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

The Variability of the Black Hole Image in M87 at the Dynamical Timescale Kaushik Satapathy 1,2 , Dimitrios Psaltis 1 , Feryal Özel 1 , Lia Medeiros 3,127 , Sean T. Dougall 1 , Chi-Kwan Chan 1 , Maciek Wielgus 4,5 , Ben S. Prather 6 , George N. Wong 6 , Charles F. Gammie 6,7 , Kazunori Akiyama 4,8,9 , Antxon Alberdi 10 , Walter Alef 11 , Juan Carlos Algaba 12 , Richard Anantua 4,5 , Keiichi Asada 13 , Rebecca Azulay 11,14,15 , Anne-Kathrin Baczko 11 , David Ball 16 , Mislav Baloković 4,5 , John Barrett 8 , Bradford A. Benson 17,18 , Dan Bintley 19 , Lindy Blackburn 4,5 , Raymond Blundell 5 , Wilfred Boland 20 , Katherine L. Bouman 4,5,21 , Geoffrey C. Bower 22 , Hope Boyce 23,24 , Michael Bremer 25 , Christiaan D. Brinkerink 26 , Roger Brissenden 4,5 , Silke Britzen 11 , Avery E. Broderick 27,28,29 , Dominique Broguiere 25 , Thomas Bronzwaer 26 , Sandra Bustamente 30 , Do-Young Byun 31,32 , John E. Carlstrom 18,33,34,35 , Andrew Chael 36,128 , Koushik Chatterjee 37 , Shami Chatterjee 38 , Ming-Tang Chen 22 , Yongjun Chen (陈永军) 39,40 , Ilje Cho 31,32 , Pierre Christian 41 , John E. Conway 42 , James M. Cordes 38 , Thomas M. Crawford 18,33 , Geoffrey B. Crew 8 , Alejandro Cruz-Osorio 43 , Yuzhu Cui 44,45 , Jordy Davelaar 26,46,47 , Mariafelicia De Laurentis 43,48,49 , Roger Deane 50,51,52 , Jessica Dempsey 19 , Gregory Desvignes 53 , Jason Dexter 54 , Sheperd S. Doeleman 4,5 , Ralph P. Eatough 11,55 , Heino Falcke 26 , Joseph Farah 4,5,56 , Vincent L. Fish 8 , Ed Fomalont 57 , H. Alyson Ford 16 , Raquel Fraga-Encinas 26 , Per Friberg 19 , Christian M. Fromm 4,5,43 , Antonio Fuentes 10 , Peter Galison 4,58,59 , Roberto García 25 , Olivier Gentaz 25 , Boris Georgiev 28,29 , Ciriaco Goddi 26,60 , Roman Gold 27,61 , Arturo I. Gómez-Ruiz 62,63 , José L. Gómez 64 , Minfeng Gu (顾敏峰) 39,65 , Mark Gurwell 5 , Kazuhiro Hada 44,45 , Daryl Haggard 23,24 , Michael H. Hecht 8 , Ronald Hesper 66 , Luis C. Ho (何子山) 67,68 , Paul Ho 13 , Mareki Honma 44,45,69 , Chih-Wei L. Huang 13 , Lei Huang (黄磊) 39,65 , David H. Hughes 62 , Shiro Ikeda 9,70,71,72 , Makoto Inoue 13 , Sara Issaoun 26 , David J. James 4,5 , Buell T. Jannuzi 16 , Michael Janssen 11 , Britton Jeter 28,29 , Wu Jiang (江悟) 39 , Alejandra Jimenez-Rosales 26 , Michael D. Johnson 4,5 , Svetlana Jorstad 73,74 , Taehyun Jung 31,32 , Mansour Karami 27,28 , Ramesh Karuppusamy 11 , Tomohisa Kawashima 75 , Garrett K. Keating 5 , Mark Kettenis 76 , Dong-Jin Kim 11 , Jae-Young Kim 11,31 , Jongsoo Kim 31 , Junhan Kim 16,21 , Motoki Kino 9,77 , Jun Yi Koay 13 , Yutaro Kofuji 44,69 , Patrick M. Koch 13 , Shoko Koyama 13,78 , Carsten Kramer 25 , Michael Kramer 11 , Thomas P. Krichbaum 11 , Cheng-Yu Kuo 13,79 , Tod R. Lauer 80 , Sang-Sung Lee 31 , Aviad Levis 21 , Yan-Rong Li (李彦荣) 81 , Zhiyuan Li (李志远) 82,83 , Michael Lindqvist 42 , Rocco Lico 10,11 , Greg Lindahl 5 , Jun Liu (刘俊) 11 , Kuo Liu 11 , Elisabetta Liuzzo 84 , Wen-Ping Lo 13,85 , Andrei P. Lobanov 11 , Laurent Loinard 86,87 , Colin Lonsdale 8 , Ru-Sen Lu (路如森) 11,88,89 , Nicholas R. MacDonald 11 , Jirong Mao (毛基荣) 90,91,92 , Nicola Marchili 11,84 , Sera Markoff 37,93 , Daniel P. Marrone 16 , Alan P. Marscher 73 , Iván Martí-Vidal 14,15 , Satoki Matsushita 13 , Lynn D. Matthews 8 , Karl M. Menten 11 , Izumi Mizuno 19 , Yosuke Mizuno 43,94 , James M. Moran 4,5 , Kotaro Moriyama 8,44 , Monika Moscibrodzka 26 , Cornelia Müller 11,26 , Alejandro Mus Mejías 14,15 , Gibwa Musoke 26,37 , Hiroshi Nagai 9,45 , Neil M. Nagar 95 , Masanori Nakamura 13,96 , Ramesh Narayan 4,5 , Gopal Narayanan 30 , Iniyan Natarajan 50,52,97 , Antonios Nathanail 43 , Joey Neilsen 98 , Roberto Neri 25 , Chunchong Ni 28,29 , Aristeidis Noutsos 11 , Michael A. Nowak 99 , Hiroki Okino 44,69 , Héctor Olivares 26 , Gisela N. Ortiz-León 11 , Tomoaki Oyama 44 , Daniel C. M. Palumbo 4,5 , Jongho Park 13,129 , Nimesh Patel 5 , Ue-Li Pen 27,100,101,102 , Dominic W. Pesce 4,5 , Vincent Piétu 25 , Richard Plambeck 103 , Aleksandar PopStefanija 30 , Oliver Porth 37,43 , Felix M. Pötzl 11 , Jorge A. Preciado-López 27 , Hung-Yi Pu 13,27,104 , Venkatessh Ramakrishnan 95 , Ramprasad Rao 22 , Mark G. Rawlings 19 , Alexander W. Raymond 4,5 , Luciano Rezzolla 43,105,106 , Bart Ripperda 47,107 , Freek Roelofs 5,26 , Alan Rogers 8 , Eduardo Ros 11 , Mel Rose 16 , Arash Roshanineshat 16 , Helge Rottmann 11 , Alan L. Roy 11 , Chet Ruszczyk 8 , Kazi L. J. Rygl 84 , Salvador Sánchez 108 , David Sánchez-Arguelles 62,109 , Mahito Sasada 44,110 , Tuomas Savolainen 11,111,112 , F. Peter Schloerb 30 , Karl-Friedrich Schuster 25 , Lijing Shao 11,68 , Zhiqiang Shen (沈志强) 39,40 , Des Small 76 , Bong Won Sohn 31,32,113 , Jason SooHoo 8 , He Sun (孙赫) 21 , Fumie Tazaki 44 , Alexandra J. Tetarenko 19 , Paul Tiede 28,29 , Remo P. J. Tilanus 16,26,114,115 , Michael Titus 8 , Kenji Toma 116,117 , Pablo Torne 11,108 , Efthalia Traianou 11 , Tyler Trent 16 , Sascha Trippe 118 , Ilse van Bemmel 76 , Huib Jan van Langevelde 76,119 , Daniel R. van Rossum 26 , Jan Wagner 11 , Derek Ward-Thompson 120 , John Wardle 121 , Jonathan Weintroub 4,5 , Norbert Wex 11 , Robert Wharton 11 , Kaj Wiik 122 , Qingwen Wu (吴庆文) 123 , Doosoo Yoon 37 , André Young 26 , Ken Young 5 , Ziri Younsi 43,124 , Feng Yuan (袁峰) 39,65,125 , Ye-Fei Yuan (袁业飞) 126 , J. Anton Zensus 11 , Guang-Yao Zhao 10 , and Shan-Shan Zhao 39 1 Astronomy Department, University of Arizona, 933 N. Cherry Ave, Tucson, AZ 85721, USA 2 Department of Physics, University of Arizona, 1118 E 4th St, Tucson, AZ 85721, USA 3 School of Natural Sciences, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA 4 Black Hole Initiative at Harvard University, 20 Garden Street, Cambridge, MA 02138, USA 5 Center for Astrophysics, Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA 6 Illinois Center for Advanced Studies of the Universe, Department of Physics, University of Illinois, 1110 West Green Street, Urbana, IL 61801, USA 7 Department of Astronomy, University of Illinois, 1002 West Green Street, Urbana, IL 61801, USA 8 Massachusetts Institute of Technology Haystack Observatory, 99 Millstone Road, Westford, MA 01886, USA 9 National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan 10 Instituto de Astrofísica de Andalucía-CSIC, Glorieta de la Astronomía s/n, E-18008 Granada, Spain The Astrophysical Journal, 925:13 (19pp), 2022 January 20 https://doi.org/10.3847/1538-4357/ac332e © 2022. The Author(s). Published by the American Astronomical Society. 1 11 Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany 12 Department of Physics, Faculty of Science, University of Malaya, 50603 Kuala Lumpur, Malaysia 13 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. 14 Departament d’Astronomia i Astrofísica, Universitat de València, C. Dr. Moliner 50, E-46100 Burjassot, València, Spain 15 Observatori Astronòmic, Universitat de València, C. Catedrático José Beltrán 2, E-46980 Paterna, València, Spain 16 Steward Observatory and Department of Astronomy, University of Arizona, 933 N. Cherry Ave., Tucson, AZ 85721, USA 17 Fermi National Accelerator Laboratory, MS209, P.O. Box 500, Batavia, IL 60510, USA 18 Department of Astronomy and Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA 19 East Asian Observatory, 660 N. A’ohoku Place, Hilo, HI 96720, USA 20 Nederlandse Onderzoekschool voor Astronomie (NOVA), PO Box 9513, 2300 RA Leiden, The Netherlands 21 California Institute of Technology, 1200 East California Boulevard, Pasadena, CA 91125, USA 22 Institute of Astronomy and Astrophysics, Academia Sinica, 645 N. A’ohoku Place, Hilo, HI 96720, USA 23 Department of Physics, McGill University, 3600 rue University, Montréal, QC, H3A 2T8, Canada 24 McGill Space Institute, McGill University, 3550 rue University, Montréal, QC, H3A 2A7, Canada 25 Institut de Radioastronomie Millimétrique, 300 rue de la Piscine, F-38406 Saint Martin d’Hères, France 26 Department of Astrophysics, Institute for Mathematics, Astrophysics and Particle Physics (IMAPP), Radboud University, P.O. 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Gobetti 101, I-40129 Bologna, Italy 2 The Astrophysical Journal, 925:13 (19pp), 2022 January 20 Satapathy et al. 85 Department of Physics, National Taiwan University, No.1, Sect. 4, Roosevelt Rd., Taipei 10617, Taiwan, R.O.C 86 Instituto de Radioastronomía y Astrofísica, Universidad Nacional Autónoma de México, Morelia 58089, México 87 Instituto de Astronomía, Universidad Nacional Autónoma de México, CdMx 04510, México 88 Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, Peopleʼs Republic of China 89 Key Laboratory of Radio Astronomy, Chinese Academy of Sciences, Nanjing 210008, Peopleʼs Republic of China 90 Yunnan Observatories, Chinese Academy of Sciences, 650011 Kunming, Yunnan Province, Peopleʼs Republic of China 91 Center for Astronomical Mega-Science, Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing, 100012, Peopleʼs Republic of China 92 Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese Academy of Sciences, 650011 Kunming, Peopleʼs Republic of China 93 Gravitation Astroparticle Physics Amsterdam (GRAPPA)Institute, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands 94 Tsung-Dao Lee Institute and School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, 200240, Peopleʼs Republic of China 95 Astronomy Department, Universidad de Concepción, Casilla 160-C, Concepción, Chile 96 National Institute of Technology, Hachinohe College, 16-1 Uwanotai, Tamonoki, Hachinohe City, Aomori 039-1192, Japan 97 South African Radio Astronomy Observatory, Observatory 7925, Cape Town, South Africa 98 Villanova University, Mendel Science Center Rm. 263B, 800 E Lancaster Ave, Villanova, PA 19085, USA 99 Physics Department, Washington University CB 1105, St Louis, MO 63130, USA 100 Canadian Institute for Theoretical Astrophysics, University of Toronto, 60 St. George Street, Toronto, ON, M5S 3H8, Canada 101 Dunlap Institute for Astronomy and Astrophysics, University of Toronto, 50 St. George Street, Toronto, ON, M5S 3H4, Canada 102 Canadian Institute for Advanced Research, 180 Dundas St West, Toronto, ON, M5G 1Z8, Canada 103 Radio Astronomy Laboratory, University of California, Berkeley, CA 94720, USA 104 Department of Physics, National Taiwan Normal University, No. 88, Sec. 4, Tingzhou Rd., Taipei 116, Taiwan, R.O.C. 105 Frankfurt Institute for Advanced Studies, Ruth-Moufang-Strasse 1, D-60438 Frankfurt, Germany 106 School of Mathematics, Trinity College, Dublin 2, Ireland 107 Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ 08544, USA 108 Instituto de Radioastronomía Milimétrica, IRAM, Avenida Divina Pastora 7, Local 20, E-18012, Granada, Spain 109 Consejo Nacional de Ciencia y Tecnología, Av. Insurgentes Sur 1582, 03940, Ciudad de México, México 110 Hiroshima Astrophysical Science Center, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526, Japan 111 Aalto University Department of Electronics and Nanoengineering, PL 15500, FI-00076 Aalto, Finland 112 Aalto University Metsähovi Radio Observatory, Metsähovintie 114, FI-02540 Kylmälä, Finland 113 Department of Astronomy, Yonsei University, Yonsei-ro 50, Seodaemun-gu, 03722 Seoul, Republic of Korea 114 Leiden Observatory—Allegro, Leiden University, P.O. Box 9513, 2300 RA Leiden, The Netherlands 115 Netherlands Organisation for Scientific Research (NWO), Postbus 93138, 2509 AC Den Haag, The Netherlands 116 Frontier Research Institute for Interdisciplinary Sciences, Tohoku University, Sendai 980-8578, Japan 117 Astronomical Institute, Tohoku University, Sendai 980-8578, Japan 118 Department of Physics and Astronomy, Seoul National University, Gwanak-gu, Seoul 08826, Republic of Korea 119 Leiden Observatory, Leiden University, Postbus 2300, 9513 RA Leiden, The Netherlands 120 Jeremiah Horrocks Institute, University of Central Lancashire, Preston, PR1 2HE, UK 121 Physics Department, Brandeis University, 415 South Street, Waltham, MA 02453, USA 122 Tuorla Observatory, Department of Physics and Astronomy, University of Turku, Finland 123 School of Physics, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, Peopleʼs Republic of China 124 Mullard Space Science Laboratory, University College London, Holmbury St. Mary, Dorking, Surrey, RH5 6NT, UK 125 School of Astronomy and Space Sciences, University of Chinese Academy of Sciences, No. 19A Yuquan Road, Beijing 100049, Peopleʼs Republic of China 126 Astronomy Department, University of Science and Technology of China, Hefei 230026, Peopleʼs Republic of China Received 2021 June 3; revised 2021 October 21; accepted 2021 October 24; published 2022 January 20 Abstract The black hole images obtained with the Event Horizon Telescope (EHT)are expected to be variable at the dynamical timescale near their horizons. For the black hole at the center of the M87 galaxy, this timescale (5–61 days)is comparable to the 6 day extent of the 2017 EHT observations. Closure phases along baseline triangles are robust interferometric observables that are sensitive to the expected structural changes of the images but are free of stationbased atmospheric and instrumental errors. We explored the day-to-day variability in closure-phase measurements on all six linearly independent nontrivial baseline triangles that can be formed from the 2017 observations. We showed that three triangles exhibit very low day-to-day variability, with a dispersion of ∼3°–5°. The only triangles that exhibit substantially higher variability (∼90°–180°)are the ones with baselines that cross the visibility amplitude minima on the u–vplane, as expected from theoretical modeling. We used two sets of general relativistic magnetohydrodynamic simulations to explore the dependence of the predicted variability on various black hole and accretion-flow parameters. We found that changing the magnetic field configuration, electron temperature model, or black hole spin has a marginal effect on the model consistency with the observed level of variability. On the other hand, the most discriminating image characteristic of models is the fractional width of the bright ring of emission. Models that best reproduce the observed small level of variability are characterized by thin ring-like images with structures dominated by gravitational lensing effects and thus least affected by turbulence in the accreting plasmas. Unified Astronomy Thesaurus concepts: Black hole physics (159);High energy astrophysics (739) 1. Introduction The Event Horizon Telescope (EHT)has produced horizonscale images of the black hole in the center of the M87 galaxy using Very Long Baseline Interferometry (VLBI)at a wavelength of 1.3 mm (EHT Collaboration et al. 2019a,2019b, 2019c,2019d,2019e,2019f). Together with upcoming studies 127 NSF Astronomy and Astrophysics Postdoctoral Fellow. 128 NASA Hubble Fellowship Program, Einstein Fellow. 129 EACOA fellow. Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s)and the title of the work, journal citation and DOI. 3 The Astrophysical Journal, 925:13 (19pp), 2022 January 20 Satapathy et al. of the black hole in the center of the Milky Way, Sgr A * , these images have opened up numerous avenues to studying physics at event-horizon scales related to both accretion processes and gravity. The structure of the accretion disks and jets observed around the black holes will improve our understanding of the behavior of plasma in these environments (EHT Collaboration et al. 2019e,2019f). The measurements of the sizes and shapes of the black hole shadows give insights into deviations of the black hole spacetime from the Kerr metric (Psaltis 2019; Psaltis et al.2020). Accretion flows, which make up the horizon-scale environments of black holes, are expected to be highly variable by nature. The radiation observed from these regions at 1.3 mm is dominated by synchrotron emission from electrons that are heated and accelerated by magnetohydrodynamic turbulence (EHT Collaboration et al. 2019e). The accretion flow is also expected to evolve at the dynamical timescales near the innermost stable circular orbit (ISCO), which range from 4 to 56 minutes for Sgr A * (for M BH =4.148 ×10 6 M e ; see Gravity Collaboration et al. 2019)and5to61daysforM87(depending on the black hole spin; Bardeen et al. 1972). The timescales suggest that there can be substantial variability in the source structure over a single observing night for Sgr A * , which makes it difficult to obtain a static image. However, we expect substantial source variability in M87 only for observations that span about a week or longer. Evidence for long-term variability in the image structure of M87 has been reported recently, based on VLBI observations spanning nearly a decade (Wielgus & The EHT Collaboration 2020). The EHT observed M87 in 2017 April for four nights with a maximum separation of six calendar days between observations. For a black hole of the inferred mass (M BH = 6.5 ×10 9 M e ; Gebhardt et al. 2011; EHT Collaboration et al. 2019f), the six days correspond to 16 GM/c 3 . This is approximately equal to ∼3.5 dynamical timescales at the ISCO for a maximally spinning black hole. Over this observation span, only a limited amount of variability in source structure was inferred between days, reflected as a small change in the thickness or azimuthal brightness distribution of the observed bright emission ring (EHT Collaboration et al. 2019d). Our goal in this paper is to use general relativistic magnetohydrodynamic (GRMHD)simulations of accretion flows, followed by general relativistic ray tracing and radiative transfer, with parameters that are relevant to the M87 black hole, in order to explore the short-term variability in the source structure as it is imprinted in the interferometric observables. The EHT, being an interferometer, measures complex visibilities, which denote the two-dimensional Fourier transform of the source image on the sky. The measurements are typically decomposed into visibility amplitudes and phases (Thompson et al. 2017). Image reconstruction techniques enable us to use these variables and create the map of the source brightness on the plane of the sky (see EHT Collaboration et al. 2019d). The visibility measurements in VLBI are affected by timedependent atmospheric and instrumental errors of the telescopes, particularly challenging for the short radio wavelengths (Thompson et al. 2017). The custom-built EHT data reduction pipeline addresses the specific requirements of the millimeter VLBI (EHT Collaboration et al. 2019c). The pipeline uses Atacama Large Millimeter/submillimeter Array (ALMA)as an anchor station, leveraging its extreme sensitivity for the calibration of the entire EHT array (Blackburn et al. 2019; Janssen et al. 2019). As a result, the measured visibilities indicate sufficient phase-stability to enable long coherent averaging and building up the high signal-to-noise ratio. Remaining station-based errors can be modeled as complex gains multiplying the visibilities. In order to eliminate the effects of these errors in the data, one can construct closure quantities with amplitudes and phases (Jennison 1958; Kulkarni 1989; Thompson et al. 2017; Blackburn et al. 2020). For a set of three baselines forming a closure triangle, the closure phase is defined as the sum of the measured visibility phases in each of these baselines. This cancels out the station-based phase measurement errors for each station. As a result, the closure phases are optimal and robust probes of the intrinsic structure of the source. Closure phases have indeed proven to be a powerful tool to study and quantify variability in synthetic data from GRMHD simulations. Medeiros et al. (2018)studied the dependence of variability in closure phases on the orientations and baseline lengths of the closure triangles (see also Roelofs et al. 2017). Triangles that involve large baselines (that probe small length scales in the source image)were shown to have a high degree of variability in closure phases. On the other hand, triangles that involve small baselines (that probe the overall size of the ring and source structure)exhibit less variability. The exception to this rule are triangles involving a baseline close to a deep visibility minimum. The visibility phases in regions of visibility minima are extremely sensitive to minor changes in source structure. This localization of variability in the Fourier space is vital in understanding variability in GRMHD simulations and in observations. In this paper, we first present a data-driven model to quantify the variability of observed closure phases in the various linearly independent triangles of the M87 observations across the six days of observations in 2017. We apply this algorithm to the 2017 EHT data on M87 and identify three closure triangles that show a remarkably small degree of variability. We then explore the degree of variability produced in a large set of GRMHD simulations, with different magnetic field configurations and prescriptions of the plasma physics, and understand the effects of various model parameters on the variability in the models. Finally, we compare the predictions of the GRMHD simulations to the observations and discuss how the latter constrain the physical properties of the accretion flow in M87. 2. Closure-phase Observations of M87 The EHT measures complex visibilities, given by () () () () Vuv dxdyIxy e,,,1 iux vy2 òò =p-+ where I(x,y)represents the total intensity of radiation at a given spatial location (x,y)on the image plane and (u,v)denote the Fourier frequencies corresponding to the (x,y)coordinates. However, due to atmospheric and instrumental effects, the measured visibilities do not represent the actual visibilities of the source. The measured visibilities are related to the source visibilities through complex gains. The measured visibility between the ith and the jth telescopes, ij V , can be written as ()gg V,2 ij ijij =* V where g i and g j are the complex gains associated with the two telescopes, and the star superscript indicates complex conjugates. We define the closure phase as the argument of the 4 The Astrophysical Journal, 925:13 (19pp), 2022 January 20 Satapathy et al. complex bispectrum along a closed triangle of three telescopes (Jennison 1958). The closure phase ijk CP f for a closure triangle formed by the ith, the jth, and the kth telescopes is hence given by [] [] [∣ ∣∣ ∣∣ ∣ ] [] () gg V gg V g g V gggVVV VV V Arg Arg Arg Arg ,3 ijk ij jk ki ijij jkjk kiki ijk ij jk ki ij jk ki ij jk ki CP 22 2 f fff = = = = =+ + ** * VV V where f ij denotes the visibility phase between the ith and the jth telescopes. Equation (3)shows that the measured closure phase in a triangle is equal to the actual closure phase represented by the source. The EHT observed M87 on four nights in 2017 April 5, 6, 10, and 11. The observations involved seven stations spread across five geographical locations (EHT Collaboration et al. 2019b): the Atacama Large Millimeter Array (ALMA)and the Atacama Pathfinder Experiment (APEX)in Chile; the James Clerk Maxwell Telescope (JCMT)and the Submillimeter Array (SMA)in Hawaii; the Large Millimeter Telescope (LMT)in Mexico; the Submillimeter Telescope Observatory (SMT)in Arizona; and the IRAM 30 m telescope in Pico Veleta (PV)in Spain. On each night of observation, the rotation of the Earth implies that each baseline traces out an elliptical path in the u–v space with the progression of time. Figure 1shows the u–vcoverage for M87 during one of the observing days (left panel)as well as the visibility amplitudes measured as a function of baseline length on the same day (right panel). Two deep minima in visibility amplitude can be seen at baseline lengths of ∼3.4 and ∼8.3 Gλencountered in three baselines (LMT-SMA, SMT-SMA, and PV-SMA)along the east–west orientation (indicated in blue). In contrast, the minimum encountered by the ALMA-LMT baseline is only marginally deep (right panel of Figure 1). We construct closure-phase quantities using all the baselines shown in Figure 1. However, closure triangles with two telescopes at the same geographical location yield trivial closure phases, with deviations from zero arising due to changes in the large scale source structure alongside systematic and thermal errors (EHT Collaboration et al. 2019c). While these triangles are usefultoquantifysystematic errors in closure phases, they do not capture any information about the source structure at horizon scales. Hence, we only choose triangles with telescopes at different geographical locations for our study. Since Chile and Hawaii have two stations each, we pick the station with the higher signal-tonoise ratio. We, therefore, consider five stations—ALMA, SMA, LMT, SMT, and PV, out of which six linearly independent closure phases can be constructed. We list the six triangles in Table 1. We choose the triangles such that all of them involve ALMA, which has the highest signal-to-noise ratio, and hence smaller uncertainties in closure-phase measurements. In Figure 2, we show the evolution of closure phases with time in Greenwich Mean Sidereal Time (GMST)coordinates over all four days of observations in three triangles that exhibit substantial day-to-day variability in closure phases over the span of the 2017 campaign (see also EHT Collaboration et al. 2019c, Section 7.3.2): Figure 1. Left: the u–vcoverage traced by the EHT for the M87 observations on 2017 April 11, with the dashed circles indicating locations of the two observed visibility minima. The colors indicate different orientations of the u–vcoverage. Right: the visibility amplitude as a function of baseline length on the same day. Table 1 Closure Triangles Used with Baseline Lengths and Inferred Variability in Closure Phases Triangle a Baseline Lengths (Gλ) b Inferred Variability c 123 ALMA-SMA-LMT 6.7–7.2 3.1–4.5 3.4–4.1 ∼30°–60° ALMA-LMT-SMT 3.4–4.2 1.3–1.5 4.8–5.5 3°.4 ALMA-PV-LMT 6.0–6.6 5.1–6.4 3.4–4.2 4°.6 ALMA-SMA-SMT 6.7–7.2 2.4–3.5 4.8–5.5 ∼180° ALMA-PV-SMA 6.0–6.2 8.2–8.3 6.8–7.0 L ALMA-PV-SMT 6.0–6.6 5.5–6.4 5.3–5.5 5°.4 Notes. a The bold baselines indicate the triangles that cross one of the deep visibility minima in the E-W orientation at 3.4 and 8.3 Gλ. b Baselines labeled 1, 2, and 3 in a triangle labeled as A-B-C correspond to the baselines A-B, B-C, and C-A, respectively. c An estimated range of variation in the high-variability triangles, and the maximum likelihood value of the inferred Gaussian variability (Equation (7)) in the low-variability triangles. 5 The Astrophysical Journal, 925:13 (19pp), 2022 January 20 Satapathy et al. (i)The ALMA-SMA-LMT triangle shows a change of 30°– 60°between the first two days and the later two days of observation. (ii)The ALMA-SMA-SMT triangle shows a swing of 180° on each day of observation. In addition to this, the direction of this swing flips between the first two days and the later two days of observation. (iii)In the ALMA-PV-SMA triangle, it is difficult to quantify the degree of variability because of the reduced complex visibility amplitudes between PV and SMA baselines, and the low signal-to-noise ratio in some data points. However, a persistent structure in the evolution of closure phases is not observed. We note that each of these closure triangles involves one of the baselines (i.e., LMT-SMA, SMT-SMA, and PV-SMA)that cross a deep visibility minimum in the E-W orientation. In addition, we note that, in the ALMA-SMA-SMT triangle, the closure phases show little day-to-day variability until ∼19 GMST on each day. We attribute this behavior to the fact that the SMT-SMA baseline encounters the deep visibility minimum at ∼19 GMST. We plan to analyze the dependence of variability in the observations on the depth of the visibility minima in a future study. The closure phases in the remaining triangles (i.e., ALMALMT-SMT, ALMA-PV-LMT, and ALMA-PV-SMT)show a persistent and continuous evolution with time during each day of observation (see Figure 3). This behavior represents the evolution of closure phases as the various baselines trace their paths on the u–vplane following the rotation of the Earth. However, the same evolution with time is repeated in all days of observations; there is little scatter around the general trend set by the rotation of the baselines. The presence of substantial scatter from day to day would have been a signature of structure variability in the image. In order to compare the observations to theoretical models, we quantify the degree of closure-phase day-to-day variability observed in this last set of triangles. Upon examination of the closure phases in the maximal set of closure triangles, we find that, outside the set of linearly independent triangles that we choose, only one triangle (LMT-SMT-PV)shows a low degree of day-to-day variability. We conclude that it is optimal and sufficient to quantify the day-to-day variability in the lowvariability triangles that we choose in Table 1, since the LMTSMT-PV triangle can be obtained by a linear combination of these three triangles. We also note that the comparison between observations and models could, in principle, be carried out for both highvariability and low-variability triangles. However, as we will discuss in Section 4, images from theoretical models uniformly show a high degree of phase variability in baselines that cross a visibility minimum and, as a result, a comparison with the high-variability triangles in the data turns out not to have much discriminating power between various models. The degree of variability exhibited on the low-variability triangles identified in the data, on the other hand, is a more significant challenge to the GRMHD models and proves to be a useful tool to distinguish between them. The quantification of day-to-day variability in the set of three low-variable triangles is not straightforward, because closure phases evolve with time on each observing day but the individual scans on each day are not aligned at the exact same times. As a result, a difference in closure phases measured in two scans separated by almost (but not exactly)one sidereal day incorporates both the change due to the slightly different locations on the u–vplane probed by the two scans and the change due to the structural changes in the image. In order to disentangle the two sources of variability, we employ a data-driven analytic model for the evolution of the closure phase with time during a single day of observation. This is meant to capture the change in closure phases introduced by the changing location on the u–vplane of the Figure 2. The evolution of closure phases plotted with time for all four days of observation for closure triangles in which baselines are known to encounter regions of deep visibility minima. All three triangles exhibit a high level of variability in closure phases across six days of observation. 6 The Astrophysical Journal, 925:13 (19pp), 2022 January 20 Satapathy et al. closure triangles. We then compare the overall change of the parameters of this model from day to day, in order to quantify the variability due to structural changes of the image. For our data-driven model, f model (t), we use a second-order polynomial in the observation time t(in GMST coordinates), given by () ( )tcctct.4 model 01 2 2 f=+ + In order to account for potential structural changes from day to day, we allow for an intrinsic Gaussian spread in the zerothorder coefficient of this polynomial (c 0 ), with a standard deviation of c0 s . In this prescription, we do not assign any physical significance to the polynomial coefficients. Since the maximum separation between any two nights of observations is six days, c0 s acts as a constraint on the overall change in closure phase at a given time in a given triangle across six days. Using this model, we define the likelihood of making a particular closure-phase measurement using a Gaussian mixture model as () () {()} () () Lc cc t ,,, 1 2 exp 1 2,5 ic ic ii ic 012 22 model 2 22 0 0 0 s ps s ff ss º+ ´-- + ⎧ ⎨ ⎩ ⎫ ⎬ ⎭ ⎡ ⎣ ⎢⎤ ⎦ ⎥ Figure 3. The parameters and functional forms of the data-driven closure-phase evolution model, applied to the closure triangles that exhibit low variability across the six days of 2017 EHT observations (see Equation (4)). Left: the posterior projected on the –c c0 0 s parameter plane, where σ 0 is a measure of the variability across the six days. The red dots indicate the most likely values. The contours indicate the levels containing 68% and 95% of the posterior probability. The dashed horizontal lines indicate the systematic error of 2°in closure-phase observations. Right: closure-phase observations plotted as a function of time for the same triangles. The cyan band indicates the most likely value of the width of the model c0 s . 7 The Astrophysical Journal, 925:13 (19pp), 2022 January 20 Satapathy et al. where f i is the closure phase in a triangle at an observation time t i , and σ i denotes the uncertainty in f i . Note that we have assumed here Gaussian statistics for the measurements of closure phases. This assumption is not expected to introduce any significant changes in our quantitative results since the triangles we will be applying this method to are characterized by high signal-to-noise measurements. Applying this mixture model to data with large errors would require using an appropriate distribution for the closure-phase errors and their correlations (see Christian & Psaltis 2020). The likelihood that nclosure-phase observations in a given triangle across all four days of observation follow the model f model (t)is ()()()Lc c c L c c c,,, ,,, 6 c i n ic012 1 012 00  ss= = () {()} () () t 1 2 exp 1 2. 7 i n ic i n ii ic122 1 model 2 22 00 å ps s ff ss =+-- + == ⎧ ⎨ ⎩ ⎫ ⎬ ⎭⎡ ⎣ ⎢⎤ ⎦ ⎥ Assuming flat priors in the polynomial coefficients and in c0 s , we map the parameter space using the Markov-chain Monte Carlo sampling. Figure 3shows the posteriors over two of the model parameters, c 0 and c0 s , for each of the three triangles that exhibits low day-to-day variability. The left panels show the 68% and 95% contours of the posteriors; the right panels show the data-driven model with the most likely value of the parameters, overplotted on the closure-phase observations in these triangles. The most likely standard deviations of variability across the six days of observations are ∼3°–5°. This is comparable to the systematic error in closure-phase observations of 2°as reported in EHT Collaboration et al. (2019c)and is remarkably small. 3. GRMHD Simulations and Library A large suite of GRMHD simulations has been generated to model the accretion flow around M87. The simulations have been performed using the algorithms harm (see Gammie et al. 2003),BHAC (Olivares Sánchez et al. 2018),KORAL (Sądowski et al. 2014), etc. The simulations have been initialized with two magnetic field configurations that led to different field structures in the flows: Standard and Normal Evolution (SANE; see Igumenshchev et al. 2003)and Magnetically Arrested Disk (MAD; see Narayan et al. 2012 and Sądowski et al. 2013).A comprehensive comparison of the GRMHD algorithms is presented in Porth et al. (2019). The set of observables that results from these simulations has been obtained by general relativistic ray tracing and radiative transfer algorithms, such as GRay (Chan et al. 2013)and ipole (Mościbrodzka & Gammie 2018). In this paper, we analyze the GRMHD simulations that were run using harm (see Gammie et al. 2003; Narayan et al. 2012). The simulations include SANE and MAD configurations of the magnetic fields and different black hole spins with corotating or counterrotating accretion disks. The flow properties, as calculated in the GRMHD simulations, scale with the mass of the black hole and, therefore, the latter does not enter the calculation. The dependence of the electron temperature on the local value of the plasma βparameter aims to capture the effects of sub-grid electron physics that are not resolved in the GRMHD simulations (see Chan et al. 2015). The accretion flows in the simulations are modeled as collisionless plasmas, where the ratio of ion temperature to electron temperature (R≡T i /T e )is given by ()RR1 1 1.8 high 2 22 b bb =+++ Here, β≡P gas /P mag is the ratio of gas pressure (P gas )to the magnetic pressure (P mag ). This prescription aims to simulate the effects of sub-grid electron heating, using this simple, physics-motivated parametric form (Mościbrodzka et al. 2016; and EHT Collaboration et al. 2019e). General relativistic ray tracing and radiative transfer is performed on the simulated system at the wavelength of 1.3 mm in order to generate the images of the black hole using the different snapshots of the simulations. For radiative transfer, the black hole mass (M BH )and the overall electron number-density scale in the accretion disk (n e )set a scale for the mass and the optical depth in the accretion disk, which determines the mass accretion rate and the flux of emission in the source. It follows from the scaling properties of the emissivity of the accretion disk at 1.3 mm and from the natural scaling of the GRMHD simulations (see Chan et al. 2015)that n e and M BH are degenerate quantities with the resulting images and, therefore, the various interferometric observables, depending only on the product Mn e BH 2(see a derivation and discussion in the Appendix). We use two sets of GRMHD models for studying closurephase variability: (i)The first set of models (henceforth Set A)is aimed at understanding the dependence of the closure-phase variability on the properties of the accretion flow, i.e., MAD versus SANE magnetic field configuration, the plasma prescription, and the electron number-density scale (n e ). In particular, they include SANE and MAD models for a single spin, three different values of R high , and five different values of n e (defined at a fiducial mass of M BH =6.5 ×10 9 M e ). For this value of the black hole mass, this range allows us to explore compact fluxes that are up to a factor of a few above and below the nominal value of 0.5 Jy. Each model is used to generate 1024 images with a time cadence of 10 GM/c 3 , amounting to a total of ∼30,000 images in this set. Ray tracing and radiative transfer calculations for this simulation set are carried out using GRay. (ii)The second set of models (henceforth Set B)explores a comprehensive set of black hole spins and R high of the accretion plasma, which are used in EHT Collaboration et al. (2019e)in the interpretation of the EHT observations. The electron number-density scale (n e )in this simulation set is tuned a priori such that the total flux of radiation at 1.3 mm is equal to 0.5 Jy for a fiducial mass of M BH =6.2 ×10 9 M e . Each model in this simulation set is used to generate 600–1000 images at a time cadence of 5 GM/c 3 , amounting to ∼50,000 images in this set. Ray tracing and radiative transfer calculations for this simulation set are carried out using ipole. The ray tracing and radiative transfer tracing calculations on the GRMHD simulations performed using ipole and GRay ignore the effects of finite light travel time. Bronzwaer et al. (2018)showed that the effect of the approximation on the lightcurve of flux density in a simulation is restricted to only a few percent. Depending upon the nature of the spatial 8 The Astrophysical Journal, 925:13 (19pp), 2022 January 20 Satapathy et al. correlations in structural variability in the simulation, the computed images may carry either time-correlated or timeuncorrelated features. These features will have an effect on the amplitude of variability inferred. In this study, we ignore these effects in inferring the variability in the images. The parameter spaces studied in sets A and B are listed in Table 2. Although the mass of the black hole M BH does not enter as an independent parameter in the calculation of the black hole images, it affects our interpretation of the observations in the following ways: (i)The characteristic time unit (τ)in the GRMHD simulations is set by the mass of the black hole as τ=GM BH /c 3 . A higher mass of the black hole implies that an observation span of 6 days would translate to a shorter elapsed time in units of τ. (ii)The angular size of the image as seen by a distant observer depends on the black hole mass (Δθ∼λ/M BH ), which implies that the EHT probes different regions in the Fourier space for black holes of different masses. A higher mass of the black hole translates to the EHT probing structures at smaller length scales. The mass of M87 has been measured using two methods prior to the EHT observations (EHT Collaboration et al. 2019e). Gebhardt et al. (2011)measured a mass of M BH =6.6 ± 0.4 ×10 9 M e using stellar dynamics. Walsh et al. (2013) measured a mass of  MM3.5 10 BH 0.7 0.9 9 =´ - +by studying emission lines from ionized gas. The GRMHD models in simulation sets A and B give images for which the ring sizes and widths correspond to masses in the range inferred by the stellar dynamics measurement of Gebhardt et al. (2011;seeEHT Collaboration et al. 2019f). Because of this, we allow the black hole mass to vary in the range 5 ×10 9 M e M BH 9.5 ×10 9 M e in both sets of the simulations. The orientation of the black hole spin axis has been constrained by long-wavelength observations of its jet at a position angle of ∼288°north of east and at an inclination of 17°with respect to the line of sight (see discussion in EHT Collaboration et al. 2019f). We trace the Set A models at an inclination of 17°. In Set B, we trace the positive spin models at an inclination of 163°and the negative spin models at 17°. The simulations typically run for lengths that correspond to ∼10 4 GM/c 3 . Here, we only use images obtained from the relaxed turbulent state with a slowly varying mass accretion rate. This corresponds to simulation times in the range of 5×10 3 t10 4 GM/c 3 . 4. Closure-phase Variability in GRMHD Simulations In order to analyze visibility phase and closure-phase variability in GRMHD models, we first transform the GRMHD snapshots into the visibility space by performing a twodimensional Fourier transform on each of the images. We then compute the amplitude and phase from the complex visibilities at each location in the u–vplane. Since the visibility phase is a directional quantity, we need to employ directional statistics to infer the standard deviation in a time-series. For a given time-series of nphases θ i , we compute its directional dispersion as {( ¯)} ( )Dn 11cos , 9 i n i 1 åqq=-- = where ¯ q represents the circular mean, defined as ¯() () ()tan sin cos .10 i ni i ni 11 1 qq q =å å -= = ⎧ ⎨ ⎩⎫ ⎬ ⎭ In the limit of small deviations in the time-series from the circular mean ¯ q , the directional dispersion can be approximated as ()D2,11 2 s » where σis inferred as a standard deviation of the time-series θ i for small fluctuations about ¯ q . We use this quantity to construct heat maps of phase variability in order to understand the dependence of variability on the u–vcoordinates (see also Medeiros et al. 2017).We present an example of such a heat map for one of the simulations in Set B in Figure 4. The left panel shows the mean visibility amplitude on the u–vplane computed across the simulation run and reveals deep visibility minima that are aligned with the spin axis of the black hole. The right panel shows a heat map of the visibility phase dispersion. There are hot regions of visibility phase variability, i.e., localized regions in u–vspace that show large dispersions. These regions lie primarily along the spin axis of the black hole and at baseline lengths that correspond to the deep minima in visibility amplitudes. In fact, a natural anticorrelation between phase variability and mean visibility amplitude is observed in all the GRMHD models, which was also explored in Medeiros et al. (2017; see their Figures 3 and 4). Based on this understanding, one expects the closure-phase variability computed along various triangles to depend on the locations of the three vertices in the u–vspace. In particular, the localized nature of variability implies that closure triangles that have baselines crossing the hot regions are expected to exhibit high variability in closure phases, whereas triangles with vertices in quiet parts of the u–vspace should show a low level of variability. In Figure 4, we show examples of two such triangles: the ALMA-SMA-LMT triangle has a baseline on a hot region and is expected to be highly variable, while a lowvariability closure triangle (ALMA-LMT-SMT)avoids the hot regions. The azimuthal extent of the hot regions of phase dispersion in the u–vplane depends on the degree of symmetry of the Table 2 GRMHD Simulation Model Parameters Parameter a Parameter Space Set A Set B Model type MAD, SANE MAD, SANE Black hole spin b +0.9 0.0, ±0.5, ±0.94 Plasma R high 1, 20, 80 1, 10, 20, 40, 80, 160 Electron number density (n e ) c 1,2.5,5,7.5,10 (×10 5 cm −3 )Tuned to constant flux Time cadence 10 GM/c 3 5GM/c 3 Notes. a The u–vcoordinates of the baselines are rotated to align the spin axis of the black hole at a position angle of 288°east of north. b The positive spin models are ray traced at an inclination of 163°and negative spin models at 17°in Set A. Set B models are ray traced at an inclination of 17°. c The electron number density is defined at a fiducial mass of 6.5 ×10 9 M e for Set A and 6.2 ×10 9 M e for Set B. 9 The Astrophysical Journal, 925:13 (19pp), 2022 January 20 Satapathy et al. The fraction η ν /χ ν is the source function S ν and, for thermal emission, is equal to the blackbody function, which depends solely on temperature T. We denote this as ()T h cº nnn B. Throughout this work, we use the analytic approximation devised by Leung et al. (2011)for synchrotron emissivity from a thermal distribution of relativistic electrons () ()()()ne Kc XXX 2 31 2exp,A3 e e 2s 2 1 2 11 12 1 6 2 1 3 hpn =Q+- n where X≡ν/ν s ,()29 sin e sc 2 n nqºQ ,ν c ≡eB/(2πm e c),Θ e ≡ k b T e /(m e c 2 )is the dimensionless electron temperature, and K 2 denotes the modified Bessel function of the second kind. In the these expressions, θdenotes the angle between the magnetic field and the emitted photon, Bdenotes the magnetic field strength, and T e denotes the electron temperature. We use the constants m e ,e,k b , and cto denote the electron mass, electron charge, the Boltzmann’s constant, and the speed of light respectively. It is explicit in this expression that the synchrotron emissivity and opacity, for a given electron temperature and magnetic field, scale proportionally to the electron number density n e . We show here that, for the parameters of the black hole at the center of the M87 galaxy, the synchrotron opacity and emissivity at a wavelength of λ=1.3 mm also scale proportionally to a power of the magnetic field, i.e., ∝B α , with α∼2. We estimate the properties of the plasma in the inner accretion flow of M87 using the following analytic model. The electron density at a given radius ris given by the continuity equation, assuming a spherical accretion rate  Mvia the relation ()  Mr h rmnu4, A4 per2 p=⎛ ⎝⎞ ⎠ where h/ris the scale height of the accretion flow, m p is the proton mass (assuming fully ionized hydrogen), and u r is the radial component of the accretion flow. We take the latter to be a fraction ξof the freefall velocity, i.e., ()uGM r.A5 rBH 12 x=⎛ ⎝⎞ ⎠ We also express the accretion rate as  MmM E º, in terms of the Eddington accretion rate defined by ()  ML c GM m c 4,A6 EE 2 BH p T p s º=  where L E is the Eddington critical luminosity, òis the radiative efficiency of the flow, σ T is the Thomson cross section, and m is the mass accretion rate in units of the Eddington accretion rate. Under these conditions, the electron density is ()    nc GM mh r rc GM M M m rc GM 710 6.5 10 2 10 5cm , A7 e 2 BH T 11 12 BH 32 5BH 9 1 5 2 BH 32 3 sx= =´ ´´ -- -- - - - - ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎛ ⎝⎞ ⎠⎛ ⎝⎞ ⎠⎛ ⎝⎞ ⎠ ⎛ ⎝⎞ ⎠⎛ ⎝⎞ ⎠ ⎛ ⎝⎞ ⎠  where in the last expression we have used typical values for the M87 black hole and set ξ=ò=0.1 and h/r=0.4. Because of the radiatively inefficient character of the accretion flow around the M87 black hole, the ion temperature T i is a fraction of the virial temperature ()TGM m kr3.A8 p v BH B = It is also customary to write the electron temperature as a fraction 1/Rof the ion temperature, i.e., ()TR GM m kr T T 1 3.A9 i e BH p Bv = ⎜⎟⎜⎟ ⎛ ⎝⎞ ⎠⎛ ⎝⎞ ⎠ Finally, we write the magnetic field in terms of the plasma β parameter as () ()nk T T B 8A10 eieB 2 bp += such that () ()    A11 Bc m GM mR R T T h r rc GM M M mrc GM 22 3 1 96.5 10 210 5G, i 2p BH T 12 12 v 12 12 2 BH 54 BH 9 12 5 12 2 BH 54 p sxb =+ ´ =´´ -- - - - ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎛ ⎝⎞ ⎠⎡ ⎣ ⎢⎤ ⎦ ⎥ ⎛ ⎝⎞ ⎠⎛ ⎝⎞ ⎠⎛ ⎝⎞ ⎠ ⎛ ⎝⎞ ⎠⎛ ⎝⎞ ⎠⎛ ⎝⎞ ⎠  and we have set T i /T v =1/3 and β=10. Figure 9shows the ratio η ν /n e evaluated at the observed frequency of the EHT (ν=230 GHz, λ=1.33 mm)and at a radial distance r=5GM BH /c 2 , as a function of the strength of the magnetic field B, for the values of the various parameters used in the previous equations, i.e., for the conditions of the inner accretion flow around the black hole at the center of M87. This figure demonstrates that, over a broad range of magnetic field strengths (spanning more than an order of magnitude), the synchrotron emissivity scales approximately as η ν ∼n e B α , with α;2. This allows us to write the emissivity as () ( ) ( )rnB Tf T,, A12 e hnann  and the opacity as () ()rnBf T,, A13 e cnan where f ν (T,r)captures the scaling of the opacity with temperature Tand the location in the accretion flow (see similar arguments in Chan et al. 2015). In order to calculate the various simulated images, we used the plasma properties from long GRMHD simulations. Nonradiative GRMHD simulations are invariant to a rescaling of the density by a factor , as long as the magnetic field is also rescaled by a factor of 12 and the internal energy by a factor of . In other words, B ne 12 ~and the Alfvén speed () B mn ep12 are the quantity that remains invariant under rescaling. Combined with the approximate expressions (A12)– (A13)derived above, this implies that the synchrotron emissivity and opacity evaluated using the simulation outputs are invariant to rescaling, as long as the product n Bn ee 12 ~ aa+remains constant. Finally, the integration of the transfer equation is performed on a coordinate system in which the distances are expressed in 16 The Astrophysical Journal, 925:13 (19pp), 2022 January 20 Satapathy et al. terms of the length scale set by the mass of the black hole, i.e., ()ds ds GM c.A14 BH 2 º¢ ⎛ ⎝⎞ ⎠ Rewriting Equation (A2)using the scaling (A12)–(A14),we find () [() ] ()r dI ds nfT GM cTI,A15 e 12 BH 2 ¢=- a+B ()()[()]()rnM G cfT T I,.A16 e 12 BH 2 =- a+B We drop the νin subscript to indicate quantities calculated at ν=230 GHz. From this last expression, it is apparent that the electron number density n e and the black hole mass M BH are degenerate quantities in the solution of the radiative transfer problem at wavelengths where the dominant source of opacity is due to synchrotron processes, with a degeneracy in the product n M e 12 B H a+and with α;2. ORCID iDs Kaushik Satapathy https://orcid.org/0000-0003-0433-3585 Lia Medeiros https://orcid.org/0000-0003-2342-6728 Sean T. Dougall https://orcid.org/0000-0002-3769-1314 Chi-Kwan Chan https://orcid.org/0000-0001-6337-6126 Maciek Wielgus https://orcid.org/0000-0002-8635-4242 George N. Wong https://orcid.org/0000-0001-6952-2147 Charles F. Gammie https://orcid.org/0000-0001-7451-8935 Kazunori Akiyama https://orcid.org/0000-0002-9475-4254 Antxon Alberdi https://orcid.org/0000-0002-9371-1033 Juan Carlos Algaba https://orcid.org/0000-0001-6993-1696 Richard Anantua https://orcid.org/0000-0003-3457-7660 Rebecca Azulay https://orcid.org/0000-0002-2200-5393 Anne-Kathrin Baczko https://orcid.org/0000-00033090-3975 Mislav Balokovićhttps://orcid.org/0000-0003-0476-6647 John Barrett https://orcid.org/0000-0002-9290-0764 Bradford A. Benson https://orcid.org/0000-00025108-6823 Lindy Blackburn https://orcid.org/0000-0002-9030-642X Raymond Blundell https://orcid.org/0000-0002-5929-5857 Katherine L. Bouman https://orcid.org/0000-00030077-4367 Geoffrey C. Bower https://orcid.org/0000-0003-4056-9982 Hope Boyce https://orcid.org/0000-0002-6530-5783 Christiaan D. Brinkerink https://orcid.org/0000-00022322-0749 Roger Brissenden https://orcid.org/0000-0002-2556-0894 Silke Britzen https://orcid.org/0000-0001-9240-6734 Avery E. Broderick https://orcid.org/0000-00023351-760X Thomas Bronzwaer https://orcid.org/0000-0003-1151-3971 Do-Young Byun https://orcid.org/0000-0003-1157-4109 Andrew Chael https://orcid.org/0000-0003-2966-6220 Koushik Chatterjee https://orcid.org/0000-0002-2825-3590 Shami Chatterjee https://orcid.org/0000-0002-2878-1502 Ilje Cho https://orcid.org/0000-0001-6083-7521 Pierre Christian https://orcid.org/0000-0001-6820-9941 John E. Conway https://orcid.org/0000-0003-2448-9181 James M. Cordes https://orcid.org/0000-0002-4049-1882 Thomas M. Crawford https://orcid.org/0000-00019000-5013 Geoffrey B. Crew https://orcid.org/0000-0002-2079-3189 Alejandro Cruz-Osorio https://orcid.org/0000-00023945-6342 Yuzhu Cui https://orcid.org/0000-0001-6311-4345 Jordy Davelaar https://orcid.org/0000-0002-2685-2434 Mariafelicia De Laurentis https://orcid.org/0000-00029945-682X Roger Deane https://orcid.org/0000-0003-1027-5043 Jessica Dempsey https://orcid.org/0000-0003-1269-9667 Gregory Desvignes https://orcid.org/0000-0003-3922-4055 Jason Dexter https://orcid.org/0000-0003-3903-0373 Sheperd S. Doeleman https://orcid.org/0000-00029031-0904 Ralph P. Eatough https://orcid.org/0000-0001-6196-4135 Heino Falcke https://orcid.org/0000-0002-2526-6724 Joseph Farah https://orcid.org/0000-0003-4914-5625 Vincent L. Fish https://orcid.org/0000-0002-7128-9345 Ed Fomalont https://orcid.org/0000-0002-9036-2747 H. Alyson Ford https://orcid.org/0000-0002-9797-0972 Raquel Fraga-Encinas https://orcid.org/0000-00025222-1361 Per Friberg https://orcid.org/0000-0002-8010-8454 Antonio Fuentes https://orcid.org/0000-0002-8773-4933 Peter Galison https://orcid.org/0000-0002-6429-3872 Roberto García https://orcid.org/0000-0002-6584-7443 Boris Georgiev https://orcid.org/0000-0002-3586-6424 Ciriaco Goddi https://orcid.org/0000-0002-2542-7743 Roman Gold https://orcid.org/0000-0003-2492-1966 Arturo I. Gómez-Ruiz https://orcid.org/0000-00019395-1670 José L. Gómez https://orcid.org/0000-0003-4190-7613 Minfeng Gu (顾敏峰)https://orcid.org/0000-00024455-6946 Mark Gurwell https://orcid.org/0000-0003-0685-3621 Kazuhiro Hada https://orcid.org/0000-0001-6906-772X Daryl Haggard https://orcid.org/0000-0001-6803-2138 Ronald Hesper https://orcid.org/0000-0003-1918-6098 Figure 9. The ratio of the thermal synchrotron emissivity η ν to the electron density n e , as a function of the strength of the magnetic field, evaluated at an observing frequency of the EHT, ν=230 GHz, and for parameters appropriate to the inner accretion flow in the black hole at the center of the M87 galaxy (see text). The vertical dashed line shows the inferred strength of the magnetic field near the black hole horizon. The dashed line shows the approximate ∼B 2 scaling. 17 The Astrophysical Journal, 925:13 (19pp), 2022 January 20 Satapathy et al. Luis C. Ho (何子山)https://orcid.org/0000-00016947-5846 Mareki Honma https://orcid.org/0000-0003-4058-9000 Chih-Wei L. Huang https://orcid.org/0000-0001-5641-3953 Lei Huang (黄磊)https://orcid.org/0000-0002-1923-227X Shiro Ikeda https://orcid.org/0000-0002-2462-1448 Sara Issaoun https://orcid.org/0000-0002-5297-921X David J. James https://orcid.org/0000-0001-5160-4486 Michael Janssen https://orcid.org/0000-0001-8685-6544 Britton Jeter https://orcid.org/0000-0003-2847-1712 Wu Jiang (江悟)https://orcid.org/0000-0001-7369-3539 Michael D. Johnson https://orcid.org/0000-0002-4120-3029 Svetlana Jorstad https://orcid.org/0000-0001-6158-1708 Taehyun Jung https://orcid.org/0000-0001-7003-8643 Mansour Karami https://orcid.org/0000-0001-7387-9333 Tomohisa Kawashima https://orcid.org/0000-00018527-0496 Garrett K. Keating https://orcid.org/0000-0002-3490-146X Mark Kettenis https://orcid.org/0000-0002-6156-5617 Dong-Jin Kim https://orcid.org/0000-0002-7038-2118 Jae-Young Kim https://orcid.org/0000-0001-8229-7183 Jongsoo Kim https://orcid.org/0000-0002-1229-0426 Junhan Kim https://orcid.org/0000-0002-4274-9373 Motoki Kino https://orcid.org/0000-0002-2709-7338 Jun Yi Koay https://orcid.org/0000-0002-7029-6658 Patrick M. Koch https://orcid.org/0000-0003-2777-5861 Shoko Koyama https://orcid.org/0000-0002-3723-3372 Carsten Kramer https://orcid.org/0000-0002-4908-4925 Michael Kramer https://orcid.org/0000-0002-4175-2271 Thomas P. Krichbaum https://orcid.org/0000-00024892-9586 Cheng-Yu Kuo https://orcid.org/0000-0001-6211-5581 Tod R. Lauer https://orcid.org/0000-0003-3234-7247 Sang-Sung Lee https://orcid.org/0000-0002-6269-594X Aviad Levis https://orcid.org/0000-0001-7307-632X Yan-Rong Li (李彦荣)https://orcid.org/0000-00015841-9179 Zhiyuan Li (李志远)https://orcid.org/0000-00030355-6437 Michael Lindqvist https://orcid.org/0000-0002-3669-0715 Rocco Lico https://orcid.org/0000-0001-7361-2460 Greg Lindahl https://orcid.org/0000-0002-6100-4772 Jun Liu (刘俊)https://orcid.org/0000-0002-7615-7499 Kuo Liu https://orcid.org/0000-0002-2953-7376 Elisabetta Liuzzo https://orcid.org/0000-0003-0995-5201 Laurent Loinard https://orcid.org/0000-0002-5635-3345 Ru-Sen Lu (路如森)https://orcid.org/0000-00027692-7967 Nicholas R. MacDonald https://orcid.org/0000-00026684-8691 Jirong Mao (毛基荣)https://orcid.org/0000-00027077-7195 Nicola Marchili https://orcid.org/0000-0002-5523-7588 Sera Markoff https://orcid.org/0000-0001-9564-0876 Daniel P. Marrone https://orcid.org/0000-0002-2367-1080 Alan P. Marscher https://orcid.org/0000-0001-7396-3332 Iván Martí-Vidal https://orcid.org/0000-0003-3708-9611 Satoki Matsushita https://orcid.org/0000-0002-2127-7880 Karl M. Menten https://orcid.org/0000-0001-6459-0669 Izumi Mizuno https://orcid.org/0000-0002-7210-6264 Yosuke Mizuno https://orcid.org/0000-0002-8131-6730 James M. Moran https://orcid.org/0000-0002-3882-4414 Kotaro Moriyama https://orcid.org/0000-0003-1364-3761 Monika Moscibrodzka https://orcid.org/0000-00024661-6332 Cornelia Müller https://orcid.org/0000-0002-2739-2994 Alejandro Mus Mejías https://orcid.org/0000-00030329-6874 Gibwa Musoke https://orcid.org/0000-0003-1984-189X Hiroshi Nagai https://orcid.org/0000-0003-0292-3645 Neil M. Nagar https://orcid.org/0000-0001-6920-662X Masanori Nakamura https://orcid.org/0000-00016081-2420 Ramesh Narayan https://orcid.org/0000-0002-1919-2730 Iniyan Natarajan https://orcid.org/0000-0001-8242-4373 Joey Neilsen https://orcid.org/0000-0002-8247-786X Roberto Neri https://orcid.org/0000-0002-7176-4046 Chunchong Ni https://orcid.org/0000-0003-1361-5699 Aristeidis Noutsos https://orcid.org/0000-0002-4151-3860 Michael A. Nowak https://orcid.org/0000-0001-6923-1315 Héctor Olivares https://orcid.org/0000-0001-6833-7580 Gisela N. Ortiz-León https://orcid.org/0000-00022863-676X Daniel C. M. Palumbo https://orcid.org/0000-00027179-3816 Jongho Park https://orcid.org/0000-0001-6558-9053 Ue-Li Pen https://orcid.org/0000-0003-2155-9578 Dominic W. Pesce https://orcid.org/0000-0002-5278-9221 Richard Plambeck https://orcid.org/0000-0001-6765-9609 Oliver Porth https://orcid.org/0000-0002-4584-2557 Felix M. Pötzl https://orcid.org/0000-0002-6579-8311 Jorge A. Preciado-López https://orcid.org/0000-00024146-0113 Hung-Yi Pu https://orcid.org/0000-0001-9270-8812 Venkatessh Ramakrishnan https://orcid.org/0000-00029248-086X Ramprasad Rao https://orcid.org/0000-0002-1407-7944 Mark G. Rawlings https://orcid.org/0000-0002-6529-202X Luciano Rezzolla https://orcid.org/0000-0002-1330-7103 Bart Ripperda https://orcid.org/0000-0002-7301-3908 Freek Roelofs https://orcid.org/0000-0001-5461-3687 Eduardo Ros https://orcid.org/0000-0001-9503-4892 Mel Rose https://orcid.org/0000-0002-2016-8746 Alan L. Roy https://orcid.org/0000-0002-1931-0135 Chet Ruszczyk https://orcid.org/0000-0001-7278-9707 Kazi L. J. Rygl https://orcid.org/0000-0003-4146-9043 David Sánchez-Arguelles https://orcid.org/0000-00027344-9920 Mahito Sasada https://orcid.org/0000-0001-5946-9960 Tuomas Savolainen https://orcid.org/0000-0001-6214-1085 Lijing Shao https://orcid.org/0000-0002-1334-8853 Zhiqiang Shen (沈志强)https://orcid.org/0000-00033540-8746 Des Small https://orcid.org/0000-0003-3723-5404 Bong Won Sohn https://orcid.org/0000-0002-4148-8378 Jason SooHoo https://orcid.org/0000-0003-1938-0720 He Sun (孙赫)https://orcid.org/0000-0003-1526-6787 Fumie Tazaki https://orcid.org/0000-0003-0236-0600 Alexandra J. Tetarenko https://orcid.org/0000-00033906-4354 Paul Tiede https://orcid.org/0000-0003-3826-5648 Remo P. J. Tilanus https://orcid.org/0000-0002-6514-553X Michael Titus https://orcid.org/0000-0002-3423-4505 Kenji Toma https://orcid.org/0000-0002-7114-6010 18 The Astrophysical Journal, 925:13 (19pp), 2022 January 20 Satapathy et al. 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