# Resolved Multifrequency Radio Observations of GG Tau

Sean M. Andrews11affiliation: Harvard-Smithsonian Center for Astrophysics, 60
Garden Street, Cambridge, MA 02138, USA , Claire J. Chandler22affiliation:
National Radio Astronomy Observatory, P.O. Box O, Socorro, NM 87801, USA ,
Andrea Isella33affiliation: California Institute of Technology, 1200 East
California Blvd., Pasadena, CA 91125, USA , T. Birnstiel11affiliation:
Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA
02138, USA , K. A. Rosenfeld11affiliation: Harvard-Smithsonian Center for
Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA , D. J.
Wilner11affiliation: Harvard-Smithsonian Center for Astrophysics, 60 Garden
Street, Cambridge, MA 02138, USA , L. M. Pérez22affiliation: National Radio
Astronomy Observatory, P.O. Box O, Socorro, NM 87801, USA , L.
Ricci33affiliation: California Institute of Technology, 1200 East California
Blvd., Pasadena, CA 91125, USA , J. M. Carpenter33affiliation: California
Institute of Technology, 1200 East California Blvd., Pasadena, CA 91125, USA ,
N. Calvet44affiliation: Department of Astronomy, University of Michigan, 500
Church Street, Ann Arbor, MI 48109, USA , S. A. Corder55affiliation: Joint
ALMA Observatory, Avenida Alonso de Córdova 3107, Vitacura, Santiago, Chile ,
A. T. Deller66affiliation: The Netherlands Institute for Radio Astronomy
(ASTRON), 7990-AA Dwingeloo, Netherlands , C. P. Dullemond77affiliation:
Heidelberg University, Center for Astronomy, Albert Ueberle Str 2, Heidelberg,
Germany , J. S. Greaves88affiliation: University of St. Andrews, Physics and
Astronomy, North Haugh, St. Andrews KY16 9SS, UK , R. J. Harris99affiliation:
Department of Astronony, University of Illinois, Urbana, IL 61810, USA , Th.
Henning1010affiliation: Max Planck Institut für Astronomie, Königstuhl 17,
69117 Heidelberg, Germany , W. Kwon1111affiliation: SRON Netherlands Institute
for Space Research, Landleven 12, 9747 AD Groningen, The Netherlands ,
J. Lazio1212affiliation: Jet Propulsion Laboratory, California Institute of
Technology, 4800 Oak Grove Dr., Pasadena, CA 91106, USA , H.
Linz1010affiliation: Max Planck Institut für Astronomie, Königstuhl 17, 69117
Heidelberg, Germany , L. G. Mundy1313affiliation: Department of Astronomy,
University of Maryland, College Park, MD 20742, USA , A. I.
Sargent33affiliation: California Institute of Technology, 1200 East California
Blvd., Pasadena, CA 91125, USA , S. Storm1313affiliation: Department of
Astronomy, University of Maryland, College Park, MD 20742, USA , L.
Testi1414affiliation: European Southern Observatory, Karl Schwarzschild Str 2,
85748, Garching, Germany 1515affiliation: INAF-Osservatorio Astrofisico di
Arcetri, Largo E. Fermi 5, 50125 Firenze, Italy<EMAIL_ADDRESS>
###### Abstract

We present sub-arcsecond resolution observations of continuum emission
associated with the GG Tau quadruple star system at wavelengths of 1.3, 2.8,
7.3, and 50 mm. These data confirm that the GG Tau A binary is encircled by a
circumbinary ring at a radius of 235 AU with a FWHM width of $\sim$60 AU. We
find no clear evidence for a radial gradient in the spectral shape of the
ring, suggesting that the particle size distribution is spatially homogeneous
on angular scales $\gtrsim$0$\farcs$1. A central point source, likely
associated with the primary component (GG Tau Aa), exhibits a composite
spectrum from dust and free-free emission. Faint emission at 7.3 mm is
observed toward the low-mass star GG Tau Ba, although its origin remains
uncertain. Using these measurements of the resolved, multifrequency emission
structure of the GG Tau A system, models of the far-infrared to radio spectrum
are developed to place constraints on the grain size distribution and dust
mass in the circumbinary ring. The non-negligible curvature present in the
ring spectrum implies a maximum particle size of 1–10 mm, although we are
unable to place strong constraints on the distribution shape. The
corresponding dust mass is 30–300 $M_{\oplus}$, at a temperature of 20–30 K.
We discuss how this significant concentration of relatively large particles in
a narrow ring at a large radius might be produced in a local region of higher
gas pressures (i.e., a particle “trap”) located near the inner edge of the
circumbinary disk.

protoplanetary disks — radio continuum: planetary systems — stars: individual
(GG Tau) — ISM: dust

##  1 Introduction

The first step of planet formation — the collisional growth of $\mu$m-sized
dust grains into $>$km-sized planetesimals, the building blocks of terrestrial
planets and the cores of giant planets — is fundamental, but physically
complicated and fraught with theoretical uncertainty. A substantial effort
with numerical simulations and laboratory experiments is converging on a basic
model framework for the growth and migration of solids embedded in a
protoplanetary gas disk (see the recent reviews by Testi et al., 2014;
Johansen et al., 2014), but direct astronomical observations of these solids
are required to test and refine it. Thermal continuum emission at mm/radio
wavelengths is well-suited for that task, as a (relatively) bright and
optically thin tracer of solid particles with sizes up to $\sim$10 cm. The
spectral behavior of this emission is diagnostic of the particle size
distribution (e.g., Beckwith & Sargent, 1991; Miyake & Nakagawa, 1993; Henning
& Stognienko, 1996; D’Alessio et al., 2001; Draine, 2006; Ricci et al., 2010b,
a). Therefore, spatially resolved measurements of the mm/radio “colors” can be
used to map out how the particle growth and transport efficiencies vary as a
function of the local physical conditions in the gas disk (Isella et al.,
2010; Banzatti et al., 2011; Guilloteau et al., 2011; Pérez et al., 2012;
Trotta et al., 2013; Menu et al., 2014).

These preliminary studies of the resolved multifrequency continuum emission
from disks indicate that the inward radial transport of mm/cm-sized solids is
a crucial factor for explaining the observed color gradients (cf., Birnstiel
et al., 2012). Birnstiel & Andrews (2014) suggested that this same radial
drift, induced by aerodynamic drag on particles that are partially coupled to
the gas in its sub-Keplerian velocity field (Adachi et al., 1976;
Weidenschilling, 1977), is also responsible for the observed discrepancies
between the sizes of the line and continuum emission in some disks (e.g.,
Panić et al., 2009; Andrews et al., 2012; Rosenfeld et al., 2013b). However,
there is a fundamental issue with the transport timescales: in these idealized
models, radial drift is much too efficient (Takeuchi & Lin, 2002, 2005; Brauer
et al., 2007). Perhaps the most promising option for slowing (or stopping)
this transport mechanism is with a “bump” in the radial gas pressure profile
(e.g., Whipple, 1972), either locally and stochastically in over-densities
generated by turbulence (e.g., Klahr & Henning, 1997; Pinilla et al., 2012b)
or globally and with long duration in density concentrations produced near
sharp ionization boundaries (just outside a “dead” zone; e.g., Dzyurkevich et
al., 2010) or through dynamical interactions with a companion (e.g., Pinilla
et al., 2012a).

The most obvious case in which the latter scenario is relevant is for a
circumbinary disk, where dynamical interactions between the stars and gas
reservoir clear the disk material inside a radius $\sim$3$\times$ larger than
the binary separation (e.g., Artymowicz & Lubow, 1994). The steep gas pressure
gradient created by this clearing will trap particles in a circumbinary
“ring”, which itself might have significantly enhanced pressures due to the
dynamical excitation of density waves (e.g., Artymowicz & Lubow, 1996; Kley &
Dirksen, 2006; Hayasaki & Okazaki, 2009). GG Tau is the canonical example of a
close young stellar pair ($\sim$0$\farcs$25 projected separation; Leinert et
al., 1991; Ghez et al., 1993) with a prominent circumbinary ring, which has
been resolved and extensively studied in mm-wave continuum and molecular line
emission (Dutrey et al., 1994; Guilloteau et al., 1999; Piétu et al., 2011) as
well as scattered light in the optical and near-infrared (Roddier et al.,
1996; Silber et al., 2000; Krist et al., 2002, 2005; McCabe et al., 2002; Itoh
et al., 2002; Duchêne et al., 2004). The (unresolved) radio spectrum indicates
that $\sim$mm/cm-sized particles are present in the GG Tau circumbinary ring
(e.g., Guilloteau et al., 1999; Rodmann et al., 2006; Scaife, 2013), lending
some additional support to the idea that radial drift is halted near the ring
edge.

Here we present resolved measurements of continuum emission from the GG Tau
system at wavelengths of 1.3, 2.8, 7.3, and 50 mm, in an effort to
characterize the dust population in the GG Tau A circumbinary ring. The
observations and data calibration are presented in Section 2. Models of the
resolved emission and broadband spectrum are developed and analyzed in Section
3\. The results are discussed in the context of current ideas about the
evolution of disk solids in Section 4.

##  2 Observations and Data Reduction

GG Tau was observed with the 15-element ($6\times 10.4$ m and $9\times 6.1$ m
antennas) Combined Array for Research in Millimeter Astronomy (CARMA) in its C
and B configurations (30–350 m and 100–1000 m baselines, respectively) with
the 1 mm receivers on 2007 September 17 and November 26, and in the B
configuration with the 3 mm receivers on 2008 January 8, January 17, and
February 15. In the former, the correlator was set up to process two 500 MHz
basebands with coarse spectral resolution in each sideband (2 GHz of total
bandwidth), with a local oscillator frequency of 228 GHz ($\lambda=1.31$ mm).
For the latter, a third 500 MHz baseband was added to each sideband (3 GHz
bandwidth), and the receivers were tuned to 106 GHz ($\lambda=2.83$ mm). The
observations alternated between GG Tau and the nearby quasars J0530+135,
J0510+180, J0431+206, J0449+113, and 3C 111, with a source–calibrator cycle
time of 12–15 minutes. Additional observations of Uranus were made for
calibration purposes. The atmospheric conditions were generally good, with 230
GHz opacities of $\sim$0.15–0.20 and 0.2–0.3 during the 1.3 and 2.8 mm
observations, respectively.

The raw visibilities were calibrated and subsequently imaged using standard
tasks in the MIRIAD software package. The passband shape across the coarse
continuum channels was calibrated using observations of J0530+135, and the
(antenna-based) complex gain response of the array to both instrumental and
atmospheric variations was determined from repeated observations of J0530+135
or 3C 111. Observations of J0510+180, J0431+206, and J0449+113 were used to
assess the quality of the gain calibration; the contribution of decoherence
due to small baseline errors and atmospheric phase noise is found to be small,
representing a “seeing” disk with FWHM $\leq$0$\farcs$1\. The absolute
amplitude scales were set by bootstrapping J0530+135 flux densities from
observations of Uranus, with systematic uncertainties of $\sim$10%. Wideband
continuum visibilities were created by averaging the spectra across the
sampled passbands. These calibrated continuum visibilities were Fourier
inverted assuming natural weighting, deconvolved with the CLEAN algorithm, and
restored with a synthesized beam to produce the emission maps shown in Figure
1. The 1.3 mm visibilities were tapered with a 0$\farcs$1 FWHM Gaussian kernel
before inversion to improve the resulting image quality. The 1.3 mm continuum
map shown in Figure 1 has a $0\farcs 67\times 0\farcs 57$ synthesized beam (at
P.A. = 132°) and an RMS noise level of 2.3 mJy beam-1; the corresponding 2.8
mm map has a $1\farcs 19\times 0\farcs 76$ beam (at P.A. = 118°) and an RMS
noise level of 0.5 mJy beam-1.

Figure 1: Synthesized continuum images of the GG Tau field at (from left to
right) wavelengths of 1.3, 2.8, 7.3, and 50 mm. Contours are drawn at 3
$\sigma$ intervals in each panel (7, 1.5, 0.04, and 0.02 mJy beam-1 from left
to right), and synthesized beam dimensions are marked in the lower left
corners. The primary beam for the CARMA 1.3 mm image is shown as a dotted gray
curve. The key emission components are labeled in the VLA Q-band (7.3 mm)
image, including the circumbinary dust ring around GG Tau Aab, faint radio
emission toward the low-mass binary GG Tau Bab (from component Ba), and the
bright radio emission from the background galaxy GG Tau/N.

New observations of the GG Tau field were also made as part of the
“Disks@EVLA” key project (project code AC982) with the 27-element (25 m
diameter antennas) Karl G. Jansky Very Large Array (VLA), employing the Q-band
receivers in the C configuration (35 m to 3.4 km baselines) on 2010 November
27, and the C-band receivers in the A configuration (0.7–36.4 km baselines) on
2011 July 26. For the Q-band observations, the recently upgraded correlator
was configured to process two contiguous 1 GHz basebands centered around 41.1
GHz ($\lambda=7.29$ mm), each comprising eight 128 MHz-wide spectral windows
with 64 channels. The C-band correlator configuration had a similar setup,
with the 1 GHz basebands centered at 4.5 and 7.5 GHz, for a mean frequency of
6 GHz ($\lambda=5.0$ cm). The observations cycled between GG Tau and the
nearby calibrator J0431+2037 at $\sim$3 and 10 minute intervals for the Q- and
C-bands, respectively. The bright quasars 3C 84 and 3C 147 were also observed
for calibration purposes.

The raw visibilities were calibrated and imaged using the “Disks@EVLA”
pipeline in the CASA software package (now the standard pipeline for high
frequency VLA observations111see https://science.nrao.edu/facilities/vla/data-
processing/pipeline). After flagging the data for radio frequency interference
and other minor issues, the observations of 3C 84 were used to calibrate the
spectral bandpass response of the system after bootstrapping flux densities in
each spectral window from observations of 3C 147 (to properly treat the shape
of the 3C 84 spectrum over the wide relative bandwidth). The Q-band
visibilities were then spectrally averaged into 16 pseudo-continuum sub-bands
(one per 128 MHz spectral window); the C-band data were not averaged, to
minimize bandwidth-smearing. Complex gain variations were calibrated with the
frequent observations of J0431+2037, and the absolute amplitude scale was
determined using a frequency-dependent emission model for the standard flux
density calibrator 3C 147 (Perley & Butler, 2013). The systematic uncertainty
in the amplitude scale is $\sim$10% at Q-band and 5% at C-band. The calibrated
visibilities were Fourier inverted assuming natural weighting, deconvolved
with the multi-frequency synthesis version of CLEAN, and restored with a
synthesized beam to make the composite continuum maps shown in Figure 1. The
Q-band map has a $0\farcs 86\times 0\farcs 61$ synthesized beam (at P.A. =
127°) and an RMS noise level of 13 $\mu$Jy beam-1, and the C-band map has a
$0\farcs 51\times 0\farcs 36$ beam (at P.A. = 127°) with an RMS noise level of
6 $\mu$Jy beam-1. The C-band image reconstruction required substantial extra
care, including imaging of the entire primary beam and faceting on
exceptionally bright background sources, to minimize artifacts at the field
center. The images shown in Figure 1 are not corrected for the response of the
primary beam.

##  3 Analysis and Results

The multifrequency continuum images in Figure 1 show emission from three
distinct components: (1) an extended ring around the close binary GG Tau A at
1.3, 2.8, and 7.3 mm, along with point-like emission at its center detected at
1.3, 7.3, and 50 mm;222It is worth pointing out that we do not detect the
“streamer” identified by Piétu et al. (2011); its estimated surface brightness
is comparable to the RMS noise level in our 1.3 mm map, and would likely lie
well below the noise floor at longer wavelengths if its origin is thermal dust
emission. (2) faint, unresolved emission at 7.3 mm associated with the low-
mass star GG Tau Ba (no emission is found toward its $\sim$1$\farcs$5
companion Bb); and (3) bright, unresolved emission at 7.3 and 50 mm from the
(presumed) extragalactic interloper GG Tau/N. The composite flux densities,
$S_{\nu}$, or upper limits for each of these components are compiled in Table
1 and displayed together in Figure 2 (note that the measurements for the GG
Tau B and N components in this figure and table were determined from images
that were corrected for the response of the primary beam). For completeness,
we also include literature measurements of the radio spectra for each
component in this figure.

Figure 2: The mm/radio spectra for the three emission components in the GG Tau
field: A (left; see Section 3.1 for more details), B (middle), and N (right),
where the emission from the latter two components were estimated from images
that have been corrected for the response of the primary beam. Our
measurements are shown in black, and flux densities from the literature are
marked in green (Table 1 lists $S_{\nu}$ for each component). The error bars
represent the formal statistical uncertainties and the systematic calibration
uncertainties, added in quadrature. Upper limits (at 3 $\sigma$) are marked
with a horizontal line and a downward-pointing arrow.

The radio spectral index of GG Tau/N, $\alpha\approx-0.7$ (defined as
$S_{\nu}\propto\nu^{\,\alpha}$), and its non-detection at mm wavelengths
indicate a non-thermal emission mechanism (e.g., Bieging et al., 1984). Given
this index, GG Tau/N is likely a background extragalactic source (AGN)
emitting an optically thin synchrotron spectrum; however, it is worth noting
that there is no confirmed counterpart at any optical or infrared wavelength.
GG Tau B is a low-mass T Tauri binary (White et al., 1999) with a modest
infrared excess (Luhman et al., 2010); the 7.3 mm emission from the primary
(Ba; spectral type M5) found here is the first detection longward of 24
$\mu$m. The nature of this Q-band emission from Ba is not clear, given the
non-detections at other wavelengths. The derived C-band upper limit allows
$\alpha\gtrsim 0.6$, consistent with origins in a magnetically active corona
(White et al., 1994; Cranmer et al., 2013) or dense wind (Reynolds, 1986).
Likewise, non-detections at 1.3 and 2.8 mm suggest that $\alpha\lesssim 2.3$,
which is also commensurate with thermal emission from a relatively cold disk
(as might be expected around such a low-mass host star; e.g., Andrews et al.,
2013) or optically thin emission from large dust particles (e.g., Ricci et
al., 2010b). Variable non-thermal radio emission is an additional (and not
mutually exclusive) possibility.

We focus here on the multifrequency, resolved emission structure from GG Tau
A, itself composed of a narrow circumbinary ring and a point-like contribution
associated with one or both components of the close binary (Dutrey et al.,
1994; Guilloteau et al., 1999; Piétu et al., 2011). In the following sections,
we describe a simple model to quantify these emission components as a function
of the observing frequency, and use those results to assess the emission
origins. First, we establish the basic spatial structure of the GG Tau A
emission (in Section 3.1), and then we employ that resolved information to
model the spectrum and extract physical constraints on the properties of the
constituent solid particles (e.g., mass, temperature, size distribution; in
Section 3.2).

###  3.1 Models for Resolved Emission from GG Tau A

We adopt a simple model prescription for the brightness distribution of the GG
Tau A emission consisting of an azimuthally-symmetric Gaussian ring and a
central point source. The ring is described by seven parameters: a mean radius
$\mu_{r}$, width $\sigma_{r}$, integrated flux density $S_{\nu,r}$ ($=\int
I_{\nu,r}\,d\Omega$), inclination angle $i$ (0° is face-on), minor axis
position angle $\varphi$ (representing the sky-projected orientation of the
ring rotation axis), and two nuisance parameters to account for the ring
center location {$\Delta\alpha_{r}$, $\Delta\delta_{r}$} (defined as arcsecond
offsets in right ascension and declination from the observed phase center).
The point source component includes three additional parameters: a flux
density $S_{\nu,c}$ and projected offsets {$\Delta\alpha_{c}$,
$\Delta\delta_{c}$} relative to the ring center. We assume a distance of 140
pc (e.g., Torres et al., 2009), and compute models for different frequencies
independently.

For a given set of these 10 parameters, we calculate synthetic complex
visibilities, $V_{\nu}$, sampled at the same spatial frequencies, $(u,v)$, as
the relevant observations. We then evaluate a Cauchy log-likelihood function
(cf., Sivia & Skilling, 2006) to represent the probability of the model,
{$V^{\rm M}_{\nu}(u,v)$}, given the observed complex visibilities, {$V^{\rm
D}_{\nu}(u,v)$}, and their uncertainties, {$\sigma^{\rm D}_{\nu}(u,v)$},

$\mathcal{L}\propto\sum_{k}\ln\left(\frac{1-e^{-R_{k}^{2}/2}}{R_{k}^{2}}\right);\,\,\,\,{\rm
where}\,\,\,\,R_{k}=\left(\frac{V^{\rm D}_{\nu}(u_{k},v_{k})-V^{\rm
M}_{\nu}(u_{k},v_{k})}{\sigma^{\rm D}_{\nu}(u_{k},v_{k})}\right).$ (1)

This log-likelihood function was preferred over its more familiar Gaussian
counterpart (where $\mathcal{L}\propto-\chi^{2}/2=\sum_{k}R_{k}^{2}/2$)
because it is more forgiving of outliers (mitigating parameter bias due to
phase calibration systematics on long baselines) and more conservative (which
is important, since our error estimates are derived solely from the visibility
weights, and therefore do not treat scatter due to issues like pointing
errors, atmospheric phase noise, etc.). The posterior probability distribution
function (PDF) was sampled with Monte Carlo Markov Chain (MCMC) calculations,
using the Goodman & Weare (2010) ensemble sampler as implemented by Foreman-
Mackey et al. (2013).

A set of initial MCMC calculations was conducted with uniform priors on all
parameters. However, given the very weak emission from the central point
source component at 1.3 and 2.8 mm, convergence on the relative offset
parameters {$\Delta\alpha_{c}$, $\Delta\delta_{c}$} was prohibitively slow
(and could therefore lead us to biased inferences of $S_{\nu,c}$). In these
initial calculations, we found that the 1.3 mm relative offsets were entirely
consistent with the well-constrained values at 7.3 mm. Therefore, new MCMC
calculations were made with (independent) Gaussian priors on these offsets,
centered on the 7.3 mm values and with standard deviations corresponding to
the inferred 68% marginal widths of their posterior PDFs. In any case, this
had no impact on the inferences for other parameters.

Figure 3: Summary of the posterior PDFs inferred from MCMC calculations in
reference to resolved interferometer data at 1.3, 2.8, and 7.3 mm, assuming a
model composed of a central point source and a Gaussian ring. The staircase
plot to the left shows marginalized two-parameter posterior PDF surfaces, with
contours drawn at 68 and 95% confidence intervals. Marginalized posterior PDFs
for each parameter are shown along the diagonal. The ring and point source
flux densities, $S_{\nu,r}$ and $S_{\nu,c}$ respectively, are normalized by
their best-fit values for clarity. Note that the 4 directional offset
parameters ($\Delta\alpha_{r}$, $\Delta\delta_{r}$, $\Delta\alpha_{c}$, and
$\Delta\delta_{c}$) are not shown, to simplify the plot. The top right panel
is a visual representation of the (normalized) radial surface brightness
profiles derived from this analysis; the widths of these profiles represent
the 68% confidence intervals.  Figure 4: Comparisons of the data and best-fit
models. The left-hand panels show the data and the corresponding images
synthesized from the model and residual visibilities in the same way as the
data. Contour levels are as in Figure 1. A cross marks the ring center and
orientation in the residuals panel. The rightmost panel in each row shows the
azimuthally-averaged (real) visibility profiles, deprojected according to the
derived ring geometry (data in black, models in red).

Figure 3 is a representation of the sampled posterior PDFs in the form of a
staircase diagram, showing both pair-wise parameter covariances (contours are
drawn at the 68 and 95% confidence intervals) and marginalized posterior PDFs
for individual parameters (the four offset parameters are not shown, and the
flux density parameters are normalized to the peaks of their marginal
posterior PDFs, for the sake of clarity). The panel in the upper right corner
is a visualization of the (area-normalized) radial surface brightness profiles
reconstructed from random draws to the joint posterior PDFs, where the shaded
widths of each profile are representative of the 68% (i.e., $\sim$1 $\sigma$)
confidence intervals. Table 2 summarizes the modeling results, listing the
“best-fit” (peaks of the marginal posterior PDFs) parameter values and their
68% confidence intervals at each observing frequency. A direct comparison of
the data and best-fit models is shown in Figure 4, in both the image plane and
with the azimuthally-averaged (and deprojected) visibility profiles.

Within the uncertainties, the GG Tau A circumbinary ring has the same mean
radius and width (as well as inclination, orientation, and center) at all
observed frequencies; $\mu_{r}\approx 230$–240 AU ($\sim$1$\farcs$7) and
$\sigma_{r}\approx 20$–30 AU (corresponding to a FWHM of 0.3–0.5″; i.e., it is
only marginally resolved). There is a hint that the ring is slightly narrower
(at the $\sim$1 $\sigma$ level) at 7.3 mm, although the difference is not
statistically (or practically) significant. Taken together, this indicates
that the spectral behavior of the dust continuum emission does not vary
radially across the ring on the angular scales and at the spectral sensitivity
currently available. If we assume that a Gaussian distribution is an
appropriate spatial model and a power-law with frequency is a reasonable
spectral model, we can very roughly estimate from the fitting results derived
here that $\Delta\alpha\lesssim 0.3$ on (radial) angular scales
$\gtrsim$0$\farcs$1\. Variations at finer scales are possible, and perhaps
likely (see Section 4).

There is some non-negligible curvature in the ring spectrum, with a steeper
spectral index at lower frequencies: $\alpha$(41–106 GHz) $\approx 3.7\pm 0.2$
and $\alpha$(106–228 GHz) $\approx 2.6\pm 0.2$ (the quoted uncertainties
account for the $\sim$10% systematics introduced in the amplitude
calibrations). The peak brightness temperature of the ring at 1.3 mm is only
$\sim$0.2 K, confirming that the emission is optically thin. Therefore, the
observed spectral curvature is a by-product of the intrinsic shape of the dust
opacity spectrum (and perhaps cool temperatures; see Section 3.2 for more
details). The ring is not detected at a wavelength of 5 cm. Assuming it has
the same emission morphology as at shorter wavelengths, a limit on its
integrated flux density can be made based on the measured RMS noise level (6
$\mu$Jy beam-1) in the C-band map: we estimate $S_{\nu,r}<40$ $\mu$Jy (3
$\sigma$).

The central point-like component, originally identified at 1.4 and 1.1 mm by
Guilloteau et al. (1999) and Piétu et al. (2011), is also detected here at
1.3, 7.3, and 50 mm. Models with a relatively faint central source at 2.8 mm
(as listed in Table 2) provide better overall matches to the visibility data,
although formally the inference on $S_{\nu,c}$ is only marginally significant
(greater than zero at the $\sim$2.7 $\sigma$ level): alternatively, we could
quote a 3 $\sigma$ upper limit as $S_{\nu,c}<4.0$ mJy. The (sparse) radio
spectrum of this central source includes contributions from different emission
mechanisms. The spectral index at high frequencies is steep, $\alpha$(41-228
GHz) $\approx 2.1\pm 0.2$, and consistent with optically thick thermal
emission from a warm dust disk. The low-frequency radio spectrum is
considerably more shallow, $\alpha$(6–41 GHz) $\approx 1.3\pm 0.1$; when
combined with the thermal spectrum, the radio emission is best explained with
an intrinsically flat spectrum ($\alpha\approx 0$), suggesting a contribution
from optically thin free-free radiation. We find no evidence that this central
component is resolved: models of the 7.3 mm emission that assume a Gaussian
emission distribution (rather than a point source) indicate a radial width
$<$12 AU (3 $\sigma$; this corresponds to a HWHM $<0\farcs 2$). However, our
models suggest that this component is marginally offset from the ring center,
with a sky-projected separation of $90\pm 30$ mas to the southeast (at P.A.
$\approx 149\pm 1$°) measured from the 7.3 mm data. Assuming the recent orbit
calculations of Köhler (2011), and associating the ring center with the
projected binary center of mass (with the $\sim$0.9:1 mass ratio of White et
al., 1999), these constraints on the scale and orientation of the offsets
suggest that the emission likely originates from the primary component GG Tau
Aa. A similar inference was made for the 1.1 mm emission detected by Piétu et
al. (2011), based on the orbit calculations of Beust & Dutrey (2005).

###  3.2 Models of the GG Tau A Spectrum

Having established an empirical model of the resolved multifrequency emission
from GG Tau A, we shift focus to develop a more physically-motivated model of
the entire far-infrared to radio spectrum. The basic goal is to help
characterize the dust population in the circumbinary ring. The emission
structure of GG Tau A that we derived in the previous section is unusually
simple in the context of protoplanetary disks. The circumbinary ring has a
well-defined mean radius and width, and is spatially isolated from the central
component. The absence of a spatial gradient in the ring spectrum, along with
its apparently low optical depth, suggest that the continuum emission is a
reasonably good tracer of the dust column density distribution. Moreover, the
ring itself is sufficiently narrow and distant from the central heating
sources that we expect it should have a near-constant radial temperature
profile (irradiation heating would produce a variation of $\lesssim$ 15%,
roughly 2–3 K, across the ring; see below). So unlike a typical disk, where
large and uncertain gradients in temperature and density can act as severe
obstacles, we have an interesting opportunity to use the mm/radio spectrum and
our structural constraints from the resolved data to extract some constraints
on the size distribution of the solids in the circumbinary ring.

To that end, we define a spectrum model as the sum of two components: thermal
dust emission in the ring (with flux densities $S_{\nu,r}$) and a composite
emission origin associated with the central point source ($S_{\nu,c}$). For
the latter, we assume a double power-law spectrum,

$S^{\rm M}_{\nu,c}=S_{\nu,0}^{\rm dust}\left(\frac{\nu}{{\rm
10\,GHz}}\right)^{\alpha_{\rm dust}}+S_{\nu,0}^{\rm ff}\left(\frac{\nu}{{\rm
10\,GHz}}\right)^{\alpha_{\rm ff}}.$ (2)

And for the dust emission in the ring, we use the classic and simple one-
dimensional thermal continuum model often adopted for disks (Adams et al.,
1987; Beckwith et al., 1990),

$S^{\rm M}_{\nu,r}=\frac{2\pi\cos{i}}{d^{2}}\int
dr\,r\,B_{\nu}(T_{r})\,(1-e^{-\kappa_{\nu}\Sigma_{r}/\cos{i}}),$ (3)

where $d=140$ pc, $i$ is the ring inclination, $B_{\nu}(T_{r})$ is the Planck
function at a given temperature, $\Sigma_{r}$ is a Gaussian surface density
profile with mean $\mu_{r}$, width $\sigma_{r}$, and peak value $\Sigma_{0}$,
and $\kappa_{\nu}$ is the dust opacity spectrum. Model opacity spectra were
derived assuming a power-law grain size ($a$) distribution, with index $q$
(where $dn/da\propto a^{-q}$) and maximum size $a_{\rm max}$ (the minimum size
was set to 0.1 $\mu$m). For easier comparisons with related work, we employed
the Ricci et al. (2010b) dust mixture333From a material composition
standpoint, this mixture is similar to the one advocated by Pollack et al.
(1994)., with volume fractions of 30% vacuum (porosity), 7% silicates, 21%
carbonaceous materials, and 42% water ice, using optical constants from
Weingartner & Draine (2001), Zubko et al. (1996), and Warren (1984),
respectively. The optical constants for the mixture were determined with the
Bruggeman rule, and the corresponding $\kappa_{\nu}$ for any particle size
were computed with a Mie code. Altogether, the model has 11 parameters,
{$S_{\nu,0}^{\rm dust}$, $\alpha_{\rm dust}$, $S_{\nu,0}^{\rm ff}$,
$\alpha_{\rm ff}$, $i$, $T_{r}$, $\Sigma_{0}$, $\mu_{r}$, $\sigma_{r}$,
$a_{\rm max}$, $q$}.

For any parameter combination, we define two residual terms at each frequency
that record the fit quality relative to the resolved measurements of the ring
and point source flux densities,

$R_{\nu,r}=\left(\frac{S^{\rm D}_{\nu,r}-S^{\rm M}_{\nu,r}}{\sigma^{\rm
D}_{\nu,r}}\right)\,\,{\rm and}\,\,\,\,R_{\nu,c}=\left(\frac{S^{\rm
D}_{\nu,c}-S^{\rm M}_{\nu,c}}{\sigma^{\rm D}_{\nu,c}}\right),$ (4)

respectively; the “observed” flux densities, $S^{\rm D}_{\nu}$, and their
associated uncertainties, $\sigma^{\rm D}_{\nu}$, can be found in Table
2.444Here we also add in quadrature a systematic calibration uncertainty term
at each frequency; see Section 2. Moreover, we made use of the unresolved
photometry in the literature to compare with the sum of the model components,
with an additional residual term

$R_{\nu,{\rm tot}}=\left(\frac{S^{\rm D}_{\nu,{\rm tot}}-[S^{\rm
M}_{\nu,r}+S^{\rm M}_{\nu,c}]}{\sigma^{\rm D}_{\nu,{\rm tot}}}\right),$ (5)

at each frequency, where {$S^{\rm D}_{\nu,{\rm tot}}$, $\sigma^{\rm
D}_{\nu,{\rm tot}}$} are compiled in Table 1. These terms are treated as a
combined residual, $R_{\nu}=\\{R_{\nu,r},R_{\nu,c},R_{\nu,{\rm tot}}\\}$, in
evaluating a log-likelihood function as in Eq. (1) (now the summation is over
$\nu$). The same MCMC algorithm utilized in Section 3.1 was employed to
optimize the model and sample the posterior PDFs for the model parameters.
Because there are more parameters than constraints describing the point
source, we adopted Gaussian priors on {$\log{S_{\nu,0}^{\rm dust}}$,
$\alpha_{\rm dust}$, $\log{S_{\nu,0}^{\rm ff}}$, $\alpha_{\rm ff}$}, with
means {-4.65, 2.1, -4.25, 0.0} and widths {0.2, 0.2, 0.2, 0.1} (the
normalizations are in log Jy units) informed by the examination of the
$S_{\nu,c}$ spectrum described above. To incorporate the measurements of the
resolved emission structure found in Section 3.1, we assumed Gaussian priors
for {$\mu_{r}$, $\sigma_{r}$, $i$} with means {235 AU, 25 AU, 37°} and widths
{5 AU, 5 AU, 1°} (see Table 2). Uniform priors were assumed for the other
parameters, {$T_{r}$, $\Sigma_{0}$, $a_{\rm max}$, $q$}.

Figure 5: The far-infrared to radio spectrum of GG Tau A, decomposed into the
circumbinary ring (blue) and central point source (red), along with their
combination (gray). The model spectrum contributions from each component,
based on random draws from the posterior PDF, are overlaid on the data; their
widths represent the 95% ($\sim$2 $\sigma$) confidence boundaries.

The parameter inferences from these fits are summarized in Table 3. The
corresponding model spectra are compared with the data in Figure 5. In terms
of the physical conditions in the ring, we infer a dust temperature of 20–30
K, comparable to the 35 K determined from the CO spectral line emission by
Guilloteau et al. (1999). The difference is plausibly a manifestation of the
modest temperature inversion that would be expected between the cooler
midplane (where the dust emission is generated) and the warmer atmosphere
traced by the CO (e.g., Dartois et al., 2003; Rosenfeld et al., 2013a). The
dust surface density at the peak of the ring is only 0.01–0.03 g cm-2,
implying a total dust mass of $\sim$30–90 M⊕ (or 1–3$\times 10^{-4}$ M⊙). We
find that relatively top-heavy grain size distributions, $q\approx 1.4$–2.7,
provide substantially better fits than the typical assumptions based on models
of collisional cascades (Dohnanyi, 1969) or measurements in the diffuse
interstellar medium (Mathis et al., 1977), where $q\approx 3.5$. With more
mass concentrated near the maximum particle size, $a_{\rm max}\approx 1$–2 mm,
the corresponding dust opacity spectrum ends up having substantial curvature
at wavelengths near $a_{\rm max}$, reproducing well the observed shape of the
mm-wave ring spectrum. A reconstruction of the inferred $\kappa_{\nu}$ is
shown in Figure 6: the 1.3 mm opacity lies in the range of 5–10 cm2 g-1,
$\sim$2–4$\times$ larger than the standard Beckwith et al. (1990) opacity
prescription for disks. Note that the ring is optically thin at all
frequencies of interest here.

Figure 6: Constraints on the dust opacity spectrum ($\sim$2 $\sigma$
confidence intervals) in the GG Tau A circumbinary ring, reconstructed from
random draws of the joint posterior PDF derived from modeling the far-infrared
to radio spectrum. Overlaid as a dashed curve is the standard opacity
prescription used for protoplanetary disks, originally advocated by Beckwith
et al. (1990).

The best-fit model has a $\chi^{2}\approx 62$, and a reduced
$\tilde{\chi}^{2}\approx 1.8$ (46 datapoints, 11 free parameters).555Note that
we do not double-count the ring+central source photometry measured here in the
“combined” model fit, despite listing those measurements in Table 1 (for the
sake of completeness). Some of these residuals are likely due to under-
estimates of flux density uncertainties, particularly for older single-dish
photometers at challenging submillimeter wavelengths. It is worth explicitly
pointing out that the favored models systematically under-predict the Ku-band
(16 GHz) flux density reported by Scaife (2013) at the $\sim$3 $\sigma$ level
(this datapoint alone drives $\chi^{2}$ up by $\sim$10). Perhaps this is due
to the difficulty of disentangling the GG Tau A and N emission in those
data,666As well as any (relatively) small contribution from GG Tau B. where
the resolution was nearly 3$\times$ the A–N separation and the contrast ratio
is high (in her Figure 1, Scaife indicates that N is roughly an order of
magnitude brighter than A at this frequency). Alternatively, it is possible
that the measurement uncertainties are fine, and instead some of the
assumptions made in the modeling are responsible for the inferred (modest)
residuals. To explore that possibility further, we re-fit the data with
several modifications to our critical assumptions.

In motivating the simplicity of the GG Tau A circumbinary ring structure, we
suggested that its dust temperature is roughly constant. However, a radial
temperature gradient could be present, particularly if the inner part of the
ring intercepts a substantial amount of incident irradiation from the central
stars (analogous to the dust sublimation boundaries of most disks, or the
“wall” features just outside transition disk cavities; e.g., Dullemond et al.,
2001; Calvet et al., 2002). As an extreme counter-example, we re-modeled the
spectrum assuming that the dust temperatures vary inversely with radius in the
ring ($T_{r}\propto 1/r$). No significant differences in the model parameters
were found, as might be expected given the narrowness of the ring. This check
validates the simplified assumption that reasonable temperature gradients have
negligible impact on our results.

The model optimization described above currently requires relatively stringent
priors on the central point source parameters (given the sparse data). But
there is considerable leeway in those prior assumptions that could still offer
reasonable fits to the data. To assess the impact of these priors on the main
physical parameters of the circumbinary ring, we re-modeled the spectrum with
a steeper combination of Gaussian priors on the spectral indices, with means
{$\alpha_{\rm dust}$, $\alpha_{\rm ff}$} = {2.5, 1.0}, and adjustments to
their corresponding normalizations (at 10 GHz), {$\log{S^{\rm dust}_{\nu,0}}$,
$\log{S^{\rm ff}_{\nu,0}}$} = {-5.25, -4.25} (in Jy units; the widths of the
priors were the same as used above). We find that adopting these alternative
priors has no notable effect on the key ring parameters; the point source is
simply too faint to matter in this regard (nor does it improve the fit
quality, even for the Ku-band point noted above). That said, the uncertain
origins of the central point source emission will remain an open question
until resolved measurements at additional frequencies are available.

Perhaps the more relevant assumptions made in the modeling concern the nature
of the grains themselves, particularly their porosities and material
compositions. To investigate these issues further, we first re-modeled the GG
Tau A spectrum assuming different (fixed) porosities. Compared to our nominal
parameters (30% porosity), models with “compact” grains (0% porosity) can
explain the data reasonably well if the circumbinary ring has a similar
$T_{r}$, a slightly ($\sim$30–50%) lower dust mass, and a steeper size
distribution ($q\approx 2.5$–3.7) with $a_{\rm max}$ just under 1 mm. For
higher porosities (60%), we instead infer a (5–8$\times$) higher dust mass and
shallow size distribution ($q\approx 1.0$–2.8) up to a maximum size of a few
mm to $\sim$1 cm. Since $a_{\rm max}$ is comparable to the wavelengths of
interest, resonances preferentially enhance the mm-wave opacities for grains
with higher filling factors (lower porosities; e.g., Miyake & Nakagawa, 1993;
Kataoka et al., 2013, but see Cuzzi et al. 2014): a higher $\kappa_{\nu}$
drives us to find lower $\Sigma_{0}$ (and therefore dust mass) for a fixed
spectrum (and vice versa). These same resonances also naturally introduce an
intrinsic curvature into the mm/cm-wave opacity spectrum that is reflected in
the inferred size distribution index ($q$): since compact grains then do not
require the added $\kappa_{\nu}$ curvature produced by a top-heavy size
distribution, it makes sense that we infer a higher $q$ when the porosity is
low (and vice versa). For a fixed composition, grains with $\sim$20–40%
porosity do produce better fits to the data.

Next, we performed a similar experiment that varied the material composition
of the grains, by re-fitting the data with extreme silicate- or carbon-rich
mixtures.777Since water ice plays only a minor role relative to silicates and
carbonaceous material in setting $\kappa_{\nu}$ in the mm/cm wavelength range,
its volume fraction is left fixed (at 42%; cf., Pollack et al., 1994) in this
experiment. Compared to our adopted mixture (with 7% silicates and 21%
carbonaceous material by volume), models with C-rich grains (28% carbon, no
silicates) in the GG Tau A ring have a comparable dust mass and temperature,
and a modestly steeper grain size distribution ($q\approx 2.2$–3.5) up to
$a_{\rm max}\approx 1$ mm. In the opposite case, models with Si-rich grains
tend toward ($\sim$10$\times$) higher dust masses and shallow size
distributions ($q\approx 1.1$–2.6) with $a_{\rm max}\approx 2$–8 mm. Silicate-
rich grains have mm/cm-wave opacity spectra that exhibit less curvature and
are preferentially reduced compared to C-rich grains, properties that
naturally account for our inferences of top-heavy size distributions and
higher dust masses when C is depleted, respectively. That said, models relying
on these extreme compositions produce poor fits to the data; the experiment is
only intended to convey the sense of the variation.

While there are many alternative assumptions along these lines that could be
made in the modeling, the investigations described above demonstrate that most
of the key parameters related to the dust in the GG Tau A circumbinary ring
are relatively robust. Within a reasonably large range of temperature
gradients, point source emission models, grain porosities and compositional
variations, we find dust temperatures of $\sim$20–30 K, dust masses of
$\sim$1–10$\times$10-4 $M_{\odot}$, and maximum grain sizes of $\sim$1–10 mm
(in all cases with preference near the lower end of the quoted ranges).
However, it should be clear that these data do not provide a strong
quantitative constraint on the power-law index of the grain size distribution,
$q$. Finally, as a point of reference, we find that the dust opacity at 1.3 mm
is restricted to $\sim$0.2–20 cm2 g-1 in all the models explored here.

##  4 Discussion and Conclusions

We have obtained and analyzed high angular resolution observations of
continuum emission associated with GG Tau, a young quadruple star system, at
wavelengths of 1.3, 2.8, 7.3, and 50 mm. These data confirm that GG Tau/N is a
background synchrotron source, and identify faint 7.3 mm emission associated
with the low-mass star GG Tau Ba, although the origin of the latter is
unclear. As had been noted previously (e.g., Guilloteau et al., 1999), we find
a bright emission ring and central peak associated with GG Tau A (although the
ring is undetected at a wavelength of 5 cm). The visibility data were modeled
with a simple surface brightness prescription. We found that the emission at
all frequencies could be described well using a (slightly offset) central
point source and a (projected) circular Gaussian ring with a mean radius
($\mu_{r}$) of $235\pm 5$ AU and width ($\sigma_{r}$) of $25\pm 5$ AU (a FWHM
of $60\pm 10$ AU), reasonably consistent with previous constraints based on
slightly different assumptions (Dutrey et al., 1994; Guilloteau et al., 1999;
Piétu et al., 2011). These morphological constraints indicate that there is
negligible radial variation (on angular scales $\gtrsim 0\farcs 1$) of the
mm/cm-wavelength spectrum ($\alpha\lesssim 0.3$, from 1.3–7.3 mm) across the
circumbinary ring.

The spectrum of the point source flattens considerably at the longest radio
wavelengths, suggesting emission contributions from both dust and free-free
radiation. The integrated spectrum of the ring structure exhibits substantial
curvature, becoming steeper at longer wavelengths, which we suggest is a
manifestation of the intrinsic shape of the dust opacity spectrum. We
developed some simple physical models of the observed GG Tau A spectrum, and
found reasonably good fits for dust temperatures ($T_{r}$) of 20–30 K, dust
masses of 30–60 $M_{\oplus}$ (1–3$\times 10^{-4}$ $M_{\odot}$), and relatively
top-heavy grain size distributions ($dn/da\propto a^{-q}$, with $q\approx
1.4$–2.7) up to maximum particle sizes ($a_{\rm max}$) of $\sim$1–2 mm, for
the grain composition of Ricci et al. (2010b) (effectively the Pollack et al.
1994 mixture with 30% porosity). Alternative assumptions about the
mineralogical makeup and porosities of the grains permit a much wider range of
the size index $q$, but still suggest similar temperatures and a relatively
narrow range of masses ($\sim$1–10$\times 10^{-4}$ $M_{\odot}$) and maximum
particle sizes ($\sim$1–10 mm). In any case, the opacity spectrum inferred in
the ring is not described well with a single power-law in the mm/cm wavelength
range; the standard assumption that $\kappa_{\nu}\propto\nu^{\beta}$ is not
appropriate here. Although we have estimated significantly lower masses (up to
a factor of $\sim$5 compared to, e.g., Guilloteau et al., 1999; Andrews &
Williams, 2005) and maximum grain sizes ($\sim$4–40$\times$ smaller than
inferred by Scaife, 2013) in the ring than in previous studies, these
properties still seem remarkably high given the large separation from the
central stars, narrow radial width of the ring, and age of the system
($\sim$1–3 Myr; or rather time available for particle growth).

A promising way to concentrate and grow dust particles up to $\sim$mm sizes at
such large distances, as well as to halt their normally fast inward migration
due to radial drift, relies on creating a “trap” in a local enhancement of the
gas pressure profile (cf., Whipple, 1972). For GG Tau A, this local pressure
maximum would likely be induced by dynamical interactions between the binary
and disk edge (e.g., Artymowicz & Lubow, 1994). But models of the GG Tau A
stellar orbit highlight a potential issue with this interpretation, since the
ring edge predicted by dynamical simulations lies well inside (at a radius of
roughly $\sim$100 AU) the edge location inferred from dust observations (e.g.,
McCabe et al., 2002; Beust & Dutrey, 2005; Köhler, 2011). Pinilla et al.
(2012a) have suggested that a dust trap could reside substantially beyond the
nominal disk “edge” if the gas pressures decrease (relatively) gradually
toward the inner part of the ring. To schematically illustrate this point in
the case of GG Tau, we used the treatment of Birnstiel et al. (2010) to
simulate the evolution of dust particles — both spatially and in size, subject
to growth, fragmentation, viscous diffusion, and radial drift — in a gas disk
that has a (static) power-law surface density profile with a Gaussian tapered
edge at a radius of 100 AU. Tuning the shape of this taper (i.e., the width of
the Gaussian) can push the gas pressure maximum back to a radius comparable to
the observed dust ring location. For a reasonable turbulent viscosity
parameter ($\alpha\approx 0.002$) and density normalization (a total gas mass
of $\sim$0.1 $M_{\odot}$), we can also reproduce the inferred maximum particle
size and width of the dust ring on appropriate timescales ($\sim$1 Myr). This
simulation is also in qualitative agreement with the infrared scattered light
geometry of the GG Tau A ring (e.g., Duchêne et al., 2004), in that it
predicts a similar distribution of $\mu$m-sized grains produced via
fragmentation in the pressure maximum. Figure 7 summarizes this demonstrative
(although by no means unique) example.

Figure 7: A schematic snapshot for an illustrative model of dust evolution in
a truncated gas disk with a gradually tapered inner edge, tuned to the
relevant parameters of the GG Tau A circumbinary disk. We assumed a static
power-law gas surface density profile ($\Sigma_{\rm gas}\propto 1/r$; red
curve), with an “edge” near the circumbinary disk truncation radius expected
from estimates of the GG Tau A orbit, and a gradual Gaussian taper (with mean
100 AU and width 120 AU). The total gas mass was 0.13 $M_{\odot}$ ($\sim$10%
of the binary mass), consistent with the estimate of Guilloteau et al. (1999).
A population of small (0.1–1 $\mu$m, with $dn/da\propto a^{-3.5}$) grains with
radially constant dust-to-gas mass ratio (0.01) was evolved in size and space
for 1 Myr, following the Birnstiel et al. (2010) prescription for a constant
turbulent viscosity parameter ($\alpha=0.002$) and fragmentation velocity (10
m s-1). The surface density distributions of $\mu$m-sized (0.5–5 $\mu$m;
green) and mm/cm-sized (500 $\mu$m $<a<$ 5 cm; blue) particles predicted by
this model are in reasonably good agreement with both infrared scattered light
observations (e.g., Duchêne et al., 2004) and our estimates of $\Sigma_{\rm
dust}$ inferred from the modeling in Section 3 (the gray shaded region
corresponds to the 2 $\sigma$ uncertainties).

Of course, it is not clear if such “soft” edges are physically realistic for
gas-rich circumbinary disks like the one around GG Tau A; detailed
hydrodynamic simulations (e.g., Bate, 2000; Günther & Kley, 2002; Kley et al.,
2008) tuned to this specific case would be required to assess the feasibility
and internal consistency of the basic scenario proposed above (although see
Beust & Dutrey, 2005). Regardless, the key point is that we should be actively
considering the coupled evolution of gas and solids in this and similar disks
when attempting to reconcile models of their tidal truncation with the (still
uncertain) stellar orbital configurations. In principle, one could test the
proposed explanation with sensitive observations of an optically thin line
tracer that tracks the gas densities near and interior to the dust disk edge
at $\sim$0$\farcs$1 resolution.888It is worth noting that this would also
require a proper accounting of the gas temperatures, which might not be
trivial in such a relatively dust-poor environment (see Bruderer, 2013). In
that context, it is compelling to note that Guilloteau et al. (1999) found
that the 13CO emission extends slightly inside the continuum ring in their
study of the GG Tau A disk; similar data at higher resolution would be
valuable. These dust trap models also predict that larger particles should be
concentrated at the maximum pressure, although the implied spectral gradient
in this particular example would not be resolved with our data. In that sense,
pushing the angular resolution of multifrequency continuum observations in
this and other cases should be considered a high priority.

Although the GG Tau A ring is a particularly spectacular (and useful) example,
the dust in circumbinary disks more generally could serve as important test
cases for studying particle traps induced by dynamical interactions with
companions. The so-called “transition” disks, where the companion is
speculated to be a giant planet, exhibit similar features — narrow dust
continuum rings (e.g., Andrews et al., 2011), often having more extended gas
disks with relatively lower dust masses (e.g., Isella et al., 2012; Rosenfeld
et al., 2013b) and occasionally showing direct (Rosenfeld et al., 2012, 2014;
Casassus et al., 2013; van der Marel et al., 2013; Bruderer et al., 2014) or
indirect (e.g., Dong et al., 2012; Follette et al., 2013) evidence for gas
well inside those dust rings. Given that similarity, we suggest that there is
great value in analyzing resolved multifrequency radio observations of dust
rings (and gas structures) in both circumbinary and transition disks,
analogous to the approach presented here. Ultimately, linking these disk
targets – which are undergoing dynamical interactions with a broad range of
companion masses – could offer important insights on how solids grow inside
gas pressure maxima with a wide diversity of strengths and shapes.

We thank Mark Reid for valuable discussions about data modeling and an
anonymous referee for helpful suggestions. S. M. A. and T. B. acknowledge
support from NASA Origins of Solar Systems grant NNX12AJ04G. A. I., L. M. P.,
and J. M. C. acknowledge support from NSF award AST-1109334. Ongoing CARMA
development and operations are supported by the National Science Foundation
under a cooperative agreement, and by the CARMA partner universities. The VLA
is run by the National Radio Astronomy Observatory, a facility of the National
Science Foundation operated under cooperative agreement by Associated
Universities, Inc.

Table 1: Radio Spectra of GG Tau Components

Component |  $\lambda$ (mm) |  $S_{\nu}$ (mJy) | Ref.
---|---|---|---
GG Tau A | 0.10 | $5158\pm 1410$ | IRAS
| 0.10 | $6560\pm 1469$ | Howard et al. (2013)
| 0.14 | $8995\pm 2291$ | AKARI
| 0.15 | $7620\pm 3048$ | Howard et al. (2013)
| 0.16 | $8600\pm 3440$ | Howard et al. (2013)
| 0.16 | $6076\pm 1974$ | AKARI
| 0.18 | $8220\pm 3288$ | Howard et al. (2013)
| 0.19 | $7130\pm 2853$ | Howard et al. (2013)
| 0.35 | $6528\pm 1639$ | Andrews & Williams (2005)
| 0.44 | $4540\pm 766$ | Moriarty-Schieven & Butner (1997)
| 0.44 | $4160\pm 665$ | Moriarty-Schieven & Butner (1997)
| 0.44 | $2726\pm 726$ | Andrews & Williams (2005)
| 0.62 | $1370\pm 382$ | Beckwith & Sargent (1991)
| 0.77 | $1250\pm 323$ | Beckwith & Sargent (1991)
| 0.79 | $1110\pm 163$ | Moriarty-Schieven et al. (1994)
| 0.79 | $1710\pm 176$ | Moriarty-Schieven & Butner (1997)
| 0.79 | $1590\pm 170$ | Moriarty-Schieven & Butner (1997)
| 0.79 | $1650\pm 183$ | Moriarty-Schieven & Butner (1997)
| 0.87 | $1255\pm 138$ | Andrews & Williams (2005)
| 1.06 | $800\pm 206$ | Beckwith & Sargent (1991)
| 1.09 | $740\pm 141$ | Moriarty-Schieven et al. (1994)
| 1.09 | $1070\pm 111$ | Moriarty-Schieven & Butner (1997)
| 1.09 | $830\pm 88$ | Moriarty-Schieven & Butner (1997)
| 1.09 | $850\pm 90$ | Moriarty-Schieven & Butner (1997)
| 1.12 | $770\pm 78$ | Piétu et al. (2011)
| 1.25 | $593\pm 130$ | Beckwith et al. (1990)
| 1.26 | $690\pm 75$ | Moriarty-Schieven & Butner (1997)
| 1.26 | $630\pm 66$ | Moriarty-Schieven & Butner (1997)
| 1.26 | $630\pm 70$ | Moriarty-Schieven & Butner (1997)
| 1.31 | $558\pm 58$ | this paper
| 1.33 | $557\pm 56$ | Harris et al. (2012)
| 1.40 | $604\pm 61$ | Guilloteau et al. (1999)
| 1.92 | $320\pm 68$ | Moriarty-Schieven & Butner (1997)
| 2.68 | $85\pm 10$ | Dutrey et al. (1994)
| 2.78 | $73\pm 15$ | Looney et al. (2000)
| 2.83 | $79\pm 9$ | this paper
| 3.06 | $41\pm 9$ | Ohashi et al. (1991)
| 3.40 | $38\pm 4$ | Guilloteau et al. (1999)
| 6.92 | $3.24\pm 0.77$ | Rodmann et al. (2006)
| 7.29 | $2.67\pm 0.29$ | this paper
| 19.09 | $0.25\pm 0.05$ | Scaife (2013)
| 50.00 | $0.036\pm 0.007$ | this paper
GG Tau Ba | 1.31 | $<9$ | this paper
| 1.33 | $<7$ | Harris et al. (2012)
| 2.83 | $<1.7$ | this paper
| 7.29 | $0.058\pm 0.014$ | this paper
| 50.00 | $<0.02$ | this paper
GG Tau Bb | 1.31 | $<9$ | this paper
| 1.33 | $<7$ | Harris et al. (2012)
| 2.83 | $<1.7$ | this paper
| 7.29 | $<0.04$ | this paper
| 50.00 | $<0.02$ | this paper
GG Tau/N | 1.31 | $<14$ | this paper
| 2.83 | $<2.0$ | this paper
| 7.29 | $0.71\pm 0.07$ | this paper
| 19.09 | $2.23\pm 0.12$ | Scaife (2013)
| 20.03 | $3\pm 1$ | Bieging et al. (1984)
| 50.00 | $2.84\pm 0.14$ | this paper
| 61.37 | $3.7\pm 0.4$ | Bieging et al. (1984)

Note. — The uncertainties on the flux densities include both statistical and
systematic calibration terms (added in quadrature). Upper limits are quoted at
the 3 $\sigma$ level, assuming point source emission. The GG Tau B and N flux
densities were determined after correction for the primary beam responses in
each observation.

Table 2: Inferred Visibility Model Parameters

Parameter | 1.3 mm | 2.8 mm | 7.3 mm
---|---|---|---
$S_{\nu,r}$ (mJy) | $543\pm 21$ | $77\pm 4$ | $2.23\,^{+0.08}_{-0.12}$
$\mu_{r}$ (AU) | $232\pm 3$ | $235\pm 3$ | $234\pm 3$
$\sigma_{r}$ (AU) | $26\pm 5$ | $29\,^{+4}_{-5}$ | $17\,^{+4}_{-8}$
$i$ (°) | $37\pm 1$ | $37\pm 1$ | $37\pm 1$
$\varphi$ (°) | $7\pm 2$ | $7\pm 2$ | $7\pm 2$
$S_{\nu,c}$ (mJy) | $15\,^{+3}_{-7}$ | $2.2\,^{+0.6}_{-0.9}$ | $0.44\,^{+0.02}_{-0.04}$
$\Delta\alpha_{c}$ (″) | $+0.03\pm 0.04$ | $+0.06\pm 0.05$ | $+0.05\pm 0.04$
$\Delta\delta_{c}$ (″) | $-0.10\pm 0.05$ | $-0.09\pm 0.06$ | $-0.08\pm 0.03$

Note. — The quoted uncertainties correspond to 68% ($\sim$1 $\sigma$)
confidence intervals; the associated $\sim$10% calibration uncertainty is not
applied to the flux density parameters. The nuisance parameters describing the
offset of the ring center from the observed phase center are not included.

Table 3: Inferred Spectrum Model Parameters

Parameter | Best-Fit Value ($\pm$1 $\sigma$)
---|---
$\Sigma_{0}$ (g cm-2) | $0.02\pm 0.01$
$\mu_{r}$ (AU) | ($235\pm 5$)
$\sigma_{r}$ (AU) | ($25\pm 5$)
$T_{r}$ (K) | $25\pm 5$
$a_{\rm max}$ (mm) | $1.5\,^{+0.3}_{-0.9}$
$q$ | $2.4\,^{+0.3}_{-1.0}$
$\log{S_{\nu,0}^{\rm dust}}$ (Jy) | (-$4.65\pm 0.20$)
$\alpha^{\rm dust}$ | ($2.1\pm 0.2$)
$\log{S_{\nu,0}^{\rm ff}}$ (Jy) | (-$4.50\pm 0.20$)
$\alpha^{\rm ff}$ | ($0.0\pm 0.1$)

Note. — These parameters are valid for the Ricci et al. (2010b) dust mixture
with 30% porosity. See the text for a discussion of alternative assumptions.
The values in parenthesis reflect the Gaussian priors we assumed for these
fits. The formal reduced $\chi^{2}$ statistic for the best-fit model is
$\sim$1.8 (see text).

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