# An asymmetric dust ring around a very low mass star ZZ Tau IRS

Jun Hashimoto Astrobiology Center, National Institutes of Natural Sciences,
2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan  Subaru Telescope, National
Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan  Department
of Astronomy, School of Science, Graduate University for Advanced Studies
(SOKENDAI), Mitaka, Tokyo 181-8588, Japan  Ruobing Dong Department of Physics
& Astronomy, University of Victoria, Victoria, BC, V8P 1A1, Canada  Takayuki
Muto  Division of Liberal Arts, Kogakuin University, 1-24-2, Nishi-Shinjuku,
Shinjuku-ku, Tokyo 163-8677, Japan

###### Abstract

We present Atacama Large Millimeter/submillimeter Array (ALMA) gas and dust
observations at band 7 (339 GHz: 0.89 mm) of the protoplanetary disk around a
very low mass star ZZ Tau IRS with a spatial resolution of 0$\farcs$25\. The
12CO $J=3\rightarrow 2$ position–velocity diagram suggests a dynamical mass of
ZZ Tau IRS of $\sim$0.1–0.3 $M_{\sun}$. The disk has a total flux density of
273.9 mJy, corresponding to an estimated mass of 24–50 $M_{\oplus}$ in dust.
The dust emission map shows a ring at $r=$ 58 au and an azimuthal asymmetry at
$r=$ 45 au with a position angle of 135°. The properties of the asymmetry,
including radial width, aspect ratio, contrast, and contribution to the total
flux, were found to be similar to the asymmetries around intermediate mass
stars ($\sim$2 $M_{\sun}$) such as MWC 758 and IRS 48. This implies that the
asymmetry in the ZZ Tau IRS disk shares a similar origin with others, despite
the star being $\sim$10 times less massive. Our observations also suggest that
the inner and outer parts of the disk may be misaligned. Overall, the ZZ Tau
IRS disk shows evidence of giant planet formation at $\sim$10 au scale at a
few Myr. If confirmed, it will challenge existing core accretion models, in
which such planets have been predicted to be extremely hard to form around
very low mass stars.

protoplanetary disks — planet–disk interactions — planets and satellites:
formation — stars: individual (ZZ Tau IRS)

##  1 Introduction

Core accretion planet formation theories predict that only terrestrial and icy
planets can form around very low mass (VLM) stars ($\lesssim$0.1–0.2
$M_{\sun}$), and the formation of gas giant planets is essentially prohibited
(e.g., Liu et al., 2019, 2020; Matsumoto et al., 2020; Miguel et al., 2020).
The reasons are multifold. First, a low stellar mass results in longer
dynamical timescale and longer growth timescale of dust, delaying the
formation of planetary cores. Secondly, the mass of dust in protoplanetary
disks, the building blocks of planets, is correlated with stellar mass (e.g.,
Ansdell et al., 2017). A low dust mass in disks around VLM stars leads to low
mass of protoplanets, suppressing pebble accretion (Ormel & Liu, 2018) and
preventing cores from ever reaching the critical mass to trigger runaway gas
accretion. Last but not least, observed stellar accretion rates suggest that
disks around VLM stars evolve faster than those around higher mass stars,
further daunting the task of planet formation (Manara et al., 2012; Liu et
al., 2020). In addition, water snowline has been proposed to be the preferred
site of planetesimal and planet formation due to its enhanced local dust-to-
gas ratio (Ormel, 2017). Around VLM stars, snowline is close to the star at
$r\sim$0.1 au. Therefore compact systems with rocky planets like the
TRAPPIST-1 (Gillon et al., 2016) are expected, while planet formation at
$r\gtrsim$1 au is not (Miguel et al., 2020).

Indeed, statistical studies show that at small orbital separations rocky and
icy planets are common around M dwarfs (e.g., Dressing & Charbonneau, 2015),
while the occurrence rate of gas giant planets dives from 14% around 2 M☉
stars to 3% around 0.5 M☉ stars (Johnson et al., 2010). Around VLM stars, only
a handful of gas giant planets have ever been found (e.g., GJ 3512; Morales et
al., 2019), and their formation mechanism is a mystery. Since their host stars
are mature mean sequence stars that have finished planet formation long
before, it is unclear where and when do those planets form. Protoplanetary
disks around VLM stars serve as excellent laboratories for investigating
planet formation in these extreme environments.

The Atacama Large Millimeter/submillimeter Array (ALMA) has revealed a variety
of sub-structures in protoplanetary disks, such as rings, gaps, cavities,
spirals, and crescent structures. While rings/gaps are the most common (e.g.,
Andrews et al., 2018; Long et al., 2018; Cieza et al., 2020), roughly 10 disks
show azimuthal asymmetries (e.g., Francis & van der Marel, 2020; van der Marel
et al., 2020; Tsukagoshi et al., 2019; Pérez et al., 2018; Dong et al., 2018b;
Cieza et al., 2017). Disks around VLM stars also harbor rings and gaps (e.g.,
Kurtovic et al., 2020); however, azimuthal asymmetries have not yet been
reported. Asymmetric dust crescents may be local dust traps at gas pressure
maxima (e.g., Raettig et al., 2015; Ragusa et al., 2017), thus ideal locations
of planet formation (e.g., Chang & Oishi, 2010). Three possible origins of
asymmetries have been discussed by van der Marel et al. (2020): (a) long-lived
anticyclonic vortices at gap edges, possibly opened by companions (e.g.,
Raettig et al., 2015), (b) gas horseshoes at the edge of eccentric cavities
curved by massive companions (e.g., Ragusa et al., 2017), and (c) part of
spiral arms (van der Marel et al., 2016) triggered by companions. The main
difference between the former two is in the mass of the companions: a vortex
can be produced at the edges of gaps opened by planets as low mass as super-
Earths (Dong et al., 2018a), whereas a horseshoe needs to be triggered by a
much more massive companion, i.e., a brown dwarf. Overall, all three
mechanisms suggest the presence of companions (planets) inside the cavity or
gap.

In this paper, we report a ring and an asymmetric structure around the VLM
star ZZ Tau IRS (spectral type: M4.5 to M5, $T_{\rm eff}$: 3015 to 3125 K,
$L_{\rm star}$: 0.02 to 0.13 $L_{\sun}$, $M_{\rm*}$: 0.09 to 0.16 $M_{\odot}$,
White & Hillenbrand, 2004; Andrews et al., 2013; Herczeg & Hillenbrand, 2014)
in the Taurus star forming region at an assumed distance of 130.7 pc111We do
not use the GAIA distance of 105.7 pc because the astrometric solution is poor
with a value of the renormalized unit weight error (RUWE) of 2.490 (GAIA EDR3;
Gaia Collaboration et al., 2020). taken from Akeson et al. (2019). The stellar
luminosity is 0.017 to 0.113 $L_{\sun}$ (White & Hillenbrand, 2004; Andrews et
al., 2013; Herczeg & Hillenbrand, 2014, scaled by the updated distance), with
the large uncertainty mainly due to factors including flux calibration and
extinction (see the discussions in Herczeg & Hillenbrand, 2014). With these
$T_{\rm eff}$ and $L_{\rm star}$, the Baraffe et al. (2015) isochrone yields a
stellar mass around 0.1 $M_{\odot}$ at an age of 1 to 12 Myr consistently. ZZ
Tau IRS is a single star (i.e., not identified as a spectroscopic binary;
Kounkel et al., 2019) while its companion ZZ Tau is 35′′ away. Its spectral
energy distribution (SED; Figure 5 in Appendix A) does not show any deficits
at IR wavelengths $\lambda\lesssim$10 $\mu$m, and thus ZZ Tau IRS was not
classified as having a cavity/gap structure in its disk. The large IR excess
relative to the stellar flux in the SED (Figure 5) was interpreted as an edge-
on disk (Furlan et al., 2011), and high-resolution optical spectroscopy of ZZ
Tau IRS also suggested the existence of a close to edge-on disk based on its
narrow emission lines from the optical jet (White & Hillenbrand, 2004). The
total disk mass (gas $+$ dust) was estimated to be 0.014 $M_{\odot}$ by
observations at millimeter wavelengths using the Submillimeter Array (Andrews
et al., 2013; Akeson et al., 2019).

##  2 ALMA archive data

We used the ALMA archive data at band 7 (program ID: 2016.1.01511.S, PI: J.
Patience), summarized in Table 1. The data were calibrated by the Common
Astronomy Software Applications (CASA) package (McMullin et al., 2007),
following the calibration scripts provided by ALMA. We performed a self-
calibration of the visibilities. The phases were self-calibrated once, with
fairly long solution intervals (solint = ‘inf’) that combined all spectral
windows. To facilitate analyses later (§ 4), we shifted the observed dust
continuum images to minimize the asymmetry relative to the phase center using
the procedure described in Appendix B.

Table 1: ALMA observations and imaging parameters Observations |  |  |  |
---|---|---|---|---
Observing date (UT) | 2017.Jul.6
Project code | 2016.1.01511.S (PI: J. Patience)
Time on source (min) | 4.5
Number of antennas | 42
Baseline lengths | 16.7 m to 2.6 km
Baseband Freqs. (GHz) | 331.9 (cont.) | 333.8 (cont.) | 343.9 (cont.) | 345.8 (12CO $J=3\rightarrow 2$)
Channel width (MHz) | 15.6 | 15.6 | 15.6 | 0.488
Continuum band width (GHz) | 7.5
Bandpass calibrator | J0510$+$1800
Flux calibrator | J0510$+$1800
Phase calibrator | J0438$+$3004
Typical PWV (mm) | 0.60
Imaging | Dust continuum |  12CO $J=3\rightarrow 2$
Robust clean parameter | $-2.0$ | 2.0
Outer $uv$-taper parameter | 0.4 $\times$ 500.0 M$\lambda$ with 48.0$\arcdeg$ |  $\cdots$
Beam shape | 254 $\times$ 248 mas at PA of $-$61.7$\arcdeg$ | 283 $\times$ 156 mas at PA of $-$50.2$\arcdeg$
r.m.s. noise (mJy/beam) | 0.268 | 9.56 at 0.5 km/s bin

The final synthesized dust continuum image of the combined data is shown in
Figure 1. In the `CLEAN` task, we set the $uv$-taper (0.4 $\times$ 500.0
M$\lambda$ at PA of 48°) with a robust parameter of $-$2.0 to obtain a nearly
circular beam, and we did not use the ‘multi-scale’ option. The r.m.s. noise
in the region far from the source is 268 $\mu$Jy/beam with a beam size of 254
$\times$ 248 mas at a position angle (PA) of $-$61.7°.

The 12CO $J=3\rightarrow 2$ line data (Table 1) were extracted by subtracting
the continuum in the $uv$ plane using the `uvcontsub` task in the CASA tools.
A line image cube with a channel width of 0.5 km/s was produced by the `CLEAN`
task. The integrated line flux map (moment 0) and the intensity-weighted
velocity map (moment 1) are shown in Figure 2. The channel maps at $+$1.0 to
$+$13.0 km/s are shown in Figure 7 in the Appendix C. The r.m.s. noise in the
moment 0 map is 61.4 mJy/beam$\cdot$km/s with a beam size of 283 $\times$ 156
mas at a PA of $-$50.2$\arcdeg$, and the r.m.s. noise in the moment 1 map is
9.56 mJy/beam at 0.5 km/s bin. The peak signal-to-noise ratio (SNR) is 24.1 in
the channel map of $+$4.0 km/s.

##  3 Results

###  3.1 Dust continuum emission

Figure 1 shows dust continuum images of the ZZ Tau IRS disk in band 7, as well
as its radial and azimuthal profiles. We found two structures: a ring at
$r\sim$50 au (hereafter we refer to the ZZ Tau IRS disk as a ‘ring’) in Figure
1(b) and an azimuthal asymmetry with a contrast ratio of 1.39 $\pm$ 0.05
between its peak and the opposite side on the ring in Figure 1(c). The peak
SNR in the ring is 78.4. The dust continuum image synthesized with only the
imaginary part (which efficiently detects asymmetric structures) indicates
that the asymmetric structure in the ring is significant with a peak SNR of
34.3 in the central panel in Figure 6 in the Appendix B. In the azimuthal
profile (Figure 1c), we found two peaks at PAs of $\sim$130$\arcdeg$ and
$\sim$310$\arcdeg$. Such a two-peak structure along a disk major axis is
commonly found in low spatial resolution images of rings
($\sim$0$\farcs$2–0$\farcs$4; e.g., Ansdell et al., 2016). Our visibility
analyses in § 4 suggest that the peak at a PA of $\sim$310$\arcdeg$ is caused
by beam effects. Hereafter we refer to the peak at a PA of $\sim$130$\arcdeg$
as the asymmetry.

The total flux density of the dust continuum derived from visibility analyses
in § 4 is 273.9 $\pm$ 27.4 mJy (Table 2), assuming a 10 % uncertainty in the
absolute flux calibration, consistent with previous single-disk observations
(300 $\pm$ 40 mJy, Di Francesco et al., 2008). The peak brightness temperature
in the ring calculated from the best-fit model in the visibility analyses (§
4) is 14.9 $\pm$ 0.9 K assuming a 10 % uncertainty in the absolute flux
calibration.

We now estimate the physical properties of the dust ring. The optical depth
$\tau_{\nu}$ is calculated as:

$\displaystyle I_{\nu}={\color[rgb]{0,0,0}B_{\nu}(T_{\rm
mid})}\\{1-{\exp}(-\tau_{\nu})\\},$ (1)

where $I_{\nu}$, $B_{\nu}$, and $T_{\rm mid}$ are the intensity, the Planck
function, and the midplane temperature, respectively. We use the midplane
temperature profile with a simplified expression for a passively heated,
flared disk in radiative equilibrium (e.g., Dullemond et al., 2001):

$\displaystyle T_{\rm mid}(r)=\left(\frac{\phi L_{*}}{8\pi r^{2}\sigma_{\rm
SB}}\right)^{0.25},$ (2)

where $L_{*}$ is the stellar luminosity (0.017 to 0.113 $L_{\odot}$), $\phi$
is the flaring angle, and $\sigma_{\rm SB}$ is the Stefan–Boltzmann constant.
If the starlight-absorbing small dust grains (sub-micron-sized) have a similar
spatial distribution to the large dust grains (millimeter-sized) traced by
millimeter emission (similar to the PDS 70 disk; Hashimoto et al. 2012;
Keppler et al. 2019), the inner wall of the ring is directly exposed to
stellar radiation, and $\phi$ reaches unity. In this case, at $r=$ 45 au where
the asymmetry is located (see visibility analyses in § 4), Equation (2) yields
$T_{\rm mid}=17.8$–$28.6$ K (uncertainty due to the stellar luminosity). The
corresponding optical depth $\tau_{\nu}$ is 0.5–1.4 based on Equation (1).
Therefore, the asymmetry is marginally optically thin. The rest of the ring is
expected to be optically thinner. Approximating the entire disk as an
optically thin structure, the total flux corresponds to a total mass (gas $+$
dust) of 7.6–15.7 $M_{\rm Jup}$ assuming the dust temperature of $T_{\rm
mid}=17.8$–$28.6$ K, an opacity per unit dust mass of $\kappa_{\nu}=$3.45 cm2
g-1 at 345 GHz (Beckwith & Sargent, 1991), and a gas-to-dust mass ratio of
100. We note that the dust mass estimate is a lower limit since the actual
dust temperature can be lower. The large dust grains are expected to
concentrate at the pressure peak outside the cavity wall and reach a
temperature lower than that estimated using Equation 2 with $\phi\sim$unity.
In the extreme case, the observed cavity in large dust grains may even be
filled with small dust grains (e.g., “missing cavities”; Dong et al. 2012),
resulting in substantially lower temperature at the ALMA ring location.

Figure 1: Synthesized image of the dust continuum for the ZZ Tau IRS ring
obtained from band 7. (a) Entire image. The r.m.s. noise measured in the
region far from the object is 0.268 mJy/beam with a beam size of 254 $\times$
248 mas at a PA of $-$61.7°. (b) Radial cut along the disk major axis at a PA
of 135$\arcdeg$ in panel (a). (c) Azimuthal profile along the elliptical ring
in panel (a). Two peaks are found at PAs of $\sim$130$\arcdeg$ and
$\sim$310$\arcdeg$. Only the peak at a PA of $\sim$130$\arcdeg$ is a real
asymmetry, whereas that at PA of $\sim$310$\arcdeg$ is an apparent structure
due to a low spatial resolution effect (see § 4).

###  3.2 Gas emission

The integrated line flux map of 12CO $J=3\rightarrow 2$ in Figure 2(a) shows a
single-peak structure with a peak flux of 1.0 $\pm$ 0.1 Jy/beam$\cdot$km/s at
16.3 $\sigma$ and a total flux above 3 $\sigma$ of 1.9 $\pm$ 0.19
Jy$\cdot$km/s assuming a 10 % uncertainty in the absolute flux calibration. As
shown in the channel maps in Figure 7 in the Appendix C, the 12CO emission is
absorbed at 6.5–7.0 km/s, and thus the total flux could be a lower limit. The
brightness temperature map of 12CO converted from the moment 8 map (maximum
values of the spectrum) is shown in Figure 2(b). Since 12CO $J=3\rightarrow 2$
could be generally optically thick, it could be a good tracer for the gas
temperature in the emitting region. At the location of the asymmetry, the
brightness temperature of 12CO is $\sim$35 K, higher than the 17.8–28.6 K
estimated by Equation (2). This could be because the emitting surface of 12CO
is in the upper hotter layer of the disk. Figure 2(d) shows the 12CO
position–velocity (PV) diagram (channel maps: Figure 7 in the Appendix C)
along PA $=$ 135$\arcdeg$. We overplotted the loci of the peak emission of a
Keplerian disk with an inclination of 60$\arcdeg$ (see § 4) around a central
star with masses of 0.1, 0.2, and 0.3 $M_{\odot}$, and found that the
dynamical mass of ZZ Tau IRS is $\sim$0.1–0.3 $M_{\sun}$. This value is
consistent with the spectroscopically determined mass (0.1–0.2 $M_{\sun}$;
Andrews et al., 2013; Herczeg & Hillenbrand, 2014). Note that we assume that
the systemic velocity of ZZ Tau IRS is 6.5 km/s in the PV diagram.

Figure 2: Synthesized images of the 12CO $J=3\rightarrow 2$ emission line for
the ZZ Tau IRS ring. (a) 12CO moment 0 map. The r.m.s. noise is 61.4
mJy/beam$\cdot$km/s with a beam size of 283 $\times$ 156 mas at a PA of
$-$50.2$\arcdeg$. (b) 12CO moment 8 map converted to the brightness
temperature. (c) 12CO moment 1 map. The r.m.s. noise is 9.56 mJy/beam at 0.5
km/s bin. The dotted contours represent the dust continuum at 10, 30, and 50
$\sigma$. (d) PV diagram along a PA of 135$\arcdeg$ taken from § 4. The three
lines denote loci of the peak emission of a Keplerian disk with a disk
inclination of 60$\arcdeg$ around 0.1 to 0.3 $M_{\odot}$ stars at $d=$ 130.7
pc. The systemic velocity (white dotted line) is assumed to be 6.5 km/s.

##  4 Visibility analyses

Since the contrast of the asymmetry in the ZZ Tau IRS ring is of order unity,
an imaging artifact such as a sidelobe of the bright ring could generate an
apparent asymmetry. To characterize the disk structure, we performed forward
modelling in which the observed visibilities are reproduced with a parametric
model of the ring in the visibility domain.

The ZZ Tau IRS disk is assumed to consist of a ring and an asymmetry. The
model ring consists of a narrow Gaussian ring (ring A) superposed on a wide
Gaussian ring (ring B) as shown in Figure 3(a), and is described as follows:

$\displaystyle I_{\rm ring}(r)\propto{\rm exp}\left(-\frac{(r-r_{\rm
peak,ringA})^{2}}{2\sigma^{2}_{r,{\rm ringA}}}\right)+\delta\ {\rm
exp}\left(-\frac{(r-r_{\rm peak,ringB})^{2}}{2\sigma^{2}_{r,{\rm
ringB}}}\right),$ (3)

where $r_{\rm peak,ringA}$, $\sigma_{r,\rm{ringA}}$, $r_{\rm peak,ringB}$,
$\sigma_{r,\rm{ringB}}$, and $\delta$ are the radial peak positions and
standard deviations of two Gaussian rings and a scaling factor between the
two, respectively. The width of the ring (FWHM) is calculated as
2.355$\sigma_{\rm ring}$.

The asymmetry is defined as an elliptical Gaussian function in polar
coordinates as follows:

$\displaystyle I_{\rm asym}(r,\theta)\propto{\rm exp}\left(-\frac{(r-r_{\rm
peak,asym})^{2}}{2\sigma^{2}_{r{\rm,asym}}}-\frac{(\theta-\theta_{\rm
peak,asym})^{2}}{2\sigma^{2}_{\theta{\rm,asym}}}\right),$ (4)

where $r_{\rm peak,asym}$, $\theta_{\rm peak,asym}$, $\sigma_{r\rm{,asym}}$,
and $\sigma_{\theta{\rm,asym}}$ are the radial peak position, azimuthal peak
position, radial standard deviation, and azimuthal standard deviation,
respectively. The flux for the asymmetry is normalized to $F_{\rm asym}$. The
combined model image is magnified and rotated with an inclination ($i$) and a
PA. The total flux for the combined model image is normalized to $F_{\rm
total}$. The radial and azimuthal widths (FWHM) of the asymmetry are
2.355$\sigma_{r,\rm{asym}}$ and 2.355$\sigma_{\theta,\rm{asym}}$,
respectively. In total, there are 13 free parameters in our model ($F_{\rm
total}$, $F_{\rm asym}$, $r_{\rm peak,ringA}$, $\sigma_{r,\rm{ringA}}$,
$r_{\rm peak,ringB}$, $\sigma_{r\rm{,ringB}}$, $\delta$, $r_{\rm peak,asym}$,
$\theta_{\rm peak,asym}$, $\sigma_{r,\rm{asym}}$, $\sigma_{\theta{\rm,asym}}$,
$i$, PA).

The model image was converted to complex visibilities with the public Python
code `vis_sample` (Loomis et al., 2017), in which model visibilities are
samples with the same ($u$, $v$) grid points as the observations. The model
visibilities are deprojected222Visibilities are deprojected in the $uv$-plane
using the following equations (e.g., Zhang et al., 2016): $u^{\prime}=(u\,{\rm
cos}\,{\rm PA}-v\,{\rm sin}\,{\rm PA})\times{\rm cos}\,i,v^{\prime}=(u\,{\rm
sin}\,{\rm PA}-v\,{\rm cos}\,{\rm PA}),$ where $i$ and PA are free parameters
in our visibility analyses in § 4. with the system PA and $i$ as free
parameters. The fitting is performed with a Markov chain Monte Carlo (MCMC)
method in the `emcee` package (Foreman-Mackey et al., 2013). The log-
likelihood function ln$L$ in visibility analyses is

$\displaystyle{\rm ln}L=-0.5\sum\left[W_{j}\\{({\rm Re}V_{j}^{\rm obs}-{\rm
Re}V_{j}^{\rm model})^{2}+({\rm Im}V_{j}^{\rm obs}-{\rm Im}V_{j}^{\rm
model})^{2}\\}\right],$ (5)

where the subscript $j$ represents the $j$-th data. $V_{j}^{\rm obs}$,
$V_{j}^{\rm model}$, and $W_{j}$ are observed and model visibilities and
weights corresponding to 1/$\sigma_{j}^{2}$ ($\sigma_{j}$ is the rms noise of
a given visibility. See details in CASA
Guides333https://casaguides.nrao.edu/index.php/DataWeightsAndCombination.),
respectively. The values of $V_{j}^{\rm obs}$ and $W_{j}$ (i.e.,
1/$\sigma_{j}^{2}$) are provided in the measurement set of ALMA data. Our
calculations used flat priors with the parameter ranges summarized in Table 2.
We ran 3,000 steps with 100 walkers, and discarded the initial 1,000 steps as
the burn-in phase based on trace plots in Figure 8 in Appendix D.

The fitting results with uncertainties computed from the 16th and 84th
percentiles, the radial profile of best-fit surface brightness for the ring
with a raw resolution, the best-fit model image, and the probability
distributions for the MCMC posteriors are shown in Table 2, Figure 3(a),
Figure 3(d), and Figure 9 in the Appendix E, respectively. We subtracted
modeled visibilities from observed visibilities (Figure 10 in Appendix F), and
made a CLEANed image (Figure 3e) in which we did not find significant
residuals. To calculate the value of the reduced-$\chi^{2}$, we derived a
factor between the weights and standard deviations (referred to as stddev) in
the visibilities by calculating the standard deviations of the real and
imaginary parts in 2 k$\lambda$ bins along the azimuthal direction in the
visibility domain. The values of weights $W_{j}$ are provided in the
measurement set of ALMA data while the standard deviations (stddev) are
calculated in our fitting. The visibilities were deprojected with $i=$
60$\arcdeg$ and PA $=$ 135$\arcdeg$. Figure 11 in Appendix G shows a
comparison between the weights and standard deviations (stddev-2). We found
that the weights are typically 4.0$\times$ higher than the standard
deviations. Finally, we calculated a reduced-$\chi^{2}$ of 1.91. We also
conducted visibility analyses using a model with one Gaussian ring $+$ one
asymmetry (i.e., $\delta=$ 0), and found significant residuals. Thus, the
model with two Gaussian rings $+$ one asymmetry is necessary to reproduce the
observed dust continuum image.

Table 2: Results of visibility analyses and parameter ranges

$r_{\rm peak,ringA}$ | FWHMringA | $r_{\rm peak,ringB}$ | FWHMringB | $\delta$ | $i$ | PA | $F_{\rm total}$ | $F_{\rm asym}$ | $r_{\rm peak,asym}$ | $\theta_{\rm peak,asym}$ | FWHMr,asym | FWHMθ,asym
---|---|---|---|---|---|---|---|---|---|---|---|---
(au) | (au) | (au) | (au) |  | (deg) | (deg) | (mJy) | (mJy) | (au) | (deg) | (au) | (deg)
57.51${}^{+0.19}_{-0.18}$ | 41.08${}^{+0.63}_{-0.60}$ | 83.62${}^{+1.61}_{-1.67}$ | 133.65${}^{+1.59}_{-1.62}$ | 0.34${}^{+0.01}_{-0.01}$ | 60.16${}^{+0.07}_{-0.07}$ | 134.73${}^{+0.09}_{-0.09}$ |  | 14.51${}^{+0.25}_{-0.24}$ | 45.15${}^{+0.22}_{-0.21}$ |  $-$18.17${}^{+0.52}_{-0.52}$ | 11.31${}^{+0.98}_{-1.01}$ | 103.07${}^{+1.59}_{-1.60}$
{0.0 .. 117.6} | {0.0 .. 78.4} | {0.0 .. 117.6} | {78.4 .. 196.1} | {0.0 .. 1.0} | {50.0 .. 70.0} | {125.0 .. 145.0} |  | {0.0 .. 100.0} | {0.0 .. 117.6} | {$-$45.0 .. $+$45.0} | {0.0 .. 117.6} | {0.0 .. 360.0}

Note. — Parentheses describe parameter ranges in our MCMC calculations.

Figure 3: (a) Azimuthally averaged radial surface brightness for the best-fit
model with a raw resolution. The asymmetry is not included in this panel. (b
and c) Radial and azimuthal profiles of the observations and the best-fit
model convolved with the beam in observations of 254 $\times$ 248 mas at a PA
of $-$61.7°. Observed data are same with those in Figure 1(b) and (c). (d)
Best-fit model image (3′′ $\times$ 3′′). (e) Residual image (6′′ $\times$
6′′). The reduced-$\chi^{2}$ is 1.91. The black dotted contours represent the
dust continuum image at 10, 30, and 50 $\sigma$.

Though the observed dust continuum image in Figure 1(a) shows two peaks along
the ring major axis, our visibility analyses suggest that only the one at PA
$=$ 130$\arcdeg$ is real.

The radial ring position (i.e., the peak position) and its FWHM (Ring A $+$
Ring B) in the model image were measured to be 58 and 49 au in Figure 3(a).
The contrast in the asymmetry, which is the ratio between the peak brightness
at the asymmetry and that at its opposite side on the ring, is measured to be
3.0 in Figure 3(d).

##  5 Discussion

###  5.1 Azimuthal asymmetry

Azimuthal asymmetries in rings have been identified in roughly ten
protoplanetary disks (e.g., van der Marel et al., 2020). At band 7, their
shape parameters (e.g., the radial FWHM and the aspect ratio defined as the
azimuthal FWHM divided by the radial FWHM) are summarized in Table 3444The
physical quantities of other asymmetries are taken from van der Marel et al.
(2020). V1247 Ori shows a crescent structure in ALMA band 7 (Kraus et al.,
2017). However, we do not include V1247 Ori because the structure does not
have a Gaussian profile (Kraus et al., 2017). Though HD 142527 also shows a
crescent structure in ALMA band 7 (Fukagawa et al., 2013), the visibility
analyses in van der Marel et al. (2020) did not find convergence; thus, we do
not include HD 142527.. All but ZZ Tau IRS are intermediate mass stars
($\sim$2 $M_{\sun}$). In Figure 4 we compare various properties of these
asymmetries as a function of the peak radial location. Though ZZ Tau IRS is a
VLM star, its asymmetry has similar properties to other asymmetries around
more massive stars, implying a similar origin.

We explore the correlation in Figure 4 by calculating Pearson’s correlation
coefficient $r$ for logarithmic numbers except the aspect ratio. The 5%
significance values for Pearson’s $r$ with $n=$7 and 8 are 0.75 and 0.71,
respectively (Pugh & Winslow, 1966). We found two (marginally) statistically
significant relations: radial location v.s. radial FWHM ($r=$0.89), and radial
location v.s. azimuthal FWHM ($r=$0.69; top panels in Figure 4). Asymmetric
structures have been interpreted as dust traps at pressure maxima in gas
vortices (e.g., Raettig et al., 2015) or horseshoes (e.g., Ragusa et al.,
2017). The radial width of a gas pressure bump is expected to be comparable to
the gas scale height (e.g., Dullemond et al., 2018), typically prescribed as
$H(r)\propto r^{q}$ with $q\sim$ unity (e.g., assuming $T(r)\propto r^{-0.5}$,
$q=1.25$). The most significant correlation we see in Figure 4, radial
location v.s. radial FWHM, may simply trace the dependence of the radial size
of gas pressure structure on $r$.

Another interpretation of asymmetries is that they are part of spiral arms
induced by companions (e.g., van der Marel et al., 2016). One way to
distinguish these mechanisms is to derive the spectral index from multi-
wavelength observations, which may constrain the size of the dust grains
(e.g., Testi et al., 2014). Dust trapping in gas vortices and horseshoes
results in a bigger dust size, while spiral arms are not expected to trap dust
particles as they deferentially rotate with respect to the local disk material
(Papaloizou et al., 2007; Li et al., 2020). Measuring the pattern speed of
asymmetries may also shed light on their origin, as vortices and horseshoes
are expected to move at the local Keplerian velocity while spirals corotate
with their drivers.

Table 3: Stellar mass and physical quantities for asymmetries

Object | Stellar mass | Radial location | Radial FWHM | Azimuthal FWHM | Aspect ratio | Contrast | Flux ratio | Refs
---|---|---|---|---|---|---|---|---
| ($M_{\sun}$) | (au) | (au) | (deg; au) |  |  | (%) |
ZZ Tau IRS | 0.2 | 45 | 11.3 | 103; 80.9 | 7.2 | 3.0 | 5.3 | a,a
MWC 758 | 1.9 | 50 | 7.5 | 49; 42.7 | 5.7 | 4.4 | 6.4 | b,c
|  | 90 | 15 | 47; 73.8 | 4.9 | 11.7 | 20.0 | b,c
HD 34282 | 2.0 | 137 | 110 | 52; 124.3 | 1.1 | 1.7 | 4.8 | b,d
SR 21 | 2.1 | 55 | 19 | 82; 78.7 | 4.1 | 2.1 | 12.5 | b,c
|  | 58 | 19 | 165; 166.9 | 8.8 | 2.0 | 26.0 | b,c
AB Aur | 2.3 | 170 | 96 | 122; 361.8 | 3.8 | 2.7 | 39.5 | b,c
IRS 48 | 2.2 | 70 | 29 | 58; 70.8 | 2.4 | — | — | b,c

Note. — (1) Object name. MWC 758 and SR 21 have two asymmetries (van der Marel
et al., 2020). (2) Stellar mass. (3) Radial location. (4) Radial FWHM
calculated by 2.355$\sigma_{r}$. The radial FWHM of the IRS 48 asymmetry is
calculated with 2.17$\sigma_{r}$ because the radial profile for IRS 48 was
found to be best fit with a 4th power in 2D Gaussian in van der Marel et al.
(2020). (5) Azimuthal FWHM. (6) The aspect ratio defined as the azimuthal FWHM
divided by the radial FWHM. (7) Contrast measured in the model between the
peak brightness and the opposite side of the ring. The model images are
constructed with the parameters from the literature in column (9). For SR 21,
since the azimuthal positions of the two asymmetries are located at almost
opposite sides, the contrast of the each asymmetry was measured for a model
containing only one asymmetry and the ring. For IRS 48, the contrast is not
measured because the model contains only the asymmetry without the ring (van
der Marel et al., 2020). (8) Ratio of the asymmetry to total flux in the
model. For IRS 48, this value is not measured because the model contains only
the asymmetry without the ring (van der Marel et al., 2020). (9) References
for information of stellar masses and properties of asymmetries: a) this work,
b) Garufi et al. (2018), c) van der Marel et al. (2020), d) van der Plas et
al. (2017).

Figure 4: Comparison of the properties of the ZZ Tau IRS asymmetry with other
asymmetries around intermediate mass stars as a function of the radial
location. The red circle represents ZZ Tau IRS. The physical quantities and
their brief explanations are summarized in Table 3. Pearson’s correlation
coefficient $r$ is inserted in panels. The 5% significance values for
Pearson’s $r$ with $n=$7 and 8 are 0.75 and 0.71, respectively (Pugh &
Winslow, 1966).

###  5.2 Gravitational stability

ZZ Tau IRS’ disk is one of the most massive among VLM stars (cf., Kurtovic et
al., 2020). We examine its gravitational stability. Theoretical studies (e.g.,
Durisen et al., 2007) suggest that if the Toomre (1964) Q-parameter, defined
as

$\displaystyle Q=\frac{c_{\rm s}\Omega_{\rm K}}{\pi G\Sigma_{\rm disk}},$ (6)

where $c_{\rm s}$, $\Omega_{\rm K}$, and $\Sigma_{\rm disk}$ are the sound
speed, the Keplerian angular velocity, and the disk (gas) surface density,
respectively, is of order of unity, the disk may be subject to gravitational
instability (GI). With the scale height of $H\sim c_{\rm s}/\Omega_{\rm K}$,
Toomre’s $Q$-parameter can be described as $Q\sim H/r\times M_{\rm
star}/M_{\rm disk}$, where $M_{\rm disk}=\pi r^{2}\Sigma_{\rm disk}$.
Therefore, if the disk’s aspect ratio is $H/r\sim 0.1$, GI occurs in massive
disks with $M_{\rm disk}/M_{\rm star}\gtrsim$0.1. The disk-to-star mass ratio
is $\sim$0.04–0.08 in ZZ Tau IRS (§ 3.1), and thus the disk may be globally
gravitationally stable. Meanwhile the Q-value at the asymmetry is 1.7–6.2
assuming a local surface density $\Sigma_{\rm disk}={\rm
cos}\,i\,\tau_{\nu}$/$\kappa_{\nu}=$ 7–20 g/cm2 ($i=60\arcdeg$ and
$\tau_{\nu}=$0.5–1.4, § 3.1), an opacity per unit dust mass of $\kappa_{\nu}=$
3.45 cm2 g-1 at 345 GHz (Beckwith & Sargent, 1991), a temperature of 17.8–28.6
K (§ 3.1), and a gas-to-dust mass ratio of 100. Hence, the asymmetry may also
be marginally stable.

###  5.3 Possible warped inner disk

ZZ Tau IRS has been suggestsed to have an edge-on inner disk (White &
Hillenbrand, 2004; Furlan et al., 2011), because the IR excess is larger than
the optical stellar flux due to partial disk obscuration of the star (Figure 5
in the Appendix A). Our visibility analyses suggest that the outer ring at
$r=$ 50 au has an inclination ($i$) of 60$\arcdeg$ (§ 4). Radiative transfer
calculations for protoplanetary disks (full disks) show that typically the
central star starts to be significantly obscured by the disk once the
inclination is larger than $\sim$70$\arcdeg$ (e.g., see Figure 2 in Whitney et
al., 2013). Hence ZZ Tau IRS could possess an inner disk misaligned with the
outer ring. Such warped inner disks have been discovered in twisted gas flow
patterns (e.g., Rosenfeld et al., 2014; Mayama et al., 2018) and shadows on
the outer disk in NIR scattering images (e.g., Marino et al., 2015). For ZZ
Tau IRS, current ALMA gas observations in Figure 2 are insufficient to confirm
a twisted gas flow due to low spatial & spectral resolution. Furthermore, high
spatial resolution observations of NIR scattering images have not yet been
carried out. Another possible explanation of the observed SED is that the
source is a “flat spectrum” young stellar object surrounded by an infalling
dusty envelope (Calvet et al., 1994). In this case, an extended reflection
nebulae around the source is expected in NIR scattered light observations
(e.g., similar to T Tau, Whitney & Hartmann, 1993). Future observations are
necessary to test these hypotheses.

A few possible mechanisms have been proposed to explain warped disks. An
exciting one is disk–companion interactions (e.g., Zhu, 2019), in which a
massive companion with a companion-star mass ratio $q\gtrsim 0.01$ (or $M_{\rm
p}\gtrsim M_{\rm J}$ around a 0.1$M_{\odot}$ star) on an inclined orbit
induces both a deep wide gap and a misaligned inner disk. Such a system with a
massive companion and a misaligned inner disk has been reported in HD 142527
(Biller et al., 2012; Marino et al., 2015). Future detection of massive
companions in the gap would serve as a test for the planet origin hypothesis.

##  6 Conclusion

We analyzed ALMA band 7 archive data (dust continuum and 12CO $J=3\rightarrow
2$ emission) of ZZ Tau IRS. The source has been classified as a VLM star with
a mass of $\sim$0.1–0.2 $M_{\sun}$ in the literature. The observed 12CO
$J=3\rightarrow 2$ kinematics suggest a dynamical mass of $\sim$0.1–0.3
$M_{\sun}$, consistent with previous estimates. The ALMA dust continuum image
at a spatial resolution of 0$\farcs$25 shows a ring and an asymmetric
structures around ZZ Tau IRS. Our visibility analysis confirmed that the
observed asymmetry is real rather than apparent.

Our best-fit model of the ALMA dust observations shows that the asymmetry has
a radial FWHM of 11.3 au, an aspect ratio of 7.2 (ratio of azimuthal FWHM to
azimuthal FWHM), a brightness contrast of 3.0, and the asymmetry contributes
5% of the total disk flux. These metrics are comparable with asymmetries seen
around intermediate mass stars ($\sim$2 $M_{\sun}$), implying a similar
origin, despite ZZ Tau IRS being 10 times less massive. While future high
spatial resolution observations in multiple wavelengths with ALMA and JVLA are
needed to determine the exact origin of the asymmetry, the three leading
hypotheses all invoke companions inside the gap.

The inner disk around ZZ Tau IRS, not visible in our observations, has been
inferred to be edge-on based on the SED. Our observations showed that the
inclination of the outer disk at $r=$ 50 au is 60°. These results imply that
the ZZ Tau IRS inner disk may be misaligned relative to the outer ring, and it
is possible that the misalignment (and the gap) is produced by a giant planet
with $M_{\rm p}\gtrsim M_{\rm J}$ on an inclined orbit inside the gap. Future
scattered light observations may test this hypothesis by searching for shadows
on the outer disk cast by the inner disk.

Overall, the structures found in the ZZ Tau IRS disk suggest the presence of
massive companions including gas giant planets at $\sim 10$ au. This is
surprising as the core accretion theories have generally prohibited the
formation of such planets around VLM stars (Miguel et al., 2020; Liu et al.,
2020, § 1). Hence, ZZ Tau IRS will be a prime target in the investigation of
planet formation around VLM stars.

We thanks the anonymous referee for a helpful review of the manuscript. We
also thanks Beibei Liu for discussions. This work was supported by JSPS
KAKENHI Grant Numbers 19H00703, 19H05089, and 19K03932, 18H05441, 17H01103.
R.D. acknowledges support from the Alfred P. Sloan Foundation and the Natural
Sciences and Engineering Research Council of Canada. This paper makes use of
the following ALMA data: ADS/JAO.ALMA#2016.1.01511.S. ALMA is a partnership of
ESO (representing its member states), NSF (USA), and NINS (Japan), together
with NRC (Canada), NSC (Taiwan), 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. This publication makes use of data
products from the Two Micron All Sky Survey, which is a joint project of the
University of Massachusetts and the Infrared Processing and Analysis
Center/California Institute of Technology, funded by the National Aeronautics
and Space Administration and the National Science Foundation. This publication
makes use of data products from the Wide-field Infrared Survey Explorer, which
is a joint project of the University of California, Los Angeles, and the Jet
Propulsion Laboratory/California Institute of Technology, funded by the
National Aeronautics and Space Administration. Software: `vis_sample` (Loomis
et al., 2017), `CASA` (McMullin et al., 2007), `emcee` (Foreman-Mackey et al.,
2013)

##  Appendix A SED of ZZ Tau IRS

Figure 5 shows the SED of ZZ Tau IRS. Extinction correction is not applied to
show the large IR excess relative to the stellar flux. The photometric data
without extinction correction are summarized in Table 4.

Figure 5: SED of ZZ Tau IRS without extinction correction. The photometric
data are summarized in Table 4. Table 4: Photometric data used in the SED of
ZZ Tau IRS

Wavelength | $\lambda F_{\lambda}$ | Reference
---|---|---
($\mu$m) | (10-12 erg s-1 cm-2) |
0.44 | 1.25 | USNO-B1.0 (Monet et al., 2003)
0.64 | 20.5 | USNO-B1.0 (Monet et al., 2003)
0.80 | 16.1 | USNO-B1.0 (Monet et al., 2003)
1.23 | 28.3 $\pm$ 0.6 | 2MASS (Cutri et al., 2003)
1.66 | 49.2 $\pm$ 1.0 | 2MASS (Cutri et al., 2003)
2.16 | 69.4 $\pm$ 1.5 | 2MASS (Cutri et al., 2003)
3.35 | 100.4 $\pm$ 2.0 | WISE (Cutri & et al., 2014)
4.60 | 164.5 $\pm$ 3.2 | WISE (Cutri & et al., 2014)
11.6 | 124.8 $\pm$ 1.6 | WISE (Cutri & et al., 2014)
22.1 | 193.8 $\pm$ 2.9 | WISE (Cutri & et al., 2014)
70 | 162.9 $\pm$ 34.3 | Herschel (Ribas et al., 2017)
100 | 93.0 $\pm$ 18.0 | Herschel (Ribas et al., 2017)
160 | 61.9 $\pm$ 13.1 | Herschel (Ribas et al., 2017)
250 | 31.2 $\pm$ 6.0 | Herschel (Ribas et al., 2017)
350 | 18.0 $\pm$ 3.4 | Herschel (Ribas et al., 2017)
500 | 9.0 $\pm$ 1.8 | Herschel (Ribas et al., 2017)
885 | 0.928 $\pm$ 0.093 | ALMA (this work)
1300 | 0.244 $\pm$ 0.003 | SMA (Andrews et al., 2013)

Note. — Extinction correction is not applied here.

##  Appendix B Dust continuum images synthesized with only the imaginary part

We shifted the observed dust continuum images to minimize the asymmetry
relative to the phase center as follows. First, the proper motions of ZZ Tau
IRS were calculated with the function `EPOCH_PROP` in GAIA ADQL (Astronomical
Data Query Language555https://gea.esac.esa.int/archive/). The phase centers
were corrected to (4h30m51.7346s, $+$24d41m47.175s) in ICRS using `fixvis` in
the CASA tools. Next, in order to make the analyses of the disk structure more
effective, we further shift the center of the image by searching for the
location where the disk asymmetry is minimized. We subtracted the 180°-rotated
image in the visibility domain. This procedure corresponds to producing a
synthesized image with only the imaginary part of the visibilities. This
method has been utilized in the analyses of DM Tau (Hashimoto et al., 2021)
and WW Cha (Kanagawa et al., 2021). Since the visibility $V$ is a Fourier
transform of the surface brightness distribution of the sky, which is a real
quantity, the visibility at $(-u,-v)$ in the $uv$-plane is a complex conjugate
of that at $(u,v)$,

$\displaystyle\overline{V}(u,v)=V(-u,-v).$ (B1)

The 180°-rotation of the image with respect to the origin $(x,y)=(0,0)$ on sky
corresponds to flipping the signs of both $(x,y)$ coordinates, i.e.,
$(x,y)\rightarrow(-x,-y)$. In the visibility domain, this is mathematically
equivalent to flipping the signs of $(u,v)$. Therefore, the visibility of
180°-rotated image is

$\displaystyle V(u,v)\rightarrow V(-u,-v)=\overline{V}(u,v).$ (B2)

Thus, subtracting the 180°-rotated image is mathematically equivalent to
setting the real part to zero and doubling the value of the imaginary part as
follows

$\displaystyle V(u,v)-V(-u,-v)=V(u,v)-\overline{V}(u,v)=2{\rm Im}(u,v).$ (B3)

In other words, the real part contains information on both the symmetric and
asymmetric structures of objects, whereas the imaginary part contains only
information on the asymmetries. Therefore, by synthesizing the image with only
the imaginary part, we selectively remove symmetric structures and reveal only
asymmetric structures. We determined the minimum r.m.s. in the central region
of the images by shifting the images relative to the new GAIA phase center in
the visibility domain by a phase shift of $\exp\left[2\pi i\left(u\,\Delta{\rm
R.A.}+v\,\Delta{\rm DEC}\right)\right]$, where $u$ and $v$ are the spatial
frequencies and $\Delta$R.A. and $\Delta$DEC are the shift values. Figure 6
shows dust continuum images synthesized with only the imaginary part,
including the image with the minimum r.m.s. We found shift values for the
minimum r.m.s. ($\Delta$RA, $\Delta$DEC) of ($-$28 mas, $-$23 mas) relative to
the phase center determined by GAIA. The final phase center in the ICRS
coordinate is (4h30m51.7325, $+$24d41m47.152).

Figure 6: Dust continuum images synthesized with only the imaginary part of
the long baseline data by shifting by 10 mas in the R.A. and DEC directions.
The r.m.s. value is measured inside the black dotted circle with a diameter of
2$\farcs$25\. The 1-$\sigma$ noise is 0.268 mJy/beam measured in a region far
from the object. The beam size is 254 $\times$ 248 mas at a PA of $-$61.7°.

##  Appendix C 12CO $J=3-2$ channel maps

Figure 7 shows the 12CO $J=3-2$ channel maps at $+$1.0 to $+$13.0 km/s.

Figure 7: Channel maps of 12CO $J=3-2$. The r.m.s noise at the 0.5 km/s bin is
9.56 mJy/beam with a beam size of 283 $\times$ 156 mas at a PA of $-$50.2°.

##  Appendix D Trace plots in MCMC calculations

Figure 8 shows trace plots for 100 walkers for 13 parameters in the model in
our visibility analyses (§ 4). The burn-in phase is set as the initial 1,000
steps.

Figure 8: Trace plots for 100 walkers for 13 parameters in the model. The
initial 1,000 steps are set as the burn-in phase and are discarded in the
histograms of the marginal distributions of the MCMC posteriors in Figure 9.

##  Appendix E Histogram of the marginal distributions for the MCMC posteriors

Figure 9 shows a corner plot of the MCMC posteriors calculated in visibility
analyses in § 4.

Figure 9: Corner plot of the MCMC posteriors calculated in visibility analyses
in § 4. The histograms on the diagonal are marginal distributions of 13 free
parameters. The parameter ranges in each parameter are described in Table 2.
The vertical dashed lines in the histograms represent the median values and
the 1 $\sigma$ confidence intervals for parameters computed from the 16th and
84th percentiles. The off-diagonal plots show the correlation for
corresponding pairs of parameters.

##  Appendix F Observed and modeled visibilities

Figure 10 shows observed and modeled visibilities in our visibility analyses
(§ 4).

Figure 10: Real (a) and imaginary (b) parts of the visibilities for the
observations (red dots) and the best-fit model (black line) in the top panel
in visibility analyses in § 4. The bottom panel shows residual visibilities
between observations and the best-fit model. The reduced-$\chi^{2}$ is 1.91.

##  Appendix G Weight values and standard deviations in the real and imaginary
parts

Figure 11 shows a comparison of the values of weights and standard deviations
in the real and imaginary parts. The values of weights ($W_{j}$) are provided
in the measurement set of ALMA data while the standard deviations (stddev) are
calculated in each 2 k$\lambda$ bin in the deprojected visibilities of the
real and imaginary parts with $i=$ 60$\arcdeg$ and PA $=$ 135$\arcdeg$. We
found that the weights are typically 4.0$\times$ higher than 1/stddev2 of the
real and imaginary parts at $\sim$100–600 k$\lambda$.

Figure 11: Comparison of weights and standard deviations (stddev) in real and
imaginary parts. The weights are typically 4.0$\times$ higher than 1/stddev2
of the real and imaginary parts at 100–600 k$\lambda$.

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