EDP Sciences
Free Access
Issue
A&A
Volume 587, March 2016
Article Number A106
Number of page(s) 15
Section Interstellar and circumstellar matter
DOI https://doi.org/10.1051/0004-6361/201526597
Published online 26 February 2016

© ESO, 2016

1. Introduction

Understanding the physical processes that govern star formation is fundamental to improving our knowledge of galaxy formation and evolution across cosmic time. An essential step towards reaching a better comprehension of star formation inside our Galaxy is to establish the relationship between the star-formation rate (SFR) and the physical properties of the interstellar gas. Schmidt (1959) conjectured that “the SFR depends on the gas density and [...] varies with a power law of the gas density”. Schmidt argued that the index of the power law in the solar neighbourhood was ~ 2. Kennicutt (1998) expressed the law as , where Σgas and ΣSFR are the surface densities of the gas and of star-formation rate. Kennicutt tested the Schmidt law in a heterogeneous sample of star-forming galaxies and showed that a power-law scaling relation exists between them and is characterised by an index β = 1.6. Using observations of local disc and starburst galaxies, Bigiel et al. (2008) found a complex scaling between ΣSFR and Σgas, which could not be described by a single power law. On smaller, sub-kpc scales, Lada et al. (2013) showed that there is no Schmidt scaling relation between local giant molecular clouds, but a series of observational studies demonstrated that a Schmidt scaling law exists within some local molecular clouds (Heiderman et al. 2010; Gutermuth et al. 2011; Lombardi et al. 2013; Lada et al. 2013; Evans et al. 2014; Harvey et al. 2014).

thumbnail Fig. 1

Composite three-colour image showing Herschel/SPIRE intensities for the region considered, where available (with the 250 μm; 350 μm; 500 μm bands shown in blue, green, and red). For regions outside the Herschel coverage, we used the Planck/IRAS dust model (τ850, T, β) to predict the intensity that would be observed at the SPIRE passbands. Toggle labels

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To be able to understand the detailed interplay between the structure of molecular clouds and their local star formation rates (as opposed to the global one that is often considered), we carry out a coordinated study of molecular clouds in the Gould belt. We use a combination of Planck and Herschel dust emission data, calibrated using near-infrared (NIR) dust extinction. This technique guarantees an optimal resolution of the cloud column density maps (which corresponds with the Herschel/SPIRE 500 36 arcsec resolution), better than that of standard NIR extinction maps in the region of the sky we consider. In our first paper we considered the Orion giant molecular cloud (Lombardi et al. 2014); here we focus on Perseus.

The Perseus cloud was first observed by Barnard, who catalogued a portion of it as Barnard 1-5 and Barnard 202-206. Since then, the cloud has been extensively studied using molecular line emission (Ridge et al. 2006), star count extinction (Bachiller & Cernicharo 1984), and dust continuum emission (Hatchell et al. 2005; Kirk et al. 2006; Enoch et al. 2006). Distance estimates for Perseus vary (Herbig & Jones 1983; Cernis 1990, 1993; Bally et al. 2008; Hirota et al. 2008; Schlafly et al. 2014); we use 240 pc for consistency with Lombardi et al. (2010). Perseus is the prototype intermediate mass star-forming region, with young B stars and two clusters, IC 348 (Muench et al. 2007) and NGC 1333 (Lada et al. 1996). The complex seems to be associated with the Perseus OB2 association (de Zeeuw et al. 1999). Recently, Herschel observations were used to characterise small regions of the cloud. Pezzuto et al. (2012) studied two star-forming dust condensations, B1-bS and B1-bN, in the B1 region (see Fig. 1). They conclude that these two sources could be good examples of the first hydrostatic core phase. Sadavoy et al. (2012) present observations of the B1-E region and propose that it may be forming the first generation of dense cores. Finally, Sadavoy et al. (2014) identified and characterised 28 candidate Class 0 protostars in the whole cloud, four of which are newly discovered. They also found that the star-formation efficiency of the clumps, as traced by Class 0 protostars, correlates strongly with the flatness of their respective column density distribution at high values. A global study of the Perseus cloud properties with Herschel data is still missing, and constitutes the main goal of this paper.

Table 1

Herschel parallel mode observation used.

This paper is organised as follows. In Sect. 2 we present the data used in this work; in Sect. 3 we briefly describe the data-reduction process; in Sect. 4 we apply the method to study the Perseus molecular cloud and present the column density and temperature maps1; in Sect. 5 we derive the local Schmidt law; finally, in Sect. 6 we present a summary.

2. Data

Perseus was observed as part of the Gould Belt Survey (André et al. 2010), one of the Herschel satellite key projects. The cloud was observed using the two photometers PACS (Poglitsch et al. 2010) and SPIRE (Griffin et al. 2010), in five bands, centered approximately at 70 μm; 160 μm; 250 μm; 350 μm; 500 μm. Table 1 gives an overview of the observation used in this work. Dust maps derived by the Planck Collaboration (Planck Collaboration XI 2014), and the NIR extinction maps that were produced by Lombardi et al. (2010) through the NICEST method (Lombardi 2009) are also used. We selected the region (corresponding to the Perseus molecular cloud) with galactic coordinates (1)Additionally, we employed the AllWISE data release (Wright et al. 2010; Mainzer et al. 2011) to select young stellar objects (YSOs) from their colours. The WISE satellite observed the whole sky in four infrared bands, often referred to as W1, W2, W3, and W4, with wavelengths centred at 3.4 μm; 4.6 μm; 12 μm; 22 μm. We retrieved 1.43 million sources from the WISE point source catalog in the science field and 1.63 million sources in the control field. We rejected any source with contamination and confusion flags and we further restricted the selection to those measurements with errors σ< 0.15 mag in all the WISE bands. The AllWISE source catalog contains associations with the 2MASS Point Source Catalog (Skrutskie et al. 2006). The precision adopted by the AllWISE collaboration for the 2MASS association is 3′′. In addition, we required photometric errors in the 2MASS bands to be less than 0.1 mag.

3. Method

The reduction technique is very close to the one presented by Lombardi et al. (2014) and therefore we only briefly summarise it in this section.

3.1. Dust model

Dust is optically thin at the Herschel observation frequencies (at least for λ> 160 μm), and therefore its emission can be modelled as a modified black body: (2)Here is the black-body function at the temperature T and the optical thickness τν is taken to be a power law of the frequency ν: (3)The frequency ν0 is an arbitrary reference frequency that we set as ν0 = 353 GHz (corresponding to λ = 850 μm), which is similar to what was done by the Planck collaboration (Planck Collaboration XI 2014).

thumbnail Fig. 2

Combined optical depth-temperature map for Perseus. The image shows the optical depth as intensity and the temperature as hue, with red (blue) corresponding to low (high) temperatures.

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3.2. Spectral energy distribution fitting

The aim of our work is to infer the effective dust colour temperature and the optical depth from the fluxes measured by Herschel at 250 μm; 350 μm; 500 μm, following Lombardi et al. (2014), and also PACS 160 μm where available. For this purpose, we first convolved all the Herschel data to the beam size of SPIRE 500 μm, i.e. to 36 arcsec and then we fitted the observed spectral energy distribution (SED) with the function (2), that was integrated over the Herschel bandpasses. The fitting procedure is described in detail in Lombardi et al. (2014). We report here, however, a brief outline of the basic steps used to generate the maps:

  • For each SPIRE band we multiply by the correcting factorC = K4e/K4p. Since the extended source calibration method changed as a consequence of the work by Griffin et al. (2013), the correcting factors are not the same as those used in Lombardi et al. (2014). Specifically, C is now equal to (0.9986,1.0015,0.9993), while before it was C = (0.9828,0.9834,0.9710) for the 250 μm; 350 μm; 500 μm bands respectively. However, this change doesn’t have a significant impact on the final maps.

  • We perform an absolute flux calibration for the Herschel bands, using Planck maps.

  • We assume a modified black-body SED and we compute the expected flux at each reference passband.

  • We modify the SED until we obtain a good match between the observed and theoretical fluxes. For this step we use a simple χ2 minimization technique that takes into account the calibration errors.

For the analysis presented here, we used the HIPE (Ott 2010) extended emission level 2.5 products for SPIRE data and Unimaps for PACS data. For PACS data, it is possible to use Scanamorphos. However there is no significant difference between the two maps.

Figure 1 shows a colour-composite image of the combined reduced Herschel/SPIRE data for the region we considered, together with the predicted fluxes from Planck at the three SPIRE passbands for locations outside the Herschel coverage.

3.3. Optical depth and temperature maps

Figure 2 shows the combined optical depth-temperature map: the effective dust temperature is represented using different values of hue, while the intensity is proportional to the optical depth. Figures 3 and 4 individually show the temperature and optical depth maps.

Note that Planck data have lower resolution (5 arcmin instead of 36 arcsec) and a significantly lower noise. In particular, the error in the regions covered by Planck data is much smaller than that on the Herschel areas. Figures 3 and 4 highlight the differences in optical depth and temperature that characterise the cloud.

thumbnail Fig. 3

Optical depth map of the field and, on a different layer, the corresponding error map. This figure, as for the following one (Fig. 3), is produced by the method described in the text, using the reduced fluxes of SPIRE and the expected fluxes at SPIRE frequencies that were deduced from the Planck maps (Planck Collaboration XI 2014).

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thumbnail Fig. 4

Temperature map of the field and, on a different layer, the corresponding error map.

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Referring to the temperature map (Fig. 4), we note that the regions IC 348 and G159.6-18.5 present a higher temperature than the rest of the cloud, and that the ring of dust that surrounds the star HD 278942 (Andersson et al. 2000) is particularly visible. The temperature ranges between ~ 10 K and ~ 36 K.

thumbnail Fig. 5

Relationship between submillimitre optical depth and NIR extinction in Perseus. The image shows the best linear fit, used to calibrate the data (solid line), together with the expected 3σ region (dashed lines), as calculated from direct error propagation in the extinction map. The contours enclose the 68% and the 95% of the points, respectively.

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3.4. Extinction conversion

To compare the results obtained here with other observations, we converted the optical depth τ850 to the extinction AK. For this purpose, we smoothed the optical depth map to the same resolution as the extinction map (2.5 arcmin) of the region (Lombardi et al. 2010). We assumed that within the range 0−2 × 10-4, the law that describes the relation between τ850 and AK is linear (see Fig. 5): (4)The slope γ is proportional to the opacity κ850 and to the extinction coefficient in the K-band, C2.2. Since the extinction in K-band is defined as (5)the relationship between γ and C2.2 is simply γ ≃ 1.0857 C2.2/κ850. The coefficient δ is associated with uncertainties in the absolute flux calibration of Herschel, or to uncertainties in the extinction maps, or both. Through a χ2 minimization method, we obtained the following values for the two parameters: (6)The fact that δ is close to zero confirms the goodness of the calibration of Herschel data and of the extinction map. The value obtained for the slope γ is quite close to those found by Lombardi et al. (2014) for Orion A and B (γOrion A = 2640 mag and γOrion B = 3460 mag). Being proportional to the extinction coefficient C2.2 and the opacity at 850 μm, the coefficient γ is related to the dust composition and grain distribution, and therefore differences in the values of γ are likely related to differences in these quantities.

If we consider a wider range of values (see Fig. A.3), with τ × 104 ≤ 10, the relation unexpectedly deviates from the linear law. In this case, the law between τ850 and AK can be fitted by the empirical relation: (7)The best fit values for the three parameters are (8)The deviation from the linear regime might be due to either the inclusion of regions in the extinction map where a large number of embedded sources are present or the lack of background stars at highest extinction. To better understand the problem, we created a map (Fig. 6) that shows the difference between the extinction values evaluated from the extinction map and from the Herschel data using the linear scaling of Eq. (4). The highest discrepancies roughly coincide with the position of the Class 0 sources reported by Sadavoy et al. (2014). Since in these regions the difference is positive, it seems that the extinction map produced through 2MASS data with the NICEST method underestimates the extinction values in the densest regions, where the number of background stars is low and contamination from embedded sources may be present. For this reason we only consider the linear case Eq. (4)to convert optical depth values to extinction for our Herschel optical depths.

thumbnail Fig. 6

Difference between the extinction values evaluated with the 2MASS/ NICEST map and the Herschel opacity map. The contour represents the AK = 0.5 mag level.

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4. Results

To characterise the maps, we produced the integral area function of the cloud, S( >AK), i.e. the surface of the cloud above a certain extinction threshold as a function of that threshold. Lada et al. (2013) observed that the shape of the function S (>AK) and of its derivative, S′ (>AK), influences the variation in the rate of star formation of the cloud, especially in the high-extinction regions.

thumbnail Fig. 7

Integral area-extinction relation for Perseus, i.e., the physical cloud area above a given extinction threshold as a function of that threshold. The solid black line shows the result for the entire field, while the solid gray line shows the same quantity for the Herschel regions. The dashed line shows again the same quantity, but for the 2MASS/NICEST data. The red line shows the slope of the power law .

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The plot of S(>AK) is shown in Fig. 7 for different extinction measurements (Herschel, Herschel and Planck, 2MASS/ NICEST). In the same figure, a red line with slope −2 is represented. Operationally, this plot is built by just counting the sky area above different values of AK. As such, the specific spacing used for the threshold AK values is irrelevant (in Fig. 7 we used a log-scale in AK). The law appears to be an excellent description of the area function over two order of magnitudes, from AK ~ 0.1 to AK ~ 10 mag. The apparent break in this law at AK ~ 15 mag might either be genuine or due to systematic effects in the Herschel maps (e.g. unresolved structures, large temperature gradients, flux contamination from point sources). At the other extreme, the break at AK< 0.15 mag can be due to various effects, including contamination by unrelated foreground and background material, and inappropriate definition of the cloud boundaries. In reality, as shown by Lombardi et al. (2015), fundamental constraints, such as contamination by unrelated foreground and background dust emission, severely limit our ability to measure the area function below AK< 0.15 mag.

Figure 8 shows S′( >AK) as a function of AK, where AK is evaluated through Eq. (4). This function is proportional to the probability distribution function (PDF) of the column density map and follows a power law with index equal to −3. Note that since we directly derived the function S( >AK) using a 3-point Lagrangian interpolation (where the spacing AK is taken into account), this operation effectively produces a result that is proportional to a linearly-binned PDF. We stress that this differs from the choice adopted by Lombardi et al. (2015), where PDFs were logarithmically binned. Consequently, the slopes evaluated in this way differ by −1 to those presented by Lombardi et al. (2015), but the results are equivalent.

thumbnail Fig. 8

Function S′( >AK), i.e. the probability distribution function of the measured column density for Perseus. In this plot, a lognormal distribution would appear as a parabola and a power law as a straight line. The red line shows the slope of the power law .

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thumbnail Fig. 9

Integral mass-extinction relation, i.e. the cloud mass above a given extinction threshold. The line codes follow the same convention as in Fig. 7. The values of the mass above a certain extinction threshold are listed in Table 2.

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thumbnail Fig. 10

Function M′ (>AK), i.e. the derivative of the integral mass function.

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Table 2

Mass values above an extinction threshold, normalized at the mass at 0.1 mag, M0.1.

Figures 9 and 10 show the integral function of the mass M (>AK) as a function of AK and its derivative M′ (>AK). The gas surface density, Σgas, can be expressed as a mass surface density through the following relation: (9)where μ = 1.37 is the mean molecular weight, corrected for the helium abundance, mp = 1.67 × 10-24 g is the proton mass and is the gas-to-dust ratio. The agreement between the solid line (Herschel and Planck data) and the dashed line (2MASS/ NICEST) in Figs. 7 and 9 is good until AK ≃ 0.1 mag, while the discrepancy we observe at high extinction values is due to the higher levels of extinction reached by Herschel. Similar to Lombardi et al. (2014), we can define the mass of the cloud as the surface density integrated over the area (10)from which (11)We note how the power-law trend observed in Fig. 9 is a direct consequence of the power law observed in Fig. 7, by replacing with in Eq. (11).

5. The local Schmidt law

Another possible application of the column density maps is to study the validity of the Schmidt law (Schmidt 1959) in the Perseus molecular cloud. Half a century ago, Schmidt conjectured that the rate of star formation, ΣSFR, depends on the (projected) gas density, Σgas, and in particular is a simple power law. Schmidt argued that the index of the power law was ~2. Recently, a series of observational studies (Lada 2015 and references therein) that use observations of protostellar objects, demonstrate that a Schmidt scaling law exists within a sample of nearby molecular clouds and it is typically characterised by an index ~2. In what follows we investigate the Schmidt law using, in particular, Class I and Class 0 protostars2 to estimate the protostar surface density. Following Lada et al. (2013), we decided to consider only these types of sources, since they are still likely to be at, or close to, their original birth place. To estimate the protostar surface density, ΣYSO, we used already existing catalogs based on Spitzer data and we built a catalog of YSOs through an analysis of the WISE satellite data. We used the WISE data mainly to enhance and improve the Spitzer-based catalogs. Indeed, even though WISE resolution (6.1′′, 6.4′′, 6.5′′, and 12.0′′ in the four bands) is lower than Spitzer, its all-sky coverage enables us to include sources detected in areas that were not surveyed by Spitzer. To make our classification as complete as possible, we also included the new sources detected by Sadavoy et al. (2014).

5.1. Identifying YSO candidates

The procedure adopted to search for YSO candidates is based on a comparison of the source distribution in several colourcolour diagrams between the science and a control field. We choose a control field in the region defined by (12)We selected this area based on two requirements: it is characterised by low extinction values and the background object distribution is similar to that of the science field since it has the same galactic latitude of the science field, which corresponds to the region defined previously for 2MASS data. To classify the sources in the science field, we approximately followed the scheme proposed by Koenig et al. (2012), which in turn is based on the scheme by Gutermuth et al. (2008, 2009). Koenig analysed several WISE colourcolour diagrams of Spitzer-selected YSOs in the Taurus region. In addition, they characterised the contaminating sources by studying the distribution of objects with declination (celestial north pole) in the W1W2 vs. W2W3 colourcolour diagram. By comparing the science and the control field colourcolour diagrams, the presence in the science field of sources with colour excess can be noted: these may be reddened background objects or embedded YSOs. As Rebull et al. (2011) noted, no colourcolour diagram can perfectly find all the YSOs and remove all the contaminants: the contamination rate for any colour selection is expected to be large and ancillary data from other observations are often needed to choose the most likely candidates. Koenig et al. (2012) estimate a contamination rate for “typical” star-forming regions of about 2.4 objects resembling Class I YSOs, 3.8 objects resembling Class II YSOs, and 1.8 objects resembling transition discs per square degree. As a result, in our field, which is ~ 100 deg2, the contaminants expected are respectively ~ 240, ~ 380, and ~ 180.

thumbnail Fig. 11

W1−W2 ([3.4] − [4.6]) vs. W2−W3 ([4.6] − [12]) diagram for the science and the control field. Red points represent candidate Class II objects, green points represent candidate Class I objects, while black points represent the other sources in the field. The red ellipse and the green lines have equations specified in the text.

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thumbnail Fig. 12

W1−W2 vs. W2−W4 diagram of the science and the control field. Red points represent candidate Class II YSOs, green points represent candidate Class I protostars, while black points represent the other sources in the field. The red ellipse and the green slopes have equations specified in the text.

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Figure 11 shows the colourcolour diagram in W1, W2, and W3 bands of the science (left) and the control field (right). By comparing them, we observed a significant excess of sources inside the marked red ellipse with centre (13)and axes a = 1 mag and b = 0.5 mag, and between the two blue slopes with equations (14)Therefore, we primarily select 213 objects inside these two regions. The reddening of the objects inside the ellipse is low, and we can therefore suppose that they are Class II YSOs. Instead, the objects lying between the two slopes (22) are highly reddened and can be classified as Class I protostars. We observe that the selected Class I protostars satisfy Koenig requirements. Furthermore, our classification is stricter since we remove all sources in the AGN/galaxy region of the diagram, and the selection criteria of Class II YSOs are close to those chosen by Koenig et al. (2012). Indeed, the minimum colour values that were accepted to include a source in the selection are (15)Figure 12 shows the diagrams [3.4] − [4.6] vs. [4.6] − [22] and is analogous to the previous one. Class II YSOs lie within the red ellipse, with centre (16)and axes a = 2.5 mag and b = 0.5 mag, while Class I protostars are included between the slopes with equations (17)The total number of sources is 130, with 117 of them classified as Class II YSOs, while the remaining 13 are classified as Class I protostars. Furthermore, we can observe the similarity with Koenig criteria.

thumbnail Fig. 13

Spatial distribution of candidate Class I objects (orange crosses) selected through the [3.4] − [4.6] vs. [4.6] − [12] colourcolour diagram and Class I objects (purple circles) selected through the [3.4] − [4.6] vs. [4.6] − [22] colourcolour diagram. The map shown here is the optical-depth map presented before.

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To find YSO candidates that are associated with known objects, we cross-checked our selection with the SIMBAD astronomical database and with the catalogs by Evans et al. (2009), Gutermuth et al. (2008, 2009), Jørgensen et al. (2006), Kirk & Myers (2011), Winston et al. (2009) and Young et al. (2015), based on Spitzer data. Many sources selected through WISE data analysis have already been included in other catalogs. Nevertheless, the complete sky coverage of the WISE satellite allows us to study regions that are parts of the cloud but that were never observed by Spitzer. The final selection consists of previously identified Spitzer sources that are merged with newly identified WISE objects. Using WISE data, 61 new sources were selected, of which 17 belong to IC 348 and just one to NGC 1333.

We now focus on the candidate Class I protostars, that were selected previously with our colourcolour diagrams. To further anlayse these sources, in particular to test whether our classification scheme is reliable, it is useful to study their spatial distribution. Figure 13 shows the positions of the Class I protostars that were selected through the [3.4] − [4.6] vs. [4.6] − [12] and [3.4] − [4.6] vs. [4.6] − [22] diagrams. All sources (except one) are placed in the highest opacity regions: this agrees well with the hypothesis that they are located close to their original place of formation. Most of Class I YSOs selected through the [3.4] − [4.6] vs. [4.6] − [12] diagram are situated in the NGC 1333 region. This could imply, as Jørgensen et al. (2006) observed, that the age of NGC 1333 is slightly lower than the age of IC 348: indeed, it is estimated that IC 348 is 2 Myr old (Muench et al. 2007), while NGC 1333 does not exceed 1 Myr (Lada et al. 1996). We already noted that the number of sources selected with the [3.4] − [4.6] vs. [4.6] − [22] is lower than for the [3.4] − [4.6] vs. [4.6] − [12] diagram. Class I protostars in [3.4] − [4.6] vs. [4.6] − [12] diagram correspond to Class I protostars in [3.4] − [4.6] vs. [4.6] − [22] diagram, except for three sources that are not detected in the first diagram. Surprisingly, the [3.4] − [4.6] vs. [4.6] − [22] diagram does not detect any Class I protostar in NGC 1333. We evaluated the cloud temperature and optical depth at each object position. We excluded from the WISE selection all the objects in areas with optical depth τ< 3.141 × 10-5 (corresponding to AK< 0.1 mag), supposing that this level of optical depth corresponds to the cloud boundaries. Figure 14 shows the final selection of Class I and Class 0 protostars (139 sources). Only two of the new sources selected by WISE are classified as Class I protostars. All the other sources in the figure have been previously classified using Spitzer data. We note that for some sources our classification differs to the one obtained by Spitzer catalogs. In these cases, we decided to follow the Spitzer classification.

thumbnail Fig. 14

Spatial distribution of the candidate Class I (pink crosses) and Class 0 objects (orange circles, see Sadavoy et al. 2014) used to estimate the Schmidt law.

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thumbnail Fig. 15

Schmidt law in the Perseus molecular cloud. Here, , and AK ∝ Σgas.

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5.2. Determining the Schmidt law

We computed the extinction at each source position using our Herschel-derived extinction map and binned them in log-spaced bins. Then we evaluated the cloud area between two consecutive extinction levels and the number of sources included within them. In this way, we evaluated the protostar surface density, ΣYSO, as a function of the extinction, as the number of sources between two consecutive extinction levels divided by the corresponding area. Since the gas surface density Σgas is proportional to the K-band extinction (see Eq. (9)), it is possible to express the Schmidt law as (18)where ΣYSO is simply equal to ΣSFR × τ, and τ is the mean age of a Class I/0 protostar. Figure 15 represents the Schmidt law for the Perseus molecular cloud. The values obtained for the parameters are (19)The value we found for β is steeper (although compatible) to what is reported by Lombardi et al. (2014) for the Orion molecular cloud complex and by Lada et al. (2013) for the Taurus molecular cloud. Indeed, Lombardi et al. (2014) find β = 1.99 ± 0.05 for Orion A and β = 2.16 ± 0.10 for Orion B, while Lada et al. (2013) find β = 2.09 ± 0.14 for Taurus. Even though the Schmidt law is a useful tool to understand the star formation inside the cloud, it does not completely describe the whole process. For this purpose, we considered the function that represents the cumulative number of sources above a certain extinction AK as a function of that extinction, as shown in Fig. 16. Even though the Schmidt law predicts that the surface density of protostars steeply grows with extinction, the effective number of protostars decreases as the extinction increases, similar to observations of the Orion molecular cloud in Lada et al. (2013).

thumbnail Fig. 16

Cumulative protostellar function in the Perseus cloud as a function of AK, i.e. NYSO( >AK).

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The total number of protostars is defined as (Lada et al. 2013) (20)Indeed, the number of protostars at a given extinction level is the product of the area S(AK), which encompasses that extinction level, and ΣYSO(AK). The total number of protostars is given by the integral of this product over all extinction in the cloud. Changing the integration variable from S to AK gives Eq. (20). We already studied the function S( >AK) (Fig. 7) and we concluded that its trend is and therefore . Thus, the number of protostars formed within the cloud not only depends on the surface density of the gas but also on the area function and its derivative of the integral area function, i.e. the (unnormalized) PDF. As Lombardi et al. (2015) show, the PDFs of a sample of nearby molecular clouds, including Perseus, roughly follow the same profile, with slopes in logarithmic binning between ~−2 and ~−4, which correspond to slopes between ~−3 and ~−5 in the linear binning that was performed here. Therefore, it is conceivable to assume that, if the clouds have the same PDF and the same Schmidt law, then the function that describes the number of protostars above a certain extinction (normalized to the total number of protostars) is roughly the same for all the clouds.

6. Conclusions

Our main results are as follows:

  • We produced optical depth and temperature maps of the Perseusmolecular cloud, obtained using the data from the Herschel andPlanck satellites. The maps have a36 arcsec resolution for Herschel observations and a 5 arcmin resolution elsewhere.

  • We calibrated the optical depth maps using 2MASS/ NICEST extinction data and we obtained extinction maps at the resolution of Herschel with a dynamic range of 1 × 10-2 mag to 20 mag of AK. In particular, we evaluated the ratio C2.2/κ850 = 3620 ± 252, i.e. the ratio between the 2.2 μm extinction coefficient and of the 850 μm opacity.

  • We studied the cumulative and differential area functions of the data, and we showed that, starting from AK ≃ 0.1 mag, the cumulative area function follows a power law with index ≃−2.

  • We used WISE data to improve current YSO catalogs that are mostly based on Spitzer data and we built an up-to-date selection of Class I/0 protostars.

  • We evaluated the local Schmidt law, , using the Herschel/Planck maps and the new object selection. We found that β = 2.4 ± 0.6.

  • We showed that the Schmidt law does not completely describe the whole star-formation process. Indeed, the total number of protostars effectively formed within a cloud depends on the surface area function S( >AK) and its derivative.


1

The maps will be publicly available through CDS (http://cdsweb.u-strasbg.fr) and the website http://www.interstellarclouds.org

2

In the following we will refer to Class I and 0 objects as protostars, and to Class II, Class I and Class 0 objects as YSOs.

Acknowledgments

We thank the anonymous referee for comments that improved the manuscript. We also wish to thank Josefa Grossscheld and Paula Stella Teixeira for the helpful discussions. This publication would not have been possible without the data products from the Herschel satellite and the Wide-field Infrared Survey Explorer (WISE). Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator Consortia and with important participation from NASA. WISE is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. This paper is also based on observations obtained with Planck (http://www.esa.int/Planck), an ESA science mission with instruments and contributions directly funded by ESA Member States, NASA, and Canada. Furthermore, it makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. This research made use of: the SIMBAD database, operated at CDS, Strasbourg, France; the VizieR catalogue access tool, CDS, Strasbourg, France; Astropy, a community-developed core Python package for Astronomy (Astropy Collaboration et al. 2013); TOPCAT, an interactive graphical viewer and editor for tabular data (Taylor 2005).

References

Appendix A: Hidden layers of multiple layer figures

In this section we provide a “flat” version of the hidden layers of multiple-layer figures.

thumbnail Fig. A.1

Error on the optical depth map (see Fig. 3).

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thumbnail Fig. A.2

Error on the effective dust temperature map (see Fig. 4).

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thumbnail Fig. A.3

Relation between 353 GHz optical depth and the infrared NICEST extinction map, for τ850 × 104 ≤ 10. The relation deviates from the linearity for high extinction values, as shown by the fit (solid line).The dashed lines represent the expected 3σ region. We also report in this plot the linear fit shown in Fig. 5 (dotted line).

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Appendix B: Potential new star cluster

While analysing the spatial distribution of the WISE excess sources, we noticed a small group slightly offset from the cloud, whose centre is located at approximately l = 162.2°,b = −19.4°. We found no reference in the literature to this, potentially, new group of young stars. The group contains 14 members spread across an area of about one square degree. None of the excess sources is present in the Hipparcos catalog, it is therefore not possible to give an estimate of its distance or its potential relation with the Perseus molecular cloud. Follow up observations for this new group of excess sources is needed. The Gaia mission is expected to dramatically improve the distance determination for faint stars, which will allow for a better understanding of the origins and relevance of this group to the history of star formation in the Perseus region.

Appendix C: Catalog of the sources used to evaluate the Schmidt law

In this section we provide the list of protostars used to evaluate the Schmidt law. We report the position for all the sources and the WISE magnitudes for the sources detected by both Spitzer and WISE.

Table C.1

Catalog of the protostars used to determine the Schmidt law.

All Tables

Table 1

Herschel parallel mode observation used.

Table 2

Mass values above an extinction threshold, normalized at the mass at 0.1 mag, M0.1.

Table C.1

Catalog of the protostars used to determine the Schmidt law.

All Figures

thumbnail Fig. 1

Composite three-colour image showing Herschel/SPIRE intensities for the region considered, where available (with the 250 μm; 350 μm; 500 μm bands shown in blue, green, and red). For regions outside the Herschel coverage, we used the Planck/IRAS dust model (τ850, T, β) to predict the intensity that would be observed at the SPIRE passbands. Toggle labels

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In the text
thumbnail Fig. 2

Combined optical depth-temperature map for Perseus. The image shows the optical depth as intensity and the temperature as hue, with red (blue) corresponding to low (high) temperatures.

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In the text
thumbnail Fig. 3

Optical depth map of the field and, on a different layer, the corresponding error map. This figure, as for the following one (Fig. 3), is produced by the method described in the text, using the reduced fluxes of SPIRE and the expected fluxes at SPIRE frequencies that were deduced from the Planck maps (Planck Collaboration XI 2014).

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In the text
thumbnail Fig. 4

Temperature map of the field and, on a different layer, the corresponding error map.

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In the text
thumbnail Fig. 5

Relationship between submillimitre optical depth and NIR extinction in Perseus. The image shows the best linear fit, used to calibrate the data (solid line), together with the expected 3σ region (dashed lines), as calculated from direct error propagation in the extinction map. The contours enclose the 68% and the 95% of the points, respectively.

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In the text
thumbnail Fig. 6

Difference between the extinction values evaluated with the 2MASS/ NICEST map and the Herschel opacity map. The contour represents the AK = 0.5 mag level.

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In the text
thumbnail Fig. 7

Integral area-extinction relation for Perseus, i.e., the physical cloud area above a given extinction threshold as a function of that threshold. The solid black line shows the result for the entire field, while the solid gray line shows the same quantity for the Herschel regions. The dashed line shows again the same quantity, but for the 2MASS/NICEST data. The red line shows the slope of the power law .

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In the text
thumbnail Fig. 8

Function S′( >AK), i.e. the probability distribution function of the measured column density for Perseus. In this plot, a lognormal distribution would appear as a parabola and a power law as a straight line. The red line shows the slope of the power law .

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In the text
thumbnail Fig. 9

Integral mass-extinction relation, i.e. the cloud mass above a given extinction threshold. The line codes follow the same convention as in Fig. 7. The values of the mass above a certain extinction threshold are listed in Table 2.

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In the text
thumbnail Fig. 10

Function M′ (>AK), i.e. the derivative of the integral mass function.

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In the text
thumbnail Fig. 11

W1−W2 ([3.4] − [4.6]) vs. W2−W3 ([4.6] − [12]) diagram for the science and the control field. Red points represent candidate Class II objects, green points represent candidate Class I objects, while black points represent the other sources in the field. The red ellipse and the green lines have equations specified in the text.

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In the text
thumbnail Fig. 12

W1−W2 vs. W2−W4 diagram of the science and the control field. Red points represent candidate Class II YSOs, green points represent candidate Class I protostars, while black points represent the other sources in the field. The red ellipse and the green slopes have equations specified in the text.

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In the text
thumbnail Fig. 13

Spatial distribution of candidate Class I objects (orange crosses) selected through the [3.4] − [4.6] vs. [4.6] − [12] colourcolour diagram and Class I objects (purple circles) selected through the [3.4] − [4.6] vs. [4.6] − [22] colourcolour diagram. The map shown here is the optical-depth map presented before.

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In the text
thumbnail Fig. 14

Spatial distribution of the candidate Class I (pink crosses) and Class 0 objects (orange circles, see Sadavoy et al. 2014) used to estimate the Schmidt law.

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In the text
thumbnail Fig. 15

Schmidt law in the Perseus molecular cloud. Here, , and AK ∝ Σgas.

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In the text
thumbnail Fig. 16

Cumulative protostellar function in the Perseus cloud as a function of AK, i.e. NYSO( >AK).

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In the text
thumbnail Fig. A.1

Error on the optical depth map (see Fig. 3).

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In the text
thumbnail Fig. A.2

Error on the effective dust temperature map (see Fig. 4).

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In the text
thumbnail Fig. A.3

Relation between 353 GHz optical depth and the infrared NICEST extinction map, for τ850 × 104 ≤ 10. The relation deviates from the linearity for high extinction values, as shown by the fit (solid line).The dashed lines represent the expected 3σ region. We also report in this plot the linear fit shown in Fig. 5 (dotted line).

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In the text

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