© ESO 2018

1. Introduction

Star formation encompasses a broad range of physical regimes that account for the collapse of gas to form smaller structures: from star-forming galaxies to individual young stars surrounded by accreting envelopes and disks, passing through intermediate scales including giant molecular clouds, clumps, and dense cores.

Almost 60 years ago, Schmidt (1959) proposed that the star formation rate (SFR) is a power-law function of the total gas density (atomic and molecular), $\sum _{\text{SFR}}=K\sum _{\text{gas}}^N$ (Kennicutt 1998), with Σ_SFR and Σ_gas the SFR and the total gas mass (M_gas) per unit surface area, and K and N the appropriate constants. The Schmidt–Kennicutt law has been observationally confirmed many times when spanning different types of galaxies, revealing a nonlinear correlation with typical values 1 < N < 2 and significant scatter. Using the HCN molecule to trace dense gas mass (M_dg; n(H₂) > 10⁴ cm⁻³), Gao & Solomon (2004) found a roughly linear (N ~ 1) correlation between the spatially integrated SFR and M_dg for a wide range of galaxies including normal spirals, luminous, and ultra-luminous infrared galaxies (LIRGs and ULIRGs, respectively). Wu & Evans (2005) then found that the spatially integrated SFR in distant star-forming clouds in our Galaxy also correlates linearly with the amount of HCN-emitting gas (see also Jackson et al. 2013; Stephens et al. 2016), showing that this correlation also connected smoothly with the one previously found for galaxies. On this basis, Wu & Evans (2005) suggested that the formation of stars in other galaxies may be understood in terms of the basic units of star formation in the Milky Way (MW), which they identified as the clouds massive enough to fully sample the initial mass function (IMF) by hosting massive young stars. The measurements were extended to nearby molecular clouds with and without massive stars using dust extinction to more accurately trace dense gas masses and direct counts of young stars to more robustly measure the SFRs (Lada et al. 2010, 2012). These measurements confirm that the correlation is linear and spreads approximately ten orders of magnitude in each axis. In summary, strong lines of evidence support the linearity of the SFR−M_dg correlation and that this connects individual Galactic clouds and the various types of star-forming galaxies (see also the recent works in Shimajiri et al. 2017; Tan et al. 2018).

The previous view somewhat neglects the role of the actual units in star formation, which are the individual, forming stars. This can be partly attributed to the fact that the SFR, defined as the rate at which gas transforms into stars (in units of M_⊙ yr⁻¹), characterizes physical systems that typically include ~10–10¹¹ objects. However, there is an equivalent parameter for individual stars that also accounts for the gas-to-star transformation by measuring the rate at which gas falls from the circumstellar environment onto the stellar surface: the stellar mass accretion rate (Ṁ_acc, in units of M_⊙ yr⁻¹ too). Direct estimates of Ṁ_acc are only available for optically visible premain sequence objects. These are young stars in a relatively advanced stage of early stellar evolution (Class II from the classification of Lada 1987), when the optically thick envelope has dissipated and direct observations of the remaining circumstellar disks and the stellar accretion shocks are possible. Their gas disk masses (M_disk; n(H₂) > 10⁵ cm⁻³) are in turn inferred either from observations of different molecules or from submillimeter and millimeter dust continuum emission and an assumed gas-to-dust ratio. Different works indicate that so far the latter method provides more accurate measurements of M_disk (Manara et al. 2016; Rosotti et al. 2017). In fact, Ṁ_acc and dust-based M_disk measurements are also linearly correlated for large samples of optically visible young stars (Najita et al. 2007, 2015; Mendigutía et al. 2012), a result that has been unambiguously confirmed from recent Atacama Large Millimeter/submillimeter Array (ALMA) data at least for the star-forming regions Lupus and Cha I, despite the large scatter (Manara et al. 2016; Mulders et al. 2017). Although one could be tempted to interpret the Ṁ_acc−M_disk correlation of stars only in terms of small-scale physical processes in the circumstellar disks, this correlation resembles a scaled-down version of the above-mentioned linear correlation linking local clouds and other galaxies.

This paper shows that the linear correlation that links galaxies and star-forming clouds within our Galaxy (through SFR and M_dg) can indeed be connected down to individual, forming stars when physically equivalent measurements (Ṁ_acc and M_disk) are used for the latter objects. We propose that such global correlation has an underlying explanation that may ultimately contribute to a unified view of star formation across an enormous range of astronomical scale. Section 2 shows that the correlations for young stars and clouds in the MW can be connected. We discuss the origin of such a single correlation under the hypothesis that the relation observed in clouds is driven by the individual (mainly Class 0/I) stars that form them. Section 3 extends the previous finding and discussion, including data for other galaxies. Finally, Sect. 4 summarizes the main conclusions.

2. Correlation between stars and clouds

Figure 1 shows the SFR versus M_dg for 39 distant (a few Kpc) and nearby (a few hundred pc) star-forming clouds in the Galaxy. Only the 28 distant clouds with log M_dg > 2.50 (log L_HCN > 1.50) from the original sample in Wu & Evans (2010) have been considered (see Appendixes A and B), and most contain HII regions and high-mass young stars. The nearby Galactic clouds are from Lada et al. (2010). The bottom left region shows the Ṁ_acc and M_disk values for the nearly complete sample of 62 Class II, optically visible, low-mass young stars in the Lupus I and III star-forming clouds (~150 pc) from Manara et al. (2016). The axes are labeled in terms of SFR and M_dg for all sources, as often used in studies of star formation scaling laws. The original conversion factors between the observations and the physical parameters compared have been kept and the data considered are representative of each regime, which is enough for our purposes. Appendix A describes how the parameters were derived, and Appendix C shows data for additional samples.

Fig. 1. Star formation rate versus dense gas mass for distant and nearby Galactic star-forming clouds (green circles with and without horizontal error bars, respectively; Wu & Evans 2010; Lada et al. 2010) and mass accretion rate versus disk mass in Class II stars in Lupus (blue circles, upper limits as one-sided bars; Manara et al. 2016). Error bars are indicated when provided in the original works. The 3σ linear fit for stars and clouds is indicated with the green dotted region. The green solid square is the direct measurement of SFR and Mdg in Lupus. The blue solid circle is the sum of the accretion rates and disk masses of the Class II stars in Lupus, and the blue open circle includes 20 Class 0/I stars in that cloud with an average accretion rate of 1 × 10−6 M⊙ yr−1 and a minimum circumstellar mass of 0.1 M⊙, that is, a maximum displacement in the x-axis given by f = 1% of the total dense gas mass in Lupus (see text).

The overall trend in Fig. 1 is confirmed from the Spearman’s probability of false correlation using the partial correlation technique (e.g., Wall & Jenkins 2003), which accounts for the fact that the plotted values were computed from a squared-distance factor that depends on the stars and clouds. Its small value (p = 9.5 × 10⁻²³ ≪ 0.05) implies that the correlation is statistically significant and not driven by the d ² factor. Least-squares fitting including 3σ errors provides log SFR = −6.9(±0.2) + 0.80(±0.1)log M_dg, with a correlation coefficient of 0.95. That the slope for the whole sample is not exactly unity is mainly due to the fact that the intercepts for stars and clouds are different. Also relevant is that the y-axis scatter is maximum for stars (~±1.5 dex) and decreases for clouds (~±0.75 dex). Both topics are discussed in Sects. 2.2 and 3.

In order to understand the correlation, we propose the hypothesis that the trend in clouds is driven by a relation between the mass accretion rate onto the central object and the total circumstellar gas mass (M_cs hereafter) surrounding the individual stars in each cloud. The total circumstellar gas mass refers to M_disk for Class II objects, and to M_disk plus physically associated material in a surrounding dense core for Class 0/I objects. In the latter case, M_disk is usually insignificant compared to M_cs, but both M_cs and M_disk refer to the surrounding dense gas mass that is directly participating in the formation of Class 0/I and Class II stars, respectively. To develop this idea in detail we need to consider the extent to which the numbers plotted in Fig. 1 are comparable (see Appendix A). In particular, if our bottom-up hypothesis is correct, the SFR and M_dg of a given cloud should scale with the sum of the individual Ṁ_acc and M_cs values of all stars in that cloud.

2.1. Link between SFR and Ṁ_acc

The Ṁ_acc and SFR estimates in the vertical axis of Fig. 1 are compared in the following. The sum of the accretion rates of all Class II stars in Lupus from Manara et al. (2016) is ~1.7 × 10⁻⁷ M_⊙ yr⁻¹ (log(ΣṀ_acc) ~ −6.8), which is more than an order of magnitude smaller than the SFR directly estimated for that cloud (log SFR ~ −4.7; Lada et al. 2010). This difference is not only caused by the fact that Class 0/I sources are not included in the sample of Manara et al. (2016), but also because of the different timescales involved in the calculation of SFR and Ṁ_acc: the averaged one during the past ~2 Myr for the clouds, and the actual (virtually instantaneous) conversion of gas into individual young stars. The amount of gas (per time unit) transformed into a given star during its age (t_*) is given by $(1/t_{*})\times {\int }_0^{t_{*}}{\dot{M}}_{\text{acc}}\left(t\right)\mathrm{d}t=(1/t_{*})\times {\int }_0^{t_{*}}{\dot{M}}_{\text{acc}}^0{e}^{-t/{\tau }_{*}}\mathrm{d}t=({\dot{M}}_{\text{acc}}^0{\tau }_{*}/t_{*})\times (1-e^{-t_{*}/\tau }),{\dot{M}}_{\text{acc}}^0$, being the initial mass accretion rate and τ_* the dissipation timescale¹. Neglecting possible “FUOr” and “EXOr” events of episodic accretion (Audard et al. 2014) and focusing on the overall, smooth decrease, an exponential decay with τ_* typically between 1 and 3 Myr is estimated from observations of Class II stars (Fedele et al. 2010; Mendigutía et al. 2012), and at least one order of magnitude smaller for Class 0/I sources (Bontemps et al. 1996). The SFR of a cloud during the past few Myr would then be the sum of the previous expression for all stars in that cloud, which is dominated by the term ${\dot{M}}_{\text{acc}}^0$.

Direct observational estimates of ${\dot{M}}_{\text{acc}}^0$ are difficult because of the embedded nature of the stars at their initial stages. The empirically based value consistent with most star-forming regions and commonly adopted is ${10}^{-4}<{\dot{M}}_{\text{acc}}^0\left(M_{\odot }{\mathrm{yr}}^{-1}\right)<{10}^{-6}$ (e.g., Palla & Stahler 2000; Padoan et al. 2014), for which −3 < log SFR < −6. This range is in rough agreement with direct estimates for the clouds. Indeed, within a cloud the actual SFR derived from the sum of the individual Ṁ_acc values is dominated by the high accretion rate of the Class 0/I sources. This is also expected to be of the order of ${\dot{M}}_{\text{acc}}^0$ for individual protostars and, in sum, matches the direct estimates of the SFR. For instance, the typical accretion rate of a Class 0/I star with an IMF stellar mass of ~0.3 M_⊙ and age of ~0.3 Myr (e.g., Padoan et al. 2014) is given by the ratio between both values, 1 × 10⁻⁶ M_⊙ yr⁻¹. Given that Lupus has ~20 Class 0/I sources (Merín et al. 2008; Benedettini et al. 2012), the aggregate value ΣṀ_acc is ~2 × 10⁻⁵ M_⊙ yr⁻¹ (log(ΣṀ_acc) ~ −4.7), which coincides with the direct estimate of the total SFR for that region averaged over 2 Myr (Lada et al. 2010). Moreover, for three molecular clouds for which low-mass protostars have been reasonably well sampled, we find 329, 139, and 43 Class 0/I sources in Orion A, Perseus, and California (Megeath et al. 2012; Carney et al. 2016; Lewis & Lada 2016; Lada et al. 2017). This gives ΣṀ_acc = 3.3×10⁻⁴, 1.4×10⁻⁴, and 4.3×10⁻⁵M_⊙ yr⁻¹, which are within a factor ≤2 the SFRs of the same clouds estimated considering all young stars and a 2 Myr timescale: 7.1 × 10⁻⁴, 1.5 × 10⁻⁴ and 7.0 × 10⁻⁵M_⊙ yr⁻¹, respectively (Lada et al. 2010). This close agreement again supports the idea that the SFRs directly estimated for entire populations and averaged over 2 Myr are recovered from the sum of the “instantaneous” (0.3 Myr) accretion rates when those are dominated by Class 0/I stars, that is, ${\langle \text{SFR}\rangle }_{2 \text{Myr}}=\mathrm{\Sigma }{\dot{M}}_{\text{acc}}^0\equiv {\text{SFR}}_0\mathrm{.}$. In other words, the SFR is essentially constant over timescales of 2 Myr, approximately ten times the typical protostellar lifetime.

2.2. Link between M_dg and M_cs

Next we consider the gas mass estimates on the horizontal axis of Fig. 1. The sum of all individual disk masses in Lupus from Manara et al. (2016) is ~0.25 M_⊙ (log (ΣM_disk) ~ −0.6), which is again orders of magnitude smaller than the M_dg value typically estimated for Lupus (log M_dg ~ 2.4; Lada et al. 2010). However, the sample in Manara et al. (2016) refers to evolved Class II stars that are associated with relatively small amounts of gas. For a more complete accounting of gas associated with young stars in a forming cloud, we need to include the contribution from protostellar (Class 0/I) objects to the dense gas budget (similar to the situation with the SFRs discussed above). Being embedded in dense cores and surrounded by infalling material, Class 0/I sources are associated with significantly higher amounts of dense gas. Indeed, protostellar objects are almost exclusively located in regions where the extinction exceeds 0.8 mag at K band (e.g., Lombardi et al. 2014; Zari et al. 2016; Lada et al. 2017), which is the defining lower limit used by Lada et al. (2010) for their measurements of M_dg in nearby clouds.

Still, circumstellar dense cores likely only account for some fraction of the dense gas in a cloud, ΣM_cs = f M_dg. Because the boundary between a dense core and the surrounding dense gas in which it is embedded is difficult to determine or even define, the value of f is uncertain. Assuming that the bottom-up hypothesis is correct, we can extrapolate the roughly linear relation for the Lupus Class II sources to match the observed (total) SFR for the Lupus cloud and then predict the value of ΣM_cs ~ SFR × τ, where τ is the accretion timescale of the Class II stars, M_disk/ Ṁ_acc. For SFR = 2×10⁻⁵ M_⊙ yr⁻¹ (Lada et al. 2010) and τ ranging between ~0.15 and 4.25 Myr (Manara et al. 2016, see also Fig. 3 and the discussion in the next section), we find 3 < ΣM_cs/M_⊙ < 85 (i.e., 0.012 < ΣM_cs/M_dg < 0.34), which for the ~20 Class 0/I sources in the cloud would mean 0.15 < 〈M_cs〉/M_⊙ < 4.2. That is, the total mass of circumstellar gas directly participating in star formation in Lupus appears to represent between 1.2% and ~34% of the total dense gas reservoir measured in the cloud, and the gas mass around a protostar directly associated with accretion typically contains between 0.15 M_⊙ and 4.2 M_⊙. Direct measurements of some Class 0/I stars in Lupus indicate that a lower limit for 〈M_cs〉 is ~0.15M_⊙ (Mowat et al. 2017). This and additional Herschel measurements of M_cs in Lupus and other star-forming regions (Rygl et al. 2013; Könyves et al. 2015) may be suggesting that f is closer to 1% than to 35%.

Generally speaking, the gas mass of protostellar and prestellar dense cores typically ranges between 0.1 and 10 M_⊙ (e.g., Enoch et al. 2008); assuming that 3 < 〈M_cs〉/M_⊙ < 5 and that f ~ N_protostars × 〈M_cs〉/M_dg, the recent extinction maps of local molecular clouds in Lombardi et al. (2014), Zari et al. (2016), and Lada et al. (2017) indicate that 7% < f (OrionA) < 12%, 22% < f (Perseus) < 33%, and 4% < f (California) < 7%. Therefore, a f value ranging between ~1% and 35% is reasonable. Such a range means that the corresponding log(ΣṀ_acc)−log(ΣM_cs) values for the clouds should be shifted between 0.5 dex (f = 35%) and 2 dex (f = 1%) to the left from their log SFR−log M_dg values in Fig. 1 (the values for the vertical axis would remain practically equal, as previously argued). This change of intercept would not break the correlation and fits even better with the trend shown by stars (see Sect. 3). It is noted that a value of f closer to the upper limit of 35% is obtained if the prestellar cores are considered. Indeed, Herchel surveys indicate that the mass around these sources can dominate over the circumstellar mass around protostars (Könyves et al. 2015). In addition, inflow motions have been detected at least in some prestellar cores at large, sub-parsec scales (e.g., Tafalla et al. 1998).

In summary, the numbers provided should be taken as a first order approach, not only because of the uncertainties involved, but also because it has been assumed that the accretion timescale for Class 0/I sources is similar to that for Class II stars (as discussed below). In any case, our arguments strongly suggest that the sum of individual M_cs measurements of Class 0/I sources represents a non-negligible contribution to the total M_dg estimates for clouds from extinction maps, in agreement with the bottom-up hypothesis.

2.3. Some observational and theoretical implications

Given that the Ṁ_acc and M_cs values of Class 0/I sources dominate over more evolved stars, a major consequence is that the observed correlation between Ṁ_acc and M_disk in Class II stars could be the tip of the iceberg; our hypothesis implies that the correlation should continue with similar slope and intercept to Ṁ_acc and M_cs in earlier Class 0/I stages, filling the gap between the Class II stars and the clouds in Fig. 1. This means that the accretion timescale for Class 0/I stars, M_cs/ Ṁ_acc, should be roughly similar (~within ±1 dex) to that for Class II objects, M_disk/ Ṁ_acc. In other words, the decline of the accretion rate must occur at a pace comparable to that for the dissipation of the circumstellar material in order to keep a similar accretion timescale along the different stages of star formation. The latter has been observed in Class II stars (apart from the Lupus data studied here, see, e.g., Fedele et al. 2010), and has been suggested by observations (Bontemps et al. 1996) and modeling (see below) for Class 0/I sources independently. In fact, assuming a direct proportionality between accretion and ejection, a linear relation between Ṁ_acc and M_cs was inferred based on a comparative study of outflow activity in Class0/I objects (Bontemps et al. 1996).

However, a direct comparison between the Class II and Class 0/I accretion timescales is not feasible so far not only because of the observed scatter in the correlation for the Class II stars in the couple of star-forming regions that have been reliably probed (Manara et al. 2016; Mulders et al. 2017), but especially because such correlation has not been reported for Class 0/I sources, apart from indirect suggestions like the one mentioned above (Bontemps et al. 1996). Indeed, accurate Ṁ_acc and M_cs values for complete samples of Class 0/I sources are generally lacking. The extrapolation of line-luminosity to accretion-luminosity correlations (Mendigutía et al. 2015) to embedded objects are providing promising results (Pomohaci et al. 2017). In particular, mid-infrared (IR) emission lines could be useful to provide Ṁ_acc estimates of highly embedded stars (Rigliaco et al. 2015), for which the spectroscopic capabilities of the James Webb Space Telescope will be ideally suited. Ongoing submillimeter surveys devoted to characterize the accretion properties of Class 0 sources (Herczeg et al. 2017) will also be of great value. Regarding M_cs, ALMA should be able to make an improved census of the protostellar envelope masses in several star-forming regions.

Similarly, our hypothesis suggests that a relevant theoretical framework must be able to predict a Ṁ_acc−M_cs correlation at the initial Class 0/I stages. Such type of theories have already been proposed, for example, predicting the Ṁ_acc−M_disk correlation in Class II stars from the imprint of the parent, turbulent core, either from its rotational properties (Dullemond et al. 2006) or/and from gravitational instabilities and density perturbations (Vorobyov & Basu 2008). Alternatively, similar correlations for Class 0/I sources have also been reproduced in models that do not assume ad-hoc initial and boundary conditions for the cores, but emphasize the importance of the turbulent medium considering the cloud as a whole (Padoan et al. 2014). According to this view, the observed Ṁ_acc−M_disk correlation in Class II stars would not be caused by the disk physics but would be evidence of an underlying Ṁ_acc−M_cs correlation in previous Class 0/I stages. In other words, if our hypothesis is confirmed, viscous disk evolution should be dominant only when the envelopes have dissipated, but its influence on the location of Class II stars in the Ṁ_acc−M_disk plane – and thus on the corresponding accretion timescale M_disk/ Ṁ_acc – would be less relevant as this would have been mainly determined at earlier, embedded stages.

3. The global correlation

As introduced in Sect. 1, strong lines of evidence show that the SFR and M_dg values of Galactic clouds and whole galaxies are connected through a linear relation. Figure 2 extends Fig. 1, adding the SFR and M_dg values of the 65 normal spiral galaxies, LIRGS, and ULIRGS at distances between a few and a few hundred Mpc from Gao & Solomon (2004). The global correlation including all sources overlaps with that for clouds and stars derived in the previous section, is also statistically significant (p = 2.6 × 10⁻³⁶), and has a slope N close to (but still different from) unity (the 3σ fit is log SFR = −6.8(±0.2) + 0.88(±0.06) log M_dg; correlation coefficient = 0.99). As in the previous section, the values for the probability of false correlation and the linear correlation coefficient have been derived from the partial correlation technique (e.g., Wall & Jenkins 2003), indicating that the global correlation is not driven by the wide range of distances to the sources. Therefore, from the statistical perspective the SFR−M_dg correlation can be extended from galaxies and clouds to young stars when Ṁ_acc and M_disk are used for the latter objects. Once again the inclusion of additional data or the use of different scaling factors between the observations and the physical parameters compared would not have a significant effect on our discussion (Appendices A and C). Moreover, one may still think that any stellar parameter could appear to be similarly correlated on a log–log diagram when such a huge range in both axes is considered, but Appendix C shows that this is not the case.

Fig. 2. Same as Fig. 1 including galaxies (red circles; Gao & Solomon 2004). The fit for the stars and clouds is now indicated by the region enclosed within the green dotted line. The red dotted region is the 3σ global fit for all sources. The green open circle is the sum of 4 × 104 clouds with a typical SFR and Mdg, and the red solid square is the direct measurement of SFR and Mdg in the MW (see text).

Fig. 3. Accretion timescales Mcs/ Ṁacc = 0.1, 1, 5, and 10 Myr are indicated from top to bottom with the dashed lines. The original data for the clouds and the galaxies in Fig. 2 (green and red open circles) are shifted −1.3 dex in the x-axis (green and red filled circles), corresponding to a fraction of gas directly associated with star formation of 5% the total dense gas mass measured in those systems.

Wu & Evans (2005) proposed that the correlation in galaxies can be understood from the sum of basic units within each galaxy, concluding that these are clumps that host high-mass young stars. However, Appendix B shows that distance dependencies and the fact that the correlation extends to local clouds that do not generally form massive stars (Lada et al. 2010, 2012) makes this conclusion less robust (see also Stephens et al. 2016). Recently, Shimajiri et al. (2017) proposed that the relation between kiloparsec-size galaxies and parsec-size clouds may be governed by the “microphysics” of core and star formation along filaments within the clouds. Following these lines of reasoning and based on Fig. 2 and our results in the previous section, we propose that, as basic products of star formation, the (sub-)parsec-size individual protostellar systems may ultimately drive the observed correlation not only for protostellar-hosting clouds but also for cloud-hosting galaxies.

Our Galaxy follows the global correlation showing log SFR ~ 0.3 (Chomiuk & Povich 2011; Kennicutt & Evans 2012) and log M_dg ~ 8.3 (e.g., Stephens et al. 2016, and references therein). These values can be roughly recovered, for example, by summing up a reasonable value of ~4 × 10⁴ clouds with a typical log SFR = −4 and log M_dg = 3.5. Regardless of the exact numbers, the bottom line is that the SFR and M_dg estimates for the MW or any other galaxy could in principle be derived also from the sum of the accreting stellar population within each galaxy, since we have shown in the previous section that the SFR and M_dg values for the clouds scale with the sum of the Ṁ_acc and M_cs values of the stars hosted by them. For the same reason argued when the SFRs of the clouds and the instantaneous accretion rates of the stars were compared, if our bottom-up hypothesis can be extended to whole galaxies, then their SFRs should be roughly constant along a timescale of 10–100 Myr, given that such a timescale is the one usually considered to estimate galactic SFRs (Appendix A).

In addition, because estimates for clouds and galaxies are averaged over the entire systems, we show in Appendix D that our bottom-up hypothesis firstly implies that the intercepts in the linear relations derived for stars, clouds, and galaxies individually should be roughly similar, and secondly that the scatter should decrease from stars to galaxies. The second consequence is clearly observed in our data (the y-axis dispersion in galaxies decreases again to ~±0.5 dex, see also Sect. 2), in the more complete sample of clouds and galaxies studied by Stephens et al. (2016), or in the very recent work by Tan et al. (2018; but see also Lada et al. 2012; Shimajiri et al. 2017). The first consequence presents some caveats. As discussed in Appendix D, the least-squares intercepts should be similar at least if all stars have comparable Ṁ_acc/M_cs ratios (see Sect. 2.3) and if the fraction of dense gas directly involved in star formation is f ~ 100%. Under these assumptions, the gas depletion timescale derived for galaxies and clouds from least-squares fitting (M_dg/SFR) should be the same as the accretion timescale of the stars (M_cs/ Ṁ_acc), but the former should be larger than the latter if f < 100%. Indeed, Fig. 3 shows that whereas an accretion timescale ranging between ~0.1 and 10 Myr matches the Class II stars, the original data for the clouds and the galaxies fall displaced to the right, consistent with larger timescales. Shifting the clouds and galaxies between ~–0.5 and −2 dex in the x-axis provides a better match with the stars, finally resulting in a global correlation with near unity slope and an intercept more consistent with the stellar accretion timescale (e.g., for the shift shown in Fig. 3, log SFR = −6.3(±0.1) + 0.98(±0.02) log M_dg). Given that ΣM_cs = f M_dg, the previous shift range corresponds to 0.01 < f < 0.35, suggesting that the amount of gas directly participating in star formation should be roughly between 1% and 35% the total dense gas mass measured in clouds and galaxies, in agreement with the observational data discussed in the previous section. Once again the underlying assumption is that the dominating Class 0/I sources have a similar accretion timescale as observed for Class II stars, for which we remark that accurate measurements of Ṁ_acc and M_cs for Class 0/I stars are the most critical observations needed to test the bottom-up hypothesis.

4. Summary and conclusions

The gas-to-star transformation rate and the dense gas mass available for star formation show a statistically significant correlation that has been extended from galaxies and Galactic clouds to individual young stars. The correlation spreads ~16 orders of magnitude, covering spatial scales from distant (hundreds of Mpc) galaxies with sizes of a few Kpc, to nearby (~150 pc) protoplanetary disks with sizes of hundreds of au. Given the difference between the intercepts of the individual linear fits determined for protoplanetary disks, star-forming clouds, and galaxies, it is not clear how physically meaningful the global correlation is. Here we have proposed and explored one possible unifying hypothesis that could explain the global correlation.

Our bottom-up hypothesis is that the correlation between the mass accretion rate and the total circumstellar gas mass surrounding young – mainly Class 0/I – stars is the underlying cause of the global correlation on all scales. Two main lines of evidence supporting this scenario have been provided. First, the SFRs of molecular clouds are recovered from the sum of empirically based stellar accretion rates. Second, the sum of the individual circumstellar gas masses appears to represent a relevant fraction of the total dense gas mass within such clouds. If that fraction is between ~1% and 35 % of the dense gas mass in clouds and galaxies, then the global correlation has a near unity slope and intercept consistent with the stellar accretion timescale, M_cs/ Ṁ_acc. Indeed, a crucial test of our hypothesis is that the Ṁ_acc−M_disk linear correlation on astronomical unit scales for Class II stars should also be observed between Ṁ_acc and M_cs on (sub-)parsec scales for Class 0/I sources. Our hypothesis also explains that the scatter in the correlation decreases from stars to galaxies, and implies that the average SFR of galaxies remains constant on a timescale of ~10–100 Myr.

The Ṁ_acc−M_disk and SFR−M_dg correlations in stars, clouds, and galaxies have been independently studied from physical contexts relatively isolated from each other. Our result suggests a global approach and that theoretical efforts should consider all scales and physical systems involved in a single correlation.

Acknowledgments

The authors thank the anonymous referee for the comments, which have served to improve this manuscript. The authors sincerely acknowledge M. Aberasturi, J. C. Muñoz-Mateos, J. M. Mas-Hesse, P. Padoan, N. Cunningham, and B. Montesinos for their help. IM acknowledges the Government of Comunidad Autónoma de Madrid, Spain, which has funded this work through a “Talento” Fellowship (2016-T1/TIC-1890).

References

Appendix A: Estimates of the different parameters for stars, clouds, and galaxies

This appendix refers to the different ways that the parameters compared in Figs. 1 and 2 were derived.

Stars: Values of Ṁ_acc refer to the instantaneous disk-to-star accretion rates, which were derived by fitting an accretion shock model to the extinction-corrected near-ultraviolet (UV) excess. Values of M_disk were calculated from ALMA submillimeter continuum data using typical assumptions of a single dust grain opacity κ(890 μm) = 3.37 cm² g⁻¹, a single dust temperature T_dust = 20 K, and a gas-to-dust ratio of 100. Further details can be found in Manara et al. (2016) and references therein. The typical densities of protoplanetary disks are n(H₂) > 10⁵ cm⁻³, which can be as high as >10⁸ cm⁻³ in the innermost, densest regions (see, e.g., Bergin et al. 2007, and references therein).

Nearby clouds: The SFRs were derived by counting the number of young stars within each cloud (N_stars), assuming a typical IMF mass (M_* = 0.5 M_⊙), and a common age for all sources (t_* = 2 Myr). Then SFR(M_⊙ yr⁻¹) = N_stars × M_* /t_* = 0.25 × 10⁻⁶ N_stars². In turn, M_dg was calculated from Two Micron All Sky Survey (2MASS) extinction maps of the clouds with A_K > 0.8 mag and a gas-to-dust ratio of 100, which involves gas densities n(H₂) > 10⁴ cm⁻³. Further details can be found in Lada et al. (2010).

Distant clouds: SFR and M_dg values were derived from the IR luminosities (L_IR, from 8 to 1000 μm) and the HCN (1–0) luminosities (L_HCN) listed in Wu & Evans (2010), following the same procedure as for the galaxies in Gao & Solomon (2004), described in the following.

Galaxies: SFRs were derived from unresolved measurements of L_IR, assuming that this is produced primarily from dust heating by O, B, and A stars, a typical Salpeter IMF, and applying SFR (M_⊙ yr⁻¹) = 2 × 10⁻¹⁰ × (L_IR/L_⊙). This methodology implicitly assumes that the SFR is constant along a timescale of ~100 Myr. Values of M_dg were estimated from unresolved measurements of L_HCN by applying M_dg = α_HCN × L_HCN, with α_HCN = 10 M_⊙ (K km s⁻¹ pc²)⁻¹ a constant derived assuming that L_HCN is emitted by the gravitationally bound cloud cores with a brightness temperature of 35 K. The gas densities involved are also n(H₂) > 10⁴ cm⁻³. Further details can be found in Gao & Solomon (2004). It is noted that recent works claim that HCN (1–0) traces densities an order of magnitude smaller (Pety et al. 2017; Kauffmann et al. 2017b). However, Kauffmann et al. (2017b) show that even in this case the M_dg values derived from L_HCN would differ from those in Gao & Solomon (2004) by less than a factor of two, which would not have a significant effect on our results.

Despite the fact that there are differences in the methodologies, spatial, and temporal scales involved to derive the parameters, all are referred to the same two physical concepts: the rate at which gas is transformed into stars, and the mass of dense gas related to stellar formation. Thus, the comparison between them can and should be carried out. Lada et al. (2012) compared the SFRs and M_dg values in clouds and galaxies, as estimated from the different methodologies summarized above (see also, e.g., Shimajiri et al. 2017). Therefore, in our work we have mainly focused on the comparison between Ṁ_acc−SFR and M_disk−M_dg in stars and clouds, when the latter were derived from direct counts of stars and extinction maps, respectively.

Appendix B: Re-analysis of the Wu & Evans(2005) sample of distant Galactic clouds

Figures 1 and 2 do not include the distant Galactic clouds from Wu & Evans (2010) with log M_dg < 2.50 (log L_HCN < 1.50). As shown in Wu & Evans (2005), these clouds do not follow the linear correlation but have a super-linear slope. In other words, for a given M_dg below that limit, distant Galactic clouds have a smaller SFR than predicted from the trend followed by the remaining Galactic clouds and the galaxies. Figure B.1 (left) shows that the ratio SFR/M_dg of the clouds from Wu & Evans (2010) actually increases with M_dg up to the mentioned limit, remaining roughly constant from that point. Based on this change of slope, Wu & Evans (2005) proposed the existence of a basic unit in star formation: if M_dg is below the mass of that unit, SFR/M_dg rises with M_dg because clouds with low M_dg do not sample the IMF completely by excluding high-mass stars. If M_dg is above the mass of that unit, then the IMF is completely sampled and additional increases of M_dg produce more units but not further increases in SFR/M_dg. In short, Wu & Evans (2005) proposed that the star formation correlation in other galaxies can be understood from the sum of basic units within each galaxy, these units being clouds with high-mass young stars. However, Fig. B.1 (right) shows that the ratio SFR/M_dg also follows a trend with the distance to the clouds that is very similar to that with M_dg. The SFR/M_dg ratio also increases for d < 2.5 kpc, remaining roughly constant from that point. The Spearman’s probability of false correlation accounting for the distances from the partial correlation technique (e.g., Wall & Jenkins 2003) is 0.1 when only the 22 clouds with log M_dg < 2.50 from Wu & Evans (2010) are plotted in the SFR−M_dg plane. This high p-value (>0.05) indicates that the SFR−M_dg correlation for those sources is spurious since this is mainly driven by the d ² factor used to compute both parameters compared. Therefore, it cannot be concluded that the change of the SFR/M_dg ratio is caused by a physical limit in M_dg because the influence of the distance cannot be ruled out. In conclusion, the proposal of high-mass star-forming clouds as the basic units driving the linear correlation cannot be supported from that data.

Fig. B.1. SFR/Mdg ratio versus the dense gas mass (left) and the distance (right) for all distant Galactic clouds from Wu & Evans (2010). Red-filled and open squares correspond to log Mdg < and > 2.50, respectively (log LHCN < and > 1.50).

In addition, the break of the SFR−M_dg linear relation for clouds with low M_dg could be explained from the fact that L_IR was used as a tracer to measure the SFR. Most of the IR is emitted by high-mass stars and as soon as they are missing L_IR could no longer be a good proxy of the SFR. In either case, by using different tracers for M_dg and SFR Lada et al. (2010, 2012) already showed that the linear correlation is also valid for nearby clouds that do not necessarily host massive stars.

Appendix C: Is the global correlation just the result of a large-scale, log–log diagram?

One may wonder if the enormous extent covered by the axes in Fig. 2 somewhat hides the scatter and washes out significant trends within the data or, in other words, if any parameter could appear correlated simply because we are zooming out. Whereas the scale considered certainly has an effect on whether this or any other empirical trend is identified or not, not all data fit within the global correlation analyzed in this work.

We have compiled additional data on top of that analyzed in Fig. 2, which is plotted in Fig. C.1 (references included in the caption). First, recent ALMA measurements of Class II stars in the Cha I star-forming region (blue crosses) have been added to the Lupus sources, both constituting the only complete data for Class II stars available to date. Even when the origin of the large scatter in Ṁ_acc is subject to debate (accretion variability, observational limits, sample selection effects, different evolutionary stages and initial conditions in previous phases; see, e.g., Mendigutía et al. 2015; Manara et al. 2016; Mulders et al. 2017, and references therein), it has been shown that there is a statistically significant, roughly linear correlation in each star-forming region separately (Manara et al. 2016; Mulders et al. 2017). Indeed, considering the Lupus and Cha I samples together, we derive a slope ~0.9 ± 0.2. Concerning the rest of the new data in Fig. C.1, the samples are selected because they have been reported to show deviations with respect to the linear scaling relation. For instance, the additional sample of (U)LIRGS (red crosses) shows a slightly super-linear slope. On the contrary, pointings of specific regions in disks of other galaxies (orange circles, see also, e.g., Bigiel et al. 2016) show a sub-linear slope, as well as clouds close to the Galactic center (green triangles with error bars) when these are considered alone. These sub-trends are relatively washed out under the axes range considered here and are roughly incorporated to the global correlation. In addition, the central molecular zone (orange squares) and some massive clumps in our Galaxy from the Millimetre Astronomy Legacy Team 90 GHz (MALT90) survey (green crosses with ~log M_dg < 4) appear displaced from the global correlation even when such a large axes range is used. Despite the fact that the tracers and methodology used to derive the SFR and M_dg values in many of the samples included in Fig. C.1 are the same as those described in Appendix A, different approaches have also been used (see, e.g., Longmore et al. 2013; Kauffmann et al. Shimajiri et al. (2017) may solve most apparent deviations with respect to the global correlation from a careful determination of the conversion factors between the observables and the physical parameters involved (see also, e.g., Lada et al. 2012; Tan et al. 2018), which have not been considered in this work. However, even if the correlation changes in certain environments, the scope of this paper is not to explain the possible sub-trends within the global correlation but to point out that the individual stars could eventually play a role in such an explanation by means of the bottom-up hypothesis.

Fig. C.1. Data and linear fits shown in Fig. 2 including additional samples: Class II stars in the Cha I star-forming region (blue crosses; Mulders et al. 2017); massive Galactic clumps from the MALT90 survey (green crosses; Stephens et al. 2016); molecular clouds close to the Galactic center (green triangles; Kauffmann et al. 2017a); the central molecular zone (CMZ) in our Galaxy (orange squares; Longmore et al. 2013), regions within disks of other galaxies (orange circles; Usero et al. 2015), (U)LIRGS and star-forming galaxies (red crosses and red squares; García-Burillo et al. 2012; Usero et al. 2015). On the other hand, the accretion rates in Lupus and Cha I are plotted versus the stellar masses instead of the disk masses (black circles and crosses; Manara et al. 2016; Mulders et al. 2017).

In contrast, Fig. C.1 also shows that when the stellar masses are plotted instead of the disk masses (black circles and crosses), the well known Ṁ_acc−M_* correlation (e.g., Mendigutía et al. 2015; Mulders et al. 2017, and references therein) shows up with a very different slope and intercept, and can be clearly distinguished from the global correlation linking stars, clouds, and galaxies. This simple modification in one of the axes demonstrates that not any single stellar parameter appears to be correlated in a logarithmic scale plot spanning over a huge axes range.

Appendix D: Relation between least-squares intercepts for stars, clouds, and galaxies

The Ṁ_acc−M_disk linear relation observed in Class II stars can be generalized to also include the total circumstellar gas mass in Class 0/I sources. For a sample of young stars in a given cloud, this is expressed as log Ṁ_acc = log K_s + log M_cs. If least-squares linear regression is used, by definition the intercept is given by log K_s = 〈log Ṁ_acc〉−〈log M_cs〉, where the arithmetic means refer to the total number of stars in the cloud (N_s). The value of K_s can then be expressed as$$K_s={\left(\underset{i=1}{\overset{N_{\mathrm{s}}}{\mathrm{\Pi }}}\frac{{\dot{M}}_{{\text{acc}}_i}}{M_{{\text{cs}}_i}}\right)}^{1/N_{\mathrm{s}}}{=\left(\underset{i=1}{\overset{N_{\mathrm{s}}}{\mathrm{\Pi }}}{k}_{{\mathrm{s}}_i}\right)}^{1/N_{\mathrm{s}}},$$(D.1)

with k_csi = Ṁ_acci/M_csi the ratio between the accretion rate and the circumstellar gas mass for each star. Therefore, K_s is the geometric mean of the individual k_csi ratios for each star. Similarly, the least-squares intercept derived for a sample of N_c clouds provides a K_c value that is the geometric mean of the individual SFRj/M_dgj ratios of each cloud, and equivalently for a sample of galaxies. Under the hypothesis proposed in this work, the SFR of a given cloud is given by the sum of the individual accretion rates of the stars within that cloud. Similarly, the M_dg values are given by the sum of the total circumstellar gas masses of the stars within that cloud, divided by a factor f that accounts for the fraction of dense gas mass that is directly involved in star formation (i.e., for a given cloud ΣM_cs = f M_dg). Therefore,$$K_{\mathrm{c}}={\left(\underset{j=1}{\overset{N_{\mathrm{c}}}{\mathrm{\Pi }}}\frac{f_j{\langle {\dot{M}}_{\text{acc}}\rangle }_j}{{\langle M_{\text{cs}}\rangle }_j}\right)}^{1/N_{\mathrm{c}}}{=\left(\underset{j=1}{\overset{N_{\mathrm{c}}}{\mathrm{\Pi }}}{f}_j{k}_{{\mathrm{c}}_j}\right)}^{1/N_{\mathrm{c}}}\mathrm{.}$$(D.2)

Similarly, for a sample of N_g galaxies,$$K_{\mathrm{g}}={\left(\underset{k=1}{\overset{N_{\mathrm{g}}}{\mathrm{\Pi }}}\frac{f_k{\langle {\dot{M}}_{\text{acc}}\rangle }_k}{{\langle M_{\text{cs}}\rangle }_k}\right)}^{1/N_{\mathrm{g}}}{=\left(\underset{k=1}{\overset{N_{\mathrm{g}}}{\mathrm{\Pi }}}{f}_k{k}_{{\mathrm{g}}_k}\right)}^{1/N_{\mathrm{g}}}\mathrm{.}$$(D.3)

Equations (D.1)–(D.3) show that the intercepts of clouds and galaxies are determined mainly by the ratio between the accretion rate and the circumstellar gas mass of the young stars hosted by them. The simplest scenario occurs when this ratio is the same for all stars, that is, k_si = k_s = Ṁ_acc/M_cs, and the fraction of dense gas directly participating in stellar accretion is the same for all clouds and galaxies. In this case f K_s = K_c = K_g, that is, the intercepts of clouds and galaxies are the same as for the stars only if the amount of gas directly participating in star formation is 100% of the dense gas in clouds and galaxies (f = 1), and smaller than the stars otherwise (0 < f < 1). From a physical perspective, ${\tau }_{\mathrm{s}}=k_{\mathrm{s}}^{-1}$ is the accretion timescale on which all M_cs would fall onto a given star for a constant mass accretion rate Ṁ_acc. Under the assumption of equal accretion to circumstellar gas ratios and f -factors, τ_s/ f = τ_c = τ_g, which means the gas depletion timescales for clouds and galaxies are the same as the accretion timescales for the stars only if the amount of gas directly participating in star formation is 100% of the dense gas in clouds and galaxies (f = 1), and larger than the stars otherwise (0 < f < 1).

Finally, when a single cloud is considered, Eq. (D.2) reduces to $K_{\mathrm{c}}=f{k}_{\mathrm{c}}=f\langle {\dot{M}}_{\text{acc}}\rangle /\langle M_{\text{cs}}\rangle =\sum _{i=1}^{N_{\mathrm{s}}}f{k}_{{\mathrm{s}}_i}{M}_{{\text{cs}}_i}/\sum _{i=1}^{N_{\mathrm{s}}}{M}_{{\text{cs}}_i}\mathrm{.}$ For a given M_cs value, the scatter in Ṁ_acc for the stars can be quantified through a fixed value σ_ksi = σ_ks, which translates to$$\sigma k_{\mathrm{c}}=\sigma k_{\mathrm{s}}\times {\left(\frac{\sum _{i=1}^{N_{\mathrm{s}}}{M}_{{\text{cs}}_i}^2}{{\left(\sum _{i=1}^{N_{\mathrm{s}}}{M}_{{\text{cs}}_i}\right)}^2}\right)}^{1/2}<\sigma k_{\mathrm{s}}\mathrm{.}$$(D.4)

Consequently, for the galaxies, σ_kg < σ_kc. Therefore, the y-axis scatter in the global correlation should be maximum for stars and decreases from clouds to galaxies.

All Figures

Fig. 1. Star formation rate versus dense gas mass for distant and nearby Galactic star-forming clouds (green circles with and without horizontal error bars, respectively; Wu & Evans 2010; Lada et al. 2010) and mass accretion rate versus disk mass in Class II stars in Lupus (blue circles, upper limits as one-sided bars; Manara et al. 2016). Error bars are indicated when provided in the original works. The 3σ linear fit for stars and clouds is indicated with the green dotted region. The green solid square is the direct measurement of SFR and Mdg in Lupus. The blue solid circle is the sum of the accretion rates and disk masses of the Class II stars in Lupus, and the blue open circle includes 20 Class 0/I stars in that cloud with an average accretion rate of 1 × 10−6 M⊙ yr−1 and a minimum circumstellar mass of 0.1 M⊙, that is, a maximum displacement in the x-axis given by f = 1% of the total dense gas mass in Lupus (see text).

In the text

Fig. 2. Same as Fig. 1 including galaxies (red circles; Gao & Solomon 2004). The fit for the stars and clouds is now indicated by the region enclosed within the green dotted line. The red dotted region is the 3σ global fit for all sources. The green open circle is the sum of 4 × 104 clouds with a typical SFR and Mdg, and the red solid square is the direct measurement of SFR and Mdg in the MW (see text).

In the text

Fig. 3. Accretion timescales Mcs/ Ṁacc = 0.1, 1, 5, and 10 Myr are indicated from top to bottom with the dashed lines. The original data for the clouds and the galaxies in Fig. 2 (green and red open circles) are shifted −1.3 dex in the x-axis (green and red filled circles), corresponding to a fraction of gas directly associated with star formation of 5% the total dense gas mass measured in those systems.

In the text

	Fig. B.1. SFR/Mdg ratio versus the dense gas mass (left) and the distance (right) for all distant Galactic clouds from Wu & Evans (2010). Red-filled and open squares correspond to log Mdg < and > 2.50, respectively (log LHCN < and > 1.50).
In the text

Fig. C.1. Data and linear fits shown in Fig. 2 including additional samples: Class II stars in the Cha I star-forming region (blue crosses; Mulders et al. 2017); massive Galactic clumps from the MALT90 survey (green crosses; Stephens et al. 2016); molecular clouds close to the Galactic center (green triangles; Kauffmann et al. 2017a); the central molecular zone (CMZ) in our Galaxy (orange squares; Longmore et al. 2013), regions within disks of other galaxies (orange circles; Usero et al. 2015), (U)LIRGS and star-forming galaxies (red crosses and red squares; García-Burillo et al. 2012; Usero et al. 2015). On the other hand, the accretion rates in Lupus and Cha I are plotted versus the stellar masses instead of the disk masses (black circles and crosses; Manara et al. 2016; Mulders et al. 2017).

In the text