First Sagittarius A* Event Horizon Telescope Results. II. EHT and Multiwavelength Observations, Data Processing, and Calibration

The ALMA Phasing System is designed to construct a summed, formatted VLBI signal without interrupting the data stream of each individual antenna, thus allowing the ALMA correlator to compute all of the intra-ALMA visibilities at the same time (Matthews et al. 2018). The calibration information related to these intra-ALMA visibilities is needed for the correct polarimetric processing of the ALMA-VLBI signal (Martí-Vidal et al. 2016; Goddi et al. 2019, 2021).

At the spatial scales sampled by the intra-ALMA baselines, the Sgr A* field consists of the sum of two components: a point-like source located at the field center and with a time-dependent flux density (i.e., the active galactic nucleus), and an extended structure that covers several arcseconds, with a total flux of about 1.1 Jy and a surface brightness of about 0.12 Jy beam⁻¹ (Wielgus et al. 2022). This extended component is known as the "minispiral," and its emission is related to ionized gas and dust in the Galactic center (e.g., Lo & Claussen 1983).

The use of the intra-ALMA self-calibration gains to compute the corrections for the VLBI amplitudes of the ALMA-related baselines provides a very accurate determination of the relative changes in ALMA's VLBI amplitude gains (Goddi et al. 2019). However, this approach has important limitations for the correct calibration of a time-variable source like Sgr A* (see also Section 4). In particular, the use of an a priori (constant) model for the amplitude self-calibration of the Sgr A* intra-ALMA visibilities results in an incorrect estimate of the phased-ALMA amplitude scaling. To overcome this limitation, we calibrate the Sgr A* ALMA gains with the source's light curve, which is computed under the assumption that the flux density distribution of the minispiral remains stable across the extent of the observations (Wielgus et al. 2022).

We note that, even though the minispiral modeling allows us to calibrate relative ALMA gains at the ∼ 1% level, a constant ∼ 10% calibration uncertainty remains for the overall gain of the phased array, which is tied to the estimated flux density of Ganymede (the source used as primary calibrator; Goddi et al. 2019).

We perform a time-dependent variant of the network calibration (M87* Paper III) of the EHT VLBI data to a merged light curve from ALMA and the SMA (presented alongside the MWL light curves in Section 4). The ALMA light curve is computed through the minispiral method described above, and a description of the SMA light curve is given in Wielgus et al. (2022). These show small offsets at the level of a few percent between the two raw light curves, within the absolute flux calibration uncertainty. The two light curves are combined by leveling the median amplitudes in the overlapping time between ALMA and SMA and a smoothing spline interpolation in time, which corrects for these offsets. The network calibration procedure leverages the presence of colocated sites, constraining gains of telescopes with a colocated partner by assuming that intrasite baselines (ALMA–APEX and JCMT–SMA; see Figure 1) observe a point source with a time-dependent flux density, corresponding to the light curve.

Ensembles of opacity-corrected antenna temperature (${T}_{a}^{* }$) measurements are used to fit for gain curves and DPFUs (Janssen et al. 2019a; Issaoun et al. 2017). For the gain curve, we fit

$$\begin{eqnarray}&&\mathrm{gc}(E)=1-B{\left(E-{E}_{0}\right)}^{2}\end{eqnarray} \tag{ 3 }$$

to normalized ${T}_{a}^{* }$ values from quasars tracked over a wide range of elevations E. The parameters B and E₀ describe the peak and shape of the gain curve, respectively. For the DPFU, we typically fit a constant value for the entire observing track to ${T}_{a}^{* }$ measurements of solar system objects to estimate the constant aperture efficiency η_ap. A statistical ${T}_{a}^{* }$ scatter translates into uncertainties of the fitted B, E₀, and η_ap parameters of the gain curve and DPFU, respectively. The uncertainty of the planet model brightness temperature is added in quadrature for the DPFU fitting. For the Butler (2012) models, we have a 10% uncertainty for Saturn and 5% for Mars, Jupiter, and Uranus. The error contribution from ${T}_{\mathrm{sys}}^{* }$ measurements is negligible. Finally, ALMA and SMA exhibit an additional calibration uncertainty from the phasing efficiency. The performance of ALMA has been determined in the quality assurance stage 2 (QA2) ALMA interferometric reduction of data (Goddi et al. 2019). The final calibration method is described in Section 3.2.1, for which we have estimated a global 10% flux density calibration uncertainty. The small SMA dishes are well characterized. Here the phasing efficiency dominates the overall calibration uncertainty, which ranges between 5% in optimal observing conditions and 15% when the phasing efficiency is low. The a priori flux calibration parameters are summarized in Table 3. The reported uncertainties provide a good upper limit for residual gain errors. Compared to our 2017 M87* data (M87* Paper III), the DPFU values are slightly updated, which does not affect our previous results, while the gain curves stayed the same.

However, there are several loss factors that are not captured in our a priori calibration framework, most notably imperfect pointing and focus solutions of individual telescopes. The case is most severe for the large LMT dish, where the measured DPFU varies from day to day. In order to provide a conservative estimate of the gain uncertainties, a bootstrap approach was applied to the DPFU measurements on 2017 April 6 and 7. With 10,000 medians drawn from the bootstrapped samples, we are able to derive the median absolute deviation, which is then scaled to an equivalent sigma. With this method we find a ∼ 35% uncertainty on the LMT DPFU.

Additionally, the antenna gains can be characterized by amplitude self-calibration, which solves the empirical corrections for time-variable instrumental or environmental factors that cannot be measured directly. In order to avoid any variability from Sgr A* that could contaminate the antenna gain estimations, the scan-averaged visibility data of calibrators (J1924−2914 and NRAO 530) are utilized by assuming their stationarity in both source structure and flux density. For both calibrators, the fiducial images independently produced with different imaging pipelines (i.e., eht-imaging, SMILI, Difmap and DMC; Papers III and IV) are employed to improve the statistics on the gains (for more details, see S. Jorstad et al. 2022, in preparation; S. Issaoun et al. 2022, in preparation). We use the mean values of the gains obtained from each imaging pipeline. As an example, the resultant antenna gains derived by amplitude self-calibration for the low-band data sets for both calibrators are shown in Figure 4. The results were obtained after performing a gain alignment procedure meant to minimize the offset between the gains from the two calibrators within the overlapping time and flagging the data points lying beyond 3σ from the mean value. The procedure is as follows: (1) we first derive a mean gain value for each calibrator by only using the gains in the overlapping times; (2) from these gain mean values for the two calibrators, in the overlapping times, we derive an average gain value and rescale the two calibrators' mean gains with this average value; (3) the scaling factors thus obtained are then applied to the gains over the entire time range. The mean gains for each station (reported on top of each panel in Figure 4) are within their corresponding a priori DPFU error budgets (Table 3). We can therefore assume that the calibrator self-calibration method captures the known gain uncertainties that enter into our a priori flux density calibration error budget. At this point, we linearly interpolate the fiducial gains obtained from the calibrators to the Sgr A* time stamps and apply them to the Sgr A* data.

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The East Asian VLBI Network (EAVN; e.g., Wajima et al. 2016; An et al. 2018; Cui et al. 2021) consists of the seven telescopes of KaVA (KVN ¹⁶⁶ and VERA ¹⁶⁷ Array; e.g., Lee et al. 2014; Niinuma et al. 2015), and additional telescopes of the Japanese VLBI Network (JVN; e.g., Doi et al. 2006) and the Chinese VLBI Network (CVN; e.g., Zheng 2015). During the EHT 2017 window, four EAVN observations were carried out at 22 and 43 GHz (Figure 2). A single, symmetric Gaussian model was found to describe the intrinsic structure of Sgr A* at both wavelengths. The measured flux densities from Sgr A* at these frequencies are 1.07 ± 0.11 Jy and 1.35 ± 0.14 Jy, respectively (Table 2). Two of the observations on 2017 April 3 and 4 are (quasi-)simultaneous with the Global 3 mm VLBI Array observations (Section 4.1.2), as well as the EHT sessions (Cho et al. 2022). These measurements provide an estimated size and flux density of Sgr A* at 1.3 mm via extrapolation of power-law models (i.e., the intrinsic size scales with observing wavelength as a power law with an index of ∼1.2 ± 0.2).

VLBI observations of Sgr A* at 86 GHz were conducted on 2017 April 3 with the Global Millimeter VLBI Array (GMVA). ¹⁶⁸ Eight Very Long Baseline Array antennas equipped with 86 GHz receivers, the Robert C. Byrd Green Bank Telescope (GBT), the Yebes 40 m telescope, the Effelsberg 100 m telescope, PV, and 37 phased-ALMA antennas participated in the observation (project code MB007, published in Issaoun et al. 2019b).

The data were recorded with a bandwidth of 256 MHz for each polarization, and fringes were detected out to 2.3 Gλ. The total on-source integration time on Sgr A* was 5.76 hr over a 12 hr track, with ALMA co-observing for 8 hr. The results of the experiment rule out jet-dominated radio emission models of Sgr A* with large viewing angles (> 20°) and provide stringent constraints on the amount of refractive noise added by the interstellar scattering screen toward the source (Issaoun et al. 2019b), discussed in more detail in Section 5.1.2. The total flux density measured at 86 GHz is 1.9 ± 0.2 Jy (Table 2).

The Paranal Observatory's Nasmyth Adaptive Optics System (NAOS) and Near-Infrared Imager and Spectrograph (CONICA) instrument on the VLT, also known as VLT/NACO, measured a K-band NIR upper limit of 3 mJy during the 2017 April 7 EHT observing run (courtesy of the MPE Galactic Center Team). Ongoing observations with the new Very Large Telescope Interferometer GRAVITY instrument (VLTI/GRAVITY; Gravity Collaboration et al. 2017) indicate that Sgr A*'s typical flux distribution in the NIR K band changes slope at a median flux density of 1.1 ± 0.3 mJy, characteristic of Sgr A*'s quiescent NIR emission (Gravity Collaboration et al. 2020).

Observations from the Swift X-ray Telescope (XRT; Gehrels et al. 2004; Burrows et al. 2005) were reprocessed with the latest calibration database files and the Swift tools contained in HEASOFT-v6.20.footnote https://heasarc.gsfc.nasa.gov/lheasoft/. Source flux in the 2–10 keV energy band is extracted from a 10''-radius circular region centered on the position of Sgr A*. Count rates are reported as measured (e.g., Degenaar et al. 2013), i.e., without any correction for the known significant absorption along the Sgr A* sight line (N_H ∼ 9 × 10²² cm⁻²).

There are 48 Swift observations of the Galactic center between 2017 April 5 and 12, with a total exposure time of 26.3 ks (Figure 2). These observations include a dedicated dense sampling schedule to coincide with the EHT observing window and two observations from the regular Galactic center monitoring program (Degenaar et al. 2013, 2015; van den Eijnden 2021). The average exposure time of the dense sampling was ∼500 s, with an average interval between observations of ∼3.5 hr.

In the 2017 April 7 Swift observation that overlaps the EHT window (Figure 6), a 2–10 keV flux is detected (0.023 counts s⁻¹) in excess of the 2017 2σ trend line (0.018 counts s⁻¹), as measured from the cumulative flux distribution observed from Sgr A*. None of the Swift observations are simultaneous with the 2017 April 11 Chandra flare described in the following section.

A series of Chandra X-ray Observatory (Weisskopf et al. 2002) exposures of Sgr A* were acquired on 2017 April 6, 7, 11, and 12 using the ACIS-S3 chip in FAINT mode with a 1/8 subarray (observations IDs 19726, 19727, 20041, 20040; PI: Garmire), for a total of ∼133 ks coordinated with the EHT campaign (Figure 2). The small subarray mitigates photon pileup during bright Sgr A* flares, as well as contamination from the magnetar SGR J1745−2900, which peaked at X-ray wavelengths in 2013 and has faded over the more than 6 yr since (Mori et al. 2013; Rea et al. 2013, 2020; Coti Zelati et al. 2015, 2017). It also achieves a frame rate of 0.44 s versus Chandra's standard rate of 3.2 s.

Chandra data reduction and analysis are performed with the CIAO v4.13 package ¹⁶⁹ (Fruscione et al. 2006), CALDB v4.9.4. We use the chandra_repro script to reprocess the level 2 events files, update the WCS coordinate system (wcs_update), and apply barycentric corrections to the event times (axbary). The 2–8 keV light curves are then extracted from a circular region of radius 125 centered on the radio position of Sgr A*. Light curves for 2017 April 6, 7, and 11 are shown in Figure 6. Using the Bayesian Blocks algorithm (Scargle 1998; Scargle et al. 2013; Williams et al. 2017), we search these light curves for flares and robustly detect one on 2017 April 11, with a second weaker detection on April 7 (orange histograms overplotted on the Chandra light curves in Figure 6).

We use specextract to extract X-ray spectra and response files from a similar 125 region, centered on Sgr A*. Since our primary interest for this data set is the flare emission, we do not extract background spectra from a separate spatial region. Instead, spectra of the quiescent off-flare intervals play the role of our background spectra.

The flare and off-flare intervals are identified by analyzing the X-ray light curves of each observation. For the easily detectable flare on 2017 April 11 (observation ID 20041), we use the direct Gaussian fitting method presented in Neilsen et al. (2013). For 2017 April 7 (observation ID 19727), we use the Bayesian Blocks decomposition (Scargle 1998; Scargle et al. 2013; Williams et al. 2017); this method is better suited to detecting the sustained low-level activity apparent toward the end of the observation.

X-ray observations from the Nuclear Spectroscopic Telescope Array (NuSTAR; Harrison et al. 2013) performed three Sgr A* observations from 2017 April 6 to 11 (observation IDs: 30302006002, 30302006004, 30302006006). These provide a total exposure time of ∼103.9 ks and were coordinated with the EHT campaign. We reduced the data using the NuSTAR Data Analysis Software NuSTARDAS-v.1.6.0 ¹⁷⁰ and HEASOFT-v.6.19, filtered for periods of high instrumental background due to South Atlantic Anomaly (SAA) passages and known bad detector pixels. Photon arrival times were corrected for onboard clock drift and processed to the solar system barycenter using the JPL-DE200 ephemeris. We used a source extraction region with 50'' radius centered on the radio position of Sgr A* and extracted 3–79 keV light curves in 100 s bins with dead time, point-spread function, and vignetting effects corrected (see Zhang et al. 2017, for further details on NuSTAR Sgr A* data reduction). For all three observations we made use of the data obtained by both focal plane modules FPMA and FPMB.

For flare spectral analysis, we used nuproducts in HEASOFT-v.6.19 to create spectra and responses from 30'' circular regions (as recommended for faint sources to minimize the background) centered on the coordinates of Sgr A*. As explained in Section 4.2.4, NuSTAR only partially detected a flare from Sgr A* on 2017 April 11 (observation ID 30302006004); hence, we focus on this observation alone in the present work. The source spectrum was extracted from the NuSTAR good time intervals (GTIs) that overlap with the Chandra flare duration. The background spectrum was extracted from off-flare time intervals in the same observation.

To obtain the best constraint on the spectrum of Sgr A* during its faint and moderate X-ray flares, we performed joint Chandra/NuSTAR spectral analysis of the two flares described in Section 4.2.2. The following analysis was performed in ISIS v1.6.2-43 and made use of Remeis isisscripts. ¹⁷¹ Chandra flare spectra were binned to a minimum of four channels and an S/N of 3 above 0.5 keV, while off-flare spectra were combined and binned to a minimum of two channels and an S/N of 3; we fit bins contained within the interval 1–9 keV. The NuSTAR FPMA/FPMB spectra of the 2017 April 11 flare were binned to a combined minimum S/N of 2 and a minimum of three channels above 3 keV. We ignored all bins not fully contained within the interval 3–79 keV. The resulting spectra are shown in Figure 7.

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Given the relatively small numbers of counts, we opted for simple absorbed power-law models for the flare spectra and compared our model to our data with Cash statistics (Cash 1979). We used the TBvarabs model with Wilms abundances and Verner cross sections (Verner et al. 1996; Wilms et al. 2000) and assumed a shared spectral index for both flares. (There is no conclusive evidence for a relationship between X-ray flare luminosity and X-ray spectral index; Neilsen et al. 2013; Zhang et al. 2017; Haggard et al. 2019.) Because a portion of the 2017 April 11 flare fell within a gap in the NuSTAR light curve, we included a cross-normalization constant between our NuSTAR and Chandra spectra (as well as between the FPMA and FPMB spectra).

For NuSTAR, we simply defined the spectrum of the off-flare interval as the background file for the 2017 April 11 flare spectrum, but we had to treat the Chandra spectra differently because they are susceptible to pileup during flares (e.g., Nowak et al. 2012), though fainter flares like those described here are not likely to be impacted. In particular, because pileup depends on the total count rate, not the background count rate, it is necessary to model the background spectrum and treat the flare emission as the sum of the background model and the absorbed power law. We fit the quiescent (off-flare) emission with a single vapec model; see Nowak et al. (2012) for more details.

Once we had a satisfactory fit to the joint flare spectra, we used the emcee Markov Chain Monte Carlo (MCMC) routine to find credible intervals for our fit parameters. For our MCMC runs, we used 10 walkers (i.e., 10 members of the ensemble) for each of our 10 free parameters and allowed them to evolve for 10,000 steps (a total of 1 million samples). These runs appear to converge within the first several tens of steps, so we discard the first 500 steps. We estimate an autocorrelation time for our parameter chains between ∼200 and 400 steps, indicating that we have 2500–5000 independent samples of each parameter.

Finally, we calculate the minimum width 90% credible interval for each parameter. We compute the cumulative distribution function for the samples for each parameter and select the smallest interval that contains 90% of the samples. The results are given in Table 5. Sgr A* flares are highly absorbed, with a column density of ${N}_{{\rm{H}}}={17.8}_{-2.5}^{+3.5}$ cm⁻²; this is a bit higher than the values found by Nowak et al. (2012) and Wang et al. (2013), but the differences are within 1σ in both cases. The photon index of the flare is ${\rm{\Gamma }}={2.1}_{-0.4}^{+0.5}$, which is not well constrained but is consistent with other analyses of X-ray flares (Porquet et al. 2008; Nowak et al. 2012; Neilsen et al. 2013; Haggard et al. 2019). The 2017 April 7 flare is only detected at 99% confidence but has an unabsorbed 2–10 keV flux of F_0704,2−10 = (0.3 ± 0.2) × 10⁻¹² erg s⁻¹ cm⁻² and a 3–79 keV flux ${F}_{\mathrm{0704,3}-79}=({0.5}_{-0.4}^{+0.8})\times {10}^{-12}$ erg s⁻¹ cm⁻². The 2017 April 11 flare has an unabsorbed 2–10 keV flux of F_1104,2−10 = $({7.8}_{-3.9}^{+2.6})\times {10}^{-12}$ erg s⁻¹ cm⁻², which rises to ${F}_{\mathrm{1104,3}-79}=({15.4}_{-7.5}^{+8.9})\times {10}^{-12}$ erg s⁻¹ cm⁻² over the interval 3–79 keV.

Table 5. Chandra/NuSTAR Joint Spectral Parameters

Parameter	Value	Units
NH	${17.8}_{-2.5}^{+3.5}$	1022 cm−2
Γ	${2.1}_{-0.4}^{+0.5}$	...
NFPMB	${1.2}_{-0.4}^{+0.9}$	...
Kvapec	${0.0012}_{-0.0005}^{+0.0009}$	...
kTvapec	${2}_{-0.5}^{+0.4}$	keV
NACIS	${1}_{-0.3}^{+0.7}$	...
F0704,2−10	${0.3}_{-0.2}^{+0.2}$	10−12 erg s−1 cm−2
F0704,3−79	${0.5}_{-0.4}^{+0.8}$	10−12 erg s−1 cm−2
F1104,2−10	${7.8}_{-3.9}^{+2.6}$	10−12 erg s−1 cm−2
F1104,3−79	${15.4}_{-7.5}^{+8.9}$	10−12 erg s−1 cm−2
FQ,2−10	${0.5}_{-0.1}^{+0.2}$	10−12 erg s−1 cm−2
FQ,3−79	${0.31}_{-0.03}^{+0.05}$	10−12 erg s−1 cm−2

Note. N_H is the X-ray absorbing column density. Γ is the flare photon index. N_FPMB is the cross-normalization of the NuSTAR FPMB relative to the FPMA. K_vapec and kT_vapec are the normalization and temperature of the quiescent vapec component. N_ACIS is the cross-normalization of the Chandra ACIS spectrum relative to NuSTAR. Flare (F₀₇₀₄, F₁₁₀₄) and quiescent fluxes (F _Q ) are quoted for the intervals 2–10 keV and 3–79 keV—the X-ray quiescent and flare fluxes presented in Table 2 are νF_ν in units of erg s⁻¹ cm⁻², and so they differ slightly from the integrated values quoted here. See Sections 4.2.4 and 5.2.2 for further details.

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As reported in Neilsen et al. (2015), the quiescent X-ray flux (F_Q,2−10) results from an admixture of two components. At the faint end of Sgr A*'s flux distribution, there are both variable and steady components, with the variable component likely arising from unresolved faint flares that contribute ∼10% of the apparent quiescent flux (see also Neilsen et al. 2013). Adopting the 2–10 keV flux from this joint fit (Table 5) and allowing a generous ∼1/3 contribution from unresolved X-ray flares, we estimate an upper limit on the median luminosity of ∼10³³ erg s⁻¹. Larger flare contributions would be inconsistent with estimates that approximately 90% of the quiescent emission from Sgr A* is in fact associated with spatially resolved emission.

The observing schedules and (u, v)-coverages of our April 6 and 7 Sgr A* observing days are shown in Figures 1 and 8, respectively. Correlated flux densities as a function of baseline length are shown in Figure 9. The measurements show a good agreement between the products of our processing pipelines. The amplitude data presented in Figure 9 are broadly consistent with the Fourier signature of a blurred ring, indicating the presence of two minima at ∼3 and ∼6.5 Gλ. The ring model is a simple description of the data, used primarily to guide the eye in Figure 9. More complex best-fitting geometric models and image reconstructions are presented in Papers III and IV.

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The calibrated Stokes ${ \mathcal I }$ VLBI data of Sgr A* are made publicly available through the EHT data portal ¹⁷² under the 2022-D02-01 code (DOI: https://doi.org/10.25739/m140-ct59). The past validation history leading up to this data release is described in M87* Paper III. The calibration procedures applied to the Sgr A* data are described in Section 3. These procedures are agnostic to the source structure and encompass the signal stabilization with the automated EHT-HOPS and rPICARD data reduction pipelines, the flux density calibration based on a priori telescope metadata, the network gain calibration of colocated stations using measured 1.3 mm light curves of Sgr A*, and the bootstrapped telescope gain calibration from the analysis of the NRAO 530 and J1924−2914 calibrator sources. The Stokes ${ \mathcal I }$ data provided are time-averaged within 10 s intervals and frequency-averaged over 1875 MHz for the 227.1 GHz low band and 229.1 GHz high band. All common radio interferometric software packages can be used to further average the data in time, depending on the needs of the data analysis method used. As described in Section 3.3, a coherent average within 120 s intervals is recommended to minimize systematics. The accompanying telescope metadata used for the flux density calibration are provided in the standard ANTAB format, ¹⁷³ and the visibilities are provided in the standard UVFITS format, ¹⁷⁴ as well as simple text files, where the visibilities are given as comma-separated values (Table 6).

Table 6. Description of the Public Sgr A* EHT Data Release Content Provided in csv Format for the Low and High Bands on 2017 April 6 and 7

Column	Unit	Description
Time	UTC	Provided as fraction of a day
T1	(string)	Antenna #1 of the baseline
T2	(string)	Antenna #2 of the baseline
U	λ	(u) baseline vector component
V	λ	(v) baseline vector component
Iamp	Jy	Stokes $\,{ \mathcal I }$ amplitude measurement
Iphase	degrees	Stokes $\,{ \mathcal I }$ phase measurement
Isigma	Jy	Stokes $\,{ \mathcal I }$ thermal noise

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Interstellar scattering is a long-known extrinsic effect on the observed radio structure of Sgr A* (Davies et al. 1976). The scattering of radio waves from Sgr A* is predominantly caused by foreground stochastic turbulent electrons along the line of the sight located far from the Galactic center (Bower et al. 2014; Dexter et al. 2017). At long observing wavelengths, the most dominant effect on the source structure is angular broadening, a chromatic effect arising from diffractive scattering (see, e.g., Narayan 1992), resulting in a Gaussian image with a size proportional to the squared wavelength for wavelengths λ ≳ 1 cm (Davies et al. 1976; van Langevelde et al. 1992; Bower et al. 2004; Shen et al. 2005; Bower et al. 2006; Johnson et al. 2018; Lu et al. 2018). Furthermore, the scattering is anisotropic, with stronger angular broadening along the east–west direction compared to the north–south direction (Frail et al. 1994; Jauncey et al. 1989). The angular broadening has an FWHM of $(1.380\pm 0.013){\lambda }_{\mathrm{cm}}^{2}\,\mathrm{mas}$ along the major axis and $(0.703\pm 0.013){\lambda }_{\mathrm{cm}}^{2}\,\mathrm{mas}$ along the minor axis, with the major axis at a position angle 819 ± 02 east of north (Johnson et al. 2018). The intrinsic source angular size is also chromatic, with a θ_src ∝ λ dependence (Johnson et al. 2018). Consequently, observations at 1.3 mm, where historical measurements of Sgr A* determined the size to be ∼50–60 μas (Doeleman et al. 2008; Fish et al. 2011; Johnson et al. 2018; Lu et al. 2018), are in the regime where the intrinsic source structure is no longer subdominant to scattering and becomes the dominant structure in the image. Angular broadening can be described by a convolution of an unscattered image with a scattering kernel, or equivalently by a multiplication of the unscattered, intrinsic interferometric visibilities by the appropriate Fourier-conjugate kernel (Goodman & Narayan 1989). More background information and reviews on interstellar scattering can be found in Rickett (1990), Narayan (1992), and Thompson et al. (2017).

Secondary effects arise from density irregularities in the interstellar medium, causing stochastic variations in the scattering, as well as diffraction effects. These variations introduce substructure in the image that is not intrinsic to the source (Goodman & Narayan 1989; Johnson & Gwinn 2015; Johnson & Narayan 2016; Psaltis et al. 2018). Scattering-induced substructure was first discovered in images of Sgr A* at 1.3 cm by Gwinn et al. (2014) and later observed at other wavelengths (Johnson et al. 2018; Issaoun et al. 2019b, 2021; Cho et al. 2022), giving additional constraints on the scattering properties of Sgr A*. This substructure is caused by modes in the scattering material on scales comparable to the image extent, so scattering models with identical scatter broadening may still exhibit strong differences in their scattering substructure. The substructure manifests in the visibility domain as "refractive noise," which is an additive complex noise component with broad correlation structure across baselines and time (Johnson & Narayan 2016). Using observations of Sgr A* from 1.3 mm to 30 cm, Johnson et al. (2018) have shown that the combined image broadening and substructure strongly constrain the power spectrum of density fluctuations, which is consistent with later observations that occurred close to the 2017 EHT observations of Sgr A* described in this work (Sections 4.1.1 and 4.1.2) and in Issaoun et al. (2019b, 2021) and Cho et al. (2022).

Calibrated EHT data sets indicate that the intrinsic structure is dominant in measured visibilities and that both diffractive and refractive scattering effects are limited, as anticipated from the empirically obtained scattering model and early EHT observations. In Figure 10, we show the scattering kernel in both the image and visibility domains based on the scattering parameters in Johnson et al. (2018), together with the calibrated EHT data. The analysis from Johnson et al. (2018) implies a non-Gaussian kernel (solid lines) more compact than the conventional Gaussian kernel adopted in early literature. Consequently, the angular broadening effect, i.e., multiplication of the intrinsic visibilities with the Fourier-conjugate kernel of scattering, causes a slight decrease in visibility amplitudes (and therefore the S/N) by a factor of a few at maximum. The observed Sgr A* visibility amplitudes exhibit a steeper decrease with baseline length than the scattering kernel, suggesting that the intrinsic source structure, larger than the scattering kernel, is clearly resolved despite angular broadening.

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The refractive substructure is likely also not dominant in Sgr A* at 1.3 mm. In Figure 10(b), we show typical refractive substructures along with the major and minor axes of the anisotropic scattering kernel expected for a reference model, adopted in Paper III, computed with eht-imaging based on the Psaltis et al. (2018) and Johnson et al. (2018) scattering model (see Paper III, Section 4.1 for details). While the rms noise levels due to the refractive substructure are dependent on the source size and shape (see, e.g., Johnson & Narayan 2016), the variations due to the deviation of the intrinsic source structure from the reference model are expected to be within a factor of a few (Paper III). Consequently, the effects of the refractive structure remain only a few percent of the total flux density at the maximum regardless of the source structure, hardly affecting the measured visibilities except for data points around the second null and/or at baselines longer than ∼6 Gλ, which are in the low-S/N regime. A more detailed analysis of the scattering properties on the observed visibilities and source structure is provided in Paper III. For the purposes of the current paper, we conclude that an estimate of the size of Sgr A* drawn from visibility data will be dominated by the intrinsic structure of the source rather than scattering.

The 1.3 mm source structure of Sgr A* has been reported to have FWHMs of several tens of microarcseconds from early EHT observations in the past decade (Doeleman et al. 2008; Fish et al. 2011; Johnson et al. 2015; Lu et al. 2018), consistent with our 2017 observations (Figure 8). The SMT–LMT baseline is the shortest intersite VLBI baseline provided in EHT observations, covering ∼0.6–1.5 Gλ corresponding to a fringe spacing of ∼140–370 μas. For a fringe spacing much larger than the source size, the visibility amplitude is well approximated by a quadratic function solely governed by the size along the baseline direction and the compact total flux density of the source (Issaoun et al. 2019a). Along with the direction of the SMT–LMT baseline, the EHT array has the ALMA–LMT baseline with intermediate baseline lengths of 2.6–4.2 Gλ (see Figure 8). The visibility amplitudes on these baselines are useful to constrain the compact total flux density and source size on VLBI scales along the directions of these baselines, together with the intrasite short baselines tracing the total flux density on arcsecond scales (see M87* Paper IV, Appendix B.1).

Here, we derive constraints on the total compact flux density and the source size in a similar manner to M87* Paper IV by making use of these baselines. Compared with M87* data presented in M87* Paper III, Sgr A* data are further calibrated using interpolated gain solutions from calibrators (Section 3.2.2). We therefore modify the equations in M87* Paper IV and derive the three constraints outlined below.

Constraint 1. The first constraint is based on the fact that the visibility amplitudes fall approximately quadratically with baseline length on short baselines, (Issaoun et al. 2019a). The intermediate to long baselines tend to measure larger correlated flux density than what is expected for a Gaussian source with equivalent quadratic amplitude behavior on short baselines. While localized visibility-domain features such as "nulls" may give lower flux densities than the equivalent Gaussian, the presence of image substructure will tend to increase the average flux density. On short baselines, we can thus express the visibility amplitudes in terms of an equivalent circular Gaussian visibility function,

$$\begin{eqnarray}&&{V}_{{\rm{G}}}({\boldsymbol{u}};{I}_{0},\theta )={I}_{0}{e}^{-\tfrac{{\left(\pi \theta | {\boldsymbol{u}}| \right)}^{2}}{4\mathrm{ln}2}},\end{eqnarray} \tag{ 5 }$$

where I₀ is the total flux density of the Gaussian source, ∣ u ∣ is the length of the baseline, and θ is its FWHM in radians. We can deduce that the measured amplitude ratio of the ALMA–LMT over SMT–LMT baselines will be larger than the corresponding ratio from a circular Gaussian source model. Consequently, the FWHM size of a circular Gaussian determined by the amplitude ratio between SMT–LMT and ALMA–LMT baselines provides an estimate of the minimum compact source size θ_cpt that is not significantly affected by the intrinsic fine-scale source structure:

$$\begin{eqnarray}&&{\theta }_{\mathrm{cpt}}\gtrsim \sqrt{\displaystyle \frac{4\mathrm{ln}2\mathrm{ln}\left(\tfrac{| {{ \mathcal V }}_{\mathrm{SMT}-\mathrm{LMT}}| }{| {{ \mathcal V }}_{\mathrm{ALMA}-\mathrm{LMT}}| }\right)}{{\pi }^{2}(| {\boldsymbol{u}}{| }_{\mathrm{ALMA}-\mathrm{LMT}}^{2}-| {\boldsymbol{u}}{| }_{\mathrm{SMT}-\mathrm{LMT}}^{2})}}.\end{eqnarray} \tag{ 6 }$$

Here ${{ \mathcal V }}_{i-j}$ denotes the true visibility on the baseline i − j. The ratio of the true visibility amplitudes is lower-bounded by

$$\begin{eqnarray}&&\displaystyle \frac{| {{ \mathcal V }}_{\mathrm{SMT}-\mathrm{LMT}}| }{| {{ \mathcal V }}_{\mathrm{ALMA}-\mathrm{LMT}}| }\geqslant \left(1-\sqrt{{\rm{\Delta }}{g}_{\mathrm{ALMA}}^{2}+{\rm{\Delta }}{g}_{\mathrm{SMT}}^{2}}\right)\displaystyle \frac{| {V}_{\mathrm{SMT}-\mathrm{LMT}}| }{| {V}_{\mathrm{ALMA}-\mathrm{LMT}}| },\end{eqnarray} \tag{ 7 }$$

where V_i−j and Δg_i denote the calibrated measured visibility and the deviation of its residual station gain from unity, respectively, after gain calibration using calibrators. Δg_ALMA and Δg_SMT are estimated to be 0.05 and 0.06, respectively, corresponding to the standard deviation of gain solutions from calibrator data across multiple imaging methods described in Section 3.2.2. We take the median of the visibility amplitude ratio from the collection of constraints derived for each single VLBI scan to derive a robust estimate of the minimum source size. With this method, we are able to derive a conservative constraint, which is agnostic to the assumed source structure (e.g., the presence of a null from a ring-like structure).

Constraint 2. The second constraint comes from the curvature of visibility amplitudes between the intrasite baselines and the LMT–SMT baseline. Since the compact flux density should not exceed the total flux density measured with the intrasite baselines, the amplitude fall from the intrasite to LMT–SMT baselines gives the maximum limit of that from the compact flux density to LMT–SMT baseline. Therefore, it gives the maximum limit of the source FWHM size with an equivalent circular Gaussian as

$$\begin{eqnarray}&&{\theta }_{\mathrm{cpt}}\leqslant \displaystyle \frac{2\sqrt{\mathrm{ln}2}}{\pi | {\boldsymbol{u}}| }\sqrt{\mathrm{ln}\displaystyle \frac{{F}_{\mathrm{tot}}}{| {{ \mathcal V }}_{\mathrm{LMT}-\mathrm{SMT}}| }}.\end{eqnarray} \tag{ 8 }$$

With the residual gain uncertainty, this maximum limit is upper-bounded by

$$\begin{eqnarray}&&\displaystyle \frac{1}{| {{ \mathcal V }}_{\mathrm{LMT}-\mathrm{SMT}}| }\leqslant \displaystyle \frac{1+\sqrt{{\rm{\Delta }}{g}_{\mathrm{LMT}}^{2}+{\rm{\Delta }}{g}_{\mathrm{SMT}}^{2}+{\rm{\Delta }}{g}_{\mathrm{tot}}^{2}}}{| {V}_{\mathrm{LMT}-\mathrm{SMT}}| }.\end{eqnarray} \tag{ 9 }$$

In the same manner as with other stations, Δg_LMT is estimated to be 0.12 after calibrator gain transfer. Δg_tot, the fractional uncertainty in the light curve, is estimated to be at most 0.2 based on the consistency of light curves obtained with different instruments / reduction pipelines (see Wielgus et al. 2022). Similarly to Constraint 1, the median of the ratio is adopted to mitigate the effects of statistical errors.

Constraint 3. The minimum compact flux density can be derived by the maximum amplitudes of the LMT–SMT baseline, since the visibility amplitudes are maximum at zero baseline length, as given by

$$\begin{eqnarray}\begin{array}{rcl}{F}_{\mathrm{cpt}} & \geqslant & | {{ \mathcal V }}_{\mathrm{SMT}-\mathrm{LMT}}| \\ & \geqslant & (1-\sqrt{{\rm{\Delta }}{g}_{\mathrm{LMT}}^{2}+{\rm{\Delta }}{g}_{\mathrm{SMT}}^{2}})| {V}_{\mathrm{LMT}-\mathrm{SMT}}| .\end{array}\end{eqnarray} \tag{ 10 }$$

In the above equation, the equality is satisfied in the extreme situation when the source is completely unresolved at the LMT–SMT baseline, whereas in reality it partially resolves the source structure. A stronger constraint is given by the equivalent circular Gaussian with the minimum source size (Constraint 1) extrapolated from the LMT–SMT baselines, described by

$$\begin{eqnarray}&&{F}_{\mathrm{cpt}}\geqslant | {{ \mathcal V }}_{\mathrm{LMT}-\mathrm{SMT}}| {e}^{\tfrac{{(\pi | {\boldsymbol{u}}| {\theta }_{\mathrm{cpt}})}^{2}}{4\mathrm{ln}2}}.\end{eqnarray} \tag{ 11 }$$

We note that the upper and lower limits on the source size obtained from Constraints 1 and 2 are along the directions of LMT–SMT and LMT–ALMA baselines. Using synthetic data from general relativistic magnetohydrodynamics (GRMHD) and semianalytic geometric models in Paper III, we have verified that these size constraints remain valid for various morphologies sharing similar profiles of visibility amplitudes with EHT Sgr A* data (see Paper III for details). The variability of the GRMHD models is often in excess of our observational measurements (Paper V). Therefore, our size constraints are also validated against a changing source structure over the course of our observations.

The derived constraints are summarized in Table 7 and Figure 11(a) for each observing day and for both days combined. As shown in Figure 11(b), the equivalent Gaussians with the minimum and maximum sizes reasonably give the upper and lower bounds of the visibility amplitudes, respectively, on short baselines. Either of the source size constraints indicates that the 2017 EHT Sgr A* data resolve compact emission of several tens of microarcseconds, consistent with the early EHT observations (Doeleman et al. 2008; Fish et al. 2011; Johnson et al. 2015, 2018; Lu et al. 2018).

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The estimated minimum fractional compact flux density is ∼70%–90%, indicating that the vast majority of the radio emission from Sgr A* arises from the horizon-scale emission resolved with the EHT. The maximum limit on the extended emission resolved out on the shortest VLBI baselines is indeed much less than for M87*, 3C 279, and Centaurus A—all of which are known to have a prominent extended jet (M87* Paper IV; Kim et al. 2020; Janssen et al. 2021)—as expected by the nondetection of an extended jet in longer-wavelength VLBI observations across many decades (see Section 5.1.2 and references therein) and early EHT observations.

Studies of the variability of Sgr A* light curves at 230 GHz in 2005–2017 (e.g., Marrone et al. 2008; Dexter et al. 2014; Bower et al. 2018) indicate that on long timescales (from a few days to years) the source fluctuates between 2.0 and 4.5 Jy. During the 2017 EHT observations, Sgr A* was in a low-luminosity state with a mean flux density of ∼2.4 ± 0.2 Jy between 213 and 229 GHz (Table 2). The 2 days for which VLBI data analysis is presented in this series of papers exhibit a low degree of variability that is typical for this source, characterized by the modulation index (standard deviation divided by mean) σ/μ < 10%. The power spectral density has a red-noise character on timescales from 1 minute to several hours, with a transition to white noise for longer timescales. On 2017 April 11 the millimeter flux rises by over 50%, following the X-ray flare maximum with a delay of ∼2 hr. The data on this day appear significantly more variable. An analysis of the 2017 April 11 VLBI data will be presented in a separate future publication. A detailed analysis of the full data set of Sgr A* light curves contemporaneous with the EHT 2017 observations is presented in the companion paper Wielgus et al. (2022).

The ${ \mathcal Q }$-metric (Roelofs et al. 2017) quantifies the variability of a compact source resolved on VLBI baselines in a series of closure phases as a function of time on a specific baseline triangle. The metric compares the observed closure phase variations (${\hat{\sigma }}^{2}$) to those expected from thermal noise (${\tilde{\epsilon }}^{2}$), after detrending the data to account for slow variations due to the evolution of baselines as Earth rotates (see Roelofs et al. 2017, for details):

$$\begin{eqnarray}&&{ \mathcal Q }=\displaystyle \frac{{\hat{\sigma }}^{2}-{\tilde{\epsilon }}^{2}}{{\hat{\sigma }}^{2}}.\end{eqnarray} \tag{ 12 }$$

A static source is expected to give a ${ \mathcal Q }$-metric value close to 0, and a variable source can give a ${ \mathcal Q }$-metric value up to 1.

Figure 12 shows the ${ \mathcal Q }$-metric for M87* and Sgr A* data. First, we have averaged the visibilities within bins of 120 s to accumulate S/N. These averaged data are used to compute closure phases. Then, we have divided the closure phases into 2 hr segments and performed the detrending with a third-order polynomial within each segment. Our analysis is therefore sensitive to structural source variability occurring on timescales between 120 s and 2 hr. The ${ \mathcal Q }$-metric is generally higher for Sgr A* than for M87*, which does not necessarily indicate that Sgr A* varies more strongly because of the different error budgets for the two sources. However, the different ${ \mathcal Q }$-metric values do indicate that intrinsic variability is detected with higher significance for Sgr A* than for M87*. The high sensitivity to thermal noise can lead to different ${ \mathcal Q }$-values on redundant triangles.

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Because of imperfections in the detrending procedure (a third-order polynomial does not generally capture any static source structure) and the particular realization of the thermal noise, a static source may also give nonzero ${ \mathcal Q }$-metric values. In order to determine on which triangles we detect significant variability, the measured ${ \mathcal Q }$-metric values are therefore compared to those simulated for a range of static source models for Sgr A*. Shown in Figure 12 are 1σ upper limits of the distributions of ${ \mathcal Q }$-metric values obtained from synthetic data generated from different single static GRMHD-GRRT snapshots (Paper V), with different realizations of either a static or moving scattering screen. We have applied random position angle rotations to 12 random frames drawn from MAD/SANE models with spins of ±0.94, ±0.5, 0; R_high = 10, 40; and inclination angles of 10°, 30°, 50°, 70° (a description of these models and their parameters is given in Paper V). For these 960 different source structures, we have generated synthetic data using 100 different thermal noise realizations for each observing day and the low- and high-frequency bands. We applied the same procedure to 11 frames of each of the source models used to calibrate the imaging methods in Paper III, which were chosen to match the measured visibility amplitudes of Sgr A*. The resulting ${ \mathcal Q }$-metric values were bootstrapped to obtain the same number of values as those from the GRMHD models.

Scattering-screen variability generally has a small effect on the ${ \mathcal Q }$-metric. Comparing the measured and simulated ${ \mathcal Q }$-metric values, we see a significant excess for Sgr A* on ALMA–LMT–SMA, ALMA–SMT–SMA, and ALMA–SMT–LMT. For these triangles, the observed closure phase variability cannot be explained by variability due to interstellar scattering or imperfections in the detrending procedure. The excess variability in degrees $\breve{{ \mathcal Q }}=\sqrt{{\hat{\sigma }}^{2}-{\tilde{\epsilon }}^{2}}$ is 16.4°, 6.8°, and 3.1° for these triangles, respectively. Hence, although the ${ \mathcal Q }$-metric does indicate intraday closure phase variability in the Sgr A* data, it only does so on a few triangles, and the variability amplitude is small.

More detailed discussions of the source variability based on static and time-dependent reconstructions of the underlying Sgr A* source model, as well as examples of typical closure phase trends, are given in Papers III and IV. Effects of phase decoherence, low S/N near interferometric nulls, residual time-dependent gain errors, stochastic interstellar scattering effects, and changing baseline vectors due to Earth rotation introduce additional scatter in the correlated flux densities shown in Figures 9, 10, and 11. Paper IV presents a model that takes these effects into account and provides a clear detection of intrinsic flux density variability on horizon scales.

The supplementary ground- and space-based data products leveraged here are published in original works and/or are available in public archives.

EAVN observations of Sgr A* at 22 and 43 GHz are available in Cho et al. (2022), GMVA 86 GHz observations are published in Issaoun et al. (2019b), and a detailed discussion of the VLT/GRAVITY NIR flux distribution can be found in Gravity Collaboration et al. (2020).

Chandra, Swift, and NuSTAR data products are available via NASA archives ^{175 , 176 , 177} and are collected in the EHT data portal ¹⁷⁸ under the 2022-D02-03 code (DOI:https://doi.org/10.25739/26fq-k306). The repository contains the following data products: