# The intrinsic structure of Sagittarius A* at 1.3 cm and 7 mm

Ilje Cho Korea Astronomy and Space Science Institute, Daedeok-daero 776,
Yuseong-gu, Daejeon 34055, Republic of Korea  University of Science and
Technology, Gajeong-ro 217, Yuseong-gu, Daejeon 34113, Republic of Korea
Instituto de Astrofísica de Andalucía - CSIC, Glorieta de la Astronomía s/n,
E-18008 Granada, Spain  Guang-Yao Zhao Instituto de Astrofísica de Andalucía -
CSIC, Glorieta de la Astronomía s/n, E-18008 Granada, Spain  Korea Astronomy
and Space Science Institute, Daedeok-daero 776, Yuseong-gu, Daejeon 34055,
Republic of Korea  Tomohisa Kawashima Institute for Cosmic Ray Research, The
University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582, Japan  Motoki
Kino National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo
181-8588, Japan  Kogakuin University of Technology & Engineering, Academic
Support Center, 2665-1 Nakano, Hachioji, Tokyo 192-0015, Japan  Kazunori
Akiyama Massachusetts Institute of Technology Haystack Observatory, 99
Millstone Road, Westford, MA 01886, USA  Black Hole Initiative at Harvard
University, 20 Garden Street, Cambridge, MA 02138, USA  National Astronomical
Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan  Michael D.
Johnson Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street,
Cambridge, MA 02138, USA  Black Hole Initiative at Harvard University, 20
Garden Street, Cambridge, MA 02138, USA  Sara Issaoun Department of
Astrophysics, Institute for Mathematics, Astrophysics and Particle Physics
(IMAPP), Radboud University, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands
Kotaro Moriyama Mizusawa VLBI Observatory, National Astronomical Observatory
of Japan, 2-12 Hoshigaoka, Mizusawa, Oshu, Iwate 023-0861, Japan
Massachusetts Institute of Technology Haystack Observatory, 99 Millstone Road,
Westford, MA 01886, USA  Xiaopeng Cheng Korea Astronomy and Space Science
Institute, Daedeok-daero 776, Yuseong-gu, Daejeon 34055, Republic of Korea
Juan-Carlos Algaba Department of Physics, Faculty of Science, University of
Malaya, 50603 Kuala Lumpur, Malaysia  Taehyun Jung Korea Astronomy and Space
Science Institute, Daedeok-daero 776, Yuseong-gu, Daejeon 34055, Republic of
Korea  University of Science and Technology, Gajeong-ro 217, Yuseong-gu,
Daejeon 34113, Republic of Korea  Bong Won Sohn Korea Astronomy and Space
Science Institute, Daedeok-daero 776, Yuseong-gu, Daejeon 34055, Republic of
Korea  University of Science and Technology, Gajeong-ro 217, Yuseong-gu,
Daejeon 34113, Republic of Korea  Department of Astronomy, Yonsei University,
Yonsei-ro 50, Seodaemun-gu, Seoul 03722, Republic of Korea  Thomas P.
Krichbaum Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121
Bonn, Germany  Maciek Wielgus Center for Astrophysics $|$ Harvard &
Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA  Black Hole Initiative
at Harvard University, 20 Garden Street, Cambridge, MA 02138, USA  Kazuhiro
Hada Mizusawa VLBI Observatory, National Astronomical Observatory of Japan,
2-12 Hoshigaoka, Mizusawa, Oshu, Iwate 023-0861, Japan  Department of
Astronomical Science, The Graduate University for Advanced Studies (SOKENDAI),
2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan  Ru-Sen Lu Shanghai Astronomical
Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030,
China  Key Laboratory of Radio Astronomy, Chinese Academy of Sciences, Nanjing
210008, China  Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69,
D-53121 Bonn, Germany  Yuzhu Cui Mizusawa VLBI Observatory, National
Astronomical Observatory of Japan, 2-12 Hoshigaoka, Mizusawa, Oshu, Iwate
023-0861, Japan  Department of Astronomical Science, The Graduate University
for Advanced Studies (SOKENDAI), 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
Satoko Sawada-Satoh The Research Institute for Time Studies, Yamaguchi
University, 1677-1 Yoshida, Yamaguchi, Yamaguchi 753-8511, Japan  Zhiqiang
Shen Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan
Road, Shanghai 200030, China  Key Laboratory of Radio Astronomy, Chinese
Academy of Sciences, Nanjing 210008, China  Jongho Park EACOA fellow
Institute of Astronomy and Astrophysics, Academia Sinica, 11F of Astronomy-
Mathematics Building, AS/NTU No. 1, Sec. 4, Roosevelt Rd, Taipei 10617,
Taiwan, R.O.C.  Department of Physics and Astronomy, Seoul National
University, Gwanak-gu, Seoul 08826, Republic of Korea  Wu Jiang Shanghai
Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road,
Shanghai 200030, China  Key Laboratory of Radio Astronomy, Chinese Academy of
Sciences, Nanjing 210008, China  Hyunwook Ro Department of Astronomy, Yonsei
University, Yonsei-ro 50, Seodaemun-gu, Seoul 03722, Republic of Korea  Korea
Astronomy and Space Science Institute, Daedeok-daero 776, Yuseong-gu, Daejeon
34055, Republic of Korea  Kunwoo Yi  Department of Physics and Astronomy,
Seoul National University, Gwanak-gu, Seoul 08826, Republic of Korea  Kiyoaki
Wajima Korea Astronomy and Space Science Institute, Daedeok-daero 776,
Yuseong-gu, Daejeon 34055, Republic of Korea  Jee Won Lee Korea Astronomy and
Space Science Institute, Daedeok-daero 776, Yuseong-gu, Daejeon 34055,
Republic of Korea  Jeffrey Hodgson  Department of Physics and Astronomy,
Sejong University, 209 Neungdong-ro, Gwangjin-gu, Seoul, South Korea  Fumie
Tazaki Mizusawa VLBI Observatory, National Astronomical Observatory of Japan,
2-12 Hoshigaoka, Mizusawa, Oshu, Iwate 023-0861, Japan  Mareki Honma Mizusawa
VLBI Observatory, National Astronomical Observatory of Japan, 2-12 Hoshigaoka,
Mizusawa, Oshu, Iwate 023-0861, Japan  Department of Astronomical Science, The
Graduate University for Advanced Studies (SOKENDAI), 2-21-1 Osawa, Mitaka,
Tokyo 181-8588, Japan  Kotaro Niinuma Graduate School of Sciences and
Technology for Innovation, Yamaguchi University, 1677-1 Yoshida, Yamaguchi,
Yamaguchi 753-8511, Japan  The Research Institute for Time Studies, Yamaguchi
University, Yoshida 1677-1, Yamaguchi, Yamaguchi 753-8511, Japan  Sascha
Trippe Department of Physics and Astronomy, Seoul National University, Gwanak-
gu, Seoul 08826, Republic of Korea  SNU Astronomy Research Center, Seoul
National University, Gwanak-gu, Seoul 08826, Republic of Korea  Tao An
Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan
Road, Shanghai 200030, China  Yingkang Zhang  Shanghai Astronomical
Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030,
China  Jeong Ae Lee  Korea Astronomy and Space Science Institute, Daedeok-
daero 776, Yuseong-gu, Daejeon 34055, Republic of Korea  Se-Jin Oh  Korea
Astronomy and Space Science Institute, Daedeok-daero 776, Yuseong-gu, Daejeon
34055, Republic of Korea  Do-Young Byun Korea Astronomy and Space Science
Institute, Daedeok-daero 776, Yuseong-gu, Daejeon 34055, Republic of Korea
University of Science and Technology, Gajeong-ro 217, Yuseong-gu, Daejeon
34113, Republic of Korea  Sang-Sung Lee Korea Astronomy and Space Science
Institute, Daedeok-daero 776, Yuseong-gu, Daejeon 34055, Republic of Korea
University of Science and Technology, Gajeong-ro 217, Yuseong-gu, Daejeon
34113, Republic of Korea  Jae-Young Kim Korea Astronomy and Space Science
Institute, Daedeok-daero 776, Yuseong-gu, Daejeon 34055, Republic of Korea
Junghwan Oh  Korea Astronomy and Space Science Institute, Daedeok-daero 776,
Yuseong-gu, Daejeon 34055, Republic of Korea  Shoko Koyama Niigata University,
8050 Ikarashi 2-no-cho, Nishi-ku, Niigata 950-2181, Japan  Institute of
Astronomy and Astrophysics, Academia Sinica, 11F of Astronomy-Mathematics
Building, AS/NTU No. 1, Sec. 4, Roosevelt Rd, Taipei 10617, Taiwan, R.O.C.
Keiichi Asada  Institute of Astronomy and Astrophysics, Academia Sinica, 11F
of Astronomy-Mathematics Building, AS/NTU No. 1, Sec. 4, Roosevelt Rd, Taipei
10617, Taiwan, R.O.C.  Xuezheng Wang  Shanghai Astronomical Observatory,
Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, China  Lang Cui
Xinjiang Astronomical Observatory, Chinese Academy of Sciences, Urumqi 830011,
China  Key Laboratory of Radio Astronomy, Chinese Academy of Sciences, Nanjing
210008, China  Yoshiaki Hagiwara Toyo University, 5-28-20 Hakusan, Bunkyo-ku,
Tokyo 112-8606, Japan  Masanori Nakamura National Institute of Technology,
Hachinohe College, Yubinbango Aomori Prefecture Hachinohe Oaza Tamonoki
character Ueno flat 16-1, 039-1192, Japan  Institute of Astronomy and
Astrophysics, Academia Sinica, 11F of Astronomy-Mathematics Building, AS/NTU
No. 1, Sec. 4, Roosevelt Rd, Taipei 10617, Taiwan, R.O.C.  Mieko Takamura
National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo
181-8588, Japan  Tomoya Hirota National Astronomical Observatory of Japan,
2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan  Koichiro Sugiyama National
Astronomical Research Institute of Thailand (Public Organization), 260 Moo 4,
T. Donkaew, A. Maerim, Chiangmai, 50180, Thailand  Noriyuki Kawaguchi National
Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
Hideyuki Kobayashi  Mizusawa VLBI Observatory, National Astronomical
Observatory of Japan, 2-12 Hoshigaoka, Mizusawa, Oshu, Iwate 023-0861, Japan
Tomoaki Oyama  Mizusawa VLBI Observatory, National Astronomical Observatory of
Japan, 2-12 Hoshigaoka, Mizusawa, Oshu, Iwate 023-0861, Japan  Yoshinori
Yonekura Center for Astronomy, Ibaraki University, 2-1-1 Bunkyo, Mito, Ibaraki
310-8512, Japan  Jongsoo Kim Korea Astronomy and Space Science Institute,
Daedeok-daero 776, Yuseong-gu, Daejeon 34055, Republic of Korea  Ju-Yeon Hwang
Korea Astronomy and Space Science Institute, Daedeok-daero 776, Yuseong-gu,
Daejeon 34055, Republic of Korea  Dong-Kyu Jung  Korea Astronomy and Space
Science Institute, Daedeok-daero 776, Yuseong-gu, Daejeon 34055, Republic of
Korea  Hyo-Ryoung Kim  Korea Astronomy and Space Science Institute, Daedeok-
daero 776, Yuseong-gu, Daejeon 34055, Republic of Korea  Jeong-Sook Kim  Basic
Science Research Institute, Chungbuk National University, Chungdae-ro 1,
Seowon-Gu, Cheongju, Chungbuk 28644, Korea  Chung-Sik Oh  Korea Astronomy and
Space Science Institute, Daedeok-daero 776, Yuseong-gu, Daejeon 34055,
Republic of Korea  Duk-Gyoo Roh  Korea Astronomy and Space Science Institute,
Daedeok-daero 776, Yuseong-gu, Daejeon 34055, Republic of Korea  Jae-Hwan Yeom
Korea Astronomy and Space Science Institute, Daedeok-daero 776, Yuseong-gu,
Daejeon 34055, Republic of Korea  Bo Xia  Shanghai Astronomical Observatory,
Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, China  Key
Laboratory of Radio Astronomy, Chinese Academy of Sciences, Nanjing 210008,
China  Weiye Zhong  Shanghai Astronomical Observatory, Chinese Academy of
Sciences, 80 Nandan Road, Shanghai 200030, China  Key Laboratory of Radio
Astronomy, Chinese Academy of Sciences, Nanjing 210008, China  Bin Li
Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan
Road, Shanghai 200030, China  Key Laboratory of Radio Astronomy, Chinese
Academy of Sciences, Nanjing 210008, China  Rongbing Zhao  Shanghai
Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road,
Shanghai 200030, China  Key Laboratory of Radio Astronomy, Chinese Academy of
Sciences, Nanjing 210008, China  Jinqing Wang  Shanghai Astronomical
Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030,
China  Key Laboratory of Radio Astronomy, Chinese Academy of Sciences, Nanjing
210008, China  Qinghui Liu  Shanghai Astronomical Observatory, Chinese Academy
of Sciences, 80 Nandan Road, Shanghai 200030, China  Key Laboratory of Radio
Astronomy, Chinese Academy of Sciences, Nanjing 210008, China  Zhong Chen
Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan
Road, Shanghai 200030, China  Key Laboratory of Radio Astronomy, Chinese
Academy of Sciences, Nanjing 210008, China

(Accepted December 8, 2021)

###### Abstract

Sagittarius A* (Sgr A∗ ), the Galactic Center supermassive black hole (SMBH),
is one of the best targets to resolve the innermost region of SMBH with very
long baseline interferometry (VLBI). In this study, we have carried out
observations toward Sgr A∗ at 1.349 cm (22.223 GHz) and 6.950 mm (43.135 GHz)
with the East Asian VLBI Network, as a part of the multi-wavelength campaign
of the Event Horizon Telescope (EHT) in 2017 April. To mitigate scattering
effects, the physically motivated scattering kernel model from Psaltis et al.
(2018) and the scattering parameters from Johnson et al. (2018) have been
applied. As a result, a single, symmetric Gaussian model well describes the
intrinsic structure of Sgr A∗ at both wavelengths. From closure amplitudes,
the major-axis sizes are $\sim 704\,\pm\,102\,\mu$as (axial ratio$\sim
1.19^{+0.24}_{-0.19}$) and $\sim 300\,\pm\,25\,\mu$as (axial ratio$\sim
1.28\,\pm\,0.2$) at 1.349 cm and 6.95 mm respectively. Together with a quasi-
simultaneous observation at 3.5 mm (86 GHz) by Issaoun et al. (2019), we show
that the intrinsic size scales with observing wavelength as a power-law, with
an index $\sim 1.2\pm 0.2$. Our results also provide estimates of the size and
compact flux density at 1.3 mm, which can be incorporated into the analysis of
the EHT observations. In terms of the origin of radio emission, we have
compared the intrinsic structures with the accretion flow scenario, especially
the radiatively inefficient accretion flow based on the Keplerian shell model.
With this, we show that a nonthermal electron population is necessary to
reproduce the source sizes.

Accretion (14), Galactic Center (565), Supermassive black holes (1663), Very
long baseline interferometry (1769)

††journal: ApJ

##  1 Introduction

The supermassive black hole (SMBH) at our Galactic Center, Sagittarius A* (Sgr
A∗ ), is the closest SMBH to the Earth, with a mass $\sim 4.1\times
10^{6}\,M_{\odot}$ (Ghez et al., 2008; Genzel et al., 2010). Thanks to its
proximity, $\sim$8.1 kpc (e.g., Gravity Collaboration et al., 2019), Sgr A∗
has the largest angular size of the Schwarzschild radius ($R_{s}$) among all
black holes, $\sim 10\,\mu$as. Therefore it is one of the best laboratories to
explore mass accretion onto SMBHs, and it is the other main target besides M
87 for Event Horizon Telescope (EHT) to resolve the black hole shadow (EHT
Collaboration, 2019a, b, c, d, e, f).

In very long baseline interferometry (VLBI) observations at centimeter (cm)
wavelengths, the structure of Sgr A∗ is dominated by scatter broadening caused
by the ionized interstellar scattering medium (ISM; see, e.g., Rickett 1990;
Narayan 1992) which is located at $\sim 5.8\,$kpc from Earth (Bower et al.,
2014a). As a result, the observed sizes are proportional to $\lambda^{2}$
where $\lambda$ is the observing wavelength (Davies et al., 1976; van
Langevelde et al., 1992; Lo et al., 1998; Bower et al., 2004; Shen et al.,
2005; Bower et al., 2006; Johnson et al., 2018). The shape has been found to
be an asymmetric Gaussian, elongated towards the east-west (i.e., stronger
angular broadening; Lo et al. 1985, 1998; Alberdi et al. 1993; Frail et al.
1994; Bower & Backer 1998). While the major and minor axis sizes are sub-
dominant to scattering at $\gtrsim 1\,$cm, there are marginal detections of
the measured sizes larger than the $\lambda^{2}$ scaling at millimeter (mm)
wavelengths which indicate the intrinsic size of Sgr A∗ (Krichbaum et al.,
1998; Lo et al., 1998; Doeleman et al., 2001; Lu et al., 2011a). Especially at
1.3 mm, the persistent asymmetric structure of Sgr A∗ has been also suggested
from the proto-EHT observations (Fish et al., 2016; Lu et al., 2018).

In addition to the angular broadening by the diffractive scattering,
refractive scattering effects introduce substructure in the image which is
caused by the density irregularities in the ionized ISM. This substructure
appears as “refractive noise” in interferometric visibilities (Goodman &
Narayan, 1989; Narayan & Goodman, 1989; Johnson & Gwinn, 2015; Johnson &
Narayan, 2016), and it was discovered at 1.3 cm observations of Sgr A∗ by
Gwinn et al. (2014). At 7 and 3 mm, non-zero closure phases have been detected
which can be interpreted as either the imprint of the intrinsic asymmetry of
Sgr A∗ or the refractive substructure (Rauch et al., 2016; Ortiz-León et al.,
2016; Brinkerink et al., 2016, 2019). To better constrain the scattering
effects, recently, Psaltis et al. (2018) (hereafter P18) suggested a
physically motivated scattering model for anisotropic scattering. They
introduced the magnetohydro dynamics (MHD) turbulence with a finite inner
scale and a wandering transverse magnetic field direction to their simplified
model and derived analytic models of expected observational properties, such
as the scatter broadening and refractive scintillation. With this model,
Johnson et al. (2018) (hereafter J18) tightly constrained the scattering
parameters such as power-law index of the phase structure function of the
scattering screen, $\alpha$, and a finite inner scale of interstellar
turbulence, $r_{\rm in}$, using multi-wavelength observational data. After
that, Issaoun et al. (2019) (hereafter I19) showed that the scattering model
from J18 provided comparable refractive noise to their observations at 3.5 mm
using the Global Millimeter VLBI Array (GMVA) in concert with the phased
Atacama Large Millimeter/submillimeter Array (ALMA) in 2017 April.

In this study, we analyze the properties of Sgr A∗ at 1.349 cm (22.223 GHz;
hereafter 1.3 cm and 22 GHz) and 6.950 mm (43.135 GHz; hereafter 7 mm and 43
GHz) with the East Asian VLBI Network (EAVN; Hagiwara et al. 2015; Wajima et
al. 2016; An et al. 2018; Cui et al. 2021) observations in 2017 April, by
mitigating the scattering effects based on the studies from P18, J18, and I19.
The observations have been carried out within two days from the EHT (1.3 mm)
and GMVA+ALMA (3.5 mm; I19) campaign, providing a range of quasi-simultaneous
multi-wavelength observations, which is crucial for constraining the physical
parameters. To this end, we report the flux densities and intrinsic structures
of Sgr A∗ at both wavelengths.

Especially the structure of Sgr A∗ can provide an important hint of its
emission model. There is a long-debate about the dominant emission source of
Sgr A∗ , whether an accretion flow or a jet. While there are many theoretical
predictions (e.g., Mościbrodzka et al., 2014) and indirect evidence (e.g.,
Brinkerink et al., 2015) of a possible jet from Sgr A∗ , a jet-like structure
has not been resolved yet. So we first find the most probable model of an
accretion flow scenario, if it can solely explain the observed results in
which circumstances. For this purpose, we have compared the measured intrinsic
structure of Sgr A∗ with radiatively inefficient accretion flow (RIAF) based
on the Keplerian shell model (Falcke et al., 2000; Broderick & Loeb, 2006; Pu
et al., 2016; Kawashima et al., 2019) and found that the thermal-nonthermal
electron population is necessary to reproduce the observational results. A
similar comparison has been carried out with spectral energy distribution
(SED; e.g., Özel et al. 2000; Yuan et al. 2003) and with theoretical structure
at 230 GHz (Chael et al., 2017; Mao et al., 2017), but has not been
investigated with the structure at lower frequency range (i.e., 22 and 43
GHz). Note that our analysis is based on a recent study of the electron energy
spectrum in an accretion flow (Broderick et al., 2011; Pu et al., 2016), so it
does not rule out the possible jet scenario which is beyond the scope of this
paper.

In this paper, we begin with the outline of our observations and data
processing in section 2. Next, in section 3, the imaging and model–fitting
procedures are described. In section 4, the overall results including the
wavelength-dependence and its implication on the EHT imaging are presented. In
section 5, the importance of nonthermal electron components for an accretion
flow model to explain our results is discussed. Lastly, we summarize our
findings in section 6.

##  2 Observations and Data reduction

Table 1: EAVN and the quasi-simultaneous observations

Array | Date (2017) |  $\lambda$ (cm) | Reference
---|---|---|---
EAVN | Apr 03 | 1.349 | This work
EAVN | Apr 04 | 0.695 | This work
GMVA+ALMA | Apr 03 | 0.35 | I19
EHT | Apr 05-11 | 0.13 | …

Note.  From left to right, the observing array, date, wavelength, and
references.

Figure 1:  $u-v$ coverage of EAVN observations at 1.3 cm (left) and 0.7 cm
(right). The KaVA observations with successful detection (S/N$>5$; cyan) and
flagged points (gray) are shown. TIA provides the longest baselines (red).
Each point has been averaged within 30 second interval.

The EAVN consists of the KaVA (KVN111Korean VLBI Network, which consists of
three 21 m telescopes in Korea: Yonsei (KYS), Ulsan (KUS), and Tamna (KTN) and
VERA222VLBI Exploration of Radio Astrometry, which consists of four 20 m
telescopes in Japan: Mizusawa (MIZ), Iriki (IRK), Ogasawara (OGA), and
Ishigakijima (ISG) Array; e.g., Niinuma et al. 2015; Hada et al. 2017; Park et
al. 2019) and additional East-Asian telescopes (e.g., Tianma-65m, Nanshan-26m,
and Hitachi-32m telescopes; Cui et al. 2021). As a part of the KaVA/EAVN Large
Program (e.g., Kino et al., 2015) which intensively monitors Sgr A∗ within
$\lesssim$ a month time interval, several EAVN observations at 1.3 cm and 7 mm
were carried out in 2017 April. In this study, we present the results of two
observations (April 3 and 4, 2017) close to the other global VLBI campaigns:
GMVA+ALMA at 3.5 mm (April 3, 2017) and EHT at 1.3 mm (April 5$-$11, 2017; see
Table 1). The EAVN data are recorded with 256 MHz (32 MHz$\,\times\,$8
channels) total bandwidth. The on-source time for Sgr A∗ and a calibrator,
NRAO 530, is $\sim$160 and $\sim$30 minutes respectively.

The KaVA and Tianma-65m (TIA) participated in the observations at both 1.3 cm
and 7 mm (Figure 1). At 1.3 cm, the Nanshan-26m (Urumqi, UR), and Hitachi-32m
(HT) telescopes additionally joined. However, UR provides the baseline length
$\sim 100-380\,{\rm M}\lambda$ where is already dominated by the refractive
scattering noises (see, section 3) so the fringes were not detected. The HT
had a severe problem with its antenna gain so it was flagged out.

The data were calibrated using NRAO Astronomical Imaging Processing System
(AIPS; Greisen 2003). The cross-power spectra were first normalized using the
auto-correlation power spectra (ACCOR in AIPS), and a multiplicative
correction factor of 1.3 was applied to all data to correct the quantization
loss in the Daejeon hardware correlator (Lee et al., 2015). For the amplitude
calibration, we used the system temperature and antenna gain curve information
(a-priori calibration method; APCAL in AIPS) for 1.3 cm data. As for the 7 mm
data, the SiO maser lines from OH 0.55$-$0.06 and VX Sgr were instead used
which provided a better gain correction (template spectrum method; ACFIT in
AIPS). Cho et al. (2017) have confirmed that the template spectrum method can
derive a more realistic antenna gain curve as a function of the elevation than
a-priori calibration method so that better accuracy of the amplitude
calibration can be achieved. Note that it is also possible to use the H2O
maser line at 1.3 cm but the nearby maser source, Sgr B2, has an extended
maser spot distribution up to $\sim 120\,$arcseconds which is much broader
than the beam size of TIA (e.g., full width half maximum, FWHM, $\sim
44\,$arcseconds at 1.3 cm) so that it may provide relatively worse results
(Cho et al., 2017). The phase calibration was implemented with three steps: 1)
the ionospheric effect and parallactic angle corrections 333Since VERA
stations have equatorial mounts, the parallactic angle corrections are applied
to KVN. (VLBATECR and VLBAPANG in AIPS), 2) instrumental phase and delay
offset correction using the best fringe fit solutions of a fringe tracer, and
3) the global fringe search (FRING in AIPS). Note that the VLBATECR was not
applied for 7 mm data since the ionospheric effect was less severe. The
correction of VLBAPANG was also small for KaVA data so that the visibility
phases were mostly consistent before and after the calibration. As a result,
the phase delay and fringe frequency solutions were successfully obtained with
a signal-to-noise ratio (S/N) $>$ 5\. The bandpass calibration was done in two
steps: 1) for the amplitudes using the total power spectra, and 2) for both
amplitudes and phases using the cross power spectra of Sgr A∗ .

After the calibrations, the visibilities at lower elevation $<5^{\circ}$ at
either telescope of a baseline were flagged. To correct the residual
amplitudes offset across the frequency channels (8 channels), both the multi-
channel data and channel-averaged data were used. First, a Gaussian model was
obtained from the channel-averaged data. The multi-channel data were then
self-calibrated with the Gaussian model and averaged across the entire
bandwidth. The frequency-averaged visibilities were coherently time averaged
over 30 seconds, and the outliers were flagged.

##  3 Imaging and model fitting

Figure 2:  Closure phases of Sgr A∗ (gray) and NRAO 530 (yellow) at 1.3 cm
(first and second rows) and 0.7 cm (third and fourth rows). From left to
right, the corresponding baselines towards Sgr A∗ in the $u-v$ plane (colored
points), closure phases as a function of observing time, and histogram of the
closure phases are shown. Each point has been averaged within 30 seconds. The
time averaged closure phase, $\tilde{\phi}_{c}$, is shown in each legend. The
uncertainty represents the standard deviation.

The scattering effects toward Sgr A∗ can be approximated by a single thin
phase screen (e.g., Narayan & Goodman, 1989; Goodman & Narayan, 1989; Narayan,
1992). The interferometric visibility of scattering kernel is
exp$\left[-\frac{1}{2}D_{\phi}(b/(1+D/R))\right]$, where $D_{\phi}$ is the
phase structure function of the scattering screen, $b$ is the baseline length,
$D$ is the distance between Earth and scattering screen, and $R$ is the
distance between Sgr A∗ and scattering screen (J18). For the baseline lengths
$b\lesssim(1+D/R)r_{\rm in}$, where $r_{\rm in}$ is a finite inner scale of
interstellar turbulence, the phase fluctuations are smooth so that the scatter
broadening provides a Gaussian blurring (i.e., diffractive scattering
dominated). At the baselines longer than this, on the other hand, both the
scattering kernel and the refractive scattering noises introduce the non-
Gaussian features (P18, J18; see also, subsection 3.3). Therefore we use the
“short” baselines where can be fitted with a Gaussian model avoiding the noise
biases. With $D=2.7\,$kpc, $R=5.4\,$kpc, $r_{\rm in}=800\,$km (J18), the
“short” baselines are found as $91$ and $176\,{\rm M}\lambda$ at 1.3 cm and 7
mm wavelengths, respectively. Note however that the non-linear imaging
reconstructions use the full baseline ranges (see subsection 3.1). In
addition, the S/N cutoffs are also applied, for instance the S/N${}_{\rm
th}>3$ and S/N${}_{\rm ref}>4$ where Nth and Nref are thermal noise and
refractive scattering noise, respectively (e.g., J18; see subsection 3.3, for
the refractive noise derivation). This corresponds to the ensemble-average
scattering limit (see Goodman & Narayan 1989; Narayan & Goodman 1989), where
the scattering effects are the convolution of a scattering kernel onto an
unscattered image.

We add 10 % of visibility amplitudes in quadrature as systematic error for all
KaVA data, which is typical amplitude gain uncertainty of cm-mm VLBI
observations (e.g., Cho et al., 2017). However, for TIA data at 1.3 cm, we add
30 % of systematic errors to account for residual bandpass, relatively poorer
system temperature measurements and larger pointing uncertainties (see, e.g.,
Cui et al. 2021 and EAVN status
report444https://radio.kasi.re.kr/eavn/status_report21/node3.html).

To confirm the single Gaussian approximation, we first check the closure
phases which are the sum of visibility phases of three baselines forming a
triangle. Since the closure phases are independent of the phase gain
uncertainties, these are robust VLBI observables together with the closure
amplitudes (e.g., Thompson et al. 2017; EHT Collaboration 2019c; Blackburn et
al. 2020; see also subsection 3.2). We compare the closure phases of Sgr A∗
with a nearby calibrator, NRAO 530. NRAO 530 shows an extended structure
towards the north and its closure phases are correspondingly deviated from
zero, especially along with triangles that include long baselines (e.g., ISG-
KYS). On the other hand, the closure phases of Sgr A∗ are consistent with zero
(Figure 2) which indicates symmetric source structure. Therefore, we model the
data with a single, elliptical Gaussian within a “short” baseline range (i.e.,
ensemble-average image).

Figure 3:  Multiplicative amplitude gain correction factors for each station
at 1.3 cm (upper) and 0.7 cm (lower). The median values from Gaussian model
fitting (red, circle) and SMILI imaging (blue, square) for Sgr A∗ are shown.
The gain corrections for NRAO 530 from SMILI imaging are also shown for
comparison (green, triangle). The uncertainties are 1$\sigma$ standard
deviation of the time dependent amplitude gain correction factors.

Figure 4:  (Upper) Normalized correlated flux density of Sgr A∗ at 1.3 cm
(left) and 0.7 cm (right). The gray circles are visibility amplitudes after
self-calibration. The blue lines are the Gaussian scattering kernel towards
the major axis (solid line) and minor axis (dashed line). The cyan-colored,
broken-dotted vertical line shows the “short” range at each wavelength. The
magenta points are the derived refractive noise, $\sigma_{\rm ref}$. (Lower)
The ratio of the residual of the visibility amplitudes and the fitted model
amplitudes, over the thermal noise. Each point has been 5 minutes averaged for
clearer visualization.

Figure 5:  Observed structure of NRAO 530 (upper) and Sgr A∗ (lower), at 1.3
cm (left) and 0.7 cm (right). The restoring beam is shown in the lower-left
side of each panel. The contour levels are (1, 2, 4, 8, 16, 32, 64) % of peak
intensity. The blue ellipses with broken-line and solid-line (with cross) at
the center of Sgr A∗ images show the best-fitted Gaussian model of ensemble-
average and intrinsic structure, respectively, derived from closure amplitudes
(see Table 2). As for the NRAO 530, the extended jet feature is well
consistent with the one seen in previous studies (e.g., Lu et al., 2011b;
Jorstad et al., 2017).

###  3.1 Self-calibration

For the self-calibration which corrects the station-based gain uncertainties,
we reconstruct the image of Sgr A∗ using two different approaches: 1) Gaussian
model fitting to the complex visibilities using the modelfit in DIFMAP
software (Shepherd et al., 1994), and 2) imaging based on both the complex
visibilities and closure quantities using the regularized maximum likelihood
(RML) method.

As for the two-dimensional (2D) Gaussian model fitting, a single, elliptical
Gaussian model is fitted within the “short” baseline range, $\lesssim\,91$ and
$\lesssim\,176\,{\rm M}\lambda$ at 1.3 cm and 7 mm wavelengths, respectively.
Note that these are $\sim$ half of the longest baseline lengths where the
fringes toward Sgr A∗ have been successfully detected. Nevertheless, we can
still derive gain corrections for all stations using only the data in this
range since every station has at least one independent closure amplitude on
most of the scans.

For the imaging via RML method, we use SMILI software
library555https://github.com/astrosmili/smili (Akiyama et al., 2017a, b,
2019). SMILI has more flexibility in the data products used for imaging, so
this enables us to use the robust closure quantities. To reconstruct the
observed structure of Sgr A∗ , therefore, both the visibility amplitudes and
closure quantities are used. For SMILI imaging, a full baseline range has been
used so that the refractive sub-structures are also reconstructed and the
visibilities are self-calibrated with this model. Note that the refractive
noises are baseline-based so that they have a minor effect on the derived
station-based gain uncertainties. The imaging parameters have been searched
for the regularizers of $\ell_{1}$ norm and total squared variation (TSV),
which represent the image sparsity and smoothness, respectively, in a range of
[0.01, 0.1, 1.0, 10, 100] (see Appendix A, for SMILI imaging and its
regularizers). The fixed values of $\ell_{1}$ prior = 2 and field-of-view =
12.8 mas (128 pixels$\,\times\,$0.1 mas/pixel) are used. Based on the
quadratic sum of $\chi^{2}$ of closure quantities, the fiducial parameters for
Sgr A∗ are found as [$\ell_{1}$-norm, TSV] = [0.1, 0.1] and [0.01, 0.1] at 1.3
and 0.7 cm, respectively. For NRAO 530, [$\ell_{1}$-norm, TSV] = [0.01, 0.1]
and [0.1, 1.0] at 1.3 and 0.7 cm, respectively.

As a result, the derived multiplicative gains for each telescope are well
consistent between two different methods (Figure 3). For comparison, the gain
solutions from NRAO 530 are also derived by the SMILI imaging. Figure 4 shows
the self-calibrated visibility amplitudes with the Gaussian model fitting.
Note that the non-Gaussian noises appear at the longer baselines. The sizes of
ensemble-average image are then obtained by Gaussian model fitting onto the
self-calibrated visibility amplitudes within the “short” range: $\sim
2.6\times 1.4$ mas (position angle $\sim\,82^{\circ}-83^{\circ}$) and $\sim
0.7\times 0.4$ mas (position angle $\sim\,83^{\circ}-85^{\circ}$) at 1.3 cm
and 7 mm, respectively (see Table 2). These are consistent with the scattering
dominated size of Sgr A∗ , with the asymptotic Gaussian scattering kernel of
$(1.380\pm 0.013)\lambda^{2}$ (mas) for major-axis and $(0.703\pm
0.013)\lambda^{2}$ (mas) for minor-axis where $\lambda$ is observing
wavelength in cm (J18). For instance, the asymptotic Gaussian scattering
kernel sizes at 1.3 and 0.7 cm are $\sim 2.51\times 1.28$ mas and $\sim
0.67\times 0.34$ mas respectively, which are slightly smaller than the
measured ensemble average sizes. Figure 5 shows the reconstructed images of
NRAO 530 and Sgr A∗ from SMILI imaging. Note that the Sgr A∗ images show
scattered, not intrinsic, structure. The contour and color maps show the beam
convolved structure. The ellipses at the center of Sgr A∗ image show the best-
fitted 2D Gaussian model of ensemble-average (broken-line) and intrinsic
(solid-line, cross) structure of Sgr A∗ , from the closure amplitudes (see
subsection 4.1 and Table 2).

###  3.2 (Log) Closure amplitudes

While the best-fitted model to visibility amplitudes can provide reasonable
antenna gain solutions, it may be still difficult to take into account all
possible gain uncertainties (Bower et al., 2014b). To overcome this, we fit
the 2D Gaussian model to the log closure amplitudes which are robust
observables free from the amplitude gain uncertainties. Note however that it
is difficult to constrain the flux density with only the closure amplitudes so
it is complementary to the self-calibration and vice-versa.

The closure amplitudes of four different stations ($i,j,k,l$) are defined as
the ratios of visibility amplitudes so that the station-based amplitude gain
errors are canceled out. Three different closure amplitudes can be formed with
any set of four stations, and two of them are independent (e.g., Chael et al.
2018),

$C_{1}=\frac{|V_{ij}||V_{kl}|}{|V_{ik}||V_{jl}|},C_{2}=\frac{|V_{ik}||V_{jl}|}{|V_{il}||V_{jk}|},C_{3}=\frac{|V_{il}||V_{jk}|}{|V_{ij}||V_{kl}|},$
(1)

where the $C$ is the closure amplitude at each quadrangle and $|V|$ is the
visibility amplitude at each baseline. The number of independent closure
amplitudes at a certain time range is $N(N-3)/2$, where the $N$ is the number
of stations. There are a number of ways to form the independent closure
amplitudes, and it is important to select the set where the S/N is high enough
to limit the correlated noise biases (Blackburn et al., 2020).

Using SMILI, we form the independent closure amplitudes from the visibilities
within the aforementioned “short” range. The closure amplitudes of 2D Gaussian
model are formed in the same way. The best fitted model is found using a Monte
Carlo method 666A Gaussian model is provided with the free parameters of major
axis size, axial ratio, and position angle. Its amplitude is then given at
each $u$, $v$ point which corresponds to the observed data. The random
sampling of the free parameters are repeated thousands times, for instance
using the No-U-Turn Sampler (NUTS) which is a particular Markov Chain Monte
Carlo (MCMC) algorithm, and finds the optimized numerical solution for the
searching parameters. onto the log closure amplitudes to avoid the biases and
to symmetrize the numerator and denominator in closure amplitudes (e.g., Chael
et al., 2018; Blackburn et al., 2020). Note that the closure amplitudes are
insensitive to the total flux density so it is not fitted but rather given as
the value obtained from the self-calibrations. Therefore three parameters of
ensemble-average image, major axis size ($\theta_{\rm maj}^{\rm en}$), axial
ratio ($r^{\rm en}\equiv\theta_{\rm maj}^{\rm en}$/$\theta_{\rm min}^{\rm
en}$), and position angle ($\theta_{\rm PA}^{\rm en}$) have been estimated.
From this, we find that the results are consistent with the amplitude self-
calibration (Table 2). This demonstrates that the reconstructed models from
imaging and 2D Gaussian model fitting properly corrects the station-based gain
uncertainties.

###  3.3 Scattering kernel model and deblurring

According to the physically motivated scattering model, the scattering kernel
is no longer a simple (anisotropic) Gaussian and not purely proportional to
the square of observing wavelengths (P18). Adopting the P18 model, therefore,
we have developed the scatter model module of the
SMILI777https://github.com/astrosmili/smili/tree/v0.1/smili/scattering. This
is almost identical with the stochastic optics module of the eht-imaging
software library888https://github.com/achael/eht-imaging (Johnson, 2016; Chael
et al., 2018), and the scattering parameters have been adopted from J18 (e.g.,
$\alpha=1.38^{+0.08}_{-0.04}$, $r_{\rm in}=800\pm 200\,$km). Then the self-
calibrated, ensemble-average visibility amplitudes and closure amplitudes are
divided by the derived scattering kernel (i.e., deblurring) which corresponds
to the deconvolution in the image domain.

To estimate the refractive noises, we first generate the synthetic images
(e.g., intrinsic structure of Sgr A∗ from J18) and scatter them $\sim$1,000
times each with a randomly generated phase screen using the stochastic optics
module. The corresponding visibilities are then obtained by Fourier transform
at each $u,v$ point of the EAVN data. As the number of iteration increases,
the standard deviation of real and imaginary parts of the visibilities
converge to the root-mean-squared (rms) refractive noise at each $u,v$ point.
The derived refractive noises are used to constrain the noise-dominated
visibilities, through the aforementioned S/Nref cutoffs.

##  4 Results

Table 2: Gaussian model fitting results

$\lambda$ (cm) | Method |  S${}_{\nu}^{\rm tot}$ (Jy) |  $\theta_{\rm maj}^{\rm en}$ ($\mu$as) |  $\theta_{\rm min}^{\rm en}$ ($\mu$as) | ${r}^{\rm en}$ |  $\theta_{\rm PA}^{\rm en}$ (deg) |  $\theta_{\rm maj}^{\rm int}$ ($\mu$as) |  $\theta_{\rm min}^{\rm int}$ ($\mu$as) | ${r}^{\rm int}$ |  $\theta_{\rm PA}^{\rm int}$ (deg)
---|---|---|---|---|---|---|---|---|---|---
1.349 | Gfit/Amp |  1.05$\,\pm\,$0.11 |  2621.1$\,\pm\,$26.4 |  1424.4$\,\pm\,$13.3 |  1.84$\,\pm\,$0.02 |  82.8$\,\pm\,$ 0.4 |  827.1$\,\pm\,$93.4 |  630.8$\,\pm\,$52.6 |  1.31$\,\pm\,$0.19 |  95.6$\,\pm\,$23.4
| SMILI/Amp |  1.04$\,\pm\,$0.10 |  2617.9$\,\pm\,$26.4 |  1412.7$\,\pm\,$13.5 |  1.85$\,\pm\,$0.02 |  82.8$\,\pm\,$ 0.4 |  814.8$\,\pm\,$93.3 |  602.6$\,\pm\,$53.1 |  1.35$\,\pm\,$0.19 |  94.6$\,\pm\,$23.4
| CA | … |  2585.1$\,\pm\,$27.9 |  1383.3$\,\pm\,$30.2 |  1.87$\,\pm\,$0.05 |  82.5$\,\pm\,$ 0.6 |  704.3$\,\pm\,$102.0 |  566.7${}^{+78.1}_{-84.8}$ |  1.19${}^{+0.24}_{-0.19}$ |  95.5$\,\pm\,$27.8
0.695 | Gfit/Amp |  1.36$\,\pm\,$0.14 |  715.6$\,\pm\,$8.6 |  414.9$\,\pm\,$8.5 |  1.73$\,\pm\,$0.04 |  84.2$\,\pm\,$0.7 |  294.6$\,\pm\,$23.6 |  229.3${}^{+12.8}_{-13.7}$ |  1.28$\,\pm\,$0.11 |  109.6$\,\pm\,$8.9
| SMILI/Amp |  1.30$\,\pm\,$0.13 |  727.2$\,\pm\,$8.6 |  425.7$\,\pm\,$8.2 |  1.71$\,\pm\,$0.04 |  85.6$\,\pm\,$0.7 |  331.1$\,\pm\,$23.4 |  235.5$\,\pm\,$12.0 |  1.40$\,\pm\,$0.11 |  112.7$\,\pm\,$7.3
| CA | … |  720.8$\,\pm\,$9.0 |  412.0$\,\pm\,$19.8 |  1.75$\,\pm\,$0.09 |  83.2$\,\pm\,$1.0 |  300.0$\,\pm\,$24.8 |  231.0${}^{+32.3}_{-34.7}$ |  1.28$\,\pm\,$0.20 |  95.2${}^{+13.6}_{-12.3}$

Note.  Overall results of total flux density, scattered and unscattered size
of Sgr A∗ at 1.3 and 0.7 cm wavelengths. From left to right: observing
wavelength, imaging/model-fitting method, total flux density, ensemble-average
(i.e., scattered) structure and intrinsic (i.e., unscattered) structure. The
results are from three different methods: self-calibrate with the Gaussian
model fitting and size fitting on the visibility amplitudes (Gfit/Amp), self-
calibrate with the SMILI imaging and size fitting on the visibility amplitudes
(SMILI/Amp), and the size fitting on the log closure amplitudes (CA). See
section 3 for more detail. The subscripts are specific frequency ($\nu$),
major axis (maj), minor axis (min), and position angle (PA). The superscripts
are total flux density (tot), ensemble-average image properties (en) and
intrinsic properties (int). The $\pm 1\,\sigma$ uncertainties are shown
considering the possible error components (see Appendix B), except the total
flux density which shows the 10 % of each measurement.

Here we summarize the observational results. The obtained intrinsic sizes of
Sgr A∗ and its wavelength-dependence are presented in subsection 4.1. In
subsection 4.2, the flux densities and spectral indices of Sgr A∗ are shown.

###  4.1 $\lambda-$dependent intrinsic size of Sgr A∗

The intrinsic structure of Sgr A∗ is found from the Gaussian model fitting
onto the deblurred visibility amplitudes or closure amplitudes. While the
“observed” size is angular broadened by the diffractive scattering, the
“intrinsic” size is scatter-deblurred so that it is free from the angular
broadening. The Monte Carlo method is used to find the optimal values of three
free parameters: “intrinsic” major axis size ($\theta_{\rm maj}^{\rm int}$),
axial ratio ($r^{\rm int}\equiv\theta_{\rm maj}^{\rm int}$/$\theta_{\rm
min}^{\rm int}$), and position angle ($\theta_{\rm PA}^{\rm int}$). Note that
the scattering deblurred amplitudes are normalized with the total flux
density, S${}_{\nu}^{\rm tot}$. The axial ratio of major and minor axis size
is $\sim 1.2-1.4$ and the PA is $\sim 95-113^{\circ}$, so the intrinsic sizes
of Sgr A∗ at both wavelengths look slightly elongated towards the east-west
direction. The PA is consistent with the previous studies (e.g., $\sim
105^{\circ}$ from Markoff et al. 2007; $\sim 95^{\circ}$ from Bower et al.
2014b), and roughly aligned with the orientation of the large-scale X-ray
emission as well (Li et al., 2013). Considering the uncertainties, however,
the axial ratios are consistent with unity within the $\lesssim 3\sigma$
uncertainty range (Table 2). See Appendix B for more details of the
uncertainty estimates. While this is consistent with the J18, Bower et al.
(2014b) reported the axial ratio $\sim 2.8$ coming from the smaller minor axis
size at 7 mm, $\sim 126\,\mu$as. This may indicate the possible time variation
of the minor axis size of Sgr A∗ , and it will be investigated through the
long-term monitoring observations (e.g., X. Cheng et al. in prep.). Note that
the results from different methods are well consistent with each other within
the uncertainties. For later discussions, the results from model fitting onto
the log closure amplitudes are used since this provides more robust
measurement and conservative uncertainties.

Together with the intrinsic size of Sgr A∗ at 3.5 mm which has been measured
in $\sim$ a day separation from our EAVN observations (I19), we fit the
wavelength-dependent intrinsic size with a power law,
$\theta_{\lambda}=\theta_{\rm 1mm}\times\lambda_{\rm mm}^{\epsilon}\,(\mu{\rm
as})$ where the $\theta_{\rm 1mm}$ is the source size at 1 mm, $\lambda_{\rm
mm}$ is the observing wavelength in mm, and $\epsilon$ is the power-law index
which characterizes the intrinsic properties of the emission region as a
function of observing wavelength. As a result, the measured sizes of Sgr A∗ at
13, 7, 3.5 mm are well fitted with a single power law (see Figure 6). The
derived power law index, $\epsilon=1.2\pm 0.2$, is consistent with the
previous experiments of $\epsilon\sim 1.2-1.3$ (e.g., Falcke et al. 2009; Lu
et al. 2011a; Ortiz-León et al. 2016; Brinkerink et al. 2019) and
$\epsilon\sim 1.0-1.1$ (e.g., Shen et al. 2005, J18). Note that the derived
$\epsilon$ from size fitting results on the visibility amplitudes, which are
self-calibrated with the Gaussian model fitting (Gfit/Amp) and SMILI imaging
(SMILI/Amp), are $1.4\pm 0.2$ and $1.3\pm 0.2$ respectively. These are
consistent with the result from closure amplitudes (CA) within each
uncertainty. The uncertainties of our fitted results are relatively large
since we only use the (quasi-) simultaneous observations so that the number of
data points is small. To better constrain the size-wavelength relation,
therefore, the intrinsic sizes at more various wavelengths are necessary
especially at 2 mm and $\gtrsim 2\,$cm. In addition, the quasi-simultaneous
campaigns at a variety of wavelengths, as we have achieved in 2017 together
with the GMVA+ALMA and EHT, are further required since the (marginal) size
variation of Sgr A∗ , especially at 7 mm, has been found (e.g., Bower et al.,
2004; Akiyama et al., 2013; Zhao et al., 2017).

From the best-fitted results, the extrapolated size at 1.3 mm is found as
$36^{+18}_{-16}\,\mu$as and $27^{+13}_{-12}\,\mu$as toward major and minor
axis, respectively. This is close to the previous measurements with EHT (e.g.,
$37\pm 13\,\mu$as from Doeleman et al. 2008; $41-44\,\mu$as from Fish et al.
2011; $51-52\,\mu$as from Lu et al. 2018; $56\pm 7\,\mu$as and
$59^{+30}_{-31}\mu$as for major and minor axis, respectively, from J18). Note
however that the structure of Sgr A∗ at 1.3 mm may be non-Gaussian (e.g.,
Johnson et al., 2015; Fish et al., 2016; Lu et al., 2018) so that the
extrapolated size can be a hint for high resolution imaging, for instance with
the EHT.

Figure 6:  Intrinsic size of Sgr A∗ as a function of wavelength. The squares
and circles show the major and minor axis sizes (i.e., full-width half-
maximum), respectively. While the filled markers are the results from this
work, the open markers show the previous experiments at 3.5 mm (I19) and 1.3
mm (J18; Doeleman et al. 2008 with green, diamond). The markers are offset
slightly in wavelength for clearer visualization. The power law fitting
results are shown: towards the major axis (red, broken-dotted line) and minor
axis (blue, broken line) sizes. Note that only the 3.5, 7, and 13 mm
wavelengths are used for the fitting which are the quasi-simultaneous
measurements in 2017 April. The 1.3 mm results are shown for comparison (e.g.,
the extrapolated sizes). The horizontal broken line (gray-colored) shows the
expected black hole shadow diameter at Sgr A∗ ($\sim 51\,\mu$as; e.g., J18).

###  4.2 Flux densities and spectral indices

At radio to sub-mm wavelengths, Sgr A∗ shows an inverted (time-averaged)
spectrum, $S_{\nu}\propto\nu^{\beta}$ ($\beta>0$; e.g., Duschl & Lesch 1994;
Morris & Serabyn 1996; Serabyn et al. 1997; Falcke et al. 1998; Krichbaum et
al. 1998; Zhao et al. 2001) where $S_{\nu}$ is specific flux density, $\nu$ is
the observing frequency, and $\beta$ is the spectral index. The spectrum peaks
and cutoffs are shown at $230-350\,$GHz (e.g., Yusef-Zadeh et al., 2006; Bower
et al., 2015), which are due to the transition of synchrotron emission from
being optically thick to thin. This implies that Sgr A∗ has a stratified,
self-absorbed geometry of plasma, either a jet or an accretion flow, and the
(sub-) mm emission arise from several $R_{\rm s}$ (e.g., Melia, 1992, 1994).
At mm/sub-mm wavelengths, in addition, there exists a break in the spectrum so
called “mm/sub-mm bump” which deviates from a single power-law index. This may
be explained by the compact components in the acceleration zone of a jet, or
the thermal electrons in an accretion flow (e.g., Lu et al., 2011a).

In our observations, the total flux densities are $\sim$1.0 and 1.3 Jy at 22
and 43 GHz, respectively (see Table 2). With the intrinsic major axis size, we
can derive a lower limit of the brightness temperature as $T_{\rm b}\gtrsim
0.4-0.5\times 10^{10}\,$K and $\gtrsim 0.8-1.0\times 10^{10}\,$K at 1.3 and
0.7 cm respectively. Together with the flux density at 86 GHz, $\sim 2\,$Jy
(I19), the spectral index of Sgr A∗ within the three frequencies is derived as
$\beta=0.39\pm 0.02$. Considering the mm/sub-mm bump at $\gtrsim 43\,$GHz, at
the same time, we derive the values of $\beta$ for 22$-$43 GHz and 43$-$86 GHz
separately and find $\beta\approx\,$0.29$\pm$0.10 and 0.44$\pm$0.16
respectively. Note that the smaller number of points leads to larger
uncertainties. Nevertheless, the results are well consistent with the
historical studies. For instance, while $\beta\sim 0.24-0.3$ up to $\sim
43\,$GHz (e.g., Krichbaum et al., 1998; Falcke et al., 1998; Lu et al., 2011a;
Bower et al., 2015), $\beta\sim 0.5$ at $\gtrsim 43\,$GHz (Falcke et al.,
1998; Bower et al., 2015).

With the derived spectral indices, the flux density at 230 GHz can be
extrapolated assuming the same spectral index up to 230 GHz and the turnover
frequency in between 230 GHz (e.g., Yusef-Zadeh et al., 2006; Bower et al.,
2015) to 1 THz (Bower et al., 2019). For the spectral index of the single
power law, $\beta\sim 0.39$, the extrapolated flux density at 230 GHz is
$\sim\,$2.73 Jy. The lower and upper limits of the extrapolated flux densities
are then estimated from the spectral indices for lower and higher frequency
ranges (i.e., $\beta\sim\,$0.29 and 0.44), respectively. With this, the
compact flux density of Sgr A∗ at 1.3 mm in 2017 April, observed with the EHT,
is expected to be $\sim 2.73^{+0.23}_{-0.59}\,$Jy.

##  5 Nonthermal electron emission models

The details of the physical nature of radio emission from Sgr A∗ remain
elusive. Here we discuss the emission models and indicatives of the existence
of nonthermal electrons to explain the wavelength-dependent intrinsic sizes of
Sgr A∗ . Pioneering work by Özel et al. (2000) discussed a possible
explanation of the prominent shoulder (excess) at $\lesssim 100\,$GHz in the
spectral energy distribution (SED) of Sgr A∗ by hybrid thermal-nonthermal
electron population. The SED discussed in Özel et al. (2000), however, was the
assembly of non-simultaneously measured radio flux values and mostly observed
by single dish data so that it potentially contained a significant amount of
emissions originated from extended region around Sgr A∗ . Therefore, it is not
entirely clear whether such a hybrid thermal-nonthermal electron population is
really required or the observed excess is just a superposition of emissions
from extended regions that cannot reflect intrinsic properties of Sgr A∗ .

In this work, we obtain the VLBI-measured quasi-simultaneous intrinsic sizes
and the corresponding flux densities at 13, 7, and 3.5 mm for the first time.
Using these data, we discuss the necessity of nonthermal electron population
in Sgr A∗ . Note that the question about the dominant radio emission model of
Sgr A∗ , either an accretion flow or a jet, is still open for debate. Since
there is no clear observational evidence of jet eruption in Sgr A∗ on VLBI-
scales yet, we will focus on the accretion flow model in this section. The
possible jet model can be compared with the results from I19 999 Comparing the
structure of Sgr A∗ at 3.5 mm with the 3D general relativistic
magnetohydrodynamic (GRMHD) simulations, I19 suggested the plausible models of
1) accretion flow dominated or 2) jet dominated with small viewing angles
$\lesssim 20^{\circ}$ (Figure 9 and 10 in their paper). With the same
simulations, our results at 13 and 7 mm disfavor the accretion flow dominated
model (for both thermal and thermal/nonthermal hybrid cases). As for the jet
dominated model, the small viewing angle is preferred, $\lesssim 30^{\circ}$..

We examine the theoretical images of radiatively inefficient accretion flow
(RIAF) based on the Keplerian shell model (Falcke et al., 2000; Broderick &
Loeb, 2006; Pu et al., 2016; Kawashima et al., 2019). The radiative transfer
is solved using general relativistic, ray-tracing radiative transfer (GRRT)
code, RAIKOU (Kawashima et al., 2019, 2021b, 2021a). The cyclo-synchrotron via
thermal electrons (Mahadevan et al., 1996) and synchrotron processes via
nonthermal electrons (see, e.g., Dexter, 2016) are incorporated. For the sake
of simplicity, the angle between the ray and magnetic field $\theta_{\rm B}$
is assumed to be $\sin\theta_{\rm B}=\pi/4$ for the calculation of
emission/absorption via the nonthermal electrons, which corresponds to the
average for the isotropic, random magnetic field.

We set the number density of thermal electrons $n_{\rm e,th}$, that of
nonthermal electrons $n_{\rm e,nth}$, electron temperature $T_{\rm e}$, and
the magnetic field $B$ at $(r,\theta)$, where $r$ and $\theta$ is the
spherical radius and the polar angle in the Schwarzschild coordinate, in the
Keplerian shell model as follows:

$\displaystyle n_{\rm e}=6{\times}10^{7}(r/r_{\rm
g})^{-1.1}\exp(-z^{2}/2H^{2})~{}[{\rm cm}^{-3}],$ (2) $\displaystyle n_{\rm
e,nth}=2.5{\times}10^{5}(r/r_{\rm g})^{-1.75}\exp(-z^{2}/2H^{2})~{}[{\rm
cm}^{-3}],$ $\displaystyle T_{\rm e}=1.5{\times}10^{11}(r/r_{\rm
g})^{-8.4}~{}[{\rm K}],$ $\displaystyle B^{2}/8\pi=0.1n_{\rm e,th}m_{\rm
p}c^{2}r_{\rm g}/6r~{}[{\rm erg~{}cm}^{-3}].$

Here, $m_{\rm p}$ is the proton mass, $r_{\rm g}(=GM/c^{2})$ is the
gravitational radius, $G$, $M$ and $c$ are the gravitational constant, the BH
mass, and the speed of light, respectively. We assume the $r_{g}$ with its
mass $M=4.1\times 10^{6}M_{\odot}$ and distance 8.1 kpc. The scale height of
the accretion flow is set to be $H/R=0.5$, where $R$ is the cylindrical radius
$R=r\sin\theta$. The acceleration mechanism of the electrons is uncertain,
although they may be accelerated by the magneto-rotational instability (MRI)
turbulence (Lynn et al., 2014), magnetic reconnection (Hoshino, 2013; Ball et
al., 2018; Werner et al., 2018; Ripperda et al., 2020) and/or by other
mechanisms. Therefore we simply assume the energy spectrum of the nonthermal
electrons to be a single power law distribution $\propto{\gamma}^{-3.5}$ in
the range $10^{2}\leq\gamma\leq 10^{6}$, where $\gamma$ is the Lorentz factor
of the electrons whose power-law index and the minimum Lorentz factor are same
as those in Broderick et al. (2011) and Pu et al. (2016). Here, a slightly
extended spatial distribution of nonthermal electrons $n_{\rm e,nth}\propto
r^{-1.75}$ (in previous works, e.g., Broderick et al. 2011 and Pu et al. 2016,
$\propto r^{-2.02}$) is found from our observational data. Note that this is
mainly due to the smaller uncertainty of our results (Figure 7) and the main
conclusion of nonthermal electron population is not strongly affected by the
small difference. The consequent higher fluxes at 1.3 cm and 7 mm require the
spatially extended emission region due to the synchrotron emission via
nonthermal electrons. Note that this is broadly applicable, not only valid for
the parameter-set chosen above, where the Keplerian shell model holds. As is
shown later, this can be confirmed by comparing the radius of the outer edge
of the accretion flow and the observed size at each wavelength.

Figure 7:  Theoretical SEDs for the models with $i=20^{\circ}$ (red) and
$60^{\circ}$ (blue). The thick-solid and thin-dashed lines represent the
results taking into account thermal plus nonthermal electrons and thermal
electrons only, respectively. The black data points show the observation data
at 22 and 43 GHz in this work. The gray data points display the previous
observational data (Falcke et al., 1998; Bower et al., 2015).  Figure 8:
Theoretical images of accretion flow in Sgr A* observed at the wavelength 1.3
cm (left), 0.7 cm (middle), and 0.3 cm (right), based on the Keplerian shell
model. The viewing angle $i$ is (a) $20^{\circ}$ and (b) $60^{\circ}$. The
synchrotron emission/absorption processes are calculated taking into account
the thermal electrons only (top) and both of the thermal and nonthermal
electrons (bottom).

Figure 8 displays the resultant images of Sgr A* accretion flow. The intensity
is shown in the log scale. All of the models with only thermal electrons show
the very compact core images. For example, at 1.3 cm, the intensity
drastically decreases outside the diameter $\sim$ 300 $\mu$as, i.e., the
intensity outside this diameter is lower than the order of magnitude of the
peak intensity and this diameter is too small compared with the measured one.
This compact emission region is a consequence of the low synchrotron frequency
in the outer region as explained in the next paragraph. On the other hand, the
synchrotron frequency can be higher than $\sim 10$ GHz if the nonthermal
electrons are assumed, so that the resultant images for the models with
nonthermal electrons show a larger emission region whose diameter is
$\sim~{}500~{}\mu$as. This is qualitatively consistent with a pioneering
theoretical work assuming one dimensional plasma structure (Özel et al.,
2000). This indicates that our observed images provide the direct detection of
the nonthermal electrons and a sufficient amount of nonthermal electrons
should exist in the outer region, as have been proposed by the previous
theoretical works focusing on the spatially unresolved SED (e.g., Yuan et al.,
2003)101010The importance of nonthermal electrons is also pointed out to
explain the variability of the X-ray flux during the flaring state in Sgr A*,
by carrying out GRMHD simulations with injecting nonthermal electrons and
subsequent GRRT computations by taking into account synchrotron
emission/absorption processes (Ball et al., 2016).. The image enlargement due
to the synchrotron emission via the nonthermal electrons is also consistent
with horizon-scale theoretical images at 230 GHz (Mao et al., 2017; Chael et
al., 2017). Figure 7 shows a comparison between the observed flux densities
and our theoretical SEDs. We find that the nonthermal synchrotron emission can
well explain the overall excess component below 100 GHz, including 22 and 43
GHz observed by the EAVN.

Figure 9:  (Left) Peak frequency of cyclo-synchrotron emission via thermal
electrons (black) and synchrotron emission via nonthermal electrons (red).
(Right) Radial profile of magnetic field strength (solid line) and electron
temperature (dashed line) of our Keplerian shell model.

We further interpret the image size of the theoretical models shown in Figure
8, by considering the peak frequency of synchrotron and cyclo-synchrotron
emissivity. The peak frequency of synchrotron and cyclo-synchrotron emissivity
$\nu_{\rm peak}$, is generally expressed in terms of $B$, $T_{\rm e}$ (for
thermal electrons), and minimum Lorentz factor $\gamma_{\rm e,min}$ (for
nonthermal electrons with the power-law index greater than 2) in the accretion
flow. With the Keplerian shell model, the $\nu_{\rm peak}$ can then be
expressed as a function of the accretion flow radius. One can roughly estimate
the outer edge of the bright core by the radius at which an observing
frequency is equal to that of emissivity peak frequency, since the emission
will become drastically faint if the observing frequency is greater than
$\nu_{\rm peak}$. Figure 9 (left) shows the peak frequency of cyclo-
synchrotron emissivity via thermal electrons and synchrotron emissivity via
nonthermal electrons, as a function of the emitting radius according to our
Keplerian shell model. The corresponding spatial distributions of $T_{\rm e}$
and $B$ are shown in Figure 9 (right). The peak frequency of our nonthermal
synchrotron emissivity can be described as $\nu_{\rm s(nth)}=\nu_{\rm
c}\gamma_{\rm e,min}^{2}$ because of our steep power-law index $3.5(>2)$ of
nonthermal electrons. Here $\nu_{\rm c}(=3eB\sin\theta_{B}/4\pi m_{\rm e}c$)
and $e$ are the cyclotron frequency of electrons and the elementary charge,
respectively. As for the peak frequency of thermal cyclo-synchrotron
emissivity, it is approximated as $10^{2}\nu_{\rm cs(th)}\lesssim\nu_{\rm
peak}\lesssim 10^{3}\nu_{\rm cs(th)}$ for $T_{\rm e}\gtrsim 10^{10}$K where
$\nu_{{\rm cs(th)}}=(3eB/4\pi m_{\rm e}c)\Theta_{\rm e}^{2}$ (see Figure 5 and
6 in Mahadevan et al. 1996). In Figure 9 (left), we find that the expected
outer radius of the accretion flow seen at $22$ GHz for thermal cyclo-
synchrotron case corresponds to only $r\approx 20-40~{}r_{\rm g}$ ($\lesssim
100\mu{\rm as}$), while that of nonthermal synchrotron emitting accretion flow
realizes $r\approx 200~{}r_{\rm g}$ ($\sim{\rm several}\times 100\mu{\rm as}$)
by the existence of nonthermal electrons in the outer region. The same logic
can be applied to 43 GHz. Here our model, $B\propto r^{-p_{1}}$ and $T_{\rm
e}\propto r^{-p_{2}}$ with $p_{1}\sim p_{2}\sim 1$, is consistent with the
recent simulation studies (e.g., Chael et al., 2017; Ressler et al., 2020a).
This strongly infers that an unusual disk model, whose electron temperature is
high and magnetic field is strong in the outer disk region, is required to
explain the observed image size of Sgr A* at 22 and 43 GHz if we do not assume
nonthermal electrons. 111111We also examined the calculation of images
assuming the strong magnetic field, which is more than an order of magnitude
stronger than the standard magnetic field strength assumed in the Keplerian
shell model, mimicking the magnetically arrested accretion flow (Narayan et
al., 2003; Tchekhovskoy et al., 2011; McKinney et al., 2012; Narayan et al.,
2012). However, the resultant image-size assuming thermal electrons is still
small. Thus, we draw the conclusion that nonthermal electrons are necessary to
explain the observed sizes of Sgr A∗ measured at 22 and 43 GHz if the disk-
dominant emission is assumed. 121212At the same time, it is fair to state the
caveat that we neglect a possible case of a jet. If this exists in Sgr A∗ , a
larger emission size might be reproduced by a pure thermal electron model
(e.g., Mościbrodzka et al., 2014). Further observational and theoretical
studies of jets are therefore needed to draw a final conclusion.

Lastly, we discuss the dependence of the axial ratio on the viewing angle $i$
of observers. Since an extensive and detailed parameter survey of $i$ is
beyond the scope of this study, here we just investigate two representative
cases: a small viewing angle (e.g., Gravity Collaboration et al., 2018) and
large viewing angle (e.g., Markoff et al., 2007; Broderick et al., 2011). Here
we set $i=20^{\circ}$ for the small viewing angle case, while $i=60^{\circ}$
is chosen for the large viewing angle case. From the model predicted images
shown in Figure 8, we find that the morphology of the emission region is
almost isotropic for $i=20^{\circ}$, while the axial ratio is $\sim 2$ for
$i=60^{\circ}$. Therefore, the axial ratio $\simeq 1.2-1.3$ from our
measurements may suggest that a small viewing angle is preferred to a large
viewing angle. Note however that the discussion here is just one heuristic
argument and further investigations are needed to reach a final conclusion. We
also note that the change of the viewing angle possibly occurs in the near
future, as a consequence of the tilt of accretion flow due to the misalignment
of the directions of magnetic flux and averaged angular momentum of the
accreting gas fed by the Wolf-Rayet stars (Ressler et al., 2020a, b). The
successive observations are therefore needed to study the exact viewing angle
of the accretion flow and its possible variation in time.

##  6 Summary

In this study, we present the intrinsic properties of Sgr A∗ at 1.3 cm and 7
mm from the EAVN observations. Through the imaging and Gaussian model fitting,
the scattered size of Sgr A∗ in an ensemble-average limit is first derived
which is dominated by the scattering effect. Adopting the recent scattering
kernel model, the self-calibrated visibilities and closure amplitudes are
deblurred. The unscattered structure of Sgr A∗ is then obtained from the
Gaussian model fitting onto the deblurred data. As a result, we find a single,
symmetric Gaussian model well explains the structure. From the closure
amplitudes, the major axis sizes are $\sim 704\,\mu$as (axial ratio$\sim
1.19$, PA$\sim 95^{\circ}$) at 1.3 cm and $\sim 300\,\mu$as (axial ratio$\sim
1.28$, PA$\sim 95^{\circ}$) at 7 mm. Together with the 3.5 mm size which has
been quasi-simultaneously measured (I19), the wavelength dependent source size
is found with the power law index $\epsilon=1.2\pm 0.2$.

The expected size of Sgr A∗ at 1.3 mm is extrapolated to
$36^{+18}_{-16}\,\mu$as and $27^{+13}_{-12}\,\mu$as toward the major and minor
axis, respectively. From the total flux densities at three wavelengths, in
addition, the spectral index is derived as $\beta\sim 0.39$ and the
extrapolated compact flux density at 1.3 mm is $\sim 2.73$ Jy. With more
(quasi) simultaneous observations at broader wavelengths region, the
wavelength dependence may be constrained more robustly and the long-term time
variation of structure can be further investigated.

As for the dominant emission scenario, we have compared the measured intrinsic
size of Sgr A∗ with the accretion flow dominated model, especially the RIAF
based on the Keplerian shell model. In this case, the intrinsic sizes at both
wavelengths are a factor of a few larger than those predicted with purely
thermal electron distribution. We find that this size-mismatch problem can be
solved by including nonthermal electron components. The obtained axial ratio
which is almost isotropic also indicates the small viewing angle of Sgr A∗ ,
$\lesssim 30-40^{\circ}$. This is consistent with the previous study of
measuring the rotating hot spot (Gravity Collaboration et al., 2018) and the
3D GRMHD simulations with the jet-dominated model (I19). To discriminate each
scenario, additional multi-wavelength observations will be of great help, for
instance investigating the frequency-dependent radio core position shifts (I.
Cho et al. in prep.).

We thank the anonymous referee and the EHT Collaboration internal reviewers &
Publication Committee, G.C. Bower, L. Loinard, and K. Rygl, who reviewed the
manuscript. We are grateful to all staff members in EAVN who helped to operate
the array and to correlate the data. The KVN is a facility operated by Korea
Astronomy and Space Science Institute (KASI) and VERA is a facility operated
by the National Astronomical Observatory of Japan (NAOJ) in collaboration with
associated universities in Japan. TIA is operated by Shanghai Astronomical
Observatory. UR is operated by Xinjiang Astronomical Observatory. HT is
operated by NAOJ and Ibaraki University. The KVN operations are supported by
Korea Research Environment Open NETwork (KREONET) which is managed and
operated by Korea Institute of Science and Technology Information (KISTI). The
theoretical calculations were carried out on the XC50 at the Center for
Computational Astrophysics, National Astronomical Observatory of Japan. I.C.
and J.P. are supported by the National Research Foundation of Korea (NRF) via
a Global PhD Fellowship Grant (NRF-2015H1A2A1033752 and NRF-2014H1A2A1018695,
respectively). I.C. acknowledges financial support by the Spanish Ministerio
de Economía y Competitividad (grants PID2019-108995GB-C21). G.-Y.Z. and T.J.
are supported by Korea Research Fellowship Program through the NRF
(NRF-2015H1D3A1066561). G.-Y.Z. acknowledges financial support from the State
Agency for Research of the Spanish MCIU through the “Center of Excellence
Severo Ochoa” award to the Instituto de Astrofísica de Andalucía
(SEV-2017-0709). K.A. is supported by the National Science Foundation (NSF)
through grants AST-1440254, AST-1614868, and AST-2034306. This work is
partially supported by JSPS KAKENHI Grant Numbers JP18K03656 (M.K.),
JP18H03721 (K.N., M.K., K.H.), JP18K13594 (T.K.), JP18KK0090 (K.H.),
JP19H01943 (K.H.), JP21H01137 (K.H., M.K.), and the Mitsubishi Foundation
grant No. 201911019 (K.H.). J.-C.A acknowledges support from the Malaysian
Fundamental Research Grant Scheme (FRGS) FRGS/1/2019/STG02 /UM/02/6. M.D.J.
acknowledges support from the National Science Foundation (AST-1716536,
AST-1935980) and the Gordon and Betty Moore Foundation (GBMF-5278). X.-P.C.
and B.-W.S. are supported by Brain Pool Program through the National Research
Foundation of Korea (NRF) funded by the Ministry of Science and ICT
(2019H1D3A1A01102564). R.-S.L. is supported by the Max Planck Partner Group of
the MPG and the CAS and acknowledges the support by the Key Program of the
National Natural Science Foundation of China (grant No. 11933007) and Research
Program of Fundamental and Frontier Sciences, CAS (grant No. ZDBS-LY-SLH011).
J.P. acknowledges financial support through the EACOA Fellowship awarded by
the East Asia Core Observatories Association, which consists of the Academia
Sinica Institute of Astronomy and Astrophysics, the National Astronomical
Observatory of Japan, Center for Astronomical Mega-Science, Chinese Academy of
Sciences, and the Korea Astronomy and Space Science Institute. S.T.
acknowledges financial support from the NRF via Basic Research Grant
2019R1F1A1059721. This work was also supported in part by MEXT SPIRE, MEXT as
“Priority Issue on post-K computer” (Elucidation of the Fundamental Laws and
Evolution of the Universe) and as “Program for Promoting Researches on the
Supercomputer Fugaku” (High-energy astrophysical phenomena in black holes and
supernovae). $Software$: AIPS (Greisen, 2003), Astropy (Astropy Collaboration
et al., 2018), DIFMAP (Shepherd et al., 1994), eht-imaging (Chael et al.,
2018), Matplotlib (Hunter, 2007), NumPy (Oliphant, 2006), PyMC (Salvatier et
al., 2016), SciPy (Jones et al., 2001), SMILI (Akiyama et al., 2017a, b,
2019),

##  Appendix A SMILI imaging

SMILI imaging is one of the regularized maximum likelihood (RML) methods. This
finds an image ($I$) which minimizes a specified objective function,

$\displaystyle J(I)=\sum_{\mathclap{\text{data
terms}}}\alpha_{\textrm{D}}\chi^{2}_{\textrm{D}}\left(I\right)-\sum_{\mathclap{\text{regularizers}}}\beta_{\textrm{R}}S_{\textrm{R}}\left(I\right),$
(A1)

where the first and second term corresponds to a measure of the inconsistency
of the image (e.g., goodness-of-fit functions, $\chi^{2}_{D}$) and
regularization ($S_{R}$), respectively. These terms often have opposite
preferences for a fiducial image, so their relative impact in the minimization
process is specified with the coefficients of regularization terms (i.e.,
hyperparameters, $\alpha_{R}$ and $\beta_{R}$). Regularizers in SMILI are
explored to constrain the image characteristics, for instance sparsity
($\ell_{1}$ norm) and smoothness (total variation and total squared
variation), and both of them can be simultaneously favored in the minimization
of the objective function. Note that the fiducial values of regularizers can
be different for different observational data towards the same target source,
since they are found by minimizing the inconsistency term at the same time.
For more details of regularizer definitions, see Appendix A of EHT
Collaboration (2019d).

##  Appendix B Error estimates

The size uncertainties of Sgr A∗ are estimated with the goodness-of-fit from
Monte Carlo method ($\sigma_{\rm fit}$) and the stochastic random phase screen
within the error range of scattering parameters ($\sigma_{\rm scat}$) (Table
3). To test the latter effects, the best fitted Gaussian models are scattered
with the random phase screen using the eht-imaging library and its
visibilities are generated with corresponding observations. Then the same
procedures of elliptical Gaussian model fitting for an ensemble-average image,
scattering kernel deblurring, and intrinsic size fitting with a single
Gaussian model are applied. While the fixed scattering parameters,
$\alpha=1.38$ and $r_{\rm in}=800\,$km, are used for ensemble-average image,
different scattering parameters within its possible ranges (e.g.,
$\alpha=1.38^{+0.08}_{-0.04}$, $r_{\rm in}=800\,\pm\,200\,$km; J18) have been
tested with randomly selected $\sim 150$ numbers for the intrinsic size of Sgr
A∗ . To derive reasonable deviations, these processes are repeated 1,000 times
for each scattering parameter. As a result, the standard deviations of the
sizes are larger towards major axis with a consistent position angle with the
best-fitted model (see Figure 10). Note that the peak intrinsic size is well
consistent with the input model (i.e., the best-fitted Gaussian) so that no
systematic biases are found.

Table 3: Size uncertainties

$\lambda$ (cm) | Axis |  $\sigma^{\rm en}_{\rm fit}$ ($\mu$as) |  $\sigma^{\rm en}_{\rm scat}$ ($\mu$as) |  $\sigma^{\rm int}_{\rm fit}$ ($\mu$as) |  $\sigma^{\rm int}_{\rm scat}$ ($\mu$as)
---|---|---|---|---|---
1.349 | Major |  $\pm\,$10.8 |  $\pm\,$25.7 |  $\pm\,$45.9 |  $\pm\,$91.1
| Minor |  $\pm\,$29.2 |  $\pm\,$7.4 | ${}^{+61.8}_{-70.1}$ |  $\pm\,$47.8
0.695 | Major |  $\pm\,$3.4 |  $\pm\,$8.3 |  $\pm\,$10.2 |  $\pm\,$22.5
| Minor |  $\pm\,$19.6 |  $\pm\,$2.4 | ${}^{+31.8}_{-34.2}$ |  $\pm\,$5.8

Note.  From left to right, observing wavelength, the axis of Gaussian
structure, the uncertainty of ensemble-average image size (from fitting and
stochastic random phase screen), and intrinsic image size (from fitting and
overall scattering effects). The Monte Carlo fitting error is slightly
different for each measurement (i.e., Gfit/Amp, SMILI/Amp, and CA), and here
we show the uncertainty from CA. The overall uncertainty of size measurement
is obtained by the quadratic sum of each error component (Table 2).

Figure 10:  Two-dimensional histogram density of the derived intrinsic sizes
of Sgr A∗ by deconvolving the random phase screen: 1.3 (left) and 0.7 cm
(right). The input model (i.e., best-fitted model) is first given (red square)
and convolved with the random phase screen. After that, deblur the scattering
kernel and fit a single Gaussian to estimate the sizes. The same procedure is
repeated with $\sim 150$ number of different scattering parameters and 1,000
times of random phase screen for each scattering parameter (gray contours; 5,
10, 20, 40, 80 % of the peak number are shown). As a result, the possible
deviation of the size measurement due to the scattering effects is derived
(blue circle with error bar). Note that no significant biases are found from
the peak value.

##  Appendix C Intensity map of the Keplerian shell model in one-dimensional
slice of the observer screen

We present the one-dimensional (1D) slices of the intensity map shown in
Figure 8. Since it is uncertain what is the true peak-intensity in the
observed blurred images due to the limited beam size, we qualitatively discuss
the theoretical intensity map.

Figure 11 displays the 1D intensity map in the observer screen. For the model
with $i=20^{\circ}$, the 1D intensity profile in the $x$-direction at $y=0$
(solid line) is almost identical to that in $y$-direction at $x=0$ (dashed
line), i.e., the nearly isotropic emission feature can be confirmed in this
figure. On the other hand, for the model with $i=60^{\circ}$, the 1D intensity
profile is significantly anisotropic, because of the finite scale height of
the accretion flow.

The sharp intensity peak appeared in the model with $i=20^{\circ}$ due to the
emission from the inner part of the disk, while it disappeared for the model
with $i=60^{\circ}$ in the horizontal direction, because the photons emitted
from the inner accretion flows are obscured by the outer accretion flow (i.e.,
the self-occultation effect). We also note that the peak intensity for the
model with $i=60^{\circ}$ appeared in $y>0$ rather than $y<0$ because of the
more significant effect of the self-occultation in $y<0$ (observer side) than
in $y>0$ (counter side).

Figure 11:  One-dimensional slice of the intensity map of theoretical image
shown in Figure 8 with the viewing angle $i=20^{\circ}$ (left) and
$60^{\circ}$ (right). The photon frequency, $\nu$, is 22 GHz. The solid and
dashed lines show the distribution of intensity in $x$ and $y$ direction at
$y=0$ and $x=0$, respectively. The results for the model with thermal and
nonthermal electrons (red) and with thermal electrons only (black) are
displayed. The intensity is normalized by the maximum value in the $x$-$y$
plane of the screen (i.e., by the peak value in the entire region of the
screen).

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