A Salpeter-like filament linear density function across nearby molecular clouds

Abstract: Filamentary structures in molecular clouds are thought to play a central role in star formation, yet the statistical distribution of their mass per unit length -- and any connection to the stellar initial mass function (IMF) -- remains poorly constrained observationally. Here we present a systematic analysis of the filament linear density function (FLDF) across seven nearby molecular clouds spanning $\sim$100--1000\,pc in distance and a wide range of star-forming environments, from quiescent to massive, using the multiscale extraction method $getsf$. The median linear densities of filaments increase approximately linearly with spatial scale, $\tildeΛ \propto Y$, and the fraction of supercritical filaments varies widely among clouds, from a few per cent to over 50%. Only when integrating over the full hierarchy of spatial scales do the combined linear densities across all clouds yield a FLDF that follows a power law ${\rm d}N/{\rm d}\logΛ\propto Λ<sup>{-α}$ with $α\approx 1.30$--$1.34$, mirroring the Salpeter IMF slope of $1.35$. Our results suggest that the stellar mass spectrum is pre-encoded in the filamentary structure of molecular clouds, providing a direct observational link between the large-scale distribution of interstellar gas and the universal slope of the stellar IMF.

• The paper demonstrates that filament linear densities across seven molecular clouds follow a Salpeter-like power-law, confirming a robust statistical link with the stellar IMF.
• The study employs advanced SED fitting and multiscale filament extraction with getsf to quantify median linear densities scaling nearly linearly with spatial resolution.
• The findings imply that hierarchical filament fragmentation underpins the universality of the IMF and informs predictive models of star formation efficiency.

Filament Linear Density Functions and the Salpeter IMF in Nearby Molecular Clouds

Context and Motivation

The stellar initial mass function (IMF), especially its Salpeter power-law slope ($\psi = 1.35$ for $dN/d\log M \propto M^{-\psi}$), represents a fundamental observational constraint in star formation theory. Recent Herschel observations uncovered the dominance of filamentary structures in the mass budget and star-forming activity within molecular clouds, motivating the filament fragmentation paradigm for the origin of dense cores and stellar mass distributions. A direct, robust statistical connection between filamentary linear densities and the IMF slope has remained elusive due to limited multiscale and multi-cloud analyses.

Data, Methods, and Filament Extraction

The study presents a rigorous statistical analysis of the filament linear density function (FLDF) across seven molecular clouds (Taurus, Ophiuchus, Perseus, Orion~A, California, IC~5146, Vela~C), spanning $\sim$140–920 pc distances and encompassing both quiescent and massive star-formation regimes.

Surface density maps were constructed from Herschel PACS and SPIRE data using spectral energy distribution (SED) fitting (hires algorithm) with corrections for zero-level offsets via Planck data. Filamentary skeletons were extracted with getsf—an advanced multiscale decomposition technique guaranteeing unbiased detection at scales from $14^{\prime\prime}$ to $216^{\prime\prime}$, with robust segment-by-segment surface density profile fitting incorporating finite boundary effects.

Figure 1: Overview of the seven studied molecular clouds, showing $13.5^{\prime\prime}$ resolution surface density maps with prominent filamentary networks.

Filament skeletons at multiple scales reveal a hierarchically nested structure, with individual segments providing statistically significant distributions of mass per unit length.

Figure 2: Surface densities in Taurus, Orion~A, and Vela~C, overlaid with multiscale getsf-extracted filament skeletons (14–216$^{\prime\prime}$ scales).

Scale-dependent Linear Density Distributions

A fundamental outcome is the quantification of scale-dependent distributions $\Lambda_k$ (mass per unit length) for all segments and filaments:

• Median linear densities ($\tilde{\Lambda}_k$) increase nearly linearly with detection scale, $\tilde{\Lambda} \propto Y_k^{1.01 \pm 0.18}$, ranging from $dN/d\log M \propto M^{-\psi}$0\,pc$dN/d\log M \propto M^{-\psi}$1 on $dN/d\log M \propto M^{-\psi}$2 to $dN/d\log M \propto M^{-\psi}$3\,pc$dN/d\log M \propto M^{-\psi}$4 on $dN/d\log M \propto M^{-\psi}$5.
• Composite FLDFs across all scales and clouds follow $dN/d\log M \propto M^{-\psi}$6, with $dN/d\log M \propto M^{-\psi}$7.

Figure 3: Scale-dependent histogram distributions of $dN/d\log M \propto M^{-\psi}$8 for all filaments and segments, including differential and cumulative forms. Power-law fit slopes are indicated above median thresholds.

Figure 4: Median linear density $dN/d\log M \propto M^{-\psi}$9 as a function of scale $\sim$0 for each cloud, highlighting nearly linear scaling.

The cumulative FLDF slope increases with scale, confirming a transition from turbulence-dominated fragmentation (shallower slopes at small scales) to gravity-dominated regimes (steeper, Salpeter-like slopes at large scales).

Figure 5: Dependence of the FLDF slope $\sim$1 on spatial scale $\sim$2 for both segments and whole filaments; best-fit power-law exponents shown.

Gravitational Stability and Star Formation Correlates

Assessment of gravitational stability incorporates the classical criterion, $\sim$3\,pc$\sim$4 at $\sim$5 K, with uncertainty margin ($\sim$6 and $\sim$7) to account for deviations from idealized models.

• The fraction of supercritical filaments ($\sim$8) rises sharply with spatial scale and varies significantly across clouds (e.g., 14\% in Ophiuchus vs. 95\% in Vela~C on $\sim$9).
• Environmentally, supercritical fractions are directly correlated with indicators of star formation efficiency.

Figure 6: Supercritical fraction of filament segments as a function of scale $14^{\prime\prime}$0 for different gravitational thresholds, stratified by cloud.

Robustness and Observational Bias Tests

FLDF power-law slopes demonstrate insensitivity to resolution degradation and distance-dependent blending, ensuring observational robustness:

• Across a factor of $14^{\prime\prime}$1 in cloud distance, measured slopes remain consistent and close to Salpeter.
• Blending only raises measured values of $14^{\prime\prime}$2, not slope $14^{\prime\prime}$3.

Figure 7: Panels (a–g): FLDF slope $14^{\prime\prime}$4 as a function of resolution for each cloud; Panel (h): Composite slopes for distance-binned cloud samples, all consistent with Salpeter.

Implications for the Stellar IMF and Core Mass Function

The core mass function (CMF) can be related to the FLDF through fragmentation physics. Under the assumption that fragmentation length $14^{\prime\prime}$5, the CMF slope is $14^{\prime\prime}$6.

• Observed scaling of filament width vs. linear density ($14^{\prime\prime}$7) is consistent with $14^{\prime\prime}$8 in high-mass regions, potentially explaining shallower CMF slopes observed by ALMA-IMF (e.g., $14^{\prime\prime}$9).
• For $216^{\prime\prime}$0, fragmentation length is independent of $216^{\prime\prime}$1, yielding CMF slope matching FLDF, $216^{\prime\prime}$2.
• Direct simultaneous measurements of FLDF and CMF in the same protoclusters are essential for resolving observed discrepancies at high masses.

The Salpeter-like composite FLDF emerges uniquely from hierarchical filament populations spanning all scales, suggesting universality of the IMF slope is a consequence of integrated, multi-scale gas fragmentation processes rather than from local environmental variations.

Theoretical and Practical Significance

The study provides the first multi-cloud, multiscale measurement confirming a Salpeter-like FLDF, quantitatively validating filament fragmentation paradigms and offering a unified framework linking large-scale filament gas distributions to dense core and stellar mass functions. Observational constraints on the parameter $216^{\prime\prime}$3 can directly probe fragmentation physics. Practical implications extend to predictive models for star formation rates and efficiencies in galactic environments, as well as refinement of filament detection algorithms.

Future theoretical developments should focus on:

• Joint FLDF–CMF analyses in high-mass star-forming regions.
• Inclusion of magnetic field and turbulence effects in fragmentation physics.
• Integration with simulation-based synthetic observations for calibration.
• Direct investigation of the efficiency variation of fragmentation with linear density.

Conclusion

This work establishes the statistical robustness of a Salpeter-like power-law in filament linear density distributions across diverse molecular clouds. The scaling of median linear density with spatial scale and the emergence of the Salpeter slope only through multi-scale integration provide strong evidence that the IMF's universality is pre-encoded in the filamentary gas structure. These results facilitate the quantitative exploration of fragmentation physics and star formation pathways, bridging large-scale ISM structure and stellar population outcomes (2604.14093).