# Multiwavelength Observations of Sgr A*. II. 2019 July 21 and 26

Joseph M. Michail Department of Physics and Astronomy, Northwestern
University, 2145 Sheridan Rd, Evanston, IL 60208, USA  Center for
Interdisciplinary Exploration and Research in Astrophysics (CIERA),
Northwestern University, 1800 Sherman Ave, Evanston, IL 60201, USA  Farhad
Yusef-Zadeh  Department of Physics and Astronomy, Northwestern University,
2145 Sheridan Rd, Evanston, IL 60208, USA  Center for Interdisciplinary
Exploration and Research in Astrophysics (CIERA), Northwestern University,
1800 Sherman Ave, Evanston, IL 60201, USA  Mark Wardle Astrophysics and Space
Technologies Research Centre and School of Mathematics and Physical Sciences,
Macquarie University, Sydney, NSW 2109, Australia  Devaky Kunneriath National
Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, VA 22903, USA
Joseph L. Hora Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden
Street, MS-65, Cambridge, MA 02138-1516, USA  Howard Bushouse Space Telescope
Science Institute, Baltimore, MD 21218, USA  Giovanni G. Fazio Center for
Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street, MS-65, Cambridge, MA
02138-1516, USA  Sera Markoff Anton Pannekoek Institute for Astronomy,
University of Amsterdam, Science Park 904, 1098 XH, Amsterdam, The Netherlands
Gravitation and Astroparticle Physics Amsterdam (GRAPPA) Institute, University
of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands  Howard A.
Smith  Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street,
MS-65, Cambridge, MA 02138-1516, USA Joseph M. Michail
<EMAIL_ADDRESS>
###### Abstract

We report on the final two days of a multiwavelength campaign of Sgr A*
observing in the radio, submillimeter, infrared, and X-ray bands in July 2019.
Sgr A* was remarkably active, showing multiple flaring events across the
electromagnetic spectrum. We detect a transient $\sim 35$-minute periodicity
feature in Spitzer light curves on 21 July 2019. Time-delayed emission was
detected in ALMA light curves, suggesting a hotspot within the accretion flow
on a stable orbit. On the same night, we observe a decreased flux in the
submillimeter light curve following an X-ray flare detected by Chandra and
model the feature with an adiabatically expanding synchrotron hotspot
occulting the accretion flow. The event is produced by a plasma
$0.55~{}R_{\text{S}}$ in radius with an electron spectrum $p=2.84$. It is
threaded by a $\sim 130$ Gauss magnetic field and expands at $0.6\%$ the speed
of light. Finally, we reveal an unambiguous flare in the infrared,
submillimeter, and radio, demonstrating that the variable emission is
intrinsically linked. We jointly fit the radio and submillimeter light curves
using an adiabatically expanding synchrotron hotspot and find it is produced
by a plasma with an electron spectrum $p=0.59$, $187$ Gauss magnetic field,
and radius $0.47~{}R_{\text{S}}$ that expands at $0.029c$. In both cases, the
uncertainty in the appropriate lower and upper electron energy bounds may
inflate the derived equipartition field strengths by a factor of 2 or more.
Our results confirm that both synchrotron- and adiabatic-cooling processes are
involved in the variable emission’s evolution at submillimeter and infrared
wavelengths.

Galactic center (565), Active galactic nuclei (16), Supermassive black holes
(1663)

††journal: ApJ††facilities: VLA, ALMA, HST (WFC3), Spitzer, CXO
(ACIS)††software: AIPS, astropy (Astropy Collaboration et al., 2013, 2018),
BBlocks (Williams et al., 2017), CASA (McMullin et al., 2007), CIAO (Fruscione
et al., 2006), LMFIT (Newville et al., 2023), matplotlib (Hunter, 2007), numpy
(Harris et al., 2020), pandas (McKinney, 2010), pyCCF (Peterson et al., 1998;
Sun et al., 2018), scipy (Virtanen et al., 2020), sklearn (Pedregosa et al.,
2011)

##  1 Introduction

The Milky Way Galaxy hosts Sagittarius A* (Sgr A*), a $4.152\times
10^{6}~{}\text{M}_{\odot}$ supermassive black hole, at its center. At a
distance of $8178$ pc (Boehle et al., 2016; Gravity Collaboration et al.,
2019), it is one of only two black holes whose angular extent is large enough
to be directly imaged by Earth-sized very long baseline interferometry (VLBI)
experiments like the Event Horizon Telescope (EHT; Event Horizon Telescope
Collaboration et al., 2022a). One complication in imaging its near-event-
horizon-scale features is its variability. Sgr A* has a gravitational
timescale $t_{g}=r_{g}/c=GM/c^{3}\sim 20$ seconds, and the Keplerian orbital
period at the innermost stable circular orbit (ISCO) is $\sim 30$ minutes if
Sgr A* does not spin (Gravity Collaboration et al., 2018), making it variable
over a short duration. Imaging near-event-horizon features requires
observations many times longer than these variability timescales, which
corrupts the visibilities produced by intrinsic structures and smears them in
the final image. Statistical attempts have been used to mitigate some of its
variable nature and create a best-fit event-horizon-scale image (Broderick et
al., 2022). However, understanding the cause of the variability will
ultimately benefit future imaging attempts of this source.

Sgr A*’s variability has been observed in the radio (i.e., Lu et al., 2011;
Yusef-Zadeh et al., 2011; Brinkerink et al., 2015), submillimeter/millimeter
(submm/mm; i.e., Dexter et al., 2014; Subroweit et al., 2017; Iwata et al.,
2020), infrared (IR; i.e., Hora et al., 2014; Witzel et al., 2018), and X-rays
(i.e., Baganoff et al., 2001; Markoff et al., 2001; Neilsen et al., 2015;
Ponti et al., 2015; Yuan & Wang, 2016). Large-amplitude hourly timescale
variability, which we denote “flares,” occurs roughly once to a few times per
day. At radio and (sub)mm, these flares appear in addition to slowly varying
“quiescent”-like emission from an accretion flow near the event horizon. At
radio frequencies, this emission is optically thick. Toward submm frequencies,
its optical depth decreases, and the emission is either marginally optically
thick or optically thin at 230 GHz and optically thin at 345 GHz (i.e., Bower
et al., 2019). Wielgus et al. (2022) suggest that the submm quiescent emission
at 230 GHz and higher is optically thin, but there may be regions within it
that have higher-than-average optical depth. The causes of the flares and the
slowly varying quiescent emission are unknown; multiwavelength observations
are crucial to unraveling this mystery.

Coordinated multiwavelength observations have led to insights into the flaring
emission. The time of the peak flaring emission at lower frequencies is
delayed relative to those at high frequencies. Between radio and submm, the
time delays are on the order of tens of minutes to an hour (i.e., Brinkerink
et al., 2015; Mossoux & Grosso, 2017; Michail et al., 2021a). Similar lags
have been discovered across a single radio frequency band using the NSF’s Karl
G. Jansky Very Large Array (VLA; Brinkerink et al., 2021; Michail et al.,
2021b). At IR, the emission has been observed to lead or simultaneously occur
with submm (i.e., Yusef-Zadeh et al., 2006b; Fazio et al., 2018). The
appearance or lack of delays depends on the observing frequency (i.e., 230 vs.
345 GHz) and the emission’s physical properties.

There is a curious connection between flares in the IR and X-rays. First,
every X-ray flare has an IR counterpart, but the opposite is not true (Dodds-
Eden et al., 2009). These “missing” flares may be caused by limited
observational sensitivity in the X-rays, as there is large-scale quiescent
X-ray emission near Sgr A*. The missing X-ray flares may not be sufficiently
bright to overcome this large-scale emission, leaving only the strongest
flares to be observed. Second, the time delay between IR and X-ray light
curves has not been well-established. Boyce et al. (2019) suggested the X-ray
might lead the IR by 10-20 minutes after analyzing more than 100 hours of
coordinated IR and X-ray monitoring. This ensemble-average value is only
significant at a 68% confidence interval. The authors conclude additional data
are required to constrain the correlation.

One model that attempts to explain the connection between IR, submm, and radio
flaring is the adiabatically-expanding synchrotron hotspot model (van der
Laan, 1966; Yusef-Zadeh et al., 2006a). It describes the time- and frequency-
dependent evolution of an initially optically-thick synchrotron-emitting
plasma as it expands. The time delay naturally explains the dynamic expansion
as the emission transitions from optically thick to thin. This picture
predicts unique continuum and polarimetric light curves, which depend on
properties such as the embedded magnetic field, electron spectrum, and
expansion speed. In a series of earlier papers, we explored the
multiwavelength and polarimetric nature of these hotspots. Michail et al.
(2023) tested this picture using a full-radiative transfer prescription to
model 235 GHz full-polarization light curves of a detected flare from ALMA.
For the first time, we constrained the three-dimensional orientation of the
embedded magnetic field. In Michail et al. (2021a, further referred to as
Paper I), we used this picture to connect the IR and submm variability and
predict the resulting radio emission.

In July 2019, we participated in organizing a ground- and space-based campaign
to simultaneously monitor Sgr A* from radio to X-ray energies. Several papers
have analyzed these data but do not focus on the complementary radio and
Hubble Space Telescope (HST) observations. GRAVITY Collaboration et al. (2021)
analyzed a remarkable flare on 18 July 2019, which occurred only a few months
after the brightest-ever IR flare with a dereddened $2.15~{}\mu$m peak flux of
$\sim 60$ mJy (Do et al., 2019). Using IR (Spitzer Space Telescope and
GRAVITY) and X-ray (Chandra X-ray Observatory and NuSTAR) observations, they
show its mean and time-resolved spectral energy distributions are consistent
with synchrotron self-Compton or a pure synchrotron spectrum with a cooling
break. The synchrotron self-Compton model was unable to reproduce any
significant submm emission (first reported in Paper I), while the latter
produced submm emission at levels consistent with the historic average.
However, properties of this model, such as higher-than-inferred electron
densities in the accretion flow, were drawbacks. Boyce et al. (2022) reported
the remaining two days of ALMA, Spitzer, and Chandra light curves from the
July 2019 campaign and remodeled the first day’s results to include the ALMA
data using an adiabatically-expanding source region. Their models rejected the
optically-thin case reported in Paper I where both the submm and IR flares
were produced by a single synchrotron component, as the IR spectral index was
not consistent with the observations and could not reproduce the X-ray
emission using synchrotron self-Compton. The emission was consistent with an
adiabatically-expanding model such that the IR and X-ray emission were both
produced by synchrotron self-Compton processes and the submm was produced by
synchrotron only. However, such a model predicted similarly high electron
densities ($n_{e}\sim 10^{9-10}$ cm-3) like those modeled by GRAVITY
Collaboration et al. (2021).

Boyce et al. (2022) focused only on modeling multiwavelength features on the
first day of observation and did not include contemporaneous HST (IR) and VLA
(radio) observations during this campaign. Here, we present the remaining two
days of monitoring with all participating observatories. We discuss data
reduction and calibration for each instrument in Section 2. In Section 3, we
present the time delay analysis between each pair of instruments and the
periodogram analysis of the IR data. In Section 5, we use the adiabatic-
expansion picture (described in Section 4) to model the detected features and
summarize our conclusions in Section 6. Our adopted mass of Sgr A* throughout
this paper corresponds to a Schwarzschild radius $R_{\text{S}}=1.23\times
10^{12}$ cm.

##  2 Observations and Data Reduction

Here, we describe the observing details and calibration procedures for the
data used in this analysis. Paper I explained the techniques in greater detail
for the ALMA, Spitzer, and Chandra data; we give summaries for these
instruments below. We present detailed explanations for the VLA and HST data.

###  2.1 Radio: Very Large Array

The VLA observed Sgr A* on three days: 18 July 2019 (Project ID: 19A-490, PI:
Yusef-Zadeh), 21 July 2019, and 26 July 2019 (Project ID: 19A-229, PI: Yusef-
Zadeh). Observations on 18 July 2019 were completed in the hybrid BnA array at
Q-band (43 GHz). On the 21st and 26th, the VLA was in the A configuration and
used the Q and Ka (35 GHz) bands, respectively. The 3-bit correlator was used
to observe over 8 GHz bandwidth and is composed of 63 intermediate frequencies
(IFs), each with a bandwidth of 128 MHz. Every IF is subdivided into 64 2-MHz
wide channels recording full-polarization products.

Observations on 18 and 21 July were not successfully completed owing to
several compounding issues. The cycle times between Sgr A* and the phase
calibrator were lengthier than required in the most extended configurations
($\sim 20$ minutes used versus the recommended $\sim 2$ minutes).
Additionally, the pointing calibrations were not completed often enough for
high-frequency observing, nominally at a cadence of $45-60$ minutes. Finally,
strong winds above the specifications for Q-band observing occurred during
these two nights. Despite our best efforts, these data were not salvageable to
extract light curves. While the 26 July observations were set up similarly,
better weather conditions and the wider primary beam at Ka-band reduced many
of the pointing issues. Therefore, we only make use of the Ka-band
observations in this analysis.

The two nights of observations follow the same procedure with three
calibrators and Sgr A*. 3C286 is the flux and polarization angle calibrator.
J1733-1304 is the bandpass, secondary complex gain, and primary instrumental
polarization calibrator, while J1744-3116 (hitherto denoted J1744) is the
primary complex gain and secondary instrumental polarization calibrator. Scans
of Sgr A* were interleaved with J1744, and J1733-1304 was observed
approximately every hour. The data were reduced in CASA (McMullin et al.,
2007, v5.6.2) with the default VLA pipeline. To calibrate the polarization
products, we follow the appropriate polarization calibration
guide111https://casaguides.nrao.edu/index.php?title=CASA_Guides:Polarization_Calibration_based_on_CASA_pipeline_standard_reduction:_The_radio_galaxy_3C75-CASA5.6.2.
The polarization-calibrated visibilities were exported from CASA and imported
into AIPS for additional baseline-based flagging and point-source phase self-
calibration. We extract light curves for Sgr A* and J1744 using DFTPL on
projected baselines $\geq 100~{}\text{k}\lambda$ at 60-second intervals for
each IF. DFTPL averages the real portion of the visibilities at the phase
center to determine light curves without the need for other (e.g., imaging-
based) techniques. The 26 July 2019 light curves at Ka-band are shown in
Figure 1.

Figure 1: VLA light curves of Sgr A* (in blue) and J1744 (in red) on 26 July
2019 at Ka-band. The error bars are smaller than the points. We bin data at
60-second intervals using projected baselines $\geq 100$ k$\lambda$. Figure 2:
ALMA light curves of Sgr A* (blue) and J1700-2610 (red) for 21 July 2019 (a)
and 26 July 2019 (b). All light curves are made with a $60$-second binning
time on projected baselines $\geq 20$ k$\lambda$.

###  2.2 Submillimeter: ALMA

Three nights of ALMA observations were taken as part of the ALMA Compact Array
(ACA) Cycle 6 filler program 2018.A.00050.T in band 7 ($\sim 330$ GHz) with
projected baselines ranging from $10-55~{}\text{k}\lambda$. We use the
scriptForPI.py code to restore the calibrated data from the automatic ALMA
pipeline in CASA version 5.4.0-70 (Pipeline-CASA54-P1-B, r42254). Four IFs
with 2 GHz of bandwidth each were composed of 1024 1.953-MHz channels
recording dual polarization products. One IF is centered on the $\approx 346$
GHz CO(3-2, v=0) transition. J1337-1257 and J1924-2914 are the bandpass and
amplitude calibrators; J1700-2610 (J1700) and J1717-3342 are the phase
calibrators for each observation. Like the VLA data, we phase self-calibrated
Sgr A* and J1700, assuming each was a point source to mitigate residual
baseline-based phase errors. Finally, we import the data into AIPS and use
DFTPL to extract the light curves with a $1$ minute binning time on baselines
$\geq 20~{}\text{k}\lambda$. We chose this limit after viewing the visibility
amplitude as a function of projected baseline distance, finding that sizes
below this threshold are dominated by the extended emission from the
Circumnuclear Disk.

We take additional steps to post-process the light curves. We discarded the
last execution block on 21 July 2019 due to amplitude-decorrelation on several
antennas caused by the low altitude of our calibrators. The calculated flux of
the phase calibrator changed (on the order of a few percent) between execution
blocks. We follow Paper I to correct this by using the first block as our
reference. The phase calibrator’s average flux in subsequent execution blocks
was used to calculate multiplicative correction factors. These scale factors
are applied to the phase calibrator and Sgr A*. We show the final ALMA light
curves in Figure 2.

Figure 3: The dereddened HST light curves for Sgr A* (blue) and the background
aperture (red) on 21 July 2019 (a) and 26 July 2019 (b) at $1.53$ µm. Each
orbit is separately plotted to show short-term flux variability clearly.

###  2.3 Near-infrared: Hubble Space Telescope

We obtained near-IR observations of Sgr A* with the HST Wide Field Camera 3
(WFC3) under program 15667 (http://dx.doi.org/10.17909/aqd3-3642 (catalog
10.17909/aqd3-3642)) over two different visits occurring on 21 July (2 orbits)
and 26 July (6 orbits). During each visit, Sgr A* was visible to HST at 40–50
minute intervals per orbit. We give a brief explanation of the data
calibration here and refer to Mossoux et al. (2016) for further details.

WFC3 observations were taken with the F153M filter with an effective
wavelength of $\lambda_{\mathrm{eff}}=1.532\ \mu$m taken in continuous
accumulation mode. The instantaneous field-of-view is approximately
$2\farcm{}26$ per side with a pixel size of $\sim 0\farcs{}13$. The
accumulation mode reads out the WFC3 detector every 25 seconds during a total
exposure time of 275 seconds. HST was dithered around Sgr A*’s location to
improve WFC3’s point spread function. We calibrate the data with the default
STScI calwf3 calibration pipeline. We corrected the astrometry of each
exposure using the radio location of the nearby IRS-16C source. We used the
photutils package of astropy to complete aperture photometry of Sgr A* using a
circular aperture of 4 pixels in diameter centered on this source’s known
radio location at 25-second intervals (the cadence of WFC3’s readout mode).

In our initial analysis, the photometry of other stars in the field in each
exposure declined by $\sim 3\%$, affecting the final light curve of Sgr A*.
Assuming stars near Sgr A* did not change over a single exposure, we corrected
this effect and applied them to Sgr A*’s light curve. We revalidated the light
curves for each star and found that they were constant over each exposure.
Finally, we applied an aperture correction using several apertures at
different sizes by calculating the curve of growth for isolated stars and
applying it to Sgr A*’s final light curve.

Using a reference star, we estimate a noise level of 0.004 mJy per exposure.
We deredden the measured fluxes with the prescription in Fritz et al. (2011).
The extinction in the near-IR band is modeled by:

$\text{A}_{\lambda}=2.62\left(\dfrac{\lambda}{2.166~{}\micron}\right)^{-2.11}.$
(1)

For $1.53$ µm, $\text{A}_{1.53~{}\mu\text{m}}=5.46$ mag, giving a flux
correction factor of $f_{1.53~{}\mu\text{m}}=2.512^{5.46}=152.8$. We show the
dereddened photometry for Sgr A* and the background aperture in Figure 3.

###  2.4 Mid-infrared: Spitzer Space Telescope

The Spitzer observations and calibration were outlined in Hora et al. (2014)
and Witzel et al. (2018), while a more detailed explanation of these data was
given in GRAVITY Collaboration et al. (2021). Three observations of Sgr A*
lasting approximately 16 hours each were taken as part of program ID 14026. In
contrast to previous Spitzer analyses, we bin the data into 1- and 5-minute
averages to obtain higher signal-to-noise light curves by averaging all fluxes
within one-half bin width. We converted the barycentric UT timestamps into UTC
using an online tool222https://astroutils.astronomy.osu.edu/time/bjd2utc.html
(Eastman et al., 2010). Fritz et al. (2011) listed a $4.5~{}\micron$
extinction of $1.06$ mag toward the Galactic Center. Therefore, we scale the
Spitzer light curves by $2.512^{1.06}=2.65$ to deredden the light curves.
Boyce et al. (2022) denoted low signal-to-noise-ratio periods in their
analysis of these Spitzer data, which we incorporate here. The light curves
for 21 July 2019 and 26 July 2019 are shown in Figure 4.

Figure 4: The dereddened Spitzer light curves at $4.5$ µm for Sgr A* during 21
and 26 July 2019 (a and b, respectively). The 1-minute binned data are shown
in black, and the 5-minute smoothed data are in red. The gray bands highlight
periods of low signal-to-noise fluxes given in Boyce et al. (2022).

###  2.5 X-ray: Chandra X-Ray Observatory

Three days of simultaneous Chandra and Spitzer observations were completed on
18, 21, and 26 July 2019. For 21 July 2019 and 26 July 2019, the observation
IDs are 20446 and 20447 (PI: Fazio), respectively. Each night’s observation
lasted 52 ks in FAINT mode with Sgr A* centered on the ACIS S3 chip in the 1/8
subarray setting. We reprocessed the data with CIAO version 4.13 and CALDB
version 4.9.4. We refine the astrometric accuracy using wcs_match and
wcs_update with three nearby Galactic Center sources from the Chandra Source
Catalog (CSC, Evans et al., 2010).

Following Capellupo et al. (2017), we determine background-subtracted light
curves of Sgr A* by placing a $1\farcs 25$ radius aperture at the radio
location of Sgr A* ($17^{\text{h}}~{}45^{\text{m}}~{}40\fs
04,-29^{\circ}~{}00\arcmin~{}28\farcs 18$). Our background aperture is
centered on $17^{\text{h}}~{}45^{\text{m}}~{}40\fs
30,-28\arcdeg~{}59\arcmin~{}52\farcs 66$ with a $10\farcs 0$ radius. This
background aperture contains no known or obvious sources (Paper I). To
minimize photons from the diffuse X-ray background, we only use events within
the $2-8$ keV range (Neilsen et al., 2013). We used dmextract to obtain the
light curves of Sgr A* at a 5-minute binning time and ran the open-source
Bayesian block algorithm,
bblocks333https://pwkit.readthedocs.io/en/latest/science/pwkit-bblocks/, on
the light curves (Scargle, 1998; Scargle et al., 2013; Williams et al., 2017)
with a false positive rate of $p_{0}=0.05$. The Bayesian blocks method
accounts for counting noise in Poissonian-like systems, where the change in
count rate must be statistically significant at a level of $1-p_{0}$. Since
every flare requires two changes (an increase and a decrease; Nowak et al.,
2012), this ensures a flare is significant at a confidence of
$1-p_{0}^{2}=99.75\%$. We show the Chandra light curves and the Bayesian block
results in Figure 5.

Figure 5: 2-8 keV Chandra X-ray Observatory light curves of Sgr A* on 21 (a)
and 26 July 2019 (b). The light curves are binned in 300-second increments and
are background subtracted. Sgr A* light curves (black) and the Bayesian Block
results (red) setting $p_{0}=0.05$ are shown.

##  3 Analysis

###  3.1 21 July 2019

Figure 6: Cross correlations for each pair of observations on 21 July 2019.
(a): The cross correlation histogram across the ALMA bandwidth. (b): The cross
correlation histogram between the lowest ALMA frequency and Spitzer. (c): The
cross correlation histogram between the (inverted) ALMA and X-ray light
curves. (d): Stacked X-ray, IR, and submm light curves during ALMA coverage.
Vertical, gray dashed lines indicate the timestamps of peaks in the X-ray/IR
light curves. In the IR plot, the green dots and solid black line represent
the 1-minute and smoothed 5-minute light curves, respectively.

Sgr A* was remarkably active on 21 July 2019 in the submm, IR, and X-ray. The
Bayesian blocks algorithm (Figure 5a) detected an increased quiescent plateau
and flare in the X-ray light curves, which then decays to a count rate
consistent with Neilsen et al. (2015, $\sim 5$ cts/ks). To detect flaring
events in the HST data, we use a root-mean-square-based (RMS-based)
calculation. For each orbit, we calculate the RMS flux of Sgr A* and the
background RMS error (RMSE) to quantify the intrinsic scatter in the data.
Then, we determine the z-score for each associated flux and error measurement
using Sgr A*’s average flux and the background RMSE in the given orbit. We
define a flare-like event when the z-score is $\geq 5\sigma$ (the typical HST
RMSE per orbit is $\sigma\sim 0.2$ mJy). For 21 July 2019, we do not detect
any flares in the $1.53$ µm data. Therefore, we only focus on the ALMA,
Spitzer, and Chandra light curves.

In Figure 6, we show cross-correlations histograms between different pairs of
the data (panels a-c) and a subset of these three light curves during the ALMA
observations (panel d). We detect several corresponding features that occur in
both the Spitzer and ALMA light curves following a single X-ray flare. The
lack of Spitzer data at the peak of the X-ray emission is due to low-SNR data
(gray band), so we are unable to definitively conclude if there is an IR
flare. Multiwavelength campaigns have shown X-ray flares always appear paired
to an IR peak (Dodds-Eden et al., 2010). Therefore, we assume a weak IR flare
occurs but is not detected.

####  3.1.1 Cross Correlations

We complete cross-correlation analyses to check for time-delayed emission
using pyCCF (Peterson et al., 1998; Sun et al., 2018). For all cross-
correlations in this paper, we use 5000 Monte Carlo realizations that remove a
random number of light curve points each time and build a time delay histogram
between two light curves. We report the mean delay value and the $95\%$
($2\sigma$) confidence interval. We consider that the light curves are delayed
if zero is not contained within the 95% interval.

We show the three cross-correlation histograms for 21 July in Figure 6a-c. We
report no statistically significant time delay across the ALMA bandwidth
($\sim 14$ GHz). Iwata et al. (2020) and Wielgus et al. (2022) suggest the
presence of a small (but detectable) time delay across their lower frequency
230 GHz bandwidth. Using Equation 2 and the values listed in Iwata et al.
(2020), we calculate an expected time delay across the 345 GHz ALMA band on
the order of $\sim 30$ seconds, assuming a range of electron indices ($p$)
between $1-3$. Since we average the light curve in 60-second increments, any
time delays shorter than this will be washed out.

Cross correlations are sensitive to peaks in emission (i.e., when the
correlation coefficient is near $1$) instead of a peak and trough (i.e., when
the correlation coefficient is near $-1$). We invert the ALMA light curve
before completing the cross-correlation between the submm and X-ray light
curves to determine the time delay between the X-ray flare peak and the $\sim
0.7$ Jy drop in submm flux. We find the X-ray data lead the submm light curves
by approximately $26$ minutes. We mark the X-ray peak flare time with a black
dashed line in Figure 6d to clearly show this detected delay. Such a feature
has been detected previously several times (i.e., Marrone et al., 2008; Meyer
et al., 2008; Yusef-Zadeh et al., 2008; Capellupo et al., 2017) and will be
further discussed in Section 5.1.2.

Finally, we complete the cross-correlation between the ALMA and Spitzer light
curves (Figure 6b); the time delay histogram shows a much different structure
than in the other two pairs of light curves. We find three groups of time
delay peaks (centered near $\pm 35$ and $0$ minutes). The majority of the 5000
realizations prefer the positive time delay where the IR leads the submm. The
negative time delay peak (where the submm leads the IR) appears because pyCCF
is model-agnostic and corresponds to the third IR peak corresponding to the
first observed one in the submm. For our analysis, it has no physical meaning.

Due to the lack of ALMA data near 3.5 hr UTC, we cannot rule out the
possibility of simultaneity. To determine if peaks in the submm and IR are
simultaneous, we follow Paper I in using a Gaussian process regression to
interpolate the ALMA data between execution blocks. We use the Gaussian
process toolkit, part of the sklearn Python package, and choose a compound
constant + radial-basis kernel. In Figure 7a,b, we show the Gaussian process-
regressed ALMA and Spitzer data. The purple band denotes the 95% confidence
interval of the Gaussian process fit, and the black marks the mean fit. The
red points are the original (non-interpolated) data. We again use pyCCF to
search for time lags between the interpolated ALMA and Spitzer data; the
cross-correlation histogram is shown in Figure 7c. We utilize the 68%
($1\sigma$) confidence interval as the interpolated data error inputs to pyCCF
instead of the 95% interval for consistency with Figure 6. From this
interpolated analysis, we find the IR data slightly lead the submm peaks by
$4.66^{+2.56}_{-2.61}$ minutes and suggests the presence of an ALMA flare near
3.5 hr UTC occurring between execution blocks.

Figure 7: Submm and IR light curves of Sgr A* on 21 July 2019. (a): Spitzer
light curve during the significant period of variability present in both the
IR and submm data. The 1-minute and smoothed 5-minute light curves are shown
in green and black, respectively. (b): The raw (red) and mean interpolated
(black) ALMA light curves during the same time range as above. We interpolated
the raw light curves using a Gaussian process, and the 95% confidence interval
(CI) is shaded in purple. (c): The pyCCF lag time histogram between the
Spitzer and interpolated ALMA data. We find the submm data are delayed $\delta
t=4.66^{+2.56}_{-2.61}$ minutes relative to the IR data.

####  3.1.2 Periodograms

Figure 8: Monte Carlo Fourier transform power spectra for 35,000 realizations
of Sgr A* and red noise during all three days of observation. The black dashed
line is the mean power spectra from the Sgr A* light curve, while the gray
shaded region shows the 68% confidence interval. $3\sigma$ (green solid) and
$4\sigma$ (red dot-dash) upper-limit power spectra are shown for red-noise
light curves with the same statistics as Sgr A*. We detect a transient
periodicity at $>4\sigma$ significance at $\sim 35$ minutes on 21 July 2019
and a slightly significant $\sim 40$ minute peak on 18 July 2019. The dotted
magenta line in the middle panel denotes the statistical peak of the
periodogram.

We now focus on the repeating flares present in the mid-IR and submm light
curves. The initial cross-correlation results (Figure 6) between these two
wavelengths suggest periodic emission occurring every $\approx 40$ minutes.
The Spitzer data were binned on equal intervals of 1 minute, so we use numpy
fast Fourier transforms (FFT) to detect evidence for transient periodicity.
However, these algorithms have neither error-weighting or significance-testing
built in. We discuss the approach to each of these topics below.

To account for the photometric scatter in Spitzer’s light curve, we complete a
Monte Carlo process to “error weight” the power spectra. To do this, we
randomly resample each of the observed points within their (normally
distributed) flux uncertainties and redo the FFT on the “new” light curve.
Completing this process 35,000 times allows us to probe a probability of $\sim
1/35000\approx 4\sigma$; we report the mean and $1\sigma$ CI power spectra as
the error-weighted power spectrum.

To determine if any peaks in the power spectrum are significant, we follow the
prescription in Do et al. (2009) to compare the measured FFT to a red-noise-
dominated power spectrum. We generate 35,000 red-noise light curves (using the
stochastic444https://stochastic.readthedocs.io/en/stable/index.html Python
package) with a spectral density exponent $\alpha=2.75$ (PSD $\propto
f^{-2.75}$; Weldon et al., 2023). These red-noise light curves have the same
temporal sampling as our light curves. We scale each of the generated light
curves to have the same mean and standard deviation as the Sgr A* light curve.
To account for measurement noise in the light curve, we re-sample each “flux”
point with the corresponding uncertainty in the Spitzer data. Finally, we
calculate the power spectra for each generated light curve and report the
$3\sigma$ and $4\sigma$ upper limits on the power. We consider a statistically
significant detection in the periodogram when Sgr A*’s mean spectral power is
greater than the $4\sigma$ red-noise spectrum. In Figure 8, we show the power
spectra for Sgr A* during all three days using all data (outside of the low-
SNR ranges) compared to the generated red noise light curves.

On 21 July 2019, we find a significant peak at a $\sim 35$ minute period with
a similar (but weaker) peak at $\sim 40$ minutes on 18 July 2019. These
timescales are similar to Spitzer’s quasi-periodic pointing wobble with a
period of $\sim 40$ minutes, which would cause photometric amplitude
variations of a couple of percent for a point source (see, e.g., Grillmair et
al., 2012, 2014, for more complete descriptions of the wobble). This effect
was present for earlier observations of Sgr A* but was largely corrected,
mitigating the resulting photometric response. Figures 8 and 2 in Hora et al.
(2014) show the photometric changes caused by the wobble and the light curve
of Sgr A* after correction, respectively.

There are several key reasons why we trust the validity of this transient
periodicity. First, the statistically significant signal only occurs in one of
the three days, specifically on the same observation where an identical peaked
structure is observed in the ALMA light curves. Second, a more comprehensive
catalog of Spitzer light curves was published in Witzel et al. (2021), where
no quasi-periodicity was detected. A wobble with a sustained period would have
been detected in their analysis at higher significance since they used Spitzer
data beginning in late 2013. Additionally, near the end of Spitzer’s mission,
the heating duty cycle was changed because it was in a different thermal state
where temperature cycling did not occur, rendering this pointing wobble
absent. Finally, in the absence the FFT results, the cross correlation plots
between ALMA and Spitzer (Figure 6b) and their light curves (Figures 6d and
7a,b) still show a similar $35$ minute timescale. Therefore, the sequential
peaks observed in Spitzer on the 21st may be suggestive of several
explanations, such as footprints of flux tubes within the accretion flow
(i.e., Porth et al., 2021) or short-lived, orbiting, transient hotspot within
the accretion flow. We discuss the latter model further in Section 5.1.1.

###  3.2 26 July 2019

Figure 9: Same as Figure 6, but for the 26 July 2019 observations. (a): Cross
correlation histogram across the VLA bandwidth. (b): Cross correlation between
the most widely separated VLA and ALMA frequencies. (c): Cross correlation
across the ALMA bandwidth. (d): Cross correlation between the ALMA and Spitzer
light curves. (e): Stacked Spitzer , ALMA, and VLA light curves centered on
the dominant observed flare. The vertical black dashed line corresponds to the
peak IR flux. The gray dash-dotted lines denote times of IR flares where there
are possible dips in the radio light curve; we do not study these unconfirmed
events here. The 1-minute and smoothed 5-minute Spitzer data are displayed by
dots and the solid green line, respectively.

We use the same RMSE-based analysis in Section 3.1 to detect flares in the HST
data on this day (Figure 3b). Aside from two points ($t\approx 11.95,14.5$ UTC
hrs), the variations in the 1.53 µm light curves are within $5\sigma$ RMSE of
the mean flux during each orbit. These data points are below the mean and do
not correspond to variations in the Spitzer light curves. Therefore, we find
no evidence for flaring events at $1.53$ µm. These non-detections allow us to
place an upper limit on the IR flare spectral index using Spitzer and HST.
Orbits 5 and 6 on this day (Figure 3b) overlap with moderately weak Spitzer
flares of $\sim 5$ mJy at 4.5 µm. The upper limits of $1$ mJy from HST (i.e.,
$\sim 5\times$RMSE per orbit) imply a spectral index of $\alpha\lesssim-1.5$
between 1.53 and 4.5 µm. This value is steeper than the canonical
$\alpha=-0.7$ found between 1.6 and 2.2 µm (Bremer et al., 2011), but
comparable to the spectral index between 1.6 and 4.5 µm at the beginning of
the bright flare observed on 18 July 2019 ($\alpha\sim-1.45$; GRAVITY
Collaboration et al., 2021).

We detect no X-ray flaring events in the 2-8 keV energy range from the
Bayesian blocks analysis (Figure 5b). To check for lower significance flares,
we test values of $p_{0}\in\left[0.05,0.95\right]$ in increments of $0.05$;
however, none were detected. This places an upper limit of $1.45$ flares day-1
and matches recent published ranges ($\sim 1.1-1.6$ flares day-1; Neilsen et
al., 2013; Ponti et al., 2015; Yuan & Wang, 2016; Mossoux & Grosso, 2017;
Bouffard et al., 2019).

The Spitzer light curves show a great deal of variability, particularly during
the latter half of the observation (Figure 4b). However, since the radio and
submm light curves occur during the first half, we do not focus on these later
variations. A flare event in the radio and submm is clearly present (Figures 1
and 2b, respectively).

####  3.2.1 Cross Correlations

We again use pyCCF to check for time-delayed emission in four different
frequency pairs: radio-radio, radio-submm, submm-submm, and submm-mid-IR. The
histograms for the cross-correlations are shown in Figure 9a-d. We do not find
evidence for a time delay across the radio and submm frequency bandwidths.
However, the lowest frequency radio emission lags $\delta
t=22.45^{+3.92}_{-4.16}$ minutes behind the highest frequency submm light
curve. We further find the submm peak delayed $\delta t=9.02^{+4.98}_{-5.82}$
minutes behind the mid-IR peak. In Figure 9e, we present the stacked mid-IR,
submm, and radio light curves, which clearly show the time delay evolution of
the flare.

We examined the lack of time delay across the ALMA bandwidth in Section 3.1.1.
However, using the values in Iwata et al. (2020) would suggest a $\sim 9$
minute time delay across the VLA bandwidth, which we do not detect (Figure
9a), but should be measurable. Time delays across a single VLA band have been
previously detected, such as the 20-minute lag between 8 and 10 GHz found by
Michail et al. (2021b). Recalculating the expected time delay using Equation 2
from Iwata et al. (2020) with the measured 22.5-minute lag between 31.45 GHz
and 347.87 GHz (Figure 9b) yields a theoretical $\sim 3$ minute delay across
the VLA bandwidth (assuming $p$ ranges from $1-3$). We argue that the absent
radio time delay is caused by observational effects in the VLA data rather
than a breakdown in the adiabatically-expanding hotspot picture. In Section
2.1, we note the numerous weather- and setup-related issues that plagued the
VLA observations; these will tend to increase the scatter (noise) in the light
curves. Since a 3-minute time delay relies on only three points per light
curve (60-second averaging), this, compounded with the increased scatter,
restricts our ability to detect delayed emission. The $2\sigma$ scatter of the
VLA’s cross correlation histogram (Figure 9a) matches the expected 3-minute
delay, implying the lag may be hidden in the noise.

####  3.2.2 Additional Light Curve Features

We note the presence of two additional features (marked with gray dash-dotted
lines in Figure 9e) in the multiwavelength light curves aside from the
prominent flare in Figure 9. Near $4.1$ hr UTC, an IR flare occurs, coincident
with a slight ($\sim 100$ mJy) increase in the submm flux but a drop ($\sim
50$ mJy) in the radio flux. It is unclear if the radio feature indicates an
occultation-like event (where the flare hides a portion of the accretion flow
as seen from the Earth) or an issue with the amplitude scaling between
observing blocks.

A tertiary feature appears in the radio light curve near $7.25$ hr UTC,
coincident with an increase in the IR. It is unclear if this is another
occultation-like event or the beginning of an additional flare. The lack of
submm coverage, the proximity of the IR feature to the beginning of a low-SNR
part of the light curve (gray band), and not observing the radio peak all blur
its final classification. Because of these issues, we do not further consider
this feature in our discussion.

##  4 Models of Observed Events in the Light Curves

In this Section, we aim to model the submm flux decrease event observed on 21
July 2019 (Figure 6) and the time-delayed flare detected in the IR, submm, and
radio regimes on 26 July 2019 (Figure 9). While several interpretations aim to
describe the anti-correlation between submm and IR/X-ray light curves, we will
focus on the occultation model. Throughout the descriptions below, we assume
that Sgr A* is comprised of two components, namely the variable and background
(quiescent) emission. The variable component generates the flares, while the
quiescent component is responsible for the non-zero flux offset and low-
amplitude, long-timescale variability.

Figure 10: The two possible setups for an occultation to occur. The accretion
flow (orange) is shown surrounding Sgr A* (black circle). $S_{q}(\nu)$ denotes
the accretion disk flux at frequency $\nu$. The size of the adiabatically-
expanding hotspot at several times is shown in gray. The resulting light
curves, as seen by an observer who is face-on to the page, are shown in the
diagrams on the right. Top: A concentric occultation occurs when the hotspot
is fully embedded within the accretion flow. Bottom: An offset occultation is
shown, where the hotspot does not begin to cover the accretion flow until some
time later. Both a flare peak (number 2) and dip (number 4) occur.

###  4.1 Occultation of the Accretion Flow by Adiabatically-Expanding
Synchrotron Hotspots

In Figure 6, we document another instance of decreasing submm flux during an
IR or X-ray flare. Yusef-Zadeh et al. (2010) models this anti-correlation as a
geometric occultation of Sgr A*’s accretion disk by a flare that is optically
thick in the submm. The flaring region is in front of the accretion disk seen
from the Earth, eclipsing a fraction of its projected area on the sky, causing
a dip in the submm light curve. The general setup for this model is depicted
in the top panel of Figure 10. The initial description of the model only
accounted for a specific electron power-law index; here, we generalize their
model for all values.

####  4.1.1 Hotspot Equations

We assume the flare is produced by self-absorbed synchrotron emission from an
electron spectrum $n(E)~{}dE\propto E^{-p}~{}dE$. We further assume the
hotspot is a homogeneous sphere with an initial radius $R$ and contains an
initial magnetic field strength $B$. For a power-law electron spectrum, the
synchrotron source function, $J_{\nu}$, is

$\displaystyle
J_{\nu}=\dfrac{\sqrt{2}}{(p+1)}\frac{\Gamma\left(\frac{3p-1}{12}\right)\Gamma\left(\frac{3p+19}{12}\right)}{\Gamma\left(\frac{3p+2}{12}\right)\Gamma\left(\frac{3p+22}{12}\right)}\dfrac{\nu^{2}}{c^{2}}E_{\nu},$
(2)

where $E_{\nu}$ is the characteristic electron energy

$\displaystyle E_{\nu}=\left(\dfrac{4\pi
m_{e}c\nu}{3Bq}\right)^{1/2}m_{e}c^{2}.$ (3)

$m_{e}$, $c$, and $q$ represent the electron mass, speed of light, and
(absolute) electron charge, respectively. The hotspot of radius $R$ spans a
solid angle $\Omega=\pi\left(R/d\right)^{2}$, where $d$ is the distance to Sgr
A*.

We define $\tau_{\nu}$, the frequency-dependent optical depth, as
$\tau_{\nu}=(4/3)\kappa_{\nu}R$. The $4/3$ prefactor accounts for the
“effective” optical depth of all lines of sight through the plasma. This is
different than the canonical $\tau_{\nu}=2\kappa_{\nu}R$, which is only true
for the optical depth through the plasma’s center. $\kappa_{\nu}$ is the
synchrotron absorption coefficient,

$\displaystyle\kappa_{\nu}=\dfrac{\sqrt{3}Bq^{3}}{4\pi
m_{e}\nu^{2}}\left(2^{p/2-1}\Gamma\left(\dfrac{3p+2}{12}\right)\Gamma\left(\dfrac{3p+22}{12}\right)\right)~{}n(E).$
(4)

We have set the pitch angles to $90^{\circ}$ in Equations 3 and 4. To adopt a
fixed ratio between the magnetic field energy density ($U_{B}=B^{2}/8\pi$) and
the electron energy density ($U_{e}=\int_{E_{1}}^{E_{2}}En(E)~{}dE$), we
define the electron number density between initial energies $E_{1}$ and
$E_{2}$ as

$\displaystyle n(E)=\dfrac{B^{2}}{8\pi\varphi
E_{0}^{2}}\left(\dfrac{E}{E_{0}}\right)^{-p}.$ (5)

$\varphi$ is the equipartition factor and controls the ratio of the energy
densities between the magnetic and particle fields. We derive $\varphi$ and
the prefactor for Equation 5 in Appendix A. As the occulting plasma expands,
the power-law bounds scale inversely with radius (i.e.,
$E_{1}^{\prime}=E_{1}(R/R_{0})^{-1}$, van der Laan, 1966), caused by adiabatic
cooling of the ultra-relativistic electron gas. For this analysis, we assume
$E_{0}=E_{1}=1$ MeV and $E_{2}=1$ GeV. We note the special constraint for the
energy bounds where $E\nless 0.5$ MeV (electron rest mass), which we account
for in our analysis.

The measured flux of the hotspot at any radius and frequency:

$S_{\nu}(R)=\Omega J_{\nu}\left(1-e^{-\tau_{\nu}}\right).$ (6)

To convert between radius and UTC time, we assume a linear expansion of the
hotspot with expansion speed $v_{\text{exp}}$:

$\dfrac{R}{R_{0}}=1+v_{\text{exp}}(t-t_{0}),$ (7)

where $t_{0}$ is the UTC time when $R=R_{0}$.

####  4.1.2 Quiescent Emission Model

In this geometric model, the fraction of the projected accretion flow’s area
occulted by the hotspot is denoted $f$. From previous analyses, the intrinsic
area of the accretion flow is modeled by a frequency-dependent power-law,
which we define as $A_{q}(\nu)$. We use the results of Cho et al. (2022) to
find $A_{q}(\nu)=6.1(\nu/300\text{~{}GHz})^{-2.4}~{}R_{\text{S}}^{2}$ assuming
a Gaussian profile having area $A_{q}(\nu)=\left(2\pi/8\ln
2\right)\theta_{\text{major}}(\nu)\theta_{\text{minor}}(\nu)$. The choice of
$A_{q}$ will determine if there is an explicit inclination term. The Cho et
al. (2022) analysis gave the intrinsic size of Sgr A* projected on the sky,
removing the need for a specific inclination term. Placing the hotspot at the
center of the accretion flow lets us define the ratio $f\equiv\pi R^{2}/A_{q}$
if $R\leq\sqrt{A_{q}/\pi}$ and $f=1$ otherwise, measuring the fraction of the
accretion flow eclipsed by the hotspot. We note Cho et al. (2022) use only one
epoch of data to determine the frequency-dependent size of Sgr A*. Cheng et
al. (2023) observed Sgr A* over 10 months, finding a variable intrinsic area
of $\approx 13\%$, while the mean was roughly $10-20\%$ higher than the Cho et
al. (2022) single-epoch results. However, this difference would not
substantially affect our results.

In addition to the intrinsic size, the accretion flow has a characteristic
flux at frequency $\nu$ denoted by $S_{q}(\nu)$. This model could be further
generalized to include a time-dependent term, but that is outside the scope of
this analysis. We assume the accretion flow is uniformly smooth across the
entire projected area.

####  4.1.3 Total Intensity Light Curve

There are three sources of flux present in this model: the unobscured and
obscured accretion flow and the hotspot. The unobscured flow has flux
$\left(1-f\right)S_{q}$ and the hotspot $\Omega
J_{\nu}\left(1-e^{-\tau_{\nu}}\right)$. The obscured flow flux passes through
the optical depth produced from the hotspot yielding a flux
$fS_{q}e^{-\tau_{\nu}}$. Altogether, the net sum of these terms yields a total
observed flux of

$\displaystyle S(\nu)$ $\displaystyle=S_{q}(\nu)+\left(\Omega
J_{\nu}-fS_{q}(\nu)\right)\left(1-e^{-\tau_{\nu}}\right).$ (8)

We note the parameters
$B,\Omega,J_{\nu},f,\tau_{\nu},\varphi,E_{1},\rm{and}~{}E_{2}$ all vary with
radius, and, therefore, time (Equation 7). When modeling this time-dependent
nature of the occultation, the secular changes in these parameters must be
accounted for.

###  4.2 Flaring Emission Produced by Adiabatically-Expanding Synchrotron
Hotspots

We model the multiwavelength flare using an adiabatically-expanding
synchrotron hotspot as described by van der Laan (1966) and Yusef-Zadeh et al.
(2006a). To characterize both the quiescent and variable emission, we use a
joint fit of accounting for secular variations in the accretion flow and the
time-dependent nature of the flaring emission.

####  4.2.1 Flaring Emission Model

The variable model is characterized by four parameters: $S_{0}$, $t_{0}$, $p$,
and $v_{\rm{exp}}$. $S_{0}$ is the peak flux density at frequency $\nu_{0}$,
which is produced by an electron spectrum $n(E)~{}dE\propto E^{-p}~{}dE$
between energies $E_{\rm{min}}$ and $E_{\rm{max}}$. The flaring region expands
at a (normalized) velocity $v_{\rm{exp}}$. To convert between UTC time and
normalized radius, we use Equation 7.

The flux density of the flare at any radius and frequency $\nu$ is given by

$\displaystyle
S_{\nu}(t)=S_{0}\left(\dfrac{\nu}{\nu_{0}}\right)^{2.5}\left(\dfrac{R}{R_{0}}\right)^{3}\dfrac{1-e^{-\tau_{\nu}}}{1-e^{-\tau_{0}}},$
(9)

where $\tau_{\nu}$ and $\tau_{0}$ are the frequency/radius-dependent and
critical optical depths, respectively. For a given $p$, the critical optical
depth can be determined from the following transcendental relationship:

$\displaystyle e^{\tau_{0}}-\left(\dfrac{2p}{3}+1\right)\tau_{0}-1=0.$ (10)

$\tau_{\nu}$ at any frequency and radius is determined via

$\displaystyle\tau_{\nu}=\tau_{0}\left(\dfrac{R}{R_{0}}\right)^{-(2p+3)}\left(\dfrac{\nu}{\nu_{0}}\right)^{-(p+4)/2}.$
(11)

####  4.2.2 Quiescent Emission Model

To model time-dependent secular and frequency-dependent variations in the
accretion flow, we use a linear model of the form

$\displaystyle I_{q}(\nu,t)=\left(I_{0}^{\nu}+\delta
I_{t}^{\nu}\left(t-t_{0}\right)\right)\left(\dfrac{\nu}{\nu_{0}}\right)^{\alpha_{\nu}}.$
(12)

Here, $I_{0}^{\nu}$ is the quiescent emission flux density at frequency
$\nu_{0}$ at time $t_{0}$, $\delta I_{t}^{\nu}$ is the time-slope of the
quiescent flux, and $\alpha_{\nu}$ is the quiescent emission’s spectral index.

##  5 Discussion

In this Section, we consider the periodic signal detected in the IR and hinted
at in the submm in the context of previous analyses. Using the frameworks
described in Section 4, we then model the occultation event and the detected
multiwavelength flare. Finally, we discuss a caveat in our modeling and
quantify how it affects our results.

###  5.1 21 July 2019

####  5.1.1 Transient Periodicity

Figure 11: Radius (relative to the innermost circular stable orbit; ISCO) of a
35-minute orbital period as a function of the normalized spin parameter
($a_{\ast}$) for Sgr A* (blue). The gray shaded region denotes the range of
normalized spins from dynamical considerations (Fragione & Loeb, 2020, 2022).
The red line shows the rough lower limit of Sgr A*’s spin comparing GRMHD
simulations with event-horizon scale imaging (Event Horizon Telescope
Collaboration et al., 2022a, b).

The transient $\sim 35$-minute peak (Figure 8) corresponds to similar
timescale detections in near-IR (Gravity Collaboration et al., 2018) and submm
(Iwata et al., 2020) analyses of Sgr A*’s light curves. Gravity Collaboration
et al. (2018) observed a changing polarization angle consistent with an
orbiting hotspot having period $P=40\pm 8$ minutes. Similarly, Iwata et al.
(2020) used several epochs of ALMA light curves ($230$ GHz) and found $\sim
35$-minute quasi-periodic variations at $>5\sigma$ statistical significance.
In the context of a non-spinning Schwarzschild black hole ($a_{\ast}=0$),
these results are consistent with variability occurring near the ISCO of Sgr
A*. However, there is tension between constraints in Sgr A*’s spin. Fragione &
Loeb (2020, 2022) argued for a low spin ($\left|a_{\ast}\right|\lesssim 0.1$)
using the dynamics of the S-star population orbiting Sgr A*. The S-stars
reside in two orthogonal disks centered on Sgr A* (Ali et al., 2020); on
timescales longer than $\sim 1$ Myr, the Lense-Thirring Effect should have
randomized these orbits if $|a_{\ast}|\approx 0.5$. This is in stark contrast
to Event Horizon Telescope Collaboration et al. (2022a, b), who preferred
$a_{\ast}\gtrsim 0.5$ when comparing event-horizon scale images to general
relativistic magnetohydrodynamic (GRMHD) simulations of Sgr A*.

Due to this discrepancy, we cannot definitively determine the orbital radius
where a $\sim 35$ minute periodicity would originate. Instead, we calculate
the orbital radius corresponding to a $35$-minute period for the full range of
$a_{\ast}$ and show it in Figure 11. We use the period-radius relationship
given in Gravity Collaboration et al. (2018, Appendix B):

$P~{}\left[\rm{min}\right]=2.137~{}\left(\dfrac{M}{4.14\times
10^{6}~{}M_{\odot}}\right)~{}\left(a_{\ast}+\left(\dfrac{2R}{R_{\rm{S}}}\right)^{3/2}\right).$
(13)

The ISCO radius is given by

$\displaystyle R_{\rm{ISCO}}$
$\displaystyle=\dfrac{R_{\rm{S}}}{2}\left(3+z_{2}\mp\sqrt{\left(3-z_{1}\right)\left(3+z_{1}+2z_{2}\right)}\right);$
(14a) $\displaystyle z_{1}$
$\displaystyle=1+\sqrt[3]{1-a_{\ast}^{2}}\left(\sqrt[3]{1+a_{\ast}}+\sqrt[3]{1-a_{\ast}}\right),$
(14b) $\displaystyle z_{2}$ $\displaystyle=\sqrt{3a_{\ast}^{2}+z_{1}^{2}},$
(14c)

where $\mp$ correspond to prograde and retrograde orbits, respectively
(Bardeen et al., 1972). If this quasi-periodicity is caused by orbital motion,
the hotspot is on a stable orbit ($R\geq R_{\rm{ISCO}}$) given
$a_{\ast}\gtrsim-0.15$. Viscous timescales are unlikely to be the cause as the
accretion flow would require a scale height (H/R) $\sim 2.2$ for a 35-minute
viscous timescale at $3~{}R_{\rm{S}}$ (see Dexter et al., 2014).

Several non-orbital causes may be the origin of this structure. Boyce et al.
(2022) noted this periodic structure in their analysis of the Spitzer data but
argued it may be formed in processes that can be modeled by correlated red
noise. Witzel et al. (2012) and Subroweit et al. (2017) found Sgr A*’s submm
variability is also consistent with red noise. If this periodic feature was a
statistical product of Sgr A*’s red noise fluctuations, observing it in the
submm and IR strongly indicates the variability at these two frequency regimes
originates from a single electron population.

However, physical causes are shown in GRMHD modeling. Ripperda et al. (2020)
found magnetic reconnection within magnetically-arrested disks (MAD), such as
those around Sgr A*, can form “plasmoid” hotspots at periodic intervals, and
the period of their formation depends on the reconnection rate. Such plasmoids
undergo physical expansion, causing a decrease in their average internal
magnetic field strength (e.g., Nathanail et al., 2020). Additionally, Porth et
al. (2021) and Chatterjee & Narayan (2022) studied orbiting flux tubes,
regions within the accretion flow with a decreased magnetic field strength
compared to the local environment, which could also produce periodic
signatures in light curves.

Plasmoids are similar to adiabatically-expanding hotspots, which describe
time-delayed emission at different frequencies as the change in optical depth
as the hotspot expands. In Figure 7, we detect a $\sim 5$ minute time delay
detected between the Spitzer and ALMA light curves. Taken together, this
feature may have been formed by a series of magnetic reconnections within the
accretion flow, whose subsequent dynamic expansion caused time-delayed
emission between these two frequency regimes. However, plasmoids cause dips in
the submm flux from depleted magnetization in the accretion flow rather than a
flare occulting it (see Section 5.1.2 below). If such events are recorded at
different wavelengths, the predicted multiwavelength signatures may be able to
distinguish between the two models.

####  5.1.2 Occultation Modeling

Figure 12: (a): Corner plot showing the physical and observable parameters of
the constrained models. The diagonal plots are histograms of the four
parameters. We denote the 16th, 50th, and 84th quantiles and are labeled at
the top of each plot. The off-diagonal plots are the two-dimensional
histograms between the pairs of parameters. (b): The range of models
consistent with the occultation. The black line shows the median model flux
during the event, while the shaded region shows the total modeled flux range.

Assuming that there was an IR flare (Figure 9), Equations 2–5 could be
manipulated to solve for the equipartition magnetic field strength, allowing
us to directly fit the submm light curve with the occultation model. However,
the Spitzer data during the period of interest is low SNR, and no discernible
flare was detected. While an X-ray flare was detected, the exact mechanism
producing this emission is still an open question (see, for example, Markoff
et al., 2001; Yusef-Zadeh et al., 2006c; Dodds-Eden et al., 2009; Ponti et
al., 2017; GRAVITY Collaboration et al., 2021). Therefore, we cannot
accurately extrapolate the IR flaring flux. Additionally, there is an
insufficient range in bandwidth for us to use all four ALMA light curves to
directly fit the data.

Instead, we create a grid of initial physical conditions to determine the IR
and submm properties of the plasma. We make grid of $500^{3}$ realizations
linearly sampling the electron energy index ($p\in\left[0.34,5\right]$), the
magnetic field strength ($B\in\left[0.1,150\right]$ Gauss), and the initial
radius ($R\in\left[0.01,10\right]~{}\rm{R}_{\rm{S}}$). For each model, we
calculate the optically thin IR flux ($S_{\text{IR}}=\Omega
J_{\nu}\tau_{\nu}$) and the (absolute) maximum change in the flux at 334 GHz
flux (Equation 8). For this calculation, we assume $S_{q}=4.48$ Jy, which is
the average flux before the occultation (Figure 6d). Determining the maximum
change in the submm flux is analytically complicated as most of the parameters
vary with the radius of the plasma. Therefore, we numerically simulate the
light curve of each realization by varying the plasma size between
$1-10\times$ its initial value and determine the radius where the (absolute)
maximum submm flux change occurs.

We limit the possible models using several constraints. First, we require the
size of the occulting plasma at the maximum change to be larger than its
initial size; this constraint stems from the non-zero time delay between the
X-ray flare and the submm dip (Figure 6). Next, we require the change in the
submm flux to be negative. Note the submm flux may increase or decrease
depending on the model conditions. An increase in the submm flux occurs when
the plasma’s brightness temperature is greater than the accretion flow’s,
producing a normal submm flare (Section 4.2). From Figure 7, we calculate a
flux drop of $\sim 610$ mJy at 334 GHz but consider models where the flux
difference is within $20\%$ of this value ($488-732$ mJy) to account for
scatter within the ALMA light curve.

While we do not detect the IR flare, we can place statistical upper- and lower
limits on its flux. As an upper limit, we calculate $5\times$RMSE (RMSE $\sim
3.2$ mJy) of the Spitzer data during the low SNR period (about 1-2.1 hr UTC).
The corresponding IR flare hidden within the noise might increase the RMSE,
affecting our upper limit. However, since we do not attempt a true fit to the
data, we do not revise this upper limit. The lower limit comes from Witzel et
al. (2021), who completed a joint-variability analysis of Sgr A* from submm to
X-ray frequencies. They determine each Spitzer flux measurement has $\sim
0.65$ mJy measurement noise. Since Spitzer was unlikely to measure flares
dimmer than this, we use $5\times$ this value as our lower limit on the IR
flux.

We find only $\sim 0.02\%~{}(27533)$ of our initial conditions are consistent
with the data. A corner plot of the initial conditions is shown in Figure 12.
While we cannot directly fit the light curves, we use the realizations
consistent with the data to determine the statistical properties of the flare.
We find the drop is consistent with a plasma of initial radius $0.55$ RS
threaded by a uniform $\sim 130$ Gauss magnetic field expanding at $\sim
0.6\%$ of the speed of light. This emission is produced by an electron
spectrum with index $p\sim 2.8$, and we estimate it yields a peak $4.5~{}\mu$m
flare flux $\sim 8$ mJy.

The “pseudo-posterior” magnetic field distribution (histograms along the
diagonal) is roughly flat, suggesting that we have not probed the best field
strengths consistent with the data. This motivates us to test additional
models, where we set an unrealistic maximum strength of $500$ Gauss. Even at
these strengths, the distribution of values consistent with the data is still
flat (not shown). The average magnetic field strength probed by this initial
analysis is stronger than typical values found in observation (e.g., $\sim
10-100$ G; Ponti et al., 2017; GRAVITY Collaboration et al., 2021; Michail et
al., 2021a, b; Witzel et al., 2021) and MHD simulations (e.g., $\sim 100$ G;
Ressler et al., 2020a). However, we note Ressler et al. (2020b) have shown
that average magnetic field strengths above $200$ Gauss are possible near the
event horizon (according to full-GRMHD modeling).

In the context of the given observation, a strong magnetic field and flat
distribution is not surprising. Occultations occur when the brightness
temperature of the responsible hotspot is lower than that of the accretion
flow. The brightness temperature for synchrotron emission scales as
$T_{B}\propto J_{\nu}/\left(\Omega\nu^{2}\right)$, where $J_{\nu}$ is the
synchrotron source function. $J_{\nu}$ itself scales as
$J_{\nu}\propto\nu^{2.5}/\sqrt{B}$ (Equation 2); therefore,
$T_{B}\propto\sqrt{\nu/B}$. This necessarily requires high magnetic field
strengths to balance the value of $\nu$ at submm frequencies. Additionally,
the flat distribution of magnetic field strengths is an artifact of degeneracy
since we only used one frequency to model the occultation. For one given
frequency, there are an infinite number of magnetic field strengths that will
produce a hotspot with a brightness temperature lower than the surrounding
accretion flow. Observing the event at a different band constrains the
required magnetic field strength to produce this feature at multiple
frequencies, where the accretion flow’s brightness temperature is also
different.

Additionally, the elevated magnetic field strength may be caused by our choice
of the electron energy spectrum’s lower energy bound, which we assume is $1$
MeV. We find a total relativistic electron density of $\sim 1.9\times 10^{8}$
cm-3 between 1-1000 MeV for our parameters. This $\sim 40\times$ higher than
the electron density found by GRAVITY Collaboration et al. (2021) during the
flare on 18 July 2019 but similar to the Boyce et al. (2022) value when
considering the same flare undergoing adiabatic expansion. We assume magnetic
equipartition with the electron energy density while modeling the flare (see
Equation 5 and Appendix A). Electrons responsible for the majority of
synchrotron emission at frequency $\nu$ and magnetic field strength $B$ have
Lorentz factors near:

$\gamma=\sqrt{~{}3578\left(\dfrac{\nu}{\text{GHz}}\right)\left(\dfrac{\text{Gauss}}{B}\right)}$
(15)

(Condon & Ransom, 2016). For 334 GHz and 130 Gauss, $\gamma\sim 100$
corresponds to an electron energy $\sim 50$ MeV. If we grossly underestimate
the lower energy bound of the electron spectrum, this overestimates the
electron energy density and the magnetic field strength. By way of example,
changing the lower energy bound to $\gamma=10$ ($5$ MeV) for an electron
spectrum with $p=2.84$ decreases the electron energy density by a factor $\sim
4$, reducing the magnetic field strength by a factor $\sim 2$
($B\propto\sqrt{U_{e}}$).

Obtaining at least one light curve in the radio would constrain the lower
energy bound of the electron spectrum ($E_{1}$). The dip in the radio flux is
lessened (since the intrinsic accretion flow size increases at lower
frequencies, decreasing $f$; Cho et al., 2022). However, the time delay of the
dip relative to submm depends on the critical optical depth of the plasma,
which is solely a function of $p$ (Equation 11). If the predicted time delay
at radio frequencies did not match the value extrapolated from the submm, it
could suggest curvature in the electron spectrum between energies probed at
the two frequencies, changing the intrinsic optical depth.

####  5.1.3 Signatures of Occulting Hotspots Offset from Accretion Flow

Our definition of $f$ implies the occulting hotspot is fully embedded in the
accretion flow at any frequency, which is a central assumption in our model
and affects the physical parameters. In Figure 10, we show both possibilities
where the flare is fully and partially embedded in the accretion flow. In the
concentric case, the total observed flux of the system is $S_{q}(\nu)$ (number
1) before the flare forms. As the plasma expands, it is optically thick and
occults part of the accretion flow (number 2) until it reaches the critical
optical depth (number 3). The continued expansion causes the emission to
become optically thin, lessening the extinction the accretion flow undergoes
(number 4) until only the flow remains (number 5).

In the offset model (bottom panels), an interesting feature occurs. As in the
previous model, the total emission measured is the accretion flow flux (number
1). However, as the plasma begins to expand, it does not occult any part of
the flow, raising the overall flux of the system (number 2). Eventually, the
hotspot’s size is large enough, and its optical depth is sufficient to begin
shadowing the accretion flow (number 3). As in the previous case, the
continuing expansion will cause a dip in the light curve (number 4) and
eventually will fade away until only the accretion flow remains (number 5). We
note such offset flares require a precise combination of distance from the
accretion flow and optical depth for the occultation feature to occur.
Additionally, if the hotspot’s distance from the accretion flow is
sufficiently far, the hotspot becomes optically thin before it causes an
occultation, rendering only a flare in the light curve. Therefore, the offset
also explains the lack of occultation events during every flaring event.

The frequency-dependent size of Sgr A* may also play a role in determining
which occultation profile occurs. At radio frequencies, the intrinsic size of
this source is larger than the reported hotspot sizes. Cho et al. (2022) found
Sgr A*’s intrinsic extent to be $\sim 30~{}R_{\rm{S}}\times 23~{}R_{\rm{S}}$
at 43 GHz. Using the relationship given in their paper, the 334 GHz size of
Sgr A* is $\sim 2.3~{}R_{\rm{S}}\times 1.7~{}R_{\rm{S}}$, comparable to the
hotspot sizes. The Gaussian-process-interpolated ALMA light curve (Figure 7)
suggests this offset profile was detected on 21 July 2019. Marrone et al.
(2008) also may have detected a similar case on 17 July 2006 during
simultaneous X-ray and submm observations (see their Figure 3). They saw
increases in the $850~{}\mu$m and $1.3$ mm light curves at the beginning of an
X-ray flare, then reached a minimum flux near its peak. Future multiwavelength
campaigns should observe complementary radio frequencies to test the
frequency-dependent nature of this occultation model.

###  5.2 26 July 2019

Figure 13: The joint radio-submm adiabatic-expansion hotspot fits. The red
points are the observed emission, while the black lines are the modeled light
curves during the fitted ranges. The corresponding fit parameter values are
given in Table 1.

Using the adiabatic-cooling model described in Section 4.2, we jointly fit the
radio and submm light curves shown in Figure 9. Modeling the IR light curve is
complicated as synchrotron cooling dominates adiabatic cooling at these high
frequencies. For example, the synchrotron cooling timescale at $4.5~{}\mu$m in
a 50 G field is $\sim 6$ minutes, while the adiabatic cooling is at
$\sim$hourly timescales. Attempting to fit all three frequency light curves
simultaneously requires a more substantial model to account for both adiabatic
and synchrotron cooling, which is beyond the scope of this analysis.
Therefore, we do not model the IR light curve, as the modeled parameters will
have little physical meaning. Instead, they will be over-biased such that the
“best-fit” adiabatic expansion picture mimics a synchrotron cooling profile.

The VLA and ALMA light curves have sufficient spectral windows to
simultaneously fit the flare and quiescent emission. The flaring emission is
fit using a single set of hotspot parameters. On the other hand, we use two
different sets of quiescent parameters to independently model the radio and
submm emission, which is necessary for two reasons. First, the quiescent model
includes a phenomenological parameter to model the time-slope of the accretion
flow at each frequency range. As each band probes different radii within the
accretion flow (Cho et al., 2022), there is no guarantee these values will be
equivalent. Second, Sgr A*’s spectral index flattens from radio to submm
frequencies (i.e., Bower et al., 2015), which would require fitting three
spectral indices (intra-radio, intra-submm, and radio-submm). By fitting one
quiescent model per frequency band, we can determine the intra-radio and
intra-submm indices and then infer the one from radio to submm using the
quiescent flux densities at each reference frequency (i.e., $I_{0}^{\nu}$,
Equation 12). The result of this joint fit is shown in Figure 13, and the
parameter values are given in Table 1.

Table 1: Fitted parameters for the joint submm and radio light curve modeling shown in Figure 13. Quoted errors are given at $1\sigma$ significance. Parameter | Description | Unit | Value
---|---|---|---
Hotspot Parameters
$S_{0}$ | Peak Flare Flux at 334 GHz | mJy | $686\pm 6$
$t_{0}$ | Flare UTC Peak Time at 334 GHz | hr UTC | $4.58\pm 0.01$
$p$ | Electron Spectrum Index | – | $0.59\pm 0.01$
$v_{\text{exp}}$ | Normalized Radial Expansion Speed | hr-1 | $5.36\pm 0.05$
Radio Quiescent Parameters
$I_{0}^{\rm{radio}}$ | Quiescent Flux at 37.54 GHz at $t=t_{0}$ | mJy | $1305.6\pm 0.6$
$\delta I_{t}^{\rm{radio}}$ | Quiescent Time-Slope at 37.54 GHz | mJy hr-1 | $-12.2\pm 0.3$
$\alpha_{\rm{radio}}$ | Ka-band Quiescent Spectral Index | – | $0.074\pm 0.003$
Submm Quiescent Parameters
$I_{0}^{\rm{submm}}$ | Quiescent Flux at 334 GHz at $t=t_{0}$ | mJy | $3083\pm 2$
$\delta I_{t}^{\rm{submm}}$ | Quiescent Time-Slope at 334 GHz | mJy hr-1 | $-79.9\pm 0.1$
$\alpha_{\rm{submm}}$ | Band 7 Quiescent Spectral Index | – | $-0.099\pm 0.023$

####  5.2.1 Quiescent Properties

We find a spectral index that is roughly flat ($\alpha\sim 0.07$) at Ka-band.
Herrnstein et al. (2004) find a variable spectral index at similar wavelengths
that range from $\sim-0.1$ to $0.7$. We note, however, that their observations
do not account for flaring emission and are an ensemble average of both the
quiescent and variable components. The (ensemble-average) spectral energy
distribution of Sgr A* from Brinkerink et al. (2015) suggested a similarly
flat spectrum and comparable flux density at radio frequencies as ours here.
Similarly, our fitted submm spectral index and flux are well within the range
of those observed (e.g., Wielgus et al., 2022). Using the fitted quiescent
fluxes, we also determine the average spectral index between 31.54 and 334 GHz
is $\alpha\sim 0.36$, which is consistent with the canonical value of
$\alpha=0.33$ for optically-thick accretion flows (Pringle, 1981) and those in
the literature (e.g., Michail et al., 2021b).

The consistency between our fitted quiescent emission in 2019 and previous
analyses is somewhat shocking. IR variability suggests that Sgr A* was more
variable in 2019 than in the past $\sim 20$ years (Do et al., 2019; Weldon et
al., 2023), which they concluded occurs from enhanced accretion onto the event
horizon. As flaring events occur within the accretion flow, one would expect
similar outliers in the quiescent submm and radio emission. However, this
analysis does not show similar drastic changes, even at 334 GHz, where the
emission primarily occurs near the event horizon, agreeing with Murchikova &
Witzel (2021), who find no change in either the 230 GHz flux or variability
statistics in the aftermath of this bright IR flare. Additional radio
observations of Sgr A* during 2019 should be analyzed to determine if a
similar change in the radio continuum properties occurred.

####  5.2.2 Hotspot Properties

The first four rows in Table 1 list the fitted parameters for the hotspot. The
plasma produces a $686$ mJy flare at 334 GHz from a shallow electron energy
spectrum with an index $p=0.59$. The low value of $p$ is caused by neglecting
synchrotron cooling in the adiabatically-expanding hotspot model. Values
typically range from $p\approx 1-3$ (i.e., Yusef-Zadeh et al., 2009; Eckart et
al., 2009; Michail et al., 2021a, 2023; Witzel et al., 2021; Boyce et al.,
2022), where $p=1$ indicates an extremely strong and efficient shock (Bell,
1978a, b). We discuss the low value of $p$ in Section 5.3.

Given these values, we can calculate an equipartition magnetic field strength
and radius for the observed flare. Once again, we assume the $E^{-0.59}$
electron energy spectrum is valid from energies between 1 to 1000 MeV
consistent with Section 5.1.2. We determine the hotspot to have a radius
$0.47$ $R_{\text{S}}$ at the time of the peak submm emission, which is
threaded by a magnetic field $\sim 187$ Gauss. The mass is $3.7\times 10^{18}$
grams, and the electron density is $2.82\times 10^{6}$ cm-3. Assuming linear
expansion of the hotspot yields an expansion velocity of $0.029c$. This
combination of magnetic field strength and electron number density is similar
to the time-averaged conditions within a few $R_{\text{S}}$ of a strongly-
jetted accretion flow around Sgr A*, as modeled with GRMHD (Chatterjee et al.,
2021).

Subroweit et al. (2017) complete a statistical analysis of 100 and 345 GHz
light curves of Sgr A*. By relating the flaring flux distribution with the
adiabatically expanding hotspot model, they suggest the majority of flares
have size $\sim 0.67~{}R_{\text{S}}$ and expand with “intermediate” velocities
of order $0.01c$. The expansion velocities have generally been found to lie in
the range of $0.001-0.1c$ (e.g., Yusef-Zadeh et al., 2008). Similarly, the
inferred radius of these hotspots are order $\sim R_{\text{S}}$, with most of
the GRAVITY IR interferometry results preferring $0.5-1.5~{}R_{\text{S}}$
(e.g., GRAVITY Collaboration et al., 2020a, b, 2021).

As with the models consistent with the occultation event, the magnetic field
strength here is much stronger than previously observed by a factor $\sim 2$.
This might be attributed to the incorrect upper energy bound rather than the
lower one described during the occultation. The sensitivity of the magnetic
field strength on the upper or lower bound depends on the electron energy
index. Equation A2 gives the electron energy density as a function of $p$,
$E_{2}$, and $E_{1}$ (the maximum and minimum electron energies,
respectively). For $p>2$ (in the case of the occultation), the value of
$U_{e}$ is more sensitive to $E_{1}$ instead of $E_{2}$ since
$E_{2}^{2-p}<E_{1}^{2-p}$. For $p\in[0,2)$, the opposite is true as in our
case of $p=0.59$, $U_{e}\propto E_{2}^{1.41}-E_{1}^{1.41}$. Using Equation 15
for 334 GHz at 180 Gauss yields $\gamma\approx 80$ (an electron energy of $40$
MeV). Reducing the original upper energy bound to 175 MeV ($\gamma=350$,
consistent with the modeling in Boyce et al., 2022) yields an equipartition
magnetic field strength $94$ Gauss, electron density $3.8\times 10^{6}$ cm-3,
mass $2.9\times 10^{18}$ grams, radius $0.39~{}R_{\text{S}}$, and expansion
speed $0.024c$.

###  5.3 Effects of Synchrotron Cooling

Throughout the models presented in Section 4, we have ignored any contribution
of synchrotron cooling to the light curves. However, at the submm frequencies
probed here, it somewhat affects the observed profiles. In the occultation
modeling (Section 5.1.2), the adiabatic and synchrotron cooling times are
roughly equal. If we define the adiabatic timescale as
$t_{\text{ad}}=R_{\text{max}}/v_{\text{exp}}$, we find this value to be $\sim
1$ hour, roughly double the synchrotron cooling timescale at 334 GHz in a 130
Gauss field, $t_{\text{sc}}\sim 0.5$ hr. Ignoring this effect will tend to
over-predict the magnetic field strength as the electrons will have lower
characteristic energies than calculated via adiabatic expansion alone. Since
the characteristic electron energy $E_{\nu}\propto B^{-1/2}$ (Equation 3), the
fitting results will tend to bias toward higher magnetic field strengths
(lower energy) to account for this discrepancy.

Likewise, the effect of synchrotron cooling is visible in the submm light
curve in Figure 13, as the predicted adiabatic model peak flux is higher than
that of the observed value. This also tends to increase the equipartition
magnetic field strength as above; however, it also affects the fitted value of
$p$. At the peak submm emission, a sufficient proportion of electrons have
cooled faster via adiabatic and synchrotron processes, which decreases the
optical depth of the plasma, suppressing the peak flux as prescribed by the
adiabatic picture.

As mentioned in Section 5.2.2, the low value of $p$ measured in the joint
radio-submm plasmon fit is caused by ignoring synchrotron cooling in the
evolution of the submm flare. The relationship between the peak flux of an
adiabatically-expanding hotspot at two different frequencies is
$S_{\nu_{2}}=S_{\nu_{1}}\left(\nu_{2}/\nu_{1}\right)^{\frac{7p+3}{4p+6}}$ (van
der Laan, 1966). As discussed above, synchrotron cooling will tend to decrease
the peak flux where it is comparable or dominant to adiabatic cooling. $p$
will be biased towards lower values to fit this dimmer-than-expected submm
peak with the radio light curves. To confirm this, we independently fit the
radio and submm data with a single adiabatically-cooling hotspot (not shown).
We find $p\sim 1.3$ for both the radio and submm fits. By way of example,
assume a $100$ mJy flare occurs at $31.54$ GHz (as estimated from Figures 9d
and 13). For $p=0.6$ and $p=1.3$, we find peak $334$ GHz fluxes of $757$ and
$1282$ mJy, respectively. The lower value of $p$ better models the observed
submm light curve; however, by ignoring synchrotron cooling, we have
effectively obtained an unrealistic value of $p$.

Overall, ignoring synchrotron cooling causes a biased value of $p$ and tends
to overpredict the calculated equipartition magnetic field strength compared
to previous observational and theoretical values. Future analyses using this
adiabatic cooling model (especially when synchrotron cooling is dominant to
adiabatic cooling) need to account for this additional effect. We note
synchrotron cooling is not the only missing process throughout this work. For
example, the lack of general relativistic effects, such as lensing, may also
affect our results. However, including these additional processes is outside
the scope of this current work.

##  6 Summary and Conclusions

We present and analyze the data during the final two days of a multiwavelength
campaign to observe Sgr A*, which occurred in July 2019 and was the last such
campaign with the Spitzer Space Telescope. Analysis of the first day was
presented in Paper I. The participating instruments ranged in energy from
radio to X-ray energies. The observations showed Sgr A* to be remarkably
active. On 26 July, we observed a flare in the IR, submm, and radio, which
presents the most evident causal link between the variability mechanisms in
these frequency regimes. We model the radio and submm light curves using an
adiabatically-expanding synchrotron hotspot. We find a transient $\sim 35$
minute periodicity in the Spitzer light curve on 21 July, corresponding to
slightly time-delayed peaks observed with ALMA, suggesting a hotspot on a
stable orbit around Sgr A*. We detect a drop in the submm flux shortly after
an X-ray flare on the same day. We modeled this feature using an
adiabatically-expanding synchrotron hotspot, which occults the accretion flow
during its dynamic expansion. We present the model in this paper after
generalizing it from a specific case for $p=2$ (Yusef-Zadeh et al., 2010).

During both days, we find the magnetic field strength estimates are uncertain
because of the lack of constraints in the upper and lower energy bounds of the
electron spectrum and the omission of synchrotron cooling. The absence of
lower frequency radio light curves does not allow for a robust lower energy
bound. Synchrotron cooling plays a crucial role in the profiles of the submm
light curves. The magnetic field strengths we estimate are uncertain on both
days because of the lack of data constraining the bound of the electron energy
spectrum and neglecting synchrotron cooling, which will tend to increase the
equipartition magnetic field strength required to model the observed features.
Modeling flares using adiabatically-expanding synchrotron hotspots with future
high-frequency submm and IR observations will require accounting for
synchrotron cooling losses during the expansion phase, placing a strong bound
on the upper energy limit. Supplementing high-quality radio frequency light
curves with such observations will constrain the bounds of the electron
spectrum and better bound possible values of the magnetic field strength
responsible for the emission.

## Acknowledgments

We are grateful to Sean Carey, Dan Marrone, and Steven Willner for useful
comments that strengthened this manuscript. We additionally thank the referee
for their very helpful and constructive feedback. This work is partially
supported by NSF grant AST-2305857 and through SOSP21A-003 and SOSPADA-026
from the NRAO. SM is grateful for support from the Dutch Research Council
(NWO) VICI award (grant 639.043.513) and the European Research Council (ERC)
SyG “BlackHolistic” (grant 101071643). GGF, JLH, and HAS acknowledge support
for this work from the NASA ADAP program under NASA grant 80NSSC18K0416. The
National Radio Astronomy Observatory is a facility of the National Science
Foundation operated under cooperative agreement by Associated Universities,
Inc. This paper makes use of the following ALMA data:
ADS/JAO.ALMA#2018.A.00050.T. ALMA is a partnership of ESO (representing its
member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST
and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the
Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and
NAOJ. Based on observations made with the NASA/ESA Hubble Space Telescope,
obtained from the data archive at the Space Telescope Science Institute. STScI
is operated by the Association of Universities for Research in Astronomy, Inc.
under NASA contract NAS 5-26555. This work is based on archival data obtained
with the Spitzer Space Telescope, which was operated by the Jet Propulsion
Laboratory, California Institute of Technology under a contract with NASA. The
scientific results reported in this article are based in part on data obtained
from the Chandra Data Archive. This research has made use of software provided
by the Chandra X-ray Center (CXC) in the application packages CIAO. This
research has made use of data obtained from the Chandra Source Catalog,
provided by the Chandra X-ray Center (CXC) as part of the Chandra Data
Archive. This research was supported in part through the computational
resources and staff contributions provided for the Quest high-performance
computing facility at Northwestern University which is jointly supported by
the Office of the Provost, the Office for Research, and Northwestern
University Information Technology.

##  Appendix A Deriving the Equipartition and Electron Spectrum Factors

For an electron distribution $n(E)\propto E^{-p}$ between energies $E_{1}$ and
$E_{2}$, the electron energy density is:

$\displaystyle
U_{e}=\int_{E_{1}}^{E_{2}}E~{}n(E)~{}dE=\int_{E_{1}}^{E_{2}}n_{0}E\left(\dfrac{E}{E_{0}}\right)^{-p}~{}dE.$
(A1)

This integration yields

$\displaystyle
U_{e}(p)=\begin{cases}\dfrac{n_{0}E_{0}^{2}}{2-p}\left(\left(\dfrac{E_{2}}{E_{0}}\right)^{2-p}-\left(\dfrac{E_{1}}{E_{0}}\right)^{2-p}\right),&p\geq
0,p\neq 2,\\\ \\\
n_{0}E_{0}^{2}\ln\left(\dfrac{E_{2}}{E_{1}}\right),&p=2,\end{cases}$ (A2)

where $n_{0}$ has units of $\rm{erg}^{-1}\rm{cm}^{-3}$. Writing $U_{B}$ and
$U_{e}$ as $U_{B}=\left(U_{B}/U_{e}\right)U_{e}$, plugging in for $U_{e}$ from
Equation A2, and substituting $U_{B}=B^{2}/8\pi$:

$\displaystyle\dfrac{B^{2}}{8\pi}=\begin{cases}\left(\dfrac{U_{B}}{U_{e}}\right)\dfrac{n_{0}E_{0}^{2}}{2-p}\left(\left(\dfrac{E_{2}}{E_{0}}\right)^{2-p}-\left(\dfrac{E_{1}}{E_{0}}\right)^{2-p}\right),&p\geq
0,p\neq 2\\\ \\\
\left(\dfrac{U_{B}}{U_{e}}\right)n_{0}E_{0}^{2}\ln\left(\dfrac{E_{2}}{E_{1}}\right),&p=2.\end{cases}$
(A3)

Solving for $n_{0}$:

$\displaystyle n_{0}=\begin{cases}\dfrac{B^{2}}{8\pi
E_{0}^{2}}\dfrac{2-p}{\left(\dfrac{U_{B}}{U_{e}}\right)\left(\left(\dfrac{E_{2}}{E_{0}}\right)^{2-p}-\left(\dfrac{E_{1}}{E_{0}}\right)^{2-p}\right)},&p\geq
0,p\neq 2\\\ \\\ \dfrac{B^{2}}{8\pi
E_{0}^{2}}\dfrac{1}{\left(\dfrac{U_{B}}{U_{e}}\right)\ln\left(\dfrac{E_{2}}{E_{1}}\right)},&p=2.\end{cases}$
(A4)

By defining,

$\displaystyle\varphi(p)\equiv\begin{cases}\left(\dfrac{U_{B}}{U_{e}}\right)\dfrac{\left(\dfrac{E_{2}}{E_{0}}\right)^{2-p}-\left(\dfrac{E_{1}}{E_{0}}\right)^{2-p}}{2-p},&p\geq
0,p\neq 2\\\ \\\
\left(\dfrac{U_{B}}{U_{e}}\right)\ln\left(\dfrac{E_{2}}{E_{1}}\right),&p=2,\end{cases}$
(A5)

we obtain

$n(E)=\dfrac{B^{2}}{8\pi\varphi E_{0}^{2}}\left(\dfrac{E}{E_{0}}\right)^{-p},$
(A6)

which is Equation 5 above. Due to adiabatic cooling of the ultra-relativistic
electron gas, $E_{1}$ and $E_{2}$ scale inversely with radius. Therefore,
$\varphi(p)\propto R^{2-p}$, which must be properly accounted for while
modeling the overall occultation light curves.

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