We present a simple dimensional argument to illustrate the impact of nonthermal support from turbulent velocity dispersion on the shape of the prestellar core mass function (CMF), precursor of the stellar initial mass function (IMF). The argument dem
onstrates the need to invoke such support to recover the Salpeter slope in the high-mass part of the CMF/IMF, whereas pure thermal support leads to a much steeper slope. This simple dimensional argument clearly highlights the results obtained in the complete Hennebelle-Chabrier theory of the IMF.
We present a simple statistical analysis of recent numerical simulations exploring the correlation between the core mass function obtained from the fragmentation of a molecular cloud and the stellar mass function which forms from these collapsing cor
es. Our analysis shows that the distributions of bound cores and sink particles obtained in the simulations are consistent with the sinks being formed predominantly from their parent core mass reservoir, with a statistical dispersion of the order of one third of the core mass. Such a characteristic dispersion suggests that the stellar initial mass function is relatively tightly correlated to the parent core mass function, leading to two similar distributions, as observed. This in turn argues in favor of the IMF being essentially determined at the early stages of core formation and being only weakly affected by the various environmental factors beyond the initial core mass reservoir, at least in the mass range explored in the present study. Accordingly, the final IMF of a star forming region should be determined reasonably accurately, statistically speaking, from the initial core mass function, provided some uniform efficiency factor. The calculations also show that these statistical fluctuations, due e.g. to variations among the core properties, broaden the low-mass tail of the IMF compared with the parent CMF, providing an explanation for the fact that this latter appears to underestimate the number of pre brown dwarf cores compared with the observationally-derived brown dwarf IMF.
We derive an analytical theory of the prestellar core initial mass function based on an extension of the Press-Schechter statistical formalism. With the same formalism, we also obtain the mass spectrum for the non self-gravitating clumps produced in
supersonic flows. The mass spectrum of the self-gravitating cores reproduces very well the observed initial mass function and identifies the different mechanisms responsible for its behaviour. The theory predicts that the shape of the IMF results from two competing contributions, namely a power-law at large scales and an exponential cut-off (lognormal form) centered around the characteristic mass for gravitational collapse. The cut-off exists already in the case of pure thermal collapse, provided that the underlying density field has a lognormal distribution. Whereas pure thermal collapse produces a power-law tail steeper than the Salpeter value, dN/dlog Mpropto M^{-x}, with x=1.35, this latter is recovered exactly for the (3D) value of the spectral index of the velocity power spectrum, nsimeq 3.8, found in observations and in numerical simulations of isothermal supersonic turbulence. Indeed, the theory predicts that x=(n+1)/(2n-4) for self-gravitating structures and x=2-n/3 for non self-gravitating structures, where n is the power spectrum index of log(rho). We show that, whereas supersonic turbulence promotes the formation of both massive stars and brown dwarfs, it has an overall negative impact on star formation, decreasing the star formation efficiency. This theory provides a novel theoretical foundation to understand the origin of the IMF and to infer its behaviour in different environments. It also provides a complementary approach and useful guidance to numerical simulations exploring star formation, while making testable predictions.
