#  First Sagittarius A* Event Horizon Telescope Results. I. The Shadow of the
Supermassive Black Hole in the Center of the Milky Way

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  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  Juan Carlos Algaba
Department of Physics, Faculty of Science, Universiti Malaya, 50603 Kuala
Lumpur, Malaysia  Richard Anantua Black Hole Initiative at Harvard University,
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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  Keiichi Asada Institute of Astronomy and
Astrophysics, Academia Sinica, 11F of Astronomy-Mathematics Building, AS/NTU
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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,
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dem Hügel 69, D-53121 Bonn, Germany  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.
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& Astrophysics, Yale University, 52 Hillhouse Avenue, New Haven, CT 06511, USA
John Barrett Massachusetts Institute of Technology Haystack Observatory, 99
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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  Lindy
Blackburn Black Hole Initiative at Harvard University, 20 Garden Street,
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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
Christiaan D. Brinkerink Department of Astrophysics, Institute for
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of Waterloo, 200 University Avenue West, Waterloo, ON, N2L 3G1, Canada
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University of Massachusetts, 01003, Amherst, MA, USA  Do-Young Byun Korea
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34055, Republic of Korea  University of Science and Technology, Gajeong-ro
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201210, People’s Republic of China  Mizusawa VLBI Observatory, National
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023-0861, Japan  Department of Astronomical Science, The Graduate University
for Advanced Studies (SOKENDAI), 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
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University, No.1, Sect.4, Roosevelt Rd., Taipei 10617, Taiwan, R.O.C  Andrei
P. Lobanov Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121
Bonn, Germany  Laurent Loinard Instituto de Radioastronomía y Astrofísica,
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Objects, Chinese Academy of Sciences, 650011 Kunming, People’s Republic of
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Markoff Anton Pannekoek Institute for Astronomy, University of Amsterdam,
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and Astrophysics, Academia Sinica, 11F of Astronomy-Mathematics Building,
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dem Hügel 69, D-53121 Bonn, Germany  F. Peter Schloerb  Department of
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Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany
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Brown  East Asian Observatory, 660 N. A’ohoku Place, Hilo, HI 96720, USA
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###### Abstract

We present the first Event Horizon Telescope (EHT) observations of Sagittarius
A* (Sgr A∗), the Galactic center source associated with a supermassive black
hole. These observations were conducted in 2017 using a global interferometric
array of eight telescopes operating at a wavelength of $\lambda=1.3\,{\rm
mm}$. The EHT data resolve a compact emission region with intrahour
variability. A variety of imaging and modeling analyses all support an image
that is dominated by a bright, thick ring with a diameter of $51.8\pm 2.3$
$\mathrm{\mu as}$ (68% credible interval). The ring has modest azimuthal
brightness asymmetry and a comparatively dim interior. Using a large suite of
numerical simulations, we demonstrate that the EHT images of Sgr A∗ are
consistent with the expected appearance of a Kerr black hole with mass
${\sim}4\times 10^{6}\,{\rm M}_{\odot}$, which is inferred to exist at this
location based on previous infrared observations of individual stellar orbits
as well as maser proper motion studies. Our model comparisons disfavor
scenarios where the black hole is viewed at high inclination ($i>50^{\circ}$),
as well as non-spinning black holes and those with retrograde accretion disks.
Our results provide direct evidence for the presence of a supermassive black
hole at the center of the Milky Way galaxy, and for the first time we connect
the predictions from dynamical measurements of stellar orbits on scales of
$10^{3}-10^{5}$ gravitational radii to event horizon-scale images and
variability. Furthermore, a comparison with the EHT results for the
supermassive black hole M87∗ shows consistency with the predictions of general
relativity spanning over three orders of magnitude in central mass.

galaxies: individual: Sgr A∗– Galaxy: center – black hole physics –
techniques: high angular resolution – techniques: interferometric

$S/N$

    signal-to-noise ratio
VLBI

    very long baseline interferometry
GRMHD

    general relativistic magnetohydrodynamics
MHD

    magnetohydrodynamics
as

    arcseconds
AGN

    Active Galactic Nuclei
LLAGN

    low-luminosity AGN
Cen A

    Centaurus A
Sgr A*

    Sagittarius A*
AGN

    active galactic nuclei
EHT

    Event Horizon Telescope
BHC

    BlackHoleCam
TANAMI

    Tracking Active Galactic Nuclei with Austral Milliarcsecond Interferometry
SMBH

    supermassive black hole
ALMA

    Atacama Large Millimeter/submillimeter Array
APEX

    Atacama Pathfinder Experiment
PV

    IRAM 30 m Telescope
JCMT

    James Clerk Maxwell Telescope
LMT

    Large Millimeter Telescope Alfonso Serrano
SPT

    South Pole Telescope
SMA

    Submillimeter Array
SMT

    Submillimeter Telescope
Chandra

    Chandra X-ray Observatory
c

    speed of light
pc

    parsec
GR

    general relativity
aips

    Astronomical Image Processing System
CASA

    Common Astronomy Software Applications
symba

    SYnthetic Measurement creator for long Baseline Arrays
Rpicard

    Radboud PIpeline for the Calibration of high Angular Resolution Data
HOPS

    Haystack Observatory Postprocessing System
HBT

    Hanbury Brown and Twiss
TOV

    Tolman-Oppenheimer-Volkoff
MRI

    magnetorotational instability
ADAF

    advection-dominated accretion flow
ADIOS

    adiabatic inflow-outflow solution
CDAF

    convection-dominated accretion flow
BZ

    Blandford-Znajek
BP

    Blandford-Payne
EM

    electromagnetic
BLR

    broad-line region
NLR

    narrow-line region
ISM

    interstellar medium
eDF

    electron distribution function
PIC

    particle-in-cell
SANE

    standard and normal evolution
MAD

    magnetically arrested disk
JIVE

    Joint Institute for VLBI ERIC
NRAO

    National Radio Astronomy Observatory
$\mu$as

    microarcseconds
Jy

    jansky
PA

    position angle
SRMHD

    special relativistic magnetohydrodynamics
FOV

    field of view
SEFD

    system equivalent flux density

##  1 Introduction

Black holes are among the boldest and most profound predictions of Einstein’s
theory of General Relativity (GR; Einstein, 1915). Originally studied as a
mathematical consequence of GR rather than as physically relevant objects
(Schwarzschild, 1916), they are now believed to be generic and sometimes
inevitable outcomes of gravitational collapse (Oppenheimer & Snyder, 1939;
Penrose, 1965). In GR, the spacetime around astrophysical black holes is
predicted to be uniquely described by the Kerr metric, which is entirely
specified by the black hole’s mass and angular momentum or “spin” (Kerr,
1963).

The first empirical evidence for their existence was through stellar-mass
black holes, beginning with observations of X-ray binary orbits (Bolton, 1972;
Webster & Murdin, 1972; McClintock & Remillard, 1986) and culminating in the
detection of gravitational waves from merging stellar-mass black holes (Abbott
et al., 2016). In parallel, the discovery that quasars are not stellar in
nature but are rather extremely luminous, compact objects located in the
centers of distant galaxies (Schmidt, 1963) led to an intensive effort to
identify and measure the supermassive black holes (SMBHs) energetically
favored to power them (Lynden-Bell, 1969). Observations now suggest that SMBHs
not only lie at the center of nearly every galaxy (Richstone et al., 1998) but
also may play a role in their evolution (see, e.g., Magorrian et al., 1998;
Fabian, 2012; Kormendy & Ho, 2013), though how exactly the ebbs and flows of
black hole activity and growth proceed is a major outstanding question in the
field.

With the advent of the Event Horizon Telescope (EHT), SMBHs can now be studied
with direct imaging (Event Horizon Telescope Collaboration et al., 2019a, b,
c, d, e, f, 2021a, 2021b, hereafter M87∗ Paper I-VIII). The combination of an
event horizon and strong lensing near black holes is predicted to produce
distinctive gravitational signatures in their images (e.g., Hilbert, 1917;
Bardeen, 1973; Luminet, 1979; Jaroszynski & Kurpiewski, 1997; Falcke et al.,
2000). In particular, simulated images of black holes typically have a central
brightness depression encircled by a bright emission ring. The ring usually
lies near the gravitationally lensed photon orbits that define the boundary of
what we hereafter refer to as the black hole “shadow.” The shadow has an
angular diameter $d_{\rm sh}\approx 10GM/(c^{2}D)\equiv 10\theta_{\rm g}$,
where $G$ is the gravitational constant, $c$ is the speed of light, $M$ is the
black hole mass, and $D$ is the black hole distance.

From the first realization that SMBHs could power bright radio cores in many
galactic nuclei (Lynden-Bell, 1969, and references therein), the search has
been on to identify them. Within our own galaxy, the compact source Sgr A∗ has
been intensely studied as a candidate SMBH since its discovery as a bright
source of radio emission located near the Galactic Center (Balick & Brown,
1974; Ekers et al., 1975; Lo et al., 1975). Decades of monitoring its proper
motion, as well as of motions of individual stars in orbit around it, have
revealed Sgr A∗ to be an extremely dense concentration of mass ($M\approx
4\times 10^{6}M_{\odot}$) that is located at and nearly motionless with
respect to the dynamical center of the Galaxy ($D\approx 8\,{\rm kpc}$),
providing strong evidence that it is the nuclear SMBH in our Galaxy (e.g.,
Gravity Collaboration et al., 2019; Do et al., 2019; Reid & Brunthaler, 2020).
As the nearest supermassive black hole, Sgr A∗ provides a unique opportunity
to directly image such an object, together with its accretion system, in the
most common, quiescent state of SMBHs across the Universe. It also provides
the chance to elucidate some of the drivers of observed cycles in accretion
power and jet launching, via comparison with the more “active” galactic
nucleus M87∗.

In this paper, we present the first EHT observations of Sgr A∗ and put them
into context with our previous results on M87∗. In section 2, we describe what
was previously known about the physical properties of Sgr A∗ and compare them
to M87∗. We then summarize our Sgr A∗ observations with the EHT and other
observatories in section 3 and discuss its variability in section 4. In
section 5, we present the first EHT images of Sgr A∗ and analyze its event-
horizon-scale structure. In section 6, we discuss the astrophysical
interpretation of these results using an extensive suite of general
relativistic magnetohydrodynamic (GRMHD) simulations, and in section 7 we
present the constraints that these results give for GR and black hole
alternatives. We provide the overall conclusions and outlook in section 8. The
five companion papers of this series provide a more comprehensive discussion
of all these topics (Event Horizon Telescope Collaboration et al., 2022a, b,
c, d, e, hereafter Papers II-VI).

##  2 Sgr A* and M87*

Decades of observations have provided a picture of our local SMBH that is
unmatched in any other galaxy (for details about the full spectrum, see Paper
II, ). Sgr A∗ has been detected from long radio wavelengths ($\sim$1 m) to the
hard X-ray band, excepting approximately 1 $\mu$m to 1 nm due to extinction
from dust in the galactic plane. Sgr A∗ is remarkable for its feeble emission,
producing a bolometric luminosity of ${\mathrel{\raise
1.29167pt\hbox{$<$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}}10^{36}\,{\rm
erg/s}$, only $\sim$100 times that of the Sun. Were it located in another
galaxy, it would likely go undetected. Nevertheless, by observing its spectrum
and variability, its environment, and its influence on surrounding bodies, a
great deal has been learned about this source specifically, and about the
astrophysical processes that operate around supermassive black holes. In this
section we describe how we assembled our current knowledge of Sgr A∗, discuss
important theoretical uncertainties about its accretion and outflows, and
compare it with the other horizon-scale EHT target, M87∗.

###  2.1 Properties of Sgr A*

The proximity of Sgr A∗ permits precise measurements of its gravitating mass
via the monitoring of resolved individual stellar orbits. High-resolution
infrared (IR) observations, using increasingly sophisticated instrumentation
and analyses, have traced out the three dimensional orbits of several stars
within the innermost arcsecond around Sgr A∗ (Schödel et al., 2002; Ghez et
al., 2003, 2008; Gillessen et al., 2009; Gravity Collaboration et al., 2018a;
Do et al., 2019; Gravity Collaboration et al., 2019). These orbits jointly
determine the mass and distance to Sgr A∗ to high precision, particularly the
ratio $M/D$ that determines the angular size of the black hole on the sky. As
discussed in Paper II and Paper VI, the current values for the mass and
distance suggest an angular shadow diameter close to 50 $\mu{\rm as}$,
comparable to that of M87∗. The closest orbital periapses confine the mass to
within $\sim$1,000 Schwarzschild radii ($R_{\rm S}=2GM/c^{2}$).

Radio observations of Sgr A∗ have provided a significant motivation for the
development of the EHT experiment. Sgr A∗ shows a flat/inverted radio
spectrum, which often arises from compact jet emission (Blandford & Königl,
1979) in other low-luminosity active galactic nuclei (LLAGN; Ho 1999; Nagar et
al. 2000). Such a spectrum can, however, result from any stratified, self-
absorbed synchrotron source, where successively higher frequencies are
produced at increasingly smaller scales, even without a jet (e.g., Narayan et
al., 1995). Sgr A∗ shows an excess of millimeter emission above the flat
centimeter-wave spectrum, the so-called “submillimeter bump,” that was
inferred to indicate the presence of a very compact emitting region at these
wavelengths (e.g., Zylka et al., 1992; Falcke et al., 1998).

To clarify the nature of this source, the structure of Sgr A∗ was investigated
using very-long-baseline interferometry (VLBI) at progressively shorter
wavelengths (see Paper II, and references therein). For wavelengths longer
than several cm, the observed source size is entirely determined by scatter-
broadening in the ionized interstellar medium, scaling with a wavelength
dependence of $\lambda^{2}$. At wavelengths of 7 mm and shorter, the imprint
of the intrinsic structure of Sgr A∗ became discernible through the scattering
(e.g., Rogers et al., 1994; Lo et al., 1998; Doeleman et al., 2001; Bower et
al., 2004; Shen et al., 2005; Bower et al., 2006). The source grows more
compact at shorter wavelengths, as expected (e.g., Özel et al., 2000), though
it does not present a clear jet structure. For wavelengths as short as $\sim$1
mm, Sgr A∗ is only slightly blurred by scattering, and Falcke et al. (2000)
predicted that submillimeter VLBI could directly image a brightness depression
within this region related to the black hole shadow.

In parallel with these developments and motivated by the goal to study black
holes, the capabilities of mm-VLBI improved rapidly. At 1.4 mm, Padin et al.
(1990) reported the first VLBI fringes and Krichbaum et al. (1998) obtained
the first VLBI detections and associated source size measurements for Sgr A∗,
each using a 2-element interferometer. Doeleman et al. (2008) observed Sgr A∗
at 1.3 mm with a 3-element VLBI array and reported the discovery of an
intrinsic source size comparable to the expected angular diameter of the black
hole shadow. These observations provided important constraints for theoretical
models (e.g., Broderick et al., 2009; Dexter et al., 2009; Mościbrodzka et
al., 2009) and strongly suggested that 1.3 mm VLBI has a clear view into the
innermost region around the Sgr A∗ black hole. Subsequent VLBI studies of Sgr
A∗ at 1.3 mm with progressively enhanced arrays revealed the compact emission
to be variable and significantly polarized, with measured “closure phases”
that are indicative of persistent asymmetry in the image structure (Fish et
al., 2011; Johnson et al., 2015; Lu et al., 2018).

Observations of Sgr A∗ outside the radio band were important for completing
the picture of this source as an LLAGN. When X-ray and gamma-ray instruments
ROSAT and Sigma/GRANAT (Predehl & Truemper, 1994; Goldwurm et al., 1994) could
not identify a bright central source, it became clear that Sgr A∗ must either
be obscured or anomalously faint compared to other known LLAGN. The first
identification of Sgr A∗ as a compact and variable source in the X-rays was
achieved with the Chandra X-ray Observatory (Chandra) (Baganoff et al., 2001,
2003). This detection, together with the ${\sim}4\times 10^{6}$ M⊙ mass of Sgr
A∗ determined from stellar orbits, sets a maximum luminosity scale for the
source, the so-called Eddington luminosity,111The Eddington luminosity is an
idealized estimate of the maximum power for an accreting black hole: $L_{\rm
Edd}\equiv 4\pi GMcm_{\rm p}/\sigma_{\rm T}\approx
10^{38}\left(M/M_{\odot}\right)$ erg/s, where $\sigma_{\rm T}$ is the Thomson
cross section and $m_{\rm p}$ is the mass of a proton. and the X-ray
measurements confirmed that Sgr A∗ is approximately 9 orders magnitude less
luminous than its $L_{\rm Edd}$ — the lowest Eddington ratio ($L/L_{\rm Edd})$
observed for any black hole.

This particularly faint high-energy emission stimulated a lively debate
regarding the nature of Sgr A∗’s accretion/outflow properties, and motivated
new theoretical developments. If the emission primarily originates in the
accretion inflow, either it must be exceptionally radiatively inefficient
(Narayan et al., 1995), or the accretion rate onto the black hole must be very
small compared to what is captured, or possibly some combination of the two
(Blandford & Begelman, 1999; Narayan et al., 2000; Quataert & Gruzinov, 2000a;
Yuan & Narayan, 2014). Alternately the emission could be dominated by an
outflow, in which case a small accretion rate would also be favored (Falcke et
al., 1993), even during flares (Markoff et al., 2001). Fortunately, the
proximity of Sgr A∗ enables unparalleled study of the accretion flow.
Observations with Chandra marginally resolve thermal bremsstrahlung emission
from near the gas capture radius, leading to an estimate of the captured
accretion rate of ${\sim}10^{-6}-10^{-5}M_{\odot}$/yr at ${\sim}10^{5}R_{\rm
S}$ (Quataert, 2002; Baganoff et al., 2003). The source of this gas can be
connected to the observable stellar winds of ${\sim}30$ individual massive
stars found in the inner parsec (Coker et al., 1999; Russell et al., 2017;
Ressler et al., 2020). The near-horizon accretion rate has also been estimated
to be $10^{-9}-10^{-7}$ $M_{\odot}$/yr through measurements of the Faraday
rotation of polarized millimeter emission from Sgr A∗ (Bower et al., 2003;
Marrone et al., 2006, 2007). This Faraday rotation is orders of magnitude
smaller than would be expected if the accretion rate at the gas capture radius
persisted to small radii (Bower et al., 1999; Agol, 2000; Quataert & Gruzinov,
2000b; Marrone et al., 2006). This conclusion is further supported by radially
resolved X-ray spectroscopy (Wang et al., 2013) suggesting that only
${\sim}1\%$ of the captured mass makes it to the SMBH. Taken together, the low
luminosity, low radiative efficiency, and weak Faraday rotation are consistent
with a weakly bound, magnetized accretion flow so diffuse that the electron
and ion temperatures are unable to remain strongly coupled.

Figure 1: The 2017 EHT array as seen from Sgr A∗. The array included eight
observatories at six locations: the ALMA and the APEX on the Llano de
Chajnantor in Chile, the LMT on Volcán Sierra Negra in Mexico, the JCMT and
SMA on Maunakea in Hawai’i, the Institut de Radioastronomie Millimétrique 30-m
telescope (PV) on Pico Veleta in Spain, the SMT on Mt. Graham in Arizona, and
the SPT in Antarctica.  Figure 2:  EHT observations of Sgr A∗ on April 7. Top-
Left: EHT baseline coverage, where dimensionless coordinates $\vec{u}=(u,v)$
give the projected baseline vector for each antenna pair in units of the
observing wavelength $\lambda$. Top-Right: Calibrated visibility amplitudes of
Sgr A∗ as a function of projected baseline length $|\vec{u}|$. Error bars show
${\pm}1\sigma$ thermal (statistical) uncertainties. Diamonds denote baselines
to APEX and JCMT to distinguish them from baselines to their co-located
observatories, ALMA and SMA, respectively. The visibilities have been
coherently averaged in 120 second intervals. For comparison, the gray dashed
line shows the Fourier transform of a thin ring with diameter $54\,\mu{\rm
as}$ that has been convolved with a circular Gaussian kernel of FWHM
$23\,\mu{\rm as}$. The red line and shaded region show the root-mean-square
variability and associated 68% credible interval over the range of baselines
for which it can be accurately measured (see Paper IV, ), while the blue
horizontal ticks at zero baseline length show the range of variations in the
total flux density. Bottom: The full light curve of Sgr A∗ on April 7,
measured using ALMA and the SMA as stand-alone interferometers.

Finally, Sgr A∗ exhibits flaring emission at most wavelengths, including
continuous variability at millimeter wavelengths. In particular, approximately
daily flares are observed at X-ray and NIR wavelengths that are often (but not
always) simultaneous (e.g., Eckart et al., 2004) and that change on timescales
as short as minutes (e.g. Baganoff et al., 2001; Genzel et al., 2003). These
timescales suggest an origin from within ${\sim}5R_{\rm S}$ of the SMBH, which
is consistent with astrometry of NIR flares using the GRAVITY instrument on
the Very Large Telescope Interferometer (Gravity Collaboration et al., 2018b).

###  2.2 Comparison to M87∗

M87∗ and Sgr A∗ have the largest angular sizes of any known SMBHs, making them
the primary EHT targets. However, they differ in several important ways.
First, they have substantially different luminosities and accretion rates,
both in absolute terms and when scaled by mass. M87∗, roughly 1500$\times$
more massive ($M=(6.5\pm 0.7)\times 10^{9}$ $M_{\odot}$; M87∗ Paper I, ), has
an inferred accretion rate of $\dot{M}\sim 10^{-3}\,M_{\odot}/{\rm yr}$ and a
bolometric luminosity of $L_{\rm bol}\approx 10^{42}\,{\rm erg/s}$ measured in
2017 (EHT MWL Science Working Group et al., 2021). Estimates of the total
kinetic power are typically $\sim 10-100$ times larger, thus conservatively
$L/L_{\rm Edd}\gtrsim 10^{-5}$ (M87∗ Paper VIII). Most likely this higher
power indicates that M87∗ is being fed directly from a larger reservoir of gas
rather than a trickle from stellar winds. This difference may underlie another
substantial divergence between the sources: the prominent, powerful jet
launched from M87∗, which can be traced at multiple wavelengths and across
nearly eight orders of magnitude in size (EHT MWL Science Working Group et
al., 2021). The M87∗ jet provides firm constraints on the source orientation
with respect to the line of sight; thus, an inclination of $\sim 20^{\circ}$
was used for numerical simulations in M87∗ Paper V, while for Sgr A∗ we do not
have any such constraints.

The difference in the masses of Sgr A∗ and M87∗ implies a similar difference
in their variability timescales. The period of the innermost stable circular
orbit (ISCO), which depends on the mass and spin of the black hole, serves as
an approximate dynamical timescale. For prograde orbits, this period ranges
from $4\pi t_{\rm g}$ (maximal spin) to $12\sqrt{6}\pi t_{\rm g}$ (zero spin),
where $t_{\rm g}\equiv GM/c^{3}$. For M87∗, this range corresponds to 5 days
to 1 month, so the source structure is expected to be effectively unchanged
over the course of an observing night. Indeed, EHT images of M87∗
reconstructed on consecutive nights are almost identical (M87∗ Paper IV).
However, for Sgr A∗, the range is only 4–30 min, so the source structure can
evolve within a single night.

##  3 EHT Observations and Data Processing

In this section we summarize the EHT observations and data processing for Sgr
A∗. We refer the reader to M87∗ Paper II for a comprehensive discussion of the
EHT instrument, M87∗ Paper III for details on the 2017 observing campaign and
data processing, and Paper II for additional details specific to the Sgr A∗
data processing.

In 2017, the EHT observed Sgr A∗ on five nights between April 5 and 11 using
an array with 8 participating observatories (see Figure 1). Weather conditions
were good or excellent at all sites on all five observing nights. The most
sensitive element in the array, the Atacama Large Millimeter/submillimeter
Array (ALMA), only observed Sgr A∗ on April 6, 7, and 11; the IRAM 30 m
Telescope (PV) only observed Sgr A∗ on April 7. In this initial series of
papers, we focus on the observation with the best baseline coverage: April 7.
In addition, we utilize the April 6 observations for testing, validation, and
selected multi-day analyses. Our multi-wavelength coverage indicated that
there was an X-ray flare on April 11, accompanied by increased 1.3 mm
variability; we will consider that more complex data set in future work.

Each site, except the JCMT and ALMA, received data in two circular
polarizations simultaneously. The JCMT received a single circular
polarization, and ALMA received orthogonal linear polarizations that are
converted to a circular basis in post processing. All EHT sites recorded data
in two frequency bands at 227.1 and 229.1 GHz, referred to as the “low” and
“high” bands, respectively. The total recording rate at each fully-outfitted
station is $32\,{\rm gigabits/second}$. The data were written to arrays of
hard drives at each site, that were then brought from all sites to common
locations where we computed the complex cross-correlation in the electric
fields measured for each pair of stations.

Following the initial computation of these correlations, residual phase and
bandpass errors were corrected with two independent processing pipelines, EHT-
HOPS (Blackburn et al., 2019) and rPICARD (Janssen et al., 2019). The data
were then a priori calibrated using the system equivalent flux densities
(SEFDs) for each telescope. Multiplication by the geometric mean of the SEFDs
of the two stations on a given baseline converts dimensionless correlation
coefficients to flux densities. SEFDs ranged from 60 Jy at ALMA to $5\times
10^{4}$ Jy at low elevation at SMT. Further corrections are still required, as
some sites do not measure the SEFD continuously, and in any case the SEFD does
not capture many amplitude corrupting effects such as pointing and focus
errors. A “network calibration” process, which uses redundancies in the array
(e.g., co-located telescopes) to provide more accurate time-variable gain
normalization for sites with co-located partners, was performed. Sgr A∗
presents a special case for calibration, as it varies significantly in time,
and is surrounded by extended emission that corrupts the visibility amplitude
for baselines within local arrays like ALMA and SMA. Wielgus et al. (2022)
discusses the techniques used to estimate the time-resolved flux density of
Sgr A∗ for this calibration. For the remaining stations, gain corrections were
computed using observations of the calibrator targets J1924-2914 and NRAO 530
(Paper II).

Figure 2 shows the EHT baseline coverage and interferometric visibility
amplitudes (“visibilities”) for Sgr A∗. The longest baselines have an
interferometric fringe spacing of $1/|\vec{u}|\approx 24\,\mu{\rm as}$, which
defines the diffraction-limited angular resolution of the EHT. The visibility
amplitudes have two deep minima, the first at $|\vec{u}|\approx 3.0\,{\rm
G}\lambda$ and the second at $|\vec{u}|\approx 6.5\,{\rm G}\lambda$. The
amplitudes have a baseline dependence that is similar to that of an
infinitesimally thin ring with a $54\,\mu{\rm as}$ diameter that has been
blurred with a circular Gaussian kernel with $23\,\mu{\rm as}$ full width at
half maximum (FWHM). The ring diameter is primarily constrained by the minima
locations, while the width is determined by the amplitude of the secondary
visibility peak between the minima. For instance, to be consistent with both a
first minimum falling between $2.5-3.5\,{\rm G}\lambda$ and a second minimum
falling between $6-7\,{\rm G}\lambda$, the ring must have a diameter of
${\sim}50-60\,\mu$as.

However, the EHT data show evidence for complex and asymmetric source
structure beyond a simple ring model, including a strong dependence on
visibility amplitudes with the baseline position angle (especially near the
first visibility minimum) and in the closure phases measured on triangles of
EHT baselines, which differ significantly from $0^{\circ}$ and $180^{\circ}$.
The EHT visibility amplitudes also show indications of residual calibration
errors. To assess and quantify the source morphology of Sgr A∗ more generally,
we applied a variety of imaging and modeling methods using both the
interferometric visibility amplitude and phase information (section 5). These
methods included a variety of approaches to account for residual calibration
errors, including iterative self-calibration, simultaneous fitting of a model
and the residual station gains, and analyses that used only closure
quantities.

##  4 Horizon-Scale Variability in Sgr A*

To characterize the spectrum and variability of Sgr A∗, the EHT observing
campaign was supported by parallel observations at several other
observatories. Longer-wavelength VLBI observations, where interstellar
scattering prevents direct observation of structural changes, were arranged to
occur within a few days of the EHT campaign. IR and X-ray observations were
arranged to be as simultaneous as possible with the EHT tracks on several
days, and they resulted in one confirmed X-ray flare on April 11.

For the observations of April 6 and 7, when no strong flares were observed in
the parallel observations, we must evaluate whether Sgr A∗ has structural
variations within our observation periods. The simplest evidence for
variability at $\lambda=1.3\,{\rm mm}$ comes from analysis of the total flux
density (the “light curve”), which is measured during EHT observations using
the ALMA and SMA connected element arrays. As discussed in Wielgus et al.
(2022), the fractional variability across all days is approximately $9\%$,
with variations of $4-13\%$ seen within individual nights (see, e.g., Figure
2). This variability is an order of magnitude stronger than is expected from
interstellar scintillation. Thus, the light curves provide evidence for
intrinsic image variability in Sgr A∗.

Each EHT baseline provides information about the detailed structure of this
variability on an angular scale determined by the baseline length. On several
triangles, EHT closure phases show slightly more variation than is expected
from their thermal noise, the changing of projected baselines (from the source
view) with Earth rotation, and interstellar scintillation (Paper II).
Comparison of interferometric visibility amplitudes for nearby baselines also
reveals variability that significantly exceeds what is expected from thermal
noise, calibration uncertainties, and baseline evolution (Paper IV).

To quantify the variability, we developed a simple parametric model for the
spatio-temporal power spectrum for the variability of Sgr A∗ (Georgiev et al.,
2022). This model represents the variance in the visibility amplitude as a
function of the radial distance in the $(u,v)$-plane, taking the form of a
broken power law, as motivated by studies of GRMHD simulations. The source-
integrated light curve is divided out in order to isolate structural variation
from overall changes in flux density. The variability power spectrum was also
empirically estimated by analyzing variations in visibility amplitudes on
nearby baselines (Broderick et al., 2022). This comparison included the same
baselines sampled on different days, as well as nearby or crossing baseline
tracks (e.g., SPT-LMT and SPT-SMA sample nearly identical baselines, but at
different times). This analysis revealed that the fractional variability can
be order unity for EHT baselines located near the two deep visibility minima
(Figure 2), even after normalizing the data to remove light curve
fluctuations, and that it significantly exceeds the variability expected from
interstellar scintillation (Paper IV).

Figure 3:  Representative EHT image of Sgr A∗ from observations on 2017 April
7. This image is an average over different reconstruction methodologies
(CLEAN, RML, and Bayesian) and reconstructed morphologies. Color denotes the
specific intensity, shown in units of brightness temperature. The inset circle
shows the restoring beam used for CLEAN image reconstructions ($20\,\mu{\rm
as}$ FWHM). The bottom panels show average images within subsets with similar
morphologies, with their prevalence indicated by the inset bars. The
multiplicity of image modes reflects uncertainty due to the sparse baseline
coverage; it does not correspond to different snapshots of the variable
source. Nearly all reconstructed images show a prominent ring morphology.
While the diameter and thickness of the ring are generally consistent across
the reconstructions, the azimuthal structure of the ring is poorly
constrained.

##  5 Horizon-Scale Structure in Sgr A*

Each interferometric visibility samples a single complex Fourier component of
the image on the sky (e.g., Thompson et al., 2017). Interferometric imaging
algorithms seek to produce images from this sparse Fourier-domain information
that are consistent with the data and physically plausible. Techniques such as
the classical CLEAN algorithm and regularized maximum likelihood (RML) methods
successfully produced EHT images of M87∗, with remarkable agreement among
methods (M87∗ Paper IV). The EHT baseline coverage for Sgr A∗ is substantially
better than for M87∗, primarily because of the additional telescope (SPT) with
mutual visibility of the source. Moreover, at $\lambda=1.3\,{\rm mm}$, Sgr A∗
has a compact flux density that is approximately four times larger than that
of M87∗, with no appreciable contribution to the short-baseline visibilities
from an extended jet. However, producing an image of Sgr A∗ requires
additional assumptions because of the rapid source variability and
interstellar scattering.

Specifically, VLBI imaging typically relies on Earth-rotation aperture
synthesis, in which the projection of each baseline sweeps out an arc in the
$(u,v)$-plane as the Earth rotates, allowing a sparse array of telescopes to
obtain the $(u,v)$-coverage necessary for the imaging of a static source
(Thompson et al., 2017). To account for the source structural variability, we
used a parametric model discussed in section 4. By incorporating this
variability error budget, imaging and modeling methods designed for a static
source can be applied to analyze data from a variable source.

To account for the interstellar scattering, we used two approaches (Paper
III). The first, “on-sky imaging,” applies no modifications to the data or
images. In this approach, the algorithms simply reconstruct the scattered
image of the source. The second, “descattered imaging,” adds an error budget
to interferometric visibilities to account for stochastic scattering
substructure before deconvolving the ensemble-average scattering kernel. Both
the ensemble-average kernel and the power spectrum of scattering are used
(Psaltis et al., 2018), each of which is precisely known from an analysis
combining of decades of observations of Sgr A∗ at centimeter wavelengths
(Johnson et al., 2018).

To test these imaging techniques and to select appropriate imaging parameters,
we developed a suite of synthetic observations of seven geometric models that
share the scattering and variability properties of Sgr A∗. This suite included
models with widely varying morphologies: rings, disks, a crescent, a double
source, and a point-like source with an extended halo. Each model was selected
to produce visibility amplitudes that were similar to those of Sgr A∗, with
two deep visibility minima, a physical scattering model applied, and
stochastic temporal evolution generated by a statistical model (Lee & Gammie,
2021).

We then selected the sets of imaging parameters that accurately reconstruct
images across the entire test suite, including both ring and non-ring data
sets. These “top set” parameter choices yield a corresponding collection of
reconstructed images of Sgr A∗ that provide both a representative average
image and a measure of its uncertainty. In addition, we used a new Bayesian
imaging method, which simultaneously estimates both the reconstructed image
and its associated variability noise model (Broderick et al., 2020). This
method does not require training on synthetic data, although we used the same
test suite for comparison and validation of this method.

When applied to the Sgr A∗ data, over $95\%$ of the top set images have a
prominent ring morphology. For an analysis using the combination of April 6
and 7 data, all samples from the Bayesian imaging posterior show a ring
morphology. In addition, geometric modeling of the EHT data shows a consistent
statistical preference for ring morphologies over alternatives with comparable
complexity. The ring has a diameter, width, and central brightness depression
that are consistent across the different choices of imaging methods and
parameters (see Paper III, ). However, the reconstructed images show diversity
in their specific attributes, particularly the azimuthal intensity
distribution around the ring. This uncertainty is a consequence of the limited
EHT baseline coverage, compounded by the challenges of imaging a variable
source. We categorized the reconstructed images into four clusters spanning
the primary reconstructed structures: three clusters are ring modes with
varying position angle, while the fourth is a comparatively small set of
reconstructed images with diverse non-ring morphologies. Figure 3 shows a
representative average image of Sgr A∗ on April 7, as well as the average
image for each of these clusters along with their relative occurrence
frequency.

To quantify the ring parameters in a complementary way, we used several
geometrical modeling methods, the parametrization of which were guided by the
reconstructed images of Sgr A∗. These models are defined by a thick ring with
azimuthal variations determined by low-order Fourier coefficients and an
additional Gaussian brightness floor. Because these simple geometric models
have a small number of parameters, they can be constrained using instantaneous
snapshots of data. Hence, we used two modeling approaches. With “snapshot”
modeling, we aggregate a series of independent fits to two-minute data
segments. This approach does not require a variability model. With “full-
track” modeling, we fit both a static geometric source model and a variability
noise model simultaneously to the entire 12-hour observation. Table 1
summarizes the consensus ring parameters measured using these methods (for
detailed results from individual methods, see Paper IV, ).

In Paper III, we also used dynamic imaging and snapshot ring modeling to
analyze the intraday image variability in Sgr A∗. We applied these analysis
methods to the 100-minute intervals on April 6 and 7 with the best sampling
(Farah et al., 2022), adopting a strong ring prior to counteract the limited
baseline coverage. On April 6, most dynamic imaging and modeling methods
recover a nearly static image, while many reconstructions on April 7 find an
evolving image. However, the results on April 7 are strongly affected by the
underlying prior assumptions; different parameters in the dynamic imaging
method result in different modes of position angle evolution in the
reconstructed images, including some reconstructions that are nearly static.
Thus, while the EHT data show detectable signs of image variability, we cannot
reliably constrain the underlying image evolution.

Table 1: Measured Parameters of Sgr A∗ Parameter | EHT Estimate
---|---
Emission ring:222 The orientation and magnitude of the ring’s brightness asymmetry is poorly constrained; it varies significantly among the reconstructed image modes and among different modeling and imaging methods. For details, see Paper IV. |
Diameter, $d$ | $51.8\pm 2.3\,\mu{\rm as}$
Fractional width, $W/d$ | ${\sim}\,30-50\%$
Orientation, $\eta$ | —
Brightness asymmetry, $A$ | ${\sim}\,0.04-0.3$
Angular gravitational radius,2 $\theta_{\rm g}$ | $4.8_{-0.7}^{+1.4}\,\mu{\rm as}$
Black hole mass,333To translate our estimate of $\theta_{\rm g}$ into an estimated mass of Sgr A∗, we use the distance to Sgr A∗ estimated using trigonometric VLBI parallaxes and proper motions of molecular masers in spiral arms of the Milky Way (Reid et al., 2019). For details, see Paper IV. $M$ | $4.0_{-0.6}^{+1.1}\times 10^{6}\,M_{\odot}$
Angular shadow diameter,444Estimates of $d_{\rm sh}$ are determined solely from EHT data, but estimates of $\delta$ use priors for $\theta_{\rm g}$ from resolved stellar orbits as indicated. For details, see Paper VI. $d_{\rm sh}$ | $48.7\pm 7.0\,\mu{\rm as}$
Schwarzschild shadow deviation,4 $\delta$ |  $-0.08^{+0.09}_{-0.09}$ (VLTI)
$-0.04^{+0.09}_{-0.10}$ (Keck)
Parameter | Previous Estimate
Angular gravitational radius, $\theta_{\rm g}$: |
Stellar orbits (VLTI)555Gravity Collaboration et al. (2022) | $5.125\pm 0.009\pm 0.020\,\mu{\rm as}$
Stellar orbits (Keck)666Do et al. (2019) | $4.92\pm 0.03\pm 0.01\,\mu{\rm as}$
Black hole distance, $D$: |
Stellar orbits (VLTI)5 | $8277\pm 9\pm 33\,{\rm pc}$
Stellar orbits (Keck)6 | $7935\pm 50\pm 32\,{\rm pc}$
Masers (cm VLBI)777Reid et al. (2019) Notes. Stated uncertainties correspond to 68% credible intervals. | $8150\pm 150\,{\rm pc}$
Black hole mass, $M$: |
Stellar orbits (VLTI)5 | $(4.297\pm 0.013)\times 10^{6}M_{\odot}$
Stellar orbits (Keck)6 | $(3.951\pm 0.047)\times 10^{6}M_{\odot}$

##  6 Implications for Accretion and Outflow Physics

What can we learn from these images and their variability properties? Focusing
first on the astrophysics of the accretion process and jet launching, we can
explore which physical scenarios are most consistent with our results, under
the assumption that Sgr A* is a Kerr black hole with mass and distance
accurately determined from stellar orbits. Our constraints on the properties
of the black hole and potential deviations from GR are explored in Section 7.

We assume that the accretion structure around Sgr A∗ is approximately governed
by ideal GRMHD, as was done for M87∗, which is common in the literature for
modeling SMBHs (see, e.g., Gammie et al., 2003). Decades of observations and
semi-analytical modeling (see Paper V for details, references, and caveats)
constrain the average plasma properties close to the event horizon of Sgr A∗,
allowing us to make several additional simplifying approximations. In
particular, for Sgr A∗ we can assume that radiative cooling does not strongly
affect the dynamics, and that the electrons and ions are weakly coupled by
Coulomb collisions, so that ions and electrons can have distinct temperatures
in some parts of the flow.

Because we model the plasma as a fluid with a single temperature, one of our
main sources of uncertainty is the treatment of the electrons, whose presence
is not explicitly accounted for in the simulation evolution equations. We
explore several parameterized models to assign the electron distribution
function (eDF), assuming that the electron temperature is proportional to the
proton temperature, with a proportionality that depends on the gas-to-magnetic
pressure ratio (Chan et al., 2015). The eDFs include thermal and variations of
non-thermal, the latter of which were not explored in the M87∗ 2017 papers.
Our fiducial thermal models employ the same eDF prescription as for the M87∗
papers, using only one free parameter, ${R_{\mathrm{high}}}$, to specify the
proton-to-electron temperature ratio in regions where gas pressure dominates
the magnetic pressure (Mościbrodzka et al., 2016). This ratio is typically
larger in the disk midplane than in the jet/outflow. Since the radiation is
produced by electrons, increasing ${R_{\mathrm{high}}}$ effectively increases
the brightness of the jets/outflow region relative to the disk, and changes
the resulting images/spectra (M87∗ Paper V). Compared to M87∗, we also allow
the inclination angle to vary.

We employ five different ideal GRMHD codes to explore a large swath of
overlapping parameter space, in some cases with very similar set-ups allowing
for consistency checks. In other cases, with a more exploratory sampling of
parameter space, we also allow differences in, for example, adiabatic index,
resolution, and size of the tori and/or computational domain. Most models are
initialized with an orbiting torus of plasma with characteristic radius $\sim
20GM/c^{2}$. The torus is seeded with a weak, poloidal magnetic field, and can
be either prograde or retrograde with respect to the black hole spin (a free
parameter). We also consider a limited set of exploratory models such as those
with ‘tilt’ where the black hole spin axis is misaligned from the rotation
axis of the torus (Liska et al., 2018; Chatterjee et al., 2020; White &
Quataert, 2022), as well as a model initiated on a very large grid using a
more realistic set-up for the outer boundary conditions in which the accretion
flow is directly fed by winds from orbiting stars (Ressler et al., 2020). Both
of these are more realistic physical scenarios but also allow a much larger
range of parameter space than we could fully explore.

We classify the fiducial models as being in either the magnetically arrested
(MAD; Narayan et al., 2003) or standard and normal evolution (SANE; Narayan et
al., 2012) modes. In MAD models the ordered magnetic fields significantly
affect the dynamics of the flow, episodically halting accretion onto the black
hole, while SANE models have weaker, more turbulent magnetic fields. Because
the dynamical timescale in Sgr A∗ is short compared to a night of
observations, it is important to run each model for enough time to capture the
range of spectral and structural variations. The simulations are typically run
for $30{,}000t_{\rm g}$, while some are run for more than $100{,}000t_{\rm g}$
in order to sample a broader distribution of behavior.

Figure 4:  Summary of constraints on our 200 fiducial GRMHD simulations. Color
indicates combined EHT constraints apart from structural or flux variability,
and hatching indicates combined non-EHT constraints. For each constraint
category and parameter combination, we delineate whether all of the three
simulation codes run with those parameters pass, whether only some pass, or
whether none pass. These exclusions leave only two models, each a MAD with
prograde spin, $30^{\circ}$ inclination, and ${R_{\mathrm{high}}}=160$. For
details, see Paper V.

For each time-dependent GRMHD simulation, with an eDF prescription and
inclination with respect to the line-of-sight, we calculate a sequence of
model images (movies) using ray tracing and including synchrotron emission and
absorption. We also calculate spectra including synchrotron emission and
absorption, bremsstrahlung emission and, using Monte Carlo methods, Compton
scattering. These synthetic data sets are then used to generate simulated EHT
images as well as multi-wavelength light curves and spectral energy
distributions (SEDs) for comparison with the Sgr A∗ data described in Paper
II. We scale all images to a benchmark average flux density of 2.4 Jy at 230
GHz to match the average synchrotron flux density of Sgr A∗ (see Paper II and
Wielgus et al. 2022).

We evaluate the simulations against three types of observational constraints:
EHT interferometric measurements, emission at other wavelengths, and
variability. The EHT constraints include 1) a measure of the image size, 2)
salient features from the visibility amplitudes, such as the location of the
first deep minimum, and 3) the diameter, asymmetry, and width of simplified
ring models fitted to well-sampled portions of the April 7 visibility data.
The constraints from other wavelengths include the flux densities at 86 GHz,
$2.2\,\mu{\rm m}$ in the NIR, and X-ray, and the major axis source size at 86
GHz, constrained from observations with the Global mm-VLBI Array. Finally, the
variability constraints are 1) the fractional 230 GHz variability on 3 h
timescales, derived from more than a decade of measurements, and 2) the
structural variability of the source, calculated at a baseline length of
4G$\lambda$ after fitting a parameterized model to the visibility amplitude
variation versus baseline length. See Paper V for the full ranges of tests and
pass/fail conditions.

Compared to M87∗ Paper V, we explore a larger range of models and model
parameter space, and we also include some additional observational
constraints. These include the degree of intrinsic variability as well as the
broadband spectral constraints given above. Accordingly, we find that all our
models fail at least one of the observational constraints. These results
indicate the power of combining interferometric data with other observational
constraints to narrow down the viable physical parameter space. We now
summarize our main results and their implications for our understanding of Sgr
A∗’s accretion state and geometry.

We primarily focus on a set of “fiducial” simulations, which use aligned
(prograde or retrograde) accretion flows and thermal eDFs defined via the
$R_{\rm high}$ prescription. We declare a model to fully pass a set of
constraints only when all GRMHD simulations with those parameters pass. This
approach helps to ensure that our selection of favored models is resilient to
small variations in the GRMHD simulation choices and software. Figure 4
summarizes these results.

All edge-on (high inclination) models fail the combined set of EHT-only
constraints for at least one simulation, and almost all retrograde models
($a_{*}<0$) fail. There are two interesting groupings of models that pass all
EHT constraints for all simulations: both have positive/prograde spin
($a_{*}=0.5$, 0.94) with lower (${\leq}\,50^{\circ}$) inclination, but some
are MAD (10) and some are SANE (8). With only ${\sim}10\%$ pass rate for all
models, it is clear that EHT imaging data alone are capable of strongly down-
selecting the potential model space. The more heterogeneous non-EHT
constraints prefer a rather different set of models. However even with all
these constraints, 11/200 models pass for all simulations; all of these are
MADs with all but one having $R_{\rm high}=160$. On the other hand, they cover
a wider range of spin compared to EHT-only constraints, with a slight
preference for $a_{*}\leq 0$ and higher inclinations, and including retrograde
and edge-on models.

Figure 5:  Simulated images of Sgr A∗. Left: A single snapshot image of a
numerical simulation of Sgr A∗ that passes 10 out of the 11 observational
criteria described in Paper V. Middle: The average of this simulation with
time sampling that matches the EHT observational cadence on April 7. Right:
Representative image reconstruction using synthetic visibilities generated
from the simulation in the adjacent panels (see Appendix H in Paper III, ).
This image has been averaged across methodologies and reconstructed
morphologies, as in Figure 3. Each panel is shown on a linear brightness scale
that is normalized to its peak.

None of the 200 fiducial models pass all 11 constraints in combination, the
most strict of which are the 86 GHz size, the ring diameter and the light
curve variability on 3-hour timescales. Of these, the light curve variability
turns out to be the most stringent constraint, passing only 4% of fiducial
models. SANE models, which are less variable than MADs, are preferred, while
all the other EHT/non-EHT constraints generally favor MADs. If we consider the
set of models that pass 10/11 constraints, it is notable that for both MADs
and SANEs, there is still a marked preference for models with prograde spin, a
lower inclination and ${R_{\mathrm{high}}}>40$ with several clustered at the
maximum of ${R_{\mathrm{high}}}=160$.

The “exploratory” models cover less of a range in parameter space but still
indicate important trends. In particular, including non-thermal electrons
(which are especially important for modeling flares) tends to push the limits
of allowed NIR flux and, to some extent, also the X-ray flux. Because the non-
thermal particles also enhance the 230 GHz flux density, rescaling to a fixed
flux density results in a smaller accretion rate. The smaller resulting
opacity then affects the image properties, such as producing a somewhat
narrower ring morphology at 230 GHz and a larger image at 86 GHz.
Nevertheless, the addition of non-thermal models does not drastically change
the preferred parameter space, and passing models also favor prograde spin and
lower inclination. Increased tilt of the accretion flow tends to increase
variability and the NIR flux, and thus leads to model failures. Similarly,
neither of the two wind-fed models passed, but this is not surprising as they
only model a single spin ($a_{*}=0$) and two instances of the thermal
${R_{\mathrm{high}}}$ eDF. With such sparse sampling of parameter space, these
classes of models require a more focused study to draw conclusions.

Overall, very few of our models are as quiet as the data. Although this wasn’t
investigated in M87∗ Paper V, subsequent work suggests a similar result for
M87∗ simulations (Satapathy et al., 2022). In general SANE models are less
variable than MADs, and face-on models are somewhat less variable than edge-
on. However, there are limitations of the modeling that may affect the
variability. For instance, collisionless effects, radiative cooling, and
improved electron heating models could all potentially reduce variability.
Thus, our variability constraints may be too strict.

There are only two models that pass everything but the variability
constraints. The models have similar parameters: both are MAD, both are
prograde spin (one has $a_{*}=0.5$, and the other has $a_{*}=0.94$), and both
have $i=30^{\circ}$ and ${R_{\mathrm{high}}}=160$. Figure 5 shows a
representative snapshot from one of these simulations and its corresponding
image reconstruction. If we take these two models as indicative, there is a
preference for accretion rates at the lower end of the range found by other
constraints, $10^{-9}-10^{-8}M_{\odot}/{\rm yr}$, and an outflow power of
${\sim}10^{38}$ erg/s, which is comparable to the bolometric luminosity of
stellar-mass black holes in X-ray binaries. The combination of high
${R_{\mathrm{high}}}$ and relatively low inclination satisfies the spectral
constraints without their associated jet structure producing too much
asymmetry to satisfy the EHT constraints. A low inclination is also consistent
with the independent estimate based on tracking the motions of NIR flaring
structures using the VLTI (Gravity Collaboration et al., 2018b). Our results
highlight the value of continued EHT plus multiwavelength monitoring of Sgr
A∗, and there is clearly much more exploration to conduct, including improved
theoretical models and numerical simulations, full SED modeling, and
polarimetric imaging, all of which will yield deeper physical insights (see,
e.g., Event Horizon Telescope Collaboration et al., 2021b).

##  7 Implications for Black Holes and General Relativity

As demonstrated using the suite of GRMHD simulations discussed in section 6,
the EHT images of Sgr A∗ are consistent with the expected appearance of a Kerr
black hole. Moreover, our images are consistent with an angular gravitational
radius $\theta_{\rm g}$ that matches the expectations from nearly Keplerian
orbits of stars on scales of $(10^{3}-10^{5})R_{\rm S}$.

To quantify this consistency, we compute $\theta_{\rm g}$ using EHT
measurements alone (for details, see Paper IV, ). Following the procedure
developed in M87∗ Paper VI, we estimate $\theta_{\rm g}$ by calibrating the
observed emission ring diameter $d$ to the known angular gravitational scale
$\theta_{\rm g}$ for a suite of synthetic datasets produced from the GRMHD
library presented in Paper V. Specifically, we generate 100 datasets from
GRMHD simulations with varying intrinsic $\theta_{\rm g}$ and that span the
explored range in accretion states, black hole spins, viewing inclinations,
position angles, and phenomenological electron heating parameters
${R_{\mathrm{high}}}$. Each synthetic dataset matches the baseline coverage
and sensitivity of the EHT observations. We then estimate the ring diameter
using multiple geometric modeling and imaging methods, deriving a separate
calibration factor $\alpha\equiv d/\theta_{\rm g}$ for each method and for
every dataset to attain a method-dependent scaling relationship and associated
uncertainty.

With this approach, we estimate $\theta_{\rm g}=4.8_{-0.7}^{+1.4}\,\mu{\rm
as}$ for Sgr A∗, where the uncertainties correspond to a 68% credible
interval. This estimate is consistent with, but much less constraining than,
measurements of $\theta_{\rm g}$ using resolved stellar orbits (see Table 1).
This procedure assumes the validity of the Kerr metric and relies on our suite
of GRMHD simulations to provide a reasonable proxy for the space of viable
emission models. The fractional uncertainty in our estimate of $\theta_{\rm
g}$ for Sgr A∗ is broader than our estimate for M87∗ ($\theta_{\rm g}=3.8\pm
0.4\,\mu{\rm as}$), primarily because of the increased calibration uncertainty
associated with the unknown inclination of Sgr A∗ and because of the increased
diameter measurement uncertainty associated with intrinsic variability of Sgr
A∗.

We also used EHT images to constrain potential deviations from the Kerr metric
and to test the nature of the compact object in Sgr A∗ (for details, see Paper
VI, ). For instance, the brightness depression has a contrast $f_{\rm
c}\mathrel{\raise 1.29167pt\hbox{$<$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}0.3$, which provides support for the existence of an
event horizon in Sgr A∗. If the compact object instead had an absorptive
boundary that radiated the thermalized energy of infalling material
$\dot{M}c^{2}$, it could still produce a depression in the EHT images but
would generate IR emission that greatly exceeds the measured spectrum of Sgr
A∗ (Broderick & Narayan, 2006; Narayan & McClintock, 2008). Alternatively, a
partially reflecting surface would reduce the depth of the depression; the EHT
images directly constrain the albedo of such a reflecting surface to be
${\mathrel{\raise 1.29167pt\hbox{$<$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}}\,0.3$. The brightness depression also rules out
several specific black hole alternatives, including some models of naked
singularities (e.g., Joshi et al., 2014) and some models of boson stars (e.g.,
Olivares et al., 2020), although other horizonless black-hole mimickers
produce apparent shadows that are consistent with the observed depression
(e.g., Shaikh et al., 2019).

The measured ring diameter also provides constraints on the spacetime metric.
To obtain these constraints, we generate a series of synthetic images to
relate a measured emission ring diameter to that of the underlying black hole
shadow. These images included a broad range of GRMHD simulations from Paper V,
as well as images for which the GRMHD and underlying metric assumptions are
relaxed. Specifically, we generated sets of images of analytic models for
accretion onto 1) a Kerr black hole, 2) a black hole with parametric
deviations from Kerr given by the Johannsen-Psaltis metric (Johannsen &
Psaltis, 2011), and 3) a non-Kerr black hole defined by the Kerr-Sen metric
(García et al., 1995). For each analytic model, we allow the emission
prescriptions to vary within physically plausible limits (for details, see
Özel et al., 2021; Younsi et al., 2021). We find a similar relationship
between the diameter of the emission ring and that of the black hole shadow in
all these cases, indicating that this relationship is insensitive to the
details of the underlying spacetime.

Selecting a subset of these models, we then generate 145 synthetic datasets.
We apply imaging and modeling methods to compute the ring diameter for each
dataset to evaluate method-dependent measurement uncertainties and biases. We
then use this calibration together with the EHT measurements of the ring
diameter to determine the angular diameter of the black hole shadow for Sgr
A∗: $d_{\rm sh}=48.7\pm 7.0\,\mu{\rm as}$. This result is tighter than the
range for $\theta_{\rm g}$ derived above because of differences in the
calibration datasets and procedures used in Paper IV and Paper VI.
Specifically, Paper IV uses a calibration suite that consists entirely of
dynamic GRMHD models, and it derives a scale factor between $\theta_{\rm g}$
and the angular diameter of the emission ring. In contrast, Paper VI uses a
calibration suite containing both static Kerr and non-Kerr images and dynamic
GRMHD models, and it derives a scale factor between $d_{\rm sh}$ and the
angular diameter of the emission ring. Paper VI also excludes datasets for
which the range of reconstructed diameters is more than $2-3$ times the range
that is measured using the Sgr A∗ data. Thus, comparison of these results
provides a measure of the impact of our assumptions and procedures on the
inference of the black hole properties.

By comparing this shadow diameter with stellar dynamical measurements of the
mass of Sgr A∗, we also determine a deviation parameter $\delta$, which
quantifies the fractional difference between the inferred shadow diameter and
its expected value for a non-spinning (Schwarzschild) black hole (for details,
see Paper VI, ). We find $\delta=-0.08^{+0.09}_{-0.09}$ when using VLTI
measurements of $\theta_{\rm g}$ and $\delta=-0.04^{+0.09}_{-0.10}$ when using
Keck measurements of $\theta_{\rm g}$ (see Table 1). For comparison, a
spinning (Kerr) black hole has $-0.08\leq\delta\leq 0$.

Figure 6:  Comparison of posterior distributions for the Schwarzschild
fractional shadow deviation parameter $\delta$ measured by the EHT for Sgr A∗
and M87∗. For Sgr A∗, $\delta$ is computed relative the expected shadow size
from monitoring stellar orbits; for M87∗, $\delta$ is computed for both the
expected shadow from stellar dynamics and from gas dynamics (for details, see
M87∗ Paper VI, ). The gray band shows the expected range of $\delta$ for the
Kerr metric. For the stellar-dynamical prior mass estimates, the EHT
measurements show close consistency with the same black hole metric over three
orders of magnitude in black hole mass.

Under the assumption that this range of calibration factors applies
generically to all non-Kerr spacetimes that have black hole shadows, we can
translate measurements of $\delta$ into constraints on parameters of these
spacetimes (Psaltis et al., 2020). In particular, our measurements exclude
specific non-Kerr solutions for Sgr A∗, such as the traversible Morris-Thorne
wormhole and the naked singularities in the Reissner-Nordström metric, which
predict shadows that are significantly smaller than those of Kerr black holes.

Relative to M87∗, the primary strengths of testing GR with EHT observations of
Sgr A∗ are the tight prior constraints on its mass-to-distance ratio
$\theta_{\rm g}$ and its shorter gravitational timescale that allows EHT
observations to span many dynamical times. Together, these results show SMBHs
consistent with predictions of the Kerr metric over a spread of three orders
of magnitude in their mass (see Figure 6). The image of Sgr A∗ probes a
similar gravitational potential to M87∗ but spacetime curvature $\xi\propto