The structure of molecular clouds – II. Column density and mass distributions

Abstract

1 INTRODUCTION

The study of giant molecular clouds (GMCs) is of great importance to understand star formation. Their properties (density distribution, dynamics, temperature, etc.) are thought to be vital in determining whether, where and how stars are forming within them. Turbulent motions inside a cloud cause fragmentation and hence determine the density distribution (Padoan, Jones & Nordlund 1997; Ballesteros-Paredes, Hartmann & Vázquez-Semadeni 1999). This in turn should have an influence on which mode of star formation (isolated or clustered) is occurring (Klessen, Heitsch & Mac Low 2000). It is, however, unclear if this is the sole determinant of the star formation mode. Or are there other important causes such as the environment, feedback and/or magnetic fields?

One way to try to answer these questions is to investigate the column density structure of a large number of GMCs in a systematic, comparable and unbiased fashion. This means ideally one should use the same tracer of column density for all clouds. Such a tracer should be as bias-free as possible and all clouds should be investigated at the same physical spatial resolution. There are a number of techniques to determine the column density of clouds. These include molecular line emission, dust continuum emission, scattered infrared light or extinction measurements. The latter on the bases of star counts, colour excess or combined methods. See Froebrich & Rowles (2010) for a brief summary and discussion of the advantages and disadvantages of the various techniques. It has been shown by Goodman, Pineda & Schnee (2008) that near-infrared extinction mapping is the best tracer of the ‘real’ column density of a cloud, based only on the assumption of a constant gas to dust ratio.

A number of large-scale maps of the column density distribution of clouds have been made to date, which could be used to compare the structure of a number of GMCs. These are for example the maps by Cambrésy (1999), Schlegel, Finkbeiner & Davis (1998), Dobashi et al. (2005), Froebrich et al. (2007) and Rowles & Froebrich (2009). Some of them do not use the ideal tracer (extinction) for the column density. Others use star counts, which result in a distance dependent bias in the column density distribution (Froebrich & del Burgo 2006), similar to using the mean colour excess instead of the median. One bias they all have in common is that clouds at different distances are mapped at different spatial resolutions. This will naturally lead to a change in the observed column density distribution since for more distant clouds one averages over larger physical scales. This will lead to the effect that small-scale high-extinction regions are averaged out and the column density distributions are skewed. In order to compare the structure of GMCs with each other, one has to determine the column density distribution at similar physical scales and using the same ‘good’ tracer for all investigated clouds.

Only then can we attempt to draw conclusions about similarities and differences in their structures. This can be done by analysing the column density and mass distribution (the focus of this paper) or determining structure functions (the focus of the next paper in the series) and comparing it to model predictions such as published by Kolmogorov (1941), She & Leveque (1994) and Boldyrev (2002). In our previous paper (Rowles & Froebrich 2009, hereafter Paper I) we presented all-sky extinction maps determined using data from the Two Micron All Sky Survey (2MASS) (Skrutskie et al. 2006). These were created using an adaptation of the Near Infrared Colour Excess (NICE) method (Lada et al. 1994), median colour excess determination and variable spatial resolution to obtain a constant signal-to-noise ratio. Here we present and analyse additional maps, determined using a range of constant spatial resolutions to map different clouds at similar spatial scales.

In Section 2 we briefly describe the new maps created for the project, and explain the data analysis methods. In Section 3 we present the results for our selection of nearby GMCs. We discuss these results and draw conclusions in Section 4.

2 DATA ANALYSIS

2.1 Cloud selection

For our analyses we selected a number of nearby GMCs. Since we are interested in their column density distribution, we selected only clouds that are not situated directly in the Galactic plane. This avoids that there are two clouds along the same line of sight, rendering the column density analysis difficult. Furthermore, only sufficiently large (or nearby) clouds are selected. This ensures that we have enough area (pixels) in our maps to analyse the column density distribution. Finally, only GMCs with a well-known distance are included in our investigations. This selection process leaves 16 cloud complexes. Their names, coordinate ranges, distances and references are listed in Table 1. To allow a comparison to the earlier work by Froebrich et al. (2007) we adopt the names for the regions from this paper. In the case of the Lupus complex, we treat several of the small clouds as one, in order to have enough area to be analysed. One should hence keep in mind when interpreting this data that they are composed of several clouds.

Table 1

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Summary of the clouds analysed in this paper. We list the cloud name, the range in galactic coordinates (l, b) covered by the region and the distance (d) adopted in this paper.

Name	l range	b range	d (pc)
Auriga 1a	156° to 173°	to	450 ± 231
Auriga 2	175° to 186°	to	140 ± 282
Cepheus	100° to 120°	to	390 ± 1253
Chamaeleon	296° to 304°	to	150 ± 304
Circinus	316° to 319°	to	700 ± 3505
Corona Australis	359° to 1°	to	170 ± 344
λ-Ori	188° to 201°	to	400 ± 806
Lupus 1, 2	334° to 342°	to	155 ± 87
Lupus 3, 4, 5, 6	335° to 344°	to	155 ± 87
Monoceros	212° to 222°	to	830 ± 508
Ophiuchus	350° to 360°	to	119 ± 67
Orion A	208° to 219°	to	410 ± 806
Orion B	201° to 211°	to	410 ± 806
Perseus	156° to 163°	to	310 ± 659
Serpens	30° to 32°	to	260 ± 1010
Taurus	164° to 178°	to	140 ± 282

Name	l range	b range	d (pc)
Auriga 1a	156° to 173°	to	450 ± 231
Auriga 2	175° to 186°	to	140 ± 282
Cepheus	100° to 120°	to	390 ± 1253
Chamaeleon	296° to 304°	to	150 ± 304
Circinus	316° to 319°	to	700 ± 3505
Corona Australis	359° to 1°	to	170 ± 344
λ-Ori	188° to 201°	to	400 ± 806
Lupus 1, 2	334° to 342°	to	155 ± 87
Lupus 3, 4, 5, 6	335° to 344°	to	155 ± 87
Monoceros	212° to 222°	to	830 ± 508
Ophiuchus	350° to 360°	to	119 ± 67
Orion A	208° to 219°	to	410 ± 806
Orion B	201° to 211°	to	410 ± 806
Perseus	156° to 163°	to	310 ± 659
Serpens	30° to 32°	to	260 ± 1010
Taurus	164° to 178°	to	140 ± 282

^a The cloud Auriga 1 is referred to as the California Molecular Cloud in Lada, Lombardi & Alves (2009).

The references for the adopted distances are the following: ¹Lada et al. (2009); ²Kenyon, Gomez & Whitney (2008); ³Kun, Kiss & Balog (2008); ⁴Knude & Hog (1998); ⁵Bally et al. (1999); ⁶Muench et al. (2008); ⁷Lombardi et al. (2008a); ⁸Carpenter & Hodapp (2008); ⁹Motte & André (2001); ¹⁰Straižys, Černis & Bartašiútė (1996).

Table 1

Open in new tab

Summary of the clouds analysed in this paper. We list the cloud name, the range in galactic coordinates (l, b) covered by the region and the distance (d) adopted in this paper.

Name	l range	b range	d (pc)
Auriga 1a	156° to 173°	to	450 ± 231
Auriga 2	175° to 186°	to	140 ± 282
Cepheus	100° to 120°	to	390 ± 1253
Chamaeleon	296° to 304°	to	150 ± 304
Circinus	316° to 319°	to	700 ± 3505
Corona Australis	359° to 1°	to	170 ± 344
λ-Ori	188° to 201°	to	400 ± 806
Lupus 1, 2	334° to 342°	to	155 ± 87
Lupus 3, 4, 5, 6	335° to 344°	to	155 ± 87
Monoceros	212° to 222°	to	830 ± 508
Ophiuchus	350° to 360°	to	119 ± 67
Orion A	208° to 219°	to	410 ± 806
Orion B	201° to 211°	to	410 ± 806
Perseus	156° to 163°	to	310 ± 659
Serpens	30° to 32°	to	260 ± 1010
Taurus	164° to 178°	to	140 ± 282

Name	l range	b range	d (pc)
Auriga 1a	156° to 173°	to	450 ± 231
Auriga 2	175° to 186°	to	140 ± 282
Cepheus	100° to 120°	to	390 ± 1253
Chamaeleon	296° to 304°	to	150 ± 304
Circinus	316° to 319°	to	700 ± 3505
Corona Australis	359° to 1°	to	170 ± 344
λ-Ori	188° to 201°	to	400 ± 806
Lupus 1, 2	334° to 342°	to	155 ± 87
Lupus 3, 4, 5, 6	335° to 344°	to	155 ± 87
Monoceros	212° to 222°	to	830 ± 508
Ophiuchus	350° to 360°	to	119 ± 67
Orion A	208° to 219°	to	410 ± 806
Orion B	201° to 211°	to	410 ± 806
Perseus	156° to 163°	to	310 ± 659
Serpens	30° to 32°	to	260 ± 1010
Taurus	164° to 178°	to	140 ± 282

^a The cloud Auriga 1 is referred to as the California Molecular Cloud in Lada, Lombardi & Alves (2009).

The references for the adopted distances are the following: ¹Lada et al. (2009); ²Kenyon, Gomez & Whitney (2008); ³Kun, Kiss & Balog (2008); ⁴Knude & Hog (1998); ⁵Bally et al. (1999); ⁶Muench et al. (2008); ⁷Lombardi et al. (2008a); ⁸Carpenter & Hodapp (2008); ⁹Motte & André (2001); ¹⁰Straižys, Černis & Bartašiútė (1996).

2.2 Constant-resolution extinction maps

The determination of the new constant-resolution extinction maps has been performed in exactly the same way as described in Paper I. We determine median 〈J−H〉 and 〈H−K〉 colour excess maps, with the same position-dependent zero-point as in Paper I. These maps are converted into optical extinction maps and are averaged (following equation 1 and using β= 1.7 as in Paper I). Similarly, maps of the uncertainties are calculated. Already in Paper I one constant-resolution map has been determined. There the spatial resolution changed from 0.5 to 2.0 arcmin, depending on the galactic latitude. Only stars within this radius were included in the extinction determination. In contrast to this, the new maps consist of square-shaped pixels with no oversampling. In other words, the pixel values in those maps are completely independent of the neighbouring pixels.

1

For the purpose of this paper, we have calculated a number of further constant spatial resolution maps, in total four, with a different resolution. We used the resolution of the maps in Paper I for the first map, and then increased the pixel size in steps of a factor of 2. This ensures we have an extinction map of each cloud with spatial resolutions covering almost an order of magnitude. Since the range of distances for our clouds varies by less than that, there will be one common physical spatial resolution where each cloud has been observed.

The new all-sky extinction maps available are listed in Table 2. They are labelled maps 1–4 depending on the pixel sizes used. Note the dependence of the actual pixel size of each map with galactic latitude.

2.3 Cloud structure analysis

We can now determine the column density distribution for each cloud at four different spatial scales. As a first step we extract the extinction values for each cloud from the maps 1–4. We then plot histograms of the number of pixels with a given extinction value, or the integrated number of pixels above a given extinction – i.e. a measure of the mass distribution.

The width of the histograms bins was varied. We used 1/8th, 1/4th, 1/2 and 1 mag of optical extinction. All subsequent analyses were conducted for each bin size, in order to check for systematic dependence of the results on this. It turned out that, as long as there are a sufficient number of pixels in each bin, the results do not depend on the bin size. For small clouds and large spatial resolution the first bin width (1/8 mag) might contain not enough data points. We hence excluded these outliers and averaged the results obtained from the other histogram bin widths to calculate the final result.

2.3.1 Lognormal fits to the column density distribution

The first method of analysis performed was to fit an analytic function to the column density distribution histogram showing the number of pixels N with an optical extinction A_V. Following Lombardi, Lada & Alves (2008b) we fit our clouds with a lognormal distribution of the form

2

If there are only a small number of clouds along the line of sight (certainly valid due to our selection criteria of clouds; see Section 2.1) and the underlying density distribution is lognormal (as predicted e.g. by Vázquez-Semadeni & García 2001) a good fit should be obtained.

2.3.2 The log(N) versus A_V column density distribution

A plot of the optical extinction versus log(N), where N is the number of pixels with the given A_V value, is to a large extent linear. This is expected since the column density distribution is caused by turbulent fragmentation and turbulence is intrinsically self-similar. We can hence use the slope (γ) of this distribution to characterize the column density distribution.

For almost all clouds there are two ranges of extinction where the plot is linear. Both possess different slopes γ and typically the change of slope happens between 5 and 10 mag of A_V. We hence determine for each cloud two slope values, one for the lower extinction region (γ_low) and one for the high A_V material (γ_high). The extinction ranges where the slopes are constant are set for each cloud manually.

2.3.3 The log(M) versus A_V mass distribution

Similar to the analysis of the column density distribution we can calculate the mass M in the cloud at extinction values higher than a given A_V value. In other words we integrate the extinction values above A_V and convert to masses using the pixel size and distance to the cloud. Like for the investigation of the column density, this mass distribution shows, over a range of A_V values, an exponential behaviour. We can hence fit the slope (δ) in the log(M) versus A_V diagram to characterize the mass distribution.

Again, for most clouds there are two regimes with different slopes. There is the lower A_V region (characterized by δ_low), whose mass distribution is governed by the turbulent properties of the cloud material. And the high A_V region (characterized by δ_high), where gravity becomes important and changes the mass distribution. The extinction value, where this change of behaviour is observed (A_V,SF), can be seen as the extinction threshold for star formation, first described by Johnstone, Di Francesco & Kirk (2004) in Ophiuchus.

3 RESULTS

3.1 Scale-dependent effects

We are planning to compare parameters of cloud structure for different clouds. Hence, all parameters need to be determined at a physical resolution common to all investigated clouds. Since most of our clouds are reasonably nearby, we chose 0.1 pc, which corresponds to the Jeans length of a core with a temperature of 15 K and a density of GMCs of 5 × 10⁴ cm⁻³.

There are in principle the following ways to determine the structure parameters at this common scale for all clouds: (i) determine the extinction maps for each cloud at the appropriate spatial resolution; (ii) determine the maps for all clouds at a sufficiently high spatial resolution and rebin them to the correct resolution before performing the data analysis; (iii) determine the cloud properties at a number of spatial resolutions and interpolate them to 0.1 pc. Option (i) would certainly be the most desirable, however the most laborious. Below we will show that option (ii) should be performed with great care since it can lead to erroneous results. Option (iii) does allow us not only to determine the cloud properties at the common physical scale, but also to investigate the properties at other spatial resolutions.

In the left-hand panel of Fig. 1 we show how the slope γ_low for the cloud Auriga 1 depends on the spatial resolution. At each spatial resolution we determine the slope four times using different histogram widths, as discussed in Section 2.3. As one can see, these slopes agree very well and can be averaged (solid line in the right-hand panel of Fig. 1). For the other clouds a similar behaviour is found (see Appendix A for the remaining plots). About half the clouds show no dependence of the slope on the scale and the other half a significant decrease of γ_low with decreasing resolution.

Figure 1

Left-hand panel: a plot showing measured slopes γ_low against spatial scale for the Auriga 1 cloud. The four data points at each resolution correspond to the four histogram bin widths used. Right-hand panel: the solid line shows the averaged slopes γ_low from the left-hand panel. The dotted line shows the slopes γ_low if the highest resolution image is simply rebinned to the lower resolutions.

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The general trend for Auriga 1 is that the slope γ_low decreases towards larger spatial scales. On can understand this trend simply by assuming that the cloud contains a number of very small, high-extinction cores which are simply not picked up at the larger spatial scales. This is not the case for all clouds investigated. In some cases we find more or less constant slopes with changing spatial scale. In those clouds the fraction of very small high-extinction cores is probably smaller.

We investigated what happens when we simply rebin the highest spatial resolution image to the lower resolution and we redetermine the slopes γ_low. The results for Auriga 1 are shown in the right-hand panel in Fig. 1 as a dotted line. One finds that the general trend of the slope values is retained. However, individual γ_low values in the rebinned images can differ by almost the 1σ uncertainties of the values obtained in the maps determined at the respective spatial resolution. See Appendix A for the effects the rebinning has on the γ_low and γ_high values. In general the differences are well below the 1σ uncertainties. However, in a few cases, such as the above quoted example of Auriga 1, larger differences can be found.

As a consequence of this result we chose to perform option (iii) to determine our cloud structure parameters for all clouds at 0.1-pc resolution. We calculate all parameters at each of the four spatial resolutions available to us, and then interpolate to obtain the values at 0.1 pc. All subsequent analysis is performed this way.

3.2 Lognormal fits to the column density distribution

Using the technique described in Section 2.3.1 we obtained fit parameters for each cloud we analysed with equation (2). In Table 3 we summarize the fit parameters and the root mean square deviation (rms) values obtained for the various spatial resolutions and histogram bin sizes for the Auriga 1 cloud as an example. There is a general trend visible for all parameters. In particular the width of the distribution increases with spatial scale. This is expected since more and more small-scale high-extinction cores are not detected anymore at these coarse resolutions.

In Fig. 2 we show the normalized column density distribution for the Auriga 1 cloud as a solid line (shown are the data for the spatial resolution closest to 0.1 pc). Overplotted is a fit with parameters scaled to 0.1 pc spatial scale. Similar plots for all individual clouds can be seen in Appendix A. We list the fit parameters and rms values (scaled to 0.1-pc resolution) for all clouds in Table 4.

Figure 2

Top: plot of the best lognormal fit (dotted line) to the normalized column density distribution (solid line) for the Auriga 1 cloud. The fit parameters are interpolated to a spatial scale of 0.1 pc, and the data for the spatial resolution closest to 0.1 pc are shown. Bottom: plot of the log N versus A_V column density distribution for the Auriga 1 cloud (symbols). Overplotted are the two fits obtained for the low and high column density region.

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Table 4

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Parameters of best fit for all clouds, at a scale of 0.1 pc. We list the parameters from the fit of the lognormal distribution, the slopes γ and δ, as well as the cloud masses, star formation threshold, extinction of a Jeans mass core and the mass fraction of clouds involved in star formation (MSF).

Name .	a (mag) .	A0 (mag) .	A1 (mag) .	σ (mag) .	rms (σ) .	γlow .	γhigh .	δlow .	δhigh .	M1 mag (103 M⊙) .	AV,SF (mag) .	(mag) .	MSF (per cent) .
Auriga 1	15.9	−14.2	15.3	1.10	4.7	−0.48	−0.24	−0.36	−0.13	268	7.4	32	0.19
Auriga 2	5.5	−4.6	5.6	1.14	2.5	−0.71	−0.32	−0.51	−0.21	13	4.9	15	0.33
Cepheus	32.9	−31.8	32.8	1.04	9.0	−0.50	−0.23	−0.37	−0.15	256	6.7	29	0.26
Chamaeleon	3.3	−2.7	3.4	1.27	3.2	−0.39	−0.18	−0.25	−0.13	45	7.3	24	0.81
Circinus	9.8	−7.5	9.9	1.17	4.5	−0.36	−0.17	−0.24	−0.16	113	6.8	30	1.4
Corona Australis	2.2	−1.7	2.5	1.46	1.6	−0.40	−0.16	−0.20	−0.15	1	3.7	23	9.6
λ-Ori	12.8	−12.3	12.9	1.11	4.6	−0.51	−0.34	−0.40	−0.13	122	7.2	28	0.12
Lupus 1, 2	8.0	−7.4	8.0	1.08	3.4	−0.73	−0.20	−0.47	−0.13	4.6	3.4	23	2.4
Lupus 3, 4, 5, 6	4.2	−2.7	4.3	1.22	2.0	−0.73	−0.21	−0.47	−0.12	24	5.1	25	0.40
Monoceros	12.5	−11.9	12.6	1.11	6.7	−0.42	−0.18	−0.32	−0.12	74	4.8	35	2.0
Ophiuchus	1.5	−0.3	1.5	1.48	7.0	−0.27	−0.11	−0.21	−0.13	9.3	8.6	27	0.78
Orion A	3.9	−3.0	4.1	1.39	4.2	−0.28	−0.15	−0.19	−0.12	44	5.4	37	5.0
Orion B	6.4	−5.5	6.5	1.23	4.3	−0.39	−0.13	−0.26	−0.11	78	6.8	38	1.1
Perseus	5.8	−5.0	5.9	1.23	3.8	−0.42	−0.22	−0.28	−0.17	29	4.8	25	3.0
Serpens	12.3	−7.3	12.4	1.13	2.1	−0.32	−0.16	−0.21	−0.14	18	7.7	31	1.3
Taurus	4.5	−3.8	4.6	1.26	3.2	−0.34	−0.15	−0.24	−0.14	19	4.4	28	4.8

Name .	a (mag) .	A0 (mag) .	A1 (mag) .	σ (mag) .	rms (σ) .	γlow .	γhigh .	δlow .	δhigh .	M1 mag (103 M⊙) .	AV,SF (mag) .	(mag) .	MSF (per cent) .
Auriga 1	15.9	−14.2	15.3	1.10	4.7	−0.48	−0.24	−0.36	−0.13	268	7.4	32	0.19
Auriga 2	5.5	−4.6	5.6	1.14	2.5	−0.71	−0.32	−0.51	−0.21	13	4.9	15	0.33
Cepheus	32.9	−31.8	32.8	1.04	9.0	−0.50	−0.23	−0.37	−0.15	256	6.7	29	0.26
Chamaeleon	3.3	−2.7	3.4	1.27	3.2	−0.39	−0.18	−0.25	−0.13	45	7.3	24	0.81
Circinus	9.8	−7.5	9.9	1.17	4.5	−0.36	−0.17	−0.24	−0.16	113	6.8	30	1.4
Corona Australis	2.2	−1.7	2.5	1.46	1.6	−0.40	−0.16	−0.20	−0.15	1	3.7	23	9.6
λ-Ori	12.8	−12.3	12.9	1.11	4.6	−0.51	−0.34	−0.40	−0.13	122	7.2	28	0.12
Lupus 1, 2	8.0	−7.4	8.0	1.08	3.4	−0.73	−0.20	−0.47	−0.13	4.6	3.4	23	2.4
Lupus 3, 4, 5, 6	4.2	−2.7	4.3	1.22	2.0	−0.73	−0.21	−0.47	−0.12	24	5.1	25	0.40
Monoceros	12.5	−11.9	12.6	1.11	6.7	−0.42	−0.18	−0.32	−0.12	74	4.8	35	2.0
Ophiuchus	1.5	−0.3	1.5	1.48	7.0	−0.27	−0.11	−0.21	−0.13	9.3	8.6	27	0.78
Orion A	3.9	−3.0	4.1	1.39	4.2	−0.28	−0.15	−0.19	−0.12	44	5.4	37	5.0
Orion B	6.4	−5.5	6.5	1.23	4.3	−0.39	−0.13	−0.26	−0.11	78	6.8	38	1.1
Perseus	5.8	−5.0	5.9	1.23	3.8	−0.42	−0.22	−0.28	−0.17	29	4.8	25	3.0
Serpens	12.3	−7.3	12.4	1.13	2.1	−0.32	−0.16	−0.21	−0.14	18	7.7	31	1.3
Taurus	4.5	−3.8	4.6	1.26	3.2	−0.34	−0.15	−0.24	−0.14	19	4.4	28	4.8

Table 4

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Parameters of best fit for all clouds, at a scale of 0.1 pc. We list the parameters from the fit of the lognormal distribution, the slopes γ and δ, as well as the cloud masses, star formation threshold, extinction of a Jeans mass core and the mass fraction of clouds involved in star formation (MSF).

Name .	a (mag) .	A0 (mag) .	A1 (mag) .	σ (mag) .	rms (σ) .	γlow .	γhigh .	δlow .	δhigh .	M1 mag (103 M⊙) .	AV,SF (mag) .	(mag) .	MSF (per cent) .
Auriga 1	15.9	−14.2	15.3	1.10	4.7	−0.48	−0.24	−0.36	−0.13	268	7.4	32	0.19
Auriga 2	5.5	−4.6	5.6	1.14	2.5	−0.71	−0.32	−0.51	−0.21	13	4.9	15	0.33
Cepheus	32.9	−31.8	32.8	1.04	9.0	−0.50	−0.23	−0.37	−0.15	256	6.7	29	0.26
Chamaeleon	3.3	−2.7	3.4	1.27	3.2	−0.39	−0.18	−0.25	−0.13	45	7.3	24	0.81
Circinus	9.8	−7.5	9.9	1.17	4.5	−0.36	−0.17	−0.24	−0.16	113	6.8	30	1.4
Corona Australis	2.2	−1.7	2.5	1.46	1.6	−0.40	−0.16	−0.20	−0.15	1	3.7	23	9.6
λ-Ori	12.8	−12.3	12.9	1.11	4.6	−0.51	−0.34	−0.40	−0.13	122	7.2	28	0.12
Lupus 1, 2	8.0	−7.4	8.0	1.08	3.4	−0.73	−0.20	−0.47	−0.13	4.6	3.4	23	2.4
Lupus 3, 4, 5, 6	4.2	−2.7	4.3	1.22	2.0	−0.73	−0.21	−0.47	−0.12	24	5.1	25	0.40
Monoceros	12.5	−11.9	12.6	1.11	6.7	−0.42	−0.18	−0.32	−0.12	74	4.8	35	2.0
Ophiuchus	1.5	−0.3	1.5	1.48	7.0	−0.27	−0.11	−0.21	−0.13	9.3	8.6	27	0.78
Orion A	3.9	−3.0	4.1	1.39	4.2	−0.28	−0.15	−0.19	−0.12	44	5.4	37	5.0
Orion B	6.4	−5.5	6.5	1.23	4.3	−0.39	−0.13	−0.26	−0.11	78	6.8	38	1.1
Perseus	5.8	−5.0	5.9	1.23	3.8	−0.42	−0.22	−0.28	−0.17	29	4.8	25	3.0
Serpens	12.3	−7.3	12.4	1.13	2.1	−0.32	−0.16	−0.21	−0.14	18	7.7	31	1.3
Taurus	4.5	−3.8	4.6	1.26	3.2	−0.34	−0.15	−0.24	−0.14	19	4.4	28	4.8

Name .	a (mag) .	A0 (mag) .	A1 (mag) .	σ (mag) .	rms (σ) .	γlow .	γhigh .	δlow .	δhigh .	M1 mag (103 M⊙) .	AV,SF (mag) .	(mag) .	MSF (per cent) .
Auriga 1	15.9	−14.2	15.3	1.10	4.7	−0.48	−0.24	−0.36	−0.13	268	7.4	32	0.19
Auriga 2	5.5	−4.6	5.6	1.14	2.5	−0.71	−0.32	−0.51	−0.21	13	4.9	15	0.33
Cepheus	32.9	−31.8	32.8	1.04	9.0	−0.50	−0.23	−0.37	−0.15	256	6.7	29	0.26
Chamaeleon	3.3	−2.7	3.4	1.27	3.2	−0.39	−0.18	−0.25	−0.13	45	7.3	24	0.81
Circinus	9.8	−7.5	9.9	1.17	4.5	−0.36	−0.17	−0.24	−0.16	113	6.8	30	1.4
Corona Australis	2.2	−1.7	2.5	1.46	1.6	−0.40	−0.16	−0.20	−0.15	1	3.7	23	9.6
λ-Ori	12.8	−12.3	12.9	1.11	4.6	−0.51	−0.34	−0.40	−0.13	122	7.2	28	0.12
Lupus 1, 2	8.0	−7.4	8.0	1.08	3.4	−0.73	−0.20	−0.47	−0.13	4.6	3.4	23	2.4
Lupus 3, 4, 5, 6	4.2	−2.7	4.3	1.22	2.0	−0.73	−0.21	−0.47	−0.12	24	5.1	25	0.40
Monoceros	12.5	−11.9	12.6	1.11	6.7	−0.42	−0.18	−0.32	−0.12	74	4.8	35	2.0
Ophiuchus	1.5	−0.3	1.5	1.48	7.0	−0.27	−0.11	−0.21	−0.13	9.3	8.6	27	0.78
Orion A	3.9	−3.0	4.1	1.39	4.2	−0.28	−0.15	−0.19	−0.12	44	5.4	37	5.0
Orion B	6.4	−5.5	6.5	1.23	4.3	−0.39	−0.13	−0.26	−0.11	78	6.8	38	1.1
Perseus	5.8	−5.0	5.9	1.23	3.8	−0.42	−0.22	−0.28	−0.17	29	4.8	25	3.0
Serpens	12.3	−7.3	12.4	1.13	2.1	−0.32	−0.16	−0.21	−0.14	18	7.7	31	1.3
Taurus	4.5	−3.8	4.6	1.26	3.2	−0.34	−0.15	−0.24	−0.14	19	4.4	28	4.8

3.3 The log(N) versus A_V column density distribution

As described in Section 2.3.2 we calculate gradients γ for each cloud in our sample. As discussed, there are usually at least two distinct regions with different slopes. One region, at low extinction values (γ_low), characterizes the general turbulence of the cloud. At higher extinction values (γ_high) gravity becomes important and changes the column density away from a lognormal distribution. As an example we show the log(N) versus A_V for Auriga 1 in Fig. 2. The plots for the other clouds are shown in Appendix A. We also show how the slopes γ_low and γ_high depend on the spatial resolution for all clouds in Fig. 4.

Figure 4

Top left-hand panel: a plot showing measured slope gradients γ_low against scale for all the clouds studied. Top right-hand panel: as in the top left-hand panel but for the high A_V region. Bottom panels: as in the top panels but for the slope gradients δ of the mass distribution in the clouds. All plots are shown to the same scale to show the differences in the scatter.

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The gradients for all clouds, averaged and interpolated to 0.1 pc resolution, are summarized in Table 4. In general the γ_low values are about twice as negative as the γ_high values. More importantly the scatter for the slopes in the high column density regions of the clouds is smaller than in the low A_V regions. For the averages and scatter for all clouds we find 〈γ_low〉=−0.45 ± 0.15 and 〈γ_high〉=−0.20 ± 0.06. This indicates that once gravity becomes important enough to influence the column density distribution (i.e. star formation starts), then we will find far fewer differences between the various clouds. While at low column densities, where external factors (proximity to supernovae, etc.) determine the turbulent motions, much larger cloud to cloud differences are seen.

3.4 The log(M) versus A_V mass distribution

As for the column density distribution we determine the slopes δ of the log(M) versus A_V mass distribution for low and high column densities. We show as an example the mass distribution of the Auriga 1 cloud in Fig. 3. Overplotted are the linear fits. The values for all slopes δ for different spatial resolutions are listed in Table 3.

Figure 3

Plot of the mass distribution in the cloud Auriga 1 together with the two fits to the low and high column density regime.

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The slopes for all clouds averaged and interpolated to 0.1 pc are summarized in Table 4, the mass distributions are shown in Appendix A. As for the values of γ, the scatter of the slopes differs for the low and high column density material. For the averages of all clouds we find 〈δ_low〉=−0.21 ± 0.11 and 〈δ_high〉=−0.14 ± 0.025. Again, we find that the star-forming (high column density) parts of the clouds are very similar, while the low A_V (turbulence-dominated) regions show a larger scatter. We show how the slopes δ_low and δ_high depend on the spatial resolution for all clouds in Fig. 4.

We extrapolate the fit to the low A_V regions, which leads to the total mass of the cloud. For the purpose of this paper, we calculate the mass in the cloud at a column density of above 1 mag of optical extinction (M_{1 mag}). This ensures we do not include noise and also only integrate over material that is above the threshold for self-shielding from ultraviolet radiation and molecular hydrogen formation (Hartmann, Ballesteros-Paredes & Bergin 2001). Extrapolating the high A_V fit towards one solar mass, one can find the extinction value a core with the Jeans mass would have (⁠⁠). Hence, this characterizes the maximum extinction in the cloud before collapse starts. Furthermore, we determine the intercept between the low and high A_V region, to characterize above which column density (A_V,SF) material is more likely involved in star formation than being characterized by the turbulence in the cloud. This can be seen as the star formation threshold, i.e. the column density above which star formation occurs in the cloud. Finally, the ratio of the mass in the cloud above an extinction A_V,SF (the mass currently associated with star formation) to the total mass of the cloud above 1 mag A_V is determined. Assuming that about 1/3 of the total mass associated with star formation is transformed into stars (Alves, Lombardi & Lada 2007) we can estimate the overall fraction of mass involved in star formation (MSF) an indicator of the star formation efficiency. All these values for each cloud are summarized in Table 4.

4 DISCUSSION AND CONCLUSIONS

We investigated the column density and mass distribution of a number of GMCs based on near-infrared median colour excess extinction maps determined from 2MASS. Our multiscale extinction mapping approach enabled us to determine the cloud properties homogeneously at a common scale of 0.1 pc for all clouds. Furthermore, it allowed us to investigate how the cloud properties depend on the spatial scale.

We find that for about half the investigated clouds the slope of the column density and mass distribution changes to more negative (steeper) values with increasing spatial scale. For the other half no change is found over a scale range of almost an order of magnitude. The former can be understood by the fact that at larger scales, small-scale high-extinction cores are not detected anymore, naturally leading to a steeper mass and column density distribution. In the latter case this does not happen. This could mean that there are no small-scale high-extinction regions. But this does not seem plausible, in particular since the clouds in this group are e.g. Taurus, Perseus, Ophiuchus and Orion. Another possibility is that there is no significant structure in those clouds at the smallest scales, and hence no change over the observed range of scales. We will analyse this in more detail in a forthcoming paper where we will determine the structure functions of all clouds.

Our fits to the column density distributions using a lognormal function resulted in variable outcomes. Some clouds, in particular Auriga 2, Corona Australis, Lupus 3, 4, 5, 6 and Serpens can be fitted very well (rms lower than 3σ). Other clouds (Cepheus, Monoceros, Ophiuchus) cannot be fitted properly by a lognormal function. This is in agreement with earlier works on some of these clouds, e.g. by Lombardi et al. (2008b) and more recently by Kainulainen et al. (2009). All investigated clouds show an excess of column density compared to a lognormal distribution at higher A_V values. The quality of the fit is hence a measure of how much excess there is in a particular cloud. This excess material has decoupled from the general turbulent field and is dominated by gravity, in other words more likely actively involved in star formation. However, in our sample we could not find any significant correlations of the achieved quality of the lognormal fit with other cloud properties or the amount of young stars in the cloud.

The analysis of the mass distribution enables us to draw further conclusions about the structure of the clouds and their star formation properties. For all clouds the change in the slope of the mass distribution is more pronounced than in the column density distribution. It is hence easier to determine the threshold at which gravity becomes the dominant force in shaping the structure. This value ranges from 3.4 mag in Lupus 1, 2 to 8.6 mag of optical extinction in Ophiuchus. We associate this with the extinction threshold for star formation, originally found to be about 7 mag of A_V in Ophiuchus by Johnstone et al. (2004), in agreement with our value. On average for all clouds we find 〈A_V,SF〉= 6.0 ± 1.5 mag optical extinction, a rather small range.

In contrast, the mass fraction of each cloud which is currently involved in star formation varies by almost two orders of magnitude. It ranges from 0.12 per cent in λ-Ori to almost 10 per cent in Corona Australis. We note that these values are not an estimate of the star formation efficiency in these clouds. They rather should be seen as the potential of the cloud to form new stars in the next few 10⁶ yr. The wide range of values indicates that our sample contains clouds at different stages in their evolution. We see clouds that currently only form a small number of stars (e.g. Auriga 1 and Corona Australis) but with a completely different fraction of mass involved in star formation. In Auriga 1 only a small fraction of the total material seems to be available for future star formation. On the contrary in Corona Australis a much larger fraction of material is expected to be forming stars. While other regions that currently form stars (Orion, Taurus and Perseus) still possess a significant fraction of material in a state which is expected to continue star formation. One example for a cloud which might have reached the end of star formation seems to be Ophiuchus. It is currently forming a large number of stars, but has only less than 1 per cent of material in a gravity-dominated form.

We have searched amongst our determined cloud properties for further correlations. In particular we hoped to find a link between the predominant mode of star formation in the cloud (clustered versus isolated) with one or several cloud properties. No such correlation could be found. This is partly expected since our extinction maps show only how much material is potentially involved in star formation, and not how much gas and dust (and its properties) have lead to the star formation mode we currently see in these clouds. One could expect that the properties of the turbulence-dominated part of the clouds (out of which the denser parts are formed) show some dependence on the star formation mode. Either because different turbulent properties cause different modes of star formation, or that feedback from young star clusters and isolated YSOs causes different turbulent properties. But again, no such dependence could be found in our data.

However, there are clear differences in the properties of the clouds at low column densities compared to high A_V values. The scatter of the γ_low and δ_low values between clouds is a factor of 2.5 (for γ) and 4.5 (for δ) larger than for the high-extinction regions. Essentially this paints the following picture of the cloud structure. The low A_V and turbulence-dominated regions differ from cloud to cloud. Their column density and mass distributions are determined by the environment of the cloud and the feedback they have experienced from the star formation processes within them. Above a (column) density threshold of about 6 mag A_V, which is independent of the cloud, gravity becomes the dominant force in shaping the structure. This part of the cloud is then more and more decoupled from the influences of its surrounding turbulent field, and thus the column density and mass distributions for all clouds are virtually identical above the extinction threshold. This also implies that local feedback from young clusters or stars has no significant influence in shaping the column density and mass distributions.

JR acknowledges a University of Kent scholarship. This publication makes use of data products from 2MASS, 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.

REFERENCES

Appendix

APPENDIX A: DEPENDENCE OF γ_low ON SCALE

This is a sample of the full appendix, which is available with the electronic version of the article (see Supporting Information).

SUPPORTING INFORMATION

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Appendix A: Dependence of γlow on scale

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