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A&A
Volume 591, July 2016
Article Number L3
Number of page(s) 6
Section Letters
DOI https://doi.org/10.1051/0004-6361/201628549
Published online 06 June 2016

© ESO, 2016

1. Introduction

The evolution of a protoplanetary disk significantly influences the planetary system that is formed. The final mass distribution of planets resembles the evolution of the surface density of gas in the disk (e.g., Thommes et al. 2008) and more massive disks lead to systems with more massive planets (Mordasini et al. 2012). The evolution of the disk structure is mainly driven by processes happening in the disk, such as dust evolution (Testi et al. 2014), and by interaction between the disk and the central star through viscous accretion and winds (Alexander et al. 2014).

In the context of viscously evolving protoplanetary disks, the mass accretion rate onto the central star (Ṁacc) and the mass of the disk (Mdisk) should be directly correlated (e.g., Eq. (7) of Hartmann et al. 1998). The ratio between these quantities is related to the viscous timescale (tν) at the outer radius of the disk (Rout) and the assumptions about the disk viscous properties. Overall, it is expected that Ṁacc~Mdisk/tν(Rout) with a coefficient of order unity (e.g., Jones et al. 2012). In disks that evolved viscously, the ratio Mdisk/Ṁacc must be comparable to the age of the system independent of the initial conditions and value of the viscosity parameter α. The tight correlation between Ṁacc and stellar masses (Ṁacc∝M⋆1.8, e.g., Muzerolle et al. 2003; Natta et al. 2006; Alcalá et al. 2014; Manara et al. 2016) was also explained by Dullemond et al. (2006) as a consequence of the initial rotation rate of the cores where disks formed. They predict a strong dependence of Mdisk on and a tight correlation of Mdisk with Ṁacc as a consequence of viscous evolution. This theoretical relation between Mdisk and Ṁacc has been empirically investigated, but previous studies were unable to find any significant correlation (e.g., Andrews et al. 2010; Ricci et al. 2010).

In this Letter we present a study of an almost complete and homogeneous dataset of young stars in the ~1–3 Myr old Lupus star-forming region (Comerón 2008, d = 150–200 pc). We collected Ṁacc measured from ultraviolet (UV) excess with the VLT/X-Shooter spectrograph, and Mdisk measured both from sub-mm continuum and CO line emission with Atacama Large Millimeter/submillimeter Array (ALMA). We look for correlations between Ṁacc and Mdisk, as predicted by viscous theory.

2. Data sample

The sample analyzed here includes Class II and transition disk (TD) young stellar objects (YSOs) with 0.1 < M⋆/M⊙ < 2.2, thus including only the TTauri stars of the ALMA sample. Both the ALMA and X-Shooter surveys are complete at the ~95% level. In total, there are 66 objects with ALMA and X-Shooter data available. The list of targets included in the analysis is reported in Table A.1.

The ALMA data are presented by Ansdell et al. (2016, hereafter AW16). The setting includes continuum emission at 335.8 GHz (890 μm) at a resolution of ~0.34′′×0.28′′ (~25 × 20 AU radius at 150 pc) and the 13CO and C18O 3–2 transitions. From the continuum emission, detected for 54 of the targets included here, AW16 derive disk dust mass (Mdisk,dust) using typical assumptions of a single dust grain opacity κ(890 μm) = 3.37 cm2/g and a single dust temperature Tdust = 20 K. From the CO emission lines, AW16 derive the disk gas mass (Mdisk,gas) for 29 disks in our sample, but for 22 of these the lower bound of the acceptable values of Mdisk,gas is unconstrained because of the lack of C18O detection. Upper limits are calculated for the other 37 targets.

We obtain Ṁacc from the X-Shooter spectra (Alcalá et al. 2014, and in prep.). Briefly, the stellar and accretion parameters are derived by finding the best fit among a grid of models including photospheric templates, a slab model for the accretion spectrum, and reddening (Manara et al. 2013). We use the UV excess as a main tracer of accretion and the broad wavelength range covered by X-Shooter (λλ ~ 330−2500) to constrain both the spectral type of the target and the extinction. Among the objects discussed here, 57 have Ṁacc derived from X-Shooter measurements, while 5 have an accretion rate compatible with chromospheric noise (non accretors), and 4 targets are observed edge-on, thus their Ṁacc are underestimated. The evolutionary models by Siess et al. (2000) are used to determine M⋆ and, thus, Ṁacc.

Finally, the sample includes several resolved binaries. All of these binaries have separations ≳2″ and are indicated in Table A.1.

3. Results

3.1. Disk dust mass

thumbnail Fig. 1

Logarithm of Ṁacc vs. logarithm of Mdisk,dust. Green filled squares are used for measured values, open squares for edge-on objects, and downward pointing open triangle for objects with accretion compatible with chromospheric noise. Transition disks are indicated with a circle. We show fit results obtained using the Bayesian fitting procedure by Kelly (2007), which considers errors on both axes and is only applied to detected targets. The assumed best fit is represented with a red solid line, while the light red lines are a subsample of the results of some chains. The best fitting with this procedure overlaps with the least-squares best fit.

The values of Mdisk,dust derived by AW16 (see Sect. 2) are a measure of the bulk dust mass in the disk. Figure 1 shows the values of Ṁacc measured for our targets as a function of Mdisk,dust1.

We first search for a correlation between the two quantities running a least-squares linear regression on the targets with both Mdisk,dust and Ṁacc measurements and find a moderate correlation with r = 0.53 and a two-sided p-value of 1.5 × 10-4 for the null hypothesis that the slope of this correlation is zero. The best fit obtained with this method has a slope of 0.7 and a standard deviation of the fit of 0.2. Then, we compute the linear regression coefficients using the fully Bayesian method by Kelly (2007)2, which allows us to include uncertainties on both axes in the fitting procedure. Uniform priors are used for the linear regression coefficients. We include some single chain results in Fig. 1, as well as the best fit obtained with this method, which has the same slope and intercept of the least-squares fit relation. We adopt the median of the results of the chains as best fit values. We refer to Appendix B for the corner plots with the posterior analysis results. The best fit obtained with this method has a slope of 0.7 ± 0.2, a standard deviation of 0.4 ± 0.1, and a correlation coefficient of 0.56 ± 0.12. We also verified that the two quantities are still correlated when upper limits on Mdisk,dust are properly considered using the same tool. The correlation coefficient increases to 0.7 ± 0.1, while the slope is larger (1.2 ± 0.2) but compatible with that obtained using detections only. The same slope is obtained including upper limits and using the emmethod and buckleyjames method in ASURV. However, the slope estimated when including upper limits is not well constrained (Kelly 2007) and should be considered with caution. We then find a probability lower than 10-4 of no-correlation using the Cox hazard test for censored data in ASURV (Lavalley et al. 1992) including upper limits on Mdisk,dust.

thumbnail Fig. 2

Logarithm of Lacc vs. logarithm of continuum emission normalized to a distance of 150 pc. Symbols are as in Fig. 1.

We show the dependence of the accretion luminosity (Lacc) on the sub-mm continuum flux normalized to a distance of 150 pc in Fig. 2 to confirm that the correlation is not induced by the conversion from Lacc to Ṁacc. Indeed, Lacc is directly measured from the spectra, while the conversion to Ṁacc depends on M⋆, which is derived from evolutionary models. A correlation is found with r = 0.6, a slope of 0.8 ± 0.2, and a standard deviation of 0.5 ± 0.1.

We then test for the robustness of the correlation, given our assumptions to convert the continuum emission in Mdisk,dust. First, we assumed a single disk opacity and gas-to-dust ratio for all disks. To test whether a random variation of these parameters would affect our results, we perform the same statistical tests on the same targets after randomly displacing the values of Mdisk,dust within a uniform distribution with size ±1 dex and centered on the measured value to mimic the uncertainties. We perform this test ten times, and the correlation is still present in nine out of ten realizations. Then, we test the effect of our assumption of a single Tdust by modifying our Mdisk,dust values assuming (Andrews et al. 2010). The correlation becomes less robust, but still significant (r = 0.3, p-value = 0.03). We conclude that there is a statistically significant relation between the logarithm of Ṁacc and the logarithm of Mdisk,dust. This relation has a slope slightly smaller than unity.

The location of TDs in Fig. 4 is also highlighted. All but one of the TDs are found to be below the best-fit relation in agreement with, for example, Najita et al. (2015). This suggests they have either lower Ṁacc, or larger disk mass, or a different gas-to-dust ratio, than typical full disks.

3.2. Disk gas mass

thumbnail Fig. 3

Logarithm of Ṁacc vs. logarithm of the disk mass derived from CO emission. Symbols are as in Fig. 1. No correlation is found between these quantities.

thumbnail Fig. 4

Logarithm of Ṁacc vs. logarithm of Mdisk= 100 ·Mdisk,dust. Symbols are as in Fig. 1. Also here, the best fit with the procedure by Kelly (2007) overlaps with the least-squares best fit. The dashed lines represent different ratios of Mdisk/Ṁacc, as labeled.

The lower detection rates of CO lines than continuum emission (AW16) implies that we only measure Mdisk,gas for a few objects. Figure 3 reports the Ṁacc vs. Mdisk,gas plot.

We perform the same statistical tests as for the Ṁacc – Mdisk,dust relation. We find no correlation between the logarithm of Ṁacc and the logarithm of Mdisk,gas using the least-squares linear regression on the targets with measured Mdisk,gas (r = 0.2, p-value = 0.3). When considering uncertainties on the measurements we find a value for the correlation coefficient of 0.5, and thus we find no correlation. We obtain the same statistically insignificant value for the correlation coefficient, which points toward no correlation, even when we include the upper limits. Finally, the Cox hazard test for censored data gives a probability of 0.25 that the two quantities are not correlated. We then conclude that we do not detect any correlation between these two quantities. The large number of upper limits compared to detection is probably a limiting factor in studying this relation and the large error bars of the measurements are another limiting factor. Deeper ALMA surveys of CO emission in protoplanetary disks are needed to further study this relation. In Fig. 3, the TDs are mixed with full disks.

4. Discussion

As mentioned in the Introduction, viscous evolution theory predicts that Ṁacc∝Mdisk/tν(Rout). The evolution of the surface density of the disk (Σ) can be analytically described provided that the viscosity (ν) is known (e.g., Pringle 1981; Lodato 2008).

As described by Jones et al. (2012), a similarity solution (Mdisk ∝ t− σ) is reached for times much larger than the viscous timescales under the simple assumptions ν ∝ Rn or ν ∝ ΣmRn, where R is the disk radius. By differentiating this solution, one obtains Ṁacc ∝ σt− (1 + σ) and thus it is possible to define the “viscous disk age” as tdisk = Mdisk/Ṁacc = t/σ. Measurements of the decline of Ṁacc with time suggest that σ ~ 0.5 (e.g., Hartmann et al. 1998; Sicilia-Aguilar et al. 2010). For viscously evolving disks, this implies that the age of the objects should be within a factor ~2 of the ratio Mdisk/Ṁacc. Jones et al. (2012) have also shown that more complex assumptions on the disk viscosity, such as different values for the α viscosity description, lead to the same asymptotic behavior with Mdisk/Ṁacc ratios usually larger than the age of the objects by a factor 2–3, but always less than 10.

Other processes happening during disk evolution, such as a layered accretion, photoevaporation, and even planet formation, all lead to very similar values of Mdisk/Ṁacc at late times, and these values are always higher than the age of the object (Jones et al. 2012). Thus, a disk that evolved only from internal processes has a Mdisk/Ṁacc ratio similar or larger than its age regardless of the assumption on the disk viscosity. The only means by which a disk might have an Mdisk/Ṁacc ratio smaller than its age is if the disk is externally truncated (Rosotti et al., in prep.).

We compare our results with these theoretical expectations by showing in Figs. 3 and 4 the Mdisk/Ṁacc ratios for three different values of Mdisk/Ṁacc = 0.1 Myr, 1 Myr, and 3 Myr. We assume that the total disk mass (Mdisk) is Mdisk = Mdisk,gas in Fig. 3, while Mdisk = 100·Mdisk,dust in Fig. 4. Indeed, to convert Mdisk,dust to Mdisk one needs to know the gas-to-dust ratio. We assume an interstellar medium value of 100 for the gas-to-dust ratio, as is commonly done (e.g., Andrews et al. 2010; Ricci et al. 2010). If the gas-to-dust ratio has no dependence on Mdisk, this has no impact on the correlation between Ṁacc and Mdisk but is instructive for the discussion. The typical age of Lupus targets is ~1–3 Myr with a spread of 1–2 Myr (e.g., Alcalá et al. 2014).

The location of the targets in Fig. 4 is in general agreement with the aforementioned theoretical expectations. Most of the targets (60%) have positions between or compatible with the 1 and 3 Myr lines. However, several of the targets in Fig. 3 do not match the expectations from viscous evolution theory as they lie above the Mdisk/Ṁacc = 1 Myr line.

The lack of correlation between Ṁacc and Mdisk,gas is contrary to expectations from viscous evolution theory. When assuming Mdisk = 100 · Mdisk,dust, however, we find a correlation between Mdisk and Ṁacc and also Mdisk/Ṁacc ratios that are compatible with expectations from theory. Thus, we are inclined to conclude that the total disk mass Mdisk ∝ Mdisk,dust, as with this assumption the correlation is present. This in turn suggests that Mdisk,gas measured from CO emission is possibly lower than the total Mdisk, at least for the more massive disks. A possible explanation for this might be that carbon is processed in more complex molecules (e.g., Bergin et al. 2014; Kama et al. 2016) or that more detailed modeling of CO lines is needed, but this discussion is out of the scope of this paper.

The slope of the observed correlation between Ṁacc and Mdisk, as measured from dust emission, is consistent with being linear, as expected if all disks evolve viscously, however, we cannot exclude that it is actually shallower. The exact slope can be derived with a better handle on the uncertainty in the Mdisk estimate, namely the gas-to-dust ratio, disk grain opacity, disk temperature, and their dependence on the stellar properties. More constraints on these values are awaited from future ALMA survey of disks with higher sensitivity, multiple band observations, and targeting several molecules in order to better determine the chemical properties of the disks. The interest in further constraining this slope is related to the fact that this relation can tell us what evolutionary processes dominate at different stellar masses.

5. Conclusions

In this Letter we compared the most complete and homogeneous datasets of properties of young stars and their disks to date. We used accretion rates onto the central star determined from UV excess with the VLT/X-Shooter spectrograph and disk masses from both sub-mm continuum and CO line emission measured by ALMA.

We detected a statistically significant correlation between Ṁacc and Mdisk with a slope that is slightly smaller than 1. This is found when assuming that the total disk mass is proportional to the disk dust mass, but not when using the disk gas mass. The latter result could be due to large uncertainties in Mdisk,gas estimate and low number statistics. For this reason, deeper surveys of gas emission in disks are needed. When measuring Mdisk from dust emission, transitional disks are found to have either a smaller Ṁacc or a larger Mdisk than full disks.

We compared the observed Mdisk derived from dust emission and Ṁacc with basic predictions from viscous evolution theory and we found a good agreement with the expected Mdisk/Ṁacc ratios for our targets.

Future studies should look for the Mdisk – Ṁacc correlation for objects with different ages and in different environments.

1

Objects observed edge-on or with accretion compatible with chromospheric noise are shown in the plot, but they are not included in the analysis of correlations between Ṁacc and Mdisk,dust.

Acknowledgments

We thank Cathie Clarke and Phil Armitage for insightful discussions. We thank the anonymous referee for insightful comments that helped to improve the presentation of the results. C.F.M. gratefully acknowledges an ESA Research Fellowship. G.R. is supported by the DISCSIM project, grant agreement 341137 funded by the European Research Council under ERC-2013-ADG. A.N. would like to acknowledge funding from Science Foundation Ireland (Grant 13/ERC/I2907). Leiden is supported by the European Union A-ERC grant 291141 CHEMPLAN, by the Netherlands Research School for Astronomy (NOVA), and by grant 614.001.352 from the Netherlands Organization for Scientific Research (NWO). JPW and MA were supported by NSF and NASA grants AST-1208911 and NNX15AC92G.

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Appendix A: Sample

Table A.1

Targets included in the analysis.

Appendix B: Corner plots of fit

We show in Fig. B.1 the corner plots showing the intercept, slope, standard deviation, and correlation coefficient for the fit

thumbnail Fig. B.1

Corner plot of the log Ṁacc vs. log(100·Mdisk,dust) fit done using the linmix routine (Kelly 2007).

of the log Ṁacc vs. log(100 · Mdisk,dust) relation (see Sect. 3.1 and Fig. 4). This is done using linmix (Kelly 2007), only data with detection on both axes, and including the errors on both axes.

All Tables

Table A.1

Targets included in the analysis.

In the text

All Figures

thumbnail Fig. 1

Logarithm of Ṁacc vs. logarithm of Mdisk,dust. Green filled squares are used for measured values, open squares for edge-on objects, and downward pointing open triangle for objects with accretion compatible with chromospheric noise. Transition disks are indicated with a circle. We show fit results obtained using the Bayesian fitting procedure by Kelly (2007), which considers errors on both axes and is only applied to detected targets. The assumed best fit is represented with a red solid line, while the light red lines are a subsample of the results of some chains. The best fitting with this procedure overlaps with the least-squares best fit.

In the text
thumbnail Fig. 2

Logarithm of Lacc vs. logarithm of continuum emission normalized to a distance of 150 pc. Symbols are as in Fig. 1.

In the text
thumbnail Fig. 3

Logarithm of Ṁacc vs. logarithm of the disk mass derived from CO emission. Symbols are as in Fig. 1. No correlation is found between these quantities.

In the text
thumbnail Fig. 4

Logarithm of Ṁacc vs. logarithm of Mdisk= 100 ·Mdisk,dust. Symbols are as in Fig. 1. Also here, the best fit with the procedure by Kelly (2007) overlaps with the least-squares best fit. The dashed lines represent different ratios of Mdisk/Ṁacc, as labeled.

In the text
thumbnail Fig. B.1

Corner plot of the log Ṁacc vs. log(100·Mdisk,dust) fit done using the linmix routine (Kelly 2007).

In the text

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