Essay: Molecular clouds

Abstract
Molecular clouds are dark, cold, dense regions of high extinction within the ISM. They confine the majority of the mass in the ISM and these clouds are the primary sites of star formation within the galaxy. Molecular clouds have a characteristic filamentary structure that is omnipresent across the ISM within both star forming regions and quiescent clouds, and these filaments share a high degree of universality in their properties. This however has not always been the case, as before the launch of the Herschel space telescope (1970-2010), Molecular clouds were thought to exhibit a discrete clump and core substructure. Filamentary properties had been observed in molecular clouds as early as the 1970’s, however, filamentary structure was not thought to be the fundamental morphology of the cold ISM; the pervading theory was one of hierarchical cloud structure, from clouds to clumps to cores. The formation of molecular clouds, their observed morphology and star forming regions is attributed to a number of factors in addition to their self-gravitating nature. Magnetic fields, like filaments, are known to be ubiquitous across the ISM and magnetism plays a fundamental role within the molecular ISM in both the formation of clouds and filaments and their subsequent evolution into clumped/star forming regions.
Introduction
Molecular clouds (MC’s) were first discovered in 1970 through spectroscopic observations of CO in the 2.6mm band in the Orion Cluster (Williams 1970) and the Lyman resonance-absorption bands of interstellar molecular hydrogen (H2) in Perseus (Carruthers 1970); observations of these star forming regions highlighted a hierarchical structure within the interstellar medium (ISM) that was first thought to comprise of cores embedded within clumps that were embedded within the larger MC’s. Clumps are an intermediate substructure with higher column densities than the parent MC, and cores are localised high column density structures within clumps. In recent decades, advances in spatial resolution has highlighted a filamentary substructure of MC’s that has redefined the morphology of gas in the ISM, irrespective of the presence of star forming regimes within them. The unprecedented discovery of the ubiquitous nature of filaments by the Herschel Space Telescope suggests there is an important connection between the structure within the ISM and the formation of pre-stellar cores and protostars, and that the formation of filaments is the first stage of the process of galactic stellar evolution. (Arzoumanian & Andre 2011) The presence of interstellar filaments and prestellar cores represent 2 two fundamental steps in the star formation process: i) turbulence within MC’s stirs up gas, giving rise to a universal web-like structure, then ii) gravity takes over to control further fragmentation into prestellar cores and protostars. (Andre et al 2017)
Filaments are elongated molecular structures with large aspect ratios and central volume densities (n) of 102cm-3 to 105cm-3 (and a corresponding column density that exceeds the local background), with minimal overall curvature and are unidirectional over their lengths.
(Arzoumanian & Andre 2018)
Whilst earlier papers suggested that the presence of filaments within MC’s would require a radical change in theories of cloud structure and formation, integrating filaments into the hierarchy of MC’s allows for decades of research to be combined to form an extensive picture of the structure of the ISM and how stars are formed within these regions.
The primary constituent of MC’s and filaments is molecular hydrogen (H2); the abundance of H2 is produced via two hydrogen atoms (H) within clouds that combine to form H2 in a two-step process on the surface of interstellar dust grains. H atoms cannot form H2 via two-body collisions, as the probability of forming a stable molecule is very small. H2 is instead formed on the surface of dust grains, (Heyer 2015) where the grain absorbs the recombination energy that would cause the combining H atoms to bounce apart. (Carruthers 1969) CO is also found in high abundance within MC’s and the chemical pathways used in the formation of CO, depend on the presence of an FUV field and the density and temperature within the cloud; the presence of OH and CO+ are necessary precursors for its formation. Both H2 and CO are subject to dissociation by UV photons that enter MC’s, therefore the coextensive abundance of H2 and CO in MCs is dependent on the rate of formation and destruction of the molecules. The abundance of molecular gas in MC’s is possible due to the surrounding outer envelope that comprises of atomic hydrogen (HI), H2, 12CO, 13CO and interstellar dust observed to encompass all MC’s,; this self-shielding nature acts to protect the inner molecular volume of the cloud from photo-dissociation by cosmic rays and UV radiation and has an observed diameter of 150-200pcs.(Blitz 1990) In particular, the line opacity for H2, 12CO & 13CO contribute to the effectiveness of the layer, with dust providing the highest element of opacity. The self-shielding layer creates a defined edge to MCs within which the bulk volume of the cloud is contained, and it is within this boundary that rotational transitions in CO (that build are observed (see §1.1) (Heyer 2015)
NEED SOMETHING HERE AS A LINK
This paper will discuss the observational methods of detecting molecular clouds, filaments and the magnetic fields within them (§), the physical and mathematical properties of MC’s and their substructure (§) and the role played by magnetism in the geometry and fragmentation of filaments (§).
Observational Methodology
MC’s, irrespective of their geometrical substructure, emit the bulk of their radiation in mm, infrared (IR) and far-infrared (FIR) wavelengths (). Observations at these ’s allow MC density structure, mass and velocity dispersions to be derived as well as identifying substructures such as filaments and cores. The observational details associated with deriving MC properties and how techniques have advanced are discussed below (§1.1,1.2) as well as the method for identifying the properties, geometry and strength of B-fields (§1.3).
1.1 – Observing Molecular Clouds
The masses of MC’s are dominated by molecular gases, with an ≈26% contribution from Helium (He). (Ballesteros Paredes 2013). The primary constituent H2 is directly unobservable with the ISM due to its homonuclear nature. H2 is a diatomic molecule and so has no permanent dipole moment; any transitions in the energy levels only occur at significantly high temperatures or in strong UV radiation fields, therefore within MC’s where T≈10K, H2 will never be directly observable. Therefore, before the discovery of MC’s in 1970, H2 was thought to be absent from the ISM; its presence could only be inferred through theories that the excess visual extinction (>1.5mag) observed in MC’s had to be due to the presence of a molecular component. (Carruthers 1969)
Molecular tracers are instead used to infer the morphology of clouds and determine further properties such as mass and density.
The low energy rotational transition of Carbon Monoxide (CO) is the primary tracer of H2 in the ISM; it is a heteronuclear diatomic molecule with a transition at =2.6mm (=115 GHz) and a high spatial resolution. The J=1 state in the transition of the J=10 lies at E=4.8×10-4eV above the ground state, corresponding to an excitation temperature of 5.5K, therefore unlike H2, the rotational transitions of CO are readily excited even in quiescent clouds; in extension, the J=2 rotational band can be excited at only 16K. CO is favourable as the J-band transitions are strong emitters of radiation as the rotations decay quickly in the molecule. The CO observed from MC’s originates from the central bulk of the cloud that is protected by the self-shielding layer.
CO as a tracer has many advantages as: i) it emits in mm/sun-mm band and can therefore be directly observed by ground based telescopes, ii) it is abundant within the ISM and is readily observable in both emission and absorption spectra (Mckee), iii) the ratio of CO to H2 is found to be near constant (Dickman 1970)
1.1.1 – The H2 – CO Conversion Factor
The masses, surface densities (∑H2) and column densities (NH2) of H2 in MC’s is derived from measured values for CO using a conversion factor XCO or CO.
The initial empirical relation between CO and H2 was derived by Dickman (1970) and is given as:
█(N_H2=(5.0±2.5) × 〖10〗^5 N_13 #(1.1) )
where N13 is the column density of 13CO.
Eq. 1.1 was limited in that it did not account for visual extinction in the clouds or the presence of helium, and was only valid up to a column density of NH2 ≈1016cm-2; it is generally shown that the column density of MC’s is much higher than this threshold for the majority of clouds.
Sandstrom (2013) proposed a modified empirical derivation of Eq. 1.1 that makes use of a conversion factor, Xco, when calculating NH2 and a conversion factor, CO, when calculating ∑H2. It is valid for all cloud density and visual extinctions.
The hydrogen column density is given as:
█(N_H2=X_CO W((_^12)C (_^16)O) for J=1→0) #(1.2) )
where W is the integrated line intensity (Kkms-1) of 12CO,
and XCO is a conversion factor = 2 x 1020 cm-2 (Kkms-1)-1, with an associated uncertainty of ±30%.
The hydrogen surface density is given as:
█(Σ_H2=α_CO I_CO #(1.3) )
where α_CO is a conversion factor = 3.1M0pc-2 (Kkms-1), where XCO = 4.6×1019CO
and Ico replaces ‘W’ above as the integrated line intensity (Kkms-1) of 12CO
As well as the radio spectroscopy of molecular lines (CO), the structure of MC’s can be mapped through dust continuum emission and stellar absorption. (Blitz and Williams 1999). The associated velocity dispersion in MC’s is given from the optically thin tracer 13CO J=10, which produces a line width profile reflective of the velocities throughout the cloud. (Heyer 2015) Substructure within MC’s is identified via higher density tracers and higher resolution imaging.
1.2 – Filaments
Further substructure of MC’s has been noted since as early as the 1970’s, but limits in available observational technology have meant that only recently has the filamentary nature of MC’s been deemed ubiquitous across the ISM. The launch of the Herschel space telescope and subsequent observations of filamentary structure defined the beginning of a new era of observational techniques for studying MC’s and their substructure.
1.2.1 – The Herschel Space Telescope
The Herschel Space Observatory was launched in May 2009, and was the largest telescope of its kind, carrying a passively cooled 3.5 diameter mirror and the only space observatory to cover from FIR to sub-mm wavebands (Pilbratt 2010). The telescope instrumentation comprises of two direct detection cameras/spectrometers; the Photodetector Array Camera and Spectrometer (PACS) which measures 55-210m in 3 bands and was used to map regions with visual extinction Av >6 and the Spectral and Photometric Imaging Receiver (SPIRE), which measures from 200-670m in 3 bands and was used to map regions of Av>3. However, both can map regions down to Av≈1. (Andre 2010a)
The final component of the telescope is a high resolution heterodyne spectrometer (HIFI), working in a wavelength range of 157-625m, which is used to translate the frequency range.
During its 4-year operation time, 95% of the available Herschel Space Observatory time was allotted to 21 key programs (KP), one of which was The Herschel Gould Belt Survey (HGBS). The HBGS is of primary relevance to this dissertation, and results published from this KP will be used in further discussion of the filamentary nature of molecular clouds.
The Gould Belt is a relatively close (<0.5kpc) molecular cloud complex located in the local spiral arm of the Milky Way. The HGBS focused on 15 of the densest cloud complexes within the region, it is a region of Galactic space where Herschel imaging can be best used to characterise the structure/early stages of star formation as molecular clouds and cores are known to emit the bulk of radiation in the Herschel observation range. (Andre 2010b) The molecular clouds in this region span a wide range of physical conditions; from quiescent non-star-forming regions such as the Polaris Flare which is a high-latitude translucent cloud, to low mass star forming regions e.g. the Pipe Nebula & Taurus, to the most active cluster-forming/high-mass star forming regions such as Aquila & Orion where 350-500 pre-stellar cores have been identified as well as 45-60 class 0 objects. (Andre 2010a)
From the observational data, filaments are identified in images from Herschel using algorithms such as the DisPerSE algorithm. DisPerSE traces filaments by connecting critical points in the pixel distribution with integral lines following gradient in a map, where the critical points are positions where the gradient is zero. (Arzoumanian & Andre 2018a)
1.2.2 – Filament Observations
MC’s and the objects associated with them are known to emit the bulk of their radiation in the Herschel observation range that previously had been inaccessible from ground-based telescopes. Pre-Herschel, filaments were observed by HI self-absorption (HISA) and through molecular line mapping/dust maps, however these methods were limited by lower resolution and the scale over which observations were mapped. (Andre 2013). The dynamic range of Herschel allows both the structure of parent clouds and their dense cores to be mapped on large (>1pc) and small (<0.1pc) scales respectively as well as mapping the morphology of interstellar filaments and describing their column densities.
Ground based telescopes are now used to compliment Herschel Data with molecular line observations of C18O and N2H+ (where N2H+ is a molecular tracer of dense gas) which give kinematic data of the filamentary structure, such as the velocity dispersion (). (Arzoumanian & Andre et al 2013)
Substructure within filaments, such as embedded cores are identified through several algorithms such as getsources, clumpfind and csar, but the (mechanics) of these algorithms and methods in identifying cores will not be discussed within this dissertation. A core is considered a robust detection if found by two or more of the listed algorithms independently. (Arzoumanian 2017)
1.3 – Observations of Magnetic Fields
Finally, we will look at the methods used to observe the B-field in MC’s. The interstellar B-field was first detected in 1949 (Hiltner 1949) and is now measured and observed via the Zeeman effect and polarisation respectively; observations of the interstellar B field are constrained by each of the fore mentioned methods ability to directly measure/observe only one property of the B-field – the field strength or geometry.
Observations of the Gould Belt have given evidence of an ordered B-field across all spatial scales, however the formation of cores and stars occurs on a length scale significantly smaller than the resolution of the 1-100pc transverse B-field observations offered by telescopes such as ALMA, SMA and the Plank satellite. The behaviour of the field on small scales within dense clumps and cores is yet to be probed by even the most modern of technology.
1.3.1 – Zeeman Effect
The Zeeman effect is the only available method to directly measure the strength of the B-field in MC’s, and gives the field strength along the line of sight (LOS) (Goodman 2003). The effect arises from the orbital motion and spin transitions of atomic orbital electrons that induce a magnetic moment, , that is proportional to the total angular momentum, J, of the atom. J is quantised and given by the angular momentum quantum number, L, which is proportional to the orbital quantum number, l. The energy of the atom is quantised through degeneracy of the energy levels, where there are multiple possible atomic orbits of equal energy, but with different values of angular momentum.
The presence of an interstellar B field lifts this degeneracy through the exertion of a torque on the atom that changes the orbit; the Zeeman effect describes the splitting of molecular emission lines into magnetic sub-levels in the presence of a B-field, where the total field strength is inferred from the degree of separation of the line splitting. (Han 2017)
The first detection of magnetically-driven line splitting was seen in the 21cm absorption line of HI in the supernova remnant Cassiopeia A (Verschuur 1968). Further detections in other molecules such as OH were not seen until 1983 (Crutcher 1983). The splitting of the spectral transition is independent of the spectral-line frequency and is proportional to a factor ‘Z’. (Crutcher 2012)
The Zeeman effect is limited, as it can only be observed in three tracer molecules – HI, OH and CN; these radicals are the only molecules with sufficiently high  at ISM/MC field strengths to allow the polarisation to be detectible at radio wavelengths. (Girart 2011) Turbulence within the ISM is a further limiting factor that makes magnetic splitting difficult to observe, as Zeeman lines get ‘lost’ within observed spectrums of Doppler shifted lines. Further to this, the magnetic vector B is observable along the LOS only, and so the Zeeman effect defines the lower limit of the total magnetic field strength.
Whilst the traceability of the Zeeman effect may be restricted, the tracers are robust in their ability to sample the field strength over a range of densities seen across all cloud geometries. HI emission is seen in both the warm and cold, neutral, diffuse ISM, and can sample densities from 1-100cm-3. Both HI and OH give measurements at densities of magnitude 102-104cm-2, and finally the 3mm line of CN is used to determine the field strength in regions of density of 105-106cm-2.
Masers are compact regions within molecular clouds, within which the spectral line emission from molecules is significantly amplified to produce emission that can provide detectable polarisation measurements of B fields, however there is a large uncertainty in their intrinsic properties which can be an issue when interpreting results.
Figure 1 – Diffuse and molecular cloud Zeeman measurements along the line of sight. Magnetic field strength (B) is plotted against the number density of hydrogen. The trend line gives the most probable maximum values of the total B field. (Crutcher et al 2010)
Values of the B-field strength for typical regions of molecular clouds such as filaments and cores can be calculated from Figure 1. The data is a compilation of 4 data sets of the three molecular tracers, with the fit from a Bayesian analysis shown in blue. Using the figure, the magnetic field strength for filaments/clouds where nH ≈ 102cm-2, B= 10G. With increasing densities, the B-field within cores where nH ≈ 106cm-3 increases to 103G and above.
1.3.2 – Polarisation
Interstellar dust within clouds produces observable thermal emission and extinction of light from background stars. Linear polarisation of this emitted radiation is the primary method of probing the magnetic field morphology of the ISM, and it has been found that the percentage of polarisation remains constant irrespective of extinction. (Goodman, Lada & Myers 1995)
Selective extinction in non-spherical interstellar dust grains aligned with the magnetic field causes starlight becomes polarised. The field morphology can be determined from the effect of these dust grains as the polarisation angle of starlight is parallel to the orientation of the transverse B-field and the dust grain is perpendicular to the magnetic field along its major axis.
Polarised thermal emission is detectable in millimetre, submillimetre and far-IR bands. (Han 2017)
Dust grains in outer cloud regions are expected to have a higher grain alignment efficiency with the B field and will absorb more photons from the local interstellar radiation field, these warmer grains therefore contribute a larger polarisation intensity (Pattle & Fissel 2018) than grains in depolarised regions in cooler, deeply embedded regions. (Fissel et al 2016)
As well as depolarisation in grain alignment due to regions of higher column density, there are also appreciably irregular fluctuations in the polarisation direction in grains that are not aligned with the field that arise from variations in the magnetic field strength. These irregular polarisation angles are seen in clouds with a weaker B-field, as the field is sufficiently weak that the field lines frozen into the cloud can be distorted by turbulent motions. These turbulent motions arise from the action of hydromagnetic waves. (Panopoulou 2016) It follows that a stronger B-field will greater resist the bending effect of turbulence and there will be a smaller observed irregularity in the polarisation angles of dust grains. (Chandrasekhar and Fermi 1953)
Whilst polarisation gives only the geometry of the observed field, the magnetic field strength can be estimated in the plane of the sky (POS) using a method developed by Chandrasekhar and Fermi in 1953, aptly named the Chandrasekhar-Fermi (CF) method.
The field strength in the plane of the sky is estimated as:
█(B_POS=Q√4πρ δV/δ #)(4)
where:
 is the gas density
dV is the 1D velocity dispersion in plane of the sky assumed to be equal to the LOS
d is the dispersion in the polarisation angles
Q is a correction factor  0 < Q < 1 (Myers and Goodman 1991 that takes into account any ordered structures on scales smaller than the telescope beam, and emission from turbulent cells within the beam including along the LOS (Pattle & Fissel – under review). For dense, self-gravitating filaments and cores in which little field substructure is expected, Q ≈ 0.5 (Nutter 2004)
The field strength has been estimated using the CF method for several clouds within the Gould Belt. Within the Taurus dark cloud complex, BPOS was found to vary with the cloud density. Low density regions gave estimations of BPOS ≈10G, sub-regions such as B213-L1485 have B ≈ 25G and for high density regions such as the Heiles Cloud 2 Regions, the magnetic field strength is estimated as B ≈42 G. (Crutcher 2012)
Polarisation studies of B-fields have a greater sample of results than studies using the Zeeman effect, as polarisation measurements are significantly easier than measuring magnetically split spectral lines. However, polarisation is limited by regions of high density (MC cores), at which the alignment between dust grains and the B-field is thought to break down (Jones et al 2015) Future techniques require a concentrated effort for Zeeman observations and polarisation methods to be used in tandem.
Properties of Molecular Clouds and Filaments
2.1 – Properties of MC’s
The ISM hierarchy is characterised depending on intrinsic properties of clouds such as mass, density, size and morphology. The hierarchy is headed by super-clouds; regions containing large complexes of MC’s such as the Gould Belt (GB), which is a relatively close (<0.5kpc) MC complex located in the local spiral arm of the Milky Way. Within MC complexes are individual clouds, which contain clumps that fragment/collapse to form filaments, within which prestellar cores and protostellar cores are embedded. (Dudorov 2017). Where a prestellar core is a self-gravitating condensation of gas and dust that can produce stellar formation via gravitational collapse (Andre et al 2000), and a protostellar core is a dense core within which a young star has formed. (Andre, P (2013a)
Table 1 (Dudorov 2017) summarises the properties of each of the hierarchical components of the ISM, highlighting the observed complex material distribution. (Kutner et al 1997)
Table 1 – The table summarises the properties of each constituent of the MC hierarchy. In column 1, T = is temperature (K), n is volume density (cm-3), M is mass of the cloud (M0), R is cloud size (pc), and v is the 1D velocity dispersion (km/s). The properties for both the warm and cold components of the ISM is shown.
MC’s are defined in 3 groups dependant on their size and location within the galaxy. The largest MC’s are self-gravitating and known as “Giant Molecular Clouds (GMCs)” with smaller clouds described as “Small Molecular Clouds (SMC’s)” or “High Latitude Clouds (HLC’s)” if they lie above/below the major plane of the galaxy (Blitz 1990). The molecular cloud column in table 1 describes the properties of GMC’s, where the associated column density (NH2) ≈ 1021cm-2, (Blitz 1993) and the cloud separation distance is ≈500pc. (Blitz & Williams 1999) SMC’s have much smaller masses, M<102M0 and are generally unbound structures and whilst there has been some evidence for stellar evolution within these regions, their contribution to the galactic star formation rate is negligible (Magnani et al 1995).
In extension to classification based on their physical properties, MC’s can be categorised by three evolutionary classes: i) Class 1 clouds have no signature of star formation, ii) Class 2 have a limited number of embedded HII regions where L < 1037 ergs-1, iii) Class 3 harbour young stellar clusters where L > 1037ergs-1. (Heyer 2015).
Pre-Herschel, the hierarchy of MC’s consisted of clouds, clumps and cores, where an increase in density is observed through the hierarchy, and regions of higher density are shown through localised peaks of molecular line emission. An example for the pre-Hershel view of MC’s is shown in Fig. 2, which shows the density contours of a highly studied GMC known as the Rosette Molecular Cloud (RMC). Extinction in regions with size ≈30%(cloud) gave further evidence of an internal-substructure in discrete units. (Scalo 1985)
Figure 2: Figure shows a contour map of a molecular cloud…Each panel = cloud, clump, core. Cores found using higher density tracers (CS). C18O is an optically thin line that gives a higher resolution image of cloud structure. The example is the RMC which is a highly observed GMC of the Milky Way (Blitz & Williams 1990)
The clumps shown within fig.2, panel 2, are transient structures with a higher column and volume density than their parent clouds (Heyer 2015) and can be bound or unbound, where unbound clumps are pressure confined. (Blitz & Williams 1999) Cores are the densest regions of MC’s the properties of which are given by columns 6&7 of table (1). Core regions have column densities, NH2, > 1021cm-2, however the size of cores within pre-Herschel molecular clouds were derived to have an average diameter of ¬≈1pc. (Mundy 1994)
The self-gravitating nature of GMC’s is highlighted by the observed pressure equilibrium curves of clouds. For example, a cloud where T=10K, and n=106cm-3, the associated pressure (P/K) is of the order of 107 P/K, since the average pressure of the ISM is ≈ 3000 P/K, there has to be a source of self-gravity throughout observable clouds or MC’s would be dispersed by thermal over-pressure. Clumps within MC’s are generally found to be gravitationally unbound as they are instead confined by the pressure of the cloud. (Blitz & Williams 1999)
2.2 – Filament Properties
The physical properties of filaments are summarised in table 1. The densest filaments within MC’s are sites of star formation and observed to have column densities exceeding ≈ 7 x 1021cm-2; filaments that are known to be self-gravitating have higher observed column density > 1022cm-2. Non-star forming filaments such as the Polaris Flare have column densities of only up to a few 1021cm-2. (Andre 2010c)
Filaments within the ISM are modelled as isothermal cylinders in near hydrostatic equilibrium. The radial profile and then by extension the density profile is well described by a Plummer profile: (Whitworth & Ward-Thompson 2001)
ρ_p (r)=(ρ_c R_flat)/〖[1+(r⁄R_flat )^2]〗^((p-1)/2) (2.1) ⇒ Σ_p (r)=A_p (ρ_c R_flat)/[1+(r⁄R_flat )^2 ]^((p-1)/2) (2.2)
where: : ρ_c is the central density of the filament, Rflat is the radius of the inner region and p=2 and is the power law exponent for large radii (r>>Rflat). For isothermal gas cylinders, the expected power law has a value of p=4 (Ostriker 1964), but is empirically derived as p=2 for filaments in the ISM. (Goodman et al 1998) The observed density profiles approach a power law and at large radii p(r) ≈ r-2 (Hill et al 2012). A power law where p=2 suggests that filaments are not hydrostatic equilibrium except through their cross section, i.e. equilibrium conditions fall away with increasing radius from the central filament axis.
In addition to the discovery of the ubiquitous nature of filaments in the ISM, a remarkable result of the HGBS is the quasi-universal inner width of filaments (2xRflat); irrespective of filament length, column density, the presence of star forming regions and other components such as turbulence or magnetic fields, the FWHM of the radial profiles of filaments is centred at ≈0.1pc. (Goodman et al 1998) The distinctive median value of 0.1pc contrasts greatly to the much broader distributions of filament length/local central Jeans length (Arzoumanian & Andre 2018which has been found to vary 0.02pc to 1.3pc (Arzoumanian & Andre et al 2013b) and is inversely proportional to the central column densities of filaments. (see fig. 3)
(image source – first results from Herschel on nearby clouds – Andre 2010)
The origin of a distinctive inner width is not well understood, and there is still significant debate among the astrophysical community whether 0.1pc is the characteristic filament width. (Arzoumanian & Andre 2018) The Larson power law for linewidth vs MC size (  R) is a proposed as the origin of the FWHM value of 0.1pc; this width corresponds to the sonic scale below which the non-thermal interstellar turbulence will become subsonic for a non-star forming gas. (Arzoumanian & Andre 2011) As the power law breaks down the clouds will transition between supersonic and subsonic turbulent gas motion (Goodman et al 1998). A second theory is that FWHM≈0.1pc corresponds to a cut-off Alfven wavelength for magnetohydrodynamic (MHD) waves in low density molecular clouds (Mouschovias 1991) given by: (Andre 2012c )
█(λ_A≈0.1pc ×(B/10G)×(n_H2/(〖10〗^3 〖cm〗^(-3) ))^(-1) #(2.3) )
where B ≈10G (Crutcher 2012)
Gravitational substructure within filaments, such as dense cores, adhere to the inner width of filaments ranging from 0.05-0.1pc in size. (Gomez et al 2012)
The continual accretion of lower density matter in the surrounding clump onto filaments gives rise to a continuous flow of material towards troughs in the gravitational potential both locally and globally, leading to a regime of collapses within collapses along the filament. The accretion material flows through striations that are perpendicular to self-gravitating filaments (Palmeirim et al 2013). Remarkably, gravitational substructure within filaments, such as dense cores, adhere to the inner width of filaments ranging from 0.05-0.1pc in size.
Dense cores and filamentary substructure are also found at intersections between filamentary structures are regions of higher column density. Hub filament structure (HFS) describes a regime where several filaments converge toward a central intersection point (hub), within which the resulting column density is significant enough to lead to star formation. Small hubs have a rounded structure with a few radiating filaments and a lower number of protostellar cores. Larger hubs are elongated parsec scale regions with 5-10+ filaments radiating outward and contain a higher proportion of stars. HFS has been observed in each of the low mass young stellar groups within 300pcs (Myers 2009)
Hubs describe the brightest supercritical filaments in regions where Av ≈ 8; this is in support of earlier ground based studies (Onishi, T., Mizuno, A., Kawamura, A. et al. 1998) that suggest there is a NH2/AV threshold required for the formation of prestellar cores of Av≈5-10. (Arzoumanian & Andre 2011)
Dense cores within filaments are defined as individual fragments of local over-density corresponding to a local minimum in gravitational potential; the boundaries of filament cores are determined by breaks in the column density distribution around the core. In general, the density structures of pre-stellar cores are well described by models for a self-gravitating Bonnor-Ebert (BE) isothermal sphere, where the BE mass is given as:
█(M_BE= 1.18 ^4/(G^3 P_0 )^(1/2) #(2.4) )
█(M_BE=5.8(T/10K)^(3/2) ((〖10〗^3 〖cm〗^(-3))/n)^(1/2) M_⨀ #(2.5) )
Where  is the 1D velocity dispersion/isothermal sound speed, cs [(=kBT/mH)1/2]. (Moushovias 1991) T is the temperature, n is the volume density and P0 is the external pressure.
Their properties are summarised in columns 6&7 of table 1.
In addition to the BE model, the collapse criterion for MC’s/filaments can be given by several parameters – the Jeans Mass/Length and the Virial Theorem, where the descriptions given below are for the non-magnetised cloud case. The Jeans Mass (MJ) is a minimum mass threshold in MC’s above which a thermally supported cloud can overcome the internal radiation pressure and undergo gravitational collapse, where:
█(M_J≈〖10〗^5 T^(3/2)/(^2 n^(1/2) ) M_⨀ #(2.6) )
where T is the temperature (K),  is the mean molecular mass within the cloud, and n is the volume density (m-3).
The Jeans length (RJ) describes the critical radius above which a thermally supported cloud will overcome the repulsive gas pressure forces and undergo gravitational collapse, where:
█(R_J≈ 〖10〗^4 (T/(^2 n))^(1/2) parsecs #(2.7) )
MJ and RJ are dependent on temperature fluctuations within MC’s such that cold clouds are more susceptible to collapse. However, Whilst MC’s/filaments are considered to be cold (¬≈10K) regions of the ISM, they are actually much warmer than the diffuse ISM (≈ 2.7K) suggesting there must be a heating mechanism within MC’s. Cosmic rays consisting primarily of relativistic protons are produced via particle acceleration within magnetised shocks of supernovae remnants. Upon entering MC’s, cosmic ray protons (CRP) with E < 1GeV interact with ambient nuclei; the ionisation of H2 is the most common result of a CRP impact.
p^++H_2 〖 H〗_2^++e^-+p^+
The electron produced from the ionisation event provide the heating affect within the MC through the dissociation of H2:
e^-+H_2  H+H+e^-