Online material

Appendix A: A catalog of dense cores identified with Herschel in the Aquila cloud complex

Based on our Herschel SPIRE/PACS parallel-mode imaging survey of the Aquila cloud complex, we identified a total of 749 dense cores, including 685 starless cores and 64 protostellar cores. (Among these, 34 starless cores shown as yellow triangles in Fig. 1, as well as 6 protostellar cores, were excluded from the scientific discussion of Sect. 5 due to likely contamination by more distant, background objects – see Sect. 4.3.) The master

catalog listing the observed properties of all of these Herschel cores is available in Table A.1. A template of this online catalog is provided below to illustrate its form and content.

The derived properties (physical radius, mass, SED dust temperature, peak column density at the resolution of the 500 μm data, average column density, peak volume density, and average density) are given in Table A.2 for each core. A portion of this online table is also provided below. The derived properties of the Herschel-detected protostars and YSOs will be published in a forthcoming paper.

Appendix B: Completeness of HGBS prestellar core extractions in Aquila

To estimate the completeness of our census of prestellar cores in Aquila, we used several sets of simulated data on the one hand (Appendix B.1), and a simple model of the core extraction process and completeness problem on the other (Appendix B.2).

Appendix B.1: Monte-Carlo simulations

To simulate real core extractions, we first constructed clean maps of the background emission at all Herschel wavelengths (including a column density plane), by subtracting the emission of the compact cores identified with getsources in the observed data (cf. Sects. 4.4 and 4.5). We then inserted several sets of model Bonnor-Ebert-like cores throughout the clean-background images in order to generate a full set of synthetic Herschel and column density images of the region. In the example illustrated in Figs. B.1 and B.2, for instance, we used a population of 5622 model starless cores with a flat input mass distribution (dN/dlog M ∝ M^-0.7, similar to the mass distribution of CO clumps) from 0.02 M_⊙ to ~ 30 M_⊙. This example is particularly useful as it allowed us to test the robustness of the conclusion that the observed prestellar CMF is significantly steeper than the mass distribution of CO clumps. The model cores had positions in a mass versus size diagram consistent with critical Bonnor-Ebert isothermal spheres at effective gas temperatures ~7–20 K. The dust continuum emission from the synthetic Bonnor-Ebert cores in all Herschel bands was simulated using an extensive grid of spherical dust radiative transfer models constructed by us with the MODUST code (e.g., Bouwman et al. 2000; Bouwman 2001). In particular, each of the synthetic prestellar cores was given a realistic dust temperature profile with a significant drop in dust temperature toward core center, as observed in the case of spatially-resolved starless cores (cf. Roy et al. 2014). The synthetic cores were spatially distributed randomly over the regions of the column density map where cm^-2 (containing most, if not all, of the observed prestellar cores in the real data – see Sect. 5.2), with no particular mass segregation. Once satisfactory synthetic skies resembling the observed images had been generated, compact source extraction and core selection/classification were performed with getsources in the same way as for the real data (see Sects. 4.4 and 4.5).

As mentioned in Sect. 4.8 and shown in Fig. 8, the results of these Monte-Carlo simulations suggest that our Herschel census of prestellar cores in the Aquila cloud complex is ~ 90% complete down to ~ 0.3 M_⊙ in true core mass. Figure B.1 further illustrates that the core mass function can be reliably determined down to the completeness mass limit. In this example, a Kolmogorov-Smirnov (K-S) test shows that the derived CMF is statistically indistinguishable (at the ~ 90% confidence level) from the input mass function above the completeness limit. In particular, the best-fit power-law function to the derived CMF (black solid line in Fig. B.1) is identical to the input dN/dlogM ∝ M^-0.7 power law. This test therefore confirms that the best-fit power law to the observed CMF (dN/dlogM ∝ M^{− 1.33 ± 0.06} – see Sect. 5.5) is significantly steeper than the typical mass distribution of CO clumps/clouds (dN/dlogM ∝ M^-0.7 – e.g. Blitz 1993; Kramer et al. 1998) and that this cannot be an artifact of the core extraction process.

The same Monte-Carlo simulations were also used to assess the accuracy of the main derived parameters (e.g. core mass, radius, and dust temperature) by comparing the estimated values after core extraction to the intrinsic input values of the model cores. Figure B.2a shows that the derived core masses tend to underestimate the true core masses by ~20–30% on average, and Fig. B.2b shows that the derived SED temperatures tend to overestimate the intrinsic mass-averaged dust temperatures of the cores by typically ~ 1 K. A similar plot for the core sizes (Fig. B.3) suggests that the derived core sizes (prior to deconvolution) are quite reliable and remain within ~ 5% of the true convolved core sizes on average. We interpret the mass effect (Fig. B.2a) as a direct consequence of the temperature effect (Fig. B.2b) since overestimating the dust temperatures leads to underestimating the core masses. The temperature effect arises from the fact that the dust temperature derived from a global fit to the SED of a starless core overestimates the mass-averaged dust temperature owing to a distribution of dust temperatures along the line of sight (see Roy et al. 2014, and Sect. 4.6).

Taking the ~20–30% mass effect into account, we conclude that the ~ 90% completeness limit at ~ 0.3 M_⊙ in true core mass corresponds to ~ 0.2 M_⊙ in observed core mass.

Appendix B.2: Model of the completeness problem

The Monte-Carlo simulations described above provide an estimate of the global completeness limit of the core survey. The completeness level of the core extractions is, however, expected to be background dependent. To assess the importance of this dependence, we constructed a simplified model of the core extraction process.

Owing to the high sensitivity and quality of the Herschel images, the HGBS survey is not limited by instrumental noise but by confusion arising from small-scale cloud structure, an effect commonly referred to as “cirrus confusion noise” in the literature (e.g., see Gautier et al. 1992; Kiss et al. 2001; Roy et al. 2010). To estimate the level of such cirrus confusion noise from the Herschel data, we measured the rms level of background fluctuations in a sliding box 1′ × 1′ in size²⁴ over the entire column density map of the Aquila complex after subtracting the sources identified by getsources. Correlating the resulting map of rms fluctuations with the input background column density map led to Fig. B.4, which clearly shows that the level of column density fluctuations increases with background column density approximately as a power law: (B.1)The power-law index of 1.6 derived here from Herschel data is very similar to that reported in earlier papers discussing cirrus noise (e.g. Gautier et al. 1992; Kiss et al. 2001; Roy et al. 2010). Since the level of background fluctuations increases with column density, one expects core extraction to be increasingly more difficult and thus survey completeness to decrease significantly in higher column density areas within the field.

The model we used to estimate the magnitude of this effect and get around the problem of a background-dependent completeness level was based on the following assumptions:

• •

  A dense core is defined as the immediate vicinity of a column density peak departing significantly, i.e., by more than 5 × N_H2,rms from the field of background cloud fluctuations.

• A prestellar core, i.e., a self-gravitating starless core, can be approximately modeled as a critical Bonnor-Ebert spheroid of mass M_BE and outer radius R_BE, bounded by the gravitational pressure of the background cloud (McKee & Tan 2003), where Σ_back = μm_H × N_H2,back. The mean intrinsic column density contrast of such a model prestellar core is Σ_BE/ Σ_back ~ 1.5, where .

• The ability to detect a core in the Herschel data depends primarily on the apparent column density significance of the core defined as Σ_core,obs/ Σ_rms, where Σ_core,obs is the apparent (observed) column density of the core after convolution with the observing beam, i.e., , and Σ_rms = μm_H × N_H2,rms. The Monte-Carlo simulations of Appendix B.1 are consistent with this assumption and suggest that the completeness level is ≳90% for cores with an apparent column density significance larger than 5 (see Fig. B.5).

In outline, our simplified model of the completeness problem may be described as follows:

• Two effects, beam dilution and temperature dilution, can make the apparent column density contrast Σ_BE,obs/ Σ_back of a model core smaller than its intrinsic column density contrast of 1.5: where ν_eff is a fiducial Herschel observing frequency which we take to correspond to λ ~ 350 μm. Taking advantage of the fact that the column density distribution of a Bonnor-Ebert core with outer radius R_BE is well approximated by a Gaussian distribution of FWHM ~ R_BE, the observed radius of the core is approximately (where corresponds to the half-power beam width resolution of the column density map projected at the distance of the Aquila cloud), and the beam dilution factor can thus be expressed as .

• The apparent column density significance can be written as the product of the apparent column density contrast and a cirrus noise factor, Σ_BE,obs/ Σ_rms = (Σ_BE,obs/ Σ_back) × (Σ_back/ Σ_rms), where the cirrus noise factor is: according to Eq. (B.1).

• Assuming that the fundamental completeness curve is the completeness function of apparent column density significance shown in Fig. B.5, completeness can be estimated as a function of core mass and background column density as . The corresponding function of M_BE is shown for five values of the background column density N_H2,back in Fig. B.6. Figure B.6 shows how the completeness of prestellar core extractions is expected to decrease as background cloud column density and cirrus noise increase.

  	Fig. B.5 Completeness curve as a function of apparent column density significance over local background cloud fluctuations derived from the Monte-Carlo simulations described in Appendix B.1.
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  	Fig. B.6 Model completeness curves of Herschel prestellar core extractions in Aquila for five values of the background cloud column density expressed in units of visual extinction from AV,back = 5 to AV,back = 20.
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• To estimate a global completeness curve for our census of prestellar cores in the Aquila complex, we used the observed distribution of mass in the cloud as a function of background column density (cf. Figs. 5a,b) and took advantage of the existence of a column density “threshold” at A_V,back ~ 5–7, above which the bulk of core and star formation is believed to occur (cf. Sect. 5.2 and Fig. 11) and the column density PDF is well fitted by a power-law distribution. We also assumed that the number of prestellar cores in the cloud scales linearly with cloud mass above the threshold. This assumption is consistent with recent infrared studies which find that the global star formation rate tends to be linearly proportional to the mass of dense gas above the threshold (e.g., Heiderman et al. 2010; Lada et al. 2010; Gao & Solomon 2004). It is also consistent with the roughly constant prestellar core formation efficiency found here above the threshold (see Fig. 12). The global completeness curve was thus computed as a weighted average of the individual completeness curves at fixed background column densities:

The resulting global completeness curve, which represents the best estimate of the completeness of our Herschel survey for prestellar cores in Aquila according to our model, is shown in Fig. B.7. It can be seen that this global completeness curve is very similar to the individual completeness curves for background column densities close to the threshold (see A_V,back = 5–10 curves in Fig. B.6). It is also very similar to the empirical completeness curve derived from Monte-Carlo simulations (see Sect. 4.8). The model completeness curve is almost flat above a true core mass level of 0.3 M_⊙. Using this model curve to correct the observed CMFs of candidate and robust prestellar cores for incompleteness would only have a minimal effect in Fig. 16 above an observed core mass level of ~ 0.2 M_⊙. (The corrected CMFs differ from the uncorrected CMFs only below ~ 1 M_⊙ and by much less than the uncertainty area displayed in light blue in Fig. 16.)

Appendix C: Effect of distance uncertainty

As mentioned in Sect. 2, there is some ambiguity concerning the distance to the Aquila molecular cloud complex. A number of arguments, presented by Bontemps et al. (2010) and

summarized in Sect. 2, suggest that the bulk of the region studied here and shown in Fig. 1 corresponds to a coherent cloud complex at d₋ = 260 pc (see also Gutermuth et al. 2008), which is the default distance adopted in the present paper. Other studies in the literature (see references in Sect. 2), however, place the complex at the larger distance, d₊ = 415 pc, of the Serpens Main cloud (Dzib et al. 2010). It is thus worth discussing how our results would be affected if we had adopted the larger distance estimate, d₊, instead of d₋. The core mass estimates, which scale as S_νd²/ [ B_ν(T_d) κ_ν ] where S_ν is integrated flux density and B_ν(T_d) is the Planck function, would systematically increase by a factor of 2.5. This would shift the CMFs shown in Figs. 16 and 17 to the right and thus lower the efficiency ϵ_core from to 0.2 ± 0.1. In comparison, the core size estimates, which scale linearly with distance d, would increase by only 60%. The BE mass ratio α_BE = M_BE,crit/M_obs, listed in Col. 17 of Table A.2, scales as d^-1 and would decrease by 60% for all cores. Accordingly, all cores would move upward as indicated by an arrow in the mass versus size diagram of Fig. 7, which would increase the fraction of prestellar cores among starless cores from 60% ± 10% to 70% ± 10%. More precisely, the number of candidate prestellar cores would increase from 446 to 565 and the number of robust prestellar cores would increase from 292 to 391, while the total number of starless cores (651) would remain the same. Accordingly, the estimated lifetime of candidate prestellar cores would also slightly increase from ~ 1.4 Myr to ~ 1.8 Myr, and that of robust prestellar cores from ~ 0.9 Myr to ~ 1.3 Myr (see Sect. 5.1), leading to t_pre = 1.5 ± 0.3 Myr. The prestellar core formation efficiency (CFE) as a function of background column density (cf. Fig. 12), and in particular the roughly constant value CFE_max ≡ f_pre ~ 15% at high column densities, would not change. Our corresponding estimate of the “efficiency” of the star formation process in dense gas (cf. Sect. 5.6), SFR /M_dense = f_pre × ϵ_core/t_pre, would however decrease from 5 × 10^-8 yr^-1 to 2 × 10^-8 yr^-1, becoming closer to the efficiency value reported by Evans et al. (2014) and Gao & Solomon (2004) than to the value found by Lada et al. (2010). Finally, the column density maps shown in Figs. 1, 3, 4, 13, and 14, as well as the spatial correspondence between cores and filaments, would remain unchanged. The scaling of our column density maps in terms of mass per unit length along the filaments would however change by ~ 60% upward, since the characteristic physical width of the filaments would increase by ~ 60%. As a consequence, the white areas which highlight supercritical filaments in Figs. 3, 4, 13, and 14 would slightly expand, improving the correspondence between the spatial distribution of prestellar cores/protostars and that of supercritical filaments. To summarize, our main conclusions do not depend strongly on the adopted distance.

© ESO, 2015