A statistical study of the mass and density structure of Infrared Dark Clouds

While low mass stars dominate the mass of galaxies, massive stars regulate their energy budget. Understanding how and when the mass distribution of stars is determined is therefore essential in establishing a comprehensive picture of galactic evolution, and star formation, throughout the Universe.

Since stars form in molecular clouds the comparison of the internal structure of the clouds and the initial mass function (IMF) of stars can provide insights on the processes responsible for the formation of stars. The mass distribution of molecular clouds, and cores within them have been extensively studied in the past twenty years. Until recently, it was believed that the mass distribution of CO clumps was described by $\Delta N_{\rm{CO}}/\Delta\log M=M^{-\alpha}$ with $\alpha=0.7\pm 0.2$ for the Milky Way (Kramer et al. 1998, Rosolowski 2004). The mass distribution of prestellar cores, the direct progenitors of stars and stellar systems, as observed in dust continuum is much steeper, resembling the Salpeter IMF with a power law index of $\alpha=1.35$ (Motte et al. 1998; Enoch et al. 2008). However, recent papers questioned the impact of the source extraction scheme used to segment the data on the final mass distribution shape (Pineda et al. 2009). Buckle et al. (2010) found a steeper mass distribution for small scale CO clumps. Also, in most cases, different tracers are required to trace different structures such as dense cores and molecular clumps, raising the question of detection biases. Statistics is often a problem too, binning small number of objects introduce artifacts (Reid & Wilson 2006). Therefore some confusion exists on what is the real mass structure of molecular clouds.

Another important physical aspect of molecular cloud structure is the probability density function (PDF) of the gas volume density. This quantity has received only little attention (e.g. Dring et al. 1996 for HI; Smith & Scalo 2009 for CO) but potentially contains crucial information on the processes at the origin of the density fluctuations. For instance, turbulence-driven fragmentation models develop initial lognormal density fluctuations (e.g. Padoan et al. 1997), which could be the main driver of the lognormal part of the IMF (Chabrier 2003). Studying the density distribution of fragments within molecular clouds could set important constraints on such models.

To perform such studies, we decided to focus on a specific type of molecular clouds, i.e. infrared dark clouds (IRDCs). IRDCs are dense molecular clouds seen in silhouette against the bright emission of the galactic plane (e.g. Perault et al. 1996; Teyssier et al. 2002; Rathborne et al. 2006; Simon et al. 2006). They are cold and only slightly processed by star formation activity, still containing the initial conditions of star formation. Peretto & Fuller (2009; hereafter PF09) recently constructed the column density maps of more than 11,000 of such IRDCs, the largest database of such structures to date. This catalogue provides the opportunity to probe molecular clouds in the Galaxy over a wide range of size scales and column density at high angular resolution using the 8$\mu$m dust absorption and a new source extraction scheme.

In section 2 of the present paper we discuss the dataset we used. In section 3 we describe and re-analyze previous results on the distance distribution of IRDCs, while in Section 4 we estimate completeness limits. Section 5 displays our main results on the size, density and mass distributions of IRDCs and fragments. Discussion is in Section 6. And finally we summarize the main findings of this paper in Section 7.

The analysed IRDCs come from a new catalogue of clouds identified in the Spitzer GLIMPSE data (PF09). IRDCs were defined as connected structures with column density peaks above $N_{\rm H_{2}}=2\times 10^{22}$ cm^-2 and boundaries defined by the contour at $N_{\rm H_{2}}=1\times 10^{22}$ cm^-2. Single peaked structures lying within the IRDCs were identified as fragments. The boundary of a fragment being defined by the contour of the local minimum between a fragment and its closest neighbour, the same criterion used to define the leaves of the dendogram analysis of Rosolowsky et al. (2008). As column density peaks, these fragments are particularly important in the context of star formation since they are the likely birth place of the future generation of stars. The catalogue includes opacity maps at 4^′′ resolution and physical properties for over 11,000 IRDCs. Extracting the densest structures, a total of $\sim$50,000 fragments have been catalogued within the full sample of clouds (PF09). Figure 1a shows a Spitzer three colour image of a region containing three filamentary IRDCs from the PF09 catalogue. Figure 1b shows the column density map of these IRDCs and identifies the fragments within the clouds.

To calculate the density and mass of the clouds the distance of each IRDC is required, however this is not yet known for most of the 11,000 IRDCs. For this analysis we have therefore adopted a statistical approach based on previous measurements of the distances to samples of IRDCs.

Several studies have measured the distance distribution of subsamples of IRDCs in both the 1st and 4th quadrant of the Galactic plane (Simon et al. 2006; Jackson et al. 2008; Marshall et al. 2009). Both kinematic and dust extinction techniques have been used to infer these distances, and although they lead to similar results, there are some differences (Fig. 4). The properties of these distance distributions are summarised in Table 1. Using dust extinction, Marshall et al. (2009) found a centrally peaked, Gaussian-like distance distribution very similar for both the 1st and 4th quadrant of the Galaxy (Fig. 4 - bottom panel). In a similar way, kinematic distances¹¹1These kinematic distances have been recalculated by taking the CS(2-1) and ¹³CO(1-0) velocities published in Jackson et al. (2008) and Simon et al. (2006), respectively, and using the Reid et al. (2009) revised galactic rotation model. shown in Fig. 4 (top panel) show a good agreement between the 1st and 4th quadrant . Although in the 1st quadrant a tail at 5 kpc clearly emerges. However most significant difference between the extinction and kinematic distances is the position of the peak, being located at $\sim 3$ kpc in one case and at $\sim 5$ kpc in the other. Both techniques have their own biases and advantages, it is therefore difficult to favor one distance distribution over another. However a Gaussian distribution is a rather good approximation to the distance distribution in both quadrants.

Fig. 4 (right) shows the average distance to the IRDCs as a function of galactic longitude for the sources with distances measured by these two techniques. Any systematic trend with longitude could introduce a bias in analysis adopting a statistical distribution for the distance of IRDCs. It is clear that there is very little variation of the IRDC distances with respect to the galactic longitude. The only region where there may be such an effect is towards galactic centre seems, an area which is not covered in our IRDC Spitzer catalogue which only extends into $l=\pm 10\hbox{${}^{\circ}$}$. It is worth noting that distance variations around the mean as a function of longitude are very similar for both methods, emphasising that is it predominantly only the average distance which differs between the two methods.

Another possible bias is with respect to the size of the IRDCs. The studies shown in Fig. 4 do not contain IRDCs as small as those in the Spitzer based sample and so it is possible that the small and large clouds have different distance distributions. However, recent observations with ATNF Mopra in CS J=1-0 (Peretto et al. in prep) of a square degree of the galactic plane ($29.8\hbox{${}^{\circ}$}<l<31.8\hbox{${}^{\circ}$}$, $-0.27\hbox{${}^{\circ}$}<b<0.27\hbox{${}^{\circ}$}$) can be used to investigate this possibility. This area covers 196 Spitzer IRDCs in total and more than 80$\%$ of the clouds can be associated with CS emission.

The distances of these clouds can be calculated using the Reid et al. (2009) galactic rotation model. Figure 5 shows the distance distribution of the Spitzer IRDCs detected in CS. In this figure, the IRDCs have been divided int to three size ranges. The 80 smallest IRDCs, those with $R_{eq}$ (PF09) less than 15^′′ have a mean distance and standard deviation of $5.2$ kpc and $0.8$ kpc. This is indistinguishable from the values of $5.3$ kpc and $0.7$ kpc and $5.3$ kpc and $0.8$ kpc for the IRDCs in the next two size ranges, $15^{\prime\prime}<R_{eq}<30^{\prime\prime}$, $R_{eq}>30^{\prime\prime}$, which contain $52$ and $39$ objects respectively. These distributions therefore show no indication that large and small IRDCs have different distributions of distance.

The mass distributions of IRDCs and fragments plotted in Fig. 8 clearly shows a steepening, from large structures to smaller fragments. While the mass distribution of IRDCs is similar to that of CO clumps, the fragment mass distribution has a slope at high masses which is reminiscent of the slope of the Salpeter IMF, although it is best fitted with a lognormal function. However, two biases could affect the shape of the high-mass end of the distribution and the interpretation that its slope is related to the Salpeter IMF. The detailed structure of this part of the distribution may be particularly sensitive to the adopted Gaussian distance distribution. Also high-mass fragments might evolve more rapidly than their low-mas analogues and therefore be under represented in extinction observations at 8$\mu$m (c.f. Hatchell & Fuller 2008).

To the first order, the mass distributions are in agreement with the theoretical work of Hennebelle & Chabrier (2008) who interpreted the transition from a flat mass distribution to a steeper one as the transition from turbulence-generated structures to gravity dominated structures. In this context it is interesting to measure the gravitational binding of IRDCs and fragments. To investigate this, we have used Larson’s relation (Larson 1981) to compute the kinetic support, and calculate the virial mass. Following Hennebelle & Chabrier (2008) we assume the effective velocity dispersion is given by

where $c_{s}$ is the sound speed $V_{0}$ is the normalization velocity of Larson’s relation, $\eta$ is the power law index of Larson’s relation, and $d$ is the size of the structure. We can then use this to compute the corresponding virial mass M_vir for every IRDC/fragment. Figure 10 shows the fraction of IRDCs and fragments having a ratio $M_{\rm H_{2}}/M_{\rm vir}>0.5$, assuming $c_{s}=0.2$ km/s (T=10 K), $V_{0}=1$ km/s, and $\eta=0.4$. Above the completeness limit all the IRDCs appear gravitationally bound, as do the majority of the fragments. Of course large uncertainties exist on the use of Larson’s relation and its normalization. However, different normalizations still give similar conclusions about the fraction of IRDCs and fragments which are bound.

A consequence of the IRDCs being gravitationally bound is that the observed change of slope of the mass distributions shown in Fig. 8 does not represent the transition from turbulence dominated to gravity dominated structures: most of the IRDCs down to the completion limit are bound. However, as shown in Fig. 10 a significant fraction of the fragments lying above the completeness limit are unbound. In other words, even if globally gravitationally bound, IRDCs may contain turbulence-generated over-densities which will probably disperse and not form stars. The physical properties of these unbound fragments are likely to be representative of the initial conditions of star formation within IRDCs.

Using the previously defined ratio to separate bound fragments to unbound ones we constructed the density distributions for both type of fragments as shown in Fig. 11. The mass ranges are the same as in Fig. 9. The unbound fragments have all very similar distributions, independent of their mass range. In particular, the high mass end of the distribution is well fit by the following relation $\Delta N/\Delta\log(n)\propto n^{-4.0\pm 0.5}$, the error bar arising from the uncertainties on Larson’s relation. The location of the peak seems to move with the completeness limit and is therefore questionable. From these plots we cannot exclude a possible lognormal distribution for unbound fragments, but the peak of such distribution would have to be below n$\sim 10^{3}$ cm^-3.

In contrast, the density distribution of the gravitationally bound fragments show a well defined peak between n=$10^{3}$ cm^-3 and $10^{4}$ cm^-3 in each mass range and a shape which broadens to lower densities as the mass range increase. This could result from the higher mass fragments of a given density evolving to being bound more rapidly that lower mass fragments.

We used the largest sample of IRDC column density maps to date in order to better characterize the size, mass, and density structure of dense molecular clouds. The large number of objects, 11,000 IRDCs and 50,000 fragments, allows a detailed analysis of the completeness of the sample. Using a statistically attribute distances to each IRDC, we have demonstrated that above the completeness limit the mass distribution of the IRDCs are consistent with a power-law $\Delta N_{\rm IRDC}/\Delta\log(M)=M^{-0.8}$, where $N_{\rm IRDC}$ is the number of clouds. For the fragments the high mass end of the mass distribution shows a steeper slope, consistent with the slope of the Salpeter IMF, with the overall distribution well fitted by a lognormal function.

Using Larson’s law to estimate the linewidth of each IRDC and fragment, we have shown that above our completeness limit all the IRDCs and the majority of fragments are likely to be bound. This implies that the transition in the shape of the mass distribution does not reflect a transition from unbound to graviationally bound structures. Looking at the distribution of fragment density shows that bound fragments dominate the high density ($n\gtrsim 10^{4}$ cm^-3) end of the distribution for all mass ranges, and dominate the whole distribution for the highest range of fragment masses. There is also a distinct broadening of the distribution with increasing fragment mass. This could be a result of the higher mass fragments evolving to being bound more rapidly that lower mass fragments. The number of unbound fragments as a function of number density has the form $\Delta N_{\rm f}/\Delta\log(n)\propto n^{-4.0\pm 0.5}$ (where $N_{\rm f}$ is the number of fragments) down to a density of $\sim 10^{3}$ cm^-3 where the completeness limit is reached.

The absence of bright infrared sources embedded in IRDCs indicates that the mass distributions and density distributions as a function of mass and degree of gravitational binding derived here are representative of the initial conditions of star formation within dense molecular clouds. These results should serve as constraints on theoretical and numerical models in order to identify and characterize the physical processes responsible for the formation and early fragmentation of molecular clouds.