The role of molecular filaments in the origin of the prestellar core mass function and stellar initial mass function

The origin of the stellar initial mass function (IMF) is one of the most debated issues in astrophysics. Here, we explore the possible link between the quasi-universal filamentary structure of star-forming molecular clouds and the origin of the IMF. Based on our recent comprehensive study of filament properties from Herschel Gould Belt survey observations (Arzoumanian et al.), we derive, for the first time, a good estimate of the filament mass function (FMF) and filament line mass function (FLMF) in nearby molecular clouds. We use the observed FLMF to propose a simple toy model for the origin of the prestellar core mass function (CMF), relying on gravitational fragmentation of thermally supercritical but virialized filaments. We find that the FMF and the FLMF have very similar shapes and are both consistent with a Salpeter-like power-law function (d$N$/dlog$M_{\rm line} \propto M_{\rm line}^{-1.5\pm0.1}$) in the regime of thermally supercritical filaments ($M_{\rm line}>16\, M_\odot$/pc). This is a remarkable result since, in contrast, the mass distribution of molecular clouds and clumps is known to be significantly shallower than the Salpeter power-law IMF, with d$N$/dlog$M_{\rm cl} \propto M_{\rm cl}^{-0.7}$. Since the vast majority of prestellar cores appear to form in thermally transcritical or supercritical filaments, we suggest that the prestellar CMF and by extension the stellar IMF are at least partly inherited from the FLMF through gravitational fragmentation of individual filaments.

The role of molecular filaments in the origin of the prestellar core mass function and stellar initial mass function Ph. André, D. Arzoumanian, V. Könyves, Y. Shimajiri, P. Palmeirim

Introduction
The origin of the stellar initial mass function (IMF) is a fundamental problem in modern astrophysics which is still highly debated (e.g., Offner et al. 2014). Two major features of the IMF are a fairly robust power-law slope at the high-mass end (Salpeter 1955) and a broad peak around ∼0.3 M corresponding to a characteristic stellar mass scale (e.g., Larson 1985). In recent years, the dominant theoretical model proposed to account for these features has been the "gravo-turbulent fragmentation" picture (e.g., Padoan & Nordlund 2002;Hennebelle & Chabrier 2008), whereby the properties of supersonic interstellar turbulence lead to the Salpeter power law, while gravity and thermal physics set the characteristic mass scale (see Larson 2005). This picture is deterministic in the sense that stellar masses are directly inherited from the distribution of prestellar core masses resulting from cloud fragmentation prior to protostellar collapse, in agreement with the observed similarity between the prestellar core mass function (CMF) and the system IMF (e.g., Motte et al. 1998; Alves et al. 2007; Könyves et al. 2015).
In contrast, a major alternative view posits that stellar masses are essentially unrelated to initial prestellar core masses and result entirely from stochastic competitive accretion and dynamical interactions between protocluster seeds at the protostellar stage (Class 0/Class I) of young stellar object evolution (Bonnell et al. 2001;Bate et al. 2003). Here, we discuss modifications to the gravo-turbulent picture based on Herschel results in nearby molecular clouds, which emphasize the importance of filaments in the star formation process and potentially the CMF and IMF (e.g., André et al. 2010).
Herschel imaging observations have shown that filamentary structures are truly ubiquitous in the cold interstellar medium (ISM) of the Milky Way (Molinari et al. 2010), dominate the mass budget of Galactic molecular clouds at high densities ( 10 4 cm −3 ) (Schisano et al. 2014;Könyves et al. 2015), and feature a high degree of universality in their properties. In particular, detailed analysis of the radial column density profiles indicates that, at least in the nearby clouds of the Gould Belt, molecular filaments are characterized by a narrow distribution of crest-averaged inner widths with a typical full width at half maximum (FWHM) value W fil ∼ 0.1 pc and a dispersion of less than a factor of ∼2 (Arzoumanian et al. 2011Koch & Rosolowsky 2015). Another major result from Herschel (e.g., André et al. 2010;Könyves et al. 2015;Marsh et al. 2016) is that the vast majority (>75%) of prestellar cores are found in dense "transcritical" or "supercritical" filaments for which the mass per unit length, M line , is close to or exceeds the critical line mass of nearly isothermal, long cylinders (e.g., Inutsuka & Miyama 1997), M line,crit = 2 c 2 s /G ∼ 16 M pc −1 , where c s ∼ 0.2 km s −1 is the isothermal sound speed for molecular gas at T ∼ 10 K. Moreover, most prestellar cores lie very close to the crests (i.e., within the inner 0.1 pc portion) of their parent filaments (e.g., Könyves et al. 2019;Ladjelate et al. 2019). These findings support a filamentary paradigm in which lowmass star formation occurs in two main steps (André et al. 2014;Inutsuka et al. 2015): (1) multiple large-scale compressions of cold interstellar material in supersonic magneto-hydrodynamic (MHD) flows generate a cobweb of ∼0.1 pc-wide filaments within sheet-like or shell-like molecular gas layers in the ISM and (2) the densest molecular filaments fragment into prestellar cores (and then protostars) by gravitational instability near or above the critical line mass, M line,crit , corresponding to Σ crit gas ∼ M line,crit /W fil ∼ 160 M pc −2 in gas surface density (A V ∼ 7.5) or n H 2 ∼ 2 × 10 4 cm −3 in volume density. This paradigm differs from the classical gravo-turbulent picture (Mac Low & Klessen 2004) in that it relies on the anisotropic formation of dense structures (such as shells, filaments, cores) in the cold ISM and the unique properties of filamentary geometry (see Larson 2005).
In the present paper, we exploit the results of our recent comprehensive study of filament properties from Herschel Gould Belt survey (HGBS) observations ) and argue that the distribution of filament masses per unit length may directly connect to the CMF and by extension the IMF. Section 2 presents our observational results on the filament line mass function. Section 3 discusses potential implications of these results for the origin of the prestellar CMF. Section 4 discusses the possible origin of the filament line mass function and concludes the paper. Arzoumanian et al. (2019) recently presented a census of filament structures observed with Herschel in eight nearby regions covered by the HGBS: IC5146, Orion B, Aquila, Musca, Polaris, Pipe, Taurus L1495, and Ophiuchus. Using the DisPerSE algorithm (Sousbie 2011) to trace filaments in the HGBS column density maps of these eight clouds 1 , they identified a total of 1310 filamentary structures, including a selected sample of 599 robust filaments with aspect ratio (length/width) >3 and central column density contrast δΣ fil /Σ cloud > 30% (where δΣ fil is the background-subtracted gas surface density of the filament and Σ cloud the surface density of the parent cloud). Performing an extensive set of tests on synthetic data, Arzoumanian et al. (2019, see their Appendix A) estimated their selected sample of 599 filaments to be more than 95% complete (and contaminated by less than 5% of spurious detections) for filaments with column density contrast ≥100%. For reference, the column density contrast of isothermal model filaments in pressure equilibrium with their parent cloud is δΣ fil /Σ cloud ≈ 1.18

Observations of the filament line mass function
Thermally transcritical filaments with M line,crit /2 M line < M line,crit (i.e., f cyl 0.5) are therefore expected to have column density contrasts 100%, while thermally supercritical filaments with well-developed power-law density profiles reach column density contrasts 100%. The selected sample of Arzoumanian et al. (2019) is thus estimated to be >95% complete to thermally supercritical filaments with M line > M line,crit ∼ 16 M pc −1 .
The differential distribution of average masses per unit length, or filament line mass function (FLMF), derived from Herschel data for the 599 filaments of this sample is shown in Fig. 1a. It can be seen that the FLMF is consistent with a power-law distribution in the supercritical mass per unit length 1 The corresponding column density maps and derived filament skeleton maps are available in fits format from http:// gouldbelt-herschel.cea.fr/archives 2 Equilibrium model filaments exist only for subcritical masses per unit length, i.e., f cyl ≤ 1.
L4, page 2 of 8 line , at a Kolmogorov-Smirnov (K-S) significance level of 92%. The error bar on the power-law exponent was derived by performing a non-parametric K-S test (see, e.g., Press et al. 1992) on the cumulative distribution of masses per unit length N(>M line ), and corresponds to the range of exponents for which the K-S significance level is larger than 68% (equivalent to 1σ in Gaussian statistics). Remarkably, the FLMF function observed above M line,crit ∼ 16 M pc −1 is very similar to the Salpeter power-law IMF (Salpeter 1955), which scales as dN/dlog M ∝ M −1.35 in the same format.
The right panel of Fig. 1 shows the distribution of total masses, integrated over filament length, for the same sample of filaments. As can be seen in Fig

Role of filaments in the prestellar CMF
At least in terms of mass, most prestellar cores appear to form just above the fiducial column density "threshold" at A V ∼ 7.5, corresponding to marginally thermally supercritical filaments with M line 16 M pc −1 see also Fig. 2). In the observationally driven filamentary paradigm of star formation supported by Herschel results (see Sect. 1), the dense cores making up the peak of the prestellar CMF, presumably related to the peak of the IMF, originate from gravitational fragmentation of filaments near the critical threshold for cylindrical gravitational instability (André et al. 2014). In this picture, the characteristic prestellar core mass roughly corresponds to the local Jeans mass in transcritical or marginally supercritical filaments. The thermal Jeans or critical Bonnor-Ebert mass (e.g., Bonnor 1956) is M BE,th ≈ 1.18 c 4 s /(G 3/2 P 1/2 cl ), where P cl is the local pressure of the ambient cloud. The latter may be expressed as a function of cloud column density, Σ cl , as P cl ≈ 0.88 G Σ 2 cl (McKee & Tan 2003). Within a critical ∼0.1 pc-wide filament at ∼10 K with M line ≈ M line,crit ∼ 16 M pc −1 and surface density Σ fil ≈ Σ crit gas ∼ 160 M pc −2 (see Sect. 1), the local Bonnor-Ebert mass is thus This corresponds very well to the peak of the prestellar CMF at ∼0.6 M observed in the Aquila cloud (Könyves et al. 2015) and is also consistent within a factor of <2 with the CMF peak found with Herschel in other nearby regions such as Taurus L1495 (Marsh et al. 2016) or Ophiuchus (Ladjelate et al. 2019).
The fragmentation of purely thermal equilibrium filaments may be expected to result in a narrow (δ-like) prestellar CMF sharply peaked at the median thermal Jeans mass (see Lee et al. 2017). However, at least two effects contribute to broadening the observed CMF. First, the filament formation process through multiple large-scale compressions generates a field of initial density fluctuations within star-forming filaments (Inutsuka 2001;Inutsuka et al. 2015). Based on a study of the density fluctuations observed with Herschel along a sample of 80 subcritical or marginally supercritical filaments in three nearby clouds, Roy et al. (2015) found that the power spectrum of line-mass fluctuations is well fitted by a power law, P(k) ∝ k α with α = −1.6 ± 0.3. This is consistent with the 1D power spectrum generated by subsonic Kolmogorov turbulence (α = −5/3). Starting from such an initial power spectrum, the theoretical analysis by Inutsuka (2001) shows that the density perturbations quickly line , assumed in the toy model and consistent with the observed FLMF in the supercritical regime (see Fig. 1). The green dashed curve shows the system IMF (Chabrier 2005, see also Kroupa 2001). The two vertical dashed lines mark the estimated 80% completeness limits of the Herschel census of prestellar cores in Orion B at low and high background column densities respectively (see Könyves et al. 2019). The CMF extends to higher masses at higher column densities, i.e., higher M line filaments in both the toy model and the observations. evolve (in about two free-fall times or ∼0.5 Myr for a critical 0.1 pc-wide filament) from a mass distribution similar to that of CO clumps (Kramer et al. 1998) to a population of protostellar cores whose mass distribution approaches the Salpeter power law at the high-mass end. However, this process alone is unlikely to produce a CMF with a well-developed Salpeter-like powerlaw tail since very long filaments would be required.
A second broadening effect is due to the power-law distribution of filament masses per unit length (FLMF) in the supercritical regime (Fig. 1a). Given the typical filament width W fil ∼ 0.1 pc (Arzoumanian et al. 2011  (1)). Hence, both higher-and lower-mass cores may form in higher M line filaments. In agreement with this expected trend, dense cores of median mass ∼10 M , i.e., an order of magnitude higher that the peak of the prestellar CMF in low-mass nearby filaments (see above and Fig. 3), have recently been detected with ALMA in the NGC 6334 main filament, which is an order of magnitude denser and more massive (M line ∼ 1000 M pc −1 ) than the Taurus B211/B213 filament and other Gould Belt filaments (Shimajiri et al. 2019a). Furthermore, observations indicate that the prestellar CMF tends 3 Assuming rough equipartition between magnetic energy and kinetic energy, thermally supercritical filaments may also be close to magnetohydrostatic equilibrium since the magnetic critical line mass M mag line,crit may largely exceed M line,crit (see Tomisaka 2014). to be broader at higher ambient cloud column densities, i.e., in denser parent filaments ; see also Fig. 3). Since the characteristic fragmentation mass M BE,eff scales linearly with M line , we may expect the Salpeter-like distribution of line masses observed above M line,crit (Fig. 1a) to directly translate into a Salpeter-like power-law distribution of characteristic core masses. In detail, the global prestellar CMF results from the convolution of the CMF produced by individual filaments with the FLMF (see Lee et al. 2017).
Based on the Herschel results and these qualitative considerations, we propose the following observationally driven quantitative scenario to illustrate the potential key role of the FLMF in the origin of the global prestellar CMF in molecular clouds. We assume that all prestellar cores form in thermally transcritical or supercritical (but virialized) filaments and that the outcome of filament fragmentation depends only on the line mass of the parent filament. We denote by f M line (m) ≡ dN M line /dlog m the differential CMF (per unit log mass, where m represents core mass) in a filament of line mass M line . While the exact form of f M line (m) is observationally quite uncertain, the foregoing arguments suggest that it should present a peak around the effective Bonnor-Ebert mass M BE,eff and may have a characteristic width scaling roughly as the ratio M BE,eff /M BE,th . We thus make the minimal assumption that f M line (m) follows a lognormal distribution centered at M BE,eff (M line ) and of standard deviation σ M line (M BE,eff /M BE,th ) in log m: We tested various simple functional forms for σ M line (M BE,eff /M BE,th )andadoptedσ 2 M line = 0.4 2 +0.3 [log (M BE,eff / M BE,th )] 2 as an illustrative fiducial form providing a reasonable good match to the observational constraints (see Fig. 3 and Appendix B).
Denoting by g(M line ) ≡ dN/dlog M line the differential FLMF per unit log line mass, the global prestellar CMF per unit log mass ξ(m) ≡ dN tot /dlog m may be obtained as a weighted integration over line mass of the CMFs in individual filaments: The global prestellar CMF expected in the framework of this toy model, and the CMFs expected in thermally transcritical filaments and slightly supercritical filaments are shown in Fig. 3 as a black solid, blue solid, and red solid curve, respectively. For comparison, the black, blue, and red histograms with error bars represent the corresponding CMFs observed with Herschel in Orion B . A good, overall agreement can be seen. More importantly, it can be seen in Fig. 3 that the global prestellar CMF approaches the powerlaw shape of the FLMF at the high-mass end. We note that the empirical toy model described here is only meant to quantify the links between the FLMF and the CMF/IMF. It may also provide useful guidelines that will help develop a self-consistent physical model for the origin of the CMF/IMF in filaments in the future.

Concluding remarks
Our discussion of the Herschel observations in Sect. 2 indicates that both the filament line mass function (FLMF) and FMF are consistent with a steep Salpeter-like power law (dN/dlog M line ∝ M −1.6 line and dN/dlog M tot ∝ M −1.4 tot , respectively) in the regime of thermally supercritical filaments (M line > 16 M pc −1 ). This is a remarkable result since, in contrast, the mass distribution of molecular clouds and clumps is observed to be significantly shallower than the Salpeter power-law IMF, namely dN/dlog M cl ∝ M −0.7 cl (Blitz 1993;Kramer et al. 1998). Theoretically, the latter is reasonably well understood in terms of the mass function of both "bound objects on the largest self-gravitating scale" (Hopkins 2012) and non-self-gravitating structures (Hennebelle & Chabrier 2008) generated by supersonic interstellar turbulence. Thus, filamentary structures in molecular clouds appear to differ from standard clumps in a fundamental way and may represent the key evolutionary step at which the steep slope of the prestellar CMF originates (and by extension that of the stellar IMF) (see Sect. 3).
In the context of the filament paradigm summarized in Sect. 1, we speculate that the observed FLMF arises from a combination of two effects. First, a spectrum of large-scale compression flows in the cold ISM produces a network of filamentary structures with an initial line mass distribution dN/dlog M line ∝ M −1 line , determined by the power spectrum of interstellar turbulence (Iwasaki, priv. comm.). Turbulence is known to generate essentially self-similar, fractal structures in interstellar clouds (e.g., Larson 1992;Elmegreen & Falgarone 1996), and this leads to a mass distribution of substructures with equal mass contribution per logarithmic interval of mass (i.e., dN/dlog M ∝ M −1 ) independent of the fractal dimension (Elmegreen 1997;Padoan & Nordlund 2002). Second, thermally supercritical filaments accrete mass from the parent molecular cloud (Arzoumanian et al. 2013;Shimajiri et al. 2019b) due to their gravitational potential ∝G M line (Hennebelle & André 2013). Therefore, they grow in mass per unit length at a ratė M line ∝ √ G M line on a characteristic timescale τ acc = M line /Ṁ line ∝ √ M line , while fragmenting and forming cores on a comparable timescale (see Heitsch 2013). The accretion timescale is on the order of 1-2 Myr for a Taurus-like filament with M line ∼ 50 M pc −1 (Palmeirim et al. 2013). As shown in Appendix C, starting from an initial line mass spectrum dN/dlog M line ∝ M −1 line , this accretion process leads to a steepening of the distribution of supercritical masses per unit length on a similar timescale (Fig. C.2), and thus to a reasonable agreement with the observed FLMF (see Fig. C.3b).
Given the empirical toy model of Sect. 3 for the CMF produced by a collection of molecular filaments and its reasonably good match to observations (Fig. 3), we conclude that the filament paradigm for star formation provides a promising conceptual framework for understanding the origin of the prestellar CMF and by extension the stellar IMF. In this appendix, we show the distribution of filament lengths in the Arzoumanian et al. (2019) sample (Fig. A.1a) and the linear correlation between filament mass and filament mass per unit length (Fig. A.1b), consistent with a roughly uniform length L eff ∼ 0.55 pc independent of M line .

Appendix B: Observational constraints on the core mass function in individual filaments
The    Fig. 1), and the black dash-dotted line shows the typical mass distribution of CO clumps (Kramer et al. 1998). in core masses also increases with M line ( ), see also Fig. B.1). In agreement with this observational trend, the qualitative arguments presented in Sect. 3 suggest that the characteristic prestellar core mass should scale with the effective Bonnor-Ebert mass M BE,eff in the parent filament and that the L4, page 6 of 8 dispersion in core masses may scale with the ratio M BE,eff /M BE,th . The blue lines in Fig. B.1 show how the median core mass and the dispersion in core masses vary with M line in the toy model of Sect. 3, which assumes a lognormal shape for f M line (m) with standard deviation σ M line = 0.4 2 + 0.3 log (M BE,eff /M BE,th ) 2 . The latter expression for σ M line corresponds to the quadratic sum of two terms: the first term represents the intrinsic spread in the core masses generated by transcritical filaments (which have M BE,eff /M BE,th ∼ 1), while the second term represents the spread in characteristic fragmentation masses within supercritical but virialized filaments (which have M BE,eff /M BE,th > 1; see Sect. 3). It can be seen in Fig. B.1 that these simple assumptions about f M line (m) and σ M line match the observational constraints reasonably well.
We also note that the high-mass end of the global prestellar CMF in our toy model is primarily driven by the power-law shape of the FLMF and depends only weakly on the detailed form assumed for f M line (m) ≡ dN M line /dlog m. This is illustrated in Fig   line ) determined by interstellar turbulence. The green, red, and blue solid curves show the model FLMF at three time steps,t = 0.2,t = 0.4,t = 0.6 after the accretion process is "switched on" att = 0, where ∆t = 0.4 roughly corresponds to the time it takes for a critical filament to double its mass per unit length (∼ 0.5-1 Myr). The median logarithmic slope of the model FLMF for 16 < M line < 500 M pc −1 is −1, −1.14, −1.30, and −1.50 att = 0, t = 0.2,t = 0.4, andt = 0.6, respectively. The vertical red and blue lines correspond to the line mass M line,critt 2 accreted by filaments with M line (0) ≈ 0 att = 0.4 andt = 0.6, respectively.
In the context of this model, we can derive the time evolution of the FLMF following an approach similar to that employed by Zinnecker (1982)