Aims . Filamentary infrared dark clouds (IRDCs) are believed to represent the initial conditions for massive star and cluster formation. Methods . We investigated the IRDC G035.39-00.33 using SiO, H 13 CO + , CH 3 OH, and CS emission observed with ALMA at 3.5″ resolution (∼0.05 pc). The analysis of the SiO emission provides a record of shock activity within the cloud, offering insights into both the current level of star formation and the cloud’s formation mechanisms. Results . We identify several regions with broad SiO emission clearly associated with outflows, pinpointing the locations of ongoing star formation across the cloud. The ALMA images also reveal a series of spatially extended SiO emission spots with narrow line profiles aligned along an arc-like path that is also seen in CS and CH 3 OH emission. While the broad SiO emission is mainly associated with the main cloud filament, as seen in visual extinction, the narrow SiO arch is located at the edge of the cloud, far from the identified sites of star formation activity. The presence of these arc-like morphologies suggests that large-scale shocks may have compressed the gas in the surroundings of the G035.39-00.33 cloud, shaping its filamentary structure. By inspecting the large-scale radio continuum emission around G035.39-00.33, we find that this IRDC is part of a larger star-forming complex where the densest and coolest material appears at the interacting regions between a supernova remnant (SNR) and an expanding HII region. In particular, we hypothesise that this IRDC may be spatially coincident with the ionised expanding gas associated with the previously identified SNR G35.6-0.4. Conclusions . We suggest that collisions between giant molecular clouds and expanding gas flows from interacting SNRs and HII regions may be responsible for the observed arc-like structures. Such shock compressions could play an important role in the formation of IRDCs and in the potential triggering of star formation.

The Taurus Molecular Cloud TMC1 is a nearby molecular cloud filament exhibiting several dense cores. It has been intensely studied in particular because of its very rich gas phase chemistry. Here, we present NIKA2 observations with the IRAM 30m telescope of its dust emission at 1 and 2mm wavelengths. Its vicinity of only 140 pc allows to study variations of its grain properties, i.e. its dust emissivity index β, on scales of only 2430 a.u. (18″). The NIKA2 maps are combined with Planck data to retrieve the emission at scales larger than the NIKA2 field-of-view. Maps of the 1mm/2mm flux ratio are then combined with a dust temperature map derived from Herschel data to create a map of the β index. Several of the observed cores exhibit a significant increase of β from ~ 1.2 in the outskirts at optical extinctions of ~ 5 mag to ~ 1.8 in the centers at ~ 18 mag. Grain models show that this steepening of the mm/submm emissivity is consistent with the gradual build-up of ice layers on the grain surfaces due to freeze-out of molecules in the cold, dense interiors of these cores.

Abstract Stars primarily form in galactic spiral arms within dense, filamentary molecular clouds. The largest and most elongated of these molecular clouds are referred to as “bones,” which are massive, velocity-coherent filaments (lengths ∼20 to >100 pc, widths ∼1–2 pc) that run approximately parallel and in close proximity to the Galactic plane. While these bones have been generally well characterized, the importance and structure of their magnetic fields (B-fields) remain largely unconstrained. Through the Stratospheric Observatory for Infrared Astronomy Legacy program FIlaments Extremely Long and Dark: a Magnetic Polarization Survey (FIELDMAPS), we mapped the B-fields of 10 bones in the Milky Way. We found that their B-fields are varied, with no single preferred alignment along the entire spine of the bones. At higher column densities, the spines of the bones are more likely to align perpendicularly to the B-fields, although this is not ubiquitous, and the alignment shows no strong correlation with the locations of identified young stellar objects. We estimated the B-field strengths across the bones and found them to be ∼30–150 μ G at parsec scales. Despite the generally low virial parameters, the B-fields are strong compared to the local gravity, suggesting that B-fields play a significant role in resisting global collapse. Moreover, the B-fields may slow and guide gas flow during dissipation. Recent star formation within the bones may be due to high-density pockets at smaller scales, which could have formed before or simultaneously with the bones.

Cloud optical properties and precipitation, which are crucial to weather and climate, are strongly influenced by cloud microphysical properties that are still poorly understood. Here, we develop a high-resolution time-correlated single-photon-counting lidar and apply it to observe cloud microphysical properties at one-centimeter range resolution in a convection chamber under well-controlled conditions. Together with concurrent in-situ measurements and theoretical analysis, our lidar observations indicate that although turbulent mixing tends to homogenize the cloud in the bulk region, entrainment and sedimentation cause inhomogeneities in droplet concentrations near the cloud top. Specifically, the topmost region is directly affected by entrainment, and lidar profiles show clear evidence of entrained air and detrained cloud filament. The transition region below exhibits vertical size sorting of cloud droplets caused by sedimentation. Our results suggest that using a single sedimentation velocity for all cloud droplets, as is done in many atmospheric models, overlooks key physics relevant to the microphysical structure near the cloud top. Our conceptual model used to describe these measurements can serve as a step toward improving the current modeling of processes in the cloud top region.

Context . It is generally accepted that high-mass stars form through a hierarchical, multi-scale fragmentation process that range from molecular clouds down to individual protostars, involving intermediate scales such as filaments. However, a comprehensive understanding of this process remains limited due to the lack of high-resolution, multi-scale observational studies that would simultaneously probe the physical conditions across the full hierarchy of star-forming structures. Aims . We aim to understand a coherent picture of the physical processes connecting filament formation, fragmentation, and dynamical scenario of high-mass star formation in the IRAS 19074+0752 (hereafter I19074) region. Methods . Primarily using new 1.3 mm continuum mosaicked observations, as part of the ALMA-INFANT survey, we analyzed the S-shaped filamentary cloud I19074 at a ∼6000 AU resolution. Leveraging the multi-scale information, we investigated the filament and clump fragmentation properties, such as core separations and masses. Results . ALMA 1.3 mm dust continuum emission reveals that the S-shaped filament consists of two physically connected components: a southern (Fs) and a northern (Fn) segment. Fn is associated with an infrared (IR)-bright HII region, while Fs appears IR-dark. The total filament length is ∼2.8 pc, with Fn and Fs spanning ∼1.0 pc and ∼1.8 pc, respectively. Their masses are ∼250−910 M ⊙ , while their line masses (∼250−360 M ⊙ pc −1 ) exceed the critical value for turbulence support, indicating they are gravitationally bound. The S-shaped morphology likely results from the expansion of the HII region, which swept up and compressed the northern part of the pre-existing filament into an arc-like structure in Fn; meanwhile, Fs retained a more linear form due to its greater distance from the ionized gas. Accordingly, a hybrid scenario could be responsible for Fn formation, which would combine the compression of a preexisting filament by the HII region with fresh gas accumulation into the shocked-compression layer. We extracted 26 dense cores from 1.3 mm emission with masses between 1.0 and 22.9 M ⊙ , with most (92%) being gravitationally bound (α vir ≤ 2). The core separations lack periodicity; instead, four core groups define four clumps (clumps 1-4) with masses of 110−620 M ⊙ . In the Fs segment, clump 1 at its southern end could be a product of edge fragmentation, while Fn exhibits hierarchical fragmentation modes: the filamentary mode responsible for clump formation within Fn and the spherical Jeans-like mode for core formation within clumps. Hierarchical fragmentation mechanisms are identified as shocked turbulence-driven within Fn and gravity-driven inside the clumps. Most cores have high mass surface densities of Σ core ≥ 1 g cm −2 , but with no robust identification of high-mass prestellar candidates. This favors dynamical clump-fed accretion-type over core-fed accretion-type models for high-mass star formation in I19074. Conclusions . The S-shaped filament in the I19074 region likely formed through the interaction with an expanding H II region, with the shocked-shell fragmentation mechanism in Fn and edge fragmentation in Fs serving as pathways for producing massive, star-forming clumps. Both mechanisms contribute to high-mass star formation via a dynamical clump-fed accretion process within their respective filamentary segments.

Abstract We carried out polarimetric observations of the bright-rimmed cloud IC 1396E/SFO 38 with SCUBA-2/POL-2 to study the effect of ultraviolet (UV) light on its structure and magnetic field. This bright-rimmed cloud appears optically to be one single cloud illuminated by the UV light from the exciting star of IC 1396. However our Stokes I image and $^{13}$CO (J = 3–2) archival data suggest that this cloud is not a simple, single structure, but appears to be composed of two parts on first glance; a head part with wings and a tail, and a north-west extension part. Since molecular clouds are generally filamentary and it seems likely that the initial structures of bright-rimmed clouds are also expected to be generally elongated, we examined the possibility that the structure was created from a single elongated cloud by the UV impact. We compared the cloud structure with a simulation study that investigated the evolution of prolate clouds exposed to UV radiation from various directions and found that this apparent two-part structure could be reproduced in a situation where a single filamentary cloud is obliquely illuminated by UV light. The magnetic field directions of the cloud are different from the ambient field direction, demonstrating the field reconfiguration. A distortion or pinch of the magnetic field is seen toward the cloud head, where an intermediate-mass star cluster is located, suggesting gravitational contraction. We roughly estimated the magnetic strength and stability in three parts of the cloud and found that the cloud head is most likely to be supercritical.

Abstract We study the structure, kinematics, and star-forming activity of filamentary cloud F-NE associated with AGAL323.444+0.096 using 13 CO ( J = 2–1) molecular line and continuum data from the far-infrared to near-infrared. The cloud comprises two elongated subfilaments, F-NE-north and F-NE-south, each spanning 25 pc, with systemic velocities of −65.25 and −67.38 km s −1 , respectively. F-NE-north and F-NE-south seem to be merging by collision. They exhibit signatures of cloud–cloud collisions, including U-shaped and arc-like morphologies, complementary components, and bridge features, suggesting ongoing collisions. The interacting zones show enhanced column densities and elevated velocity dispersion, further supporting the collision scenario. All these features are consistent with the “fray and gather” model. Dense clumps and young stellar objects are predominantly concentrated in these collision regions, implying that filament merging and localized collisions drive the formation of massive dusty clumps capable of nurturing massive stars. These results demonstrate that filament merging by collision is crucial for both filament structural evolution and the formation of massive stars and clusters.

Abstract We present a detailed study of the magnetic field structure in the G111 molecular cloud, a ring-like filamentary cloud within the NGC 7538 region. Our analysis combines multiwavelength polarization data and molecular-line observations to investigate the magnetic field’s role in the cloud’s formation and evolution. We utilized interstellar dust polarization from the Planck telescope to trace large-scale field orientations, starlight extinction polarization from the Kanata telescope to probe the cloud’s magnetic field after foreground subtraction, and velocity gradients derived from CO isotopologues observed with the IRAM 30 m telescope to examine dense regions. Our results reveal a coherent yet spatially varying magnetic field within G111. The alignment between Planck-derived orientations and starlight extinction polarization highlights significant foreground dust contamination, which we correct through careful subtraction. The global alignment of the magnetic field with density structures suggests that the field is dynamically important in shaping the cloud. Variations in CO-derived orientations further suggest that local dynamical effects, such as gravitational interactions and turbulence, influence the cloud’s structure. The curved magnetic field along the dense ridges, coinciding with mid-infrared emission in WISE data, indicates shock compression, likely driven by stellar feedback or supernova remnants. Our findings support a scenario where G111’s morphology results from turbulent shock-driven compression, rather than simple gravitational contraction. The interplay between magnetic fields and external forces is crucial in shaping molecular clouds and regulating star formation. Future high-resolution observations will be essential to further constrain the magnetic field’s role in cloud evolution.

Abstract Star cluster formation and assembly occur inside filamentary and turbulent molecular clouds, which imprint both spatial and kinematic substructure on the young cluster. In this paper, we quantify the amount and evolution of this substructure in simulations of star cluster formation that include radiation magnetohydrodynamical evolution of the gas, coupled with detailed stellar dynamics, binary formation and evolution, and stellar feedback. We find that both spatial and kinematic substructure are present at early times. Both are erased as the cluster assembles through the formation of new stars as well as the merger of subclusters. Spatial substructure is erased over a timescale of approximately 2.5 times the initial freefall time of the cloud. Kinematic substructure persists for longer and is still present to the end of our simulations. We also explored our simulations for evidence of early dynamical mass segregation and concluded that the presence of a population of binary stars can accelerate and enhance the mass segregation process.

Abstract The filamentary dark cloud complex in Norma reveals signs of active low-mass star formation including protostars, Hα emission line stars, Herbig–Haro objects, and the eruptive FU Orionis-like star V346 Nor. We present results of the first pointed X-ray observations of the Norma dark cloud, focusing on the westernmost Sandqvist 187 region. Chandra detected 75 X-ray sources, and a complementary XMM-Newton observation detected 92 sources within the Chandra field of view, of which 46 are cross-matched to Chandra, yielding 121 unique X-ray sources. We present a catalog of X-ray sources along with basic X-ray properties and candidate IR and optical counterparts. Existing near-IR photometry reveals several X-ray sources with color excesses typical of young stars with disks. Gaia parallaxes single out foreground stars and X-ray sources at distances of 500–1000 pc that are probable cloud members. The known emission line stars Sz 136 and Sz 137 were detected but V346 Nor was not. Interestingly, the optical and IR counterparts of the brightest Chandra source are not known with certainty but the prime suspects are very faint. Thus, the nature of the object responsible for the bright X-ray emission remains speculative. The X-ray observations presented here will serve as a pathfinder for identifying and characterizing the young stellar population in the Norma dark cloud.

The mixing properties of vapor content, temperature, and particle fields are of paramount importance in cloud turbulence as they pertain to essential processes, such as cloud water droplet evaporation and entrainment. Our study examines the mixing of a single cloudy air (which implies droplet-laden) filament with its clear air environment, a characteristic process at the cloud edge, in two ways. The first consists of three-dimensional combined Euler-Lagrangian direct numerical simulations which describe the scalar supersaturation as an Eulerian field and the individual cloud water droplets as an ensemble of Lagrangian tracers. The second way builds on the recently developed diffuselet method, a kinematic Lagrangian framework that decomposes a scalar filament into a collection of small sections subject to deformation by local stirring and cross-sheet diffusion. The Schmidt number is Sc=0.7. The entrainment process causes a deformation of the supersaturated cloud filament in combination with diffusion until the system reaches a well-mixed equilibrium state, which implies for the present configuration that all droplets are evaporated. We compare the time dependence of the mean square and probability density function of the supersaturation field. For the initial period of the mixing process, they agree very well; at later stages, deviations caused by nonzero mean of the conserved scalar are observed. For the cases including cloud water droplets, we also investigate the impact of droplet number density and condensation growth response. Turbulence causes deviations from the d2 law similar to recent experiments in sprays. A simulation at a Schmidt number that is by a factor of 100 larger than in clouds improves the agreement between simulation and diffuselet method significantly. The latter result promotes the diffuselet framework as an efficient parametrization for turbulent high-Sc mixing which can reduce the resolution efforts of the viscous-convective range of scalar turbulence.

ABSTRACT We explore the Schmidt–Kennicutt (SK) relations and the star formation efficiency per free-fall time ($\epsilon _{\rm ff}$), mirroring observational studies, in numerical simulations of filamentary molecular clouds undergoing gravitational contraction. We find that (a) collapsing clouds accurately replicate the observed SK relations for Galactic clouds and (b) $\epsilon _{\rm ff}$ is small and constant in space and in time, with values similar to those found in local clouds. We propose that this constancy arises from the similar radial scaling of the free-fall time ($\tau _{\rm ff}$) and the internal mass in density structures with spherically averaged density profiles near $r^{-2}$. We additionally show that (c) the star formation rate (SFR) increases rapidly in time; (d) the low values of $\epsilon _{\rm ff}$ result from evaluating ${\tau _{\rm ff}}$ and the characteristic star formation time-scale over different time intervals, combined with the increasing SFR, and (e) the fact that star clusters are significantly denser than the gas clumps from which they form is a natural consequence of the rapidly increasing SFR, the continuous replenishment of the star-forming gas by the accretion flow, and the near $r^{-2}$ density profile induced by the collapse. Finally, we argue that interpreting $\epsilon _{\rm ff}$ as an efficiency is problematic since it is not bounded by unity, and because the gas mass in clouds evolves. Instead, we propose that viewing $\epsilon _{\rm ff}$ as the ratio of the actual SFR to the gas free-fall rate. In summary, our results show that the SK relation, the low values of $\epsilon _{\rm ff}$, and the mass density of stellar clusters arise naturally from gravitational contraction.

Abstract Observations indicate that dense molecular filamentary clouds are sites of star formation. The filament width determines the most unstable scale for self-gravitational fragmentation and influences the stellar mass. Therefore, constraining the evolution of filaments and the origin of their properties is important for understanding star formation. Although some observations show a universal width of 0.1 pc, many theoretical studies predict the contraction of thermally supercritical filaments (>17 M ⊙ pc−1) due to radial collapse. Through nonideal magnetohydrodynamics simulations with ambipolar diffusion, we explore the formation and evolution of filaments via slow-shock instability at the front of accretion flows. We reveal that ambipolar diffusion allows the gas in the filament to flow across the magnetic fields around the shock front, forming dense blobs behind the concave points of the shock front. The blobs transfer momentum that drives internal turbulence. We name this mechanism “the STORM”(Slow-shock-mediated Turbulent flOw Reinforced by Magnetic diffusion). The persistence and efficiency of the turbulence inside the filament are driven by the magnetic field and the ambipolar diffusion effect, respectively. The STORM mechanism sustains the width even when the filament reaches very large line masses (∼100 M ⊙ pc−1).

Abstract We present a multiwavelength study of the filamentary cloud G47 (d ∼ 4.44 kpc), which hosts the mid-infrared bubbles N98, B1, and B2. The SMGPS 1.3 GHz continuum map detects ionized emission toward all the bubbles, marking the first detection of ionized emission toward the B2 bubble. Analysis of the unWISE 12.0 μm image, the Spitzer 8.0 μm image, and the Herschel column density and temperature maps reveals two previously unreported hub–filament system candidates associated with the H ii regions B2 and N98, which are powered by massive OB stars. This indirectly favors the applicability of a global nonisotropic collapse scenario for massive star formation in N98 and B2. The position–position–velocity diagram of FUGIN 13CO (1–0) shows significant velocity variations from 61 to 53 km s−1 toward areas between B2 and N98, where the magnetic field morphology exhibits significant curvature and high velocity dispersion (i.e., 2.3–3.1 km s−1) is observed. This may be explained by the expansion of the H ii regions B2 and N98. The energy budget of the cloud, estimated using SOFIA/HAWC+ and molecular line data, suggests that the magnetic field dominates over turbulence and gravity in G47. Furthermore, the radial column density and velocity profiles of G47 display signatures of converging flows in a sheet-like structure. The relative orientations between the magnetic field and local gravity suggest that G47 may undergo gravitational contraction along the magnetic field lines once it becomes magnetically supercritical.

We searched for potential “birthmarks” left from the formation of filamentary molecular clouds in the Ophiuchus complex. We used high dynamic range column density and temperature maps derived from Herschel, Planck, and 2MASS/NICEST extinction data. We found two distinct types of filaments based on their orientation relative to nearby massive stars: radial (R-type) and tangential (T-type). R-type filaments exhibit decreasing mass profiles away from massive stars, while T-type filaments show flat but structured profiles. We propose a scenario where the two filament types originate from the dynamic interplay of compression and stretching forces exerted by a fast outflow emanating from the OB association. The two formation mechanisms leave distinct observable birthmarks (namely, filament orientation, mass distribution, and star formation location) on each filament type. Our results illustrate a complex phase in molecular cloud evolution with two simultaneous yet contrasting processes: the formation of filaments and stars via the dispersal of residual gas from a previous massive star formation event. Our approach highlights the importance of taking into account the wider context of a star-forming complex rather than concentrating exclusively on particular subregions.

Abstract Recent observations of the solar atmosphere in cool extreme-ultraviolet lines have reported the prevalence of coronal rain falling from coronal cloud filaments that are associated with the magnetic dips of coronal X-point structures. These filaments mysteriously appear as clouds of mass in the corona that subsequently shrink and disappear due to mass losses that drain as coronal rain along arced field lines. Using a two-and-a-half-dimensional magnetohydrodynamic model, we investigated evaporation-condensation as the formation mechanism of the subset of coronal cloud filaments that form above coronal X-points. Our simulation included the effects of field-aligned thermal conduction and optically thin radiation, and used the state-of-the-art transition region adaptive conduction (TRAC) method to model the formation, maintenance, and mass loss of a filament above a coronal X-point. This paper presents a physical model that demonstrates magnetic reconnection as a filament loss mechanism, producing hybrid filament/coronal rain via mass losses through the X-point. A detailed analysis of how the mass of the filament forces the field to reconnect is also presented, revealing three phases that characterize the evolution of the reconnecting current sheet and associated mass losses. We conclude that the formation of certain coronal cloud filaments and subsequent mass losses via coronal rain can be explained by the evaporation-condensation model combined with filament mass losses forced by magnetic reconnection. We also report that rebound shocks generated by the impact of coronal rain condensations on the chromosphere together with retractive upflows can cause upward-propagating condensations to form through a dynamic thermal runaway process.

Abstract Polarization of starlight induced by dust grains aligned with the magnetic field (hereafter B-field) is widely used to measure the 2D B-fields projected onto the plane-of-sky. Here, we introduce a new method to infer 3D B-fields using starlight polarization. We show that the inclination angle or line-of-sight component of B-fields can be constrained by the starlight polarization efficiency from observations, the alignment degree provided by the magnetically enhanced radiative torque (MRAT) alignment theory, and the effect of B-field tangling. We first perform synthetic observations of starlight polarization of magnetohydrodynamic (MHD) simulations of a filamentary cloud with our updated POLARIS code incorporating the modern MRAT theory. We test the new technique with synthetic observations and find that the B-field inclination angles can be accurately determined by the synthetic starlight polarization efficiency once the effects of grain alignment, dust properties, and B-field fluctuations are well characterized. The technique can provide an accurate constraint on B-field inclination angles using optical polarization in low-density regions A V < 3 with efficient MRAT alignment, whereas the technique can infer further to high-density regions with significant alignment loss at A V ∼ 8–30 by using near-infrared polarization. Our new technique unlocks the full potential of tracing 3D B-fields and constraining dust properties and grain alignment physics on multiple scales of the diffuse interstellar medium and star-forming regions using multiwavelength starlight polarization observations.

Abstract Filamentary molecular clouds are an essential intermediate stage in the star formation process. To test whether these structures are universal throughout cosmic star formation history, it is crucial to study low-metallicity environments within the Local Group. We present an analysis of Atacama Large Millimeter/submillimeter Array (ALMA) archival data at the spatial resolution of ~0.1 pc for 17 massive young stellar objects (YSOs) in the Small Magellanic Cloud (SMC; Z ~ 0.2 Z ⊙). This sample represents approximately 30% of the YSOs confirmed by Spitzer spectroscopy. Early ALMA studies of the SMC have shown that the CO emission line traces an H2 number density of ≳104 cm−3, an order of magnitude higher than in typical Galactic environments. Using the CO(J = 3–2) data, we investigate the spatial and velocity distribution of molecular clouds. Our analysis shows that about 60% of the clouds have steep radial profiles from the spine of the elongated structures, while the remaining clouds have a smooth distribution and are characterized by lower brightness temperatures. We categorize the former as filaments and the latter as nonfilaments. Some of the filamentary clouds are associated with YSOs with outflows and exhibit higher temperatures, likely reflecting their formation conditions, suggesting that these clouds are younger than the nonfilamentary ones. This indicates that even if filaments form during star formation, their steep structures may become less prominent and transition to a lower-temperature state. Such transitions in structure and temperature have not been reported in metal-rich regions, highlighting a key behavior for characterizing the evolution of the interstellar medium and star formation in low-metallicity environments.

Abstract In this study, we perform 3D magnetohydrodynamics (MHD) simulations of filamentary molecular clouds. We then generate synthetic observations based on the simulation results. Using these, we investigate how the new polarization data analysis method recently introduced by Doi et al. (2021, ApJ, 923, L9) reflects the magnetic field structure in turbulent filamentary molecular clouds. Doi et al. proposed that the $R_{\rm {FWHM}}$, the ratio of the full width at half maximum (FWHM) of the polarized intensity ($PI$) to that of the total intensity ($I$) can be used to probe the three-dimensional structure of the magnetic field. We calculate the $R_{\rm {FWHM}}$ from the density and magnetic field structure obtained in the 3D-MHD simulations. We find that the mean and variance of $R_{\rm {FWHM}}$ within a filament are smaller and larger, respectively, with a larger inclination of the magnetic field to the plane-of-sky. We also find that both small-scale ($\lt $0.1 pc) and large-scale ($\gtrsim$0.1 pc) turbulence affect the polarized intensity of the dust thermal emission. We conclude that future extensive observations of $R_{\rm {FWHM}}$ may be able to quantify the inclination of the magnetic field to the plane-of-sky in the filamentary molecular clouds.

VolDen: A tool to extract number density from the column density of filamentary molecular clouds