11institutetext: Università degli Studi di Milano, Via Celoria 16, 20133
Milano, Italy
11email<EMAIL_ADDRESS>22institutetext: European
Southern Observatory, Karl-Schwarzschild-Strasse 2, 85748 Garching bei
München, Germany
22email<EMAIL_ADDRESS>33institutetext: Kavli Institute for Astronomy and Astrophysics, Peking University, Yiheyuan 5, Haidian Qu, 100871 Beijing, China 44institutetext: Center for Astrophysics | Harvard & Smithsonian, 60 Garden St., Cambridge, MA 02138, USA 55institutetext: Institute of Astronomy, University of Cambridge, Madingley Road, CB3 0HA Cambridge, UK 66institutetext: Observatoire de Paris, PSL University, Sorbonne University, CNRS, LERMA, 61 Av. de l’Observatoire, 75014 Paris, France 77institutetext: Academia Sinica Institute of Astronomy and Astrophysics, 11F of Astro-Math Bldg, 1, Sec. 4, Roosevelt Rd, Taipei 10617, Taiwan 88institutetext: Institute of Astronomy, Department of Physics, National Tsing Hua University, Hsinchu, Taiwan 99institutetext: Univ. Grenoble Alpes, CNRS, IPAG, F-38000 Grenoble, France 1010institutetext: Max-Planck-Institut für Astronomie, Königstuhl 17, 69117, Heidelberg, Germany. 1111institutetext: Mullard Space Science Laboratory, University College London, Holmbury St Mary, Dorking, Surrey RH5 6NT, UK. 1212institutetext: School of Physics and Astronomy, University of Leicester, Leicester LE1 7RH, UK 1313institutetext: Univ Lyon, Univ Lyon1, Ens de Lyon, CNRS, Centre de Recherche Astrophysique de Lyon UMR5574, F-69230, Saint-Genis-Laval, France

# Observational constraints on disc sizes
in protoplanetary discs in multiple systems in the Taurus region

II. Gas disc sizes

A.A. Rota Observational constraints on disc sizes  in protoplanetary discs in
multiple systems in the Taurus regionObservational constraints on disc sizes
in protoplanetary discs in multiple systems in the Taurus regionObservational
constraints on disc sizes  in protoplanetary discs in multiple systems in the
Taurus regionObservational constraints on disc sizes  in protoplanetary discs
in multiple systems in the Taurus region C.F. Manara Observational constraints
on disc sizes  in protoplanetary discs in multiple systems in the Taurus
regionObservational constraints on disc sizes  in protoplanetary discs in
multiple systems in the Taurus region A. Miotello Observational constraints on
disc sizes  in protoplanetary discs in multiple systems in the Taurus
regionObservational constraints on disc sizes  in protoplanetary discs in
multiple systems in the Taurus region G. Lodato Observational constraints on
disc sizes  in protoplanetary discs in multiple systems in the Taurus
regionObservational constraints on disc sizes  in protoplanetary discs in
multiple systems in the Taurus region S. Facchini Observational constraints on
disc sizes  in protoplanetary discs in multiple systems in the Taurus
regionObservational constraints on disc sizes  in protoplanetary discs in
multiple systems in the Taurus region M. Koutoulaki Observational constraints
on disc sizes  in protoplanetary discs in multiple systems in the Taurus
regionObservational constraints on disc sizes  in protoplanetary discs in
multiple systems in the Taurus region G. Herczeg Observational constraints on
disc sizes  in protoplanetary discs in multiple systems in the Taurus
regionObservational constraints on disc sizes  in protoplanetary discs in
multiple systems in the Taurus region F. Long Observational constraints on
disc sizes  in protoplanetary discs in multiple systems in the Taurus
regionObservational constraints on disc sizes  in protoplanetary discs in
multiple systems in the Taurus region M. Tazzari Observational constraints on
disc sizes  in protoplanetary discs in multiple systems in the Taurus
regionObservational constraints on disc sizes  in protoplanetary discs in
multiple systems in the Taurus region S. Cabrit Observational constraints on
disc sizes  in protoplanetary discs in multiple systems in the Taurus
regionObservational constraints on disc sizes  in protoplanetary discs in
multiple systems in the Taurus region D. Harsono Observational constraints on
disc sizes  in protoplanetary discs in multiple systems in the Taurus
regionObservational constraints on disc sizes  in protoplanetary discs in
multiple systems in the Taurus regionObservational constraints on disc sizes
in protoplanetary discs in multiple systems in the Taurus regionObservational
constraints on disc sizes  in protoplanetary discs in multiple systems in the
Taurus region F. Ménard Observational constraints on disc sizes  in
protoplanetary discs in multiple systems in the Taurus regionObservational
constraints on disc sizes  in protoplanetary discs in multiple systems in the
Taurus region P. Pinilla Observational constraints on disc sizes  in
protoplanetary discs in multiple systems in the Taurus regionObservational
constraints on disc sizes  in protoplanetary discs in multiple systems in the
Taurus regionObservational constraints on disc sizes  in protoplanetary discs
in multiple systems in the Taurus regionObservational constraints on disc
sizes  in protoplanetary discs in multiple systems in the Taurus region G. van
der Plas Observational constraints on disc sizes  in protoplanetary discs in
multiple systems in the Taurus regionObservational constraints on disc sizes
in protoplanetary discs in multiple systems in the Taurus region E. Ragusa
Observational constraints on disc sizes  in protoplanetary discs in multiple
systems in the Taurus regionObservational constraints on disc sizes  in
protoplanetary discs in multiple systems in the Taurus regionObservational
constraints on disc sizes  in protoplanetary discs in multiple systems in the
Taurus regionObservational constraints on disc sizes  in protoplanetary discs
in multiple systems in the Taurus region Hsi-Wei Yen Observational constraints
on disc sizes  in protoplanetary discs in multiple systems in the Taurus
regionObservational constraints on disc sizes  in protoplanetary discs in
multiple systems in the Taurus region

(Received April 9, 2021; accepted -)

The formation of multiple stellar systems is a natural by-product of the star-
formation process, and its impact on the properties of protoplanetary discs
and on the formation of planets is still to be fully understood. To date, no
detailed uniform study of the gas emission from a sample of protoplanetary
discs around multiple stellar systems has been performed. Here we analyse new
ALMA observations at a $\sim$21 au resolution of the molecular CO gas emission
targeting discs in eight multiple stellar systems in the Taurus star-forming
regions. 12CO gas emission is detected around all primaries and in seven
companions. With these data, we estimate the inclination and the position
angle for all primary discs and for five secondary or tertiary discs, and
measure the gas disc radii of these objects with a cumulative flux technique
on the spatially resolved zeroth moment images. When considering the radius
including 95% of the flux as a metric, the estimated gas disc size in multiple
stellar systems is found to be on average $\sim 4.2$ times larger than the
dust disc size. This ratio is higher than what was recently found in a
population of more isolated and single systems. On the contrary, when
considering the radius including 68% of the flux, no difference between
multiple and single discs is found in the distribution of ratios. This
discrepancy is due to the sharp truncation of the outer dusty disc observed in
multiple stellar systems. The measured gas disc sizes are consistent with
tidal truncation models in multiple stellar systems assuming eccentricities of
$\sim 0.15$-$0.5$, as expected in typical binary systems.

###### Key Words.:

Protoplanetary disks - binaries: visual - binaries: general - Stars: formation
- Stars: variables: T Tauri, Herbig Ae/Be

##  1 Introduction

The collapse and fragmentation of molecular cloud cores frequently leads to
the formation of binary stars or higher order multiple systems (e.g., Bate
2018; Reipurth et al. 2014). In a tight multiple system, the outer parts of
the individual discs in the system are affected by tidal forces from the
companions, usually resulting in disc truncation. The effect of tidal
disruption of discs on the ability of protoplanetary discs in multiple systems
to form planets is still a matter of investigation.

From an observational point of view, exoplanet detection surveys were
initially strongly biased towards binary systems with separation $\geq 200$ au
(Eggenberger & Udry 2010), because these searches focused on stellar
environments as similar as possible to the solar system. However, in 2003 the
first exoplanet in a close binary was detected in the $\gamma$ Cephei system
(Hatzes et al. 2003) and exoplanets have been detected in many multiple
systems (Hatzes 2016), also in close-binaries with separation $\sim 20$ AU. In
particular, 217 exoplanets are known in binary-star systems and 51 in higher
multiplicity systems (updated to Sep 2021)111The data of all detected planets
are collected in the Exoplanet-catalogue maintained by J. Schneider
(http://exoplanet.eu); whereas the binary and higher multiplicity systems can
be found separately in the catalogue of exoplanets in binary star systems
maintained by R. Schwarz (Schwarz et al. 2016;
http://www.univie.ac.at/adg/schwarz/multiple.html)..

In multiple systems, dynamical interactions experienced by circumstellar discs
affect their evolution (Papaloizou & Pringle 1977; Artymowicz & Lubow 1994;
Rosotti & Clarke 2018; Zagaria et al. 2021a) and may have a negative impact on
the formation and evolution of planetary systems. Tidal effects due to
interactions may lead the outer parts of the discs to become strongly
misaligned or warped (Lodato & Facchini 2013), impacting planet migration and
orbit stability and evolution (Holman & Wiegert 1999; Jensen & Akeson 2014).
Moreover, disc truncation may affect the availability of the building blocks
of planets (Zsom et al. 2011; Kraus et al. 2016) and may reduce disc lifetimes
of either or both the disc around the primary or secondary component (Monin et
al. 2007). These considerations could suggest that multiplicity reduces the
probability of planet formation, since there is insufficient mass to form
planets or insufficient time for planetesimal formation to operate before the
disc is dispersed.

The detection of planets in multiple stellar systems and in close-binaries has
triggered studies investigating how such planets could form and, more
generally, about how planet-formation is affected by binarity (Thebault &
Haghighipour 2015). In particular, to investigate these two aspects, the
observation of circumstellar discs in multiple systems is crucial to study how
discs evolve and how interactions between discs in multiple systems affect
their evolution.

Surveys of multiple stellar systems conducted with the Submillimeter Array
(SMA) and the Atacama Large Millimeter/submillimeter Array (ALMA) are
providing constraints on the theory of disc evolution in multiple systems. In
particular, observations of discs around multiple stars in the Taurus and
$\rho$-Ophiucus regions (Harris et al. 2012; Akeson & Jensen 2014; Cox et al.
2017; Akeson et al. 2019; Manara et al. 2019; Long et al. 2019; Zurlo et al.
2021) show that discs in multiple systems are on average fainter in the
(sub)millimeter at any given stellar mass than those in single systems. The
dusty disc around the primary component, the more massive star, is usually
brighter and more massive than the disc around the secondary component.

Manara et al. (2019) discussed a sample of ten multiple stellar systems in the
Taurus star-forming region, composed by eight binaries and two higher-order
multiple systems $-$ UZ Tau (e.g., White & Ghez 2001) and T Tau (e.g., Köhler
et al. 2016). The sample was part of a high angular resolution ($\sim
0\aas@@fstack{\prime\prime}12$) and high sensitivity ($\sim
50\leavevmode\nobreak\ $\mathrm{\SIUnitSymbolMicro}$$Jy beam-1 at 225 GHz -
1.33 mm) survey of 32 protoplanetary discs around stars with spectral type
earlier than M3 (Long et al. 2018, 2019) and covers a wide range of system
parameters. Manara et al. (2019) showed that, in general, the dust radii of
the discs around secondary or tertiary components of multiple systems are
statistically smaller than the sizes of the discs around primary stars in
multiple systems, which are in turn smaller than the discs around single stars
and whose brightness profiles present a steeper outer edge than singles. The
typical measured ratio between $R_{\mathrm{disc,dust}}$ and the projected
separation ($a_{\mathrm{p}}$) was $\lesssim 0.1$. This result can be
reconciled with tidal truncation models (e.g., Papaloizou & Pringle 1977;
Artymowicz & Lubow 1994) only in the highly improbable case that the observed
binary systems were all on highly eccentric ($e>$ 0.7) orbits. Assuming that
dust radii are smaller than gas radii by factors $\gtrsim 2-3$ (e.g., Ansdell
et al. 2018), possibly due to a more effective drift of the dust probed by 1.3
mm observations, would result in values of eccentricities in agreement with
the expected ones, $0\lesssim e\lesssim 0.5$ (e.g., Duchêne & Kraus 2013).
However, the analysis above was performed using the dust disc radii estimates,
since the data available did not cover the gas emission in these discs, which
is however the component that we need to compare with current models of tidal
truncation as this is the one that better responds to the disc dynamics. By
considering the effect of dust growth and drift in truncated systems, Zagaria
et al. (2021b) recently showed that the results by Manara et al. (2019) can be
reconciled with tidal truncation models without the need to invoke extremely
high eccentric orbits. This result must however be confirmed also
observationally by measuring the gas size of discs around multiple systems.

In this work, we analyse new high-resolution ALMA observations of line and
continuum emission in the same sample of multiple stellar systems in the
Taurus star-forming region as of Manara et al. (2019). Our aim is to test
tidal truncation models by measuring for the first time the gas size of discs
in multiple stellar systems in a uniform way, with the same instrument and a
similar noise level at high spatial resolution.

The paper is organised as follows. In Sect. 2 we present the sample analysed
in our work. In Sect. 3 we discuss the observations, and how the data were
calibrated. In Sect. 4 we analyse the new ALMA observations, deriving the
geometrical properties of the discs and the disc radii. We then discuss the
results in Sect. 5, comparing the measured radii to theoretical models to
constrain the eccentricities of the orbits expected with our data. Finally, we
draw our conclusions in Sect. 6.

Figure 1: 12CO zeroth moment images of the discs around multiple stars in the
Taurus star-forming region observed in the new ALMA observations. All bars in
the bottom right of each panel are 1′′long, which is $\sim 140$ au at the
typical distance of the Taurus region. The FWHM beam size is shown in the
bottom left of each panel. In each image, the color scale has the peak flux as
the maximum, and two times the image rms as minimum. White contours show 5
times the rms of the continuum emission. The components of the systems are
labelled. The label for any undetected secondary components in the continuum
emission is not shown.

##  2 Sample properties

The sample of targets analysed in this work comprises eight targets. Seven
binary systems – CIDA 9, DH Tau, DK Tau, HK Tau, HN Tau, RW Aur, V710 Tau, and
one quadruple system – UZ Tau. The primary components of the CIDA 9 and UZ Tau
systems are very bright, and their circumprimary discs show ring-like
structures, while the other systems have smooth discs around the primary stars
(Long et al. 2018).

Eleven systems were originally selected for the apparent multiplicity of their
stars (10 systems from Manara et al. 2019 and GK Tau from Long et al. 2019).
New ALMA observations were requested to detect CO lines in nine targets out of
ten of Manara et al. (2019) (i.e. CIDA 9, DH Tau, DK Tau, HK Tau, HN Tau, T
Tau, UY Aur, UZ Tau, V710 Tau) and in GK Tau. For the last target of Manara et
al. (2019), RW Aur, CO observations with similar resolution and sensitivity
were already available in the ALMA archive (Rodriguez et al. 2018). Three
sources – GK Tau, UY Aur, and T Tau – were then excluded from the analysis
performed in this work. GK Tau and GI Tau are possibly part of a bound system.
However, considering the high separation of the targets ($\sim 1700$ au;
Hartigan et al. 1994, Herczeg & Hillenbrand 2014)) the sizes of both GI Tau
and GK Tau are not affected by tidal truncation (Pearce et al. 2020), and are
thus excluded from the analysis in this work. Finally, the observation of UY
Aur and T Tau do not show a velocity pattern compatible with Keplerian
rotation (see Section 3), making it difficult to determine if the gas emission
is reflecting the disc dynamics, and therefore to estimate the radius of
individual discs. The data for these two objects will be analysed in future
work.

Table 2 reports information on the eight multiple stellar systems analysed in
this work. We assumed the stellar masses inferred by Manara et al. (2019). In
that work, the effective temperature was used with the stellar luminosity
corrected assuming the Gaia DR2 distances of the targets (Gaia Collaboration
et al. 2018), and masses were obtained assuming the evolutionary models by
Baraffe et al. (2015) and the nonmagnetic models by Feiden (2016), as already
done in Pascucci et al. (2016) and Long et al. (2019). Only in two cases the
stellar masses are derived from orbital dynamics and not as just described: UZ
Tau E (Simon et al. 2000) and HN Tau A (Simon et al. 2017).

The projected separations ($a_{\mathrm{p}}$, assumed from Manara et al. 2019,
and references therein) of the individual components in the systems range from
$\sim 49$ au to 461 au, the mass ratios ($q=M_{2}/M_{1}$) from $\sim 0.9$ to
$\sim 0.1$, and the mass parameters ($\mu=M_{1}M_{2}/(M_{1}+M_{2})$) from
$\sim 0.1$ to $\sim 0.5$.

Table 1: Information on the stellar and multiplicity properties of the targets
analysed in this work

|  $a_{\mathrm{p}}$ [′′] |  $d$ [pc] | SpT1 | SpT2 |  $M_{1}$ [$M_{\odot}$] |  $M_{2}$ [$M_{\odot}$] | $q$ | $\mu$
---|---|---|---|---|---|---|---|---
CIDA 9 | 2.35 | 171 | M2 | M4.5 | 0.43${}^{+0.15}_{-0.10}$ | 0.19$\pm$0.1 | 0.44 | 0.31
DH Tau | 2.34 | 135 | M2.5 | M7.5 | 0.37${}^{+0.13}_{-0.10}$ | 0.04$\pm$0.2 | 0.11 | 0.10
DK Tau | 2.38 | 128 | K8.5 | M1.5 | 0.60${}^{+0.16}_{-0.13}$ | 0.44$\pm$0.2 | 0.73 | 0.42
HK Tau | 2.32 | 133 | M1.5 | M2 | 0.44${}^{+0.14}_{-0.11}$ | 0.37$\pm$0.2 | 0.84 | 0.46
HN Tau | 3.16 | 136 | K3 | M5 | 1.53$\pm$0.15 | 0.16$\pm$0.1 | 0.10 | 0.09
RW Aur | 1.49 | 163 | K0 | K6.5 | 1.20${}^{+0.18}_{-0.13}$ | $0.81\pm 0.2$ | 0.67 | 0.41
UZ Tau∗ | 3.52 | 131 | M2 | M3 | 1.23$\pm$0.07 | 0.58$\pm$0.2 | 0.47 | 0.32
UZ Tau W | 0.375 | 131 | M3 | M3 | 0.30$\pm$0.04 | 0.28$\pm$0.2 | 0.93 | 0.48
V710 Tau | 3.22 | 142 | M2 | M3.5 | 0.42${}^{+0.13}_{-0.11}$ | 0.25$\pm$0.1 | 0.60 | 0.37
222 We report the projected separations $a_{\mathrm{p}}$, the distance $d$
from the observer, the spectral type of the primary (SpT1) and of the
secondary (SpT2) stars, the masses of the primary and the secondary stars
($M_{1}$ and $M_{2}$ respectively) the mass ratios $q$, and the mass
parameters $\mu$. ∗ For UZ Tau, $M_{1}$ refers to the total mass of the UZ Tau
E spectroscopic binary, while $M_{2}$ is the sum of the masses of UZ Tau Wa
and UZ Tau Wb.

##  3 Observations and data reduction

The ALMA observations presented here were obtained in program 2018.1.00771.S,
(PI: Manara). The observations were conducted in ALMA Band 6 with an angular
resolution of $0\aas@@fstack{\prime\prime}15$ $-\sim 21$ au at the distance of
Taurus, and with an integration time of $\sim 40$ min per source. These
observations present an angular resolution slightly lower and a longer
integration time than the previous Band 6 observations presented by Long et
al. (2018, 2019) and Manara et al. (2019), i.e.
$0\aas@@fstack{\prime\prime}12$ and $\sim 4-9$ min/source, respectively. In
fact, the aim of the new observations was to detect different molecular
emission lines with a higher signal-to-noise (SNR). The spectral set-up was
chosen to include one continuum band (centered at 233 GHz), and the following
molecular emission lines: 12CO $(J=2-1)$, 13CO $(J=2-1)$, C18O $(J=2-1)$, N2D
$(J=3-2)$, DCN $(v=0,J=3-2)$, and H2CO $3(0,3)-2(0,2)$. Two configurations
were requested, C43-6 and C43-3, but only observations in the former
configuration were executed. Therefore, the maximum recoverable scale of the
observations was only $\sim$1.8′′.

The targets were observed in two Scheduling Blocks (SBs), one including HN
Tau, T Tau, and V710 Tau, and the other one including the remaining seven
targets of the program. Each science target was observed multiple times in
different Execution Blocks (EBs), three and six for each SB respectively. The
observations of the first SB, including three science targets, were executed
between August 17, 2019 and September 19, 2019, while those of the second SB,
including seven targets, were carried out between September 21, 2019 and
September 27, 2019. J0510+1800 was used as flux, bandpass, pointing, and
atmosphere calibrator, while J0440+1437 and J0438+3004 were used as phase
calibrators.

The calibrated data were retrieved using the script for PI provided by the
observatory. In order to improve the SNR, the data were self-calibrated, in
each EB separately. This step was performed using the Common Astronomy
Software Applications package (CASA, version 5.6.1, McMullin et al. 2007). To
speed up the calibration, we applied a channel averaging of 125MHz in each
spectral window (spw): this led to 8 channels in the continuum spw, centered
on 233.0 GHz with a bandwidth of 1875.00 MHz, and one channel in the other
five spws, used for the line observations. The channel averaging was done
after flagging any channel in which emission lines were found. In each EB we
applied from one to four rounds of phase-only self-calibration, stopping the
calibration when no significant improvement was measured in the SNR from one
step to the next one. We then concatenated all EBs and applied one round of
amplitude self-calibration. The final improvement in SNR has always been
larger than $100\%$ for all targets, thanks to both a drop in the noise and an
improvement in the observed flux. As final step, we have applied the phase and
amplitude calibration to the spectral windows containing emission lines.

Continuum and line images were created with the CASA task tclean. Continuum
images were created using both the continuum spectral window and the line-free
channels of the three spws in which CO isotopologues emission were detected.
The cleaning was performed with Briggs weighting and a robust parameter of
$+0.5$ using an interactive cleaning, except for the line images of HN Tau, GK
Tau, UY Aur and T Tau that were created through the CASA auto-multithresh
algorithm with Briggs weighting and a robust parameter of $+0.5$. The cleaning
was performed with a channel width of $0.1$ km/s for 12CO emission, except in
the case of HN Tau where a $0.2$ km/s channel width was used. The 13CO and
C18O lines were cleaned with channel width of $0.2-0.5$ km/s.

Table 2: Information on the cleaned images Target |  | rms | Beam size
---|---|---|---
|  | [mJy/beam per channel] |  [$\hbox{${}^{\prime\prime}$}\times\hbox{${}^{\prime\prime}$}$]
CIDA 9 |  continuum | 3.0e-02 | $0.25\times 0.19$
|  12CO | 4.9 | $0.27\times 0.20$
|  13CO | 5.0 | $0.28\times 0.21$
|  C18O | 2.3 | $0.28\times 0.21$
DH Tau |  continuum | 2.7e-02 | $0.23\times 0.15$
|  12CO | 4.6 | $0.25\times 0.16$
|  13CO | 3.6 | $0.27\times 0.17$
|  C18O | 2.0 | $0.27\times 0.17$
DK Tau |  continuum | 2.7e-02 | $0.28\times 0.14$
|  12CO | 4.4 | $0.24\times 0.15$
|  13CO | 2.5 | $0.26\times 0.15$
|  C18O | 2.1 | $0.26\times 0.15$
GK Tau |  continuum | 3.4e-02 | $0.24\times 0.15$
|  12CO | 12.2 | $0.26\times 0.16$
|  13CO | 5.1 | $0.27\times 0.17$
|  C18O | 4.1 | $0.27\times 0.17$
HK Tau |  continuum | 2.8e-02 | $0.22\times 0.14$
|  12CO | 4.7 | $0.24\times 0.15$
|  13CO | 3.7 | $0.25\times 0.15$
|  C18O | 1.7 | $0.25\times 0.15$
HN Tau |  continuum | 2.7e-02 | $0.19\times 0.12$
|  12CO | 6.9 | $0.19\times 0.12$
|  13CO | 3.3 | $0.20\times 0.13$
|  C18O | 2.4 | $0.20\times 0.13$
T Tau |  continuum | 4.3e-02 | $0.13\times 0.12$
|  12CO | 41 | $0.14\times 0.14$
|  13CO | 4.6 | $0.15\times 0.14$
|  C18O | 3.7 | $0.15\times 0.14$
UY Aur |  continuum | 2.7e-02 | $0.24\times 0.13$
|  12CO | 14 | $0.26\times 0.15$
|  13CO | 4.5 | $0.28\times 0.15$
|  C18O | 3.7 | $0.27\times 0.15$
UZ Tau |  continuum | 2.8e-02 | $0.23\times 0.14$
|  12CO | 4.2 | $0.24\times 0.15$
|  13CO | 4.6 | $0.25\times 0.15$
|  C18O | 2.4 | $0.25\times 0.15$
V710 Tau |  continuum | 3.4e-02 | $0.27\times 0.23$
|  12CO | 5.6 | $0.28\times 0.25$
|  13CO | 4.5 | $0.29\times 0.26$
|  C18O | 3.1 | $0.30\times 0.26$

In this work, in order to estimate disc radii and test tidal truncation
models, we analyse the 12CO emission. In Table 2 we report the rms of the
resulting continuum image and of the zeroth moment image of the CO
isotopologues, and the beam sizes. Figure 1 shows the 12CO zeroth moment
images of the 10 systems observed in our program, of which three (GK Tau, UY
Aur, and T Tau) are excluded from the analysis provided in this paper (see
Sect. 2). The figure shows 5 times the rms of the continuum emission with
white contours.

We detect and spatially resolve discs around all primary stars both in
continuum and 12CO emission. For discs around the secondary stars, three discs
(V710 Tau B, HN Tau B, DH Tau B) are not detected with neither dust continuum
nor CO emission, CIDA 9 B disc does not show any CO emission despite hosting a
dusty disc detected in the continuum, and both components are detected in the
remaining sample. Table 3 shows for each systems which components are detected
in the continuum emission and in CO emission.

Table 3: Detections of continuum and CO isotopologues.

| Cont det |  12CO det |  13CO det |  C18O det
---|---|---|---|---
CIDA 9 |  A, B∗ | A | A | A
DH Tau | A | A | A |  A∗
DK Tau | A, B | A, B |  A† | -
GK Tau | A | A | A | -
HK Tau | A, B | A, B | A, B | -
HN Tau | A | A | - | -
T Tau | N, Sa, Sb | N, Sa, Sb | N, Sa, Sb | N, Sa, Sb
UY Aur | A, B | A, B | A, B | -
UZ Tau |  E, Wa∗, Wb∗ | E, Wa, Wb | E, Wa, Wb |  E∗, Wa∗
V710 Tau | A | A | A | A
333For each system we indicate the detected components. Note that GK Tau is a
wide binary system with GI Tau: the label ‘A’ refers to the disc around GK
Tau. ∗Not resolved; † marginally resolved.

Figure 2 shows the velocity maps of the observed targets, including the three
systems not analysed in this work. In particular, the “quadratic” method to
image the velocity maps, created with bettermoments tools (Teague & Foreman-
Mackey 2018) and applying a 4 sigma clipping process, is shown. Excluding T
Tau and UY Aur systems, a pattern compatible with Keplerian rotation is
observed in all detected circumprimary and circumsecondary discs, a few of
which are already well known from the literature (e.g. HK Tau discs, Jensen &
Akeson 2014). The spectra and the first moment images of all targets in the
sample are presented in Appendix A.

Figure 2: 12CO velocity map images of the discs created with bettermoments
quadratic method (Teague & Foreman-Mackey 2018). A 4 sigma clipping process
was applied to all images. The FWHM beam size is shown in the bottom left of
each panel. Each image is scaled so that the maximum and the minimum are equal
to the systemic velocity of the primary target $\pm 2$ km/s. Green contours
show 5 times the rms of the continuum emission. The components of the systems
are labeled. The label for any undetected secondary components in the
continuum emission is not shown.

To this sample, we add the high angular resolution observation of the tidally
disrupted protoplanetary discs of the RW Aurigae system analysed by Rodriguez
et al. (2018). As part of ALMA Cycle 3 and 4 projects 2015.1.01506.S (PI
Rodriguez) and 2016.1.00877.S (PI Rodriguez), RW Aur was observed in Band 6
(225 GHz) with the 12m array for a total integration time of 311.17 min and
with the 7m array for a total additional integration time of 217.23 min.
Additionally, RW Aur was observed in Band 7 (338 GHz) for a total integration
time of 22.8 min. The data were calibrated by the NAASC, and two rounds of
phase-only self-calibration were applied to each set of observations. The 12CO
2-1 and 3-2 lines were imaged at a velocity resolution of 0.5 km/s, using a
Briggs weighting with a robust value of 0.5, and with a uv-taper of
$0\aas@@fstack{\prime\prime}6\times 0\aas@@fstack{\prime\prime}25$, resulting
in a synthesised beam of $0\aas@@fstack{\prime\prime}30\times
0\aas@@fstack{\prime\prime}25$ and an rms of 1.5 mJy/beam. Figure 3 shows the
12CO zeroth moment image (left), the 12CO first moment map (central) and the
12CO velocity map imaged with bettermoments quadratic method (right) for RW
Aur system.

Figure 3: Maps of the RW Aur system observed by Rodriguez et al. (2018). Left:
12CO zeroth moment. The color scale has the peak flux as the maximum, and two
times the image rms as minimum. Right: 12CO first velocity maps created with
bettermoments quadratic method (Teague & Foreman-Mackey 2018) and with a 4
sigma clipping process. Image scaled so that the maximum and the minimum are
equal to the systemic velocity of the primary target $\pm 2$ km/s.

The channel maps and the spectra of the 12CO emission for all targets in the
sample are reported in Appendix A and are useful to check for absorption due
to foreground cloud material in the data. Signatures of partial absorption
around the systemic velocity is observed in almost all discs; absorption in
the red-shifted velocities is observed in the disc around CIDA 9 A, while the
red-shifted velocities around DH Tau A are totally absorbed by the cloud below
the continuum level. These absorption features are in line with previous
observations (e.g., Guilloteau et al. 2013; Akeson & Jensen 2014; Czekala et
al. 2019). Table 4 summarizes the absorbed velocities in the disc 12CO
emission.

Table 4: Systemic velocities (in km/s; second column) and absorbed velocities
(in km/s; third and fourth columns) in the disc 12CO emission.

| Systemic Velocity | Primary | Secondary
---|---|---|---
CIDA 9 | $5.93^{+0.010}_{-0.02}$ |  -∗ ‡ | non resolved
DH Tau | $5.014^{+0.013}_{-0.013}$ |  5.4-† | non detected
DK Tau | $5.081^{+0.002}_{-0.003}$ |  -‡ | 3.0-7.0
HK Tau | $5.34^{+0.12}_{-0.12}$ |  5.2-7.2 ∗ | 5.5-6.9
HN Tau | $5.320^{+0.004}_{-0.002}$ | 3.8-5.6 | non detected
RW Aur | $\sim 6.0$ |  -‡ |  -‡
UZ Tau | $4.95^{+0.03}_{-0.03}$ | 5.5-6.3 | Wa,Wb: 5.4-7.2
V710 Tau | $6.467^{+0.005}_{-0.001}$ | 5.8-6.9 | non detected
444The systemic velocities shown in the first column are estimated from the
Keplerian modeling of the circumprimary disc velocity maps (see Section 4.1
and Table 5 for further details). ∗ Sign of partial absorption in the red-
shifted velocities. † Red-shifted velocities totally absorbed. ‡ Sign of
partial absorption around the systemic velocity.

##  4 Data analysis

In this Section, we estimate the radii of the discs. In order to measure this
quantity, one needs first to determine the position angle and inclination of
the discs. Once these disc parameters are derived an analysis on the image
plane is performed in order to calculate the disc radii.

###  4.1 Geometrical properties of discs from Keplerian modeling

The geometrical properties of interest for each disc are the disc inclination
$i$, its position angle PA, and the target centre (ra,dec). To estimate these
properties, we used the eddy tool (Teague 2019), that aims at reconstructing
the rotational profile of a gas emission line by fitting a Keplerian rotation
pattern to the velocity map of the observations, taking into account the
relative uncertainties calculated with the bettermoments tool (Teague &
Foreman-Mackey 2018) and assuming a geometrically thin disc.

Since there is a degeneracy between the inclination $i$ and the stellar mass
$M_{\star}$, the stellar mass was fixed to the values reported in Table 2. In
this way, eddy would fit five free parameters (ra, dec, PA, $i$ and the
systemic velocity $v_{\mathrm{LSR}}$), given two fixed parameter ($M_{\star}$
and the distance $d$ of the disc from the observer). To account for the impact
of the uncertainty on $M_{\star}$ on the estimate of the parameters of the
discs, we performed for each target three fits, one adopting the assumed value
of $M_{\star}$, and one each for the $\pm$1$\sigma$ values of $M_{\star}$
given the uncertainty (see Table 2). The fit parameters are initialised using
moment 0 value, for the target center, and the values by Manara et al. (2019),
for inclination and position angle. Flat priors for all parameters were
chosen, with Gaussian priors only for the target center. The number of walkers
and steps needed to achieve the convergence of the chain were typically $200$
walkers and $\sim 1000$-$2000$ steps, with the last $\sim 800$-$1000$ steps
used to sample the posterior distribution.

We account for cloud absorption in discs excluding from the fit the velocities
absorbed by the cloud (see Table 4). Since the disc around DH Tau A is heavily
absorbed in the red-shifted velocities, we only fit the 13CO velocity map of
DH Tau.

We did not fit the disc around the secondary star of the HK Tau system. This
disc is almost edge-on, so that the assumption of geometrically thin disc
emission cannot be applied and the vertical height of the disc must be
considered. Since the HK Tau system is well known in the literature, we
assumed the position angle and the inclination estimated in a previous work by
McCabe et al. (2011) (see also Jensen & Akeson 2014): PA=$(42\pm
0)$$\mathrm{\SIUnitSymbolDegree}$ and $i=(85\pm
1)$\mathrm{\SIUnitSymbolDegree}$$.

To avoid issues with the convergence of the eddy fit, we estimated the source
centre of the circumsecondary discs fitting two elliptical Gaussian components
to the 12CO image with the CASA task imfit. All other parameters of
circumsecondary discs were estimated through eddy. Table 5 reports the
estimated disc properties for each disc.

The agreement between the source centres obtained here with those derived by
Manara et al. (2019) is good when accounting for the typical proper motion of
the targets. We also compared the position angle and inclination obtained
fitting the gas emission with eddy with the values obtained by Manara et al.
(2019) fitting the continuum data in the uv-plane. This comparison is shown in
Figure 4. The values obtained with the two methods are in agreement within the
(3$\sigma$) uncertainties, with the exception of the circumsecondary discs
around DK Tau B, RW Aur B and UZ Tau Wa, where the estimates with both method
are uncertain due to the small size of the targets. Another exception,
probably due to some issues in the Galario fitting, is the circumprimary disc
around HN Tau A. The agreement within the uncertainties of the estimates with
the two different methods confirms that the assumptions on the stellar masses
are reasonable, except possibly in the four cases just mentioned. Indeed,
Manara et al. (2019) did not need any assumption on stellar masses to fit the
continuum, whereas in the Keplerian modelling performed here a value of
$M_{\star}$ was assumed.

Figure 4: Position angle (top) and inclination (bottom) estimated through eddy
fitting on 12CO first moment maps compered with that one estimate by Manara et
al. (2019) through $(u,v)$-plane modelling of the continuum data with Galario.
Red circles refer to circumprimary discs, while blue squares refer to
circumsecondary discs.  Table 5: Source centres (ra,dec), inclination $i$,
position angle PA and systemic velocity $v_{LSR}$ for all discs in the
analysed sample.

| ra | dec | inc [deg] | PA [deg] |  $v_{lsr}$ [m/s]
---|---|---|---|---|---
CIDA 9 A | 05h05m22.8260 | +25d31m30.5550 | $56.0^{+1.4}_{-3.5}$ | $284.2^{+0.7}_{-0.4}$ | $5930^{+10}_{-20}$
DH Tau A | 04h29m41.5610 | +26d32m57.7442 | $21.4^{+4.2}_{-3.0}$ | $183.3^{+1.0}_{-1.3}$ | $5014^{+13}_{-13}$
DK Tau A | 04h30m44.2509 | +26d01m24.2994 | $21.1^{+3.3}_{-2.7}$ | $188.2^{+0.1}_{-0.1}$ | $5081^{+2}_{-3}$
DK Tau B | 04h30m44.4048 | +26d01m23.1589 | $64.0^{+4.6}_{-5.8}$ | $181.6^{+1.7}_{-2.6}$ | $6000^{+390}_{-240}$
HK Tau A | 04h31m50.5769 | +24d24m17.3400 | $59.48^{+0.6}_{-1.0}$ | $358.1^{+2.4}_{-4.5}$ | $5340^{+120}_{-120}$
HK Tau B | 04h31m50.6122 | +24d24m15.1045 | $85^{+1}_{-1}$ | $42.0^{+0.0}_{-0.0}$ | $\sim 6000$
HN Tau A | 04h33m39.3782 | +17d51m51.9698 | $41.6^{+2.7}_{-2.4}$ | $92.07^{+0.04}_{-0.07}$ | $5320_{-2}^{+4}$
RW Aur A | 05h07m49.5716 | +30d24m04.7380 | $61.7^{+0.5}_{-0.8}$ | $225.38^{+0.03}_{-0.11}$ | $\sim 6000$
RW Aur B | 05h07m49.4597 | +30d24m04.3309 | $38.6^{+4.7}_{-6.6}$ | $232.2^{+0.4}_{-0.4}$ | $\sim 5200$
UZ Tau E | 04h32m43.0742 | +25d52m30.6460 | $59.4^{+0.1}_{-0.1}$ | $276.96^{+0.05}_{-0.03}$ | $4950^{+30}_{-30}$
UZ Tau Wa | 04h32m42.8236 | +25d52m31.1959 | $40.5^{+3.0}_{-2.5}$ | $264.7^{+0.2}_{-0.3}$ | $6571.3^{+0.6}_{-0.5}$
UZ Tau Wb | 04h32m42.8159 | +25d52m30.8347 | $50.4^{+4.7}_{-13.1}$ | $267.9^{+0.4}_{-0.4}$ | $6390^{+50}_{-30}$
V710 Tau A | 04h31m57.8081 | +18d21m37.6092 | $45.1^{+3.8}_{-5.5}$ | $266.049^{+0.70}_{-0.03}$ | $6467_{-1}^{+5}$
555The position angle PA is defined as the angle to the redshifted disc major-
axis counting from North to East. The target centres for circumsecondary discs
and for RW Aur A disc were estimated through the CASA tool imfit. The disc
inclination and position angle of HK Tau B are those reported in McCabe et al.
(2011). The systemic velocity of HK Tau B is estimated from the spectrum (see
Figure 19), while $v_{\mathrm{lsr}}$ of RW Aur discs are assumed from
Rodriguez et al. (2018). All other parameters were estimated through eddy
tool.

###  4.2 Disc radii estimate: the cumulative flux technique

To estimate the disc radii, we perform an image plane analysis using the
cumulative flux technique (for example, see Ansdell et al. 2018)666For
reference, this method has been called ”curve of growth” by Ansdell et al.
(2018)., computing aperture photometry on the zeroth moment images of the 12CO
emission and on the continuum images using the CASA tool imstat. This
technique is applied on the continuum emission (both on new data and on data
from Manara et al. 2019) and on the 12CO zeroth moment image of each target,
both primary and secondary (tertiary), when detected.

The cumulative flux technique consists of calculating the flux of the source
at increasingly larger radii. In particular, for each target the flux is
calculated summing over concentric annuli centred in the source centre and
corrected for the projected position of the disc (PA and inclination from
Table 5). The annuli are increased in step of $\mathrm{d}r=$
0$\aas@@fstack{\prime\prime}$05 (one-third of the angular resolution of the
data), and the flux uncertainty in each annulus is estimated as the standard
deviation of the fluxes in 100 random annuli selected well outside the disc.
We consider the error on the disc parameters calculating three cumulative flux
curves assuming in each of these either the maximum or minimum allowed values
for the position angle and inclination of the ellipses given the errors on
these two quantities. In this way, both the uncertainty due to the
(statistical) standard deviation of fluxes (random annuli in the field) and to
the uncertainty on the stellar masses (uncertainty on PA and $i$) are
considered.

Once the maximum flux is calculated, we estimate the disc radius
$R_{\mathrm{disc}}$ (or $R_{95}$) as the radius containing $95\%$ of the total
flux and the effective disc radius $R_{\mathrm{eff}}$ (or $R_{68}$) as that
containing $68\%$ of the total flux. The uncertainty on the radius is
calculated considering the statistical error on the maximum flux and, since
the error in the radius estimate must be at least half of the resolution, we
summed in quadrature this error with 0$\aas@@fstack{\prime\prime}$075.

A different procedure than the above described was applied in the case of the
HK Tau B disc. For this target, known also from the literature to be a edge-on
disc (e.g., McCabe et al. 2011; Jensen & Akeson 2014), the vertical structure
in the gas emission is more extended than the continuum. Assuming an
inclination of $\sim 85$$\mathrm{\SIUnitSymbolDegree}$ (McCabe et al. 2011;
Jensen & Akeson 2014) would lead to an estimated gas radius much larger than
the separation between the components of the HK Tau system. To correct for
this effect, the disc radius is measured along the major axis, centering on
the source rectangles with increasing width and fixed height (i.e. fixed to
the HK Tau beam size in that direction – $\sim
0\aas@@fstack{\prime\prime}25$), and rotated by the position angle of the
disc. No significant change in the estimated disc radius was found modifying
the height of the rectangles from $0\aas@@fstack{\prime\prime}10$ to
$0\aas@@fstack{\prime\prime}25$. The dust disc size is instead affected by
optical depth effects. Being an almost edge-on disc, the brightness profile of
HK Tau B is less peaked at small radii (near the disc centre) because of dust
self-absorption (Villenave et al. 2020). The observed profile is dominated by
the emission from larger radii, where the disc is colder, and thus resulting
in a larger estimate of the dust disc radius (see Section 5.1 for further
details).

Since RW Aur system is known to experience interaction between the components
(Rodriguez et al. 2018), to avoid considering the emission due to the
interaction, the 12CO total flux of the RW Aur A and B discs was assumed to be
the enclosed flux closest to the one estimated through the CASA tool imfit
(two-Gaussian fit) – $3.661\pm 0.094$ Jy km/s and $0.949\pm 0.076$ Jy km/s,
respectively.

Since the southern part of the 12CO emission of DH Tau circumprimary disc is
absorbed by the cloud (Figure 19), we applied the cumulative flux technique
only on the northern part, masking the emission from the southern half of the
disc. Since the position angle of DH Tau A disc is $\sim
180$\mathrm{\SIUnitSymbolDegree}$$, we masked the emission observed below a
line with PA$=90$\mathrm{\SIUnitSymbolDegree}$$ centred on the source centre.
We also masked the southern part of the UZ Tau Wa disc and the northern part
of the UZ Tau Wb disc, both in the continuum and 12CO emission, to exclude
emission arising from the other component of the system. Since the position
angle of UZ Tau Wa and Wb discs is $\sim 270$\mathrm{\SIUnitSymbolDegree}$$,
we drew two lines with PA$=90$\mathrm{\SIUnitSymbolDegree}$$ centred on UZ Tau
Wa disc centre and UZ Tau Wb disc centre and we masked the emission observed
below and above each line, respectively.

Figure 5 and 6 show the cumulative flux curves for each disc in the sample.
The first columns show the cumulative sum of the fluxes measured on the
continuum emission from the data by Manara et al. (2019) and the new data; the
second, third and fourth columns show the cumulative flux relative to the
12CO, 13CO, and C18O emission, respectively. The blue lines show the total
fluxes (see Table 8), while the red lines show the disc radii ($R_{95}$, see
Table 7).

Figure 5: Cumulative fluxes as a function of aperture radius estimated using
the cumulative flux technique. Each panel shows the fluxes for a different
data-set/image (the name of the target for each row is shown in the lower
right corner of the first column). The first column shows the continuum
emission of the data from Manara et al. (2019) (green dots) and the continuum
emission from the new data (black dots). The second, third and fourth columns
show the 12CO, 13CO and C18O moment 0 emission from the new data,
respectively. The blue dashed lines show the estimated disc total fluxes; the
red solid lines show the disc radii, defined as radii containing the $95\%$ of
the total flux (see values reported in Table 8 and 7). In the first column,
only the total fluxes and disc radii estimated from the new continuum data are
shown. Figure 6: Cumulative flux curves as a function of aperture radius
estimated using the cumulative flux technique. Same as in Figure 5 for the
remaining discs in the sample. The total fluxes and disc radii estimated from
the continuum data from (Manara et al. 2019) are shown in the case of RW Aur A
and B discs (first column, last two rows). In all the other cases, the total
fluxes and disc radii estimated from new continuum data are shown. Figure 7:
12CO zeroth moment images of the discs with overlapped ellipses including the
$95\%$ (solid) and $68\%$ (dashed) of the total flux. The ellipses are centred
on the target source centre. All bars in the bottom right of each panel are
1$\aas@@fstack{\prime\prime}$0 long, which is $\sim 140$ au at the distance of
Taurus. The beam FWHM size is shown in the bottom left of each panel. In each
image, the color scale has the peak flux as the maximum, and two times the
image rms as minimum. White contours show 5 times the rms of the continuum
emission.

Table 7 shows the estimated radii and their error for each disc, while Figure
7 shows the ellipses including the $95\%$ (solid) and $68\%$ (dashed) of the
total flux overlapped to the zeroth moment maps. As shown by the red solid
ellipses, the radii estimated with the cumulative flux technique get the
really outer extent of the discs, where little emission is detected, giving
the possibility to test tidal truncation models. Table 8 shows the total
fluxes we estimated for each discs with the cumulative flux technique applied
on the continuum image and on the 12CO, 13CO and C18O zeroth moment images.

The effective disc radii estimated for the RW Aur system are in very good
agreement with the ones reported in Rodriguez et al. (2018), estimated using
CASA built-in measurement tool (McMullin et al. 2007), both for the
circumprimary and the circumsecondary discs $-$ $\sim 58$ au and $\sim 38$,
respectively, at a distance of 140 pc (assumed by Rodriguez et al. 2018).

Table 6: Radii estimated with the cumulative flux technique

Target |  | 12CO [′′] | Continuum [′′] | Continuum from
---|---|---|---|---
|  Manara et al. (2019) [′′]
CIDA 9 A | $R_{68}$ | $1.10^{+0.08}_{-0.09}$ | $0.40^{+0.08}_{-0.08}$ | $0.35^{+0.08}_{-0.08}$
$R_{95}$ | $1.70_{-0.09}^{+0.13}$ | $0.650^{+0.08}_{-0.08}$ | $0.50^{+0.08}_{-0.08}$
DH Tau A | $R_{68}$ | $0.25^{+0.08}_{-0.08}$ | $0.15^{+0.08}_{-0.08}$ | $0.10^{+0.08}_{-0.08}$
$R_{95}$ | $0.55_{-0.08}^{+0.08}$ | $0.25_{-0.08}^{+0.08}$ | $0.20_{-0.08}^{+0.08}$
DK Tau A | $R_{68}$ | $0.20_{-0.08}^{+0.08}$ | $0.15_{-0.08}^{+0.08}$ | $0.10_{-0.08}^{+0.08}$
$R_{95}$ | $0.40^{+0.13}_{-0.13}$ | $0.25^{+0.08}_{-0.08}$ | $0.20^{+0.08}_{-0.08}$
DK Tau B | $R_{68}$ | $0.25_{-0.08}^{+0.09}$ | $0.20_{-0.09}^{+0.08}$ | $0.10_{-0.08}^{+0.08}$
$R_{95}$ | $0.50_{-0.13}^{+0.09}$ | $0.30_{-0.08}^{+0.09}$ | $0.20_{-0.08}^{+0.09}$
HK Tau A | $R_{68}$ | $0.50_{-0.09}^{+0.08}$ | $0.20_{-0.08}^{+0.08}$ | $0.15_{-0.08}^{+0.08}$
$R_{95}$ | $0.80_{-0.09}^{+0.09}$ | $0.35_{-0.08}^{+0.08}$ | $0.30_{-0.09}^{+0.08}$
HK Tau B | $R_{68}$ | $0.25_{-0.08}^{+0.08}$ | $0.20_{-0.08}^{+0.08}$ | $0.10_{-0.08}^{+0.08}$
$R_{95}$ | $0.55_{-0.09}^{+0.13}$ | $0.40_{-0.08}^{+0.08}$ | $0.20_{-0.08}^{+0.09}$
HN Tau A | $R_{68}$ | $0.25_{-0.09}^{+0.08}$ | $0.15_{-0.08}^{+0.08}$ | $0.10_{-0.08}^{+0.08}$
$R_{95}$ | $0.40_{-0.09}^{+0.09}$ | $0.25_{-0.08}^{+0.08}$ | $0.20_{-0.08}^{+0.08}$
RW Aur A | $R_{68}$ | $0.50_{-0.09}^{+0.08}$ | - | $0.15_{-0.08}^{+0.08}$
$R_{95}$ | $0.85_{-0.09}^{+0.13}$ | - | $0.30_{-0.08}^{+0.08}$
RW Aur B | $R_{68}$ | $0.30_{-0.08}^{+0.09}$ | - | $0.10_{-0.08}^{+0.08}$
$R_{95}$ | $0.55_{-0.13}^{+0.13}$ | - | $0.25_{-0.09}^{+0.09}$
UZ Tau E | $R_{68}$ | $1.50_{-0.09}^{+0.08}$ | $0.50_{-0.08}^{+0.08}$ | $0.45_{-0.08}^{+0.08}$
$R_{95}$ | $2.40_{-0.13}^{+0.13}$ | $0.80_{-0.09}^{+0.09}$ | $0.70_{-0.08}^{+0.09}$
UZ Tau Wa | $R_{68}$ | $0.30_{-0.08}^{+0.08}$ | $0.15_{-0.08}^{+0.08}$ | $0.10_{-0.08}^{+0.08}$
$R_{95}$ | $0.50_{-0.08}^{+0.09}$ | $0.25_{-0.08}^{+0.08}$ | $0.20_{-0.08}^{+0.08}$
UZ Tau Wb | $R_{68}$ | $0.20_{-0.08}^{+0.08}$ | $0.10_{-0.08}^{+0.08}$ | $0.10_{-0.08}^{+0.08}$
$R_{95}$ | $0.30_{-0.08}^{+0.08}$ | $0.20_{-0.08}^{+0.09}$ | $0.15_{-0.08}^{+0.08}$
V710 Tau A | $R_{68}$ | $0.45_{-0.08}^{+0.08}$ | $0.25_{-0.08}^{+0.08}$ | $0.20_{-0.08}^{+0.08}$
$R_{95}$ | $0.70_{-0.08}^{+0.09}$ | $0.45_{-0.08}^{+0.08}$ | $0.35_{-0.08}^{+0.08}$

777We report the effective disc radius $R_{68}$ and the disc radius $R_{95}$
calculated applying the cumulative flux technique on the 12CO and continuum
emission from new observations, and on the continuum emission data by Manara
et al. (2019).

Table 7: Total fluxes measured with the cumulative flux technique.

Target | $F_{\rm cont}$ | $F(^{12}\rm CO)$ | $F(^{13}\rm CO)$ | $F(\rm C^{18}O)$
---|---|---|---|---
CIDA 9 A | $36.17^{+0.11}_{-0.10}$ | $1685.1^{+55.9}_{-39.8}$ | $491.4^{+20.6}_{-17.4}$ | $115.0^{+12.7}_{-10.0}$
DH Tau A | $26.71_{+0.06}^{-0.07}$ | $409.02^{+0.06}_{-0.07}$ | $101.3^{+7.3}_{-7.2}$ | -
DK Tau A | $31.94^{+0.07}_{-0.07}$ | $583.7^{+19.4}_{-19.5}$ | - | -
DK Tau B | $2.87^{+0.07}_{-0.10}$ | $275.1^{+12.9}_{-15.5}$ | - | -
HK Tau A | $35.50^{+0.08}_{-0.08}$ | $1233.5^{+31.6}_{-25.5}$ | $181.8^{+18.0}_{-18.4}$ | -
HK Tau B | $4.51^{+0.02}_{-0.02}$ | $265.8^{+5.3}_{-5.3}$ | $48.8^{+3.6}_{-3.6}$ | -
HN Tau A | $13.06^{+0.08}_{-0.09}$ | $873.4^{+45.7}_{-50.8}$ | - | -
RW Aur A | - | $3678^{+159}_{-54}$ | - | -
RW Aur B | - | $987.8^{100.1}_{-68.2}$ | - | -
UZ Tau E | $132.1^{+1.7}_{-1.7}$ | $7584^{+100}_{-101}$ | $1333^{+38}_{-38}$ | -
UZ Tau Wa | $14.9^{+5.3}_{-3.6}$ | $550.6^{+1.3}_{-1.9}$ | $94.2^{+1.4}_{-2.3}$ | -
UZ Tau Wb | $12.9^{+1.6}_{-0.2}$ | $415.5^{+8.5}_{-8.5}$ | $112.0^{+2.6}_{-2.6}$ | -
V710 Tau A | $55.33^{+0.10}_{-0.09}$ | $939.5^{+16.3}_{-15.8}$ | $300.8^{+15.0}_{-18.2}$ | $119.1^{+12.3}_{-11.0}$
888Continuum fluxes are reported in mJy, line fluxes in mJy km/s.

Figure 8: 12CO estimated radii as a function of dust radii obtained with
(u,v)-plane modelling of the continuum data by Manara et al. (2019). Dust and
gas radii estimated as the radii including the 68% (right) and the 95% (left)
of the total flux. The red circles are for circumprimary discs, while blue
squares for the circumsecondary.  Figure 9: Gas-dust size ratios
($R_{\mathrm{95,gas}}/R_{\mathrm{95,dust}}$) as a function of the projected
separation. The red circles are for circumprimary discs, while blue squares
for the circumsecondary.

##  5 Discussion

###  5.1 Comparing dust and gas disc radii

The emission arising from the dusty component of discs, and thus the dust disc
radius measurement, is regulated by several processes, like growth,
fragmentation, and transport of dust. On the contrary, the gas emission extent
is regulated by chemical processes and by the dynamics of the disc, thus in
our case by the tidal interaction of the components of the multiple systems we
are studying. Here we investigate whether the ratio between the size of the
discs (defined as the radius containing the 95% of the total flux) measured in
the 12CO gas component and in the dust component ($R_{\rm disc,gas}$/$R_{\rm
disc,dust}$) is different in systems where tidal truncation is at work with
respect to more isolated systems.

The ratio $R_{\rm disc,gas}$/$R_{\rm disc,dust}$ has been studied in a limited
number of targets in Taurus (Kurtovic et al. 2021), and in a larger sample in
the Lupus star-forming region, firstly analysed by Ansdell et al. (2018), and
recently revised by Sanchis et al. (2021). The ratio $R_{\rm
disc,gas}$/$R_{\rm disc,dust}$ in mostly isolated or wide binary systems in
the Lupus star-forming region has a median and a mean value of $\sim$2.5 and
$\sim$2.8 respectively, both when considering the radii at 68% or 95% of the
total flux. In general, the larger gas sizes can be explained as a combination
of the effect of radial drift of dust, and of the different opacity between
the two components (e.g., Facchini et al. 2017; Trapman et al. 2020). Sanchis
et al. (2021) noted that the gas-dust size ratios and the stellar masses are
uncorrelated, contrary to expectations from theoretical and observational
works suggesting that the radial drift is more effective in discs around low-
mass stars (e.g., Pascucci et al. 2016; Pinilla et al. 2013). For the sake of
this work, the lack of any correlation between these two quantities assures
that we can carry out a comparison between our results with those by Sanchis
et al. (2021) with no biases due to different stellar mass distributions.

For the following discussion, we use the values of $R_{\mathrm{dust}}$
obtained by Manara et al. (2019) by modelling the continuum data in the
visibility plane using the Galario library (Tazzari et al. 2018) because this
makes the results directly comparable with Sanchis et al. (2021), who
performed a similar analysis.

Our sample of multiple system reveals that the gas disc radii are typically
$\sim 2-4$ times larger than the dust disc radii, for both the effective gas-
to-dust radii $R_{68,\mathrm{gas}}/R_{68,\mathrm{dust}}$ and the gas-to-dust
disc radii $R_{95,\mathrm{gas}}/R_{95,\mathrm{dust}}$ (Figure 8). As shown by
Figure 8, HK Tau B disc is the biggest outlier in the gas-to-dust size ratio
distribution, with $R_{95,\text{gas}}/R_{95,\text{dust}}\sim 1$. Since it is
an almost edge-on disc, this small ratio may be due to optical depth effects,
resulting in an overestimate of the dust radius and, as a consequence, in an
underestimate of the gas-to-dust disc radius. We thus decided to exclude HK
Tau B disc from the statistical analysis discussed below.

The average value (and relative standard deviation) of
$R_{68,\mathrm{gas}}/R_{68,\mathrm{dust}}$ is $3.2\pm 1.1$ ($3.1\pm 1.1$ and
$3.4\pm 0.9$ for circumprimary and circumsecondary discs, respectively), and
$R_{95,\mathrm{gas}}/R_{95,\mathrm{dust}}=4.2\pm 1.6$ (with average values of
$3.9\pm 1.2$ for circumprimary discs and $5.0\pm 1.9$ for the
circumsecondary). The Kolmogorov-Smirnov (K-S) two-sided test for the null
hypothesis that the samples of gas-to-dust size ratios around primary stars
and around secondary stars are drawn from the same continuous distribution
leads to a p-value $\sim 0.06$ (for both the 68% and the 95% ratios), thus
confirming that the circumprimary and circumsecondary discs are likely to be
drawn from the same distribution (see Figure 11).

Figure 9 shows the ratio $R_{\rm disc,gas}$/$R_{\rm disc,dust}$ as a function
of the projected separation of the components in the systems. No evidence of a
correlation between the size ratio and the projected separation is shown,
suggesting that the gas-dust size ratio does not depend on the distance
between the components in the system. Other features, such as the eccentricity
of the orbits, need to be considered to explain the observed distribution of
size ratios (see Section 5.2).

Figures 10 and 11 show the comparison between the 95% and the 68 % gas-to-dust
disc size ratios measured in our sample of multiple stellar systems in the
Taurus region, and the one measured by Sanchis et al. (2021) for 42 discs in
the Lupus star-forming regions, which are mainly singles or, in two cases, in
wide binary systems with separation $\sim 1000$ au, high enough that discs are
not affected by tidal truncation (Pearce et al. 2020).

The effective ratio $R_{68,\mathrm{gas}}/R_{68,\mathrm{dust}}$ estimated in
Taurus multiple system is in very good agreement with the average ratio
obtained in the sample of discs around single stars by Sanchis et al. (2021)
($\sim 3.2$ vs $\sim 2.8$, respectively). The K-S test performed on these two
samples confirms that the gas-to-dust size ratios in Taurus multiples are
likely to be drawn by the same distribution of Lupus single discs (p-value
$\sim 0.13$). On the contrary, the $R_{95,\mathrm{gas}}/R_{95,\mathrm{dust}}$
average ratio in Taurus multiple systems is larger than the ratio estimated by
Sanchis et al. (2021), with the K-S test confirming the two samples to be
statistically different (p-value $\sim 0.012$).

The difference in the values of $R_{95,\mathrm{gas}}/R_{95,\mathrm{dust}}$
estimated in discs around binaries and in singles is possibly due to the sharp
truncation of the outer dusty discs in binary systems (Manara et al. 2019).
However, this difference may also be affected by the method through which the
gas disc radii have been estimated. As discussed, we applied the cumulative
flux technique directly on the zeroth moment image, with no parametric form to
model the emission profile, while Sanchis et al. (2021) modelled the CO
emission of each disc by fitting the integrated line map to a Gaussian (or a
Nuker) profile in the image plane, and then they applied the cumulative flux
technique on the modelled emission profile. The latter method may lead to an
underestimate of the 95% disc radii, smoothing the outer emission of the
discs.

In general, we conclude that the ratio of the gas to dust disc size in
multiple stellar systems is found to be on the higher side of the distribution
of values when compared to a population of more isolated systems and
considering the 95% disc radius, which is more sensitive to the full disc
size.

Figure 10: Comparison between the gas-to-dust size ratio for discs in the
Taurus binaries (red and blue triangles) and for single discs in Lupus (grey
diamonds). Effective disc radii $R_{68}$ (left) and disc radii $R_{95}$
(right) are shown.

Figure 11: Normalized cumulative distribution function for the gas-to-dust
effective disc sizes $R_{68,\mathrm{gas}}/R_{68,\mathrm{dust}}$ (top) and for
the gas-to-dust disc sizes $R_{95,\mathrm{gas}}/R_{95,\mathrm{dust}}$ (bottom)
for discs around single stars in Lupus (grey, Sanchis et al. 2021) and discs
around multiple stellar system in Taurus (black). Discs around primary stars
in Taurus are shown in red, while circumsecondary discs in Taurus are shown in
blue.

###  5.2 Comparison of gas disc radii with tidal truncation models

We now compare the observed 12CO disc radii with the theoretical expectations
for tidal truncation predicted by Artymowicz & Lubow (1994). As discussed by
Manara et al. (2019), and references therein, it can be analytically computed
that the truncation radius for a disc in a binary system with a circular orbit
inclined along the plane of the sky (eccentricity $e=0$, mass ratio $q$ = 1,
disc orbit misalignment $=0^{\circ}$) is $\sim 0.33\leavevmode\nobreak\ a$,
where $a$ is the semi-major axis of the binary orbit. The value of the
truncation radius becomes larger for the circumprimary disc when the mass
ratio $q$ is smaller than 1, and smaller for the circumsecondary disc for
smaller values of $q$. The truncation radii instead decrease with higher
eccentricities both in circumprimary and circumsecondary discs. For systems
with $e\gtrsim 0.9$, more violent processes than the disc-satellite
interaction, such as the collision between the disc and the companion star,
would occur.

Figure 12: Observed 12CO gas radii as a function of the projected separation.
Blue squares refer to the circumsecondary discs, while red circles to the
circumprimary discs.

Let us first assume that the projected separation $a_{\mathrm{p}}$ is $\sim
a$, which again is correct if the orbit has low eccentricity and the plane of
the orbit is aligned close to the plane of the sky, and compare the estimated
gas disc radii $R_{95,gas}$ with the projected separation between the
components (reported in Table 2). Since the UZ Tau system is a quadruple
system (with UZ Tau E being a spectroscopic binary), when comparing
observations with analytical models, we consider the UZ Tau system as two
binaries. The first binary is composed by UZ Tau E and UZ Tau Wab (considering
the sum of the masses of UZ Tau Wa and Wb), and has a projected separation of
$3\aas@@fstack{\prime\prime}52$ (assuming the position of UZ Tau Wab as the
center of mass of the two stars); the second binary is composed by UZ Tau Wa
and UZ Tau Wb, and has $a_{\mathrm{p}}=0\aas@@fstack{\prime\prime}375$.

Figure 12 shows that the typical observed ratio
$R_{95,\mathrm{gas}}/a_{\mathrm{p}}$ in our sample is $\sim 0.15-0.35$, and
more specifically $>0.1$ in all targets, contrary to the typical ratio of
$\lesssim 0.1$ measured for the dust radii in Manara et al. (2019). This
typical observed ratio is due to the fact that the gas disc radii are larger
than the dust disc radii, as discussed in Section 5.1, and already suggested
in Manara et al. (2019). However, this is the first time that this value
$R_{95,\mathrm{gas}}/a_{\mathrm{p}}$ is measured in a large sample of objects.

The observed distribution of gas disc radii points to the fact that the data
do not agree with a simple description, as the ratio
$R_{95,\mathrm{gas}}/a_{\mathrm{p}}$ typically differs from the expected value
of 0.33 (only 2/13 discs show a ratio $\sim 0.33$). Therefore, a more detailed
comparison is to be considered.

To better quantify this result, we compare
$R_{95,\mathrm{gas}}/a_{\mathrm{p}}$ with theoretical models by Artymowicz &
Lubow (1994), using the equation derived by Manara et al. (2019) under the
assumption that the discs and the binary orbit are co-planar:

$\frac{R_{\rm trunc}}{a_{\rm p}}=\frac{0.49\cdot q^{2/3}_{i}}{0.6\cdot
q_{i}^{2/3}+\ln(1+q_{i}^{1/3})}\left(b\cdot
e^{c}+0.88\mu^{0.01}\right)\cdot\\\
\cdot\left[\frac{1-e^{2}}{1+e\cdot\cos\nu}\sqrt{1-\sin^{2}(\omega+\nu)\sin^{2}i}\right]^{-1},$
(1)

where $\nu$ the true anomaly, $\omega$ the longitude of periastron, and i the
inclination of the plane of the orbit with respect to the line of sight,
$q_{i}$ is the mass ratio (either $q_{1}=M_{1}/M_{2}$ or
$q_{2}=q=M_{2}/M_{1}$), and $b$ and $c$ are the parameters derived by Manara
et al. (2019) that depend on the disc viscosity or equivalently on the
Reynolds number, $\mathcal{R}$. Figure 13 shows this comparison in the case of
zero-eccentricity orbits ($e=0$) and inclination of the orbit $i=0$. Red
circles and blue squares show the observed ratio for circumprimary and
circumsecondary discs, respectively. Red crosses and blue pluses show for each
target the ratio expected from model in the case $e=0$. The longer the dashed
line that links observed radii with models is, the less is the measured ratio
in agreement with zero-eccentricity model predictions. Excluding the discs
around UZ Tau E and CIDA 9 A and the disc around RW Aur A, the measured values
of the observed ratios point to disc radii that are typically smaller than
expectations from tidal truncation models at zero eccentricity.

Figure 13: 12CO gas radii as a function of the projected separation. Blue
squares and red circles show the observed ratio in the circumsecondary discs
and the circumprimary discs, respectively. Blue pluses and red crosses show
the expected ratio from tidal truncation model in the case of circular orbits
($e=0$) for circumsecondary discs and circumprimary discs, respectively. The
longer the dashed line that links observed radii with models is, the less is
the agreement with zero-eccentricity model.

As discussed in Manara et al. (2019), at a given inclination of the orbital
plane $i$, the truncation radius $R_{\mathrm{t}}$ in units of the projected
separation $a_{\mathrm{p}}$ has a minimum when the secondary star is observed
at the apoastron $-$ $a_{\mathrm{p}}=a(1+e)$ $-$ and has a maximum at the
periastron $-$ $a_{\mathrm{p}}=a(1-e)$:

$\frac{R_{\mathrm{t}}}{a}\frac{1}{(1+e)\cos
i}<\frac{R_{\mathrm{t}}}{a_{\mathrm{p}}}<\frac{R_{\mathrm{t}}}{a}\frac{1}{(1-e)\cos
i}.$ (2)

The minimum ratio is found at $i=0$\mathrm{\SIUnitSymbolDegree}$$ and
$R_{\mathrm{t}}/a_{\mathrm{p}}$ increases for higher orbital inclination. In
Figure 14, we plot the ratio of the truncation radius to the projected
separation of the orbit as a function of eccentricity for all the targets in
the sample, assuming an orbital inclination
$i=0$\mathrm{\SIUnitSymbolDegree}$$ with respect to the line of sight. The
intersection between the observed $R_{\mathrm{t}}/a_{\mathrm{p}}$ (red bands
in Figure 14) and the theoretical model assuming that the secondary is
observed at apoastron (lower black curves in the figure) represents the
minimum possible value of the eccentricity for the system. This statement
holds even in case that the orbital plane of the binary is misaligned with
respect to the disc ($i>0$\mathrm{\SIUnitSymbolDegree}$$), since the black
curves in Figure 14 will move to higher values leading to a larger minimum
eccentricity. As an example, we refer to the V710 A disc (second row, last
column in the figure). To be compatible with tidal truncation models, the
observed ratio of disc radius to separation must lie between the bottom and
the top black curves. The observed ratio is shown as a red band in the plot,
from which we see that, in this case, the eccentricity must be larger than
$\simeq 0.2$. For inclined orbits, the minimum eccentricity must be even
larger. The dominant uncertainty on the eccentricity estimated in this way
comes from the lack of information on the local Reynolds number, and hence
different Reynolds numbers are assumed – $\mathcal{R}=10^{4},10^{5},10^{6}$.

Figure 14: Condition on the eccentricity of the stellar orbits for which the
estimated truncation radii of the discs in the sample are in agreement with
theoretical predictions (see Sect. 5.2). Red bands show the estimated
$R_{95,\mathrm{gas}}/a_{\mathrm{p}}$ with their uncertainties. Black curves
show the expected truncation radius in unit of the projected separation for
different Reynolds numbers – dashed curves for $\mathcal{R}=10^{4}$, solid for
$\mathcal{R}=10^{5}$, and dotted for $\mathcal{R}=10^{6}$. The lower black
curves are obtained assuming that the secondary is observed at apoastron,
while black curves at the top assuming that the secondary is located at
periastron. The expected $R_{\mathrm{t}}/a_{\mathrm{p}}$ increases for higher
orbital inclination, and thus the black curves would move to higher values in
the plot if the assumption of face-on orbits is not fulfilled (see Equation
2).

For each target, Table 9 shows the ratio between disc radii and separations
predicted by the models assuming $e=0$ and the observed ratio between disc
radii $R_{\mathrm{95,gas}}$ and projected separations. Finally, the minimum
eccentricities of the orbits estimated assuming zero orbital inclination and
physical separation $a$ between the components that matches the projected
separation $a_{\mathrm{p}}$ are shown in the last column in Table 9 and in
Figure 15.

Table 8: Comparison between observed disc radii and tidal truncation models.

Target | $(R_{\mathrm{disc}}/a)\mathrm{byL\&A94}$ | $R_{\mathrm{95,gas}}/a_{\mathrm{p}}$ | $e_{\mathrm{min}}$
---|---|---|---
CIDA 9 A | $0.39$ | $0.72$ | -
DH Tau A | $0.49$ | $0.24$ | $0.31^{+0.15}_{-0.09}$
DK Tau A | $0.35$ | $0.17$ | $0.32^{+0.23}_{-0.16}$
DK Tau B | $0.31$ | $0.21$ | $0.16^{+0.18}_{-0.10}$
HK Tau A | $0.34$ | $0.34$ | $0.00^{+0.06}_{-0.00}$
HK Tau B | $0.32$ | $0.24$ | $0.12^{+0.12}_{-0.10}$
HN Tau A | $0.49$ | $0.13$ | $0.57^{+0.19}_{-0.13}$
RW Aur A | $0.36$ | $0.57$ | 0‡
RW Aur B | $0.30$ | $0.37$ | 0†
UZ Tau E | $0.39$ | $0.68$ | -
UZ Tau Wa | $0.34$ | $1.33$ | -
UZ Tau Wb | $0.33$ | $0.80$ | 0∗
V710 Tau A | $0.37$ | $0.22$ | $0.22^{+0.11}_{-0.08}$
999Second column: expected ratio between disc radii and separation as
predicted by Artymowicz & Lubow (1994) when assuming circular orbits ($e=0$).
Third column: observed ratio between disc radii $R_{\mathrm{95,gas}}$ and
projected separation. Fourth column: minimum eccentricities of the orbits
estimated assuming zero orbital inclination and physical separation $a$
between the components that matches the projected separation $a_{\mathrm{p}}$.
Agreement with analytical model with zero eccentricity † within one sigma; ∗
within 1.5 sigmas; ‡ within 3.5 sigmas (see Figure 14).

The tidal truncation models assuming face-on inclination of the orbit cannot
explain the observed large values of $R_{\mathrm{t}}/a_{\mathrm{p}}$ in the
CIDA 9, UZ Tau, and RW Aur systems, except assuming unrealistic parameters
($e\sim 1$ and targets observed at the periastron); this can be explained
either by a high inclination of the orbit with respect to the plane of the sky
and by the fact that the disc sizes are regulated by other additional
processes (e.g. substructures and interaction between the components) – not
included in the model – on top of tidal truncation. Excluding these systems,
the estimated minimum eccentricities show typical values of $\sim 0.15-0.5$
and an average value of $\sim 0.3$, in very good agreement with the observed
distribution of eccentricities for main-sequence binary systems of low-mass
(e.g., Duchêne & Kraus 2013). This confirms observationally the result by
Zagaria et al. (2021b) that no highly eccentric orbits must be invoked to
explain the observed small dust radii of discs in multiple stellar systems.
When comparing both the circumprimary and circumsecondary disc sizes to the
models – comparison possible only in DK Tau and HK Tau systems, a very good
agreement with the two estimated eccentricities is found.

We finally explore whether more eccentric orbits would have an impact on the
relative sizes of the gas and dust components of discs, as predicted by
previous theoretical works (e.g., Clarke & Pringle 1993). Figure 16 shows that
no correlation is observed between the observed gas-dust size ratios and the
estimated eccentricity of the orbits. However, to definitively refute a
dependence of the gas-dust size ratio on the eccentricity, it is necessary to
significantly increase the sample of systems with measured size ratio and
eccentricity.

Figure 15: Estimated minimum eccentricities for each target. Red circles refer
to circumprimary discs, while blue squares refer to circumsecondary discs.
Figure 16: Gas-dust size ratios ($R_{\mathrm{95,gas}}/R_{\mathrm{95,dust}}$)
as a function of the minimum eccentricities of the orbits. Red circles refer
to circumprimary discs, while blue squares refer to circumsecondary discs.

###  5.3 Disc misalignment

Figure 17: Comparison between the position angles (top) and the inclination
(bottom) for the discs around the primary and the secondary in each multiple
stellar system where both discs are detected and resolved. In the case of UZ
Tau system, we considered UZ Tau system as a two binaries, comparing UZ Tau E
parameters with UZ Tau Wab parameters (comparison labeled as UZ_Wa and UZ_Wb
for the two UZ Tau Wab discs) and UZ Tau Wa parameters with UZ Tau Wb
parameters (comparison labeled as UZ_Wab). In the other cases, the first two
letters of the target name are labelled.  Figure 18: Relative inclination of
the discs in each multiple stellar systems as a function of their projected
separation. In the case of UZ Tau system, we considered UZ Tau system as a two
binaries, comparing UZ Tau E parameters with UZ Tau Wab parameters (comparison
labeled as UZ_Wa and UZ_Wb for the two UZ Tau Wab discs) and UZ Tau Wa
parameters with UZ Tau Wb parameters (comparison labeled as UZ_Wab). In the
other cases, the first two letters of the target name are labelled.

The amount of alignment of the plane of rotation of disc in a multiple system
can be used as a constraint to star formation models. In the most simplified
case, disc fragmentation would tend to form close binaries with aligned
angular momentum vectors, while turbulent fragmentation would produce more
randomly distributed binary orientation (e.g., Bate 2000; Offner et al. 2010;
Kratter et al. 2010; Bate 2018). It is therefore instructive to explore the
observed values of the relative inclinations for the discs in our sample to be
used to constrain disc evolution models in multiple systems.

Figure 17 shows the alignment of discs around the components of the multiple
systems in which both components are detected and resolved. The position
angles and inclinations used to make this comparison are those reported in
Table 5. The first two letters of the system name are labeled. In the case of
the UZ Tau system, we compare the UZ Tau E disc with the UZ Tau Wab discs,
separately (labeled as UZ_Wa and UZ_Wb), and the UZ Tau Wa disc with the UZ
Tau Wb disc. The inclination of the components of the multiple systems appear
to be different in most cases (top panel), with only the East and West
components of UZ Tau showing some agreement. The position angles of the discs
are also usually similar, with the exception of the HK Tau discs, which show
different values (bottom panel).

However, the degree of alignment should be explored using the relative
inclination of the discs, computed using the spherical law of cosines:
$\cos(\Delta
i)=\cos(i_{1})\cdot\cos(i_{2})+\sin(i_{1})\sin(i_{2})\cos(\Omega_{1}-\Omega_{2})$,
with $i$ the inclination (ambiguous in sign, so that it leads to two families
of solutions) and $\Omega$ the position angle. The result is shown in Figure
18. The discs in the UZ Tau system show relative inclination differences
$\lesssim 0-20$\mathrm{\SIUnitSymbolDegree}$$, while the RW Aur, DK Tau and HK
Tau systems show a larger relative inclination ($\sim
25$\mathrm{\SIUnitSymbolDegree}$$, $\sim 40$\mathrm{\SIUnitSymbolDegree}$$,
and $\sim 50$\mathrm{\SIUnitSymbolDegree}$$, respectively). For the latter,
this is in line with the earlier results of Jensen & Akeson (2014). We
conclude that most of the observed binaries appear not to be coplanar,
possibly a result of turbulent fragmentation, rather than disc fragmentation.

##  6 Conclusions

We have presented here new Band 6 ALMA observations of CO line and continuum
emission in ten multiple stellar systems in the Taurus star-forming region.
The observed sample comprises seven binaries, one triple, one quadruple and
one star part of a very wide binary system, and was observed at a spatial
resolution of $\sim 0\aas@@fstack{\prime\prime}15$ and with an integration
time of $\sim 40$ min/target. Three systems (GK Tau, UY Aur and T Tau) were
not analysed here. One additional multiple system, RW Aur, was also included
in the analysis using ALMA archival data. Of this sample, eight multiple
stellar systems were analysed in this work.

We derived the disc geometrical properties using the eddy tool (Teague 2019),
and then performed an image plane analysis of the data, to estimate the disc
total fluxes with a cumulative flux technique. Assuming as disc radii those
enclosing a certain fraction of the total disc fluxes (68% or 95%), we
compared the gas radii to the dust radii estimated by Manara et al. (2019),
and derived the gas-to-dust size ratio to be $\sim 2-4$. The effective (68%)
gas-to-dust size ratio distribution is found to be statistically compatible to
the ratio estimated by Sanchis et al. (2021) in a population of single discs.
On the contrary, considering the 95% disc radius, the gas-to-dust size ratio
is found to be on the high-end of the distribution of the gas-dust size ratios
measured in a population of more isolated systems, possibly due to the sharp
truncation of the outer dusty discs in binary systems (Manara et al. 2019).

We compared our estimates with analytical predictions for the tidal truncation
on disc sizes in binary systems (Artymowicz & Lubow 1994). In general, the 95%
gas disc radii are $\sim 0.15-0.35$ times the projected separation of the
binaries, suggesting that the systems are not in circular orbits. Exception to
this typical ratio are the discs around UZ Tau E, CIDA 9 A, and RW Aur that
show very large value of this ratio possibly due to projection effects or
additional processes regulating disc truncation (e.g. substructure for the
first two systems and interaction between the component for RW Aur). When
comparing these ratios with the theoretical predictions, the minimum
eccentricities of the orbits are $0\lesssim e\lesssim 0.5$, with an average
value of $\sim 0.3$, in good agreement with expectations (e.g., Duchêne &
Kraus 2013). Finally, the discs in multiple systems appear to be misaligned,
possibly a result of turbulent fragmentation.

This study shows the importance of deep ALMA observations of line emission
from discs in multiple stellar systems to derive their sizes and constrain
models of tidal interactions. Future studies targeting multiple systems should
aim at larger samples and should cover a larger span of binary separations, in
particular focusing on the shorter separations.

###### Acknowledgements.

We thank the anonymous referee for the comments that greatly improved this
paper. This paper makes use of the following ALMA data:
ADS/JAO.ALMA#2018.1.00771.S, ADS/JAO.ALMA#2016.1.00877.S,
ADS/JAO.ALMA#015.1.01506.S. ALMA is a partnership of ESO (representing its
member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST
and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the
Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and
NAOJ. The authors are thankful to Joey Rodriguez for sharing the reduced CO
cubes for RW Aur, to Enrique Sanchis for sharing the values of the gas disc
radii prior to publication, and to Leonardo Testi for very helpful tips on the
self-calibration of the data. A.A.R. thanks Cristiano Longarini for useful
discussions. A.A.R. acknowledges support from ESO for a six month visit
through the SSDF Funding. This project has received funding from the European
Union’s Horizon 2020 research and innovation programme under the Marie
Sklodowska-Curie grant agreement No 823823 (DUSTBUSTERS). This work was partly
supported by the Deutsche Forschungs-Gemeinschaft (DFG, German Research
Foundation) - Ref no. FOR 2634/1 TE 1024/1-1. M.T. has been supported by the
UK Science and Technology research Council (STFC) via the consolidated grant
ST/S000623/1, and by the European Union’s Horizon 2020 research and innovation
programme under the Marie Sklodowska-Curie grant agreement No. 823823 (RISE
DUSTBUSTERS project). FMe, GvdP acknowledge funding from ANR of France under
contract number ANR-16-CE31-0013. H.-W.Y. acknowledges support from Ministry
of Science and Technology (MOST) in Taiwan through the grant MOST 108-
2112-M-001-003-MY2. P.P. acknowledges support provided by the Alexander von
Humboldt Foundation in the framework of the Sofja Kovalevskaja Award endowed
by the Federal Ministry of Education and Research. E.R. acknowledges financial
support from the European Research Council (ERC) under the European Union’s
Horizon 2020 research and innovation programme (grant agreement No. 681601 and
No. 864965). F.L. acknowledges support from the Smithsonian Institution as the
Submillimeter Array (SMA) Fellow. D.H. is supported by CICA through a grant
and grant number 110J0353I9 from the Ministry of Education of Taiwan.

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##  Appendix A Maps and spectra for all sample

In this section, we report the maps, the spectra, and the channel maps of all
analysed and detected targets. Although in this work we focus on the 12CO
emission, the targeted emission lines include also CO isotopologues $-$ 13CO
and C18O.

The continuum and CO zeroth and first moment maps are shown with the
integrated spectra for all targets in Figure 19. The detected discs (see Table
3) show a rotation pattern, that is fitted with eddy (Section 4.1).

The channel maps of shown in Figures 20-26 are used also to identify the
channels affected by absorption due to the foreground molecular cloud,
particularly pronounced in the 12CO emission. The velocity ranges affected by
cloud absorption are reported in Table 4, and excluded from the analysis with
eddy.

Figure 19: Images and spectra of all target in the analysed sample. Each row
refers to a system in the sample. The first columns show the continuum images
with the names of the targets indicated in the top left. The second, fourth
and sixth columns show the zeroth moment images for the 12CO, the 13CO, and
the C18O emission, respectively – scaled so that the maximum is equal to the
peak flux and the minimum is clipped at the image rms. The third, fifth and
seventh columns show the first moment images (calculated through the
bettermoments tool) for the 12CO, the 13CO, and the C18O emission,
respectively – scaled so that the maximum and the minimum are equal to the
systemic velocity of the target $\pm 2$ km/s. The FWHM beam size is shown in
the bottom left of each panel. All bars in the bottom right of each panel are
0$\aas@@fstack{\prime\prime}$5 long. The last two columns show the 12CO and
13CO spectra of the circumprimary (red), circumsecondary (blue), and
circumtertiary (green) discs (when applicable), with vertical black lines
showing the systemic velocities fitted with the eddy tool and grey band
showing the excluded absorbed velocities in eddy fitting on circumprimary disc
rotation maps (see Table 4). Figure 20: Channel maps of the 12CO emission in
CIDA 9 A disc. Each panel shows the velocity of the channel in the upper left
corner. The FWHM beam size is shown in the bottom left. The images are scale
so that the maximum is equal to the peak flux and the minimum is clipped at
the image rms. White contours show 5, 30, and 100 times the rms of the
continuum emission.  Figure 21: Channel maps of the 12CO emission in DH Tau A
disc. Same panels and symbols as in Figure 20.  Figure 22: Channel maps of the
12CO emission in DK Tau discs. Same panels and symbols as in Figure 20.
Figure 23: Channel maps of the 12CO emission in HK Tau discs. Same panels and
symbols as in Figure 20.  Figure 24: Channel maps of the 12CO emission in HN
Tau A disc. Same panels and symbols as in Figure 20.  Figure 25: Channel maps
of the 12CO emission in UZ Tau discs. Same panels and symbols as in Figure 20.
Figure 26: Channel maps of the 12CO emission in V710 Tau A disc. Same panels
and symbols as in Figure 20.

##  Appendix B Comparison between new continuum data and older continuum data

As described in Sect. 2, the new data analyzed here have been taken for the
same sample as the one presented by Manara et al. (2019) and Long et al.
(2019). The main differences of these samples, on top of the presence in the
new one of the 12CO emission, are the spatial resolution and the sensitivity
of the observations. The new data have slightly worst spatial resolution
($0\aas@@fstack{\prime\prime}15$ vs $0\aas@@fstack{\prime\prime}12$) and
better sensitivity ($\sim$30 $\mu$Jy rms vs $\sim
50$\mathrm{\SIUnitSymbolMicro}$$Jy beam-1). Whereas this work focuses on the
analysis of the gas emission, we discuss here the differences between the two
data-sets in the continuum emission, and motivate the choices of which data-
sets and analysis techniques have been used to discuss our results.

We first show in Figure 27 that the measured flux of the continuum emission
measured on the data presented by Manara et al. (2019) agrees with the total
flux measured on the new data within the 10% uncertainty that is usually
assumed on ALMA flux measurements. Exception to this good agreement are the
circumsecondary discs around the stars in UZ Tau systems, which fluxes differ
by $\sim 20-30\%$.

Figure 27: Comparisons between the measured total disc fluxes on the new
continuum observation and on the data presented by Manara et al. (2019). The
left panel shows the ratio between the total fluxes calculated on the data by
Manara et al. (2019) ($F_{\mathrm{Manara,c.f.t.}}$) and on the new continuum
data ($F_{\mathrm{new}}$), both estimated through the cumulative flux
technique. The right-hand panel, instead, reports the ratio using the fluxes
calculated by Manara et al. (2019) with the uv-modelling method
($F_{\mathrm{Manara,Galario}}$), and on the new continuum data with the
cumulative flux technique. Red circles are used for circumprimary discs, blue
squares for circumsecondary ones.

We then compare the measured sizes of the discs on the old and new data,
accounting for the differences in resolution and method. Images of
interferometric data are always the result of the convolution of the effective
beam of the interferometer with the intensity profile of the science target,
i.e., the disc. The size of the beam is thus a bigger effect when it is close
to the physical size of the target. In order to estimate the impact of the
beam on our measurements, we first neglect the fact that the beam is
elliptical with beam major axis BMAJ and minor axis BMIN, and we make the
assumption that the beam can be modelled as a one-dimensional Gaussian:

$\mathcal{B}(x)=\frac{1}{\sqrt{2\pi}\sigma_{\mathrm{res}}}\exp\biggl{(}-\frac{1}{2}\frac{(x-\mu)}{\sigma_{\mathrm{res}}^{2}}\biggr{)},$
(3)

where $\mu$ is the source centre and the standard deviation
$\sigma_{\mathrm{res}}$ is linked to the beam size $\Gamma$ (FWHM beam) as

$\sigma_{\mathrm{res}}=\frac{\Gamma}{2\sqrt{2\ln{2}}}.$ (4)

The convolution of the beams size with the intrinsic (“true”) intensity
profile of the disc is then calculated under the corresponding assumption that
the disc can be described with a one-dimensional Gaussian. Hence, the standard
deviation $\sigma_{\mathrm{true}}$ of the Gaussian is equal to the effective
disc radius $R_{68,\mathrm{true}}$ (since the effective disc radius is defined
as the radius containing the $68\%$ of the total flux). The convolution
between these two Gaussians is a one-dimensional Gaussian with standard
deviation equal to

$\sigma_{\mathrm{obs}}^{2}=\sigma_{\mathrm{true}}^{2}+\sigma_{\mathrm{res}}^{2}=R_{68,\mathrm{true}}^{2}+\sigma_{\mathrm{res}}^{2},$
(5)

which represents the observed effective disc radius:
$\sigma_{\mathrm{obs}}\equiv R_{68,\mathrm{obs}}$.

|  Beam size [ $\hbox{${}^{\prime\prime}$}\times\hbox{${}^{\prime\prime}$}$]
---|---
CIDA 9 | $0.130\times 0.099$
DH Tau | $0.132\times 0.107$
DK Tau | $0.129\times 0.107$
HK Tau | $0.122\times 0.107$
HN Tau | $0.142\times 0.100$
RW Aur | $0.158\times 0.100$
T Tau | $0.138\times 0.100$
UY Aur | $0.152\times 0.099$
UZ Tau | $0.127\times 0.105$
V710 Tau | $0.139\times 0.099$
Table 9: Beam sizes of the observations by Manara et al. (2019).

The data analyzed here have a spatial resolution of
$0\aas@@fstack{\prime\prime}15$, whereas the observations analysed by Manara
et al. (2019) had an angular resolution of $0\aas@@fstack{\prime\prime}12$.
Tables 2 and 9 show for each target the synthesised beams of the observations
respectively for the new data and the data by Manara et al. (2019). Since we
are assuming a one-dimensional Gaussian intensity profile and beam, we are
neglecting the position angle effect both of the disc and of the beam. Hence,
the minimum BMIN gives an estimate of the best beam resolution, while the
maximum BMAJ gives an estimate of the worst beam resolution. In the continuum
data by Manara et al. (2019), the minimum BMIN and the maximum BMAJ are
respectively $\Gamma_{\mathrm{min,old}}=0\aas@@fstack{\prime\prime}099$ and
$\Gamma_{\mathrm{max,old}}=0\aas@@fstack{\prime\prime}142$ (BMIN of CIDA 9 and
BMAJ of HN Tau, respectively). In the new continuum data, the minimum BMIN and
the maximum BMAJ are respectively
$\Gamma_{\mathrm{min,new}}=0\aas@@fstack{\prime\prime}120$ and
$\Gamma_{\mathrm{max,new}}=0\aas@@fstack{\prime\prime}282$ (BMIN of HN Tau and
BMAJ of V710 Tau, respectively). In this way, the maximum effect on the ratio
of the beam size between the data by Manara et al. (2019) and the new data is
given by

$\frac{\sigma_{\mathrm{obs,new}}}{\sigma_{\mathrm{obs,old}}}\biggl{|}_{\mathrm{max}}(\sigma_{\mathrm{true}})=\sqrt{\frac{\sigma_{\mathrm{true}}^{2}+\bigl{[}\Gamma_{\mathrm{max,new}}/(2\sqrt{2\ln{2}})\bigr{]}^{2}}{\sigma_{\mathrm{true}}^{2}+\bigr{[}\Gamma_{\mathrm{min,old}}/(2\sqrt{2\ln{2}})\bigr{]}^{2}}},$
(6)

where $\sigma_{\mathrm{true}}$ is the “true” effective disc radius, i.e.
$R_{68,\mathrm{true}}$. On the contrast, the minimum effect is given by

$\frac{\sigma_{\mathrm{obs,new}}}{\sigma_{\mathrm{obs,old}}}\biggl{|}_{\mathrm{min}}(\sigma_{\mathrm{true}})=\sqrt{\frac{\sigma_{\mathrm{true}}^{2}+\bigl{[}\Gamma_{\mathrm{min,new}}/(2\sqrt{2\ln{2}})\bigr{]}^{2}}{\sigma_{\mathrm{true}}^{2}+\bigr{[}\Gamma_{\mathrm{max,old}}/(2\sqrt{2\ln{2}})\bigr{]}^{2}}}.$
(7)

If the disc is described by a one-dimensional Gaussian intensity profile,
since $R_{95,\mathrm{true}}$ is defined as the radius containing the $95\%$ of
the total flux, $R_{95}\equiv 2R_{68}$. Therefore, when estimating the
maximum/minimum effect of the difference in the beam size as a function of the
disc radii $R_{95}$ the factor two simplifies, and Equation (6) and (7) hold
with $\sigma_{\mathrm{true}}$ expressing the dust disc radii $R_{95}$.

Figure 28: Observed ratio between the dust radii estimated with the cumulative
flux technique applied on the data by Manara et al. (2019) and the dust radii
from the new continuum data ($R_{\mathrm{new}}/R_{\mathrm{Manara}}$) as a
function of the new dust radii ($R_{\mathrm{new}}$). Left: dust radii
estimated as the radii including the 68% of the total flux (i.e. effective
disc radii). Right: dust radii estimated as the radii including the 95% of the
total flux. Black lines show the maximum (solid) and minimum (dashed) effect
on the ratio of the beam size between the data by Manara et al. (2019) and the
new data (see Equations (6) and (7)). Blue squares refer to the
circumsecondary disc and red circles to the circumprimary discs.

Figure 28 shows these maximum (black solid line) and minimum (black dashed
line) effects and compares these estimates with the observed ratio between the
dust effective radii ($R_{68}$, left panel) and the dust radii ($R_{95}$,
right panel) estimated with the cumulative flux technique applied on the data
by Manara et al. (2019) and the new continuum data. In general, as shown by
the black lines, the smaller the disc size is, the greater the beam-size
effect is, while, as expected, for larger discs the effect due to the
different beam size tends towards 1. The general trend of the observed ratio
(red circles for circumprimary discs and blue squares for circumsecondary
discs) shows a good agreement with the estimates (black lines). Figure 28
shows also a systematic shift on the $y$-direction of the observed ratio both
between $R_{68}$ and $R_{95}$. We would expect such a systematic shift in the
ratio between the dust radii $R_{95}$ (and not between $R_{68}$), due to the
difference in integration time between the observations by Manara et al.
(2019) and the new observations $-\sim 4-9$ min/target versus $\sim 40$
min/target, respectively. This difference makes the new observations deeper,
i.e. more sensitive, than the ones by Manara et al. (2019). However, as
mentioned, Figure 27 shows that the dust fluxes estimated on the new data and
on the data by Manara et al. (2019) are in good agreement with the 10%
uncertainty usually assumed on ALMA fluxes measurements. Therefore, in most
systems, this systematic shift probably is not due to a sensitivity effect and
the reason why it is observed must be further investigated. In particular, the
systematic shift might be due to the assumption on the intensity profile of
the disc, which actually is not a Gaussian, but a profile with less peaked
emission in the centre, meaning that the 68% (and 95%) of the emission is
distributed over a larger radius.

Finally, we compare the radii estimated with the analysis carried out in the
$(u,v)$ plane (Manara et al. 2019) with those obtained with the method used in
this work (Sect. 4). This is shown in Figure 29. With the visibility plane
analysis a resolution down to one third of the beam can be achieved (e.g.,
Jennings et al. 2020). On the other hand, the cumulative flux technique is
performed in the image plane, thus the convolution of the beam with the
intensity profile of the disc places an intrinsic resolution limit.

Figure 29: Comparison between dust disc radii estimated with the cumulative
flux technique (c.f.t.) applied on the data by Manara et al. (2019) and with
those estimated by Manara et al. (2019) with Galario-modelling
($R_{\mathrm{c.f.t}}/R_{\mathrm{Galario}}$). Left: dust radii estimated as the
radii including the 68% of the total flux (i.e. effective disc radii). Right:
dust radii estimated as the radii including the 95% of the total flux. Red
circles are used for circumprimary discs, blue square for circumsecondary
discs.

The left-hand panel in Figure 29 shows the comparison between the effective
dust radii $R_{68}$ estimated with the two methods. We cannot identify any
particular difference between the two radii estimates, with the only exception
being RW Aur discs and DK Tau B disc. We think that the fit with Galario for
RW Aur and the fit with eddy for DK Tau B could have led to wrong results.

On the contrary, in the right panel showing the comparison between the dust
radii $R_{95}$ it is observed that the dust radii estimated from the new
observations are typically slightly larger than those obtained with the data
by Manara et al. (2019). This is due to the cumulative flux technique, that
gets the really outer extent of the discs, where little emission is detected
(see Figure 7).

Since in the visibility plane analysis a resolution under the beam size is
achievable, the dust radii estimated using Galario are more reliable,
especially for the more compact discs.