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The identification of filaments on far infrared and submillim iter images.Morphology, physical conditions and relation with star formation of

filamentary structure

E. Schisano1

Infrared Processing and Analysis Center, California Institute of Technology, Pasadena, CA, 91125, USA.

eugenio@ipac.caltech.edu

K. L. J. Rygl1

European Space Research and Technology Centre (ESA-ESTEC), Keplerlaan 1, P.O. Box 299, 2200 AG Noordwijk,The Netherlands

S. Molinari and G. Busquet and D. Elia and M. PestalozziIstituto di Astrofisica e Planetologia Spaziali, INAF-IAPS, Via Fosso del Cavaliere 100, 00133, Roma, Italy.

D. Polychroni1

University of Athens, Departement of Astrophysics, Astronomy and Mechanics, Faculty of Physics, Panepistimiopolis,15784 Zografos, Athens, Greece.

N. BillotInstituto de RadioAstronomıa Milimetrica Avenida Divina Pastora, 7, Nucleo Central E 18012 Granada, Spain

S. Carey and R. PaladiniInfrared Processing and Analysis Center, California Institute of Technology, Pasadena, CA, 91125, USA.

A. Noriega-CrespoSpace Telescope Science Institute, Baltimore, 21218, USA

T. J. T. MooreAstrophysics Research Institute, Liverpool John Moores University, 146 Brownlow Hill, Liverpool, L3 5RF UK.

R. PlumeDepartment of Physics and Astronomy and the Institute for Space Imaging Sciences, University of Calgary, Calgary,

AB T2N IN4, Canada.

S. C. O. Glover1Istituto di Astrofisica e Planetologia Spaziali, INAF-IAPS, Via Fosso del Cavaliere 100, 00133, Roma, Italy.

1

Zentrum fur Astronomie, Institut fur Theoretische Astrophysik, Universitat Heidelberg, Albert-Ueberle-Str. 2,D-69120 Heidelberg, Germany.

and

E. Vazquez-SemadeniCentro de Radioastronomıa y Astrofısica (CRyA), Universidad Nacional Autonoma de Mexico, CP 58190 Morelia,

Michoacan, Mexico.

ABSTRACT

Observations of molecular clouds reveal a complex structure, with gas and dust often arranged infilamentary rather than spherical geometries. The associations of pre- and proto- stellar cores with thefilaments suggest a direct link with the process of star formation. Any study of the properties of such fila-ments requires a representative samples from different enviroments and so an unbiased detection method.We developed such an approach using the Hessian matrix of a surface-brightness distribution to identifyfilaments and determine their physical and morphological properties. After testing the method on simu-lated, but realistic filaments, we apply the algorithms to column-density maps computed fromHerschelobservations of the Galactic Plane obtained by the Hi-GAL project. We identified∼ 500 filaments, in thelongitude range ofl=216.5o to l=225.5o, with lengths from∼1 pc up to∼30 pc and widths between 0.1 pcand 2.5 pc. Average column densities are between 1020 cm−2and 1022 cm−2. Filaments include the ma-jority of dense material withNH2>6×1021cm−2. We find that the pre- and proto-stellar compact sourcesalready identified in the same region are mostly associated with filaments. However, surface densities inexcess of the expected critical values for high-mass star formation are only found on the filaments, indi-cating that these structures are necessary to channel material into the clumps. Furthermore, we analyzethe gravitational stability of filaments and discuss their relationship with star formation.

Subject headings:Star: formation – ISM: clouds - structure

1. Introduction

Molecular clouds are the birthplaces of stars. Ob-servations at different wavelengths and using differentmolecular tracers of the relatively best studied andnearby star-forming regions suggest complex mor-phologies, with the dust and gas arranged mostlyalong elongated, almost one-dimensional, filamen-tary structures (e. g., Hartmann 2002; Hatchell et al.2005; Myers 2009). TheHerschelSpace Observa-tory (Pilbratt et al. 2010), thanks to its superior spatialresolution and sensitivity in the far-infrared, is nowshowing that the filamentary organization of the denseinterstellar material is much more pervasive than wasinitially thought. From sub-parsec scales in nearbystar formation regions (Andre et al. 2010) to tens-of-parsecs scale along spiral arms (Molinari et al. 2010a),filaments appear to be key-structures required to build

the densities necessary for star formation. The abun-dance of compact star-forming seeds along these struc-tures (see Elia et al. 2010; Henning et al. 2010), frompre-stellar to protostellar young condensations, indi-cates that it is in filaments where the initial conditionsfor star formation may be set.

Despite the ubiquity of filaments in star form-ing regions, it is still unclear how they form andwhat their real relationship is with the mechanismsof star formation. Recent theoretical modeling ofmolecular cloud formation tends to produce fila-mentary structures formed by different mechanisms,like decaying supersonic turbulence (Padoan et al.2007), cooling in the post-shock regions of large-scale colliding flows (Heitsch & Hartmann 2008;Vazquez-Semadeni et al. 2011), or global gravita-tional instabilities (Hartmann & Burkert 2007). Whilethese predictions seem in qualitative agreement with

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observed morphologies, a detailed quantitative com-parison has yet to be done. Significant advancesare now possible with the availability of completepanoramic surveys of both the Galactic Plane, the Hi-GAL project (Molinari et al. 2010a), and of nearbystar-forming regions, like the Gould Belt project(Andre et al. 2010), carried out with theHerschelsatellite that, in principle, allow an unbiased character-ization of filaments over a wide range of spatial scalesand physical conditions.

Visual selection methods used in the recent past toidentify the most obvious structures that appear elon-gated and the subsequent manual analyses to selectedportions of these filaments (e. g., Hartmann 2002;Hatchell et al. 2005; Busquet et al. 2013) become im-pratical when applied to large data sets. For example,the Hi-GAL project, data from which has been usedfor this work, mapped withHerschelthe entire Galac-tic Plane at 70, 160, 250, 350, and 500µm coveringa total area of∼720 square degrees (Molinari et al.2010b). Filaments are found everywhere in Hi-GALmaps. Therefore, the quantitative and qualitativeorder-of-magnitude improvement in available datasetsbrought byHerschel, requires a change of perspec-tives when it comes to analysis methodologies. Theproblem of identifying specific patterns in imageshas already been faced in other scientific fields, inparticular in computer engineering (see for exampleGonzalez & Woods 2002), usingad hocconvolutionwith optimal filtering (like the Canny detector; Canny1986) or studies of the local properties (topology) ofthe images (Hessian matrix studies, Skeleton, MorseTheory, Shapefinders; Sheth et al. 2003). More relatedto astrophysics is the issue of determining the filamen-tary pattern from cosmological N-body simulationsof the dark matter distributions (the cosmic web) orfrom the observed large-scale distribution of galaxies.To accomplish these goals, different approaches havebeen developed (see Aragon-Calvo et al. 2007, for areview) reaching different degrees of complexity.

More recently, Sousbie (2011) has presented aspecific formalism (DisPerSE) based on the discreteMorse theory, which is able to recognize salient fea-tures of the large scale cosmic web. The corre-sponding software has been already applied success-fully to column density maps computed from the far-infrared/sub-millimeter data (e. g., Arzoumanian et al.2011; Hill et al. 2011; Peretto et al. 2012). Neverthe-less, the key issue to be addressed when identifyingparticular patterns is a definition of the feature to be

identified. Given a precise definition for the desiredpattern, it is possible to define the best method to high-light the defined structures. As an example in theskeleton approach, as well as in DisPerSE, a filamentis defined as the one-dimensional segment given bythe central denser region of extended elliptical struc-ture. For this reason, the skeleton is determined bychoosing among all the paths that connect the saddlepoints of the density (intensity) field to the local max-ima, the one that, point by point, shows the smallestvariation in the gradient (Novikov et al. 2006; Sousbie2011). Such a definition allows the correct tracing ofthe ridge of the filaments.

In this paper, we consider filaments not as a one-dimensional structures for which we simply trace themain ridge, or spine, but as an extended 2-dimensionalfeature that covers a portion of the map. Our aim isthen to identify on the map the regions that belong tothe filamentary structure in order to derive its morpho-logical and physical properties. To this end, we startby defining a filament asan elongated region with arelatively higher brightness contrast with respect to itssurrounding, formalizing the intuitive idea of what afilament looks like based on what the eye sees on amap. Hence, instead of an approach involving the localextrema, we prefer to focus on a differential method,specifically the investigation of the eigenvalues of theHessian matrix of the intensity (density) field, directlyrelated to the contrast. Understanding where a filamen-tary structure merges into the surrounding backgroundis the main critical point to be addressed, because itnot only determines the extent of the region but alsoallows a realistic estimate of the background withoutwhich a reliable determination of the properties of thefilament is difficult to obtain.

We present here a method to detect and extract com-plex filamentary structures of variable intensity from2D maps in the presence of high and variable back-ground. In section 2 we describe our methods, in sec-tion 3 we apply the algorithm to realistic simulations offilaments superimposed on real observed backgroundfields, proving the strength and the reliability of themethod to identify filaments. Finally, in section 4 weapply the method to real data, extracting physical pa-rameters of the filaments, and we list our main conclu-sions in section 5.

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2. Identifying filaments: The methodology

Differential methods have already been proved to beuseful to highlight structures like compact sources, e.g.the photometry code CuTEx (Molinari et al. 2011), orfilaments (see Fig. 3 of Molinari et al. 2010a). In Cu-TEx the multidirectional second derivatives are usedto enhance the portion of the map with the strongestcurvature of the intensity field along four fixed direc-tions (x, y and the diagonals), corresponding to thecompact source centers due to their particular symme-tries. Molinari et al. (2011) have shown that the sameoperators qualitatively also trace the edges of extendedstructures like filaments. However, unlike sources, fil-aments are not strongly highlighted in the derivativealong the four directions adopted by CuTEx. There-fore, we generalized the approach adopted with Cu-TEx by Molinari et al. (2011) to the specific case of fil-aments. To such aim, we initially follow the prescrip-tion described by Bond et al. (2010) for the classifica-tion of features, like filaments, voids and walls, presentin smoothed galaxy distributions. In their work, theauthors noticed that each feature has a particular “fin-gerprint” of the curvature along the principal axis. Forexample, on first approximation, filaments can be con-sidered as cylinder-like patterns that are more convexalong one direction with respect to the orthogonal one;in particular, the difference in curvature between thetwo directions would be the highest if the chosen direc-tions are along the cylinder axis, which would have aflat curvature) and the orthogonal radially directed one,that would have the larger convexity. As the curvatureof an intensity map along any direction is proportionalto its directional second derivative, the Hessian opera-tor is most suited to characterize the spatial properties.Differently than the fixed directions adopted for Cu-TEx, the eigendecomposition of the Hessian matrix,H(x,y), of the emission intensity fieldI(x,y) (dust ther-mal emission in the far infrared in this particular case)immediately gives the directions of the principal axesat each position(x, y) of the observed map, by meansof the two eigenvectors,A1 and A2. The two eigen-valuesλ1 andλ2 are proportional to the curvature at(x,y)along these direction. As we are focusing on thedetection of emission features, we will be interestedin convex morphologies, i.e.λ1≤ λ2≤0. In this nota-tion we will assume that direction 1, being the one ofmaximum absolute curvature, will identify the cross-filament direction. A simple analysis of these eigen-values can in principle give the direction, shape andthe contrast of the local structure. For the case of a

filament andnear its axisthe relationship:

λ1 ≪ λ2 ≤ 0 (1)

should hold, with the filament axis defined by thedirection of A2. Although useful for tracing the fea-tures on simulations, or relatively smooth data, thisapproach has some drawbacks when applied to mapsof the interstellar medium. In fact, the above relationdoes not hold close to strong overdensities like, for ex-ample, compact clumps or cores found along the fila-ment. In these casesλ1≃ λ2≤ 0 and, therefore, it is notpossible to define the principal directions with enoughaccuracy. Moreover, equation 1 holds only near theridge of the filament, with the predominance of oneeigenvalue with respect to the other weakening as onemoves radially away from the filament center. Al-though we have to relax the formal criteria in Eq. 1 toidentify filaments, maps of the second derivatives havethe advantage that they filter out the large scale emis-sion and emphasize the more concentrated emissionfrom compact sources and filaments (Molinari et al.2010a, 2011), due to its ability to pinpoint strong vari-ation in the gradient (i.e. change in the contrast) ofthe intensity distribution. Hence, whole filamentaryregions, and not only their axis (hereafter “spine”) areincluded in the regions defined by a simple, conve-niently chosen, thresholding of the second derivativemap.

The pixel-to-pixel noise has a strong impact on thespatial regularity of the Hessian matrix, even for a rel-atively high signal to noise (S/N) map. In fact, thenoise is amplified in theH(x,y) by the derivative fil-ter, that is by construction a high-frequency passbandfilter, and then it affects the estimation of the correctlocal eigenvaluesλ1andλ2. The amplitude of the in-crease of the noise depends on how the differentiationis implemented. For the case of a 5-point derivative,see formula (3) of Molinari et al. (2011), we estimatean increase in the noise level of∼20% on the secondderivative images and a further increase of∼15% inthe eigenvalue maps. Smoothing reduces the noise,but it also blurs the map, damping the variations onthe small spatial scales and hence the contrast of thefilament. Our tests indicate that smoothing through agaussian with HWHM of the order of an instrumentalbeam represents a reasonable compromise between theneed for noise suppression and blurring of the struc-ture. With such a choise the pixel-to-pixel noise is re-duced roughly by a factor∼2 while the variation in the

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contrast decreases at the most by 20% with respect tothe unsmoothed value.

Thus, for a given intensity map we then computethe Hessian matrix, diagonalize and sort the eigenval-ues in each pixel, producing two maps of eigenval-uesλ1(x,y) ≤ λ2(x,y). We exclude from the analysisthe pixels whereλ1(x,y) ≥ 0, which identify concaveshapes in the emission map. Two possibilities can oc-cur for the remaining pixels:λ2(x,y)< 0, that identifiesconvex regions, orλ2(x,y) ≥ 0 for saddle points. Bothcases occur in typical filamentary features with modu-lated emission along the axis.

The next step is to threshold the eigenvalue mapwith the highest absolute value ofλ1(x,y). The adoptedthreshold defines the lowest contrast that a regionshould exhibit to be considered as belonging to a fila-mentary structure. The optimal choice of the thresholddepends on the condition of the map on which the useris working; in particular it depends on the strength ofthe diffuse background emission and of the pixel-to-pixel noise.

We apply a morphological closing operator witha structural element half as wide as the beam tosmooth the edges of the identified regions on scalessmaller than the beam, similar to what was done byRosolowsky et al. (2010). Then, we proceed by iden-tifying connected regions of thresholded image pix-els and label them by progressive numbers; these re-gions are called Regions of Interest (RoI, hereafter).The border of each RoI represents a first rough esti-mate for the edges of the filament. However, sincewe use relaxed criteria with respect to Bond et al.(2010), different types of structures might contami-nate the sample of candidate filaments. In particularrelatively roundish structures like large and elongatedcompact clumps, or clusters of compact objects lyingon a strong intensity field, might be selected as well.To remove this contamination we carry out an ellipsefit to each candidate RoI and discard all regions forwhich the axis ratio of the fitted ellipse is above a fidu-cial value of 0.75. Additionally, we also require thatthe major axis has a minimum length of three timesthe instrumental point-spread function of the image toexclude slightly elongated sources that cannot be con-sidered as filaments. However, it is observed that theremay be cases where filaments may intersect and gener-ate a web-like structure that, depending on the contrastthreshold adopted, may be catalogued as one single re-gion; if the overall shape of the RoI happens to be moreor less roundish, it would be discarded by the above

criteria. To prevent this from happening, we also com-pute the filling factor of the RoI, as the ratio betweenthe area of the RoI and the area of the fitted ellipse;regions whose filling factor is less than a fiducial valueof 0.8 are kept as candidate filamentary structures.The fiducial ellipticity and filling factor threshold val-ues adopted to identify and discard “roundish” clumpstructures have been determined from tests carried outon Hi-GAL maps; the values can, however, be modi-fied as an input to the detection code.

Once the list of candidate filamentary RoIs has beendecontaminated from compact roundish clumps or un-dersized elongated structures, we proceed to iden-tify the spine of the filament by applying a morpho-logical operator of “thinning” on each region (seeGonzalez & Woods 1992). In short, a “thinning” op-erator on a RoI works by correlating each pixel andits surrounding with specific binary masks definingspecifical patterns. These patterns are designed to de-termine if a pixel belongs to the RoIs boundary; insuch a case the pixel is removed. We adopted a 3×3binary mask, so the classification of a pixel as a bound-ary depends strictly on its closest neighbours. Thesame approach has already been applied in a differ-ent field like the identification of filaments on the Sun(Qu et al. 2005). Under the assumption that the fila-ment is symmetric in its profile, by repeating the pro-cedure iteratively until no further pixels can be re-moved, the surviving points constitute the spine of thefilament. In the case of slightly elliptical blobs, thestructure would reduce to few pixels, or even to onepoint in the circular case, that are filtered out whenapplying again our criteria on minimum length. Thespine pixels are then connected through a “MinimumSpanning Tree” (MST), implemented with the Primalgorithm (Cormen et al. 2009), to define the uniquepath that joins them together. This allows us to iden-tify nodal points where multiple branches depart andimmediately enables the classification of structures inmain hubs of peripheral branches. An example of themethod applied over a very simple and bright simu-lated filament is given in Fig. 1. The simulated fila-ment has variable intensity along its spine with peri-odical fluctuation of amplitude equal to 20% and fewcompact sources of different size distributed along itsaxis; although this seems an idealized situation, thepresence of the three sources would have caused theoriginal method of Bond et al. (2010) to break the fila-ment into three portions.

Our main goal in this exercise is to obtain the phys-

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ical characterization of the filamentary structures. Forexample, we would be interested not only in knowingwhere filament spines are, but also what their massesare. To this aim, we also need to estimate the cross-spine size of the filament as well as of the underly-ing background level that we need to subtract to ob-tain the true contribution to the emission of the fila-ment material alone. To do so, for each spine pointwe fit the brightness profile in the direction orthogonalto the spine with a Gaussian function and compute themedian of all the FWHM values obtained; an associ-ated uncertainty is provided by the standard deviationof the individual width estimates along the filament(see Fig. 2). A new region mask is then created, sym-metrical around the spine and with total width equalto twice the median filament FWHM. For very com-plex features where filaments are organized in web-like structures, the cross-spine profile fitting often failsto converge as there are not enough background pixelsto reliably constrain the fit.

We then provide an additional measure of the fil-ament width by adopting the initial RoIs identifiedby the Hessian eigenvalues thresholding, and enlarg-ing them with the morphological “dilation” operator(Gonzalez & Woods 1992) applied three times in se-quence. The merit in doing this is that whatever thethreshold adopted, the thresholding is always doneover the map of minimumbut negativeeigenvalues;in other words, the pixels selected will always belongto regions where the curvature of the brightness pro-file in the maximum curvature direction is within theconvexity region. This implies in general a conser-vative identification of the filament region, because itwould neglect the wings of the filaments where theemission profile changes concavity before joining thebackground emission. In cases of isolated filamentswhere the cross-spine Gaussian fitting converges, wewill show (see below) that the two width estimates arein very good agreement.

The measurement of the background level on whichthe filamentary structure is sitting is very important fora reliable measurement of the total emission of the fila-ment, or of the total mass in case the filament detectionis run on a column density map as we will show be-low. Once the filament RoI and spine have been deter-mined, we proceed as follows. For each spine point wecompute the direction perpendicular to the local spine,and we select the pixels intersecting the 2-pixels-wideboundary region surrounding the filament RoI on eachside of the spine along such a direction. These two

sets of pixels (one on each side of the filament) are fit-ted with a line producing a reliable local estimate forthe background. This is repeated for all filament spinepoints, producing a detailed estimate of how the back-ground varies along the filament.

3. Code performances on simulated and real fila-ments

To characterize the performance of the software inconditions that more closely resemble real situations,we carried out an extensive series of tests using setsof simulated filamentary structures superimposed onmaps of the ISM emission showing a strongly variablebackground. As it is best to test in the most realisticconditions possible, we used maps from the Hi-GALsurvey.

While the simplest and ideal shape of a filament isan homogeneous straight cylinder shape, those condi-tions are rarely (if ever) found in observations. Grav-ity, turbulence and other evolutionary effects twist theorientation of the “ideal” shape and also gather thematter in different places along the original structure.Moreover, what we generally see is a 2D projection ofa 3D structure that, depending on the viewing angle,may significantly amplify any departure from the idealelongated cylinder shape. A realistic filament is gener-ated by first computing randomly twisted curves as the“spine” for the structure, with variable profile alongsuch curves up to 20% with respect to the mean in-tensity . The brightness distribution in the cross-spinedirection is assumed to be Gaussian, and in some caseswe added compact sources of different sizes at differ-ent positions along the spine. Filaments were thenrandomly rotated to avoid any bias from specific ori-entations. Regridding of the simulation has been alsoincluded to estimate the effect of the pixelization onthe performance of the method. We produced variousset of simulations, with different numbers of filamentsand degree of clustering, divided in two groups definedby the width of the structure simulated: unresolved,”thin”, and resolved, ”thick”, filaments.

We want to stress here that our method is suitedto identify extended, but still concentrated, emission.The emission from the large scale structures is stronglydampened in the derivative maps, so broad filamentscan be detected only if they have strong central inten-sities. In fact, the dampening in the derivative map in-creases with the spatial scales following a power-lawbehaviour with an exponent of 2 for scales≥6 pixels,

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i.e. 2 times the PSF for Nyquist sampled maps, seealso Molinari (2014). Therefore, for example, struc-tures with a typical scale of∼2.5 times the PSF havetheir central intensities dampened by a factor of∼10,instead the ones with scales of∼7.5 times the PSF willappear∼100 times fainter in the derivative map than inthe original intensity map. Hence, for a fixed threholdlevel on the same background, it is possible to iden-tify, if they exist, structures with scales of the orderof ∼2.5 times the PSF and intensities 10 times fainterthan the ones with scales of∼7.5 times the beam. Itis clear that the closer the width of the structure willbe to the scales of the background emission, the harderthe structure will be distinguishable. They will standout on the derivative images only if their intensities arecomparable with that of the smallest scales present inthe background component.

3.1. Unresolved simulated filaments

In Fig. 3 we show an example of a simulation: thepresented case has 25 filaments all having a cross-spine size with FWHM of∼3 pixels, namely the sizeof 1 PSF, corresponding essentially to unresolved fil-aments assuming a fully Nyquist sampled map, likethe Hi-GAL maps (Molinari et al. 2010a). The set offilaments was distributed over three different patchesof diffuse emission extracted from Hi-GAL 250µmmaps to try and make results independent from a spe-cific local background condition. For each case, wenormalized the mean intensity along the spine of thesimulated filaments to the median value of the back-ground in the patch to achieve a contrast level equalto 1. Moreover, we also generated images where thebrightness of the filaments was decreased by a factorof 2 and 4 with respect to the background image, tosimulate filaments with different contrast levels (as anexample we present the simulation with contrast 0.5 inFig. 3, top right). The filament extraction method isthen applied over all simulated fields (25 filaments, forthree different background configurations, for 3 differ-ent filament/background contrast ratio), using 4 differ-ent extraction thresholds. The code performances arecharacterized by comparing the length, width and areaof the recovered filaments with those of the input sim-ulated ones.

Figure 4 reports the results for the recovered fila-ment length as a function of the input length. Resultsare shown for the lowest (top row) and the highest(bottom row) extraction thresholds, and for decreas-ing filament/background contrasts (from left to right).

On average the results are very good, with recoveredlength that in most situations agrees with the input val-ues within 20% for the range of thresholds adopted.In the case of the lower extraction threshold, we see ageneral trend to obtain lengths that are systematicallyoverestimated by about 20% irrespective of the fila-ment contrast. This can be explained by the fact thatwith low thresholds on the minimum Hessian eigen-value the regions initially selected by the threshold-ing are larger, and the subsequent “thinning” system-atically produces longer spines. The situation clearlyimproves going to higher thresholds for nominal andhalved contrast ratios (Fig. 4d ande) independent ofthe type of background used; if, however, the higherthresholds are used and the contrast gets too low (panelf, a factor 4 less with respect to the situation depictedin Fig. 3), then the code starts to break the filamentsup into shorter portions depending on the backgroundwhere the simulated filament falls. However, for in-termediate thresholds lengths of the structures are stillrecovered within 20% accuracy even for this faint case.

The code behavior in recovering the average widthof the filaments is more regular, independent of the ex-traction threshold, background type and contrast: therecovered widths mostly agree better than 20% withrespect to the input values.

An accurate estimate of the filament background iscritical for a reliable measurement of the intrinsic fila-ment emission, whether it is total flux or total columndensity (if run on a column density map). In Fig. 5 weshow an example of the reliability of our backgroundestimates. We take a real field over the Galactic Plane(top-left) and superimpose a set of simulated filaments.We normalized the filaments to have constrast 1, 0.5and 0.25, (only the case with contrast 0.5 is shown inthe top-right panel). For the patch of background pre-sented in Fig. 5 the mean intensity along the filamentspine is 35, 18 and 9 arbitrary units respectively forcontrast 1, 0.5 and 0.25. We then extract the filamen-tary structures, identify the ones that correspond to thesimulated filaments (these are the only ones for whichwe have a truth table) and subtract only them. In thebottom-left panel we show the difference between ourestimate of the background after the subtraction of thefilaments and the initial background (top-left panel).The distribution of such differences between the in-put and the filament-subtracted backgrounds, for allthe pixels where a simulated filament was inserted, isshown in the bottom-right of Fig. 5 with a gaussian fitoverplotted in red. The distribution is centered around

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zero difference, with 95% of the pixels falling in thegaussian fit with a FWHM of 5 in arbitrary units. Theremaining 5% of pixels shows residuals as large as 30arbitrary units and are generally located at the brightposition of the original background map (with valuesas large as 80 arbitrary units), sometimes on real com-pact sources, or where multiple filaments nest eachother. Very similar distributions are found for all thecontrast cases and depends mostly on the background,with residuals generally small with respect to the dis-tribution of background intensities at the filament po-sition. In other words, the code delivers reliable esti-mates of the filament underlying backgrounds.

3.2. Resolved simulated filaments

We further carried out simulations where the fila-ments are assumed to be resolved. Fig. 6 is the ana-logue of Fig. 3 for a different filament distribution, butin this case the filaments have a FWHM three timeslarger. As the intrinsic curvature will be lower for theseextended structures, we expect the code performanceto degrade accordingly. In fact, while the backgroundhas not been changed, the filaments have a shallowerintensity variation along the radial direction, so theyare less prominent in the derivative image.

The recovered lengths for the retrieved filaments areshown in Fig. 7, where the meaning of the symbolsand of the different panels is the same as in Fig. 4.The code continues to perform very well on average,showing very similar behavior as for the unresolvedfeatures, up to moderate contrast between filament andbackground. For the lowest contrasted filaments sim-ulated here, the code has more trouble recovering thecorrect length for low detection thresholds, producinga larger scatter of values (Fig. 7c) than the unresolvedcase. In addition, at higher thresholds, the moderateand low contrast filaments are also undetected, andthe few detected ones are broken into smaller portions(Fig. 7f ). As expected, therefore, this method basedon the second derivatives that are computed over a dis-crete set of pixels performs less and less reliably theshallower the structure. It is fair to point out that thisbreak-down in performance is experienced for veryunfavorable conditions where the filament/backgroundcontrast is 4 times less than what appears in Fig. 6. Ifthe contrast is decreased by only a factor of 2 (Fig. 7bande) the code performs much better in recovering thelength.

The situation is worse when one considers the

widths of the filaments for the resolved case. As thefeatures are much shallower than the unresolved fil-ament case, while the spatial dynamical range of thebackground has not changed, the Gaussian fits per-formed at all spine positions are less constrained. Theaverage result is that in the best contrast situations thewidth is systematically underestimated by about 20%.For lower contrast filaments the determination is muchmore noisy and the width is recovered with an uncer-tainty of the order of 30-40%.

3.3. Filaments widths and background estimateson a real filament

To prove in more detail the ability of our approachto recover a correct estimate of the filament width andbackground level, we illustrate the algorithm perfor-mance results on a real filament extracted from themore general results that will be presented in the fol-lowing section. Fig. 8 shows a typical real situationfor a relatively isolated filament. We see that the av-erage width of the filament as it would be estimatedfrom the Gaussian fitting of the radial profile as ex-plained in§2 (the dotted vertical line in Fig. 8-bottompanel) is in excellent agreement with the cross-spinesize of the final filament RoI (after applying the dila-tion operator), that corresponds to the left boundary ofthe grey shaded area in Fig. 8 (bottom panel), and tothe black line in Fig. 8 (top panel). The shaded areacorresponds to the radial distance spanned by the pix-els that in Fig. 8 (top panel) are enclosed between thefull and dashed black lines, where the background isestimated.

We point out that our method is totally consistentwith classifingas filamentaryall the pixels within theborders defined by the flattening of the radial profile.Such definition has already been used in previous workon filaments, i.e. Hennemann et al. (2012).

3.4. Performance evaluation

The results of the extended set of simulations illus-trated in the previous sections build confidence in thefilament extraction method that we have developed.The method has been proved to identify easily struc-tures that are as large as 3 times hypothetical spatialresolution element of the test maps. It is clear that thebest situation is for unresolved filaments, where thecurvature of the brightness distribution is higher. Fil-aments lengths and widths are in general recovered towithin a 20% uncertainty with respect to input values,

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unless the filament/background contrast is very low.As expected the situation gets worse when shallowerfilaments are used in the simulations. While thesefainter structures are still identified, the estimation oftheir morphological and physical parameters becomeunreliable. Similar uncertainities arise for wider sim-ulated filaments but with intensities comparable withthat of the background component. Structures widerthan 3 times the PSF are identified only if they are rel-atively bright with respect to the background. In sucha case the estimation of the parameters is satisfactorydue to the high contrast filament/background.

To summarize, the output of method is very reli-able for structures as wide as 3 times the spatial reso-lution element of the map. The performances quicklydegrade for wider structures which, due to the intrinsicdegeneracy in the method between the width and thecentral intensities, can be identified only if they are asbrigth as the background. For a fixed threshold, thewidest structure that can be identified depends on thebackground properties (its smallest scale and the rela-tive intensity).

Another relevant point coming out of the sim-ulations is that different thresholds are appropri-ate to highlight different kinds of structures, re-solved/unresolved and with different contrast inten-sity, over different background values. While a highthreshold value is able to properly recover unresolvedfilaments with high/moderate contrast with respect tothe background, it splits faint structures into multiplesegments of shorter lengths. However, adopting a lowthreshold enlarges the identified RoIs and artificiallyincreases the filament lengths in the high/moderatecase. Ideally, we would like to apply a higher thresh-old on regions where the filament variations dominateover the background and a lower threshold where theyare shallower and fainter.

Hence, we adopt as a local estimator for the thresh-old the standard deviation of the minimum eigenvaluecomputed on map regions 61×61 pixels wide. Increas-ing the region size does not change substantially theminimum threhsold value. This is expected since byenlarging the region where the threshold is computed,we are including the contribution from larger scales,which is neglegible for scales greater than∼60 pixels.In fact, those scales are dampened up to≤0.5% of theiroriginal value. With such a choice, the threshold willbe higher in regions with large and intense fluctuationsof the emission, eventually dominated by the presenceof the filaments, while it will decrease in detecting re-

gions with a shallower contrast where the variationsare smaller.

Finally, it is worth noticing that while it is straight-forward to identify filaments as elongated structuresin the isolated cases, it clearly becomes difficult whenmultiple objects overlap each other like in the cases ofour simulation. On real data, multiple filaments can bephysically connected and converge toward larger struc-tures, called “hubs” (Myers 2009) or crossing eachother due to line of sight effects. From this point onwe will call one filamenta whole region correspond-ing to one identified RoI. However, in the case of com-plex RoIs the axis is not a simple segment, but oftenit can be composed of multiple segments connected toeach other in nodal positions. We will call each oneof those segments abranch. These branches reflectasymmetries of the RoI and they have two differentphysical interpretations:a) they represent the portionof a larger filament between two local overdensitiesinside the structure,b) they are physically separatedfilaments, connected to the main structure by our al-gorithm since there is not a strong discontinuity in thecontrast variation.

As an example in Fig. 9 we show a 32’×17’ wideregion of the column density map centered at(l,b) =(58.3917, 0.4235) computed fromHerschelHi-GALobservations. In this figure there are five filaments,only three of which, indicated byB, C, D, are com-posed of a single axis. The two remaining, identifiedasA andE, are composed by multiple branches start-ing from the nodal points, indicated with a grey circlesin the figure, and trace the main remaining structures.

The physical quantities for each branch, like massor column densities, are computed on the subregions ofthe original filament mask determined by associatingwith each branch all the pixels that are closer to therelative branch spine.

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4. Discussion of the filamentary structures in theOuter Galaxy

Our filament identification algorithm was run onour column density map which was calculated fromthe fourHerschelHi-GAL maps in the Galactic lon-gitude range ofl=216.5o to l=225.5o, hereafter indi-cated asl217–224, at the wavelengths 160, 250, 350and 500µm. These maps are the first Outer Galaxytiles published from the Hi-GAL survey. Thel217–224 70µm to 500µm and column density maps havebeen presented in Elia et al. (2013), along with a com-pact source catalog. The adopted dust opacity lawwask0( ν

ν0)β with k0 = 0.1 cm2 g−1 at ν0 = 1250 GHz

(250µm) (Hildebrand 1983) andβ was assumed to be2. It is important to notice that the dust opacity pa-rameters adopted to compute the column density mapare rather uncertain. The column density values de-pend on the assumptions in the dust emission model.For example, the fixed spectral indexβ = 2 might bewrong for the cold and denser regions, where a highervalue for β is expected. A lower value forβ influ-ence the grey-body fit outputs with an overestimationof the temperature and an underestimation of the col-umn density. Furthermore, a more realistic dust modelcan be adopted for the more diffuse material, see forexample Compiegne (2010). We estimate that the un-certainity on the dust emission model can affect ourestimate of the column density map by a factor of∼2.

Elia et al. (2013) determined the kinematic dis-tances of the compact sources inl217–224 from theCO (1–0) emission observed with the NANTEN tele-scope. Clump distances range from 370 pc up to8.5 kpc and, as shown in Figure 3 of Elia et al. (2013),the degree of contamination from kinematically sep-arated regions along the line of sight in thel217–224longitude range is very low.

The filament extraction was run with a threshold ofthree times the local standard deviation of the mini-mum eigenvalue (see Sec. 3.4). Moreover, we filteredout any regions with a length smaller than 4 times thebeam, i.e.∼12 pixels or∼2’, as further constraint onthe ellipticity of the structure, in addition to the onedescribed in Sec. 2. With such a choice we are ex-cluding short, but more distant, structures. Hence, weexpect that our sample will be incomplete in terms ofwide angular sizes. Moreover, even narrow structurescan be missed by the detection algorithm, if their con-trast variation is lower than the adopted threshold. Far-ther filaments have a shallower gradient of the con-

trast along their profile due to beam dilution. Thus,the sample will lack also of faint and narrow sourceswhose variation is closer to the one of the background.Despite the incompleteness of the sample, our aimshere are to give a first estimate of the statistical prop-erties of structures thatlook filamentary on Herschelmaps. The shorter and fainter structures are statisti-cally represented in our sample by the nearby struc-tures. However, we remark that since the region stud-ied in this work is mostly dominated by the emissioncoming from distances less than 1.5 kpc (Fig. 3 andTable 1 of Elia et al. (2013)) the incompleteness of thesample will have a minor impact on our results.

4.1. Morphological properties of filaments

The algorithm identified∼500 filaments containingin total∼2000 branches spread across the Galactic lon-gitude range. The detected filaments are shown on thecolumn density map in Figure 10. A visual inspectionof the result indicates that all the major filaments iden-tifiable by eye are traced by the algorithms.

We cross correlated each filament RoI with theclump positions from Elia et al. (2013) and, for the fil-aments with a match, we assigned the distance givenby the mean value of the clump distances found withintheir border. We found that 40% of the detected fila-ments have at least one associated clump and their dis-tances range between 500 pc to 8.5 kpc with a mediandistance of 1.1 kpc (shown in Fig. 11, panela) corre-sponding to the average distance of the CMa OB1 as-sociation (Ruprecht 1966). The distance distribution iscompatible with the two main Galactic arm structuresalong this line of sight: the Orion spur locate at dis-tance of≤1 kpc and the Perseus arm at∼ 2 kpc. Fewfilaments might be associated with the Outer arm, lo-cated at a distance∼5 kpc, however we do not find adefined separation between filaments in such a struc-ture and the one in the Perseus arm. The remaining60% of the filament sample lacking of a clear dis-tance association were assumed to be at the 1.1 kpc,i.e. the median of the distribution (not shown in panela). In the remaining panels of Fig. 11 we plot sepa-rately the filaments with a kinematic distance (i.e., fil-aments with clumps) and these without (i.e., filamentswithout clumps) in solid and dotted lines, respectively.We point out that the percentage of filaments withouta clump detection inside their border is affected by thecriteria adopted by Elia et al. (2013). In fact, the cat-alog presented by these authors includes the clumpsidentified on the Hi-GAL maps for which they could

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determine a distance estimation, through a detectionin the NANTEN CO observations. Hence, two effectscontribute to the number of non detected clumps in-side the filament: the NANTEN observationsa) do notcover the whole area surveyed byHerschelHi-GALdata (see Fig. 4 of Elia et al. 2013), andb) have a lowsensitivity. We found that∼10% of the detected fila-ments fall outside the NANTEN coverage area. Fur-thermore, we compared the maximum column densityfound in each filament RoI and estimated that another8% of the sample are structures that might be unde-tected in the CO data.

The histogram of the filament spine length (forthose filaments with a kinematic distance) peaks ataround 2 pc, and despite the presence of a significanttail that extends up to 60 pc, most of the filamentshave lengths between 1.5 pc and 9 pc, with a medianvalue of 2.45 pc (panelb, solid black line). The fil-aments whose distance was assigned to the medianvalue, show a different distribution (panelb, dashedblack line): their lengths strongly peak at 1 pc and thendrop off quickly. The cut we have adopted in our se-lection criteria translates to an artificial length-cutoffat 0.74 pc for the median filament distance of 1.1 kpc.The lengths are estimated from the map and are liableto projection effects due to a possible inclination effect.No information is available for possible inclination ofthese structures along the line of sight. Assuming arandom uniform distribution for the inclination of thefilaments, the observed mean value of the inclinationangle with respect to the line of sight would be∼ 57o,implying that the intrinsic filament length would be∼19% longer. However, due to projection effects wedo not identify all the filaments that have a small anglebetween their axis and the line of sight, hence the truemean inclination would be larger. The net result is thatthe length distribution is closer to the intrinsic one.

In panelc of Fig. 11 we show the distribution of themeasured width for the identifed structures. Almost allthe filaments are resolved in their radial direction (seeFig. 12) and only 8% of the filaments with a reliabledistance have widths that are compatible within the er-rors with the beam size. We stress that the majorityof detected filaments have width∼1.9 times the beam,despite the simulations having shown that the methodis more sensitive to unresolved “thin” filaments withrespect to the resolved ones independently of the con-trast. There is no apparent reason that far, unresolved,structures should not be detected by the algorithm, aslong they are not so shallow to be confused with the

variations of the background. We have checked thatthe filtering of the sample on lengths, ellipticities andfilling factors does not systematically remove only thefilaments with width of the order of the beam findingthat there is no net effect on the width distribution ofthe sample. Nevertheless, Fig.12 still show a selec-tion effect with the larger structures identifed at fartherdistances. If the same filament population detected atabout∼1 kpc would be shifted toward larger distances,we should detect a larger number of unresolved struc-tures. Instead, if we compare the width distributions ofthe filaments separating the sample into distance bins,we found effectively a lack of narrow structures. Partof the reason is to be attributed to the beam dilutionthat smooth more the variation of the density gradientfor more distant objects as discussed in Sect.4 affect-ing the ability to detect the filaments with respect tothe their surrounding emission. Furthermore, we an-alyzed the SPIRE maps and the column density mapcomputed from them that we adopted in this studyto search for structures with sizes of the order of thebeam. We found that either for filaments and for com-pact sources there is a low number of objects whosewidth is close to the theoretical SPIRE beam. For thecase of the compact sources identified by Elia et al.(2013) we found that the large majority of sources havea size that is∼1.2−−1.3 times the theoretical beam. Asimilar result is found almost everywhere in the galac-tic plane (see also Molinari (2014)). We explain such aresult measured directly on the maps as an effect intro-duced by the local halo in which the sources are em-bedded that broaden the radial profile. In fact we notethat the few cases compatible with the beam are gen-erally well isolated objects on a very low backgroundemission. A similar effect is found also for the fila-ments, the structures usually embedded in dense ex-tended enviroment have a broader profile than the fewisolated filaments. If we consider as unresolved all thefilaments with a width within 1.3 times the theoreticalbeam size we find that∼ 20% of the whole sample arecompatible within the errors with a structure not re-solved in the radial direction. A similar percentage isfound if the sample is split among filaments closer andfarther than 1.5 kpc showing that there is no significantstatistical difference in the sample with the distance.

We computed the deconvolved widths for the re-solved filaments and found a median width equal to0.3 pc, a factor of∼3 larger than the one identified innearby clouds (Arzoumanian et al. 2011).

The distribution of the aspect ratio, defined as the

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ratio between the filament length and its deconvolvedwidth, is presented in Fig. 11 (paneld). The medianvalue for the whole sample of filaments is 7.5, with thebulk of filaments having aspect ratios between 2 and40.

The filaments we have identified on the Hi-GALmaps are typically longer and have higher aspect ra-tios than the ones found by Hacar et al. (2013) in theL1495/B213 Taurus star-forming region, which havelengths ranging between 0.2 pc and 0.6 pc and aspectratio between 2 and 7. The Hi-GAL filaments are in-stead similar to the filaments identified by ammoniaemission in more distant massive star-forming regions(i.e. Busquet et al. 2013 with lengths of 0.6–3.0pc andaspect ratios of 5–20). It is not unexpected to findstructures of different lengths when analyzing a wholeportion of the Galactic plane. However, we stress that,at least for the lower-contrast filaments, the measuredlengths might be underestimated. Based on our sim-ulations we found that some of the shorter filamentsmight belong to longer structures that were split intosmaller portions by the adopted threshold in the fila-ment extraction.

Our sample of filaments is spread over a wide rangeof distances, with the majority located around 1 kpc.Given the almost bimodal distance distribution around1 and 2 kpc, we divided the sample, for which we knowthe distance through the association with clumps, into“near” distance for filaments withd < 1.5 kpc and“far” distancesd > 1.5 kpc (see also the blue dashedline in panel (a) of Fig. 11). There are almost twiceas many filaments at “near” distances (121), than at“far” distances (70). The distribution of the “near” fil-ament lengths has a mean of 2.6 pc with a standard de-viation of 2.1 pc, while for “far” filaments the meanlength is 6.9 pc and the standard deviation is 8.0 pc.“Near” filaments have more constrained spine lengthsand are well represented by the main distribution seenin panel (b). The spine length of farther filaments hasa larger spread all across the histogram (panel (b)),however such an effect is mostly due to the cut on thesize of the structure we have imposed in the extrac-tion. In fact, the distribution shows a cut-off at ∼1.4pc, corresponding to our filter length of∼ 2’ for themedian distance of 2.45 kpc if we consider only thesample at the “far” distance. Our filter length impliesthat, depending on the distance, we are missing struc-tures with lengths between 1.4 to 4.6 pc, the latter be-ing the shortest filament we would keep for the dis-tance of 8.5 kpc. The filament (deconvolved) widths

show a similar trend: nearby filaments widths are nar-rower and well confined, the distribution has a meanof 0.26 pc and a standard deviation of 0.16 pc, fartherfilaments have wider widths with a larger spread, witha mean of 0.82 pc and standard deviation of 0.57 pc.Again the effect of the distance justifies the two differ-ent shapes of the distribution.

4.2. Probability density functions of column den-sity

Fig. 10 strongly suggests that the filaments aredenser structures with a certain morphology, some-times embedded in a less dense molecular cloud.Hence, before discussing the average physical prop-erties of the identified filament sample, we discussthe probability density functions (PDFs) of columndensity to quantify the difference between the filamen-tary structure and the more diffuse material. PDFsare a useful tool for detecting the presence of den-sity structures, e.g. clumps and cores, in molec-ular clouds. A lognormal distribution of columndensities is usually taken as proof of an isothermalmedium where significant large-scale turbulent mo-tions are taking place (Vazquez-Semadeni & Garcıa2001), while the departures from that, generally iden-tified as power-law tails in the high column end of thedistribution, are a sign that self-gravity is starting totake hold (Kainulainen et al. 2009) or, equally feasi-ble, for the presence of a non-isothermal turbulence(Passot & Vazquez-Semadeni 1998).

The global PDF of thel217–224 region has alreadybeen discussed in Elia et al. (2013), here we continuethe analysis dividing the region into on and off fila-ment. Furthermore, we separate the PDFs for pixelscontaining clumps from those without for both fila-mentary and non-filamentary regions, by flagging foreach clump all the pixels within a HPBW of the clumpposition. Such a separation quantifies how much theclumps contribute at the high column density end ofthe distribution, where we expect a strong contributionfrom gravitationally bound structures.

Fig. 13 shows that the shape of the distributionsof filamentary (without clumps) and non-filamentary(with clumps) regions are clearly different. The off-filament pixels follow a lognormal distribution at lowcolumn densities with a power-law at higher col-umn densities. The peak of the distribution is atNH2∼2.5×1021cm−2 representing the column densi-ties of the diffuse galactic material in the outer galaxy

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(see also Fig. 10). While the filaments’pixels havea less pronounced lognormal distribution peaking athigher column densities (NH2∼4×1021cm−2), theyhave a much more dominant power-law tail at high col-umn densities. Below the limit ofNH2≤4×1021cm−2

only 1% or less of the pixels in the column den-sity map lie in filaments. Column densities with4×1021cm−2 ≤ NH2≤6×1021 cm−2 are clearly domi-nated by the non-filamentary molecular cloud emis-sion. ForNH2≥6×1021cm−2 the majority of the pixelsfall in filamentary regions.

Clumps on filaments dominate the high-columndensity end, as expected, but do not account for theentire power-law tail. Hence, the dense material con-tained in the filament, but not in the clumps, may in-dicate that the filament itself is not dominantly madeof isothermal material. A possible explanation is thepresence of some clumps not identified in the previousanalysis, while a more suggestive hypothesis would bethat self-gravity is taking over not only in the clumpsbut also in some portions of the filament. In otherwords, the presence of such high density regions mightindicate that filaments are part of a globally collapsingflow. Our data are not conclusive to determine if suchhypothesis is correct and further investigation throughspectroscopic data is needed. In fact, Schneider et al.(2010) and Kirk et al. (2013), analyzing the molecularline profiles, found hints of global collapse and accre-tion onto the filaments in nearby star forming regions.

The clump pixel distribution off filaments does notdominate at the high column density end (NH2≥ 2×1022cm−2),but at intermediate column densities (3×1021cm−2 ≤

NH2≤ 2×1022 cm−2). It is likely that the high-columndensity end of the off filament distribution belongs tothe dense molecular cloud surrounding the filamentarystructures (see also Fig. 14 and Fig. 15).

4.3. Filaments column densities

We have identified filaments in theHerscheldataas isolated structures as well as parts of a more com-plex structure: the denser parts of a molecular cloud,see Fig. 10. For every pixel in each filament we de-fined the contribution to the measured column densityfrom the filamentas the difference between the pixelvalue, given by the greybody fit of the 160–500µmfluxes, and the local estimated background given bythe interpolation along the direction orthogonal to thefilament spine (see also Sec. 3.3). Fig. 14 and Fig. 15show some examples of filaments (in the left panels)

and the estimated background (in the right panels).Panelsa) andb) in Fig. 14 (the latter shows the samefilament of Fig. 8) are isolated filaments which includethe majority of the material, while in panelsc) andd)in Fig. 15 the identified filaments are deeply embeddedin the cloud. Denser filaments are found in denser en-viroment, as shown also in Fig. 16, suggesting a scal-ing relationship between the mean density of the back-ground and the matter accumulated into the filament.

We show in Fig. 17 the histogram of themeanvalueof the filament contribution to the column density,adopted in the following as an estimate of the aver-age column density of the filament. Our filament sam-ple covers a range of average column densities from1019 cm−2 up to 1022 cm−2. We divide the sample intothree groups: ”A” filaments with at least one associatedclump and therefore with a kinematic distance, ”B” fil-aments without any association, lacking distances, and”C” filaments with unknown association (see Sec. 4.1),still lacking a distance determination.

Filaments with clumps are clearly denser (medianvalue is 4.8 × 1020 cm−2) than those without (medianequal to 1.7 × 1020 cm−2). The distribution of fila-ments in group ”A” and ”B” are very different andthe probabilityPAB, obtained with the Kolmogorov-Smirnoff (KS) statistic, that the populations are drawnfrom the same distribution is very low (PAB≪1×10−5).Moreover, we computed either the probability that thepopulation with unknown association ”C” could bedrawn from the distribution of filaments with clumps,PCA, and the probability that ”C” could be drawnfrom the filaments without clumps,PCB. We foundthat the population ”C” is significantly different fromboth,PCA≪PCB∼ 10−4, however, the large differencebetween the two probabilities seems to indicate thatthis sample is composed mostly by filaments withoutclumps. We conclude that the differences between thepopulations ”B” and ”A” are real and not due to thelack of distance determination. The filaments in pop-ulation ”B” did not form clumps because the columndensities were too low.

Despite the column density being adistance inde-pendentquantity, it can be influenced by beam dilutionin unresolved or low filling factor sources. Given thatone observes a structure with the same physical size,one expects lower column densities at farther distanceswhen beam dilution plays a role. As in Sec. 4.1, wedivided the sample into the “near” and “far” popula-tions and overplotted the corresponding histograms inFig. 17. Indeed, we found that the average column den-

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sities for nearby filaments are higher and with a largerspread (the distribution has a mean of 9.7×1020cm−2

and a standard deviation of 14.0×1020cm−2), whileat larger distances filaments have lower column den-sities and with a smaller spread (the distribution hasa mean of 6.8×1020cm−2 and a standard deviation of7.5×1020cm−2). Given the small differences betweenthe mean values of the two samples, it appears thatbeam dilution has a small effect onthe average columndensity estimator. This is especially true consideringthat almost all the filaments in our sample are resolvedin the radial direction and have lengths several timeslarger than the beam.

Our estimates of the column densities are gener-ally lower than the ones found by Arzoumanian et al.(2011) in nearby star-forming complexes. However,we point out that they report the central column den-sities, which are always higher than the mean value ofthe overall structure. To compare these quantities wealso estimated the central column densities from thepixels of the filament spine thatdo not belong to in-dividual compact sources, after the subtraction of theestimated background. The maximum central columndensities measured are a factor 5.8±2.4 higher than theaverage filament column densities. However, the max-imum values for the column densities along the spinemight be influenced by undetected sources and/or bylocal density enhancement. In the same way, the meanand the median are affected by the low column den-sity pixels, still traced by the code, connecting differ-ent potions of the filament through regions where thematerial has been partially removed. If we adopt asestimator the 3rd quartile of the distribution of the col-umn densities along the spine, background subtractedand not belonging to the sources, we would find thatthe central column densities are higher by a factor of3.1±1.2 than the average column densities. With theadoption of these factors, we again find that our centralcolumn densities are comparable with the ones foundwith Herschelby Arzoumanian et al. (2011) in nearbystar-forming complexes (NH2∼1021 cm−2).

Finally, we want to emphasize some caveats re-lated to our estimation of the filament contribution tothe column density. First, in every pixel the columndensity values are affected by beam dilution, whichsmooths the density enhancements in the central partof the filaments. Second, when we compute the fila-ment contribution after subtracting an estimate of thebackground, consisting of diffuse emission from theGalactic plane and/or the underlying surrounding ma-

terial, we are implicitly assuming that the backgroundand filament add linearly to give the calculated columndensity. Strictly speaking, this is only true when bothcomponents have roughly the same temperature. How-ever, this condition is generally not satisfied, expe-cially in the denser regions, which are typically colder.The overall effect is that the assumed filamentary col-umn density contribution in asingle pixelis an under-estimate of the real column density, with larger dis-crepancies found at higher column densities.

Both beam dilution and the uncertainity due to thesingle temperature approximation affect the estimateof the central column density with respect to the av-erage column density defined at the beginning of thissection. Hence, in the following, all the derived quan-tities related to column density have been estimatedusing the average value.

4.4. High-mass star formation in filaments

Elia et al. (2013) performed a thorough study of thestar-forming content of thel217–224 longitude rangeof the Hi-GAL data. They identified the compactsources (clumps), and classified them as protostellar,prestellar, or unbound clumps. Protostellar objects areobjects which contain 22 and/or 70 µm emission in-dicating the presence of young stellar objects (YSOs).The remaining starless objects, which do not containa YSO (no 22 or 70µm emission), can be dividedinto prestellar objects, which are gravitationally boundobjects evolutionarily younger than protostellar ones,and unbound starless objects, by comparing their mass,M, with their Bonnor-Ebert mass,MBE (see Elia et al.(2013) for a detailed discussion). Here we adopted thecriteria of M > MBE to separate the prestellar objectsfrom the unbound starless ones.

We correlate the above clump classification withour filament sample to understand the impact of fila-mentary structure on the star formation activity. Theoverall result shown by Fig. 17 is that star forma-tion is found preferably on the densest filaments, withhigher average column densities (distribution peak isat 0.5–1×1021cm−2) than those without star forma-tion, whose average column densities peak at 2×1020 cm−2. The differentation starts at 4–6×1020cm−2

(see Fig. 17). Moreover we found that not all fila-ments have clumps. If we exclude from the samplethe filaments for which the clump detection might havebeen biased due to the NANTEN observation cover-age and sensitivity, we found that 50% of the detected

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filaments do not have clumps. These filaments with-out clumps are in a very early stage of evolution or,alternatively, they might be transient structures, onlyconfined by the external pressure.

Not all the remaining filaments show signs of ongo-ing star formation, in fact filaments with only unboundstarless clumps (4% of the sample) do not contributeto it.

Polychroni et al. (2013) investigated the fraction ofsourcesonandoff filament in the L 1641 clouds in theOrion A complex and found that 67% of the prestellarand protostellar sources are located on a filament. Inour case, that includes several molecular clouds, wefind a similar fraction with the majority, 74%, of theclumps reported by Elia et al. (2013) falling within ourfilament sample. However, there are still a significantnumber of clumps detected off filaments.

We computed the clump surface densities from themasses and radii reported in Elia et al. (2013) andcompared clumps located on filaments,Σon, and thesenot on filaments,Σo f f (see Fig. 18). The distributionof Σon peaks around 0.1 g cm−2 and reaches values upto 10 g cm−2, whileΣo f f peaks below 0.1 g cm−2 and itreaches a maximum value of∼0.8 g cm−2. The shapeof the two distributions are statistically different witha more evident tail toward higher surface densities forsources on filaments. The different shapes might beexplained by the larger uncertainities ofΣon due to thedifficulties in decoupling of the compact source con-tributions from the underlying structure. However, weestimate that such uncertainities are up to∼30%, whileto match the two distributions is would be needed thatΣon is overestimated by a factor of∼3. Therefore, itis very likely that the sources on the filaments havelarger surface densities than the ones outside. Moreimportant, we observe that all the sources lying out-side of the filamentary regions have surface densitiessmaller thanΣ ∼ 1 g cm−2. Such value is advocatedin theory of the star formation through turbulent ac-cretion as the threshold limit below which themassivestars cannot form due to fragmentation. Recent obser-vations indicate that such a threshold limit should berevised to a lower value of∼0.2 g cm−2(Butler & Tan2012). Even with this revised limit, our results sug-gest that it is favourable to form massive stars in thefilamentary regions.

Similar conclusions were reached by Polychroni et al.(2013) from the analysis of the clump mass functions(CMFs) for on andoff filament sources. They found,indeed, that the CMF ofon-filament sources peaks

at higher masses (∼ 4 M⊙) than theoff-filament ones(∼ 0.8M⊙), suggesting that the discrepancy is causedby the larger reservoir of material available locally onfilaments in respect to the isolated clumps.

4.5. Stability of filaments

Given that filaments can be roughly approximatedas cylinders, theory shows that such structures havea maximum linear density, or mass per unit lengthMline, above which the system would not be in equi-librium against its self-gravity. For the simplifiedcase of a cylinder infinitely extended in thez-directionwith support given only by thermal pressure, the crit-ical mass per unit length,Mline,crit, is only a functionof temperature (Ostriker 1964; Larson 1985). It be-comes a more complicated function when other ef-fects like turbulence and/or magnetic fields are takeninto account (Fiege & Pudritz 2000), in this case theMline,crit will increase by a small factor. Structureswith Mlineabove theMline,critwill start collapsing alongthe radial direction. The critical mass per unit lengthscales linearly with the temperature and its value isaroundMline,crit∼16 M⊙ pc−1 for the typical tempera-ture in molecular clouds ofT ∼10 K, (see for exam-ple Andre et al. 2010). We estimated the mass per unitlength for each detected filament from the average col-umn density in the RoI, given as the sum of the contri-bution from the filament (see Sec. 4.3) divided by thearea of the RoI, times the mean width. In the simplecase of a straight filament, aligned on the plane of thesky, with no density variations along the spine and con-stant radial profile along the structure, the estimatordefined above equals the integration along the radialprofile divided by the length of the structure, i.e. themass per unit length. The unknown inclination of thestructure affects our estimate ofMline. However, whileon the one hand the area of the filament projected onthe plane of the sky is reduced due to projection ef-fects, on the other the measured column density willincrease by∼60% for the same reason. The two ef-fects partially balance each other.

For the more complex filaments detected in thisfield there might be small discrepancies. We stress thatour definition entails an average (global)Mline for thewhole filamentary structure. In other words, we are as-sumingthe total mass measured in the entire structurewas initially uniformly distributed along the filamentwhen it formed. Therefore, even if we determine lowvalues ofMline we cannot exclude that locally, in asmall portion of the filament, the density is enough to

15

become gravitationally unstable.

Figure 19 shows the mass of the clumps identi-fied in the filaments with respect toMline. We haveexcluded from our analysis the filaments with star-less clumps that have no contribution in terms of starformation. We additionally show the distribution ofMline for these filaments without clumps (green line):all filaments without clumps haveMline<10 M⊙ pc−1,i. e., smaller thanMline,crit∼16M⊙ pc−1. Filaments withclumps span a much wider range of values, withMline

up to 100M⊙ pc−1. Many filaments found in thel217–224 field haveMline<16 M⊙ pc−1, and hence they aresubcritical (Mline<Mline,crit), even though they containclumps. We stress that the value of 16M⊙ pc−1shouldnot be taken as a strict limit since real filamentsa)might not be correctly described by isothermal modeland b) have a finite extension along thez-direction.Hence, even in the case of finite slightly “subcritical”filaments it is expected that both the external pressureand the gravity play a role.

The distribution ofMline for filaments with clumps(see Fig.19) indicates that many filaments did not haveinitially supercriticalMline to begin the clump collapseand star formation. If we assume thatMline remainedunchanged during the onset of star formation, then thisimplies that gravity alone was insufficient to cause aglobal collapse into clumps. Moreover, the ability toform clumps with lower or higher masses is inherent tothe initial Mline of the filament: more massive clumpsform in the more critical (and supercritical) filaments.Since this last result might be biased by including thecontribution of the clumps to the average column den-sity, we estimated theMline excluding the clump masscontribution to the identified filament. This estimatorrepresents the stability of the remaining material in thefilament against gravitational collapse and for a fila-ment where the star formation process is complete itcan be lower thanMline,crit. The trend shown in Fig. 19does not change with removing the clumps from theMline calculation, indicating that the relation betweenMline and the clump mass is real.

Finding filaments hosting clumps with an averageMline lower than the critical value is not completelysurprising. Our filaments are comparable to infrareddark clouds (IRDCs) and high-extinction clouds thatoften display a filamentary morphology and are gen-erally found to have distances of 2–4 kpc (see for ex-ample Rathborne et al. 2006; Rygl et al. 2010). Re-cently, Hernandez & Tan (2011) and Hernandez et al.(2012) investigated the dynamical state of two IRDCs

and foundMline≃0.2–0.5Mline,crit. While their anal-ysis is based on molecular line data, taking into ac-count the (stabilizing) non-thermal contribution to theMline,crit, they find clear signs of star formation ac-tivity, through the presence of 8µm and/or 24 µmpoint sources, in gravitationally stable structures.More generally molecular clouds, and also filamen-tary molecular clouds, are found overall to be gravita-tionally unbound when their masses are compared tothe virial mass despite the fact they contain gravita-tionally bound clumps and star formation (Rygl et al.2010; Hernandez & Tan 2011). The general idea isthat these large scale structures are not far from virialequilibrium and that external pressure or flows couldhave initiated the star formation activity (Tan 2000).Therefore, while the external pressure is confining thelarger structures, the smaller scales (found locally)have to be supercritical to show hints of star forma-tion activity (Andre et al. 2010). The sweeping upof interstellar material and its accumulation in largescale filamentary structures through converging flows(Heitsch & Hartmann 2008; Vazquez-Semadeni et al.2011) is compatible with such results, forming thelarge structure and the inital local overdensities at thesame time. Our results indicate that these processeshave to act quickly, on timescales shorter than the fila-ment and clump free-fall time scales. If that were notthe case, supercritical filaments withMline> Mline,crit

would have a larger number of clumps, since furtherclumps forms as result of the filament contraction andfragmentation. We do not find such an evidence inour sample. Even though it is difficult to confirm theconverging flow scenario without kinematical infor-mation or shock tracing molecules, such as narrowSiO emission, (Jimenez-Serra et al. 2010), our data en-courage the further investigation of these convergingflows. Furthermore, we expect that all the supercrit-ical filaments are characterized by a state of globalgravitational collapse, such as the DR 21 filament(Schneider et al. 2010), where molecular line observa-tions strongly suggest the convergence of large scaleflows as the cause of its formation (Schneider et al.2010; Csengeri et al. 2011).

4.6. The nature of filaments

In the previous section we concentrated on averageglobal quantities measured on the whole filamentarystructure. However, as explained above, the evidenceof subcritical filaments with hints of star formation re-quires the presence of local instabilities inside the fila-

16

ment. Therefore we focus the following analysis on fil-ament branches (see Sec. 3.4 for the definition) whichgive more local information than the global averagesdescribed so far.

Figure 20 shows the histograms of the branchMline

after separating the branches into filaments hostingclumps from the branches in filaments without clumps.The branches belonging to filaments with clumps havehigher linear densities (medianMline ∼10 M⊙ pc−1)than the ones without (medianMline ∼3.3 M⊙ pc−1).Furthermore, we split the branches within filamentswith clumps on the basis of their local association withclumps (in green) or not (in magenta). We found thatthe branches without clumps, but belonging to fila-ments with clumps, have still larger linear densities(with a medianMline ∼10 M⊙ pc−1) than the brancheswithout any clumps in their surroundings. It is un-likely that this difference is due to the distance asso-ciation, even if theMline depends linearly on the dis-tance, since we found that a similar relationship existsfor the branch average column densities. The brancheswith clumps, instead, are denser with a medianMline

∼ 17M⊙ pc−1. They dominate the distribution forMline≥16M⊙ pc−1 (the equilibrium limit against frag-mentation for an isothermal cylinder at 10 K, dottedline), despite the fact there are still a few brancheswithout clump closer toMline,crit.

We have further refined the division in Fig. 20 byspecifying the branches that contain protostellar ob-jects, prestellar objects, and branches without clumpsin the top panel of Fig. 21. For the latter we tookonly the branches where we can determine a distancethrough filament association to avoid any bias thatmight result from the assumed distance. The classi-fication of a branch is based on the most evolved ob-ject found within, hence, branches that contain bothprestellar and protostellar objects have been consid-ered as branches with protostellar objects. With thisdefinition, the total number of branches with an as-sociated clump divides into 20% classified as proto-stellar and 80% classified as prestellar. We foundthat branches with protostellar clumps have the high-est Mline with a median value of∼60M⊙ pc−1, wellabove the critical mass per unit length of 16M⊙ pc−1.Branches with prestellar clumps have a medianMline

∼15M⊙ pc−1. We further checked if our results are af-fected by systematic effects due to the distance sincewe are integrating on larger volumes for more distantfilaments. In the bottom panel of Fig. 21 we select onlythe branches of the filaments that falls in the range be-

tween 700 and 1400 pc. We chose such an intervalto have a statistically significant number of objects.The difference inMlinebetween branches with proto-stellar clumps and the ones with prestellar clumps ismantained. In such a distance range we found 180branches (84% of the total) with prestellar clumps and36 (16% of the total) with protostellar. We tested dif-ferent distance ranges and found that the branches withprestellar clumps always have a distribution with a me-dian between 10 and 14M⊙ pc−1, while the median ofthe distribution of branches with protostellar clumpsvaries between∼40 and 70M⊙ pc−1. The number frac-tion of branches classified as prestellar is always 4-5times larger than the one classified protostellar. Weconclude that distance selection effects are not affect-ing our result.

The analysis of the localMline confirms that al-most all the branches hosting protostellar clumps arelocally unstable against gravity despite the possibil-ity that the overall filamentary structure being po-tentially subcritical. The branches belonging to fila-ments with clumps have a higherMline with respectto the branches in filaments without clumps, with val-ues closer to virial equilibrium. Thus, clump forma-tion is somehow linked to the properties of the largescale filament and, although this result might be af-fected by undetected clumps on the filament, the fil-aments hosting clumps are locally different than theones without. Furthermore, we interpret the strongdifferentiation in Mline for branches with prestellarand protostellar objects shown in Fig. 21 in terms ofan evolutionary scenario, in which the protostellarbranches are intrinsically more evolved than prestel-lar branches. This result indicates thatMline mightbenot constant during the onset of the star formationas also suggested by Heitsch et al. (2009). In sucha scenario the branches and their filaments increasetheir linear density with time by contracting and/or ac-creting of material. The idea of mass accumulationwith time is consistent with observations of velocityshifts along filaments (Peretto et al. 2013; Kirk et al.2013). Moreover, Herschel observations of filamentsin the Taurus star forming region revealed substruc-tures (“striations”) connected to the filament along per-pendicular directions with respect to the filament axis(Palmeirim et al. 2013). These authors suggested thatthe presence of such structures are a hint that accumu-lation of mass is going on.

Our data indicate that the branches would have toincrease their linear density by∼ 45M⊙ pc−1 (given

17

by the difference between the medians of the twodistribution) on the timescale of prestellar evolutionuntil the first protostars forms. Unfortunately prestel-lar lifetimes are very uncertain, Motte et al. (2007)suggested that it lasts for 103–104yr depending onthe mass of the clumps and with the assumption ofa constant accretion rate. However, observations ofnearby clouds indicate that there are a similar num-ber prestellar and protostellar cores suggesting theyhave a similar lifetime of the order of 4.5×105yr(Ward-Thompson et al. 2007; Enoch et al. 2008). Thesame lifetime of∼105yr is estimated from numeri-cal hydrodynamical simulation (Galvan-Madrid et al.2007; Gong & Ostriker 2011). Hence, our results re-quire an accretion rate of∼ 10−4M⊙ pc−1 yr−1 to matchthe measured higher linear densities in the filamen-tary region with protostellar clumps. Such accretionrates are possible and they are lower than that esti-mated by Kirk et al. (2013), both for accretionalongthe filament or for accretionfrom the enviroment. Re-cently Gomez & Vazquez-Semadeni (2014) analyzedfilaments that form in simulations where collidingflows are responsable for the inital cloud formation.They found that filaments accrete from the enviromentand simultaneously accrete onto the clumps withinthem. They determine linear densities increments upto ∼ 3 × 10−5M⊙ pc−1 yr−1, a few times lower thanthe value we estimate. However, we point out theyare analyzing structures that are∼15 pc long, whileour results are estimated from the measure of the localbranches that are smaller portion of the filaments.

The observed increase in linear density in our sam-ple could be explained by a contraction of the filamentsby gravity. In such a case the shrinking of the fila-ments would be evident from the structures with highercentral column density having smaller widths. Fig. 22shows the branch deconvolved width,W, as a functionof the average column density along the branch spineexcluding the source contributions,Nc

H2. No hints of

correlation between the width and theNcH2

is found,regardless of whether we analyze branches with orwithout clumps. We obtain similar results if we se-lect only the branches in a small range of distances tominimize the effect that we might be missing narrow,unresolved, or wide, but faint, structures. Hence, werule out that contraction of the filament is responsi-ble for the increase in linear density. A similar resultwas found by Arzoumanian et al. (2011) for the fila-ment sizes in nearby star forming regions and theseauthors adopted it as evidence in favour of the large-

scale turbulence scenario for theformationof the fila-ments. In fact, if filaments were formed as a result ofgravitational collapse and fragmentation of anisolatedsheet(Inutsuka & Miyama 1992), thenW should beanticorrelated withNc

H2and should be equal to the ther-

mal Jeans lengthλJ = c2s/(GµmHNc

H2), inconsistent

with what is found. In contrast to Arzoumanian et al.(2011) whose average width is∼0.1 pc, we measurean average filament width of∼0.5 pc in accordancewith the sizes measured for some filamentary IRDCs(Jackson et al. 2010). Such a larger average width hasdynamical implications for these structures. If we as-sume that the filament width is roughly equal to the ef-fective Jean lengthλe f f

J =σ2tot/(GµmHNc

H2), whereσtot

takes into account both thermal and non thermal con-tributions, we do expect that the supercritical filamentsneeds a larger non-thermal contribution to hold theirsizes in comparison to the smaller structures identi-fied in nearby star forming region (Arzoumanian et al.2013). An additional contribution to the velocity dis-persion with respect to the one initially given by thelarge-scale turbulence in the interstellar medium wasalready suggested by Arzoumanian et al. (2013). Inour case, either these structures are formed throughthe turbulent scenario with larger non-thermal sup-port or they require a larger contribution from the ac-cretion. We want to stress that, sinceMline∼Nc

H2×

W, the larger average widths imply a lower value ofcentral column density to reach the critical value ofMline,crit≈16M⊙ pc−1. Thus, we estimate that the criti-cal central column density, excluding any non-thermalsupport, is∼1.8×1021cm−2, implying that accretionhas to be present in all regions of the filaments withNc

H2above such value. However, such results have to

be confirmed through direct measurements ofσtot in-side these regions, that is beyond the reach of the cur-rent NANTEN data.

Finally, in Fig. 23 we plot theL/M ratio of theclumps versus the mass per unit length, separatingthe protostellar clumps (blue squares) from prestel-lar clumps (red squares) to provide an evolutionarymarker for the objects forming in the branches. TheL/M ratio is usually correlated with the evolutionarystate of a forming star (Molinari et al. 2008). In thisstar-formation model, two phases are considered: 1)the mass accretion phase, in which the stellar objectsaccretes from its massive envelope and increases itsluminosity. This phase ends when the stellar objectarrives at the zero age main sequence and becomes astar; 2) the envelope dispersion phase, in which the en-

18

velope mass decreases while the luminosity stays con-stant. Assuming that all the objects belong to the sameinitial mass function, and that each clump forms a sin-gle object, one can use theL/M ratio to distinguishbetween more and less evolved objects.

The prestellar clumps have a constant meanL/Mratio of∼0.06 covering a range of localMline between1–100 M⊙ pc−1. The L/M dispersion increases forMline> 10M⊙ pc−1 due to the presence of prestellarclumps withL/M> 0.1. Almost no prestellar clumpsare found in branches withMline ≥ 100 M⊙ pc−1. Theprotostellar clumps have a higherL/M ratio with me-dian value∼1 and present a larger scatter than theprestellar clumps. All the branches hosting the pro-tostellar clumps haveMline≥ 10M⊙ pc−1.

If evolution is the only mechanism in action, wedo expect to find higherL/M ratios in branches withhigherMline. A tentative increase ofL/M can be drawnat least for supercritical branches that appear to bemore evolved than the subcritical ones. However, thelarge dispersion in theL/M suggests that the scenariois not complete. In particular, we cannot exclude thepossibility that the branches have a different star for-mation rate and, therefore, the clumps would have dif-ferent values ofL/M despite having a similar evolutionhistory.

We found very few branches that are highly super-critical (Mline≥100M⊙ pc−1). Surprisingly, they hostvery few prestellar clumps, while we would expecta larger number due to the filament fragmentation,since they evolve on faster timescales than the fila-ment (Toala et al. 2012). It is possible that the alreadyformed clumps have a strong effect on the filamentarystructure through outflows and other feedback mecha-nisms while the filaments themselves are still accret-ing material from the surroundings. Such feedbackchanges the local properties of the filament and pre-vents the further formation of clumps. The end resultis that the filament would be dissipated rather quickly.This would explain the larger number of branches withprestellar objects versus the ones with protostellars.Therefore, filamentary structures are rather short-livedentities.

5. Conclusions

In this paper we have described a method to identifyfilaments of variable intensity from 2D in an automaticand unbiased way, taking into account the extendednature of these structures. The method has been op-

timized to work for the typical properties of filamentsobserved by theHerschel, with filaments overlapping,a strong and variable background and hosting com-pact sources. With the help of simulations, we haveshown the strengths of our approach not only as a wayto detect, but as a tool to determine the morphologi-cal and physical properties of the filaments. Lengthsand widths are typically recovered within 20% of theexpected value. Larger discrepancies are found in thecase of fainter, less contrasted, structures. We stressthe need for a good estimate of the background foraccurate filament mass measurements and we haveshown that our approach allows us to decouple the fil-ament contribution from the background with errorsestimated around∼15-20% (dispersion of the resid-uals) in the case of features with moderate surface-brightness contrast.

We applied our method to the column density mapcalculated fromHerschelobservations of the OuterGalaxy in the Galactic longitude range ofl = 216.5o

to l = 225.5o (l217–224) to measure the filament prop-erties and attempt to determine their role in the starformation process. We found that filaments are foundat various distances between 0.5 to 7.5 kpc. Theycan be identified at various spatial scales, from lengthsas short as∼0.5 pc up to 30 pc and widths between0.1 pc to 3 pc, most of which are typically resolved intheHerschelobservations.The measured aspect ratiosrange between 3 to 30. Distances appear not to haveany selection effect if not that the shorter filaments areundetected at the farther distances.

The column density PDF indicates that almost allthe dense material withNH2≥6×1021cm−2 is arrangedinto filamentary structure. However, not all high den-sity material is associated with clumpshostedby thefilamentary structure; a significative fraction are lo-cal density enhancements on the filaments hinting ofa state of global collapse.

The majority of the clumps (74%) identified inprevious studies are located within the borders ofour filament sample. Nevertheless we still foundstar formation going onoutside the filaments. Itis unlikely that these objects form in the filamentsand are successively dispersed. However, we find asignificative difference between the surface densityof clumps on the filaments versus the one outside,with the clumps on the filaments showing higher sur-face densities.Furthermore, we observe that the ma-jority of the clumps outside the filamentary regionshave surface densities below the value necessary for

19

high-mass star formation (Krumholz & McKee 2008;Butler & Tan 2012), hence it is very likely that theyfinally fragment into cluster of low-mass stars. Thisseems not to be the case for the clumps on the fila-ments where the higher surface densities overcome theassumed threshold limit. It is worth noting that so farobservation of regions with high mass clumps havegenerally highlighted a larger filamentary structurein which they are embedded (Schneider et al. 2010;Hill et al. 2011; Peretto et al. 2013; Polychroni et al.2013; Nguyen Luong et al. 2011). Hence, even if ourdata are not conclusive, the filamentary shape seemsto be an important vehicle to channel enough materialinto small regions and to allow the formation of high-mass stars. Observations with higher spatial resolutiontoward the clumps outside our filament sample woulddetermine if massive star formation happens also with-out filaments and, if that is the case, the relative num-ber of those sources with respect to the number ofmassive clumps inside filaments.

On the other hand, a significant number of filamentsdo not host any clumps. We estimated the mass perunit length,Mline, for all the filaments in our sampleand found that the structures without clumps all havea Mline≤16M⊙ pc−1, the critical value for a filamentsustained by the thermal pressure exterted by materialwith T ∼10 K. Hence, all these filaments are transientstructures that are kept together by external pressurefrom the interstellar medium. The filaments hostingclumps, instead, span a large range ofMline, between 1to 100M⊙ pc−1. Such a result is puzzling, since if theclumps are a direct result of filament collapse and frag-mentation due to gravity, we would not expect to findclumps on any filament withMline≤16M⊙ pc−1. Anynon-thermal contribution, not accounted for by ouranalysis, would increase the value of theMline,critandso the number of filaments with no clumps. Such re-sults agree with what is found for a few IRDCs in theinner Galaxy. Moreover, we found that the supercriti-cal filaments,Mline∼80-90M⊙ pc−1 have more materialaggregated in clumps, hosting also the most massiveones.

Thus, we suggest a possible scenario for fast for-mation of the filaments, where these structures and theinitial seed for the clumps are formed at the same time.In such scenario the global structure can be in equilib-rium (or close to it), and the clump formation starts onlocal scales, induced by processes like flows or exter-nal pressures that locally enhance the linear density.We studied the local scales in the filament, by ana-

lyzing the branches in which the filament can be di-vided. We confirmed that the branchMline values aretypically higher than those of the whole structure. Thisresult is also in agreement with the protostellar objectsonly found on the supercritical filaments. Structuresare created with a range of masses depending on theamount of surrounding material. In the most denseones a mini-“starburst” process starts with a higher starformation rate that can form protostellar objects veryquickly. This scenario effectively decouples the clumpformation from the filament evolution (at least in theirearly stages).

However, when we compareMline of the localsubstructures we found a statistically significant dif-ference between the filamentary subregions hostingprestellar clumps versus the one with protostellarclumps. In particular, we found higher values of lin-ear density in the subregions with protostellar clumpswith respect to the ones with only prestellar clumps.While these results are still consistent with the fastformation scenario, it suggests that the differentationmight be set by the evolution of the structure. Fila-ments, after their formation, increase theirMline andprogress to form prestellar clumps that evolve intoprotostars on timescales faster than the filament evo-lution. It is unlikely that the enhancement ofMlineisdue to the shrinking of the filament due to self-gravity,but our results play in favour of an accretion of mate-rial from the surrounding (within the filament itself orfrom the enviroment). Our data requires moderate in-crease of linear density with time∼10−4 M⊙ pc−1 yr−1,a rate compatible with the one measured for filamentsin nearby molecular clouds.

Following Arzoumanian et al. (2013) we expectthat all the filaments with widths larger than the Jeanslength should be in a state of global collapse due totheir gravity, since the internal thermal pressure aloneis not able to sustain the structure. For the mean widthof our filament sample of 0.5 pc, we expect that allthe structure with a central column density higherthan∼1.8×1021cm−2would be in dynamical collapse.However, since we do not measure any shrinking of thefilament, hence we expect an increase of non-thermalsupport for those more condensed filaments. If such aresult is really hinting at filaments in a state of globalcollapse, we do expect a larger number of those withrespect to the one identified for the studies of nearbyclouds Arzoumanian et al. (2013). This result requiresadditional confirmation for future molecular spectro-scopic observations.

20

The authors are grateful to the anonymous refereefor the useful comments which improved the presen-tation of the work. E.S. acknowledge support fromthe NASA Astrophysics Data Analysis Program grantNNX12AE18G. K.L.J.R. and G.B. are supported byan Italian Space Agency (ASI) fellowship under con-tract number I/005/11/0. DP is funded through the Op-erational Program “Education and Lifelong Learning”that is co-financed by the European Union (EuropeanSocial Fund) and Greek national funds. S.C.O.G. ac-knowledges support from the Deutsche Forschungsge-meinschaft via SFB 881 “The Milk Way System” (sub-projects B1 and B2). The authors are also grateful toM. Pereira-Santaella and N. Marchili for helpful sug-gestions.E.S. is very grateful to Antonia Pierni and wishes tothank her for all the moments they spent together whilehe was working on this research.

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This 2-column preprint was prepared with the AAS LATEX macrosv5.2.

22

Fig. 1.— Example of our method applied to a sin-gle filament (top-left panel) with a 20% flux modula-tion and three bright clumps distributed along its axis.The top-right panel shows the image of the minimumeigenvalues of the Hessian matrix; the green contoursmark the region of interest resulting form the thresh-olding at two different levels, a high and a low thresh-old, while the blue line represent the estimate of thespine of the filament after the ”thinning” of the RoI.In the lower panel are shown the simulated filamentspine profile (black) and the extracted profile for thelow threshold (blue).

Fig. 2.— Example of the ability to retrieve the widthof a simulated filament. In the left panel we see thatthe method retrieves (blue line) exactly the cross-spinewidth of the filament in input (black line) when thefilament has no sources along its axis. In the rightpanel we show the results for the same filament in Fig.1; the ensemble of the cross spine profiles (the set ofgrey points) shows more dispersion with respect to thewidth of the input filament (black line). The fit to thesepoints (the blue line) represents the average filamentcross-spine profile and it is overestimated with respectto the true value (although consistent within the un-certainty bars that reflect the spread of filament widthalong the spine).

23

Fig. 3.— Results of the filament detection over aset of 25 simulated filaments with FWHM of 3 pix-els in the top-left panel, overlayed on a patch ofemission from Hi-GAL 250µm (top-right panel) usedas a background. Only the highest contrast fila-ment/background situation simulated is reported in thefigures. The bottom-left panel shows the minimumeigenvalues image used for thresholding, while thebottom-right shows the masks of the recovered fila-mentary structures; this shows more output filamentsthan the 25 input filaments because the backgroundimage used is realHerscheldata from the Hi-GAL sur-vey and as such contains real filaments.

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over

ed F

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ent L

engt

h (a

rcse

c)

Fig. 4.— Results for the recovered filaments lengthsin the case of unresolved filaments versusf the truelength. Each symbol represents one filament, and thethree different symbol types are for the three back-ground types (see Fig. 3). The top row is for the low-est used extraction threshold, which is more sensitiveto relatively fainter structures; the bottom row is forthe highest used threshold, which is less sensitive tofainter emission. For each row, the three panels showthe results for the three filament/background contrastratios used: in the left panel is the nominal situationthat is represented in Fig. 3, the center panel is for con-trast reduced by a factor 2, while the right panel is forcontrast reduced by a factor 4. In all panels the dottedline represents the identity line, while the dashed linesindicate a 20% discrepancy.

24

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Fig. 5.— A typical field of the Galactic Plane at250 µm (top-left), with a superimposed set of simu-lated filaments(top-right); the color-scale provides theabsolute signal levels of the map in arbitrary units.The simulated filaments shown here have a mean in-tensity value along the spine equal to 18 arbitraryunits. The bottom-left panel shows the difference be-tween the background estimates at the filament posi-tions and the original background. The bottom-rightpanel shows the intensity distribution of the differencesof the bottom-left panel only for the pixels belongingto filaments. In red line we overlap the gaussian fit tosuch distribution.

Fig. 6.— Same as Fig. 3, but with filaments threetimes wider and differently distributed on the back-ground image.

25

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Fig. 7.— Same of Fig. 4, but for the case of resolvedfilaments.

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Fig. 8.— Top: subsection of the larger column den-sity map of Elia et al. (2013) with a typical filament.The dotted line encloses the RoI selected by the initialthresholding on the Hessian eigenvalues; the full blackline is the RoI after applying the dilation operator threetimes; the dashed line encloses the area over which thebackground is estimated.Bottom: cross-spine radialprofile of the column density (blue line) for the regionenclosed in the yellow-line rectangular area in the toppanel. The full red line is the Gaussian fit, with thedotted vertical line marking a typical radial distanceof 1 FWHM from the central spine. The full magentaline is the background profile estimated from the col-umn density corresponding to radial distances fallinginto the shaded area.

26

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Fig. 10.— Column density map computed from the Hi-GAL maps inthe Galactic longitude range ofl=217 tol=224presented in Elia et al. (2013). The thick black lines indicate the main spine of each detected filament. The detectedfilaments are the ones that show a variation of their contrasthigher than 3 times the standard deviation computedlocally on regions that are wide 11.7× 11.7 arcmin2 in size.

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Fig. 9.— Column density map of a portion of the Hi-GAL observation centered at(l,b) = (58.3917, 0.4235)and wide 32’× 17’. The axis of five filamentary re-gions, indicated by the letters fromA to E, are shownas thick black lines. FilamentsA andE are composedrespectively by 5 and 12 branches connected to nodalpoints (indicated by grey circles in the figure).

27

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Fig. 11.— Distributions of filament properties: fil-ament kinematic distance (panela), filament length(panelb), measured width (panelc), aspect ratio (paneld) in the∼500 filaments identified in thel =217–224longitude range. The cyan vertical line depicts themedian value of filaments distance (1.1 kpc), length(2.5 pc), aspect ratio (7.5). The red line in the panela indicates the 1.5 kpc separation mark between the“near” and “far” sample (see text). In panels (b) to (d)we plotted in solid lines the spine lengths, the widthand the aspect ratio’s for the filaments with a distanceestimate while the dashed line shows the histogramof the spine lengths and aspect ratios of the filamentsfor which we assumed the median distance of 1.1 kpc.Furthermore, we divided the sample depending theirestimated distance in near (d< 1.5 kpc, plotted inred)and far objects (d≥ 1.5 kpc, plotted inblue). The greenvertical line shows the length cutoff for filaments at1.1 kpc.

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Fig. 12.— Measured width as a function of the dis-tance for filaments with a reliable distance. For com-parison, we also indicate with green lines the apparentsize of an unresolved object with size equal to the theo-retical beam (solid line on bottom), 1.3 times the theo-retical beam (see the text for details - solid line on top)and five times such a value (dashed line) as a functionof the distance. The blue square represent the mediansize of the filaments identifed by Arzoumanian et al.(2011) in the IC5146 molecular cloud at a distance of460 pc.

28

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Fig. 13.— Column density map pixel distributions.Pixels associated with a filament constitute the violethistogram, Pixels not associated with filaments consti-tute the blue histogram.

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Fig. 14.— Examples of filaments. In left panels wepresent the original column density map where the fila-ments are identified, while in right panels we show theestimated background on the filaments. The top pan-elsa) andb) are examples of isolated filaments, whilepanelc) andd) are filaments embedded in the denserenviroment of a molecular cloud.

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Fig. 15.— As in figure 15, but for the case of fila-ments embedded in the denser enviroment of a molec-ular cloud.

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Fig. 16.— Filament background column densityagainst average filament column density after subtract-ing the background, red squares indicate filamentswith distance d< 1.5 kpc, while blue triangles filamentsd≥1.5kpc. The black dots indicate the filaments with-out an associated distance arbitrary assigned at dis-tance d= 1.1 kpc (see the text for details).

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Fig. 17.— Histogram of filament column density forfilaments with and without clumps. The sample hasbeen divided in three groups, filaments wih clumps,filaments without clumps and filaments where the as-sociation is uncertain (see text). Furthermore, the fil-aments with clumps have been separated into “near”(d<1.5 kpc - plotted in red) and “far” (d≥1.5kpc - plot-ted in blue) filaments. The black dots represent fil-aments without a distance estimate for which we as-sumed a distance d= 1.5 kpc.

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Fig. 18.— Clump surface density distribution forclumps located within filaments and clumps not lo-cated within filaments.

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Fig. 19.— Mass of the clumps inside a filament asfunction of the hosting filament mass per unit length.Small dots are individual clump masses divided in redfor near filaments (distance d<1.5 kpc) and blue forfar ones (distance d≥1.5 kpc). Triangles depict thetotal mass in clumps on that filament. For compari-son, we also plot the filaments without clumps (greenhistogram) to show their distribution in mass per unitlength. The vertical dashed line marks the critical massper unit lengthMline,crit∼16 M⊙ pc−1 for T ∼10 K.

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Fig. 20.— Histograms of branch mass per unit lengthsfor the branches belonging to filament with clumps(solid black line) and filament without clumps (dashedblack line) with their median represented as verticallines. The branches belonging to filament with clumpsare further splitted into branches hosting clumps them-selves (magenta line) and branches without clumps intheir local surroundng (green line). The dotted line in-dicate theMline,critvalue of 16M⊙ pc−1.

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Fig. 21.—Top panel: Histogram of masses for unitlength of all the branches with protostellar clumps(blue solid line), with prestellar clumps (red solid line),and branches without clumps (black dot dashed line).Bottom panel:Distribution of masses for unit lengthof the branches with associated distances between 700and 1400 pc with protostellar clumps (blue solid line)and prestellar clumps (red solid line).

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Fig. 22.— Mean column density measured along thecentral region of the branch excluding the overdensi-ties due to the clumps versus the deconvolved widthfor branches belonging to filaments without clumps(black empty squares), branches without clumps butbelonging to filaments with clumps (green triangles)and branches with clumps (blue diamonds).

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Fig. 23.— Branch mass per unit length versus clumpL/M ratio, with a separation between the brancheshosting only prestellar clumps and the one with at leastone protostellar clump. The lines represent the meanof the L/M distribution in logarithmic bin of size 0.2dex.

32