We derive an analytical theory of the prestellar core initial mass function ( IMF) based on an extension of the PressSchechter statistical formalism. Our approach relies on the general concept of the gravothermal and gravoturbulent collapse of a molecular cloud, with a selection criterion based on the thermal or turbulent Jeans mass, which yields the derivation of the mass spectrum of self-gravitating objects in a quiescent or a turbulent environment. The same formalism also yields the mass spectrum of non-self-gravitating clumps produced in supersonic flows. The mass spectrum of the self-gravitating cores reproduces well the observed IMF. The theory predicts that the shape of the IMF results from two competing contributions, namely, a power law at large scales and an exponential cutoff ( lognormal form) centered around the characteristic mass for gravitational collapse. The cutoff exists both in the case of thermal or turbulent 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/d log M / M Àx with x ' 1:35, the latter is recovered exactly for the (three-dimensional) value of the spectral index of the velocity power spectrum, n ' 3:8, found in observations and in numerical simulations of isothermal supersonic turbulence. Indeed, the theory predicts that x ¼ (n þ 1)/(2n À 4) for self-gravitating structures and x ¼ 2 À n 0 /3 for non-self-gravitating structures, where n 0 is the power spectrum index of log . 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 provides useful guidance to numerical simulations exploring star formation, while making testable predictions.

Context. Theoretical studies of collapsing clouds have found that even a relatively weak magnetic field may prevent the formation of disks and their fragmentation. However, most previous studies have been limited to cases where the magnetic field and the rotation axis of the cloud are aligned. Aims. We study the transport of angular momentum, and its effects on disk formation, for non-aligned initial configurations and a range of magnetic intensities. Methods. We perform three-dimensional, adaptive mesh, numerical simulations of magnetically supercritical collapsing dense cores using the magneto-hydrodynamic code Ramses. We compute the contributions of all the relevant processes transporting angular momentum, in both the envelope and the region of the disk. We clearly define centrifugally supported disks and thoroughly study their properties. Results. At variance with earlier analyses, we show that the transport of angular momentum acts less efficiently in collapsing cores with non-aligned rotation and magnetic field. Analytically, this result can be understood by taking into account the bending of field lines occurring during the gravitational collapse. For the transport of angular momentum, we conclude that magnetic braking in the mean direction of the magnetic field tends to dominate over both the gravitational and outflow transport of angular momentum. We find that massive disks, containing at least 10% of the initial core mass, can form during the earliest stages of star formation even for mass-to-flux ratios as small as three to five times the critical value. At higher field intensities, the early formation of massive disks is prevented. Conclusions. Given the ubiquity of Class I disks, and because the early formation of massive disks can take place at moderate magnetic intensities, we speculate that for stronger fields, disks will form later, when most of the envelope will have been accreted. In addition, we speculate that some observed early massive disks may actually be outflow cavities, mistaken for disks by projection effects.