Free Access
Issue
A&A
Volume 544, August 2012
Article Number A141
Number of page(s) 11
Section Interstellar and circumstellar matter
DOI https://doi.org/10.1051/0004-6361/201219558
Published online 15 August 2012

© ESO, 2012

1. Introduction

For a long time, filaments have been recognised as a common feature of interstellar clouds that is also closely related to the star formation process (e.g., Elmegreen & Elmegreen 1979; Schneider & Elmegreen 1979; Bally et al. 1987). Recent Herschel studies and ground-based surveys have revealed the ubiquity of filamentary structures within dense interstellar clouds, and pre-stellar cores and protostars along dense, gravitationally unstable filaments (e.g., Men’shchikov et al. 2010; André et al. 2010; Arzoumanian et al. 2011; Juvela et al. 2012). Therefore, the structure and formation of the filaments are important parts of star formation studies. The filaments are naturally formed as a result of interstellar turbulence (e.g., Padoan & Nordlund 2011), but significant contribution must exist from immediate triggering by supernova explosions and the radiation and stellar winds from massive stars (e.g., Peretto et al. 2012). The filaments are expected to undergo gravitational fragmentation that will lead to star formation (e.g., Inutsuka & Miyama 1997; Myers 2009; Heitsch et al. 2011).

Many recent studies have addressed the question of the typical shape of the filament cross-sections. The observed column-density profiles are fitted with the so-called Plummer function, partly because of the theoretical predictions associated with this functional form. In particular, an infinite isothermal cylinder in hydrostatic equilibrium is expected to follow this profile (Stodólkiewicz 1963; Ostriker 1964), while, for example, magnetic support, deviations from the isothermal conditions, and initiated collapse would all cause characteristic effects on the parameters of the profile function (see Arzoumanian et al. 2011). If the parameters of the profile function can be determined with high accuracy, this will provide important information on the formation and the current state of the filaments.

The central question is, can we accurately measure the structure and mass of the filaments. Our knowledge is mostly based on molecular lines and dust emission. The kinematic information provided by line observations is crucial but the use of molecules for a structural analysis is complicated by abundance and excitation variations that are difficult to measure and model. The situation is exacerbated in the dense environments where many molecules are severely depleted as they begin to freeze onto dust grains.

Dust emission is considered a reliable probe of dense clouds, especially at sub-millimetre wavelengths where the dependence on the dust temperature is relatively weak. However, the line-of-sight temperature variations can still play a significant role and, in the case of dense cores, can render the column density estimates uncertain by a factor of several (Malinen et al. 2011). Furthermore, dust is expected to undergo evolution in dense and cold environments. This may include accretion of ice mantles, agglomeration of dust particles (leading to the disappearance of small grains), and temperature-dependent changes in the dust optical properties (e.g., Ossenkopf & Henning 1994; Ormel et al. 2011; Meny et al. 2007; Köhler et al. 2011). The analysis of sub-millimetre observations is particularly affected by changes in the dust spectral index β and by the possible increase in the dust opacity κ. The value of κ may vary by a factor of several between diffuse and dense regions. This is important for the study of cloud structure, because the derived column densities are directly proportional to 1/κ. These effects could be expected to be less severe in filaments than in the cloud cores because of the lower density of the filaments and the consequently longer timescale of grain growth. However, significant changes in dust sub-millimetre opacity have been seen also towards filaments. For example, a factor of  ~3.4 increase was reported for a filament in Taurus with AV < 10m (Stepnik et al. 2003). In addition to the uncertainty in κ, a wrong assumption about β will bias the mass estimates through its effect on the colour temperature estimates (Malinen et al. 2011; Juvela & Ysard 2012; Ysard et al. 2012).

In this paper, we investigate the possible differences between the real density structure of the filaments and the one deduced from typical sub-millimetre observations of dust emission. The cloud models are based on magnetohydrodynamic simulations with distinct filamentary structures. The emphasis is not on the nature of the filaments themselves but on the uncertainties in deriving their properties from observations. We carry out radiative transfer modelling to produce synthetic surface-brightness maps and estimate the profile functions of selected filaments. The results are compared to the real column-density structure read directly from the model.

The structure of the paper is the following. In Sect. 2, we present the cloud models and the calculations leading to synthetic surface-brightness maps. In Sect. 3, we describe the selected filaments and the procedures used to fit them with Plummer profiles. The main results are shown in Sect. 4, where we list the results for the filament profile fits, both using the true column density and the column density estimated from the synthetic observations. We discuss the results in Sect. 5 before presenting our final conclusions in Sect. 6.

thumbnail Fig. 1

AV maps of the selected twelve filaments in model A. The dashed lines indicate the filament segments included in the analysis. The AV values are obtained directly from the model cloud where the 1000  ×  1000 pixel image has been convolved with a beam of FWHM equal to three pixels.

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Table 1

Main parameters of the model clouds.

2. Simulated clouds and observations

The modelling is based on a magnetohydrodynamic (MHD) simulation carried out on a regular grid of 10003 cells. The root mean square (rms) sonic Mach number is  ~9 and the rms Alfvénic Mach number  ~2.8. The data are the same as those used in Malinen et al. (2011) and we refer to that paper for details. The MHD calculations assume an isothermal equation of state. The simulation was run first without gravity and the stirring continued for several shock crossing-times to develop turbulence well-separated from the initial conditions. The simulation was continued with gravity switched on and the data used here corresponds to the situation during early collapse (model I in Malinen et al. 2011). We use the unigrid run (Padoan & Nordlund 2011) rather than the adaptive mesh refinement (AMR) MHD data because we need high spatial resolution not only for the cores but also for the lower density filaments.

The cloud size was scaled so that the mean visual extinction AV through the cloud is 1.0m or 2m. In the following, these are called models A and B. Assuming a linear cloud size of 6 pc, these correspond to mean hydrogen densities of 111 cm-3 and 222 cm-3, respectively.

For the radiative transfer calculations, we built an adaptive grid from the 10003 full resolution model. The adaptive grid is constructed by combining neighbouring cells that are optically thin, and have similar densities. The maximum allowed V-band optical thickness of a cell is τmax = 0.04 in model A. The maximum density contrast was set to ρmax/ρmin = 30. The resulting adaptive model has approximately 160 × 106 cells and consists of 839 grids. The same grid was used for model B. The gridding ensures that moderately dense regions, the filaments in particular, are also sampled at the best resolution.

We use the radiative transfer program described in Lunttila & Juvela (2012; see also Juvela & Padoan 2003; Juvela 2005). The cloud is illuminated externally by the isotropic interstellar radiation-field (Mathis et al. 1983). The dust is assumed to have the properties of the standard Milky Way diffuse interstellar dust (RV = 3.1) described in Draine (2003). Because of the low dust temperatures and the modest optical thickness of the model clouds, the absorption of the thermal dust emission does not contribute significantly to the dust heating. Therefore, the dust temperature was calculated in a single step without iteration.

In the third model C, the average visual extinction is set to AV ~ 4m. Assuming a linear cloud size of 6 pc, the mean density is n(H) = 450 cm-3. In model C, we use two dust components. The second one corresponds to a modification where the dust sub-millimetre opacity is increased. This is implemented by multiplying the original opacities with a factor (λ/20   μm)0.5 at wavelengths longer than 20 μm. The resulting dust-opacity spectral index is thus β ~ 1.5 in the sub-millimetre and, as a result, the sub-millimetre opacity κν is higher than for the unmodified dust component. For example, at 250 μm the κν has increased by a factor of 3.5. The relative abundance of the two dust components is varied according to the local density so that the abundances are equal at a density of n(H) ~ 5 × 104 cm-3 (see Fig. 5 in Malinen et al. 2011). At low densities, the relative abundance of the modified dust component goes to zero. By the time the density reaches 106 cm-3, practically all the dust is in the modified form. The calculations for model C were carried out at a resolution of 5003 cells where the grid was obtained by direct downsampling the original 10003 data.

The radiative transfer calculations (Juvela & Padoan 2003; Lunttila & Juvela 2012) solve the three-dimensional (3D) temperature structure of the model clouds and result in synthetic surface-brightness maps of 1000  ×  1000 (models A and B) or 500  ×  500 pixels (model C). Surface brightness maps were calculated at 160, 250, 350, and 500 μm which correspond to the wavelengths of Herschel observations. The full-resolution images were convolved with Gaussian beams of full width at half maximum (FWHM) corresponding to 3, 6, or 12 pixels (in model C, the beam widths in pixels are half of these values). Before the convolution, we add to the maps white noise that corresponds to 3.7, 1.20, 0.85, 0.35 MJy/sr per original Herschel beam size of 12, 18, 25, and 37″, respectively. The values were obtained using the Herschel observations planning tool HSPOT1 and they correspond to the instrumental noise levels of typical Herschel mapping observations (as in Malinen et al. 2011). Most actual Herschel observations are confusion-limited, particularly at wavelengths λ ≥ 250   μm. In our simulations, the confusion results only from structures inside the modelled volume. In more realistic simulations, especially for low surface-brightness values, one might also have to consider the confusion by extragalactic point sources. The final common resolution of all the maps (i.e., the size of the beam used in the convolution) is 40″. When the data are convolved with a beam equal to 3 pixels, the pixel size is thus 13.3″ and the map size is 3.7 degrees. With the assumed cloud size of 6 pc, the FWHM beam sizes of 3, 6, and 12 pixels translate to the cloud distances of 93, 186, and 371 pc (in the following these are called the near, medium, and far distances). For the near distance, the filaments are clearly larger than the beam. For the medium distance, the filaments are still slightly larger than the beam, but in the case of the far distance, the observed filament width is dominated by the convolution.

3. Methods of analysis

The synthetic surface-brightness observations were converted to maps of column density. The dust colour temperature was calculated pixel by pixel by fitting the observed 160, 250, 350, and 500 μm intensities with a modified black-body curve. The data points were weighted according to the noise levels of the simulation. In the fits, the opacity spectral index was fixed to a constant value of β = 1.8 or β = 2.0. These values differ from the actual value in the dust model (Milky Way dust with RV = 3.1), where the spectral index between the wavelengths of 160 μm and 500 μm is 2.09.

With the dust colour temperature known, the column density is calculated as (1)where Sν is the intensity of the fitted modified black-body curve, Bν is the Planck function, and κν is the dust opacity. The κν value was taken directly from the dust model. Therefore, the derived column density NObs differs from the true column density NTrue only because of noise and systematic errors in the colour temperature TC, the latter being caused by line-of-sight temperature variations.

We selected four filaments from each of the three column-density images calculated towards three orthogonal directions (see Fig. 1). The filament segments were selected by eye and the path of each filament was set to follow the peak of the filament as seen in the NTrue maps obtained from the cloud model.

The procedure used in analysing the filaments is basically the same as in Juvela et al. (2012). The column density profiles perpendicular to the filament path were extracted at 20″ intervals. The individual profiles were aligned so that their peaks match before calculating the median profile of the whole filament. Thus, the location of the filament was allowed to differ from the originally selected path. A fixed range of [− 0.3 pc,  + 0.3 pc] around the peak of the column density profile was included in the subsequent analysis. The filament FWHM was measured by fitting the profile with a Gaussian function plus a constant background. The filament FWHM values were determined for each cross-section separately and for the final combined median profile. The median profile of each filament was also fitted with a model consisting of a linear background plus a Plummer-like column-density profile (2)(see Arzoumanian et al. 2011; Nutter et al. 2008; Juvela et al. 2012). The factor Ap is calculated using the formula , where we assume an inclination angle of i = 0 (Arzoumanian et al. 2011). In addition to the two parameters describing the background, the free parameters of the Plummer function are the central density of the filament ρc, the size of the inner flat part Rflat, and the power-law exponent p. The finite resolution of the observations was taken into account by fitting the observed profile not directly with the theoretical Plummer profile but with a Plummer profile convolved with the beam used in the simulated observations. The convolution weights take into account that the actual convolution takes place over a two-dimensional map. This allows us to derive deconvolved parameters for even unresolved filaments.

In addition to the fitted parameters, we also look at some derived quantities. In particular, the mass per unit length of filament, Mline, can be calculated by integrating over the fitted Plummer profile.

4. Results

The twelve filaments were analysed using the true column density NTrue and the column densities NObs estimated from synthetic observations. The analysis was repeated with an actual resolution equal to 3, 6, and 12 pixels of the original 1000  ×  1000 pixel maps, corresponding to the near, medium, and far cloud distances. Owing to the lower resolution of the model C, these correspond to convolution with a beam with a FWHM equal to 1.5, 3, and 6 pixels, respectively.

Figure 2 shows examples of the column-density profiles, comparing the true column density profiles to those estimated from the synthetic observations. These already indicate that in models A and B the column densities are underestimated by  ~ 10% at the centre of the filament, because the estimated colour temperature TC overestimates the mass-averaged dust temperature. That the analysis is carried out with a slightly too low value of the spectral index, β = 2.0, instead of 2.09, contributes in the same direction but is a smaller factor. In model C, the sub-millimetre dust opacity was increased by a factor of a few in the densest parts of the cloud. This is directly reflected in the column density estimates that are more than twice the correct value at the centre of the filaments. The figure also shows that, apart from the filament itself, the profiles are affected by other emission components, i.e., by background confusion.

Each filament was fitted with a Plummer function. Figure 3 shows the results for model A in the case where the observed column densities were estimated with β = 2.0 and the cloud distance was set to d = 93 pc, which corresponds to convolution with a beam of FWHM equal to 3 pixels. The figure demonstrates that the profiles can be well-fitted using the Plummer function. In Fig. 3, the parameter values show a large scatter. Malinen et al. (2012) noted that the fitting of Plummer profiles is sometimes sensitive to the initial parameter values of the least squares minimisation. Visual inspection shows that all the fits in Fig. 3 are good. However, in addition to intrinsic variations between the filaments, some of the scatter is caused by the instrumental noise and especially by the confusion caused by other emission along the line-of-sight.

thumbnail Fig. 2

The profile of the filament number 1 as seen in the models A − C (frames a)c), respectively). The solid thick line shows the actual column-density profile in the model cloud. The other curves show the profiles derived from the synthetic observations when the cloud is at a near distance of 93 pc (thin solid line) or 371 pc (dotted line).

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thumbnail Fig. 3

Fits of the Plummer function to profiles extracted from the observed column-density maps (β = 2.0) of model A cloud at the near distance. The solid black curves show the estimated median profiles of the filaments and the dashed cyan lines the fitted functions. The parameters ρc (central number density of H2 in units of cm-3), p, and Mline (in units of M/pc) are shown in the frames, as well as the rms value of the fit (in units of 1021 cm-2) and the FWHM values of the filaments obtained from fits with Gaussian functions.

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Figure 4 shows a comparison of the estimated values of ρc, p, and Mline for all three beam sizes (3, 6, and 12 pixels) for model A. On the x axis, we plot the parameters obtained using the true column densities, and on the y axis the values derived from the analysis of synthetic observations with and without observational noise and with β equal to either 1.8 or 2.0. The effect of the Δβ = 0.2 is relatively small and is seen mainly as a systematic effect in the density ρc and mass MLine values where it amounts to an underestimation of 20–30%. Without noise, the parameter values obtained using the observed and true column densities are similar to within a few percent. The effect of noise is often small but can also lead to tens of percent differences in the parameter values. The loss of resolution (i.e., convolution with a larger beam) has a clear impact when noise is included and leads to a larger scatter especially in the estimates of the central density, which are underestimated in most cases, by even up to  ~2 orders of magnitude. In the fitting routine, the maximum allowed value for the parameter p is 10. The largest bias in p values occurs when the value derived from either the true or observed column density (but not both) is  ~10. Compared to the other parameters (especially the central density ρc), the mass MLine is less affected by the noise and the resolution.

The corresponding results for the higher column-density model B are presented in Fig. 5. We also note that the parameter values obtained using the observed and true column densities without instrumental noise are in most cases similar to within a few percent.

The scatter in the central density ρc values (from NObs including noise vs. those from NTrue) appears to be smaller than for model A. Using the observations with beam sizes equal to 3 and 6 pixels (corresponding to the near and medium cloud distances) the density and mass are mostly either correct or somewhat underestimated. However, when low resolution with a beam size of 12 pixels is used, the scatter increases and the parameters can be also overestimated. For the MLine values, the error is up to a factor of  ~2. We note that this is the error between the masses derived from either the true or the estimated column density. We return later to the comparison with the actual mass of the filaments that can be obtained directly from the model cloud.

thumbnail Fig. 4

Comparison of the Plummer parameters derived from true column density and the values estimated from the synthetic maps of model A. The rows correspond to parameters ρc, p, and Mline. The columns correspond to beam sizes equal to 3, 6, and 12 pixels, i.e., to the near, medium and far cloud distances. On the x-axis, we always plot the parameter values from NTrue and on the y-axis the values from NObs derived with β = 2.0 and without observational noise (solid circles), with β = 2.0 and noise (open circles), and with β = 1.8 and noise (triangles).

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thumbnail Fig. 5

As Fig. 4, but for the model B that has twice as high a column density.

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In model C, the abundance of modified dust, i.e., the component with a higher sub-millimetre opacity, increases with density. Figure 6 shows the relative abundance of the modified dust as projected values in three orthogonal directions. The locations of the selected filaments are marked in the figure. The figure shows that, while modified dust is the dominant component towards the centre of many filaments, its abundance remains below 20% in most of the area included in the Plummer fits, i.e., at distances of a few arcmin from the filament centre. Thus, the spatial variation in the dust properties should be able to bias both the NObs values and the observed shape of the profiles, and hence affect the parameters of the Plummer fits.

thumbnail Fig. 6

The fractional abundance of modified dust in the model C. The three frames correspond to three orthogonal directions of observations. The white circles indicate the ends of the filament segments included in the analysis.

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Figure 7 presents some examples of the column density profiles of the model C filaments, comparing NObs and NTrue. Because the column densities were estimated assuming a constant dust opacity, κν, the column densities were overestimated whenever the dust opacity had been increased. In the figure, the NTrue values have been scaled by a factor of 1.5. This provides rough agreement with NObs in the outer parts of the filaments, while at the centre the synthetic observations lead to still higher column densities by a factor of  ~ 2. For example, the effect on the observed filament FWHM depends on whether the emission at the half value point is produced mainly by the normal dust or the modified dust. In the latter case, the observed filament FWHM would not be affected, even though the column density can be strongly overestimated.

thumbnail Fig. 7

Profiles of the filaments in model C (at the near cloud distance). The thick lines correspond to the true column density NTrue of the model cloud multiplied by a factor of 1.5. The thin lines denote the column density derived from synthetic observations.

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The values of the Plummer parameters obtained with model C are shown in Fig. 8. The optical depth of the cloud is about twice as high as in the case of model B. Therefore, in addition to the spatial variation in the dust properties, the radiative transfer effects also should have a stronger effect on the results. The observations are seen to overestimate the filament mass by up to a factor of several. Thus, the increase in the average κν dominates over the opposite effect resulting from the dispersion of dust temperatures along the line of sight. The volume density estimates can be  ~0.5–1 orders of magnitude too high and are more biased when the resolution of the observations is low. The scatter is larger than in the previous models, especially when the data are convolved with the largest 12 pixel beam. The smaller spectral index β = 1.8 still leads to lower MLine values but the scatter in the values suggests that the fits are less well-constrained. The p values can be divided into two groups: those close to p ~ 10 and those with values p ~ 2–4. At low p, the values obtained from synthetic observations do not show any systematic bias. In contrast to models A and B, in model C there are also cases without instrumental noise where the p value derived from either the observed column density or the true column density (but not both) is close to 10.

The FWHM values of the filaments were calculated with Gaussian fits and Fig. 9 compares the values in the three models. The values derived from NTrue should be identical in all three models. However, because of the coarser initial resolution of model C, the values of that model differ slightly from those of the others. In models A and B, the values obtained from synthetic observations are always within a couple of percent of the NTrue results. In model B, the filament optical depths are above AV = 10m and one could expect the central column density to be underestimated relative to the lower density outer regions. However, the effect on the beam convolution is still almost negligible. More dispersion is seen in model C. Because the enhanced emissivity increases the signal at the centre of the filaments, the estimated FWHM values of the filaments is expected to be smaller. This is also the observed trend, the analysis of NObs leading to  ~10–20% smaller values than NTrue. The effect is not quite constant, probably because of the additional effect of instrumental noise and the confusion by dense background structures. The largest differences are  ~40% (filaments number 1 and 12).

thumbnail Fig. 8

As Fig. 4, but for model C containing two dust components with a spatially varying abundance ratio. The beam size (3, 6, or 12 pixels corresponding to the near, medium, or far cloud distance, respectively) is indicated on top of each column of frames.

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We compare the p values of the twelve filaments in the three models in Fig. 10, using the medium cloud distance of 186 pc (beam size equal to 6 pixels) and β = 2.0. When the parameter values are derived from the true column density, NTrue (frame c), all the models give similar results, regardless of the value. In four out of twelve cases, the p value is above 8, indicating that the parameters are not well-constrained, i.e., the minimum χ2 value is reached with parameter values that are far from those obtained in the absence of noise. Most of the other twelve filaments have p values in the range 2–4. When the parameter values are derived from the observed NObs without instrumental noise (frame a), the results are mostly similar (to within Δp ~  0.5) to the parameter values derived from the true NTrue. Only in two cases is the p value in model C biased by a factor of  ~2, notably to opposite directions. However, when using the observed NObs with noise (frame b), the situation gets more complicated. Now only half of the filaments have similar p values (to within Δp ~  0.5) as in the case for the true NTrue. This applies to all three models. The bias can now also be seen in all models, particularly model A. These results indicate that the effect of instrumental noise on the derived p parameter is more prominent than the radiative transfer effects.

thumbnail Fig. 9

The FWHM values of the twelve filaments. The frames a)c) correspond to the models A–C, respectively. The squares and the triangles correspond to the near cloud distance of 93 pc and the medium cloud distance of 186 pc, respectively. The solid symbols are values derived from NTrue and the open symbols values from NObs. The dashed lines indicate the median values for the FWHM values derived from NObs. Upper line corresponds to the distance of 186 pc and lower line to the distance of 93 pc.

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thumbnail Fig. 10

The p values of the twelve filaments using medium cloud distance of 186 pc (beam size of 6 pixels) and β = 2.0. Squares indicate model A, triangles model B, and circles model C. a) Values are derived from the observed NObs without noise. b) Values are derived from the observed NObs with noise. c) Values are derived from the true NTrue.

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The results of Figs. 4,  5, and  8, are summarised in Fig. 11. The figure shows for the three model clouds and the three cloud distances the distributions of the relative parameter errors using boxplots2. The errors are calculated as the difference between the values derived from NObs (with β = 2.0) and NTrue, normalised with the latter parameter estimate. The figure includes the error distributions for the logarithm of the central density ρc, parameter p, and the mass Mline.

thumbnail Fig. 11

Boxplot for the relative errors in lg   ρc, p, and Mline for the three cloud models and the three cloud distances (beam sizes of 3, 6, and 12 pixels corresponding to the near, medium, and far distances). The errors are calculated by comparing the values derived from NObs to those obtained from NTrue (see Figs. 4, 5, and 8). For each parameter, the box extends between the first and third quartile with a horizontal line indicating the median value. The vertical lines (whiskers) extend from the box to the last data point within a distance equal to 1.5 times the distance between the third and the first quartile. Points outside this range are drawn with crosses. The number of outliers falling outside the plotted range is indicated with numbers near the upper and lower borders.

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thumbnail Fig. 12

The dependence of the extracted Plummer parameters ρc, p, and Mline on the noise level in the case of model A and the near cloud distance. The level of instrumental noise is either 0.5, 1.0, 2.0, or 4.0 times the default value (see Sect. 2). The plot shows for all the twelve filaments the median values and, as vertical bars, the inter-quartile intervals of the estimated parameters that are obtained from 50 noise realisations of the surface brightness maps. The coloured lines connect the median values. The vertical bars plotted for the individual filaments are shifted slightly along the x axis to avoid overlap.

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thumbnail Fig. 13

Filament 1 seen from two orthogonal viewing directions. The object was identified from observations towards the x direction. The figure shows the object from the y and z directions (frames a) and b), respectively). The circles denote volume density maxima for a few lines of sight through the centre of the filament as seen towards the x direction. The background image is the AV of the model cloud.

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thumbnail Fig. 14

Comparison of the ρc and filament mass estimates from the Plummer fits (open symbols) and the actual values of maximum line-of-sight densities and masses in the cloud model (solid squares). The error bars on the true densities correspond to the 1σ dispersion of the peak density values along the filament. In the mass estimates the solid squares correspond to the total mass within a projected distance of 0.1 pc of the filament. Model A: frames a) and d), model B: frames b) and e), model C: frames c) and f).

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5. Discussion

The estimated central density of the filament ρc, the power-law exponent p, and the mass per length unit MLine were examined in Sect. 4. We have shown that the filaments of our MHD model can be well-fitted with the Plummer function. However, the fitting is sometimes sensitive to the initial values of the fitting parameters (not all initial values leading to a good fit) and the parameters extracted from apparently good fits can have a large scatter, especially in the case of filaments that are not clearly resolved. This indicates that the parameters depend on the amount of instrumental noise added and thus very likely on the confusion with background structures. All the fits used in this paper visually look good, i.e., the least squares procedure appears to have converged to an apparently valid solution. However, different initial values might lead to slightly different combinations of parameters values because, for example, Rflat and p, are intrinsically correlated. Malinen et al. (2012) also noted that small changes in the input data can sometimes have a large effect on the estimated Plummer parameters.

In models A and B, when observational (i.e., instrumental) noise was not included, the parameters derived from NTrue and NObs were similar to within a few percent. Furthermore, in the case of the near distance cloud (beam size equal to 3 pixels), the results did not depend on the level of instrumental noise. When the resolution of the observations was poor, in particular for the far cloud distance of 371 pc corresponding to convolution with a beam of FWHM equal to 12 pixels, noise did cause significant scatter. In the central density ρc the errors could occasionally increase by up to two orders of magnitude. In this case, the beam is larger than the width of the filament and the observations provided weak constraints on the central profile. Occasionally, the result is a very narrow and high density peak that, when convolved with the large beam, is still consistent with the observations.

Compared to model A, model B was much less affected by the noise because of its higher column density and thus higher signal-to-noise ratios. In model C, with modified dust, central density was overestimated by up to  ~0.5–1 orders of magnitude, or even more with the lowest resolution. Similarly, the mass estimates could be incorrect by a factor of several. The scatter was much larger than in the models A and B, especially for the medium and far cloud distances. For the clouds at the far distance (lowest resolution), the total width of the profile is dominated by the convolution and the values of the derived Plummer parameters are more uncertain. This is clearly demonstrated by Fig. 11. The typical errors increase by a factor of several between the near and far distances. On the other hand, in the case of model B the errors are only half of those found in model A. For model C, the errors are larger. The mass estimates are directly affected by the wrong assumption made about the dust opacity but the other parameters do not appear to be very significantly biased.

The central question is, are the fitted parameter values, in particular p, so reliable that theoretical conclusions can be made based on them. In all our models, the p parameter values have a bipolar behaviour. The lower values (~2–4) are mostly within a few percent of those derived from the true column density. The largest errors occur when the p value derived from either the observed or true column density is near the maximum allowed value of 10. In models A and B, this occurs only when instrumental noise is included, but in model C also in the absence of noise. Noise and low resolution clearly increase the probability of getting a high value for p, which implies that the parameters are not well-constrained. Malinen et al. (2012) demonstrated that the confidence regions of the Plummer parameters can be wide and the parameters (e.g., p and Rflat) strongly correlated. We also compared the p parameter values derived from our three models for individual filaments (Fig. 10). The effect of noise appears to be much stronger than the radiative transfer effects.

To separate the effects of instrumental noise from those of spatial resolution, we carried out tests where only the noise levels were varied. Figure 12 shows the results for model A at the near distance. The noise is scaled between 0.5 and 4.0 times the values used above. The figure shows the median parameter values and their inter-quartile ranges for each filament and each noise level, as determined from 50 noise realisations. With the default noise, all the parameters shown are still well-determined. When the noise is increased, the scatter in the parameters derived from the noise realisations increases rapidly. The noise increase also causes a bias in the case of some filaments. However, this varies from filament to filament and, for example, the median value of Mline can either increase or decrease. The figure also implies that as the cloud distance is increased, part of the increase in the scatter in the parameter values is caused by noise. At the far distance, the apparent size of the filament is four time smaller than at the near distance and the total signal-to-noise ratio (integrated over the source) is correspondingly smaller. At the relative noise value of 4.0 the value of p is well-constrained (to within 50%) for fewer than half of the selected filaments. In the fits, p was always restricted to values below 10.0.

We have concentrated on the differences between the true and the observed column densities and their effect on the estimated filament parameters. However, we can note some additional factors affecting the interpretation of these parameters. As in our models, the line-of-sight confusion can be a problem in real observations. For example, some of the occasionally large values of the parameter p are suspected to be caused by background structures. Those can be confused with the filament itself or they may affect the fits by modifying the background estimates (see Fig. 3).

The line-of-sight confusion can be even more severe and some of the identified filaments can be the result of a chance alignment of unrelated (or related but still physically separate) density structures. Of the four filaments selected from the x direction maps, none is seen as a clear isolated filament when examined from the other two orthogonal directions. Figure 13 illustrates this in the case of filament 1. The filament was identified from the image towards the direction x. We determined the locations of the volume density maxima for a few positions along the filament along the line of sight. The images calculated for the other two orthogonal viewing-directions show that the filament is actually a superposition of at least two independent density structures. This may also be a common scenario in the case of real clouds, as suggested, for example, by the identification of several kinematically separate components in some of the Taurus filaments (Tafalla 2010, priv. comm.).

Figure 14 shows the central densities of filaments derived from the observations. For comparison, the figure includes the actual maximum line-of-sight density that is obtained from the model clouds and averaged along the length of the projected filament. For model C, the densities have been overestimated because of the difference between the true and the assumed κν values. In models A and B, there is less bias but even in model B the densities are sometimes overestimated by a factor of a few. This is the case for example for the filament number 1 that was already seen to consist of separate density structures along the line of sight. This result reminds us that continuum observations cannot directly measure the volume density. The uncertainty in the ρc values is several tens of percent.

Figure 14 also compares the mass estimates Mline with the actual cloud mass within a projected distance of 0.1 pc of the filament centre. Apart from model C, the true masses are on average slightly higher than the values derived from the fits to the filament profiles. To some extent, this may reflect the presence of an extended cloud component that in the Plummer fits is modelled as a separate component and thus not included in the Mline estimates. In some cases, there is a clear anticorrelation between the ρc and Mline values (e.g., filaments 4 and 8). When the fitted profile has a narrow peak, a larger fraction of the mass gets attributed to the background component. However, the background is usually low (see Fig. 3) and cannot always explain the factor of  ~ 2 differences in the models A and B.

The bias in the mass estimates can be traced back to the assumptions of the column density calculations, i.e., a constant line-of-sight temperature and a dust spectral index β = 2.0 (instead of the slightly larger actual value of the dust model). Although the bias in the column density values should increase towards the centre of the filament, this was not reflected in the parameters ρc or p as a systematic error. In model C, the inclusion of dust with higher sub-millimetre opacity causes a bias mainly in the mass estimates, although the central density and the parameter p should also be affected. That the column densities were overestimated more towards the centre of the filaments should lead to an overestimation of the parameter ρc. This is also seen in the results, the central densities being overestimated by a factor of 2–3. This is consistent with the modified dust being responsible for most of the signal along the central line of sight. The relative abundance of the dust with higher κν increased as a function of density and this might have been expected to lead to higher p values. In practice, no clear bias, however, is seen. Together with the high column density, the high dust emissivity increased the signal-to-noise ratio in model C far above that of models A and B. Nevertheless, in model C the scatter in the parameter values was also the largest.

6. Conclusions

We have examined the reliability of the parameters that can be derived for interstellar cloud filaments using dust emission data. The study is based on synthetic observations calculated with both MHD simulations of filamentary clouds and radiative transfer modelling of the sub-millimetre dust emission. The observational parameters correspond to typical Herschel observations. We have examined the estimated FWHM widths of the filaments and the parameters of the Plummer functions used to describe the filament profiles. The results have led to the following conclusions:

  • For clouds at a distance of  ~100 pc, the profile parameters obtained from the observations and from the true column density are usually similar to within a few percent. However, differences of up to tens of percent are observed in the case of a few filaments.

  • The results are affected by the instrumental noise. In the higher density models, the signal-to-noise ratios were higher and the errors were correspondingly smaller. If the instrumental noise was increased from the assumed default values, the errors increased rapidly. When the noise was increased by a factor of four, the filament parameters were no longer well-constrained, even for nearby clouds.

  • The uncertainties caused by the background confusion and the typical instrumental noise outweigh the effects of radiative transfer.

  • The uncertainties increase rapidly when the resolution of the observations is decreased. At a distance of  ~400 pc, the filaments are barely resolved, the typical errors are larger by a factor of a few, and there are more cases where the profile fit leads to completely wrong parameter values.

  • We examined a model where the dust opacity increases as a function of density. In the analysis, the mass estimates were naturally affected by wrong assumptions about the dust opacity. The errors were also larger for the other parameters but they are not strongly biased.

  • The derived filament properties are affected by the line-of-sight confusion. In our study, some of the filaments that were identified from the column density maps are discontinuous structures of the 3D density field.


Acknowledgments

The authors acknowledge the support of the Academy of Finland grants Nos. 127015 and 250741. J.M. also acknowledges a grant from Magnus Ehrnrooth Foundation.

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All Tables

Table 1

Main parameters of the model clouds.

All Figures

thumbnail Fig. 1

AV maps of the selected twelve filaments in model A. The dashed lines indicate the filament segments included in the analysis. The AV values are obtained directly from the model cloud where the 1000  ×  1000 pixel image has been convolved with a beam of FWHM equal to three pixels.

Open with DEXTER
In the text
thumbnail Fig. 2

The profile of the filament number 1 as seen in the models A − C (frames a)c), respectively). The solid thick line shows the actual column-density profile in the model cloud. The other curves show the profiles derived from the synthetic observations when the cloud is at a near distance of 93 pc (thin solid line) or 371 pc (dotted line).

Open with DEXTER
In the text
thumbnail Fig. 3

Fits of the Plummer function to profiles extracted from the observed column-density maps (β = 2.0) of model A cloud at the near distance. The solid black curves show the estimated median profiles of the filaments and the dashed cyan lines the fitted functions. The parameters ρc (central number density of H2 in units of cm-3), p, and Mline (in units of M/pc) are shown in the frames, as well as the rms value of the fit (in units of 1021 cm-2) and the FWHM values of the filaments obtained from fits with Gaussian functions.

Open with DEXTER
In the text
thumbnail Fig. 4

Comparison of the Plummer parameters derived from true column density and the values estimated from the synthetic maps of model A. The rows correspond to parameters ρc, p, and Mline. The columns correspond to beam sizes equal to 3, 6, and 12 pixels, i.e., to the near, medium and far cloud distances. On the x-axis, we always plot the parameter values from NTrue and on the y-axis the values from NObs derived with β = 2.0 and without observational noise (solid circles), with β = 2.0 and noise (open circles), and with β = 1.8 and noise (triangles).

Open with DEXTER
In the text
thumbnail Fig. 5

As Fig. 4, but for the model B that has twice as high a column density.

Open with DEXTER
In the text
thumbnail Fig. 6

The fractional abundance of modified dust in the model C. The three frames correspond to three orthogonal directions of observations. The white circles indicate the ends of the filament segments included in the analysis.

Open with DEXTER
In the text
thumbnail Fig. 7

Profiles of the filaments in model C (at the near cloud distance). The thick lines correspond to the true column density NTrue of the model cloud multiplied by a factor of 1.5. The thin lines denote the column density derived from synthetic observations.

Open with DEXTER
In the text
thumbnail Fig. 8

As Fig. 4, but for model C containing two dust components with a spatially varying abundance ratio. The beam size (3, 6, or 12 pixels corresponding to the near, medium, or far cloud distance, respectively) is indicated on top of each column of frames.

Open with DEXTER
In the text
thumbnail Fig. 9

The FWHM values of the twelve filaments. The frames a)c) correspond to the models A–C, respectively. The squares and the triangles correspond to the near cloud distance of 93 pc and the medium cloud distance of 186 pc, respectively. The solid symbols are values derived from NTrue and the open symbols values from NObs. The dashed lines indicate the median values for the FWHM values derived from NObs. Upper line corresponds to the distance of 186 pc and lower line to the distance of 93 pc.

Open with DEXTER
In the text
thumbnail Fig. 10

The p values of the twelve filaments using medium cloud distance of 186 pc (beam size of 6 pixels) and β = 2.0. Squares indicate model A, triangles model B, and circles model C. a) Values are derived from the observed NObs without noise. b) Values are derived from the observed NObs with noise. c) Values are derived from the true NTrue.

Open with DEXTER
In the text
thumbnail Fig. 11

Boxplot for the relative errors in lg   ρc, p, and Mline for the three cloud models and the three cloud distances (beam sizes of 3, 6, and 12 pixels corresponding to the near, medium, and far distances). The errors are calculated by comparing the values derived from NObs to those obtained from NTrue (see Figs. 4, 5, and 8). For each parameter, the box extends between the first and third quartile with a horizontal line indicating the median value. The vertical lines (whiskers) extend from the box to the last data point within a distance equal to 1.5 times the distance between the third and the first quartile. Points outside this range are drawn with crosses. The number of outliers falling outside the plotted range is indicated with numbers near the upper and lower borders.

Open with DEXTER
In the text
thumbnail Fig. 12

The dependence of the extracted Plummer parameters ρc, p, and Mline on the noise level in the case of model A and the near cloud distance. The level of instrumental noise is either 0.5, 1.0, 2.0, or 4.0 times the default value (see Sect. 2). The plot shows for all the twelve filaments the median values and, as vertical bars, the inter-quartile intervals of the estimated parameters that are obtained from 50 noise realisations of the surface brightness maps. The coloured lines connect the median values. The vertical bars plotted for the individual filaments are shifted slightly along the x axis to avoid overlap.

Open with DEXTER
In the text
thumbnail Fig. 13

Filament 1 seen from two orthogonal viewing directions. The object was identified from observations towards the x direction. The figure shows the object from the y and z directions (frames a) and b), respectively). The circles denote volume density maxima for a few lines of sight through the centre of the filament as seen towards the x direction. The background image is the AV of the model cloud.

Open with DEXTER
In the text
thumbnail Fig. 14

Comparison of the ρc and filament mass estimates from the Plummer fits (open symbols) and the actual values of maximum line-of-sight densities and masses in the cloud model (solid squares). The error bars on the true densities correspond to the 1σ dispersion of the peak density values along the filament. In the mass estimates the solid squares correspond to the total mass within a projected distance of 0.1 pc of the filament. Model A: frames a) and d), model B: frames b) and e), model C: frames c) and f).

Open with DEXTER
In the text

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