Protostellar chimney flues: are jets and outflows lifting submillimetre dust grains from discs into envelopes?

Low dust opacity spectral indices ($\beta<1$) measured in the inner envelopes of class 0/I young stellar objects (age $\sim 10^{4-5}$ yr) have been interpreted as the presence of (sub-)millimetre dust grains in these environments. The density conditions and the lifetimes of collapsing envelopes have proven unfavorable for the growth of solids up to millimetre sizes. As an alternative, magneto-hydrodynamical simulations suggest that protostellar jets and outflows might lift grains from circumstellar discs and diffuse them in the envelope. We reframe available data for the CALYPSO sample of Class 0/I sources and show tentative evidence for an anti-correlation between the value of $\beta_{1-3mm}$ measured in the inner envelope and the mass loss rate of their jets and outflows, supporting a connection between the two. We discuss the implications that dust transport from the disc to the inner envelope might have for several aspects of planet formation. Finally, we urge for more accurate measurements of both correlated quantities and extension of this work to larger samples, necessary to further test the transport scenario.


INTRODUCTION
The formation of terrestrial planets and of the rocky cores of giant planets is thought to happen in a coreaccretion scenario, a process spanning ten orders of magnitude in size, where interstellar medium, sub-micron dust grains grow into km-sized objects.While dust growth has been long thought to take place exclusively in isolated, evolved protoplanetary discs revolving around class II young stellar objects (YSOs), recent observations indicate that dust growth up to millimeter sizes might start in collapsing protostellar envelopes, thus much earlier and further away from host stars than previously thought.Observationally, the slope α of the spectral energy distribution (SED) across (sub-)millimetric wavelengths is a means to interpret inter-stellar dust properties and its size.Specifically, if (i) dust opacity scales as a power law (κ ∝ ν β ), (ii) the emission is optically thin, and (iii) the Rayleigh-Jeans (RJ) approximation holds, then β = α − 2 (Natta et al. 2007, Testi et al. 2014).In turn, β depends on dust properties, and strongly on the maximum grain size of the dust population.For the interstellar medium, typically β ∼ 1.7 (Weingartner & Draine 2001).In Class II objects, β < 1 suggests the presence of millimetre dust grains (e.g., Testi et al. 2014, Tazzari et al. 2021, Macías et al. 2021).
cm −3 ) and short timescales (a few 10 5 yr) that characterize these environments (Ormel et al. 2009, Lebreuilly et al. 2023, Silsbee et al. 2022).It will be crucial for next simulations to test the effects of generally disregarded processes, like the dust back-reaction on the turbulence through gas-dust friction and dust-magnetic-field interaction (Hennebelle & Lebreuilly 2023), to check whether growth remains a viable scenario.
Alternative or concomitant processes must be considered that could contribute to explain the observed low β.For example, Wong et al. (2016) first presented a simple analytical model to argue that millimetre dust from the disc could be entrained by protostellar outflows and transported to the envelope.Giacalone et al. (2019) also presented an analytical model for the entraiment of dust grains along magnetohydrodynamical (MHD) disc winds, and concluded that grains of ∼10 µm can be lifted by MHD winds and be transported outwards in the disc of T Tauri and Herbig Ae/Be objects.However, their model assumes typical evolved mass outflow rates of ∼ 10 −8 M ⊙ /yr.Since the maximum grain sizes lifted in the envelopes depend linearly on this quantity, it can be much larger in young Class 0/I objects, for which the mass loss rates are orders of magnitude higher.These findings might have found recent confirmation thanks to exquisite JWST observations of the Tau 042021 edge-on disc, for which Duchene et al. (2023) reported an Xshaped feature in dust scattered whose spatial location is consistent with ALMA CO line emission tracing an outflow.Their observations suggest the entrainment of ≳ 10µm grains even beyond 300 au.Finally, the findings of these models are supported by Tsukamoto et al. (2021) and Lebreuilly et al. (2023), who arrived to con-sistent conclusions via three-dimensional MHD simulations.In particular, Tsukamoto et al. (2021) proposed the expression ash-fall1 , referring to the dust grains decoupling from the entrainment outflow and their subsequent fall back in the disc.
Thus, as outflows represent in principle a means to transport submillimetre grains to envelopes, we here explore the unique CALYPSO sample (Maury et al. 2019) to test whether a correlation holds between the observed power of jets and outflows in Class 0 protostars and the dust opacity index in their envelopes.

THE SAMPLE
The sources that make up our sample are part of the Continuum And Lines in Young ProtoStellar Objects IRAM-PdBI Large Program (CALYPSO2 ; Maury et al. 2019).CALYPSO is a survey of 16 Class 0 sources, located in different star forming regions (d≤450 pc), observed in three spectral setups (centered at ∼ 94, 219 and 231 GHz).The observations were carried out with the IRAM Plateu de Bure Interferometer (PdBI).See Maury et al. (2019) for further details.Out of the sixteen sources, nine can be fully characterized for our purposes.Only for these, in fact, a reliable measure of β and of the jets/outflows mass loss rates could be performed ( ṀJ , ṀOF ) (Table 1).Among the sources considered in this study, seven are in binary or multiple systems.We report considerations on their multiplicity in Appendix A. While this is a low number statistics, we note that CA-LYPSO is unique in its uniformity and is the only survey for which the SiO (5−4) transition is systematically targeted to detect the high-velocity jets of a sample of protostars (see Section 4).This allows us to perform the jets/outflows analysis as explained in Sect. 4 and have deep enough continuum datasets with which Galametz et al. (2019) measured dust optical properties.Finally, we note that the CALYPSO data for the sources we consider here have been self-calibrated as explained in Section 2 of Maury et al. (2019), and the self-calibrated data has been later used in Galametz et al. (2019) and Podio et al. (2021), i.e. the works that have measured the dust opacity spectral index β and the jets mass loss rates ṀJ that we also consider in this work.

THE DUST OPACITY SPECTRAL INDEX
As stated in Section 1, the dust opacity spectral index β can be derived starting from the radio spectrum of a source and carries dependencies on the properties of interstellar dust, such as the maximum grain size of the distribution.Galametz et al. (2019) used CALYPSO 1.3 and 3.2 millimetre continuum observations to constrain β and infer the maximum dust grain sizes in the protostellar envelopes of the sources we consider in this work, up to ∼2000 au radial distances from the central protostars.While we report on the details of their measurements in Appendix B for completeness, we here briefly summarise them for the reader's convenience.
They measured β including a temperature correction that accounts for discrepancies from the Rayleigh-Jeans approximation due to low envelope temperatures: where ν 2 = 231 GHz and ν 1 = 94 GHz are the representative frequencies of the PdBI observations, F ν is the flux at each frequency.The term B ν (T ) is the value of the Planck function at a temperature T that depends on the radial distance from the central protostar3 (T ∝ r −0.4 ), evaluated at frequency ν i .For each envelope, they measured β across scales and reported best-fit linear models to extrapolate β at any other scale.We report the final estimated envelope-onlyβ values at 500 au in Table 1.
We note that β also depends on the extent of porosity and on the composition of both grain bulk and ice mantles (e.g., Natta et al. 2007).However, the lowest values observed by Galametz et al. (2019) are only reconcil-able with laboratory experiments in which the sizes of the dust grains are ≳ 100µm, regardless of ice mantles properties (Köhler et al. 2015, Ysard et al. 2019).The effect of porosity would also not affect the interpretation of low β as due to large grain sizes (e.g., Birnstiel et al. 2018).

JETS AND OUTFLOWS MASS LOSS RATES
In this section, we report previous measurements of jets energetics for the CALYPSO sample and present new ones for their low-velocity outflow counterparts.We note that we are interested in the instantaneous mass loss rates of these components rather than their total ejected mass, hence we do not complement the CALYPSO observations with single dish data.This is the case because we aim to investigate a link between the presently observed low β values of Galametz et al. (2019) and the continuous flow of material along jets and outflows.The mass loss rates are constant along the jets/outflows extension since mass needs to be conserved, and we measure the latter (outflow) as Podio et al. (2021) measured the former (jet), i.e., at the first peak of the respective tracer emission, to minimize the contribution of possible gas entrained along the jet.The positions of these peaks are in Tab. 4 of Podio et al. (2021).For the reader's convenience, we note that the maximum recoverable scale of the observations is reported to be about 8. ′′ 0 (Podio et al. 2021).Based on the SiO (5−4) transition at the innermost knots of the blueand red-shifted lobes, Podio et al. (2021) defined the high-velocity (HV) ranges, where the emission probes the jet, for all sources associated with SiO (see their  4).In this work, we define the outflow as the emission on the complementary low-velocity (LV) ranges.In Fig. 1, we show the spatial distribution of CO (2−1) towards L1448-C, obtained integrating on the LV and HV ranges: HV CO traces the collimated jet, which is believed to originate from the inner disc region, while LV CO probes the wide-angle outflow, which is likely to arise from a more extended disc region.The HV ranges for all the CALYPSO sources are listed in Table 4 of Podio et al. (2021).
At this point, Podio et al. (2021) estimated the ṀJ , in the blue (B) and red (R) lobes.Here, we apply the same methodology to infer the LV outflows Ṁ OF of the sources in the CALYPSO sample, for the first time.The beamaveraged CO column densities in the jet and outflow, N CO , are derived from the integrated line intensities on HV and LV, respectively.We assume local thermodynamic equilibrium (LTE) at a fixed excitation temperature, T K = 100 K for HV jets (e.g., Cabrit et al. 2007), and T K = 20 K for LV outflows (e.g., Bachiller et al. 2001), and that the emission is optically thin.The jet and outflow mass-loss rates are computed as (Lee et al. 2007, Podio et al. 2015): where 1/ √ C accounts for compression in the shocks (C = 3), m H2 is the mass of molecular hydrogen, X CO = 10 −4 the assumed CO abundance with respect to H 2 , b t the beam size perpendicular to the jet, and V tan the tangential jet/outflow velocity.The latter is obtained by correcting for inclination the jet/outflow velocities, assumed to be 100 km s −1 for the HV jet, and 10 km s −1 for the LV outflow.The inclination is derived from the ratio between the assumed jet velocity and its radial component from the HV spectra (see Table 4 of Podio et al. 2021).
For the HV jets, Podio et al. (2021) identified the sources for which ṀJ is a lower limit by comparing CO and SiO spectra.The estimated rates carry a factor 3-10 of uncertainty due to the calibration of the parameters of Eq. 2. The LV outflow emission is likely optically thick, therefore, the estimated ṀOF must be considered as lower limits.We can estimate the uncertainty introduced by optical depth using 13 CO emission in the assumption that it is optically thin.For two sources only (IRAS 4A1 and IRAS 4B1), 13 CO is detected along the jet (see the maps in Maret et al. (2020)).For these two sources we can reliably estimate the 12 CO/ 13 CO ratio, hence opacity.We find τ (R) IRAS4B1 = 7.These values imply ṀOF higher by a factor at least ∼6-18.Since we cannot repeat this analysis for all sources, we here stress that the derived ṀOF (in Table 1 and Fig. 2) are lower limits and we consider the jets to be a more robust proxy of the effective mass loss rates of each protostar.Observations of optically thin tracers of the low-velocity outflows will be key to further test the correlation we propose.

A TENTATIVE ANTI-CORRELATION
Modern theoretical efforts have shown how growing dust grains in protostellar envelopes is problematic due to the lifetimes and densities of these environments (Ormel et al. 2009, Bate 2022, Silsbee et al. 2022, Lebreuilly et al. 2023).If millimeter dust, implied by recent measurements of low dust opacity spectral indices in envelopes (Miotello et al. 2014, Galametz et al. 2019), cannot grow at envelope scales, alternative processes might explain their presence therein.We here show a tentative anti-correlation between β with ṀJ and ṀOF , potentially supporting a scenario in which protostars launching powerful outflows can lift millimeter grains into their envelopes.Fig. 2 show the β indices found by Galametz et al. (2019) as a function of ṀJ and ṀOF summed over the blue and red lobes (see Section 2).The values are reported in Table 1.We do not include SerpM-S68N because SiO (5−4) emission in Podio et al. ( 2021) is only at low velocities, likely due to the system inclination, thus impeding the identification of the LV and HV.
The resulting Pearson correlation coefficients are: We evaluate the statistical significance of such a correlation by means of a two-tailed Student's t-test, where the null hypothesis is that ρ = 0 (against ρ ̸ = 0).We reject the null hypothesis at p < 0.04 level in the jet case, and at the p < 0.06 level for the outflows.These tentative correlations might support a dust transport scenario from young discs to their embedding envelopes.
Alternative explanations to the observed tentative correlation are possible in case these two share correlations with other quantities.Bontemps et al. (1996) found a correlation between the envelope mass of Class 0/I YSOs and the CO momentum flux of their outflows.Since Galametz et al. (2019) observed a correlation between β and envelope mass of CALYPSO sources, then the tentative correlation we show in Fig. 2 might be the combined result of these underlying relationships.However, it remains unclear whether the fundamental causal correlation is the one between the dust opacity spectral index and envelope mass or the mass loss reates, as presented here.Moreover, the β − ṀOF correlation in Fig. 2 might be caused by an underlying ṀJ − ṀOF correlation.Such a correlation cannot be quantified here, given that the estimated ṀOF are lower limits.To rule out possible contamination of the correlation from any dependence of the measured β and mass loss rates on the inclination of the source (disc/jet), we reject possible underlying correlations in Appendix D.

DISCUSSION
We here further discuss our findings, and the conditions that need to be met in order for the proposed dust transport to happen.

When and where do transported grains grow?
If outflows are lifting millimetre (or larger) dust grains into the envelopes of Class 0 objects, these must have first grown in the disc.Brauer et al. (2008) studied dust coagulation in the first 1 Myr of disc evolution at representative 1, 10 and 100 au scales and found that millimetre dust grains dominate the dust size distribution already after few 10 3 yr in the inner 1 au of the disc.Lebreuilly et al. (2023) considered several dust size distributions and simulated their early evolution during protostellar collapse under the effects of turbulent, brownian and radial motions.They found that millimetre grains are formed at ≤ 0.1 au scales in few years after the first Larson core formation start.Laboratory experiments have been performed to constrain the stickiness of dust grains in the disc inner regions.When heated at 1000 K, dust grains become super-dry and their stickiness can increase up to a factor 100, thus providing the conditions to grow even larger agglomerates (Bogdan et al. 2020, Pillich et al. 2021).These temperatures are typically reached in the inner ∼ 0.1 au of low-mass protostellar discs.At these distances, both jets and outflows could lift grains.Indeed, the typical foot-point of jet is much closer to the star than the outflows'.For example, Lee et al. (2017) measured a 0.05−0.3au footpoint radius for the high-velocity SiO jet in the Class 0 HH212 source.Low-velocity outflows, instead, likely extend to a wider disc region out to radii of even 20−40 au (Bjerkeli et al. 2016, Tabone et al. 2017, Lee et al. 2018), and thus could entrain grains from a larger reservoir.6.2.Can outflows lift millimetre grains?Wong et al. (2016) and Giacalone et al. (2019) presented an analytical treatment in which they explored the conditions for the uplifting of dust grains along outflows.Wong et al. (2016) presented the critical mass of the protostar for which, if M * < M cr , grains of a given size could be entrained against gravity (see their Eq.7).Another analytical model, by Giacalone et al. (2019), reported an equation to compute the maximum grain size a max that a given wind can uplift against the gravity of a star of mass M * .We report the latter for the reader's convenience: where Ṁ⊙ /yr is the mass loss rate of the outflow, T gas is the gas temperature, r is the launching footpoint, z/r the disc flaring ratio, r + /r − the ratio between disc's outer and inner edge.See Giacalone et al. (2019) for the details.We notice that the three sources of our sample with the largest outflows mass loss rates (≳ 2 • 10 −7 M ⊙ /yr) have β < 0.8.If we consider this value in Eq. 3, and we fix M * = 1M ⊙ , T gas = 20K at the outflow's base, r = 1 au, z/r = 0.1, r + = 50 au (typical Class 0 disc radius, e.g., Maury et al. 2022) and r − = 0.1 au, we obtain a max ≳ 150µm.Since outflows mass loss rates are lower limits due to optical depth effects, a max could be higher by even an order of magnitude.We refrain from evaluating Eq. 3 source by source since it was derived for class II objects rather than class 0/I, and because most parameters suffer from large uncertainties for young sources.Thus, at face value, assuming standard parameters, grains larger than 100 µm could be lifted for the sources with highest mass loss rates (and lowest betas).Similar findings for the maximum sizes of dust grains entrainable by outflows were reported by Lebreuilly et al. (2020) and Tsukamoto et al. (2021).They both performed magneto-hydrodynamical simulations.Lebreuilly et al. (2020) ran their setup including large grains to account for growth that might have happened at earlier times, while Tsukamoto et al. ( 2021) models dust coagulation.They both found that large grains in the inner region of disc (a few 100 µm to 1 cm) can be entrained.These grains then decouple from the gas and are ejected from the outflow into the envelope, before falling back into the disc like ash fall, as coined by Tsukamoto et al. (2021).

Do grains survive the transport?
Given their lower velocities and temperatures, as well as a wider entraining base, outflows seem to be the preferred mechanism to lift dust grains from protoplanetary discs to the inner envelopes of young protostars (Wong et al. 2016, Lebreuilly et al. 2020, Tsukamoto et al. 2021).The tentative β − ṀOF correlation we present in Fig. 2 might support this scenario.The observed β − ṀJ correlation might either mean that jets are contributing to the mechanism or that they share an underlying correlation with the outflows.We thus here discuss if lifted dust grains would survive the transport along jets.Given the much lower speeds and temperatures of outflows, their survival to transport along the latter is a consequence.
The destruction of silicon-bearing dust grains in shocks has been identified as the mechanism that enriches SiO in the interstellar medium and makes of this molecule a key jet tracer (e.g., Flower & Forêts 1994;Caselli et al. 1997;Schilke et al. 1997).However, shock models predict that only a small fraction (< 10%) of grains is destroyed in the mild shocks along jets, with typical velocities of 20 − 50 km s −1 and pre-shock gas densities of 10 4 −10 6 cm −3 (e.g., Gusdorf et al. 2008a,b;Guillet et al. 2011).
In the wide grid of models explored by Gusdorf et al. (2008a), where the shock velocities range 20 km s −1 < v s < 50 km s −1 and the pre-shock gas densities are in the interval 10 4 cm −3 < n H < 10 6 cm −3 , no more than 5% of Si is released in the gas phase by sputtering.Taking into account shattering and vaporisation of the grains in grain-grain collisions may enhance the fraction of grains destroyed to ∼ 8% (Guillet et al. 2011).These shock models reproduce the typical SiO abundances estimated in protostellar shocks which span from a few 10 −8 and a few 10 −7 (e.g., Gibb et al. 2004;Bachiller & Pérez Gutiérrez 1997;Tafalla et al. 2010).Recent high angular resolution observations, e.g. in CALYPSO, indicate that SiO may reach abundances > 5 × 10 −6 in jets, which requires either dust-free jets or the fraction of grains sputtered in shocks being larger than 10% (for [Si/H] ⊙ ∼ 3.5 × 10 −5 , Holweger 2001).
Finally, Wong et al. (2016) studied whether (sub-)millimetre dust seeds would survive grain-grain collisions in the envelope, after reaching the transport limit velocity (v ∼ 0.5 km/s), given by the gravity-drag equilibrium along the outflow.Making use of the shattering model of Kobayashi & Tanaka (2010), they concluded that millimetre-sized dust grains could survive in the envelope environment: only a fraction as small as 10% might be destroyed.
Thus, it seems reasonable that a large percentage of dust grains could survive the transport along outflows and even jets, being only partially eroded by collisions with both other grains and gas molecules in the latter.However, we note that there is a strong necessity for dust laboratory and modeling studies to assess the effects of high temperatures in the inner disc if submillimeter dust were lifted from inner outflows footpoints.In particular, it will be crucial to test whether the high temperatures would sublimate grain's mantles materials causing them to further shrink in size.

Potential implications
The possibility that protostellar outflows lift large millimetre grains from the disc into the envelopes of young stellar objects can have several implications for the evolution of dust throughout the system.The outwards transport of dust can extend the timescales of grain growth in discs, limited by the meter barrier problem; it can affect the physical properties of grains as they are transported upwards away from the optically thick disc; and it can contribute to explain mixing of the mineralogy of outer discs, like the one found in meteorites in the Solar System.
First, the orbital dynamics of dust grains orbiting in a disc depends on their stokes number, defined by their composition, density and size.When particles grow in size, they experience a larger and larger headwind that slows them down and cause an inward orbit shift, known as radial drift.In a typical disc orbiting a 1 M ⊙ star, radial drift velocities of solids at 1 au reach a maximum for meter-sized boulders, thus causing intermediate solids to rapidly fall towards the central star in timescales much faster than the ones estimated for planet cores formation (Weidenschilling 1977).At larger radii, this peak velocity is reached for even smaller pebbles.If outflows were uplifting grains in a continuous recycle of dust to the outer disc, this would setback grown millimetre grains in its outskirts and contribute to stretch the available time-span to form larger agglomerates, as already proposed by Tsukamoto et al. (2021).Moreover, if young protoplanetary discs harbor ring substructures that act like dust traps (as is the case for, e.g., GY91 from Sheehan & Eisner 2018 or IRS63 from Segura-Cox et al. 2020), then outwards transported grains will be halted on their drift back towards the inner disc at one of these substructures, potential birthplaces of planetesimals via streaming instabilities (Chambers 2021;Carrera et al. 2021).
Secondly, transported dust grains would undergo physical and chemical reprocessing once in the envelope.While they are partially shielded from the radiation of the star in the dense disc, they are going to be lifted in the much thinner envelope and the different energy and intensity of stellar radiation impinging onto them could change their structural and compositional properties.Furthermore, the grains would be transported from the warm inner disc to the colder envelopes where molecular freeze-out could form ice mantles.
Lastly, the uplifting and outward transport of inner disc grains represents a potential explanation for the discovery of cristalline grains in the outskirts of protoplanetary discs, where the temperatures are too low to explain spectral observations of silicate lines (e.g., Apai et al. 2005, Sargent et al. 2009).Along the same line, Trinquier et al. (2009) and Williams et al. (2020) observed anomalous abundances of 46 Ti, 50 Ti and 54 Cr isotopes in outer Solar System chondrules (mm-sized meteorite inclusions).Since Calcium-Aluminum Inclusions (CAIs), which formed in the inner Solar System, consistently show high abundances in both isotopes, they proposed a mixing of solar nebula material in the early stages of formation.In the same direction are the recent findings of Hellmann et al. (2023), who show that carbonaceous chondrites display correlation in different isotopes abundances which can be explained by mixing of refractory inclusions, chondrules, and chondrite-like matrix.They thus highlight the need for a mechanism to transport these constituents from the inner disc to its outskirts and trap them in rings where the meteorites would form.If dynamical barriers to outwards viscous transport were present, such as the core of a Jupiterlike planet, protostellar outflows might have played this transport role: the grains extracted by outflows from inner disc regions will later fall back onto the disc out to larger radii.

CONCLUSIONS
Recently, extremely low dust opacity indices have been observed at few hundred au scales in the envelopes of Class 0 sources, and have been interpreted as the presence of millimetric dust grains (Galametz et al. 2019).Since theoretical models seem to discard the possibility of growing millimetre grains at the densities typical of protostellar envelopes (e.g.Ormel et al. 2009, Silsbee et al. 2022), we propose here a possible observational test to an alternative explanation, the transport of dust from the disc into envelopes via protostellar outflows.The mechanism has been studied analytically by Wong et al. (2016) and Giacalone et al. (2019) and is supported by numerical simulations of Lebreuilly et al. (2020) and Tsukamoto et al. (2021).
We show a tentative anti-correlation between protostellar envelopes β and their mass loss rates driven by jets and outflows.Such a correlation can be interpreted as supporting a scenario in which protostellar outflows transport large disc grains into the envelopes of young sources.
If protostellar outflows are indeed lifting millimetre grains in the envelopes of young sources, implications are important for the meter-size barrier problem, the reprocessing of dust during its life cycle, and for material mixing throughout planetary systems, as already suggested for the Solar System (see Sect. 6).While further measurements of both dust opacity index and mass loss rates will be key in either confirming or disproving such a correlation, we here stress how we explored this possibility with a unique sample in this regard, for which uniform observations, reduction and analyses were carried out.ALMA and JWST synergies will be key to better constrain both dust properties and jets/outflows energetics in a larger sample of young sources.man Excellence Strategy, and from the German Ministry for Economic Affairs and Climate Action in project "MAINN" (funding ID 50OO2206).LC thank Chris Ormel and Sebastian Krijt for insightful discussions on this topic.We thank the referees for their helpful com-ments, which helped us improved the content and presentation of this work.

APPENDIX
A. BINARY PROTOSTARS When stars form from the collapse of gas clouds, fragmentation of dense cores often leads to binary or multiple systems.It is estimated that the fraction of stars with at least one companion in the Galaxy is between ∼20% for M-type sources up to ∼90% for O-type ones (Offner et al. 2023).The protostars of the CALYPSO sample are no exception.The PdBI observations beam allowed Maury et al. (2019) to separate systems in the maps with separations larger than ∼60 au in Taurus, ∼90 au in Perseus and 132 au in Serpens South.For Serpens Main, the systems (SerpM-SMM4, SerpM-S68N) are probed down to distances smaller than 160 au.These spatial resolutions are based on distance measurements from Zucker et al. (2019) for Taurus (140 pc), Ortiz-León et al. (2018a) for Perseus (290 pc), Ortiz-León et al. (2023) for Serpens South (441 pc) and Ortiz-León et al. (2018b) for Serpens Main (436 pc).Moreover, on the large-scale end, they were sensitive to companions up to ∼1500-2800 au, depending on the region.They finally classified IRAM04191, L1521F, L1527, L1157, GF9-2, SerpS-MM22 as single sources.On the contrary, L1448-2A, L1448-N, L1448-C, IRAS4A, IRAS4B, SerpS-MM18, SVS13B, IRAS2A were classified as having a companion.For each protostar considered in this work, we report the distance of their comapnion(s), if any, in Table 2.We note that the tightest binary systems have not been considered here since either a measurement of β, ṀJ , or ṀOF was impractical in the studies of Galametz et al. (2019), Podio et al. (2021) or our own, respectively.While most sources of this study are well resolved binaries, their separation is usually closer than the extent of their envelopes, thus they share a common envelope.The only exceptions are SVS13B and IRAS4B1, for which the companion(s) have much wider separations.For all the sources considered in this work, and that enter the tentative correlation described in section 5, the source of jets and outflows was well resolved (e.g., see Fig. 1) and the measurement of β could be performed after model subtraction of the secondaries.Furthermore, we note that the low β of Galametz et al. (2019)  tostar and thus far from possible contamination of the much larger common envelope (see section 3).
B. DETAILS ON β MEASUREMENTS OF GALAMETZ ET AL. ( 2019) Galametz et al. (2019) measured the dust opacity spectral index in a sample of Class 0/I protostellar envelopes.First, for β to be a trustworthy proxy of the maximum grain size of a dust distribution, the emission over which the radio spectrum is sampled needs to be optically thin.Hence, they estimated the envelopes optical depths and found τ well below 0.1 at few hundred au scales for every source (see their Fig. 2).To make sure the measured β would be representative of the envelope alone, Galametz et al. (2019) subtracted both the emission of binary companions (see Section A) and circumstellar discs.The companions were subtracted by fitting and removing a gaussian model centered on the secondary sources from the visibilities (further details in Maury et al. (2019)).Secondly, the contribution of the circumstellar disc orbiting the target protostar was evaluated in the uv space as the mean of the amplitudes after 200 kλ and subtracted from the shorter-baseline visibili- 7 M⊙/yr) (10 −7 M⊙/yr) (10 −8 M⊙/yr) (10 −8 M⊙/yr) (M⊙) L1527

Figure 1 .
Figure 1.The L1448-C jet and the outflow, as imaged with PdBI in CO (2 − 1) and SiO (5 − 4).Left panels: CO (blue, cyan, magenta, and red) and SiO (black) spectra at the position of the blue-shifted and red-shifted SiO knots located closest to the driving source (from Podio et al. 2021).The horizontal and vertical solid lines indicate the baseline and systemic velocity, Vsys = +5.1 km s −1 , respectively.The vertical black dotted lines indicate the high-velocity (HV, blue/red) and low-velocity (LV, cyan/magenta) ranges, which trace the jet and the outflow, respectively.The definition of LV and HV ranges is based on the SiO (5 − 4) emission (see text).Right panels: Maps of CO (2 − 1), integrated on the LV (top panel) and HV (bottom panel) ranges.First contours and steps are 5σ, corresponding to 1 Jy km s −1 beam −1 .The black stars indicate the positions of the protostars L1448-C (at the center) and L1448-CS.The black solid line shows the jet/outflow PA (Podio et al. 2021).The beam size is in the bottom-left corner.

Figure 2 .
Figure2.The total (red plus blue lobe) jet mass loss rates (cyan points, upper x-axis) and outflows mass loss rates (magenta points, lower x-axis) around young Class 0 sources anti-correlates with β (y-axis) of their inner envelopes (β500au).For each source, a dotted gray line connects the corresponding jet and outflow rates.Source names are on top of the corresponding magenta point.The best-fit linear relations are shown in cyan and magenta, for outflows and jets respectively.
Fig. C.1 and Table

Figure 3 .
Figure 3. Scatter plot of the jets mass loss rates for the CALYPSO sample with inclinations of their jets (left), and same for the outflows mass loss rates (center) and dust opacity spectral index (right).The Pearson correlation coefficient and related p-value are reported in the lower right of each panel.

Table 2 .
are measured in the inner envelope of each pro-Stellar companions associated to the protostars considered in this work.The separations are reported in Maury et al. (2019).(a): Note that the physical separation of the SerpS-MM18 reported therein should instead be 4420 au, given the most upto-date distance measurements of the Serpens South region Ortiz-León et al. 2023.