Scale-dependent surface and volume density properties of filaments in molecular clouds

We present a systematic analysis of scale-dependent properties of filamentary structures in seven nearby molecular clouds $-$ Taurus, Ophiuchus, Perseus, Orion A, California, IC 5146, Vela C $-$ using the multiscale extraction method $getsf$. Alongside the usual surface density profiles $\Sigma(r)$, we derived volume density profiles $\rho(r)$ for a large sample of filaments, providing new observational constraints on their three-dimensional structure. The half-maximum widths $H$ and $h$ of the surface and volume density profiles, respectively, systematically increase with the spatial scale, following power laws $\tilde{H} \propto Y^{0.50}$ and $\tilde{h} \propto Y^{0.37}$, with distributions spanning $\sim 0.01 - 1$ pc across all scales, challenging the notion of a universal filament width of $\sim 0.1$ pc. The median volume density slopes $\tilde{\beta} \approx 2.1 - 2.4$ are systematically lower than the value $\beta = 4$ expected for an isothermal cylinder in hydrostatic equilibrium. For shallow profiles with $\beta \lesssim 1$, the volume density width $h$ falls below the surface density width $H$ by one to two orders of magnitude, demonstrating that surface density widths overestimate the true physical extent of filaments with shallow profiles. Volume density contrasts are substantially higher than surface density contrasts ($\tilde{C}_{\rho} \approx 17 - 52$ versus $\tilde{C}_{\Sigma} \approx 1.1 - 2.7$), confirming that filaments are substantially more prominent in three dimensions than their projected appearance suggests. Measured filament widths and slopes systematically depend on the angular resolution and distance, highlighting the importance of accounting for resolution bias in comparative filament studies.