circumstellar disk we obtain $c_{s}$ = 0.9-1.3 km s-1 (for $T$ = 140-300 K),
$H$ = 2.4-3.5 au (at the disk outer radius of 12 au), and $Q$ = 22-81.
Finally, for the VLA 4B circumstellar disk we obtain $c_{s}$ = 0.9-1.6 km s-1
(for $T$ = 140-450 K), $H$ = 1.1-1.9 au (at the disk outer radius of 9 au),
and $Q$ = 5-25. Therefore, all the disks are expected to be stable and self-
gravity is not likely important in the disks of SVS 13 at the present time.
This does not exclude, however, that the binary could have originated through
the fragmentation of a circumstellar disk, which in the past was more massive
and gravitationally unstable.

###  3.6 Chemistry of the Circumstellar and Circumbinary Environment

####  3.6.1 Hot Corinos in the SVS 13 Protobinary System

Complex Organic Molecules (COMs, molecules with six or more atoms with at
least one carbon atom; Herbst & van Dishoeck 2009) are usually detected in
association with the so-called hot molecular cores (HMC), which are hot and
dense regions that characterize one of the first stages in the formation of
massive stars (Osorio et al., 1999). However, in a few cases ($\sim$20
currently known), these complex molecules have been also detected in
association with low-mass protostars (e.g., IRAS 16293$-$2422: van Dishoeck et
al. 1995; Schöier et al. 2002; Jørgensen et al. 2016, 2018, NGC1333 IRAS 4A:
Sahu et al. 2019; De Simone et al. 2020), and these regions are called “hot
corinos”.

Hot corinos are small, dense regions very close to their host protostar with
temperatures higher than the water ice desorption temperature of $\sim$100 K,
and are considered the low-mass counterparts of HMCs. COMs are the trademark
of hot corinos. COMs have been detected previously toward SVS 13 with poorer
angular resolution (from $0\farcs 3$ to few arcseconds) observations that do
not resolve well the emission of the two components (López-Sepulcre et al.,
2015; Lefèvre et al., 2017; De Simone et al., 2017; Bianchi et al., 2019;
Belloche et al., 2020b). Since in these observations the peak emission of the
detected COMs was found to be closer to the compact source VLA 4A, Lefèvre et
al. (2017) proposed that this component of the binary was the hot corino. With
our higher angular resolution observations ($0\farcs 08\times 0\farcs$16), we
are able to detect COMs associated with both components of the binary, VLA 4A
and VLA 4B, as well as in the circumbinary material (Table 8).

We confirm the previous detection of acetaldehyde, ethanol (Bianchi et al.,
2019; Belloche et al., 2020b), formamide (López-Sepulcre et al., 2015; Bianchi
et al., 2019), ethylene glycol, methanol (Lefèvre et al., 2017; Belloche et
al., 2020b), and methyl formate (Lefèvre et al., 2017; Bianchi et al., 2019;
Belloche et al., 2020b) toward SVS 13. We additionally report the detection,
for the first time in this source, of acetone (CH3COCH3), ethyl cyanide
(C2H5CN), ethylene oxide (c-C2H4O), methyl cyanide (CH313CN), deuterated
acetaldehyde (CH3CDO), deuterated methanol (CH2DOH)222The detection of the
CH3OD isomer was reported by Lefèvre et al. (2017)., and deuterated methyl
cyanide (CH2DCN).

In general, these COMs emit in a region that appears to be associated with
circumbinary gas. In a few transitions (marked 4A in Table 8), the emitting
region is restricted to a smaller region surrounding VLA 4A (e.g., Fig. 13).
Interestingly, deuterated acetaldehyde is only detected toward the position
between the two protostars (marked as CB in Table 8).

We find that, in general, the COMs detected toward VLA 4A are also present
toward VLA 4B. The weaker transitions are not detected toward VLA 4B, and a
few transitions are only detected toward VLA 4A. This suggests that the
chemistry in VLA 4A could be much richer than in VLA 4B, as previous studies
with lower angular resolution also suggest (e.g., Lefèvre et al. 2017). As
noted in Section 3.4.1, the velocity dispersion toward VLA 4A is much larger
than toward VLA 4B; also, VLA 4A appears to be the origin of the main spiral
arm. All these facts together suggest that a higher kinematical and heating
activity is taking place in VLA 4A. Alternatively, the lack of detection of
several molecular transitions toward VLA 4B could reflect the fact that the
continuum emission peak toward VLA 4B has a very high optical depth that makes
difficult the observation of line emission at this position. Recently, De
Simone et al. (2020) have suggested that some reported differences in the COM
detections toward binary protostars that were attributed to differences in the
chemical activity of the two binary components, could, in fact, be due to a
high dust opacity that would hide the molecular lines. This effect could be
taking place in the SVS 13 binary leading to a reduction, or even a
suppression, of the detectable continuum-subtracted molecular line emission
toward VLA 4B. However, the high sensitivity of our observations allow us to
confirm that COMs are present in both components of the SVS 13 binary and,
therefore, that both components are “bona fide” hot corinos.

We note that there is a 3-$\sigma$ negative contour toward VLA 4B in the left
and middle top panels of Figure 13, and in Figure 14. The present analysis
does not allow us to distinguish whether this is a weak absorption feature or
just the result of a somewhat imperfect continuum subtraction. Detailed
analysis of these possible absorption features will be investigated in a
future work.

####  3.6.2 Chemical Abundances

We detected four transitions of the main isotopologue of methanol (CH3OH).
However, these transitions appear to be optically thick, so we cannot derive
reliable values of the molecular column density directly from these data.
Fortunately, we detected the isotopologues 13CH3OH and CH318OH, which are
optically thinner, and are therefore more suitable for deriving column
densities. However, the number of detected transitions of these isotopologues
is insufficient to construct rotational diagrams for a good and reliable
temperature and column density fitting. So, we have adopted a temperature of
140 K, derived from other molecular species, to obtain the column densities
that best reproduce the observed spectra, assuming optically thin emission and
LTE. Finally, the methanol column density can be obtained assuming isotopic
ratios (12C/13C = 77 and 16O/18O = 560; Wilson & Rood 1994).

In Table 6 we report the values of the 13CH3OH and CH318OH column densities
toward the center of the circumbinary disk (CB), at a position midway between
VLA 4A and VLA 4B, as well as the adopted column density of CH3OH. The CH3OH
methanol column density derived from the 13CH3OH isotopologue is
0.6$\times$$10^{19}$ cm-2, while the value derived from CH318OH is
1.7$\times$$10^{19}$ cm-2. If the isotopic ratios were as assumed (Wilson &
Rood, 1994), and the emission of both isotopologues were optically thin, we
would expect the observed 13CH3OH line intensity (as well as the estimated
column density) to be a factor 560/77 $\simeq$ 7 higher than that of the
CH318OH, resulting in the same derived CH3OH column density in both cases.
However, the ratio of the observed isotopologue line intensities is $\sim$3, a
value smaller than expected, indicating that either the 13CH3OH emission is
optically thick or the isotopic ratios are somewhat different than assumed.

To clarify this issue, we obtained additional spectra of the two isotopologues
toward positions slightly offset from CB, where the column density has
decreased and the emission is expected to become optically thinner. We found
that the intensities drop by a factor of $\sim$6-8 and $\sim$3-4 toward
positions offset $0\farcs 2$ north (close to the S3 feature) and south (close
to spiral S1) of CB, respectively. However, the isotopologue line intensity
ratio toward these positions is $\sim$2 and $\sim$4, respectively, not much
different from the ratio of $\sim$3 observed toward CB. Therefore, we conclude
that the discrepancy in the inferred methanol column densities does not seem
to be due to opacity effects, and that the 13C isotope is probably somewhat
less abundant and/or the 18O more abundant than we have initially assumed;
that is, 12C/13C $\gtrsim$ 77 and/or 16O/18O $\lesssim$ 560\. Thus, we adopt
in Table 6 an intermediate value of $\sim$1$\times 10^{19}$ cm-2 for the
methanol column density toward CB.

Belloche et al. (2020b) found a lower value of $2\times 10^{18}$ cm-2 for the
methanol column density in SVS 13. As discussed above (e.g., Sect. 3.5.1), we
attribute this discrepancy in SVS 13 to the larger beam of those observations,
probing scales of $\sim$100 au, as compared with our data that trace smaller
scales of $\sim$30 au. On the other hand, methanol column densities of
$\sim$$10^{19}$ cm-2, similar to the value we obtain for SVS 13, have been
reported in other hot corinos, even at larger scales of $\sim$60 au in IRAS
16293$-$2422B (Jørgensen et al., 2016, 2018) and $\sim$120 au in L483
(Jacobsen et al., 2019).

One of the molecules for which we derived a rotational temperature (Fig. 15)
is methyl formate (CH3OCHO). Since the CH3OCHO column density toward CB is
$2.3\times 10^{17}$ cm-2 (Table 6), we infer an abundance relative to methanol
[CH3OCHO/CH3OH] $\simeq$ 0.023 in SVS 13. Our value is lower than the value of
0.10 reported by Belloche et al. (2020b) in SVS 13, likely because the column
density of methanol obtained by these authors is lower (see above). The
CH3OCHO abundances relative to methanol reported for other hot corinos span
two orders of magnitude (see Fig. 7 of Bergner et al. 2019 or Fig. 17 of Yang
et al. 2021 for a comparison). Our value, similar to the 0.026 value found in
IRAS 16293$-$2422B by Jørgensen et al. (2016, 2018), lies in the middle of the
range. Additionally, from the CH3OH and CH2DOH column densities toward CB
(Table 6) we obtain a deuteration fraction D/H = 0.05, similar to values found
in other low-mass protostars (Jørgensen et al., 2018; Taquet et al., 2019).

The analysis toward the positions of the individual protostars is more
challenging since the 13CH3OH and CH318OH lines are wider and difficult to
separate from other lines there. To obtain abundances relative to methanol
toward these positions we infer the methanol column density from the
deuterated methanol, CH2DOH, which is detected toward all the positions (Table
6), using the deuteration fraction. In this way, for aGg’-(CH2OH)2, which we
only detect toward the position of VLA 4A, we obtain an abundance relative to
methanol of $7.1\times 10^{-3}$, which is quite similar to the value of
$6.4\times 10^{-3}$ reported by Belloche et al. (2020b).

Abundance ratios and their variation across the source can be obtained for
other detected molecules and a more complete analysis can be performed from
the data collected with ALMA. This further chemical analysis is postponed for
a separate paper.

Nevertheless, a rough estimate of the absolute abundances, i.e., relative to
H2, can be obtained for methanol and the other molecules listed in Table 6,
toward the central position (CB) using the beam-averaged column density
obtained from the dust emission. As discussed in Section 3.2.2, the dust
emission is optically thin toward this position, and we can further assume
that there the dust temperature coincides with the rotational temperature of
140 K obtained from molecular observations. We cannot make a similar
assumption toward the positions of VLA 4A and VLA 4B, because it is unclear
whether toward these positions the dust originates from the same region as the
observed molecular emission since, as discussed in Section 3.2.2, at least
part of the dust emission appears to come from a different, more compact and
hotter region. Thus, assuming a gas-to-dust ratio of 100, and a dust opacity
$\kappa_{\nu}$(0.9 mm) = 1.65 cm2 g-1 (per unit mass of dust), as in Section
3.5.2, we estimate an H2 column density of 1.8$\times$1024 cm-2 toward CB from
which we infer the abundances relative to H2 of the molecules listed in Table
6.

We obtain an abundance of methanol relative to H2 of 5.6$\times$10-6 toward
the central position (CB). This abundance of methanol is consistent with the
results of Bianchi et al. (2019), who report a value of the 13CH3OH abundance
of 3$\times$10-8 in SVS 13, similar to the value of 4.4$\times$10-8 listed in
our Table 6 for the same isotopologue. The methanol abundance we have
estimated in SVS 13 is of the same order than the values obtained in other
protobinaries (e.g., $<$3$\times$10-6 in IRAS 16193$-$2422; Jørgensen et al.
2016). We note that the presence of temperature gradients, uncertainties in
the dust opacity and in the gas-to-dust ratio can be important and difficult
to estimate. Therefore the values of the abundances relative to H2 should be
taken with caution.

Nevertheless, we can still measure variations from one position to another as
they are less affected by the uncertainties in the absolute values. We find
evidence of a decrease of the methanol abundance toward the north of the
central position CB, since when moving 0$\farcs$2 (60 au) to the north of this
central position, the estimated column densities of the 13CH3OH and CH318OH
(and therefore, the methanol abundance) decrease by a larger factor than the
H2 column density. Assuming that there are no significant variations in the
dust properties at these scales, we infer that the methanol abundance relative
to H2 toward this northern position decreases by a factor 2.5-3.2 with respect
to the CB position. We do not see a similar behavior, and the abundance
remains the same, when moving the same distance to the south of the central
position. We speculate on whether the presence of more prominent spiral arms
connected to the southern part of the circumbinary disk could be related with
an overabundance of methanol relative to the northern part.

##  4 Summary and Conclusions

We carried out a comprehensive study of the protobinary system associated with
the SVS 13 star in the star-forming region of NGC 1333. We have clarified the
geometry of the system and the physical properties, even at very small scales
of $\sim$10-20 au, thanks to sensitive, high-angular resolution VLA and ALMA
observations. These observations provided us with proper motions measurements
in the plane of the sky and radial velocities along the line-of-sight,
allowing us to infer the kinematical and dynamical properties of the system.
Overall, temperatures, masses and other physical parameters have been inferred
to characterize the SVS 13 system.

Despite the uncertainties involved, we favor the scenario of a single
component for VLA 4A and for VLA 4B. The SVS 13 system would, then, consist of
two protostars, with masses of $\sim$0.26 $M_{\odot}$ and $\sim$0.6
$M_{\odot}$ for VLA 4A and VLA 4B respectively, each of them surrounded by a
circumstellar protoplanetary disk, and encompassed by a larger molecular and
dust structure associated with at least two spiral arms that we interpret as
an incipient circumbinary disk (see Fig. 12). The mass of the whole system is
dominated by the protostars, with a total mass of 1 $M_{\odot}$, as suggested
both by the orbital motions and the dynamical calculations of the rotating
and/or infalling gas for each individual protostar. The mass in the disks is
much smaller ($<10\%$). In this configuration, VLA 4B is likely the primary
and VLA 4A the secondary, with a ratio between the secondary and primary
masses, q$\simeq$0.26/0.6=0.4. However, uncertainties are still large, and a
configuration with two objects of similar mass (or even with VLA 4A being the
more massive) cannot be discarded with the current data. This result can be
substantially improved with sensitive new observations of higher angular
resolution that can better resolve the PV diagrams and potentially reveal a
Keplerian rotation profile. Obtaining molecular data of high angular
resolution but at lower frequencies, where the dust contribution should be
smaller, could help to minimize the difficulties in detecting molecular
emission from the circumstellar disk of VLA 4B, which has been hindered in our
observation by the very high opacity of the continuum. These observations will
be easily performed with the next generation Very Large Array, that will
provide unprecedented sensitivity and angular resolution in the 40 to 120 GHz
regime.

Table 7: Summary of Physical Parameters of the SVS 13 Protobinary System

Parameter | VLA 4A | VLA 4B | Circumbinary | Notes
---|---|---|---|---
Disk Size (dust) | 24$\pm$1 au $\times$ 10$\pm$6 au | 18$\pm$1 au $\times$ 14$\pm$3 au | 750$\pm$75 au $\times$ 600$\pm$75 au | 1
Disk PA (dust) | 83$\arcdeg$$\pm$8$\arcdeg$ | 93$\arcdeg$$\pm$17$\arcdeg$ |  $\sim$90$\arcdeg$ | 1
$R_{\rm disk}$, $i$ (dust) | 12$\pm$1 au, 64$\arcdeg$$\pm$16$\arcdeg$ | 9$\pm$1 au, 40$\arcdeg$$\pm$12$\arcdeg$ | 375$\pm$38 au, 34$\arcdeg$$\pm$12$\arcdeg$ | 2
$T_{\rm dust}$ | 260$\pm$60 K | 430$\pm$80 K |  $\cdots$ | 3
Disk Size (gas) | 67$\pm$3 au $\times$ 62$\pm$3 au |  $\cdots$ | 240$\pm$30 au $\times$ 180$\pm$30 au | 4
Disk PA (gas) |  $\sim$90$\arcdeg$ |  $\cdots$ |  $\sim$90$\arcdeg$ | 4
$R_{\rm disk}$, $i$ (gas) | 33$\pm$2 au, 22$\arcdeg$$\pm$7$\arcdeg$ |  $\cdots$ | 120$\pm$15 au, 38$\arcdeg$$\pm$14$\arcdeg$ | 5
$M_{*}$ (EthyGly) | 0.25$\pm$0.15 $M_{\sun}$ |  $\cdots$ |  $\cdots$ | 6
$M_{*}$ (CS) | 0.27$\pm$0.10 $M_{\sun}$ | 0.60$\pm$0.20 $M_{\sun}$ |  $\cdots$ | 7
$T_{\rm rot}$ | 126-145 K | 90-160 K | 138-139 K | 8
$M_{\rm disk}$ | 0.004-0.009 $M_{\sun}$ | 0.009-0.030 $M_{\sun}$ | 0.052 $M_{\sun}$ | 9
$\mu_{\alpha}\cos\delta$ | 8.22$\pm$0.63 mas yr-1 | 8.56$\pm$0.34 mas yr-1 |  | 10
$\mu_{\delta}$ |  $-$9.19$\pm$0.41 mas yr-1 |  $-$10.79$\pm$0.44 mas yr-1 |  | 10
$V_{\rm LSR}$ | 7.36$\pm$0.21 km s-1 | 9.33$\pm$0.18 km s-1 | 8.77$\pm$0.16 km s-1 | 11
$\vec{V}_{\rm rel}$(B$-$A) | (0.48$\pm$1.02 km s-1, $-$2.28$\pm$0.82 km s-1, 1.97$\pm$0.28 km s-1) |  | 12
$i$ (orbit) | 41.7$\arcdeg$$\pm$9.5$\arcdeg$ |  | 13
$a$ (orbit) | 90 au |  | 14
$M_{*}$ (orbit) | 1.0$\pm$0.4 $M_{\sun}$ |  | 15
$P$ (orbit) | 850 yr |  | 16
$c_{s}$ | 0.90-1.3 km s-1 | 0.9-1.6 km s-1 | 0.90 km s-1 | 17
$H(R_{\rm disk})$ | 2.4-3.5 au | 1.1-1.9 au | 40 au | 18
$Q(R_{\rm disk})$ | 22-81 | 5-25 | 13 | 19

11footnotetext:  VLA 4A and VLA 4B: Deconvolved FWHM size obtained from a
Gaussian fit to the continuum image made using baselines $>$750 k$\lambda$,
where the extended emission has been filtered out (Fig. 8). The errors given
are the formal errors of the fits. Similar sizes of 30$\pm$1 au $\times$
16$\pm$4 au, PA=80$\arcdeg$$\pm$7$\arcdeg$ (VLA 4A) and 20$\pm$1 au $\times$
14$\pm$5 au, PA=93$\arcdeg$$\pm$20$\arcdeg$ (VLA 4B) are obtained from a fit
to an image made using baselines $>$500 k$\lambda$. Circumbinary emission:
Size and PA of the emission above a 6-$\sigma$ level, measured on the image
made using all the visibilities (Fig. 7 in Sect. 3.2.2), including the
emission of the spirals.

22footnotetext:  Disk radius and inclination angle. The disk radius is
obtained as half the major axis; since the sources are elongated roughly along
the direction of the minor axis of the beam, where the angular resolution is
higher, the disk radii are well constrained. The disk inclination angle
($i$=0$\arcdeg$ means face-on) is obtained from the aspect ratio of the
emission (see note 1); uncertainties are derived as explained in Appendix C,
but they may be significantly underestimated for the circumstellar disks since
their minor axes are poorly resolved because of the elongated beam.

33footnotetext:  Dust temperature in the circumstellar disks, taken as the
brightness temperature derived from Gaussian fits to the highest angular
resolution 0.9 mm continuum images (Table 5 in Section 3.2.2). The values
listed are those with the smallest uncertainties, obtained from the image made
using baselines $>$500 k$\lambda$ for VLA 4A and $>$750 k$\lambda$ for VLA 4B.
The dust emission in the circumbinary disk appears to be optically thin and
the brightness temperature is not useful as a constraint on the dust
temperature.

44footnotetext:  VLA 4A: Derived from a model fitting of a disk to ethylene
glycol transitions (Fig. 13 in Sect. 3.4.3; see Appendix D). VLA 4B: No
suitable molecular emission imaged. Circumbinary: From a Gaussian fit to the
zero-order moments of the H13CN and HC15N isotopologues; a slightly larger
size of $\sim$300 $\times$ 240 au is obtained from the CS(7-6) transition,
which is probably optically thicker (Fig. 11 in Sect. 3.4.1).

55footnotetext:  VLA 4A: Disk radius and inclinaton angle derived from the
model fitting (Appendix D). Circumbinary: See notes 2 and 4.

66footnotetext:  Stellar mass derived from the model fitting of a thin
Keplerian disk to the ethylene glycol transitions associated with VLA 4A
(Appendix D and Sect. 3.5.3).

77footnotetext:  Stellar mass derived from the model fitting of a spherical
infalling envelope to the CS(7-6) emission associated with VLA 4A and VLA 4B
(Appendix E and Sect. 3.5.3).

88footnotetext:  Rotational temperature derived from a LTE fit to several
molecular transitions (see Sect. 3.5.1).

99footnotetext:  Total mass (gas+dust) of the circumstellar/circumbinary
material, obtained from the dust emission assuming a gas-to-dust ratio of 100
and optically thin emission at 0.9 mm (circumbinary) and 7 mm (circumstellar).
For the circumbinary mass we assume a dust temperature of 140 K. The lower
limits for the VLA 4A and VLA 4B circumstellar disk masses have been obtained
for temperatures of 300 K and 450 K, respectively, while the upper limits have
been obtained using 140 K (Sect. 3.5.2).

1010footnotetext:  Absolute proper motions, in right ascension and
declination, of the two protostars (Sect. 3.1.3).

1111footnotetext:  LSR line-of-sight velocity derived from molecular spectra
toward the positions of VLA 4A, VLA 4B, and toward the geometric center of the
system (CB) that we adopt as representative of the circumbinary disk (Fig.
22). We give the mean velocity and standard deviation of 20 lines toward CB
and 15 lines toward VLA 4A and VLA 4B. The assumed systemic velocity of the
ambient cloud is $V_{\rm sys}=8.5$ km s-1.

1212footnotetext:  Components of the 3D relative velocity of VLA 4B with
respect to VLA 4A, along the right ascension, declination and line-of-sight
axes, obtained from the observed differences in the proper motions and in the
LSR velocity (Table 4 and Sect. 3.5.3).

1313footnotetext:  Inclination of the orbital plane with respect to the plane
of the sky, derived from the 3D relative velocity of the two protostars.
Uncertainties are estimated as explained in Appendix C. The orbital parameters
are such that the observed orbital motion is counter-clockwise and the angular
momentum points toward the Earth (Fig. 12).

1414footnotetext:  Projected distance between VLA 4A and VLA 4B, taken as the
separation between the two components of the binary. The small relative proper
motion in right ascension as compared with the proper motion in declination
(Fig. 5 in Sect. 3.1.3), and the alignment of the sources along the major axes
of the disks (PA $\simeq$ 90$\arcdeg$; this table) suggests that this is a
good approximation.

1515footnotetext:  Total mass of the stellar system, derived from the orbital
motions (Sect. 3.5.3).

1616footnotetext:  Orbital period of the binary, derived from the orbital
motions (Sect. 3.5.3).

1717footnotetext:  Sound speed in the disks. See note 8 for the adopted
temperatures.

1818footnotetext:  Scale height (Sect. 3.5.3) at the disk radius (12 au for
VLA 4A, 9 au for VLA 4B, and 120 au for the circumbinary). See note 8 for the
adopted temperatures.

1919footnotetext:  Toomre Q parameter (see Sect. 3.5.3). All these disks are
found to be stable ($Q>1$).

Note. — Assuming $d$ = 300 pc.

In Table 7 we provide a characterization of this protobinary system in terms
of its physical properties. In a forthcoming paper we will compare these
properties, and other parameters inferred from them with the predictions of
existing numerical simulations. This will provide a deeper understanding of
the binary formation process and will guide future theoretical and
observational works.

Here we summarize our main conclusions:

  1. 1.

We confirm that SVS 13 is a multiple protostellar system with at least two
embedded protostars, separated by 90 au (for an adopted distance of 300 pc),
known as VLA 4A and VLA 4B. We cannot rule out a higher order multiple system,
as each of the two protostars could have a companion at scales below 20-30 au.
However, we favor the interpretation in terms of a binary system.

  2. 2.

Both protostars are associated with free-free emission (likely from jets),
circumstellar dust and molecular gas (likely from disks), and complex organic
molecules characteristic of hot corinos. The radii (half the FWHM obtained
from a Gaussian fit) of the VLA 4A and VLA 4B circumstellar disks of dust are
$\sim$12 au and $\sim$9 au, respectively. In both disks, the PA of the major
axis is $\sim$90∘. However, we cannot reliably obtain the inclination angle
from the aspect ratio because of insufficient angular resolution in the
direction of the minor axes due to a very elongated synthesized beam.

  3. 3.

In addition to the circumstellar disks, this protostellar system has an
associated circumbinary structure that appears to be a circumbinary disk in
the early stages of formation. Its radius, as inferred from the molecular
emission, is $\sim$120 au. The circumbinary disk image appears elongated along
PA$\sim$90∘, similar to the circumstellar disks, and from the aspect ratio we
infer an inclination angle of the molecular disk plane with respect to the
plane of the sky of $\sim$40$\arcdeg$. The circumbinary dust emission presents
a larger radius of $\sim$375 au, with prominent spiral arms extending to
distances of $\sim$500 au.

  4. 4.

The dust emission appears more intense and compact toward VLA 4B (the eastern
component), suggesting a very high column density and optical depth, at very
small scales, that makes difficult the detection of molecular lines toward the
stellar position. The dust column density toward VLA 4A is lower than toward
VLA 4B, resulting in a lower dust opacity and stronger observed molecular
lines. However, the dust emission associated with VLA 4A is more extended,
resulting in a larger total amount of dust in its surroundings.

  5. 5.

Molecular transitions typical of hot corinos are detected toward both VLA 4A
and VLA 4B, as well as in the circumbinary disk. We are able to estimate
rotational temperatures and molecular column densities, indicating warm
temperatures of $\sim$140 K for the molecular gas across the various
components of the source, and a rich chemistry. The dust emission suggests
still higher temperatures of $\sim$400 K at very small radii of $\sim$10 au.

  6. 6.

We detect line-of-sight velocity gradients in both the circumstellar and
circumbinary disks. These, together with the spiral arms and the stellar
proper motions, all indicate a rotation in a counter-clockwise direction as
observed from the Earth, with the northern half of the disk closer to the
observer. The angular momentum vector therefore points toward the observer.

  7. 7.

The orbital proper motions have been inferred from high-resolution continuum
data taken over 30 years. From these data and the line-of-sight velocities,
the 3D relative orbital velocity is obtained. From this, a total mass
(dominated by the stellar component) of 1 $M_{\sun}$, an orbital period of 850
yr, and an inclination angle of the orbit with respect to the plane of the sky
of 42$\arcdeg$ are derived. Most of the positions used in the proper motion
analysis were obtained from free-free emission that can be affected by
episodic ejections of blobs of ionized material that reduce the accuracy of
the measurements. The emission at 7 mm and shorter wavelengths, which traces
circumstellar dust and is not affected by this issue, is strong enough in SVS
13 that future observations at these wavelengths will give more accurate
positions to more tightly constrain the orbital parameters and mass of this
young system. These will provide important tests of models for the formation
of binary star systems.

  8. 8.

We identified several ethylene glycol transitions, which are associated only
with VLA 4A, showing a clear east-west velocity gradient. By fitting a
rotating disk, we consistently get a VLA 4A stellar mass of 0.25$\pm$0.15
$M_{\sun}$, a disk radius of 33$\pm$2 au, and an inclination angle of
$22\arcdeg\pm 7\arcdeg$, from independent fits to the different observed
transitions. The molecular disk radius appears to be larger than the dust disk
radius, as is often observed in other protoplanetary disks.

  9. 9.

The CS (7-6) position-velocity plots show indications of gravitational
acceleration. From a simple model of spherical infall onto each of the
protostars, we obtain rough estimates of their stellar masses of 0.27$\pm$0.10
$M_{\sun}$ for VLA 4A and 0.60$\pm$0.20 $M_{\sun}$ for VLA 4B, which are
consistent with the total mass inferred from the orbital motion and the mass
obtained from the ethylene glycol fitting of the VLA 4A circumstellar disk.
These calculations suggest that VLA 4B is the primary, but the uncertainties
are still large.

  10. 10.

From the dust emission we estimate the masses of the circumstellar and the
circumbinary disks, obtaining 0.004-0.009 $M_{\sun}$ for the VLA 4A
circumstellar disk, 0.009-0.030 $M_{\sun}$ for the VLA 4B circumstellar disk,
and 0.052 $M_{\sun}$ for the circumbinary disk. We find that the disks have
masses of the order of 1.5-5% the mass of the stars they surround, and the
Toomre parameter is Q$>1$ for all the disks, indicating that they are stable
at the present time. However, this does not exclude the possibility that the
binary could have originated through the past fragmentation of a circumstellar
disk around one of the stars.

  11. 11.

We report the detection, to the best of our knowledge for the first time in
SVS 13, of acetone (CH3COCH3), ethyl cyanide (C2H5CN), ethylene oxide
(c-C2H4O), methyl cyanide (CH313CN), deuterated acetaldehyde (CH3CDO),
deuterated methanol (CH2DOH), and deuterated methyl cyanide (CH2DCN).

  12. 12.

We estimate a methanol column density of $\sim$1$\times$1019 cm-2 and an
abundance relative to H2 of $\sim$6$\times$10-6 toward the central position,
in between VLA 4A and VLA 4B, with signs of detectable variations in abundance
at scales of $\sim$60 au. Similar values of the methanol column density and
abundance have been reported in the literature in other protobinary sources.
However, lower values have been inferred from previous observations toward SVS
13 performed with poorer angular resolution. We also derive a deuteration
fraction D/H = 0.05, similar to values found in other low-mass protostars.
Finally, we infer an abundance of methyl formate relative to methanol of
$\sim$0.023 in SVS 13, similar to the value reported in the literature toward
the IRAS 16293$-$2422B protostar, which also belongs to a multiple system.

In conclusion, SVS 13 appears to be an excellent testbed for testing numerical
simulations of the earliest stages in the formation of binary and multiple
stellar systems.

We thank Jim Moran, who tackled some years ago the “divide problem”, for his
helpful comments and inspiration. We also thank the referee for a constructive
and helpful report. G.A., G.B.-C., I.dG.-M., A.K.D.-R., R.E., G.A.F., J.F.G.,
M.O. and J.M.T. acknowledge support from the Spanish MINECO/AEI through the
AYA2017-84390-C2 grant (co-funded by FEDER) and MCIN/AEI/10.13039/501100011033
through the PID2020-114461GB-I00 and PID2020-117710G-I00 grants. G.A.,
G.B.-C., A.K.D.-R., G.A.F., J.F.G., and M.O. acknowledge financial support
from the State Agency for Research of the Spanish MCIU through the “Center of
Excellence Severo Ochoa” award for the Instituto de Astrofísica de Andalucía
(SEV-2017-0709). G.B.-C. also acknowledges support from the Spanish MICINN/AEI
through the PRE2018-086111 (SEV-2017-0709-18-1) grant (co-funded by ESF).
G.A.F also acknowledges support from the Collaborative Research Centre 956,
funded by the Deutsche Forschungsgemeinschaft (DFG) project ID 184018867.
L.A.Z. acknowledges financial support from CONACyT-280775 and UNAM-PAPIIT
IN110618 grants, México. This paper makes use of the following ALMA data:
ADS/JAO.ALMA#2015.1.01229.S, ADS/JAO.ALMA#2016.1.01305.S. ALMA is a
partnership of ESO (representing its member states), NSF (USA) and NINS
(Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan) and KASI
(Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA
Observatory is operated by ESO, AUI/NRAO and NAOJ. Based on analysis carried
out with the CASSIS software (Vastel et al., 2015) and JPL, CDMS and VASTEL
databases. CASSIS has been developed by IRAP-UPS/CNRS. This research made use
of Astropy a community-developed core Python package for Astronomy (Astropy
Collaboration et al., 2013, 2018).

##  Appendix A VLA Images of Other Radio Sources in the Field

Figures 17, 18, 19, 20, and 21 show, respectively, the VLA images of sources
VLA 3, VLA 20, VLA 17, VLA 2, and VLA 22 (Table 3) in order of increasing
(projected) distance to SVS 13.

Figure 17: VLA images of source VLA 3 at X (3 cm), K (1.3 cm), Ka (9 mm), and
Q (7 mm) bands. Contour levels are $-$3, 3, 4, 6, 8, 10, 12, and 14 times the
rms of each image. The synthesized beams and the rms are shown in the bottom
left and right corners, respectively, of the images. We used Briggs weighting
with the robust parameter set to 0 for the image at X band, and natural
weighting for the remaining images. The synthesized beams are $0\farcs
22\times 0\farcs 16$, PA = $-80.3\arcdeg$ at X band; $0\farcs 11\times 0\farcs
10$, PA = $-80.7\arcdeg$ at K band; $0\farcs 10\times 0\farcs 07$, PA =
$-72.9\arcdeg$ at Ka band; and $0\farcs 17\times 0\farcs 12$, PA =
$60.9\arcdeg$ at Q band.

Figure 18: Same as Fig. 17 but for source VLA 20. Contour levels are $-$3, 3,
4, 5, 6, 7, 8, 9, and 11 times the rms of each image.

Figure 19: Same as Fig. 17 but for source VLA 17. Contour levels are $-$3, 3,
4, 5, 6, 8, 10, 12, 15, 18, 25, 35, and 50 times the rms of each image.

Figure 20: Same as Fig. 17 but for source VLA 2. Contour levels are $-$3, 3,
4, 5, 7, 9, 12, 15, 20, 25, 30, 40, 50, and 60 times the rms of each image.

Figure 21: VLA images of source VLA 22 in left (LCP; left column) and right
circular polarization (RCP; right column). The total intensity can be obtained
as $I=(LCP+RCP)/2$. Contour levels are $-$3, 3, 4, 5, 6, 7, 8, and 9 times the
average rms of the RCP and LCP images, 6.7 $\mu$Jy beam-1 (X band), 15.5
$\mu$Jy beam-1 (K band), and 47 $\mu$Jy beam-1 (Ka band); and $-$3, $-$2, 2,
and 3 times 150 $\mu$Jy beam-1 for the Q band image. Plus signs indicate the
positions of VLA 22N and VLA 22S, obtained from the K-band image (Table 3). We
used Briggs weighting with the robust parameter set to 0.5 and set to 0 for
the X- and Ka-band images, respectively, and natural weighting for the
remaining images. The synthesized beams are $0\farcs 20\times 0\farcs 14$, PA
= $-81.3\arcdeg$ (X band); $0\farcs 11\times 0\farcs 10$, PA = $-80.7\arcdeg$
(K band); $0\farcs 25\times 0\farcs 15$, PA = $-73.6\arcdeg$ (Ka band); and
$0\farcs 06\times 0\farcs 05$, PA = $-83.4\arcdeg$ (Q band).

##  Appendix B Spectra and Identification of Molecular Transitions

Figure 22 shows the observed spectra toward the positions of VLA 4A, VLA 4B
and the central position (CB). In Table 8 we list the identified molecular
transitions.

Figure 22: Spectra taken in beam-size ($0\farcs 16\times 0\farcs 08$, PA =
0$\arcdeg$) regions toward VLA 4A (blue), VLA 4B (red), and the geometric
center of the components (black). VLA 4A and VLA 4B spectra are offset by 25 K
and $-$25 K, respectively, for illustrative purposes. The bottom x-axis shows
the observed frequency, and the top x-axis shows the velocity referred to the
frequency of 13CO (3-2) (a), SO2 (11-12) (b), CS (7-6) (c), HC15N (4-3) (d),
H13CN (4-3) (e), and 12CO (3-2) (f). The most prominent lines have been
labeled; the ethylene glycol transitions associated only with VLA 4A (Fig. 13)
are labeled following the Appendix D nomenclature.

Table 8: Molecular Transitions Detected toward SVS 13aaFrom spectra obtained in a beam-size ($0\farcs 16\times 0\farcs 08$, PA = 0$\arcdeg$) region toward VLA 4A, VLA 4B, and toward the geometric center of the components (CB) that we adopt as representative of the circumbinary disk. |  |  | Rest FrequencybbFor identified transitions, this is the laboratory frequency reported in the catalogs used (JPL, CDMS, VASTEL). For unidentified transitions, we list the frequency derived from a Gaussian fit to the line and the formal error of the fit. In general, we used for the fit the spectrum taken toward the geometric center of the two components, VLA 4A and VLA 4B, assuming $V_{\rm LSR}$ = 8.77 km s-1. In the cases where the unidentified line is detected only toward VLA 4A, $V_{\rm LSR}$ = 7.36 km s-1 is used. When the unidentified line is blended with known lines, a multiple Gaussian fit is performed assuming that the value of the FWHM is the same for all the lines. We only report intense ($>10\sigma$) unidentified lines. | E${}_{u}/k$ | Aul |
---|---|---|---|---|---|---
Species | Formula | Quantum Numbers | (MHz) | (K) | (s-1) | Notescc4A: Detected only toward VLA 4A; CB: Detected only in the circumbinary material, toward the geometric center of the two components (CB); bl: Blended with another line; db: Doubtful identification; $<$5$\sigma$: Low ($<5\sigma$) signal-to-noise ratio; nLTE: Not well reproduced by the LTE model (differences $\gtrsim 50\%$); wide: Wide line with probable blending of several lines.
Acetaldehyde | CH3CHO, vt=0 | 5 3 2 2 - 4 2 2 2 | 330566.705 | 34.26 | 1.15E-4 | nLTE
|  | 17 1 16 3 - 16 1 15 3 | 330580.420 | 352.489 | 1.25E-3 |
|  | 17 1 16 2 - 16 1 15 2 | 331602.178 | 146.645 | 1.25E-3 |
|  | 17 1 16 0 - 16 1 15 0 | 331680.380 | 146.6 | 1.25E-3 |
| CH3CDO, vt=0 | 17 2 15 0 - 16 2 14 0 | 331490.785 | 149.161 | 1.28E-3 | CB
Acetone | CH3COCH3, v=0 | 17 15 3 1 - 16 14 2 1 | 330506.310 | 138.132 | 1.28E-3 |
|  | 17 15 2 1 - 16 14 3 1 | 330512.291 | 138.132 | 1.28E-3 | bl
|  | 17 15 3 1 - 16 14 3 2 | 330549.327 | 138.023 | 1.28E-3 |
|  | 15 8 7 0 - 14 7 8 1 | 331525.975 | 95.682 | 1.30E-4 | bl
|  | 17 17 1 0 - 16 16 0 0 | 342780.035 | 147.049 | 1.85E-3 |
|  | 17 17 0 0 - 16 16 1 0 | 342780.036 | 147.049 | 1.85E-3 |
|  | 41 3 39 0 - 41 2 40 1 | 342836.464 | 447.674 | 1.53E-4 | bl, db
|  | 41 2 39 0 - 41 1 40 1 | 342836.464 | 447.674 | 1.53E-4 | bl, db
|  | 18 12 6 1 - 17 11 7 1 | 342836.912 | 141.302 | 4.35E-4 | bl, db
|  | 18 12 6 0 - 17 11 7 0 | 342864.593 | 141.23 | 4.41E-4 |  $<$5$\sigma$
|  | 18 12 6 0 - 17 11 7 1 | 342896.394 | 141.271 | 4.35E-4 | bl
|  | 18 15 4 0 - 17 14 3 0 | 345256.037 | 150.827 | 1.29E-3 | bl
|  | 18 15 3 0 - 17 14 4 0 | 345305.365 | 150.828 | 1.29E-3 |
|  | 24 12 12 1 - 23 13 11 1 | 345875.586 | 235.239 | 7.51E-4 | bl, $<$5$\sigma$
|  | 24 12 12 1 - 23 13 11 2 | 345877.896 | 235.239 | 7.52E-4 | bl, $<$5$\sigma$
Butadiynyl |  13CCCCH | 36 36.5 37 36.5 - 35 35.5 36 35.5 | 330637.439 | 293.629 | 1.57E-4 | bl
|  | 36 36.5 37 37.5 - 35 35.5 36 36.5 | 330637.439 | 293.628 | 1.57E-4 | bl
|  | 36 36.5 36 36.5 - 35 35.5 35 35.5 | 330638.331 | 293.618 | 1.57E-4 | bl
|  | 36 36.5 36 35.5 - 35 35.5 35 34.5 | 330638.355 | 293.618 | 1.57E-4 | bl
|  | 36 35.5 36 35.5 - 35 34.5 35 34.5 | 330672.730 | 293.685 | 1.57E-4 | bl
|  | 36 35.5 36 36.5 - 35 34.5 35 35.5 | 330672.754 | 293.685 | 1.57E-4 | bl
|  | 36 35.5 35 34.5 - 35 34.5 34 33.5 | 330673.647 | 293.692 | 1.57E-4 | bl
|  | 36 35.5 35 35.5 - 35 34.5 34 34.5 | 330673.647 | 293.693 | 1.57E-4 | bl
Carbon Monoxide | CO, v=0 | 3 - 2 | 345795.990 | 33.192 | 2.50E-6 |
|  13CO, v=0 | 3 - 2 | 330587.965 | 31.732 | 2.19E-6 |
Carbon Monosulfide | CS, v=0 | 7 0 - 6 0 | 342882.850 | 65.828 | 8.40E-4 |
Ethanol | C2H5OH, v=0 | 19 4 16 0 - 18 4 15 0 | 330478.629 | 234.672 | 3.23E-4 |
|  | 14 5 9 1 - 14 4 11 0 | 330599.812 | 179.62 | 1.04E-4 | bl
|  | 20 0 20 2 - 19 1 19 2 | 330621.123 | 169.18 | 3.82E-4 |
|  | 9 6 3 0 - 8 5 3 1 | 331469.617 | 138.161 | 1.61E-4 |
|  | 9 6 4 0 - 8 5 4 1 | 331469.664 | 138.161 | 1.61E-4 | bl
|  | 11 5 7 1 - 11 4 7 0 | 331648.876 | 147.183 | 1.02E-4 | bl
|  | 11 3 8 2 - 10 2 9 2 | 331655.211 | 66.945 | 1.26E-4 |
|  | 21 1 21 0 - 20 1 20 0 | 345229.285 | 241.552 | 3.71E-4 | bl
|  | 21 1 21 1 - 20 1 20 1 | 345295.355 | 246.22 | 3.67E-4 |
|  | 22 3 19 1 - 21 4 17 0 | 345311.658 | 286.007 | 7.58E-5 |
|  | 21 0 21 0 - 20 0 20 0 | 345333.442 | 241.539 | 3.72E-4 |
|  | 21 0 21 1 - 20 0 20 1 | 345408.165 | 246.207 | 3.67E-4 | bl
Ethyl Cyanide | C2H5CN, v=0 | 37 12 25 - 36 12 24 | 331484.515 | 462.06 | 2.77E-3 | bl
|  | 37 12 26 - 36 12 25 | 331484.515 | 462.06 | 2.77E-3 | bl
|  | 37 11 26 - 36 11 25 | 331487.287 | 436.589 | 2.83E-3 |
|  | 37 11 27 - 36 11 26 | 331487.287 | 436.589 | 2.83E-3 |
|  | 37 10 28 - 36 10 27 | 331523.317 | 413.325 | 2.88E-3 | bl
|  | 37 10 27 - 36 10 26 | 331523.317 | 413.325 | 2.88E-3 | bl
|  | 37 14 23 - 36 14 22 | 331549.416 | 519.581 | 2.66E-3 | CB, $<$5$\sigma$
|  | 37 14 24 - 36 14 23 | 331549.416 | 519.581 | 2.66E-3 | CB, $<$5$\sigma$
|  | 37 9 29 - 36 9 28 | 331605.594 | 392.28 | 2.92E-3 | bl
|  | 37 9 28 - 36 9 27 | 331605.594 | 392.28 | 2.92E-3 | bl
|  | 37 15 22 - 36 15 21 | 331608.406 | 551.61 | 2.59E-3 | bl
|  | 37 15 23 - 36 15 22 | 331608.406 | 551.61 | 2.59E-3 | bl
|  | 37 2 35 - 36 2 34 | 331662.287 | 310.75 | 3.08E-3 |
|  | 10 6 5 - 9 5 4 | 344278.941 | 63.667 | 2.03E-4 |
|  | 10 6 4 - 9 5 5 | 344278.941 | 63.667 | 2.03E-4 |
Ethylene Glycol | aGg’-(CH2OH)2 | 22 6 17 0 - 21 5 17 1 | 330510.922 | 142.638 | 3.93E-5 | bl
|  | 18 8 10 0 - 17 7 10 1 | 330539.249 | 115.544 | 2.45E-5 | bl
|  | 18 8 11 0 - 17 7 11 1 | 330544.758 | 115.544 | 2.44E-5 | bl
|  | 33 18 15 0 - 32 18 14 1 | 331479.210 | 434.667 | 6.27E-4 | 4A, bl
|  | 33 18 16 0 - 32 18 15 1 | 331479.210 | 434.667 | 6.27E-4 | 4A, bl
|  | 35 3 33 0 - 34 3 32 1 | 331553.542 | 304.273 | 8.86E-4 | 4A, bl
|  | 35 2 33 0 - 34 2 32 1 | 331554.797 | 304.273 | 8.86E-4 | 4A, bl
|  | 33 17 16 0 - 32 17 15 1 | 331608.371 | 417.526 | 6.56E-4 | bl
|  | 33 17 17 0 - 32 17 16 1 | 331608.371 | 417.526 | 6.56E-4 | bl
|  | 34 4 31 1 - 33 4 30 0 | 342800.519 | 297.127 | 9.61E-4 | 4A, nLTE
|  | 34 3 31 1 - 33 3 30 0 | 342850.907 | 297.124 | 9.81E-4 | 4A, nLTE
|  | 22 5 17 0 - 21 4 18 0 | 342899.671 | 138.24 | 4.26E-5 | bl
|  | 34 13 22 0 - 33 13 21 1 | 342920.095 | 375.646 | 8.43E-4 | 4A, bl
|  | 34 13 21 0 - 33 13 20 1 | 342920.100 | 375.646 | 8.43E-4 | 4A, bl
|  | 20 4 16 1 - 19 2 17 0 | 342941.447 | 113.509 | 8.79E-6 | db, $<$5$\sigma$
|  | 34 5 30 1 - 33 4 29 1 | 344137.936 | 305.126 | 2.42E-4 | bl
|  | 33 18 15 1 - 32 18 14 0 | 344226.046 | 434.972 | 7.02E-4 | bl
|  | 33 18 16 1 - 32 18 15 0 | 344226.046 | 434.972 | 7.02E-4 | bl
|  | 20 17 3 1 - 20 16 5 0 | 344228.068 | 245.387 | 8.84E-6 | bl
|  | 20 17 4 1 - 20 16 4 0 | 344228.068 | 245.387 | 8.84E-6 | bl
|  | 33 19 14 1 - 32 19 13 0 | 344231.354 | 453.099 | 6.68E-4 | bl
|  | 33 19 15 1 - 32 19 14 0 | 344231.354 | 453.099 | 6.68E-4 | bl
|  | 33 17 16 1 - 32 17 15 0 | 344253.269 | 417.828 | 7.35E-4 | 4A
|  | 33 17 17 1 - 32 17 16 0 | 344253.269 | 417.828 | 7.35E-4 | 4A
|  | 33 20 13 1 - 32 20 12 0 | 344264.330 | 472.207 | 6.33E-4 | 4A
|  | 33 20 14 1 - 32 20 13 0 | 344264.330 | 472.207 | 6.33E-4 | 4A
|  | 33 16 17 1 - 32 16 16 0 | 344319.573 | 401.672 | 7.65E-4 | bl
|  | 33 16 18 1 - 32 16 17 0 | 344319.573 | 401.672 | 7.65E-4 | bl
|  | 33 21 12 1 - 32 21 11 0 | 344321.301 | 492.291 | 5.95E-4 | bl
|  | 33 21 13 1 - 32 21 12 0 | 344321.301 | 492.291 | 5.95E-4 | bl
|  | 33 12 22 1 - 32 12 21 0 | 345224.705 | 347.051 | 8.75E-4 | 4A, bl
|  | 33 12 21 1 - 32 12 20 0 | 345224.778 | 347.051 | 8.75E-4 | 4A, bl
|  | 21 7 15 0 - 20 6 14 0 | 345266.087 | 137.904 | 8.68E-5 | 4A, bl
|  | 35 4 31 0 - 34 4 30 1 | 345278.673 | 321.646 | 8.99E-4 | bl
|  | 33 11 23 1 - 32 11 22 0 | 345737.011 | 335.95 | 9.00E-4 | 4A, bl
|  | 33 11 22 1 - 32 11 21 0 | 345738.443 | 335.95 | 9.01E-4 | 4A, bl
|  | 34 9 26 0 - 33 9 25 1 | 345812.719 | 333.572 | 9.36E-4 | bl
|  | 34 8 27 0 - 33 8 26 1 | 345832.886 | 325.787 | 5.43E-4 |
Ethylene Oxide | c-C2H4O | 9 5 5 - 8 4 4 | 345688.322 | 90.408 | 3.37E-4 | bl
Formaldehyde | H2C18O | 5 0 5 - 4 0 4 | 345881.039 | 49.871 | 1.19E-3 | bl
| p-D2CO | 6 1 6 - 5 1 5 | 330674.347 | 53.025 | 1.02E-3 | bl
Formamide | HC(O)NH2, v=0 | 16 3 14 - 16 2 15 | 331685.873 | 165.588 | 7.87E-5 | 4A, bl, db
|  | 16 1 15 - 15 1 14 | 345325.391 | 145.155 | 3.02E-3 |
Hydrogen Cyanide | HC15N, v=0 | 4 - 3 | 344200.109 | 41.298 | 1.88E-3 | bl
| H13CN, v=0 | 4 - 3 | 345339.769 | 41.435 | 1.90E-3 | bl
Methanol | CH3OH, vt=0 | 11 1 10 0 - 11 0 11 0 | 331502.319 | 169.008 | 3.93E-4 | nLTE
|  | 18 2 17 1 - 17 3 15 1 | 344109.039 | 419.401 | 6.80E-5 | bl
|  | 16 1 15 0 - 15 2 14 0 | 345903.916 | 332.651 | 1.04E-4 |
| CH3OH, vt=1 | 10 2 9 5 - 11 3 9 5 | 344312.267 | 491.913 | 1.77E-4 | bl
| CH318OH, vt=0 | 13 0 13 0 - 12 1 12 0 | 331482.670 | 202.281 | 1.99E-4 | bl
|  13CH3OH, vt=0 | 7 2 5 0 - 6 2 4 0 | 330535.222 | 85.799 | 1.44E-4 | bl
|  | 7 -2 6 0 - 6 -2 5 0 | 330535.890 | 89.448 | 1.46E-4 | bl
| CH2DOH | 5 3 2 1 - 6 2 5 0 | 330606.339 | 77.042 | 7.70E-6 |
|  | 9 0 9 1 - 8 2 6 0 | 330686.419 | 109.536 | 4.06E-7 | db
|  | 15 7 9 1 - 16 6 10 1 | 331647.951 | 458.992 | 9.48E-6 | bl, $<$5$\sigma$
|  | 15 7 8 1 - 16 6 11 1 | 331647.959 | 458.992 | 9.48E-6 | bl, $<$5$\sigma$
|  | 17 2 16 0 - 17 1 17 0 | 342935.645 | 342.828 | 1.67E-4 |
|  | 20 2 18 1 - 19 3 16 2 | 344138.882 | 478.684 | 7.97E-5 | bl
|  | 8 2 7 2 - 7 3 5 0 | 345284.157 | 112.579 | 1.88E-6 | bl
|  | 16 2 14 0 - 15 3 13 0 | 345398.904 | 310.213 | 5.83E-5 |
|  | 23 4 19 1 - 23 3 21 2 | 345701.510 | 662.813 | 1.34E-4 |
|  | 3 2 1 1 - 2 1 2 1 | 345718.718 | 39.435 | 4.23E-5 | bl, nLTE
|  | 19 1 19 1 - 18 2 17 2 | 345820.793 | 418.03 | 2.94E-5 | bl
|  | 22 4 19 1 - 22 3 19 2 | 345850.485 | 613.622 | 1.29E-4 |
Methyl Cyanide | CH313CN, v=0 | 18 6 - 17 6 | 330679.886 | 407.876 | 2.80E-3 |
| CH2DCN | 20 1 20 - 19 1 19 | 345685.380 | 179.63 | 3.60E-3 | bl
Methyl Formate | CH3OCHO v=0 | 27 13 15 4 - 26 13 14 4 | 330602.640 | 522.121 | 4.24E-4 | bl
|  | 27 11 16 5 - 26 11 15 5 | 330634.841 | 490.367 | 4.59E-4 |
|  | 28 3 25 2 - 27 4 24 1 | 330641.527 | 247.741 | 5.56E-5 |
|  | 28 3 25 0 - 27 4 24 0 | 330653.474 | 247.738 | 5.56E-5 |
|  | 27 12 16 3 - 26 12 15 3 | 330701.651 | 505.429 | 4.43E-4 |
|  | 27 12 15 3 - 26 12 14 3 | 330701.651 | 505.429 | 4.43E-4 |
|  | 28 3 25 0 - 27 3 24 0 | 331469.465 | 247.739 | 5.36E-4 | bl
|  | 27 10 17 5 - 26 10 16 5 | 331483.804 | 476.502 | 4.78E-4 | bl
|  | 27 22 5 2 - 26 22 4 2 | 331539.684 | 543.597 | 1.88E-4 | db, nLTE
|  | 27 21 6 0 - 26 21 5 0 | 331564.665 | 515.157 | 2.21E-4 |
|  | 27 21 7 0 - 26 21 6 0 | 331564.665 | 515.157 | 2.21E-4 |
|  | 27 21 6 2 - 26 21 5 2 | 331586.163 | 515.149 | 2.21E-4 |  $<$5$\sigma$
|  | 27 21 7 1 - 26 21 6 1 | 331589.894 | 515.146 | 2.21E-4 |  $<$5$\sigma$
|  | 27 20 7 0 - 26 20 6 0 | 331625.030 | 488.026 | 2.52E-4 |
|  | 27 20 8 0 - 26 20 7 0 | 331625.030 | 488.026 | 2.52E-4 |
|  | 27 20 8 1 - 26 20 7 1 | 331647.933 | 488.017 | 2.52E-4 | bl
|  | 27 20 7 2 - 26 20 6 2 | 331648.683 | 488.018 | 2.52E-4 | bl
|  | 28 10 18 5 - 27 10 17 5 | 344108.640 | 493.017 | 5.42E-4 | bl, nLTE
|  | 28 18 10 0 - 27 18 9 0 | 344170.930 | 454.246 | 3.66E-4 |
|  | 28 18 11 0 - 27 18 10 0 | 344170.930 | 454.246 | 3.66E-4 |
|  | 28 18 10 2 - 27 18 9 2 | 344187.246 | 454.239 | 3.66E-4 |
|  | 28 18 11 1 - 27 18 10 1 | 344197.339 | 454.238 | 3.66E-4 | bl
|  | 23 13 10 2 - 23 12 11 2 | 344206.436 | 274.953 | 4.27E-5 | bl
|  | 23 13 11 1 - 23 12 12 1 | 344210.812 | 274.944 | 4.27E-5 | bl
|  | 23 13 10 0 - 23 12 11 0 | 344237.391 | 274.954 | 4.27E-5 |
|  | 23 13 11 0 - 23 12 12 0 | 344237.414 | 274.954 | 4.27E-5 |
|  | 28 11 18 4 - 27 11 17 4 | 344300.259 | 506.713 | 5.27E-4 |
|  | 28 17 11 0 - 27 17 10 0 | 344322.992 | 431.091 | 3.94E-4 | bl
|  | 28 17 12 0 - 27 17 11 0 | 344322.992 | 431.091 | 3.94E-4 | bl
|  | 18 13 5 2 - 18 12 6 2 | 345230.296 | 212.973 | 3.22E-5 | bl
|  | 18 13 6 1 - 18 12 7 1 | 345241.935 | 212.959 | 3.22E-5 |
|  | 28 10 19 4 - 27 10 18 4 | 345248.210 | 492.904 | 5.48E-4 |
|  | 18 13 5 0 - 18 12 6 0 | 345281.320 | 212.973 | 3.22E-5 | bl
|  | 18 13 6 0 - 18 12 7 0 | 345281.320 | 212.973 | 3.22E-5 | bl
|  | 17 13 4 2 - 17 12 5 2 | 345351.333 | 202.36 | 2.89E-5 |
|  | 17 13 5 1 - 17 12 6 1 | 345364.235 | 202.345 | 2.89E-5 |
|  | 17 13 4 0 - 17 12 5 0 | 345405.869 | 202.36 | 2.89E-5 | bl
|  | 17 13 5 0 - 17 12 6 0 | 345405.869 | 202.36 | 2.89E-5 | bl
|  | 16 13 3 2 - 16 12 4 2 | 345451.103 | 192.34 | 2.51E-5 | bl, $<$5$\sigma$
|  | 13 13 0 0 - 13 12 1 0 | 345714.339 | 165.831 | 8.37E-6 | bl, $<$5$\sigma$
|  | 13 13 1 0 - 13 12 2 0 | 345714.339 | 165.831 | 8.37E-6 | bl, $<$5$\sigma$
|  | 9 9 0 0 - 8 8 1 0 | 345718.662 | 80.32 | 9.86E-5 | bl
|  | 9 9 1 0 - 8 8 0 0 | 345718.662 | 80.32 | 9.86E-5 | bl
|  | 28 6 23 4 - 27 6 22 4 | 345828.557 | 451.764 | 5.99E-4 | bl
|  | 9 9 0 3 - 8 8 1 3 | 345908.676 | 268.174 | 1.00E-4 | bl
|  | 9 9 1 3 - 8 8 0 3 | 345908.676 | 268.174 | 1.00E-4 | bl
Nitrogen Sulfide | NS | 8 1 1 8.5-7 1 -1 7.5 | 345823.288 | 70.795 | 7.36E-4 | bl
|  | 8 1 1 7.5-7 1 -1 6.5 | 345823.288 | 70.797 | 7.23E-4 | bl
|  | 8 1 1 6.5-7 1 -1 5.5 | 345824.130 | 70.799 | 7.21E-4 | bl
Sulfur Dioxide | SO2, v=0 | 11 6 6 - 12 5 7 | 331580.244 | 148.954 | 4.35E-5 |
|  | 13 2 12 - 12 1 11 | 345338.538 | 92.984 | 2.38E-4 | bl
|  34SO2 | 21 2 20 - 21 1 21 | 330667.565 | 218.583 | 1.45E-4 |  $<$5$\sigma$
|  | 10 4 6 - 10 3 7 | 344245.346 | 88.384 | 2.96E-4 |
|  | 9 4 6 - 9 3 7 | 345285.620 | 79.203 | 2.88E-4 | bl
Sulfur Monoxide | SO, v=0 | 8 8 - 7 7 | 344310.612 | 87.482 | 5.19E-4 | bl
Thioformaldehyde | H2CS | 10 0 10 - 9 0 9 | 342946.424 | 90.592 | 6.08E-4 |
| H213CS | 10 2 8 - 9 2 7 | 330544.304 | 140.012 | 5.23E-4 | bl
Unknown-1 |  $\cdots$ |  $\cdots$ | 330505.15$\pm$0.11 |  $\cdots$ |  $\cdots$ | bl
Unknown-2 |  $\cdots$ |  $\cdots$ | 330557.55$\pm$0.04 |  $\cdots$ |  $\cdots$ |
Unknown-3 |  $\cdots$ |  $\cdots$ | 330568.61$\pm$0.05 |  $\cdots$ |  $\cdots$ | bl
Unknown-4 |  $\cdots$ |  $\cdots$ | 330622.86$\pm$0.03 |  $\cdots$ |  $\cdots$ | bl
Unknown-5 |  $\cdots$ |  $\cdots$ | 330648.68$\pm$0.06 |  $\cdots$ |  $\cdots$ |
Unknown-6 |  $\cdots$ |  $\cdots$ | 330664.961$\pm$0.022 |  $\cdots$ |  $\cdots$ | bl
Unknown-7 |  $\cdots$ |  $\cdots$ | 331562.17$\pm$0.06 |  $\cdots$ |  $\cdots$ | bl
Unknown-8 |  $\cdots$ |  $\cdots$ | 331694.10$\pm$0.06 |  $\cdots$ |  $\cdots$ | CB
Unknown-9 |  $\cdots$ |  $\cdots$ | 342861.82$\pm$0.04 |  $\cdots$ |  $\cdots$ | bl
Unknown-10 |  $\cdots$ |  $\cdots$ | 342905.71$\pm$0.06 |  $\cdots$ |  $\cdots$ |
Unknown-11 |  $\cdots$ |  $\cdots$ | 342995.72$\pm$0.06 |  $\cdots$ |  $\cdots$ | bl
Unknown-12 |  $\cdots$ |  $\cdots$ | 342998.21$\pm$0.05 |  $\cdots$ |  $\cdots$ |
Unknown-13 |  $\cdots$ |  $\cdots$ | 344123.71$\pm$0.04 |  $\cdots$ |  $\cdots$ |
Unknown-14 |  $\cdots$ |  $\cdots$ | 344280.85$\pm$0.07 |  $\cdots$ |  $\cdots$ | bl
Unknown-15 |  $\cdots$ |  $\cdots$ | 345302.06$\pm$0.13 |  $\cdots$ |  $\cdots$ |
Unknown-16 |  $\cdots$ |  $\cdots$ | 345319.621$\pm$0.020 |  $\cdots$ |  $\cdots$ |
Unknown-17 |  $\cdots$ |  $\cdots$ | 345411.10$\pm$0.06 |  $\cdots$ |  $\cdots$ | bl
Unknown-18 |  $\cdots$ |  $\cdots$ | 345445.60$\pm$0.02 |  $\cdots$ |  $\cdots$ |
Unknown-19 |  $\cdots$ |  $\cdots$ | 345700.70$\pm$0.09 |  $\cdots$ |  $\cdots$ | bl
Unknown-20 |  $\cdots$ |  $\cdots$ | 345911.585$\pm$0.025 |  $\cdots$ |  $\cdots$ |
Unknown-21 |  $\cdots$ |  $\cdots$ | 342832.08$\pm$0.13 |  $\cdots$ |  $\cdots$ | 4A, bl
Unknown-22 |  $\cdots$ |  $\cdots$ | 342905.38$\pm$0.07 |  $\cdots$ |  $\cdots$ | 4A, bl
Unknown-23 |  $\cdots$ |  $\cdots$ | 342909.54$\pm$0.06 |  $\cdots$ |  $\cdots$ | 4A, bl
Unknown-24 |  $\cdots$ |  $\cdots$ | 342913.42$\pm$0.10 |  $\cdots$ |  $\cdots$ | 4A, bl
Unknown-25 |  $\cdots$ |  $\cdots$ | 342962.01$\pm$0.08 |  $\cdots$ |  $\cdots$ | 4A, bl
Unknown-26 |  $\cdots$ |  $\cdots$ | 344213.68$\pm$0.22 |  $\cdots$ |  $\cdots$ | 4A, bl
Unknown-27 |  $\cdots$ |  $\cdots$ | 345354.29$\pm$0.14 |  $\cdots$ |  $\cdots$ | 4A, bl
Unknown-28 |  $\cdots$ |  $\cdots$ | 345430.67$\pm$0.17 |  $\cdots$ |  $\cdots$ | 4A, bl, wide

##  Appendix C Error estimate of the inclination of a disk

The inclination of the disks in SVS 13 is estimated from the ratio of the
minor axis, $a_{\mathrm{min}}$, and major axis, $a_{\mathrm{maj}}$, of the
projection of the disk on the sky,

$i=\arccos\frac{a_{\mathrm{min}}}{a_{\mathrm{maj}}}.$ (C1)

The inclination of the orbit is estimated from the ratio of the components of
the orbital velocity along the line-of-sight (the difference in the LSR
velocity of the VLA 4B and VLA 4A line emission), $v_{\mathrm{los}}$, and on
the plane of the sky (the difference in the proper motion velocities of VLA 4B
and VLA 4A), $v_{\mathrm{sky}}$,

$i=\arctan\frac{v_{\mathrm{los}}}{v_{\mathrm{sky}}}.$ (C2)

In both cases, the estimation of the inclination involves the ratio of two
magnitudes known with a given uncertainty. Let us call the two magnitudes $x$
and $y$, which we assume are independent, normally-distributed (Gaussian)
random variables with expected values $\mu_{x}$ and $\mu_{y}$, and standard
deviations $\sigma_{x}$ and $\sigma_{y}$. In the following we will calculate
an approximation of the expected value and standard deviation of the ratio
$z=x/y$, and after this, the expected value and standard deviation of the
inclination, $\mu_{i}$ and $\sigma_{i}$.

###  C.1 Error of the ratio

The study of the error of the ratio of two normal random variables is not
easy. For instance, the case of variables with zero mean is presented in
Papoulis & Pillai (2002). In the general case of non-zero mean, the
probability density function of the ratio is obtained by Díaz-Francés & Rubio
(2012), but the expression is cumbersome and the integrals have to be
evaluated numerically. Another approach, based on the expansion up to second
or third order in $\sigma_{x}/\mu_{x}$ and $\sigma_{y}/\mu_{y}$, is more
practical.

####  C.1.1 Expected value of the ratio

The probability density function of the Gaussian random variable $x$ is given
by

$f(x)=\frac{1}{\sqrt{2\pi}\,\sigma_{x}}\exp\left[-\frac{1}{2}\left(\frac{x-\mu_{x}}{\sigma_{x}}\right)^{2}\right],$
(C3)

and a similar expression for $y$.

The expected value of the ratio $z=x/y$ is defined as

$E[z]=\int_{-\infty}^{+\infty}\\!\\!\\!\int_{-\infty}^{+\infty}\frac{x}{y}\frac{1}{2\pi\sigma_{x}\sigma_{y}}\exp\left[-\frac{1}{2}\left(\frac{x-\mu_{x}}{\sigma_{x}}\right)^{2}\right]\exp\left[-\frac{1}{2}\left(\frac{y-\mu_{y}}{\sigma_{y}}\right)^{2}\right]dx\,dy.$
(C4)

The integral over $x$ gives the expected value of $x$, so

$E[z]=\mu_{x}\int_{-\infty}^{+\infty}\frac{1}{y}\frac{1}{\sqrt{2\pi}\,\sigma_{y}}\exp\left[-\frac{1}{2}\left(\frac{y-\mu_{y}}{\sigma_{y}}\right)^{2}\right]\,dy.$
(C5)

Let us use the standard normally distributed variable

$\epsilon=\frac{y-\mu_{y}}{\sigma_{y}},$ (C6)

defined with zero expected value $\mu_{\epsilon}=0$ and unit standard
deviation $\sigma_{\epsilon}=1$. The change of variable gives

$E[z]=\frac{\mu_{x}}{\mu_{y}}\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}\frac{1}{1+(\sigma_{y}/\mu_{y})\epsilon}\exp\left(-\frac{\epsilon^{2}}{2}\right)d\epsilon.$
(C7)

Let us expand $1/[1+(\sigma_{y}/\mu_{y})\epsilon]$ up to order 2,

$\frac{1}{1+(\sigma_{y}/\mu_{y})\epsilon}\simeq
1-\frac{\sigma_{y}}{\mu_{y}}\,\epsilon+\frac{\sigma_{y}^{2}}{\mu_{y}^{2}}\,\epsilon^{2}+\ldots$
(C8)

The odd terms of the expansion have zero integral (and thus the expansion is
correct up to third order). The integral of the constant term is 1, and that
of the quadratic term is $\sigma_{y}^{2}/\mu_{y}^{2}$ (because
$\sigma_{\epsilon}=1$). The final result is

$\mu_{z}\equiv
E[z]\simeq\frac{\mu_{x}}{\mu_{y}}\left(1+\frac{\sigma_{y}^{2}}{\mu_{y}^{2}}\right).$
(C9)

It is noticeable that there is a bias in the expected value of the ratio, so
that $E[x/y]\neq E[x]/E[y]$.

####  C.1.2 Standard deviation of the ratio

The same procedure can be applied to estimate the variance of the ratio
$z=x/y$. The expected value of $z^{2}$ is

$E[z^{2}]=\int_{-\infty}^{+\infty}\\!\\!\\!\int_{-\infty}^{+\infty}\frac{x^{2}}{y^{2}}\frac{1}{2\pi\sigma_{x}\sigma_{y}}\exp\left[-\frac{1}{2}\left(\frac{x-\mu_{x}}{\sigma_{x}}\right)^{2}\right]\exp\left[-\frac{1}{2}\left(\frac{y-\mu_{y}}{\sigma_{y}}\right)^{2}\right]dx\,dy.$
(C10)

The integral over $x$ gives the expected value of $x^{2}$,
$E[x^{2}]=\mu_{x}^{2}+\sigma_{x}^{2}$, so

$E[z^{2}]=(\mu_{x}^{2}+\sigma_{x}^{2})\int_{-\infty}^{+\infty}\frac{1}{y^{2}}\frac{1}{\sqrt{2\pi}\,\sigma_{y}}\exp\left[-\frac{1}{2}\left(\frac{y-\mu_{y}}{\sigma_{y}}\right)^{2}\right]\,dy.$
(C11)

The same change of variables as before gives

$E[z^{2}]=\frac{\mu_{x}^{2}+\sigma_{x}^{2}}{\mu_{y}^{2}}\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}\frac{1}{[1+(\sigma_{y}/\mu_{y})\epsilon]^{2}}\exp\left(-\frac{\epsilon^{2}}{2}\right)\,d\epsilon.$
(C12)

The expansion of $1/[1+(\sigma_{y}/\mu_{y})\epsilon]^{2}$ is

$\frac{1}{[1+(\sigma_{y}/\mu_{y})\epsilon]^{2}}\simeq
1-2\frac{\sigma_{y}}{\mu_{y}}\,\epsilon+3\frac{\sigma_{y}^{2}}{\mu_{y}^{2}}\,\epsilon^{2}+\ldots$
(C13)

The integral of the constant term is 1, that of the quadratic term is
$3\sigma_{y}^{2}/\mu_{y}^{2}$, and the result is

$E[z^{2}]\simeq\frac{\mu_{x}^{2}+\sigma_{x}^{2}}{\mu_{y}^{2}}\left(1+3\frac{\sigma_{y}^{2}}{\mu_{y}^{2}}\right).$
(C14)

The variance is $\sigma_{z}^{2}=E[z^{2}]-E^{2}[z]$. The square of $E[z]$,
neglecting terms of fourth order in $\delta_{x}$ and $\delta_{y}$, is

$E^{2}[z]\simeq\frac{\mu_{x}^{2}}{\mu_{y}^{2}}\left(1+2\frac{\sigma_{y}^{2}}{\mu_{y}^{2}}\right),$
(C15)

resulting in a difference, neglecting again terms of fourth order,

$\sigma_{z}^{2}\simeq\frac{\mu_{x}^{2}}{\mu_{y}^{2}}\left(\frac{\sigma_{x}^{2}}{\mu_{x}^{2}}+\frac{\sigma_{y}^{2}}{\mu_{y}^{2}}\right),$
(C16)

and the standard deviation is

$\sigma_{z}\simeq\frac{\mu_{x}}{\mu_{y}}\sqrt{\frac{\sigma_{x}^{2}}{\mu_{x}^{2}}+\frac{\sigma_{y}^{2}}{\mu_{y}^{2}}}.$
(C17)

In the limiting case $\sigma_{y}=0$ the expression reduces to the usual value
$\sigma_{z}=\sigma_{x}/\mu_{y}$.

###  C.2 Error of $i=\arccos z$

Let us now calculate an approximation to the expected value and standard
deviation of the inclination

$i=\arccos z,$ (C18)

given the expected value, $\mu_{z}$, and standard deviation, $\sigma_{z}$, of
$z$.

####  C.2.1 Expected value of $i=\arccos z$

The expected value $E[i]$ is

$\mu_{i}\equiv E[i]=\int_{-\infty}^{+\infty}\\!\\!\\!\arccos
z\frac{1}{\sqrt{2\pi}\,\sigma_{z}}\exp\left[-\frac{1}{2}\left(\frac{z-\mu_{z}}{\sigma_{z}}\right)^{2}\right]\,dz.$
(C19)

The expansion of $\arccos$ around the expected value $\mu_{z}$, using as
before the standard normally distributed variable
$\epsilon=(z-\mu_{z})/\sigma_{z}$, is

$\arccos
z\simeq\arccos\mu_{z}-\frac{\sigma_{z}}{\sqrt{1-\mu_{z}^{2}}}\epsilon-\frac{1}{2!}\,\frac{\mu_{z}\sigma_{z}^{2}}{(1-\mu_{z}^{2})^{3/2}}\epsilon^{2}+\ldots$
(C20)

The integral is not null for the even terms of the expansion only, and thus
the approximation is correct up to third order. The result of the integration
is

$E[i]\simeq\arccos\mu_{z}-\frac{1}{2}\,\frac{\mu_{z}\sigma_{z}^{2}}{(1-\mu_{z}^{2})^{3/2}}.$
(C21)

It is noticeable that there is a bias in the expected value of the
inclination, because $E[\arccos z]\neq\arccos(E[z])$.

####  C.2.2 Standard deviation of $i=\arccos z$

In a similar way, let us calculate $E[i^{2}]$,

$E[i^{2}]=\int_{-\infty}^{+\infty}\\!\\!\\!\arccos^{2}z\frac{1}{\sqrt{2\pi}\,\sigma_{z}}\exp\left[-\frac{1}{2}\left(\frac{z-\mu_{z}}{\sigma_{z}}\right)^{2}\right]\,dz.$
(C22)

The expansion of $\arccos^{2}$ is

$\arccos^{2}z\simeq\arccos^{2}\mu_{z}-2\frac{\sigma_{z}\arccos\mu_{z}}{\sqrt{1-\mu_{z}^{2}}}\epsilon-\frac{\sigma_{z}^{2}}{1-\mu_{z}^{2}}\left(\frac{\mu_{z}\arccos\mu_{z}}{\sqrt{1-\mu_{z}^{2}}}-1\right)\epsilon^{2}+\ldots$
(C23)

The integration gives non-zero values for the even terms, resulting in

$E[i^{2}]\simeq\arccos^{2}\mu_{z}-\frac{\sigma_{z}^{2}}{1-\mu_{z}^{2}}\left(\frac{\mu_{z}\arccos\mu_{z}}{\sqrt{1-\mu_{z}^{2}}}-1\right).$
(C24)

The square of $E[i]$, discarding terms of fourth order, is

$E^{2}[i]\simeq\arccos^{2}\mu_{z}-\frac{\sigma_{z}^{2}}{1-\mu_{z}^{2}}\,\frac{\mu_{z}\arccos\mu_{z}}{\sqrt{1-\mu_{z}^{2}}}.$
(C25)

The difference gives us the variance $\sigma_{i}^{2}=E[i^{2}]-E^{2}[i]$,
resulting in a standard deviation that is simply

$\sigma_{i}\simeq\frac{\sigma_{z}}{\sqrt{1-\mu_{z}^{2}}}.$ (C26)

###  C.3 Error of $i=\arctan z$

Let us now calculate an approximation to the expected value and standard
deviation of the inclination

$i=\arctan z,$ (C27)

given the expected value, $\mu_{z}$, and standard deviation, $\sigma_{z}$, of
$z$. The procedure is the same as that used in the former section.

####  C.3.1 Expected value of $i=\arctan z$

The expansion of $\arctan$ around the expected value $\mu_{z}$, using the
standard normally distributed variable $\epsilon=(z-\mu_{z})/\sigma_{z}$, is

$\arctan
z\simeq\arctan\mu_{z}+\frac{\sigma_{z}}{(1+\mu_{z}^{2})^{2}}\epsilon-\frac{\mu_{z}\sigma_{z}^{2}}{(1+\mu_{z}^{2})^{2}}\epsilon^{2}+\ldots$
(C28)

The expected value of $i$, neglecting terms of order higher than three is

$E[i]\simeq\arctan\mu_{z}-\frac{\mu_{z}\sigma_{z}^{2}}{(1+\mu_{z}^{2})^{2}}.$
(C29)

Like in the former case, there is a bias in the expected value of the
inclination, because $E[\arctan z]\neq\arctan(E[z])$.

####  C.3.2 Standard deviation of $i=\arctan z$

To calculate $E[i^{2}]$ we need the expansion

$\arctan^{2}z\simeq\arctan^{2}\mu_{z}+2\frac{\sigma_{z}\arctan\mu_{z}}{1+\mu_{z}^{2}}\epsilon+\frac{\sigma_{z}^{2}}{(1+\mu_{z}^{2})^{2}}(1-2\mu_{z}\arctan\mu_{z})\epsilon^{2}+\ldots$
(C30)

resulting in

$E[i^{2}]\simeq\arctan^{2}\mu_{z}+\frac{\sigma_{z}^{2}}{(1+\mu_{z}^{2})^{2}}(1-2\mu_{z}\arctan\mu_{z}).$
(C31)

The difference, $\sigma_{i}^{2}=E[i^{2}]-E^{2}[i]$, neglecting terms of fourth
order, gives

$\sigma_{i}\simeq\frac{\sigma_{z}}{1+\mu_{z}^{2}}.$ (C32)

###  C.4 Summary

Given two independent Gaussian random variables $x$ and $y$, with expected
values $\mu_{x}$ and $\mu_{y}$, and standard deviations $\sigma_{x}$ and
$\sigma_{y}$, the expected value and the standard deviation of the ratio
$z=x/y$ are

$\displaystyle\mu_{z}$ $\displaystyle\simeq$
$\displaystyle\displaystyle\frac{\mu_{x}}{\mu_{y}}\left(1+\frac{\sigma_{y}^{2}}{\mu_{y}^{2}}\right),$
$\displaystyle\sigma_{z}$ $\displaystyle\simeq$
$\displaystyle\frac{\mu_{x}}{\mu_{y}}\sqrt{\frac{\sigma_{x}^{2}}{\mu_{x}^{2}}+\frac{\sigma_{y}^{2}}{\mu_{y}^{2}}}.$
(C33)

Given the ratio of the minor and major axes, $a_{\mathrm{min}}$ and
$a_{\mathrm{maj}}$, of the projection of the disk on the plane of the sky,
$z=a_{\mathrm{min}}/a_{\mathrm{maj}}$, the expected value and the standard
deviation of the disk inclination, $i=\arccos z$, are

$\displaystyle\mu_{i}$ $\displaystyle\simeq$
$\displaystyle\arccos\mu_{z}-\displaystyle\frac{1}{2}\,\frac{\mu_{z}\sigma_{z}^{2}}{(1-\mu_{z}^{2})^{3/2}},$
$\displaystyle\sigma_{i}$ $\displaystyle\simeq$
$\displaystyle\displaystyle\frac{\sigma_{z}}{\sqrt{1-\mu_{z}^{2}}}.$ (C34)

Given the ratio of the components of the orbital velocity along the line-of-
sight and on the plane of the sky, $z=v_{\mathrm{los}}/v_{\mathrm{sky}}$, the
expected value and the standard deviation of the orbit inclination, $i=\arctan
z$, are

$\displaystyle\mu_{i}$ $\displaystyle\simeq$
$\displaystyle\displaystyle\arctan\mu_{z}-\frac{\mu_{z}\sigma_{z}^{2}}{(1+\mu_{z}^{2})^{2}},$
$\displaystyle\sigma_{i}$ $\displaystyle\simeq$
$\displaystyle\displaystyle\frac{\sigma_{z}}{1+\mu_{z}^{2}}.$ (C35)

##  Appendix D Fitting of a Keplerian Thin Disk Model

We use the model of a thin disk with a Keplerian rotation developed by R.
Estalella (see Appendix B in Zapata et al. 2019 and references therein for
details) to constrain the properties of the VLA 4A source. The model consists
of a geometrically thin disk with an inner radius $R_{\rm inn}$ and an outer
radius $R_{\rm out}$, where the intensity and velocity are described by power
laws of the radius with index $-$1 and $-$0.5, respectively. The kinematics of
the disk is computed by considering a cube composed of a grid of points in the
plane of the disk, and for each point of the grid, a number of velocity
channels. For each point of the grid, the projection of its rotation velocity
onto the line-of-sight is calculated. A Gaussian line profile, of width
$\Delta V$ (assumed constant for the whole disk), and proportional to the
estimated intensity is added to the channels associated with the grid point.
After this first cube is computed, its projection onto the plane of the sky is
estimated. Finally, each channel of the plane-of-the-sky cube is convolved
with a Gaussian with the size of the beam. The intensity depends on a scale
factor, which is estimated by minimizing the sum of the squared differences
between the data cube and the model cube. The parameter space is searched for
the minimum of the rms fit residual for all the channel maps using a pseudo-
random sequence as described in Estalella et al. (2012), and Estalella (2017).
Once a minimum of the rms fit residual is found, the uncertainty in the
parameters fitted is found as described in Wall & Jenkins (2003) and Estalella
(2017).

###  D.1 VLA 4A Circumstellar Disk traced by Ethylene Glycol

The ethylene glycol lines detected only toward VLA 4A (Table 9) were used to
fit the rotation of a geometrically thin disk. The model was fitted to the
spectral data, assuming as position of its center that of VLA 4A, and a
reference velocity $V_{\mathrm{sys}}=7.36$ km s-1 (see Table 10). The beam
used to convolve the model intensity was that of the observation, that is,
$0\farcs 154\times 0\farcs 079$ ($\mathrm{PA}=-1.9^{\circ}$). The free
parameters of the model were the linewidth (without rotational broadening),
$\Delta V$, the inclination of the disk, $i$, the inner and outer radii,
$R_{\mathrm{inn}}$ and $R_{\mathrm{out}}$, and the rotation velocity at a
reference radius of $1^{\prime\prime}$, times the sinus of the inclination,
$V_{0}\sin i$.

Table 9: Ethylene glycol (aGg’-(CH2OH)2) transitions.

|  | Frequency
---|---|---
Identifier | Transition | (GHz)
EthyGly1 | (34 04 31 1 – 33 04 30 0) | 342.801
EthyGly2 | (34 03 31 1 – 33 03 30 0) | 342.851
EthyGly3 | (33 11 23 1 – 32 11 22 0) | 345.737a
| (33 11 22 1 – 33 11 21 0) | 345.738a

a The frequency difference between the two transitions is 1.432 MHz (1.242 km
s-1) and both appear blended.

Table 10: Positions and Reference Velocities.

| $\alpha(J2000)$ | $\delta(J2000)$ | $V_{\mathrm{sys}}$
---|---|---|---
Object | (h:m:s) | (d:m:s) | (km s-1)
VLA 4A | 03:29:03.7460 | +31:16:03.769 | $7.36\pm 0.21$
VLA 4B | 03:29:03.7691 | +31:16:03.761 | $9.33\pm 0.18$
Ambient Cloud | $\cdots$ | $\cdots$ | 8.5a

a Adopted (Sect. 3.4.1).

The results from the fit are given in Table 11, and the derived parameters,
for a distance to SVS 13 of 300 pc, are given in Table 12.

Table 11: VLA 4A: Best Fit of a Geometrically-Thin Disk with Keplerian
Rotation.

| $\Delta V$ | $i$ | $R_{\mathrm{inn}}$ | $R_{\mathrm{out}}$ | ${V_{0}\sin i}^{\,a}$
---|---|---|---|---|---
Line | (km s-1) | (deg) | (arcsec) | (arcsec) | (km s-1)
EthyGly1 | $1.53\pm 0.18$ | $22.6\pm 5.6$ | $0.011\pm 0.003$ | $0.109\pm 0.004$ | $0.312\pm 0.018$
EthyGly2 | $1.87\pm 0.22$ | $21.8\pm 6.7$ | $0.006\pm 0.003$ | $0.112\pm 0.010$ | $0.327\pm 0.019$
EthyGly3 | $2.74\pm 0.13$ | $22.3\pm 3.2$ | $0.010\pm 0.003$ | $0.110\pm 0.004$ | $0.255\pm 0.017$

a At a radius of $1^{\prime\prime}$.

Table 12: VLA 4A: Parameters Derived from the Best Fit to the Ethylene Glycol
Lines.

| $R_{\mathrm{inn}}$ | $R_{\mathrm{out}}$ | ${V_{0}}^{a}$ | ${M_{\ast}}^{b}$
---|---|---|---|---
Line | (au) | (au) | (km s-1) | ($M_{\odot}$)
EthyGly1 | $3.3\pm 0.9$ | $32.8\pm 1.3$ | $0.81\pm 0.20$ | $0.22\pm 0.11$
EthyGly2 | $1.9\pm 0.8$ | $33.5\pm 2.9$ | $0.88\pm 0.26$ | $0.26\pm 0.16$
EthyGly3 | $2.9\pm 0.9$ | $33.1\pm 1.1$ | $0.67\pm 0.10$ | $0.15\pm 0.05$

a At a radius of $1^{\prime\prime}$ (300 au). b Central mass.

Figure 23 shows a comparison of the zeroth-order and first-order moment maps
of the EthyGly1 and EthyGly2 transitions with the thin-disk model. Figures 24
and 25 show a comparison of the observed and thin-disk model PV plots and
channel maps, respectively.

Figure 23:  VLA 4A circumstellar disk: First-order moment (intensity weighted
velocity in km s-1; color scale) overlaid on the zeroth-order moment
(integrated intensity; contours) of EthyGly1 (_left_), EthyGly2 (_center_),
and the thin-disk model of EthyGly2 (_right_). For the three panels, contours
range from 15% to 90% in steps of 15% of the peak (0.120, 0.168 and 0.150 Jy
beam-1 km s-1 for EthyGly1, EthyGly2 and the model of EthyGly2, respectively).
Offsets are relative to the position of VLA 4A (Table 10), in right ascension
(X axis) and declination (Y axis), positive toward east and north,
respectively.

Figure 24:  VLA 4A circumstellar disk: Position-velocity cuts for EthyGly1
(_top panels_), EthyGly2 (_middle panels_), and the thin-disk model (_bottom
panels_). The left panels show the cuts along the major axis (east-west),
while the right panels show the perpendicular direction. Offsets are positive
westward and northward, respectively. The model is for a central mass
$M_{\ast}=0.25~{}M_{\odot}$, $R_{\mathrm{inn}}=1.8$ au, and
$R_{\mathrm{out}}=33.5$ au, with an inclination $i=22^{\circ}$. Contours start
at 10% of the peak of the EthyGly2 emission (peak = 0.047 Jy beam-1),
increasing in steps of 10%. The curves in the major axis cuts are the
Keplerian rotation curves for central masses $M_{\ast}=0.10$, 0.25, and 0.40
$M_{\odot}$, which are plotted as a reference.

Figure 25:  VLA 4A circumstellar disk: Channel maps around the reference
velocity of the source, $V_{\mathrm{sys}}=7.36$ km s-1, of EthyGly2 (_three
leftmost columns_) and the thin-disk model (_three rightmost columns_).
Velocities range from 4.76 km s-1 (top-left corner) to 9.45 km s-1 (bottom-
right corner), in steps of 0.43 km s-1. Contours start at 10% of the peak
(0.047 Jy beam-1), increasing in steps of 13%. Offsets are relative to the
position of VLA 4A (Table 10), in right ascension (X axis) and declination (Y
axis), positive toward east and north respectively.

The results obtained for the three lines are remarkably consistent. The
linewidth of EthyGly3 is consistent with those of EthyGly1 and EthyGly2 if we
take into account the separation of the two blended lines in EthyGly3 (see
Table 9). In summary, the ethylene glycol data indicate the presence of a
rotating disk around VLA 4A, with an inner radius of $\sim 2$ au, an outer
radius of 33 au, an inclination of $22^{\circ}$ ($0^{\circ}$ for a face-on
disk), a Keplerian rotation velocity with a value of 0.9 km s-1 at a reference
radius of 300 au, corresponding to 2.7 km s-1 at $r=30$ au, or 9 km s-1 at 3
au. The Keplerian rotation implies a central mass of $\sim 0.25$ $M_{\odot}$.

##  Appendix E Fitting of an Optically Thin Spherical Infalling Envelope Model

###  E.1 Infalling Spherical Envelope Centered on VLA 4A traced by CS (7–6)

The integrated CS (7–6) emission is shown in Figure 11 (left column panels).
The PV cuts through the position of VLA 4A show high velocities near its
center, suggesting gravitational acceleration (Fig. 16). The morphology of the
PV cuts is quite similar for all the position angles (see Figs. 16 and 26),
suggesting that assuming spherical symmetry is an acceptable approximation.
The excess emission at offsets $\sim-0\farcs 5$ in the cuts at PA =
$-90^{\circ}$ and $-110^{\circ}$ is at the position of VLA 4B, and is
associated with its own envelope. We assume that this emission does not belong
to the envelope around VLA 4A.

Figure 26:  VLA 4A envelope: Position-velocity cuts of the CS (7–6) emission,
centered on the position of VLA 4A, for different position angles. The
position offsets are distances from VLA 4A along the position angle, which is
indicated for each panel. Positive offsets correspond to points north of VLA
4A in the cut at $\mathrm{PA}=0^{\circ}$ and to points west of VLA 4A in the
cuts from $\mathrm{PA}=-20^{\circ}$ to $-150^{\circ}$. Contours start at 10%
of the peak (0.120 Jy beam-1), increasing in steps of 10%.

Thus, the CS emission seems to be tracing an infalling envelope. We compared
the observed emission from that of a spherically symmetric envelope in infall,
assuming that the line emission is optically thin (perhaps only for the wings
of the line), with infall velocity and temperature proportional to $r^{-1/2}$
(characteristic of the main accretion phase of infall), with no rotation,
inner radius $R_{\mathrm{inn}}$, outer radius $R_{\mathrm{out}}$, and constant
linewidth $\Delta V$. The volume density of the envelope, as a consequence of
the continuity equation, must be $\rho\propto r^{-3/2}$. For such a model we
computed the emission as a function of position and velocity, and calculated
the 2-D convolution of the emission with a Gaussian beam. In particular, the
intensity as a function of the projected radius was calculated at each line-
of-sight velocity as proportional to the product of the temperature and
density of the intersection of the isovelocity surface with the line-of-sight,
and to the derivative of the line-of-sight distance with respect to the line-
of-sight velocity (R. Estalella et al. in preparation).

A satisfactory agreement was obtained qualitatively for the range of
parameters shown in Table 13. The model PV cut is shown in Figure 27. The
inner radius of the envelope is $R_{\mathrm{inn}}=10$–20 au, and the outer
radius is $R_{\mathrm{out}}=90$–120 au. The infall velocity is $V_{0}=1.0$–1.5
km s-1 at a reference radius of $1^{\prime\prime}$ (300 au), corresponding to
1.7–2.6 km s-1 at 100 au. This range of infall velocities corresponds to the
free-fall onto a central mass $M_{\ast}=0.17$–0.38 $M_{\odot}$.

Table 13: VLA 4A: Infalling Spherical Envelope traced by the CS (7–6) line.

Parameter | Value | Units
---|---|---
Linewidth $\Delta V$ | 0.5–1.5 | km s-1
Inner radius $R_{\mathrm{inn}}$ | 0.04–0.07 | arcsec
| 10–20 | au
Outer radius $R_{\mathrm{out}}$ | 0.3–0.4 | arcsec
| 90–120 | au
Infall velocity ${V_{0}}^{\,a}$ | 1.0–1.5 | km s-1
Central mass $M_{\ast}$ | 0.17–0.38 | $M_{\odot}$

a At a radius of $1^{\prime\prime}$ (300 au).

Figure 27:  VLA 4A envelope: Position-velocity cut of a model of spherically
symmetric envelope in free-fall, with $\Delta V=1$ km s-1,
$R_{\mathrm{inn}}=0\farcs 05$ (15 au), $R_{\mathrm{out}}=0\farcs 35$ (105 au),
$V_{0}=1.2$ km s-1 (at a reference radius of 300 au), giving a central mass
$M_{\ast}=0.26$ $M_{\odot}$. Contours start at 10% of the peak (arbitrary
intensity), increasing in steps of 10%.

###  E.2 Infalling Spherical Envelope Centered on VLA 4B traced by CS (7–6)

Because of the presence of the large envelope around VLA 4A (Fig. 11), in the
fitting of the VLA 4B data, we consider only positions east of VLA 4B, away
from the position of VLA 4A. Also, there is a lack of emission to the south of
VLA 4B, which could be due to the disruption of the southern half of the
envelope by the blue-shifted jet (Fig. 2). Therefore, we restrict our fitting
to position angles in the range $0^{\circ}$ to 90∘. In Figure 28 we show
position-velocity cuts centered on the position of VLA 4B, for position angles
in this range. As can be seen in the figure, the red-shifted velocities,
$V>9.3$ km s-1, show a pattern similar to that expected for an infalling
spherical envelope.

We fitted the model of an infalling envelope with optically thin emission,
with infall velocity and temperature proportional to $r^{-1/2}$, with no
rotation, inner radius $R_{\mathrm{inn}}$, outer radius $R_{\mathrm{out}}$,
and constant linewidth $\Delta V$. The range of values of the parameters that
fit better the pattern observed in the PV cuts are given in Table 14. The
model PV cut is shown in the rightmost panel of Figure 28. The inner radius of
the envelope is $R_{\mathrm{inn}}\simeq 12$ au, and the outer radius is
$R_{\mathrm{out}}\simeq 150$ au. The infall velocity is $V_{0}=$ 1.5-2.2 km
s-1 at a reference radius of $1^{\prime\prime}$ (300 au), corresponding to
2.6-3.8 km s-1 at 100 au. These infall velocities correspond to the free-fall
onto a central mass $M_{\ast}\simeq 0.6$ $M_{\odot}$.

Figure 28:  VLA 4B envelope: (_Leftmost panels_) Position-velocity cuts of the
CS (7–6) emission, through the position of VLA 4B, for different position
angles, from $\mathrm{PA}=0^{\circ}$ to $90^{\circ}$. (_Rightmost panel_)
Position-velocity cut of a model of a spherically symmetric envelope in free-
fall, with $\Delta V=0.3$ km s-1, $R_{\mathrm{inn}}=0\farcs 04$ (12 au),
$R_{\mathrm{out}}=0\farcs 5$ (150 au), $V_{0}=1.8$ km s-1 (at a reference
radius of 300 au), giving a central mass $M_{\ast}=0.6$ $M_{\odot}$. Contours
start at 10% of the peak (0.100 Jy beam-1), increasing in steps of 10%. The
position offsets are distances from VLA 4B along the position angle, which is
indicated for each panel.  Table 14: VLA 4B. Infalling spherical envelope
traced by the CS (7–6) line.

Parameter | Value | Units
---|---|---
Linewidth $\Delta V$ | $\sim 0.3$ | km s-1
Inner radius $R_{\mathrm{inn}}$ | $\sim 0.04$ | arcsec
| $\sim 12$ | au
Outer radius $R_{\mathrm{out}}$ | $\sim 0.5$ | arcsec
| $\sim 150$ | au
Infall velocity ${V_{0}}^{\,a}$ | 1.5–2.2 | km s-1
Central mass $M_{\ast}$ | 0.38–0.82 | $M_{\odot}$

a At a radius of $1^{\prime\prime}$ (300 au).

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