The emergence and development of a power-law tail (PLT) at the high-density end of the observed column-density distribution are thought to be indicative for advanced evolution of star-forming molecular clouds. As shown from many numerical simulations, it corresponds to a morphologically analogous evolution of the mass-density distribution ( -PDF). The latter may display also a second, shallower PLT at the stage of collapse of the first formed protostellar cores. It is difficult to estimate the parameters of a possible second PLT due to resolution constraints. To address the issue, we extend the method for the extraction of single PLTs from arbitrary density distributions suggested by Veltchev et al. (MNRAS, 489, 788, 2019) to detect a second PLT. The technique is elaborated through tests on an analytic ρ-PDF and applied to a set of hydrodynamical high-resolution simulations of isothermal self-gravitating clouds. In all but one case two PLTs were detected-the first slope is always steeper and the second one is typically ln V∕ ln ∼ −1. These results are in a good agreement with numerical and theoretical works and do suggest that the technique extracts correctly double PLTs from smooth PDFs.

The mass function of clumps observed in molecular clouds raises interesting theoretical issues, especially in its relation to the stellar initial mass function. We propose a statistical model of the mass function of prestellar cores (CMF), formed in self-gravitating isothermal clouds at a given stage of their evolution. The latter is characterized by the mass-density probability distribution function (ρ-PDF), which is a power-law with slope q. The variety of MCs is divided in ensembles according to the PDF slope and each ensemble is represented by a single spherical cloud. The cores are considered as elements of self-similar structure typical for fractal clouds and are modeled by spherical objects populating each cloud shell. Our model assumes relations between size, mass and density of the statistical cores. Out of them a core mass-density relationship ρ ∝ m x is derived where x = 1/(1+q). We found that q determines the existence or non-existence of a threshold density for core collapse. The derived general CMF is a power law of slope −1 while the CMF of gravitationally unstable cores has a slope (−1 + x/2), comparable with the slopes of the high-mass part of the stellar initial mass function and of observational CMFs.