Harvard University, 20 Garden Street, Cambridge, MA 02138, USA
Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA

0000-0003-4058-2837]Norbert Wex
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

0000-0002-7416-5209]Robert Wharton
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

0000-0002-8635-4242]Maciek Wielgus
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

0000-0002-0862-3398]Kaj Wiik
Tuorla Observatory, Department of Physics and Astronomy, University of Turku, Finland

0000-0003-2618-797X]Gunther Witzel
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

0000-0002-6894-1072]Michael F. Wondrak
Department of Astrophysics, Institute for Mathematics, Astrophysics and Particle Physics (IMAPP), Radboud University, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands
Radboud Excellence Fellow of Radboud University, Nijmegen, The Netherlands

0000-0001-6952-2147]George N. Wong

School of Natural Sciences, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA
Princeton Gravity Initiative, Princeton University, Princeton, New Jersey 08544, USA

0000-0003-4773-4987]Qingwen Wu (吴庆文)

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

0000-0002-6017-8199]Paul Yamaguchi
Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA

0000-0001-8694-8166]Doosoo Yoon
Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904, 1098 XH, Amsterdam, The Netherlands

0000-0003-0000-2682]André Young
Department of Astrophysics, Institute for Mathematics, Astrophysics and Particle Physics (IMAPP), Radboud University, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands

0000-0002-3666-4920]Ken Young
Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA

0000-0001-9283-1191]Ziri Younsi
Mullard Space Science Laboratory, University College London, Holmbury St. Mary, Dorking, Surrey, RH5 6NT, UK
Institut für Theoretische Physik, Goethe-Universität Frankfurt, Max-von-Laue-Straße 1, D-60438 Frankfurt am Main, Germany

0000-0003-3564-6437]Feng Yuan (袁峰)

Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, People's Republic of China
Key Laboratory for Research in Galaxies and Cosmology, Chinese Academy of Sciences, Shanghai 200030, People's Republic of China
School of Astronomy and Space Sciences, University of Chinese Academy of Sciences, No. 19A Yuquan Road, Beijing 100049, People's Republic of China

0000-0002-7330-4756]Ye-Fei Yuan (袁业飞)
Astronomy Department, University of Science and Technology of China, Hefei 230026, People's Republic of China

0000-0001-7470-3321]J. Anton Zensus
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

0000-0002-2967-790X]Shuo Zhang
Bard College, 30 Campus Road, Annandale-on-Hudson, NY, 12504

0000-0002-4417-1659]Guang-Yao Zhao
Instituto de Astrofísica de Andalucía-CSIC, Glorieta de la Astronomía s/n, E-18008 Granada, Spain

0000-0002-9774-3606]Shan-Shan Zhao (赵杉杉)
Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, People's Republic of China

Dominic O. Chang
Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA
Black Hole Initiative at Harvard University, 20 Garden Street, Cambridge, MA 02138, USA

0The Event Horizon Telescope Collaboration
The EHT Collaboration et al.

In this paper we quantify the temporal variability and image morphology of the horizon-scale emission from , as observed by the EHT in 2017 April at a wavelength of 1.3 mm.  We find that the data exhibit variability that exceeds what can be explained by the uncertainties in the data or by the effects of interstellar scattering.  The magnitude of this variability can be a substantial fraction of the correlated flux density, reaching $\sim$100% on some baselines.  Through an exploration of simple geometric source models, we demonstrate that ring-like morphologies provide better fits to the data than do other morphologies with comparable complexity. We develop two strategies for fitting static geometric ring models to the time-variable data; one strategy fits models to short segments of data over which the source is static and averages these independent fits, while the other fits models to the full dataset using a parametric model for the structural variability power spectrum around the average source structure.  Both geometric modeling and image-domain feature extraction techniques determine the ring diameter to be (68% credible intervals), with the ring thickness constrained to have an FWHM between $\sim$30% and 50% of the ring diameter. To bring the diameter measurements to a common physical scale, we calibrate them using synthetic data generated from GRMHD simulations.  This calibration constrains the angular size of the gravitational radius to be , which we combine with an independent distance measurement from maser parallaxes to determine the mass of to be .

§ INTRODUCTION

Sagittarius A$^*$ (), the radio source associated with the supermassive black hole (SMBH) at the center of the Milky Way, is thought to subtend the largest angular size of all black holes in the sky.  At a distance of $D \approx 8$ kpc and with a mass of $M \approx 4 \times 10^6$ <cit.>, has a Schwarzschild radius of $\sim$10 .  Models of optically thin spherical accretion flows around SMBHs generically predict that they will appear to distant observers as bright rings of emission surrounding a darker central “shadow” <cit.>, and a variety of more general accretion flow simulations have demonstrated that the diameter of this ring is typically $\sim$5 times larger than the Schwarzschild radius <cit.>.  The Event Horizon Telescope (EHT) collaboration provided observational verification of this picture, using a global very long baseline interferometry (VLBI) network of radio telescopes observing at a frequency of $\sim$230 GHz to resolve the $\sim$40 ring of emission around the 87 SMBH <cit.>.

The predicted ring diameter for is $\sim$50 , about 25% larger than what the EHT observed for 87.  However, because is more than three orders of magnitude less massive than 87, all dynamical timescales in the system are correspondingly shorter.  In particular, the typical gravitational timescale for is $G M / c^3 \approx 20$ s, implying that the source structure can vary substantially over the several-hour duration of a single EHT observation.  Consistent with this expectation, exhibits broadband variability on timescales of minutes to hours <cit.>.  The multiwavelength properties of during the 2017 EHT observing campaign are described in <cit.>.

The potential for rapid structural variability complicates the analysis of EHT observations of .  A standard strategy for ameliorating the sparsity of VLBI data sets is Earth-rotation aperture synthesis, whereby Fourier coverage of the array is accumulated as the Earth rotates and baselines change their orientation with respect to the source <cit.>.  This strategy is predicated on the source remaining static throughout the observing period, in which case the accumulated data measure a single image structure.  However, violates this assumption on timescales as short as minutes.  After several hours, the variable components of the image structure in are expected to be uncorrelated <cit.>.
Thus, image reconstructions from the EHT data are focused on reconstructing time-averaged source structures <cit.>.

Despite the necessity of reconstructing an average source structure, the data collected within a single multihour observation epoch are associated with many specific instances of the variable emission from , i.e., they represent an amalgam of observations of instantaneous images. The imaging strategy pursued for the EHT observations of aims to mitigate the impact of this changing source structure through the introduction of a “variability noise budget,” which absorbs the structural evolution into inflated uncertainties and thereby permits imaging algorithms to reconstruct a time-averaged image under the usual static source assumption.[Note that this procedure ignores the subhour correlations that are present in the data; the implications of these correlations are discussed in <ref>.] The image reconstruction procedure is described in detail in , and the results confirm that the data are consistent with being produced by a ring-like emission structure with a diameter of $\sim$50 .

For the EHT observations of 87, morphological properties of the observed ring (e.g., diameter, thickness, orientation) were quantified using both imaging and geometrical modeling analyses , and the measured ring diameter was calibrated using general relativistic magnetohydrodynamic (GRMHD) simulations from  to constrain the mass of the SMBH.  The current paper applies a conceptually similar strategy to the analysis of the EHT data, though significant alterations have been made to meet the new challenges posed by and to tailor the analyses appropriately.  In this paper, we first characterize the variability seen in the data, and we develop a framework for mitigating the impact of variability when imaging or modeling the data.  We then make measurements of the ring size and other structural properties using both imaging and geometrical modeling analyses, and we derive and apply a GRMHD-based calibration to bring ring size measurements made using different techniques to a common physical scale.

This paper is organized as follows.  <ref> provides an overview of the observations and data processing.  In <ref>, we quantify the variability on different spatial scales, and we outline the strategies used to mitigate its impact during imaging and modeling.  In <ref>, we discuss salient data properties in the context of a ring-like emission structure, and we describe our procedure for using GRMHD simulations to calibrate different ring size measurement techniques to a common physical scale.  Sections <ref>, <ref>, and <ref> detail our three primary strategies for measuring the ring size and describe their application to the data.  Our results are presented in <ref>, and we summarize and conclude in <ref>.  This paper is the fourth in a series that describes the analysis of the 2017 EHT observations of .  The series is summarized in <cit.>.  The data processing and calibration are described in , imaging is carried out in , physical simulations are described in <cit.>, and tests of gravity are presented in <cit.>.

§ OBSERVATIONS AND DATA PRODUCTS

In this section, we briefly review the interferometric data products used for analyses in this paper (<ref>), and we summarize the observations (<ref>) and data processing (<ref>) that precede these analyses. A more comprehensive description of the data collection, correlation, and calibration can be found in , , and references therein.

§.§ VLBI data products

As a radio interferometer, the EHT is natively sensitive to the Fourier transform of the sky-plane emission structure.  For a source of emission $I(\boldsymbol{x},t)$, the complex visibility $\mathcal{V}(\boldsymbol{u},t)$ is given by

\begin{equation}
\mathcal{V}(\boldsymbol{u},t) = \iint e^{- 2 \pi i \boldsymbol{u} \cdot \boldsymbol{x}} I(\boldsymbol{x},t) \text{d}^2\boldsymbol{x} ,
\end{equation}

where $t$ is time, $\boldsymbol{x} = (x,y)$ are angular coordinates on the sky, and $\boldsymbol{u} = (u,v)$ are projected baseline coordinates in units of the observing wavelength <cit.>.

The ideal visibilities $\mathcal{V}$ are not directly observable because they are corrupted by both statistical errors and a variety of systematic effects.  For the EHT, the dominant systematics are complex station-based gain corruptions.  The relationship between an ideal visibility $\mathcal{V}_{ij}$ and the observed visibility $V_{ij}$ on a baseline connecting stations $i$ and $j$ is given by

\begin{equation}
V_{ij} = g_i g_j^* \mathcal{V}_{ij} + \sigma_{\text{th},ij} \equiv |V_{ij}| e^{i \phi_{ij}}, \label{eqn:VisibilityCorruptions}
\end{equation}

where $\sigma_{\text{th},ij}$ is the statistical (or “thermal”) error on the baseline, $g_i$ and $g_j$ are the station gains, and we have defined the visibility amplitude $|V_{ij}|$ and phase $\phi_{ij}$.  The statistical error is well described as a zero-mean circularly symmetric complex Gaussian random variable with a variance determined (per the radiometer equation) by the station sensitivities, integration time, and frequency bandwidth <cit.>.  The station gains vary in time at every site and must in general be either calibrated out or determined alongside the source structure.

The presence of station-based systematics motivates the construction and use of “closure quantities” that are invariant to such corruptions.  A closure phase $\psi_{ijk}$ <cit.> is the sum of visibility phases around a closed triangle of baselines connecting stations $i$, $j$, and $k$,

\begin{equation}
\psi_{ijk} = \phi_{ij} + \phi_{jk} + \phi_{ki} . \label{eqn:ClosurePhase}
\end{equation}

Closure phases are invariant to station-based phase corruptions, such that the measured closure phase is equal to the ideal closure phase, up to statistical errors.  Similarly, a closure amplitude $A_{ijk\ell}$ <cit.> is the ratio of pairs of visibility amplitudes on a closed quadrangle of baselines connecting stations $i$, $j$, $k$, and $\ell$,

\begin{equation}
A_{ijk\ell} = \frac{|V_{ij}| |V_{k\ell}|}{|V_{ik}| |V_{j\ell}|} .
\end{equation}

Analogous with closure phases, closure amplitudes are invariant to station-based amplitude corruptions.  Because closure quantities are constructed from nonlinear combinations of complex visibilities, they have correlated and non-Gaussian error statistics; a detailed discussion is provided in <cit.>.

§.§ EHT observations of Sgr A*

The EHT observed on 2017 April 5, 6, 7, 10, and 11 with the phased Atacama Large  Millimeter/submillimeter Array (ALMA) and the Atacama Pathfinder Experiment (APEX) on the Llano de Chajnantor in Chile, the Large Millimeter Telescope Alfonso Serrano (LMT) on Volcán Sierra Negra in Mexico, the James Clerk Maxwell Telescope (JCMT) and phased Submillimeter Array (SMA) on Maunakea in , the IRAM 30 m telescope (PV) on Pico Veleta in Spain, the Submillimeter Telescope (SMT) on Mt. Graham in Arizona, and the South Pole Telescope (SPT) in Antarctica .  Only the April 6, 7, and 11 observations included the highly sensitive ALMA station, and the April 11 light curve exhibits strong variability <cit.> that is presumably associated with an X-ray flare that occurred shortly before the start of the track .  In this paper, we thus analyze primarily the April 6 and April 7 data sets.  We note that while  focuses on the April 7 data set, with the April 6 data set used for secondary validation, most of the analyses carried out in this paper instead focus on a joint data set that combines the April 6 and April 7 data.

At each site the data were recorded in two 1.875 GHz wide frequency bands, centered around sky frequencies of 227.1 GHz (low band; LO) and 229.1 GHz (high band; HI), and in each of two polarization modes.  For all telescopes except ALMA and JCMT, the data were recorded in a dual circular polarization mode: right-hand circular polarization (RCP; R) and left-hand circular polarization (LCP; L).  ALMA recorded using linear feeds, and the data were later converted to a circular polarization basis during the DiFX <cit.> correlation <cit.>.  The JCMT observed only a single hand of circular polarization at a time, with the specific handedness (RCP or LCP) changing from day to day.  All other stations observed in a standard dual-polarization mode, which allows the construction of RR, RL, LR, and LL correlation products. The analyses in this paper use only the parallel-hand correlations (i.e., RR and LL), which are averaged to form Stokes $I$ data products.  Because JCMT records only a single hand at a time, we instead form “pseudo-$I$” data products for JCMT baselines, using whichever parallel-hand correlation is available as a stand-in for Stokes $I$.[The “pseudo-$I$” formation is a good approximation for Stokes $I$ when the magnitude of the Stokes $V$ contribution is small.  We expect this condition to be met for the 2017 EHT observations of <cit.>, and the impact of residual Stokes $V$ is captured by the systematic error budget .]

§.§ Data reduction

After correlation, residual phase and bandpass errors are corrected with two independent processing pipelines: EHT-HOPS <cit.> producing “HOPS” <cit.> data and rPICARD <cit.> producing “CASA” <cit.> data.  Relative phase gains between RCP and LCP have been corrected based on the assumption of zero circular polarization on baselines between ALMA and other EHT stations.  Absolute flux density scales are based on a priori measurements of each station's sensitivity, resulting in a ${\sim}10$% typical uncertainty in the amplitude gains .  The amplitude gains of the colocated ALMA/APEX and SMA/JCMT stations have been further refined via time-variable network calibration  using a light curve of the compact flux measured by ALMA and SMA <cit.>.  For the remaining stations, gross amplitude gain errors have been corrected by a transfer of gain solutions from the J1924-2914 and NRAO 530 calibrator sources as described in .

Following the completion of the above calibration pipelines, additional preprocessing of the data has been carried out as described in , including calibration of the LMT and JCMT station gains and normalization of the visibility amplitudes by the total light curve.  The characterization of residual calibration effects (e.g., polarization leakage) into a systematic error budget, as well as a more comprehensive description of the overall EHT data reduction, is provided in .

§ VARIABILITY EXTRACTION AND MITIGATION

The statistical errors quoted in  and summarized in the preceding section do not account for three additional sources of uncertainty that can otherwise substantially bias any analysis efforts.  First, unaccounted-for nonclosing (i.e., baseline-based) systematic errors are present in the data at a level that is on the order of $\sim$1% of the visibility amplitude, which is often larger than the formal statistical errors .  Second, significant refractive scattering in the interstellar medium produces additional substructure within the image that is not present in the intrinsic emission map <cit.>.  Third, there is intraday variability in the source itself.  Source variability is theoretically expected to arise on a broad range of timescales, and it is explicitly seen in GRMHD simulations on timescales as short as minutes <cit.>.  Such variability was also observed in the light curve of during the 2017 EHT campaign on timescales from 1 minute to several hours <cit.>.

In this section, we summarize the theoretical expectations for and characteristics of the variability based on GRMHD simulations, present an estimate for the degree of structural variability in directly from the visibility amplitude data, and describe the strategies pursued here and in  to mitigate the impact of the three components of additional error listed above.

§.§ Expectations from theory

In low-luminosity SMBH systems such as , we expect the emission to originate from the immediate vicinity of the black hole, i.e., on scales comparable to the event horizon size. Here, all characteristic speeds of the hot relativistic gas approach the speed of light. The timescales associated with these processes are therefore set by the gravitational timescale, $G M/c^3$, which is $\sim$20 s for . This timescale is $\sim$3 orders of magnitude shorter than the nightly observations carried out by the EHT, so a single observation contains many realizations of the underlying source variability.

GRMHD simulations can model the dynamical processes in and, using ray-tracing and radiative transfer, provide a theoretical expectation for the observed emission.  provides a library of GRMHD simulations and associated movies, which have been scaled to the conditions during the EHT 2017 observations (e.g., the average total 230 GHz flux is set to the EHT measurement).  We use the variability characteristics of these simulations as our expectation for the variability seen by the EHT.

GRMHD simulations are universally described by a “red-red” power spectrum, with the largest fluctuations in the emission occurring on the longest timescales and the largest spatial scales <cit.>. Spatially, the largest scale for variability is limited to the size of the emitting region, which for an observing frequency of 230 GHz is typically several $G M/c^2$ and which for the EHT data is constrained to be $\lesssim$87 .  Temporally, the simulations exhibit a red power spectrum that flattens on timescales ${\gtrsim}1000\ GM/c^3$. Observations of the total flux variability in corroborate this expectation, finding a red-noise spectrum extending to timescales of several hours and flattening on longer timescales <cit.>.

We can, without loss of generality, express the time-variable image structure $I$ in terms of some static mean image $I_{\text{avg}}$ and a zero-mean time-variable component $\delta I$ that captures all of the variation,

\begin{equation}
I(\boldsymbol{x},t) = I_{\text{avg}}(\boldsymbol{x}) + \delta I(\boldsymbol{x},t) .
\end{equation}

The linearity of the Fourier transform ensures that an analogous decomposition holds for $\mathcal{V}$, which is thus simply the sum of an analogous $\mathcal{V}_0$ and $\delta\mathcal{V}$. The variation $\delta\mathcal{V}$ represents the component of the data we wish to mitigate.

The EHT stations ALMA and SMA are themselves interferometric arrays capable of separating out extended structure <cit.> from the light curve,

\begin{equation}
L(t)=\mathcal{V}(\boldsymbol{0},t)=\iint I(\boldsymbol{x},t) d^2 \boldsymbol{x} ,
\end{equation}

on the largest spatial scales, predicted by GRMHD simulations to be the most variable.
Using this motivation, the light-curve-normalized image is defined to be

\begin{equation}
\hat{I}(\boldsymbol{x},t) \equiv \frac{I(\boldsymbol{x},t)}{L(t)}
\end{equation}

with $\hat{I}_{\text{avg}}$ and $\delta \hat{I}$ similarly defined; here, the “hat” diacritic denotes light-curve normalization.
From GRMHD simulations, the expected noise is well approximated by a broken power law,

\begin{equation}
\sigma_{\text{var}}^2
\equiv
\<\delta \hat{\mathcal{V}}^2\>
\approx
\frac{a^2 \left( |\boldsymbol{u}| / u_0 \right)^c}{1 + \left( |\boldsymbol{u}| / u_0 \right)^{b+c}} ,
\label{eq:noise-model}
\end{equation}

along any radial direction <cit.>. This broken power law is described by four parameters: a break at $u_0$, an amplitude $a$ representing the amount of noise at the break location, and long- and short-baseline power-law indices $b$ and $c$, respectively.
Typically, we expect that $c\gtrsim2$, due to the compact nature of the source.

In <ref>, red lines show $\sigma_{\text{var}}^2$ measured for an example GRMHD simulation about average images that have been constructed on observationally relevant timescales. The variability has been averaged in azimuth and across different black hole spin orientations.  As the timescale over which the average image is constructed increases, the location of the break $u_0$ decreases and the amount of power at the break
increases.[This behavior is generic across a large number of simulations of and explored in detail in <cit.>.] This behavior can intuitively be understood as the GRMHD simulations changing less for short timescales.  For comparison, we show the thermal, systematic, and refractive scattering noise. For timescales longer than $\sim$10 min, the variability noise dominates on EHT VLBI baselines.

The amount of excess variance in the visibility amplitudes expected for a GRMHD simulation (red lines, $\sigma_\text{var}^2$), after subtraction of the average image and normalization of the flux density by the light curve. Shown also are the typical thermal noise (black dashed line) and a 1% fractional systematic noise (green band) proportional to the mean image visibility amplitudes. The expected degree of refractive scattering  is shown by the purple bands, with purple lines evaluated for a Gaussian source at the projected location of EHT data . The variability is shown about a mean image constructed on different observationally relevant timescales. The fractional systematic and variability noise have been averaged over azimuth and over the position angle of the diffractive screen.

§.§ Intraday variability in the Sgr A* data

Location and detrended visibility amplitudes for crossing and following tracks from the observations on April 5, 6, 7, and 10.  Left: the locations of all data points in the $(u,v)$-plane.  Individual baselines correspond to the curving tracks seen in this plot, and we have highlighted the crossing and following tracks using colored points.  For reference, circles of constant baseline length are shown by the dashed gray lines.  The central time stamps for the highlighted tracks are labeled in the corresponding colors. Right: linearly detrended and light-curve-normalized visibility amplitudes for the crossing and following baselines highlighted in the left panel, shown in corresponding point colors.  For crossing tracks (bottom panels), the points within $1~{\rm G}\lambda$ of the crossing point are shown. Error bars show the thermal errors. For following tracks (top panels) all points for which the baselines overlap are shown.  All highlighted points are employed in the linear detrending for each panel.  The associated estimates of the mean and standard deviation are shown by the gray dashed line and horizontal band in each panel.  The April 7 visibility amplitudes are indicated by open circles.

The intraday variability expected from theoretical considerations can be observed directly in the data.
<ref> shows the combined baseline coverage for the EHT's 2017 campaign, including the observations on April 5, 6, 7, and 10.
The upper limit on the source size of 87  implies that the complex visibilities will be correlated in regions of the -plane smaller than ${\sim}2$ .
In practice, the visibility amplitudes exhibit variations on scales smaller than this and otherwise appear strongly correlated on scales of $1$ (see , Figure 3).
Therefore, among the baseline tracks in <ref> there are four regions where the -coverage is redundant, i.e., multiple baselines pass within 1 of the same -position.
We separate the redundant baseline combinations into “crossing tracks,” in which two baseline tracks intersect at a single -point, and “following tracks,” in which two baselines follow a nearly identical extended track in the -plane.  Both sets of redundant baselines provide an opportunity to directly probe the degree of intraday variability in the visibilities at specific locations in the -plane.

Prior to making comparisons, we apply the data preprocessing steps outlined in <ref> to mitigate unphysical sources of variability.  To avoid addressing the unknown atmospheric phase delays, we focus exclusively on visibility amplitudes.
Because source structure will produce additional variations in the visibility amplitudes that are hard to visualize in projection and obscure the relative degree of variability, we detrend the visibility amplitudes with a linear model.
The crossing and following tracks discussed below are shown in the top and bottom subpanels of <ref>, respectively.

Chile–PV vs. Chile–SPT: The first crossing track we consider contains baselines between the Chile stations (ALMA, APEX) and PV and SPT, which both cross near $(u,v) = ($4 , 3.5 ) at times separated by 6.2 hr.  The concurrent ALMA and APEX baselines are consistent within the reported statistical errors, and thus there is no evidence for unaddressed baseline-specific dominant systematic errors. The normalized visibility amplitudes for the Chile–PV and Chile–SPT baselines individually vary smoothly with time.  Nevertheless, they differ significantly at the crossing point, and this difference is consistent in magnitude with the variation found across days (indicated by the gray band in the relevant panel of <ref>).

Chile–SMT vs. Chile–SPT: The second crossing track we consider contains baselines between the Chile stations (ALMA, APEX) and SMT and SPT, which both cross near $(u,v) = ($3 , 4.5 ) at times separated by 5.2 hr. Again, we find excellent agreement between ALMA and APEX baselines, individually smooth variations on the Chile–SMT and Chile–SPT baselines, and significant differences in the visibility amplitudes between those baselines.

SMA–SPT vs. LMT–SPT: The first following track we consider contains baselines between the SPT, which is located at the South Pole, and SMA and LMT, which have similar latitudes.  Because the baseline tracks are coincident across a large range of locations in the -plane, this following track permits many direct comparisons at a baseline length of 8 at times separated by 3.4 hr.  As with both crossing tracks, significant differences exist between the two sets of baselines, consistent with the range across multiple days.

SMT–SPT vs. PV–SPT: The final following track we consider again involves the SPT, and now the SMT and PV, which also have similar latitudes.  This is the longest set of baselines that we consider, with a length of roughly 8.5 and covering similar regions in the -plane at times separated by 6.7 hr.  Again, significant variations are exhibited, consistent with those across days.

In summary, intraday variability is observed on multiple baselines with lengths ranging from 5 to 8.5 and on timescales as short as 3.4 hr.  In all cases, this variability is broadly consistent with that observed on interday timescales.
Furthermore, the variability behavior is consistent with theoretical expectations from GRMHD simulations and empirical expectations from the light curve, both of which imply that the variable elements of the emission should be uncorrelated beyond a timescale of a few hours <cit.>.  Any average image of reconstructed from data spanning a time range longer than several hours captures the long-timescale asymptotic source structure; the intrinsic image averaged over a single day or multiple days is thus expected to exhibit similar structure.

§.§ Model-agnostic variability quantification

To quantify the variability observed in the EHT data, we make use of the procedure described in <cit.>.  This procedure provides an estimate of the excess variability – i.e., the visibility amplitude variance in excess of that caused by known sources, such as average source structure, statistical and systematic uncertainties, and scattering – as a function of baseline length.  We apply the same data preparation steps summarized in <ref> and described in <cit.>, combining the visibility amplitudes measured on April 5, 6, 7, and 10 in both observing bands.  All data points are weighted equally.

Illustration of detrended visibility amplitudes and associated variance estimate.
Top: scan-averaged tracks in -coordinates with a circular region of diameter 1 superposed (red disk), centered at $(-2.1$ , 4.7 ).  Scans within the region are dark red, while those outside are blue.
Middle: light-curve-normalized visibility amplitudes as a function of $u$, projected in $v$ (limited to points within the top panel).
Bottom: light-curve-normalized visibility amplitudes after detrending with a linear model defined by the scans within the 1 circular region.  The estimated mean and standard deviation are shown by the orange dashed line and horizontal band.

Model-agnostic estimate of the azimuthally averaged excess variance of the visibility amplitudes, after subtracting the variance from the reported statistical errors, as a function of baseline length.  Nonparametric estimates (filled and open black circles) are obtained across April 5, 6, 7, and 10 and using both high- and low-band data.  The filled black circles indicate significant detections of source variability, while the open black circles indicate variance measurements that are dominated by the other sources of uncertainty; only the former are used in the parametric fitting.  Uncertainties associated with the thermal errors, uncertain station gains, and polarization leakage are indicated by the error bars.  Azimuthally averaged thermal errors are shown by the gray triangles and provide an approximate lower limit on the range of accurate variance estimates. For comparison, the magnitudes of the variance induced by refractive scattering are shown in purple along the minor (top) and major (bottom) axes of the diffractive scattering kernel ; the variance along individual tracks on April 7, as well as a $\sim$10 mJy floor (assuming a fixed 2.5 Jy total flux), is shown by the solid and dashed purple lines, respectively.  The orange band indicates the 95th-percentile range of fits to the filled variance estimates shown by filled points by the broken power law of the form in <ref>, with a handful of specific examples shown explicitly.

The procedure is illustrated in <ref>.  We again make use of the strong correlations induced by the finite source size, and for every location in the -plane we consider only those data points falling within a circular region of diameter 1 centered at that point (red circular region in the top panel of <ref>).  Within each such region containing at least three data points, we linearly detrend the light-curve-normalized visibility amplitudes with respect to $u$ and $v$ to remove variations due to physical structure (bottom panel of <ref>), and we compute the variance of the residuals.  This variance is then debiased to remove the contributions from the reported statistical errors, as described in <cit.>.  Finally, the variances from all regions having a common baseline length are averaged to produce an azimuthally averaged set of variances.  The uncertainty in the variance estimates is obtained via Monte Carlo sampling of the unknown gains, leakage terms, and statistical errors.

<ref> shows the results of applying this procedure to the data, with the normalized visibility amplitude variance measurements given by the black points.  For baselines shorter than $2.5$ , the LMT calibration procedure precludes an accurate estimate of the variance, and thus these baselines have been excluded.  For baselines between ${\sim}2.5$ and 6 in length, our empirical estimates of the noise exceed the typical contributions from statistical errors and refractive scattering, indicating the presence of an additional source of structural variability.  The degree of inferred variability is consistent with that seen in prior millimeter-VLBI data sets, which is discussed further in <ref>.  For baselines longer than 6 , our measurements are consistent with the degree of variability expected from the statistical uncertainties in the data; we thus do not directly constrain the source variability on these long baselines.

To characterize the variability behavior within the -plane, we fit a broken power law of the form in <ref> to the normalized variance measurements.  As indicated by the filled black circles in <ref>, significant measurements exist only in the range of baselines with lengths $\sim$2.5–6 ; on baselines longer than 6 , we are unable to distinguish the variability from its associated measurement uncertainties.  We thus perform the broken power-law fit only to the $\sim$2.5–6 range of baselines, where we have significant measurements, and we find no evidence for a break in the power law in this region.  As a result, only an upper limit can be placed on $u_0$, and we are not able to constrain the short-baseline power-law index, $c$.
The range of permitted broken power-law fits is illustrated in <ref> by the orange shaded region, with several samples from the posterior distribution explicitly plotted as orange lines.

Joint posteriors of the constrained parameters after fitting a broken power law to the model-agnostic normalized variances estimates.  Because the amplitude is well constrained within the range of baseline lengths for which good estimates of the variability exist, we set the normalization at $|\boldsymbol{u}|=4~{\rm G}\lambda$, denoted as $a_4$. Contours show the enclosed 50th, 90th and 99th percentiles.  The purple bands indicate the ranges used as priors during the full-track modeling, associated with the interquartile ranges.

Because the location of the broken power-law break is poorly constrained, the parameters $u_0$ and $a$ (describing the location of the break and the amplitude of the power law at the break, respectively) are strongly correlated and highly uncertain.  However, it is clear from the orange shaded region in <ref> that there is only a narrow range of variances permitted over the 2.5–6 range of baselines over which the data are constraining.  We thus choose to characterize the amplitude of the excess variability noise at $|\boldsymbol{u}|=4\,{\rm G}\lambda$, which we denote as $a_4$. Joint posteriors for $a_4$, the break location $u_0$, and the long-baseline power-law index $b$ are shown in <ref>.  These constraints are used to inform the prior distributions for the full-track geometric modeling described in <ref>; the associated prior ranges on each parameter are indicated by the purple shaded regions in <ref>.

§.§ Description of variability mitigation approaches

Having established the existence of structural variability and quantified its magnitude in the data,
we now turn to strategies for mitigating its impact on downstream analyses.
We employ the light-curve-normalized visibility data, which eliminates large-scale variations and correlations by construction.
In principle, there are four methods that we might pursue to address the remaining structural variability:

* Analyze time-averaged data products.

* Employ explicitly time-variable models.

* Analyze short time segments of the data and combine the results afterward to characterize the average source structure.

* Simultaneously reconstruct the average source structure and a statistical characterization of the structural variability.

The first of these options is complicated substantially by the uncertain visibility phases, which limit our ability to coherently average the data on timescales longer than several minutes.  The second option can be employed either when a descriptive low-dimensional model for the source structure can be constructed <cit.>, or when there is sufficient -coverage for nonparametric dynamical imaging algorithms to be successful <cit.>.  The latter approach is explored in the dynamical imaging analyses described in , ultimately demonstrating that the -coverage is insufficient to permit unambiguous reconstructions of the variable source structure.

We dub the third option “snapshot” modeling, whereby a simple geometric model of the source structure is fit to segments of the data that are short enough in duration (${\lesssim}3$ minutes) for the impact of structural variability to be subdominant to other sources of visibility uncertainty (e.g., refractive scattering; see <ref>).  Though the data sparsity is exacerbated by restricting the reconstructions to only a single snapshot at a time, the model itself is also correspondingly restricted in its parameterization of the source structure.  The results of the fits to each individual snapshot are then combined across the entire data set, effectively averaging over the source variability.  Details of our snapshot modeling analyses as applied to are presented in <ref>.

The fourth option we refer to as “full-track” modeling, which aims to simultaneously reconstruct both the average source structure and a set of parameters describing the contribution of the structural variability to the visibility data variances <cit.>.  In contrast to the snapshot modeling, full-track modeling considers the entire data set at once and uses a parameterized “variability noise” model to appropriately modify the data uncertainties as part of the fitting procedure.  In this way, the full-track modeling retains access to sufficient -coverage to permit fitting a nonparametric image model to the data , though in <ref> we also pursue full-track geometric modeling to provide a cross-comparison with the results from the snapshot geometric modeling.  Our parameterization of the variability noise follows <ref>, with the amplitude specified at a baseline length of 4 as described in <ref>.  A detailed description of our full-track modeling approach as applied to is presented in <ref>.

Both the snapshot and full-track modeling approaches focus on describing the average source structure and treating the structural variability in a statistical manner.  This goal is formally mismatched with what the EHT data measure for a single day, which is instead a collection of complex visibilities that sample different instantaneous realizations of the intrinsic source structure.  The nature of this mismatch impacts the full-track analyses significantly.

The variability mitigation scheme employed by full-track modeling presumes that the variability may be modeled as excess uncorrelated fluctuations in the complex visibility data.  This assumption is well justified on timescales exceeding a few hours, but significant correlations between visibilities exist on shorter timescales.  Within a single day, subhour correlations that are localized in the -plane can induce significant biases in the source structure reconstructed from the sparse EHT -coverage of . The noise model is thus fundamentally misspecified for EHT data, with the level of misspecification increasing as shorter-in-time segments of data are analyzed; <ref> describes pathological behavior that can arise when analyzing EHT data from only a single day.  While prominent artifacts associated with these subhour correlations are present in the April 6 reconstructions shown in <ref> and <ref>, we note that the underlying origin of these artifacts is no less present on April 7.

The impact of unmodeled correlations on the reconstructed source structure can be ameliorated by combining multiple days, which provides visibility samples associated with independent realizations of the source structure.  This additional sampling rapidly brings the statistical properties of the data into better agreement with the assumptions underpinning the full-track analyses; even the combination of just 2 days is often sufficient to mitigate the subhour correlations in analysis experiments that make use of GRMHD simulation data.  For this reason, we combine both the April 6 and April 7 data sets during the analysis of the data.  For comparison, <ref> presents the results of equivalent analyses applied to the April 6 and April 7 data sets individually.

§ RING CHARACTERIZATION AND CALIBRATION

We have a strong prior expectation – from both prior millimeter-VLBI observations of a different black hole  and theoretical simulations of the accretion flow around itself  – that ought to contain a ring of emission, and we thus aim to determine the characteristics of the ring-like image structure that best describes the data.  In this section, we first review the evidence from the data for a ring-like image structure, and we then present a geometric model for fitting parameters of interest and describe our procedure for bringing ring size measurements made using different techniques to a common physical scale.

§.§ Evidence for a ring

In reconstructing images of ,  explores a large space of imaging algorithms and associated assumptions. The resulting “top sets” of images
contain primarily “ring-like” image structures, though a small fraction of the images are morphologically ambiguous.  These “nonring” images still nominally provide a reasonable fit to the data and so are not ruled out from the  results.

We can quantify the preference for a ring-like image structure by fitting the data with a set of simple geometrical models. Employing the snapshot geometric modeling technique detailed in <ref>, we compare the Bayesian evidence, $\mathcal{Z}$, between these different geometric models.
The value of $\mathcal{Z}$ serves as a model comparison metric that naturally balances improvements in fit quality against increases in model complexity, with larger values of $\mathcal{Z}$ indicating preferred models <cit.>. <ref> shows the results of a survey over simple geometric models with varying complexity, captured here by the number of parameters required to specify the model.  At all levels of complexity, ring-like models outperform the other tested models.  This disparity is most stark for the simplest models but continues to hold as the models increase in complexity.

Comparison of the relative Bayesian evidence, $\Delta \ln \mathcal{Z}$, for a series of increasingly complex geometric models fitted using closure amplitudes and closure phases within the snapshot modeling formalism described in detail in <ref>.  The fits have been carried out using on the the HOPS April 7 data, and each point in the figure is colored according to the number of free parameters in the model; the number of free parameters in each model is also indicated in the horizontal axis labels.  The panel on the right shows a zoom-in to the highest-evidence region of the left panel.  Ring-like models are indicated with circles, and nonring models are indicated with crosses.  All Bayesian evidence values are quoted relative to the highest value attained across all models. The parameter counts reflect the fact that all models are normalized to have unit total flux density and are centered at the image origin. The crescent model consists of a smaller disk subtracted from an offset larger disk. In the crescent+floor model, the smaller disk may have a nonzero flux density. The m-ring and mG-ring models are defined in <ref>.  The maximum value of $\ln\mathcal{Z}$ among the models explored in this figure is obtained for an $m=2$ model, in agreement with the  analysis described in <ref>.

The remainder of this paper proceeds with analyses that presuppose a ring-like emission structure for .

§.§ Salient features in the context of a ring model

The overall structure of the visibility amplitudes (see the left panel of <ref>) exhibits at least three distinct regions:

* A “short-baseline” region containing baselines shorter than $\sim$2 .  The effects of data calibration and pre-processing – particularly the light-curve normalization and LMT calibration procedures  – are evident in the unit total flux density and the Gaussian structure of the visibility amplitudes in this region.

* An “intermediate-baseline” region containing baselines between $\sim$2 and 6 .  The visibility amplitudes in this region exhibit a general rise and then fall with increasing baseline length, peaking at a flux density approximately $\sim$20% of the total at a baseline length of $\sim$4 .

* A “long-baseline” region containing baselines with lengths in excess of $\sim$6 .  The visibility amplitudes in this region generally rise with increasing baseline length from a deep minimum near $\sim$6.5 , approximately flattening out at longer baselines to a level that is $\sim$3%–10% of the total flux density.

The visibility amplitudes exhibit indications of asymmetric source structure, particularly on baselines with lengths of $\sim$3 that fall near the first minimum.  Here, the baselines between the SMT and stations (oriented approximately in the east–west direction) have systematically higher correlated flux densities than the similar-length baselines between the LMT and Chile stations (oriented approximately in the north–south direction).  The implication for the source morphology is that we would expect to see more symmetric structure in the north–south than in the east–west direction.  Detailed geometric modeling analyses that are able to capture this asymmetry are described in <ref> and <ref>; here, we consider only a simple azimuthally symmetric toy model that captures some salient features of interest.

We attempt to understand the visibility behavior in light of expectations for a ring-like emitting structure.  Specifically, we consider a geometric construction whereby an infinitesimally thin circular ring bordering an inner disk of emission is convolved with a Gaussian blurring kernel.  The visibility function $V$ produced by such an emission structure is given by

\begin{equation}
\begin{aligned}
V & = F_0 V_{\text{Gauss}} \left[ f_d V_{\text{disk}} + (1 - f_d) V_{\text{ring}} \right] \\
& = F_0 \exp\left( - \frac{w^2 \xi^2}{4 \ln(2)} \right) \left[ \frac{2 f_d}{\xi} J_1(\xi) + (1 - f_d) J_0(\xi) \right] ,
\end{aligned}
\end{equation}

where $F_0$ is the total flux in the image, $f_d$ is the fraction of that flux that is contained in the disk component, $w = W / d$ is a fractional ring width, $W$ is the FWHM of the Gaussian convolving kernel, $d$ is the diameter of the ring and disk components, $\xi \equiv \pi |\boldsymbol{u}| d$ is a normalized radial visibility-domain coordinate, and $J_n(\xi)$ is a Bessel function of the first kind of order $n$.

An illustration of how the observed visibilities constrain the source structure, within the context of a simple geometric model (see <ref>).  The left panel shows the debiased visibility amplitudes from April 7 (black points) after normalizing by the light curve, averaging in time across scans, and deconvolving the diffractive scattering kernel.  Only data points with a signal-to-noise ratio greater than 3 are shown.  The ranges of baseline lengths corresponding to the locations of the first and second visibility minima are highlighted in cyan and purple, respectively.  The locations of these minima constrain the diameter of an emitting ring, as shown in the top right panel.  Similarly, the ranges of normalized amplitudes corresponding to the first and second visibility maxima are highlighted in red and orange, respectively.  Both the absolute fractional amplitudes and their relative values (shown in green) constrain a combination of the fractional ring width and the fractional central flux.  The “best-fit” model visibility amplitudes are shown by the gray curve in the left panel, with the corresponding parameters marked by the gray line in the top right panel and the gray cross in the bottom right panel.  The image structure corresponding to this model is shown in the upper right corner of the left panel, with a 20 scale bar shown in white.

The three regions of data identified above are separated by apparent minima in the visibility amplitudes, and they can be approximately characterized by the baseline locations of those minima and the peak flux density levels achieved at the visibility maxima between them. <ref> illustrates how this characterization manifests as constraints on the defining parameters of the geometric toy model.  The cyan and purple shaded regions in the left panel indicate the approximate ranges of baseline lengths corresponding to the locations of the first and second visibility minima, respectively.  The locations of these minima constrain the diameter of the emitting structure, as shown in the top right panel of <ref>.  To be consistent with both a first minimum falling between $\sim$2.5 and 3.5 and a second minimum falling between $\sim$6 and 7 , the emitting region must be between $\sim$50 and 60 across.  The amplitudes of two visibility maxima – one falling between the first and second visibility minima, and the second following the second minimum – constrain a combination of the fractional disk flux $f_d$ and the fractional ring width $w$.  The bottom right panel of <ref> shows the constraints from the first and second visibility maxima in red and orange, respectively, and from the ratio of the two in green.

Taken together, even these few, simple, and only modestly constrained visibility features result in a rather narrow permitted range of model parameter values for $d$, $w$, and $f_d$; an example of a “best-fit” model from within the permitted range is shown by the gray curve in the left panel of <ref>.  However, we stress that the above constraints only strictly hold within the context of the specific toy model used to derive them.  More general and robust constraints on the emission structure require a model that can accommodate more than just the gross features; such models are produced as part of the imaging ( and <ref>) and geometric modeling analyses (see <ref> and <ref>) carried out in this paper series.

§.§ Geometric ring model specification

The ring-like images reconstructed in  are not azimuthally symmetric, but instead show pronounced azimuthal brightness variations that we would like to capture in our geometric modeling analyses.  In this section, we specify the “” model that we use in <ref> and <ref> to quantify the morphological properties of the observed emission.

§.§.§ Image-domain representation of model

Adopting the construction developed by <cit.>, we can model an infinitesimally thin circular ring with azimuthal brightness variations using a sum over angular Fourier modes indexed by integer $k$,

\begin{align}
I_{\text{ring}}(r,\phi)=\frac{F_{\rm ring}}{\pi d} \, \delta\!\left(r-\frac{d}{2}\right) \sum_{k=-m}^m\beta_k e^{i k \phi}. \label{eqn:RingImage}
\end{align}

Here $r$ is the image radial coordinate, $\phi$ is the azimuthal coordinate (east of north), $d$ is the ring diameter, $\{ \beta_k \}$ are the set of (dimensionless) complex azimuthal mode coefficients, and $m$ sets the order of the expansion.  Because the image is real, $\beta_{-k} = \beta_k^\ast$; we enforce $\beta_0 \equiv 1$ so that $F_{\rm ring}$ sets the total flux density of the ring.  Given that the images from  show a ring of radius ${\sim}25$ and the diffraction-limited EHT resolution is ${\sim}20$ , we expect the data to primarily constrain ring modes with $m \lsim \pi \left(25/20\right) \approx 4$.  We refer to this asymmetric ring as an “m-ring” of order $m$.

For the purposes of constraining additional image structures, we augment this m-ring in two ways.  First, we convolve the m-ring with a circular Gaussian kernel of FWHM $W$,

\begin{equation}
I_{\text{ring}}(r,\phi;W) = I_{\text{ring}}(r,\phi) \ast \left[\frac{4 \ln(2)}{\pi W^2} \exp\!\left( - \frac{4 \ln(2) r^2}{W^2} \right)\right]. \label{eqn:ModelImage}
\end{equation}

Second, we add a circular Gaussian component that is concentric with the ring, which serves to provide a nonzero brightness floor interior to the ring. The Gaussian component has a total flux density of $F_{\text{Gauss}}$ and an FWHM of $W_{\text{Gauss}}$,

\begin{align}
I_{\text{Gauss}}(r,\phi) = \frac{4 \ln(2) F_{\text{Gauss}}}{\pi W_{\text{Gauss}}^2} \exp\!\left( - \frac{4 \ln(2) r^2}{W_{\text{Gauss}}^2} \right). \label{eqn:GaussImage}
\end{align}

We refer to the resulting composite model $I(r,\phi)$, where

\begin{equation}
I(r,\phi) = I_{\text{ring}}(r,\phi; W) + I_{\text{Gauss}}(r,\phi) ,
\end{equation}

as an “.”  An example is shown in <ref>.

Example model. The model consists of an m-ring of diameter $d$, with azimuthal variations determined by Fourier coefficients $\beta_m$ and with thickness determined by convolution with a circular Gaussian of FWHM $W$. The model also includes a circular Gaussian of FWHM $W_{\text{Gauss}}$ and fractional relative flux density $f_{\rm Gauss}$ (<ref>). The position angle $\eta$ of the ring is determined by the phase of the  mode (<ref>), while the magnitude of its asymmetry $A$ is determined by the amplitude of the  mode (<ref>).  The red curve in the middle panel shows a radial profile in the horizontal direction; the orange curve in the bottom panel shows the azimuthal profile and its decomposition into its three modes.  The plotted model has $w = 0.3$, $f_{\rm Gauss} = 0.2$, $W_{\text{Gauss}}/d=0.8$, $\beta_1 = -0.3i$, and $\beta_2 = 0.1+0.1i$.

An of order $m$ has $5 + 2 m$ model parameters: the flux density in the ring ($F_{\text{ring}}$), the diameter of the ring ($d$), the flux density in the central Gaussian ($F_{\text{Gauss}}$), the FWHM of the central Gaussian ($W_{\text{Gauss}}$), the FWHM of the ring convolving kernel ($W$), and two parameters for each complex Fourier coefficient $\beta_k$ with $1\leq k \leq m$.

§.§.§ Visibility-domain representation of model

To aid in efficient parameter space exploration, the model is intentionally constructed using components and transformations that permit analytic Fourier transformations. The Fourier transform of the m-ring image (<ref>) is given by

\begin{equation}
V_{\text{ring}}(|\boldsymbol{u}|,\phi_u) = F_{\text{ring}} \sum_{k=-m}^m \beta_k J_k(\pi |\boldsymbol{u}| d) e^{ik (\phi_u - \pi/2)} ,
\end{equation}

where $(|\boldsymbol{u}|,\phi_u)$ are polar coordinates in the Fourier domain.
The convolution with a circular Gaussian in the image plane corresponds to multiplication of this function by the Fourier transform of the convolving kernel,

\begin{equation}
V_{\text{ring}}(|\boldsymbol{u}|,\phi_u;W) = \exp\left( -\frac{\pi^2 W^2 |\boldsymbol{u}|^2}{4 \ln(2)} \right) V_{\text{ring}}(|\boldsymbol{u}|,\phi_u) .
\end{equation}

The Fourier transform of the Gaussian image (<ref>) is given by

\begin{equation}
V_{\text{Gauss}}(|\boldsymbol{u}|,\phi_u) = F_{\text{Gauss}} \exp\left( - \frac{\pi^2 W_{\text{Gauss}}^2 |\boldsymbol{u}|^2}{4 \ln(2)} \right) .
\end{equation}

By the linearity of the Fourier transform, the visibility-domain representation of the model is then simply the sum of these two components,

\begin{equation}
V(|\boldsymbol{u}|,\phi_u) = V_{\text{ring}}(|\boldsymbol{u}|,\phi_u;W) + V_{\text{Gauss}}(|\boldsymbol{u}|,\phi_u) .
\end{equation}

When interpreting model-fitting results in subsequent sections, we are interested in a number of derivative quantities.  We will typically work with the fractional thickness of the ring, $w$, defined to be

\begin{equation}
w \equiv \frac{W}{d} .
\end{equation}

Similarly, we are typically interested in fractional representations of flux densities. We define

\begin{equation*}
F_0 \equiv F_{\text{ring}} + F_{\text{Gauss}}
\end{equation*}

to be the total flux density, and then

\begin{equation}
f_{\text{ring}} \equiv \frac{F_{\text{ring}}}{F_0}
\end{equation}

\begin{equation}
f_{\text{Gauss}} \equiv \frac{F_{\text{Gauss}}}{F_0} \label{eqn:FractionalGaussFlux}
\end{equation}

are the fraction of the total flux density that is contained in the ring and in the Gaussian components, respectively.  Note that $F_0$ is typically close to or fixed to unity as a consequence of normalizing the data by the light curve.  We also define a fractional central flux as

\begin{equation}
f_{\text{c}} \equiv \frac{F_{\text{Gauss}}(r<d/2)}{F_{\text{ring}} + F_{\text{Gauss}}(r<d/2)} , \label{eqn:FractionalCentralFlux}
\end{equation}

where $F_{\text{Gauss}}(r<d/2)$ is the integrated flux density of the central Gaussian component interior to the ring radius, given by

\begin{equation}
F_{\text{Gauss}}(r< d/2) = F_{\text{Gauss}}  \left[ 1 - \exp\left( - \frac{d^2 \ln(2)}{W_{\text{Gauss}}^2} \right) \right] .
\end{equation}

Following , the m-ring position angle $\eta$ and degree of azimuthal asymmetry $A$ are both determined by the coefficient of the $m=1$ mode,

\begin{align}
\eta &\equiv {\rm arg}\left( \int_0^{2\pi} I(\phi) e^{i \phi} d\phi \right) \nonumber \\
&= -{\rm arg}\left( \beta_1 \right) , \label{eqn:PositionAngle}\\
A &\equiv \frac{ \left| \int_0^{2\pi} I(\phi) e^{i \phi} d\phi \right| }{\int_0^{2\pi} I(\phi) e^{i \phi} d\phi} \nonumber \\
&= |\beta_1| . \label{eqn:Asymmetry}
\end{align}

A number of these derivative quantities are illustrated in the example shown in <ref>.

§.§ Calibrating ring size measurements to a common physical scale

The parameters returned by the geometric modeling and feature extraction analyses used in this paper to describe the emission structure do not correspond directly to physical quantities.  Instead, the relationship between measured and physical quantities must be calibrated using data for which we know the correct underlying physical system's defining parameters.  For ring size measurements, the associated physical quantity of interest is related to the angular size of the gravitational radius,

\begin{equation}
\theta_g = \frac{G M}{c^2 D} . \label{eqn:ThetaG}
\end{equation}

which sets the absolute scale of the system.

Under the assumption that the emission near the black hole originates from some “typical” radius, a measurement of the angular diameter $d$ of the emitting region will be related to $\theta_g$ by a scaling factor $\alpha$,

\begin{equation}
d = \alpha \theta_g . \label{eqn:Alpha}
\end{equation}

If the observations were directly sensitive to the critical curve bounding the black hole shadow, then $\alpha$ could be determined analytically and would take on a value ranging from $\sim$9.6 to 10.4 depending on the black hole spin and inclination <cit.>.  For more realistic emission structures and measurement strategies, the value of $\alpha$ cannot be determined from first principles and must instead be calibrated.

Our $\alpha$ calibration strategy generally follows the procedure developed in .  Using the library of GRMHD simulations described in , we generate a suite of 100 synthetic data sets that emulate the cadence and sensitivity of the 2017 EHT observations and that contain a realistic character and magnitude of data corruption; <ref> describes the generation of these synthetic data sets.  In the analyses described in Sections <ref>, <ref>, and <ref>, 90 of these 100 synthetic data sets are used to derive the $\alpha$ calibration for each analysis pathway, while the remaining 10 data sets are used to validate the calibration.

After carrying out ring size measurements on each of the data sets in the suite, we determine $\alpha$ (for each specific combination of data set and measurement technique) by dividing the measured ring diameter by the known value of $\theta_g$ (per <ref>).  For a given measurement technique, the distribution of $\alpha$ values that results from applying this procedure to the entire suite of synthetic data sets then provides a measure of $\alpha$ and its theoretical uncertainty.  The $\alpha$ value associated with each measurement technique can then be used to translate ring size measurements into their corresponding $\theta_g$ constraints.  We note that this calibration strategy assumes that the images contained in the GRMHD library provide a reliable representation of the emission structure in the vicinity of ; a separate calibration strategy that relaxes this GRMHD assumption is presented in .

<ref> describes elements of the calibration and validation strategy that are specific to each of the analysis pathways detailed in Sections <ref>, <ref>, and <ref>.

§ IMAGE-DOMAIN FEATURE EXTRACTION

The imaging carried out in  permits very flexible emission structures to be reconstructed from the data, but the majority of these images exhibit a ring-like morphology whose properties we seek to characterize.  In this section, we describe our image-domain feature extraction (IDFE) procedure, which uses a topological classification scheme to identify the presence of a ring-like structure in an image and quantifies the parameters that best describe this ring using two different algorithms.  We apply this IDFE procedure to the image reconstructions from .

§.§ Imaging methods and products

The imaging analyses carried out in  use four different algorithms classified into three categories: one sampling-based posterior exploration algorithm (, ), one CLEAN-based deconvolution algorithm (, ), and two “regularized maximum likelihood” (RML) algorithms (, ; and , ).
All methods produce image reconstructions using band-combined data (i.e., both low band and high band) and the latter three are run on two versions of the data: a “descattered” version that attempts to deconvolve the effects of the diffractive scattering kernel from the data, and a “scattered” version that applies no such deconvolution.  The posterior exploration imaging method instead applies the effects of diffractive scattering as part of its internal forward model, rather than deconvolving the data; the analogous “scattered” and “descattered” versions of the images thus correspond simply to those for which the scattering kernel has been applied or not, respectively.  The posterior exploration imaging jointly reconstructs the combined April 6 and April 7 data sets (see <ref>), while the CLEAN and RML imaging reconstructs each day individually, focusing primarily on the April 7 data and using the April 6 data for cross-validation.  Example fits and residuals for each of the imaging pipelines are shown in <ref>[Note that the preprocessing and data products used during imaging are not the same across imaging pipelines; and fit to complex visibilities, while and fit iteratively to different combinations of data products that include visibility amplitudes, closure phases, and log closure amplitudes .  For clarity in <ref>, we simply show residual complex visibilities for each imaging pipeline using a representative image from that pipeline's top set or posterior.], and $\chi^2$ statistics for each image are provided in <ref>; detailed descriptions of the data preprocessing and imaging procedures for each imaging algorithm are provided in .

Representative examples of imaging results for each of the four imaging pipelines used in ; is shown in the top left panel, in the top right panel, in the bottom left panel, and in the bottom right panel.  The top section of each panel shows the light-curve-normalized complex visibility data (in blue) as a function of baseline length; the light-curve-normalized visibilities are denoted as $\hat{V}$.  The real parts of the complex visibilities are plotted as filled markers, and the imaginary parts of the complex visibilities are plotted as open markers; the corresponding model visibilities are overplotted as red points.  The plotted data have been through the pre-analysis and pre-imaging calibration procedures described in , , and .  The bottom section of each panel shows the normalized residuals – i.e., the difference between the model and data visibilities, normalized by the data uncertainties – as a function of baseline length.  The solid red horizontal line marks zero residual, and the two dotted horizontal red lines mark $\pm$ one standard deviation.  The blue histogram on the right side of each bottom panel shows the distribution of normalized residuals, with the solid red curve showing a unit-variance normal distribution and the dotted green curve showing a normal distribution with variance equal to that of the normalized residuals.  We note that the visibilities for the , , and pipelines have been “descattered” and so have somewhat larger typical amplitudes than the visibilities for the pipeline (for which the scattering is incorporated as part of the forward model; see <ref> and ).  We also note that the different imaging pipelines make different choices about data averaging: and average the data over 60 s intervals, averages over 120 s intervals, and averages over scans.  Detailed descriptions of each of the imaging methods are provided in .

For the CLEAN and RML imaging methods, there are a number of tunable hyperparameters associated with each algorithm whose values are determined through extensive “parameter surveys” carried out on synthetic data sets. During a parameter survey, images of each synthetic data set are reconstructed using a broad range of possible values for each hyperparameter. Settings that produce high-fidelity image reconstructions across all synthetic data sets are collected into a “top set” of hyperparameters, and these settings are then applied for imaging the data. The resulting top sets of images capture emission structures that are consistent with the data, and we use these top-set images for the feature extraction analyses in this paper.

The imaging algorithm explores a posterior distribution over the image structure, and there are no hyperparameters that require synthetic data surveys to determine.  Rather than producing a top set of images, instead produces a sample of images drawn from the posterior determined from the data. We use these posterior image samples for the feature extraction analyses in this paper.

Morphological parameter distributions from IDFE analyses, applied to the descattered top-set and posterior images corresponding to the combined LO+HI band data from the HOPS calibration pipeline. The distributions shown correspond to combined April 6+7 results for posterior imaging and April 7 data for top-set imaging. No -based filtering has been applied.

§.§ Image-domain feature extraction methods

Given the top-set and posterior images from , we carry out IDFE analyses using two separate tools: and .  An independent cross-validation of both IDFE tools has been carried out in <cit.>.  In this section, we provide a brief overview of each method and specify the details relevant for the analyses presented in this paper.

The Ring Extractor () is an IDFE tool for quantifying the morphological properties of ring-like images. It is available as part of the software library and is described in detail in <cit.>. was the main tool used in  to extract ring properties from the 87 images, and detailed definitions of the various parameters are provided in that paper.

For the majority of the -derived ring parameters, we retain the same definitions as used in .  first defines a ring center $(x_0,y_0)$, which is determined to be the point in the image from which radial intensity profiles have a minimum dispersion in their peak intensity radii.  The ring radius, $r_0$, is then taken to be the average of these peak intensity radii over all angles, and the ring thickness $w$ is taken to be the angular average of the FWHM about the peak measured along each radial intensity profile.  To avoid biases associated with a nonzero floor to the image brightness outside of the ring, we subtract out the quantity

\begin{equation}
I_{\text{floor}} = \frac{1}{2\pi} \int_0^{2\pi} I(r_{\mathrm{max}}=60\,\uas,\phi) \text{d}\phi
\end{equation}

when computing the FWHM, i.e., we compute the average FWHM of $I(r,\phi) - I_{\text{floor}}$.
For all other ring parameters, the definitions remain the same as those used in .

defines the ring position angle $\eta$ and asymmetry $A$ as the argument and amplitude, respectively, of the first circular mode,

\begin{equation}
\beta_1 = \left\langle \frac{\int_0^{2\pi} I(\phi) \cos(\phi) \text{d}\phi}{\int_0^{2\pi} I(\phi) \text{d}\phi} \right\rangle_r ,
\end{equation}

where the angled brackets denote a radial average between $r_0 - w/2$ and $r_0 + w/2$.[We note that this definition for the position angle $\eta$ does not necessarily return the azimuthal location of the intensity peak; rather, it tracks the circular mean of the azimuthal intensity profile.]  These definitions are analogous to those used to define the corresponding position angle and asymmetry of the model (<ref> and <ref>, respectively).  The fractional central brightness $f_c$ is defined to be the ratio of the mean brightness within 5 of the center to the azimuthally averaged brightness along the ring (i.e., along $r = r_0$).

As in , we replace the negative pixels in images with zero values before performing analyses.

Comparison of two ring classification procedures.  Each panel shows a mean ring and nonring image for a single imaging pipeline, with the top row showing how the images are classified by  in the “permissive” mode and the bottom row showing the classification determined by the clustering analysis from .  All of the images have been produced using descattered data from the HOPS calibration pipeline.  The results correspond to combined April 6 and 7 data for posterior imaging and April 7 data for top-set imaging.  All of the images share a common brightness color scale; the absolute brightness scale is arbitrary because each image has been normalized to have unit total flux density.

Variational Image-Domain Analysis <cit.> is an IDFE tool for quantifying the parameters describing a specifiable image morphology; it is written in Julia <cit.> and contained in the package [<https://github.com/ptiede/VIDA.jl>].  employs a template-matching approach for image analysis, using parameterized templates to approximate an image and adjusting the parameters of the templates
until a specified cost function is minimized.  Within , the cost function takes the form of a probability divergence, which provides a distance metric between the image and template; the template parameters that minimize this divergence are taken to provide the best description of the image.  The optimization strategy and additional details are provided in <cit.>.

For the IDFE analyses in this paper, we use 's  template and the least-squares divergence (for details, see Section 8 of ).  This template describes an image structure that is similar to the model (<ref>), and it is characterized by a ring center ($x_0$, $y_0$), a ring diameter $d=2r_0$, an FHWM fractional ring thickness $w$, and a cosine expansion describing the azimuthal brightness distribution $S(\phi)$,

\begin{align}
S(\phi) = 1 - 2\sum_{k=1}^m A_k \cos\left[k (\phi - \eta_k)\right] .
\end{align}

To maintain consistency with the geometric modeling analyses (see <ref> and <ref>), we use $m=4$.  We also restrict the value of the $A_1$ parameter to be $<0.5$ to avoid negative flux in the template.  As with the model, the orientation $\eta$ is equal to the first-order phase $\eta_1$, and the asymmetry $A$ is equal to the first-order coefficient $A_1$.

To permit the presence of a central brightness floor, the  template contains an additional component in the form of a circular disk whose center point is fixed to coincide with that of the ring. The disk radius is fixed to be $r_0$. A Gaussian falloff is stitched to the outer edge of the disk, such that for radii larger than $r_0$ the intensity profile becomes a Gaussian with mean $r_0$ and an FWHM that matches the ring thickness. The flux of this disk component is a free parameter in the template.  We then retain the same definition of the fractional central brightness $f_c$ as used by .

§.§ Identifying rings via topological classification

The output of the IDFE analysis is a set of distributions for the ring parameters from each imaging method; <ref> shows an example set of results from applying both IDFE software packages to the descattered posterior and top-set images.  However, both and implicitly assume that the images fed into them contain a ring-like emission structure. If the input image does not contain a ring, then the output measurements may not be meaningful.  For each input image, we thus wish to determine both whether the image contains a ring-like structure and how sensitive the IDFE results are to the specific manner in which “a ring-like structure” is defined.

To determine whether the images we are analyzing with and contain ring-like structures, we use [<https://github.com/focisrc/metronization>], a software that preprocesses the images into a form suitable for topological analysis and extracts topologically relevant features with the help of the open-source computational topology code [<https://mrzv.org/software/dionysus2>] <cit.>. A detailed description of  can be found in <cit.>.

The  preprocessing procedure  consists of the following steps:

* First, the image undergoes a “robust” thresholding step, in which the pixels are sorted by brightness in a cumulative sequence, and all pixels below a certain threshold in this sequence have their values set to zero and the rest are set to a value of one.

* Next, in a process called “skeletonization,” the Boolean image produced in the first step is reduced to its topological skeleton that preserves the topological characteristics of the original shape. This step thins large contiguous areas of flagged pixels and enlarges the “holes.”

* The topological skeleton is rebinned and downsampled.  Holes smaller than the rebinning resolution are preserved by the skeletonization in the previous step.

* The downsampled image undergoes skeletonization once more.