Context : Herschel observations of interstellar clouds support a paradigm for star formation in which molecular filaments play a central role .
One of the foundations of this paradigm is the finding , based on detailed studies of the transverse column density profiles observed with Herschel , that nearby molecular filaments share a common inner width of \sim 0.1 pc .
The existence of a characteristic filament width has been recently questioned , however , on the grounds that it seems inconsistent with the scale-free nature of the power spectrum of interstellar cloud images .

Aims : In an effort to clarify the origin of this apparent discrepancy , we examined the power spectra of the Herschel /SPIRE 250 \mu m images of the Polaris , Aquila , and Taurus–L1495 clouds in detail and performed a number of simple numerical experiments by injecting synthetic filaments in both the Herschel images and synthetic background images .

Methods : We constructed several populations of synthetic filaments of 0.1 pc width with realistic area filling factors ( A _ { fil } ) and distributions of column density contrasts ( \delta _ { c } ) .
After adding synthetic filaments to the original Herschel images , we re-computed the image power spectra and compared the results with the original , essentially scale-free power spectra .
We used the \chi ^ { 2 } _ { variance } of the residuals between the best power-law fit and the output power spectrum in each simulation as a diagnostic of the presence ( or absence ) of a significant departure from a scale-free power spectrum .

Results : We found that \chi ^ { 2 } _ { variance } depends primarily on the combined parameter \delta _ { c } ^ { 2 } A _ { fil } .
According to our numerical experiments , a significant departure from a scale-free behavior and thus the presence of a characteristic filament width become detectable in the power spectrum when \delta _ { c } ^ { 2 } A _ { fil } \gtrapprox 0.1 for synthetic filaments with Gaussian profiles and \delta _ { c } ^ { 2 } A _ { fil } \gtrapprox 0.4 for synthetic filaments with Plummer-like density profiles .
Analysis of the real Herschel 250 \mu m data suggests that \delta _ { c } ^ { 2 } A _ { fil } is \sim 0.01 in the case of the Polaris cloud and \sim 0.016 in the Aquila cloud , significantly below the fiducial detection limit of \delta _ { c } ^ { 2 } A _ { fil } \sim 0.1 in both cases .
In both clouds , the observed filament contrasts and area filling factors are such that the filamentary structure contributes only \sim 1 / 5 of the power in the image power spectrum at angular frequencies where an effect of the characteristic filament width is expected .

Conclusions : We conclude that the essentially scale-free power spectra of Herschel images remain consistent with the existence of a characteristic filament width \sim 0.1 pc and do not invalidate the conclusions drawn from studies of the filament profiles .