The initial and boundary conditions of the Galactic star formation in molecular clouds are not well understood. In an effort to shed new light on this long-standing problem, we measured the properties of dense cores and filamentary structures in the Vela C molecular cloud, observed with Herschel. We used the hires algorithm to create high-resolution surface densities (11.7″) from the Herschel multiwavelength dust continuum. We applied the getsf extraction method to separate the components of sources and filaments from each other and their backgrounds before detecting, measuring, and cataloging the structures. The cores and filamentary structures constitute 40% of the total mass of Vela C; most of the material is in the low-density molecular background cloud. We selected 570 reliable cores, of which 149 are the protostellar cores and 421 are the starless cores. Almost 78% (329 of 421) of the starless cores were identified with the gravitationally bound prestellar cores. The exponent of the CMF (α = 1.35) is identical to that of the Salpeter IMF. The high-resolution surface density image helped us determine and subtract backgrounds and measure the sizes of the structures. We selected 68 filaments with at least one side that appeared not blended with adjacent structures. The filament widths are in the range from 0.15 pc to 0.63 pc, and have a median value of W = 0.3 ± 0.11 pc. The surface densities of filaments are well correlated with their contrasts and linear densities. Within uncertainties of the filament instability criterion, many filaments (39) may be both supercritical and subcritical. A large fraction of filaments (29), in which are found 94 prestellar cores, 83 protostellar cores, and only 1 unbound starless core, can definitely be considered supercritical. Taking into account the uncertainties, the supercritical filaments contain only prestellar and protostellar cores. Our findings support the idea that there is a direct relationship between the CMF and IMF and that filaments play a key role in the formation of prestellar cores, which is consistent with the previous Herschel results.

A simple Gaussian size deconvolution method is routinely used to remove the blur of observed images caused by insufficient angular resolutions of existing telescopes, thereby to estimate the physical sizes of extracted sources and filaments. To ensure that the physical conclusions derived from observations are correct, it is necessary to know the inaccuracies and biases of the size deconvolution method, which is expected to work when the structures, as well as the telescope beams, have Gaussian shapes. This study employed model images of the spherical and cylindrical objects with Gaussian and power-law shapes, representing the dense cores and filaments observed in star-forming regions. The images were convolved to a wide range of angular resolutions to probe various degrees of resolvedness of the model objects. Simplified shapes of the flat, convex, and concave backgrounds were added to the model images, then planar backgrounds across the footprints of the structures are subtracted and sizes of the sources and filaments were measured and deconvolved. When background subtraction happens to be inaccurate, the observed structures acquire profoundly non-Gaussian profiles. The deconvolved half maximum sizes can be strongly under- or overestimated, by factors of up to ~20 when the structures are unresolved or partially resolved. For resolved structures, the errors are generally within a factor of ~2; although, the deconvolved sizes can be overestimated by factors of up to ~6 for some power-law models. The results show that Gaussian size deconvolution cannot be applied to unresolved structures, whereas it can only be applied to the Gaussian-like structures, including the critical Bonnor-Ebert spheres, when they are at least partially resolved. The deconvolution method must be considered inapplicable for the power-law sources and filaments with shallow profiles. This work also reveals subtle properties of convolution for structures of different geometry. When convolved with different kernels, spherical objects and cylindrical filaments with identical profiles obtain different widths and shapes. In principle, a physical filament, imaged by the telescope with a non-Gaussian point-spread function, could appear substantially shallower than the structure is in reality, even when it is resolved.