A Steeper than Linear Disk Mass-Stellar Mass Scaling Relation
I. Pascucci, L. Testi, G. J. Herczeg, F. Long, C. F. Manara, N. Hendler, G. D. Mulders, S. Krijt, F. Ciesla, Th. Henning, S. Mohanty, E. Drabek-Maunder, D. Apai, L. Szucs, G. Sacco, J. Olofsson
aa r X i v : . [ a s t r o - ph . E P ] A ug A Steeper than Linear Disk Mass-Stellar Mass Scaling Relation
I. Pascucci Lunar and Planetary Laboratory, The University of Arizona, Tucson, AZ 85721, USA [email protected]
L. Testi , European Southern Observatory, Karl-Schwarzschild-Strasse 2, D-85748 Garching bei M¨unchen, Germany
G. J. Herczeg and F. Long
Kavli Institute for Astronomy and Astrophysics, Peking University, Yi He Yuan Lu 5, Haidian Qu, 100871Beijing, China
C. F. Manara
Scientific Support Office, Directorate of Science, European Space Research and Technology Centre(ESA/ESTEC), Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands
N. Hendler and G. D. Mulders Lunar and Planetary Laboratory, The University of Arizona, Tucson, AZ 85721, USA
S. Krijt and F. Ciesla Department of the Geophysical Sciences, The University of Chicago, Chicago, IL 60637, USA
Th. Henning
Max Planck Institute for Astronomy, K¨onigstuhl 17, D-69117 Heidelberg, Germany
S. Mohanty and E. Drabek-Maunder
Imperial College London, 1010 Blackett Lab, Prince Consort Rd., London SW7 2AZ, UK
D. Apai , Steward Observatory, The University of Arizona, Tucson, AZ 85721, USA
L. Sz˝ucs Earths in Other Solar Systems Team, NASA Nexus for Exoplanet System Science INAF-Arcetri, Largo E. Fermi 5, I-50125 Firenze, Italy Gothenburg Center for Advance Studies in Science and Technology, Chalmers University of Technology and University ofGothenburg, SE-412 96 Gothenburg, Sweden Lunar and Planetary Laboratory, The University of Arizona, Tucson, AZ 85721, USA ax-Planck-Institut f¨ur extraterrestrische Physik, Giessenbachstrasse 1, D-85748 Garching, Germany G. Sacco
INAF-Arcetri, Largo E. Fermi 5, I-50125 Firenze, Italy
J. Olofsson
Instituto de Fisica y Astronomia, Facultad de Ciencias, Universidad de Valparaiso, Playa Ancha,Valparaiso, Chile
ABSTRACT
The disk mass is among the most important input parameter for every planet formation modelto determine the number and masses of the planets that can form. We present an ALMA 887 µ msurvey of the disk population around objects from ∼ M ⊙ in the nearby ∼ µ m flux densitiesinto dust disk masses, hereafter M dust . We find that the M dust − M ∗ relation is steeper thanlinear and of the form M dust ∝ ( M ∗ ) . − . , where the range in the power law index reflects twoextremes of the possible relation between the average dust temperature and stellar luminosity.By re-analyzing all millimeter data available for nearby regions in a self-consistent way, we showthat the 1-3 Myr-old regions of Taurus, Lupus, and Chamaeleon I share the same M dust − M ∗ relation, while the 10 Myr-old Upper Sco association has a steeper relation. Theoretical modelsof grain growth, drift, and fragmentation reproduce this trend and suggest that disks are in thefragmentation-limited regime. In this regime millimeter grains will be located closer in aroundlower-mass stars, a prediction that can be tested with deeper and higher spatial resolution ALMAobservations. Subject headings: planetary systems:protoplanetary disks, stars:pre-main sequence
1. Introduction
The number of known exoplanets has expo-nentially grown in the past decade, revealingsystems that are unlike our Solar System (e.g.Winn & Fabrycky 2015). While there is clearlya large diversity in planetary architectures, sev-eral trends with the mass of the central starare emerging. These include: i) a positive cor-relation between stellar mass and the occur-rence rate of Jovian planets within a few AU(e.g. Johnson et al. 2010; Howard et al. 2012;Bonfils et al. 2013), although no correlation ispresent for the population of hot Jupiters withina 10 days period (Obermeier et al. 2016) and ii) alarger occurrence rate of close-in Earth-sized plan-ets around M dwarfs than around sun-like stars (Dressing & Charbonneau 2013; Mulders et al.2015a). These trends are likely the result of stellarmass-dependent disk properties. Indeed, planetformation models find that the disk mass stronglyimpacts the frequency and location of planetsthat can form, from giants down to Earth-size(e.g. Raymond et al. 2007; Alibert et al. 2011;Mordasini et al. 2012). Therefore, the scaling ofdisk mass versus stellar mass will yield a stellarmass dependence for the planet population.Measuring gas disk masses is notoriously chal-lenging both in the early ( ∼ −
10 Myr) protoplan-etary phase (e.g. Kamp et al. 2011; Miotello et al.2014) and in the late debris disk phase (e.g.Pascucci et al. 2006; Mo´or et al. 2015). The disk2ass in solids, up to mm-cm in size, is better con-strained via continuum mm-cm wavelength obser-vations since the emission from most dust grainsis optically thin at these wavelengths. Still, in-dividual dust disk masses can have an order ofmagnitude uncertainty because the absolute valueof the dust opacity, which depends both on thegrain composition and size distribution, is notknown (e.g. Beckwith et al. 2000).Pre-ALMA millimeter surveys of nearby star-forming regions provided dust disk masses forover a hundred young stars, primarily with Kand early M spectral types (see Williams & Cieza2011 and Testi et al. 2014 for reviews). In spiteof a large scatter in disk masses at any stellarmass, the data were consistent with a linear diskmass ( M dust ) − stellar mass ( M ∗ ) scaling rela-tion (Andrews et al. 2013; Mohanty et al. 2013),as hinted earlier on by the detection of a few brightdisks around sub-stellar objects (Klein et al. 2003;Scholz et al. 2006; Harvey et al. 2012). However,these studies were dominated by upper limits be-low the M0 spectral type, meaning that they onlyprobed the upper envelope of disk masses in thelow stellar mass end. This left open the possi-bility of a steeper M dust − M ∗ relation buriedin the non-detections. This suspicion was cor-roborated by the observation that stellar ac-cretion rates ( ˙ M ), tracing the gas disk compo-nent, display a steeper dependence with stellarmass when the population of low-mass stars iswell sampled (e.g. Natta et al. 2006; Fang et al.2009; Rigliaco et al. 2011; Alcal´a et al. 2014). Ifthe steeper relation is due to the way disks vis-cously evolve and disperse (e.g. Hartmann et al.2006; Alexander & Armitage 2006; Ercolano et al.2014) and if M dust somehow traces the total(gas+dust) disk mass, ˙ M and M dust should scalesimilarly with stellar mass.The increased sensitivity of ALMA is now en-abling us to survey entire star-forming regionsand to probe the millimeter luminosity of young( ∼ −
10 Myr) protoplanetary disks identifiedin previous infrared images. The 1.3 mm sur-vey of the Orion OMC1 detected continuum emis-sion toward 49 cluster members and reported nocorrelation between M dust and M ∗ (Eisner et al.2016). However, as also pointed out by the au-thors, the statistical significance of this result islimited given the small number of ALMA detec- tions and that spectroscopically-determined stel-lar masses in the OMC1 are only available for lessthan half of the ALMA-detected sources. Thesurvey of the 5 −
10 Myr old Upper Sco asso-ciation (Slesnick et al. 2008) covered all knowndisks around stars from ∼ M ⊙ andreported a steeper than linear relation between M dust and M ∗ (Barenfeld et al. 2016). After re-moving debris/evolved transitional disks, they alsofound that the M dust /M ∗ ratio in Upper Sco is ∼ ∼ M dust − M ∗ relation in Lupus is similar to that in Taurus andshallower than that in Upper Sco.Here, we present an ALMA 887 µ m survey ofthe ∼ M ⊙ down to the sub-stellar regime (Sections 2and 3). We demonstrate that the M dust − M ∗ relation in Chamaeleon I is steeper than linear,under a broad range of assumptions made to con-vert flux densities into dust disk masses (Sections 4and 5). By re-analyzing in a self-consistent way allthe sub-mm fluxes and stellar properties availablefor other nearby star-forming regions we also showthat Taurus, Lupus, and Chamaeleon I have thesame M dust − M ∗ relation, within the inferred un-certainties, and confirm that the one in Upper Scois steeper (Section 6). We discuss the possibilitythat the steeper relation traces either the growthof pebbles into larger solids that become unde-tectable by ALMA or a more efficient inward driftin disks around the lowest mass stars (Section 6).
2. The Chamaeleon I sample
In previous studies our group has assembledthe stellar properties and spectral energy dis-tribution (SED) of each Chamaeleon I memberand used continuum radiative transfer codes tomodel disk structures down to the substellarregime (Sz˝ucs et al. 2010; Mulders et al. 2012;Olofsson et al. 2013). Our modeling included op-3ical, 2MASS, Spitzer, WISE, and, when available,Herschel and mm photometry. We did not includeany spectroscopic data, e.g. Spitzer IRS spectra.Only objects displaying excess emission at morethan one wavelength were included in our ALMAsurvey. In this way we excluded all Class IIIobjects (Luhman et al. 2008). In addition, weremoved the few known Class 0 and I sources(Luhman et al. 2008; Belloche et al. 2011). Thesecriteria result in 93 objects with dust disks, mostlyClass II, but see later for sub-groups. Table 1includes their 2MASS designations, other com-monly used names, multiplicity information fromthe literature, and the spectral types (SpTy) fromLuhman (2007, 2008). This latter informationwas also used to set the exposure times (see Sec-tion 3). We note that our sample is not com-plete in the sub-stellar regime (SpTy later thanM6). For instance, the well known disk around theM7.75 brown dwarf Cha H α ≤
40 AU that are small enough to af-fect disk evolution (Kraus et al. 2012).The SEDs of 87 of our ALMA targets are clas-sified in Luhman et al. (2008) and Manoj et al.(2011) using the spectral slope α = d log( λF λ ) /d log λ between ∼ µ m (2MASS K-band photometry) and24 µ m ( Spitzer /MIPS photometry in the first con-tribution and
Spitzer /IRS spectroscopy in the sec-ond). As discussed in Manoj et al. (2011) thetwo SED classifications are in good agreement.Six of our ALMA targets were not observedwith Spitzer but all have WISE photometry at12 µ m (W3 channel, Cutri et al. 2012). We usethe following approach to classify them. First,we plot the de-reddened α − versus α − forall Chamaeleon I members that have 2MASS K-band, WISE 12 µ m, and MIPS 24 µ m photometry.From this plot we find that the two quantities The six unclassified targets are: J11160287-7624533,J11085367-7521359, J10561638-7630530, J11071181-7625501, J11175211-7629392, and J11004022-7619280. To de-redden the magnitudes we used the A J extinctionsprovided in Luhman (2007) and the Mathis (1990) red-dening law because all of our sources have low extinction,A J < . are well correlated and the best fit relationshipis: α − = 1 . ± . × α − + 0 . ± . α − from the measured α − for the 6 unclassifiedsources. The inferred α − spectral indices arebetween -1.7 and -0.9, all Class II SED follow-ing Manoj et al. (2011). The transitional disks(Class II/T) are identified as having a deficit offlux at wavelengths less than 8 µ m compared withthe Class II median and comparable or higher ex-cess emission beyond ∼ µ m following Kim et al.(2009) and Manoj et al. (2011). By excludingthe IRS Spitzer spectroscopy from our analysiswe missed the Class II/T disk around the M0 starSz 18, also known as T25 (Kim et al. 2009). Its in-frared excess is only pronounced beyond ∼ µ mand the source was outside the MIPS 24 µ m fieldof view (Luhman et al. 2008), thus appearing asa Class III source based on the available pho-tometry. In summary, our sample includes 3 flatspectra (FS), 82 Class II, and 8 Class II/T.As part of a parallel effort to simultaneouslyderive stellar parameters, extinction, and massaccretion rates, our group has obtained VLT X-Shooter spectra for 89 out of 93 of our ALMAChamaeleon I targets. The observations, data re-duction, and properties inferred from the VLTspectra are summarized in Manara et al. (2014,2016a,c). For eight sources, typically late Mdwarfs, these new spectra were either not ac-quired or lacked enough signal-to-noise (hereafter,S/N) to reliably derive stellar and accretion prop-erties, hence we adopt here the spectral typeclassification and stellar properties reported inLuhman (2007, 2008), see Table 1. As discussed inManara et al. (2016a,c) the difference between thenew and literature spectral type is in most casesless than a spectral subclass. The largest differ-ence occurs for the K-type stars and is thoughtto arise from the lack of good temperature diag-nostics in the low-resolution red spectra used inprevious studies for spectral classification. To derive stellar masses (and ages) we followedthe standard approach of comparing empirical ef-fective temperatures and stellar luminosities tothose predicted by pre-main sequence evolution-ary models. Effective temperatures and luminosi-ties for our ALMA Chamaeleon I sample are taken4 .8 3.7 3.6 3.5 3.4log T * (K)-3-2-101 l og L * ( L s un ) . M y r M y r M y r M y r M y r Fig. 1.— H-R diagram of our ALMAChamaeleon I sample (each source is representedby an empty circle). The non-magnetic evolution-ary tracks from Feiden (2016) are plotted for ef-fective temperatures greater than 3,700 K (SpTyM1 and earlier) and masses greater than 0.5 M ⊙ .For effective temperatures lower than 4,200 K andmasses lower than 0.5 M ⊙ we plot the evolutionarytracks from Baraffe et al. (2015). Note the similar-ity of the two sets of isochrones in the overlappingeffective temperature region for ages ≥ ⊙ down to the substellarregime . The Feiden (2016) tracks cover from 0.09 to 5.7 M ⊙ while Following Andrews et al. (2013), we adopt aBayesian inference approach to assign a stellarmass, an age, and associated uncertainties to eachof our ALMA targets. The first step in this ap-proach is to interpolate the Baraffe et al. (2015)and Feiden (2016) models on a common, finelysampled, age grid. Based on the Chamaeleon IH-R diagram in Figure 1, we include the earli-est isochrones at 0.5 Myr through to 50 Myr-oldisochrones with a step of 0.01 in log scale. Stel-lar masses are also sampled with the same spac-ing in log scale. We use the Baraffe et al. (2015)tracks for all objects with effective temperatures ≤ ∼ T ∗ and luminosity L ∗ in the H-R diagram,we compute a conditional likelihood function, as-suming uniform priors on the model parameters,as: F ( ˆ T , ˆ L | T ∗ , L ∗ ) = 12 πσ T ∗ σ L ∗ exp ( − . × [ ( T ∗ − ˆ T ) σ T ∗ + ( L ∗ − ˆ L ) σ L ∗ ])(1) where ˆ T and ˆ L are the model grid temperaturesand luminosities, while σ T ∗ and σ L ∗ are the un-certainties associated with T ∗ and L ∗ . The un-certainty in log( T ∗ ) is assumed to be 0.02 forSpTy earlier than M3 and 0.01 for later SpTywhile the uncertainty in log( L ∗ ) is taken to be0.1 (see Manara et al. 2016c). We then integrate F ( ˆ T , ˆ L | T ∗ , L ∗ ) over the age and mass covered bythe model grids and obtain two marginal proba-bility density functions, see the curves in Figure 2.The best fit mass and age are the peaks of thesefunctions and the uncertainties are the values thatencompass 68% of the area under the functions.This approach could be applied to all but 9sources for which age estimates are found to be atthe boundary of our grid. For the four sources forwhich our method identifies the youngest 0.5 Myrisochrone and appear over-luminous in the H-R the Baraffe et al. (2015) from 0.015 to 1.4 M ⊙ , hence theyare the only ones available in the sub-stellar regime. , we choose this isochrone and computethe stellar mass based solely on the stellar effectivetemperature. For the other five sources for whichour method gives the oldest isochrone of 50 Myrwe take the median age of our Chamaeleon Isources and again compute stellar masses basedsolely on stellar effective temperatures. Threeout of these five ’old’ sources (J10533978-7712338,J11111083-7641574, and J11160287-7624533) haveSED and/or spatially resolved imagery suggestingthat the central star is surrounded by an edge-on disk (Luhman 2007; Robberto et al. 212), thusexplaining why they appear under-luminous in theH-R diagram. We note that our ALMA sample hasa median age of 3.5 Myr, slightly older than thepreviously computed median age (Luhman 2007).The resulting masses and their uncertainties, whenavailable, are reported in the last column of Ta-ble 1. -1.0 -0.5 0.0log M * (M sun )6.06.57.0 l og t ( y r) J11044258-7741571(0.24 M sun - 3.3 Myr)
Fig. 2.— Example of the likelihood function usedto estimate stellar masses and ages. The best fitparameters for J11044258-7741571 are listed onthe top right of the panel. The 68% confidenceintervals are the red regions of the marginal prob-ability density functions. These regions are calcu-lated from the cumulative integral such that thearea above and below the best fit parameter areeach 0.34. J11065906-7718535, J11094260-7725578, J11105597-7645325, and J11183572-7935548 J10533978-7712338, J11063945-7736052, J11082570-7716396, J11111083-7641574, and J11160287-7624533
3. Observations and data reduction
Our observations were carried out as part ofthe ALMA Cycle 2 campaign on 2014 May 1-3UTC (54 sources) and on 2015 May 18-19 UTC (39sources). The 2014 observations included all starswith SpTy from Luhman equal or earlier than M3(hereafter,
Hot sample) while in 2015 we observedthe remaining later SpTy sources (hereafter,
Cool sample).All observations were obtained in Band 7 witha spatial resolution of 0 . ′′ × . ′′ , see Table 2 fordetails on the number of 12m antennas, baselines,and calibrators. Each science block (SB), com-prising either all Hot or Cool sources plus any cal-ibrator, was executed twice. The correlator wasconfigured to record dual polarization with threecontinuum basebands of 5.6 GHz aggregated band-width centered at 330.0, 341.1, and 343.0 GHz foran average frequency of 338 GHz (887 µ m). Thefourth baseband was devoted to the serendipitousdetection of gas lines and was split in two sub-bands of 0.1 GHz each centered at 329.3 GHz and330.6 GHz to cover the C O (3-2) and the CO(3-2) transitions. This paper focuses on the con-tinuum data, the reduction and analysis of the COdata will be presented in a separate contribution(Long et al. in prep.). Exposure times for the
Hot sample were set to achieve a 1 σ rms of 1 mJy/beamin the aggregated continuum bandwidth, while forthe Cool sample we required 0.2 mJy/beam. As acomparison previous single dish mm observationsof the Chamaeleon I star-forming region had 1 σ sensitivities greater than 10 mJy over a beam of ∼ ′′ (Henning et al. 1993; Belloche et al. 2011).The ALMA data were calibrated using theCASA software package. The initial reductionscripts were provided by the North AmericanALMA Science Center and included phase, band-pass, and flux calibration. We re-ran the scriptsusing CASA 4.3.1. We used Pallas as the fluxcalibrator for the Hot sample SBs, Ganymede forthe first
Cool sample SB, and the quasar J1107-448 for the second
Cool sample SB. The flux scalewas within 5% and 8% of the two SBs for the
Hot and
Cool samples respectively. For both samples,we used the average of the two SB fluxes in thecalibration script. In the analysis that follows weadopt a conservative 1 σ uncertainty of 10% on theabsolute flux scale.6irty continuum images were created from thecalibrated visibilities using CASA v4.4 and nat-ural weighting and by averaging the three con-tinuum basebands (see Figures 16 to 21 in theelectronic version of the paper). We computedthe rms of each image in a region outside the ex-pected target location and found a median of 0.99and 0.23 mJy/beam for the Hot and
Cool samples,very close to the requested sensitivities. We alsocomputed an initial flux density at the target lo-cation by integrating within the 3 rms closed con-tour. This flux density, in combination with theimage rms and visual inspection, was used to de-cide if a source is detected. With this approachwe classified 45/54
Hot and 21/39
Cool targets asdetected.We also identified ten bright
Hot and two bright
Cool sources with S/N ranging from 36 to 100and rms larger than 2 times the median rms thatwould benefit from self-calibration. For these 12sources we followed the steps suggested by theNorth American ALMA Science Center for thebrightest of our targets, J11100010-7634578. Fromeach of the 12 measurement sets we produced animage with Briggs robust weighting parameter ofzero and cell size 0 . ′′ Hot or Cold sample, and saved the model in the measure-ment set header. We then calibrated the phasesusing the model data column, applied the new cal-ibration to the measurement set, and produced anew image from the better-calibrated data. We re-peated the cycle of cleaning and phase calibrationa second time starting from the new image and byapplying a deeper cleaning, down to about 3 timesthe median rms of the
Hot or Cold sample. Theimage produced in this second cycle was cleaned athird time, with phases and amplitudes calibratedand applied to the original measurement set. Withthis approach we found that the final image rmsalways improved, reaching the median value of ∼ Hot and ∼ The 10
Hot sources that require self-calibration areJ10581677-7717170, J10590699-7701404, J11022491-7733357, J11040909-7627193, J11074366-7739411,J11080297-7738425, J11081509-7733531, J11092379-7623207, J11094742-7726290, and J11100010-7634578while the two
Cold sources are J11004022-7619280 andJ11062554-7633418. the
Cold samples even for the brightest of oursources, J11100010-7634578, whose initial imagerms was ∼
24 mJy/beam. The 12 phase and am-plitude calibrated measurement sets are used inthe following steps to compute the source param-eters.
4. Results
To compute the flux densities and to deter-mine whether the emission is spatially resolvedwe rely on the visibility data as, e.g. discussedin Carpenter et al. (2014). First, we fit all of our66 detections with an elliptical Gaussian using the uvmodelfit task in CASA. This model has 6 free pa-rameters: the integrated flux density; the offsetsin right ascension and declination from the phasecenter; the FWHM; the aspect ratio; and the posi-tion angle. With the underlying assumption thatthe model describes well the data, we scale the un-certainties on the fitted parameters by the factorneeded to produce a reduced χ of 1. If the ra-tio of the FWHM to its uncertainty is less than 2,which happens for 32 sources, we also fit the vis-ibility data with a point source model which hasonly 3 free parameters: the integrated flux densityand the offsets in right ascension and declinationfrom the phase center. For 25 out of 32 sources wefind that the reduced χ of the point source modelis less than that of the Gaussian model, hence weadopt the point source fits. Even for the 7 sourceswhere the reduced χ of the Gaussian model islower than that of a point source model, we adoptthe point source fits because the difference in themodels’ reduced χ is much smaller than the un-certainty on their values, which is approximately p /N for the over 7,000 visibility points that arefitted. Finally, for the 27 sources that are not de-tected we also fit a point source model keeping theoffsets in right ascension and declination fixed to-0 . ′′ . ′′
0, respectively, the median values fromthe sources that are detected.To visualize the goodness of the fits we comparethe best fit model (solid line) to the real compo-nent of the observed visibilities (filled circles) asa function of projected baseline length (UV dis-tance), see Figure 3 as an example, all other figuresare available in the electronic version. In thesefigures all visibilities are re-centered to the contin-uum centroids found with uvmodelfit , each visibil-7ty point is the average of the visibilities within a30 k λ range, and the error bars are the standarddeviation divided by √ N − N is the num-ber of visibility points in the same range. Abouthalf of the detected sources have spatially resolvedemission, as evidenced by visibilities that declinein amplitude with increasing UV distance. Amongthem, J10563044-7711393 and J10581677-7717170have resolved dust cavities, hence the Gaussian fitdiscussed above does not provide a good estimatefor the source flux density. For these two sourceswe compute flux densities within the 3 σ contour inthe deconvolve image , see Figure 4. J10581677-7717170 is a known transition disk with an esti-mated dust cavity of ∼
30 AU in radius (Kim et al.2009). On the contrary, J10563044-7711393 hasnot been classified as a transition disk based onits infrared photometry but a
Spitzer /IRS spec-trum could not be extracted for this source dueto its faintness (Manoj et al. 2011). The radius ofboth cavities is ∼
45 AU as measured from the im-ages and from the location of the first null in thevisibility plot (see eq. A9 in Hughes et al. 2007).Overall, we have identified two sources withdust disk cavities, 32 sources whose mm emis-sion is resolved (elliptical Gaussian model), 32sources with unresolved mm emission (pointsource model), and 27 sources with too faint orabsent mm emission to be detected in our survey.Among the resolved mm sources 23 belong to the
Hot sample and 9 to the
Cool sample implyingthat ∼
51% and 39% of the detected sources areresolved in the two samples respectively. Table 3summarizes the measured continuum flux densi-ties ( F ν ) and uncertainties, offsets from the phasecenter in right ascension and declination for thedetected sources (∆ α and ∆ δ ), and FWHMs forthe resolved mm sources. In the analysis thatfollows we calculate upper limits for sources thatare not detected as 3 times the uncertainty on F ν which is also reported in Table 3.Flux densities and upper limits as a func-tion of stellar masses are shown in Figure 5 ina log-log plot, circles for detections and down-ward pointing triangles for non-detections. Notethat the SED-identified transition disks are not We remind the reader that J10581677-7717170 was one ofthe sources that required self-calibration (see Section 3),hence the flux density is computed on the final phase andamplitude calibrated image. among the brightest mm disks. Two of them,J11071330-7743498 (SpTy M3.5) and J11124268-7722230 (SpTy G8), remain undetected at oursensitivity. However, the latter source has alsoa ∼ M ⊙ companion at a projected distanceof 38 AU (Daemgen et al. 2013) that might havetidally truncated the disk of the primary, leadingto a lower than average mm flux. The disks aroundJ11100704-7629376 and J11103801-7732399, twoK-type stars with companions at ∼
20 AU and27 AU distance respectively, also appear fainterthan disks around stars of similar stellar mass andmight have been truncated. Stars in Taurus withcompanions at tens of AU have also fainter disksthan expected for their mass (Harris et al. 2012).At the other extreme, the star J11100010-7634578has a companion at 65 mas and the brightest mmdisk, in this case a circumbinary disk. Circumbi-nary disks are also found to be among the brightestmm disks in Taurus (Harris et al. 2012). -1.5 -1.0 -0.5 0.0 0.5log M * (M sun )-10123 l og F mm ( m J y ) Fig. 5.— Flux densities ( F mm ) as a function ofstellar masses ( M ∗ ). Circles are sources with de-tected mm flux while downward pointing trian-gles represent non detections. Sources markedwith an ’X’ are FS disks, a green dot within themain symbol denotes Class II/T SEDs while ared color denotes ’close’ binaries (projected sep-aration ≤
40 AU). The dashed line gives the bestfit relationship using a Bayesian approach that ac-counts for censored data. The median errorbar inlog( M ∗ ) and log( F mm ) is shown in the upper leftcorner of the plot and corresponds to ± . ± .
02 dex respectively.Figure 5 demonstrates that mm fluxes have8 spread of more than a dex at a given stel-lar mass, part of which, as mentioned above,may be attributed to stellar multiplicity. Inspite of the spread, flux densities are stronglycorrelated with stellar mass. This trend is notunique to the Chamaeleon I star-forming re-gion (Andrews et al. 2013; Mohanty et al. 2013;Barenfeld et al. 2016; Ansdell et al. 2016). As-suming a linear relationship in the log-log plane,we can determine the best fit using the Bayesianmethod developed by Kelly (2007) that properlyaccounts for the measurement uncertainties, non-detections, and intrinsic scatter. This Bayesianmethod assumes Gaussian measurement errors,hence we have adopted the full range of the stel-lar mass uncertainty, covering 68% of the areaunder the marginal probability density function,and divided it by two as the error on each stel-lar mass. For the 10 sources where we hadto fix the isochrone we use the median uncer-tainty in log( M ∗ ) of ± . F mm /mJy)=1.9( ± . × log( M ∗ /M ⊙ )+1.6( ± . σ confidence intervals in stel-lar mass are often not symmetric around the bestvalue, they are still small enough that the as-sumed Gaussian distribution does not affect theBayesian fit. We tested that only when the er-ror on log( M ∗ ) becomes larger than 2.5 times themedian value, the best fit relation is no longerconsistent with the one reported here and the in-ferred slope steepens. This means that the intrin-sic scatter in mm fluxes drives the best fit givenour measurement errors in log( F mm ) and log( M ∗ ).This is also confirmed by other regression meth-ods that do not account for measurement errorsbut recover the same slope and intercept of theBayesian approach within the quoted uncertainties(see Appendix A). To further test the robustnessof this relation, we also compute stellar massesusing the effective temperatures and luminositiesin Luhman (2007) and find the following best fitlog( F mm /mJy)=2.1( ± . × log( M ∗ /M ⊙ )+1.7( ± . F mm − M ∗ relationin Chamaeleon I is much steeper than linear andthe mm flux scales almost with the square of thestellar mass.The 1.9-2.1( ± F mm − M ∗ slope in Taurus is 1.5 ± .
2, lower butstill marginally consistent with the one we find inChamaeleon I. We caution that lower values canalso result from low sensitivity at the lower stellarmass end. As a test we degrade our sensitivityto the typical 850 µ m 1 σ sensitivity of ∼ Hot and
Cool sam-ples in Chamaeleon I. The best fit slope of thisdegraded dataset is only 1.3 ± − stellar mass dependence.
5. Dust disk masses
Dust disk emission at millimeter wavelengths ismostly optically thin, hence continuum flux den-sities can be used to estimate dust disk masses(e.g. Beckwith et al. 1990). We adopt the sim-plified approach commonly used in the field (e.g.Natta et al. 2000) and assume isothermal and op-tically thin emission to compute disk masses asfollows: log M dust = log F ν +2 log d − log κ ν − log B ν ( T dust ) (2) where F ν is the flux density at 338 GHz (887 µ m), d is the distance (160 pc for Chamaeleon I,Luhman 2008), κ ν is the dust opacity, and B ν ( T dust ) is the Planck function at the tempera-ture T dust . We adopt a dust opacity of 2.3 cm g − at 230 GHz with a frequency dependence of ν . ,the same as in Andrews et al. (2013) for Tau-rus and Carpenter et al. (2014) for Upper Sco.The average dust temperature responsible forthe mm emission ( T dust ) is poorly constrained.Andrews et al. (2013) performed 2D continuumradiative transfer calculations for a representativegrid of disk models and proposed the followingscaling relation for stars in the 0.1 to 100 L ⊙ lu-minosity range: T dust = 25K × ( L ∗ /L ⊙ ) . . How-ever, van der Plas et al. (2016) and Hendler et al.(2016) show that a weaker T dust − L ∗ dependencecan be reached by adjusting some of the disk9nput parameters used in Andrews et al. (2013),most notably the outer disk radius. In particular,Hendler et al. (2016) find that if lower mass starshave smaller dust disks then the T dust − L ∗ rela-tion flattens out, becoming almost independent ofstellar luminosity if the dust disk radius scales lin-early with stellar mass. As discussed in Section 4,the percentage of resolved disks is higher in the Hot than in the
Cool sample, perhaps hinting onsmaller dust disks around lower mass stars. How-ever, this could be also due to low S/N at the lowend of the stellar mass spectrum. Because a S/Non the continuum ≥
30 is needed to properly esti-mate dust disk sizes (Tazzari et al. 2016), deeperALMA observations are needed to pin down if andhow the disk size scales with stellar mass.Given the uncertainty in the T dust − L ∗ re-lation, we compute dust disk masses for twoextreme cases: a) a constant T dust fixed to20 K to directly compare our results to recentALMA surveys of other star-forming regions(e.g. Lupus, Ansdell et al. 2016) and b) a vary-ing T dust with stellar luminosity as proposedby Andrews et al. (2013). Several studies haveapplied a plateau of ∼
10 K to the outer disktemperature (Mohanty et al. 2013; Ricci et al.2014; Testi et al. 2016), given that this is thevalue reached by dust grains heated by the in-terstellar radiation field in giant molecular clouds(Mathis et al. 1983). We have decided not to ap-ply this plateau in our study for two reasons.First, continuum radiative transfer models showthat the interstellar radiation field has a negligibleeffect on the dust disk temperature and outer diskscan be colder than 10 K (van der Plas et al. 2016;Hendler et al. 2016). Second, Guilloteau et al.(2016) note that the edge-on disk of the FlyingSaucer absorbs radiation from CO backgroundclouds and infer very low dust temperatures of 5-7 K at ∼
100 AU in this disk. The lowest luminositysource in our Chamaeleon I sample, J11082570-7716396 with L bol = 0 . L ⊙ , has a T dust of4.8 K with our prescription. Such a value is below10 K but still consistent with the lower temper-atures found in disk models and in the FlyingSaucer disk.Figure 6 summarizes our findings with blackand orange symbols for case a) and b) respec-tively. A lower T dust for lower luminosity (typi-cally lower mass) objects results in a lower Planck function hence in a higher dust mass estimate.When applying to these two extreme cases thesame Bayesian approach described in Section 4 wefind the following best fits:log( M dust /M ⊕ )=1.9( ± . × log( M ∗ /M ⊙ )+1.1( ± . T dust andlog( M dust /M ⊕ )=1.3( ± . × log( M ∗ /M ⊙ )+1.1( ± . T dust decreasing with stellar luminosity. Thestandard deviation (hereafter, dispersion) aboutthe regression is 0.8 ± . M dust − M ∗ rela-tion is the same as that of the F mm – M ∗ relationfor the assumption of constant temperature whileit is flatter when the temperature decreases withstellar luminosity. Importantly, even the flatter re-lation is steeper than the linear one inferred frompre-ALMA disk surveys (Andrews et al. 2013;Mohanty et al. 2013) and from ALMA surveyswith a limited coverage of stellar masses (e.g.Carpenter et al. 2014 and Section 4). Most likelythe M dust − M ∗ relation is steeper than 1.3( ± . M dust − M ∗ relation will requiremeasuring how dust disk sizes scale with stellarmasses.
6. Discussion6.1. The disk-stellar mass scaling relationin nearby regions
The four nearby regions of Taurus ( d = 140 pc ,age ∼ d = 140 pc ,age ∼ d =160 pc , age ∼ d = 145, age ∼ µ m),decreases from ∼
65% in Taurus to ∼
50% in Lu-pus and Chamaeleon I and drops to only ∼
15% inUpper Sco (Ribas et al. 2014). Over the same agerange there is tentative evidence for an increase in10 * (M sun )-2-1012 l og M du s t ( M ea r t h ) Fig. 6.— Dust disk masses ( M dust ) as a function ofstellar masses ( M ∗ ). Black symbols are for a con-stant dust disk temperature of 20 K while orangesymbols use the T dust − L ∗ scaling relation pro-posed by Andrews et al. (2013). The dashed linesare the best fits for these two cases. Note that thescaling relation proposed by Andrews et al. (2013)flattens the disk-stellar mass relation.the frequency of Class II/T SEDs relative to thetotal disk population, just a few % at ages ≤ ∼
10% at older ages (Espaillat et al. 2014).These observations trace the depletion/dispersalof small micron-sized grains within a few AU fromthe star and support a scenario in which proto-planetary material is cleared from inside out (seeAlexander et al. 2014 for a recent review on diskdispersal timescales and mechanisms). Millime-ter observations probe the population of largermm/cm sized-grains at radial distances &
10 AU.Thanks to the exquisite sensitivity of ALMA thereare now millimeter surveys that parallel those atinfrared wavelengths in sample size, thus enablingtesting if significant evolution occurs in the outerdisk over the ∼ Hot and
Cool sam-ples, respectively. To compare their M dust − M ∗ relations we re-analyze all the datasets in a self- consistent manner: we re-compute all the stellarmasses as discussed in Section 2.1 using the sameevolutionary tracks and then apply the approachdescribed in Section 5 to account for mm detec-tions and upper limits. The first step is importantbecause, as pointed out in Andrews et al. (2013),different evolutionary tracks can result in slightlydifferent M dust − M ∗ relations. We note thatthe adopted spectral type-effective temperaturescale is essentially the same in all 4 regions witha small difference of only ∼
10 K in the M7-M8range where there are only a few, if any, sourcesin each region. For Upper Sco we only considerdisks classified as ’Full’ and ’Transitional’ in Ta-ble 1 of Barenfeld et al. (2016), equivalent to theClass II and II/T SEDs in Chamaeleon I. Moreevolved/debris disks, Class III-type, are not in-cluded in the Taurus, Lupus, and Chamaeleon Imillimeter surveys. These disks most likelyrepresent a different evolutionary stage whenmost of the gas disk has been dispersed (e.g.Pascucci et al. 2006) and the millimeter emissionarises from second generation dust produced inthe collision of larger asteroid-size bodies. The re-sulting M dust − M ∗ relations for these four regionsare summarized in Table 4 and plotted in Fig-ure 7 for the case of constant dust temperature.This case is essentially equivalent to comparingsub-millimeter luminosities as a function of M ∗ indifferent star-forming regions (see also Section 5).Taurus has the shallowest M dust − M ∗ rela-tion among these regions. However, as discussedin Section 4, the lower sensitivity of the sur-vey can account for the apparent difference withChamaeleon I. Lupus has the same slope but ap-pears to have slightly more massive disks thanTaurus and Chamaeleon I. However, given the few ∼ M ⊙ stars in Lupus the intercept is less welldetermined than in Taurus and Chamaeleon I. In-deed, adding the 20 obscured Lupus sources byrandomly assigning a stellar mass reduces the in-tercept by 0.3 making the M dust − M ∗ relation ofLupus the same as the one of Taurus (Ansdell pri-vate communication) and Chamaeleon I. Hence,we conclude that the same M dust − M ∗ relationis shared by star-forming regions that are 1-3 Myrold. We also note that the relation is steeper thanlinear. As already pointed out in Barenfeld et al.(2016) and Ansdell et al. (2016) the disk mass dis-tribution in the ∼ with the cendiff commandin the NADA R package, we find that the diskmass distribution in Chamaeleon I is indistin-guishable from that of Taurus (p=52%) and Lu-pus (p=8%) but different from that of Upper Sco(p=0.0001%) within the same ∼ . − . M ⊙ stellar-mass range. The mean dust disk mass is ∼ M ⊕ for Chamaeleon I but only ∼ M ⊕ forUpper Sco in the assumption of constant dust tem-perature and with our value for the dust opac-ity. Table 4 shows that the M dust − M ∗ relation isalso steeper in Upper Sco than in the other threeyounger regions (see also Fig. 6 in Ansdell et al.2016). Based on the inferred relations, it appearsthat disks around 0.5 M ⊙ have depleted their dustdisk mass in mm grains by a factor of 2.5 by ∼
10 Myr, while disks around 0.1 M ⊙ by an evenlarger factor of 5. To further corroborate ourfinding, we perform the same Wilcoxon test onthe disk mass distribution for stars more and lessmassive than ∼ . M ⊙ . The probability thatChamaeleon I and Upper Sco have the same diskmass distribution is as high as 52% for > . M ⊙ stars while it is only 0.02% for the lower stellarmass bin with average masses that are a factorof 2 lower in Upper Sco than in Chamaeleon I.With lowering the stellar mass value to create thetwo disk mass samples, the probability that thehigh-stellar mass bin in Chamaeleon I and Up-per Sco have the same disk mass distribution alsodecreases reaching 1% at 0 . M ⊙ . This demon-strates that differences in the two distributions aremore pronounced toward the lower stellar massend, well in line with a steeper M dust − M ∗ rela-tion in Upper Sco than in Chamaeleon I.Finally, it is interesting to note that the disper-sion around the M dust − M ∗ relations is very sim-ilar in the four regions and amounts to ∼ The null hypothesis is that two groups have the same dis-tribution, p denotes the probability to reject the null hy-pothesis. Censored data are included in cendiff. flecting a range of initial conditions which might,at least in part, account for the diversity of plan-etary systems.
In the previous Section we showed that the 1-3 Myr-old star-forming regions of Taurus, Lupus,and Chamaeleon I share the same M dust − M ∗ re-lation while the older Upper Sco association hasa steeper relation. What is the physical processleading to a steepening of the M dust − M ∗ relationwith time? One possibility would be to invoke astellar-mass dependent conversion to larger grains,in that disks around lower mass stars would con-vert more mm grains into larger cm grains thatgo undetected. Alternatively, the higher depletionof mm-sized grains toward lower-mass stars couldresult from more efficient inward drift, i.e. mm-sized grains would be still orbiting the star but inthe inner and not in the outer disk where opticaldepth effects might hide them.To test these scenarios we use the Lagrangiancode developed by Krijt et al. (2016) and simulatethe evolution of dust disk grains subject to: a)growth and fragmentation; b) growth and radialdrift; and c) growth, radial drift, and fragmenta-tion. In all models, the dust disk initially extendsfrom 2 to 200 AU with a power-law surface den-sity with index -1.5, the dust-to-gas mass ratio is0.01, the total mass equals 1% of the central starmass, the turbulence is characterized by α = 0 . M ⊙ (y-axis) and the other 2 M ⊙ (x-axis). Inthe left panel the dust disk temperature is assumedto be fixed to 20 K while in the right panel itvaries radially and equals the gas disk tempera-ture which is prescribed to decrease with radiusand be higher around high-mass stars: T gas =280 K × ( r/AU ) − . ( M ∗ /M ⊙ ) . . While the re-sulting mm fluxes depend on the assumed dustdisk temperature, as highlighted in Section 5, theevolutionary behavior is the same. More specifi-12ally, growth and fragmentation (red dashed lineand symbols) do not change the initial flux ra-tio of the two disks, hence cannot explain thesteepening of the M dust − M ∗ relation with time.Growth and drift (light blue dot-dashed line andsymbols) are faster in denser disks around higher-mass stars, hence these disks are depleted fasterof mm grains and become mm faint sooner thandisks around lower mass stars. This is oppositeto what is observed. Finally, the more realisticcase of growth, radial drift, and fragmentation(black dotted line and symbols) shows a behav-ior consistent with the observations, in that thedisk around the 0.2 M ⊙ reduces its mm flux fasterthan the disk around the 2 M ⊙ star. This is be-cause the timescale on which radial drift removesthe largest grains is shorter around low-mass stars.As a result, the disk around the 2 M ⊙ star canremain mm bright longer. To first order thetimescale over which dust is removed is the inverseof the Stokes number ( St ) of the largest grainswhich, in the Epstein regime, scales as St − ∝ α ( c s /v frag ) ∝ ( M ∗ ) . in the fragmentation-limited case and as St − ∝ ( c s /v K ) ∝ ( M ∗ ) − . in the drift-limited case (Birnstiel et al. 2012).Thus, dust removal is faster around lower-massstars only in the fragmentation-limited case. Forthe specific models shown in Figure 8, the max-imum grain size is < ∼
50 AUaround the 2 M ⊙ star and outside of ∼
15 AUaround the 0.2 M ⊙ star.In summary, the comparison between modelsand observations suggests that the maximum grainsize in the outer disk is fragmentation-limited,rather than drift-limited. As already pointed outin the literature (e.g. Pinilla et al. 2013), a re-duced drift efficiency, perhaps caused by radialpressure bumps, is necessary in all models tomatch the observed lifetime of disks at mm wave-lengths.The scenarios discussed above can be testedwith future millimeter observations. If graingrowth from mm to cm in size is responsible forthe steepening of the M dust − M ∗ relation withtime (Barenfeld et al. 2016), we should expect astellar-mass- and time-dependent power law in-dex β of the dust opacity. More specifically, olderdisks around lower-mass stars should have a lower β than disks around younger higher-mass stars. Adependence of β with stellar mass is not seen for T d =20K mm (t)(mJy) for 2 M sun F mm ( t )( m J y ) f o r . M s un Growth+fragmentationGrowth+driftGrowth+fragmentation+driftt~0Myrt=0.1Myrt=0.5Myrt=1Myr T d =T gas
10 100F mm (t)(mJy) for 2 M sun Growth+fragmentationGrowth+driftGrowth+fragmentation+driftt~0Myrt=0.1Myrt=0.5Myrt=1Myr
Fig. 8.— Evolution of the 850 µ m flux densityfor a disk around a 2 M ⊙ star (x-axis) and a diskaround a 0.2 M ⊙ star (y-axis). For both stars,the initial disk mass is set equal to 1% of thestellar mass. The left panel assumes a constantdust temperature through the disk while the rightpanel a radially decreasing dust temperature setequal to the gas temperature, see text for details.The growth+fragmentation+drift case can qual-itatively explain the observed steepening in the M dust − M ∗ relation with time.Taurus disks around ∼ . − . M ⊙ stars and fora few disks around sub-stellar objects (Ricci et al.2010, 2014), but it should be tested if it arises overa statistically significant sample of disks spanninga broad range in stellar masses and at later evo-lutionary times. If instead the maximum grainsize is fragmentation limited as we suggest, the β dependence with stellar mass would be oppositebecause higher-mass stars would have, on aver-age, larger grains in their disks than lower-massstars. In addition, there would not be a time-dependence because the fragmentation-limitedregime is insensitive to the surface density evo-lution (Birnstiel et al. 2012).Another prediction of this scenario is thatdisks around lower-mass stars would be smallerin size than disks around higher-mass stars. Pre-vious work has pointed out that dust disk radiicorrelate positively with mm fluxes for T Tauristars (Isella et al. 2009; Andrews et al. 2010;Guilloteau et al. 2011) but the scatter is largeand what is really needed is to demonstrate acorrelation with stellar mass. In the sub-stellar13egime, there are only five disks whose dust diskradii at mm wavelengths can be reliably inferred.The three in Taurus are rather large (50 −
100 AU,Ricci et al. 2014) while the two in ρ Oph are muchsmaller ( <
25 AU, Testi et al. 2016). A system-atic ALMA survey with high spatial resolutionand sufficient S/N is missing.Finally, we would like to comment on the find-ing of a longer disk lifetime around low-mass starsbased on infrared observations. Carpenter et al.(2006) found that the fraction of optically thickdisks in Upper Sco is higher for K+M dwarfs( ∼ . − M ⊙ ) than for earlier spectral type stars.Expanding upon this, Bayo et al. (2012) reporteda higher fraction of optical thick disks around starsless massive than ∼ . M ⊙ in the 5-12 Myr-oldCollinder 69 cluster. These results demonstratethat the inner disk of low-mass stars is not de-pleted of micron-sized grains but do not place anyconstraint on the outer disk. On the contrary, theALMA observations presented in this paper tracethe population of mm grains in the outer disk. In-ward drifting mm grains that collide and replenishthe inner disk of smaller sub-micron grains mightexplain both the apparent lack of mm grains inthe outer disk and the longer lived optically thickdisks around low-mass stars. In the classical paradigm of disk evolution, theaccretion of disk gas onto the star is thought toresult from the coupling of the stellar magneticfield with ions in so-called active layers of thedisk (magneto-rotational instability model, e.g.,Gammie 1996). However, in this standard picturethe accretion rate is independent from the mass ofthe central star. Hartmann et al. (2006) showedthat a weak linear dependence can be recoveredwhen including stellar irradiation as a disk heat-ing mechanism in addition to viscous accretion.Further steepening the relation would be possibleif disks around very low-mass stars are less mas-sive, fully magnetically active, and as such havingviscously evolved substantially (Hartmann et al.2006). Alternatively, Ercolano et al. (2014) haveproposed that the ˙ M − M ∗ relation is flatter forspectral types earlier than M due to a specificdisk dispersal mechanism, star-driven X-ray pho-toevaporation. Looking at the complete stellar mass range, Dullemond et al. (2006) have shownthat a steep ˙ M ∼ ( M ∗ ) . relation arises naturallyif the centrifugal radius of the parent core is in-dependent of the mass of the core and the spreadin ˙ M at any stellar mass would reflect an initialdistribution of core rotation rates. In all cases, ˙ M should scale linearly with the disk mass, implyingthat the ˙ M − M ∗ relation should be the same asthe disk mass-stellar mass relation.The ˙ M − M ∗ relation has been determinedfor Taurus, Lupus, and Chamaeleon I whileit is not available for Upper Sco. For thesethree young regions the relation is close toa power law of two: ˙ M ∝ ( M ∗ ) . ± . forTaurus (Herczeg & Hillenbrand 2008); ˙ M ∝ ( M ∗ ) . ± . for Lupus (Alcal´a et al. 2014); and˙ M ∝ ( M ∗ ) . ± . for Chamaeleon I (Manara et al.2016a). While we do not have total (gas+dust)disk masses, it is interesting to note that M dust displays the same steep relation with M ∗ in thesethree regions if the average dust temperature isconstant, while the relation is slightly shallowerfor a dust temperature scaling with stellar lumi-nosity (see Table 4). A more robust way to testthe basic prediction of a linear relation between ˙ M and disk mass is to directly relate these quantitiesfor the same large sample of objects belonging tothe same star-forming region. This could be re-cently achieved for the Lupus clouds. Assuming aconstant dust temperature to convert millimeterfluxes into dust disk masses, Manara et al. (2016b)showed that ˙ M and M dust are correlated in Lupusin a way that is compatible with viscous evolutionmodels. Interestingly, the gas disk mass inferredfrom CO isotopologues does not show a similarcorrelation with ˙ M . This may be the result ofCO not being a good tracer of the total gas diskmass because carbon can be sequestered in morecomplex molecules on icy grains (e.g. Bergin et al.2014) and/or because of complex isotope-selectiveprocesses (Miotello et al. 2014, 2016). It would beinteresting to extend such studies to other regions,especially Upper Sco, where the M dust − M ∗ rela-tion is even steeper than in younger star-formingregions. Given the relevance of disk masses to planet for-mation models, we discuss here the uncertainties14n estimating total disk masses, whether disks ap-pear to be close to being gravitationally unstable,and how dust disk masses compare to the amountof solids locked into exoplanets.As discussed in Section 5 the average disk tem-perature tracing mm emission affects the absolutevalue of the dust disk mass, as well as the disk-stellar mass scaling relation, with cooler temper-atures leading to higher disk mass estimates. Forthe two temperature relations adopted here theaverage difference in dust disk masses amountsto a factor of ∼
3. An even larger uncertaintyis introduced by the dust opacity which dependson grain composition as well as size distribution(see e.g. Testi et al. 2014), which are both stillpoorly constrained. Silicates constitute the mainsource of opacity at ∼ µ m (Pollack et al.1994; Henning & Stognienko 1996; Semenov et al.2003). Even assuming a fixed dust composition,the 1.3 mm opacity can vary by a factor of ∼ / gdust opacity we have adopted is close to the onefor a grain size distribution extending to 1 cm.This means that if the true grain size distribu-tion were truncated at 1 mm the dust disk masseswould be a factor of 4 lower than those we report.Given that our choice of dust opacity maximizesdust disk masses over the range of grain sizes ex-pected/detectable in the outer disk, we will con-tinue our discussion adopting the dust disk massesobtained with a constant dust temperature which,instead, minimizes the disk masses toward lower-mass stars.Figure 9 shows the distribution of M disk /M ∗ where M disk is simply the dust disk mass mul-tiplied by the ISM gas-to-dust ratio of 100. Al-though recent gas mass estimates using rotationallines from CO isotopologues have claimed gas-to-dust ratios well below the ISM value in youngdisks (Williams & Best 2014; Ansdell et al. 2016),detailed physico-chemical disk models need tobe carried out to properly account for isotope-selective processes (Miotello et al. 2014, 2016). In addition, carbon can be extracted from CO viareactions with He + and form hydrocarbons thatfreeze-out, thus reducing the CO abundance inthe disk atmosphere (Favre et al. 2013). Indeed,the only disk with an independent mass estimate,using the HD (J=1-0) transition at far-infraredwavelengths, has a gas-to-dust mass ratio consis-tent with the ISM value and confirms that massesusing CO isotopologues can be off by up to afactor of 100 (Bergin et al. 2013). With thesecaveats it is interesting to compare the inferreddistribution of M disk /M ∗ ratios to the limitingmass ratio above which gravitational instabili-ties set in ( M disk /M ∗ ∼ .
1, e.g., Lodato et al.2005, dashed line in Figure 9). While the median M disk /M ∗ value of ∼ .
04 is well below 0.1, thebrightest source in our sample is close to the grav-itational instability boundary. In addition, sixother sources, ranging in stellar mass from ∼ . M ⊙ , have ratios only a factor of 4 lowerthan the gravitational instability limit and appearto delineate an upper horizontal boundary. It isinteresting to speculate that this upper bound-ary is the one set by gravitational instability, butindependent observations of the gas content arenecessary to make any firm conclusion. -1.5 -1.0 -0.5 0.0 0.5log M * (M sun )-4-3-2-1 l og M d i s k / M * grav. instability Fig. 9.— Disk to stellar mass ratios as a func-tion of stellar masses ( M ∗ ) for our Chamaeleon Iregion. Disk masses are dust masses (using a con-stant dust disk temperature of 20 K) multiplied by100.How do disk masses compare with the masslocked up in exoplanets around other stars?Najita & Kenyon (2014) used a Monte Carlo ap-15roach to create ensembles of systems with planetsand debris disks at their known incidence rates andcompared them to the Taurus protoplanetary diskmasses from Andrews et al. (2013). They foundthat the mass in solids in Class II sources arebarely enough to account for the known popula-tion of Kepler and RV planets plus debris disks andseems to fall short for the 5-30 M ⊕ planets at 0.5-10 AU discovered by microlensing. Mulders et al.(2015b) focused on stellar mass dependencies inthe amount of solids from the well-characterized Kepler survey, probing planets with periods within50 days ( ∼ ∼ ∼ . M ⊙ ) appear to be already short in solids bya factor of at least 2 to reproduce the average massin exoplanets. At ∼
10 Myr the deficit amounts tomore than a factor of 5 as shown by the Upper Scoregion. Recently, Gillon et al. (2016) reported thediscovery of 3 close-in ( < . M ⊙ star TRAPPIST-1. Inter-estingly, the largest dust disk mass that we canobtain from the relations in Table 4 for such a staris only 1.6 M ⊕ , not enough to reproduce the totalmass in the TRAPPIST-1 planetary system. Evenif half of the disk mass is already converted intoplanetesimals in ∼
7. Conclusions
We presented an ALMA 887 µ m survey ofthe disk population around objects from ∼ * (M sun )5101520 M du s t ( M ea r t h ) C h a I du s t d i s k s WRF16 WM14
Fig. 10.— Dust disk masses in Chamaeleon I(black dashed and dotted lines) compared to theaverage mass in solids (red and blue squares)from the
Kepler exoplanets as computed inMulders et al. (2015b). The planet mass-radiusrelation by WRF16 (Wolfgang et al. 2016) givesa higher average mass than that by WM14(Weiss & Marcy 2014). Regardless of the assumedrelation, low-mass stars ( ∼ . M ⊙ ) have dust diskmasses lower than the average mass locked up inclose-in exoplanets.0.03 M ⊙ in the nearby ∼ • We detect thermal dust emission from 66 outof 93 disks, spatially resolve 34 of them, andidentify two disks with large dust cavities( ∼
45 AU in radius). • We find that the disk-stellar mass scaling re-lation in Chamaeleon I is steeper than linear: M dust ∝ ( M ∗ ) . − . , where the range in thepower law index reflects two extreme rela-tions between the average dust temperatureand stellar luminosity. • By re-analyzing in a self-consistent way allmillimeter data available for nearby regions,we show that the 1-3 Myr-old regions of Tau-rus, Lupus, and Chamaeleon I have the same M dust − M ∗ relation while the 10 Myr-old Up-16er Sco association has an even steeper re-lation. • The dispersion around the M dust − M ∗ rela-tion is very similar among regions with ages ∼ −
10 Myr hinting at a range of initialconditions which might partly account forthe diversity of planetary systems. • The slopes of the M dust − M ∗ and of the˙ M − M ∗ relations are the same for Taurus,Chamaeleon I, and Lupus when assuminga constant dust temperature, in agreementwith the basic expectation from viscous diskmodels.By comparing our results with theoretical modelsof grain growth, drift, and fragmentation we showthat a steeping of the M dust − M ∗ relation withtime occurs if outer disks are in the fragmentation-limited regime. This is because when fragmenta-tion sets the largest grain size, radial drift will oc-cur at shorter timescales around lower-mass stars.This scenario of redistributing mass in the disk canalso account for the apparent lack of solids in Myr-old disks around low-mass stars ( ≤ . M ⊙ ) whencompared to the average mass of solids locked intoclose-in exoplanets. Such a scenario results in astellar-mass-dependent but not a time-dependentpower-law index of the dust opacity. It also impliesa stellar-mass-dependent disk size for mm grains.Deeper and higher resolution millimeter observa-tions are needed to test the predicted trends. Es-tablishing if and how the size of dust disks scaleswith stellar mass will also enable to measure thedependence between the average dust temperatureand stellar luminosity which is crucial to pin downthe exact M dust − M ∗ relation.The authors thank the anonymous referee andthe statistic editor for insightful comments thathelped improving the manuscript. IP thanksMegan Ansdell and John Carpenter for sharingsome of their results in advance of publicationand for clarifying their procedures to analyze theALMA data. IP also acknowledges support from aNSF Astronomy & Astrophysics Research Grant(ID: 1515392). GH and LF are supported bygeneral grant 11473005 awarded by the NationalScience Foundation of China. CFM gratefullyacknowledges an ESA Research Fellowship. LT acknowledges partial support from Italian Minis-tero dell ' Istruzione, Universit`a e Ricerca throughthe grant Progetti Premiali 2012 – iALMA (CUPC52I13000140001) and from Gothenburg Centreof Advanced Studies in Science and Technologythrough the program
Origins of habitable plan-ets . This material is based upon work supportedby the National Aeronautics and Space Adminis-tration under Agreement No. NNX15AD94G forthe program
Earths in Other Solar Systems . Theresults reported herein benefitted from collabora-tions and/or information exchange within NASA ' sNexus for Exoplanet System Science (NExSS) re-search coordination network sponsored by NASA ' sScience Mission Directorate. Facilities:
ALMA.17 -5051015
J10533978
J10555973 -202468
J10561638 -50050100150
J10563044 -50510152025
J10574219
J10580597 -1000100200300400
J10581677
J10590108
J10590699
J11004022 -50050100150200250
J11022491 -15-10-5051015
J11023265 λ ]-4-2024 R ea l V i s [ m J y ] J11025504
J11040425
J11040909
J11044258
Fig. 3.— Real part of the observed visibility (circles) as a function of the projected baseline using a samplingof 30 k λ . The best model fit to the data (red solid line) is shown for all targets except the two disks withresolved cavities, see text for details. Similar figures for all other targets are available online.18ig. 4.— The two disks in our Chamaeleon I sample with spatially resolved dust cavities. J10581677 is aknown transition disk based on its infrared photometry and spectroscopy.19 * (M sun )-2-1012 l og M du s t ( M ea r t h ) T dust fixed to 20KTaurus Lupus ChaI UpperSco Fig. 7.— The M dust − M ∗ relation in four different regions: Taurus (green solid line), Lupus (red dot-dashedline), Chamaeleon I (black dashed line), and Upper Sco (light blue dotted line). These relations are obtainedassuming a fixed dust temperature of 20 K (see also Table 4). For Chamaeleon I we also plot the individualdust disk masses. Note that the ∼
10 Myr-old Upper Sco has a steeper M dust − M ∗ relation than the otherstar-forming regions. 20 . Comparison of linear regression methods Here, we compare different linear regression methods to fit the F mm − M ∗ relation in the log-log plane.We will show how the intrinsic scatter in the relation and censored values (upper limits to the millimeterflux density) contribute to the best fit slope and intercept.We start by comparing the results from two IDL routines (fitexy and mpfitexy) that do not account forupper limits, i.e. we only fit the 66 sources with measured flux densities. Both routines assume symmetricmeasurement errors in x [log( M ∗ )] and y [log( F mm )] and use the Nukers’ estimator to find the best fit (seee.g. Tremaine et al. 2002). The main difference is that mpfitexy accounts for the intrinsic scatter and canautomatically adjust it to ensure a reduced χ of unity. Indeed, this is necessary for our dataset where F mm has a large spread at each stellar mass and confirmed by the fact that fitexy cannot find a good fit, the χ isgreater than 550 and the probability that the model is correct is zero. The mpfitexy requires a scatter aboutthe relation of 0.5 dex to obtain χ ∼
1. In addition, the uncertainties in the slope and intercept from fitexyare unrealistically low and the best fit is dominated by a few precise measurements when not accounting forthe scatter thus biasing the derived slope and intercept (see Table 5). These issues are well documented inTremaine et al. (2002).Next, we compare three different routines that account for censored data using different methods. The oneutilized throughout the paper is the linmix err routine (IDL version) written by Kelly (2007) and alreadyused in several other astronomical applications. This routine accounts for both measurement errors andintrinsic scatter while the other two routines from the R statistical package (censReg and cenken) do notinclude individual measurement errors. The fact that they all provide the same slope and intercept withinthe quoted uncertainties (Table 5) again confirms that the intrinsic scatter about the relation drives the bestfit. In what follows we briefly summarize the methods used in these routines and additional lessons learnedfrom the comparison.The linmix err routine uses a Bayesian approach assuming a normal linear regression model, i.e. theconditional distribution is a normal density, and computes the likelihood function of the data by integratingthe conditional distribution. The measurement errors and the intrinsic scatter about the line are all assumedto be normally distributed. A Markov chain Monte Carlo method is used to compute the uncertainties onthe slope and intercept. Further details about the approach are summarized in Kelly (2007).The censReg R routine is based on the parametric Maximum likelihood estimation and assumes a normaldistribution of the error term (see e.g. Greene 2008). As mentioned above individual measurement errorson x and y are not taken into account and one single left censoring (upper limit) is considered. Because oursurvey has different upper limits for the
Hot and
Cool samples, we had to use the less stringent one, theone from the
Hot sample. In other words, the results reported in Table 5 are from treating 58 datapointsas uncensored (detections) and 35 as upper limits, 27 of them are true upper limits while 8 are additionaldetections below the upper limit set by the
Hot sample.Finally, the cenken R routine uses the non-parametric Akritas-Thiel-Sen line with the Turnbull estimateof intercept (Akritas et al. 1995). The advantage of this method is that it does not make any assumptionabout the distribution of the data. While measurement errors on x and y are not included, upper limits canbe specified individually meaning that both the
Hot and
Cool sample upper limits can be properly takeninto account.As summarized in Table 5 the three routines treating censored data find the same slope and interceptfor the F mm − M ∗ relation. As expected, the slope is steeper and the intercept is lower than that obtainedconsidering only uncensored data but properly accounting for the scatter (mpfitexy). The slightly lowerslope from censReg probably reflects that the Cool sample upper limits are not treated (see also Section 4for a similar effect when applying an even shallower cutoff as in the Taurus survey). Finally, the fact thatparametric and non-parametric approaches reach the same results suggest that the slope and intercept ofthe F mm − M ∗ relation are not affected by the underlining assumptions on the distribution of the data.21 EFERENCES
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This 2-column preprint was prepared with the AAS L A TEXmacros v5.2. able 1Source Properties M ∗ )Name ( ′′ ) Luhman Sample adopted ( M ⊙ )J10533978-7712338 M2.75 II Hot M2 M16b -0.41 † J10555973-7724399 T3 2.210 D13 M0 II Hot K7 M16a -0.13 (-0.17,-0.07)J10561638-7630530 ESOH α
553 M5.6 II Cool M6.5 M16b -0.96 (-1.03,-0.89)J10563044-7711393 T4 M0.5 II Hot K7 M16a -0.07 (-0.15, 0.19)J10574219-7659356 T5 0.160 N12 M3.25 II Hot M3 M16b -0.52 (-0.57,-0.47)J10580597-7711501 M5.25 II Cool M5.5 M16b -0.96 (-1.06,-0.86)J10581677-7717170 SzCha 5.120 D13 K0 II/T Hot K2 M14 0.10 ( 0.06, 0.14)J10590108-7722407 TWCha K2 II Hot K7 M16a -0.07 (-0.14, 0.17)J10590699-7701404 CRCha K2 II Hot K0 M16a 0.23 ( 0.18, 0.28)J11004022-7619280 T10 M3.75 II Cool M4 M16b -0.62 (-0.69,-0.54)J11022491-7733357 CSCha K6 II/T Hot K2 M14 0.13 ( 0.09, 0.20)J11023265-7729129 CHXR71 0.560 D13 M3 II Hot M3 M16b -0.52 (-0.58,-0.45)J11025504-7721508 T12 M4.5 II Cool M4.5 M16a -0.74 (-1.23,-0.68)J11040425-7639328 CHSM1715 M4.25 II Cool M4.5 M16b -0.74 (-0.83,-0.65)J11040909-7627193 CTChaA 2.670 D13 K5 II Hot K5 M16a -0.06 (-0.16, 0.04)J11044258-7741571 ISO52 M4 II Cool M4 M16a -0.62 (-0.69,-0.54)J11045701-7715569 T16 M3 II Hot M3 M16b -0.53 (-0.59,-0.47)J11062554-7633418 ESOH α
559 M5.25 II Cool M5.5 M16b -0.91 (-1.01,-0.81)J11062942-7724586 M6 II Cool M6 L07 -1.12 (-1.66,-1.00)J11063276-7625210 CHSM7869 M6 II Cool M6.5 M16b -1.13 (-1.25,-0.97)J11063945-7736052 ISO79 M5.25 II Cool M5 M16b -0.78 † J11064180-7635489 Hn5 M4.5 II Cool M5 M16a -0.78 (-0.86,-0.68)J11064510-7727023 CHXR20 28.46 KH07 K6 II Hot K6 M16b -0.03 (-0.10, 0.22)J11065906-7718535 T23 M4.25 II Cool M4.5 M16a -0.71 † J11065939-7530559 M5.25 II Cool M5.5 M16b -0.97 (-1.07,-0.87)J11070925-7718471 M3 II Hot M3 L07 -0.52 (-0.58,-0.45)J11071181-7625501 CHSM9484 M5.25 II Cool M5.5 M16b -0.97 (-1.07,-0.87)J11071206-7632232 T24 M0.5 II Hot M0 M16a -0.23 (-0.34,-0.12)J11071330-7743498 CHXR22E M3.5 II/T Hot M4 M14 -0.63 (-0.71,-0.55)J11071860-7732516 ChaH α α ∗ T30 M2.5 II Hot M3 M16b -0.51 (-0.58,-0.44)J11080002-7717304 CHXR30A 0.460 La08 K8 II Hot K7 M16b -0.18 (-0.28,-0.07)J11080148-7742288 VWCha,T31 0.660 D13 K8 II Hot K7 M16a -0.20 (-0.30,-0.11)J11080297-7738425 ESOH α
562 0.280 D13 M1.25 FS Hot M1 M16a -0.20 (-0.29, 0.04)J11081509-7733531 T33A 2.400 D13 G7 FS Hot K0 M16a 0.12 ( 0.08, 0.16)J11081850-7730408 ∗ ISO138 M6.5 II Cool M6.5 M16b -1.14 (-1.24,-1.03)J11082238-7730277 ISO143 18.16 KH07 M5 II Cool M5.5 M16a -0.90 (-0.99,-0.79)J11082570-7716396 M8 II Cool M8 L07 -1.51 † J11082650-7715550 ISO147 M5.75 II Cool M5.5 M16b -0.96 (-1.06,-0.86)J11083905-7716042 Sz27 K8 II/T Hot K7 M14 -0.08 (-0.15, 0.16)J11083952-7734166 ChaH α † J11094621-7634463 Hn10e 19.17 KH07 M3.25 II Hot M3 M16b -0.47 (-0.54,-0.39)J11094742-7726290 B43,ISO207 M3.25 II Hot M1 M16b -0.22 (-0.30, 0.02)J11095215-7639128 ISO217 M6.25 II Cool M6.25 L07 -1.20 (-1.76,-1.08)J11095336-7728365 ISO220 M5.75 II Cool M5.5 M16b -0.96 (-1.06,-0.86)J11095340-7634255 T42 K5 II Hot K7 M16b -0.12 (-0.21,-0.03)J11095407-7629253 T43 0.780 D13 M2 II Hot M1 M16b -0.21 (-0.30, 0.03)J11095873-7737088 WXCha 0.740 D13 M1.25 II Hot M0.5 M16b -0.29 (-0.39,-0.19)J11100010-7634578 WWCha 0.006 A15 K5 II Hot K0 M16a 0.21 ( 0.17, 0.27)J11100369-7633291 Hn11 K8 II Hot M0 M16b -0.14 (-0.23, 0.12)J11100469-7635452 FNCha M1 II Hot K7 M16a -0.08 (-0.15, 0.16)J11100704-7629376 T46 0.120 N12 M0 II Hot K7 M16b -0.14 (-0.24,-0.05)J11100785-7727480 ISO235 M5.5 II Cool M5.5 M16b -0.89 (-0.99,-0.79)J11101141-7635292 ISO237 28.32 KH07 K5.5 II Hot K5 M16a 0.00 (-0.06, 0.26)J11103801-7732399 CHXR47 0.170 D13 K3 II Hot K4 M16b 0.05 (-0.04, 0.14)J11104141-7720480 ISO252 M6 II Cool M5.5 M16b -0.96 (-1.06,-0.86)J11104959-7717517 T47 12.09 KH07 M2 II Hot M2 M16b -0.38 (-0.51,-0.25)J11105333-7634319 T48 M3.75 II Hot M3 M16b -0.51 (-0.58,-0.44)J11105359-7725004 ISO256 M4.5 II Cool M5 M16b -0.81 (-0.90,-0.72)J11105597-7645325 Hn13 0.130 La08 M5.75 II Cool M6.5 M16b -0.98 † J11111083-7641574 ESOH α
569 M2.5 II Hot M1 M16b -0.31 † J11113965-7620152 T49 24.38 KH07 M2 II Hot M3.5 M16a -0.59 (-0.65,-0.53)J11114632-7620092 CHXN18N K6 II Hot K2 M16a 0.09 ( 0.05, 0.13)J11120351-7726009 ISO282 M4.75 II Cool M5.5 M16b -0.89 (-0.98,-0.81)J11120984-7634366 T50 M5 II Cool M5 M16b -0.78 (-0.84,-0.72)J11122441-7637064 T51 1.970 D13 K3.5 II Hot K2 M16a 0.04 ( 0.00, 0.10)J11122772-7644223 T52 11.18 KH07 G9 II Hot K0 M16a 0.20 ( 0.15, 0.25)J11123092-7644241 ∗ T53 M1 II Hot M0.5 M16b -0.17 (-0.26, 0.09)J11124268-7722230 T54A 0.240 D13 G8 II/T Hot K0 M16a 0.20 ( 0.16, 0.25)J11124861-7647066 Hn17 M4 II Cool M4.5 M16a -0.69 (-0.77,-0.60) able 1— Continued M ∗ )Name ( ′′ ) Luhman Sample adopted ( M ⊙ )J11132446-7629227 ∗ Hn18 M3.5 II Hot M4 M16a -0.62 (-0.69,-0.54)J11142454-7733062 Hn21W 5.480 D13 M4 II Cool M4.5 M16a -0.71 (-0.79,-0.63)J11160287-7624533 ESOH α
574 K8 II Hot K8 L07 -0.19 † J11173700-7704381 Sz45 M0.5 II/T Hot M0.5 M14 -0.28 (-0.39,-0.17)J11175211-7629392 M4.5 II Cool M4.5 L07 -0.69 (-0.77,-0.61)J11183572-7935548 M4.75 II/T Cool M5 M16b -0.77 † J11241186-7630425 M5 II/T Cool M5.5 M16b -0.90 (-0.99,-0.79)J11432669-7804454 M5 II Cool M5.5 M16b -0.86 (-0.93,-0.79) † For these stars we fixed the isochrone, hence there are no uncertainties associated with the estimated stellar mass, see Section 2.1. ∗ T30 is the secondary of T31 at a separation of 16 . ′′
52. ISO 138 is the secondary of ISO 143 at 18 . ′′
16. T53 is the secondary of T52 at11 . ′′
18. Hn18 is the secondary of CHXR60 (not included in our ALMA survey) at a separation of 28 . ′′ Table 2ALMA Observations
UTC Date Number Baseline Range pwv CalibratorsAntennas (m) (mm) Flux Passband Phase2014 May 1-3 37 17-558 0.6 Pallas J1427-4206 J1058-80032015 May 18-19 39 21-556 0.6 Ganymede,J1107-448 J0538-4405,J1337-1257 J1058-8003 able 3Measured Continuum Flux Densities ν ∆ α ∆ δ FWHM(mJy) (arcsec) (arcsec) (arcsec)J10533978-7712338 4.60 ± ± ± ± ± ± × ± ± ± † ± ± ± ± ± ± ± † ± ± ± ± × ± ± ± × ± ± ± × ± ± ± × ± ± ± ± ± ± ± ± ± ± × ∗ ± ± ± ± ± ± ± ± ± × ± ± ± ± ± ± ± ± ± ± × ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± × ± ± ± ± × ± ± ± × ± ± ± ± ± ± ± ± ± × ± ± ± ± ± ± ± ± ± ± × ± ± ± × ± ∗ ± ± ± ± ± ± ± ± ± ± ± × ± ± ± ± ± ± ± ± ± ± ± × ± ± ± ± ± ± ± × ± ± ± ± ± × ± ± ± × ± ± ± ± ± ± × ± ± ± ± ± ± ± ± ± ± ± ± ± × ± ± ± ± ± ± ± × ± ± ± × ± ± ± × ± ± ± ± ± ± × ± ± ± ± ± ± × ± ± ± ± ± ± ± ± ± ± × ∗ ± ± ± ± ± able 3— Continued ν ∆ α ∆ δ FWHM(mJy) (arcsec) (arcsec) (arcsec)J11132446-7629227 8.07 ± ± ± ± ± ± × ± ± ± × ± ± ± × ± ± ± ± × ± ± ± ± ± ± × Note.—
Sources with a FWHM reported in the last column of the table are thosethat were fitted with an elliptical gaussian. Undetected sources have ellipses inall columns following the flux density column. For these sources flux densitiesare measured assuming a point source model and fixed ∆ α and ∆ δ to the medianvalues of the detected sources. † Sources with rings. Integrated flux density measured on image within the 3 σ contour. ∗ Sources that have additional mm detections in their exposures: J11044258-7741571 (ISO 52) at ∼ ′′ , coordinates (11:04:40.59;-77:41:56.9); J11082238-7730277 (ISO 143) at ∼ ′′ , coordinates (11:08:21.11;-77:30:18.9); and J11123092-7644241 (T53) at ∼ ′′ , coordinates (11:12:27.7;-76:44:22.3). In the first twocases there is no object in the SIMBAD Astronomical Database associated withthe mm emission. In the case of T53 we detect the disk from the companion T52.Fluxes from these additional detections are not reported in the table. Table 4 M dust − M ∗ relations. Region Age (Myr) αT βT α β DispersionTaurus 1-2 1.6(0.2) 1.2(0.1) 1.1(0.2) 1.0(0.1) 0.7(0.1)Lupus ∗ Note.—
The listed α and β values (uncertainties in parenthesis) are the slope and interceptof the following linear relation: log( M dust /M ⊕ )= α × log( M ∗ /M ⊙ )+ β . The first two entriesare obtained assuming a fixed dust temperature of 20 K while the other entries assume a dustdisk temperature scaling with stellar luminosity (see text for more details). ∗ There are twenty sources in Lupus that do not have stellar masses (Ansdell et al. 2016;Alcal´a et al. 2016). While Ansdell et al. (2016) have assigned masses in a MC fashion fol-lowing the distribution of the Lupus I-IV YSOs, we do not include these sources in our fits.The slope and dispersion reported in Ansdell et al. (2016) are the same as those reportedhere.
Table 5Summary of methods for the F mm − M ∗ relation in Chamaeleon I. Routine Method Censored Slope Interceptfitexy (IDL) Nukers n 2.43(0.08) 2.16(0.04)mpfitexy (IDL) Nukers (with scatter) n 1.5(0.2) 1.8(0.1)censReg (R) Maximum Likelihood y 1.8(0.2) 1.6(0.1)cenken (R) Akritas-Thiel-Sen y 1.9 1.7linmix err (IDL) Bayesian y 1.9(0.2) 1.6(0.1)
Note.—