Multi-wavelength Observations of Sgr A*. I. 2019 July 18
  We present and analyze ALMA submillimeter observations from a
multi-wavelength campaign of Sgr A* during 18 July 2019. In addition to the
submillimeter, we utilize concurrent mid-IR (Spitzer) and X-ray (Chandra)
observations. The submillimeter emission lags less than $\delta t\approx30$
minutes behind the mid-IR data. However, the entire submillimeter flare was not
observed, raising the possibility that the time delay is a consequence of
incomplete sampling of the light curve. The decay of the submillimeter emission
is not consistent with synchrotron cooling. Therefore, we analyze these data
adopting an adiabatically expanding synchrotron source that is initially
optically thick or thin in the submillimeter, yielding time-delayed or
synchronous flaring with the IR, respectively. The time-delayed model is
consistent with a plasma blob of radius $0.8~R_{\text{S}}$ (Schwarzschild
radius), electron power-law index $p=3.5$ ($N(E)\propto E^{-p}$), equipartition
magnetic field of $B_{\text{eq}}\approx90$ Gauss, and expansion velocity
$v_{\text{exp}}\approx0.004c$. The simultaneous emission is fit by a plasma
blob of radius $2~R_{\text{S}}$, $p=2.5$, $B_{\text{eq}}\approx27$ Gauss, and
$v_{\text{exp}}\approx0.014c$. Since the submillimeter time delay is not
completely unambiguous, we cannot definitively conclude which model better
represents the data. This observation presents the best evidence for a unified
flaring mechanism between submillimeter and X-ray wavelengths and places
significant constraints on the source size and magnetic field strength. We show
that concurrent observations at lower frequencies would be able to determine if
the flaring emission is initially optically thick or thin in the submillimeter.
